20 ( α , β , and γ 1 pockets), and hence the extrapolation from the Landau fan plots may have biggish uncertainty. In order to validate the topological nature, more solid evidences are needed. Other parameters for these four bands, such as the Fermi energy E F , extremal cross-sectional areas A F , Fermi momentum k F , Fermi velocity v F , cyclotron effective mass m ∗ , and Dingle temperature T D , are calculated and summarized in Table 1 . Table 1 Parameters derived from quantum oscillations in different magnetic field range for EuAs 3 . Full size table In TSMs, in addition to nonzero Berry phase, the n -LMR induced by the chiral anomaly can also serve as a smoking gun for nontrivial topology [19] , [20] , [21] . Figure 2e displays the n -LMR of EuAs 3 with magnetic field parallel to the electric current I . The kinks in the n -LMR curves below the ordering temperature arise from the field-induced transitions [15] , [16] , [17] , i.e., from a collinear AFM phase to incommensurate and commensurate spiral phases [15] , [16] , [17] . Negative MR in magnetic systems is not uncommon [19] , [20] , [21] when the applied magnetic field suppresses the inelastic magnetic scattering from local moments or magnetic impurities, leading to a negative MR for charge transport along all directions [19] , [20] , [21] . However, in EuAs 3 we only observed a n -LMR when the magnetic field is applied parallel to the electric current (Supplementary Fig. 2 ). Furthermore, if the applied external magnetic field has a strong effect on the magnetic scattering and induces a n -LMR, the changes in the n -LMR will occur predominantly below and above the ordering temperature. This is not observed. Instead, we find several minor kinks arising from the magnetic transitions, on top of a much larger signal. Therefore, the suppression of magnetic scattering can be excluded as the origin of the n -LMR in EuAs 3 . The n -LMR also displays a wide variety of temperature dependences, ruling out current jetting effects and the weak localization effect [19] , [20] , [21] . Since nontrivial band topology has been suggested in EuAs 3 , the chiral anomaly arising from Weyl fermions is the most likely mechanism behind the n -LMR. The n -LMR induced by the chiral anomaly in TSMs can be analyzed through the Adler–Bell–Jackiw (ABJ) chiral anomaly equation [19] , [20] , [21] : \\({{{{{\\rm{\\sigma }}}}}}\\left(B\\right)=\\left({1+C}_{{{{{{\\rm{w}}}}}}}{B}^{2}\\right)\\left({\\sigma }_{0}+a\\sqrt{B}\\right)+{{{{{{\\rm{\\sigma }}}}}}}_{{{{{{\\rm{N}}}}}}}\\) , where σ 0 , C w , and \\({\\sigma }_{{{{{{\\rm{N}}}}}}}^{-1}={{\\rho }}_{0}+A{B}^{2}\\) denote the conductivity at zero field, a temperature-dependent positive parameter originating from chiral anomaly, and the conventional nonlinear band contribution around Fermi energy, respectively. Figure 2f shows the conductivity and the fit to the data below 3 T at various temperatures. The data above the ordering temperature are well described by the ABJ equation, while the data at lower temperatures do not fit as well, which may be ascribed to magnetic transitions or topological transitions. The inset to Fig. 2f shows the temperature dependence of C w . At 2 K, C w is 0.253(7) T −2 . With increasing temperature, a clear anomaly in C w around the ordering temperature can be observed, verifying the proposal above. When T > T N , C w decreases monotonically, as observed in SrAs 3 [7] . Taken together, these results demonstrate that EuAs 3 is a magnetic topological massive Dirac metal in its AFM ground state. Topological state in the spin-polarized state We now turn to the exploration of topology in the spin-polarized state, where in Fig. 2b we have already observed a clear change in the quantum oscillations. Figure 3a plots the MR of EuAs 3 in higher magnetic field, and an unsaturated XMR ~2 × 10 5 % at 1.8 K and 28.3 T is observed. By analyzing the oscillatory components above B M (inset in Fig. 3b ), frequency components are identified at F = 93, 158, 346, and 597 T, which are referred to here as the ξ , α ′, ε , and η bands, respectively (Fig. 3b ). These four bands are different from those in lower field (Fig. 2c ), indicating that they are likely rooted in different band structure. This is unsurprising since the unit cell is no longer doubled by antiferromagnetism, but the field-induced spin polarization can also play a significant role. We thus conducted band calculations for the field-polarized state (Supplementary Fig. 3 ) and the paramagnetic state (Supplementary Fig. 4 ), and these are indeed quite different, as we discuss in more detail in Supplementary Notes 3 and 4 . Fig. 3: Quantum oscillation study in the spin-polarized state of EuAs 3 and Hall resistivity measurements. a Magnetoresistance measurements of EuAs 3 single crystal under higher magnetic field up to 28.3 T. b FFT results at various temperatures, yielding the four bands ξ , α ′, ε , and η . The inset displays the oscillatory component ρ xx above B M . c Landau index n plotted against 1/ B for the SdH oscillations at 1.8 K. The left inset shows the extrapolation of 1/ B to zero. The right inset shows the normalized FFT amplitude (Amp. nor ) as a function of temperature, and the solid lines represent the Lifshitz–Kosevich formula fit. d SdH oscillations at different angles; the inset is a schematic illustration of the experimental geometry and the angle θ . For θ = 0°, the magnetic field is parallel to the c axis. For θ = 90°, the magnetic field is applied along the [110] direction. e The oscillatory component ρ xx as a function of 1/ B . Angular dependence of f the FFT frequencies, where error bars represent the full widths at half maximum of the FFT peaks, and g the Landau level index intercepts. The error bar of the intercepts is determined from fitting. h Hall resistivity at various temperatures. i Carrier concentration and mobility as a function of temperature, and error bar is determined from fitting. The shadow area represents the temperature interval where a Lifshitz transition takes place. Full size image In the field-polarized state, we find four Fermi surface sheets—two electron and two hole sheets and double nodal loops at the Y point, one each for the spin-up and spin-down channels (Supplementary Fig. 3d ). To identify the topological nature of the four bands seen in quantum oscillations, a Landau index fan diagram is plotted in Fig. 3c , and the intercepts are −0.0(1), 0.67(3), 0.34(4), and 0.61(4) for the ξ , α ′, ε , and η bands, respectively. Although the intercepts from the fit with large Landau index number cannot serve as a smoking gun for topology, we still use them, because it is difficult for us to evaluate the topological nature in the fully spin-polarized state above 11.0 T (the critical field is deduced from specific heat in Supplementary Fig. 1c ). The intercepts give a hint that the ξ band is topologically protected, while the α ′ and η bands are topologically trivial. The intermediate value for ε is suggestive of a possible nontrivial Berry phase but does not allow a strong conclusion and will require further verification. The cyclotron effective masses m * for these four pockets can be obtained by fitting the temperature dependence of the normalized FFT amplitude, as shown in the right inset to Fig. 3c . Other parameters can be also extracted, and all values are summarized in Table 1 . To better reveal the Fermi surface anisotropy and topology of EuAs 3 , angle-dependent MR measurements have been performed at 1.8 K, in the experimental geometry shown in the inset to Fig. 3d . Upon rotating sample from 0° to 90°, the magnitude of the MR is reduced monotonically, as also seen in a polar plot of the MR (Supplementary Fig. 2b ). We extract the frequency components for the ξ , α ′, ε , and η bands by analyzing the oscillatory component (Fig. 3e ) and summarize the results in Fig. 3f . The angle dependence of the ξ , α ′, and η bands is of 3D character, while the ε band is well described below 50° by the formula F = F 3D + F 2D /cos( θ ), where F 2D and F 3D denote 2D and 3D components, respectively. The ratio between the 2D and 3D components derived from the fit ( F 2D / F 3D ) is ∼ 1.76, suggesting that the ε pocket exhibits mainly 2D character although a 3D component also exists. Now, we turn to the angle dependence of the Berry phase. As shown in Fig. 3g , the intercept for the topological ξ band shows strong angle dependence, similar to results in other systems such as Cd 3 As 2 [22] , ZrSiM (M = Se, Te) [23] , or ZrTe 5 [24] . For θ < 30° and θ ≥ 60°, the intercept falls between −1/8 and 1/8, while it falls between 3/8 and 5/8 for 30° ≤ θ < 60°. For the ε band, the intercept from 0° to 70° fluctuates between 1/8 and 5/8, averaging to 0.34(5), which is suggestive of trivial topology. However, when θ reaches 80° and 90°, this intercept falls between −1/8 and 1/8, implying nonzero Berry phase. For η and α ′ bands, the intercepts at all angles remain between 1/8 and 5/8, averaging 0.4(2) and 0.5(2), respectively, indicating trivial topology. Figure 3h, i show Hall results, which will be discussed later. TNL structure in the paramagnetic state Since our band structure calculations identify nodal loops at Y and our quantum oscillation data indicate nontrivial band topology, we also directly investigated the band structure with ARPES (Fig. 4 ). Momentum analysis in this technique is incompatible with magnetic field, so we investigated the paramagnetic rather than the field-polarized state; however, as shown in more detail in Supplementary Figs. 3 and 4 , a closed nodal loop persists at the Y point in the paramagnetic state. In order to visualize the nodal loop in EuAs 3 , the photon energy dependence of the electronic structure along the k y direction was investigated at 12 K within the vertical plane of the (010) cleaved surface, as sketched in Fig. 4c . Fig. 4: Verification of topological nodal-line structure by ARPES measurements in the paramagnetic state of EuAs 3 . a Photon energy-dependent plot of photoemission intensity in the k z – k y plane taken at E F −0.2 eV. b Photoemission intensity map of constant energy contours at 0.5 eV below E F in the k x – k z plane, the data were collected using photons with hν = 55 eV. c The Brillouin zone (BZ) of EuAs 3 , with high-symmetry points and (010) surface labeled. d The band dispersions along k y direction probed by different photon energies. The calculations plotted by the black dotted curves superimposed on the experimental electronic structure. The red ellipse illustrates the topological nontrivial nodal loop schematically. e The band dispersions along cuts 1–4 as indicated in b , respectively. f Corresponding energy-distribution curves (EDCs) taken at different photon energies. Full size image From the intensity plot of the Fermi surface at 12 K in the k y – k z plane (Fig. 4a ) taken at E F −0.2 eV, the pocket centered at the Y point (54 eV) can be easily identified, and two nodes arising from the crossing of the electron-like and hole-like bands can be also observed in Fig. 4b , which agrees with the band calculations. As observed in SrAs 3 , the drumhead-like surface state of EuAs 3 is buried in the bulk state, so it cannot be resolved by ARPES. For ARPES cuts away from the Y point, the band-crossing area shrinks gradually and finally disappears, and the topological nontrivial nodal loop encircled the Y point, illustrated by the red ellipse in Fig. 4d . Besides, the k y -dependent evolution of the band structure shows a good agreement with the calculations (the black dotted curves in Fig. 4d ) and the corresponding energy-distribution curves could further confirm the nodes introduced by the band crossing and their k y -dependent evolution in Fig. 4f (dashed line is a guide for the eyes to trace the dispersions). The evolution of the nodes along the k x -direction is presented in Fig. 4e , which shows the band dispersions along cuts 1–4 indicated in Fig. 4b (photoemission intensity map of constant energy contours at 0.5 eV below E F ). We noticed both the electron- and hole-like bands deplete their spectral weight from 1 to 4, consistent with the band-crossing scenario. However, whether or not a gap opens in the nodes away from k x = 0 remains vague due to the intrinsic broadness of the electron-like band. We also measure the electronic structure of another sample at 18 K, as shown in Supplementary Fig. 5 , and obtain the same results. And we also estimate the Fermi momenta k F and Fermi velocities v F to be k F = 0.12 and 0.14 Å − 1 and v F = 3.7 × 10 5 and 1.17 × 10 5 m/s, respectively, for the hole and electron bands, the same order of magnitude as for the ε and η bands (see Table 1 ). These data are extremely similar to what was found in SrAs 3 [9] . The verification of the nodal-line structure in the paramagnetic state by utilizing ARPES measurements and density functional theory (DFT) calculations shows remarkable agreement between the theoretical and experimental values. For the spin-polarized state, which is predicted to hosts closely similar but spin–split band structure to the paramagnetic state, nodal-line structure is thus strongly expected to exist in spin-polarized EuAs 3 . Very recently, lifted degeneracy of the Bloch bands was observed in the paramagnetic phase of EuCd 2 As 2 , producing a spin-fluctuation-induced Weyl semimetal state [25] . The magnetic susceptibility in EuAs 3 reveals a positive Curie–Weiss temperature T CW of 4.4 K for magnetic fields applied within the ab plane (Supplementary Fig. 6 ), suggestive of ferromagnetic fluctuations deep in the paramagnetic phase. The ferromagnetic correlations in EuAs 3 may induce band splitting within the paramagnetic phase, which may be resolvable with higher-resolution ARPES, such as laser ARPES."},{"section":"Discussion","content":"It is clear from our transport measurements that the electronic structure in the antiferromagnetically ordered state is very different from that found in the field-polarized or paramagnetic states. This is a consequence of the doubling of the unit cell due to AFM order and the coupling of this magnetic order to the electronic structure and is well described by our band calculations. However, a possible additional Lifshitz transition below 3 K has also been suggested. Figure 3h shows the Hall resistivity ( ρ xy ) from 0.3 to 30 K. The ρ xy curves are clearly nonlinear, implying the coexistence of two types of carriers, as predicted by band structure calculations. On cooling, the slope of the curve changes from positive to negative, indicating an increased contribution from electron carriers. The carrier concentration and mobility are extracted by fitting the low-field Hall conductivity with a two-carrier model, and results are summarized in Fig. 3i . For 3 ≤ T ≤ 30 K, the concentration of hole carriers is larger than that of electron carriers. Upon decreasing the temperature <3 K, the concentration of electron carriers is suddenly enhanced, accompanied by a sharp increase in the mobility of hole carriers. These indicate a possible Lifshitz transition. Temperature-induced Lifshitz transitions are also observed in other TSMs, for example, MTe 5 (M = Zr, Hf) [26] , [27] , InTe 1-δ [28] , ZrSiSe [29] , WTe 2 [30] , and TaIrTe 4 [31] . Anomalies in both longitudinal resistivity and Hall resistivity/coefficient can be found in MTe 5 (M = Zr, Hf) [26] , [27] , InTe 1-δ [28] , and ZrSiSe [29] , but not in WTe 2 [30] and TaIrTe 4 [31] . Since we have observed the change of Hall resistivity in EuAs 3 (Fig. 3h ), we wonder how the longitudinal resistivity evolves with decreasing temperature. The inset of Fig. 2a shows the low-temperature resistivity from 0.3 to 2.5 K in zero field, and we did not observe any distinct anomaly. Considering that the variations in resistivity may be very weak and the temperature range from 0.3 to 2.5 K is not appropriate, we measured two more samples (denoted as Sample 4 and Sample 5 in Supplementary Fig. 7a ), and found that there is a very weak anomaly at ~2.3 K in resistivity for both samples (see Supplementary Fig. 7b ). Since the Lifshitz transition should also manifest in quantum oscillations, we further check the low-field MR data below B M in Fig. 2b and the FFT in Fig. 2c . One can see that, due to the limited oscillatory periods and/or noise at 3 K, the Lifshitz transition cannot be resolved from the low-field quantum oscillation data. To verify it, other low-temperature probes are needed, for example ARPES and scanning tunneling microscopy/scanning tunneling spectroscopy. Now, we turn to the high-field state > B M . The temperature dependence of Hall coefficient measured at 9 T for Sample 5 is shown in Supplementary Fig. 7c . With decreasing temperature, a small peak at ~3.6 K arises and Hall coefficient changes its sign from positive to negative at ~2.3 K. We then analyze the oscillatory component (∆ ρ xy ) > B M , and a new oscillation frequency of 374 T (denoted as the φ band) can be clearly distinguished at 0.3 K, as shown in Supplementary Fig. 8a . The trivial topology nature for φ band has also been demonstrated (see Supplementary Fig. 8b ). Therefore, a temperature-induced Lifshitz transition likely exists in both AFM and spin-polarized states, although the change of Fermi surface topology with temperature in these two states may be different. Generally speaking, Lifshitz transitions are related to electronic transitions at zero temperature and involve abrupt changes of the Fermi surface topology. However, in topological materials, Lifshitz transitions can also involve other types of zero-energy modes, such as Weyl or Dirac nodes, nodal lines, flat bands, Majorana modes, etc. [32] . It has been proposed that multiple types of novel Lifshitz transitions involving Weyl points are possible depending on how they connect Fermi surfaces and pockets. For instance, the Lifshitz transition can correspond to the transfer of Berry flux between Fermi pockets connected by type-II Weyl points [33] . To understand the physics behind the low-temperature Lifshitz transition in EuAs 3 , more work is needed. According to the conventional charge-carrier compensation picture for XMR, the ratio n h / n e should be unity [1] , [2] . At 0.3 K, n h / n e for EuAs 3 is ∼ 1.0, consistent with this picture. However, for 3 ≤ T ≤ 30 K, n h /n e varies between 1.5 and 2.5 while the MR remains large and unsaturated, evidently excluding the charge-compensation picture for EuAs 3 . XMR is also frequently encountered in the cases of topologically protected electronic band structure and when open orbits contribute [1] , [2] , [34] , [35] . According to the open-orbit effect, the unsaturated XMR is only observed for current along the open orbits [35] . However, the observation of the unsaturated XMR with different current direction in EuAs 3 excludes the open-orbit effect (see Supplementary Fig. 9 ). Besides, for both the charge-carrier compensation picture and open-orbit effect, a B 2 dependence of MR is suggested [34] , [35] , which is different from the situation of EuAs 3 reported here (Supplementary Fig. 9c ). Since we have verified nontrivial band topology in EuAs 3 , we consider this the more likely explanation. In summary, combining band calculations, electrical transport, and ARPES measurements on the magnetic compound EuAs 3 , we report a magnetism-induced topological transition from a TNL semimetal in the paramagnetic or the spin-polarized state to a topological massive Dirac metal in the AFM ground state. The paramagnetic and spin-polarized states differ by the splitting of a topological nodal line associated with the spin splitting of the band structure. An XMR of ∼ 2 × 10 5 % and an as-yet-unexplained temperature-induced Lifshitz transition <3 K have also been revealed. These results indicate that magnetic EuAs 3 could serve as a unique platform to explore exotic physics at the interface of magnetism and topology."},{"section":"Methods","content":"Sample synthesis Eu (99.95%, Alfa Aesar), As (99.999%, PrMat), and Bi (99.9999%, Aladdin) blocks were mixed in a molar ratio of 1:3:26 and placed into an alumina crucible. The crucible was sealed in a quartz ampoule under vacuum and subsequently heated to 900 °C in 15 h. After reaction at this temperature for 20 h, the ampoule was cooled to 700 °C over 20 h and then slowly cooled to 450 °C at − 1 °C/h. The excess Bi flux was then removed using a centrifuge, revealing EuAs 3 single crystals with black shiny metallic luster. Electrical transport and thermodynamic measurements For electrical transport measurements, a single crystal was cut into a bar shape. A standard four-probe method was used for the longitudinal resistivity measurement. Data were collected in a 3 He and a 4 He cryostat. Magnetic susceptibility and specific heat measurements were performed in a magnetic property measurement system (MPMS, Quantum Design) and a physical property measurement system (PPMS, Quantum Design), respectively. High-field measurements were performed at the Steady High Magnetic Field Facilities, High Magnetic Field Laboratory, Chinese Academy of Sciences in Hefei. ARPES measurements ARPES measurements were performed at beam line BL13U at the National Synchrotron Radiation Laboratory (NSRL), China (photon energy hν = 12-38 eV); beam line BL03U of Shanghai Synchrotron Radiation Facility (SSRF), China (photon energy hν = 34-90 eV). The samples were cleaved in situ at 18 K (12 K) and measured in ultrahigh vacuum with a base pressure of better than 3.5 × 10 − 11 (5 × 10 −11 ) mbar at NSRL (SSRF). Data were recorded by a Scienta R4000 at NSRL and SSRF. The energy and momentum resolution were 10 meV and 0.2°, respectively. DFT calculations First-principles calculations were carried out within the framework of the projector augmented wave method [36] , [37] and employed the generalized gradient approximation (GGA) [38] with Perdew–Burke–Ernzerhof formula [39] , as implemented in the Vienna ab initio Simulation Package [40] . Two unit cells repeated along the b axis were adopted to simulate the AFM configuration indicated by neutron diffraction experiment for EuAs 3 [14] . The energy cutoff was chosen to be 500 eV. A Γ-centered 8 × 6 × 14 Monkhorst–Pack k -point grid was used to produce the well-converged results for the AFM phase. For the spin-polarized and paramagnetic band calculations, the same unit cell was used. Γ-centered 10 × 10 × 10 and 6 × 6 × 6 grids were used in the first BZ for the unit cell and supercell magnetic structures, respectively. The convergence criterion of energy in relaxation was set to be 10 − 6 eV and the atomic positions were fully relaxed until the maximum force on each atom was <0.02 eV/Å. The electronic correlations of Eu-4 f states were treated by the GGA + U method [41] . SOC was considered in a self-consistent manner. The Wannier90 package [42] was adopted to construct Wannier functions from the first-principles results. The WannierTools code [43] was used to investigate the topological features of surface state spectra."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"Topological semimetals (TSMs), including Dirac, Weyl, nodal-line, and triple-point semimetals, can be divided into two categories—non-magnetic and magnetic TSMs depending on whether magnetism is involved [1] , [2] , [3] . Compared with better-known non-magnetic TSMs, magnetic TSMs have unique properties due to their broken time-reversal symmetry (TRS): for example, nonzero net Berry curvatures that can induce anomalous Hall or Nernst effects, only one pair of Weyl nodes for some magnetic Weyl semimetals, and a good ability to manipulate the spin for spintronics applications [3] . Moreover, when magnetism is involved, interactions of the external magnetic field with the magnetic moments can result in exotic properties, such as Weyl states induced by magnetic exchange [4] , [5] . However, in contrast to non-magnetic TSMs, theoretical predictions and experimental studies on magnetic TSMs are rarer and more difficult due to the complexity of the magnetic configuration for calculations and the difficulty in synthesis of single crystals [3] . In fact, the very nature of the interaction between magnetism and topology in magnetic TSMs remains mysterious. Given how few such compounds are known, seeking and fully characterizing new magnetic TSMs is a priority for the new light they may shed on these issues. Recently, the non-magnetic CaP 3 family of materials was proposed as potential host of topological nodal-line (TNL) semimetals [6] , among which SrAs 3 possesses a TNL feature at ambient pressure [7] , [8] , [9] and exotic properties under high pressure [10] . Isostructural with SrAs 3 , EuAs 3 orders in an incommensurate antiferromagnetic (AFM) state at T N = 11 K, and then undergoes an incommensurate–commensurate lock-in phase transition at T L = 10.3 K, producing a collinear AFM ground state [11] , [12] , [13] , [14] , [15] , [16] . Previous electrical transport studies found an extremely anisotropic magnetoresistance (MR), which is strongly related to the magnetic configuration of EuAs 3 [17] . However, experiments sensitive to the topology have not been reported on EuAs 3 . In this paper, we demonstrate a magnetism-induced topological transition from a TNL semimetal in the paramagnetic or the spin-polarized state to a topological massive Dirac metal in the AFM ground state. First, we explore the band structure in the AFM ground state through band calculations and transport measurements, demonstrating that EuAs 3 is a magnetic topological massive Dirac metal. Second, Shubnikov–de Haas (SdH) oscillations and band calculations in the spin-polarized state are displayed, yielding a proposal that EuAs 3 is a TNL semimetal with an extremely large magnetoresistance (XMR) of ∼ 2 × 10 5 % at 1.8 K and 28.3 T. Third, our angle-resolved photoemission spectroscopy (ARPES) results in the paramagnetic state verify the nodal-line structure as predicted by band calculations. Ultimately, the origin of the XMR and a temperature-induced Lifshitz transition are revealed.\",\n \"references\": {\n \"indices\": [\n 1\n ],\n \"infos\": [\n {\n \"idx\": \"1\",\n \"title\": \"Armitage, N. P., Mele, E. J. & Vishwanath, A. Weyl and Dirac semimetals in three-dimensional solids.Rev. Mod. Phys.90, 015001 (2018).\",\n \"link\": \"https://doi.org/10.1103%2FRevModPhys.90.015001\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Topological massive Dirac metal state in the AFM state EuAs 3 crystallizes in a monoclinic structure (space group C 2/ m , No. 12), and the magnetic moments of Eu 2+ are oriented parallel and antiparallel to the monoclinic b axis [11] , [12] , [13] , [14] , [15] , [16] , as plotted in Fig. 1a . Figure 1b shows the bulk and (110) surface Brillouin zones (BZs) of EuAs 3 in the doubled unit cell corresponding to its AFM ground state. The calculated band structure including spin–orbit coupling (SOC) in this magnetic ground state as determined by neutron diffraction experiments [12] for EuAs 3 is displayed in Fig. 1c . In addition to topological bands around the Γ point, several trivial bands cross the Fermi level, indicating that EuAs 3 is a metal rather than a semimetal. In magnetic systems, TRS is broken. To preserve the Dirac node, extra symmetries, for example, the combination of inversion ( I ) and time-reversal ( T ) symmetries, i.e., IT , are necessary [3] . The Dirac band crossing is not topologically protected, and it can be gapped out by SOC to turn into a gapped dispersion of massive Dirac fermions [3] . Following this clue, two massive Dirac points around Γ point are identified, as shown in the inset to Fig. 1c . The complicated Fermi surface is composed of two hole sheets and one electron sheet in the AFM state (Fig. 1d ), all of three-dimensional (3D) character. The electron sheet consists of two individual pockets, i.e., electron_1 and electron_2, which can both be detected by quantum oscillations. Fig. 1: Topological massive Dirac metal state in the antiferromagnetic state of EuAs 3 , revealed by band calculations. a Schematic for the crystal structure of EuAs 3 in the doubled magnetic unit cell. The arrows on Eu 2+ represent the spin directions, which are parallel and antiparallel to the b axis. b Bulk and (110)-projected surface BZs of the doubled magnetic unit cell of EuAs 3 with several high-symmetry points marked. c Band structure of EuAs 3 from GGA + SOC + U ( U = 5 eV) calculations for the AFM ground state. The inset shows the massive Dirac point with a small gap. d Fermi surfaces of EuAs 3 derived from the band structure. e Projected band structure of EuAs 3 , where the symbol size represents the projected weight of Bloch states onto the As- p x , p y , and p z orbitals as labeled. Band inversion can be observed at the Γ point. The Wannier charge center is calculated in the f k z = 0 and g k z = 0.5 planes. h Calculated surface states on the (010) surface. The nontrivial topological surface states are clearly visible. Full size image Projected band structure analysis shows that the low energy states near the Fermi level are dominated by As-4 p states (Fig. 1e ). There are clear signatures of band inversion between As- p x,y and As- p z orbitals at the Γ point. To identify the topological character, we calculated the Z 2 invariant by employing the Willson loop method [18] . Figure 1f, g show the evolution of the Wannier charge center on two representative planes of the bulk BZ. From the calculations, the Z 2 invariant for the k z = 0 plane is 1, whereas Z 2 is 0 for the k z = 0.5 plane, providing strong evidence for nontrivial topology. Moreover, topologically protected surface states are expected, and we can unambiguously identify nontrivial surface states in the surface spectrum for the semi-infinite (010) surface, as displayed in Fig. 1h , confirming further the nontrivial topology in the AFM state. To verify the predictions from band calculations, we conducted electrical transport measurements. Resistivity in zero magnetic field is plotted in Fig. 2a , which displays typical metallic behavior with a low-temperature peak corresponding to the magnetic transitions. The magnetic transitions were also found by thermodynamic measurements (Supplementary Fig. 1 ) to be consistent with previous reports [11] , [12] , [13] , [14] , [15] , [16] . The inset to Fig. 2a shows the fit to the resistivity data below 2.5 K using a power law: ρ = ρ 0 + AT 2 , where ρ 0 is the residual resistivity and A the electronic scattering coefficient. The fit gives a residual resistivity ρ 0 of 2.6 µΩcm, and the residual resistivity ratio ρ 300K / ρ 0 is ~72. Figure 2b shows the low-field MR data with evident SdH oscillations. B M in Fig. 2b denotes the critical field, above which the spins are fully polarized by the external magnetic field. The SdH oscillation amplitude can be described by the Lifshitz–Kosevich formula [1] , [2] : \\\\(\\\\,\\\\Delta {{\\\\rho }}_{{xx}}{{{{{\\\\rm{\\\\propto }}}}}}\\\\,\\\\frac{2{{{{{{\\\\rm{\\\\pi }}}}}}}^{2}{k}_{{{{{{\\\\rm{B}}}}}}}T/{\\\\hslash {\\\\omega}}_{{{{{{\\\\rm{c}}}}}}}}{{{\\\\sinh }}\\\\left(2{{{{{{\\\\rm{\\\\pi }}}}}}}^{2}{k}_{{{{{{\\\\rm{B}}}}}}}T/{\\\\hslash {\\\\omega}}_{{{{{{\\\\rm{c}}}}}}}\\\\right)}{e}^{2{{{{{{\\\\rm{\\\\pi }}}}}}}^{2}{k}_{{{{{{\\\\rm{B}}}}}}}{T}_{{{{{{\\\\rm{D}}}}}}}/{\\\\hslash {\\\\omega}}_{{{{{{\\\\rm{c}}}}}}}}{{\\\\cos }}2{{{{{\\\\rm{\\\\pi }}}}}}\\\\,(\\\\frac{F}{B}-{\\\\gamma }+{\\\\delta }),\\\\) where ω c = eB / m * is the cyclotron frequency and T D is the Dingle temperature. \\\\({\\\\gamma }=\\\\frac{1}{2}-\\\\left(\\\\frac{1}{2{\\\\pi }}\\\\right){\\\\phi }_{{{{{{\\\\rm{B}}}}}}}\\\\) (0 ≤ γ ≤ 1) is the Onsager phase factor, and ϕ B is a geometrical phase known as the Berry phase. For a topological system with peculiar electron state degeneracy and intra-band coupling, a π Berry phase will be observed. 2 πδ is an additional phase shift resulting from the curvature of the Fermi surface in the third direction, where δ varies from 0 to ±1/8 for a quasi-two-dimensional (quasi-2D) cylindrical Fermi surface and a corrugated 3D Fermi surface, respectively [7] , [8] . The cyclotron effective mass m * can be obtained from the thermal damping factor \\\\({{R}}_{T}=\\\\frac{2{{\\\\pi }}^{2}{k}_{{{{{{\\\\rm{B}}}}}}}T/{\\\\hslash {\\\\omega}}_{{{{{{\\\\rm{c}}}}}}}}{{{\\\\sinh }}\\\\left(2{{\\\\pi }}^{2}{k}_{{{{{{\\\\rm{B}}}}}}}T/{\\\\hslash {\\\\omega}}_{{{{{{\\\\rm{c}}}}}}}\\\\right)}\\\\) . Fig. 2: Quantum oscillations and negative longitudinal magnetoresistance ( n -LMR) in the antiferromagnetic state of EuAs 3 . a Resistivity of EuAs 3 single crystal in zero magnetic field. The inset shows the fit to the low-temperature data. b MR accompanied by distinct SdH oscillations. B M represents the critical magnetic field, which induces a magnetic transition from a collinear antiferromagnetic phase to a polarized ferromagnetic phase. c FFT results at various temperatures. The inset displays the oscillatory component ρ xx below B M . Four bands, i.e., α , β , γ 1 , and γ 2 , can be distinguished. The latter two construct one electron sheet. d Landau index n plotted against 1/ B for the SdH oscillations at 0.3 K. Lines represent linear fits. The right inset shows the extrapolation of 1/ B to zero. The left inset shows the normalized FFT amplitude (Amp. nor ) as a function of temperature, and the solid lines represent fits to the Lifshitz–Kosevich formula. e n -LMR measured with magnetic field parallel to the electric current I at various temperatures. f Longitudinal conductivity at various temperatures fit to the Adler–Bell–Jackiw chiral anomaly equation. The inset shows the emergence of a positive parameter originating from the chiral anomaly C w , and error bar of the data is determined from fitting. Full size image By analyzing the oscillatory component (inset to Fig. 2c ) below B M via fast Fourier transform (FFT), four bands are uncovered, i.e., 156, 185, 217, and 7 T, referred to as α , β , γ 1 , and γ 2 , respectively, in line with the band calculations. To check their topological nature, a Landau index fan diagram is plotted in Fig. 2d , yielding intercepts of 0.6(2), 0.5(1), −0.03(9), and 0.07(8) for α , β , γ 1 , and γ 2 , respectively. Throughout this paper, we assign integer indices to the ∆ ρ xx peak positions in 1/ B and half integer indices to the ∆ ρ xx valleys. According to the Lifshitz–Onsager quantization rule for a corrugated 3D Fermi surface, intercepts falling between −1/8 and 1/8 suggest nonzero Berry phase, while intercepts in the range 3/8–5/8 indicate trivial band topology. Therefore, the γ 1 and γ 2 bands may be topologically protected, while the other two are trivial. However, the index number is >20 ( α , β , and γ 1 pockets), and hence the extrapolation from the Landau fan plots may have biggish uncertainty. In order to validate the topological nature, more solid evidences are needed. Other parameters for these four bands, such as the Fermi energy E F , extremal cross-sectional areas A F , Fermi momentum k F , Fermi velocity v F , cyclotron effective mass m ∗ , and Dingle temperature T D , are calculated and summarized in Table 1 . Table 1 Parameters derived from quantum oscillations in different magnetic field range for EuAs 3 . Full size table In TSMs, in addition to nonzero Berry phase, the n -LMR induced by the chiral anomaly can also serve as a smoking gun for nontrivial topology [19] , [20] , [21] . Figure 2e displays the n -LMR of EuAs 3 with magnetic field parallel to the electric current I . The kinks in the n -LMR curves below the ordering temperature arise from the field-induced transitions [15] , [16] , [17] , i.e., from a collinear AFM phase to incommensurate and commensurate spiral phases [15] , [16] , [17] . Negative MR in magnetic systems is not uncommon [19] , [20] , [21] when the applied magnetic field suppresses the inelastic magnetic scattering from local moments or magnetic impurities, leading to a negative MR for charge transport along all directions [19] , [20] , [21] . However, in EuAs 3 we only observed a n -LMR when the magnetic field is applied parallel to the electric current (Supplementary Fig. 2 ). Furthermore, if the applied external magnetic field has a strong effect on the magnetic scattering and induces a n -LMR, the changes in the n -LMR will occur predominantly below and above the ordering temperature. This is not observed. Instead, we find several minor kinks arising from the magnetic transitions, on top of a much larger signal. Therefore, the suppression of magnetic scattering can be excluded as the origin of the n -LMR in EuAs 3 . The n -LMR also displays a wide variety of temperature dependences, ruling out current jetting effects and the weak localization effect [19] , [20] , [21] . Since nontrivial band topology has been suggested in EuAs 3 , the chiral anomaly arising from Weyl fermions is the most likely mechanism behind the n -LMR. The n -LMR induced by the chiral anomaly in TSMs can be analyzed through the Adler–Bell–Jackiw (ABJ) chiral anomaly equation [19] , [20] , [21] : \\\\({{{{{\\\\rm{\\\\sigma }}}}}}\\\\left(B\\\\right)=\\\\left({1+C}_{{{{{{\\\\rm{w}}}}}}}{B}^{2}\\\\right)\\\\left({\\\\sigma }_{0}+a\\\\sqrt{B}\\\\right)+{{{{{{\\\\rm{\\\\sigma }}}}}}}_{{{{{{\\\\rm{N}}}}}}}\\\\) , where σ 0 , C w , and \\\\({\\\\sigma }_{{{{{{\\\\rm{N}}}}}}}^{-1}={{\\\\rho }}_{0}+A{B}^{2}\\\\) denote the conductivity at zero field, a temperature-dependent positive parameter originating from chiral anomaly, and the conventional nonlinear band contribution around Fermi energy, respectively. Figure 2f shows the conductivity and the fit to the data below 3 T at various temperatures. The data above the ordering temperature are well described by the ABJ equation, while the data at lower temperatures do not fit as well, which may be ascribed to magnetic transitions or topological transitions. The inset to Fig. 2f shows the temperature dependence of C w . At 2 K, C w is 0.253(7) T −2 . With increasing temperature, a clear anomaly in C w around the ordering temperature can be observed, verifying the proposal above. When T > T N , C w decreases monotonically, as observed in SrAs 3 [7] . Taken together, these results demonstrate that EuAs 3 is a magnetic topological massive Dirac metal in its AFM ground state. Topological state in the spin-polarized state We now turn to the exploration of topology in the spin-polarized state, where in Fig. 2b we have already observed a clear change in the quantum oscillations. Figure 3a plots the MR of EuAs 3 in higher magnetic field, and an unsaturated XMR ~2 × 10 5 % at 1.8 K and 28.3 T is observed. By analyzing the oscillatory components above B M (inset in Fig. 3b ), frequency components are identified at F = 93, 158, 346, and 597 T, which are referred to here as the ξ , α ′, ε , and η bands, respectively (Fig. 3b ). These four bands are different from those in lower field (Fig. 2c ), indicating that they are likely rooted in different band structure. This is unsurprising since the unit cell is no longer doubled by antiferromagnetism, but the field-induced spin polarization can also play a significant role. We thus conducted band calculations for the field-polarized state (Supplementary Fig. 3 ) and the paramagnetic state (Supplementary Fig. 4 ), and these are indeed quite different, as we discuss in more detail in Supplementary Notes 3 and 4 . Fig. 3: Quantum oscillation study in the spin-polarized state of EuAs 3 and Hall resistivity measurements. a Magnetoresistance measurements of EuAs 3 single crystal under higher magnetic field up to 28.3 T. b FFT results at various temperatures, yielding the four bands ξ , α ′, ε , and η . The inset displays the oscillatory component ρ xx above B M . c Landau index n plotted against 1/ B for the SdH oscillations at 1.8 K. The left inset shows the extrapolation of 1/ B to zero. The right inset shows the normalized FFT amplitude (Amp. nor ) as a function of temperature, and the solid lines represent the Lifshitz–Kosevich formula fit. d SdH oscillations at different angles; the inset is a schematic illustration of the experimental geometry and the angle θ . For θ = 0°, the magnetic field is parallel to the c axis. For θ = 90°, the magnetic field is applied along the [110] direction. e The oscillatory component ρ xx as a function of 1/ B . Angular dependence of f the FFT frequencies, where error bars represent the full widths at half maximum of the FFT peaks, and g the Landau level index intercepts. The error bar of the intercepts is determined from fitting. h Hall resistivity at various temperatures. i Carrier concentration and mobility as a function of temperature, and error bar is determined from fitting. The shadow area represents the temperature interval where a Lifshitz transition takes place. Full size image In the field-polarized state, we find four Fermi surface sheets—two electron and two hole sheets and double nodal loops at the Y point, one each for the spin-up and spin-down channels (Supplementary Fig. 3d ). To identify the topological nature of the four bands seen in quantum oscillations, a Landau index fan diagram is plotted in Fig. 3c , and the intercepts are −0.0(1), 0.67(3), 0.34(4), and 0.61(4) for the ξ , α ′, ε , and η bands, respectively. Although the intercepts from the fit with large Landau index number cannot serve as a smoking gun for topology, we still use them, because it is difficult for us to evaluate the topological nature in the fully spin-polarized state above 11.0 T (the critical field is deduced from specific heat in Supplementary Fig. 1c ). The intercepts give a hint that the ξ band is topologically protected, while the α ′ and η bands are topologically trivial. The intermediate value for ε is suggestive of a possible nontrivial Berry phase but does not allow a strong conclusion and will require further verification. The cyclotron effective masses m * for these four pockets can be obtained by fitting the temperature dependence of the normalized FFT amplitude, as shown in the right inset to Fig. 3c . Other parameters can be also extracted, and all values are summarized in Table 1 . To better reveal the Fermi surface anisotropy and topology of EuAs 3 , angle-dependent MR measurements have been performed at 1.8 K, in the experimental geometry shown in the inset to Fig. 3d . Upon rotating sample from 0° to 90°, the magnitude of the MR is reduced monotonically, as also seen in a polar plot of the MR (Supplementary Fig. 2b ). We extract the frequency components for the ξ , α ′, ε , and η bands by analyzing the oscillatory component (Fig. 3e ) and summarize the results in Fig. 3f . The angle dependence of the ξ , α ′, and η bands is of 3D character, while the ε band is well described below 50° by the formula F = F 3D + F 2D /cos( θ ), where F 2D and F 3D denote 2D and 3D components, respectively. The ratio between the 2D and 3D components derived from the fit ( F 2D / F 3D ) is ∼ 1.76, suggesting that the ε pocket exhibits mainly 2D character although a 3D component also exists. Now, we turn to the angle dependence of the Berry phase. As shown in Fig. 3g , the intercept for the topological ξ band shows strong angle dependence, similar to results in other systems such as Cd 3 As 2 [22] , ZrSiM (M = Se, Te) [23] , or ZrTe 5 [24] . For θ < 30° and θ ≥ 60°, the intercept falls between −1/8 and 1/8, while it falls between 3/8 and 5/8 for 30° ≤ θ < 60°. For the ε band, the intercept from 0° to 70° fluctuates between 1/8 and 5/8, averaging to 0.34(5), which is suggestive of trivial topology. However, when θ reaches 80° and 90°, this intercept falls between −1/8 and 1/8, implying nonzero Berry phase. For η and α ′ bands, the intercepts at all angles remain between 1/8 and 5/8, averaging 0.4(2) and 0.5(2), respectively, indicating trivial topology. Figure 3h, i show Hall results, which will be discussed later. TNL structure in the paramagnetic state Since our band structure calculations identify nodal loops at Y and our quantum oscillation data indicate nontrivial band topology, we also directly investigated the band structure with ARPES (Fig. 4 ). Momentum analysis in this technique is incompatible with magnetic field, so we investigated the paramagnetic rather than the field-polarized state; however, as shown in more detail in Supplementary Figs. 3 and 4 , a closed nodal loop persists at the Y point in the paramagnetic state. In order to visualize the nodal loop in EuAs 3 , the photon energy dependence of the electronic structure along the k y direction was investigated at 12 K within the vertical plane of the (010) cleaved surface, as sketched in Fig. 4c . Fig. 4: Verification of topological nodal-line structure by ARPES measurements in the paramagnetic state of EuAs 3 . a Photon energy-dependent plot of photoemission intensity in the k z – k y plane taken at E F −0.2 eV. b Photoemission intensity map of constant energy contours at 0.5 eV below E F in the k x – k z plane, the data were collected using photons with hν = 55 eV. c The Brillouin zone (BZ) of EuAs 3 , with high-symmetry points and (010) surface labeled. d The band dispersions along k y direction probed by different photon energies. The calculations plotted by the black dotted curves superimposed on the experimental electronic structure. The red ellipse illustrates the topological nontrivial nodal loop schematically. e The band dispersions along cuts 1–4 as indicated in b , respectively. f Corresponding energy-distribution curves (EDCs) taken at different photon energies. Full size image From the intensity plot of the Fermi surface at 12 K in the k y – k z plane (Fig. 4a ) taken at E F −0.2 eV, the pocket centered at the Y point (54 eV) can be easily identified, and two nodes arising from the crossing of the electron-like and hole-like bands can be also observed in Fig. 4b , which agrees with the band calculations. As observed in SrAs 3 , the drumhead-like surface state of EuAs 3 is buried in the bulk state, so it cannot be resolved by ARPES. For ARPES cuts away from the Y point, the band-crossing area shrinks gradually and finally disappears, and the topological nontrivial nodal loop encircled the Y point, illustrated by the red ellipse in Fig. 4d . Besides, the k y -dependent evolution of the band structure shows a good agreement with the calculations (the black dotted curves in Fig. 4d ) and the corresponding energy-distribution curves could further confirm the nodes introduced by the band crossing and their k y -dependent evolution in Fig. 4f (dashed line is a guide for the eyes to trace the dispersions). The evolution of the nodes along the k x -direction is presented in Fig. 4e , which shows the band dispersions along cuts 1–4 indicated in Fig. 4b (photoemission intensity map of constant energy contours at 0.5 eV below E F ). We noticed both the electron- and hole-like bands deplete their spectral weight from 1 to 4, consistent with the band-crossing scenario. However, whether or not a gap opens in the nodes away from k x = 0 remains vague due to the intrinsic broadness of the electron-like band. We also measure the electronic structure of another sample at 18 K, as shown in Supplementary Fig. 5 , and obtain the same results. And we also estimate the Fermi momenta k F and Fermi velocities v F to be k F = 0.12 and 0.14 Å − 1 and v F = 3.7 × 10 5 and 1.17 × 10 5 m/s, respectively, for the hole and electron bands, the same order of magnitude as for the ε and η bands (see Table 1 ). These data are extremely similar to what was found in SrAs 3 [9] . The verification of the nodal-line structure in the paramagnetic state by utilizing ARPES measurements and density functional theory (DFT) calculations shows remarkable agreement between the theoretical and experimental values. For the spin-polarized state, which is predicted to hosts closely similar but spin–split band structure to the paramagnetic state, nodal-line structure is thus strongly expected to exist in spin-polarized EuAs 3 . Very recently, lifted degeneracy of the Bloch bands was observed in the paramagnetic phase of EuCd 2 As 2 , producing a spin-fluctuation-induced Weyl semimetal state [25] . The magnetic susceptibility in EuAs 3 reveals a positive Curie–Weiss temperature T CW of 4.4 K for magnetic fields applied within the ab plane (Supplementary Fig. 6 ), suggestive of ferromagnetic fluctuations deep in the paramagnetic phase. The ferromagnetic correlations in EuAs 3 may induce band splitting within the paramagnetic phase, which may be resolvable with higher-resolution ARPES, such as laser ARPES.\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"It is clear from our transport measurements that the electronic structure in the antiferromagnetically ordered state is very different from that found in the field-polarized or paramagnetic states. This is a consequence of the doubling of the unit cell due to AFM order and the coupling of this magnetic order to the electronic structure and is well described by our band calculations. However, a possible additional Lifshitz transition below 3 K has also been suggested. Figure 3h shows the Hall resistivity ( ρ xy ) from 0.3 to 30 K. The ρ xy curves are clearly nonlinear, implying the coexistence of two types of carriers, as predicted by band structure calculations. On cooling, the slope of the curve changes from positive to negative, indicating an increased contribution from electron carriers. The carrier concentration and mobility are extracted by fitting the low-field Hall conductivity with a two-carrier model, and results are summarized in Fig. 3i . For 3 ≤ T ≤ 30 K, the concentration of hole carriers is larger than that of electron carriers. Upon decreasing the temperature <3 K, the concentration of electron carriers is suddenly enhanced, accompanied by a sharp increase in the mobility of hole carriers. These indicate a possible Lifshitz transition. Temperature-induced Lifshitz transitions are also observed in other TSMs, for example, MTe 5 (M = Zr, Hf) [26] , [27] , InTe 1-δ [28] , ZrSiSe [29] , WTe 2 [30] , and TaIrTe 4 [31] . Anomalies in both longitudinal resistivity and Hall resistivity/coefficient can be found in MTe 5 (M = Zr, Hf) [26] , [27] , InTe 1-δ [28] , and ZrSiSe [29] , but not in WTe 2 [30] and TaIrTe 4 [31] . Since we have observed the change of Hall resistivity in EuAs 3 (Fig. 3h ), we wonder how the longitudinal resistivity evolves with decreasing temperature. The inset of Fig. 2a shows the low-temperature resistivity from 0.3 to 2.5 K in zero field, and we did not observe any distinct anomaly. Considering that the variations in resistivity may be very weak and the temperature range from 0.3 to 2.5 K is not appropriate, we measured two more samples (denoted as Sample 4 and Sample 5 in Supplementary Fig. 7a ), and found that there is a very weak anomaly at ~2.3 K in resistivity for both samples (see Supplementary Fig. 7b ). Since the Lifshitz transition should also manifest in quantum oscillations, we further check the low-field MR data below B M in Fig. 2b and the FFT in Fig. 2c . One can see that, due to the limited oscillatory periods and/or noise at 3 K, the Lifshitz transition cannot be resolved from the low-field quantum oscillation data. To verify it, other low-temperature probes are needed, for example ARPES and scanning tunneling microscopy/scanning tunneling spectroscopy. Now, we turn to the high-field state > B M . The temperature dependence of Hall coefficient measured at 9 T for Sample 5 is shown in Supplementary Fig. 7c . With decreasing temperature, a small peak at ~3.6 K arises and Hall coefficient changes its sign from positive to negative at ~2.3 K. We then analyze the oscillatory component (∆ ρ xy ) > B M , and a new oscillation frequency of 374 T (denoted as the φ band) can be clearly distinguished at 0.3 K, as shown in Supplementary Fig. 8a . The trivial topology nature for φ band has also been demonstrated (see Supplementary Fig. 8b ). Therefore, a temperature-induced Lifshitz transition likely exists in both AFM and spin-polarized states, although the change of Fermi surface topology with temperature in these two states may be different. Generally speaking, Lifshitz transitions are related to electronic transitions at zero temperature and involve abrupt changes of the Fermi surface topology. However, in topological materials, Lifshitz transitions can also involve other types of zero-energy modes, such as Weyl or Dirac nodes, nodal lines, flat bands, Majorana modes, etc. [32] . It has been proposed that multiple types of novel Lifshitz transitions involving Weyl points are possible depending on how they connect Fermi surfaces and pockets. For instance, the Lifshitz transition can correspond to the transfer of Berry flux between Fermi pockets connected by type-II Weyl points [33] . To understand the physics behind the low-temperature Lifshitz transition in EuAs 3 , more work is needed. According to the conventional charge-carrier compensation picture for XMR, the ratio n h / n e should be unity [1] , [2] . At 0.3 K, n h / n e for EuAs 3 is ∼ 1.0, consistent with this picture. However, for 3 ≤ T ≤ 30 K, n h /n e varies between 1.5 and 2.5 while the MR remains large and unsaturated, evidently excluding the charge-compensation picture for EuAs 3 . XMR is also frequently encountered in the cases of topologically protected electronic band structure and when open orbits contribute [1] , [2] , [34] , [35] . According to the open-orbit effect, the unsaturated XMR is only observed for current along the open orbits [35] . However, the observation of the unsaturated XMR with different current direction in EuAs 3 excludes the open-orbit effect (see Supplementary Fig. 9 ). Besides, for both the charge-carrier compensation picture and open-orbit effect, a B 2 dependence of MR is suggested [34] , [35] , which is different from the situation of EuAs 3 reported here (Supplementary Fig. 9c ). Since we have verified nontrivial band topology in EuAs 3 , we consider this the more likely explanation. In summary, combining band calculations, electrical transport, and ARPES measurements on the magnetic compound EuAs 3 , we report a magnetism-induced topological transition from a TNL semimetal in the paramagnetic or the spin-polarized state to a topological massive Dirac metal in the AFM ground state. The paramagnetic and spin-polarized states differ by the splitting of a topological nodal line associated with the spin splitting of the band structure. An XMR of ∼ 2 × 10 5 % and an as-yet-unexplained temperature-induced Lifshitz transition <3 K have also been revealed. These results indicate that magnetic EuAs 3 could serve as a unique platform to explore exotic physics at the interface of magnetism and topology.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Sample synthesis Eu (99.95%, Alfa Aesar), As (99.999%, PrMat), and Bi (99.9999%, Aladdin) blocks were mixed in a molar ratio of 1:3:26 and placed into an alumina crucible. The crucible was sealed in a quartz ampoule under vacuum and subsequently heated to 900 °C in 15 h. After reaction at this temperature for 20 h, the ampoule was cooled to 700 °C over 20 h and then slowly cooled to 450 °C at − 1 °C/h. The excess Bi flux was then removed using a centrifuge, revealing EuAs 3 single crystals with black shiny metallic luster. Electrical transport and thermodynamic measurements For electrical transport measurements, a single crystal was cut into a bar shape. A standard four-probe method was used for the longitudinal resistivity measurement. Data were collected in a 3 He and a 4 He cryostat. Magnetic susceptibility and specific heat measurements were performed in a magnetic property measurement system (MPMS, Quantum Design) and a physical property measurement system (PPMS, Quantum Design), respectively. High-field measurements were performed at the Steady High Magnetic Field Facilities, High Magnetic Field Laboratory, Chinese Academy of Sciences in Hefei. ARPES measurements ARPES measurements were performed at beam line BL13U at the National Synchrotron Radiation Laboratory (NSRL), China (photon energy hν = 12-38 eV); beam line BL03U of Shanghai Synchrotron Radiation Facility (SSRF), China (photon energy hν = 34-90 eV). The samples were cleaved in situ at 18 K (12 K) and measured in ultrahigh vacuum with a base pressure of better than 3.5 × 10 − 11 (5 × 10 −11 ) mbar at NSRL (SSRF). Data were recorded by a Scienta R4000 at NSRL and SSRF. The energy and momentum resolution were 10 meV and 0.2°, respectively. DFT calculations First-principles calculations were carried out within the framework of the projector augmented wave method [36] , [37] and employed the generalized gradient approximation (GGA) [38] with Perdew–Burke–Ernzerhof formula [39] , as implemented in the Vienna ab initio Simulation Package [40] . Two unit cells repeated along the b axis were adopted to simulate the AFM configuration indicated by neutron diffraction experiment for EuAs 3 [14] . The energy cutoff was chosen to be 500 eV. A Γ-centered 8 × 6 × 14 Monkhorst–Pack k -point grid was used to produce the well-converged results for the AFM phase. For the spin-polarized and paramagnetic band calculations, the same unit cell was used. Γ-centered 10 × 10 × 10 and 6 × 6 × 6 grids were used in the first BZ for the unit cell and supercell magnetic structures, respectively. The convergence criterion of energy in relaxation was set to be 10 − 6 eV and the atomic positions were fully relaxed until the maximum force on each atom was <0.02 eV/Å. The electronic correlations of Eu-4 f states were treated by the GGA + U method [41] . SOC was considered in a self-consistent manner. The Wannier90 package [42] was adopted to construct Wannier functions from the first-principles results. The WannierTools code [43] was used to investigate the topological features of surface state spectra.\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"The core research idea is to demonstrate a magnetism-induced topological transition from a topological nodal-line semimetal in the paramagnetic or spin-polarized state to a topological massive Dirac metal in the antiferromagnetic ground state in EuAs₃, through a combination of band calculations, transport measurements, Shubnikov–de Haas oscillations, and angle-resolved photoemission spectroscopy, while also revealing the origin of the extremely large magnetoresistance and a temperature-induced Lifshitz transition."},"entities":{"kind":"list like","value":["magnetism-induced topological transition","topological nodal-line semimetal","paramagnetic state","spin-polarized state","topological massive Dirac metal","antiferromagnetic ground state","EuAs₃","band calculations","transport measurements","Shubnikov–de Haas oscillations","angle-resolved photoemission spectroscopy","extremely large magnetoresistance","temperature-induced Lifshitz transition"],"string":"[\n \"magnetism-induced topological transition\",\n \"topological nodal-line semimetal\",\n \"paramagnetic state\",\n \"spin-polarized state\",\n \"topological massive Dirac metal\",\n \"antiferromagnetic ground state\",\n \"EuAs₃\",\n \"band calculations\",\n \"transport measurements\",\n \"Shubnikov–de Haas oscillations\",\n \"angle-resolved photoemission spectroscopy\",\n \"extremely large magnetoresistance\",\n \"temperature-induced Lifshitz transition\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1.","title":"Armitage, N. P., Mele, E. J. & Vishwanath, A. Weyl and Dirac semimetals in three-dimensional solids.Rev. Mod. Phys.90, 015001 (2018).","link":"https://doi.org/10.1103%2FRevModPhys.90.015001"}],"string":"[\n {\n \"idx\": \"1.\",\n \"title\": \"Armitage, N. P., Mele, E. J. & Vishwanath, A. Weyl and Dirac semimetals in three-dimensional solids.Rev. Mod. Phys.90, 015001 (2018).\",\n \"link\": \"https://doi.org/10.1103%2FRevModPhys.90.015001\"\n }\n]","isConversational":false}}},{"rowIdx":188,"cells":{"unique_id":{"kind":"string","value":"ncomms9590"},"subfield":{"kind":"string","value":"Astronomy and planetary science"},"title":{"kind":"string","value":"Wave-driven butterfly distribution of Van Allen belt relativistic electrons"},"abstract":{"kind":"string","value":"Van Allen radiation belts consist of relativistic electrons trapped by Earth's magnetic field. Trapped electrons often drift azimuthally around Earth and display a butterfly pitch angle distribution of a minimum at 90° further out than geostationary orbit. This is usually attributed to drift shell splitting resulting from day–night asymmetry in Earth’s magnetic field. However, direct observation of a butterfly distribution well inside of geostationary orbit and the origin of this phenomenon have not been provided so far. Here we report high-resolution observation that a unusual butterfly pitch angle distribution of relativistic electrons occurred within 5 Earth radii during the 28 June 2013 geomagnetic storm. Simulation results show that combined acceleration by chorus and magnetosonic waves can successfully explain the electron flux evolution both in the energy and butterfly pitch angle distribution. The current provides a great support for the mechanism of wave-driven butterfly distribution of relativistic electrons."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"The Earth’s outer Van Allen radiation belt is composed of trapped electrons with relativistic energy ( E k ≥1 MeV). Fluxes and pitch angle distributions of relativistic electrons often exhibit dramatic and highly dynamic changes during geomagnetic storms or substorms, which are associated with different physical mechanisms. Those mechanisms include transport, energization and loss processes. Wave–particle interaction plays an important role in energy exchange between various modes of plasma waves and Van Allen radiation belt relativistic electrons. Two types of plasma waves, chorus and magnetosonic (MS) wave (also known as ‘equatorial noise’), can efficiently accelerate electrons up to relativistic energies. Relativistic electrons can pose serious damage to satellites and astronauts in space, it is therefore important to understand and ultimately predict Van Allen radiation belt electron dynamic variations. Radiation belt trapped electrons experience azimuthal drift-motion around Earth when they bounce between northern and southern magnetic mirror points. Due to the day–night azimuthal asymmetry in Earth’s magnetic field, their drift shells which are traced out by their guiding centres can separate radially for different pitch angles, which is known as drift shell splitting [1] . Significant drift shell splitting occurs only in the outer radiation belt L >6 (with L being the radial distance in Earth radii R E ), where the asymmetry becomes substantial. Previous theoretical [1] , [2] and observational [3] , [4] , [5] works have long confirmed drift shell splitting. Drift shell splitting separates the high and low pitch angle particles in nightside injections as they move to the dayside magnetosphere. The higher pitch angle particles drift to larger radial distance beyond the magnetopause on the dayside and may consequently be lost from the distribution [6] . Therefore, drift shell splitting can be interpreted in terms of butterfly pitch angle distributions, namely, a sharp dropout in the flux of 90° electrons. Another drift shell splitting is in combination with a negative radial flux gradient as equatorially mirroring particles drift around Earth at locations L =6–12 (refs 7 , 8 ). Drift shell splitting will most easily affect trapped particle populations further out than geosynchronous orbit. Occurrence of drift shell splitting inside of geosynchronous orbit would need a very large magnetopause compression during very large geomagnetic storms. A fundamental problem, both theoretically and observationally, is that there is any butterfly pitch angle distribution of relativistic electrons below L =5 associated with a new mechanism instead of drift shell splitting. Previous studies have analysed the characteristics and evolution of pitch angle distributions of the outer radiation belt electrons [9] , [10] . Chorus and MS waves were proposed to produce butterfly distributions by preferentially accelerating off-equator electrons [11] , [12] . Adiabatic processes could potentially yield the formation of butterfly distributions at locations L ≥6 (refs 13 , 14 ). Direct confirmation of wave-driven butterfly distributions below L =5 requires simultaneous high-resolution data but this is generally unavailable before the launch of NASA’s Van Allen Radiation Belt Storm Probes in 2012 (ref. 15 ). Fortunately, the unique events on the 28 June 2013 geomagnetic storm observed from Van Allen Probes provide an excellent opportunity to identify such mechanisms. Here we report the formation of a unusual butterfly pitch angle distribution of relativistic electrons around L =4.8, corresponding to the occurrence of strong chorus and MS waves at the same time. Using two-dimensional simulation, we show that flux enhancements of relativistic electrons are most pronounced between the medium pitch angles 30° and 60° due to the dominant acceleration process for combined chorus and MS waves. Meanwhile, the pitch angle distributions close to 90° are increased due to relatively larger acceleration process for chorus alone. The combined acceleration by chorus and MS waves substantially modifies the whole population of relativistic electrons, producing the butterfly distribution. Our detailed modelling, together with the correlated Van Allen Probes data, presents a further understanding on how chorus and MS waves play different roles in Earth’s outer radiation belts.","references":{"indices":[1,2,3,4,5,6,10,11,12],"infos":[{"idx":"1","title":"Roederer, J. G. On the adiabatic motion of energetic particles in a model magnetosphere.J. Geophys. Res.72, 981–992 (1967).","link":"https://doi.org/10.1029%2FJZ072i003p00981"},{"idx":"2","title":"Stone, E. C. The physical significance of and application ofL,B0 andR0 to geomagnetically trapped particles.J. Geophys. Res.68, 4157–4166 (1963).","link":"https://doi.org/10.1029%2FJZ068i014p04157"},{"idx":"3","title":"West, H. I., Buck, R. M. Jr & Walton, J. R. Electron pitch angle distributions throughout the magnetosphere as observed on Ogo 5.J. Geophys. Res.78, 1064–1081 (1973).","link":"https://doi.org/10.1029%2FJA078i007p01064"},{"idx":"4","title":"Baker, D. N., Higbie, P. R., Hones, E. W. Jr & Belian, R. D. High-resolution energetic particle measurements at 6.6RE. Low-energy electron anisotropies and short-term substorm predictions.J. Geophys. Res.83, 4863–4868 (1978).","link":"https://doi.org/10.1029%2FJA083iA10p04863"},{"idx":"5","title":"Takahashi, K. et al. Drift-shell splitting of energetic ions injected at pseudo-substorm onsets.J. Geophys. Res.102, 22117–22130 (1997).","link":"https://doi.org/10.1029%2F97JA01870"},{"idx":"6","title":"Selesnick, R. S. & Blake, J. B. Relativistic electron drift shell splitting.J. Geophys. Res.1079, 1265 (2002).","link":"https://doi.org/10.1029%2F2001JA009179"},{"idx":"10","title":"Gannon, J. L., Li, X. & Heynderickx, D. Pitch angle distribution analysis of radiation belt electrons based on combined release and radiation effects satellite medium electrons A data.J. Geophys. Res.112, A05212 (2007).","link":"https://doi.org/10.1029%2F2005JA011565"},{"idx":"11","title":"Horne, R. B. et al. Timescale for radiation belt electron acceleration by whistler mode chorus waves.J. Geophys. Res.110, A03225 (2005).","link":"https://doi.org/10.1029%2F2004JA010811"},{"idx":"12","title":"Horne, R. B. et al. Electron acceleration in the Van Allen radiation belts by fast magnetosonic waves.Geophys. Res. Lett.34, L17107 (2007).","link":"https://doi.org/10.1029%2F2007GL030267"}]}},{"section":"Results","content":"Correlated Van Allen Probe data On 28 June 2013, a moderate storm with Dst≈−100 nT ( Fig. 1a ) was triggered by an interplanetary coronal mass ejection [16] . This is a relatively minor magnetopause compression, thus drift shell splitting is not expected to occur inside of geosynchronous orbit. A large negative B z (the z component of the interplanetary magnetic field) occurs during the period 1100 hours on June 28 to 1200 UT on June 29 ( Fig. 1b ), leading to efficient coupling with Earth’s magnetosphere and prolonged geomagnetic activity [17] . Distinct whistler-mode chorus and MS emissions ( Fig. 1d ) were observed by the Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS) Waves instrument [18] , [19] onboard Van Allen Probe A. Chorus and MS waves are right hand polarized electromagnetic waves. Chorus waves are excited by the injection of energetic (tens of keV) plasma sheet electrons into the inner magnetosphere during the period of enhanced plasma convection associated with the intervals of negative B z (refs 20 , 21 , 22 ). MS waves are generated by a ring distribution of low-energy ( ∼ 10 keV) protons at frequencies close to the harmonics of the proton gyrofrequency [23] . Figure 1: Van Allen Probe data during the 28–30 June 2013 storm. ( a ) The Dst index. ( b ) The interplanetary magnetic field B z . ( c ) Solar wind dynamic pressure. ( d ) Magnetic spectral intensity of chorus and MS waves from the EMFISIS instrument on Van Allen Probe A. The black and white lines denote the lower hybrid resonance frequency f lhr and 0.1 f ce ( f ce being the electron gyrofrequency). ( e – h ) Relativistic electron differential fluxes from the REPT instrument onboard the Van Allen Probe A as a function of L showing a rapid increase in the radial range 3.5< L <5.5 where strong chorus and MS waves occur. The light grey shading area indicates the simulation period. Full size image The fluxes of relativistic (2–3.6 MeV) electrons ( Fig. 1e–h ) from the Relativistic Electron-Proton Telescope (REPT) instrument [24] onboard Van Allen Probes decreased during the main phase of the storm when the magnetopause was compressed down to L =7.6, and dramatically increased during the recovery phase after 29 June when the solar wind dynamic pressure decreased and the magnetopause moved beyond L =10 ( Fig. 1c ). In Fig. 2 , we show data during the period 000 to 1600 UT on 29 June 2013 (corresponding to the shaded area in Fig. 1 ) from the EMFISIS instrument for magnetic and electric spectral intensity of chorus and MS waves ( Fig. 2a,b ), wave normal angle θ ( Fig. 2c ) and wave ellipticity ( Fig. 2d ). The observed chorus wave has a low normal angle ( θ ≈0°) and circular polarization (ellipticity≈0) [25] . Meanwhile, the observed MS wave is highly oblique ( θ ≈90°) and linearly polarized (ellipticity≈0) [26] . This indicates that the MS or chorus wave k vector is approximately perpendicular or parallel to the ambient magnetic field direction [12] . Figure 2: Formation of butterfly distribution of relativistic electrons. Data from the EMFISIS instrument for magnetic and electric spectral intensity (in unit of log 10 ) of chorus and MS waves ( a , b ), wave normal angle ( c ), the angle between Earth’s magnetic field and the normal to the plane of the wave, and wave ellipticity ( d ), the degree of elliptical polarization. Chorus wave occurs above 0.1 f ce (the white dashed line). MS wave occur as a series of narrow tones, spaced at multiples of the proton gyrofrequency f cp up to f lhr (the white solid line). ( c , d ) The observed MS or chorus wave has a high ( θ ≈90°) or low ( θ ≈0°) normal angle and a high (ellipticity≈0) or low (ellipticity≈1) degree of elliptical polarization. ( e – h ) Pitch angle distribution for different indicated energies (2–3.6 MeV) over ∼ 40 min duration from the REPT instrument. Fluxes (in the same unit as that in Fig. 1 ) of relativistic electrons have minima around pitch angle 90°, and peaks around the pitch angle range of 30°–60° or 120°–150°. Full size image A very interesting feature here is that distinct butterfly distributions of fluxes of relativistic (2–3.6 MeV) electrons from the REPT instrument occurred around L =4.8 over the interval 1228-1312 UT ( Fig. 2e–h ), corresponding to the occurrence of enhanced chorus and MS waves. Electron fluxes have minima around pitch angle 90°, and peaks around the pitch angle range of 30°–60° or 120°–150°. Observations of butterfly distributions at locations L <5 are unlikely due to drift shell splitting as Earth’s magnetic field is more dipolar and only distorted/stretched under extreme geomagnetic conditions. As this was a relatively minor compression and only a moderate geomagnetic storm, it is unlikely that drift shell splitting is the cause of these observations. Here we perform a two-dimensional simulation, together with the high-resolution observations from the Van Allen Probes, to examine whether chorus and MS waves can be responsible for evolution of both the energy and butterfly pitch angle distribution of the observed relativistic electron flux increase. Numerical modelling To model the temporal evolution of the electron distribution function, we need to solve the Fokker–Planck diffusion equation, which is associated with pitch-angle and momentum diffusion coefficients driven by wave–particle interaction. Then evaluation of diffusion coefficients needs specific information of wave amplitudes and spectral properties. In this event, due to the brief UT intervals of Van Allen Probes, the in situ measurements of chorus or MS waves were confined to a limited range of magnetic local time (MLT) and L -shells: L =4–6 and MLT=22–24 for chorus; L =2–3.5 and MLT=10–18 for MS. Previous works have demonstrated that MS waves can propagate over a broad region of MLT and L -shells [23] , [27] , [28] . Here using the previously developed programmes [29] , [30] , we present ray-tracing results of MS waves based on the wave data. MS waves are launched at L =5.6 for different parameters and MLT regions at the geomagnetic equator ( Supplementary Table 1 ). The modelled results ( Supplementary Fig. 1 ) confirm that MS waves can propagate either into or out of the plasmasphere through the plasmapause, covering a broad region of L =2–5.6, particularly the observed butterfly distribution location L =4.8 and MLT=19. In this event, Van Allen Probes stayed relatively deep inside the plasmasphere in the day–evening sector. Consequently, those MS waves which potentially occurred in the day–evening sector can not be directly observed by Van Allen Probes in their orbits. Since relativistic electrons drift eastward around Earth approximately in a circular orbit, the observed butterfly distribution should come from the whole contribution of resonances with those MS waves in different MLT regions. We have also checked the Van Allen Probes data on 28 July 2013 and found that MS waves were indeed observed at L =4.8 and MLT=19. However, Van Allen Probes only passed the observed location in a brief UT interval every day, they may not observe the MS waves each time. In addition, using the Gaussian fitting method (not shown for brevity), we find that those MS waves display a similar Gaussian distribution to those at L =3 and MLT=17.3 on 29 July. It is therefore possibly reasonable to expect that the MS waves potentially existing at L =4.8 and MLT=19 on 29 July to follow the similar Gaussian distribution to that at L =3 and MLT=17.3. Meanwhile, for calculating the diffusion coefficients of MS waves, we assume that MS waves propagate very highly oblique ( θ =86°–89 ° ), distributed in a standard Gaussian form ( X =tan θ ) with a peak value X m =tan89°. We noted that a concise modelling method by using maxima wave intensities was adopted in the previous work [26] . Moreover, chorus occurs over a broad range of MLT from the nightside through dawn to the dayside [31] , [32] , potentially producing efficient scattering precipitation of energetic (tens of keV) electrons into the atmosphere. Considering that the ratio between the precipitated and trapped electron fluxes measured by Polar Orbiting Environmental Satellites (POES) is approximately proportional to the chorus power spectral intensity [33] , previous studies have developed a novel technique to obtain a dynamic global model of chorus wave amplitudes as a function of L , MLT and time by converting the measured POES flux ratios at different MLT [34] , [35] . Here we use the same approach (described in Methods), together with the previous parametric study [36] , to obtain the global distribution of chorus wave amplitude as a function of L , MLT in this event. Specifically, we remove the proton contamination by the same method in the previous work [37] . The model parameters listed in Table 1 are adopted to calculate diffusion coefficients for chorus waves. However, the current POES model does not incorporate other loss mechanisms and further improvements are needed in the future. Considering that observations of a global distribution of chorus waves are still very limited, this model should move a step forward in constructing an event-dependent global dynamic model of chorus waves. Table 1 Adopted model parameters for chorus a . Full size table In general, chorus waves which consist of substructures are coherent at the equator but become less coherent off the equator, resonating with relativistic electrons [38] . The consequences of coherent wave–particle interactions involving relativistic electrons and chorus waves off the equator have been presented in the previous work [39] . Here the quasi-linear theory to treat wave–particle interaction is adopted and the corresponding conditions will be discussed below. We assume chorus waves to obey the same Gaussian distributions in wave frequency and wave normal angle at and off the equator. Considering that relativistic electrons move along the geomagnetic field line and bounce back between mirror points, we consider wave–particle interaction at each location and calculate bounce-averaged diffusion coefficients. The obtained bounce-averaged diffusion coefficients ( Fig. 3 ) cover an entire region of pitch angles 0°–90° for chorus but a limited region 30°–60° for MS waves. This can allow for significant increases in fluxes of relativistic (2–3.6 MeV) electrons at medium pitch angles 30°–60° by MS waves [12] and at high pitch angles up to 90° by chorus waves [40] , leading to the resultant butterfly distribution on a timescale comparable to 10 h. Figure 3: Diffusion rates. Bounce-averaged pitch angle (〈 D αα 〉, ( a , d )), momentum (〈 D pp 〉, ( b , e )) and cross (〈 D αp 〉, ( c , f )) diffusion rates (in unit of s −1 ) for resonant interactions between chorus (left) or MS (right) wave with electrons. Diffusion rates cover an entire region of pitch angles 0°–90° for chorus but a limited region 30°–60° for MS waves. Combination of chorus and MS waves leads to efficient acceleration of electrons between 2 and 3.6 MeV at medium pitch angles 30°–60° and high pitch angles up to 90°, yielding the resultant butterfly distribution on a timescale comparable to 10 h. Full size image Using aforementioned bounce-averaged diffusion coefficients, we calculate phase space density (PSD) f t evolution of electrons in solving a two-dimensional Fokker–Planck diffusion equation [41] . The evolution of differential flux j is then simulated in the pitch angle region 0°–90° by the subsequent conversion j = p 2 f t and the results are extended to the range 90°–180° due to the mirror symmetry. We show in Fig. 4a remarkable agreement between the simulated pitch angle distribution ( Fig. 4c,d ) and the REPT observation ( Fig. 4a,b ) during the acceleration interval. The most pronounced flux enhancement occurs over the medium pitch angles 30°–60°, where momentum diffusion rates for combined chorus and MS waves dominate ( Fig. 3 ). Furthermore, the pitch angle distributions at higher pitch angles close to 90° are enhanced due to larger momentum diffusion rates of chorus waves. Finally, the combined acceleration by chorus and MS waves significantly alters the whole population of relativistic electrons, yielding the butterfly distribution in about 9 h. Figure 4: Comparison of Fokker–Planck simulation results with observations. ( a , b ) Observed evolution of the relativistic electron fluxes as a function of pitch angle at the same location (shown inside panels) during the simulation. ( c , d ) Starting with an initial condition representative of relativistic electrons ( a ), we show the relativistic (2–3.6 MeV) electron fluxes due to acceleration by chorus and MS waves from a numerical solution to the two-dimensional Fokker–Planck diffusion equation. Acceleration is most pronounced within the medium pitch angle region 30°–60° or 120°–150° produced by both chorus and MS waves, and noticeable around the high pitch angles induced by chorus waves. The combined acceleration by chorus and MS waves leads to the butterfly distribution both in the magnitude and the time scale ( d ) comparable to the observation ( b ). The current simulation provides a further support for wave-driven butterfly distribution of relativistic electrons during this storm. Full size image"},{"section":"Discussion","content":"The present modelling provides a further confirmation of wave-driven butterfly pitch angle distribution of relativistic electrons observed by the REPT instrument at lower locations ( L <5) of the outer radiation belt. This is in contrast to the formation of butterfly distributions at higher locations ( L >6) induced by drift shell splitting due to local magnetic field asymmetry. The associated physical processes are schematically presented in Fig. 5 . Relativistic electrons at L <5 drift azimuthally around Earth inside the magnetosphere without loss to the magnetopause, continuously resonating with chorus and MS waves. The combined acceleration by chorus and MS waves occurs preferably between the medium pitch angles 30° and 60°, leading to the unusual butterfly distribution. The combined acceleration process described here is a universal physical process, which should also be effective in the magnetospheres of Jupiter, Saturn and other magnetized plasma environments in the cosmos. Figure 5: Schematic diagram of formation of butterfly distributions. The long black curves indicate Earth’s magnetic field lines. The grey area denote the region of dense cold plasma known as plasmasphere. The blue and yellow areas stand for primary occurrences of chorus and MS waves. The wavy line represents chorus (blue) and MS (yellow) waves. Relativistic electrons originating from the plasma sheet azimuthally drift eastward around Earth. (1) Higher pitch angle electrons at L >6 drift to larger radial distance beyond the dayside magnetopause and are consequently lost due to the day–night magnetic field asymmetry (2), yielding the regular butterfly distribution (3). (4) Electrons at L <5 drift inside the magnetosphere without loss to the dayside magnetopause because of a very small day–night magnetic field asymmetry, continuously resonating with chorus and MS waves. (5) Wave acceleration can enhance electron PSD primarily within the medium pitch angles, leading to the formation of the unusual butterfly distribution (6). Full size image It should be mentioned that, in a departure from the coherent chorus–electron interaction approach [38] , we use the quasi-linear theory of wave–particle interaction, which has been frequently adopted by radiation belt research community to treat wave–particle interaction [12] , [40] , [42] . Previous work [43] has shown that energetic electron pitch angle scattering by coherent chorus waves is 3 orders more rapid than by incoherent chorus waves. Coherent chorus waves can produce particle diffusion, phase trapping and/or phase bunching, which is determined primarily by the competing effects of wave amplitude and ambient magnetic field inhomogeneity at resonance. However, application of coherent wave–particle interaction to the strom-time global evolution of energetic particle distributions has not yet been established so far. We therefore expect the adopted quasi-linear theory here to be a basis for comparison with future developments in nonlinear modelling. The basic conditions for the quasi-linear theory are that each individual particle randomly moves in velocity space, resonates with a succession of uncorrelated and small amplitude waves, and is scattered in a small amount in pitch angle and energy each time. Those conditions are basically satisfied in Earth’s radiation belts for naturally generated MS and chorus waves, where the MS wave bandwidth is generally above the proton gyrofrequency up to the lower hybrid frequency and the chorus wave roughly lies in the frequency range 0.1–0.8 f ce ( f ce being the electron gyrofrequency). In general, there could be a distorted/stretched geomagnetic field on the nightside during a geomagnetic storm. We have examined the geomagnetic field data from Van Allen Probes and found that the observed data are close to the dipole field model during this storm period. The largest distortion of the dipole field is about 20% in the simulation period. Moreover, we perform a test-particle simulation of trajectories of trapped relativistic (2 MeV) electrons by the TS04 magnetic model [44] for different high pitch angles. Simulation results ( Fig. 6 ) show that, starting at the location L =4.8 and MLT=24, relativistic electrons drift eastward around Earth and approach a farthest location L ≈5.1 on the dayside. Then they pass the observed location without loss to the magnetopause because the magnetopause locates above L =7.6 in the simulation period. Simulations for different energies or pitch angles (not shown for brevity) indeed show similar results. This indicates that drift shell splitting is unlikely to play a role in the observed butterfly electron distribution in this event. Figure 6: Trajectories of trapped electrons. Test-particle simulations of relativistic (2 MeV) electron trajectories by the TS04 magnetic model for different pitch angles 70° (green), 80° (blue) and 90° (black). The input parameters for TS04 magnetic model are based on the observations. Starting at the location (the symbols S ): L =4.8 and MLT=24, relativistic electrons drift eastward around Earth and approach a farthest location L ≈5.1 on the dayside. Then they pass the observed location (the symbol O ) without loss to the magnetopause because the magnetopause locates above L =7.6 in the simulation period. Full size image A basic assumption is adopted here that the cold plasma density remains constant and unchanging during the 9-h simulation period. Using the upper hybrid resonance frequency observation from Van Allen Probes, we infer the ambient electron density at locations along Van Allen Probes’ orbits and find that the electron density is comparable to the adopted plasma density and does not change much in 9 h. Analogous to the previous work [40] , this assumption is probably reasonable in the absence of simultaneous and continuous local time observations of cold plasma density in the 9-h period. To check whether the plasmaspheric plume exists, we analyse the potential data from the Time History of Events and Macroscale Interactions During Substorms (THEMIS) spacecraft at L =4.8, and find no distinct presence of plasmaspheric plume except a slightly higher electron density in the region around/after MLT=16 in the simulation period. The plasmaspheric plume resulting from time-varying convection generally forms with a more structured distribution of plasma in the dusk region in the storm main phase. Meanwhile, the plume moves eastward (towards later MLTs) in Earth’s rotation eastward, and gradually fades/erodes as the strength of convection decreases. As shown in Fig. 1 , the simulation period is in the storm recovery phase on 29 June, >24 h after the onset of the storm on 28 June. Hence, the plume probably already faded and moved to the later MLT region. However, further research is still required in the future because there are numerous uncertainties in accurate determination of either the upper hybrid resonance frequency from Van Allen Probes or the THEMIS spacecraft potential."},{"section":"Methods","content":"Ray tracing of MS waves We perform the ray tracing of MS waves with wave vector k and frequency ω by the following standard Hamiltonian equations [45] : where R is the position vector of a point on the ray path, t is the group time, D represents the standard wave dispersion relation obeying D ( R , ω , k )=0 at every point along the ray path. The spatial variation in D can be written: where B 0 is the ambient magnetic field and N c is the background plasma density. We adopt a dipole magnetic field model and the MLT-dependent plasmatrough density model [46] . The Earth-centred Cartesian and local Cartesian coordinate systems for the ray-tracing calculation are described in Supplementary Note 1 . Calculation of the global chorus wave amplitude Recent works [34] , [35] have constructed a global chorus wave model based on the data of low-altitude electron population collected by multiple POES satellites. The Medium Energy Proton and Electron Detector (MEPED) onboard POES has two electron solid-state detector (0° and 90°) telescopes to measure electron fluxes in three energy channels (> 30 keV,> 100 keV and>300 keV) [47] . The 90° telescope largely measures the trapped flux over the invariant latitude range between 55° and 70°, and the 0° telescope measures precipitating flux inside the bounce loss cone at L >1.4 (ref. 48 ). Using the electron distribution function near the loss cone [33] , the chorus wave amplitude can be calculated from the ratio between the measured precipitated and trapped electron fluxes (30–100 keV and 100–300 keV). The electron energy spectrum is assumed to follow a kappa-type function [49] , [50] . The basic equation for linking the ratio of electron count rates measured by the 0° and 90° telescopes to chorus wave amplitude is shown in Supplementary Note 2 . The ratios of precipitated and trapped electron fluxes measured by multiple POES satellites are used to construct the global chorus wave model over a broad range in L and MLT, and the ratios obtained in four distinct MLT sectors are shown in Supplementary Fig. 2 . Calculation of diffusion coefficients due to chorus and MS waves We assume that the wave spectral density B f 2 follows a typical Gaussian frequency distribution with a center f m , a half-width δf , a band between f 1 and f 2 (ref. 51 ). here is the wave amplitude in units of Tesla and erf is the error function. To allow the data modelling to be more reliable, we average the observed wave magnetic field intensity over the indicated time period in this event and then apply the corresponding Gaussian fit for MS waves as shown in Supplementary Fig. 3 . We choose the wave normal angle distribution to also satisfy a standard Gaussian form: where X =tan θ ( θ 1 ≤ θ ≤ θ 2 , X 1,2 =tan θ 1,2 ), with a half-width X ω and a peak X m . Based on the observation, we choose X m =tan 89°, X ω =tan 86°, X 1 = X m − X ω , X 2 = X m + X ω ; and the maximum latitude for the presence of MS wave λ m =10°. Based on the ray-tracing results ( Supplementary Fig. 1 ), we assume the MS wave spectral intensity at L =4.8 and MLT=19 hours to follow the similar Gaussian distribution to that as shown in Supplementary Fig. 3 , which are then used to calculate the MS-driven bounce-averaged diffusion coefficients. For the chorus waves, we adopt the wave parameters as shown in Table 1 to calculate the chorus-driven bounce-averaged diffusion coefficients at the location L =4.8. We consider contribution from harmonic resonances up to n =±5 for both chorus and MS waves. The ambient plasma density is obtained by the MLT-dependent plasmatrough density model [46] and further assumed to remain latitudinally constant and unchanged during the 9-h simulation period. Note that there is no realistic data available either in 08–12 or in 16–20 MLT. We assume the chorus wave power in 08–12 MLT to follow the latitude-dependent model [36] . The corresponding adopted wave amplitude, wave spectrum and latitudinal occurrence of chorus waves, the ambient electron density and the equatorial ratio of plasma frequency to gyrofrequency ( f pe / f ce ) are shown in Table 1 . Evaluation of the relativistic electron PSD evolution The evolution of the electron PSD f t is calculated by solving the bounce-averaged pitch angle and momentum diffusion equation here p is the electron momentum, G = p 2 T ( α e )sin α e cos α e with α e being the equatorial pitch angle, the normalized bounce time T ( α e )≈1.30−0.56sin α e ; 〈 D αα 〉, 〈 D pp 〉, and 〈 D αp 〉=〈 D pα 〉 denote bounce-averaged diffusion coefficients in pitch angle, momentum and cross pitch angle–momentum. The explicit expressions of those bounce-averaged diffusion coefficients can be found in the previous work. [41] The initial condition is modelled by a bi-modal kappa-type distribution function of energetic electrons [49] , [50] : Each component, with variable weighting parameters a 1 and a 2 , is expressed as: here n h is the number density of energetic electrons, l indicates the loss-cone index, Γ is the gamma function, κ and are the spectral index and effective thermal energy scaled by the electron rest mass energy m e c 2 ( ∼ 0.5 MeV). For the pitch angle boundary condition, f t =0 at the loss-cone α e = α L (sin α L = L −3/2 (4−3/ L ) −1/4 ) to simulate a rapid precipitation of electrons inside the loss cone ( Fig. 2e–h ), and ∂ f t /∂ α e =0 at α e =90°. For the energy diffusion boundary conditions, at the lower boundary 0.2 MeV, and at the upper boundary 10 MeV. Based on the observation, we choose the following values of parameters: ( ∼ 75 keV), κ =4, n h =0.16 cm −3 ; a 1 =0.6, l 1 =0.6; a 2 =0.4, l 2 =1.5. We solve the diffusion equation(6) using the recently developed hybrid difference method [41] , which is efficient, stable and easily parallel programmed. The numerical grid is set to be 91 × 101 and uniform in pitch angle and natural logarithm of momentum. The drifting average is taken 25% for MS wave and 1/6 for chorus wave in each aforementioned MLT sector ( Table 1 )."},{"section":"Additional information","content":"How to cite this article: Xiao, F. et al . Wave-driven butterfly distribution of Van Allen belt relativistic electrons. Nat. Commun. 6:8590 doi: 10.1038/ncomms9590 (2015)."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"The Earth’s outer Van Allen radiation belt is composed of trapped electrons with relativistic energy ( E k ≥1 MeV). Fluxes and pitch angle distributions of relativistic electrons often exhibit dramatic and highly dynamic changes during geomagnetic storms or substorms, which are associated with different physical mechanisms. Those mechanisms include transport, energization and loss processes. Wave–particle interaction plays an important role in energy exchange between various modes of plasma waves and Van Allen radiation belt relativistic electrons. Two types of plasma waves, chorus and magnetosonic (MS) wave (also known as ‘equatorial noise’), can efficiently accelerate electrons up to relativistic energies. Relativistic electrons can pose serious damage to satellites and astronauts in space, it is therefore important to understand and ultimately predict Van Allen radiation belt electron dynamic variations. Radiation belt trapped electrons experience azimuthal drift-motion around Earth when they bounce between northern and southern magnetic mirror points. Due to the day–night azimuthal asymmetry in Earth’s magnetic field, their drift shells which are traced out by their guiding centres can separate radially for different pitch angles, which is known as drift shell splitting [1] . Significant drift shell splitting occurs only in the outer radiation belt L >6 (with L being the radial distance in Earth radii R E ), where the asymmetry becomes substantial. Previous theoretical [1] , [2] and observational [3] , [4] , [5] works have long confirmed drift shell splitting. Drift shell splitting separates the high and low pitch angle particles in nightside injections as they move to the dayside magnetosphere. The higher pitch angle particles drift to larger radial distance beyond the magnetopause on the dayside and may consequently be lost from the distribution [6] . Therefore, drift shell splitting can be interpreted in terms of butterfly pitch angle distributions, namely, a sharp dropout in the flux of 90° electrons. Another drift shell splitting is in combination with a negative radial flux gradient as equatorially mirroring particles drift around Earth at locations L =6–12 (refs 7 , 8 ). Drift shell splitting will most easily affect trapped particle populations further out than geosynchronous orbit. Occurrence of drift shell splitting inside of geosynchronous orbit would need a very large magnetopause compression during very large geomagnetic storms. A fundamental problem, both theoretically and observationally, is that there is any butterfly pitch angle distribution of relativistic electrons below L =5 associated with a new mechanism instead of drift shell splitting. Previous studies have analysed the characteristics and evolution of pitch angle distributions of the outer radiation belt electrons [9] , [10] . Chorus and MS waves were proposed to produce butterfly distributions by preferentially accelerating off-equator electrons [11] , [12] . Adiabatic processes could potentially yield the formation of butterfly distributions at locations L ≥6 (refs 13 , 14 ). Direct confirmation of wave-driven butterfly distributions below L =5 requires simultaneous high-resolution data but this is generally unavailable before the launch of NASA’s Van Allen Radiation Belt Storm Probes in 2012 (ref. 15 ). Fortunately, the unique events on the 28 June 2013 geomagnetic storm observed from Van Allen Probes provide an excellent opportunity to identify such mechanisms. Here we report the formation of a unusual butterfly pitch angle distribution of relativistic electrons around L =4.8, corresponding to the occurrence of strong chorus and MS waves at the same time. Using two-dimensional simulation, we show that flux enhancements of relativistic electrons are most pronounced between the medium pitch angles 30° and 60° due to the dominant acceleration process for combined chorus and MS waves. Meanwhile, the pitch angle distributions close to 90° are increased due to relatively larger acceleration process for chorus alone. The combined acceleration by chorus and MS waves substantially modifies the whole population of relativistic electrons, producing the butterfly distribution. Our detailed modelling, together with the correlated Van Allen Probes data, presents a further understanding on how chorus and MS waves play different roles in Earth’s outer radiation belts.\",\n \"references\": {\n \"indices\": [\n 1,\n 2,\n 3,\n 4,\n 5,\n 6,\n 10,\n 11,\n 12\n ],\n \"infos\": [\n {\n \"idx\": \"1\",\n \"title\": \"Roederer, J. G. On the adiabatic motion of energetic particles in a model magnetosphere.J. Geophys. Res.72, 981–992 (1967).\",\n \"link\": \"https://doi.org/10.1029%2FJZ072i003p00981\"\n },\n {\n \"idx\": \"2\",\n \"title\": \"Stone, E. C. The physical significance of and application ofL,B0 andR0 to geomagnetically trapped particles.J. Geophys. Res.68, 4157–4166 (1963).\",\n \"link\": \"https://doi.org/10.1029%2FJZ068i014p04157\"\n },\n {\n \"idx\": \"3\",\n \"title\": \"West, H. I., Buck, R. M. Jr & Walton, J. R. Electron pitch angle distributions throughout the magnetosphere as observed on Ogo 5.J. Geophys. Res.78, 1064–1081 (1973).\",\n \"link\": \"https://doi.org/10.1029%2FJA078i007p01064\"\n },\n {\n \"idx\": \"4\",\n \"title\": \"Baker, D. N., Higbie, P. R., Hones, E. W. Jr & Belian, R. D. High-resolution energetic particle measurements at 6.6RE. Low-energy electron anisotropies and short-term substorm predictions.J. Geophys. Res.83, 4863–4868 (1978).\",\n \"link\": \"https://doi.org/10.1029%2FJA083iA10p04863\"\n },\n {\n \"idx\": \"5\",\n \"title\": \"Takahashi, K. et al. Drift-shell splitting of energetic ions injected at pseudo-substorm onsets.J. Geophys. Res.102, 22117–22130 (1997).\",\n \"link\": \"https://doi.org/10.1029%2F97JA01870\"\n },\n {\n \"idx\": \"6\",\n \"title\": \"Selesnick, R. S. & Blake, J. B. Relativistic electron drift shell splitting.J. Geophys. Res.1079, 1265 (2002).\",\n \"link\": \"https://doi.org/10.1029%2F2001JA009179\"\n },\n {\n \"idx\": \"10\",\n \"title\": \"Gannon, J. L., Li, X. & Heynderickx, D. Pitch angle distribution analysis of radiation belt electrons based on combined release and radiation effects satellite medium electrons A data.J. Geophys. Res.112, A05212 (2007).\",\n \"link\": \"https://doi.org/10.1029%2F2005JA011565\"\n },\n {\n \"idx\": \"11\",\n \"title\": \"Horne, R. B. et al. Timescale for radiation belt electron acceleration by whistler mode chorus waves.J. Geophys. Res.110, A03225 (2005).\",\n \"link\": \"https://doi.org/10.1029%2F2004JA010811\"\n },\n {\n \"idx\": \"12\",\n \"title\": \"Horne, R. B. et al. Electron acceleration in the Van Allen radiation belts by fast magnetosonic waves.Geophys. Res. Lett.34, L17107 (2007).\",\n \"link\": \"https://doi.org/10.1029%2F2007GL030267\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Correlated Van Allen Probe data On 28 June 2013, a moderate storm with Dst≈−100 nT ( Fig. 1a ) was triggered by an interplanetary coronal mass ejection [16] . This is a relatively minor magnetopause compression, thus drift shell splitting is not expected to occur inside of geosynchronous orbit. A large negative B z (the z component of the interplanetary magnetic field) occurs during the period 1100 hours on June 28 to 1200 UT on June 29 ( Fig. 1b ), leading to efficient coupling with Earth’s magnetosphere and prolonged geomagnetic activity [17] . Distinct whistler-mode chorus and MS emissions ( Fig. 1d ) were observed by the Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS) Waves instrument [18] , [19] onboard Van Allen Probe A. Chorus and MS waves are right hand polarized electromagnetic waves. Chorus waves are excited by the injection of energetic (tens of keV) plasma sheet electrons into the inner magnetosphere during the period of enhanced plasma convection associated with the intervals of negative B z (refs 20 , 21 , 22 ). MS waves are generated by a ring distribution of low-energy ( ∼ 10 keV) protons at frequencies close to the harmonics of the proton gyrofrequency [23] . Figure 1: Van Allen Probe data during the 28–30 June 2013 storm. ( a ) The Dst index. ( b ) The interplanetary magnetic field B z . ( c ) Solar wind dynamic pressure. ( d ) Magnetic spectral intensity of chorus and MS waves from the EMFISIS instrument on Van Allen Probe A. The black and white lines denote the lower hybrid resonance frequency f lhr and 0.1 f ce ( f ce being the electron gyrofrequency). ( e – h ) Relativistic electron differential fluxes from the REPT instrument onboard the Van Allen Probe A as a function of L showing a rapid increase in the radial range 3.5< L <5.5 where strong chorus and MS waves occur. The light grey shading area indicates the simulation period. Full size image The fluxes of relativistic (2–3.6 MeV) electrons ( Fig. 1e–h ) from the Relativistic Electron-Proton Telescope (REPT) instrument [24] onboard Van Allen Probes decreased during the main phase of the storm when the magnetopause was compressed down to L =7.6, and dramatically increased during the recovery phase after 29 June when the solar wind dynamic pressure decreased and the magnetopause moved beyond L =10 ( Fig. 1c ). In Fig. 2 , we show data during the period 000 to 1600 UT on 29 June 2013 (corresponding to the shaded area in Fig. 1 ) from the EMFISIS instrument for magnetic and electric spectral intensity of chorus and MS waves ( Fig. 2a,b ), wave normal angle θ ( Fig. 2c ) and wave ellipticity ( Fig. 2d ). The observed chorus wave has a low normal angle ( θ ≈0°) and circular polarization (ellipticity≈0) [25] . Meanwhile, the observed MS wave is highly oblique ( θ ≈90°) and linearly polarized (ellipticity≈0) [26] . This indicates that the MS or chorus wave k vector is approximately perpendicular or parallel to the ambient magnetic field direction [12] . Figure 2: Formation of butterfly distribution of relativistic electrons. Data from the EMFISIS instrument for magnetic and electric spectral intensity (in unit of log 10 ) of chorus and MS waves ( a , b ), wave normal angle ( c ), the angle between Earth’s magnetic field and the normal to the plane of the wave, and wave ellipticity ( d ), the degree of elliptical polarization. Chorus wave occurs above 0.1 f ce (the white dashed line). MS wave occur as a series of narrow tones, spaced at multiples of the proton gyrofrequency f cp up to f lhr (the white solid line). ( c , d ) The observed MS or chorus wave has a high ( θ ≈90°) or low ( θ ≈0°) normal angle and a high (ellipticity≈0) or low (ellipticity≈1) degree of elliptical polarization. ( e – h ) Pitch angle distribution for different indicated energies (2–3.6 MeV) over ∼ 40 min duration from the REPT instrument. Fluxes (in the same unit as that in Fig. 1 ) of relativistic electrons have minima around pitch angle 90°, and peaks around the pitch angle range of 30°–60° or 120°–150°. Full size image A very interesting feature here is that distinct butterfly distributions of fluxes of relativistic (2–3.6 MeV) electrons from the REPT instrument occurred around L =4.8 over the interval 1228-1312 UT ( Fig. 2e–h ), corresponding to the occurrence of enhanced chorus and MS waves. Electron fluxes have minima around pitch angle 90°, and peaks around the pitch angle range of 30°–60° or 120°–150°. Observations of butterfly distributions at locations L <5 are unlikely due to drift shell splitting as Earth’s magnetic field is more dipolar and only distorted/stretched under extreme geomagnetic conditions. As this was a relatively minor compression and only a moderate geomagnetic storm, it is unlikely that drift shell splitting is the cause of these observations. Here we perform a two-dimensional simulation, together with the high-resolution observations from the Van Allen Probes, to examine whether chorus and MS waves can be responsible for evolution of both the energy and butterfly pitch angle distribution of the observed relativistic electron flux increase. Numerical modelling To model the temporal evolution of the electron distribution function, we need to solve the Fokker–Planck diffusion equation, which is associated with pitch-angle and momentum diffusion coefficients driven by wave–particle interaction. Then evaluation of diffusion coefficients needs specific information of wave amplitudes and spectral properties. In this event, due to the brief UT intervals of Van Allen Probes, the in situ measurements of chorus or MS waves were confined to a limited range of magnetic local time (MLT) and L -shells: L =4–6 and MLT=22–24 for chorus; L =2–3.5 and MLT=10–18 for MS. Previous works have demonstrated that MS waves can propagate over a broad region of MLT and L -shells [23] , [27] , [28] . Here using the previously developed programmes [29] , [30] , we present ray-tracing results of MS waves based on the wave data. MS waves are launched at L =5.6 for different parameters and MLT regions at the geomagnetic equator ( Supplementary Table 1 ). The modelled results ( Supplementary Fig. 1 ) confirm that MS waves can propagate either into or out of the plasmasphere through the plasmapause, covering a broad region of L =2–5.6, particularly the observed butterfly distribution location L =4.8 and MLT=19. In this event, Van Allen Probes stayed relatively deep inside the plasmasphere in the day–evening sector. Consequently, those MS waves which potentially occurred in the day–evening sector can not be directly observed by Van Allen Probes in their orbits. Since relativistic electrons drift eastward around Earth approximately in a circular orbit, the observed butterfly distribution should come from the whole contribution of resonances with those MS waves in different MLT regions. We have also checked the Van Allen Probes data on 28 July 2013 and found that MS waves were indeed observed at L =4.8 and MLT=19. However, Van Allen Probes only passed the observed location in a brief UT interval every day, they may not observe the MS waves each time. In addition, using the Gaussian fitting method (not shown for brevity), we find that those MS waves display a similar Gaussian distribution to those at L =3 and MLT=17.3 on 29 July. It is therefore possibly reasonable to expect that the MS waves potentially existing at L =4.8 and MLT=19 on 29 July to follow the similar Gaussian distribution to that at L =3 and MLT=17.3. Meanwhile, for calculating the diffusion coefficients of MS waves, we assume that MS waves propagate very highly oblique ( θ =86°–89 ° ), distributed in a standard Gaussian form ( X =tan θ ) with a peak value X m =tan89°. We noted that a concise modelling method by using maxima wave intensities was adopted in the previous work [26] . Moreover, chorus occurs over a broad range of MLT from the nightside through dawn to the dayside [31] , [32] , potentially producing efficient scattering precipitation of energetic (tens of keV) electrons into the atmosphere. Considering that the ratio between the precipitated and trapped electron fluxes measured by Polar Orbiting Environmental Satellites (POES) is approximately proportional to the chorus power spectral intensity [33] , previous studies have developed a novel technique to obtain a dynamic global model of chorus wave amplitudes as a function of L , MLT and time by converting the measured POES flux ratios at different MLT [34] , [35] . Here we use the same approach (described in Methods), together with the previous parametric study [36] , to obtain the global distribution of chorus wave amplitude as a function of L , MLT in this event. Specifically, we remove the proton contamination by the same method in the previous work [37] . The model parameters listed in Table 1 are adopted to calculate diffusion coefficients for chorus waves. However, the current POES model does not incorporate other loss mechanisms and further improvements are needed in the future. Considering that observations of a global distribution of chorus waves are still very limited, this model should move a step forward in constructing an event-dependent global dynamic model of chorus waves. Table 1 Adopted model parameters for chorus a . Full size table In general, chorus waves which consist of substructures are coherent at the equator but become less coherent off the equator, resonating with relativistic electrons [38] . The consequences of coherent wave–particle interactions involving relativistic electrons and chorus waves off the equator have been presented in the previous work [39] . Here the quasi-linear theory to treat wave–particle interaction is adopted and the corresponding conditions will be discussed below. We assume chorus waves to obey the same Gaussian distributions in wave frequency and wave normal angle at and off the equator. Considering that relativistic electrons move along the geomagnetic field line and bounce back between mirror points, we consider wave–particle interaction at each location and calculate bounce-averaged diffusion coefficients. The obtained bounce-averaged diffusion coefficients ( Fig. 3 ) cover an entire region of pitch angles 0°–90° for chorus but a limited region 30°–60° for MS waves. This can allow for significant increases in fluxes of relativistic (2–3.6 MeV) electrons at medium pitch angles 30°–60° by MS waves [12] and at high pitch angles up to 90° by chorus waves [40] , leading to the resultant butterfly distribution on a timescale comparable to 10 h. Figure 3: Diffusion rates. Bounce-averaged pitch angle (〈 D αα 〉, ( a , d )), momentum (〈 D pp 〉, ( b , e )) and cross (〈 D αp 〉, ( c , f )) diffusion rates (in unit of s −1 ) for resonant interactions between chorus (left) or MS (right) wave with electrons. Diffusion rates cover an entire region of pitch angles 0°–90° for chorus but a limited region 30°–60° for MS waves. Combination of chorus and MS waves leads to efficient acceleration of electrons between 2 and 3.6 MeV at medium pitch angles 30°–60° and high pitch angles up to 90°, yielding the resultant butterfly distribution on a timescale comparable to 10 h. Full size image Using aforementioned bounce-averaged diffusion coefficients, we calculate phase space density (PSD) f t evolution of electrons in solving a two-dimensional Fokker–Planck diffusion equation [41] . The evolution of differential flux j is then simulated in the pitch angle region 0°–90° by the subsequent conversion j = p 2 f t and the results are extended to the range 90°–180° due to the mirror symmetry. We show in Fig. 4a remarkable agreement between the simulated pitch angle distribution ( Fig. 4c,d ) and the REPT observation ( Fig. 4a,b ) during the acceleration interval. The most pronounced flux enhancement occurs over the medium pitch angles 30°–60°, where momentum diffusion rates for combined chorus and MS waves dominate ( Fig. 3 ). Furthermore, the pitch angle distributions at higher pitch angles close to 90° are enhanced due to larger momentum diffusion rates of chorus waves. Finally, the combined acceleration by chorus and MS waves significantly alters the whole population of relativistic electrons, yielding the butterfly distribution in about 9 h. Figure 4: Comparison of Fokker–Planck simulation results with observations. ( a , b ) Observed evolution of the relativistic electron fluxes as a function of pitch angle at the same location (shown inside panels) during the simulation. ( c , d ) Starting with an initial condition representative of relativistic electrons ( a ), we show the relativistic (2–3.6 MeV) electron fluxes due to acceleration by chorus and MS waves from a numerical solution to the two-dimensional Fokker–Planck diffusion equation. Acceleration is most pronounced within the medium pitch angle region 30°–60° or 120°–150° produced by both chorus and MS waves, and noticeable around the high pitch angles induced by chorus waves. The combined acceleration by chorus and MS waves leads to the butterfly distribution both in the magnitude and the time scale ( d ) comparable to the observation ( b ). The current simulation provides a further support for wave-driven butterfly distribution of relativistic electrons during this storm. Full size image\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"The present modelling provides a further confirmation of wave-driven butterfly pitch angle distribution of relativistic electrons observed by the REPT instrument at lower locations ( L <5) of the outer radiation belt. This is in contrast to the formation of butterfly distributions at higher locations ( L >6) induced by drift shell splitting due to local magnetic field asymmetry. The associated physical processes are schematically presented in Fig. 5 . Relativistic electrons at L <5 drift azimuthally around Earth inside the magnetosphere without loss to the magnetopause, continuously resonating with chorus and MS waves. The combined acceleration by chorus and MS waves occurs preferably between the medium pitch angles 30° and 60°, leading to the unusual butterfly distribution. The combined acceleration process described here is a universal physical process, which should also be effective in the magnetospheres of Jupiter, Saturn and other magnetized plasma environments in the cosmos. Figure 5: Schematic diagram of formation of butterfly distributions. The long black curves indicate Earth’s magnetic field lines. The grey area denote the region of dense cold plasma known as plasmasphere. The blue and yellow areas stand for primary occurrences of chorus and MS waves. The wavy line represents chorus (blue) and MS (yellow) waves. Relativistic electrons originating from the plasma sheet azimuthally drift eastward around Earth. (1) Higher pitch angle electrons at L >6 drift to larger radial distance beyond the dayside magnetopause and are consequently lost due to the day–night magnetic field asymmetry (2), yielding the regular butterfly distribution (3). (4) Electrons at L <5 drift inside the magnetosphere without loss to the dayside magnetopause because of a very small day–night magnetic field asymmetry, continuously resonating with chorus and MS waves. (5) Wave acceleration can enhance electron PSD primarily within the medium pitch angles, leading to the formation of the unusual butterfly distribution (6). Full size image It should be mentioned that, in a departure from the coherent chorus–electron interaction approach [38] , we use the quasi-linear theory of wave–particle interaction, which has been frequently adopted by radiation belt research community to treat wave–particle interaction [12] , [40] , [42] . Previous work [43] has shown that energetic electron pitch angle scattering by coherent chorus waves is 3 orders more rapid than by incoherent chorus waves. Coherent chorus waves can produce particle diffusion, phase trapping and/or phase bunching, which is determined primarily by the competing effects of wave amplitude and ambient magnetic field inhomogeneity at resonance. However, application of coherent wave–particle interaction to the strom-time global evolution of energetic particle distributions has not yet been established so far. We therefore expect the adopted quasi-linear theory here to be a basis for comparison with future developments in nonlinear modelling. The basic conditions for the quasi-linear theory are that each individual particle randomly moves in velocity space, resonates with a succession of uncorrelated and small amplitude waves, and is scattered in a small amount in pitch angle and energy each time. Those conditions are basically satisfied in Earth’s radiation belts for naturally generated MS and chorus waves, where the MS wave bandwidth is generally above the proton gyrofrequency up to the lower hybrid frequency and the chorus wave roughly lies in the frequency range 0.1–0.8 f ce ( f ce being the electron gyrofrequency). In general, there could be a distorted/stretched geomagnetic field on the nightside during a geomagnetic storm. We have examined the geomagnetic field data from Van Allen Probes and found that the observed data are close to the dipole field model during this storm period. The largest distortion of the dipole field is about 20% in the simulation period. Moreover, we perform a test-particle simulation of trajectories of trapped relativistic (2 MeV) electrons by the TS04 magnetic model [44] for different high pitch angles. Simulation results ( Fig. 6 ) show that, starting at the location L =4.8 and MLT=24, relativistic electrons drift eastward around Earth and approach a farthest location L ≈5.1 on the dayside. Then they pass the observed location without loss to the magnetopause because the magnetopause locates above L =7.6 in the simulation period. Simulations for different energies or pitch angles (not shown for brevity) indeed show similar results. This indicates that drift shell splitting is unlikely to play a role in the observed butterfly electron distribution in this event. Figure 6: Trajectories of trapped electrons. Test-particle simulations of relativistic (2 MeV) electron trajectories by the TS04 magnetic model for different pitch angles 70° (green), 80° (blue) and 90° (black). The input parameters for TS04 magnetic model are based on the observations. Starting at the location (the symbols S ): L =4.8 and MLT=24, relativistic electrons drift eastward around Earth and approach a farthest location L ≈5.1 on the dayside. Then they pass the observed location (the symbol O ) without loss to the magnetopause because the magnetopause locates above L =7.6 in the simulation period. Full size image A basic assumption is adopted here that the cold plasma density remains constant and unchanging during the 9-h simulation period. Using the upper hybrid resonance frequency observation from Van Allen Probes, we infer the ambient electron density at locations along Van Allen Probes’ orbits and find that the electron density is comparable to the adopted plasma density and does not change much in 9 h. Analogous to the previous work [40] , this assumption is probably reasonable in the absence of simultaneous and continuous local time observations of cold plasma density in the 9-h period. To check whether the plasmaspheric plume exists, we analyse the potential data from the Time History of Events and Macroscale Interactions During Substorms (THEMIS) spacecraft at L =4.8, and find no distinct presence of plasmaspheric plume except a slightly higher electron density in the region around/after MLT=16 in the simulation period. The plasmaspheric plume resulting from time-varying convection generally forms with a more structured distribution of plasma in the dusk region in the storm main phase. Meanwhile, the plume moves eastward (towards later MLTs) in Earth’s rotation eastward, and gradually fades/erodes as the strength of convection decreases. As shown in Fig. 1 , the simulation period is in the storm recovery phase on 29 June, >24 h after the onset of the storm on 28 June. Hence, the plume probably already faded and moved to the later MLT region. However, further research is still required in the future because there are numerous uncertainties in accurate determination of either the upper hybrid resonance frequency from Van Allen Probes or the THEMIS spacecraft potential.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Ray tracing of MS waves We perform the ray tracing of MS waves with wave vector k and frequency ω by the following standard Hamiltonian equations [45] : where R is the position vector of a point on the ray path, t is the group time, D represents the standard wave dispersion relation obeying D ( R , ω , k )=0 at every point along the ray path. The spatial variation in D can be written: where B 0 is the ambient magnetic field and N c is the background plasma density. We adopt a dipole magnetic field model and the MLT-dependent plasmatrough density model [46] . The Earth-centred Cartesian and local Cartesian coordinate systems for the ray-tracing calculation are described in Supplementary Note 1 . Calculation of the global chorus wave amplitude Recent works [34] , [35] have constructed a global chorus wave model based on the data of low-altitude electron population collected by multiple POES satellites. The Medium Energy Proton and Electron Detector (MEPED) onboard POES has two electron solid-state detector (0° and 90°) telescopes to measure electron fluxes in three energy channels (> 30 keV,> 100 keV and>300 keV) [47] . The 90° telescope largely measures the trapped flux over the invariant latitude range between 55° and 70°, and the 0° telescope measures precipitating flux inside the bounce loss cone at L >1.4 (ref. 48 ). Using the electron distribution function near the loss cone [33] , the chorus wave amplitude can be calculated from the ratio between the measured precipitated and trapped electron fluxes (30–100 keV and 100–300 keV). The electron energy spectrum is assumed to follow a kappa-type function [49] , [50] . The basic equation for linking the ratio of electron count rates measured by the 0° and 90° telescopes to chorus wave amplitude is shown in Supplementary Note 2 . The ratios of precipitated and trapped electron fluxes measured by multiple POES satellites are used to construct the global chorus wave model over a broad range in L and MLT, and the ratios obtained in four distinct MLT sectors are shown in Supplementary Fig. 2 . Calculation of diffusion coefficients due to chorus and MS waves We assume that the wave spectral density B f 2 follows a typical Gaussian frequency distribution with a center f m , a half-width δf , a band between f 1 and f 2 (ref. 51 ). here is the wave amplitude in units of Tesla and erf is the error function. To allow the data modelling to be more reliable, we average the observed wave magnetic field intensity over the indicated time period in this event and then apply the corresponding Gaussian fit for MS waves as shown in Supplementary Fig. 3 . We choose the wave normal angle distribution to also satisfy a standard Gaussian form: where X =tan θ ( θ 1 ≤ θ ≤ θ 2 , X 1,2 =tan θ 1,2 ), with a half-width X ω and a peak X m . Based on the observation, we choose X m =tan 89°, X ω =tan 86°, X 1 = X m − X ω , X 2 = X m + X ω ; and the maximum latitude for the presence of MS wave λ m =10°. Based on the ray-tracing results ( Supplementary Fig. 1 ), we assume the MS wave spectral intensity at L =4.8 and MLT=19 hours to follow the similar Gaussian distribution to that as shown in Supplementary Fig. 3 , which are then used to calculate the MS-driven bounce-averaged diffusion coefficients. For the chorus waves, we adopt the wave parameters as shown in Table 1 to calculate the chorus-driven bounce-averaged diffusion coefficients at the location L =4.8. We consider contribution from harmonic resonances up to n =±5 for both chorus and MS waves. The ambient plasma density is obtained by the MLT-dependent plasmatrough density model [46] and further assumed to remain latitudinally constant and unchanged during the 9-h simulation period. Note that there is no realistic data available either in 08–12 or in 16–20 MLT. We assume the chorus wave power in 08–12 MLT to follow the latitude-dependent model [36] . The corresponding adopted wave amplitude, wave spectrum and latitudinal occurrence of chorus waves, the ambient electron density and the equatorial ratio of plasma frequency to gyrofrequency ( f pe / f ce ) are shown in Table 1 . Evaluation of the relativistic electron PSD evolution The evolution of the electron PSD f t is calculated by solving the bounce-averaged pitch angle and momentum diffusion equation here p is the electron momentum, G = p 2 T ( α e )sin α e cos α e with α e being the equatorial pitch angle, the normalized bounce time T ( α e )≈1.30−0.56sin α e ; 〈 D αα 〉, 〈 D pp 〉, and 〈 D αp 〉=〈 D pα 〉 denote bounce-averaged diffusion coefficients in pitch angle, momentum and cross pitch angle–momentum. The explicit expressions of those bounce-averaged diffusion coefficients can be found in the previous work. [41] The initial condition is modelled by a bi-modal kappa-type distribution function of energetic electrons [49] , [50] : Each component, with variable weighting parameters a 1 and a 2 , is expressed as: here n h is the number density of energetic electrons, l indicates the loss-cone index, Γ is the gamma function, κ and are the spectral index and effective thermal energy scaled by the electron rest mass energy m e c 2 ( ∼ 0.5 MeV). For the pitch angle boundary condition, f t =0 at the loss-cone α e = α L (sin α L = L −3/2 (4−3/ L ) −1/4 ) to simulate a rapid precipitation of electrons inside the loss cone ( Fig. 2e–h ), and ∂ f t /∂ α e =0 at α e =90°. For the energy diffusion boundary conditions, at the lower boundary 0.2 MeV, and at the upper boundary 10 MeV. Based on the observation, we choose the following values of parameters: ( ∼ 75 keV), κ =4, n h =0.16 cm −3 ; a 1 =0.6, l 1 =0.6; a 2 =0.4, l 2 =1.5. We solve the diffusion equation(6) using the recently developed hybrid difference method [41] , which is efficient, stable and easily parallel programmed. The numerical grid is set to be 91 × 101 and uniform in pitch angle and natural logarithm of momentum. The drifting average is taken 25% for MS wave and 1/6 for chorus wave in each aforementioned MLT sector ( Table 1 ).\"\n },\n {\n \"section\": \"Additional information\",\n \"content\": \"How to cite this article: Xiao, F. et al . Wave-driven butterfly distribution of Van Allen belt relativistic electrons. Nat. Commun. 6:8590 doi: 10.1038/ncomms9590 (2015).\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"The core research idea is to investigate how combined chorus and magnetosonic wave acceleration generates an unusual butterfly pitch angle distribution of relativistic electrons around L = 4.8 in Earth's outer Van Allen radiation belt, as observed during the 28 June 2013 geomagnetic storm and supported by two-dimensional simulations showing distinct flux enhancements at medium and near-90° pitch angles."},"entities":{"kind":"list like","value":["combined chorus and magnetosonic wave acceleration","unusual butterfly pitch angle distribution","relativistic electrons","L = 4.8","Earth's outer Van Allen radiation belt","28 June 2013 geomagnetic storm","two-dimensional simulations","medium pitch angles","near-90° pitch angles","flux enhancements"],"string":"[\n \"combined chorus and magnetosonic wave acceleration\",\n \"unusual butterfly pitch angle distribution\",\n \"relativistic electrons\",\n \"L = 4.8\",\n \"Earth's outer Van Allen radiation belt\",\n \"28 June 2013 geomagnetic storm\",\n \"two-dimensional simulations\",\n \"medium pitch angles\",\n \"near-90° pitch angles\",\n \"flux enhancements\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1","title":"Roederer, J. G. On the adiabatic motion of energetic particles in a model magnetosphere.J. Geophys. Res.72, 981–992 (1967).","link":"https://doi.org/10.1029%2FJZ072i003p00981"},{"idx":"2","title":"Stone, E. C. The physical significance of and application ofL,B0 andR0 to geomagnetically trapped particles.J. Geophys. Res.68, 4157–4166 (1963).","link":"https://doi.org/10.1029%2FJZ068i014p04157"},{"idx":"3","title":"West, H. I., Buck, R. M. Jr & Walton, J. R. Electron pitch angle distributions throughout the magnetosphere as observed on Ogo 5.J. Geophys. Res.78, 1064–1081 (1973).","link":"https://doi.org/10.1029%2FJA078i007p01064"},{"idx":"4","title":"Baker, D. N., Higbie, P. R., Hones, E. W. Jr & Belian, R. D. High-resolution energetic particle measurements at 6.6RE. Low-energy electron anisotropies and short-term substorm predictions.J. Geophys. Res.83, 4863–4868 (1978).","link":"https://doi.org/10.1029%2FJA083iA10p04863"},{"idx":"5","title":"Takahashi, K. et al. Drift-shell splitting of energetic ions injected at pseudo-substorm onsets.J. Geophys. Res.102, 22117–22130 (1997).","link":"https://doi.org/10.1029%2F97JA01870"},{"idx":"6","title":"Selesnick, R. S. & Blake, J. B. Relativistic electron drift shell splitting.J. Geophys. Res.1079, 1265 (2002).","link":"https://doi.org/10.1029%2F2001JA009179"},{"idx":"7","title":"Garcia, H. A. & Spjeldvik, W. N. Anisotropy characteristics of geomagnetically trapped ions.J. Geophys. Res.90, 347–358 (1985).","link":"https://doi.org/10.1029%2FJA090iA01p00347"},{"idx":"8","title":"Sibeck, D. G., McEntire, R. W., Lui, A. T. Y., Lopez, R. E. & Krimigis, S. M. Magnetic field drift shell splitting: Cause of unusual dayside particle pitch angle distributions during storms and substorms.J. Geophys. Res.92, 13345–13497 (1987).","link":"https://doi.org/10.1029%2FJD092iD11p13345"},{"idx":"10","title":"Gannon, J. L., Li, X. & Heynderickx, D. Pitch angle distribution analysis of radiation belt electrons based on combined release and radiation effects satellite medium electrons A data.J. Geophys. Res.112, A05212 (2007).","link":"https://doi.org/10.1029%2F2005JA011565"},{"idx":"11","title":"Horne, R. B. et al. Timescale for radiation belt electron acceleration by whistler mode chorus waves.J. Geophys. Res.110, A03225 (2005).","link":"https://doi.org/10.1029%2F2004JA010811"},{"idx":"12","title":"Horne, R. B. et al. Electron acceleration in the Van Allen radiation belts by fast magnetosonic waves.Geophys. Res. Lett.34, L17107 (2007).","link":"https://doi.org/10.1029%2F2007GL030267"},{"idx":"13","title":"Su, Z., Xiao, F., Zheng, H. & Wang, S. Combined radial diffusion and adiabatic transport of radiation belt electrons with arbitrary pitch angles.J. Geophys. Res.115, A10249 (2010).","link":"https://doi.org/10.1029%2F2010JD014286"},{"idx":"15","title":"Mauk, B. H. et al. Science objectives and rationale for the radiation belt storm probes mission.Space Sci. Rev.179, 3–27 (2012).","link":"https://link.springer.com/doi/10.1007/s11214-012-9908-y"},{"idx":"16","title":"Gonzalez, W. D., Tsurutani, B. T. & Gonzalez, A. L. C. Interplanetary origin of geomagnetic storms.Space Sci. Rev.88, 529–562 (1999).","link":"https://doi.org/10.1023%2FA%3A1005160129098"},{"idx":"17","title":"Gonzalez, W. D. et al. What is a geomagnetic storm.J. Geophys. Res.99, 5771–5792 (1994).","link":"https://doi.org/10.1029%2F93JA02867"},{"idx":"18","title":"Kletzing, C. A. et al. The electric and magnetic field instrument suite and integrated science (EMFISIS) on RBSP.Space Sci. Rev.179, 127–181 (2013).","link":"https://link.springer.com/doi/10.1007/s11214-013-9993-6"},{"idx":"19","title":"Wygant, J. R. et al. The electric field and waves (EFW) instruments on the radiation belt storm probes mission.Space Sci. Rev.179, 183–220 (2013).","link":"https://link.springer.com/doi/10.1007/s11214-013-0013-7"},{"idx":"22","title":"Summers, D., Tong, R. & Thorne, R. M. Limit on stably trapped particle fluxes in planetary magnetospheres.J. Geophys. Res.114, A10210 (2009).","link":"https://doi.org/10.1029%2F2009JA014428"},{"idx":"23","title":"Chen, L., Thorne, R. M., Jordanova, V. K., Thomsen, M. F. & Horne, R. B. Magnetosonic wave instability analysis for proton ring distributions observed by the LANL magnetospheric plasma analyser.J. Geophys. Res116, A03223 (2011).","link":"https://doi.org/10.1029%2F2011JD016244"},{"idx":"24","title":"Baker, D. N. et al. The relativistic electron-proton telescope (REPT) instrument on board the Radiation Belt Storm Probes (RBSP) spacecraft: Characterization of Earth’s radiation belt high-energy particle populations.Space Sci. Rev.179, 337–381 (2012).","link":"https://link.springer.com/doi/10.1007/s11214-012-9950-9"},{"idx":"26","title":"Tsurutani, B. T. et al. Extremely intense ELF magnetosonicwaves: A survey of polar observations.J. Geophys. Res. Space Phys.119, 964–977 (2014).","link":"https://doi.org/10.1002%2F2013JA019284"},{"idx":"27","title":"Xiao, F., Zhou, Q., He, Z. & Tang, L. Three-dimensional ray tracing of fast magnetosonic waves.J. Geophys. Res.117, A06208 (2012).","link":"https://doi.org/10.1029%2F2011JG001889"},{"idx":"28","title":"Ma, Q., Li, W., Thorne, R. M. & Angelopoulos, V. Global distribution of equatorial magnetosonic waves observed by THEMIS.Geophys. Res. Lett.40, 1895–1901 (2013).","link":"https://doi.org/10.1002%2Fgrl.50434"},{"idx":"29","title":"Horne, R. B. Path-integrated growth of electrostatic waves: The generation of terrestrial myriametric radiation.J. Geophys. Res.94, 8895–8909 (1989).","link":"https://doi.org/10.1029%2FJA094iA07p08895"},{"idx":"30","title":"Xiao, F., Chen, L., Zheng, H. & Wang, S. A parametric ray tracing study of superluminous auroral kilometric radiation wave modes.J. Geophys. Res.112, A10214 (2007).","link":"https://doi.org/10.1029%2F2007JD008956"},{"idx":"31","title":"Meredith, N. P. et al. Global models of lower band and upper band chorus from multiple satellite observations.J. Geophys. Res.117, A10225 (2012).","link":"https://doi.org/10.1029%2F2012JA017978"},{"idx":"33","title":"Kennel, C. F. & Petschek, H. E. Limit on stably trapped particle fluxes.J. Geophys. Res.71, 1–28 (1966).","link":"https://doi.org/10.1029%2FJZ071i001p00001"},{"idx":"34","title":"Li, W. et al. Constructing the global distribution of chorus wave intensity using measurements of electrons by the POES satellites and waves by the Van Allen Probes.Geophys. Res. Lett.40, 4526–4532 (2013).","link":"https://doi.org/10.1002%2Fgrl.50920"},{"idx":"35","title":"Ni, B. et al. A novel technique to construct the global distribution of whistler mode chorus wave intensity using low-altitude POES electron data.J. Geophys. Res. Space Phys.119, 5685–5699 (2014).","link":"https://doi.org/10.1002%2F2014JA019935"},{"idx":"36","title":"Shprits, Y. Y., Meredith, N. P. & Thorne, R. M. Parameterization of radiation belt electron loss timescales due to interactions with chorus waves.Geophys. Res. Lett.34, L11110 (2007).","link":"https://doi.org/10.1029%2F2006GL029050"},{"idx":"37","title":"Lam, M. M. et al. Origin of energetic electron precipitation >30 keV into the atmosphere.J. Geophys. Res.115, A00F08 (2010).","link":"https://doi.org/10.1029%2F2009JA014619"},{"idx":"39","title":"Tsurutani, B. T., Lakhina, G. S. & Verkhoglyadova, O. P. Energetic electron (>10 keV) microburst precipitation,∼5-15 s X-ray pulsations, chorus, and wave–particle interactions: A review.J. Geophys. Res. Space Phys.118, 2296–2312 (2013).","link":"https://doi.org/10.1002%2Fjgra.50264"},{"idx":"40","title":"Thorne, R. M. et al. Rapid local acceleration of relativistic radiation-belt electrons by magnetospheric chorus.Nature504, 411–414 (2013).","link":"https://doi.org/10.1038%2Fnature12889"},{"idx":"42","title":"Horne, R. B. et al. Wave acceleration of electrons in the Van Allen radiation belts.Nature437, 227–230 (2005).","link":"https://doi.org/10.1038%2Fnature03939"},{"idx":"43","title":"Lakhina, G. S., Tsurutani, B. T., Verkhoglyadova, O. P. & Pickett, J. S. Pitch angle transport of electrons due to cyclotron interactions with the coherent chorus subelements.J. Geophys. Res.115, A00F15 (2010).","link":"https://doi.org/10.1029%2F2009JA014885"},{"idx":"44","title":"Tsyganenko, N. A. & Sitnov, M. I. Modelling the dynamics of the inner magnetosphere during strong geomagnetic storms.J. Geophys. Res.110, A03208 (2005).","link":"https://doi.org/10.1029%2F2004JA010798"},{"idx":"45","title":"Suchy, K. Real Hamilton equations of geomagnetic optics for media with moderate absorption.Radio Sci.16, 1179–1182 (1981).","link":"https://doi.org/10.1029%2FRS016i006p01179"},{"idx":"46","title":"Sheeley, B. W., Moldwin, M. B., Rassoul, H. K. & Anderson, R. R. An empirical plasmasphere and trough density model: CRRES observations.J. Geophys. Res.106, 25631–25641 (2001).","link":"https://doi.org/10.1029%2F2000JA000286"},{"idx":"50","title":"Vasyliunas, V. M. A survey of low-energy electrons in the evening sector of the magnetosphere with ogo 1 and ogo 3.J. Geophys. Res.73, 2839–2884 (1968).","link":"https://doi.org/10.1029%2FJA073i009p02839"},{"idx":"51","title":"Lyons, L. R., Thorne, R. M. & Kennel, C. F. Pitch angle diffusion of radiation belt electrons within the plasmasphere.J. Geophys. Res.77, 3455–3474 (1972).","link":"https://doi.org/10.1029%2FJA077i019p03455"}],"string":"[\n {\n \"idx\": \"1\",\n \"title\": \"Roederer, J. G. On the adiabatic motion of energetic particles in a model magnetosphere.J. Geophys. Res.72, 981–992 (1967).\",\n \"link\": \"https://doi.org/10.1029%2FJZ072i003p00981\"\n },\n {\n \"idx\": \"2\",\n \"title\": \"Stone, E. C. The physical significance of and application ofL,B0 andR0 to geomagnetically trapped particles.J. Geophys. Res.68, 4157–4166 (1963).\",\n \"link\": \"https://doi.org/10.1029%2FJZ068i014p04157\"\n },\n {\n \"idx\": \"3\",\n \"title\": \"West, H. I., Buck, R. M. Jr & Walton, J. R. Electron pitch angle distributions throughout the magnetosphere as observed on Ogo 5.J. Geophys. Res.78, 1064–1081 (1973).\",\n \"link\": \"https://doi.org/10.1029%2FJA078i007p01064\"\n },\n {\n \"idx\": \"4\",\n \"title\": \"Baker, D. N., Higbie, P. R., Hones, E. W. Jr & Belian, R. D. High-resolution energetic particle measurements at 6.6RE. Low-energy electron anisotropies and short-term substorm predictions.J. Geophys. Res.83, 4863–4868 (1978).\",\n \"link\": \"https://doi.org/10.1029%2FJA083iA10p04863\"\n },\n {\n \"idx\": \"5\",\n \"title\": \"Takahashi, K. et al. Drift-shell splitting of energetic ions injected at pseudo-substorm onsets.J. Geophys. Res.102, 22117–22130 (1997).\",\n \"link\": \"https://doi.org/10.1029%2F97JA01870\"\n },\n {\n \"idx\": \"6\",\n \"title\": \"Selesnick, R. S. & Blake, J. B. Relativistic electron drift shell splitting.J. Geophys. Res.1079, 1265 (2002).\",\n \"link\": \"https://doi.org/10.1029%2F2001JA009179\"\n },\n {\n \"idx\": \"7\",\n \"title\": \"Garcia, H. A. & Spjeldvik, W. N. Anisotropy characteristics of geomagnetically trapped ions.J. Geophys. Res.90, 347–358 (1985).\",\n \"link\": \"https://doi.org/10.1029%2FJA090iA01p00347\"\n },\n {\n \"idx\": \"8\",\n \"title\": \"Sibeck, D. G., McEntire, R. W., Lui, A. T. Y., Lopez, R. E. & Krimigis, S. M. Magnetic field drift shell splitting: Cause of unusual dayside particle pitch angle distributions during storms and substorms.J. Geophys. Res.92, 13345–13497 (1987).\",\n \"link\": \"https://doi.org/10.1029%2FJD092iD11p13345\"\n },\n {\n \"idx\": \"10\",\n \"title\": \"Gannon, J. L., Li, X. & Heynderickx, D. Pitch angle distribution analysis of radiation belt electrons based on combined release and radiation effects satellite medium electrons A data.J. Geophys. Res.112, A05212 (2007).\",\n \"link\": \"https://doi.org/10.1029%2F2005JA011565\"\n },\n {\n \"idx\": \"11\",\n \"title\": \"Horne, R. B. et al. Timescale for radiation belt electron acceleration by whistler mode chorus waves.J. Geophys. Res.110, A03225 (2005).\",\n \"link\": \"https://doi.org/10.1029%2F2004JA010811\"\n },\n {\n \"idx\": \"12\",\n \"title\": \"Horne, R. B. et al. Electron acceleration in the Van Allen radiation belts by fast magnetosonic waves.Geophys. Res. Lett.34, L17107 (2007).\",\n \"link\": \"https://doi.org/10.1029%2F2007GL030267\"\n },\n {\n \"idx\": \"13\",\n \"title\": \"Su, Z., Xiao, F., Zheng, H. & Wang, S. Combined radial diffusion and adiabatic transport of radiation belt electrons with arbitrary pitch angles.J. Geophys. Res.115, A10249 (2010).\",\n \"link\": \"https://doi.org/10.1029%2F2010JD014286\"\n },\n {\n \"idx\": \"15\",\n \"title\": \"Mauk, B. H. et al. Science objectives and rationale for the radiation belt storm probes mission.Space Sci. Rev.179, 3–27 (2012).\",\n \"link\": \"https://link.springer.com/doi/10.1007/s11214-012-9908-y\"\n },\n {\n \"idx\": \"16\",\n \"title\": \"Gonzalez, W. D., Tsurutani, B. T. & Gonzalez, A. L. C. Interplanetary origin of geomagnetic storms.Space Sci. Rev.88, 529–562 (1999).\",\n \"link\": \"https://doi.org/10.1023%2FA%3A1005160129098\"\n },\n {\n \"idx\": \"17\",\n \"title\": \"Gonzalez, W. D. et al. What is a geomagnetic storm.J. Geophys. Res.99, 5771–5792 (1994).\",\n \"link\": \"https://doi.org/10.1029%2F93JA02867\"\n },\n {\n \"idx\": \"18\",\n \"title\": \"Kletzing, C. A. et al. The electric and magnetic field instrument suite and integrated science (EMFISIS) on RBSP.Space Sci. Rev.179, 127–181 (2013).\",\n \"link\": \"https://link.springer.com/doi/10.1007/s11214-013-9993-6\"\n },\n {\n \"idx\": \"19\",\n \"title\": \"Wygant, J. R. et al. The electric field and waves (EFW) instruments on the radiation belt storm probes mission.Space Sci. Rev.179, 183–220 (2013).\",\n \"link\": \"https://link.springer.com/doi/10.1007/s11214-013-0013-7\"\n },\n {\n \"idx\": \"22\",\n \"title\": \"Summers, D., Tong, R. & Thorne, R. M. Limit on stably trapped particle fluxes in planetary magnetospheres.J. Geophys. Res.114, A10210 (2009).\",\n \"link\": \"https://doi.org/10.1029%2F2009JA014428\"\n },\n {\n \"idx\": \"23\",\n \"title\": \"Chen, L., Thorne, R. M., Jordanova, V. K., Thomsen, M. F. & Horne, R. B. Magnetosonic wave instability analysis for proton ring distributions observed by the LANL magnetospheric plasma analyser.J. Geophys. Res116, A03223 (2011).\",\n \"link\": \"https://doi.org/10.1029%2F2011JD016244\"\n },\n {\n \"idx\": \"24\",\n \"title\": \"Baker, D. N. et al. The relativistic electron-proton telescope (REPT) instrument on board the Radiation Belt Storm Probes (RBSP) spacecraft: Characterization of Earth’s radiation belt high-energy particle populations.Space Sci. Rev.179, 337–381 (2012).\",\n \"link\": \"https://link.springer.com/doi/10.1007/s11214-012-9950-9\"\n },\n {\n \"idx\": \"26\",\n \"title\": \"Tsurutani, B. T. et al. Extremely intense ELF magnetosonicwaves: A survey of polar observations.J. Geophys. Res. Space Phys.119, 964–977 (2014).\",\n \"link\": \"https://doi.org/10.1002%2F2013JA019284\"\n },\n {\n \"idx\": \"27\",\n \"title\": \"Xiao, F., Zhou, Q., He, Z. & Tang, L. Three-dimensional ray tracing of fast magnetosonic waves.J. Geophys. Res.117, A06208 (2012).\",\n \"link\": \"https://doi.org/10.1029%2F2011JG001889\"\n },\n {\n \"idx\": \"28\",\n \"title\": \"Ma, Q., Li, W., Thorne, R. M. & Angelopoulos, V. Global distribution of equatorial magnetosonic waves observed by THEMIS.Geophys. Res. Lett.40, 1895–1901 (2013).\",\n \"link\": \"https://doi.org/10.1002%2Fgrl.50434\"\n },\n {\n \"idx\": \"29\",\n \"title\": \"Horne, R. B. Path-integrated growth of electrostatic waves: The generation of terrestrial myriametric radiation.J. Geophys. Res.94, 8895–8909 (1989).\",\n \"link\": \"https://doi.org/10.1029%2FJA094iA07p08895\"\n },\n {\n \"idx\": \"30\",\n \"title\": \"Xiao, F., Chen, L., Zheng, H. & Wang, S. A parametric ray tracing study of superluminous auroral kilometric radiation wave modes.J. Geophys. Res.112, A10214 (2007).\",\n \"link\": \"https://doi.org/10.1029%2F2007JD008956\"\n },\n {\n \"idx\": \"31\",\n \"title\": \"Meredith, N. P. et al. Global models of lower band and upper band chorus from multiple satellite observations.J. Geophys. Res.117, A10225 (2012).\",\n \"link\": \"https://doi.org/10.1029%2F2012JA017978\"\n },\n {\n \"idx\": \"33\",\n \"title\": \"Kennel, C. F. & Petschek, H. E. Limit on stably trapped particle fluxes.J. Geophys. Res.71, 1–28 (1966).\",\n \"link\": \"https://doi.org/10.1029%2FJZ071i001p00001\"\n },\n {\n \"idx\": \"34\",\n \"title\": \"Li, W. et al. Constructing the global distribution of chorus wave intensity using measurements of electrons by the POES satellites and waves by the Van Allen Probes.Geophys. Res. Lett.40, 4526–4532 (2013).\",\n \"link\": \"https://doi.org/10.1002%2Fgrl.50920\"\n },\n {\n \"idx\": \"35\",\n \"title\": \"Ni, B. et al. A novel technique to construct the global distribution of whistler mode chorus wave intensity using low-altitude POES electron data.J. Geophys. Res. Space Phys.119, 5685–5699 (2014).\",\n \"link\": \"https://doi.org/10.1002%2F2014JA019935\"\n },\n {\n \"idx\": \"36\",\n \"title\": \"Shprits, Y. Y., Meredith, N. P. & Thorne, R. M. Parameterization of radiation belt electron loss timescales due to interactions with chorus waves.Geophys. Res. Lett.34, L11110 (2007).\",\n \"link\": \"https://doi.org/10.1029%2F2006GL029050\"\n },\n {\n \"idx\": \"37\",\n \"title\": \"Lam, M. M. et al. Origin of energetic electron precipitation >30 keV into the atmosphere.J. Geophys. Res.115, A00F08 (2010).\",\n \"link\": \"https://doi.org/10.1029%2F2009JA014619\"\n },\n {\n \"idx\": \"39\",\n \"title\": \"Tsurutani, B. T., Lakhina, G. S. & Verkhoglyadova, O. P. Energetic electron (>10 keV) microburst precipitation,∼5-15 s X-ray pulsations, chorus, and wave–particle interactions: A review.J. Geophys. Res. Space Phys.118, 2296–2312 (2013).\",\n \"link\": \"https://doi.org/10.1002%2Fjgra.50264\"\n },\n {\n \"idx\": \"40\",\n \"title\": \"Thorne, R. M. et al. Rapid local acceleration of relativistic radiation-belt electrons by magnetospheric chorus.Nature504, 411–414 (2013).\",\n \"link\": \"https://doi.org/10.1038%2Fnature12889\"\n },\n {\n \"idx\": \"42\",\n \"title\": \"Horne, R. B. et al. Wave acceleration of electrons in the Van Allen radiation belts.Nature437, 227–230 (2005).\",\n \"link\": \"https://doi.org/10.1038%2Fnature03939\"\n },\n {\n \"idx\": \"43\",\n \"title\": \"Lakhina, G. S., Tsurutani, B. T., Verkhoglyadova, O. P. & Pickett, J. S. Pitch angle transport of electrons due to cyclotron interactions with the coherent chorus subelements.J. Geophys. Res.115, A00F15 (2010).\",\n \"link\": \"https://doi.org/10.1029%2F2009JA014885\"\n },\n {\n \"idx\": \"44\",\n \"title\": \"Tsyganenko, N. A. & Sitnov, M. I. Modelling the dynamics of the inner magnetosphere during strong geomagnetic storms.J. Geophys. Res.110, A03208 (2005).\",\n \"link\": \"https://doi.org/10.1029%2F2004JA010798\"\n },\n {\n \"idx\": \"45\",\n \"title\": \"Suchy, K. Real Hamilton equations of geomagnetic optics for media with moderate absorption.Radio Sci.16, 1179–1182 (1981).\",\n \"link\": \"https://doi.org/10.1029%2FRS016i006p01179\"\n },\n {\n \"idx\": \"46\",\n \"title\": \"Sheeley, B. W., Moldwin, M. B., Rassoul, H. K. & Anderson, R. R. An empirical plasmasphere and trough density model: CRRES observations.J. Geophys. Res.106, 25631–25641 (2001).\",\n \"link\": \"https://doi.org/10.1029%2F2000JA000286\"\n },\n {\n \"idx\": \"50\",\n \"title\": \"Vasyliunas, V. M. A survey of low-energy electrons in the evening sector of the magnetosphere with ogo 1 and ogo 3.J. Geophys. Res.73, 2839–2884 (1968).\",\n \"link\": \"https://doi.org/10.1029%2FJA073i009p02839\"\n },\n {\n \"idx\": \"51\",\n \"title\": \"Lyons, L. R., Thorne, R. M. & Kennel, C. F. Pitch angle diffusion of radiation belt electrons within the plasmasphere.J. Geophys. Res.77, 3455–3474 (1972).\",\n \"link\": \"https://doi.org/10.1029%2FJA077i019p03455\"\n }\n]","isConversational":false}}},{"rowIdx":189,"cells":{"unique_id":{"kind":"string","value":"s41467-021-27149-z"},"subfield":{"kind":"string","value":"Physics"},"title":{"kind":"string","value":"Second sound in the crossover from the Bose-Einstein condensate to the Bardeen-Cooper-Schrieffer superfluid"},"abstract":{"kind":"string","value":"Second sound is an entropy wave which propagates in the superfluid component of a quantum liquid. Because it is an entropy wave, it probes the thermodynamic properties of the quantum liquid. Here, we study second sound propagation for a large range of interaction strengths within the crossover between a Bose-Einstein condensate (BEC) and the Bardeen-Cooper-Schrieffer (BCS) superfluid, extending previous work at unitarity. In particular, we investigate the strongly-interacting regime where currently theoretical predictions only exist in terms of an interpolation in the crossover. Working with a quantum gas of ultracold fermionic 6 Li atoms with tunable interactions, we show that the second sound speed varies only slightly in the crossover regime. By varying the excitation procedure, we gain deeper insight on sound propagation. We compare our measurement results with classical-field simulations, which help with the interpretation of our experiments."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"Second sound is a transport phenomenon of quantum liquids that emerges below the critical temperature for superfluidity T c [1] , [2] , [3] . It was experimentally discovered [4] in 1944 in He II [5] and was described with a hydrodynamic two-fluid model [2] , [6] , [7] , [8] which treats He II as a mixture of a superfluid (SF) and a normal fluid (NF). The SF component has no entropy and flows without dissipation. The NF component carries all the entropy and has non-zero viscosity. In the limit of vanishing temperature T → 0, the two-fluid model predicts that first sound (i.e., standard sound waves) corresponds to a propagating pressure oscillation with constant entropy, while second sound is an entropy oscillation propagating at constant pressure [8] . The properties of a SF naturally depend on parameters such as its temperature and the interaction strength between its particles. With the advent of ultracold quantum gases, with tunable interactions, these dependencies can now be studied. In particular, an ultracold fermionic quantum gas with a tunable Feshbach resonance offers a unique opportunity to access various sorts of superfluidity in one system, ranging continuously between a Bose-Einstein condensate (BEC) of bosonic molecules, a resonant SF, and a SF gas of Cooper pairs (BCS superfluid) [9] , [10] , [11] . In the experiment, this is done by tuning the interaction parameter \\({({k}_{{{{{{{{\\rm{F}}}}}}}}}a)}^{-1}\\) , where a is the scattering length, \\({k}_{{{{{{{{\\rm{F}}}}}}}}}=\\sqrt{2m{E}_{{{{{{{{\\rm{F}}}}}}}}}}/\\hslash\\) the Fermi wavenumber, E F is the Fermi energy, and m the atomic mass. A large range of thermodynamical properties of the BEC-BCS crossover has been studied, e.g., in refs. [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] . This includes experiments on first sound (see e.g., ref. [16] ) and second sound. Second sound has recently been observed by Sidorenkov et al. [20] in a unitary Fermi gas and by Ville et al. [21] and Christodoulou et al. [22] in a two-dimensional bosonic SF. Second sound was also possibly present in an experiment by Meppelink et al. [23] as pointed out in ref. [3] . Here, we experimentally investigate how second sound changes across the BEC–BCS crossover. This is important, because full theoretical calculations are not yet available for the entire strongly interacting regime. Nevertheless, comparing our measurements to existing calculations and interpolations we find reasonable agreement. In particular, c-field simulations in the BEC regime match quite well the observed wave dynamics in experiments at interaction strengths of up to \\({({k}_{{{{{{{{\\rm{F}}}}}}}}}a)}^{-1}=1\\) . In addition, we investigate experimentally and theoretically the system response when modifying the excitation scheme. As second sound is mainly an entropy wave and first sound is mainly a pressure wave, different excitation schemes give rise to different responses for first and second sound. This helps for separating the generally weak second sound signals from the first sound ones. We find that this separation works especially well when first sound is excited as a density dip wave packet. For this case, we were able to quantitatively compare the amplitudes of first and second sound and compare the results with a prediction.","references":{"indices":[1,6,9,12,13,14,15,16,17,18,19,20,21,22,23],"infos":[{"idx":"1","title":"Halperin, W. P. Eighty years of superfluidity.Nature553, 413–414 (2018).","link":"https://doi.org/10.1038%2Fd41586-018-00417-7"},{"idx":"6","title":"Tisza, L. Transport phenomena in helium II.Nature141, 913 (1938).","link":"https://doi.org/10.1038%2F141913a0"},{"idx":"9","title":"Giorgini, S., Pitaevskii, L. P. & Stringari, S. Theory of ultracold atomic Fermi gases.Rev. Mod. Phys.80, 1215–1275 (2008).","link":"https://doi.org/10.1103%2FRevModPhys.80.1215"},{"idx":"12","title":"Nascimbène, S., Navon, N., Jiang, K. J., Chevy, F. & Salomon, C. Exploring the thermodynamics of a universal Fermi gas.Nature463, 1057–1060 (2010).","link":"https://doi.org/10.1038%2Fnature08814"},{"idx":"13","title":"Ku, M. J. H., Sommer, A. T., Cheuk, L. W. & Zwierlein, M. W. Revealing the superfluid lambda transition in the universal thermodynamics of a unitary Fermi gas.Science335, 563 (2012).","link":"https://doi.org/10.1126%2Fscience.1214987"},{"idx":"14","title":"Shin, Y., Schunck, C. H., Schirotzek, A. & Ketterle, W. Phase diagram of a two-component Fermi gas with resonant interactions.Nature451, 689–693 (2008).","link":"https://doi.org/10.1038%2Fnature06473"},{"idx":"15","title":"Sagi, Y., Drake, T. E., Paudel, R., Chapurin, R. & Jin, D. S. Breakdown of the Fermi liquid description for strongly interacting fermions.Phys. Rev. Lett.114, 075301 (2015).","link":"https://doi.org/10.1103%2FPhysRevLett.114.075301"},{"idx":"16","title":"Joseph, J. et al. Measurement of Sound Velocity in a Fermi Gas near a Feshbach Resonance.Phys. Rev. Lett.98, 170401 (2007).","link":"https://doi.org/10.1103%2FPhysRevLett.98.170401"},{"idx":"17","title":"Taylor, E. et al. First and second sound in a strongly interacting Fermi gas.Phys. Rev. A80, 053601 (2009).","link":"https://doi.org/10.1103%2FPhysRevA.80.053601"},{"idx":"18","title":"Perali, A., Pieri., P., Pisani., L. & Calvanese Strinati, G. BCS-BEC crossover at finite temperature for superfluid trapped Fermi atoms.Phys. Rev. Lett.92, 220404 (2004).","link":"https://doi.org/10.1103%2FPhysRevLett.92.220404"},{"idx":"19","title":"Navon, N., Nascimbène, S., Chevy, S. & Salomon, C. The equation of state of a low-temperature Fermi gas with tunable interactions.Science328, 729–732 (2010).","link":"https://doi.org/10.1126%2Fscience.1187582"},{"idx":"20","title":"Sidorenkov, L. et al. Second sound and the superfluid fraction in a Fermi gas with resonant interactions.Nature498, 78–82 (2013).","link":"https://doi.org/10.1038%2Fnature12136"},{"idx":"21","title":"Ville, J. L. et al. Sound propagation in a uniform superfluid two-dimensional Bose gas.Phys. Rev. Lett.121, 145301 (2018).","link":"https://doi.org/10.1103%2FPhysRevLett.121.145301"},{"idx":"22","title":"Christodoulou, P. et al. Observation of first and second sound in a BKT superfluid.Nature594, 191–194 (2021).","link":"https://doi.org/10.1038%2Fs41586-021-03537-9"},{"idx":"23","title":"Meppelink, R., Koller, S. B. & van der Straten, P. Sound propagation in a Bose-Einstein condensate at finite temperatures.Phys. Rev. A80, 043605 (2009).","link":"https://doi.org/10.1103%2FPhysRevA.80.043605"}]}},{"section":"Results","content":"Experimental details Our experiments are carried out with a balanced, two-component ultracold gas of fermionic 6 Li atoms in the two lowest hyperfine states \\(\\left|F,{m}_{F}\\right\\rangle =\\left|1/2,\\pm \\!1/2\\right\\rangle\\) of the electronic ground state. The gas is confined by a combined magnetic and optical dipole trap with a trap depth of U 0 ≈ 1 μK × k B , for details see ref. [24] , [25] . The trap is nearly harmonic and cylindrically symmetric with trapping frequencies ω r = 2 π × 305 Hz and ω x = 2 π × 21 Hz. The temperature and the particle density are controlled by evaporative cooling. In the experiments the temperature ranges approximately from 0.13 T F to 0.30 T F , where \\({T}_{{{{{{{{\\rm{F}}}}}}}}}={E}_{{{{{{{{\\rm{F}}}}}}}}}/{k}_{{{{{{{{\\rm{B}}}}}}}}}=\\hslash \\bar{\\omega }{(3N)}^{1/3}/{k}_{{{{{{{{\\rm{B}}}}}}}}}\\) is the Fermi temperature, \\(\\bar{\\omega }={\\left({\\omega }_{x}{\\omega }_{r}^{2}\\right)}^{1/3}\\) is the geometric mean of the trapping frequencies and N is the total number of atoms. The scattering length a is tunable with an external magnetic field B via a magnetic Feshbach resonance at 832 G [26] . To excite sound modes in the system, we focus a blue-detuned 532 nm laser onto the trap center following the approaches introduced in refs. [16] , [20] , [27] (Fig. 1 a). The laser beam is aligned perpendicularly to the optical dipole trap and produces a repulsive potential barrier of U ex ≈ 0.2 U 0 . At its focus, the beam has a waist of about 20 μm, which is comparable to the cloud size in the radial direction. To excite sound waves, the height of this additional potential is modulated. The excited sound modes generally exhibit contributions from both first and second sound [28] , [29] , [30] . However, it is possible to generate preferentially either one of the two sound modes by adapting the excitation method. Fig. 1: Sound excitation in a trapped superfluid Fermi gas in the vicinity of the BEC-BCS crossover. a Set-up: A focussed, intensity-modulated, blue-detuned laser beam excites sound waves in the cigar-shaped atom cloud. b Two different modulation sequences of the laser intensity. Purple dashed line: step excitation. Green solid line: heat pulse. The time t is given in units of the axial trapping period 2 π / ω x . c Sketch of a bimodal density distribution of a trapped BEC (purple line) at y = z = 0. At the center of the trap a blue detuned beam produces a dimple in the potential. Modulating the beam intensity produces first sound waves (red arrows) and second sound (orange arrows) waves. Second sound reduces the local density of the cloud, while for first sound a density peak emerges. The thin black line shows the profile of the unperturbed cloud. Please note that the crests and troughs of the waves are shown in an exaggerated way for better visibility. d The false color plot shows the measured local change in the density \\({{\\Delta }}\\bar{n}(x,t)\\) as a function of axial position x and time t . Here, \\({({k}_{{{{{{{{\\rm{F}}}}}}}}}a)}^{-1}=(1.61\\pm 0.05)\\) at B = 735 G and T / T c = (0.80 ± 0.15). After excitation, two wave packets (bright traces, marked with red arrows) propagate with first sound velocity u 1 towards the edges of the cloud. The excitation method predominantly excites first sound. Second sound is present as well but is barely discernible here. e Propagation of first sound waves (bright traces, marked with red arrow) and second sound waves (dark traces, marked with orange arrows) after excitation with sinusoidal pulse of ( b ). All other settings are the same as in ( d ). f Simulated sound propagation for the same parameters as in ( e ). The orange arrows mark the propagating second sound and the red arrows the first sound, respectively. Full size image To excite primarily first sound, we abruptly switch on the excitation laser beam (Fig. 1 b), similarly as for the first experiments on sound propagation in a dilute BEC [27] . This applies pressure on the cold cloud on both sides of the laser beam and creates two density wave packets (Fig. 1 c) which propagate out in opposite directions along the axial trap axis with the speed u 1 . In the experiments, we detect these waves with the help of absorption imaging by measuring the density distribution of the atomic cloud as a function of time. Figure 1 d shows such density waves for an experiment at \\({({k}_{{{{{{{{\\rm{F}}}}}}}}}a)}^{-1}\\approx (1.61\\pm 0.05)\\) , B = 735 G and a temperature of T = (220 ± 30) nK = (0.30 ± 0.06) T F , which corresponds to T = (0.80 ± 0.15) T c , where T c is the critical temperature. For the given interaction strength, we used T c = 0.37 T F (see Supplementary Note 1 ). Figure 1 d is a time-ordered stack of one-dimensional line density profiles of the atom cloud [31] . Each profile was obtained by integrating the measured column density of the atomic cloud along the transverse (i.e., y -) direction (see “Methods” for details). The time-ordered stack shows the propagation of the sound waves along the axial direction x as a function of time. We observe two density wave packets which propagate with first sound velocity from the trap center towards the edge of the cloud (two bright traces, marked with red arrows). The propagating waves produce a density modulation of only a few percent of the peak density and can be considered as a weak perturbation of the system. To obtain the speed of sound, we examine how the center position of each wave packet changes with time. The center positions are determined via a Gaussian fit. Please note that the central sound speed is determined over a local area of the cloud where the density varies by about 30%. As a consequence, the measured sound speed should be considered a mean value. From Fig. 1 d we obtain u 1 = (17.2 ± 3) mm/s near the trap center. Our analysis shows that the sound propagation slows down as the pulse approaches the edge of the cloud where the particle density decreases. In the following, we focus on the sound speed close to the trap center. To primarily excite second sound, we sinusoidally modulate the intensity of the excitation beam for 7 ms with a modulation frequency of ω ex = 2 π × 570 Hz ≈ 2 ω r and a modulation amplitude of Δ U ≈ 0.2 U 0 . This parametrically heats the gas in radial direction (Fig. 1 b). Subsequent thermalization via collisions occurs within a few milliseconds. As a consequence a local depletion of the SF density is created, which is filled with normal gas, forming a region of increased entropy (Fig. 1 c). This gives rise to two wave packets which propagate outwards along the axial direction with the speed of second sound. Figure 1 e shows corresponding experimental data where we measure the local density distribution as in Fig. 1 d. The second sound wave appears here as a density dip (dark traces, marked with orange arrows). A clear indication that the dark trace corresponds to second sound is the fact that it vanishes at the Thomas-Fermi radius R TF ≈ 110 μm where the SF fraction vanishes. Second sound only propagates inside the SF phase. Besides a second sound wave, the excitation also produces a first sound wave (bright traces, marked with red arrows) which propagates faster than the second sound wave and travels beyond the Thomas-Fermi radius. The first sound wave is broader than in Fig. 1 d, which can be mainly explained by the longer excitation pulse. To obtain u 2 we measure the time-dependent position of the minimum of each dark trace, which is again determined via a Gaussian fit. For Fig. 1 e we obtain u 2 = (5.1 ± 1.1) mm/s. We note that for the small excitation amplitudes in our experiments (see colorbar in Fig. 1 d–e), we do not observe the asymmetric sound wave distortions reported in ref. [23] . These distortions were also absent in ref. [20] . Figure 1 f shows numerical simulations of our experiment applying a dynamical c-field method [32] (see Supplementary Note 2 for detailed information on the method). The dimer scattering length [33] is a dd = 0.6 a and we assume all fermionic atoms to be paired up in molecules. To compare the simulations with the experimental results we choose the same values of \\({({k}_{{{{{{{{\\rm{F}}}}}}}}}a)}^{-1}\\) and the same central density of the trapped gas as in the experiment. The theory value for u 2 is (5.7 ± 0.05) mm/s in agreement with the experimental value (5.1 ± 1.1) mm/s. Interaction strength dependence of second sound We now perform measurements of second sound speed in the region \\((-0.22\\pm 0.04) \\, < \\,{({k}_{{{{{{{{\\rm{F}}}}}}}}}a)}^{-1} \\, < \\,(1.61\\pm 0.05)\\) of the BCS-BEC crossover. These are shown in Fig. 2 along with theoretical predictions. The second sound velocity u 2 is given in units of the Fermi velocity \\({v}_{{{{{{{{\\rm{F}}}}}}}}}=\\hslash {k}_{{{{{{{{\\rm{F}}}}}}}}}^{{{\\rm{hom}}}}/m\\) . Here, the Fermi wavenumber \\({k}_{{{{{{{{\\rm{F}}}}}}}}}^{{{\\rm{hom}}}}= {\\left(3{\\pi }^{2}{n}_{0}\\right)}^{1/3}\\) is determined from the 3D peak density close to the trap center. The peak density is deduced using the inverse Abel transformation [34] and a self-consistent mean field calculation (see Supplementary Note 4 ). We have verified that in the BEC regime our mean field calculation gives similar results for the peak density when we input the trapping frequencies, the temperature, the scattering length and the total number of particles. The blue dash-dotted line is a calculation from ref. [28] , based on a hydrodynamic description in a homogeneous gas for the limiting cases of the BEC and the BCS regime and unitarity. To connect these regimes, the results are interpolated across the crossover, bridging the range \\(|{({k}_{{{{{{{{\\rm{F}}}}}}}}}a)}^{-1}| \\, < \\, 1\\) . The blue solid and the brown solid lines are our analytic hydrodynamic calculations which are valid in the BCS and BEC limit, respectively (see Supplementary Note 3 ). For comparison, we show the results of the numerical c-field simulations (green squares), which agree with both, analytic description and experimental results. Despite the large error bars the measurements indicate an increase of u 2 when approaching unitarity from the BEC side, in agreement with the theoretical results. Fig. 2: Second sound velocity u 2 as a function of interaction strength. The purple circles depict measured data for temperatures in the range T = 105−230 nK which corresponds to T / T c = 0.66−0.84 (see Supplementary Note 1 ). The error bars are due to statistical uncertainties. The brown and blue solid line show hydrodynamic predictions for the BEC and BCS regime at T = 0.75 T c , respectively (see Supplementary Note 3 ). The shaded areas mark the second sound velocity in the temperature range of the experiments. The blue dash-dotted line shows a theoretical prediction of second sound in the crossover [28] for a homogeneous gas at T / T c = 0.75. It interpolates between the results from hydrodynamic theory in the BEC and BCS regime. The green squares are results of our numerical c-field simulations which are consistent with both, analytic and experimental results. For comparison we also show the second sound velocity on the resonance measured in ref. [20] at the temperatures T / T c = 0.65 (blue triangle), T / T c = 0.75 (brown triangle), and T / T c = 0.85 (red triangle). Full size image In general, second sound can only propagate in the SF phase of the gas. It is therefore natural to ask how the SF density n s and the speed of second sound u 2 are related. Using our measurements for u 2 we can roughly determine the relationship between n s and u 2 for our temperature T and \\({({k}_{{{{{{{{\\rm{F}}}}}}}}}a)}^{-1} \\, > \\, 1.5\\) , since in this regime the SF density can be estimated. For this, we carry out self-consistent mean field calculations to determine the density distributions of the SF and the NF for an interacting BEC in the trap (see Supplementary Note 4 ). As an important input into these calculations we make use of the Thomas-Fermi radius which we have measured in the second sound experiments (the measured Thomas-Fermi radii can be found in Supplementary Note 1 ). As an example, from the measurement at \\({({k}_{{{{{{{{\\rm{F}}}}}}}}}a)}^{-1}=(1.61\\pm 0.05)\\) we determine the peak SF fraction to be n s0 / n 0 = 0.98 close to the trap center at maximum density (see Supplementary Note 4 ), where the local \\({({k}_{{{{{{{{\\rm{F}}}}}}}}}^{{{\\rm{hom}}}}a)}^{-1}=(1.06\\pm 0.05)\\) and \\(T/{T}_{{{{{{{{\\rm{c}}}}}}}}}^{{{\\rm{hom}}}}=(0.40\\pm 0.15)\\) , with \\({T}_{{{{{{{{\\rm{c}}}}}}}}}^{{{\\rm{hom}}}}= 0.21{T}_{{{{{{{{\\rm{F}}}}}}}}}^{{{\\rm{hom}}}}\\) and \\({T}_{{{{{{{{\\rm{F}}}}}}}}}^{{{\\rm{hom}}}}={\\hslash }^{2}{({k}_{{{{{{{{\\rm{F}}}}}}}}}^{{{\\rm{hom}}}})}^{2}/2m{k}_{{{{{{{{\\rm{B}}}}}}}}}\\) . For comparison, for a homogeneous weakly interacting BEC with a SF fraction close to unity the temperature would need to be \\(T\\ll {T}_{{{{{{{{\\rm{c}}}}}}}}}^{{{\\rm{hom}}}}\\) , according to \\({n}_{{{{{{{{\\rm{s}}}}}}}}}/n=1-{(T/{T}_{{{{{{{{\\rm{c}}}}}}}}}^{{{\\rm{hom}}}})}^{3/2}\\) . At unitarity, by contrast, the SF fraction reaches unity already at \\(T/{T}_{{{{{{{{\\rm{c}}}}}}}}}^{{{\\rm{hom}}}}\\approx 0.55\\) , as shown by Sidorenkov et al. [20] . As expected, this comparison shows that for a given \\(T/{T}_{{{{{{{{\\rm{c}}}}}}}}}^{{{\\rm{hom}}}}\\) the SF fraction grows with interaction strength. Tuning the sound mode excitation In the following we investigate how the SF gas responds to different excitation protocols [28] , [29] , [30] . For this, we vary the excitation scheme, the excitation frequency and amplitude. By observing the corresponding response of the system we gain additional insights into the nature of first and second sound. In Fig. 3 a we show the evolution of the system after a step pulse excitation at B = 735 G and Δ U = 0.3 U 0 , in which both, first and second sound are excited. In contrast to the experiment in Fig. 1 d, the laser beam is abruptly switched off - not on. As a consequence, the wave packets of both first and second sound now correspond to dips in the particle density. In Fig. 3 b we show the density distribution for the time and position range indicated by the purple rectangle in Fig. 3 a. From a fit of two Gaussian dips to the two wave packets, we determine an amplitude ratio of W 2 / W 1 ≈ 1.1. This result approximately matches the predictions of refs. [29] , [30] (see also Supplementary Note 3 ), where the response of both, a weakly and a strongly interacting molecular Bose gas has been studied. For an interaction parameter of \\({({k}_{{{{{{{{\\rm{F}}}}}}}}}a)}^{-1}=2\\) , the prediction yields W 2 / W 1 = 0.9, which is of similar magnitude as our result. Fig. 3: Comparing signal strength of first and second sound. a Sound excitation experiment at \\({({k}_{{{{{{{{\\rm{F}}}}}}}}}a)}^{-1}=(1.61\\pm 0.05)\\) and at a temperature of T / T c = (0.80 ± 0.15). In contrast to Fig. 1 d, first sound (red arrows) and second sound (orange arrows) are now visible simultaneously. For t ω x /2 π < 0.15 first and second sound waves overlap and therefore cannot be distinguished from each other. b shows \\({{\\Delta }}\\bar{n}\\) for tω x /2 π = 0.43. We fit the center position of each of the two sound waves using a Gaussian function (solid line). Full size image Next, we study the system response when modifying the excitation scheme. For this, we vary the excitation frequency and the number of modulation cycles. Figure 4 a–d shows the system response for a modulation frequency of ω ex = 0.61 ω r , so that parametric heating is rather suppressed and coupling to first sound is enhanced as compared to the measurement shown in Fig. 1 e. While Fig. 4 a, b corresponds to a 1.5 cycle modulation, Fig. 4 c, d corresponds to a 1 cycle modulation. Fig. 4: Sound excitation with different modulation sequences. a , c \\({{\\Delta }}\\bar{n}(x,t)\\) data at ω ex = 0.61 ω r , Δ U = 0.3 U 0 and \\({\\left({k}_{{{{{{{{\\rm{F}}}}}}}}}a\\right)}^{-1}=(1.61\\pm 0.05)\\) for a modulation of 1.5 cycles and of 1 cycle, respectively. The excitation pulse excites both, first and second sound waves. b , d \\({{\\Delta }}\\bar{n}(x,t)\\) from numerical c-field simulations. Top row: False color images of \\({{\\Delta }}\\bar{n}(x,t)\\) . First and second sound waves are marked with red and orange arrows, respectively. The inset in ( d ) is an enlargement, showing how the calculated bright first sound wave (black arrow) is damped when it crosses the second sound wave (orange arrow). Mid row: Shown is \\({{\\Delta }}\\bar{n}\\) for tω x /2 π = 0 ( a , b ) and for tω x /2 π = 0.15 ( c , d ). Bottom row: Applied excitation scheme. Full size image The numerical simulations in Fig. 4 b, d demonstrate how the excitation pattern produces a corresponding wave train of first sound. Once the simulated waves of first sound have propagated beyond the Thomas-Fermi radius, they diffuse out and lose signal strength. In the simulation, the diffusion of the first sound wave train is stronger than in the experiment. This might be explained by the choice of the discretization length which was used in the simulations (see Supplementary Note 2B ). The first sound wave train is always followed by a single dark second sound wave packet. The experimental data in Fig. 4 a agree reasonably well with the simulation in Fig. 4 b. Comparing Fig. 4 c and d, there seems to be a discrepancy. In the experiment clearly a first sound wave train, consisting of two bright traces and one dark trace, propagates beyond the Thomas-Fermi radius. In the simulation, however, the second bright trace of the wave train, which is clearly visible in the center, disappears before it reaches the Thomas-Fermi radius (black arrow). The corresponding damping happens when the bright sound trace crosses the dark second sound trace (see also insert in Fig. 4 d). The reason for the damping is that first sound waves diffuse in a thermal environment. This diffusion, however, is apparently too strong in the simulation results, which is probably a consequence of constraints in the numerical resolution (for more details see Supplementary Note 2B ). In conclusion, we have studied second sound propagation in an ultracold Fermi gas of 6 Li atoms throughout the BEC-BCS crossover for a finite temperature of T ≈ 0.75 T c . We find the second sound velocity to vary only slightly across the BCS-BEC crossover, which is in agreement with an interpolation of hydrodynamic theory [28] . In the BEC regime, the results match numerical predictions based on c-field simulations. In addition, we investigate the response of the SF gas on various excitation pulse shapes, ranging from gentle local heating to an abrupt kick. While a sinusoidal excitation pulse leads to a corresponding wave train for the first sound, it only produces a single pulse for the second sound. In the future, it will be useful to extend our measurements in the strongly interacting regime to a larger range of temperatures below T c . Since the second sound velocity is related to the local SF density, these measurements can help to determine the correlation between the SF density and the temperature in the strongly interacting regime."},{"section":"Methods","content":"Calculating \\({{\\Delta }}\\bar{n}\\) from the density profiles Each of the experimental sound propagation images in Figs. 1d–e , 3a , 4a and c is a time-ordered stack of one-dimensional line density profiles \\({{\\Delta }}\\bar{n}(x,t)\\) of the atom cloud. A one-dimensional line density profile n ( x , t ) is produced as follows: After sound excitation has ended and after an additional propagation time t , we take an absorption image of the atom cloud to obtain the column density distribution n ex ( x , y , t ). Next, we integrate the absorption image along the y- axis (which is perpendicular to the symmetry axis of the cigar-shaped atom cloud) to obtain a one-dimensional line density profile n ex ( x , t ). To reduce noise, we average 15 density profiles and obtain \\({\\bar{n}}_{{{{{{{{\\rm{ex}}}}}}}}}(x,t)\\) . We repeat this procedure for an unperturbed cloud to obtain \\(\\bar{n}(x)\\) . By subtracting the two density profiles from each other we obtain \\({{\\Delta }}\\bar{n}(x,t)=({\\bar{n}}_{{{{{{{{\\rm{ex}}}}}}}}}(x,t)-\\bar{n}(x))/\\bar{n}(0)\\) ."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"Second sound is a transport phenomenon of quantum liquids that emerges below the critical temperature for superfluidity T c [1] , [2] , [3] . It was experimentally discovered [4] in 1944 in He II [5] and was described with a hydrodynamic two-fluid model [2] , [6] , [7] , [8] which treats He II as a mixture of a superfluid (SF) and a normal fluid (NF). The SF component has no entropy and flows without dissipation. The NF component carries all the entropy and has non-zero viscosity. In the limit of vanishing temperature T → 0, the two-fluid model predicts that first sound (i.e., standard sound waves) corresponds to a propagating pressure oscillation with constant entropy, while second sound is an entropy oscillation propagating at constant pressure [8] . The properties of a SF naturally depend on parameters such as its temperature and the interaction strength between its particles. With the advent of ultracold quantum gases, with tunable interactions, these dependencies can now be studied. In particular, an ultracold fermionic quantum gas with a tunable Feshbach resonance offers a unique opportunity to access various sorts of superfluidity in one system, ranging continuously between a Bose-Einstein condensate (BEC) of bosonic molecules, a resonant SF, and a SF gas of Cooper pairs (BCS superfluid) [9] , [10] , [11] . In the experiment, this is done by tuning the interaction parameter \\\\({({k}_{{{{{{{{\\\\rm{F}}}}}}}}}a)}^{-1}\\\\) , where a is the scattering length, \\\\({k}_{{{{{{{{\\\\rm{F}}}}}}}}}=\\\\sqrt{2m{E}_{{{{{{{{\\\\rm{F}}}}}}}}}}/\\\\hslash\\\\) the Fermi wavenumber, E F is the Fermi energy, and m the atomic mass. A large range of thermodynamical properties of the BEC-BCS crossover has been studied, e.g., in refs. [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] . This includes experiments on first sound (see e.g., ref. [16] ) and second sound. Second sound has recently been observed by Sidorenkov et al. [20] in a unitary Fermi gas and by Ville et al. [21] and Christodoulou et al. [22] in a two-dimensional bosonic SF. Second sound was also possibly present in an experiment by Meppelink et al. [23] as pointed out in ref. [3] . Here, we experimentally investigate how second sound changes across the BEC–BCS crossover. This is important, because full theoretical calculations are not yet available for the entire strongly interacting regime. Nevertheless, comparing our measurements to existing calculations and interpolations we find reasonable agreement. In particular, c-field simulations in the BEC regime match quite well the observed wave dynamics in experiments at interaction strengths of up to \\\\({({k}_{{{{{{{{\\\\rm{F}}}}}}}}}a)}^{-1}=1\\\\) . In addition, we investigate experimentally and theoretically the system response when modifying the excitation scheme. As second sound is mainly an entropy wave and first sound is mainly a pressure wave, different excitation schemes give rise to different responses for first and second sound. This helps for separating the generally weak second sound signals from the first sound ones. We find that this separation works especially well when first sound is excited as a density dip wave packet. For this case, we were able to quantitatively compare the amplitudes of first and second sound and compare the results with a prediction.\",\n \"references\": {\n \"indices\": [\n 1,\n 6,\n 9,\n 12,\n 13,\n 14,\n 15,\n 16,\n 17,\n 18,\n 19,\n 20,\n 21,\n 22,\n 23\n ],\n \"infos\": [\n {\n \"idx\": \"1\",\n \"title\": \"Halperin, W. P. Eighty years of superfluidity.Nature553, 413–414 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fd41586-018-00417-7\"\n },\n {\n \"idx\": \"6\",\n \"title\": \"Tisza, L. Transport phenomena in helium II.Nature141, 913 (1938).\",\n \"link\": \"https://doi.org/10.1038%2F141913a0\"\n },\n {\n \"idx\": \"9\",\n \"title\": \"Giorgini, S., Pitaevskii, L. P. & Stringari, S. Theory of ultracold atomic Fermi gases.Rev. Mod. Phys.80, 1215–1275 (2008).\",\n \"link\": \"https://doi.org/10.1103%2FRevModPhys.80.1215\"\n },\n {\n \"idx\": \"12\",\n \"title\": \"Nascimbène, S., Navon, N., Jiang, K. J., Chevy, F. & Salomon, C. Exploring the thermodynamics of a universal Fermi gas.Nature463, 1057–1060 (2010).\",\n \"link\": \"https://doi.org/10.1038%2Fnature08814\"\n },\n {\n \"idx\": \"13\",\n \"title\": \"Ku, M. J. H., Sommer, A. T., Cheuk, L. W. & Zwierlein, M. W. Revealing the superfluid lambda transition in the universal thermodynamics of a unitary Fermi gas.Science335, 563 (2012).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1214987\"\n },\n {\n \"idx\": \"14\",\n \"title\": \"Shin, Y., Schunck, C. H., Schirotzek, A. & Ketterle, W. Phase diagram of a two-component Fermi gas with resonant interactions.Nature451, 689–693 (2008).\",\n \"link\": \"https://doi.org/10.1038%2Fnature06473\"\n },\n {\n \"idx\": \"15\",\n \"title\": \"Sagi, Y., Drake, T. E., Paudel, R., Chapurin, R. & Jin, D. S. Breakdown of the Fermi liquid description for strongly interacting fermions.Phys. Rev. Lett.114, 075301 (2015).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.114.075301\"\n },\n {\n \"idx\": \"16\",\n \"title\": \"Joseph, J. et al. Measurement of Sound Velocity in a Fermi Gas near a Feshbach Resonance.Phys. Rev. Lett.98, 170401 (2007).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.98.170401\"\n },\n {\n \"idx\": \"17\",\n \"title\": \"Taylor, E. et al. First and second sound in a strongly interacting Fermi gas.Phys. Rev. A80, 053601 (2009).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevA.80.053601\"\n },\n {\n \"idx\": \"18\",\n \"title\": \"Perali, A., Pieri., P., Pisani., L. & Calvanese Strinati, G. BCS-BEC crossover at finite temperature for superfluid trapped Fermi atoms.Phys. Rev. Lett.92, 220404 (2004).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.92.220404\"\n },\n {\n \"idx\": \"19\",\n \"title\": \"Navon, N., Nascimbène, S., Chevy, S. & Salomon, C. The equation of state of a low-temperature Fermi gas with tunable interactions.Science328, 729–732 (2010).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1187582\"\n },\n {\n \"idx\": \"20\",\n \"title\": \"Sidorenkov, L. et al. Second sound and the superfluid fraction in a Fermi gas with resonant interactions.Nature498, 78–82 (2013).\",\n \"link\": \"https://doi.org/10.1038%2Fnature12136\"\n },\n {\n \"idx\": \"21\",\n \"title\": \"Ville, J. L. et al. Sound propagation in a uniform superfluid two-dimensional Bose gas.Phys. Rev. Lett.121, 145301 (2018).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.121.145301\"\n },\n {\n \"idx\": \"22\",\n \"title\": \"Christodoulou, P. et al. Observation of first and second sound in a BKT superfluid.Nature594, 191–194 (2021).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-021-03537-9\"\n },\n {\n \"idx\": \"23\",\n \"title\": \"Meppelink, R., Koller, S. B. & van der Straten, P. Sound propagation in a Bose-Einstein condensate at finite temperatures.Phys. Rev. A80, 043605 (2009).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevA.80.043605\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Experimental details Our experiments are carried out with a balanced, two-component ultracold gas of fermionic 6 Li atoms in the two lowest hyperfine states \\\\(\\\\left|F,{m}_{F}\\\\right\\\\rangle =\\\\left|1/2,\\\\pm \\\\!1/2\\\\right\\\\rangle\\\\) of the electronic ground state. The gas is confined by a combined magnetic and optical dipole trap with a trap depth of U 0 ≈ 1 μK × k B , for details see ref. [24] , [25] . The trap is nearly harmonic and cylindrically symmetric with trapping frequencies ω r = 2 π × 305 Hz and ω x = 2 π × 21 Hz. The temperature and the particle density are controlled by evaporative cooling. In the experiments the temperature ranges approximately from 0.13 T F to 0.30 T F , where \\\\({T}_{{{{{{{{\\\\rm{F}}}}}}}}}={E}_{{{{{{{{\\\\rm{F}}}}}}}}}/{k}_{{{{{{{{\\\\rm{B}}}}}}}}}=\\\\hslash \\\\bar{\\\\omega }{(3N)}^{1/3}/{k}_{{{{{{{{\\\\rm{B}}}}}}}}}\\\\) is the Fermi temperature, \\\\(\\\\bar{\\\\omega }={\\\\left({\\\\omega }_{x}{\\\\omega }_{r}^{2}\\\\right)}^{1/3}\\\\) is the geometric mean of the trapping frequencies and N is the total number of atoms. The scattering length a is tunable with an external magnetic field B via a magnetic Feshbach resonance at 832 G [26] . To excite sound modes in the system, we focus a blue-detuned 532 nm laser onto the trap center following the approaches introduced in refs. [16] , [20] , [27] (Fig. 1 a). The laser beam is aligned perpendicularly to the optical dipole trap and produces a repulsive potential barrier of U ex ≈ 0.2 U 0 . At its focus, the beam has a waist of about 20 μm, which is comparable to the cloud size in the radial direction. To excite sound waves, the height of this additional potential is modulated. The excited sound modes generally exhibit contributions from both first and second sound [28] , [29] , [30] . However, it is possible to generate preferentially either one of the two sound modes by adapting the excitation method. Fig. 1: Sound excitation in a trapped superfluid Fermi gas in the vicinity of the BEC-BCS crossover. a Set-up: A focussed, intensity-modulated, blue-detuned laser beam excites sound waves in the cigar-shaped atom cloud. b Two different modulation sequences of the laser intensity. Purple dashed line: step excitation. Green solid line: heat pulse. The time t is given in units of the axial trapping period 2 π / ω x . c Sketch of a bimodal density distribution of a trapped BEC (purple line) at y = z = 0. At the center of the trap a blue detuned beam produces a dimple in the potential. Modulating the beam intensity produces first sound waves (red arrows) and second sound (orange arrows) waves. Second sound reduces the local density of the cloud, while for first sound a density peak emerges. The thin black line shows the profile of the unperturbed cloud. Please note that the crests and troughs of the waves are shown in an exaggerated way for better visibility. d The false color plot shows the measured local change in the density \\\\({{\\\\Delta }}\\\\bar{n}(x,t)\\\\) as a function of axial position x and time t . Here, \\\\({({k}_{{{{{{{{\\\\rm{F}}}}}}}}}a)}^{-1}=(1.61\\\\pm 0.05)\\\\) at B = 735 G and T / T c = (0.80 ± 0.15). After excitation, two wave packets (bright traces, marked with red arrows) propagate with first sound velocity u 1 towards the edges of the cloud. The excitation method predominantly excites first sound. Second sound is present as well but is barely discernible here. e Propagation of first sound waves (bright traces, marked with red arrow) and second sound waves (dark traces, marked with orange arrows) after excitation with sinusoidal pulse of ( b ). All other settings are the same as in ( d ). f Simulated sound propagation for the same parameters as in ( e ). The orange arrows mark the propagating second sound and the red arrows the first sound, respectively. Full size image To excite primarily first sound, we abruptly switch on the excitation laser beam (Fig. 1 b), similarly as for the first experiments on sound propagation in a dilute BEC [27] . This applies pressure on the cold cloud on both sides of the laser beam and creates two density wave packets (Fig. 1 c) which propagate out in opposite directions along the axial trap axis with the speed u 1 . In the experiments, we detect these waves with the help of absorption imaging by measuring the density distribution of the atomic cloud as a function of time. Figure 1 d shows such density waves for an experiment at \\\\({({k}_{{{{{{{{\\\\rm{F}}}}}}}}}a)}^{-1}\\\\approx (1.61\\\\pm 0.05)\\\\) , B = 735 G and a temperature of T = (220 ± 30) nK = (0.30 ± 0.06) T F , which corresponds to T = (0.80 ± 0.15) T c , where T c is the critical temperature. For the given interaction strength, we used T c = 0.37 T F (see Supplementary Note 1 ). Figure 1 d is a time-ordered stack of one-dimensional line density profiles of the atom cloud [31] . Each profile was obtained by integrating the measured column density of the atomic cloud along the transverse (i.e., y -) direction (see “Methods” for details). The time-ordered stack shows the propagation of the sound waves along the axial direction x as a function of time. We observe two density wave packets which propagate with first sound velocity from the trap center towards the edge of the cloud (two bright traces, marked with red arrows). The propagating waves produce a density modulation of only a few percent of the peak density and can be considered as a weak perturbation of the system. To obtain the speed of sound, we examine how the center position of each wave packet changes with time. The center positions are determined via a Gaussian fit. Please note that the central sound speed is determined over a local area of the cloud where the density varies by about 30%. As a consequence, the measured sound speed should be considered a mean value. From Fig. 1 d we obtain u 1 = (17.2 ± 3) mm/s near the trap center. Our analysis shows that the sound propagation slows down as the pulse approaches the edge of the cloud where the particle density decreases. In the following, we focus on the sound speed close to the trap center. To primarily excite second sound, we sinusoidally modulate the intensity of the excitation beam for 7 ms with a modulation frequency of ω ex = 2 π × 570 Hz ≈ 2 ω r and a modulation amplitude of Δ U ≈ 0.2 U 0 . This parametrically heats the gas in radial direction (Fig. 1 b). Subsequent thermalization via collisions occurs within a few milliseconds. As a consequence a local depletion of the SF density is created, which is filled with normal gas, forming a region of increased entropy (Fig. 1 c). This gives rise to two wave packets which propagate outwards along the axial direction with the speed of second sound. Figure 1 e shows corresponding experimental data where we measure the local density distribution as in Fig. 1 d. The second sound wave appears here as a density dip (dark traces, marked with orange arrows). A clear indication that the dark trace corresponds to second sound is the fact that it vanishes at the Thomas-Fermi radius R TF ≈ 110 μm where the SF fraction vanishes. Second sound only propagates inside the SF phase. Besides a second sound wave, the excitation also produces a first sound wave (bright traces, marked with red arrows) which propagates faster than the second sound wave and travels beyond the Thomas-Fermi radius. The first sound wave is broader than in Fig. 1 d, which can be mainly explained by the longer excitation pulse. To obtain u 2 we measure the time-dependent position of the minimum of each dark trace, which is again determined via a Gaussian fit. For Fig. 1 e we obtain u 2 = (5.1 ± 1.1) mm/s. We note that for the small excitation amplitudes in our experiments (see colorbar in Fig. 1 d–e), we do not observe the asymmetric sound wave distortions reported in ref. [23] . These distortions were also absent in ref. [20] . Figure 1 f shows numerical simulations of our experiment applying a dynamical c-field method [32] (see Supplementary Note 2 for detailed information on the method). The dimer scattering length [33] is a dd = 0.6 a and we assume all fermionic atoms to be paired up in molecules. To compare the simulations with the experimental results we choose the same values of \\\\({({k}_{{{{{{{{\\\\rm{F}}}}}}}}}a)}^{-1}\\\\) and the same central density of the trapped gas as in the experiment. The theory value for u 2 is (5.7 ± 0.05) mm/s in agreement with the experimental value (5.1 ± 1.1) mm/s. Interaction strength dependence of second sound We now perform measurements of second sound speed in the region \\\\((-0.22\\\\pm 0.04) \\\\, < \\\\,{({k}_{{{{{{{{\\\\rm{F}}}}}}}}}a)}^{-1} \\\\, < \\\\,(1.61\\\\pm 0.05)\\\\) of the BCS-BEC crossover. These are shown in Fig. 2 along with theoretical predictions. The second sound velocity u 2 is given in units of the Fermi velocity \\\\({v}_{{{{{{{{\\\\rm{F}}}}}}}}}=\\\\hslash {k}_{{{{{{{{\\\\rm{F}}}}}}}}}^{{{\\\\rm{hom}}}}/m\\\\) . Here, the Fermi wavenumber \\\\({k}_{{{{{{{{\\\\rm{F}}}}}}}}}^{{{\\\\rm{hom}}}}= {\\\\left(3{\\\\pi }^{2}{n}_{0}\\\\right)}^{1/3}\\\\) is determined from the 3D peak density close to the trap center. The peak density is deduced using the inverse Abel transformation [34] and a self-consistent mean field calculation (see Supplementary Note 4 ). We have verified that in the BEC regime our mean field calculation gives similar results for the peak density when we input the trapping frequencies, the temperature, the scattering length and the total number of particles. The blue dash-dotted line is a calculation from ref. [28] , based on a hydrodynamic description in a homogeneous gas for the limiting cases of the BEC and the BCS regime and unitarity. To connect these regimes, the results are interpolated across the crossover, bridging the range \\\\(|{({k}_{{{{{{{{\\\\rm{F}}}}}}}}}a)}^{-1}| \\\\, < \\\\, 1\\\\) . The blue solid and the brown solid lines are our analytic hydrodynamic calculations which are valid in the BCS and BEC limit, respectively (see Supplementary Note 3 ). For comparison, we show the results of the numerical c-field simulations (green squares), which agree with both, analytic description and experimental results. Despite the large error bars the measurements indicate an increase of u 2 when approaching unitarity from the BEC side, in agreement with the theoretical results. Fig. 2: Second sound velocity u 2 as a function of interaction strength. The purple circles depict measured data for temperatures in the range T = 105−230 nK which corresponds to T / T c = 0.66−0.84 (see Supplementary Note 1 ). The error bars are due to statistical uncertainties. The brown and blue solid line show hydrodynamic predictions for the BEC and BCS regime at T = 0.75 T c , respectively (see Supplementary Note 3 ). The shaded areas mark the second sound velocity in the temperature range of the experiments. The blue dash-dotted line shows a theoretical prediction of second sound in the crossover [28] for a homogeneous gas at T / T c = 0.75. It interpolates between the results from hydrodynamic theory in the BEC and BCS regime. The green squares are results of our numerical c-field simulations which are consistent with both, analytic and experimental results. For comparison we also show the second sound velocity on the resonance measured in ref. [20] at the temperatures T / T c = 0.65 (blue triangle), T / T c = 0.75 (brown triangle), and T / T c = 0.85 (red triangle). Full size image In general, second sound can only propagate in the SF phase of the gas. It is therefore natural to ask how the SF density n s and the speed of second sound u 2 are related. Using our measurements for u 2 we can roughly determine the relationship between n s and u 2 for our temperature T and \\\\({({k}_{{{{{{{{\\\\rm{F}}}}}}}}}a)}^{-1} \\\\, > \\\\, 1.5\\\\) , since in this regime the SF density can be estimated. For this, we carry out self-consistent mean field calculations to determine the density distributions of the SF and the NF for an interacting BEC in the trap (see Supplementary Note 4 ). As an important input into these calculations we make use of the Thomas-Fermi radius which we have measured in the second sound experiments (the measured Thomas-Fermi radii can be found in Supplementary Note 1 ). As an example, from the measurement at \\\\({({k}_{{{{{{{{\\\\rm{F}}}}}}}}}a)}^{-1}=(1.61\\\\pm 0.05)\\\\) we determine the peak SF fraction to be n s0 / n 0 = 0.98 close to the trap center at maximum density (see Supplementary Note 4 ), where the local \\\\({({k}_{{{{{{{{\\\\rm{F}}}}}}}}}^{{{\\\\rm{hom}}}}a)}^{-1}=(1.06\\\\pm 0.05)\\\\) and \\\\(T/{T}_{{{{{{{{\\\\rm{c}}}}}}}}}^{{{\\\\rm{hom}}}}=(0.40\\\\pm 0.15)\\\\) , with \\\\({T}_{{{{{{{{\\\\rm{c}}}}}}}}}^{{{\\\\rm{hom}}}}= 0.21{T}_{{{{{{{{\\\\rm{F}}}}}}}}}^{{{\\\\rm{hom}}}}\\\\) and \\\\({T}_{{{{{{{{\\\\rm{F}}}}}}}}}^{{{\\\\rm{hom}}}}={\\\\hslash }^{2}{({k}_{{{{{{{{\\\\rm{F}}}}}}}}}^{{{\\\\rm{hom}}}})}^{2}/2m{k}_{{{{{{{{\\\\rm{B}}}}}}}}}\\\\) . For comparison, for a homogeneous weakly interacting BEC with a SF fraction close to unity the temperature would need to be \\\\(T\\\\ll {T}_{{{{{{{{\\\\rm{c}}}}}}}}}^{{{\\\\rm{hom}}}}\\\\) , according to \\\\({n}_{{{{{{{{\\\\rm{s}}}}}}}}}/n=1-{(T/{T}_{{{{{{{{\\\\rm{c}}}}}}}}}^{{{\\\\rm{hom}}}})}^{3/2}\\\\) . At unitarity, by contrast, the SF fraction reaches unity already at \\\\(T/{T}_{{{{{{{{\\\\rm{c}}}}}}}}}^{{{\\\\rm{hom}}}}\\\\approx 0.55\\\\) , as shown by Sidorenkov et al. [20] . As expected, this comparison shows that for a given \\\\(T/{T}_{{{{{{{{\\\\rm{c}}}}}}}}}^{{{\\\\rm{hom}}}}\\\\) the SF fraction grows with interaction strength. Tuning the sound mode excitation In the following we investigate how the SF gas responds to different excitation protocols [28] , [29] , [30] . For this, we vary the excitation scheme, the excitation frequency and amplitude. By observing the corresponding response of the system we gain additional insights into the nature of first and second sound. In Fig. 3 a we show the evolution of the system after a step pulse excitation at B = 735 G and Δ U = 0.3 U 0 , in which both, first and second sound are excited. In contrast to the experiment in Fig. 1 d, the laser beam is abruptly switched off - not on. As a consequence, the wave packets of both first and second sound now correspond to dips in the particle density. In Fig. 3 b we show the density distribution for the time and position range indicated by the purple rectangle in Fig. 3 a. From a fit of two Gaussian dips to the two wave packets, we determine an amplitude ratio of W 2 / W 1 ≈ 1.1. This result approximately matches the predictions of refs. [29] , [30] (see also Supplementary Note 3 ), where the response of both, a weakly and a strongly interacting molecular Bose gas has been studied. For an interaction parameter of \\\\({({k}_{{{{{{{{\\\\rm{F}}}}}}}}}a)}^{-1}=2\\\\) , the prediction yields W 2 / W 1 = 0.9, which is of similar magnitude as our result. Fig. 3: Comparing signal strength of first and second sound. a Sound excitation experiment at \\\\({({k}_{{{{{{{{\\\\rm{F}}}}}}}}}a)}^{-1}=(1.61\\\\pm 0.05)\\\\) and at a temperature of T / T c = (0.80 ± 0.15). In contrast to Fig. 1 d, first sound (red arrows) and second sound (orange arrows) are now visible simultaneously. For t ω x /2 π < 0.15 first and second sound waves overlap and therefore cannot be distinguished from each other. b shows \\\\({{\\\\Delta }}\\\\bar{n}\\\\) for tω x /2 π = 0.43. We fit the center position of each of the two sound waves using a Gaussian function (solid line). Full size image Next, we study the system response when modifying the excitation scheme. For this, we vary the excitation frequency and the number of modulation cycles. Figure 4 a–d shows the system response for a modulation frequency of ω ex = 0.61 ω r , so that parametric heating is rather suppressed and coupling to first sound is enhanced as compared to the measurement shown in Fig. 1 e. While Fig. 4 a, b corresponds to a 1.5 cycle modulation, Fig. 4 c, d corresponds to a 1 cycle modulation. Fig. 4: Sound excitation with different modulation sequences. a , c \\\\({{\\\\Delta }}\\\\bar{n}(x,t)\\\\) data at ω ex = 0.61 ω r , Δ U = 0.3 U 0 and \\\\({\\\\left({k}_{{{{{{{{\\\\rm{F}}}}}}}}}a\\\\right)}^{-1}=(1.61\\\\pm 0.05)\\\\) for a modulation of 1.5 cycles and of 1 cycle, respectively. The excitation pulse excites both, first and second sound waves. b , d \\\\({{\\\\Delta }}\\\\bar{n}(x,t)\\\\) from numerical c-field simulations. Top row: False color images of \\\\({{\\\\Delta }}\\\\bar{n}(x,t)\\\\) . First and second sound waves are marked with red and orange arrows, respectively. The inset in ( d ) is an enlargement, showing how the calculated bright first sound wave (black arrow) is damped when it crosses the second sound wave (orange arrow). Mid row: Shown is \\\\({{\\\\Delta }}\\\\bar{n}\\\\) for tω x /2 π = 0 ( a , b ) and for tω x /2 π = 0.15 ( c , d ). Bottom row: Applied excitation scheme. Full size image The numerical simulations in Fig. 4 b, d demonstrate how the excitation pattern produces a corresponding wave train of first sound. Once the simulated waves of first sound have propagated beyond the Thomas-Fermi radius, they diffuse out and lose signal strength. In the simulation, the diffusion of the first sound wave train is stronger than in the experiment. This might be explained by the choice of the discretization length which was used in the simulations (see Supplementary Note 2B ). The first sound wave train is always followed by a single dark second sound wave packet. The experimental data in Fig. 4 a agree reasonably well with the simulation in Fig. 4 b. Comparing Fig. 4 c and d, there seems to be a discrepancy. In the experiment clearly a first sound wave train, consisting of two bright traces and one dark trace, propagates beyond the Thomas-Fermi radius. In the simulation, however, the second bright trace of the wave train, which is clearly visible in the center, disappears before it reaches the Thomas-Fermi radius (black arrow). The corresponding damping happens when the bright sound trace crosses the dark second sound trace (see also insert in Fig. 4 d). The reason for the damping is that first sound waves diffuse in a thermal environment. This diffusion, however, is apparently too strong in the simulation results, which is probably a consequence of constraints in the numerical resolution (for more details see Supplementary Note 2B ). In conclusion, we have studied second sound propagation in an ultracold Fermi gas of 6 Li atoms throughout the BEC-BCS crossover for a finite temperature of T ≈ 0.75 T c . We find the second sound velocity to vary only slightly across the BCS-BEC crossover, which is in agreement with an interpolation of hydrodynamic theory [28] . In the BEC regime, the results match numerical predictions based on c-field simulations. In addition, we investigate the response of the SF gas on various excitation pulse shapes, ranging from gentle local heating to an abrupt kick. While a sinusoidal excitation pulse leads to a corresponding wave train for the first sound, it only produces a single pulse for the second sound. In the future, it will be useful to extend our measurements in the strongly interacting regime to a larger range of temperatures below T c . Since the second sound velocity is related to the local SF density, these measurements can help to determine the correlation between the SF density and the temperature in the strongly interacting regime.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Calculating \\\\({{\\\\Delta }}\\\\bar{n}\\\\) from the density profiles Each of the experimental sound propagation images in Figs. 1d–e , 3a , 4a and c is a time-ordered stack of one-dimensional line density profiles \\\\({{\\\\Delta }}\\\\bar{n}(x,t)\\\\) of the atom cloud. A one-dimensional line density profile n ( x , t ) is produced as follows: After sound excitation has ended and after an additional propagation time t , we take an absorption image of the atom cloud to obtain the column density distribution n ex ( x , y , t ). Next, we integrate the absorption image along the y- axis (which is perpendicular to the symmetry axis of the cigar-shaped atom cloud) to obtain a one-dimensional line density profile n ex ( x , t ). To reduce noise, we average 15 density profiles and obtain \\\\({\\\\bar{n}}_{{{{{{{{\\\\rm{ex}}}}}}}}}(x,t)\\\\) . We repeat this procedure for an unperturbed cloud to obtain \\\\(\\\\bar{n}(x)\\\\) . By subtracting the two density profiles from each other we obtain \\\\({{\\\\Delta }}\\\\bar{n}(x,t)=({\\\\bar{n}}_{{{{{{{{\\\\rm{ex}}}}}}}}}(x,t)-\\\\bar{n}(x))/\\\\bar{n}(0)\\\\) .\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"The core research idea is to experimentally investigate how second sound, an entropy wave distinct from first sound, changes across the BEC–BCS crossover in a tunable ultracold fermionic quantum gas, and to compare these measurements with existing theoretical models and simulations in order to better understand the behavior of second sound in strongly interacting superfluid systems."},"entities":{"kind":"list like","value":["second sound","entropy wave","first sound","BEC–BCS crossover","ultracold fermionic quantum gas","theoretical models","simulations","superfluid systems","experimental investigation","tunable quantum gas"],"string":"[\n \"second sound\",\n \"entropy wave\",\n \"first sound\",\n \"BEC–BCS crossover\",\n \"ultracold fermionic quantum gas\",\n \"theoretical models\",\n \"simulations\",\n \"superfluid systems\",\n \"experimental investigation\",\n \"tunable quantum gas\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1.","title":"Halperin, W. 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Sound propagation in a Bose-Einstein condensate at finite temperatures.Phys. Rev. A80, 043605 (2009).","link":"https://doi.org/10.1103%2FPhysRevA.80.043605"},{"idx":"24.","title":"Paintner, T. et al. Pair fraction in a finite-temperature Fermi gas on the BEC side of the BCS-BEC crossover.Phys. Rev. A99, 053617 (2019).","link":"https://doi.org/10.1103%2FPhysRevA.99.053617"},{"idx":"25.","title":"Hoffmann, D. K., Paintner, T., Limmer, W., Petrov, D. S. & Hecker Denschlag, J. Reaction kinetics of ultracold molecule-molecule collisions.Nat. Commun.9, 5244 (2018).","link":"https://doi.org/10.1038%2Fs41467-018-07576-1"},{"idx":"26.","title":"Zürn, G. et al. Precise characterization of6Li Feshbach resonances using trap-sideband-resolved RF spectroscopy of weakly bound molecules.Phys. Rev. Lett.110, 135301 (2013).","link":"https://doi.org/10.1103%2FPhysRevLett.110.135301"},{"idx":"27.","title":"Andrews, M. R. et al. Propagation of sound in a Bose-Einstein condensate.Phys. Rev. Lett.79, 553 (1997).","link":"https://doi.org/10.1103%2FPhysRevLett.79.553"},{"idx":"28.","title":"Heiselberg, H. Sound modes at the BCS-BEC crossover.Phys. Rev. A73, 013607 (2006).","link":"https://doi.org/10.1103%2FPhysRevA.73.013607"},{"idx":"29.","title":"Arahata, E. & Nikuni, T. Propagation of second sound in a superfluid Fermi gas in the unitary limit.Phys. Rev. A80, 043613 (2009).","link":"https://doi.org/10.1103%2FPhysRevA.80.043613"},{"idx":"30.","title":"Hu, H., Taylor, E., Liu, X.-J., Stringari, S. & Griffin, A. Second sound and the density response function in uniform superfluid atomic gases.N. J. Phys.12, 043040 (2010).","link":"https://doi.org/10.1088%2F1367-2630%2F12%2F4%2F043040"},{"idx":"32.","title":"Singh, V. P. et al. Probing superfluidity of Bose-Einstein condensates via laser stirring.Phys. Rev. A93, 023634 (2016).","link":"https://doi.org/10.1103%2FPhysRevA.93.023634"},{"idx":"33.","title":"Petrov, D. S., Salomon, C. & Shlyapnikov, G. V. Weakly bound dimers of fermionic atoms.Phys. Rev. Lett.93, 090404 (2004).","link":"https://doi.org/10.1103%2FPhysRevLett.93.090404"},{"idx":"34.","title":"Partridge, G. B. et al. Deformation of a trapped Fermi gas with unequal spin populations.Phys. Rev. Lett.97, 190407 (2006).","link":"https://doi.org/10.1103%2FPhysRevLett.97.190407"}],"string":"[\n {\n \"idx\": \"1.\",\n \"title\": \"Halperin, W. P. Eighty years of superfluidity.Nature553, 413–414 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fd41586-018-00417-7\"\n },\n {\n \"idx\": \"6.\",\n \"title\": \"Tisza, L. Transport phenomena in helium II.Nature141, 913 (1938).\",\n \"link\": \"https://doi.org/10.1038%2F141913a0\"\n },\n {\n \"idx\": \"9.\",\n \"title\": \"Giorgini, S., Pitaevskii, L. P. & Stringari, S. Theory of ultracold atomic Fermi gases.Rev. Mod. 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Breakdown of the Fermi liquid description for strongly interacting fermions.Phys. Rev. Lett.114, 075301 (2015).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.114.075301\"\n },\n {\n \"idx\": \"16.\",\n \"title\": \"Joseph, J. et al. Measurement of Sound Velocity in a Fermi Gas near a Feshbach Resonance.Phys. Rev. Lett.98, 170401 (2007).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.98.170401\"\n },\n {\n \"idx\": \"17.\",\n \"title\": \"Taylor, E. et al. First and second sound in a strongly interacting Fermi gas.Phys. Rev. A80, 053601 (2009).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevA.80.053601\"\n },\n {\n \"idx\": \"18.\",\n \"title\": \"Perali, A., Pieri., P., Pisani., L. & Calvanese Strinati, G. BCS-BEC crossover at finite temperature for superfluid trapped Fermi atoms.Phys. Rev. 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Observation of first and second sound in a BKT superfluid.Nature594, 191–194 (2021).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-021-03537-9\"\n },\n {\n \"idx\": \"23.\",\n \"title\": \"Meppelink, R., Koller, S. B. & van der Straten, P. Sound propagation in a Bose-Einstein condensate at finite temperatures.Phys. Rev. A80, 043605 (2009).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevA.80.043605\"\n },\n {\n \"idx\": \"24.\",\n \"title\": \"Paintner, T. et al. Pair fraction in a finite-temperature Fermi gas on the BEC side of the BCS-BEC crossover.Phys. Rev. A99, 053617 (2019).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevA.99.053617\"\n },\n {\n \"idx\": \"25.\",\n \"title\": \"Hoffmann, D. K., Paintner, T., Limmer, W., Petrov, D. S. & Hecker Denschlag, J. Reaction kinetics of ultracold molecule-molecule collisions.Nat. Commun.9, 5244 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-018-07576-1\"\n },\n {\n \"idx\": \"26.\",\n \"title\": \"Zürn, G. et al. Precise characterization of6Li Feshbach resonances using trap-sideband-resolved RF spectroscopy of weakly bound molecules.Phys. Rev. Lett.110, 135301 (2013).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.110.135301\"\n },\n {\n \"idx\": \"27.\",\n \"title\": \"Andrews, M. R. et al. Propagation of sound in a Bose-Einstein condensate.Phys. Rev. Lett.79, 553 (1997).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.79.553\"\n },\n {\n \"idx\": \"28.\",\n \"title\": \"Heiselberg, H. Sound modes at the BCS-BEC crossover.Phys. Rev. A73, 013607 (2006).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevA.73.013607\"\n },\n {\n \"idx\": \"29.\",\n \"title\": \"Arahata, E. & Nikuni, T. Propagation of second sound in a superfluid Fermi gas in the unitary limit.Phys. Rev. A80, 043613 (2009).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevA.80.043613\"\n },\n {\n \"idx\": \"30.\",\n \"title\": \"Hu, H., Taylor, E., Liu, X.-J., Stringari, S. & Griffin, A. Second sound and the density response function in uniform superfluid atomic gases.N. J. Phys.12, 043040 (2010).\",\n \"link\": \"https://doi.org/10.1088%2F1367-2630%2F12%2F4%2F043040\"\n },\n {\n \"idx\": \"32.\",\n \"title\": \"Singh, V. P. et al. Probing superfluidity of Bose-Einstein condensates via laser stirring.Phys. Rev. A93, 023634 (2016).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevA.93.023634\"\n },\n {\n \"idx\": \"33.\",\n \"title\": \"Petrov, D. S., Salomon, C. & Shlyapnikov, G. V. Weakly bound dimers of fermionic atoms.Phys. Rev. Lett.93, 090404 (2004).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.93.090404\"\n },\n {\n \"idx\": \"34.\",\n \"title\": \"Partridge, G. B. et al. Deformation of a trapped Fermi gas with unequal spin populations.Phys. Rev. Lett.97, 190407 (2006).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.97.190407\"\n }\n]","isConversational":false}}},{"rowIdx":190,"cells":{"unique_id":{"kind":"string","value":"s41467-020-16364-9"},"subfield":{"kind":"string","value":"Nanoscience and technology"},"title":{"kind":"string","value":"Ultrathin high-κantimony oxide single crystals"},"abstract":{"kind":"string","value":"Ultrathin oxides have been reported to possess excellent properties in electronic, magnetic, optical, and catalytic fields. However, the current and primary approaches toward the preparation of ultrathin oxides are only applicable to amorphous or polycrystalline oxide nanosheets or films. Here, we successfully synthesize high-quality ultrathin antimony oxide single crystals via a substrate-buffer-controlled chemical vapor deposition strategy. The as-obtained ultrathin antimony oxide single crystals exhibit high dielectric constant (~100) and large breakdown voltage (~5.7 GV m −1 ). Such a strategy can also be utilized to fabricate other ultrathin oxides, opening up an avenue in broadening the applicaitons of ultrathin oxides in many emerging fields."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"Ultrathin oxides have been reported to possess excellent properties in electronic, magnetic, optical, and catalytic fields [1] , [2] , [3] , [4] , [5] , [6] . However, the ways to simply synthesize ultrathin oxides through a general and facile method is a big challenge. The current and primary approaches toward the preparation of ultrathin oxides arise from wet-chemistry synthesis, where a variety of ultrathin metal oxides can be attained [7] , [8] , [9] . In addition, the liquid metal-based reaction route, which requires metals that can be co-alloyed into the melt, has been utilized to obtain low-dimensional metal oxides [1] . However, these methods are only applicable to amorphous or polycrystalline oxide nanosheets or films with degraded properties [1] , [7] , [8] , [9] . Recently, Li et al. presented an approach for the preparation of ultrathin high- κ oxide, which showed the potential for two-dimensional electronic devices [10] . However, it is prerequisite to grow a molecular crystal as a seeding layer for the uniform deposition of the ultrathin oxide. Thus far, the research for high-quality and reproducible ultrathin high- κ oxide single crystals has not been achieved yet, albeit this is highly desirable. In this work, we realize the growth of ultrathin antimony oxide single crystals (down to 1.8 nm) based on a specially substrate-buffer-controlled chemical vapor deposition (SBC–CVD) strategy. After the growth, synchrotron-based X-ray diffraction (XRD) and high-resolution transmission electron microscopy (HRTEM) are conducted to elucidate the structure of such ultrathin antimony oxide single crystals. The SbO 1.93 single crystals exhibit excellent insulating capabilities with high dielectric constant (~100) and large breakdown electric field (~5.7 GV m −1 ). Such ultrathin antimony oxide single crystals will facilitate the ongoing research of the applications of ultrathin oxides.","references":{"indices":[1,2,3,4,5,6,7,8,9,10],"infos":[{"idx":"1","title":"Zavabeti, A. et al. A liquid metal reaction environment for the room-temperature synthesis of atomically thin metal oxides.Science358, 332–335 (2017).","link":"https://doi.org/10.1126%2Fscience.aao4249"},{"idx":"2","title":"Fiori, G. et al. Electronics based on two-dimensional materials.Nat. Nanotechnol.9, 768–779 (2014).","link":"https://doi.org/10.1038%2Fnnano.2014.207"},{"idx":"3","title":"Li, L., Richter, C., Mannhart, J. & Ashoori, R. C. Coexistence of magnetic order and two-dimensional superconductivity at LaAlO3/SrTiO3interfaces.Nat. Phys.7, 762–766 (2011).","link":"https://doi.org/10.1038%2Fnphys2080"},{"idx":"4","title":"Bert, J. A. et al. Direct imaging of the coexistence of ferromagnetism and superconductivity at the LaAlO3/SrTiO3interface.Nat. Phys.7, 767–771 (2011).","link":"https://doi.org/10.1038%2Fnphys2079"},{"idx":"5","title":"Zhao, C., Zhang, H. T., Si, W. J. & Wu, H. Mass production of two-dimensional oxides by rapid heating of hydrous chlorides.Nat. Commun.7, 12543 (2016).","link":"https://doi.org/10.1038%2Fncomms12543"},{"idx":"6","title":"Hu, Z. M. et al. Rapid mass production of two-dimensional metal oxides and hydroxides via the molten salts method.Nat. Commun.8, 15630 (2017).","link":"https://doi.org/10.1038%2Fncomms15630"},{"idx":"7","title":"Sun, Y. et al. Pits confined in ultrathin cerium(IV) oxide for studying catalytic centers in carbon monoxide oxidation.Nat. Commun.4, 2899 (2013).","link":"https://doi.org/10.1038%2Fncomms3899"},{"idx":"8","title":"Sun, Z. et al. Generalized self-assembly of scalable two-dimensional transition metal oxide nanosheets.Nat. Commun.5, 3813 (2014).","link":"https://doi.org/10.1038%2Fncomms4813"},{"idx":"9","title":"Zhao, H. W. et al. A generalized strategy for the synthesis of large-size ultrathin two-dimensional metal oxide nanosheets.Angew. Chem. Int. Ed.56, 8766–8770 (2017).","link":"https://doi.org/10.1002%2Fanie.201703871"},{"idx":"10","title":"Li, W. S. et al. Uniform and ultrathin high-κgate dielectrics for two-dimensional electronic devices.Nat. Electron.2, 563–571 (2019).","link":"https://doi.org/10.1038%2Fs41928-019-0334-y"}]}},{"section":"Results","content":"The growth and characterizations of ultrathin antimony oxide single crystals The strategy for the growth of ultrathin antimony oxide single crystals is schematically illustrated in Fig. 1a . Before the growth, a re-solidified Ag(111) substrate is prepared by a designed high-temperature (1000 °C) annealing process (details in Supplementary Figs. 1 – 6 ). Then, to achieve the growth of ultrathin antimony oxide single crystals, commercial antimony powder is placed upstream to provide antimony vapor, and the Ag substrates are placed in the downstream area with temperature T = 750 °C. The growth starts while O 2 is introduced into the system, and the growth process is maintained for a few minutes. Typical optical microscope (OM) image of the ultrathin antimony oxide is shown in Fig. 1b , which reveals that these triangular crystals exhibit a uniform size of about 5 μm, in agreement with the scanning electron microscope (SEM) data (Supplementary Fig. 7a ). Raman spectra of the as-grown triangular crystals are shown in Fig. 1c , revealing that antimony oxide with varying thickness corresponds to the homologous peaks. The only difference is that the Raman spectra intensity shows a linear relation with the thickness. As shown in Supplementary Fig. 7b , Raman intensity maps using the A 1g mode demonstrate a uniform response over the entire crystal, which proves the uniformity of the ultrathin antimony oxide crystal. Then, X-ray photoelectron spectroscopy (XPS) is implemented to further characterize the chemical composition of the as-prepared ultrathin antimony oxide crystals (Fig. 1d ). The peaks at 530.3 and 539.6 eV are attributed to the Sb 3d 5/2 and 3d 3/2 states, and the peak at 531.3 eV is corresponding to the O 1 s state, respectively [11] . Due to that the reaction Gibbs free energy of Sb with O 2 (−110.97 kJ mol −1 ) is much lower than that of Ag (−7.84 kJ mol −1 ) [12] , Sb will react with oxygen first. But the amount of O 2 must be limited to avoid the oxidation of the Ag substrate. To achieve the complete oxidation of Sb to form ultrathin antimony oxide single crystals, the amount of elementary Sb should also be controlled. Here, the Ag substrate serves as a buffer to achieve a modest supply of Sb. Such SBC–CVD mechanism is confirmed by the XPS depth profile. From the valence analysis of Sb along a surface normal-direction (Supplementary Fig. 8 ), it can be seen that, with the removal of the surface layer, the Sb 0 appears and keeps at a constant amount within the whole Ag substrate, which demonstrates that the excess Sb is buffered within the substrate, and it is very critical for the complete oxidation of Sb and the subsequent growth of ultrathin antimony oxide crystals. Fig. 1: Growth and spectral characterizations of ultrathin antimony oxide crystals. a Schematic of the strategy for the growth of ultrathin antimony oxide. b OM image of ultrathin antimony oxide crystals grown on the re-solidified Ag substrate. c Raman spectra of ultrathin antimony oxide crystals with different thicknesses. d XPS spectra of the ultrathin antimony oxide. Full size image Structure elucidation of ultrathin antimony oxide To further reveal the structure of ultrathin antimony oxide, synchrotron-based grazing-incidence XRD (GI-XRD) is conducted. As shown in Fig. 2a , the diffraction patterns are recorded on the image plate for the ultrathin antimony oxide. Then, owing to the similarity between the GI-XRD pattern of ultrathin antimony oxide and the simulated XRD pattern of β-antimony (Supplementary Fig. 9 ), the structure of β-antimony is used as the initial model for the structure creation of ultrathin antimony oxide. The rough crystal cell parameters can be obtained, and the preliminary cell structure can be parsed out. Then, the preliminary cell structure is further refined by Generalized Structure and Analysis Software (GSAS) package [13] (Supplementary Note 1). Figure 2b shows the experimental and the fitted data, where there is only a slight difference between them. From the diffraction result, the structure of the ultrathin antimony oxide can be elucidated, which is shown in Fig. 2c . The atomic ratio of Sb to O is 1:1.93, then the chemical formula can be defined as SbO 1.93 . And the top view and side view are demonstrated in Fig. 2d and e , respectively. Specifically, the crystal data and structure refinement for SbO 1.93 are listed in Supplementary Table 1 . Atomic coordinates, occupancy factors ( f occ ) and isotropic displacement parameters ( U iso ) are listed in Supplementary Table 2 . Bond lengths and bond angles are listed in Supplementary Table 3 . The unit-cell parameters and lattice plane parameters are listed in Supplementary Table 4 . To verify the rationality of the final structure, we compared the as-obtained XRD pattern of the SbO 1.93 sample with the simulated XRD patterns of β-antimony and theoretical SbO 1.93 structure. As shown in Supplementary Fig. 9 , the XRD pattern of SbO 1.93 sample is well consistent with the simulated XRD pattern of theoretical SbO 1.93 structure. Owing to oxygen-containing structure, both the peak position and the peak intensity of SbO 1.93 ’s XRD pattern have changed. The comparison between the structure of β-antimony and that of SbO 1.93 is exhibited in Supplementary Fig. 10 , from which the similar atomic configuration can be observed in both vertical and side views. Moreover, as shown in Supplementary Fig. 10 , it can be found that both β-antimony and SbO 1.93 present a hexagonal symmetry with R \\({\\bar{3}}\\) m space group (no. 166). Fig. 2: Structure elucidation of ultrathin antimony oxide. a Synchrotron-based GI-XRD image of ultrathin antimony oxide. b The fitting of GI-XRD data. c – e Schematic of the fitted SbO 1.93 structure along the [001] ( c ), [120] ( d ), and [110] ( e ) directions. Full size image We further use TEM and energy dispersive X-ray spectroscopy (EDS) to confirm the composition and the crystal structure of the ultrathin SbO 1.93 . Figure 3a shows the TEM image of an ultrathin SbO 1.93 crystal that is transferred onto the copper grid through water-soluble polyvinyl alcohol (PVA) mediator [14] . As displayed in Fig. 3b , the EDS mapping reveals the uniform spatial distribution of Sb and O in the crystal. Meanwhile, the EDS spectrum also confirms the existence of the element of Sb and O in the crystal (Fig. 3c ). Figure 3d shows a typical high-resolution TEM (HRTEM) image, which clearly resolves a \\(\\bar{1}20\\) lattice (2.18 Å) with a periodic atomic distribution. This result is in a good agreement with the fitted lattice in Supplementary Table 4 and it further support the \\({\\mathrm{R}}{\\bar{3}}{\\mathrm{m}}\\) symmetry of the ultrathin SbO 1.93 , which was totally different from the structures of other common antimony oxide (Supplementary Fig. 11 and Supplementary Note 2). Besides, we have conducted cross-sectional TEM characterizations to confirm the arrangement in the direction perpendicular to the Sb–O triangular lattice plane. In detail, the ultrathin SbO 1.93 crystal was first protected by a Pt layer, and then the cross-section TEM sample was prepared by focused ion beam (FIB). After that, cross-sectional TEM characterization was conducted, and corresponding results are displayed in Fig. 3e–h . Figure 3e shows the cross-sectional TEM image of the ultrathin SbO 1.93 lamella along the [210] direction, with a clearly resolved interlayer distance of 0.38 nm, which corresponds to the distance of the (003) lattice. From the schematic of the fitted SbO 1.93 structure viewing from [210] direction (as shown in Fig. 3f ), it can be concluded that the interlayer distance is one third than the distance of the c-axis lattice, which is in good agreement with the cross-sectional TEM results. Besides, the fast Fourier transform (FFT) pattern exhibited in Fig. 3g is well consistent with the simulated selected area electron diffraction (SAED) pattern along the [210] zone axis (Fig. 3h ), which further confirms the as-proposed arrangement in the perpendicular direction. Furthermore, Supplementary Fig. 12 shows the experimental SAED images along [001], [110], and [100] directions, which match well with the simulated SAED patterns, confirming that the as-simulated structure is correct and reasonable. Fig. 3: Atomic-scale analysis of the ultrathin SbO 1.93 crystal. a TEM image of an ultrathin SbO 1.93 crystal. b STEM image and corresponding EDS elemental mapping of O, Sb, and their overlap for the SbO 1.93 crystal. c The TEM-EDS spectrum of the SbO 1.93 crystal. d HRTEM image of the R-3m SbO 1.93 crystal. e Cross-sectional TEM image of the SbO 1.93 lamella along the [210] direction. f Schematic of the fitted SbO 1.93 structure viewing from [210] direction. g FFT from the area in e . h The simulated SAED pattern of SbO 1.93 along the [210] zone axis. Full size image The insulating property of the ultrathin SbO 1.93 To further characterize the insulating property of the ultrathin SbO 1.93 crystals, we conduct atomic force microscopy (AFM) investigation of the as-prepared ultrathin SbO 1.93 crystals on the Ag substrate. The ultrathin SbO 1.93 crystals show a smooth appearance under the AFM imaging. The different thickness (1.8 nm, 6 nm, 8 nm, and 16 nm) of the ultrathin SbO 1.93 crystals can be seen in Supplementary Fig. 13 . After that, conductive AFM characterizations are implemented to investigate the insulating property of the as-grown ultrathin SbO 1.93 crystals. Due to that silver is a good conductor, the as-prepared ultrathin SbO 1.93 crystals on the Ag substrate is directly used for the conductive AFM test (Fig. 4a ). As shown in Fig. 4b , the tested ultrathin SbO 1.93 crystal is completely insulating and pinhole-free, despite that it is only ~1.5 nm thick (Fig. 4c ). The breakdown electric field, defined as the field at which the current rises above the noise level, is measured to be ~5.7 GV m −1 , as determined by the current–voltage (I–V) characterization of the ultrathin SbO 1.93 crystal (Fig. 4d ). This value is one order of magnitude higher than the breakthrough field for CVD-grown multilayer h -BN [15] , highlighting the excellent quality of the ultrathin SbO 1.93 crystals. Fitting the I–V curve to the Schottky emission model allows us to determine a dielectric constant of ~100 for the ultrathin SbO 1.93 crystals (inset of Fig. 4d; Supplementary Note 3) [16] , [17] . Furthermore, we analyze the low-loss region of the electron energy loss spectrum (EELS) of the ultrathin SbO 1.93 crystal and found that the band gap of the ultrathin SbO 1.93 is ~6.3 eV (Supplementary Fig. 14 ; Supplementary Note 4). It is noteworthy that, compared with other oxides (such as SiO 2 , Al 2 O 3 , ZrO 2 , Ta 2 O 5 , La 2 O 3 , HfO 2 , and graphite oxide (GO)) [18] , [19] , [20] and h -BN [15] , SbO 1.93 has a better electrical isolation property (Fig. 4e ), which holds substantial promise for future miniaturized electronic and optoelectronic devices. Fig. 4: Conductive AFM test for the ultrathin SbO 1.93 . a Schematic of the peak force tunneling AFM (PF-TUNA) setup. b AFM current maps for the edge region of an ultrathin SbO 1.93 crystal on Ag substrate. c The height (top) and current (bottom) profiles correspond to the region in b indicated by the yellow line. The Ag region and the SbO 1.93 region are corresponding to the left and right sides, respectively. d Current–voltage curve measured through the ultrathin SbO 1.93 crystal. Inset: fit to the Schottky emission model and the determined dielectric constant. e Dielectric constant and breakdown voltage of typical high- κ materials, graphene oxide, h -BN, and SbO 1.93 in this work. Full size image To probe whether such ultrathin antimony oxide could act as a compatible oxide layer for antimonene, a hydrogen-reduction process is conducted. The reduction process is shown schematically in Supplementary Fig. 15a . The Raman spectrum of antimonene/antimony oxide is different from that of the pure ultrathin antimony oxide, and the peak at 152 cm −1 corresponds to the A 1g of antimonene, confirming that antimonene has been obtained after reduction (Supplementary Fig. 15b ) [21] . The results are further confirmed by XPS. As shown in Supplementary Fig. 15c , peaks at 528.0 and 537.3 eV are attributed to Sb 0 . In comparison with the result of pure antimony oxide crystals, the relative intensity of these two peaks increase significantly, which proved the existence of antimonene. Supplementary Fig. 15d shows the photoluminescence (PL) properties of the antimonene/antimony oxide under 355-nm laser excitation, and the emission peak of antimonene is located at 560 nm (2.21 eV), which is consistent with the theoretical prediction of the monolayer antimonene [21] . Time-resolved photoluminescence (TRPL) decays monitored at 560-nm emission peak (excitation wavelength 400 nm) are collected to provide further insights. The PL decays shown in Supplementary Fig. 15e can be fitted by a single exponential function, and τ is 2.8 ns. This suggests the existence of one pathway for the electron–hole radiative recombination from one emission species, which is corresponding to the emission of monolayer antimonene. The electronic property of antimonene/antimony oxide is also investigated. The electrical contact is defined by standard electron beam lithography followed by metal deposition of 10-nm-thick chromium and 30-nm-thick gold. The electrical characterization of the fabricated antimonene/antimony oxide devices is performed using a semiconductor parameter analyzer at room temperature. Schematic illustration of the device structure of antimonene/antimony oxide is shown in Supplementary Fig. 15f . Supplementary Fig. 15g shows the current–voltage (I ds –V ds ) characteristics of antimonene/antimony oxide. Different from the electrical isolation properties of ultrathin antimony oxide (Supplementary Fig. 16 ), the antimonene/antimony oxide shows semiconducting feature, like the intrinsic antimonene."},{"section":"Discussion","content":"In conclusion, we report the growth of high-quality ultrathin antimony oxide single crystals based on the SBC–CVD strategy. To elucidate the structure of ultrathin antimony oxide, synchrotron-based XRD and HRTEM are conducted. This type of high-quality ultrathin insulators has a high dielectric constant (~100) and a large breakdown electric field (~5.7 GV m −1 ). Furthermore, such strategy is not limited to antimony oxide, allowing to fabricate other oxide, such as bismuth oxide, germanium oxide and tin oxide (details in Supplementary Figs. 17 – 19 and Supplementary Table 5 ). We envision that this strategy can be expanded to construct more other ultrathin oxide crystals, thereby advancing the practical application of ultrathin oxides."},{"section":"Methods","content":"CVD growth of ultrathin antimony oxide crystals To achieve the growth of ultrathin antimony oxide crystals, we utilized a re-solidified Ag substrate. At first, a piece of Ag wire was placed on Co foil. Under the protection of an Ar/H 2 atmosphere, the Ag spread evenly over the entire foil by annealing at 1000°C for ∼ 20 min. After that, the as-obtained Ag(111) substrate was utilized for the CVD growth reaction. During the synthesis process, the antimony powder was placed in the upstream, and the furnace was heated up to 750 °C. The vapor of Sb was carried downstream by 20 sccm Ar, 5 sccm H 2 and O 2 . And then Sb atoms react with O 2 to grow ultrathin antimony oxide on the substrates. After the growth process, the furnace was cooled down fast to room temperature. The transfer of ultrathin antimony oxide to the copper grid and SiO 2 /Si substrate At first, the 9 wt% PVA (Alfa Aesar, 98–99% hydrolyzed, high-molecular-weight) aqueous solution was spin-coated on the top of a smooth base. The pre-spin speed is 1000 rpm (time = 10 s) and then the spin speed is 3000 rpm (time = 60 s), followed by baking at 100 °C for 60 s on a heating stage. After the baking process, the polymer mediator has been obtained. Then the PVA film was covered on the sample and heated to 80 °C for half an hour. After that, the PVA film was transferred from the growth substrate to the target substrate. The polymer mediator was removed by dissolving in deionized water at 50 °C for ∼ 2 h, and finally, the ultrathin antimony oxide single crystals were left on the target substrate. Characterization Optical images were taken with an optical microscope (Olympus DX51). Raman spectra with an excitation wavelength of 532 nm were carried out using a Renishaw inVia, and Raman mapping was taken by WITec alpha 300 R with an excitation wavelength of 488 nm. SEM images were obtained from a ZEISS Merlin Compact SEM with EDS spectra collected by X-MaxN Oxford EDS. X-ray photoelectron spectroscopy was performed on a Thermo Scientific, ESCALAB 250Xi. The binding energies were calibrated by referencing the C 1 s peak (284.8 eV). The thickness was measured on a NT-MDT Ntegra Spectra atomic force microscope. The conductive AFM characterizations were also conducted on NT-MDT Ntegra Spectra. TEM images were taken by FEI Titan themis 200 operated at an accelerating voltage of 80 kV. The antimony oxide lamellas for the cross-sectional TEM characterization were prepared via FIB treating by FEI Helios NanoLab 450 S DualBeam-FIB with UHREM FEG-SEM. The information of element distribution was also collected with EDS (INCAPentalFETx3 Oxford). The GI-XRD of the ultrathin antimony oxide were recorded at the BL14B1 beamline of Shanghai Synchrotron Radiation Facility (SSRF), and the data were calibrated with the LaB 6 standard sample. The incident photon energy was 10 keV (wavelength = 1.2398 Å) at an incident angle of 0.5°, and the diffracted X-rays were collected at an interval of 80 s using the 2D Mar225 CCD detector. XRD measurements for the re-solidified Ag substrates were performed using a Rigaku MiniFlex600 with Cu–K α radiation over the range of 2 θ = 20–80°. AES was performed using a PHI-700 system in an ultrahigh vacuum (UHV) system under a base pressure below 3.9 × 10 −9 Torr."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"Ultrathin oxides have been reported to possess excellent properties in electronic, magnetic, optical, and catalytic fields [1] , [2] , [3] , [4] , [5] , [6] . However, the ways to simply synthesize ultrathin oxides through a general and facile method is a big challenge. The current and primary approaches toward the preparation of ultrathin oxides arise from wet-chemistry synthesis, where a variety of ultrathin metal oxides can be attained [7] , [8] , [9] . In addition, the liquid metal-based reaction route, which requires metals that can be co-alloyed into the melt, has been utilized to obtain low-dimensional metal oxides [1] . However, these methods are only applicable to amorphous or polycrystalline oxide nanosheets or films with degraded properties [1] , [7] , [8] , [9] . Recently, Li et al. presented an approach for the preparation of ultrathin high- κ oxide, which showed the potential for two-dimensional electronic devices [10] . However, it is prerequisite to grow a molecular crystal as a seeding layer for the uniform deposition of the ultrathin oxide. Thus far, the research for high-quality and reproducible ultrathin high- κ oxide single crystals has not been achieved yet, albeit this is highly desirable. In this work, we realize the growth of ultrathin antimony oxide single crystals (down to 1.8 nm) based on a specially substrate-buffer-controlled chemical vapor deposition (SBC–CVD) strategy. After the growth, synchrotron-based X-ray diffraction (XRD) and high-resolution transmission electron microscopy (HRTEM) are conducted to elucidate the structure of such ultrathin antimony oxide single crystals. The SbO 1.93 single crystals exhibit excellent insulating capabilities with high dielectric constant (~100) and large breakdown electric field (~5.7 GV m −1 ). Such ultrathin antimony oxide single crystals will facilitate the ongoing research of the applications of ultrathin oxides.\",\n \"references\": {\n \"indices\": [\n 1,\n 2,\n 3,\n 4,\n 5,\n 6,\n 7,\n 8,\n 9,\n 10\n ],\n \"infos\": [\n {\n \"idx\": \"1\",\n \"title\": \"Zavabeti, A. et al. A liquid metal reaction environment for the room-temperature synthesis of atomically thin metal oxides.Science358, 332–335 (2017).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.aao4249\"\n },\n {\n \"idx\": \"2\",\n \"title\": \"Fiori, G. et al. Electronics based on two-dimensional materials.Nat. Nanotechnol.9, 768–779 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fnnano.2014.207\"\n },\n {\n \"idx\": \"3\",\n \"title\": \"Li, L., Richter, C., Mannhart, J. & Ashoori, R. C. Coexistence of magnetic order and two-dimensional superconductivity at LaAlO3/SrTiO3interfaces.Nat. Phys.7, 762–766 (2011).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys2080\"\n },\n {\n \"idx\": \"4\",\n \"title\": \"Bert, J. A. et al. Direct imaging of the coexistence of ferromagnetism and superconductivity at the LaAlO3/SrTiO3interface.Nat. Phys.7, 767–771 (2011).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys2079\"\n },\n {\n \"idx\": \"5\",\n \"title\": \"Zhao, C., Zhang, H. T., Si, W. J. & Wu, H. Mass production of two-dimensional oxides by rapid heating of hydrous chlorides.Nat. Commun.7, 12543 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms12543\"\n },\n {\n \"idx\": \"6\",\n \"title\": \"Hu, Z. M. et al. Rapid mass production of two-dimensional metal oxides and hydroxides via the molten salts method.Nat. Commun.8, 15630 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms15630\"\n },\n {\n \"idx\": \"7\",\n \"title\": \"Sun, Y. et al. Pits confined in ultrathin cerium(IV) oxide for studying catalytic centers in carbon monoxide oxidation.Nat. Commun.4, 2899 (2013).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms3899\"\n },\n {\n \"idx\": \"8\",\n \"title\": \"Sun, Z. et al. Generalized self-assembly of scalable two-dimensional transition metal oxide nanosheets.Nat. Commun.5, 3813 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms4813\"\n },\n {\n \"idx\": \"9\",\n \"title\": \"Zhao, H. W. et al. A generalized strategy for the synthesis of large-size ultrathin two-dimensional metal oxide nanosheets.Angew. Chem. Int. Ed.56, 8766–8770 (2017).\",\n \"link\": \"https://doi.org/10.1002%2Fanie.201703871\"\n },\n {\n \"idx\": \"10\",\n \"title\": \"Li, W. S. et al. Uniform and ultrathin high-κgate dielectrics for two-dimensional electronic devices.Nat. Electron.2, 563–571 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41928-019-0334-y\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"The growth and characterizations of ultrathin antimony oxide single crystals The strategy for the growth of ultrathin antimony oxide single crystals is schematically illustrated in Fig. 1a . Before the growth, a re-solidified Ag(111) substrate is prepared by a designed high-temperature (1000 °C) annealing process (details in Supplementary Figs. 1 – 6 ). Then, to achieve the growth of ultrathin antimony oxide single crystals, commercial antimony powder is placed upstream to provide antimony vapor, and the Ag substrates are placed in the downstream area with temperature T = 750 °C. The growth starts while O 2 is introduced into the system, and the growth process is maintained for a few minutes. Typical optical microscope (OM) image of the ultrathin antimony oxide is shown in Fig. 1b , which reveals that these triangular crystals exhibit a uniform size of about 5 μm, in agreement with the scanning electron microscope (SEM) data (Supplementary Fig. 7a ). Raman spectra of the as-grown triangular crystals are shown in Fig. 1c , revealing that antimony oxide with varying thickness corresponds to the homologous peaks. The only difference is that the Raman spectra intensity shows a linear relation with the thickness. As shown in Supplementary Fig. 7b , Raman intensity maps using the A 1g mode demonstrate a uniform response over the entire crystal, which proves the uniformity of the ultrathin antimony oxide crystal. Then, X-ray photoelectron spectroscopy (XPS) is implemented to further characterize the chemical composition of the as-prepared ultrathin antimony oxide crystals (Fig. 1d ). The peaks at 530.3 and 539.6 eV are attributed to the Sb 3d 5/2 and 3d 3/2 states, and the peak at 531.3 eV is corresponding to the O 1 s state, respectively [11] . Due to that the reaction Gibbs free energy of Sb with O 2 (−110.97 kJ mol −1 ) is much lower than that of Ag (−7.84 kJ mol −1 ) [12] , Sb will react with oxygen first. But the amount of O 2 must be limited to avoid the oxidation of the Ag substrate. To achieve the complete oxidation of Sb to form ultrathin antimony oxide single crystals, the amount of elementary Sb should also be controlled. Here, the Ag substrate serves as a buffer to achieve a modest supply of Sb. Such SBC–CVD mechanism is confirmed by the XPS depth profile. From the valence analysis of Sb along a surface normal-direction (Supplementary Fig. 8 ), it can be seen that, with the removal of the surface layer, the Sb 0 appears and keeps at a constant amount within the whole Ag substrate, which demonstrates that the excess Sb is buffered within the substrate, and it is very critical for the complete oxidation of Sb and the subsequent growth of ultrathin antimony oxide crystals. Fig. 1: Growth and spectral characterizations of ultrathin antimony oxide crystals. a Schematic of the strategy for the growth of ultrathin antimony oxide. b OM image of ultrathin antimony oxide crystals grown on the re-solidified Ag substrate. c Raman spectra of ultrathin antimony oxide crystals with different thicknesses. d XPS spectra of the ultrathin antimony oxide. Full size image Structure elucidation of ultrathin antimony oxide To further reveal the structure of ultrathin antimony oxide, synchrotron-based grazing-incidence XRD (GI-XRD) is conducted. As shown in Fig. 2a , the diffraction patterns are recorded on the image plate for the ultrathin antimony oxide. Then, owing to the similarity between the GI-XRD pattern of ultrathin antimony oxide and the simulated XRD pattern of β-antimony (Supplementary Fig. 9 ), the structure of β-antimony is used as the initial model for the structure creation of ultrathin antimony oxide. The rough crystal cell parameters can be obtained, and the preliminary cell structure can be parsed out. Then, the preliminary cell structure is further refined by Generalized Structure and Analysis Software (GSAS) package [13] (Supplementary Note 1). Figure 2b shows the experimental and the fitted data, where there is only a slight difference between them. From the diffraction result, the structure of the ultrathin antimony oxide can be elucidated, which is shown in Fig. 2c . The atomic ratio of Sb to O is 1:1.93, then the chemical formula can be defined as SbO 1.93 . And the top view and side view are demonstrated in Fig. 2d and e , respectively. Specifically, the crystal data and structure refinement for SbO 1.93 are listed in Supplementary Table 1 . Atomic coordinates, occupancy factors ( f occ ) and isotropic displacement parameters ( U iso ) are listed in Supplementary Table 2 . Bond lengths and bond angles are listed in Supplementary Table 3 . The unit-cell parameters and lattice plane parameters are listed in Supplementary Table 4 . To verify the rationality of the final structure, we compared the as-obtained XRD pattern of the SbO 1.93 sample with the simulated XRD patterns of β-antimony and theoretical SbO 1.93 structure. As shown in Supplementary Fig. 9 , the XRD pattern of SbO 1.93 sample is well consistent with the simulated XRD pattern of theoretical SbO 1.93 structure. Owing to oxygen-containing structure, both the peak position and the peak intensity of SbO 1.93 ’s XRD pattern have changed. The comparison between the structure of β-antimony and that of SbO 1.93 is exhibited in Supplementary Fig. 10 , from which the similar atomic configuration can be observed in both vertical and side views. Moreover, as shown in Supplementary Fig. 10 , it can be found that both β-antimony and SbO 1.93 present a hexagonal symmetry with R \\\\({\\\\bar{3}}\\\\) m space group (no. 166). Fig. 2: Structure elucidation of ultrathin antimony oxide. a Synchrotron-based GI-XRD image of ultrathin antimony oxide. b The fitting of GI-XRD data. c – e Schematic of the fitted SbO 1.93 structure along the [001] ( c ), [120] ( d ), and [110] ( e ) directions. Full size image We further use TEM and energy dispersive X-ray spectroscopy (EDS) to confirm the composition and the crystal structure of the ultrathin SbO 1.93 . Figure 3a shows the TEM image of an ultrathin SbO 1.93 crystal that is transferred onto the copper grid through water-soluble polyvinyl alcohol (PVA) mediator [14] . As displayed in Fig. 3b , the EDS mapping reveals the uniform spatial distribution of Sb and O in the crystal. Meanwhile, the EDS spectrum also confirms the existence of the element of Sb and O in the crystal (Fig. 3c ). Figure 3d shows a typical high-resolution TEM (HRTEM) image, which clearly resolves a \\\\(\\\\bar{1}20\\\\) lattice (2.18 Å) with a periodic atomic distribution. This result is in a good agreement with the fitted lattice in Supplementary Table 4 and it further support the \\\\({\\\\mathrm{R}}{\\\\bar{3}}{\\\\mathrm{m}}\\\\) symmetry of the ultrathin SbO 1.93 , which was totally different from the structures of other common antimony oxide (Supplementary Fig. 11 and Supplementary Note 2). Besides, we have conducted cross-sectional TEM characterizations to confirm the arrangement in the direction perpendicular to the Sb–O triangular lattice plane. In detail, the ultrathin SbO 1.93 crystal was first protected by a Pt layer, and then the cross-section TEM sample was prepared by focused ion beam (FIB). After that, cross-sectional TEM characterization was conducted, and corresponding results are displayed in Fig. 3e–h . Figure 3e shows the cross-sectional TEM image of the ultrathin SbO 1.93 lamella along the [210] direction, with a clearly resolved interlayer distance of 0.38 nm, which corresponds to the distance of the (003) lattice. From the schematic of the fitted SbO 1.93 structure viewing from [210] direction (as shown in Fig. 3f ), it can be concluded that the interlayer distance is one third than the distance of the c-axis lattice, which is in good agreement with the cross-sectional TEM results. Besides, the fast Fourier transform (FFT) pattern exhibited in Fig. 3g is well consistent with the simulated selected area electron diffraction (SAED) pattern along the [210] zone axis (Fig. 3h ), which further confirms the as-proposed arrangement in the perpendicular direction. Furthermore, Supplementary Fig. 12 shows the experimental SAED images along [001], [110], and [100] directions, which match well with the simulated SAED patterns, confirming that the as-simulated structure is correct and reasonable. Fig. 3: Atomic-scale analysis of the ultrathin SbO 1.93 crystal. a TEM image of an ultrathin SbO 1.93 crystal. b STEM image and corresponding EDS elemental mapping of O, Sb, and their overlap for the SbO 1.93 crystal. c The TEM-EDS spectrum of the SbO 1.93 crystal. d HRTEM image of the R-3m SbO 1.93 crystal. e Cross-sectional TEM image of the SbO 1.93 lamella along the [210] direction. f Schematic of the fitted SbO 1.93 structure viewing from [210] direction. g FFT from the area in e . h The simulated SAED pattern of SbO 1.93 along the [210] zone axis. Full size image The insulating property of the ultrathin SbO 1.93 To further characterize the insulating property of the ultrathin SbO 1.93 crystals, we conduct atomic force microscopy (AFM) investigation of the as-prepared ultrathin SbO 1.93 crystals on the Ag substrate. The ultrathin SbO 1.93 crystals show a smooth appearance under the AFM imaging. The different thickness (1.8 nm, 6 nm, 8 nm, and 16 nm) of the ultrathin SbO 1.93 crystals can be seen in Supplementary Fig. 13 . After that, conductive AFM characterizations are implemented to investigate the insulating property of the as-grown ultrathin SbO 1.93 crystals. Due to that silver is a good conductor, the as-prepared ultrathin SbO 1.93 crystals on the Ag substrate is directly used for the conductive AFM test (Fig. 4a ). As shown in Fig. 4b , the tested ultrathin SbO 1.93 crystal is completely insulating and pinhole-free, despite that it is only ~1.5 nm thick (Fig. 4c ). The breakdown electric field, defined as the field at which the current rises above the noise level, is measured to be ~5.7 GV m −1 , as determined by the current–voltage (I–V) characterization of the ultrathin SbO 1.93 crystal (Fig. 4d ). This value is one order of magnitude higher than the breakthrough field for CVD-grown multilayer h -BN [15] , highlighting the excellent quality of the ultrathin SbO 1.93 crystals. Fitting the I–V curve to the Schottky emission model allows us to determine a dielectric constant of ~100 for the ultrathin SbO 1.93 crystals (inset of Fig. 4d; Supplementary Note 3) [16] , [17] . Furthermore, we analyze the low-loss region of the electron energy loss spectrum (EELS) of the ultrathin SbO 1.93 crystal and found that the band gap of the ultrathin SbO 1.93 is ~6.3 eV (Supplementary Fig. 14 ; Supplementary Note 4). It is noteworthy that, compared with other oxides (such as SiO 2 , Al 2 O 3 , ZrO 2 , Ta 2 O 5 , La 2 O 3 , HfO 2 , and graphite oxide (GO)) [18] , [19] , [20] and h -BN [15] , SbO 1.93 has a better electrical isolation property (Fig. 4e ), which holds substantial promise for future miniaturized electronic and optoelectronic devices. Fig. 4: Conductive AFM test for the ultrathin SbO 1.93 . a Schematic of the peak force tunneling AFM (PF-TUNA) setup. b AFM current maps for the edge region of an ultrathin SbO 1.93 crystal on Ag substrate. c The height (top) and current (bottom) profiles correspond to the region in b indicated by the yellow line. The Ag region and the SbO 1.93 region are corresponding to the left and right sides, respectively. d Current–voltage curve measured through the ultrathin SbO 1.93 crystal. Inset: fit to the Schottky emission model and the determined dielectric constant. e Dielectric constant and breakdown voltage of typical high- κ materials, graphene oxide, h -BN, and SbO 1.93 in this work. Full size image To probe whether such ultrathin antimony oxide could act as a compatible oxide layer for antimonene, a hydrogen-reduction process is conducted. The reduction process is shown schematically in Supplementary Fig. 15a . The Raman spectrum of antimonene/antimony oxide is different from that of the pure ultrathin antimony oxide, and the peak at 152 cm −1 corresponds to the A 1g of antimonene, confirming that antimonene has been obtained after reduction (Supplementary Fig. 15b ) [21] . The results are further confirmed by XPS. As shown in Supplementary Fig. 15c , peaks at 528.0 and 537.3 eV are attributed to Sb 0 . In comparison with the result of pure antimony oxide crystals, the relative intensity of these two peaks increase significantly, which proved the existence of antimonene. Supplementary Fig. 15d shows the photoluminescence (PL) properties of the antimonene/antimony oxide under 355-nm laser excitation, and the emission peak of antimonene is located at 560 nm (2.21 eV), which is consistent with the theoretical prediction of the monolayer antimonene [21] . Time-resolved photoluminescence (TRPL) decays monitored at 560-nm emission peak (excitation wavelength 400 nm) are collected to provide further insights. The PL decays shown in Supplementary Fig. 15e can be fitted by a single exponential function, and τ is 2.8 ns. This suggests the existence of one pathway for the electron–hole radiative recombination from one emission species, which is corresponding to the emission of monolayer antimonene. The electronic property of antimonene/antimony oxide is also investigated. The electrical contact is defined by standard electron beam lithography followed by metal deposition of 10-nm-thick chromium and 30-nm-thick gold. The electrical characterization of the fabricated antimonene/antimony oxide devices is performed using a semiconductor parameter analyzer at room temperature. Schematic illustration of the device structure of antimonene/antimony oxide is shown in Supplementary Fig. 15f . Supplementary Fig. 15g shows the current–voltage (I ds –V ds ) characteristics of antimonene/antimony oxide. Different from the electrical isolation properties of ultrathin antimony oxide (Supplementary Fig. 16 ), the antimonene/antimony oxide shows semiconducting feature, like the intrinsic antimonene.\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"In conclusion, we report the growth of high-quality ultrathin antimony oxide single crystals based on the SBC–CVD strategy. To elucidate the structure of ultrathin antimony oxide, synchrotron-based XRD and HRTEM are conducted. This type of high-quality ultrathin insulators has a high dielectric constant (~100) and a large breakdown electric field (~5.7 GV m −1 ). Furthermore, such strategy is not limited to antimony oxide, allowing to fabricate other oxide, such as bismuth oxide, germanium oxide and tin oxide (details in Supplementary Figs. 17 – 19 and Supplementary Table 5 ). We envision that this strategy can be expanded to construct more other ultrathin oxide crystals, thereby advancing the practical application of ultrathin oxides.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"CVD growth of ultrathin antimony oxide crystals To achieve the growth of ultrathin antimony oxide crystals, we utilized a re-solidified Ag substrate. At first, a piece of Ag wire was placed on Co foil. Under the protection of an Ar/H 2 atmosphere, the Ag spread evenly over the entire foil by annealing at 1000°C for ∼ 20 min. After that, the as-obtained Ag(111) substrate was utilized for the CVD growth reaction. During the synthesis process, the antimony powder was placed in the upstream, and the furnace was heated up to 750 °C. The vapor of Sb was carried downstream by 20 sccm Ar, 5 sccm H 2 and O 2 . And then Sb atoms react with O 2 to grow ultrathin antimony oxide on the substrates. After the growth process, the furnace was cooled down fast to room temperature. The transfer of ultrathin antimony oxide to the copper grid and SiO 2 /Si substrate At first, the 9 wt% PVA (Alfa Aesar, 98–99% hydrolyzed, high-molecular-weight) aqueous solution was spin-coated on the top of a smooth base. The pre-spin speed is 1000 rpm (time = 10 s) and then the spin speed is 3000 rpm (time = 60 s), followed by baking at 100 °C for 60 s on a heating stage. After the baking process, the polymer mediator has been obtained. Then the PVA film was covered on the sample and heated to 80 °C for half an hour. After that, the PVA film was transferred from the growth substrate to the target substrate. The polymer mediator was removed by dissolving in deionized water at 50 °C for ∼ 2 h, and finally, the ultrathin antimony oxide single crystals were left on the target substrate. Characterization Optical images were taken with an optical microscope (Olympus DX51). Raman spectra with an excitation wavelength of 532 nm were carried out using a Renishaw inVia, and Raman mapping was taken by WITec alpha 300 R with an excitation wavelength of 488 nm. SEM images were obtained from a ZEISS Merlin Compact SEM with EDS spectra collected by X-MaxN Oxford EDS. X-ray photoelectron spectroscopy was performed on a Thermo Scientific, ESCALAB 250Xi. The binding energies were calibrated by referencing the C 1 s peak (284.8 eV). The thickness was measured on a NT-MDT Ntegra Spectra atomic force microscope. The conductive AFM characterizations were also conducted on NT-MDT Ntegra Spectra. TEM images were taken by FEI Titan themis 200 operated at an accelerating voltage of 80 kV. The antimony oxide lamellas for the cross-sectional TEM characterization were prepared via FIB treating by FEI Helios NanoLab 450 S DualBeam-FIB with UHREM FEG-SEM. The information of element distribution was also collected with EDS (INCAPentalFETx3 Oxford). The GI-XRD of the ultrathin antimony oxide were recorded at the BL14B1 beamline of Shanghai Synchrotron Radiation Facility (SSRF), and the data were calibrated with the LaB 6 standard sample. The incident photon energy was 10 keV (wavelength = 1.2398 Å) at an incident angle of 0.5°, and the diffracted X-rays were collected at an interval of 80 s using the 2D Mar225 CCD detector. XRD measurements for the re-solidified Ag substrates were performed using a Rigaku MiniFlex600 with Cu–K α radiation over the range of 2 θ = 20–80°. AES was performed using a PHI-700 system in an ultrahigh vacuum (UHV) system under a base pressure below 3.9 × 10 −9 Torr.\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"This research proposes a substrate-buffer-controlled chemical vapor deposition (SBC–CVD) strategy to achieve the growth of ultrathin antimony oxide single crystals with high dielectric constant and excellent insulating properties, aiming to overcome the limitations of existing methods and enable the development of high-quality, reproducible ultrathin oxide materials for advanced two-dimensional electronic devices."},"entities":{"kind":"list like","value":["substrate-buffer-controlled chemical vapor deposition","SBC–CVD strategy","ultrathin antimony oxide single crystals","high dielectric constant","excellent insulating properties","existing methods","high-quality","reproducible ultrathin oxide materials","advanced two-dimensional electronic devices"],"string":"[\n \"substrate-buffer-controlled chemical vapor deposition\",\n \"SBC–CVD strategy\",\n \"ultrathin antimony oxide single crystals\",\n \"high dielectric constant\",\n \"excellent insulating properties\",\n \"existing methods\",\n \"high-quality\",\n \"reproducible ultrathin oxide materials\",\n \"advanced two-dimensional electronic devices\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1.","title":"Zavabeti, A. et al. A liquid metal reaction environment for the room-temperature synthesis of atomically thin metal oxides.Science358, 332–335 (2017).","link":"https://doi.org/10.1126%2Fscience.aao4249"},{"idx":"2.","title":"Fiori, G. et al. Electronics based on two-dimensional materials.Nat. Nanotechnol.9, 768–779 (2014).","link":"https://doi.org/10.1038%2Fnnano.2014.207"},{"idx":"3.","title":"Li, L., Richter, C., Mannhart, J. & Ashoori, R. C. Coexistence of magnetic order and two-dimensional superconductivity at LaAlO3/SrTiO3interfaces.Nat. Phys.7, 762–766 (2011).","link":"https://doi.org/10.1038%2Fnphys2080"},{"idx":"4.","title":"Bert, J. A. et al. Direct imaging of the coexistence of ferromagnetism and superconductivity at the LaAlO3/SrTiO3interface.Nat. Phys.7, 767–771 (2011).","link":"https://doi.org/10.1038%2Fnphys2079"},{"idx":"5.","title":"Zhao, C., Zhang, H. T., Si, W. J. & Wu, H. Mass production of two-dimensional oxides by rapid heating of hydrous chlorides.Nat. Commun.7, 12543 (2016).","link":"https://doi.org/10.1038%2Fncomms12543"},{"idx":"6.","title":"Hu, Z. M. et al. Rapid mass production of two-dimensional metal oxides and hydroxides via the molten salts method.Nat. Commun.8, 15630 (2017).","link":"https://doi.org/10.1038%2Fncomms15630"},{"idx":"7.","title":"Sun, Y. et al. Pits confined in ultrathin cerium(IV) oxide for studying catalytic centers in carbon monoxide oxidation.Nat. Commun.4, 2899 (2013).","link":"https://doi.org/10.1038%2Fncomms3899"},{"idx":"8.","title":"Sun, Z. et al. Generalized self-assembly of scalable two-dimensional transition metal oxide nanosheets.Nat. Commun.5, 3813 (2014).","link":"https://doi.org/10.1038%2Fncomms4813"},{"idx":"9.","title":"Zhao, H. W. et al. A generalized strategy for the synthesis of large-size ultrathin two-dimensional metal oxide nanosheets.Angew. Chem. Int. Ed.56, 8766–8770 (2017).","link":"https://doi.org/10.1002%2Fanie.201703871"},{"idx":"10.","title":"Li, W. S. et al. Uniform and ultrathin high-κgate dielectrics for two-dimensional electronic devices.Nat. Electron.2, 563–571 (2019).","link":"https://doi.org/10.1038%2Fs41928-019-0334-y"},{"idx":"11.","title":"Zeng, D. W., Xie, C. S., Zhu, B. L. & Song, W. L. Characteristics of Sb2O3nanoparticles synthesized from antimony by vapor condensation method.Mater. Lett.58, 312–315 (2004).","link":"https://doi.org/10.1016%2FS0167-577X%2803%2900476-2"},{"idx":"12.","title":"Yao, S., Kaku, Y. & Kozuka, Z. The activity-coefficients of oxygen in liquid Ag-In, Ag-Sn, Ag-Sb and Ag-Ge alloys.Trans. Jpn. Inst. Met.27, 616–622 (1986).","link":"https://doi.org/10.2320%2Fmatertrans1960.27.616"},{"idx":"13.","title":"Toby, B. H. EXPGUI, a graphical user interface for GSAS.J. Appl. Crystallogr.34, 210–213 (2001).","link":"https://doi.org/10.1107%2FS0021889801002242"},{"idx":"14.","title":"Lu, Z. X. et al. Universal transfer and stacking of chemical vapor deposition grown two-dimensional atomic layers with water-soluble polymer mediator.ACS Nano10, 5237–5242 (2016).","link":"https://doi.org/10.1021%2Facsnano.6b00961"},{"idx":"15.","title":"Kim, S. M. et al. Synthesis of large-area multilayer hexagonal boron nitride for high material performance.Nat. Commun.6, 8662 (2015).","link":"https://doi.org/10.1038%2Fncomms9662"},{"idx":"16.","title":"Jahanmir, J. & West, P. Evidence of Schottky emission in scanning tunneling microscopes operated in ambient air.Appl. Phys. Lett.52, 2086–2088 (1988).","link":"https://doi.org/10.1063%2F1.99582"},{"idx":"18.","title":"Standley, B., Mendez, A., Schmidgall, E. & Bockrath, M. Graphene–graphite oxide field-effect transistors.Nano Lett.12, 1165–1169 (2012).","link":"https://doi.org/10.1021%2Fnl2028415"},{"idx":"19.","title":"Choi, J. H., Mao, Y. & Chang, J. P. Development of hafnium based high-κmaterials—a review.Mater. Sci. Eng. Rep.72, 97–136 (2011).","link":"https://doi.org/10.1016%2Fj.mser.2010.12.001"},{"idx":"20.","title":"Osada, M. & Sasaki, T. Two-dimensional dielectric nanosheets: novel nanoelectronics from nanocrystal building blocks.Adv. Mater.24, 210–228 (2012).","link":"https://doi.org/10.1002%2Fadma.201103241"},{"idx":"21.","title":"Zhang, S. L., Yan, Z., Li, Y. F., Chen, Z. F. & Zeng, H. B. Atomically thin arsenene and antimonene: semimetal-semiconductor and indirect-direct band-gap transitions.Angew. Chem. Int. Ed.54, 3112–3115 (2015).","link":"https://doi.org/10.1002%2Fanie.201411246"}],"string":"[\n {\n \"idx\": \"1.\",\n \"title\": \"Zavabeti, A. et al. A liquid metal reaction environment for the room-temperature synthesis of atomically thin metal oxides.Science358, 332–335 (2017).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.aao4249\"\n },\n {\n \"idx\": \"2.\",\n \"title\": \"Fiori, G. et al. Electronics based on two-dimensional materials.Nat. Nanotechnol.9, 768–779 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fnnano.2014.207\"\n },\n {\n \"idx\": \"3.\",\n \"title\": \"Li, L., Richter, C., Mannhart, J. & Ashoori, R. C. Coexistence of magnetic order and two-dimensional superconductivity at LaAlO3/SrTiO3interfaces.Nat. Phys.7, 762–766 (2011).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys2080\"\n },\n {\n \"idx\": \"4.\",\n \"title\": \"Bert, J. A. et al. Direct imaging of the coexistence of ferromagnetism and superconductivity at the LaAlO3/SrTiO3interface.Nat. Phys.7, 767–771 (2011).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys2079\"\n },\n {\n \"idx\": \"5.\",\n \"title\": \"Zhao, C., Zhang, H. T., Si, W. J. & Wu, H. Mass production of two-dimensional oxides by rapid heating of hydrous chlorides.Nat. Commun.7, 12543 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms12543\"\n },\n {\n \"idx\": \"6.\",\n \"title\": \"Hu, Z. M. et al. Rapid mass production of two-dimensional metal oxides and hydroxides via the molten salts method.Nat. Commun.8, 15630 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms15630\"\n },\n {\n \"idx\": \"7.\",\n \"title\": \"Sun, Y. et al. Pits confined in ultrathin cerium(IV) oxide for studying catalytic centers in carbon monoxide oxidation.Nat. Commun.4, 2899 (2013).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms3899\"\n },\n {\n \"idx\": \"8.\",\n \"title\": \"Sun, Z. et al. Generalized self-assembly of scalable two-dimensional transition metal oxide nanosheets.Nat. Commun.5, 3813 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms4813\"\n },\n {\n \"idx\": \"9.\",\n \"title\": \"Zhao, H. W. et al. A generalized strategy for the synthesis of large-size ultrathin two-dimensional metal oxide nanosheets.Angew. Chem. Int. Ed.56, 8766–8770 (2017).\",\n \"link\": \"https://doi.org/10.1002%2Fanie.201703871\"\n },\n {\n \"idx\": \"10.\",\n \"title\": \"Li, W. S. et al. Uniform and ultrathin high-κgate dielectrics for two-dimensional electronic devices.Nat. Electron.2, 563–571 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41928-019-0334-y\"\n },\n {\n \"idx\": \"11.\",\n \"title\": \"Zeng, D. W., Xie, C. S., Zhu, B. L. & Song, W. L. Characteristics of Sb2O3nanoparticles synthesized from antimony by vapor condensation method.Mater. Lett.58, 312–315 (2004).\",\n \"link\": \"https://doi.org/10.1016%2FS0167-577X%2803%2900476-2\"\n },\n {\n \"idx\": \"12.\",\n \"title\": \"Yao, S., Kaku, Y. & Kozuka, Z. The activity-coefficients of oxygen in liquid Ag-In, Ag-Sn, Ag-Sb and Ag-Ge alloys.Trans. Jpn. Inst. Met.27, 616–622 (1986).\",\n \"link\": \"https://doi.org/10.2320%2Fmatertrans1960.27.616\"\n },\n {\n \"idx\": \"13.\",\n \"title\": \"Toby, B. H. EXPGUI, a graphical user interface for GSAS.J. Appl. Crystallogr.34, 210–213 (2001).\",\n \"link\": \"https://doi.org/10.1107%2FS0021889801002242\"\n },\n {\n \"idx\": \"14.\",\n \"title\": \"Lu, Z. X. et al. Universal transfer and stacking of chemical vapor deposition grown two-dimensional atomic layers with water-soluble polymer mediator.ACS Nano10, 5237–5242 (2016).\",\n \"link\": \"https://doi.org/10.1021%2Facsnano.6b00961\"\n },\n {\n \"idx\": \"15.\",\n \"title\": \"Kim, S. M. et al. Synthesis of large-area multilayer hexagonal boron nitride for high material performance.Nat. Commun.6, 8662 (2015).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms9662\"\n },\n {\n \"idx\": \"16.\",\n \"title\": \"Jahanmir, J. & West, P. Evidence of Schottky emission in scanning tunneling microscopes operated in ambient air.Appl. Phys. Lett.52, 2086–2088 (1988).\",\n \"link\": \"https://doi.org/10.1063%2F1.99582\"\n },\n {\n \"idx\": \"18.\",\n \"title\": \"Standley, B., Mendez, A., Schmidgall, E. & Bockrath, M. Graphene–graphite oxide field-effect transistors.Nano Lett.12, 1165–1169 (2012).\",\n \"link\": \"https://doi.org/10.1021%2Fnl2028415\"\n },\n {\n \"idx\": \"19.\",\n \"title\": \"Choi, J. H., Mao, Y. & Chang, J. P. Development of hafnium based high-κmaterials—a review.Mater. Sci. Eng. Rep.72, 97–136 (2011).\",\n \"link\": \"https://doi.org/10.1016%2Fj.mser.2010.12.001\"\n },\n {\n \"idx\": \"20.\",\n \"title\": \"Osada, M. & Sasaki, T. Two-dimensional dielectric nanosheets: novel nanoelectronics from nanocrystal building blocks.Adv. Mater.24, 210–228 (2012).\",\n \"link\": \"https://doi.org/10.1002%2Fadma.201103241\"\n },\n {\n \"idx\": \"21.\",\n \"title\": \"Zhang, S. L., Yan, Z., Li, Y. F., Chen, Z. F. & Zeng, H. B. Atomically thin arsenene and antimonene: semimetal-semiconductor and indirect-direct band-gap transitions.Angew. Chem. Int. Ed.54, 3112–3115 (2015).\",\n \"link\": \"https://doi.org/10.1002%2Fanie.201411246\"\n }\n]","isConversational":false}}},{"rowIdx":191,"cells":{"unique_id":{"kind":"string","value":"s41467-019-08948-x"},"subfield":{"kind":"string","value":"Mathematics and computing"},"title":{"kind":"string","value":"Percolation thresholds for photonic quantum computing"},"abstract":{"kind":"string","value":"Despite linear-optical fusion (Bell measurement) being probabilistic, photonic cluster states for universal quantum computation can be prepared without feed-forward by fusing small n -photon entangled clusters, if the success probability of each fusion attempt is above a threshold, \\({\\mathrm{\\lambda }}_{\\mathrm{c}}^{(n)}\\) . We prove a general bound \\({\\mathrm{\\lambda }}_{\\mathrm{c}}^{(n)} \\ge 1/(n - 1)\\) , and develop a conceptual method to construct long-range-connected clusters where \\({\\mathrm{\\lambda }}_{\\mathrm{c}}^{(n)}\\) becomes the bond percolation threshold of a logical graph. This mapping lets us find constructions that require lower fusion success probabilities than currently known, and settle a heretofore open question by showing that a universal cluster state can be created by fusing 3-photon clusters over a 2D lattice with a fusion success probability that is achievable with linear optics and single photons, making this attractive for integrated-photonic realizations."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"Optical qubits are a promising candidate for scalable universal quantum computing because of the relative ease of generating photons, fabricating photonic circuits [1] , [2] , [3] , and low decoherence rates. Assembling a universal photonic cluster state, e.g., by fusing small entangled clusters using linear optical interferometers into progressively larger ones, is limited by probabilistic operations because of the limitations of linear optics. For example, a linear–optical circuit for a two-qubit Bell-state measurement (BSM)—the smallest primitive to “fuse” two cluster states—can succeed with at most probability 1/2 = 0.5 [4] , [5] . However, since photons are the base resource to encode qubits anyway, it is reasonable to allow photons to be used as an ancilla resource to boost the success probability of linear–optic BSMs [6] , [7] , [8] . The maximum known BSM success probability attainable using linear optics, boosted by injecting eight ancilla single photons, is 25/32 = 0.78125 [8] . It is not known if a higher BSM success probability is possible with linear optics boosted with more single photons. Very little is understood about similar success-probability limits associated with linear–optical realizations of larger (three or more qubits) projective measurements, and those for creating small photonic entangled clusters [9] . The lack of our understanding of the maximum efficiency in creating universal photonic cluster states goes back to the mathematical structure of manipulations of multiphoton entangled states using linear optics and photon detection being complex, as evidenced by the hardness of boson sampling [10] . Linear optical quantum computing (LOQC) in its widely known circuit-model form—coined by the seminal KLM paper [11] —employs single-photon dual-rail qubits (| 0 〉 ≡ |10〉, | 1 〉 ≡ |01〉), probabilistic linear–optical gates, single-photon detection, and measurement-induced feedforward. Even though KLM’s scheme in principle can do universal quantum computing, the overheads are astronomically high. Many variants of KLM were proposed [12] , [13] , [14] that use entangled ancillas to reduce the overheads, but the hardness was pushed into creating those ancillas. Schemes for efficient generation of these ancillas from simpler resources such as single photons remain unknown. Furthermore, the need for detection-induced feedback creates additional experimental issues. This resulted in LOQC being largely disfavored in lieu of other qubit technologies, such as superconducting and trapped-ion qubits. Kieling, Rudolph, and Eisert proposed a cluster-model version of LOQC [15] , which mitigates these issues. Their scheme places d + 1 photon “star” clusters at nodes of a regular lattice G with degree- d nodes and stitches them together using two-photon fusions (BSMs) along the lattice edges (see Fig. 1 ). If the success probability of each fusion attempt λ is above the bond percolation threshold p c (G) of the underlying lattice G, the resulting cluster state will be a long-range-connected (random) subgraph of G, which can then be renormalized into a universal cluster state by identifying a subarray of nodes in the percolated lattice—the intersection points of vertical and horizontal “percolation highways” [16] , [17] —to serve as the nodes of a logical universal lattice and identifying paths connecting them through other nodes of the percolated subgraph as edges of this logical lattice (see Fig. 2a ). The inspiration behind this model is that if small d + 1 photon clusters can be created, and a lattice can be identified whose bond- percolation threshold is below a practically attainable fusion success probability, then the clusters can be linear–optically fused into a large universal cluster state in a one-shot (or “ballistic”) fashion, with no detection-induced feedforward. In order to produce conservative estimates for the fusion success probability required to produce a renormalizable cluster state, this work disregarded the “failure modes” of the fusion attempts, i.e., the leftover cluster states when fusion attempts fail. Fig. 1 Creating a percolated square lattice with fusion. Fusing five-photon star clusters into a percolated 2D square lattice is possible with two-qubit fusion operations, if each fusion attempt succeeds with the probability more than 0.5, the bond-percolation threshold of the lattice [15] Full size image Fig. 2 Bond-percolation threshold of a square lattice. a Each bond in a regular lattice of m bonds is activated with probability p . If p < p c (the subcritical regime), where p c is the bond-percolation threshold, the size of the largest connected component is O (log m ). At p = p c , a sharp transition occurs, and for 1 ≥ p > p c , the size of the largest connected component is O ( m ). In this (supercritical) regime, O ( m ) criss-crossing edge-disjoint paths (percolation highways) exist from left to right and from top to bottom [17] , using which one can identify a logical lattice whose nodes are the intersection points of the highways, and edges are paths connecting those nodes. b Size of the largest connected component, expressed as a fraction of m , as a function of p ∈ [0, 1] Full size image Significant progress in this model of LOQC came recently from Rudolph and collaborators, who by paying closer attention to the failure modes, and a clever design of 3D lattices with a low-percolation threshold, e.g., the diamond [18] and the pyrochlore lattice [19] , were able to show that ballistic creation of a universal cluster state is possible starting with a large supply of three-photon-entangled (GHZ state) clusters, with two-photon fusions that succeed with a probability ≈0.625. Since 0.625 < 25/32 = 0.78125, the best-known success probability attainable with linear optics boosted with unentangled single-photon ancillas [8] , this gave the first realizable ballistic construction of a universal cluster if three-photon GHZ clusters were available as a starting point. The results of Gimeno-Segovia et al. and Zaidi et al. [18] , [19] on ballistic (feedback-free) creation of universal clusters, in conjunction with a host of progress in experimental proposals to directly create small photonic entangled clusters using quantum emitters [20] , [21] , [22] (rather than starting with single photons and fusing them probabilistically using heralded methods into larger clusters), have resurrected the optimism in scalable optical quantum computing. However, the fundamental limits of efficiency and resource overhead associated with this philosophy—that of direct generation of a universal cluster state using small photon clusters and linear optics—remain unanswered. Refs. [18] , [19] present two (clever, yet ad hoc) examples of constructions that use three-photon clusters as the starting point. The goal of this paper is to explore this fundamental limit. We address the following general question: Given an unlimited supply of n -photon-entangled dual-rail-basis clusters, and repeated use of a linear–optical circuit that realizes two-qubit fusion with probability of success λ, but with no restrictions on how the clusters to be fused are aligned in space and time, what is the minimum value \\({\\lambda }_{\\mathrm{c}}^{(n)}\\) of each fusion’s success probability that will allow for ballistic creation of a universal cluster state? Before we proceed, it should be clear that the two examples we discussed above—one in Fig. 1 [15] and that in ref. [18] —establish the following bounds: $${\\lambda }_{\\mathrm{c}}^{(5)} \\le 0.5,$$ (1) and $${\\lambda }_{\\mathrm{c}}^{(3)} \\,\\, \\lesssim \\,\\, 0.625,$$ (2) where we use ≲ sign instead of ≤ in Eq. ( 2 ), since the 0.625 threshold was an approximate numerical estimation of the threshold using a small-sized diamond lattice [18] . We focus on Bell-measurement-based fusion operations. Although Browne and Rudolph’s original paper that coined the term “fusion” [13] also proposed an alternative linear optical fusion circuit (called fusion-I), subsequent work has focussed on Bell-measurement-based fusion (fusion-II of ref. [13] ) because of its natural loss tolerance and recent progress in boosting its success probability by injecting unentangled single-photon ancillas [8] . In this paper, we prove that starting with n -photon clusters, a fusion success probability of at least 1/( n −1) is required to create a resource state for universal quantum computation without feedforward. We map a planned set of fusion operations on photon pairs among a regular spatial array of n -photon clusters—the cluster-fusion lattice, G 1 —to an instance of bond percolation on a logical lattice G 2 . We use this map to find new constructions to produce universal photonic cluster states starting with three-photon cluster states as the initial resource, without feedforward, that require a lower fusion success probability than was previously known. Finally, we settle a heretofore open question by showing that a universal cluster state can be created by fusing three-photon clusters over a 2D lattice with a fusion success probability that is achievable with linear optics and single photons, making this attractive for integrated-photonic realizations.","references":{"indices":[1,2,3,4,5,6,7,8,9,11,12,13,14,16,18,19,21],"infos":[{"idx":"1","title":"Reck, M. & Zeilinger, A. Experimental realization of any discrete unitary operator.Phys. Rev. Lett.73, 58–61 (1994).","link":"https://doi.org/10.1103%2FPhysRevLett.73.58"},{"idx":"2","title":"Carolan, J. et al. 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We develop a systematic graphical construction of a universal cluster, wherein \\({\\lambda }_{\\mathrm{c}}^{(n)}\\) becomes the usual bond-percolation threshold of a logical graph that we specify, while accounting for the fusion failure modes. This is a key realization that was missing in both Kieling et al.’s construction [15] as well as in refs. [18] , [19] . This not only enabled us to readily generate achievable thresholds (upper bounds on \\({\\lambda }_{\\mathrm{c}}^{(n)}\\) ) using existing literature on bond percolation, but also helped us to prove the aforesaid lower bound. We believe that this logical construction also naturally generalizes to future extensions that may employ multi-qubit fusions (projective measurements). We describe this construction systematically in the section “Reconciling prior results in a new framework”, and present several improved thresholds using this new formalism. However, we develop a fully axiomatic construction of this logical graph for the most general case if left open for future work. 3. Using our logical construction, we prove \\({\\lambda }_{\\mathrm{c}}^{(3)} \\le 0.5898\\) as an achievable threshold, which improves upon the best existing result in ref. [18] . As a by-product of our logical construction allowing a standard bond-percolation interpretation, we were able to reinterpret the construction of ref. [18] in our formalism and used Newman and Ziff’s efficient Monte Carlo algorithm to refine their threshold (2) to \\({\\lambda }_{\\mathrm{c}}^{(3)} \\le 0.627\\) . 4. We settle a heretofore open question posed by Rudolph and collaborators [18] , [19] , [23] by showing that a universal cluster can be created by linear–optically fusing three-photon GHZ clusters over a 2D (brickwork) lattice with fusion success probability 0.746, which is below 25/32, the maximum known fusion success probability achievable with a single-photon-boosted linear–optic scheme [8] , making this attractive for integrated-photonic realizations. In the next section, we will begin with a discussion of the high-level problem setup and the constraints therein, following which we will present a logical technical progression leading to our main results and the key intuitions. Detailed derivations and technical graphical constructions not central to following the main arguments in the paper are presented in the Supplementary Note 2 . Generality and constraints of our problem setup Figure 3 and its caption explains our high-level problem setup. It follows from our discussion above that we are limiting ourselves to only those multimode linear–optical unitaries U that can be pieced together using a sequence of two-qubit linear–optic BSMs. Our problem formulation, however, subsumes the special cases of the constructions in refs. [15] , [18] , [19] . Furthermore, our formulation allows for yet-undiscovered fusion circuits that may be boosted with up to n -photon entangled clusters (which may attain a success probability greater than the currently best known, 25/32). It also admits generalizations to multi-qubit fusion, i.e., m -qubit GHZ-state projections, the theory behind optimal linear–optical realizations of which it remains not well understood. Fig. 3 Ballistic photonic cluster state generation for quantum computing. A steady stream of entangled clusters of n -photons or less ( n = 3 shown) is incident on a linear–optical interferometer (i.e., a multimode unitary transformation U ), which produces an entangled cluster of photons at its output. If the percolation condition is met, the output can be renormalized into a fully connected logical cluster for universal cluster-model quantum computing. The detector outputs are not used to herald whether or not the undetected outputs are in a universal cluster state. They always are. The detector outputs carry complete information on the location of missing edges in the random graph, and thus how to renormalize it Full size image The detectors shown in Fig. 3 are those corresponding to the linear–optical fusion gates pushed (deferred) all the way to the right with no loss of generality. Similarly, any photons or photon clusters used to boost any fusion gate are pushed to the left of U . This way, one can envision the cluster preparation as a multimode linear optical circuit presented with individual photons and photon clusters as input, and producing (deterministically) a renormalizable universal cluster state as its output. The individual detector outputs are random, but if we design U in a way that the percolation condition is met, the undetected outputs of U would, with high probability, carry a random instance of a universal long-range-connected (percolated) cluster and the vector of detector outputs would simply tell us what cluster state got created on the undetected outputs, and thus give us the necessary data to compute criss-crossing edge-disjoint paths through the percolated lattice to renormalize it into a universal logical cluster state [24] . General lower bounds In this section, we present an intuitive explanation of our lower bound on \\({\\lambda }_{\\mathrm{c}}^{(n)}\\) . A detailed formal proof is provided in Supplementary Note 1 . We have the following general lower bound: $${\\lambda }_{\\mathrm{c}}^{(n)} \\ge \\frac{1}{{n - 1}},\\forall n \\ge 2.$$ (3) Proof sketch—starting with n -photon clusters, and any sequence of fusion attempts, the resulting instance of the logical graph is a bond-percolation instance, with bond success probability λ , on some graph of maximum degree n . Of all infinite graphs of maximum degree n , the minimum bond-percolation threshold is that of the degree- n Bethe lattice, and equals 1/( n −1). This implies that if λ < 1/( n −1), the percolation condition cannot be met no matter what. Hence, \\({\\lambda }_{\\mathrm{c}}^{(n)} \\ge 1/(n - 1)\\) . Since the Bethe lattice is a tree and since trees are not a universal resource, this does not prove 1/( n −1) to be an achievable threshold for universal cluster creation (see Supplementary Note 3 for more details). However, we conjecture (and provide evidence in its favor) that λ = [1/( n −1)] + ε with any ε > 0 that may suffice to create a universal cluster state. This conjecture is supported by the fact that starting with N copies of photon clusters of size n and probabilistic operations that succeed with any probability greater than 1/( n −1), it is possible to obtain an O ( N )-sized tree graph (detailed in Supplementary Note 3 ). We also present a random graph construction, which provides intuition for why probabilistic operations on photon clusters of size n may provide enough connectivity to obtain a large connected cluster when λ > 1/( n −1) (Supplementary Note 3 ). We also show that if m -qubit fusions are used to fuse n -qubit clusters for m ≥ 2, the optimum threshold on the fusion success probability required to generate a universal cluster ballistically must satisfy $${\\lambda }_{\\mathrm{c}}^{(n,m)} \\ge \\frac{1}{{(n - 1)(m - 1)}},\\forall n \\ge 2,m \\ge 2.$$ (4) Very little is known about linear–optical circuits for m > 2 qubit fusion [9] (e.g., projecting three qubits to one of the eight orthogonal three-qubit GHZ states), their associated success probabilities, and various failure outcomes. Thus, it remains unclear if the above bound on \\({\\lambda }_{\\mathrm{c}}^{(n,m)}\\) is tight. Reconciling prior results in a new framework Our next contribution is a reinterpretation of the constructions in refs. [15] , [18] , [19] as a standard bond-percolation problem on a logical lattice, which lets us identify (and sharpen) the 0.627 threshold found in ref. [18] as the standard bond-percolation threshold of a 3D-modified (10,3)-b lattice [25] . We first describe the mapping of a spatially regular arrangement of fusion attempts among photons in n -photon clusters—the cluster-fusion lattice G 1 —to a logical lattice G 2 , each of whose nodes correspond to an n -photon cluster. We then show that the standard bond-percolation threshold of G 2 equals the threshold on the success probability of fusion attempts in G 1 exceeding, which generates, with high probability, a long-range-connected (random) photonic lattice \\({\\mathrm{G}}_1^\\prime\\) among the unmeasured photons in G 1 , which is renormalizable for universal quantum computing. The mapping is illustrated via an example shown in Fig. 4 . We will restrict, for simplicity, to n = 3 photon-line clusters as the initial resource to explain the mapping. But the G 1 -to-G 2 mapping we describe below, holds for clusters of any size n and shape. Fig. 4 Mapping two-photon fusions with many three-photon clusters to bond percolation on a logical graph. a , b Examples of two types of two-photon fusion operations, and their post-measurement states for success and failure outcomes. The linear–optical circuits for these (blue and green) fusions are shown in Figs. 2 and 3 of ref. [18] , respectively. From a graph connectivity point of view, two-photon fusion can be interpreted as coloring the measured photons black, and if the fusion succeeds, drawing a new bond between the black photons. c We map three-photon clusters to nodes of a logical graph of color based on how many photons in the cluster are left unmeasured. d An example of a cluster-fusion lattice G 1 with both kinds of fusions, and ( e ) the corresponding logical graph G 2 . f The bond-percolation curve and threshold ( λ c ≈ 0.672) of G 2 . If each fusion attempt in ( d ) succeeds with probability λ > λ c , it results in a random graph \\({\\mathrm{G}}_2^\\prime\\) which is a giant-connected component (GCC) consisting of red nodes of G 2 . This implies the creation of a random photonic cluster state \\({\\mathrm{G}}_1^\\prime\\) , which is a GCC of G 1 comprising the unmeasured photons—a resource renormalizable for universal cluster-model photonic quantum computing Full size image G 1 represents the pattern of pairwise fusion attempts we intend to perform on a subset of photons across many three-photon clusters. To construct G 1 , we begin with (1) a (two- or higher-dimensional) spatial arrangement of three-photon line clusters, (2) identifying pairs of photons (belonging to distinct clusters), upon which fusions will be attempted, and (3) one of two types for each of those fusions: blue or green, linear optical circuits to realize, which are shown in Figs. 2 and 3 of ref. [18] , respectively. Success and failure outcomes of an example green and blue fusion are shown in Fig. 4a, b . We thus form the cluster-fusion lattice G 1 : The nodes of G 1 are photons, colored white or black: referring to whether the photons will be left unmeasured vs. will be measured (and hence destroyed) in fusion attempts. The edges of G 1 are of three kinds: black corresponding to the pre-existing entanglement between photons within a three-photon cluster, and, blue and green (dashed) edges connecting black nodes corresponding to two types of (planned) fusion attempts. Now, we construct the logical graph G 2 , each of whose nodes correspond to one three-photon cluster of G 1 . The color of a node of G 2 —white, red, blue, and black, is determined based on the number of black (measured) photons in the corresponding photonic cluster in G 1 : (i) zero, (ii) one, (iii) two, or (iv) three, respectively (see Fig. 4c) . There are two kinds of edges in G 2 : blue or green, corresponding to the two types of fusion attempts. We impose four conditions on the graphs we consider, which are not required for the above G 1 → G 2 mapping, but needed for our percolation-threshold equivalence to hold. Each condition can be stated either for G 1 or G 2 . Condition 1: Any loop in G 2 must contain at least three nonblack nodes. Condition 2: One endpoint of a blue-dashed edge in G 1 must be the degree-2 (middle) node of a three-photon cluster. We will call this node A. The other end node is a degree-1 (leaf) node of a three-photon cluster. We call this node B. Condition 3: Both endpoints of a green-dashed edge in G 1 must be degree-1 (leaf) nodes of two three-photon clusters. Condition 4: For each cluster in G 1 , all whose photons are measured in fusion attempts, there is at least one cluster with one unmeasured photon at a constant distance (number of hops) away. Translated to G 2 , this implies that every black node has a nonblack node at a constant distance from it (e.g., one bond away for the example shown in Fig. 4e ). We now describe the action of the blue and green fusion operations on photonic cluster states. In the following discussion, we will use “cluster edge” and “cluster node” to refer to edges and nodes in a photonic cluster state, to avoid any confusion with the logical graph. A “cluster edge” refers to entanglement between neighboring photons (e.g., black edges in G 1 ) and not fusion attempts (e.g., blue and green-dashed edges of G 1 ). A “cluster node” is a photon. After the fusion attempts in G 1 have been made, all the black cluster nodes in G 1 are destroyed. If a fusion attempt is successful, then additional cluster edges (depicting a newly created entanglement) appear among cluster nodes that were former neighbors of the measured (black) cluster nodes. Each fusion attempt succeeds or fails with probability λ or 1− λ , respectively. Depending upon the success–failure outcomes of all the fusion attempts, a new photonic lattice is created—a random graph state \\({\\mathrm{G}}_1^\\prime\\) —whose cluster nodes are the white cluster nodes (unmeasured photons) of G 1 . In order to describe the \\({\\mathrm{G}}_1 \\to {\\mathrm{G}}_1^\\prime\\) mapping, we need to describe the success and failure outcomes of both kinds of fusion. To do so, we need a few additional definitions: Cluster edge parity (of a pair of cluster nodes): The parity of a cluster edge is 1 if the cluster edge exists, and 0 otherwise. Neighborhood inversion (on a cluster node): A unitary action on a cluster node in a photonic cluster state, which flips the cluster edge parity between every pair of neighbors of the cluster node. So, some cluster edges can be deleted, and new cluster edges can be created. Let us now describe the action of the two fusion operations. Either type of fusion attempt destroys the two photons fused, regardless of whether the fusion succeeds. Blue fusion: Fig. 4a shows an example of a blue fusion on a pair of photons in two three-photon clusters. The blue fusion is asymmetric across the two cluster nodes it acts upon. Let us label the measured cluster nodes A and B. If the fusion succeeds, it flips the cluster edge parity between every neighboring pair of A and B. In other words, if one neighboring cluster node of A and one neighboring cluster node of B had no direct cluster edge between them prior to the fusion attempt, after successful fusion, they get a direct cluster edge between them. Also, if a neighbor of A and a neighbor of B had a direct cluster edge between them before fusion, this cluster edge is removed after successful fusion [18] . Condition 1 above prevents the latter to ever happen, i.e., we never attempt to fuse two cluster nodes whose neighbors had a direct cluster edge prior to the fusion attempt. If a blue fusion fails, we first perform a neighborhood inversion on cluster node A, i.e., we flip the cluster edge parity between every pair of neighbors of cluster node A and then remove both cluster nodes A and B from the graph. If the blue fusion shown in Fig. 4a succeeds, we obtain a four-photon cluster state. If it fails, we perform a neighborhood inversion on cluster node A and remove both measured cluster nodes, leaving the two cluster fragments with two photons in each. Condition 2 above ensures that one of the two cluster nodes for blue fusions is a leaf cluster node of a cluster. This is the B cluster node, which does not undergo neighborhood inversion after a fusion attempt failure. Green fusion: The green fusion, shown in an example in Fig. 4b , has the same behavior as the blue fusion when it succeeds. As in the case of the blue fusion, we never fuse two cluster nodes whose neighbors already have a direct cluster edge, due to Condition 1 we imposed above. In the case of failure, we simply remove the two measured cluster nodes and all their cluster edges. The green fusions are symmetric, and as stated above in Condition 3, always used between a pair of leaf cluster nodes of two clusters. We introduced above the (deterministic) lattice G 1 denoting cluster-fusion attempts, and mapped it to a (deterministic) logical lattice G 2 , whose edges denote fusion attempts. The probabilistic success–failure fusion outcomes map G 1 to a (random) photonic lattice \\({\\mathrm{G}}_1^\\prime\\) among the unmeasured photons, and map G 2 to a (random) lattice \\({\\mathrm{G}}_2^\\prime\\) , which is a standard bond-percolation subgraph of G 2 where each edge is missing (independent of the other edges) with probability 1− λ . Under a given success–failure fusion outcome pattern, we will now argue that if \\({\\mathrm{G}}_2^\\prime\\) has a giant-connected component (GCC), which will happen if λ > p c (G 2 ), the bond-percolation threshold of G 2 , then \\({\\mathrm{G}}_1^\\prime\\) —the photonic lattice involving the unmeasured nodes—will also have a GCC, which is a lattice renormalizable for universal cluster-model quantum computing. The black nodes in G 2 disappear after fusion attempts (they have no photons left in them), but help provide connections between (the unmeasured photons within) the nonblack nodes. If N is the number of nodes in G 2 , bond percolation on G 2 , i.e., λ > p c (G 2 ) ensures that \\({\\mathrm{G}}_2^\\prime\\) —the random bond-percolation instance (subgraph) of G 2 —contains a unique GCC, i.e., a single subgraph of \\({\\mathrm{G}}_2^\\prime\\) has O ( N ) nodes (black and nonblack nodes combined). However, since every black node has a nonblack node at a constant distance from it (Condition 4 above), the probability of a nonblack node being connected to the nearest black node is a constant. Therefore, the presence of O ( N ) black nodes in the GCC implies the presence of O ( N ) nonblack nodes in the GCC, and hence, there must be O ( N ) nonblack nodes in the GCC of \\({\\mathrm{G}}_2^\\prime\\) . Given any success–failure pattern of all the fusion attempts, two unmeasured photons, i.e., nodes of \\({\\mathrm{G}}_1^\\prime\\) , have a connected path in \\({\\mathrm{G}}_1^\\prime\\) if and only if the corresponding white nodes in G 1 have a connected path, via black edges and successful fusion (green or blue-dashed) edges. This can be seen through the examples shown in Fig. 4a, b . This, and the fact that nonblack nodes of G 2 map to unmeasured (white) nodes of G 1 , a GCC with O ( N ) nonblack nodes in \\({\\mathrm{G}}_2^\\prime\\) implies the presence of a GCC of O ( N ) (white) nodes in \\({\\mathrm{G}}_1^\\prime\\) . Hence, if λ > p c (G 2 ), the leftover photonic lattice \\({\\mathrm{G}}_1^\\prime\\) will have a GCC, which is a lattice renormalizable for universal cluster-model quantum computing. For the G 2 in Fig. 4e , we numerically get p c (G 2 ) ≈ 0.672. See plot in Fig. 4f . This establishes that \\({\\lambda }_{\\mathrm{c}}^{(3)} \\le 0.672\\) . It is instructive to note here that if the logical graph is composed only of black and red nodes, the post-fusion random photonic cluster state \\({\\mathrm{G}}_1^\\prime\\) is a random subgraph of G 3 : the effective lattice formed by the red nodes alone (each of which are three-photon clusters with one photon unmeasured). The nodes of G 3 are the red nodes of G 2 . In the example shown in Fig. 4e , even though G 2 is a nonplanar two-layer lattice, \\({\\mathrm{G}}_1^\\prime\\) is a random subgraph of G 3 , a planar (2D) square lattice. All the logical graph examples in this paper will have the aforesaid property, i.e., composed of black and red nodes only. Further intuition regarding the development of the above construction, and an alternative way to view the generation of \\({\\mathrm{G}}_1^\\prime\\) as a modified site–bond percolation on G 1 directly, are provided in Supplementary Note 2 . Equipped with this new formalism, we construct a 3D logical lattice G 2 —a modified (10,3)-b lattice—shown in Fig. 5 . In this case, G 3 , the lattice connecting the red nodes of G 2 if all the fusions were to succeed, is the 3D diamond lattice. The bond-percolation threshold of the modified (10,3)-b lattice, and hence the threshold on fusion success probability that would ensure \\({\\mathrm{G}}_1^\\prime\\) to be long-range connected, is p c (G 2 ) ≈ 0.627. Therefore, \\({\\lambda }_{\\mathrm{c}}^{(3)} \\le 0.627\\) , which gives a better (lower) upper bound than the example we examined above. Fig. 5 Bond percolation on the modified (10,3)-b lattice. The ( x , y )-plane layers of parallel 1D line lattices of black (degree-3) nodes are stacked along the z direction. The layers alternate between the line lattices pointing in the x and y directions. Neighboring layers are straddled by a layer of red (degree-2) nodes. Along each line lattice of black nodes, the blue bonds alternate between the + z and − z directions. The adjective “modified” we use in the name of this lattice refers to the fact that in the standard (10,3)-b lattice, the red nodes are not present, i.e., the adjacent ( x , y ) planes of parallel lattices in alternating directions are directly connected via bonds Full size image Let us see how and why G 3 for this example is in the diamond lattice. Let us assume that all the fusion attempts succeed, so that the black nodes of G 2 in Fig. 5 are gone. Let us pick any red node and follow the blue-dashed edge up to the neighboring black node. The black node has two neighboring black nodes, each of which is connected to a red node via blue-dashed edges. Once the photons within the black nodes have disappeared after the fusions, the red node we started with will inherit these two red nodes as its immediate neighbors. Similarly, if we followed the blue-dashed edge down from the original red node we picked, we would find two other red-node neighbors of that red node after the fusions are done. So, if all the fusion attempts were to succeed, we will get a regular 3D lattice of red nodes, where each node has degree 4. This is the diamond lattice. The authors of ref. [18] did exactly the above, attempting to create a percolated instance of the 3D diamond lattice as a long-range-connected photonic lattice. But our reinterpretation of their (nonstandard) percolation threshold as a standard bond-percolation threshold of the modified (10,3)-b lattice not only provides a natural way to account for failure modes of fusion attempts, but allows one to exploit existing literature on efficient Monte Carlo methods (viz., the Newman–Ziff method [26] ) to calculate sharper percolation thresholds. More importantly, this interpretation also opens the door for new constructions and improved thresholds, which we describe next. Improved thresholds Leveraging this new insight, we embark upon a progression of improved achievability thresholds for n = 3, using higher-dimensional generalizations of the modified (10,3)-b lattice. First, we consider a 4D extension of the (10,3)-b lattice (see Fig. 6 ). It consists of a doubly infinite stack of ( x , y )-plane layers—of parallel 1D line lattices of black (degree-3) nodes—stacked along the z and w directions, respectively. Of the three neighboring bonds of a black node, two (green) bonds—connecting to neighboring black nodes in the line lattice to which it belongs—are in the ( x , y ) plane, whereas one (blue) bond—connecting to a red node, which in turns connects via another blue bond to a black node in a neighboring ( x , y )-plane layer—points in either the z direction or in the w direction. Along each line lattice of black nodes, the blue bonds alternate between directions + z , + w , − z , − w , …, and so on. The graph has a period of four in each of the x , y , z , and w dimensions. One period of the lattice is depicted in Fig. 6 . The inner axes represent an ( x , y ) plane at a given value of z and w . This construction results in longer loops compared with the 3D case discussed above, while retaining the 3D graph’s coordination number (average node degree), which in turn lowers the bond-percolation threshold. We find, using a Newman–Ziff simulation performed on a 4D-modified (10,3)-b lattice of size N ~ 10 7 nodes, that $${\\lambda }_{\\mathrm{c}}^{(3)} \\le 0.611.$$ (5) Fig. 6 Schematic of the 4D extension of the (10,3)-b lattice. The inner plots with x and y axes represent projections of the lattice on the ( x , y ) plane at the z and w values shown on the outer axes Full size image Next, we consider an infinite-dimensional modified (10,3)-b lattice, which becomes locally tree-like, as shown in Fig. 7 . However, note that for any finite dimension of this family of modified (10,3)-b lattices, no matter how large the dimension is, it is not a tree, and a percolated subgraph of it will give us a renormalizable universal cluster. Similar to the 3D- and 4D-modified (10,3)-b lattices, each black node has two green bonds and one blue bond (which in turn leads to a black node via a red node and another blue bond). We denote the expected number of children of a node when approached via a green bond as E 1 and the expected number of children of a node when approached via a blue bond as E 2 . When counting the number of children of a node, we only count red nodes since they are the only nodes with unmeasured qubits. Counting children from the top of Fig. 7 , each black node is labeled as 1 or 2 depending on the bond from which it is approached. Counting children at the points labeled E 1 and E 2 yields the equations E 1 = λE 1 + λ + λ 2 E 2 and E 2 = 2 λE 1 , where λ is the bond probability. For percolation, E 1 → ∞ and solving the equations with this condition, we find that \\({\\lambda }_{\\mathrm{c}} + 2{\\lambda }_{\\mathrm{c}}^3 = 1\\) , which leads to λ c ≈ 0.5898. Therefore, $${\\lambda }_{\\mathrm{c}}^{(3)} \\le 0.5898,$$ (6) thereby bringing the best-known achievable threshold with three-photon clusters as the initial resource closer to our general lower bound applied to n = 3, \\({\\lambda }_{\\mathrm{c}}^{(3)} \\ge 0.5\\) . Fig. 7 Schematic of the ∞-D extension of the (10,3)-b lattice. This lattice when used as the logical graph with node colors as shown yields λ c ≈ 0.5898. Percolation threshold was evaluated analytically Full size image Universal cluster creation on a 2D planar lattice We found a 2D logical lattice, a modified brickwork (see Fig. 8 ), for which λ c = 0.746. This gives a bound \\({\\lambda }_c^{(3)} \\le 0.746\\) , which is looser, compared with the ones we obtained from the constructions in the previous section. However, since 0.746 < 25/32, this result shows that single-photon-boosted linear optical fusion and three-photon GHZ-state clusters are sufficient to generate a universal lattice ballistically by fusing the clusters on a two-dimensional lattice. Fusion on a 2D lattice is significantly simpler, compared with 3D (or higher-dimensional) lattices from an experimental standpoint, since 2D programmable linear optics is a fast-maturing technology. Whether or not the above is possible was posed as a challenging problem by Rudolph and colleagues in refs. [18] , [19] . Fig. 8 Bond percolation on the modified 2D brickwork lattice. This 2D lattice has a bond-percolation threshold of 0.746, which is less than the fusion success probability achievable with single-photon boosting. This shows that single-photon-boosted linear optical fusion and three-photon GHZ- state clusters are sufficient to generate a universal lattice ballistically by fusing the clusters on a two-dimensional lattice Full size image Effects of photon loss on the thresholds We prove an extension of our lower bound on \\({\\lambda }_{\\mathrm{c}}^{(n)}\\) . We use a loss model inspired by proposals to produce photonic microclusters using quantum emitters [21] , [27] , [28] , which have been recently realized experimentally [22] . In these proposals, since each node of the microcluster is added sequentially, we assume that the total transmissivity of the photon in its lifetime is of the form \\(\\eta = \\eta _0^n\\) (the loss is 1− η ) where n is the size of the microcluster and η 0 represents the transmissivity in one step. Hence, the total loss seen by the photon increases with the size of the microcluster. The import fidelity of the clusters generated due to unheralded photon losses and other forms of errors (e.g., mode mismatch, detector dark clicks) have not been considered. To obtain a lower bound, we only look at loss in the photons undergoing fusion and observe that a fusion gate only succeeds if both input photons are detected. Furthermore, for our bound, we treat loss as the removal of the corresponding node from the cluster state or a measurement in the Z basis. In general, loss of a qubit is not equivalent to a Z -basis measurement. Loss results in tracing out the qubit, but treating it as a Z -basis measurement leads to a lower value of λ c ( Z -basis measurement gives us more information than tracing out over the Z basis) and is sufficient for a lower bound on \\({\\lambda }_{\\mathrm{c}}^{(n)}\\) . Hence, if \\({\\lambda }\\eta _0^{2n} < 1/(n - 1)\\) , there cannot be a giant-connected component in the cluster, and we obtain the loss-dependent lower bound of \\({\\lambda }_{\\mathrm{c}} \\ge 1/[(n - 1)\\eta _0^{2n}] \\equiv {\\lambda }_{\\mathrm{c}}^{({\\mathrm{LB}})}\\) . Figure 9 plots \\({\\lambda }_{\\mathrm{c}}^{({\\mathrm{LB}})}\\) as a function of n for different values of η 0 and we find that there is an optimum value of n for any η 0 < 1, which gives the lowest \\({\\lambda }_{\\mathrm{c}}^{({\\mathrm{LB}})}\\) . In other words, there is an optimum size of the starting microcluster, for which the required fusion probability is minimized. Fig. 9 Effect of photon loss on thresholds. a A loss-dependent lower bound \\({\\lambda }_{\\mathrm{c}}^{({\\mathrm{LB}})}\\) on the critical fusion probability λ c as a function of the input microcluster size n for different values of η 0 ; ( b ) a loss-dependent lower bound \\(\\eta _{0{\\mathrm{c}}}^{({\\mathrm{LB}})}\\) on the critical loss parameter η 0 as a function of n for different values of fusion success probability λ Full size image Conversely, for a given fusion success probability λ , there exists a threshold η 0c , s.t., if η 0 < η 0c , the post-fusion cluster cannot be percolated. We thus have a lower bound \\(\\eta _{0{\\mathrm{c}}} \\ge \\eta _{0{\\mathrm{c}}}^{{\\mathrm{(LB)}}}\\) , where \\(\\eta _{0{\\mathrm{c}}}^{{\\mathrm{(LB)}}} = \\left[ {1/{\\lambda }(n - 1)} \\right]^{1/(2n)}\\) . In Fig. 9b , we plot \\(\\eta _{0{\\mathrm{c}}}^{({\\mathrm{LB}})}\\) for different values of n , for λ = 0.75 and λ = 1. There is an optimum value of n which gives the best lower bound on loss tolerance, e.g., for λ = 0.75, six-photon microclusters give the lowest bound of \\(\\eta _{0{\\mathrm{c}}}^{{\\mathrm{(LB)}}} = 0.8957\\) which corresponds to a loss (1 − η ) of 48.36% seen by each photon. Furthermore, we find that going from λ = 0.75, which is attainable using four single- ancilla photons and (lossless) linear optics [8] to deterministic fusion ( λ = 1), \\(\\eta _{0{\\mathrm{c}}}^{({\\mathrm{LB}})}\\) only decreases slightly, i.e., the equivalent per-photon loss threshold increases from 89.6 to 87.1%. Hence, when losses are accounted for in ballistic cluster state creation, the advantage in having a fully deterministic fusion may be relatively small. It is important to note, however, that the numbers presented here are only lower bounds on η 0c and λ 0c , and may not be tight. For example, our results show that any ballistic cluster creation process that starts with six-photon microclusters and uses destructive fusion with λ = 0.75 cannot tolerate more than 48.36% loss. However, these results do not prove that 48.36% loss is sufficient for ballistic cluster creation."},{"section":"Discussion","content":"Despite recent progress in LOQC making it resurface as a strong contender to scalable quantum computing, many questions remain, whose answers will be crucial to its eventual success. A major challenge, as expounded in ref. [23] as well, is an experimental scheme to build an on-demand source of high-fidelity small entangled cluster states, e.g., three-photon GHZ states. The other big experimental challenge is to realize high-quality quantum interference of the photon clusters using a linear optical interferometer that minimizes in-line losses and mode mismatch. A few immediate problems we leave open in this paper are as follows: (a) proving our conjecture on the achievability of \\({\\lambda }_{\\mathrm{c}}^{(n)} = 1/(n - 1) + \\varepsilon\\) ; and (b) for a given n , finding the construction that gives the smallest overhead (i.e., edge length of an optimally chosen renormalized universal lattice), while not limiting only to those linear–optical unitaries U that can be assembled with two-qubit fusion gates (i.e., allow for higher-dimensional linear–optical projective operations to potentially fuse more than two clusters at once). On the theoretical front, some of the most important open problems are as follows: (1) extending this work to include mode mismatch within the linear optic fusion circuits; (2) finding the minimum size clusters and the most resource-efficient schemes for ballistic cluster creation that are immune to unheralded losses. In this paper, we only considered losses in photons undergoing fusion. This was sufficient for obtaining bounds on \\({\\lambda }_{\\mathrm{c}}^{(n)}\\) and η 0c . However, any losses in photons that do not undergo fusion are unheralded, i.e., we do not know which photons were lost. The strategy of renormalizing by finding criss-crossing highways and choosing the intersection of these highways to be logical qubits, which gives us a perfect universal cluster state in the lossless case, fails in the presence of such unheralded loss. This is because even if we have a percolated cluster, we need to know the obtained graph state exactly in order to compute the highways. In order to correct for these unheralded photon losses (and other forms of qubit errors), an additional layer of error correction is needed [29] , [30] , [31] , [32] , [33] . Let us consider five-photon GHZ states as a starting point, which correspond to five node clusters in the “star” configuration, and place them at the nodes of a Raussendorf lattice (which has degree-4 nodes). The central photon of the star is placed on a node of the Raussendorf lattice and the four leaf nodes of the star are placed along the four neighboring bonds. We now attempt two-qubit fusion on each bond, attempting to construct a bond-percolated instance of the Raussendorf lattice following the recipe in ref. [15] . If each fusion attempt succeeds with probability λ, we obtain a subgraph of the Raussendorf lattice where each bond is present independently with probability λ. Recent work has shown that a subgraph of the Raussendorf lattice with 0.145 or a smaller fraction of missing bonds [33] (or with 0.249 or a smaller fraction of missing qubit sites [30] ) can be used for universal quantum computation while correcting for unheralded qubit errors. The amount of unheralded error on each physical qubit that can be corrected using such a thinned Rausendorf lattice depends upon how far below 0.145 the actual fraction of missing bonds is. While unheralded photon losses can be accounted for with such a construction starting with five-photon clusters as described above, it requires a fusion probability exceeding 0.85, which is greater than the best known linear optic Bell measurements boosted with single photons or squeezing. Future work should look at ways of reducing the required fusion probability to within 0.78—which is achievable with single-photon boosting—by using more efficient constructions and with smaller clusters as the starting point, and one should investigate experimentally realizable methods that improve upon the achievable fusion success probability beyond 0.78. It will also be important to extend the error-correction method to account for mode-mismatch errors in the linear optic mixing in the fusion gates, as well as detector excess noise, such as dark clicks. Code availability All the numerical data presented in this paper are the results of C simulations conducted by M.P. The code used to generate thiese data will be made available to the interested reader upon reasonable request."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"Optical qubits are a promising candidate for scalable universal quantum computing because of the relative ease of generating photons, fabricating photonic circuits [1] , [2] , [3] , and low decoherence rates. Assembling a universal photonic cluster state, e.g., by fusing small entangled clusters using linear optical interferometers into progressively larger ones, is limited by probabilistic operations because of the limitations of linear optics. For example, a linear–optical circuit for a two-qubit Bell-state measurement (BSM)—the smallest primitive to “fuse” two cluster states—can succeed with at most probability 1/2 = 0.5 [4] , [5] . However, since photons are the base resource to encode qubits anyway, it is reasonable to allow photons to be used as an ancilla resource to boost the success probability of linear–optic BSMs [6] , [7] , [8] . The maximum known BSM success probability attainable using linear optics, boosted by injecting eight ancilla single photons, is 25/32 = 0.78125 [8] . It is not known if a higher BSM success probability is possible with linear optics boosted with more single photons. Very little is understood about similar success-probability limits associated with linear–optical realizations of larger (three or more qubits) projective measurements, and those for creating small photonic entangled clusters [9] . The lack of our understanding of the maximum efficiency in creating universal photonic cluster states goes back to the mathematical structure of manipulations of multiphoton entangled states using linear optics and photon detection being complex, as evidenced by the hardness of boson sampling [10] . Linear optical quantum computing (LOQC) in its widely known circuit-model form—coined by the seminal KLM paper [11] —employs single-photon dual-rail qubits (| 0 〉 ≡ |10〉, | 1 〉 ≡ |01〉), probabilistic linear–optical gates, single-photon detection, and measurement-induced feedforward. Even though KLM’s scheme in principle can do universal quantum computing, the overheads are astronomically high. Many variants of KLM were proposed [12] , [13] , [14] that use entangled ancillas to reduce the overheads, but the hardness was pushed into creating those ancillas. Schemes for efficient generation of these ancillas from simpler resources such as single photons remain unknown. Furthermore, the need for detection-induced feedback creates additional experimental issues. This resulted in LOQC being largely disfavored in lieu of other qubit technologies, such as superconducting and trapped-ion qubits. Kieling, Rudolph, and Eisert proposed a cluster-model version of LOQC [15] , which mitigates these issues. Their scheme places d + 1 photon “star” clusters at nodes of a regular lattice G with degree- d nodes and stitches them together using two-photon fusions (BSMs) along the lattice edges (see Fig. 1 ). If the success probability of each fusion attempt λ is above the bond percolation threshold p c (G) of the underlying lattice G, the resulting cluster state will be a long-range-connected (random) subgraph of G, which can then be renormalized into a universal cluster state by identifying a subarray of nodes in the percolated lattice—the intersection points of vertical and horizontal “percolation highways” [16] , [17] —to serve as the nodes of a logical universal lattice and identifying paths connecting them through other nodes of the percolated subgraph as edges of this logical lattice (see Fig. 2a ). The inspiration behind this model is that if small d + 1 photon clusters can be created, and a lattice can be identified whose bond- percolation threshold is below a practically attainable fusion success probability, then the clusters can be linear–optically fused into a large universal cluster state in a one-shot (or “ballistic”) fashion, with no detection-induced feedforward. In order to produce conservative estimates for the fusion success probability required to produce a renormalizable cluster state, this work disregarded the “failure modes” of the fusion attempts, i.e., the leftover cluster states when fusion attempts fail. Fig. 1 Creating a percolated square lattice with fusion. Fusing five-photon star clusters into a percolated 2D square lattice is possible with two-qubit fusion operations, if each fusion attempt succeeds with the probability more than 0.5, the bond-percolation threshold of the lattice [15] Full size image Fig. 2 Bond-percolation threshold of a square lattice. a Each bond in a regular lattice of m bonds is activated with probability p . If p < p c (the subcritical regime), where p c is the bond-percolation threshold, the size of the largest connected component is O (log m ). At p = p c , a sharp transition occurs, and for 1 ≥ p > p c , the size of the largest connected component is O ( m ). In this (supercritical) regime, O ( m ) criss-crossing edge-disjoint paths (percolation highways) exist from left to right and from top to bottom [17] , using which one can identify a logical lattice whose nodes are the intersection points of the highways, and edges are paths connecting those nodes. b Size of the largest connected component, expressed as a fraction of m , as a function of p ∈ [0, 1] Full size image Significant progress in this model of LOQC came recently from Rudolph and collaborators, who by paying closer attention to the failure modes, and a clever design of 3D lattices with a low-percolation threshold, e.g., the diamond [18] and the pyrochlore lattice [19] , were able to show that ballistic creation of a universal cluster state is possible starting with a large supply of three-photon-entangled (GHZ state) clusters, with two-photon fusions that succeed with a probability ≈0.625. Since 0.625 < 25/32 = 0.78125, the best-known success probability attainable with linear optics boosted with unentangled single-photon ancillas [8] , this gave the first realizable ballistic construction of a universal cluster if three-photon GHZ clusters were available as a starting point. The results of Gimeno-Segovia et al. and Zaidi et al. [18] , [19] on ballistic (feedback-free) creation of universal clusters, in conjunction with a host of progress in experimental proposals to directly create small photonic entangled clusters using quantum emitters [20] , [21] , [22] (rather than starting with single photons and fusing them probabilistically using heralded methods into larger clusters), have resurrected the optimism in scalable optical quantum computing. However, the fundamental limits of efficiency and resource overhead associated with this philosophy—that of direct generation of a universal cluster state using small photon clusters and linear optics—remain unanswered. Refs. [18] , [19] present two (clever, yet ad hoc) examples of constructions that use three-photon clusters as the starting point. The goal of this paper is to explore this fundamental limit. We address the following general question: Given an unlimited supply of n -photon-entangled dual-rail-basis clusters, and repeated use of a linear–optical circuit that realizes two-qubit fusion with probability of success λ, but with no restrictions on how the clusters to be fused are aligned in space and time, what is the minimum value \\\\({\\\\lambda }_{\\\\mathrm{c}}^{(n)}\\\\) of each fusion’s success probability that will allow for ballistic creation of a universal cluster state? Before we proceed, it should be clear that the two examples we discussed above—one in Fig. 1 [15] and that in ref. [18] —establish the following bounds: $${\\\\lambda }_{\\\\mathrm{c}}^{(5)} \\\\le 0.5,$$ (1) and $${\\\\lambda }_{\\\\mathrm{c}}^{(3)} \\\\,\\\\, \\\\lesssim \\\\,\\\\, 0.625,$$ (2) where we use ≲ sign instead of ≤ in Eq. ( 2 ), since the 0.625 threshold was an approximate numerical estimation of the threshold using a small-sized diamond lattice [18] . We focus on Bell-measurement-based fusion operations. Although Browne and Rudolph’s original paper that coined the term “fusion” [13] also proposed an alternative linear optical fusion circuit (called fusion-I), subsequent work has focussed on Bell-measurement-based fusion (fusion-II of ref. [13] ) because of its natural loss tolerance and recent progress in boosting its success probability by injecting unentangled single-photon ancillas [8] . In this paper, we prove that starting with n -photon clusters, a fusion success probability of at least 1/( n −1) is required to create a resource state for universal quantum computation without feedforward. We map a planned set of fusion operations on photon pairs among a regular spatial array of n -photon clusters—the cluster-fusion lattice, G 1 —to an instance of bond percolation on a logical lattice G 2 . We use this map to find new constructions to produce universal photonic cluster states starting with three-photon cluster states as the initial resource, without feedforward, that require a lower fusion success probability than was previously known. 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From three-photon Greenberger-Horne-Zeilinger states to ballistic universal quantum computation.Phys. Rev. Lett.115, 020502 (2015).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.115.020502\"\n },\n {\n \"idx\": \"19\",\n \"title\": \"Zaidi, H. A., Dawson, C., Van Loock, P. & Rudolph, T. Near-deterministic creation of universal cluster states with probabilistic Bell measurements and three-qubit resource states.Phys. Rev. A - At., Mol., Opt. Phys.91, 042301 (2015).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevA.91.042301\"\n },\n {\n \"idx\": \"21\",\n \"title\": \"Rao, D. D. B., Yang, S. & Wrachtrup, J. Generation of entangled photon strings using NV centers in diamond.Phys. Rev. B - Condens. Matter Mater. Phys.92, 081301 (2015).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.92.081301\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Summary of the main results The main results in this paper are as follows: 1. We prove a general lower bound \\\\({\\\\lambda }_{\\\\mathrm{c}}^{(n)} > 1/(n - 1)\\\\) , and provide evidence in favor of our conjecture that it is tight (achievable as closely as we wish). 2. We develop a systematic graphical construction of a universal cluster, wherein \\\\({\\\\lambda }_{\\\\mathrm{c}}^{(n)}\\\\) becomes the usual bond-percolation threshold of a logical graph that we specify, while accounting for the fusion failure modes. This is a key realization that was missing in both Kieling et al.’s construction [15] as well as in refs. [18] , [19] . This not only enabled us to readily generate achievable thresholds (upper bounds on \\\\({\\\\lambda }_{\\\\mathrm{c}}^{(n)}\\\\) ) using existing literature on bond percolation, but also helped us to prove the aforesaid lower bound. We believe that this logical construction also naturally generalizes to future extensions that may employ multi-qubit fusions (projective measurements). We describe this construction systematically in the section “Reconciling prior results in a new framework”, and present several improved thresholds using this new formalism. However, we develop a fully axiomatic construction of this logical graph for the most general case if left open for future work. 3. Using our logical construction, we prove \\\\({\\\\lambda }_{\\\\mathrm{c}}^{(3)} \\\\le 0.5898\\\\) as an achievable threshold, which improves upon the best existing result in ref. [18] . As a by-product of our logical construction allowing a standard bond-percolation interpretation, we were able to reinterpret the construction of ref. [18] in our formalism and used Newman and Ziff’s efficient Monte Carlo algorithm to refine their threshold (2) to \\\\({\\\\lambda }_{\\\\mathrm{c}}^{(3)} \\\\le 0.627\\\\) . 4. We settle a heretofore open question posed by Rudolph and collaborators [18] , [19] , [23] by showing that a universal cluster can be created by linear–optically fusing three-photon GHZ clusters over a 2D (brickwork) lattice with fusion success probability 0.746, which is below 25/32, the maximum known fusion success probability achievable with a single-photon-boosted linear–optic scheme [8] , making this attractive for integrated-photonic realizations. In the next section, we will begin with a discussion of the high-level problem setup and the constraints therein, following which we will present a logical technical progression leading to our main results and the key intuitions. Detailed derivations and technical graphical constructions not central to following the main arguments in the paper are presented in the Supplementary Note 2 . Generality and constraints of our problem setup Figure 3 and its caption explains our high-level problem setup. It follows from our discussion above that we are limiting ourselves to only those multimode linear–optical unitaries U that can be pieced together using a sequence of two-qubit linear–optic BSMs. Our problem formulation, however, subsumes the special cases of the constructions in refs. [15] , [18] , [19] . Furthermore, our formulation allows for yet-undiscovered fusion circuits that may be boosted with up to n -photon entangled clusters (which may attain a success probability greater than the currently best known, 25/32). It also admits generalizations to multi-qubit fusion, i.e., m -qubit GHZ-state projections, the theory behind optimal linear–optical realizations of which it remains not well understood. Fig. 3 Ballistic photonic cluster state generation for quantum computing. A steady stream of entangled clusters of n -photons or less ( n = 3 shown) is incident on a linear–optical interferometer (i.e., a multimode unitary transformation U ), which produces an entangled cluster of photons at its output. If the percolation condition is met, the output can be renormalized into a fully connected logical cluster for universal cluster-model quantum computing. The detector outputs are not used to herald whether or not the undetected outputs are in a universal cluster state. They always are. The detector outputs carry complete information on the location of missing edges in the random graph, and thus how to renormalize it Full size image The detectors shown in Fig. 3 are those corresponding to the linear–optical fusion gates pushed (deferred) all the way to the right with no loss of generality. Similarly, any photons or photon clusters used to boost any fusion gate are pushed to the left of U . This way, one can envision the cluster preparation as a multimode linear optical circuit presented with individual photons and photon clusters as input, and producing (deterministically) a renormalizable universal cluster state as its output. The individual detector outputs are random, but if we design U in a way that the percolation condition is met, the undetected outputs of U would, with high probability, carry a random instance of a universal long-range-connected (percolated) cluster and the vector of detector outputs would simply tell us what cluster state got created on the undetected outputs, and thus give us the necessary data to compute criss-crossing edge-disjoint paths through the percolated lattice to renormalize it into a universal logical cluster state [24] . General lower bounds In this section, we present an intuitive explanation of our lower bound on \\\\({\\\\lambda }_{\\\\mathrm{c}}^{(n)}\\\\) . A detailed formal proof is provided in Supplementary Note 1 . We have the following general lower bound: $${\\\\lambda }_{\\\\mathrm{c}}^{(n)} \\\\ge \\\\frac{1}{{n - 1}},\\\\forall n \\\\ge 2.$$ (3) Proof sketch—starting with n -photon clusters, and any sequence of fusion attempts, the resulting instance of the logical graph is a bond-percolation instance, with bond success probability λ , on some graph of maximum degree n . Of all infinite graphs of maximum degree n , the minimum bond-percolation threshold is that of the degree- n Bethe lattice, and equals 1/( n −1). This implies that if λ < 1/( n −1), the percolation condition cannot be met no matter what. Hence, \\\\({\\\\lambda }_{\\\\mathrm{c}}^{(n)} \\\\ge 1/(n - 1)\\\\) . Since the Bethe lattice is a tree and since trees are not a universal resource, this does not prove 1/( n −1) to be an achievable threshold for universal cluster creation (see Supplementary Note 3 for more details). However, we conjecture (and provide evidence in its favor) that λ = [1/( n −1)] + ε with any ε > 0 that may suffice to create a universal cluster state. This conjecture is supported by the fact that starting with N copies of photon clusters of size n and probabilistic operations that succeed with any probability greater than 1/( n −1), it is possible to obtain an O ( N )-sized tree graph (detailed in Supplementary Note 3 ). We also present a random graph construction, which provides intuition for why probabilistic operations on photon clusters of size n may provide enough connectivity to obtain a large connected cluster when λ > 1/( n −1) (Supplementary Note 3 ). We also show that if m -qubit fusions are used to fuse n -qubit clusters for m ≥ 2, the optimum threshold on the fusion success probability required to generate a universal cluster ballistically must satisfy $${\\\\lambda }_{\\\\mathrm{c}}^{(n,m)} \\\\ge \\\\frac{1}{{(n - 1)(m - 1)}},\\\\forall n \\\\ge 2,m \\\\ge 2.$$ (4) Very little is known about linear–optical circuits for m > 2 qubit fusion [9] (e.g., projecting three qubits to one of the eight orthogonal three-qubit GHZ states), their associated success probabilities, and various failure outcomes. Thus, it remains unclear if the above bound on \\\\({\\\\lambda }_{\\\\mathrm{c}}^{(n,m)}\\\\) is tight. Reconciling prior results in a new framework Our next contribution is a reinterpretation of the constructions in refs. [15] , [18] , [19] as a standard bond-percolation problem on a logical lattice, which lets us identify (and sharpen) the 0.627 threshold found in ref. [18] as the standard bond-percolation threshold of a 3D-modified (10,3)-b lattice [25] . We first describe the mapping of a spatially regular arrangement of fusion attempts among photons in n -photon clusters—the cluster-fusion lattice G 1 —to a logical lattice G 2 , each of whose nodes correspond to an n -photon cluster. We then show that the standard bond-percolation threshold of G 2 equals the threshold on the success probability of fusion attempts in G 1 exceeding, which generates, with high probability, a long-range-connected (random) photonic lattice \\\\({\\\\mathrm{G}}_1^\\\\prime\\\\) among the unmeasured photons in G 1 , which is renormalizable for universal quantum computing. The mapping is illustrated via an example shown in Fig. 4 . We will restrict, for simplicity, to n = 3 photon-line clusters as the initial resource to explain the mapping. But the G 1 -to-G 2 mapping we describe below, holds for clusters of any size n and shape. Fig. 4 Mapping two-photon fusions with many three-photon clusters to bond percolation on a logical graph. a , b Examples of two types of two-photon fusion operations, and their post-measurement states for success and failure outcomes. The linear–optical circuits for these (blue and green) fusions are shown in Figs. 2 and 3 of ref. [18] , respectively. From a graph connectivity point of view, two-photon fusion can be interpreted as coloring the measured photons black, and if the fusion succeeds, drawing a new bond between the black photons. c We map three-photon clusters to nodes of a logical graph of color based on how many photons in the cluster are left unmeasured. d An example of a cluster-fusion lattice G 1 with both kinds of fusions, and ( e ) the corresponding logical graph G 2 . f The bond-percolation curve and threshold ( λ c ≈ 0.672) of G 2 . If each fusion attempt in ( d ) succeeds with probability λ > λ c , it results in a random graph \\\\({\\\\mathrm{G}}_2^\\\\prime\\\\) which is a giant-connected component (GCC) consisting of red nodes of G 2 . This implies the creation of a random photonic cluster state \\\\({\\\\mathrm{G}}_1^\\\\prime\\\\) , which is a GCC of G 1 comprising the unmeasured photons—a resource renormalizable for universal cluster-model photonic quantum computing Full size image G 1 represents the pattern of pairwise fusion attempts we intend to perform on a subset of photons across many three-photon clusters. To construct G 1 , we begin with (1) a (two- or higher-dimensional) spatial arrangement of three-photon line clusters, (2) identifying pairs of photons (belonging to distinct clusters), upon which fusions will be attempted, and (3) one of two types for each of those fusions: blue or green, linear optical circuits to realize, which are shown in Figs. 2 and 3 of ref. [18] , respectively. Success and failure outcomes of an example green and blue fusion are shown in Fig. 4a, b . We thus form the cluster-fusion lattice G 1 : The nodes of G 1 are photons, colored white or black: referring to whether the photons will be left unmeasured vs. will be measured (and hence destroyed) in fusion attempts. The edges of G 1 are of three kinds: black corresponding to the pre-existing entanglement between photons within a three-photon cluster, and, blue and green (dashed) edges connecting black nodes corresponding to two types of (planned) fusion attempts. Now, we construct the logical graph G 2 , each of whose nodes correspond to one three-photon cluster of G 1 . The color of a node of G 2 —white, red, blue, and black, is determined based on the number of black (measured) photons in the corresponding photonic cluster in G 1 : (i) zero, (ii) one, (iii) two, or (iv) three, respectively (see Fig. 4c) . There are two kinds of edges in G 2 : blue or green, corresponding to the two types of fusion attempts. We impose four conditions on the graphs we consider, which are not required for the above G 1 → G 2 mapping, but needed for our percolation-threshold equivalence to hold. Each condition can be stated either for G 1 or G 2 . Condition 1: Any loop in G 2 must contain at least three nonblack nodes. Condition 2: One endpoint of a blue-dashed edge in G 1 must be the degree-2 (middle) node of a three-photon cluster. We will call this node A. The other end node is a degree-1 (leaf) node of a three-photon cluster. We call this node B. Condition 3: Both endpoints of a green-dashed edge in G 1 must be degree-1 (leaf) nodes of two three-photon clusters. Condition 4: For each cluster in G 1 , all whose photons are measured in fusion attempts, there is at least one cluster with one unmeasured photon at a constant distance (number of hops) away. Translated to G 2 , this implies that every black node has a nonblack node at a constant distance from it (e.g., one bond away for the example shown in Fig. 4e ). We now describe the action of the blue and green fusion operations on photonic cluster states. In the following discussion, we will use “cluster edge” and “cluster node” to refer to edges and nodes in a photonic cluster state, to avoid any confusion with the logical graph. A “cluster edge” refers to entanglement between neighboring photons (e.g., black edges in G 1 ) and not fusion attempts (e.g., blue and green-dashed edges of G 1 ). A “cluster node” is a photon. After the fusion attempts in G 1 have been made, all the black cluster nodes in G 1 are destroyed. If a fusion attempt is successful, then additional cluster edges (depicting a newly created entanglement) appear among cluster nodes that were former neighbors of the measured (black) cluster nodes. Each fusion attempt succeeds or fails with probability λ or 1− λ , respectively. Depending upon the success–failure outcomes of all the fusion attempts, a new photonic lattice is created—a random graph state \\\\({\\\\mathrm{G}}_1^\\\\prime\\\\) —whose cluster nodes are the white cluster nodes (unmeasured photons) of G 1 . In order to describe the \\\\({\\\\mathrm{G}}_1 \\\\to {\\\\mathrm{G}}_1^\\\\prime\\\\) mapping, we need to describe the success and failure outcomes of both kinds of fusion. To do so, we need a few additional definitions: Cluster edge parity (of a pair of cluster nodes): The parity of a cluster edge is 1 if the cluster edge exists, and 0 otherwise. Neighborhood inversion (on a cluster node): A unitary action on a cluster node in a photonic cluster state, which flips the cluster edge parity between every pair of neighbors of the cluster node. So, some cluster edges can be deleted, and new cluster edges can be created. Let us now describe the action of the two fusion operations. Either type of fusion attempt destroys the two photons fused, regardless of whether the fusion succeeds. Blue fusion: Fig. 4a shows an example of a blue fusion on a pair of photons in two three-photon clusters. The blue fusion is asymmetric across the two cluster nodes it acts upon. Let us label the measured cluster nodes A and B. If the fusion succeeds, it flips the cluster edge parity between every neighboring pair of A and B. In other words, if one neighboring cluster node of A and one neighboring cluster node of B had no direct cluster edge between them prior to the fusion attempt, after successful fusion, they get a direct cluster edge between them. Also, if a neighbor of A and a neighbor of B had a direct cluster edge between them before fusion, this cluster edge is removed after successful fusion [18] . Condition 1 above prevents the latter to ever happen, i.e., we never attempt to fuse two cluster nodes whose neighbors had a direct cluster edge prior to the fusion attempt. If a blue fusion fails, we first perform a neighborhood inversion on cluster node A, i.e., we flip the cluster edge parity between every pair of neighbors of cluster node A and then remove both cluster nodes A and B from the graph. If the blue fusion shown in Fig. 4a succeeds, we obtain a four-photon cluster state. If it fails, we perform a neighborhood inversion on cluster node A and remove both measured cluster nodes, leaving the two cluster fragments with two photons in each. Condition 2 above ensures that one of the two cluster nodes for blue fusions is a leaf cluster node of a cluster. This is the B cluster node, which does not undergo neighborhood inversion after a fusion attempt failure. Green fusion: The green fusion, shown in an example in Fig. 4b , has the same behavior as the blue fusion when it succeeds. As in the case of the blue fusion, we never fuse two cluster nodes whose neighbors already have a direct cluster edge, due to Condition 1 we imposed above. In the case of failure, we simply remove the two measured cluster nodes and all their cluster edges. The green fusions are symmetric, and as stated above in Condition 3, always used between a pair of leaf cluster nodes of two clusters. We introduced above the (deterministic) lattice G 1 denoting cluster-fusion attempts, and mapped it to a (deterministic) logical lattice G 2 , whose edges denote fusion attempts. The probabilistic success–failure fusion outcomes map G 1 to a (random) photonic lattice \\\\({\\\\mathrm{G}}_1^\\\\prime\\\\) among the unmeasured photons, and map G 2 to a (random) lattice \\\\({\\\\mathrm{G}}_2^\\\\prime\\\\) , which is a standard bond-percolation subgraph of G 2 where each edge is missing (independent of the other edges) with probability 1− λ . Under a given success–failure fusion outcome pattern, we will now argue that if \\\\({\\\\mathrm{G}}_2^\\\\prime\\\\) has a giant-connected component (GCC), which will happen if λ > p c (G 2 ), the bond-percolation threshold of G 2 , then \\\\({\\\\mathrm{G}}_1^\\\\prime\\\\) —the photonic lattice involving the unmeasured nodes—will also have a GCC, which is a lattice renormalizable for universal cluster-model quantum computing. The black nodes in G 2 disappear after fusion attempts (they have no photons left in them), but help provide connections between (the unmeasured photons within) the nonblack nodes. If N is the number of nodes in G 2 , bond percolation on G 2 , i.e., λ > p c (G 2 ) ensures that \\\\({\\\\mathrm{G}}_2^\\\\prime\\\\) —the random bond-percolation instance (subgraph) of G 2 —contains a unique GCC, i.e., a single subgraph of \\\\({\\\\mathrm{G}}_2^\\\\prime\\\\) has O ( N ) nodes (black and nonblack nodes combined). However, since every black node has a nonblack node at a constant distance from it (Condition 4 above), the probability of a nonblack node being connected to the nearest black node is a constant. Therefore, the presence of O ( N ) black nodes in the GCC implies the presence of O ( N ) nonblack nodes in the GCC, and hence, there must be O ( N ) nonblack nodes in the GCC of \\\\({\\\\mathrm{G}}_2^\\\\prime\\\\) . Given any success–failure pattern of all the fusion attempts, two unmeasured photons, i.e., nodes of \\\\({\\\\mathrm{G}}_1^\\\\prime\\\\) , have a connected path in \\\\({\\\\mathrm{G}}_1^\\\\prime\\\\) if and only if the corresponding white nodes in G 1 have a connected path, via black edges and successful fusion (green or blue-dashed) edges. This can be seen through the examples shown in Fig. 4a, b . This, and the fact that nonblack nodes of G 2 map to unmeasured (white) nodes of G 1 , a GCC with O ( N ) nonblack nodes in \\\\({\\\\mathrm{G}}_2^\\\\prime\\\\) implies the presence of a GCC of O ( N ) (white) nodes in \\\\({\\\\mathrm{G}}_1^\\\\prime\\\\) . Hence, if λ > p c (G 2 ), the leftover photonic lattice \\\\({\\\\mathrm{G}}_1^\\\\prime\\\\) will have a GCC, which is a lattice renormalizable for universal cluster-model quantum computing. For the G 2 in Fig. 4e , we numerically get p c (G 2 ) ≈ 0.672. See plot in Fig. 4f . This establishes that \\\\({\\\\lambda }_{\\\\mathrm{c}}^{(3)} \\\\le 0.672\\\\) . It is instructive to note here that if the logical graph is composed only of black and red nodes, the post-fusion random photonic cluster state \\\\({\\\\mathrm{G}}_1^\\\\prime\\\\) is a random subgraph of G 3 : the effective lattice formed by the red nodes alone (each of which are three-photon clusters with one photon unmeasured). The nodes of G 3 are the red nodes of G 2 . In the example shown in Fig. 4e , even though G 2 is a nonplanar two-layer lattice, \\\\({\\\\mathrm{G}}_1^\\\\prime\\\\) is a random subgraph of G 3 , a planar (2D) square lattice. All the logical graph examples in this paper will have the aforesaid property, i.e., composed of black and red nodes only. Further intuition regarding the development of the above construction, and an alternative way to view the generation of \\\\({\\\\mathrm{G}}_1^\\\\prime\\\\) as a modified site–bond percolation on G 1 directly, are provided in Supplementary Note 2 . Equipped with this new formalism, we construct a 3D logical lattice G 2 —a modified (10,3)-b lattice—shown in Fig. 5 . In this case, G 3 , the lattice connecting the red nodes of G 2 if all the fusions were to succeed, is the 3D diamond lattice. The bond-percolation threshold of the modified (10,3)-b lattice, and hence the threshold on fusion success probability that would ensure \\\\({\\\\mathrm{G}}_1^\\\\prime\\\\) to be long-range connected, is p c (G 2 ) ≈ 0.627. Therefore, \\\\({\\\\lambda }_{\\\\mathrm{c}}^{(3)} \\\\le 0.627\\\\) , which gives a better (lower) upper bound than the example we examined above. Fig. 5 Bond percolation on the modified (10,3)-b lattice. The ( x , y )-plane layers of parallel 1D line lattices of black (degree-3) nodes are stacked along the z direction. The layers alternate between the line lattices pointing in the x and y directions. Neighboring layers are straddled by a layer of red (degree-2) nodes. Along each line lattice of black nodes, the blue bonds alternate between the + z and − z directions. The adjective “modified” we use in the name of this lattice refers to the fact that in the standard (10,3)-b lattice, the red nodes are not present, i.e., the adjacent ( x , y ) planes of parallel lattices in alternating directions are directly connected via bonds Full size image Let us see how and why G 3 for this example is in the diamond lattice. Let us assume that all the fusion attempts succeed, so that the black nodes of G 2 in Fig. 5 are gone. Let us pick any red node and follow the blue-dashed edge up to the neighboring black node. The black node has two neighboring black nodes, each of which is connected to a red node via blue-dashed edges. Once the photons within the black nodes have disappeared after the fusions, the red node we started with will inherit these two red nodes as its immediate neighbors. Similarly, if we followed the blue-dashed edge down from the original red node we picked, we would find two other red-node neighbors of that red node after the fusions are done. So, if all the fusion attempts were to succeed, we will get a regular 3D lattice of red nodes, where each node has degree 4. This is the diamond lattice. The authors of ref. [18] did exactly the above, attempting to create a percolated instance of the 3D diamond lattice as a long-range-connected photonic lattice. But our reinterpretation of their (nonstandard) percolation threshold as a standard bond-percolation threshold of the modified (10,3)-b lattice not only provides a natural way to account for failure modes of fusion attempts, but allows one to exploit existing literature on efficient Monte Carlo methods (viz., the Newman–Ziff method [26] ) to calculate sharper percolation thresholds. More importantly, this interpretation also opens the door for new constructions and improved thresholds, which we describe next. Improved thresholds Leveraging this new insight, we embark upon a progression of improved achievability thresholds for n = 3, using higher-dimensional generalizations of the modified (10,3)-b lattice. First, we consider a 4D extension of the (10,3)-b lattice (see Fig. 6 ). It consists of a doubly infinite stack of ( x , y )-plane layers—of parallel 1D line lattices of black (degree-3) nodes—stacked along the z and w directions, respectively. Of the three neighboring bonds of a black node, two (green) bonds—connecting to neighboring black nodes in the line lattice to which it belongs—are in the ( x , y ) plane, whereas one (blue) bond—connecting to a red node, which in turns connects via another blue bond to a black node in a neighboring ( x , y )-plane layer—points in either the z direction or in the w direction. Along each line lattice of black nodes, the blue bonds alternate between directions + z , + w , − z , − w , …, and so on. The graph has a period of four in each of the x , y , z , and w dimensions. One period of the lattice is depicted in Fig. 6 . The inner axes represent an ( x , y ) plane at a given value of z and w . This construction results in longer loops compared with the 3D case discussed above, while retaining the 3D graph’s coordination number (average node degree), which in turn lowers the bond-percolation threshold. We find, using a Newman–Ziff simulation performed on a 4D-modified (10,3)-b lattice of size N ~ 10 7 nodes, that $${\\\\lambda }_{\\\\mathrm{c}}^{(3)} \\\\le 0.611.$$ (5) Fig. 6 Schematic of the 4D extension of the (10,3)-b lattice. The inner plots with x and y axes represent projections of the lattice on the ( x , y ) plane at the z and w values shown on the outer axes Full size image Next, we consider an infinite-dimensional modified (10,3)-b lattice, which becomes locally tree-like, as shown in Fig. 7 . However, note that for any finite dimension of this family of modified (10,3)-b lattices, no matter how large the dimension is, it is not a tree, and a percolated subgraph of it will give us a renormalizable universal cluster. Similar to the 3D- and 4D-modified (10,3)-b lattices, each black node has two green bonds and one blue bond (which in turn leads to a black node via a red node and another blue bond). We denote the expected number of children of a node when approached via a green bond as E 1 and the expected number of children of a node when approached via a blue bond as E 2 . When counting the number of children of a node, we only count red nodes since they are the only nodes with unmeasured qubits. Counting children from the top of Fig. 7 , each black node is labeled as 1 or 2 depending on the bond from which it is approached. Counting children at the points labeled E 1 and E 2 yields the equations E 1 = λE 1 + λ + λ 2 E 2 and E 2 = 2 λE 1 , where λ is the bond probability. For percolation, E 1 → ∞ and solving the equations with this condition, we find that \\\\({\\\\lambda }_{\\\\mathrm{c}} + 2{\\\\lambda }_{\\\\mathrm{c}}^3 = 1\\\\) , which leads to λ c ≈ 0.5898. Therefore, $${\\\\lambda }_{\\\\mathrm{c}}^{(3)} \\\\le 0.5898,$$ (6) thereby bringing the best-known achievable threshold with three-photon clusters as the initial resource closer to our general lower bound applied to n = 3, \\\\({\\\\lambda }_{\\\\mathrm{c}}^{(3)} \\\\ge 0.5\\\\) . Fig. 7 Schematic of the ∞-D extension of the (10,3)-b lattice. This lattice when used as the logical graph with node colors as shown yields λ c ≈ 0.5898. Percolation threshold was evaluated analytically Full size image Universal cluster creation on a 2D planar lattice We found a 2D logical lattice, a modified brickwork (see Fig. 8 ), for which λ c = 0.746. This gives a bound \\\\({\\\\lambda }_c^{(3)} \\\\le 0.746\\\\) , which is looser, compared with the ones we obtained from the constructions in the previous section. However, since 0.746 < 25/32, this result shows that single-photon-boosted linear optical fusion and three-photon GHZ-state clusters are sufficient to generate a universal lattice ballistically by fusing the clusters on a two-dimensional lattice. Fusion on a 2D lattice is significantly simpler, compared with 3D (or higher-dimensional) lattices from an experimental standpoint, since 2D programmable linear optics is a fast-maturing technology. Whether or not the above is possible was posed as a challenging problem by Rudolph and colleagues in refs. [18] , [19] . Fig. 8 Bond percolation on the modified 2D brickwork lattice. This 2D lattice has a bond-percolation threshold of 0.746, which is less than the fusion success probability achievable with single-photon boosting. This shows that single-photon-boosted linear optical fusion and three-photon GHZ- state clusters are sufficient to generate a universal lattice ballistically by fusing the clusters on a two-dimensional lattice Full size image Effects of photon loss on the thresholds We prove an extension of our lower bound on \\\\({\\\\lambda }_{\\\\mathrm{c}}^{(n)}\\\\) . We use a loss model inspired by proposals to produce photonic microclusters using quantum emitters [21] , [27] , [28] , which have been recently realized experimentally [22] . In these proposals, since each node of the microcluster is added sequentially, we assume that the total transmissivity of the photon in its lifetime is of the form \\\\(\\\\eta = \\\\eta _0^n\\\\) (the loss is 1− η ) where n is the size of the microcluster and η 0 represents the transmissivity in one step. Hence, the total loss seen by the photon increases with the size of the microcluster. The import fidelity of the clusters generated due to unheralded photon losses and other forms of errors (e.g., mode mismatch, detector dark clicks) have not been considered. To obtain a lower bound, we only look at loss in the photons undergoing fusion and observe that a fusion gate only succeeds if both input photons are detected. Furthermore, for our bound, we treat loss as the removal of the corresponding node from the cluster state or a measurement in the Z basis. In general, loss of a qubit is not equivalent to a Z -basis measurement. Loss results in tracing out the qubit, but treating it as a Z -basis measurement leads to a lower value of λ c ( Z -basis measurement gives us more information than tracing out over the Z basis) and is sufficient for a lower bound on \\\\({\\\\lambda }_{\\\\mathrm{c}}^{(n)}\\\\) . Hence, if \\\\({\\\\lambda }\\\\eta _0^{2n} < 1/(n - 1)\\\\) , there cannot be a giant-connected component in the cluster, and we obtain the loss-dependent lower bound of \\\\({\\\\lambda }_{\\\\mathrm{c}} \\\\ge 1/[(n - 1)\\\\eta _0^{2n}] \\\\equiv {\\\\lambda }_{\\\\mathrm{c}}^{({\\\\mathrm{LB}})}\\\\) . Figure 9 plots \\\\({\\\\lambda }_{\\\\mathrm{c}}^{({\\\\mathrm{LB}})}\\\\) as a function of n for different values of η 0 and we find that there is an optimum value of n for any η 0 < 1, which gives the lowest \\\\({\\\\lambda }_{\\\\mathrm{c}}^{({\\\\mathrm{LB}})}\\\\) . In other words, there is an optimum size of the starting microcluster, for which the required fusion probability is minimized. Fig. 9 Effect of photon loss on thresholds. a A loss-dependent lower bound \\\\({\\\\lambda }_{\\\\mathrm{c}}^{({\\\\mathrm{LB}})}\\\\) on the critical fusion probability λ c as a function of the input microcluster size n for different values of η 0 ; ( b ) a loss-dependent lower bound \\\\(\\\\eta _{0{\\\\mathrm{c}}}^{({\\\\mathrm{LB}})}\\\\) on the critical loss parameter η 0 as a function of n for different values of fusion success probability λ Full size image Conversely, for a given fusion success probability λ , there exists a threshold η 0c , s.t., if η 0 < η 0c , the post-fusion cluster cannot be percolated. We thus have a lower bound \\\\(\\\\eta _{0{\\\\mathrm{c}}} \\\\ge \\\\eta _{0{\\\\mathrm{c}}}^{{\\\\mathrm{(LB)}}}\\\\) , where \\\\(\\\\eta _{0{\\\\mathrm{c}}}^{{\\\\mathrm{(LB)}}} = \\\\left[ {1/{\\\\lambda }(n - 1)} \\\\right]^{1/(2n)}\\\\) . In Fig. 9b , we plot \\\\(\\\\eta _{0{\\\\mathrm{c}}}^{({\\\\mathrm{LB}})}\\\\) for different values of n , for λ = 0.75 and λ = 1. There is an optimum value of n which gives the best lower bound on loss tolerance, e.g., for λ = 0.75, six-photon microclusters give the lowest bound of \\\\(\\\\eta _{0{\\\\mathrm{c}}}^{{\\\\mathrm{(LB)}}} = 0.8957\\\\) which corresponds to a loss (1 − η ) of 48.36% seen by each photon. Furthermore, we find that going from λ = 0.75, which is attainable using four single- ancilla photons and (lossless) linear optics [8] to deterministic fusion ( λ = 1), \\\\(\\\\eta _{0{\\\\mathrm{c}}}^{({\\\\mathrm{LB}})}\\\\) only decreases slightly, i.e., the equivalent per-photon loss threshold increases from 89.6 to 87.1%. Hence, when losses are accounted for in ballistic cluster state creation, the advantage in having a fully deterministic fusion may be relatively small. It is important to note, however, that the numbers presented here are only lower bounds on η 0c and λ 0c , and may not be tight. For example, our results show that any ballistic cluster creation process that starts with six-photon microclusters and uses destructive fusion with λ = 0.75 cannot tolerate more than 48.36% loss. However, these results do not prove that 48.36% loss is sufficient for ballistic cluster creation.\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"Despite recent progress in LOQC making it resurface as a strong contender to scalable quantum computing, many questions remain, whose answers will be crucial to its eventual success. A major challenge, as expounded in ref. [23] as well, is an experimental scheme to build an on-demand source of high-fidelity small entangled cluster states, e.g., three-photon GHZ states. The other big experimental challenge is to realize high-quality quantum interference of the photon clusters using a linear optical interferometer that minimizes in-line losses and mode mismatch. A few immediate problems we leave open in this paper are as follows: (a) proving our conjecture on the achievability of \\\\({\\\\lambda }_{\\\\mathrm{c}}^{(n)} = 1/(n - 1) + \\\\varepsilon\\\\) ; and (b) for a given n , finding the construction that gives the smallest overhead (i.e., edge length of an optimally chosen renormalized universal lattice), while not limiting only to those linear–optical unitaries U that can be assembled with two-qubit fusion gates (i.e., allow for higher-dimensional linear–optical projective operations to potentially fuse more than two clusters at once). On the theoretical front, some of the most important open problems are as follows: (1) extending this work to include mode mismatch within the linear optic fusion circuits; (2) finding the minimum size clusters and the most resource-efficient schemes for ballistic cluster creation that are immune to unheralded losses. In this paper, we only considered losses in photons undergoing fusion. This was sufficient for obtaining bounds on \\\\({\\\\lambda }_{\\\\mathrm{c}}^{(n)}\\\\) and η 0c . However, any losses in photons that do not undergo fusion are unheralded, i.e., we do not know which photons were lost. The strategy of renormalizing by finding criss-crossing highways and choosing the intersection of these highways to be logical qubits, which gives us a perfect universal cluster state in the lossless case, fails in the presence of such unheralded loss. This is because even if we have a percolated cluster, we need to know the obtained graph state exactly in order to compute the highways. In order to correct for these unheralded photon losses (and other forms of qubit errors), an additional layer of error correction is needed [29] , [30] , [31] , [32] , [33] . Let us consider five-photon GHZ states as a starting point, which correspond to five node clusters in the “star” configuration, and place them at the nodes of a Raussendorf lattice (which has degree-4 nodes). The central photon of the star is placed on a node of the Raussendorf lattice and the four leaf nodes of the star are placed along the four neighboring bonds. We now attempt two-qubit fusion on each bond, attempting to construct a bond-percolated instance of the Raussendorf lattice following the recipe in ref. [15] . If each fusion attempt succeeds with probability λ, we obtain a subgraph of the Raussendorf lattice where each bond is present independently with probability λ. Recent work has shown that a subgraph of the Raussendorf lattice with 0.145 or a smaller fraction of missing bonds [33] (or with 0.249 or a smaller fraction of missing qubit sites [30] ) can be used for universal quantum computation while correcting for unheralded qubit errors. The amount of unheralded error on each physical qubit that can be corrected using such a thinned Rausendorf lattice depends upon how far below 0.145 the actual fraction of missing bonds is. While unheralded photon losses can be accounted for with such a construction starting with five-photon clusters as described above, it requires a fusion probability exceeding 0.85, which is greater than the best known linear optic Bell measurements boosted with single photons or squeezing. Future work should look at ways of reducing the required fusion probability to within 0.78—which is achievable with single-photon boosting—by using more efficient constructions and with smaller clusters as the starting point, and one should investigate experimentally realizable methods that improve upon the achievable fusion success probability beyond 0.78. It will also be important to extend the error-correction method to account for mode-mismatch errors in the linear optic mixing in the fusion gates, as well as detector excess noise, such as dark clicks. Code availability All the numerical data presented in this paper are the results of C simulations conducted by M.P. The code used to generate thiese data will be made available to the interested reader upon reasonable request.\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"The core research idea is to determine the minimum success probability threshold required for ballistic creation of a universal photonic cluster state through linear-optical fusion of n-photon-entangled clusters, without the need for measurement-induced feedforward, and to demonstrate that this threshold can be achieved using three-photon clusters and linear optics with single-photon ancillas, enabling scalable and experimentally feasible optical quantum computing."},"entities":{"kind":"list like","value":["minimum success probability threshold","ballistic creation","universal photonic cluster state","linear-optical fusion","n-photon-entangled clusters","measurement-induced feedforward","three-photon clusters","linear optics","single-photon ancillas","scalable optical quantum computing","experimentally feasible optical quantum computing"],"string":"[\n \"minimum success probability threshold\",\n \"ballistic creation\",\n \"universal photonic cluster state\",\n \"linear-optical fusion\",\n \"n-photon-entangled clusters\",\n \"measurement-induced feedforward\",\n \"three-photon clusters\",\n \"linear optics\",\n \"single-photon ancillas\",\n \"scalable optical quantum computing\",\n \"experimentally feasible optical quantum computing\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1.","title":"Reck, M. & Zeilinger, A. 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Loss tolerance in one-way quantum computation via counterfactual error correction.Phys. Rev. Lett.97, 120501 (2006).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.97.120501\"\n },\n {\n \"idx\": \"30.\",\n \"title\": \"Barrett, S. D. & Stace, T. M. Fault tolerant quantum computation with very high threshold for loss errors.Phys. Rev. Lett.105, 200502 (2010).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.105.200502\"\n },\n {\n \"idx\": \"32.\",\n \"title\": \"Herr, D., Paler, A., Devitt, S. J. & Nori, F. A local and scalable lattice renormalization method for ballistic quantum computation.npj Quantum Inf.4, 27 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41534-018-0076-0\"\n },\n {\n \"idx\": \"33.\",\n \"title\": \"Auger, J. M., Anwar, H., Gimeno-Segovia, M., Stace, T. M. & Browne, D. E. Fault-tolerant quantum computation with nondeterministic entangling gates.Phys. Rev. A.97, 030301 (2018).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevA.97.030301\"\n }\n]","isConversational":false}}},{"rowIdx":192,"cells":{"unique_id":{"kind":"string","value":"s41467-024-55205-x"},"subfield":{"kind":"string","value":"Materials science"},"title":{"kind":"string","value":"Distinguishing local isomorphism classes in quasicrystals by high-order harmonic spectroscopy"},"abstract":{"kind":"string","value":"Electron diffraction spectroscopy is a fundamental tool for investigating quasicrystal structures, which unveils the quasiperiodic long-range order. Nevertheless, it falls short in effectively distinguishing separate local isomorphism classes. This is a long outstanding problem. Here, we study the high-order harmonic generation in two-dimensional generalized Penrose quasicrystals to optically resolve different local isomorphism classes. The results reveal that: (i) harmonic spectra from different parts of a quasicrystal are identical, even though their atomic arrangements vary significantly. (ii) The harmonic yields of diverse local isomorphism classes exhibit variations, providing a way to distinguish local isomorphism classes. (iii) The rotational symmetry of harmonic yield can serve as a characteristic of quasicrystal harmonics and is consistent with the orientation order. Our results not only pave the way for confirming the experimental reproducibility of quasicrystal harmonics and identifying quasicrystal local isomorphism classes, but also shed light on comprehending electron dynamics influenced by the vertex environments."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"Choices of the unit cell and packing methods in crystals exhibit a strict one-to-one correspondence to form a single undefeated periodic structure. Very different from crystals, two or more unit cells can be assembled into an infinite number of various quasicrystals, which can be further subdivided into local isomorphism classes (LICs) [1] , [2] . Two quasicrystals are local isomorphic if and only if every local arrangement with any finite size in one is also present in the other. Since the discovery of quasicrystals [3] , electron diffraction spectroscopy (EDS) has been a gold standard for extracting the symmetry and finding new quasicrystal materials. Two quasicrystals have identical diffraction patterns if and only if they are local isomorphic [1] . Diffraction peaks of distinct LICs have the same locations but slightly different intensities. However, we will see that, as illustrated below in the two-dimensional (2D) generalized Penrose quasicrystals, the subtle EDS disparity between diverse LICs limits the discrimination. Since different LICs exhibit significant disparities in physical properties, including free energy, entropy [1] , [2] , electron localization [4] , and superuniformity [5] , [6] , an alternative approach is requisite to distinguish various quasicrystals. High-order harmonic generation (HHG) from gases is one of the classical products of nonlinear laser-matter interactions [7] , [8] . It is important in compressing laser pulse width [9] and obtaining attosecond light sources [10] . Crystals [11] , [12] , [13] , [14] , [15] , [16] and liquids [17] , [18] have recently garnered significant interest as sources of HHG. Decoding HHG to retrieve the microstructure and electron properties of matter is one of the crucial topics in ultrafast optics. Significant progress has been made in atomic orbital imaging [19] , [20] , [21] , chiral molecule inspection [22] , [23] , [24] , crystal band reconstruction [15] , [25] , valence electron imaging [26] , and topological phase transition detection [27] , [28] . Compared to EDS, lasers afford more control freedoms, such as polarization, waveform, ellipticity, and so on, which offer distinctive avenues for imaging quasicrystal structures. Nonetheless, research in this field is currently rare. Quasicrystals, lying between periodic crystals and disordered amorphous materials (like liquids), are renowned for their unique order, which may lead to novel harmonic generation mechanisms and a new type of HHG source [29] , [30] , [31] . On the other hand, the orientation-dependent HHG in quasicrystals can be expected to reveal their forbidden rotational symmetries, serving as a fingerprint for experimental identification. Furthermore, HHGs capture ultrafast electron dynamics and offer a potential all-optical method to explore both structures and electron properties of quasicrystals. As we all know, imperfect symmetry and disorder are always present in real solids, such as glass, ceramics, polycrystals, and crystals at room temperature. Research on these aspects is very important for the development of attosecond science, but most of them are skipped due to the complexity of the system. From this perspective, the study of quasicrystal HHG is very relevant for understanding the harmonic generation mechanism in complex systems, which allows us to study complex electron dynamic processes in an ideal ordered system. Demonstrating distinctive rotational symmetry in EDS, 2D quasicrystals have simpler spatial structures compared to three-dimensional settings and have real physical counterparts, like 30 ∘ twisted bilayer graphene [32] and the alloys with 8 [33] , 10 [34] , and 12-fold [35] symmetries. Therefore, studying HHG in 2D quasicrystals is both more feasible and essential. Recently, the topological properties [36] , [37] , [38] , and cold atom simulation [39] , [40] , [41] of 2D quasicrystals have also received extensive attention. As the classical paradigm and pioneer of quasicrystals, the static electron properties of 2D Penrose tiling [42] have been widely studied in the density of states [43] , [44] and the localization of wave functions [45] . To generate Penrose tiling, various methods have been developed, including matching rules [42] , [46] , self-similarity transformations [45] , the generalized dual method (GDM) [1] , [2] , and superspace projection [47] , [48] , [49] , [50] , [51] , [52] . Notably, including Penrose tilings, GDM and five-dimensional superspace projection can generate various quasiperiodic tilings, which are collectively termed generalized Penrose tilings [49] , [50] , [51] , [52] . These tilings share the same fat and thin rhombic unit cells but can belong to different LICs. In this work, we take 2D generalized Penrose quasicrystals as an example to distinguish different LICs by HHG. We observed distinct differences in HHG yield and orientation dependence among LICs, allowing for their differentiation. These variations arise from differences in vertex environments (VEs), linked to the system’s hyperuniformity. Furthermore, the orientation-dependent HHG retains the crystallography-forbidden tenfold rotational symmetry, characteristic of quasicrystal harmonic radiation. Atomic units (a.u.) are used throughout unless stated otherwise.","references":{"indices":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,24,25,26,27,29,30,31,32,33,34,35,36,38,39,40,41,42,43,44,45,48,49,50,51,52],"infos":[{"idx":"1","title":"Levine, D. & Steinhardt, P. J. Definition and structure.Phys. Rev. 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As exemplified by the 0th grid lines marked with red dashed lines, the lines are numbered in each grid to denote the open regions split by the 5-grid into five integers ( k 0 , k 1 , ⋯ , k 4 ) [1] , [2] . The tiling and rhombuses vertices \\(\\left\\{{{{\\bf{t}}}}\\right\\}\\) can be obtained by the dual transformation, \\({{{\\bf{t}}}}={\\sum }_{i=0}^{4}{k}_{i}{{{{\\bf{e}}}}}_{i}\\) . The algebraic distance between the 0th line of i th grid and the origin is defined as the parameter γ i , which determines the grid plane and its dual. Socolar et al. [2] proved a critical theorem: If and only if \\(\\left\\{{\\gamma }_{i}\\right\\}\\) and \\(\\left\\{{\\gamma }_{i}^{{\\prime} }\\right\\}\\) of two 5-grids are satisfied, \\({\\sum }_{i=0}^{4}{\\gamma }_{i}-\\mathop{\\sum }_{i=0}^{4}{\\gamma }_{i}^{{\\prime} }=m,m\\in {\\mathbb{Z}}\\) , their dual tilings belong to the same LIC. Hence, the parameter γ , the fractional part of \\(\\mathop{\\sum }_{i=0}^{4}{\\gamma }_{i}\\) , dictates the LIC of configurations, and its absolute value can be consistently mapped to the interval [0, 0.5]. In particular, when γ = 0, the tiling belongs to the Penrose LIC and can be obtained by the “matching rule” [42] , [46] . Varying \\(\\left\\{{\\gamma }_{i}\\right\\}\\) and enlarging the rhombuses side length l to 8 a.u., we obtain a series of tilings, exhibited in Fig. 1 . Configurations 1 and 2 belong to LIC1 ( γ = 0), and configurations 3–5 are included in LIC2 ( γ = 0.5). The localized Gaussian potentials are adopted to describe the atomic potential located on the rhombus vertices [18] . Configurations’ energy spectrum and eigenstates can be obtained by diagonalizing the field-free Hamiltonian. Taking configuration 1 as an example, we recognize the local, critical, and extended eigenstates, which are consistent with the tight-binding central model [45] . The light localization in photonic quasicrystal [4] is also reproduced by the electron states of configuration 3. This evidence confirms the rationality of our model (see Supplementary Note 1 and Supplementary Figs. 1 and 2 ). From the energy spectrum (Supplementary Fig. 1 ), our models have a bandgap of about 5 eV, similar to typical insulators (>2 eV). We will see that the interband harmonics play a major role in distinguishing LICs. With suitable laser conditions, our results should be applicable to other gapped quasicrystal models. Fig. 1: Several 2D generalized Penrose quasicrystal configurations obtained by GDM. a Schematic diagram of the periodic 5-grid in the GDM. Black arrows and gray grids represent the star vectors and the 5-grid, respectively, with the 0th grid line indicated by red dashed lines. b – f respectively denote the configurations 1–5 obtained by the GDM, including fat and thin rhombuses represented in light blue and red. The number of atoms N atom (or vertices) in each configuration is successively 991, 981, 975, 991, and 981. Configurations 1 and 2, belonging to LIC1 (the Penrose LIC), have grid parameters \\(\\left\\{{\\gamma }_{i}^{1}=-0.2\\right\\}\\) and \\(\\left\\{{\\gamma }_{0}^{2}=-0.48,{\\gamma }_{1}^{2}=-0.39,{\\gamma }_{2}^{2}=-0.50,{\\gamma }_{3}^{2}=-0.29,{\\gamma }_{4}^{2}=-0.34\\right\\}\\) , respectively. Configurations 3–5 belong to LIC2 for \\(\\lfloor \\mathop{\\sum }_{i=0}^{4}{\\gamma }_{i}\\rfloor -\\mathop{\\sum }_{i=0}^{4}{\\gamma }_{i}=0.5\\) ( ⌊ ⌋ is the “floor” operator), and the grid parameters are \\(\\left\\{{\\gamma }_{0}^{3}=-0.15,{\\gamma }_{1}^{3}=-0.22,{\\gamma }_{2}^{3}=-0.23,{\\gamma }_{3}^{3}=-0.42,{\\gamma }_{4}^{3}=-0.48\\right\\},\\left\\{{\\gamma }_{i}^{4}=-0.3\\right\\}\\) , and \\(\\left\\{{\\gamma }_{i}^{5}=-0.5\\right\\}\\) , respectively. If γ 0 = γ 1 = γ 2 = γ 3 = γ 4 , the dual tiling will have a symmetric center. Source data are provided as a Source Data file. Full size image Electron diffraction spectroscopy Since our model involves vertex-decoration-only structures with a single atom type at the vertices, the EDS (or structure factor) can be simply obtained via Fourier transform of the atomic positions (see Supplementary Note 2 for detail), without requiring complex corrections for atom types, distributions, or disorder [53] . EDSs of different configurations in the two kinds of LICs have been calculated (see Supplementary Note 2 and Supplementary Fig. 3 ). Although our models possess finite size, we can still arrive at the same conclusion as elucidated by Levine et al. [1] . The local isomorphic configurations possess nearly identical EDS, while varying LICs exhibit subtle variations in diffraction peak intensities, underscoring that EDS is ineffective in discriminating different LICs. Simulation of electron dynamics in quasicrystals The laser-quasicrystal interaction is simulated by solving the time-dependent Schrödinger equation (TDSE). Similar to the time-dependent density functional theory with frozen Kohn–Sham potentials [27] , [54] , N atom valence states below zero energy are selected as initial states and evolve independently. The HHG spectra are obtained by the Fourier transform of the laser-induced currents, and the yield of the n th order harmonic is I n . Influence of laser spot position As illustrated in Fig. 2 a, unlike periodic crystals, spot irradiation at different positions (A or B) on the quasicrystal may encounter diverse atomic structures. In two independent experiments, the lasers always cover distinct atomic patterns. Investigating the impact of spot position variations on quasicrystal HHG will aid in confirming experimental reproducibility, which can be achieved by comparing harmonic responses among local isomorphic configurations, according to the definition of LIC. As depicted in the schematic diagram in Fig. 2 b, we investigate the scenario where a linearly polarized laser illuminates a 2D quasicrystal vertically. First, we fix the polarization angle θ at 0 ∘ and configure the laser with wavelength λ = 2.6 μm and total duration L t = 10 T ( T is the optical period). Short pulse duration and low photon energy mean a small absorbing cross-section, enabling higher laser intensities for nonperturbative effects without damaging the target. In Fig. 2 c, d, HHGs and the phases (see Supplementary Fig. 4 ) of configurations in one LIC with apparent structure differences are almost identical. In fact, this conclusion is robust for the laser’s intensity, wavelength, and polarization angle (see Supplementary Note 3 and Supplementary Figs. 5 – 7 ). In a circular polarization scenario, when the number of atoms reaches about 6100, the harmonic selection rule [55] , [56] , [57] of each configuration aligns with 10 k ± 1 ( \\(k\\in {\\mathbb{N}}\\) ) (see Supplementary Fig. 8 ), consistent with quasicrystal orientational order (determined by the star vector in Fig. 1 a) and reciprocal lattice symmetry. Consequently, we predict identical harmonic radiation in actual experiments when the laser irradiates different positions on the quasicrystal, irrespective of linear, circular polarization, or any ellipticity in between. The unexpected insensitivity of quasicrystal harmonics to laser irradiation position can be clarified by considering the electron motion environment and the concept of LICs. In quasicrystals, HHG occurs as electrons initially localized at vertices are driven by a laser field in a finite region and scatter with atoms. Different initial vertices lead to distinct finite regions with varying atomic arrangements, which are shared among local isomorphic configurations. As noted, different configurations within the same LIC should produce nearly identical harmonic emissions. Focusing the laser on different parts of the quasicrystals reveals distinct atomic patterns, yet these irradiated regions belong to the same LIC, sharing the same electron motion environments and harmonic spectra. This will be elaborated upon in the following parts. Fig. 2: Harmonic spectra comparison across different configurations. a Schematic diagram of laser irradiation at various positions on the quasicrystal. b Schematic diagram of the interaction between generalized Penrose quasicrystals and linearly polarized laser field. The HHG spectra of configurations in ( c ) LIC1 and ( d ) LIC2, driven by a linearly polarized laser with an intensity of I 0 = 3 × 10 13 W cm −2 , and sin 2 envelope. e Comparing 11th to 19th order harmonic spectra from configurations 1 and 3 of different LICs in linear coordinates. The harmonic spectra of configurations 1 to 5 are shown as solid or dashed lines in black, magenta, red, blue, and green, respectively. Source data are provided as a Source Data file. Full size image HHG of different LICs Now, let’s revisit HHG’s role in distinguishing LICs, with a primary focus on linear polarization driving lasers. To drive electrons far enough in the quasicrystals, we used an intensity of I 0 = 3 × 10 13 W cm −2 . Similar laser intensity can be found in ref. [58] . In Fig. 2 e, we compared the HHG of configurations 1 and 3 from two different LICs and found a difference in yield: The 11th–19th order (the plateau region) harmonic yield of configuration 1 is higher than configuration 3, whose stability has been verified for I 0 > 2 × 10 13 W cm −2 and λ > 1800 nm (see Supplementary Figs. 5 and 6 ). We conclude that different LICs exhibit obvious variations in harmonic yield. Therefore, the HHG provides an alternative to determine whether two configurations belong to the same LIC. In the following parts, we will observe that variations in harmonic yield among different LICs correlate with the diversity of VEs. This affects the coherence of electron motion and harmonic radiation, characterizable through superuniformity. Furthermore, we compare the orientation-dependent harmonic yields of diverse LICs. Figure 3 a–f shows the orientation-dependent yield of the 11th to 21st-order harmonics for configuration 1 and configuration 3. Firstly, we find that the θ -dependent harmonic spectra of LIC1 and LIC2 exhibit distinct intensity differences, enabling the identification of different LICs. As shown in Fig. 3 a–c, the 11th, 13th, and 15th order harmonics of configuration 1 are notably stronger than configuration 3 across all orientations. In Fig. 3 d and e, as polarization direction varies, the two non-local isomorphic configurations alternate in dominating the 17th and 19th-order harmonic radiation intensities. In Fig. 3 f, configuration 3’s 21st harmonic intensity exceeds that of configuration 1 for all polarization directions. Additional comparisons of orientation-dependent harmonic yields in various configurations are provided (see Supplementary Note 4 and Supplementary Figs. 9 and 10 ), further validating the effectiveness of θ -dependent HHG in distinguishing LICs. We also compared the lower-order (e.g., 3rd-order) harmonics below the plateau region and found minimal differences. Thus, lower-order harmonics (in the perturbative regime) are not suitable for distinguishing LICs. Fig. 3: Orientation-dependent harmonic yield. a – f The θ -dependent harmonic yield I n of the 11th–21st order in configuration 1 (LIC1, the solid black line) and configuration 3 (LIC2, the solid red line), with the radial representing harmonic yield (in arb. units). Source data are provided as a Source Data file. Full size image Secondly, let’s delve into the symmetry of orientation-dependent HHG. As depicted in Fig. 3 , the θ -dependent harmonic yield of configurations 1 and 3 approximately exhibits 10-fold rotational symmetry, despite the absence of C 10 rotational symmetry in these configurations. This counterintuitive phenomenon can also be understood via the definition of the LIC and the finite electron motion range. Configurations 3 and 5 belong to the same LIC2, with the latter exhibiting 10-fold rotational symmetry (see Fig. 1 ). According to the LIC definition, in sufficiently large systems, any finite-size atomic arrangement in configuration 5 can be found in configuration 3. Under laser driving, the finite atomic arrangement experienced by an electron initially located at a vertice is shared by configurations 1 and 5, emitting identical harmonics. Hence, in sufficiently large systems, the orientation-dependent HHG of configuration 3 should align with configuration 5, exhibiting C 10 symmetry. Relative to configuration 1, we can identify another local isomorphic configuration \\(\\bar{1}\\) (with grid parameter \\(\\left\\{{\\gamma }_{i}^{\\bar{1}}=-0.8\\right\\}\\) ) whose atomic structure is equivalent to rotating configuration 1 by 180° along the symmetry axis. Hence, any finite region in configuration 1 always appears in the configuration \\(\\bar{1}\\) in a rotated form by 180°. As configurations 1 and \\(\\bar{1}\\) are local isomorphic, any finite region of configuration \\(\\bar{1}\\) is always present in configuration 1. Thus, in sufficiently large configuration 1, there is always a pair of electrons whose finite motion regions rotated 180° relative to each other, resulting in a C 10 symmetry for the orientation-dependent HHG of configuration 1 with C 5 symmetry. In our simulations, finite-size effects can induce subtle fluctuations in the harmonic spectra intensities of distinct configurations within an LIC, which leads to a slight deviation in the symmetry of the orientation-dependent HHG from C 10 and can be mitigated by expanding the system. Therefore, we conclude that for sufficiently large systems (such as the actual laser spot size), the θ -dependent harmonic spectra of LIC1 and LIC2 will tend towards C 10 , which is required by the quasicrystal’s orientation order (determined via star vectors) and conforms to the EDS’s symmetry. Another key element is that the orientation-dependent electron excitation probability and order-dependent electron motion trajectory together lead to different θ -dependent shapes of harmonic yield across different orders. It’s noteworthy that in our discussion, all configurations possess identical star vectors, a prerequisite for distinguishing different LICs. As depicted in Fig. 3 , certain harmonics (e.g., 21st order) exhibit high yields in the star vectors and their opposite directions, offering a means to determine orientation order experimentally. As shown in the next two sections, differences in harmonic yield between LICs stem from the diversity of VEs, whose spatial distribution is governed by orientational and quasiperiodic order (modulated by γ ), explaining the distinct HHG angular dependencies across LICs. Real-space electron motion perspective It is known that in semiconductors and insulators, interband and intraband transitions play an important role in HHG [59] , [60] , [61] . Interband harmonics are often described as follows. The excited electron and hole wave packets are driven by the laser field to travel, and then they recombine at specific moments, emitting high-energy photons with gap energy. However, without the shallow-lattice approximation [41] , our models’ finite size and electron localization limit the application of energy bands in k space. We can always sort the eigenvalues and divide the energy spectrum into the valence band (below zero) and conduction band (above zero) in the energy space (see Supplementary Fig. 1 ). By projecting the time-dependent wave function onto time-independent eigenstates [62] , we distinguish intraband and interband harmonics in energy space. We find that the HHG plateau region, relevant for differentiating LICs, is mainly dominated by interband mechanisms (see Supplementary Note 5 and Supplementary Figs. 11 and 12 ), which are associated with the long-range motion of electrons [61] . So, we qualitatively analyze the relationship between harmonic radiation and LICs from the view of the excursion-electron dynamic environment. Taking the partial enlargement of valence electron density of configuration 1 in Fig. 4 a as an example, the initial electrons are mainly distributed at the vertices, and the VE [4] , [5] , [52] composed of rhombus tiles around a vertex will affect the dynamic process and harmonic radiation. Driven by a laser field, an electron from a vertex undergoes a complex motion trajectory and emits harmonic radiation through scattering with atoms on the vertices. The intricate multiple-scattering processes involving numerous atoms are challenging to track accurately. If electrons from different vertices observe different vertex patterns, the emitted harmonic radiations will exhibit suppressed coherence. According to the definition of LIC, a finite atomic arrangement centered around one vertex is consistently found in another configuration. So, as illustrated in Fig. 4 b, a moiré pattern and the overlap regions (the light blue areas) can be generated by appropriately translating two isomorphic tilings sharing the same VEs. The overlap regions cover the vast majority of vertices in the two tilings, and the electrons in overlap regions undergo uniform scattering processes, which makes the almost uniform harmonic radiation. Thus, quasicrystal HHG remains unaffected by laser irradiation position, holding true across different laser conditions, including intensity, wavelength, orientation, and polarization. Fig. 4: Localized electron density and moiré patterns. a Partial enlargement of the spatial distribution of electron density in configuration 1. b The moiré pattern built by local isomorphic configurations 1 (the black tiling) and 2 (the magenta tiling) under appropriate translation. The light blue region indicates the overlap between the two configurations. Source data are provided as a Source Data file. Full size image On the other hand, the mismatch between the two patterns in Fig. 4 b indeed corresponds to a phason flip for VEs Q or D [53] (see Fig. 5 a and Supplementary Fig. 13 ). Phason flips between local isomorphic configurations affect only specific patches, not all Q and D. This contrasts with random phason flips applied to each patch, which introduce a tiling disorder [53] . The perspective on phason flips is enlightening. Perhaps we could relate HHG with the problem of phasonic disorder. Here, we try to investigate the impact of the phasonic disorder on harmonic yield in configuration 1 (see Supplementary Note 6 ). We found that as the disorder increases (controlled by flip iterations), the yields of some orders in the plateau region initially decrease and then recover, which can be qualitatively explained by the types and proportions of VEs (see Supplementary Fig. 14 ). A more detailed analysis of phasonic disorder’s effects on HHG will be presented in future work. Fig. 5: The γ -dependent VEs frequency, superuniformity, and harmonic yield. a 16 potential VEs in the generalized Penrose quasicrystal. b VEs frequency F as a function of γ . c A localized enlargement of the lower shaded region in ( b ). d Quasicrystal hyperuniformity parameter \\(\\bar{{{\\Lambda }}}(\\infty )\\) as a function of γ . Black and red lines represent results considering 400 and 10 local isomorphic configurations at each γ , respectively. e γ -dependence of 11th to 19th harmonic yields, shown as solid lines in black, blue, magenta, green, and purple, respectively. Laser parameters align with Fig. 2 . Source data are provided as a Source Data file. Full size image VEs and hyperuniformity In this part, we aim to uncover the reasons behind the differences in harmonic intensity among various LICs. In generalized Penrose quasicrystals, fat and thin rhombi collectively create 16 distinct VEs [4] , [5] , [52] , as illustrated in Fig. 5 a. It is worth noting that within these VEs, M (defined by two fat and two thin rhombuses) consists of two inequivalent forms that are mirror images and cannot be transformed into each other by translation or rotation in the 2D plane. The two patterns of M always appear in pairs in the discussed configurations and contribute to the harmonic radiation simultaneously. In the following discussion, we will refer to both forms collectively as M. The frequency F [4] , [5] , [52] , defined as the ratio of different VEs to the total vertices, varies with changes in LICs or parameter γ , as shown in Fig. 5 b and c The Penrose LIC (LIC1) with γ = 0 is distinguished for having a fewest types of VEs, potentially explaining the higher harmonic yield observed in Fig. 2 e. Taking the endpoint p of the VE Q in Fig. 5 a as an example, different LICs restrict the possible evolutions of p . In LIC1, p has six choices (Q, K, J, S, T, V), while in LIC2, there are ten choices (Q, K, M, J, R, S, U, W, Y, Z). It shows that in different LICs, the VEs diversity at scattering point p is different when electrons originate from Q. The analysis also applies to D or other VEs. In fact, all VEs in the system are interconnected through complex electron scattering and collectively contribute to harmonic radiation. Thus, greater VE diversity and a more uniform F distribution will lead to lower motion environment similarity for electrons originating from the same VEs but different positions, which reduces the HHG coherence and yield (as we will see in Fig. 5 e). Using Voronoi cells as a metric, Lin et al. [5] emphasized that the local distribution of VEs governs quasicrystals’ hyperuniformity, characterized by the suppression of long-wavelength density fluctuations. We expect that the variation of the hyperuniformity [5] , [6] will manifest in the harmonic spectrum of different LICs. Let’s discuss hyperuniformity briefly: considering quasicrystal atoms as a point pattern in d -dimensional Euclidean space \\({{\\mathbb{R}}}^{d}\\) , the number of points within a spherical window centered at x i with radius R is denoted as N ( R ; x i ), a random variable about x i with variance σ 2 ( R ). If σ 2 ( R ) grows slower than the window volume, i.e., σ 2 ( R ) ~ R α , α < d , the system is hyperuniform. In the generalized Penrose quasicrystal ( α = 1), σ 2 ( R ) is expressed as σ 2 ( R ) ~ Λ( R ) R + o ( R ), and o ( R ) denotes terms of lower order than R [5] . Following ref. [5] , we use the cumulative moving average \\(\\bar{{{\\Lambda }}}(R)=1/R\\int_{0}^{R}{{\\Lambda }}({R}^{{\\prime} })d{R}^{{\\prime} }\\) to eliminate small-scale variations and take the asymptotic average \\(\\bar{{{\\Lambda }}}(\\infty )=\\mathop{\\lim }_{R\\to \\infty }\\bar{{{\\Lambda }}}(R)\\) to describe the hyperuniformity. In calculations, quasicrystal atoms ( N atom ≈ 980) concentrate within a circle (radius R 0 = 16 l ) centered at the origin. M = 10,000 sampling window centers x i are uniformly distributed within a concentric circular region (radius R max = R 0 /2). The black dotted line in Fig. 5 d illustrates the γ -dependent hyperuniformity parameter \\(\\bar{{{\\Lambda }}}(\\infty )\\) , averaging over N con = 400 local isomorphic configurations for each γ point to mitigate finite-size fluctuations. Our results align with ref. [5] , showing that \\(\\bar{{{\\Lambda }}}(\\infty )\\) tends to increase monotonically with higher γ values. At γ = 0, LIC1 exhibits the global minimum of \\(\\bar{{{\\Lambda }}}(\\infty )\\) , with the highest level of hyperuniformity. As γ increases, hyperuniformity decreases for different LICs. The HHG yield VS the hyperuniformity of LICs While simulating the harmonic yield dependence on the parameter γ , to balance computational feasibility and accuracy, we focused on six distinct LICs at γ = 0, 0.1, 0.2, 0.3, 0.4, and 0.5. For each γ , we averaged \\(\\bar{{{\\Lambda }}}(\\infty )\\) over N con = 10 random configurations. The result exhibits a monotonic increase and closely agrees with the results obtained by averaging over 400 configurations, as shown by the red dotted line in Fig. 5 d. The current was averaged over the 10 configurations to obtain the HHG for each LIC. As shown in Fig. 5 e, with increasing γ , the yields of the 11th to 15th harmonics monotonically decrease, consistent with the hyperuniformity. For the 17th and 19th harmonics, an anomalous enhancement occurs for γ > 0.3. We speculate that this is due to the presence of Z and ST VEs at γ = 0.3 in Fig. 5 c. These VEs, characterized by the second-highest and highest occurrences of thin rhombi, exhibit high atomic density, leading to some electrons being bound. The bounded electrons will undergo rapid oscillations along the short diagonal of the thin rhombus under the external field, potentially causing an increase in high-frequency harmonic yield. Hence, we conclude that quasicrystal hyperuniformity can be characterized by HHG yields. Hyperuniformity also affects electron transport, notably reflected in current amplitude variations. In our simulation (see Supplementary Note 7 and Supplementary Fig. 15 a), increasing the γ parameter reduces hyperuniformity, leading to a gradual decline in current amplitudes for distinct LICs. This is also attributed to diverse VE choices, influencing the similarity of electron motion environments and affecting both the coherence and amplitude of current. Special VEs (X, Y, Z, and ST) [4] , with higher proportions of thin rhombi and atomic density, modify the electron momentum more strongly. Due to finite size constraints, the proportions of these VEs fluctuate within different configurations of one LIC, which is also reflected in the current amplitudes (see Supplementary Fig. 15 b)."},{"section":"Discussion","content":"Distinguishing LICs in quasicrystals is an important task since they have many distinct physical properties. Traditional EDS relies on the scattering near the nuclei, resulting in almost the same pattern for different LICs. HHG comes from the laser-induced electron motions with a wide range in quasicrystals, which encodes more information than EDS. Here we proposed to use high-order harmonic spectroscopy with more freedom to identify different LICs. When a linearly polarized laser illuminates the quasicrystal, the HHG yield of different LICs can have obvious distinctions. In addition, for our quasicrystal models, the orientation-dependent harmonic yield tends to exhibit C 10 symmetry (forbidden by crystallography), which is consistent with the orientation order and can take the role of the quasicrystal HHG fingerprint. Therefore, different LICs can be distinguished by HHG in yield and orientation-dependence. Furthermore, we use harmonic yields to reveal the quasicrystal hyperuniformity, which is dependent on the VE distribution of various LICs and impacts electron transport properties, such as current amplitude. Our work establishes a link between quasicrystal HHG and LIC, reveals the effect of VE distribution on electron scattering, and provides a new all-optical scheme for studying the structural properties of quasicrystal. It may also be applied to study the topological properties of quasicrystals in the future. Further, we note that the quasicrystal can be approximated by a periodic system with tunable unit cell sizes [4] . This approach will aid in understanding how periodicity breaking and electron incoherent scattering affect quasicrystal harmonic yield in future studies."},{"section":"Methods","content":"Initial state acquisition Before the laser field arrives, the electronic structure of the 2D generalized Penrose quasicrystal, including eigenenergies ε n and eigenstates ψ n , can be described by the stationary Schrödinger equation, $${\\hat{H}}_{0}{\\psi }_{n}={\\varepsilon }_{n}{\\psi }_{n}.$$ (1) Here, \\({\\hat{H}}_{0}\\) is the system’s stationary Hamiltonian, $${\\hat{H}}_{0}=\\frac{{\\hslash }^{2}}{2m}{{{{\\boldsymbol{\\nabla }}}}}^{2}+{V}_{ion}({{{\\bf{r}}}}).$$ (2) Similar to the tight-binding vertex model (considering interactions between atoms at rhombus vertices), the localized Gaussian potentials are adopted to describe the atomic Coulomb potential, $${V}_{{{{\\rm{ion}}}}}({{{\\bf{r}}}})=-\\sum _{v=1}^{{N}_{{{{\\rm{atom}}}}}}\\,{V}_{0}\\exp \\left[-\\frac{{\\left({{{\\bf{r}}}}-{{{{\\bf{r}}}}}_{v}\\right)}^{2}}{2{\\alpha }^{2}}\\right],$$ (3) where V 0 = 1 a.u., α = 0.8 a.u., and r v is the atomic position determined by the GDM. As shown in Fig. 1 , the N atom atoms of the 2D quasicrystal configurations are distributed within a circle of radius R 0 = 16 l , where l = 8 a.u. is the rhombus edge length. In the simulation, we construct the Hamiltonian matrix H 0 in discrete 2D real space using the finite difference method, then diagonalize it to obtain the eigenenergies and eigenstates, applying open boundary conditions. The number of initially occupied valence electrons equals the total number of atoms, N atom . As shown in Supplementary Note 1 and Supplementary Figs. 1 and 2 , when studying the system’s static properties (such as localization and energy spectrum), we use a circular computational box with a radius of R 0 to avoid interference from free electron states near the boundary. To simulate harmonic radiation from the laser-quasicrystal interaction, we construct the H 0 matrix in a larger rectangular box with side lengths 2 R 0 + 2 L a b and diagonalize it to obtain the N atom initial states (with ε n < 0). During electron evolution, an absorption function of the form \\({\\cos }^{1/8}\\) , with width L a b = 20 a.u., is applied near the box boundary to prevent electron reflection. Time-dependent Schrödinger equation The electron dynamics in the laser-quasicrystal interaction are described by the 2D TDSE in the length gauge, $$i\\hslash \\frac{\\partial }{\\partial t}\\psi ({{{\\bf{r}}}},t)=\\left[{\\hat{H}}_{0}+{{{\\bf{E}}}}(t)\\cdot {{{\\bf{r}}}}\\right]\\psi ({{{\\bf{r}}}},t).$$ (4) E ( t ) is the laser electric field. When the angle between the laser principal axis and the x -axis is θ (see Fig. 2 b), the components in x and y directions of the electric field can be written as, $${E}_{x}(t)={E}_{x,\\theta=0}(t)\\cos \\theta -{E}_{y,\\theta=0}(t)\\sin \\theta,$$ (5) $${E}_{y}(t)={E}_{x,\\theta=0}(t)\\sin \\theta+{E}_{y,\\theta=0}(t)\\cos \\theta .$$ (6) Here, $${E}_{x,\\theta=0}(t)={E}_{0}/\\sqrt{1+{\\epsilon }^{2}}f(t)\\cos ({\\omega }_{0}t),$$ (7) $${E}_{y,\\theta=0}(t)=\\epsilon {E}_{0}/\\sqrt{1+{\\epsilon }^{2}}f(t)\\sin ({\\omega }_{0}t).$$ (8) E 0 , ϵ , f ( t ), ω 0 , and L t , respectively, denote the field strength, ellipticity, envelope, carrier frequency, and total duration. In the calculation, we apply the split-operator method to evolve the N atom initial states ψ i independently to ψ i ( r , t ). The χ (= x or y ) direction harmonic spectrum H χ ( ω ) is given by the Fourier transform \\({H}_{\\chi }(\\omega )=| {{{\\mathscr{F}}}}[{J}_{\\chi }(t)]{| }^{2}\\) . J χ ( t ) is the total laser-induced current in the χ direction, with $${J}_{\\chi }(t)=\\sum\\limits_{i}^{{N}_{atom}}{j}_{\\chi,i}(t),$$ (9) and j χ , i ( t ) = \\(\\langle {\\psi }_{i}({{{\\bf{r}}}},t)| {\\hat{p}}_{\\chi }| {\\psi }_{i}({{{\\bf{r}}}},t)\\rangle\\) . The total HHG is H t o t ( ω ) = H x ( ω ) + H y ( ω ), and the yield I n of the n th harmonic is evaluated by, $${I}_{n}=\\int_{(n-0.5){\\omega }_{0}}^{(n+0.5){\\omega }_{0}}{H}_{tot}(\\omega )d\\omega .$$ (10) In the actual calculation, we set the time step d t = 0.25 a.u., and the grid spacing d x = d y = 0.25 a.u."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"Choices of the unit cell and packing methods in crystals exhibit a strict one-to-one correspondence to form a single undefeated periodic structure. Very different from crystals, two or more unit cells can be assembled into an infinite number of various quasicrystals, which can be further subdivided into local isomorphism classes (LICs) [1] , [2] . Two quasicrystals are local isomorphic if and only if every local arrangement with any finite size in one is also present in the other. Since the discovery of quasicrystals [3] , electron diffraction spectroscopy (EDS) has been a gold standard for extracting the symmetry and finding new quasicrystal materials. Two quasicrystals have identical diffraction patterns if and only if they are local isomorphic [1] . Diffraction peaks of distinct LICs have the same locations but slightly different intensities. However, we will see that, as illustrated below in the two-dimensional (2D) generalized Penrose quasicrystals, the subtle EDS disparity between diverse LICs limits the discrimination. Since different LICs exhibit significant disparities in physical properties, including free energy, entropy [1] , [2] , electron localization [4] , and superuniformity [5] , [6] , an alternative approach is requisite to distinguish various quasicrystals. High-order harmonic generation (HHG) from gases is one of the classical products of nonlinear laser-matter interactions [7] , [8] . It is important in compressing laser pulse width [9] and obtaining attosecond light sources [10] . Crystals [11] , [12] , [13] , [14] , [15] , [16] and liquids [17] , [18] have recently garnered significant interest as sources of HHG. Decoding HHG to retrieve the microstructure and electron properties of matter is one of the crucial topics in ultrafast optics. Significant progress has been made in atomic orbital imaging [19] , [20] , [21] , chiral molecule inspection [22] , [23] , [24] , crystal band reconstruction [15] , [25] , valence electron imaging [26] , and topological phase transition detection [27] , [28] . Compared to EDS, lasers afford more control freedoms, such as polarization, waveform, ellipticity, and so on, which offer distinctive avenues for imaging quasicrystal structures. Nonetheless, research in this field is currently rare. Quasicrystals, lying between periodic crystals and disordered amorphous materials (like liquids), are renowned for their unique order, which may lead to novel harmonic generation mechanisms and a new type of HHG source [29] , [30] , [31] . On the other hand, the orientation-dependent HHG in quasicrystals can be expected to reveal their forbidden rotational symmetries, serving as a fingerprint for experimental identification. Furthermore, HHGs capture ultrafast electron dynamics and offer a potential all-optical method to explore both structures and electron properties of quasicrystals. As we all know, imperfect symmetry and disorder are always present in real solids, such as glass, ceramics, polycrystals, and crystals at room temperature. Research on these aspects is very important for the development of attosecond science, but most of them are skipped due to the complexity of the system. From this perspective, the study of quasicrystal HHG is very relevant for understanding the harmonic generation mechanism in complex systems, which allows us to study complex electron dynamic processes in an ideal ordered system. Demonstrating distinctive rotational symmetry in EDS, 2D quasicrystals have simpler spatial structures compared to three-dimensional settings and have real physical counterparts, like 30 ∘ twisted bilayer graphene [32] and the alloys with 8 [33] , 10 [34] , and 12-fold [35] symmetries. Therefore, studying HHG in 2D quasicrystals is both more feasible and essential. Recently, the topological properties [36] , [37] , [38] , and cold atom simulation [39] , [40] , [41] of 2D quasicrystals have also received extensive attention. As the classical paradigm and pioneer of quasicrystals, the static electron properties of 2D Penrose tiling [42] have been widely studied in the density of states [43] , [44] and the localization of wave functions [45] . To generate Penrose tiling, various methods have been developed, including matching rules [42] , [46] , self-similarity transformations [45] , the generalized dual method (GDM) [1] , [2] , and superspace projection [47] , [48] , [49] , [50] , [51] , [52] . Notably, including Penrose tilings, GDM and five-dimensional superspace projection can generate various quasiperiodic tilings, which are collectively termed generalized Penrose tilings [49] , [50] , [51] , [52] . These tilings share the same fat and thin rhombic unit cells but can belong to different LICs. In this work, we take 2D generalized Penrose quasicrystals as an example to distinguish different LICs by HHG. We observed distinct differences in HHG yield and orientation dependence among LICs, allowing for their differentiation. These variations arise from differences in vertex environments (VEs), linked to the system’s hyperuniformity. Furthermore, the orientation-dependent HHG retains the crystallography-forbidden tenfold rotational symmetry, characteristic of quasicrystal harmonic radiation. 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As exemplified by the 0th grid lines marked with red dashed lines, the lines are numbered in each grid to denote the open regions split by the 5-grid into five integers ( k 0 , k 1 , ⋯ , k 4 ) [1] , [2] . The tiling and rhombuses vertices \\\\(\\\\left\\\\{{{{\\\\bf{t}}}}\\\\right\\\\}\\\\) can be obtained by the dual transformation, \\\\({{{\\\\bf{t}}}}={\\\\sum }_{i=0}^{4}{k}_{i}{{{{\\\\bf{e}}}}}_{i}\\\\) . The algebraic distance between the 0th line of i th grid and the origin is defined as the parameter γ i , which determines the grid plane and its dual. Socolar et al. [2] proved a critical theorem: If and only if \\\\(\\\\left\\\\{{\\\\gamma }_{i}\\\\right\\\\}\\\\) and \\\\(\\\\left\\\\{{\\\\gamma }_{i}^{{\\\\prime} }\\\\right\\\\}\\\\) of two 5-grids are satisfied, \\\\({\\\\sum }_{i=0}^{4}{\\\\gamma }_{i}-\\\\mathop{\\\\sum }_{i=0}^{4}{\\\\gamma }_{i}^{{\\\\prime} }=m,m\\\\in {\\\\mathbb{Z}}\\\\) , their dual tilings belong to the same LIC. Hence, the parameter γ , the fractional part of \\\\(\\\\mathop{\\\\sum }_{i=0}^{4}{\\\\gamma }_{i}\\\\) , dictates the LIC of configurations, and its absolute value can be consistently mapped to the interval [0, 0.5]. In particular, when γ = 0, the tiling belongs to the Penrose LIC and can be obtained by the “matching rule” [42] , [46] . Varying \\\\(\\\\left\\\\{{\\\\gamma }_{i}\\\\right\\\\}\\\\) and enlarging the rhombuses side length l to 8 a.u., we obtain a series of tilings, exhibited in Fig. 1 . Configurations 1 and 2 belong to LIC1 ( γ = 0), and configurations 3–5 are included in LIC2 ( γ = 0.5). The localized Gaussian potentials are adopted to describe the atomic potential located on the rhombus vertices [18] . Configurations’ energy spectrum and eigenstates can be obtained by diagonalizing the field-free Hamiltonian. Taking configuration 1 as an example, we recognize the local, critical, and extended eigenstates, which are consistent with the tight-binding central model [45] . The light localization in photonic quasicrystal [4] is also reproduced by the electron states of configuration 3. This evidence confirms the rationality of our model (see Supplementary Note 1 and Supplementary Figs. 1 and 2 ). From the energy spectrum (Supplementary Fig. 1 ), our models have a bandgap of about 5 eV, similar to typical insulators (>2 eV). We will see that the interband harmonics play a major role in distinguishing LICs. With suitable laser conditions, our results should be applicable to other gapped quasicrystal models. Fig. 1: Several 2D generalized Penrose quasicrystal configurations obtained by GDM. a Schematic diagram of the periodic 5-grid in the GDM. Black arrows and gray grids represent the star vectors and the 5-grid, respectively, with the 0th grid line indicated by red dashed lines. b – f respectively denote the configurations 1–5 obtained by the GDM, including fat and thin rhombuses represented in light blue and red. The number of atoms N atom (or vertices) in each configuration is successively 991, 981, 975, 991, and 981. Configurations 1 and 2, belonging to LIC1 (the Penrose LIC), have grid parameters \\\\(\\\\left\\\\{{\\\\gamma }_{i}^{1}=-0.2\\\\right\\\\}\\\\) and \\\\(\\\\left\\\\{{\\\\gamma }_{0}^{2}=-0.48,{\\\\gamma }_{1}^{2}=-0.39,{\\\\gamma }_{2}^{2}=-0.50,{\\\\gamma }_{3}^{2}=-0.29,{\\\\gamma }_{4}^{2}=-0.34\\\\right\\\\}\\\\) , respectively. Configurations 3–5 belong to LIC2 for \\\\(\\\\lfloor \\\\mathop{\\\\sum }_{i=0}^{4}{\\\\gamma }_{i}\\\\rfloor -\\\\mathop{\\\\sum }_{i=0}^{4}{\\\\gamma }_{i}=0.5\\\\) ( ⌊ ⌋ is the “floor” operator), and the grid parameters are \\\\(\\\\left\\\\{{\\\\gamma }_{0}^{3}=-0.15,{\\\\gamma }_{1}^{3}=-0.22,{\\\\gamma }_{2}^{3}=-0.23,{\\\\gamma }_{3}^{3}=-0.42,{\\\\gamma }_{4}^{3}=-0.48\\\\right\\\\},\\\\left\\\\{{\\\\gamma }_{i}^{4}=-0.3\\\\right\\\\}\\\\) , and \\\\(\\\\left\\\\{{\\\\gamma }_{i}^{5}=-0.5\\\\right\\\\}\\\\) , respectively. If γ 0 = γ 1 = γ 2 = γ 3 = γ 4 , the dual tiling will have a symmetric center. Source data are provided as a Source Data file. Full size image Electron diffraction spectroscopy Since our model involves vertex-decoration-only structures with a single atom type at the vertices, the EDS (or structure factor) can be simply obtained via Fourier transform of the atomic positions (see Supplementary Note 2 for detail), without requiring complex corrections for atom types, distributions, or disorder [53] . EDSs of different configurations in the two kinds of LICs have been calculated (see Supplementary Note 2 and Supplementary Fig. 3 ). Although our models possess finite size, we can still arrive at the same conclusion as elucidated by Levine et al. [1] . The local isomorphic configurations possess nearly identical EDS, while varying LICs exhibit subtle variations in diffraction peak intensities, underscoring that EDS is ineffective in discriminating different LICs. Simulation of electron dynamics in quasicrystals The laser-quasicrystal interaction is simulated by solving the time-dependent Schrödinger equation (TDSE). Similar to the time-dependent density functional theory with frozen Kohn–Sham potentials [27] , [54] , N atom valence states below zero energy are selected as initial states and evolve independently. The HHG spectra are obtained by the Fourier transform of the laser-induced currents, and the yield of the n th order harmonic is I n . Influence of laser spot position As illustrated in Fig. 2 a, unlike periodic crystals, spot irradiation at different positions (A or B) on the quasicrystal may encounter diverse atomic structures. In two independent experiments, the lasers always cover distinct atomic patterns. Investigating the impact of spot position variations on quasicrystal HHG will aid in confirming experimental reproducibility, which can be achieved by comparing harmonic responses among local isomorphic configurations, according to the definition of LIC. As depicted in the schematic diagram in Fig. 2 b, we investigate the scenario where a linearly polarized laser illuminates a 2D quasicrystal vertically. First, we fix the polarization angle θ at 0 ∘ and configure the laser with wavelength λ = 2.6 μm and total duration L t = 10 T ( T is the optical period). Short pulse duration and low photon energy mean a small absorbing cross-section, enabling higher laser intensities for nonperturbative effects without damaging the target. In Fig. 2 c, d, HHGs and the phases (see Supplementary Fig. 4 ) of configurations in one LIC with apparent structure differences are almost identical. In fact, this conclusion is robust for the laser’s intensity, wavelength, and polarization angle (see Supplementary Note 3 and Supplementary Figs. 5 – 7 ). In a circular polarization scenario, when the number of atoms reaches about 6100, the harmonic selection rule [55] , [56] , [57] of each configuration aligns with 10 k ± 1 ( \\\\(k\\\\in {\\\\mathbb{N}}\\\\) ) (see Supplementary Fig. 8 ), consistent with quasicrystal orientational order (determined by the star vector in Fig. 1 a) and reciprocal lattice symmetry. Consequently, we predict identical harmonic radiation in actual experiments when the laser irradiates different positions on the quasicrystal, irrespective of linear, circular polarization, or any ellipticity in between. The unexpected insensitivity of quasicrystal harmonics to laser irradiation position can be clarified by considering the electron motion environment and the concept of LICs. In quasicrystals, HHG occurs as electrons initially localized at vertices are driven by a laser field in a finite region and scatter with atoms. Different initial vertices lead to distinct finite regions with varying atomic arrangements, which are shared among local isomorphic configurations. As noted, different configurations within the same LIC should produce nearly identical harmonic emissions. Focusing the laser on different parts of the quasicrystals reveals distinct atomic patterns, yet these irradiated regions belong to the same LIC, sharing the same electron motion environments and harmonic spectra. This will be elaborated upon in the following parts. Fig. 2: Harmonic spectra comparison across different configurations. a Schematic diagram of laser irradiation at various positions on the quasicrystal. b Schematic diagram of the interaction between generalized Penrose quasicrystals and linearly polarized laser field. The HHG spectra of configurations in ( c ) LIC1 and ( d ) LIC2, driven by a linearly polarized laser with an intensity of I 0 = 3 × 10 13 W cm −2 , and sin 2 envelope. e Comparing 11th to 19th order harmonic spectra from configurations 1 and 3 of different LICs in linear coordinates. The harmonic spectra of configurations 1 to 5 are shown as solid or dashed lines in black, magenta, red, blue, and green, respectively. Source data are provided as a Source Data file. Full size image HHG of different LICs Now, let’s revisit HHG’s role in distinguishing LICs, with a primary focus on linear polarization driving lasers. To drive electrons far enough in the quasicrystals, we used an intensity of I 0 = 3 × 10 13 W cm −2 . Similar laser intensity can be found in ref. [58] . In Fig. 2 e, we compared the HHG of configurations 1 and 3 from two different LICs and found a difference in yield: The 11th–19th order (the plateau region) harmonic yield of configuration 1 is higher than configuration 3, whose stability has been verified for I 0 > 2 × 10 13 W cm −2 and λ > 1800 nm (see Supplementary Figs. 5 and 6 ). We conclude that different LICs exhibit obvious variations in harmonic yield. Therefore, the HHG provides an alternative to determine whether two configurations belong to the same LIC. In the following parts, we will observe that variations in harmonic yield among different LICs correlate with the diversity of VEs. This affects the coherence of electron motion and harmonic radiation, characterizable through superuniformity. Furthermore, we compare the orientation-dependent harmonic yields of diverse LICs. Figure 3 a–f shows the orientation-dependent yield of the 11th to 21st-order harmonics for configuration 1 and configuration 3. Firstly, we find that the θ -dependent harmonic spectra of LIC1 and LIC2 exhibit distinct intensity differences, enabling the identification of different LICs. As shown in Fig. 3 a–c, the 11th, 13th, and 15th order harmonics of configuration 1 are notably stronger than configuration 3 across all orientations. In Fig. 3 d and e, as polarization direction varies, the two non-local isomorphic configurations alternate in dominating the 17th and 19th-order harmonic radiation intensities. In Fig. 3 f, configuration 3’s 21st harmonic intensity exceeds that of configuration 1 for all polarization directions. Additional comparisons of orientation-dependent harmonic yields in various configurations are provided (see Supplementary Note 4 and Supplementary Figs. 9 and 10 ), further validating the effectiveness of θ -dependent HHG in distinguishing LICs. We also compared the lower-order (e.g., 3rd-order) harmonics below the plateau region and found minimal differences. Thus, lower-order harmonics (in the perturbative regime) are not suitable for distinguishing LICs. Fig. 3: Orientation-dependent harmonic yield. a – f The θ -dependent harmonic yield I n of the 11th–21st order in configuration 1 (LIC1, the solid black line) and configuration 3 (LIC2, the solid red line), with the radial representing harmonic yield (in arb. units). Source data are provided as a Source Data file. Full size image Secondly, let’s delve into the symmetry of orientation-dependent HHG. As depicted in Fig. 3 , the θ -dependent harmonic yield of configurations 1 and 3 approximately exhibits 10-fold rotational symmetry, despite the absence of C 10 rotational symmetry in these configurations. This counterintuitive phenomenon can also be understood via the definition of the LIC and the finite electron motion range. Configurations 3 and 5 belong to the same LIC2, with the latter exhibiting 10-fold rotational symmetry (see Fig. 1 ). According to the LIC definition, in sufficiently large systems, any finite-size atomic arrangement in configuration 5 can be found in configuration 3. Under laser driving, the finite atomic arrangement experienced by an electron initially located at a vertice is shared by configurations 1 and 5, emitting identical harmonics. Hence, in sufficiently large systems, the orientation-dependent HHG of configuration 3 should align with configuration 5, exhibiting C 10 symmetry. Relative to configuration 1, we can identify another local isomorphic configuration \\\\(\\\\bar{1}\\\\) (with grid parameter \\\\(\\\\left\\\\{{\\\\gamma }_{i}^{\\\\bar{1}}=-0.8\\\\right\\\\}\\\\) ) whose atomic structure is equivalent to rotating configuration 1 by 180° along the symmetry axis. Hence, any finite region in configuration 1 always appears in the configuration \\\\(\\\\bar{1}\\\\) in a rotated form by 180°. As configurations 1 and \\\\(\\\\bar{1}\\\\) are local isomorphic, any finite region of configuration \\\\(\\\\bar{1}\\\\) is always present in configuration 1. Thus, in sufficiently large configuration 1, there is always a pair of electrons whose finite motion regions rotated 180° relative to each other, resulting in a C 10 symmetry for the orientation-dependent HHG of configuration 1 with C 5 symmetry. In our simulations, finite-size effects can induce subtle fluctuations in the harmonic spectra intensities of distinct configurations within an LIC, which leads to a slight deviation in the symmetry of the orientation-dependent HHG from C 10 and can be mitigated by expanding the system. Therefore, we conclude that for sufficiently large systems (such as the actual laser spot size), the θ -dependent harmonic spectra of LIC1 and LIC2 will tend towards C 10 , which is required by the quasicrystal’s orientation order (determined via star vectors) and conforms to the EDS’s symmetry. Another key element is that the orientation-dependent electron excitation probability and order-dependent electron motion trajectory together lead to different θ -dependent shapes of harmonic yield across different orders. It’s noteworthy that in our discussion, all configurations possess identical star vectors, a prerequisite for distinguishing different LICs. As depicted in Fig. 3 , certain harmonics (e.g., 21st order) exhibit high yields in the star vectors and their opposite directions, offering a means to determine orientation order experimentally. As shown in the next two sections, differences in harmonic yield between LICs stem from the diversity of VEs, whose spatial distribution is governed by orientational and quasiperiodic order (modulated by γ ), explaining the distinct HHG angular dependencies across LICs. Real-space electron motion perspective It is known that in semiconductors and insulators, interband and intraband transitions play an important role in HHG [59] , [60] , [61] . Interband harmonics are often described as follows. The excited electron and hole wave packets are driven by the laser field to travel, and then they recombine at specific moments, emitting high-energy photons with gap energy. However, without the shallow-lattice approximation [41] , our models’ finite size and electron localization limit the application of energy bands in k space. We can always sort the eigenvalues and divide the energy spectrum into the valence band (below zero) and conduction band (above zero) in the energy space (see Supplementary Fig. 1 ). By projecting the time-dependent wave function onto time-independent eigenstates [62] , we distinguish intraband and interband harmonics in energy space. We find that the HHG plateau region, relevant for differentiating LICs, is mainly dominated by interband mechanisms (see Supplementary Note 5 and Supplementary Figs. 11 and 12 ), which are associated with the long-range motion of electrons [61] . So, we qualitatively analyze the relationship between harmonic radiation and LICs from the view of the excursion-electron dynamic environment. Taking the partial enlargement of valence electron density of configuration 1 in Fig. 4 a as an example, the initial electrons are mainly distributed at the vertices, and the VE [4] , [5] , [52] composed of rhombus tiles around a vertex will affect the dynamic process and harmonic radiation. Driven by a laser field, an electron from a vertex undergoes a complex motion trajectory and emits harmonic radiation through scattering with atoms on the vertices. The intricate multiple-scattering processes involving numerous atoms are challenging to track accurately. If electrons from different vertices observe different vertex patterns, the emitted harmonic radiations will exhibit suppressed coherence. According to the definition of LIC, a finite atomic arrangement centered around one vertex is consistently found in another configuration. So, as illustrated in Fig. 4 b, a moiré pattern and the overlap regions (the light blue areas) can be generated by appropriately translating two isomorphic tilings sharing the same VEs. The overlap regions cover the vast majority of vertices in the two tilings, and the electrons in overlap regions undergo uniform scattering processes, which makes the almost uniform harmonic radiation. Thus, quasicrystal HHG remains unaffected by laser irradiation position, holding true across different laser conditions, including intensity, wavelength, orientation, and polarization. Fig. 4: Localized electron density and moiré patterns. a Partial enlargement of the spatial distribution of electron density in configuration 1. b The moiré pattern built by local isomorphic configurations 1 (the black tiling) and 2 (the magenta tiling) under appropriate translation. The light blue region indicates the overlap between the two configurations. Source data are provided as a Source Data file. Full size image On the other hand, the mismatch between the two patterns in Fig. 4 b indeed corresponds to a phason flip for VEs Q or D [53] (see Fig. 5 a and Supplementary Fig. 13 ). Phason flips between local isomorphic configurations affect only specific patches, not all Q and D. This contrasts with random phason flips applied to each patch, which introduce a tiling disorder [53] . The perspective on phason flips is enlightening. Perhaps we could relate HHG with the problem of phasonic disorder. Here, we try to investigate the impact of the phasonic disorder on harmonic yield in configuration 1 (see Supplementary Note 6 ). We found that as the disorder increases (controlled by flip iterations), the yields of some orders in the plateau region initially decrease and then recover, which can be qualitatively explained by the types and proportions of VEs (see Supplementary Fig. 14 ). A more detailed analysis of phasonic disorder’s effects on HHG will be presented in future work. Fig. 5: The γ -dependent VEs frequency, superuniformity, and harmonic yield. a 16 potential VEs in the generalized Penrose quasicrystal. b VEs frequency F as a function of γ . c A localized enlargement of the lower shaded region in ( b ). d Quasicrystal hyperuniformity parameter \\\\(\\\\bar{{{\\\\Lambda }}}(\\\\infty )\\\\) as a function of γ . Black and red lines represent results considering 400 and 10 local isomorphic configurations at each γ , respectively. e γ -dependence of 11th to 19th harmonic yields, shown as solid lines in black, blue, magenta, green, and purple, respectively. Laser parameters align with Fig. 2 . Source data are provided as a Source Data file. Full size image VEs and hyperuniformity In this part, we aim to uncover the reasons behind the differences in harmonic intensity among various LICs. In generalized Penrose quasicrystals, fat and thin rhombi collectively create 16 distinct VEs [4] , [5] , [52] , as illustrated in Fig. 5 a. It is worth noting that within these VEs, M (defined by two fat and two thin rhombuses) consists of two inequivalent forms that are mirror images and cannot be transformed into each other by translation or rotation in the 2D plane. The two patterns of M always appear in pairs in the discussed configurations and contribute to the harmonic radiation simultaneously. In the following discussion, we will refer to both forms collectively as M. The frequency F [4] , [5] , [52] , defined as the ratio of different VEs to the total vertices, varies with changes in LICs or parameter γ , as shown in Fig. 5 b and c The Penrose LIC (LIC1) with γ = 0 is distinguished for having a fewest types of VEs, potentially explaining the higher harmonic yield observed in Fig. 2 e. Taking the endpoint p of the VE Q in Fig. 5 a as an example, different LICs restrict the possible evolutions of p . In LIC1, p has six choices (Q, K, J, S, T, V), while in LIC2, there are ten choices (Q, K, M, J, R, S, U, W, Y, Z). It shows that in different LICs, the VEs diversity at scattering point p is different when electrons originate from Q. The analysis also applies to D or other VEs. In fact, all VEs in the system are interconnected through complex electron scattering and collectively contribute to harmonic radiation. Thus, greater VE diversity and a more uniform F distribution will lead to lower motion environment similarity for electrons originating from the same VEs but different positions, which reduces the HHG coherence and yield (as we will see in Fig. 5 e). Using Voronoi cells as a metric, Lin et al. [5] emphasized that the local distribution of VEs governs quasicrystals’ hyperuniformity, characterized by the suppression of long-wavelength density fluctuations. We expect that the variation of the hyperuniformity [5] , [6] will manifest in the harmonic spectrum of different LICs. Let’s discuss hyperuniformity briefly: considering quasicrystal atoms as a point pattern in d -dimensional Euclidean space \\\\({{\\\\mathbb{R}}}^{d}\\\\) , the number of points within a spherical window centered at x i with radius R is denoted as N ( R ; x i ), a random variable about x i with variance σ 2 ( R ). If σ 2 ( R ) grows slower than the window volume, i.e., σ 2 ( R ) ~ R α , α < d , the system is hyperuniform. In the generalized Penrose quasicrystal ( α = 1), σ 2 ( R ) is expressed as σ 2 ( R ) ~ Λ( R ) R + o ( R ), and o ( R ) denotes terms of lower order than R [5] . Following ref. [5] , we use the cumulative moving average \\\\(\\\\bar{{{\\\\Lambda }}}(R)=1/R\\\\int_{0}^{R}{{\\\\Lambda }}({R}^{{\\\\prime} })d{R}^{{\\\\prime} }\\\\) to eliminate small-scale variations and take the asymptotic average \\\\(\\\\bar{{{\\\\Lambda }}}(\\\\infty )=\\\\mathop{\\\\lim }_{R\\\\to \\\\infty }\\\\bar{{{\\\\Lambda }}}(R)\\\\) to describe the hyperuniformity. In calculations, quasicrystal atoms ( N atom ≈ 980) concentrate within a circle (radius R 0 = 16 l ) centered at the origin. M = 10,000 sampling window centers x i are uniformly distributed within a concentric circular region (radius R max = R 0 /2). The black dotted line in Fig. 5 d illustrates the γ -dependent hyperuniformity parameter \\\\(\\\\bar{{{\\\\Lambda }}}(\\\\infty )\\\\) , averaging over N con = 400 local isomorphic configurations for each γ point to mitigate finite-size fluctuations. Our results align with ref. [5] , showing that \\\\(\\\\bar{{{\\\\Lambda }}}(\\\\infty )\\\\) tends to increase monotonically with higher γ values. At γ = 0, LIC1 exhibits the global minimum of \\\\(\\\\bar{{{\\\\Lambda }}}(\\\\infty )\\\\) , with the highest level of hyperuniformity. As γ increases, hyperuniformity decreases for different LICs. The HHG yield VS the hyperuniformity of LICs While simulating the harmonic yield dependence on the parameter γ , to balance computational feasibility and accuracy, we focused on six distinct LICs at γ = 0, 0.1, 0.2, 0.3, 0.4, and 0.5. For each γ , we averaged \\\\(\\\\bar{{{\\\\Lambda }}}(\\\\infty )\\\\) over N con = 10 random configurations. The result exhibits a monotonic increase and closely agrees with the results obtained by averaging over 400 configurations, as shown by the red dotted line in Fig. 5 d. The current was averaged over the 10 configurations to obtain the HHG for each LIC. As shown in Fig. 5 e, with increasing γ , the yields of the 11th to 15th harmonics monotonically decrease, consistent with the hyperuniformity. For the 17th and 19th harmonics, an anomalous enhancement occurs for γ > 0.3. We speculate that this is due to the presence of Z and ST VEs at γ = 0.3 in Fig. 5 c. These VEs, characterized by the second-highest and highest occurrences of thin rhombi, exhibit high atomic density, leading to some electrons being bound. The bounded electrons will undergo rapid oscillations along the short diagonal of the thin rhombus under the external field, potentially causing an increase in high-frequency harmonic yield. Hence, we conclude that quasicrystal hyperuniformity can be characterized by HHG yields. Hyperuniformity also affects electron transport, notably reflected in current amplitude variations. In our simulation (see Supplementary Note 7 and Supplementary Fig. 15 a), increasing the γ parameter reduces hyperuniformity, leading to a gradual decline in current amplitudes for distinct LICs. This is also attributed to diverse VE choices, influencing the similarity of electron motion environments and affecting both the coherence and amplitude of current. Special VEs (X, Y, Z, and ST) [4] , with higher proportions of thin rhombi and atomic density, modify the electron momentum more strongly. Due to finite size constraints, the proportions of these VEs fluctuate within different configurations of one LIC, which is also reflected in the current amplitudes (see Supplementary Fig. 15 b).\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"Distinguishing LICs in quasicrystals is an important task since they have many distinct physical properties. Traditional EDS relies on the scattering near the nuclei, resulting in almost the same pattern for different LICs. HHG comes from the laser-induced electron motions with a wide range in quasicrystals, which encodes more information than EDS. Here we proposed to use high-order harmonic spectroscopy with more freedom to identify different LICs. When a linearly polarized laser illuminates the quasicrystal, the HHG yield of different LICs can have obvious distinctions. In addition, for our quasicrystal models, the orientation-dependent harmonic yield tends to exhibit C 10 symmetry (forbidden by crystallography), which is consistent with the orientation order and can take the role of the quasicrystal HHG fingerprint. Therefore, different LICs can be distinguished by HHG in yield and orientation-dependence. Furthermore, we use harmonic yields to reveal the quasicrystal hyperuniformity, which is dependent on the VE distribution of various LICs and impacts electron transport properties, such as current amplitude. Our work establishes a link between quasicrystal HHG and LIC, reveals the effect of VE distribution on electron scattering, and provides a new all-optical scheme for studying the structural properties of quasicrystal. It may also be applied to study the topological properties of quasicrystals in the future. Further, we note that the quasicrystal can be approximated by a periodic system with tunable unit cell sizes [4] . This approach will aid in understanding how periodicity breaking and electron incoherent scattering affect quasicrystal harmonic yield in future studies.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Initial state acquisition Before the laser field arrives, the electronic structure of the 2D generalized Penrose quasicrystal, including eigenenergies ε n and eigenstates ψ n , can be described by the stationary Schrödinger equation, $${\\\\hat{H}}_{0}{\\\\psi }_{n}={\\\\varepsilon }_{n}{\\\\psi }_{n}.$$ (1) Here, \\\\({\\\\hat{H}}_{0}\\\\) is the system’s stationary Hamiltonian, $${\\\\hat{H}}_{0}=\\\\frac{{\\\\hslash }^{2}}{2m}{{{{\\\\boldsymbol{\\\\nabla }}}}}^{2}+{V}_{ion}({{{\\\\bf{r}}}}).$$ (2) Similar to the tight-binding vertex model (considering interactions between atoms at rhombus vertices), the localized Gaussian potentials are adopted to describe the atomic Coulomb potential, $${V}_{{{{\\\\rm{ion}}}}}({{{\\\\bf{r}}}})=-\\\\sum _{v=1}^{{N}_{{{{\\\\rm{atom}}}}}}\\\\,{V}_{0}\\\\exp \\\\left[-\\\\frac{{\\\\left({{{\\\\bf{r}}}}-{{{{\\\\bf{r}}}}}_{v}\\\\right)}^{2}}{2{\\\\alpha }^{2}}\\\\right],$$ (3) where V 0 = 1 a.u., α = 0.8 a.u., and r v is the atomic position determined by the GDM. As shown in Fig. 1 , the N atom atoms of the 2D quasicrystal configurations are distributed within a circle of radius R 0 = 16 l , where l = 8 a.u. is the rhombus edge length. In the simulation, we construct the Hamiltonian matrix H 0 in discrete 2D real space using the finite difference method, then diagonalize it to obtain the eigenenergies and eigenstates, applying open boundary conditions. The number of initially occupied valence electrons equals the total number of atoms, N atom . As shown in Supplementary Note 1 and Supplementary Figs. 1 and 2 , when studying the system’s static properties (such as localization and energy spectrum), we use a circular computational box with a radius of R 0 to avoid interference from free electron states near the boundary. To simulate harmonic radiation from the laser-quasicrystal interaction, we construct the H 0 matrix in a larger rectangular box with side lengths 2 R 0 + 2 L a b and diagonalize it to obtain the N atom initial states (with ε n < 0). During electron evolution, an absorption function of the form \\\\({\\\\cos }^{1/8}\\\\) , with width L a b = 20 a.u., is applied near the box boundary to prevent electron reflection. Time-dependent Schrödinger equation The electron dynamics in the laser-quasicrystal interaction are described by the 2D TDSE in the length gauge, $$i\\\\hslash \\\\frac{\\\\partial }{\\\\partial t}\\\\psi ({{{\\\\bf{r}}}},t)=\\\\left[{\\\\hat{H}}_{0}+{{{\\\\bf{E}}}}(t)\\\\cdot {{{\\\\bf{r}}}}\\\\right]\\\\psi ({{{\\\\bf{r}}}},t).$$ (4) E ( t ) is the laser electric field. When the angle between the laser principal axis and the x -axis is θ (see Fig. 2 b), the components in x and y directions of the electric field can be written as, $${E}_{x}(t)={E}_{x,\\\\theta=0}(t)\\\\cos \\\\theta -{E}_{y,\\\\theta=0}(t)\\\\sin \\\\theta,$$ (5) $${E}_{y}(t)={E}_{x,\\\\theta=0}(t)\\\\sin \\\\theta+{E}_{y,\\\\theta=0}(t)\\\\cos \\\\theta .$$ (6) Here, $${E}_{x,\\\\theta=0}(t)={E}_{0}/\\\\sqrt{1+{\\\\epsilon }^{2}}f(t)\\\\cos ({\\\\omega }_{0}t),$$ (7) $${E}_{y,\\\\theta=0}(t)=\\\\epsilon {E}_{0}/\\\\sqrt{1+{\\\\epsilon }^{2}}f(t)\\\\sin ({\\\\omega }_{0}t).$$ (8) E 0 , ϵ , f ( t ), ω 0 , and L t , respectively, denote the field strength, ellipticity, envelope, carrier frequency, and total duration. In the calculation, we apply the split-operator method to evolve the N atom initial states ψ i independently to ψ i ( r , t ). The χ (= x or y ) direction harmonic spectrum H χ ( ω ) is given by the Fourier transform \\\\({H}_{\\\\chi }(\\\\omega )=| {{{\\\\mathscr{F}}}}[{J}_{\\\\chi }(t)]{| }^{2}\\\\) . J χ ( t ) is the total laser-induced current in the χ direction, with $${J}_{\\\\chi }(t)=\\\\sum\\\\limits_{i}^{{N}_{atom}}{j}_{\\\\chi,i}(t),$$ (9) and j χ , i ( t ) = \\\\(\\\\langle {\\\\psi }_{i}({{{\\\\bf{r}}}},t)| {\\\\hat{p}}_{\\\\chi }| {\\\\psi }_{i}({{{\\\\bf{r}}}},t)\\\\rangle\\\\) . The total HHG is H t o t ( ω ) = H x ( ω ) + H y ( ω ), and the yield I n of the n th harmonic is evaluated by, $${I}_{n}=\\\\int_{(n-0.5){\\\\omega }_{0}}^{(n+0.5){\\\\omega }_{0}}{H}_{tot}(\\\\omega )d\\\\omega .$$ (10) In the actual calculation, we set the time step d t = 0.25 a.u., and the grid spacing d x = d y = 0.25 a.u.\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"This research proposes that high-order harmonic generation (HHG) can serve as a novel and effective method to distinguish between different local isomorphism classes (LICs) of two-dimensional generalized Penrose quasicrystals by exploiting distinct differences in HHG yield and orientation dependence, which arise from variations in vertex environments linked to hyperuniformity, while also preserving the quasicrystal's characteristic tenfold rotational symmetry in the harmonic radiation."},"entities":{"kind":"list like","value":["high-order harmonic generation","HHG","local isomorphism classes","LICs","two-dimensional generalized Penrose quasicrystals","vertex environments","hyperuniformity","tenfold rotational symmetry","harmonic radiation","orientation dependence","HHG yield","quasicrystal","distinct differences","exploitation","preservation"],"string":"[\n \"high-order harmonic generation\",\n \"HHG\",\n \"local isomorphism classes\",\n \"LICs\",\n \"two-dimensional generalized Penrose quasicrystals\",\n \"vertex environments\",\n \"hyperuniformity\",\n \"tenfold rotational symmetry\",\n \"harmonic radiation\",\n \"orientation dependence\",\n \"HHG yield\",\n \"quasicrystal\",\n \"distinct differences\",\n \"exploitation\",\n \"preservation\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1.","title":"Levine, D. & Steinhardt, P. 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B110, 014209 (2024).","link":"https://doi.org/10.1103%2FPhysRevB.110.014209"},{"idx":"32.","title":"Yao, W. et al. Quasicrystalline 30∘twisted bilayer graphene as an incommensurate superlattice with strong interlayer coupling.Proc. Natl Acad. Sci. USA115, 6928–6933 (2018).","link":"https://doi.org/10.1073%2Fpnas.1720865115"},{"idx":"33.","title":"Wang, N., Chen, H. & Kuo, K. H. Two-dimensional quasicrystal with eightfold rotational symmetry.Phys. Rev. Lett.59, 1010–1013 (1987).","link":"https://doi.org/10.1103%2FPhysRevLett.59.1010"},{"idx":"34.","title":"Bendersky, L. Quasicrystal with one-dimensional translational symmetry and a tenfold rotation axis.Phys. Rev. Lett.55, 1461–1463 (1985).","link":"https://doi.org/10.1103%2FPhysRevLett.55.1461"},{"idx":"35.","title":"Ishimasa, T., Nissen, H.-U. & Fukano, Y. New ordered state between crystalline and amorphous in Ni-Cr particles.Phys. Rev. Lett.55, 511–513 (1985).","link":"https://doi.org/10.1103%2FPhysRevLett.55.511"},{"idx":"36.","title":"Tran, D. T., Dauphin, A., Goldman, N. & Gaspard, P. Topological Hofstadter insulators in a two-dimensional quasicrystal.Phys. Rev. B91, 085125 (2015).","link":"https://doi.org/10.1103%2FPhysRevB.91.085125"},{"idx":"38.","title":"Huang, H. & Liu, F. Quantum spin Hall effect and spin bott index in a quasicrystal lattice.Phys. Rev. Lett.121, 126401 (2018).","link":"https://doi.org/10.1103%2FPhysRevLett.121.126401"},{"idx":"39.","title":"Viebahn, K., Sbroscia, M., Carter, E., Yu, J. & Schneider, U. Matter-wave diffraction from a quasicrystalline optical lattice.Phys. Rev. Lett.122, 110404 (2019).","link":"https://doi.org/10.1103%2FPhysRevLett.122.110404"},{"idx":"40.","title":"Sbroscia, M. et al. Observing localization in a 2D quasicrystalline optical lattice.Phys. Rev. Lett.125, 200604 (2020).","link":"https://doi.org/10.1103%2FPhysRevLett.125.200604"},{"idx":"41.","title":"Spurrier, S. & Cooper, N. R. Semiclassical dynamics, Berry curvature, and spiral holonomy in optical quasicrystals.Phys. Rev. A97, 043603 (2018).","link":"https://doi.org/10.1103%2FPhysRevA.97.043603"},{"idx":"42.","title":"Penrose, R. Pentaplexity A class of non-periodic tilings of the plane.Math. Intell.2, 32–37 (1979).","link":"https://link.springer.com/doi/10.1007/BF03024384"},{"idx":"43.","title":"Choy, T. C. Density of states for a two-dimensional Penrose lattice: evidence of a strong Van-Hove singularity.Phys. Rev. Lett.55, 2915–2918 (1985).","link":"https://doi.org/10.1103%2FPhysRevLett.55.2915"},{"idx":"44.","title":"Odagaki, T. & Nguyen, D. Electronic and vibrational spectra of two-dimensional quasicrystals.Phys. Rev. B33, 2184–2190 (1986).","link":"https://doi.org/10.1103%2FPhysRevB.33.2184"},{"idx":"45.","title":"Ma, P. & Liu, Y. Inflation rules, band structure, and localization of electronic states in a two-dimensional Penrose lattice.Phys. Rev. B39, 9904–9911 (1989).","link":"https://doi.org/10.1103%2FPhysRevB.39.9904"},{"idx":"48.","title":"Elser, V. Indexing problems in quasicrystal diffraction.Phys. Rev. B32, 4892–4898 (1985).","link":"https://doi.org/10.1103%2FPhysRevB.32.4892"},{"idx":"49.","title":"Pavlovitch, A. & Kleman, M. Generalised 2D Penrose tilings: structural properties.J. Phys. A Math. Gen.20, 687 (1987).","link":"https://doi.org/10.1088%2F0305-4470%2F20%2F3%2F031"},{"idx":"50.","title":"Ishihara, K. N. & Yamamoto, A. Penrose patterns and related structures. I. Superstructure and generalized Penrose patterns.Acta Crystallogr. A44, 508–516 (1988).","link":"https://doi.org/10.1107%2FS0108767388002958"},{"idx":"51.","title":"Chodyn, M., Kuczera, P. & Wolny, J. Generalized Penrose tiling as a quasilattice for decagonal quasicrystal structure analysis.Acta Crystallogr. A71, 161–168 (2015).","link":"https://doi.org/10.1107%2FS2053273314024917"},{"idx":"52.","title":"Zobetz, E. & Preisinger, A. Vertex frequencies in generalized Penrose patterns.Acta Crystallogr. A46, 962–970 (1990).","link":"https://doi.org/10.1107%2FS0108767390008479"},{"idx":"53.","title":"Bugański, I., Strzalka, R. & Wolny, J. A new approach to phason disorder for a decagonal quasicrystal: the moment series expansion of the tiling distribution function for AlCuRh.J. App. Cryst.53, 904–913 (2020).","link":"https://doi.org/10.1107%2FS1600576720006251"},{"idx":"54.","title":"Guan, M. et al. Theoretical insights into ultrafast dynamics in quantum materials.Ultrafast Sci.2022, 9767251 (2022).","link":"https://doi.org/10.34133%2F2022%2F9767251"},{"idx":"55.","title":"Alon, O. E., Averbukh, V. & Moiseyev, N. Selection rules for the high harmonic generation spectra.Phys. Rev. Lett.80, 3743–3746 (1998).","link":"https://doi.org/10.1103%2FPhysRevLett.80.3743"},{"idx":"56.","title":"Liu, X. et al. Selection rules of high-order-harmonic generation: symmetries of molecules and laser fields.Phys. Rev. A94, 033410 (2016).","link":"https://doi.org/10.1103%2FPhysRevA.94.033410"},{"idx":"57.","title":"Saito, N. et al. Observation of selection rules for circularly polarized fields in high-harmonic generation from a crystalline solid.Optica4, 1333–1336 (2017).","link":"https://doi.org/10.1364%2FOPTICA.4.001333"},{"idx":"58.","title":"Apostolova, T. & Obreshkov, B. High harmonic generation from bulk diamond driven by intense femtosecond laser pulse.Diam. Relat. Mater.82, 165–172 (2018).","link":"https://doi.org/10.1016%2Fj.diamond.2017.12.013"},{"idx":"59.","title":"Golde, D., Meier, T. & Koch, S. W. High harmonics generated in semiconductor nanostructures by the coupled dynamics of optical inter- and intraband excitations.Phys. Rev. B77, 075330 (2008).","link":"https://doi.org/10.1103%2FPhysRevB.77.075330"},{"idx":"60.","title":"Schubert, O. et al. Sub-cycle control of terahertz high-harmonic generation by dynamical Bloch oscillations.Nat. Photon.8, 119–123 (2014).","link":"https://doi.org/10.1038%2Fnphoton.2013.349"},{"idx":"61.","title":"Vampa, G. et al. Theoretical analysis of high-harmonic generation in solids.Phys. Rev. Lett.113, 073901 (2014).","link":"https://doi.org/10.1103%2FPhysRevLett.113.073901"},{"idx":"62.","title":"Guan, Z., Zhou, X. & Bian, X. High-order-harmonic generation from periodic potentials driven by few-cycle laser pulses.Phys. Rev. A93, 033852 (2016).","link":"https://doi.org/10.1103%2FPhysRevA.93.033852"}],"string":"[\n {\n \"idx\": \"1.\",\n \"title\": \"Levine, D. & Steinhardt, P. J. Definition and structure.Phys. Rev. B34, 596–616 (1986).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.34.596\"\n },\n {\n \"idx\": \"2.\",\n \"title\": \"Socolar, J. E. S. & Steinhardt, P. J. 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Lett.124, 203901 (2020).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.124.203901\"\n },\n {\n \"idx\": \"19.\",\n \"title\": \"Itatani, J. et al. Tomographic imaging of molecular orbitals.Nature432, 867–871 (2004).\",\n \"link\": \"https://doi.org/10.1038%2Fnature03183\"\n },\n {\n \"idx\": \"20.\",\n \"title\": \"Shafir, D., Mairesse, Y., Villeneuve, D. M., Corkum, P. B. & Dudovich, N. Atomic wavefunctions probed through strong-field light-matter interaction.Nat. Phys.5, 412–416 (2009).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys1251\"\n },\n {\n \"idx\": \"21.\",\n \"title\": \"Haessler, S. et al. Attosecond imaging of molecular electronic wavepackets.Nat. Phys.6, 200–206 (2010).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys1511\"\n },\n {\n \"idx\": \"22.\",\n \"title\": \"Cireasa, R. et al. Probing molecular chirality on a sub-femtosecond timescale.Nat. Phys.11, 654–658 (2015).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys3369\"\n },\n {\n \"idx\": \"24.\",\n \"title\": \"Ayuso, D., Ordonez, A. F., Ivanov, M. & Smirnova, O. Ultrafast optical rotation in chiral molecules with ultrashort and tightly focused beams.Optica8, 1243–1246 (2021).\",\n \"link\": \"https://doi.org/10.1364%2FOPTICA.423618\"\n },\n {\n \"idx\": \"25.\",\n \"title\": \"Li, L. et al. Determination of electron band structure using temporal interferometry.Phys. Rev. Lett.124, 157403 (2020).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.124.157403\"\n },\n {\n \"idx\": \"26.\",\n \"title\": \"Lakhotia, H. et al. Laser picoscopy of valence electrons in solids.Nature583, 55–59 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-020-2429-z\"\n },\n {\n \"idx\": \"27.\",\n \"title\": \"Bauer, D. & Hansen, K. K. High-harmonic generation in solids with and without topological edge states.Phys. Rev. Lett.120, 177401 (2018).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.120.177401\"\n },\n {\n \"idx\": \"29.\",\n \"title\": \"Liu, J.-Q. & Bian, X.-B. Multichannel high-order harmonic generation from fractal bands in Fibonacci quasicrystals.Phys. Rev. Lett.127, 213901 (2021).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.127.213901\"\n },\n {\n \"idx\": \"30.\",\n \"title\": \"Liu, J.-Q. & Bian, X.-B. Detection of edge states by high-order harmonic generation in the Harper model.Phys. Rev. B105, 054308 (2022).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.105.054308\"\n },\n {\n \"idx\": \"31.\",\n \"title\": \"Dziurawiec, M. et al. Unraveling multifractality and mobility edges in quasiperiodic Aubry-André-Harper chains through high-harmonic generation.Phys. Rev. B110, 014209 (2024).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.110.014209\"\n },\n {\n \"idx\": \"32.\",\n \"title\": \"Yao, W. et al. Quasicrystalline 30∘twisted bilayer graphene as an incommensurate superlattice with strong interlayer coupling.Proc. Natl Acad. Sci. USA115, 6928–6933 (2018).\",\n \"link\": \"https://doi.org/10.1073%2Fpnas.1720865115\"\n },\n {\n \"idx\": \"33.\",\n \"title\": \"Wang, N., Chen, H. & Kuo, K. H. Two-dimensional quasicrystal with eightfold rotational symmetry.Phys. Rev. Lett.59, 1010–1013 (1987).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.59.1010\"\n },\n {\n \"idx\": \"34.\",\n \"title\": \"Bendersky, L. Quasicrystal with one-dimensional translational symmetry and a tenfold rotation axis.Phys. Rev. Lett.55, 1461–1463 (1985).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.55.1461\"\n },\n {\n \"idx\": \"35.\",\n \"title\": \"Ishimasa, T., Nissen, H.-U. & Fukano, Y. New ordered state between crystalline and amorphous in Ni-Cr particles.Phys. Rev. Lett.55, 511–513 (1985).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.55.511\"\n },\n {\n \"idx\": \"36.\",\n \"title\": \"Tran, D. T., Dauphin, A., Goldman, N. & Gaspard, P. Topological Hofstadter insulators in a two-dimensional quasicrystal.Phys. Rev. B91, 085125 (2015).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.91.085125\"\n },\n {\n \"idx\": \"38.\",\n \"title\": \"Huang, H. & Liu, F. Quantum spin Hall effect and spin bott index in a quasicrystal lattice.Phys. Rev. Lett.121, 126401 (2018).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.121.126401\"\n },\n {\n \"idx\": \"39.\",\n \"title\": \"Viebahn, K., Sbroscia, M., Carter, E., Yu, J. & Schneider, U. Matter-wave diffraction from a quasicrystalline optical lattice.Phys. Rev. Lett.122, 110404 (2019).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.122.110404\"\n },\n {\n \"idx\": \"40.\",\n \"title\": \"Sbroscia, M. et al. Observing localization in a 2D quasicrystalline optical lattice.Phys. Rev. Lett.125, 200604 (2020).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.125.200604\"\n },\n {\n \"idx\": \"41.\",\n \"title\": \"Spurrier, S. & Cooper, N. R. Semiclassical dynamics, Berry curvature, and spiral holonomy in optical quasicrystals.Phys. Rev. A97, 043603 (2018).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevA.97.043603\"\n },\n {\n \"idx\": \"42.\",\n \"title\": \"Penrose, R. Pentaplexity A class of non-periodic tilings of the plane.Math. Intell.2, 32–37 (1979).\",\n \"link\": \"https://link.springer.com/doi/10.1007/BF03024384\"\n },\n {\n \"idx\": \"43.\",\n \"title\": \"Choy, T. C. Density of states for a two-dimensional Penrose lattice: evidence of a strong Van-Hove singularity.Phys. Rev. Lett.55, 2915–2918 (1985).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.55.2915\"\n },\n {\n \"idx\": \"44.\",\n \"title\": \"Odagaki, T. & Nguyen, D. Electronic and vibrational spectra of two-dimensional quasicrystals.Phys. Rev. B33, 2184–2190 (1986).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.33.2184\"\n },\n {\n \"idx\": \"45.\",\n \"title\": \"Ma, P. & Liu, Y. Inflation rules, band structure, and localization of electronic states in a two-dimensional Penrose lattice.Phys. Rev. B39, 9904–9911 (1989).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.39.9904\"\n },\n {\n \"idx\": \"48.\",\n \"title\": \"Elser, V. Indexing problems in quasicrystal diffraction.Phys. Rev. B32, 4892–4898 (1985).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.32.4892\"\n },\n {\n \"idx\": \"49.\",\n \"title\": \"Pavlovitch, A. & Kleman, M. Generalised 2D Penrose tilings: structural properties.J. Phys. A Math. Gen.20, 687 (1987).\",\n \"link\": \"https://doi.org/10.1088%2F0305-4470%2F20%2F3%2F031\"\n },\n {\n \"idx\": \"50.\",\n \"title\": \"Ishihara, K. N. & Yamamoto, A. Penrose patterns and related structures. I. Superstructure and generalized Penrose patterns.Acta Crystallogr. A44, 508–516 (1988).\",\n \"link\": \"https://doi.org/10.1107%2FS0108767388002958\"\n },\n {\n \"idx\": \"51.\",\n \"title\": \"Chodyn, M., Kuczera, P. & Wolny, J. Generalized Penrose tiling as a quasilattice for decagonal quasicrystal structure analysis.Acta Crystallogr. A71, 161–168 (2015).\",\n \"link\": \"https://doi.org/10.1107%2FS2053273314024917\"\n },\n {\n \"idx\": \"52.\",\n \"title\": \"Zobetz, E. & Preisinger, A. Vertex frequencies in generalized Penrose patterns.Acta Crystallogr. A46, 962–970 (1990).\",\n \"link\": \"https://doi.org/10.1107%2FS0108767390008479\"\n },\n {\n \"idx\": \"53.\",\n \"title\": \"Bugański, I., Strzalka, R. & Wolny, J. A new approach to phason disorder for a decagonal quasicrystal: the moment series expansion of the tiling distribution function for AlCuRh.J. App. Cryst.53, 904–913 (2020).\",\n \"link\": \"https://doi.org/10.1107%2FS1600576720006251\"\n },\n {\n \"idx\": \"54.\",\n \"title\": \"Guan, M. et al. Theoretical insights into ultrafast dynamics in quantum materials.Ultrafast Sci.2022, 9767251 (2022).\",\n \"link\": \"https://doi.org/10.34133%2F2022%2F9767251\"\n },\n {\n \"idx\": \"55.\",\n \"title\": \"Alon, O. E., Averbukh, V. & Moiseyev, N. Selection rules for the high harmonic generation spectra.Phys. Rev. Lett.80, 3743–3746 (1998).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.80.3743\"\n },\n {\n \"idx\": \"56.\",\n \"title\": \"Liu, X. et al. Selection rules of high-order-harmonic generation: symmetries of molecules and laser fields.Phys. Rev. A94, 033410 (2016).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevA.94.033410\"\n },\n {\n \"idx\": \"57.\",\n \"title\": \"Saito, N. et al. Observation of selection rules for circularly polarized fields in high-harmonic generation from a crystalline solid.Optica4, 1333–1336 (2017).\",\n \"link\": \"https://doi.org/10.1364%2FOPTICA.4.001333\"\n },\n {\n \"idx\": \"58.\",\n \"title\": \"Apostolova, T. & Obreshkov, B. High harmonic generation from bulk diamond driven by intense femtosecond laser pulse.Diam. Relat. Mater.82, 165–172 (2018).\",\n \"link\": \"https://doi.org/10.1016%2Fj.diamond.2017.12.013\"\n },\n {\n \"idx\": \"59.\",\n \"title\": \"Golde, D., Meier, T. & Koch, S. W. High harmonics generated in semiconductor nanostructures by the coupled dynamics of optical inter- and intraband excitations.Phys. Rev. B77, 075330 (2008).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.77.075330\"\n },\n {\n \"idx\": \"60.\",\n \"title\": \"Schubert, O. et al. Sub-cycle control of terahertz high-harmonic generation by dynamical Bloch oscillations.Nat. Photon.8, 119–123 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fnphoton.2013.349\"\n },\n {\n \"idx\": \"61.\",\n \"title\": \"Vampa, G. et al. Theoretical analysis of high-harmonic generation in solids.Phys. Rev. Lett.113, 073901 (2014).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.113.073901\"\n },\n {\n \"idx\": \"62.\",\n \"title\": \"Guan, Z., Zhou, X. & Bian, X. High-order-harmonic generation from periodic potentials driven by few-cycle laser pulses.Phys. Rev. A93, 033852 (2016).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevA.93.033852\"\n }\n]","isConversational":false}}},{"rowIdx":193,"cells":{"unique_id":{"kind":"string","value":"s41467-020-20519-z"},"subfield":{"kind":"string","value":"Nanoscience and technology"},"title":{"kind":"string","value":"Spontaneous sparse learning for PCM-based memristor neural networks"},"abstract":{"kind":"string","value":"Neural networks trained by backpropagation have achieved tremendous successes on numerous intelligent tasks. However, naïve gradient-based training and updating methods on memristors impede applications due to intrinsic material properties. Here, we built a 39 nm 1 Gb phase change memory (PCM) memristor array and quantified the unique resistance drift effect. On this basis, spontaneous sparse learning (SSL) scheme that leverages the resistance drift to improve PCM-based memristor network training is developed. During training, SSL regards the drift effect as spontaneous consistency-based distillation process that reinforces the array weights at the high-resistance state continuously unless the gradient-based method switches them to low resistance. Experiments show that the SSL not only helps the convergence of network with better performance and sparsity controllability without additional computation in handwritten digit classification. This work promotes the learning algorithms with the intrinsic properties of memristor devices, opening a new direction for development of neuromorphic computing chips."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"Recently, the field of artificial intelligence (AI) has witnessed tremendous advances in deep neural networks (DNNs) [1] , [2] , [3] , [4] , [5] , [6] , [7] . In DNNs, as a connectionist approach to AI, knowledge is represented by a hierarchically distributed pattern of activation nodes and a large number of weights storing the connection strength between the nodes [8] , [9] , [10] . Since weights are trainable depending on some relevant data without explicit rules, ‘learning’ can be defined naturally [11] . However, to gain powerful generalization capabilities, DNNs usually have massive tunable weights, thus the network training and inference are such data-intensive that the computation efficiency is usually limited by the memory access [12] , [13] . These networks are highly amenable to computation via large and dense matrix–vector multiplications that can be highly parallelized [14] , [15] . This has led to tremendous opportunities for hardware acceleration. First, DNN accelerators and neuromorphic chips were designed with the application-specific integrated circuits to alleviate the memory access bottleneck to speed up the computation [16] . However, using this design approach, the synaptic weights are still stored in random access memories rather than directly encoded by the states of emerging analog devices and computed at the memory locations. Such analog devices assembled systems enable fundamental Non-von Neumann scheme and are believed to achieve significant speedup and power reduction for on-chip implementation of DNNs [17] , [18] , [19] , [20] , [21] . One of the most attractive analog devices is the two-terminal memristor, such as phase change memory (PCM) and resistive random access memory (RRAM), which offers the advantages of high density, fast operation speed, and low-power consumption [22] , [23] . The weight elements of DNNs can be represented by the conductance of memristors, which can be programmed by update pulses or read by low-amplitude reading pulses. Therefore, the weights in memristor networks could be accessed and tuned locally, making it extremely suitable for the DNN hardware representation [24] . When implemented in the crossbar structure, memristor networks are ideal substrates that directly perform multiply-accumulate operations (MACs) at the weight locations [25] . This parallel computing property speeds up the training and inference of DNNs with significant low-energy consumption, providing a promising computing paradigm for neuromorphic computing systems [26] , [27] , [28] . However, there is still a huge discrepancy between the ideal unrestricted weight change in DNNs and the actual conductance change in memristors, resulting in the imprecise encoding of network weights and the corresponding computation. The conductance change of memristors is not only non-linear, asymmetric, and precision-limited, but also stochastic because of the device-to-device and pulse-to-pulse variations [29] , [30] , [31] . These properties generally lead to non-convergence when training real memristor neural networks. A promising solution to this issue is developing a co-optimization design that combines device properties (hardware insights) with DNN training techniques (software insights), such as quantization, pruning, and sparsification. Several approaches have been reported to restore the properties of the ideal weight, which ensures more stable performance but results in inefficiency and impracticality [17] , [32] , [33] . Our studies indicated that an adequate learning scheme able to exploit the stochasticity in PCM and incorporate brain-inspired approaches to enhance DNNs is essential to overcome this issue [34] , [35] . At present, the development of memristors is still insufficient to meet the ideal requirements for DNN training. Comparing the emerging memories for synapse device, PCM exhibits advantages on on/off ratio, endurance, and retention than RRAM, and on/off ratio than MRAM [36] . In addition, PCM is more mature technology as it has been manufacturing on a foundry basis. The physical mechanism of PCM is also well understood, which facilitates IC design of chip products. Thus, it is very practical to overcome the disadvantages of PCM and exploit the high reliability and the suitability for DNN applications. However, a critical issue for realization of PCM-based memristor neural network is that even if the weights represented by the resistance are precisely tuned to the ideal weights, its most cumbersome feature, the resistance drift issue, still remains, which has not been considered in previous studies [31] , [37] . Resistance can deviate continuously from the initial intended values because PCM undergoes a spontaneous resistance increase due to the structural relaxation after amorphization [38] . The resistance drift not only causes a major reliability issue of PCM, but also exacerbates the imprecise encoding of network weights in memristors. The conventional approach is to minimize the resistance drift by implementing error correction codes or adding specific periphery circuits, such as customized circuitry for rapid iterative write, modulation coding for drift-resilient, coding to new readout schemes that are drift resilient [39] . However, such approaches result in significant increase of complexity and power consumptions, and decrease of throughput. Here we built a 39 nm 1 Gb PCM-based memristor neural network and quantified its non-ideal characteristics, especially the unique resistance drift effect. PCM weights play a unique role in in-memory computations (e.g., measuring time stayed in a weight state and modifying weight non-linearly according to the time) but the resistance drift effect is usually thought to be a problem for application. In this work, we make use of the unique resistance drift effect to develop a new learning scheme for PCM-based memristor neural network, and a multi-layer perceptron (MLP) with 784-256-10 structure and 203,264 weights was built. It has been reported that resistance drift increases the effect of image classification in SNNs, but it is explained as an effect of improving the weight representation and has not been linked to a new learning method [39] . In some neuromorphic computing systems, weights encoded by memristors are binary-valued (+1 or –1) for feasibility and thus weights stay in either +1 or –1 until an update process. To embody such abstraction in PCM-based memristor DNNs, we utilize the power of spontaneity of PCM memristor, that is, the electrical properties of the resistance drift that can be described by power law using a defect model [40] , [41] , [42] , [43] . In this setup, we fill the gap between actual behavior and unrealistic expectation in a PCM-based memristor DNN accelerator. At the same time, we incorporate sparsification based on weight consistency in a very natural way, i.e., discriminating a PCM weight staying consistently in high resistance with spontaneous resistance increase. During training, the negative weight (–1) corresponding to the crystalline state decreases the weighted sum (i.e., sum of multiplications between each input and weight) but the positive weight (+1 + δ , δ ≥ 0) corresponding to amorphous state increases the weighted sum for the non-negative inputs. Because δ is getting larger as long as a weight remains in the positive state, therefore, a prior knowledge that the consistency of positive weights increases the final output values potentially is incorporated in this PCM-based memristor neural network. Interestingly, the final output is selectively increased when the prediction is correct. That is, the consistency correlates with accuracy improvement. Weight consistency map reveals that the new learning scheme system prevents weights from being stuck in the one of weight states during the whole training, which controls the discrimination of the one weight state from the other weight state, i.e., the sparsity. It is noticeable that the weight sparsity is changed differently between input-side weights and output-side weights. The consistency boosts the tendency of sparsity enhancement by increasing the number of negative input-side weights but suppresses the increase of the number of negative output-side weights. Thus, SSL manifests the negative contribution of some input data to the accuracy by increasing the sparsity and utilize the minimize uncontrollable negative contribution to the output. A possible outline of PCM-based memristor neural network system, device implementation, operation schemes, and solution to resistance drift even after training is carefully addressed. Our approach not only improves the classification accuracy of the PCM-based memristor neural network on MNIST handwritten digit dataset from 89.6% to 93.2%, but also controls the weight sparsity differently between layers (+1.4% for the input-side layer of about 200k weights while –4.5% for the output-side layer of about 2.5k weights) without any additional computation and operation. The demonstration based on the statistical parameters extracted from a 1 Gb PCM array also indicates higher feasibility. Unlike the passive approach of suppressing the drift effect, our SSL promotes the fusion of computer-based neural algorithms with the intrinsic properties of memristor devices [44] , [45] , [46] . To our best knowledge, this is the first learning method leveraging the PCM resistance drift to improve the memristor-based neural network performance. This opens up a new path for the development of PCM-based memristor neuromorphic accelerators.","references":{"indices":[1,2,5,6,8,9,10,14,15,17,18,20,21,22,23,24,25,27,28,31,32,34,35,36,38,39,40,42,43,45],"infos":[{"idx":"1","title":"LeCun, Y., Bengio, Y. & Hinton, G. Deep learning.Nature521, 436–444 (2015).","link":"https://doi.org/10.1038%2Fnature14539"},{"idx":"2","title":"Schmidhuber, J. Deep learning in neural networks: an overview.Neural Netw.61, 85–117 (2015).","link":"https://doi.org/10.1016%2Fj.neunet.2014.09.003"},{"idx":"5","title":"Yao, X. A review of evolutionary artificial neural networks.Int. J. Intell. Syst.8, 539–567 (1993).","link":"https://doi.org/10.1002%2Fint.4550080406"},{"idx":"6","title":"Auda, G. & Kamel, M. Modular neural networks: a survey.Int. J. 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After the training, the positive weights are pinned to a fixed value to be used for the inference without weight variation. BPU has its own full precision weight storage to be updated, like conventional neural network models for weight quantization. The weight is quantized and then stored in FFUs for the feedforward process. Although the storage is just replaced to PCM memristor in this neural network, the consistency is stored as additional resistance increase in PCM memristor automatically as shown in Fig. 1b . Fig. 1: A PCM-based memristor neural network incorporating the resistance drift for the consistency-induced weight increase under distributed computing and weight precision reduction for new learning method. a The conventional neural network is depicted as a black box, which is divided into BPU (back propagation with 32-bit weight storage) and FFU (feed forward with R-drift incorporated PCM weight storage). The training process is completed with the PCM followed by the pinning operation to prevent unwanted R-drift for inference. b In conventional, weight change pattern does not affect itself directly. However, it can be possible through the resistance drift. For example, if a PCM remains in amorphous states consistently without switching, the weight is increased continuously according to the time. Then, weight change patterns can be reflected to weight value. c The high- and low-resistance are mapped to +1 and −1 weight state, respectively (see Fig. 2 ). The resistance can be in one of three states of ‘staying’ in crystalline state, ‘alternating’ between amorphous and crystalline, and ‘drifting’ in amorphous. Without priors about the weight, the weight will be +1 or −1 with 1/2 and 1/2 probability, respectively, in the conventional neural network. The probability that the final weight value is +1 is higher than 1/2 based on the likelihood in +1 state, like the Bayesian inference. The higher probability is encoded as greater weight value (1 + δ ) like the gain coding in neuroscience, which indicates the increase of neural activity in amplitude as the input stimulus becomes certain. d Continuous resistance read shows different behaviors between amorphous (DRIFT) and crystalline states (ON). The resistance in amorphous is increasing gradually while the resistance in crystalline is almost constant. The resistance drift can be described by the power law. R ( t 0 ) is the first resistance measurement after the time t 0 . R ( t ) is resistance at time t ( t > t 0 ) and d is the drift coefficient which can be determined from a curve fitting. e SEM corner view of the PCM (39 nm, 1 Gbit) array used. f TEM images of some cells in the yz and xz planes reveal the damascene structure of GST layer. Full size image A possible weight change in PCM memristor weight storage can be represented as a combination of three patterns; staying in –1, alternating between +1 and –1, increasing in +1 + δ (left column in Fig. 1c ). The probability distributions of the corresponding full precision weights are simplified through the binarization (middle column). As it is binarized, it discards a lot of information about the full precision weight. However, Bayesian inference can be considered as a method to reconstruct information about the probability distribution of full precision weights, which predicts posterior probability based on prior probability and likelihood. Since it is not known how the full precision weight will be formed, +1, –1 can be thought of as having a prior probability of 50:50. This is a method of predicting the distribution of full precision weights based on likelihood where +1 and –1 appear when binarizing. We can infer that the distribution of full precision weights is shifted toward the positive side by reflecting the likelihood of +1. This inferred probability distribution is called the posterior probability distribution, and in the next inference, the posterior probability of the previous step becomes the prior probability, and then inference continues according to likelihood. There is no way to perfectly reflect this post-probability in a 1-bit memristor, but PCM can encode the increased probability with an increase in weight value using the drift effect. Then, according to the prior (i.e., the significance of positive weights), the posterior is determined by increasing the weights inspired from the gain coding (right column) [48] , [49] . Thus the weight increase based on the weight history can be rationalized by Bayesian inference and the gain coding. The resistance drift of a PCM memristor is compared after two different operations, crystallization, and amorphization. Continuous resistance read shows different behaviors between amorphous (DRIFT) and crystalline states (ON), as shown in Fig. 1d . The resistance of the amorphous state is increasing gradually according to the power law, while the resistance of the crystalline state is almost constant [41] , [42] , [50] . Thus, the prior can be incorporated into the weights naturally. Scanning electron microscopy (SEM) and tunneling electron microscopy (TEM) reveal the PCM array and the cell structure as shown in Fig. 1e, f (more details of PCM array and cell in “Methods” section) PCM memristor operation In our neural network, although the conventional weight storage is replaced with PCM memristor, the operation of the weight storage is the same as the conventional one. Some memristors are switched (S) and others are not (N). When a PCM memristor is not switched and remains in the amorphous state, the resistance is increased due to the spontaneous resistance drift. Figure 2a shows an example of how the resistance state of the PCM memristor changes as a sequence of specific S and N is delivered. The PCM memristor is switched occasionally in the middle of continual resistance read. The resistance read at the amorphous state shows a slight increase in resistance until it is switched back to the crystalline state. From 200 to 1200 resistance reads, the resistance drift continues without switching the amorphous state. In this way, the previously defined consistency in the pattern of weight evolutions is recorded as a resistance drift. Fig. 2: Mapping methods and corresponding circuits. a (Top) The schematic indicates the process of switching the PCM weight storage in FFUs according to the binarization of updated full-precision weight in BPU. (Bottom) A explanatory pattern of the resistance change is shown with corresponding sequence of S (Switch) and N (Non-switch). b (Top) The equation represents a mapping from resistance to weight. Instead of implementing MAC, resistance R ji is read cell by cell in this mapping and the asymmetry in resistance variation (large variation in large resistance) is mitigated. (Bottom) The plot shows the results of the mapping from the resistance change in a . c (Top) The schematic indicates the mapping method defined in b . (Bottom) The resultant weight from the explanatory pattern of the resistance change in a through the mapping scheme in b . d (Top) The equation represents a mapping for the inference after the weight pinning process. (Bottom) A possible circuit to implement the feedforward acceleration (i.e., multiply-and-accumulate, MAC) is represented. The calculated results ( I i ) from the input ( V j ’s) in this circuit is identical to MAC process. Full size image Dual weight mapping strategy To implement spontaneous weight modification from the resistance drift, we must consider how to map different physical quantities (e.g., high and low resistances) with positive values to typical weight values (e.g., +1 and –1, respectively). This can be solved by some practical mapping methods known as 2-memristor cell [31] , [45] or dummy-cell methods [51] , [52] . Also, the mapping from conductance to weight values helps accelerate feedforward propagation, i.e., matrix–vector multiplication or MAC. However, training with resistance drift in conductance is not effective because some modifications are required to make the resistance drift effect equivalent in the conductance scheme. Therefore, we propose a resistance scheme for the training and conversion to conductance scheme for the inference. For the training, the PCM memristor is written and read memristor cell-by-memristor cell, but is read as an analog memory in our practice. The computations of the feedforward propagation are shared with FFUs for several training data though MAC is not introduced. Thus, it is freer for processing the resistance values, e.g., taking the logarithmic value of the resistance to equalize the magnitude of the variations in high and low resistance (denoted by, R h and R l , respectively). As shown in Fig. 2b , we obtained \\(W_{{ji}}^{\\mathrm{{train}}}\\) from the measurement of \\(I_{R_{ji}}\\) and some arithmetic operations. The dummy-cell method was used to assign the weight increase property only to the positive weights. T denotes the translation (highlighted by red color) to make a pair of R h and R l centered at the zero point and S denotes the scaling (highlighted by blue color) to make the translated values +1 and –1, respectively. As a result, we can obtain the intended weight values as shown in Fig. 2c . Then, the weighted sum is calculated separately for the feedforward propagation for the training. To accelerate the feedforward propagation during the inference after the training, it is required to map the weight values from the conductance. Because we propose the weight pinning, two distinguished static conductance states are required. Considering the reuse of the PCM memristor weight, all high resistance memristors (including resistance drift) are switched to low resistance (equivalently, high conductance denoted by G h ) while all low resistance memristors are switched to high resistance (equivalently, low conductance denoted by G l ). Then, G h and G l are mapped to +1 + δ pin and –1, respectively. The mapping for the inference is similar to that for the training process except T and S . In this scheme, the resistance drift is negligible due to the reciprocal relation between the resistance and the conductance. To implement MAC, we just measure the total current I i when adequate V j ’s are supplied according to the input data. The measured current can be formulated from the circuit and the result is identical to the indented weighted sum as depicted in Fig. 2d . Classification results The most prominent result observed in this research is that the classification accuracy increased up to ~93.2% after the resistance drift effect was introduced. Figure 3a shows the training (inset) and testing accuracy (from 10 repeats) for the network with the conventional binary weight (denoted by ‘conventional’) and the network with the spontaneously increasing weight (denoted by ‘drift’). The remarkable fact is that the improvement in accuracy requires little additional computation cost and operation energy. Interestingly, drift demonstrated a larger deviation earlier, but a steady increase in accuracy with deviation reduced, unlike conventional (error bars) networks. This is because the consistency for the weight change could not be sufficiently assessed at the early stage of training and then the consistency is getting established and incorporated well. Fig. 3: Classification accuracy. a The classification accuracy in the conventional and drift neural network is compared. The line is the mean value and the bar is standard deviation from 10 runs. Comparing with the conventional in both train and test accuracy, the drift exhibits smaller deviation in latter epochs while it does larger deviation in former epochs. The final test accuracies are 89.6% and 93.2% in conventional and drift, respectively. b Confusion matrix reveals the discrepancies between the true label and the prediction. The diagonal and off-diagonal components indicate the number of correct and incorrect predictions, respectively. The components in the additional row vector at the top and column vector at the right show the aggregated number of incorrect predictions along each column and true label along each row, respectively. The off-diagonal yellowish component indicates significant incorrect prediction in the conventional. The mostly mispredicted class is ‘8’ (red boxes) and the sum is 222. Also, the prediction as ‘3’ is mostly incorrect (black boxes) and the sum is 205. c Digit-wise accuracy reveals the difference of accuracy between digits. The digit ‘1’ seems to be mostly accurate while ‘8’ shows poor accuracy. d Digit ‘3’ and ‘8’ have top-2 attempt/success ratio of classification. However, the classification of ‘3’ is decent in the accuracy against the high attempt/success ratio while the classification of ‘8’ is not. e In the drift, the most misprediction of ‘8’ is mostly improved from 222 to 76. Also, the most misprediction as ‘3’ is mostly improved from 205 to 109, too. f Almost digits are improved in accuracy. Accuracy of digit ‘8’ and ‘9’ is greatly improved. Digit ‘0’ and ‘1’ still show high accuracy. Full size image To identify the improvement in accuracy, we examined the confusion matrix as shown in Fig. 3b and e . The confusion matrix indicated how many times the neural network classified each input digit image into each class. The diagonal number indicates correct classifications for each digit while the off-diagonal number means the total occurrence of the confusion (i.e., the discrepancy between the prediction and the truth). The numbers at the top and right (single row and column, respectively) are the sum of each row and column except diagonal value. For example, the highest value ‘205’ (sum of values surrounded by black lines in the confusion matrix) at the top means that the neural network mispredicted the input digit images as ‘3’ mostly (denoted by ‘misprediction as ‘3”), and the highest value ‘222’ (sum of values surrounded by red lines in confusion matrix) at the right means that the class ‘8’ is mostly mispredicted by the neural network (denoted by ‘misprediction of ‘8”). Because some specific digits like ‘3’ and ‘8’ showed distinctive results in confusion matrix, we compared a digit-wise accuracy during the training. The digit-wise accuracy is defined as the number of correct predictions divided by the number of actual images for each digit. Digit ‘0’ and ‘1’ outperform in accuracy while digit ‘8’ shows low accuracy overall in Fig. 3c . Although ‘3’ and ‘8’ show the top-2 highest attempt/success ratio (shown in Fig. 3d ), the accuracy of ‘3’ is much higher than the accuracy of ‘8’. The improved result (denoted by ‘Drift’) shows that the number of misprediction as ‘3’ is reduced from 205 to 109 and the number of misprediction of ‘8’ is reduced from 222 to 76 significantly. The SSL shows accuracy improvement for most of the digits, especially better for ‘8’ as found in Fig. 3f . Drift effect during training To reveal two contradicting results related to the output node ‘8’, that is, the weakest drift effect and the greatest improvement in accuracy, the impact of the drift effect on the final output in the feedforward process is analyzed in Fig. 4 . Here, the actual 256 nodes in the hidden layer and 10 output nodes were simplified to five ( n j = 5, n i = 5) for convenience as shown in Fig. 4a . Consequently, there are five classes (e.g., hand-written digits from 0, 1, 2, 3, and 4) to be distinguished and the weight has a 5 by 5 dimension. We introduced the drift effect as an additional term, \\(\\delta _{ji}^{\\left( {23} \\right)}\\left( t \\right)\\) to the weight connecting i th node in the second (hidden) layer and j th node in the third (output) layer. It should be noted that \\(\\delta _{ji}^{\\left( {23} \\right)}\\) ( t ) is non-zero positive only for j and i satisfying \\(w_{ji}^{\\left( {23} \\right){\\mathrm{{bin}}}}\\) = +1. In this schematic, the input vector \\(x_j^{\\left( 2 \\right)}\\) and weight matrix \\(w_{ji}^{\\left( {23} \\right){\\mathrm{{bin}}}}\\) are identical to those in other cases, respectively, but \\(\\delta _{ji}^{\\left( {23} \\right)}\\) ( t ) is different for all the cases. We compared three different cases as shown in Fig. 4b . Among weights with a value of +1 in the weight matrix, the weight with a drift value of 0 indicates the PCM cell that has just switched to +1 state without time to drift. Weights with zero drift were arbitrarily selected. Those positive weights that are just switched will have various drift values. The top panel is a conventional drift-less feedforward process so the weighted sum and the softmax output are calculated as \\(s_i^{\\left( 3 \\right)} = \\mathop {\\sum}\\nolimits_j {x_j^{\\left( 2 \\right)}} w_{ji}^{\\left( {23} \\right){\\mathrm{{bin}}}}\\) and \\(x_i^{\\left( 3 \\right)}\\) = softmax( \\(s_i^{\\left( 3 \\right)}\\) ), respectively. Because \\(x_i^{\\left( 3 \\right)}\\) has the highest value (0.44) at i = 3, the conventional neural network infers that the input is \\({\\mathrm{{class}}}_{i = 3}\\) . The middle and bottom panels indicate different effects of \\(\\delta _{ji}^{\\left( {23} \\right)}\\) ( t ), and thus weights experience the drift differently. For instance, some elements of the weighted sum are increased or decreased according to which weights experience the drift effect greater or lesser. The effect of drift to be emphasized is that the change in each \\(s_i^{\\left( 3 \\right)}\\) had a direct influence on every \\(x_i^{\\left( 3 \\right)}\\) because of the mutually exclusive property of the softmax function. In the middle panel, large \\(d_{i = 3}^{\\left( 3 \\right)}\\) (=1.70−1.10 = 0.6) made \\(x_{i = 3}^{\\left( 3 \\right){\\mathrm{{drift}}}}\\) (=softmax( \\(s_{i = 3}^{\\left( 3 \\right)}\\) + \\(d_{i = 3}^{\\left( 3 \\right)}\\) ) = 0.57) larger than \\(x_{i = 2}^{\\left( 3 \\right)}\\) (=0.44) and \\(x_{i = 2}^{\\left( 3 \\right){\\mathrm{{drift}}}}\\) (=softmax( \\(s_{i = 2}^{\\left( 3 \\right)}\\) + \\(d_{i = 2}^{\\left( 3 \\right)}\\) )=0.28) less than \\(x_{i = 2}^{\\left( 3 \\right)}=0.36\\) even for non-zero \\(d_{i = 2}^{\\left( 3 \\right)}\\) (=1.01−0.90 = −0.11) simultaneously. Therefore, loss ( \\(E = - \\mathop {\\sum}\\nolimits_{i = 1}^{n_i} {t_i\\log \\left( {x_i} \\right)}\\) ) will be reduced from 0.36 to 0.24 in the middle panel because the index i of the largest \\(d_i^{\\left( 3 \\right)}\\) is 3, which is identical to the index i of the true label of the input (on the other hand, \\(\\delta _{ji}^{(23)}\\left( t \\right)\\) has the opposite (0.4 and 0.1 were swapped from the middle panel) pattern in the bottom panel, so the prediction is incorrect (i.e., the indices are not identical each other) and the loss increases to 0.40. Fig. 4: Feedforward explains the relation between the weight increase and the output values. a The network model in use is schematically depicted. The components x, s, w , and t denote for input, weighted sum, weight, and target, respectively. b Three different cases are compared to identify weight increase effects on the final output with some simplifications. All parameters are identical except the pattern of the weight increase term, \\(\\delta _{ji}^{\\left( {23} \\right)}(t)\\) . The first panel explains the conventional feedforward process without the consistency-induced weight increase. The only weights between the hidden and output layers are considered. The input, weights, and the output are 1 × 5 row vector, 5 × 5 matrix, and 1 × 5 row vector, respectively in this schematic for convenience. The feedforward process can be calculated straightforwardly. For the input and weights, the final \\(x_{i = 3}^{\\left( 3 \\right)}\\) has the greatest value and thus the neural network classify the input as \\({\\mathrm{{class}}}_{i = 3}\\) . In this case, the loss is 0.36 if the true label is [0 0 1 0 0]. In the middle panel comparing with the bottom panel, \\(\\delta _{ji}^{\\left( {23} \\right)}(t)\\) has the opposite pattern, i.e., the highest (0.4) and the lowest (0.1) values are swapped. According to the \\(\\delta _{ji}^{\\left( {23} \\right)}(t)\\) , \\(d_i^{\\left( 3 \\right)}\\) is added to the original \\(s_i^{\\left( 3 \\right)}\\) in a , so some components are increased or decreased. Full size image Because \\(\\delta _{ji}\\left( t \\right)\\) is determined internally during the training process, we can only estimate it heuristically through the above two extreme cases. If the drift effect improves the prediction, the loss will be reduced and the weight update becomes smaller than the conventional case. On the contrary, if the drift effect makes the prediction worse, the loss will increase and the weight update becomes larger than the conventional case correspondingly. In this way, some weights that have a significant contribution to minimizing the loss function will most likely experience the drift effect. Weight consistency Although it is hard to interpret the optimization of the drift effect during the training explicitly, the weight consistency reveals the pattern of weight changes. Figure 5a shows the results made from the subtraction of two heatmaps which implies the weight consistency in the conventional and drift, respectively. Thus, larger values indicate more significant drift effects on the weight consistency. The numbers in the horizontal axis indicates the digits (0–9) to be classified. The numbers in vertical axis (−10 to +10) denote how many times a weight stays in the same state for the first 10 epochs. The 256 weights can be considered to be associated with the output node ‘0’. The consistency of these weights can be reflected by adding each weight value once for each epoch. For example, if a weight has been in the −1 state for 10 epochs, the added weight value is −10. Thus, all of 256 weights assigned to each digit falls into one of the values between −10 and +10 for both the conventional and drift networks. The negative values at the bottom (deep bluish squares) indicates the slight reduction of the number of weights stuck in negative states. During the last 10 epochs, the negative consistent weights are significantly reduced, and some weights remain in the positive states long instead in Fig. 5b . For the input-side weights, the consistency of 28 × 28 weights connected to each node in hidden layer of the neural network is accumulated. There is no noticeable pattern during the first 10 epochs, but some weights connected to the central region of the input images show stronger negative contribution to the accuracy during the last 10 epochs as shown in Fig. 5c and d. At the early of training, there is no significant difference in weight consistency. As the training progresses, some weights stuck in either positive or negative states are remarkably reduced and some weights changing between two states are increased. Interestingly, the input-side and output-side weights show opposite patterns. Some weights stuck in the positive state are reduced for the input-side weights but some weights in the negative state are reduced for the output-side weights. Therefore, the accuracy-oriented consistency leads to the perturbation of some weights. Fig. 5: Weight consistency. a The graph explains how the drift effect affects the output-side weight consistency. The numbers in horizontal axis indicates the digit (0~9) to be classified. The numbers in vertical axis (−10 to +10) means how many times a weight stays in the same state for the first 10 epochs. Thus, all of 256 weights assigned to each digit fall into one of the values between −10 and +10 for both the conventional and drift networks. The values in the heatmap is the difference between the conventional and drift. The negative values at the bottom (deep blueish squares) indicates the slight reduction of the number of weights stuck in negative states. b During the last 10 epochs, the negative consistent weights is significantly reduced and some weights remain in the positive states long instead. c For the input-side weights, the consistency of 28 × 28 weights connected to each node in hidden layer of the neural network is accumulated. There is no noticeable pattern during the first 10 epochs. d During the last 10 epochs, some weights connected to the central region of the input images show stronger negative contribution to the accuracy. Full size image Sparsity Because the perturbation of some weights is not reflected in the accuracy calculated at a certain time, the weight sparsity (i.e., the number of negative weights over the number of total weights) is examined. In the conventional, the number of negative weights is increased for both the input-side (Fig. 6a ) and output-side (Fig. 6b ) weights during the training. On the contrary, in the drift, the number of negative weights is further increased for the input-side weights (Fig. 6c ) by about 1.4%, but the number of negative weights is decreased for the output-side weights (Fig. 6d ) by about 4.5%. The blue and red lines indicate the proportion of the total negative and positive weights, respectively. The output-side weights are grouped by the output digits and the input-side weights are grouped by each node in hidden layers (i.e., 256 groups and 28 × 28 weights per group). Fig. 6: Weight sparsity. a For input-side weights, the number of negative weights is increased in the conventional. b For output-side weights, the sparsity shows the gradual increase of negative weights in the conventional. c The negative contribution is promoted in the drift. d The sparsity is fluctuated slightly and not changed significantly in the drift. Thus, weight sparsity exhibits different trends between output-side and input-side weights. Full size image Considering the introduction of the accuracy-oriented consistency, the consistency of some weights compromising the accuracy will be excluded mostly. We can imagine an overlap between the consistency and accuracy. The weights can be subdivided by the position in the neural network (i.e., input-side or output-side), the consistency, and the sparsity. Therefore, the overlap portion (denoted by selectivity) determines the effectiveness of SSL scheme. Because the drift effect increases the output due to the non-negative contribution to the weighted sum basically, the similarity of the output vector to a one-hot vector indicates the effectiveness of SSL. Selectivity The maximum component in an output vector is <1 and the rests are larger than 0 as shown in Fig. 7a . We can define the digit-wise difficulty of classification based on the output vector, because the neural networks try to make the output like a one-hot vector as much as possible. The classification of a digit can be considered much difficult as more as the component of the digit in output vectors deviates from 1 or 0. In this perspective, the digit ‘8’ is much difficult to be distinguished from others than ‘1’ in the conventional. Surprisingly, the drift effect discriminates the digit ‘1’ output as two subgroups, though the average output of ‘1’ is close to 1 in Fig. 7a . In each group, the output of ‘1’ in the conventional is totally different according to whether a certain digit ‘1’ input image is classified correctly in the drift or not (two top-right bluish lines) as shown in Fig. 7b . That is, the output of ‘1’ in the conventional is extremely deviated from 1 for some images classified incorrectly in the drift. However, digit ‘8’ does not show such dependence on the correctness in the drift (two top-left reddish lines). The drift effect makes the output of ‘8’ close to a one-hot vector eventually (orange lines). In Fig. 7c , the average output is improved and then shows the promising output like a one-hot vector. The accuracy-oriented consistency subdivides both the digit ‘8’ and ‘0’ according to the difficulty. Then, SSL exploits the decision boundary of classification to maximize the accuracy. Fig. 7: Selectivity. a Average output for the test image is compared among digits. The results show the greatest deviation from 1 and 0 for the digit ‘8’, which indicates ‘8’ is hard to train in conventional. Digit ‘1’ is the closest to 1. b Although the average output of ‘1’ is close to 1 in a , however, drift effect discriminates the ‘0’ input images two subgroups. The output of ‘1’ in the conventional is totally different according to whether the input image is classified correctly in the drift (two top-right bluish lines). That is, the output of ‘1’ in the conventional is extremely deviated from 1 for some images classified incorrectly in the drift. However, digit ‘8’ do not show the dependence on the correctness in the drift (two top-left reddish lines). The drift effect makes the output of ‘8’ close to an one-hot vector eventually (orange lines). c Average output is improved and then shows the promising output like a one-hot vector. d Number of increases and decreases of the maximum values of \\(x_i^{(3){\\mathrm{{drift}}}}\\) with respect to \\(x_i^{(3)}\\) and their ratio demonstrates the selectivity of the output modification. The overall average is 6.29. When the drift neural network predicts correctly, the ratio is increased to 10.67 (for the test images predicted incorrectly in the conventional neural network) and 9.68 (for the test images predicted correctly in the conventional neural network). In the contrary, the ratio is far below the average when the drift neural network predicts incorrectly. Full size image The number of increases and decreases of Max( \\(x_i^{{\\mathrm{{drift}}}}\\) ) compared with Max( x i ) are plotted with respect to the correctness, as shown in Fig. 7d . The total number of increases or decreases can be divided into four cases, I/I, I/C, C/I, and C/C, where I and C denote ‘incorrect’ and ‘correct’ respectively, and the former and the latter refer to ‘conventional/drift’. The total number of increases is larger than that of decreases essentially and the ratio (i.e., the number of increases over the number of decreases) is 6.29. It is noticeable that the ratio is remarkably higher when the classification of the drift case is correct. Simultaneously, the ratio is reduced considerably when the result is incorrect. The ratios are 10.7 and 9.68 for I/C and C/C (above the overall ratio), whereas they are 1.06 and 2.85 for C/I and I/I (below the overall ratio), respectively. The results indicate that the selectivity of the output increase is originated from the consistency and can be supported analytically by tracing the backpropagation process. Weight pinning One important issue of the spontaneous weight modification utilizing resistance drift is that the optimized weight will undergo inevitable further change after training. However, a suitable constant weight value can be found to represent all individually changed weights due to the drift without degrading performance in the inference process. After training, all positive weights can be regarded as a fixed positive constant like a conventional digital memory device. Then we can find the constant that maximizes the accuracy of the test images. When the fixed positive constant is changed from 1.05 to 1.70 by 0.05 step, we compared the accuracy and then found that it is compatible to the accuracy just after the training when w pin = 1.4 (name after weight pinning) as shown in Fig. 8a . Fig. 8: Weight pinning and summary. a Due to the spontaneity of the resistance drift, the weight should be pinned not to lose the learned from the consistency. After the training, we can find a suitable weight value w pin (e.g., by scanning from 1.05 to 1.70) to maximize the accuracy near the best obtained for the training. w pin is 1.4 for the neural network and corresponding parameters used in this work. b The effect of the consistency-induced weight increase can be summarized as a weight sparsification based on the input-wise classification difficulty. Full size image"},{"section":"Discussion","content":"From the digit-wise analysis, we found some results. (1) Digit ‘8’ classification is mostly inaccurate. (2) The accuracy-oriented consistency (i.e., drift effect) is introduced. (3) The consistency controls the weight sparsity, so the weights are regrouped based on the digit-wise (via output-side weights) and image-wise (via input-side weights) manners. (4) By drift effect, digit ‘1’ input images are easily subdivided into two parts while digit ‘8’ input images are not. (5) Then, the network increases the classification accuracy of ‘8’ mostly by exploiting the room for weight modification without degrading the classification accuracy of ‘1’. The accuracy improvement due to consistency-induced weight increase is summarized as shown in Fig. 8b . Our approach is rationalized in that the difficulty of classification is related to the difficulty of weight determination (i.e., consistency) [53] , [54] , [55] . The difficulty-based accuracy improvement is similar to the experimental results demonstrating the decision confidence in the brain [56] , [57] , [58] , [59] . In animal experiments, different neural activity according to the difficulty of the decision is the most important feature of the confidence [49] , [60] . For example, for a mixture of two stimuli (e.g., scent A and scent B) with a different ratio, neural activity is most active for incorrect decision in easy decision tasks (A: 100% +B: 0% or A: 0% +B: 100%) and least active for correct decision in easy decision tasks. In the contrary, for the decision in difficult decision tasks (A: 51% +B: 49% or A: 49% +B: 51%), neural activity is moderately active regardless of the correctness. For this reason, the process of confidence formation is to classify easy tasks definitely, which is similar to the classification of ‘1’ under the drift effect. It is generally known that the reset current decreases as the cell size decreases [61] , [62] . In addition, a lower reset current is associated with a decrease in amorphous volume and consequent higher amorphous resistance [63] . Interestingly, the resistance in the amorphous state correlates with the drift coefficient [64] . The higher the amorphous resistance, the greater the drift coefficient. To our best knowledge, there is no publication on the impact of technology scaling on the resistance drift yet. Various experimental results have explained resistance drift in different ways and share a commonality that it is related to structural relaxation and stress release. Therefore, the electronic state of the material has a direct effect on the resistance drift rather than the technology node scaling or device structure. When PCM cell is not fully reset, the drift coefficient as well as resistance becomes smaller. The drift coefficient of the fully reset GST is ~0.09–0.11 [39] . In our case, PCM is almost fully reset (~0.10) and thus the impact due to technology node reduction is negligible. Even if the drift coefficient becomes greater with the size scaling or structure changes, we can slightly increase the reset current to decrease the reset resistance, thus maintaining the drift coefficient to a similar level. We tested the classification accuracy over a wide range of drift coefficient means and deviations as shown in Supplementary Fig. 7 . The consistently improved accuracy indicates that our learning scheme is very robust to the changes in drift effect caused by technology scaling or environmental temperatures. Interestingly, it is known that the drift coefficient does not depend on the constant temperature [42] , [65] . If the temperature rises suddenly, the drift coefficient temporarily increases. As shown in Supplementary Fig. 7c , the accuracy remains increasingly improved until the drift coefficient increases to ~1. Since the drift factor in the fully reset state is bounded, it does not have a value much higher than 0.12 for GST-based PCM [43] , [66] . Therefore, the accuracy of proposed scheme will not deteriorate at a high temperature as long as it is below the glass transition temperature of phase change materials. In this work, we built an ensemble of a mature 1 Gb PCM array with 39 nm technology, and by leveraging the statistical parameters obtained from the measurement of resistance drift, demonstrated a spontaneous sparse learning scheme in a PCM-based memristor neural network. By encoding the consistency of weight changes into the weight increases spontaneously, the neural network demonstrates the discriminated weight modification depending on the classification difficulty. Without any additional computation and operation costs, this new training scheme with sparsity controlled (+1.4% for output-side weights and −4.5% for input-side weights) improves the accuracy (93.2%) of a handwritten digit classification application (MNIST dataset) compared with the baseline neural network (89.6%). Additionally, due to the compatibility of the consistency-induced weight increase with any PCM-based memristor neural network, our results have potential to be related to evolutionary neural networks [4] , [5] inspired by the spontaneous behavior and modular neural networks [6] , [7] , [67] . Our work provides a potential prototype neural network that implements a metacognitive function, such as confidence, with low computation workloads for lightweight devices. Furthermore, our approach can be applied to RRAM as well as PCM, and improved learning effects are expected in more complex networks using multi-level cells. Although the resistance drift is weak in RRAM, the resistance drift appears distinctly in the conductive-bridge RAM, one of RRAM branches, due to the lateral diffusion of the metal constituting the bridge [68] . Since the drift coefficient is roughly observed within the range of 0.001–0.1 similar to PCM depending on the resistance of the initial resistance, the method presented in this study is expected to be applicable to those RRAM devices [69] . Our work may provide inspirations to RRAM research, i.e., leveraging the unique material characteristics (e.g., conductive-bridge modulation) to improve the performance/efficiency of memristor networks or to incorporate another learning scheme into memristor networks. The basic operation of our PCM array is well evaluated up to the 2-bit 4 level. The proposed SSL algorithm is expected to further improve the performance of network in the case of multilevel operation. Our proposed learning scheme utilizes the intrinsic resistance drift to improve the learning accuracy without additional cost, which opens up a new path to develop learning schemes using the materials’ properties."},{"section":"Methods","content":"Device fabrication Cross-point PCM arrays of cells with n +-Si/n-Si/ p -Si/TiN/Ge 2 Sb 2 Te 5 (GST)/TiW/W structures were fabricated on a 300-mm Si wafer. A highly doped n +-Si layer was prepared with B at a doping concentration exceeding 10 22 atoms/cm 2 . The n -Si/ p -Si diode was epitaxially grown on the wafer by chemical vapor deposition (CVD) with B at a doping concentration of 10 21 atoms/cm 2 for n -type and P at a doping concentration of 10 20 atoms/cm 2 for p -type. TiN films were deposited on the diodes by physical vapor deposition (PVD) as the bottom electrode (BE). The stack was dry-etched by etching along the y -direction and then along the x -direction to form pillar structures. The stack of n-Si/p-Si/TiN was etched into y -axis-oriented parallel lines using a hard mask of SiO 2 . The gaps between the etched lines were filled with SiO 2 . The gap-filled structure was flattened by chemical–mechanical polishing (CMP) and etched again with x -axis-oriented parallel lines. This formed word lines (WL) of n+-Si at the bottom of the diode. The gaps between diodes were then filled with SiO 2 and flattened by CMP. Then, interconnects and trenches were formed in the SiO 2 with lithography along the y -direction to build GST and a top electrode (TE) on the TiN through the dual damascene process. The interconnects and trenches were filled with GST and then TiW was deposited as the TE. After CMP, a line of W metal was formed using lithography on the TiW as the bit line. Finally, the cross-point PCM array was fabricated. The contact area between the GST and bottom electrode was ~40 nm × 80 nm. PCM Array information The PCM Array used in the research is an array with 39 nm technology in the process of developing PCM arrays with 90, 50, 45, 39 and 20 nm technologies [70] , [71] , [72] (officially reported for some technology only). Various operations and evaluation data including temperature and resistance drift for each technology were partially disclosed [73] , [74] . For example, resistance distributions for four resistance levels of (00), (01), (10), and (11) after an elapse of 400 h (1.4 × 10 6 s) and an additional thermal annealing at 130 °C for 12 h bake were provided for 90 nm PCM array. Also, for 45 nm PCM array, analysis of the drift effect was presented in depth to improve multi-level operational characteristics and verify practicality. The total array sizes fabricated by 90 and 45 nm technology are 512 Mb (16 banks of 32 Mb, 8 blocks/bank) and 1 Gbit (16 banks of 64 Mb, 8 blocks), respectively. For the 39 nm PCM Array, the temperature and drift characteristics were not officially disclosed at the wafer-level, but they are similar to those of the 45 nm PCM array. For the 39 nm PCM array, 2-PCM synapse behavior was evaluated [75] . Although the operation circuit was slightly deformed according to the characteristics of each technology to implement ancillary functions for performance improvement, such as high read or write throughput, the characteristics of each PCM array were well evaluated as can be seen in the literature provided. Measurement Each PCM cell is in the low resistance state at about 10 4 Ω which was measured by using 1 V pulse with 5 ns rising time, 60 ns duration, and 5 ns falling time. The cell resistance is increased to about 10 7 Ω using a pulse of 5 V height with 5 ns rising time, 60 ns duration, and 5 ns falling time. We then make the cell have a few 10 4 Ω by applying a voltage of 3 V with 5 ns rising time, 400 ns duration, and 300 ns falling time. The PCM cell is switched 20 times between set and reset states with 10 resistance read after each switching. Then the resistance is measured 1000 times just after the reset switching to measure the resistance drift as depicted in Supplementary Fig. 1a . A pulse generator (Keysight 33600A) was used to provide electrical pulse to the cell. A source meter (Keithley 2600) was used to measure the cell resistance. Pulse is applied to the TE while bottom electrode is ground through the contact pads. Supplementary Fig. 1b shows the wafer being measured. Parameter extraction We extracted resistance distribution and deviation from over 100 cells. For each cell, the set and the reset resistance were measured over 100 data points and then the resistance drift was measured for 1000 data point with 1 read-operation per 1 s. Then, the sample mean and the sample standard deviation of the set resistance, the reset resistance, and the drift parameter ( d in Eq. (1)) were extracted and denoted by \\(\\mu _{{\\mathrm{{SET}}}_i}^{\\mathrm{{s}}}\\) , \\(\\sigma _{{\\mathrm{{SET}}}_i}^{\\mathrm{{s}}}\\) , \\(\\mu _{{\\mathrm{{RESET}}}_i}^{\\mathrm{{s}}}\\) , \\(\\sigma _{{\\mathrm{{RESET}}}_i}^{\\mathrm{{s}}}\\) , \\(\\mu _{{\\mathrm{{DRIFT}}}_i}^{\\mathrm{{s}}}\\) , and \\(\\sigma _{{\\mathrm{{DRIFT}}}_i}^{\\mathrm{{s}}}\\) , respectively, for the \\(i{{\\mathrm{th}}}\\) cell. Then, the statistical parameter for the resistance distribution, i.e., population mean ( \\(\\mu _{{\\mathrm{{SET}}}_i}^{\\mathrm{{p}}}\\) , \\(\\mu _{{\\mathrm{{RESET}}}_i}^{\\mathrm{{p}}}\\) , and \\(\\mu _{{\\mathrm{{DRIFT}}}_i}^{\\mathrm{{p}}}\\) ) and population standard deviation ( \\(\\sigma _{{\\mathrm{{SET}}}_i}^{\\mathrm{{p}}}\\) , \\(\\sigma _{{\\mathrm{{RESET}}}_i}^{\\mathrm{{p}}}\\) , and \\(\\sigma _{{\\mathrm{{DRIFT}}}_i}^{\\mathrm{{p}}}\\) ) with an error term depending on the number of sampling data and the distribution model were calculated for the i th cell. Supposed that the resistance distribution is a normal distribution, \\(\\mu _i^{\\mathrm{{p}}}\\) can be estimated as \\(\\mu _i^{\\mathrm{{s}}}\\) with standard error of \\(\\frac{{\\mu _i^{\\mathrm{{s}}}}}{{\\sqrt n }}\\) while \\(\\sigma _i^{\\mathrm{{p}}}\\) can be estimated as \\(\\sigma _i^{\\mathrm{{s}}}\\) with standard error of \\(\\sqrt {\\frac{2}{{n - 1}}} \\left( {\\sigma _i^{\\mathrm{{s}}}} \\right)^2\\) . This process is depicted in Supplementary Fig. 2 (i and ii) for R ij , j th measurement of resistance of i th cell. In addition, the cell-to-cell population mean ( M p ) and population standard deviation ( Σ p ) were determined for both distributions of \\(\\mu _i^{\\mathrm{{p}}}\\) ’s and \\(\\sigma _i^{\\mathrm{{p}}}\\) ’s as shown in Supplementary Fig. 3 (iii and iv), which are denoted by \\(M_\\mu ^{\\mathrm{{p}}}\\) , \\(\\Sigma _\\mu ^{\\mathrm{{p}}}\\) and \\(M_\\sigma ^{\\mathrm{{p}}}\\) , \\(\\Sigma _\\sigma ^{\\mathrm{{p}}}\\) . By sampling a value from the normal distribution defined by \\(M_\\mu ^{\\mathrm{{p}}}\\) and \\(\\Sigma _\\mu ^{\\mathrm{{p}}}\\) , we could determine a mean resistance ( μ * ) of a possible PCM cell. Then, the resistance deviation ( α * ) that corresponds with μ * is determined as a value that has the same cumulated probability in the normal distribution defined by \\(M_\\sigma ^{\\mathrm{{p}}}\\) , \\(\\Sigma _\\sigma ^{\\mathrm{{p}}}\\) . Thus, six unique parameters ( \\(\\mu _{{\\mathrm{{set}}}_i}^ \\ast\\) , \\(\\sigma _{{\\mathrm{{set}}}_i}^ \\ast\\) , \\(\\mu _{{\\mathrm{{reset}}}_i}^ \\ast\\) , \\(\\sigma _{{\\mathrm{{reset}}}_i}^ \\ast\\) , \\(\\mu _{{\\mathrm{{drift}}}_i}^ \\ast\\) , and \\(\\sigma _{{\\mathrm{{drift}}}_i}^ \\ast\\) ) were assigned to each PCM cell for set, reset, and drift independently and randomly. Finally, \\(R_{{\\mathrm{{set}}}}\\) , \\(R_{{\\mathrm{{reset}}}}\\) , and d drift were generated and renewed every switching based on the uniquely assigned parameters to PCM cells. This process is depicted in Supplementary Fig. 4 (v, vi, and vii) and key statistical data that is determined from measurement are represented in bottom of the figure. Neural network We used a neural network with 784 nodes in input layer (denoted by \\(x_k^{\\left( 1 \\right)}\\) ), 256 nodes in one hidden layer (denoted by \\(x_j^{\\left( 2 \\right)}\\) ), and 10 nodes in output layer (denoted by \\(x_i^{\\left( 3 \\right)}\\) ), for the MNIST hand-written digit classification, as shown in Supplementary Fig. 5 . The layers and the weights between the layers are distinguished by the superscript. All nodes are fully connected and any regularization process such as batch normalization and dropout are excluded. Each 28 × 28 greyscale input image is normalized to [0, 1] and denoted by \\(x_k^{\\left( 1 \\right)}\\) . The weights are initialized by the variance scaling initialization. We applied ReLU activation in the hidden layer and softmax in the output layer, and then cross-entropy function was used as a loss function. To exclude any adaptive factor, we used stochastic gradient descent optimizer, constant learning rate of 0.001, and mini-batch size of 100. Because we propose PCM as a weight device, the full precision weights are binarized and then the results are written in PCM cells. Some PCM cells which are in amorphous state and not updated experiences the drift effect. The modified weights increase the weighting of some inputs and affects the whole training process. The algorithm is described in Supplementary Fig. 6 ."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"Recently, the field of artificial intelligence (AI) has witnessed tremendous advances in deep neural networks (DNNs) [1] , [2] , [3] , [4] , [5] , [6] , [7] . In DNNs, as a connectionist approach to AI, knowledge is represented by a hierarchically distributed pattern of activation nodes and a large number of weights storing the connection strength between the nodes [8] , [9] , [10] . Since weights are trainable depending on some relevant data without explicit rules, ‘learning’ can be defined naturally [11] . However, to gain powerful generalization capabilities, DNNs usually have massive tunable weights, thus the network training and inference are such data-intensive that the computation efficiency is usually limited by the memory access [12] , [13] . These networks are highly amenable to computation via large and dense matrix–vector multiplications that can be highly parallelized [14] , [15] . This has led to tremendous opportunities for hardware acceleration. First, DNN accelerators and neuromorphic chips were designed with the application-specific integrated circuits to alleviate the memory access bottleneck to speed up the computation [16] . However, using this design approach, the synaptic weights are still stored in random access memories rather than directly encoded by the states of emerging analog devices and computed at the memory locations. Such analog devices assembled systems enable fundamental Non-von Neumann scheme and are believed to achieve significant speedup and power reduction for on-chip implementation of DNNs [17] , [18] , [19] , [20] , [21] . One of the most attractive analog devices is the two-terminal memristor, such as phase change memory (PCM) and resistive random access memory (RRAM), which offers the advantages of high density, fast operation speed, and low-power consumption [22] , [23] . The weight elements of DNNs can be represented by the conductance of memristors, which can be programmed by update pulses or read by low-amplitude reading pulses. Therefore, the weights in memristor networks could be accessed and tuned locally, making it extremely suitable for the DNN hardware representation [24] . When implemented in the crossbar structure, memristor networks are ideal substrates that directly perform multiply-accumulate operations (MACs) at the weight locations [25] . This parallel computing property speeds up the training and inference of DNNs with significant low-energy consumption, providing a promising computing paradigm for neuromorphic computing systems [26] , [27] , [28] . However, there is still a huge discrepancy between the ideal unrestricted weight change in DNNs and the actual conductance change in memristors, resulting in the imprecise encoding of network weights and the corresponding computation. The conductance change of memristors is not only non-linear, asymmetric, and precision-limited, but also stochastic because of the device-to-device and pulse-to-pulse variations [29] , [30] , [31] . These properties generally lead to non-convergence when training real memristor neural networks. A promising solution to this issue is developing a co-optimization design that combines device properties (hardware insights) with DNN training techniques (software insights), such as quantization, pruning, and sparsification. Several approaches have been reported to restore the properties of the ideal weight, which ensures more stable performance but results in inefficiency and impracticality [17] , [32] , [33] . Our studies indicated that an adequate learning scheme able to exploit the stochasticity in PCM and incorporate brain-inspired approaches to enhance DNNs is essential to overcome this issue [34] , [35] . At present, the development of memristors is still insufficient to meet the ideal requirements for DNN training. Comparing the emerging memories for synapse device, PCM exhibits advantages on on/off ratio, endurance, and retention than RRAM, and on/off ratio than MRAM [36] . In addition, PCM is more mature technology as it has been manufacturing on a foundry basis. The physical mechanism of PCM is also well understood, which facilitates IC design of chip products. Thus, it is very practical to overcome the disadvantages of PCM and exploit the high reliability and the suitability for DNN applications. However, a critical issue for realization of PCM-based memristor neural network is that even if the weights represented by the resistance are precisely tuned to the ideal weights, its most cumbersome feature, the resistance drift issue, still remains, which has not been considered in previous studies [31] , [37] . Resistance can deviate continuously from the initial intended values because PCM undergoes a spontaneous resistance increase due to the structural relaxation after amorphization [38] . The resistance drift not only causes a major reliability issue of PCM, but also exacerbates the imprecise encoding of network weights in memristors. The conventional approach is to minimize the resistance drift by implementing error correction codes or adding specific periphery circuits, such as customized circuitry for rapid iterative write, modulation coding for drift-resilient, coding to new readout schemes that are drift resilient [39] . However, such approaches result in significant increase of complexity and power consumptions, and decrease of throughput. Here we built a 39 nm 1 Gb PCM-based memristor neural network and quantified its non-ideal characteristics, especially the unique resistance drift effect. PCM weights play a unique role in in-memory computations (e.g., measuring time stayed in a weight state and modifying weight non-linearly according to the time) but the resistance drift effect is usually thought to be a problem for application. In this work, we make use of the unique resistance drift effect to develop a new learning scheme for PCM-based memristor neural network, and a multi-layer perceptron (MLP) with 784-256-10 structure and 203,264 weights was built. It has been reported that resistance drift increases the effect of image classification in SNNs, but it is explained as an effect of improving the weight representation and has not been linked to a new learning method [39] . In some neuromorphic computing systems, weights encoded by memristors are binary-valued (+1 or –1) for feasibility and thus weights stay in either +1 or –1 until an update process. To embody such abstraction in PCM-based memristor DNNs, we utilize the power of spontaneity of PCM memristor, that is, the electrical properties of the resistance drift that can be described by power law using a defect model [40] , [41] , [42] , [43] . In this setup, we fill the gap between actual behavior and unrealistic expectation in a PCM-based memristor DNN accelerator. At the same time, we incorporate sparsification based on weight consistency in a very natural way, i.e., discriminating a PCM weight staying consistently in high resistance with spontaneous resistance increase. During training, the negative weight (–1) corresponding to the crystalline state decreases the weighted sum (i.e., sum of multiplications between each input and weight) but the positive weight (+1 + δ , δ ≥ 0) corresponding to amorphous state increases the weighted sum for the non-negative inputs. Because δ is getting larger as long as a weight remains in the positive state, therefore, a prior knowledge that the consistency of positive weights increases the final output values potentially is incorporated in this PCM-based memristor neural network. Interestingly, the final output is selectively increased when the prediction is correct. That is, the consistency correlates with accuracy improvement. Weight consistency map reveals that the new learning scheme system prevents weights from being stuck in the one of weight states during the whole training, which controls the discrimination of the one weight state from the other weight state, i.e., the sparsity. It is noticeable that the weight sparsity is changed differently between input-side weights and output-side weights. The consistency boosts the tendency of sparsity enhancement by increasing the number of negative input-side weights but suppresses the increase of the number of negative output-side weights. Thus, SSL manifests the negative contribution of some input data to the accuracy by increasing the sparsity and utilize the minimize uncontrollable negative contribution to the output. A possible outline of PCM-based memristor neural network system, device implementation, operation schemes, and solution to resistance drift even after training is carefully addressed. Our approach not only improves the classification accuracy of the PCM-based memristor neural network on MNIST handwritten digit dataset from 89.6% to 93.2%, but also controls the weight sparsity differently between layers (+1.4% for the input-side layer of about 200k weights while –4.5% for the output-side layer of about 2.5k weights) without any additional computation and operation. The demonstration based on the statistical parameters extracted from a 1 Gb PCM array also indicates higher feasibility. Unlike the passive approach of suppressing the drift effect, our SSL promotes the fusion of computer-based neural algorithms with the intrinsic properties of memristor devices [44] , [45] , [46] . To our best knowledge, this is the first learning method leveraging the PCM resistance drift to improve the memristor-based neural network performance. 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Electron Devices59, 2206–2214 (2012).\",\n \"link\": \"https://doi.org/10.1109%2FTED.2012.2197951\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Description of the working system Our approach is based on the distributed computing and the data quantization. A neural network is separated into several feedforward units (FFUs) and a central backpropagation unit (BPU) as shown in Fig. 1a [47] . BPU calculates weight modifications to update the old weights in high precision (e.g., float32) according to the feedforward output received from FFUs and supervised labels. Then the new weights are binarized in BPU and sent to FFUs. FFUs update their own PCM weight storage and send the loss for the next input to BPU again. In this configuration, the multiple FFUs share the computation workloads. Also, the data transfer from BPU to FFUs is reduced by the data quantization. After the training, the positive weights are pinned to a fixed value to be used for the inference without weight variation. BPU has its own full precision weight storage to be updated, like conventional neural network models for weight quantization. The weight is quantized and then stored in FFUs for the feedforward process. Although the storage is just replaced to PCM memristor in this neural network, the consistency is stored as additional resistance increase in PCM memristor automatically as shown in Fig. 1b . Fig. 1: A PCM-based memristor neural network incorporating the resistance drift for the consistency-induced weight increase under distributed computing and weight precision reduction for new learning method. a The conventional neural network is depicted as a black box, which is divided into BPU (back propagation with 32-bit weight storage) and FFU (feed forward with R-drift incorporated PCM weight storage). The training process is completed with the PCM followed by the pinning operation to prevent unwanted R-drift for inference. b In conventional, weight change pattern does not affect itself directly. However, it can be possible through the resistance drift. For example, if a PCM remains in amorphous states consistently without switching, the weight is increased continuously according to the time. Then, weight change patterns can be reflected to weight value. c The high- and low-resistance are mapped to +1 and −1 weight state, respectively (see Fig. 2 ). The resistance can be in one of three states of ‘staying’ in crystalline state, ‘alternating’ between amorphous and crystalline, and ‘drifting’ in amorphous. Without priors about the weight, the weight will be +1 or −1 with 1/2 and 1/2 probability, respectively, in the conventional neural network. The probability that the final weight value is +1 is higher than 1/2 based on the likelihood in +1 state, like the Bayesian inference. The higher probability is encoded as greater weight value (1 + δ ) like the gain coding in neuroscience, which indicates the increase of neural activity in amplitude as the input stimulus becomes certain. d Continuous resistance read shows different behaviors between amorphous (DRIFT) and crystalline states (ON). The resistance in amorphous is increasing gradually while the resistance in crystalline is almost constant. The resistance drift can be described by the power law. R ( t 0 ) is the first resistance measurement after the time t 0 . R ( t ) is resistance at time t ( t > t 0 ) and d is the drift coefficient which can be determined from a curve fitting. e SEM corner view of the PCM (39 nm, 1 Gbit) array used. f TEM images of some cells in the yz and xz planes reveal the damascene structure of GST layer. Full size image A possible weight change in PCM memristor weight storage can be represented as a combination of three patterns; staying in –1, alternating between +1 and –1, increasing in +1 + δ (left column in Fig. 1c ). The probability distributions of the corresponding full precision weights are simplified through the binarization (middle column). As it is binarized, it discards a lot of information about the full precision weight. However, Bayesian inference can be considered as a method to reconstruct information about the probability distribution of full precision weights, which predicts posterior probability based on prior probability and likelihood. Since it is not known how the full precision weight will be formed, +1, –1 can be thought of as having a prior probability of 50:50. This is a method of predicting the distribution of full precision weights based on likelihood where +1 and –1 appear when binarizing. We can infer that the distribution of full precision weights is shifted toward the positive side by reflecting the likelihood of +1. This inferred probability distribution is called the posterior probability distribution, and in the next inference, the posterior probability of the previous step becomes the prior probability, and then inference continues according to likelihood. There is no way to perfectly reflect this post-probability in a 1-bit memristor, but PCM can encode the increased probability with an increase in weight value using the drift effect. Then, according to the prior (i.e., the significance of positive weights), the posterior is determined by increasing the weights inspired from the gain coding (right column) [48] , [49] . Thus the weight increase based on the weight history can be rationalized by Bayesian inference and the gain coding. The resistance drift of a PCM memristor is compared after two different operations, crystallization, and amorphization. Continuous resistance read shows different behaviors between amorphous (DRIFT) and crystalline states (ON), as shown in Fig. 1d . The resistance of the amorphous state is increasing gradually according to the power law, while the resistance of the crystalline state is almost constant [41] , [42] , [50] . Thus, the prior can be incorporated into the weights naturally. Scanning electron microscopy (SEM) and tunneling electron microscopy (TEM) reveal the PCM array and the cell structure as shown in Fig. 1e, f (more details of PCM array and cell in “Methods” section) PCM memristor operation In our neural network, although the conventional weight storage is replaced with PCM memristor, the operation of the weight storage is the same as the conventional one. Some memristors are switched (S) and others are not (N). When a PCM memristor is not switched and remains in the amorphous state, the resistance is increased due to the spontaneous resistance drift. Figure 2a shows an example of how the resistance state of the PCM memristor changes as a sequence of specific S and N is delivered. The PCM memristor is switched occasionally in the middle of continual resistance read. The resistance read at the amorphous state shows a slight increase in resistance until it is switched back to the crystalline state. From 200 to 1200 resistance reads, the resistance drift continues without switching the amorphous state. In this way, the previously defined consistency in the pattern of weight evolutions is recorded as a resistance drift. Fig. 2: Mapping methods and corresponding circuits. a (Top) The schematic indicates the process of switching the PCM weight storage in FFUs according to the binarization of updated full-precision weight in BPU. (Bottom) A explanatory pattern of the resistance change is shown with corresponding sequence of S (Switch) and N (Non-switch). b (Top) The equation represents a mapping from resistance to weight. Instead of implementing MAC, resistance R ji is read cell by cell in this mapping and the asymmetry in resistance variation (large variation in large resistance) is mitigated. (Bottom) The plot shows the results of the mapping from the resistance change in a . c (Top) The schematic indicates the mapping method defined in b . (Bottom) The resultant weight from the explanatory pattern of the resistance change in a through the mapping scheme in b . d (Top) The equation represents a mapping for the inference after the weight pinning process. (Bottom) A possible circuit to implement the feedforward acceleration (i.e., multiply-and-accumulate, MAC) is represented. The calculated results ( I i ) from the input ( V j ’s) in this circuit is identical to MAC process. Full size image Dual weight mapping strategy To implement spontaneous weight modification from the resistance drift, we must consider how to map different physical quantities (e.g., high and low resistances) with positive values to typical weight values (e.g., +1 and –1, respectively). This can be solved by some practical mapping methods known as 2-memristor cell [31] , [45] or dummy-cell methods [51] , [52] . Also, the mapping from conductance to weight values helps accelerate feedforward propagation, i.e., matrix–vector multiplication or MAC. However, training with resistance drift in conductance is not effective because some modifications are required to make the resistance drift effect equivalent in the conductance scheme. Therefore, we propose a resistance scheme for the training and conversion to conductance scheme for the inference. For the training, the PCM memristor is written and read memristor cell-by-memristor cell, but is read as an analog memory in our practice. The computations of the feedforward propagation are shared with FFUs for several training data though MAC is not introduced. Thus, it is freer for processing the resistance values, e.g., taking the logarithmic value of the resistance to equalize the magnitude of the variations in high and low resistance (denoted by, R h and R l , respectively). As shown in Fig. 2b , we obtained \\\\(W_{{ji}}^{\\\\mathrm{{train}}}\\\\) from the measurement of \\\\(I_{R_{ji}}\\\\) and some arithmetic operations. The dummy-cell method was used to assign the weight increase property only to the positive weights. T denotes the translation (highlighted by red color) to make a pair of R h and R l centered at the zero point and S denotes the scaling (highlighted by blue color) to make the translated values +1 and –1, respectively. As a result, we can obtain the intended weight values as shown in Fig. 2c . Then, the weighted sum is calculated separately for the feedforward propagation for the training. To accelerate the feedforward propagation during the inference after the training, it is required to map the weight values from the conductance. Because we propose the weight pinning, two distinguished static conductance states are required. Considering the reuse of the PCM memristor weight, all high resistance memristors (including resistance drift) are switched to low resistance (equivalently, high conductance denoted by G h ) while all low resistance memristors are switched to high resistance (equivalently, low conductance denoted by G l ). Then, G h and G l are mapped to +1 + δ pin and –1, respectively. The mapping for the inference is similar to that for the training process except T and S . In this scheme, the resistance drift is negligible due to the reciprocal relation between the resistance and the conductance. To implement MAC, we just measure the total current I i when adequate V j ’s are supplied according to the input data. The measured current can be formulated from the circuit and the result is identical to the indented weighted sum as depicted in Fig. 2d . Classification results The most prominent result observed in this research is that the classification accuracy increased up to ~93.2% after the resistance drift effect was introduced. Figure 3a shows the training (inset) and testing accuracy (from 10 repeats) for the network with the conventional binary weight (denoted by ‘conventional’) and the network with the spontaneously increasing weight (denoted by ‘drift’). The remarkable fact is that the improvement in accuracy requires little additional computation cost and operation energy. Interestingly, drift demonstrated a larger deviation earlier, but a steady increase in accuracy with deviation reduced, unlike conventional (error bars) networks. This is because the consistency for the weight change could not be sufficiently assessed at the early stage of training and then the consistency is getting established and incorporated well. Fig. 3: Classification accuracy. a The classification accuracy in the conventional and drift neural network is compared. The line is the mean value and the bar is standard deviation from 10 runs. Comparing with the conventional in both train and test accuracy, the drift exhibits smaller deviation in latter epochs while it does larger deviation in former epochs. The final test accuracies are 89.6% and 93.2% in conventional and drift, respectively. b Confusion matrix reveals the discrepancies between the true label and the prediction. The diagonal and off-diagonal components indicate the number of correct and incorrect predictions, respectively. The components in the additional row vector at the top and column vector at the right show the aggregated number of incorrect predictions along each column and true label along each row, respectively. The off-diagonal yellowish component indicates significant incorrect prediction in the conventional. The mostly mispredicted class is ‘8’ (red boxes) and the sum is 222. Also, the prediction as ‘3’ is mostly incorrect (black boxes) and the sum is 205. c Digit-wise accuracy reveals the difference of accuracy between digits. The digit ‘1’ seems to be mostly accurate while ‘8’ shows poor accuracy. d Digit ‘3’ and ‘8’ have top-2 attempt/success ratio of classification. However, the classification of ‘3’ is decent in the accuracy against the high attempt/success ratio while the classification of ‘8’ is not. e In the drift, the most misprediction of ‘8’ is mostly improved from 222 to 76. Also, the most misprediction as ‘3’ is mostly improved from 205 to 109, too. f Almost digits are improved in accuracy. Accuracy of digit ‘8’ and ‘9’ is greatly improved. Digit ‘0’ and ‘1’ still show high accuracy. Full size image To identify the improvement in accuracy, we examined the confusion matrix as shown in Fig. 3b and e . The confusion matrix indicated how many times the neural network classified each input digit image into each class. The diagonal number indicates correct classifications for each digit while the off-diagonal number means the total occurrence of the confusion (i.e., the discrepancy between the prediction and the truth). The numbers at the top and right (single row and column, respectively) are the sum of each row and column except diagonal value. For example, the highest value ‘205’ (sum of values surrounded by black lines in the confusion matrix) at the top means that the neural network mispredicted the input digit images as ‘3’ mostly (denoted by ‘misprediction as ‘3”), and the highest value ‘222’ (sum of values surrounded by red lines in confusion matrix) at the right means that the class ‘8’ is mostly mispredicted by the neural network (denoted by ‘misprediction of ‘8”). Because some specific digits like ‘3’ and ‘8’ showed distinctive results in confusion matrix, we compared a digit-wise accuracy during the training. The digit-wise accuracy is defined as the number of correct predictions divided by the number of actual images for each digit. Digit ‘0’ and ‘1’ outperform in accuracy while digit ‘8’ shows low accuracy overall in Fig. 3c . Although ‘3’ and ‘8’ show the top-2 highest attempt/success ratio (shown in Fig. 3d ), the accuracy of ‘3’ is much higher than the accuracy of ‘8’. The improved result (denoted by ‘Drift’) shows that the number of misprediction as ‘3’ is reduced from 205 to 109 and the number of misprediction of ‘8’ is reduced from 222 to 76 significantly. The SSL shows accuracy improvement for most of the digits, especially better for ‘8’ as found in Fig. 3f . Drift effect during training To reveal two contradicting results related to the output node ‘8’, that is, the weakest drift effect and the greatest improvement in accuracy, the impact of the drift effect on the final output in the feedforward process is analyzed in Fig. 4 . Here, the actual 256 nodes in the hidden layer and 10 output nodes were simplified to five ( n j = 5, n i = 5) for convenience as shown in Fig. 4a . Consequently, there are five classes (e.g., hand-written digits from 0, 1, 2, 3, and 4) to be distinguished and the weight has a 5 by 5 dimension. We introduced the drift effect as an additional term, \\\\(\\\\delta _{ji}^{\\\\left( {23} \\\\right)}\\\\left( t \\\\right)\\\\) to the weight connecting i th node in the second (hidden) layer and j th node in the third (output) layer. It should be noted that \\\\(\\\\delta _{ji}^{\\\\left( {23} \\\\right)}\\\\) ( t ) is non-zero positive only for j and i satisfying \\\\(w_{ji}^{\\\\left( {23} \\\\right){\\\\mathrm{{bin}}}}\\\\) = +1. In this schematic, the input vector \\\\(x_j^{\\\\left( 2 \\\\right)}\\\\) and weight matrix \\\\(w_{ji}^{\\\\left( {23} \\\\right){\\\\mathrm{{bin}}}}\\\\) are identical to those in other cases, respectively, but \\\\(\\\\delta _{ji}^{\\\\left( {23} \\\\right)}\\\\) ( t ) is different for all the cases. We compared three different cases as shown in Fig. 4b . Among weights with a value of +1 in the weight matrix, the weight with a drift value of 0 indicates the PCM cell that has just switched to +1 state without time to drift. Weights with zero drift were arbitrarily selected. Those positive weights that are just switched will have various drift values. The top panel is a conventional drift-less feedforward process so the weighted sum and the softmax output are calculated as \\\\(s_i^{\\\\left( 3 \\\\right)} = \\\\mathop {\\\\sum}\\\\nolimits_j {x_j^{\\\\left( 2 \\\\right)}} w_{ji}^{\\\\left( {23} \\\\right){\\\\mathrm{{bin}}}}\\\\) and \\\\(x_i^{\\\\left( 3 \\\\right)}\\\\) = softmax( \\\\(s_i^{\\\\left( 3 \\\\right)}\\\\) ), respectively. Because \\\\(x_i^{\\\\left( 3 \\\\right)}\\\\) has the highest value (0.44) at i = 3, the conventional neural network infers that the input is \\\\({\\\\mathrm{{class}}}_{i = 3}\\\\) . The middle and bottom panels indicate different effects of \\\\(\\\\delta _{ji}^{\\\\left( {23} \\\\right)}\\\\) ( t ), and thus weights experience the drift differently. For instance, some elements of the weighted sum are increased or decreased according to which weights experience the drift effect greater or lesser. The effect of drift to be emphasized is that the change in each \\\\(s_i^{\\\\left( 3 \\\\right)}\\\\) had a direct influence on every \\\\(x_i^{\\\\left( 3 \\\\right)}\\\\) because of the mutually exclusive property of the softmax function. In the middle panel, large \\\\(d_{i = 3}^{\\\\left( 3 \\\\right)}\\\\) (=1.70−1.10 = 0.6) made \\\\(x_{i = 3}^{\\\\left( 3 \\\\right){\\\\mathrm{{drift}}}}\\\\) (=softmax( \\\\(s_{i = 3}^{\\\\left( 3 \\\\right)}\\\\) + \\\\(d_{i = 3}^{\\\\left( 3 \\\\right)}\\\\) ) = 0.57) larger than \\\\(x_{i = 2}^{\\\\left( 3 \\\\right)}\\\\) (=0.44) and \\\\(x_{i = 2}^{\\\\left( 3 \\\\right){\\\\mathrm{{drift}}}}\\\\) (=softmax( \\\\(s_{i = 2}^{\\\\left( 3 \\\\right)}\\\\) + \\\\(d_{i = 2}^{\\\\left( 3 \\\\right)}\\\\) )=0.28) less than \\\\(x_{i = 2}^{\\\\left( 3 \\\\right)}=0.36\\\\) even for non-zero \\\\(d_{i = 2}^{\\\\left( 3 \\\\right)}\\\\) (=1.01−0.90 = −0.11) simultaneously. Therefore, loss ( \\\\(E = - \\\\mathop {\\\\sum}\\\\nolimits_{i = 1}^{n_i} {t_i\\\\log \\\\left( {x_i} \\\\right)}\\\\) ) will be reduced from 0.36 to 0.24 in the middle panel because the index i of the largest \\\\(d_i^{\\\\left( 3 \\\\right)}\\\\) is 3, which is identical to the index i of the true label of the input (on the other hand, \\\\(\\\\delta _{ji}^{(23)}\\\\left( t \\\\right)\\\\) has the opposite (0.4 and 0.1 were swapped from the middle panel) pattern in the bottom panel, so the prediction is incorrect (i.e., the indices are not identical each other) and the loss increases to 0.40. Fig. 4: Feedforward explains the relation between the weight increase and the output values. a The network model in use is schematically depicted. The components x, s, w , and t denote for input, weighted sum, weight, and target, respectively. b Three different cases are compared to identify weight increase effects on the final output with some simplifications. All parameters are identical except the pattern of the weight increase term, \\\\(\\\\delta _{ji}^{\\\\left( {23} \\\\right)}(t)\\\\) . The first panel explains the conventional feedforward process without the consistency-induced weight increase. The only weights between the hidden and output layers are considered. The input, weights, and the output are 1 × 5 row vector, 5 × 5 matrix, and 1 × 5 row vector, respectively in this schematic for convenience. The feedforward process can be calculated straightforwardly. For the input and weights, the final \\\\(x_{i = 3}^{\\\\left( 3 \\\\right)}\\\\) has the greatest value and thus the neural network classify the input as \\\\({\\\\mathrm{{class}}}_{i = 3}\\\\) . In this case, the loss is 0.36 if the true label is [0 0 1 0 0]. In the middle panel comparing with the bottom panel, \\\\(\\\\delta _{ji}^{\\\\left( {23} \\\\right)}(t)\\\\) has the opposite pattern, i.e., the highest (0.4) and the lowest (0.1) values are swapped. According to the \\\\(\\\\delta _{ji}^{\\\\left( {23} \\\\right)}(t)\\\\) , \\\\(d_i^{\\\\left( 3 \\\\right)}\\\\) is added to the original \\\\(s_i^{\\\\left( 3 \\\\right)}\\\\) in a , so some components are increased or decreased. Full size image Because \\\\(\\\\delta _{ji}\\\\left( t \\\\right)\\\\) is determined internally during the training process, we can only estimate it heuristically through the above two extreme cases. If the drift effect improves the prediction, the loss will be reduced and the weight update becomes smaller than the conventional case. On the contrary, if the drift effect makes the prediction worse, the loss will increase and the weight update becomes larger than the conventional case correspondingly. In this way, some weights that have a significant contribution to minimizing the loss function will most likely experience the drift effect. Weight consistency Although it is hard to interpret the optimization of the drift effect during the training explicitly, the weight consistency reveals the pattern of weight changes. Figure 5a shows the results made from the subtraction of two heatmaps which implies the weight consistency in the conventional and drift, respectively. Thus, larger values indicate more significant drift effects on the weight consistency. The numbers in the horizontal axis indicates the digits (0–9) to be classified. The numbers in vertical axis (−10 to +10) denote how many times a weight stays in the same state for the first 10 epochs. The 256 weights can be considered to be associated with the output node ‘0’. The consistency of these weights can be reflected by adding each weight value once for each epoch. For example, if a weight has been in the −1 state for 10 epochs, the added weight value is −10. Thus, all of 256 weights assigned to each digit falls into one of the values between −10 and +10 for both the conventional and drift networks. The negative values at the bottom (deep bluish squares) indicates the slight reduction of the number of weights stuck in negative states. During the last 10 epochs, the negative consistent weights are significantly reduced, and some weights remain in the positive states long instead in Fig. 5b . For the input-side weights, the consistency of 28 × 28 weights connected to each node in hidden layer of the neural network is accumulated. There is no noticeable pattern during the first 10 epochs, but some weights connected to the central region of the input images show stronger negative contribution to the accuracy during the last 10 epochs as shown in Fig. 5c and d. At the early of training, there is no significant difference in weight consistency. As the training progresses, some weights stuck in either positive or negative states are remarkably reduced and some weights changing between two states are increased. Interestingly, the input-side and output-side weights show opposite patterns. Some weights stuck in the positive state are reduced for the input-side weights but some weights in the negative state are reduced for the output-side weights. Therefore, the accuracy-oriented consistency leads to the perturbation of some weights. Fig. 5: Weight consistency. a The graph explains how the drift effect affects the output-side weight consistency. The numbers in horizontal axis indicates the digit (0~9) to be classified. The numbers in vertical axis (−10 to +10) means how many times a weight stays in the same state for the first 10 epochs. Thus, all of 256 weights assigned to each digit fall into one of the values between −10 and +10 for both the conventional and drift networks. The values in the heatmap is the difference between the conventional and drift. The negative values at the bottom (deep blueish squares) indicates the slight reduction of the number of weights stuck in negative states. b During the last 10 epochs, the negative consistent weights is significantly reduced and some weights remain in the positive states long instead. c For the input-side weights, the consistency of 28 × 28 weights connected to each node in hidden layer of the neural network is accumulated. There is no noticeable pattern during the first 10 epochs. d During the last 10 epochs, some weights connected to the central region of the input images show stronger negative contribution to the accuracy. Full size image Sparsity Because the perturbation of some weights is not reflected in the accuracy calculated at a certain time, the weight sparsity (i.e., the number of negative weights over the number of total weights) is examined. In the conventional, the number of negative weights is increased for both the input-side (Fig. 6a ) and output-side (Fig. 6b ) weights during the training. On the contrary, in the drift, the number of negative weights is further increased for the input-side weights (Fig. 6c ) by about 1.4%, but the number of negative weights is decreased for the output-side weights (Fig. 6d ) by about 4.5%. The blue and red lines indicate the proportion of the total negative and positive weights, respectively. The output-side weights are grouped by the output digits and the input-side weights are grouped by each node in hidden layers (i.e., 256 groups and 28 × 28 weights per group). Fig. 6: Weight sparsity. a For input-side weights, the number of negative weights is increased in the conventional. b For output-side weights, the sparsity shows the gradual increase of negative weights in the conventional. c The negative contribution is promoted in the drift. d The sparsity is fluctuated slightly and not changed significantly in the drift. Thus, weight sparsity exhibits different trends between output-side and input-side weights. Full size image Considering the introduction of the accuracy-oriented consistency, the consistency of some weights compromising the accuracy will be excluded mostly. We can imagine an overlap between the consistency and accuracy. The weights can be subdivided by the position in the neural network (i.e., input-side or output-side), the consistency, and the sparsity. Therefore, the overlap portion (denoted by selectivity) determines the effectiveness of SSL scheme. Because the drift effect increases the output due to the non-negative contribution to the weighted sum basically, the similarity of the output vector to a one-hot vector indicates the effectiveness of SSL. Selectivity The maximum component in an output vector is <1 and the rests are larger than 0 as shown in Fig. 7a . We can define the digit-wise difficulty of classification based on the output vector, because the neural networks try to make the output like a one-hot vector as much as possible. The classification of a digit can be considered much difficult as more as the component of the digit in output vectors deviates from 1 or 0. In this perspective, the digit ‘8’ is much difficult to be distinguished from others than ‘1’ in the conventional. Surprisingly, the drift effect discriminates the digit ‘1’ output as two subgroups, though the average output of ‘1’ is close to 1 in Fig. 7a . In each group, the output of ‘1’ in the conventional is totally different according to whether a certain digit ‘1’ input image is classified correctly in the drift or not (two top-right bluish lines) as shown in Fig. 7b . That is, the output of ‘1’ in the conventional is extremely deviated from 1 for some images classified incorrectly in the drift. However, digit ‘8’ does not show such dependence on the correctness in the drift (two top-left reddish lines). The drift effect makes the output of ‘8’ close to a one-hot vector eventually (orange lines). In Fig. 7c , the average output is improved and then shows the promising output like a one-hot vector. The accuracy-oriented consistency subdivides both the digit ‘8’ and ‘0’ according to the difficulty. Then, SSL exploits the decision boundary of classification to maximize the accuracy. Fig. 7: Selectivity. a Average output for the test image is compared among digits. The results show the greatest deviation from 1 and 0 for the digit ‘8’, which indicates ‘8’ is hard to train in conventional. Digit ‘1’ is the closest to 1. b Although the average output of ‘1’ is close to 1 in a , however, drift effect discriminates the ‘0’ input images two subgroups. The output of ‘1’ in the conventional is totally different according to whether the input image is classified correctly in the drift (two top-right bluish lines). That is, the output of ‘1’ in the conventional is extremely deviated from 1 for some images classified incorrectly in the drift. However, digit ‘8’ do not show the dependence on the correctness in the drift (two top-left reddish lines). The drift effect makes the output of ‘8’ close to an one-hot vector eventually (orange lines). c Average output is improved and then shows the promising output like a one-hot vector. d Number of increases and decreases of the maximum values of \\\\(x_i^{(3){\\\\mathrm{{drift}}}}\\\\) with respect to \\\\(x_i^{(3)}\\\\) and their ratio demonstrates the selectivity of the output modification. The overall average is 6.29. When the drift neural network predicts correctly, the ratio is increased to 10.67 (for the test images predicted incorrectly in the conventional neural network) and 9.68 (for the test images predicted correctly in the conventional neural network). In the contrary, the ratio is far below the average when the drift neural network predicts incorrectly. Full size image The number of increases and decreases of Max( \\\\(x_i^{{\\\\mathrm{{drift}}}}\\\\) ) compared with Max( x i ) are plotted with respect to the correctness, as shown in Fig. 7d . The total number of increases or decreases can be divided into four cases, I/I, I/C, C/I, and C/C, where I and C denote ‘incorrect’ and ‘correct’ respectively, and the former and the latter refer to ‘conventional/drift’. The total number of increases is larger than that of decreases essentially and the ratio (i.e., the number of increases over the number of decreases) is 6.29. It is noticeable that the ratio is remarkably higher when the classification of the drift case is correct. Simultaneously, the ratio is reduced considerably when the result is incorrect. The ratios are 10.7 and 9.68 for I/C and C/C (above the overall ratio), whereas they are 1.06 and 2.85 for C/I and I/I (below the overall ratio), respectively. The results indicate that the selectivity of the output increase is originated from the consistency and can be supported analytically by tracing the backpropagation process. Weight pinning One important issue of the spontaneous weight modification utilizing resistance drift is that the optimized weight will undergo inevitable further change after training. However, a suitable constant weight value can be found to represent all individually changed weights due to the drift without degrading performance in the inference process. After training, all positive weights can be regarded as a fixed positive constant like a conventional digital memory device. Then we can find the constant that maximizes the accuracy of the test images. When the fixed positive constant is changed from 1.05 to 1.70 by 0.05 step, we compared the accuracy and then found that it is compatible to the accuracy just after the training when w pin = 1.4 (name after weight pinning) as shown in Fig. 8a . Fig. 8: Weight pinning and summary. a Due to the spontaneity of the resistance drift, the weight should be pinned not to lose the learned from the consistency. After the training, we can find a suitable weight value w pin (e.g., by scanning from 1.05 to 1.70) to maximize the accuracy near the best obtained for the training. w pin is 1.4 for the neural network and corresponding parameters used in this work. b The effect of the consistency-induced weight increase can be summarized as a weight sparsification based on the input-wise classification difficulty. Full size image\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"From the digit-wise analysis, we found some results. (1) Digit ‘8’ classification is mostly inaccurate. (2) The accuracy-oriented consistency (i.e., drift effect) is introduced. (3) The consistency controls the weight sparsity, so the weights are regrouped based on the digit-wise (via output-side weights) and image-wise (via input-side weights) manners. (4) By drift effect, digit ‘1’ input images are easily subdivided into two parts while digit ‘8’ input images are not. (5) Then, the network increases the classification accuracy of ‘8’ mostly by exploiting the room for weight modification without degrading the classification accuracy of ‘1’. The accuracy improvement due to consistency-induced weight increase is summarized as shown in Fig. 8b . Our approach is rationalized in that the difficulty of classification is related to the difficulty of weight determination (i.e., consistency) [53] , [54] , [55] . The difficulty-based accuracy improvement is similar to the experimental results demonstrating the decision confidence in the brain [56] , [57] , [58] , [59] . In animal experiments, different neural activity according to the difficulty of the decision is the most important feature of the confidence [49] , [60] . For example, for a mixture of two stimuli (e.g., scent A and scent B) with a different ratio, neural activity is most active for incorrect decision in easy decision tasks (A: 100% +B: 0% or A: 0% +B: 100%) and least active for correct decision in easy decision tasks. In the contrary, for the decision in difficult decision tasks (A: 51% +B: 49% or A: 49% +B: 51%), neural activity is moderately active regardless of the correctness. For this reason, the process of confidence formation is to classify easy tasks definitely, which is similar to the classification of ‘1’ under the drift effect. It is generally known that the reset current decreases as the cell size decreases [61] , [62] . In addition, a lower reset current is associated with a decrease in amorphous volume and consequent higher amorphous resistance [63] . Interestingly, the resistance in the amorphous state correlates with the drift coefficient [64] . The higher the amorphous resistance, the greater the drift coefficient. To our best knowledge, there is no publication on the impact of technology scaling on the resistance drift yet. Various experimental results have explained resistance drift in different ways and share a commonality that it is related to structural relaxation and stress release. Therefore, the electronic state of the material has a direct effect on the resistance drift rather than the technology node scaling or device structure. When PCM cell is not fully reset, the drift coefficient as well as resistance becomes smaller. The drift coefficient of the fully reset GST is ~0.09–0.11 [39] . In our case, PCM is almost fully reset (~0.10) and thus the impact due to technology node reduction is negligible. Even if the drift coefficient becomes greater with the size scaling or structure changes, we can slightly increase the reset current to decrease the reset resistance, thus maintaining the drift coefficient to a similar level. We tested the classification accuracy over a wide range of drift coefficient means and deviations as shown in Supplementary Fig. 7 . The consistently improved accuracy indicates that our learning scheme is very robust to the changes in drift effect caused by technology scaling or environmental temperatures. Interestingly, it is known that the drift coefficient does not depend on the constant temperature [42] , [65] . If the temperature rises suddenly, the drift coefficient temporarily increases. As shown in Supplementary Fig. 7c , the accuracy remains increasingly improved until the drift coefficient increases to ~1. Since the drift factor in the fully reset state is bounded, it does not have a value much higher than 0.12 for GST-based PCM [43] , [66] . Therefore, the accuracy of proposed scheme will not deteriorate at a high temperature as long as it is below the glass transition temperature of phase change materials. In this work, we built an ensemble of a mature 1 Gb PCM array with 39 nm technology, and by leveraging the statistical parameters obtained from the measurement of resistance drift, demonstrated a spontaneous sparse learning scheme in a PCM-based memristor neural network. By encoding the consistency of weight changes into the weight increases spontaneously, the neural network demonstrates the discriminated weight modification depending on the classification difficulty. Without any additional computation and operation costs, this new training scheme with sparsity controlled (+1.4% for output-side weights and −4.5% for input-side weights) improves the accuracy (93.2%) of a handwritten digit classification application (MNIST dataset) compared with the baseline neural network (89.6%). Additionally, due to the compatibility of the consistency-induced weight increase with any PCM-based memristor neural network, our results have potential to be related to evolutionary neural networks [4] , [5] inspired by the spontaneous behavior and modular neural networks [6] , [7] , [67] . Our work provides a potential prototype neural network that implements a metacognitive function, such as confidence, with low computation workloads for lightweight devices. Furthermore, our approach can be applied to RRAM as well as PCM, and improved learning effects are expected in more complex networks using multi-level cells. Although the resistance drift is weak in RRAM, the resistance drift appears distinctly in the conductive-bridge RAM, one of RRAM branches, due to the lateral diffusion of the metal constituting the bridge [68] . Since the drift coefficient is roughly observed within the range of 0.001–0.1 similar to PCM depending on the resistance of the initial resistance, the method presented in this study is expected to be applicable to those RRAM devices [69] . Our work may provide inspirations to RRAM research, i.e., leveraging the unique material characteristics (e.g., conductive-bridge modulation) to improve the performance/efficiency of memristor networks or to incorporate another learning scheme into memristor networks. The basic operation of our PCM array is well evaluated up to the 2-bit 4 level. The proposed SSL algorithm is expected to further improve the performance of network in the case of multilevel operation. Our proposed learning scheme utilizes the intrinsic resistance drift to improve the learning accuracy without additional cost, which opens up a new path to develop learning schemes using the materials’ properties.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Device fabrication Cross-point PCM arrays of cells with n +-Si/n-Si/ p -Si/TiN/Ge 2 Sb 2 Te 5 (GST)/TiW/W structures were fabricated on a 300-mm Si wafer. A highly doped n +-Si layer was prepared with B at a doping concentration exceeding 10 22 atoms/cm 2 . The n -Si/ p -Si diode was epitaxially grown on the wafer by chemical vapor deposition (CVD) with B at a doping concentration of 10 21 atoms/cm 2 for n -type and P at a doping concentration of 10 20 atoms/cm 2 for p -type. TiN films were deposited on the diodes by physical vapor deposition (PVD) as the bottom electrode (BE). The stack was dry-etched by etching along the y -direction and then along the x -direction to form pillar structures. The stack of n-Si/p-Si/TiN was etched into y -axis-oriented parallel lines using a hard mask of SiO 2 . The gaps between the etched lines were filled with SiO 2 . The gap-filled structure was flattened by chemical–mechanical polishing (CMP) and etched again with x -axis-oriented parallel lines. This formed word lines (WL) of n+-Si at the bottom of the diode. The gaps between diodes were then filled with SiO 2 and flattened by CMP. Then, interconnects and trenches were formed in the SiO 2 with lithography along the y -direction to build GST and a top electrode (TE) on the TiN through the dual damascene process. The interconnects and trenches were filled with GST and then TiW was deposited as the TE. After CMP, a line of W metal was formed using lithography on the TiW as the bit line. Finally, the cross-point PCM array was fabricated. The contact area between the GST and bottom electrode was ~40 nm × 80 nm. PCM Array information The PCM Array used in the research is an array with 39 nm technology in the process of developing PCM arrays with 90, 50, 45, 39 and 20 nm technologies [70] , [71] , [72] (officially reported for some technology only). Various operations and evaluation data including temperature and resistance drift for each technology were partially disclosed [73] , [74] . For example, resistance distributions for four resistance levels of (00), (01), (10), and (11) after an elapse of 400 h (1.4 × 10 6 s) and an additional thermal annealing at 130 °C for 12 h bake were provided for 90 nm PCM array. Also, for 45 nm PCM array, analysis of the drift effect was presented in depth to improve multi-level operational characteristics and verify practicality. The total array sizes fabricated by 90 and 45 nm technology are 512 Mb (16 banks of 32 Mb, 8 blocks/bank) and 1 Gbit (16 banks of 64 Mb, 8 blocks), respectively. For the 39 nm PCM Array, the temperature and drift characteristics were not officially disclosed at the wafer-level, but they are similar to those of the 45 nm PCM array. For the 39 nm PCM array, 2-PCM synapse behavior was evaluated [75] . Although the operation circuit was slightly deformed according to the characteristics of each technology to implement ancillary functions for performance improvement, such as high read or write throughput, the characteristics of each PCM array were well evaluated as can be seen in the literature provided. Measurement Each PCM cell is in the low resistance state at about 10 4 Ω which was measured by using 1 V pulse with 5 ns rising time, 60 ns duration, and 5 ns falling time. The cell resistance is increased to about 10 7 Ω using a pulse of 5 V height with 5 ns rising time, 60 ns duration, and 5 ns falling time. We then make the cell have a few 10 4 Ω by applying a voltage of 3 V with 5 ns rising time, 400 ns duration, and 300 ns falling time. The PCM cell is switched 20 times between set and reset states with 10 resistance read after each switching. Then the resistance is measured 1000 times just after the reset switching to measure the resistance drift as depicted in Supplementary Fig. 1a . A pulse generator (Keysight 33600A) was used to provide electrical pulse to the cell. A source meter (Keithley 2600) was used to measure the cell resistance. Pulse is applied to the TE while bottom electrode is ground through the contact pads. Supplementary Fig. 1b shows the wafer being measured. Parameter extraction We extracted resistance distribution and deviation from over 100 cells. For each cell, the set and the reset resistance were measured over 100 data points and then the resistance drift was measured for 1000 data point with 1 read-operation per 1 s. Then, the sample mean and the sample standard deviation of the set resistance, the reset resistance, and the drift parameter ( d in Eq. (1)) were extracted and denoted by \\\\(\\\\mu _{{\\\\mathrm{{SET}}}_i}^{\\\\mathrm{{s}}}\\\\) , \\\\(\\\\sigma _{{\\\\mathrm{{SET}}}_i}^{\\\\mathrm{{s}}}\\\\) , \\\\(\\\\mu _{{\\\\mathrm{{RESET}}}_i}^{\\\\mathrm{{s}}}\\\\) , \\\\(\\\\sigma _{{\\\\mathrm{{RESET}}}_i}^{\\\\mathrm{{s}}}\\\\) , \\\\(\\\\mu _{{\\\\mathrm{{DRIFT}}}_i}^{\\\\mathrm{{s}}}\\\\) , and \\\\(\\\\sigma _{{\\\\mathrm{{DRIFT}}}_i}^{\\\\mathrm{{s}}}\\\\) , respectively, for the \\\\(i{{\\\\mathrm{th}}}\\\\) cell. Then, the statistical parameter for the resistance distribution, i.e., population mean ( \\\\(\\\\mu _{{\\\\mathrm{{SET}}}_i}^{\\\\mathrm{{p}}}\\\\) , \\\\(\\\\mu _{{\\\\mathrm{{RESET}}}_i}^{\\\\mathrm{{p}}}\\\\) , and \\\\(\\\\mu _{{\\\\mathrm{{DRIFT}}}_i}^{\\\\mathrm{{p}}}\\\\) ) and population standard deviation ( \\\\(\\\\sigma _{{\\\\mathrm{{SET}}}_i}^{\\\\mathrm{{p}}}\\\\) , \\\\(\\\\sigma _{{\\\\mathrm{{RESET}}}_i}^{\\\\mathrm{{p}}}\\\\) , and \\\\(\\\\sigma _{{\\\\mathrm{{DRIFT}}}_i}^{\\\\mathrm{{p}}}\\\\) ) with an error term depending on the number of sampling data and the distribution model were calculated for the i th cell. Supposed that the resistance distribution is a normal distribution, \\\\(\\\\mu _i^{\\\\mathrm{{p}}}\\\\) can be estimated as \\\\(\\\\mu _i^{\\\\mathrm{{s}}}\\\\) with standard error of \\\\(\\\\frac{{\\\\mu _i^{\\\\mathrm{{s}}}}}{{\\\\sqrt n }}\\\\) while \\\\(\\\\sigma _i^{\\\\mathrm{{p}}}\\\\) can be estimated as \\\\(\\\\sigma _i^{\\\\mathrm{{s}}}\\\\) with standard error of \\\\(\\\\sqrt {\\\\frac{2}{{n - 1}}} \\\\left( {\\\\sigma _i^{\\\\mathrm{{s}}}} \\\\right)^2\\\\) . This process is depicted in Supplementary Fig. 2 (i and ii) for R ij , j th measurement of resistance of i th cell. In addition, the cell-to-cell population mean ( M p ) and population standard deviation ( Σ p ) were determined for both distributions of \\\\(\\\\mu _i^{\\\\mathrm{{p}}}\\\\) ’s and \\\\(\\\\sigma _i^{\\\\mathrm{{p}}}\\\\) ’s as shown in Supplementary Fig. 3 (iii and iv), which are denoted by \\\\(M_\\\\mu ^{\\\\mathrm{{p}}}\\\\) , \\\\(\\\\Sigma _\\\\mu ^{\\\\mathrm{{p}}}\\\\) and \\\\(M_\\\\sigma ^{\\\\mathrm{{p}}}\\\\) , \\\\(\\\\Sigma _\\\\sigma ^{\\\\mathrm{{p}}}\\\\) . By sampling a value from the normal distribution defined by \\\\(M_\\\\mu ^{\\\\mathrm{{p}}}\\\\) and \\\\(\\\\Sigma _\\\\mu ^{\\\\mathrm{{p}}}\\\\) , we could determine a mean resistance ( μ * ) of a possible PCM cell. Then, the resistance deviation ( α * ) that corresponds with μ * is determined as a value that has the same cumulated probability in the normal distribution defined by \\\\(M_\\\\sigma ^{\\\\mathrm{{p}}}\\\\) , \\\\(\\\\Sigma _\\\\sigma ^{\\\\mathrm{{p}}}\\\\) . Thus, six unique parameters ( \\\\(\\\\mu _{{\\\\mathrm{{set}}}_i}^ \\\\ast\\\\) , \\\\(\\\\sigma _{{\\\\mathrm{{set}}}_i}^ \\\\ast\\\\) , \\\\(\\\\mu _{{\\\\mathrm{{reset}}}_i}^ \\\\ast\\\\) , \\\\(\\\\sigma _{{\\\\mathrm{{reset}}}_i}^ \\\\ast\\\\) , \\\\(\\\\mu _{{\\\\mathrm{{drift}}}_i}^ \\\\ast\\\\) , and \\\\(\\\\sigma _{{\\\\mathrm{{drift}}}_i}^ \\\\ast\\\\) ) were assigned to each PCM cell for set, reset, and drift independently and randomly. Finally, \\\\(R_{{\\\\mathrm{{set}}}}\\\\) , \\\\(R_{{\\\\mathrm{{reset}}}}\\\\) , and d drift were generated and renewed every switching based on the uniquely assigned parameters to PCM cells. This process is depicted in Supplementary Fig. 4 (v, vi, and vii) and key statistical data that is determined from measurement are represented in bottom of the figure. Neural network We used a neural network with 784 nodes in input layer (denoted by \\\\(x_k^{\\\\left( 1 \\\\right)}\\\\) ), 256 nodes in one hidden layer (denoted by \\\\(x_j^{\\\\left( 2 \\\\right)}\\\\) ), and 10 nodes in output layer (denoted by \\\\(x_i^{\\\\left( 3 \\\\right)}\\\\) ), for the MNIST hand-written digit classification, as shown in Supplementary Fig. 5 . The layers and the weights between the layers are distinguished by the superscript. All nodes are fully connected and any regularization process such as batch normalization and dropout are excluded. Each 28 × 28 greyscale input image is normalized to [0, 1] and denoted by \\\\(x_k^{\\\\left( 1 \\\\right)}\\\\) . The weights are initialized by the variance scaling initialization. We applied ReLU activation in the hidden layer and softmax in the output layer, and then cross-entropy function was used as a loss function. To exclude any adaptive factor, we used stochastic gradient descent optimizer, constant learning rate of 0.001, and mini-batch size of 100. Because we propose PCM as a weight device, the full precision weights are binarized and then the results are written in PCM cells. Some PCM cells which are in amorphous state and not updated experiences the drift effect. The modified weights increase the weighting of some inputs and affects the whole training process. The algorithm is described in Supplementary Fig. 6 .\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"The core research idea is to develop a novel learning scheme for phase change memory (PCM)-based memristor neural networks that leverages the intrinsic resistance drift effect—a typically undesirable property—by incorporating it into the training process to improve classification accuracy and control weight sparsity across different layers without additional computation, thereby transforming a hardware limitation into a functional advantage for neuromorphic computing."},"entities":{"kind":"list like","value":["novel learning scheme","phase change memory","PCM-based memristor neural networks","resistance drift effect","training process","classification accuracy","weight sparsity","neuromorphic computing","hardware limitation","functional advantage"],"string":"[\n \"novel learning scheme\",\n \"phase change memory\",\n \"PCM-based memristor neural networks\",\n \"resistance drift effect\",\n \"training process\",\n \"classification accuracy\",\n \"weight sparsity\",\n \"neuromorphic computing\",\n \"hardware limitation\",\n \"functional advantage\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1.","title":"LeCun, Y., Bengio, Y. & Hinton, G. Deep learning.Nature521, 436–444 (2015).","link":"https://doi.org/10.1038%2Fnature14539"},{"idx":"2.","title":"Schmidhuber, J. Deep learning in neural networks: an overview.Neural Netw.61, 85–117 (2015).","link":"https://doi.org/10.1016%2Fj.neunet.2014.09.003"},{"idx":"5.","title":"Yao, X. A review of evolutionary artificial neural networks.Int. J. Intell. Syst.8, 539–567 (1993).","link":"https://doi.org/10.1002%2Fint.4550080406"},{"idx":"6.","title":"Auda, G. & Kamel, M. Modular neural networks: a survey.Int. J. Neural Syst.9, 129–151 (1999).","link":"https://doi.org/10.1142%2FS0129065799000125"},{"idx":"8.","title":"Mira, J. M. Symbols versus connections: 50 years of artificial intelligence.Neurocomputing71, 671–680 (2008).","link":"https://doi.org/10.1016%2Fj.neucom.2007.06.009"},{"idx":"9.","title":"Smolensky, P. Connectionist AI, symbolic AI, and the brain.Artif. Intell. Rev.1, 95–109 (1987).","link":"https://link.springer.com/doi/10.1007/BF00130011"},{"idx":"10.","title":"Maia, T. V. & Cleeremans, A. Consciousness: converging insights from connectionist modeling and neuroscience.Trends Cogn. Sci.9, 397–404 (2005).","link":"https://doi.org/10.1016%2Fj.tics.2005.06.016"},{"idx":"14.","title":"Indiveri, G., Linares-Barranco, B., Legenstein, R., Deligeorgis, G. & Prodromakis, T. Integration of nanoscale memristor synapses in neuromorphic computing architectures.Nanotechnology24, 384010 (2013).","link":"https://doi.org/10.1088%2F0957-4484%2F24%2F38%2F384010"},{"idx":"15.","title":"Hu, M. et al. Memristor crossbar-based neuromorphic computing system: a case study.IEEE Trans. Neural Netw. Learn Syst.25, 1864–1878 (2014).","link":"https://doi.org/10.1109%2FTNNLS.2013.2296777"},{"idx":"17.","title":"Boybat, I. et al. Neuromorphic computing with multi-memristive synapses.Nat. Commun.9, 2514 (2018).","link":"https://doi.org/10.1038%2Fs41467-018-04933-y"},{"idx":"18.","title":"Narayanan, P. et al. Toward on-chip acceleration of the backpropagation algorithm using nonvolatile memory.IBM J. Res. Dev.61, 11:11–11:11 (2017).","link":"https://doi.org/10.1147%2FJRD.2017.2716579"},{"idx":"20.","title":"Li, C. et al. Analogue signal and image processing with large memristor crossbars.Nat. Electron.1, 52–59 (2017).","link":"https://doi.org/10.1038%2Fs41928-017-0002-z"},{"idx":"21.","title":"Yao, P. et al. Face classification using electronic synapses.Nat. Commun.8, 1–8 (2017).","link":"https://doi.org/10.1038%2Fncomms15199"},{"idx":"22.","title":"Jeong, D. S., Kim, K. M., Kim, S., Choi, B. J. & Hwang, C. S. Memristors for energy-efficient new computing paradigms.Adv. Electron. Mater.2, 1600090 (2016).","link":"https://doi.org/10.1002%2Faelm.201600090"},{"idx":"23.","title":"Kim, K. H. et al. A functional hybrid memristor crossbar-array/CMOS system for data storage and neuromorphic applications.Nano Lett.12, 389–395 (2012).","link":"https://doi.org/10.1021%2Fnl203687n"},{"idx":"24.","title":"Ielmini, D. & Wong, H.-S. P. In-memory computing with resistive switching devices.Nat. Electron.1, 333–343 (2018).","link":"https://doi.org/10.1038%2Fs41928-018-0092-2"},{"idx":"25.","title":"Xia, Q. & Yang, J. J. Memristive crossbar arrays for brain-inspired computing.Nat. Mater.18, 309–323 (2019).","link":"https://doi.org/10.1038%2Fs41563-019-0291-x"},{"idx":"27.","title":"Chabi, D., Wang, Z., Bennett, C., Klein, J.-O. & Zhao, W. Ultrahigh density memristor neural crossbar for on-chip supervised learning.IEEE Trans. Nanotechnol.14, 954–962 (2015).","link":"https://doi.org/10.1109%2FTNANO.2015.2448554"},{"idx":"28.","title":"Zhang, Y., Wang, X., Li, Y. & Friedman, E. G. Memristive model for synaptic circuits.IEEE Trans. Circuits Syst. II64, 767–771 (2017).","link":"https://doi.org/10.1109%2FTCSII.2016.2605069"},{"idx":"31.","title":"Burr, G. W. et al. Experimental demonstration and tolerancing of a large-scale neural network (165,000 synapses) using phase-change memory as the synaptic weight element.IEEE Trans. Electron Devices62, 3498–3507 (2015).","link":"https://doi.org/10.1109%2FTED.2015.2439635"},{"idx":"32.","title":"Chang, C.-C. et al. Mitigating asymmetric nonlinear weight update effects in hardware neural network based on analog resistive synapse.IEEE J. Emerg. Sel. Top. Circuits Syst.8, 116–124 (2018).","link":"https://doi.org/10.1109%2FJETCAS.2017.2771529"},{"idx":"34.","title":"Lee, J. H., Lim, D. H., Jeong, H., Ma, H. M. & Shi, L. P. Exploring cycle-to-cycle and device-to-device variation tolerance in MLC storage-based neural network training.IEEE Trans. Electron Devices66, 2172–2178 (2019).","link":"https://doi.org/10.1109%2FTED.2019.2906249"},{"idx":"35.","title":"Pei, J. et al. Towards artificial general intelligence with hybrid Tianjic chip architecture.Nature572, 106–111 (2019).","link":"https://doi.org/10.1038%2Fs41586-019-1424-8"},{"idx":"36.","title":"Ielmini, D. & Ambrogio, S. Emerging neuromorphic devices.Nanotechnology31, 092001 (2019).","link":"https://doi.org/10.1088%2F1361-6528%2Fab554b"},{"idx":"38.","title":"Raty, J. Y. et al. Aging mechanisms in amorphous phase-change materials.Nat. Commun.6, 7467 (2015).","link":"https://doi.org/10.1038%2Fncomms8467"},{"idx":"39.","title":"Oh, S., Shi, Y., Liu, X., Song, J. & Kuzum, D. Drift-enhanced unsupervised learning of handwritten digits in spiking neural network with PCM synapses.IEEE Electron Device Lett.39, 1768–1771 (2018).","link":"https://doi.org/10.1109%2FLED.2018.2872434"},{"idx":"40.","title":"Boniardi, M. et al. Statistics of resistance drift due to structural relaxation in phase-change memory arrays.IEEE Trans. Electron Devices57, 2690–2696 (2010).","link":"https://doi.org/10.1109%2FTED.2010.2058771"},{"idx":"42.","title":"Boniardi, M. & Ielmini, D. Physical origin of the resistance drift exponent in amorphous phase change materials.Appl. Phys. Lett.98, 243506 (2011).","link":"https://doi.org/10.1063%2F1.3599559"},{"idx":"43.","title":"Pirovano, A. et al. Low-field amorphous state resistance and threshold voltage drift in chalcogenide materials.IEEE Trans. Electron Devices51, 714–719 (2004).","link":"https://doi.org/10.1109%2FTED.2004.825805"},{"idx":"45.","title":"Bichler, O. et al. Visual pattern extraction using energy-efficient “2-PCM synapse” neuromorphic architecture.IEEE Trans. Electron Devices59, 2206–2214 (2012).","link":"https://doi.org/10.1109%2FTED.2012.2197951"},{"idx":"48.","title":"Knill, D. C. & Pouget, A. The Bayesian brain: the role of uncertainty in neural coding and computation.Trends Neurosci.27, 712–719 (2004).","link":"https://doi.org/10.1016%2Fj.tins.2004.10.007"},{"idx":"49.","title":"Pouget, A., Drugowitsch, J. & Kepecs, A. Confidence and certainty: distinct probabilistic quantities for different goals.Nat. Neurosci.19, 366–374 (2016).","link":"https://doi.org/10.1038%2Fnn.4240"},{"idx":"51.","title":"Chang, C.-C. et al. Mitigating asymmetric nonlinear weight update effects in hardware neural network based on analog resistive synapse.IEEE J. Emerg. Sel. Top. Circuits Syst.8, 116–124 (2017).","link":"https://doi.org/10.1109%2FJETCAS.2017.2771529"},{"idx":"56.","title":"Rolls, E. T., Grabenhorst, F. & Deco, G. Choice, difficulty, and confidence in the brain.NeuroImage53, 694–706 (2010).","link":"https://doi.org/10.1016%2Fj.neuroimage.2010.06.073"},{"idx":"57.","title":"Perry, C. J. & Barron, A. B. Honey bees selectively avoid difficult choices.Proc. Natl Acad. Sci. USA110, 19155–19159 (2013).","link":"https://doi.org/10.1073%2Fpnas.1314571110"},{"idx":"58.","title":"Gherman, S. & Philiastides, M. G. Neural representations of confidence emerge from the process of decision formation during perceptual choices.NeuroImage106, 134–143 (2015).","link":"https://doi.org/10.1016%2Fj.neuroimage.2014.11.036"},{"idx":"59.","title":"Kiani, R. & Shadlen, M. N. Representation of confidence associated with a decision by neurons in the parietal cortex.Science324, 759–764 (2009).","link":"https://doi.org/10.1126%2Fscience.1169405"},{"idx":"60.","title":"Kepecs, A., Uchida, N., Zariwala, H. A. & Mainen, Z. F. Neural correlates, computation and behavioural impact of decision confidence.Nature455, 227–231 (2008).","link":"https://doi.org/10.1038%2Fnature07200"},{"idx":"63.","title":"Kim, K. & Jeong, G. The prospects of non-volatile phase-change RAM.Microsyst. Technol.13, 145–147 (2006).","link":"https://link.springer.com/doi/10.1007/s00542-006-0150-y"},{"idx":"66.","title":"Ielmini, D., Lacaita, A. L. & Mantegazza, D. Recovery and drift dynamics of resistance and threshold voltages in phase-change memories.IEEE Trans. Electron Devices54, 308–315 (2007).","link":"https://doi.org/10.1109%2FTED.2006.888752"},{"idx":"68.","title":"Woo, J. Y. et al. Role of local chemical potential of Cu on data retention properties of Cu-based conductive-bridge RAM.IEEE Electron Device Lett.37, 173–175 (2016).","link":"https://doi.org/10.1109%2FLED.2015.2507178"},{"idx":"75.","title":"Kang, D.-H., Jun, H.-G., Ryoo, K.-C., Jeong, H. & Sohn, H. Emulation of spike-timing dependent plasticity in nano-scale phase change memory.Neurocomputing155, 153–158 (2015).","link":"https://doi.org/10.1016%2Fj.neucom.2014.12.036"}],"string":"[\n {\n \"idx\": \"1.\",\n \"title\": \"LeCun, Y., Bengio, Y. & Hinton, G. Deep learning.Nature521, 436–444 (2015).\",\n \"link\": \"https://doi.org/10.1038%2Fnature14539\"\n },\n {\n \"idx\": \"2.\",\n \"title\": \"Schmidhuber, J. Deep learning in neural networks: an overview.Neural Netw.61, 85–117 (2015).\",\n \"link\": \"https://doi.org/10.1016%2Fj.neunet.2014.09.003\"\n },\n {\n \"idx\": \"5.\",\n \"title\": \"Yao, X. A review of evolutionary artificial neural networks.Int. J. Intell. Syst.8, 539–567 (1993).\",\n \"link\": \"https://doi.org/10.1002%2Fint.4550080406\"\n },\n {\n \"idx\": \"6.\",\n \"title\": \"Auda, G. & Kamel, M. Modular neural networks: a survey.Int. J. Neural Syst.9, 129–151 (1999).\",\n \"link\": \"https://doi.org/10.1142%2FS0129065799000125\"\n },\n {\n \"idx\": \"8.\",\n \"title\": \"Mira, J. M. Symbols versus connections: 50 years of artificial intelligence.Neurocomputing71, 671–680 (2008).\",\n \"link\": \"https://doi.org/10.1016%2Fj.neucom.2007.06.009\"\n },\n {\n \"idx\": \"9.\",\n \"title\": \"Smolensky, P. Connectionist AI, symbolic AI, and the brain.Artif. Intell. Rev.1, 95–109 (1987).\",\n \"link\": \"https://link.springer.com/doi/10.1007/BF00130011\"\n },\n {\n \"idx\": \"10.\",\n \"title\": \"Maia, T. V. & Cleeremans, A. Consciousness: converging insights from connectionist modeling and neuroscience.Trends Cogn. Sci.9, 397–404 (2005).\",\n \"link\": \"https://doi.org/10.1016%2Fj.tics.2005.06.016\"\n },\n {\n \"idx\": \"14.\",\n \"title\": \"Indiveri, G., Linares-Barranco, B., Legenstein, R., Deligeorgis, G. & Prodromakis, T. Integration of nanoscale memristor synapses in neuromorphic computing architectures.Nanotechnology24, 384010 (2013).\",\n \"link\": \"https://doi.org/10.1088%2F0957-4484%2F24%2F38%2F384010\"\n },\n {\n \"idx\": \"15.\",\n \"title\": \"Hu, M. et al. Memristor crossbar-based neuromorphic computing system: a case study.IEEE Trans. Neural Netw. Learn Syst.25, 1864–1878 (2014).\",\n \"link\": \"https://doi.org/10.1109%2FTNNLS.2013.2296777\"\n },\n {\n \"idx\": \"17.\",\n \"title\": \"Boybat, I. et al. Neuromorphic computing with multi-memristive synapses.Nat. Commun.9, 2514 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-018-04933-y\"\n },\n {\n \"idx\": \"18.\",\n \"title\": \"Narayanan, P. et al. Toward on-chip acceleration of the backpropagation algorithm using nonvolatile memory.IBM J. Res. Dev.61, 11:11–11:11 (2017).\",\n \"link\": \"https://doi.org/10.1147%2FJRD.2017.2716579\"\n },\n {\n \"idx\": \"20.\",\n \"title\": \"Li, C. et al. Analogue signal and image processing with large memristor crossbars.Nat. Electron.1, 52–59 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fs41928-017-0002-z\"\n },\n {\n \"idx\": \"21.\",\n \"title\": \"Yao, P. et al. Face classification using electronic synapses.Nat. Commun.8, 1–8 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms15199\"\n },\n {\n \"idx\": \"22.\",\n \"title\": \"Jeong, D. S., Kim, K. M., Kim, S., Choi, B. J. & Hwang, C. S. Memristors for energy-efficient new computing paradigms.Adv. Electron. Mater.2, 1600090 (2016).\",\n \"link\": \"https://doi.org/10.1002%2Faelm.201600090\"\n },\n {\n \"idx\": \"23.\",\n \"title\": \"Kim, K. H. et al. A functional hybrid memristor crossbar-array/CMOS system for data storage and neuromorphic applications.Nano Lett.12, 389–395 (2012).\",\n \"link\": \"https://doi.org/10.1021%2Fnl203687n\"\n },\n {\n \"idx\": \"24.\",\n \"title\": \"Ielmini, D. & Wong, H.-S. P. In-memory computing with resistive switching devices.Nat. Electron.1, 333–343 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41928-018-0092-2\"\n },\n {\n \"idx\": \"25.\",\n \"title\": \"Xia, Q. & Yang, J. J. Memristive crossbar arrays for brain-inspired computing.Nat. Mater.18, 309–323 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41563-019-0291-x\"\n },\n {\n \"idx\": \"27.\",\n \"title\": \"Chabi, D., Wang, Z., Bennett, C., Klein, J.-O. & Zhao, W. Ultrahigh density memristor neural crossbar for on-chip supervised learning.IEEE Trans. Nanotechnol.14, 954–962 (2015).\",\n \"link\": \"https://doi.org/10.1109%2FTNANO.2015.2448554\"\n },\n {\n \"idx\": \"28.\",\n \"title\": \"Zhang, Y., Wang, X., Li, Y. & Friedman, E. G. Memristive model for synaptic circuits.IEEE Trans. Circuits Syst. II64, 767–771 (2017).\",\n \"link\": \"https://doi.org/10.1109%2FTCSII.2016.2605069\"\n },\n {\n \"idx\": \"31.\",\n \"title\": \"Burr, G. W. et al. Experimental demonstration and tolerancing of a large-scale neural network (165,000 synapses) using phase-change memory as the synaptic weight element.IEEE Trans. Electron Devices62, 3498–3507 (2015).\",\n \"link\": \"https://doi.org/10.1109%2FTED.2015.2439635\"\n },\n {\n \"idx\": \"32.\",\n \"title\": \"Chang, C.-C. et al. Mitigating asymmetric nonlinear weight update effects in hardware neural network based on analog resistive synapse.IEEE J. Emerg. Sel. Top. Circuits Syst.8, 116–124 (2018).\",\n \"link\": \"https://doi.org/10.1109%2FJETCAS.2017.2771529\"\n },\n {\n \"idx\": \"34.\",\n \"title\": \"Lee, J. H., Lim, D. H., Jeong, H., Ma, H. M. & Shi, L. P. Exploring cycle-to-cycle and device-to-device variation tolerance in MLC storage-based neural network training.IEEE Trans. Electron Devices66, 2172–2178 (2019).\",\n \"link\": \"https://doi.org/10.1109%2FTED.2019.2906249\"\n },\n {\n \"idx\": \"35.\",\n \"title\": \"Pei, J. et al. Towards artificial general intelligence with hybrid Tianjic chip architecture.Nature572, 106–111 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-019-1424-8\"\n },\n {\n \"idx\": \"36.\",\n \"title\": \"Ielmini, D. & Ambrogio, S. Emerging neuromorphic devices.Nanotechnology31, 092001 (2019).\",\n \"link\": \"https://doi.org/10.1088%2F1361-6528%2Fab554b\"\n },\n {\n \"idx\": \"38.\",\n \"title\": \"Raty, J. Y. et al. Aging mechanisms in amorphous phase-change materials.Nat. Commun.6, 7467 (2015).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms8467\"\n },\n {\n \"idx\": \"39.\",\n \"title\": \"Oh, S., Shi, Y., Liu, X., Song, J. & Kuzum, D. Drift-enhanced unsupervised learning of handwritten digits in spiking neural network with PCM synapses.IEEE Electron Device Lett.39, 1768–1771 (2018).\",\n \"link\": \"https://doi.org/10.1109%2FLED.2018.2872434\"\n },\n {\n \"idx\": \"40.\",\n \"title\": \"Boniardi, M. et al. Statistics of resistance drift due to structural relaxation in phase-change memory arrays.IEEE Trans. Electron Devices57, 2690–2696 (2010).\",\n \"link\": \"https://doi.org/10.1109%2FTED.2010.2058771\"\n },\n {\n \"idx\": \"42.\",\n \"title\": \"Boniardi, M. & Ielmini, D. Physical origin of the resistance drift exponent in amorphous phase change materials.Appl. Phys. Lett.98, 243506 (2011).\",\n \"link\": \"https://doi.org/10.1063%2F1.3599559\"\n },\n {\n \"idx\": \"43.\",\n \"title\": \"Pirovano, A. et al. Low-field amorphous state resistance and threshold voltage drift in chalcogenide materials.IEEE Trans. Electron Devices51, 714–719 (2004).\",\n \"link\": \"https://doi.org/10.1109%2FTED.2004.825805\"\n },\n {\n \"idx\": \"45.\",\n \"title\": \"Bichler, O. et al. Visual pattern extraction using energy-efficient “2-PCM synapse” neuromorphic architecture.IEEE Trans. Electron Devices59, 2206–2214 (2012).\",\n \"link\": \"https://doi.org/10.1109%2FTED.2012.2197951\"\n },\n {\n \"idx\": \"48.\",\n \"title\": \"Knill, D. C. & Pouget, A. The Bayesian brain: the role of uncertainty in neural coding and computation.Trends Neurosci.27, 712–719 (2004).\",\n \"link\": \"https://doi.org/10.1016%2Fj.tins.2004.10.007\"\n },\n {\n \"idx\": \"49.\",\n \"title\": \"Pouget, A., Drugowitsch, J. & Kepecs, A. Confidence and certainty: distinct probabilistic quantities for different goals.Nat. Neurosci.19, 366–374 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fnn.4240\"\n },\n {\n \"idx\": \"51.\",\n \"title\": \"Chang, C.-C. et al. Mitigating asymmetric nonlinear weight update effects in hardware neural network based on analog resistive synapse.IEEE J. Emerg. Sel. Top. Circuits Syst.8, 116–124 (2017).\",\n \"link\": \"https://doi.org/10.1109%2FJETCAS.2017.2771529\"\n },\n {\n \"idx\": \"56.\",\n \"title\": \"Rolls, E. T., Grabenhorst, F. & Deco, G. Choice, difficulty, and confidence in the brain.NeuroImage53, 694–706 (2010).\",\n \"link\": \"https://doi.org/10.1016%2Fj.neuroimage.2010.06.073\"\n },\n {\n \"idx\": \"57.\",\n \"title\": \"Perry, C. J. & Barron, A. B. Honey bees selectively avoid difficult choices.Proc. Natl Acad. Sci. USA110, 19155–19159 (2013).\",\n \"link\": \"https://doi.org/10.1073%2Fpnas.1314571110\"\n },\n {\n \"idx\": \"58.\",\n \"title\": \"Gherman, S. & Philiastides, M. G. Neural representations of confidence emerge from the process of decision formation during perceptual choices.NeuroImage106, 134–143 (2015).\",\n \"link\": \"https://doi.org/10.1016%2Fj.neuroimage.2014.11.036\"\n },\n {\n \"idx\": \"59.\",\n \"title\": \"Kiani, R. & Shadlen, M. N. Representation of confidence associated with a decision by neurons in the parietal cortex.Science324, 759–764 (2009).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1169405\"\n },\n {\n \"idx\": \"60.\",\n \"title\": \"Kepecs, A., Uchida, N., Zariwala, H. A. & Mainen, Z. F. Neural correlates, computation and behavioural impact of decision confidence.Nature455, 227–231 (2008).\",\n \"link\": \"https://doi.org/10.1038%2Fnature07200\"\n },\n {\n \"idx\": \"63.\",\n \"title\": \"Kim, K. & Jeong, G. The prospects of non-volatile phase-change RAM.Microsyst. Technol.13, 145–147 (2006).\",\n \"link\": \"https://link.springer.com/doi/10.1007/s00542-006-0150-y\"\n },\n {\n \"idx\": \"66.\",\n \"title\": \"Ielmini, D., Lacaita, A. L. & Mantegazza, D. Recovery and drift dynamics of resistance and threshold voltages in phase-change memories.IEEE Trans. Electron Devices54, 308–315 (2007).\",\n \"link\": \"https://doi.org/10.1109%2FTED.2006.888752\"\n },\n {\n \"idx\": \"68.\",\n \"title\": \"Woo, J. Y. et al. 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Recent tunneling experiments indicate a transition between nodal and gapped pairing with electron filling, which is not naturally understood within current theory. We demonstrate that the coexistence of superconductivity and flavor polarization leads to pairing channels that are guaranteed by symmetry to be entirely band-off-diagonal, with a variety of consequences: most notably, the pairing invariant under all symmetries can have Bogoliubov Fermi surfaces in the superconducting state with protected nodal lines, or may be fully gapped, depending on parameters, and the band-off-diagonal chiral p -wave state exhibits transitions between gapped and nodal regions upon varying the doping. We demonstrate that band-off-diagonal pairing can be the leading state when only phonons are considered, and is also uniquely favored by fluctuations of a time-reversal-symmetric intervalley coherent order motivated by recent experiments. Consequently, band-off-diagonal superconductivity allows for the reconciliation of several key experimental observations in graphene moiré systems."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"The fascinating physics [1] , [2] of correlated graphene moiré superlattices, such as twisted-bilayer (TBG) and twisted-trilayer graphene (TTG), has generated extensive efforts to uncover the mysteries of their phase diagrams. Much progress has been made towards understanding their normal-state physics, including the correlated insulating phases [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] and the reset behavior [19] , [20] ; the latter, which is believed to be associated with the onset of flavor polarization, appears in the same density range and can coexist with superconductivity [13] , [19] , [20] , [21] , [22] , [23] , [24] , [25] , [26] , [27] , [28] , [29] , [30] , [31] , [32] , [33] , [34] . However, the form and symmetry of the superconducting order parameter and the pairing glue are still unknown, despite significant theoretical efforts [27] , [28] , [29] , [30] , [33] , [35] , [36] , [37] , [38] , [39] , [40] , [41] , [42] , [43] , [44] , [45] , [46] , [47] . Tunneling conductance measurements taken within the superconducting state reveal a V-shaped density of states (DOS) [48] , [49] which can become U -shaped at other electron concentrations [49] . Setting aside the possibility of thermal fluctuations as origin [50] , this is most naturally interpreted as a transition from nodal to fully gapped superconductivity. For a consistent microscopic theoretical understanding, this provides the following challenges: (i) electron–phonon coupling—a widely discussed [33] , [35] , [36] , [37] , [38] , [39] , [40] pairing mechanism in TBG and TTG—will typically mediate an entirely attractive interaction in the Cooper channel, with the leading pairing state that transforms trivially under all symmetries and is thus fully gapped [51] , [52] . (ii) Even when the low-energy interactions favor an irreducible presentation (IR), e.g., E of C 3 , with nodal basis functions ( p - or d -wave), the generically fully gapped chiral configuration wins over the nodal nematic one within mean-field. (iii) Even if we assume that the nodal state is energetically favored, e.g., due to significant corrections beyond mean-field [27] , [53] , [54] , [55] , one is still left to explain why there is a transition to another, fully gapped superconductor upon changing the filling. In this work, we show that the combination of flavor polarization and the representations of the symmetries in the flat bands of TBG and TTG allow for pairing channels that are completely off-diagonal in the flat bands and that such band-off-diagonal states can naturally reconcile all three key challenges (i–iii). More specifically, we find two distinct band-off-diagonal states: one of them transforms under the trivial representation A of the system’s point group C 6 (or one of A 1,2 of D 6 if we set the displacement field to zero) but can nonetheless have symmetry-protected nodal lines, akin to Bogoliubov Fermi surfaces discussed in refs. [56] , [57] , see Fig. 1 a–c for an intuitive visual explanation. The surprising possibility of the existence of such Bogliubov Fermi surfaces without an external magnetic field is unique to twisted graphene systems in that it follows as a direct consequence of both the symmetry and relative flatness of their normal-state bands. The second off-diagonal state transforms under a two-dimensional IR ( E 2 of C 6 ). Its associated chiral state, E 2 (1, i ), which is favored in the mean-field over the nematic one, has the unique property of exhibiting nodal lines or being fully gapped depending on the filling fraction, even when the order parameter is kept fixed. We supplement our general symmetry arguments and phenomenological models with Hartree-Fock (HF) calculations on the continuum model, studying a variety of different pairing mechanisms. We find that nodal band-off-diagonal pairing is favored by the optical A 1 and B 1 phonon modes and by fluctuations of a time-reversal symmetric intervalley coherent (T-IVC) state (the T-IVC state has Kekulé order on the graphene scale [58] , [59] , [60] ). Evidence for the former has been provided by a recent photoemission study [61] while evidence for the latter comes from recent STM experiments [7] . Furthermore, also fluctuations of a time-reversal-symmetric sublattice polarized state (SLP+) are attractive in the band-off-diagonal channel (see Table 2 for a formal definition of the order parameters). We also show that fluctuations of both T-IVC and of a nematic, time-reversal symmetric IVC order [62] favor either the band-off-diagonal A or an E 1 state with band-diagonal components, which may also be nodal; the winner is determined by the relative amount of nematic IVC and T-IVC fluctuations. Fig. 1: Spectral properties of interband pairing. While for band-diagonal pairing a small superconducting order parameter can immediately open up a gap as time-reversal symmetry guarantees that the associated avoided crossings [gray regions in a ] in the Bogoliubov spectrum are at the Fermi level, this is not the case for band-off-diagonal pairing ( b ). Here, a sufficiently strong order-parameter value is required to establish a full gap, see c . Its k dependence according to Eq. ( 3 ) is shown in d , where the red line indicates nodal points. If the band structure has Dirac points, there will be a point on the horizontal axis (blue cross). Consequently, if there is another momentum point located above the red line (blue circle), continuity of the Hamiltonian implies a nodal point on any path connecting the two momenta. e Gap of the isotropic A 2 state and δ k , ϵ k (zeros indicated in red) for the normal-state toy model defined in the text. BEC/BI refers to the Bose-Einstein condensate/band insulator limit. f Complex phase \\({\\varphi }_{{{{{{{{\\boldsymbol{k}}}}}}}}}=\\arg ({X}_{{{{{{{{\\boldsymbol{k}}}}}}}}}+i{Y}_{{{{{{{{\\boldsymbol{k}}}}}}}}})\\) for leading basis function with small subleading corrections. g Shows the gap of the chiral p -wave E 2 (1, i ) state with Δ 0 = 1.5 t and the value of \\({D}_{{{{{{{{{\\boldsymbol{k}}}}}}}}}_{j}}: \\!\\!=| {\\delta }_{{{{{{{{{\\boldsymbol{k}}}}}}}}}_{j}}| -| {\\epsilon }_{{{{{{{{{\\boldsymbol{k}}}}}}}}}_{j}}-\\mu |\\) for k j at the three symmetry-in-equivalent vortices in f as a function of μ . We took \\({t}^{{\\prime} }=-2.2t,t \\, > \\, 0\\) , in b , d . Full size image","references":{"indices":[1,2,3,4,5,6,8,19,20,21,22,23,25,26,27,29,31,32,33,34,35,36,37,38,40,42,43,47,48,49,51,52,53,54,55,58,59,60,62],"infos":[{"idx":"1","title":"Andrei, E. Y. & MacDonald, A. H. Graphene bilayers with a twist.Nat. 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A general pairing order parameter in the inter-valley channel couples as $${{{{{{{{\\mathcal{H}}}}}}}}}_{{{{{{{{\\rm{p}}}}}}}}}=\\mathop{\\sum}\\limits_{{{{{{{{\\boldsymbol{k}}}}}}}},\\eta= \\pm ,\\alpha,{\\alpha }^{{\\prime} }}{c}_{{{{{{{{\\boldsymbol{k}}}}}}}},\\alpha,\\eta }^{{{{\\dagger}}} }{\\left({{{\\Delta }}}_{{{{{{{{\\boldsymbol{k}}}}}}}},\\eta }\\right)}_{\\alpha,{\\alpha }^{{\\prime} }}{c}_{-{{{{{{{\\boldsymbol{k}}}}}}}},{\\alpha }^{{\\prime} },-\\eta }^{{{{\\dagger}}} }+\\,{{\\mbox{H.c.}}}\\,,$$ (1) where the order parameter \\({{{\\Delta }}}_{{{{{{{{\\boldsymbol{k}}}}}}}},\\eta }=-{{{\\Delta }}}_{-{{{{{{{\\boldsymbol{k}}}}}}}},-\\eta }^{T}\\) is a matrix in band space. The physical spin texture of the superconductor is entirely determined by the form of the underlying normal-state’s polarization: if the spins are aligned in the two valleys, the superconductor is a non-unitary triplet, while anti-alignment [24] , [28] leads to a singlet-triplet admixed state [13] , [27] , [28] . In both cases, all of the following states are well defined, with the aforementioned spin structures and symmetries given by appropriate combinations of spinless operations and spin rotations (see Supplementary Appendix A1). We will classify the pairing states according to the irreducible representations (IRs) of the system’s point group D 6 , which is generated by six-fold rotations ( C 6 z ) along the z axis and two-fold rotation symmetry ( C 2 x ) along the in-plane x axis. Note a displacement field ( D 0 ≠ 0) breaks the in-plane rotations leading to the point group C 6 . Importantly, all IRs of D 6 and C 6 are either even or odd under C 2 z . Choosing the phases of the Bloch states such that C 2 z acts as c k , α , η → c − k , α ,− η , it holds $${C}_{2z}:\\quad {{{\\Delta }}}_{{{{{{{{\\boldsymbol{k}}}}}}}},\\eta }\\quad \\longrightarrow \\quad {{{\\Delta }}}_{-{{{{{{{\\boldsymbol{k}}}}}}}},-\\eta }=-{{{\\Delta }}}_{{{{{{{{\\boldsymbol{k}}}}}}}},\\eta }^{T}.$$ (2) This immediately implies that the pairing states in all IRs even under C 2 z ( A 1 , A 2 , E 2 of D 6 ) must be anti-symmetric in band space and, thus, entirely band-off-diagonal, whereas the order parameters of the other IRs ( B 1 , B 2 , E 1 ) are symmetric and can contain both band-diagonal and band-off-diagonal components. While superconducting order parameters with finite band-off-diagonal components are rather common in multi-band systems, the existence of pairing states that are constrained to be entirely band-off-diagonal is rather unique and follows from the combination of C 2 z symmetry and the spin polarization in the normal state. Importantly, this is unaffected by strain or nematic order breaking C 3 z as long as C 2 z remains, which guarantees that there are IRs with entirely band-off-diagonal order parameters. Choosing the phase conventions of the Bloch states such that C 2 x and C 3 z act as \\({c}_{{{{{{{{\\boldsymbol{k}}}}}}}},\\alpha,\\eta }\\to {({\\sigma }_{z})}_{\\alpha \\alpha }{c}_{({k}_{x},-{k}_{y}),\\alpha,\\eta }\\) and \\({c}_{{{{{{{{\\boldsymbol{k}}}}}}}},\\alpha,\\eta }\\to {c}_{{C}_{3z}{{{{{{{\\boldsymbol{k}}}}}}}},\\alpha,\\eta }\\) , respectively, the resulting candidate order parameters are summarized in Table 1 . Note that a momentum-independent representation of C 2 x must be σ z due to the bands’ eigenvalues at the Γ-M line, which in turn are connected to the topological obstruction of the flat bands [63] . The reality (Hermiticity) constraint in Table 1 on χ , X , and Y ( \\(\\hat{\\chi },\\, \\hat{X}\\) , and \\(\\hat{Y}\\) ) comes from the residual spinless time-reversal symmetry Θ of the normal state [64] , [65] . The two two-dimensional IRs E 1,2 are each associated with three pairing states—two nematic phases E 1,2 (1, 0), E 1,2 (0, 1) and one chiral state E 1,2 (1, i ). Table 1 Summary of pairing states in spin-polarized flat bands Full size table Spectral properties We here have the rather unique situation that there are pairing channels, associated with the IRs A 1,2 and E 2 , where the pairing is constrained by C 2 z to be entirely band-off-diagonal. One immediate very unusual consequence is that the superconducting order-parameter transforming under the trivial representation ( A 1 ) has a symmetry-imposed line of zeros along the Γ-M line, and hence a nodal point in the spectrum. This is related to the topology-induced non-trivial representation of C 2 x in band space. We refer to ref. [39] for the discussion of other topological nodal points for pairing in obstructed TBG bands. As we will show next, band-off-diagonal pairing leads to additional unusual spectral properties with far-reaching consequences for graphene moiré systems. To this end, consider the following effective Hamiltonian, \\({{{{{{{{\\mathcal{H}}}}}}}}}_{{\\sigma }_{y}}={\\sum }_{{{{{{{{\\boldsymbol{k}}}}}}}}}{c}_{{{{{{{{\\boldsymbol{k}}}}}}}},\\alpha,\\eta }^{{{{\\dagger}}} }{c}_{{{{{{{{\\boldsymbol{k}}}}}}}},\\alpha,\\eta }{\\xi }_{\\eta \\cdot {{{{{{{\\boldsymbol{k}}}}}}}},\\alpha }+{\\sum }_{{{{{{{{\\boldsymbol{k}}}}}}}}}[{{{\\Delta }}}_{{{{{{{{\\boldsymbol{k}}}}}}}}}\\,{c}_{{{{{{{{\\boldsymbol{k}}}}}}}},+}^{{{{\\dagger}}} }{\\sigma }_{y}{c}_{-{{{{{{{\\boldsymbol{k}}}}}}}},-}^{{{{\\dagger}}} }+\\,{{\\mbox{H.c.}}}\\,]\\) , where the scalar function Δ k describes the form of pairing. We will here study two cases that are conventionally considered to be fully gapped, (i) a momentum-independent “ s -wave state” ( A 2 or A pairing in Table 1 ) where Δ k = Δ 0 and (ii) a “chiral p -wave” state, or more precisely an E 2 (1, i ) state, where Δ k = Δ 0 ( X k + i Y k ) with ( X k , Y k ) being smooth, MBZ-periodic functions transforming as ( x , y ) under C 3 z . Furthermore, we parameterize the dispersion, ξ η ⋅ k , α , of the two flat bands ( α = ± ) in valley η = ± as ξ k , α = ϵ k − μ + α δ k , where ϵ k and δ k are C 3 z (and, for D 0 = 0, C 2 x ) symmetric functions. The Bogoliubov spectrum of \\({{{{{{{{\\mathcal{H}}}}}}}}}_{{\\sigma }_{y}}\\) has four bands, given by \\(\\pm{\\delta }_{{{{{{{{\\boldsymbol{k}}}}}}}}}\\pm \\sqrt{{({\\epsilon }_{{{{{{{{\\boldsymbol{k}}}}}}}}}-\\mu )}^{2}+| {{{\\Delta }}}_{{{{{{{{\\boldsymbol{k}}}}}}}}}{| }^{2}}\\) . Consequently, the excitation gap at momentum k reads as $${{\\Delta }}{E}_{{{{{{{{\\boldsymbol{k}}}}}}}}}=\\Big | | {\\delta }_{{{{{{{{\\boldsymbol{k}}}}}}}}}| -\\sqrt{{({\\epsilon }_{{{{{{{{\\boldsymbol{k}}}}}}}}}-\\mu )}^{2}+| {{{\\Delta }}}_{{{{{{{{\\boldsymbol{k}}}}}}}}}{| }^{2}}\\Big|,$$ (3) which is shown in Fig. 1 d, and therefore exhibits nodes where \\(| {\\delta }_{{{{{{{{\\boldsymbol{k}}}}}}}}}|=\\sqrt{{({\\epsilon }_{{{{{{{{\\boldsymbol{k}}}}}}}}}-\\mu )}^{2}+| {{{\\Delta }}}_{{{{{{{{\\boldsymbol{k}}}}}}}}}{| }^{2}}\\) . As long as the band structure has Dirac points, there are points k D in the Brillouin zone with \\({\\delta }_{{{{{{{{{\\boldsymbol{k}}}}}}}}}_{D}}=0\\) , associated with the blue cross in Fig. 1 d. Furthermore, for a metallic normal state, μ must be within the bandwidth and, hence, there must be a region R in momentum space where ∣ δ k ∣ > ∣ ϵ k − μ ∣ . For the momentum-independent A 2 state, Δ k = Δ 0 , this implies that there exists \\({{{\\Delta }}}_{0}^{c} > 0\\) such that there is k * ∈ R with parameters (such as the blue circle) above the red solid line in Fig. 1 d as long as \\(| {{{\\Delta }}}_{0}| < {{{\\Delta }}}_{0}^{c}\\) . By continuity, this means that there must be a nodal point on any line connecting k D and k * . Consequently, for μ within the bandwidth and \\({\\delta }_{{{{{{{{{\\boldsymbol{k}}}}}}}}}_{D}}=0\\) for some k D , the A 2 will always have a nodal line if ∣ Δ 0 ∣ is sufficiently small, consistent with the intuitive picture based on the Bogoliubov spectrum in Fig. 1 a–c. We illustrate this further in Fig. 1 e using a toy model with \\({\\delta }_{{{{{{{{\\boldsymbol{k}}}}}}}}}=t\\left|1+{e}^{i{{{{{{{{\\boldsymbol{a}}}}}}}}}_{1}\\cdot {{{{{{{\\boldsymbol{k}}}}}}}}}+{e}^{-i{{{{{{{{\\boldsymbol{a}}}}}}}}}_{2}\\cdot {{{{{{{\\boldsymbol{k}}}}}}}}}\\right|\\) and \\({\\epsilon }_{{{{{{{{\\boldsymbol{k}}}}}}}}}={t}^{{\\prime} }\\mathop{\\sum }\\nolimits_{j=1}^{3}\\cos {{{{{{{{\\boldsymbol{a}}}}}}}}}_{j}\\cdot {{{{{{{\\boldsymbol{k}}}}}}}},\\, {{{{{{{{\\boldsymbol{a}}}}}}}}}_{j}={[{C}_{3z}]\\,}^{j-1}{(\\sqrt{3},0)}^{T}\\) . This leads to the second unexpected conclusion that, for any pairing mechanism, including conventional electron–phonon coupling, the leading instability either has nodal lines in a finite region below T c or transforms non-trivially under the symmetries of the normal state. For electron–phonon pairing (or pairing mediated by the fluctuations of any time-reversal-symmetric order parameter [52] , such as the T-IVC state) this is particularly unexpected since it is generally believed to always lead to a fully gapped state that transforms trivially under all symmetries. In fact, this can be proven in general terms [51] , [52] , even for spin-orbit-split Fermi surfaces and beyond mean-field theory [52] . The crucial difference to these works, however, is that spinfull time-reversal is broken in our case such that the Fermi-Dirac constraint is inconsistent with a non-sign-changing, band-diagonal pairing state. This leads to the unique situation that although electron–phonon coupling will lead to entirely attractive interactions in the Cooper channel, the superconducting energetics is frustrated: the dominant pairing state is determined by whether the energetic loss due to non-resonant band-off-diagonal Cooper pairs ( A 2 pairing) or the costs from sign changes of the order parameter (such as B 1 ) are less harmful. We will demonstrate this explicitly by a model calculation in sec. “electron–phonon coupling below, where either A 2 or B 1 is dominant, depending on the form of the electron–phonon coupling”. Let us first, however, discuss the general spectral properties of the “chiral p -wave” state which is canonically expected to be fully gapped as long as the Fermi surfaces do not cross the zeros of X k + i Y k . Three of these zeros have to be at the Γ, K , and \\({K}^{{\\prime} }\\) points as a consequence of C 3 z symmetry. In the absence of fine-tuning, X k + i Y k will have vortices at these points with vorticity v = + 1. As can be seen in Fig. 1 f, where we show the phase of X k + i Y k using an admixture of the two lowest-order terms, the net vorticity of + 3 at these high-symmetry points has to be compensated by anti-vortices at generic momenta. The lowest possible number is three C 3 z -related vortices, which appear near the M points in Fig. 1 f. If it holds ∣ δ k ∣ > ∣ ϵ k − μ ∣ at any of these zeros k = k j , we obtain a point above the red line in Fig. 1 d and, thus, a nodal point along any contour between that k j and k D ; as opposed to the A 2 state, this holds irrespective of the value of Δ 0 and therefore all the way to zero temperature. In summary, we find that also the E 2 (1, i ) “chiral p -wave” state is not generically fully gapped but instead will exhibit a nodal line encircling any zero k j of X k + i Y k with \\(| {\\delta }_{{{{{{{{{\\boldsymbol{k}}}}}}}}}_{j}}| > | {\\epsilon }_{{{{{{{{{\\boldsymbol{k}}}}}}}}}_{j}}-\\mu |\\) . This leads to an interesting filling dependence of the superconducting gap, as we illustrate in our toy model in Fig. 1 g along with the criterion \\({D}_{{{{{{{{{\\boldsymbol{k}}}}}}}}}_{j}}:=| {\\delta }_{{{{{{{{{\\boldsymbol{k}}}}}}}}}_{j}}| -| {\\epsilon }_{{{{{{{{{\\boldsymbol{k}}}}}}}}}_{j}}-\\mu | > 0\\) evaluated at the vortices at Γ, K/K’, and near M. Depending on μ , D k is positive only near the Γ point or only in a region surrounding the vortices close to the M points, leading to nodal lines encircling Γ and near the M points, respectively, as shown in the inset of Fig. 1 g. These regimes are separated by a fully gapped region where D k < 0 for all k , which could explain the fully gapped to nodal transition seen in tunneling experiments [49] when the filling fraction is changed. Note that \\({D}_{{{{{{{{{\\boldsymbol{k}}}}}}}}}_{j}}=-| {\\epsilon }_{{{{{{{{{\\boldsymbol{k}}}}}}}}}_{j}}-\\mu | \\le 0\\) for k j at the K and K’ points. In Fig. 1 g, \\({D}_{K}={D}_{{K}^{{\\prime} }}\\) vanishes close to the top of the band, which simply means that the Fermi surfaces cross the K, K’ points and the superconductor has nodal points for this fine-tuned value of the chemical potential. Fluctuation-induced pairing Having discussed the unique energetics of pairing and spectral properties of the resulting superconductors in spin-polarized quasi-flat-bands with Dirac cones on a general level, we next study these aspects more explicitly by solving the superconducting self-consistency equations in the flat bands common to alternating-twist graphene systems. We will start with pairing induced by fluctuations of a nearby symmetry-broken phase. To this end, we will couple the low-energy electrons introduced in Eq. ( 1 ) to a collective bosonic field \\({\\phi }_{j}({{{{{{{\\boldsymbol{q}}}}}}}})={\\phi }_{j}^{{{{\\dagger}}} }(-{{{{{{{\\boldsymbol{q}}}}}}}})\\) via $${{{{{{{{\\mathcal{H}}}}}}}}}_{\\phi }=\\mathop{\\sum}\\limits_{{{{{{{{\\boldsymbol{k}}}}}}}},{{{{{{{\\boldsymbol{q}}}}}}}},j}{c}_{{{{{{{{\\boldsymbol{k}}}}}}}}+{{{{{{{\\boldsymbol{q}}}}}}}},\\alpha,\\eta }^{{{{\\dagger}}} }{\\lambda }_{\\alpha,\\eta ;{\\alpha }^{{\\prime} },{\\eta }^{{\\prime} }}^{j}{c}_{{{{{{{{\\boldsymbol{k}}}}}}}},{\\alpha }^{{\\prime} },{\\eta }^{{\\prime} }}{\\phi }_{j}({{{{{{{\\boldsymbol{q}}}}}}}}),$$ (4) where the Hermitian matrices λ j capture the nature of the correlated insulating phase; we here choose and normalize λ j such that \\({({\\lambda }^{j})}^{2}={\\mathbb{1}}\\) . Both for twisted bi- [9] and trilayer graphene [14] , [15] , [29] , the stable phases emerging out of the U (4) × U (4) [9] manifold in the chiral-flat (decoupled) limit are natural candidates. Integrating out the bosonic modes, we obtain an effective electronic interaction which in the for superconductivity relevant intervalley Cooper channel reads as $${{{{{{{{\\mathcal{H}}}}}}}}}_{{{{{{{{\\rm{int}}}}}}}}}^{\\phi }=\t -\\mathop{\\sum}\\limits_{{{{{{{{\\boldsymbol{k}}}}}}}},{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} }}{\\chi }_{{{{{{{{\\boldsymbol{k}}}}}}}}-{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} }}{{{{{{{{\\mathcal{V}}}}}}}}}_{(\\eta,\\alpha,\\beta ),({\\eta }^{{\\prime} },{\\alpha }^{{\\prime} },{\\beta }^{{\\prime} })}\\\\ \t \\times {c}_{-{{{{{{{\\boldsymbol{k}}}}}}}},\\beta,-\\eta }^{{{{\\dagger}}} }{c}_{{{{{{{{\\boldsymbol{k}}}}}}}},\\alpha,\\eta }^{{{{\\dagger}}} }{c}_{{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} },{\\alpha }^{{\\prime} },{\\eta }^{{\\prime} }}{c}_{-{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} },{\\beta }^{{\\prime} },-{\\eta }^{{\\prime} }},$$ (5) with vertex $${{{{{{{{\\mathcal{V}}}}}}}}}_{(\\eta,\\alpha,\\beta ),({\\eta }^{{\\prime} },{\\alpha }^{{\\prime} },{\\beta }^{{\\prime} })}={t}_{\\phi }\\mathop{\\sum}\\limits_{j}{\\left[{\\lambda }_{\\beta,\\eta ;{\\beta }^{{\\prime} },{\\eta }^{{\\prime} }}^{j}\\right]}^{*}{\\lambda }_{\\alpha,\\eta ;{\\alpha }^{{\\prime} },{\\eta }^{{\\prime} }}^{j},$$ (6) t ϕ = ± 1 encoding whether the order parameter is even or odd under time-reversal, Θ ϕ j ( q )Θ † = t ϕ ϕ j ( q ), and χ q > 0 denoting the (static) susceptibility of ϕ j . Before discussing numerical results for the full model, we first focus on perfectly flat bands. In this limit, the leading superconducting instability within mean-field theory is given by the largest eigenvalue of \\({{{{{{{\\mathcal{V}}}}}}}}\\) in Eq. ( 6 ) viewed as a matrix in the multi-index ( η , α , β ). Furthermore, if there is an anti-symmetric, valley-off-diagonal matrix D obeying (see Methods) $${[D{\\eta }_{x},{\\lambda \\,}^{j}]}_{-{t}_{\\phi }}\\equiv D{\\eta }_{x}{\\lambda }^{j}-{t}_{\\phi }{\\lambda }^{j}D{\\eta }_{x}=0,$$ (7) the associated leading superconducting order parameter in Eq. ( 1 ) is given by \\({({{{\\Delta }}}_{{{{{{{{\\boldsymbol{k}}}}}}}},\\eta })}_{\\alpha,{\\alpha }^{{\\prime} }}={\\delta }_{{{{{{{{\\boldsymbol{k}}}}}}}}}{(D{\\eta }_{x})}_{\\alpha,\\eta ;{\\alpha }^{{\\prime} }\\eta }\\) with δ k > 0; here η j denotes Pauli matrices in valley space and the precise form of δ k is determined by χ ( q ). T-IVC fluctuations Motivated by recent experiments [7] providing direct evidence for T-IVC order, we start with T-IVC fluctuations as a pairing glue. In the U (4) × U (4) symmetric limit, the T-IVC state is associated with λ j = σ 0 η j , j = x , y , within our conventions. Since t ϕ = + 1, we are looking for D η x that commutes with λ j . Interestingly, there is a unique anti-symmetric, valley-off-diagonal matrix D ∝ σ y η x with that property, implying that the leading pairing state has the form Δ k , η = σ y δ k , δ k > 0. This is exactly the A 2 state in Table 1 , which, as discussed above, will have nodal lines at least in the vicinity of T c when a finite band dispersion is taken into account. Intuitively, the fact that A 2 pairing is favored can be understood by noticing that the valley-off-diagonal form of λ j leads to an attractive interaction across the valleys, which penalizes the B 1 state with its sign change between the two valleys. In fact, it holds \\({{{{{{{{\\mathcal{V}}}}}}}}}_{(\\eta,\\alpha,\\beta ),({\\eta }^{{\\prime} },{\\alpha }^{{\\prime} },{\\beta }^{{\\prime} })}=(1-\\eta \\,{\\eta }^{{\\prime} })\\mathop{\\sum }\\nolimits_{\\mu=0}^{3}{({\\sigma }_{\\mu }^{*})}_{\\alpha,\\beta }{({\\sigma }_{\\mu })}_{{\\alpha }^{{\\prime} },{\\beta }^{{\\prime} }}\\) showing explicitly that it is repulsive (attractive) in the B 1 ( A 2 ) channel. To go beyond the flat-band limit, we solve the superconducting mean-field equations numerically. We take the flat TBG bands from the continuum model [66] as the starting point. To capture the spin-polarized normal state, we supplement it with Coulomb repulsion and perform HF calculation (see Supplementary Appendix A for details). As can be seen in the resulting band structure shown in Fig. 2 a with interaction renormalization assuming filling fraction ν = 2, this not only pushes one spin flavor below the Fermi level but also induces significant band renormalizations. For our subsequent study of superconducticity, we project onto the two bands at the Fermi level and associate them with the creation operators c k , α in the interactions in Eqs. ( 4 ) and ( 5 ). In our numerical computations, we choose \\(\\chi ({{{{{{{\\boldsymbol{q}}}}}}}})=\\frac{1}{{A}_{m}}\\frac{V}{{\\alpha }^{2}+| {{{{{{{\\boldsymbol{q}}}}}}}}{| }^{2}/{k}_{\\theta }^{2}}\\) where A m is the real space area of a moiré unit cell, and take α = 0.05 for concreteness, although we checked our main conclusion do not crucially depend on this form. In all of our numerics, we work at doping ν = 2.5. Fig. 2: Pairing mediated by T-IVC fluctuations. We show a the band structure of the normal state with spin polarization (K, K', and Γ label the high-symmetry points of the moiré scale Brillouin zone) and b the critical temperature T c (in units of the maximum band splitting W 0 ≃ 9.4 meV) as a function of coupling strength V measured in units of the critical coupling V c ,1 = 105 meV ⋅ nm 2 obtained from the linearized gap equation. The band structure (with color indicating the band-projected value of the anomalous correlator) of the A 2 state and its order parameter are shown in c and d . The DOS of the T = 0 superconductor for several different values of coupling strength V is plotted in e . The DOS was computed as \\({\\sum }_{{{{{{{{\\boldsymbol{k}}}}}}}}}\\delta \\left({E}_{{{{{{{{\\boldsymbol{k}}}}}}}}}-\\omega \\right)\\) , replacing the δ function with Lorentzians with half width at half max 0.3 meV (much smaller than the typical superconducting order parameter). The critical coupling V c ,2 where the nodal lines disappear is V c ,2 ≃ 1.4 V c ,1 . Full size image As expected, we indeed find that the A 2 state dominates, both right at the critical temperature T c , obtained from the linearized gap equation, and at T = 0 as we show by iteratively solving the full self-consistency equation (see Supplementary Appendix C). One crucial effect of the finite dispersion and splitting between the bands is that a finite interaction strength, V > V c ,1 , is required to stabilize the superconducting phase, as can be seen in the plot of T c in Fig. 2 b. Superconductivity ceases to be a weak-coupling instability as the Bloch states ( k , α , η ) and \\((-{{{{{{{\\boldsymbol{k}}}}}}}},\\, {\\alpha }^{{\\prime} },-\\eta )\\) are not degenerate for \\(\\alpha \\, \\ne \\, {\\alpha }^{{\\prime} }\\) , cutting off the logarithmic divergence known from BCS theory. The quasi-particle spectrum and order parameter of superconductivity from T = 0 numerics are shown in Fig. 2 c, d. In accordance with our general discussion above, we observe that the order parameter only has finite components proportional to σ y , which do not mix with the band-even contributions ∝ σ 0, x , z as a result of C 2 z symmetry. Furthermore, it does not change sign as a function of k and, for sufficiently small V but still with V > V c ,1 , the nodal lines in the superconducting spectrum persist all the way to T = 0, while the nodal line is gapped out at low T < T c if V > V c ,2 . The interaction-strength-dependence of the superconducting gap can be more clearly seen in Fig. 2 e, where we show the DOS for the self-consistent solution at T = 0. For large V , the superconductor becomes fully gapped at T = 0, leading to a U -shaped DOS. With smaller V , the magnitude of the order-parameter decreases and the superconductor eventually exhibits nodal lines, as explained above. In the regime just before these nodal lines appear, there is an increase in the DOS near the Fermi level, roughly when the order parameter and the maximal band splitting are comparable, leading to a V-shaped DOS (green line). The lifetime parameter used to compute the DOS is 0.3 meV; this choice was based on our k -grid spacing. While it is not necessarily small with respect to the tunneling gap (which vanishes at V c ,2 ), it is small with respect to Δ( k ), which is of order 5 meV just as the state is becoming fully gapped for our choice of normal state. This behavior of the DOS with interaction strength may offer a natural explanation for the U -shaped tunneling conductance measurements near ν = 2 and V-shaped tunneling conductance measurements near ν = 3 observed in TTG [49] ; if we are considering T-IVC fluctuations of the insulator at ν = 2, then it may be reasonable to expect the coupling to these fluctuations could grow weaker as we dope towards ν = 3, in line with the experimentally observed ν dependence. Note that the regime we call V-shaped here is strictly speaking fully gapped. However, the crucial difference to the BCS state is that the gap is much smaller than the order-parameter magnitude as a result of the different Bogoliubov spectrum in Eq. ( 3 ). This is why, depending not only on the magnitude of the pairing but also on the precise form of the normal state, the resulting tunneling spectra can resemble those observed experimentally [48] , [49] , such as the green curve in Fig. 2 e, making the A 2 state an attractive candidate. The regime of small V where stable superconductivity with true Bogoliubov Fermi surfaces is observed can further exhibit a peak at ω = 0 which is due to a Van Hove singularity crossing the Fermi level, see blue curve in Fig. 2 e; while this peak has not been observed experimentally, its presence crucially depends on details of the normal-state band structure and is only found to be energetically favored in a very small regime of V in our model. Electron–phonon coupling To illustrate that the off-diagonal A 2 state is more generally favored beyond just T-IVC fluctuations, we next discuss electron–phonon coupling, which is frequently considered a plausible pairing mechanism for twisted moiré systems [33] , [35] , [36] , [37] , [38] , [39] . Similar to ref. [35] , we use that the optical A 1 , B 1 , and E 2 phonon modes are known [67] to dominate the electron–phonon coupling in single-layer graphene. As these are optical phonons, we further assume that the impact of the interlayer coupling on the phonons can be neglected and arrive at $${{{{{{{{\\mathcal{H}}}}}}}}}_{EP}=\t \\int\\,{{{{{{{\\rm{d}}}}}}}}{{{{{{{\\boldsymbol{r}}}}}}}}\\,{\\psi }_{\\ell,s}^{{{{\\dagger}}} }({{{{{{{\\boldsymbol{r}}}}}}}})\\left[{g}_{{A}_{1}}{{{\\Lambda }}}_{{A}_{1}}{u}_{{A}_{1},\\mu }({{{{{{{\\boldsymbol{r}}}}}}}})\\right. \\\\ \t \\left.+{g}_{{B}_{1}}{{{\\Lambda }}}_{{B}_{1}}{u}_{{B}_{1},\\mu }({{{{{{{\\boldsymbol{r}}}}}}}})+{g}_{{E}_{2}}{{{{{{{{\\boldsymbol{\\Lambda }}}}}}}}}_{{E}_{2}}\\cdot {{{{{{{{\\boldsymbol{u}}}}}}}}}_{{E}_{2},\\mu }({{{{{{{\\boldsymbol{r}}}}}}}})\\right]{({{{{{{{{\\boldsymbol{v}}}}}}}}}_{\\mu })}_{\\ell }{\\psi }_{\\ell,s}({{{{{{{\\boldsymbol{r}}}}}}}})$$ (8) for the electron–phonon coupling, where v μ encode the layer structure of the modes (see Methods). Symmetry dictates that the vertices Λ g are given by \\({{{\\Lambda }}}_{{A}_{1}}={\\eta }_{x}{\\rho }_{x},\\, {{{\\Lambda }}}_{{B}_{1}}={\\eta }_{y}{\\rho }_{x}\\) , and \\({{{{{{{{\\boldsymbol{\\Lambda }}}}}}}}}_{{E}_{2}}=({\\eta }_{z}{\\rho }_{y},-{\\rho }_{x})\\) where ρ acts on the microscopic sublattice basis. Integrating out the phonons and projecting to the flat bands, we obtain an effective electron-electron interaction (see Methods) $${{{{{{{\\mathcal{H}}}}}}}}_{{{{{{{{\\rm{int}}}}}}}}}^{C}=\t -\\mathop{\\sum}\\limits_{{{{{{{{\\boldsymbol{k}}}}}}}},{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} }}{V}_{g}{\\left[{\\lambda }_{{{{{{{{\\boldsymbol{k}}}}}}}},\\beta,\\eta ;{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} },{\\beta }^{{\\prime} },{\\eta }^{{\\prime} }}^{g,j,\\mu }\\right]}^{*}{\\lambda }_{{{{{{{{\\boldsymbol{k}}}}}}}},\\alpha,\\eta ;{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} },{\\alpha }^{{\\prime} },{\\eta }^{{\\prime} }}^{g,j,\\mu }\\\\ \t \\times {c}_{-{{{{{{{\\boldsymbol{k}}}}}}}},\\beta,-\\eta }^{{{{\\dagger}}} }{c}_{{{{{{{{\\boldsymbol{k}}}}}}}},\\alpha,\\eta }^{{{{\\dagger}}} }{c}_{{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} },{\\alpha }^{{\\prime} },{\\eta }^{{\\prime} }}{c}_{-{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} },{\\beta }^{{\\prime} },-{\\eta }^{{\\prime} }},$$ (9) where the coupling constants V g of the three different phonon modes g = A 1 , B 1 , E 2 are estimated to obey \\({V}_{{A}_{1}}={V}_{{B}_{1}}\\simeq 1.33{V}_{{E}_{2}}\\) for parallel spins in the two valleys, while \\({V}_{{A}_{1}}={V}_{{B}_{1}}=0\\) for anti-parallel spins. From Eq. ( 9 ), it is clear that the induced interaction would be always completely attractive if we focused on intra-band pairing, \\(\\alpha={\\alpha }^{{\\prime} }=\\beta={\\beta }^{{\\prime} }\\) , which in spinful systems generically favors the trivial pairing channel [51] , [52] . In our case, the combination of two energetically close bands and the trivial pairing being purely band-off-diagonal leads to competition between different superconductors, even with electron–phonon coupling alone. To demonstrate this, we study intra-valley pairing within the mean-field approximation and parametrize the relative strength of the different phonon modes with an angle variable θ ph according to \\({V}_{{A}_{1}}={V}_{{B}_{1}}={V}_{0}\\cos {\\theta }_{{{{{{{{\\rm{ph}}}}}}}}},{V}_{{E}_{2}}={V}_{0}\\sin {\\theta }_{{{{{{{{\\rm{ph}}}}}}}}}\\) . The results of the mean-field calculation are summarized in Fig. 3 . We see that the A 2 pairing state is favored by the intervalley phonons ( θ ph = 0) inspite of its band-off-diagonal nature leading to a suppressed gap [see Fig. 3 a]. This is natural as these phonons mediate an attractive interaction between the two valleys which disfavors the B 1 state, similar to T-IVC fluctuations. In fact, focusing on the leading, momentum-independent term, \\({\\lambda }_{{{{{{{{\\boldsymbol{k}}}}}}}},{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} }}^{g,+}\\to {\\lambda }^{g,+},g={A}_{1},{B}_{1}\\) , symmetry dictates \\({\\lambda }^{{A}_{1},+}\\propto {\\sigma }_{0}{\\eta }_{1}\\) and \\({\\lambda }^{{B}_{1},+}\\propto {\\sigma }_{0}{\\eta }_{2}\\) in the chiral limit (see Supplementary Appendix D3). This maps the problem exactly to that of T-IVC fluctuations, immediately explaining why the order parameter has a fixed sign in Fig. 3 b. As θ ph is increased, the B 1 state is favored (roughly for θ ph > π /4) as can be seen in Fig. 3 c. This is expected since the intra-valley E 2 phonon mediates an attractive interaction within each valley such that the energy gain due to the enhanced gap [Fig. 3 d], associated with the band-diagonal matrix elements of the B 1 state, will overcompensate the energetic loss due to the sign change of B 1 ’s order-parameter between the two valleys. This picture is consistent with the dominant and non-sign-changing nature of the band-diagonal components of the B 1 state, see Fig. 3 e–g. Finally, this behavior can also be understood by applying the commutator criterion in Eq. ( 7 ) in the microscopic sublattice basis, see Supplementary Appendix D1. Fig. 3: Pairing from electron–phonon coupling. We show a the band structure and b the self-consistent order parameter of the A 2 pairing for θ ph = 0 and T = 0. The eigenvalues corresponding to the A 2 and B 1 pairings in the linearized gap equation at T = 5 K, which is close to their T c , are shown in c as a function of θ ph . We show an example of the band structure ( d ) of the B 1 pairing and its order parameter ( e , f , g ). In accordance with symmetry, the A 2 ( B 1 ) state only has order-parameter components ∝ σ y ( ∝ σ 0, x , z ). We took ν = 2.5 and V 0 = 250 meV ⋅ (nm) 2 with a continuum-model bandwidth ≃ 2 meV. We point out that if A 1 phonons are dominant, as suggested by recent experimental work [61] and past theoretical study in mono-layer graphene [68] , we would expect our A 2 pairing to dominate assuming the pairing potential is sufficiently large. We also emphasize that although the pairing functions for A 2 pairing (b) when θ = 0 and B 1 pairing when θ = π /2 ( e , f , g ) are roughly equal, the excitation spectra show the B 1 state with a band gap on the order of the pairing strength ( d ) while the A 1 state’s band gap is nearly zero, see a . Full size image This shows that, as opposed to the conventional scenario [51] , [52] , there are two possible leading superconducting states and the superconducting pairing state does not transform trivially under the symmetries of the system even when phonons alone provide the pairing glue. We have checked in our T = 0 numerics that a 60–70 meV ⋅ (nm) 2 coupling to A 1 and B 1 phonons (based on ref. [67] ) is roughly of the order needed to stabilize the A 2 pairing, assuming the normal state is the flat bands of the un-renormalized continuum model, which in our case has a bandwidth of 2 meV. However, we note that if the interaction-renormalized band splitting is much larger than the continuum-model bandwidth, or if the normal state has anti-parallel spins in either valley, additional particle-hole fluctuations, such as those of T-IVC order, will also be required for pairing. An interesting scenario arises for anti-parallel spins in the two valleys as a magnetic field will cant the spins and, hence, increase the projection of the intervalley phonon matrix elements to the flat bands. At least in TTG, with the suppressed orbital coupling, this could give rise to re-entrant superconductivity at high fields [25] . Other particle-hole fluctuations Finally, we discuss pairing induced by fluctuations of other particle-hole instabilities. In Table 2 , we list the resulting leading superconductors taking λ j in Eq. ( 4 ) to be any of the different strong-coupling candidate order parameters [9] , [13] , [14] , [15] , [29] . In particular, in addition to the T-IVC, we will consider the time-reversal-odd Kramers intervalley coherent state (K-IVC), and time-reversal-odd and -even sublattice polarized states (SLP− and SLP+). To analyze how sensitive our conclusions are to the precise form of the coupling of the strong-coupling fluctuating orders to the electrons, we also perform numerics by projecting momentum-independent coupling vertices in the microscopic basis with the correct symmetries (see, e.g., Table II in ref. [13] ), listed as \\({\\bar{\\lambda }}^{j}\\) in Table 2 , to the flat bands. In the band basis, this leads to momentum-dependent coupling vertices, cf. Eq. ( 9 ). Motivated by recent experiments [7] , we will also consider fluctuations of an additional nematic, time-reversal symmetric, layer-odd, intervalley coherent state (N-IVC) [62] which is not a candidate ground state in the strong-coupling limit; unlike the other strong-coupling ground states, the N-IVC has no momentum-independent representation in the flat-band basis but does have a momentum-independent matrix order parameter in the sublattice basis which takes the form \\({\\lambda }^{(j,{j}^{{\\prime} })}={({\\eta }_{x},\\, {\\eta }_{y})}_{j}{({\\rho }_{0},\\, {\\rho }_{z})}_{{j}^{{\\prime} }}\\) . The results for fluctuations of the projected strong-coupling orders \\({\\bar{\\lambda }}^{j}\\) in Table 2 and of the projected N-IVC state are shown in Fig. 4 , where we use the angle θ fluc. to tune the relative strength between T-IVC and any of the other type of fluctuation-induced interactions by multiplying the T-IVC interaction potential with \\(\\cos ({\\theta }_{{{{{{{{\\rm{fluc}}}}}}}}.})\\) and the other fluctuation potential with \\(\\sin ({\\theta }_{{{{{{{{\\rm{fluc}}}}}}}}.})\\) . In our microscopic numerics, we have taken a potential form \\(\\chi ({{{{{{{\\boldsymbol{q}}}}}}}})=\\frac{1}{{A}_{m}}\\frac{V}{{\\alpha }^{2}+| {{{{{{{\\boldsymbol{q}}}}}}}}{| }^{2}/{k}_{\\theta }^{2}}\\) again with α = 0.2 and with V = 4200 meV ⋅ (nm) 2 . We chose the value of V such that the transitions between the different pairing states are clearly visible in Fig. 4 when varying θ fluc. . In accordance with the prediction for \\({\\bar{\\lambda }}^{j}\\) in Table 2 , SLP + fluctuations further stabilize the A 2 superconductor, see Fig. 4 a. As such, the band-diagonal B 1 superconducting channel, where SLP + fluctuations are also attractive, can become the leading channel (favored over A 2 as a result of the finite bandwidth) only very close to θ fluc. = π /2. K-IVC fluctuations, however, are repulsive for A 2 pairing and favor the B 1 state more strongly. Table 2 Leading superconducting states in the flat-band limit, following from Eq. ( 7 ), for pairing mediated by fluctuations of the indicated orders, defined by using λ j in Eq. ( 4 ) Full size table Fig. 4: Pairing for different particle-hole fluctuations. These are defined by the coupling matrices \\({\\bar{\\lambda }}^{j}\\) listed in Table 2 . Similar to Fig. 3 c, we show T c of the leading pairing states, where θ fluc. tunes the relative strength between T-IVC-induced interactions ( \\(\\propto \\cos {\\theta }_{{{{{{{{\\rm{fluc}}}}}}}}.}\\) ) and interactions ( \\(\\propto \\sin {\\theta }_{{{{{{{{\\rm{fluc}}}}}}}}.}\\) ) coming from fluctuations of a SLP+, b K-IVC, c SLP−, and d N-IVC fluctuations. Full size image So far, the strong-coupling ( λ j ) and sublattice ( \\({\\bar{\\lambda }}^{j}\\) ) form of the couplings in Table 2 lead to the same conclusions. This is different for SLP− fluctuations (Fig. 4 d), where the projection-induced momentum dependence in the band basis can stabilize the E 1 superconductor. This can be understood by applying Eq. ( 7 ) in a sublattice basis (see Supplementary Appendix D1). We also find the E 1 state when fluctuations of the N-IVC state of ref. [62] dominate. Examples of the E 1 nematic and B 2 order parameters that emerge for SLP− fluctuations or N-IVC fluctuations are shown in Supplementary Appendix F. We point out that the nematic E 1 pairing is also an interesting candidate given that despite having nonzero pairing in the σ 0 , σ x , σ z channels, it will be nodal as long as the σ x components do not gap out the nodes in the band-diagonal parts."},{"section":"Discussion","content":"Taken together, we see that the proposed band-off-diagonal A 2 superconductor is an especially attractive candidate for TBG and TTG: first, it can lead to both V-shaped or U -shaped DOS, depending on lifetime parameters, the normal state, and the coupling strength V , see Fig. 2 e. As these parameters might vary from sample to sample and within a sample (e.g., V is expected to decrease upon doping further away from the insulator), this can naturally explain the tunneling data of [48] , [49] . We emphasize however that at least at the level of our mean-field numerics, we only expect a V-shape in the regime where the superconducting pairing is of the order of the bandwidth; this is the regime, where although the pairing is finite and can be quite large, the gap in the superconducting spectrum is either just closing or very small relative to the pairing. Increasing the pairing further will lead to an evolution from V to U -shaped while decreasing the pairing will eventually lead to a nodal Fermi surface and presumably a peak at zero energy in the DOS. Second, despite its interband nature, A 2 is the unique pairing state that is favored by fluctuations of two out of the four strong-coupling candidates we consider for the correlated insulator, see Fig. 4 a–c. What is more, this includes the T-IVC state, signatures of which are observed in recent experiments [7] . Finally, it is also favored by the likely dominant [61] , [68] optical intervalley phonon modes. We emphasize that, both in the case of fluctuating correlated insulators and phonons, the minimum attractive coupling needed to stabilize a purely band-off-diagonal state depends on the energy splitting between the two flat bands in the normal state; if the bands of our normal state are closer to degenerate, irrespective of the total bandwidth, the needed coupling to stabilize the A 2 pairing in mean-field will decrease. The other band-off-diagonal superconductor we identify transforms under the IR E 2 , i.e., can be thought of as a p -wave state. Its spectral properties also agree well with the experiment as the chiral configurations, E 2 (1, i ), which is favored within mean-field theory over a nematic E 2 state, can also have nodal regions, depending on filling. As can be seen in Fig. 1 g, this can lead to a transition from gapped to nodal when increasing the electron filling starting at ν ≃ 2. However, as opposed to the A 2 state, E 2 does not naturally appear as leading instability when considering optical phonons or fluctuations of any of the strong-coupling order parameters of the correlated insulator. While this makes it energetically less natural than A 2 , we cannot exclude it since its phenomenology agrees well with the experiment and since the precise form of the coupling of the dominant low-energy collective excitations are not known—significant momentum dependencies beyond λ j and \\({\\bar{\\lambda }}^{j}\\) in Table 2 could stabilize E 2 pairing as well. We also find in our numerics a nematic E 1 state which may be preferred over its chiral version in the presence of sufficient strain or due to fluctuation corrections [27] , [53] , [54] , [55] . We find the E 1 state is the leading instability of nematic IVC fluctuations and SLP − fluctuations, and is a subleading instability of T-IVC fluctuations. The E 1 state is interesting in its own right, as it can also be nodal. As superconductivity might further coexist with T-IVC order [7] , we have checked (see Supplementary Appendix E) that this does not alter our main observation: the preserved C 2 z symmetry still allows for entirely band-off-diagonal states, with transitions from nodal to full gapped, which are stabilized (among other fluctuations) by intervalley phonons. For the future, it will be interesting to go beyond mean-field and analyze the competition of our band-off-diagonal states with odd-frequency pairing, which we study in a follow-up work [69] . It also seems promising to study Andreev reflection [48] , [49] for our interband pairing scenario. On a more general level, our work shows that the observation of nodal pairing in twisted graphene systems does not immediately exclude a chiral superconducting state nor an entire electron–phonon-based pairing mechanism. It illustrates that a microscopic understanding of the superconducting states in graphene moiré systems requires taking into account their intrinsically multi-band nature. Note added . Just before posting our work, ref. [70] appeared online, which discusses pairing induced by A 1 phonons in spinful TBG bands."},{"section":"Methods","content":"Flat-band limit To derive Eq. ( 7 ), we take the flat-band limit, ξ k ,± → 0, in the linearized gap equation. For the interaction defined in Eqs. ( 4 )–( 6 )), we get (with moiré cell area A m ) $${({{{\\Delta }}}_{{{{{{{{\\boldsymbol{k}}}}}}}},\\eta })}_{\\beta,{\\beta }^{{\\prime} }}=\t {t}_{\\phi }\\frac{1}{4{A}_{m}T}\\mathop{\\sum}\\limits_{{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} }}{\\chi }_{{{{{{{{\\boldsymbol{k}}}}}}}}-{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} }} \\\\ \t \\times \\mathop{\\sum}\\limits_{j}{\\left[{\\lambda }_{{\\beta }^{{\\prime} },\\eta ;{\\alpha }^{{\\prime} },{\\eta }^{{\\prime} }}^{j}\\right]}^{*}{\\lambda }_{\\beta,\\eta ;\\alpha,{\\eta }^{{\\prime} }}^{j}{({{{\\Delta }}}_{{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} },{\\eta }^{{\\prime} }})}_{\\alpha,{\\alpha }^{{\\prime} }}.$$ (10) We define \\({({\\hat{{{\\Delta }}}}_{{{{{{{{\\boldsymbol{k}}}}}}}}})}_{\\alpha,\\eta ;{\\alpha }^{{\\prime} },{\\eta }^{{\\prime} }}:={({{{\\Delta }}}_{{{{{{{{\\boldsymbol{k}}}}}}}},\\eta })}_{\\alpha,{\\alpha }^{{\\prime} }}{\\delta }_{\\eta,{\\eta }^{{\\prime} }}\\) and note that finding the leading superconducting state according to Eq. ( 10 ) is equivalent to determining \\({\\hat{{{\\Delta }}}}_{{{{{{{{\\boldsymbol{k}}}}}}}}}\\) that maximizes the functional $${{{{{{{\\mathcal{F}}}}}}}}[{\\hat{{{\\Delta }}}}_{{{{{{{{\\boldsymbol{k}}}}}}}}}]:=\\frac{{\\sum }_{{{{{{{{\\boldsymbol{k}}}}}}}},{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} },j} \\, {\\chi }_{{{{{{{{\\boldsymbol{k}}}}}}}}-{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} }}\\,{t}_{\\phi }\\,{{\\mbox{tr}}}\\,[{\\lambda }^{j}{\\hat{{{\\Delta }}}}_{{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} }}{({\\lambda }^{j})}^{{{{\\dagger}}} }{\\hat{{{\\Delta }}}}_{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{{{\\dagger}}} }]}{{\\sum }_{{{{{{{{\\boldsymbol{k}}}}}}}}}\\,{{\\mbox{tr}}}\\,[{\\hat{{{\\Delta }}}}_{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{{{\\dagger}}} }{\\hat{{{\\Delta }}}}_{{{{{{{{\\boldsymbol{k}}}}}}}}}]}.$$ (11) Since \\({\\chi }_{{{{{{{{\\boldsymbol{k}}}}}}}}-{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} }} \\, > \\, 0\\) , the maximum value will be reached if we can maximize \\({t}_{\\phi }\\,{{\\mbox{tr}}}\\,[{\\lambda }^{j}{\\hat{{{\\Delta }}}}_{{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} }}{({\\lambda }^{j})}^{{{{\\dagger}}} }{\\hat{{{\\Delta }}}}_{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{{{\\dagger}}} }]\\) for each \\({{{{{{{\\boldsymbol{k}}}}}}}},\\, {{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} },\\, j\\) separately. As the Frobenius inner product \\({\\left\\langle A,B\\right\\rangle }_{F}=\\,{{\\mbox{tr}}}\\,[{A}^{{{{\\dagger}}} }B]\\) reaches its maximum (minimum) at fixed \\(\\left\\langle A,\\, A\\right\\rangle\\) and \\(\\left\\langle B,\\, B\\right\\rangle\\) , if A = c B with c > 0 ( c < 0), \\({t}_{\\phi }\\,{{\\mbox{tr}}}\\,[{\\lambda }^{j}{\\hat{{{\\Delta }}}}_{{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} }}{({\\lambda }^{j})}^{{{{\\dagger}}} }{\\hat{{{\\Delta }}}}_{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{{{\\dagger}}} }]\\) is maximized if \\({\\hat{{{\\Delta }}}}_{{{{{{{{\\boldsymbol{k}}}}}}}}}={t}_{\\phi }{c}_{{{{{{{{\\boldsymbol{k}}}}}}}},{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} }}{\\lambda }^{j}{\\hat{{{\\Delta }}}}_{{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} }}{({\\lambda }^{j})}^{{{{\\dagger}}} }\\) with \\({c}_{{{{{{{{\\boldsymbol{k}}}}}}}},{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} }} \\, > \\, 0\\) . For the ansatz \\({\\hat{{{\\Delta }}}}_{{{{{{{{\\boldsymbol{k}}}}}}}}}={\\delta }_{{{{{{{{\\boldsymbol{k}}}}}}}}}D{\\eta }_{x}\\) (and assuming for now that δ k has a fixed sign for all k ), this is obeyed if $$D{\\eta }_{x}={t}_{\\phi }{\\lambda }^{j}D{\\eta }_{x}{({\\lambda }^{j})}^{{{{\\dagger}}} },\\quad \\forall j.$$ (12) We state Eq. ( 12 ) as the (anti)commutator condition ( 7 ) in the main text [equivalent if \\({({\\lambda }^{j})}^{2}={\\mathbb{1}}\\) ], not only because it highlights the simple algebraic and basis independent nature of the condition but also since it emphasizes the similarities to the generalized Anderson theorem of [71] , [72] . If we can find a solution to Eq. ( 12 ), we know that the maximum (or at least one of the possibly degenerate maxima) of \\({{{{{{{\\mathcal{F}}}}}}}}[{\\hat{{{\\Delta }}}}_{{{{{{{{\\boldsymbol{k}}}}}}}}}]\\) is of the form of \\({\\hat{{{\\Delta }}}}_{{{{{{{{\\boldsymbol{k}}}}}}}}}={\\delta }_{{{{{{{{\\boldsymbol{k}}}}}}}}}D{\\eta }_{x}\\) where δ k is obtained as the maximum of the reduced functional $$\\widetilde{{{{{{{{\\mathcal{F}}}}}}}}}[{\\delta }_{{{{{{{{\\boldsymbol{k}}}}}}}}}]:=\\frac{{\\sum }_{{{{{{{{\\boldsymbol{k}}}}}}}},{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} }} \\, {\\chi }_{{{{{{{{\\boldsymbol{k}}}}}}}}-{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} }}{\\delta }_{{{{{{{{\\boldsymbol{k}}}}}}}}}^{*}{\\delta }_{{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} }}^{}}{{\\sum }_{{{{{{{{\\boldsymbol{k}}}}}}}}}| {\\delta }_{{{{{{{{\\boldsymbol{k}}}}}}}}}{| }^{2}},$$ (13) or equivalently as the largest eigenvector of \\({\\chi }_{{{{{{{{\\boldsymbol{k}}}}}}}}-{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} }}\\) viewed as a matrix in k and \\({{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} }\\) . As \\({\\chi }_{{{{{{{{\\boldsymbol{k}}}}}}}}-{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} }} > 0\\) (due to stability), the Perron-Frobenium theorem then immediately implies δ k > 0, in line with out assumption above and as stated in the main text. Electron–phonon coupling To present more details on the electron–phonon coupling, the associated displacement operators in Eq. ( 8 ) can be expressed in terms of canonical bosons, b g , α , μ , q , $${({u}_{g,\\mu }({{{{{{{\\boldsymbol{r}}}}}}}}))}_{j}=\\mathop{\\sum}\\limits_{{{{{{{{\\boldsymbol{q}}}}}}}}}\\frac{{b}_{g,j,\\mu,{{{{{{{\\boldsymbol{q}}}}}}}}}{e}^{i{{{{{{{\\boldsymbol{q}}}}}}}}\\cdot {{{{{{{\\boldsymbol{r}}}}}}}}}+\\,{{\\mbox{H.c.}}}\\,}{\\sqrt{2NM{\\omega }_{g}({{{{{{{\\boldsymbol{q}}}}}}}})}},$$ (14) where j refers to the two components for the E 2 phonon (is idle for A 1 , B 1 ), M is the carbon mass, and ω g ( q ) is the phonon dispersion, characterizing the phononic part of the Hamiltonian, \\({{{{{{{{\\mathcal{H}}}}}}}}}_{P}={\\sum }_{{{{{{{{\\boldsymbol{q}}}}}}}}}{\\omega }_{g}({{{{{{{\\boldsymbol{q}}}}}}}}){b}_{g,j,\\mu,{{{{{{{\\boldsymbol{q}}}}}}}}}^{{{{\\dagger}}} }{b}_{g,j,\\mu,{{{{{{{\\boldsymbol{q}}}}}}}}}\\) . As for \\({({{{{{{{{\\boldsymbol{v}}}}}}}}}_{\\mu })}_{\\ell }\\) in Eq. ( 8 ), ℓ = 1, 2 refers to the physical graphene layer in the case of TBG. One can, in principle, choose any orthonormal basis; we will find it convenient to use the layer-exchange even and odd states, \\({{{{{{{{\\boldsymbol{v}}}}}}}}}_{\\pm }={(1,\\pm 1)}^{T}/\\sqrt{2}\\) . For TTG, the situation is more involved (see Supplementary Appendix D2), but our arguments about which phonons are attractive in which pairing channels will hold for both systems. We project \\({{{{{{{{\\mathcal{H}}}}}}}}}_{EP}\\) in Eq. ( 8 ) onto the two flat bands ( α = ± ) in each valley η of the spin-polarized continuum-model, leading to a coupling term similar to Eq. ( 4 ) with momentum-dependent coupling matrices, \\({\\lambda }^{j}\\to {\\lambda }_{{{{{{{{\\boldsymbol{k}}}}}}}},{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} }}^{g,j,\\mu }\\) . Investigating the matrix elements \\({\\lambda }_{{{{{{{{\\boldsymbol{k}}}}}}}},{{{{{{{{\\boldsymbol{k}}}}}}}}}^{{\\prime} }}^{g,j,\\mu }\\) , we notice that they almost vanish for the layer-odd intervalley ( A 1 , B 1 ) phonons, which can be understood as a consequence of chiral and particle-hole symmetry (see Supplementary Appendix D3). The situation is the reverse for the intra-valley ( E 2 ) phonons, where the layer-even matrix elements are numerically small and the layer-odd matrix elements dominate. We therefore focus on layer-even (odd) intervalley (intra-valley) phonon couplings. Neglecting the momentum dependence in the phonon frequencies and retardation effects, the resulting electron-electron interaction in the inter-valley Cooper channel obtained by integrating out the phonons is given by Eq. ( 9 ). Here, \\({V}_{g}={g}_{g}^{2}/(2N{\\omega }_{g}^{2}) \\, > \\, 0\\) and \\({V}_{{A}_{1}}={V}_{{B}_{1}}\\simeq 1.33{V}_{{E}_{2}}\\) results from \\({g}_{{A}_{1}}={g}_{{B}_{1}}\\simeq {g}_{{E}_{2}}\\) and the phonon frequencies estimated in ref. [67] . Importantly, this only holds for parallel spins in the two valleys. For anti-parallel spins, the projection of the coupling matrices to the flat bands vanishes for the intervalley phonon modes A 1 and B 1 such that \\({V}_{{A}_{1}}={V}_{{B}_{1}}=0\\) ."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"The fascinating physics [1] , [2] of correlated graphene moiré superlattices, such as twisted-bilayer (TBG) and twisted-trilayer graphene (TTG), has generated extensive efforts to uncover the mysteries of their phase diagrams. Much progress has been made towards understanding their normal-state physics, including the correlated insulating phases [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] and the reset behavior [19] , [20] ; the latter, which is believed to be associated with the onset of flavor polarization, appears in the same density range and can coexist with superconductivity [13] , [19] , [20] , [21] , [22] , [23] , [24] , [25] , [26] , [27] , [28] , [29] , [30] , [31] , [32] , [33] , [34] . However, the form and symmetry of the superconducting order parameter and the pairing glue are still unknown, despite significant theoretical efforts [27] , [28] , [29] , [30] , [33] , [35] , [36] , [37] , [38] , [39] , [40] , [41] , [42] , [43] , [44] , [45] , [46] , [47] . Tunneling conductance measurements taken within the superconducting state reveal a V-shaped density of states (DOS) [48] , [49] which can become U -shaped at other electron concentrations [49] . Setting aside the possibility of thermal fluctuations as origin [50] , this is most naturally interpreted as a transition from nodal to fully gapped superconductivity. For a consistent microscopic theoretical understanding, this provides the following challenges: (i) electron–phonon coupling—a widely discussed [33] , [35] , [36] , [37] , [38] , [39] , [40] pairing mechanism in TBG and TTG—will typically mediate an entirely attractive interaction in the Cooper channel, with the leading pairing state that transforms trivially under all symmetries and is thus fully gapped [51] , [52] . (ii) Even when the low-energy interactions favor an irreducible presentation (IR), e.g., E of C 3 , with nodal basis functions ( p - or d -wave), the generically fully gapped chiral configuration wins over the nodal nematic one within mean-field. (iii) Even if we assume that the nodal state is energetically favored, e.g., due to significant corrections beyond mean-field [27] , [53] , [54] , [55] , one is still left to explain why there is a transition to another, fully gapped superconductor upon changing the filling. In this work, we show that the combination of flavor polarization and the representations of the symmetries in the flat bands of TBG and TTG allow for pairing channels that are completely off-diagonal in the flat bands and that such band-off-diagonal states can naturally reconcile all three key challenges (i–iii). More specifically, we find two distinct band-off-diagonal states: one of them transforms under the trivial representation A of the system’s point group C 6 (or one of A 1,2 of D 6 if we set the displacement field to zero) but can nonetheless have symmetry-protected nodal lines, akin to Bogoliubov Fermi surfaces discussed in refs. [56] , [57] , see Fig. 1 a–c for an intuitive visual explanation. The surprising possibility of the existence of such Bogliubov Fermi surfaces without an external magnetic field is unique to twisted graphene systems in that it follows as a direct consequence of both the symmetry and relative flatness of their normal-state bands. The second off-diagonal state transforms under a two-dimensional IR ( E 2 of C 6 ). Its associated chiral state, E 2 (1, i ), which is favored in the mean-field over the nematic one, has the unique property of exhibiting nodal lines or being fully gapped depending on the filling fraction, even when the order parameter is kept fixed. We supplement our general symmetry arguments and phenomenological models with Hartree-Fock (HF) calculations on the continuum model, studying a variety of different pairing mechanisms. We find that nodal band-off-diagonal pairing is favored by the optical A 1 and B 1 phonon modes and by fluctuations of a time-reversal symmetric intervalley coherent (T-IVC) state (the T-IVC state has Kekulé order on the graphene scale [58] , [59] , [60] ). Evidence for the former has been provided by a recent photoemission study [61] while evidence for the latter comes from recent STM experiments [7] . Furthermore, also fluctuations of a time-reversal-symmetric sublattice polarized state (SLP+) are attractive in the band-off-diagonal channel (see Table 2 for a formal definition of the order parameters). We also show that fluctuations of both T-IVC and of a nematic, time-reversal symmetric IVC order [62] favor either the band-off-diagonal A or an E 1 state with band-diagonal components, which may also be nodal; the winner is determined by the relative amount of nematic IVC and T-IVC fluctuations. Fig. 1: Spectral properties of interband pairing. While for band-diagonal pairing a small superconducting order parameter can immediately open up a gap as time-reversal symmetry guarantees that the associated avoided crossings [gray regions in a ] in the Bogoliubov spectrum are at the Fermi level, this is not the case for band-off-diagonal pairing ( b ). Here, a sufficiently strong order-parameter value is required to establish a full gap, see c . Its k dependence according to Eq. ( 3 ) is shown in d , where the red line indicates nodal points. If the band structure has Dirac points, there will be a point on the horizontal axis (blue cross). Consequently, if there is another momentum point located above the red line (blue circle), continuity of the Hamiltonian implies a nodal point on any path connecting the two momenta. e Gap of the isotropic A 2 state and δ k , ϵ k (zeros indicated in red) for the normal-state toy model defined in the text. BEC/BI refers to the Bose-Einstein condensate/band insulator limit. f Complex phase \\\\({\\\\varphi }_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}=\\\\arg ({X}_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}+i{Y}_{{{{{{{{\\\\boldsymbol{k}}}}}}}}})\\\\) for leading basis function with small subleading corrections. g Shows the gap of the chiral p -wave E 2 (1, i ) state with Δ 0 = 1.5 t and the value of \\\\({D}_{{{{{{{{{\\\\boldsymbol{k}}}}}}}}}_{j}}: \\\\!\\\\!=| {\\\\delta }_{{{{{{{{{\\\\boldsymbol{k}}}}}}}}}_{j}}| -| {\\\\epsilon }_{{{{{{{{{\\\\boldsymbol{k}}}}}}}}}_{j}}-\\\\mu |\\\\) for k j at the three symmetry-in-equivalent vortices in f as a function of μ . 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Electric-field-tunable electronic nematic order in twisted double-bilayer graphene.2D Mater.8, 034005 (2021).\",\n \"link\": \"https://doi.org/10.1088%2F2053-1583%2Fabfcd6\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Possible pairing states Let us begin by classifying the superconducting instabilities in graphene moiré systems in the limit where the low-energy bands are spin-polarized but allow for multiple bands. We denote the spinless low-energy fermionic creation operators by \\\\({c}_{{{{{{{{\\\\boldsymbol{k}}}}}}}},\\\\alpha,\\\\eta }^{{{{\\\\dagger}}} }\\\\) with momentum k in valley η = ± , and of band index α labeling the upper ( α = + ) and lower ( α = − ) quasi-flat bands. As a result of two-fold rotational symmetry, C 2 z , along the out-of-plane ( z ) direction or effective spinless time-reversal symmetry, Θ, the non-interacting band structure ξ k , α , η obeys ξ k , α , η = ξ − k , α ,− η ≡ ξ η ⋅ k , α and intervalley pairing is expected to dominate. A general pairing order parameter in the inter-valley channel couples as $${{{{{{{{\\\\mathcal{H}}}}}}}}}_{{{{{{{{\\\\rm{p}}}}}}}}}=\\\\mathop{\\\\sum}\\\\limits_{{{{{{{{\\\\boldsymbol{k}}}}}}}},\\\\eta= \\\\pm ,\\\\alpha,{\\\\alpha }^{{\\\\prime} }}{c}_{{{{{{{{\\\\boldsymbol{k}}}}}}}},\\\\alpha,\\\\eta }^{{{{\\\\dagger}}} }{\\\\left({{{\\\\Delta }}}_{{{{{{{{\\\\boldsymbol{k}}}}}}}},\\\\eta }\\\\right)}_{\\\\alpha,{\\\\alpha }^{{\\\\prime} }}{c}_{-{{{{{{{\\\\boldsymbol{k}}}}}}}},{\\\\alpha }^{{\\\\prime} },-\\\\eta }^{{{{\\\\dagger}}} }+\\\\,{{\\\\mbox{H.c.}}}\\\\,,$$ (1) where the order parameter \\\\({{{\\\\Delta }}}_{{{{{{{{\\\\boldsymbol{k}}}}}}}},\\\\eta }=-{{{\\\\Delta }}}_{-{{{{{{{\\\\boldsymbol{k}}}}}}}},-\\\\eta }^{T}\\\\) is a matrix in band space. The physical spin texture of the superconductor is entirely determined by the form of the underlying normal-state’s polarization: if the spins are aligned in the two valleys, the superconductor is a non-unitary triplet, while anti-alignment [24] , [28] leads to a singlet-triplet admixed state [13] , [27] , [28] . In both cases, all of the following states are well defined, with the aforementioned spin structures and symmetries given by appropriate combinations of spinless operations and spin rotations (see Supplementary Appendix A1). We will classify the pairing states according to the irreducible representations (IRs) of the system’s point group D 6 , which is generated by six-fold rotations ( C 6 z ) along the z axis and two-fold rotation symmetry ( C 2 x ) along the in-plane x axis. Note a displacement field ( D 0 ≠ 0) breaks the in-plane rotations leading to the point group C 6 . Importantly, all IRs of D 6 and C 6 are either even or odd under C 2 z . Choosing the phases of the Bloch states such that C 2 z acts as c k , α , η → c − k , α ,− η , it holds $${C}_{2z}:\\\\quad {{{\\\\Delta }}}_{{{{{{{{\\\\boldsymbol{k}}}}}}}},\\\\eta }\\\\quad \\\\longrightarrow \\\\quad {{{\\\\Delta }}}_{-{{{{{{{\\\\boldsymbol{k}}}}}}}},-\\\\eta }=-{{{\\\\Delta }}}_{{{{{{{{\\\\boldsymbol{k}}}}}}}},\\\\eta }^{T}.$$ (2) This immediately implies that the pairing states in all IRs even under C 2 z ( A 1 , A 2 , E 2 of D 6 ) must be anti-symmetric in band space and, thus, entirely band-off-diagonal, whereas the order parameters of the other IRs ( B 1 , B 2 , E 1 ) are symmetric and can contain both band-diagonal and band-off-diagonal components. While superconducting order parameters with finite band-off-diagonal components are rather common in multi-band systems, the existence of pairing states that are constrained to be entirely band-off-diagonal is rather unique and follows from the combination of C 2 z symmetry and the spin polarization in the normal state. Importantly, this is unaffected by strain or nematic order breaking C 3 z as long as C 2 z remains, which guarantees that there are IRs with entirely band-off-diagonal order parameters. Choosing the phase conventions of the Bloch states such that C 2 x and C 3 z act as \\\\({c}_{{{{{{{{\\\\boldsymbol{k}}}}}}}},\\\\alpha,\\\\eta }\\\\to {({\\\\sigma }_{z})}_{\\\\alpha \\\\alpha }{c}_{({k}_{x},-{k}_{y}),\\\\alpha,\\\\eta }\\\\) and \\\\({c}_{{{{{{{{\\\\boldsymbol{k}}}}}}}},\\\\alpha,\\\\eta }\\\\to {c}_{{C}_{3z}{{{{{{{\\\\boldsymbol{k}}}}}}}},\\\\alpha,\\\\eta }\\\\) , respectively, the resulting candidate order parameters are summarized in Table 1 . Note that a momentum-independent representation of C 2 x must be σ z due to the bands’ eigenvalues at the Γ-M line, which in turn are connected to the topological obstruction of the flat bands [63] . The reality (Hermiticity) constraint in Table 1 on χ , X , and Y ( \\\\(\\\\hat{\\\\chi },\\\\, \\\\hat{X}\\\\) , and \\\\(\\\\hat{Y}\\\\) ) comes from the residual spinless time-reversal symmetry Θ of the normal state [64] , [65] . The two two-dimensional IRs E 1,2 are each associated with three pairing states—two nematic phases E 1,2 (1, 0), E 1,2 (0, 1) and one chiral state E 1,2 (1, i ). Table 1 Summary of pairing states in spin-polarized flat bands Full size table Spectral properties We here have the rather unique situation that there are pairing channels, associated with the IRs A 1,2 and E 2 , where the pairing is constrained by C 2 z to be entirely band-off-diagonal. One immediate very unusual consequence is that the superconducting order-parameter transforming under the trivial representation ( A 1 ) has a symmetry-imposed line of zeros along the Γ-M line, and hence a nodal point in the spectrum. This is related to the topology-induced non-trivial representation of C 2 x in band space. We refer to ref. [39] for the discussion of other topological nodal points for pairing in obstructed TBG bands. As we will show next, band-off-diagonal pairing leads to additional unusual spectral properties with far-reaching consequences for graphene moiré systems. To this end, consider the following effective Hamiltonian, \\\\({{{{{{{{\\\\mathcal{H}}}}}}}}}_{{\\\\sigma }_{y}}={\\\\sum }_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}{c}_{{{{{{{{\\\\boldsymbol{k}}}}}}}},\\\\alpha,\\\\eta }^{{{{\\\\dagger}}} }{c}_{{{{{{{{\\\\boldsymbol{k}}}}}}}},\\\\alpha,\\\\eta }{\\\\xi }_{\\\\eta \\\\cdot {{{{{{{\\\\boldsymbol{k}}}}}}}},\\\\alpha }+{\\\\sum }_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}[{{{\\\\Delta }}}_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}\\\\,{c}_{{{{{{{{\\\\boldsymbol{k}}}}}}}},+}^{{{{\\\\dagger}}} }{\\\\sigma }_{y}{c}_{-{{{{{{{\\\\boldsymbol{k}}}}}}}},-}^{{{{\\\\dagger}}} }+\\\\,{{\\\\mbox{H.c.}}}\\\\,]\\\\) , where the scalar function Δ k describes the form of pairing. We will here study two cases that are conventionally considered to be fully gapped, (i) a momentum-independent “ s -wave state” ( A 2 or A pairing in Table 1 ) where Δ k = Δ 0 and (ii) a “chiral p -wave” state, or more precisely an E 2 (1, i ) state, where Δ k = Δ 0 ( X k + i Y k ) with ( X k , Y k ) being smooth, MBZ-periodic functions transforming as ( x , y ) under C 3 z . Furthermore, we parameterize the dispersion, ξ η ⋅ k , α , of the two flat bands ( α = ± ) in valley η = ± as ξ k , α = ϵ k − μ + α δ k , where ϵ k and δ k are C 3 z (and, for D 0 = 0, C 2 x ) symmetric functions. The Bogoliubov spectrum of \\\\({{{{{{{{\\\\mathcal{H}}}}}}}}}_{{\\\\sigma }_{y}}\\\\) has four bands, given by \\\\(\\\\pm{\\\\delta }_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}\\\\pm \\\\sqrt{{({\\\\epsilon }_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}-\\\\mu )}^{2}+| {{{\\\\Delta }}}_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}{| }^{2}}\\\\) . Consequently, the excitation gap at momentum k reads as $${{\\\\Delta }}{E}_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}=\\\\Big | | {\\\\delta }_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}| -\\\\sqrt{{({\\\\epsilon }_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}-\\\\mu )}^{2}+| {{{\\\\Delta }}}_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}{| }^{2}}\\\\Big|,$$ (3) which is shown in Fig. 1 d, and therefore exhibits nodes where \\\\(| {\\\\delta }_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}|=\\\\sqrt{{({\\\\epsilon }_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}-\\\\mu )}^{2}+| {{{\\\\Delta }}}_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}{| }^{2}}\\\\) . As long as the band structure has Dirac points, there are points k D in the Brillouin zone with \\\\({\\\\delta }_{{{{{{{{{\\\\boldsymbol{k}}}}}}}}}_{D}}=0\\\\) , associated with the blue cross in Fig. 1 d. Furthermore, for a metallic normal state, μ must be within the bandwidth and, hence, there must be a region R in momentum space where ∣ δ k ∣ > ∣ ϵ k − μ ∣ . For the momentum-independent A 2 state, Δ k = Δ 0 , this implies that there exists \\\\({{{\\\\Delta }}}_{0}^{c} > 0\\\\) such that there is k * ∈ R with parameters (such as the blue circle) above the red solid line in Fig. 1 d as long as \\\\(| {{{\\\\Delta }}}_{0}| < {{{\\\\Delta }}}_{0}^{c}\\\\) . By continuity, this means that there must be a nodal point on any line connecting k D and k * . Consequently, for μ within the bandwidth and \\\\({\\\\delta }_{{{{{{{{{\\\\boldsymbol{k}}}}}}}}}_{D}}=0\\\\) for some k D , the A 2 will always have a nodal line if ∣ Δ 0 ∣ is sufficiently small, consistent with the intuitive picture based on the Bogoliubov spectrum in Fig. 1 a–c. We illustrate this further in Fig. 1 e using a toy model with \\\\({\\\\delta }_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}=t\\\\left|1+{e}^{i{{{{{{{{\\\\boldsymbol{a}}}}}}}}}_{1}\\\\cdot {{{{{{{\\\\boldsymbol{k}}}}}}}}}+{e}^{-i{{{{{{{{\\\\boldsymbol{a}}}}}}}}}_{2}\\\\cdot {{{{{{{\\\\boldsymbol{k}}}}}}}}}\\\\right|\\\\) and \\\\({\\\\epsilon }_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}={t}^{{\\\\prime} }\\\\mathop{\\\\sum }\\\\nolimits_{j=1}^{3}\\\\cos {{{{{{{{\\\\boldsymbol{a}}}}}}}}}_{j}\\\\cdot {{{{{{{\\\\boldsymbol{k}}}}}}}},\\\\, {{{{{{{{\\\\boldsymbol{a}}}}}}}}}_{j}={[{C}_{3z}]\\\\,}^{j-1}{(\\\\sqrt{3},0)}^{T}\\\\) . This leads to the second unexpected conclusion that, for any pairing mechanism, including conventional electron–phonon coupling, the leading instability either has nodal lines in a finite region below T c or transforms non-trivially under the symmetries of the normal state. For electron–phonon pairing (or pairing mediated by the fluctuations of any time-reversal-symmetric order parameter [52] , such as the T-IVC state) this is particularly unexpected since it is generally believed to always lead to a fully gapped state that transforms trivially under all symmetries. In fact, this can be proven in general terms [51] , [52] , even for spin-orbit-split Fermi surfaces and beyond mean-field theory [52] . The crucial difference to these works, however, is that spinfull time-reversal is broken in our case such that the Fermi-Dirac constraint is inconsistent with a non-sign-changing, band-diagonal pairing state. This leads to the unique situation that although electron–phonon coupling will lead to entirely attractive interactions in the Cooper channel, the superconducting energetics is frustrated: the dominant pairing state is determined by whether the energetic loss due to non-resonant band-off-diagonal Cooper pairs ( A 2 pairing) or the costs from sign changes of the order parameter (such as B 1 ) are less harmful. We will demonstrate this explicitly by a model calculation in sec. “electron–phonon coupling below, where either A 2 or B 1 is dominant, depending on the form of the electron–phonon coupling”. Let us first, however, discuss the general spectral properties of the “chiral p -wave” state which is canonically expected to be fully gapped as long as the Fermi surfaces do not cross the zeros of X k + i Y k . Three of these zeros have to be at the Γ, K , and \\\\({K}^{{\\\\prime} }\\\\) points as a consequence of C 3 z symmetry. In the absence of fine-tuning, X k + i Y k will have vortices at these points with vorticity v = + 1. As can be seen in Fig. 1 f, where we show the phase of X k + i Y k using an admixture of the two lowest-order terms, the net vorticity of + 3 at these high-symmetry points has to be compensated by anti-vortices at generic momenta. The lowest possible number is three C 3 z -related vortices, which appear near the M points in Fig. 1 f. If it holds ∣ δ k ∣ > ∣ ϵ k − μ ∣ at any of these zeros k = k j , we obtain a point above the red line in Fig. 1 d and, thus, a nodal point along any contour between that k j and k D ; as opposed to the A 2 state, this holds irrespective of the value of Δ 0 and therefore all the way to zero temperature. In summary, we find that also the E 2 (1, i ) “chiral p -wave” state is not generically fully gapped but instead will exhibit a nodal line encircling any zero k j of X k + i Y k with \\\\(| {\\\\delta }_{{{{{{{{{\\\\boldsymbol{k}}}}}}}}}_{j}}| > | {\\\\epsilon }_{{{{{{{{{\\\\boldsymbol{k}}}}}}}}}_{j}}-\\\\mu |\\\\) . This leads to an interesting filling dependence of the superconducting gap, as we illustrate in our toy model in Fig. 1 g along with the criterion \\\\({D}_{{{{{{{{{\\\\boldsymbol{k}}}}}}}}}_{j}}:=| {\\\\delta }_{{{{{{{{{\\\\boldsymbol{k}}}}}}}}}_{j}}| -| {\\\\epsilon }_{{{{{{{{{\\\\boldsymbol{k}}}}}}}}}_{j}}-\\\\mu | > 0\\\\) evaluated at the vortices at Γ, K/K’, and near M. Depending on μ , D k is positive only near the Γ point or only in a region surrounding the vortices close to the M points, leading to nodal lines encircling Γ and near the M points, respectively, as shown in the inset of Fig. 1 g. These regimes are separated by a fully gapped region where D k < 0 for all k , which could explain the fully gapped to nodal transition seen in tunneling experiments [49] when the filling fraction is changed. Note that \\\\({D}_{{{{{{{{{\\\\boldsymbol{k}}}}}}}}}_{j}}=-| {\\\\epsilon }_{{{{{{{{{\\\\boldsymbol{k}}}}}}}}}_{j}}-\\\\mu | \\\\le 0\\\\) for k j at the K and K’ points. In Fig. 1 g, \\\\({D}_{K}={D}_{{K}^{{\\\\prime} }}\\\\) vanishes close to the top of the band, which simply means that the Fermi surfaces cross the K, K’ points and the superconductor has nodal points for this fine-tuned value of the chemical potential. Fluctuation-induced pairing Having discussed the unique energetics of pairing and spectral properties of the resulting superconductors in spin-polarized quasi-flat-bands with Dirac cones on a general level, we next study these aspects more explicitly by solving the superconducting self-consistency equations in the flat bands common to alternating-twist graphene systems. We will start with pairing induced by fluctuations of a nearby symmetry-broken phase. To this end, we will couple the low-energy electrons introduced in Eq. ( 1 ) to a collective bosonic field \\\\({\\\\phi }_{j}({{{{{{{\\\\boldsymbol{q}}}}}}}})={\\\\phi }_{j}^{{{{\\\\dagger}}} }(-{{{{{{{\\\\boldsymbol{q}}}}}}}})\\\\) via $${{{{{{{{\\\\mathcal{H}}}}}}}}}_{\\\\phi }=\\\\mathop{\\\\sum}\\\\limits_{{{{{{{{\\\\boldsymbol{k}}}}}}}},{{{{{{{\\\\boldsymbol{q}}}}}}}},j}{c}_{{{{{{{{\\\\boldsymbol{k}}}}}}}}+{{{{{{{\\\\boldsymbol{q}}}}}}}},\\\\alpha,\\\\eta }^{{{{\\\\dagger}}} }{\\\\lambda }_{\\\\alpha,\\\\eta ;{\\\\alpha }^{{\\\\prime} },{\\\\eta }^{{\\\\prime} }}^{j}{c}_{{{{{{{{\\\\boldsymbol{k}}}}}}}},{\\\\alpha }^{{\\\\prime} },{\\\\eta }^{{\\\\prime} }}{\\\\phi }_{j}({{{{{{{\\\\boldsymbol{q}}}}}}}}),$$ (4) where the Hermitian matrices λ j capture the nature of the correlated insulating phase; we here choose and normalize λ j such that \\\\({({\\\\lambda }^{j})}^{2}={\\\\mathbb{1}}\\\\) . Both for twisted bi- [9] and trilayer graphene [14] , [15] , [29] , the stable phases emerging out of the U (4) × U (4) [9] manifold in the chiral-flat (decoupled) limit are natural candidates. Integrating out the bosonic modes, we obtain an effective electronic interaction which in the for superconductivity relevant intervalley Cooper channel reads as $${{{{{{{{\\\\mathcal{H}}}}}}}}}_{{{{{{{{\\\\rm{int}}}}}}}}}^{\\\\phi }=\\t -\\\\mathop{\\\\sum}\\\\limits_{{{{{{{{\\\\boldsymbol{k}}}}}}}},{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} }}{\\\\chi }_{{{{{{{{\\\\boldsymbol{k}}}}}}}}-{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} }}{{{{{{{{\\\\mathcal{V}}}}}}}}}_{(\\\\eta,\\\\alpha,\\\\beta ),({\\\\eta }^{{\\\\prime} },{\\\\alpha }^{{\\\\prime} },{\\\\beta }^{{\\\\prime} })}\\\\\\\\ \\t \\\\times {c}_{-{{{{{{{\\\\boldsymbol{k}}}}}}}},\\\\beta,-\\\\eta }^{{{{\\\\dagger}}} }{c}_{{{{{{{{\\\\boldsymbol{k}}}}}}}},\\\\alpha,\\\\eta }^{{{{\\\\dagger}}} }{c}_{{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} },{\\\\alpha }^{{\\\\prime} },{\\\\eta }^{{\\\\prime} }}{c}_{-{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} },{\\\\beta }^{{\\\\prime} },-{\\\\eta }^{{\\\\prime} }},$$ (5) with vertex $${{{{{{{{\\\\mathcal{V}}}}}}}}}_{(\\\\eta,\\\\alpha,\\\\beta ),({\\\\eta }^{{\\\\prime} },{\\\\alpha }^{{\\\\prime} },{\\\\beta }^{{\\\\prime} })}={t}_{\\\\phi }\\\\mathop{\\\\sum}\\\\limits_{j}{\\\\left[{\\\\lambda }_{\\\\beta,\\\\eta ;{\\\\beta }^{{\\\\prime} },{\\\\eta }^{{\\\\prime} }}^{j}\\\\right]}^{*}{\\\\lambda }_{\\\\alpha,\\\\eta ;{\\\\alpha }^{{\\\\prime} },{\\\\eta }^{{\\\\prime} }}^{j},$$ (6) t ϕ = ± 1 encoding whether the order parameter is even or odd under time-reversal, Θ ϕ j ( q )Θ † = t ϕ ϕ j ( q ), and χ q > 0 denoting the (static) susceptibility of ϕ j . Before discussing numerical results for the full model, we first focus on perfectly flat bands. In this limit, the leading superconducting instability within mean-field theory is given by the largest eigenvalue of \\\\({{{{{{{\\\\mathcal{V}}}}}}}}\\\\) in Eq. ( 6 ) viewed as a matrix in the multi-index ( η , α , β ). Furthermore, if there is an anti-symmetric, valley-off-diagonal matrix D obeying (see Methods) $${[D{\\\\eta }_{x},{\\\\lambda \\\\,}^{j}]}_{-{t}_{\\\\phi }}\\\\equiv D{\\\\eta }_{x}{\\\\lambda }^{j}-{t}_{\\\\phi }{\\\\lambda }^{j}D{\\\\eta }_{x}=0,$$ (7) the associated leading superconducting order parameter in Eq. ( 1 ) is given by \\\\({({{{\\\\Delta }}}_{{{{{{{{\\\\boldsymbol{k}}}}}}}},\\\\eta })}_{\\\\alpha,{\\\\alpha }^{{\\\\prime} }}={\\\\delta }_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}{(D{\\\\eta }_{x})}_{\\\\alpha,\\\\eta ;{\\\\alpha }^{{\\\\prime} }\\\\eta }\\\\) with δ k > 0; here η j denotes Pauli matrices in valley space and the precise form of δ k is determined by χ ( q ). T-IVC fluctuations Motivated by recent experiments [7] providing direct evidence for T-IVC order, we start with T-IVC fluctuations as a pairing glue. In the U (4) × U (4) symmetric limit, the T-IVC state is associated with λ j = σ 0 η j , j = x , y , within our conventions. Since t ϕ = + 1, we are looking for D η x that commutes with λ j . Interestingly, there is a unique anti-symmetric, valley-off-diagonal matrix D ∝ σ y η x with that property, implying that the leading pairing state has the form Δ k , η = σ y δ k , δ k > 0. This is exactly the A 2 state in Table 1 , which, as discussed above, will have nodal lines at least in the vicinity of T c when a finite band dispersion is taken into account. Intuitively, the fact that A 2 pairing is favored can be understood by noticing that the valley-off-diagonal form of λ j leads to an attractive interaction across the valleys, which penalizes the B 1 state with its sign change between the two valleys. In fact, it holds \\\\({{{{{{{{\\\\mathcal{V}}}}}}}}}_{(\\\\eta,\\\\alpha,\\\\beta ),({\\\\eta }^{{\\\\prime} },{\\\\alpha }^{{\\\\prime} },{\\\\beta }^{{\\\\prime} })}=(1-\\\\eta \\\\,{\\\\eta }^{{\\\\prime} })\\\\mathop{\\\\sum }\\\\nolimits_{\\\\mu=0}^{3}{({\\\\sigma }_{\\\\mu }^{*})}_{\\\\alpha,\\\\beta }{({\\\\sigma }_{\\\\mu })}_{{\\\\alpha }^{{\\\\prime} },{\\\\beta }^{{\\\\prime} }}\\\\) showing explicitly that it is repulsive (attractive) in the B 1 ( A 2 ) channel. To go beyond the flat-band limit, we solve the superconducting mean-field equations numerically. We take the flat TBG bands from the continuum model [66] as the starting point. To capture the spin-polarized normal state, we supplement it with Coulomb repulsion and perform HF calculation (see Supplementary Appendix A for details). As can be seen in the resulting band structure shown in Fig. 2 a with interaction renormalization assuming filling fraction ν = 2, this not only pushes one spin flavor below the Fermi level but also induces significant band renormalizations. For our subsequent study of superconducticity, we project onto the two bands at the Fermi level and associate them with the creation operators c k , α in the interactions in Eqs. ( 4 ) and ( 5 ). In our numerical computations, we choose \\\\(\\\\chi ({{{{{{{\\\\boldsymbol{q}}}}}}}})=\\\\frac{1}{{A}_{m}}\\\\frac{V}{{\\\\alpha }^{2}+| {{{{{{{\\\\boldsymbol{q}}}}}}}}{| }^{2}/{k}_{\\\\theta }^{2}}\\\\) where A m is the real space area of a moiré unit cell, and take α = 0.05 for concreteness, although we checked our main conclusion do not crucially depend on this form. In all of our numerics, we work at doping ν = 2.5. Fig. 2: Pairing mediated by T-IVC fluctuations. We show a the band structure of the normal state with spin polarization (K, K', and Γ label the high-symmetry points of the moiré scale Brillouin zone) and b the critical temperature T c (in units of the maximum band splitting W 0 ≃ 9.4 meV) as a function of coupling strength V measured in units of the critical coupling V c ,1 = 105 meV ⋅ nm 2 obtained from the linearized gap equation. The band structure (with color indicating the band-projected value of the anomalous correlator) of the A 2 state and its order parameter are shown in c and d . The DOS of the T = 0 superconductor for several different values of coupling strength V is plotted in e . The DOS was computed as \\\\({\\\\sum }_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}\\\\delta \\\\left({E}_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}-\\\\omega \\\\right)\\\\) , replacing the δ function with Lorentzians with half width at half max 0.3 meV (much smaller than the typical superconducting order parameter). The critical coupling V c ,2 where the nodal lines disappear is V c ,2 ≃ 1.4 V c ,1 . Full size image As expected, we indeed find that the A 2 state dominates, both right at the critical temperature T c , obtained from the linearized gap equation, and at T = 0 as we show by iteratively solving the full self-consistency equation (see Supplementary Appendix C). One crucial effect of the finite dispersion and splitting between the bands is that a finite interaction strength, V > V c ,1 , is required to stabilize the superconducting phase, as can be seen in the plot of T c in Fig. 2 b. Superconductivity ceases to be a weak-coupling instability as the Bloch states ( k , α , η ) and \\\\((-{{{{{{{\\\\boldsymbol{k}}}}}}}},\\\\, {\\\\alpha }^{{\\\\prime} },-\\\\eta )\\\\) are not degenerate for \\\\(\\\\alpha \\\\, \\\\ne \\\\, {\\\\alpha }^{{\\\\prime} }\\\\) , cutting off the logarithmic divergence known from BCS theory. The quasi-particle spectrum and order parameter of superconductivity from T = 0 numerics are shown in Fig. 2 c, d. In accordance with our general discussion above, we observe that the order parameter only has finite components proportional to σ y , which do not mix with the band-even contributions ∝ σ 0, x , z as a result of C 2 z symmetry. Furthermore, it does not change sign as a function of k and, for sufficiently small V but still with V > V c ,1 , the nodal lines in the superconducting spectrum persist all the way to T = 0, while the nodal line is gapped out at low T < T c if V > V c ,2 . The interaction-strength-dependence of the superconducting gap can be more clearly seen in Fig. 2 e, where we show the DOS for the self-consistent solution at T = 0. For large V , the superconductor becomes fully gapped at T = 0, leading to a U -shaped DOS. With smaller V , the magnitude of the order-parameter decreases and the superconductor eventually exhibits nodal lines, as explained above. In the regime just before these nodal lines appear, there is an increase in the DOS near the Fermi level, roughly when the order parameter and the maximal band splitting are comparable, leading to a V-shaped DOS (green line). The lifetime parameter used to compute the DOS is 0.3 meV; this choice was based on our k -grid spacing. While it is not necessarily small with respect to the tunneling gap (which vanishes at V c ,2 ), it is small with respect to Δ( k ), which is of order 5 meV just as the state is becoming fully gapped for our choice of normal state. This behavior of the DOS with interaction strength may offer a natural explanation for the U -shaped tunneling conductance measurements near ν = 2 and V-shaped tunneling conductance measurements near ν = 3 observed in TTG [49] ; if we are considering T-IVC fluctuations of the insulator at ν = 2, then it may be reasonable to expect the coupling to these fluctuations could grow weaker as we dope towards ν = 3, in line with the experimentally observed ν dependence. Note that the regime we call V-shaped here is strictly speaking fully gapped. However, the crucial difference to the BCS state is that the gap is much smaller than the order-parameter magnitude as a result of the different Bogoliubov spectrum in Eq. ( 3 ). This is why, depending not only on the magnitude of the pairing but also on the precise form of the normal state, the resulting tunneling spectra can resemble those observed experimentally [48] , [49] , such as the green curve in Fig. 2 e, making the A 2 state an attractive candidate. The regime of small V where stable superconductivity with true Bogoliubov Fermi surfaces is observed can further exhibit a peak at ω = 0 which is due to a Van Hove singularity crossing the Fermi level, see blue curve in Fig. 2 e; while this peak has not been observed experimentally, its presence crucially depends on details of the normal-state band structure and is only found to be energetically favored in a very small regime of V in our model. Electron–phonon coupling To illustrate that the off-diagonal A 2 state is more generally favored beyond just T-IVC fluctuations, we next discuss electron–phonon coupling, which is frequently considered a plausible pairing mechanism for twisted moiré systems [33] , [35] , [36] , [37] , [38] , [39] . Similar to ref. [35] , we use that the optical A 1 , B 1 , and E 2 phonon modes are known [67] to dominate the electron–phonon coupling in single-layer graphene. As these are optical phonons, we further assume that the impact of the interlayer coupling on the phonons can be neglected and arrive at $${{{{{{{{\\\\mathcal{H}}}}}}}}}_{EP}=\\t \\\\int\\\\,{{{{{{{\\\\rm{d}}}}}}}}{{{{{{{\\\\boldsymbol{r}}}}}}}}\\\\,{\\\\psi }_{\\\\ell,s}^{{{{\\\\dagger}}} }({{{{{{{\\\\boldsymbol{r}}}}}}}})\\\\left[{g}_{{A}_{1}}{{{\\\\Lambda }}}_{{A}_{1}}{u}_{{A}_{1},\\\\mu }({{{{{{{\\\\boldsymbol{r}}}}}}}})\\\\right. \\\\\\\\ \\t \\\\left.+{g}_{{B}_{1}}{{{\\\\Lambda }}}_{{B}_{1}}{u}_{{B}_{1},\\\\mu }({{{{{{{\\\\boldsymbol{r}}}}}}}})+{g}_{{E}_{2}}{{{{{{{{\\\\boldsymbol{\\\\Lambda }}}}}}}}}_{{E}_{2}}\\\\cdot {{{{{{{{\\\\boldsymbol{u}}}}}}}}}_{{E}_{2},\\\\mu }({{{{{{{\\\\boldsymbol{r}}}}}}}})\\\\right]{({{{{{{{{\\\\boldsymbol{v}}}}}}}}}_{\\\\mu })}_{\\\\ell }{\\\\psi }_{\\\\ell,s}({{{{{{{\\\\boldsymbol{r}}}}}}}})$$ (8) for the electron–phonon coupling, where v μ encode the layer structure of the modes (see Methods). Symmetry dictates that the vertices Λ g are given by \\\\({{{\\\\Lambda }}}_{{A}_{1}}={\\\\eta }_{x}{\\\\rho }_{x},\\\\, {{{\\\\Lambda }}}_{{B}_{1}}={\\\\eta }_{y}{\\\\rho }_{x}\\\\) , and \\\\({{{{{{{{\\\\boldsymbol{\\\\Lambda }}}}}}}}}_{{E}_{2}}=({\\\\eta }_{z}{\\\\rho }_{y},-{\\\\rho }_{x})\\\\) where ρ acts on the microscopic sublattice basis. Integrating out the phonons and projecting to the flat bands, we obtain an effective electron-electron interaction (see Methods) $${{{{{{{\\\\mathcal{H}}}}}}}}_{{{{{{{{\\\\rm{int}}}}}}}}}^{C}=\\t -\\\\mathop{\\\\sum}\\\\limits_{{{{{{{{\\\\boldsymbol{k}}}}}}}},{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} }}{V}_{g}{\\\\left[{\\\\lambda }_{{{{{{{{\\\\boldsymbol{k}}}}}}}},\\\\beta,\\\\eta ;{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} },{\\\\beta }^{{\\\\prime} },{\\\\eta }^{{\\\\prime} }}^{g,j,\\\\mu }\\\\right]}^{*}{\\\\lambda }_{{{{{{{{\\\\boldsymbol{k}}}}}}}},\\\\alpha,\\\\eta ;{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} },{\\\\alpha }^{{\\\\prime} },{\\\\eta }^{{\\\\prime} }}^{g,j,\\\\mu }\\\\\\\\ \\t \\\\times {c}_{-{{{{{{{\\\\boldsymbol{k}}}}}}}},\\\\beta,-\\\\eta }^{{{{\\\\dagger}}} }{c}_{{{{{{{{\\\\boldsymbol{k}}}}}}}},\\\\alpha,\\\\eta }^{{{{\\\\dagger}}} }{c}_{{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} },{\\\\alpha }^{{\\\\prime} },{\\\\eta }^{{\\\\prime} }}{c}_{-{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} },{\\\\beta }^{{\\\\prime} },-{\\\\eta }^{{\\\\prime} }},$$ (9) where the coupling constants V g of the three different phonon modes g = A 1 , B 1 , E 2 are estimated to obey \\\\({V}_{{A}_{1}}={V}_{{B}_{1}}\\\\simeq 1.33{V}_{{E}_{2}}\\\\) for parallel spins in the two valleys, while \\\\({V}_{{A}_{1}}={V}_{{B}_{1}}=0\\\\) for anti-parallel spins. From Eq. ( 9 ), it is clear that the induced interaction would be always completely attractive if we focused on intra-band pairing, \\\\(\\\\alpha={\\\\alpha }^{{\\\\prime} }=\\\\beta={\\\\beta }^{{\\\\prime} }\\\\) , which in spinful systems generically favors the trivial pairing channel [51] , [52] . In our case, the combination of two energetically close bands and the trivial pairing being purely band-off-diagonal leads to competition between different superconductors, even with electron–phonon coupling alone. To demonstrate this, we study intra-valley pairing within the mean-field approximation and parametrize the relative strength of the different phonon modes with an angle variable θ ph according to \\\\({V}_{{A}_{1}}={V}_{{B}_{1}}={V}_{0}\\\\cos {\\\\theta }_{{{{{{{{\\\\rm{ph}}}}}}}}},{V}_{{E}_{2}}={V}_{0}\\\\sin {\\\\theta }_{{{{{{{{\\\\rm{ph}}}}}}}}}\\\\) . The results of the mean-field calculation are summarized in Fig. 3 . We see that the A 2 pairing state is favored by the intervalley phonons ( θ ph = 0) inspite of its band-off-diagonal nature leading to a suppressed gap [see Fig. 3 a]. This is natural as these phonons mediate an attractive interaction between the two valleys which disfavors the B 1 state, similar to T-IVC fluctuations. In fact, focusing on the leading, momentum-independent term, \\\\({\\\\lambda }_{{{{{{{{\\\\boldsymbol{k}}}}}}}},{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} }}^{g,+}\\\\to {\\\\lambda }^{g,+},g={A}_{1},{B}_{1}\\\\) , symmetry dictates \\\\({\\\\lambda }^{{A}_{1},+}\\\\propto {\\\\sigma }_{0}{\\\\eta }_{1}\\\\) and \\\\({\\\\lambda }^{{B}_{1},+}\\\\propto {\\\\sigma }_{0}{\\\\eta }_{2}\\\\) in the chiral limit (see Supplementary Appendix D3). This maps the problem exactly to that of T-IVC fluctuations, immediately explaining why the order parameter has a fixed sign in Fig. 3 b. As θ ph is increased, the B 1 state is favored (roughly for θ ph > π /4) as can be seen in Fig. 3 c. This is expected since the intra-valley E 2 phonon mediates an attractive interaction within each valley such that the energy gain due to the enhanced gap [Fig. 3 d], associated with the band-diagonal matrix elements of the B 1 state, will overcompensate the energetic loss due to the sign change of B 1 ’s order-parameter between the two valleys. This picture is consistent with the dominant and non-sign-changing nature of the band-diagonal components of the B 1 state, see Fig. 3 e–g. Finally, this behavior can also be understood by applying the commutator criterion in Eq. ( 7 ) in the microscopic sublattice basis, see Supplementary Appendix D1. Fig. 3: Pairing from electron–phonon coupling. We show a the band structure and b the self-consistent order parameter of the A 2 pairing for θ ph = 0 and T = 0. The eigenvalues corresponding to the A 2 and B 1 pairings in the linearized gap equation at T = 5 K, which is close to their T c , are shown in c as a function of θ ph . We show an example of the band structure ( d ) of the B 1 pairing and its order parameter ( e , f , g ). In accordance with symmetry, the A 2 ( B 1 ) state only has order-parameter components ∝ σ y ( ∝ σ 0, x , z ). We took ν = 2.5 and V 0 = 250 meV ⋅ (nm) 2 with a continuum-model bandwidth ≃ 2 meV. We point out that if A 1 phonons are dominant, as suggested by recent experimental work [61] and past theoretical study in mono-layer graphene [68] , we would expect our A 2 pairing to dominate assuming the pairing potential is sufficiently large. We also emphasize that although the pairing functions for A 2 pairing (b) when θ = 0 and B 1 pairing when θ = π /2 ( e , f , g ) are roughly equal, the excitation spectra show the B 1 state with a band gap on the order of the pairing strength ( d ) while the A 1 state’s band gap is nearly zero, see a . Full size image This shows that, as opposed to the conventional scenario [51] , [52] , there are two possible leading superconducting states and the superconducting pairing state does not transform trivially under the symmetries of the system even when phonons alone provide the pairing glue. We have checked in our T = 0 numerics that a 60–70 meV ⋅ (nm) 2 coupling to A 1 and B 1 phonons (based on ref. [67] ) is roughly of the order needed to stabilize the A 2 pairing, assuming the normal state is the flat bands of the un-renormalized continuum model, which in our case has a bandwidth of 2 meV. However, we note that if the interaction-renormalized band splitting is much larger than the continuum-model bandwidth, or if the normal state has anti-parallel spins in either valley, additional particle-hole fluctuations, such as those of T-IVC order, will also be required for pairing. An interesting scenario arises for anti-parallel spins in the two valleys as a magnetic field will cant the spins and, hence, increase the projection of the intervalley phonon matrix elements to the flat bands. At least in TTG, with the suppressed orbital coupling, this could give rise to re-entrant superconductivity at high fields [25] . Other particle-hole fluctuations Finally, we discuss pairing induced by fluctuations of other particle-hole instabilities. In Table 2 , we list the resulting leading superconductors taking λ j in Eq. ( 4 ) to be any of the different strong-coupling candidate order parameters [9] , [13] , [14] , [15] , [29] . In particular, in addition to the T-IVC, we will consider the time-reversal-odd Kramers intervalley coherent state (K-IVC), and time-reversal-odd and -even sublattice polarized states (SLP− and SLP+). To analyze how sensitive our conclusions are to the precise form of the coupling of the strong-coupling fluctuating orders to the electrons, we also perform numerics by projecting momentum-independent coupling vertices in the microscopic basis with the correct symmetries (see, e.g., Table II in ref. [13] ), listed as \\\\({\\\\bar{\\\\lambda }}^{j}\\\\) in Table 2 , to the flat bands. In the band basis, this leads to momentum-dependent coupling vertices, cf. Eq. ( 9 ). Motivated by recent experiments [7] , we will also consider fluctuations of an additional nematic, time-reversal symmetric, layer-odd, intervalley coherent state (N-IVC) [62] which is not a candidate ground state in the strong-coupling limit; unlike the other strong-coupling ground states, the N-IVC has no momentum-independent representation in the flat-band basis but does have a momentum-independent matrix order parameter in the sublattice basis which takes the form \\\\({\\\\lambda }^{(j,{j}^{{\\\\prime} })}={({\\\\eta }_{x},\\\\, {\\\\eta }_{y})}_{j}{({\\\\rho }_{0},\\\\, {\\\\rho }_{z})}_{{j}^{{\\\\prime} }}\\\\) . The results for fluctuations of the projected strong-coupling orders \\\\({\\\\bar{\\\\lambda }}^{j}\\\\) in Table 2 and of the projected N-IVC state are shown in Fig. 4 , where we use the angle θ fluc. to tune the relative strength between T-IVC and any of the other type of fluctuation-induced interactions by multiplying the T-IVC interaction potential with \\\\(\\\\cos ({\\\\theta }_{{{{{{{{\\\\rm{fluc}}}}}}}}.})\\\\) and the other fluctuation potential with \\\\(\\\\sin ({\\\\theta }_{{{{{{{{\\\\rm{fluc}}}}}}}}.})\\\\) . In our microscopic numerics, we have taken a potential form \\\\(\\\\chi ({{{{{{{\\\\boldsymbol{q}}}}}}}})=\\\\frac{1}{{A}_{m}}\\\\frac{V}{{\\\\alpha }^{2}+| {{{{{{{\\\\boldsymbol{q}}}}}}}}{| }^{2}/{k}_{\\\\theta }^{2}}\\\\) again with α = 0.2 and with V = 4200 meV ⋅ (nm) 2 . We chose the value of V such that the transitions between the different pairing states are clearly visible in Fig. 4 when varying θ fluc. . In accordance with the prediction for \\\\({\\\\bar{\\\\lambda }}^{j}\\\\) in Table 2 , SLP + fluctuations further stabilize the A 2 superconductor, see Fig. 4 a. As such, the band-diagonal B 1 superconducting channel, where SLP + fluctuations are also attractive, can become the leading channel (favored over A 2 as a result of the finite bandwidth) only very close to θ fluc. = π /2. K-IVC fluctuations, however, are repulsive for A 2 pairing and favor the B 1 state more strongly. Table 2 Leading superconducting states in the flat-band limit, following from Eq. ( 7 ), for pairing mediated by fluctuations of the indicated orders, defined by using λ j in Eq. ( 4 ) Full size table Fig. 4: Pairing for different particle-hole fluctuations. These are defined by the coupling matrices \\\\({\\\\bar{\\\\lambda }}^{j}\\\\) listed in Table 2 . Similar to Fig. 3 c, we show T c of the leading pairing states, where θ fluc. tunes the relative strength between T-IVC-induced interactions ( \\\\(\\\\propto \\\\cos {\\\\theta }_{{{{{{{{\\\\rm{fluc}}}}}}}}.}\\\\) ) and interactions ( \\\\(\\\\propto \\\\sin {\\\\theta }_{{{{{{{{\\\\rm{fluc}}}}}}}}.}\\\\) ) coming from fluctuations of a SLP+, b K-IVC, c SLP−, and d N-IVC fluctuations. Full size image So far, the strong-coupling ( λ j ) and sublattice ( \\\\({\\\\bar{\\\\lambda }}^{j}\\\\) ) form of the couplings in Table 2 lead to the same conclusions. This is different for SLP− fluctuations (Fig. 4 d), where the projection-induced momentum dependence in the band basis can stabilize the E 1 superconductor. This can be understood by applying Eq. ( 7 ) in a sublattice basis (see Supplementary Appendix D1). We also find the E 1 state when fluctuations of the N-IVC state of ref. [62] dominate. Examples of the E 1 nematic and B 2 order parameters that emerge for SLP− fluctuations or N-IVC fluctuations are shown in Supplementary Appendix F. We point out that the nematic E 1 pairing is also an interesting candidate given that despite having nonzero pairing in the σ 0 , σ x , σ z channels, it will be nodal as long as the σ x components do not gap out the nodes in the band-diagonal parts.\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"Taken together, we see that the proposed band-off-diagonal A 2 superconductor is an especially attractive candidate for TBG and TTG: first, it can lead to both V-shaped or U -shaped DOS, depending on lifetime parameters, the normal state, and the coupling strength V , see Fig. 2 e. As these parameters might vary from sample to sample and within a sample (e.g., V is expected to decrease upon doping further away from the insulator), this can naturally explain the tunneling data of [48] , [49] . We emphasize however that at least at the level of our mean-field numerics, we only expect a V-shape in the regime where the superconducting pairing is of the order of the bandwidth; this is the regime, where although the pairing is finite and can be quite large, the gap in the superconducting spectrum is either just closing or very small relative to the pairing. Increasing the pairing further will lead to an evolution from V to U -shaped while decreasing the pairing will eventually lead to a nodal Fermi surface and presumably a peak at zero energy in the DOS. Second, despite its interband nature, A 2 is the unique pairing state that is favored by fluctuations of two out of the four strong-coupling candidates we consider for the correlated insulator, see Fig. 4 a–c. What is more, this includes the T-IVC state, signatures of which are observed in recent experiments [7] . Finally, it is also favored by the likely dominant [61] , [68] optical intervalley phonon modes. We emphasize that, both in the case of fluctuating correlated insulators and phonons, the minimum attractive coupling needed to stabilize a purely band-off-diagonal state depends on the energy splitting between the two flat bands in the normal state; if the bands of our normal state are closer to degenerate, irrespective of the total bandwidth, the needed coupling to stabilize the A 2 pairing in mean-field will decrease. The other band-off-diagonal superconductor we identify transforms under the IR E 2 , i.e., can be thought of as a p -wave state. Its spectral properties also agree well with the experiment as the chiral configurations, E 2 (1, i ), which is favored within mean-field theory over a nematic E 2 state, can also have nodal regions, depending on filling. As can be seen in Fig. 1 g, this can lead to a transition from gapped to nodal when increasing the electron filling starting at ν ≃ 2. However, as opposed to the A 2 state, E 2 does not naturally appear as leading instability when considering optical phonons or fluctuations of any of the strong-coupling order parameters of the correlated insulator. While this makes it energetically less natural than A 2 , we cannot exclude it since its phenomenology agrees well with the experiment and since the precise form of the coupling of the dominant low-energy collective excitations are not known—significant momentum dependencies beyond λ j and \\\\({\\\\bar{\\\\lambda }}^{j}\\\\) in Table 2 could stabilize E 2 pairing as well. We also find in our numerics a nematic E 1 state which may be preferred over its chiral version in the presence of sufficient strain or due to fluctuation corrections [27] , [53] , [54] , [55] . We find the E 1 state is the leading instability of nematic IVC fluctuations and SLP − fluctuations, and is a subleading instability of T-IVC fluctuations. The E 1 state is interesting in its own right, as it can also be nodal. As superconductivity might further coexist with T-IVC order [7] , we have checked (see Supplementary Appendix E) that this does not alter our main observation: the preserved C 2 z symmetry still allows for entirely band-off-diagonal states, with transitions from nodal to full gapped, which are stabilized (among other fluctuations) by intervalley phonons. For the future, it will be interesting to go beyond mean-field and analyze the competition of our band-off-diagonal states with odd-frequency pairing, which we study in a follow-up work [69] . It also seems promising to study Andreev reflection [48] , [49] for our interband pairing scenario. On a more general level, our work shows that the observation of nodal pairing in twisted graphene systems does not immediately exclude a chiral superconducting state nor an entire electron–phonon-based pairing mechanism. It illustrates that a microscopic understanding of the superconducting states in graphene moiré systems requires taking into account their intrinsically multi-band nature. Note added . Just before posting our work, ref. [70] appeared online, which discusses pairing induced by A 1 phonons in spinful TBG bands.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Flat-band limit To derive Eq. ( 7 ), we take the flat-band limit, ξ k ,± → 0, in the linearized gap equation. For the interaction defined in Eqs. ( 4 )–( 6 )), we get (with moiré cell area A m ) $${({{{\\\\Delta }}}_{{{{{{{{\\\\boldsymbol{k}}}}}}}},\\\\eta })}_{\\\\beta,{\\\\beta }^{{\\\\prime} }}=\\t {t}_{\\\\phi }\\\\frac{1}{4{A}_{m}T}\\\\mathop{\\\\sum}\\\\limits_{{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} }}{\\\\chi }_{{{{{{{{\\\\boldsymbol{k}}}}}}}}-{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} }} \\\\\\\\ \\t \\\\times \\\\mathop{\\\\sum}\\\\limits_{j}{\\\\left[{\\\\lambda }_{{\\\\beta }^{{\\\\prime} },\\\\eta ;{\\\\alpha }^{{\\\\prime} },{\\\\eta }^{{\\\\prime} }}^{j}\\\\right]}^{*}{\\\\lambda }_{\\\\beta,\\\\eta ;\\\\alpha,{\\\\eta }^{{\\\\prime} }}^{j}{({{{\\\\Delta }}}_{{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} },{\\\\eta }^{{\\\\prime} }})}_{\\\\alpha,{\\\\alpha }^{{\\\\prime} }}.$$ (10) We define \\\\({({\\\\hat{{{\\\\Delta }}}}_{{{{{{{{\\\\boldsymbol{k}}}}}}}}})}_{\\\\alpha,\\\\eta ;{\\\\alpha }^{{\\\\prime} },{\\\\eta }^{{\\\\prime} }}:={({{{\\\\Delta }}}_{{{{{{{{\\\\boldsymbol{k}}}}}}}},\\\\eta })}_{\\\\alpha,{\\\\alpha }^{{\\\\prime} }}{\\\\delta }_{\\\\eta,{\\\\eta }^{{\\\\prime} }}\\\\) and note that finding the leading superconducting state according to Eq. ( 10 ) is equivalent to determining \\\\({\\\\hat{{{\\\\Delta }}}}_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}\\\\) that maximizes the functional $${{{{{{{\\\\mathcal{F}}}}}}}}[{\\\\hat{{{\\\\Delta }}}}_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}]:=\\\\frac{{\\\\sum }_{{{{{{{{\\\\boldsymbol{k}}}}}}}},{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} },j} \\\\, {\\\\chi }_{{{{{{{{\\\\boldsymbol{k}}}}}}}}-{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} }}\\\\,{t}_{\\\\phi }\\\\,{{\\\\mbox{tr}}}\\\\,[{\\\\lambda }^{j}{\\\\hat{{{\\\\Delta }}}}_{{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} }}{({\\\\lambda }^{j})}^{{{{\\\\dagger}}} }{\\\\hat{{{\\\\Delta }}}}_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{{{\\\\dagger}}} }]}{{\\\\sum }_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}\\\\,{{\\\\mbox{tr}}}\\\\,[{\\\\hat{{{\\\\Delta }}}}_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{{{\\\\dagger}}} }{\\\\hat{{{\\\\Delta }}}}_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}]}.$$ (11) Since \\\\({\\\\chi }_{{{{{{{{\\\\boldsymbol{k}}}}}}}}-{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} }} \\\\, > \\\\, 0\\\\) , the maximum value will be reached if we can maximize \\\\({t}_{\\\\phi }\\\\,{{\\\\mbox{tr}}}\\\\,[{\\\\lambda }^{j}{\\\\hat{{{\\\\Delta }}}}_{{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} }}{({\\\\lambda }^{j})}^{{{{\\\\dagger}}} }{\\\\hat{{{\\\\Delta }}}}_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{{{\\\\dagger}}} }]\\\\) for each \\\\({{{{{{{\\\\boldsymbol{k}}}}}}}},\\\\, {{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} },\\\\, j\\\\) separately. As the Frobenius inner product \\\\({\\\\left\\\\langle A,B\\\\right\\\\rangle }_{F}=\\\\,{{\\\\mbox{tr}}}\\\\,[{A}^{{{{\\\\dagger}}} }B]\\\\) reaches its maximum (minimum) at fixed \\\\(\\\\left\\\\langle A,\\\\, A\\\\right\\\\rangle\\\\) and \\\\(\\\\left\\\\langle B,\\\\, B\\\\right\\\\rangle\\\\) , if A = c B with c > 0 ( c < 0), \\\\({t}_{\\\\phi }\\\\,{{\\\\mbox{tr}}}\\\\,[{\\\\lambda }^{j}{\\\\hat{{{\\\\Delta }}}}_{{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} }}{({\\\\lambda }^{j})}^{{{{\\\\dagger}}} }{\\\\hat{{{\\\\Delta }}}}_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{{{\\\\dagger}}} }]\\\\) is maximized if \\\\({\\\\hat{{{\\\\Delta }}}}_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}={t}_{\\\\phi }{c}_{{{{{{{{\\\\boldsymbol{k}}}}}}}},{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} }}{\\\\lambda }^{j}{\\\\hat{{{\\\\Delta }}}}_{{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} }}{({\\\\lambda }^{j})}^{{{{\\\\dagger}}} }\\\\) with \\\\({c}_{{{{{{{{\\\\boldsymbol{k}}}}}}}},{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} }} \\\\, > \\\\, 0\\\\) . For the ansatz \\\\({\\\\hat{{{\\\\Delta }}}}_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}={\\\\delta }_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}D{\\\\eta }_{x}\\\\) (and assuming for now that δ k has a fixed sign for all k ), this is obeyed if $$D{\\\\eta }_{x}={t}_{\\\\phi }{\\\\lambda }^{j}D{\\\\eta }_{x}{({\\\\lambda }^{j})}^{{{{\\\\dagger}}} },\\\\quad \\\\forall j.$$ (12) We state Eq. ( 12 ) as the (anti)commutator condition ( 7 ) in the main text [equivalent if \\\\({({\\\\lambda }^{j})}^{2}={\\\\mathbb{1}}\\\\) ], not only because it highlights the simple algebraic and basis independent nature of the condition but also since it emphasizes the similarities to the generalized Anderson theorem of [71] , [72] . If we can find a solution to Eq. ( 12 ), we know that the maximum (or at least one of the possibly degenerate maxima) of \\\\({{{{{{{\\\\mathcal{F}}}}}}}}[{\\\\hat{{{\\\\Delta }}}}_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}]\\\\) is of the form of \\\\({\\\\hat{{{\\\\Delta }}}}_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}={\\\\delta }_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}D{\\\\eta }_{x}\\\\) where δ k is obtained as the maximum of the reduced functional $$\\\\widetilde{{{{{{{{\\\\mathcal{F}}}}}}}}}[{\\\\delta }_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}]:=\\\\frac{{\\\\sum }_{{{{{{{{\\\\boldsymbol{k}}}}}}}},{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} }} \\\\, {\\\\chi }_{{{{{{{{\\\\boldsymbol{k}}}}}}}}-{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} }}{\\\\delta }_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{*}{\\\\delta }_{{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} }}^{}}{{\\\\sum }_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}| {\\\\delta }_{{{{{{{{\\\\boldsymbol{k}}}}}}}}}{| }^{2}},$$ (13) or equivalently as the largest eigenvector of \\\\({\\\\chi }_{{{{{{{{\\\\boldsymbol{k}}}}}}}}-{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} }}\\\\) viewed as a matrix in k and \\\\({{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} }\\\\) . As \\\\({\\\\chi }_{{{{{{{{\\\\boldsymbol{k}}}}}}}}-{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} }} > 0\\\\) (due to stability), the Perron-Frobenium theorem then immediately implies δ k > 0, in line with out assumption above and as stated in the main text. Electron–phonon coupling To present more details on the electron–phonon coupling, the associated displacement operators in Eq. ( 8 ) can be expressed in terms of canonical bosons, b g , α , μ , q , $${({u}_{g,\\\\mu }({{{{{{{\\\\boldsymbol{r}}}}}}}}))}_{j}=\\\\mathop{\\\\sum}\\\\limits_{{{{{{{{\\\\boldsymbol{q}}}}}}}}}\\\\frac{{b}_{g,j,\\\\mu,{{{{{{{\\\\boldsymbol{q}}}}}}}}}{e}^{i{{{{{{{\\\\boldsymbol{q}}}}}}}}\\\\cdot {{{{{{{\\\\boldsymbol{r}}}}}}}}}+\\\\,{{\\\\mbox{H.c.}}}\\\\,}{\\\\sqrt{2NM{\\\\omega }_{g}({{{{{{{\\\\boldsymbol{q}}}}}}}})}},$$ (14) where j refers to the two components for the E 2 phonon (is idle for A 1 , B 1 ), M is the carbon mass, and ω g ( q ) is the phonon dispersion, characterizing the phononic part of the Hamiltonian, \\\\({{{{{{{{\\\\mathcal{H}}}}}}}}}_{P}={\\\\sum }_{{{{{{{{\\\\boldsymbol{q}}}}}}}}}{\\\\omega }_{g}({{{{{{{\\\\boldsymbol{q}}}}}}}}){b}_{g,j,\\\\mu,{{{{{{{\\\\boldsymbol{q}}}}}}}}}^{{{{\\\\dagger}}} }{b}_{g,j,\\\\mu,{{{{{{{\\\\boldsymbol{q}}}}}}}}}\\\\) . As for \\\\({({{{{{{{{\\\\boldsymbol{v}}}}}}}}}_{\\\\mu })}_{\\\\ell }\\\\) in Eq. ( 8 ), ℓ = 1, 2 refers to the physical graphene layer in the case of TBG. One can, in principle, choose any orthonormal basis; we will find it convenient to use the layer-exchange even and odd states, \\\\({{{{{{{{\\\\boldsymbol{v}}}}}}}}}_{\\\\pm }={(1,\\\\pm 1)}^{T}/\\\\sqrt{2}\\\\) . For TTG, the situation is more involved (see Supplementary Appendix D2), but our arguments about which phonons are attractive in which pairing channels will hold for both systems. We project \\\\({{{{{{{{\\\\mathcal{H}}}}}}}}}_{EP}\\\\) in Eq. ( 8 ) onto the two flat bands ( α = ± ) in each valley η of the spin-polarized continuum-model, leading to a coupling term similar to Eq. ( 4 ) with momentum-dependent coupling matrices, \\\\({\\\\lambda }^{j}\\\\to {\\\\lambda }_{{{{{{{{\\\\boldsymbol{k}}}}}}}},{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} }}^{g,j,\\\\mu }\\\\) . Investigating the matrix elements \\\\({\\\\lambda }_{{{{{{{{\\\\boldsymbol{k}}}}}}}},{{{{{{{{\\\\boldsymbol{k}}}}}}}}}^{{\\\\prime} }}^{g,j,\\\\mu }\\\\) , we notice that they almost vanish for the layer-odd intervalley ( A 1 , B 1 ) phonons, which can be understood as a consequence of chiral and particle-hole symmetry (see Supplementary Appendix D3). The situation is the reverse for the intra-valley ( E 2 ) phonons, where the layer-even matrix elements are numerically small and the layer-odd matrix elements dominate. We therefore focus on layer-even (odd) intervalley (intra-valley) phonon couplings. Neglecting the momentum dependence in the phonon frequencies and retardation effects, the resulting electron-electron interaction in the inter-valley Cooper channel obtained by integrating out the phonons is given by Eq. ( 9 ). Here, \\\\({V}_{g}={g}_{g}^{2}/(2N{\\\\omega }_{g}^{2}) \\\\, > \\\\, 0\\\\) and \\\\({V}_{{A}_{1}}={V}_{{B}_{1}}\\\\simeq 1.33{V}_{{E}_{2}}\\\\) results from \\\\({g}_{{A}_{1}}={g}_{{B}_{1}}\\\\simeq {g}_{{E}_{2}}\\\\) and the phonon frequencies estimated in ref. [67] . Importantly, this only holds for parallel spins in the two valleys. For anti-parallel spins, the projection of the coupling matrices to the flat bands vanishes for the intervalley phonon modes A 1 and B 1 such that \\\\({V}_{{A}_{1}}={V}_{{B}_{1}}=0\\\\) .\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"The core research idea is to demonstrate that the combination of flavor polarization and symmetry representations in the flat bands of twisted-bilayer and twisted-trilayer graphene enables the existence of band-off-diagonal superconducting pairing states—specifically, one transforming under the trivial representation with symmetry-protected nodal lines and another under a two-dimensional irreducible representation that can exhibit either nodal or fully gapped behavior depending on filling—thereby addressing key theoretical challenges in understanding the nature of superconductivity in these systems."},"entities":{"kind":"list like","value":["flavor polarization","symmetry representations","flat bands","twisted-bilayer graphene","twisted-trilayer graphene","band-off-diagonal superconducting pairing states","trivial representation","symmetry-protected nodal lines","two-dimensional irreducible representation","nodal behavior","fully gapped behavior","filling","theoretical challenges","superconductivity"],"string":"[\n \"flavor polarization\",\n \"symmetry representations\",\n \"flat bands\",\n \"twisted-bilayer graphene\",\n \"twisted-trilayer graphene\",\n \"band-off-diagonal superconducting pairing states\",\n \"trivial representation\",\n \"symmetry-protected nodal lines\",\n \"two-dimensional irreducible representation\",\n \"nodal behavior\",\n \"fully gapped behavior\",\n \"filling\",\n \"theoretical challenges\",\n \"superconductivity\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1.","title":"Andrei, E. Y. & MacDonald, A. H. Graphene bilayers with a twist.Nat. Mater.19, 1265 (2020).","link":"https://doi.org/10.1038%2Fs41563-020-00840-0"},{"idx":"2.","title":"Balents, L., Dean, C. R., Efetov, D. K. & Young, A. F. Superconductivity and strong correlations in moiré flat bands.Nat. Phys.16, 725 (2020).","link":"https://doi.org/10.1038%2Fs41567-020-0906-9"},{"idx":"3.","title":"Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices.Nature556, 80 (2018).","link":"https://doi.org/10.1038%2Fnature26154"},{"idx":"4.","title":"Lu, X. et al. Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene.Nature574, 653 (2019).","link":"https://doi.org/10.1038%2Fs41586-019-1695-0"},{"idx":"5.","title":"Sharpe, A. L. et al. Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene.Science365, 605 (2019).","link":"https://doi.org/10.1126%2Fscience.aaw3780"},{"idx":"6.","title":"Nuckolls, K. P. et al. Strongly correlated chern insulators in magic-angle twisted bilayer graphene.Nature588, 610 (2020).","link":"https://doi.org/10.1038%2Fs41586-020-3028-8"},{"idx":"8.","title":"Kang, J. & Vafek, O. Strong coupling phases of partially filled twisted bilayer graphene narrow bands.Phys. Rev. Lett.122, 246401 (2019).","link":"https://doi.org/10.1103%2FPhysRevLett.122.246401"},{"idx":"19.","title":"Wong, D. et al. Cascade of electronic transitions in magic-angle twisted bilayer graphene.Nature582, 198 (2020).","link":"https://doi.org/10.1038%2Fs41586-020-2339-0"},{"idx":"20.","title":"Zondiner, U. et al. Cascade of phase transitions and dirac revivals in magic-angle graphene.Nature582, 203 (2020).","link":"https://doi.org/10.1038%2Fs41586-020-2373-y"},{"idx":"21.","title":"Park, J. M., Cao, Y., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Tunable strongly coupled superconductivity in magic-angle twisted trilayer graphene.Nature590, 249 (2021).","link":"https://doi.org/10.1038%2Fs41586-021-03192-0"},{"idx":"22.","title":"Hao, Z. et al. Electric field–tunable superconductivity in alternating-twist magic-angle trilayer graphene.Science371, 1133 (2021).","link":"https://doi.org/10.1126%2Fscience.abg0399"},{"idx":"23.","title":"Lin, J.-X. et al. Zero-field superconducting diode effect in small-twist-angle trilayer graphene.Nat. Phys.18, 1221 (2022).","link":"https://doi.org/10.1038%2Fs41567-022-01700-1"},{"idx":"25.","title":"Cao, Y., Park, J. M., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Pauli-limit violation and re-entrant superconductivity in moirégraphene.Nature595, 526 (2021).","link":"https://doi.org/10.1038%2Fs41586-021-03685-y"},{"idx":"26.","title":"Ojajärvi, R., Hyart, T., Silaev, M. A. & Heikkilä, T. T. Competition of electron-phonon mediated superconductivity and stoner magnetism on a flat band.Phys. 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Lett.129, 187001 (2022).","link":"https://doi.org/10.1103%2FPhysRevLett.129.187001"},{"idx":"43.","title":"Fischer, A., Klebl, L., Honerkamp, C. & Kennes, D. M. Spin-fluctuation-induced pairing in twisted bilayer graphene.Phys. Rev. B103, L041103 (2021).","link":"https://doi.org/10.1103%2FPhysRevB.103.L041103"},{"idx":"47.","title":"You, Y.-Z. & Vishwanath, A. Superconductivity from valley fluctuations and approximate so(4) symmetry in a weak coupling theory of twisted bilayer graphene.npj Quant. Mater.4, 16 (2019).","link":"https://doi.org/10.1038%2Fs41535-019-0153-4"},{"idx":"48.","title":"Oh, M. et al. Evidence for unconventional superconductivity in twisted bilayer graphene.Nature600, 240 (2021).","link":"https://doi.org/10.1038%2Fs41586-021-04121-x"},{"idx":"49.","title":"Kim, H. et al. Evidence for unconventional superconductivity in twisted trilayer graphene.Nature606, 494 (2022).","link":"https://doi.org/10.1038%2Fs41586-022-04715-z"},{"idx":"51.","title":"Brydon, P. M. 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B99, 144507 (2019).","link":"https://doi.org/10.1103%2FPhysRevB.99.144507"},{"idx":"55.","title":"Gali, V. & Fernandes, R. M. Role of electromagnetic gauge-field fluctuations in the selection between chiral and nematic superconductivity.Phys. Rev. B106, 094509 (2022).","link":"https://doi.org/10.1103%2FPhysRevB.106.094509"},{"idx":"58.","title":"Read, N. & Sachdev, S. Spin-Peierls, valence-bond solid, and Néel ground states of low-dimensional quantum antiferromagnets.Phys. Rev. B42, 4568 (1990). Appendix B.","link":"https://doi.org/10.1103%2FPhysRevB.42.4568"},{"idx":"59.","title":"Lee, J. & Sachdev, S. Wess-Zumino-Witten terms in graphene Landau levels.Phys. Rev. Lett.114, 226801 (2015).","link":"https://doi.org/10.1103%2FPhysRevLett.114.226801"},{"idx":"60.","title":"Liu, X. et al. Visualizing broken symmetry and topological defects in a quantum Hall ferromagnet.Science375, 321 (2022).","link":"https://doi.org/10.1126%2Fscience.abm3770"},{"idx":"62.","title":"Samajdar, R. et al. 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Phys.16, 725 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41567-020-0906-9\"\n },\n {\n \"idx\": \"3.\",\n \"title\": \"Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices.Nature556, 80 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fnature26154\"\n },\n {\n \"idx\": \"4.\",\n \"title\": \"Lu, X. et al. Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene.Nature574, 653 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-019-1695-0\"\n },\n {\n \"idx\": \"5.\",\n \"title\": \"Sharpe, A. L. et al. Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene.Science365, 605 (2019).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.aaw3780\"\n },\n {\n \"idx\": \"6.\",\n \"title\": \"Nuckolls, K. P. et al. Strongly correlated chern insulators in magic-angle twisted bilayer graphene.Nature588, 610 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-020-3028-8\"\n },\n {\n \"idx\": \"8.\",\n \"title\": \"Kang, J. & Vafek, O. Strong coupling phases of partially filled twisted bilayer graphene narrow bands.Phys. Rev. Lett.122, 246401 (2019).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.122.246401\"\n },\n {\n \"idx\": \"19.\",\n \"title\": \"Wong, D. et al. Cascade of electronic transitions in magic-angle twisted bilayer graphene.Nature582, 198 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-020-2339-0\"\n },\n {\n \"idx\": \"20.\",\n \"title\": \"Zondiner, U. et al. Cascade of phase transitions and dirac revivals in magic-angle graphene.Nature582, 203 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-020-2373-y\"\n },\n {\n \"idx\": \"21.\",\n \"title\": \"Park, J. M., Cao, Y., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Tunable strongly coupled superconductivity in magic-angle twisted trilayer graphene.Nature590, 249 (2021).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-021-03192-0\"\n },\n {\n \"idx\": \"22.\",\n \"title\": \"Hao, Z. et al. Electric field–tunable superconductivity in alternating-twist magic-angle trilayer graphene.Science371, 1133 (2021).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.abg0399\"\n },\n {\n \"idx\": \"23.\",\n \"title\": \"Lin, J.-X. et al. Zero-field superconducting diode effect in small-twist-angle trilayer graphene.Nat. Phys.18, 1221 (2022).\",\n \"link\": \"https://doi.org/10.1038%2Fs41567-022-01700-1\"\n },\n {\n \"idx\": \"25.\",\n \"title\": \"Cao, Y., Park, J. M., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Pauli-limit violation and re-entrant superconductivity in moirégraphene.Nature595, 526 (2021).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-021-03685-y\"\n },\n {\n \"idx\": \"26.\",\n \"title\": \"Ojajärvi, R., Hyart, T., Silaev, M. A. & Heikkilä, T. T. Competition of electron-phonon mediated superconductivity and stoner magnetism on a flat band.Phys. Rev. B98, 054515 (2018).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.98.054515\"\n },\n {\n \"idx\": \"27.\",\n \"title\": \"Scheurer, M. S. & Samajdar, R. Pairing in graphene-based moiré superlattices.Phys. Rev. Research2, 033062 (2020).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevResearch.2.033062\"\n },\n {\n \"idx\": \"29.\",\n \"title\": \"Christos, M., Sachdev, S. & Scheurer, M. S. Superconductivity, correlated insulators, and Wess-Zumino-Witten terms in twisted bilayer graphene.Proc. Natl Acad. Sci. USA117, 29543 (2020).\",\n \"link\": \"https://doi.org/10.1073%2Fpnas.2014691117\"\n },\n {\n \"idx\": \"31.\",\n \"title\": \"Khalaf, E., Ledwith, P. & Vishwanath, A. Symmetry constraints on superconductivity in twisted bilayer graphene: Fractional vortices, 4econdensates, or nonunitary pairing.Phys. Rev. B105, 224508 (2022).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.105.224508\"\n },\n {\n \"idx\": \"32.\",\n \"title\": \"Scammell, H. D., Li, J. I. A. & Scheurer, M. S. Theory of zero-field superconducting diode effect in twisted trilayer graphene.2D Materials9, 025027 (2022).\",\n \"link\": \"https://doi.org/10.1088%2F2053-1583%2Fac5b16\"\n },\n {\n \"idx\": \"33.\",\n \"title\": \"Shavit, G., Berg, E., Stern, A. & Oreg, Y. Theory of correlated insulators and superconductivity in twisted bilayer graphene.Phys. Rev. Lett.127, 247703 (2021).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.127.247703\"\n },\n {\n \"idx\": \"34.\",\n \"title\": \"Crépel, V., Cea, T., Fu, L. & Guinea, F. Unconventional superconductivity due to interband polarization.Phys. Rev. B105, 094506 (2022).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.105.094506\"\n },\n {\n \"idx\": \"35.\",\n \"title\": \"Wu, F., MacDonald, A. H. & Martin, I. Theory of phonon-mediated superconductivity in twisted bilayer graphene.Phys. Rev. Lett.121, 257001 (2018).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.121.257001\"\n },\n {\n \"idx\": \"36.\",\n \"title\": \"Lian, B., Wang, Z. & Bernevig, B. A. Twisted bilayer graphene: a phonon-driven superconductor.Phys. Rev. Lett.122, 257002 (2019).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.122.257002\"\n },\n {\n \"idx\": \"37.\",\n \"title\": \"Lewandowski, C., Chowdhury, D. & Ruhman, J. Pairing in magic-angle twisted bilayer graphene: Role of phonon and plasmon umklapp.Phys. Rev. B103, 235401 (2021).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.103.235401\"\n },\n {\n \"idx\": \"38.\",\n \"title\": \"Lewandowski, C., Nadj-Perge, S. & Chowdhury, D. Does filling-dependent band renormalization aid pairing in twisted bilayer graphene?npj Quant. Mater.6, 82 (2021).\",\n \"link\": \"https://doi.org/10.1038%2Fs41535-021-00379-6\"\n },\n {\n \"idx\": \"40.\",\n \"title\": \"Phong, Vo. T., Pantaleón, P. A., Cea, T. & Guinea, F. Band structure and superconductivity in twisted trilayer graphene.Phys. Rev. B104, L121116 (2021).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.104.L121116\"\n },\n {\n \"idx\": \"42.\",\n \"title\": \"Huang, C., Wei, N., Qin, W. & MacDonald, A. H. Pseudospin paramagnons and the superconducting dome in magic angle twisted bilayer graphene.Phys. Rev. Lett.129, 187001 (2022).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.129.187001\"\n },\n {\n \"idx\": \"43.\",\n \"title\": \"Fischer, A., Klebl, L., Honerkamp, C. & Kennes, D. M. Spin-fluctuation-induced pairing in twisted bilayer graphene.Phys. Rev. B103, L041103 (2021).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.103.L041103\"\n },\n {\n \"idx\": \"47.\",\n \"title\": \"You, Y.-Z. & Vishwanath, A. Superconductivity from valley fluctuations and approximate so(4) symmetry in a weak coupling theory of twisted bilayer graphene.npj Quant. Mater.4, 16 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41535-019-0153-4\"\n },\n {\n \"idx\": \"48.\",\n \"title\": \"Oh, M. et al. Evidence for unconventional superconductivity in twisted bilayer graphene.Nature600, 240 (2021).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-021-04121-x\"\n },\n {\n \"idx\": \"49.\",\n \"title\": \"Kim, H. et al. Evidence for unconventional superconductivity in twisted trilayer graphene.Nature606, 494 (2022).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-022-04715-z\"\n },\n {\n \"idx\": \"51.\",\n \"title\": \"Brydon, P. M. R., Das Sarma, S., Hui, H.-Y. & Sau, J. D. Odd-parity superconductivity from phonon-mediated pairing: application to CuxBi2Se3.Phys. Rev. B90, 184512 (2014).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.90.184512\"\n },\n {\n \"idx\": \"52.\",\n \"title\": \"Scheurer, M. S. Mechanism, time-reversal symmetry, and topology of superconductivity in noncentrosymmetric systems.Phys. Rev. B93, 174509 (2016).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.93.174509\"\n },\n {\n \"idx\": \"53.\",\n \"title\": \"Anderson, P. W. & Brinkman, W. F. Anisotropic superfluidity in3He: a possible interpretation of its stability as a spin-fluctuation effect.Phys. Rev. Lett.30, 1108 (1973).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.30.1108\"\n },\n {\n \"idx\": \"54.\",\n \"title\": \"Kozii, V., Isobe, H., Venderbos, J. W. F. & Fu, L. Nematic superconductivity stabilized by density wave fluctuations: possible application to twisted bilayer graphene.Phys. Rev. B99, 144507 (2019).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.99.144507\"\n },\n {\n \"idx\": \"55.\",\n \"title\": \"Gali, V. & Fernandes, R. M. Role of electromagnetic gauge-field fluctuations in the selection between chiral and nematic superconductivity.Phys. Rev. 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Res.2, 023140 (2020).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevResearch.2.023140\"\n }\n]","isConversational":false}}},{"rowIdx":195,"cells":{"unique_id":{"kind":"string","value":"s41467-019-10864-z"},"subfield":{"kind":"string","value":"Mathematics and computing"},"title":{"kind":"string","value":"Detection of cell-type-specific risk-CpG sites in epigenome-wide association studies"},"abstract":{"kind":"string","value":"In epigenome-wide association studies, the measured signals for each sample are a mixture of methylation profiles from different cell types. Current approaches to the association detection claim whether a cytosine-phosphate-guanine (CpG) site is associated with the phenotype or not at aggregate level and can suffer from low statistical power. Here, we propose a statistical method, HIgh REsolution (HIRE), which not only improves the power of association detection at aggregate level as compared to the existing methods but also enables the detection of risk-CpG sites for individual cell types."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"Epigenome-wide association studies (EWAS) aim to identify cytosine-phosphate-guanine (CpG) sites associated with phenotypes of interest, such as disease status [1] , [2] , [3] , smoking history [4] , [5] , body mass index [6] , and age [7] , [8] . However, because the samples in EWAS are measured at the bulk level rather than at the single-cell level, the obtained methylome for each sample shows the signals aggregated from distinct cell types [3] , [9] , [10] , which leads to two main challenges in the analysis of EWAS data. On the one hand, the cell type compositions differ among samples and can be associated with phenotypes [3] , [10] . Both binary phenotypes, such as the diseased or normal status [3] , and continuous phenotypes, such as age [10] , have been found to affect the cell type compositions. As a result, ignoring the cellular heterogeneity in EWAS can lead to many spurious associations [10] , [11] , [12] , [13] . On the other hand, the phenotype may change the methylation level of a CpG site in some but not all of the cell types. Identification of the exact cell types that carry the risk-CpG sites can deepen our understandings of disease mechanisms. However, such identification is challenging because only the aggregated-level signals can be observed. To the best of our knowledge, no existing statistical method for EWAS can detect cell-type-specific associations despite active research to account for cell-type heterogeneity. The existing approaches can be categorized into two schools [14] : reference-based and reference-free methods. The reference-based methods [9] , [15] require the reference methylation profiles for each cell type to be known a priori, and they regress the aggregated methylation levels observed from each sample on the same set of references to learn the sample’s cellular compositions. However, because samples have different attributes, such as age and gender, the methylation levels of a given cell type can vary among samples. It is thus problematic to assume that all of the samples have the same set of reference profiles [10] , [14] . Furthermore, high-quality references are difficult to obtain for most EWAS due to the existence of unknown cell types, the high cost of cell sorting, and confounding effects [14] . Consequently, a large amount of recent EWAS literature was devoted to identification of risk-CpG sites without the need for the reference methylation profiles. The reference-free methods can generally be further divided into two classes according to whether they estimate the cell-type mixing proportions directly. The direct-decomposition-based procedures consist of two stages. In the first stage, they simultaneously estimate the cellular compositions of each sample and the cell-type-specific reference methylomes via quadratic programming [16] ; and in the second stage, they treat the estimated cell-type proportions as covariates with additive effects in the linear models to conduct association tests. However, when estimating cellular compositions during the first stage, the direct-decomposition-based methods also do not consider samples’ phenotype information, thus suffering from the same problem of biasing the cellular composition estimates as the reference-based approaches [9] . Moreover, similar to tumor purity [17] , we argue that the estimated cellular composition has a multiplicative rather than an additive effect on the observed methylation level (Methods). The second class of methods, which is exemplified by SVA [18] , RefFreeEWAS [19] , and ReFACTor [13] , does not carry out cell-type decompositions. They resort to singular value decomposition, which includes the principal component analysis, to construct surrogates for the underlying cell-type composition. EWASher, a linear mixed model, also belongs to this class because it is equivalent to the use of principal components as fixed-effect covariates [11] . However, the use of principal components as the covariates in the regression undergoes the same issue of additive effects as the direct-decomposition-based methods. Therefore, the existing reference-free methods have low power in detecting risk-CpG sites [12] . Although the existing methods aim to address the cellular heterogeneity problem in EWAS and claim whether a CpG site is associated with phenotypes at the aggregate level, none of them can identify the risk-CpG sites for each individual cell type, thus missing the opportunity to obtain finer-grained results in EWAS. Here, we propose a method, HIRE, to identify the association in EWAS at a HIgh REsolution: detecting whether a CpG site has any associations with the phenotypes in each cell type (Methods). The keys to HIRE’s success are twofold. First, HIRE links the underlying cell-type-specific methylation profiles for each sample to the sample’s phenotypes, thus avoiding the bias in estimating the cellular composition by the reference-based and direct-decomposition-based methods. Second, HIRE correctly characterizes the cellular compositions as the multiplicative effects, whereas the existing methods inappropriately treat the cell proportions as additive effects (Methods). HIRE is applicable to EWAS with binary phenotypes, continuous phenotypes, or both. By helping researchers understand in which cell types the CpG sites are affected by a disease, HIRE can ultimately facilitate the development of epigenetic therapies by targeting the specifically affected cell types.","references":{"indices":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],"infos":[{"idx":"1","title":"Rakyan, V. K., Down, T. A., Balding, D. J. & Beck, S. Epigenome-wide association studies for common human diseases.Nat. Rev. Genet.12, 529–541 (2011).","link":"https://doi.org/10.1038%2Fnrg3000"},{"idx":"2","title":"Verma, M. Epigenome-wide association studies (EWAS) in cancer.Curr. Genom.13, 308–313 (2012).","link":"https://doi.org/10.2174%2F138920212800793294"},{"idx":"3","title":"Liu, Y. et al. Epigenome-wide association data implicate DNA methylation as an intermediary of genetic risk in rheumatoid arthritis.Nat. Biotechnol.31, 142–147 (2013).","link":"https://doi.org/10.1038%2Fnbt.2487"},{"idx":"4","title":"Gao, X., Jia, M., Zhang, Y., Breitling, L. P. & Brenner, H. DNA methylation changes of whole blood cells in response to active smoking exposure in adults: a systematic review of DNA methylation studies.Clin. Epigenetics7, 113 (2015).","link":"https://link.springer.com/doi/10.1186/s13148-015-0148-3"},{"idx":"5","title":"Joehanes, R. et al. Epigenetic signatures of cigarette smoking.Circ. Cardiovasc. Genet.9, 436–447 (2016).","link":"https://doi.org/10.1161%2FCIRCGENETICS.116.001506"},{"idx":"6","title":"Wahl, S. et al. Epigenome-wide association study of body mass index, and the adverse outcomes of adiposity.Nature541, 81–86 (2017).","link":"https://doi.org/10.1038%2Fnature20784"},{"idx":"7","title":"Teschendorff, A. E. et al. Age-dependent DNA methylation of genes that are suppressed in stem cells is a hallmark of cancer.Genome Res.20, 440–446 (2010).","link":"https://doi.org/10.1101%2Fgr.103606.109"},{"idx":"8","title":"Horvath, S. DNA methylation age of human tissues and cell types.Genome Biol.14, 3156 (2013).","link":"https://link.springer.com/doi/10.1186/gb-2013-14-10-r115"},{"idx":"9","title":"Houseman, E. A. et al. DNA methylation arrays as surrogate measures of cell mixture distribution.BMC Bioinformatics13, 86 (2012).","link":"https://link.springer.com/doi/10.1186/1471-2105-13-86"},{"idx":"10","title":"Jaffe, A. E. & Irizarry, R. A. Accounting for cellular heterogeneity is critical in epigenome-wide association studies.Genome Biol.15, R31 (2014).","link":"https://link.springer.com/doi/10.1186/gb-2014-15-2-r31"},{"idx":"11","title":"Zou, J., Lippert, C., Heckerman, D., Aryee, M. & Listgarten, J. Epigenome-wide association studies without the need for cell-type composition.Nat. Methods11, 309–311 (2014).","link":"https://doi.org/10.1038%2Fnmeth.2815"},{"idx":"12","title":"McGregor, K. et al. An evaluation of methods correcting for cell-type heterogeneity in DNA methylation studies.Genome Biol.17, 84 (2016).","link":"https://link.springer.com/doi/10.1186/s13059-016-0935-y"},{"idx":"13","title":"Rahmani, E. et al. Sparse PCA corrects for cell type heterogeneity in epigenome-wide association studies.Nat. Methods13, 443–445 (2016).","link":"https://doi.org/10.1038%2Fnmeth.3809"},{"idx":"14","title":"Teschendorff, A. E. & Relton, C. L. Statistical and integrative system-level analysis of DNA methylation data.Nat. Rev. Genet.19, 129–147 (2017).","link":"https://doi.org/10.1038%2Fnrg.2017.86"},{"idx":"15","title":"Accomando, W. P., Wiencke, J. K., Houseman, E. A., Nelson, H. H. & Kelsey, K. T. Quantitative reconstruction of leukocyte subsets using DNA methylation.Genome Biol.15, R50 (2014).","link":"https://link.springer.com/doi/10.1186/gb-2014-15-3-r50"},{"idx":"16","title":"Houseman, E. A. et al. Reference-free deconvolution of DNA methylation data and mediation by cell composition effects.BMC Bioinformatics17, 259 (2016).","link":"https://link.springer.com/doi/10.1186/s12859-016-1140-4"},{"idx":"17","title":"Zheng, X., Zhang, N., Wu, H.-J. & Wu, H. Estimating and accounting for tumor purity in the analysis of DNA methylation data from cancer studies.Genome Biol.18, 17 (2017).","link":"https://link.springer.com/doi/10.1186/s13059-016-1143-5"},{"idx":"18","title":"Leek, J. T. & Storey, J. D. Capturing heterogeneity in gene expression studies by surrogate variable analysis.PLoS Genet.3, e161 (2007).","link":"https://doi.org/10.1371%2Fjournal.pgen.0030161"},{"idx":"19","title":"Houseman, E. A., Molitor, J. & Marsit, C. J. Reference-free cell mixture adjustments in analysis of DNA methylation data.Bioinformatics30, 1431–1439 (2014).","link":"https://doi.org/10.1093%2Fbioinformatics%2Fbtu029"}]}},{"section":"Results","content":"Method overview HIRE is a hierarchical model that closely follows the data generation process. Its elaborate modeling depicts how phenotypes affect the methylation levels of each sample. Here, we briefly introduce the method. The technical details are provided in the Methods section and the Supplementary Methods . Let us first review the cornerstone in most EWAS approaches. These methods model the observed methylation levels of the m CpG sites for sample i , O i = ( O 1 i , O 2 i , …, O mi ) T , as the weighted average of the methylation profiles of K cell types, u i = ( u i 1 , u i 2 , …, u iK ). The weights are the cellular compositions p i = ( p 1 i , p 2 i , …, p Ki ) T of sample i (see the top panel of Fig. 1a ). However, regardless of whether the reference is known a priori or not, the existing methods assume that the cell-type-specific methylation profiles u i s remain the same for all samples: u i = M , for i = 1, …, n . Unfortunately, because the methylation levels can actually change with covariates such as age and disease status, ignoring the covariates’ effects and enforcing static reference methylomes can bias the estimation of p i and thus affect all downstream analyses [14] . More importantly, the assumption that cell-type-specific methylation profiles are the same for each sample prevents the detection of cell-type-specific risk-CpG sites. Fig. 1 A simple cartoon illustration of the HIRE model with three cell types ( K = 3) and two phenotypes (disease status and age; q = 2). a Data generation procedure for the observed methylation vector O i for sample i ( i = 1, …, n ). In the top panel, O i is the convolution of cell-type-specific methylation profiles u i with cellular compositions p i . Both u i and p i depend on the attributes of sample i . The bottom panel describes how sample i ’s phenotypes affect u i via two phenotype-effect matrices B 1 and B 2 . In B 1 and B 2 , the white square represents zero, which indicates that the phenotype exerts no influence on the corresponding methylation level in u i . b Inputs and outputs of HIRE. We input the observed methylation matrix O , the phenotype data matrix X , and a predetermined cell type number K into HIRE, and HIRE outputs the estimates for the cellular compositions \\(\\widehat {\\mathbf{p}}\\) , the baseline methylation profiles \\(\\widehat {\\boldsymbol{\\mu }}\\) , the phenotype effects \\(\\widehat {\\mathbf{B}}_\\ell \\) , and the penalized BIC value. In addition, HIRE tests whether there is any association between CpG site j and phenotype \\(\\ell \\) in cell type k — \\(H_0:\\beta _{jk\\ell } = 0\\) vs \\(H_1:\\beta _{jk\\ell } \\ \\ne\\ 0\\) —and provides the p-values Full size image For association detection at the aggregate level, after estimation of p i using the deconvolution-based approach or its surrogates from principal component-based methods, the existing methods examine a linear model in which the phenotypes \\({\\mathbf{x}}_i = (x_{i1}, \\ldots ,x_{i\\ell }, \\ldots ,x_{iq})^T\\) and the cellular proportions p i exert additive effects on the methylation level O i : (1) A CpG-site j is then associated with phenotype \\(\\ell \\) if we reject the null hypothesis that the covariate coefficient \\(T_{j\\ell }\\) equals zero. In contrast, HIRE further models the effect of each phenotype on each cell type as shown in the bottom panel of Fig. 1a . In cell type k , sample i ’s cell-type-specific methylation profile, u ik , is the summation of the corresponding baseline cell-type-specific methylation levels, μ k , and the phenotype effects \\({\\mathbf{B}}_{k\\ell }x_{i\\ell }\\) on sample i from all the l = 1, …, q phenotypes: \\({\\mathbf{u}}_{ik} = {\\boldsymbol{\\mu }}_k + \\mathop {\\sum}_{l = 1}^q {{\\mathbf{B}}_{k\\ell }} x_{i\\ell }\\) , where \\(x_{i\\ell }\\) is the phenotype \\(\\ell \\) of sample i and \\({\\mathbf{B}}_{k\\ell } = (\\beta _{1k\\ell }, \\ldots ,\\beta _{mk\\ell })^T\\) —the k th column of \\({\\mathbf{B}}_\\ell \\) —reflects the association of phenotype \\(\\ell \\) with each of the m CpG sites in cell type k . Thus, by collecting the baseline cell-type-specific methylation profiles to μ = ( μ 1 , …, μ k ) and denoting the m by K phenotype coefficient matrix \\((\\beta _{jk\\ell }:1 \\le j \\le m,1 \\le k \\le K)\\) by \\({\\mathbf{B}}_\\ell \\) , we now have: (2) A comparison of \\({\\mathrm{x}}_i\\) in Eq. ( 1 ) and \\({x_{i \\ell }}{\\boldsymbol{p}}_i ,{\\ell} = 1,…,q\\) in Eq. ( 2 ) reveals that via the two-layer hierarchical model HIRE correctly captures the multiplicative effects of the cellular compositions on the phenotype effects (see also Methods and the Supplementary Methods ). As a result, HIRE achieves greater statistical power for association detection at the aggregate level and enables the fine-scale resolutions that were previously infeasible. We mathematically prove that the HIRE model is identifiable under mild conditions that are easily met in reality (see Theorem 1 and its proof in Methods). Figure 1b summarizes the inputs and outputs of HIRE. Given the methylation measurements at the aggregate level of n samples, HIRE can estimate all parameters of interests — p i ( i = 1, …, n ), μ , and \\({\\mathbf{B}}_\\ell \\) ( \\(\\ell = 1, \\ldots ,q\\) ). HIRE then determines whether any association exists between CpG site j and phenotype \\(\\ell \\) in each individual cell type by testing the hypotheses \\(H_0:\\beta _{jk\\ell } = 0\\) versus \\(H_1:\\beta _{jk\\ell } \\ \\ne\\ 0\\) . When the null hypothesis \\(H_0:\\beta _{jk\\ell } = 0\\) is rejected, HIRE calls CpG site j as a risk-CpG site for phenotype \\(\\ell \\) in cell type k . The detection of cell-type-specific risk-CpG sites cannot be performed with any of the existing state-of-the-art methods. Moreover, HIRE allows users to prespecify the number of cell types K . When K is unknown, HIRE selects the number of cell types according to the penalized Bayesian information criterion (pBIC) [20] ( Supplementary Methods ). Simulation As the definition of the gold standard for real data is debatable [21] , [22] , we designed extensive simulation studies to evaluate the performance of HIRE and compared it with commonly used methods—unadjusted analysis, SVA, RefFreeEWAS, EWASHer, and ReFACTor (Methods). We generated datasets in which the observed methylation was a mixture of several cell types and each sample was accompanied with a diseased or normal status and a continuous age attribute. We deliberately designed some cell types to have similar baseline methylation profiles to mimic cell types from the same cell lineage. We set the sample size n to 180, 300, and 600 and let the underlying cell type number K be 3, 5, and 7. For each pair of ( n , K ), we investigated two scenarios in which (1) all phenotype effects \\(\\beta _{jk\\ell }\\) s are zero—the true null case—to compare the ability of each method to control false positives; and (2) a small portion of \\(\\beta _{jk\\ell }\\) s are non-zero—the true alternative case—to study each method’s power to detect risk-CpG sites. Under the true alternative, both the binary and the continuous phenotypes were assumed to have cell-type-specific risk-CpG sites and to affect the cell-type proportions among the samples [10] . We further simulated phenotype effects with various directions and magnitudes. Under the true null, HIRE, EWASHer, and ReFACTor control the false positive rates (FPRs) very well: none are greater than 0.05% (Table 1 and Supplementary Figs. 1 – 9 ). In comparison, RefFreeEWAS often has FPRs greater than 0.1% and thus does not perform as well as HIRE, and the unadjusted analysis and SVA further suffer from the dramatic inflation of false positives. For the true alternative settings, given that the FPRs are well-controlled, with FPRs below 0.05%, HIRE achieves the highest true positive rates (TPR) of all methods in every simulation setting (see also Fig. 2a and Supplementary Figs. 10 – 17 ). As expected, as the sample size increases, HIRE’s power increases. For example, when the data include five cell types, HIRE can identify 89.6% of the risk-CpG sites with 300 samples, and HIRE can detect almost all risk-CpG sites when the sample size reaches 600, which is a typical sample size for EWAS. Although EWASHer and ReFACTor have low FPRs, they miss a large proportion of risk-CpG sites. EWASHer’s maximum TPR is only 35.33%, and ReFACTor’s maximum TPR is slightly over 60%. However, in those cases, HIRE’s power is greater than 95%. Consistent with the true null scenario, in the true alternative, RefFreeEWAS has inflated FPRs compared to HIRE, and the unadjusted analysis and SVA always have huge false positives. Therefore, HIRE substantially improves the power of association detection at the aggregate level compared with existing methods. Table 1 Performance of HIRE and other competing methods in simulation studies Full size table Fig. 2 Association detection performance of HIRE and commonly used methods in the true alternative setting with K = 3 and n = 180. Source data are provided as a Source Data file. In all figures, red corresponds to HIRE; yellow indicates the unadjusted analysis; brown represents SVA; purple refers to RefFreeEWAS; dark blue indicates EWASher; and light blue corresponds to ReFACTor. a ROC curves of HIRE and commonly used methods. HIRE has the largest area under the curve among all of the methods. b True cell-type-specific association pattern with disease status for 10,000 simulated CpG sites; columns correspond to cell types, and the rows represent the CpG sites. Dark cells correspond to risk-CpG sites, and grey cells are CpG sites not associated with the disease status. c Detected cell-type-specific association pattern with disease status by HIRE. Darkness represents \\(-{\\mathrm{log}}_{10}(p - {\\mathrm{value}})\\) d – i The p -value density plots for association with disease status in the simulation dataset for d HIRE, e unadjusted analysis, f SVA, g RefFreeEWAS, h EWASHer, and i ReFACTor. j – o The Q-Q plots for association with disease status for j HIRE, k unadjusted analysis, l SVA, m RefFreeEWAS, n EWASHer, and o ReFACTor Full size image In the multiple hypothesis testing, the p-values from the truly null features should follow a uniform distribution on (0, 1), whereas those for the truly alternative features are concentrated near zero [23] . Both the histograms (Fig. 2d–i ) and the Q-Q plots (Fig. 2j–o ) show that the p-value distribution of HIRE is the best fit to the underlying truth—there are only a small proportion of signals, followed by RefFreeEWAS and ReFACTor. EWASHer easily overcorrects signals with its p-value density having a dip near zero (Fig. 2h ), thus failing to detect the true associations. In contrast, the unadjusted analysis and SVA generate very small p-values clustered near zero, resulting in inflated type I errors. In addition to the traditional association detection at the aggregate level, HIRE can identify the association for each CpG site with the phenotypes under each cell type. Table 2 shows the FPR and TPR of HIRE for each cell type in various simulation settings. Such fine analysis is not possible with the other methods. Consistent with association detection at the aggregate level, HIRE always controls the FPR well. When K = 3 and n = 180, HIRE accurately detects the risk-CpG sites associated with disease status a TPR of greater than 83% and an FPR of 0.01% or less in all three cell types. Similarly, most of the CpG sites affected by age are also correctly identified in each cell type. HIRE’s learned cell-type-specific association patterns closely matches the underlying true associations (see Fig. 2b, c and Supplementary Figs. 18 – 26 ). Once again, HIRE’s power decreases with the number of cell types and increases with the sample size. When the samples consist of seven cell types and the proportion of the least abundant cell type is as low as 4.2%, given a typical current EWAS with around 600 samples, HIRE can detect most cell-type-specific risk-CpG sites reasonably well. Moreover, HIRE’s estimates for the baseline methylation profiles, cellular compositions, and phenotype effects have little bias (Supplementary Figs. 27 – 62 ); therefore, HIRE can provide accurate estimates and is powerful in detecting cell-type-specific risk-CpG sites. Table 2 Performance of HIRE in detecting cell-type-specific risk-CpG sites Full size table In the HIRE model, we assume that different CpG sites are independent, and we investigate the performance of HIRE when such a model assumption is violated and dependences exist among nearby CpG sites. Specifically, we assume that every 50 consecutive CpG sites belongs to a block. For CpG sites within the same block, their random noises \\({\\boldsymbol{\\epsilon }}\\) follow a multivariate normal distribution with mean zero and 50 × 50 covariance matrix Σ, and Σ’s corresponding correlation matrix has its ( i , j ) entry equal to ρ | i − j | . We vary ρ to 0.8, 0.6, and 0.4. A comparison of Supplementary Tables 1 – 3 with Supplementary Table 4 shows that even when strong correlations exit among nearby CpG sites, HIRE still provides good performances in controlling the FPR and detecting the risk-CpG sites under the model misspecification setting. To further evaluate HIRE’s performance on experimentally mixed samples, we conducted another semi-simulated dataset that includes six samples mixed with six purified cell types in predetermined proportions [24] . Once again, HIRE successfully recovers the six underlying reference cell types and estimates the cellular compositions well (see Methods). Real data analysis HIRE also provides greater insight into real data than previous studies. The rheumatoid arthritis (RA) dataset [3] contains methylation profiles collected from the whole blood of 354 patients with RA and 335 normal participants. In addition to the RA status, other attributes such as gender, smoking history, age, and batch information are available. We first corrected the batch effects and then applied HIRE to the dataset (Methods). Figure 3a displays the p-values regarding the association with the RA status for each CpG site in each cell type, in which HIRE selected six cell types (Supplementary Fig. 63a ), consistent with the number of cell types in the previous study [13] . Despite potential batch effects and biological variability, three of the six cell types can be matched to known blood cell references—cell type 1 was matched to CD4+ T cells, cell types 2 and 4 were matched to neutrophils, and the remaining three cell types cannot be aligned to the references (Methods and Supplementary Fig. 64 ). HIRE detected 63 risk-CpG sites in cell type 3—the largest number of associations across all cell types—but no risk-CpG sites in cell type 1 (Supplementary Table 5 ). Therefore, the disease status affected some but not necessarily all cell types. Note that the significant CpG site cg06373940 called by HIRE is located on gene ERCC3 . The level of ERCC3 ’s corresponding protein has been reported to increase in RA synovium [25] . Moreover, we found that five CpG sites had a significant association with smoking history (Supplementary Fig. 65 and Supplementary Table 6 ). One of them is cg05575921, which was recently linked to smoking in two other independent studies of blood samples [26] , [27] . However, these findings were missed by the association detection at the aggregate level in previous analyses of the same dataset [11] , [13] . The p-value density plots and Q-Q plots for the commonly used methods are also displayed in Fig. 3c–n ; they present patterns similar to those observed in the simulation study except for an obvious overcorrection by ReFACTor. Fig. 3 Application of HIRE and commonly used methods to two real methylation datasets: RA and GALA II. Source data are provided as a Source Data file. a Cell-type-specic association pattern with RA status detected by HIRE in the RA dataset. Darkness represents the −log10(p−value). b Cell-type-specic association pattern with gender detected by HIRE in the GALA II dataset. The darkness represents the −log10(p−value). c – h The p-value density plots for association with RA status in the RA dataset for c HIRE, d unadjusted analysis, e SVA, f RefFreeEWAS, g EWASHer, and h ReFACTor. i - n Q-Q plots for association with RA status in the RA dataset for i HIRE, j unadjusted analysis, k SVA, l RefFreeEWAS, m EWASHer, and n ReFACTor Full size image The high resolution provided by HIRE makes it a powerful tool for EWAS studies. Rahmani et al. used ReFACTor [13] to analyze the GALA II blood methylation dataset [28] , which consists of 573 samples collected from a pediatric Latino population. Each sample includes the gender information and belongs to one of the following four populations: Mexican, Mixed Latino, Puerto Rican, and Other Latino. We applied HIRE to the dataset to investigate whether any cell-type-specific CpG sites were associated with gender and ethnicity. We created three dummy variables to represent the four ethnic groups. By taking the indicators of ethnicity as phenotypes in the model, HIRE automatically and simultaneously accounts for the population differences in cell composition and cell-type-specific methylation levels. HIRE correctly selected the number of cell types as six as reported in the previous study [13] (Supplementary Fig. 63b ). According to cell-type alignment, cell types 1 and 5 can be annotated as CD4+ T cells; cell types 2, 3, and 4 belong to neutrophils; and cell type 6 was annotated as CD56+ natural killer cell (CD56+ NK) using the references (Supplementary Fig. 66 ). HIRE found that 1936 CpG sites were associated with ethnicity across all cell types (Supplementary Fig. 67 ) and identified 14, 52, 155, 15, 18, and 14 risk-CpG sites for gender in cell types 1–6, respectively (Fig. 3b ). Gene set enrichment analysis showed that the genes that harbored risk-CpG sites for gender were significantly enriched in seven canonical pathways (Supplementary Table 7 ), of which the PID_CMYB_PATHWAY was ranked the highest. The transcription factor c − MYB in the PID_CMYB_PATHWAY enhances the progression of breast cancer [29] ; therefore, the different occurrence rates of breast cancers in men and women may be linked to the differences at the epigenome level. In comparison, only one pathway was found to be enriched with the genes that host the risk-CpG sites claimed by ReFACTor at the aggregate level (Supplementary Table 8 ). All of these observations highlight the importance of the finer-scale resolutions of HIRE."},{"section":"Discussion","content":"In reality, the phenotype may affect a risk-CpG site in some but not all of the cell types. HIRE can detect the cell-type-specific association pattern with each phenotype for EWAS. The identification of cell-type-specific risk-CpG sites will help epigenetic therapies to target the affected cell types in a more effective manner. Statistically, instead of assuming fixed reference methylomes for all samples as the existing methods do [9] , [13] , [16] , HIRE allows each sample’s cell-type-specific methylation profiles to depend on its phenotypes. As a result, HIRE correctly models the multiplicative effects of the cellular compositions on the observed methylation levels, whereas the existing approaches all misspecify the cellular compositions as additive effects (Methods). As a result, HIRE enables the detection of cell-type-specific risk-CpG sites that cannot be feasibly detected with existing state-of-the-art methods. As a byproduct, HIRE also improves the statistical power of association detection at the aggregate level relative to existing state-of-the-art methods. Computationally, the time complexity of one iteration by HIRE is O ( nmKp + nK 3 ), which thus provides fast convergence when K is moderate. The statistical and computational advantages equip HIRE to be scaled up for large-cohort EWAS. So far, in the EWAS community, no gold-standard exists for the comparison of various methods. Ideally, we would like to have epigenetic spike-in experiments in which purified cell types are isolated, CpG-sites are epigenetically edited on a per-cell-type basis, and cell types are finally mixed in predetermined proportions. Given such experiments, the underlying knowledge of which CpGs are differentially methylated in each cell type and the cell mixing proportions for each sample are known. However, biotechnologies for epigenetic editing, such as CRISPR-Cas, are still not mature at this stage, with many off-target modifications [30] . Therefore, most computational EWAS studies refer to numerical simulation studies rather than to experimental studies when evaluating the performance of their algorithms [12] , [13] . Here, we follow the example of previous comparative studies and design our simulation studies to serve as the computational counterpart of experimental spike-in studies. With the rapid advances in epigenetic editing, we hope the community can devote greater effort in the near future to the creation of a gold-standard dataset, such as those generated in the early years for gene expression microarray studies [31] . The beta-values that represent methylation levels always lie between zero and one. As previous approaches to EWAS often assume normal distribution for the beta-values and show good performances in real applications [9] , [13] , in HIRE, we also assume that the beta-values follow a normal distribution. Consequently, the fitted methylation level may lie outside the range of [0, 1]. Nevertheless, we do in fact constrain the baseline methylation profiles μ jk s to the closed interval [0, 1] and force the cellular compositions p ki s to be non-negative and to add up to one: \\(\\mathop {\\sum}_{k = 1}^K {p_{ki}} = 1\\) . As a result, because the phenotypes have no effect on most CpG sites, most observations, O ji s, have their means \\(\\mathop {\\sum}_{k = 1}^K {\\mu _{jk}} p_{ki}\\) s in [0, 1]. In fact, for both the RA dataset and the GALA II dataset, more than 99.99% of the fitted methylation values \\(\\hat O_{ji}\\) s based on HIRE estimates lie between zero and one. Therefore, the normal assumption fits the data reasonably well and does not have a large effect on the performance of HIRE. One major issue for all of the cell-type deconvolution methods is that deconvolution cannot be achieved if the cellular compositions do not vary among samples. For example, assuming that the samples are mixtures of two cell types and p i = p for all of the samples, then the observed methylation profile O i equals u i 1 p 1 + u i 2 p 2 = ( u i 1 + p 2 C ) p 1 + ( u i 2 − p 1 C ) p 2 := \\(\\widetilde {\\mathbf{u}}_{i1}p_1 + \\widetilde {\\mathbf{u}}_{i2}p_2\\) for any constant C . As a result, u i 1 and u i 2 are not estimable. In our paper, we show mathematically that HIRE is identifiable under mild conditions in Theorem 1 and that condition (b) of Theorem 1 formulates the requirement for the variability of the cellular compositions (Methods). HIRE can accurately estimate cellular compositions of tissues with great cellular heterogeneity, such as blood. Although the mild conditions in Theorem 1 are easily met for real DNA methylation data, identification of both sufficient and necessary conditions for model identifiability is a theoretically interesting and challenging statistical problem that we will investigate in a future study. HIRE requires a moderate sample size to obtain precise estimates because HIRE needs to learn (1 + 2 K + qK ) m + ( K − 1) n parameters with a total of mn observed values. Our simulation studies show that HIRE performs very well at the aggregate level with 180 samples (Table 1 ). If the sample size drops below 150, say to 120, HIRE can still control the FPR well but begins to lose power (Supplementary Table 9 ). For small sample sizes, we have also developed a special case of HIRE by reparameterizing all \\(\\sigma _{gk}^2\\) s as one single parameter σ 2 , and we found that such a variance-stabilized approach can achieve even better inflation control (see Supplementary Figs. 71 – 76 ) and power comparable to HIRE (see Supplementary Table 10 ). Like the two datasets analyzed in the real application, a typical sample size for a current EWAS exceeds 500, thus guaranteeing a high TPR for HIRE. Given the decreasing cost of EWAS, we recommend that researchers collect at least 200 samples for their studies for association detection at the aggregate level and 600 samples for identification of cell-type-specific risk-CpG sites. A larger sample size can further boost the power. With the popularity of EWAS, we believe that HIRE will be widely applied, and we hope that HIRE can motivate more researchers to mine out finer-scale results from EWAS."},{"section":"Methods","content":"Multiplicative effects of cellular composition on methylation In this section, we illustrate that the effects of the cell-type composition are actually multiplicative. Let us assume that the beta-values that represent the methylation levels are observed across m CpG sites for n samples. As the measured sample comprises cells of various types, the observed beta-value is a weighted average of the mean methylation levels of distinct cell types, and the weights correspond to the proportions of each cell type. Let O ji denote the measurement at CpG site j for sample i . If we assume that there exist K cell types in all samples and that the mean methylation level for CpG site j in cell type k is μ jk , then $$O_{ji} = \\mathop {\\sum}\\limits_{k = 1}^K {\\mu _{jk}} p_{ki} + \\epsilon _{ji},$$ where p ki is the proportion of cell type k in sample i with a natural constraint \\(\\mathop {\\sum}_{k = 1}^K {p_{ki}} = 1\\) , and \\(\\epsilon _{ji}\\) is a random error. Let us consider a case-control EWAS. Without loss of generality, we assume that CpG site j is differentially methylated between cases and controls in cell type 1 with a mean shift δ j 1 and that it is not differentially methylated in the remaining cell types. As a result, for case samples, $$O_{ji} = (\\mu _{j1} + \\delta _{j1})p_{1i} + \\mathop {\\sum}\\limits_{k = 2}^K {\\mu _{jk}} p_{ki} + \\epsilon _{ji} = \\delta _{j1}p_{1i} + \\mathop {\\sum}\\limits_{k = 1}^K {\\mu _{jk}} p_{ki} + \\epsilon _{ji}.$$ If we then use Z i to indicate the case-control status of sample i , the observed methylation level becomes $$O_{ji} = \\delta _{j1}p_{1i}Z_i + \\mathop {\\sum}\\limits_{k = 1}^K {\\mu _{jk}} p_{ki} + \\epsilon _{ji}.$$ (3) Therefore, the proportions of cell type 1— p 1 i , i = 1, …, n —have multiplicative effects rather than additive effects on the mean difference between the case and control samples. The existing methods, which either estimate the cell type proportions explicitly or approximate them implicitly with surrogate variables, add the estimated proportions and the case-control indicator Z i as the covariates to the regression as follows: $$O_{ji} = \\alpha _j + \\tau _jZ_i + \\mathop {\\sum}\\limits_{k = 1}^{K - 1} {b_{jk}} \\hat p_{ki} + \\epsilon _{ji},$$ (4) where b jk s are the regression coefficients. As a result, CpG site j is called differentially methylated on the basis of hypothesis testing for τ j = 0. In general, τ j in Eq. ( 4 ) is not equal to δ j 1 in Eq. ( 3 ). Please see the Supplementary Notes for a numerical example. Moreover, testing for τ j = 0 loses the information regarding cell type in which CpG site j may be at risk. To account for the multiplicative effects, we propose the HIRE model that conserves the individual cell-type level information, which is introduced in the next section. The HIRE model HIRE uses a hierarchical model to closely follow the data generation process for the EWAS data. To begin, we assume that the baseline methylation level for CpG site j in cell type k is μ jk . For sample i with phenotypes x i = ( x i 1 , …, x iq ), the mean methylation value for CpG site j in cell type k is assumed to be \\(\\mu _{jk} + \\mathop {\\sum}_{\\ell = 1}^q {\\beta _{jk\\ell }} x_{i\\ell }\\) . In other words, the phenotypes have linear effects where \\(\\beta _{jk\\ell }\\) characterizes the influence of phenotype \\(\\ell \\) on CpG site j in cell type k . Let u ijk represent the signal from CpG site j in cell type k for sample i with x i . We assume that u ijk follows a normal distribution with mean \\(\\mu _{jk} + \\mathop {\\sum}_{\\ell = 1}^q {\\beta _{jk\\ell }} x_{i\\ell }\\) and standard deviation σ jk , $$u_{ijk}\\sim N \\left (\\mu _{jk} + \\mathop {\\sum}\\limits_{\\ell = 1}^q {\\beta _{jk\\ell }} x_{i\\ell },\\sigma _{jk}^2\\right).$$ (5) After u ijk s are generated for all of the K cell types, the observed methylation value O ji is sampled as follows: $$O_{ji}\\sim N\\left(\\mathop {\\sum}\\limits_{k = 1}^K {u_{ijk}} p_{ki},\\sigma _{\\epsilon j}^2\\right).$$ (6) Collectively, O = { O ji : 1 ≤ j ≤ m , 1 ≤ i ≤ n } denote the observed data; u = {( u ij 1 , …, u ijK ) T : 1 ≤ i ≤ n , 1 ≤ j ≤ m } are the missing data; and μ j = ( μ j 1 , …, μ jK ) T , \\({\\mathbf{B}}^{(j)} = (\\beta _{jk\\ell })_{K \\times q}\\) , \\(\\sigma _{\\epsilon j}^2\\) , the diagonal matrix \\(\\Sigma _j = diag(\\sigma _{j1}^2, \\ldots ,\\sigma _{jK}^2)\\) for j = 1, …, m , and p i = ( p 1 i , …, p Ki ) T for i = 1, …, n are the parameters. With \\({\\mathbf{\\Theta }} = \\{ {\\mathbf{p}}_i,{\\boldsymbol{\\mu }}_j,{\\mathbf{B}}^{(j)},\\Sigma _j,\\sigma _{\\epsilon ,j}^2:1 \\le j \\le m,1 \\le i \\le n\\} \\) , the complete data log-likelihood function, l c , can be expressed as follows: $$\\begin{array}{l}l_c({\\mathbf{\\Theta }}|{\\mathbf{O}},{\\mathbf{u}}) = \\mathop {\\sum}\\limits_{i = 1}^n {\\mathop {\\sum}\\limits_{j = 1}^m {\\left\\{ { - \\frac{1}{2}{\\mathrm{log}}\\sigma _{\\epsilon ,j}^2 - \\frac{{(O_{ji} - {\\mathbf{u}}_{ij}^T{\\mathbf{p}}_i)^2}}{{2\\sigma _{\\epsilon ,j}^2}}} \\right.} } - \\frac{1}{2}\\mathop {\\sum}\\limits_{k = 1}^K {\\mathrm{log}}\\sigma _{jk}^2\\\\ \\left. { - \\frac{1}{2}({\\mathbf{u}}_{ij} - {\\boldsymbol{\\mu }}_j - {\\mathbf{B}}^{(j)}{\\mathbf{x}}_i)^T\\Sigma _j^{ - 1}({\\mathbf{u}}_{ij} - {\\boldsymbol{\\mu }}_j - {\\mathbf{B}}^{(j)}{\\mathbf{x}}_i)} \\right\\} + Constant.\\end{array}$$ Accordingly, we develop a generalized expectation-maximization algorithm [32] to estimate the parameters. In the expectation-maximization algorithm, a good initialization can lead to faster convergence than random starts. We adopt the cellular composition estimations from the methylation matrix decomposition algorithm [16] with slight modifications as the initializations. The initial values for the baseline methylation profiles μ jk are accordingly estimated by simple linear regressions. As the number of risk-CpG sites is often small, all of the phenotype effects \\(\\beta _{jk\\ell }\\) are set to zero at the beginning. For the standard deviations, the initial values are randomly sampled from inverse gamma distributions with small means. We choose the number of cell types K by using a variant of the penalized Bayesian information criterion (pBIC) [20] (see details in Supplementary Methods ). For each phenotype \\(\\ell \\) , we can conduct the hypothesis test \\(H_0:\\beta _{jk\\ell } = 0\\) versus \\(H_1:\\beta _{jk\\ell } \\ \\ne \\ 0\\) for any cell type k and any CpG site j . Combining Eqs. ( 5 ) and ( 6 ), we obtain the following equations: $$E\\left[ {O_{ji}} \\right] = \\mu _{j1} + \\mathop {\\sum}\\limits_{k = 2}^K {(\\mu _{jk} - \\mu _{j1})} p_{ki} + \\mathop {\\sum}\\limits_{k = 1}^K {\\mathop {\\sum}\\limits_{\\ell = 1}^q {\\beta _{jk\\ell }} } x_{i\\ell }p_{ki},\\:\\:\\:i = 1, \\ldots ,n.$$ (7) We can then take ( O j 1 , …, O jn ) as the response vector and concatenate 1 n , ( p k 1 , …, p kn ) ( k = 2, …, K ) and \\((x_{1\\ell }p_{k1}, \\ldots ,x_{n\\ell }p_{kn})\\) \\((\\ell = 1, \\ldots ,q;k = 1, \\ldots ,K)\\) to a n × ( p + 1) · K design matrix in the linear regression. We plug in the estimated cellular compositions \\(\\hat p_{ki\\ell }\\) and conduct the hypothesis test for \\(\\beta _{jk\\ell } = 0\\) using the two-sided t-tests in the linear models. We claim that CpG site j has an association with phenotype \\(\\ell \\) at the aggregate level if phenotype \\(\\ell \\) affects CpG site j in at least one of the K cell types. Note that in the regression we incorporate the estimated cellular compositions into the linear model as multiplicative effects rather than additive effects. More technical details of the method and the algorithm are available in the Supplementary Methods . Data simulation We compared the performance of HIRE with five previous methods—unadjusted analysis, SVA, RefFreeEWAS, EWASHer, and ReFACTor—in 18 simulation settings. We set the sample size n to 180, 300, and 600 and let the underlying cell type number K be 3, 5, and 7. For each pair of ( n , K ), we investigated the true null case and the true alternative case. As a result, we have in total 3 (the number of sample sizes) × 3 (the number of cell types) × 2 (the true null case and the true alternative case) = 18 simulation settings. For each setting, we considered 10,000 CpG sites and simultaneously accounted for the following factors. Cell lineage. We first constructed the baseline methylation matrix μ = ( μ jk ) m × K , in which each column corresponds to the baseline methylation levels of a cell type. To mimic the phenomenon in which cell types from the same lineage have similar methylation profiles, we assumed that K sim of the total K cell types were similar. Specifically, without loss of generality, we assumed that the first K sim cell types came from the same cell lineage and that the remaining K − K sim cell types are irrelevant to one another. We set K sim to 2, 2, and 3 for K = 3, 5, and 7, respectively. We generated μ jk for cell types k = 1, K sim + 1, …, K from the beta distribution beta (3, 6) on each CpG site j independently. For each of the remaining cell types k ′ = 2, …, K sim , we randomly selected 20% of the CpG sites and drew their μ jk ′ s independently from beta (3, 6); and for the remaining 80% of CpG sites, we let their μ jk ′ be μ j 1 plus a very small randomness, thus inducing the similarities among cell types 1 to K sim . Discrete and continuous phenotypes. We further generated a discrete and a continuous phenotype x = ( x 1 , x 2 ) T for each individual i ( i = 1, …, n ). We let the first n /3 individuals be the control samples with x i 1 = 0 for i = 1, …, n /3 and the remaining 2 n /3 individuals serve as cases with x i 1 = 1 for i = n /3 + 1, …, n . The continuous phenotypes x 2 = ( x 12 , …, x i 2 , …, x n 2 ) T were independently drawn from a Unif (20, 50) to act as age. Phenotype effects with different magnitudes and directions. We then simulated the phenotype effect \\(\\beta _{jk\\ell }\\) of each phenotype \\(\\ell \\) on CpG site j in cell type k . For the true null cases, all of the \\(\\beta _{jk\\ell }\\) s are zero. For a true alternative setting, we set nonzero phenotype effects as follows. For phenotype 1—the case/control status, we let it affect the first 10 CpG sites in all of the cell types: β jk 1 ≠ 0 for j = 1, …, 10 and k = 1, …, K . We then assumed that the next 10 CpG sites were influenced by the disease status in the first K sim cell types which come from the same lineage but not the other cell types: β jk 1 ≠ 0 ( k = 1, …, K sim ) and β jk 1 = 0 ( k = K sim + 1, …, K ) for any j = 11, …, 20. Furthermore, for cell type k ∈ { K sim + 1, …, K }, we let the disease status affect CpG sites j = 20 + 10( k − K sim − 1) + 1, …, 20 + 10( k − K sim ) only in cell type k . We generated the cell-type-specific effects of age in a similar fashion for CpG site loci 21 to 40 + 10( K − K sim ). For each nonzero β jk 1 , we let β jk 1 = r jk · ω jk , where ω jk ~ Unif (0.07, 0.15) and r jk takes values of 1 and −1 with equal probabilities. Thus, β jk 1 s can have both positive and negative effects. In the same spirit, we generated nonzero β jk 2 s with \\(r_{jk}^\\prime \\) s and \\(\\omega _{jk}^\\prime \\) s where \\(\\omega _{jk}^\\prime \\sim Unif(0.007,0.015)\\) . Association between phenotypes and cellular compositions. Notice that the phenotypes may be associated with the cellular composition. Therefore, when K = 3, we drew p i = ( p 1 i , …, p Ki ) from a Dirichlet distribution Dir (4, 4, 2 + 0.1 x i 2 ) if sample i is a control and p i ~ Dir (4, 4, 5 + 0.1 x i 2 ) if it is a case; when K = 5, we let p i ~ Dir (3, 3, 3, 3, 2 + 0.1 x i 2 ) for a control sample and p i ~ Dir (3, 3, 3, 3, 5 + 0.1 x i 2 ) for a case sample; and when K = 7, we sampled p i ~ Dir (1, 3, 3, 3, 2, 2, 2 + 0.1 x i 2 ) for controls and p i ~ Dir (1, 3, 3, 3, 2, 2, 5 + 0.1 x i 2 ) for cases. Finally, we generated the observed value O ji for CpG site j of sample i as follows: sample u ijk from N ( μ jk + β jk 1 x i 1 + β jk 2 x i 2 , 0.01 2 ) for k = 1, …, K ; and sample O ji from \\(N(\\mathop {\\sum}_{k = 1}^K {u_{ijk}} p_{ki},0.01^2)\\) . In case O ji lies outside the interval (0, 1), we truncate it to zero if O ji is lower than zero and to one if O ji is greater than one. Semi-simulated dataset including samples with known cell mix proportions The GEO dataset GSE110554 [24] contains purified cell-type-specific methylation profiles for six cell types: neutrophils, monocytes, B cells, CD4+ T, CD8+ T, and NK. Moreover, GSE110554 includes mixed samples whose methylation signals were aggregated from the six cell types with predetermined cell mix proportions. Therefore, because of the known cell type and cellular proportion information, GSE110554 is an ideal dataset with which to test HIRE’s performance. In GSE110554 , the number of mixed samples is much smaller than the typical size of an EWAS and, as discussed in the manuscript, HIRE usually requires hundreds of samples to obtain accurate and stable results. Therefore, to increase the sample size, we first generated a simulated methylation dataset with 600 samples using the purified methylation profiles. We focused on 10k CpG sites, including the 450 IDOL CpG sites, which were previously identified as the optimal library of CpG sites for estimation of leukocyte subtype proportions [24] , and another 9550 CpG sites whose methylation values across the purified cell types fell within the range of [0.2, 0.8] and had large standard deviations [11] . We then combined the 600 samples and six mixed samples (generated by method A) [24] available in GSE110554 to compose a semi-simulated dataset. After applying HIRE to the semi-simulated data, we annotated the estimated cell types based on the methylation profiles from GSE110554 . Supplementary Figure 69 shows the heatmap for the Pearson correlation matrix between inferred cell types and the underlying truth. The correlation signals on the diagonal are the strongest in each row. HIRE successfully recovers the six underlying cell types. We also compared the estimated cellular compositions with the underlying true proportions for the six mixed samples. Each panel in Supplementary Fig. 70 displays a scatter plot between the cellular proportion estimates and the true mix proportions for a given cell type; they all indicate that HIRE obtains good estimates for cellular compositions. Cell type matching protocol Assume that we have the reference methylation profiles for the H annotated cell types. We first denote the methylation profile for cell type h as ϕ h = ( ϕ 1 h , …, ϕ mh ). We aim to annotate μ k using the references. Following the previous study [33] , first, we calculate the cosine similarity, the Pearson correlation, and the Spearman correlation between μ k and ϕ h for each cell type h ∈ {1, …, H }. Notice that the three similarity measures lie between −1 and 1, and a high positive value indicates great similarity between two vectors. Second, for each similarity measure \\(\\ell \\) ( \\(\\ell = 1,2,3\\) ), we identify the cell type \\(h_\\ell \\) that has the maximal degree of similarity with μ k . If at least two out of the three similarity measures identify the same reference cell type \\(\\tilde h\\) and their corresponding similarity values are greater than 0.5, then we annotate μ k with the reference cell type \\(\\tilde h\\) . Otherwise, μ k is believed to belong to a new cell type that is not included in the references. We repeat the above process for each methylation profile μ k estimated from HIRE. Blood cell references The two real data sets analyzed in our applications were obtained from whole blood. Therefore, we prepared the references from a whole blood methylation study [34] with GEO accession code GSE35069 . The study included seven isolated blood cell subpopulations—CD4+ T cells, CD8+ T cells, CD14+ monocytes, CD19+ B cells, CD56+ NK cells, neutrophils, and eosinophils—for six individuals. Accordingly, we define the reference profile ϕ h for cell type h as the average methylation profile of these individuals, i.e., \\({\\boldsymbol{\\phi }}_h: = \\frac{1}{6}\\mathop {\\sum}_{i = 1}^6 {{\\boldsymbol{\\phi }}_{hi}} \\) . Data preprocessing The RA dataset is publicly available in GEO with accession number GSE42861 . The dataset measures the methylation levels of the whole blood. The methylation data have been normalized by Illumina’s control probe scaling procedure (see Liu et al. [3] “Illumina 450K microarray data preprocessing” section for details). The dataset includes 689 samples, and the RA status, age, gender, smoking history, and batch information are available for each sample. We removed two samples GSM1051535 and GSM1051691 because their smoking information is missing. CpG sites with a high methylation mean (>0.8) and a low methylation mean (<0.2) were discarded [11] , [13] . We adjusted the data for batch effects using COMBAT [35] . The correction process was justified because we did not observe a high degree of co-linearity between the RA status and the batches (Supplementary Fig. 68 ). The 10,000 most variable CpG sites were kept. For the RA status, we denoted RA patients with 1 and the normal control subjects with 0; we represented men with 1 and women with 0; for the smoking history, we used (0, 0, 0) to refer to “never,” (1, 0, 0) to “ex,” (0, 1, 0) to “current,” and (0, 0, 1) to “occasional” smokers. We downloaded the GALA II dataset from Gene Expression Omnibus (GEO) with accession number GSE77716 . The dataset contains the whole-blood DNA methylation beta-values from 573 samples. The beta-values have been normalized by SWAN [36] and corrected for batch effects by COMBAT [35] . There are two types of covariates: gender and ethnicity. Ethnicity includes Mexican, Mixed Latino, Puerto Rican, and Other Latino. Out of the 573 samples, one sample “GSM2057284” has no gender information, so we removed it. As suggested by previous studies [11] , [13] , CpG sites with a mean methylation value of less than 0.2 or higher than 0.8 were filtered out. We selected the 10,000 most variable of the remaining CpG sites. For gender, we denoted men with 1 and women with 0. For the ethnicity variables, we used three dummy variables to represent the four ethnicity categories. In particular, (0, 0, 0), (1, 0, 0), (0, 1, 0), and (0, 0, 1) corresponded to Mexican, Mixed Latino, Puerto Rican, and Other Latino, respectively. For ReFACTor and EWASHer, according to their rules, we first filtered out CpG sites that were consistently hypomethylated or consistently hypermethylated and then regressed out the known covariates. We finally used the residuals to perform their analysis. Note that in their software these steps are processed automatically. For RefFreeEWAS, SVA, and the unadjusted analysis, the phenotypes and the covariates were regarded as the fixed effects in the regression model. In detail, for ReFACTor, in both GALA II and RA datasets, the cell type number “K” was specified to be six, which was the same as in their paper [13] . For RefFreeEWAS, we fixed the dimensionality of latent space “d” at six in the real data. For SVA, we also fixed the number of surrogate variables to six. Gene enrichment analysis was carried out on the Broad Institute website http://software.broadinstitute.org/gsea/msigdb/annotate.jsp . The canonical pathways were selected as the basis gene sets, and only pathways with a false discovery rate of less than 0.05 were reported. Identifiability of HIRE Although the non-negative matrix factorization (NNMF) O = μ P has been widely applied in cell type deconvolution [16] , where O is the observed methylation matrix, μ is the unknown methylation profile, and P is the unknown cellular compositions, model identifiability is rarely discussed. During the review period of our paper, Rahmani et al. [37] provided a setting under which the NMMF model is not identifiable. Why then does NNMF always provide satisfactory cell type deconvolution results in real practice, and why can HIRE estimate all those parameters well? Here, we show mathematically that the HIRE model is identifiable under mild conditions that are easily met in reality. Let us first introduce some notations and definitions. In the HIRE model, the whole parameter set is denoted by \\({\\mathbf{\\Theta }}: = \\{ {\\mathbf{P}}_i,{\\boldsymbol{\\mu }}_j,{\\mathbf{B}}_\\ell ^{(j)},\\sigma _{jk}^2,\\sigma _{\\epsilon j}^2:1 \\le j \\le m,1 \\le i \\le n,1 \\le k \\le K,1 \\le \\ell \\le q\\} \\) , where p i is the cellular composition vector of sample i , μ j is the baseline methylation vector of CpG site j , \\({\\mathbf{B}}_\\ell ^{(j)}\\) is the phenotype \\(\\ell \\) effect vector on CpG site j , \\(\\sigma _{jk}^2\\) is the cell-type- k noise variance on CpG site j , and \\(\\sigma _{\\epsilon j}^2\\) is the overall noise variance on CpG site j . The observed data in our study are the methylation matrix O = { O ij :1 ≤ i ≤ n , 1 ≤ j ≤ m } and the covariate matrix \\({\\mathbf{X}} = ({\\mathbf{x}}_1, \\ldots ,{\\mathbf{x}}_\\ell , \\ldots ,{\\mathbf{x}}_q)\\) , where \\({\\mathbf{x}}_\\ell \\) is the column vector that indicates phenotype- \\(\\ell \\) for the n samples. The observed likelihood function \\(({\\mathbf{\\Theta }}|{\\mathbf{O}}) = \\mathop {\\prod}_{i = 1}^n {\\mathop {\\prod}_{j = 1}^m N } (O_{ji}:{\\mathbf{P}}_i^T {\\boldsymbol{\\mu }}_{j} + \\mathop {\\sum}_{\\ell = 1}^q {x_{i\\ell }} {\\mathbf{P}}_i^T {\\mathbf{B}}_\\ell ^{(j)},\\mathop {\\sum}_{k = 1}^K {\\sigma _{jk}^2} \\, P_{ik}^2 + \\sigma _{\\epsilon j}^2)\\) (see Eq. (S7) in the Supplementary Methods ), where N ( O : η , τ 2 ) indicates the normal density with mean η and variance τ 2 at value O . We further define 1 K = (1, 1, …, 1) T as a K -dimension column vector with all entries being one, an n by K matrix J 1 as \\({\\mathbf{1}}_n{\\mathbf{1}}_K^T\\) , and an n by K matrix \\({\\mathbf{J}}_{x_\\ell }\\) as \\({\\mathbf{x}}_\\ell {\\mathbf{1}}_K^T\\) for each \\(1 \\le \\ell \\le q\\) . We use ⊙ to represent the entry-wise matrix product for two matrices M and N with the same dimension, i.e., \\(({\\mathbf{M}} \\odot {\\mathbf{N}})_{ij}: = {\\mathbf{M}}_{ij}{\\mathbf{N}}_{ij}\\) . Theorem 1. If (a) for each cell type k, there exists a CpG site r k such that \\({\\mathbf{B}}_\\ell ^{(r_k)} = 0\\) for any phenotype \\(\\ell \\) and \\(\\mu _{r_kk} = 1\\) while \\(\\mu _{r_kk{\\prime}} = 0\\) for k ′ ≠ k , and (b) the cellular compositions P satisfies that \\(rank(({\\mathbf{J}}_1 \\odot {\\mathbf{P}}^T,{\\mathbf{J}}_{x_1} \\odot {\\mathbf{P}}^T, \\ldots ,{\\mathbf{J}}_{x_\\ell } \\odot {\\mathbf{P}}^T, \\ldots ,{\\mathbf{J}}_{x_q} \\odot {\\mathbf{P}}^T)) = (q + 1)K\\) and \\(rank(({\\mathbf{1}}_n,{\\mathbf{P}}^T) \\odot ({\\mathbf{1}}_n,{\\mathbf{P}}^T)) = K + 1\\) , then the HIRE model is identifiable. In other words, \\(L({\\mathbf{\\Theta }}|{\\mathbf{O}}) = L(\\widetilde {\\mathbf{\\Theta }}|{\\mathbf{O}})\\) for any O implies \\({\\mathbf{\\Theta }} = \\widetilde {\\mathbf{\\Theta }}\\) . Proof: First, by integrating out all O elements except O ji , \\(L({\\mathbf{\\Theta }}|{\\mathbf{O}}) = L(\\widetilde {\\mathbf{\\Theta }}|{\\mathbf{O}})\\) implies \\(N(O_{ji}:{\\mathbf{P}}_i^T {\\boldsymbol{\\mu }}_{j} + \\mathop {\\sum}_{\\ell = 1}^q {x_{i\\ell }} {\\mathbf{P}}_i^T {\\mathbf{B}}_\\ell ^{(j)},\\mathop {\\sum}_{k = 1}^K {\\sigma _{jk}^2} P_{ki}^2 + \\sigma _{\\epsilon j}^2)\\) = \\(N(O_{ji}:\\widetilde {\\mathbf{P}}_i^T \\widetilde \\mu_{j} + \\mathop {\\sum}_{\\ell = 1}^q {x_{i\\ell }} \\widetilde {\\mathbf{P}}_i^T\\widetilde {\\mathbf{B}}_\\ell ^{(j)},\\mathop {\\sum}_{k = 1}^K {\\tilde \\sigma _{jk}^2} \\tilde P_{ki}^2 + \\tilde \\sigma _{\\epsilon j}^2)\\) . Because the univariate normal distribution is identifiable, we have $${\\mathbf{P}}_i^T{\\boldsymbol{\\mu }}_j + \\mathop {\\sum}\\limits_{\\ell = 1}^q {x_{i\\ell }} {\\mathbf{P}}_i^T{\\mathbf{B}}_\\ell ^{(j)} = \\widetilde {\\mathbf{P}}_i^T\\widetilde {\\boldsymbol{\\mu }}_j + \\mathop {\\sum}\\limits_{\\ell = 1}^q {x_{i\\ell }} \\widetilde {\\mathbf{P}}_i^T\\widetilde {\\mathbf{B}}_\\ell ^{(j)},$$ (8) $$\\mathop {\\sum}\\limits_{k = 1}^K {\\sigma _{jk}^2} P_{ki}^2 + \\sigma _{\\epsilon j}^2 = \\mathop {\\sum}\\limits_{k = 1}^K {\\tilde \\sigma _{jk}^2} \\tilde P_{ki}^2 + \\tilde \\sigma _{\\epsilon j}^2.$$ (9) Taking j = r k in Eq. ( 8 ), we have \\(LHS = {\\mathbf{P}}_i^T{\\boldsymbol{\\mu }}_{r_k} + \\mathop {\\sum}_{\\ell = 1}^q {x_{i\\ell }} {\\mathbf{P}}_i^T{\\mathbf{B}}_\\ell ^{(r_k)} = {\\mathbf{P}}_i^T{\\boldsymbol{\\mu }}_{r_k} = 0 + P_{ki}\\cdot 1 + 0 = P_{ki}\\) and similarly \\(RHS = \\tilde P_{ki}\\) , so \\(P_{ki} = \\tilde P_{ki}\\) , which holds for any i and k . Hence, we obtain \\({\\mathbf{P}} = \\widetilde {\\mathbf{P}}\\) . Next, we rewrite Eq. ( 8 ) into a matrix form. $$({\\mathbf{P}}_i^T,x_{i1}{\\mathbf{P}}_i^T, \\ldots ,x_{iq}{\\mathbf{P}}_i^T)\\left( {\\begin{array}{*{20}{c}} {{\\boldsymbol{\\mu }}_j} \\\\ {{\\mathbf{B}}_1^{(j)}} \\\\ \\vdots \\\\ {{\\mathbf{B}}_q^{(j)}} \\end{array}} \\right) = ({\\mathbf{P}}_i^T,x_{i1}{\\mathbf{P}}_i^T, \\ldots ,x_{iq}{\\mathbf{P}}_i^T)\\left( {\\begin{array}{*{20}{c}} {\\widetilde {\\boldsymbol{\\mu }}_j} \\\\ {\\widetilde {\\mathbf{B}}_1^{(j)}} \\\\ \\vdots \\\\ {\\widetilde {\\mathbf{B}}_q^{(j)}} \\end{array}} \\right),\\;\\;i = 1, \\ldots ,n.$$ By combining these n equations, it follows that $$\\begin{array}{l}({\\mathbf{J}}_1 \\odot {\\mathbf{P}}^T,{\\mathbf{J}}_{x_1} \\odot {\\mathbf{P}}^T, \\ldots ,{\\mathbf{J}}_{x_\\ell } \\odot {\\mathbf{P}}^T, \\ldots ,{\\mathbf{J}}_{x_q} \\odot {\\mathbf{P}}^T)\\left( {\\begin{array}{*{20}{c}} {{\\boldsymbol{\\mu }}_j} \\\\ {{\\mathbf{B}}_1^{(j)}} \\\\ \\vdots \\\\ {{\\mathbf{B}}_q^{(j)}} \\end{array}} \\right)\\\\ = ({\\mathbf{J}}_1 \\odot {\\mathbf{P}}^T,{\\mathbf{J}}_{x_1} \\odot {\\mathbf{P}}^T, \\ldots ,{\\mathbf{J}}_{x_\\ell } \\odot {\\mathbf{P}}^T, \\ldots ,{\\mathbf{J}}_{x_q} \\odot {\\mathbf{P}}^T)\\left( {\\begin{array}{*{20}{c}} {\\widetilde {\\boldsymbol{\\mu }}_j} \\\\ {\\widetilde {\\mathbf{B}}_1^{(j)}} \\\\ \\vdots \\\\ {\\widetilde {\\mathbf{B}}_q^{(j)}} \\end{array}} \\right).\\end{array}$$ (10) Because the rank of \\(A: = ({\\mathbf{J}}_1 \\odot {\\mathbf{P}}^T,{\\mathbf{J}}_{x_1} \\odot {\\mathbf{P}}^T, \\ldots ,{\\mathbf{J}}_{x_\\ell } \\odot {\\mathbf{P}}^T, \\ldots ,{\\mathbf{J}}_{x_q} \\odot {\\mathbf{P}}^T)\\) is ( q + 1) K (full column rank), A has a left inverse A −1 . Multiplying Eq. ( 10 ) by A −1 from the left on both sides, we obtain \\({\\boldsymbol{\\mu }}_j = \\widetilde {\\boldsymbol{\\mu }}_j\\) and \\({\\mathbf{B}}_\\ell ^{(j)} = \\widetilde {\\mathbf{B}}_\\ell ^{(j)}\\) for \\(1 \\le \\ell \\le q\\) . Therefore, we have \\({\\boldsymbol{\\mu }} = \\widetilde {\\boldsymbol{\\mu }}\\) , \\({\\mathbf{B}} = \\widetilde {\\mathbf{B}}\\) . In addition, because Eq. ( 9 ) holds for any i , we can also rewrite it into a matrix form. $$\\left( {\\begin{array}{*{20}{c}} 1 & {P_{11}^2} & \\ldots & {P_{K1}^2} \\\\ 1 & {P_{12}^2} & \\ldots & {P_{K2}^2} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 1 & {P_{1n}^2} & \\ldots & {P_{Kn}^2} \\end{array}} \\right)\\left( {\\begin{array}{*{20}{c}} {\\sigma _{\\epsilon j}^2} \\\\ {\\sigma _{j1}^2} \\\\ \\vdots \\\\ {\\sigma _{jK}^2} \\end{array}} \\right) = \\left( {\\begin{array}{*{20}{c}} 1 & {P_{11}^2} & \\ldots & {P_{K1}^2} \\\\ 1 & {P_{12}^2} & \\ldots & {P_{K2}^2} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 1 & {P_{1n}^2} & \\ldots & {P_{Kn}^2} \\end{array}} \\right)\\left( {\\begin{array}{*{20}{c}} {\\tilde \\sigma _{\\epsilon j}^2} \\\\ {\\tilde \\sigma _{j1}^2} \\\\ \\vdots \\\\ {\\tilde \\sigma _{jK}^2} \\end{array}} \\right)$$ The left matrix is equal to \\(({\\mathbf{1}}_n,{\\mathbf{P}}^T) \\odot ({\\mathbf{1}}_n,{\\mathbf{P}}^T)\\) which has a full column rank; therefore, it has a left inverse. Consequently, \\(\\sigma _{\\epsilon j}^2 = \\tilde \\sigma _{\\epsilon j}^2\\) and \\(\\sigma _{jk}^2 = \\tilde \\sigma _{jk}^2\\) . As a result, \\({\\mathbf{\\Theta }} = \\widetilde {\\mathbf{\\Theta }}\\) , and we have proven the identifiability of HIRE. \\(\\square \\) Conditions (a) and (b) are easily met for DNA methylation data. Condition (a) requires that for each cell type k , there exists a CpG site that is not associated with any phenotype and is only methylated in cell type k but not methylated in any other cell type. Given the 450K CpG sites assayed by the microarray, we can expect that such CpG sites are not absent at all. Moreover, condition (a) can also be relaxed to the condition that for each cell type k , there exists a CpG site r k such that \\({\\mathbf{B}}_\\ell ^{(r_k)} = 0\\) for any phenotype \\(\\ell \\) and \\(\\mu _{r_kk} = 1\\) while \\(\\mu _{r_kk\\prime } = 0\\) for k ′ ≠ k or there exists a CpG site r k such that \\({\\mathbf{B}}_\\ell ^{(r_k)} = 0\\) for any phenotype \\(\\ell \\) and \\(\\mu _{r_kk} = 0\\) while \\(\\mu _{r_kk{\\prime}} = 1\\) for k ′ ≠ k . The proof follows in a similar manner. For condition (b), intuitively, the rank requirement of \\(({\\mathbf{1}}_n,{\\mathbf{P}}^T) \\odot ({\\mathbf{1}}_n,{\\mathbf{P}}^T)\\) asks the cellular compositions to vary across subjects, which guards against the case in which all the subjects have the same cellular compositions and hence no cell type deconvolution is possible; the rank requirement on \\(({\\mathbf{J}}_1 \\odot {\\mathbf{P}}^T,{\\mathbf{J}}_{x_1} \\odot {\\mathbf{P}}^T, \\ldots ,{\\mathbf{J}}_{x_\\ell } \\odot {\\mathbf{P}}^T, \\ldots ,{\\mathbf{J}}_{x_q} \\odot {\\mathbf{P}}^T)\\) is the same requirement as those in a standard linear regression, which requires that no collinearity exists among the covariates. Because the sample size n is much larger than the underlying cell type number K and the phenotype number q , the two rank requirements can commonly be satisfied in reality. Reporting summary Further information on research design is available in the Nature Research Reporting Summary linked to this article."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"Epigenome-wide association studies (EWAS) aim to identify cytosine-phosphate-guanine (CpG) sites associated with phenotypes of interest, such as disease status [1] , [2] , [3] , smoking history [4] , [5] , body mass index [6] , and age [7] , [8] . However, because the samples in EWAS are measured at the bulk level rather than at the single-cell level, the obtained methylome for each sample shows the signals aggregated from distinct cell types [3] , [9] , [10] , which leads to two main challenges in the analysis of EWAS data. On the one hand, the cell type compositions differ among samples and can be associated with phenotypes [3] , [10] . Both binary phenotypes, such as the diseased or normal status [3] , and continuous phenotypes, such as age [10] , have been found to affect the cell type compositions. As a result, ignoring the cellular heterogeneity in EWAS can lead to many spurious associations [10] , [11] , [12] , [13] . On the other hand, the phenotype may change the methylation level of a CpG site in some but not all of the cell types. Identification of the exact cell types that carry the risk-CpG sites can deepen our understandings of disease mechanisms. However, such identification is challenging because only the aggregated-level signals can be observed. To the best of our knowledge, no existing statistical method for EWAS can detect cell-type-specific associations despite active research to account for cell-type heterogeneity. The existing approaches can be categorized into two schools [14] : reference-based and reference-free methods. The reference-based methods [9] , [15] require the reference methylation profiles for each cell type to be known a priori, and they regress the aggregated methylation levels observed from each sample on the same set of references to learn the sample’s cellular compositions. However, because samples have different attributes, such as age and gender, the methylation levels of a given cell type can vary among samples. It is thus problematic to assume that all of the samples have the same set of reference profiles [10] , [14] . Furthermore, high-quality references are difficult to obtain for most EWAS due to the existence of unknown cell types, the high cost of cell sorting, and confounding effects [14] . Consequently, a large amount of recent EWAS literature was devoted to identification of risk-CpG sites without the need for the reference methylation profiles. The reference-free methods can generally be further divided into two classes according to whether they estimate the cell-type mixing proportions directly. The direct-decomposition-based procedures consist of two stages. In the first stage, they simultaneously estimate the cellular compositions of each sample and the cell-type-specific reference methylomes via quadratic programming [16] ; and in the second stage, they treat the estimated cell-type proportions as covariates with additive effects in the linear models to conduct association tests. However, when estimating cellular compositions during the first stage, the direct-decomposition-based methods also do not consider samples’ phenotype information, thus suffering from the same problem of biasing the cellular composition estimates as the reference-based approaches [9] . Moreover, similar to tumor purity [17] , we argue that the estimated cellular composition has a multiplicative rather than an additive effect on the observed methylation level (Methods). The second class of methods, which is exemplified by SVA [18] , RefFreeEWAS [19] , and ReFACTor [13] , does not carry out cell-type decompositions. They resort to singular value decomposition, which includes the principal component analysis, to construct surrogates for the underlying cell-type composition. EWASher, a linear mixed model, also belongs to this class because it is equivalent to the use of principal components as fixed-effect covariates [11] . However, the use of principal components as the covariates in the regression undergoes the same issue of additive effects as the direct-decomposition-based methods. Therefore, the existing reference-free methods have low power in detecting risk-CpG sites [12] . Although the existing methods aim to address the cellular heterogeneity problem in EWAS and claim whether a CpG site is associated with phenotypes at the aggregate level, none of them can identify the risk-CpG sites for each individual cell type, thus missing the opportunity to obtain finer-grained results in EWAS. Here, we propose a method, HIRE, to identify the association in EWAS at a HIgh REsolution: detecting whether a CpG site has any associations with the phenotypes in each cell type (Methods). The keys to HIRE’s success are twofold. First, HIRE links the underlying cell-type-specific methylation profiles for each sample to the sample’s phenotypes, thus avoiding the bias in estimating the cellular composition by the reference-based and direct-decomposition-based methods. Second, HIRE correctly characterizes the cellular compositions as the multiplicative effects, whereas the existing methods inappropriately treat the cell proportions as additive effects (Methods). HIRE is applicable to EWAS with binary phenotypes, continuous phenotypes, or both. By helping researchers understand in which cell types the CpG sites are affected by a disease, HIRE can ultimately facilitate the development of epigenetic therapies by targeting the specifically affected cell types.\",\n \"references\": {\n \"indices\": [\n 1,\n 2,\n 3,\n 4,\n 5,\n 6,\n 7,\n 8,\n 9,\n 10,\n 11,\n 12,\n 13,\n 14,\n 15,\n 16,\n 17,\n 18,\n 19\n ],\n \"infos\": [\n {\n \"idx\": \"1\",\n \"title\": \"Rakyan, V. K., Down, T. A., Balding, D. J. & Beck, S. Epigenome-wide association studies for common human diseases.Nat. Rev. Genet.12, 529–541 (2011).\",\n \"link\": \"https://doi.org/10.1038%2Fnrg3000\"\n },\n {\n \"idx\": \"2\",\n \"title\": \"Verma, M. Epigenome-wide association studies (EWAS) in cancer.Curr. Genom.13, 308–313 (2012).\",\n \"link\": \"https://doi.org/10.2174%2F138920212800793294\"\n },\n {\n \"idx\": \"3\",\n \"title\": \"Liu, Y. et al. 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Epigenome-wide association study of body mass index, and the adverse outcomes of adiposity.Nature541, 81–86 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fnature20784\"\n },\n {\n \"idx\": \"7\",\n \"title\": \"Teschendorff, A. E. et al. Age-dependent DNA methylation of genes that are suppressed in stem cells is a hallmark of cancer.Genome Res.20, 440–446 (2010).\",\n \"link\": \"https://doi.org/10.1101%2Fgr.103606.109\"\n },\n {\n \"idx\": \"8\",\n \"title\": \"Horvath, S. DNA methylation age of human tissues and cell types.Genome Biol.14, 3156 (2013).\",\n \"link\": \"https://link.springer.com/doi/10.1186/gb-2013-14-10-r115\"\n },\n {\n \"idx\": \"9\",\n \"title\": \"Houseman, E. A. et al. DNA methylation arrays as surrogate measures of cell mixture distribution.BMC Bioinformatics13, 86 (2012).\",\n \"link\": \"https://link.springer.com/doi/10.1186/1471-2105-13-86\"\n },\n {\n \"idx\": \"10\",\n \"title\": \"Jaffe, A. E. & Irizarry, R. A. Accounting for cellular heterogeneity is critical in epigenome-wide association studies.Genome Biol.15, R31 (2014).\",\n \"link\": \"https://link.springer.com/doi/10.1186/gb-2014-15-2-r31\"\n },\n {\n \"idx\": \"11\",\n \"title\": \"Zou, J., Lippert, C., Heckerman, D., Aryee, M. & Listgarten, J. Epigenome-wide association studies without the need for cell-type composition.Nat. Methods11, 309–311 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fnmeth.2815\"\n },\n {\n \"idx\": \"12\",\n \"title\": \"McGregor, K. et al. An evaluation of methods correcting for cell-type heterogeneity in DNA methylation studies.Genome Biol.17, 84 (2016).\",\n \"link\": \"https://link.springer.com/doi/10.1186/s13059-016-0935-y\"\n },\n {\n \"idx\": \"13\",\n \"title\": \"Rahmani, E. et al. Sparse PCA corrects for cell type heterogeneity in epigenome-wide association studies.Nat. Methods13, 443–445 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fnmeth.3809\"\n },\n {\n \"idx\": \"14\",\n \"title\": \"Teschendorff, A. E. & Relton, C. L. Statistical and integrative system-level analysis of DNA methylation data.Nat. Rev. Genet.19, 129–147 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fnrg.2017.86\"\n },\n {\n \"idx\": \"15\",\n \"title\": \"Accomando, W. P., Wiencke, J. K., Houseman, E. A., Nelson, H. H. & Kelsey, K. T. Quantitative reconstruction of leukocyte subsets using DNA methylation.Genome Biol.15, R50 (2014).\",\n \"link\": \"https://link.springer.com/doi/10.1186/gb-2014-15-3-r50\"\n },\n {\n \"idx\": \"16\",\n \"title\": \"Houseman, E. A. et al. Reference-free deconvolution of DNA methylation data and mediation by cell composition effects.BMC Bioinformatics17, 259 (2016).\",\n \"link\": \"https://link.springer.com/doi/10.1186/s12859-016-1140-4\"\n },\n {\n \"idx\": \"17\",\n \"title\": \"Zheng, X., Zhang, N., Wu, H.-J. & Wu, H. Estimating and accounting for tumor purity in the analysis of DNA methylation data from cancer studies.Genome Biol.18, 17 (2017).\",\n \"link\": \"https://link.springer.com/doi/10.1186/s13059-016-1143-5\"\n },\n {\n \"idx\": \"18\",\n \"title\": \"Leek, J. T. & Storey, J. D. Capturing heterogeneity in gene expression studies by surrogate variable analysis.PLoS Genet.3, e161 (2007).\",\n \"link\": \"https://doi.org/10.1371%2Fjournal.pgen.0030161\"\n },\n {\n \"idx\": \"19\",\n \"title\": \"Houseman, E. A., Molitor, J. & Marsit, C. J. Reference-free cell mixture adjustments in analysis of DNA methylation data.Bioinformatics30, 1431–1439 (2014).\",\n \"link\": \"https://doi.org/10.1093%2Fbioinformatics%2Fbtu029\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Method overview HIRE is a hierarchical model that closely follows the data generation process. Its elaborate modeling depicts how phenotypes affect the methylation levels of each sample. Here, we briefly introduce the method. The technical details are provided in the Methods section and the Supplementary Methods . Let us first review the cornerstone in most EWAS approaches. These methods model the observed methylation levels of the m CpG sites for sample i , O i = ( O 1 i , O 2 i , …, O mi ) T , as the weighted average of the methylation profiles of K cell types, u i = ( u i 1 , u i 2 , …, u iK ). The weights are the cellular compositions p i = ( p 1 i , p 2 i , …, p Ki ) T of sample i (see the top panel of Fig. 1a ). However, regardless of whether the reference is known a priori or not, the existing methods assume that the cell-type-specific methylation profiles u i s remain the same for all samples: u i = M , for i = 1, …, n . Unfortunately, because the methylation levels can actually change with covariates such as age and disease status, ignoring the covariates’ effects and enforcing static reference methylomes can bias the estimation of p i and thus affect all downstream analyses [14] . More importantly, the assumption that cell-type-specific methylation profiles are the same for each sample prevents the detection of cell-type-specific risk-CpG sites. Fig. 1 A simple cartoon illustration of the HIRE model with three cell types ( K = 3) and two phenotypes (disease status and age; q = 2). a Data generation procedure for the observed methylation vector O i for sample i ( i = 1, …, n ). In the top panel, O i is the convolution of cell-type-specific methylation profiles u i with cellular compositions p i . Both u i and p i depend on the attributes of sample i . The bottom panel describes how sample i ’s phenotypes affect u i via two phenotype-effect matrices B 1 and B 2 . In B 1 and B 2 , the white square represents zero, which indicates that the phenotype exerts no influence on the corresponding methylation level in u i . b Inputs and outputs of HIRE. We input the observed methylation matrix O , the phenotype data matrix X , and a predetermined cell type number K into HIRE, and HIRE outputs the estimates for the cellular compositions \\\\(\\\\widehat {\\\\mathbf{p}}\\\\) , the baseline methylation profiles \\\\(\\\\widehat {\\\\boldsymbol{\\\\mu }}\\\\) , the phenotype effects \\\\(\\\\widehat {\\\\mathbf{B}}_\\\\ell \\\\) , and the penalized BIC value. In addition, HIRE tests whether there is any association between CpG site j and phenotype \\\\(\\\\ell \\\\) in cell type k — \\\\(H_0:\\\\beta _{jk\\\\ell } = 0\\\\) vs \\\\(H_1:\\\\beta _{jk\\\\ell } \\\\ \\\\ne\\\\ 0\\\\) —and provides the p-values Full size image For association detection at the aggregate level, after estimation of p i using the deconvolution-based approach or its surrogates from principal component-based methods, the existing methods examine a linear model in which the phenotypes \\\\({\\\\mathbf{x}}_i = (x_{i1}, \\\\ldots ,x_{i\\\\ell }, \\\\ldots ,x_{iq})^T\\\\) and the cellular proportions p i exert additive effects on the methylation level O i : (1) A CpG-site j is then associated with phenotype \\\\(\\\\ell \\\\) if we reject the null hypothesis that the covariate coefficient \\\\(T_{j\\\\ell }\\\\) equals zero. In contrast, HIRE further models the effect of each phenotype on each cell type as shown in the bottom panel of Fig. 1a . In cell type k , sample i ’s cell-type-specific methylation profile, u ik , is the summation of the corresponding baseline cell-type-specific methylation levels, μ k , and the phenotype effects \\\\({\\\\mathbf{B}}_{k\\\\ell }x_{i\\\\ell }\\\\) on sample i from all the l = 1, …, q phenotypes: \\\\({\\\\mathbf{u}}_{ik} = {\\\\boldsymbol{\\\\mu }}_k + \\\\mathop {\\\\sum}_{l = 1}^q {{\\\\mathbf{B}}_{k\\\\ell }} x_{i\\\\ell }\\\\) , where \\\\(x_{i\\\\ell }\\\\) is the phenotype \\\\(\\\\ell \\\\) of sample i and \\\\({\\\\mathbf{B}}_{k\\\\ell } = (\\\\beta _{1k\\\\ell }, \\\\ldots ,\\\\beta _{mk\\\\ell })^T\\\\) —the k th column of \\\\({\\\\mathbf{B}}_\\\\ell \\\\) —reflects the association of phenotype \\\\(\\\\ell \\\\) with each of the m CpG sites in cell type k . Thus, by collecting the baseline cell-type-specific methylation profiles to μ = ( μ 1 , …, μ k ) and denoting the m by K phenotype coefficient matrix \\\\((\\\\beta _{jk\\\\ell }:1 \\\\le j \\\\le m,1 \\\\le k \\\\le K)\\\\) by \\\\({\\\\mathbf{B}}_\\\\ell \\\\) , we now have: (2) A comparison of \\\\({\\\\mathrm{x}}_i\\\\) in Eq. ( 1 ) and \\\\({x_{i \\\\ell }}{\\\\boldsymbol{p}}_i ,{\\\\ell} = 1,…,q\\\\) in Eq. ( 2 ) reveals that via the two-layer hierarchical model HIRE correctly captures the multiplicative effects of the cellular compositions on the phenotype effects (see also Methods and the Supplementary Methods ). As a result, HIRE achieves greater statistical power for association detection at the aggregate level and enables the fine-scale resolutions that were previously infeasible. We mathematically prove that the HIRE model is identifiable under mild conditions that are easily met in reality (see Theorem 1 and its proof in Methods). Figure 1b summarizes the inputs and outputs of HIRE. Given the methylation measurements at the aggregate level of n samples, HIRE can estimate all parameters of interests — p i ( i = 1, …, n ), μ , and \\\\({\\\\mathbf{B}}_\\\\ell \\\\) ( \\\\(\\\\ell = 1, \\\\ldots ,q\\\\) ). HIRE then determines whether any association exists between CpG site j and phenotype \\\\(\\\\ell \\\\) in each individual cell type by testing the hypotheses \\\\(H_0:\\\\beta _{jk\\\\ell } = 0\\\\) versus \\\\(H_1:\\\\beta _{jk\\\\ell } \\\\ \\\\ne\\\\ 0\\\\) . When the null hypothesis \\\\(H_0:\\\\beta _{jk\\\\ell } = 0\\\\) is rejected, HIRE calls CpG site j as a risk-CpG site for phenotype \\\\(\\\\ell \\\\) in cell type k . The detection of cell-type-specific risk-CpG sites cannot be performed with any of the existing state-of-the-art methods. Moreover, HIRE allows users to prespecify the number of cell types K . When K is unknown, HIRE selects the number of cell types according to the penalized Bayesian information criterion (pBIC) [20] ( Supplementary Methods ). Simulation As the definition of the gold standard for real data is debatable [21] , [22] , we designed extensive simulation studies to evaluate the performance of HIRE and compared it with commonly used methods—unadjusted analysis, SVA, RefFreeEWAS, EWASHer, and ReFACTor (Methods). We generated datasets in which the observed methylation was a mixture of several cell types and each sample was accompanied with a diseased or normal status and a continuous age attribute. We deliberately designed some cell types to have similar baseline methylation profiles to mimic cell types from the same cell lineage. We set the sample size n to 180, 300, and 600 and let the underlying cell type number K be 3, 5, and 7. For each pair of ( n , K ), we investigated two scenarios in which (1) all phenotype effects \\\\(\\\\beta _{jk\\\\ell }\\\\) s are zero—the true null case—to compare the ability of each method to control false positives; and (2) a small portion of \\\\(\\\\beta _{jk\\\\ell }\\\\) s are non-zero—the true alternative case—to study each method’s power to detect risk-CpG sites. Under the true alternative, both the binary and the continuous phenotypes were assumed to have cell-type-specific risk-CpG sites and to affect the cell-type proportions among the samples [10] . We further simulated phenotype effects with various directions and magnitudes. Under the true null, HIRE, EWASHer, and ReFACTor control the false positive rates (FPRs) very well: none are greater than 0.05% (Table 1 and Supplementary Figs. 1 – 9 ). In comparison, RefFreeEWAS often has FPRs greater than 0.1% and thus does not perform as well as HIRE, and the unadjusted analysis and SVA further suffer from the dramatic inflation of false positives. For the true alternative settings, given that the FPRs are well-controlled, with FPRs below 0.05%, HIRE achieves the highest true positive rates (TPR) of all methods in every simulation setting (see also Fig. 2a and Supplementary Figs. 10 – 17 ). As expected, as the sample size increases, HIRE’s power increases. For example, when the data include five cell types, HIRE can identify 89.6% of the risk-CpG sites with 300 samples, and HIRE can detect almost all risk-CpG sites when the sample size reaches 600, which is a typical sample size for EWAS. Although EWASHer and ReFACTor have low FPRs, they miss a large proportion of risk-CpG sites. EWASHer’s maximum TPR is only 35.33%, and ReFACTor’s maximum TPR is slightly over 60%. However, in those cases, HIRE’s power is greater than 95%. Consistent with the true null scenario, in the true alternative, RefFreeEWAS has inflated FPRs compared to HIRE, and the unadjusted analysis and SVA always have huge false positives. Therefore, HIRE substantially improves the power of association detection at the aggregate level compared with existing methods. Table 1 Performance of HIRE and other competing methods in simulation studies Full size table Fig. 2 Association detection performance of HIRE and commonly used methods in the true alternative setting with K = 3 and n = 180. Source data are provided as a Source Data file. In all figures, red corresponds to HIRE; yellow indicates the unadjusted analysis; brown represents SVA; purple refers to RefFreeEWAS; dark blue indicates EWASher; and light blue corresponds to ReFACTor. a ROC curves of HIRE and commonly used methods. HIRE has the largest area under the curve among all of the methods. b True cell-type-specific association pattern with disease status for 10,000 simulated CpG sites; columns correspond to cell types, and the rows represent the CpG sites. Dark cells correspond to risk-CpG sites, and grey cells are CpG sites not associated with the disease status. c Detected cell-type-specific association pattern with disease status by HIRE. Darkness represents \\\\(-{\\\\mathrm{log}}_{10}(p - {\\\\mathrm{value}})\\\\) d – i The p -value density plots for association with disease status in the simulation dataset for d HIRE, e unadjusted analysis, f SVA, g RefFreeEWAS, h EWASHer, and i ReFACTor. j – o The Q-Q plots for association with disease status for j HIRE, k unadjusted analysis, l SVA, m RefFreeEWAS, n EWASHer, and o ReFACTor Full size image In the multiple hypothesis testing, the p-values from the truly null features should follow a uniform distribution on (0, 1), whereas those for the truly alternative features are concentrated near zero [23] . Both the histograms (Fig. 2d–i ) and the Q-Q plots (Fig. 2j–o ) show that the p-value distribution of HIRE is the best fit to the underlying truth—there are only a small proportion of signals, followed by RefFreeEWAS and ReFACTor. EWASHer easily overcorrects signals with its p-value density having a dip near zero (Fig. 2h ), thus failing to detect the true associations. In contrast, the unadjusted analysis and SVA generate very small p-values clustered near zero, resulting in inflated type I errors. In addition to the traditional association detection at the aggregate level, HIRE can identify the association for each CpG site with the phenotypes under each cell type. Table 2 shows the FPR and TPR of HIRE for each cell type in various simulation settings. Such fine analysis is not possible with the other methods. Consistent with association detection at the aggregate level, HIRE always controls the FPR well. When K = 3 and n = 180, HIRE accurately detects the risk-CpG sites associated with disease status a TPR of greater than 83% and an FPR of 0.01% or less in all three cell types. Similarly, most of the CpG sites affected by age are also correctly identified in each cell type. HIRE’s learned cell-type-specific association patterns closely matches the underlying true associations (see Fig. 2b, c and Supplementary Figs. 18 – 26 ). Once again, HIRE’s power decreases with the number of cell types and increases with the sample size. When the samples consist of seven cell types and the proportion of the least abundant cell type is as low as 4.2%, given a typical current EWAS with around 600 samples, HIRE can detect most cell-type-specific risk-CpG sites reasonably well. Moreover, HIRE’s estimates for the baseline methylation profiles, cellular compositions, and phenotype effects have little bias (Supplementary Figs. 27 – 62 ); therefore, HIRE can provide accurate estimates and is powerful in detecting cell-type-specific risk-CpG sites. Table 2 Performance of HIRE in detecting cell-type-specific risk-CpG sites Full size table In the HIRE model, we assume that different CpG sites are independent, and we investigate the performance of HIRE when such a model assumption is violated and dependences exist among nearby CpG sites. Specifically, we assume that every 50 consecutive CpG sites belongs to a block. For CpG sites within the same block, their random noises \\\\({\\\\boldsymbol{\\\\epsilon }}\\\\) follow a multivariate normal distribution with mean zero and 50 × 50 covariance matrix Σ, and Σ’s corresponding correlation matrix has its ( i , j ) entry equal to ρ | i − j | . We vary ρ to 0.8, 0.6, and 0.4. A comparison of Supplementary Tables 1 – 3 with Supplementary Table 4 shows that even when strong correlations exit among nearby CpG sites, HIRE still provides good performances in controlling the FPR and detecting the risk-CpG sites under the model misspecification setting. To further evaluate HIRE’s performance on experimentally mixed samples, we conducted another semi-simulated dataset that includes six samples mixed with six purified cell types in predetermined proportions [24] . Once again, HIRE successfully recovers the six underlying reference cell types and estimates the cellular compositions well (see Methods). Real data analysis HIRE also provides greater insight into real data than previous studies. The rheumatoid arthritis (RA) dataset [3] contains methylation profiles collected from the whole blood of 354 patients with RA and 335 normal participants. In addition to the RA status, other attributes such as gender, smoking history, age, and batch information are available. We first corrected the batch effects and then applied HIRE to the dataset (Methods). Figure 3a displays the p-values regarding the association with the RA status for each CpG site in each cell type, in which HIRE selected six cell types (Supplementary Fig. 63a ), consistent with the number of cell types in the previous study [13] . Despite potential batch effects and biological variability, three of the six cell types can be matched to known blood cell references—cell type 1 was matched to CD4+ T cells, cell types 2 and 4 were matched to neutrophils, and the remaining three cell types cannot be aligned to the references (Methods and Supplementary Fig. 64 ). HIRE detected 63 risk-CpG sites in cell type 3—the largest number of associations across all cell types—but no risk-CpG sites in cell type 1 (Supplementary Table 5 ). Therefore, the disease status affected some but not necessarily all cell types. Note that the significant CpG site cg06373940 called by HIRE is located on gene ERCC3 . The level of ERCC3 ’s corresponding protein has been reported to increase in RA synovium [25] . Moreover, we found that five CpG sites had a significant association with smoking history (Supplementary Fig. 65 and Supplementary Table 6 ). One of them is cg05575921, which was recently linked to smoking in two other independent studies of blood samples [26] , [27] . However, these findings were missed by the association detection at the aggregate level in previous analyses of the same dataset [11] , [13] . The p-value density plots and Q-Q plots for the commonly used methods are also displayed in Fig. 3c–n ; they present patterns similar to those observed in the simulation study except for an obvious overcorrection by ReFACTor. Fig. 3 Application of HIRE and commonly used methods to two real methylation datasets: RA and GALA II. Source data are provided as a Source Data file. a Cell-type-specic association pattern with RA status detected by HIRE in the RA dataset. Darkness represents the −log10(p−value). b Cell-type-specic association pattern with gender detected by HIRE in the GALA II dataset. The darkness represents the −log10(p−value). c – h The p-value density plots for association with RA status in the RA dataset for c HIRE, d unadjusted analysis, e SVA, f RefFreeEWAS, g EWASHer, and h ReFACTor. i - n Q-Q plots for association with RA status in the RA dataset for i HIRE, j unadjusted analysis, k SVA, l RefFreeEWAS, m EWASHer, and n ReFACTor Full size image The high resolution provided by HIRE makes it a powerful tool for EWAS studies. Rahmani et al. used ReFACTor [13] to analyze the GALA II blood methylation dataset [28] , which consists of 573 samples collected from a pediatric Latino population. Each sample includes the gender information and belongs to one of the following four populations: Mexican, Mixed Latino, Puerto Rican, and Other Latino. We applied HIRE to the dataset to investigate whether any cell-type-specific CpG sites were associated with gender and ethnicity. We created three dummy variables to represent the four ethnic groups. By taking the indicators of ethnicity as phenotypes in the model, HIRE automatically and simultaneously accounts for the population differences in cell composition and cell-type-specific methylation levels. HIRE correctly selected the number of cell types as six as reported in the previous study [13] (Supplementary Fig. 63b ). According to cell-type alignment, cell types 1 and 5 can be annotated as CD4+ T cells; cell types 2, 3, and 4 belong to neutrophils; and cell type 6 was annotated as CD56+ natural killer cell (CD56+ NK) using the references (Supplementary Fig. 66 ). HIRE found that 1936 CpG sites were associated with ethnicity across all cell types (Supplementary Fig. 67 ) and identified 14, 52, 155, 15, 18, and 14 risk-CpG sites for gender in cell types 1–6, respectively (Fig. 3b ). Gene set enrichment analysis showed that the genes that harbored risk-CpG sites for gender were significantly enriched in seven canonical pathways (Supplementary Table 7 ), of which the PID_CMYB_PATHWAY was ranked the highest. The transcription factor c − MYB in the PID_CMYB_PATHWAY enhances the progression of breast cancer [29] ; therefore, the different occurrence rates of breast cancers in men and women may be linked to the differences at the epigenome level. In comparison, only one pathway was found to be enriched with the genes that host the risk-CpG sites claimed by ReFACTor at the aggregate level (Supplementary Table 8 ). All of these observations highlight the importance of the finer-scale resolutions of HIRE.\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"In reality, the phenotype may affect a risk-CpG site in some but not all of the cell types. HIRE can detect the cell-type-specific association pattern with each phenotype for EWAS. The identification of cell-type-specific risk-CpG sites will help epigenetic therapies to target the affected cell types in a more effective manner. Statistically, instead of assuming fixed reference methylomes for all samples as the existing methods do [9] , [13] , [16] , HIRE allows each sample’s cell-type-specific methylation profiles to depend on its phenotypes. As a result, HIRE correctly models the multiplicative effects of the cellular compositions on the observed methylation levels, whereas the existing approaches all misspecify the cellular compositions as additive effects (Methods). As a result, HIRE enables the detection of cell-type-specific risk-CpG sites that cannot be feasibly detected with existing state-of-the-art methods. As a byproduct, HIRE also improves the statistical power of association detection at the aggregate level relative to existing state-of-the-art methods. Computationally, the time complexity of one iteration by HIRE is O ( nmKp + nK 3 ), which thus provides fast convergence when K is moderate. The statistical and computational advantages equip HIRE to be scaled up for large-cohort EWAS. So far, in the EWAS community, no gold-standard exists for the comparison of various methods. Ideally, we would like to have epigenetic spike-in experiments in which purified cell types are isolated, CpG-sites are epigenetically edited on a per-cell-type basis, and cell types are finally mixed in predetermined proportions. Given such experiments, the underlying knowledge of which CpGs are differentially methylated in each cell type and the cell mixing proportions for each sample are known. However, biotechnologies for epigenetic editing, such as CRISPR-Cas, are still not mature at this stage, with many off-target modifications [30] . Therefore, most computational EWAS studies refer to numerical simulation studies rather than to experimental studies when evaluating the performance of their algorithms [12] , [13] . Here, we follow the example of previous comparative studies and design our simulation studies to serve as the computational counterpart of experimental spike-in studies. With the rapid advances in epigenetic editing, we hope the community can devote greater effort in the near future to the creation of a gold-standard dataset, such as those generated in the early years for gene expression microarray studies [31] . The beta-values that represent methylation levels always lie between zero and one. As previous approaches to EWAS often assume normal distribution for the beta-values and show good performances in real applications [9] , [13] , in HIRE, we also assume that the beta-values follow a normal distribution. Consequently, the fitted methylation level may lie outside the range of [0, 1]. Nevertheless, we do in fact constrain the baseline methylation profiles μ jk s to the closed interval [0, 1] and force the cellular compositions p ki s to be non-negative and to add up to one: \\\\(\\\\mathop {\\\\sum}_{k = 1}^K {p_{ki}} = 1\\\\) . As a result, because the phenotypes have no effect on most CpG sites, most observations, O ji s, have their means \\\\(\\\\mathop {\\\\sum}_{k = 1}^K {\\\\mu _{jk}} p_{ki}\\\\) s in [0, 1]. In fact, for both the RA dataset and the GALA II dataset, more than 99.99% of the fitted methylation values \\\\(\\\\hat O_{ji}\\\\) s based on HIRE estimates lie between zero and one. Therefore, the normal assumption fits the data reasonably well and does not have a large effect on the performance of HIRE. One major issue for all of the cell-type deconvolution methods is that deconvolution cannot be achieved if the cellular compositions do not vary among samples. For example, assuming that the samples are mixtures of two cell types and p i = p for all of the samples, then the observed methylation profile O i equals u i 1 p 1 + u i 2 p 2 = ( u i 1 + p 2 C ) p 1 + ( u i 2 − p 1 C ) p 2 := \\\\(\\\\widetilde {\\\\mathbf{u}}_{i1}p_1 + \\\\widetilde {\\\\mathbf{u}}_{i2}p_2\\\\) for any constant C . As a result, u i 1 and u i 2 are not estimable. In our paper, we show mathematically that HIRE is identifiable under mild conditions in Theorem 1 and that condition (b) of Theorem 1 formulates the requirement for the variability of the cellular compositions (Methods). HIRE can accurately estimate cellular compositions of tissues with great cellular heterogeneity, such as blood. Although the mild conditions in Theorem 1 are easily met for real DNA methylation data, identification of both sufficient and necessary conditions for model identifiability is a theoretically interesting and challenging statistical problem that we will investigate in a future study. HIRE requires a moderate sample size to obtain precise estimates because HIRE needs to learn (1 + 2 K + qK ) m + ( K − 1) n parameters with a total of mn observed values. Our simulation studies show that HIRE performs very well at the aggregate level with 180 samples (Table 1 ). If the sample size drops below 150, say to 120, HIRE can still control the FPR well but begins to lose power (Supplementary Table 9 ). For small sample sizes, we have also developed a special case of HIRE by reparameterizing all \\\\(\\\\sigma _{gk}^2\\\\) s as one single parameter σ 2 , and we found that such a variance-stabilized approach can achieve even better inflation control (see Supplementary Figs. 71 – 76 ) and power comparable to HIRE (see Supplementary Table 10 ). Like the two datasets analyzed in the real application, a typical sample size for a current EWAS exceeds 500, thus guaranteeing a high TPR for HIRE. Given the decreasing cost of EWAS, we recommend that researchers collect at least 200 samples for their studies for association detection at the aggregate level and 600 samples for identification of cell-type-specific risk-CpG sites. A larger sample size can further boost the power. With the popularity of EWAS, we believe that HIRE will be widely applied, and we hope that HIRE can motivate more researchers to mine out finer-scale results from EWAS.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Multiplicative effects of cellular composition on methylation In this section, we illustrate that the effects of the cell-type composition are actually multiplicative. Let us assume that the beta-values that represent the methylation levels are observed across m CpG sites for n samples. As the measured sample comprises cells of various types, the observed beta-value is a weighted average of the mean methylation levels of distinct cell types, and the weights correspond to the proportions of each cell type. Let O ji denote the measurement at CpG site j for sample i . If we assume that there exist K cell types in all samples and that the mean methylation level for CpG site j in cell type k is μ jk , then $$O_{ji} = \\\\mathop {\\\\sum}\\\\limits_{k = 1}^K {\\\\mu _{jk}} p_{ki} + \\\\epsilon _{ji},$$ where p ki is the proportion of cell type k in sample i with a natural constraint \\\\(\\\\mathop {\\\\sum}_{k = 1}^K {p_{ki}} = 1\\\\) , and \\\\(\\\\epsilon _{ji}\\\\) is a random error. Let us consider a case-control EWAS. Without loss of generality, we assume that CpG site j is differentially methylated between cases and controls in cell type 1 with a mean shift δ j 1 and that it is not differentially methylated in the remaining cell types. As a result, for case samples, $$O_{ji} = (\\\\mu _{j1} + \\\\delta _{j1})p_{1i} + \\\\mathop {\\\\sum}\\\\limits_{k = 2}^K {\\\\mu _{jk}} p_{ki} + \\\\epsilon _{ji} = \\\\delta _{j1}p_{1i} + \\\\mathop {\\\\sum}\\\\limits_{k = 1}^K {\\\\mu _{jk}} p_{ki} + \\\\epsilon _{ji}.$$ If we then use Z i to indicate the case-control status of sample i , the observed methylation level becomes $$O_{ji} = \\\\delta _{j1}p_{1i}Z_i + \\\\mathop {\\\\sum}\\\\limits_{k = 1}^K {\\\\mu _{jk}} p_{ki} + \\\\epsilon _{ji}.$$ (3) Therefore, the proportions of cell type 1— p 1 i , i = 1, …, n —have multiplicative effects rather than additive effects on the mean difference between the case and control samples. The existing methods, which either estimate the cell type proportions explicitly or approximate them implicitly with surrogate variables, add the estimated proportions and the case-control indicator Z i as the covariates to the regression as follows: $$O_{ji} = \\\\alpha _j + \\\\tau _jZ_i + \\\\mathop {\\\\sum}\\\\limits_{k = 1}^{K - 1} {b_{jk}} \\\\hat p_{ki} + \\\\epsilon _{ji},$$ (4) where b jk s are the regression coefficients. As a result, CpG site j is called differentially methylated on the basis of hypothesis testing for τ j = 0. In general, τ j in Eq. ( 4 ) is not equal to δ j 1 in Eq. ( 3 ). Please see the Supplementary Notes for a numerical example. Moreover, testing for τ j = 0 loses the information regarding cell type in which CpG site j may be at risk. To account for the multiplicative effects, we propose the HIRE model that conserves the individual cell-type level information, which is introduced in the next section. The HIRE model HIRE uses a hierarchical model to closely follow the data generation process for the EWAS data. To begin, we assume that the baseline methylation level for CpG site j in cell type k is μ jk . For sample i with phenotypes x i = ( x i 1 , …, x iq ), the mean methylation value for CpG site j in cell type k is assumed to be \\\\(\\\\mu _{jk} + \\\\mathop {\\\\sum}_{\\\\ell = 1}^q {\\\\beta _{jk\\\\ell }} x_{i\\\\ell }\\\\) . In other words, the phenotypes have linear effects where \\\\(\\\\beta _{jk\\\\ell }\\\\) characterizes the influence of phenotype \\\\(\\\\ell \\\\) on CpG site j in cell type k . Let u ijk represent the signal from CpG site j in cell type k for sample i with x i . We assume that u ijk follows a normal distribution with mean \\\\(\\\\mu _{jk} + \\\\mathop {\\\\sum}_{\\\\ell = 1}^q {\\\\beta _{jk\\\\ell }} x_{i\\\\ell }\\\\) and standard deviation σ jk , $$u_{ijk}\\\\sim N \\\\left (\\\\mu _{jk} + \\\\mathop {\\\\sum}\\\\limits_{\\\\ell = 1}^q {\\\\beta _{jk\\\\ell }} x_{i\\\\ell },\\\\sigma _{jk}^2\\\\right).$$ (5) After u ijk s are generated for all of the K cell types, the observed methylation value O ji is sampled as follows: $$O_{ji}\\\\sim N\\\\left(\\\\mathop {\\\\sum}\\\\limits_{k = 1}^K {u_{ijk}} p_{ki},\\\\sigma _{\\\\epsilon j}^2\\\\right).$$ (6) Collectively, O = { O ji : 1 ≤ j ≤ m , 1 ≤ i ≤ n } denote the observed data; u = {( u ij 1 , …, u ijK ) T : 1 ≤ i ≤ n , 1 ≤ j ≤ m } are the missing data; and μ j = ( μ j 1 , …, μ jK ) T , \\\\({\\\\mathbf{B}}^{(j)} = (\\\\beta _{jk\\\\ell })_{K \\\\times q}\\\\) , \\\\(\\\\sigma _{\\\\epsilon j}^2\\\\) , the diagonal matrix \\\\(\\\\Sigma _j = diag(\\\\sigma _{j1}^2, \\\\ldots ,\\\\sigma _{jK}^2)\\\\) for j = 1, …, m , and p i = ( p 1 i , …, p Ki ) T for i = 1, …, n are the parameters. With \\\\({\\\\mathbf{\\\\Theta }} = \\\\{ {\\\\mathbf{p}}_i,{\\\\boldsymbol{\\\\mu }}_j,{\\\\mathbf{B}}^{(j)},\\\\Sigma _j,\\\\sigma _{\\\\epsilon ,j}^2:1 \\\\le j \\\\le m,1 \\\\le i \\\\le n\\\\} \\\\) , the complete data log-likelihood function, l c , can be expressed as follows: $$\\\\begin{array}{l}l_c({\\\\mathbf{\\\\Theta }}|{\\\\mathbf{O}},{\\\\mathbf{u}}) = \\\\mathop {\\\\sum}\\\\limits_{i = 1}^n {\\\\mathop {\\\\sum}\\\\limits_{j = 1}^m {\\\\left\\\\{ { - \\\\frac{1}{2}{\\\\mathrm{log}}\\\\sigma _{\\\\epsilon ,j}^2 - \\\\frac{{(O_{ji} - {\\\\mathbf{u}}_{ij}^T{\\\\mathbf{p}}_i)^2}}{{2\\\\sigma _{\\\\epsilon ,j}^2}}} \\\\right.} } - \\\\frac{1}{2}\\\\mathop {\\\\sum}\\\\limits_{k = 1}^K {\\\\mathrm{log}}\\\\sigma _{jk}^2\\\\\\\\ \\\\left. { - \\\\frac{1}{2}({\\\\mathbf{u}}_{ij} - {\\\\boldsymbol{\\\\mu }}_j - {\\\\mathbf{B}}^{(j)}{\\\\mathbf{x}}_i)^T\\\\Sigma _j^{ - 1}({\\\\mathbf{u}}_{ij} - {\\\\boldsymbol{\\\\mu }}_j - {\\\\mathbf{B}}^{(j)}{\\\\mathbf{x}}_i)} \\\\right\\\\} + Constant.\\\\end{array}$$ Accordingly, we develop a generalized expectation-maximization algorithm [32] to estimate the parameters. In the expectation-maximization algorithm, a good initialization can lead to faster convergence than random starts. We adopt the cellular composition estimations from the methylation matrix decomposition algorithm [16] with slight modifications as the initializations. The initial values for the baseline methylation profiles μ jk are accordingly estimated by simple linear regressions. As the number of risk-CpG sites is often small, all of the phenotype effects \\\\(\\\\beta _{jk\\\\ell }\\\\) are set to zero at the beginning. For the standard deviations, the initial values are randomly sampled from inverse gamma distributions with small means. We choose the number of cell types K by using a variant of the penalized Bayesian information criterion (pBIC) [20] (see details in Supplementary Methods ). For each phenotype \\\\(\\\\ell \\\\) , we can conduct the hypothesis test \\\\(H_0:\\\\beta _{jk\\\\ell } = 0\\\\) versus \\\\(H_1:\\\\beta _{jk\\\\ell } \\\\ \\\\ne \\\\ 0\\\\) for any cell type k and any CpG site j . Combining Eqs. ( 5 ) and ( 6 ), we obtain the following equations: $$E\\\\left[ {O_{ji}} \\\\right] = \\\\mu _{j1} + \\\\mathop {\\\\sum}\\\\limits_{k = 2}^K {(\\\\mu _{jk} - \\\\mu _{j1})} p_{ki} + \\\\mathop {\\\\sum}\\\\limits_{k = 1}^K {\\\\mathop {\\\\sum}\\\\limits_{\\\\ell = 1}^q {\\\\beta _{jk\\\\ell }} } x_{i\\\\ell }p_{ki},\\\\:\\\\:\\\\:i = 1, \\\\ldots ,n.$$ (7) We can then take ( O j 1 , …, O jn ) as the response vector and concatenate 1 n , ( p k 1 , …, p kn ) ( k = 2, …, K ) and \\\\((x_{1\\\\ell }p_{k1}, \\\\ldots ,x_{n\\\\ell }p_{kn})\\\\) \\\\((\\\\ell = 1, \\\\ldots ,q;k = 1, \\\\ldots ,K)\\\\) to a n × ( p + 1) · K design matrix in the linear regression. We plug in the estimated cellular compositions \\\\(\\\\hat p_{ki\\\\ell }\\\\) and conduct the hypothesis test for \\\\(\\\\beta _{jk\\\\ell } = 0\\\\) using the two-sided t-tests in the linear models. We claim that CpG site j has an association with phenotype \\\\(\\\\ell \\\\) at the aggregate level if phenotype \\\\(\\\\ell \\\\) affects CpG site j in at least one of the K cell types. Note that in the regression we incorporate the estimated cellular compositions into the linear model as multiplicative effects rather than additive effects. More technical details of the method and the algorithm are available in the Supplementary Methods . Data simulation We compared the performance of HIRE with five previous methods—unadjusted analysis, SVA, RefFreeEWAS, EWASHer, and ReFACTor—in 18 simulation settings. We set the sample size n to 180, 300, and 600 and let the underlying cell type number K be 3, 5, and 7. For each pair of ( n , K ), we investigated the true null case and the true alternative case. As a result, we have in total 3 (the number of sample sizes) × 3 (the number of cell types) × 2 (the true null case and the true alternative case) = 18 simulation settings. For each setting, we considered 10,000 CpG sites and simultaneously accounted for the following factors. Cell lineage. We first constructed the baseline methylation matrix μ = ( μ jk ) m × K , in which each column corresponds to the baseline methylation levels of a cell type. To mimic the phenomenon in which cell types from the same lineage have similar methylation profiles, we assumed that K sim of the total K cell types were similar. Specifically, without loss of generality, we assumed that the first K sim cell types came from the same cell lineage and that the remaining K − K sim cell types are irrelevant to one another. We set K sim to 2, 2, and 3 for K = 3, 5, and 7, respectively. We generated μ jk for cell types k = 1, K sim + 1, …, K from the beta distribution beta (3, 6) on each CpG site j independently. For each of the remaining cell types k ′ = 2, …, K sim , we randomly selected 20% of the CpG sites and drew their μ jk ′ s independently from beta (3, 6); and for the remaining 80% of CpG sites, we let their μ jk ′ be μ j 1 plus a very small randomness, thus inducing the similarities among cell types 1 to K sim . Discrete and continuous phenotypes. We further generated a discrete and a continuous phenotype x = ( x 1 , x 2 ) T for each individual i ( i = 1, …, n ). We let the first n /3 individuals be the control samples with x i 1 = 0 for i = 1, …, n /3 and the remaining 2 n /3 individuals serve as cases with x i 1 = 1 for i = n /3 + 1, …, n . The continuous phenotypes x 2 = ( x 12 , …, x i 2 , …, x n 2 ) T were independently drawn from a Unif (20, 50) to act as age. Phenotype effects with different magnitudes and directions. We then simulated the phenotype effect \\\\(\\\\beta _{jk\\\\ell }\\\\) of each phenotype \\\\(\\\\ell \\\\) on CpG site j in cell type k . For the true null cases, all of the \\\\(\\\\beta _{jk\\\\ell }\\\\) s are zero. For a true alternative setting, we set nonzero phenotype effects as follows. For phenotype 1—the case/control status, we let it affect the first 10 CpG sites in all of the cell types: β jk 1 ≠ 0 for j = 1, …, 10 and k = 1, …, K . We then assumed that the next 10 CpG sites were influenced by the disease status in the first K sim cell types which come from the same lineage but not the other cell types: β jk 1 ≠ 0 ( k = 1, …, K sim ) and β jk 1 = 0 ( k = K sim + 1, …, K ) for any j = 11, …, 20. Furthermore, for cell type k ∈ { K sim + 1, …, K }, we let the disease status affect CpG sites j = 20 + 10( k − K sim − 1) + 1, …, 20 + 10( k − K sim ) only in cell type k . We generated the cell-type-specific effects of age in a similar fashion for CpG site loci 21 to 40 + 10( K − K sim ). For each nonzero β jk 1 , we let β jk 1 = r jk · ω jk , where ω jk ~ Unif (0.07, 0.15) and r jk takes values of 1 and −1 with equal probabilities. Thus, β jk 1 s can have both positive and negative effects. In the same spirit, we generated nonzero β jk 2 s with \\\\(r_{jk}^\\\\prime \\\\) s and \\\\(\\\\omega _{jk}^\\\\prime \\\\) s where \\\\(\\\\omega _{jk}^\\\\prime \\\\sim Unif(0.007,0.015)\\\\) . Association between phenotypes and cellular compositions. Notice that the phenotypes may be associated with the cellular composition. Therefore, when K = 3, we drew p i = ( p 1 i , …, p Ki ) from a Dirichlet distribution Dir (4, 4, 2 + 0.1 x i 2 ) if sample i is a control and p i ~ Dir (4, 4, 5 + 0.1 x i 2 ) if it is a case; when K = 5, we let p i ~ Dir (3, 3, 3, 3, 2 + 0.1 x i 2 ) for a control sample and p i ~ Dir (3, 3, 3, 3, 5 + 0.1 x i 2 ) for a case sample; and when K = 7, we sampled p i ~ Dir (1, 3, 3, 3, 2, 2, 2 + 0.1 x i 2 ) for controls and p i ~ Dir (1, 3, 3, 3, 2, 2, 5 + 0.1 x i 2 ) for cases. Finally, we generated the observed value O ji for CpG site j of sample i as follows: sample u ijk from N ( μ jk + β jk 1 x i 1 + β jk 2 x i 2 , 0.01 2 ) for k = 1, …, K ; and sample O ji from \\\\(N(\\\\mathop {\\\\sum}_{k = 1}^K {u_{ijk}} p_{ki},0.01^2)\\\\) . In case O ji lies outside the interval (0, 1), we truncate it to zero if O ji is lower than zero and to one if O ji is greater than one. Semi-simulated dataset including samples with known cell mix proportions The GEO dataset GSE110554 [24] contains purified cell-type-specific methylation profiles for six cell types: neutrophils, monocytes, B cells, CD4+ T, CD8+ T, and NK. Moreover, GSE110554 includes mixed samples whose methylation signals were aggregated from the six cell types with predetermined cell mix proportions. Therefore, because of the known cell type and cellular proportion information, GSE110554 is an ideal dataset with which to test HIRE’s performance. In GSE110554 , the number of mixed samples is much smaller than the typical size of an EWAS and, as discussed in the manuscript, HIRE usually requires hundreds of samples to obtain accurate and stable results. Therefore, to increase the sample size, we first generated a simulated methylation dataset with 600 samples using the purified methylation profiles. We focused on 10k CpG sites, including the 450 IDOL CpG sites, which were previously identified as the optimal library of CpG sites for estimation of leukocyte subtype proportions [24] , and another 9550 CpG sites whose methylation values across the purified cell types fell within the range of [0.2, 0.8] and had large standard deviations [11] . We then combined the 600 samples and six mixed samples (generated by method A) [24] available in GSE110554 to compose a semi-simulated dataset. After applying HIRE to the semi-simulated data, we annotated the estimated cell types based on the methylation profiles from GSE110554 . Supplementary Figure 69 shows the heatmap for the Pearson correlation matrix between inferred cell types and the underlying truth. The correlation signals on the diagonal are the strongest in each row. HIRE successfully recovers the six underlying cell types. We also compared the estimated cellular compositions with the underlying true proportions for the six mixed samples. Each panel in Supplementary Fig. 70 displays a scatter plot between the cellular proportion estimates and the true mix proportions for a given cell type; they all indicate that HIRE obtains good estimates for cellular compositions. Cell type matching protocol Assume that we have the reference methylation profiles for the H annotated cell types. We first denote the methylation profile for cell type h as ϕ h = ( ϕ 1 h , …, ϕ mh ). We aim to annotate μ k using the references. Following the previous study [33] , first, we calculate the cosine similarity, the Pearson correlation, and the Spearman correlation between μ k and ϕ h for each cell type h ∈ {1, …, H }. Notice that the three similarity measures lie between −1 and 1, and a high positive value indicates great similarity between two vectors. Second, for each similarity measure \\\\(\\\\ell \\\\) ( \\\\(\\\\ell = 1,2,3\\\\) ), we identify the cell type \\\\(h_\\\\ell \\\\) that has the maximal degree of similarity with μ k . If at least two out of the three similarity measures identify the same reference cell type \\\\(\\\\tilde h\\\\) and their corresponding similarity values are greater than 0.5, then we annotate μ k with the reference cell type \\\\(\\\\tilde h\\\\) . Otherwise, μ k is believed to belong to a new cell type that is not included in the references. We repeat the above process for each methylation profile μ k estimated from HIRE. Blood cell references The two real data sets analyzed in our applications were obtained from whole blood. Therefore, we prepared the references from a whole blood methylation study [34] with GEO accession code GSE35069 . The study included seven isolated blood cell subpopulations—CD4+ T cells, CD8+ T cells, CD14+ monocytes, CD19+ B cells, CD56+ NK cells, neutrophils, and eosinophils—for six individuals. Accordingly, we define the reference profile ϕ h for cell type h as the average methylation profile of these individuals, i.e., \\\\({\\\\boldsymbol{\\\\phi }}_h: = \\\\frac{1}{6}\\\\mathop {\\\\sum}_{i = 1}^6 {{\\\\boldsymbol{\\\\phi }}_{hi}} \\\\) . Data preprocessing The RA dataset is publicly available in GEO with accession number GSE42861 . The dataset measures the methylation levels of the whole blood. The methylation data have been normalized by Illumina’s control probe scaling procedure (see Liu et al. [3] “Illumina 450K microarray data preprocessing” section for details). The dataset includes 689 samples, and the RA status, age, gender, smoking history, and batch information are available for each sample. We removed two samples GSM1051535 and GSM1051691 because their smoking information is missing. CpG sites with a high methylation mean (>0.8) and a low methylation mean (<0.2) were discarded [11] , [13] . We adjusted the data for batch effects using COMBAT [35] . The correction process was justified because we did not observe a high degree of co-linearity between the RA status and the batches (Supplementary Fig. 68 ). The 10,000 most variable CpG sites were kept. For the RA status, we denoted RA patients with 1 and the normal control subjects with 0; we represented men with 1 and women with 0; for the smoking history, we used (0, 0, 0) to refer to “never,” (1, 0, 0) to “ex,” (0, 1, 0) to “current,” and (0, 0, 1) to “occasional” smokers. We downloaded the GALA II dataset from Gene Expression Omnibus (GEO) with accession number GSE77716 . The dataset contains the whole-blood DNA methylation beta-values from 573 samples. The beta-values have been normalized by SWAN [36] and corrected for batch effects by COMBAT [35] . There are two types of covariates: gender and ethnicity. Ethnicity includes Mexican, Mixed Latino, Puerto Rican, and Other Latino. Out of the 573 samples, one sample “GSM2057284” has no gender information, so we removed it. As suggested by previous studies [11] , [13] , CpG sites with a mean methylation value of less than 0.2 or higher than 0.8 were filtered out. We selected the 10,000 most variable of the remaining CpG sites. For gender, we denoted men with 1 and women with 0. For the ethnicity variables, we used three dummy variables to represent the four ethnicity categories. In particular, (0, 0, 0), (1, 0, 0), (0, 1, 0), and (0, 0, 1) corresponded to Mexican, Mixed Latino, Puerto Rican, and Other Latino, respectively. For ReFACTor and EWASHer, according to their rules, we first filtered out CpG sites that were consistently hypomethylated or consistently hypermethylated and then regressed out the known covariates. We finally used the residuals to perform their analysis. Note that in their software these steps are processed automatically. For RefFreeEWAS, SVA, and the unadjusted analysis, the phenotypes and the covariates were regarded as the fixed effects in the regression model. In detail, for ReFACTor, in both GALA II and RA datasets, the cell type number “K” was specified to be six, which was the same as in their paper [13] . For RefFreeEWAS, we fixed the dimensionality of latent space “d” at six in the real data. For SVA, we also fixed the number of surrogate variables to six. Gene enrichment analysis was carried out on the Broad Institute website http://software.broadinstitute.org/gsea/msigdb/annotate.jsp . The canonical pathways were selected as the basis gene sets, and only pathways with a false discovery rate of less than 0.05 were reported. Identifiability of HIRE Although the non-negative matrix factorization (NNMF) O = μ P has been widely applied in cell type deconvolution [16] , where O is the observed methylation matrix, μ is the unknown methylation profile, and P is the unknown cellular compositions, model identifiability is rarely discussed. During the review period of our paper, Rahmani et al. [37] provided a setting under which the NMMF model is not identifiable. Why then does NNMF always provide satisfactory cell type deconvolution results in real practice, and why can HIRE estimate all those parameters well? Here, we show mathematically that the HIRE model is identifiable under mild conditions that are easily met in reality. Let us first introduce some notations and definitions. In the HIRE model, the whole parameter set is denoted by \\\\({\\\\mathbf{\\\\Theta }}: = \\\\{ {\\\\mathbf{P}}_i,{\\\\boldsymbol{\\\\mu }}_j,{\\\\mathbf{B}}_\\\\ell ^{(j)},\\\\sigma _{jk}^2,\\\\sigma _{\\\\epsilon j}^2:1 \\\\le j \\\\le m,1 \\\\le i \\\\le n,1 \\\\le k \\\\le K,1 \\\\le \\\\ell \\\\le q\\\\} \\\\) , where p i is the cellular composition vector of sample i , μ j is the baseline methylation vector of CpG site j , \\\\({\\\\mathbf{B}}_\\\\ell ^{(j)}\\\\) is the phenotype \\\\(\\\\ell \\\\) effect vector on CpG site j , \\\\(\\\\sigma _{jk}^2\\\\) is the cell-type- k noise variance on CpG site j , and \\\\(\\\\sigma _{\\\\epsilon j}^2\\\\) is the overall noise variance on CpG site j . The observed data in our study are the methylation matrix O = { O ij :1 ≤ i ≤ n , 1 ≤ j ≤ m } and the covariate matrix \\\\({\\\\mathbf{X}} = ({\\\\mathbf{x}}_1, \\\\ldots ,{\\\\mathbf{x}}_\\\\ell , \\\\ldots ,{\\\\mathbf{x}}_q)\\\\) , where \\\\({\\\\mathbf{x}}_\\\\ell \\\\) is the column vector that indicates phenotype- \\\\(\\\\ell \\\\) for the n samples. The observed likelihood function \\\\(({\\\\mathbf{\\\\Theta }}|{\\\\mathbf{O}}) = \\\\mathop {\\\\prod}_{i = 1}^n {\\\\mathop {\\\\prod}_{j = 1}^m N } (O_{ji}:{\\\\mathbf{P}}_i^T {\\\\boldsymbol{\\\\mu }}_{j} + \\\\mathop {\\\\sum}_{\\\\ell = 1}^q {x_{i\\\\ell }} {\\\\mathbf{P}}_i^T {\\\\mathbf{B}}_\\\\ell ^{(j)},\\\\mathop {\\\\sum}_{k = 1}^K {\\\\sigma _{jk}^2} \\\\, P_{ik}^2 + \\\\sigma _{\\\\epsilon j}^2)\\\\) (see Eq. (S7) in the Supplementary Methods ), where N ( O : η , τ 2 ) indicates the normal density with mean η and variance τ 2 at value O . We further define 1 K = (1, 1, …, 1) T as a K -dimension column vector with all entries being one, an n by K matrix J 1 as \\\\({\\\\mathbf{1}}_n{\\\\mathbf{1}}_K^T\\\\) , and an n by K matrix \\\\({\\\\mathbf{J}}_{x_\\\\ell }\\\\) as \\\\({\\\\mathbf{x}}_\\\\ell {\\\\mathbf{1}}_K^T\\\\) for each \\\\(1 \\\\le \\\\ell \\\\le q\\\\) . We use ⊙ to represent the entry-wise matrix product for two matrices M and N with the same dimension, i.e., \\\\(({\\\\mathbf{M}} \\\\odot {\\\\mathbf{N}})_{ij}: = {\\\\mathbf{M}}_{ij}{\\\\mathbf{N}}_{ij}\\\\) . Theorem 1. If (a) for each cell type k, there exists a CpG site r k such that \\\\({\\\\mathbf{B}}_\\\\ell ^{(r_k)} = 0\\\\) for any phenotype \\\\(\\\\ell \\\\) and \\\\(\\\\mu _{r_kk} = 1\\\\) while \\\\(\\\\mu _{r_kk{\\\\prime}} = 0\\\\) for k ′ ≠ k , and (b) the cellular compositions P satisfies that \\\\(rank(({\\\\mathbf{J}}_1 \\\\odot {\\\\mathbf{P}}^T,{\\\\mathbf{J}}_{x_1} \\\\odot {\\\\mathbf{P}}^T, \\\\ldots ,{\\\\mathbf{J}}_{x_\\\\ell } \\\\odot {\\\\mathbf{P}}^T, \\\\ldots ,{\\\\mathbf{J}}_{x_q} \\\\odot {\\\\mathbf{P}}^T)) = (q + 1)K\\\\) and \\\\(rank(({\\\\mathbf{1}}_n,{\\\\mathbf{P}}^T) \\\\odot ({\\\\mathbf{1}}_n,{\\\\mathbf{P}}^T)) = K + 1\\\\) , then the HIRE model is identifiable. In other words, \\\\(L({\\\\mathbf{\\\\Theta }}|{\\\\mathbf{O}}) = L(\\\\widetilde {\\\\mathbf{\\\\Theta }}|{\\\\mathbf{O}})\\\\) for any O implies \\\\({\\\\mathbf{\\\\Theta }} = \\\\widetilde {\\\\mathbf{\\\\Theta }}\\\\) . Proof: First, by integrating out all O elements except O ji , \\\\(L({\\\\mathbf{\\\\Theta }}|{\\\\mathbf{O}}) = L(\\\\widetilde {\\\\mathbf{\\\\Theta }}|{\\\\mathbf{O}})\\\\) implies \\\\(N(O_{ji}:{\\\\mathbf{P}}_i^T {\\\\boldsymbol{\\\\mu }}_{j} + \\\\mathop {\\\\sum}_{\\\\ell = 1}^q {x_{i\\\\ell }} {\\\\mathbf{P}}_i^T {\\\\mathbf{B}}_\\\\ell ^{(j)},\\\\mathop {\\\\sum}_{k = 1}^K {\\\\sigma _{jk}^2} P_{ki}^2 + \\\\sigma _{\\\\epsilon j}^2)\\\\) = \\\\(N(O_{ji}:\\\\widetilde {\\\\mathbf{P}}_i^T \\\\widetilde \\\\mu_{j} + \\\\mathop {\\\\sum}_{\\\\ell = 1}^q {x_{i\\\\ell }} \\\\widetilde {\\\\mathbf{P}}_i^T\\\\widetilde {\\\\mathbf{B}}_\\\\ell ^{(j)},\\\\mathop {\\\\sum}_{k = 1}^K {\\\\tilde \\\\sigma _{jk}^2} \\\\tilde P_{ki}^2 + \\\\tilde \\\\sigma _{\\\\epsilon j}^2)\\\\) . Because the univariate normal distribution is identifiable, we have $${\\\\mathbf{P}}_i^T{\\\\boldsymbol{\\\\mu }}_j + \\\\mathop {\\\\sum}\\\\limits_{\\\\ell = 1}^q {x_{i\\\\ell }} {\\\\mathbf{P}}_i^T{\\\\mathbf{B}}_\\\\ell ^{(j)} = \\\\widetilde {\\\\mathbf{P}}_i^T\\\\widetilde {\\\\boldsymbol{\\\\mu }}_j + \\\\mathop {\\\\sum}\\\\limits_{\\\\ell = 1}^q {x_{i\\\\ell }} \\\\widetilde {\\\\mathbf{P}}_i^T\\\\widetilde {\\\\mathbf{B}}_\\\\ell ^{(j)},$$ (8) $$\\\\mathop {\\\\sum}\\\\limits_{k = 1}^K {\\\\sigma _{jk}^2} P_{ki}^2 + \\\\sigma _{\\\\epsilon j}^2 = \\\\mathop {\\\\sum}\\\\limits_{k = 1}^K {\\\\tilde \\\\sigma _{jk}^2} \\\\tilde P_{ki}^2 + \\\\tilde \\\\sigma _{\\\\epsilon j}^2.$$ (9) Taking j = r k in Eq. ( 8 ), we have \\\\(LHS = {\\\\mathbf{P}}_i^T{\\\\boldsymbol{\\\\mu }}_{r_k} + \\\\mathop {\\\\sum}_{\\\\ell = 1}^q {x_{i\\\\ell }} {\\\\mathbf{P}}_i^T{\\\\mathbf{B}}_\\\\ell ^{(r_k)} = {\\\\mathbf{P}}_i^T{\\\\boldsymbol{\\\\mu }}_{r_k} = 0 + P_{ki}\\\\cdot 1 + 0 = P_{ki}\\\\) and similarly \\\\(RHS = \\\\tilde P_{ki}\\\\) , so \\\\(P_{ki} = \\\\tilde P_{ki}\\\\) , which holds for any i and k . Hence, we obtain \\\\({\\\\mathbf{P}} = \\\\widetilde {\\\\mathbf{P}}\\\\) . Next, we rewrite Eq. ( 8 ) into a matrix form. $$({\\\\mathbf{P}}_i^T,x_{i1}{\\\\mathbf{P}}_i^T, \\\\ldots ,x_{iq}{\\\\mathbf{P}}_i^T)\\\\left( {\\\\begin{array}{*{20}{c}} {{\\\\boldsymbol{\\\\mu }}_j} \\\\\\\\ {{\\\\mathbf{B}}_1^{(j)}} \\\\\\\\ \\\\vdots \\\\\\\\ {{\\\\mathbf{B}}_q^{(j)}} \\\\end{array}} \\\\right) = ({\\\\mathbf{P}}_i^T,x_{i1}{\\\\mathbf{P}}_i^T, \\\\ldots ,x_{iq}{\\\\mathbf{P}}_i^T)\\\\left( {\\\\begin{array}{*{20}{c}} {\\\\widetilde {\\\\boldsymbol{\\\\mu }}_j} \\\\\\\\ {\\\\widetilde {\\\\mathbf{B}}_1^{(j)}} \\\\\\\\ \\\\vdots \\\\\\\\ {\\\\widetilde {\\\\mathbf{B}}_q^{(j)}} \\\\end{array}} \\\\right),\\\\;\\\\;i = 1, \\\\ldots ,n.$$ By combining these n equations, it follows that $$\\\\begin{array}{l}({\\\\mathbf{J}}_1 \\\\odot {\\\\mathbf{P}}^T,{\\\\mathbf{J}}_{x_1} \\\\odot {\\\\mathbf{P}}^T, \\\\ldots ,{\\\\mathbf{J}}_{x_\\\\ell } \\\\odot {\\\\mathbf{P}}^T, \\\\ldots ,{\\\\mathbf{J}}_{x_q} \\\\odot {\\\\mathbf{P}}^T)\\\\left( {\\\\begin{array}{*{20}{c}} {{\\\\boldsymbol{\\\\mu }}_j} \\\\\\\\ {{\\\\mathbf{B}}_1^{(j)}} \\\\\\\\ \\\\vdots \\\\\\\\ {{\\\\mathbf{B}}_q^{(j)}} \\\\end{array}} \\\\right)\\\\\\\\ = ({\\\\mathbf{J}}_1 \\\\odot {\\\\mathbf{P}}^T,{\\\\mathbf{J}}_{x_1} \\\\odot {\\\\mathbf{P}}^T, \\\\ldots ,{\\\\mathbf{J}}_{x_\\\\ell } \\\\odot {\\\\mathbf{P}}^T, \\\\ldots ,{\\\\mathbf{J}}_{x_q} \\\\odot {\\\\mathbf{P}}^T)\\\\left( {\\\\begin{array}{*{20}{c}} {\\\\widetilde {\\\\boldsymbol{\\\\mu }}_j} \\\\\\\\ {\\\\widetilde {\\\\mathbf{B}}_1^{(j)}} \\\\\\\\ \\\\vdots \\\\\\\\ {\\\\widetilde {\\\\mathbf{B}}_q^{(j)}} \\\\end{array}} \\\\right).\\\\end{array}$$ (10) Because the rank of \\\\(A: = ({\\\\mathbf{J}}_1 \\\\odot {\\\\mathbf{P}}^T,{\\\\mathbf{J}}_{x_1} \\\\odot {\\\\mathbf{P}}^T, \\\\ldots ,{\\\\mathbf{J}}_{x_\\\\ell } \\\\odot {\\\\mathbf{P}}^T, \\\\ldots ,{\\\\mathbf{J}}_{x_q} \\\\odot {\\\\mathbf{P}}^T)\\\\) is ( q + 1) K (full column rank), A has a left inverse A −1 . Multiplying Eq. ( 10 ) by A −1 from the left on both sides, we obtain \\\\({\\\\boldsymbol{\\\\mu }}_j = \\\\widetilde {\\\\boldsymbol{\\\\mu }}_j\\\\) and \\\\({\\\\mathbf{B}}_\\\\ell ^{(j)} = \\\\widetilde {\\\\mathbf{B}}_\\\\ell ^{(j)}\\\\) for \\\\(1 \\\\le \\\\ell \\\\le q\\\\) . Therefore, we have \\\\({\\\\boldsymbol{\\\\mu }} = \\\\widetilde {\\\\boldsymbol{\\\\mu }}\\\\) , \\\\({\\\\mathbf{B}} = \\\\widetilde {\\\\mathbf{B}}\\\\) . In addition, because Eq. ( 9 ) holds for any i , we can also rewrite it into a matrix form. $$\\\\left( {\\\\begin{array}{*{20}{c}} 1 & {P_{11}^2} & \\\\ldots & {P_{K1}^2} \\\\\\\\ 1 & {P_{12}^2} & \\\\ldots & {P_{K2}^2} \\\\\\\\ \\\\vdots & \\\\vdots & \\\\ddots & \\\\vdots \\\\\\\\ 1 & {P_{1n}^2} & \\\\ldots & {P_{Kn}^2} \\\\end{array}} \\\\right)\\\\left( {\\\\begin{array}{*{20}{c}} {\\\\sigma _{\\\\epsilon j}^2} \\\\\\\\ {\\\\sigma _{j1}^2} \\\\\\\\ \\\\vdots \\\\\\\\ {\\\\sigma _{jK}^2} \\\\end{array}} \\\\right) = \\\\left( {\\\\begin{array}{*{20}{c}} 1 & {P_{11}^2} & \\\\ldots & {P_{K1}^2} \\\\\\\\ 1 & {P_{12}^2} & \\\\ldots & {P_{K2}^2} \\\\\\\\ \\\\vdots & \\\\vdots & \\\\ddots & \\\\vdots \\\\\\\\ 1 & {P_{1n}^2} & \\\\ldots & {P_{Kn}^2} \\\\end{array}} \\\\right)\\\\left( {\\\\begin{array}{*{20}{c}} {\\\\tilde \\\\sigma _{\\\\epsilon j}^2} \\\\\\\\ {\\\\tilde \\\\sigma _{j1}^2} \\\\\\\\ \\\\vdots \\\\\\\\ {\\\\tilde \\\\sigma _{jK}^2} \\\\end{array}} \\\\right)$$ The left matrix is equal to \\\\(({\\\\mathbf{1}}_n,{\\\\mathbf{P}}^T) \\\\odot ({\\\\mathbf{1}}_n,{\\\\mathbf{P}}^T)\\\\) which has a full column rank; therefore, it has a left inverse. Consequently, \\\\(\\\\sigma _{\\\\epsilon j}^2 = \\\\tilde \\\\sigma _{\\\\epsilon j}^2\\\\) and \\\\(\\\\sigma _{jk}^2 = \\\\tilde \\\\sigma _{jk}^2\\\\) . As a result, \\\\({\\\\mathbf{\\\\Theta }} = \\\\widetilde {\\\\mathbf{\\\\Theta }}\\\\) , and we have proven the identifiability of HIRE. \\\\(\\\\square \\\\) Conditions (a) and (b) are easily met for DNA methylation data. Condition (a) requires that for each cell type k , there exists a CpG site that is not associated with any phenotype and is only methylated in cell type k but not methylated in any other cell type. Given the 450K CpG sites assayed by the microarray, we can expect that such CpG sites are not absent at all. Moreover, condition (a) can also be relaxed to the condition that for each cell type k , there exists a CpG site r k such that \\\\({\\\\mathbf{B}}_\\\\ell ^{(r_k)} = 0\\\\) for any phenotype \\\\(\\\\ell \\\\) and \\\\(\\\\mu _{r_kk} = 1\\\\) while \\\\(\\\\mu _{r_kk\\\\prime } = 0\\\\) for k ′ ≠ k or there exists a CpG site r k such that \\\\({\\\\mathbf{B}}_\\\\ell ^{(r_k)} = 0\\\\) for any phenotype \\\\(\\\\ell \\\\) and \\\\(\\\\mu _{r_kk} = 0\\\\) while \\\\(\\\\mu _{r_kk{\\\\prime}} = 1\\\\) for k ′ ≠ k . The proof follows in a similar manner. For condition (b), intuitively, the rank requirement of \\\\(({\\\\mathbf{1}}_n,{\\\\mathbf{P}}^T) \\\\odot ({\\\\mathbf{1}}_n,{\\\\mathbf{P}}^T)\\\\) asks the cellular compositions to vary across subjects, which guards against the case in which all the subjects have the same cellular compositions and hence no cell type deconvolution is possible; the rank requirement on \\\\(({\\\\mathbf{J}}_1 \\\\odot {\\\\mathbf{P}}^T,{\\\\mathbf{J}}_{x_1} \\\\odot {\\\\mathbf{P}}^T, \\\\ldots ,{\\\\mathbf{J}}_{x_\\\\ell } \\\\odot {\\\\mathbf{P}}^T, \\\\ldots ,{\\\\mathbf{J}}_{x_q} \\\\odot {\\\\mathbf{P}}^T)\\\\) is the same requirement as those in a standard linear regression, which requires that no collinearity exists among the covariates. Because the sample size n is much larger than the underlying cell type number K and the phenotype number q , the two rank requirements can commonly be satisfied in reality. Reporting summary Further information on research design is available in the Nature Research Reporting Summary linked to this article.\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"The core research idea of this study is to propose a novel method called HIRE that addresses the limitations of existing epigenome-wide association study (EWAS) approaches by identifying cell-type-specific associations between CpG sites and phenotypes through a model that accounts for cellular heterogeneity as multiplicative effects, rather than additive ones, thereby enabling higher-resolution insights into the epigenetic mechanisms underlying diseases and phenotypic traits."},"entities":{"kind":"list like","value":["HIRE","epigenome-wide association study","EWAS","cell-type-specific associations","CpG sites","phenotypes","cellular heterogeneity","multiplicative effects","additive effects","higher-resolution insights","epigenetic mechanisms","diseases","phenotypic traits"],"string":"[\n \"HIRE\",\n \"epigenome-wide association study\",\n \"EWAS\",\n \"cell-type-specific associations\",\n \"CpG sites\",\n \"phenotypes\",\n \"cellular heterogeneity\",\n \"multiplicative effects\",\n \"additive effects\",\n \"higher-resolution insights\",\n \"epigenetic mechanisms\",\n \"diseases\",\n \"phenotypic traits\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1.","title":"Rakyan, V. 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Epigenetic signatures of cigarette smoking.Circ. Cardiovasc. Genet.9, 436–447 (2016).","link":"https://doi.org/10.1161%2FCIRCGENETICS.116.001506"},{"idx":"6.","title":"Wahl, S. et al. Epigenome-wide association study of body mass index, and the adverse outcomes of adiposity.Nature541, 81–86 (2017).","link":"https://doi.org/10.1038%2Fnature20784"},{"idx":"7.","title":"Teschendorff, A. E. et al. Age-dependent DNA methylation of genes that are suppressed in stem cells is a hallmark of cancer.Genome Res.20, 440–446 (2010).","link":"https://doi.org/10.1101%2Fgr.103606.109"},{"idx":"8.","title":"Horvath, S. DNA methylation age of human tissues and cell types.Genome Biol.14, 3156 (2013).","link":"https://link.springer.com/doi/10.1186/gb-2013-14-10-r115"},{"idx":"9.","title":"Houseman, E. A. et al. DNA methylation arrays as surrogate measures of cell mixture distribution.BMC Bioinformatics13, 86 (2012).","link":"https://link.springer.com/doi/10.1186/1471-2105-13-86"},{"idx":"10.","title":"Jaffe, A. E. & Irizarry, R. A. Accounting for cellular heterogeneity is critical in epigenome-wide association studies.Genome Biol.15, R31 (2014).","link":"https://link.springer.com/doi/10.1186/gb-2014-15-2-r31"},{"idx":"11.","title":"Zou, J., Lippert, C., Heckerman, D., Aryee, M. & Listgarten, J. Epigenome-wide association studies without the need for cell-type composition.Nat. Methods11, 309–311 (2014).","link":"https://doi.org/10.1038%2Fnmeth.2815"},{"idx":"12.","title":"McGregor, K. et al. An evaluation of methods correcting for cell-type heterogeneity in DNA methylation studies.Genome Biol.17, 84 (2016).","link":"https://link.springer.com/doi/10.1186/s13059-016-0935-y"},{"idx":"13.","title":"Rahmani, E. et al. Sparse PCA corrects for cell type heterogeneity in epigenome-wide association studies.Nat. Methods13, 443–445 (2016).","link":"https://doi.org/10.1038%2Fnmeth.3809"},{"idx":"14.","title":"Teschendorff, A. E. & Relton, C. L. Statistical and integrative system-level analysis of DNA methylation data.Nat. Rev. Genet.19, 129–147 (2017).","link":"https://doi.org/10.1038%2Fnrg.2017.86"},{"idx":"15.","title":"Accomando, W. P., Wiencke, J. K., Houseman, E. A., Nelson, H. H. & Kelsey, K. T. Quantitative reconstruction of leukocyte subsets using DNA methylation.Genome Biol.15, R50 (2014).","link":"https://link.springer.com/doi/10.1186/gb-2014-15-3-r50"},{"idx":"16.","title":"Houseman, E. A. et al. Reference-free deconvolution of DNA methylation data and mediation by cell composition effects.BMC Bioinformatics17, 259 (2016).","link":"https://link.springer.com/doi/10.1186/s12859-016-1140-4"},{"idx":"17.","title":"Zheng, X., Zhang, N., Wu, H.-J. & Wu, H. Estimating and accounting for tumor purity in the analysis of DNA methylation data from cancer studies.Genome Biol.18, 17 (2017).","link":"https://link.springer.com/doi/10.1186/s13059-016-1143-5"},{"idx":"18.","title":"Leek, J. T. & Storey, J. D. Capturing heterogeneity in gene expression studies by surrogate variable analysis.PLoS Genet.3, e161 (2007).","link":"https://doi.org/10.1371%2Fjournal.pgen.0030161"},{"idx":"19.","title":"Houseman, E. A., Molitor, J. & Marsit, C. J. Reference-free cell mixture adjustments in analysis of DNA methylation data.Bioinformatics30, 1431–1439 (2014).","link":"https://doi.org/10.1093%2Fbioinformatics%2Fbtu029"},{"idx":"21.","title":"Zheng, S. C. et al. Correcting for cell-type heterogeneity in epigenome-wide association studies: revisiting previous analyses.Nat. Methods14, 216–217 (2017).","link":"https://doi.org/10.1038%2Fnmeth.4187"},{"idx":"22.","title":"Rahmani, E. et al. Correcting for cell-type heterogeneity in DNA methylation: a comprehensive evaluation.Nat. Methods14, 218–219 (2017).","link":"https://doi.org/10.1038%2Fnmeth.4190"},{"idx":"23.","title":"Storey, J. D. & Tibshirani, R. Statistical significance for genomewide studies.Proc. Natl Acad. Sci. USA100, 9440–9445 (2003).","link":"https://doi.org/10.1073%2Fpnas.1530509100"},{"idx":"24.","title":"Salas, L. A. et al. An optimized library for reference-based deconvolution of whole-blood biospecimens assayed using the illumina humanmethylationepic beadarray.Genome Biol.19, 64 (2018).","link":"https://link.springer.com/doi/10.1186/s13059-018-1448-7"},{"idx":"25.","title":"Neumann, E. et al. Identification of differentially expressed genes in rheumatoid arthritis by a combination of complementary DNA array and rna arbitrarily primed-polymerase chain reaction.Arthritis Rheumatol.46, 52–63 (2002).","link":"https://doi.org/10.1002%2F1529-0131%28200201%2946%3A1%3C52%3A%3AAID-ART10048%3E3.0.CO%3B2-1"},{"idx":"26.","title":"Fasanelli, F. et al. Hypomethylation of smoking-related genes is associated with future lung cancer in four prospective cohorts.Nat. Commun.6, 10192 (2015).","link":"https://doi.org/10.1038%2Fncomms10192"},{"idx":"27.","title":"Ambatipudi, S. et al. Tobacco smoking-associated genome-wide DNA methylation changes in the epic study.Epigenomics8, 599–618 (2016).","link":"https://doi.org/10.2217%2Fepi-2016-0001"},{"idx":"28.","title":"Pino-Yanes, M. et al. Genetic ancestry influences asthma susceptibility and lung function among latinos.J. Allergy Clin. Immunol.135, 228–235 (2015).","link":"https://doi.org/10.1016%2Fj.jaci.2014.07.053"},{"idx":"29.","title":"Li, Y. et al. c-Myb enhances breast cancer invasion and metastasis through the wnt/β-catenin/axin2 pathway.Cancer Res.76, 3364–3375 (2016).","link":"https://doi.org/10.1158%2F0008-5472.CAN-15-2302"},{"idx":"30.","title":"Zhang, X.-H., Tee, L. Y., Wang, X.-G., Huang, Q.-S. & Yang, S.-H. Off-target effects in CRISPR/Cas9-mediated genome engineering.Mol. Ther. Nucleic Acids4, e264 (2015).","link":"https://doi.org/10.1038%2Fmtna.2015.37"},{"idx":"31.","title":"Bolstad, B. M., Irizarry, R. A., Åstrand, M. & Speed, T. P. A comparison of normalization methods for high density oligonucleotide array data based on variance and bias.Bioinformatics19, 185–193 (2003).","link":"https://doi.org/10.1093%2Fbioinformatics%2F19.2.185"},{"idx":"33.","title":"Kiselev, V. Yu, Yiu, A. & Hemberg, M. scmap: projection of single-cell RNA-seq data across data sets.Nat. Methods15, 359 (2018).","link":"https://doi.org/10.1038%2Fnmeth.4644"},{"idx":"34.","title":"Reinius, L. E. et al. Differential DNA methylation in purified human blood cells: implications for cell lineage and studies on disease susceptibility.PLoS ONE,7, e41361 (2012).","link":"https://doi.org/10.1371%2Fjournal.pone.0041361"},{"idx":"35.","title":"Johnson, W. E., Li, C. & Rabinovic, A. Adjusting batch effects in microarray expression data using empirical Bayes methods.Biostatistics8, 118–127 (2007).","link":"https://doi.org/10.1093%2Fbiostatistics%2Fkxj037"},{"idx":"36.","title":"Maksimovic, J., Gordon, L. & Oshlack, A. Swan: Subset-quantile within array normalization for illumina infinium humanmethylation450 beadchips.Genome Biol.13, R44 (2012).","link":"https://link.springer.com/doi/10.1186/gb-2012-13-6-r44"},{"idx":"37.","title":"Rahmani, E. et al. BayesCCE: a Bayesian framework for estimating cell-type composition from DNA methylation without the need for methylation reference.Genome Biol.19, 141 (2018).","link":"https://link.springer.com/doi/10.1186/s13059-018-1513-2"}],"string":"[\n {\n \"idx\": \"1.\",\n \"title\": \"Rakyan, V. K., Down, T. A., Balding, D. J. & Beck, S. Epigenome-wide association studies for common human diseases.Nat. Rev. Genet.12, 529–541 (2011).\",\n \"link\": \"https://doi.org/10.1038%2Fnrg3000\"\n },\n {\n \"idx\": \"2.\",\n \"title\": \"Verma, M. Epigenome-wide association studies (EWAS) in cancer.Curr. Genom.13, 308–313 (2012).\",\n \"link\": \"https://doi.org/10.2174%2F138920212800793294\"\n },\n {\n \"idx\": \"3.\",\n \"title\": \"Liu, Y. et al. Epigenome-wide association data implicate DNA methylation as an intermediary of genetic risk in rheumatoid arthritis.Nat. Biotechnol.31, 142–147 (2013).\",\n \"link\": \"https://doi.org/10.1038%2Fnbt.2487\"\n },\n {\n \"idx\": \"4.\",\n \"title\": \"Gao, X., Jia, M., Zhang, Y., Breitling, L. P. & Brenner, H. DNA methylation changes of whole blood cells in response to active smoking exposure in adults: a systematic review of DNA methylation studies.Clin. Epigenetics7, 113 (2015).\",\n \"link\": \"https://link.springer.com/doi/10.1186/s13148-015-0148-3\"\n },\n {\n \"idx\": \"5.\",\n \"title\": \"Joehanes, R. et al. Epigenetic signatures of cigarette smoking.Circ. Cardiovasc. Genet.9, 436–447 (2016).\",\n \"link\": \"https://doi.org/10.1161%2FCIRCGENETICS.116.001506\"\n },\n {\n \"idx\": \"6.\",\n \"title\": \"Wahl, S. et al. Epigenome-wide association study of body mass index, and the adverse outcomes of adiposity.Nature541, 81–86 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fnature20784\"\n },\n {\n \"idx\": \"7.\",\n \"title\": \"Teschendorff, A. E. et al. Age-dependent DNA methylation of genes that are suppressed in stem cells is a hallmark of cancer.Genome Res.20, 440–446 (2010).\",\n \"link\": \"https://doi.org/10.1101%2Fgr.103606.109\"\n },\n {\n \"idx\": \"8.\",\n \"title\": \"Horvath, S. DNA methylation age of human tissues and cell types.Genome Biol.14, 3156 (2013).\",\n \"link\": \"https://link.springer.com/doi/10.1186/gb-2013-14-10-r115\"\n },\n {\n \"idx\": \"9.\",\n \"title\": \"Houseman, E. A. et al. DNA methylation arrays as surrogate measures of cell mixture distribution.BMC Bioinformatics13, 86 (2012).\",\n \"link\": \"https://link.springer.com/doi/10.1186/1471-2105-13-86\"\n },\n {\n \"idx\": \"10.\",\n \"title\": \"Jaffe, A. E. & Irizarry, R. A. Accounting for cellular heterogeneity is critical in epigenome-wide association studies.Genome Biol.15, R31 (2014).\",\n \"link\": \"https://link.springer.com/doi/10.1186/gb-2014-15-2-r31\"\n },\n {\n \"idx\": \"11.\",\n \"title\": \"Zou, J., Lippert, C., Heckerman, D., Aryee, M. & Listgarten, J. Epigenome-wide association studies without the need for cell-type composition.Nat. Methods11, 309–311 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fnmeth.2815\"\n },\n {\n \"idx\": \"12.\",\n \"title\": \"McGregor, K. et al. An evaluation of methods correcting for cell-type heterogeneity in DNA methylation studies.Genome Biol.17, 84 (2016).\",\n \"link\": \"https://link.springer.com/doi/10.1186/s13059-016-0935-y\"\n },\n {\n \"idx\": \"13.\",\n \"title\": \"Rahmani, E. et al. Sparse PCA corrects for cell type heterogeneity in epigenome-wide association studies.Nat. Methods13, 443–445 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fnmeth.3809\"\n },\n {\n \"idx\": \"14.\",\n \"title\": \"Teschendorff, A. E. & Relton, C. L. Statistical and integrative system-level analysis of DNA methylation data.Nat. Rev. Genet.19, 129–147 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fnrg.2017.86\"\n },\n {\n \"idx\": \"15.\",\n \"title\": \"Accomando, W. P., Wiencke, J. K., Houseman, E. A., Nelson, H. H. & Kelsey, K. T. Quantitative reconstruction of leukocyte subsets using DNA methylation.Genome Biol.15, R50 (2014).\",\n \"link\": \"https://link.springer.com/doi/10.1186/gb-2014-15-3-r50\"\n },\n {\n \"idx\": \"16.\",\n \"title\": \"Houseman, E. A. et al. Reference-free deconvolution of DNA methylation data and mediation by cell composition effects.BMC Bioinformatics17, 259 (2016).\",\n \"link\": \"https://link.springer.com/doi/10.1186/s12859-016-1140-4\"\n },\n {\n \"idx\": \"17.\",\n \"title\": \"Zheng, X., Zhang, N., Wu, H.-J. & Wu, H. Estimating and accounting for tumor purity in the analysis of DNA methylation data from cancer studies.Genome Biol.18, 17 (2017).\",\n \"link\": \"https://link.springer.com/doi/10.1186/s13059-016-1143-5\"\n },\n {\n \"idx\": \"18.\",\n \"title\": \"Leek, J. T. & Storey, J. D. Capturing heterogeneity in gene expression studies by surrogate variable analysis.PLoS Genet.3, e161 (2007).\",\n \"link\": \"https://doi.org/10.1371%2Fjournal.pgen.0030161\"\n },\n {\n \"idx\": \"19.\",\n \"title\": \"Houseman, E. A., Molitor, J. & Marsit, C. J. Reference-free cell mixture adjustments in analysis of DNA methylation data.Bioinformatics30, 1431–1439 (2014).\",\n \"link\": \"https://doi.org/10.1093%2Fbioinformatics%2Fbtu029\"\n },\n {\n \"idx\": \"21.\",\n \"title\": \"Zheng, S. C. et al. Correcting for cell-type heterogeneity in epigenome-wide association studies: revisiting previous analyses.Nat. Methods14, 216–217 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fnmeth.4187\"\n },\n {\n \"idx\": \"22.\",\n \"title\": \"Rahmani, E. et al. Correcting for cell-type heterogeneity in DNA methylation: a comprehensive evaluation.Nat. Methods14, 218–219 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fnmeth.4190\"\n },\n {\n \"idx\": \"23.\",\n \"title\": \"Storey, J. D. & Tibshirani, R. Statistical significance for genomewide studies.Proc. Natl Acad. Sci. USA100, 9440–9445 (2003).\",\n \"link\": \"https://doi.org/10.1073%2Fpnas.1530509100\"\n },\n {\n \"idx\": \"24.\",\n \"title\": \"Salas, L. A. et al. An optimized library for reference-based deconvolution of whole-blood biospecimens assayed using the illumina humanmethylationepic beadarray.Genome Biol.19, 64 (2018).\",\n \"link\": \"https://link.springer.com/doi/10.1186/s13059-018-1448-7\"\n },\n {\n \"idx\": \"25.\",\n \"title\": \"Neumann, E. et al. Identification of differentially expressed genes in rheumatoid arthritis by a combination of complementary DNA array and rna arbitrarily primed-polymerase chain reaction.Arthritis Rheumatol.46, 52–63 (2002).\",\n \"link\": \"https://doi.org/10.1002%2F1529-0131%28200201%2946%3A1%3C52%3A%3AAID-ART10048%3E3.0.CO%3B2-1\"\n },\n {\n \"idx\": \"26.\",\n \"title\": \"Fasanelli, F. et al. Hypomethylation of smoking-related genes is associated with future lung cancer in four prospective cohorts.Nat. Commun.6, 10192 (2015).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms10192\"\n },\n {\n \"idx\": \"27.\",\n \"title\": \"Ambatipudi, S. et al. Tobacco smoking-associated genome-wide DNA methylation changes in the epic study.Epigenomics8, 599–618 (2016).\",\n \"link\": \"https://doi.org/10.2217%2Fepi-2016-0001\"\n },\n {\n \"idx\": \"28.\",\n \"title\": \"Pino-Yanes, M. et al. Genetic ancestry influences asthma susceptibility and lung function among latinos.J. Allergy Clin. Immunol.135, 228–235 (2015).\",\n \"link\": \"https://doi.org/10.1016%2Fj.jaci.2014.07.053\"\n },\n {\n \"idx\": \"29.\",\n \"title\": \"Li, Y. et al. c-Myb enhances breast cancer invasion and metastasis through the wnt/β-catenin/axin2 pathway.Cancer Res.76, 3364–3375 (2016).\",\n \"link\": \"https://doi.org/10.1158%2F0008-5472.CAN-15-2302\"\n },\n {\n \"idx\": \"30.\",\n \"title\": \"Zhang, X.-H., Tee, L. Y., Wang, X.-G., Huang, Q.-S. & Yang, S.-H. Off-target effects in CRISPR/Cas9-mediated genome engineering.Mol. Ther. Nucleic Acids4, e264 (2015).\",\n \"link\": \"https://doi.org/10.1038%2Fmtna.2015.37\"\n },\n {\n \"idx\": \"31.\",\n \"title\": \"Bolstad, B. M., Irizarry, R. A., Åstrand, M. & Speed, T. P. A comparison of normalization methods for high density oligonucleotide array data based on variance and bias.Bioinformatics19, 185–193 (2003).\",\n \"link\": \"https://doi.org/10.1093%2Fbioinformatics%2F19.2.185\"\n },\n {\n \"idx\": \"33.\",\n \"title\": \"Kiselev, V. Yu, Yiu, A. & Hemberg, M. scmap: projection of single-cell RNA-seq data across data sets.Nat. Methods15, 359 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fnmeth.4644\"\n },\n {\n \"idx\": \"34.\",\n \"title\": \"Reinius, L. E. et al. Differential DNA methylation in purified human blood cells: implications for cell lineage and studies on disease susceptibility.PLoS ONE,7, e41361 (2012).\",\n \"link\": \"https://doi.org/10.1371%2Fjournal.pone.0041361\"\n },\n {\n \"idx\": \"35.\",\n \"title\": \"Johnson, W. E., Li, C. & Rabinovic, A. Adjusting batch effects in microarray expression data using empirical Bayes methods.Biostatistics8, 118–127 (2007).\",\n \"link\": \"https://doi.org/10.1093%2Fbiostatistics%2Fkxj037\"\n },\n {\n \"idx\": \"36.\",\n \"title\": \"Maksimovic, J., Gordon, L. & Oshlack, A. Swan: Subset-quantile within array normalization for illumina infinium humanmethylation450 beadchips.Genome Biol.13, R44 (2012).\",\n \"link\": \"https://link.springer.com/doi/10.1186/gb-2012-13-6-r44\"\n },\n {\n \"idx\": \"37.\",\n \"title\": \"Rahmani, E. et al. BayesCCE: a Bayesian framework for estimating cell-type composition from DNA methylation without the need for methylation reference.Genome Biol.19, 141 (2018).\",\n \"link\": \"https://link.springer.com/doi/10.1186/s13059-018-1513-2\"\n }\n]","isConversational":false}}},{"rowIdx":196,"cells":{"unique_id":{"kind":"string","value":"ncomms3781"},"subfield":{"kind":"string","value":"Physics"},"title":{"kind":"string","value":"Efficient separation of the orbital angular momentum eigenstates of light"},"abstract":{"kind":"string","value":"Orbital angular momentum (OAM) of light is an attractive degree of freedom for fundamental studies in quantum mechanics. In addition, the discrete unbounded state-space of OAM has been used to enhance classical and quantum communications. Unambiguous measurement of OAM is a key part of all such experiments. However, state-of-the-art methods for separating single photons carrying a large number of different OAM values are limited to a theoretical separation efficiency of about 77%. Here we demonstrate a method which uses a series of unitary optical transformations to enable the measurement of light's OAM with an experimental separation efficiency of >92%. Furthermore, we demonstrate the separation of modes in the angular position basis, which is mutually unbiased with respect to the OAM basis. The high degree of certainty achieved by our method makes it particularly attractive for enhancing the information capacity of multi-level quantum cryptography systems."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"It has long been known that the polarization state of the optical field can be associated with the spin angular momentum [1] , [2] . In 1992, it was shown by Allen et al. [3] that in the paraxial regime, an vortex phase structure of a circularly symmetric beam corresponds to units of orbital angular momentum (OAM). Since then, OAM of light has been found as a useful tool in a variety of applications [4] , [5] , [6] , [7] , [8] . OAM modes with different values form a large orthonormal set of functions that can be used to encode information. It has been suggested that OAM encoding can be used alongside polarization to increase the channel capacity of communication systems [9] , [10] . In addition, the use of a multi-level encoding basis such as the OAM basis can increase the tolerance of quantum key distribution (QKD) systems to eavesdropping [11] . Clearly, applications of this sort will require a method of sorting photons carrying OAM with a high separation efficiency, which is a measure for the degree of separation of the sorted OAM modes, as well as a high power transmission efficiency, which is an indicator of the optical throughput of the system in presence of loss. Although the two eigenstates of polarization can be separated easily by using a polarizing beam splitter, separation of OAM eigenstates has proven to be challenging until recently [12] , [13] , [14] . A forked hologram can be used to add or remove spiral phase structures in its diffraction orders [15] . Similarly, a q-plate can be used to create OAM modes by utilizing the spin–orbit coupling in inhomogeneous anisotropic media [16] , [17] , [18] . Such elements can be used for measuring the OAM content of a beam by performing a series of projection measurements [4] , [8] . In this method, an input laser beam illuminates multiple forked holograms or q-plates with different quantum numbers and the transmitted light is collected by multiple single-mode fibres which correspond to different OAM values. If an OAM beam with quantum number enters this system, its phase structure will only be corrected by the element that adds a phase of − and a detection happens at the corresponding detector. This method can be used to distinguish between different OAM modes with high accuracy [8] , [19] . However, projection measurements are always limited by a success rate of 1/ N , where N is the number of the OAM modes involved in the experiment. This property makes this method unsuitable for single-photon experiments that involve a large number of OAM modes. More sophisticated hologram designs can be made to achieve separation of several OAM modes using a single phase-screen [15] . However, none of these designs have proven to be efficient enough for separation of OAM modes at the single-photon level. Similarly, the results achieved using thick holograms have been only marginally better [20] . Leach et al. [14] have presented a technique for separating OAM modes at the single-photon level based on using a Mach–Zehdner interferometer with a Dove prism in each arm. This set-up allows the separation of single photons based on their parity. Although in principle this method works with a 100% efficiency, separating N modes requires N –1 cascaded interferometers. The challenge in designing and aligning such a complicated system considerably limits its applications. Berkhout et al. [12] have recently demonstrated a very successful method for sorting OAM modes that employs a Cartesian to log-polar transformation. Under this transformation, the azimuthal phase profile of an OAM mode is mapped to a tilted planar wavefront. As a result, a set of OAM modes with different quantum numbers is converted to a set of truncated plane waves with wavefront tilts which are proportional to . These plane waves are then separated by a single lens. This method of sorting OAM modes is exclusively based on unitary transformations and in theory can achieve a power transmission efficiency of unity. However, the degree of separation of two neighbouring OAM modes is limited by diffraction due to the finite size of the unwrapped modes. In this work, we experimentally implement an architecture that combines log-polar coordinate mapping with refractive beam copying to substantially enhance the separation of the transformed OAM modes. Furthermore, we use a variation of this method for sorting angular (ANG) modes, which form an unbiased basis set with respect to the OAM modes. We achieve a separation efficiency of 92.1±0.7 for a set of 25 OAM modes and 92.7±0.5 for the corresponding 25 ANG modes, which are equivalent to mutual information values of 4.18 and 4.16 bits per detected photons, respectively.","references":{"indices":[1,2,3,4,5,7,8,9,10,11,12,13,14,15,16,17,18,19,20],"infos":[{"idx":"1","title":"Poynting, J. H. The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light.Proc. R. Soc. London, Ser. A82, 560–567 (1909).","link":"https://doi.org/10.1098%2Frspa.1909.0060"},{"idx":"2","title":"Beth, R. Mechanical detection and measurement of the angular momentum of light.Phys. Rev50, 115–125 (1936).","link":"https://doi.org/10.1103%2FPhysRev.50.115"},{"idx":"3","title":"Allen, L., Beijersbergen, M. W., Spreeuw, R. & Woerdman, J. P. Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes.Phys. Rev. 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Quantum key distribution in a high-dimensional state space: exploiting the transverse degree of freedom of the photon.Proc. SPIE7948, 79480L (2011).","link":"https://doi.org/10.1117%2F12.873491"},{"idx":"10","title":"Malik, M. et al. Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding.Opt. Express20, 13195–13200 (2012).","link":"https://doi.org/10.1364%2FOE.20.013195"},{"idx":"11","title":"Bourennane, M., Karlsson, A. & Björk, G. Quantum key distribution using multilevel encoding.Phys. Rev. A64, 012306 (2001).","link":"https://doi.org/10.1103%2FPhysRevA.64.012306"},{"idx":"12","title":"Berkhout, G., Lavery, M., Courtial, J., Beijersbergen, M. & Padgett, M. Efficient sorting of orbital angular momentum states of light.Phys. Rev. Lett.105, 153601 (2010).","link":"https://doi.org/10.1103%2FPhysRevLett.105.153601"},{"idx":"13","title":"Lavery, M. P. J. et al. Refractive elements for the measurement of the orbital angular momentum of a single photon.Opt. Express20, 2110–2115 (2012).","link":"https://doi.org/10.1364%2FOE.20.002110"},{"idx":"14","title":"Leach, J. et al. Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon.Phys. Rev. Lett.92, 013601 (2004).","link":"https://doi.org/10.1103%2FPhysRevLett.92.013601"},{"idx":"15","title":"Gibson, G. et al. Free-space information transfer using light beams carrying orbital angular momentum.Opt. Express12, 5448–5456 (2004).","link":"https://doi.org/10.1364%2FOPEX.12.005448"},{"idx":"16","title":"Marrucci, L., Manzo, C. & Paparo, D. Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media.Phys. Rev. Lett.96, 163905 (2006).","link":"https://doi.org/10.1103%2FPhysRevLett.96.163905"},{"idx":"17","title":"Nagali, E. et al. Quantum information transfer from spin to orbital angular momentum of photons.Phys. Rev. Lett.103, 013601 (2009).","link":"https://doi.org/10.1103%2FPhysRevLett.103.013601"},{"idx":"18","title":"Marrucci, L. et al. Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications.J. Opt.13, 064001 (2011).","link":"https://doi.org/10.1088%2F2040-8978%2F13%2F6%2F064001"},{"idx":"19","title":"Karimi, E., Piccirillo, B., Nagali, E., Marrucci, L. & Santamato, E. Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates.Appl. Phys. Lett.94, 231124 (2009).","link":"https://doi.org/10.1063%2F1.3154549"},{"idx":"20","title":"Gruneisen, M. T., Dymale, R. C., Stoltenberg, K. E. & Steinhoff, N. Optical vortex discrimination with a transmission volume hologram.New J. Phys.13, 0830309 (2011).","link":"https://doi.org/10.1088%2F1367-2630%2F13%2F8%2F083030"}]}},{"section":"Results","content":"Mode overlap in log-polar coordinate mapping To achieve the coordinate transformation, the beam’s amplitude a ( x , y ) is multiplied by a phase-only transmittance function ϕ 1 ( x , y ). The amplitude of the field in the conjugate Fourier plane can be calculated by using the stationary phase approximation, resulting in a geometrical map between the input and output planes [21] , [22] . This mapping is accompanied by an undesired varying phase term. This phase distortion arises from variations in the optical path length and can be removed by introducing a second phase plate ϕ 2 ( u , v ) at the Fourier plane ( u , v ) of the input plane. A coordinate mapping of v = a arctan( y / x ) and can be achieved using the following phase elements [12] , [23] : It is easy to check that the azimuthal phase profile of an OAM mode is mapped to a tilted planar wavefront under this transformation. The set of plane waves resulting from the transformation of OAM modes with different quantum numbers are then separated by a single lens positioned in the ( u , v ) plane. Figure 1a demonstrate the situation where the plane waves resulting from the transformation of two neighbouring OAM modes are focused to diffraction-limited spots which are partially overlapping. In Ref. 13 , the authors have measured the fraction of intensity overlap between the neighbouring modes to be about 20% (see Fig. 1e ). This is in fact an inherent limitation of this method and cannot be avoided by magnifying the transformed plane waves. Magnifying a plane wave results in a decrease in the tilt angle across its wavefront. Such magnified plane waves can be focused to narrower spots, but the spacing between the spots gets reduced accordingly such that the degree of overlap between them remains constant ( Fig. 1b ). Figure 1: Effect of beam copying on mode overlap. ( a ) Two tilted plane waves resulting from the transformation of two neighbouring OAM modes are focused to overlapping spots by a lens. ( b ) Magnified plane waves can be focused to narrower spots that are spaced closer to each other albeit with no change in overlap. ( c ) Multiple coherent copies of the transformed plane waves are placed next to each other resulting in focused spots with reduced overlap. ( d ) Experimental data for OAM modes focused after the log-polar mapping. The data for seven modes =−3:3 are super-imposed on top of each other. The =−2 mode is displayed in false colour to mark the overlap with the neighbouring modes. ( e ) Experimental data for the modes from part d when enhanced by making seven copies in the Fourier domain. Full size image Motivated by a suggestion of Berkhout et al. [12] , we recently investigated the effects of a beam-copying device on the degree of separation of transformed OAM modes [24] . Our previous simulation work along with the preliminary experimental results suggested that the use of such a device may dramatically reduce mode overlap. In the present paper, we describe the full implementation of this idea. This method works by mapping each OAM mode to multiple copies of the tilted planar wavefronts. In this situation, the width of the transformed beam has been increased, whereas the angle of the tilt across it remains constant. Note that the periodic phase jumps between the neighbouring copies of the truncated plane waves are equal to integer multiples of 2 π , so the resulting larger beam will have a smooth wavefront. These converted modes can now be focused to a series of spots which have the same spacing as before but are much narrower in width [24] ( Fig. 1c,e ). Refractive beam copying We have used a periodic phase-only hologram known as a fan-out element to make multiple coherent copies of the unwrapped OAM modes. The phase structure of such a device can be described as [26] : where 2 N +1 is the number of copies of the beam, s is the angular separation between them and x is the transverse dimension along which the copies are made (see Fig. 2a,b ). Here, γ m and α m are relative phase and intensity parameters associated with different diffraction orders (see Supplementary Note 1 and Supplementary Table S1 ). These parameters can be optimized using numerical or analytical methods to uniformly distribute >99% of the incident light between the copies [25] , [26] . Figure 2: Holograms for diffractive beam copying. ( a ) The hologram for creating seven copies (shown on the left) along with the corresponding hologram for correcting the relative phase between the copies (shown on the right). ( b ) The fan-out and phase corrector holograms for creating nine copies. Full size image Sorting OAM modes In our experiment, we generate OAM modes using a spatial light modulator (SLM) and a 4f system of lenses [27] . As shown in Fig. 3 , the OAM beam then propagates through the refractive elements R1 and R2 which perform the log-polar coordinate transformation [13] . A fan-out element realized on SLM1 generates multiple copies of this truncated plane wave mode. For clarity, Fig. 3 shows a fan-out element only creating three copies. However, we have used a fan-out design which creates seven copies of the beam [26] (see Supplementary Fig. S1 and Supplementary Table S1 ). SLM2 is used to correct the relative phase between these copies at the Fourier plane of the fan-out element. The holograms for the fan-out element and the corresponding phase corrector are shown in Fig. 2a . Finally, a lens is used to focus the extended plane-wave to a spot, where a charge-coupled-device (CCD) camera measures its intensity profile. Figure 3: The schematic of the mode sorting set-up. The optical transformation converts the OAM modes to plane waves as they go through the first and second refractive phase elements. An SLM is used to create multiple copies of the unwrapped beam. A lens focuses the resulting wide beam into a spot, after the phase distortions are removed by the second SLM. (To simplify the demonstration, a fan-out element creating only three copies has been displayed. For further simplification, we have also eliminated the Fourier transform lens between the fan-out and the phase corrector element.) Full size image We have measured OAM modes with quantum numbers of =−12 to =12 using our method. We divide the area on the CCD into 25 non-overlapping adjacent spatial bins. Each of the bins corresponds to the central position of the spot resulting from the transformation of an OAM mode. By sending a known OAM mode through the system and measuring the power in all 25 bins, we have constructed the conditional probability matrix for detection of all 25 OAM modes ( Fig. 4a ). The average probability of correctly detecting an OAM mode is calculated to be 92.1±0.7%. This is slightly lower than the theoretical maximum of 97% [24] owing to the non-ideal behaviour of a phase-only fan-out element, undesired effects from pixelation of the SLMs and small misalignments in the system. However, this is substantially higher than the maximum separation efficiency of 77% achieved by just the log-polar mapping method [12] . We have compared these two methods quantitatively in Supplementary Note 2 . Figure 4: The conditional probability of detection. The horizontal axes indicate the indices for the sent and received ( a ) OAM modes and ( b ) ANG modes. For spatial profile of the modes, see Supplementary Fig. S3 and Supplementary Fig. S4 . The details of calculations are presented in Supplementary Note 3 . Full size image Another common measure of quantizing the separation efficiency of a mode sorter is the mutual information between the set of sent and detected modes. For the data presented here, the mutual information is calculated to be 4.18 bits per detected photon, as compared with the theoretical upper limit of 4.64 bits per detected photon for a set of 25 modes (see Supplementary Note 3 and Supplementary Fig. S2 ). It should be emphasized that these values are calculated for the detected photons and they exclude the technical loss in the experimental set-up. The use of this metric can be justified considering the fact that the losses are caused by the limitations in technology and they are not inherent to our method. The refractive elements that perform the log-polar mapping have a combined transmission efficiency of 85% [13] . The SLMs that realize the fan-out element and its phase corrector are each measured to have a diffraction efficiency of 42% (for a complete analysis of the effect of loss on the overall information capacity of the system, please see Supplementary Note 3 ). QKD and angular modes In order to guarantee security against eavesdropping in QKD, at least two mutually unbiased bases (MUBs) are required [28] . A MUB for OAM can be constructed from the superposition of OAM modes with equal weights and a relative phase [10] , [24] : Here, an OAM mode is written as and L is the maximum OAM quantum number which is used in forming the basis. Members of this MUB are referred to as the angular (ANG) modes, due to angular localization of their intensity patterns ( Fig. 5 ). It is easy to check that if an ANG mode is measured in the OAM basis, all the component OAM modes in the range will be detected with equal probabilities. ANG modes with different indices are orthogonal in term of their amplitudes; however, the intensity profiles of these modes have side lobes around their central angular component which creates substantial overlap between any two neighbouring modes (see Fig. 5 ). A detection scheme based on the intensity confinement of ANG mode has the rate of error of 23% (ref. 24 ). Figure 5: Generation of OAM and ANG modes. Examples of holograms (top line) for generating OAM and ANG modes along with the intensity profile (bottom line) of the resulting spatial modes. Full size image Sorting ANG modes An efficient method of sorting both OAM and ANG modes is needed to realize an OAM-based QKD system. Previous methods of sorting OAM modes are incapable of sorting ANG modes [10] . However, a variation of the sorting method described above can be used to sort these modes [24] . As the unwrapped OAM and ANG modes are related via a Fourier transform in the horizontal direction, we can sort ANG modes by removing the Fourier-transforming lens L1 after the second refractive element R2. Instead, the unwrapped mode is imaged directly onto the fan-out element. We use a fan-out element creating nine copies (see Supplementary Note 1 and Supplementary Table S1 ). The holograms for the fan-out element and the corresponding phase corrector element are shown in Fig. 2b . The ANG modes are transformed to a series of separated spots at the focal plane of the lens L2. The conditional probability matrix for 25 ANG modes is shown in Fig. 4b . The average probability for correct detection of an ANG mode in this set is measured to be 92.7±0.6%. The mutual information calculated from the measured conditional probability matrix equals 4.16 bits per detected photon, demonstrating the capability of sorting ANG modes with approximately the same separation efficiency as that of the OAM modes. In this experiment, we have demonstrated a method for unambiguous discrimination of OAM modes. A log-polar optical transformation is used along with a refractive beam-copying method to map OAM modes to a set of extended plane waves which are then separated using a lens. Using our technique, we have measured a channel capacity of 4.18 bits per detected photon for a free-space communication system using 25 OAM modes. In addition, we have demonstrated a comparable ability to efficiently sort 25 modes in the mutually unbiased basis of angular position. Our technique will enable the development of high-dimensional QKD systems with record channel capacities of >4 bits per detected photon. In addition, this method will allow for detailed studies of fundamental quantum phenomena in the discrete unbounded state-space of OAM."},{"section":"Methods","content":"Equipments and experimental techniques A Helium Neon laser beam (632.8 nm) is spatially filtered using a single-mode fibre. The output of the fibre is collimated and illuminates a high-definition Holoeye PLUTO phase-only SLM. The OAM and ANG modes are created using computer generated holograms [27] (see Supplementary Note 4 ). An aperture in used in between a 4f system to pick up the first-order diffracted beam. Machined PMMA (Poly methyl methacrylate) refractive elements mounted in a Thorlabs cage system are used for log-polar coordinate mapping [13] . Two Holoeye PLUTO phase-only SLMs are used for realizing the hologram for the fan-out element and its corresponding phase-correcting element. A telescope system is used for coherent imaging of ANG modes to the fan-out hologram. Single images were obtained on a Canon 5D mark III CCD camera providing 14 bits of dynamic range. A Semrock 632.8 nm MaxLine filter is used in front of the camera to block stray light. An exposure time of 125 μs is chosen to further suppress the background light."},{"section":"Additional information","content":"How to cite this article: Mirhosseini, M. et al. Efficient separation of the orbital angular momentum eigenstates of light. Nat. Commun. 4:2781 doi: 10.1038/ncomms3781 (2013)."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"It has long been known that the polarization state of the optical field can be associated with the spin angular momentum [1] , [2] . In 1992, it was shown by Allen et al. [3] that in the paraxial regime, an vortex phase structure of a circularly symmetric beam corresponds to units of orbital angular momentum (OAM). Since then, OAM of light has been found as a useful tool in a variety of applications [4] , [5] , [6] , [7] , [8] . OAM modes with different values form a large orthonormal set of functions that can be used to encode information. It has been suggested that OAM encoding can be used alongside polarization to increase the channel capacity of communication systems [9] , [10] . In addition, the use of a multi-level encoding basis such as the OAM basis can increase the tolerance of quantum key distribution (QKD) systems to eavesdropping [11] . Clearly, applications of this sort will require a method of sorting photons carrying OAM with a high separation efficiency, which is a measure for the degree of separation of the sorted OAM modes, as well as a high power transmission efficiency, which is an indicator of the optical throughput of the system in presence of loss. Although the two eigenstates of polarization can be separated easily by using a polarizing beam splitter, separation of OAM eigenstates has proven to be challenging until recently [12] , [13] , [14] . A forked hologram can be used to add or remove spiral phase structures in its diffraction orders [15] . Similarly, a q-plate can be used to create OAM modes by utilizing the spin–orbit coupling in inhomogeneous anisotropic media [16] , [17] , [18] . Such elements can be used for measuring the OAM content of a beam by performing a series of projection measurements [4] , [8] . In this method, an input laser beam illuminates multiple forked holograms or q-plates with different quantum numbers and the transmitted light is collected by multiple single-mode fibres which correspond to different OAM values. If an OAM beam with quantum number enters this system, its phase structure will only be corrected by the element that adds a phase of − and a detection happens at the corresponding detector. This method can be used to distinguish between different OAM modes with high accuracy [8] , [19] . However, projection measurements are always limited by a success rate of 1/ N , where N is the number of the OAM modes involved in the experiment. This property makes this method unsuitable for single-photon experiments that involve a large number of OAM modes. More sophisticated hologram designs can be made to achieve separation of several OAM modes using a single phase-screen [15] . However, none of these designs have proven to be efficient enough for separation of OAM modes at the single-photon level. Similarly, the results achieved using thick holograms have been only marginally better [20] . Leach et al. [14] have presented a technique for separating OAM modes at the single-photon level based on using a Mach–Zehdner interferometer with a Dove prism in each arm. This set-up allows the separation of single photons based on their parity. Although in principle this method works with a 100% efficiency, separating N modes requires N –1 cascaded interferometers. The challenge in designing and aligning such a complicated system considerably limits its applications. Berkhout et al. [12] have recently demonstrated a very successful method for sorting OAM modes that employs a Cartesian to log-polar transformation. Under this transformation, the azimuthal phase profile of an OAM mode is mapped to a tilted planar wavefront. As a result, a set of OAM modes with different quantum numbers is converted to a set of truncated plane waves with wavefront tilts which are proportional to . These plane waves are then separated by a single lens. This method of sorting OAM modes is exclusively based on unitary transformations and in theory can achieve a power transmission efficiency of unity. However, the degree of separation of two neighbouring OAM modes is limited by diffraction due to the finite size of the unwrapped modes. In this work, we experimentally implement an architecture that combines log-polar coordinate mapping with refractive beam copying to substantially enhance the separation of the transformed OAM modes. Furthermore, we use a variation of this method for sorting angular (ANG) modes, which form an unbiased basis set with respect to the OAM modes. We achieve a separation efficiency of 92.1±0.7 for a set of 25 OAM modes and 92.7±0.5 for the corresponding 25 ANG modes, which are equivalent to mutual information values of 4.18 and 4.16 bits per detected photons, respectively.\",\n \"references\": {\n \"indices\": [\n 1,\n 2,\n 3,\n 4,\n 5,\n 7,\n 8,\n 9,\n 10,\n 11,\n 12,\n 13,\n 14,\n 15,\n 16,\n 17,\n 18,\n 19,\n 20\n ],\n \"infos\": [\n {\n \"idx\": \"1\",\n \"title\": \"Poynting, J. H. The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light.Proc. R. Soc. London, Ser. A82, 560–567 (1909).\",\n \"link\": \"https://doi.org/10.1098%2Frspa.1909.0060\"\n },\n {\n \"idx\": \"2\",\n \"title\": \"Beth, R. Mechanical detection and measurement of the angular momentum of light.Phys. Rev50, 115–125 (1936).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRev.50.115\"\n },\n {\n \"idx\": \"3\",\n \"title\": \"Allen, L., Beijersbergen, M. W., Spreeuw, R. & Woerdman, J. P. Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes.Phys. Rev. A45, 8185–8189 (1992).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevA.45.8185\"\n },\n {\n \"idx\": \"4\",\n \"title\": \"Mair, A., Vaziri, A., Weihs, G. & Zeilinger, A. Entanglement of the orbital angular momentum states of photons.Nature412, 313–316 (2001).\",\n \"link\": \"https://doi.org/10.1038%2F35085529\"\n },\n {\n \"idx\": \"5\",\n \"title\": \"Molina-Terriza, G., Torres, J. P. & Torner, L. Twisted photons.Nat. Phys.3, 305–310 (2007).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys607\"\n },\n {\n \"idx\": \"7\",\n \"title\": \"Barreiro, J. T., Wei, T.-C. & Kwiat, P. G. Beating the channel capacity limit for linear photonic superdense coding.Nat. Photon.4, 282–286 (2008).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys919\"\n },\n {\n \"idx\": \"8\",\n \"title\": \"Wang, J. et al. Terabit free-space data transmission employing orbital angular momentum multiplexing.Nat. Photon.6, 488–496 (2012).\",\n \"link\": \"https://doi.org/10.1038%2Fnphoton.2012.138\"\n },\n {\n \"idx\": \"9\",\n \"title\": \"Boyd, R. W. et al. Quantum key distribution in a high-dimensional state space: exploiting the transverse degree of freedom of the photon.Proc. SPIE7948, 79480L (2011).\",\n \"link\": \"https://doi.org/10.1117%2F12.873491\"\n },\n {\n \"idx\": \"10\",\n \"title\": \"Malik, M. et al. Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding.Opt. Express20, 13195–13200 (2012).\",\n \"link\": \"https://doi.org/10.1364%2FOE.20.013195\"\n },\n {\n \"idx\": \"11\",\n \"title\": \"Bourennane, M., Karlsson, A. & Björk, G. Quantum key distribution using multilevel encoding.Phys. Rev. A64, 012306 (2001).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevA.64.012306\"\n },\n {\n \"idx\": \"12\",\n \"title\": \"Berkhout, G., Lavery, M., Courtial, J., Beijersbergen, M. & Padgett, M. Efficient sorting of orbital angular momentum states of light.Phys. Rev. Lett.105, 153601 (2010).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.105.153601\"\n },\n {\n \"idx\": \"13\",\n \"title\": \"Lavery, M. P. J. et al. Refractive elements for the measurement of the orbital angular momentum of a single photon.Opt. Express20, 2110–2115 (2012).\",\n \"link\": \"https://doi.org/10.1364%2FOE.20.002110\"\n },\n {\n \"idx\": \"14\",\n \"title\": \"Leach, J. et al. Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon.Phys. Rev. Lett.92, 013601 (2004).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.92.013601\"\n },\n {\n \"idx\": \"15\",\n \"title\": \"Gibson, G. et al. Free-space information transfer using light beams carrying orbital angular momentum.Opt. Express12, 5448–5456 (2004).\",\n \"link\": \"https://doi.org/10.1364%2FOPEX.12.005448\"\n },\n {\n \"idx\": \"16\",\n \"title\": \"Marrucci, L., Manzo, C. & Paparo, D. Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media.Phys. Rev. Lett.96, 163905 (2006).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.96.163905\"\n },\n {\n \"idx\": \"17\",\n \"title\": \"Nagali, E. et al. Quantum information transfer from spin to orbital angular momentum of photons.Phys. Rev. Lett.103, 013601 (2009).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.103.013601\"\n },\n {\n \"idx\": \"18\",\n \"title\": \"Marrucci, L. et al. Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications.J. Opt.13, 064001 (2011).\",\n \"link\": \"https://doi.org/10.1088%2F2040-8978%2F13%2F6%2F064001\"\n },\n {\n \"idx\": \"19\",\n \"title\": \"Karimi, E., Piccirillo, B., Nagali, E., Marrucci, L. & Santamato, E. Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates.Appl. Phys. Lett.94, 231124 (2009).\",\n \"link\": \"https://doi.org/10.1063%2F1.3154549\"\n },\n {\n \"idx\": \"20\",\n \"title\": \"Gruneisen, M. T., Dymale, R. C., Stoltenberg, K. E. & Steinhoff, N. Optical vortex discrimination with a transmission volume hologram.New J. Phys.13, 0830309 (2011).\",\n \"link\": \"https://doi.org/10.1088%2F1367-2630%2F13%2F8%2F083030\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Mode overlap in log-polar coordinate mapping To achieve the coordinate transformation, the beam’s amplitude a ( x , y ) is multiplied by a phase-only transmittance function ϕ 1 ( x , y ). The amplitude of the field in the conjugate Fourier plane can be calculated by using the stationary phase approximation, resulting in a geometrical map between the input and output planes [21] , [22] . This mapping is accompanied by an undesired varying phase term. This phase distortion arises from variations in the optical path length and can be removed by introducing a second phase plate ϕ 2 ( u , v ) at the Fourier plane ( u , v ) of the input plane. A coordinate mapping of v = a arctan( y / x ) and can be achieved using the following phase elements [12] , [23] : It is easy to check that the azimuthal phase profile of an OAM mode is mapped to a tilted planar wavefront under this transformation. The set of plane waves resulting from the transformation of OAM modes with different quantum numbers are then separated by a single lens positioned in the ( u , v ) plane. Figure 1a demonstrate the situation where the plane waves resulting from the transformation of two neighbouring OAM modes are focused to diffraction-limited spots which are partially overlapping. In Ref. 13 , the authors have measured the fraction of intensity overlap between the neighbouring modes to be about 20% (see Fig. 1e ). This is in fact an inherent limitation of this method and cannot be avoided by magnifying the transformed plane waves. Magnifying a plane wave results in a decrease in the tilt angle across its wavefront. Such magnified plane waves can be focused to narrower spots, but the spacing between the spots gets reduced accordingly such that the degree of overlap between them remains constant ( Fig. 1b ). Figure 1: Effect of beam copying on mode overlap. ( a ) Two tilted plane waves resulting from the transformation of two neighbouring OAM modes are focused to overlapping spots by a lens. ( b ) Magnified plane waves can be focused to narrower spots that are spaced closer to each other albeit with no change in overlap. ( c ) Multiple coherent copies of the transformed plane waves are placed next to each other resulting in focused spots with reduced overlap. ( d ) Experimental data for OAM modes focused after the log-polar mapping. The data for seven modes =−3:3 are super-imposed on top of each other. The =−2 mode is displayed in false colour to mark the overlap with the neighbouring modes. ( e ) Experimental data for the modes from part d when enhanced by making seven copies in the Fourier domain. Full size image Motivated by a suggestion of Berkhout et al. [12] , we recently investigated the effects of a beam-copying device on the degree of separation of transformed OAM modes [24] . Our previous simulation work along with the preliminary experimental results suggested that the use of such a device may dramatically reduce mode overlap. In the present paper, we describe the full implementation of this idea. This method works by mapping each OAM mode to multiple copies of the tilted planar wavefronts. In this situation, the width of the transformed beam has been increased, whereas the angle of the tilt across it remains constant. Note that the periodic phase jumps between the neighbouring copies of the truncated plane waves are equal to integer multiples of 2 π , so the resulting larger beam will have a smooth wavefront. These converted modes can now be focused to a series of spots which have the same spacing as before but are much narrower in width [24] ( Fig. 1c,e ). Refractive beam copying We have used a periodic phase-only hologram known as a fan-out element to make multiple coherent copies of the unwrapped OAM modes. The phase structure of such a device can be described as [26] : where 2 N +1 is the number of copies of the beam, s is the angular separation between them and x is the transverse dimension along which the copies are made (see Fig. 2a,b ). Here, γ m and α m are relative phase and intensity parameters associated with different diffraction orders (see Supplementary Note 1 and Supplementary Table S1 ). These parameters can be optimized using numerical or analytical methods to uniformly distribute >99% of the incident light between the copies [25] , [26] . Figure 2: Holograms for diffractive beam copying. ( a ) The hologram for creating seven copies (shown on the left) along with the corresponding hologram for correcting the relative phase between the copies (shown on the right). ( b ) The fan-out and phase corrector holograms for creating nine copies. Full size image Sorting OAM modes In our experiment, we generate OAM modes using a spatial light modulator (SLM) and a 4f system of lenses [27] . As shown in Fig. 3 , the OAM beam then propagates through the refractive elements R1 and R2 which perform the log-polar coordinate transformation [13] . A fan-out element realized on SLM1 generates multiple copies of this truncated plane wave mode. For clarity, Fig. 3 shows a fan-out element only creating three copies. However, we have used a fan-out design which creates seven copies of the beam [26] (see Supplementary Fig. S1 and Supplementary Table S1 ). SLM2 is used to correct the relative phase between these copies at the Fourier plane of the fan-out element. The holograms for the fan-out element and the corresponding phase corrector are shown in Fig. 2a . Finally, a lens is used to focus the extended plane-wave to a spot, where a charge-coupled-device (CCD) camera measures its intensity profile. Figure 3: The schematic of the mode sorting set-up. The optical transformation converts the OAM modes to plane waves as they go through the first and second refractive phase elements. An SLM is used to create multiple copies of the unwrapped beam. A lens focuses the resulting wide beam into a spot, after the phase distortions are removed by the second SLM. (To simplify the demonstration, a fan-out element creating only three copies has been displayed. For further simplification, we have also eliminated the Fourier transform lens between the fan-out and the phase corrector element.) Full size image We have measured OAM modes with quantum numbers of =−12 to =12 using our method. We divide the area on the CCD into 25 non-overlapping adjacent spatial bins. Each of the bins corresponds to the central position of the spot resulting from the transformation of an OAM mode. By sending a known OAM mode through the system and measuring the power in all 25 bins, we have constructed the conditional probability matrix for detection of all 25 OAM modes ( Fig. 4a ). The average probability of correctly detecting an OAM mode is calculated to be 92.1±0.7%. This is slightly lower than the theoretical maximum of 97% [24] owing to the non-ideal behaviour of a phase-only fan-out element, undesired effects from pixelation of the SLMs and small misalignments in the system. However, this is substantially higher than the maximum separation efficiency of 77% achieved by just the log-polar mapping method [12] . We have compared these two methods quantitatively in Supplementary Note 2 . Figure 4: The conditional probability of detection. The horizontal axes indicate the indices for the sent and received ( a ) OAM modes and ( b ) ANG modes. For spatial profile of the modes, see Supplementary Fig. S3 and Supplementary Fig. S4 . The details of calculations are presented in Supplementary Note 3 . Full size image Another common measure of quantizing the separation efficiency of a mode sorter is the mutual information between the set of sent and detected modes. For the data presented here, the mutual information is calculated to be 4.18 bits per detected photon, as compared with the theoretical upper limit of 4.64 bits per detected photon for a set of 25 modes (see Supplementary Note 3 and Supplementary Fig. S2 ). It should be emphasized that these values are calculated for the detected photons and they exclude the technical loss in the experimental set-up. The use of this metric can be justified considering the fact that the losses are caused by the limitations in technology and they are not inherent to our method. The refractive elements that perform the log-polar mapping have a combined transmission efficiency of 85% [13] . The SLMs that realize the fan-out element and its phase corrector are each measured to have a diffraction efficiency of 42% (for a complete analysis of the effect of loss on the overall information capacity of the system, please see Supplementary Note 3 ). QKD and angular modes In order to guarantee security against eavesdropping in QKD, at least two mutually unbiased bases (MUBs) are required [28] . A MUB for OAM can be constructed from the superposition of OAM modes with equal weights and a relative phase [10] , [24] : Here, an OAM mode is written as and L is the maximum OAM quantum number which is used in forming the basis. Members of this MUB are referred to as the angular (ANG) modes, due to angular localization of their intensity patterns ( Fig. 5 ). It is easy to check that if an ANG mode is measured in the OAM basis, all the component OAM modes in the range will be detected with equal probabilities. ANG modes with different indices are orthogonal in term of their amplitudes; however, the intensity profiles of these modes have side lobes around their central angular component which creates substantial overlap between any two neighbouring modes (see Fig. 5 ). A detection scheme based on the intensity confinement of ANG mode has the rate of error of 23% (ref. 24 ). Figure 5: Generation of OAM and ANG modes. Examples of holograms (top line) for generating OAM and ANG modes along with the intensity profile (bottom line) of the resulting spatial modes. Full size image Sorting ANG modes An efficient method of sorting both OAM and ANG modes is needed to realize an OAM-based QKD system. Previous methods of sorting OAM modes are incapable of sorting ANG modes [10] . However, a variation of the sorting method described above can be used to sort these modes [24] . As the unwrapped OAM and ANG modes are related via a Fourier transform in the horizontal direction, we can sort ANG modes by removing the Fourier-transforming lens L1 after the second refractive element R2. Instead, the unwrapped mode is imaged directly onto the fan-out element. We use a fan-out element creating nine copies (see Supplementary Note 1 and Supplementary Table S1 ). The holograms for the fan-out element and the corresponding phase corrector element are shown in Fig. 2b . The ANG modes are transformed to a series of separated spots at the focal plane of the lens L2. The conditional probability matrix for 25 ANG modes is shown in Fig. 4b . The average probability for correct detection of an ANG mode in this set is measured to be 92.7±0.6%. The mutual information calculated from the measured conditional probability matrix equals 4.16 bits per detected photon, demonstrating the capability of sorting ANG modes with approximately the same separation efficiency as that of the OAM modes. In this experiment, we have demonstrated a method for unambiguous discrimination of OAM modes. A log-polar optical transformation is used along with a refractive beam-copying method to map OAM modes to a set of extended plane waves which are then separated using a lens. Using our technique, we have measured a channel capacity of 4.18 bits per detected photon for a free-space communication system using 25 OAM modes. In addition, we have demonstrated a comparable ability to efficiently sort 25 modes in the mutually unbiased basis of angular position. Our technique will enable the development of high-dimensional QKD systems with record channel capacities of >4 bits per detected photon. In addition, this method will allow for detailed studies of fundamental quantum phenomena in the discrete unbounded state-space of OAM.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Equipments and experimental techniques A Helium Neon laser beam (632.8 nm) is spatially filtered using a single-mode fibre. The output of the fibre is collimated and illuminates a high-definition Holoeye PLUTO phase-only SLM. The OAM and ANG modes are created using computer generated holograms [27] (see Supplementary Note 4 ). An aperture in used in between a 4f system to pick up the first-order diffracted beam. Machined PMMA (Poly methyl methacrylate) refractive elements mounted in a Thorlabs cage system are used for log-polar coordinate mapping [13] . Two Holoeye PLUTO phase-only SLMs are used for realizing the hologram for the fan-out element and its corresponding phase-correcting element. A telescope system is used for coherent imaging of ANG modes to the fan-out hologram. Single images were obtained on a Canon 5D mark III CCD camera providing 14 bits of dynamic range. A Semrock 632.8 nm MaxLine filter is used in front of the camera to block stray light. An exposure time of 125 μs is chosen to further suppress the background light.\"\n },\n {\n \"section\": \"Additional information\",\n \"content\": \"How to cite this article: Mirhosseini, M. et al. Efficient separation of the orbital angular momentum eigenstates of light. Nat. Commun. 4:2781 doi: 10.1038/ncomms3781 (2013).\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"The core research idea is to experimentally implement an architecture that combines log-polar coordinate mapping with refractive beam copying to significantly enhance the separation efficiency of orbital angular momentum (OAM) modes, achieving a separation efficiency of 92.1±0.7 for 25 OAM modes and extending the method to sort angular (ANG) modes with similar efficiency, thereby enabling high-performance OAM-based communication and quantum information systems."},"entities":{"kind":"list like","value":["log-polar coordinate mapping","refractive beam copying","orbital angular momentum modes","separation efficiency","25 OAM modes","angular modes","OAM-based communication","quantum information systems"],"string":"[\n \"log-polar coordinate mapping\",\n \"refractive beam copying\",\n \"orbital angular momentum modes\",\n \"separation efficiency\",\n \"25 OAM modes\",\n \"angular modes\",\n \"OAM-based communication\",\n \"quantum information systems\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1","title":"Poynting, J. H. The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light.Proc. R. Soc. London, Ser. A82, 560–567 (1909).","link":"https://doi.org/10.1098%2Frspa.1909.0060"},{"idx":"2","title":"Beth, R. Mechanical detection and measurement of the angular momentum of light.Phys. Rev50, 115–125 (1936).","link":"https://doi.org/10.1103%2FPhysRev.50.115"},{"idx":"3","title":"Allen, L., Beijersbergen, M. W., Spreeuw, R. & Woerdman, J. P. Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes.Phys. Rev. A45, 8185–8189 (1992).","link":"https://doi.org/10.1103%2FPhysRevA.45.8185"},{"idx":"4","title":"Mair, A., Vaziri, A., Weihs, G. & Zeilinger, A. Entanglement of the orbital angular momentum states of photons.Nature412, 313–316 (2001).","link":"https://doi.org/10.1038%2F35085529"},{"idx":"5","title":"Molina-Terriza, G., Torres, J. P. & Torner, L. Twisted photons.Nat. Phys.3, 305–310 (2007).","link":"https://doi.org/10.1038%2Fnphys607"},{"idx":"7","title":"Barreiro, J. T., Wei, T.-C. & Kwiat, P. G. Beating the channel capacity limit for linear photonic superdense coding.Nat. Photon.4, 282–286 (2008).","link":"https://doi.org/10.1038%2Fnphys919"},{"idx":"8","title":"Wang, J. et al. Terabit free-space data transmission employing orbital angular momentum multiplexing.Nat. Photon.6, 488–496 (2012).","link":"https://doi.org/10.1038%2Fnphoton.2012.138"},{"idx":"9","title":"Boyd, R. W. et al. Quantum key distribution in a high-dimensional state space: exploiting the transverse degree of freedom of the photon.Proc. SPIE7948, 79480L (2011).","link":"https://doi.org/10.1117%2F12.873491"},{"idx":"10","title":"Malik, M. et al. Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding.Opt. Express20, 13195–13200 (2012).","link":"https://doi.org/10.1364%2FOE.20.013195"},{"idx":"11","title":"Bourennane, M., Karlsson, A. & Björk, G. Quantum key distribution using multilevel encoding.Phys. Rev. A64, 012306 (2001).","link":"https://doi.org/10.1103%2FPhysRevA.64.012306"},{"idx":"12","title":"Berkhout, G., Lavery, M., Courtial, J., Beijersbergen, M. & Padgett, M. Efficient sorting of orbital angular momentum states of light.Phys. Rev. Lett.105, 153601 (2010).","link":"https://doi.org/10.1103%2FPhysRevLett.105.153601"},{"idx":"13","title":"Lavery, M. P. J. et al. Refractive elements for the measurement of the orbital angular momentum of a single photon.Opt. Express20, 2110–2115 (2012).","link":"https://doi.org/10.1364%2FOE.20.002110"},{"idx":"14","title":"Leach, J. et al. Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon.Phys. Rev. Lett.92, 013601 (2004).","link":"https://doi.org/10.1103%2FPhysRevLett.92.013601"},{"idx":"15","title":"Gibson, G. et al. Free-space information transfer using light beams carrying orbital angular momentum.Opt. Express12, 5448–5456 (2004).","link":"https://doi.org/10.1364%2FOPEX.12.005448"},{"idx":"16","title":"Marrucci, L., Manzo, C. & Paparo, D. Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media.Phys. Rev. Lett.96, 163905 (2006).","link":"https://doi.org/10.1103%2FPhysRevLett.96.163905"},{"idx":"17","title":"Nagali, E. et al. Quantum information transfer from spin to orbital angular momentum of photons.Phys. Rev. Lett.103, 013601 (2009).","link":"https://doi.org/10.1103%2FPhysRevLett.103.013601"},{"idx":"18","title":"Marrucci, L. et al. Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications.J. Opt.13, 064001 (2011).","link":"https://doi.org/10.1088%2F2040-8978%2F13%2F6%2F064001"},{"idx":"19","title":"Karimi, E., Piccirillo, B., Nagali, E., Marrucci, L. & Santamato, E. Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates.Appl. Phys. Lett.94, 231124 (2009).","link":"https://doi.org/10.1063%2F1.3154549"},{"idx":"20","title":"Gruneisen, M. T., Dymale, R. C., Stoltenberg, K. E. & Steinhoff, N. Optical vortex discrimination with a transmission volume hologram.New J. Phys.13, 0830309 (2011).","link":"https://doi.org/10.1088%2F1367-2630%2F13%2F8%2F083030"},{"idx":"21","title":"Bryngdahl, O. Geometrical transformations in optics.J. Opt. Soc. Am. A64, 1092–1099 (1974).","link":"https://doi.org/10.1364%2FJOSA.64.001092"},{"idx":"22","title":"Bryngdahl, O. Optical map transformations.Opt. Commun.10, 164–168 (1974).","link":"https://doi.org/10.1016%2F0030-4018%2874%2990046-7"},{"idx":"23","title":"Saito, Y., Komatsu, S.-I. & Ohzu, H. Scale and rotation invariant real time optical correlator using computer generated hologram.Opt. Commun.47, 8–11 (1983).","link":"https://doi.org/10.1016%2F0030-4018%2883%2990326-7"},{"idx":"24","title":"O'Sullivan, M. N., Mirhosseini, M., Malik, M. & Boyd, R. W. Near-perfect sorting of orbital angular momentum and angular position states of light.Opt. Express.20, 24444–24449 (2012).","link":"https://doi.org/10.1364%2FOE.20.024444"},{"idx":"25","title":"Prongue, D., Herzig, H. P., Dandliker, R. & Gale, M. T. Optimized kinoform structures for highly efficient fan-out elements.Appl. Opt.31, 5706–5711 (1992).","link":"https://doi.org/10.1364%2FAO.31.005706"},{"idx":"26","title":"Romero, L. A. & Dickey, F. M. Theory of optimal beam splitting by phase gratings. I. One-dimensional gratings.J. Opt. Soc. Am. A24, 2280–2295 (2007).","link":"https://doi.org/10.1364%2FJOSAA.24.002280"},{"idx":"27","title":"Arrizon, V., Ruiz, U., Carrada, R. & Gonzalez, L. A. Pixelated phase computer holograms for the accurate encoding of scalar complex fields.J. Opt. Soc. Am. A24, 3500–3507 (2007).","link":"https://doi.org/10.1364%2FJOSAA.24.003500"}],"string":"[\n {\n \"idx\": \"1\",\n \"title\": \"Poynting, J. H. The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light.Proc. R. Soc. London, Ser. A82, 560–567 (1909).\",\n \"link\": \"https://doi.org/10.1098%2Frspa.1909.0060\"\n },\n {\n \"idx\": \"2\",\n \"title\": \"Beth, R. Mechanical detection and measurement of the angular momentum of light.Phys. Rev50, 115–125 (1936).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRev.50.115\"\n },\n {\n \"idx\": \"3\",\n \"title\": \"Allen, L., Beijersbergen, M. W., Spreeuw, R. & Woerdman, J. P. Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes.Phys. Rev. A45, 8185–8189 (1992).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevA.45.8185\"\n },\n {\n \"idx\": \"4\",\n \"title\": \"Mair, A., Vaziri, A., Weihs, G. & Zeilinger, A. Entanglement of the orbital angular momentum states of photons.Nature412, 313–316 (2001).\",\n \"link\": \"https://doi.org/10.1038%2F35085529\"\n },\n {\n \"idx\": \"5\",\n \"title\": \"Molina-Terriza, G., Torres, J. P. & Torner, L. Twisted photons.Nat. Phys.3, 305–310 (2007).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys607\"\n },\n {\n \"idx\": \"7\",\n \"title\": \"Barreiro, J. T., Wei, T.-C. & Kwiat, P. G. Beating the channel capacity limit for linear photonic superdense coding.Nat. Photon.4, 282–286 (2008).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys919\"\n },\n {\n \"idx\": \"8\",\n \"title\": \"Wang, J. et al. Terabit free-space data transmission employing orbital angular momentum multiplexing.Nat. Photon.6, 488–496 (2012).\",\n \"link\": \"https://doi.org/10.1038%2Fnphoton.2012.138\"\n },\n {\n \"idx\": \"9\",\n \"title\": \"Boyd, R. W. et al. Quantum key distribution in a high-dimensional state space: exploiting the transverse degree of freedom of the photon.Proc. SPIE7948, 79480L (2011).\",\n \"link\": \"https://doi.org/10.1117%2F12.873491\"\n },\n {\n \"idx\": \"10\",\n \"title\": \"Malik, M. et al. Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding.Opt. Express20, 13195–13200 (2012).\",\n \"link\": \"https://doi.org/10.1364%2FOE.20.013195\"\n },\n {\n \"idx\": \"11\",\n \"title\": \"Bourennane, M., Karlsson, A. & Björk, G. Quantum key distribution using multilevel encoding.Phys. Rev. A64, 012306 (2001).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevA.64.012306\"\n },\n {\n \"idx\": \"12\",\n \"title\": \"Berkhout, G., Lavery, M., Courtial, J., Beijersbergen, M. & Padgett, M. Efficient sorting of orbital angular momentum states of light.Phys. Rev. Lett.105, 153601 (2010).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.105.153601\"\n },\n {\n \"idx\": \"13\",\n \"title\": \"Lavery, M. P. J. et al. Refractive elements for the measurement of the orbital angular momentum of a single photon.Opt. Express20, 2110–2115 (2012).\",\n \"link\": \"https://doi.org/10.1364%2FOE.20.002110\"\n },\n {\n \"idx\": \"14\",\n \"title\": \"Leach, J. et al. Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon.Phys. Rev. Lett.92, 013601 (2004).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.92.013601\"\n },\n {\n \"idx\": \"15\",\n \"title\": \"Gibson, G. et al. Free-space information transfer using light beams carrying orbital angular momentum.Opt. Express12, 5448–5456 (2004).\",\n \"link\": \"https://doi.org/10.1364%2FOPEX.12.005448\"\n },\n {\n \"idx\": \"16\",\n \"title\": \"Marrucci, L., Manzo, C. & Paparo, D. Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media.Phys. Rev. Lett.96, 163905 (2006).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.96.163905\"\n },\n {\n \"idx\": \"17\",\n \"title\": \"Nagali, E. et al. Quantum information transfer from spin to orbital angular momentum of photons.Phys. Rev. Lett.103, 013601 (2009).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.103.013601\"\n },\n {\n \"idx\": \"18\",\n \"title\": \"Marrucci, L. et al. Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications.J. Opt.13, 064001 (2011).\",\n \"link\": \"https://doi.org/10.1088%2F2040-8978%2F13%2F6%2F064001\"\n },\n {\n \"idx\": \"19\",\n \"title\": \"Karimi, E., Piccirillo, B., Nagali, E., Marrucci, L. & Santamato, E. Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates.Appl. Phys. Lett.94, 231124 (2009).\",\n \"link\": \"https://doi.org/10.1063%2F1.3154549\"\n },\n {\n \"idx\": \"20\",\n \"title\": \"Gruneisen, M. T., Dymale, R. C., Stoltenberg, K. E. & Steinhoff, N. Optical vortex discrimination with a transmission volume hologram.New J. Phys.13, 0830309 (2011).\",\n \"link\": \"https://doi.org/10.1088%2F1367-2630%2F13%2F8%2F083030\"\n },\n {\n \"idx\": \"21\",\n \"title\": \"Bryngdahl, O. Geometrical transformations in optics.J. Opt. Soc. Am. A64, 1092–1099 (1974).\",\n \"link\": \"https://doi.org/10.1364%2FJOSA.64.001092\"\n },\n {\n \"idx\": \"22\",\n \"title\": \"Bryngdahl, O. Optical map transformations.Opt. Commun.10, 164–168 (1974).\",\n \"link\": \"https://doi.org/10.1016%2F0030-4018%2874%2990046-7\"\n },\n {\n \"idx\": \"23\",\n \"title\": \"Saito, Y., Komatsu, S.-I. & Ohzu, H. Scale and rotation invariant real time optical correlator using computer generated hologram.Opt. Commun.47, 8–11 (1983).\",\n \"link\": \"https://doi.org/10.1016%2F0030-4018%2883%2990326-7\"\n },\n {\n \"idx\": \"24\",\n \"title\": \"O'Sullivan, M. N., Mirhosseini, M., Malik, M. & Boyd, R. W. Near-perfect sorting of orbital angular momentum and angular position states of light.Opt. Express.20, 24444–24449 (2012).\",\n \"link\": \"https://doi.org/10.1364%2FOE.20.024444\"\n },\n {\n \"idx\": \"25\",\n \"title\": \"Prongue, D., Herzig, H. P., Dandliker, R. & Gale, M. T. Optimized kinoform structures for highly efficient fan-out elements.Appl. Opt.31, 5706–5711 (1992).\",\n \"link\": \"https://doi.org/10.1364%2FAO.31.005706\"\n },\n {\n \"idx\": \"26\",\n \"title\": \"Romero, L. A. & Dickey, F. M. Theory of optimal beam splitting by phase gratings. I. One-dimensional gratings.J. Opt. Soc. Am. A24, 2280–2295 (2007).\",\n \"link\": \"https://doi.org/10.1364%2FJOSAA.24.002280\"\n },\n {\n \"idx\": \"27\",\n \"title\": \"Arrizon, V., Ruiz, U., Carrada, R. & Gonzalez, L. A. Pixelated phase computer holograms for the accurate encoding of scalar complex fields.J. Opt. Soc. Am. A24, 3500–3507 (2007).\",\n \"link\": \"https://doi.org/10.1364%2FJOSAA.24.003500\"\n }\n]","isConversational":false}}},{"rowIdx":197,"cells":{"unique_id":{"kind":"string","value":"s41467-019-13371-3"},"subfield":{"kind":"string","value":"Nanoscience and technology"},"title":{"kind":"string","value":"Templated dewetting of single-crystal sub-millimeter-long nanowires and on-chip silicon circuits"},"abstract":{"kind":"string","value":"Large-scale, defect-free, micro- and nano-circuits with controlled inter-connections represent the nexus between electronic and photonic components. However, their fabrication over large scales often requires demanding procedures that are hardly scalable. Here we synthesize arrays of parallel ultra-long (up to 0.75 mm), monocrystalline, silicon-based nano-wires and complex, connected circuits exploiting low-resolution etching and annealing of thin silicon films on insulator. Phase field simulations reveal that crystal faceting and stabilization of the wires against breaking is due to surface energy anisotropy. Wires splitting, inter-connections and direction are independently managed by engineering the dewetting fronts and exploiting the spontaneous formation of kinks. Finally, we fabricate field-effect transistors with state-of-the-art trans-conductance and electron mobility. Beyond the first experimental evidence of controlled dewetting of patches featuring a record aspect ratio of \\(\\sim\\) 1/60000 and self-assembled \\(\\sim\\) mm long nano-wires, our method constitutes a distinct and promising approach for the deterministic implementation of atomically-smooth, mono-crystalline electronic and photonic circuits."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"Semiconductor nanowires exhibit superior and configurable electronic and optical properties [1] , [2] . Their disruptive potential has been demonstrated in photonics [3] (e.g. for lasers [4] or quantum optics [5] ), electronics [6] , thermoelectricity [7] , energy storage with batteries [8] , gas [9] and mechanical [10] sensing, topological quantum states [11] , and much more. For these reasons their growth has been tackled with a plethora of techniques aiming at the production of controlled, ultra-long structures matching the needs of high yield, scalability (e.g. integration of a large number of devices on the same monolithic nano-wire) and material quality (e.g. smooth interfaces). As such, the range of available approaches to grow elongated crystalline structures steadily increases [1] , from direct top-down design, as, e.g., three-dimensional mesoscale lithography [12] , superlattice nanowire pattern transfer [13] up to the exploitation of natural phenomena such as the renowned Plateau-Rayleigh instability [14] . Several bottom-up, self-assembly methods can be employed to obtain high-quality parallel wires [15] . Nevertheless, a full control over their morphology, size, position, direction, inter-connection, and electrical isolation remains a challenge as current techniques are not versatile and are limited to a few micrometers length. On the other side, lithographic top-down methods can be used to precisely prepare a substrate for further engineering, as for instance shown for the cases of controlled wetting [16] or nano-imprint lithography [17] . Hybrid top-down/bottom-up approaches [18] , [19] , [20] , [21] can marry the crystalline quality of epitaxial self-assembly with the ultimate control of nano-structures position and size and create quantum-grade materials, such as complex assemblies of connected wires and membranes. However, their exploitation in scalable devices is often hindered by the complexity of their implementation requiring too many, cumbersome nano-fabrication steps. In fact, these approaches often need high-resolution etching (e.g. in order to set the final size of the epitaxial structures) and provide out-of-plane objects eventually requiring further processing before their implementation in a device [18] , [22] . These methods are hardly scalable as they exploit complex epitiaxial growth steps, often involving a metallic catalyst and are limited to structures extending only over a few micrometers. Here we fabricate arrays of in-plane, ultra-long nano-wires (up to 0.75 mm) and complex inter-connected circuits of monocrystalline silicon using a catalyst- and epitaxy-free, hybrid top-down/bottom-up approach based on the natural morphological evolution of thin solid films. We control the metamorphosis of a commercial (001)-oriented ultra-thin silicon film on insulator (UT-SOI) relying only on low-resolution etching and annealing, directly transforming it in monocrystalline nano-wires. The final structures have a lateral size up to five times smaller than the etched patch width, are obtained with size- and position-control and are electrically-isolated from the substrate. Phase field simulations of surface-diffusion-limited kinetics elucidate the key role of the surface-energy anisotropy in stabilizing the dewetting outcome against breaking. They quantitatively reproduce the main features of the morphological evolution of the patches. Exploiting the orientation-dependent edge faceting promoted by surface energy minimization [23] which hinders the onset of the typical Rayleigh-like instability along the wire axis [24] we extend these results to arbitrary in-plane crystallographic directions, building complex circuits of wires featuring splitting, changes in their directions and inter-connections. Finally, with a simple spin-on-dopant post-fabrication method, we render the wires conductive, demonstrating the possibility to use them as field-effect transistor [6] with trans-conductance and electron mobility similar to state-of-the-art nanowires devices.","references":{"indices":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],"infos":[{"idx":"1","title":"Xia, Y. et al. One-dimensional nanostructures: synthesis, characterization, and applications.Adv. Mater.15, 353–389 (2003).","link":"https://doi.org/10.1002%2Fadma.200390087"},{"idx":"2","title":"Law, M., Goldberger, J. & Yang, P. Semiconductor nanowires and nanotubes.Annu. Rev. Mater. Res.34, 83–122 (2004).","link":"https://doi.org/10.1146%2Fannurev.matsci.34.040203.112300"},{"idx":"3","title":"Yan, R., Gargas, D. & Yang, P. Nanowire photonics.Nat. Photonics3, 569 (2009).","link":"https://doi.org/10.1038%2Fnphoton.2009.184"},{"idx":"4","title":"Saxena, D. et al. 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Although silicon dewetting is also possible starting from an amorphous UT-SOI [29] , [30] , the need of a precise and controlled dewetting front in order to form regular nano-architectures requires mono-crystalline wafers where the dewetting process only affects the shape of the layer which always remains a mono-crystal [31] , [32] , [33] . The potential of templated dewetting was showcased for a mono-crystalline 12 nm thick UT-SOI where, in analogy with metals [34] , arrays of complex nano-architectures of islands and wires (hundred nanometer high and few micrometer long, circa) were reported [28] . The key tool used to enhance the stability of the dewetting outcome against breaking leading to reproducible patterns was the creation of ad hoc dewetting fronts triggering the formation of opposite rims that move one towards the other. So far, this approach was limited to patches extending over a few \\(\\upmu\\) m (aspect ratio \\(\\sim\\) 1/400) and, due to the anisotropy of surface energy, strictly oriented along the stable dewetting fronts (e.g. [110]) [28] . Following this concept, we tested the stability against annealing of a 12 nm thick UT-SOI (at temperature between 720 and 775 \\({}^{\\circ }\\) C, for periods ranging from 15 min to 2 h) patterned by electron-beam lithography and reactive ion etching in long trenches with variable pitch ( \\({d}_{{\\rm{LL}}}\\) = 0.5 up to 4 \\(\\upmu\\) m) defining patches featuring a width \\(w\\) ranging from \\(\\sim\\) 400 nm up to \\(\\sim\\) 3.8 \\(\\upmu\\) m. We first consider patches oriented along the stable dewetting front [110] (Figs. 1 – 3 ). Figure 1 a describes the method while a more detailed description is provided in the Methods section and reference [28] . Fig. 1: Ultra-long nano-wires formation. a Scheme of the sample fabrication: long patches are created in a 12 nm thick ultra-thin silicon on insulator (UT-SOI) on buried oxide (BOX, SiO \\({}_{2}\\) , 25 nm) with a pitch of \\({d}_{{\\rm{LL}}}\\) and width \\(w\\) . b Scanning electron microscope (SEM) image of 21 parallel nano-wires of length 0.75 mm, \\({d}_{{\\rm{LL}}}\\) = 1.3 \\(\\upmu\\) m, and \\(w\\) = 700 nm (a high-resolution image with the full wire length is provided in the high-resolution Supplementary Data 1 ). Annealing temperature was 775 \\({}^{\\circ }\\) C for 30′ in ultra-high vacuum (10 \\({}^{-10}\\) Torr, circa). The yellow areas highlight the etched trenches. Bottom right inset: blow up of a free-propagating edge and a dewetted wire. The distance covered by a freely receding rim is highlighted by \\(\\Delta x\\) . c Top panel: detail of the extremity of the wires. Bottom-right panel: blow-up of a wire extremity. Full size image Fig. 2: Atomic force microscopy of parallel wires oriented along the stable dewetting front. a – c Atomic force microscopy (AFM) images of wires arrays dewetted at 720 \\({}^{\\circ }\\) C for 120′ for \\({d}_{{\\rm{LL}}}\\) = 1, 2 and 4 \\(\\upmu\\) m respectively, and a length of 75 \\(\\upmu\\) m. The width of the etched trenches was about 100 nm (highlighted by vertical, white dotted lines). Horizontal dashed lines highlight the height profile shown in d . Vertical dashed lines highlight the sagittal height profile shown in f . d Transverse height profiles (from a – c , respectively top, central and bottom panel). The shaded areas highlight the original UT-SOI profile before annealing. e Statistical distributions of the full width at half maximum (FWHM) of the wires shown in a – c . The continuous lines are Gaussian envelops of the statistical distributions. f Sagittal profiles of the wires shown in a – c . g Statistical distributions of the wires height shown in f after subtraction of the height average value. The continuous lines are Gaussian envelops of the statistical distributions. Full size image Fig. 3: Phase field simulations of anisotropic and isotropic patches. a – c Left panel: cross sections at different times of phase field simulations of dewetting of a patch with initial aspect ratio h/w = 1.2/100, 1.2/200, and 1.4/200, respectively, taking into account the crystal anisotropy. Right panels: same as the left panel for the isotropic case. The full dynamics is shown in the Supplementary Movies 1 and 2 for a left and right panels respectively; 3 and 4 for b left and right panels, respectively; 5 and 6 for c left and right panels respectively). d Comparison between the experimental data (symbols, from Fig. 2 d ) and phase field simulations (continuous lines). The shaded areas highlight the buried oxide (BOX) underneath the dewetted structures. Full size image In optimized conditions (annealing temperature and time, and patterns width) we observe the formation of extremely long wires, with a length limited only by the patch design (up to 0.75 mm, Fig. 1 b, c and high-resolution Supplementary Data 1 ). Wires with no breaks and perfectly homogeneous height and width over their full length can be formed with a 100 \\(\\%\\) success rate. All the UT-SOI available in the patch ( \\(w\\) = 700 nm) collapsed in individual wires featuring a base of about 160 nm ( \\(\\sim\\) 4 times smaller than \\(w\\) ) and a height of 50 nm. In these conditions of annealing temperature and time, a simple stable dewetting front freely receding (semi-infinite UT-SOI) covers a distance of about \\(\\Delta x\\) = 650 nm (bottom-right inset in Fig. 1 b), a length comparable to the overall patch width \\(w\\) . In a similar sample, \\(\\sim\\) 0.9 and \\(\\sim\\) 1.9 \\(\\upmu\\) m wide patches ( \\({d}_{{\\rm{LL}}}\\) = 1 and 2 \\(\\upmu\\) m, respectively) collapsed in an individual wire, whereas \\(\\sim\\) 3.9 \\(\\upmu\\) m patches ( \\({d}_{{\\rm{LL}}}\\) = 4 \\(\\upmu\\) m) were partially dewetted in two parallel counter-propagating rims, as revealed by AFM measurements (Fig. 2 a–d). These features are attributed to a faster dewetting dynamics (and in turn to an earlier onset of the morphological instabilities) associated to smaller radius (i.e. larger curvature at the surface [35] , [36] ) even when considering a film in contact with a substrate [37] , [38] , [39] , [40] : the dewetting process for the larger pitches (lower overall curvature with respect to smaller pitch), is not concluded at the end of the annealing step. It is worth noting that, although the process for \\({d}_{{\\rm{LL}}}\\) = 1 \\(\\upmu\\) m is faster than that one relative to \\({d}_{{\\rm{LL}}}\\) = 2 \\(\\upmu\\) m, also in the former case the wires did not yet break into islands (the Plateau-Rayleigh instability along the wire did not take place). In all investigated cases, the fluctuations around the average full width at half maximum (FWHM) and heights of the wires were only a few nanometers (Fig. 2 f–g) accounting for the remarkable control of the dewetting process. We also observe that for longer annealing time and higher temperature, patches of any width produced isolated islands (eventually elongated), as expected from the conventional Plateau-Rayleigh instability (not shown) [14] . Phase field simulations The former results are compared to 2D phase field simulations of surface diffusion [41] , [42] , including surface-energy anisotropy [43] , [44] solved by a finite element approach [45] , [46] (see the Supplementary Note 1 for more details on the simulation method in use). They are performed mimicking the evolution in time of the cross-section of the experimental cases (Fig. 3 , Supplementary Movies 1 – 6 ). For each investigated aspect ratio ( \\(h/w\\) = 1.2/100, 1.2/200 and 1.2/400) we reproduce the dewetting dynamics, including the typical facets of the equilibrium shape of Si ({ \\(001\\) }, { \\(113\\) }, { \\(111\\) }, and { \\(110\\) }) by means of the corresponding surface-energy values [47] , [48] (Fig. 3 a–c, left panels). We systematically compare these results with the isotropic counterpart, by averaging the energies of different orientations (Fig. 3 a–c, right panels). The relevant features emerging from this analysis are summarized as follows: (i) For simulations reproducing \\(h/w\\) = 1.2/100 and 1.2/200 (corresponding to \\({d}_{{\\rm{LL}}}\\) = 1 \\(\\upmu\\) m and 2 \\(\\upmu\\) m, respectively Fig. 3 a, b, left panels), the patch effectively collapses in a single wire, directly reproducing the corresponding experiments (Fig. 3 d, left and central panels), thus providing a confirmation of the diffusion-limited kinetics at play. Discrepancies between experiments and theory for \\({d}_{{\\rm{LL}}}\\) = 1 \\(\\upmu\\) m can be attributed to convolution effects with the AFM tip leading to an overestimation of the wires FWHM also reflected in the large discrepancy between the wires width shown in Figs. 1 c, 2 e. (ii) The simulations for \\(h/w\\) = 1.2/400 (corresponding to \\({d}_{{\\rm{LL}}}\\) = 4 \\(\\upmu\\) m, Fig. 3 c, left panel) show first the formation of two parallel, counter-propagating rims, leading finally to a breakup in two parallel wires as final stage (not observed in the experiment, as the dewetting process is not complete, Fig. 2 c). A good agreement with experiments is found when focusing on the intermediate time steps (Fig. 3 d, right panel). (iii) Surface faceting is found to play a central role in determining a quantitative outcome of the process [49] . For \\(h/w\\) = 1.2/100 a single wire is obtained with and without anisotropy, whereas the case \\(h/w\\) = 1.2/200 deviates from experiments, providing two parallel wires as final state of the process, when neglecting preferential orientations (Fig. 3 b, right panel). Also for larger patches ( \\(h/w\\) = 1.2/400, Fig. 3 c, d, right panel) the isotropic case predicts three islands against the two found in the anisotropic case, confirming the tendency of the surface anisotropy forces to stabilize the patch against break-up. (iv) For the largest patch, the experimental rims are smaller than the prediction by phase field simulations and the valleys next to the rims are not visible. This feature is attributed to an effective larger stiffness/anisotropy of the real structures with respect to those considered in these simulations. This could be readily accounted for by phase field simulations [50] at the cost of a significant increase in the computational budget without however, delivering relevant, additional information than those discussed so far. Templated dewetting along arbitrary fronts In a different sample, similar patch arrangements are etched with a slight mis-cut of 2° circa, with respect to a stable dewetting front. We now consider patches size of 800 nm in width ( \\({d}_{{\\rm{LL}}}\\) = 1 \\(\\upmu\\) m, Fig. 4 a). Fig. 4: Wires tilt and connected circuits. a SEM image of wires obtained from \\(\\sim\\) 800 nm wide patches, \\(\\sim\\) 2° mis-cut with respect to the main crystallographic axis [1-10] (highlighted by white, dashed lines) and annealed at 740 \\({}^{\\circ }\\) C for 15 min. The yellow areas highlight the etched parts. Bottom inset: blow-up of a wire. White arrows highlight small tips at the wires edges; yellow arrows highlight tips in correspondence of the wire kinks. The equilibrium facets of the wire are highlighted with colored rectangles (green: { \\(111\\) }; blue: { \\(113\\) }; red: { \\(001\\) }. b SEM images of connections between parallel wires. The yellow areas highlight the etched parts. In each column, the two panels show different realizations of the connectors between two parallel wires oriented at different angles with respect to the [110] crystallographic direction. Respectively from the left to the right: 90°, 0°, and 15°. The full set of images of complex circuits of three connected wires oriented at different angles with respect to the crystallographic axes is provided in the Supplementary Note 2 and high-resolution Supplementary Data 1 – 7 in their full 33 \\(\\upmu\\) m length. Full size image The mis-cut does not impede the formation of well-ordered arrays of parallel and uniform wires, mostly intact and following the macroscopic direction imposed by the etching. Small kinks are formed during edge retraction in analogy to what was observed in the metallic counterpart [23] , [51] , [52] , [53] . However, in our case, the periodicity of the edge undulation of the Si film is not as regular as those found in metals and is linked to the presence of small tips at the BOX surface (highlighted by white and yellow arrows in the bottom panel of Fig. 4 a) on the denuded BOX and at the sides of the wires, where they touch the BOX (these are also found in spontaneously dewetted samples and are thus not attributed to the lithographic process, see the Supplementary Note 2 ). All the kinks form in presence of an impurity at the wires/box interface (yellow arrows in the bottom inset of Fig. 4 a) whereas in some cases only a wrinkling of the { \\(113\\) } and { \\(111\\) } facets is observed (white arrows in the bottom inset of Fig. 4 a). The top { \\(001\\) } facet is instead always flat. The mechanism illustrated in Fig. 4 a helps in setting arbitrary orientations of the etched profile and obtaining slightly curved structures. This can be expanded to more complex patterns leading to connected networks of wires. We address this point by etching parallel patches (700 nm large and 33 \\(\\upmu\\) m long) and studying the effect of their orientation with respect to the crystallographic axes on the stability against breaking (Fig. 4 b). Between the patches (etching highlighted by yellow areas in Fig. 4 ) we added several connectors with variable size and respective alignment. This design is repeated with 15° steps with respect to the [110] direction in order to cover 360° (Supplementary Note 2 and high-resolution Supplementary Data 2 – 7 ). As found for metals [23] , [51] we observe a general tendency to more frequent break-up of the patches along the unstable axis [100] whereas along the stable one [110] we find one individual, elongated island as previously shown for simple wires (Supplementary Note 2 and high-resolution Supplementary Data 2 ). Nonetheless, for short enough annealing time, well connected structures featuring a limited number of breaks can be found in a large range of patch orientations. Up to 15° with respect to [110] the structures are not broken along the wires nor at the level of large and small connectors (respectively 1.5 \\(\\upmu\\) m and 0.8 \\(\\upmu\\) m wide, Fig. 4 b). Finally, for the larger connectors the structure is robust against breaking up to 45° (high-resolution Supplementary Data 2 – 7 ). This demonstrates that it is possible to control the continuity, connectivity and curvature of the wires up to several degrees of misalignment with respect to the stable dewetting front without any optimization. A more appropriate choice of etching design and experimental conditions (e.g. annealing time, patch width and shape as well as ad hoc additional features etched within [28] ) may improve the quality of the final outcome. Crystalline structure of templated dewetted wires In order to rule out the presence of crystalline defects in the dewetted structures we performed atomic-resolution scanning transmission electron microscopy (STEM) imaging on a wire (Fig. 5 ). In line with previous evidences in Si- and SiGe-based islands [27] , [31] , we observe the typical crystalline structure of bulk Si and the absence of extended dislocations. A slight crystalline disorder can be observed in some part of the wire body, at the interface with the original UT-SOI substrate (at about 12 nm from the BOX, Fig. 5 c, d). This feature has been previously observed in STEM images [27] , [31] and we ascribe it to residual defectivity on the UT-SOI substrate, possibly due to a non ideal cleaning of the surface. For the sake of thoroughness we mention that geometric phase analysis performed on the full wire section does not reveal any strain in the crystal structure (not shown). Fig. 5: Atomic-resolution scanning transmission electron microscopy imaging. a High Angle Annular Dark Field (HAADF) scanning transmission electron micrograph (STEM) of the section of a dewetted wire. The original UT-SOI is highlighted by a dashed rectangle. b Blow-up of a (yellow rectangle) highlighting the crystalline structure of the top part of the silicon wire (c-Si) and the native oxide surrounding it (thickness about 2.5 nm). c Blow-up of the bottom-left part of the wire from a (blue rectangle) highlighting its crystalline structure and the presence of some alloy disorder. d Blow up of c (green rectangle) highlighting the alloy disorder. Full size image Electric conduction from parallel wires arrays Through the templated dewetting process we demonstrated crystalline silicon wires formed directly on an insulating substrate. To show the potential of these structures for electronic circuits a doping procedure involving phosphorus spin-on-dopant (SOD) deposition and rapid thermal annealing treatment has been carried out on 70 \\(\\upmu\\) m long nanowires (base size \\(\\sim\\) 150 nm, height \\(\\sim\\) 75 nm) to render them conductive (Fig. 6 and Methods) [54] . All metal contacts (source, drain and gate, respectively S, D, and G) are 130 nm thick gold and were placed by e-beam deposition on 15 parallel nano-wires. S and D partially wet the silicon wires whereas G is separated from the wires by a 100 nm thick silicon dioxide (deposited via e-beam evaporation) covering the wires for about 30 \\(\\upmu\\) m (Fig. 6 ). The oxide leakage characteristics is about 10 pA (not shown). Fig. 6: Electrical properties of parallel wires. a Scheme of the field effect transistor (FET) wires device (doping about 3 \\(\\times\\) 10 \\({}^{18}\\) cm \\({}^{-3}\\) , n-type). b SEM images of the electric contacts (left part, source, gate and drain, respectively S, D, and G) and a detail of the S contact (right). The gold contact (Au) and the SiO \\({}_{2}\\) partially covering the wires are highlighted. c Electrical transport characteristics of 15 parallel wires (S-D current versus S-D voltage curves recorded at different G voltage. Inset: S-D current versus G voltage recorded at source-drain voltage \\({V}_{{\\rm{SD}}}\\) = 2 V for 15 parallel wires. Full size image S-D current curves as a function of the S-D voltage are registered for different G voltages, demonstrating a typical behavior of a field-effect transistor (FET). This transistor works in enhancement mode, where the saturation and the S-D voltage are characteristic of a n-channel FET [55] . From the I–V curves obtained on 15 parallel nanowires for different G tension, we estimate a trans-conductance ( \\({G}_{{\\rm{NW}}}\\) = \\(\\Delta {I}_{{\\rm{SD}}}/\\Delta {V}_{{\\rm{G}}}\\) , where \\(\\Delta {I}_{{\\rm{SD}}}\\) is the source-drain current modulation against the corresponding change in gate tension \\(\\Delta {V}_{{\\rm{G}}}\\) ) of the order of \\(\\sim \\upmu\\) S per wire (Fig. 6 ). Adopting the common approximations for nanowire-based transistors [56] and neglecting spurious effects [57] , we can estimate the electron mobility with the formula \\({\\upmu }_{{\\rm{e}}}\\) = \\(({L}^{2}{G}_{{\\rm{NW}}})/({V}_{{\\rm{SD}}}{C}_{{\\rm{NW}}})\\) , where \\(L\\) is the gate contact length, \\({V}_{{\\rm{SD}}}\\) is the source-drain tension and \\({C}_{{\\rm{NW}}}\\) is the gate capacitance. In a cylindrical geometry, this latter characteristic can be expressed as \\({C}_{{\\rm{NW}}}\\) = \\(2\\pi {\\epsilon }_{{\\rm{0}}}{\\epsilon }_{{\\rm{r}}}L/{\\mathrm{cos}}{h}^{-1}(t/R)\\) , with \\({\\epsilon }_{{\\rm{0}}}\\) vacuum permittivity, \\({\\epsilon }_{{\\rm{r}}}\\) the static dielectric constant of the gate oxide, \\({t}_{{\\rm{tot}}}={t}_{{\\rm{ox}}}+R\\) distance between wire center and metallic contact, where \\({t}_{{\\rm{ox}}}\\) is the oxide thickness ad \\(R\\) the wire radius. Assuming \\({\\epsilon }_{{\\rm{r}}} \\sim\\) 3.9, \\(L \\sim\\) 30 nm, \\({t}_{{\\rm{ox}}} \\sim\\) 100 nm, \\(R\\) \\(\\sim\\) 60 nm (estimated as average between height and base size), we obtain a gate capacitance \\({C}_{{\\rm{NW}}} \\sim\\) 5 fF. With these values, we can roughly estimate an electron mobility ranging between 0.5 and 5 \\(\\times\\) 10 \\({}^{3}\\) cm \\({}^{2}\\) V \\({}^{-1}\\) s \\({}^{-1}\\) ."},{"section":"Discussion","content":"There are several differences and advantages of our wires with respect to previous demonstrations of similar structures implemented via dewetting and other fabrication techniques, particularly regarding the simplicity and versatility of implementation, improved comprehension of the dewetting mechanism and improved quality of the structures enhancing the electrical properties of the implemented devices. In what follows we discuss all these features with respect to the existing state-of-the-art. In this work, we extend the coherent control of dewetting by more than 2 order of magnitude, going from patches of a few \\(\\upmu\\) m [28] to 0.75 mm (only limited by the pattern etched prior to annealing set by the writing field of the e-beam lithography in use). This is a record aspect ratio of a \\(\\sim\\) 1/60000 patch providing uniform and reproducible nanowires. In optimal annealing conditions and patch size we obtain a 100 \\(\\%\\) success in forming perfect faceted nano-wires with no breaks along their length and size fluctuations in the few per cent range. An important difference with respect to previous reports of Si dewetting [28] is the control of patch evolution for patterns oriented along unstable fronts. The formation of kinks during dewetting (mediated by small defects at the BOX/wire interface) allows to adjust the macroscopic directions of the final structures without breaking. This feature has never been reported so far in semiconductor dewetting and it allows to curve, split and connect the wires ad libitum in order to form complex circuits. The limit of this method is evident when considering patches larger than a few \\(\\upmu\\) m. Although the fluctuation of the wires width and height is always well below 10 \\(\\%\\) , larger patches ( d \\({}_{{\\rm{LL}}}\\) = 4 \\(\\upmu\\) m, Fig. 2 ) provide wires showing a spread of their width about three times larger than those formed from smaller patches ( d \\({}_{{\\rm{LL}}}\\) = 1 and 2 \\(\\upmu\\) m, Fig. 2 ). This lower level of control over the rim evolution for larger structures was reported for the case of simple square patches [28] : above 3 \\(\\upmu\\) m, disorder effects play an important role leading to marked deviations of the dewetting dynamics with respect to what expected for ideal systems (as those shown in the simulations, Fig. 3 and reference [28] ). These spurious effects could be attributed to extrinsic disorder locally perturbing the rim evolution and affecting in a more marked way larger patches with respect to smaller ones (e.g. native oxide not properly removed or other impurities present in the ultra-high vacuum used for silicon dewetting). Furthermore, small tips found at the rim/BOX interface can perturb the dewetting dynamics [28] . Thus, although these defects allow the formation of kinks along the wires and thus to curve them with respect to the preferential axes orientations, they may be a limit for a coherent control of larger patches (e.g. those that would result in two parallel wires instead of an individual one). In the present case of 12 nm thick UT-SOI, a coherent control of the patch evolution (in absence of additional features in the initial patch design) is limited to <3 \\(\\upmu\\) m. In the present work, we also managed to compare real systems with realistic models taking into account anisotropic surface diffusion. So far, simulations of templated dewetting of UT-SOI based on a phase field approach considered patches featuring an aspect ratio of, at most, 1/160 [28] . This was in stark contrast with the real systems featuring a much smaller value of \\(\\sim\\) 1/400. Furthermore, this attempt to reproduce the experimental outcomes did not take into account the underlying crystal anisotropy. A reasonable agreement between experiments and simulations was found for short time evolution while showing marked discrepancies for longer times. More generally, in the last few years several theoretical works tried to tackle the anisotropic dewetting dynamic with sharp interface models for both cases of weak [58] and strong [50] anisotropy. However, in all these cases the patch aspect ratio was at most 1/60 which is pretty far from the actual experimental conditions used for metal and semiconductor dewetting. Here we used a phase field model taking into account surface diffusion and surface-energy anisotropy for a 1/1 scale case (aspect ratio up to 1/330). Our novel theoretical understanding of the anisotropic dewetting problem allows therefore to correctly predict long-time evolution of the main features observed in experiments showing that the presence of facets (due to anisotropic surface energy) stabilizes the dewetting outcome against breaking. Isotropic models (e.g. showing two parallel wires instead of only one) fails in this task, at least for larger patches. From a fabrication standpoint our approach offers several advantages with respect to other bottom-up methods. We implemented our wires on commercial UT-SOI wafers, available in a large set of device thickness, orientation, doping, composition (e.g. SiGe and Ge on insulator), BOX thickness, and up to 12 inches in size. These features are not matched by other methods implemented on, at most, a few inches and expensive epilayers [19] , [20] , [21] (e.g. for III–V-based structures). Our structures are implemented only in two steps, etching and annealing. Thus, templated dewetting of wires offers a versatility in directions, splitting and connections not attainable with catalyst-based approaches for in-plane wires growth [59] (where all the structures were parallel) and an easier implementation with respect to recent demonstrations were the wires were bound together with complex patterning and epitaxial nano-fabrication methods [18] , [19] , [20] , [21] , [22] , [60] , [61] (eventually requiring cumbersome post-growth processing for the implementation of an electronic device). Other self-assembly methods are not suitable for the formation of crystalline structures on amorphous SiO \\({}_{2}\\) . Bottom-up methods for semiconductor-based 3D structures rely on strain-induced assembly (e.g. via Stranski Krastanov for III–V and IV–IV) [62] , droplet epitaxy and droplet etching (only for III–V) [63] , vapor-liquid-solid growth via gold catalyst [59] , or more advanced hybrid top-down/bottom-up approaches [19] . All these strategies can be exploited only when a precise epitaxial relation holds between the substrate and the deposited material. As such, they require a crystalline support. In contrast to this, we directly produce well-ordered, monocrystalline nano-architectures on amorphous SiO \\({}_{2}\\) without any epitaxial relation. A straightforward consequence is that silicon dewetting provides electrical isolation of the three-dimensional nano-achitectures from the substrate, a clear advantage for the implementation of electronic devices (e.g. the field-effect transistors shown here) with respect to recent in-plane nano-architectures epitaxially grown on a III–V substrate [18] , [21] that becomes insulating only at very low temperature [19] . Another important difference between dewetting and recent reports of advanced epitaxial structures based on selective area growth, is the lack of strain and alloy disorder that is known to complicate the interpretation of the device behavior [19] , [20] . The width of our wires is 4–5 times smaller than the initial patch width, implying that a low-resolution etching (e.g. based on optical or nano-imprint lithography) can in principle be exploited. This constitutes an advantage with respect to current hybrid top-down/bottom-up approaches, where the resolution of the lithography directly sets the final size of the structures [19] , [21] . In this respect, it is worth stressing that a high resolution for electron-beam lithography inevitably requires a reduced writing field, limiting the size of nanowires to rather short channel length (a few \\(\\upmu\\) m for both top-down [57] , [64] and bottom-up [18] , [19] , [20] , [21] approaches). In our case, a low resolution allows for mm scale structures providing an aspect ratio of the order of 1/10 \\({}^{4}\\) . Our approach allows to tune the wires height on the same substrate in contrast with top-down approaches that lead to structures with a height fixed by the thickness of the UT-SOI in use or by the etching depth. We achieved this by setting the initial patch lateral width (e.g. from \\(w\\) = 0.8 \\(\\upmu\\) m up to \\(w\\) = 2 \\(\\upmu\\) m) providing wires having a base between 150 and 400 nm and height ranging from 50 to 100 nm. In order to obtain smaller wires, it is in principle possible to use thinner UT-SOI layers (e.g. by etching a few nm of the top silicon layer). One of the most important features of nano-wires is their electrical properties. For instance, the changes in conduction of wire-based transistors can be efficiently exploited for sensing [64] or for thermoelectricity [7] with wires having rough interfaces. Nevertheless, most works on nanowires have focused on studying and optimizing their surface smoothness. Surface defects are very common and lead to electron-surface roughness scattering influencing the charge carrier density in the underlying silicon matrix [64] modifying the electronic properties of a device (e.g. drop of the electron mobility) [65] . Thus, trans-conductance and electron mobility are important figures of merits that account for the quality of the interfaces and the limits of a wire-based device. In our wires, the measured value of trans-conductance is \\({G}_{{\\rm{NW}}} \\sim\\) 1–9 \\(\\upmu\\) S. This is quite similar to state-of-the art FET transistors based on Si-nanowires grown via conventional bottom-up methods. In fact, for these devices, \\({G}_{{\\rm{NW}}}\\) is at most a few \\(\\upmu\\) S [66] , [67] even when considering wires having size close to those shown here [6] . From the electric characterization, a rough estimation of electron mobility in our FET transistors provides \\({\\upmu }_{{\\rm{e}}}\\) = 0.5–5 \\(\\times\\) 10 \\({}^{3}\\) cm \\({}^{2}\\) V \\({}^{-1}\\) s \\({}^{-1}\\) . These values are in line with those found in Si-based bottom-up nanowires devices [6] , [10] , [66] . For top-down FET wires (usually fabricated via e-beam lithography and reactive ion etching) typical values of \\({\\upmu }_{{\\rm{e}}}\\) lie in few hundreds of cm \\({}^{2}\\) V \\({}^{-1}\\) s \\({}^{-1}\\) [56] , [68] . Larger values of \\({\\upmu }_{{\\rm{e}}}\\) , up to \\(\\sim\\) 10 \\({}^{3}\\) cm \\({}^{2}\\) V \\({}^{-1}\\) s \\({}^{-1}\\) , can be reached with top-down wires at the price of cumbersome fabrication methods (e.g. for smoothing the wires walls via oxidation), Ge alloying or strain engineering [69] , [70] . Thus, even without any kind of optimization of our devices, we can reach relatively high electron mobility. We interpret this feature as a possible consequence of the reduced surface roughness in our structures with respect to those obtained via vapor-liquid-solid growth and top-down methods. In conclusion, we showed that dewetting, a spontaneous shape instability common to many different thin films of organic and inorganic substances, can be efficiently controlled in order to form extremely long and connected circuits of monocrystalline silicon wires on SiO \\({}_{2}\\) . We extended the control of this process to silicon patches having the record aspect ratio of 1/60000 (to be compared with the case of metals 1/100 [34] , and semiconductors 1/400 [28] ) forming extremely elongated silicon mono-crystals. We directly compare the experimental outcomes with 1/1 scale phase field simulations of surface diffusion that quantitatively reproduce their morphological evolution. We show a clear evidence of the key role played by faceting in stabilizing the dewetting outcome against breaking and thus for the reproducibility of the process and the stability of the final structures. Phosphorous-doped conductive Si wires are implemented showcasing the possibility to fabricate conducting nano-channels and transistors on an insulating substrate. Since the proposed approach is very general, it can be adapted to tune the Si wires aspect ratio by choosing suitable UT-SOI thickness and pattern periodicity combined with more complex, connected nano-architectures towards a full exploitation of their record length and atomically smooth facets. Owing to a similar dewetting dynamic ruled by surface-diffusion-limited kinetics observed in SiGe alloys [32] similar results can be extended to these materials rendering this method attractive for wires formation with different materials with the perspective of band-gap engineering and carrier mobility enhancement."},{"section":"Methods","content":"Sample preparation A 12 nm-thick UT-SOI on a 25 nm thick buried oxide (BOX) was etched by electron-beam lithography and reactive etching with parallel trenches, from 0.75 mm to 70 \\(\\upmu\\) m long, with variable pitch (0.5–4 \\(\\upmu\\) m line-to-line distance, d \\({}_{{\\rm{LL}}}\\) ) and orientations with respect to the crystallographic axes. Thus, the samples were processed by plasma and wet chemical cleaning in N \\({}_{2}\\) atmosphere for 20–30 s in a 5–10 \\(\\%\\) HF. Finally, they were annealed in the ultrahigh vacuum ( \\(\\sim\\) 10 \\({}^{-10}\\) Torr) of a molecular beam epitaxy reactor. Sample doping Deposition of Spin on dopant (SOD, OD P508) was performed by spin-coating at 4000 rpm for 1 min and baking on a hot plate (10 min at 120 \\({}^{\\circ }\\) C). Thermal diffusion of dopant was induced by rapid thermal annealing (30″ at 850 \\({}^{\\circ }\\) C in N \\({}_{2}\\) atmosphere). The SOD was removed by wet etching for 60″ in dilute HF (1:10). The metal pads (5 nm Ti/150 nm Au), after the definition of the contact design process by electron-beam lithography, are deposited by e-beam deposition. Electronic imaging Scanning transmission electron micrograph have been acquired in z-contrast with a Cs probe corrected JEOL ARM 200F operated at 60 keV."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"Semiconductor nanowires exhibit superior and configurable electronic and optical properties [1] , [2] . Their disruptive potential has been demonstrated in photonics [3] (e.g. for lasers [4] or quantum optics [5] ), electronics [6] , thermoelectricity [7] , energy storage with batteries [8] , gas [9] and mechanical [10] sensing, topological quantum states [11] , and much more. For these reasons their growth has been tackled with a plethora of techniques aiming at the production of controlled, ultra-long structures matching the needs of high yield, scalability (e.g. integration of a large number of devices on the same monolithic nano-wire) and material quality (e.g. smooth interfaces). As such, the range of available approaches to grow elongated crystalline structures steadily increases [1] , from direct top-down design, as, e.g., three-dimensional mesoscale lithography [12] , superlattice nanowire pattern transfer [13] up to the exploitation of natural phenomena such as the renowned Plateau-Rayleigh instability [14] . Several bottom-up, self-assembly methods can be employed to obtain high-quality parallel wires [15] . Nevertheless, a full control over their morphology, size, position, direction, inter-connection, and electrical isolation remains a challenge as current techniques are not versatile and are limited to a few micrometers length. On the other side, lithographic top-down methods can be used to precisely prepare a substrate for further engineering, as for instance shown for the cases of controlled wetting [16] or nano-imprint lithography [17] . Hybrid top-down/bottom-up approaches [18] , [19] , [20] , [21] can marry the crystalline quality of epitaxial self-assembly with the ultimate control of nano-structures position and size and create quantum-grade materials, such as complex assemblies of connected wires and membranes. However, their exploitation in scalable devices is often hindered by the complexity of their implementation requiring too many, cumbersome nano-fabrication steps. In fact, these approaches often need high-resolution etching (e.g. in order to set the final size of the epitaxial structures) and provide out-of-plane objects eventually requiring further processing before their implementation in a device [18] , [22] . These methods are hardly scalable as they exploit complex epitiaxial growth steps, often involving a metallic catalyst and are limited to structures extending only over a few micrometers. Here we fabricate arrays of in-plane, ultra-long nano-wires (up to 0.75 mm) and complex inter-connected circuits of monocrystalline silicon using a catalyst- and epitaxy-free, hybrid top-down/bottom-up approach based on the natural morphological evolution of thin solid films. We control the metamorphosis of a commercial (001)-oriented ultra-thin silicon film on insulator (UT-SOI) relying only on low-resolution etching and annealing, directly transforming it in monocrystalline nano-wires. The final structures have a lateral size up to five times smaller than the etched patch width, are obtained with size- and position-control and are electrically-isolated from the substrate. Phase field simulations of surface-diffusion-limited kinetics elucidate the key role of the surface-energy anisotropy in stabilizing the dewetting outcome against breaking. They quantitatively reproduce the main features of the morphological evolution of the patches. Exploiting the orientation-dependent edge faceting promoted by surface energy minimization [23] which hinders the onset of the typical Rayleigh-like instability along the wire axis [24] we extend these results to arbitrary in-plane crystallographic directions, building complex circuits of wires featuring splitting, changes in their directions and inter-connections. Finally, with a simple spin-on-dopant post-fabrication method, we render the wires conductive, demonstrating the possibility to use them as field-effect transistor [6] with trans-conductance and electron mobility similar to state-of-the-art nanowires devices.\",\n \"references\": {\n \"indices\": [\n 1,\n 2,\n 3,\n 4,\n 5,\n 6,\n 7,\n 8,\n 9,\n 10,\n 11,\n 12,\n 13,\n 14,\n 15,\n 16,\n 17,\n 18,\n 19,\n 20,\n 21,\n 22,\n 23,\n 24\n ],\n \"infos\": [\n {\n \"idx\": \"1\",\n \"title\": \"Xia, Y. et al. One-dimensional nanostructures: synthesis, characterization, and applications.Adv. Mater.15, 353–389 (2003).\",\n \"link\": \"https://doi.org/10.1002%2Fadma.200390087\"\n },\n {\n \"idx\": \"2\",\n \"title\": \"Law, M., Goldberger, J. & Yang, P. Semiconductor nanowires and nanotubes.Annu. Rev. Mater. Res.34, 83–122 (2004).\",\n \"link\": \"https://doi.org/10.1146%2Fannurev.matsci.34.040203.112300\"\n },\n {\n \"idx\": \"3\",\n \"title\": \"Yan, R., Gargas, D. & Yang, P. Nanowire photonics.Nat. Photonics3, 569 (2009).\",\n \"link\": \"https://doi.org/10.1038%2Fnphoton.2009.184\"\n },\n {\n \"idx\": \"4\",\n \"title\": \"Saxena, D. et al. Optically pumped room-temperature gaas nanowire lasers.Nat. Photonics7, 963 (2013).\",\n \"link\": \"https://doi.org/10.1038%2Fnphoton.2013.303\"\n },\n {\n \"idx\": \"5\",\n \"title\": \"Claudon, J. et al. A highly efficient single-photon source based on a quantum dot in a photonic nanowire.Nat. Photonics4, 174 (2010).\",\n \"link\": \"https://doi.org/10.1038%2Fnphoton.2009.287x\"\n },\n {\n \"idx\": \"6\",\n \"title\": \"Cui, Y., Zhong, Z., Wang, D., Wang, W. U. & Lieber, C. M. High performance silicon nanowire field effect transistors.Nano Lett.3, 149–152 (2003).\",\n \"link\": \"https://doi.org/10.1021%2Fnl025875l\"\n },\n {\n \"idx\": \"7\",\n \"title\": \"Hochbaum, A. I. et al. 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Nanotechnol.13, 192 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41565-017-0032-8\"\n },\n {\n \"idx\": \"12\",\n \"title\": \"Luo, Z. et al. Atomic gold–enabled three-dimensional lithography for silicon mesostructures.Science348, 1451–1455 (2015).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1257278\"\n },\n {\n \"idx\": \"13\",\n \"title\": \"Melosh, N. A. et al. Ultrahigh-density nanowire lattices and circuits.Science300, 112–115 (2003).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1081940\"\n },\n {\n \"idx\": \"14\",\n \"title\": \"Day, R. W. et al. Plateau-rayleigh crystal growth of periodic shells on one-dimensional substrates.Nat. Nanotechnol.10, 345 (2015).\",\n \"link\": \"https://doi.org/10.1038%2Fnnano.2015.23\"\n },\n {\n \"idx\": \"15\",\n \"title\": \"Jin, F. H., Peter, W. & Margit, Z. Semiconductor nanowires: from SelfâĂŘOrganization to Patterned Growth.Small2, 700–717 (2006).\",\n \"link\": \"https://doi.org/10.1002%2Fsmll.200500322\"\n },\n {\n \"idx\": \"16\",\n \"title\": \"Cavallini, M., Gentili, D., Greco, P., Valle, F. & Biscarini, F. Micro-and nanopatterning by lithographically controlled wetting.Nat. Protoc.7, 1668 (2012).\",\n \"link\": \"https://doi.org/10.1038%2Fnprot.2012.094\"\n },\n {\n \"idx\": \"17\",\n \"title\": \"Spinelli, P., Verschuuren, M. A. & Polman, A. Broadband omnidirectional antireflection coating based on subwavelength surface mie resonators.Nat. Commun.3, 692 (2012).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms1691\"\n },\n {\n \"idx\": \"18\",\n \"title\": \"Gazibegovic, S. et al. Epitaxy of advanced nanowire quantum devices.Nature548, 434 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fnature23468\"\n },\n {\n \"idx\": \"19\",\n \"title\": \"Krizek, F. et al. Field effect enhancement in buffered quantum nanowire networks.Phys. Rev. Materials2, 093401 (2018).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevMaterials.2.093401\"\n },\n {\n \"idx\": \"20\",\n \"title\": \"Friedl, M. et al. Template-assisted scalable nanowire networks.Nano Lett.18, 2666–2671 (2018).\",\n \"link\": \"https://doi.org/10.1021%2Facs.nanolett.8b00554\"\n },\n {\n \"idx\": \"21\",\n \"title\": \"Vaitiekėnas, S. et al. Selective-area-grown semiconductor-superconductor hybrids: A basis for topological networks.Phys. Rev. Lett.121, 147701 (2018).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.121.147701\"\n },\n {\n \"idx\": \"22\",\n \"title\": \"Plissard, S. R. et al. Formation and electronic properties of insb nanocrosses.Nat. Nanotechnol8, 859 (2013).\",\n \"link\": \"https://doi.org/10.1038%2Fnnano.2013.198\"\n },\n {\n \"idx\": \"23\",\n \"title\": \"Ye, J. & Thompson, C. V. Anisotropic edge retraction and hole growth during solid-state dewetting of single crystal nickel thin films.Acta Mater.59, 582–589 (2011).\",\n \"link\": \"https://doi.org/10.1016%2Fj.actamat.2010.09.062\"\n },\n {\n \"idx\": \"24\",\n \"title\": \"Rayleigh, L. On the instability of jets.P. Lond. Math. Soc.1, 4–13 (1878).\",\n \"link\": \"https://doi.org/10.1112%2Fplms%2Fs1-10.1.4\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Templated dewetting along stable dewetting fronts Dewetting of monocrystalline thin silicon films is a spontaneous phenomenon where capillary forces drive mass transport via surface-diffusion-limited kinetics [25] , [26] . It leads to a complete metamorphosis of the flat layer in three-dimensional structures through hole nucleation, rims formation (where mass accumulates while receding), followed by finger-like structures and finally, in isolated islands. This natural shape evolution can, however, be controlled by engineering the dewetting fronts by patterning the UT-SOI prior to annealing [27] , [28] . Although silicon dewetting is also possible starting from an amorphous UT-SOI [29] , [30] , the need of a precise and controlled dewetting front in order to form regular nano-architectures requires mono-crystalline wafers where the dewetting process only affects the shape of the layer which always remains a mono-crystal [31] , [32] , [33] . The potential of templated dewetting was showcased for a mono-crystalline 12 nm thick UT-SOI where, in analogy with metals [34] , arrays of complex nano-architectures of islands and wires (hundred nanometer high and few micrometer long, circa) were reported [28] . The key tool used to enhance the stability of the dewetting outcome against breaking leading to reproducible patterns was the creation of ad hoc dewetting fronts triggering the formation of opposite rims that move one towards the other. So far, this approach was limited to patches extending over a few \\\\(\\\\upmu\\\\) m (aspect ratio \\\\(\\\\sim\\\\) 1/400) and, due to the anisotropy of surface energy, strictly oriented along the stable dewetting fronts (e.g. [110]) [28] . Following this concept, we tested the stability against annealing of a 12 nm thick UT-SOI (at temperature between 720 and 775 \\\\({}^{\\\\circ }\\\\) C, for periods ranging from 15 min to 2 h) patterned by electron-beam lithography and reactive ion etching in long trenches with variable pitch ( \\\\({d}_{{\\\\rm{LL}}}\\\\) = 0.5 up to 4 \\\\(\\\\upmu\\\\) m) defining patches featuring a width \\\\(w\\\\) ranging from \\\\(\\\\sim\\\\) 400 nm up to \\\\(\\\\sim\\\\) 3.8 \\\\(\\\\upmu\\\\) m. We first consider patches oriented along the stable dewetting front [110] (Figs. 1 – 3 ). Figure 1 a describes the method while a more detailed description is provided in the Methods section and reference [28] . Fig. 1: Ultra-long nano-wires formation. a Scheme of the sample fabrication: long patches are created in a 12 nm thick ultra-thin silicon on insulator (UT-SOI) on buried oxide (BOX, SiO \\\\({}_{2}\\\\) , 25 nm) with a pitch of \\\\({d}_{{\\\\rm{LL}}}\\\\) and width \\\\(w\\\\) . b Scanning electron microscope (SEM) image of 21 parallel nano-wires of length 0.75 mm, \\\\({d}_{{\\\\rm{LL}}}\\\\) = 1.3 \\\\(\\\\upmu\\\\) m, and \\\\(w\\\\) = 700 nm (a high-resolution image with the full wire length is provided in the high-resolution Supplementary Data 1 ). Annealing temperature was 775 \\\\({}^{\\\\circ }\\\\) C for 30′ in ultra-high vacuum (10 \\\\({}^{-10}\\\\) Torr, circa). The yellow areas highlight the etched trenches. Bottom right inset: blow up of a free-propagating edge and a dewetted wire. The distance covered by a freely receding rim is highlighted by \\\\(\\\\Delta x\\\\) . c Top panel: detail of the extremity of the wires. Bottom-right panel: blow-up of a wire extremity. Full size image Fig. 2: Atomic force microscopy of parallel wires oriented along the stable dewetting front. a – c Atomic force microscopy (AFM) images of wires arrays dewetted at 720 \\\\({}^{\\\\circ }\\\\) C for 120′ for \\\\({d}_{{\\\\rm{LL}}}\\\\) = 1, 2 and 4 \\\\(\\\\upmu\\\\) m respectively, and a length of 75 \\\\(\\\\upmu\\\\) m. The width of the etched trenches was about 100 nm (highlighted by vertical, white dotted lines). Horizontal dashed lines highlight the height profile shown in d . Vertical dashed lines highlight the sagittal height profile shown in f . d Transverse height profiles (from a – c , respectively top, central and bottom panel). The shaded areas highlight the original UT-SOI profile before annealing. e Statistical distributions of the full width at half maximum (FWHM) of the wires shown in a – c . The continuous lines are Gaussian envelops of the statistical distributions. f Sagittal profiles of the wires shown in a – c . g Statistical distributions of the wires height shown in f after subtraction of the height average value. The continuous lines are Gaussian envelops of the statistical distributions. Full size image Fig. 3: Phase field simulations of anisotropic and isotropic patches. a – c Left panel: cross sections at different times of phase field simulations of dewetting of a patch with initial aspect ratio h/w = 1.2/100, 1.2/200, and 1.4/200, respectively, taking into account the crystal anisotropy. Right panels: same as the left panel for the isotropic case. The full dynamics is shown in the Supplementary Movies 1 and 2 for a left and right panels respectively; 3 and 4 for b left and right panels, respectively; 5 and 6 for c left and right panels respectively). d Comparison between the experimental data (symbols, from Fig. 2 d ) and phase field simulations (continuous lines). The shaded areas highlight the buried oxide (BOX) underneath the dewetted structures. Full size image In optimized conditions (annealing temperature and time, and patterns width) we observe the formation of extremely long wires, with a length limited only by the patch design (up to 0.75 mm, Fig. 1 b, c and high-resolution Supplementary Data 1 ). Wires with no breaks and perfectly homogeneous height and width over their full length can be formed with a 100 \\\\(\\\\%\\\\) success rate. All the UT-SOI available in the patch ( \\\\(w\\\\) = 700 nm) collapsed in individual wires featuring a base of about 160 nm ( \\\\(\\\\sim\\\\) 4 times smaller than \\\\(w\\\\) ) and a height of 50 nm. In these conditions of annealing temperature and time, a simple stable dewetting front freely receding (semi-infinite UT-SOI) covers a distance of about \\\\(\\\\Delta x\\\\) = 650 nm (bottom-right inset in Fig. 1 b), a length comparable to the overall patch width \\\\(w\\\\) . In a similar sample, \\\\(\\\\sim\\\\) 0.9 and \\\\(\\\\sim\\\\) 1.9 \\\\(\\\\upmu\\\\) m wide patches ( \\\\({d}_{{\\\\rm{LL}}}\\\\) = 1 and 2 \\\\(\\\\upmu\\\\) m, respectively) collapsed in an individual wire, whereas \\\\(\\\\sim\\\\) 3.9 \\\\(\\\\upmu\\\\) m patches ( \\\\({d}_{{\\\\rm{LL}}}\\\\) = 4 \\\\(\\\\upmu\\\\) m) were partially dewetted in two parallel counter-propagating rims, as revealed by AFM measurements (Fig. 2 a–d). These features are attributed to a faster dewetting dynamics (and in turn to an earlier onset of the morphological instabilities) associated to smaller radius (i.e. larger curvature at the surface [35] , [36] ) even when considering a film in contact with a substrate [37] , [38] , [39] , [40] : the dewetting process for the larger pitches (lower overall curvature with respect to smaller pitch), is not concluded at the end of the annealing step. It is worth noting that, although the process for \\\\({d}_{{\\\\rm{LL}}}\\\\) = 1 \\\\(\\\\upmu\\\\) m is faster than that one relative to \\\\({d}_{{\\\\rm{LL}}}\\\\) = 2 \\\\(\\\\upmu\\\\) m, also in the former case the wires did not yet break into islands (the Plateau-Rayleigh instability along the wire did not take place). In all investigated cases, the fluctuations around the average full width at half maximum (FWHM) and heights of the wires were only a few nanometers (Fig. 2 f–g) accounting for the remarkable control of the dewetting process. We also observe that for longer annealing time and higher temperature, patches of any width produced isolated islands (eventually elongated), as expected from the conventional Plateau-Rayleigh instability (not shown) [14] . Phase field simulations The former results are compared to 2D phase field simulations of surface diffusion [41] , [42] , including surface-energy anisotropy [43] , [44] solved by a finite element approach [45] , [46] (see the Supplementary Note 1 for more details on the simulation method in use). They are performed mimicking the evolution in time of the cross-section of the experimental cases (Fig. 3 , Supplementary Movies 1 – 6 ). For each investigated aspect ratio ( \\\\(h/w\\\\) = 1.2/100, 1.2/200 and 1.2/400) we reproduce the dewetting dynamics, including the typical facets of the equilibrium shape of Si ({ \\\\(001\\\\) }, { \\\\(113\\\\) }, { \\\\(111\\\\) }, and { \\\\(110\\\\) }) by means of the corresponding surface-energy values [47] , [48] (Fig. 3 a–c, left panels). We systematically compare these results with the isotropic counterpart, by averaging the energies of different orientations (Fig. 3 a–c, right panels). The relevant features emerging from this analysis are summarized as follows: (i) For simulations reproducing \\\\(h/w\\\\) = 1.2/100 and 1.2/200 (corresponding to \\\\({d}_{{\\\\rm{LL}}}\\\\) = 1 \\\\(\\\\upmu\\\\) m and 2 \\\\(\\\\upmu\\\\) m, respectively Fig. 3 a, b, left panels), the patch effectively collapses in a single wire, directly reproducing the corresponding experiments (Fig. 3 d, left and central panels), thus providing a confirmation of the diffusion-limited kinetics at play. Discrepancies between experiments and theory for \\\\({d}_{{\\\\rm{LL}}}\\\\) = 1 \\\\(\\\\upmu\\\\) m can be attributed to convolution effects with the AFM tip leading to an overestimation of the wires FWHM also reflected in the large discrepancy between the wires width shown in Figs. 1 c, 2 e. (ii) The simulations for \\\\(h/w\\\\) = 1.2/400 (corresponding to \\\\({d}_{{\\\\rm{LL}}}\\\\) = 4 \\\\(\\\\upmu\\\\) m, Fig. 3 c, left panel) show first the formation of two parallel, counter-propagating rims, leading finally to a breakup in two parallel wires as final stage (not observed in the experiment, as the dewetting process is not complete, Fig. 2 c). A good agreement with experiments is found when focusing on the intermediate time steps (Fig. 3 d, right panel). (iii) Surface faceting is found to play a central role in determining a quantitative outcome of the process [49] . For \\\\(h/w\\\\) = 1.2/100 a single wire is obtained with and without anisotropy, whereas the case \\\\(h/w\\\\) = 1.2/200 deviates from experiments, providing two parallel wires as final state of the process, when neglecting preferential orientations (Fig. 3 b, right panel). Also for larger patches ( \\\\(h/w\\\\) = 1.2/400, Fig. 3 c, d, right panel) the isotropic case predicts three islands against the two found in the anisotropic case, confirming the tendency of the surface anisotropy forces to stabilize the patch against break-up. (iv) For the largest patch, the experimental rims are smaller than the prediction by phase field simulations and the valleys next to the rims are not visible. This feature is attributed to an effective larger stiffness/anisotropy of the real structures with respect to those considered in these simulations. This could be readily accounted for by phase field simulations [50] at the cost of a significant increase in the computational budget without however, delivering relevant, additional information than those discussed so far. Templated dewetting along arbitrary fronts In a different sample, similar patch arrangements are etched with a slight mis-cut of 2° circa, with respect to a stable dewetting front. We now consider patches size of 800 nm in width ( \\\\({d}_{{\\\\rm{LL}}}\\\\) = 1 \\\\(\\\\upmu\\\\) m, Fig. 4 a). Fig. 4: Wires tilt and connected circuits. a SEM image of wires obtained from \\\\(\\\\sim\\\\) 800 nm wide patches, \\\\(\\\\sim\\\\) 2° mis-cut with respect to the main crystallographic axis [1-10] (highlighted by white, dashed lines) and annealed at 740 \\\\({}^{\\\\circ }\\\\) C for 15 min. The yellow areas highlight the etched parts. Bottom inset: blow-up of a wire. White arrows highlight small tips at the wires edges; yellow arrows highlight tips in correspondence of the wire kinks. The equilibrium facets of the wire are highlighted with colored rectangles (green: { \\\\(111\\\\) }; blue: { \\\\(113\\\\) }; red: { \\\\(001\\\\) }. b SEM images of connections between parallel wires. The yellow areas highlight the etched parts. In each column, the two panels show different realizations of the connectors between two parallel wires oriented at different angles with respect to the [110] crystallographic direction. Respectively from the left to the right: 90°, 0°, and 15°. The full set of images of complex circuits of three connected wires oriented at different angles with respect to the crystallographic axes is provided in the Supplementary Note 2 and high-resolution Supplementary Data 1 – 7 in their full 33 \\\\(\\\\upmu\\\\) m length. Full size image The mis-cut does not impede the formation of well-ordered arrays of parallel and uniform wires, mostly intact and following the macroscopic direction imposed by the etching. Small kinks are formed during edge retraction in analogy to what was observed in the metallic counterpart [23] , [51] , [52] , [53] . However, in our case, the periodicity of the edge undulation of the Si film is not as regular as those found in metals and is linked to the presence of small tips at the BOX surface (highlighted by white and yellow arrows in the bottom panel of Fig. 4 a) on the denuded BOX and at the sides of the wires, where they touch the BOX (these are also found in spontaneously dewetted samples and are thus not attributed to the lithographic process, see the Supplementary Note 2 ). All the kinks form in presence of an impurity at the wires/box interface (yellow arrows in the bottom inset of Fig. 4 a) whereas in some cases only a wrinkling of the { \\\\(113\\\\) } and { \\\\(111\\\\) } facets is observed (white arrows in the bottom inset of Fig. 4 a). The top { \\\\(001\\\\) } facet is instead always flat. The mechanism illustrated in Fig. 4 a helps in setting arbitrary orientations of the etched profile and obtaining slightly curved structures. This can be expanded to more complex patterns leading to connected networks of wires. We address this point by etching parallel patches (700 nm large and 33 \\\\(\\\\upmu\\\\) m long) and studying the effect of their orientation with respect to the crystallographic axes on the stability against breaking (Fig. 4 b). Between the patches (etching highlighted by yellow areas in Fig. 4 ) we added several connectors with variable size and respective alignment. This design is repeated with 15° steps with respect to the [110] direction in order to cover 360° (Supplementary Note 2 and high-resolution Supplementary Data 2 – 7 ). As found for metals [23] , [51] we observe a general tendency to more frequent break-up of the patches along the unstable axis [100] whereas along the stable one [110] we find one individual, elongated island as previously shown for simple wires (Supplementary Note 2 and high-resolution Supplementary Data 2 ). Nonetheless, for short enough annealing time, well connected structures featuring a limited number of breaks can be found in a large range of patch orientations. Up to 15° with respect to [110] the structures are not broken along the wires nor at the level of large and small connectors (respectively 1.5 \\\\(\\\\upmu\\\\) m and 0.8 \\\\(\\\\upmu\\\\) m wide, Fig. 4 b). Finally, for the larger connectors the structure is robust against breaking up to 45° (high-resolution Supplementary Data 2 – 7 ). This demonstrates that it is possible to control the continuity, connectivity and curvature of the wires up to several degrees of misalignment with respect to the stable dewetting front without any optimization. A more appropriate choice of etching design and experimental conditions (e.g. annealing time, patch width and shape as well as ad hoc additional features etched within [28] ) may improve the quality of the final outcome. Crystalline structure of templated dewetted wires In order to rule out the presence of crystalline defects in the dewetted structures we performed atomic-resolution scanning transmission electron microscopy (STEM) imaging on a wire (Fig. 5 ). In line with previous evidences in Si- and SiGe-based islands [27] , [31] , we observe the typical crystalline structure of bulk Si and the absence of extended dislocations. A slight crystalline disorder can be observed in some part of the wire body, at the interface with the original UT-SOI substrate (at about 12 nm from the BOX, Fig. 5 c, d). This feature has been previously observed in STEM images [27] , [31] and we ascribe it to residual defectivity on the UT-SOI substrate, possibly due to a non ideal cleaning of the surface. For the sake of thoroughness we mention that geometric phase analysis performed on the full wire section does not reveal any strain in the crystal structure (not shown). Fig. 5: Atomic-resolution scanning transmission electron microscopy imaging. a High Angle Annular Dark Field (HAADF) scanning transmission electron micrograph (STEM) of the section of a dewetted wire. The original UT-SOI is highlighted by a dashed rectangle. b Blow-up of a (yellow rectangle) highlighting the crystalline structure of the top part of the silicon wire (c-Si) and the native oxide surrounding it (thickness about 2.5 nm). c Blow-up of the bottom-left part of the wire from a (blue rectangle) highlighting its crystalline structure and the presence of some alloy disorder. d Blow up of c (green rectangle) highlighting the alloy disorder. Full size image Electric conduction from parallel wires arrays Through the templated dewetting process we demonstrated crystalline silicon wires formed directly on an insulating substrate. To show the potential of these structures for electronic circuits a doping procedure involving phosphorus spin-on-dopant (SOD) deposition and rapid thermal annealing treatment has been carried out on 70 \\\\(\\\\upmu\\\\) m long nanowires (base size \\\\(\\\\sim\\\\) 150 nm, height \\\\(\\\\sim\\\\) 75 nm) to render them conductive (Fig. 6 and Methods) [54] . All metal contacts (source, drain and gate, respectively S, D, and G) are 130 nm thick gold and were placed by e-beam deposition on 15 parallel nano-wires. S and D partially wet the silicon wires whereas G is separated from the wires by a 100 nm thick silicon dioxide (deposited via e-beam evaporation) covering the wires for about 30 \\\\(\\\\upmu\\\\) m (Fig. 6 ). The oxide leakage characteristics is about 10 pA (not shown). Fig. 6: Electrical properties of parallel wires. a Scheme of the field effect transistor (FET) wires device (doping about 3 \\\\(\\\\times\\\\) 10 \\\\({}^{18}\\\\) cm \\\\({}^{-3}\\\\) , n-type). b SEM images of the electric contacts (left part, source, gate and drain, respectively S, D, and G) and a detail of the S contact (right). The gold contact (Au) and the SiO \\\\({}_{2}\\\\) partially covering the wires are highlighted. c Electrical transport characteristics of 15 parallel wires (S-D current versus S-D voltage curves recorded at different G voltage. Inset: S-D current versus G voltage recorded at source-drain voltage \\\\({V}_{{\\\\rm{SD}}}\\\\) = 2 V for 15 parallel wires. Full size image S-D current curves as a function of the S-D voltage are registered for different G voltages, demonstrating a typical behavior of a field-effect transistor (FET). This transistor works in enhancement mode, where the saturation and the S-D voltage are characteristic of a n-channel FET [55] . From the I–V curves obtained on 15 parallel nanowires for different G tension, we estimate a trans-conductance ( \\\\({G}_{{\\\\rm{NW}}}\\\\) = \\\\(\\\\Delta {I}_{{\\\\rm{SD}}}/\\\\Delta {V}_{{\\\\rm{G}}}\\\\) , where \\\\(\\\\Delta {I}_{{\\\\rm{SD}}}\\\\) is the source-drain current modulation against the corresponding change in gate tension \\\\(\\\\Delta {V}_{{\\\\rm{G}}}\\\\) ) of the order of \\\\(\\\\sim \\\\upmu\\\\) S per wire (Fig. 6 ). Adopting the common approximations for nanowire-based transistors [56] and neglecting spurious effects [57] , we can estimate the electron mobility with the formula \\\\({\\\\upmu }_{{\\\\rm{e}}}\\\\) = \\\\(({L}^{2}{G}_{{\\\\rm{NW}}})/({V}_{{\\\\rm{SD}}}{C}_{{\\\\rm{NW}}})\\\\) , where \\\\(L\\\\) is the gate contact length, \\\\({V}_{{\\\\rm{SD}}}\\\\) is the source-drain tension and \\\\({C}_{{\\\\rm{NW}}}\\\\) is the gate capacitance. In a cylindrical geometry, this latter characteristic can be expressed as \\\\({C}_{{\\\\rm{NW}}}\\\\) = \\\\(2\\\\pi {\\\\epsilon }_{{\\\\rm{0}}}{\\\\epsilon }_{{\\\\rm{r}}}L/{\\\\mathrm{cos}}{h}^{-1}(t/R)\\\\) , with \\\\({\\\\epsilon }_{{\\\\rm{0}}}\\\\) vacuum permittivity, \\\\({\\\\epsilon }_{{\\\\rm{r}}}\\\\) the static dielectric constant of the gate oxide, \\\\({t}_{{\\\\rm{tot}}}={t}_{{\\\\rm{ox}}}+R\\\\) distance between wire center and metallic contact, where \\\\({t}_{{\\\\rm{ox}}}\\\\) is the oxide thickness ad \\\\(R\\\\) the wire radius. Assuming \\\\({\\\\epsilon }_{{\\\\rm{r}}} \\\\sim\\\\) 3.9, \\\\(L \\\\sim\\\\) 30 nm, \\\\({t}_{{\\\\rm{ox}}} \\\\sim\\\\) 100 nm, \\\\(R\\\\) \\\\(\\\\sim\\\\) 60 nm (estimated as average between height and base size), we obtain a gate capacitance \\\\({C}_{{\\\\rm{NW}}} \\\\sim\\\\) 5 fF. With these values, we can roughly estimate an electron mobility ranging between 0.5 and 5 \\\\(\\\\times\\\\) 10 \\\\({}^{3}\\\\) cm \\\\({}^{2}\\\\) V \\\\({}^{-1}\\\\) s \\\\({}^{-1}\\\\) .\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"There are several differences and advantages of our wires with respect to previous demonstrations of similar structures implemented via dewetting and other fabrication techniques, particularly regarding the simplicity and versatility of implementation, improved comprehension of the dewetting mechanism and improved quality of the structures enhancing the electrical properties of the implemented devices. In what follows we discuss all these features with respect to the existing state-of-the-art. In this work, we extend the coherent control of dewetting by more than 2 order of magnitude, going from patches of a few \\\\(\\\\upmu\\\\) m [28] to 0.75 mm (only limited by the pattern etched prior to annealing set by the writing field of the e-beam lithography in use). This is a record aspect ratio of a \\\\(\\\\sim\\\\) 1/60000 patch providing uniform and reproducible nanowires. In optimal annealing conditions and patch size we obtain a 100 \\\\(\\\\%\\\\) success in forming perfect faceted nano-wires with no breaks along their length and size fluctuations in the few per cent range. An important difference with respect to previous reports of Si dewetting [28] is the control of patch evolution for patterns oriented along unstable fronts. The formation of kinks during dewetting (mediated by small defects at the BOX/wire interface) allows to adjust the macroscopic directions of the final structures without breaking. This feature has never been reported so far in semiconductor dewetting and it allows to curve, split and connect the wires ad libitum in order to form complex circuits. The limit of this method is evident when considering patches larger than a few \\\\(\\\\upmu\\\\) m. Although the fluctuation of the wires width and height is always well below 10 \\\\(\\\\%\\\\) , larger patches ( d \\\\({}_{{\\\\rm{LL}}}\\\\) = 4 \\\\(\\\\upmu\\\\) m, Fig. 2 ) provide wires showing a spread of their width about three times larger than those formed from smaller patches ( d \\\\({}_{{\\\\rm{LL}}}\\\\) = 1 and 2 \\\\(\\\\upmu\\\\) m, Fig. 2 ). This lower level of control over the rim evolution for larger structures was reported for the case of simple square patches [28] : above 3 \\\\(\\\\upmu\\\\) m, disorder effects play an important role leading to marked deviations of the dewetting dynamics with respect to what expected for ideal systems (as those shown in the simulations, Fig. 3 and reference [28] ). These spurious effects could be attributed to extrinsic disorder locally perturbing the rim evolution and affecting in a more marked way larger patches with respect to smaller ones (e.g. native oxide not properly removed or other impurities present in the ultra-high vacuum used for silicon dewetting). Furthermore, small tips found at the rim/BOX interface can perturb the dewetting dynamics [28] . Thus, although these defects allow the formation of kinks along the wires and thus to curve them with respect to the preferential axes orientations, they may be a limit for a coherent control of larger patches (e.g. those that would result in two parallel wires instead of an individual one). In the present case of 12 nm thick UT-SOI, a coherent control of the patch evolution (in absence of additional features in the initial patch design) is limited to <3 \\\\(\\\\upmu\\\\) m. In the present work, we also managed to compare real systems with realistic models taking into account anisotropic surface diffusion. So far, simulations of templated dewetting of UT-SOI based on a phase field approach considered patches featuring an aspect ratio of, at most, 1/160 [28] . This was in stark contrast with the real systems featuring a much smaller value of \\\\(\\\\sim\\\\) 1/400. Furthermore, this attempt to reproduce the experimental outcomes did not take into account the underlying crystal anisotropy. A reasonable agreement between experiments and simulations was found for short time evolution while showing marked discrepancies for longer times. More generally, in the last few years several theoretical works tried to tackle the anisotropic dewetting dynamic with sharp interface models for both cases of weak [58] and strong [50] anisotropy. However, in all these cases the patch aspect ratio was at most 1/60 which is pretty far from the actual experimental conditions used for metal and semiconductor dewetting. Here we used a phase field model taking into account surface diffusion and surface-energy anisotropy for a 1/1 scale case (aspect ratio up to 1/330). Our novel theoretical understanding of the anisotropic dewetting problem allows therefore to correctly predict long-time evolution of the main features observed in experiments showing that the presence of facets (due to anisotropic surface energy) stabilizes the dewetting outcome against breaking. Isotropic models (e.g. showing two parallel wires instead of only one) fails in this task, at least for larger patches. From a fabrication standpoint our approach offers several advantages with respect to other bottom-up methods. We implemented our wires on commercial UT-SOI wafers, available in a large set of device thickness, orientation, doping, composition (e.g. SiGe and Ge on insulator), BOX thickness, and up to 12 inches in size. These features are not matched by other methods implemented on, at most, a few inches and expensive epilayers [19] , [20] , [21] (e.g. for III–V-based structures). Our structures are implemented only in two steps, etching and annealing. Thus, templated dewetting of wires offers a versatility in directions, splitting and connections not attainable with catalyst-based approaches for in-plane wires growth [59] (where all the structures were parallel) and an easier implementation with respect to recent demonstrations were the wires were bound together with complex patterning and epitaxial nano-fabrication methods [18] , [19] , [20] , [21] , [22] , [60] , [61] (eventually requiring cumbersome post-growth processing for the implementation of an electronic device). Other self-assembly methods are not suitable for the formation of crystalline structures on amorphous SiO \\\\({}_{2}\\\\) . Bottom-up methods for semiconductor-based 3D structures rely on strain-induced assembly (e.g. via Stranski Krastanov for III–V and IV–IV) [62] , droplet epitaxy and droplet etching (only for III–V) [63] , vapor-liquid-solid growth via gold catalyst [59] , or more advanced hybrid top-down/bottom-up approaches [19] . All these strategies can be exploited only when a precise epitaxial relation holds between the substrate and the deposited material. As such, they require a crystalline support. In contrast to this, we directly produce well-ordered, monocrystalline nano-architectures on amorphous SiO \\\\({}_{2}\\\\) without any epitaxial relation. A straightforward consequence is that silicon dewetting provides electrical isolation of the three-dimensional nano-achitectures from the substrate, a clear advantage for the implementation of electronic devices (e.g. the field-effect transistors shown here) with respect to recent in-plane nano-architectures epitaxially grown on a III–V substrate [18] , [21] that becomes insulating only at very low temperature [19] . Another important difference between dewetting and recent reports of advanced epitaxial structures based on selective area growth, is the lack of strain and alloy disorder that is known to complicate the interpretation of the device behavior [19] , [20] . The width of our wires is 4–5 times smaller than the initial patch width, implying that a low-resolution etching (e.g. based on optical or nano-imprint lithography) can in principle be exploited. This constitutes an advantage with respect to current hybrid top-down/bottom-up approaches, where the resolution of the lithography directly sets the final size of the structures [19] , [21] . In this respect, it is worth stressing that a high resolution for electron-beam lithography inevitably requires a reduced writing field, limiting the size of nanowires to rather short channel length (a few \\\\(\\\\upmu\\\\) m for both top-down [57] , [64] and bottom-up [18] , [19] , [20] , [21] approaches). In our case, a low resolution allows for mm scale structures providing an aspect ratio of the order of 1/10 \\\\({}^{4}\\\\) . Our approach allows to tune the wires height on the same substrate in contrast with top-down approaches that lead to structures with a height fixed by the thickness of the UT-SOI in use or by the etching depth. We achieved this by setting the initial patch lateral width (e.g. from \\\\(w\\\\) = 0.8 \\\\(\\\\upmu\\\\) m up to \\\\(w\\\\) = 2 \\\\(\\\\upmu\\\\) m) providing wires having a base between 150 and 400 nm and height ranging from 50 to 100 nm. In order to obtain smaller wires, it is in principle possible to use thinner UT-SOI layers (e.g. by etching a few nm of the top silicon layer). One of the most important features of nano-wires is their electrical properties. For instance, the changes in conduction of wire-based transistors can be efficiently exploited for sensing [64] or for thermoelectricity [7] with wires having rough interfaces. Nevertheless, most works on nanowires have focused on studying and optimizing their surface smoothness. Surface defects are very common and lead to electron-surface roughness scattering influencing the charge carrier density in the underlying silicon matrix [64] modifying the electronic properties of a device (e.g. drop of the electron mobility) [65] . Thus, trans-conductance and electron mobility are important figures of merits that account for the quality of the interfaces and the limits of a wire-based device. In our wires, the measured value of trans-conductance is \\\\({G}_{{\\\\rm{NW}}} \\\\sim\\\\) 1–9 \\\\(\\\\upmu\\\\) S. This is quite similar to state-of-the art FET transistors based on Si-nanowires grown via conventional bottom-up methods. In fact, for these devices, \\\\({G}_{{\\\\rm{NW}}}\\\\) is at most a few \\\\(\\\\upmu\\\\) S [66] , [67] even when considering wires having size close to those shown here [6] . From the electric characterization, a rough estimation of electron mobility in our FET transistors provides \\\\({\\\\upmu }_{{\\\\rm{e}}}\\\\) = 0.5–5 \\\\(\\\\times\\\\) 10 \\\\({}^{3}\\\\) cm \\\\({}^{2}\\\\) V \\\\({}^{-1}\\\\) s \\\\({}^{-1}\\\\) . These values are in line with those found in Si-based bottom-up nanowires devices [6] , [10] , [66] . For top-down FET wires (usually fabricated via e-beam lithography and reactive ion etching) typical values of \\\\({\\\\upmu }_{{\\\\rm{e}}}\\\\) lie in few hundreds of cm \\\\({}^{2}\\\\) V \\\\({}^{-1}\\\\) s \\\\({}^{-1}\\\\) [56] , [68] . Larger values of \\\\({\\\\upmu }_{{\\\\rm{e}}}\\\\) , up to \\\\(\\\\sim\\\\) 10 \\\\({}^{3}\\\\) cm \\\\({}^{2}\\\\) V \\\\({}^{-1}\\\\) s \\\\({}^{-1}\\\\) , can be reached with top-down wires at the price of cumbersome fabrication methods (e.g. for smoothing the wires walls via oxidation), Ge alloying or strain engineering [69] , [70] . Thus, even without any kind of optimization of our devices, we can reach relatively high electron mobility. We interpret this feature as a possible consequence of the reduced surface roughness in our structures with respect to those obtained via vapor-liquid-solid growth and top-down methods. In conclusion, we showed that dewetting, a spontaneous shape instability common to many different thin films of organic and inorganic substances, can be efficiently controlled in order to form extremely long and connected circuits of monocrystalline silicon wires on SiO \\\\({}_{2}\\\\) . We extended the control of this process to silicon patches having the record aspect ratio of 1/60000 (to be compared with the case of metals 1/100 [34] , and semiconductors 1/400 [28] ) forming extremely elongated silicon mono-crystals. We directly compare the experimental outcomes with 1/1 scale phase field simulations of surface diffusion that quantitatively reproduce their morphological evolution. We show a clear evidence of the key role played by faceting in stabilizing the dewetting outcome against breaking and thus for the reproducibility of the process and the stability of the final structures. Phosphorous-doped conductive Si wires are implemented showcasing the possibility to fabricate conducting nano-channels and transistors on an insulating substrate. Since the proposed approach is very general, it can be adapted to tune the Si wires aspect ratio by choosing suitable UT-SOI thickness and pattern periodicity combined with more complex, connected nano-architectures towards a full exploitation of their record length and atomically smooth facets. Owing to a similar dewetting dynamic ruled by surface-diffusion-limited kinetics observed in SiGe alloys [32] similar results can be extended to these materials rendering this method attractive for wires formation with different materials with the perspective of band-gap engineering and carrier mobility enhancement.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Sample preparation A 12 nm-thick UT-SOI on a 25 nm thick buried oxide (BOX) was etched by electron-beam lithography and reactive etching with parallel trenches, from 0.75 mm to 70 \\\\(\\\\upmu\\\\) m long, with variable pitch (0.5–4 \\\\(\\\\upmu\\\\) m line-to-line distance, d \\\\({}_{{\\\\rm{LL}}}\\\\) ) and orientations with respect to the crystallographic axes. Thus, the samples were processed by plasma and wet chemical cleaning in N \\\\({}_{2}\\\\) atmosphere for 20–30 s in a 5–10 \\\\(\\\\%\\\\) HF. Finally, they were annealed in the ultrahigh vacuum ( \\\\(\\\\sim\\\\) 10 \\\\({}^{-10}\\\\) Torr) of a molecular beam epitaxy reactor. Sample doping Deposition of Spin on dopant (SOD, OD P508) was performed by spin-coating at 4000 rpm for 1 min and baking on a hot plate (10 min at 120 \\\\({}^{\\\\circ }\\\\) C). Thermal diffusion of dopant was induced by rapid thermal annealing (30″ at 850 \\\\({}^{\\\\circ }\\\\) C in N \\\\({}_{2}\\\\) atmosphere). The SOD was removed by wet etching for 60″ in dilute HF (1:10). The metal pads (5 nm Ti/150 nm Au), after the definition of the contact design process by electron-beam lithography, are deposited by e-beam deposition. Electronic imaging Scanning transmission electron micrograph have been acquired in z-contrast with a Cs probe corrected JEOL ARM 200F operated at 60 keV.\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"The core research idea is to fabricate arrays of in-plane, ultra-long, monocrystalline silicon nanowires and complex interconnected circuits using a catalyst- and epitaxy-free hybrid top-down/bottom-up approach based on the controlled morphological evolution of thin solid films through low-resolution etching and annealing, enabling size- and position-controlled, electrically isolated nanostructures that can be further doped for high-performance field-effect transistor applications."},"entities":{"kind":"list like","value":["in-plane ultra-long monocrystalline silicon nanowires","complex interconnected circuits","catalyst-free approach","epitaxy-free approach","hybrid top-down/bottom-up approach","controlled morphological evolution","thin solid films","low-resolution etching","annealing","size-controlled nanostructures","position-controlled nanostructures","electrically isolated nanostructures","doping","high-performance field-effect transistors"],"string":"[\n \"in-plane ultra-long monocrystalline silicon nanowires\",\n \"complex interconnected circuits\",\n \"catalyst-free approach\",\n \"epitaxy-free approach\",\n \"hybrid top-down/bottom-up approach\",\n \"controlled morphological evolution\",\n \"thin solid films\",\n \"low-resolution etching\",\n \"annealing\",\n \"size-controlled nanostructures\",\n \"position-controlled nanostructures\",\n \"electrically isolated nanostructures\",\n \"doping\",\n \"high-performance field-effect transistors\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1.","title":"Xia, Y. et al. 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Therefore, although high-pressure changes the physical state of all condensed matter, most of the extraordinary properties discovered vanish after decompression and cannot be utilized. Here, we demonstrate sublattice decoupling in a mixed-anion chalcohalide Rb 6 Re 6 S 8 I 8 upon compression, in which the [Rb 6 I 2 ] 4+ framework is soft and plastic, while the [Re 6 S 8 I 6 ] 4- clusters are hard and elastic. This discrepancy in the rigidity allows the applied pressure to selectively amorphize the framework while maintaining the ordered state in the cluster, leading to intriguing photocurrent generation and enhancement upon compression. These high-pressure properties are retained at ambience, permitting scalable synthesis of the decompressed samples using a large-volume press, followed by further fabrication into self-powered broadband photodetectors with a response time of ~ 10 2 μs and a specific detectivity of ~ 10 11 Jones. This study subverts the stereotype that pressure engineering is hardly to be employed for device applications."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"Using pressure to manipulate the physical state of condensed matter is as fundamental as changing temperature or tuning chemical composition for the same purpose [1] . Therefore, pressure is a distinctive dimension for seeking improved or even unprecedented properties in compressed materials, as evidenced by numerous reports on metallic hydrogen, high-temperature superconductors, photovoltaic semiconductors, and so on [2] , [3] , [4] , [5] , [6] . However, retaining and utilizing properties attainable at high pressure for ambient applications has been challenging because of the degradation of the high-pressure-induced properties after decompression and the inevitable trade-off between sample quantity and the magnitude of attainable pressure [7] , [8] . Two strategies have been proposed to overcome this formidable obstacle. In the first approach, inspired by natural mineral inclusions [9] , [10] , [11] , nanostructured diamond capsules (NDCs) have been developed to confine samples and preserve their high pressure-induced properties [12] . However, man-made NDCs can encapsulate only gaseous and volatile samples because of their softness and compressibility. In addition, this technique requires high temperatures to convert graphite to diamond for in situ preservation of target materials, thus showing limited applicability for organics or materials with low decomposition temperature. Moreover, after decompression, the target material is isolated in nanostructured diamond and forbidden to form a pure material, thus impeding contact-based material analysis or large-scale device applications. Another route is to engineer the kinetic barriers in the phase transformation to obtain ambient-pressure metastable materials [13] , [14] , [15] . This method has been realized in specific nanomaterials with interconnected nanocrystal networks [13] . However, no device applications using such metastable materials have been reported to date. A potential opportunity for ambient retention of high pressure-induced properties lies in tailoring the coupling between sublattices. Conventionally, a sublattice is a simple mathematical concept related to a subset of the translations of a lattice [16] . Sublattice modeling has now been extensively adopted to describe a variety of materials, ranging from alloys to ionic liquids [17] , [18] . All of these successful studies indicate that the sublattices in these systems are strongly coupled through their structural periodicity; thus, their responses under stimuli should be synchronized. By contrast, if the material is designed such that individual sublattices are weakly coupled through nonchemical bonding and a large discrepancy in the rigidity is present between various sublattices, pressure can be used to decouple the structural periodicity. More importantly, the properties acquired at high pressure become irreversibly frozen during decompression because of the nonsynchronized structural changes between the sublattices. Very recently, the Rb 6 Re 6 S 8 I 8 chalcohalide inorganic compound has been reported [19] , which is structurally characterized by two sublattices: the [Re 6 S 8 I 6 ] 4- (Re-S-I(1)) cluster and the [Rb 6 I 2 ] 4+ (Rb-I(2)) electrostatic framework. These two sublattices are mutually coupled by electrostatic attraction forces that are significantly weaker than those of chemical bonds. At ambient pressure, this material shows strong photoluminescence (PL) but does not exhibit a photocurrent response. However, upon compression, there is a significant pressure-generated photocurrent in this material, which is retained at ambient pressure after the applied pressure is completely removed. As a proof for device application, a large-volume press (LVP) is employed to pressurize and depressurize a scalable quantity of Rb 6 Re 6 S 8 I 8 inheriting the pressure-induced photoelectric properties. This material is then used to build ambience-operational photodetectors. Notable photocurrent responses are observed under excitation by both ultraviolet and visible light. These results, together with the elucidated thermodynamic and kinetic mechanisms, enable high-pressure properties to be employed in optoelectronics that must operate under ambient conditions.","references":{"indices":[1,2,3,4,5,6,7,8,9,10,12,13,14,15,17,18,19],"infos":[{"idx":"1","title":"Mao, H.-k & Hemley, R. J. Ultrahigh-pressure transitions in solid hydrogen.Rev. Mod. 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Owing to lattice shrinkage, the high-pressure X-ray diffraction (XRD) pattern implies monotonic shifts of Bragg reflections towards larger 2 θ angles from 1 atm to ~10–12 GPa, above which the pattern becomes diffusive for lattice distortion, and the structure cannot maintain long-range ordered atomic arrangements (Fig. 1a ). Using GSAS software, we extract the pressure dependence of interatomic distances, where the framework-related Rb-Rb, Rb-I(1) and Rb-I(2) undergo monotonic distance reductions, yet the cluster-related bond lengths of Re-I(1), Re-Re and Re-S, as well as the I(1)-Re-S, S-Re-S and Re-S-Re bond angles, are nearly unchanged; thus, the clusters remain undisturbed at higher pressures (Fig. 1 b, c , and Supplementary Figs. 2 and 3 ), and the root cause of amorphization is the collapse of the Rb-I(2) framework. Fig. 1: Structural evolutions of Rb 6 Re 6 S 8 I 8 upon compression. a Pressure-dependent in situ synchrotron-based XRD pattern, showing structural amorphization starting at ~12 GPa. The XRD pattern collected at ambient pressure is highlighted in gray, and indicates that no crystalline phase transition occurred prior to amorphization. b and c Evolution of the interatomic distance and bond angles under pressure, suggesting differential rigidity. We identify two distinct compression behaviors dominated by the significantly soft Rb-I(2) framework and the cage-structured Re-S-I(1) sublattice that is extremely insensitive to pressure. d Pressure dependence of the Raman spectra upon compression. e Pressure dependence of the full width at half maximum (FWHM) of the representative Raman modes, which are nearly unchanged upon compression. f First and second derivatives of the Re L 3 edge \\(\\mu (E)\\) plotted versus the photon energy of Re 6 Rb 6 S 8 I 8 at different pressures upon compression. g Constant edge energy ( E 0 ) and white line energy ( E w ) during the investigated compression process. h Pressure dependence of the XAS Fourier transform magnitude as a function of interatomic distance and fittings based on nearest neighbor (Re-S) at selected pressures. i Fitted interatomic Re-S distance. The low R-factors, which are on the order of 10 -4 show the high goodness of fit. During the whole compression process up to 25.4 GPa, no obvious change occurs in the interatomic distance within the [Re 6 S 8 I 6 ] 4- cluster. The error bars in ( g and i ) denote the fitting errors. Full size image The hard rigidity of the cluster sublattice is further supported by the results of two independent experiments, namely high-pressure Raman spectroscopy (Supplementary Fig. 4 ) and X-ray absorption spectroscopy (XAS) (Supplementary Fig. 5 ). According to the calculated results of the vibrational mode assignment, all of the initial Raman modes are based on the cluster rather than on the framework. The pressure dependence of the Raman spectra clearly shows that these modes are distinguished up to 20 GPa (Fig. 1d ), with no noticeable increase in the full width at half maximum (FWHM) (Fig. 1e ), showing short-range order characteristics. Such sharp modes exhibit a sustained blueshift, even after the process of amorphization is completed, as revealed by XRD (Fig. 1a ), demonstrating a specific decoupling between long-range crystalline disorder and short-range local order. Using XAS as another quantitative validation tool, it is observed that the edge energy ( E 0 ) and line energy ( E w ), which are defined by the peaks of the first and second derivatives of \\(\\mu (E)\\) , respectively, remain constant at ~10.537 keV and ~10.539 keV as the pressure increases (Fig. 1 f, g ), indicating that there is no change in the valence state or coordination environment of Re. This pressure independence of the energy position reflects the robust nature of the Re-S-I cluster. Furthermore, the Fourier transform magnitude of the XAS spectra suggests that the Re-S interatomic distance does not vary significantly (Fig. 1 h and i), again verifying the nonsynchronous lattice disorder processes and the difference in rigidity between cluster-structured [Re 6 S 8 I 6 ] 4- and electrostatic [Rb 6 I 2 ] 4+ sublattices. The difference in the compressibility between the cluster and electrostatic frameworks provides a structural basis for the adjustable and irreversible optical and electronic properties discussed below. Pressure-induced emission transitions We then focused on the high-pressure-driven functional response of Rb 6 Re 6 S 8 I 8 . The different compressibility of the Re-S-I(1) cluster and the Rb-I(2) electrostatic framework can lead to attenuated photoluminescence (PL) but also an emergent photocurrent response of Rb 6 Re 6 S 8 I 8 . As shown in Fig. 2a and Supplementary Fig. 6 , while pressure-dependent PL generally results in a reduction in the emission intensity at higher pressures, slight increases in the PL intensity and prolongation of the carrier lifetime are observed at a mild pressure of ~0.1 GPa (Fig. 2 b– g ; Supplementary Figs. 7 – 9 ). On the other hand, the PL peak wavelength displays a monotonic blueshift with increasing pressure (Fig. 2c ), which is in contrast to the trend observed for the pressure-dependent UV-Vis absorbance. In addition, the full width at half maximum (FWHM) of the PL spectra also decreases with increasing pressure (Fig. 2d ), which is additionally corroborated by the second central moment ( m 2 ), which also displays a similar decreasing trend (Fig. 2e ), indicating that the energy density of states increases upon compression [20] . We also note that the PL annihilation is associated with the shortened PL lifetimes upon compression (Fig. 2 f, g ). Fig. 2: Pressure-dependent optical properties of Rb 6 Re 6 S 8 I 8 during compression. a Pressure-dependent PL contour plot from 1 atm to 12 GPa, where the PL emission is fully annihilated. b – e Pressure dependences of the PL intensity, PL peak wavelength, and full width at half maximum (FWHM) of the PL peak and second central moments ( m 2 ), respectively. Here, for a Gaussian-type emission profile, the FWHM is conveniently quantified by m 2 as m 2 = (FWHM) 2 /8ln2, where the error bar in d shows the fitting errors. Note that two individual samples, 1 and 2, were employed to confirm the validity of the experimental results. f Experimental time-resolved PL decays (dots) and fittings (solid lines) from 1 atm to 10.2 GPa. g The extracted carrier lifetimes obtained by fitting the time-resolved PL decay traces in f . h Comparison of the on-off photocurrent response and magnitudes from 1 atm to 24.7 GPa. i Evolution of the photocurrents extracted from ( h ). j Pressure-dependent UV-vis absorbance near bandgap regions from 1 atm to 30 GPa, where a drastic narrowing of the bandgap can be observed at approximately 17 GPa. Therefore, the onset pressures for abrupt bandgap narrowing, drastic photocurrent increase, and long-range disorder are coincident, thus indicating that structural amorphization is the common origin for the property changes. The sizes of the symbols shown in ( b , c , e , g and i )cover the experimental or fitting errors. Full size image The PL properties and exciton/carrier dynamics of similar compounds are highly dependent on the properties of structurally isolated clusters [20] , [21] , [22] , and the lifetime is strongly affected by ligands of rhenium. Upon compression, hydrostatic pressure first compresses the Rb-I(2) electrostatic framework owing to its much softer nature, with the compressed Rb-I(2) framework eventually tuning atomic connections within the Re-S-I(1) cluster structure. Then, the original crystal field of the Re-S-I(1) cluster is modified due to the perturbed electron clouds of the S/I ligands under lattice compression. Therefore, PL attenuation at higher pressures can be considered the result of increased long-range interactions between the Re-S-I(1) cluster and the Rb-I(2) framework. This process is critically influenced by spin-spin interactions, which are governed by the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction and further modulated by crystal field effects [23] , [24] . To delve deeper into the mechanisms underlying emission transitions, we investigate the role of self-trapped excitons (STEs). In crystalline materials, excitons interact with phonons via electron-phonon coupling. When this coupling is weak, excitons are primarily scattered by phonons. In contrast, strong coupling can lead to the self-trapping of excitons. Specifically, an STE is formed because photoinduced local lattice distortion occurs due to strong electron-phonon coupling [25] . For the studied Rb 6 Re 6 S 8 I 8 material, the broad sub-bandgap emission with a large Stokes shift observed at ambience is the fingerprint of the STE. Two additional pieces of evidence further support the existence of STEs in Rb 6 Re 6 S 8 I 8 : 1) Soft lattice. According to the XRD results shown in Supplementary Fig. 2 , the bulk modulus of Rb 6 Re 6 S 8 I 8 is ~11 GPa, which is significantly lower than that of the conventional semiconductor Si and even lower than that of organic-inorganic hybrid perovskites ( ~ 15‒20 GPa) [26] . This soft lattice is conducive to lattice deformation, which is crucial for the formation of an STE [27] . 2) Low electronic dimensionality. From the calculated band structure results shown in Supplementary Fig. 10 , we observe flat conduction bands (CBs) and valence bands (VBs), indicating localized zero-dimensional features. Such low electronic dimensionality is a prerequisite for the formation of STEs [28] . For excitons to become self-trapped, neither photogenerated electrons nor holes, or both, can exhibit high delocalization; that is, the effective mass of at least one carrier must be sufficiently high. In other words, low electronic dimensionality, which reflects the extent of orbital overlap, is essential for the formation of STEs and, consequently, for broadband emission. All of these observations demonstrated that STE is the origin of broad PL, and we can see that the evolution of STEs controls the emission transitions observed at pressures. Upon compression, the STE in Rb 6 Re 6 S 8 I 8 is suppressed and weakened, which can be easily understood from the fact that its lattice becomes more rigid at higher pressure; therefore, photoinduced local lattice distortion and electron-phonon coupling become difficult. This argument is also supported by the reduced FWHM of the PL spectra upon increased pressure, as shown in Fig. 2d , indicating that the STE feature is weakened at high pressure. In addition, we further studied the pressure-dependent self-trapping energy ( \\({E}_{{{{\\rm{st}}}}}\\) ), which is estimated as \\({E}_{{{{\\rm{st}}}}}=\\,{E}_{{{{\\rm{g}}}}}-\\,{E}_{{{{\\rm{PL}}}}}\\) , where \\({E}_{{{{\\rm{g}}}}}\\) is the bandgap energy, and \\({E}_{{{{\\rm{PL}}}}}\\) is the PL emission energy. In the studied material, rhenium chalcohalide clusters are completely isolated by the large [Rb 6 I 2 ] 4+ framework, forming a nominal 0D structure, as confirmed by our band-structure calculations (Supplementary Fig. 10 ). In 0D systems, the energy barrier between the free exciton and STE is considered not to exist [27] , [29] . We also note that the VB of this material is flat, particularly at low pressure, so that the lattice deformation energy should be very small. Supplementary Fig. 11 shows the pressure-dependent bandgap energy and PL energy upon compression, from which it is observed that from ambient pressure to 10 GPa, the bandgap energy is first relatively stable, and then decreases, while the PL energy increases monotonically. Therefore, we can conclude that a reduction of \\({E}_{{{{\\rm{st}}}}}\\) , rendering the STE unstable; this implies that the STE cannot act as an effective PL mechanism at high pressures. Pressure-induced photocurrent generation and enhancement The pressure-induced photocurrent transitions, where the STE again plays a crucial role, are of particular interest. As shown in Fig. 2 h and i, we can identify two stages of the evolution of the pressure-induced photocurrent. The first stage is the initial photocurrent generation occurring at ~2 GPa, prior to which the photocurrent is undetectable. The second stage occurs at ~15 GPa, after which the photocurrent is greatly enhanced upon compression. The self-trapping energy describes the extent of binding of the STE. Therefore, the reduced \\({E}_{{{{\\rm{st}}}}}\\) of Rb 6 Re 6 S 8 I 8 at higher pressures facilitates the dissociation of STEs into free carriers. From the calculated band structure (Supplementary Fig. 10 ), one can expect that owing to the fluctuation of the VB at higher pressure, the effective mass will decrease upon compression, leading to enhanced carrier mobility. Then, the pressure-induced greater mobility allows these free carriers to more easily migrate towards the current-collecting electrodes of the photodetector devices. Therefore, a detectable photocurrent is achieved via pressure-reduced excitonic behavior at ~2 GPa. For the pressure-enhanced photocurrent above ~15 GPa, the increasing trend of the photocurrent agrees well with the redshift in the optical bandgap (Fig. 2j ; Supplementary Figs. 12 – 15 ), indicating their common structural origin. Structurally, compression results in amorphization, causing the formation of a chemical bond between Rb and I(2) that originally interacted through Coulomb attraction. This argument can be verified by extending the graph of the Rb-I(2) interatomic distance (Fig. 1b ) to a higher pressure, e.g., 15 GPa, from which a 3.6 Å interatomic distance that is very close to the Rb-I ionic bond length (3.66 Å) can be extracted. Thus, it is reasonably believed that there is a high probability that Rb-I forms chemical bonds in Rb 6 Re 6 S 8 I 8 within the pressure range beyond 15 GPa; this facilitates an electron transition from the VB to the CB, thereby dramatically narrowing the optical bandgap. The higher carrier density therefore leads to a photocurrent enhancement of over three orders of magnitude within the 15–20 GPa pressure range. Irreversible structural transition In addition to the different compression behaviors of the Re-S-I(1) cluster and the Rb-I(2) framework, these two structural moieties also exhibit distinct irreversibility after decompression, and the ambient structure of the recovered sample is strongly associated with the peak applied pressure (Fig. 3 ; Supplementary Figs. 16 – 19 ). Generally, Rb 6 Re 6 S 8 I 8 shows long-range disorder as it is decompressed from higher peak pressures, rendering it amorphous without structural robustness (Fig. 3a ). The ambient structures of the as-synthesized Rb 6 Re 6 S 8 I 8 sample and the Rb 6 Re 6 S 8 I 8 among samples decompressed from different peak pressures were further compared using in situ Raman spectroscopy (Fig. 3b ). For the samples decompressed from 25 GPa, distinctive modes are clearly shown, suggesting an ordered Re-S-I(1) cluster structure. Nonetheless, even for the sample that underwent high-pressure treatment and then was decompressed from a relatively lower peak pressure of 18 GPa, the XRD pattern already clearly shows a disordered long-range structure (Fig. 3a ). This phenomenon indicates the different structural reversibility of the Re-S-I(1) cluster and the Rb-I(2) framework. Fig. 3: Compression-decompression dependent structural transitions of Rb 6 Re 6 S 8 I 8 . a XRD patterns of samples of powders ground from the as-prepared crystal and samples recovered from different peak pressures. b Raman spectra of samples of powders ground from the as-prepared crystal and samples recovered from different peak pressures. c XAS Fourier transform magnitudes as a function of interatomic distance and fittings (dashed line) based on nearest neighbor (Re-S) at ambient pressure as recovered from 1.5 GPa and 25 GPa peak pressures. d Schematic illustration of hydrostatic compression of Rb 6 Re 6 S 8 I 8 , which leads to nonsynchronous lattice disorder of the [Re 6 S 8 I 6 ] 4- cluster and the Rb-I electrostatic framework, also showing the plasticity of the Rb-I framework, resulting in incomplete recovery from the high-pressure disorder state. The nonsynchronous amorphization processes of the Re-S-I(1) and Rb-I(2) sublattices, along with their different extents of recrystallization, indicate the irreversibility of high-pressure amorphization. Full size image Therefore, one can visualize a decoupled Re-S-I(1) cluster and Rb-I(2) framework, where the former component is mechanically more resilient to pressure and does not affect structural periodicity to a large extent, whereas Rb-I(2) is spatially widespread, and directly regulates the long-range order of the Rb 6 Re 6 S 8 I 8 lattice; this means that structural plasticity exists in Rb 6 Re 6 S 8 I 8 , which is dictated by the least irreversible recovery of the Rb-I(2) framework from the high-pressure treatment cycle. Furthermore, by fitting the Fourier transformed XAS magnitude with the nearest neighbor (Re-S), it was revealed that after the compression-decompression cycle with a peak pressure of 25 GPa, the bond length of Re-S was 2.27 Å, showing only negligible deviation from the 2.33 Å Re-S bond length obtained with a peak pressure of 1.5 GPa (Fig. 3c ). Figure 3d shows the structural evolution of the Re-S-I(1) cluster and the Rb-I(2) framework before and after high-pressure treatment. The structural irreversibility therein gives rise to a permanent modification of functional properties, which is promising for engineered device design. We also note that even after the compression-decompression cycle with a low peak-pressure of 2 GPa, although no phase transition is observed, the microstructure of [Re 6 S 8 I 6 ] 4- is irreversibly modified, leading to a PL enhancement (Supplementary Fig. 16 ). Pressure-cycle-induced photocurrent retainable at ambience Benefiting from irreversible structural transitions, recovered Rb 6 Re 6 S 8 I 8 retains its high-pressure properties that depend on the magnitude of peak applied pressure (Supplementary Figs. 20 – 23 ). Specifically, it even exhibits monotonically increased photocurrents with larger number of compression-decompression cycles (Fig. 4a ). To explore the structural origin of the retained photocurrents at ambience, we conducted molecular dynamics simulations of the mean squared distances (MSDs) of the Rb, I, S, and Re atoms and observed more intensive fluctuations in the Rb-Rb distance as a function of simulation time, indicating that the polarizability and plasticity of the framework atoms are much greater than those of the other structural components (Fig. 4b ; Supplementary Fig. 24 ). The larger deformability of the Rb-I(2) framework thereby explains the observed drastic bandgap redshift occurring at 17 GPa (Fig. 2j ) along with the concomitant photocurrent enhancement (Fig. 2i ). Fig. 4: Retainable photocurrent at ambience, large-scale sample preparation, and photodetector device. a Comparison of ambient on-off photocurrent responses before compression (gray), after first (light red) and second (dark red) compression-decompresion cycles, showing notable generation and increase in the photocurrent after pressure treatments. b , Mean squared distances of the Rb, I, S, and Re atoms. The fluctuations stabilized with increasing time in the molecular dynamics simulations. c Scheme of the LVP utilized in applying pressure treatment to a large-scale Rb 6 Re 6 S 8 I 8 sample. Scale bar, 1 mm. d Sketch of a Schottky-type thin-film Rb 6 Re 6 S 8 I 8 photodetector device. e Good solution processability of the decompressed Rb 6 Re 6 S 8 I 8 sample. The optical picture of LVP-treated Rb 6 Re 6 S 8 I 8 dissolved in DMF confirmed that the material is dispersible, where the rubidium iodide ([Rb 6 I 2 ] 4+ ) framework is stable, rather than behaving as Rb + and I - in DMF. f Photograph of the LVP-treated Rb 6 Re 6 S 8 I 8 -based photodetector device and the on-device interdigitated Pt electrode imaged under a microscope. g The I-V curve of devices under 367-nm illumination. h On-off photocurrents under 367-nm illumination. i The I-V curve of devices under 520-nm illumination. j On-off photocurrents under 520-nm illumination. For all tests, no bias voltage is involved, and the devices are self-powered. k Response time and specific detectivity ( D * ) with both 367 nm and 520 nm illuminations. Here, τ r and τ d denote the rise and decay response times, respectively. l Schematic diagram of the device ultraviolet and visible imaging device. The photocurrent mappings are observed with the specific pattern obtained by irradiating the device with the “HPSTAR” logo mask using the 367 nm ultraviolet and 520 nm visible lasers. “Max” and “Min” in the color scale denote the relative magnitudes of photocurrent activity. As such, the adoption of LVP in combination with permanent photocurrent generation from high-pressure treatment thus allows practical and scalable device applications. Full size image We also compared the bandgap difference between the initial state of the materials before compression and the decompressed state from high-pressure cycles, with the data shown in Supplementary Fig. 20 . It is observed that the optical bandgap cannot recover to the magnitude before compression; however, it exhibits a redshift after the first compression-decompression cycle and another redshift after the second cycle. Additionally, it is observed from the pressure dependence of the Urbach energy upon compression-decompression cycles that prior to the compression-decompression cycles, the absorbance curve exhibits a sharper profile, indicating a relatively ordered structure, whereas after the compression-decompression cycles, there is a significant increase in the magnitude of the Urbach energy, indicating a more disordered structure (Supplementary Fig. 20 ) [30] . All of these results are consistent with the argument that amorphization disorder plays a crucial role in photocurrent enhancement at high-pressure region over 15 GPa. In addition, the magnitude of the self-trapping energy can be estimated by the difference between the bandgap energy and the emission energy. Considering that the emission energy recovers to the initial state before compression (Supplementary Fig. 21 ), we can conclude that the self-trapping energy decreases after pressure treatment because the bandgap redshifts, meaning that the STE becomes unstable, leading to PL quenching (Supplementary Fig. 21 ) and the generation of photocurrent due to the dissociation of the STEs into free carriers. Scalable decompressed sample for device fabrication To demonstrate the practical device applicability of the photocurrent enhancement, we use a large volume press (LVP) to prepare large-scale samples with high-pressure treatment (Fig. 4c ). The sample pressure-treated under LVP was then fabricated into a Schottky-type Pt/Rb 6 Re 6 S 8 I 8 /Pt photodetector on a substrate (Fig. 4d ), by taking the advantage of the good solution processability of the decompressed samples. Figure 4e shows the picture of N,N-dimethylformamide (DMF) solution of decompressed Rb 6 Re 6 S 8 I 8 sample, from which one can see the sample is totally dissolved in DMF solvent. We further let the solvent evaporate to induce a recrystallization of Rb 6 Re 6 S 8 I 8 , and conducted XRD experiment on this sample. Although structural diffraction results in identifiable Bragg peaks, it shows rather diffusive background in the XRD pattern (Supplementary Fig. 25a ), thus indicating a partially amorphous nature of the solution-processed sample, confirming the metastable phase is preserved. Here, the rubidium iodide ([Rb 6 I 2 ] 4+ ) framework is stable in DMF, rather than behaving as Rb + and I - , which can be understood by the effect of ion association. Such a demonstration is further supported by the study of particle size in DMF solvent. We conducted dynamic light scattering (DLS) experiments on three samples in DMF with the same mol% concentration: as-prepared Rb 6 Re 6 S 8 I 8 ; large volume press (LVP)-treated Rb 6 Re 6 S 8 I 8 , and RbI. Supplementary Fig. 25b shows that the average particle size of LVP-treated Rb 6 Re 6 S 8 I 8 as dissolved in DMF is slightly larger than that of as-prepared Rb 6 Re 6 S 8 I 8 without pressure treatment, and significantly larger than that of RbI. The clear discrepancy of average particle sizes among these three specimens suggests that framework and cluster can be preserved within the LVP-treated Rb 6 Re 6 S 8 I 8 after dissolution in organic solvent, thereby laying the foundation for all subsequent device fabrication that utilizes the amorphized crystal structure for its permanently enhanced photocurrents. The detailed device setup with connected probes and interdigitated Pt electrodes optically characterized as shown in Fig. 4f . The fabricated self-powered photodetector shows a functional current response within a −1.0 to +1.0 V current-voltage scan under 367 nm (Fig. 4 g, h ) and 520 nm (Fig. 4 i, j ) illuminations, demonstrating a large on-off ratio with photocurrent magnitudes of ~60 pA and ~8 nA, respectively. In addition to the fast photocurrent dynamics on the order of 10 2 μs, the detector also exhibits considerable specific detectivity of ~10 11 Jones (Fig. 4k ). Using the photocurrent magnitude as an indicator, we finally achieve highly spatial-resolved red‒green-blue (RGB) imaging, in which the word “HPSTAR” is clearly visible (Fig. 4l )."},{"section":"Discussion","content":"We have investigated the photoluminescent to optoelectronic transition occurring in a compressed compound with isolated clusters. Using a combination of high-pressure experiments, we have examined whether such a transition can be retained after the applied pressure was completely released, and found a retention of high-pressure properties, as manifested by the greatly increased photocurrent arising from the initially non-optoelectronic state. These results can guide further applications for the construction of crystalline-amorphous hybrid structures and provide crucial clues regarding the materials that may show high-pressure memory at ambience. In the future, more research is needed to better understand the mechanism by which amorphization leads to the sharp increase in photocurrent, especially on the electronic structure of disorder phase. Our study also has revealed the potential of pressure treatment in practical applications, enabling the integration of high-pressure-treated materials into functional devices with high-pressure-enhanced performance. Through industrially accessible routes, we have realized large-scale sample preparation and efficient device fabrication of high-performance photodetectors. These results provide a practical demonstration of the feasibility of using pressure-treated materials in real-world optoelectronic applications. To guide future research, we present the following discussion and perspective. Differences in the rigidity between weakly coupled sublattices is crucial for achieving amorphization upon compression. Pressure-induced amorphization (PIA) can be understood in terms of mechanical collapse (known as the spinodal limit) and a thermodynamically metastable transition, which delimit the upper and lower bounds of this phenomenon [31] , [32] . In compounds featuring rigid isolated clusters embedded within soft frameworks, applied pressure can markedly reduce the inter-cluster spacing before inducing interactions. However, due to their inherent rigidity, the highly symmetrical clusters impose stringent local symmetry constraints. Consequently, the interplay and balance between these two opposing effects can generate significant shear or other forms of stress, prompting large-scale diffusion of framework atoms and ultimately driving the system into an amorphous state. Such pressure-induced phase transitions have been experimentally observed in compounds characterized by structures comprising isolated polyhedral units such as molybdates [33] , tungstates [33] , silicides [34] , and orthosilicates [35] . We note that a simple soft lattice cannot ensure easy PIA. Pure RbI does not undergo long-range lattice disorder until 67 GPa [36] , although it has a bulk modulus on the order of 10.49 GPa [37] , which is even slightly lower than that of Rb 6 Re 6 S 8 I 8. ( ~ 11 GPa). The trade-off between a large sample amount and high pressure achieved is a fundamental and unresolved problem in all pressure-assisted material preparations. Because PIA plays an essential role in discovering exotic and unprecedented material properties under high-pressure conditions, identifying good PIA candidates based on their initial structures is important, because it enables the realization of these properties without the costly use of ultra-high pressure. Differential sublattice rigidity is also a critical factor for the ambient retention of high-pressure properties. Rb 6 Re 6 S 8 I 8 can retain its notable high-pressure photocurrent behavior at ambience because it maintains partial amorphization after full depressurization, exhibiting a thermodynamically metastable and kinetically trapped state. Owing to the relatively weak electrostatic interaction rather than strong chemical bonding between the cluster and framework, upon compression, the framework atoms of Rb and I(2) can deviate from the equilibrium positions to a large extent, thus phenomenologically forming an equipotential surface with small energy variations [32] , which provides the basis for multiple metastable states. Then, the relaxation effect, which is useful for understanding the transformations of metastable states, should be analyzed, where the relaxation time is an essential factor in determining the dynamics. In principle, the in-lab decompression process should be much faster than the recovery of materials; otherwise, no irreversibility will be obtained. With the relaxation time ( τ ) equal to the viscosity ( η ) divided by the modulus ( G ∞ ), as regulated by Maxwell’s expression [38] , the soft nature (small modulus) of the Rb-I(2) framework results in a phenomenally large timescale. By contrast, the relaxation time for the Re-S-I(1) cluster, owing to its rigidity, is much shorter. In addition, upon compression, some Rb-I(2) chemical bonds form during amorphization. Regardless of whether these bonds break after depressurization, it is difficult for such a Rb-I(2) framework to rapidly recover to its original lattice positions. In other words, within the weakly interacting cluster-framework sublattices, the hard clusters can restore the order state, whereas the soft and plastic nature of the Rb-I(2) framework leads to a largely extended relaxation time period. Furthermore, the notable steric hindrance effect, which originates from the bulky Re-S-I(1) clusters, impedes the recovery process of Rb-I(2) that must overcome high barriers and has a large timescale. As such, both timescale and spatial effects jointly contribute to the partial amorphization of Rb 6 Re 6 S 8 I 8 after pressure treatment, resulting in irreversible structural and property changes. We further consider the structural characteristics that can facilitate irreversible structural and functional transitions after pressure engineering. Some features, such as the differential sublattice rigidity revealed in this work, structural gradients, including heterojunctions and homojunctions, two-dimensional layered structures with weak interlayer interactions, superlattices, and meticulously designed structural defects, can potentially give rise to both thermodynamic and kinetic discrepancies between various domains in a material, and should be further examined. In addition, it is necessary to consider the time factor, particularly the sample history and the kinetics of compression/decompression, because the generation and conversion of metastable states is a non-traversal process developing under nonequilibrium conditions [39] , [40] , [41] ."},{"section":"Methods","content":"Crystals growth High-quality Rb 6 Re 6 S 8 I 8 single crystals were synthesized using the salt flux method [19] . High-purity RbI (99.9%), Re (99.99%), S (99.99%) powders, and I 2 spheres (99.99%) were obtained from Aladdin and combined in a molar ratio of 20:6:8:4 under an argon atmosphere. The mixture was placed in a alumina crucible, which was subsequently sealed in a quartz tube under 10 -3 Pa. The assembly was then heated to 800 °C for 10 hours and then kept at this temperature for another 10 hours before slowly cooling to 625 °C at 1 °C·h -1 . Finally, the furnace was turned off, allowing it to cool naturally to room temperature. Through mechanical separation, millimeter-scale orange Rb 6 Re 6 S 8 I 8 single crystals were obtained. Structural characterization at ambient environment The single-crystal X-ray diffraction data were collected on a Bruker D8 Venture diffractometer equipped with a Mo K α radiation source ( λ = 0.71073 Å). The structure of Rb 6 Re 6 S 8 I 8 was refined using the SHELXL21 refinement package and the Olex2 software package [42] , [43] . Using Oxford X-Max N (Oxford Instruments), we obtained morphology and confirmed the chemical composition of Rb 6 Re 6 S 8 I 8 . Manufacturing of high-pressure environment The high-pressure environment was generated using a symmetrical diamond anvil cell (DAC). Pressure calibration was performed using the ruby fluorescence method [44] , with mineral oil serving as the pressure-transmitting medium (PTM) unless otherwise specified. High-pressure synchrotron X-ray diffraction The crystal structures of Rb 6 Re 6 S 8 I 8 under various pressures were determined through in situ synchrotron XRD measurements. Experiments were conducted at beamline 15U of the Shanghai Synchrotron Radiation Facility (SSRF) using a monochromatic X-ray beam focused to dimensions of 11 µm (horizontal) × 3 µm (vertical). Two-dimensional (2D) diffraction patterns were collected and integrated into one-dimensional (1D) data using the Dioptas software package [45] . Structural refinements were performed using the Rietveld method implemented in the General Structure Analysis System-II (GSAS-II) software [46] . The elastic modulus was derived by fitting the experimental data to the third-order Birch-Murnaghan equation of state (Eq. 1 ) [47] : $$P\\left(V\\right)=\\,\\frac{{3B}_{0}}{2}\\left[{\\left(\\frac{{V}_{0}}{V}\\right)}^{\\frac{7}{3}}-\\,{\\left(\\frac{{V}_{0}}{V}\\right)}^{\\frac{5}{3}}\\,\\right]\\left\\{1+\\frac{3}{4}\\left({B}^{{\\prime} }-4\\right)\\left[{\\left(\\frac{{V}_{0}}{V}\\right)}^{\\frac{2}{3}}-1\\right]\\right\\}$$ (1) where V 0 represents the unit cell volume at ambient pressure, determined to be 3757.0(7) Å 3 , and B 0 is the bulk modulus, calculated as 11.0(3) GPa. During the fitting process, the first derivative of the bulk modulus, B ′, was fixed at 12.0. A pair of 300-µm Ia-type diamonds were employed for the measurements. High-pressure X-ray absorption spectroscopy X-ray absorption spectra (XAS) at the Re L 3 -edge were collected at the ODE beamline of the SOLEIL synchrotron radiation facility. Data were acquired in transmission mode, with energy calibration performed using a standard Re foil as a reference. Data processing and fitting were carried out using the Athena and Artemis software packages [48] . A pair of nano-polycrystalline diamond anvils with a 300-µm culet size were employed. High pressure is generated by squeezing DAC through a membrane device. High-pressure UV-visible absorption spectra A custom-designed micro-region spectra system (Gora-UVN-FL, assembled by Ideaoptics) equipped with a Mitutoyo 20× NUV objective was employed to measure UV-visible absorption spectra. A deuterium/halogen light source was used, which provided a wavelength range spanning from 300 nm to 1100 nm. The bandgap of Rb 6 Re 6 S 8 I 8 was determined using the Tauc plot method, which is based on the Eq. ( 2 ) [49] : $${\\alpha }^{1/2}={{{\\rm{A}}}}\\left(h\\nu -{E}_{{{{\\rm{g}}}}}\\right)$$ (2) where α represents the absorption coefficient, hν signifies the photon energy, E g denotes the bandgap, and A is a proportionality constant. The bandgap value can be accurately determined by plotting ( αhν ) 1/2 against hν and extrapolating the linear portion of the curve to the energy axis. The intercept provides the precise bandgap value. For the measurements, a pair of 300-μm IIa-type diamonds were used, with silicone oil serving as the PTM. High-pressure photoluminescence Static photoluminescence (PL) spectra were acquired using a 405 nm laser as the excitation source. The emission signals were collected using a NOVA refrigeration spectrometer coupled to a MIcrochrome 5 Pro CMOS detector, with a spectral resolution of 1 nm. High-pressure time-resolved photoluminescence For time-resolved PL dynamics measurements, sample was photoexcited by a 405 nm laser diode with a pulse width of 20 ps and a repetition rate of 20 kHz. All PL and time-resolved PL experiments were performed using a modified Mao-type symmetrical diamond anvil cell (DAC) equipped with IIa-type diamond anvils, each with a culet size of 300 μm. For PL imaging, the sample was excited by UV light filtered from a xenon lamp source. Silicone oil was employed as the pressure-transmitting medium (PTM). High-pressure photocurrent The high-pressure photocurrent test was conducted on a Keithley 4200A-SCS Parameter Analyzer. A xenon lamp was employed as the simulated visible light source. A symmetrical Sb-type DAC consisting of a pair of 300-μm Ia diamonds was employed. A pre-pressed stainless-steel sheet was drilled with a 280-μm hole, which was subsequently filled with cubic boron nitride (cBN) and subjected to further pressurization. The gasket was then insulated using insulating glue after drilling a 120-μm hole. Two electrodes were fabricated using Pt wires, and the photocurrent was measured under an applied voltage of 3 V for 3 cycles, with each cycle consisting of a 60 seconds light source on period followed by a 60 seconds light source off period. High-pressure Raman spectra Raman spectra were acquired utilizing a home-designed low-wavenumber Raman system equipped with a 785 nm laser. The Raman signals were collected by an IsoPlane 320 spectrograph (Princeton Instruments) coupled to a PyLon100BR_eXcelon detector (Princeton Instruments) cooled with liquid nitrogen. High-pressure middle-infrared transmission spectra The Bruker vertex 70 v FT-IR spectrometer was employed to measure the MIR transmission spectra. A symmetrical Sb-type DAC consisting of a pair of 200-μm IIa diamonds was used. The transmission signal of the chamber containing silicone oil was collected to serve as a reference signal. Rb 6 Re 6 S 8 I 8 (about 5 μm thick) covers the hole, and then the transmission signal was measured under different pressures. The pressure is calibrated by measuring the high-frequency edge of the first-order Raman band at the center of the culet [50] . High-pressure temperature dependence of resistance The RT curves were measured on physical property measurement system (PPMS, Quantum Design) using a Be-Cu piston-cylinder press containing a pair of 150-μm Ia diamonds. The experimental procedure involved pre-pressing a Be-Cu gasket, drilling a 130-μm hole, filling it with cBN, and subjecting it to further pre-pressing. Afterward, a 65-μm hole was drilled and packed with Rb 6 Re 6 S 8 I 8 . Four Pt wires were utilized to establish a connection between the copper wires of the external circuit and create a four-electrode configuration. The pressure is calibrated by measuring the high-frequency edge of the first-order Raman band at the center of the culet [50] . First-principles calculation First-principles calculations were performed within the framework of density functional theory (DFT) using the Vienna Ab Initio Simulation Package (VASP) [51] , [52] . The exchange-correlation functional was treated using the generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) parameterization optimized for solids. The valence electron configurations were described by projector augmented wave (PAW) potentials, with the following electronic states included: Rb (4 s 2 4p 6 5 s 1 ), Re (5 d 5 6 s 2 ), S (3 s 2 3p 4 ), and I (5 s 2 5p 5 ). A plane-wave basis set with a kinetic energy cutoff of 550 eV was employed, ensuring convergence of the forces on each atom to less than 0.01 electron volt per angstrom. The Brillouin zone was sampled using a Monkhorst-Pack k -point mesh with a spacing of 0.2 Å -1 . The electronic band structures are obtained from the refined structures through synchrotron XRD and the Brillouin zones are sampled along high-symmetry points. Infrared intensities and Raman off-resonant activity were calculated using density functional perturbation theory [53] , [54] , [55] , respectively. The discrete intensity points are further convolved with the Lorentz function to obtain the spectrum. The kinetic stabilities are checked by running first-principles molecular dynamic simulation (FPMD). The system is one unit cell, which is consist of 24 Re, 32 I, 24 Rb, and 32 S atoms. We first run equilibrium for 2 picoseconds (ps) at the target P - T with the NVT ensemble (constant number of atoms, volume, and temperature). Atomic trajectories were recorded with a time step of 1 fs, and the total simulation time was set to 10 ps for each pressure-temperature ( P - T ) condition. Temperature control was achieved using a Nosé–Hoover thermostat [56] . Pair-distribution function was calculated and averaged from the last 2000 steps of the trajectory. The mean squared displacement is calculated along the trajectory of atomic motions through the Python package Kinisi [57] . Pressurization treatment with large volume press The Rb 6 Re 6 S 8 I 8 powder was compacted into a cylindrical ZrO 2 cavity, which was subsequently incorporated into an Al 2 O 3 /MgO 2 octahedral structure. This octahedron was centrally situated within a cubic arrangement of eight WC anvils and then placed in the LVP (Kawai-type multi-anvil apparatus, SHTech-2000, Max Voggenreiter). The pressure was systematically increased to 15 GPa over 6 hours, sustained for 2 hours, and then gradually released to ambient pressure within 10 hours. Fabrication and measurements of photodetector devices The wafer cleaning process involved sequential treatments with detergent, acetone and isopropyl alcohol, and deionized water for 20 minutes each. Then, the wafers were exposed to UV light for 10 minutes and spin coating was performed in a vacuum glove box. A solution consisting of 15 mg LVP-treated Rb 6 Re 6 S 8 I 8 in 1.5 mL N,N-dimethylformamide (DMF) was prepared and spin-coated at 3000 rpm for 30 seconds, and then annealed at 100 °C for 10 minutes before cooling down to room temperature. Subsequently, the Pt electrode was deposited by magnetron sputtering through a mask plate in a vacuum of 9 × 10 -4 Pa using a 99.99% Pt target with a device area of 0.5 cm 2 . The optoelectronic properties of the prepared devices were characterized using a semiconductor parameter analyzer (Keithley 2400, Tektronix) and light sources with wavelengths of 367 nm and 520 nm. The lighter power density is 50 mW·cm -2 for both excitations. To determine the response rate, the LED light source was connected to a function generator (TDS2022B, Tektronix) to emit light pulses at different frequencies. In addition, the devices were imaged with specific photocurrent mapping patterns using the assembled imaging system."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"Using pressure to manipulate the physical state of condensed matter is as fundamental as changing temperature or tuning chemical composition for the same purpose [1] . Therefore, pressure is a distinctive dimension for seeking improved or even unprecedented properties in compressed materials, as evidenced by numerous reports on metallic hydrogen, high-temperature superconductors, photovoltaic semiconductors, and so on [2] , [3] , [4] , [5] , [6] . However, retaining and utilizing properties attainable at high pressure for ambient applications has been challenging because of the degradation of the high-pressure-induced properties after decompression and the inevitable trade-off between sample quantity and the magnitude of attainable pressure [7] , [8] . Two strategies have been proposed to overcome this formidable obstacle. In the first approach, inspired by natural mineral inclusions [9] , [10] , [11] , nanostructured diamond capsules (NDCs) have been developed to confine samples and preserve their high pressure-induced properties [12] . However, man-made NDCs can encapsulate only gaseous and volatile samples because of their softness and compressibility. In addition, this technique requires high temperatures to convert graphite to diamond for in situ preservation of target materials, thus showing limited applicability for organics or materials with low decomposition temperature. Moreover, after decompression, the target material is isolated in nanostructured diamond and forbidden to form a pure material, thus impeding contact-based material analysis or large-scale device applications. Another route is to engineer the kinetic barriers in the phase transformation to obtain ambient-pressure metastable materials [13] , [14] , [15] . This method has been realized in specific nanomaterials with interconnected nanocrystal networks [13] . However, no device applications using such metastable materials have been reported to date. A potential opportunity for ambient retention of high pressure-induced properties lies in tailoring the coupling between sublattices. Conventionally, a sublattice is a simple mathematical concept related to a subset of the translations of a lattice [16] . Sublattice modeling has now been extensively adopted to describe a variety of materials, ranging from alloys to ionic liquids [17] , [18] . All of these successful studies indicate that the sublattices in these systems are strongly coupled through their structural periodicity; thus, their responses under stimuli should be synchronized. By contrast, if the material is designed such that individual sublattices are weakly coupled through nonchemical bonding and a large discrepancy in the rigidity is present between various sublattices, pressure can be used to decouple the structural periodicity. More importantly, the properties acquired at high pressure become irreversibly frozen during decompression because of the nonsynchronized structural changes between the sublattices. Very recently, the Rb 6 Re 6 S 8 I 8 chalcohalide inorganic compound has been reported [19] , which is structurally characterized by two sublattices: the [Re 6 S 8 I 6 ] 4- (Re-S-I(1)) cluster and the [Rb 6 I 2 ] 4+ (Rb-I(2)) electrostatic framework. These two sublattices are mutually coupled by electrostatic attraction forces that are significantly weaker than those of chemical bonds. At ambient pressure, this material shows strong photoluminescence (PL) but does not exhibit a photocurrent response. However, upon compression, there is a significant pressure-generated photocurrent in this material, which is retained at ambient pressure after the applied pressure is completely removed. As a proof for device application, a large-volume press (LVP) is employed to pressurize and depressurize a scalable quantity of Rb 6 Re 6 S 8 I 8 inheriting the pressure-induced photoelectric properties. This material is then used to build ambience-operational photodetectors. Notable photocurrent responses are observed under excitation by both ultraviolet and visible light. These results, together with the elucidated thermodynamic and kinetic mechanisms, enable high-pressure properties to be employed in optoelectronics that must operate under ambient conditions.\",\n \"references\": {\n \"indices\": [\n 1,\n 2,\n 3,\n 4,\n 5,\n 6,\n 7,\n 8,\n 9,\n 10,\n 12,\n 13,\n 14,\n 15,\n 17,\n 18,\n 19\n ],\n \"infos\": [\n {\n \"idx\": \"1\",\n \"title\": \"Mao, H.-k & Hemley, R. J. Ultrahigh-pressure transitions in solid hydrogen.Rev. Mod. Phys.66, 671–692 (1994).\",\n \"link\": \"https://doi.org/10.1103%2FRevModPhys.66.671\"\n },\n {\n \"idx\": \"2\",\n \"title\": \"Philip, D.-S., Ross, T. H. & Eugene, G. Evidence for a new phase of dense hydrogen above 325 gigapascals.Nature529, 63–67 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fnature16164\"\n },\n {\n \"idx\": \"3\",\n \"title\": \"José, A. F.-L. et al. A perspective on conventional high-temperature superconductors at high pressure: Methods and materials.Phys. Rep.856, 1–78 (2020).\",\n \"link\": \"https://doi.org/10.1016%2Fj.physrep.2020.02.003\"\n },\n {\n \"idx\": \"4\",\n \"title\": \"Capitani, F. et al. Spectroscopic evidence of a new energy scale for superconductivity in H3S.Nat. Phys.13, 859–863 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys4156\"\n },\n {\n \"idx\": \"5\",\n \"title\": \"Kong, L. et al. Simultaneous band-gap narrowing and carrier-lifetime prolongation of organic–inorganic trihalide perovskites.Proc. Natl Acad. Sci. USA3, 8910–8915 (2016).\",\n \"link\": \"https://doi.org/10.1073%2Fpnas.1609030113\"\n },\n {\n \"idx\": \"6\",\n \"title\": \"Tang, H. et al. Toughening oxide glasses through paracrystallization.Nat. Mater.22, 1189–1195 (2023).\",\n \"link\": \"https://doi.org/10.1038%2Fs41563-023-01625-x\"\n },\n {\n \"idx\": \"7\",\n \"title\": \"Shen, G. & Mao, H.-k High-pressure studies with x-rays using diamond anvil cells.Rep. Prog. Phys.80, 016101 (2017).\",\n \"link\": \"https://doi.org/10.1088%2F1361-6633%2F80%2F1%2F016101\"\n },\n {\n \"idx\": \"8\",\n \"title\": \"Mao, H.-k., Chen, X.-J., Ding, Y., Li, B. & Wang, L. Solids, liquids, and gases under high pressure.Rev. Mod. Phys.90, 015007 (2018).\",\n \"link\": \"https://doi.org/10.1103%2FRevModPhys.90.015007\"\n },\n {\n \"idx\": \"9\",\n \"title\": \"Mao, H.-k. & Mao, W. Geomimicry—Liberating high-pressure research by encapsulation.Matter Radiat. Extremes7, 068102 (2022).\",\n \"link\": \"https://doi.org/10.1063%2F5.0126898\"\n },\n {\n \"idx\": \"10\",\n \"title\": \"Tschauner, O. et al. Discovery of davemaoite, CaSiO3-perovskite, as a mineral from the lower mantle.Science374, 891–894 (2021).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.abl8568\"\n },\n {\n \"idx\": \"12\",\n \"title\": \"Zeng, Z. et al. Preservation of high-pressure volatiles in nanostructured diamond capsules.Nature608, 513–517 (2022).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-022-04955-z\"\n },\n {\n \"idx\": \"13\",\n \"title\": \"Xiao, T. et al. Nanocrystals with metastable high-pressure phases under ambient conditions.Science377, 870–874 (2022).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.abq7684\"\n },\n {\n \"idx\": \"14\",\n \"title\": \"Mao, W. & Lin, Y. Making the most of metastability.Science377, 814–815 (2022).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.add5433\"\n },\n {\n \"idx\": \"15\",\n \"title\": \"Meng, L. et al. Theoretical and Experimental Advances in High-Pressure Behaviors of Nanoparticles.Chem. Rev.123, 10206–10257 (2023).\",\n \"link\": \"https://doi.org/10.1021%2Facs.chemrev.3c00169\"\n },\n {\n \"idx\": \"17\",\n \"title\": \"Sundman, B. O. & Ågren, J. The Sublattice Model.MRS Online Proc. Libr.19, 115–127 (1982).\",\n \"link\": \"https://doi.org/10.1557%2FPROC-19-115\"\n },\n {\n \"idx\": \"18\",\n \"title\": \"Singh, S. Phase transitions in liquid crystals.Phys. Rep.324, 107–269 (2000).\",\n \"link\": \"https://doi.org/10.1016%2FS0370-1573%2899%2900049-6\"\n },\n {\n \"idx\": \"19\",\n \"title\": \"Laing, C. C. et al. Solution-processable mixed-anion cluster chalcohalide Rb6Re6S8I8in a light-emitting diode.Nat. Mater.23, 230–236 (2024).\",\n \"link\": \"https://doi.org/10.1038%2Fs41563-023-01740-9\"\n }\n ]\n }\n },\n {\n \"section\": \"Results and discussion\",\n \"content\": \"Discovery of differential sublattice rigidity We first prepared high-quality Rb 6 Re 6 S 8 I 8 single crystals (Supplementary Fig. 1 and Supplementary Table 1 ). As revealed by a combination of in situ high-pressure structural characterizations (Fig. 1 ), the rigidities of two sublattices, the [Re 6 S 8 I 6 ] 4- cluster and the [Rb 6 I 2 ] 4+ framework, are significantly different. Owing to lattice shrinkage, the high-pressure X-ray diffraction (XRD) pattern implies monotonic shifts of Bragg reflections towards larger 2 θ angles from 1 atm to ~10–12 GPa, above which the pattern becomes diffusive for lattice distortion, and the structure cannot maintain long-range ordered atomic arrangements (Fig. 1a ). Using GSAS software, we extract the pressure dependence of interatomic distances, where the framework-related Rb-Rb, Rb-I(1) and Rb-I(2) undergo monotonic distance reductions, yet the cluster-related bond lengths of Re-I(1), Re-Re and Re-S, as well as the I(1)-Re-S, S-Re-S and Re-S-Re bond angles, are nearly unchanged; thus, the clusters remain undisturbed at higher pressures (Fig. 1 b, c , and Supplementary Figs. 2 and 3 ), and the root cause of amorphization is the collapse of the Rb-I(2) framework. Fig. 1: Structural evolutions of Rb 6 Re 6 S 8 I 8 upon compression. a Pressure-dependent in situ synchrotron-based XRD pattern, showing structural amorphization starting at ~12 GPa. The XRD pattern collected at ambient pressure is highlighted in gray, and indicates that no crystalline phase transition occurred prior to amorphization. b and c Evolution of the interatomic distance and bond angles under pressure, suggesting differential rigidity. We identify two distinct compression behaviors dominated by the significantly soft Rb-I(2) framework and the cage-structured Re-S-I(1) sublattice that is extremely insensitive to pressure. d Pressure dependence of the Raman spectra upon compression. e Pressure dependence of the full width at half maximum (FWHM) of the representative Raman modes, which are nearly unchanged upon compression. f First and second derivatives of the Re L 3 edge \\\\(\\\\mu (E)\\\\) plotted versus the photon energy of Re 6 Rb 6 S 8 I 8 at different pressures upon compression. g Constant edge energy ( E 0 ) and white line energy ( E w ) during the investigated compression process. h Pressure dependence of the XAS Fourier transform magnitude as a function of interatomic distance and fittings based on nearest neighbor (Re-S) at selected pressures. i Fitted interatomic Re-S distance. The low R-factors, which are on the order of 10 -4 show the high goodness of fit. During the whole compression process up to 25.4 GPa, no obvious change occurs in the interatomic distance within the [Re 6 S 8 I 6 ] 4- cluster. The error bars in ( g and i ) denote the fitting errors. Full size image The hard rigidity of the cluster sublattice is further supported by the results of two independent experiments, namely high-pressure Raman spectroscopy (Supplementary Fig. 4 ) and X-ray absorption spectroscopy (XAS) (Supplementary Fig. 5 ). According to the calculated results of the vibrational mode assignment, all of the initial Raman modes are based on the cluster rather than on the framework. The pressure dependence of the Raman spectra clearly shows that these modes are distinguished up to 20 GPa (Fig. 1d ), with no noticeable increase in the full width at half maximum (FWHM) (Fig. 1e ), showing short-range order characteristics. Such sharp modes exhibit a sustained blueshift, even after the process of amorphization is completed, as revealed by XRD (Fig. 1a ), demonstrating a specific decoupling between long-range crystalline disorder and short-range local order. Using XAS as another quantitative validation tool, it is observed that the edge energy ( E 0 ) and line energy ( E w ), which are defined by the peaks of the first and second derivatives of \\\\(\\\\mu (E)\\\\) , respectively, remain constant at ~10.537 keV and ~10.539 keV as the pressure increases (Fig. 1 f, g ), indicating that there is no change in the valence state or coordination environment of Re. This pressure independence of the energy position reflects the robust nature of the Re-S-I cluster. Furthermore, the Fourier transform magnitude of the XAS spectra suggests that the Re-S interatomic distance does not vary significantly (Fig. 1 h and i), again verifying the nonsynchronous lattice disorder processes and the difference in rigidity between cluster-structured [Re 6 S 8 I 6 ] 4- and electrostatic [Rb 6 I 2 ] 4+ sublattices. The difference in the compressibility between the cluster and electrostatic frameworks provides a structural basis for the adjustable and irreversible optical and electronic properties discussed below. Pressure-induced emission transitions We then focused on the high-pressure-driven functional response of Rb 6 Re 6 S 8 I 8 . The different compressibility of the Re-S-I(1) cluster and the Rb-I(2) electrostatic framework can lead to attenuated photoluminescence (PL) but also an emergent photocurrent response of Rb 6 Re 6 S 8 I 8 . As shown in Fig. 2a and Supplementary Fig. 6 , while pressure-dependent PL generally results in a reduction in the emission intensity at higher pressures, slight increases in the PL intensity and prolongation of the carrier lifetime are observed at a mild pressure of ~0.1 GPa (Fig. 2 b– g ; Supplementary Figs. 7 – 9 ). On the other hand, the PL peak wavelength displays a monotonic blueshift with increasing pressure (Fig. 2c ), which is in contrast to the trend observed for the pressure-dependent UV-Vis absorbance. In addition, the full width at half maximum (FWHM) of the PL spectra also decreases with increasing pressure (Fig. 2d ), which is additionally corroborated by the second central moment ( m 2 ), which also displays a similar decreasing trend (Fig. 2e ), indicating that the energy density of states increases upon compression [20] . We also note that the PL annihilation is associated with the shortened PL lifetimes upon compression (Fig. 2 f, g ). Fig. 2: Pressure-dependent optical properties of Rb 6 Re 6 S 8 I 8 during compression. a Pressure-dependent PL contour plot from 1 atm to 12 GPa, where the PL emission is fully annihilated. b – e Pressure dependences of the PL intensity, PL peak wavelength, and full width at half maximum (FWHM) of the PL peak and second central moments ( m 2 ), respectively. Here, for a Gaussian-type emission profile, the FWHM is conveniently quantified by m 2 as m 2 = (FWHM) 2 /8ln2, where the error bar in d shows the fitting errors. Note that two individual samples, 1 and 2, were employed to confirm the validity of the experimental results. f Experimental time-resolved PL decays (dots) and fittings (solid lines) from 1 atm to 10.2 GPa. g The extracted carrier lifetimes obtained by fitting the time-resolved PL decay traces in f . h Comparison of the on-off photocurrent response and magnitudes from 1 atm to 24.7 GPa. i Evolution of the photocurrents extracted from ( h ). j Pressure-dependent UV-vis absorbance near bandgap regions from 1 atm to 30 GPa, where a drastic narrowing of the bandgap can be observed at approximately 17 GPa. Therefore, the onset pressures for abrupt bandgap narrowing, drastic photocurrent increase, and long-range disorder are coincident, thus indicating that structural amorphization is the common origin for the property changes. The sizes of the symbols shown in ( b , c , e , g and i )cover the experimental or fitting errors. Full size image The PL properties and exciton/carrier dynamics of similar compounds are highly dependent on the properties of structurally isolated clusters [20] , [21] , [22] , and the lifetime is strongly affected by ligands of rhenium. Upon compression, hydrostatic pressure first compresses the Rb-I(2) electrostatic framework owing to its much softer nature, with the compressed Rb-I(2) framework eventually tuning atomic connections within the Re-S-I(1) cluster structure. Then, the original crystal field of the Re-S-I(1) cluster is modified due to the perturbed electron clouds of the S/I ligands under lattice compression. Therefore, PL attenuation at higher pressures can be considered the result of increased long-range interactions between the Re-S-I(1) cluster and the Rb-I(2) framework. This process is critically influenced by spin-spin interactions, which are governed by the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction and further modulated by crystal field effects [23] , [24] . To delve deeper into the mechanisms underlying emission transitions, we investigate the role of self-trapped excitons (STEs). In crystalline materials, excitons interact with phonons via electron-phonon coupling. When this coupling is weak, excitons are primarily scattered by phonons. In contrast, strong coupling can lead to the self-trapping of excitons. Specifically, an STE is formed because photoinduced local lattice distortion occurs due to strong electron-phonon coupling [25] . For the studied Rb 6 Re 6 S 8 I 8 material, the broad sub-bandgap emission with a large Stokes shift observed at ambience is the fingerprint of the STE. Two additional pieces of evidence further support the existence of STEs in Rb 6 Re 6 S 8 I 8 : 1) Soft lattice. According to the XRD results shown in Supplementary Fig. 2 , the bulk modulus of Rb 6 Re 6 S 8 I 8 is ~11 GPa, which is significantly lower than that of the conventional semiconductor Si and even lower than that of organic-inorganic hybrid perovskites ( ~ 15‒20 GPa) [26] . This soft lattice is conducive to lattice deformation, which is crucial for the formation of an STE [27] . 2) Low electronic dimensionality. From the calculated band structure results shown in Supplementary Fig. 10 , we observe flat conduction bands (CBs) and valence bands (VBs), indicating localized zero-dimensional features. Such low electronic dimensionality is a prerequisite for the formation of STEs [28] . For excitons to become self-trapped, neither photogenerated electrons nor holes, or both, can exhibit high delocalization; that is, the effective mass of at least one carrier must be sufficiently high. In other words, low electronic dimensionality, which reflects the extent of orbital overlap, is essential for the formation of STEs and, consequently, for broadband emission. All of these observations demonstrated that STE is the origin of broad PL, and we can see that the evolution of STEs controls the emission transitions observed at pressures. Upon compression, the STE in Rb 6 Re 6 S 8 I 8 is suppressed and weakened, which can be easily understood from the fact that its lattice becomes more rigid at higher pressure; therefore, photoinduced local lattice distortion and electron-phonon coupling become difficult. This argument is also supported by the reduced FWHM of the PL spectra upon increased pressure, as shown in Fig. 2d , indicating that the STE feature is weakened at high pressure. In addition, we further studied the pressure-dependent self-trapping energy ( \\\\({E}_{{{{\\\\rm{st}}}}}\\\\) ), which is estimated as \\\\({E}_{{{{\\\\rm{st}}}}}=\\\\,{E}_{{{{\\\\rm{g}}}}}-\\\\,{E}_{{{{\\\\rm{PL}}}}}\\\\) , where \\\\({E}_{{{{\\\\rm{g}}}}}\\\\) is the bandgap energy, and \\\\({E}_{{{{\\\\rm{PL}}}}}\\\\) is the PL emission energy. In the studied material, rhenium chalcohalide clusters are completely isolated by the large [Rb 6 I 2 ] 4+ framework, forming a nominal 0D structure, as confirmed by our band-structure calculations (Supplementary Fig. 10 ). In 0D systems, the energy barrier between the free exciton and STE is considered not to exist [27] , [29] . We also note that the VB of this material is flat, particularly at low pressure, so that the lattice deformation energy should be very small. Supplementary Fig. 11 shows the pressure-dependent bandgap energy and PL energy upon compression, from which it is observed that from ambient pressure to 10 GPa, the bandgap energy is first relatively stable, and then decreases, while the PL energy increases monotonically. Therefore, we can conclude that a reduction of \\\\({E}_{{{{\\\\rm{st}}}}}\\\\) , rendering the STE unstable; this implies that the STE cannot act as an effective PL mechanism at high pressures. Pressure-induced photocurrent generation and enhancement The pressure-induced photocurrent transitions, where the STE again plays a crucial role, are of particular interest. As shown in Fig. 2 h and i, we can identify two stages of the evolution of the pressure-induced photocurrent. The first stage is the initial photocurrent generation occurring at ~2 GPa, prior to which the photocurrent is undetectable. The second stage occurs at ~15 GPa, after which the photocurrent is greatly enhanced upon compression. The self-trapping energy describes the extent of binding of the STE. Therefore, the reduced \\\\({E}_{{{{\\\\rm{st}}}}}\\\\) of Rb 6 Re 6 S 8 I 8 at higher pressures facilitates the dissociation of STEs into free carriers. From the calculated band structure (Supplementary Fig. 10 ), one can expect that owing to the fluctuation of the VB at higher pressure, the effective mass will decrease upon compression, leading to enhanced carrier mobility. Then, the pressure-induced greater mobility allows these free carriers to more easily migrate towards the current-collecting electrodes of the photodetector devices. Therefore, a detectable photocurrent is achieved via pressure-reduced excitonic behavior at ~2 GPa. For the pressure-enhanced photocurrent above ~15 GPa, the increasing trend of the photocurrent agrees well with the redshift in the optical bandgap (Fig. 2j ; Supplementary Figs. 12 – 15 ), indicating their common structural origin. Structurally, compression results in amorphization, causing the formation of a chemical bond between Rb and I(2) that originally interacted through Coulomb attraction. This argument can be verified by extending the graph of the Rb-I(2) interatomic distance (Fig. 1b ) to a higher pressure, e.g., 15 GPa, from which a 3.6 Å interatomic distance that is very close to the Rb-I ionic bond length (3.66 Å) can be extracted. Thus, it is reasonably believed that there is a high probability that Rb-I forms chemical bonds in Rb 6 Re 6 S 8 I 8 within the pressure range beyond 15 GPa; this facilitates an electron transition from the VB to the CB, thereby dramatically narrowing the optical bandgap. The higher carrier density therefore leads to a photocurrent enhancement of over three orders of magnitude within the 15–20 GPa pressure range. Irreversible structural transition In addition to the different compression behaviors of the Re-S-I(1) cluster and the Rb-I(2) framework, these two structural moieties also exhibit distinct irreversibility after decompression, and the ambient structure of the recovered sample is strongly associated with the peak applied pressure (Fig. 3 ; Supplementary Figs. 16 – 19 ). Generally, Rb 6 Re 6 S 8 I 8 shows long-range disorder as it is decompressed from higher peak pressures, rendering it amorphous without structural robustness (Fig. 3a ). The ambient structures of the as-synthesized Rb 6 Re 6 S 8 I 8 sample and the Rb 6 Re 6 S 8 I 8 among samples decompressed from different peak pressures were further compared using in situ Raman spectroscopy (Fig. 3b ). For the samples decompressed from 25 GPa, distinctive modes are clearly shown, suggesting an ordered Re-S-I(1) cluster structure. Nonetheless, even for the sample that underwent high-pressure treatment and then was decompressed from a relatively lower peak pressure of 18 GPa, the XRD pattern already clearly shows a disordered long-range structure (Fig. 3a ). This phenomenon indicates the different structural reversibility of the Re-S-I(1) cluster and the Rb-I(2) framework. Fig. 3: Compression-decompression dependent structural transitions of Rb 6 Re 6 S 8 I 8 . a XRD patterns of samples of powders ground from the as-prepared crystal and samples recovered from different peak pressures. b Raman spectra of samples of powders ground from the as-prepared crystal and samples recovered from different peak pressures. c XAS Fourier transform magnitudes as a function of interatomic distance and fittings (dashed line) based on nearest neighbor (Re-S) at ambient pressure as recovered from 1.5 GPa and 25 GPa peak pressures. d Schematic illustration of hydrostatic compression of Rb 6 Re 6 S 8 I 8 , which leads to nonsynchronous lattice disorder of the [Re 6 S 8 I 6 ] 4- cluster and the Rb-I electrostatic framework, also showing the plasticity of the Rb-I framework, resulting in incomplete recovery from the high-pressure disorder state. The nonsynchronous amorphization processes of the Re-S-I(1) and Rb-I(2) sublattices, along with their different extents of recrystallization, indicate the irreversibility of high-pressure amorphization. Full size image Therefore, one can visualize a decoupled Re-S-I(1) cluster and Rb-I(2) framework, where the former component is mechanically more resilient to pressure and does not affect structural periodicity to a large extent, whereas Rb-I(2) is spatially widespread, and directly regulates the long-range order of the Rb 6 Re 6 S 8 I 8 lattice; this means that structural plasticity exists in Rb 6 Re 6 S 8 I 8 , which is dictated by the least irreversible recovery of the Rb-I(2) framework from the high-pressure treatment cycle. Furthermore, by fitting the Fourier transformed XAS magnitude with the nearest neighbor (Re-S), it was revealed that after the compression-decompression cycle with a peak pressure of 25 GPa, the bond length of Re-S was 2.27 Å, showing only negligible deviation from the 2.33 Å Re-S bond length obtained with a peak pressure of 1.5 GPa (Fig. 3c ). Figure 3d shows the structural evolution of the Re-S-I(1) cluster and the Rb-I(2) framework before and after high-pressure treatment. The structural irreversibility therein gives rise to a permanent modification of functional properties, which is promising for engineered device design. We also note that even after the compression-decompression cycle with a low peak-pressure of 2 GPa, although no phase transition is observed, the microstructure of [Re 6 S 8 I 6 ] 4- is irreversibly modified, leading to a PL enhancement (Supplementary Fig. 16 ). Pressure-cycle-induced photocurrent retainable at ambience Benefiting from irreversible structural transitions, recovered Rb 6 Re 6 S 8 I 8 retains its high-pressure properties that depend on the magnitude of peak applied pressure (Supplementary Figs. 20 – 23 ). Specifically, it even exhibits monotonically increased photocurrents with larger number of compression-decompression cycles (Fig. 4a ). To explore the structural origin of the retained photocurrents at ambience, we conducted molecular dynamics simulations of the mean squared distances (MSDs) of the Rb, I, S, and Re atoms and observed more intensive fluctuations in the Rb-Rb distance as a function of simulation time, indicating that the polarizability and plasticity of the framework atoms are much greater than those of the other structural components (Fig. 4b ; Supplementary Fig. 24 ). The larger deformability of the Rb-I(2) framework thereby explains the observed drastic bandgap redshift occurring at 17 GPa (Fig. 2j ) along with the concomitant photocurrent enhancement (Fig. 2i ). Fig. 4: Retainable photocurrent at ambience, large-scale sample preparation, and photodetector device. a Comparison of ambient on-off photocurrent responses before compression (gray), after first (light red) and second (dark red) compression-decompresion cycles, showing notable generation and increase in the photocurrent after pressure treatments. b , Mean squared distances of the Rb, I, S, and Re atoms. The fluctuations stabilized with increasing time in the molecular dynamics simulations. c Scheme of the LVP utilized in applying pressure treatment to a large-scale Rb 6 Re 6 S 8 I 8 sample. Scale bar, 1 mm. d Sketch of a Schottky-type thin-film Rb 6 Re 6 S 8 I 8 photodetector device. e Good solution processability of the decompressed Rb 6 Re 6 S 8 I 8 sample. The optical picture of LVP-treated Rb 6 Re 6 S 8 I 8 dissolved in DMF confirmed that the material is dispersible, where the rubidium iodide ([Rb 6 I 2 ] 4+ ) framework is stable, rather than behaving as Rb + and I - in DMF. f Photograph of the LVP-treated Rb 6 Re 6 S 8 I 8 -based photodetector device and the on-device interdigitated Pt electrode imaged under a microscope. g The I-V curve of devices under 367-nm illumination. h On-off photocurrents under 367-nm illumination. i The I-V curve of devices under 520-nm illumination. j On-off photocurrents under 520-nm illumination. For all tests, no bias voltage is involved, and the devices are self-powered. k Response time and specific detectivity ( D * ) with both 367 nm and 520 nm illuminations. Here, τ r and τ d denote the rise and decay response times, respectively. l Schematic diagram of the device ultraviolet and visible imaging device. The photocurrent mappings are observed with the specific pattern obtained by irradiating the device with the “HPSTAR” logo mask using the 367 nm ultraviolet and 520 nm visible lasers. “Max” and “Min” in the color scale denote the relative magnitudes of photocurrent activity. As such, the adoption of LVP in combination with permanent photocurrent generation from high-pressure treatment thus allows practical and scalable device applications. Full size image We also compared the bandgap difference between the initial state of the materials before compression and the decompressed state from high-pressure cycles, with the data shown in Supplementary Fig. 20 . It is observed that the optical bandgap cannot recover to the magnitude before compression; however, it exhibits a redshift after the first compression-decompression cycle and another redshift after the second cycle. Additionally, it is observed from the pressure dependence of the Urbach energy upon compression-decompression cycles that prior to the compression-decompression cycles, the absorbance curve exhibits a sharper profile, indicating a relatively ordered structure, whereas after the compression-decompression cycles, there is a significant increase in the magnitude of the Urbach energy, indicating a more disordered structure (Supplementary Fig. 20 ) [30] . All of these results are consistent with the argument that amorphization disorder plays a crucial role in photocurrent enhancement at high-pressure region over 15 GPa. In addition, the magnitude of the self-trapping energy can be estimated by the difference between the bandgap energy and the emission energy. Considering that the emission energy recovers to the initial state before compression (Supplementary Fig. 21 ), we can conclude that the self-trapping energy decreases after pressure treatment because the bandgap redshifts, meaning that the STE becomes unstable, leading to PL quenching (Supplementary Fig. 21 ) and the generation of photocurrent due to the dissociation of the STEs into free carriers. Scalable decompressed sample for device fabrication To demonstrate the practical device applicability of the photocurrent enhancement, we use a large volume press (LVP) to prepare large-scale samples with high-pressure treatment (Fig. 4c ). The sample pressure-treated under LVP was then fabricated into a Schottky-type Pt/Rb 6 Re 6 S 8 I 8 /Pt photodetector on a substrate (Fig. 4d ), by taking the advantage of the good solution processability of the decompressed samples. Figure 4e shows the picture of N,N-dimethylformamide (DMF) solution of decompressed Rb 6 Re 6 S 8 I 8 sample, from which one can see the sample is totally dissolved in DMF solvent. We further let the solvent evaporate to induce a recrystallization of Rb 6 Re 6 S 8 I 8 , and conducted XRD experiment on this sample. Although structural diffraction results in identifiable Bragg peaks, it shows rather diffusive background in the XRD pattern (Supplementary Fig. 25a ), thus indicating a partially amorphous nature of the solution-processed sample, confirming the metastable phase is preserved. Here, the rubidium iodide ([Rb 6 I 2 ] 4+ ) framework is stable in DMF, rather than behaving as Rb + and I - , which can be understood by the effect of ion association. Such a demonstration is further supported by the study of particle size in DMF solvent. We conducted dynamic light scattering (DLS) experiments on three samples in DMF with the same mol% concentration: as-prepared Rb 6 Re 6 S 8 I 8 ; large volume press (LVP)-treated Rb 6 Re 6 S 8 I 8 , and RbI. Supplementary Fig. 25b shows that the average particle size of LVP-treated Rb 6 Re 6 S 8 I 8 as dissolved in DMF is slightly larger than that of as-prepared Rb 6 Re 6 S 8 I 8 without pressure treatment, and significantly larger than that of RbI. The clear discrepancy of average particle sizes among these three specimens suggests that framework and cluster can be preserved within the LVP-treated Rb 6 Re 6 S 8 I 8 after dissolution in organic solvent, thereby laying the foundation for all subsequent device fabrication that utilizes the amorphized crystal structure for its permanently enhanced photocurrents. The detailed device setup with connected probes and interdigitated Pt electrodes optically characterized as shown in Fig. 4f . The fabricated self-powered photodetector shows a functional current response within a −1.0 to +1.0 V current-voltage scan under 367 nm (Fig. 4 g, h ) and 520 nm (Fig. 4 i, j ) illuminations, demonstrating a large on-off ratio with photocurrent magnitudes of ~60 pA and ~8 nA, respectively. In addition to the fast photocurrent dynamics on the order of 10 2 μs, the detector also exhibits considerable specific detectivity of ~10 11 Jones (Fig. 4k ). Using the photocurrent magnitude as an indicator, we finally achieve highly spatial-resolved red‒green-blue (RGB) imaging, in which the word “HPSTAR” is clearly visible (Fig. 4l ).\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"We have investigated the photoluminescent to optoelectronic transition occurring in a compressed compound with isolated clusters. Using a combination of high-pressure experiments, we have examined whether such a transition can be retained after the applied pressure was completely released, and found a retention of high-pressure properties, as manifested by the greatly increased photocurrent arising from the initially non-optoelectronic state. These results can guide further applications for the construction of crystalline-amorphous hybrid structures and provide crucial clues regarding the materials that may show high-pressure memory at ambience. In the future, more research is needed to better understand the mechanism by which amorphization leads to the sharp increase in photocurrent, especially on the electronic structure of disorder phase. Our study also has revealed the potential of pressure treatment in practical applications, enabling the integration of high-pressure-treated materials into functional devices with high-pressure-enhanced performance. Through industrially accessible routes, we have realized large-scale sample preparation and efficient device fabrication of high-performance photodetectors. These results provide a practical demonstration of the feasibility of using pressure-treated materials in real-world optoelectronic applications. To guide future research, we present the following discussion and perspective. Differences in the rigidity between weakly coupled sublattices is crucial for achieving amorphization upon compression. Pressure-induced amorphization (PIA) can be understood in terms of mechanical collapse (known as the spinodal limit) and a thermodynamically metastable transition, which delimit the upper and lower bounds of this phenomenon [31] , [32] . In compounds featuring rigid isolated clusters embedded within soft frameworks, applied pressure can markedly reduce the inter-cluster spacing before inducing interactions. However, due to their inherent rigidity, the highly symmetrical clusters impose stringent local symmetry constraints. Consequently, the interplay and balance between these two opposing effects can generate significant shear or other forms of stress, prompting large-scale diffusion of framework atoms and ultimately driving the system into an amorphous state. Such pressure-induced phase transitions have been experimentally observed in compounds characterized by structures comprising isolated polyhedral units such as molybdates [33] , tungstates [33] , silicides [34] , and orthosilicates [35] . We note that a simple soft lattice cannot ensure easy PIA. Pure RbI does not undergo long-range lattice disorder until 67 GPa [36] , although it has a bulk modulus on the order of 10.49 GPa [37] , which is even slightly lower than that of Rb 6 Re 6 S 8 I 8. ( ~ 11 GPa). The trade-off between a large sample amount and high pressure achieved is a fundamental and unresolved problem in all pressure-assisted material preparations. Because PIA plays an essential role in discovering exotic and unprecedented material properties under high-pressure conditions, identifying good PIA candidates based on their initial structures is important, because it enables the realization of these properties without the costly use of ultra-high pressure. Differential sublattice rigidity is also a critical factor for the ambient retention of high-pressure properties. Rb 6 Re 6 S 8 I 8 can retain its notable high-pressure photocurrent behavior at ambience because it maintains partial amorphization after full depressurization, exhibiting a thermodynamically metastable and kinetically trapped state. Owing to the relatively weak electrostatic interaction rather than strong chemical bonding between the cluster and framework, upon compression, the framework atoms of Rb and I(2) can deviate from the equilibrium positions to a large extent, thus phenomenologically forming an equipotential surface with small energy variations [32] , which provides the basis for multiple metastable states. Then, the relaxation effect, which is useful for understanding the transformations of metastable states, should be analyzed, where the relaxation time is an essential factor in determining the dynamics. In principle, the in-lab decompression process should be much faster than the recovery of materials; otherwise, no irreversibility will be obtained. With the relaxation time ( τ ) equal to the viscosity ( η ) divided by the modulus ( G ∞ ), as regulated by Maxwell’s expression [38] , the soft nature (small modulus) of the Rb-I(2) framework results in a phenomenally large timescale. By contrast, the relaxation time for the Re-S-I(1) cluster, owing to its rigidity, is much shorter. In addition, upon compression, some Rb-I(2) chemical bonds form during amorphization. Regardless of whether these bonds break after depressurization, it is difficult for such a Rb-I(2) framework to rapidly recover to its original lattice positions. In other words, within the weakly interacting cluster-framework sublattices, the hard clusters can restore the order state, whereas the soft and plastic nature of the Rb-I(2) framework leads to a largely extended relaxation time period. Furthermore, the notable steric hindrance effect, which originates from the bulky Re-S-I(1) clusters, impedes the recovery process of Rb-I(2) that must overcome high barriers and has a large timescale. As such, both timescale and spatial effects jointly contribute to the partial amorphization of Rb 6 Re 6 S 8 I 8 after pressure treatment, resulting in irreversible structural and property changes. We further consider the structural characteristics that can facilitate irreversible structural and functional transitions after pressure engineering. Some features, such as the differential sublattice rigidity revealed in this work, structural gradients, including heterojunctions and homojunctions, two-dimensional layered structures with weak interlayer interactions, superlattices, and meticulously designed structural defects, can potentially give rise to both thermodynamic and kinetic discrepancies between various domains in a material, and should be further examined. In addition, it is necessary to consider the time factor, particularly the sample history and the kinetics of compression/decompression, because the generation and conversion of metastable states is a non-traversal process developing under nonequilibrium conditions [39] , [40] , [41] .\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Crystals growth High-quality Rb 6 Re 6 S 8 I 8 single crystals were synthesized using the salt flux method [19] . High-purity RbI (99.9%), Re (99.99%), S (99.99%) powders, and I 2 spheres (99.99%) were obtained from Aladdin and combined in a molar ratio of 20:6:8:4 under an argon atmosphere. The mixture was placed in a alumina crucible, which was subsequently sealed in a quartz tube under 10 -3 Pa. The assembly was then heated to 800 °C for 10 hours and then kept at this temperature for another 10 hours before slowly cooling to 625 °C at 1 °C·h -1 . Finally, the furnace was turned off, allowing it to cool naturally to room temperature. Through mechanical separation, millimeter-scale orange Rb 6 Re 6 S 8 I 8 single crystals were obtained. Structural characterization at ambient environment The single-crystal X-ray diffraction data were collected on a Bruker D8 Venture diffractometer equipped with a Mo K α radiation source ( λ = 0.71073 Å). The structure of Rb 6 Re 6 S 8 I 8 was refined using the SHELXL21 refinement package and the Olex2 software package [42] , [43] . Using Oxford X-Max N (Oxford Instruments), we obtained morphology and confirmed the chemical composition of Rb 6 Re 6 S 8 I 8 . Manufacturing of high-pressure environment The high-pressure environment was generated using a symmetrical diamond anvil cell (DAC). Pressure calibration was performed using the ruby fluorescence method [44] , with mineral oil serving as the pressure-transmitting medium (PTM) unless otherwise specified. High-pressure synchrotron X-ray diffraction The crystal structures of Rb 6 Re 6 S 8 I 8 under various pressures were determined through in situ synchrotron XRD measurements. Experiments were conducted at beamline 15U of the Shanghai Synchrotron Radiation Facility (SSRF) using a monochromatic X-ray beam focused to dimensions of 11 µm (horizontal) × 3 µm (vertical). Two-dimensional (2D) diffraction patterns were collected and integrated into one-dimensional (1D) data using the Dioptas software package [45] . Structural refinements were performed using the Rietveld method implemented in the General Structure Analysis System-II (GSAS-II) software [46] . The elastic modulus was derived by fitting the experimental data to the third-order Birch-Murnaghan equation of state (Eq. 1 ) [47] : $$P\\\\left(V\\\\right)=\\\\,\\\\frac{{3B}_{0}}{2}\\\\left[{\\\\left(\\\\frac{{V}_{0}}{V}\\\\right)}^{\\\\frac{7}{3}}-\\\\,{\\\\left(\\\\frac{{V}_{0}}{V}\\\\right)}^{\\\\frac{5}{3}}\\\\,\\\\right]\\\\left\\\\{1+\\\\frac{3}{4}\\\\left({B}^{{\\\\prime} }-4\\\\right)\\\\left[{\\\\left(\\\\frac{{V}_{0}}{V}\\\\right)}^{\\\\frac{2}{3}}-1\\\\right]\\\\right\\\\}$$ (1) where V 0 represents the unit cell volume at ambient pressure, determined to be 3757.0(7) Å 3 , and B 0 is the bulk modulus, calculated as 11.0(3) GPa. During the fitting process, the first derivative of the bulk modulus, B ′, was fixed at 12.0. A pair of 300-µm Ia-type diamonds were employed for the measurements. High-pressure X-ray absorption spectroscopy X-ray absorption spectra (XAS) at the Re L 3 -edge were collected at the ODE beamline of the SOLEIL synchrotron radiation facility. Data were acquired in transmission mode, with energy calibration performed using a standard Re foil as a reference. Data processing and fitting were carried out using the Athena and Artemis software packages [48] . A pair of nano-polycrystalline diamond anvils with a 300-µm culet size were employed. High pressure is generated by squeezing DAC through a membrane device. High-pressure UV-visible absorption spectra A custom-designed micro-region spectra system (Gora-UVN-FL, assembled by Ideaoptics) equipped with a Mitutoyo 20× NUV objective was employed to measure UV-visible absorption spectra. A deuterium/halogen light source was used, which provided a wavelength range spanning from 300 nm to 1100 nm. The bandgap of Rb 6 Re 6 S 8 I 8 was determined using the Tauc plot method, which is based on the Eq. ( 2 ) [49] : $${\\\\alpha }^{1/2}={{{\\\\rm{A}}}}\\\\left(h\\\\nu -{E}_{{{{\\\\rm{g}}}}}\\\\right)$$ (2) where α represents the absorption coefficient, hν signifies the photon energy, E g denotes the bandgap, and A is a proportionality constant. The bandgap value can be accurately determined by plotting ( αhν ) 1/2 against hν and extrapolating the linear portion of the curve to the energy axis. The intercept provides the precise bandgap value. For the measurements, a pair of 300-μm IIa-type diamonds were used, with silicone oil serving as the PTM. High-pressure photoluminescence Static photoluminescence (PL) spectra were acquired using a 405 nm laser as the excitation source. The emission signals were collected using a NOVA refrigeration spectrometer coupled to a MIcrochrome 5 Pro CMOS detector, with a spectral resolution of 1 nm. High-pressure time-resolved photoluminescence For time-resolved PL dynamics measurements, sample was photoexcited by a 405 nm laser diode with a pulse width of 20 ps and a repetition rate of 20 kHz. All PL and time-resolved PL experiments were performed using a modified Mao-type symmetrical diamond anvil cell (DAC) equipped with IIa-type diamond anvils, each with a culet size of 300 μm. For PL imaging, the sample was excited by UV light filtered from a xenon lamp source. Silicone oil was employed as the pressure-transmitting medium (PTM). High-pressure photocurrent The high-pressure photocurrent test was conducted on a Keithley 4200A-SCS Parameter Analyzer. A xenon lamp was employed as the simulated visible light source. A symmetrical Sb-type DAC consisting of a pair of 300-μm Ia diamonds was employed. A pre-pressed stainless-steel sheet was drilled with a 280-μm hole, which was subsequently filled with cubic boron nitride (cBN) and subjected to further pressurization. The gasket was then insulated using insulating glue after drilling a 120-μm hole. Two electrodes were fabricated using Pt wires, and the photocurrent was measured under an applied voltage of 3 V for 3 cycles, with each cycle consisting of a 60 seconds light source on period followed by a 60 seconds light source off period. High-pressure Raman spectra Raman spectra were acquired utilizing a home-designed low-wavenumber Raman system equipped with a 785 nm laser. The Raman signals were collected by an IsoPlane 320 spectrograph (Princeton Instruments) coupled to a PyLon100BR_eXcelon detector (Princeton Instruments) cooled with liquid nitrogen. High-pressure middle-infrared transmission spectra The Bruker vertex 70 v FT-IR spectrometer was employed to measure the MIR transmission spectra. A symmetrical Sb-type DAC consisting of a pair of 200-μm IIa diamonds was used. The transmission signal of the chamber containing silicone oil was collected to serve as a reference signal. Rb 6 Re 6 S 8 I 8 (about 5 μm thick) covers the hole, and then the transmission signal was measured under different pressures. The pressure is calibrated by measuring the high-frequency edge of the first-order Raman band at the center of the culet [50] . High-pressure temperature dependence of resistance The RT curves were measured on physical property measurement system (PPMS, Quantum Design) using a Be-Cu piston-cylinder press containing a pair of 150-μm Ia diamonds. The experimental procedure involved pre-pressing a Be-Cu gasket, drilling a 130-μm hole, filling it with cBN, and subjecting it to further pre-pressing. Afterward, a 65-μm hole was drilled and packed with Rb 6 Re 6 S 8 I 8 . Four Pt wires were utilized to establish a connection between the copper wires of the external circuit and create a four-electrode configuration. The pressure is calibrated by measuring the high-frequency edge of the first-order Raman band at the center of the culet [50] . First-principles calculation First-principles calculations were performed within the framework of density functional theory (DFT) using the Vienna Ab Initio Simulation Package (VASP) [51] , [52] . The exchange-correlation functional was treated using the generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) parameterization optimized for solids. The valence electron configurations were described by projector augmented wave (PAW) potentials, with the following electronic states included: Rb (4 s 2 4p 6 5 s 1 ), Re (5 d 5 6 s 2 ), S (3 s 2 3p 4 ), and I (5 s 2 5p 5 ). A plane-wave basis set with a kinetic energy cutoff of 550 eV was employed, ensuring convergence of the forces on each atom to less than 0.01 electron volt per angstrom. The Brillouin zone was sampled using a Monkhorst-Pack k -point mesh with a spacing of 0.2 Å -1 . The electronic band structures are obtained from the refined structures through synchrotron XRD and the Brillouin zones are sampled along high-symmetry points. Infrared intensities and Raman off-resonant activity were calculated using density functional perturbation theory [53] , [54] , [55] , respectively. The discrete intensity points are further convolved with the Lorentz function to obtain the spectrum. The kinetic stabilities are checked by running first-principles molecular dynamic simulation (FPMD). The system is one unit cell, which is consist of 24 Re, 32 I, 24 Rb, and 32 S atoms. We first run equilibrium for 2 picoseconds (ps) at the target P - T with the NVT ensemble (constant number of atoms, volume, and temperature). Atomic trajectories were recorded with a time step of 1 fs, and the total simulation time was set to 10 ps for each pressure-temperature ( P - T ) condition. Temperature control was achieved using a Nosé–Hoover thermostat [56] . Pair-distribution function was calculated and averaged from the last 2000 steps of the trajectory. The mean squared displacement is calculated along the trajectory of atomic motions through the Python package Kinisi [57] . Pressurization treatment with large volume press The Rb 6 Re 6 S 8 I 8 powder was compacted into a cylindrical ZrO 2 cavity, which was subsequently incorporated into an Al 2 O 3 /MgO 2 octahedral structure. This octahedron was centrally situated within a cubic arrangement of eight WC anvils and then placed in the LVP (Kawai-type multi-anvil apparatus, SHTech-2000, Max Voggenreiter). The pressure was systematically increased to 15 GPa over 6 hours, sustained for 2 hours, and then gradually released to ambient pressure within 10 hours. Fabrication and measurements of photodetector devices The wafer cleaning process involved sequential treatments with detergent, acetone and isopropyl alcohol, and deionized water for 20 minutes each. Then, the wafers were exposed to UV light for 10 minutes and spin coating was performed in a vacuum glove box. A solution consisting of 15 mg LVP-treated Rb 6 Re 6 S 8 I 8 in 1.5 mL N,N-dimethylformamide (DMF) was prepared and spin-coated at 3000 rpm for 30 seconds, and then annealed at 100 °C for 10 minutes before cooling down to room temperature. Subsequently, the Pt electrode was deposited by magnetron sputtering through a mask plate in a vacuum of 9 × 10 -4 Pa using a 99.99% Pt target with a device area of 0.5 cm 2 . The optoelectronic properties of the prepared devices were characterized using a semiconductor parameter analyzer (Keithley 2400, Tektronix) and light sources with wavelengths of 367 nm and 520 nm. The lighter power density is 50 mW·cm -2 for both excitations. To determine the response rate, the LED light source was connected to a function generator (TDS2022B, Tektronix) to emit light pulses at different frequencies. In addition, the devices were imaged with specific photocurrent mapping patterns using the assembled imaging system.\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"The core research idea is to achieve ambient retention of high-pressure-induced photoelectric properties in materials by designing systems with weakly coupled sublattices that decouple under pressure and irreversibly retain their modified structural and functional characteristics upon decompression, as demonstrated in the Rb₆Re₆S₈I₈ chalcohalide compound where electrostatically coupled sublattices enable pressure-induced photocurrent generation that persists at ambient conditions for scalable optoelectronic device applications."},"entities":{"kind":"list like","value":["ambient retention","high-pressure-induced photoelectric properties","weakly coupled sublattices","structural and functional characteristics","Rb₆Re₆S₈I₈ chalcohalide compound","electrostatically coupled sublattices","pressure-induced photocurrent generation","scalable optoelectronic device applications","decompression","irreversible retention","material design","pressure decoupling","functional materials","ambient conditions","photoelectric material systems"],"string":"[\n \"ambient retention\",\n \"high-pressure-induced photoelectric properties\",\n \"weakly coupled sublattices\",\n \"structural and functional characteristics\",\n \"Rb₆Re₆S₈I₈ chalcohalide compound\",\n \"electrostatically coupled sublattices\",\n \"pressure-induced photocurrent generation\",\n \"scalable optoelectronic device applications\",\n \"decompression\",\n \"irreversible retention\",\n \"material design\",\n \"pressure decoupling\",\n \"functional materials\",\n \"ambient conditions\",\n \"photoelectric material systems\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1.","title":"Mao, H.-k & Hemley, R. 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Res.: Solid Earth91, 4673 (1986).","link":"https://doi.org/10.1029%2FJB091iB05p04673"},{"idx":"45.","title":"Prescher, C. & Prakapenka, V. B. DIOPTAS: a program for reduction of two-dimensional X-ray diffraction data and data exploration.High. Press. Res.35, 223 (2015).","link":"https://doi.org/10.1080%2F08957959.2015.1059835"},{"idx":"46.","title":"Toby, B. H. & Von Dreele, R. B. GSAS-II: the genesis of a modern open-source all purpose crystallography software package.J. Appl. Crystallogr.46, 544 (2013).","link":"https://doi.org/10.1107%2FS0021889813003531"},{"idx":"47.","title":"Katsura, T. & Tange, Y. A simple derivation of the Birch-Murnaghan equations of state (EOSs) and comparison with EOSs derived from other definitions of finite strain.Minerals9, 745 (2019).","link":"https://doi.org/10.3390%2Fmin9120745"},{"idx":"48.","title":"Ravel, B. & Newville, M. ATHENA and ARTEMIS: interactive graphical data analysis using IFEFFIT.Phys. Scr.115, 1007–1010 (2005).","link":"https://doi.org/10.1238%2FPhysica.Topical.115a01007"},{"idx":"49.","title":"Tauc, J. Optical properties and electronic structure of amorphous Ge and Si.Mater. Res. Bull.3, 37–46 (1968).","link":"https://doi.org/10.1016%2F0025-5408%2868%2990023-8"},{"idx":"50.","title":"Akahama, Y. & Kawamura, H. Pressure calibration of diamond anvil Raman gauge to 310 GPa.J. Appl. Phys.100, 043516 (2006).","link":"https://doi.org/10.1063%2F1.2335683"},{"idx":"51.","title":"Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple.Phys. Rev. Lett.77, 3865–3868 (1996).","link":"https://doi.org/10.1103%2FPhysRevLett.77.3865"},{"idx":"52.","title":"Perdew, J. P. et al. Restoring the density-gradient expansion for exchange in solids and surfaces.Phys. Rev. Lett.100, 136406 (2008).","link":"https://doi.org/10.1103%2FPhysRevLett.100.136406"},{"idx":"53.","title":"Giannozzi, P. & Baroni, S. Vibrational and dielectric properties of C60 from density-functional perturbation theory.J. Chem. Phys.100, 8537–8539 (1994).","link":"https://doi.org/10.1063%2F1.466753"},{"idx":"55.","title":"Porezag, D. & Pederson, M. R. Infrared intensities and Raman-scattering activities within density-functional theory.Phys. Rev. B54, 7830–7836 (1996).","link":"https://doi.org/10.1103%2FPhysRevB.54.7830"},{"idx":"56.","title":"Evans, D. J. & Holian, B. L. The Nose-Hoover thermostat.J. Chem. Phys.83, 4069–4074 (1985).","link":"https://doi.org/10.1063%2F1.449071"}],"string":"[\n {\n \"idx\": \"1.\",\n \"title\": \"Mao, H.-k & Hemley, R. J. Ultrahigh-pressure transitions in solid hydrogen.Rev. Mod. Phys.66, 671–692 (1994).\",\n \"link\": \"https://doi.org/10.1103%2FRevModPhys.66.671\"\n },\n {\n \"idx\": \"2.\",\n \"title\": \"Philip, D.-S., Ross, T. H. & Eugene, G. 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Local elastic expansion model for viscous-flow activation energies of glass-forming molecular liquids.Phys. Rev. B53, 2171 (1996).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.53.2171\"\n },\n {\n \"idx\": \"39.\",\n \"title\": \"Mel’Cuk, A. I., Ramos, R. A., Gould, H., Klein, W. & Mountain, R. D. Long-lived structures in fragile glass-forming liquids.Phys. Rev. Lett.75, 2522 (1992).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.75.2522\"\n },\n {\n \"idx\": \"40.\",\n \"title\": \"Debenedetti, P. G. & Stillinger, F. H. Supercooled liquids and the glass transition.Nature410, 259–267 (2001).\",\n \"link\": \"https://doi.org/10.1038%2F35065704\"\n },\n {\n \"idx\": \"41.\",\n \"title\": \"Angell, C. A. Formation of glasses from liquids and biopolymers.Science267, 1924–1935 (1995).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.267.5206.1924\"\n },\n {\n \"idx\": \"42.\",\n \"title\": \"Sheldrick, G. M. Crystal structure refinement with SHELXL.Acta Crystallogr. C.71, 3–8 (2015).\",\n \"link\": \"https://doi.org/10.1107%2FS2053229614024218\"\n },\n {\n \"idx\": \"43.\",\n \"title\": \"Dolomanov, O. V., Bourhis, L. J., Gildea, R. J., Howard, J. A. K. & Puschmann, H. OLEX2: a complete structure solution, refinement and analysis program.J. Appl. Crystallogr.42, 339–341 (2009).\",\n \"link\": \"https://doi.org/10.1107%2FS0021889808042726\"\n },\n {\n \"idx\": \"44.\",\n \"title\": \"Mao, H.-k., Xu, J. & Bell, P. M. Calibration of the ruby pressure gauge to 800 kbar under quasi‐hydrostatic conditions.J. Geophys. Res.: Solid Earth91, 4673 (1986).\",\n \"link\": \"https://doi.org/10.1029%2FJB091iB05p04673\"\n },\n {\n \"idx\": \"45.\",\n \"title\": \"Prescher, C. & Prakapenka, V. B. DIOPTAS: a program for reduction of two-dimensional X-ray diffraction data and data exploration.High. Press. Res.35, 223 (2015).\",\n \"link\": \"https://doi.org/10.1080%2F08957959.2015.1059835\"\n },\n {\n \"idx\": \"46.\",\n \"title\": \"Toby, B. H. & Von Dreele, R. B. GSAS-II: the genesis of a modern open-source all purpose crystallography software package.J. Appl. Crystallogr.46, 544 (2013).\",\n \"link\": \"https://doi.org/10.1107%2FS0021889813003531\"\n },\n {\n \"idx\": \"47.\",\n \"title\": \"Katsura, T. & Tange, Y. A simple derivation of the Birch-Murnaghan equations of state (EOSs) and comparison with EOSs derived from other definitions of finite strain.Minerals9, 745 (2019).\",\n \"link\": \"https://doi.org/10.3390%2Fmin9120745\"\n },\n {\n \"idx\": \"48.\",\n \"title\": \"Ravel, B. & Newville, M. ATHENA and ARTEMIS: interactive graphical data analysis using IFEFFIT.Phys. Scr.115, 1007–1010 (2005).\",\n \"link\": \"https://doi.org/10.1238%2FPhysica.Topical.115a01007\"\n },\n {\n \"idx\": \"49.\",\n \"title\": \"Tauc, J. Optical properties and electronic structure of amorphous Ge and Si.Mater. Res. Bull.3, 37–46 (1968).\",\n \"link\": \"https://doi.org/10.1016%2F0025-5408%2868%2990023-8\"\n },\n {\n \"idx\": \"50.\",\n \"title\": \"Akahama, Y. & Kawamura, H. Pressure calibration of diamond anvil Raman gauge to 310 GPa.J. Appl. Phys.100, 043516 (2006).\",\n \"link\": \"https://doi.org/10.1063%2F1.2335683\"\n },\n {\n \"idx\": \"51.\",\n \"title\": \"Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple.Phys. Rev. Lett.77, 3865–3868 (1996).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.77.3865\"\n },\n {\n \"idx\": \"52.\",\n \"title\": \"Perdew, J. P. et al. Restoring the density-gradient expansion for exchange in solids and surfaces.Phys. Rev. Lett.100, 136406 (2008).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.100.136406\"\n },\n {\n \"idx\": \"53.\",\n \"title\": \"Giannozzi, P. & Baroni, S. Vibrational and dielectric properties of C60 from density-functional perturbation theory.J. Chem. Phys.100, 8537–8539 (1994).\",\n \"link\": \"https://doi.org/10.1063%2F1.466753\"\n },\n {\n \"idx\": \"55.\",\n \"title\": \"Porezag, D. & Pederson, M. R. Infrared intensities and Raman-scattering activities within density-functional theory.Phys. Rev. B54, 7830–7836 (1996).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.54.7830\"\n },\n {\n \"idx\": \"56.\",\n \"title\": \"Evans, D. J. & Holian, B. L. The Nose-Hoover thermostat.J. Chem. Phys.83, 4069–4074 (1985).\",\n \"link\": \"https://doi.org/10.1063%2F1.449071\"\n }\n]","isConversational":false}}},{"rowIdx":199,"cells":{"unique_id":{"kind":"string","value":"s41467-024-47018-9"},"subfield":{"kind":"string","value":"Nanoscience and technology"},"title":{"kind":"string","value":"Rewritable printing of ionic liquid nanofilm utilizing focused ion beam induced film wetting"},"abstract":{"kind":"string","value":"Manipulating liquid flow over open solid substrate at nanoscale is important for printing, sensing, and energy devices. The predominant methods of liquid maneuvering usually involve complicated surface fabrications, while recent attempts employing external stimuli face difficulties in attaining nanoscale flow control. Here we report a largely unexplored ion beam induced film wetting (IBFW) technology for open surface nanofluidics. Local electrostatic forces, which are generated by the unique charging effect of Helium focused ion beam (HFIB), induce precursor film of ionic liquid and the disjoining pressure propels and stabilizes the nanofilm with desired patterns. The IBFW technique eliminates the complicated surface fabrication procedures to achieve nanoscale flow in a controllable and rewritable manner. By combining with electrochemical deposition, various solid materials with desired patterns can be produced."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"Programmable control of fluid motion over open solid surfaces at small scales [1] , [2] are crucial for countless technological applications such as printing [3] , [4] , biosensing [5] , [6] , energy generating [7] , [8] , air water harvesting [9] , and chemical synthesis [10] etc. The predominant methods resort to either topographical or chemical gradients fabricated on solid surfaces, such as micro(nano)channels [11] , [12] , bioinspired surface structures [13] , [14] , [15] , [16] , [17] , [18] , [19] , chemical modifications [20] , [21] , etc. However, such methods permanently alter the solid substrate. Once fabrication finished, the flow pattern can barely be changed. Moreover, the cost of flow control increases significantly with the complexity of surface modifications. To improve the controllability and flexibility, external-field-stimuli have been recently adopted including temperature [22] , [23] , light [24] , [25] , magnetic [26] , [27] or electric fields [4] , [28] , [29] . The current external-field-stimuli methods mainly resort to the field induced surface-tension-gradients (STGs), which works well for system size below the capillary length ( \\({\\lambda }_{{{{{{\\rm{capillary}}}}}}} \\sim \\sqrt{\\gamma /\\rho g} \\sim {10}^{-3}{{{{{\\rm{m}}}}}}\\) ) since the surface energy dominates the free energy of micrometer systems. However, the spatial resolutions of such methods are restricted at microscale and fail to achieve nanoscale flow control. For instances, the reported thermal capillary films [23] , [24] , [25] have thickness above 5 μm and minimum line width about 30 μm. More importantly, the finger instability is usually inevitable [30] in STG based methods and jeopardizes the patterning performances. Actually, when the liquid film thickness approaches nanometer scale, the interfacial overlapping dominates the system free energy [31] , [32] , [33] instead of surface tension. Such phenomenon usually termed as disjoining pressure, arising from intermolecular forces, has been employed as a simplified mechanical description of liquid nanofilm [34] , [35] , [36] . Therefore, to achieve patterning flow at nanoscale, the liquid film thickness needs to reach nanometer to harness the disjoining pressure to stabilize the film pattern, and also requires the spatial resolution of external-field-stimuli to attain nanometer size. Compared with thermal [25] and pyroelectrical [26] , [37] fields, electron/ion beams which can be readily focused to sub-nanometer size [38] , [39] emerge as a promising option of external-field-stimulus. For example, 120 keV electron beam has been reported to induce the stick-slip motion of water nanodroplets [39] . However, the flow speeds were slow (around 3−20 nm/s), the travel lengths were short (around 189 nm), and more importantly, the flow directions and trajectories were uncontrollable while the physical mechanism remained ambiguous. Despite of these, their results still revealed that the interaction between electron/ion beams and liquid is a prospective candidate for liquid maneuvering and can be harnessed for open surface nanofluidics. Here, we present an unexplored concept of open surface nanofluidics called ion beam induced film wetting (IBFW) and achieve the nanoscale patterning of ionic liquid film without resorting to any surface fabrications or special electric circuits. The Helium focused ion beam (HFIB) is generated by a state-of-the-art Helium ion microscope [38] (HIM, ORION NanoFab, Zeiss), which can provide a beam spot of diameter 0.5 nm, and a wide range of dose densities. Since the HIM only operates in an ultra-high vacuum, we select ionic liquids (ILs) as the liquid material due to its ultra-low vapor pressure. We investigate the interaction between the HFIB and IL under different dose densities and discover an unreported liquid inducing mode of HFIB. The liquid inducing mode is attributed to electrical field induced ion emission and disjoining pressure which propels and stabilizes the liquid film pattern. Based on the IBFW inducing mode, we develop a nano-printing technique of IL, with film thickness down to 20−30 nm, minimal line width about 100 nm and corner radius down to 20 nm, and compare its performances with the reported methods. Besides, ILs are also known for their unique properties such as wide electrochemical potential window, high ionic conductivity, low toxicity and thermostability. These features make ILs increasingly important as electrolytes for lithium battery [40] , [41] and electrodepositions of various materials ranging from metal nanoparticles [42] , [43] , metal organic complexes [44] to conducting polymer films [45] . We further demonstrate the IBFW as a versatile tool for various application fields including gas sensing circuit, in-situ chemical reaction chip, and electrochemical deposition of solid materials with desired patterns. The simplicity and versatility of IBFW technique suggests prospect in a range of liquid manipulation applications. By combining with electrochemical procedures, such technique can not only produce patterned liquid film but also solid materials which reveals possibility in nano-transistors fabrication [46] , energy devices [40] and immunosensor circuit printing [47] . We expect this technique can open a new avenue for applications in nano-printing and nano-circuit manufacturing.","references":{"indices":[1,2,3,4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47],"infos":[{"idx":"1","title":"Dai, H., Dong, Z. & Jiang, L. Directional liquid dynamics of interfaces with superwettability.Sci. Adv.6, eabb5528 (2020).","link":"https://doi.org/10.1126%2Fsciadv.abb5528"},{"idx":"2","title":"Liu, M., Wang, S. & Jiang, L. Nature-inspired superwettability systems.Nat. Rev. Mater.2, 1–17 (2017).","link":"https://doi.org/10.1038%2Fnatrevmats.2017.36"},{"idx":"3","title":"Lee, W. et al. Universal assembly of liquid metal particles in polymers enables elastic printed circuit board.Science378, 637–641 (2022).","link":"https://doi.org/10.1126%2Fscience.abo6631"},{"idx":"4","title":"Ferraro, P., Coppola, S., Grilli, S., Paturzo, M. & Vespini, V. 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We systematically investigate the interaction between HFIB and the IL employing different combinations of dwell time, τ , and scan spot pixel spacing, s , at constant beam current, I = 1 pA (Fig. 1a ). When dose density \\(D=I\\cdot \\tau \\cdot {s}^{-2} < 2.5{{{{{\\rm{ions}}}}}}\\cdot {{{{{{\\rm{nm}}}}}}}^{-2}\\) , the HFIB exerts negligible influence on both solid and liquid samples which is the imaging mode of HIM. At extremely high dose density ( \\(D > 500{{{{{\\rm{ions}}}}}}\\cdot {{{{{{\\rm{nm}}}}}}}^{-2}\\) ), the HFIB decomposes/etches the samples and Helium bubbling may occur (Supplementary Fig. 1 ). However, at moderate dose density range ( \\(2.5{{{{{\\rm{ions}}}}}}\\cdot {{{{{{\\rm{nm}}}}}}}^{-2}\\le D < 200{{{{{\\rm{ions}}}}}}\\cdot {{{{{{\\rm{nm}}}}}}}^{-2}\\) ), the ILs are induced to flow into the irradiated area while the solid substrates remain intact (Supplementary Note 1 ) which is the liquid inducing mode of HIM. After the HFIB with appropriate parameters (Supplementary Table 1 ) scans outwards from the contact line (CTL) of reservoir (Fig. 1b ), IL flows through local protrusions of CTL and branches off to form hierarchical rivulets (Supplementary Fig. 2 , Supplementary Movie 1 ). The liquid inducing mode of HIM has not been reported to the best of our knowledge, and can be harnessed to develop a nano-printing technique. Fig. 1: Helium focused ion beam liquid inducing mode and working procedure of ion-beam-induced-film-wetting nano-printing. a Phase diagram of three working modes under different HFIB dose density, manipulated by the dwell time and the scan spot density. The background color map shows the dose density ranges from 10 −2 −10 4 ions/nm 2 . The horizontal axis corresponds to the reciprocal of the square of the scan spot spacing, s −2 , which represents the number of scan spots within 1 nm 2 . The imaging mode exerts no influence to samples (blue squares); the inducing mode can induce directional liquid flow (red circles); and the damaging mode can etch samples and induce Helium bubbling (cyan triangle). b The hierarchical branched flow pattern of ionic liquid after a frame of HFIB scan. The IL will eventually fill the entire irradiated area in 30 s. c Contact line region of the IL reservoir and nano-patterning-and-visualization-engine pattern design. Left is a schematic diagram, and right is an HIM image. The white arrow indicates the pattern direction and area in NPVE. The dashed yellow box represents the CTL region chosen for HFIB scan. d The rectangular film pattern (50 μm × 1 μm) fabricated by IBFW. The insets are the details to manifest small corner radius and well reproduction of the design, the yellow box edges are 2 μm. e The IBFW fabricated ‘PKU COE’ pattern demonstrates its potential in on-demand printing of complex liquid film networks. f The minimal line width that IBFW can fabricate is around 106 nm. Full size image As an example of IBFW printing technology (Fig. 1c, d ), we induce small amount of liquid from the reservoir to fabricate a rectangular film pattern (50 μm × 1 μm). Step 1: Sample preparation. A SiO 2 wafer is placed horizontally as the substrate, Fig. 1c . No surface fabrication or physical mask is needed. A 0.1 μL droplet of IL is deposited on substrate as the liquid reservoir. The contact angle of the IL on the substrate is approximately 60° (Supplementary Fig. 3 ). Then the sample is transferred into the chamber of HIM. Step 2: CTL identification and pattern design. Under the imaging mode of the HIM, the CTL of the droplet is identified. As the yellow dotted box marked in Fig. 1c , part of the CTL is chosen as the starting position of the film pattern (50 μm × 1 μm) which is drawn employing software Nano Patterning and Visualization Engine (NPVE). The NPVE rasterize-fills the pattern with scan spot arrays (Supplementary Fig. 4 ). Step 3: HFIB irradiation and liquid inducing. Employing inducing mode of the HIM, the HFIB scans the designed area. The ion beam vector-scans the spot arrays point by point with serpentine scan style and specific beam parameters. Noteworthily, some other conditions other than beam parameters also influence the IBFW performances. The starting position of scan should be located close to or stuck into the CTL, and the scan direction should be outwards from the reservoir, while other configurations cannot achieve the best inducing performance (Supplementary Fig. 5 and Supplementary Movie 2 , 3 ). The speed differences between liquid film propagation and ion beam scanning can explain the scan direction effects on IBFW inducing performance (Supplementary Movie 4 ). Also, the flood gun should be turned on. Since the helium ions carry positive charges, continuous irradiation would cause too much positive charges in the sample surface. The excess surface charges would alter the direction of the subsequent Helium ions and finally make the liquid film pattern tilted (Supplementary Fig. 6 ). When the point-by-point irradiation finishes, the thin liquid film is fabricated with designed pattern (Fig. 1d ). As shown in Fig. 1d , IBFW fabricated nanofilm reproduces not only the size of the designed pattern (50 μm × 1 μm), but also the straight edges and the 90° corners. Moreover, liquid film networks can be easily rewritten on the same solid substrate after the liquid is dissolved by acetone. As a result, the liquid film pathways can be reprogrammed in an on-demand manner. To further demonstrate the prospect in complex patterning of IBFW, we fabricate a liquid film pattern making up the words “PKU COE” (Peking University College of Engineering) in Fig. 1e . The word patterns are all drawn employing NPVE without physical masks. Throughout the patterning process, the liquid is induced to flow along arbitrary geometrical tracks with curved lanes, crisscross junctions, and corners to form complicated liquid networks. The liquid film well reproduces the designed pattern. The width and corner radius of the liquid film barely change during the propagation regardless of the complexity of the pattern. Figure 1f presents an example of the minimal line width, 100 nm, IBFW can achieve. By assembly of separate scan patterns sequentially, the IBFW technique can also fabricate liquid film channel up to hundreds μm in length. Apart from continuous pattern, discontinuous liquid pattern can be fabricated by the introduction of damaging mode of HFIB. As shown in Supplementary Fig. 7a PKU pattern is separated from the flow channel that connect to the droplet reservoir. Working principles of IBFW When HFIB irradiates at or near the CTL of droplet, the Helium ions generate special charges distribution [48] in the SiO 2 substrate, the positive surface charges induce the primary ion emission from the IL reservoir (Fig. 2a ). HFIB exhibits two distinguishable features compared with electron beam and other ion beams [48] , [49] . Firstly, the divergence of HFIB interacting with solid samples is much smaller than electron beam due to its larger mass [49] , which lead to a more localized surface charging area and consequently a programmable control over liquid flow (Supplementary Fig. 8 ). Secondly, HFIB tends to penetrate sample and induce less damaging compared with Gallium beams [48] . The penetration depth exerts significant influence on IBFW’s application potential in electrochemistry field. For example, 30 keV He beams with hundreds nm stopping range can easily penetrate a 10 nm Au and 5 nm Ti electrode deposited on SiO 2 wafer and induce patterned liquid flow without devastating effect to Au surface (Supplementary Fig. 9 ), while 30 keV Ga with stopping range less than 20 nm [48] is hard to penetrate metal films and can easily cause damage to the electrode surface. During the HFIB irradiation, He ions interact with solid atoms and excite holes-electrons in the sample, then the He ions lose kinetic energy and rest within the stopping range. Since the excited electrons in amorphous SiO 2 only survive 10 ns or less before the recombination takes place [50] , [51] , the positive charges dominate the surface charging and account for the primary anion emission. Monte Carlo simulation results [48] reveal that the lateral projection length of 30 kV He ions irradiated on SiO 2 (also SiO 2 with 10 nm Au and 5 nm Ti layers) ranges from 90 nm to 110 nm (Supplementary Fig. 9a, b ). The consequent positive charges (ions, holes) distribute over 10 2 nm in lateral direction and determine the ion emission and IBFW spatial resolution. Fig. 2: Working principles of ion-beam-induced-film-wetting. a − c Schematic diagrams of the IBFW working principle. a When helium focused ion beam irradiates the contact line, the positive surface charging induces the primary anion emission. b When HFIB ceases to scan, surface charging dissipates and the emitted anions induce the secondary cation emission. c The emitted ions from previous stages form an ultra-thin precursor film, and the consequent disjoining pressure, Π , propels and stabilizes liquid film. d The mechanical balance between the surface charge density induced electrostatic force and the surface tension of liquid-vacuum interface. The distance between HFIB scan spot and contact line is d ; the surface charging uniformly distributes over a region with length scale, l 0 ; the surface-charge exerts electrostatic force and distorts the liquid-vacuum interface, balanced by the surface tension γ ; α is the angle between substrate and the connecting line of surface-charge center and ion emission center; r * is the ion emission critical radius, and is also the vertical distance between ion emission center and substrate. Full size image When the HFIB ceases to irradiate, the surface charges dissipate due to the drainage current and the electron-hole recombination (Fig. 2b ) then the emitted anions induce the secondary cation emission. Both ions meet ahead of the CTL and form an ultra-thin precursor film with thickness comparable to ion size at the irradiated area. As a matter of fact, when the electric field of surface charge is strong enough, not only separate ions but clusters or even tiny droplets contain both ions are emitted [52] to scanned area and make up the precursor films. In both cases, the ultralow thickness of precursor film gives rise to the high disjoining pressure (10 5~6 Pa) that irrigates and thickens the precursor film until be balanced by the capillary force (10 2~3 Pa) and a continuous liquid film is formed (Fig. 2c ). The surface charging process of dielectric materials (SiO 2 for example) under the irradiation of focus ion beams (Ga + or He + ) was thoroughly discussed in literatures [53] , [54] , [55] , and the charging accumulation and dissipation is manipulated by the following factors: (1) generation of electron-hole pairs in the solid by incident ions; (2) neutralization of the incident ions by the excited free electrons; (3) sputtering of the surface atoms; (4) charging due to the secondary ion-electron emission; (5) leakage of mobile electron-hole pairs to the silicon substrate; (6) induced shallow traps by the incident ions and a consequent preferred trapping relative to the deep traps. The surface charge density (SCD) of SiO 2 at HFIB irradiation [54] can be expressed as a function of time (Supplementary Note 2 ): $$\\frac{dQ(t)}{{dt}}=P\\left(1+{\\gamma }_{e}\\right)\\cdot I\\left(t\\right)-k\\frac{Q\\left(t\\right)}{{\\epsilon }_{r}{\\epsilon }_{0}}-\\frac{7}{4}{YI}\\left(t\\right)\\cdot {\\varOmega }_{0}\\frac{Q\\left(t\\right)}{{R}_{p}}-{\\int }_{0}^{t}J(t){dt}.$$ (1) The RHS composes of 4 terms. The first is the ion incident term which represents the electron-hole pairs accumulation induced by the ion incident and secondary emission, where P is the probability factor accounts for the electron-hole recombination, γ e is the secondary electron emission yield of SiO 2 , and I ( t ) is the beam current density of HFIB. The second term is the leakage current term, where k is the conductivity of the substrate, \\({\\epsilon }_{r}\\) is the substrate relative permittivity, and \\({\\epsilon }_{0}\\) is the vacuum permittivity constant. The third term is the sputtering yield induced charge reduction, where Y is the sputtering yield acquired from SRIM simulation, Ω 0 is the atomic volume which can be estimated by the average density of SiO 2 , R p is the ions stopping range from SRIM. The last term accounts for the accumulation of emitted counterions, where J ( t ) is the ion emission rate at current SCD, which is often described as a kinetic process in which ions evaporate from liquid-vacuum interface. The emission current density reads [56] : $${j}_{e}=\\frac{J}{{\\pi}{r}^{*2}}=\\frac{{k}_{B}T}{h}\\sigma \\exp \\left(-\\frac{\\Delta G-G({{{{{{\\bf{E}}}}}}}_{{{{{{\\bf{n}}}}}}}^{{{{{{\\bf{v}}}}}}})}{{k}_{B}T}\\right),$$ (2) where j e is the current emitted per unit IL-vacuum surface area, k B is Boltzmann’s constant, T is the liquid temperature, h is Planck’s constant, σ is the local net charge density at the liquid-vacuum interface, ∆ G is the Gibbs free energy barrier for an ion to be emitted, \\({{{{{{\\bf{E}}}}}}}_{{{{{{\\bf{n}}}}}}}^{{{{{{\\bf{v}}}}}}}\\) is the local vacuum electric field normal to the interface. \\(G({{{{{{\\bf{E}}}}}}}_{{{{{{\\bf{n}}}}}}}^{{{{{{\\bf{v}}}}}}})\\) is the reduction of solvation energy barrier due to the external electric field, assumed to take the form \\(G\\left({{{{{{\\bf{E}}}}}}}_{{{{{{\\bf{n}}}}}}}^{{{{{{\\bf{v}}}}}}}\\right)=\\sqrt{\\frac{{q}^{3}{{{{{{\\bf{E}}}}}}}_{{{{{{\\bf{n}}}}}}}^{{{{{{\\bf{v}}}}}}}}{4\\pi {\\epsilon }_{0}}\\frac{{\\epsilon }_{r}-1}{{\\epsilon }_{r}+1}}\\) by the Schottky hump, where q is the ion’s charge. The solvation energy of emitted ion can be estimated by the Born model as \\(\\Delta G={\\left(\\frac{27}{4}\\pi \\right)}^{1/3}\\frac{{\\gamma }^{1/3}{q}^{4/3}{({1-\\epsilon }_{r})}^{2/3}}{{({4\\pi \\epsilon }_{0})}^{2/3}}\\) , where γ is the liquid–vacuum surface tension. Adopting the mechanical model proposed in the following paragraph, the electric field of SCD, \\({{{{{{\\bf{E}}}}}}}_{{{{{{\\bf{n}}}}}}}^{{{{{{\\bf{v}}}}}}}\\) , can help to calculate the ion emission rate and the SCD. A schematic diagram is shown in Fig. 2d to model the mechanical balance between IL surface tension and the electrostatic force exerted by the surface charge. The IL–vacuum surface is distorted by the electric field of surface charges and forms a bumping meniscus. When the meniscus is distorted to be hemispherical, the vertical component of surface tension reaches maximum. Once the surface charge continues to increase, a significant ion emission would take place during which both ions, clusters and tiny droplets may emit from the interface [52] , [56] , [57] . The distance between the scan spot (center of surface charge) and the contact line ranges from 10 0 to 10 2 nm, as long as the surface charge is strong enough to induce ion emission. The critical SCD that can induce significant ion emission depends on the distance between the starting scan spot and the reservoir CTL (more detailed deduction can be found in Supplementary Note 3 ): $${Q}_{{{{{{\\rm{surf}}}}}}}\\cong \\frac{{(d+{r}^{*})}^{2}+{l}_{0}(d+{r}^{*})}{{k}_{0}{\\cos }^{3}\\alpha }{{{{{{\\bf{E}}}}}}}^{{{{{{\\boldsymbol{*}}}}}}}.$$ (3) Where d is the distance between the scan spot and the CTL; \\({r}^{*}=\\frac{{q}^{6}\\gamma }{4{\\pi }^{2}{\\epsilon }_{0}^{3}{\\left(\\Delta G\\right)}^{4}} \\sim {10}^{-8}{{{{{\\rm{m}}}}}}\\) is the characteristic ion emission radius derived in literature [52] ; l 0 is the surface charging area length scale, which represents the lateral distribution of positive surface charges [48] ; k 0 is the Coulomb constant; α is the ion emission angle as depicted in Fig. 2d ; and \\({{{{{{\\bf{E}}}}}}}^{{{{{{\\boldsymbol{*}}}}}}} \\sim {10}^{9 \\sim 11}{{{{{\\rm{V}}}}}}/{{{{{\\rm{m}}}}}}\\) is the characteristic electric field [56] for significant ion emission. At given separation distance d , the critical SCD can be calculated by Eq. ( 3 ). By invoking Eqs. ( 1 ) and ( 2 ), the dosage of HFIB required for the critical SCD can be calculated. The relationship between critical HFIB dose density to induce IBFW, and the distance between starting scan spot and droplet CTL, d , is calculated employing Eqs. ( 1 – 3 ) with the results depicted in Fig. 3a by blue line. The beam current employed in calculation is 1 pA, scan spacing is 1 nm, and the dwell time determines the calculated dose density. Here, we assume that the critical dose density that takes to induce significant ion emission coincides with the dose to achieve IBFW. The experiments are conducted to measure the critical HFIB dose density with 1 pA beam current and 1 nm spacing to induce continuous liquid film from reservoir with the results shown in Fig. 3a by orange rhombuses. The calculation results agree with the experimental critical dose density qualitatively, and confirm our hypothesis that the surface charging induced ion emission accounts for the IBFW. Fig. 3: Theoretical analysis, experimental and simulation results of ion-beam-induced-film-wetting mechanism. a Experimental helium focused ion beam dosage density to induce ionic liquid flow (ions/nm 2 ) and calculated results of critical dose density to induce significant ion emission. The inset shows when the scan position is separated too far from the contact line, the consecutive liquid film degenerates to the local protrusion flow as shown in Fig. 1b . The scale bar is 5 μm. b IBFW induced liquid film lengths (μm) on substrates with different conductivities. Experimental film length data of distinct samples are presented as mean ± s.d. n = 5. The right ordinate represents the overall ion emission number calculated by the beam parameters and sample characters. In the horizontal axis, TOX stands for thermal oxidized, and CVD stands for chemical vapor deposited. c Molecular dynamics simulation of [EMIM][DCA] droplet (640 ion pairs) deposited on fused silica substrate going through surface charges injection and removal. The arrows indicate the most directed movements of ions: the pale blue arrows at the beginning stage represent the surface charge induced primary anion emission; the purple arrows of cations represent the emitted anions induced secondary cation emission. d IBFW film flow speed as a function of liquid film length. The single spot experiments are conducted with a line pattern of scan spots, at constant beam current 1 pA, dwell time 10 μs, spacing 1 nm. The change width experiments are conducted with rectangular pattern with constant length 20 μm and different widths, while keeping the beam current 1 pA, dwell time 2 μs, spacing 1 nm. Data of change width method are presented as mean ± s.d. n = 12. The blue shaded region is the calculated velocity range due to the range of slip lengths [58] , [59] , the deep blue line is the calculated result employing the average slip length [58] . Full size image Apart from the HFIB configuration, the substrate characteristics also influence the IBFW behavior. As shown by Eq. ( 1 ), the electrical conductivity is easy to manipulate while exerts significant effect on the liquid inducing performance. In Fig. 3b , we select 7 different solid substrates vary from well conductors to dielectrics. The orange rectangles and black triangles represent the lengths of liquid film that can be induced on different substrates under identical HFIB treatment. IL film cannot be induced on pure conducting samples such as Au and Cu, liquid extends slightly on semiconductor (Si), whereas liquid film propagates a long distance on insulated substrates e.g ., quartz and mica. Employing Eqs. ( 1 – 3 ), we can calculate the numbers of emitted ions during a single spot scan of HFIB (Supplementary Note 2 , Supplementary Fig. 10 ), and is depicted in Fig. 3b as blue circles. As can be seen, the ion emission model we propose herein can qualitatively explain the different effects of IBFW on conducting, semi-conducting and insulating substrates. Yet the IBFW effect is also influenced by other substrate situations (roughness, for example), and the differences between various insulated substrates are failed to be captured by the simplified 2D model. Not only the solid substrates, the ILs employed also influence the IBFW performances. Four kinds of ILs are tested on SiO 2 substrate with the results shown in Supplementary Fig. 11 . The results show that liquid viscosity has significant influence on IBFW liquid film, by raising the flow friction which works as a counterpart of the electrostatic driving forces. The relation between wetting status and IBFW effect, however, is ambiguous, indicating that the electrostatic forces of IBFW is not quite relevant with the wetting status. It is probably due to the fact that the IBFW electrostatic forces are much greater than the wetting van der Waals forces at the contact line. Molecular Dynamics simulation is employed to reveal the initiating stage of IBFW (Fig. 3c ). The yellow spheres are atoms of cations, and blue spheres are atoms of anions as shown in the top-left legend. The bottom-left color scale of the trajectory lines represents time steps in simulation. The primary and secondary ion emissions are identified by the trajectories of anions which are represented by the white-to-blue lines and cations represented by the red-to-purple lines. The arrows indicate the most directed movements of ions. The pale blue arrows with more unified orientation at the beginning stage represent the surface charge induced primary anion emission. The purple arrows of cations represent the emitted anions induced secondary cation emission with less orientability. The red shaded regions represent the surface area where positive charges are injected and then removed. We also observe an ultra-thin precursor film in MD simulation, see Supplementary Fig. 12 . We next verify that the disjoining pressure propels and stabilizes the nanofilm. The propagation speed of IBFW liquid film decreases monotonically with the increase of film length: $${{{{{\\bf{U}}}}}} \\sim \\frac{{h}^{2}+3{bh}}{3\\mu }\\cdot \\frac{{{{{{\\boldsymbol{\\Pi }}}}}}\\left({h}_{\\min }\\right)-\\gamma {{{{{\\boldsymbol{\\kappa }}}}}}}{L}.$$ (4) Where U is the average flow speed, h is equilibrium film thickness, b is the slip length of IL-SiO 2 interface [58] , [59] , [60] , μ is IL viscosity, Π ( h min ) is disjoining pressure at minimum film thickness h min , L is film length, γ is IL surface tension, and κ is curvature of IL-vacuum interface at the conjunction of film and reservoir (Supplementary Table. 2 , Supplementary Note 4 , Supplementary Fig. 13 ). We depict the calculation results in Fig. 3d . Since the boundary slip length of IL-SiO 2 interface depends on the combined surface conditions and ranges from 2 nm to over 16 nm [59] , we employ the lower (2 nm) and upper (16 nm) limits of slip length to give an estimation on the possible range of flow speed in Fig. 3d as the blue shaded region, and the calculation result of average value 10 nm [58] is shown by the deep blue line. The relationship between flow velocity and film length can also be measured experimentally. As shown in Supplementary Fig. 4 , the HFIB scans the designed pattern row by row, so the beam speed vertical to the CTL can be calculated as: $${{{{{{\\bf{v}}}}}}}_{{{{{{\\bf{beam}}}}}}}=\\frac{{{{{{{\\bf{s}}}}}}}_{{{{{{\\bf{vertical}}}}}}}}{{N}_{{{{{{\\rm{row}}}}}}}\\cdot \\tau+{V}_{{{{{{\\rm{refresh}}}}}}}},$$ (5) Where, s vertical is the vertical scan spacing, N row is the number of scan spots in one row, τ is the dwell time that HFIB stays at a single spot, V refresh is a small time (10 μs) that NPVE takes to reset the HFIB for next row of scan. If the vertical speed of HFIB exceeds the film velocity, the distance between the scan spot and the liquid film CTL would increases until the scan spot is too far ahead of the film and the liquid would cease to flow. The critical interaction distance with given beam parameters can be determined experimentally (Supplementary Fig. 14a ). Due to the pronounced impact of dose density on the flow velocity of liquid film (Supplementary Fig. 14b ), the flow velocity measurements are conducted under the same dose density by keeping the beam current I = 1 pA, scan spot spacing s = 1 nm and dwell time τ = 2 μs constant and only alter the width of the rectangle pattern. The pattern width controls the N row and consequently alters the beam vertical speed. By scanning rectangle patterns with same length but different widths outwards from the reservoir CTL, the vertical speed of beam can be changed at constant HFIB dose density. The IBFW film length decreases with the beam speed increasing (Supplementary Fig. 14c ), and consequently the average flow velocity of films with different length can be measured. The change width measurement results are shown in Fig. 3d by the hollow orange circles. The results of film speed at extremely long film lengths are acquired by first fabricating a long liquid film (300 μm × 10 μm, 600 μm × 10 μm, and 900 μm × 10 μm respectively) from the reservoir, then the change width measurements are conducted at the front of the long film. Since the fabrication of extreme long film can be time consuming, these data are only measured once. The other experiments are repeated for twelve times with the average value and standard error shown in Fig. 3d . The shortage of the change-width method is that the NPVE scan pattern assembling limits the maximum velocity the beam can move vertically. To overcome such limitation, we adopt single spot scan method. In which, a line pattern made up by a series of scan spots is used. The scan speed is altered by changing the vertical refresh time between each scan spot, while keeping beam current 1 pA, dwell time 10 μs and spacing 1 nm all constant. The dwell time is elongated to compensate the dose density reduction, since the scan area is influenced by neighboring scan spots in a rectangular pattern. All single spot measurements are repeated at least five times. The results of single spot scan are represented by the green triangles in Fig. 3d . The consistency between the calculation and experiments suggests that disjoining pressure can explain the propagation of IBFW film. The discrepancy at extreme long film length may be due to the HFIB irradiation history. The fabrication of film with hundreds of μm length usually takes hours of HFIB irradiation. The accumulated positive charges lead to a higher surface potential and a boundary slip length that exceeds the upper limit in literature [60] , which give rise to an unexpected higher film speed. Base on the film length-speed relationship curve (Fig. 3d ), the fabrication limit of film length for IBFW technique can be determined to fulfill specific requirements for fabrication efficiency. For example, the longest film we fabricated reaches 800 μm to 900 μm, with the film velocity decreases to 10 −2 μm/s (the in-situ chemical reaction chip in Applications section). Besides, the equilibrium film thickness predicted by the balance between disjoining pressure [61] , [62] and Young-Laplace pressure agrees with the AFM measured average film thickness (Supplementary Note 5 , Supplementary Fig. 15 ). Hence, we prove that disjoining pressure is important in the propelling and stabilizing of IBFW nanofilm. We also exclude the potential roles played by HFIB induced surface morphological or chemical modification effects and heating effect induced Marangoni flow in Supplementary Note 1 , Supplementary Figs. 9 , 16 , and 17 . More details of MD can be found in Supplementary Note 6. Since the temperature increasing effect is less than 1 K based on our calculation. The thermal energy difference, \\({k}_{b}\\Delta T \\, < \\, 1{{{{{\\rm{K}}}}}}\\cdot {k}_{b}=8.62\\times {10}^{-5}{{{{{\\rm{eV}}}}}}\\) , induced by HFIB is far less than the energy barrier for ions to overcome to evaporate from the liquid phase. The evaporation effect may also be insignificant in current experiments. Working performances of IBFW We employ atomic force microscope (AFM) to manifest the nanoscale flow control of IBFW. Figure 4a shows the front 7 μm of a 28 μm × 1 μm IL film with thickness around 30−40 nm (an average of 35.4 ± 1.7 nm). The film thickness remains unchanged along the flow path (Fig. 4a ). The film width is 1 μm, coincides with designed pattern. The liquid-air interface is much smoother than solid substrate (RMS roughness 7.9 ± 5.7 nm). The minimal line width of IBFW film reaches 106 nm (Supplementary Fig. 18 ), and is limited by the surface charges spatial distribution. If the surface charges can be trapped within a narrower spatial range, the ideal line width limitation may be comparable with film thickness. Fig. 4: Working performances of ion-beam-induced-film-wetting. a Tapping mode atomic force microscope image of the front 7 μm of a 28 μm × 1 μm ionic liquid film. The height profile at y = 4.9 μm is projected onto the x-z plane, and the height profile at x = 5.2 μm is projected onto the y-z plane. b The rewritable test results of IBFW method on the same substrate. The same micro hole is adopted to guarantee all experiments are conducted at the same place of the same substrate. The induced film lengths of a 20 μm × 1 μm scan pattern keep constant throughout 10 times of rewritable tests. c The liquid film line width and thickness compared with published results [23] , [24] . The inset show part of the IBFW liquid film imaged by AFM. The corner radiuses between the fabricated liquid film and reservoir are compared in ( d ) with the inset illustrates a IBFW film with corner radius down to 20 nm. e The flow velocities at different liquid film lengths are compared with literatures [23] , [24] . The data of IBFW are presented as mean ± s.d. n = 5. f System free energy scenario of a liquid film with different film thickness, and a comparison of different flow control strategies. The left vertical axis represents the ratio that different energy terms contribute to the total free energy. The yellow vertical line represents capillary length and the red vertical line represents overlapping length ( \\({\\lambda }_{{{{{{\\rm{overlapping}}}}}}} \\sim \\sqrt{\\frac{{A}_{{slv}}}{\\gamma }}\\) ). Different symbols represent different strategies, the IBFW pushes the limit towards nanoscale while most counterparts fall in the surface tension region. Full size image As shown in Fig. 4b , the rewritability of IBFW liquid film is tested for ten times on the same SiO 2 wafer. During the experiments, a 20 μm × 1 μm pattern is repeatedly fabricated at the same position each time. Then the substrate is transferred for AFM characterization (Supplementary Fig. 19 ) before the liquid film and droplet is rinsed for next test. Through the experiments, the film length, geometrical features and the smooth liquid-air interfaces are barely changed. Such results verify that IBFW is a rewritable open surface nanofluidic technique. The performances of IBFW are compared with published methods [23] , [24] in Fig. 4c−e . The IBFW film achieves a minimal line width and thickness which are two orders of magnitude lower than those induced by thermal and optical fields (Fig. 4c ). Moreover, the IBFW prevents the formation of menisci between the film and the reservoir, which are common in previous studies where the corners between the film and the reservoir CTL usually have a radius of 20−40 μm. However, we achieve sharp corners (90°) with a corner radius of 20 nm (the inset of Fig. 4d ). IBFW also achieves relative high flow speed without substrate confinement. As shown in Fig. 4e , the flow speed is at least 2 μm/s at 20 μm film length at 1 pA beam current, which is one order of magnitude higher than reported methods (lower than 0.5 μm/s). The disjoining pressure harnessed in current work explains the better performances of IBFW. We delineate a system free energy ratio scenario for a liquid film system with unit length/ width and thickness vary from 1 nm to 4 mm in Fig. 4f , and the film thickness and corner radius are compared with published results [16] , [18] , [23] , [24] . System free energy composes of the volumetric term (gravity, electrostatic, etc.), the surface tension term, and the disjoining pressure term. The surface tension remains fixed magnitude of \\({10}^{1}\\,{{{{{\\rm{mN}}}}}}/{{{{{\\rm{m}}}}}}\\) , and its contribution to the system is almost constant with the thickness variation. When system size exceeds capillary length, λ capillary , the volumetric term contributes most of the system free energy. For any system with characteristic length below capillary length, however, the influence of volumetric term (gravity etc.) can be neglected. At millimeter to micrometer range, the surface tension dominates, and the majority of traditional microfluidics methods belong to such region, with the spatial resolution difficult to approach nanoscale. When the system size reduces to nanometer region, the interfacial overlapping [31] , [33] emerges. The disjoining pressure increases rapidly when film thickness approaches sub-nanometer and dominates system energy. Following the capillary length, an overlapping length can be defined as \\({\\lambda }_{{{{{{\\rm{overlapping}}}}}}} \\sim \\sqrt{\\frac{{A}_{{slv}}}{\\gamma }} \\sim {10}^{-9 \\sim -8}{{{{{\\rm{m}}}}}}\\) , where A slv is the Hamaker constant of solid-liquid-vacuum interfaces system. Below such size the overlapping energy contribution far exceeds any other terms, and can be termed as the characteristic length scale for nanofluidics. Neither the traditional external stimuli or surface fabrication methods can acquire nanoscale control of flow, while the IBFW approaches the limit. Applications of IBFW As discussed previously, IL film pattern prepared by IBFW technology manifests three distinguishable features. First, the ultralow film thickness down to 30 nm indicates a high surface-volume ratio which is a key role in improving the gas sensing circuit sensitivity. Second, the capability of fabricating liquid film with desired pattern in a programmable and rewritable manner, which is important for in-situ chemical reaction and microfluidics chips. Third, the ILs are widely used in electrochemistry and reveal the possibility of transforming liquid film pattern into various solid materials ranges from organic to inorganic compounds. In Fig. 5a, b and Supplementary Fig. 20 , we present a room-humidity-sensing circuit utilizing IBFW technique. The water molecules are adsorbed by the IL then diffuse to and react at the electrode surfaces, generating a reaction current I ds , at constant voltage 1 V. The source-drain currents of a IBFW nanofilm circuit and a micrometer-size droplet circuit are measured within the same chamber with the relative humidity ranging from 40% to 70%. The current in nanofilm linearly depends on the humidity, while no significant change can be observed for the micrometer droplet. The competition between the adsorption at liquid-air interface and the diffusion within liquid circuit can explain such differences. For the droplet circuit, the long distance for the water molecules to diffuse to the electrode surface restricts the reaction rate, therefore, the change of current with humidity is not obvious. Nevertheless, the thin liquid film circuit with much shorter diffusion distance and relaxation time largely accelerates the diffusion process, which is the rate-determining step of the current experiment. Moreover, the much higher surface-volume ratio benefits the adsorption of the vapor molecules and further improves the reaction current sensitivity. Both the sensitivity and response speed are greatly enhanced due to the size effect endowed by IBFW nano film. The simple device presented here can manifest the feasibility of the IBFW circuit in sensing circuit manufacturing. Fig. 5: Application examples of ion-beam-induced-film-wetting. a , b Proof-of-concept gas sensing prototype. a Optical image of IBFW film that connects two droplets settled on Au electrodes. b Results of IBFW nano-circuit humidity sensor compared with a micrometer droplet circuit. The inset shows the schematic of the IBFW humidity sensor. The dash lines are the linear fitting results of drainage-source currents of microdroplet and nanofilm. The voltage exerted is 1 V. c In-situ chemical reaction chip. The left is the schematics, crosshair channels connect four droplets which are used to inject reagent water solutions. 0.1 μL of 0.1 M NaSCN, 0.1 mM Co 2+ , 0.1 mM Fe 3+ and 0.1 mM Cu 2+ are injected clockwise into four droplets. The inset on the bottom-left shows the microchip on fingertip, with red circle shows the four droplets, the edge of the SiO 2 substrate is 10 mm. The right part is the optical image of the reaction chip that has been stored in vacuum chamber for 48 h after the injection, the purple boxes are 80 μm × 80 μm. d Schematics of a three-electrode electrochemistry experiment. Analytes are dissolved in the droplet with IBFW film pattern printed onto the working electrode. 10 nm Au and 5 nm Ti serve as the working electrode, Pt probe and silver-plated probe stuck into the droplet are the counter and pseudo-reference electrodes respectively. e Optical images of the patterned solid particles deposited on Au electrode surface. The upper part shows Ag particles (20 μm × 3 μm rectangle with PKU letters) deposited at −0.2 V. The lower part shows blue AgTCNQ particles (50 μm × 10 μm) deposited at −0.1 V. Both are potentiostatically deposited for 180 s. Cyclic-voltammograms of ( f ) 10 mM Ag + ([EMIM][NTf 2 ]) and ( g ) 6 mM Ag + and 5 mM TCNQ ([EMIM][NTf 2 ]) solution. f The reduction peak appears at −0.295 V, with peak current decreases with cycles. g The first reduction peak (Ag + to Ag) appears at 0.095 V. The +300 mV shift at present of TCNQ is consistent with literature [44] . All optical images are acquired by Leica DM4B microscope. Full size image To demonstrate the potential of IBFW for microfluidics chip fabrication, we design an in-situ chemical reaction micro-chip. In Fig. 5c , the schematic diagram shows a crosshair shaped micro fluid channel connects four separated droplets in four directions. Four square expansion windows are made on each part of the channel for the convenience of observation. The bottom-left inset shows the chip with four droplets on finger-tip. After the IBFW fabrications, 0.1 μL of sodium thiocyanate solution (NaSCN 0.1 M in deionized water) is injected into the top droplet, and serves as the colorimetric reagent for the detection of and in-situ reaction with different metal ions. After the injection of NaSCN, 0.1 μL of 0.1 mM Fe 3+ solution, 0.1 mM Cu 2+ solution and 0.1 mM Co 2+ are injected into the bottom, left and right droplets respectively. The microchip is rested in atmosphere for 20 min for the metal ions fully diffuse into the channels and react with SCN - within different square windows with the ion names printed previously. Then the sample is transferred into vacuum chamber for 48 h to diminish the water content in the solution system, which will alter the hydration status of the metal ion complexes and improve the colorimetric visibility. The red complex Fe(SCN) 3 deposits in the bottom window. The gray deposition in the left window is complex Cu(SCN) 2 . And the blue deposition in the right window is complex Co(SCN) 2 . As shown in Fig. 5c , the SCN − participates into 3 different reactions within several hundreds of micrometers flow channel. The time series pictures are shown in Supplementary Fig. 21 , and the liquid film patterns remain unchanged during the experiments which last for over one month. The IBFW fluid channel exhibits great stabilities against vacuum/air transferring, the injection of solutions into droplet reservoir, and gravity. Such behavior demonstrates the robustness of the IBFW liquid film. More importantly, all reagents are dissolved in deionized water then injected into the IL droplets and diffuse into the IL flow channel. These experiments demonstrate that IBFW liquid film pattern can act as a stable flow channel for later injection of analytes dissolved in water, ethanol, and various molecular liquids due to the amphiphilicity of ILs. Such results greatly broaden the potential application fields for IBFW. Due to their unique properties, ILs have been proved to be an important category of solvent and electrolytes. Here we demonstrate that, by further combining with electrochemical procedure, the IBFW also manifests the capability of transforming liquid film pattern into various solid materials. Figure 5d shows the schematics of a three-electrode electrochemistry experiment. A droplet of 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide ([EMIM][NTf 2 ]) serves as the solvent of possible analytes, which are silver bis (trifluoromethylsulfonyl)imide (Ag[NTf 2 ]) or a mixture of Ag + and 7,7,8,8-tetracyanoquinodimethane (TCNQ) in our experiments. The liquid film pattern is fabricated by the IBFW technology onto the thin gold electrode and would be subsequently transformed into nanoparticles. A 10 nm Au and 5 nm Ti film deposited onto SiO 2 serves as the working electrode and is connected to the workstation by a Pt probe. As revealed by the MC simulation results (Supplementary Fig. 9 b), the He ions with vertical stopping-range exceeds 290 nm can easily penetrate the 15 nm metal film and deposit positive charges into the 300 nm SiO 2 layer underneath. Therefore, the IBFW can be achieved on an ultra-thin metal film deposited on insulating substrate, and is not contradict with the conclusion that pure conducting substrates lead to the failure of IBFW shown in Fig. 3b . The counter electrode is a Pt probe emerged in IL, and a silver-plated probe serves as the pseudo reference electrode. In Fig. 5e , we present an example of solid particles deposited from IBFW liquid film. The upper part shows silver nanoparticles with designed pattern (a 20 μm × 3 μm channel and PKU letters) potentiostatically deposited at −0.2 V (vs. Ag) for 180 s onto the gold electrode surface. And the lower part shows blue AgTCNQ particles on gold surface make up a 50 μm × 10 μm rectangular pattern deposited at −0.1 V (vs. Ag). Noteworthily, the electrowetting phenomenon is ubiquitous in IL-electrode systems with contact angle hysteresis ranges from several to tens of degrees [63] . When the voltage applied between the IL and electrode surface is large enough to overcome contact angle hysteresis, the contact line of the IL droplet will be shifted and the IBFW patterns will be jeopardized. According to ref. [64] , a negative bias voltage exceeds −1 V would induce significant contact angle decreases for [EMIM][NTf 2 ] on gold with contact line spreading forwards, which is also observed in our experiments. A negative voltage smaller than −0.5 V, on the contrary, does not influence the contact angle obviously. Therefore, the deposition voltages adopted in current work does not influence the contact line position or the IBFW liquid film. Four cycles of cyclic voltammogram (CV) of 10 mM Ag + in IL is shown in Fig. 5f . The reduction peak of Ag + takes place at −0.29 V (vs. Ag) with the peak current decreases as the scan cycles increases. We believe that the micro litter droplet with limited analyte dissolved lead to such results. The Ag + concentration decreases quickly after each cycle of CV scan, and the electron transfer is slower for the oxidation of Ag metal. In Fig. 5g , we test the CV curves of five scan cycles of IL droplet with 6 mM Ag + and 5 mM TCNQ. Two reduction peaks can be distinguished, one at 0.095 V (vs. Ag), and the other at −0.21 V (vs. Ag). The first peak corresponds to the reduction of Ag + to Ag 0 (metal) [44] , and the second peak is related to the formation of AgTCNQ complex (solid). Finally, we conceptually validate that IBFW technology is capable of transferring liquid film pattern into various solid materials and reveal the possibilities can be produced by combining IBFW with electrochemical procedures. The three-electrode experiment configuration, example of open-circuit-potential V-t curve and potentiostatic deposit I-t curve are shown in Supplementary Fig. 22 . As summarized in Fig. 6 , the IBFW technology manifests several intriguing features that can be harnessed for a variety of application fields. The IBFW technique can fabricate patterned ILs film with 30 nm thickness, 100 nm spatial resolution and over hundreds of μm film length on insulating substrates (or coated with conducting metal films with 10 1 nm thickness). The surface-volume ratio endowed by the nanometer scale thickness can largely enhance the sensitivity of the IBFW film, and can be utilized in gas sensing circuit. The good solubility and biocompatibility of ILs make them suitable for the dissolve of various analytes. The IBFW film also exhibits robustness against air environment exposure, gravity, and physical contact to droplet reservoir. Such features reveal that the IBFW film can act as stable flow channel for the analyte solutions injected to the reservoir, and can largely simplify the fabrication procedures of micro/nanofluidic chips. Last but not least, IBFW liquid film with wide potential window can be combined with electrochemical procedures and the patterned liquid film can be transformed into different solid particles. Such results demonstrate the IBFW as a versatile tool for both nanofluidics and liquid/solid materials printing. Fig. 6: Application potential of ion-beam-induced-film-wetting technology. The distinct features of ionic liquid and IBFW technique manifest potential application fields. The nanometer scale film thickness makes IBFW suitable for gas sensing circuit manufacturing. The good solubility of IL and the on-demand patterning capability make IBFW suitable for nanofluidic and reaction chip fabrication. The wide potential window and stability of IL reveal possibility in transforming IBFW liquid film pattern into various solid materials. Full size image In summary, we have demonstrated a novel strategy to control ionic liquid flow on dielectric substrates at nanoscale leveraging a novel Ion Beam induced Film Wetting mechanism. Distinct from the prevailing approaches, IBFW manipulates liquid film with minimal modification to both solid substrates and liquid. The performances of IBFW are manifested and are imparted by the utilizing of long-range intermolecular forces to stimulate and stabilize the nanofilm. IBFW can be applied to fabricate rewritable liquid circuits with intricate desired patterns for sensing and reaction applications. The combination of electrochemistry and IBFW can even transfer the liquid film pattern into solid ranges from organic to inorganic materials. Our findings will open a new avenue for versatile application fields such as on-demand manufacture of nano circuit, in-situ combinational chemistry, efficient environmental gases sensing and detection."},{"section":"Methods","content":"IBFW experiment The fabrication details of the experiments used SiO 2 wafers are described below. An amorphous SiO 2 layer with a thickness of approximately 3 μm is deposited by plasma-enhanced chemical vapor deposition (PECVD) on a 500 μm thickness quartz substrate surface. Prior to the experiment, chips are cleaned with acetone, ethanol, and ultrapure water sequentially with ultrasonication. To remove any surface organic pollutants, the wafers are annealed in a tubing furnace at 320 °C with a mixture gas flow of hydrogen and argon for at least 3 h, a SUNJUNE oxygen plasma cleaner can also fulfill the purpose. The ionic liquids used in experiments are bought from Lanzhou Greenchem ILs, LICP. Prior to the experiment, the ionic liquid is dried in a vacuum chamber at 80 °C for 24 h to remove the dissolved water. The tested ILs include 1-ethyl-3-methylimidazolium dicyanamide [EMIM][DCA] (98%), 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide [EMIM][NTf 2 ] (98%), 1-butyl-3-methylimidazolium tetrafluoroborate [BMIM][BF 4 ] (98%), 1-ethyl-3-methylimidazolium tetrafluoroborate [EMIM][BF 4 ] (95%) purchased from Tianjin Heowns. A drop of ionic liquid (~100 μm diameter) is deposited by the capillary tube on the top surface of the SiO 2 substrate acting as a liquid reservoir. Then, the sample is placed in a specimen holder and transferred into the vacuum chamber of a Zeiss helium ion microscope. The images are secondary electron images acquired by the raster scans of 30 kV He focused ion beam. The proof-of-concept and the rewritable tests of IBFW technology employed one single PECVD SiO 2 wafer with preparation procedures discussed previously and a micro droplet of [EMIM][DCA] ionic liquid was deposited on the silica wafer repeatedly. After sample preparation, the solid substrate with liquid reservoir settled on the top is transferred into the HIM chamber. The CTL of droplet reservoir is identified under imaging mode, and NPVE software is employed to fabricate a 20 μm × 1 μm film pattern with the flood gun turned on to guarantee the fabricated film reproduces the pattern (the reason is discussed in Supplementary Fig. S8 ). After IBFW fabrication, the sample is transferred and the film pattern is characterized under the tapping mode of AFM. Once characterization is done, the solid substrate is merged into a beaker filled with acetone and placed inside a fume cupboard for 24 h. After dissolving the IL droplet and film with acetone, an ultrasonic cleaner and deionized water are employed to eliminate the residual acetone of the substrate. Such procedures were repeated for ten times in the rewritable tests. Wetting status and liquid viscosity effect on IBFW The liquid viscosity influences on the IBFW performances are tested with four different ILs, [EMIM][DCA], [EMIM][BF 4 ], [BMIM][PF 6 ] and [EMIM][NTf 2 ]. Contact angles of the 4 ILs on three substrates PECVD SiO 2 , TOX SiO 2 , and 10 nm Au 5 nm Ti on 300 nm SiO 2 . The contact angle measurements are conducted with a homemade platform, compose of a commercialized CCD camera (SZ-CTV, OLYMPUS), a light source and a customized metal sample platform which can adjust height and tilting angle. 1 μL of ILs droplet is deposited onto the solid surface with 1 mm capillary tube. Molecular dynamics simulations The molecular dynamics simulation details can be found in Supplementary Note 6 . Gas sensing liquid nano-circuit The IL nanofilm circuit for gas sensing is fabricated as follow. Electrodes (Ti/Au, 5/50 nm) are deposited on doped Si wafer covered with 300 nm of SiO 2 by means of lithography, electron beam evaporation, and lift off procedure. Two droplets of [EMIM][DCA] IL with radius ~250 μm are deposited on the electrodes acting as the liquid reservoirs. IL film connecting two reservoirs on the electrodes is obtained by the IBFW method to construct a circuit with 0.5 μm width and 80 μm length. All the transient measurements are performed at room temperature (25 °C) employing a Keithley Sourcemeter2636B, biased with a constant voltage between the source and drain electrodes. The relative humidity (RH) data are measured by a commercialized air sensor (ClearGrass air monitor, Qing Ping) with an error ±0.5%. In-situ chemical reaction chip Sodium thiocyanate (NaSCN, 98.5%), cobalt(II) sulfate hydrate (100%) are all purchased from Macklin. Copper sulfate (CuSO 4 , 98%), iron(III) chloride (FeCl 3 , 98%) are purchased from Beijing Tongguang chemistry.The [EMIM][NTF 2 ] RTIL is chosen and four IL droplets are deposited onto a silicon dioxide chip by capillary tube to serve as the liquid reservoir for the fabrication of fluid channels employing IBFW technology. After the fabrication is finished, freshly prepared solutions are injected into the ILs droplets that have previously been deposited and connected by IBFW liquid channel. First, 0.1 μL solution of 0.1 M NaSCN in DI water is injected into the top IL droplet, and wait for 15 min for SCN − to diffuse into the entire flow channel. Then, 0.1 mM Fe 3+ solution, 0.1 mM Cu 2+ solution are injected into the bottom and left IL droplet separately. Finally, 0.1 mM Co 2+ solution is injected into the right droplet. All the injections employ a micropump. Right after the injection, the reaction chip is observed employing an optical microscope (Leica DM4 B, which is employed for all optical microscope imaging). Then the chip is stored in a vacuum chamber for 48 h, and is observed again. The customized sample holder can adjust the tilting angle 0−180°. The reaction chip is mounted to the tilting platform with 120°, and is stored in a sample box for over one week to test the IBFW film pattern stability against gravity. Electrochemical depositions Silver bis(trifluoromethylsulfonyl)imide (AgNTf 2 98% EP) is purchased from Amethyst. 7,7,8,8-tetracyanoquinodimethane (TCNQ, 98% EP) is purchased from Bidepharm. [EMIM][NTf 2 ] is chosen as the solvent and electrolyte for the cyclic voltammogram and potentiostatic deposition experiments. All the electrochemical measurements are undertaken under nitrogen atmosphere at 20−25 °C with a CHI1000C Electrochemical Workstation (CH Instrument, Texas, USA) employing a three-electrode configuration. A homemade electrode probe platform is employed to test the micro droplet. Working electrodes (Ti/Au, 5/10 nm) are deposited on doped Si wafer covered with 300 nm SiO 2 by means of lithography, electron beam evaporation, and lift off procedure. A platinum probe is stuck into the IL droplet and serves as the counter electrode. An Ag plating probe serves as the pseudo-reference electrode. We do not choose a commercialized reference electrode due to their size usually far exceed the droplets used in our experiments. Prior to experiments, the working electrodes are washed with acetone, ethanol, and ultrapure water sequentially with ultrasonication, the counter electrode and pseudo reference electrode are rinsed with ethanol and DI water. Before experiment, the IL is dried in vacuum chamber at 80 °C for 24 h to eliminate water. Then the 10 mM AgNTf 2 and 6 mM Ag + with 5 mM TCNQ solutions are prepared and deposited onto the silicon wafer with Au working electrodes. All these procedures are conducted in a glove box under nitrogen atmosphere. Then the silicon wafers with analyte solution droplets are transferred into the vacuum chamber of HIM for IBFW experiments. After the solution patterns are fabricated, the wafers are transferred to the electrochemical workstation under nitrogen atmosphere. Reporting summary Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"Programmable control of fluid motion over open solid surfaces at small scales [1] , [2] are crucial for countless technological applications such as printing [3] , [4] , biosensing [5] , [6] , energy generating [7] , [8] , air water harvesting [9] , and chemical synthesis [10] etc. The predominant methods resort to either topographical or chemical gradients fabricated on solid surfaces, such as micro(nano)channels [11] , [12] , bioinspired surface structures [13] , [14] , [15] , [16] , [17] , [18] , [19] , chemical modifications [20] , [21] , etc. However, such methods permanently alter the solid substrate. Once fabrication finished, the flow pattern can barely be changed. Moreover, the cost of flow control increases significantly with the complexity of surface modifications. To improve the controllability and flexibility, external-field-stimuli have been recently adopted including temperature [22] , [23] , light [24] , [25] , magnetic [26] , [27] or electric fields [4] , [28] , [29] . The current external-field-stimuli methods mainly resort to the field induced surface-tension-gradients (STGs), which works well for system size below the capillary length ( \\\\({\\\\lambda }_{{{{{{\\\\rm{capillary}}}}}}} \\\\sim \\\\sqrt{\\\\gamma /\\\\rho g} \\\\sim {10}^{-3}{{{{{\\\\rm{m}}}}}}\\\\) ) since the surface energy dominates the free energy of micrometer systems. However, the spatial resolutions of such methods are restricted at microscale and fail to achieve nanoscale flow control. For instances, the reported thermal capillary films [23] , [24] , [25] have thickness above 5 μm and minimum line width about 30 μm. More importantly, the finger instability is usually inevitable [30] in STG based methods and jeopardizes the patterning performances. Actually, when the liquid film thickness approaches nanometer scale, the interfacial overlapping dominates the system free energy [31] , [32] , [33] instead of surface tension. Such phenomenon usually termed as disjoining pressure, arising from intermolecular forces, has been employed as a simplified mechanical description of liquid nanofilm [34] , [35] , [36] . Therefore, to achieve patterning flow at nanoscale, the liquid film thickness needs to reach nanometer to harness the disjoining pressure to stabilize the film pattern, and also requires the spatial resolution of external-field-stimuli to attain nanometer size. Compared with thermal [25] and pyroelectrical [26] , [37] fields, electron/ion beams which can be readily focused to sub-nanometer size [38] , [39] emerge as a promising option of external-field-stimulus. For example, 120 keV electron beam has been reported to induce the stick-slip motion of water nanodroplets [39] . However, the flow speeds were slow (around 3−20 nm/s), the travel lengths were short (around 189 nm), and more importantly, the flow directions and trajectories were uncontrollable while the physical mechanism remained ambiguous. Despite of these, their results still revealed that the interaction between electron/ion beams and liquid is a prospective candidate for liquid maneuvering and can be harnessed for open surface nanofluidics. Here, we present an unexplored concept of open surface nanofluidics called ion beam induced film wetting (IBFW) and achieve the nanoscale patterning of ionic liquid film without resorting to any surface fabrications or special electric circuits. The Helium focused ion beam (HFIB) is generated by a state-of-the-art Helium ion microscope [38] (HIM, ORION NanoFab, Zeiss), which can provide a beam spot of diameter 0.5 nm, and a wide range of dose densities. Since the HIM only operates in an ultra-high vacuum, we select ionic liquids (ILs) as the liquid material due to its ultra-low vapor pressure. We investigate the interaction between the HFIB and IL under different dose densities and discover an unreported liquid inducing mode of HFIB. The liquid inducing mode is attributed to electrical field induced ion emission and disjoining pressure which propels and stabilizes the liquid film pattern. Based on the IBFW inducing mode, we develop a nano-printing technique of IL, with film thickness down to 20−30 nm, minimal line width about 100 nm and corner radius down to 20 nm, and compare its performances with the reported methods. Besides, ILs are also known for their unique properties such as wide electrochemical potential window, high ionic conductivity, low toxicity and thermostability. These features make ILs increasingly important as electrolytes for lithium battery [40] , [41] and electrodepositions of various materials ranging from metal nanoparticles [42] , [43] , metal organic complexes [44] to conducting polymer films [45] . We further demonstrate the IBFW as a versatile tool for various application fields including gas sensing circuit, in-situ chemical reaction chip, and electrochemical deposition of solid materials with desired patterns. The simplicity and versatility of IBFW technique suggests prospect in a range of liquid manipulation applications. By combining with electrochemical procedures, such technique can not only produce patterned liquid film but also solid materials which reveals possibility in nano-transistors fabrication [46] , energy devices [40] and immunosensor circuit printing [47] . 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We systematically investigate the interaction between HFIB and the IL employing different combinations of dwell time, τ , and scan spot pixel spacing, s , at constant beam current, I = 1 pA (Fig. 1a ). When dose density \\\\(D=I\\\\cdot \\\\tau \\\\cdot {s}^{-2} < 2.5{{{{{\\\\rm{ions}}}}}}\\\\cdot {{{{{{\\\\rm{nm}}}}}}}^{-2}\\\\) , the HFIB exerts negligible influence on both solid and liquid samples which is the imaging mode of HIM. At extremely high dose density ( \\\\(D > 500{{{{{\\\\rm{ions}}}}}}\\\\cdot {{{{{{\\\\rm{nm}}}}}}}^{-2}\\\\) ), the HFIB decomposes/etches the samples and Helium bubbling may occur (Supplementary Fig. 1 ). However, at moderate dose density range ( \\\\(2.5{{{{{\\\\rm{ions}}}}}}\\\\cdot {{{{{{\\\\rm{nm}}}}}}}^{-2}\\\\le D < 200{{{{{\\\\rm{ions}}}}}}\\\\cdot {{{{{{\\\\rm{nm}}}}}}}^{-2}\\\\) ), the ILs are induced to flow into the irradiated area while the solid substrates remain intact (Supplementary Note 1 ) which is the liquid inducing mode of HIM. After the HFIB with appropriate parameters (Supplementary Table 1 ) scans outwards from the contact line (CTL) of reservoir (Fig. 1b ), IL flows through local protrusions of CTL and branches off to form hierarchical rivulets (Supplementary Fig. 2 , Supplementary Movie 1 ). The liquid inducing mode of HIM has not been reported to the best of our knowledge, and can be harnessed to develop a nano-printing technique. Fig. 1: Helium focused ion beam liquid inducing mode and working procedure of ion-beam-induced-film-wetting nano-printing. a Phase diagram of three working modes under different HFIB dose density, manipulated by the dwell time and the scan spot density. The background color map shows the dose density ranges from 10 −2 −10 4 ions/nm 2 . The horizontal axis corresponds to the reciprocal of the square of the scan spot spacing, s −2 , which represents the number of scan spots within 1 nm 2 . The imaging mode exerts no influence to samples (blue squares); the inducing mode can induce directional liquid flow (red circles); and the damaging mode can etch samples and induce Helium bubbling (cyan triangle). b The hierarchical branched flow pattern of ionic liquid after a frame of HFIB scan. The IL will eventually fill the entire irradiated area in 30 s. c Contact line region of the IL reservoir and nano-patterning-and-visualization-engine pattern design. Left is a schematic diagram, and right is an HIM image. The white arrow indicates the pattern direction and area in NPVE. The dashed yellow box represents the CTL region chosen for HFIB scan. d The rectangular film pattern (50 μm × 1 μm) fabricated by IBFW. The insets are the details to manifest small corner radius and well reproduction of the design, the yellow box edges are 2 μm. e The IBFW fabricated ‘PKU COE’ pattern demonstrates its potential in on-demand printing of complex liquid film networks. f The minimal line width that IBFW can fabricate is around 106 nm. Full size image As an example of IBFW printing technology (Fig. 1c, d ), we induce small amount of liquid from the reservoir to fabricate a rectangular film pattern (50 μm × 1 μm). Step 1: Sample preparation. A SiO 2 wafer is placed horizontally as the substrate, Fig. 1c . No surface fabrication or physical mask is needed. A 0.1 μL droplet of IL is deposited on substrate as the liquid reservoir. The contact angle of the IL on the substrate is approximately 60° (Supplementary Fig. 3 ). Then the sample is transferred into the chamber of HIM. Step 2: CTL identification and pattern design. Under the imaging mode of the HIM, the CTL of the droplet is identified. As the yellow dotted box marked in Fig. 1c , part of the CTL is chosen as the starting position of the film pattern (50 μm × 1 μm) which is drawn employing software Nano Patterning and Visualization Engine (NPVE). The NPVE rasterize-fills the pattern with scan spot arrays (Supplementary Fig. 4 ). Step 3: HFIB irradiation and liquid inducing. Employing inducing mode of the HIM, the HFIB scans the designed area. The ion beam vector-scans the spot arrays point by point with serpentine scan style and specific beam parameters. Noteworthily, some other conditions other than beam parameters also influence the IBFW performances. The starting position of scan should be located close to or stuck into the CTL, and the scan direction should be outwards from the reservoir, while other configurations cannot achieve the best inducing performance (Supplementary Fig. 5 and Supplementary Movie 2 , 3 ). The speed differences between liquid film propagation and ion beam scanning can explain the scan direction effects on IBFW inducing performance (Supplementary Movie 4 ). Also, the flood gun should be turned on. Since the helium ions carry positive charges, continuous irradiation would cause too much positive charges in the sample surface. The excess surface charges would alter the direction of the subsequent Helium ions and finally make the liquid film pattern tilted (Supplementary Fig. 6 ). When the point-by-point irradiation finishes, the thin liquid film is fabricated with designed pattern (Fig. 1d ). As shown in Fig. 1d , IBFW fabricated nanofilm reproduces not only the size of the designed pattern (50 μm × 1 μm), but also the straight edges and the 90° corners. Moreover, liquid film networks can be easily rewritten on the same solid substrate after the liquid is dissolved by acetone. As a result, the liquid film pathways can be reprogrammed in an on-demand manner. To further demonstrate the prospect in complex patterning of IBFW, we fabricate a liquid film pattern making up the words “PKU COE” (Peking University College of Engineering) in Fig. 1e . The word patterns are all drawn employing NPVE without physical masks. Throughout the patterning process, the liquid is induced to flow along arbitrary geometrical tracks with curved lanes, crisscross junctions, and corners to form complicated liquid networks. The liquid film well reproduces the designed pattern. The width and corner radius of the liquid film barely change during the propagation regardless of the complexity of the pattern. Figure 1f presents an example of the minimal line width, 100 nm, IBFW can achieve. By assembly of separate scan patterns sequentially, the IBFW technique can also fabricate liquid film channel up to hundreds μm in length. Apart from continuous pattern, discontinuous liquid pattern can be fabricated by the introduction of damaging mode of HFIB. As shown in Supplementary Fig. 7a PKU pattern is separated from the flow channel that connect to the droplet reservoir. Working principles of IBFW When HFIB irradiates at or near the CTL of droplet, the Helium ions generate special charges distribution [48] in the SiO 2 substrate, the positive surface charges induce the primary ion emission from the IL reservoir (Fig. 2a ). HFIB exhibits two distinguishable features compared with electron beam and other ion beams [48] , [49] . Firstly, the divergence of HFIB interacting with solid samples is much smaller than electron beam due to its larger mass [49] , which lead to a more localized surface charging area and consequently a programmable control over liquid flow (Supplementary Fig. 8 ). Secondly, HFIB tends to penetrate sample and induce less damaging compared with Gallium beams [48] . The penetration depth exerts significant influence on IBFW’s application potential in electrochemistry field. For example, 30 keV He beams with hundreds nm stopping range can easily penetrate a 10 nm Au and 5 nm Ti electrode deposited on SiO 2 wafer and induce patterned liquid flow without devastating effect to Au surface (Supplementary Fig. 9 ), while 30 keV Ga with stopping range less than 20 nm [48] is hard to penetrate metal films and can easily cause damage to the electrode surface. During the HFIB irradiation, He ions interact with solid atoms and excite holes-electrons in the sample, then the He ions lose kinetic energy and rest within the stopping range. Since the excited electrons in amorphous SiO 2 only survive 10 ns or less before the recombination takes place [50] , [51] , the positive charges dominate the surface charging and account for the primary anion emission. Monte Carlo simulation results [48] reveal that the lateral projection length of 30 kV He ions irradiated on SiO 2 (also SiO 2 with 10 nm Au and 5 nm Ti layers) ranges from 90 nm to 110 nm (Supplementary Fig. 9a, b ). The consequent positive charges (ions, holes) distribute over 10 2 nm in lateral direction and determine the ion emission and IBFW spatial resolution. Fig. 2: Working principles of ion-beam-induced-film-wetting. a − c Schematic diagrams of the IBFW working principle. a When helium focused ion beam irradiates the contact line, the positive surface charging induces the primary anion emission. b When HFIB ceases to scan, surface charging dissipates and the emitted anions induce the secondary cation emission. c The emitted ions from previous stages form an ultra-thin precursor film, and the consequent disjoining pressure, Π , propels and stabilizes liquid film. d The mechanical balance between the surface charge density induced electrostatic force and the surface tension of liquid-vacuum interface. The distance between HFIB scan spot and contact line is d ; the surface charging uniformly distributes over a region with length scale, l 0 ; the surface-charge exerts electrostatic force and distorts the liquid-vacuum interface, balanced by the surface tension γ ; α is the angle between substrate and the connecting line of surface-charge center and ion emission center; r * is the ion emission critical radius, and is also the vertical distance between ion emission center and substrate. Full size image When the HFIB ceases to irradiate, the surface charges dissipate due to the drainage current and the electron-hole recombination (Fig. 2b ) then the emitted anions induce the secondary cation emission. Both ions meet ahead of the CTL and form an ultra-thin precursor film with thickness comparable to ion size at the irradiated area. As a matter of fact, when the electric field of surface charge is strong enough, not only separate ions but clusters or even tiny droplets contain both ions are emitted [52] to scanned area and make up the precursor films. In both cases, the ultralow thickness of precursor film gives rise to the high disjoining pressure (10 5~6 Pa) that irrigates and thickens the precursor film until be balanced by the capillary force (10 2~3 Pa) and a continuous liquid film is formed (Fig. 2c ). The surface charging process of dielectric materials (SiO 2 for example) under the irradiation of focus ion beams (Ga + or He + ) was thoroughly discussed in literatures [53] , [54] , [55] , and the charging accumulation and dissipation is manipulated by the following factors: (1) generation of electron-hole pairs in the solid by incident ions; (2) neutralization of the incident ions by the excited free electrons; (3) sputtering of the surface atoms; (4) charging due to the secondary ion-electron emission; (5) leakage of mobile electron-hole pairs to the silicon substrate; (6) induced shallow traps by the incident ions and a consequent preferred trapping relative to the deep traps. The surface charge density (SCD) of SiO 2 at HFIB irradiation [54] can be expressed as a function of time (Supplementary Note 2 ): $$\\\\frac{dQ(t)}{{dt}}=P\\\\left(1+{\\\\gamma }_{e}\\\\right)\\\\cdot I\\\\left(t\\\\right)-k\\\\frac{Q\\\\left(t\\\\right)}{{\\\\epsilon }_{r}{\\\\epsilon }_{0}}-\\\\frac{7}{4}{YI}\\\\left(t\\\\right)\\\\cdot {\\\\varOmega }_{0}\\\\frac{Q\\\\left(t\\\\right)}{{R}_{p}}-{\\\\int }_{0}^{t}J(t){dt}.$$ (1) The RHS composes of 4 terms. The first is the ion incident term which represents the electron-hole pairs accumulation induced by the ion incident and secondary emission, where P is the probability factor accounts for the electron-hole recombination, γ e is the secondary electron emission yield of SiO 2 , and I ( t ) is the beam current density of HFIB. The second term is the leakage current term, where k is the conductivity of the substrate, \\\\({\\\\epsilon }_{r}\\\\) is the substrate relative permittivity, and \\\\({\\\\epsilon }_{0}\\\\) is the vacuum permittivity constant. The third term is the sputtering yield induced charge reduction, where Y is the sputtering yield acquired from SRIM simulation, Ω 0 is the atomic volume which can be estimated by the average density of SiO 2 , R p is the ions stopping range from SRIM. The last term accounts for the accumulation of emitted counterions, where J ( t ) is the ion emission rate at current SCD, which is often described as a kinetic process in which ions evaporate from liquid-vacuum interface. The emission current density reads [56] : $${j}_{e}=\\\\frac{J}{{\\\\pi}{r}^{*2}}=\\\\frac{{k}_{B}T}{h}\\\\sigma \\\\exp \\\\left(-\\\\frac{\\\\Delta G-G({{{{{{\\\\bf{E}}}}}}}_{{{{{{\\\\bf{n}}}}}}}^{{{{{{\\\\bf{v}}}}}}})}{{k}_{B}T}\\\\right),$$ (2) where j e is the current emitted per unit IL-vacuum surface area, k B is Boltzmann’s constant, T is the liquid temperature, h is Planck’s constant, σ is the local net charge density at the liquid-vacuum interface, ∆ G is the Gibbs free energy barrier for an ion to be emitted, \\\\({{{{{{\\\\bf{E}}}}}}}_{{{{{{\\\\bf{n}}}}}}}^{{{{{{\\\\bf{v}}}}}}}\\\\) is the local vacuum electric field normal to the interface. \\\\(G({{{{{{\\\\bf{E}}}}}}}_{{{{{{\\\\bf{n}}}}}}}^{{{{{{\\\\bf{v}}}}}}})\\\\) is the reduction of solvation energy barrier due to the external electric field, assumed to take the form \\\\(G\\\\left({{{{{{\\\\bf{E}}}}}}}_{{{{{{\\\\bf{n}}}}}}}^{{{{{{\\\\bf{v}}}}}}}\\\\right)=\\\\sqrt{\\\\frac{{q}^{3}{{{{{{\\\\bf{E}}}}}}}_{{{{{{\\\\bf{n}}}}}}}^{{{{{{\\\\bf{v}}}}}}}}{4\\\\pi {\\\\epsilon }_{0}}\\\\frac{{\\\\epsilon }_{r}-1}{{\\\\epsilon }_{r}+1}}\\\\) by the Schottky hump, where q is the ion’s charge. The solvation energy of emitted ion can be estimated by the Born model as \\\\(\\\\Delta G={\\\\left(\\\\frac{27}{4}\\\\pi \\\\right)}^{1/3}\\\\frac{{\\\\gamma }^{1/3}{q}^{4/3}{({1-\\\\epsilon }_{r})}^{2/3}}{{({4\\\\pi \\\\epsilon }_{0})}^{2/3}}\\\\) , where γ is the liquid–vacuum surface tension. Adopting the mechanical model proposed in the following paragraph, the electric field of SCD, \\\\({{{{{{\\\\bf{E}}}}}}}_{{{{{{\\\\bf{n}}}}}}}^{{{{{{\\\\bf{v}}}}}}}\\\\) , can help to calculate the ion emission rate and the SCD. A schematic diagram is shown in Fig. 2d to model the mechanical balance between IL surface tension and the electrostatic force exerted by the surface charge. The IL–vacuum surface is distorted by the electric field of surface charges and forms a bumping meniscus. When the meniscus is distorted to be hemispherical, the vertical component of surface tension reaches maximum. Once the surface charge continues to increase, a significant ion emission would take place during which both ions, clusters and tiny droplets may emit from the interface [52] , [56] , [57] . The distance between the scan spot (center of surface charge) and the contact line ranges from 10 0 to 10 2 nm, as long as the surface charge is strong enough to induce ion emission. The critical SCD that can induce significant ion emission depends on the distance between the starting scan spot and the reservoir CTL (more detailed deduction can be found in Supplementary Note 3 ): $${Q}_{{{{{{\\\\rm{surf}}}}}}}\\\\cong \\\\frac{{(d+{r}^{*})}^{2}+{l}_{0}(d+{r}^{*})}{{k}_{0}{\\\\cos }^{3}\\\\alpha }{{{{{{\\\\bf{E}}}}}}}^{{{{{{\\\\boldsymbol{*}}}}}}}.$$ (3) Where d is the distance between the scan spot and the CTL; \\\\({r}^{*}=\\\\frac{{q}^{6}\\\\gamma }{4{\\\\pi }^{2}{\\\\epsilon }_{0}^{3}{\\\\left(\\\\Delta G\\\\right)}^{4}} \\\\sim {10}^{-8}{{{{{\\\\rm{m}}}}}}\\\\) is the characteristic ion emission radius derived in literature [52] ; l 0 is the surface charging area length scale, which represents the lateral distribution of positive surface charges [48] ; k 0 is the Coulomb constant; α is the ion emission angle as depicted in Fig. 2d ; and \\\\({{{{{{\\\\bf{E}}}}}}}^{{{{{{\\\\boldsymbol{*}}}}}}} \\\\sim {10}^{9 \\\\sim 11}{{{{{\\\\rm{V}}}}}}/{{{{{\\\\rm{m}}}}}}\\\\) is the characteristic electric field [56] for significant ion emission. At given separation distance d , the critical SCD can be calculated by Eq. ( 3 ). By invoking Eqs. ( 1 ) and ( 2 ), the dosage of HFIB required for the critical SCD can be calculated. The relationship between critical HFIB dose density to induce IBFW, and the distance between starting scan spot and droplet CTL, d , is calculated employing Eqs. ( 1 – 3 ) with the results depicted in Fig. 3a by blue line. The beam current employed in calculation is 1 pA, scan spacing is 1 nm, and the dwell time determines the calculated dose density. Here, we assume that the critical dose density that takes to induce significant ion emission coincides with the dose to achieve IBFW. The experiments are conducted to measure the critical HFIB dose density with 1 pA beam current and 1 nm spacing to induce continuous liquid film from reservoir with the results shown in Fig. 3a by orange rhombuses. The calculation results agree with the experimental critical dose density qualitatively, and confirm our hypothesis that the surface charging induced ion emission accounts for the IBFW. Fig. 3: Theoretical analysis, experimental and simulation results of ion-beam-induced-film-wetting mechanism. a Experimental helium focused ion beam dosage density to induce ionic liquid flow (ions/nm 2 ) and calculated results of critical dose density to induce significant ion emission. The inset shows when the scan position is separated too far from the contact line, the consecutive liquid film degenerates to the local protrusion flow as shown in Fig. 1b . The scale bar is 5 μm. b IBFW induced liquid film lengths (μm) on substrates with different conductivities. Experimental film length data of distinct samples are presented as mean ± s.d. n = 5. The right ordinate represents the overall ion emission number calculated by the beam parameters and sample characters. In the horizontal axis, TOX stands for thermal oxidized, and CVD stands for chemical vapor deposited. c Molecular dynamics simulation of [EMIM][DCA] droplet (640 ion pairs) deposited on fused silica substrate going through surface charges injection and removal. The arrows indicate the most directed movements of ions: the pale blue arrows at the beginning stage represent the surface charge induced primary anion emission; the purple arrows of cations represent the emitted anions induced secondary cation emission. d IBFW film flow speed as a function of liquid film length. The single spot experiments are conducted with a line pattern of scan spots, at constant beam current 1 pA, dwell time 10 μs, spacing 1 nm. The change width experiments are conducted with rectangular pattern with constant length 20 μm and different widths, while keeping the beam current 1 pA, dwell time 2 μs, spacing 1 nm. Data of change width method are presented as mean ± s.d. n = 12. The blue shaded region is the calculated velocity range due to the range of slip lengths [58] , [59] , the deep blue line is the calculated result employing the average slip length [58] . Full size image Apart from the HFIB configuration, the substrate characteristics also influence the IBFW behavior. As shown by Eq. ( 1 ), the electrical conductivity is easy to manipulate while exerts significant effect on the liquid inducing performance. In Fig. 3b , we select 7 different solid substrates vary from well conductors to dielectrics. The orange rectangles and black triangles represent the lengths of liquid film that can be induced on different substrates under identical HFIB treatment. IL film cannot be induced on pure conducting samples such as Au and Cu, liquid extends slightly on semiconductor (Si), whereas liquid film propagates a long distance on insulated substrates e.g ., quartz and mica. Employing Eqs. ( 1 – 3 ), we can calculate the numbers of emitted ions during a single spot scan of HFIB (Supplementary Note 2 , Supplementary Fig. 10 ), and is depicted in Fig. 3b as blue circles. As can be seen, the ion emission model we propose herein can qualitatively explain the different effects of IBFW on conducting, semi-conducting and insulating substrates. Yet the IBFW effect is also influenced by other substrate situations (roughness, for example), and the differences between various insulated substrates are failed to be captured by the simplified 2D model. Not only the solid substrates, the ILs employed also influence the IBFW performances. Four kinds of ILs are tested on SiO 2 substrate with the results shown in Supplementary Fig. 11 . The results show that liquid viscosity has significant influence on IBFW liquid film, by raising the flow friction which works as a counterpart of the electrostatic driving forces. The relation between wetting status and IBFW effect, however, is ambiguous, indicating that the electrostatic forces of IBFW is not quite relevant with the wetting status. It is probably due to the fact that the IBFW electrostatic forces are much greater than the wetting van der Waals forces at the contact line. Molecular Dynamics simulation is employed to reveal the initiating stage of IBFW (Fig. 3c ). The yellow spheres are atoms of cations, and blue spheres are atoms of anions as shown in the top-left legend. The bottom-left color scale of the trajectory lines represents time steps in simulation. The primary and secondary ion emissions are identified by the trajectories of anions which are represented by the white-to-blue lines and cations represented by the red-to-purple lines. The arrows indicate the most directed movements of ions. The pale blue arrows with more unified orientation at the beginning stage represent the surface charge induced primary anion emission. The purple arrows of cations represent the emitted anions induced secondary cation emission with less orientability. The red shaded regions represent the surface area where positive charges are injected and then removed. We also observe an ultra-thin precursor film in MD simulation, see Supplementary Fig. 12 . We next verify that the disjoining pressure propels and stabilizes the nanofilm. The propagation speed of IBFW liquid film decreases monotonically with the increase of film length: $${{{{{\\\\bf{U}}}}}} \\\\sim \\\\frac{{h}^{2}+3{bh}}{3\\\\mu }\\\\cdot \\\\frac{{{{{{\\\\boldsymbol{\\\\Pi }}}}}}\\\\left({h}_{\\\\min }\\\\right)-\\\\gamma {{{{{\\\\boldsymbol{\\\\kappa }}}}}}}{L}.$$ (4) Where U is the average flow speed, h is equilibrium film thickness, b is the slip length of IL-SiO 2 interface [58] , [59] , [60] , μ is IL viscosity, Π ( h min ) is disjoining pressure at minimum film thickness h min , L is film length, γ is IL surface tension, and κ is curvature of IL-vacuum interface at the conjunction of film and reservoir (Supplementary Table. 2 , Supplementary Note 4 , Supplementary Fig. 13 ). We depict the calculation results in Fig. 3d . Since the boundary slip length of IL-SiO 2 interface depends on the combined surface conditions and ranges from 2 nm to over 16 nm [59] , we employ the lower (2 nm) and upper (16 nm) limits of slip length to give an estimation on the possible range of flow speed in Fig. 3d as the blue shaded region, and the calculation result of average value 10 nm [58] is shown by the deep blue line. The relationship between flow velocity and film length can also be measured experimentally. As shown in Supplementary Fig. 4 , the HFIB scans the designed pattern row by row, so the beam speed vertical to the CTL can be calculated as: $${{{{{{\\\\bf{v}}}}}}}_{{{{{{\\\\bf{beam}}}}}}}=\\\\frac{{{{{{{\\\\bf{s}}}}}}}_{{{{{{\\\\bf{vertical}}}}}}}}{{N}_{{{{{{\\\\rm{row}}}}}}}\\\\cdot \\\\tau+{V}_{{{{{{\\\\rm{refresh}}}}}}}},$$ (5) Where, s vertical is the vertical scan spacing, N row is the number of scan spots in one row, τ is the dwell time that HFIB stays at a single spot, V refresh is a small time (10 μs) that NPVE takes to reset the HFIB for next row of scan. If the vertical speed of HFIB exceeds the film velocity, the distance between the scan spot and the liquid film CTL would increases until the scan spot is too far ahead of the film and the liquid would cease to flow. The critical interaction distance with given beam parameters can be determined experimentally (Supplementary Fig. 14a ). Due to the pronounced impact of dose density on the flow velocity of liquid film (Supplementary Fig. 14b ), the flow velocity measurements are conducted under the same dose density by keeping the beam current I = 1 pA, scan spot spacing s = 1 nm and dwell time τ = 2 μs constant and only alter the width of the rectangle pattern. The pattern width controls the N row and consequently alters the beam vertical speed. By scanning rectangle patterns with same length but different widths outwards from the reservoir CTL, the vertical speed of beam can be changed at constant HFIB dose density. The IBFW film length decreases with the beam speed increasing (Supplementary Fig. 14c ), and consequently the average flow velocity of films with different length can be measured. The change width measurement results are shown in Fig. 3d by the hollow orange circles. The results of film speed at extremely long film lengths are acquired by first fabricating a long liquid film (300 μm × 10 μm, 600 μm × 10 μm, and 900 μm × 10 μm respectively) from the reservoir, then the change width measurements are conducted at the front of the long film. Since the fabrication of extreme long film can be time consuming, these data are only measured once. The other experiments are repeated for twelve times with the average value and standard error shown in Fig. 3d . The shortage of the change-width method is that the NPVE scan pattern assembling limits the maximum velocity the beam can move vertically. To overcome such limitation, we adopt single spot scan method. In which, a line pattern made up by a series of scan spots is used. The scan speed is altered by changing the vertical refresh time between each scan spot, while keeping beam current 1 pA, dwell time 10 μs and spacing 1 nm all constant. The dwell time is elongated to compensate the dose density reduction, since the scan area is influenced by neighboring scan spots in a rectangular pattern. All single spot measurements are repeated at least five times. The results of single spot scan are represented by the green triangles in Fig. 3d . The consistency between the calculation and experiments suggests that disjoining pressure can explain the propagation of IBFW film. The discrepancy at extreme long film length may be due to the HFIB irradiation history. The fabrication of film with hundreds of μm length usually takes hours of HFIB irradiation. The accumulated positive charges lead to a higher surface potential and a boundary slip length that exceeds the upper limit in literature [60] , which give rise to an unexpected higher film speed. Base on the film length-speed relationship curve (Fig. 3d ), the fabrication limit of film length for IBFW technique can be determined to fulfill specific requirements for fabrication efficiency. For example, the longest film we fabricated reaches 800 μm to 900 μm, with the film velocity decreases to 10 −2 μm/s (the in-situ chemical reaction chip in Applications section). Besides, the equilibrium film thickness predicted by the balance between disjoining pressure [61] , [62] and Young-Laplace pressure agrees with the AFM measured average film thickness (Supplementary Note 5 , Supplementary Fig. 15 ). Hence, we prove that disjoining pressure is important in the propelling and stabilizing of IBFW nanofilm. We also exclude the potential roles played by HFIB induced surface morphological or chemical modification effects and heating effect induced Marangoni flow in Supplementary Note 1 , Supplementary Figs. 9 , 16 , and 17 . More details of MD can be found in Supplementary Note 6. Since the temperature increasing effect is less than 1 K based on our calculation. The thermal energy difference, \\\\({k}_{b}\\\\Delta T \\\\, < \\\\, 1{{{{{\\\\rm{K}}}}}}\\\\cdot {k}_{b}=8.62\\\\times {10}^{-5}{{{{{\\\\rm{eV}}}}}}\\\\) , induced by HFIB is far less than the energy barrier for ions to overcome to evaporate from the liquid phase. The evaporation effect may also be insignificant in current experiments. Working performances of IBFW We employ atomic force microscope (AFM) to manifest the nanoscale flow control of IBFW. Figure 4a shows the front 7 μm of a 28 μm × 1 μm IL film with thickness around 30−40 nm (an average of 35.4 ± 1.7 nm). The film thickness remains unchanged along the flow path (Fig. 4a ). The film width is 1 μm, coincides with designed pattern. The liquid-air interface is much smoother than solid substrate (RMS roughness 7.9 ± 5.7 nm). The minimal line width of IBFW film reaches 106 nm (Supplementary Fig. 18 ), and is limited by the surface charges spatial distribution. If the surface charges can be trapped within a narrower spatial range, the ideal line width limitation may be comparable with film thickness. Fig. 4: Working performances of ion-beam-induced-film-wetting. a Tapping mode atomic force microscope image of the front 7 μm of a 28 μm × 1 μm ionic liquid film. The height profile at y = 4.9 μm is projected onto the x-z plane, and the height profile at x = 5.2 μm is projected onto the y-z plane. b The rewritable test results of IBFW method on the same substrate. The same micro hole is adopted to guarantee all experiments are conducted at the same place of the same substrate. The induced film lengths of a 20 μm × 1 μm scan pattern keep constant throughout 10 times of rewritable tests. c The liquid film line width and thickness compared with published results [23] , [24] . The inset show part of the IBFW liquid film imaged by AFM. The corner radiuses between the fabricated liquid film and reservoir are compared in ( d ) with the inset illustrates a IBFW film with corner radius down to 20 nm. e The flow velocities at different liquid film lengths are compared with literatures [23] , [24] . The data of IBFW are presented as mean ± s.d. n = 5. f System free energy scenario of a liquid film with different film thickness, and a comparison of different flow control strategies. The left vertical axis represents the ratio that different energy terms contribute to the total free energy. The yellow vertical line represents capillary length and the red vertical line represents overlapping length ( \\\\({\\\\lambda }_{{{{{{\\\\rm{overlapping}}}}}}} \\\\sim \\\\sqrt{\\\\frac{{A}_{{slv}}}{\\\\gamma }}\\\\) ). Different symbols represent different strategies, the IBFW pushes the limit towards nanoscale while most counterparts fall in the surface tension region. Full size image As shown in Fig. 4b , the rewritability of IBFW liquid film is tested for ten times on the same SiO 2 wafer. During the experiments, a 20 μm × 1 μm pattern is repeatedly fabricated at the same position each time. Then the substrate is transferred for AFM characterization (Supplementary Fig. 19 ) before the liquid film and droplet is rinsed for next test. Through the experiments, the film length, geometrical features and the smooth liquid-air interfaces are barely changed. Such results verify that IBFW is a rewritable open surface nanofluidic technique. The performances of IBFW are compared with published methods [23] , [24] in Fig. 4c−e . The IBFW film achieves a minimal line width and thickness which are two orders of magnitude lower than those induced by thermal and optical fields (Fig. 4c ). Moreover, the IBFW prevents the formation of menisci between the film and the reservoir, which are common in previous studies where the corners between the film and the reservoir CTL usually have a radius of 20−40 μm. However, we achieve sharp corners (90°) with a corner radius of 20 nm (the inset of Fig. 4d ). IBFW also achieves relative high flow speed without substrate confinement. As shown in Fig. 4e , the flow speed is at least 2 μm/s at 20 μm film length at 1 pA beam current, which is one order of magnitude higher than reported methods (lower than 0.5 μm/s). The disjoining pressure harnessed in current work explains the better performances of IBFW. We delineate a system free energy ratio scenario for a liquid film system with unit length/ width and thickness vary from 1 nm to 4 mm in Fig. 4f , and the film thickness and corner radius are compared with published results [16] , [18] , [23] , [24] . System free energy composes of the volumetric term (gravity, electrostatic, etc.), the surface tension term, and the disjoining pressure term. The surface tension remains fixed magnitude of \\\\({10}^{1}\\\\,{{{{{\\\\rm{mN}}}}}}/{{{{{\\\\rm{m}}}}}}\\\\) , and its contribution to the system is almost constant with the thickness variation. When system size exceeds capillary length, λ capillary , the volumetric term contributes most of the system free energy. For any system with characteristic length below capillary length, however, the influence of volumetric term (gravity etc.) can be neglected. At millimeter to micrometer range, the surface tension dominates, and the majority of traditional microfluidics methods belong to such region, with the spatial resolution difficult to approach nanoscale. When the system size reduces to nanometer region, the interfacial overlapping [31] , [33] emerges. The disjoining pressure increases rapidly when film thickness approaches sub-nanometer and dominates system energy. Following the capillary length, an overlapping length can be defined as \\\\({\\\\lambda }_{{{{{{\\\\rm{overlapping}}}}}}} \\\\sim \\\\sqrt{\\\\frac{{A}_{{slv}}}{\\\\gamma }} \\\\sim {10}^{-9 \\\\sim -8}{{{{{\\\\rm{m}}}}}}\\\\) , where A slv is the Hamaker constant of solid-liquid-vacuum interfaces system. Below such size the overlapping energy contribution far exceeds any other terms, and can be termed as the characteristic length scale for nanofluidics. Neither the traditional external stimuli or surface fabrication methods can acquire nanoscale control of flow, while the IBFW approaches the limit. Applications of IBFW As discussed previously, IL film pattern prepared by IBFW technology manifests three distinguishable features. First, the ultralow film thickness down to 30 nm indicates a high surface-volume ratio which is a key role in improving the gas sensing circuit sensitivity. Second, the capability of fabricating liquid film with desired pattern in a programmable and rewritable manner, which is important for in-situ chemical reaction and microfluidics chips. Third, the ILs are widely used in electrochemistry and reveal the possibility of transforming liquid film pattern into various solid materials ranges from organic to inorganic compounds. In Fig. 5a, b and Supplementary Fig. 20 , we present a room-humidity-sensing circuit utilizing IBFW technique. The water molecules are adsorbed by the IL then diffuse to and react at the electrode surfaces, generating a reaction current I ds , at constant voltage 1 V. The source-drain currents of a IBFW nanofilm circuit and a micrometer-size droplet circuit are measured within the same chamber with the relative humidity ranging from 40% to 70%. The current in nanofilm linearly depends on the humidity, while no significant change can be observed for the micrometer droplet. The competition between the adsorption at liquid-air interface and the diffusion within liquid circuit can explain such differences. For the droplet circuit, the long distance for the water molecules to diffuse to the electrode surface restricts the reaction rate, therefore, the change of current with humidity is not obvious. Nevertheless, the thin liquid film circuit with much shorter diffusion distance and relaxation time largely accelerates the diffusion process, which is the rate-determining step of the current experiment. Moreover, the much higher surface-volume ratio benefits the adsorption of the vapor molecules and further improves the reaction current sensitivity. Both the sensitivity and response speed are greatly enhanced due to the size effect endowed by IBFW nano film. The simple device presented here can manifest the feasibility of the IBFW circuit in sensing circuit manufacturing. Fig. 5: Application examples of ion-beam-induced-film-wetting. a , b Proof-of-concept gas sensing prototype. a Optical image of IBFW film that connects two droplets settled on Au electrodes. b Results of IBFW nano-circuit humidity sensor compared with a micrometer droplet circuit. The inset shows the schematic of the IBFW humidity sensor. The dash lines are the linear fitting results of drainage-source currents of microdroplet and nanofilm. The voltage exerted is 1 V. c In-situ chemical reaction chip. The left is the schematics, crosshair channels connect four droplets which are used to inject reagent water solutions. 0.1 μL of 0.1 M NaSCN, 0.1 mM Co 2+ , 0.1 mM Fe 3+ and 0.1 mM Cu 2+ are injected clockwise into four droplets. The inset on the bottom-left shows the microchip on fingertip, with red circle shows the four droplets, the edge of the SiO 2 substrate is 10 mm. The right part is the optical image of the reaction chip that has been stored in vacuum chamber for 48 h after the injection, the purple boxes are 80 μm × 80 μm. d Schematics of a three-electrode electrochemistry experiment. Analytes are dissolved in the droplet with IBFW film pattern printed onto the working electrode. 10 nm Au and 5 nm Ti serve as the working electrode, Pt probe and silver-plated probe stuck into the droplet are the counter and pseudo-reference electrodes respectively. e Optical images of the patterned solid particles deposited on Au electrode surface. The upper part shows Ag particles (20 μm × 3 μm rectangle with PKU letters) deposited at −0.2 V. The lower part shows blue AgTCNQ particles (50 μm × 10 μm) deposited at −0.1 V. Both are potentiostatically deposited for 180 s. Cyclic-voltammograms of ( f ) 10 mM Ag + ([EMIM][NTf 2 ]) and ( g ) 6 mM Ag + and 5 mM TCNQ ([EMIM][NTf 2 ]) solution. f The reduction peak appears at −0.295 V, with peak current decreases with cycles. g The first reduction peak (Ag + to Ag) appears at 0.095 V. The +300 mV shift at present of TCNQ is consistent with literature [44] . All optical images are acquired by Leica DM4B microscope. Full size image To demonstrate the potential of IBFW for microfluidics chip fabrication, we design an in-situ chemical reaction micro-chip. In Fig. 5c , the schematic diagram shows a crosshair shaped micro fluid channel connects four separated droplets in four directions. Four square expansion windows are made on each part of the channel for the convenience of observation. The bottom-left inset shows the chip with four droplets on finger-tip. After the IBFW fabrications, 0.1 μL of sodium thiocyanate solution (NaSCN 0.1 M in deionized water) is injected into the top droplet, and serves as the colorimetric reagent for the detection of and in-situ reaction with different metal ions. After the injection of NaSCN, 0.1 μL of 0.1 mM Fe 3+ solution, 0.1 mM Cu 2+ solution and 0.1 mM Co 2+ are injected into the bottom, left and right droplets respectively. The microchip is rested in atmosphere for 20 min for the metal ions fully diffuse into the channels and react with SCN - within different square windows with the ion names printed previously. Then the sample is transferred into vacuum chamber for 48 h to diminish the water content in the solution system, which will alter the hydration status of the metal ion complexes and improve the colorimetric visibility. The red complex Fe(SCN) 3 deposits in the bottom window. The gray deposition in the left window is complex Cu(SCN) 2 . And the blue deposition in the right window is complex Co(SCN) 2 . As shown in Fig. 5c , the SCN − participates into 3 different reactions within several hundreds of micrometers flow channel. The time series pictures are shown in Supplementary Fig. 21 , and the liquid film patterns remain unchanged during the experiments which last for over one month. The IBFW fluid channel exhibits great stabilities against vacuum/air transferring, the injection of solutions into droplet reservoir, and gravity. Such behavior demonstrates the robustness of the IBFW liquid film. More importantly, all reagents are dissolved in deionized water then injected into the IL droplets and diffuse into the IL flow channel. These experiments demonstrate that IBFW liquid film pattern can act as a stable flow channel for later injection of analytes dissolved in water, ethanol, and various molecular liquids due to the amphiphilicity of ILs. Such results greatly broaden the potential application fields for IBFW. Due to their unique properties, ILs have been proved to be an important category of solvent and electrolytes. Here we demonstrate that, by further combining with electrochemical procedure, the IBFW also manifests the capability of transforming liquid film pattern into various solid materials. Figure 5d shows the schematics of a three-electrode electrochemistry experiment. A droplet of 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide ([EMIM][NTf 2 ]) serves as the solvent of possible analytes, which are silver bis (trifluoromethylsulfonyl)imide (Ag[NTf 2 ]) or a mixture of Ag + and 7,7,8,8-tetracyanoquinodimethane (TCNQ) in our experiments. The liquid film pattern is fabricated by the IBFW technology onto the thin gold electrode and would be subsequently transformed into nanoparticles. A 10 nm Au and 5 nm Ti film deposited onto SiO 2 serves as the working electrode and is connected to the workstation by a Pt probe. As revealed by the MC simulation results (Supplementary Fig. 9 b), the He ions with vertical stopping-range exceeds 290 nm can easily penetrate the 15 nm metal film and deposit positive charges into the 300 nm SiO 2 layer underneath. Therefore, the IBFW can be achieved on an ultra-thin metal film deposited on insulating substrate, and is not contradict with the conclusion that pure conducting substrates lead to the failure of IBFW shown in Fig. 3b . The counter electrode is a Pt probe emerged in IL, and a silver-plated probe serves as the pseudo reference electrode. In Fig. 5e , we present an example of solid particles deposited from IBFW liquid film. The upper part shows silver nanoparticles with designed pattern (a 20 μm × 3 μm channel and PKU letters) potentiostatically deposited at −0.2 V (vs. Ag) for 180 s onto the gold electrode surface. And the lower part shows blue AgTCNQ particles on gold surface make up a 50 μm × 10 μm rectangular pattern deposited at −0.1 V (vs. Ag). Noteworthily, the electrowetting phenomenon is ubiquitous in IL-electrode systems with contact angle hysteresis ranges from several to tens of degrees [63] . When the voltage applied between the IL and electrode surface is large enough to overcome contact angle hysteresis, the contact line of the IL droplet will be shifted and the IBFW patterns will be jeopardized. According to ref. [64] , a negative bias voltage exceeds −1 V would induce significant contact angle decreases for [EMIM][NTf 2 ] on gold with contact line spreading forwards, which is also observed in our experiments. A negative voltage smaller than −0.5 V, on the contrary, does not influence the contact angle obviously. Therefore, the deposition voltages adopted in current work does not influence the contact line position or the IBFW liquid film. Four cycles of cyclic voltammogram (CV) of 10 mM Ag + in IL is shown in Fig. 5f . The reduction peak of Ag + takes place at −0.29 V (vs. Ag) with the peak current decreases as the scan cycles increases. We believe that the micro litter droplet with limited analyte dissolved lead to such results. The Ag + concentration decreases quickly after each cycle of CV scan, and the electron transfer is slower for the oxidation of Ag metal. In Fig. 5g , we test the CV curves of five scan cycles of IL droplet with 6 mM Ag + and 5 mM TCNQ. Two reduction peaks can be distinguished, one at 0.095 V (vs. Ag), and the other at −0.21 V (vs. Ag). The first peak corresponds to the reduction of Ag + to Ag 0 (metal) [44] , and the second peak is related to the formation of AgTCNQ complex (solid). Finally, we conceptually validate that IBFW technology is capable of transferring liquid film pattern into various solid materials and reveal the possibilities can be produced by combining IBFW with electrochemical procedures. The three-electrode experiment configuration, example of open-circuit-potential V-t curve and potentiostatic deposit I-t curve are shown in Supplementary Fig. 22 . As summarized in Fig. 6 , the IBFW technology manifests several intriguing features that can be harnessed for a variety of application fields. The IBFW technique can fabricate patterned ILs film with 30 nm thickness, 100 nm spatial resolution and over hundreds of μm film length on insulating substrates (or coated with conducting metal films with 10 1 nm thickness). The surface-volume ratio endowed by the nanometer scale thickness can largely enhance the sensitivity of the IBFW film, and can be utilized in gas sensing circuit. The good solubility and biocompatibility of ILs make them suitable for the dissolve of various analytes. The IBFW film also exhibits robustness against air environment exposure, gravity, and physical contact to droplet reservoir. Such features reveal that the IBFW film can act as stable flow channel for the analyte solutions injected to the reservoir, and can largely simplify the fabrication procedures of micro/nanofluidic chips. Last but not least, IBFW liquid film with wide potential window can be combined with electrochemical procedures and the patterned liquid film can be transformed into different solid particles. Such results demonstrate the IBFW as a versatile tool for both nanofluidics and liquid/solid materials printing. Fig. 6: Application potential of ion-beam-induced-film-wetting technology. The distinct features of ionic liquid and IBFW technique manifest potential application fields. The nanometer scale film thickness makes IBFW suitable for gas sensing circuit manufacturing. The good solubility of IL and the on-demand patterning capability make IBFW suitable for nanofluidic and reaction chip fabrication. The wide potential window and stability of IL reveal possibility in transforming IBFW liquid film pattern into various solid materials. Full size image In summary, we have demonstrated a novel strategy to control ionic liquid flow on dielectric substrates at nanoscale leveraging a novel Ion Beam induced Film Wetting mechanism. Distinct from the prevailing approaches, IBFW manipulates liquid film with minimal modification to both solid substrates and liquid. The performances of IBFW are manifested and are imparted by the utilizing of long-range intermolecular forces to stimulate and stabilize the nanofilm. IBFW can be applied to fabricate rewritable liquid circuits with intricate desired patterns for sensing and reaction applications. The combination of electrochemistry and IBFW can even transfer the liquid film pattern into solid ranges from organic to inorganic materials. Our findings will open a new avenue for versatile application fields such as on-demand manufacture of nano circuit, in-situ combinational chemistry, efficient environmental gases sensing and detection.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"IBFW experiment The fabrication details of the experiments used SiO 2 wafers are described below. An amorphous SiO 2 layer with a thickness of approximately 3 μm is deposited by plasma-enhanced chemical vapor deposition (PECVD) on a 500 μm thickness quartz substrate surface. Prior to the experiment, chips are cleaned with acetone, ethanol, and ultrapure water sequentially with ultrasonication. To remove any surface organic pollutants, the wafers are annealed in a tubing furnace at 320 °C with a mixture gas flow of hydrogen and argon for at least 3 h, a SUNJUNE oxygen plasma cleaner can also fulfill the purpose. The ionic liquids used in experiments are bought from Lanzhou Greenchem ILs, LICP. Prior to the experiment, the ionic liquid is dried in a vacuum chamber at 80 °C for 24 h to remove the dissolved water. The tested ILs include 1-ethyl-3-methylimidazolium dicyanamide [EMIM][DCA] (98%), 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide [EMIM][NTf 2 ] (98%), 1-butyl-3-methylimidazolium tetrafluoroborate [BMIM][BF 4 ] (98%), 1-ethyl-3-methylimidazolium tetrafluoroborate [EMIM][BF 4 ] (95%) purchased from Tianjin Heowns. A drop of ionic liquid (~100 μm diameter) is deposited by the capillary tube on the top surface of the SiO 2 substrate acting as a liquid reservoir. Then, the sample is placed in a specimen holder and transferred into the vacuum chamber of a Zeiss helium ion microscope. The images are secondary electron images acquired by the raster scans of 30 kV He focused ion beam. The proof-of-concept and the rewritable tests of IBFW technology employed one single PECVD SiO 2 wafer with preparation procedures discussed previously and a micro droplet of [EMIM][DCA] ionic liquid was deposited on the silica wafer repeatedly. After sample preparation, the solid substrate with liquid reservoir settled on the top is transferred into the HIM chamber. The CTL of droplet reservoir is identified under imaging mode, and NPVE software is employed to fabricate a 20 μm × 1 μm film pattern with the flood gun turned on to guarantee the fabricated film reproduces the pattern (the reason is discussed in Supplementary Fig. S8 ). After IBFW fabrication, the sample is transferred and the film pattern is characterized under the tapping mode of AFM. Once characterization is done, the solid substrate is merged into a beaker filled with acetone and placed inside a fume cupboard for 24 h. After dissolving the IL droplet and film with acetone, an ultrasonic cleaner and deionized water are employed to eliminate the residual acetone of the substrate. Such procedures were repeated for ten times in the rewritable tests. Wetting status and liquid viscosity effect on IBFW The liquid viscosity influences on the IBFW performances are tested with four different ILs, [EMIM][DCA], [EMIM][BF 4 ], [BMIM][PF 6 ] and [EMIM][NTf 2 ]. Contact angles of the 4 ILs on three substrates PECVD SiO 2 , TOX SiO 2 , and 10 nm Au 5 nm Ti on 300 nm SiO 2 . The contact angle measurements are conducted with a homemade platform, compose of a commercialized CCD camera (SZ-CTV, OLYMPUS), a light source and a customized metal sample platform which can adjust height and tilting angle. 1 μL of ILs droplet is deposited onto the solid surface with 1 mm capillary tube. Molecular dynamics simulations The molecular dynamics simulation details can be found in Supplementary Note 6 . Gas sensing liquid nano-circuit The IL nanofilm circuit for gas sensing is fabricated as follow. Electrodes (Ti/Au, 5/50 nm) are deposited on doped Si wafer covered with 300 nm of SiO 2 by means of lithography, electron beam evaporation, and lift off procedure. Two droplets of [EMIM][DCA] IL with radius ~250 μm are deposited on the electrodes acting as the liquid reservoirs. IL film connecting two reservoirs on the electrodes is obtained by the IBFW method to construct a circuit with 0.5 μm width and 80 μm length. All the transient measurements are performed at room temperature (25 °C) employing a Keithley Sourcemeter2636B, biased with a constant voltage between the source and drain electrodes. The relative humidity (RH) data are measured by a commercialized air sensor (ClearGrass air monitor, Qing Ping) with an error ±0.5%. In-situ chemical reaction chip Sodium thiocyanate (NaSCN, 98.5%), cobalt(II) sulfate hydrate (100%) are all purchased from Macklin. Copper sulfate (CuSO 4 , 98%), iron(III) chloride (FeCl 3 , 98%) are purchased from Beijing Tongguang chemistry.The [EMIM][NTF 2 ] RTIL is chosen and four IL droplets are deposited onto a silicon dioxide chip by capillary tube to serve as the liquid reservoir for the fabrication of fluid channels employing IBFW technology. After the fabrication is finished, freshly prepared solutions are injected into the ILs droplets that have previously been deposited and connected by IBFW liquid channel. First, 0.1 μL solution of 0.1 M NaSCN in DI water is injected into the top IL droplet, and wait for 15 min for SCN − to diffuse into the entire flow channel. Then, 0.1 mM Fe 3+ solution, 0.1 mM Cu 2+ solution are injected into the bottom and left IL droplet separately. Finally, 0.1 mM Co 2+ solution is injected into the right droplet. All the injections employ a micropump. Right after the injection, the reaction chip is observed employing an optical microscope (Leica DM4 B, which is employed for all optical microscope imaging). Then the chip is stored in a vacuum chamber for 48 h, and is observed again. The customized sample holder can adjust the tilting angle 0−180°. The reaction chip is mounted to the tilting platform with 120°, and is stored in a sample box for over one week to test the IBFW film pattern stability against gravity. Electrochemical depositions Silver bis(trifluoromethylsulfonyl)imide (AgNTf 2 98% EP) is purchased from Amethyst. 7,7,8,8-tetracyanoquinodimethane (TCNQ, 98% EP) is purchased from Bidepharm. [EMIM][NTf 2 ] is chosen as the solvent and electrolyte for the cyclic voltammogram and potentiostatic deposition experiments. All the electrochemical measurements are undertaken under nitrogen atmosphere at 20−25 °C with a CHI1000C Electrochemical Workstation (CH Instrument, Texas, USA) employing a three-electrode configuration. A homemade electrode probe platform is employed to test the micro droplet. Working electrodes (Ti/Au, 5/10 nm) are deposited on doped Si wafer covered with 300 nm SiO 2 by means of lithography, electron beam evaporation, and lift off procedure. A platinum probe is stuck into the IL droplet and serves as the counter electrode. An Ag plating probe serves as the pseudo-reference electrode. We do not choose a commercialized reference electrode due to their size usually far exceed the droplets used in our experiments. Prior to experiments, the working electrodes are washed with acetone, ethanol, and ultrapure water sequentially with ultrasonication, the counter electrode and pseudo reference electrode are rinsed with ethanol and DI water. Before experiment, the IL is dried in vacuum chamber at 80 °C for 24 h to eliminate water. Then the 10 mM AgNTf 2 and 6 mM Ag + with 5 mM TCNQ solutions are prepared and deposited onto the silicon wafer with Au working electrodes. All these procedures are conducted in a glove box under nitrogen atmosphere. Then the silicon wafers with analyte solution droplets are transferred into the vacuum chamber of HIM for IBFW experiments. After the solution patterns are fabricated, the wafers are transferred to the electrochemical workstation under nitrogen atmosphere. Reporting summary Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"The core research idea is to develop a novel nanofluidic technique called ion beam induced film wetting (IBFW) that utilizes a Helium focused ion beam to achieve programmable, high-resolution patterning of ionic liquid films at the nanoscale by exploiting ion beam-induced electrical fields and disjoining pressure, enabling precise control of liquid motion on open solid surfaces without surface modifications or external circuits, and demonstrating its applicability in nano-printing, electrochemical deposition, and functional nanodevice fabrication."},"entities":{"kind":"list like","value":["nanofluidic technique","ion beam induced film wetting","Helium focused ion beam","ionic liquid films","ion beam-induced electrical fields","disjoining pressure","programmable high-resolution patterning","liquid motion control","open solid surfaces","surface modifications","external circuits","nano-printing","electrochemical deposition","functional nanodevice fabrication"],"string":"[\n \"nanofluidic technique\",\n \"ion beam induced film wetting\",\n \"Helium focused ion beam\",\n \"ionic liquid films\",\n \"ion beam-induced electrical fields\",\n \"disjoining pressure\",\n \"programmable high-resolution patterning\",\n \"liquid motion control\",\n \"open solid surfaces\",\n \"surface modifications\",\n \"external circuits\",\n \"nano-printing\",\n \"electrochemical deposition\",\n \"functional nanodevice fabrication\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1.","title":"Dai, H., Dong, Z. & Jiang, L. 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Interfacial nanostructure and asymmetric electrowetting of ionic liquids.Langmuir33, 9539–9547 (2017).\",\n \"link\": \"https://doi.org/10.1021%2Facs.langmuir.7b00082\"\n },\n {\n \"idx\": \"65.\",\n \"title\": \"Fox, D. et al. Helium ion microscopy of graphene: beam damage, image quality and edge contrast.Nanotechnology24, 335702 (2013).\",\n \"link\": \"https://doi.org/10.1088%2F0957-4484%2F24%2F33%2F335702\"\n },\n {\n \"idx\": \"66.\",\n \"title\": \"Ziegler, J. F., Ziegler, M. D. & Biersack, J. P. SRIM – The stopping and range of ions in matter (2010).Nucl. Instrum. Methods Phys. Res. Sect. B Beam Interact. Mater.268, 1818–1823 (2010).\",\n \"link\": \"https://doi.org/10.1016%2Fj.nimb.2010.02.091\"\n },\n {\n \"idx\": \"67.\",\n \"title\": \"Li, Y. et al. Ultrafast diameter-dependent water evaporation from nanopores.ACS Nano13, 3363–3372 (2019).\",\n \"link\": \"https://doi.org/10.1021%2Facsnano.8b09258\"\n },\n {\n \"idx\": \"68.\",\n \"title\": \"Magnani, M. & Gamero-Castaño, M. Energy barrier for ion field emission from a dielectric liquid sphere.Phys. Rev. E105, 054802 (2022).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevE.105.054802\"\n },\n {\n \"idx\": \"69.\",\n \"title\": \"Martínez, L., Andrade, R., Birgin, E. G. & Martínez, J. M. PACKMOL: a package for building initial configurations for molecular dynamics simulations.J. Comput. Chem.30, 2157–2164 (2009).\",\n \"link\": \"https://doi.org/10.1002%2Fjcc.21224\"\n },\n {\n \"idx\": \"70.\",\n \"title\": \"Canongia Lopes, J. N., Deschamps, J. & Pádua, A. A. H. Modeling ionic liquids using a systematic all-atom force field.J. Phys. Chem. B108, 2038–2047 (2004).\",\n \"link\": \"https://doi.org/10.1021%2Fjp0362133\"\n },\n {\n \"idx\": \"71.\",\n \"title\": \"Penkov, O. V. et al. Ion-beam irradiation of DLC-based nanocomposite: creation of a highly biocompatible surface.Appl. Surf. Sci.469, 896–903 (2019).\",\n \"link\": \"https://doi.org/10.1016%2Fj.apsusc.2018.11.109\"\n },\n {\n \"idx\": \"72.\",\n \"title\": \"Canongia Lopes, J. N., Pádua, A. A. H. & Shimizu, K. Molecular force field for ionic liquids IV: Trialkylimidazolium and Alkoxycarbonyl-Imidazolium Cations; Alkylsulfonate and Alkylsulfate Anions.J. Phys. Chem. B112, 5039–5046 (2008).\",\n \"link\": \"https://doi.org/10.1021%2Fjp800281e\"\n },\n {\n \"idx\": \"73.\",\n \"title\": \"Jorgensen, W. L., Maxwell, D. S. & Tirado-Rives, J. Development and testing of the OPLS all-atom force field on conformational energetics and properties of organic liquids.J. Am. Chem. Soc.118, 11225–11236 (1996).\",\n \"link\": \"https://doi.org/10.1021%2Fja9621760\"\n },\n {\n \"idx\": \"74.\",\n \"title\": \"Stukowski, A. Visualization and analysis of atomistic simulation data with OVITO–the open visualization tool.Model. Simul. Mater. Sci. Eng.18, 015012 (2009).\",\n \"link\": \"https://doi.org/10.1088%2F0965-0393%2F18%2F1%2F015012\"\n },\n {\n \"idx\": \"75.\",\n \"title\": \"Klomfar, J., Součková, M. & Pátek, J. Temperature dependence of the surface tension and density at 0.1 MPa for 1-Ethyl- and 1-Butyl-3-methylimidazolium Dicyanamide.J. Chem. Eng. Data56, 3454–3462 (2011).\",\n \"link\": \"https://doi.org/10.1021%2Fje200502j\"\n },\n {\n \"idx\": \"76.\",\n \"title\": \"Chen, Y., Zhao, Z. & Liu, Y. Wettability characteristic of PTFE and glass surface irradiated by keV ions.Appl. Surf. Sci.254, 5497–5500 (2008).\",\n \"link\": \"https://doi.org/10.1016%2Fj.apsusc.2008.02.097\"\n },\n {\n \"idx\": \"77.\",\n \"title\": \"Canongia Lopes, J. N. & Pádua, A. A. H. Molecular force field for ionic liquids III: Imidazolium, Pyridinium, and Phosphonium Cations; Chloride, Bromide, and Dicyanamide Anions.J. Phys. Chem. B110, 19586–19592 (2006).\",\n \"link\": \"https://doi.org/10.1021%2Fjp063901o\"\n },\n {\n \"idx\": \"78.\",\n \"title\": \"Canongia Lopes, J. N. & Pádua, A. A. H. Molecular force field for ionic liquids composed of Triflate or Bistriflylimide Anions.J. Phys. Chem. B108, 16893–16898 (2004).\",\n \"link\": \"https://doi.org/10.1021%2Fjp0476545\"\n },\n {\n \"idx\": \"79.\",\n \"title\": \"Allen, D. et al. An investigation of the radiochemical stability of ionic liquids.Green. Chem.4, 152–158 (2002).\",\n \"link\": \"https://doi.org/10.1039%2Fb111042j\"\n },\n {\n \"idx\": \"80.\",\n \"title\": \"Xue, Z., Qin, L., Jiang, J., Mu, T. & Gao, G. Thermal, electrochemical and radiolytic stabilities of ionic liquids.Phys. Chem. Chem. Phys.20, 8382–8402 (2018).\",\n \"link\": \"https://doi.org/10.1039%2FC7CP07483B\"\n }\n]","isConversational":false}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":1,"numItemsPerPage":100,"numTotalItems":11341,"offset":100,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6IjVHZDBvd0g5MTM2eDZjc1FvbE1zcktNWUZoRVFoUm5PVVVybEpjOUhaUEEifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc4NTc1MjAxNywianRpIjoiYjkzODA3YTQtYTAzMS00MDg5LThjNjEtY2Q1ODk2NmVhNzYzIiwic3ViIjoiL2RhdGFzZXRzL1hpYW8tWW91dGgvTkNfUGh5c2ljcyIsImV4cCI6MTc4NTc1NTYxNywiaXNzIjoiaHR0cHM6Ly9odWdnaW5nZmFjZS5jbyJ9.6Dcs75btgETUIM2rCdIQDnsivD1Zj4Fz_EX24jdLAtq5lFU4gVQIi5eN7lH_VjNh3Z4vLh-NHsdA0e_Klnz8Dw","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":11341,"numBytesParquet":615937106},{"config":"default","split":"test","numRows":100,"numBytesParquet":5378114}]},"discussionsStats":{"closed":0,"open":1,"total":1},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
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s41467-019-12076-x | Physics | Generating keV ion distributions for nuclear reactions at near solid-density using intense short-pulse lasers | Our understanding of a large range of astrophysical phenomena depends on a precise knowledge of charged particle nuclear reactions that occur at very low rates, which are difficult to measure under relevant plasma conditions. Here, we describe a method for generating dense plasmas at effective ion temperatures >20 keV,... | [
{
"section": "Introduction",
"content": "Our understanding of a large range of astrophysical phenomena depends on a precise knowledge of charged particle nuclear reactions that occur at very low rates [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] . Historically, measuring these reactions in astrophysi... | The core research idea is to use a laser-driven target-normal sheath acceleration (TNSA) mechanism to generate a near-solid density, isotropic plasma in unstructured nanofoam targets—protected by a solid layer from direct laser irradiation—for the controlled study of nuclear reaction rates under astrophysical condition... | [
"laser-driven target-normal sheath acceleration",
"TNSA mechanism",
"near-solid density plasma",
"isotropic plasma",
"unstructured nanofoam targets",
"solid layer",
"direct laser irradiation",
"nuclear reaction rates",
"astrophysical conditions",
"cross sections",
"traditional accelerator-based ... | [
{
"idx": "1.",
"title": "Bethe, H. A. Energy production in stars.Phys. Rev.55, 434 (1939).",
"link": "https://doi.org/10.1103%2FPhysRev.55.434"
},
{
"idx": "2.",
"title": "Burbidge, E. M., Burbidge, G. R., Fowler, W. A. & Hoyle, F. Synthesis of the elements in stars.Rev. Mod. Phys.29, 547 (1... |
ncomms2810 | Physics | Tomonaga–Luttinger physics in electronic quantum circuits | In one-dimensional conductors, interactions result in correlated electronic systems. At low energy, a hallmark signature of the so-called Tomonaga–Luttinger liquids is the universal conductance curve predicted in presence of an impurity. A seemingly different topic is the quantum laws of electricity, when distinct quan... | [
{
"section": "Introduction",
"content": "Despite a very large number of strongly interacting electrons, the low-lying electronic excitations in conventional bulk metals can be described as weakly interacting Fermi quasiparticles, as attested by the remarkable success of Landau’s Fermi liquid theory [1] . Th... | The core research idea is to investigate experimentally the dynamical Coulomb blockade (DCB) conductance suppression across a quantum coherent conductor embedded in a dissipative circuit, and to demonstrate that this phenomenon can be rigorously mapped onto the transport properties of a Tomonaga-Luttinger liquid (TLL) ... | [
"dynamical Coulomb blockade",
"DCB conductance suppression",
"quantum coherent conductor",
"dissipative circuit",
"Tomonaga-Luttinger liquid",
"TLL physics",
"impurity",
"transport properties",
"universal phenomenological scaling law",
"circuit parameters",
"quantum conductors"
] | [
{
"idx": "3",
"title": "Tomonaga, S. Remarks on Bloch’s Method of Sound Waves applied to Many-Fermion Problem.Prog. Theor. Phys.5, 544–569 (1950).",
"link": "https://doi.org/10.1143%2Fptp%2F5.4.544"
},
{
"idx": "4",
"title": "Luttinger, J. M. An exactly soluble model of a many-Fermion System... |
ncomms14107 | Nanoscience and technology | Interfacing broadband photonic qubits to on-chip cavity-protected rare-earth ensembles | Ensembles of solid-state optical emitters enable broadband quantum storage and transduction of photonic qubits, with applications in high-rate quantum networks for secure communications and interconnecting future quantum computers. To transfer quantum states using ensembles, rephasing techniques are used to mitigate fa... | [
{
"section": "Introduction",
"content": "Ensembles of rare-earth ions doped in crystals exhibit outstanding quantum coherence properties and large inhomogeneous linewidths [1] that are suitable for quantum information transfer with broadband photons in high-speed optical quantum communication networks [2] ,... | The core research idea is to demonstrate strong cavity protection against decoherence in the optical domain using a dense ensemble of neodymium atoms coupled to a nanophotonic cavity, enabling an efficient quantum light-matter interface with ~50 GHz bandwidth for future quantum networks. | [
"cavity protection",
"decoherence",
"optical domain",
"dense ensemble",
"neodymium atoms",
"nanophotonic cavity",
"quantum light-matter interface",
"50 GHz bandwidth",
"quantum networks"
] | [
{
"idx": "1",
"title": "Thiel, C., Böttger, T. & Cone, R. L. Rare-earth-doped materials for applications in quantum information storage and signal processing.J. Lumin.131, 353–361 (2011).",
"link": "https://doi.org/10.1016%2Fj.jlumin.2010.12.015"
},
{
"idx": "2",
"title": "Northup, T. E. & B... |
s41467-020-14399-6 | Energy science and technology | Stretchable fabric generates electric power from woven thermoelectric fibers | Assembling thermoelectric modules into fabric to harvest energy from body heat could one day power multitudinous wearable electronics. However, the invalid 2D architecture of fabric limits the application in thermoelectrics. Here, we make the valid thermoelectric fabric woven out of thermoelectric fibers producing an u... | [
{
"section": "Introduction",
"content": "With the escalating use of portable electronics, wearable thermoelectric generators (TEGs) are highly desired as a potential [1] power supply to harvest electricity from human body heat [2] , [3] . Intense effort has been devoted to develop flexible TE devices over t... | The core research idea is to develop a novel wearable thermoelectric generator (TEG) architecture using π-type carbon nanotube fiber-based modules interlocked in a substrate-free, three-dimensional woven design that aligns with the direction of heat flow, offering high power density, stretchability, flexibility, and sc... | [
"Wearable thermoelectric generator",
"π-type carbon nanotube fiber-based modules",
"substrate-free architecture",
"three-dimensional woven design",
"heat flow alignment",
"high power density",
"stretchability",
"flexibility",
"scalability",
"conventional two-dimensional architectures",
"substrat... | [
{
"idx": "1.",
"title": "Snyder, G. J. & Toberer, E. S. Complex thermoelectric materials.Nat. Mater.7, 105–114 (2008).",
"link": "https://doi.org/10.1038%2Fnmat2090"
},
{
"idx": "2.",
"title": "Venkatasubramanian, R., Siivola, E., Colpitts, T. & O’Quinn, B. Thin-film thermoelectric devices w... |
ncomms9611 | Astronomy and planetary science | Multiradionuclide evidence for the solar origin of the cosmic-ray events ofAD774/5 and 993/4 | The origin of two large peaks in the atmospheric radiocarbon ( 14 C) concentration at AD 774/5 and 993/4 is still debated. There is consensus, however, that these features can only be explained by an increase in the atmospheric 14 C production rate due to an extraterrestrial event. Here we provide evidence that these p... | [
{
"section": "Introduction",
"content": "The sun irregularly expels large amounts of energetic particles into the interplanetary space and into the vicinity of the Earth which can be observed as so-called solar proton events (SPE). In the context of modern society, this poses a threat to communication, elec... | By analyzing annually resolved measurements of cosmogenic radionuclides such as beryllium-10 and chlorine-36 from ice cores, this study aims to constrain the magnitude and energy spectrum of exceptional solar proton events like those in AD 775 and 994, which are hypothesized to have been significantly larger than any r... | [
"annually resolved measurements",
"cosmogenic radionuclides",
"beryllium-10",
"chlorine-36",
"ice cores",
"solar proton events",
"AD 775",
"AD 994",
"satellite era",
"fluence spectrum",
"high fluxes",
"protons",
"100 MeV kinetic energy"
] | [
{
"idx": "1",
"title": "Schrijver, C. et al. Estimating the frequency of extremely energetic solar events, based on solar, stellar, lunar, and terrestrial records.J. Geophys. Res.117, A08103 (2012).",
"link": "https://doi.org/10.1029%2F2012JA017706"
},
{
"idx": "2",
"title": "Shea, M. & Smar... |
ncomms11493 | Optics and photonics | 0.5-keV Soft X-ray attosecond continua | Attosecond light pulses in the extreme ultraviolet have drawn a great deal of attention due to their ability to interrogate electronic dynamics in real time. Nevertheless, to follow charge dynamics and excitations in materials, element selectivity is a prerequisite, which demands such pulses in the soft X-ray region, a... | [
{
"section": "Introduction",
"content": "The availability of waveform-controlled pulsed coherent radiation covering the entire water window represents a current frontier in the development of ultrafast soft X-ray sources since this spectral range contains the K-shell absorption edges of carbon (284 eV), nit... | The core research idea is to develop a high-flux, phase-matched, CEP-controlled high harmonic generation source that covers the entire water window spectral range (200–543 eV), enabling simultaneous attosecond-resolution soft X-ray absorption measurements at multiple key absorption edges—such as carbon K-shell (284 eV)... | [
"high-flux",
"phase-matched",
"CEP-controlled",
"high harmonic generation source",
"water window spectral range",
"attosecond-resolution",
"soft X-ray absorption measurements",
"carbon K-shell",
"nitrogen absorption edge",
"oxygen absorption edge",
"charge dynamics",
"electron correlation",
... | [
{
"idx": "1",
"title": "Krausz, F. & Ivanov, M. Y. Attosecond physics.Rev. Mod. Phys.81, 163–23497 (2009).",
"link": "https://doi.org/10.1103%2FRevModPhys.81.163"
},
{
"idx": "2",
"title": "Belshaw, L. et al. Observation of ultrafast charge migration in an amino acid.Phys. Chem. Lett.3, 3751... |
s41467-022-33652-8 | Nanoscience and technology | Metamaterial-enabled asymmetric negative refraction of GHz mechanical waves | Wave refraction at an interface between different materials is a basic yet fundamental phenomenon, transversal to several scientific realms – electromagnetism, gas and fluid acoustics, solid mechanics, and possibly also matter waves. Under specific circumstances, mostly enabled by structuration below the wavelength sca... | [
{
"section": "Introduction",
"content": "Coherent phenomena in wave systems strongly depend on the underlying symmetries of the potential energy landscape. Noether’s theorem [1] and its extensions link together symmetries and conserved physical quantities, ultimately allowing scientists to tailor specific e... | The core research idea is to achieve full control of mechanical wave propagation at GHz frequencies on chip-scale devices through the deliberate breaking of in-plane geometrical symmetries in GaAs metamaterials, enabling the observation and utilization of topological effects such as asymmetric and negative refraction f... | [
"mechanical wave propagation",
"GHz frequencies",
"chip-scale devices",
"in-plane geometrical symmetries",
"GaAs metamaterials",
"topological effects",
"asymmetric refraction",
"negative refraction",
"phonon routing",
"quantum technological applications"
] | [
{
"idx": "2.",
"title": "Joannopoulos, J. D., Villeneuve, P. R. & Fan, S. Photonic crystals: putting a new twist on light.Nature386, 143 (1997).",
"link": "https://doi.org/10.1038%2F386143a0"
},
{
"idx": "3.",
"title": "Portalupi, S. L. et al. Planar photonic crystal cavities with far-field ... |
s41467-020-19758-x | Materials science | Voltage-driven motion of nitrogen ions: a new paradigm for magneto-ionics | Magneto-ionics, understood as voltage-driven ion transport in magnetic materials, has largely relied on controlled migration of oxygen ions. Here, we demonstrate room-temperature voltage-driven nitrogen transport ( i.e ., nitrogen magneto-ionics) by electrolyte-gating of a CoN film. Nitrogen magneto-ionics in CoN is co... | [
{
"section": "Introduction",
"content": "Magneto-ionics [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , i.e., the change in the magnetic properties of materials due to electric field-induced ion motion, is acquiring a leading role, among other ma... | The core research idea is to demonstrate nitrogen magneto-ionics as an improved alternative to oxygen magneto-ionics by showing that voltage-driven transport of structural nitrogen in CoN films enables lower operating voltages, enhanced cyclability, and overall superior magneto-ionic effects compared to oxygen-based sy... | [
"nitrogen magneto-ionics",
"oxygen magneto-ionics",
"voltage-driven transport",
"structural nitrogen",
"CoN films",
"lower operating voltages",
"enhanced cyclability",
"magneto-ionic effects",
"nitrogen's lower electronegativity",
"weaker bonding with cobalt",
"neuromorphic computing",
"micro-... | [
{
"idx": "1.",
"title": "Molinari, A., Hahn, H. & Kruk, R. Voltage-control of magnetism in all-solid-state and solid/liquid magnetoelectric composites.Adv. Mater.31, 1806662 (2019).",
"link": "https://doi.org/10.1002%2Fadma.201806662"
},
{
"idx": "2.",
"title": "Bauer, U. et al. Magneto-ioni... |
s41467-024-53627-1 | Physics | Polar charge density wave in a superconductor with crystallographic chirality | Symmetry plays an important role in determining the physical properties in condensed matter physics, as the symmetry operations of any physical property must include the symmetry operations of the point group of the crystal. As a consequence, crystallographic polarity and chirality are expected to have an impact on the... | [
{
"section": "Introduction",
"content": "The concept of ‘ferroelectric metals’ was first proposed by Anderson and Blount more than half a century ago [1] . Ferroelectricity and metallicity are conventionally considered incompatible as itinerant electrons tend to screen long-range Coulomb interactions and we... | This research proposes that Mo₃Al₂C represents a new category of superconductor with a structural polarity and chirality induced by a charge-density wave (CDW) order, which emerges below a phase transition temperature of 155 K, and exhibits superconductivity at 8 K, making it a unique material that combines structural ... | [
"Mo₃Al₂C",
"superconductor",
"structural polarity",
"chirality",
"charge-density wave order",
"phase transition temperature",
"155 K",
"superconductivity",
"8 K",
"material",
"single system"
] | [
{
"idx": "1.",
"title": "Anderson, P. W. & Blount, E. I. Symmetry considerations on martensitic transformations: “ferroelectric\" metals?Phys. Rev. Lett.14, 217–219 (1965).",
"link": "https://doi.org/10.1103%2FPhysRevLett.14.217"
},
{
"idx": "2.",
"title": "Benedek, N. A. & Birol, T. ‘ferroe... |
s41467-022-33919-0 | Optics and photonics | Continuous entanglement distribution over a transnational 248 km fiber link | Reliable long-distance distribution of entanglement is a key technique for many quantum applications, most notably quantum key distribution. Here, we present a continuously working, trusted-node free international link between Austria and Slovakia, directly distributing polarization-entangled photon pairs via 248 km of... | [
{
"section": "Introduction",
"content": "Entanglement is the basis of many quantum applications [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] . The technically most mature of them, quantum key distribution (QKD) [9] , [10] , [11] , harnesses quantum correlations of entangled photons to produce cryptographic... | The core research idea is to demonstrate continuously operated, long-distance entanglement distribution over 248 km of deployed telecom fiber, connecting Bratislava and St. Pölten via Vienna, using state-of-the-art equipment and nonlocal polarization stabilization to achieve high duty cycle and secure key rates despite... | [
"entanglement distribution",
"deployed telecom fiber",
"Bratislava",
"St. Pölten",
"Vienna",
"state-of-the-art equipment",
"nonlocal polarization stabilization",
"high duty cycle",
"secure key rates",
"high losses",
"fiber-based quantum key distribution",
"national borders",
"trusted nodes"
... | [
{
"idx": "1.",
"title": "Bennett, C. H. & Wiesner, S. J. Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states.Phys. Rev. Lett.69, 2881 (1992).",
"link": "https://doi.org/10.1103%2FPhysRevLett.69.2881"
},
{
"idx": "2.",
"title": "Bouwmeester, D. et al. Experimen... |
s41467-018-03471-x | Optics and photonics | Photonic chip-based soliton frequency combs covering the biological imaging window | Dissipative Kerr solitons (DKS) in optical microresonators provide a highly miniaturised, chip-integrated frequency comb source with unprecedentedly high repetition rates and spectral bandwidth. To date, such frequency comb sources have been successfully applied in the optical telecommunication band for dual-comb spect... | [
{
"section": "Introduction",
"content": "Microresonator-based frequency combs are optical frequency combs generated from a continuous-wave (CW) laser via parametric four-wave mixing processes in high- Q microresonators [1] . They have attracted significant interest owing to their compactness, CMOS-compatibl... | This research proposes that Si₃N₄ microresonators can overcome the challenges of generating dissipative Kerr soliton (DKS)-based frequency combs in the near-infrared (NIR) and visible wavelength regions—where increased normal group-velocity dispersion, scattering losses, and Raman-Kerr competition typically inhibit sol... | [
"Si₃N₄ microresonators",
"dissipative Kerr soliton",
"frequency combs",
"near-infrared",
"visible wavelength regions",
"normal group-velocity dispersion",
"scattering losses",
"Raman-Kerr competition",
"dispersion engineering",
"hybridized microresonator modes",
"avoided modal crossings",
"ano... | [
{
"idx": "1.",
"title": "Del’Haye, P. et al. Optical frequency comb generation from a monolithic microresonator.Nature450, 1214–1217 (2007).",
"link": "https://doi.org/10.1038%2Fnature06401"
},
{
"idx": "2.",
"title": "Kippenberg, T. J., Holzwarth, R. & Diddams, S. Microresonator-based optic... |
s41467-022-30472-8 | Physics | Proton stopping measurements at low velocity in warm dense carbon | Ion stopping in warm dense matter is a process of fundamental importance for the understanding of the properties of dense plasmas, the realization and the interpretation of experiments involving ion-beam-heated warm dense matter samples, and for inertial confinement fusion research. The theoretical description of the i... | [
{
"section": "Introduction",
"content": "Ion stopping in warm dense matter (WDM) is an important topic in inertial confinement fusion (ICF) for the ignition of small-margin ICF targets by α -particle self-heating [1] , [2] and for ICF schemes using ion beams as the main driver, like heavy-ion fusion [3] , [... | The core research idea is to experimentally investigate ion stopping power in warm dense matter under conditions of moderate to strong electron coupling and low to moderate projectile velocity ratios (v_p/v_th 3–10) using laser-generated proton beams and precisely characterized warm dense carbon targets, in order to te... | [
"ion stopping power",
"warm dense matter",
"moderate to strong electron coupling",
"low to moderate projectile velocity ratios",
"v_p/v_th 3–10",
"laser-generated proton beams",
"precisely characterized warm dense carbon targets",
"first-principles stopping-power models",
"experimental data",
"α-p... | [
{
"idx": "1.",
"title": "Hurricane, O. et al. Approaching a burning plasma on the NIF.Phys. Plasmas26, 052704 (2019).",
"link": "https://doi.org/10.1063%2F1.5087256"
},
{
"idx": "2.",
"title": "Zylstra, A. et al. On alpha-particle transport in inertial fusion.Phys. Plasmas26, 800–806 (2019).... |
ncomms1545 | Materials science | Selective dispersion of high purity semiconducting single-walled carbon nanotubes with regioregular poly(3-alkylthiophene)s | Conjugated polymers, such as polyfluorene and poly(phenylene vinylene), have been used to selectively disperse semiconducting single-walled carbon nanotubes (sc-SWNTs), but these polymers have limited applications in transistors and solar cells. Regioregular poly(3-alkylthiophene)s (rr-P3ATs) are the most widely used m... | [
{
"section": "Introduction",
"content": "Single-walled carbon nanotubes (SWNTs) have a wide range of extraordinary mechanical, chemical and electronic properties [1] , making them ideal candidates for various applications, such as composites [2] , energy storage [3] , biological and chemical sensors [4] , [... | The core research idea is to achieve highly selective dispersion of semiconducting single-walled carbon nanotubes (sc-SWNTs) using regioregular poly(3-dodecylthiophene) (rr-P3DDT) through rational selection of polymer, solvent, and temperature, enabling direct film preparation after simple centrifugation and resulting ... | [
"semiconducting single-walled carbon nanotubes",
"regioregular poly(3-dodecylthiophene)",
"polymer selection",
"solvent selection",
"temperature selection",
"centrifugation",
"film preparation",
"SWNT network transistors",
"charge-carrier mobilities",
"on/off ratios"
] | [
{
"idx": "2",
"title": "Sekitani, T. et al. A rubberlike stretchable active matrix using elastic conductors.Science321, 1468–1472 (2008).",
"link": "https://doi.org/10.1126%2Fscience.1160309"
},
{
"idx": "3",
"title": "Berson, S., de Bettignies, R., Bailly, S., Guillerez, S. & Jousselme, B. ... |
s41467-022-29776-6 | Optics and photonics | Two octave supercontinuum generation in a non-silica graded-index multimode fiber | The generation of a two-octave supercontinuum from the visible to mid-infrared (700–2800 nm) in a non-silica graded-index multimode fiber is reported. The fiber design is based on a nanostructured core comprised of two types of drawn lead-bismuth-gallate glass rods with different refractive indices. This yields an effe... | [
{
"section": "Introduction",
"content": "Understanding the physics of complex nonlinear optical systems has been the focus of intense research in recent years and, in this context, the generation of broadband supercontinuum (SC) in graded-index multimode fibers (GRIN MMFs) has attracted particular attention... | The core research idea is to extend the generation of broadband supercontinuum sources into the mid-infrared regime by utilizing a non-silica graded-index multimode fiber made of lead-bismuth-gallate glass, which enables a two-octave supercontinuum spanning from 700 nm to 2800 nm through nonlinear spatio-temporal dynam... | [
"Broadband supercontinuum sources",
"mid-infrared regime",
"non-silica graded-index multimode fiber",
"lead-bismuth-gallate glass",
"two-octave supercontinuum",
"nonlinear spatio-temporal dynamics",
"self-phase modulation",
"geometric parametric instabilities",
"soliton dynamics",
"self-cleaning b... | [
{
"idx": "1.",
"title": "Renninger, W. H. & Wise, F. W. Optical solitons in graded-index multimode fibres.Nat. Commun.4, 1719 (2013).",
"link": "https://doi.org/10.1038%2Fncomms2739"
},
{
"idx": "2.",
"title": "Wright, L. G., Renninger, W. H., Christodoulides, D. N. & Wise, F. W. Spatiotempo... |
ncomms11797 | Materials science | Wafer-scale fabrication and growth dynamics of suspended graphene nanoribbon arrays | Adding a mechanical degree of freedom to the electrical and optical properties of atomically thin materials can provide an excellent platform to investigate various optoelectrical physics and devices with mechanical motion interaction. The large scale fabrication of such atomically thin materials with suspended structu... | [
{
"section": "Introduction",
"content": "Atomically thin materials such as graphene [1] , [2] and other two-dimensional (2D) materials [3] , [4] attract intense attention because of their prominent electrical and optical properties [5] , [6] , [7] . Because of their extraordinarily small thickness, large su... | This study proposes a novel bottom–up synthesis method for achieving wafer-scale, high-yield fabrication of suspended graphene nanoribbon (GNR) arrays with uniform edge structures and high electrical conductivity, enabling the integration of these GNRs into scalable, locally gate-tunable devices for advanced optoelectr... | [
"bottom–up synthesis method",
"wafer-scale fabrication",
"high-yield fabrication",
"suspended graphene nanoribbon arrays",
"uniform edge structures",
"high electrical conductivity",
"locally gate-tunable devices",
"optoelectronic applications",
"chemical applications",
"biological applications",
... | [
{
"idx": "1",
"title": "Novoselov, K. S. et al. Electric field effect in atomically thin carbon films.Science306, 666–669 (2004).",
"link": "https://doi.org/10.1126%2Fscience.1102896"
},
{
"idx": "2",
"title": "Geim, A. K. & Novoselov, K. S. The rise of graphene.Nat. Mater.6, 183–191 (2007).... |
s41467-022-28606-z | Chemistry | Ammonia pools in zeolites for direct fabrication of catalytic centers | Reduction process is a key step to fabricate metal-zeolite catalysts in catalytic synthesis. However, because of the strong interaction force, metal oxides in zeolites are very difficult to be reduced. Existing reduction technologies are always energy-intensive, and inevitably cause the agglomeration of metallic partic... | [
{
"section": "Introduction",
"content": "Since the first synthetic preparation of zeolites by Barrer in the 1940’s, zeolites as microporous materials represent a “New Frontier” of solid-state chemistry [1] . Up to now, several kinds of zeolites have been applied as catalysts in the oil refining and petroche... | The core research idea is to develop a novel, energy-efficient, and universally applicable method for fabricating robust metal-zeolite catalysts through an in situ self-reduction process—termed the "ammonia pool effect" (APE)—where ammonia molecules released during the calcination of NH4-zeolites reduce supported metal... | [
"metal-zeolite catalysts",
"in situ self-reduction process",
"ammonia pool effect",
"NH4-zeolites",
"calcination",
"ammonia molecules",
"supported metal oxides",
"metallic particles",
"catalytic activity",
"selectivity",
"zeolite structures",
"metal types",
"external reducing agents"
] | [
{
"idx": "1.",
"title": "Ozin, G. A., Kuperman, A. & Stein, A. Advanced zeolite, materials science.Angew. Chem. Int. Ed. Engl.28, 359–376 (1989).",
"link": "https://doi.org/10.1002%2Fanie.198903591"
},
{
"idx": "2.",
"title": "Primo, A. & Garcia, H. Zeolites as catalysts in oil refining.Chem... |
ncomms6841 | Materials science | Microwave gain medium with negative refractive index | Artificial effective media are attractive because of the fantastic applications they may enable, such as super lensing and electromagnetic invisibility. However, the inevitable loss due to their strongly dispersive nature is one of the fundamental challenges preventing such applications from becoming a reality. In this... | [
{
"section": "Introduction",
"content": "Artificial effective media are attractive because of the fantastic applications such as super lens and electromagnetic invisibility [1] , [2] . However, the inevitable loss due to their strongly dispersive nature is one of the fundamental challenges preventing such a... | This study experimentally demonstrates an active microwave metamaterial that simultaneously exhibits negative refractive index and net gain, achieved through the integration of negative-resistance devices, thereby addressing the fundamental challenge of loss in traditional metamaterials while maintaining causality and ... | [
"active microwave metamaterial",
"negative refractive index",
"net gain",
"negative-resistance devices",
"traditional metamaterials",
"loss",
"causality",
"practical applications",
"perfect lenses",
"invisibility cloaks"
] | [
{
"idx": "1",
"title": "Pendry, J. B. Negative refraction makes a perfect lens.Phys. Rev. Lett.85, 3966 (2000).",
"link": "https://doi.org/10.1103%2FPhysRevLett.85.3966"
},
{
"idx": "2",
"title": "Schurig, D. et al. Metamaterial electromagnetic cloak at microwave frequencies.Science314, 977–... |
s41467-023-43154-w | Physics | Melting and defect transitions in FeO up to pressures of Earth’s core-mantle boundary | The high-pressure melting curve of FeO controls key aspects of Earth’s deep interior and the evolution of rocky planets more broadly. However, existing melting studies on wüstite were conducted across a limited pressure range and exhibit substantial disagreement. Here we use an in-situ dual-technique approach that comb... | [
{
"section": "Introduction",
"content": "Wüstite (Fe 1- x O) has long been recognized for its prominent role in controlling the properties and evolution of Earth and other rocky planetary bodies [1] , [2] , [3] , [4] . In particular, FeO represents an end-member component in Earth’s major mineralogical syst... | This study investigates the high-pressure and high-temperature behavior of Fe0.94O, focusing on its phase diagram and potential defect order-disorder transitions, using a multi-technique approach combining synchrotron x-ray diffraction and synchrotron Mössbauer spectroscopy to address uncertainties in the melting curve... | [
"Fe0.94O",
"phase diagram",
"defect order-disorder transitions",
"synchrotron x-ray diffraction",
"synchrotron Mössbauer spectroscopy",
"melting curve",
"FeO",
"Earth's deep interior processes",
"high-pressure behavior",
"high-temperature behavior",
"multi-technique approach"
] | [
{
"idx": "1.",
"title": "Mao, H., Shu, J., Fei, Y., Hu, J. & Hemley, R. J. The wüstite enigma.Phys. Earth Planet. Inter.96, 135–145 (1996).",
"link": "https://doi.org/10.1016%2F0031-9201%2896%2903146-9"
},
{
"idx": "2.",
"title": "Hazen, R. M. & Jeanloz, R. Wüstite (Fe1-xO): a review of its ... |
s41467-020-20382-y | Nanoscience and technology | Missing Shapiro steps in topologically trivial Josephson junction on InAs quantum well | Josephson junctions hosting Majorana fermions have been predicted to exhibit a 4 π periodic current phase relation. One experimental consequence of this periodicity is the disappearance of odd steps in Shapiro steps experiments. Experimentally, missing odd Shapiro steps have been observed in a number of materials syste... | [
{
"section": "Introduction",
"content": "Recently, the drive to understand and control the order parameter characterizing the collective state of electrons in quantum heterostructures has intensified. New physical behavior can emerge that is absent in the isolated constituent materials. With regards to supe... | The core research idea is to demonstrate that missing Shapiro steps in Josephson junctions can arise from Landau–Zener transitions of high-transparency modes in multi-mode junctions, providing a robust mechanism for the fractional a.c. Josephson effect that does not require topological superconductivity, as evidenced b... | [
"Shapiro steps",
"Josephson junctions",
"Landau–Zener transitions",
"high-transparency modes",
"multi-mode junctions",
"fractional a.c. Josephson effect",
"topological superconductivity",
"high-quality junctions",
"topologically trivial phase"
] | [
{
"idx": "1.",
"title": "Kitaev, A. Y. Unpaired Majorana fermions in quantum wires.Phys. Usp.44, 131–136 (2001).",
"link": "https://doi.org/10.1070%2F1063-7869%2F44%2F10S%2FS29"
},
{
"idx": "2.",
"title": "Fu, L. & Kane, C. L. Superconducting proximity effect and Majorana fermions at the sur... |
s41467-024-52576-z | Physics | Metastable water at several compression rates and its freezing kinetics into ice VII | Water can be dynamically over-compressed well into the stability field of ice VII. Whether water then transforms into ice VII, vitreous ice or a metastable novel crystalline phase remained uncertain. We report here the freezing of over-compressed water to ice VII by time-resolved X-ray diffraction. Quasi-isothermal dyn... | [
{
"section": "Introduction",
"content": "Water under pressure is both ubiquitous in natural phenomena and a complex system. The quasi-static pressure exploration of H 2 O phase diagram, under various temperatures, has revealed a rich polymorphism, with 19 crystalline forms of ice identified so far [1] , [2]... | The core research idea is to investigate, at the microscopic level, the solidification process of metastable water under dynamic compression, specifically determining whether the freezing state at high compression rates above 2 GPa is ice VII or a metastable structure, and explaining why the phase transition sequence r... | [
"solidification process",
"metastable water",
"dynamic compression",
"freezing state",
"high compression rates",
"ice VII",
"metastable structure",
"phase transition sequence",
"rapid cooling",
"compression rates",
"time-resolved X-ray diffraction",
"complementary techniques",
"microscopic l... | [
{
"idx": "1.",
"title": "Hansen, T. C. The everlasting hunt for new ice phases.Nat. Commun.12, 3161 (2021).",
"link": "https://doi.org/10.1038%2Fs41467-021-23403-6"
},
{
"idx": "2.",
"title": "Loerting, T., Fuentes-Landete, V., Tonauer, C. M. & Gasser, T. M. Open questions on the structures ... |
s41467-018-05404-0 | Optics and photonics | Optomechanical time-domain reflectometry | Optical fibres constitute an exceptional sensing platform. However, standard fibres present an inherent sensing challenge: they confine light to an inner core. Consequently, distributed fibre sensors are restricted to the measurement of conditions that prevail within the core. This work presents distributed analysis of... | [
{
"section": "Introduction",
"content": "Optical fibre sensors cover hundreds of kilometres, are readily embedded within structures, are immune to electro-magnetic interference, and suitable for hazardous environments [1] , [2] . Fibres support distributed measurements, in which each segment becomes an inde... | The core research idea is to propose and demonstrate a new sensing concept called optomechanical time-domain reflectometry (OM-TDR) that enables distributed chemical and mechanical sensing along standard optical fibres by exploiting forward scattering induced by guided acoustic waves, thereby overcoming the limitations... | [
"optomechanical time-domain reflectometry",
"OM-TDR",
"distributed chemical sensing",
"distributed mechanical sensing",
"standard optical fibres",
"forward scattering",
"guided acoustic waves",
"traditional backscatter-based methods",
"spatially resolved measurements",
"fibre cladding",
"high pr... | [
{
"idx": "3.",
"title": "Barnoski, M. K., Rourke, M. D., Jensen, S. M. & Melville, R. T. Optical time domain reflectometer.Appl. Opt.16, 2375–2379 (1977).",
"link": "https://doi.org/10.1364%2FAO.16.002375"
},
{
"idx": "4.",
"title": "Dakin, J. P., Pratt, D. J., Bibby, G. W. & Ross, J. N. Dis... |
s41467-022-29913-1 | Mathematics and computing | Nanosecond optical switching and control system for data center networks | Electrical switching based data center networks have an intrinsic bandwidth bottleneck and, require inefficient and power-consuming multi-tier switching layers to cope with the rapid growing traffic in data centers. With the benefits of ultra-large bandwidth, high-efficient cost and power consumption, switching traffic... | [
{
"section": "Introduction",
"content": "The escalation of traffic-boosting applications and the scale out of powerful servers have significantly increased the traffic volume inside the data centers (DCs) [1] . Consequently, each aggregation switching node in the data center network (DCN) has to handle mult... | This work proposes and experimentally demonstrates a nanosecond optical switching and control system for data center networks (DCNs) that integrates a label control and synchronization mechanism for ultra-fast switch control, an Optical Flow Control protocol to resolve packet contention, and a precise clock distributio... | [
"nanosecond optical switching",
"data center networks",
"label control",
"synchronization mechanism",
"ultra-fast switch control",
"Optical Flow Control protocol",
"packet contention",
"precise clock distribution",
"nanosecond data packet recovery",
"burst-mode clock and data recovery receivers",
... | [
{
"idx": "2.",
"title": "Terzenidis, N., Moralis-Pegios, M., Mourgias-Alexandris, G., Vyrsokinos, K. & Pleros, N. High-port low-latency optical switch architecture with optical feed-forward buffering for 256-node disaggregated data centers.Opt. Express26, 8756–8766 (2018).",
"link": "https://doi.org/10.... |
s41467-019-09577-0 | Physics | Spin-current-mediated rapid magnon localisation and coalescence after ultrafast optical pumping of ferrimagnetic alloys | Sub-picosecond magnetisation manipulation via femtosecond optical pumping has attracted wide attention ever since its original discovery in 1996. However, the spatial evolution of the magnetisation is not yet well understood, in part due to the difficulty in experimentally probing such rapid dynamics. Here, we find evi... | [
{
"section": "Introduction",
"content": "Spin dynamics upon femtosecond optical pumping [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] have been intensely studied during the last two decades both because of potential applications for information storage and the... | The core research idea is to investigate the fundamental physics of spatially varying magnetisation dynamics in ferrimagnetic Gd0.24Fe0.665Co0.095 alloys after ultrafast optical pumping by identifying and characterising two distinct dynamic processes—magnon localisation and magnon coalescence—through time-resolved reso... | [
"fundamental physics",
"spatially varying magnetisation dynamics",
"ferrimagnetic Gd0.24Fe0.665Co0.095 alloys",
"ultrafast optical pumping",
"magnon localisation",
"magnon coalescence",
"time-resolved resonant X-ray scattering experiments",
"multiscale atomistic-micromagnetic simulations",
"nonlinea... | [
{
"idx": "1.",
"title": "Beaurepaire, E., Merle, J.-C., Daunois, A. & Bigot, J.-Y. Ultrafast spin dynamics in ferromagnetic nickel.Phys. Rev. Lett.76, 4250 (1996).",
"link": "https://doi.org/10.1103%2FPhysRevLett.76.4250"
},
{
"idx": "2.",
"title": "Stanciu, C. D. et al. All-optical magnetic... |
s41467-024-48850-9 | Energy science and technology | Pd(II)/Pd(IV) redox shuttle to suppress vacancy defects at grain boundaries for efficient kesterite solar cells | Charge loss at grain boundaries of kesterite Cu 2 ZnSn(S, Se) 4 polycrystalline absorbers is an important cause limiting the performance of this emerging thin-film solar cell. Herein, we report a Pd element assisted reaction strategy to suppress atomic vacancy defects in GB regions. The Pd, on one hand in the form of P... | [
{
"section": "Introduction",
"content": "Kesterite solar cell has emerged as one of the most promising thin-film photovoltaics since the absorber, Cu 2 ZnSn(S, Se) 4 (CZTSSe), is a nontoxic material composed of earth-abundant elements and has excellent thermodynamic stability and moreover this cell technolo... | The core research idea is to address the challenge of vacancy defects caused by Se or Sn volatilization at grain boundaries in CZTSSe absorbers by developing a local reaction strategy that utilizes palladium to form non-volatile PdSex compounds, which cover the grain boundaries and act as a redox shuttle to suppress el... | [
"Vacancy defects",
"Se volatilization",
"Sn volatilization",
"Grain boundaries",
"CZTSSe absorbers",
"Local reaction strategy",
"Palladium",
"Non-volatile PdSex compounds",
"Redox shuttle",
"Element loss",
"Defect density",
"High-efficiency kesterite solar cells"
] | [
{
"idx": "1.",
"title": "Wang, W. et al. Device characteristics of CZTSSe thin-film solar cells with 12.6% efficiency.Adv. Energy Mater.4, 1301465 (2014).",
"link": "https://doi.org/10.1002%2Faenm.201301465"
},
{
"idx": "2.",
"title": "Yan, C. et al. Cu2ZnSnS4solar cells with over 10% power ... |
ncomms12297 | Nanoscience and technology | Two-dimensional chiral topological superconductivity in Shiba lattices | The chiral p-wave superconductor is the archetypal example of a state of matter that supports non-Abelian anyons, a highly desired type of exotic quasiparticle. With this, it is foundational for the distant goal of building a topological quantum computer. While some candidate materials for bulk chiral superconductors e... | [
{
"section": "Introduction",
"content": "The chiral p-wave superconductor in two dimensions (2D) and the closely related fractional quantum Hall Pfaffian state at filling fraction ν =5/2 are the archetypal examples of topologically ordered states of matter that support non-Abelian anyonic excitations [1] , ... | The core research idea is to propose and analyze a strategy for realizing a two-dimensional (2D) topological chiral p-wave superconductor with tunable Chern numbers and chiral Majorana edge modes by depositing magnetic adatoms, such as transition metal ions like Fe, in a 2D lattice on the surface of a superconducting m... | [
"2D topological chiral p-wave superconductor",
"tunable Chern numbers",
"chiral Majorana edge modes",
"magnetic adatoms",
"transition metal ions",
"Fe",
"2D lattice",
"superconducting material",
"strong spin-orbit coupling",
"Pb(111)",
"exchange coupling",
"induced superconductivity",
"magne... | [] |
s41467-020-17253-x | Materials science | Phase-controllable growth of ultrathin 2D magnetic FeTe crystals | Two-dimensional (2D) magnets with intrinsic ferromagnetic/antiferromagnetic (FM/AFM) ordering are highly desirable for future spintronic devices. However, the direct growth of their crystals is in its infancy. Here we report a chemical vapor deposition approach to controllably grow layered tetragonal and non-layered he... | [
{
"section": "Introduction",
"content": "Recently, 2D magnets have drawn much attention because they provide an ideal platform for exploring magnetism down to atomic-layer thicknesses [1] , [2] . The discovery of long-range magnetic order in 2D van der Waals (vdW) crystals is vital to the understanding of t... | This research proposes the controlled synthesis of antiferromagnetic and ferromagnetic 2D FeTe nanoplates with tunable structural phases—tetragonal (layered) and hexagonal (non-layered)—on SiO2/Si substrates through a temperature-regulated CVD method, aiming to investigate the relationship between structural distortion... | [
"controlled synthesis",
"antiferromagnetic",
"ferromagnetic",
"2D FeTe nanoplates",
"structural phases",
"tetragonal phase",
"hexagonal phase",
"SiO2/Si substrates",
"temperature-regulated CVD method",
"structural distortion",
"magnetic properties",
"ferromagnetism",
"hexagonal phase (non-la... | [
{
"idx": "1.",
"title": "Burch, K. S., Mandrus, D. & Park, J.-G. Magnetism in two-dimensional van der Waals materials.Nature563, 47–52 (2018).",
"link": "https://doi.org/10.1038%2Fs41586-018-0631-z"
},
{
"idx": "2.",
"title": "Sethulakshmi, N. et al. Magnetism in two-dimensional materials be... |
s41467-018-04822-4 | Optics and photonics | Reconfigurable photonic generation of broadband chirped waveforms using a single CW laser and low-frequency electronics | Broadband radio-frequency chirped waveforms (RFCWs) with dynamically tunable parameters are of fundamental interest to many practical applications. Recently, photonic-assisted solutions have been demonstrated to overcome the bandwidth and flexibility constraints of electronic RFCW generation techniques. However, state-... | [
{
"section": "Introduction",
"content": "Radio-frequency chirped waveforms (RFCW), or “RF chirps” are broadband waveforms whose instantaneous frequency exhibits well-defined temporal variations. Waveforms with linear frequency modulation, or “linear RF chirps” are of primary importance for applications such... | The core research idea is to propose a novel photonic approach for generating broadband linear radio-frequency chirped waveforms (RFCWs) using a simple and compact system based on a single continuous-wave (CW) laser and standard low-frequency electronics, enabling high reconfigurability, exceptional time-bandwidth prod... | [
"photonic approach",
"broadband linear radio-frequency chirped waveforms",
"continuous-wave laser",
"standard low-frequency electronics",
"high reconfigurability",
"time-bandwidth product",
"real-time control",
"chirp parameters",
"mutual coherence",
"mode-locked lasers",
"long dispersive fiber ... | [
{
"idx": "5.",
"title": "Neill, J. L. et al. Segmented chirped-pulse Fourier transform sub- millimeter spectroscopy for broadband gas analysis.Opt. Express21, 19743–19749 (2013).",
"link": "https://doi.org/10.1364%2FOE.21.019743"
},
{
"idx": "6.",
"title": "Colombo, P., Zhou, Y., Prozument, ... |
s41467-024-46633-w | Physics | Revealing Fermi surface evolution and Berry curvature in an ideal type-II Weyl semimetal | In type-II Weyl semimetals (WSMs), the tilting of the Weyl cones leads to the coexistence of electron and hole pockets that touch at the Weyl nodes. These electrons and holes experience the Berry curvature generated by the Weyl nodes, leading to an anomalous Hall effect that is highly sensitive to the Fermi level posit... | [
{
"section": "Introduction",
"content": "The Weyl semimetal (WSM) is a three-dimensional gapless topological phase characterized by energy bands that intersect at points in momentum space known as Weyl nodes [1] , [2] , [3] , [4] . The stability of these band crossings relies on the breaking of either time ... | The core research idea is to identify MnBi₂₋ₓSbₓTe₄ as an ideal type-II Weyl semimetal with a single pair of Weyl nodes by experimentally tuning the Fermi level across the Weyl nodes through Sb substitution, thereby observing the evolution of Fermi-surface sections and the anomalous Hall conductivity to isolate the top... | [
"MnBi₂₋ₓSbₓTe₄",
"type-II Weyl semimetal",
"Weyl nodes",
"Fermi level",
"Sb substitution",
"Fermi-surface sections",
"anomalous Hall conductivity",
"topological contributions",
"transport behavior",
"experimentally tuning",
"single pair",
"evolution",
"theoretical prediction",
"ideal type-... | [
{
"idx": "1.",
"title": "Armitage, N. P., Mele, E. J. & Vishwanath, A. Weyl and Dirac semimetals in three-dimensional solids.Rev. Mod. Phys.90, 015001 (2018).",
"link": "https://doi.org/10.1103%2FRevModPhys.90.015001"
},
{
"idx": "2.",
"title": "Yan, B. & Felser, C. Topological materials: We... |
s41467-025-57406-4 | Materials science | Signatures of paracrystallinity in amorphous silicon from machine-learning-driven molecular dynamics | The structure of amorphous silicon has been studied for decades. The two main theories are based on a continuous random network and on a ‘paracrystalline’ model, respectively—the latter defined as showing localized structural order resembling the crystalline state whilst retaining an overall amorphous network. However,... | [
{
"section": "Introduction",
"content": "Amorphous silicon (a-Si) is one of the most widely studied disordered network solids [1] , [2] , [3] , [4] , owing in equal parts to fundamental interest and to its range of applications. In particular, a-Si has a larger band gap than its crystalline counterpart, whi... | The core research idea is to investigate the structural and energetic properties of amorphous silicon (a-Si) using a machine-learning-based molecular dynamics approach in order to explore the existence of a middle ground between fully disordered continuous random network (CRN) and crystalline structures, with the aim o... | [
"amorphous silicon",
"machine learning molecular dynamics",
"continuous random network",
"crystalline structures",
"paracrystalline model",
"structural properties",
"energetic properties",
"experimental data",
"coexistence",
"CRN",
"paracrystalline phases"
] | [
{
"idx": "1.",
"title": "Lewis, L. J. Fifty years of amorphous silicon models: the end of the story?J. Non Cryst. Solids580, 121383 (2022).",
"link": "https://doi.org/10.1016%2Fj.jnoncrysol.2021.121383"
},
{
"idx": "2.",
"title": "Treacy, M. M. J. & Borisenko, K. B. The local structure of am... |
s41467-017-01465-9 | Materials science | Charge-density-wave order takes over antiferromagnetism in Bi2Sr2−xLaxCuO6superconductors | Superconductivity appears in the cuprates when a spin order is destroyed, while the role of charge is less known. Recently, charge density wave (CDW) was found below the superconducting dome in YBa 2 Cu 3 O y when a high magnetic field is applied perpendicular to the CuO 2 plane, which was suggested to arise from incip... | [
{
"section": "Introduction",
"content": "High transition-temperature ( T c ) superconductivity is obtained by doping carriers to destroy an antiferromagnetic (AF) spin ordered Mott insulating phase. Although it is generally believed that the interaction responsible for the spin order is important for the su... | The core research idea is to investigate the relationship between intertwined antiferromagnetic spin order, charge density wave (CDW), pseudogap, and superconductivity in Bi2Sr2−xLaxCuO6+δ by applying high magnetic fields parallel to the Cu−O bond direction, aiming to disentangle these orders and reveal that long-range... | [
"antiferromagnetic spin order",
"charge density wave",
"pseudogap",
"superconductivity",
"Bi2Sr2−xLaxCuO6+δ",
"high magnetic fields",
"Cu−O bond direction",
"long-range CDW",
"superconducting dome",
"critical doping level",
"vortex core effects",
"CuO chains"
] | [
{
"idx": "1.",
"title": "Lee, P. A., Nagaosa, N. & Wen, X.-G. Doping a Mott insulator: physics of high-temperature superconductivity.Rev. Mod. Phys.78, 17–85 (2006).",
"link": "https://doi.org/10.1103%2FRevModPhys.78.17"
},
{
"idx": "2.",
"title": "Keimer, B., Kivelson, S. A., Norman, M. R.,... |
s41467-018-07778-7 | Materials science | Geometric quenching of orbital pair breaking in a single crystalline superconducting nanomesh network | In a superconductor Cooper pairs condense into a single state and in so doing support dissipation free charge flow and perfect diamagnetism. In a magnetic field the minimum kinetic energy of the Cooper pairs increases, producing an orbital pair breaking effect. We show that it is possible to significantly quench the or... | [
{
"section": "Introduction",
"content": "Fueled by the rapid advances in nanoscale materials engineering, nanoscale superconductivity has emerged [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21] , [22] , [23] , [24] , [25... | The core research idea is to determine whether it is possible to engineer a superconducting nanostructure that exhibits enhanced critical magnetic fields in both parallel and perpendicular field orientations while maintaining long-range phase coherence, and to explore the properties of such a structure using epitaxiall... | [
"superconducting nanostructure",
"enhanced critical magnetic fields",
"parallel field orientation",
"perpendicular field orientation",
"long-range phase coherence",
"epitaxially grown ultra-thin single-crystalline Pb nanomeshes",
"tunable geometries"
] | [
{
"idx": "1.",
"title": "Alexander, S. Superconductivity of networks. A percolation approach to effects of disorder.Phys. Rev. B27, 1541–1557 (1983).",
"link": "https://doi.org/10.1103%2FPhysRevB.27.1541"
},
{
"idx": "2.",
"title": "van der Zant, H. S.., Webster, M. N., Romijn, J. & Mooij, J... |
s41467-020-15860-2 | Physics | Non-Markovian recovery makes complex networks more resilient against large-scale failures | Non-Markovian spontaneous recovery processes with a time delay (memory) are ubiquitous in the real world. How does the non-Markovian characteristic affect failure propagation in complex networks? We consider failures due to internal causes at the nodal level and external failures due to an adverse environment, and deve... | [
{
"section": "Introduction",
"content": "The dynamics of failure propagation on complex networks constitute an active area of research in network science and engineering with significant and broad applications. This is because the functioning of a modern society relies on the cooperative working of many net... | This research investigates how non-Markovian recovery processes, which incorporate memory effects in node recovery, influence cascading failure propagation on complex networks, with the hypothesis that such memory effects enhance network resilience against large-scale failures compared to Markovian recovery processes w... | [
"non-Markovian recovery processes",
"memory effects",
"node recovery",
"cascading failure propagation",
"complex networks",
"network resilience",
"large-scale failures",
"Markovian recovery processes",
"constant recovery rates",
"failure dynamics",
"resilience enhancement",
"memory-dependent r... | [
{
"idx": "1.",
"title": "Motter, A. E. & Lai, Y.-C. Cascade-based attacks on complex networks.Phys. Rev. E66, 065102 (2002).",
"link": "https://doi.org/10.1103%2FPhysRevE.66.065102"
},
{
"idx": "2.",
"title": "Zhao, L., Park, K. & Lai, Y.-C. Attack vulnerability of scale-free networks due to... |
s41467-023-37477-x | Mathematics and computing | Explainable multi-task learning for multi-modality biological data analysis | Current biotechnologies can simultaneously measure multiple high-dimensional modalities (e.g., RNA, DNA accessibility, and protein) from the same cells. A combination of different analytical tasks (e.g., multi-modal integration and cross-modal analysis) is required to comprehensively understand such data, inferring how... | [
{
"section": "Introduction",
"content": "Recent advances in single-cell biotechnology make it possible to simultaneously measure gene expression along with other high-dimensional modalities for the same cells [1] , [2] , [3] . For example, the patch-seq technique simultaneously measures cell gene expression... | The core research idea is to develop an explainable multi-task deep neural network with an encoder-decoder-discriminator structure that simultaneously addresses joint group identification and cross-modal prediction in single-cell multi-modality omics data, while quantifying cell-type-specific, cross-modal feature-to-fe... | [
"explainable multi-task deep neural network",
"encoder-decoder-discriminator structure",
"single-cell multi-modality omics data",
"joint group identification",
"cross-modal prediction",
"cell-type-specific cross-modal feature-to-feature relevance",
"unified framework",
"biological insight discovery",
... | [
{
"idx": "2.",
"title": "Argelaguet, R., Cuomo, A. S. E., Stegle, O. & Marioni, J. C. Computational principles and challenges in single-cell data integration.Nat. Biotechnol.39, 1202–1215 (2021).",
"link": "https://doi.org/10.1038%2Fs41587-021-00895-7"
},
{
"idx": "3.",
"title": "Zhu, C., Pr... |
s41467-024-45293-0 | Materials science | Rapid synthesis of phosphor-glass composites in seconds based on particle self-stabilization | Phosphor-glass composites (PGC) are excellent candidates for highly efficient and stable photonic converters; however, their synthesis generally requires harsh procedures and long time, resulting in additional performance loss and energy consumption. Here we develop a rapid synthetic route to PGC within about 10 second... | [
{
"section": "Introduction",
"content": "Rare earth phosphor materials as photonic converters, exampled by commercial Y 3 Al 5 O 12 :Ce 3+ (YAG:Ce), CaAlSiN 3 :Eu 2+ , and Sr[LiAl 3 N 4 ]:Eu 2+ , etc., have received broad attentions in lighting and display [1] , [2] , [3] , [4] , sensor [5] , [6] , and phot... | This research proposes a fast synthesis strategy for phosphor-glass composites based on a particle self-stabilization model in glass melt, which utilizes the wettability between molten glass and phosphor particles to create an energy barrier that prevents particle sintering and enables the rapid (within ~10 seconds) fa... | [
"fast synthesis strategy",
"phosphor-glass composites",
"particle self-stabilization model",
"glass melt",
"wettability",
"energy barrier",
"particle sintering",
"YAG:Ce phosphor composites",
"quantum efficiency",
"absorption coefficient",
"luminous performance",
"blue laser excitation"
] | [
{
"idx": "1.",
"title": "Pust, P. et al. Narrow-band red-emitting Sr[LiAl3N4]:Eu2+as a next-generation LED-phosphor material.Nat. Mater.13, 891–896 (2014).",
"link": "https://doi.org/10.1038%2Fnmat4012"
},
{
"idx": "2.",
"title": "Hoerder, G. J. et al. Sr[Li2Al2O2N2]:Eu2+—A high performance ... |
s41467-017-00532-5 | Optics and photonics | Airway surface liquid pH is not acidic in children with cystic fibrosis | Modulation of airway surface liquid (ASL) pH has been proposed as a therapy for cystic fibrosis (CF). However, evidence that ASL pH is reduced in CF is limited and conflicting. The technical challenges associated with measuring ASL pH in vivo have precluded accurate measurements in humans. In order to address this defi... | [
{
"section": "Introduction",
"content": "Cystic fibrosis (CF) is one of the most common lethal genetic diseases, affecting ∼ 30,000 people in the United States and over 70,000 people worldwide [1] , [2] , [3] , [4] . Mortality is predominantly caused by progressive airway disease resulting in respiratory fa... | The core research idea is to test the hypothesis that, contrary to findings in newborn CF pigs, baseline airway surface liquid pH in the lower airways of young children with cystic fibrosis is not reduced compared to children without CF, and that the previously proposed CFTR-mediated reduction in ASL pH as a central pa... | [
"baseline airway surface liquid pH",
"lower airways",
"young children",
"cystic fibrosis",
"CF pigs",
"CFTR-mediated reduction",
"ASL pH",
"paracellular acid/base shunting",
"physiological bicarbonate conditions",
"early CF lung disease"
] | [
{
"idx": "1.",
"title": "Farrell, P. M. The prevalence of cystic fibrosis in the European Union.J. Cyst. Fibros.7, 450–453 (2008).",
"link": "https://doi.org/10.1016%2Fj.jcf.2008.03.007"
},
{
"idx": "5.",
"title": "O’Sullivan, B. P. & Freedman, S. D. Cystic fibrosis.Lancet373, 1891–1904 (200... |
s41467-023-41202-z | Physics | Exploring non-equilibrium processes and spatio-temporal scaling laws in heated egg yolk using coherent X-rays | The soft-grainy microstructure of cooked egg yolk is the result of a series of out-of-equilibrium processes of its protein-lipid contents; however, it is unclear how egg yolk constituents contribute to these processes to create the desired microstructure. By employing X-ray photon correlation spectroscopy, we investiga... | [
{
"section": "Introduction",
"content": "Egg yolk is widely utilised as a culinary component due to its high nutritious value [1] and excellent emulsifying and gelling abilities [2] , [3] , [4] , [5] . When heated, it undergoes a solution-to-gel transition and although this looks very simple from a cooking ... | The core research idea is to investigate the nanoscopic non-equilibrium processes involved in egg yolk gelation—specifically the temporal evolution and scaling behavior of protein gelation, LDL aggregation, and microstructural formation under heat induction—using low-dose X-ray photon correlation spectroscopy (XPCS) in... | [
"nanoscopic non-equilibrium processes",
"egg yolk gelation",
"temporal evolution",
"scaling behavior",
"protein gelation",
"LDL aggregation",
"microstructural formation",
"heat induction",
"low-dose X-ray photon correlation spectroscopy",
"ultra-small angle X-ray scattering geometry",
"time-temp... | [] |
s41467-024-45185-3 | Optics and photonics | Statistics of modal condensation in nonlinear multimode fibers | Optical pulses traveling through multimode optical fibers encounter the influence of both linear disturbances and nonlinearity, resulting in a complex and chaotic redistribution of power among different modes. In our research, we explore the phenomenon where multimode fibers reach stable states marked by the concentrat... | [
{
"section": "Introduction",
"content": "Multimode (MM) optical fibers [1] , [2] , [3] have experienced a resurgence in interest over the last decade. This renewed enthusiasm is driven by the potential to enhance transmission capacity through spatial-division multiplexing (SDM) techniques [4] , [5] , as wel... | The core research idea is to investigate the thermodynamic behavior of beam propagation in graded-index multimode optical fibers across different energy regimes—ranging from the linear regime to the soliton regime—by employing a weighted Bose-Einstein equation to model entropy extrema and analyze the emergence of local... | [
"thermodynamic behavior",
"beam propagation",
"graded-index multimode optical fibers",
"energy regimes",
"linear regime",
"soliton regime",
"weighted Bose-Einstein equation",
"entropy extrema",
"energy condensation states",
"solitons",
"quasi-solitons",
"glassy states",
"experimental observa... | [
{
"idx": "1.",
"title": "Gloge, D. & Marcatili, E. A. J. Multimode theory of graded-core fibers.Bell Syst. Tech. J.52, 1563–1578 (1973).",
"link": "https://doi.org/10.1002%2Fj.1538-7305.1973.tb02033.x"
},
{
"idx": "2.",
"title": "Marcuse, D. Derivation of coupled power equations.Bell Syst. T... |
s41467-021-22047-w | Materials science | A flexible ultrasensitive optoelectronic sensor array for neuromorphic vision systems | The challenges of developing neuromorphic vision systems inspired by the human eye come not only from how to recreate the flexibility, sophistication, and adaptability of animal systems, but also how to do so with computational efficiency and elegance. Similar to biological systems, these neuromorphic circuits integrat... | [
{
"section": "Introduction",
"content": "The human visual system is essential for both survival and learning. It is an efficient process in which the retina detects light stimuli and pre-processes image information in parallel before the brain conducts more complex actions [1] , [2] , [3] . In recent years,... | The core research idea is to develop a flexible optoelectronic sensor array using a combination of carbon nanotubes (CNTs) and all-inorganic perovskite CsPbBr3 quantum dots (QDs) as active materials, aiming to create a high-performance neuromorphic vision system that integrates image sensing, memory, and data preproces... | [
"flexible optoelectronic sensor array",
"carbon nanotubes",
"all-inorganic perovskite CsPbBr3 quantum dots",
"high-performance neuromorphic vision system",
"image sensing",
"memory",
"data preprocessing",
"exceptional sensitivity to light",
"ultra-high responsivity",
"detectivity",
"real-time pa... | [
{
"idx": "1.",
"title": "Pocock, D. C. D. Sight and knowledge.Trans. Inst. Br. Geogr.6, 385–393 (1981).",
"link": "https://doi.org/10.2307%2F621875"
},
{
"idx": "2.",
"title": "Kolb, H. How the retina works: much of the construction of an image takes place in the retina itself through the us... |
s41467-024-49153-9 | Mathematics and computing | Development and deployment of a histopathology-based deep learning algorithm for patient prescreening in a clinical trial | Accurate identification of genetic alterations in tumors, such as Fibroblast Growth Factor Receptor, is crucial for treating with targeted therapies; however, molecular testing can delay patient care due to the time and tissue required. Successful development, validation, and deployment of an AI-based, biomarker-detect... | [
{
"section": "Introduction",
"content": "Comprehensive genetic and molecular testing of cancer tissue is critical for providing physicians with actionable insights to drive precise selection of targeted cancer treatments [1] . Currently, Fibroblast Growth Factor Receptor ( FGFR ) alteration testing is used ... | This research proposes the development, validation, and clinical deployment of a deep learning-based algorithm that uses H&E-stained whole slide images from patients with advanced urothelial cancer to predict the presence of specific, clinically actionable FGFR alterations, with the goal of improving patient screening ... | [
"deep learning-based algorithm",
"H&E-stained whole slide images",
"advanced urothelial cancer",
"FGFR alterations",
"patient screening efficiency",
"molecular testing",
"targeted therapies",
"clinical trials",
"algorithm validation",
"clinical deployment",
"actionable genetic alterations",
"i... | [
{
"idx": "1.",
"title": "Malone, E. R., Oliva, M., Sabatini, P. J. B., Stockley, T. L. & Siu, L. L. Molecular profiling for precision cancer therapies.Genome Med.12, 8 (2020).",
"link": "https://link.springer.com/doi/10.1186/s13073-019-0703-1"
},
{
"idx": "2.",
"title": "Loriot, Y. et al. Er... |
s41467-023-41371-x | Energy science and technology | Multistage coupling water-enabled electric generator with customizable energy output | Constant water circulation between land, ocean and atmosphere contains great and sustainable energy, which has been successfully employed to generate electricity by the burgeoning water-enabled electric generator. However, water in various forms (e.g. liquid, moisture) is inevitably discharged after one-time use in cur... | [
{
"section": "Introduction",
"content": "Water constantly circulates between land, ocean and atmosphere in the form of gas, liquid and solid on the earth, which contains great and sustainable energy to be exploited [1] , [2] , [3] , [4] , [5] , [6] . Recently, the emerging water-enabled electric generator (... | The core research idea is to develop a multistage coupling water-enabled electric generator (mc-WEG) that integrates multifunctional layers and a water diversion design to simultaneously utilize internal liquid flow and moisture diffusion for continuous and efficient electricity generation, thereby overcoming the limit... | [
"multistage coupling water-enabled electric generator",
"mc-WEG",
"multifunctional layers",
"water diversion design",
"internal liquid flow",
"moisture diffusion",
"electricity generation",
"single-stage water-enabled generators",
"customizable power",
"high-output power",
"electronic devices"
] | [
{
"idx": "2.",
"title": "Zhang, Z. et al. Emerging hydrovoltaic technology.Nat. Nanotechnol.13, 1109–1119 (2018).",
"link": "https://doi.org/10.1038%2Fs41565-018-0228-6"
},
{
"idx": "3.",
"title": "Kleidon, A. & Renner, M. Thermodynamic limits of hydrologic cycling within the earth system: c... |
s41467-017-02127-6 | Materials science | Polymorphic design of DNA origami structures through mechanical control of modular components | Scaffolded DNA origami enables the bottom-up fabrication of diverse DNA nanostructures by designing hundreds of staple strands, comprised of complementary sequences to the specific binding locations of a scaffold strand. Despite its exceptionally high design flexibility, poor reusability of staples has been one of the ... | [
{
"section": "Introduction",
"content": "The extensive design space of scaffolded DNA origami [1] , [2] , [3] comes from the availability of drawing a unique scaffold pathway with corresponding sequence design of staple strands for each structure. By utilizing that, a number of different structures were cre... | This research proposes a modular design method for scaffolded DNA origami based on selective replacement of staple strands in specific modules of a conventional 12-helix bundle structure, enabling polymorphic variation in bending stiffness, included angle, and folding characteristics while maintaining compatibility wit... | [
"modular design method",
"scaffolded DNA origami",
"selective replacement",
"staple strands",
"12-helix bundle structure",
"polymorphic variation",
"bending stiffness",
"included angle",
"folding characteristics",
"lattice-packing rules",
"design software",
"scaffold pathway rigidity",
"cost... | [
{
"idx": "1.",
"title": "Rothemund, P. W. K. Folding DNA to create nanoscale shapes and patterns.Nature440, 297–302 (2006).",
"link": "https://doi.org/10.1038%2Fnature04586"
},
{
"idx": "2.",
"title": "Shih, W. M. & Lin, C. Knitting complex weaves with DNA origami.Curr. Opin. Struct. Biol.20... |
s41467-019-08324-9 | Physics | Reconstructing the quantum critical fan of strongly correlated systems using quantum correlations | Albeit occurring at zero temperature, quantum critical phenomena have a huge impact on the finite-temperature phase diagram of strongly correlated systems, giving experimental access to their observation. Indeed, the existence of a gapless, zero-temperature quantum critical point induces the existence of an extended re... | [
{
"section": "Introduction",
"content": "Quantum critical (QC) phenomena [1] , [2] , [3] , [4] , [5] represent possibly the most dramatic manifestation of quantum mechanics at the macroscopic scale. Their typical setting involves an Hamiltonian \\({\\cal H} = {\\cal H}_0 + gV\\) in which the competition bet... | The core research idea is to propose a constructive definition of the quantum critical regime based on observables that do not admit any classical analog—specifically quantum coherence measures—capturing quantum correlations and fluctuations in generic mixed states, and to demonstrate, through quantum Monte Carlo and e... | [
"quantum critical regime",
"quantum coherence measures",
"quantum correlations",
"quantum fluctuations",
"generic mixed states",
"quantum Monte Carlo",
"exact diagonalization",
"quantum phase transition models",
"power-law scaling",
"quantum critical fan",
"spectroscopy",
"strongly correlated ... | [
{
"idx": "1.",
"title": "Sondhi, S. L., Girvin, S. M., Carini, J. P. & Shahar, D. Continuous quantum phase transitions.Rev. Mod. Phys.69, 315–333 (1997).",
"link": "https://doi.org/10.1103%2FRevModPhys.69.315"
},
{
"idx": "6.",
"title": "Kaul, R. K., Melko, R. G. & Sandvik, A. W. Bridging la... |
s41467-021-26655-4 | Materials science | MOFs-based nanoagent enables dual mitochondrial damage in synergistic antitumor therapy via oxidative stress and calcium overload | Targeting subcellular organelle with multilevel damage has shown great promise for antitumor therapy. Here, we report a core-shell type of nanoagent with iron (III) carboxylate metal-organic frameworks (MOFs) as shell while upconversion nanoparticles (UCNPs) as core, which enables near-infrared (NIR) light-triggered sy... | [
{
"section": "Introduction",
"content": "The traditional single-modality antitumor strategy, such as chemotherapy and radiotherapy, generally encounters challenges such as adverse side effects and limited therapeutic efficacy [1] , [2] , [3] , [4] . These challenges highlight the vital significance of devel... | This research proposes a novel mitochondria-targeted antitumor strategy using a folic acid-decorated, core–shell MOF-based nanoagent (FMUP) that leverages NIR light-triggered photoacidification and Fe²⁺ release to synergistically enhance Fenton reaction-driven chemodynamic therapy and induce mitochondrial calcium overl... | [
"Mitochondria-targeted antitumor strategy",
"folic acid-decorated nanoagent",
"core–shell MOF-based nanoagent",
"FMUP",
"NIR light-triggered photoacidification",
"Fe²⁺ release",
"Fenton reaction",
"chemodynamic therapy",
"mitochondrial calcium overload",
"spatiotemporally controlled damage",
"du... | [
{
"idx": "1.",
"title": "Dantzer, R., Meagher, M. W. & Cleeland, C. S. Translational approaches to treatment-induced symptoms in cancer patients.Nat. Rev. Clin. Oncol.9, 414–426 (2012).",
"link": "https://doi.org/10.1038%2Fnrclinonc.2012.88"
},
{
"idx": "2.",
"title": "Cleeland, C. S. et al.... |
s41467-024-46025-0 | Nanoscience and technology | Targeted delivery of Fc-fused PD-L1 for effective management of acute and chronic colitis | The PD-1/PD-L1 pathway in mucosal immunity is currently actively explored and considered as a target for inflammatory bowel disease (IBD) treatment. However, systemic PD-L1 administration may cause unpredictable adverse effects due to immunosuppression. Here we show that reactive oxygen species (ROS)-responsive nanopar... | [
{
"section": "Introduction",
"content": "The incidence of inflammatory bowel disease (IBD) in Asia has increased in recent years, and patients’ quality of life is seriously affected. The pathogenesis of IBD is considered to be related to heredity, environmental factors, immunity and the intestinal flora [1]... | The core research idea is to develop and evaluate ROS-responsive nanoparticles carrying Fc-fused PD-L1 (PD-L1-Fc/Oxi-αCD nanoparticles) for targeted delivery to inflamed intestinal tissues in order to effectively alleviate colitis symptoms by modulating immune responses, inflammatory cytokines, and gut microbiota while... | [
"ROS-responsive nanoparticles",
"Fc-fused PD-L1",
"PD-L1-Fc/Oxi-αCD nanoparticles",
"inflamed intestinal tissues",
"colitis symptoms",
"immune responses",
"inflammatory cytokines",
"gut microbiota",
"biosafety",
"DSS-induced mouse models"
] | [
{
"idx": "1.",
"title": "Agrawal, M., Allin, K. H., Petralia, F., Colombel, J. F. & Jess, T. Multiomics to elucidate inflammatory bowel disease risk factors and pathways.Nat. Rev. Gastroenterol. Hepatol.19, 399–409 (2022).",
"link": "https://doi.org/10.1038%2Fs41575-022-00593-y"
},
{
"idx": "2."... |
s41467-021-27531-x | Materials science | Electroresistance in multipolar antiferroelectric Cu2Se semiconductor | Electric field-induced changes in the electrical resistance of a material are considered essential and enabling processes for future efficient large-scale computations. However, the underlying physical mechanisms of electroresistance are currently remain largely unknown. Herein, an electrically reversible resistance ch... | [
{
"section": "Introduction",
"content": "Electroresistance is a phenomenon whereby the resistance of a substance can be electrically tuned by an application of electric field [1] , [2] , [3] . Reversible alterations (switching) of the resistance in electronic devices called memristors is viewed as the essen... | The core research idea is to investigate the relationship between electrically driven crystalline domain reorientation along multiple polar directions and resistance change in thin α-Cu2Se crystals, proposing that the material exhibits multipole antiferroelectric behavior with localized "antiferroelectric-like" dipoles... | [
"electrically driven crystalline domain reorientation",
"multiple polar directions",
"resistance change",
"thin α-Cu2Se crystals",
"multipole antiferroelectric behavior",
"localized antiferroelectric-like dipoles",
"electroresistance effect",
"dynamic evolution",
"Cu+ ion positions",
"applied elec... | [
{
"idx": "1.",
"title": "Garcia, V. et al. Giant tunnel electroresistance for non-destructive readout of ferroelectric states.Nature460, 81–84 (2009).",
"link": "https://doi.org/10.1038%2Fnature08128"
},
{
"idx": "2.",
"title": "Zhuravlev, M. Y., Sabirianov, R. F., Jaswal, S. S. & Tsymbal, E... |
ncomms1097 | Nanoscience and technology | Structural bases for the interaction of frataxin with the central components of iron–sulphur cluster assembly | Reduced levels of frataxin, an essential protein of as yet unknown function, are responsible for causing the neurodegenerative pathology Friedreich's ataxia. Independent reports have linked frataxin to iron–sulphur cluster assembly through interactions with the two central components of this machinery: desulphurase Nfs... | [
{
"section": "Introduction",
"content": "Frataxin is an essential iron-binding protein that is highly conserved in most organisms, from bacteria to humans [1] . Reduced expression levels of this protein are sufficient to induce Friedreich's ataxia, a relentless and currently incurable neurodegenerative dise... | The core research idea is to determine the specific molecular interactions and functional role of frataxin, particularly its binding to components of the Nfs1/Isu complex and the influence of conserved residues—such as a surface-exposed tryptophan—in modulating iron–sulphur cluster assembly, in order to resolve the ong... | [
"frataxin",
"Nfs1/Isu complex",
"iron–sulphur cluster assembly",
"surface-exposed tryptophan",
"structural approaches",
"biochemical approaches",
"mutagenesis approaches",
"iron chaperone",
"iron scavenger",
"gatekeeper",
"cluster assembly machinery",
"in vitro studies",
"in vivo mechanisms"... | [
{
"idx": "1",
"title": "Pandolfo, M. & Pastore, A. The pathogenesis of Friedreich ataxia and the structure and function of frataxin.J. Neurol.256(Suppl 1), 9–17 (2009).",
"link": "https://link.springer.com/doi/10.1007/s00415-009-1003-2"
},
{
"idx": "2",
"title": "Cossee, M. et al. Inactivati... |
ncomms11356 | Physics | Scalable photonic network architecture based on motional averaging in room temperature gas | Quantum interfaces between photons and atomic ensembles have emerged as powerful tools for quantum technologies. Efficient storage and retrieval of single photons requires long-lived collective atomic states, which is typically achieved with immobilized atoms. Thermal atomic vapours, which present a simple and scalable... | [
{
"section": "Introduction",
"content": "Quantum systems can potentially enable powerful quantum computation [1] , [2] , [3] and highly secure quantum networks [4] , [5] , [6] , [7] . Especially for the latter, it is essential that the information can be stored in quantum memories for processing [6] , [8] .... | The core research idea is to introduce room temperature microcells with spin-protecting coatings as a scalable system for discrete variable, ensemble-based quantum information processing by theoretically and experimentally demonstrating how motional averaging can circumvent the detrimental effects of atomic motion, the... | [
"room temperature microcells",
"spin-protecting coatings",
"discrete variable",
"ensemble-based quantum information processing",
"motional averaging",
"atomic motion",
"efficient light–atom interactions",
"coherent light–atom interactions",
"single-photon level",
"scalable photonic quantum network... | [
{
"idx": "1",
"title": "Ladd, T. D. et al. Quantum computers.Nature464, 45–53 (2010).",
"link": "https://doi.org/10.1038%2Fnature08812"
},
{
"idx": "2",
"title": "Feynman, R. P. Simulating physics with computers.Int. J. Theor. Phys.21, 467–488 (1982).",
"link": "https://link.springer.com... |
s41467-024-44765-7 | Materials science | Unconventionally fast transport through sliding dynamics of rodlike particles in macromolecular networks | Transport of rodlike particles in confinement environments of macromolecular networks plays crucial roles in many important biological processes and technological applications. The relevant understanding has been limited to thin rods with diameter much smaller than network mesh size, although the opposite case, of whic... | [
{
"section": "Introduction",
"content": "The transport of guest particles in confinement environments of macromolecular networks holds significant importance in various fields, including biological science, materials science, and soft matter physics [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] . It serves ... | This research proposes to systematically investigate the transport dynamics of rodlike particles with diameters comparable to or larger than the mesh size of cross-linked macromolecular networks, aiming to uncover novel diffusion mechanisms—such as nonmonotonic translational diffusion and sliding dynamics—and establish... | [
"rodlike particles",
"cross-linked macromolecular networks",
"transport dynamics",
"diffusion mechanisms",
"nonmonotonic translational diffusion",
"sliding dynamics",
"particle transport",
"biological network systems",
"synthetic network systems",
"mesh size",
"translational diffusion",
"optim... | [
{
"idx": "1.",
"title": "Button, B. et al. Periciliary brush promotes lung health by separating the mucus layer from airway epithelia.Science337, 937–941 (2012).",
"link": "https://doi.org/10.1126%2Fscience.1223012"
},
{
"idx": "2.",
"title": "Orell, A., Fröls, S. & Albers, S. V. Archaeal Bi... |
s41467-025-57010-6 | Materials science | Local halide heterogeneity drives surface wrinkling in mixed-halide wide-bandgap perovskites | Compositional heterogeneity in wide-bandgap (1.8 − 2.1 eV) mixed-halide perovskites is a key bottleneck in the processing of high-quality solution-processed thin films and prevents their application in efficient multijunction solar cells. Notably, mixed-cation (formamidinium-methylammonium) wide-bandgap perovskite film... | [
{
"section": "Introduction",
"content": "Mixed-halide wide-bandgap ABX 3 (A is a monovalent cation, B is lead or tin, and X is a halide ion (iodide or bromide)) perovskite semiconductors are promising candidates for use in monolithic multijunction photovoltaic devices where the use of complementary absorber... | This research investigates how morphological disorder and compositional heterogeneity in mixed-cation (FA–MA) mixed-halide (I–Br) perovskite thin films—formed during antisolvent-based deposition—are linked to crystallization dynamics, local halide distribution, and defect behavior, and how these factors collectively im... | [
"morphological disorder",
"compositional heterogeneity",
"mixed-cation (FA–MA)",
"mixed-halide (I–Br)",
"perovskite thin films",
"antisolvent-based deposition",
"crystallization dynamics",
"local halide distribution",
"defect behavior",
"photoluminescence",
"charge-carrier dynamics",
"photosta... | [
{
"idx": "1.",
"title": "Hörantner, M. T. et al. The potential of multijunction perovskite solar cells.ACS Energy Lett.2, 2506–2513 (2017).",
"link": "https://doi.org/10.1021%2Facsenergylett.7b00647"
},
{
"idx": "2.",
"title": "Lin, R. et al. All-perovskite tandem solar cells with 3D/3D bila... |
s41467-019-08355-2 | Materials science | Ingestible hydrogel device | Devices that interact with living organisms are typically made of metals, silicon, ceramics, and plastics. Implantation of such devices for long-term monitoring or treatment generally requires invasive procedures. Hydrogels offer new opportunities for human-machine interactions due to their superior mechanical complian... | [
{
"section": "Introduction",
"content": "The integration of technology with the human body requires devices that are biocompatible, mechanically flexible, and robust over time in biological organisms [1] , [2] , [3] , [4] , [5] . For instance, devices that reside in the stomach for days to months can enable... | The core research idea is to develop a pufferfish-inspired ingestible hydrogel device that rapidly swells by imbibing water—rather than slow diffusion—while maintaining long-term mechanical robustness in the gastric environment, enabling prolonged physiological monitoring, drug delivery, and on-demand removal through c... | [
"pufferfish-inspired ingestible hydrogel device",
"rapid swelling",
"water imbibition",
"slow diffusion",
"long-term mechanical robustness",
"gastric environment",
"prolonged physiological monitoring",
"drug delivery",
"controlled shrinking",
"ingestible device",
"hydrogel material",
"water ab... | [
{
"idx": "1.",
"title": "Bettinger, C. J. Materials advances for next-generation ingestible electronic medical devices.Trends Biotechnol.33, 575–585 (2015).",
"link": "https://doi.org/10.1016%2Fj.tibtech.2015.07.008"
},
{
"idx": "3.",
"title": "Liu, X. et al. Stretchable living materials and... |
s41467-023-41458-5 | Energy science and technology | Impacts of climate change, population growth, and power sector decarbonization on urban building energy use | Climate, technologies, and socio-economic changes will influence future building energy use in cities. However, current low-resolution regional and state-level analyses are insufficient to reliably assist city-level decision-making. Here we estimate mid-century hourly building energy consumption in 277 U.S. urban areas... | [
{
"section": "Introduction",
"content": "Residential and commercial buildings together are responsible for 39% of the U.S. energy consumption and 28% of the U.S. greenhouse gas emissions [1] . In densely populated urban areas, the share of energy use and emissions attributable to buildings can be even highe... | This study proposes a hybrid bottom-up modeling approach that integrates high-resolution physics-based building energy simulations with statistical models and downscaled CMIP6 climate projections to quantify the impacts of climate change, population growth, and power sector decarbonization on city-scale building energy... | [
"hybrid bottom-up modeling approach",
"high-resolution physics-based building energy simulations",
"statistical models",
"downscaled CMIP6 climate projections",
"climate change",
"population growth",
"power sector decarbonization",
"city-scale building energy use intensity",
"277 urban areas",
"co... | [] |
s41467-023-42075-y | Astronomy and planetary science | Soluble organic matter Molecular atlas of Ryugu reveals cold hydrothermalism on C-type asteroid parent body | The sample from the near-Earth carbonaceous asteroid (162173) Ryugu is analyzed in the context of carbonaceous meteorites soluble organic matter. The analysis of soluble molecules of samples collected by the Hayabusa2 spacecraft shines light on an extremely high molecular diversity on the C-type asteroid. Sequential so... | [
{
"section": "Introduction",
"content": "The soluble organic matter (SOM) in carbonaceous chondrites is chemically highly diverse and integrates in its chemistry much of the history on temperature events and water-rock interaction on the parent body [1] , [2] , [3] . Two touch-down samplings by the Hayabusa... | The core research idea is to analyze the molecular characteristics of soluble organic matter (SOM) extracted from the surface sample A0106 of the C-type asteroid (162173) Ryugu using ultrahigh-resolution mass spectrometry, and to compare its organic diversity with that of 36 carbonaceous chondrites that have undergone ... | [
"soluble organic matter",
"surface sample A0106",
"C-type asteroid (162173) Ryugu",
"ultrahigh-resolution mass spectrometry",
"organic diversity",
"carbonaceous chondrites",
"moderate temperatures",
"partial aqueous alteration",
"hydrothermal processes",
"meteorites"
] | [
{
"idx": "1.",
"title": "Ph. Schmitt-Kopplin, Z. et al. High molecular diversity of extraterrestrial organic matter in Murchison meteorite revealed 40 years after its fall.Proc. Natl Acad. Sci. USA107, 2763–2768 (2010).",
"link": "https://doi.org/10.1073%2Fpnas.0912157107"
},
{
"idx": "2.",
... |
s41467-022-30206-w | Physics | Study of the doubly charmed tetraquark\({{{{{{\rm{T}}}}}}}_{{{{{{\rm{c}}}}}}{{{{{\rm{c}}}}}}}^{+}\) | Quantum chromodynamics, the theory of the strong force, describes interactions of coloured quarks and gluons and the formation of hadronic matter. Conventional hadronic matter consists of baryons and mesons made of three quarks and quark-antiquark pairs, respectively. Particles with an alternative quark content are kno... | [
{
"section": "Introduction",
"content": "Hadrons with quark content other than that seen in mesons ( \\({{{{{{\\rm{q}}}}}}}_{1}{\\bar{{{{{{\\rm{q}}}}}}}}_{2}\\) ) and baryons (q 1 q 2 q 3 ) have been actively discussed since the birth of the quark model [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] . Since ... | The core research idea is to investigate the properties of the ${{{{{{\rm{T}}}}}}}_{{{{{{\rm{c}}}}}}{{{{{\rm{c}}}}}}}^{+}$ tetraquark state with ${{{{{{\rm{c}}}}}}{{{{{\rm{c}}}}}}\overline{{{{{{\rm{u}}}}}}}\overline{{{{{{\rm{d}}}}}}}$ quark content by constructing a dedicated amplitude model that accounts for its decay... | [
"Tc c c quark content",
"tetraquark state",
"amplitude model",
"decay channels",
"mass spectra",
"production-related observables",
"proton–proton collision data",
"internal structure",
"quark content",
"dedicated amplitude model",
"c c u bar d bar quark content",
"Tc c c + tetraquark state",
... | [
{
"idx": "1.",
"title": "Gell-Mann, M. A schematic model of baryons and mesons.Phys. Lett.8, 214 (1964).",
"link": "https://doi.org/10.1016%2FS0031-9163%2864%2992001-3"
},
{
"idx": "6.",
"title": "Rossi, G. C. & Veneziano, G. A possible description of baryon dynamics in dual and gauge theori... |
s41467-023-36426-y | Materials science | Polymer-dispersed liquid crystal elastomers as moldable shape-programmable material | The current development of soft shape-memory materials often results in materials that are typically limited to the synthesis of thin-walled specimens and usually rely on complex, low-yield manufacturing techniques to fabricate macro-sized, solid three-dimensional objects. However, such geometrical limitations and slow... | [
{
"section": "Introduction",
"content": "A great number of emerging material-related technologies strive to enhance commonplace materials with additional functionalities in order to expand their usage beyond the fundamental scope. By combining the inherent characteristics of several constituents into one mu... | The core research idea is to develop a moldable shape-memory composite material, termed 'polymer-dispersed liquid crystal elastomers' (PDLCEs), that combines the intrinsic glassiness of nematic microstructured liquid crystal elastomer (MC-LCE) particles with a poly(dimethylsiloxane) (PDMS) matrix to enable reprogrammab... | [
"moldable shape-memory composite material",
"polymer-dispersed liquid crystal elastomers",
"nematic microstructured liquid crystal elastomer particles",
"poly(dimethylsiloxane) matrix",
"reprogrammable shape-memory behavior",
"temperature-controlled morphing",
"solid 3D objects",
"external magnetic fi... | [
{
"idx": "1.",
"title": "Shen, Z., Chen, F., Zhu, X., Yong, K.-T. & Gu, G. Stimuli-responsive functional materials for soft robotics.J. Mater. Chem. B8, 8972–8991 (2020).",
"link": "https://doi.org/10.1039%2FD0TB01585G"
},
{
"idx": "2.",
"title": "Liu, J. A.-C., Gillen, J. H., Mishra, S. R.,... |
s41467-025-56490-w | Materials science | Valley charge-transfer insulator in twisted double bilayer WSe2 | In flat-band systems, emergent physics can be substantially modified by the presence of another nearby electronic band. For example, a Mott ˘ Hubbard insulator can turn into a charge transfer insulator if other electronic states enter between the upper and lower Hubbard bands. Here, we introduce twisted double bilayer ... | [
{
"section": "Introduction",
"content": "Electron transport in solids with a flat dispersion can be dominated by Coulomb interactions if the kinetic energy, set by the bandwidth, is relatively much smaller. From a local Hubbard interaction picture, adding electrons is hindered by the large repulsive Coulomb... | The core research idea is to investigate a novel valley charge-transfer insulator state in a twisted transition metal dichalcogenide (TDB-WSe₂) moiré flat band system, where correlated phases and tunable insulating behavior emerge from valley-selective charge transfer between the K-valley and upper Hubbard Γ-valley, di... | [
"valley charge-transfer insulator state",
"twisted transition metal dichalcogenide",
"TDB-WSe₂",
"moiré flat band system",
"correlated phases",
"tunable insulating behavior",
"valley-selective charge transfer",
"K-valley",
"upper Hubbard Γ-valley",
"conventional Mott-Hubbard insulator",
"charge-... | [
{
"idx": "1.",
"title": "Mott, N. F. Metal-insulator transition.Rev. Mod. Phys.40, 677 (1968).",
"link": "https://doi.org/10.1103%2FRevModPhys.40.677"
},
{
"idx": "2.",
"title": "Zaanen, J., Sawatzky, G. & Allen, J. Band gaps and electronic structure of transition-metal compounds.Phys. Rev. ... |
s41467-022-28296-7 | Engineering | Granular porous landslide tsunami modelling – the 2014 Lake Askja flank collapse | Subaerial landslides and volcano flank collapses can generate tsunamis with devastating consequences. The lack of comprehensive models incorporating both the landslide and the wave mechanics represents a gap in providing consistent predictions of real events. Here, we present a novel three-dimensional granular landslid... | [
{
"section": "Introduction",
"content": "Tsunamis induced by giant landslides impacting water bodies can be catastrophic [1] , [2] , as recently manifested by the 2018 Anak Krakatau volcanic sector collapse that caused 430 fatalities in the Sunda Strait [3] , [4] , [5] . More than six other events during th... | This research proposes a novel multiphase granular flow and tsunami model based on the μ(I), ϕ(I)-rheology, complemented by critical state theory and pore fluid flow, to provide a unified and predictive framework that accurately captures both landslide dynamics and tsunami generation across laboratory and real-world sc... | [
"multiphase granular flow",
"tsunami model",
"μ(I)",
"ϕ(I)-rheology",
"critical state theory",
"pore fluid flow",
"unified framework",
"predictive framework",
"landslide dynamics",
"tsunami generation",
"laboratory scale",
"real-world scale"
] | [
{
"idx": "1.",
"title": "Weiss, R., Fritz, H. M. & Wünnemann, K. Hybrid modeling of the mega-tsunami runup in Lituya Bay after half a century.Geophys. Res. Lett.36, L09602 (2009).",
"link": "https://doi.org/10.1029%2F2009GL037814"
},
{
"idx": "2.",
"title": "Harbitz, C. B. et al. Rockslide t... |
ncomms6839 | Materials science | Nanoscale sub-100 picosecond all-optical magnetization switching in GdFeCo microstructures | Ultrafast magnetization reversal driven by femtosecond laser pulses has been shown to be a promising way to write information. Seeking to improve the recording density has raised intriguing fundamental questions about the feasibility of combining ultrafast temporal resolution with sub-wavelength spatial resolution for ... | [
{
"section": "Introduction",
"content": "The ever-increasing demands for faster and denser magnetic recording has been continuously fueling the search for ways to control magnetization in a medium by means other than magnetic fields. Several approaches based on excitation by intense Teraherz pulses [1] , [2... | The core research idea is to experimentally demonstrate nanoscale and sub-100 ps all-optical magnetization switching using linearly polarized laser pulses by exploiting the intrinsic focusing effects of laser pulse propagation within magnetic structures, as revealed through finite-difference time-domain simulations and... | [
"nanoscale",
"sub-100 ps all-optical magnetization switching",
"linearly polarized laser pulses",
"intrinsic focusing effects",
"laser pulse propagation",
"magnetic structures",
"finite-difference time-domain simulations",
"time-resolved photo-emission electron microscopy",
"X-ray magnetic circular ... | [
{
"idx": "1",
"title": "Tudosa, I. et al. The ultimate speed of magnetic switching in granular recording media.Nature428, 831–833 (2004).",
"link": "https://doi.org/10.1038%2Fnature02438"
},
{
"idx": "2",
"title": "Kampfrath, T. et al. Coherent terahertz control of antiferromagnetic spin wav... |
ncomms2951 | Nanoscience and technology | Photocurrent in graphene harnessed by tunable intrinsic plasmons | Graphene’s optical properties in the infrared and terahertz can be tailored and enhanced by patterning graphene into periodic metamaterials with sub-wavelength feature sizes. Here we demonstrate polarization-sensitive and gate-tunable photodetection in graphene nanoribbon arrays. The long-lived hybrid plasmon–phonon mo... | [
{
"section": "Introduction",
"content": "Graphene photodetectors rely on graphene for the absorption of light as well as the generation of an electrical signal through photovoltaic [1] , [2] , [3] , [4] , [5] , thermoelectric [6] , [7] , [8] , [9] or bolometric [10] , [11] , [12] mechanisms. However, the li... | The core research idea is to develop graphene photodetectors that utilize intrinsic graphene plasmons, excited via arrays of graphene nanoribbons, which couple with surface polar phonons of the SiO₂ substrate to form tunable hybrid plasmon–phonon modes that, upon decay, generate a photocurrent through distinct electron... | [
"graphene photodetectors",
"intrinsic graphene plasmons",
"graphene nanoribbons",
"surface polar phonons",
"SiO₂ substrate",
"tunable hybrid plasmon–phonon modes",
"photocurrent",
"electron temperature effects",
"phonon temperature effects",
"mid-infrared light detection",
"ambient conditions"
] | [
{
"idx": "1",
"title": "Lee, E. J. H., Balasubramanian, K., Weitz, R. T., Burghard, M. & Kern, K. . Contact and edge effects in graphene devices.Nat. Nanotech.3, 486–490 (2008).",
"link": "https://doi.org/10.1038%2Fnnano.2008.172"
},
{
"idx": "2",
"title": "Mueller, T., Xia, F., Freitag, M.,... |
s41467-024-49775-z | Engineering | Machine learning-enabled forward prediction and inverse design of 4D-printed active plates | Shape transformations of active composites (ACs) depend on the spatial distribution of constituent materials. Voxel-level complex material distributions can be encoded by 3D printing, offering enormous freedom for possible shape-change 4D-printed ACs. However, efficiently designing the material distribution to achieve ... | [
{
"section": "Introduction",
"content": "Active composites (ACs), consisting of active materials that respond differently to external stimuli, can transform their shapes once stimulated. Typical stimuli include heat [1] , [2] , [3] , [4] , light [5] , [6] , water [7] , [8] , and magnetic field [9] , [10] , ... | This research proposes a machine learning (ML)-enabled, voxel-level inverse design approach for 4D-printed active composite (AC) plates that integrates a deep residual network (ResNet)-based forward model with gradient-based and gradient-free optimization algorithms to efficiently predict and achieve complex 3D shape c... | [
"machine learning",
"voxel-level inverse design",
"4D-printed active composite plates",
"deep residual network",
"ResNet-based forward model",
"gradient-based optimization",
"gradient-free optimization",
"complex 3D shape changes",
"high-dimensional material distributions",
"large design space",
... | [
{
"idx": "1.",
"title": "Mao, Y. Q. et al. Sequential self-folding structures by 3D printed digital shape memory polymers.Sci. Rep.5, 13616 (2015).",
"link": "https://doi.org/10.1038%2Fsrep13616"
},
{
"idx": "2.",
"title": "Roach, D. J. et al. Novel ink for ambient condition printing of liqu... |
s41467-023-39114-z | Physics | Spectral signatures of a unique charge density wave in Ta2NiSe7 | Charge Density Waves (CDW) are commonly associated with the presence of near-Fermi level states which are separated from others, or “nested", by a wavector of q . Here we use Angle-Resolved Photo Emission Spectroscopy (ARPES) on the CDW material Ta 2 NiSe 7 and identify a total absence of any plausible nesting of state... | [
{
"section": "Introduction",
"content": "Charge density wave (CDW) instabilities, originally formulated by Peierls for a single 1D band at half-filling, can be naively generalised to real materials by the concept of Fermi surface nesting (FSN) [1] , [2] , [3] . If some portions of the Fermi surface can be m... | The core research idea is to investigate the unique charge density wave (CDW) instability in Ta₂NiSe₇, which defies the conventional Fermi surface nesting paradigm, by examining its three-dimensional Fermi surface and the unusual band backfolding at wavevectors q and 2q, suggesting a novel microscopic mechanism involvi... | [
"charge density wave instability",
"Ta₂NiSe₇",
"Fermi surface nesting paradigm",
"three-dimensional Fermi surface",
"band backfolding",
"wavevectors q",
"wavevectors 2q",
"novel microscopic mechanism",
"CDW formation scenarios"
] | [
{
"idx": "1.",
"title": "Wilson, J., Di Salvo, F. & Mahajan, S. Charge-density waves and superlattices in the metallic layered transition metal dichalcogenides.Adv. Phys.24, 117–201 (1975).",
"link": "https://doi.org/10.1080%2F00018737500101391"
},
{
"idx": "2.",
"title": "Whangbo, M.-H., Ca... |
s41467-023-39980-7 | Engineering | Plug & play origami modules with all-purpose deformation modes | Three basic deformation modes of an object (bending, twisting, and contraction/extension) along with their various combinations and delicate controls lead to diverse locomotion. As a result, seeking mechanisms to achieve simple to complex deformation modes in a controllable manner is a focal point in related engineerin... | [
{
"section": "Introduction",
"content": "Three basic motion types, namely, bending, twisting, and contraction/extension, as well as their combinations, constitute the primary locomotion of creatures in nature [1] , [2] . Thus, the tireless pursuit of mimicking these simple to complex motion types is the foc... | The core research idea is to develop a versatile soft actuation module based on a modified Kresling origami pattern with individually controllable pneumatic pouches, enabling the decoupled realization of three fundamental motion types—bending, twisting, and contraction/extension—as well as their combinations, thereby a... | [
"soft actuation module",
"modified Kresling origami pattern",
"individually controllable pneumatic pouches",
"bending",
"twisting",
"contraction/extension",
"motion modes",
"soft robotic systems"
] | [
{
"idx": "1.",
"title": "Rus, D. & Sung, C. Spotlight on origami robots.Sci. Robot.3, aat0938 (2018).",
"link": "https://doi.org/10.1126%2Fscirobotics.aat0938"
},
{
"idx": "2.",
"title": "Rus, D. & Tolley, M. T. Design, fabrication and control of soft robots.Nature521, 467–475 (2015).",
... |
ncomms15227 | Physics | Inferring personal economic status from social network location | It is commonly believed that patterns of social ties affect individuals’ economic status. Here we translate this concept into an operational definition at the network level, which allows us to infer the economic well-being of individuals through a measure of their location and influence in the social network. We analys... | [
{
"section": "Introduction",
"content": "The long-standing problem of how the network of social contacts [1] , [2] , [3] influences the economic status of individuals has drawn large attention due to its importance in a diversity of socioeconomic issues ranging from policy to marketing [4] , [5] , [6] , [7]... | The core research idea is to investigate how an individual's location in a social network, measured by the collective influence (CI) metric, can accurately predict their socioeconomic status at the personal level, and to demonstrate that individuals with higher economic status exhibit distinct and optimal network patte... | [
"social network",
"individual",
"socioeconomic status",
"collective influence (CI) metric",
"network patterns",
"economic status",
"micro-targeting",
"marketing",
"social interventions",
"personal level",
"optimal network patterns",
"high CI values",
"effective micro-targeting"
] | [
{
"idx": "1",
"title": "Newman, M. E. The structure and function of complex networks.SIAM Rev.45, 167–256 (2003).",
"link": "https://doi.org/10.1137%2FS003614450342480"
},
{
"idx": "4",
"title": "Granovetter, M. S. The strength of weak ties.Am. J. Sociol.78, 1360–1380 (1973).",
"link": "... |
s41467-025-58198-3 | Materials science | Polarity modulation in compositionally tunable Bi2O2Se thin films | Two-dimensional (2D) materials have emerged as one of most promising candidates to meet the demands of beyond-silicon technology. Among 2D semiconductors, Bi 2 O 2 Se (BOSe) stands out as a channel material for advanced electronic applications, due to its high electron mobility and the formation of a native high-k diel... | [
{
"section": "Introduction",
"content": "Silicon has been the dominating semiconductor for microelectronics since the invention of integrated circuits (ICs) in the late 1950s. Its moderate bandgap (~1.1 eV), excellent chemical stability, and abundance in the lithosphere enable the complex procedure and furt... | This research proposes a homogeneous integration of n-type and p-type Bi₂O₂Se (BOSe) semiconductors through site-specific chemical doping with Zn²+ to enable complementary transistor characteristics, demonstrating planar and vertical BOSe-based PN homojunctions with atomic-sharp interfaces, rectification behavior, and ... | [
"homogeneous integration",
"n-type Bi₂O₂Se",
"p-type Bi₂O₂Se",
"Zn²+ chemical doping",
"complementary transistor characteristics",
"planar BOSe-based PN homojunctions",
"vertical BOSe-based PN homojunctions",
"atomic-sharp interfaces",
"rectification behavior",
"high-performance photodetection",
... | [
{
"idx": "1.",
"title": "Ortiz-Young, D., Chiu, H. C., Kim, S., Voïtchovsky, K. & Riedo, E. The interplay between apparent viscosity and wettability in nanoconfined water.Nat. Commun.4, 2482 (2013).",
"link": "https://doi.org/10.1038%2Fncomms3482"
},
{
"idx": "2.",
"title": "Chen, M. L. et a... |
s41467-021-22646-7 | Physics | A time-domain phase diagram of metastable states in a charge ordered quantum material | Metastable self-organized electronic states in quantum materials are of fundamental importance, displaying emergent dynamical properties that may be used in new generations of sensors and memory devices. Such states are typically formed through phase transitions under non-equilibrium conditions and the final state is r... | [
{
"section": "Introduction",
"content": "Short optical or electrical pulses can create non-equilibrium conditions that lead to electronic self-organization in quantum materials which can be usefully applied in various devices [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] . Recent rapid progress in the... | The core research idea is to construct a temporal phase diagram of non-equilibrium metastable states in the quasi-2D transition metal dichalcogenide TaS₂ by combining multi-pulse femtosecond time-resolved coherent phonon spectroscopy with scanning tunneling microscopy, focusing on the dynamics of photoinduced hidden an... | [
"temporal phase diagram",
"non-equilibrium metastable states",
"quasi-2D transition metal dichalcogenide TaS₂",
"multi-pulse femtosecond time-resolved coherent phonon spectroscopy",
"scanning tunneling microscopy",
"photoinduced hidden states",
"photoinduced amorphous states",
"charged-lattice-gas mod... | [
{
"idx": "1.",
"title": "Vaskivskyi, I. et al. Fast electronic resistance switching involving hidden charge density wave states.Nat. Commun.7, 11442 (2016).",
"link": "https://doi.org/10.1038%2Fncomms11442"
},
{
"idx": "2.",
"title": "Vaskivskyi, I. et al. Controlling the metal-to-insulator ... |
ncomms13001 | Astronomy and planetary science | Earth’s magnetosphere and outer radiation belt under sub-Alfvénic solar wind | The interaction between Earth’s magnetic field and the solar wind results in the formation of a collisionless bow shock 60,000–100,000 km upstream of our planet, as long as the solar wind fast magnetosonic Mach (hereafter Mach) number exceeds unity. Here, we present one of those extremely rare instances, when the solar... | [
{
"section": "Introduction",
"content": "Under normal solar wind conditions, a bow shock forms sunward of Earth with a subsolar distance of 11–14 Earth radii ( R E =6,371 km). Behind it, in the magnetosheath, the flow becomes slower than the fast magnetosonic speed. The magnetosheath is separated from the m... | The core research idea is to investigate how Earth's magnetosphere and radiation belts respond to sub-fast solar wind conditions, which are extremely rare at Earth but potentially common for extrasolar planets, by analyzing a unique event on 17–18 January 2013 when the solar wind Mach number dropped below 1, allowing s... | [
"Earth's magnetosphere",
"radiation belts",
"sub-fast solar wind conditions",
"extrasolar planets",
"solar wind Mach number",
"dayside magnetosphere",
"magnetopause",
"radiation belt dynamics",
"unique event",
"17–18 January 2013",
"simultaneous observations",
"extreme conditions",
"space we... | [
{
"idx": "1",
"title": "Shue, J.-H. et al. Magnetopause location under extreme solar wind conditions.J. Geophys. Res.103, 17691–17700 (1998).",
"link": "https://doi.org/10.1029%2F98JA01103"
},
{
"idx": "2",
"title": "Farris, M. H. & Russell, C. T. Determining the standoff distance of the bow... |
s41467-019-13378-w | Materials science | Approaching diamond’s theoretical elasticity and strength limits | Diamond is the hardest natural material, but its practical strength is low and its elastic deformability extremely limited. While recent experiments have demonstrated that diamond nanoneedles can sustain exceptionally large elastic tensile strains with high tensile strengths, the size- and orientation-dependence of the... | [
{
"section": "Introduction",
"content": "The ideal strength of a crystalline solid is defined as the maximum stress a perfect crystal lattice can withstand at 0 K [1] , [2] . However, virtually no practical bulk materials survive under the ideal maximum stress because of the presence of internal defects and... | This research proposes to investigate the size, orientation, and surface-defect dependence of elastic deformation in high-surface-quality diamond nanoneedles with various diameters and crystallographic orientations (<100>, <110>, <111>), using a combination of nanomechanical bending experiments, finite element method s... | [
"diamond nanoneedles",
"elastic deformation",
"surface-defect dependence",
"nanomechanical bending experiments",
"finite element method simulations",
"first principles calculations",
"local strain",
"tensile properties",
"crystallographic orientations",
"<100> orientation",
"<110> orientation",
... | [
{
"idx": "1.",
"title": "Liu, F., Ming, P. & Li, J. Ab initio calculation of ideal strength and phonon instability of graphene under tension.Phys. Rev. B76, 064120 (2007).",
"link": "https://doi.org/10.1103%2FPhysRevB.76.064120"
},
{
"idx": "2.",
"title": "Zhu, T. & Li, J. Ultra-strength mat... |
s41467-023-40152-w | Optics and photonics | Harnessing microcomb-based parallel chaos for random number generation and optical decision making | Optical chaos is vital for various applications such as private communication, encryption, anti-interference sensing, and reinforcement learning. Chaotic microcombs have emerged as promising sources for generating massive optical chaos. However, their inter-channel correlation behavior remains elusive, limiting their p... | [
{
"section": "Introduction",
"content": "Chaos is a fundamental phenomenon in physics that exhibits random behaviors due to its great sensitivity to small changes of conditions [1] . It has been playing key roles behind a wide range of applications for modern society: in communications, the generation of ch... | The core research idea is to investigate and demonstrate the use of an optical chaotic comb generated in a high-quality, high-nonlinear optical microcavity as a massively parallel chaotic source with low inter-channel correlation, enabling scalable and integrated applications in secure communications, high-speed random... | [
"optical chaotic comb",
"high-quality optical microcavity",
"high-nonlinear optical microcavity",
"massively parallel chaotic source",
"low inter-channel correlation",
"secure communications",
"high-speed random number generation",
"optical computations",
"interference-free ranging technologies"
] | [
{
"idx": "2.",
"title": "Argyris, A. et al. Chaos-based communications at high bit rates using commercial fibre-optic links.Nature438, 343–346 (2005).",
"link": "https://doi.org/10.1038%2Fnature04275"
},
{
"idx": "3.",
"title": "Wang, L. et al. Scheme of coherent optical chaos communication.... |
ncomms13454 | Optics and photonics | Proposal and proof-of-principle demonstration of non-destructive detection of photonic qubits using a Tm:LiNbO3waveguide | Non-destructive detection of photonic qubits is an enabling technology for quantum information processing and quantum communication. For practical applications, such as quantum repeaters and networks, it is desirable to implement such detection in a way that allows some form of multiplexing as well as easy integration ... | [
{
"section": "Introduction",
"content": "The possibility to detect the presence of a visible or telecommunication-wavelength photon in a non-destructive [1] and state-preserving manner is highly desirable for photonic quantum information processing [2] and quantum networks [3] , [4] , [5] as it makes it pos... | The core research idea is to propose and experimentally validate a solid-state scheme for non-destructive detection of photonic time-bin qubits using an impurity-doped crystal, which preserves the qubit state, allows for multiplexing, and is compatible with existing quantum information processing components, with poten... | [
"solid-state scheme",
"non-destructive detection",
"photonic time-bin qubits",
"impurity-doped crystal",
"qubit state preservation",
"multiplexing",
"quantum information processing components",
"single-photon level implementation",
"experimental validation"
] | [
{
"idx": "1",
"title": "Imoto, N., Haus, H. A. & Yamamoto, Y. Quantum nondemolition measurement of the photon number via the optical Kerr effect.Phys. Rev. A32, 2287–2292 (1985).",
"link": "https://doi.org/10.1103%2FPhysRevA.32.2287"
},
{
"idx": "2",
"title": "Nemoto, K. & Munro, W. J. Nearl... |
s41467-024-45156-8 | Nanoscience and technology | A nanoparticle-based sonodynamic therapy reducesHelicobacter pyloriinfection in mouse without disrupting gut microbiota | Infection by Helicobacter pylori , a prevalent global pathogen, currently requires antibiotic-based treatments, which often lead to antimicrobial resistance and gut microbiota dysbiosis. Here, we develop a non-antibiotic approach using sonodynamic therapy mediated by a lecithin bilayer-coated poly(lactic-co-glycolic) n... | [
{
"section": "Introduction",
"content": "H elicobacter pylori ( H. pylori ) is a human-transmittable pathogen that infects over 50% of the world’s population [1] . For example, in China, the H. pylori infection rate for adults is 40–50% [2] . H. pylori infection can lead to various complications including c... | The core research idea is to develop a non-antibiotic sonodynamic therapy using FDA-approved nanoparticles (Ver-PLGA@Lecithin) combined with low-intensity ultrasound (0.5 W/cm² for 10 min) to effectively eradicate gastric H. pylori infection while preserving gut microbiota integrity, eliminating the virulence factor Va... | [
"non-antibiotic sonodynamic therapy",
"FDA-approved nanoparticles",
"Ver-PLGA@Lecithin",
"low-intensity ultrasound",
"gastric H. pylori infection",
"gut microbiota integrity",
"virulence factor VacA",
"antimicrobial resistance",
"microbiota dysbiosis",
"conventional antibiotic therapies",
"0.5 W... | [
{
"idx": "1.",
"title": "Hooi, J. K. Y. et al. Global prevalence of helicobacter pylori infection: systematic review and meta-analysis.Gastroenterology153, 420–429 (2017).",
"link": "https://doi.org/10.1053%2Fj.gastro.2017.04.022"
},
{
"idx": "2.",
"title": "O’Connor, A., O’Morain, C. A. & F... |
ncomms12927 | Optics and photonics | Interplay between evanescence and disorder in deep subwavelength photonic structures | Deep subwavelength features are expected to have minimal impact on wave transport. Here we show that in contrast to this common understanding, disorder can have a dramatic effect in a one-dimensional disordered optical system with spatial features a thousand times smaller than the wavelength. We examine a unique regime... | [
{
"section": "Introduction",
"content": "Localization due to disorder was originally proposed by Anderson [1] in 1958 in the context of the metal insulator transition in solid-state physics. Over the years, it has been shown that localization phenomena are fundamental and ubiquitous [2] , [3] , manifested i... | The core research idea is to investigate whether Anderson localization can be induced by multiple scattering of evanescent waves in disordered multilayer structures with extreme subwavelength layer thicknesses (d ∼ λ/1,000), and to demonstrate that such localization—driven by Goos–Hanchen phase shifts—results in expone... | [
"Anderson localization",
"multiple scattering",
"evanescent waves",
"disordered multilayer structures",
"extreme subwavelength layer thicknesses",
"Goos–Hanchen phase shifts",
"exponentially localized modes",
"short localization lengths",
"high-Q resonances",
"enhanced transmission",
"nanoscale ... | [] |
ncomms4577 | Optics and photonics | Optofluidic sorting of material chirality by chiral light | The lack of mirror symmetry, chirality, plays a fundamental role in physics, chemistry and life sciences. The passive separation of entities that only differ by their handedness without need of a chiral material environment remains a challenging task with attractive scientific and industrial benefits. To date, only a f... | [
{
"section": "Introduction",
"content": "Harnessing material handedness is a topical challenge for many industries as it is a basic requirement towards the synthesis of drugs, additives or pesticides, to name a few. This implies the development of chiral sorting strategies that correspond to the ability to ... | The core research idea is to experimentally demonstrate a passive optofluidic chiral sorting strategy that enables the separation of mirror-imaged chiral microparticles based solely on their opposite handedness by exploiting the intrinsic chirality of light-matter interactions, without requiring chiral-shaped particles... | [
"passive optofluidic chiral sorting strategy",
"mirror-imaged chiral microparticles",
"opposite handedness",
"intrinsic chirality",
"light-matter interactions",
"chiral-shaped particles",
"chiral recognition",
"chiral light fields",
"fluidic environment"
] | [
{
"idx": "1",
"title": "Gübitz, G. & Schmid, M. G. Chiral separation principles in chromatographic and electromigration techniques.Mol. Biotechnol.32, 159–179 (2006).",
"link": "https://doi.org/10.1385%2FMB%3A32%3A2%3A159"
},
{
"idx": "2",
"title": "Ward, T. J. & Baker, B. A. Chiral separati... |
ncomms13485 | Materials science | Spin injection and inverse Edelstein effect in the surface states of topological Kondo insulator SmB6 | There has been considerable interest in exploiting the spin degrees of freedom of electrons for potential information storage and computing technologies. Topological insulators (TIs), a class of quantum materials, have special gapless edge/surface states, where the spin polarization of the Dirac fermions is locked to t... | [
{
"section": "Introduction",
"content": "Spintronics aims to use the spin degrees of freedom for information technologies [1] , [2] , [3] . The injection of spin-polarized carriers into two-dimensional quantum materials, including graphene and the surface states of topological insulators (TIs), is particula... | The core research idea is to demonstrate spin injection into the surface states of SmB6, a topological Kondo insulator, using spin pumping and to observe the inverse Edelstein effect, thereby confirming the spin–momentum locking properties of these topologically protected surface states. | [
"spin injection",
"surface states",
"SmB6",
"topological Kondo insulator",
"spin pumping",
"inverse Edelstein effect",
"spin–momentum locking",
"topologically protected surface states"
] | [] |
ncomms14468 | Physics | Ubiquitous strong electron–phonon coupling at the interface of FeSe/SrTiO3 | The observation of replica bands in single-unit-cell FeSe on SrTiO 3 (STO)(001) by angle-resolved photoemission spectroscopy (ARPES) has led to the conjecture that the coupling between FeSe electrons and the STO phonons are responsible for the enhancement of T c over other FeSe-based superconductors. However the recent... | [
{
"section": "Introduction",
"content": "The discovery of high-temperature superconductivity in single-unit-cell (1 UC) FeSe/STO(001) continues to attract a great deal of interest [1] . In particular, a recent mutual inductance measurement has shown an onset of diamagnetism at the same temperature (65 K) as... | The core research idea is to investigate whether the enhanced superconducting transition temperature (Tc) in single-unit-cell FeSe films grown on SrTiO3 (STO) substrates, particularly on the (110) surface, is caused by a generic electron–phonon coupling mechanism similar to that observed at the (001) interface, where s... | [
"superconducting transition temperature",
"single-unit-cell FeSe films",
"SrTiO3 substrates",
"(110) surface",
"electron–phonon coupling mechanism",
"(001) interface",
"FeSe electrons",
"STO phonons",
"replica bands",
"bulk FeSe"
] | [
{
"idx": "1",
"title": "Wang, Q.-Y. et al. Interface-induced high-temperature superconductivity in single unit-cell FeSe films on SrTiO3 .Chin. Phys. Lett.29, 037402 (2012).",
"link": "https://doi.org/10.1088%2F0256-307X%2F29%2F3%2F037402"
},
{
"idx": "2",
"title": "Zhang, Z. et al. Onset of... |
s41467-023-42026-7 | Physics | Spin-filtered measurements of Andreev bound states in semiconductor-superconductor nanowire devices | Semiconductor nanowires coupled to superconductors can host Andreev bound states with distinct spin and parity, including a spin-zero state with an even number of electrons and a spin-1/2 state with odd-parity. Considering the difference in spin of the even and odd states, spin-filtered measurements can reveal the unde... | [
{
"section": "Introduction",
"content": "Low-dimensional III-V semiconductors proximitized via coupling to superconductors have been researched extensively in recent decades [1] , [2] . Interest in these hybrid systems is a result of their gate-tunability, strong response to magnetic fields, and large spin-... | The core research idea is to investigate the spin-polarized excitation spectrum of an Andreev bound state (ABS) in a hybrid InSb-Al nanowire using a three-terminal setup with a spin-polarized quantum dot and a conventional tunnel junction, aiming to demonstrate complete spin polarization of the ABS, its reversal under ... | [
"spin-polarized excitation spectrum",
"Andreev bound state",
"hybrid InSb-Al nanowire",
"three-terminal setup",
"spin-polarized quantum dot",
"conventional tunnel junction",
"complete spin polarization",
"magnetic field reversal",
"transport blockade",
"persistent spin-polarized zero-bias peak"
] | [
{
"idx": "1.",
"title": "Aghaee, M. et al. InAs-Al hybrid devices passing the topological gap protocol.Phys. Rev. B107, 245423 (2023).",
"link": "https://doi.org/10.1103%2FPhysRevB.107.245423"
},
{
"idx": "2.",
"title": "Prada, E. et al. From Andreev to Majorana bound states in hybrid superc... |
s41467-024-46748-0 | Nanoscience and technology | Velocities of transmission eigenchannels and diffusion | The diffusion model is used to calculate both the time-averaged flow of particles in stochastic media and the propagation of waves averaged over ensembles of disordered static configurations. For classical waves exciting static disordered samples, such as a layer of paint or a tissue sample, the flux transmitted throug... | [
{
"section": "Introduction",
"content": "The diffusion equation describes the flow of particles, waves and energy from neutrons, electrical charge, molecules and microscopic particles to light, sound, and heat [1] , [2] , [3] , [4] . The diffusion model begins with the assumption that scattering is local—th... | This study investigates how the velocity distributions of transmission eigenchannels on the input and output surfaces of multiply scattering media are preserved rather than randomized by multiple scattering, and how the interplay between transmission eigenvalues and longitudinal velocity components affects energy densi... | [
"velocity distributions",
"transmission eigenchannels",
"multiply scattering media",
"multiple scattering",
"transmission eigenvalues",
"longitudinal velocity components",
"energy density",
"diffusion coefficient",
"boundary conditions",
"conductance",
"sample geometry",
"transverse mode prope... | [
{
"idx": "6.",
"title": "Lagendijk, A., Vreeker, R. & De Vries, P. Influence of internal reflection on diffusive transport in strongly scattering media.Phys. Lett. A136, 81–88 (1989).",
"link": "https://doi.org/10.1016%2F0375-9601%2889%2990683-X"
},
{
"idx": "7.",
"title": "Zhu, J. X., Pine,... |
s41467-019-13727-9 | Materials science | Biointerfacial self-assembly generates lipid membrane coated bacteria for enhanced oral delivery and treatment | The gut microbiota represents a huge community of microorganisms that play essential roles in immune modulation and homeostasis maintenance. Microbiota transplantation is an important approach to prevent and treat disease as it can inhibit pathogen colonization and positively modulate bacterial composition. However, th... | [
{
"section": "Introduction",
"content": "The gut microbiota represents a huge community of microorganisms including archaea, bacteria, fungi, viruses and microeukaryotes that occupy the gastrointestinal (GI) tract [1] , [2] . Intensive research on the gut microbial community has provided growing evidence th... | The core research idea is to develop a novel method for creating super gut microbes by coating bacteria with a self-assembled lipid membrane, which enhances their survival in harsh gastrointestinal conditions, maintains their bioactivity and colonization ability, and improves their therapeutic efficacy in disease preve... | [
"super gut microbes",
"self-assembled lipid membrane",
"bacteria",
"gastrointestinal conditions",
"bioactivity",
"colonization ability",
"therapeutic efficacy",
"disease prevention",
"disease treatment",
"mouse models",
"infection",
"colitis"
] | [
{
"idx": "1.",
"title": "Sekirov, I., Russell, S. L., Antunes, L. C. & Finlay, B. B. Gut microbiota in health and disease.Physiol. Rev.90, 859–904 (2010).",
"link": "https://doi.org/10.1152%2Fphysrev.00045.2009"
},
{
"idx": "2.",
"title": "Brussow, H. Biome engineering-2020.Microb. Biotechno... |
s41467-018-03407-5 | Optics and photonics | Experimental observation of photonic nodal line degeneracies in metacrystals | Nodal line semimetals (NLS) are three-dimensional (3D) crystals that support band crossings in the form of one-dimensional rings in the Brillouin zone. In the presence of spin–orbit coupling or lowered crystal symmetry, NLS may transform into Dirac semimetals, Weyl semimetals, or 3D topological insulators. In the photo... | [
{
"section": "Introduction",
"content": "Recent progress in the research on topological phases of matter has led to the exciting findings of Chern insulators [1] , topological insulators [2] , [3] , Weyl semimetals [4] , [5] , [6] , Dirac semimetals [7] , etc. The concept of nontrivial topological physics i... | The core research idea is to experimentally realize a clean nodal line in a cut-wire metacrystal, thereby introducing nodal line degeneracies into photonic metacrystals and metamaterials for the first time, which opens a pathway to investigate unique topological features such as drumhead surface states, topological pha... | [
"nodal line",
"cut-wire metacrystal",
"photonic metacrystals",
"metamaterials",
"nodal line degeneracies",
"drumhead surface states",
"topological phase transitions",
"Weyl degeneracies",
"resonance scattering",
"negative refraction"
] | [
{
"idx": "1.",
"title": "Haldane, F. D. M. Model for a quantum Hall effect without Landau levels: condensed-matter realization of the ‘parity anomaly’.Phys. Rev. Lett.61, 2015–2018 (1988).",
"link": "https://doi.org/10.1103%2FPhysRevLett.61.2015"
},
{
"idx": "2.",
"title": "Fu, L., Kane, C. ... |
ncomms2635 | Nanoscience and technology | Quantum dot imaging platform for single-cell molecular profiling | Study of normal cell physiology and disease pathogenesis heavily relies on untangling the complexity of intracellular molecular mechanisms and pathways. To achieve this goal, comprehensive molecular profiling of individual cells within the context of microenvironment is required. Here we report the development of a mul... | [
{
"section": "Introduction",
"content": "Comprehensive and quantitative molecular profiling on the single-cell level has the potential to unravel complex molecular pathways that underlie physiological and pathological processes, and thus represents a highly desirable tool for many applications that span the... | The core research idea is to develop a multicolour multicycle molecular profiling (M3P) technology based on quantum dot-protein A (QD-SpA) probes that enables highly multiplexed, quantitative, and subcellular-resolution molecular profiling of over 100 targets in single cells while preserving specimen morphology, thereb... | [
"multicolour multicycle molecular profiling",
"M3P technology",
"quantum dot-protein A probes",
"QD-SpA probes",
"highly multiplexed molecular profiling",
"quantitative molecular profiling",
"subcellular-resolution molecular profiling",
"single cells",
"specimen morphology",
"current techniques",
... | [
{
"idx": "1",
"title": "de Souza, N. Single-cell methods.Nat. Methods9, 35 (2012).",
"link": "https://doi.org/10.1038%2Fnmeth.1819"
},
{
"idx": "2",
"title": "Liotta, L. & Petricoin, E. Molecular profiling of human cancer.Nat. Rev. Genet.1, 48–56 (2000).",
"link": "https://doi.org/10.103... |
s41467-019-13643-y | Mathematics and computing | A first-principle mechanism for particulate aggregation and self-assembly in stratified fluids | An extremely broad and important class of phenomena in nature involves the settling and aggregation of matter under gravitation in fluid systems. Here, we observe and model mathematically an unexpected fundamental mechanism by which particles suspended within stratification may self-assemble and form large aggregates w... | [
{
"section": "Introduction",
"content": "Particle sedimentation and aggregation in stratified fluids is observed throughout natural systems. Some examples include: sedimenting “marine snow” particles in lakes and oceans (central to carbon sequestration) [1] , dense microplastics in the oceans (which impact ... | This research investigates an unexplored, adhesion-independent mechanism of particle attraction and self-assembly in stratified fluids, driven by first-principle fluid dynamics phenomena, through a combination of experiments, simulations, and theoretical analysis to understand how flow structures induced by sedimenting... | [
"stratified fluids",
"particle attraction",
"self-assembly",
"adhesion-independent mechanism",
"first-principle fluid dynamics",
"sedimenting particles",
"flow structures",
"collective self-organization",
"short-range binding effects",
"experiments",
"simulations",
"theoretical analysis"
] | [
{
"idx": "1.",
"title": "MacIntyre, S., Alldredge, A. L. & Gottschalk, C. C. Accumulation of marine snow at density discontinuities in the water column.Limnol. Oceanogr.40, 449–468 (1995).",
"link": "https://doi.org/10.4319%2Flo.1995.40.3.0449"
},
{
"idx": "2.",
"title": "Jamieson, A. J., Br... |
s41467-024-54956-x | Mathematics and computing | Globally scalable glacier mapping by deep learning matches expert delineation accuracy | Accurate global glacier mapping is critical for understanding climate change impacts. Despite its importance, automated glacier mapping at a global scale remains largely unexplored. Here we address this gap and propose Glacier-VisionTransformer-U-Net (GlaViTU), a convolutional-transformer deep learning model, and five ... | [
{
"section": "Introduction",
"content": "Glaciers are highly sensitive to alterations in temperature and precipitation, making them important indicators of climate change, and recognised as an essential climate variable within the Global Climate Observing System (GCOS) programme [1] . In the last decades, t... | This study proposes the development and application of a convolutional-transformer deep learning model, combined with a comprehensive multimodal satellite dataset, to enable automated, accurate, and globally generalisable glacier mapping across diverse environmental conditions and time periods, while also quantifying m... | [
"convolutional-transformer deep learning model",
"multimodal satellite dataset",
"glacier mapping",
"environmental conditions",
"time periods",
"mapping uncertainty",
"automated mapping",
"accurate mapping",
"globally generalisable mapping",
"transparent decision-making",
"deep learning",
"sat... | [
{
"idx": "4.",
"title": "Rounce, D. R. et al. Global glacier change in the 21st century: every increase in temperature matters.Science379, 78–83 (2023).",
"link": "https://doi.org/10.1126%2Fscience.abo1324"
},
{
"idx": "5.",
"title": "Zemp, M. et al. Global glacier mass changes and their con... |
s41467-022-29170-2 | Materials science | Current-induced Néel order switching facilitated by magnetic phase transition | Terahertz (THz) spin dynamics and vanishing stray field make antiferromagnetic (AFM) materials the most promising candidate for the next-generation magnetic memory technology with revolutionary storage density and writing speed. However, owing to the extremely large exchange energy barriers, energy-efficient manipulati... | [
{
"section": "Introduction",
"content": "Antiferromagnetic materials are promising candidates for the high-density and fast-speed spintronic devices in the future [1] , [2] , [3] , [4] , the analogy to current ferromagnet (FM)-based spintronic applications such as hard disk drive and magnetic random-access ... | The core research idea is to enhance the energy efficiency of current-driven Néel vector switching in antiferromagnetic materials by utilizing the magnetic phase transition between antiferromagnetic and ferromagnetic states in FeRh, where Joule heating-induced switching to the ferromagnetic phase allows efficient manip... | [
"energy efficiency",
"current-driven Néel vector switching",
"antiferromagnetic materials",
"magnetic phase transition",
"antiferromagnetic state",
"ferromagnetic state",
"FeRh",
"Joule heating-induced switching",
"ferromagnetic phase",
"spin-orbit torque",
"switched magnetic state",
"antiferr... | [
{
"idx": "1.",
"title": "Baltz, V. et al. Antiferromagnetic spintronics.Rev. Mod. Phys.90, 015005 (2018).",
"link": "https://doi.org/10.1103%2FRevModPhys.90.015005"
},
{
"idx": "2.",
"title": "Jungwirth, T., Marti, X., Wadley, P. & Wunderlich, J. Antiferromagnetic spintronics.Nat. Nanotechno... |
s41467-023-41263-0 | Energy science and technology | A systematic discrepancy between the short circuit current and the integrated quantum efficiency in perovskite solar cells | Halide perovskites solar cells are now approaching commercialisation. In this transition from academic research towards industrialisation, standardized testing protocols and reliable dissemination of performance metrics are crucial. In this study, we analyze data from over 16,000 publications in the Perovskite Database... | [
{
"section": "Introduction",
"content": "In the last decade, halide perovskites have emerged as a class of promising solar cell materials. During this time, record efficiencies have surpassed 25% [1] , [2] and the research has gone from basic research to gradually also containing more technology-oriented de... | The core research idea is to investigate the reliability of perovskite solar cell performance metrics by comparing the short-circuit current obtained from standard JV measurements with that derived from integrated external quantum efficiency measurements, in order to address measurement discrepancies caused by factors ... | [
"perovskite solar cell",
"short-circuit current",
"JV measurements",
"integrated external quantum efficiency measurements",
"calibration uncertainties",
"spectral mismatch",
"hysteresis",
"transient effects",
"power conversion efficiencies"
] | [
{
"idx": "1.",
"title": "Green, M. A. et al. Solar cell efficiency tables (Version 58).Prog. Photovoltaics: Res. Appl.29, 657–667 (2021).",
"link": "https://doi.org/10.1002%2Fpip.3444"
},
{
"idx": "2.",
"title": "Min, H. et al. Perovskite solar cells with atomically coherent interlayers on S... |
s41467-025-57111-2 | Nanoscience and technology | Magnetic Bloch states at integer flux quanta induced by super-moiré potential in graphene aligned with twisted boron nitride | Two-dimensional electron systems in both magnetic fields and periodic potentials are described by the Hofstadter butterfly, a fundamental problem of solid-state physics. While moiré systems provide a powerful method to realize this type of spectrum, previous experiments have been limited to fractional flux quanta regim... | [
{
"section": "Introduction",
"content": "The Hofstadter butterfly is a self-similar recursive energy spectrum to describe the quantum behavior of an electron moving in both a two-dimensional (2D) periodic potential and a magnetic field [1] , which is a fundamental problem of solid-state physics. At commensu... | The core research idea is to construct a super-moiré structure by aligning monolayer graphene with two twisted hexagonal boron nitride layers, creating a large-period moiré potential that enables the experimental realization of magnetic Bloch states at integer flux quanta ($\phi /{\phi }_{0}=1-9$), thereby allowing the... | [
"super-moiré structure",
"monolayer graphene",
"twisted hexagonal boron nitride layers",
"large-period moiré potential",
"magnetic Bloch states",
"integer flux quanta",
"Brown-Zak oscillations",
"Hofstadter butterfly",
"topological phases",
"two-dimensional superlattices"
] | [
{
"idx": "1.",
"title": "Hofstadter, D. R. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields.Phys. Rev. B14, 2239 (1976).",
"link": "https://doi.org/10.1103%2FPhysRevB.14.2239"
},
{
"idx": "2.",
"title": "Brown, E. Bloch electrons in a uniform mag... |
s41467-023-37210-8 | Nanoscience and technology | Superfluid response of an atomically thin gate-tuned van der Waals superconductor | A growing number of two-dimensional superconductors are being discovered in the family of exfoliated van der Waals materials. Due to small sample volume, the superfluid response of these materials has not been characterized. Here, we use a local magnetic probe to directly measure this key property of the tunable, gate-... | [
{
"section": "Introduction",
"content": "The two defining properties of a superconductor are a vanishing electrical resistance and the expulsion of magnetic fields below a characteristic critical temperature, T c . Typically, superconductivity is first identified in a material by observing a sharp drop in t... | The core research idea is to directly measure the magnetic response of the gate-induced superconducting state in few-layer MoS₂ to probe the superfluid stiffness and characterize the nature of superconductivity, including potential unconventional behavior, in this atomically thin van der Waals material. | [
"magnetic response",
"gate-induced superconducting state",
"few-layer MoS₂",
"superfluid stiffness",
"nature of superconductivity",
"unconventional behavior",
"atomically thin van der Waals material"
] | [
{
"idx": "2.",
"title": "Uemura, Y. J. et al. Universal correlations between Tc and ns/m* (Carrier density over effective mass) in High-Tccuprate superconductors.Phys. Rev. Lett.62, 2317–2320 (1989).",
"link": "https://doi.org/10.1103%2FPhysRevLett.62.2317"
},
{
"idx": "3.",
"title": "Lember... |
ncomms4908 | Physics | Optical hyperpolarization and NMR detection of129Xe on a microfluidic chip | Optically hyperpolarized 129 Xe gas has become a powerful contrast agent in nuclear magnetic resonance (NMR) spectroscopy and imaging, with applications ranging from studies of the human lung to the targeted detection of biomolecules. Equally attractive is its potential use to enhance the sensitivity of microfluidic NM... | [
{
"section": "Introduction",
"content": "Owing to the ease with which it may be placed in a non-equilibrium spin polarization state, 129 Xe gas has found extensive use in recent nuclear magnetic resonance (NMR) spectroscopy and magnetic resonance imaging (MRI) experiments, greatly enhancing the resultant NM... | The core research idea is to develop a microfabricated, portable, and low-cost source of hyperpolarized 129 Xe gas that integrates with microfluidic systems and enables fully chip-based, high-sensitivity nuclear magnetic resonance (NMR) instrumentation through the combination of optical hyperpolarization, optical magne... | [
"microfabricated source",
"hyperpolarized 129 Xe gas",
"microfluidic systems",
"chip-based NMR instrumentation",
"optical hyperpolarization",
"optical magnetometry",
"microfluidic technologies",
"portable source",
"low-cost source",
"nuclear magnetic resonance instrumentation",
"gas source",
"... | [
{
"idx": "1",
"title": "Walker, T. G. & Happer, W. Spin-exchange optical pumping of noble-gas nuclei.Rev. Mod. Phys.69, 629–642 (1997).",
"link": "https://doi.org/10.1103%2FRevModPhys.69.629"
},
{
"idx": "2",
"title": "Goodson, B. M. Nuclear magnetic resonance of laser-polarized noble gases ... |
s41467-021-22909-3 | Physics | Nanoscale real-time detection of quantum vortices at millikelvin temperatures | Since we still lack a theory of classical turbulence, attention has focused on the conceptually simpler turbulence in quantum fluids. Reaching a better understanding of the quantum case may provide additional insight into the classical counterpart. That said, we have hitherto lacked detectors capable of the real-time, ... | [
{
"section": "Introduction",
"content": "Despite the long history of classical turbulence, and its very significant impact on human affairs we still lack a theory, since the governing Navier-Stokes equations are only soluble in quite specific simple situations. The importance of a general solution to these ... | The core research idea is to demonstrate the detection of single quantum vortices in superfluid helium-4 in real time using nanobeam resonators, enabling the study of vortex tangle dynamics at the single-vortex level and on the scale of vortex separation, thereby advancing our understanding of quantum turbulence and it... | [
"single quantum vortices",
"superfluid helium-4",
"nanobeam resonators",
"vortex tangle dynamics",
"single-vortex level",
"vortex separation",
"quantum turbulence",
"classical turbulence"
] | [
{
"idx": "3.",
"title": "Griffiths, D. J. & Allen, J. F. The behaviour of vortex lines near a wire in liquid helium ii.Proc. R. Soc. Lond. Ser. A277, 214–236 (1964).",
"link": "https://doi.org/10.1098%2Frspa.1964.0018"
},
{
"idx": "4.",
"title": "Zieve, R. J., Frei, C. M. & Wolfson, D. L. En... |
s41467-024-50574-9 | Optics and photonics | Room-temperature quantum nanoplasmonic coherent perfect absorption | Light-matter superposition states obtained via strong coupling play a decisive role in quantum information processing, but the deleterious effects of material dissipation and environment-induced decoherence inevitably destroy coherent light-matter polaritons over time. Here, we propose the use of coherent perfect absor... | [
{
"section": "Introduction",
"content": "Quantum electrodynamic strong coupling lies at the heart of contemporary research efforts in condensed-matter quantum optics, representing both a fundamentally profound and technologically compelling regime of light-matter interaction [1] , [2] , [3] . In the strong ... | The core research idea is to propose and theoretically explore a novel strategy for selectively initializing and temporally preserving plasmon-exciton polariton states in a nanocavity-emitter system at room temperature through quantum nanoplasmonic coherent perfect absorption (qnCPA), which leverages coherent, unidirec... | [
"plasmon-exciton polariton states",
"nanocavity-emitter system",
"quantum nanoplasmonic coherent perfect absorption",
"qnCPA",
"coherent energy transfer",
"unidirectional energy transfer",
"non-perturbing energy transfer",
"intrinsic losses",
"strongly coupled light-matter states",
"room temperatu... | [
{
"idx": "1.",
"title": "Ebbesen, T. W. Hybrid light-matter states in a molecular and material science perspective.Acc. Chem. Res.49, 2403–2412 (2016).",
"link": "https://doi.org/10.1021%2Facs.accounts.6b00295"
},
{
"idx": "2.",
"title": "Flick, J., Rivera, N. & Narang, P. Strong light-matte... |
s41467-021-26482-7 | Physics | Magnetism-induced topological transition in EuAs3 | The nature of the interaction between magnetism and topology in magnetic topological semimetals remains mysterious, but may be expected to lead to a variety of novel physics. We systematically studied the magnetic semimetal EuAs 3 , demonstrating a magnetism-induced topological transition from a topological nodal-line ... | [
{
"section": "Introduction",
"content": "Topological semimetals (TSMs), including Dirac, Weyl, nodal-line, and triple-point semimetals, can be divided into two categories—non-magnetic and magnetic TSMs depending on whether magnetism is involved [1] , [2] , [3] . Compared with better-known non-magnetic TSMs,... | The core research idea is to demonstrate a magnetism-induced topological transition from a topological nodal-line semimetal in the paramagnetic or spin-polarized state to a topological massive Dirac metal in the antiferromagnetic ground state in EuAs₃, through a combination of band calculations, transport measurements,... | [
"magnetism-induced topological transition",
"topological nodal-line semimetal",
"paramagnetic state",
"spin-polarized state",
"topological massive Dirac metal",
"antiferromagnetic ground state",
"EuAs₃",
"band calculations",
"transport measurements",
"Shubnikov–de Haas oscillations",
"angle-reso... | [
{
"idx": "1.",
"title": "Armitage, N. P., Mele, E. J. & Vishwanath, A. Weyl and Dirac semimetals in three-dimensional solids.Rev. Mod. Phys.90, 015001 (2018).",
"link": "https://doi.org/10.1103%2FRevModPhys.90.015001"
}
] |
ncomms9590 | Astronomy and planetary science | Wave-driven butterfly distribution of Van Allen belt relativistic electrons | Van Allen radiation belts consist of relativistic electrons trapped by Earth's magnetic field. Trapped electrons often drift azimuthally around Earth and display a butterfly pitch angle distribution of a minimum at 90° further out than geostationary orbit. This is usually attributed to drift shell splitting resulting f... | [
{
"section": "Introduction",
"content": "The Earth’s outer Van Allen radiation belt is composed of trapped electrons with relativistic energy ( E k ≥1 MeV). Fluxes and pitch angle distributions of relativistic electrons often exhibit dramatic and highly dynamic changes during geomagnetic storms or substorms... | The core research idea is to investigate how combined chorus and magnetosonic wave acceleration generates an unusual butterfly pitch angle distribution of relativistic electrons around L = 4.8 in Earth's outer Van Allen radiation belt, as observed during the 28 June 2013 geomagnetic storm and supported by two-dimension... | [
"combined chorus and magnetosonic wave acceleration",
"unusual butterfly pitch angle distribution",
"relativistic electrons",
"L = 4.8",
"Earth's outer Van Allen radiation belt",
"28 June 2013 geomagnetic storm",
"two-dimensional simulations",
"medium pitch angles",
"near-90° pitch angles",
"flux ... | [
{
"idx": "1",
"title": "Roederer, J. G. On the adiabatic motion of energetic particles in a model magnetosphere.J. Geophys. Res.72, 981–992 (1967).",
"link": "https://doi.org/10.1029%2FJZ072i003p00981"
},
{
"idx": "2",
"title": "Stone, E. C. The physical significance of and application ofL,B... |
s41467-021-27149-z | Physics | Second sound in the crossover from the Bose-Einstein condensate to the Bardeen-Cooper-Schrieffer superfluid | Second sound is an entropy wave which propagates in the superfluid component of a quantum liquid. Because it is an entropy wave, it probes the thermodynamic properties of the quantum liquid. Here, we study second sound propagation for a large range of interaction strengths within the crossover between a Bose-Einstein c... | [
{
"section": "Introduction",
"content": "Second sound is a transport phenomenon of quantum liquids that emerges below the critical temperature for superfluidity T c [1] , [2] , [3] . It was experimentally discovered [4] in 1944 in He II [5] and was described with a hydrodynamic two-fluid model [2] , [6] , [... | The core research idea is to experimentally investigate how second sound, an entropy wave distinct from first sound, changes across the BEC–BCS crossover in a tunable ultracold fermionic quantum gas, and to compare these measurements with existing theoretical models and simulations in order to better understand the beh... | [
"second sound",
"entropy wave",
"first sound",
"BEC–BCS crossover",
"ultracold fermionic quantum gas",
"theoretical models",
"simulations",
"superfluid systems",
"experimental investigation",
"tunable quantum gas"
] | [
{
"idx": "1.",
"title": "Halperin, W. P. Eighty years of superfluidity.Nature553, 413–414 (2018).",
"link": "https://doi.org/10.1038%2Fd41586-018-00417-7"
},
{
"idx": "6.",
"title": "Tisza, L. Transport phenomena in helium II.Nature141, 913 (1938).",
"link": "https://doi.org/10.1038%2F14... |
s41467-020-16364-9 | Nanoscience and technology | Ultrathin high-κantimony oxide single crystals | Ultrathin oxides have been reported to possess excellent properties in electronic, magnetic, optical, and catalytic fields. However, the current and primary approaches toward the preparation of ultrathin oxides are only applicable to amorphous or polycrystalline oxide nanosheets or films. Here, we successfully synthesi... | [
{
"section": "Introduction",
"content": "Ultrathin oxides have been reported to possess excellent properties in electronic, magnetic, optical, and catalytic fields [1] , [2] , [3] , [4] , [5] , [6] . However, the ways to simply synthesize ultrathin oxides through a general and facile method is a big challen... | This research proposes a substrate-buffer-controlled chemical vapor deposition (SBC–CVD) strategy to achieve the growth of ultrathin antimony oxide single crystals with high dielectric constant and excellent insulating properties, aiming to overcome the limitations of existing methods and enable the development of high... | [
"substrate-buffer-controlled chemical vapor deposition",
"SBC–CVD strategy",
"ultrathin antimony oxide single crystals",
"high dielectric constant",
"excellent insulating properties",
"existing methods",
"high-quality",
"reproducible ultrathin oxide materials",
"advanced two-dimensional electronic d... | [
{
"idx": "1.",
"title": "Zavabeti, A. et al. A liquid metal reaction environment for the room-temperature synthesis of atomically thin metal oxides.Science358, 332–335 (2017).",
"link": "https://doi.org/10.1126%2Fscience.aao4249"
},
{
"idx": "2.",
"title": "Fiori, G. et al. Electronics based... |
s41467-019-08948-x | Mathematics and computing | Percolation thresholds for photonic quantum computing | Despite linear-optical fusion (Bell measurement) being probabilistic, photonic cluster states for universal quantum computation can be prepared without feed-forward by fusing small n -photon entangled clusters, if the success probability of each fusion attempt is above a threshold, \({\mathrm{\lambda }}_{\mathrm{c}}^{(... | [
{
"section": "Introduction",
"content": "Optical qubits are a promising candidate for scalable universal quantum computing because of the relative ease of generating photons, fabricating photonic circuits [1] , [2] , [3] , and low decoherence rates. Assembling a universal photonic cluster state, e.g., by fu... | The core research idea is to determine the minimum success probability threshold required for ballistic creation of a universal photonic cluster state through linear-optical fusion of n-photon-entangled clusters, without the need for measurement-induced feedforward, and to demonstrate that this threshold can be achieve... | [
"minimum success probability threshold",
"ballistic creation",
"universal photonic cluster state",
"linear-optical fusion",
"n-photon-entangled clusters",
"measurement-induced feedforward",
"three-photon clusters",
"linear optics",
"single-photon ancillas",
"scalable optical quantum computing",
... | [
{
"idx": "1.",
"title": "Reck, M. & Zeilinger, A. Experimental realization of any discrete unitary operator.Phys. Rev. Lett.73, 58–61 (1994).",
"link": "https://doi.org/10.1103%2FPhysRevLett.73.58"
},
{
"idx": "2.",
"title": "Carolan, J. et al. Universal linear optics.Science349, 711–716 (20... |
s41467-024-55205-x | Materials science | Distinguishing local isomorphism classes in quasicrystals by high-order harmonic spectroscopy | Electron diffraction spectroscopy is a fundamental tool for investigating quasicrystal structures, which unveils the quasiperiodic long-range order. Nevertheless, it falls short in effectively distinguishing separate local isomorphism classes. This is a long outstanding problem. Here, we study the high-order harmonic g... | [
{
"section": "Introduction",
"content": "Choices of the unit cell and packing methods in crystals exhibit a strict one-to-one correspondence to form a single undefeated periodic structure. Very different from crystals, two or more unit cells can be assembled into an infinite number of various quasicrystals,... | This research proposes that high-order harmonic generation (HHG) can serve as a novel and effective method to distinguish between different local isomorphism classes (LICs) of two-dimensional generalized Penrose quasicrystals by exploiting distinct differences in HHG yield and orientation dependence, which arise from v... | [
"high-order harmonic generation",
"HHG",
"local isomorphism classes",
"LICs",
"two-dimensional generalized Penrose quasicrystals",
"vertex environments",
"hyperuniformity",
"tenfold rotational symmetry",
"harmonic radiation",
"orientation dependence",
"HHG yield",
"quasicrystal",
"distinct d... | [
{
"idx": "1.",
"title": "Levine, D. & Steinhardt, P. J. Definition and structure.Phys. Rev. B34, 596–616 (1986).",
"link": "https://doi.org/10.1103%2FPhysRevB.34.596"
},
{
"idx": "2.",
"title": "Socolar, J. E. S. & Steinhardt, P. J. Quasicrystals. II. Unit-cell configurations.Phys. Rev. B34,... |
s41467-020-20519-z | Nanoscience and technology | Spontaneous sparse learning for PCM-based memristor neural networks | Neural networks trained by backpropagation have achieved tremendous successes on numerous intelligent tasks. However, naïve gradient-based training and updating methods on memristors impede applications due to intrinsic material properties. Here, we built a 39 nm 1 Gb phase change memory (PCM) memristor array and quant... | [
{
"section": "Introduction",
"content": "Recently, the field of artificial intelligence (AI) has witnessed tremendous advances in deep neural networks (DNNs) [1] , [2] , [3] , [4] , [5] , [6] , [7] . In DNNs, as a connectionist approach to AI, knowledge is represented by a hierarchically distributed pattern... | The core research idea is to develop a novel learning scheme for phase change memory (PCM)-based memristor neural networks that leverages the intrinsic resistance drift effect—a typically undesirable property—by incorporating it into the training process to improve classification accuracy and control weight sparsity ac... | [
"novel learning scheme",
"phase change memory",
"PCM-based memristor neural networks",
"resistance drift effect",
"training process",
"classification accuracy",
"weight sparsity",
"neuromorphic computing",
"hardware limitation",
"functional advantage"
] | [
{
"idx": "1.",
"title": "LeCun, Y., Bengio, Y. & Hinton, G. Deep learning.Nature521, 436–444 (2015).",
"link": "https://doi.org/10.1038%2Fnature14539"
},
{
"idx": "2.",
"title": "Schmidhuber, J. Deep learning in neural networks: an overview.Neural Netw.61, 85–117 (2015).",
"link": "https... |
s41467-023-42471-4 | Physics | Nodal band-off-diagonal superconductivity in twisted graphene superlattices | The superconducting state and mechanism are among the least understood phenomena in twisted graphene systems. Recent tunneling experiments indicate a transition between nodal and gapped pairing with electron filling, which is not naturally understood within current theory. We demonstrate that the coexistence of superco... | [
{
"section": "Introduction",
"content": "The fascinating physics [1] , [2] of correlated graphene moiré superlattices, such as twisted-bilayer (TBG) and twisted-trilayer graphene (TTG), has generated extensive efforts to uncover the mysteries of their phase diagrams. Much progress has been made towards unde... | The core research idea is to demonstrate that the combination of flavor polarization and symmetry representations in the flat bands of twisted-bilayer and twisted-trilayer graphene enables the existence of band-off-diagonal superconducting pairing states—specifically, one transforming under the trivial representation w... | [
"flavor polarization",
"symmetry representations",
"flat bands",
"twisted-bilayer graphene",
"twisted-trilayer graphene",
"band-off-diagonal superconducting pairing states",
"trivial representation",
"symmetry-protected nodal lines",
"two-dimensional irreducible representation",
"nodal behavior",
... | [
{
"idx": "1.",
"title": "Andrei, E. Y. & MacDonald, A. H. Graphene bilayers with a twist.Nat. Mater.19, 1265 (2020).",
"link": "https://doi.org/10.1038%2Fs41563-020-00840-0"
},
{
"idx": "2.",
"title": "Balents, L., Dean, C. R., Efetov, D. K. & Young, A. F. Superconductivity and strong correl... |
s41467-019-10864-z | Mathematics and computing | Detection of cell-type-specific risk-CpG sites in epigenome-wide association studies | In epigenome-wide association studies, the measured signals for each sample are a mixture of methylation profiles from different cell types. Current approaches to the association detection claim whether a cytosine-phosphate-guanine (CpG) site is associated with the phenotype or not at aggregate level and can suffer fro... | [
{
"section": "Introduction",
"content": "Epigenome-wide association studies (EWAS) aim to identify cytosine-phosphate-guanine (CpG) sites associated with phenotypes of interest, such as disease status [1] , [2] , [3] , smoking history [4] , [5] , body mass index [6] , and age [7] , [8] . However, because th... | The core research idea of this study is to propose a novel method called HIRE that addresses the limitations of existing epigenome-wide association study (EWAS) approaches by identifying cell-type-specific associations between CpG sites and phenotypes through a model that accounts for cellular heterogeneity as multipli... | [
"HIRE",
"epigenome-wide association study",
"EWAS",
"cell-type-specific associations",
"CpG sites",
"phenotypes",
"cellular heterogeneity",
"multiplicative effects",
"additive effects",
"higher-resolution insights",
"epigenetic mechanisms",
"diseases",
"phenotypic traits"
] | [
{
"idx": "1.",
"title": "Rakyan, V. K., Down, T. A., Balding, D. J. & Beck, S. Epigenome-wide association studies for common human diseases.Nat. Rev. Genet.12, 529–541 (2011).",
"link": "https://doi.org/10.1038%2Fnrg3000"
},
{
"idx": "2.",
"title": "Verma, M. Epigenome-wide association studi... |
ncomms3781 | Physics | Efficient separation of the orbital angular momentum eigenstates of light | Orbital angular momentum (OAM) of light is an attractive degree of freedom for fundamental studies in quantum mechanics. In addition, the discrete unbounded state-space of OAM has been used to enhance classical and quantum communications. Unambiguous measurement of OAM is a key part of all such experiments. However, st... | [
{
"section": "Introduction",
"content": "It has long been known that the polarization state of the optical field can be associated with the spin angular momentum [1] , [2] . In 1992, it was shown by Allen et al. [3] that in the paraxial regime, an vortex phase structure of a circularly symmetric beam corres... | The core research idea is to experimentally implement an architecture that combines log-polar coordinate mapping with refractive beam copying to significantly enhance the separation efficiency of orbital angular momentum (OAM) modes, achieving a separation efficiency of 92.1±0.7 for 25 OAM modes and extending the metho... | [
"log-polar coordinate mapping",
"refractive beam copying",
"orbital angular momentum modes",
"separation efficiency",
"25 OAM modes",
"angular modes",
"OAM-based communication",
"quantum information systems"
] | [
{
"idx": "1",
"title": "Poynting, J. H. The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light.Proc. R. Soc. London, Ser. A82, 560–567 (1909).",
"link": "https://doi.org/10.1098%2Frspa.1909.0060"
},
{
"idx": "2",
"title": "Be... |
s41467-019-13371-3 | Nanoscience and technology | Templated dewetting of single-crystal sub-millimeter-long nanowires and on-chip silicon circuits | Large-scale, defect-free, micro- and nano-circuits with controlled inter-connections represent the nexus between electronic and photonic components. However, their fabrication over large scales often requires demanding procedures that are hardly scalable. Here we synthesize arrays of parallel ultra-long (up to 0.75 mm)... | [
{
"section": "Introduction",
"content": "Semiconductor nanowires exhibit superior and configurable electronic and optical properties [1] , [2] . Their disruptive potential has been demonstrated in photonics [3] (e.g. for lasers [4] or quantum optics [5] ), electronics [6] , thermoelectricity [7] , energy st... | The core research idea is to fabricate arrays of in-plane, ultra-long, monocrystalline silicon nanowires and complex interconnected circuits using a catalyst- and epitaxy-free hybrid top-down/bottom-up approach based on the controlled morphological evolution of thin solid films through low-resolution etching and anneal... | [
"in-plane ultra-long monocrystalline silicon nanowires",
"complex interconnected circuits",
"catalyst-free approach",
"epitaxy-free approach",
"hybrid top-down/bottom-up approach",
"controlled morphological evolution",
"thin solid films",
"low-resolution etching",
"annealing",
"size-controlled nan... | [
{
"idx": "1.",
"title": "Xia, Y. et al. One-dimensional nanostructures: synthesis, characterization, and applications.Adv. Mater.15, 353–389 (2003).",
"link": "https://doi.org/10.1002%2Fadma.200390087"
},
{
"idx": "2.",
"title": "Law, M., Goldberger, J. & Yang, P. Semiconductor nanowires and... |
s41467-025-57523-0 | Materials science | Retention of high-pressure solution-processable metastable phase to ambience via differential sublattice rigidity for broadband photodetectors | Materials science exploits only properties that are available at ambience. Therefore, although high-pressure changes the physical state of all condensed matter, most of the extraordinary properties discovered vanish after decompression and cannot be utilized. Here, we demonstrate sublattice decoupling in a mixed-anion ... | [
{
"section": "Introduction",
"content": "Using pressure to manipulate the physical state of condensed matter is as fundamental as changing temperature or tuning chemical composition for the same purpose [1] . Therefore, pressure is a distinctive dimension for seeking improved or even unprecedented propertie... | The core research idea is to achieve ambient retention of high-pressure-induced photoelectric properties in materials by designing systems with weakly coupled sublattices that decouple under pressure and irreversibly retain their modified structural and functional characteristics upon decompression, as demonstrated in ... | [
"ambient retention",
"high-pressure-induced photoelectric properties",
"weakly coupled sublattices",
"structural and functional characteristics",
"Rb₆Re₆S₈I₈ chalcohalide compound",
"electrostatically coupled sublattices",
"pressure-induced photocurrent generation",
"scalable optoelectronic device app... | [
{
"idx": "1.",
"title": "Mao, H.-k & Hemley, R. J. Ultrahigh-pressure transitions in solid hydrogen.Rev. Mod. Phys.66, 671–692 (1994).",
"link": "https://doi.org/10.1103%2FRevModPhys.66.671"
},
{
"idx": "2.",
"title": "Philip, D.-S., Ross, T. H. & Eugene, G. Evidence for a new phase of dense... |
s41467-024-47018-9 | Nanoscience and technology | Rewritable printing of ionic liquid nanofilm utilizing focused ion beam induced film wetting | Manipulating liquid flow over open solid substrate at nanoscale is important for printing, sensing, and energy devices. The predominant methods of liquid maneuvering usually involve complicated surface fabrications, while recent attempts employing external stimuli face difficulties in attaining nanoscale flow control. ... | [
{
"section": "Introduction",
"content": "Programmable control of fluid motion over open solid surfaces at small scales [1] , [2] are crucial for countless technological applications such as printing [3] , [4] , biosensing [5] , [6] , energy generating [7] , [8] , air water harvesting [9] , and chemical synt... | The core research idea is to develop a novel nanofluidic technique called ion beam induced film wetting (IBFW) that utilizes a Helium focused ion beam to achieve programmable, high-resolution patterning of ionic liquid films at the nanoscale by exploiting ion beam-induced electrical fields and disjoining pressure, enab... | [
"nanofluidic technique",
"ion beam induced film wetting",
"Helium focused ion beam",
"ionic liquid films",
"ion beam-induced electrical fields",
"disjoining pressure",
"programmable high-resolution patterning",
"liquid motion control",
"open solid surfaces",
"surface modifications",
"external ci... | [
{
"idx": "1.",
"title": "Dai, H., Dong, Z. & Jiang, L. Directional liquid dynamics of interfaces with superwettability.Sci. Adv.6, eabb5528 (2020).",
"link": "https://doi.org/10.1126%2Fsciadv.abb5528"
},
{
"idx": "2.",
"title": "Liu, M., Wang, S. & Jiang, L. Nature-inspired superwettability ... |
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