( θ LP = 90°) and five different DWP orientations ( θ DWP = 15°, 30°, 45°, 60°, 75°). The coordinate systems used are shown in the bottom-left inset. b Reflection amplitude (| r uu |) and bi r efringence (Δ ϕ vu ) calculated as a function of the nanobrick side lengths L u and L v , for the air gap T a = 400 nm, with other parameters as follows: λ = 800 nm, t m = 50 nm, Λ = 250 nm. Color map is related to the reflection amplitude, while colored lines represent constant birefringence contours. The red cross indicates the selected nanobrick dimensions L u × L v = 200 nm × 100 nm. c Reflection amplitude (| r uu | ) and bi r efringence (Δ ϕ vu ) calculated as a function of the nanobrick side length L v and air gap T a , for the other side length L u = 200 nm. Color map is related to the reflection amplitude, while colored dashed lines represent constant birefringence contours. The vertical, red dotted line corresponds to the selected nanobrick dimensions L u × L v = 200 nm × 100 nm. d Reflection amplitudes | r uu ( vv ) | (black dashed and dotted), phase ϕ uu ( vv ) (red dashed and dotted), and birefringence Δ ϕ vu (red solid) calculated as a function of the air gap T a for the selected nanobrick dimensions L u × L v = 200 nm × 100 nm. e Scanning electron microscopy (SEM) image of a part of the OMS nanobrick array fabricated on the glass substrate. The scale bar is 500 nm. Full size image","references":{"indices":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],"infos":[{"idx":"1","title":"Ding, F., Pors, A. & Bozhevolnyi, S. I. Gradient metasurfaces: a review of fundamentals and applications.Rep. Prog. Phys.81, 026401 (2018).","link":"https://doi.org/10.1088%2F1361-6633%2Faa8732"},{"idx":"2","title":"Shaltout, A. M., Shalaev, V. M. & Brongersma, M. L. 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Appl.9, 76 (2020).","link":"https://doi.org/10.1038%2Fs41377-020-0313-0"}]}},{"section":"Results","content":"MEMS-OMS-based DWP design Variable birefringence of the proposed DWP originates from the anisotropy of OMS elements placed at different distances from the MEMS mirror and can be described by considering the hybrid OMS-integrated Fabry–Pérot (FP) configuration [19] , [20] , [21] . For example, large reflection amplitudes along with large phase differences (>270°) for two orthogonal linear polarizations (LPs) can be realized for the operation wavelength λ = 800 nm at a specific air gap of T a = 400 nm for OMS nanobricks with sufficiently large aspect ratios (Fig. 1b ). By varying the gap T a , the phase difference, i.e., the DWP birefringence, can be dramatically changed, covering the whole tuning range of 0-2π, in a periodic fashion with the spatial period λ /2 = 400 nm (Fig. 1c ). Note that the selected size of the OMS unit cell is Λ = 250 nm ( Λ < λ /2), resulting in our choice of the nanobrick long side length L u = 200 nm due to the electron-beam lithography (EBL) fabrication (aspect ratio) constraints. The short side length L v is determined by maximizing reflection amplitudes for both orthogonal polarizations (i.e., | r uu | and | r vv | ) expected at all air gaps T a along with ensuring the full-range birefringence control (Fig. 1c and Complementary Fig. S1 ). The nanobrick height is also optimized to enable the full-range birefringence control and large reflection amplitudes, resulting in the choice of t m = 50 nm, a value that conforms well with the fabrication tolerances (Supplementary Fig. S1 ). Variations in the reflected field amplitudes (| r uu | , | r vv | ), phases ( ϕ uu , ϕ vv ), and birefringence ( Δϕ vu ), when changing the air gap T a for the designed DWP unit cell ( L u = 200 nm, L v = 100 nm), are shown in Fig. 1d , indicating the full 0-2π birefringence tunability and large reflection amplitudes (>0.9). Such a distinct polarization-dependent phase response of the designed DWP is achieved due to a very high OMS reflectivity for the light polarized along the long nanobrick side, so that the reflected field phase ϕ uu does not depend on the air gap T a , whereas the reflected field phase ϕ vv of the orthogonal polarization varies periodically from 0 to 2π with changing the air gap (Supplementary Fig. S2 ). Performances of the DWP component Following the simulations and design considerations presented above that indicate the possibility of realizing efficient DWPs with full-range birefringence tunability, we conducted the corresponding experimental investigations. The MEMS-OMS-based DWP for polarization conversion was assembled from a separately fabricated OMS (Fig. 1e ) and an ultra-flat MEMS mirror that were wire bonded to a printed circuit board (PCB) for electrical connection (Supplementary Figs. S3 , S4 , Methods/Fabrication). Owing to the FP nature of the designed DWP component, the overall response is inherently periodic with respect to the variable air gap T a , thus relaxing the requirement for small, nm-sized, air gaps essential for the previously developed MEMS-OMS configurations [22] and thereby simplifying the assembly process. We should however note that the access to nm-sized air gaps, although not easily gained, would allow one to operate the DWP with gap surface plasmons being generated [22] , a regime that promises a slightly broader operation wavelength range as discussed below. To quantify the air gap T a at each actuation voltage V m , we make use of the air gap-dependent MEMS-OMS operation [22] by exploiting the fact that the MEMS-OMS behaves as a mirror at periodically distributed air gaps (depending on the light wavelength used) regardless of the OMS design (Supplementary Figs. S5 , S6 ). By determining the wavelength corresponding to the mirror-like operation (no polarization transformation) at each voltage, it is possible to accurately evaluate in situ the air gap T a realized in each polarization measurement (Supplementary Fig. S7 ). For the characterization of the fabricated DWP performance, a supercontinuum laser beam with the well-defined LP state ( θ LP = 90°) was directed onto the DWP oriented at θ DWP = 45° (Fig. 2a ). The MEMS mirror was electrically actuated to tune the DWP birefringence by changing the MEMS-OMS air gap and thereby adjusting the polarization state of the reflected light. Full Stokes polarimetry [23] was performed by rotating a quarter-wave plate (QWP) and a polarizer while analyzing the reflected light with a spectrometer (Methods/Characterization, Supplementary Fig. S8 ). Recording the DWP polarization-resolved reflection spectra (Supplementary Fig. S9 ) enables the determination of wavelength-resolved Stokes parameters along with the polarization conversion efficiencies and degree of polarization (DOP) while varying the MEMS-OMS air gap (Fig. 2b and Supplementary Figs. S9, S10 ). The quality of polarization conversion in the considered configuration and its operation bandwidth can be assessed by requiring efficient modulation of Stokes parameters S 1 and S 3 , while keeping S 2 insignificant. It is seen (Fig. 2 b, c ) that the fabricated DWP performs very good, matching remarkably well the simulations, within the ~100 nm band (i.e., ~10%) around the design wavelength of λ = 800 nm. The experimental polarization conversion efficiencies (for all possible polarization conversions) of the DWP are rather high within the bandwidth (Supplementary Fig. S9 ), being estimated at ~75% at the design wavelength λ = 800 nm (Fig. 2c ), due to the full-range birefringence control realized and small absorption losses present in the MEMS-OMS configuration under off-resonance operation. Note that, as the air gap increases, the reflected light polarization changes periodically from incident LP | y > state to | r >, | x >, | l > states (Fig. 2a ) with the corresponding Stokes parameters ( S 1 , S 2 , S 3 ) changing from approximately (−1, 0, 0) to (0, 0, 1), (1, 0, 0) and (0, 0, −1), respectively (Fig. 2 b, c ). This DWP transformation is equivalent to the transformation from the mirror-like operation, or zero-wave plate (ZWP), to QWP, half-wave plate (HWP), and three-quarters-wave plate (TQWP). Note that the DOP remains being close to 1.0 during the whole DWP tuning, indicating an overall purity of polarization conversion. Slight deviations from the ideal conversion can be explained by fabricated nanobrick deviations from the design dimensions, the reflection from the front glass/air interface as well as possible mirror tilting. The polarization trajectory at λ = 800 nm follows one complete revolution around the Poincaré sphere, passing through basic polarization states | y >, | r >, | x >, and | l >, and circulating repeatedly with increasing T a (Fig. 2d and Supplementary Video S1 ). It is noticed that the polarization trajectory on the Poincaré sphere changing with the air gap T a exhibit a slight asymmetry (Fig. 2d and Supplementary Video S1 ), owing to different (also slowly varying) reflection amplitudes of the designed MEMS-OMS DWP structure for the two orthogonal linear polarizations (i.e., | u > and | v > ), as shown in Fig. 1d and Supplementary Fig. S2 . The optimized DWP orientation, for generating all four typical polarization states (i.e., | x >, | y >, | r >, | l > ) in the best way, is θ DWP = ∼ 46° (Supplementary Fig. S11 and Table 2 ), since | r uu | is overall slightly smaller than | r vv | (Fig. 1d ). The reflection from the front glass/air interface of the DWP component also matters, which, however, can be eliminated in future developments by making use of anti-reflection coating. As a final comment, in the current experimental demonstration, the DWP is operated with air gaps between 2 and 3 μm, covering two adjacent FP resonance orders and exhibiting practically the same performance. Fig. 2: Polarization transformations for the incident LP oriented at 45° with respect to the DWP axes. a Schematic of the dynamic wave plate (DWP) operation at the wavelength of 800 nm for the fixed incident linear polarization (LP) state | y > ( θ LP = 90°) and DWP orientation θ DWP = 45°, illustrating the reflected light polarizations being continuously changed from | l > , | y > , | r > to | x > with increasing the ai r gap T a . The inset is a typical optical image of the optical metasurface (OMS) array illuminated with the incident laser spot confined within the orange dashed circle. The scale bar is 10 μm. b Calculated (left) and measured (right) normalized Stokes parameters ( S 1 , S 2 , S 3 ) of the reflected light as a function of the air gap T a and light wavelength λ . c Calculated (lines) and measured (markers) normalized Stokes parameters (above) and polarization conversion efficiencies (below) as a function of the air gap T a for the light wavelength λ = 800 nm. d Calculated (line) and measured (circles) polarization trajectory on the Poincaré sphere mapping the reflected light polarization evolution shown in ( c ). As the air gap T a increases, the reflected polarization state revolves repeatedly around the sphere. The green star marks the incident LP state: | y >. The calculated (red) and measured (black) polarization ellipses illustrate the polarization contrast between the main orthogonal polarization states (| x > and | y > , | r > and | l > ) realized by tuning the air gap T a . Full size image High contrast between the main orthogonal polarization states (| x > and | y >, | r > and | l > ), corresponding to HWP/ZWP and QWP/TQWP transformations, can be visualized at both the direct image (DI) and Fourier image (FI) planes (Fig. 3a , b). The dynamic polarization conversion behavior, induced by actuating the MEMS mirror with alternating voltages at a slow switching frequency, is captured by a charge-coupled device (CCD) camera (Supplementary Videos S2, S3 ), showing good repeatability and high modulation efficiency. By actuating the MEMS mirror with a periodic rectangular signal, filtering the reflected light with analyzers (i.e., | x >, | y >, | r > or | l > ) and detecting it with a fast photodetector, we observe relatively fast switching with rise/fall times of ~0.15/0.11 and ~0.36/0.23 ms for respective ZWP/HWP and QWP/TQWP transformations (Fig. 3c , d). The switching time for the QWP/TQWP transformation is approximately twice longer due to an approximately twice longer relocation of the MEMS mirror required (Fig. 2c ), also reflected in an approximately twice larger switching voltage used (cf., Fig. 3 c, d ). Note that, by optimizing MEMS mirrors for achieving faster switching, one should be able to reach switching frequencies in the MHz range [6] , [24] . We should also remark that a baseline voltage of ~14 V is used to move the MEMS mirror closer to the OMS, which would not be necessary for devices with more accurate mounting so that the maximum applied voltage can be kept below ~6 V. Fig. 3: Polarization conversion dynamics. a , b Optical images of the reflected light at the direct image (DI) and Fourier image (FI) planes for the fixed input linear polarization (LP) state | y > ( θ LP = 90°) and dynamic wave plate (DWP) orientation ( θ DWP = 45°) at λ = 800 nm, showing polarization modulation between orthogonal ( a ) LP states (| x >, | y >), corresponding to zero-wave plate (ZWP)/half-wave plate (HWP) transformation, by changing the actuation voltage V m from 20.2 to 17.1 V, and ( b ) circular polarization (CP) states (| r >, | l >), corresponding to quarter-wave plate (QWP)/three-quarters-wave plate (TQWP) transformation, by changing the actuation voltage V m from 19.1 to 14.4 V. For the FIs, the light is filtered using an iris letting through only light reflected from inside the orange dashed circles (shown in DIs). The leftmost column in ( a , b ) indicates the polarizer orientations used for filtering the respective polarization states in the reflected light. Scale bars in the DIs and FIs are 10 μm and 0.02 k 0 , respectively. c , d Temporal evolution of the reflected light power for c ZWP/HWP and d QWP/TQWP transformations, measured by actuating the MEMS mirror with a periodic rectangular signal and filtering respective polarization states. Full size image Versatile polarization transformations with DWP In addition to evaluating the DWP polarization tunability, polarization conversion efficiencies, and temporal response, we also explored the possibility of continuous LP rotation by a hybrid QWP-DWP component consisting of a general QWP, whose fast axis is oriented along the x -direction, and a DWP oriented at θ DWP = 45° (Fig. 4a and Supplementary Fig. S12 ). In this configuration, the QWP transforms the LP basis (| u >, | v >) of the DWP eigenstates into the CP basis ((| r >, | l >) of the hybrid QWP-DWP component, thus enabling continuous LP rotation by tuning the DWP birefringence via tuning the air gap T a . The reflected light polarization states measured while changing the air gap are visualized by a polarization trajectory around the equator of the Poincaré sphere (Fig. 4a , Supplementary Fig. S12 , and Video S4 ), demonstrating the generation of LP light with arbitrary LP orientation by adjusting the actuation voltage V m (i.e., the air gap T a ). To ease steering potential practical applications, we have also investigated the polarization transformations realized for different DWP orientations ( θ DWP = 15°, 30°, 60°, 75°) and fixed incident LP state | y > (Fig. 4b , Supplementary Fig. S13 , and Video S5 ), as well as for different incident LP states ( θ LP = 60°, 75°, 105°, 120°) and fixed DWP orientation θ DWP = 45° (Fig. 4c , Supplementary Fig. S14, and Video S6 ). The polarization trajectories on the Poincaré sphere are seen to be differently tilted for different DWP orientations having the common point, the incident LP state | y > (Fig. 4b ), while being parallel to the plane ( S 1 , S 3 ) and reflecting the incident LP state (Fig. 4c ). Note that, for any given point on the Poincaré sphere, one can identify suitable orientations of the DWP and incident LP state enabling closed polarization trajectories to pass this point, so that a multitude of polarization modulation capabilities can be realized with the same DWP. Finally, we would like to emphasize that all experimental results agree exceedingly well with the simulations without any fitting parameters, demonstrating convincingly that the fabricated DWP behaves according to the design and that the modeling approach developed is well suited for use in the future MEMS-OMS component developments. Fig. 4: Versatile polarization transformations. a – c Calculated (lines) and measured (circles) polarization trajectories on the Poincaré sphere realized at the wavelength of 800 nm by tuning the air gap T a for different dynamic wave plate (DWP) orientations, illustrating the diversity of possible polarization transformations: a continuous linear polarization (LP) rotation realized by combining the DWP with a conventional quarter-wave plate (QWP) with the incident LP state | y > and DWP orientation θ DWP = 45°, b various elliptical polarization transformations realized for different DWP orientations with the fixed incident LP state | y >, and c various elliptical polarization transformations realized for different incident LP states with the fixed DWP orientation θ DWP = 45°. The green stars in ( a – c ) indicate respective incident LP states. Full size image"},{"section":"Discussion","content":"To summarize, capitalizing on our development of the MEMS-OMS platform [22] we have demonstrated the electrically driven DWP operating in reflection with high polarization conversion efficiencies (~75%), broadband operation (~100 nm near the operating wavelength of 800 nm), fast responses (<0.4 ms) and full-range birefringence control that enables completely encircling the Poincaré sphere along trajectories determined by the incident light polarization and DWP orientation. It should be noted that, given the access to nm-sized air gaps, one can exploit the same design principle to realize the DWP with gap surface plasmons being generated [22] , a regime that promises a broader operation wavelength range (~160 nm) although probably at the expense of a lower efficiency (~50%) [22] . Importantly, the general approach developed can also be applied to design a DWP operating in transmission by using a partially transmitting MEMS mirror and placing an OMS in the middle of an FP cavity [25] . Given that a multitude of polarization modulation capabilities can be realized with the same DWP, we believe that the demonstrated electrically driven DWP configuration with full-range birefringence control opens fascinating perspectives for successful integration of high-performance compact dynamic polarization components into future miniaturized reconfigurable/adaptive optical networks and systems [26] , [27] ."},{"section":"Methods","content":"Numerical calculations All numerical simulations were done using COMSOL Multiphysics version 5.6. The model is composed of a rectangular volume with a square footprint with sides Λ = 250 nm and periodic boundary conditions were employed for both u and v directions. The DWP unit cell is divided into two parts of air and glass, with one gold nanobrick placed against the glass region. The corners of the nanobrick are rounded with a 5 nm radius. The refractive index of air is set as 1 and that of glass as 1.46 for all wavelengths, while the gold permittivity was interpolated as a function of wavelength from experimental tabulated values [28] . Using this model, the complex reflection and transmission coefficients for the glass/OMS/air interface are calculated for both propagation directions (i.e., a normal incident from glass or air) and for light linearly polarized along both u and v separately. These are used to calculate the total reflection coefficient \\({r}_{{{{\\rm{FP}}}}}\\) by including the gold substrate with the FP equation [19] , [20] , [21] $${r}_{{{{\\rm{FP}}}}}={r}_{12}+\\frac{{t}_{12}{t}_{21}{r}_{23}{e}^{i2{{{\\bf{k}}}}{n}_{2}{T}_{a}}}{1-{r}_{21}{r}_{23}{e}^{i2{{{\\bf{k}}}}{n}_{2}{T}_{a}}}$$ (1) Here, r mn ( t mn ) denotes the reflection (transmission) coefficients for light incident on material n from material m and the materials are numbered 1, 2, 3 for respectively glass substrate, air, and gold substrate, n 2 represents the refractive index of the air (i.e., medium 2 between the OMS layer and gold substrate), and k is the wavenumber in a vacuum. Note that the effect of the OMS on the interface between the air and glass substrate is included in r 12 , r 21 , t 12 , and t 21 , and that even for normal incidence r FP are polarization-dependent due to the anisotropy of the OMS layer. The reflection coefficient r 23 from the air/gold interface is calculated directly using the Fresnel equation. An illustration of the DWP geometry is shown in Supplementary Fig. S2 together with plots explaining the behavior of the total reflection coefficient r FP with varying T a . For air gaps much smaller than the wavelengths, there is near-field coupling between the nanobricks and the gold substrate in addition to the FP resonances [1] , [20] , requiring numerical simulations including also the gold substrate in COMSOL. The results obtained using the FP equation were confirmed to give the same results as COMSOL simulations with the whole glass/nanobrick/air/gold substrate model when T a > 100 nm for a wavelength of λ = 800 nm. As a final comment, to compare with the measurements, the imaginary part of the gold permittivity is increased by three times in the simulations of Figs. 2 , 4 and Supplementary Figs. S7 , S9 – S14 , accounting for the surface roughness and grain boundary effects of the fabricated gold nanobricks as well as the increased damping associated with the titanium (Ti) adhesion layers between the gold/glass interface. Fabrication The OMS for developing MEMS-OMS DWP were fabricated using electron-beam lithography (EBL), thin-film deposition, and lift-off techniques. First, a 100-nm-thick poly(methyl methacrylate) (PMMA A2, MicroChem) layer and a 40-nm thick conductive polymer layer (AR-PC 5090, Allresist) were successively spin-coated on the glass substrate (Borofloat 33 wafer, Wafer Universe). Note that the glass substrate was preprocessed to have a 10-μm-high circular pedestal using optical lithography and wet etching, and the OMS pattern was defined on the pedestal using EBL (JEOL JSM-6500F field-emission SEM with a Raith Elphy Quantum lithography system). After development, the OMS layer was formulated by depositing a 1-nm Ti adhesion layer and a 50-nm gold layer (Tornado 400, Cryofox) followed by lift-off in acetone (Supplementary Fig. S4 ). The pedestal on the glass substrate is very effective for reducing the possible contaminants between the MEMS mirror and OMS surface, thus improving the stability and repeatability of the DWP components. The MEMS mirror is fabricated using standard semiconductor manufacturing processes (Supplementary Fig. S3 ), in which thin-film lead zirconate titanate (PZT) is incorporated for long-stroke, low-voltage electrical actuation. For use in the MEMS-OMS component, the ultra-flat MEMS mirror was sputtered with a 100 nm gold layer. After the gold deposition, the MEMS mirror surface is inspected with white light interferometry (Zygo NewView 6300), showing overall good flatness and roughness all over the whole MEMS mirror (i.e., ~3 mm diameter) (Supplementary Fig. S3 ). The MEM-OMS-based DWP component (Supplementary Fig. S4 ) was assembled by gluing the MEMS mirror with the glass substrate upon which OMS is structured and then glued to a printed circuit board (PCB), followed by a gold wire bonding process between the MEMS mirror and PCB for enabling simple electrical connection to a voltage controller used to actuate the MEMS mirror. Characterization The experimental setup is shown in Supplementary Fig. S8 . A collimated fiber-coupled supercontinuum laser (SuperK Extreme, NKT) was directed through an HWP (AHWP10M-980, Thorlabs), a mirror, a linear polarizer (Pol 1 ; LPNIR050-MP2, Thorlabs), two beam splitters (BS 1,2 ; CCM1-BS014, Thorlabs) successively, and then focused onto the DWP samples by an objective (Obj; M Plan Apo, ×20/0.42NA, Mitutoyo). The combination of HWP and Pol 1 is used for altering the input LP states as well as the intensity. The reflected light was collected by the same objective and passed through two beam splitters (BS 2,3 ; CCM1-BS014, Thorlabs) and a tube lens (TL; TTL200-S8, Thorlabs), generating the first direct image plane where an iris is placed for filtering out the reflected light within the DWP area. The first direct image is then transformed by a relay lens (RL; AC254-200-B-ML, f = 200 mm, Thorlabs) to the corresponding Fourier image and captured by a CCD camera (CCD; DCC1545M, Thorlabs), according to a 2 f configuration. Note that a flip lens (FL; AC254-100-B-ML, f = 100 mm, Thorlabs) is used for switching between the direct and Fourier images, and a Stokes analyzer composed of a QWP (AQWP10M-980, Thorlabs) and a linear polarizer (Pol 2 ; LPNIR050-MP2, Thorlabs) is implemented before the CCD camera for performing full Stokes polarimetry [23] . Two beam splitters are configured for cross-compensating the polarization-dependent phase shifts in the beam splitters for both incidence and reflection routes. To obtain the wavelength-resolved full Stokes parameters, we replaced the CCD camera with a fiber-coupled spectrometer (QE Pro , Ocean Optics) and conducted measurements at the Fourier image plane. By rotating the QWP and Pol 2 , we recorded polarization-resolved spectra of I x ( λ ), I y ( λ ), I a ( λ ), I b ( λ ), I r ( λ ), I l ( λ ), and the stokes parameters ( s 1 , s 2 , s 3 ) are calculated as s 1 = ( I x ( λ ) – I y ( λ ))/( I x ( λ ) + I y ( λ )), s 2 = ( I a ( λ ) – I b ( λ ))/( I x ( λ ) + I y ( λ )), s 3 = ( I r ( λ ) – I l ( λ ))/( I x ( λ ) + I y ( λ )). For a reasonable comparison with the simulations, the stokes parameters ( s 1 , s 2 , s 3 ) are normalized to the polarized proportion of the reflected light beam: \\({S}_{{{\\mathrm{1,2,3}}}}=\\frac{{s}_{{{\\mathrm{1,2,3}}}}}{{{{\\rm{DOP}}}}}\\) , and the degree of polarization (DOP) is defined as \\({{{\\rm{DOP}}}}=\\sqrt{{s}_{1}^{2}+{s}_{2}^{2}+{s}_{3}^{2}}\\) . The polarization conversion efficiency (Fig. 2 and Supplementary Figs. S9 – S11 ) is defined as the ratio of the reflected light power in a specific polarization channel (i.e., | x >, | y >, | a >, | b >, | r >, | l > ) to the incident linearly | y > polarized light power. The coordinates system used is indicated in the lower-left inset of Fig. 1a , with z being the optical axis, x and y are transverse axes in the laboratory frame of reference, while u and v are transverse axes oriented along the long and short sides of the rectangular nanobricks. The angle between u and x is denoted θ DWP , while the angle between the x -axis and the polarization direction of LP incident light is denoted θ LP . To estimate the switching speeds between different orthogonal LP and CP bases (i.e., different DWP status), the setup described above is modified by replacing the input laser and CCD camera with a cw Ti:sapphire laser (Spectra-Physics 3900 S, wavelength range: 700 to 1000 nm), and a photodetector (PD; PDA20CS-EC, Thorlabs), respectively. The signals from the PD are acquired with an oscilloscope (DSOX2024A, Keysight). In the measurement, the MEMS-OMS-based DWP is actuated with periodically alternating voltages and different polarizations (i.e., | x >, | y >, | r > and | l > ) can be filtered by the Stokes analyzer."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"Recent years brought explosive growth of OMS applications in terms of both diversity and sophistication [1] , especially with OMS configurations becoming dynamic, i.e., offering not only spatial but also temporal control over transmitted and reflected optical fields [2] . Plentiful dynamic OMSs have been demonstrated in the past decade, implementing active amplitude and phase modulation with diversified functionalities [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] . Despite significant progress achieved in dynamic control of amplitude and phase responses, OMS-enabled active polarization control in the optical range remains largely unexplored. At the same time, the polarization of light is crucial for the realization of adaptive integrated photonic systems due to the fundamental role of polarization as another intrinsic characteristic of optical waves, which is uncorrelated with the amplitude and phase. To implement active polarization control with dynamic OMSs, one should be able to dynamically tune the OMS optical anisotropy in a wide range. The corresponding OMS realizations rely on the birefringence property of natural materials [11] , [12] or metasurfaces [13] , [14] , [15] , [16] , [17] , [18] , consisting of anisotropic meta-atoms. In either case, since available OMS thickness (i.e., interaction length) and birefringence are limited, identification of a suitable OMS configuration is a rather complicated and arduous issue, with the outcome exhibiting drawbacks, such as slow responses, narrow bandwidths, limited dynamic ranges of birefringence, and low polarization conversion efficiencies (Supplementary Table 1 ). In this article, we demonstrate electrically controlled full-range birefringence by integrating a plasmonic OMS with judiciously designed anisotropic meta-atoms and a thin-film piezoelectric MEMS mirror. The resulting MEMS-OMS-based DWP operates in reflection, featuring continuously varied anisotropy and enabling complete encircling of the Poincaré sphere, e.g., the polarization conversion from linear to circular, orthogonal linear, and opposite circular polarizations (Fig. 1 ). The OMS component, which is composed of a glass substrate with an OMS layer containing a 2D array of identical rectangular gold nanobricks, is placed close to an actuated MEMS gold mirror, whose separation T a from the OMS is accurately controlled by applying the actuation voltage V m (Fig. 1a ). In the conceptual schematic, a linearly polarized light beam is incident onto the DWP, and the polarization state of the reflected light is shown continuously adjusted by changing the actuation voltage, with different polarization trajectories encircling the Poincaré sphere being available for different DWP orientations (Fig. 1a ). Fig. 1: Electrically controlled MEMS-OMS-based DWP. a Schematic of the dynamic wave plate (DWP) component consisting of an optical metasurface (OMS) on a glass substrate, mounted on a movable micro-electro-mechanical system (MEMS) mirror. By applying a voltage V m , the separation between the MEMS mirror and OMS can be changed, thereby achieving a continuously variable birefringence in reflection. The bottom-right inset is a close-up view of the MEMS-OMS unit cell representing a gold nanobrick on glass separated by an air gap T a from the MEMS gold mirror. The top-right inset illustrates, with the Poincaré sphere, different reflected field polarization trajectories that can be realized by changing V m for the fixed incident linear polarization (LP) state | y > ( θ LP = 90°) and five different DWP orientations ( θ DWP = 15°, 30°, 45°, 60°, 75°). The coordinate systems used are shown in the bottom-left inset. b Reflection amplitude (| r uu |) and bi r efringence (Δ ϕ vu ) calculated as a function of the nanobrick side lengths L u and L v , for the air gap T a = 400 nm, with other parameters as follows: λ = 800 nm, t m = 50 nm, Λ = 250 nm. Color map is related to the reflection amplitude, while colored lines represent constant birefringence contours. The red cross indicates the selected nanobrick dimensions L u × L v = 200 nm × 100 nm. c Reflection amplitude (| r uu | ) and bi r efringence (Δ ϕ vu ) calculated as a function of the nanobrick side length L v and air gap T a , for the other side length L u = 200 nm. Color map is related to the reflection amplitude, while colored dashed lines represent constant birefringence contours. The vertical, red dotted line corresponds to the selected nanobrick dimensions L u × L v = 200 nm × 100 nm. d Reflection amplitudes | r uu ( vv ) | (black dashed and dotted), phase ϕ uu ( vv ) (red dashed and dotted), and birefringence Δ ϕ vu (red solid) calculated as a function of the air gap T a for the selected nanobrick dimensions L u × L v = 200 nm × 100 nm. e Scanning electron microscopy (SEM) image of a part of the OMS nanobrick array fabricated on the glass substrate. The scale bar is 500 nm. Full size image\",\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 ],\n \"infos\": [\n {\n \"idx\": \"1\",\n \"title\": \"Ding, F., Pors, A. & Bozhevolnyi, S. I. Gradient metasurfaces: a review of fundamentals and applications.Rep. Prog. 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Ultrafast synthesis and switching of light polarization in nonlinear anisotropic metamaterials.Nat. Photon11, 628–633 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fs41566-017-0002-6\"\n },\n {\n \"idx\": \"14\",\n \"title\": \"Schirato, A. et al. Transient optical symmetry breaking for ultrafast broadband dichroism in plasmonic metasurfaces.Nat. Photon14, 723–727 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41566-020-00702-w\"\n },\n {\n \"idx\": \"15\",\n \"title\": \"Wang, K. et al. High contrast, femtosecond light polarization manipulation in epsilon-near-zero material coupled to a plasmonic nanoantenna array.ACS Photonics8, 2791–2799 (2021).\",\n \"link\": \"https://doi.org/10.1021%2Facsphotonics.1c00971\"\n },\n {\n \"idx\": \"16\",\n \"title\": \"Zanotto, S. et al. Optomechanics of chiral dielectric metasurfaces.Adv. Opt. Mater.8, 1901507 (2020).\",\n \"link\": \"https://doi.org/10.1002%2Fadom.201901507\"\n },\n {\n \"idx\": \"17\",\n \"title\": \"Shi, Z. et al. Continuous angle-tunable birefringence with freeform metasurfaces for arbitrary polarization conversion.Sci. Adv.6, eaba3367 (2020).\",\n \"link\": \"https://doi.org/10.1126%2Fsciadv.aba3367\"\n },\n {\n \"idx\": \"18\",\n \"title\": \"Zhang, X. et al. Controlling angular dispersions in optical metasurfaces.Light Sci. Appl.9, 76 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41377-020-0313-0\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"MEMS-OMS-based DWP design Variable birefringence of the proposed DWP originates from the anisotropy of OMS elements placed at different distances from the MEMS mirror and can be described by considering the hybrid OMS-integrated Fabry–Pérot (FP) configuration [19] , [20] , [21] . For example, large reflection amplitudes along with large phase differences (>270°) for two orthogonal linear polarizations (LPs) can be realized for the operation wavelength λ = 800 nm at a specific air gap of T a = 400 nm for OMS nanobricks with sufficiently large aspect ratios (Fig. 1b ). By varying the gap T a , the phase difference, i.e., the DWP birefringence, can be dramatically changed, covering the whole tuning range of 0-2π, in a periodic fashion with the spatial period λ /2 = 400 nm (Fig. 1c ). Note that the selected size of the OMS unit cell is Λ = 250 nm ( Λ < λ /2), resulting in our choice of the nanobrick long side length L u = 200 nm due to the electron-beam lithography (EBL) fabrication (aspect ratio) constraints. The short side length L v is determined by maximizing reflection amplitudes for both orthogonal polarizations (i.e., | r uu | and | r vv | ) expected at all air gaps T a along with ensuring the full-range birefringence control (Fig. 1c and Complementary Fig. S1 ). The nanobrick height is also optimized to enable the full-range birefringence control and large reflection amplitudes, resulting in the choice of t m = 50 nm, a value that conforms well with the fabrication tolerances (Supplementary Fig. S1 ). Variations in the reflected field amplitudes (| r uu | , | r vv | ), phases ( ϕ uu , ϕ vv ), and birefringence ( Δϕ vu ), when changing the air gap T a for the designed DWP unit cell ( L u = 200 nm, L v = 100 nm), are shown in Fig. 1d , indicating the full 0-2π birefringence tunability and large reflection amplitudes (>0.9). Such a distinct polarization-dependent phase response of the designed DWP is achieved due to a very high OMS reflectivity for the light polarized along the long nanobrick side, so that the reflected field phase ϕ uu does not depend on the air gap T a , whereas the reflected field phase ϕ vv of the orthogonal polarization varies periodically from 0 to 2π with changing the air gap (Supplementary Fig. S2 ). Performances of the DWP component Following the simulations and design considerations presented above that indicate the possibility of realizing efficient DWPs with full-range birefringence tunability, we conducted the corresponding experimental investigations. The MEMS-OMS-based DWP for polarization conversion was assembled from a separately fabricated OMS (Fig. 1e ) and an ultra-flat MEMS mirror that were wire bonded to a printed circuit board (PCB) for electrical connection (Supplementary Figs. S3 , S4 , Methods/Fabrication). Owing to the FP nature of the designed DWP component, the overall response is inherently periodic with respect to the variable air gap T a , thus relaxing the requirement for small, nm-sized, air gaps essential for the previously developed MEMS-OMS configurations [22] and thereby simplifying the assembly process. We should however note that the access to nm-sized air gaps, although not easily gained, would allow one to operate the DWP with gap surface plasmons being generated [22] , a regime that promises a slightly broader operation wavelength range as discussed below. To quantify the air gap T a at each actuation voltage V m , we make use of the air gap-dependent MEMS-OMS operation [22] by exploiting the fact that the MEMS-OMS behaves as a mirror at periodically distributed air gaps (depending on the light wavelength used) regardless of the OMS design (Supplementary Figs. S5 , S6 ). By determining the wavelength corresponding to the mirror-like operation (no polarization transformation) at each voltage, it is possible to accurately evaluate in situ the air gap T a realized in each polarization measurement (Supplementary Fig. S7 ). For the characterization of the fabricated DWP performance, a supercontinuum laser beam with the well-defined LP state ( θ LP = 90°) was directed onto the DWP oriented at θ DWP = 45° (Fig. 2a ). The MEMS mirror was electrically actuated to tune the DWP birefringence by changing the MEMS-OMS air gap and thereby adjusting the polarization state of the reflected light. Full Stokes polarimetry [23] was performed by rotating a quarter-wave plate (QWP) and a polarizer while analyzing the reflected light with a spectrometer (Methods/Characterization, Supplementary Fig. S8 ). Recording the DWP polarization-resolved reflection spectra (Supplementary Fig. S9 ) enables the determination of wavelength-resolved Stokes parameters along with the polarization conversion efficiencies and degree of polarization (DOP) while varying the MEMS-OMS air gap (Fig. 2b and Supplementary Figs. S9, S10 ). The quality of polarization conversion in the considered configuration and its operation bandwidth can be assessed by requiring efficient modulation of Stokes parameters S 1 and S 3 , while keeping S 2 insignificant. It is seen (Fig. 2 b, c ) that the fabricated DWP performs very good, matching remarkably well the simulations, within the ~100 nm band (i.e., ~10%) around the design wavelength of λ = 800 nm. The experimental polarization conversion efficiencies (for all possible polarization conversions) of the DWP are rather high within the bandwidth (Supplementary Fig. S9 ), being estimated at ~75% at the design wavelength λ = 800 nm (Fig. 2c ), due to the full-range birefringence control realized and small absorption losses present in the MEMS-OMS configuration under off-resonance operation. Note that, as the air gap increases, the reflected light polarization changes periodically from incident LP | y > state to | r >, | x >, | l > states (Fig. 2a ) with the corresponding Stokes parameters ( S 1 , S 2 , S 3 ) changing from approximately (−1, 0, 0) to (0, 0, 1), (1, 0, 0) and (0, 0, −1), respectively (Fig. 2 b, c ). This DWP transformation is equivalent to the transformation from the mirror-like operation, or zero-wave plate (ZWP), to QWP, half-wave plate (HWP), and three-quarters-wave plate (TQWP). Note that the DOP remains being close to 1.0 during the whole DWP tuning, indicating an overall purity of polarization conversion. Slight deviations from the ideal conversion can be explained by fabricated nanobrick deviations from the design dimensions, the reflection from the front glass/air interface as well as possible mirror tilting. The polarization trajectory at λ = 800 nm follows one complete revolution around the Poincaré sphere, passing through basic polarization states | y >, | r >, | x >, and | l >, and circulating repeatedly with increasing T a (Fig. 2d and Supplementary Video S1 ). It is noticed that the polarization trajectory on the Poincaré sphere changing with the air gap T a exhibit a slight asymmetry (Fig. 2d and Supplementary Video S1 ), owing to different (also slowly varying) reflection amplitudes of the designed MEMS-OMS DWP structure for the two orthogonal linear polarizations (i.e., | u > and | v > ), as shown in Fig. 1d and Supplementary Fig. S2 . The optimized DWP orientation, for generating all four typical polarization states (i.e., | x >, | y >, | r >, | l > ) in the best way, is θ DWP = ∼ 46° (Supplementary Fig. S11 and Table 2 ), since | r uu | is overall slightly smaller than | r vv | (Fig. 1d ). The reflection from the front glass/air interface of the DWP component also matters, which, however, can be eliminated in future developments by making use of anti-reflection coating. As a final comment, in the current experimental demonstration, the DWP is operated with air gaps between 2 and 3 μm, covering two adjacent FP resonance orders and exhibiting practically the same performance. Fig. 2: Polarization transformations for the incident LP oriented at 45° with respect to the DWP axes. a Schematic of the dynamic wave plate (DWP) operation at the wavelength of 800 nm for the fixed incident linear polarization (LP) state | y > ( θ LP = 90°) and DWP orientation θ DWP = 45°, illustrating the reflected light polarizations being continuously changed from | l > , | y > , | r > to | x > with increasing the ai r gap T a . The inset is a typical optical image of the optical metasurface (OMS) array illuminated with the incident laser spot confined within the orange dashed circle. The scale bar is 10 μm. b Calculated (left) and measured (right) normalized Stokes parameters ( S 1 , S 2 , S 3 ) of the reflected light as a function of the air gap T a and light wavelength λ . c Calculated (lines) and measured (markers) normalized Stokes parameters (above) and polarization conversion efficiencies (below) as a function of the air gap T a for the light wavelength λ = 800 nm. d Calculated (line) and measured (circles) polarization trajectory on the Poincaré sphere mapping the reflected light polarization evolution shown in ( c ). As the air gap T a increases, the reflected polarization state revolves repeatedly around the sphere. The green star marks the incident LP state: | y >. The calculated (red) and measured (black) polarization ellipses illustrate the polarization contrast between the main orthogonal polarization states (| x > and | y > , | r > and | l > ) realized by tuning the air gap T a . Full size image High contrast between the main orthogonal polarization states (| x > and | y >, | r > and | l > ), corresponding to HWP/ZWP and QWP/TQWP transformations, can be visualized at both the direct image (DI) and Fourier image (FI) planes (Fig. 3a , b). The dynamic polarization conversion behavior, induced by actuating the MEMS mirror with alternating voltages at a slow switching frequency, is captured by a charge-coupled device (CCD) camera (Supplementary Videos S2, S3 ), showing good repeatability and high modulation efficiency. By actuating the MEMS mirror with a periodic rectangular signal, filtering the reflected light with analyzers (i.e., | x >, | y >, | r > or | l > ) and detecting it with a fast photodetector, we observe relatively fast switching with rise/fall times of ~0.15/0.11 and ~0.36/0.23 ms for respective ZWP/HWP and QWP/TQWP transformations (Fig. 3c , d). The switching time for the QWP/TQWP transformation is approximately twice longer due to an approximately twice longer relocation of the MEMS mirror required (Fig. 2c ), also reflected in an approximately twice larger switching voltage used (cf., Fig. 3 c, d ). Note that, by optimizing MEMS mirrors for achieving faster switching, one should be able to reach switching frequencies in the MHz range [6] , [24] . We should also remark that a baseline voltage of ~14 V is used to move the MEMS mirror closer to the OMS, which would not be necessary for devices with more accurate mounting so that the maximum applied voltage can be kept below ~6 V. Fig. 3: Polarization conversion dynamics. a , b Optical images of the reflected light at the direct image (DI) and Fourier image (FI) planes for the fixed input linear polarization (LP) state | y > ( θ LP = 90°) and dynamic wave plate (DWP) orientation ( θ DWP = 45°) at λ = 800 nm, showing polarization modulation between orthogonal ( a ) LP states (| x >, | y >), corresponding to zero-wave plate (ZWP)/half-wave plate (HWP) transformation, by changing the actuation voltage V m from 20.2 to 17.1 V, and ( b ) circular polarization (CP) states (| r >, | l >), corresponding to quarter-wave plate (QWP)/three-quarters-wave plate (TQWP) transformation, by changing the actuation voltage V m from 19.1 to 14.4 V. For the FIs, the light is filtered using an iris letting through only light reflected from inside the orange dashed circles (shown in DIs). The leftmost column in ( a , b ) indicates the polarizer orientations used for filtering the respective polarization states in the reflected light. Scale bars in the DIs and FIs are 10 μm and 0.02 k 0 , respectively. c , d Temporal evolution of the reflected light power for c ZWP/HWP and d QWP/TQWP transformations, measured by actuating the MEMS mirror with a periodic rectangular signal and filtering respective polarization states. Full size image Versatile polarization transformations with DWP In addition to evaluating the DWP polarization tunability, polarization conversion efficiencies, and temporal response, we also explored the possibility of continuous LP rotation by a hybrid QWP-DWP component consisting of a general QWP, whose fast axis is oriented along the x -direction, and a DWP oriented at θ DWP = 45° (Fig. 4a and Supplementary Fig. S12 ). In this configuration, the QWP transforms the LP basis (| u >, | v >) of the DWP eigenstates into the CP basis ((| r >, | l >) of the hybrid QWP-DWP component, thus enabling continuous LP rotation by tuning the DWP birefringence via tuning the air gap T a . The reflected light polarization states measured while changing the air gap are visualized by a polarization trajectory around the equator of the Poincaré sphere (Fig. 4a , Supplementary Fig. S12 , and Video S4 ), demonstrating the generation of LP light with arbitrary LP orientation by adjusting the actuation voltage V m (i.e., the air gap T a ). To ease steering potential practical applications, we have also investigated the polarization transformations realized for different DWP orientations ( θ DWP = 15°, 30°, 60°, 75°) and fixed incident LP state | y > (Fig. 4b , Supplementary Fig. S13 , and Video S5 ), as well as for different incident LP states ( θ LP = 60°, 75°, 105°, 120°) and fixed DWP orientation θ DWP = 45° (Fig. 4c , Supplementary Fig. S14, and Video S6 ). The polarization trajectories on the Poincaré sphere are seen to be differently tilted for different DWP orientations having the common point, the incident LP state | y > (Fig. 4b ), while being parallel to the plane ( S 1 , S 3 ) and reflecting the incident LP state (Fig. 4c ). Note that, for any given point on the Poincaré sphere, one can identify suitable orientations of the DWP and incident LP state enabling closed polarization trajectories to pass this point, so that a multitude of polarization modulation capabilities can be realized with the same DWP. Finally, we would like to emphasize that all experimental results agree exceedingly well with the simulations without any fitting parameters, demonstrating convincingly that the fabricated DWP behaves according to the design and that the modeling approach developed is well suited for use in the future MEMS-OMS component developments. Fig. 4: Versatile polarization transformations. a – c Calculated (lines) and measured (circles) polarization trajectories on the Poincaré sphere realized at the wavelength of 800 nm by tuning the air gap T a for different dynamic wave plate (DWP) orientations, illustrating the diversity of possible polarization transformations: a continuous linear polarization (LP) rotation realized by combining the DWP with a conventional quarter-wave plate (QWP) with the incident LP state | y > and DWP orientation θ DWP = 45°, b various elliptical polarization transformations realized for different DWP orientations with the fixed incident LP state | y >, and c various elliptical polarization transformations realized for different incident LP states with the fixed DWP orientation θ DWP = 45°. The green stars in ( a – c ) indicate respective incident LP states. Full size image\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"To summarize, capitalizing on our development of the MEMS-OMS platform [22] we have demonstrated the electrically driven DWP operating in reflection with high polarization conversion efficiencies (~75%), broadband operation (~100 nm near the operating wavelength of 800 nm), fast responses (<0.4 ms) and full-range birefringence control that enables completely encircling the Poincaré sphere along trajectories determined by the incident light polarization and DWP orientation. It should be noted that, given the access to nm-sized air gaps, one can exploit the same design principle to realize the DWP with gap surface plasmons being generated [22] , a regime that promises a broader operation wavelength range (~160 nm) although probably at the expense of a lower efficiency (~50%) [22] . Importantly, the general approach developed can also be applied to design a DWP operating in transmission by using a partially transmitting MEMS mirror and placing an OMS in the middle of an FP cavity [25] . Given that a multitude of polarization modulation capabilities can be realized with the same DWP, we believe that the demonstrated electrically driven DWP configuration with full-range birefringence control opens fascinating perspectives for successful integration of high-performance compact dynamic polarization components into future miniaturized reconfigurable/adaptive optical networks and systems [26] , [27] .\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Numerical calculations All numerical simulations were done using COMSOL Multiphysics version 5.6. The model is composed of a rectangular volume with a square footprint with sides Λ = 250 nm and periodic boundary conditions were employed for both u and v directions. The DWP unit cell is divided into two parts of air and glass, with one gold nanobrick placed against the glass region. The corners of the nanobrick are rounded with a 5 nm radius. The refractive index of air is set as 1 and that of glass as 1.46 for all wavelengths, while the gold permittivity was interpolated as a function of wavelength from experimental tabulated values [28] . Using this model, the complex reflection and transmission coefficients for the glass/OMS/air interface are calculated for both propagation directions (i.e., a normal incident from glass or air) and for light linearly polarized along both u and v separately. These are used to calculate the total reflection coefficient \\\\({r}_{{{{\\\\rm{FP}}}}}\\\\) by including the gold substrate with the FP equation [19] , [20] , [21] $${r}_{{{{\\\\rm{FP}}}}}={r}_{12}+\\\\frac{{t}_{12}{t}_{21}{r}_{23}{e}^{i2{{{\\\\bf{k}}}}{n}_{2}{T}_{a}}}{1-{r}_{21}{r}_{23}{e}^{i2{{{\\\\bf{k}}}}{n}_{2}{T}_{a}}}$$ (1) Here, r mn ( t mn ) denotes the reflection (transmission) coefficients for light incident on material n from material m and the materials are numbered 1, 2, 3 for respectively glass substrate, air, and gold substrate, n 2 represents the refractive index of the air (i.e., medium 2 between the OMS layer and gold substrate), and k is the wavenumber in a vacuum. Note that the effect of the OMS on the interface between the air and glass substrate is included in r 12 , r 21 , t 12 , and t 21 , and that even for normal incidence r FP are polarization-dependent due to the anisotropy of the OMS layer. The reflection coefficient r 23 from the air/gold interface is calculated directly using the Fresnel equation. An illustration of the DWP geometry is shown in Supplementary Fig. S2 together with plots explaining the behavior of the total reflection coefficient r FP with varying T a . For air gaps much smaller than the wavelengths, there is near-field coupling between the nanobricks and the gold substrate in addition to the FP resonances [1] , [20] , requiring numerical simulations including also the gold substrate in COMSOL. The results obtained using the FP equation were confirmed to give the same results as COMSOL simulations with the whole glass/nanobrick/air/gold substrate model when T a > 100 nm for a wavelength of λ = 800 nm. As a final comment, to compare with the measurements, the imaginary part of the gold permittivity is increased by three times in the simulations of Figs. 2 , 4 and Supplementary Figs. S7 , S9 – S14 , accounting for the surface roughness and grain boundary effects of the fabricated gold nanobricks as well as the increased damping associated with the titanium (Ti) adhesion layers between the gold/glass interface. Fabrication The OMS for developing MEMS-OMS DWP were fabricated using electron-beam lithography (EBL), thin-film deposition, and lift-off techniques. First, a 100-nm-thick poly(methyl methacrylate) (PMMA A2, MicroChem) layer and a 40-nm thick conductive polymer layer (AR-PC 5090, Allresist) were successively spin-coated on the glass substrate (Borofloat 33 wafer, Wafer Universe). Note that the glass substrate was preprocessed to have a 10-μm-high circular pedestal using optical lithography and wet etching, and the OMS pattern was defined on the pedestal using EBL (JEOL JSM-6500F field-emission SEM with a Raith Elphy Quantum lithography system). After development, the OMS layer was formulated by depositing a 1-nm Ti adhesion layer and a 50-nm gold layer (Tornado 400, Cryofox) followed by lift-off in acetone (Supplementary Fig. S4 ). The pedestal on the glass substrate is very effective for reducing the possible contaminants between the MEMS mirror and OMS surface, thus improving the stability and repeatability of the DWP components. The MEMS mirror is fabricated using standard semiconductor manufacturing processes (Supplementary Fig. S3 ), in which thin-film lead zirconate titanate (PZT) is incorporated for long-stroke, low-voltage electrical actuation. For use in the MEMS-OMS component, the ultra-flat MEMS mirror was sputtered with a 100 nm gold layer. After the gold deposition, the MEMS mirror surface is inspected with white light interferometry (Zygo NewView 6300), showing overall good flatness and roughness all over the whole MEMS mirror (i.e., ~3 mm diameter) (Supplementary Fig. S3 ). The MEM-OMS-based DWP component (Supplementary Fig. S4 ) was assembled by gluing the MEMS mirror with the glass substrate upon which OMS is structured and then glued to a printed circuit board (PCB), followed by a gold wire bonding process between the MEMS mirror and PCB for enabling simple electrical connection to a voltage controller used to actuate the MEMS mirror. Characterization The experimental setup is shown in Supplementary Fig. S8 . A collimated fiber-coupled supercontinuum laser (SuperK Extreme, NKT) was directed through an HWP (AHWP10M-980, Thorlabs), a mirror, a linear polarizer (Pol 1 ; LPNIR050-MP2, Thorlabs), two beam splitters (BS 1,2 ; CCM1-BS014, Thorlabs) successively, and then focused onto the DWP samples by an objective (Obj; M Plan Apo, ×20/0.42NA, Mitutoyo). The combination of HWP and Pol 1 is used for altering the input LP states as well as the intensity. The reflected light was collected by the same objective and passed through two beam splitters (BS 2,3 ; CCM1-BS014, Thorlabs) and a tube lens (TL; TTL200-S8, Thorlabs), generating the first direct image plane where an iris is placed for filtering out the reflected light within the DWP area. The first direct image is then transformed by a relay lens (RL; AC254-200-B-ML, f = 200 mm, Thorlabs) to the corresponding Fourier image and captured by a CCD camera (CCD; DCC1545M, Thorlabs), according to a 2 f configuration. Note that a flip lens (FL; AC254-100-B-ML, f = 100 mm, Thorlabs) is used for switching between the direct and Fourier images, and a Stokes analyzer composed of a QWP (AQWP10M-980, Thorlabs) and a linear polarizer (Pol 2 ; LPNIR050-MP2, Thorlabs) is implemented before the CCD camera for performing full Stokes polarimetry [23] . Two beam splitters are configured for cross-compensating the polarization-dependent phase shifts in the beam splitters for both incidence and reflection routes. To obtain the wavelength-resolved full Stokes parameters, we replaced the CCD camera with a fiber-coupled spectrometer (QE Pro , Ocean Optics) and conducted measurements at the Fourier image plane. By rotating the QWP and Pol 2 , we recorded polarization-resolved spectra of I x ( λ ), I y ( λ ), I a ( λ ), I b ( λ ), I r ( λ ), I l ( λ ), and the stokes parameters ( s 1 , s 2 , s 3 ) are calculated as s 1 = ( I x ( λ ) – I y ( λ ))/( I x ( λ ) + I y ( λ )), s 2 = ( I a ( λ ) – I b ( λ ))/( I x ( λ ) + I y ( λ )), s 3 = ( I r ( λ ) – I l ( λ ))/( I x ( λ ) + I y ( λ )). For a reasonable comparison with the simulations, the stokes parameters ( s 1 , s 2 , s 3 ) are normalized to the polarized proportion of the reflected light beam: \\\\({S}_{{{\\\\mathrm{1,2,3}}}}=\\\\frac{{s}_{{{\\\\mathrm{1,2,3}}}}}{{{{\\\\rm{DOP}}}}}\\\\) , and the degree of polarization (DOP) is defined as \\\\({{{\\\\rm{DOP}}}}=\\\\sqrt{{s}_{1}^{2}+{s}_{2}^{2}+{s}_{3}^{2}}\\\\) . The polarization conversion efficiency (Fig. 2 and Supplementary Figs. S9 – S11 ) is defined as the ratio of the reflected light power in a specific polarization channel (i.e., | x >, | y >, | a >, | b >, | r >, | l > ) to the incident linearly | y > polarized light power. The coordinates system used is indicated in the lower-left inset of Fig. 1a , with z being the optical axis, x and y are transverse axes in the laboratory frame of reference, while u and v are transverse axes oriented along the long and short sides of the rectangular nanobricks. The angle between u and x is denoted θ DWP , while the angle between the x -axis and the polarization direction of LP incident light is denoted θ LP . To estimate the switching speeds between different orthogonal LP and CP bases (i.e., different DWP status), the setup described above is modified by replacing the input laser and CCD camera with a cw Ti:sapphire laser (Spectra-Physics 3900 S, wavelength range: 700 to 1000 nm), and a photodetector (PD; PDA20CS-EC, Thorlabs), respectively. The signals from the PD are acquired with an oscilloscope (DSOX2024A, Keysight). In the measurement, the MEMS-OMS-based DWP is actuated with periodically alternating voltages and different polarizations (i.e., | x >, | y >, | r > and | l > ) can be filtered by the Stokes analyzer.\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"The core research idea is to demonstrate electrically controlled full-range birefringence through the integration of a plasmonic optical metasurface with anisotropic meta-atoms and a thin-film piezoelectric MEMS mirror, enabling dynamic polarization control in reflection with continuously variable anisotropy and complete encircling of the Poincaré sphere."},"entities":{"kind":"list like","value":["electrically controlled full-range birefringence","plasmonic optical metasurface","anisotropic meta-atoms","thin-film piezoelectric MEMS mirror","dynamic polarization control","reflection","continuously variable anisotropy","complete encircling","Poincaré sphere"],"string":"[\n \"electrically controlled full-range birefringence\",\n \"plasmonic optical metasurface\",\n \"anisotropic meta-atoms\",\n \"thin-film piezoelectric MEMS mirror\",\n \"dynamic polarization control\",\n \"reflection\",\n \"continuously variable anisotropy\",\n \"complete encircling\",\n \"Poincaré sphere\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1.","title":"Ding, F., Pors, A. & Bozhevolnyi, S. I. Gradient metasurfaces: a review of fundamentals and applications.Rep. Prog. Phys.81, 026401 (2018).","link":"https://doi.org/10.1088%2F1361-6633%2Faa8732"},{"idx":"2.","title":"Shaltout, A. M., Shalaev, V. M. & Brongersma, M. L. Spatiotemporal light control with active metasurfaces.Science364, eaat3100 (2019).","link":"https://doi.org/10.1126%2Fscience.aat3100"},{"idx":"3.","title":"Wang, Q. et al. Optically reconfigurable metasurfaces and photonic devices based on phase change materials.Nat. Photon10, 60–65 (2016).","link":"https://doi.org/10.1038%2Fnphoton.2015.247"},{"idx":"4.","title":"Li, S. Q. et al. Phase-only transmissive spatial light modulator based on tunable dielectric metasurface.Science364, 1087–1090 (2019).","link":"https://doi.org/10.1126%2Fscience.aaw6747"},{"idx":"5.","title":"Wu, P. C. et al. Dynamic beam steering with all-dielectric electro-optic III–V multiple-quantum-well metasurfaces.Nat. Commun.10, 3654 (2019).","link":"https://doi.org/10.1038%2Fs41467-019-11598-8"},{"idx":"6.","title":"Holsteen, A. L., Cihan, A. F. & Brongersma, M. L. Temporal color mixing and dynamic beam shaping with silicon metasurfaces.Science365, 257–260 (2019).","link":"https://doi.org/10.1126%2Fscience.aax5961"},{"idx":"7.","title":"van de Groep, J. et al. Exciton resonance tuning of an atomically thin lens.Nat. Photon14, 426–430 (2020).","link":"https://doi.org/10.1038%2Fs41566-020-0624-y"},{"idx":"8.","title":"Shalaginov, M. Y. et al. Reconfigurable all-dielectric metalens with diffraction-limited performance.Nat. Commun.12, 1225 (2021).","link":"https://doi.org/10.1038%2Fs41467-021-21440-9"},{"idx":"9.","title":"Zhang, Y. et al. Electrically reconfigurable non-volatile metasurface using low-loss optical phase-change material.Nat. Nanotechnol.16, 661–666 (2021).","link":"https://doi.org/10.1038%2Fs41565-021-00881-9"},{"idx":"10.","title":"Wang, Y. et al. Electrical tuning of phase-change antennas and metasurfaces.Nat. Nanotechnol.16, 667–672 (2021).","link":"https://doi.org/10.1038%2Fs41565-021-00882-8"},{"idx":"11.","title":"Yu, P., Li, J. & Liu, N. Electrically tunable optical metasurfaces for dynamic polarization conversion.Nano Lett.21, 6690–6695 (2021).","link":"https://doi.org/10.1021%2Facs.nanolett.1c02318"},{"idx":"12.","title":"Biswas, S., Grajower, M. Y., Watanabe, K., Taniguchi, T. & Atwater, H. A. Broadband electro-optic polarization conversion with atomically thin black phosphorus.Science374, 448–453 (2021).","link":"https://doi.org/10.1126%2Fscience.abj7053"},{"idx":"13.","title":"Nicholls, L. H. et al. Ultrafast synthesis and switching of light polarization in nonlinear anisotropic metamaterials.Nat. Photon11, 628–633 (2017).","link":"https://doi.org/10.1038%2Fs41566-017-0002-6"},{"idx":"14.","title":"Schirato, A. et al. Transient optical symmetry breaking for ultrafast broadband dichroism in plasmonic metasurfaces.Nat. Photon14, 723–727 (2020).","link":"https://doi.org/10.1038%2Fs41566-020-00702-w"},{"idx":"15.","title":"Wang, K. et al. High contrast, femtosecond light polarization manipulation in epsilon-near-zero material coupled to a plasmonic nanoantenna array.ACS Photonics8, 2791–2799 (2021).","link":"https://doi.org/10.1021%2Facsphotonics.1c00971"},{"idx":"16.","title":"Zanotto, S. et al. Optomechanics of chiral dielectric metasurfaces.Adv. Opt. Mater.8, 1901507 (2020).","link":"https://doi.org/10.1002%2Fadom.201901507"},{"idx":"17.","title":"Shi, Z. et al. Continuous angle-tunable birefringence with freeform metasurfaces for arbitrary polarization conversion.Sci. Adv.6, eaba3367 (2020).","link":"https://doi.org/10.1126%2Fsciadv.aba3367"},{"idx":"18.","title":"Zhang, X. et al. Controlling angular dispersions in optical metasurfaces.Light Sci. Appl.9, 76 (2020).","link":"https://doi.org/10.1038%2Fs41377-020-0313-0"},{"idx":"19.","title":"Chen, H.-T. Interference theory of metamaterial perfect absorbers.Opt. Express20, 7165 (2012).","link":"https://doi.org/10.1364%2FOE.20.007165"},{"idx":"20.","title":"Ameling, R. & Giessen, H. Microcavity plasmonics: strong coupling of photonic cavities and plasmons.Laser Photon. Rev.7, 141–169 (2013).","link":"https://doi.org/10.1002%2Flpor.201100041"},{"idx":"21.","title":"Berkhout, A. & Koenderink, A. F. Perfect absorption and phase singularities in plasmon antenna array etalons.ACS Photonics6, 2917–2925 (2019).","link":"https://doi.org/10.1021%2Facsphotonics.9b01019"},{"idx":"22.","title":"Meng, C. et al. Dynamic piezoelectric MEMS-based optical metasurfaces.Sci. Adv.7, eabg5639 (2021).","link":"https://doi.org/10.1126%2Fsciadv.abg5639"},{"idx":"24.","title":"Wang, T., Sawada, R. & Lee, C. A piezoelectric micromachined ultrasonic transducer using piston-like membrane motion.IEEE Electron Device Lett.36, 957–959 (2015).","link":"https://doi.org/10.1109%2FLED.2015.2459075"},{"idx":"25.","title":"Berkhout, A., Wolterink, T. A. W. & Koenderink, A. F. Strong coupling to generate complex birefringence: metasurface in the middle etalons.ACS Photonics7, 2799–2806 (2020).","link":"https://doi.org/10.1021%2Facsphotonics.0c01064"},{"idx":"26.","title":"Dai, D., Bauters, J. & Bowers, J. E. Passive technologies for future large-scale photonic integrated circuits on silicon: Polarization handling, light non-reciprocity and loss reduction.Light Sci. Appl.1, e1 (2012).","link":"https://doi.org/10.1038%2Flsa.2012.1"},{"idx":"27.","title":"He, C. et al. Polarisation optics for biomedical and clinical applications: a review.Light Sci. Appl.10, 194 (2021).","link":"https://doi.org/10.1038%2Fs41377-021-00639-x"},{"idx":"28.","title":"Johnson, P. B. & Christy, R. W. Optical constants of the noble metals.Phys. Rev. B6, 4370–4379 (1972).","link":"https://doi.org/10.1103%2FPhysRevB.6.4370"}],"string":"[\n {\n \"idx\": \"1.\",\n \"title\": \"Ding, F., Pors, A. & Bozhevolnyi, S. I. Gradient metasurfaces: a review of fundamentals and applications.Rep. Prog. Phys.81, 026401 (2018).\",\n \"link\": \"https://doi.org/10.1088%2F1361-6633%2Faa8732\"\n },\n {\n \"idx\": \"2.\",\n \"title\": \"Shaltout, A. M., Shalaev, V. M. & Brongersma, M. L. Spatiotemporal light control with active metasurfaces.Science364, eaat3100 (2019).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.aat3100\"\n },\n {\n \"idx\": \"3.\",\n \"title\": \"Wang, Q. et al. Optically reconfigurable metasurfaces and photonic devices based on phase change materials.Nat. Photon10, 60–65 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fnphoton.2015.247\"\n },\n {\n \"idx\": \"4.\",\n \"title\": \"Li, S. Q. et al. Phase-only transmissive spatial light modulator based on tunable dielectric metasurface.Science364, 1087–1090 (2019).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.aaw6747\"\n },\n {\n \"idx\": \"5.\",\n \"title\": \"Wu, P. C. et al. Dynamic beam steering with all-dielectric electro-optic III–V multiple-quantum-well metasurfaces.Nat. Commun.10, 3654 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-019-11598-8\"\n },\n {\n \"idx\": \"6.\",\n \"title\": \"Holsteen, A. L., Cihan, A. F. & Brongersma, M. L. Temporal color mixing and dynamic beam shaping with silicon metasurfaces.Science365, 257–260 (2019).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.aax5961\"\n },\n {\n \"idx\": \"7.\",\n \"title\": \"van de Groep, J. et al. Exciton resonance tuning of an atomically thin lens.Nat. Photon14, 426–430 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41566-020-0624-y\"\n },\n {\n \"idx\": \"8.\",\n \"title\": \"Shalaginov, M. Y. et al. Reconfigurable all-dielectric metalens with diffraction-limited performance.Nat. Commun.12, 1225 (2021).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-021-21440-9\"\n },\n {\n \"idx\": \"9.\",\n \"title\": \"Zhang, Y. et al. Electrically reconfigurable non-volatile metasurface using low-loss optical phase-change material.Nat. Nanotechnol.16, 661–666 (2021).\",\n \"link\": \"https://doi.org/10.1038%2Fs41565-021-00881-9\"\n },\n {\n \"idx\": \"10.\",\n \"title\": \"Wang, Y. et al. Electrical tuning of phase-change antennas and metasurfaces.Nat. Nanotechnol.16, 667–672 (2021).\",\n \"link\": \"https://doi.org/10.1038%2Fs41565-021-00882-8\"\n },\n {\n \"idx\": \"11.\",\n \"title\": \"Yu, P., Li, J. & Liu, N. Electrically tunable optical metasurfaces for dynamic polarization conversion.Nano Lett.21, 6690–6695 (2021).\",\n \"link\": \"https://doi.org/10.1021%2Facs.nanolett.1c02318\"\n },\n {\n \"idx\": \"12.\",\n \"title\": \"Biswas, S., Grajower, M. Y., Watanabe, K., Taniguchi, T. & Atwater, H. A. Broadband electro-optic polarization conversion with atomically thin black phosphorus.Science374, 448–453 (2021).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.abj7053\"\n },\n {\n \"idx\": \"13.\",\n \"title\": \"Nicholls, L. H. et al. Ultrafast synthesis and switching of light polarization in nonlinear anisotropic metamaterials.Nat. Photon11, 628–633 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fs41566-017-0002-6\"\n },\n {\n \"idx\": \"14.\",\n \"title\": \"Schirato, A. et al. Transient optical symmetry breaking for ultrafast broadband dichroism in plasmonic metasurfaces.Nat. Photon14, 723–727 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41566-020-00702-w\"\n },\n {\n \"idx\": \"15.\",\n \"title\": \"Wang, K. et al. High contrast, femtosecond light polarization manipulation in epsilon-near-zero material coupled to a plasmonic nanoantenna array.ACS Photonics8, 2791–2799 (2021).\",\n \"link\": \"https://doi.org/10.1021%2Facsphotonics.1c00971\"\n },\n {\n \"idx\": \"16.\",\n \"title\": \"Zanotto, S. et al. Optomechanics of chiral dielectric metasurfaces.Adv. Opt. Mater.8, 1901507 (2020).\",\n \"link\": \"https://doi.org/10.1002%2Fadom.201901507\"\n },\n {\n \"idx\": \"17.\",\n \"title\": \"Shi, Z. et al. Continuous angle-tunable birefringence with freeform metasurfaces for arbitrary polarization conversion.Sci. Adv.6, eaba3367 (2020).\",\n \"link\": \"https://doi.org/10.1126%2Fsciadv.aba3367\"\n },\n {\n \"idx\": \"18.\",\n \"title\": \"Zhang, X. et al. Controlling angular dispersions in optical metasurfaces.Light Sci. Appl.9, 76 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41377-020-0313-0\"\n },\n {\n \"idx\": \"19.\",\n \"title\": \"Chen, H.-T. Interference theory of metamaterial perfect absorbers.Opt. Express20, 7165 (2012).\",\n \"link\": \"https://doi.org/10.1364%2FOE.20.007165\"\n },\n {\n \"idx\": \"20.\",\n \"title\": \"Ameling, R. & Giessen, H. Microcavity plasmonics: strong coupling of photonic cavities and plasmons.Laser Photon. Rev.7, 141–169 (2013).\",\n \"link\": \"https://doi.org/10.1002%2Flpor.201100041\"\n },\n {\n \"idx\": \"21.\",\n \"title\": \"Berkhout, A. & Koenderink, A. F. Perfect absorption and phase singularities in plasmon antenna array etalons.ACS Photonics6, 2917–2925 (2019).\",\n \"link\": \"https://doi.org/10.1021%2Facsphotonics.9b01019\"\n },\n {\n \"idx\": \"22.\",\n \"title\": \"Meng, C. et al. Dynamic piezoelectric MEMS-based optical metasurfaces.Sci. Adv.7, eabg5639 (2021).\",\n \"link\": \"https://doi.org/10.1126%2Fsciadv.abg5639\"\n },\n {\n \"idx\": \"24.\",\n \"title\": \"Wang, T., Sawada, R. & Lee, C. A piezoelectric micromachined ultrasonic transducer using piston-like membrane motion.IEEE Electron Device Lett.36, 957–959 (2015).\",\n \"link\": \"https://doi.org/10.1109%2FLED.2015.2459075\"\n },\n {\n \"idx\": \"25.\",\n \"title\": \"Berkhout, A., Wolterink, T. A. W. & Koenderink, A. F. Strong coupling to generate complex birefringence: metasurface in the middle etalons.ACS Photonics7, 2799–2806 (2020).\",\n \"link\": \"https://doi.org/10.1021%2Facsphotonics.0c01064\"\n },\n {\n \"idx\": \"26.\",\n \"title\": \"Dai, D., Bauters, J. & Bowers, J. E. Passive technologies for future large-scale photonic integrated circuits on silicon: Polarization handling, light non-reciprocity and loss reduction.Light Sci. Appl.1, e1 (2012).\",\n \"link\": \"https://doi.org/10.1038%2Flsa.2012.1\"\n },\n {\n \"idx\": \"27.\",\n \"title\": \"He, C. et al. Polarisation optics for biomedical and clinical applications: a review.Light Sci. Appl.10, 194 (2021).\",\n \"link\": \"https://doi.org/10.1038%2Fs41377-021-00639-x\"\n },\n {\n \"idx\": \"28.\",\n \"title\": \"Johnson, P. B. & Christy, R. W. Optical constants of the noble metals.Phys. Rev. B6, 4370–4379 (1972).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.6.4370\"\n }\n]","isConversational":false}}},{"rowIdx":11188,"cells":{"unique_id":{"kind":"string","value":"ncomms13682"},"subfield":{"kind":"string","value":"Nanoscience and technology"},"title":{"kind":"string","value":"Miniature optical planar camera based on a wide-angle metasurface doublet corrected for monochromatic aberrations"},"abstract":{"kind":"string","value":"Optical metasurfaces are two-dimensional arrays of nano-scatterers that modify optical wavefronts at subwavelength spatial resolution. They are poised to revolutionize optics by enabling complex low-cost systems where multiple metasurfaces are lithographically stacked and integrated with electronics. For imaging applications, metasurface stacks can perform sophisticated image corrections and can be directly integrated with image sensors. Here we demonstrate this concept with a miniature flat camera integrating a monolithic metasurface lens doublet corrected for monochromatic aberrations, and an image sensor. The doublet lens, which acts as a fisheye photographic objective, has a small f -number of 0.9, an angle-of-view larger than 60° × 60°, and operates at 850 nm wavelength with 70% focusing efficiency. The camera exhibits nearly diffraction-limited image quality, which indicates the potential of this technology in the development of optical systems for microscopy, photography, and computer vision."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"Optical systems such as cameras, spectrometers and microscopes are conventionally made by assembling discrete bulk optical components like lenses, gratings and filters. The optical components are manufactured separately using cutting, polishing and grinding, and have to be assembled with tight alignment tolerances, a process that is becoming more challenging as the optical systems shrink in size. Furthermore, the continuous progress of mobile, wearable, and portable consumer electronics and medical devices has rapidly increased the demand for high-performance and low-cost miniature optical systems. Optical metasurfaces offer an alternative approach for realization of optical components [1] , [2] , [3] , [4] , [5] . Recent advances have increased their efficiency and functionalities, thus allowing metasurface diffractive optical components with comparable or superior performance than conventional optical components [6] , [7] , [8] , [9] , [10] . The main advantage of metasurfaces stems from the capability to make sophisticated planar optical systems composed of lithographically stacked electronic and metasurface layers. The resulting optical system is aligned lithographically, thus eliminating the need for post-fabrication alignments. The development of the optoelectronic image sensor has been a significant step towards the on-chip integration of cameras [11] ; however, the camera lenses are yet to be fully integrated with the image sensor. The freedom in controlling the metasurface phase profiles has enabled the implementation of spherical-aberration-free flat lenses that focus normally incident light to diffraction limited spots [7] , [12] , [13] , [14] . Such lenses have been used in applications requiring focusing of an optical beam or collimating emission from an optical fibre [15] or a semiconductor laser [10] . However, the metasurface lenses suffer from other monochromatic aberrations (i.e., coma and astigmatism), which reduce their field of view and hinder their adoption in imaging applications where having a large field of view is an essential requirement. A metasurface lens can be corrected for coma if it is patterned on the surface of a sphere [16] , [17] , [18] , but direct patterning of nano-structures on curved surfaces is challenging. Although conformal metasurfaces might provide a solution [19] , the resulting device would not be flat. As we show here, another approach for correcting monochromatic aberrations of a metasurface lens is through cascading and forming a metasurface doublet lens. Here we show that a doublet lens formed by cascading two metasurfaces can be corrected over a wide range of incident angles. We also demonstrate an ultra-slim, low f -number camera, composed of two metasurface lenses placed on top of an image sensor. The camera represents an example of the optical systems enabled by the metasurface vertical integration platform.","references":{"indices":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,16,17,18,19],"infos":[{"idx":"1","title":"Kildishev, A. V., Boltasseva, A. & Shalaev, V. M. 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Simulated focal spots for such a lens are shown in Fig. 1b , exhibiting diffraction limited focusing for normal incidence and significant aberrations for incident angles as small as a few degrees. The proposed doublet lens ( Fig. 1c ) is composed of two metasurfaces behaving as polarization insensitive phase plates that are patterned on two sides of a single transparent substrate. The aberrations of two cascaded phase plates surrounded by vacuum have been studied previously in the context of holographic lenses, and it has been shown that such a combination can realize a fisheye lens with significantly reduced monochromatic aberrations [20] . We used the ray tracing approach to optimize the phase profiles of the two metasurfaces when they are separated by a 1-mm-thick fused silica substrate. Simulation results of the focal plane spot for different incident angles ( θ ) are presented in Fig. 1d , showing nearly diffraction limited focusing by the doublet up to almost 30° incident angle. The doublet lens has an input aperture diameter of 800 μm and a focal length of 717 μm corresponding to an f -number of 0.9. In the optimum design, the first metasurface operates as a corrector plate and the second one performs the significant portion of focusing; thus, we refer to them as correcting and focusing metasurfaces, respectively. The metasurfaces are designed for the operation wavelength of 850 nm, and are implemented using the dielectric nano-post metasurface platform shown in Fig. 2a (ref. 7 ). The metasurfaces are composed of hexagonal arrays of amorphous silicon nano-posts with different diameters that rest on a fused silica substrate and are covered by the SU-8 polymer. The nano-posts behave as truncated waveguides with circular cross sections supporting Fabry–Pérot resonances [7] , [9] , [19] . The high refractive index between the nano-posts and their surroundings leads to weak optical coupling among the nano-posts and allows for the implementation of any phase profile with subwavelength resolution by spatially varying the diameters of the nano-posts. Simulated intensity transmission and phase of the transmission coefficient for different nano-post diameters are presented in Fig. 2b , showing that 2 π phase coverage is achieved with an average transmission over 96% (see Methods for details). Figure 1: Focusing by metasurface singlet and doublet lenses. ( a ) Schematic illustration of focusing of on-axis and off-axis light by a spherical-aberration-free metasurface singlet lens. ( b ) Simulated focal plane intensity for different incident angles. Scale bar, 2 μm. ( c , d ) Similar illustration and simulation results as presented in a , b but for a metasurface doublet lens corrected for monochromatic aberrations. Scale bar, 2 μm. Both lenses have aperture diameter of 800 μm and focal length of 717 μm ( f -number of 0.9) and the simulation wavelength is 850 nm. See Methods for details. Full size image Figure 2: Monolithic metasurface doublet lens. ( a ) A schematic illustration of the dielectric metasurface used to implement the metasurface doublet lens. The metasurface is composed of an array of amorphous silicon nano-posts covered with a layer of SU-8 polymer and arranged in a hexagonal lattice. ( b ) Simulated intensity transmission (| t | 2 ) and the phase of transmission coefficient ( ∠ t ) of the metasurface shown in a with identical nano-posts as a function of the nano-posts’ diameter. The diameters with low transmission values, which are highlighted by two grey rectangles, are excluded from the designs. The nano-posts are 600 nm tall, the lattice constant is 450 nm, and the simulation wavelength is 850 nm. ( c ) Schematic drawing of the monolithic metasurface doublet lens composed of two metasurfaces on two sides of a 1-mm-thick fused silica substrate, an aperture stop and a field stop. The photographs of the top and bottom sides of an array of doublet lenses are also shown. ( d ) Scanning electron micrographs showing a top and an oblique view of the amorphous silicon nano-posts composing the metasurfaces. Scale bars, 1 μm. Full size image Device fabrication We fabricated the metasurfaces on both sides of a fused silica substrate by depositing amorphous silicon and defining the nano-post pattern using e-beam lithography and dry etching (see Methods for the details). First, the correcting metasurfaces were patterned on the top side of the substrate, and then the focusing metasurfaces were aligned and patterned on the substrate’s bottom side (as schematically shown in Fig. 2c ). To protect the metasurfaces while processing the other side of the substrate, the metasurfaces were cladded by a layer of cured SU-8 polymer. Aperture and field stops were formed by depositing and patterning opaque metal layers on the top and bottom sides of the substrate, respectively, and anti-reflection layers were coated on both sides of the device. Photos of the top and bottom sides of a set of fabricated metasurface doublet lenses are shown in Fig. 2c . Scanning electron microscope images of the nano-posts are shown in Fig. 2d . Focal spot and focusing efficiency characterizations We characterized the fabricated metasurface doublet by illuminating it with an 850 nm laser beam at different incident angles (as shown in Fig. 3a ), and measuring its focal spot and focusing efficiency. For comparison, a spherical-aberration-free singlet metasurface lens with the same aperture diameter and focal length as the doublet lens (phase profile , ρ : radial coordinate, f =717 μm: focal length, D =800 μm: aperture diameter) was also fabricated and characterized. The focal spots of the metasurface doublet and singlet lenses were measured with two different polarizations of incident light and are shown along with the corresponding simulation results in Fig. 3b,c , respectively (see Methods for details). The doublet lens has a nearly diffraction limited focal spot for incident angles up to more than 25° (with the criterion of Strehl ratio of larger than 0.9, see Supplementary Fig. 1 ) while the singlet exhibits significant aberrations even at incident angles of a few degrees. As Fig. 3b,c shows, simulated and measured spot shapes agree well. For the doublet lens, a small asymmetry in the 0° spot shape and slightly larger aberrations are observed in the measured spots compared with the simulation results, which we attribute to a misalignment (estimated ∼ 2 μm along both x and y directions) between the top and bottom side patterns ( Supplementary Fig. 2 ). Figure 3: Measured and simulated focal spots of the metasurface doublet and singlet lenses. ( a ) Schematic drawing of the measurement setup. ( b ) Simulated and measured focal plane intensity profiles of the metasurface doublet lens for different incident angles ( θ ). Simulation results are shown in the top row, and the measurement results for the transverse electric (TE) and the transverse magnetic (TM) polarizations are shown in the second and third rows, respectively. Simulation results are obtained using scalar approximation (that is, ignoring polarization dependence). Scale bar, 2 μm. ( c ) Simulated and measured focal plane intensity profiles for a metasurface singlet with the same aperture diameter and focal length as the metasurface doublet. For the range of angles shown, the measured intensity distributions are polarization insensitive. Scale bar, 2 μm. ( d ) Measured focusing efficiency of the metasurface doublet for TE- and TM-polarized incident light as a function of incident angle. The measured data points are shown by the symbols and the solid lines are eye guides. ( e ) Transverse location of the focal spot for the doublet lens as a function of incident angle. The measured data points are shown by the symbols, and the solid line shows the f sin( θ ) curve, where f =717 μm is the focal length of the metasurface doublet lens. Full size image The focusing efficiency (ratio of the focused power to the incident power) for the metasurface doublet lens is shown in Fig. 3d , and is ∼ 70% for normally incident light. The focusing efficiency is polarization dependent, and its value for unpolarized light drops at the rate of ∼ 1% per degree as the incident angle increases. The measured focusing efficiency at normal incidence is lower than the average of the transmission shown in Fig. 2b because of the large numerical aperture (NA) of the focusing metasurface [7] , undesired scattering due to the sidewall roughness of the nano-posts, residual reflection at the air/SU-8 interfaces, and measurement artefacts (see Methods for details). The metasurfaces are polarization insensitive at normal incidence, but their diffraction efficiency depends on the polarization of incident light for non-zero incident angles. The focusing efficiency is lower for the transverse magnetic polarized light compared with the transverse electric polarized light because of the excitation of some resonances of the nano-posts with the axial component of the electric field of the incident light [19] . This also causes the slight difference between the transverse electric and transverse magnetic spot shapes for the 30° incident light shown in Fig. 3b . We measured a focusing efficiency of ∼ 75% for the singlet, and did not observe a detectable difference between the focal spots measured with transverse electric and transverse magnetic polarizations. The measured relative location of the doublet lens focal spot as a function of incident angle is shown in Fig. 3e along with the f sin( θ ) curve. The good agreement between the measured data and the curve indicates that the metasurface doublet lens can be used as an orthographic fisheye lens or a wide angle Fourier transform lens [20] . Also, the specific f sin( θ ) fisheye distortion of the image leads to a uniform brightness over the image plane [21] . Imaging performance We characterized the imaging performance of the metasurface doublet lens using the experimental setup shown in Fig. 4a . A pattern printed on a letter-size paper was used as an object. The object was placed ∼ 25 cm away from the metasurface doublet lens and was illuminated by an LED (centre wavelength: 850 nm, bandwidth: 40 nm, spectrum shown in Supplementary Fig. 3 ). The image formed by the doublet lens was magnified by approximately × 10 using an objective and a tube lens and captured by a camera. A bandpass filter with 10 nm bandwidth (see Supplementary Fig. 3 for the spectrum) was used to spectrally filter the image and reduce the effect of chromatic aberration on the image quality. Figure 4b shows the image captured by the camera, and its insets depict the zoomed-in views of the image at 0°, 15° and 30° view angles. For comparison, an image captured using the same setup but with the metasurface singlet lens is shown in Fig. 4c . The objective lens used for magnifying the images has a smaller NA than the metasurface lenses and limits the resolution of the captured images (see Supplementary Fig. 4 for an image taken with a higher NA objective). Figure 4: Imaging with the metasurface doublet lens. ( a ) Schematic of the measurement setup. A pattern printed on a letter-size paper is used as the object. The image formed by the metasurface camera is magnified by the combination of the objective lens and the tube lens and is captured by the camera. A bandpass filter (centre wavelength: 850 nm, FWHM bandwidth: 10 nm) is placed between the objective and tube lens to reduce chromatic aberrations. ( b ) Image taken with the metasurface doublet lens and ( c ) with the spherical-aberration-free metasurface singlet lens. Scale bar, 100 μm. The insets show zoomed-in views of the images at the locations indicated by the rectangles with the same outline colours which correspond to viewing angles of 0°, 15° and 30°. Scale bar, 10 μm. ( d , e ) Modulation transfer function (MTF) of the metasurface doublet and singlet lenses, respectively. The solid and dashed lines show the MTF in the tangential plane (along x in Fig. 3b ) and sagittal plane (along y in Fig. 3b ), respectively. The diffraction limited MTF of a lens with aperture diameter of 800 μm and focal length of 717 μm is also shown for comparison. ( f ) Schematic drawing of a miniature planar camera realized using a metasurface doublet lens and a CMOS image sensor. ( g ) Imaging setup and the image captured by the miniature camera. Scale bar, 100 μm. The bandpass filter (centre wavelength: 850 nm, FWHM bandwidth: 10 nm) placed in front of the LED reduces the chromatic aberration. Full size image Any imaging system can be considered as low pass spatial filter whose transfer function varies across the field of view. For incoherent imaging systems, the transfer function for each point in the field of view can be obtained by computing the Fourier transform of the focal spot intensity. The modulus of this transfer function is referred to as the modulation transfer function (MTF) and represents the relative contrast of the image versus the spatial details of the object. The MTFs for the metasurface doublet and singlet lenses were computed using the measured focal spots ( Fig. 3b,c ) and are shown in Fig. 4d,e , respectively. Both the images and the MTFs shown in Fig. 4b–e demonstrate the effectiveness of correction achieved by cascading two metasurfaces, and the diffraction limited performance of the metasurface doublet lens over a wide field of view. Miniature metasurface camera To further demonstrate the use of this technology in imaging applications, we realized a miniature planar camera by using a metasurface doublet lens and a CMOS image sensor as schematically shown in Fig. 4f . To compensate for the light propagating through the cover glass protecting the image sensor, another doublet lens was optimized (see Metasurface Doublet Lens II in Supplementary Fig. 5 ). The total dimensions of the camera (including the image sensor) are 1.6 mm × 1.6 mm × 1.7 mm. The miniature camera was characterized using the setup shown in Fig. 4g and by imaging the object shown in Fig. 4a , which was illuminated by a filtered LED (centre wavelength: 850 nm, bandwidth: 10 nm, see Supplementary Fig. 3 for the spectrum). The image captured by the image sensor is also shown in Fig. 4g , which shows a wide field of view. The camera’s image quality is reduced by the nonuniform sensitivity of the image sensor pixels to the 850 nm light due to the colour filters, and by its larger-than-optimal pixel size. Therefore, the image quality can be improved by using a monochromatic image sensor with a smaller pixel size (the optimum pixel size for the miniature camera is ∼ 0.4 μm based on the MTFs shown in Fig. 4d ). Thus, the miniature camera benefits from the current technological trend in pixel size reduction. The intensity of the image formed by a camera only depends on the NA of its lens (it is proportional to 1/ f -number 2 =4NA 2 (ref. 22 )); therefore, the metasurface miniature camera collects a small optical power but forms a high brightness image. Furthermore, the metasurface doublet lens is telecentric in the image space, and light is incident on the image sensor with the uniform angular distribution ( Supplementary Fig. 5 ), and thus removing the need for the variable incident angle correction in the image sensor. Correcting chromatic abberations The metasurface doublet lens suffers from chromatic aberrations that reduce the image quality of the miniature camera as the illumination bandwidth increases. Simulated focal spots of the metasurface doublet lens for different illumination bandwidths and the corresponding MTFs are shown in Fig. 5a,b , respectively. See Supplementary Fig. 6 for the off-axis MTFs. As it can be seen from the MTFs, the imaging resolution decreases as the illumination bandwidth increases. This effect can be seen as reduced contrast and lower resolution in the image shown in Fig. 5c (40 nm bandwidth illumination) compared with the image shown in Fig. 5d (10 nm bandwidth illuminations). For the imaging purpose, the fractional bandwidth of a metasurface lens is proportional to ( Supplementary Note 1 ) and can be increased by reducing the NA of the metasurface lens and its focal length. Also, since the MTFs of the metasurface doublet lens shown in Fig. 5b have significant high-frequency components, the unfavourable effect of chromatic aberration can to some extent be corrected using Wiener deconvolution [23] . Figure 5e shows the deconvolution results of the image shown in Fig. 5c that is taken with a 40-nm-bandwidth illumination (see Methods for the details). As expected, the deconvolved image appears sharper and has a higher contrast than the original image; however, deconvolution also amplifies the noise, limiting its applicability for correcting the chromatic aberrations over a significantly wider bandwidth. Figure 5: Chromatic aberration of metasurface doublet lens and its correction. ( a ) Illustration of focusing of wideband light by a metasurface doublet lens. The spectrum of the incident light (shown at the bottom) is assumed as a Gaussian function centred at the design wavelength of the doublet lens (850 nm). Simulated focal plane intensity for the metasurface doublet lens as a function of incident light’s bandwidth for incident angles of 0° and 15° are shown on the right. Scale bar, 2 μm. ( b ) On-axis modulation transfer function (MTF) of the metasurface doublet lens for different bandwidth of incident light. The MTF is computed using the simulation results shown in a . ( c ) Image formed by the metasurface doublet lens with unfiltered LED illumination (40 nm FWHM bandwidth) and ( d ) with filtered LED illumination (10 nm FWHM bandwidth). Scale bar, 100 μm. See Supplementary Fig. 3 for the spectra. The images are captured using the setup shown in Fig. 4a but with a higher magnification objective ( × 20, 0.4 NA). ( e ) Result of chromatic aberration correction for the image shown in d . Scale bar, 100 μm. The insets in c – e show zoomed-in views of the images at the locations indicated by the rectangles with the same outline colours corresponding to viewing angles of 0° (red border) and 15° (green border), and the scale bars shown in the insets represent 10 μm. FWHM, full width at half maximum. Full size image"},{"section":"Discussion","content":"The metasurface doublet lens and camera can be further miniaturized by reducing the thickness of the substrate, the diameters of the metasurface lenses, the focal length of the lens, and the distance to the image sensor by the same scale factor, while using the same nano-post metasurface design presented in Fig. 2 . For example, a 10 × smaller camera (160 μm × 160 μm × 170 μm) can be designed and fabricated using a similar procedure on a 100-μm-thick fused silica substrate. Such a camera would have 10 × larger bandwidth compared with the miniature camera presented here, the same image plane intensity, but with 10 × smaller image and 100 × lower number of distinguishable pixels (94 × 94 pixels instead of 940 × 940). Compared with other miniature lenses reported previously [24] , [25] , [26] and Awaiba NanEye camera ( http://www.awaiba.com ), the metasurface doublet offers significantly smaller f -number and better correction for monochromatic aberrations that lead to brighter images with higher resolution; however, they have larger chromatic aberration (that is, narrower bandwidth). The miniature metasurface camera concept can be extended for colour and hyperspectral imaging by using a set of metasurfaces that are designed for different centre wavelengths and fabricated side by side on the same chip. Each of the metasurface doublet lenses forms an image on a portion of a single monochromatic image sensor. High-quality thin-film colour filters with different centre wavelengths can be directly deposited on the correcting metasurface of each doublet lens, and the colour filter efficiency issues associated with small size colour filters will be avoided [27] , [28] . Also, multiwavelength metasurface lenses that work at multiple discrete wavelengths have been demonstrated [29] , [30] , [31] , [32] . However, the multiwavelength metasurfaces exhibit the same chromatic dispersion (that is, ) and thus similar chromatic aberrations as the single wavelength metasurface lenses. The amorphous silicon metasurfaces have negligible absorption loss for wavelengths above 650 nm. For shorter wavelengths, materials with lower absorption loss such as polycrystalline silicon, gallium phosphide, titanium dioxide [33] , [34] or silicon nitride [35] , [36] can be used. The metasurface-enabled camera we reported here has a flat and thin form factor, small f -number, and exhibits nearly diffraction limited performance over a large field of view. From a manufacturing standpoint, the metasurface doublets have several advantages over conventional lens modules. Conventional lens modules are composed of multiple lenses that are separately manufactured and later aligned and assembled together to form the module. On the other hand, the metasurface doublets are batch manufactured with simultaneous fabrication of tens of thousands of doublets on each wafer, and with the metasurfaces aligned to each other using lithographic steps during fabrication. Furthermore, the assembly of the conventional lens modules with the image sensors has to be done in a back-end step, but the metasurface doublet can be monolithically stacked on top of image sensors. More generally, this work demonstrates a vertical on-chip integration architecture for designing and manufacturing optical systems, which is enabled through high performance metasurfaces. This architecture will enable low-cost realization of conventional optical systems (for example, spectrometers, 3D scanners, projectors, microscopes and so on), and systems with novel functionalities in a thin and planar form factor with immediate applications in medical imaging and diagnostics, surveillance and consumer electronics."},{"section":"Methods","content":"Simulation and design The phase profiles of the two metasurfaces composing the doublet lenses were obtained through the ray tracing technique using a commercial optical design software (Zemax OpticStudio, Zemax LLC). The phase profiles were defined as even order polynomials of the radial coordinate ρ as where R is the radius of the metasurface, and the coefficients a n were optimized for minimizing the focal spot size (root mean square spot size) at incident angles up to 30°. Two different metasurface doublet lenses were designed. The first doublet lens (metasurface doublet lens I) is optimized for focusing incident light in air, and was used in measurements shown in Figs 3 and 4b,d . The second doublet lens (metasurface doublet lens II) is optimized for focusing through the ∼ 445-μm-thick cover glass of the CMOS image sensor, and was used in the implementation of the miniature camera as shown in Fig. 4f,g . The optimal values of the coefficients for the two doublet lenses are listed in Supplementary Tables 1 and 2 , and the corresponding phase profiles are plotted in Supplementary Fig. 7 . The phase profile for the spherical-aberration-free metasurface singlet is given by , where f =717 μm is the focal length of the singlet (which is the same as the focal length of the doublet lens I). The simulation results shown in Figs 1b,d and 2b,c and Supplementary Fig. 2 were computed assuming the metasurfaces operate as ideal phase masks (that is, their phase profile is independent of the incident angle). Incident light was modelled as a plane wave and optical waves passing through the metasurfaces were propagated through the homogeneous regions (that is, fused silica and air) using the plane wave expansion technique [37] . The simulated focal plane intensity results for wideband incident light, which are shown Fig. 5a , were obtained by computing a weighted average of the optical intensity at several discrete wavelengths in the bandwidth of the incident light. The weights were chosen according to the power density of incident light ( Fig. 5a , bottom) assuming the diffraction efficiency of the metasurfaces is constant over the incident light’s bandwidth. This assumption is justified because the efficiency of dielectric nano-post metasurfaces does not vary significantly over ∼ 10% fractional bandwidth [7] . The simulated bandwidth-dependent modulation transfer function of the metasurface doublet lens shown in Fig. 5b and Supplementary Fig. 6 were obtained by taking the Fourier transform of the simulated focal plane intensity distributions presented in Fig. 5a . Simulated transmission data of the periodic array of amorphous silicon nano-posts presented in Fig. 2b were obtained by using the rigorous coupled wave analysis technique using a freely available software package [38] . The simulations were performed at . The amorphous silicon nano-posts (with refractive index of 3.6 at 850 nm) are 600 nm tall, rest on a fused silica substrate (refractive index of 1.45), and are cladded with a layer of the SU-8 polymer (refractive index of 1.58 at 850 nm). The imaginary part of the refractive index of amorphous silicon is smaller than 10 −4 at 850 nm and was ignored in the simulations. The nano-posts are arranged in a hexagonal lattice with the lattice constant of a =450 nm. For normal incidence, the array is non-diffractive in both SU-8 and fused silica at wavelengths longer than . The refractive indices of amorphous silicon and SU-8 polymer were obtained through variable angle spectroscopic measurements. The metasurface patterns were generated using their phase profiles and the relation between the transmission and the diameter of the nano-posts shown in Fig. 2b . The diameter of the nano-post at each lattice site ( d ) was chosen to minimize the transmission error defined as , which is the difference between the nano-post transmission t ( d ) and the desired transmission . The nano-posts diameters corresponding to low transmission, which are highlighted in Fig. 2b , are automatically avoided in this selection process, as the low transmission amplitude results in a large transmission error. Device fabrication The metasurfaces forming the doublet lenses shown in Fig. 2c were fabricated on both sides of a 1-mm-thick fused silica substrate. The substrate was cleaned using a piranha solution and an oxygen plasma. A 600-nm-thick layer of hydrogenated amorphous silicon was deposited on each side of the substrate using the plasma enhanced chemical vapour deposition technique with a 5% mixture of silane in argon at 200 °C. Next, the nano-post patterns for the correcting metasurfaces were defined on one side of the substrate as follows. First a ∼ 300-nm-thick positive electron resist (ZEP-520A) was spin coated on the substrate and baked at 180 ° C for 5 min. Then a ∼ 60-nm-thick layer of a water soluble conductive polymer (aquaSAVE, Mitsubishi Rayon) was spin coated on the resist to function as a charge dissipating layer during electron-beam patterning. The metasurface patterns and alignment marks were written on the resist using electron-beam lithography. The conductive polymer was then dissolved in water and the resist was developed in a resist developer solution (ZED-N50, Zeon Chemicals). A 70-nm-thick layer of aluminium oxide was deposited on the resist and was patterned by lifting-off the resist in a solvent (Remover PG, MicroChem). The patterned aluminium oxide was then used as the hard mask in dry etching of the underlying amorphous silicon layer. The dry etching was performed in a mixture of SF 6 and C 4 F 8 plasmas using an inductively coupled plasma reactive ion etching process. Next, the aluminium oxide mask was dissolved in a 1:1 mixture of ammonium hydroxide and hydrogen peroxide heated to 80 °C. Figure 2d shows scanning electron micrographs of the top and oblique view of the nano-posts at this step of the fabrication process. The metasurfaces were then cladded by the SU-8 polymer (SU-8 2002, MicroChem) that acts as a protective layer for the metasurfaces during the processing of the substrate’s backside. A ∼ 2-μm-thick layer of SU-8 was spin coated on the sample, baked at 90 °C for 5 min, and reflowed at 200 °C for 30 min to achieve a completely planarized surface. The SU-8 polymer was then ultraviolet exposed and cured by baking at 200 °C for another 30 min. The complete planarization of the metasurfaces, and the void-free filling of the gaps between the nano-posts were verified by cleaving a test sample fabricated using a similar procedure and inspecting the cleaved cross section using a scanning electron microscope. The focusing metasurfaces were patterned on the backside of the substrate using a procedure similar to the one used for patterning the correcting metasurfaces. To align the top and bottom metasurface patterns, a second set of alignment marks was patterned on the backside of the substrate and aligned to the topside alignment marks using optical lithography. The focusing metasurface pattern was subsequently aligned to these alignment marks. The aperture and field stops were then patterned by photo-lithography, deposition of chrome/gold (10 nm/100 nm) layers, and photoresist lift-off. To reduce the reflection at the interface between SU-8 and air, a ∼ 150-nm-thick layer of hydrogen silsesquioxane (XR-1541 from Dow Corning with refractive index of 1.4 at 850 nm) was spin coated on both sides of the substrate and baked at 180 °C for 5 min. Systematic fabrication errors due to non-optimal exposure dose in e-beam lithography, or over and under etching will generally increase or decrease the nano-post diameters almost by the same amount. To compensate for such errors, we fabricated a set of devices (as shown in Fig. 2c ) with all nano-post diameters biased by the same amount (in steps of 5 nm) from their design values. All the devices in the set showed similar focal spots, but with different focusing efficiencies. The focusing efficiency at normal incidence decreased by ∼ 2.5% for every 5 nm error in the nano-post diameters. Measurement procedure and data analysis The measurement results shown in Fig. 3b–e were obtained using the experimental setup schematically shown in Fig. 3a . An 850 nm semiconductor laser (Thorlabs L850P010, measured spectrum shown in Supplementary Fig. 3 ) was coupled to a single mode fibre. The fibre passed through a manual polarization controller and was connected to a fibre collimation package (Thorlabs F220APC-780, 1/ e 2 beam diameter: ∼ 2.3 mm). The collimated beam was passed through a linear polarizer (Thorlabs LPNIR050-MP) that sets the polarization of light incident on the doublet. The collimator and the polarizer were mounted on a rotation stage whose rotation axis coincides with the metasurface doublet lens. The focal plane of the doublet lens was imaged using an objective lens, a tube lens (Thorlabs AC254-200-B, focal length: 20 cm) and a camera (Photometrics CoolSNAP K4, pixel size: 7.4 μm). A × 100 objective lens (Olympus UMPlanFl, NA=0.95) was used in measurements shown in Fig. 3b–d , and a × 50 objective lens (Olympus LMPlanFl N, NA=0.5) with a larger field of view was used for obtaining the focal spot position data shown in Fig. 3e . A calibration sample with known feature sizes was used to accurately determine the magnification of the objective/tube lens combination for both of the objectives. The dark noise of the camera was subtracted from the measured intensity images shown in Fig. 3b,c . The focusing efficiency presented in Fig. 3d is defined as the ratio of the optical power focused by the lens to the optical power incident on the lens aperture. The focusing efficiency for the normal incidence (zero incident angle) was measured by placing a 15-μm-diameter pinhole in the focal plane of the doublet lens and measuring the optical power passed through the pinhole and dividing it by the power of the incident optical beam. For this measurement, the 1/ e 2 diameter of the incident beam was reduced to ∼ 500 μm by using a lens (Thorlabs LB1945-B, focal length: 20 cm) to ensure that more than 99% of the incident power passes through the aperture of the doublet lens (800 μm input aperture diameter). The incident and focused optical powers were measured using an optical power meter (Thorlabs PM100D with Thorlabs S122C power sensor). The pinhole was a 15-μm-diameter circular aperture formed by depositing ∼ 100 nm chrome on a fused silica substrate and had a transmission of ∼ 94% (that is, 6% of the power was reflected by the two fused silica/air interfaces), therefore the reported focusing efficiency values presented in Fig. 3d underestimate the actual values by a few percent. The focusing efficiency values for non-zero incident angles were found using the focal spot intensity distributions captured by the camera and the directly measured focusing efficiency for normal incidence. First, the focused optical powers for different incident angles were obtained by integrating the focal plane intensity distributions within a 15-μm-diameter circle centred at the intensity maximum. The intensity distributions were captured by the camera when the doublet was illuminated by a large diameter beam (1/ e 2 beam diameter of ∼ 2.3 mm) and the dark noise of the camera was subtracted from the recorded intensities before integration. Next, the focused optical powers for different incident angles were compared with the focused power at normal incidence and corrected for the smaller effective input aperture (that is, a cos( θ ) factor). The measurement was performed for transverse electric (with electric field parallel to the doublet lens’s surface) and transverse magnetic (with magnetic field parallel to the doublet lens’s surface) polarizations of the incident beam. The images presented in Fig. 4b,c were captured using the experimental setup schematically shown in Fig. 4a . A pattern was printed on a letter-size paper ( ∼ 22 cm × 28 cm) and used as an object. The object was placed in front of the metasurface lens at a distance of ∼ 25 cm from it. Three ruler marks were also printed as a part of the pattern at viewing angles of 0°, 15° and 30°. The object was illuminated by an 850 nm LED (Thorlabs LED851L, measured spectrum shown in Supplementary Fig. 3 ). The images formed by the metasurface lenses were magnified by approximately × 10 using an objective lens (Olympus UMPlanFl, × 10, NA=0.3) and a tube lens (Thorlabs AC254-200-B, focal length: 20 cm). A bandpass filter (Thorlabs FL850-10, centre wavelength: 850 nm, full width at half maximum (FWHM) bandwidth: 10 nm) was placed between the objective lens and the tube lens. The placement of the filter between the objective and the tube lens did not introduce any discernible aberrations to the optical system. The magnified images were captured using a camera (Photometrics CoolSNAP K4) with pixel size of 7.4 μm. The images shown in Fig. 5c,d and Supplementary Fig. 4 were also captured using the same setup but with different objective lenses (Olympus LMPlanFl, × 20, NA=0.4 for Fig. 5c,d and Olympus LMPlanFl N, × 50, NA=0.5 for Supplementary Fig. 4 ). The miniature camera schematically shown in Fig. 4f is composed of the metasurface doublet lens II (with parameters listed in Supplementary Table 2 ) and a low-cost colour CMOS image sensor (OmniVision OV5640, pixel size: 1.4 μm) with a cover glass thickness of 445±20 μm. An air gap of 220 μm was set between the metasurface doublet lens and the image sensor to facilitate the assembly of the camera. The metasurface doublet was mounted on a 3-axis translation stage during the measurements. To set the distance between the image sensor chip and the doublet, a far object was imaged and the distance was adjusted until the image was brought into focus. The modulation transfer functions shown in Fig. 4d,e were computed by taking the Fourier transform of the measured focal plane intensity distributions shown in Fig. 3b,c , respectively. The dark noise of the camera was first subtracted from the recorded intensity distributions. The diffraction limit curves shown in Fig. 4d,e are the simulated modulation transfer function of a diffraction limited lens (that is, Fourier transform of the diffraction limited Airy disk) with the same focal length ( f =717 μm) and aperture diameter ( D =800 μm) as the metasurface doublet and singlet lenses used in the measurements. The image shown in Fig. 5e was obtained using Wiener deconvolution [23] , and by filtering the image shown in Fig. 5c by the Wiener filter where is the spatial frequency, is the on-axis modulation transfer function of the metasurface doublet lens for illumination with 40 nm FWHM bandwidth (shown in Fig. 5b ), and and are the power spectral densities of the noise and the image, respectively. The noise was assumed to be white (that is, constant power spectral density), and was assumed to be equal to the power spectral density of an image formed with a diffraction limited imaging system with NA=0.4 (that is, the NA of the objective lens used for magnification in the experimental setup). The signal to noise ratio was found as ∼ 250 by estimating the camera’s noise, and was used to set the constant value for . Data availability The data that support the finding of this study are available from the corresponding author upon request."},{"section":"Additional information","content":"How to cite this article: Arbabi, A. et al . Miniature optical planar camera based on a wide-angle metasurface doublet corrected for monochromatic aberrations. Nat. Commun. 7, 13682 doi: 10.1038/ncomms13682 (2016). Publisher’s note : Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"Optical systems such as cameras, spectrometers and microscopes are conventionally made by assembling discrete bulk optical components like lenses, gratings and filters. The optical components are manufactured separately using cutting, polishing and grinding, and have to be assembled with tight alignment tolerances, a process that is becoming more challenging as the optical systems shrink in size. Furthermore, the continuous progress of mobile, wearable, and portable consumer electronics and medical devices has rapidly increased the demand for high-performance and low-cost miniature optical systems. Optical metasurfaces offer an alternative approach for realization of optical components [1] , [2] , [3] , [4] , [5] . Recent advances have increased their efficiency and functionalities, thus allowing metasurface diffractive optical components with comparable or superior performance than conventional optical components [6] , [7] , [8] , [9] , [10] . The main advantage of metasurfaces stems from the capability to make sophisticated planar optical systems composed of lithographically stacked electronic and metasurface layers. The resulting optical system is aligned lithographically, thus eliminating the need for post-fabrication alignments. The development of the optoelectronic image sensor has been a significant step towards the on-chip integration of cameras [11] ; however, the camera lenses are yet to be fully integrated with the image sensor. The freedom in controlling the metasurface phase profiles has enabled the implementation of spherical-aberration-free flat lenses that focus normally incident light to diffraction limited spots [7] , [12] , [13] , [14] . Such lenses have been used in applications requiring focusing of an optical beam or collimating emission from an optical fibre [15] or a semiconductor laser [10] . However, the metasurface lenses suffer from other monochromatic aberrations (i.e., coma and astigmatism), which reduce their field of view and hinder their adoption in imaging applications where having a large field of view is an essential requirement. A metasurface lens can be corrected for coma if it is patterned on the surface of a sphere [16] , [17] , [18] , but direct patterning of nano-structures on curved surfaces is challenging. Although conformal metasurfaces might provide a solution [19] , the resulting device would not be flat. As we show here, another approach for correcting monochromatic aberrations of a metasurface lens is through cascading and forming a metasurface doublet lens. Here we show that a doublet lens formed by cascading two metasurfaces can be corrected over a wide range of incident angles. We also demonstrate an ultra-slim, low f -number camera, composed of two metasurface lenses placed on top of an image sensor. 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Aberration-free imaging with an aplanatic curved diffractive element.Appl. Opt.40, 5825–5829 (2001).\",\n \"link\": \"https://doi.org/10.1364%2FAO.40.005825\"\n },\n {\n \"idx\": \"18\",\n \"title\": \"Aieta, F., Genevet, P., Kats, M. & Capasso, F. Aberrations of flat lenses and aplanatic metasurfaces.Opt. Express21, 31530–31539 (2013).\",\n \"link\": \"https://doi.org/10.1364%2FOE.21.031530\"\n },\n {\n \"idx\": \"19\",\n \"title\": \"Kamali, S. M., Arbabi, A., Arbabi, E., Horie, Y. & Faraon, A. Decoupling optical function and geometrical form using conformal flexible dielectric metasurfaces.Nat. Commun.7, 11618 (2015).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms11618\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Design and optimization of the metasurface doublet lens Figure 1a schematically shows focusing by a spherical-aberration-free metasurface lens. Simulated focal spots for such a lens are shown in Fig. 1b , exhibiting diffraction limited focusing for normal incidence and significant aberrations for incident angles as small as a few degrees. The proposed doublet lens ( Fig. 1c ) is composed of two metasurfaces behaving as polarization insensitive phase plates that are patterned on two sides of a single transparent substrate. The aberrations of two cascaded phase plates surrounded by vacuum have been studied previously in the context of holographic lenses, and it has been shown that such a combination can realize a fisheye lens with significantly reduced monochromatic aberrations [20] . We used the ray tracing approach to optimize the phase profiles of the two metasurfaces when they are separated by a 1-mm-thick fused silica substrate. Simulation results of the focal plane spot for different incident angles ( θ ) are presented in Fig. 1d , showing nearly diffraction limited focusing by the doublet up to almost 30° incident angle. The doublet lens has an input aperture diameter of 800 μm and a focal length of 717 μm corresponding to an f -number of 0.9. In the optimum design, the first metasurface operates as a corrector plate and the second one performs the significant portion of focusing; thus, we refer to them as correcting and focusing metasurfaces, respectively. The metasurfaces are designed for the operation wavelength of 850 nm, and are implemented using the dielectric nano-post metasurface platform shown in Fig. 2a (ref. 7 ). The metasurfaces are composed of hexagonal arrays of amorphous silicon nano-posts with different diameters that rest on a fused silica substrate and are covered by the SU-8 polymer. The nano-posts behave as truncated waveguides with circular cross sections supporting Fabry–Pérot resonances [7] , [9] , [19] . The high refractive index between the nano-posts and their surroundings leads to weak optical coupling among the nano-posts and allows for the implementation of any phase profile with subwavelength resolution by spatially varying the diameters of the nano-posts. Simulated intensity transmission and phase of the transmission coefficient for different nano-post diameters are presented in Fig. 2b , showing that 2 π phase coverage is achieved with an average transmission over 96% (see Methods for details). Figure 1: Focusing by metasurface singlet and doublet lenses. ( a ) Schematic illustration of focusing of on-axis and off-axis light by a spherical-aberration-free metasurface singlet lens. ( b ) Simulated focal plane intensity for different incident angles. Scale bar, 2 μm. ( c , d ) Similar illustration and simulation results as presented in a , b but for a metasurface doublet lens corrected for monochromatic aberrations. Scale bar, 2 μm. Both lenses have aperture diameter of 800 μm and focal length of 717 μm ( f -number of 0.9) and the simulation wavelength is 850 nm. See Methods for details. Full size image Figure 2: Monolithic metasurface doublet lens. ( a ) A schematic illustration of the dielectric metasurface used to implement the metasurface doublet lens. The metasurface is composed of an array of amorphous silicon nano-posts covered with a layer of SU-8 polymer and arranged in a hexagonal lattice. ( b ) Simulated intensity transmission (| t | 2 ) and the phase of transmission coefficient ( ∠ t ) of the metasurface shown in a with identical nano-posts as a function of the nano-posts’ diameter. The diameters with low transmission values, which are highlighted by two grey rectangles, are excluded from the designs. The nano-posts are 600 nm tall, the lattice constant is 450 nm, and the simulation wavelength is 850 nm. ( c ) Schematic drawing of the monolithic metasurface doublet lens composed of two metasurfaces on two sides of a 1-mm-thick fused silica substrate, an aperture stop and a field stop. The photographs of the top and bottom sides of an array of doublet lenses are also shown. ( d ) Scanning electron micrographs showing a top and an oblique view of the amorphous silicon nano-posts composing the metasurfaces. Scale bars, 1 μm. Full size image Device fabrication We fabricated the metasurfaces on both sides of a fused silica substrate by depositing amorphous silicon and defining the nano-post pattern using e-beam lithography and dry etching (see Methods for the details). First, the correcting metasurfaces were patterned on the top side of the substrate, and then the focusing metasurfaces were aligned and patterned on the substrate’s bottom side (as schematically shown in Fig. 2c ). To protect the metasurfaces while processing the other side of the substrate, the metasurfaces were cladded by a layer of cured SU-8 polymer. Aperture and field stops were formed by depositing and patterning opaque metal layers on the top and bottom sides of the substrate, respectively, and anti-reflection layers were coated on both sides of the device. Photos of the top and bottom sides of a set of fabricated metasurface doublet lenses are shown in Fig. 2c . Scanning electron microscope images of the nano-posts are shown in Fig. 2d . Focal spot and focusing efficiency characterizations We characterized the fabricated metasurface doublet by illuminating it with an 850 nm laser beam at different incident angles (as shown in Fig. 3a ), and measuring its focal spot and focusing efficiency. For comparison, a spherical-aberration-free singlet metasurface lens with the same aperture diameter and focal length as the doublet lens (phase profile , ρ : radial coordinate, f =717 μm: focal length, D =800 μm: aperture diameter) was also fabricated and characterized. The focal spots of the metasurface doublet and singlet lenses were measured with two different polarizations of incident light and are shown along with the corresponding simulation results in Fig. 3b,c , respectively (see Methods for details). The doublet lens has a nearly diffraction limited focal spot for incident angles up to more than 25° (with the criterion of Strehl ratio of larger than 0.9, see Supplementary Fig. 1 ) while the singlet exhibits significant aberrations even at incident angles of a few degrees. As Fig. 3b,c shows, simulated and measured spot shapes agree well. For the doublet lens, a small asymmetry in the 0° spot shape and slightly larger aberrations are observed in the measured spots compared with the simulation results, which we attribute to a misalignment (estimated ∼ 2 μm along both x and y directions) between the top and bottom side patterns ( Supplementary Fig. 2 ). Figure 3: Measured and simulated focal spots of the metasurface doublet and singlet lenses. ( a ) Schematic drawing of the measurement setup. ( b ) Simulated and measured focal plane intensity profiles of the metasurface doublet lens for different incident angles ( θ ). Simulation results are shown in the top row, and the measurement results for the transverse electric (TE) and the transverse magnetic (TM) polarizations are shown in the second and third rows, respectively. Simulation results are obtained using scalar approximation (that is, ignoring polarization dependence). Scale bar, 2 μm. ( c ) Simulated and measured focal plane intensity profiles for a metasurface singlet with the same aperture diameter and focal length as the metasurface doublet. For the range of angles shown, the measured intensity distributions are polarization insensitive. Scale bar, 2 μm. ( d ) Measured focusing efficiency of the metasurface doublet for TE- and TM-polarized incident light as a function of incident angle. The measured data points are shown by the symbols and the solid lines are eye guides. ( e ) Transverse location of the focal spot for the doublet lens as a function of incident angle. The measured data points are shown by the symbols, and the solid line shows the f sin( θ ) curve, where f =717 μm is the focal length of the metasurface doublet lens. Full size image The focusing efficiency (ratio of the focused power to the incident power) for the metasurface doublet lens is shown in Fig. 3d , and is ∼ 70% for normally incident light. The focusing efficiency is polarization dependent, and its value for unpolarized light drops at the rate of ∼ 1% per degree as the incident angle increases. The measured focusing efficiency at normal incidence is lower than the average of the transmission shown in Fig. 2b because of the large numerical aperture (NA) of the focusing metasurface [7] , undesired scattering due to the sidewall roughness of the nano-posts, residual reflection at the air/SU-8 interfaces, and measurement artefacts (see Methods for details). The metasurfaces are polarization insensitive at normal incidence, but their diffraction efficiency depends on the polarization of incident light for non-zero incident angles. The focusing efficiency is lower for the transverse magnetic polarized light compared with the transverse electric polarized light because of the excitation of some resonances of the nano-posts with the axial component of the electric field of the incident light [19] . This also causes the slight difference between the transverse electric and transverse magnetic spot shapes for the 30° incident light shown in Fig. 3b . We measured a focusing efficiency of ∼ 75% for the singlet, and did not observe a detectable difference between the focal spots measured with transverse electric and transverse magnetic polarizations. The measured relative location of the doublet lens focal spot as a function of incident angle is shown in Fig. 3e along with the f sin( θ ) curve. The good agreement between the measured data and the curve indicates that the metasurface doublet lens can be used as an orthographic fisheye lens or a wide angle Fourier transform lens [20] . Also, the specific f sin( θ ) fisheye distortion of the image leads to a uniform brightness over the image plane [21] . Imaging performance We characterized the imaging performance of the metasurface doublet lens using the experimental setup shown in Fig. 4a . A pattern printed on a letter-size paper was used as an object. The object was placed ∼ 25 cm away from the metasurface doublet lens and was illuminated by an LED (centre wavelength: 850 nm, bandwidth: 40 nm, spectrum shown in Supplementary Fig. 3 ). The image formed by the doublet lens was magnified by approximately × 10 using an objective and a tube lens and captured by a camera. A bandpass filter with 10 nm bandwidth (see Supplementary Fig. 3 for the spectrum) was used to spectrally filter the image and reduce the effect of chromatic aberration on the image quality. Figure 4b shows the image captured by the camera, and its insets depict the zoomed-in views of the image at 0°, 15° and 30° view angles. For comparison, an image captured using the same setup but with the metasurface singlet lens is shown in Fig. 4c . The objective lens used for magnifying the images has a smaller NA than the metasurface lenses and limits the resolution of the captured images (see Supplementary Fig. 4 for an image taken with a higher NA objective). Figure 4: Imaging with the metasurface doublet lens. ( a ) Schematic of the measurement setup. A pattern printed on a letter-size paper is used as the object. The image formed by the metasurface camera is magnified by the combination of the objective lens and the tube lens and is captured by the camera. A bandpass filter (centre wavelength: 850 nm, FWHM bandwidth: 10 nm) is placed between the objective and tube lens to reduce chromatic aberrations. ( b ) Image taken with the metasurface doublet lens and ( c ) with the spherical-aberration-free metasurface singlet lens. Scale bar, 100 μm. The insets show zoomed-in views of the images at the locations indicated by the rectangles with the same outline colours which correspond to viewing angles of 0°, 15° and 30°. Scale bar, 10 μm. ( d , e ) Modulation transfer function (MTF) of the metasurface doublet and singlet lenses, respectively. The solid and dashed lines show the MTF in the tangential plane (along x in Fig. 3b ) and sagittal plane (along y in Fig. 3b ), respectively. The diffraction limited MTF of a lens with aperture diameter of 800 μm and focal length of 717 μm is also shown for comparison. ( f ) Schematic drawing of a miniature planar camera realized using a metasurface doublet lens and a CMOS image sensor. ( g ) Imaging setup and the image captured by the miniature camera. Scale bar, 100 μm. The bandpass filter (centre wavelength: 850 nm, FWHM bandwidth: 10 nm) placed in front of the LED reduces the chromatic aberration. Full size image Any imaging system can be considered as low pass spatial filter whose transfer function varies across the field of view. For incoherent imaging systems, the transfer function for each point in the field of view can be obtained by computing the Fourier transform of the focal spot intensity. The modulus of this transfer function is referred to as the modulation transfer function (MTF) and represents the relative contrast of the image versus the spatial details of the object. The MTFs for the metasurface doublet and singlet lenses were computed using the measured focal spots ( Fig. 3b,c ) and are shown in Fig. 4d,e , respectively. Both the images and the MTFs shown in Fig. 4b–e demonstrate the effectiveness of correction achieved by cascading two metasurfaces, and the diffraction limited performance of the metasurface doublet lens over a wide field of view. Miniature metasurface camera To further demonstrate the use of this technology in imaging applications, we realized a miniature planar camera by using a metasurface doublet lens and a CMOS image sensor as schematically shown in Fig. 4f . To compensate for the light propagating through the cover glass protecting the image sensor, another doublet lens was optimized (see Metasurface Doublet Lens II in Supplementary Fig. 5 ). The total dimensions of the camera (including the image sensor) are 1.6 mm × 1.6 mm × 1.7 mm. The miniature camera was characterized using the setup shown in Fig. 4g and by imaging the object shown in Fig. 4a , which was illuminated by a filtered LED (centre wavelength: 850 nm, bandwidth: 10 nm, see Supplementary Fig. 3 for the spectrum). The image captured by the image sensor is also shown in Fig. 4g , which shows a wide field of view. The camera’s image quality is reduced by the nonuniform sensitivity of the image sensor pixels to the 850 nm light due to the colour filters, and by its larger-than-optimal pixel size. Therefore, the image quality can be improved by using a monochromatic image sensor with a smaller pixel size (the optimum pixel size for the miniature camera is ∼ 0.4 μm based on the MTFs shown in Fig. 4d ). Thus, the miniature camera benefits from the current technological trend in pixel size reduction. The intensity of the image formed by a camera only depends on the NA of its lens (it is proportional to 1/ f -number 2 =4NA 2 (ref. 22 )); therefore, the metasurface miniature camera collects a small optical power but forms a high brightness image. Furthermore, the metasurface doublet lens is telecentric in the image space, and light is incident on the image sensor with the uniform angular distribution ( Supplementary Fig. 5 ), and thus removing the need for the variable incident angle correction in the image sensor. Correcting chromatic abberations The metasurface doublet lens suffers from chromatic aberrations that reduce the image quality of the miniature camera as the illumination bandwidth increases. Simulated focal spots of the metasurface doublet lens for different illumination bandwidths and the corresponding MTFs are shown in Fig. 5a,b , respectively. See Supplementary Fig. 6 for the off-axis MTFs. As it can be seen from the MTFs, the imaging resolution decreases as the illumination bandwidth increases. This effect can be seen as reduced contrast and lower resolution in the image shown in Fig. 5c (40 nm bandwidth illumination) compared with the image shown in Fig. 5d (10 nm bandwidth illuminations). For the imaging purpose, the fractional bandwidth of a metasurface lens is proportional to ( Supplementary Note 1 ) and can be increased by reducing the NA of the metasurface lens and its focal length. Also, since the MTFs of the metasurface doublet lens shown in Fig. 5b have significant high-frequency components, the unfavourable effect of chromatic aberration can to some extent be corrected using Wiener deconvolution [23] . Figure 5e shows the deconvolution results of the image shown in Fig. 5c that is taken with a 40-nm-bandwidth illumination (see Methods for the details). As expected, the deconvolved image appears sharper and has a higher contrast than the original image; however, deconvolution also amplifies the noise, limiting its applicability for correcting the chromatic aberrations over a significantly wider bandwidth. Figure 5: Chromatic aberration of metasurface doublet lens and its correction. ( a ) Illustration of focusing of wideband light by a metasurface doublet lens. The spectrum of the incident light (shown at the bottom) is assumed as a Gaussian function centred at the design wavelength of the doublet lens (850 nm). Simulated focal plane intensity for the metasurface doublet lens as a function of incident light’s bandwidth for incident angles of 0° and 15° are shown on the right. Scale bar, 2 μm. ( b ) On-axis modulation transfer function (MTF) of the metasurface doublet lens for different bandwidth of incident light. The MTF is computed using the simulation results shown in a . ( c ) Image formed by the metasurface doublet lens with unfiltered LED illumination (40 nm FWHM bandwidth) and ( d ) with filtered LED illumination (10 nm FWHM bandwidth). Scale bar, 100 μm. See Supplementary Fig. 3 for the spectra. The images are captured using the setup shown in Fig. 4a but with a higher magnification objective ( × 20, 0.4 NA). ( e ) Result of chromatic aberration correction for the image shown in d . Scale bar, 100 μm. The insets in c – e show zoomed-in views of the images at the locations indicated by the rectangles with the same outline colours corresponding to viewing angles of 0° (red border) and 15° (green border), and the scale bars shown in the insets represent 10 μm. FWHM, full width at half maximum. Full size image\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"The metasurface doublet lens and camera can be further miniaturized by reducing the thickness of the substrate, the diameters of the metasurface lenses, the focal length of the lens, and the distance to the image sensor by the same scale factor, while using the same nano-post metasurface design presented in Fig. 2 . For example, a 10 × smaller camera (160 μm × 160 μm × 170 μm) can be designed and fabricated using a similar procedure on a 100-μm-thick fused silica substrate. Such a camera would have 10 × larger bandwidth compared with the miniature camera presented here, the same image plane intensity, but with 10 × smaller image and 100 × lower number of distinguishable pixels (94 × 94 pixels instead of 940 × 940). Compared with other miniature lenses reported previously [24] , [25] , [26] and Awaiba NanEye camera ( http://www.awaiba.com ), the metasurface doublet offers significantly smaller f -number and better correction for monochromatic aberrations that lead to brighter images with higher resolution; however, they have larger chromatic aberration (that is, narrower bandwidth). The miniature metasurface camera concept can be extended for colour and hyperspectral imaging by using a set of metasurfaces that are designed for different centre wavelengths and fabricated side by side on the same chip. Each of the metasurface doublet lenses forms an image on a portion of a single monochromatic image sensor. High-quality thin-film colour filters with different centre wavelengths can be directly deposited on the correcting metasurface of each doublet lens, and the colour filter efficiency issues associated with small size colour filters will be avoided [27] , [28] . Also, multiwavelength metasurface lenses that work at multiple discrete wavelengths have been demonstrated [29] , [30] , [31] , [32] . However, the multiwavelength metasurfaces exhibit the same chromatic dispersion (that is, ) and thus similar chromatic aberrations as the single wavelength metasurface lenses. The amorphous silicon metasurfaces have negligible absorption loss for wavelengths above 650 nm. For shorter wavelengths, materials with lower absorption loss such as polycrystalline silicon, gallium phosphide, titanium dioxide [33] , [34] or silicon nitride [35] , [36] can be used. The metasurface-enabled camera we reported here has a flat and thin form factor, small f -number, and exhibits nearly diffraction limited performance over a large field of view. From a manufacturing standpoint, the metasurface doublets have several advantages over conventional lens modules. Conventional lens modules are composed of multiple lenses that are separately manufactured and later aligned and assembled together to form the module. On the other hand, the metasurface doublets are batch manufactured with simultaneous fabrication of tens of thousands of doublets on each wafer, and with the metasurfaces aligned to each other using lithographic steps during fabrication. Furthermore, the assembly of the conventional lens modules with the image sensors has to be done in a back-end step, but the metasurface doublet can be monolithically stacked on top of image sensors. More generally, this work demonstrates a vertical on-chip integration architecture for designing and manufacturing optical systems, which is enabled through high performance metasurfaces. This architecture will enable low-cost realization of conventional optical systems (for example, spectrometers, 3D scanners, projectors, microscopes and so on), and systems with novel functionalities in a thin and planar form factor with immediate applications in medical imaging and diagnostics, surveillance and consumer electronics.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Simulation and design The phase profiles of the two metasurfaces composing the doublet lenses were obtained through the ray tracing technique using a commercial optical design software (Zemax OpticStudio, Zemax LLC). The phase profiles were defined as even order polynomials of the radial coordinate ρ as where R is the radius of the metasurface, and the coefficients a n were optimized for minimizing the focal spot size (root mean square spot size) at incident angles up to 30°. Two different metasurface doublet lenses were designed. The first doublet lens (metasurface doublet lens I) is optimized for focusing incident light in air, and was used in measurements shown in Figs 3 and 4b,d . The second doublet lens (metasurface doublet lens II) is optimized for focusing through the ∼ 445-μm-thick cover glass of the CMOS image sensor, and was used in the implementation of the miniature camera as shown in Fig. 4f,g . The optimal values of the coefficients for the two doublet lenses are listed in Supplementary Tables 1 and 2 , and the corresponding phase profiles are plotted in Supplementary Fig. 7 . The phase profile for the spherical-aberration-free metasurface singlet is given by , where f =717 μm is the focal length of the singlet (which is the same as the focal length of the doublet lens I). The simulation results shown in Figs 1b,d and 2b,c and Supplementary Fig. 2 were computed assuming the metasurfaces operate as ideal phase masks (that is, their phase profile is independent of the incident angle). Incident light was modelled as a plane wave and optical waves passing through the metasurfaces were propagated through the homogeneous regions (that is, fused silica and air) using the plane wave expansion technique [37] . The simulated focal plane intensity results for wideband incident light, which are shown Fig. 5a , were obtained by computing a weighted average of the optical intensity at several discrete wavelengths in the bandwidth of the incident light. The weights were chosen according to the power density of incident light ( Fig. 5a , bottom) assuming the diffraction efficiency of the metasurfaces is constant over the incident light’s bandwidth. This assumption is justified because the efficiency of dielectric nano-post metasurfaces does not vary significantly over ∼ 10% fractional bandwidth [7] . The simulated bandwidth-dependent modulation transfer function of the metasurface doublet lens shown in Fig. 5b and Supplementary Fig. 6 were obtained by taking the Fourier transform of the simulated focal plane intensity distributions presented in Fig. 5a . Simulated transmission data of the periodic array of amorphous silicon nano-posts presented in Fig. 2b were obtained by using the rigorous coupled wave analysis technique using a freely available software package [38] . The simulations were performed at . The amorphous silicon nano-posts (with refractive index of 3.6 at 850 nm) are 600 nm tall, rest on a fused silica substrate (refractive index of 1.45), and are cladded with a layer of the SU-8 polymer (refractive index of 1.58 at 850 nm). The imaginary part of the refractive index of amorphous silicon is smaller than 10 −4 at 850 nm and was ignored in the simulations. The nano-posts are arranged in a hexagonal lattice with the lattice constant of a =450 nm. For normal incidence, the array is non-diffractive in both SU-8 and fused silica at wavelengths longer than . The refractive indices of amorphous silicon and SU-8 polymer were obtained through variable angle spectroscopic measurements. The metasurface patterns were generated using their phase profiles and the relation between the transmission and the diameter of the nano-posts shown in Fig. 2b . The diameter of the nano-post at each lattice site ( d ) was chosen to minimize the transmission error defined as , which is the difference between the nano-post transmission t ( d ) and the desired transmission . The nano-posts diameters corresponding to low transmission, which are highlighted in Fig. 2b , are automatically avoided in this selection process, as the low transmission amplitude results in a large transmission error. Device fabrication The metasurfaces forming the doublet lenses shown in Fig. 2c were fabricated on both sides of a 1-mm-thick fused silica substrate. The substrate was cleaned using a piranha solution and an oxygen plasma. A 600-nm-thick layer of hydrogenated amorphous silicon was deposited on each side of the substrate using the plasma enhanced chemical vapour deposition technique with a 5% mixture of silane in argon at 200 °C. Next, the nano-post patterns for the correcting metasurfaces were defined on one side of the substrate as follows. First a ∼ 300-nm-thick positive electron resist (ZEP-520A) was spin coated on the substrate and baked at 180 ° C for 5 min. Then a ∼ 60-nm-thick layer of a water soluble conductive polymer (aquaSAVE, Mitsubishi Rayon) was spin coated on the resist to function as a charge dissipating layer during electron-beam patterning. The metasurface patterns and alignment marks were written on the resist using electron-beam lithography. The conductive polymer was then dissolved in water and the resist was developed in a resist developer solution (ZED-N50, Zeon Chemicals). A 70-nm-thick layer of aluminium oxide was deposited on the resist and was patterned by lifting-off the resist in a solvent (Remover PG, MicroChem). The patterned aluminium oxide was then used as the hard mask in dry etching of the underlying amorphous silicon layer. The dry etching was performed in a mixture of SF 6 and C 4 F 8 plasmas using an inductively coupled plasma reactive ion etching process. Next, the aluminium oxide mask was dissolved in a 1:1 mixture of ammonium hydroxide and hydrogen peroxide heated to 80 °C. Figure 2d shows scanning electron micrographs of the top and oblique view of the nano-posts at this step of the fabrication process. The metasurfaces were then cladded by the SU-8 polymer (SU-8 2002, MicroChem) that acts as a protective layer for the metasurfaces during the processing of the substrate’s backside. A ∼ 2-μm-thick layer of SU-8 was spin coated on the sample, baked at 90 °C for 5 min, and reflowed at 200 °C for 30 min to achieve a completely planarized surface. The SU-8 polymer was then ultraviolet exposed and cured by baking at 200 °C for another 30 min. The complete planarization of the metasurfaces, and the void-free filling of the gaps between the nano-posts were verified by cleaving a test sample fabricated using a similar procedure and inspecting the cleaved cross section using a scanning electron microscope. The focusing metasurfaces were patterned on the backside of the substrate using a procedure similar to the one used for patterning the correcting metasurfaces. To align the top and bottom metasurface patterns, a second set of alignment marks was patterned on the backside of the substrate and aligned to the topside alignment marks using optical lithography. The focusing metasurface pattern was subsequently aligned to these alignment marks. The aperture and field stops were then patterned by photo-lithography, deposition of chrome/gold (10 nm/100 nm) layers, and photoresist lift-off. To reduce the reflection at the interface between SU-8 and air, a ∼ 150-nm-thick layer of hydrogen silsesquioxane (XR-1541 from Dow Corning with refractive index of 1.4 at 850 nm) was spin coated on both sides of the substrate and baked at 180 °C for 5 min. Systematic fabrication errors due to non-optimal exposure dose in e-beam lithography, or over and under etching will generally increase or decrease the nano-post diameters almost by the same amount. To compensate for such errors, we fabricated a set of devices (as shown in Fig. 2c ) with all nano-post diameters biased by the same amount (in steps of 5 nm) from their design values. All the devices in the set showed similar focal spots, but with different focusing efficiencies. The focusing efficiency at normal incidence decreased by ∼ 2.5% for every 5 nm error in the nano-post diameters. Measurement procedure and data analysis The measurement results shown in Fig. 3b–e were obtained using the experimental setup schematically shown in Fig. 3a . An 850 nm semiconductor laser (Thorlabs L850P010, measured spectrum shown in Supplementary Fig. 3 ) was coupled to a single mode fibre. The fibre passed through a manual polarization controller and was connected to a fibre collimation package (Thorlabs F220APC-780, 1/ e 2 beam diameter: ∼ 2.3 mm). The collimated beam was passed through a linear polarizer (Thorlabs LPNIR050-MP) that sets the polarization of light incident on the doublet. The collimator and the polarizer were mounted on a rotation stage whose rotation axis coincides with the metasurface doublet lens. The focal plane of the doublet lens was imaged using an objective lens, a tube lens (Thorlabs AC254-200-B, focal length: 20 cm) and a camera (Photometrics CoolSNAP K4, pixel size: 7.4 μm). A × 100 objective lens (Olympus UMPlanFl, NA=0.95) was used in measurements shown in Fig. 3b–d , and a × 50 objective lens (Olympus LMPlanFl N, NA=0.5) with a larger field of view was used for obtaining the focal spot position data shown in Fig. 3e . A calibration sample with known feature sizes was used to accurately determine the magnification of the objective/tube lens combination for both of the objectives. The dark noise of the camera was subtracted from the measured intensity images shown in Fig. 3b,c . The focusing efficiency presented in Fig. 3d is defined as the ratio of the optical power focused by the lens to the optical power incident on the lens aperture. The focusing efficiency for the normal incidence (zero incident angle) was measured by placing a 15-μm-diameter pinhole in the focal plane of the doublet lens and measuring the optical power passed through the pinhole and dividing it by the power of the incident optical beam. For this measurement, the 1/ e 2 diameter of the incident beam was reduced to ∼ 500 μm by using a lens (Thorlabs LB1945-B, focal length: 20 cm) to ensure that more than 99% of the incident power passes through the aperture of the doublet lens (800 μm input aperture diameter). The incident and focused optical powers were measured using an optical power meter (Thorlabs PM100D with Thorlabs S122C power sensor). The pinhole was a 15-μm-diameter circular aperture formed by depositing ∼ 100 nm chrome on a fused silica substrate and had a transmission of ∼ 94% (that is, 6% of the power was reflected by the two fused silica/air interfaces), therefore the reported focusing efficiency values presented in Fig. 3d underestimate the actual values by a few percent. The focusing efficiency values for non-zero incident angles were found using the focal spot intensity distributions captured by the camera and the directly measured focusing efficiency for normal incidence. First, the focused optical powers for different incident angles were obtained by integrating the focal plane intensity distributions within a 15-μm-diameter circle centred at the intensity maximum. The intensity distributions were captured by the camera when the doublet was illuminated by a large diameter beam (1/ e 2 beam diameter of ∼ 2.3 mm) and the dark noise of the camera was subtracted from the recorded intensities before integration. Next, the focused optical powers for different incident angles were compared with the focused power at normal incidence and corrected for the smaller effective input aperture (that is, a cos( θ ) factor). The measurement was performed for transverse electric (with electric field parallel to the doublet lens’s surface) and transverse magnetic (with magnetic field parallel to the doublet lens’s surface) polarizations of the incident beam. The images presented in Fig. 4b,c were captured using the experimental setup schematically shown in Fig. 4a . A pattern was printed on a letter-size paper ( ∼ 22 cm × 28 cm) and used as an object. The object was placed in front of the metasurface lens at a distance of ∼ 25 cm from it. Three ruler marks were also printed as a part of the pattern at viewing angles of 0°, 15° and 30°. The object was illuminated by an 850 nm LED (Thorlabs LED851L, measured spectrum shown in Supplementary Fig. 3 ). The images formed by the metasurface lenses were magnified by approximately × 10 using an objective lens (Olympus UMPlanFl, × 10, NA=0.3) and a tube lens (Thorlabs AC254-200-B, focal length: 20 cm). A bandpass filter (Thorlabs FL850-10, centre wavelength: 850 nm, full width at half maximum (FWHM) bandwidth: 10 nm) was placed between the objective lens and the tube lens. The placement of the filter between the objective and the tube lens did not introduce any discernible aberrations to the optical system. The magnified images were captured using a camera (Photometrics CoolSNAP K4) with pixel size of 7.4 μm. The images shown in Fig. 5c,d and Supplementary Fig. 4 were also captured using the same setup but with different objective lenses (Olympus LMPlanFl, × 20, NA=0.4 for Fig. 5c,d and Olympus LMPlanFl N, × 50, NA=0.5 for Supplementary Fig. 4 ). The miniature camera schematically shown in Fig. 4f is composed of the metasurface doublet lens II (with parameters listed in Supplementary Table 2 ) and a low-cost colour CMOS image sensor (OmniVision OV5640, pixel size: 1.4 μm) with a cover glass thickness of 445±20 μm. An air gap of 220 μm was set between the metasurface doublet lens and the image sensor to facilitate the assembly of the camera. The metasurface doublet was mounted on a 3-axis translation stage during the measurements. To set the distance between the image sensor chip and the doublet, a far object was imaged and the distance was adjusted until the image was brought into focus. The modulation transfer functions shown in Fig. 4d,e were computed by taking the Fourier transform of the measured focal plane intensity distributions shown in Fig. 3b,c , respectively. The dark noise of the camera was first subtracted from the recorded intensity distributions. The diffraction limit curves shown in Fig. 4d,e are the simulated modulation transfer function of a diffraction limited lens (that is, Fourier transform of the diffraction limited Airy disk) with the same focal length ( f =717 μm) and aperture diameter ( D =800 μm) as the metasurface doublet and singlet lenses used in the measurements. The image shown in Fig. 5e was obtained using Wiener deconvolution [23] , and by filtering the image shown in Fig. 5c by the Wiener filter where is the spatial frequency, is the on-axis modulation transfer function of the metasurface doublet lens for illumination with 40 nm FWHM bandwidth (shown in Fig. 5b ), and and are the power spectral densities of the noise and the image, respectively. The noise was assumed to be white (that is, constant power spectral density), and was assumed to be equal to the power spectral density of an image formed with a diffraction limited imaging system with NA=0.4 (that is, the NA of the objective lens used for magnification in the experimental setup). The signal to noise ratio was found as ∼ 250 by estimating the camera’s noise, and was used to set the constant value for . Data availability The data that support the finding of this study are available from the corresponding author upon request.\"\n },\n {\n \"section\": \"Additional information\",\n \"content\": \"How to cite this article: Arbabi, A. et al . Miniature optical planar camera based on a wide-angle metasurface doublet corrected for monochromatic aberrations. Nat. Commun. 7, 13682 doi: 10.1038/ncomms13682 (2016). Publisher’s note : Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"The core research idea is to correct monochromatic aberrations in metasurface lenses by cascading two metasurfaces to form a doublet lens, enabling a wide field of view and facilitating the development of ultra-slim, low f-number cameras through vertical integration with image sensors."},"entities":{"kind":"list like","value":["monochromatic aberrations","metasurface lenses","cascading","doublet lens","wide field of view","ultra-slim cameras","low f-number cameras","vertical integration","image sensors"],"string":"[\n \"monochromatic aberrations\",\n \"metasurface lenses\",\n \"cascading\",\n \"doublet lens\",\n \"wide field of view\",\n \"ultra-slim cameras\",\n \"low f-number cameras\",\n \"vertical integration\",\n \"image sensors\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1","title":"Kildishev, A. 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Orbital angular momentum-based space division multiplexing for high-capacity underwater optical communications.Sci. Rep.6, 33306 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fsrep33306\"\n },\n {\n \"idx\": \"38\",\n \"title\": \"Liu, V. & Fan, S. S4: a free electromagnetic solver for layered periodic structures.Comput. Phys. Commun.183, 2233–2244 (2012).\",\n \"link\": \"https://doi.org/10.1016%2Fj.cpc.2012.04.026\"\n }\n]","isConversational":false}}},{"rowIdx":11189,"cells":{"unique_id":{"kind":"string","value":"s41467-020-16108-9"},"subfield":{"kind":"string","value":"Mathematics and computing"},"title":{"kind":"string","value":"Accurate deep neural network inference using computational phase-change memory"},"abstract":{"kind":"string","value":"In-memory computing using resistive memory devices is a promising non-von Neumann approach for making energy-efficient deep learning inference hardware. However, due to device variability and noise, the network needs to be trained in a specific way so that transferring the digitally trained weights to the analog resistive memory devices will not result in significant loss of accuracy. Here, we introduce a methodology to train ResNet-type convolutional neural networks that results in no appreciable accuracy loss when transferring weights to phase-change memory (PCM) devices. We also propose a compensation technique that exploits the batch normalization parameters to improve the accuracy retention over time. We achieve a classification accuracy of 93.7% on CIFAR-10 and a top-1 accuracy of 71.6% on ImageNet benchmarks after mapping the trained weights to PCM. Our hardware results on CIFAR-10 with ResNet-32 demonstrate an accuracy above 93.5% retained over a one-day period, where each of the 361,722 synaptic weights is programmed on just two PCM devices organized in a differential configuration."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"Deep neural networks (DNNs) have revolutionized the field of artificial intelligence and have achieved unprecedented success in cognitive tasks such as image and speech recognition. Platforms for deploying the trained model of such networks and performing inference in an energy-efficient manner are highly attractive for edge computing applications. In particular, internet-of-things battery-powered devices and autonomous cars could especially benefit from fast, low-power, and reliably accurate DNN inference engines. Significant progress in this direction has been made with the introduction of specialized hardware for inference operating at reduced digital precision (4–8-bit), such as Google’s tensor processing unit [1] and low-power graphical processing units such as NVIDIA T4 [2] . While these platforms are very flexible, they are based on architectures where there is a physical separation between memory and processing units. The models are typically stored in off-chip memory, leading to constant shuttling of data between memory and processing units, which limits the maximum achievable energy efficiency. In order to reduce the data transfers to a minimum in inference accelerators, a promising avenue is to employ in-memory computing using non-volatile memory devices [3] , [4] , [5] . Both charge-based storage devices, such as Flash memory [6] , and resistance-based (memristive) storage devices, such as metal-oxide resistive random-access memory [7] , [8] , [9] , [10] and phase-change memory (PCM) [11] , [12] , [13] , [14] are being investigated for this. In this approach, the network weights are encoded as the analog charge state or conductance state of these devices organized in crossbar arrays, and the matrix-vector multiplications during inference can be performed in-situ in a single time step by exploiting Kirchhoff’s circuit laws. The fact that these devices are non-volatile (the weights will be retained when the power supply is turned off) and have multi-level storage capability (a single device can encode an analog range of values as opposed to 1 bit) is very attractive for inference applications. However, due to the analog nature of the weights programmed in these devices, only limited precision can be achieved in the matrix-vector multiplications and this could limit the achievable inference accuracy of the accelerator. One potential solution to this problem is to train the network fully on hardware [13] , [14] , [15] , such that all hardware non-idealities would be de facto included as constraints during training. Another similar approach is to perform partial optimizations of the hardware weights after transferring a trained model to the chip [9] , [16] , [17] . The drawback of these approaches is that every neural network would have to be trained on each individual chip before deployment. Off-line variation-aware training schemes have also been proposed, where hardware non-idealities such as device-to-device variations [18] , [19] , defective devices [19] , or IR drop [18] are first characterized and then fed into the training algorithm running in software. However, these approaches would require characterizing and training the neural network from scratch for every chip. A more practical approach would be to have a single custom generic training algorithm run entirely in software, which would make the network immune to most of the hardware non-idealities, but at the same time would require only very little knowledge about the specific hardware it will be deployed on. In this way, the model would have to be trained only once and could be deployed on a multitude of different chips. To this end, several works have proposed to inject noise in the training algorithm to the layer inputs [20] , synaptic weights [21] , and pre-activations [22] , [23] . However, previous demonstrations have generally been limited to rather simple and shallow networks, and experimental validations of the effectiveness of the various approaches have been missing. We are aware of one recent work that analyzed more complex problems such as ImageNet classification [23] , however the hardware model used was rather abstract and no experimental validation was presented. In this work, we explore injecting noise to the synaptic weights during the training of DNNs in software as a generic method to improve the network resilience against analog in-memory computing hardware non-idealities. We focus on the ResNet convolutional neural network (CNN) architecture, and introduce a number of techniques that allow us to achieve a classification accuracy of 93.7% on the CIFAR-10 dataset and a top-1 accuracy of 71.6% on the ImageNet benchmark after mapping the trained weights to PCM synapses. In contrast to previous approaches, the noise injected during training is crudely estimated from a one-time all-around hardware characterization, and captures the combined effect of read and write noise without introducing additional noise-related training hyperparameters. We validate the training approach through hardware/software experiments, where each of the 361,722 weights of ResNet-32 is programmed on two PCM devices of a prototype chip, and the rest of the network functionality is simulated in software. We achieve an experimental accuracy of 93.75% after programming, which stays above 92.6% over a period of 1 day. To improve the accuracy retention further, we develop a method to periodically calibrate the batch normalization parameters to correct the activation distributions during inference. We demonstrate a significant improvement in the accuracy retention with this method (up to 93.5% on hardware for CIFAR-10) compared with a simple global scaling of the layers’ outputs, at the cost of additional digital computations during calibration. Finally, we discuss our training approach with respect to other methods and quantify the tradeoffs in terms of accuracy and ease of training.","references":{"indices":[5,6,7,8,9,10,11,12,13,17,20,21],"infos":[{"idx":"5","title":"Wang, Z. et al. Resistive switching materials for information processing.Nat. Rev. Mater.5, 173–195 (2020).","link":"https://doi.org/10.1038%2Fs41578-019-0159-3"},{"idx":"6","title":"Merrikh-Bayat, F. et al. High-performance mixed-signal neurocomputing with nanoscale floating-gate memory cell arrays.IEEE Trans. Neural Netw. Learn. Syst.29, 4782–4790 (2018).","link":"https://doi.org/10.1109%2FTNNLS.2017.2778940"},{"idx":"7","title":"Chen, W.-H. et al. CMOS-integrated memristive non-volatile computing-in-memory for AI edge processors.Nat. Electron.2, 420–428 (2019).","link":"https://doi.org/10.1038%2Fs41928-019-0288-0"},{"idx":"8","title":"Hu, M. et al. Memristor-based analog computation and neural network classification with a dot product engine.Adv. 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Equivalent-accuracy accelerated neural-network training using analogue memory.Nature558, 60–67 (2018).","link":"https://doi.org/10.1038%2Fs41586-018-0180-5"},{"idx":"17","title":"Gonugondla, S. K., Kang, M. & Shanbhag, N. R. A variation-tolerant in-memory machine learning classifier via on-chip training.IEEE J. Solid-State Circuits53, 3163–3173 (2018).","link":"https://doi.org/10.1109%2FJSSC.2018.2867275"},{"idx":"20","title":"Moon, S., Shin, K. & Jeon, D. Enhancing reliability of analog neural network processors.IEEE Trans. Very Large Scale Integr. Syst.27, 1455–1459 (2019).","link":"https://doi.org/10.1109%2FTVLSI.2019.2893256"},{"idx":"21","title":"Miyashita, D., Kousai, S., Suzuki, T. & Deguchi, J. A neuromorphic chip optimized for deep learning and CMOS technology with time-domain analog and digital mixed-signal processing.IEEE J. Solid State Circuits52, 2679–2689 (2017).","link":"https://doi.org/10.1109%2FJSSC.2017.2712626"}]}},{"section":"Results","content":"Problem statement For our experiments, we consider two residual networks on two different datasets: ResNet-32 on the CIFAR-10 dataset, and ResNet-34 on the ImageNet dataset [24] . As shown in Fig. 1a , ResNet-32 consists of three different ResNet blocks with ten 3 × 3 kernels each, and is used to classify 32 × 32-pixel RGB images that belong to one out of ten classes (see Methods). The network contains 361,722 synaptic weights. The ResNet-34 network used for the 1000-class ImageNet dataset is shown in Supplementary Fig. 1 . The main differences compared to ResNet-32 are the number and size of the ResNet blocks, and a larger number of input/output channels (see Methods). Fig. 1: Training and inference methodology. a ResNet-32 network architecture for CIFAR-10 image classification. The architecture of ResNet-32 used in this study is a slightly modified version of the original implementation [24] with fewer input and output channels in ResNet blocks 2 and 3. This allows each layer to fit on a single crossbar array of size 512 × 512 in an eventual computational memory accelerator. b Training and inference of an example layer of ResNet-32 according to the methodology proposed in this paper. Software training is performed by injecting a random noise term to the weights used during the forward propagation, \\(\\delta {W}_{{\\rm{tr}}}^{{{l}}}\\) , which is representative of the combined read and write noise of the memristive devices used during inference (see section Training procedure). When transferring the weights of a convolution layer to memristive crossbars for inference, they are flattened into a 2-D matrix by collapsing each filter into a single vector programmed on a crossbar column, and stacking all filters on separate columns [25] . The weights are then programmed as the differential conductance of two memristive devices. Input activations a l are applied as voltages on the crossbar rows. The output current from the column containing the G − devices is subtracted from the one from the column containing the G + devices. The differential current output I from the crossbar then routes to a digital circuitry that performs batch normalization and the corresponding rectified linear unit (ReLU) activation function, in order to obtain the input activations for the next layer a l +1 . The final softmax activation function can be performed off-chip if required. An optional correction of the batch normalization parameters can be periodically performed to improve the accuracy retention over time (see sections Hardware/software inference experiment on CIFAR-10 and Adaptive batch normalization statistics update). The input image is padded with zero values at the border to ensure that the convolution operation preserves the height and width of the image. Therefore, considering an input image of size n × n , the convolution operation can be performed in n 2 matrix-vector multiplication cycles. Full size image The weights of all convolution layers along with the fully connected layer of ResNet-32 can be mapped on memristive crossbar arrays as explained in Fig. 1b [25] . Each synaptic weight can be mapped on a differential pair of memristive devices that are located on two different columns. For a given layer l , the synaptic weight \\({W}_{{{ij}}}^{{{l}}}\\) of the ( i , j ) th synaptic element is represented by the effective synaptic conductance \\({G}_{{{ij}}}^{{{l}}}\\) given by $${G}_{{{ij}}}^{{{l}}}={G}_{{{ij}}}^{{{l}},+}-{G}_{{{ij}}}^{{{l}},-},$$ (1) where \\({G}_{{{ij}}}^{{{l}},+}\\) and \\({G}_{{{ij}}}^{{{l}},-}\\) are the conductance values of the two devices forming the differential pair. Those device conductance values are defined as the effective conductance perceived in the operation of a non-ideal memristive crossbar array, and therefore include all the circuit non-idealities from the crossbar and peripheral circuitry. The mapping between the synaptic weight \\({W}_{{{ij}}}^{{{l}}}\\) obtained after software training and the corresponding synaptic conductance is given by $${G}_{{{ij}}}^{{{l}}}={W}_{{{ij}}}^{{{l}}}\\,\\, \\times \\frac{{G}_{\\max }}{{W}_{\\max }^{{{l}}}}+\\delta {G}_{{{ij}}}^{{{l}}}={G}_{{\\rm{T}},{{ij}}}^{{{l}}}+\\delta {G}_{{{ij}}}^{{{l}}},$$ (2) where \\({G}_{\\max }\\) is the maximum reliably programmable device conductance and \\({W}_{\\max }^{{{l}}}\\) is the maximum absolute synaptic weight value of layer l . \\(\\delta {G}_{{{ij}}}^{{{l}}}\\) represents the synaptic conductance error from the ideal target conductance value \\({G}_{{\\rm{T}},{{ij}}}^{{{l}}}={W}_{{{ij}}}^{{{l}}}\\, \\times \\frac{{G}_{\\max }}{{W}_{\\max }^{{{l}}}}\\) . \\(\\delta {G}_{{{ij}}}^{{{l}}}\\) is a time-varying random variable that describes the effects of non-ideal device programming (inaccuracies associated with write) and conductance fluctuations over time (inaccuracies associated with read). Possible factors leading to such conductance errors include inaccuracies in programming the synaptic conductance to \\({G}_{{\\rm{T}},{{ij}}}^{{{l}}}\\) , 1/ f noise from memristive devices and circuits, temporal conductance drift, device-to-device variations, defective (stuck) devices, and circuit non-idealities (e.g., IR drop). Clearly, a direct mapping of the synaptic weights of a DNN trained with 32-bit floating point (FP32) precision to the same DNN with memristive synapses is expected to degrade the network accuracy due to the added error in the weights arising from \\(\\delta {G}_{{{ij}}}^{{{l}}}\\) . For existing memristive technologies, the magnitude of \\(\\delta {G}_{{{ij}}}^{{{l}}}\\) may range from 1 to 10% of the magnitude of \\({G}_{{\\rm{T}},{{ij}}}^{{{l}}}\\) [11] , which in general is not tolerable by DNNs trained with FP32 without any constrains. Imposing such errors as constraints during training can be beneficial in improving the network accuracy. In fact, quantization of the weights or activations [26] , and injecting noise on the weights [27] , activations [28] , or gradients [29] have been widely used as DNN regularizers during training to reduce overfitting on the training dataset [30] , [31] . These techniques can improve the accuracy of DNN inference when it is performed with the same model precision as during training. However, achieving baseline accuracy while performing DNN inference on a model which is inevitably different from the one obtained after training, as is the case for any analog in-memory computing hardware, is a more difficult problem and requires additional investigations. Although a large body of efficient techniques to train DNNs with reduced digital precision has been reported [32] , [33] , it is unlikely that such procedures can generally be applied as-is to analog in-memory computing hardware due to the random nature of \\(\\delta {G}_{{{ij}}}^{{{l}}}\\) . Since quantization errors coming from rounding to reduced fixed-point precision are not random, DNNs trained in this way are not a priori expected to be suitable for deployment on analog in-memory computing hardware. Techniques that inject random Gaussian noise during training are a much more natural fit to make the network robust to errors from analog in-memory computing hardware. As early as in 1994, it was shown that injecting noise on the synaptic weights during training enhances the tolerance to weight perturbations of multi-layer perceptrons, and the application of this technique to analog neural hardware was discussed [34] . Recent works have also proposed to apply noise to the layer inputs or pre-activations in order to improve the network tolerance to hardware noise [20] , [23] . In this work, we follow the original approach of Murray et al. [34] of injecting Gaussian noise to the synaptic weights during training. Next, we discuss different techniques that we introduced together with synaptic weight noise in order to improve the accuracy of inference on ResNet and achieve close to software-equivalent accuracy after transferring the weights to PCM hardware. Training procedure When performing inference with analog in-memory computing hardware, the DNN experiences errors primarily due to (i) inaccurate programming of the network weights onto the devices (write noise) and (ii) temporal fluctuations of the hardware weights (read noise). We can cast the effect of these errors into a single error term \\(\\delta {G}_{{{ij}}}^{{{l}}}\\) that distorts each synaptic weight when performing forward propagation during inference. Hence, we propose to add random noise that corresponds to the error induced by \\(\\delta {G}_{{{ij}}}^{{{l}}}\\) to the synaptic weights at each forward pass during training (see Fig. 1b ). The backward pass and weight updates are performed with weights that did not experience this noise. We found that adding noise to the weights only in the forward propagation is sufficient to achieve close to baseline accuracy for a noise magnitude comparable to that of our hardware, and adding noise during the backward propagation did not improve the results further. For simplicity, we assume that \\(\\delta {G}_{{{ij}}}^{{{l}}}\\) is Gaussian distributed, which is usually the case for analog memristive hardware. Weights are linearly mapped to the entire conductance range \\({G}_{\\max }\\) of the hardware, hence the standard deviation \\({\\sigma }_{\\delta {W}_{{\\rm{tr}}}}^{l}\\) of the Gaussian noise on weights to be applied during training, for a layer l , can be obtained as $$\\frac{{\\sigma }_{\\delta {W}_{{\\rm{tr}}}}^{{{l}}}}{{W}_{\\max }^{{{l}}}}\\equiv {\\eta }_{{\\rm{tr}}}=\\frac{{\\sigma }_{\\delta {G}}}{{G}_{\\max }},$$ (3) where σ δG is a representative standard deviation of the combined read and write noise measured from hardware. During training, the weight distribution of every layer and hence \\({W}_{\\max }^{{{l}}}\\) changes, therefore \\({\\sigma }_{\\delta {W}_{{\\rm{tr}}}}^{{{l}}}\\) is recomputed after every weight update so that \\({\\eta }_{{\\rm{tr}}}\\) stays constant throughout training. We found this to be especially important in achieving good training convergence with this method. Weight initialization can have a significant effect on DNN training [35] . Two different weight initializations can lead to completely different minima when optimizing the network objective function. The network optimum when training with additive noise could be closer to the FP32 training optimum than to a completely random initialization. So it can be beneficial to initialize weights from a pretrained baseline network and then retrain this network by injecting noise. A similar observation was reported for training ResNet with reduced digital precision [33] . For achieving high classification accuracy in our experiments, we found this strategy more helpful than random initialization. The noise injected during training according to Eq. ( 3 ) is closely related to the maximum weight of a layer, and can thus grow uncontrollably with outlier weight values. Controlling the weight distribution in a desirable range can improve the network training convergence and make the mapping of weights to hardware with limited conductance range easier. We therefore clip the synaptic weights at layer l after every weight update in the range \\([-\\alpha \\times {\\sigma }_{{W}^{{{l}}}},\\alpha \\times {\\sigma }_{{W}^{{{l}}}}]\\) , where \\({\\sigma }_{{{{W}}}^{{{l}}}}\\) is the standard deviation of weights in layer l and α is a tunable hyperparameter. In our studies, α = 2.0 and α = 2.5 worked the best for ResNet-32 and ResNet-34, respectively. DNN convergence accuracy, in general, is sensitive to the learning rate used during training. Since we initialize the network parameters from a baseline network, using the same learning rate scheduling as that of the baseline network does not guarantee accurate convergence. To choose appropriate learning rate scheduling for ResNet-32, we first forward propagate the training set on the pretrained baseline network with injected synaptic weight noise and note the resulting accuracy. We note the learning rate evolution starting from this accuracy in the baseline network training curve until convergence, and use the same learning rate evolution while retraining the network by injecting noise. We performed simulations to characterize the inference performance after training incorporating the injection of Gaussian noise in conjunction with the techniques presented above. We computed the classification accuracy for different amounts of injected noise \\({\\eta }_{{\\rm{tr}}}\\) during training. We also show how the accuracy is affected when the inference weights are perturbed by a certain amount of relative noise \\({\\eta }_{\\inf }\\equiv \\frac{{\\sigma }_{\\delta {W}_{\\inf }}^{l}}{{W}_{\\max }^{l}}\\) , where \\({\\sigma }_{\\delta {W}_{\\inf }}^{l}\\) is the standard deviation of the noise injected to the weights of layer l before performing inference on the test dataset. The test accuracy of ResNet-32 on CIFAR-10 obtained for different amounts of noise injected during training, without inducing any perturbation during inference ( \\({\\eta }_{\\inf }=0\\) ), is plotted in Fig. 2a . It can be seen that the training algorithm is able to achieve a test accuracy close to the software baseline of 93.87% with up to approximately \\({\\eta }_{{\\rm{tr}}}=8 \\%\\) . The tolerance of the networks trained with different amounts of \\({\\eta }_{{\\rm{tr}}}\\) to weight perturbations during inference, \\({\\eta }_{\\inf }\\) , is shown in Fig. 2b . For a given value of \\({\\eta }_{\\inf }\\) , in general, the highest test accuracy can be obtained for the network that has been trained with a comparable amount of synaptic weight noise, i.e. for \\({\\eta }_{{\\rm{tr}}}\\approx {\\eta }_{\\inf }\\) . The test accuracy for \\({\\eta }_{{\\rm{tr}}}={\\eta }_{\\inf }\\) is shown in Fig. 2c . It can be seen that for up to \\({\\eta }_{\\inf }=5 \\%\\) , an accuracy within 0.5% of the software baseline is achievable. The impact of the weight initialization, clipping, and learning rate scheduling on the accuracy is shown in Supplementary Fig. 2 . Not incorporating one of those three techniques results in at least 1% drop in test accuracy for \\({\\eta }_{{\\rm{tr}}}={\\eta }_{\\inf }=3.8 \\%\\) . Fig. 2: Impact of injecting weight noise during training and inference on network accuracy. a Test accuracy on CIFAR-10 obtained for different amounts of relative injected weight noise during training \\({\\eta }_{{\\rm{tr}}}\\) without inducing any perturbation during inference ( \\({\\eta }_{\\inf }=0\\) ). When \\({\\eta }_{{\\rm{tr}}}\\, > \\, 8 \\%\\) , the training convergence starts to become affected by the high noise and it is not possible anymore to reach the software baseline within the same number of epochs. The error bars represent the standard deviation over 10 training runs. b Test accuracy on CIFAR-10 obtained for the networks trained with different amounts of relative weight noise \\({\\eta }_{{\\rm{tr}}}\\) as a function of the weight noise injected during inference \\({\\eta }_{\\inf }\\) . In most cases, \\({\\eta }_{{\\rm{tr}}}\\) can be increased above \\({\\eta }_{\\inf }\\) up to a certain point and still lead to comparable or slightly higher (within ≈0.1%) test accuracy than for \\({\\eta }_{{\\rm{tr}}}={\\eta }_{\\inf }\\) . However, when \\({\\eta }_{{\\rm{tr}}}\\) becomes much higher than \\({\\eta }_{\\inf }\\) , the test accuracy decreases due to the inability of the network to achieve baseline accuracy when \\({\\eta }_{{\\rm{tr}}}\\, > \\, 8 \\%\\) . Each data point represents the average over 10 training runs and 100 inference runs. c Test accuracy on CIFAR-10 as a function of \\({\\eta }_{{\\rm{tr}}}={\\eta }_{\\inf }\\) . The error bars represent the standard deviation over 100 inference runs averaged over 10 training runs. d Top-1 accuracy on ImageNet as a function of \\({\\eta }_{{\\rm{tr}}}={\\eta }_{\\inf }\\) . The error bars represent the standard deviation over 10 inference runs on a single training run. Full size image The top-1 accuracy of ResNet-34 on ImageNet for \\({\\eta }_{{\\rm{tr}}}={\\eta }_{\\inf }\\) is shown in Fig. 2d . Consistent with previous observations [23] , [33] , we found that the network recovers high accuracy extremely quickly when retraining with additive noise due to quick updates of the batch normalization parameters (see Supplementary Note 1 ), and obtained satisfactory convergence after only 8 epochs. The accuracy on ImageNet is much more sensitive to the noise injected during training than for CIFAR-10, and when noise is injected on all layers, there is more than 0.5% accuracy drop from the baseline even down to ~1.2% relative noise. In the literature, many network compression techniques allow higher precision for the first and last layers, which are more sensitive to noise [33] , [36] . We applied the same simplification to our problem, which means that we removed the noise during training on the first convolutional layer and the last dense layer, and performed inference with the first and last layer without noise. The obtained accuracy after training, by injecting the same training and inference noise as previously, can be increased by more than 1% with this technique (see Fig. 2d ). Weight transfer to PCM-based synapses In order to validate the effectiveness of the above training methodology, we performed experiments on a prototype multi-level PCM chip comprising 1 million PCM devices fabricated in 90 nm CMOS baseline technology [37] . PCM is a memristive technology, which records data in a nanometric volume of phase-change material sandwiched between two electrodes [38] , [39] . The phase-change material is in the low-resistive crystalline phase in an as-fabricated device. By applying a current pulse of sufficient amplitude (typically referred to as the RESET pulse) an amorphous region around the narrow bottom electrode is created via a melt-quench process. The device will be in a low conductance state if the high-resistive amorphous region blocks the current path between the two electrodes. The size of the amorphous region can be modulated in an almost completely analog manner by the application of suitable electrical pulses. Hence, a continuum of conductance values can be programmed in a single PCM device over a range of more than two orders of magnitude. An optimized iterative programming algorithm was developed to program the conductance values in the PCM devices with high accuracy (see Methods). The experimental cumulative distributions of conductance values for 11 representative programmed levels, measured approximately 25 s after programming, are shown in Fig. 3a . The standard deviation of these distributions is extracted and fitted with a polynomial function of the target conductance (dashed lines in Fig. 3a ) as shown in Fig. 3b . For all levels, we achieve a standard deviation less than 1.2 μS, which is more than two times lower than that reported in previous works on nanoscale PCM arrays for a similar conductance range [40] , [41] . Fig. 3: Impact of weight transfer to PCM synapses on inference accuracy. a Cumulative distributions of 11 representative iteratively programmed conductance levels on 10,000 PCM devices per level. The vertical dashed lines denote the target conductance for each level. b Conductance standard deviation of the 11 levels as a function of target conductance. The inset shows a representative conductance distribution of one level. c Test accuracy on CIFAR-10 after software training and after weight transfer to PCM synapses for different training schemes. d Top-1 accuracy on ImageNet after software training and after weight transfer to PCM synapses for different training schemes. The weights of all layers of ResNet-34 are transferred to PCM synapses, including the first and the last. In c and d the error bars represent the standard deviation over 10 inference runs. Full size image We studied the impact of weight transfer to PCM synapses on the inference accuracy of networks trained with FP32 weights, 4-bit and ternary digital weights, and additive weight noise. The weight transfer to PCM is simulated by mapping each trained network weight to a synaptic conductance value \\({G}_{{{ij}}}^{{{l}}}\\) , computed from Eq. ( 2 ), using \\({G}_{\\max }=25\\) μS and the conductance standard deviation measured from hardware. The conductance error \\(\\delta {G}_{{{ij}}}^{{{l}}}\\) in Eq. ( 2 ) is modeled as a Gaussian distributed random variable with zero mean and standard deviation given by the fitted curve of Fig. 3b for its corresponding target conductance \\(| {G}_{{\\rm{T}},{{ij}}}^{l}|\\) . This procedure accurately reproduces the experimental conductance values obtained when iteratively programming any conductance level within the range of 0–25 μS. For all training schemes, only one differential pair is used to encode a network weight, and we do not perform any additional quantization of the weights before mapping them to PCM conductance values. The resulting test accuracy obtained after software training and after weight transfer to PCM synapses for ResNet-32 on CIFAR-10 is shown in Fig. 3c . It can be seen that the standard FP32 training without constraints performs the worst after transfer to PCM synapses. Training with 4-bit digital weights (using the method described in the ref. [33] ), which is roughly the effective precision of our PCM devices [40] , improves the performance after transfer with respect to FP32, but nevertheless the accuracy decreases by more than 1% after transferring the 4-bit weights to PCM. Training ternary digital weights [42] leads to a lower performance drop (<0.5%) when transferring weights to PCM, although we were not able to reach the FP32 baseline with ternary weights on this network. Therefore the accuracy after transfer is worse than for the 4-bit weights. When performing training by injecting Gaussian noise with \\({\\eta }_{{\\rm{tr}}}=3.8 \\%\\) , corresponding to σ δG = 0.94 μS (median of the 11 values reported in Fig. 3b ), the best overall performance after transfer to PCM is obtained. The resulting accuracy of 93.7% is less than 0.2% below the FP32 baseline. A rather broad range of values of \\({\\eta }_{{\\rm{tr}}}\\) lead to a similar resulting accuracy (see Supplementary Fig. 3 ), demonstrating that \\({\\eta }_{{\\rm{tr}}}\\) does not have to be very precisely determined for obtaining satisfactory results on PCM. The accuracy obtained without perturbing the weights after training by injecting noise is slightly higher than the FP32 baseline, which could be attributed to improved generalization resulting from the additive noise training. The top-1 accuracy for ResNet-34 on ImageNet after transfer to PCM synapses for different training procedures is shown in Fig. 3d . Training with additive noise increases the accuracy by approximately 6% on PCM compared with FP32 and 4-bit [33] training. The accuracy of 71.6% achieved with additive noise training on PCM is significantly higher than that reported in Fig. 2d with \\({\\eta }_{\\inf }=3.8 \\%\\) , which could be attributed to a high percentage of network weights mapped to low conductance values with lower standard deviation than the median of 0.94 μS. Hardware/software inference experiment on CIFAR-10 In addition to the weight transfer simulations of the previous section, we performed a hardware/software experiment in which all weights of ResNet-32 were physically programmed on the PCM chip. This allows us to address the effect of imperfect yield and temporal conductance fluctuations on the accuracy experimentally. Although we could achieve good test accuracy after weight transfer to PCM synapses, an important challenge for any analog in-memory computing hardware is to be able to retain this accuracy over time. This is especially true for PCM due to the high 1/ f noise experienced in these devices as well as temporal conductance drift. The conductance values in PCM drift over time t according to the relation \\(G(t)=G({t}_{0}){(t/{t}_{0})}^{-\\nu }\\) , where G ( t 0 ) is the conductance measured at time t 0 after programming and ν is the drift exponent, which depends on the device, phase-change material, and phase configuration of the PCM ( ν is higher for the amorphous than the crystalline phase) [43] . In our PCM devices, ν ≈ 0.06 on average. Therefore, it is essential to measure experimentally how the test accuracy evolves over time during inference with PCM. All 361,722 synaptic weights of ResNet-32 trained with \\({\\eta }_{{\\rm{tr}}}=3.8 \\%\\) were programmed individually on two PCM devices of the chip. Depending on the sign of \\({G}_{{{\\rm{T}}},{ij}}^{l}\\) , either \\({G}_{ij}^{l,+}\\) or \\({G}_{ij}^{l,-}\\) was iteratively programmed to \\(| {G}_{{\\rm{T}},{{ij}}}^{{{l}}}|\\) , and the other device was RESET close to 0 μS with a single pulse of 450 μA amplitude and 50 ns width. The iterative programming algorithm converged on 99.1% of the devices programmed to nonzero conductance, and no screening for defective devices on the chip was performed prior to the experiments. The scatter plot of the PCM weights measured approximately 25 s after programming versus the target weights \\({W}_{{{ij}}}^{{{l}}}\\) is shown in Fig. 4a . After programming, the PCM analog conductance values were periodically read from hardware over a period of 1 day, scaled to the network weights, and reported to the software that performed inference on the test dataset (see Methods). Fig. 4: Inference experiment with ResNet-32 on CIFAR-10. a Scatter plot of weights programmed in the PCM chip versus target weights obtained after training. The inset shows the distribution of the relative error between programmed and target synaptic conductance, \\(({G}_{{{ij}}}^{{{l}}}-{G}_{{\\rm{T}},{{ij}}}^{{{l}}})/{G}_{\\max }\\) , and its standard deviation σ . b Distributions of programmed devices whose target conductance fall within 0.25 μS from five representative G T values. The distributions are shown at four different times spanning the experiment duration of one day. The filled bars are the measured hardware data, the black lines are the PCM model. c Individual device conductance evolution over time of four arbitrarily picked devices from the chip programmed to four distinct G T values, along with one PCM model realization for the same G T values. d Measured test accuracy over time from the weights of the PCM chip, along with the corresponding PCM model match. A global drift compensation (GDC) procedure is performed for every layer before performing inference on the test set. The filled areas from the PCM model correspond to one standard deviation over 25 inference runs. The baseline refers to the FP32 software accuracy of 93.87%. Full size image In addition to the experiment, we developed an advanced behavioral model of the hardware in order to precisely capture the conductance evolution over time during inference (see Supplementary Note 2 ). The model is built based on an extensive experimental characterization of the array-level statistics of hardware noise and drift [44] . Conductance drift is modeled using a Gaussian distributed drift exponent across devices, whose mean and standard deviation both depend on the target conductance state \\(| {G}_{{\\rm{T}},{{ij}}}^{{{l}}}|\\) . Conductance noise with the experimentally observed 1/ f 1.21 frequency dependence is also incorporated with a magnitude that depends on the target conductance state and time. The model is able to accurately reproduce both the array-level statistics (see Fig. 4b ) and individual device behavior (see Fig. 4c ) observed over the duration of the experiment. Accurate modeling of all the complex dependencies of noise and drift as a function of time and conductance state was found to be very critical in being able to reproduce the experimental evolution of the accuracy on ResNet. The resulting accuracy on CIFAR-10 over time is shown in Fig. 4d . The test accuracy measured 25 s after programming is 93.75%, which is very similar to the result obtained in Fig. 3c . However, if nothing is done to compensate for conductance drift, the accuracy quickly decreases down to 10% (random guessing) within approximately 1000 s. This is because the magnitude of the PCM weights gradually reduces over time due to drift and this prevents the activations from properly propagating throughout the network. A simple global scaling calibration procedure can be used to compensate for the effect of drift on the matrix-vector multiplications performed with PCM crossbars. As proposed in ref. [40] , the summed current of a subset of the columns in the array can be periodically read over time at a constant voltage. The resulting total current is then divided by the summed current of the same columns but read at time t 0 . This results in a single scaling factor that can be applied to the output of the entire crossbar in order to compensate for a global conductance shift (see Methods and Supplementary Fig. 4 ). Since this factor can be combined with the batch normalization parameters, it does not incur any additional overhead when performing inference. This simple global drift compensation (GDC) procedure was implemented for every layer before carrying out inference on the test set, and the results are shown in Fig. 4d . It can be seen that GDC allows the retention of a test accuracy above 92.6% for 1 day on the PCM chip, and effectively prevents the effect of global weight decay over time as illustrated in Supplementary Fig. 4 . A good agreement of the accuracy evolution between model and experiment is obtained, hence validating its use for extrapolating results over a longer period of time and for assessing the accuracy of larger networks that cannot fit on our current hardware. Adaptive batch normalization statistics update Although GDC can compensate for a global conductance shift across the array, it cannot mitigate the effect of 1/ f noise and drift variability across devices. From the model, we observe that 1/ f noise is responsible for the random accuracy fluctuations, whereas drift variability and its dependence on the target conductance state cause the monotonous accuracy decrease over time (see Supplementary Fig. 5 ). In order to improve the accuracy retention further, we propose to leverage the batch normalization parameters to correct the activation distributions during inference such that their mean and variance match those that were optimally learned during training. During inference, batch normalization is performed by normalizing the preactivations by their corresponding running mean μ and variance σ 2 computed during training. Then, scale and shift factors ( γ and β ) that were learned through backpropagation are applied to the normalized preactivations. Since γ and β are learnable parameters, it is not desirable to change them since it would require retraining the model on the PCM devices. However, updating μ and σ 2 is more intuitive, since the mean and variance of the preactivations are affected by noise and drift. Leveraging this idea, we introduce a new compensation technique called adaptive batch normalization statistics update (AdaBS), which improves the accuracy retention beyond GDC at the cost of additional computations during the calibration phase. As described in Fig. 5a , the calibration phase consists in sending multiple mini-batches from a set of calibration images that come from the same distribution than the images seen during inference. In this study, we use the images from the training dataset as calibration images. The running mean and variance of preactivations are computed across the entire calibration dataset. The new values of μ and σ 2 computed during calibration are then used for subsequent inference. The main advantage of this technique is that it does not incur additional digital computations nor weight programming during inference, because we are only updating the batch normalization parameters μ and σ 2 when the calibration is performed. However, injecting the entire training dataset to compute μ and σ 2 in the calibration phase would bring significant overhead. When reducing the amount of injected images, the number of updates of the running statistics becomes smaller, and if the momentum used for computing μ and σ 2 is not properly tuned to account for this, the network accuracy decreases significantly. To tackle this issue, we developed a procedure to obtain the optimal momentum as a function of the number of mini-batches used for calibration (see Methods and Supplementary Note 3 ). With this method, we were able to reduce the number of calibration images down to 5.2% of the CIFAR-10 training dataset (2600 images) without affecting the accuracy. With that number of images, the overhead in terms of digital computations of the AdaBS calibration is about 52% of performing batch normalization during inference on the whole CIFAR-10 test set (see Supplementary Note 3 ). Furthermore, AdaBS requires additional off-chip memory resources to store the calibration images, leading to extra off-chip data transfers during calibration (see Supplementary Note 3 ). It may appear cumbersome to send so many images to the device to perform the calibration, however since it is only performed periodically over time when the device is idle and not every time an image is inferred by the network, the calibration cost can be amortized. The calibration overhead can be further reduced by using more efficient variants of batch normalization such as the L 1 -norm version (see Supplementary Note 3 ). Moreover, although we used AdaBS (and GDC) to compensate solely for the drift of the PCM devices, the same procedure can be applied to mitigate conductance changes due to ambient temperature variations, a critical issue for any analog in-memory computing hardware. Fig. 5: Adaptive batch normalization statistics update with PCM synapses. a The AdaBS calibration procedure consists in updating the running mean μ and variance σ 2 parameters of the batch normalization performed in the digital unit of the in-memory computing hardware. The calibration is performed periodically when the device is idle, and after calibration the values of μ and σ 2 of every layer are updated for subsequent inference. Note that during the calibration phase, batch normalization is performed using the mini-batch mean μ B and variance \\({\\sigma }_{{\\rm{B}}}^{2}\\) instead of μ and σ 2 (see Methods and Supplementary Note 3 ). b Test accuracy of ResNet-32 on CIFAR-10 with GDC and AdaBS using the PCM model, along with experimental test accuracy obtained by applying AdaBS on ResNet-32 with the measured weights from the PCM chip. The filled areas from the PCM model correspond to one standard deviation over 25 inference runs. c Top-1 accuracy of ResNet-34 on ImageNet with GDC and AdaBS computed using the PCM model. Implementations using PCM synapses for all layers as well as first and last layers in digital FP32, and PCM synapses for all other layers, are shown. In the latter, no noise is applied on the first and last layers during training. The filled areas correspond to one standard deviation over 10 inference runs. In b and c the baseline refers to the FP32 software accuracy. Full size image The resulting accuracy when performing AdaBS on ResNet-32 with hardware weights before carrying out inference on the test set is shown in Fig. 5b . AdaBS allows to retain a test accuracy above 93.5% over one day, an improvement of 0.9% compared with GDC. This improvement becomes 1.8% for one year when extrapolating the results using the PCM model. This improvement results from optimally tuning scale and shift factors for each individual column of the crossbar with AdaBS, instead of a single scale factor for all columns in the case of GDC. Because of this more extensive parameter optimization, AdaBS can effectively compensate for drift variability and, more generally, random conductance variations across devices (see Supplementary Figs. 5 – 6 ). We also applied AdaBS on the ImageNet classification task with ResNet-34, trained with \\({\\eta }_{{\\rm{tr}}}=3.8 \\%\\) , using the PCM model to simulate the weight evolution for one year. By applying the same AdaBS method as for CIFAR-10 using only 0.1% of the ImageNet training dataset for calibration (1300 images), the accuracy after one year is increased by 7% compared with GDC when all layers are implemented with PCM synapses (see Fig. 5c ). When the first and last layers are implemented in digital FP32, the initial accuracy increases to 71.9% and the retention is significantly improved. This technique, combined with AdaBS, allows the retention of an accuracy above 71% for one year. Drawbacks in efficiency when performing inference on hardware in this way have to be mentioned, but they stay limited given the small number of parameters and input size of the first and last layers (the first and last layers contain less than 3% of the network weights and are responsible for about 3.12% of the multiply-and-accumulate operations during inference with ResNet-34)."},{"section":"Discussion","content":"Combined together, the strategies developed in this study allow us to achieve the highest accuracies reported so far with analog resistive memory on the CIFAR-10 and ImageNet benchmarks with residual networks close to their original implementation [24] . Although there is still room for improvement especially on ImageNet, those accuracies are already comparable or higher than those reported on ternary weight networks [42] , for example 71.6% top-1 accuracy of ResNet-34 on ImageNet with first layer in FP32 [45] . Importantly, the accuracies we report are achieved with just two nanoscale PCM devices encoding the value of a weight. A common approach that could improve the accuracy further is to use multiple devices to encode different bits of a weight [3] , [41] , at the expense of area and energy penalty, and additional support required by the peripheral circuitry. Aligned with previous observations [23] , [33] , we notice that retraining ResNet with additive noise results mainly in adapting the batch normalization parameters, whereas the weights stay close to the full-precision weights trained without noise. Hence, retraining by injecting noise from a pretrained baseline network rather than from scratch is very effective since the network recovers high accuracy very quickly, especially for ImageNet. Although our experiments are not done on a fully-integrated chip that supports all functions of deep learning inference, the most critical effects of array-level variability, noise, and drift, are fully accounted for because each weight of the network is programmed on individual PCM devices of our array. Aspects of a fully-integrated chip that are not entirely captured in our experiments such as IR drop and additional circuit nonidealities such as offsets and noise have been studied in previous works and could be mitigated by additional retraining methods [18] , [20] . Experiments on fully-integrated memristor chips could be used to faithfully quantify their impact on the inference accuracy [7] , [9] , [46] . With respect to device yield, additional simulations showed that additive noise training also helps to mitigate the impact of devices stuck at low or high conductance on the inference accuracy (see Supplementary Fig. 7 ). Additional errors due to quantization coming from the crossbar data converters are analyzed further below. There exist many different methods of training a neural network with noise that aim to improve the resilience of the model to analog mixed-signal hardware. These include injecting additive noise on the inputs of every layer [20] , on the preactivations [22] , [23] , or just adding noise on the input data [47] . Moreover, injecting multiplicative Gaussian noise to the weights [34] ( \\({\\sigma }_{\\delta {W}_{{\\rm{tr}}},{{ij}}}^{l}\\propto | {W}_{{{ij}}}^{{{l}}}|\\) ) is also defensible regarding the observed noise on the hardware. We analyzed the four aforementioned methods, attempting to reach the same accuracy demonstrated previously after weight transfer to PCM devices, to identify their possible benefits and drawbacks (see Supplementary Note 4 ). We found that it is possible to adjust the training procedure of all four methods to achieve a similar accuracy on CIFAR-10 after transferring the weights to PCM synapses. Somewhat surprisingly, even adding noise on the input data during training, which is just a simple form of data augmentation, leads to a model which is more resilient to weight perturbations during inference. This shows that it is not necessary to train a model with very complicated noise models that imitate the observed hardware noise precisely. As long as the data propagated through the network is corrupted by a Gaussian noise of the right magnitude, the model is expected to be robust to mapping on PCM devices. However, all four methods require one or multiple noise scaling factor hyperparameters to tune in order to reach satisfactory accuracy after transfer to PCM. In contrast, our proposed methodology estimates the additive noise to inject on the weights, \\({\\eta }_{{\\rm{tr}}}\\) , from a simple hardware characterization, avoiding any hyperparameter search for noise scaling factors. The value of \\({\\eta }_{{\\rm{tr}}}\\) does not have to be very precise either, because there are a range of values that lead to similar accuracy after transfer to PCM (see Supplementary Fig. 3 ). Moreover, we found that injecting noise on weights achieves better accuracy retention over time (see Supplementary Note 4 ), which suggests that weight noise mimics the behavior of the PCM hardware better. A critical issue for in-memory computing hardware is the need for digital-to-analog (analog-to-digital) conversion every time data goes in (out) of the crossbar arrays. These data conversions lead to quantization of the activations and preactivations, respectively, which introduce additional errors in the forward propagation. Based on a recent ADC survey [23] , 8-bit data conversion is a good tradeoff between precision and energy consumption. Hence, we analyzed the effect of quantizing the input and output of every layer of ResNet-32 and ResNet-34 to 8-bit on the inference accuracy. We set the input/output quantization ranges to the 99.995th percentile of the activation/preactivation distributions that are obtained when forward propagating 10 k randomly sampled images from the training dataset through the baseline network. As shown in Supplementary Fig. 8 , even though the 8-bit quantization is not included in our training algorithm, the quantization has a minimal effect on the mean accuracy of ResNet-32 on CIFAR-10 (<0.05% drop) and ResNet-34 on ImageNet (<0.15% drop) after weight transfer to PCM synapses. The accuracy evolution over time, retaining the same quantization ranges, does not degrade significantly further and stays well within one standard deviation of that obtained without quantization. The small accuracy deviations could be potentially overcome by including the quantization in the retraining process, which will likely be necessary if less than 8-bit resolution is desired for higher energy efficiency. Although a computational memory accelerates the matrix-vector multiplication operations in a DNN, communicating activations between computational memory cores executing different layers can become a bottleneck. This bottleneck depends upon two factors, (i) the way different layers are connected to each other and (ii) the latency of the hardware implementation to transfer activations from one core to another. Designing optimal interconnectivity between the cores for state-of-the-art deep CNNs is an open research problem. Indeed, having the network weights stationary during execution in a computational memory puts limits on what portion of the computation can be forwarded to different cores. This ultimately results in long-established hardware communication fabrics being ill-fit for the task. One topology for communication fabrics that is well-suited for computational memory is proposed by Dazzi et al. [48] . It is based on a 5 parallel prism (5PP) graph topology and facilitates inter-layer pipelined execution of CNNs [3] . The proposed 5PP topology allows the mapping of all the primary connectivities of state-of-the-art neural networks, including ResNet, DenseNet and Inception-style networks [48] . As discussed in ref. [48] , the ResNet-32 implementation with 5PP can result in potentially 2× improvement in pipeline stage latency with similar bandwidth requirements compared with a standard 2D-mesh. Assuming that each layer is mapped onto a crossbar of size 512 × 512 having a computational cycle time of 100 ns, 8-bit activations, communication links with data rate of 5Gbps [49] , and a pipeline scheme identical to that proposed in ref. [3] , a single image inference latency of 52 μs and frame rate of 38,600 frames per second (FPS) for ResNet-32 on CIFAR-10 is estimated. The energy efficiency of a computational PCM core of this size implementing a layer of ResNet-32 is estimated to be 11.9 TOPS W −1 (tera operations per seconds per watt), with an area expenditure of approximately 0.57 mm 2 (see Supplementary Note 5 ). As an approximate comparison, YodaNN [50] , a digital DNN inference accelerator for binary weight networks with ultra-low power budget, achieves 434.8 FPS in high throughput mode for a 9-layer CNN (BinaryConnect [51] ) on CIFAR-10. Although not a direct comparison, the proposed topology and pipelined execution of ResNet-32 could result in 88× speedup, with a deeper network than the digital solution. In summary, we introduced strategies for training ResNet-type CNNs for deployment on analog in-memory computing hardware, as well as improving the accuracy retention on such hardware. We proposed to inject noise to the synaptic weights in proportion to the combined read and write conductance noise of the hardware during the forward pass of training. This approach combined with judicious weight initialization, clipping, and learning rate scheduling, allowed us to achieve an accuracy of 93.7% on the CIFAR-10 dataset and a top-1 accuracy on the ImageNet benchmark of 71.6% after mapping the trained weights to analog PCM synapses. Our methods introduce only a single additional hyperparameter during training, the weight clip scale α , since the magnitude of the injected noise can be easily deduced from a one-time hardware characterization. After programming the trained weights of ResNet-32 on 723,444 PCM devices of a prototype chip, the accuracy computed from the measured hardware weights stayed above 92.6% over a period of 1 day, which is to the best of our knowledge the highest accuracy experimentally reported to-date on the CIFAR-10 dataset by any analog resistive memory hardware. A global scaling procedure was used to compensate for the conductance drift of the PCM devices, which was found to be critical in improving the accuracy retention. However, global scaling could not mitigate the effect of 1/ f noise and drift variability across devices, which led to accuracy fluctuations and monotonous accuracy decrease over time, respectively. Periodically calibrating the batch normalization parameters before inference allowed to partly alleviate those issues at the cost of additional digital computations, increasing the 1-day accuracy to 93.5% on hardware. These results demonstrate the feasibility of realizing accurate inference on complex DNNs through analog in-memory computing using existing PCM devices."},{"section":"Methods","content":"Experiments on PCM hardware platform The experimental platform is built around a prototype PCM chip that comprises 3 million PCM devices. The PCM array is organized as a matrix of word lines (WL) and bit lines (BL). In addition to the PCM devices, the prototype chip integrates the circuitry for device addressing, and for write and read operations. The PCM chip is interfaced to a hardware platform comprising a field programmable gate array (FPGA) board and an analog-front-end (AFE) board. The AFE board contains the digital-to-analog converters, and provides the power supplies as well as the voltage and current reference sources to the PCM chip. The FPGA board implements the data acquisition and the digital logic to interface with the PCM device under test and with all the electronics of the AFE board. The FPGA board is also used to implement the overall system control and data management as well as the interface with the host. The experimental platform is operated from a host computer, and a Matlab environment is used to coordinate the experiments. The PCM devices are integrated into the chip in 90-nm CMOS technology using the key-hole process described in the ref. [52] . The phase-change material is doped Ge 2 Sb 2 Te 5 . The bottom electrode has a radius of ~20 nm and a height of ~50 nm. The phase-change material is ~100 nm thick and extends to the top electrode, whose radius is ~100 nm. All experiments performed in this work are done on an array containing 1 million devices accessed via transistors, which is organized as a matrix of 512 WL and 2048 BL. A PCM device is selected by serially addressing a WL and a BL. To read a PCM device, the selected BL is biased to a constant voltage (300 mV) by a voltage regulator via a voltage generated off chip. The sensed current is integrated by a capacitor, and the resulting voltage is then digitized by the on-chip 8-bit cyclic analog-to-digital converter (ADC). The total duration of applying the read pulse and converting the data with the ADC is 1 μs. The readout characteristic is calibrated via on-chip reference polysilicon resistors. To program a PCM device, a voltage generated off chip is converted on chip into a programming current. This current is then mirrored into the selected BL for the desired duration of the programming pulse. Iterative programming involving a sequence of program-and-verify steps is used to program the PCM devices to the desired conductance values [53] . The devices are initialized to a high-conductance state via a staircase-pulse sequence. The sequence starts with a RESET pulse of amplitude 450 μA and width 50 ns, followed by six pulses of amplitude decreasing regularly from 160 to 60 μA and with a constant width of 1000 ns. After initialization, each device is set to a desired conductance value through a program-and-verify scheme. The conductance of all devices in the array is read five times consecutively at a voltage of 0.3 V, and the mean conductance of these reads is used for verification. If the read conductance of a specific device does not fall within 0.25 μS from its target conductance, it receives a programming pulse where the pulse amplitude is incremented or decremented proportionally to the difference between the read and target conductance. The pulse amplitude ranges between 80 and 400 μA, and the pulse width is 1000 ns. This program-and-verify scheme is repeated for a maximum of 55 iterations. In the hardware/software inference experiments, the analog conductance values of the PCM devices encoding the network weights, \\({G}_{{{ij}}}^{{{l}},+}\\) and \\({G}_{{{ij}}}^{{{l}},-}\\) , are serially read individually with the 8-bit on-chip ADC at predefined timestamps spaced over a period of one day. The read conductance values at every timestamp are reported to a TensorFlow-based software. This software performs the forward propagation of the CIFAR-10 test set on the weights read from hardware and computes the resulting classification accuracy. The drift compensation techniques, GDC and AdaBS, are performed entirely in software at every timestamp based on the conductance values read from hardware. PCM-based deep learning inference simulator We developed a simulation framework to test the efficacy of DNN inference using PCM devices. We chose Google’s TensorFlow [54] deep learning framework for the simulator development. The large library of algorithms in TensorFlow enables us to use native implementation of required activation functions and batch normalization. Moreover, any regular TensorFlow code of a DNN can be easily ported to our simulator. As shown in Supplementary Fig. 9 , custom made TensorFlow operations are implemented that generate PCM conductance values from the behavioral model of hardware PCM devices that was developed (see Supplementary Note 2 ). All the nonidealities including conductance range, programming noise, read noise, and conductance drift are implemented in TensorFlow following the equations shown in Supplementary Note 2 . The simulator can also take the PCM conductance data measured from hardware as input, in order to perform inference on the hardware data. Data converters that simulate digital quantization of data at the input and output of crossbars are also implemented with tunable quantization ranges and precision. In this study, the data converters were turned off for all simulations except those presented in Supplementary Fig. 8 . The drift correction techniques are implemented post quantization of the crossbar output. Implementation of ResNet-32 training on CIFAR-10 ResNet-32 has 31 convolution layers with 3 × 3 kernels, 2 convolution layers with 1 × 1 kernels, and a final fully-connected layer. The network contains 361,722 synaptic weights. It consists of three different ResNet blocks with 10 3 × 3 kernels each. After the first convolution layer, there is a unity residual feed forward connection after every two convolution layers, except the 1 × 1 residual convolution connection to make output channels compatible between two layers. Each convolution layer is followed by batch normalization [55] . ReLU activation is used after every batch normalization except in case of residual connections, where the ReLU activation is computed after summation. Each residual connection with 1 × 1 convolution and first layer of ResNet blocks 2, 3 downsample the input by using a stride of 2 pixels. The 8 × 8 output of the last convolution layer is then downsampled to 1 × 1 resolution using global average pooling [56] , which is followed by a single fully-connected layer. For the last fully-connected layer, no batch normalization is performed. The architecture of ResNet-32 used in this study is a slightly modified version of the original implementation [24] with fewer input and output channels in ResNet blocks 2 and 3. This network is trained on the well-known CIFAR-10 classification dataset [57] . It has 32 × 32 pixels RGB images that belong to one of the ten classes. The network is trained on the 50,000 images of the training set, and evaluation is performed on the 10,000 images of the test set. The training is performed using stochastic gradient descent with a momentum of 0.9. The network objective is categorical cross entropy function over ten classes of the input image. Learning rate scheduling is performed to reduce learning rate by 90% at every 50th training epoch. The initial learning rate for the baseline network is 0.1 and training converges in 200 epochs with a mini-batch size of 128. Weights of all convolution and fully connected layers of the baseline network are initialized using He Normal [35] initialization. The baseline network is retrained by injecting Gaussian noise for up to 150 epochs with weight clip scale α = 2. We preprocess the training images by randomly cropping a 32 × 32 patch after padding 2 pixels along the height and width of the image. We also apply a random horizontal flip on the images from the train set. Additionally, we apply cutout [58] on the training set images. For both training and test set, we apply channel wise normalization for 0 mean and unit standard deviation. Implementation of ResNet-34 training on ImageNet The architecture of the ResNet-34 network for ImageNet classification is derived from ref. [24] . It has 32 convolution layers with 3 × 3 kernels, 3 convolution layers with 1 × 1 kernels, a first convolution layer with 7 × 7 kernels and a final fully-connected layer. The network has 21,797,672 synaptic weights. The first convolution layer downsamples the input by using a stride of 2 pixels, followed by a maxpooling layer with kernel size of 3 × 3 and stride of 2 to downsample the feature maps to the resolution of 56 × 56 pixels. Each residual connection with 1 × 1 convolution and first layer of ResNet blocks 2, 3, 4 downsample the input by using a stride of 2 pixels. A global average pooling layer before the final fully-connected layer downsamples the 7 × 7 input to 1 × 1 resolution. The final fully-connected layer computes the output prediction corresponding to 1000 classes. We trained ResNet-34 on the ImageNet [59] dataset. The ImageNet dataset has 1.3 M images in the training set and 50 k images in the test set. Images in the ImageNet dataset are preprocessed by following the same preprocessing steps as that of the Pytorch baseline model. Training images are randomly cropped to a 224 × 224 patch and then random horizontal flip is applied on the images. Channel wise normalization is performed on the images in both training and test set for 0 mean and unit standard deviation. Only for the test set, images are first resized to 256 × 256 using bilinear interpolation method and then a center crop is performed to obtain the 224 × 224 image patch. The network objective function is softmax cross entropy on network output and corresponding 1000 labels. The network objective is minimized using the stochastic gradient descent algorithm with a momentum of 0.9. We obtained our baseline network architecture and its parameters from the Pytorch model zoo ( https://pytorch.org/docs/stable/torchvision/models.html ). We use this network to perform additive noise training by injecting Gaussian noise for a total of 10 training epochs. In contrast to ResNet-32 on CIFAR-10, no learning rate scheduling was performed since the network was trained only for 10 epochs with additive noise. We use mini-batch size of 400 and learning rate of 0.001 for the additive noise training simulations. We also use L2 weight decay of 0.0001 and weight clip scale of α = 2.5 for the additive noise training. Global drift compensation (GDC) method The GDC [40] calibration phase consists of computing the summed current of L columns in each array encoding a network layer (see Supplementary Fig. 4 ). Those L columns contain devices initially programmed to known conductance values G mn ( t 0 ). By reading those column currents, I m , periodically with applied voltage V cal on all N rows, we can compensate for a global conductance shift in the array during inference. When input data is processed by the crossbar during inference, the crossbar output can be scaled by \\(1/\\hat{\\alpha }\\) , where $$\\hat{\\alpha }=\\frac{\\mathop{\\sum }\\nolimits_{m=1}^{L}{I}_{m}}{{V}_{{\\rm{cal}}}\\mathop{\\sum }\\nolimits_{n=1}^{N}\\mathop{\\sum }\\nolimits_{m=1}^{L}{G}_{mn}({t}_{0})}.$$ This procedure is especially simple because L can be chosen to be small, enough to get sufficient statistics. Moreover, \\(\\hat{\\alpha }\\) is computed from the device data itself, without resorting to any assumption on how the conductance changes nor requiring extra timing information [60] . The term \\({V}_{{\\rm{cal}}}\\mathop{\\sum }\\nolimits_{n = 1}^{N}\\mathop{\\sum }\\nolimits_{m = 1}^{L}{G}_{mn}({t}_{0})\\) needs to be computed only once, stored in the digital memory of the chip, and is reused for all calibrations. Reading the subset of L columns of the crossbar can be done while the PCM array is idle, i.e., when there are no incoming images to be processed by the device. Performing the L current summations can be implemented either with on-chip digital circuitry or in the control unit of the chip. At the end of the calibration phase, \\(1/\\hat{\\alpha }\\) is computed and stored locally in digital unit of the crossbar. The output scaling by \\(1/\\hat{\\alpha }\\) during inference can be combined with batch normalization because it is a linear operation. In our experiments, the calibration procedure was performed using all columns of each layer (e.g., L is equal to two times number of output channels) every time before inference is performed on the whole test set. Adaptive batch normalization statistics update (AdaBS) technique Batch normalization is performed differently in the training and inference phases of a DNN. During the training of a DNN, the input to the batch normalization operation x i is normalized to zero mean and unit variance by computing its mean ( μ B ) and variance ( \\({\\sigma }_{{\\rm{B}}}^{2}\\) ) over a mini-batch of m images $${\\mu }_{{\\rm{B}}}=\\frac{1}{m}\\mathop{\\sum }\\limits_{i = 1}^{m}{x}_{i}$$ (4) $${\\sigma }_{{\\rm{B}}}^{2}=\\frac{1}{m}\\mathop{\\sum }\\limits_{i = 1}^{m}{\\left({x}_{i}-{\\mu }_{{\\rm{B}}}\\right)}^{2}.$$ (5) The normalized input is then scaled and shifted by γ and β . During the training phase, γ and β are learned through backpropagation. In parallel, a global running mean ( μ ) and variance ( σ 2 ) are computed by exponentially averaging μ B and \\({\\sigma }_{{\\rm{B}}}^{2}\\) respectively, over all the training batches $$\\mu =p\\cdot \\mu +(1-p)\\cdot {\\mu }_{{\\rm{B}}}$$ (6) $${\\sigma }^{2}=p\\cdot {\\sigma }^{2}+(1-p)\\cdot {\\sigma }_{{\\rm{B}}}^{2},$$ (7) where p is the momentum. After training, the estimates of the global mean and variance μ and σ 2 are then used during the inference phase. When performing forward propagation during inference, the batch normalization coefficients μ , σ 2 , γ , and β are used for normalization, scale, and shift. The calibration phase of AdaBS consists in recomputing and updating μ and σ 2 for every layer where batch normalization is present. We recompute μ and σ 2 by feeding a randomly sampled set of mini-batches from the training dataset. In recomputing μ and σ 2 , hyper-parameters such as mini-batch size ( m ) and momentum ( p ) need to be carefully tuned to achieve the best network accuracy. For AdaBS calibration, we observed that using an optimal value of the momentum is necessary to achieve good inference accuracy evolution over time. For this, we have developed an algorithm to estimate the optimal value of momentum by an empirical analysis, which is explained in Supplementary Note 3 . Based on this analysis, the formula we used to compute the optimal momentum as a function of the number of injected mini-batches n is $$p=0.01{5}^{(1/n)}.$$ (8) Using Eq. ( 8 ) to compute the momentum, we found that with a fixed mini-batch size of m = 200 images, it is sufficient to inject n = 13 mini-batches for the AdaBS calibration of the ResNet-32 network, that is approximately 5% of the CIFAR-10 training set (2600 images). The sensitivity of the accuracy to the number of images used for AdaBS calibration is shown in Supplementary Note 3 . For ResNet-34 on ImageNet, we used mini-batch size of m = 50 and n = 26 mini-batches, that is 0.1% of the ImageNet training set (1300 images). In the experiments presented in Fig. 5 , AdaBS calibration was performed for every layer before performing inference on the test set, except the last layer because it does not have batch normalization."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"Deep neural networks (DNNs) have revolutionized the field of artificial intelligence and have achieved unprecedented success in cognitive tasks such as image and speech recognition. Platforms for deploying the trained model of such networks and performing inference in an energy-efficient manner are highly attractive for edge computing applications. In particular, internet-of-things battery-powered devices and autonomous cars could especially benefit from fast, low-power, and reliably accurate DNN inference engines. Significant progress in this direction has been made with the introduction of specialized hardware for inference operating at reduced digital precision (4–8-bit), such as Google’s tensor processing unit [1] and low-power graphical processing units such as NVIDIA T4 [2] . While these platforms are very flexible, they are based on architectures where there is a physical separation between memory and processing units. The models are typically stored in off-chip memory, leading to constant shuttling of data between memory and processing units, which limits the maximum achievable energy efficiency. In order to reduce the data transfers to a minimum in inference accelerators, a promising avenue is to employ in-memory computing using non-volatile memory devices [3] , [4] , [5] . Both charge-based storage devices, such as Flash memory [6] , and resistance-based (memristive) storage devices, such as metal-oxide resistive random-access memory [7] , [8] , [9] , [10] and phase-change memory (PCM) [11] , [12] , [13] , [14] are being investigated for this. In this approach, the network weights are encoded as the analog charge state or conductance state of these devices organized in crossbar arrays, and the matrix-vector multiplications during inference can be performed in-situ in a single time step by exploiting Kirchhoff’s circuit laws. The fact that these devices are non-volatile (the weights will be retained when the power supply is turned off) and have multi-level storage capability (a single device can encode an analog range of values as opposed to 1 bit) is very attractive for inference applications. However, due to the analog nature of the weights programmed in these devices, only limited precision can be achieved in the matrix-vector multiplications and this could limit the achievable inference accuracy of the accelerator. One potential solution to this problem is to train the network fully on hardware [13] , [14] , [15] , such that all hardware non-idealities would be de facto included as constraints during training. Another similar approach is to perform partial optimizations of the hardware weights after transferring a trained model to the chip [9] , [16] , [17] . The drawback of these approaches is that every neural network would have to be trained on each individual chip before deployment. Off-line variation-aware training schemes have also been proposed, where hardware non-idealities such as device-to-device variations [18] , [19] , defective devices [19] , or IR drop [18] are first characterized and then fed into the training algorithm running in software. However, these approaches would require characterizing and training the neural network from scratch for every chip. A more practical approach would be to have a single custom generic training algorithm run entirely in software, which would make the network immune to most of the hardware non-idealities, but at the same time would require only very little knowledge about the specific hardware it will be deployed on. In this way, the model would have to be trained only once and could be deployed on a multitude of different chips. To this end, several works have proposed to inject noise in the training algorithm to the layer inputs [20] , synaptic weights [21] , and pre-activations [22] , [23] . However, previous demonstrations have generally been limited to rather simple and shallow networks, and experimental validations of the effectiveness of the various approaches have been missing. We are aware of one recent work that analyzed more complex problems such as ImageNet classification [23] , however the hardware model used was rather abstract and no experimental validation was presented. In this work, we explore injecting noise to the synaptic weights during the training of DNNs in software as a generic method to improve the network resilience against analog in-memory computing hardware non-idealities. We focus on the ResNet convolutional neural network (CNN) architecture, and introduce a number of techniques that allow us to achieve a classification accuracy of 93.7% on the CIFAR-10 dataset and a top-1 accuracy of 71.6% on the ImageNet benchmark after mapping the trained weights to PCM synapses. In contrast to previous approaches, the noise injected during training is crudely estimated from a one-time all-around hardware characterization, and captures the combined effect of read and write noise without introducing additional noise-related training hyperparameters. We validate the training approach through hardware/software experiments, where each of the 361,722 weights of ResNet-32 is programmed on two PCM devices of a prototype chip, and the rest of the network functionality is simulated in software. We achieve an experimental accuracy of 93.75% after programming, which stays above 92.6% over a period of 1 day. To improve the accuracy retention further, we develop a method to periodically calibrate the batch normalization parameters to correct the activation distributions during inference. We demonstrate a significant improvement in the accuracy retention with this method (up to 93.5% on hardware for CIFAR-10) compared with a simple global scaling of the layers’ outputs, at the cost of additional digital computations during calibration. Finally, we discuss our training approach with respect to other methods and quantify the tradeoffs in terms of accuracy and ease of training.\",\n \"references\": {\n \"indices\": [\n 5,\n 6,\n 7,\n 8,\n 9,\n 10,\n 11,\n 12,\n 13,\n 17,\n 20,\n 21\n ],\n \"infos\": [\n {\n \"idx\": \"5\",\n \"title\": \"Wang, Z. et al. Resistive switching materials for information processing.Nat. Rev. Mater.5, 173–195 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41578-019-0159-3\"\n },\n {\n \"idx\": \"6\",\n \"title\": \"Merrikh-Bayat, F. et al. High-performance mixed-signal neurocomputing with nanoscale floating-gate memory cell arrays.IEEE Trans. Neural Netw. Learn. Syst.29, 4782–4790 (2018).\",\n \"link\": \"https://doi.org/10.1109%2FTNNLS.2017.2778940\"\n },\n {\n \"idx\": \"7\",\n \"title\": \"Chen, W.-H. et al. CMOS-integrated memristive non-volatile computing-in-memory for AI edge processors.Nat. Electron.2, 420–428 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41928-019-0288-0\"\n },\n {\n \"idx\": \"8\",\n \"title\": \"Hu, M. et al. Memristor-based analog computation and neural network classification with a dot product engine.Adv. Mater.30, 1705914 (2018).\",\n \"link\": \"https://doi.org/10.1002%2Fadma.201705914\"\n },\n {\n \"idx\": \"9\",\n \"title\": \"Yao, P. et al. Fully hardware-implemented memristor convolutional neural network.Nature577, 641–646 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-020-1942-4\"\n },\n {\n \"idx\": \"10\",\n \"title\": \"Yin, S. et al. Monolithically integrated RRAM- and CMOS-based in-memory computing optimizations for efficient deep learning.IEEE Micro39, 54–63 (2019).\",\n \"link\": \"https://doi.org/10.1109%2FMM.2019.2943047\"\n },\n {\n \"idx\": \"11\",\n \"title\": \"Le Gallo, M. et al. Mixed-precision in-memory computing.Nat. Electron.1, 246–253 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41928-018-0054-8\"\n },\n {\n \"idx\": \"12\",\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\": \"13\",\n \"title\": \"Ambrogio, S. et al. Equivalent-accuracy accelerated neural-network training using analogue memory.Nature558, 60–67 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-018-0180-5\"\n },\n {\n \"idx\": \"17\",\n \"title\": \"Gonugondla, S. K., Kang, M. & Shanbhag, N. R. A variation-tolerant in-memory machine learning classifier via on-chip training.IEEE J. Solid-State Circuits53, 3163–3173 (2018).\",\n \"link\": \"https://doi.org/10.1109%2FJSSC.2018.2867275\"\n },\n {\n \"idx\": \"20\",\n \"title\": \"Moon, S., Shin, K. & Jeon, D. Enhancing reliability of analog neural network processors.IEEE Trans. Very Large Scale Integr. Syst.27, 1455–1459 (2019).\",\n \"link\": \"https://doi.org/10.1109%2FTVLSI.2019.2893256\"\n },\n {\n \"idx\": \"21\",\n \"title\": \"Miyashita, D., Kousai, S., Suzuki, T. & Deguchi, J. A neuromorphic chip optimized for deep learning and CMOS technology with time-domain analog and digital mixed-signal processing.IEEE J. Solid State Circuits52, 2679–2689 (2017).\",\n \"link\": \"https://doi.org/10.1109%2FJSSC.2017.2712626\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Problem statement For our experiments, we consider two residual networks on two different datasets: ResNet-32 on the CIFAR-10 dataset, and ResNet-34 on the ImageNet dataset [24] . As shown in Fig. 1a , ResNet-32 consists of three different ResNet blocks with ten 3 × 3 kernels each, and is used to classify 32 × 32-pixel RGB images that belong to one out of ten classes (see Methods). The network contains 361,722 synaptic weights. The ResNet-34 network used for the 1000-class ImageNet dataset is shown in Supplementary Fig. 1 . The main differences compared to ResNet-32 are the number and size of the ResNet blocks, and a larger number of input/output channels (see Methods). Fig. 1: Training and inference methodology. a ResNet-32 network architecture for CIFAR-10 image classification. The architecture of ResNet-32 used in this study is a slightly modified version of the original implementation [24] with fewer input and output channels in ResNet blocks 2 and 3. This allows each layer to fit on a single crossbar array of size 512 × 512 in an eventual computational memory accelerator. b Training and inference of an example layer of ResNet-32 according to the methodology proposed in this paper. Software training is performed by injecting a random noise term to the weights used during the forward propagation, \\\\(\\\\delta {W}_{{\\\\rm{tr}}}^{{{l}}}\\\\) , which is representative of the combined read and write noise of the memristive devices used during inference (see section Training procedure). When transferring the weights of a convolution layer to memristive crossbars for inference, they are flattened into a 2-D matrix by collapsing each filter into a single vector programmed on a crossbar column, and stacking all filters on separate columns [25] . The weights are then programmed as the differential conductance of two memristive devices. Input activations a l are applied as voltages on the crossbar rows. The output current from the column containing the G − devices is subtracted from the one from the column containing the G + devices. The differential current output I from the crossbar then routes to a digital circuitry that performs batch normalization and the corresponding rectified linear unit (ReLU) activation function, in order to obtain the input activations for the next layer a l +1 . The final softmax activation function can be performed off-chip if required. An optional correction of the batch normalization parameters can be periodically performed to improve the accuracy retention over time (see sections Hardware/software inference experiment on CIFAR-10 and Adaptive batch normalization statistics update). The input image is padded with zero values at the border to ensure that the convolution operation preserves the height and width of the image. Therefore, considering an input image of size n × n , the convolution operation can be performed in n 2 matrix-vector multiplication cycles. Full size image The weights of all convolution layers along with the fully connected layer of ResNet-32 can be mapped on memristive crossbar arrays as explained in Fig. 1b [25] . Each synaptic weight can be mapped on a differential pair of memristive devices that are located on two different columns. For a given layer l , the synaptic weight \\\\({W}_{{{ij}}}^{{{l}}}\\\\) of the ( i , j ) th synaptic element is represented by the effective synaptic conductance \\\\({G}_{{{ij}}}^{{{l}}}\\\\) given by $${G}_{{{ij}}}^{{{l}}}={G}_{{{ij}}}^{{{l}},+}-{G}_{{{ij}}}^{{{l}},-},$$ (1) where \\\\({G}_{{{ij}}}^{{{l}},+}\\\\) and \\\\({G}_{{{ij}}}^{{{l}},-}\\\\) are the conductance values of the two devices forming the differential pair. Those device conductance values are defined as the effective conductance perceived in the operation of a non-ideal memristive crossbar array, and therefore include all the circuit non-idealities from the crossbar and peripheral circuitry. The mapping between the synaptic weight \\\\({W}_{{{ij}}}^{{{l}}}\\\\) obtained after software training and the corresponding synaptic conductance is given by $${G}_{{{ij}}}^{{{l}}}={W}_{{{ij}}}^{{{l}}}\\\\,\\\\, \\\\times \\\\frac{{G}_{\\\\max }}{{W}_{\\\\max }^{{{l}}}}+\\\\delta {G}_{{{ij}}}^{{{l}}}={G}_{{\\\\rm{T}},{{ij}}}^{{{l}}}+\\\\delta {G}_{{{ij}}}^{{{l}}},$$ (2) where \\\\({G}_{\\\\max }\\\\) is the maximum reliably programmable device conductance and \\\\({W}_{\\\\max }^{{{l}}}\\\\) is the maximum absolute synaptic weight value of layer l . \\\\(\\\\delta {G}_{{{ij}}}^{{{l}}}\\\\) represents the synaptic conductance error from the ideal target conductance value \\\\({G}_{{\\\\rm{T}},{{ij}}}^{{{l}}}={W}_{{{ij}}}^{{{l}}}\\\\, \\\\times \\\\frac{{G}_{\\\\max }}{{W}_{\\\\max }^{{{l}}}}\\\\) . \\\\(\\\\delta {G}_{{{ij}}}^{{{l}}}\\\\) is a time-varying random variable that describes the effects of non-ideal device programming (inaccuracies associated with write) and conductance fluctuations over time (inaccuracies associated with read). Possible factors leading to such conductance errors include inaccuracies in programming the synaptic conductance to \\\\({G}_{{\\\\rm{T}},{{ij}}}^{{{l}}}\\\\) , 1/ f noise from memristive devices and circuits, temporal conductance drift, device-to-device variations, defective (stuck) devices, and circuit non-idealities (e.g., IR drop). Clearly, a direct mapping of the synaptic weights of a DNN trained with 32-bit floating point (FP32) precision to the same DNN with memristive synapses is expected to degrade the network accuracy due to the added error in the weights arising from \\\\(\\\\delta {G}_{{{ij}}}^{{{l}}}\\\\) . For existing memristive technologies, the magnitude of \\\\(\\\\delta {G}_{{{ij}}}^{{{l}}}\\\\) may range from 1 to 10% of the magnitude of \\\\({G}_{{\\\\rm{T}},{{ij}}}^{{{l}}}\\\\) [11] , which in general is not tolerable by DNNs trained with FP32 without any constrains. Imposing such errors as constraints during training can be beneficial in improving the network accuracy. In fact, quantization of the weights or activations [26] , and injecting noise on the weights [27] , activations [28] , or gradients [29] have been widely used as DNN regularizers during training to reduce overfitting on the training dataset [30] , [31] . These techniques can improve the accuracy of DNN inference when it is performed with the same model precision as during training. However, achieving baseline accuracy while performing DNN inference on a model which is inevitably different from the one obtained after training, as is the case for any analog in-memory computing hardware, is a more difficult problem and requires additional investigations. Although a large body of efficient techniques to train DNNs with reduced digital precision has been reported [32] , [33] , it is unlikely that such procedures can generally be applied as-is to analog in-memory computing hardware due to the random nature of \\\\(\\\\delta {G}_{{{ij}}}^{{{l}}}\\\\) . Since quantization errors coming from rounding to reduced fixed-point precision are not random, DNNs trained in this way are not a priori expected to be suitable for deployment on analog in-memory computing hardware. Techniques that inject random Gaussian noise during training are a much more natural fit to make the network robust to errors from analog in-memory computing hardware. As early as in 1994, it was shown that injecting noise on the synaptic weights during training enhances the tolerance to weight perturbations of multi-layer perceptrons, and the application of this technique to analog neural hardware was discussed [34] . Recent works have also proposed to apply noise to the layer inputs or pre-activations in order to improve the network tolerance to hardware noise [20] , [23] . In this work, we follow the original approach of Murray et al. [34] of injecting Gaussian noise to the synaptic weights during training. Next, we discuss different techniques that we introduced together with synaptic weight noise in order to improve the accuracy of inference on ResNet and achieve close to software-equivalent accuracy after transferring the weights to PCM hardware. Training procedure When performing inference with analog in-memory computing hardware, the DNN experiences errors primarily due to (i) inaccurate programming of the network weights onto the devices (write noise) and (ii) temporal fluctuations of the hardware weights (read noise). We can cast the effect of these errors into a single error term \\\\(\\\\delta {G}_{{{ij}}}^{{{l}}}\\\\) that distorts each synaptic weight when performing forward propagation during inference. Hence, we propose to add random noise that corresponds to the error induced by \\\\(\\\\delta {G}_{{{ij}}}^{{{l}}}\\\\) to the synaptic weights at each forward pass during training (see Fig. 1b ). The backward pass and weight updates are performed with weights that did not experience this noise. We found that adding noise to the weights only in the forward propagation is sufficient to achieve close to baseline accuracy for a noise magnitude comparable to that of our hardware, and adding noise during the backward propagation did not improve the results further. For simplicity, we assume that \\\\(\\\\delta {G}_{{{ij}}}^{{{l}}}\\\\) is Gaussian distributed, which is usually the case for analog memristive hardware. Weights are linearly mapped to the entire conductance range \\\\({G}_{\\\\max }\\\\) of the hardware, hence the standard deviation \\\\({\\\\sigma }_{\\\\delta {W}_{{\\\\rm{tr}}}}^{l}\\\\) of the Gaussian noise on weights to be applied during training, for a layer l , can be obtained as $$\\\\frac{{\\\\sigma }_{\\\\delta {W}_{{\\\\rm{tr}}}}^{{{l}}}}{{W}_{\\\\max }^{{{l}}}}\\\\equiv {\\\\eta }_{{\\\\rm{tr}}}=\\\\frac{{\\\\sigma }_{\\\\delta {G}}}{{G}_{\\\\max }},$$ (3) where σ δG is a representative standard deviation of the combined read and write noise measured from hardware. During training, the weight distribution of every layer and hence \\\\({W}_{\\\\max }^{{{l}}}\\\\) changes, therefore \\\\({\\\\sigma }_{\\\\delta {W}_{{\\\\rm{tr}}}}^{{{l}}}\\\\) is recomputed after every weight update so that \\\\({\\\\eta }_{{\\\\rm{tr}}}\\\\) stays constant throughout training. We found this to be especially important in achieving good training convergence with this method. Weight initialization can have a significant effect on DNN training [35] . Two different weight initializations can lead to completely different minima when optimizing the network objective function. The network optimum when training with additive noise could be closer to the FP32 training optimum than to a completely random initialization. So it can be beneficial to initialize weights from a pretrained baseline network and then retrain this network by injecting noise. A similar observation was reported for training ResNet with reduced digital precision [33] . For achieving high classification accuracy in our experiments, we found this strategy more helpful than random initialization. The noise injected during training according to Eq. ( 3 ) is closely related to the maximum weight of a layer, and can thus grow uncontrollably with outlier weight values. Controlling the weight distribution in a desirable range can improve the network training convergence and make the mapping of weights to hardware with limited conductance range easier. We therefore clip the synaptic weights at layer l after every weight update in the range \\\\([-\\\\alpha \\\\times {\\\\sigma }_{{W}^{{{l}}}},\\\\alpha \\\\times {\\\\sigma }_{{W}^{{{l}}}}]\\\\) , where \\\\({\\\\sigma }_{{{{W}}}^{{{l}}}}\\\\) is the standard deviation of weights in layer l and α is a tunable hyperparameter. In our studies, α = 2.0 and α = 2.5 worked the best for ResNet-32 and ResNet-34, respectively. DNN convergence accuracy, in general, is sensitive to the learning rate used during training. Since we initialize the network parameters from a baseline network, using the same learning rate scheduling as that of the baseline network does not guarantee accurate convergence. To choose appropriate learning rate scheduling for ResNet-32, we first forward propagate the training set on the pretrained baseline network with injected synaptic weight noise and note the resulting accuracy. We note the learning rate evolution starting from this accuracy in the baseline network training curve until convergence, and use the same learning rate evolution while retraining the network by injecting noise. We performed simulations to characterize the inference performance after training incorporating the injection of Gaussian noise in conjunction with the techniques presented above. We computed the classification accuracy for different amounts of injected noise \\\\({\\\\eta }_{{\\\\rm{tr}}}\\\\) during training. We also show how the accuracy is affected when the inference weights are perturbed by a certain amount of relative noise \\\\({\\\\eta }_{\\\\inf }\\\\equiv \\\\frac{{\\\\sigma }_{\\\\delta {W}_{\\\\inf }}^{l}}{{W}_{\\\\max }^{l}}\\\\) , where \\\\({\\\\sigma }_{\\\\delta {W}_{\\\\inf }}^{l}\\\\) is the standard deviation of the noise injected to the weights of layer l before performing inference on the test dataset. The test accuracy of ResNet-32 on CIFAR-10 obtained for different amounts of noise injected during training, without inducing any perturbation during inference ( \\\\({\\\\eta }_{\\\\inf }=0\\\\) ), is plotted in Fig. 2a . It can be seen that the training algorithm is able to achieve a test accuracy close to the software baseline of 93.87% with up to approximately \\\\({\\\\eta }_{{\\\\rm{tr}}}=8 \\\\%\\\\) . The tolerance of the networks trained with different amounts of \\\\({\\\\eta }_{{\\\\rm{tr}}}\\\\) to weight perturbations during inference, \\\\({\\\\eta }_{\\\\inf }\\\\) , is shown in Fig. 2b . For a given value of \\\\({\\\\eta }_{\\\\inf }\\\\) , in general, the highest test accuracy can be obtained for the network that has been trained with a comparable amount of synaptic weight noise, i.e. for \\\\({\\\\eta }_{{\\\\rm{tr}}}\\\\approx {\\\\eta }_{\\\\inf }\\\\) . The test accuracy for \\\\({\\\\eta }_{{\\\\rm{tr}}}={\\\\eta }_{\\\\inf }\\\\) is shown in Fig. 2c . It can be seen that for up to \\\\({\\\\eta }_{\\\\inf }=5 \\\\%\\\\) , an accuracy within 0.5% of the software baseline is achievable. The impact of the weight initialization, clipping, and learning rate scheduling on the accuracy is shown in Supplementary Fig. 2 . Not incorporating one of those three techniques results in at least 1% drop in test accuracy for \\\\({\\\\eta }_{{\\\\rm{tr}}}={\\\\eta }_{\\\\inf }=3.8 \\\\%\\\\) . Fig. 2: Impact of injecting weight noise during training and inference on network accuracy. a Test accuracy on CIFAR-10 obtained for different amounts of relative injected weight noise during training \\\\({\\\\eta }_{{\\\\rm{tr}}}\\\\) without inducing any perturbation during inference ( \\\\({\\\\eta }_{\\\\inf }=0\\\\) ). When \\\\({\\\\eta }_{{\\\\rm{tr}}}\\\\, > \\\\, 8 \\\\%\\\\) , the training convergence starts to become affected by the high noise and it is not possible anymore to reach the software baseline within the same number of epochs. The error bars represent the standard deviation over 10 training runs. b Test accuracy on CIFAR-10 obtained for the networks trained with different amounts of relative weight noise \\\\({\\\\eta }_{{\\\\rm{tr}}}\\\\) as a function of the weight noise injected during inference \\\\({\\\\eta }_{\\\\inf }\\\\) . In most cases, \\\\({\\\\eta }_{{\\\\rm{tr}}}\\\\) can be increased above \\\\({\\\\eta }_{\\\\inf }\\\\) up to a certain point and still lead to comparable or slightly higher (within ≈0.1%) test accuracy than for \\\\({\\\\eta }_{{\\\\rm{tr}}}={\\\\eta }_{\\\\inf }\\\\) . However, when \\\\({\\\\eta }_{{\\\\rm{tr}}}\\\\) becomes much higher than \\\\({\\\\eta }_{\\\\inf }\\\\) , the test accuracy decreases due to the inability of the network to achieve baseline accuracy when \\\\({\\\\eta }_{{\\\\rm{tr}}}\\\\, > \\\\, 8 \\\\%\\\\) . Each data point represents the average over 10 training runs and 100 inference runs. c Test accuracy on CIFAR-10 as a function of \\\\({\\\\eta }_{{\\\\rm{tr}}}={\\\\eta }_{\\\\inf }\\\\) . The error bars represent the standard deviation over 100 inference runs averaged over 10 training runs. d Top-1 accuracy on ImageNet as a function of \\\\({\\\\eta }_{{\\\\rm{tr}}}={\\\\eta }_{\\\\inf }\\\\) . The error bars represent the standard deviation over 10 inference runs on a single training run. Full size image The top-1 accuracy of ResNet-34 on ImageNet for \\\\({\\\\eta }_{{\\\\rm{tr}}}={\\\\eta }_{\\\\inf }\\\\) is shown in Fig. 2d . Consistent with previous observations [23] , [33] , we found that the network recovers high accuracy extremely quickly when retraining with additive noise due to quick updates of the batch normalization parameters (see Supplementary Note 1 ), and obtained satisfactory convergence after only 8 epochs. The accuracy on ImageNet is much more sensitive to the noise injected during training than for CIFAR-10, and when noise is injected on all layers, there is more than 0.5% accuracy drop from the baseline even down to ~1.2% relative noise. In the literature, many network compression techniques allow higher precision for the first and last layers, which are more sensitive to noise [33] , [36] . We applied the same simplification to our problem, which means that we removed the noise during training on the first convolutional layer and the last dense layer, and performed inference with the first and last layer without noise. The obtained accuracy after training, by injecting the same training and inference noise as previously, can be increased by more than 1% with this technique (see Fig. 2d ). Weight transfer to PCM-based synapses In order to validate the effectiveness of the above training methodology, we performed experiments on a prototype multi-level PCM chip comprising 1 million PCM devices fabricated in 90 nm CMOS baseline technology [37] . PCM is a memristive technology, which records data in a nanometric volume of phase-change material sandwiched between two electrodes [38] , [39] . The phase-change material is in the low-resistive crystalline phase in an as-fabricated device. By applying a current pulse of sufficient amplitude (typically referred to as the RESET pulse) an amorphous region around the narrow bottom electrode is created via a melt-quench process. The device will be in a low conductance state if the high-resistive amorphous region blocks the current path between the two electrodes. The size of the amorphous region can be modulated in an almost completely analog manner by the application of suitable electrical pulses. Hence, a continuum of conductance values can be programmed in a single PCM device over a range of more than two orders of magnitude. An optimized iterative programming algorithm was developed to program the conductance values in the PCM devices with high accuracy (see Methods). The experimental cumulative distributions of conductance values for 11 representative programmed levels, measured approximately 25 s after programming, are shown in Fig. 3a . The standard deviation of these distributions is extracted and fitted with a polynomial function of the target conductance (dashed lines in Fig. 3a ) as shown in Fig. 3b . For all levels, we achieve a standard deviation less than 1.2 μS, which is more than two times lower than that reported in previous works on nanoscale PCM arrays for a similar conductance range [40] , [41] . Fig. 3: Impact of weight transfer to PCM synapses on inference accuracy. a Cumulative distributions of 11 representative iteratively programmed conductance levels on 10,000 PCM devices per level. The vertical dashed lines denote the target conductance for each level. b Conductance standard deviation of the 11 levels as a function of target conductance. The inset shows a representative conductance distribution of one level. c Test accuracy on CIFAR-10 after software training and after weight transfer to PCM synapses for different training schemes. d Top-1 accuracy on ImageNet after software training and after weight transfer to PCM synapses for different training schemes. The weights of all layers of ResNet-34 are transferred to PCM synapses, including the first and the last. In c and d the error bars represent the standard deviation over 10 inference runs. Full size image We studied the impact of weight transfer to PCM synapses on the inference accuracy of networks trained with FP32 weights, 4-bit and ternary digital weights, and additive weight noise. The weight transfer to PCM is simulated by mapping each trained network weight to a synaptic conductance value \\\\({G}_{{{ij}}}^{{{l}}}\\\\) , computed from Eq. ( 2 ), using \\\\({G}_{\\\\max }=25\\\\) μS and the conductance standard deviation measured from hardware. The conductance error \\\\(\\\\delta {G}_{{{ij}}}^{{{l}}}\\\\) in Eq. ( 2 ) is modeled as a Gaussian distributed random variable with zero mean and standard deviation given by the fitted curve of Fig. 3b for its corresponding target conductance \\\\(| {G}_{{\\\\rm{T}},{{ij}}}^{l}|\\\\) . This procedure accurately reproduces the experimental conductance values obtained when iteratively programming any conductance level within the range of 0–25 μS. For all training schemes, only one differential pair is used to encode a network weight, and we do not perform any additional quantization of the weights before mapping them to PCM conductance values. The resulting test accuracy obtained after software training and after weight transfer to PCM synapses for ResNet-32 on CIFAR-10 is shown in Fig. 3c . It can be seen that the standard FP32 training without constraints performs the worst after transfer to PCM synapses. Training with 4-bit digital weights (using the method described in the ref. [33] ), which is roughly the effective precision of our PCM devices [40] , improves the performance after transfer with respect to FP32, but nevertheless the accuracy decreases by more than 1% after transferring the 4-bit weights to PCM. Training ternary digital weights [42] leads to a lower performance drop (<0.5%) when transferring weights to PCM, although we were not able to reach the FP32 baseline with ternary weights on this network. Therefore the accuracy after transfer is worse than for the 4-bit weights. When performing training by injecting Gaussian noise with \\\\({\\\\eta }_{{\\\\rm{tr}}}=3.8 \\\\%\\\\) , corresponding to σ δG = 0.94 μS (median of the 11 values reported in Fig. 3b ), the best overall performance after transfer to PCM is obtained. The resulting accuracy of 93.7% is less than 0.2% below the FP32 baseline. A rather broad range of values of \\\\({\\\\eta }_{{\\\\rm{tr}}}\\\\) lead to a similar resulting accuracy (see Supplementary Fig. 3 ), demonstrating that \\\\({\\\\eta }_{{\\\\rm{tr}}}\\\\) does not have to be very precisely determined for obtaining satisfactory results on PCM. The accuracy obtained without perturbing the weights after training by injecting noise is slightly higher than the FP32 baseline, which could be attributed to improved generalization resulting from the additive noise training. The top-1 accuracy for ResNet-34 on ImageNet after transfer to PCM synapses for different training procedures is shown in Fig. 3d . Training with additive noise increases the accuracy by approximately 6% on PCM compared with FP32 and 4-bit [33] training. The accuracy of 71.6% achieved with additive noise training on PCM is significantly higher than that reported in Fig. 2d with \\\\({\\\\eta }_{\\\\inf }=3.8 \\\\%\\\\) , which could be attributed to a high percentage of network weights mapped to low conductance values with lower standard deviation than the median of 0.94 μS. Hardware/software inference experiment on CIFAR-10 In addition to the weight transfer simulations of the previous section, we performed a hardware/software experiment in which all weights of ResNet-32 were physically programmed on the PCM chip. This allows us to address the effect of imperfect yield and temporal conductance fluctuations on the accuracy experimentally. Although we could achieve good test accuracy after weight transfer to PCM synapses, an important challenge for any analog in-memory computing hardware is to be able to retain this accuracy over time. This is especially true for PCM due to the high 1/ f noise experienced in these devices as well as temporal conductance drift. The conductance values in PCM drift over time t according to the relation \\\\(G(t)=G({t}_{0}){(t/{t}_{0})}^{-\\\\nu }\\\\) , where G ( t 0 ) is the conductance measured at time t 0 after programming and ν is the drift exponent, which depends on the device, phase-change material, and phase configuration of the PCM ( ν is higher for the amorphous than the crystalline phase) [43] . In our PCM devices, ν ≈ 0.06 on average. Therefore, it is essential to measure experimentally how the test accuracy evolves over time during inference with PCM. All 361,722 synaptic weights of ResNet-32 trained with \\\\({\\\\eta }_{{\\\\rm{tr}}}=3.8 \\\\%\\\\) were programmed individually on two PCM devices of the chip. Depending on the sign of \\\\({G}_{{{\\\\rm{T}}},{ij}}^{l}\\\\) , either \\\\({G}_{ij}^{l,+}\\\\) or \\\\({G}_{ij}^{l,-}\\\\) was iteratively programmed to \\\\(| {G}_{{\\\\rm{T}},{{ij}}}^{{{l}}}|\\\\) , and the other device was RESET close to 0 μS with a single pulse of 450 μA amplitude and 50 ns width. The iterative programming algorithm converged on 99.1% of the devices programmed to nonzero conductance, and no screening for defective devices on the chip was performed prior to the experiments. The scatter plot of the PCM weights measured approximately 25 s after programming versus the target weights \\\\({W}_{{{ij}}}^{{{l}}}\\\\) is shown in Fig. 4a . After programming, the PCM analog conductance values were periodically read from hardware over a period of 1 day, scaled to the network weights, and reported to the software that performed inference on the test dataset (see Methods). Fig. 4: Inference experiment with ResNet-32 on CIFAR-10. a Scatter plot of weights programmed in the PCM chip versus target weights obtained after training. The inset shows the distribution of the relative error between programmed and target synaptic conductance, \\\\(({G}_{{{ij}}}^{{{l}}}-{G}_{{\\\\rm{T}},{{ij}}}^{{{l}}})/{G}_{\\\\max }\\\\) , and its standard deviation σ . b Distributions of programmed devices whose target conductance fall within 0.25 μS from five representative G T values. The distributions are shown at four different times spanning the experiment duration of one day. The filled bars are the measured hardware data, the black lines are the PCM model. c Individual device conductance evolution over time of four arbitrarily picked devices from the chip programmed to four distinct G T values, along with one PCM model realization for the same G T values. d Measured test accuracy over time from the weights of the PCM chip, along with the corresponding PCM model match. A global drift compensation (GDC) procedure is performed for every layer before performing inference on the test set. The filled areas from the PCM model correspond to one standard deviation over 25 inference runs. The baseline refers to the FP32 software accuracy of 93.87%. Full size image In addition to the experiment, we developed an advanced behavioral model of the hardware in order to precisely capture the conductance evolution over time during inference (see Supplementary Note 2 ). The model is built based on an extensive experimental characterization of the array-level statistics of hardware noise and drift [44] . Conductance drift is modeled using a Gaussian distributed drift exponent across devices, whose mean and standard deviation both depend on the target conductance state \\\\(| {G}_{{\\\\rm{T}},{{ij}}}^{{{l}}}|\\\\) . Conductance noise with the experimentally observed 1/ f 1.21 frequency dependence is also incorporated with a magnitude that depends on the target conductance state and time. The model is able to accurately reproduce both the array-level statistics (see Fig. 4b ) and individual device behavior (see Fig. 4c ) observed over the duration of the experiment. Accurate modeling of all the complex dependencies of noise and drift as a function of time and conductance state was found to be very critical in being able to reproduce the experimental evolution of the accuracy on ResNet. The resulting accuracy on CIFAR-10 over time is shown in Fig. 4d . The test accuracy measured 25 s after programming is 93.75%, which is very similar to the result obtained in Fig. 3c . However, if nothing is done to compensate for conductance drift, the accuracy quickly decreases down to 10% (random guessing) within approximately 1000 s. This is because the magnitude of the PCM weights gradually reduces over time due to drift and this prevents the activations from properly propagating throughout the network. A simple global scaling calibration procedure can be used to compensate for the effect of drift on the matrix-vector multiplications performed with PCM crossbars. As proposed in ref. [40] , the summed current of a subset of the columns in the array can be periodically read over time at a constant voltage. The resulting total current is then divided by the summed current of the same columns but read at time t 0 . This results in a single scaling factor that can be applied to the output of the entire crossbar in order to compensate for a global conductance shift (see Methods and Supplementary Fig. 4 ). Since this factor can be combined with the batch normalization parameters, it does not incur any additional overhead when performing inference. This simple global drift compensation (GDC) procedure was implemented for every layer before carrying out inference on the test set, and the results are shown in Fig. 4d . It can be seen that GDC allows the retention of a test accuracy above 92.6% for 1 day on the PCM chip, and effectively prevents the effect of global weight decay over time as illustrated in Supplementary Fig. 4 . A good agreement of the accuracy evolution between model and experiment is obtained, hence validating its use for extrapolating results over a longer period of time and for assessing the accuracy of larger networks that cannot fit on our current hardware. Adaptive batch normalization statistics update Although GDC can compensate for a global conductance shift across the array, it cannot mitigate the effect of 1/ f noise and drift variability across devices. From the model, we observe that 1/ f noise is responsible for the random accuracy fluctuations, whereas drift variability and its dependence on the target conductance state cause the monotonous accuracy decrease over time (see Supplementary Fig. 5 ). In order to improve the accuracy retention further, we propose to leverage the batch normalization parameters to correct the activation distributions during inference such that their mean and variance match those that were optimally learned during training. During inference, batch normalization is performed by normalizing the preactivations by their corresponding running mean μ and variance σ 2 computed during training. Then, scale and shift factors ( γ and β ) that were learned through backpropagation are applied to the normalized preactivations. Since γ and β are learnable parameters, it is not desirable to change them since it would require retraining the model on the PCM devices. However, updating μ and σ 2 is more intuitive, since the mean and variance of the preactivations are affected by noise and drift. Leveraging this idea, we introduce a new compensation technique called adaptive batch normalization statistics update (AdaBS), which improves the accuracy retention beyond GDC at the cost of additional computations during the calibration phase. As described in Fig. 5a , the calibration phase consists in sending multiple mini-batches from a set of calibration images that come from the same distribution than the images seen during inference. In this study, we use the images from the training dataset as calibration images. The running mean and variance of preactivations are computed across the entire calibration dataset. The new values of μ and σ 2 computed during calibration are then used for subsequent inference. The main advantage of this technique is that it does not incur additional digital computations nor weight programming during inference, because we are only updating the batch normalization parameters μ and σ 2 when the calibration is performed. However, injecting the entire training dataset to compute μ and σ 2 in the calibration phase would bring significant overhead. When reducing the amount of injected images, the number of updates of the running statistics becomes smaller, and if the momentum used for computing μ and σ 2 is not properly tuned to account for this, the network accuracy decreases significantly. To tackle this issue, we developed a procedure to obtain the optimal momentum as a function of the number of mini-batches used for calibration (see Methods and Supplementary Note 3 ). With this method, we were able to reduce the number of calibration images down to 5.2% of the CIFAR-10 training dataset (2600 images) without affecting the accuracy. With that number of images, the overhead in terms of digital computations of the AdaBS calibration is about 52% of performing batch normalization during inference on the whole CIFAR-10 test set (see Supplementary Note 3 ). Furthermore, AdaBS requires additional off-chip memory resources to store the calibration images, leading to extra off-chip data transfers during calibration (see Supplementary Note 3 ). It may appear cumbersome to send so many images to the device to perform the calibration, however since it is only performed periodically over time when the device is idle and not every time an image is inferred by the network, the calibration cost can be amortized. The calibration overhead can be further reduced by using more efficient variants of batch normalization such as the L 1 -norm version (see Supplementary Note 3 ). Moreover, although we used AdaBS (and GDC) to compensate solely for the drift of the PCM devices, the same procedure can be applied to mitigate conductance changes due to ambient temperature variations, a critical issue for any analog in-memory computing hardware. Fig. 5: Adaptive batch normalization statistics update with PCM synapses. a The AdaBS calibration procedure consists in updating the running mean μ and variance σ 2 parameters of the batch normalization performed in the digital unit of the in-memory computing hardware. The calibration is performed periodically when the device is idle, and after calibration the values of μ and σ 2 of every layer are updated for subsequent inference. Note that during the calibration phase, batch normalization is performed using the mini-batch mean μ B and variance \\\\({\\\\sigma }_{{\\\\rm{B}}}^{2}\\\\) instead of μ and σ 2 (see Methods and Supplementary Note 3 ). b Test accuracy of ResNet-32 on CIFAR-10 with GDC and AdaBS using the PCM model, along with experimental test accuracy obtained by applying AdaBS on ResNet-32 with the measured weights from the PCM chip. The filled areas from the PCM model correspond to one standard deviation over 25 inference runs. c Top-1 accuracy of ResNet-34 on ImageNet with GDC and AdaBS computed using the PCM model. Implementations using PCM synapses for all layers as well as first and last layers in digital FP32, and PCM synapses for all other layers, are shown. In the latter, no noise is applied on the first and last layers during training. The filled areas correspond to one standard deviation over 10 inference runs. In b and c the baseline refers to the FP32 software accuracy. Full size image The resulting accuracy when performing AdaBS on ResNet-32 with hardware weights before carrying out inference on the test set is shown in Fig. 5b . AdaBS allows to retain a test accuracy above 93.5% over one day, an improvement of 0.9% compared with GDC. This improvement becomes 1.8% for one year when extrapolating the results using the PCM model. This improvement results from optimally tuning scale and shift factors for each individual column of the crossbar with AdaBS, instead of a single scale factor for all columns in the case of GDC. Because of this more extensive parameter optimization, AdaBS can effectively compensate for drift variability and, more generally, random conductance variations across devices (see Supplementary Figs. 5 – 6 ). We also applied AdaBS on the ImageNet classification task with ResNet-34, trained with \\\\({\\\\eta }_{{\\\\rm{tr}}}=3.8 \\\\%\\\\) , using the PCM model to simulate the weight evolution for one year. By applying the same AdaBS method as for CIFAR-10 using only 0.1% of the ImageNet training dataset for calibration (1300 images), the accuracy after one year is increased by 7% compared with GDC when all layers are implemented with PCM synapses (see Fig. 5c ). When the first and last layers are implemented in digital FP32, the initial accuracy increases to 71.9% and the retention is significantly improved. This technique, combined with AdaBS, allows the retention of an accuracy above 71% for one year. Drawbacks in efficiency when performing inference on hardware in this way have to be mentioned, but they stay limited given the small number of parameters and input size of the first and last layers (the first and last layers contain less than 3% of the network weights and are responsible for about 3.12% of the multiply-and-accumulate operations during inference with ResNet-34).\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"Combined together, the strategies developed in this study allow us to achieve the highest accuracies reported so far with analog resistive memory on the CIFAR-10 and ImageNet benchmarks with residual networks close to their original implementation [24] . Although there is still room for improvement especially on ImageNet, those accuracies are already comparable or higher than those reported on ternary weight networks [42] , for example 71.6% top-1 accuracy of ResNet-34 on ImageNet with first layer in FP32 [45] . Importantly, the accuracies we report are achieved with just two nanoscale PCM devices encoding the value of a weight. A common approach that could improve the accuracy further is to use multiple devices to encode different bits of a weight [3] , [41] , at the expense of area and energy penalty, and additional support required by the peripheral circuitry. Aligned with previous observations [23] , [33] , we notice that retraining ResNet with additive noise results mainly in adapting the batch normalization parameters, whereas the weights stay close to the full-precision weights trained without noise. Hence, retraining by injecting noise from a pretrained baseline network rather than from scratch is very effective since the network recovers high accuracy very quickly, especially for ImageNet. Although our experiments are not done on a fully-integrated chip that supports all functions of deep learning inference, the most critical effects of array-level variability, noise, and drift, are fully accounted for because each weight of the network is programmed on individual PCM devices of our array. Aspects of a fully-integrated chip that are not entirely captured in our experiments such as IR drop and additional circuit nonidealities such as offsets and noise have been studied in previous works and could be mitigated by additional retraining methods [18] , [20] . Experiments on fully-integrated memristor chips could be used to faithfully quantify their impact on the inference accuracy [7] , [9] , [46] . With respect to device yield, additional simulations showed that additive noise training also helps to mitigate the impact of devices stuck at low or high conductance on the inference accuracy (see Supplementary Fig. 7 ). Additional errors due to quantization coming from the crossbar data converters are analyzed further below. There exist many different methods of training a neural network with noise that aim to improve the resilience of the model to analog mixed-signal hardware. These include injecting additive noise on the inputs of every layer [20] , on the preactivations [22] , [23] , or just adding noise on the input data [47] . Moreover, injecting multiplicative Gaussian noise to the weights [34] ( \\\\({\\\\sigma }_{\\\\delta {W}_{{\\\\rm{tr}}},{{ij}}}^{l}\\\\propto | {W}_{{{ij}}}^{{{l}}}|\\\\) ) is also defensible regarding the observed noise on the hardware. We analyzed the four aforementioned methods, attempting to reach the same accuracy demonstrated previously after weight transfer to PCM devices, to identify their possible benefits and drawbacks (see Supplementary Note 4 ). We found that it is possible to adjust the training procedure of all four methods to achieve a similar accuracy on CIFAR-10 after transferring the weights to PCM synapses. Somewhat surprisingly, even adding noise on the input data during training, which is just a simple form of data augmentation, leads to a model which is more resilient to weight perturbations during inference. This shows that it is not necessary to train a model with very complicated noise models that imitate the observed hardware noise precisely. As long as the data propagated through the network is corrupted by a Gaussian noise of the right magnitude, the model is expected to be robust to mapping on PCM devices. However, all four methods require one or multiple noise scaling factor hyperparameters to tune in order to reach satisfactory accuracy after transfer to PCM. In contrast, our proposed methodology estimates the additive noise to inject on the weights, \\\\({\\\\eta }_{{\\\\rm{tr}}}\\\\) , from a simple hardware characterization, avoiding any hyperparameter search for noise scaling factors. The value of \\\\({\\\\eta }_{{\\\\rm{tr}}}\\\\) does not have to be very precise either, because there are a range of values that lead to similar accuracy after transfer to PCM (see Supplementary Fig. 3 ). Moreover, we found that injecting noise on weights achieves better accuracy retention over time (see Supplementary Note 4 ), which suggests that weight noise mimics the behavior of the PCM hardware better. A critical issue for in-memory computing hardware is the need for digital-to-analog (analog-to-digital) conversion every time data goes in (out) of the crossbar arrays. These data conversions lead to quantization of the activations and preactivations, respectively, which introduce additional errors in the forward propagation. Based on a recent ADC survey [23] , 8-bit data conversion is a good tradeoff between precision and energy consumption. Hence, we analyzed the effect of quantizing the input and output of every layer of ResNet-32 and ResNet-34 to 8-bit on the inference accuracy. We set the input/output quantization ranges to the 99.995th percentile of the activation/preactivation distributions that are obtained when forward propagating 10 k randomly sampled images from the training dataset through the baseline network. As shown in Supplementary Fig. 8 , even though the 8-bit quantization is not included in our training algorithm, the quantization has a minimal effect on the mean accuracy of ResNet-32 on CIFAR-10 (<0.05% drop) and ResNet-34 on ImageNet (<0.15% drop) after weight transfer to PCM synapses. The accuracy evolution over time, retaining the same quantization ranges, does not degrade significantly further and stays well within one standard deviation of that obtained without quantization. The small accuracy deviations could be potentially overcome by including the quantization in the retraining process, which will likely be necessary if less than 8-bit resolution is desired for higher energy efficiency. Although a computational memory accelerates the matrix-vector multiplication operations in a DNN, communicating activations between computational memory cores executing different layers can become a bottleneck. This bottleneck depends upon two factors, (i) the way different layers are connected to each other and (ii) the latency of the hardware implementation to transfer activations from one core to another. Designing optimal interconnectivity between the cores for state-of-the-art deep CNNs is an open research problem. Indeed, having the network weights stationary during execution in a computational memory puts limits on what portion of the computation can be forwarded to different cores. This ultimately results in long-established hardware communication fabrics being ill-fit for the task. One topology for communication fabrics that is well-suited for computational memory is proposed by Dazzi et al. [48] . It is based on a 5 parallel prism (5PP) graph topology and facilitates inter-layer pipelined execution of CNNs [3] . The proposed 5PP topology allows the mapping of all the primary connectivities of state-of-the-art neural networks, including ResNet, DenseNet and Inception-style networks [48] . As discussed in ref. [48] , the ResNet-32 implementation with 5PP can result in potentially 2× improvement in pipeline stage latency with similar bandwidth requirements compared with a standard 2D-mesh. Assuming that each layer is mapped onto a crossbar of size 512 × 512 having a computational cycle time of 100 ns, 8-bit activations, communication links with data rate of 5Gbps [49] , and a pipeline scheme identical to that proposed in ref. [3] , a single image inference latency of 52 μs and frame rate of 38,600 frames per second (FPS) for ResNet-32 on CIFAR-10 is estimated. The energy efficiency of a computational PCM core of this size implementing a layer of ResNet-32 is estimated to be 11.9 TOPS W −1 (tera operations per seconds per watt), with an area expenditure of approximately 0.57 mm 2 (see Supplementary Note 5 ). As an approximate comparison, YodaNN [50] , a digital DNN inference accelerator for binary weight networks with ultra-low power budget, achieves 434.8 FPS in high throughput mode for a 9-layer CNN (BinaryConnect [51] ) on CIFAR-10. Although not a direct comparison, the proposed topology and pipelined execution of ResNet-32 could result in 88× speedup, with a deeper network than the digital solution. In summary, we introduced strategies for training ResNet-type CNNs for deployment on analog in-memory computing hardware, as well as improving the accuracy retention on such hardware. We proposed to inject noise to the synaptic weights in proportion to the combined read and write conductance noise of the hardware during the forward pass of training. This approach combined with judicious weight initialization, clipping, and learning rate scheduling, allowed us to achieve an accuracy of 93.7% on the CIFAR-10 dataset and a top-1 accuracy on the ImageNet benchmark of 71.6% after mapping the trained weights to analog PCM synapses. Our methods introduce only a single additional hyperparameter during training, the weight clip scale α , since the magnitude of the injected noise can be easily deduced from a one-time hardware characterization. After programming the trained weights of ResNet-32 on 723,444 PCM devices of a prototype chip, the accuracy computed from the measured hardware weights stayed above 92.6% over a period of 1 day, which is to the best of our knowledge the highest accuracy experimentally reported to-date on the CIFAR-10 dataset by any analog resistive memory hardware. A global scaling procedure was used to compensate for the conductance drift of the PCM devices, which was found to be critical in improving the accuracy retention. However, global scaling could not mitigate the effect of 1/ f noise and drift variability across devices, which led to accuracy fluctuations and monotonous accuracy decrease over time, respectively. Periodically calibrating the batch normalization parameters before inference allowed to partly alleviate those issues at the cost of additional digital computations, increasing the 1-day accuracy to 93.5% on hardware. These results demonstrate the feasibility of realizing accurate inference on complex DNNs through analog in-memory computing using existing PCM devices.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Experiments on PCM hardware platform The experimental platform is built around a prototype PCM chip that comprises 3 million PCM devices. The PCM array is organized as a matrix of word lines (WL) and bit lines (BL). In addition to the PCM devices, the prototype chip integrates the circuitry for device addressing, and for write and read operations. The PCM chip is interfaced to a hardware platform comprising a field programmable gate array (FPGA) board and an analog-front-end (AFE) board. The AFE board contains the digital-to-analog converters, and provides the power supplies as well as the voltage and current reference sources to the PCM chip. The FPGA board implements the data acquisition and the digital logic to interface with the PCM device under test and with all the electronics of the AFE board. The FPGA board is also used to implement the overall system control and data management as well as the interface with the host. The experimental platform is operated from a host computer, and a Matlab environment is used to coordinate the experiments. The PCM devices are integrated into the chip in 90-nm CMOS technology using the key-hole process described in the ref. [52] . The phase-change material is doped Ge 2 Sb 2 Te 5 . The bottom electrode has a radius of ~20 nm and a height of ~50 nm. The phase-change material is ~100 nm thick and extends to the top electrode, whose radius is ~100 nm. All experiments performed in this work are done on an array containing 1 million devices accessed via transistors, which is organized as a matrix of 512 WL and 2048 BL. A PCM device is selected by serially addressing a WL and a BL. To read a PCM device, the selected BL is biased to a constant voltage (300 mV) by a voltage regulator via a voltage generated off chip. The sensed current is integrated by a capacitor, and the resulting voltage is then digitized by the on-chip 8-bit cyclic analog-to-digital converter (ADC). The total duration of applying the read pulse and converting the data with the ADC is 1 μs. The readout characteristic is calibrated via on-chip reference polysilicon resistors. To program a PCM device, a voltage generated off chip is converted on chip into a programming current. This current is then mirrored into the selected BL for the desired duration of the programming pulse. Iterative programming involving a sequence of program-and-verify steps is used to program the PCM devices to the desired conductance values [53] . The devices are initialized to a high-conductance state via a staircase-pulse sequence. The sequence starts with a RESET pulse of amplitude 450 μA and width 50 ns, followed by six pulses of amplitude decreasing regularly from 160 to 60 μA and with a constant width of 1000 ns. After initialization, each device is set to a desired conductance value through a program-and-verify scheme. The conductance of all devices in the array is read five times consecutively at a voltage of 0.3 V, and the mean conductance of these reads is used for verification. If the read conductance of a specific device does not fall within 0.25 μS from its target conductance, it receives a programming pulse where the pulse amplitude is incremented or decremented proportionally to the difference between the read and target conductance. The pulse amplitude ranges between 80 and 400 μA, and the pulse width is 1000 ns. This program-and-verify scheme is repeated for a maximum of 55 iterations. In the hardware/software inference experiments, the analog conductance values of the PCM devices encoding the network weights, \\\\({G}_{{{ij}}}^{{{l}},+}\\\\) and \\\\({G}_{{{ij}}}^{{{l}},-}\\\\) , are serially read individually with the 8-bit on-chip ADC at predefined timestamps spaced over a period of one day. The read conductance values at every timestamp are reported to a TensorFlow-based software. This software performs the forward propagation of the CIFAR-10 test set on the weights read from hardware and computes the resulting classification accuracy. The drift compensation techniques, GDC and AdaBS, are performed entirely in software at every timestamp based on the conductance values read from hardware. PCM-based deep learning inference simulator We developed a simulation framework to test the efficacy of DNN inference using PCM devices. We chose Google’s TensorFlow [54] deep learning framework for the simulator development. The large library of algorithms in TensorFlow enables us to use native implementation of required activation functions and batch normalization. Moreover, any regular TensorFlow code of a DNN can be easily ported to our simulator. As shown in Supplementary Fig. 9 , custom made TensorFlow operations are implemented that generate PCM conductance values from the behavioral model of hardware PCM devices that was developed (see Supplementary Note 2 ). All the nonidealities including conductance range, programming noise, read noise, and conductance drift are implemented in TensorFlow following the equations shown in Supplementary Note 2 . The simulator can also take the PCM conductance data measured from hardware as input, in order to perform inference on the hardware data. Data converters that simulate digital quantization of data at the input and output of crossbars are also implemented with tunable quantization ranges and precision. In this study, the data converters were turned off for all simulations except those presented in Supplementary Fig. 8 . The drift correction techniques are implemented post quantization of the crossbar output. Implementation of ResNet-32 training on CIFAR-10 ResNet-32 has 31 convolution layers with 3 × 3 kernels, 2 convolution layers with 1 × 1 kernels, and a final fully-connected layer. The network contains 361,722 synaptic weights. It consists of three different ResNet blocks with 10 3 × 3 kernels each. After the first convolution layer, there is a unity residual feed forward connection after every two convolution layers, except the 1 × 1 residual convolution connection to make output channels compatible between two layers. Each convolution layer is followed by batch normalization [55] . ReLU activation is used after every batch normalization except in case of residual connections, where the ReLU activation is computed after summation. Each residual connection with 1 × 1 convolution and first layer of ResNet blocks 2, 3 downsample the input by using a stride of 2 pixels. The 8 × 8 output of the last convolution layer is then downsampled to 1 × 1 resolution using global average pooling [56] , which is followed by a single fully-connected layer. For the last fully-connected layer, no batch normalization is performed. The architecture of ResNet-32 used in this study is a slightly modified version of the original implementation [24] with fewer input and output channels in ResNet blocks 2 and 3. This network is trained on the well-known CIFAR-10 classification dataset [57] . It has 32 × 32 pixels RGB images that belong to one of the ten classes. The network is trained on the 50,000 images of the training set, and evaluation is performed on the 10,000 images of the test set. The training is performed using stochastic gradient descent with a momentum of 0.9. The network objective is categorical cross entropy function over ten classes of the input image. Learning rate scheduling is performed to reduce learning rate by 90% at every 50th training epoch. The initial learning rate for the baseline network is 0.1 and training converges in 200 epochs with a mini-batch size of 128. Weights of all convolution and fully connected layers of the baseline network are initialized using He Normal [35] initialization. The baseline network is retrained by injecting Gaussian noise for up to 150 epochs with weight clip scale α = 2. We preprocess the training images by randomly cropping a 32 × 32 patch after padding 2 pixels along the height and width of the image. We also apply a random horizontal flip on the images from the train set. Additionally, we apply cutout [58] on the training set images. For both training and test set, we apply channel wise normalization for 0 mean and unit standard deviation. Implementation of ResNet-34 training on ImageNet The architecture of the ResNet-34 network for ImageNet classification is derived from ref. [24] . It has 32 convolution layers with 3 × 3 kernels, 3 convolution layers with 1 × 1 kernels, a first convolution layer with 7 × 7 kernels and a final fully-connected layer. The network has 21,797,672 synaptic weights. The first convolution layer downsamples the input by using a stride of 2 pixels, followed by a maxpooling layer with kernel size of 3 × 3 and stride of 2 to downsample the feature maps to the resolution of 56 × 56 pixels. Each residual connection with 1 × 1 convolution and first layer of ResNet blocks 2, 3, 4 downsample the input by using a stride of 2 pixels. A global average pooling layer before the final fully-connected layer downsamples the 7 × 7 input to 1 × 1 resolution. The final fully-connected layer computes the output prediction corresponding to 1000 classes. We trained ResNet-34 on the ImageNet [59] dataset. The ImageNet dataset has 1.3 M images in the training set and 50 k images in the test set. Images in the ImageNet dataset are preprocessed by following the same preprocessing steps as that of the Pytorch baseline model. Training images are randomly cropped to a 224 × 224 patch and then random horizontal flip is applied on the images. Channel wise normalization is performed on the images in both training and test set for 0 mean and unit standard deviation. Only for the test set, images are first resized to 256 × 256 using bilinear interpolation method and then a center crop is performed to obtain the 224 × 224 image patch. The network objective function is softmax cross entropy on network output and corresponding 1000 labels. The network objective is minimized using the stochastic gradient descent algorithm with a momentum of 0.9. We obtained our baseline network architecture and its parameters from the Pytorch model zoo ( https://pytorch.org/docs/stable/torchvision/models.html ). We use this network to perform additive noise training by injecting Gaussian noise for a total of 10 training epochs. In contrast to ResNet-32 on CIFAR-10, no learning rate scheduling was performed since the network was trained only for 10 epochs with additive noise. We use mini-batch size of 400 and learning rate of 0.001 for the additive noise training simulations. We also use L2 weight decay of 0.0001 and weight clip scale of α = 2.5 for the additive noise training. Global drift compensation (GDC) method The GDC [40] calibration phase consists of computing the summed current of L columns in each array encoding a network layer (see Supplementary Fig. 4 ). Those L columns contain devices initially programmed to known conductance values G mn ( t 0 ). By reading those column currents, I m , periodically with applied voltage V cal on all N rows, we can compensate for a global conductance shift in the array during inference. When input data is processed by the crossbar during inference, the crossbar output can be scaled by \\\\(1/\\\\hat{\\\\alpha }\\\\) , where $$\\\\hat{\\\\alpha }=\\\\frac{\\\\mathop{\\\\sum }\\\\nolimits_{m=1}^{L}{I}_{m}}{{V}_{{\\\\rm{cal}}}\\\\mathop{\\\\sum }\\\\nolimits_{n=1}^{N}\\\\mathop{\\\\sum }\\\\nolimits_{m=1}^{L}{G}_{mn}({t}_{0})}.$$ This procedure is especially simple because L can be chosen to be small, enough to get sufficient statistics. Moreover, \\\\(\\\\hat{\\\\alpha }\\\\) is computed from the device data itself, without resorting to any assumption on how the conductance changes nor requiring extra timing information [60] . The term \\\\({V}_{{\\\\rm{cal}}}\\\\mathop{\\\\sum }\\\\nolimits_{n = 1}^{N}\\\\mathop{\\\\sum }\\\\nolimits_{m = 1}^{L}{G}_{mn}({t}_{0})\\\\) needs to be computed only once, stored in the digital memory of the chip, and is reused for all calibrations. Reading the subset of L columns of the crossbar can be done while the PCM array is idle, i.e., when there are no incoming images to be processed by the device. Performing the L current summations can be implemented either with on-chip digital circuitry or in the control unit of the chip. At the end of the calibration phase, \\\\(1/\\\\hat{\\\\alpha }\\\\) is computed and stored locally in digital unit of the crossbar. The output scaling by \\\\(1/\\\\hat{\\\\alpha }\\\\) during inference can be combined with batch normalization because it is a linear operation. In our experiments, the calibration procedure was performed using all columns of each layer (e.g., L is equal to two times number of output channels) every time before inference is performed on the whole test set. Adaptive batch normalization statistics update (AdaBS) technique Batch normalization is performed differently in the training and inference phases of a DNN. During the training of a DNN, the input to the batch normalization operation x i is normalized to zero mean and unit variance by computing its mean ( μ B ) and variance ( \\\\({\\\\sigma }_{{\\\\rm{B}}}^{2}\\\\) ) over a mini-batch of m images $${\\\\mu }_{{\\\\rm{B}}}=\\\\frac{1}{m}\\\\mathop{\\\\sum }\\\\limits_{i = 1}^{m}{x}_{i}$$ (4) $${\\\\sigma }_{{\\\\rm{B}}}^{2}=\\\\frac{1}{m}\\\\mathop{\\\\sum }\\\\limits_{i = 1}^{m}{\\\\left({x}_{i}-{\\\\mu }_{{\\\\rm{B}}}\\\\right)}^{2}.$$ (5) The normalized input is then scaled and shifted by γ and β . During the training phase, γ and β are learned through backpropagation. In parallel, a global running mean ( μ ) and variance ( σ 2 ) are computed by exponentially averaging μ B and \\\\({\\\\sigma }_{{\\\\rm{B}}}^{2}\\\\) respectively, over all the training batches $$\\\\mu =p\\\\cdot \\\\mu +(1-p)\\\\cdot {\\\\mu }_{{\\\\rm{B}}}$$ (6) $${\\\\sigma }^{2}=p\\\\cdot {\\\\sigma }^{2}+(1-p)\\\\cdot {\\\\sigma }_{{\\\\rm{B}}}^{2},$$ (7) where p is the momentum. After training, the estimates of the global mean and variance μ and σ 2 are then used during the inference phase. When performing forward propagation during inference, the batch normalization coefficients μ , σ 2 , γ , and β are used for normalization, scale, and shift. The calibration phase of AdaBS consists in recomputing and updating μ and σ 2 for every layer where batch normalization is present. We recompute μ and σ 2 by feeding a randomly sampled set of mini-batches from the training dataset. In recomputing μ and σ 2 , hyper-parameters such as mini-batch size ( m ) and momentum ( p ) need to be carefully tuned to achieve the best network accuracy. For AdaBS calibration, we observed that using an optimal value of the momentum is necessary to achieve good inference accuracy evolution over time. For this, we have developed an algorithm to estimate the optimal value of momentum by an empirical analysis, which is explained in Supplementary Note 3 . Based on this analysis, the formula we used to compute the optimal momentum as a function of the number of injected mini-batches n is $$p=0.01{5}^{(1/n)}.$$ (8) Using Eq. ( 8 ) to compute the momentum, we found that with a fixed mini-batch size of m = 200 images, it is sufficient to inject n = 13 mini-batches for the AdaBS calibration of the ResNet-32 network, that is approximately 5% of the CIFAR-10 training set (2600 images). The sensitivity of the accuracy to the number of images used for AdaBS calibration is shown in Supplementary Note 3 . For ResNet-34 on ImageNet, we used mini-batch size of m = 50 and n = 26 mini-batches, that is 0.1% of the ImageNet training set (1300 images). In the experiments presented in Fig. 5 , AdaBS calibration was performed for every layer before performing inference on the test set, except the last layer because it does not have batch normalization.\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"The core research idea is to improve the resilience of deep neural networks to hardware non-idealities in analog in-memory computing platforms by injecting noise into synaptic weights during software-based training, enabling accurate and robust deployment on a variety of chips with minimal hardware-specific characterization and without the need for chip-specific retraining."},"entities":{"kind":"list like","value":["deep neural networks","hardware non-idealities","analog in-memory computing platforms","noise injection","synaptic weights","software-based training","accurate deployment","robust deployment","chip-specific retraining","hardware-specific characterization","resilient neural networks","noise injection during training","analog computing platforms"],"string":"[\n \"deep neural networks\",\n \"hardware non-idealities\",\n \"analog in-memory computing platforms\",\n \"noise injection\",\n \"synaptic weights\",\n \"software-based training\",\n \"accurate deployment\",\n \"robust deployment\",\n \"chip-specific retraining\",\n \"hardware-specific characterization\",\n \"resilient neural networks\",\n \"noise injection during training\",\n \"analog computing platforms\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"5.","title":"Wang, Z. et al. 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Mater.30, 1705914 (2018).\",\n \"link\": \"https://doi.org/10.1002%2Fadma.201705914\"\n },\n {\n \"idx\": \"9.\",\n \"title\": \"Yao, P. et al. Fully hardware-implemented memristor convolutional neural network.Nature577, 641–646 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-020-1942-4\"\n },\n {\n \"idx\": \"10.\",\n \"title\": \"Yin, S. et al. Monolithically integrated RRAM- and CMOS-based in-memory computing optimizations for efficient deep learning.IEEE Micro39, 54–63 (2019).\",\n \"link\": \"https://doi.org/10.1109%2FMM.2019.2943047\"\n },\n {\n \"idx\": \"11.\",\n \"title\": \"Le Gallo, M. et al. Mixed-precision in-memory computing.Nat. Electron.1, 246–253 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41928-018-0054-8\"\n },\n {\n \"idx\": \"12.\",\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\": \"13.\",\n \"title\": \"Ambrogio, S. et al. Equivalent-accuracy accelerated neural-network training using analogue memory.Nature558, 60–67 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-018-0180-5\"\n },\n {\n \"idx\": \"17.\",\n \"title\": \"Gonugondla, S. K., Kang, M. & Shanbhag, N. R. A variation-tolerant in-memory machine learning classifier via on-chip training.IEEE J. Solid-State Circuits53, 3163–3173 (2018).\",\n \"link\": \"https://doi.org/10.1109%2FJSSC.2018.2867275\"\n },\n {\n \"idx\": \"20.\",\n \"title\": \"Moon, S., Shin, K. & Jeon, D. Enhancing reliability of analog neural network processors.IEEE Trans. Very Large Scale Integr. Syst.27, 1455–1459 (2019).\",\n \"link\": \"https://doi.org/10.1109%2FTVLSI.2019.2893256\"\n },\n {\n \"idx\": \"21.\",\n \"title\": \"Miyashita, D., Kousai, S., Suzuki, T. & Deguchi, J. A neuromorphic chip optimized for deep learning and CMOS technology with time-domain analog and digital mixed-signal processing.IEEE J. Solid State Circuits52, 2679–2689 (2017).\",\n \"link\": \"https://doi.org/10.1109%2FJSSC.2017.2712626\"\n },\n {\n \"idx\": \"25.\",\n \"title\": \"Gokmen, T., Onen, M. & Haensch, W. Training deep convolutional neural networks with resistive cross-point devices.Front. Neurosci.11, 1–22 (2017).\",\n \"link\": \"https://doi.org/10.3389%2Ffnins.2017.00538\"\n },\n {\n \"idx\": \"30.\",\n \"title\": \"An, G. The effects of adding noise during backpropagation training on a generalization performance.Neural Comput.8, 643–674 (1996).\",\n \"link\": \"https://doi.org/10.1162%2Fneco.1996.8.3.643\"\n },\n {\n \"idx\": \"34.\",\n \"title\": \"Murray, A. F. & Edwards, P. J. Enhanced MLP performance and fault tolerance resulting from synaptic weight noise during training.IEEE Trans. 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Vis.115, 211–252 (2015).\",\n \"link\": \"https://link.springer.com/doi/10.1007/s11263-015-0816-y\"\n }\n]","isConversational":false}}},{"rowIdx":11190,"cells":{"unique_id":{"kind":"string","value":"s41467-020-15506-3"},"subfield":{"kind":"string","value":"Astronomy and planetary science"},"title":{"kind":"string","value":"Plasmapause surface wave oscillates the magnetosphere and diffuse aurora"},"abstract":{"kind":"string","value":"Energy circulation in geospace lies at the heart of space weather research. In the inner magnetosphere, the steep plasmapause boundary separates the cold dense plasmasphere, which corotates with the planet, from the hot ring current/plasma sheet outside. Theoretical studies suggested that plasmapause surface waves related to the sharp inhomogeneity exist and act as a source of geomagnetic pulsations, but direct evidence of the waves and their role in magnetospheric dynamics have not yet been detected. Here, we show direct observations of a plasmapause surface wave and its impacts during a geomagnetic storm using multi-satellite and ground-based measurements. The wave oscillates the plasmapause in the afternoon-dusk sector, triggers sawtooth auroral displays, and drives outward-propagating ultra-low frequency waves. We also show that the surface-wave-driven sawtooth auroras occurred in more than 90% of geomagnetic storms during 2014–2018, indicating that they are a systematic and crucial process in driving space energy dissipation."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"The solar wind and embedded interplanetary magnetic field (IMF) play a crucial role in driving terrestrial magnetospheric energy dissipation. When the IMF has a southward component, the mass and energy in solar winds enter into and convect in the magnetosphere and are eventually released into the ionosphere and upper atmosphere, generating spectacular auroras in the Earth’s polar regions [1] , [2] . A large part of the electromagnetic energy is carried by ultra-low frequency (ULF; ~0.1 mHz to 10 Hz) waves that propagate throughout the system and couple different regions together. The ULF waves play important roles in generating quasi-periodic geomagnetic perturbations [3] and energizing energetic particles in the Earth’s radiation belts [4] , [5] . It is commonly accepted that ULF waves can be driven externally by solar wind perturbations [6] and magnetopause surface waves [7] , [8] , [9] , or internally by, for instance, plasma instabilities in the nightside magnetosphere [10] . Many auroral activities are related to the ULF waves in the magnetosphere, such as the substorm expansion phase onset [11] , the auroral arcs [12] , and the fluctuations of auroral intensity [13] , [14] . The ULF waves generated by the magnetopause surface wave propagate radially inwards [15] and penetrate deep inside the inner magnetosphere during geomagnetic storms [16] , while it is unknown whether the reverse process can occur. The inner part of the Earth’s magnetosphere, known as the plasmasphere, is full of cold (~1 eV) and dense plasmas that corotate with the planet [17] . The outer boundary of the plasmasphere is called the plasmapause, which separates two types of plasmas contents characterized by different temperatures and densities [18] , [19] . Energetic particles outside the plasmapause can precipitate into the middle- and high-latitude ionosphere along magnetic field lines and generate spectacular auroras in both hemispheres [20] . Under quiescent conditions, a longitudinally smooth equatorward boundary of the diffuse aurora is usually expected, corresponding to an azimuthally flat plasmapause boundary, which separates the hot ring current/inner plasma sheet from the cold plasmasphere [21] . During geomagnetic storms, especially in the main phase, the enhanced solar-wind-induced convection electric field can penetrate deep into the inner magnetosphere and erodes the outer part of the corotating flow to form a sharp plasmapause in radial direction [22] , [23] . On the basis of magnetohydrodynamic (MHD) theory, Chen and Hasegawa [24] predicted that impulses acting on the sharp plasmapause boundary may result in a discrete eigenmode, i.e., a plasmapause surface wave (PSW), at the plasmapause surface. Nevertheless, direct evidence of the PSW has not yet been found from in situ measurements and the function of PSW on magnetospheric dynamics remains unclear. Despite the fact that the diffuse auroral boundary is generally smooth, spectacular sawtooth-shaped large-scale undulations along the equatorward edge of the diffuse aurora (hereafter shortened to ‘sawtooth aurora’ (SA)) are observed in the afternoon-to-evening sector during geomagnetically disturbed periods [25] , [26] , [27] , [28] , [29] . Several plasma instability mechanisms [25] , [28] , [29] have been proposed to interpret the origin of SAs, but a conclusion is far from imminent due to the lack of conjugated observations of the SAs and their magnetospheric source regions. Since the SA and the plasmapause are located at similar L -shells and possibly linked by the same magnetic flux tubes, a physical connection between PSW and SA is naturally expected and yet to be investigated. Here we present a direct observational evidence of PSW using conjugated satellite and ground observations and demonstrate that it is a systematic ULF wave driver in the magnetosphere. We show that the SA on the equatorward edge of diffuse aurora generated by the PSW occur in more than 90% of geomagnetic storms, indicating that PSW is a systematic consequence of geomagnetic storms and has crucial impacts on energy dissipation in the ionosphere-magnetosphere coupling system.","references":{"indices":[1,2,3,5,6,7,8,9,10,11,12,13,14,15,16,18,19,20,21,22,23,24,25,26,28,29],"infos":[{"idx":"1","title":"Lyon, J. G. The solar wind-magnetosphere-ionosphere system.Science288, 1987–1991 (2000).","link":"https://doi.org/10.1126%2Fscience.288.5473.1987"},{"idx":"2","title":"Angelopoulos, V. et al. Tail reconnection triggering substorm onset.Science321, 931–935 (2008).","link":"https://doi.org/10.1126%2Fscience.1160495"},{"idx":"3","title":"Glassmeier, K.-H., Othmer, C., Cramm, R., Stellmacher, M. & Engebretson, M. Magnetospheric field line resonances: a comparative planetology approach.Surv. 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Res.115, A04210 (2010).","link":"https://doi.org/10.1029%2F2009JA014106"}]}},{"section":"Results","content":"Evidence of PSW Figure 1a shows the satellite locations from 13:00 to 15:30 UT and the schematic plasmaspheric configuration on 16 July 2017 during the main phase of a geomagnetic storm under the southward IMF condition (Supplementary Fig. 1 ). The Van Allen Probes (VAP) A spacecraft [30] , [31] (blue curve in Fig. 1a ) observed alternating distributions of the cold and hot plasmas in the afternoon-to-dusk sector (Fig. 1d–f ). Clear large-amplitude oscillations at the frequency of the upper hybrid resonance in the spectrogram of electric field (Fig. 1d ) indicate that the plasmaspheric electron density (PED) (see the Methods section, calculation of PED) was oscillating along the spacecraft trajectory. A comparison between the electron density and the spectra of electrons (Fig. 1e ) and protons (Fig. 1f ) further demonstrates that the hot plasmas periodically intruded into the cold plasma regions. Similar distributions were observed by the VAP-B spacecraft (red curve in Fig. 1a ) at the same time (Supplementary Fig. 2 ). The Exploration of energization and Radiation in Geospace (ERG, also called Arase) spacecraft [32] , [33] , [34] (pink curve in Fig. 1a ) also observed similar plasmas distributions in the dusk sector (Fig. 1g–i ). These data reveal that these spacecraft were passing over the plasmapause several times, implying a sawtooth-shaped plasmapause undulation, as illustrated in Fig. 1a . During this period, the Defense Meteorological Satellite Program (DMSP) F17 satellite [35] observed SA at the dusk side in both hemispheres (Fig. 1b, c ). The SA boundary (thick dotted line in Fig. 1c ) is mapped to the sawtooth-shaped plasmapause on the equatorial plane (white line in Fig. 1a ) along the modeled magnetic field lines [36] . Clearly, the sawtooth-shaped plasmapause structure is the manifestation of the PSW that was propagating sunward/westward, as will be detailed below. Fig. 1: Coordinated observations of PSW and associated SA on 16 July 2017. a Schematic diagram showing the geometry of the plasmasphere (deep green area), the auroral boundary (white thick curve), and spacecraft trajectories (VAP-A, VAP-B, and ERG spacecraft shown in blue, red, and pink, respectively) on the equatorial plane in the solar magnetic (SM) coordinate system with the Sun to the left (see the Methods section, coordinate system). The shape of the plasmapause is calculated from an empirical plasmaspheric model [37] . The clear sawtooth-shaped plasmapause structure visible in the afternoon-dusk sector indicates the PSW, which is manually added based on the wavelength of the PSW calculated in the main text. The closed circles in the spacecraft trajectories indicate temporal intervals of 1 h beginning at 13:00 UTC (indicated by the leftmost circles overlaid with crosses). Both VAP-A and VAP-B spacecraft were located in the southern hemisphere while ERG spacecraft crossed the magnetic equator at ~14:10 UTC. Open diamonds represent the plasmapause crossings by the spacecraft. Note that VAP-A and VAP-B crossed the plasmapause region roughly azimuthally while ERG crossed almost radially. b SA observed by DMSP F17 in the Northern Hemisphere (NH) at 13:27:45 UTC. c SA observed by DMSP F17 in the SH at 14:37:13 UTC. The dashed grid lines in b , c denote the altitude-adjusted corrected geomagnetic (AACGM) [38] latitudes and magnetic local time (MLT). The thick dotted line in c represents the boundary of the SA, which is projected onto the SM equatorial plane using the Tsyganenko 96 magnetic field model [36] and denoted by the white curve at the duskside plasmapause in ( a ). d Spectrogram of the electric field from the high frequency receiver of the EMFISIS instrument suite onboard VAP-A. The next two panels show the energy spectrograms of electrons e and protons f measured by the HOPE mass spectrometer onboard VAP-A, respectively. The plasmaspheric electron density (blue curve) is overlaid on ( e , f) . g – i The same as those of d – f but for the ERG spacecraft. Details of the instruments and data availability can be found in Methods (see the Methods section, data usage). Full size image Sunward/westward propagating PSW We evaluate the wave characteristics and phase relationships associated with the PSW in the field-aligned (FA) coordinate system, in which e p is along the background magnetic field (direction obtained from the 15-min sliding averaged data), e a (roughly eastward) is parallel to e p × R ( R is the radial vector pointing from the centre of the Earth toward the satellite), and e r (roughly radially outward) completes the orthogonal set. For VAP observations, the power of the perturbations of the radial magnetic field B r peaked at the frequency of ~1.5 mHz, and ~1.4 mHz for the azimuthal electric field perturbation E a (Fig. 2a ). For ERG observations, the perturbations of the magnetic field also peaked at ~1.4 mHz (Fig. 2f ). When the spacecraft arrived at the plasmapause boundary region, intense wave activities were measured. As the spacecraft left from the boundary to return to the plasmasphere, the waves quickly stopped being detected by the spacecraft (Supplementary Fig. 3 ). These measurements are in general agreement with the conception of a surface wave along the plasmapause boundary [24] . Fig. 2: Spectral analysis of the magnetic field perturbations measured by VAP and ERG. a Normalized power spectra of the perturbations of the magnetic field radial component of the magnetic field B r (black, thin for VAP-A and thick for VAP-B) and the electric field azimuthal component E a (red) measured by VAP-A. b Bandpass filtered (1.5 ± 0.5 mHz) perturbations of B r (red), field-aligned magnetic field B p (green) and E a (blue). c Bandpass filtered (1.5 ± 0.5 mHz) perturbations of the azimuthal magnetic field B a (red) and the radial electric field E r (blue). d Bandpass filtered (1.5 ± 0.5 mHz) perturbations of B r (red), B p (green) and radial ion velocity V r (blue) (see the Methods section, data usage). The background in b – d represents electron density with dark color (colorless) for high (low) density inside (outside) the plasmasphere. e Bandpass filtered (1.5 ± 0.5 mHz) perturbations of B r measured by VAP-A (red) and VAP-B (blue). f Normalized power spectra of the perturbations of B r (black), B a (red), and B p (blue) measured by ERG. The vertical line indicates a peak at 1.4 mHz. g Radial profile of the plasmaspheric electron density. h – j Bandpass filtered (1.4 ± 0.5 mHz) perturbations of B r , B a , and B p , respectively between 13:30 and 14:30 UTC. The thick vertical line indicates crossing of the magnetic equatorial plane by ERG with the left side in the southern hemisphere (inside plasmapause) and the right side in the northern hemisphere (outside plasmapause). Full size image Clear coherent phase relationships are found between the magnetic and electric fields in the bandpass-filtered signals (Fig. 2b–d ): B r was nearly in antiphase with E a (−164 ± 9°) and was roughly orthogonal to the field-aligned magnetic perturbation B p (108 ± 22°); B p had a ~90° phase difference with both E a (89 ± 15°) and the radial ion velocity V r (89 ± 13°); and B a was roughly in quadrature with E r (83 ± 15°). However, the phase difference was not stable at a value close to 90°. By using the simultaneously measured B r from VAP-A and VAP-B (Fig. 2e ), it is determined that the azimuthal wavelength was 10° ± 0.3°, the azimuthal propagating speed was 0.010 ± 0.001° s −1 , and the azimuthal mode number m was 36 ± 1 for the wave (see the Methods section, determination of propagating speed). The different wave amplitudes observed by VAP-A and VAP-B in Fig. 2e might be caused by either the variation of relative distance between the spacecraft and the plasmapause surface, or the time varying dynamic evolution of the PSW during sunward/westward propagation (Supplementary Fig. 3 ). The antiphase of n e with B r in Fig. 2b indicates that B r was in antiphase with the geocentric distance of the plasmapause. The magnetic perturbations exhibited a 180° phase difference on either side of the magnetic equator and also on either side of the plasmapause boundary (opposite amplitude in Fig. 2h–j ), meaning that the waves at the plasmapause region are fundamental mode PSW eigenmodes [24] , [39] . The presence of both poloidal oscillations of B r and E a and toroidal oscillations of B a and E r shows that the observed PSW manifested a mixed poloidal and toroidal modes [40] . The finite value of m leads to the coexistence and joint action of these two modes. At such a moderate high- m (~36), the poloidal perturbations are partly field-aligned guided, and the toroidal perturbations have a compressional feature [41] . The B p component was 90° out-of-phase with E a (Fig. 1b ) and V r was 90° lagged behind B p (Fig. 1d ), indicating the presence of fast compressional MHD wave modes. It is known that the slow mode wave can also be coupled with shear Alfvén waves, particularly in high-β plasma environment [42] . In the plasmapause region, however, the β parameter is very low ( ≪ 0.01), the observed features, therefore, suggest that the waves associated with the PSW are coupling between fast waves with shear Alfvén waves in consideration of the sunward/westward propagation of PSW. Conjugated sunward/westward propagating SA Owing to the modulation of the plasmapause by the PSW, the energetic electrons and protons that intruded into the low plasmaspheric density regions (Fig. 1d, g ) were possibly scattered by the electron electrostatic cyclotron harmonic (ECH) waves [20] and precipitated into the polar upper atmosphere to generate diffuse aurora. Images from the SSUSI instrument onboard the DMSP F17 satellite at two selected times (Fig. 1b, c , see Supplementary Fig. 4 for the entire image sequence) displayed giant undulations at the equatorward boundary of the diffuse aurora in both hemispheres, i.e., SAs, which were collocated with the sawtooth-shaped plasmapause on the equatorial plane via field line mapping, as shown in Fig. 1a . Within the limitations of the temporal evolution and spatial coverage of SSUSI, it is approximately estimated that the SA was possibly initiated between 11:15 and 11:45 UTC and ended at ~15:10 UTC (Supplementary Fig. 4b, f ). The DMSP F17 and F18 satellites [35] passed through the polar region successively in ~2 min, which is perfect for analysing the dynamical evolution of the SA. With a cross-correlation analysis [43] between the emission intensity profiles at AACGM latitudes of 58.5° in the Northern Hemisphere (Fig. 3a, b ) and −60.0° in the Southern Hemisphere (Fig. 3d, e ), we determined that the SA propagated sunward/westward at a phase speed of 0.01 ± 0.001° s −1 and the azimuthal wavelength was about 6.4°–10.2° for the SA (Fig. 3c, f ). In addition, the azimuthal wavelength was shown to decrease with increasing MLT. The azimuthal wavelength and sunward/westward phase speed of the SA found from the DMSP data are in agreement with the PSW (~9.7°–10.5° and 0.01° s −1 ) obtained based on the VAP measurements, unambiguously proving that the SA was driven by the PSW; in other words, the SAs are the optical atmospheric/auroral manifestation of the PSWs. Fig. 3: Determination of the phase speed and azimuthal wavelength of SA. a Auroral image obtained by the DMSP F18 satellite for 13:23–13:30 UTC in the NH with the SA captured at 13:25:17 UTC. b Auroral image obtained by the DMSP F17 satellite between 13:25–13:32 UTC in the NH with the SA captured at 13:27:45 UTC. c Emission intensity variations of the SA at the AACGM latitude of 58.5° as indicated by the blue dashed line in ( a ) and red dashed line in ( b ). d Auroral image obtained by the DMSP F18 satellite between 14:30 and 14:37 UTC in the SH with the SA captured at 14:34:48 UTC. e Auroral image obtained by the DMSP F17 satellite between 14:32 and 14:39 UTC in the SH with the SA captured at 14:37:13 UTC. f Emission intensity variations of the SA at the AACGM latitude of −60° as indicated by the blue dashed line in ( d ) and red dashed line in ( e ). Full size image Poleward and sunward/westward propagating ULF waves Apart from driving the SA, the PSW generated outward-propagating ULF waves outside the plasmapause, giving rise to the field line resonance (FLR) [3] and driving ULF geomagnetic pulsations. Figure 4 shows the analysis of magnetic perturbations observed by ground-based station chains (Fig. 4a ) from the International Monitor for Auroral Geomagnetic Effects (IMAGE) magnetometer array [44] . The power spectra of the magnetic field perturbations showed several significant signals at ~0.6 mHz, ~1.1 mHz, ~1.4 mHz, and ~2.0 mHz, with lower frequency pulsations occurring at higher latitudes (Supplementary Fig. 5 ). The wave amplitude of the N (north) component in the frequency of 1.4 ± 0.5 mHz (the closest to the frequency of PSW-associated waves in Fig. 2a ) maximized at a magnetic latitude of ~66° (MAS station), while the phase changed by nearly 180° over the amplitude maximum (Fig. 4b ), indicating the driving of FLR outside the plasmapause by the PSW. The 1.4 ± 0.5 mHz bandpass filtered N (Fig. 4c ) and Z (Fig. 4d ) components of the geomagnetic perturbations showed clear poleward propagation (i.e., radially outward in the magnetosphere) of the ULF waves starting from the OUJ station (at the AACGM latitude of 61.42° and longitude of 105.46°), which are the closest to the SA (Fig. 4c, d ), i.e., the counterpart to the sawtooth-shaped plasmapause region in the magnetosphere. This is apparently contrary to the inward (or equatorward) propagation of the ULF waves generated by external sources (e.g., magnetopause [15] or in the near-Earth magnetotail [14] , [45] ). The 1.4 ± 0.5 mHz bandpass filtered E component of the geomagnetic perturbations (Fig. 4e ) showed clear sunward/westward propagation of the ULF waves with a speed of ~0.012° s −1 and an azimuthal mode number of 36.8 ± 0.6 (see the Methods section, determination of m on ground). Both the propagating speed and the azimuthal mode number are consistent with those of the PSW and the SA. Fig. 4: ULF waves and FLR observed on the ground. a Stations selected from the IMAGE magnetometer array with the background auroral image observed by DMSP F17 satellite at 13:27:45 UTC (see also Fig. 1b ). The blue chain is aligned in almost the same magnetic longitudes of ~106 ± 4° with the magnetic latitude increasing from 54.5° at TAR to 75.1° at LYR. The red chain is aligned in almost the same magnetic latitudes of ~61.5 ± 0.5° with the magnetic longitude increasing from 93.3° at RVK to 106.1° at OUJ. The black/cyan arrow represents the footpoints of VAP-A/ERG satellite moving towards low/high latitudes between 13:40 UT and 15:00 UT with the diamond marking plasmapause crossing. b Amplitude (black) and phase difference (blue) of the N component of the blue chain plotted as a function of geomagnetic latitude. c Bandpass filtered (1.4 ± 0.5 mHz) N component (north) of the geomagnetic field perturbations. d Bandpass filtered (1.4 ± 0.5 mHz) Z component (vertical) of the geomagnetic field perturbations. Clear poleward propagation of the waves is highlighted by the dashed lines. e Bandpass filtered (1.4 ± 0.5 mHz) E component (east) of the geomagnetic field perturbations. Clear sunward/westward propagation of the waves is highlighted by the dashed lines. Full size image"},{"section":"Discussion","content":"The 16 July 2017 geomagnetic storm provided an opportunity to identify the PSW and its consequences of oscillating both the magnetosphere and the diffuse aurora. The enhanced convection electric field during the storm made the plasmapause in the afternoon-dusk sector very sharp. Inhomogeneities in the plasma density and magnetic field lead to the coupling between the shear Alfvén waves and the magnetosonic waves on closed field lines. Field oscillations in the magnetosphere usually have a continuous spectrum. Nevertheless, solution of the MHD wave equation shows the coupling between the surface wave (i.e., evanescent compressional Alfvén wave) and the shear Alfvén wave if there exists a sharp change in Alfvén speed (related to changes in magnetic field and plasma mass density), and a localized perturbation propagating along the plasma surface across which the Alfvén speed changes will suddenly possess a discrete eigenfrequency [24] . Since the resonant absorption is negligibly small [46] , a localized coupled ULF wave corresponding to the discrete eigenmode can thus be excited at such a location by an impulse which has a frequency spectrum that covers this eigenfrequency. Observations in both the topside ionosphere and the plasmasphere showed that the radial width of the plasmapause was ~0.1–0.2 R E (see Supplementary Figs. 6 , 7 ), which was much smaller than the transverse wave scale of ~0.8 R E for the PSW with m -value 36 at a geocentric distance of 4.5 R E , satisfying the requirement of the interface width in exciting surface waves [47] . The sharp discontinuity in the density led the magnetosonic wave to be undamped [24] , [46] , [48] , allowing excitation of a standing surface eigenmode, i.e., the PSW propagating sunward/westward at the plasmapause boundary and along the field lines to both the northern and southern polar ionosphere. Using the plasmapause crossing measurements, it is estimated that the frequency of the fundamental poloidal wave is 1.35–2.11 mHz, close to 1.5 mHz (see the Methods section, estimation of eigenfrequency). Above estimations indicate that the configuration of the plasmapause during the storm main phase could provide favorable conditions to excite PSW. The PSW could be excited by external perturbations such as the sudden enhancement of the plasma pressure in the duskside ring current/plasma sheet owing to fast storm time injections. The driving sources could be either monochromatic with a frequency similar to the eigenfrequency of the plasmapause surface or impulsive and broadband whose frequency range covers the eigenfrequency of the plasmapause surface. Between 11:00 and 11:30 UT, strong ion/electron injections were observed (Supplementary Figs. 8 , 9 ). The power spectrum density of the energetic ion/electron flux between 11:00 and 13:00 UT exhibits a clear peak at 1.5 mHz (Supplementary Fig. 10 ), close to the eigenfrequency of the plasmapause surface. Such a periodically enhancing particle flux led to impinging of periodically varying plasma pressure on the plasmapause surface and may actually excite the PSW at 1.5 mHz. In addition, it is noted that other types of pressure variations, such as impulsive injection (like a delta function) and continuous injection (like a step function), are both broadband in frequency domain. Take the magnetopause surface wave for an example, the magnetosheath jet’s total pressure is impulsive and broadband [9] . Therefore, impulsive or continuous injection may both contribute to the excitation of the PSW. The proposed excitation process of PSW is similar to the magnetopause surface waves excited by the sudden enhancement of solar wind pressure [6] , interplanetary shocks [49] , or magnetosheath jets [9] . The free energy should be continuously supplied to maintain the long-term evolution and propagation of the PSW. Consistency between the durations of the PSW (or SAs) and the enhanced hot plasma pressure (Supplementary Fig. 9f ) indicates that the free energy was provided by the hot plasma injections. Other internal instabilities, such as the ion drift resonances which can excite fundamental poloidal waves [50] , [51] and the Kelvin-Helmholtz instability which can excite magnetopause surface waves [7] , [8] , could certainly play a role at some point in the excitation of the ULF waves, in addition to the PSW-associated ULF waves. Drift wave perturbations or the effect from ULF waves in the vicinity of the plasmapause might also be possible interpretations to the observed waves. Multi-satellite measurements show that the observed waves propagated along the plasmapause surface with a fixed frequency, excited outward-propagating ULF waves and drove FLR outside the plasmapause. These characteristics are similar to that of the magnetopause surface waves [8] , [9] which excite earthward-propagating ULF waves and drive FLR in the magnetosphere. In these regards, we suggest that the PSW concept should be a reasonable interpretation for the observations. The equatorial plasmapause became sawtooth-shaped due to the modulation of the PSW. The low-density region of PSW was intruded by hot plasma. Scattering and precipitation of the hot plasma by waves like ECH then generated SA in the afternoon-evening sector, which had the same azimuthal wavelength and sunward/westward phase speed as the PSW. In addition, the local lower ratio of electron plasma frequency to gyrofrequency in the night sector during disturbed periods also leads to much more efficient scattering by chorus waves (effective for ECH as well) just outside the plasmapause [52] , [53] , [54] . The SAs that occur at the equatorial boundary of the diffuse aurora on the dusk side are different from other quasi-periodic auroral structures such as the torch auroral structures (or omega band) [55] and the pulsating auroral forms generated by giant pulsations [56] , both of which occur in the auroral oval and in the midnight to morning sector, although similar wave activities are found to exist during these different auroral morphologies. Like the magnetopause surface wave that generates earthward-propagating ULF waves and drives FLR in the magnetosphere [8] , the PSW generated outward-propagating ULF waves and drove FLR outside the plasmapause. Such a process has not yet been observed and interpreted in the existing literature. The amplitude and phase structures on the ground are different from the giant pulsations though they have similar wave number and structures [57] . The amplitude and phase structures observed here are due to the radially outward propagation of the ULF waves from the plasmapause boundary and resonance with local field lines on the duskside, while the giant pulsations are latitudinally localized and occur almost exclusively on the morningside with peak occurrence in the postmidnight sector [58] . The ground signatures of surface waves are not well understood, including both the PSW and the magnetopause surface wave [9] . In this study, we show observational evidence that PSW can exhibit different signals to the conjugated ground magnetometer data. Further theoretical investigations, simulations, and observations are necessary to characterize the occurrence rate and properties of PSW. Although the PSWs are hardly captured by in situ observations since satellites are required to cross the plasmapause region at a specific time and at a certain location, the SAs that visualize the PSW are frequently observed during geomagnetic storms. Besides the DMSP satellites, the recently launched Chinese Fengyun-3D (FY-3D) satellite in a low-Earth orbit can capture high-resolution global images of SAs (Fig. 5 ) with the on-board wide-field auroral imager (WAI) [59] . We have surveyed the aurora data recorded by both DMSP and FY-3D satellites during geomagnetic storms (the minimum disturbance storm-time (Dst) index less than -40 nT) from 2014 to 2018 (listed in Supplementary Table 1 ). The probability of occurrence of SAs is found to be greater than 90% during geomagnetic storms (94 out of 103, as shown in Supplementary Table 1 ). For the remaining 9 storms, it is not clear whether these auroral structures are present or not due to the spatial coverage and temporal evolution of the auroral images. Furthermore, coordinated ground-based geomagnetic data were available in the dusk sector (MLT = 16–19 h) for 24 events of the 94 observed SAs. The geomagnetic pulsations occurred simultaneously with the SAs and propagated essentially radially outward when mapping to the magnetic equator and the amplitude peaked outside the plasmapause (Supplementary Fig. 11 ). These results definitely indicate that the PSWs, as well as the SAs, are systematic and crucial consequences of geomagnetic storms. The reconfiguration and energy redistribution processes in the magnetosphere and ionosphere associated with the PSW appear to be common and regular during the geomagnetically disturbing times, and theoretical and statistical studies are needed in the future for better understanding of the rules that govern these processes. Fig. 5: SAs observed by FY-3D WAI during four geomagnetic storms in 2018. All the images observed between a 22:11–22:32 UTC on 14 March 2018, b 22:31–22:48 UTC on 5 May 2018, c 20:53–21:15 UTC on 18 March 2018, and d 23:54–00:15 UTC on 25 August 2018 are projected onto a reference sphere at a height of 110 km in the AACGM–MLT coordinate system. The SAs are highlighted by the red arrows. Detailed structural parameters of the SAs are listed in Table 1 . Full size image The high resolution auroral images from the FY-3D satellite show that the AACGM latitudes of crest, azimuthal wavelengths, and crest-to-trough amplitudes of the SAs vary with the strengths (i.e., the Dst values) of storms (Table 1 ), implying that the characteristics of the PSWs also vary with geomagnetic activity and that the following magnetospheric and ionospheric effects will also be different. Although it remains unclear what mechanisms determine the azimuthal wavelength (i.e., azimuthal mode number) of the PSW from a theoretical view, these observations could provide important implications to future theoretical investigations on this topic. Understanding the factors (e.g., configuration of the plasmapause boundary and the plasma pressure variations outside the plasmapause during storms) that trigger and control the generation and evolution of the PSW is critical in establishing the generation process. In addition, the wave-particle interactions that lie behind this process are key to understanding the resultant energy transfer and auroral activities. Table 1 Structural parameters of the SAs extracted from Fig. 5 . Full size table The generation of surface waves and ULF waves are fundamental plasma processes in space environments and can occur in other planetary magnetospheres [3] , [60] , [61] . As is well-known in the space of giant planets, the corotation breakdown (the cause of the plasmapause in the Earth) is known to be fundamental to the dynamics of the rapid rotating planetary magnetospheres (e.g., Jupiter and Saturn) [62] , [63] with the brightest auroral emissions associated with this process [64] . Therefore, the processes identified along terrestrial plasmapause boundary layer may be even more critical to the environments at rapidly rotating planets and may provide a crucial direction for investigations."},{"section":"Methods","content":"Coordinate systems The geocentric solar magnetospheric (GSM) coordinate system is defined as follows [65] . The x -axis of GSM points from the Earth to the Sun, the z -axis is the projection of dipole axis on geocentric solar ecliptic (GSE) yz plane (the x -axis of GSE points from the Earth to the Sun, the z -axis of GSE points to the ecliptic north pole, the y -axis of GSE completes the right-handed system), and the y -axis of GSM completes the right-handed system. The y -axis of the solar magnetic (SM) system is perpendicular to the plane containing the Sun-Earth line and the dipole axis, the z -axis is along the dipole axis, and the x -axis completes the right-handed system. The modified GSE (mGSE) coordinate system is a near GSE system for Van Allen Probes (VAP). The x -axis of mGSE is the spin axis unit vector in GSE coordinates, the y -axis is anti-parallel to the cross product between the x -axis of mGSE and the y -axis of GSE, and the z -axis of mGSE completes the right-handed system. Altitude-adjusted corrected geomagnetic (AACGM) [38] coordinates are an extension of corrected geomagnetic coordinates that more accurately represent the actual magnetic field. In AACGM coordinates points along a given magnetic field line are given the same coordinates and are thus a better reflection of magnetic conjugacy. Data usage Observations reported in this paper are taken from the VAP-A and VAP-B spacecraft, the DMSP F17 and F18 satellites, the ERG spacecraft, the FY-3D satellite, the Time History of Events and Macroscale Interactions during Substorms (THEMIS) E spacecraft, the Magnetospheric Multiscale (MMS) mission, the Geostationary Operational Environmental Satellite (GOES) 15 satellite, the ground-based IMAGE magnetometer array, and the SuperMAG database. Detailed information is introduced below. The spectrogram (6 s resolution, 10 kHz to 500 kHz) from the high frequency receiver of the Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS) [30] onboard VAP and the spectrogram (1 s resolution, 10 kHz to 10 MHz) from the high frequency analyzer of the Plasma Wave Experiment (PWE) [32] onboard ERG are used to determine the plasmaspheric electron density. The magnetic field (1 s resolution in GSM coordinates) from the fluxgate magnetometer of EMFISIS [30] and the electric field (32 sample/s in mGSE coordinates) from the Electric Field and Waves (EFW) instrument [66] onboard VAP and the magnetic field (8 s resolution in GSM coordinates) from the Magnetic Field Experiments (MGF) [67] onboard ERG are used together to evaluate the wave characteristics associated with the plasmapause surface wave. For VAP spacecraft, the spin axis electric field component ( E x ) is estimated using the assumption that that E • B = 0 or the parallel electric field is zero. This is used most frequently for large-scale convection electric fields, MHD structures and ULF waves, and small-scale waves for which perpendicular electric fields are larger than parallel. Then the ion velocity is calculated by V = E × B / B 2 . The omni-direction energy flux data from the Helium, Oxygen, Proton, and Electron (HOPE) mass spectrometer [31] onboard VAP, from the low-energy particle experiments–electron analyzer (LEP-e) [33] and the low-energy particle experiments–ion analyzer (LEP-i) [34] onboard ERG, from the Fly’s Eye Energetic Particle Sensor (FEEPS) [68] and the Hot Plasma Composition Analyzer (HCPA) [69] onboard MMS-1, and from the Energetic Particle Sensor (EPS) [70] onboard GOES-15 are used to evaluate the distributions and evolutions of hot plasma. The auroral disk images in N 2 Lyman–Birge–Hopfield (LBH) bands from the Special Sensor Ultraviolet Spectrographic Imager (SSUSI) [35] onboard DMSP F17 and F18 satellites are used to acquire the parameters of the SA. All the auroral images are projected onto a reference sphere at a height of 110 km. Both the DMSP F17 and F18 satellites orbit the Earth in a sun-synchronous orbit with a period of ~102 min and two disk images of aurora are obtained in each orbit. All the auroral images during the PSW event are shown in Supplementary Fig. 4 . On 16 July 2017, the DMSP F17 and F18 satellites passed the same polar region successively with a time difference of ~2 min, and therefore, only the F17 images are shown in Supplementary Fig. 4 . The densities of O + , H + , and He + measured by the Retarding Potential Analyzer (RPA) [71] on board the DMSP satellite are used to calculate the latitudinal plasma mass density profiles, which can reflect the radial profile of the plasma mass density in the plasmasphere. Examples are shown in Supplementary Fig. 7 . The wide-field auroral imager (WAI) [59] onboard FY-3D satellite, which was launched on 15 November 2017 into a sun-synchronous orbit at an altitude of ~840 km with a period of ~102 min, also provided auroral disk images in N 2 LBH bands but with larger field-of-view that can capture the global high-resolution structure of the SA (Fig. 5 ). The geomagnetic field data (10 s resolution) from the ground-based station chains (latitudinal chain from low to high latitude: TAR, NUR, HAN, OUJ, RAN, PEL, MUO, MAS, SOR, BJN, HOP, and LYR; longitudinal chain from west to east: RVK, LYC, and OUJ) of the IMAGE magnetometer array [44] and the geomagnetic field data (1 min resolution) from the ground-based station chain (from low to high latitude: OTT, T51, T30, T52, T45, T31, T44, T46, and T47) of the SuperMAG database [72] are used to evaluate the geomagnetic pulsations on the ground. The latitudinal IMAGE chain is aligned in almost the same magnetic longitudes of ~106 ± 4° (in the dusk sector at ~15:00 UTC) with the magnetic latitude increasing from 54.5° at TAR to 75.1° at LYR. The longitudinal IMAGE chain is aligned in almost the same magnetic latitudes of ~61.5 ± 0.5° with the magnetic longitude increasing from 93.3° at RVK to 106.1° at OUJ, and the SuperMAG chain is aligned around magnetic longitudes of 0 ± 4° (in the dusk sector at ~23:00 UT) with the magnetic latitude increasing from 55.0° at OTT to 71.5° at T47. All the geomagnetic field data are presented in the NEZ frame, in which horizontal components N and E point geomagnetically north and east, respectively, and Z is the vertical component. The original data are all detrended by subtracting the 1-h sliding averages. Calculation of plasmaspheric electron density The electron density ( n e , in cm –3 ) measured by VAP spacecraft [73] is calculated by n e = ( f 2 UHR – f 2 ce )/8980 2 , where f UHR is the upper hybrid resonance (UHR) frequency in Hz identified from the frequency-time spectrogram of electric field of the EMFISIS instrument, f ce = eB / m e is the electron cyclotron frequency in Hz, B is the strength of the magnetic field simultaneously measured by the EMFISIS instrument, and m e is the electron mass. The method for the ERG spacecraft is the same. The inferred electron densities are shown in Fig. 1e , f and h , i . For the THEMIS-E satellite, the spacecraft potential (refers to the potential of the spacecraft body relative to the ambient plasma) measured by the electric field instrument (EFI) [74] and the electron thermal velocities measured by the electrostatic analyzer (ESA) [75] are used to calculate the electron density, which is used to evaluate the plasmapause configuration (Supplementary Fig. 6 ). Determination of the azimuthal propagating speed of PSW We define the azimuthal wavelength of the PSW as λ PSW and the azimuthal propagating speed as v PSW . Between 13:50 and 14:40 UTC, the longitudinal separation between VAP-A and VAP-B is d λ sat = 17.3 ± 0.3° and the azimuthal angle difference between the electro density peaks observed by VAP-A is d λ peak = 3.0 ± 0.2° with a time interval of dt = 725 ± 25 s (Fig. 2b ). Since VAP-A is closer to dusk than VAP-B, the phase difference indicates a sunward/westward propagating wave, and λ PSW should be less than d λ sat and greater than d λ peak . The phase of B r,VAP-A leading B r,VAP-B by 90 ± 1° between 13:45 and 14:15 UTC (Fig. 2e ) indicated that d λ sat = (3/4 + n ) λ PSW ( n = 0, 1, 2, …). The loss of phase after 14:15 in Fig. 2f is because the VAP-B spacecraft had entered the plasmasphere and was far away from the plasmapause region. For n = 0, we get an azimuthal mode number m ≈ 16 and λ PSW = 22.5°, which dissatisfies the limitation on λ PSW . When n = 1, we get m = 36 ± 1 and λ PSW = 10 ± 0.3°, which satisfies the limitation on λ PSW . According to the relationship λ PSW = d λ peak + v PSW × dt , we obtain the sunward/westward propagating speed v PSW = 0.01 ± 0.001° s −1 . Both the azimuthal wavelength and the sunward/westward propagating speed showed agreement between the PSW (~9.7°–10.5° and 0.01° s −1 ) and the SA (~6.4°–10.4° and 0.01° s −1 ), unambiguously proving that the SAs are driven by the PSWs, i.e., the SA is the optical manifestation of the PSW. Because the PSW was observed at an earlier MLT than the SA, and the wavelength of the PSW is generally greater than that of the SA, which is consistent with the decrease in SA wavelength with increasing MLT (Fig. 3 ). Determination of the azimuthal wave number on the ground The three stations (OUJ, LYC, and RVK) are used to calculate the m -value of the sunward/westward propagating ULF waves on the ground. The azimuthal mode number m of the ULF waves is calculated by m = d φ / d ϕ , where d φ is the phase angle difference of the ULF waves between two stations and d ϕ is the magnetic longitude separation between two stations. Using cross-phase analysis between station pairs (OUJ-LYC and OUJ-RVK, d ϕ OUJ-LYC = 6.85°, d ϕ OUJ-RVK = 12.83°, d φ OUJ-LYC = 248°, and d φ OUJ-RVK = 480°) during 13:00 UT and 13:30 UT, the m -value is calculated to be 36.8 ± 0.6, consistent with the m -value of the PSW. The averaged time difference of the wave peaks and averaged magnetic longitude difference between the station pairs are dt = ~400 s and d ϕ = 6.42°, respectively. Considering the Earth’s eastward rotating speed of 0.0042° s −1 , the sunward/westward propagating speed of the ULF waves is ~0.012° s −1 . It is noted that the direction of ULF wave propagation can be affected in the data by the shape of the plasmapause, and the propagation direction and speed can be more precisely determined if more station chains at different latitudes are available at the same time. Estimation of the eigenfrequency at the plasmapause According to the dispersion equation given by Chen and Hasegawa [24] , the wave propagation speed along the magnetic field line at the plasmapause is v = 2 1/2 v A , where v A is the Alfvén speed of the magnetic field line inside the plasmapause boundary. According to THEMIS-E satellite measurement (Supplementary Fig. 6 ), the strength of the magnetic field is B = ~400 nT and the electron number density (equivalent to total number density of H + , He + , and O + ) is ~1500 cm −3 inside the plasmapause boundary. Since no ion composition measurements were available in the magnetosphere during this storm, typical values of relative ion concentrations in literature are considered, with the relative concentration of O + between 5 and 30% and He + between 5 and 10% [17] . The plasma mass density is estimated to be 4.54 × 10 −18 ~1.11 × 10 −17 kg m −3 and the resultant v is 150 ~ 240 km s −1 . The length of the field line Λ at the plasmapause is calculated to be ~5.6 × 10 4 km with the Tsyganenko 96 magnetic field model. Taking into account that half a wavelength equals to Λ for the fundamental poloidal wave, we get the fundamental frequency ω = v /(2 Λ ) = 1.35 ~ 2.11 mHz, close to 1.5 mHz. The above estimations indicate that the configuration of the plasmapause during the storm main phase supports the excitation and generation of the PSW."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"The solar wind and embedded interplanetary magnetic field (IMF) play a crucial role in driving terrestrial magnetospheric energy dissipation. When the IMF has a southward component, the mass and energy in solar winds enter into and convect in the magnetosphere and are eventually released into the ionosphere and upper atmosphere, generating spectacular auroras in the Earth’s polar regions [1] , [2] . A large part of the electromagnetic energy is carried by ultra-low frequency (ULF; ~0.1 mHz to 10 Hz) waves that propagate throughout the system and couple different regions together. The ULF waves play important roles in generating quasi-periodic geomagnetic perturbations [3] and energizing energetic particles in the Earth’s radiation belts [4] , [5] . It is commonly accepted that ULF waves can be driven externally by solar wind perturbations [6] and magnetopause surface waves [7] , [8] , [9] , or internally by, for instance, plasma instabilities in the nightside magnetosphere [10] . Many auroral activities are related to the ULF waves in the magnetosphere, such as the substorm expansion phase onset [11] , the auroral arcs [12] , and the fluctuations of auroral intensity [13] , [14] . The ULF waves generated by the magnetopause surface wave propagate radially inwards [15] and penetrate deep inside the inner magnetosphere during geomagnetic storms [16] , while it is unknown whether the reverse process can occur. The inner part of the Earth’s magnetosphere, known as the plasmasphere, is full of cold (~1 eV) and dense plasmas that corotate with the planet [17] . The outer boundary of the plasmasphere is called the plasmapause, which separates two types of plasmas contents characterized by different temperatures and densities [18] , [19] . Energetic particles outside the plasmapause can precipitate into the middle- and high-latitude ionosphere along magnetic field lines and generate spectacular auroras in both hemispheres [20] . Under quiescent conditions, a longitudinally smooth equatorward boundary of the diffuse aurora is usually expected, corresponding to an azimuthally flat plasmapause boundary, which separates the hot ring current/inner plasma sheet from the cold plasmasphere [21] . During geomagnetic storms, especially in the main phase, the enhanced solar-wind-induced convection electric field can penetrate deep into the inner magnetosphere and erodes the outer part of the corotating flow to form a sharp plasmapause in radial direction [22] , [23] . On the basis of magnetohydrodynamic (MHD) theory, Chen and Hasegawa [24] predicted that impulses acting on the sharp plasmapause boundary may result in a discrete eigenmode, i.e., a plasmapause surface wave (PSW), at the plasmapause surface. Nevertheless, direct evidence of the PSW has not yet been found from in situ measurements and the function of PSW on magnetospheric dynamics remains unclear. Despite the fact that the diffuse auroral boundary is generally smooth, spectacular sawtooth-shaped large-scale undulations along the equatorward edge of the diffuse aurora (hereafter shortened to ‘sawtooth aurora’ (SA)) are observed in the afternoon-to-evening sector during geomagnetically disturbed periods [25] , [26] , [27] , [28] , [29] . Several plasma instability mechanisms [25] , [28] , [29] have been proposed to interpret the origin of SAs, but a conclusion is far from imminent due to the lack of conjugated observations of the SAs and their magnetospheric source regions. Since the SA and the plasmapause are located at similar L -shells and possibly linked by the same magnetic flux tubes, a physical connection between PSW and SA is naturally expected and yet to be investigated. Here we present a direct observational evidence of PSW using conjugated satellite and ground observations and demonstrate that it is a systematic ULF wave driver in the magnetosphere. We show that the SA on the equatorward edge of diffuse aurora generated by the PSW occur in more than 90% of geomagnetic storms, indicating that PSW is a systematic consequence of geomagnetic storms and has crucial impacts on energy dissipation in the ionosphere-magnetosphere coupling system.\",\n \"references\": {\n \"indices\": [\n 1,\n 2,\n 3,\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 18,\n 19,\n 20,\n 21,\n 22,\n 23,\n 24,\n 25,\n 26,\n 28,\n 29\n ],\n \"infos\": [\n {\n \"idx\": \"1\",\n \"title\": \"Lyon, J. G. The solar wind-magnetosphere-ionosphere system.Science288, 1987–1991 (2000).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.288.5473.1987\"\n },\n {\n \"idx\": \"2\",\n \"title\": \"Angelopoulos, V. et al. Tail reconnection triggering substorm onset.Science321, 931–935 (2008).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1160495\"\n },\n {\n \"idx\": \"3\",\n \"title\": \"Glassmeier, K.-H., Othmer, C., Cramm, R., Stellmacher, M. & Engebretson, M. Magnetospheric field line resonances: a comparative planetology approach.Surv. Geophys.20, 61–109 (1999).\",\n \"link\": \"https://doi.org/10.1023%2FA%3A1006659717963\"\n },\n {\n \"idx\": \"5\",\n \"title\": \"Zong, Q. G., Rankin, R. & Zhou, X. Z. The interaction of ultra-low-frequency pc3-5 waves with charged particles in Earth’s magnetosphere.Rev. Mod. Plasma Phys.1, 10 (2017).\",\n \"link\": \"https://link.springer.com/doi/10.1007/s41614-017-0011-4\"\n },\n {\n \"idx\": \"6\",\n \"title\": \"Claudepierre, S. G., Hudson, M. K., Lotko, W., Lyon, J. G. & Denton, R. E. Solar wind driving of magnetospheric ULF waves: Field line resonances driven by dynamic pressure fluctuations.J. Geophys. 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Spatial and temporal characteristics of giant undulations.Geophys. Res. Lett.21, 2673–2676 (1994).\",\n \"link\": \"https://doi.org/10.1029%2F94GL02240\"\n },\n {\n \"idx\": \"28\",\n \"title\": \"Lewis, W. S. et al. Duskside auroral undulations observed by IMAGE and their possible association with large-scale structures on the inner edge of the electron plasma sheet.Geophys. Res. Lett.32, L24103 (2005).\",\n \"link\": \"https://doi.org/10.1029%2F2005GL024390\"\n },\n {\n \"idx\": \"29\",\n \"title\": \"Henderson, M. G. et al. Start-to-end global imaging of a sunward propagating, SAPS associated giant undulation event.J. Geophys. Res.115, A04210 (2010).\",\n \"link\": \"https://doi.org/10.1029%2F2009JA014106\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Evidence of PSW Figure 1a shows the satellite locations from 13:00 to 15:30 UT and the schematic plasmaspheric configuration on 16 July 2017 during the main phase of a geomagnetic storm under the southward IMF condition (Supplementary Fig. 1 ). The Van Allen Probes (VAP) A spacecraft [30] , [31] (blue curve in Fig. 1a ) observed alternating distributions of the cold and hot plasmas in the afternoon-to-dusk sector (Fig. 1d–f ). Clear large-amplitude oscillations at the frequency of the upper hybrid resonance in the spectrogram of electric field (Fig. 1d ) indicate that the plasmaspheric electron density (PED) (see the Methods section, calculation of PED) was oscillating along the spacecraft trajectory. A comparison between the electron density and the spectra of electrons (Fig. 1e ) and protons (Fig. 1f ) further demonstrates that the hot plasmas periodically intruded into the cold plasma regions. Similar distributions were observed by the VAP-B spacecraft (red curve in Fig. 1a ) at the same time (Supplementary Fig. 2 ). The Exploration of energization and Radiation in Geospace (ERG, also called Arase) spacecraft [32] , [33] , [34] (pink curve in Fig. 1a ) also observed similar plasmas distributions in the dusk sector (Fig. 1g–i ). These data reveal that these spacecraft were passing over the plasmapause several times, implying a sawtooth-shaped plasmapause undulation, as illustrated in Fig. 1a . During this period, the Defense Meteorological Satellite Program (DMSP) F17 satellite [35] observed SA at the dusk side in both hemispheres (Fig. 1b, c ). The SA boundary (thick dotted line in Fig. 1c ) is mapped to the sawtooth-shaped plasmapause on the equatorial plane (white line in Fig. 1a ) along the modeled magnetic field lines [36] . Clearly, the sawtooth-shaped plasmapause structure is the manifestation of the PSW that was propagating sunward/westward, as will be detailed below. Fig. 1: Coordinated observations of PSW and associated SA on 16 July 2017. a Schematic diagram showing the geometry of the plasmasphere (deep green area), the auroral boundary (white thick curve), and spacecraft trajectories (VAP-A, VAP-B, and ERG spacecraft shown in blue, red, and pink, respectively) on the equatorial plane in the solar magnetic (SM) coordinate system with the Sun to the left (see the Methods section, coordinate system). The shape of the plasmapause is calculated from an empirical plasmaspheric model [37] . The clear sawtooth-shaped plasmapause structure visible in the afternoon-dusk sector indicates the PSW, which is manually added based on the wavelength of the PSW calculated in the main text. The closed circles in the spacecraft trajectories indicate temporal intervals of 1 h beginning at 13:00 UTC (indicated by the leftmost circles overlaid with crosses). Both VAP-A and VAP-B spacecraft were located in the southern hemisphere while ERG spacecraft crossed the magnetic equator at ~14:10 UTC. Open diamonds represent the plasmapause crossings by the spacecraft. Note that VAP-A and VAP-B crossed the plasmapause region roughly azimuthally while ERG crossed almost radially. b SA observed by DMSP F17 in the Northern Hemisphere (NH) at 13:27:45 UTC. c SA observed by DMSP F17 in the SH at 14:37:13 UTC. The dashed grid lines in b , c denote the altitude-adjusted corrected geomagnetic (AACGM) [38] latitudes and magnetic local time (MLT). The thick dotted line in c represents the boundary of the SA, which is projected onto the SM equatorial plane using the Tsyganenko 96 magnetic field model [36] and denoted by the white curve at the duskside plasmapause in ( a ). d Spectrogram of the electric field from the high frequency receiver of the EMFISIS instrument suite onboard VAP-A. The next two panels show the energy spectrograms of electrons e and protons f measured by the HOPE mass spectrometer onboard VAP-A, respectively. The plasmaspheric electron density (blue curve) is overlaid on ( e , f) . g – i The same as those of d – f but for the ERG spacecraft. Details of the instruments and data availability can be found in Methods (see the Methods section, data usage). Full size image Sunward/westward propagating PSW We evaluate the wave characteristics and phase relationships associated with the PSW in the field-aligned (FA) coordinate system, in which e p is along the background magnetic field (direction obtained from the 15-min sliding averaged data), e a (roughly eastward) is parallel to e p × R ( R is the radial vector pointing from the centre of the Earth toward the satellite), and e r (roughly radially outward) completes the orthogonal set. For VAP observations, the power of the perturbations of the radial magnetic field B r peaked at the frequency of ~1.5 mHz, and ~1.4 mHz for the azimuthal electric field perturbation E a (Fig. 2a ). For ERG observations, the perturbations of the magnetic field also peaked at ~1.4 mHz (Fig. 2f ). When the spacecraft arrived at the plasmapause boundary region, intense wave activities were measured. As the spacecraft left from the boundary to return to the plasmasphere, the waves quickly stopped being detected by the spacecraft (Supplementary Fig. 3 ). These measurements are in general agreement with the conception of a surface wave along the plasmapause boundary [24] . Fig. 2: Spectral analysis of the magnetic field perturbations measured by VAP and ERG. a Normalized power spectra of the perturbations of the magnetic field radial component of the magnetic field B r (black, thin for VAP-A and thick for VAP-B) and the electric field azimuthal component E a (red) measured by VAP-A. b Bandpass filtered (1.5 ± 0.5 mHz) perturbations of B r (red), field-aligned magnetic field B p (green) and E a (blue). c Bandpass filtered (1.5 ± 0.5 mHz) perturbations of the azimuthal magnetic field B a (red) and the radial electric field E r (blue). d Bandpass filtered (1.5 ± 0.5 mHz) perturbations of B r (red), B p (green) and radial ion velocity V r (blue) (see the Methods section, data usage). The background in b – d represents electron density with dark color (colorless) for high (low) density inside (outside) the plasmasphere. e Bandpass filtered (1.5 ± 0.5 mHz) perturbations of B r measured by VAP-A (red) and VAP-B (blue). f Normalized power spectra of the perturbations of B r (black), B a (red), and B p (blue) measured by ERG. The vertical line indicates a peak at 1.4 mHz. g Radial profile of the plasmaspheric electron density. h – j Bandpass filtered (1.4 ± 0.5 mHz) perturbations of B r , B a , and B p , respectively between 13:30 and 14:30 UTC. The thick vertical line indicates crossing of the magnetic equatorial plane by ERG with the left side in the southern hemisphere (inside plasmapause) and the right side in the northern hemisphere (outside plasmapause). Full size image Clear coherent phase relationships are found between the magnetic and electric fields in the bandpass-filtered signals (Fig. 2b–d ): B r was nearly in antiphase with E a (−164 ± 9°) and was roughly orthogonal to the field-aligned magnetic perturbation B p (108 ± 22°); B p had a ~90° phase difference with both E a (89 ± 15°) and the radial ion velocity V r (89 ± 13°); and B a was roughly in quadrature with E r (83 ± 15°). However, the phase difference was not stable at a value close to 90°. By using the simultaneously measured B r from VAP-A and VAP-B (Fig. 2e ), it is determined that the azimuthal wavelength was 10° ± 0.3°, the azimuthal propagating speed was 0.010 ± 0.001° s −1 , and the azimuthal mode number m was 36 ± 1 for the wave (see the Methods section, determination of propagating speed). The different wave amplitudes observed by VAP-A and VAP-B in Fig. 2e might be caused by either the variation of relative distance between the spacecraft and the plasmapause surface, or the time varying dynamic evolution of the PSW during sunward/westward propagation (Supplementary Fig. 3 ). The antiphase of n e with B r in Fig. 2b indicates that B r was in antiphase with the geocentric distance of the plasmapause. The magnetic perturbations exhibited a 180° phase difference on either side of the magnetic equator and also on either side of the plasmapause boundary (opposite amplitude in Fig. 2h–j ), meaning that the waves at the plasmapause region are fundamental mode PSW eigenmodes [24] , [39] . The presence of both poloidal oscillations of B r and E a and toroidal oscillations of B a and E r shows that the observed PSW manifested a mixed poloidal and toroidal modes [40] . The finite value of m leads to the coexistence and joint action of these two modes. At such a moderate high- m (~36), the poloidal perturbations are partly field-aligned guided, and the toroidal perturbations have a compressional feature [41] . The B p component was 90° out-of-phase with E a (Fig. 1b ) and V r was 90° lagged behind B p (Fig. 1d ), indicating the presence of fast compressional MHD wave modes. It is known that the slow mode wave can also be coupled with shear Alfvén waves, particularly in high-β plasma environment [42] . In the plasmapause region, however, the β parameter is very low ( ≪ 0.01), the observed features, therefore, suggest that the waves associated with the PSW are coupling between fast waves with shear Alfvén waves in consideration of the sunward/westward propagation of PSW. Conjugated sunward/westward propagating SA Owing to the modulation of the plasmapause by the PSW, the energetic electrons and protons that intruded into the low plasmaspheric density regions (Fig. 1d, g ) were possibly scattered by the electron electrostatic cyclotron harmonic (ECH) waves [20] and precipitated into the polar upper atmosphere to generate diffuse aurora. Images from the SSUSI instrument onboard the DMSP F17 satellite at two selected times (Fig. 1b, c , see Supplementary Fig. 4 for the entire image sequence) displayed giant undulations at the equatorward boundary of the diffuse aurora in both hemispheres, i.e., SAs, which were collocated with the sawtooth-shaped plasmapause on the equatorial plane via field line mapping, as shown in Fig. 1a . Within the limitations of the temporal evolution and spatial coverage of SSUSI, it is approximately estimated that the SA was possibly initiated between 11:15 and 11:45 UTC and ended at ~15:10 UTC (Supplementary Fig. 4b, f ). The DMSP F17 and F18 satellites [35] passed through the polar region successively in ~2 min, which is perfect for analysing the dynamical evolution of the SA. With a cross-correlation analysis [43] between the emission intensity profiles at AACGM latitudes of 58.5° in the Northern Hemisphere (Fig. 3a, b ) and −60.0° in the Southern Hemisphere (Fig. 3d, e ), we determined that the SA propagated sunward/westward at a phase speed of 0.01 ± 0.001° s −1 and the azimuthal wavelength was about 6.4°–10.2° for the SA (Fig. 3c, f ). In addition, the azimuthal wavelength was shown to decrease with increasing MLT. The azimuthal wavelength and sunward/westward phase speed of the SA found from the DMSP data are in agreement with the PSW (~9.7°–10.5° and 0.01° s −1 ) obtained based on the VAP measurements, unambiguously proving that the SA was driven by the PSW; in other words, the SAs are the optical atmospheric/auroral manifestation of the PSWs. Fig. 3: Determination of the phase speed and azimuthal wavelength of SA. a Auroral image obtained by the DMSP F18 satellite for 13:23–13:30 UTC in the NH with the SA captured at 13:25:17 UTC. b Auroral image obtained by the DMSP F17 satellite between 13:25–13:32 UTC in the NH with the SA captured at 13:27:45 UTC. c Emission intensity variations of the SA at the AACGM latitude of 58.5° as indicated by the blue dashed line in ( a ) and red dashed line in ( b ). d Auroral image obtained by the DMSP F18 satellite between 14:30 and 14:37 UTC in the SH with the SA captured at 14:34:48 UTC. e Auroral image obtained by the DMSP F17 satellite between 14:32 and 14:39 UTC in the SH with the SA captured at 14:37:13 UTC. f Emission intensity variations of the SA at the AACGM latitude of −60° as indicated by the blue dashed line in ( d ) and red dashed line in ( e ). Full size image Poleward and sunward/westward propagating ULF waves Apart from driving the SA, the PSW generated outward-propagating ULF waves outside the plasmapause, giving rise to the field line resonance (FLR) [3] and driving ULF geomagnetic pulsations. Figure 4 shows the analysis of magnetic perturbations observed by ground-based station chains (Fig. 4a ) from the International Monitor for Auroral Geomagnetic Effects (IMAGE) magnetometer array [44] . The power spectra of the magnetic field perturbations showed several significant signals at ~0.6 mHz, ~1.1 mHz, ~1.4 mHz, and ~2.0 mHz, with lower frequency pulsations occurring at higher latitudes (Supplementary Fig. 5 ). The wave amplitude of the N (north) component in the frequency of 1.4 ± 0.5 mHz (the closest to the frequency of PSW-associated waves in Fig. 2a ) maximized at a magnetic latitude of ~66° (MAS station), while the phase changed by nearly 180° over the amplitude maximum (Fig. 4b ), indicating the driving of FLR outside the plasmapause by the PSW. The 1.4 ± 0.5 mHz bandpass filtered N (Fig. 4c ) and Z (Fig. 4d ) components of the geomagnetic perturbations showed clear poleward propagation (i.e., radially outward in the magnetosphere) of the ULF waves starting from the OUJ station (at the AACGM latitude of 61.42° and longitude of 105.46°), which are the closest to the SA (Fig. 4c, d ), i.e., the counterpart to the sawtooth-shaped plasmapause region in the magnetosphere. This is apparently contrary to the inward (or equatorward) propagation of the ULF waves generated by external sources (e.g., magnetopause [15] or in the near-Earth magnetotail [14] , [45] ). The 1.4 ± 0.5 mHz bandpass filtered E component of the geomagnetic perturbations (Fig. 4e ) showed clear sunward/westward propagation of the ULF waves with a speed of ~0.012° s −1 and an azimuthal mode number of 36.8 ± 0.6 (see the Methods section, determination of m on ground). Both the propagating speed and the azimuthal mode number are consistent with those of the PSW and the SA. Fig. 4: ULF waves and FLR observed on the ground. a Stations selected from the IMAGE magnetometer array with the background auroral image observed by DMSP F17 satellite at 13:27:45 UTC (see also Fig. 1b ). The blue chain is aligned in almost the same magnetic longitudes of ~106 ± 4° with the magnetic latitude increasing from 54.5° at TAR to 75.1° at LYR. The red chain is aligned in almost the same magnetic latitudes of ~61.5 ± 0.5° with the magnetic longitude increasing from 93.3° at RVK to 106.1° at OUJ. The black/cyan arrow represents the footpoints of VAP-A/ERG satellite moving towards low/high latitudes between 13:40 UT and 15:00 UT with the diamond marking plasmapause crossing. b Amplitude (black) and phase difference (blue) of the N component of the blue chain plotted as a function of geomagnetic latitude. c Bandpass filtered (1.4 ± 0.5 mHz) N component (north) of the geomagnetic field perturbations. d Bandpass filtered (1.4 ± 0.5 mHz) Z component (vertical) of the geomagnetic field perturbations. Clear poleward propagation of the waves is highlighted by the dashed lines. e Bandpass filtered (1.4 ± 0.5 mHz) E component (east) of the geomagnetic field perturbations. Clear sunward/westward propagation of the waves is highlighted by the dashed lines. Full size image\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"The 16 July 2017 geomagnetic storm provided an opportunity to identify the PSW and its consequences of oscillating both the magnetosphere and the diffuse aurora. The enhanced convection electric field during the storm made the plasmapause in the afternoon-dusk sector very sharp. Inhomogeneities in the plasma density and magnetic field lead to the coupling between the shear Alfvén waves and the magnetosonic waves on closed field lines. Field oscillations in the magnetosphere usually have a continuous spectrum. Nevertheless, solution of the MHD wave equation shows the coupling between the surface wave (i.e., evanescent compressional Alfvén wave) and the shear Alfvén wave if there exists a sharp change in Alfvén speed (related to changes in magnetic field and plasma mass density), and a localized perturbation propagating along the plasma surface across which the Alfvén speed changes will suddenly possess a discrete eigenfrequency [24] . Since the resonant absorption is negligibly small [46] , a localized coupled ULF wave corresponding to the discrete eigenmode can thus be excited at such a location by an impulse which has a frequency spectrum that covers this eigenfrequency. Observations in both the topside ionosphere and the plasmasphere showed that the radial width of the plasmapause was ~0.1–0.2 R E (see Supplementary Figs. 6 , 7 ), which was much smaller than the transverse wave scale of ~0.8 R E for the PSW with m -value 36 at a geocentric distance of 4.5 R E , satisfying the requirement of the interface width in exciting surface waves [47] . The sharp discontinuity in the density led the magnetosonic wave to be undamped [24] , [46] , [48] , allowing excitation of a standing surface eigenmode, i.e., the PSW propagating sunward/westward at the plasmapause boundary and along the field lines to both the northern and southern polar ionosphere. Using the plasmapause crossing measurements, it is estimated that the frequency of the fundamental poloidal wave is 1.35–2.11 mHz, close to 1.5 mHz (see the Methods section, estimation of eigenfrequency). Above estimations indicate that the configuration of the plasmapause during the storm main phase could provide favorable conditions to excite PSW. The PSW could be excited by external perturbations such as the sudden enhancement of the plasma pressure in the duskside ring current/plasma sheet owing to fast storm time injections. The driving sources could be either monochromatic with a frequency similar to the eigenfrequency of the plasmapause surface or impulsive and broadband whose frequency range covers the eigenfrequency of the plasmapause surface. Between 11:00 and 11:30 UT, strong ion/electron injections were observed (Supplementary Figs. 8 , 9 ). The power spectrum density of the energetic ion/electron flux between 11:00 and 13:00 UT exhibits a clear peak at 1.5 mHz (Supplementary Fig. 10 ), close to the eigenfrequency of the plasmapause surface. Such a periodically enhancing particle flux led to impinging of periodically varying plasma pressure on the plasmapause surface and may actually excite the PSW at 1.5 mHz. In addition, it is noted that other types of pressure variations, such as impulsive injection (like a delta function) and continuous injection (like a step function), are both broadband in frequency domain. Take the magnetopause surface wave for an example, the magnetosheath jet’s total pressure is impulsive and broadband [9] . Therefore, impulsive or continuous injection may both contribute to the excitation of the PSW. The proposed excitation process of PSW is similar to the magnetopause surface waves excited by the sudden enhancement of solar wind pressure [6] , interplanetary shocks [49] , or magnetosheath jets [9] . The free energy should be continuously supplied to maintain the long-term evolution and propagation of the PSW. Consistency between the durations of the PSW (or SAs) and the enhanced hot plasma pressure (Supplementary Fig. 9f ) indicates that the free energy was provided by the hot plasma injections. Other internal instabilities, such as the ion drift resonances which can excite fundamental poloidal waves [50] , [51] and the Kelvin-Helmholtz instability which can excite magnetopause surface waves [7] , [8] , could certainly play a role at some point in the excitation of the ULF waves, in addition to the PSW-associated ULF waves. Drift wave perturbations or the effect from ULF waves in the vicinity of the plasmapause might also be possible interpretations to the observed waves. Multi-satellite measurements show that the observed waves propagated along the plasmapause surface with a fixed frequency, excited outward-propagating ULF waves and drove FLR outside the plasmapause. These characteristics are similar to that of the magnetopause surface waves [8] , [9] which excite earthward-propagating ULF waves and drive FLR in the magnetosphere. In these regards, we suggest that the PSW concept should be a reasonable interpretation for the observations. The equatorial plasmapause became sawtooth-shaped due to the modulation of the PSW. The low-density region of PSW was intruded by hot plasma. Scattering and precipitation of the hot plasma by waves like ECH then generated SA in the afternoon-evening sector, which had the same azimuthal wavelength and sunward/westward phase speed as the PSW. In addition, the local lower ratio of electron plasma frequency to gyrofrequency in the night sector during disturbed periods also leads to much more efficient scattering by chorus waves (effective for ECH as well) just outside the plasmapause [52] , [53] , [54] . The SAs that occur at the equatorial boundary of the diffuse aurora on the dusk side are different from other quasi-periodic auroral structures such as the torch auroral structures (or omega band) [55] and the pulsating auroral forms generated by giant pulsations [56] , both of which occur in the auroral oval and in the midnight to morning sector, although similar wave activities are found to exist during these different auroral morphologies. Like the magnetopause surface wave that generates earthward-propagating ULF waves and drives FLR in the magnetosphere [8] , the PSW generated outward-propagating ULF waves and drove FLR outside the plasmapause. Such a process has not yet been observed and interpreted in the existing literature. The amplitude and phase structures on the ground are different from the giant pulsations though they have similar wave number and structures [57] . The amplitude and phase structures observed here are due to the radially outward propagation of the ULF waves from the plasmapause boundary and resonance with local field lines on the duskside, while the giant pulsations are latitudinally localized and occur almost exclusively on the morningside with peak occurrence in the postmidnight sector [58] . The ground signatures of surface waves are not well understood, including both the PSW and the magnetopause surface wave [9] . In this study, we show observational evidence that PSW can exhibit different signals to the conjugated ground magnetometer data. Further theoretical investigations, simulations, and observations are necessary to characterize the occurrence rate and properties of PSW. Although the PSWs are hardly captured by in situ observations since satellites are required to cross the plasmapause region at a specific time and at a certain location, the SAs that visualize the PSW are frequently observed during geomagnetic storms. Besides the DMSP satellites, the recently launched Chinese Fengyun-3D (FY-3D) satellite in a low-Earth orbit can capture high-resolution global images of SAs (Fig. 5 ) with the on-board wide-field auroral imager (WAI) [59] . We have surveyed the aurora data recorded by both DMSP and FY-3D satellites during geomagnetic storms (the minimum disturbance storm-time (Dst) index less than -40 nT) from 2014 to 2018 (listed in Supplementary Table 1 ). The probability of occurrence of SAs is found to be greater than 90% during geomagnetic storms (94 out of 103, as shown in Supplementary Table 1 ). For the remaining 9 storms, it is not clear whether these auroral structures are present or not due to the spatial coverage and temporal evolution of the auroral images. Furthermore, coordinated ground-based geomagnetic data were available in the dusk sector (MLT = 16–19 h) for 24 events of the 94 observed SAs. The geomagnetic pulsations occurred simultaneously with the SAs and propagated essentially radially outward when mapping to the magnetic equator and the amplitude peaked outside the plasmapause (Supplementary Fig. 11 ). These results definitely indicate that the PSWs, as well as the SAs, are systematic and crucial consequences of geomagnetic storms. The reconfiguration and energy redistribution processes in the magnetosphere and ionosphere associated with the PSW appear to be common and regular during the geomagnetically disturbing times, and theoretical and statistical studies are needed in the future for better understanding of the rules that govern these processes. Fig. 5: SAs observed by FY-3D WAI during four geomagnetic storms in 2018. All the images observed between a 22:11–22:32 UTC on 14 March 2018, b 22:31–22:48 UTC on 5 May 2018, c 20:53–21:15 UTC on 18 March 2018, and d 23:54–00:15 UTC on 25 August 2018 are projected onto a reference sphere at a height of 110 km in the AACGM–MLT coordinate system. The SAs are highlighted by the red arrows. Detailed structural parameters of the SAs are listed in Table 1 . Full size image The high resolution auroral images from the FY-3D satellite show that the AACGM latitudes of crest, azimuthal wavelengths, and crest-to-trough amplitudes of the SAs vary with the strengths (i.e., the Dst values) of storms (Table 1 ), implying that the characteristics of the PSWs also vary with geomagnetic activity and that the following magnetospheric and ionospheric effects will also be different. Although it remains unclear what mechanisms determine the azimuthal wavelength (i.e., azimuthal mode number) of the PSW from a theoretical view, these observations could provide important implications to future theoretical investigations on this topic. Understanding the factors (e.g., configuration of the plasmapause boundary and the plasma pressure variations outside the plasmapause during storms) that trigger and control the generation and evolution of the PSW is critical in establishing the generation process. In addition, the wave-particle interactions that lie behind this process are key to understanding the resultant energy transfer and auroral activities. Table 1 Structural parameters of the SAs extracted from Fig. 5 . Full size table The generation of surface waves and ULF waves are fundamental plasma processes in space environments and can occur in other planetary magnetospheres [3] , [60] , [61] . As is well-known in the space of giant planets, the corotation breakdown (the cause of the plasmapause in the Earth) is known to be fundamental to the dynamics of the rapid rotating planetary magnetospheres (e.g., Jupiter and Saturn) [62] , [63] with the brightest auroral emissions associated with this process [64] . Therefore, the processes identified along terrestrial plasmapause boundary layer may be even more critical to the environments at rapidly rotating planets and may provide a crucial direction for investigations.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Coordinate systems The geocentric solar magnetospheric (GSM) coordinate system is defined as follows [65] . The x -axis of GSM points from the Earth to the Sun, the z -axis is the projection of dipole axis on geocentric solar ecliptic (GSE) yz plane (the x -axis of GSE points from the Earth to the Sun, the z -axis of GSE points to the ecliptic north pole, the y -axis of GSE completes the right-handed system), and the y -axis of GSM completes the right-handed system. The y -axis of the solar magnetic (SM) system is perpendicular to the plane containing the Sun-Earth line and the dipole axis, the z -axis is along the dipole axis, and the x -axis completes the right-handed system. The modified GSE (mGSE) coordinate system is a near GSE system for Van Allen Probes (VAP). The x -axis of mGSE is the spin axis unit vector in GSE coordinates, the y -axis is anti-parallel to the cross product between the x -axis of mGSE and the y -axis of GSE, and the z -axis of mGSE completes the right-handed system. Altitude-adjusted corrected geomagnetic (AACGM) [38] coordinates are an extension of corrected geomagnetic coordinates that more accurately represent the actual magnetic field. In AACGM coordinates points along a given magnetic field line are given the same coordinates and are thus a better reflection of magnetic conjugacy. Data usage Observations reported in this paper are taken from the VAP-A and VAP-B spacecraft, the DMSP F17 and F18 satellites, the ERG spacecraft, the FY-3D satellite, the Time History of Events and Macroscale Interactions during Substorms (THEMIS) E spacecraft, the Magnetospheric Multiscale (MMS) mission, the Geostationary Operational Environmental Satellite (GOES) 15 satellite, the ground-based IMAGE magnetometer array, and the SuperMAG database. Detailed information is introduced below. The spectrogram (6 s resolution, 10 kHz to 500 kHz) from the high frequency receiver of the Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS) [30] onboard VAP and the spectrogram (1 s resolution, 10 kHz to 10 MHz) from the high frequency analyzer of the Plasma Wave Experiment (PWE) [32] onboard ERG are used to determine the plasmaspheric electron density. The magnetic field (1 s resolution in GSM coordinates) from the fluxgate magnetometer of EMFISIS [30] and the electric field (32 sample/s in mGSE coordinates) from the Electric Field and Waves (EFW) instrument [66] onboard VAP and the magnetic field (8 s resolution in GSM coordinates) from the Magnetic Field Experiments (MGF) [67] onboard ERG are used together to evaluate the wave characteristics associated with the plasmapause surface wave. For VAP spacecraft, the spin axis electric field component ( E x ) is estimated using the assumption that that E • B = 0 or the parallel electric field is zero. This is used most frequently for large-scale convection electric fields, MHD structures and ULF waves, and small-scale waves for which perpendicular electric fields are larger than parallel. Then the ion velocity is calculated by V = E × B / B 2 . The omni-direction energy flux data from the Helium, Oxygen, Proton, and Electron (HOPE) mass spectrometer [31] onboard VAP, from the low-energy particle experiments–electron analyzer (LEP-e) [33] and the low-energy particle experiments–ion analyzer (LEP-i) [34] onboard ERG, from the Fly’s Eye Energetic Particle Sensor (FEEPS) [68] and the Hot Plasma Composition Analyzer (HCPA) [69] onboard MMS-1, and from the Energetic Particle Sensor (EPS) [70] onboard GOES-15 are used to evaluate the distributions and evolutions of hot plasma. The auroral disk images in N 2 Lyman–Birge–Hopfield (LBH) bands from the Special Sensor Ultraviolet Spectrographic Imager (SSUSI) [35] onboard DMSP F17 and F18 satellites are used to acquire the parameters of the SA. All the auroral images are projected onto a reference sphere at a height of 110 km. Both the DMSP F17 and F18 satellites orbit the Earth in a sun-synchronous orbit with a period of ~102 min and two disk images of aurora are obtained in each orbit. All the auroral images during the PSW event are shown in Supplementary Fig. 4 . On 16 July 2017, the DMSP F17 and F18 satellites passed the same polar region successively with a time difference of ~2 min, and therefore, only the F17 images are shown in Supplementary Fig. 4 . The densities of O + , H + , and He + measured by the Retarding Potential Analyzer (RPA) [71] on board the DMSP satellite are used to calculate the latitudinal plasma mass density profiles, which can reflect the radial profile of the plasma mass density in the plasmasphere. Examples are shown in Supplementary Fig. 7 . The wide-field auroral imager (WAI) [59] onboard FY-3D satellite, which was launched on 15 November 2017 into a sun-synchronous orbit at an altitude of ~840 km with a period of ~102 min, also provided auroral disk images in N 2 LBH bands but with larger field-of-view that can capture the global high-resolution structure of the SA (Fig. 5 ). The geomagnetic field data (10 s resolution) from the ground-based station chains (latitudinal chain from low to high latitude: TAR, NUR, HAN, OUJ, RAN, PEL, MUO, MAS, SOR, BJN, HOP, and LYR; longitudinal chain from west to east: RVK, LYC, and OUJ) of the IMAGE magnetometer array [44] and the geomagnetic field data (1 min resolution) from the ground-based station chain (from low to high latitude: OTT, T51, T30, T52, T45, T31, T44, T46, and T47) of the SuperMAG database [72] are used to evaluate the geomagnetic pulsations on the ground. The latitudinal IMAGE chain is aligned in almost the same magnetic longitudes of ~106 ± 4° (in the dusk sector at ~15:00 UTC) with the magnetic latitude increasing from 54.5° at TAR to 75.1° at LYR. The longitudinal IMAGE chain is aligned in almost the same magnetic latitudes of ~61.5 ± 0.5° with the magnetic longitude increasing from 93.3° at RVK to 106.1° at OUJ, and the SuperMAG chain is aligned around magnetic longitudes of 0 ± 4° (in the dusk sector at ~23:00 UT) with the magnetic latitude increasing from 55.0° at OTT to 71.5° at T47. All the geomagnetic field data are presented in the NEZ frame, in which horizontal components N and E point geomagnetically north and east, respectively, and Z is the vertical component. The original data are all detrended by subtracting the 1-h sliding averages. Calculation of plasmaspheric electron density The electron density ( n e , in cm –3 ) measured by VAP spacecraft [73] is calculated by n e = ( f 2 UHR – f 2 ce )/8980 2 , where f UHR is the upper hybrid resonance (UHR) frequency in Hz identified from the frequency-time spectrogram of electric field of the EMFISIS instrument, f ce = eB / m e is the electron cyclotron frequency in Hz, B is the strength of the magnetic field simultaneously measured by the EMFISIS instrument, and m e is the electron mass. The method for the ERG spacecraft is the same. The inferred electron densities are shown in Fig. 1e , f and h , i . For the THEMIS-E satellite, the spacecraft potential (refers to the potential of the spacecraft body relative to the ambient plasma) measured by the electric field instrument (EFI) [74] and the electron thermal velocities measured by the electrostatic analyzer (ESA) [75] are used to calculate the electron density, which is used to evaluate the plasmapause configuration (Supplementary Fig. 6 ). Determination of the azimuthal propagating speed of PSW We define the azimuthal wavelength of the PSW as λ PSW and the azimuthal propagating speed as v PSW . Between 13:50 and 14:40 UTC, the longitudinal separation between VAP-A and VAP-B is d λ sat = 17.3 ± 0.3° and the azimuthal angle difference between the electro density peaks observed by VAP-A is d λ peak = 3.0 ± 0.2° with a time interval of dt = 725 ± 25 s (Fig. 2b ). Since VAP-A is closer to dusk than VAP-B, the phase difference indicates a sunward/westward propagating wave, and λ PSW should be less than d λ sat and greater than d λ peak . The phase of B r,VAP-A leading B r,VAP-B by 90 ± 1° between 13:45 and 14:15 UTC (Fig. 2e ) indicated that d λ sat = (3/4 + n ) λ PSW ( n = 0, 1, 2, …). The loss of phase after 14:15 in Fig. 2f is because the VAP-B spacecraft had entered the plasmasphere and was far away from the plasmapause region. For n = 0, we get an azimuthal mode number m ≈ 16 and λ PSW = 22.5°, which dissatisfies the limitation on λ PSW . When n = 1, we get m = 36 ± 1 and λ PSW = 10 ± 0.3°, which satisfies the limitation on λ PSW . According to the relationship λ PSW = d λ peak + v PSW × dt , we obtain the sunward/westward propagating speed v PSW = 0.01 ± 0.001° s −1 . Both the azimuthal wavelength and the sunward/westward propagating speed showed agreement between the PSW (~9.7°–10.5° and 0.01° s −1 ) and the SA (~6.4°–10.4° and 0.01° s −1 ), unambiguously proving that the SAs are driven by the PSWs, i.e., the SA is the optical manifestation of the PSW. Because the PSW was observed at an earlier MLT than the SA, and the wavelength of the PSW is generally greater than that of the SA, which is consistent with the decrease in SA wavelength with increasing MLT (Fig. 3 ). Determination of the azimuthal wave number on the ground The three stations (OUJ, LYC, and RVK) are used to calculate the m -value of the sunward/westward propagating ULF waves on the ground. The azimuthal mode number m of the ULF waves is calculated by m = d φ / d ϕ , where d φ is the phase angle difference of the ULF waves between two stations and d ϕ is the magnetic longitude separation between two stations. Using cross-phase analysis between station pairs (OUJ-LYC and OUJ-RVK, d ϕ OUJ-LYC = 6.85°, d ϕ OUJ-RVK = 12.83°, d φ OUJ-LYC = 248°, and d φ OUJ-RVK = 480°) during 13:00 UT and 13:30 UT, the m -value is calculated to be 36.8 ± 0.6, consistent with the m -value of the PSW. The averaged time difference of the wave peaks and averaged magnetic longitude difference between the station pairs are dt = ~400 s and d ϕ = 6.42°, respectively. Considering the Earth’s eastward rotating speed of 0.0042° s −1 , the sunward/westward propagating speed of the ULF waves is ~0.012° s −1 . It is noted that the direction of ULF wave propagation can be affected in the data by the shape of the plasmapause, and the propagation direction and speed can be more precisely determined if more station chains at different latitudes are available at the same time. Estimation of the eigenfrequency at the plasmapause According to the dispersion equation given by Chen and Hasegawa [24] , the wave propagation speed along the magnetic field line at the plasmapause is v = 2 1/2 v A , where v A is the Alfvén speed of the magnetic field line inside the plasmapause boundary. According to THEMIS-E satellite measurement (Supplementary Fig. 6 ), the strength of the magnetic field is B = ~400 nT and the electron number density (equivalent to total number density of H + , He + , and O + ) is ~1500 cm −3 inside the plasmapause boundary. Since no ion composition measurements were available in the magnetosphere during this storm, typical values of relative ion concentrations in literature are considered, with the relative concentration of O + between 5 and 30% and He + between 5 and 10% [17] . The plasma mass density is estimated to be 4.54 × 10 −18 ~1.11 × 10 −17 kg m −3 and the resultant v is 150 ~ 240 km s −1 . The length of the field line Λ at the plasmapause is calculated to be ~5.6 × 10 4 km with the Tsyganenko 96 magnetic field model. Taking into account that half a wavelength equals to Λ for the fundamental poloidal wave, we get the fundamental frequency ω = v /(2 Λ ) = 1.35 ~ 2.11 mHz, close to 1.5 mHz. The above estimations indicate that the configuration of the plasmapause during the storm main phase supports the excitation and generation of the PSW.\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"This study proposes that plasmapause surface waves (PSW), generated during geomagnetic storms, act as a systematic ultra-low frequency (ULF) wave driver in the magnetosphere, and that these waves are physically connected to the formation of sawtooth-shaped auroral undulations (SA) on the equatorward edge of the diffuse aurora, thereby playing a crucial role in energy dissipation within the ionosphere-magnetosphere coupling system."},"entities":{"kind":"list like","value":["plasmapause surface waves","geomagnetic storms","ultra-low frequency wave driver","magnetosphere","sawtooth-shaped auroral undulations","equatorward edge","diffuse aurora","energy dissipation","ionosphere-magnetosphere coupling system"],"string":"[\n \"plasmapause surface waves\",\n \"geomagnetic storms\",\n \"ultra-low frequency wave driver\",\n \"magnetosphere\",\n \"sawtooth-shaped auroral undulations\",\n \"equatorward edge\",\n \"diffuse aurora\",\n \"energy dissipation\",\n \"ionosphere-magnetosphere coupling system\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1.","title":"Lyon, J. 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Rev.141, 303–341 (2008).\",\n \"link\": \"https://link.springer.com/doi/10.1007/s11214-008-9469-2\"\n },\n {\n \"idx\": \"75.\",\n \"title\": \"McFadden, J. P. et al. The THEMIS ESA plasma instrument and in-flight calibration.Space Sci. Rev.141, 277–302 (2008).\",\n \"link\": \"https://link.springer.com/doi/10.1007/s11214-008-9440-2\"\n }\n]","isConversational":false}}},{"rowIdx":11191,"cells":{"unique_id":{"kind":"string","value":"s41467-023-41236-3"},"subfield":{"kind":"string","value":"Physics"},"title":{"kind":"string","value":"An electronic origin of charge order in infinite-layer nickelates"},"abstract":{"kind":"string","value":"A charge order (CO) with a wavevector \\({{{{{{{\\bf{q}}}}}}}}\\simeq \\left(\\frac{1}{3},\\, 0,\\, 0\\right)\\) is observed in infinite-layer nickelates. Here we use first-principles calculations to demonstrate a charge-transfer-driven CO mechanism in infinite-layer nickelates, which leads to a characteristic Ni 1+ -Ni 2+ -Ni 1+ stripe state. For every three Ni atoms, due to the presence of near-Fermi-level conduction bands, Hubbard interaction on Ni- d orbitals transfers electrons on one Ni atom to conduction bands and leaves electrons on the other two Ni atoms to become more localized. We further derive a low-energy effective model to elucidate that the CO state arises from a delicate competition between Hubbard interaction on Ni- d orbitals and charge transfer energy between Ni- d orbitals and conduction bands. With physically reasonable parameters, \\({{{{{{{\\bf{q}}}}}}}}=\\left(\\frac{1}{3},\\, 0,\\, 0\\right)\\) CO state is more stable than uniform paramagnetic state and usual checkerboard antiferromagnetic state. Our work highlights the multi-band nature of infinite-layer nickelates, which leads to some distinctive correlated properties that are not found in cuprates."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"Motivated by cuprates, superconductivity was proposed [1] , [2] and has been recently discovered in hole-doped infinite-layer nickelates that contain Ni 1+ ions of the similar d 9 electronic configuration [3] , [4] , [5] , [6] , [7] , [8] , [9] . While this seems to confirm the initial expectation and support the Mott scenario for high-temperature superconductivity [10] , the nickelates show very different properties even in their undoped parent compounds. Instead of an antiferromagnetic Mott insulator, the resistivity of the parent compounds exhibits metallic behavior at high temperatures and an upturn below around 70 K [3] , [8] , [11] , [12] . So far, no static long-range magnetic order has been observed down to the lowest measured temperature [13] , [14] , although spin fluctuations have been reported in experiment [15] . These are all different from cuprates and have stimulated heated debate concerning the nature of the nickelate parent state, from which superconductivity is born upon hole doping. Although several theoretical scenarios [16] , [17] , [18] , [19] , [20] have been proposed to emphasize the key role of single-band Hubbard interaction, a growing consensus is that multi-band physics is indispensable in nickelates [21] , [22] , [23] , [24] , [25] , [26] , [27] , [28] , [29] , [30] , [31] , [32] . What has not been sufficiently explored is whether this multi-band feature combined with a strong correlation on Ni- d orbitals may lead to some unique experimental consequences in infinite-layer nickelates. Recently, a charge order (CO) with broken translational symmetry has been reported independently by three experimental groups in the parent compounds of nickelate superconductors [33] , [34] , [35] . This raises a question concerning the CO origin and its competition with other phases. CO in cuprates is well known and has been intensively studied [36] , [37] . Despite the seeming similarity, the CO in nickelates has a few important differences [33] , [34] , [35] . First, it already exists and has the highest onset temperature in undoped compounds. Second, unlike cuprates where doped holes reside on oxygen atoms, holes are mainly introduced into Ni- d orbitals, as supported by the experiments [38] , [39] . Finally and most importantly, it has an ordering wavevector \\({{{{{{{\\bf{q}}}}}}}}\\simeq \\left(\\frac{1}{3},\\, 0,\\, 0\\right)\\) , compatible with a commensurate modulation with period 3 a 0 ( a 0 is the lattice constant). This stripe pattern has not been observed in cuprates [40] . Hence, these differences may imply a new mechanism for the CO in the nickelates, which is most probably associated with the multi-band nature of the parent compounds [21] , [22] , [23] , [24] , [25] , [26] , [27] , [28] , [29] , [30] , [31] , [32] . In this work, we first use density functional theory plus dynamical mean field theory (DFT+DMFT) calculations [41] , [42] , [43] , [44] to reveal a special charge-transfer-driven CO mechanism in a prototypical infinite-layer nickelate NdNiO 2 . We find that for every three Ni atoms, due to the presence of Nd- d /interstitial- s derived conduction bands near the Fermi level, electrons on one Ni atom are transferred to the conduction bands under a reasonably large Hubbard interaction U Ni on Ni- d orbitals, while electrons on the other two Ni atoms become more localized. This leads to a characteristic Ni 1+ -Ni 2+ -Ni 1+ CO stripe state. Then, we derive a low-energy effective model based on the realistic band structure of NdNiO 2 . Using the effective model, we further elucidate that the \\({{{{{{{\\bf{q}}}}}}}}=(\\frac{1}{3},\\, 0,\\, 0)\\) CO stripe state arises from a delicate competition between Hubbard interaction on Ni- d orbitals and charge transfer energy between Ni- d orbitals and conduction bands. Tuning the charge transfer energy controls not only the stability of the CO state but also its stripe pattern. With Hubbard interaction strength and charge transfer energy in a physically reasonable range, we find that the experimentally observed \\({{{{{{{\\bf{q}}}}}}}}=\\left(\\frac{1}{3},\\, 0,\\, 0\\right)\\) stripe CO state is more stable than both the uniform paramagnetic (PM) state and the usual checkerboard antiferromagnetic state.","references":{"indices":[],"infos":[]}},{"section":"Results","content":"DFT+DMFT calculations Unlike cuprate superconductors, a consensus has not been reached regarding what is the minimal model to study infinite-layer nickelates. Therefore, we first downfold the DFT-calculated band structure of NdNiO 2 to a large 17-orbital tight-binding model by using the maximally localized Wannier functions (MLWFs) [45] . The model includes five Ni- d orbitals, five Nd- d orbitals, six O- p orbitals and one interstitial- s orbital per primitive cell. Our previous studies [25] , [46] show that this 17-orbital model is sufficiently large that it can almost exactly reproduce the non-interacting band structure of NdNiO 2 in an energy window of about 15 eV around the Fermi level (see Supplementary Fig. 1 ). A Slater–Kanamori interaction is added on Ni- d orbitals. We use DMFT to solve the interacting model. To test a possible \\({{{{{{{\\bf{q}}}}}}}}=\\left(\\frac{1}{3},\\, 0,\\, 0\\right)\\) CO state, we use a 3 × 1 × 1 supercell. Since in the 17-orbital model, the MLWF for the Ni- d orbitals are very localized and atomic-like (see Supplementary Table 1 ), we use a Hubbard U Ni = 10 eV and a Hund’s exchange J Ni = 1 eV, which has proved to be reasonable for this large energy window treatment of transition metal oxides [26] , [47] , [48] , [49] . More computational details are found in the Methods. We first add perturbations on the three Ni atoms in the simulation cell. After the self-consistency loop is converged, we do find a CO state whose spectral function is shown in the Fig. 1a, b . For comparison, we also enforce symmetry on the three Ni atoms and thus obtain a uniform PM state whose spectral function is shown in Fig. 1c, d . Figure 1a, c shows the Ni- d and O- p projected spectral functions. In the CO state, we find two different types of Ni atoms. The spectral functions for Ni1- d and Ni3- d orbitals are identical (collectively referred to as Ni1- d ). Ni2- d orbital is of the other type. In the PM state, all three Ni atoms are equivalent due to the imposed symmetry. The inset shows the Ni- d occupancy. We find that compared to Ni- d occupancy in the uniform PM state, in the CO state, the Ni1- d occupancy is larger and approaches 9 e , but the Ni2- d occupancy is substantially smaller and is close to 8 e . This leads to a characteristic Ni 1+ -Ni 2+ -Ni 1+ stripe pattern. More importantly, the average Ni- d occupancy in the CO state is smaller than that in the PM state, indicating that the CO is not a simple charge modulation between the three Ni atoms, but rather there is a net charge transfer from the Ni2 atom to other atoms. Figure 1b, d shows the Nd- d and interstitial- s projected spectral functions, which are almost identical between the CO and the PM states. However, close to the Fermi level, we find appreciable differences that Nd- d and interstitial- s states are more occupied in the CO state than in the PM state. Therefore, the emergence of CO state in NdNiO 2 is accompanied by a charge transfer from Ni2 atom to Nd- d /interstitial- s orbitals that form conduction bands close to the Fermi level. We will show below that it is precisely this charge transfer that can yield an energy gain and stabilize the CO state. Fig. 1: Spectral functions of NdNiO 2 . The spectral functions are calculated by DFT+DMFT method using the 17-orbital model with U Ni = 10 eV and J Ni = 1 eV. a , b The charge-ordered state (CO). a The green, red and purple curves are Ni1- d , Ni2- d and O- p projected spectral functions, respectively. The inset shows the site-resolved Ni- d occupancy. The d occupancy of Ni2 site is substantially smaller than that of Ni1 and Ni3 sites. b The blue and orange curves are Nd- d and interstitial- s projected spectral functions, respectively. The inset shows the energy range near the Fermi level. c , d The uniform paramagnetic state (PM). c The green and purple curves are Ni- d and O- p projected spectral functions, respectively. The inset shows the site-resolved Ni- d occupancy. d identical to ( b ) but for the PM state. In all panels, the dashed line is the Fermi level, which is shifted to zero point. Source data are provided as a Source Data file. Full size image Next, we study how the charge disproportionation of the CO state depends on U Ni and J Ni . Figure 2a shows the Ni- d occupancy of both CO and PM states as a function of U Ni with J Ni being fixed at 1 eV. As U Ni exceeds 6 eV, the charge disproportionation of Ni- d occupancy develops in the CO state and becomes more pronounced with U Ni . On the other hand, changing J Ni in a reasonable range (from 0.7 to 1.0 eV) with U Ni being fixed at 10 eV does not strongly affect the charge disproportionation in the CO state. Furthermore, we also test the correlation effects from Nd- d orbitals. By considering Hubbard interaction on Nd- d orbitals, we do not find qualitative changes in the main results (see Supplementary Note 3 ). Fig. 2: Dependence of Ni- d occupancy N d on U Ni and J Ni . The dependence of N d on U Ni and J Ni is calculated by using the 17-orbital model. a Ni- d occupancy N d as a function of U Ni with J Ni = 1 eV. b Ni- d occupancy N d as a function of J Ni with U Ni = 10 eV. In both panels, the red solid symbols and blue open symbols correspond to the Ni- d occupancy in the charge-ordered (CO) and uniform paramagnetic (PM) states, respectively. Source data are provided as a Source Data file. Full size image Low-energy effective model To gain a deeper understanding of the origin and generality of the CO state that is found in the DFT+DMFT calculations of NdNiO 2 , we build a low-energy effective model. We want to make the model simple, which aims to qualitatively reproduce the key features of the CO state. The advantage of such a simple effective model is that we can obtain an accurate self-energy so as to resolve the total energy difference between different competing phases. In addition, it also helps us identify the control parameters for the CO state. The effective model, equipped with proper energy dispersions, may also be applied to the study of the charge ordering phenomenon in other strongly correlated materials. From the preceding DFT+DMFT calculations, we find that there are two key ingredients in the formation of the CO state in NdNiO 2 . One is the Hubbard interaction on Ni- d orbitals and the other is the charge transfer from Ni- d orbitals to the conduction bands that consist of Nd- d /interstitial- s orbitals. In order to describe the interplay between these two factors, we include a correlated orbital and an auxiliary orbital in the effective model. To make a connection to infinite-layer nickelates, we choose Ni- \\({d}_{{x}^{2}-{y}^{2}}\\) orbital as the correlated orbital and add local Hubbard interaction on it. We also choose an effective- s orbital as the auxiliary orbital that provides a free conduction band. We note that in real NdNiO 2 , there are multiple conduction bands that are composed of Nd- d and interstitial- s orbitals. Here the effective- s orbital (not to be confused with the interstitial- s orbital) is employed to simplify the description of conduction bands. Similar models of correlated Ni- \\({d}_{{x}^{2}-{y}^{2}}\\) orbital plus free conduction bands have been used to study the low-energy physics of infinite-layer nickelates [17] , [25] , [32] , [50] , [51] , [52] , [53] . The full Hamiltonian of our effective model takes the form: $$\\hat{H} =\t \\mathop{\\sum}\\limits_{{{{{{{{\\bf{k}}}}}}}}\\sigma }\\left[{\\epsilon }_{d}({{{{{{{\\bf{k}}}}}}}}){\\hat{d}}_{{{{{{{{\\bf{k}}}}}}}}\\sigma }^{{{{\\dagger}}} }{\\hat{d}}_{{{{{{{{\\bf{k}}}}}}}}\\sigma }+{\\epsilon }_{s}({{{{{{{\\bf{k}}}}}}}}){\\hat{s}}_{{{{{{{{\\bf{k}}}}}}}}\\sigma }^{{{{\\dagger}}} }{\\hat{s}}_{{{{{{{{\\bf{k}}}}}}}}\\sigma }+\\left({V}_{ds}({{{{{{{\\bf{k}}}}}}}}){\\hat{d}}_{{{{{{{{\\bf{k}}}}}}}}\\sigma }^{{{{\\dagger}}} }{s}_{{{{{{{{\\bf{k}}}}}}}}\\sigma }+{{{{{{{\\rm{h.c.}}}}}}}}\\right)\\right] \\\\ \t+{U}_{d}\\mathop{\\sum}\\limits_{i}{\\hat{n}}_{i\\uparrow }^{d}{\\hat{n}}_{i\\downarrow }^{d}-{\\hat{V}}^{{{{{{{{\\rm{dc}}}}}}}}}$$ (1) where \\({\\hat{d}}_{{{{{{{{\\bf{k}}}}}}}}\\sigma }^{{{{\\dagger}}} }\\) ( \\({\\hat{s}}_{{{{{{{{\\bf{k}}}}}}}}\\sigma }^{{{{\\dagger}}} }\\) ) is the creation operator on Ni- \\({d}_{{x}^{2}-{y}^{2}}\\) (effective- s orbital) with momentum k and spin σ . \\({\\hat{n}}_{i\\sigma }^{d}={\\hat{d}}_{i\\sigma }^{{{{\\dagger}}} }{\\hat{d}}_{i\\sigma }\\) is the occupancy operator of Ni- \\({d}_{{x}^{2}-{y}^{2}}\\) orbital at site i with spin σ . \\({\\hat{V}}^{{{{{{{{\\rm{dc}}}}}}}}}\\) is the double counting potential [54] . The details of Eq. ( 1 ) are found in the Methods. In order to study the CO found in NdNiO 2 , we fit all the hopping parameters in the energy dispersion and hybridization to the DFT band structure. The non-interacting part of Eq. ( 1 ) reproduces the near-Fermi-level DFT band structure of NdNiO 2 , in which there are two bands crossing the Fermi level (see Supplementary Fig. 1 ). We mention one important parameter in the energy dispersion: the bare onsite energy difference between Ni- \\({d}_{{x}^{2}-{y}^{2}}\\) and effective- s orbitals E d s = 0.70 eV (see the Methods), which we will show is another key control parameter for the CO state besides the Hubbard interaction on Ni- d orbitals. Next, we consider three competing states in the low-energy effective model (see Fig. 3a for a schematic): the uniform PM state with dynamically fluctuating spins (top), the checkerboard AFM (middle) and the CO state with an ordering wavevector \\({{{{{{{\\bf{q}}}}}}}}=\\left(\\frac{1}{3},\\, 0,\\, 0\\right)\\) (bottom). Due to the simplicity of the model Eq. ( 1 ), we can obtain a highly accurate self-energy of Ni \\({d}_{{x}^{2}-{y}^{2}}\\) orbital, which enables us to calculate the total energy of each state as a function of U d and resolve the total energy difference between various competing phases (the details of the total energy calculations are found in the Methods). The results are shown in Fig. 3b . We find that when U d exceeds a critical value, the AFM and CO states can both be stabilized. More importantly, the CO state has lower energy than the AFM state and the PM state. To find a proper U d for our model, we calculate the effective mass of the Ni- \\({d}_{{x}^{2}-{y}^{2}}\\) orbital (see Supplementary Note 4 for details). We find that U d = 3.7 eV leads to an effective mass of 5.5, which exactly reproduces the result from the full ab initio GW+EDMFT calculations [55] . At this U d value, the CO state has lower energy than the PM state by about 50 meV/f.u., a magnitude reasonably consistent with the CO onset of infinite-layer nickelates in experiment [33] , [34] , [35] . Furthermore, the CO state is also more stable than the AFM state by about 30 meV/f.u. which may provide an explanation for the lacking of long-range AFM ordering in infinite-layer nickelates despite the substantial AFM superexchange found in experiment [14] , [15] . Fig. 3: Low-energy effective model. a Schematics of the three competing states studied in the effective model: the uniform PM state (top), the checkerboard AFM state (middle) and the \\({{{{{{{\\bf{q}}}}}}}}=\\left(\\frac{1}{3},\\, 0,\\, 0\\right)\\) CO state (bottom). b The total energy E calculated using the low-energy effective model Eq. ( 1 ) (up to a constant) in the PM (red), the AFM (blue) and the CO (green) states as a function of U d . The purple dashed line highlights U d = 3.7 eV, which reproduces the Ni- \\({d}_{{x}^{2}-{y}^{2}}\\) orbital effective mass from the ab initio GW+EDMFT calculation [55] . The error bar is smaller than the symbol size (see Methods). c – h Orbital-resolved spectral functions calculated by using the low-energy effective model with U d = 3.7 eV. c In the PM state. e In the AFM state. g In the CO state. d , f and h are the enlarged plots showing the corresponding near-Fermi-level features of the spectral function. Source data are provided as a Source Data file. Full size image Figure 3c–h compares the spectral functions of the low-energy effective model Eq. ( 1 ) for the PM, AFM and CO states (calculated with U d = 3.7 eV). Figure 3c displays the orbital-resolved spectral functions of the PM state. The red (blue) curve is the Ni- \\({d}_{{x}^{2}-{y}^{2}}\\) (effective- s ) projected spectral function. Ni- \\({d}_{{x}^{2}-{y}^{2}}\\) orbital exhibits a characteristic three-peak feature: lower Hubbard band (LHB), upper Hubbard band (UHB) and a quasi-particle peak around the Fermi level. The effective- s orbital has an occupancy of about 0.1 e /f.u., due to the self-doping effect. This result is consistent with the previous studies using similar effective models [18] , [25] , [56] . Figure 3d zooms-in the Ni- \\({d}_{{x}^{2}-{y}^{2}}\\) projected spectral function close to the Fermi level, which highlights the quasi-particle peak and the partially occupied conduction band. Figure 3e, f shows the spectral function of the AFM state. Compared to the PM state, the quasi-particle peak is more pronounced but shifts away from the Fermi level. The self-doping effect is weaker in the AFM state. Figure 3g shows the orbital-resolved spectral functions of the CO state, where the red and green curves are for the two different types of Ni- \\({d}_{{x}^{2}-{y}^{2}}\\) orbital, and the blue curve is for the effective- s orbital. Similar to the PM state, the Ni1- \\({d}_{{x}^{2}-{y}^{2}}\\) orbital in the CO state (red curve) exhibits a three-peak feature, but the Ni LHB moves toward lower energy. This indicates that the Ni1- \\({d}_{{x}^{2}-{y}^{2}}\\) orbital becomes more localized (i.e. closer to half-filling) in the CO state. In contrast, the Ni2- \\({d}_{{x}^{2}-{y}^{2}}\\) orbital in the CO state is almost empty. Compared to the PM state, the effective- s orbital is more occupied. The spectral functions in Fig. 3g qualitatively reproduce the Ni 1+ -Ni 2+ -Ni 1+ stripe pattern and the charge transfer from Ni2 site to the conduction bands, which are found from the preceding DFT+DMFT calculations. Figure 3h shows an enlarged plot of the Ni- \\({d}_{{x}^{2}-{y}^{2}}\\) projected spectral functions near the Fermi level in the CO state. The Ni- \\({d}_{{x}^{2}-{y}^{2}}\\) quasi-particle peak in the CO state has a smaller peak value than in the PM state (calculated at the same U d ), indicating that in addition to local Hubbard interaction, the formation of long-range CO further suppresses the quasi-particle peak (the effect is more pronounced at larger U d , see Supplementary Note 6 ). Energetics Next, we analyze the energetics of the low-energy effective model in more detail. In particular, we study why the CO state may have a lower energy than the AFM state. Figure 4a shows a schematic electronic structure of the effective model Eq. ( 1 ). The brown arrow indicates the charge transfer from a Ni site to neighboring effective- s orbitals. The upper panel is the one before the charge transfer, while the lower panel is after the charge transfer (when U d exceeds the critical value). The CO can emerge if there is an overall energy gain from the charge transfer. However, the energy of Ni LHB is lower than the effective- s orbitals. Thus, the electron transfer from the Ni2 site to the effective- s orbitals seems to result in an energy cost. To solve this paradox, we compare the total energy of the CO and AFM states and decompose the energy difference into three contributions: E = E kin + E pot − E dc , where E kin is the kinetic energy, E pot is the potential energy and E dc is the double-counting energy (see the Methods). Fig. 4: Charge transfer and energetics. a Schematics of the interacting electronic structure of the low-energy effective model Eq. ( 1 ). The brown arrow indicates the charge transfer from a Ni site to neighboring effective- s orbitals. The upper panel is the one before the charge transfer and the lower panel is after the charge transfer (when \\({U}_{d} > {U}_{d}^{{{{{{{{\\rm{critical}}}}}}}}}\\) ). b The total energy difference between the CO and the AFM states decomposed into kinetic energy (kin, red circles), potential energy (pot, blue squares) and double-counting energy (dc, green pentagons) as a function of U d , calculated by using the low-energy effective model Eq. ( 1 ). The purple dashed line highlights U d = 3.7 eV. The error bar is smaller than the symbol size (see Methods). Source data are provided as a Source Data file. Full size image We first analyze the case of U d = 3.7 eV, as highlighted by the purple dashed line in Fig. 4b . We find that compared to the AFM state, which does not have a charge transfer, the charge transfer indeed results in an energy cost in E kin for the CO state. This is consistent with the schematics shown in Fig. 4a . However, the charge transfer also leads to an energy gain in E pot . That is because, in the AFM state, every Ni site has almost one electron on the \\({d}_{{x}^{2}-{y}^{2}}\\) orbital. Due to the repulsive Hubbard interaction, \\({E}^{{{{{{{{\\rm{pot}}}}}}}}}=\\frac{1}{2}\\,{{{{\\rm{Tr}}}}}\\,({\\hat{{{\\Sigma }}}}_{{{{{\\rm{cor}}}}}}{\\hat{G}}_{{{{{\\rm{cor}}}}}}) \\, > \\, 0\\) for each Ni. In the CO state, the Ni1 and Ni3 sites have one electron on the \\({d}_{{x}^{2}-{y}^{2}}\\) orbital. But the charge transfer depletes the electron on the Ni2- \\({d}_{{x}^{2}-{y}^{2}}\\) orbital, so that its self-energy Σ cor essentially becomes zero. This decreases the potential energy and always leads to an energy gain Δ E pot < 0 when U d exceeds the critical value. As for the double-counting energy, \\({E}^{{{{{\\rm{dc}}}}}}=\\frac{1}{2}{U}_{d}{N}_{d}({N}_{d}-1)\\) [57] . The AFM state is close to a half-filled configuration for every Ni site, while the CO state has a configuration in which Ni2 site has almost zero occupancy and Ni1/Ni3 site is half-filled. Therefore, in both states, the double counting energy E dc is small and so is their difference (compared to Δ E pot ). In short, there is a delicate competition between charge transfer energy and local Hubbard interaction. When the potential energy gain outweighs the kinetic energy cost during the charge transfer, the CO state becomes more stable than the AFM state. Then, we analyze the case of a general U d (see Fig. 4b ). We find that the trend does not change in Δ E pot and Δ E dc . However, when U d is sufficiently large, Δ E kin itself becomes negative (i.e., a net energy gain after the charge transfer). That is because as U d increases, the Ni1/Ni3 LHB moves toward the lower energy after the charge transfer (i.e. ∣ Δ 2 ∣ > ∣ Δ 1 ∣ ). This effect decreases the kinetic energy and eventually counteracts the kinetic energy cost from the charge transfer (brown arrow in Fig. 4a ). CO stability and stripe pattern The preceding analysis shows that the charge transfer from Ni- \\({d}_{{x}^{2}-{y}^{2}}\\) orbital to the near-Fermi-level conduction bands plays a crucial role in the formation of the CO state. In our effective model, the conduction bands are derived from the effective- s orbitals. Figure 5a is a schematic non-interacting electronic structure of the effective model Eq. ( 1 ). We show below that the charge transfer energy E d s , which is defined as the energy separation between the bare onsite energy of Ni- \\({d}_{{x}^{2}-{y}^{2}}\\) and effective- s orbitals, controls not only stability but also stripe pattern of the CO state in the effective model. Fig. 5: CO stability and stripe pattern. a Schematics of the non-interacting electronic structure of the low-energy effective model Eq. ( 1 ) where E d s is the energy separation between the bare energy of Ni- \\({d}_{{x}^{2}-{y}^{2}}\\) and effective- s orbitals. b The total energy (up to a constant) of the PM (red), the AFM (blue) and the CO (green) states as a function of E d s , calculated by using the low-energy effective model Eq. ( 1 ) with U d = 3.7 eV. The orange dashed line highlights E d s = 0.7 eV, which is obtained from fitting to the DFT band structure of NdNiO 2 . c The total energy of different stripe CO states with the wavevector \\({{{{{{{\\bf{q}}}}}}}}=\\left(\\frac{1}{N},\\, 0,\\, 0\\right)\\) as a function of U d , calculated by using the low-energy effective model Eq. ( 1 ) with E d s = 0.7 eV. d The total energy of different stripe CO with the wavevector \\({{{{{{{\\bf{q}}}}}}}}=\\left(\\frac{1}{N},\\, 0,\\, 0\\right)\\) as a function of E d s , calculated by using the low-energy effective model with U d = 3.7 eV. In panels ( b – d ), the error bar is smaller than the symbol size (see Methods). Source data are provided as a Source Data file. Full size image We first examine the total energy of the PM, AFM and CO states as a function of E d s using the effective model Eq. ( 1 ). For each E d s , we recalculate the total energy of each state and show the results in Fig. 5b . We use U d = 3.7 eV but find a similar trend using other values of U d (see Supplementary Note 7 ). Fitting to the DFT band structure, we obtain E d s = 0.7 eV (the orange dashed line in Fig. 5b ), at which the CO state is more stable than the AFM and PM states. As E d s increases, the total energy of the CO state first becomes higher than the AFM state ( E d s > 0.8 eV) and then even higher than the PM state ( E d s > 0.9 eV). This can be understood because increasing E d s leads to a larger kinetic energy cost in the charge transfer (as we demonstrated in Fig. 4a ). Next, we study different stripe CO patterns and compare their total energies using the effective model Eq. ( 1 ). We use a N × 1 × 1 supercell to study the stripe CO with the wavevector \\({{{{{{{\\bf{q}}}}}}}}=\\left(\\frac{1}{N},\\, 0,\\, 0\\right)\\) . Before discussing the results, we first present an intuitive picture of how the CO total energy may depend on N . As we have shown, the CO state emerges from a charge transfer of almost one electron from one Ni site to the conduction bands for every N Ni atoms. During the charge transfer process, the electron needs to overcome the charge transfer energy E d s . This results in a kinetic energy cost ∝ E d s . On the other hand, the charge transfer depletes the charges on that Ni site. Since the local Hubbard interaction is repulsive in our model, this leads to a potential energy gain ∝ U d . Thus, the total energy of the entire system ( N -Ni-atom cell) is: $${E}_{N{{{{{{{\\rm{-Ni-atom}}}}}}}}}=\\alpha {E}_{ds}-\\beta {U}_{d}+N{E}_{0}$$ (2) where α and β are both positive and E 0 is the energy contribution that does not change during the charge transfer. The tricky point is that both α and β also depend on the wavevector or the wavelength (i.e., N ). Therefore, the total energy per-Ni-atom cell is: $${E}_{{{{{{{{\\rm{per-Ni-atom}}}}}}}}}=\\frac{{E}_{N{{{{{{{\\rm{-Ni-atom}}}}}}}}}}{N}=\\alpha (N)\\frac{{E}_{ds}}{N}-\\beta (N)\\frac{{U}_{d}}{N}+{E}_{0}$$ (3) For given E d s and U d , minimizing E per-Ni-atom with respect to N yields the optimal wavelength N or equivalently optimal wavevector q . However, due to interaction, the analytical expressions of α ( N ) and β ( N ) are not known. Therefore, we numerically calculate E N -Ni-atom as a function of N for different E d s and U d . Figure 5c compares the total energy of different stripe CO states for a few representative U d at E d s = 0.7 eV (DFT-fitted value). We find that at E d s = 0.7 eV, the \\({{{{{{{\\bf{q}}}}}}}}=\\left(\\frac{1}{3},\\, 0,\\, 0\\right)\\) CO has the lowest total energy in a range of U d values. The reason that the total energy of the stripe CO state does not monotonically change with N is because with N increasing, on the one hand, the kinetic energy cost decreases; on the other hand, the potential energy gain also decreases. These two factors are competing and the calculations find that the total energy minimum appears at an optimal CO wavevector \\({{{{{{{\\bf{q}}}}}}}}=(\\frac{1}{3},\\, 0,\\, 0)\\) . While our model is simplified, this result is consistent with the experimentally observed CO stripe pattern of infinite-layer nickelates [33] , [34] , [35] . Figure 5d compares the total energy of different stripe CO states for a few representative E d s at U d = 3.7 eV. We find that as E d s increases from 0.6 to 0.8 eV, the wavevector of the lowest energy CO state evolves from \\(\\left(\\frac{1}{3},\\, 0,\\, 0\\right)\\) to \\(\\left(\\frac{1}{4},\\, 0,\\, 0\\right)\\) . This is because as E d s changes, the decreasing rate of the kinetic energy with N also changes, which leads to a different optimal wavevector for the CO state. We mention that when E d s ≥ 0.8 eV, the AFM state has lower energy than the CO state (see Fig. 5b ). This means that based on our model calculations, the CO state with a wavevector of \\({{{{{{{\\bf{q}}}}}}}}=\\left(\\frac{1}{4},\\, 0,\\, 0\\right)\\) cannot be observed. Instead, it gives way to the AFM state. To summarize, in the phase space spanned by ( U d , E d s ), our effective model Eq. ( 1 ) has a parameter range in which the stripe CO state is more stable than the AFM and PM states with an optimal wavevector \\({{{{{{{\\bf{q}}}}}}}}=(\\frac{1}{3},\\, 0,\\, 0)\\) , which reproduces the experimental observations of infinite-layer nickelates [33] , [34] , [35] ."},{"section":"Discussion","content":"We make a few comments before the conclusion. First, we note that CO or charge density wave (CDW) may have different origins. In addition to electron–electron interaction, electron–phonon interaction may also play an important role in the formation of CDW, which has been intensively studied in other material systems such as transition metal dichalcogenides [58] . A CDW state that arises from electron–phonon interaction is accompanied by a structural distortion. Thus, in addition to using the ideal lattice structure of infinite-layer nickelates, we also study possible structural distortions in the CO state [59] , [60] . Since the charge transfer breaks the original translational symmetry, oxygen atoms may move away from their ideal positions (see Supplementary Note 5 ). The movement of oxygen atoms can be modeled by changing the onsite energy of Ni- \\({d}_{{x}^{2}-{y}^{2}}\\) orbital in the effective model Eq. ( 1 ). Our calculations find that the oxygen atom movements are minute (~0.05 Å), which is in agreement with the other study [61] . The resulting energy gain in CO is less than 5 meV/f.u., about one order of magnitude smaller than the energy difference shown in Fig. 3b . The energy gain may be used as a rough estimation to quantify the relative importance of different mechanisms. Our results imply that the CO state found in infinite-layer nickelates is more likely of electronic origin, while structural distortions driven by electron–phonon interaction may play a secondary role. Second, the CO state from our calculations has a substantial charge disproportionation, driven by a charge transfer from a portion of Ni atoms to the near-Fermi-level conduction bands. A direct consequence is that the average Ni occupancy in the CO state is substantially smaller than that of a uniform PM state. Such a CO state is dissimilar to the CDW state found in the calculations of the doped single-band Hubbard model [62] , [63] , doped t - J model [64] and a 2-orbital model of ref. [53] , in which the charge profile shows a Friedel oscillation with the average occupancy per site being equal to that of uniform PM state. In those states, variation in site occupancy is due to charge transfer between neighboring correlated sites. Third, as we mentioned previously, the analytical expression of CO energy dependence on wavevector E ( q ) is not known. Therefore, we cannot find the optimal wavevector q by solving ∇ q E ( q ) = 0. Instead, we have to test as many known competing states as possible (within computational capability). In addition to the \\({{{{{{{\\bf{q}}}}}}}}=\\left(\\frac{1}{N},\\, 0,\\, 0\\right)\\) CO state, we also test a different CO state, which can be accommodated by the available simulation cell (see Supplementary Note 8 ). We use the low-energy effective model to study all these competing states (PM state, AFM state and five CO states with different wavevectors) and compare their total energy. Our calculations find that using the DFT-fitted energy dispersion for a reasonable range of U d , the \\({{{{{{{\\bf{q}}}}}}}}=\\left(\\frac{1}{3},\\, 0,\\, 0\\right)\\) CO state is the most energetically favorable one among all the states considered. Fourth, a number of studies [27] , [31] , [65] , [66] have shown that Ni \\({d}_{3{z}^{2}-{r}^{2}}\\) orbital plays an important role in infinite-layer nickelates. Comparison of the low-energy effective model calculations to the 17-orbital model calculations, which include all five Ni- d orbitals, reveals that the minimum physics needed to account for the CO phenomenon is a charge transfer from a correlated Ni- \\({d}_{{x}^{2}-{y}^{2}}\\) orbital to a conduction band that is close to the Fermi level. Adding other orbitals into the effective model may describe more physical phenomena of infinite-layer nickelates, but as far as charge ordering is concerned, the 2-orbital effective model is sufficient. Finally, we briefly discuss how one may experimentally distinguish CO states of different origins in infinite-layer nickelates. If the CO in infinite-layer nickelates arises from the revealed charge-transfer mechanism, it will exhibit a mixed valence state of Ni, a characteristic Ni 1+ -Ni 2+ -Ni 1+ stripe pattern, as well as a unidirectional Ni 2+ chain in which holes are localized. In the experiment, one may probe this mixed valence state of Ni either by x-ray absorption spectroscopy (XAS) or cross-sectional electron energy loss spectroscopy (EELS) [36] . XAS can measure the average Ni valence, whose spectrum in the CO state should be distinct from that of pure Ni 1+ or Ni 2+ . A more informative measurement is the cross-sectional EELS, which may provide a spatially resolved spectrum of Ni 1+ and Ni 2+ columns in the CO state. In addition, scanning tunneling microscopy can image the localized holes in the unidirectional Ni 2+ chain [36] . On the other hand, if the CO or CDW is driven by phonon and electron–phonon coupling, a characteristic phonon of the ideal crystal structure should become soft around the CDW transition temperature and the emergence of CDW accompanies a periodic-lattice-distortion, which lowers the total energy [58] , [67] , [68] . The vibrational mode of characteristic phonon should match the pattern of periodic-lattice-distortion observed at low temperatures [67] , [68] . The magnitude of the periodic-lattice-distortion should be such that the resulting energy gain is comparable to the CDW transition temperature [58] . In experiment, inelastic x-ray scattering or neutron scattering can measure temperature dependence of phonon dispersion and identify soft phonons, if they exist around CDW temperature [68] . The low-temperature periodic-lattice-distortion can be probed either by x-ray diffraction (XRD) or by transmission electron microscopy (TEM). The emergence of periodic-lattice-distortion lowers the crystal symmetry and thus changes the diffraction pattern in XRD measurements [69] . TEM, in particular with the recent developments of electron ptychography [70] , can achieve atomic-resolution imaging, enabling visualization of periodic-lattice-distortion in real space and accurate measurements of structural distortions. Using the lattice distortions found from TEM measurements, one can estimate the energy gain from the periodic-lattice-distortion by performing first-principles calculations and then comparing the energy gain to the experimental CDW transition temperature. In short, the experimental observation of “a mixed valence state of Ni/localized holes on unidirectional Ni 2+ chain” versus “soft phonon/periodic-lattice-distortion” may distinguish different origins of CO in infinite-layer nickelates. We hope that the two sets of experiments outlined above, as well as other experimental techniques, will identify the true underlying mechanism among various theoretical proposals. To conclude, our first-principles calculations show that a \\({{{{{{{\\bf{q}}}}}}}}=\\left(\\frac{1}{3},\\, 0,\\, 0\\right)\\) CO state, which has a characteristic Ni 1+ -Ni 2+ -Ni 1+ stripe pattern, is stabilized in infinite-layer nickelates when Hubbard interaction on Ni- d orbitals is reasonably large. For every three Ni atoms, the CO state is driven by a charge transfer from one Ni atom to near-Fermi-level conduction bands, leaving electrons on the other two Ni atoms to become more localized. We further derive a simple low-energy effective model and elucidate that stabilization of the CO state over PM and AFM states is the consequence of a delicate competition between Hubbard interaction U d on a correlated orbital and charge transfer energy E d s between the correlated orbital and the conduction band. Such a competition may induce CO states in other correlated materials besides infinite-layer nickelates. Manipulating the CO state in infinite-layer nickelates by tuning the charge transfer energy may also affect their superconductivity [71] , which deserves further experimental study."},{"section":"Methods","content":"DFT+DMFT calculations We perform DFT+DMFT calculations [41] , [42] , [43] , [44] with the aid of MLWFs [45] , [72] . The DFT method is implemented in the Vienna ab initio simulation package code [73] with the projector augmented wave method [74] . The Perdew–Burke–Ernzerhof [75] functional is used as the exchange-correlation functional in DFT calculations. The Nd-4 f orbitals are treated as core states in the pseudopotential because the 4 f shell is hidden in the core of the Nd ion, and Hubbard interaction on 4 f orbitals is very large (>10 eV). As a result, the Nd-4 f electrons are far from the Fermi level and turn into strongly localized f spins. The direct coupling between Nd-4 f electrons and itinerant electrons is minute [76] . We use an energy cutoff of 600 eV and sample the Brillouin zone by using Γ-centered k -mesh of 12 × 12 × 12 per primitive cell. The crystal structure is fully relaxed with an energy convergence criterion of 10 −6 eV, force convergence criterion of 0.01 eV/Å and strain convergence of 0.1 kbar. The DFT-optimized crystal structures are in good agreement with the experimental P 4/ m m m structures ( a DFT = 3.91 Å, \\({a}^{{{{{{{{\\rm{EXP}}}}}}}}}\\) = 3.92 Å, c DFT = 3.31 Å, \\({c}^{{{{{{{{\\rm{EXP}}}}}}}}}\\) = 3.28 Å). To describe the checkerboard antiferromagnetic ordering, we expand the cell to a \\(\\sqrt{2}\\times \\sqrt{2}\\times 1\\) supercell. To describe the \\({{{{{{{\\bf{q}}}}}}}}=\\left(\\frac{1}{N},\\, 0,\\, 0\\right)\\) CO state ( N is an integer), we expand the cell to an N × 1 × 1 supercell. We downfold the DFT-calculated band structure of NdNiO 2 to a 17-orbital tight-binding model by using the MLWF method [45] . The 17 orbitals include five Ni- d orbitals, five Nd- d orbitals, six O- p orbitals and one interstitial s orbital. This 17-orbital model can well reproduce the non-interacting band structure of NdNiO 2 in an energy window of about 15 eV around the Fermi level (see Supplementary Fig. 1 ). A Slater–Kanamori interaction is added on the Ni- d orbitals [77] . A fully localized-limit (FLL) double counting correction is applied to the full interacting model [57] . We also test other double counting correction [48] and no qualitative change is found in the main results. More details of the full Hamiltonian are found in Supplementary Note 1 . We treat the two Ni e g orbitals with DMFT method, while the filled Ni t 2 g shell is treated with a static Hartree–Fock approximation [56] , [78] . When we use the DMFT method to solve the interacting 17-orbital model, we employ the continuous-time quantum Monte Carlo (CTQMC) algorithm based on hybridization expansion [79] , [80] . The impurity solver was developed by Haule [81] . For each DMFT iteration, a total of 1 billion Monte Carlo samples are collected to converge the impurity Green function and self-energy. We set the temperature to be 116 K. We check all the main results at a lower temperature of 58 K and no significant difference is found. For the CO state, we allow all the Ni sites to be inequivalent and the DMFT self-consistent condition involves the self-energies of all inequivalent Ni atoms. To obtain the spectral functions, the imaginary axis self-energy is continued to the real axis by using the maximum entropy method [82] . Then, the real axis local Green function is calculated using the Dyson equation. A 90 × 90 × 90 k -point mesh is used to converge the spectral function of the 17-orbital model. The full charge-self-consistency does not qualitatively change the main results of charge transfer and charge disproportionation in the calculations [83] . To relieve the computational burden, the results shown in the main text are from one-shot DFT+DMFT calculations. We also apply the above relevant parameters when using DMFT to solve the low-energy effective model. Low-energy effective model The low-energy effective model to describe the CO phenomenon is a 2-orbital model, which can be compactly written as follows: $$\\hat{H}=\\mathop{\\sum}\\limits_{{{{{{{{\\bf{k}}}}}}}}\\sigma }{\\hat{{{\\Psi }}}}_{{{{{{{{\\bf{k}}}}}}}}\\sigma }^{{{{\\dagger}}} }{{{{{{{{\\mathcal{H}}}}}}}}}_{0}({{{{{{{\\bf{k}}}}}}}}){\\hat{{{\\Psi }}}}_{{{{{{{{\\bf{k}}}}}}}}\\sigma }+{U}_{d}\\mathop{\\sum}\\limits_{i}{\\hat{n}}_{i\\uparrow }^{d}{\\hat{n}}_{i\\downarrow }^{d}-{\\hat{V}}^{{{{{{{{\\rm{dc}}}}}}}}}$$ (4) where \\({\\hat{{{\\Psi }}}}_{{{{{{{{\\bf{k}}}}}}}}\\sigma }^{{{{\\dagger}}} }=({\\hat{d}}_{{{{{{{{\\bf{k}}}}}}}}\\sigma }^{{{{\\dagger}}} },\\, {\\hat{s}}_{{{{{{{{\\bf{k}}}}}}}}\\sigma }^{{{{\\dagger}}} })\\) are the creation operators on Ni- \\({d}_{{x}^{2}-{y}^{2}}\\) and effective- s orbitals with momentum k and spin σ . \\({\\hat{n}}_{i\\sigma }^{d}={\\hat{d}}_{i\\sigma }^{{{{\\dagger}}} }{\\hat{d}}_{i\\sigma }\\) is the occupancy operator of Ni- \\({d}_{{x}^{2}-{y}^{2}}\\) orbital at site i with spin σ . \\({{{{{{{{\\mathcal{H}}}}}}}}}_{0}({{{{{{{\\bf{k}}}}}}}})\\) is a 2 × 2 matrix: $${{{{{{{{\\mathcal{H}}}}}}}}}_{0}({{{{{{{\\bf{k}}}}}}}})=\\left[\\begin{array}{ll}{\\epsilon }_{d}({{{{{{{\\bf{k}}}}}}}})&{V}_{ds}({{{{{{{\\bf{k}}}}}}}})\\\\ {V}_{ds}^{*}({{{{{{{\\bf{k}}}}}}}})&{\\epsilon }_{s}({{{{{{{\\bf{k}}}}}}}})\\end{array}\\right]$$ (5) The energy dispersion and hybridization terms are: $${\\epsilon }_{d}({{{{{{{\\bf{k}}}}}}}})=\t 2{t}_{d}^{100}\\left(\\cos {k}_{x}+\\cos {k}_{y}\\right)+4{t}_{d}^{110}\\cos {k}_{x}\\cos {k}_{y} \\\\ \t+2{t}_{d}^{200}\\left(\\cos 2{k}_{x}+\\cos 2{k}_{y}\\right)$$ (6) $${\\epsilon }_{s}({{{{{{{\\bf{k}}}}}}}})=\t {E}_{ds}+2{t}_{s}^{100}\\left(\\cos {k}_{x}+\\cos {k}_{y}\\right)+2{t}_{s}^{001}\\cos {k}_{z}+4{t}_{s}^{110}\\cos {k}_{x}\\cos {k}_{y}\\\\ \t+4{t}_{s}^{101}\\left(\\cos {k}_{x}+\\cos {k}_{y}\\right)\\cos {k}_{z}+8{t}_{s}^{111}\\cos {k}_{x}\\cos {k}_{y}\\cos {k}_{z}$$ (7) $${V}_{ds}({{{{{{{\\bf{k}}}}}}}})=2{t}_{ds}^{100}\\left(\\cos {k}_{x}-\\cos {k}_{y}\\right)\\left(1+{e}^{-i{k}_{z}}\\right)$$ (8) The onsite energy difference E d s and hopping parameters are obtained by fitting to the near-Fermi-level DFT band structure of NdNiO 2 (see Supplementary Fig. 1 ). The fitted values are shown in Table 1 . We use the FLL double counting and V dc reads: $${V}^{{{{{{{{\\rm{dc}}}}}}}}}={U}_{d}\\left({N}_{d}-\\frac{1}{2}\\right)$$ (9) Table 1 The onsite energy and hopping matrix elements in the 2-orbital model Full size table Total energy For the low-energy effective model, we calculate the total energy within the DFT+DMFT method. The total energy in the DFT+DMFT calculations has the following expression [44] , [84] : $${E}^{{{{{{{{\\rm{DFT+DMFT}}}}}}}}}={E}^{{{{{{{{\\rm{DFT}}}}}}}}}+{E}^{{{{{{{{\\rm{DMFT}}}}}}}}}-{E}^{{{{{{{{\\rm{dc}}}}}}}}}$$ (10) Here E DFT is: $${E}^{{{{{{{{\\rm{DFT}}}}}}}}}={E}_{0}^{{{{{{{{\\rm{DFT}}}}}}}}}-{E}_{0}^{{{{{{{{\\rm{kin}}}}}}}}}$$ (11) where \\({E}_{0}^{{{{{{{{\\rm{DFT}}}}}}}}}\\) is the standard DFT total energy. \\({E}_{0}^{{{{{{{{\\rm{kin}}}}}}}}}\\) is the kinetic energy of the non-interaction Hamiltonian within the model space: $${E}_{0}^{{{{{{{{\\rm{kin}}}}}}}}}=\\,{{\\mbox{Tr}}}\\,\\left(\\hat{{{{{{{{{\\mathcal{H}}}}}}}}}_{0}}\\hat{{G}_{0}}\\right)=\\frac{1}{{N}_{{{{{{{{\\bf{k}}}}}}}}}}\\mathop{\\sum}\\limits_{{{{{{{{\\bf{k}}}}}}}}\\sigma }\\mathop{\\sum}\\limits_{l}{\\epsilon }_{{{{{{{{\\bf{k}}}}}}}}l}{n}_{F}({\\epsilon }_{{{{{{{{\\bf{k}}}}}}}}l}-\\mu )$$ (12) where n F is the Fermi-Dirac occupancy, l is the band index and σ is the spin. In the main text, we use the DFT-optimized crystal structure for all the states. Thus, the PM, AFM and CO states all share the same non-interacting Hamiltonian (up to a multiplication factor), we omit the E DFT in the calculation of total energy (since it is identical for all three states). The remaining total energy is $$E={E}^{{{{{{{{\\rm{DMFT}}}}}}}}}-{E}^{{{{{{{{\\rm{dc}}}}}}}}}={E}^{{{{{{{{\\rm{kin}}}}}}}}}+{E}^{{{{{{{{\\rm{pot}}}}}}}}}-{E}^{{{{{{{{\\rm{dc}}}}}}}}}$$ (13) Here, the DMFT kinetic energy E kin is defined as follows: $${E}^{{{{{\\rm{kin}}}}}}={{{{\\rm{Tr}}}}}\\,\\left(\\hat{{{{{{{{{\\mathcal{H}}}}}}}}}_{0}}\\hat{G}\\right)=\\frac{T}{{N}_{{{{{{{{\\bf{k}}}}}}}}}}\\mathop{\\sum}\\limits_{{{{{{{{\\bf{k}}}}}}}}\\sigma }\\mathop{\\sum}\\limits_{{\\omega }_{n}}\\mathop{\\sum}\\limits_{m{m}^{{\\prime} }}{\\left[{{{{{{{{\\mathcal{H}}}}}}}}}_{0}({{{{{{{\\bf{k}}}}}}}})\\right]}_{m{m}^{{\\prime} }}{\\left[{G}_{\\sigma }({{{{{{{\\bf{k}}}}}}}},\\, i{\\omega }_{n})\\right]}_{{m}^{{\\prime} }m}$$ (14) where σ is the spin, T is the temperature, \\(m,{m}^{{\\prime} }\\) are the indices of orbital basis. \\({{{{{{{{\\mathcal{H}}}}}}}}}_{0}({{{{{{{\\bf{k}}}}}}}})\\) is the non-interacting part of the effective model Eq. ( 5 ) and G is the dressed lattice Green function of the interacting model. The DMFT potential energy E pot is defined as follows: $${E}^{{{{{\\rm{pot}}}}}}=\t \\frac{1}{2}{{{{{{{\\rm{Tr}}}}}}}}\\left({\\hat{{{\\Sigma }}}}_{{{{{{{{\\rm{cor}}}}}}}}}{\\hat{G}}_{{{{{{{{\\rm{cor}}}}}}}}}\\right)=\\frac{1}{2}T\\mathop{\\sum}\\limits_{\\sigma }\\mathop{\\sum}\\limits_{{\\omega }_{n}}\\mathop{\\sum}\\limits_{m}{e}^{-i{\\omega }_{n}{0}^{-}}\\\\ \t {\\left[{{{\\Sigma }}}_{{{{{\\rm{cor}}}}}}^{\\sigma }(i{\\omega }_{n})\\right]}_{m}{\\left[{G}_{{{{{\\rm{cor}}}}}}^{\\sigma }(i{\\omega }_{n})\\right]}_{m}$$ (15) where σ is the spin, T is the temperature, m is the index of orbital basis. Σ cor and G cor are the self-energy and the local Green function in the correlated subspace. E dc is the double counting energy. We employ the FLL double counting and E dc reads: $${E}^{{{{{{{{\\rm{dc}}}}}}}}}=\\frac{1}{2}{U}_{d}{N}_{d}({N}_{d}-1)$$ (16) Taking the derivative of E dc with respect to N d yields the double-counting potential V dc Eq. ( 9 ). In the low-energy effective model, the Hund’s term vanishes because there is only one correlated orbital Ni- \\({d}_{{x}^{2}-{y}^{2}}\\) per site. We note that when we study the structural distortions in the CO state, the crystal structure changes in each calculation and so does the DFT total energy. In that case, we need to use Eq. ( 10 ) to calculate the DFT+DMFT total energy. Since the total energy is calculated using CTQMC, we average the total energy of the last ten DMFT iterations. We find that due to the highly accurate self-energy, the standard deviation of the total energy is about 1 meV/f.u., which is smaller than the symbol size in the figures."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"Motivated by cuprates, superconductivity was proposed [1] , [2] and has been recently discovered in hole-doped infinite-layer nickelates that contain Ni 1+ ions of the similar d 9 electronic configuration [3] , [4] , [5] , [6] , [7] , [8] , [9] . While this seems to confirm the initial expectation and support the Mott scenario for high-temperature superconductivity [10] , the nickelates show very different properties even in their undoped parent compounds. Instead of an antiferromagnetic Mott insulator, the resistivity of the parent compounds exhibits metallic behavior at high temperatures and an upturn below around 70 K [3] , [8] , [11] , [12] . So far, no static long-range magnetic order has been observed down to the lowest measured temperature [13] , [14] , although spin fluctuations have been reported in experiment [15] . These are all different from cuprates and have stimulated heated debate concerning the nature of the nickelate parent state, from which superconductivity is born upon hole doping. Although several theoretical scenarios [16] , [17] , [18] , [19] , [20] have been proposed to emphasize the key role of single-band Hubbard interaction, a growing consensus is that multi-band physics is indispensable in nickelates [21] , [22] , [23] , [24] , [25] , [26] , [27] , [28] , [29] , [30] , [31] , [32] . What has not been sufficiently explored is whether this multi-band feature combined with a strong correlation on Ni- d orbitals may lead to some unique experimental consequences in infinite-layer nickelates. Recently, a charge order (CO) with broken translational symmetry has been reported independently by three experimental groups in the parent compounds of nickelate superconductors [33] , [34] , [35] . This raises a question concerning the CO origin and its competition with other phases. CO in cuprates is well known and has been intensively studied [36] , [37] . Despite the seeming similarity, the CO in nickelates has a few important differences [33] , [34] , [35] . First, it already exists and has the highest onset temperature in undoped compounds. Second, unlike cuprates where doped holes reside on oxygen atoms, holes are mainly introduced into Ni- d orbitals, as supported by the experiments [38] , [39] . Finally and most importantly, it has an ordering wavevector \\\\({{{{{{{\\\\bf{q}}}}}}}}\\\\simeq \\\\left(\\\\frac{1}{3},\\\\, 0,\\\\, 0\\\\right)\\\\) , compatible with a commensurate modulation with period 3 a 0 ( a 0 is the lattice constant). This stripe pattern has not been observed in cuprates [40] . Hence, these differences may imply a new mechanism for the CO in the nickelates, which is most probably associated with the multi-band nature of the parent compounds [21] , [22] , [23] , [24] , [25] , [26] , [27] , [28] , [29] , [30] , [31] , [32] . In this work, we first use density functional theory plus dynamical mean field theory (DFT+DMFT) calculations [41] , [42] , [43] , [44] to reveal a special charge-transfer-driven CO mechanism in a prototypical infinite-layer nickelate NdNiO 2 . We find that for every three Ni atoms, due to the presence of Nd- d /interstitial- s derived conduction bands near the Fermi level, electrons on one Ni atom are transferred to the conduction bands under a reasonably large Hubbard interaction U Ni on Ni- d orbitals, while electrons on the other two Ni atoms become more localized. This leads to a characteristic Ni 1+ -Ni 2+ -Ni 1+ CO stripe state. Then, we derive a low-energy effective model based on the realistic band structure of NdNiO 2 . Using the effective model, we further elucidate that the \\\\({{{{{{{\\\\bf{q}}}}}}}}=(\\\\frac{1}{3},\\\\, 0,\\\\, 0)\\\\) CO stripe state arises from a delicate competition between Hubbard interaction on Ni- d orbitals and charge transfer energy between Ni- d orbitals and conduction bands. Tuning the charge transfer energy controls not only the stability of the CO state but also its stripe pattern. With Hubbard interaction strength and charge transfer energy in a physically reasonable range, we find that the experimentally observed \\\\({{{{{{{\\\\bf{q}}}}}}}}=\\\\left(\\\\frac{1}{3},\\\\, 0,\\\\, 0\\\\right)\\\\) stripe CO state is more stable than both the uniform paramagnetic (PM) state and the usual checkerboard antiferromagnetic state.\",\n \"references\": {\n \"indices\": [],\n \"infos\": []\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"DFT+DMFT calculations Unlike cuprate superconductors, a consensus has not been reached regarding what is the minimal model to study infinite-layer nickelates. Therefore, we first downfold the DFT-calculated band structure of NdNiO 2 to a large 17-orbital tight-binding model by using the maximally localized Wannier functions (MLWFs) [45] . The model includes five Ni- d orbitals, five Nd- d orbitals, six O- p orbitals and one interstitial- s orbital per primitive cell. Our previous studies [25] , [46] show that this 17-orbital model is sufficiently large that it can almost exactly reproduce the non-interacting band structure of NdNiO 2 in an energy window of about 15 eV around the Fermi level (see Supplementary Fig. 1 ). A Slater–Kanamori interaction is added on Ni- d orbitals. We use DMFT to solve the interacting model. To test a possible \\\\({{{{{{{\\\\bf{q}}}}}}}}=\\\\left(\\\\frac{1}{3},\\\\, 0,\\\\, 0\\\\right)\\\\) CO state, we use a 3 × 1 × 1 supercell. Since in the 17-orbital model, the MLWF for the Ni- d orbitals are very localized and atomic-like (see Supplementary Table 1 ), we use a Hubbard U Ni = 10 eV and a Hund’s exchange J Ni = 1 eV, which has proved to be reasonable for this large energy window treatment of transition metal oxides [26] , [47] , [48] , [49] . More computational details are found in the Methods. We first add perturbations on the three Ni atoms in the simulation cell. After the self-consistency loop is converged, we do find a CO state whose spectral function is shown in the Fig. 1a, b . For comparison, we also enforce symmetry on the three Ni atoms and thus obtain a uniform PM state whose spectral function is shown in Fig. 1c, d . Figure 1a, c shows the Ni- d and O- p projected spectral functions. In the CO state, we find two different types of Ni atoms. The spectral functions for Ni1- d and Ni3- d orbitals are identical (collectively referred to as Ni1- d ). Ni2- d orbital is of the other type. In the PM state, all three Ni atoms are equivalent due to the imposed symmetry. The inset shows the Ni- d occupancy. We find that compared to Ni- d occupancy in the uniform PM state, in the CO state, the Ni1- d occupancy is larger and approaches 9 e , but the Ni2- d occupancy is substantially smaller and is close to 8 e . This leads to a characteristic Ni 1+ -Ni 2+ -Ni 1+ stripe pattern. More importantly, the average Ni- d occupancy in the CO state is smaller than that in the PM state, indicating that the CO is not a simple charge modulation between the three Ni atoms, but rather there is a net charge transfer from the Ni2 atom to other atoms. Figure 1b, d shows the Nd- d and interstitial- s projected spectral functions, which are almost identical between the CO and the PM states. However, close to the Fermi level, we find appreciable differences that Nd- d and interstitial- s states are more occupied in the CO state than in the PM state. Therefore, the emergence of CO state in NdNiO 2 is accompanied by a charge transfer from Ni2 atom to Nd- d /interstitial- s orbitals that form conduction bands close to the Fermi level. We will show below that it is precisely this charge transfer that can yield an energy gain and stabilize the CO state. Fig. 1: Spectral functions of NdNiO 2 . The spectral functions are calculated by DFT+DMFT method using the 17-orbital model with U Ni = 10 eV and J Ni = 1 eV. a , b The charge-ordered state (CO). a The green, red and purple curves are Ni1- d , Ni2- d and O- p projected spectral functions, respectively. The inset shows the site-resolved Ni- d occupancy. The d occupancy of Ni2 site is substantially smaller than that of Ni1 and Ni3 sites. b The blue and orange curves are Nd- d and interstitial- s projected spectral functions, respectively. The inset shows the energy range near the Fermi level. c , d The uniform paramagnetic state (PM). c The green and purple curves are Ni- d and O- p projected spectral functions, respectively. The inset shows the site-resolved Ni- d occupancy. d identical to ( b ) but for the PM state. In all panels, the dashed line is the Fermi level, which is shifted to zero point. Source data are provided as a Source Data file. Full size image Next, we study how the charge disproportionation of the CO state depends on U Ni and J Ni . Figure 2a shows the Ni- d occupancy of both CO and PM states as a function of U Ni with J Ni being fixed at 1 eV. As U Ni exceeds 6 eV, the charge disproportionation of Ni- d occupancy develops in the CO state and becomes more pronounced with U Ni . On the other hand, changing J Ni in a reasonable range (from 0.7 to 1.0 eV) with U Ni being fixed at 10 eV does not strongly affect the charge disproportionation in the CO state. Furthermore, we also test the correlation effects from Nd- d orbitals. By considering Hubbard interaction on Nd- d orbitals, we do not find qualitative changes in the main results (see Supplementary Note 3 ). Fig. 2: Dependence of Ni- d occupancy N d on U Ni and J Ni . The dependence of N d on U Ni and J Ni is calculated by using the 17-orbital model. a Ni- d occupancy N d as a function of U Ni with J Ni = 1 eV. b Ni- d occupancy N d as a function of J Ni with U Ni = 10 eV. In both panels, the red solid symbols and blue open symbols correspond to the Ni- d occupancy in the charge-ordered (CO) and uniform paramagnetic (PM) states, respectively. Source data are provided as a Source Data file. Full size image Low-energy effective model To gain a deeper understanding of the origin and generality of the CO state that is found in the DFT+DMFT calculations of NdNiO 2 , we build a low-energy effective model. We want to make the model simple, which aims to qualitatively reproduce the key features of the CO state. The advantage of such a simple effective model is that we can obtain an accurate self-energy so as to resolve the total energy difference between different competing phases. In addition, it also helps us identify the control parameters for the CO state. The effective model, equipped with proper energy dispersions, may also be applied to the study of the charge ordering phenomenon in other strongly correlated materials. From the preceding DFT+DMFT calculations, we find that there are two key ingredients in the formation of the CO state in NdNiO 2 . One is the Hubbard interaction on Ni- d orbitals and the other is the charge transfer from Ni- d orbitals to the conduction bands that consist of Nd- d /interstitial- s orbitals. In order to describe the interplay between these two factors, we include a correlated orbital and an auxiliary orbital in the effective model. To make a connection to infinite-layer nickelates, we choose Ni- \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) orbital as the correlated orbital and add local Hubbard interaction on it. We also choose an effective- s orbital as the auxiliary orbital that provides a free conduction band. We note that in real NdNiO 2 , there are multiple conduction bands that are composed of Nd- d and interstitial- s orbitals. Here the effective- s orbital (not to be confused with the interstitial- s orbital) is employed to simplify the description of conduction bands. Similar models of correlated Ni- \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) orbital plus free conduction bands have been used to study the low-energy physics of infinite-layer nickelates [17] , [25] , [32] , [50] , [51] , [52] , [53] . The full Hamiltonian of our effective model takes the form: $$\\\\hat{H} =\\t \\\\mathop{\\\\sum}\\\\limits_{{{{{{{{\\\\bf{k}}}}}}}}\\\\sigma }\\\\left[{\\\\epsilon }_{d}({{{{{{{\\\\bf{k}}}}}}}}){\\\\hat{d}}_{{{{{{{{\\\\bf{k}}}}}}}}\\\\sigma }^{{{{\\\\dagger}}} }{\\\\hat{d}}_{{{{{{{{\\\\bf{k}}}}}}}}\\\\sigma }+{\\\\epsilon }_{s}({{{{{{{\\\\bf{k}}}}}}}}){\\\\hat{s}}_{{{{{{{{\\\\bf{k}}}}}}}}\\\\sigma }^{{{{\\\\dagger}}} }{\\\\hat{s}}_{{{{{{{{\\\\bf{k}}}}}}}}\\\\sigma }+\\\\left({V}_{ds}({{{{{{{\\\\bf{k}}}}}}}}){\\\\hat{d}}_{{{{{{{{\\\\bf{k}}}}}}}}\\\\sigma }^{{{{\\\\dagger}}} }{s}_{{{{{{{{\\\\bf{k}}}}}}}}\\\\sigma }+{{{{{{{\\\\rm{h.c.}}}}}}}}\\\\right)\\\\right] \\\\\\\\ \\t+{U}_{d}\\\\mathop{\\\\sum}\\\\limits_{i}{\\\\hat{n}}_{i\\\\uparrow }^{d}{\\\\hat{n}}_{i\\\\downarrow }^{d}-{\\\\hat{V}}^{{{{{{{{\\\\rm{dc}}}}}}}}}$$ (1) where \\\\({\\\\hat{d}}_{{{{{{{{\\\\bf{k}}}}}}}}\\\\sigma }^{{{{\\\\dagger}}} }\\\\) ( \\\\({\\\\hat{s}}_{{{{{{{{\\\\bf{k}}}}}}}}\\\\sigma }^{{{{\\\\dagger}}} }\\\\) ) is the creation operator on Ni- \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) (effective- s orbital) with momentum k and spin σ . \\\\({\\\\hat{n}}_{i\\\\sigma }^{d}={\\\\hat{d}}_{i\\\\sigma }^{{{{\\\\dagger}}} }{\\\\hat{d}}_{i\\\\sigma }\\\\) is the occupancy operator of Ni- \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) orbital at site i with spin σ . \\\\({\\\\hat{V}}^{{{{{{{{\\\\rm{dc}}}}}}}}}\\\\) is the double counting potential [54] . The details of Eq. ( 1 ) are found in the Methods. In order to study the CO found in NdNiO 2 , we fit all the hopping parameters in the energy dispersion and hybridization to the DFT band structure. The non-interacting part of Eq. ( 1 ) reproduces the near-Fermi-level DFT band structure of NdNiO 2 , in which there are two bands crossing the Fermi level (see Supplementary Fig. 1 ). We mention one important parameter in the energy dispersion: the bare onsite energy difference between Ni- \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) and effective- s orbitals E d s = 0.70 eV (see the Methods), which we will show is another key control parameter for the CO state besides the Hubbard interaction on Ni- d orbitals. Next, we consider three competing states in the low-energy effective model (see Fig. 3a for a schematic): the uniform PM state with dynamically fluctuating spins (top), the checkerboard AFM (middle) and the CO state with an ordering wavevector \\\\({{{{{{{\\\\bf{q}}}}}}}}=\\\\left(\\\\frac{1}{3},\\\\, 0,\\\\, 0\\\\right)\\\\) (bottom). Due to the simplicity of the model Eq. ( 1 ), we can obtain a highly accurate self-energy of Ni \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) orbital, which enables us to calculate the total energy of each state as a function of U d and resolve the total energy difference between various competing phases (the details of the total energy calculations are found in the Methods). The results are shown in Fig. 3b . We find that when U d exceeds a critical value, the AFM and CO states can both be stabilized. More importantly, the CO state has lower energy than the AFM state and the PM state. To find a proper U d for our model, we calculate the effective mass of the Ni- \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) orbital (see Supplementary Note 4 for details). We find that U d = 3.7 eV leads to an effective mass of 5.5, which exactly reproduces the result from the full ab initio GW+EDMFT calculations [55] . At this U d value, the CO state has lower energy than the PM state by about 50 meV/f.u., a magnitude reasonably consistent with the CO onset of infinite-layer nickelates in experiment [33] , [34] , [35] . Furthermore, the CO state is also more stable than the AFM state by about 30 meV/f.u. which may provide an explanation for the lacking of long-range AFM ordering in infinite-layer nickelates despite the substantial AFM superexchange found in experiment [14] , [15] . Fig. 3: Low-energy effective model. a Schematics of the three competing states studied in the effective model: the uniform PM state (top), the checkerboard AFM state (middle) and the \\\\({{{{{{{\\\\bf{q}}}}}}}}=\\\\left(\\\\frac{1}{3},\\\\, 0,\\\\, 0\\\\right)\\\\) CO state (bottom). b The total energy E calculated using the low-energy effective model Eq. ( 1 ) (up to a constant) in the PM (red), the AFM (blue) and the CO (green) states as a function of U d . The purple dashed line highlights U d = 3.7 eV, which reproduces the Ni- \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) orbital effective mass from the ab initio GW+EDMFT calculation [55] . The error bar is smaller than the symbol size (see Methods). c – h Orbital-resolved spectral functions calculated by using the low-energy effective model with U d = 3.7 eV. c In the PM state. e In the AFM state. g In the CO state. d , f and h are the enlarged plots showing the corresponding near-Fermi-level features of the spectral function. Source data are provided as a Source Data file. Full size image Figure 3c–h compares the spectral functions of the low-energy effective model Eq. ( 1 ) for the PM, AFM and CO states (calculated with U d = 3.7 eV). Figure 3c displays the orbital-resolved spectral functions of the PM state. The red (blue) curve is the Ni- \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) (effective- s ) projected spectral function. Ni- \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) orbital exhibits a characteristic three-peak feature: lower Hubbard band (LHB), upper Hubbard band (UHB) and a quasi-particle peak around the Fermi level. The effective- s orbital has an occupancy of about 0.1 e /f.u., due to the self-doping effect. This result is consistent with the previous studies using similar effective models [18] , [25] , [56] . Figure 3d zooms-in the Ni- \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) projected spectral function close to the Fermi level, which highlights the quasi-particle peak and the partially occupied conduction band. Figure 3e, f shows the spectral function of the AFM state. Compared to the PM state, the quasi-particle peak is more pronounced but shifts away from the Fermi level. The self-doping effect is weaker in the AFM state. Figure 3g shows the orbital-resolved spectral functions of the CO state, where the red and green curves are for the two different types of Ni- \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) orbital, and the blue curve is for the effective- s orbital. Similar to the PM state, the Ni1- \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) orbital in the CO state (red curve) exhibits a three-peak feature, but the Ni LHB moves toward lower energy. This indicates that the Ni1- \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) orbital becomes more localized (i.e. closer to half-filling) in the CO state. In contrast, the Ni2- \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) orbital in the CO state is almost empty. Compared to the PM state, the effective- s orbital is more occupied. The spectral functions in Fig. 3g qualitatively reproduce the Ni 1+ -Ni 2+ -Ni 1+ stripe pattern and the charge transfer from Ni2 site to the conduction bands, which are found from the preceding DFT+DMFT calculations. Figure 3h shows an enlarged plot of the Ni- \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) projected spectral functions near the Fermi level in the CO state. The Ni- \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) quasi-particle peak in the CO state has a smaller peak value than in the PM state (calculated at the same U d ), indicating that in addition to local Hubbard interaction, the formation of long-range CO further suppresses the quasi-particle peak (the effect is more pronounced at larger U d , see Supplementary Note 6 ). Energetics Next, we analyze the energetics of the low-energy effective model in more detail. In particular, we study why the CO state may have a lower energy than the AFM state. Figure 4a shows a schematic electronic structure of the effective model Eq. ( 1 ). The brown arrow indicates the charge transfer from a Ni site to neighboring effective- s orbitals. The upper panel is the one before the charge transfer, while the lower panel is after the charge transfer (when U d exceeds the critical value). The CO can emerge if there is an overall energy gain from the charge transfer. However, the energy of Ni LHB is lower than the effective- s orbitals. Thus, the electron transfer from the Ni2 site to the effective- s orbitals seems to result in an energy cost. To solve this paradox, we compare the total energy of the CO and AFM states and decompose the energy difference into three contributions: E = E kin + E pot − E dc , where E kin is the kinetic energy, E pot is the potential energy and E dc is the double-counting energy (see the Methods). Fig. 4: Charge transfer and energetics. a Schematics of the interacting electronic structure of the low-energy effective model Eq. ( 1 ). The brown arrow indicates the charge transfer from a Ni site to neighboring effective- s orbitals. The upper panel is the one before the charge transfer and the lower panel is after the charge transfer (when \\\\({U}_{d} > {U}_{d}^{{{{{{{{\\\\rm{critical}}}}}}}}}\\\\) ). b The total energy difference between the CO and the AFM states decomposed into kinetic energy (kin, red circles), potential energy (pot, blue squares) and double-counting energy (dc, green pentagons) as a function of U d , calculated by using the low-energy effective model Eq. ( 1 ). The purple dashed line highlights U d = 3.7 eV. The error bar is smaller than the symbol size (see Methods). Source data are provided as a Source Data file. Full size image We first analyze the case of U d = 3.7 eV, as highlighted by the purple dashed line in Fig. 4b . We find that compared to the AFM state, which does not have a charge transfer, the charge transfer indeed results in an energy cost in E kin for the CO state. This is consistent with the schematics shown in Fig. 4a . However, the charge transfer also leads to an energy gain in E pot . That is because, in the AFM state, every Ni site has almost one electron on the \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) orbital. Due to the repulsive Hubbard interaction, \\\\({E}^{{{{{{{{\\\\rm{pot}}}}}}}}}=\\\\frac{1}{2}\\\\,{{{{\\\\rm{Tr}}}}}\\\\,({\\\\hat{{{\\\\Sigma }}}}_{{{{{\\\\rm{cor}}}}}}{\\\\hat{G}}_{{{{{\\\\rm{cor}}}}}}) \\\\, > \\\\, 0\\\\) for each Ni. In the CO state, the Ni1 and Ni3 sites have one electron on the \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) orbital. But the charge transfer depletes the electron on the Ni2- \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) orbital, so that its self-energy Σ cor essentially becomes zero. This decreases the potential energy and always leads to an energy gain Δ E pot < 0 when U d exceeds the critical value. As for the double-counting energy, \\\\({E}^{{{{{\\\\rm{dc}}}}}}=\\\\frac{1}{2}{U}_{d}{N}_{d}({N}_{d}-1)\\\\) [57] . The AFM state is close to a half-filled configuration for every Ni site, while the CO state has a configuration in which Ni2 site has almost zero occupancy and Ni1/Ni3 site is half-filled. Therefore, in both states, the double counting energy E dc is small and so is their difference (compared to Δ E pot ). In short, there is a delicate competition between charge transfer energy and local Hubbard interaction. When the potential energy gain outweighs the kinetic energy cost during the charge transfer, the CO state becomes more stable than the AFM state. Then, we analyze the case of a general U d (see Fig. 4b ). We find that the trend does not change in Δ E pot and Δ E dc . However, when U d is sufficiently large, Δ E kin itself becomes negative (i.e., a net energy gain after the charge transfer). That is because as U d increases, the Ni1/Ni3 LHB moves toward the lower energy after the charge transfer (i.e. ∣ Δ 2 ∣ > ∣ Δ 1 ∣ ). This effect decreases the kinetic energy and eventually counteracts the kinetic energy cost from the charge transfer (brown arrow in Fig. 4a ). CO stability and stripe pattern The preceding analysis shows that the charge transfer from Ni- \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) orbital to the near-Fermi-level conduction bands plays a crucial role in the formation of the CO state. In our effective model, the conduction bands are derived from the effective- s orbitals. Figure 5a is a schematic non-interacting electronic structure of the effective model Eq. ( 1 ). We show below that the charge transfer energy E d s , which is defined as the energy separation between the bare onsite energy of Ni- \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) and effective- s orbitals, controls not only stability but also stripe pattern of the CO state in the effective model. Fig. 5: CO stability and stripe pattern. a Schematics of the non-interacting electronic structure of the low-energy effective model Eq. ( 1 ) where E d s is the energy separation between the bare energy of Ni- \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) and effective- s orbitals. b The total energy (up to a constant) of the PM (red), the AFM (blue) and the CO (green) states as a function of E d s , calculated by using the low-energy effective model Eq. ( 1 ) with U d = 3.7 eV. The orange dashed line highlights E d s = 0.7 eV, which is obtained from fitting to the DFT band structure of NdNiO 2 . c The total energy of different stripe CO states with the wavevector \\\\({{{{{{{\\\\bf{q}}}}}}}}=\\\\left(\\\\frac{1}{N},\\\\, 0,\\\\, 0\\\\right)\\\\) as a function of U d , calculated by using the low-energy effective model Eq. ( 1 ) with E d s = 0.7 eV. d The total energy of different stripe CO with the wavevector \\\\({{{{{{{\\\\bf{q}}}}}}}}=\\\\left(\\\\frac{1}{N},\\\\, 0,\\\\, 0\\\\right)\\\\) as a function of E d s , calculated by using the low-energy effective model with U d = 3.7 eV. In panels ( b – d ), the error bar is smaller than the symbol size (see Methods). Source data are provided as a Source Data file. Full size image We first examine the total energy of the PM, AFM and CO states as a function of E d s using the effective model Eq. ( 1 ). For each E d s , we recalculate the total energy of each state and show the results in Fig. 5b . We use U d = 3.7 eV but find a similar trend using other values of U d (see Supplementary Note 7 ). Fitting to the DFT band structure, we obtain E d s = 0.7 eV (the orange dashed line in Fig. 5b ), at which the CO state is more stable than the AFM and PM states. As E d s increases, the total energy of the CO state first becomes higher than the AFM state ( E d s > 0.8 eV) and then even higher than the PM state ( E d s > 0.9 eV). This can be understood because increasing E d s leads to a larger kinetic energy cost in the charge transfer (as we demonstrated in Fig. 4a ). Next, we study different stripe CO patterns and compare their total energies using the effective model Eq. ( 1 ). We use a N × 1 × 1 supercell to study the stripe CO with the wavevector \\\\({{{{{{{\\\\bf{q}}}}}}}}=\\\\left(\\\\frac{1}{N},\\\\, 0,\\\\, 0\\\\right)\\\\) . Before discussing the results, we first present an intuitive picture of how the CO total energy may depend on N . As we have shown, the CO state emerges from a charge transfer of almost one electron from one Ni site to the conduction bands for every N Ni atoms. During the charge transfer process, the electron needs to overcome the charge transfer energy E d s . This results in a kinetic energy cost ∝ E d s . On the other hand, the charge transfer depletes the charges on that Ni site. Since the local Hubbard interaction is repulsive in our model, this leads to a potential energy gain ∝ U d . Thus, the total energy of the entire system ( N -Ni-atom cell) is: $${E}_{N{{{{{{{\\\\rm{-Ni-atom}}}}}}}}}=\\\\alpha {E}_{ds}-\\\\beta {U}_{d}+N{E}_{0}$$ (2) where α and β are both positive and E 0 is the energy contribution that does not change during the charge transfer. The tricky point is that both α and β also depend on the wavevector or the wavelength (i.e., N ). Therefore, the total energy per-Ni-atom cell is: $${E}_{{{{{{{{\\\\rm{per-Ni-atom}}}}}}}}}=\\\\frac{{E}_{N{{{{{{{\\\\rm{-Ni-atom}}}}}}}}}}{N}=\\\\alpha (N)\\\\frac{{E}_{ds}}{N}-\\\\beta (N)\\\\frac{{U}_{d}}{N}+{E}_{0}$$ (3) For given E d s and U d , minimizing E per-Ni-atom with respect to N yields the optimal wavelength N or equivalently optimal wavevector q . However, due to interaction, the analytical expressions of α ( N ) and β ( N ) are not known. Therefore, we numerically calculate E N -Ni-atom as a function of N for different E d s and U d . Figure 5c compares the total energy of different stripe CO states for a few representative U d at E d s = 0.7 eV (DFT-fitted value). We find that at E d s = 0.7 eV, the \\\\({{{{{{{\\\\bf{q}}}}}}}}=\\\\left(\\\\frac{1}{3},\\\\, 0,\\\\, 0\\\\right)\\\\) CO has the lowest total energy in a range of U d values. The reason that the total energy of the stripe CO state does not monotonically change with N is because with N increasing, on the one hand, the kinetic energy cost decreases; on the other hand, the potential energy gain also decreases. These two factors are competing and the calculations find that the total energy minimum appears at an optimal CO wavevector \\\\({{{{{{{\\\\bf{q}}}}}}}}=(\\\\frac{1}{3},\\\\, 0,\\\\, 0)\\\\) . While our model is simplified, this result is consistent with the experimentally observed CO stripe pattern of infinite-layer nickelates [33] , [34] , [35] . Figure 5d compares the total energy of different stripe CO states for a few representative E d s at U d = 3.7 eV. We find that as E d s increases from 0.6 to 0.8 eV, the wavevector of the lowest energy CO state evolves from \\\\(\\\\left(\\\\frac{1}{3},\\\\, 0,\\\\, 0\\\\right)\\\\) to \\\\(\\\\left(\\\\frac{1}{4},\\\\, 0,\\\\, 0\\\\right)\\\\) . This is because as E d s changes, the decreasing rate of the kinetic energy with N also changes, which leads to a different optimal wavevector for the CO state. We mention that when E d s ≥ 0.8 eV, the AFM state has lower energy than the CO state (see Fig. 5b ). This means that based on our model calculations, the CO state with a wavevector of \\\\({{{{{{{\\\\bf{q}}}}}}}}=\\\\left(\\\\frac{1}{4},\\\\, 0,\\\\, 0\\\\right)\\\\) cannot be observed. Instead, it gives way to the AFM state. To summarize, in the phase space spanned by ( U d , E d s ), our effective model Eq. ( 1 ) has a parameter range in which the stripe CO state is more stable than the AFM and PM states with an optimal wavevector \\\\({{{{{{{\\\\bf{q}}}}}}}}=(\\\\frac{1}{3},\\\\, 0,\\\\, 0)\\\\) , which reproduces the experimental observations of infinite-layer nickelates [33] , [34] , [35] .\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"We make a few comments before the conclusion. First, we note that CO or charge density wave (CDW) may have different origins. In addition to electron–electron interaction, electron–phonon interaction may also play an important role in the formation of CDW, which has been intensively studied in other material systems such as transition metal dichalcogenides [58] . A CDW state that arises from electron–phonon interaction is accompanied by a structural distortion. Thus, in addition to using the ideal lattice structure of infinite-layer nickelates, we also study possible structural distortions in the CO state [59] , [60] . Since the charge transfer breaks the original translational symmetry, oxygen atoms may move away from their ideal positions (see Supplementary Note 5 ). The movement of oxygen atoms can be modeled by changing the onsite energy of Ni- \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) orbital in the effective model Eq. ( 1 ). Our calculations find that the oxygen atom movements are minute (~0.05 Å), which is in agreement with the other study [61] . The resulting energy gain in CO is less than 5 meV/f.u., about one order of magnitude smaller than the energy difference shown in Fig. 3b . The energy gain may be used as a rough estimation to quantify the relative importance of different mechanisms. Our results imply that the CO state found in infinite-layer nickelates is more likely of electronic origin, while structural distortions driven by electron–phonon interaction may play a secondary role. Second, the CO state from our calculations has a substantial charge disproportionation, driven by a charge transfer from a portion of Ni atoms to the near-Fermi-level conduction bands. A direct consequence is that the average Ni occupancy in the CO state is substantially smaller than that of a uniform PM state. Such a CO state is dissimilar to the CDW state found in the calculations of the doped single-band Hubbard model [62] , [63] , doped t - J model [64] and a 2-orbital model of ref. [53] , in which the charge profile shows a Friedel oscillation with the average occupancy per site being equal to that of uniform PM state. In those states, variation in site occupancy is due to charge transfer between neighboring correlated sites. Third, as we mentioned previously, the analytical expression of CO energy dependence on wavevector E ( q ) is not known. Therefore, we cannot find the optimal wavevector q by solving ∇ q E ( q ) = 0. Instead, we have to test as many known competing states as possible (within computational capability). In addition to the \\\\({{{{{{{\\\\bf{q}}}}}}}}=\\\\left(\\\\frac{1}{N},\\\\, 0,\\\\, 0\\\\right)\\\\) CO state, we also test a different CO state, which can be accommodated by the available simulation cell (see Supplementary Note 8 ). We use the low-energy effective model to study all these competing states (PM state, AFM state and five CO states with different wavevectors) and compare their total energy. Our calculations find that using the DFT-fitted energy dispersion for a reasonable range of U d , the \\\\({{{{{{{\\\\bf{q}}}}}}}}=\\\\left(\\\\frac{1}{3},\\\\, 0,\\\\, 0\\\\right)\\\\) CO state is the most energetically favorable one among all the states considered. Fourth, a number of studies [27] , [31] , [65] , [66] have shown that Ni \\\\({d}_{3{z}^{2}-{r}^{2}}\\\\) orbital plays an important role in infinite-layer nickelates. Comparison of the low-energy effective model calculations to the 17-orbital model calculations, which include all five Ni- d orbitals, reveals that the minimum physics needed to account for the CO phenomenon is a charge transfer from a correlated Ni- \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) orbital to a conduction band that is close to the Fermi level. Adding other orbitals into the effective model may describe more physical phenomena of infinite-layer nickelates, but as far as charge ordering is concerned, the 2-orbital effective model is sufficient. Finally, we briefly discuss how one may experimentally distinguish CO states of different origins in infinite-layer nickelates. If the CO in infinite-layer nickelates arises from the revealed charge-transfer mechanism, it will exhibit a mixed valence state of Ni, a characteristic Ni 1+ -Ni 2+ -Ni 1+ stripe pattern, as well as a unidirectional Ni 2+ chain in which holes are localized. In the experiment, one may probe this mixed valence state of Ni either by x-ray absorption spectroscopy (XAS) or cross-sectional electron energy loss spectroscopy (EELS) [36] . XAS can measure the average Ni valence, whose spectrum in the CO state should be distinct from that of pure Ni 1+ or Ni 2+ . A more informative measurement is the cross-sectional EELS, which may provide a spatially resolved spectrum of Ni 1+ and Ni 2+ columns in the CO state. In addition, scanning tunneling microscopy can image the localized holes in the unidirectional Ni 2+ chain [36] . On the other hand, if the CO or CDW is driven by phonon and electron–phonon coupling, a characteristic phonon of the ideal crystal structure should become soft around the CDW transition temperature and the emergence of CDW accompanies a periodic-lattice-distortion, which lowers the total energy [58] , [67] , [68] . The vibrational mode of characteristic phonon should match the pattern of periodic-lattice-distortion observed at low temperatures [67] , [68] . The magnitude of the periodic-lattice-distortion should be such that the resulting energy gain is comparable to the CDW transition temperature [58] . In experiment, inelastic x-ray scattering or neutron scattering can measure temperature dependence of phonon dispersion and identify soft phonons, if they exist around CDW temperature [68] . The low-temperature periodic-lattice-distortion can be probed either by x-ray diffraction (XRD) or by transmission electron microscopy (TEM). The emergence of periodic-lattice-distortion lowers the crystal symmetry and thus changes the diffraction pattern in XRD measurements [69] . TEM, in particular with the recent developments of electron ptychography [70] , can achieve atomic-resolution imaging, enabling visualization of periodic-lattice-distortion in real space and accurate measurements of structural distortions. Using the lattice distortions found from TEM measurements, one can estimate the energy gain from the periodic-lattice-distortion by performing first-principles calculations and then comparing the energy gain to the experimental CDW transition temperature. In short, the experimental observation of “a mixed valence state of Ni/localized holes on unidirectional Ni 2+ chain” versus “soft phonon/periodic-lattice-distortion” may distinguish different origins of CO in infinite-layer nickelates. We hope that the two sets of experiments outlined above, as well as other experimental techniques, will identify the true underlying mechanism among various theoretical proposals. To conclude, our first-principles calculations show that a \\\\({{{{{{{\\\\bf{q}}}}}}}}=\\\\left(\\\\frac{1}{3},\\\\, 0,\\\\, 0\\\\right)\\\\) CO state, which has a characteristic Ni 1+ -Ni 2+ -Ni 1+ stripe pattern, is stabilized in infinite-layer nickelates when Hubbard interaction on Ni- d orbitals is reasonably large. For every three Ni atoms, the CO state is driven by a charge transfer from one Ni atom to near-Fermi-level conduction bands, leaving electrons on the other two Ni atoms to become more localized. We further derive a simple low-energy effective model and elucidate that stabilization of the CO state over PM and AFM states is the consequence of a delicate competition between Hubbard interaction U d on a correlated orbital and charge transfer energy E d s between the correlated orbital and the conduction band. Such a competition may induce CO states in other correlated materials besides infinite-layer nickelates. Manipulating the CO state in infinite-layer nickelates by tuning the charge transfer energy may also affect their superconductivity [71] , which deserves further experimental study.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"DFT+DMFT calculations We perform DFT+DMFT calculations [41] , [42] , [43] , [44] with the aid of MLWFs [45] , [72] . The DFT method is implemented in the Vienna ab initio simulation package code [73] with the projector augmented wave method [74] . The Perdew–Burke–Ernzerhof [75] functional is used as the exchange-correlation functional in DFT calculations. The Nd-4 f orbitals are treated as core states in the pseudopotential because the 4 f shell is hidden in the core of the Nd ion, and Hubbard interaction on 4 f orbitals is very large (>10 eV). As a result, the Nd-4 f electrons are far from the Fermi level and turn into strongly localized f spins. The direct coupling between Nd-4 f electrons and itinerant electrons is minute [76] . We use an energy cutoff of 600 eV and sample the Brillouin zone by using Γ-centered k -mesh of 12 × 12 × 12 per primitive cell. The crystal structure is fully relaxed with an energy convergence criterion of 10 −6 eV, force convergence criterion of 0.01 eV/Å and strain convergence of 0.1 kbar. The DFT-optimized crystal structures are in good agreement with the experimental P 4/ m m m structures ( a DFT = 3.91 Å, \\\\({a}^{{{{{{{{\\\\rm{EXP}}}}}}}}}\\\\) = 3.92 Å, c DFT = 3.31 Å, \\\\({c}^{{{{{{{{\\\\rm{EXP}}}}}}}}}\\\\) = 3.28 Å). To describe the checkerboard antiferromagnetic ordering, we expand the cell to a \\\\(\\\\sqrt{2}\\\\times \\\\sqrt{2}\\\\times 1\\\\) supercell. To describe the \\\\({{{{{{{\\\\bf{q}}}}}}}}=\\\\left(\\\\frac{1}{N},\\\\, 0,\\\\, 0\\\\right)\\\\) CO state ( N is an integer), we expand the cell to an N × 1 × 1 supercell. We downfold the DFT-calculated band structure of NdNiO 2 to a 17-orbital tight-binding model by using the MLWF method [45] . The 17 orbitals include five Ni- d orbitals, five Nd- d orbitals, six O- p orbitals and one interstitial s orbital. This 17-orbital model can well reproduce the non-interacting band structure of NdNiO 2 in an energy window of about 15 eV around the Fermi level (see Supplementary Fig. 1 ). A Slater–Kanamori interaction is added on the Ni- d orbitals [77] . A fully localized-limit (FLL) double counting correction is applied to the full interacting model [57] . We also test other double counting correction [48] and no qualitative change is found in the main results. More details of the full Hamiltonian are found in Supplementary Note 1 . We treat the two Ni e g orbitals with DMFT method, while the filled Ni t 2 g shell is treated with a static Hartree–Fock approximation [56] , [78] . When we use the DMFT method to solve the interacting 17-orbital model, we employ the continuous-time quantum Monte Carlo (CTQMC) algorithm based on hybridization expansion [79] , [80] . The impurity solver was developed by Haule [81] . For each DMFT iteration, a total of 1 billion Monte Carlo samples are collected to converge the impurity Green function and self-energy. We set the temperature to be 116 K. We check all the main results at a lower temperature of 58 K and no significant difference is found. For the CO state, we allow all the Ni sites to be inequivalent and the DMFT self-consistent condition involves the self-energies of all inequivalent Ni atoms. To obtain the spectral functions, the imaginary axis self-energy is continued to the real axis by using the maximum entropy method [82] . Then, the real axis local Green function is calculated using the Dyson equation. A 90 × 90 × 90 k -point mesh is used to converge the spectral function of the 17-orbital model. The full charge-self-consistency does not qualitatively change the main results of charge transfer and charge disproportionation in the calculations [83] . To relieve the computational burden, the results shown in the main text are from one-shot DFT+DMFT calculations. We also apply the above relevant parameters when using DMFT to solve the low-energy effective model. Low-energy effective model The low-energy effective model to describe the CO phenomenon is a 2-orbital model, which can be compactly written as follows: $$\\\\hat{H}=\\\\mathop{\\\\sum}\\\\limits_{{{{{{{{\\\\bf{k}}}}}}}}\\\\sigma }{\\\\hat{{{\\\\Psi }}}}_{{{{{{{{\\\\bf{k}}}}}}}}\\\\sigma }^{{{{\\\\dagger}}} }{{{{{{{{\\\\mathcal{H}}}}}}}}}_{0}({{{{{{{\\\\bf{k}}}}}}}}){\\\\hat{{{\\\\Psi }}}}_{{{{{{{{\\\\bf{k}}}}}}}}\\\\sigma }+{U}_{d}\\\\mathop{\\\\sum}\\\\limits_{i}{\\\\hat{n}}_{i\\\\uparrow }^{d}{\\\\hat{n}}_{i\\\\downarrow }^{d}-{\\\\hat{V}}^{{{{{{{{\\\\rm{dc}}}}}}}}}$$ (4) where \\\\({\\\\hat{{{\\\\Psi }}}}_{{{{{{{{\\\\bf{k}}}}}}}}\\\\sigma }^{{{{\\\\dagger}}} }=({\\\\hat{d}}_{{{{{{{{\\\\bf{k}}}}}}}}\\\\sigma }^{{{{\\\\dagger}}} },\\\\, {\\\\hat{s}}_{{{{{{{{\\\\bf{k}}}}}}}}\\\\sigma }^{{{{\\\\dagger}}} })\\\\) are the creation operators on Ni- \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) and effective- s orbitals with momentum k and spin σ . \\\\({\\\\hat{n}}_{i\\\\sigma }^{d}={\\\\hat{d}}_{i\\\\sigma }^{{{{\\\\dagger}}} }{\\\\hat{d}}_{i\\\\sigma }\\\\) is the occupancy operator of Ni- \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) orbital at site i with spin σ . \\\\({{{{{{{{\\\\mathcal{H}}}}}}}}}_{0}({{{{{{{\\\\bf{k}}}}}}}})\\\\) is a 2 × 2 matrix: $${{{{{{{{\\\\mathcal{H}}}}}}}}}_{0}({{{{{{{\\\\bf{k}}}}}}}})=\\\\left[\\\\begin{array}{ll}{\\\\epsilon }_{d}({{{{{{{\\\\bf{k}}}}}}}})&{V}_{ds}({{{{{{{\\\\bf{k}}}}}}}})\\\\\\\\ {V}_{ds}^{*}({{{{{{{\\\\bf{k}}}}}}}})&{\\\\epsilon }_{s}({{{{{{{\\\\bf{k}}}}}}}})\\\\end{array}\\\\right]$$ (5) The energy dispersion and hybridization terms are: $${\\\\epsilon }_{d}({{{{{{{\\\\bf{k}}}}}}}})=\\t 2{t}_{d}^{100}\\\\left(\\\\cos {k}_{x}+\\\\cos {k}_{y}\\\\right)+4{t}_{d}^{110}\\\\cos {k}_{x}\\\\cos {k}_{y} \\\\\\\\ \\t+2{t}_{d}^{200}\\\\left(\\\\cos 2{k}_{x}+\\\\cos 2{k}_{y}\\\\right)$$ (6) $${\\\\epsilon }_{s}({{{{{{{\\\\bf{k}}}}}}}})=\\t {E}_{ds}+2{t}_{s}^{100}\\\\left(\\\\cos {k}_{x}+\\\\cos {k}_{y}\\\\right)+2{t}_{s}^{001}\\\\cos {k}_{z}+4{t}_{s}^{110}\\\\cos {k}_{x}\\\\cos {k}_{y}\\\\\\\\ \\t+4{t}_{s}^{101}\\\\left(\\\\cos {k}_{x}+\\\\cos {k}_{y}\\\\right)\\\\cos {k}_{z}+8{t}_{s}^{111}\\\\cos {k}_{x}\\\\cos {k}_{y}\\\\cos {k}_{z}$$ (7) $${V}_{ds}({{{{{{{\\\\bf{k}}}}}}}})=2{t}_{ds}^{100}\\\\left(\\\\cos {k}_{x}-\\\\cos {k}_{y}\\\\right)\\\\left(1+{e}^{-i{k}_{z}}\\\\right)$$ (8) The onsite energy difference E d s and hopping parameters are obtained by fitting to the near-Fermi-level DFT band structure of NdNiO 2 (see Supplementary Fig. 1 ). The fitted values are shown in Table 1 . We use the FLL double counting and V dc reads: $${V}^{{{{{{{{\\\\rm{dc}}}}}}}}}={U}_{d}\\\\left({N}_{d}-\\\\frac{1}{2}\\\\right)$$ (9) Table 1 The onsite energy and hopping matrix elements in the 2-orbital model Full size table Total energy For the low-energy effective model, we calculate the total energy within the DFT+DMFT method. The total energy in the DFT+DMFT calculations has the following expression [44] , [84] : $${E}^{{{{{{{{\\\\rm{DFT+DMFT}}}}}}}}}={E}^{{{{{{{{\\\\rm{DFT}}}}}}}}}+{E}^{{{{{{{{\\\\rm{DMFT}}}}}}}}}-{E}^{{{{{{{{\\\\rm{dc}}}}}}}}}$$ (10) Here E DFT is: $${E}^{{{{{{{{\\\\rm{DFT}}}}}}}}}={E}_{0}^{{{{{{{{\\\\rm{DFT}}}}}}}}}-{E}_{0}^{{{{{{{{\\\\rm{kin}}}}}}}}}$$ (11) where \\\\({E}_{0}^{{{{{{{{\\\\rm{DFT}}}}}}}}}\\\\) is the standard DFT total energy. \\\\({E}_{0}^{{{{{{{{\\\\rm{kin}}}}}}}}}\\\\) is the kinetic energy of the non-interaction Hamiltonian within the model space: $${E}_{0}^{{{{{{{{\\\\rm{kin}}}}}}}}}=\\\\,{{\\\\mbox{Tr}}}\\\\,\\\\left(\\\\hat{{{{{{{{{\\\\mathcal{H}}}}}}}}}_{0}}\\\\hat{{G}_{0}}\\\\right)=\\\\frac{1}{{N}_{{{{{{{{\\\\bf{k}}}}}}}}}}\\\\mathop{\\\\sum}\\\\limits_{{{{{{{{\\\\bf{k}}}}}}}}\\\\sigma }\\\\mathop{\\\\sum}\\\\limits_{l}{\\\\epsilon }_{{{{{{{{\\\\bf{k}}}}}}}}l}{n}_{F}({\\\\epsilon }_{{{{{{{{\\\\bf{k}}}}}}}}l}-\\\\mu )$$ (12) where n F is the Fermi-Dirac occupancy, l is the band index and σ is the spin. In the main text, we use the DFT-optimized crystal structure for all the states. Thus, the PM, AFM and CO states all share the same non-interacting Hamiltonian (up to a multiplication factor), we omit the E DFT in the calculation of total energy (since it is identical for all three states). The remaining total energy is $$E={E}^{{{{{{{{\\\\rm{DMFT}}}}}}}}}-{E}^{{{{{{{{\\\\rm{dc}}}}}}}}}={E}^{{{{{{{{\\\\rm{kin}}}}}}}}}+{E}^{{{{{{{{\\\\rm{pot}}}}}}}}}-{E}^{{{{{{{{\\\\rm{dc}}}}}}}}}$$ (13) Here, the DMFT kinetic energy E kin is defined as follows: $${E}^{{{{{\\\\rm{kin}}}}}}={{{{\\\\rm{Tr}}}}}\\\\,\\\\left(\\\\hat{{{{{{{{{\\\\mathcal{H}}}}}}}}}_{0}}\\\\hat{G}\\\\right)=\\\\frac{T}{{N}_{{{{{{{{\\\\bf{k}}}}}}}}}}\\\\mathop{\\\\sum}\\\\limits_{{{{{{{{\\\\bf{k}}}}}}}}\\\\sigma }\\\\mathop{\\\\sum}\\\\limits_{{\\\\omega }_{n}}\\\\mathop{\\\\sum}\\\\limits_{m{m}^{{\\\\prime} }}{\\\\left[{{{{{{{{\\\\mathcal{H}}}}}}}}}_{0}({{{{{{{\\\\bf{k}}}}}}}})\\\\right]}_{m{m}^{{\\\\prime} }}{\\\\left[{G}_{\\\\sigma }({{{{{{{\\\\bf{k}}}}}}}},\\\\, i{\\\\omega }_{n})\\\\right]}_{{m}^{{\\\\prime} }m}$$ (14) where σ is the spin, T is the temperature, \\\\(m,{m}^{{\\\\prime} }\\\\) are the indices of orbital basis. \\\\({{{{{{{{\\\\mathcal{H}}}}}}}}}_{0}({{{{{{{\\\\bf{k}}}}}}}})\\\\) is the non-interacting part of the effective model Eq. ( 5 ) and G is the dressed lattice Green function of the interacting model. The DMFT potential energy E pot is defined as follows: $${E}^{{{{{\\\\rm{pot}}}}}}=\\t \\\\frac{1}{2}{{{{{{{\\\\rm{Tr}}}}}}}}\\\\left({\\\\hat{{{\\\\Sigma }}}}_{{{{{{{{\\\\rm{cor}}}}}}}}}{\\\\hat{G}}_{{{{{{{{\\\\rm{cor}}}}}}}}}\\\\right)=\\\\frac{1}{2}T\\\\mathop{\\\\sum}\\\\limits_{\\\\sigma }\\\\mathop{\\\\sum}\\\\limits_{{\\\\omega }_{n}}\\\\mathop{\\\\sum}\\\\limits_{m}{e}^{-i{\\\\omega }_{n}{0}^{-}}\\\\\\\\ \\t {\\\\left[{{{\\\\Sigma }}}_{{{{{\\\\rm{cor}}}}}}^{\\\\sigma }(i{\\\\omega }_{n})\\\\right]}_{m}{\\\\left[{G}_{{{{{\\\\rm{cor}}}}}}^{\\\\sigma }(i{\\\\omega }_{n})\\\\right]}_{m}$$ (15) where σ is the spin, T is the temperature, m is the index of orbital basis. Σ cor and G cor are the self-energy and the local Green function in the correlated subspace. E dc is the double counting energy. We employ the FLL double counting and E dc reads: $${E}^{{{{{{{{\\\\rm{dc}}}}}}}}}=\\\\frac{1}{2}{U}_{d}{N}_{d}({N}_{d}-1)$$ (16) Taking the derivative of E dc with respect to N d yields the double-counting potential V dc Eq. ( 9 ). In the low-energy effective model, the Hund’s term vanishes because there is only one correlated orbital Ni- \\\\({d}_{{x}^{2}-{y}^{2}}\\\\) per site. We note that when we study the structural distortions in the CO state, the crystal structure changes in each calculation and so does the DFT total energy. In that case, we need to use Eq. ( 10 ) to calculate the DFT+DMFT total energy. Since the total energy is calculated using CTQMC, we average the total energy of the last ten DMFT iterations. We find that due to the highly accurate self-energy, the standard deviation of the total energy is about 1 meV/f.u., which is smaller than the symbol size in the figures.\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"core researchThe idea is to investigate whether the unique charge order observed in infinite-layer nickelates, characterized by a commensurate stripe pattern with wavevector ${\\bf{q}}\\simeq \\left(\\frac{1}{3},\\, 0,\\, 0\\right)$, arises from a charge-transfer-driven mechanism involving multi-band physics and the interplay between Hubbard interactions on Ni-$d$ orbitals and charge transfer energy to conduction bands, as revealed through DFT+DMFT calculations and an effective model in NdNiO$_2$."},"entities":{"kind":"list like","value":["infinite-layer nickelates","commensurate stripe pattern","wavevector","Hubbard interactions","Ni-$d$ orbitals","charge transfer energy","conduction bands","DFT+DMFT calculations","effective model","NdNiO$_2$","charge order","multi-band physics","charge-transfer-driven mechanism","${\\bf{q}}\\simeq \\left(\\frac{1}{3}","\\","0","\\","0\\right)$"],"string":"[\n \"infinite-layer nickelates\",\n \"commensurate stripe pattern\",\n \"wavevector\",\n \"Hubbard interactions\",\n \"Ni-$d$ orbitals\",\n \"charge transfer energy\",\n \"conduction bands\",\n \"DFT+DMFT calculations\",\n \"effective model\",\n \"NdNiO$_2$\",\n \"charge order\",\n \"multi-band physics\",\n \"charge-transfer-driven mechanism\",\n \"${\\\\bf{q}}\\\\simeq \\\\left(\\\\frac{1}{3}\",\n \"\\\\\",\n \"0\",\n \"\\\\\",\n \"0\\\\right)$\"\n]","isConversational":false},"references":{"kind":"list like","value":[],"string":"[]","isConversational":false}}},{"rowIdx":11192,"cells":{"unique_id":{"kind":"string","value":"s41467-019-13059-8"},"subfield":{"kind":"string","value":"Optics and photonics"},"title":{"kind":"string","value":"Control of the temporal and polarization response of a multimode fiber"},"abstract":{"kind":"string","value":"Control of the spatial and temporal properties of light propagating in disordered media have been demonstrated over the last decade using spatial light modulators. Most of the previous studies demonstrated spatial focusing to the speckle grain size, and manipulation of the temporal properties of the achieved focus. In this work, we demonstrate an approach to control the total temporal impulse response, not only at a single speckle grain but over all spatial degrees of freedom (spatial and polarization modes) at any arbitrary delay time through a multimode fiber. Global enhancement or suppression of the total light intensity exiting a multimode fibre is shown for arbitrary delays and polarization states. This work could benefit to applications that require pulse delivery in disordered media."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"Light propagation in disordered systems has been extensively studied over the past 40 years [1] . Transmitted light through such materials forms a complex pattern after propagation, known as speckle, that is the signature of the interference between a large number of modes within the sample. Although this speckle looks random, this mixing of light is nonetheless linear and thus deterministic. Wavefront shaping has revolutionized the spatial control of coherent light beams thanks to the use of spatial light modulators (SLM) in disordered systems [2] , [3] , such as biological tissue, white paint or multimode fibers. Different approaches have been developed to control the propagation of coherent light that has experienced scattering, such as iterative optimization [4] , [5] , optical phase conjugation [6] , and the measurement of the optical transmission matrix [7] . These methods have been further extended to the control of polarization [8] , spectral [9] , [10] , and temporal properties of light [11] , and widely used for imaging [12] , [13] , [14] , [15] and optical manipulation [16] . The spatial and temporal distortions of scattered light are coupled in disordered systems. Therefore, the temporal properties of a spatio-temporal speckle field can be adjusted with spatial-only control. In previous works, manipulating the temporal properties of scattered light was achieved with spatio-temporal focusing with a single SLM. The output pulse is focused on a single speckle grain, and the spectral properties of the focus are controlled to ensure a short duration of the output pulse. Spatio-temporal focusing of the output pulse can be reached for instance with optimization algorithms [17] , [18] , digital phase conjugation [19] , pulse shaping methods [20] , or via the knowledge of the transmission matrix of the sample [11] , [21] . However, the output pulse has only a short duration at this specified focus position, while the background speckle remains temporally elongated [22] . The temporal control of the spatio-temporal field is then limited to a specific spatial mode (speckle position or a specific spatial pattern), at the expense of undefined temporal behavior for the other spatial modes. Equivalently, control of the impulse response of only a small portion of the total power means the remaining field is effectively a loss and potentially a source of noise. Most of the total input power does not arrive at a controlled delay, it is scattered into the temporal background. A control of the full speckle pattern over a certain spectral bandwidth has been proposed based on the transmission matrix, with the time delay operator [23] , [24] , [25] . Nonetheless, the limited controlled spectral bandwidth does not allow a full temporal control of the spatio-temporal speckle. In this paper, we propose a novel method to adjust the temporal properties of the total output intensity at any arbitrary delay over all the spatial and polarization modes propagating through a multimode fiber. After experimentally measuring the Multi-Spectral Transmission Matrix (MSTM) of the fiber, we calculate the Time Resolved Transmission Matrix (TRTM) with a Fourier Transform. Exploiting the TRTM, we demonstrate for the first time both temporal enhancement and attenuation of the full beam at specific delay times at the expense of undefined behaviour in the spatial domain. In contrast with previous spatio-temporal focusing experiments where the temporal properties are adjusted only in the spatial focus position, all the output spatial modes are controlled at that specific delay. We also demonstrate control of the often neglected polarization degree of freedom, representing half of all the modes supported, which forms an unprecedented full control of the spatio-temporal output field over a large bandwidth.","references":{"indices":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],"infos":[{"idx":"1","title":"Goodman, J. W. Some fundamental properties of speckle.J Opt. Soc. Am.66, 1145–1150 (1976).","link":"https://doi.org/10.1364%2FJOSA.66.001145"},{"idx":"2","title":"Rotter, S. & Gigan, S. Light fields in complex media: mesoscopic scattering meets wave control.Rev. Mod. Phys.89, 015005 (2017).","link":"https://doi.org/10.1103%2FRevModPhys.89.015005"},{"idx":"3","title":"Mosk, A. P., Lagendijk, A., Lerosey, G. & Fink, M. Controlling waves in space and time for imaging and focusing in complex media.Nat. Photonics6, 283–292 (2012). ISSN 1749-4885.","link":"https://doi.org/10.1038%2Fnphoton.2012.88"},{"idx":"4","title":"Vellekoop, I. 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The transmission matrix has been widely used to manipulate light that has suffered from scattering or mode mixing, for focusing [7] or imaging purposes [28] . Recently, the transmission matrix has been extended to the spectral domain. The Multi Spectral Transmission Matrix [9] , [21] , also known as the optical transfer function [29] , is a stack of transmission matrices \\(U(\\lambda )\\) for a set of different input wavelength, that can be measured by sweeping the wavelength or using hyper-spectral imaging [30] . The MSTM enables a spectral control of the output field exploiting the spectral diversity of the medium, and could be used for focusing a pattern at a given wavelength [9] , [29] , pulse shaping the output focus [21] and imaging through scattering samples [31] . We exploit the setup presented in Fig. 1 to measure the Multi Spectral Transmission Matrix of a multimode fiber (MMF). The light source is a tunable continuous wave (CW) laser (New focus TLM-8700) operating between 1510 nm and 1620 nm (bandwidth \\(\\Delta \\lambda =110\\) nm \\(\\simeq 13.5\\) THz). We use a phase-only spatial light modulator (Meadowlark P1920) to generate any spatial and polarization input state in amplitude and phase [32] . The SLM is located in the Fourier plane of a multimode fiber. The fiber is a 1-m length of a 50 \\(\\mu\\) m core radius step-index multimode fiber (Thorlabs M42L01), which theoretically supports ~127 spatial modes per polarization state [33] . A step-index multimode fiber has more mode coupling and a more complex and spread impulse response than a graded index fiber with similar dimensions. The multimode fiber we use in this experiment is not in the strong mode coupling regime [34] (see Supplementary Note 2 for more details on the mode coupling in the fiber). The output field is measured with digital off-axis holography, using an external path-length matched reference beam and an InGaAs camera (Xenics Xeva-1.7-320). The two polarization states are measured on the left and the right side of the camera using a polarizer beam displacer. Fig. 1 Apparatus of the experimental setup. SMF: single mode fiber; SLM: spatial light modulator; PBS: polarizing beam splitter; MMF: multimode fiber; NPBS: non-polarizing beam splitter; (L1, L2, L3, and L4): optical lenses. Focal length: L1: 25 mm; L2: 10 mm; L3: 4 mm; and L4: 60 mm Full size image Measurement of the Transmission Matrix of a multimode fiber We use a swept-wavelength interferometer (SWI) to measure the multi spectral transmission matrix of the MMF [29] , [32] . We measure the complete transmission matrix by generating all the spatial and polarization modes (a total of \\({N}_{\\text{input}}=254\\) modes) on the input facet of the MMF using the SLM. We launch the modes of the step-index fiber in a linearly polarized basis with helical wavefronts. For each individual input mode, we sweep the laser and record the output field for all the wavelengths with the digital holography beam analysis. Through the sweep, the camera is triggered at 10.6 GHz through the sweep with an external Mach-Zehnder interferometer that acts as a k-clock. This triggering method enables the measurement of \\({N}_{\\lambda }=1273\\) monochromatic transmission matrices per sweep. Each output field is then digitally decomposed on the fly in the Laguerre-Gaussian basis ( \\({N}_{\\text{output}}=650\\) modes). For a specific input mode, the corresponding 2D slice of dimension \\({N}_{\\text{output}}\\times {N}_{\\lambda }\\) of the MSTM is measured in ~16 seconds: the full MSTM is then measured in a bit more than an hour. Additional details on the experimental setup are provided in Supplementary Note 1 . Enhancing the intensity at an arbitrary delay time With a sufficient spectral sampling, the MSTM can be Fourier transformed to access the transmission matrices as a function of delay \\(U(\\tau )\\) . The obtained Time Resolved Transmission matrix (TRTM) contains the full relationship between the spatiotemporal input field to the spatiotemporal output field. The TRTM has been exploited so far to focus light in space and time [11] and to study spatio-temporal correlations [35] . We show in Supplementary Fig. 1 a set of transmission matrices at different delay times of the multimode fiber, that we calculated from our measured MSTM. We demonstrate here how to exploit the TRTM to control the light intensity integrated over all spatial and polarization modes at an arbitrary delay \\({\\tau }_{s}\\) . The transmission matrix of the MMF at \\({\\tau }_{s}\\) , that we write \\({U}_{{\\tau }_{s}}\\) , is extracted from the TRTM. Initially developed in acoustics [36] , [37] , the Time Reversal Operator \\({U}^{\\dagger }U\\) , where the \\(\\dagger\\) operator stands for the conjugate transpose, has been extensively used in the monochromatic regime for selective focusing [38] , and the study of open channels [39] , [40] , [41] , [42] . Indeed, its eigenvalues are directly related to the total transmitted energy at the output of the disordered system [26] , [43] . In the time domain, the eigenvectors of \\({U}_{{\\tau }_{s}}^{\\dagger }{U}_{{\\tau }_{s}}\\) have been used to study optimal transmission through scattering systems [44] . For instance, in order to maximize the transmitted energy at the delay time \\({\\tau }_{s}\\) , the input state \\({E}_{\\,\\text{input}}^{\\text{max}\\,}\\) is the eigenvector of \\({U}_{{\\tau }_{s}}^{\\dagger }{U}_{{\\tau }_{s}}\\) associated to the highest eigenvalue [35] . Our experimental setup enables us to measure all spatial modes in both polarizations at both ends of the fiber within \\(U(\\lambda )\\) , and thus within \\({U}_{{\\tau }_{s}}\\) . Therefore launching \\({E}_{\\,\\text{input}}^{\\text{max}\\,}\\) is thus enhancing both the horizontal and the vertical polarization states at the output of the MMF at the delay time \\({\\tau }_{s}\\) . In contrast with digital phase conjugation of \\({U}_{{\\tau }_{s}}\\) [11] , this eigenstate is not focusing light in a certain spatial mode or in a spatial position: it is enhancing the light intensity at \\({\\tau }_{s}\\) for all the spatial and polarization modes. We experimentally demonstrate this enhancement of light intensity at an arbitrary delay time in Fig. 2 . The delay time \\(\\tau =0\\) ps corresponds to the arrival time of the first principal mode of the MMF [23] . As an illustrative example, we maximize the transmitted intensity at \\({\\tau }_{\\text{max}}=\\) 8.5 ps in Fig. 2 a. Once we display the eigenvector associated to the maximum eigenvalue of \\({U}_{{\\tau }_{\\text{max}}}^{\\dagger }{U}_{{\\tau }_{\\text{max}}}\\) onto the SLM, we measure the impulse response \\(I(\\tau )\\) of the output state. For this purpose, we measure the spectrally resolved output field \\({E}_{\\text{output}}(\\lambda )\\) with the SWI, while the fixed input state is being displayed on the SLM. The impulse response is calculated from the superposition over all the \\({N}_{\\text{output}}\\) output modes: $$I(\\tau ) = \\sum _{j=1}^{{N}_{\\text{output}}}| {\\hat{E}}_{\\text{output}}(\\tau ,j){| }^{2}$$ (1) with \\({\\hat{E}}_{\\text{output}}(\\tau ,j)\\) the Fourier transform of \\({E}_{\\text{output}}(\\lambda )\\) decomposed on the j-th output mode. The impulse response is then normalized. The calculated impulse response with Eq. 1 is then equivalent to what would be measured by a large photodetector at the end of the fiber. Fig. 2 Enhancing light intensity at a chosen delay time over all the spatial and polarization modes of a multimode fiber. a Enhancing light intensity at \\({\\tau }_{\\text{max}}=8.7\\) ps. Blue: time of flight distribution of the multimode fiber. Red: experimental impulse response of the eigenstate maximizing the light intensity at \\({\\tau }_{\\text{max}}\\) . Green: numerical propagation of this eigenstate with the experimentally measured MSTM. Inset: zoom around \\({\\tau }_{\\text{max}}\\) ; Reconstructed output intensity at \\({\\tau }_{\\text{max}}\\) for the two polarization states for the eigenstate (top row) and a random combination of modes (bottom row). b Enhancement ratio \\(\\eta\\) for different delay times. Red: enhancement ratios calculated with the experimentally measured impulse responses. Green: enhancement ratios calculated with the numerically propagated impulse responses Full size image In Fig. 2 a, we plot the experimentally measured time of flight distribution of the multimode fiber in blue color as a reference impulse response. It corresponds to the average impulse response \\({I}_{\\text{rand}}\\) over a set of \\({N}_{\\text{rand}}\\) = 20 random combinations over all spatial and polarization input states. The experimentally measured impulse response \\({I}_{{\\tau }_{\\text{max}}}\\) of the maximum eigenstate at \\({\\tau }_{\\text{max}}\\) is plotted in red in Fig. 2 a, which clearly shows a sharp peak at \\({\\tau }_{\\text{max}}\\) . This impulse response could be numerically predicted with the MSTM. For every wavelength \\(\\lambda\\) , the expected output field \\({E}_{\\,\\text{output}}^{\\text{numerical}\\,}(\\lambda )\\) reads \\({E}_{\\,\\text{output}}^{\\text{numerical}\\,}(\\lambda )\\,=\\,U(\\lambda )\\,{E}_{\\,\\text{input}}^{\\text{max}\\,}\\) . From the set of \\({E}_{\\,\\text{output}}^{\\text{numerical}\\,}(\\lambda )\\) , we can use Eq. 1 to calculate the expected impulse response \\({I}_{{\\tau }_{\\text{max}}}^{\\text{numerical}\\,}\\) . We plot \\({I}_{{\\tau }_{\\text{max}}}^{\\text{numerical}\\,}\\) in green in Fig. 2 a, which is, as expected, very similar to the experimentally measured impulse response. From the spectrally resolved field measurements, we can reconstruct the output intensity at the delay time \\({\\tau }_{\\text{max}}\\) with a Fourier transform. In the inset of Fig. 2 a, we show the reconstructed intensity at \\({\\tau }_{\\text{max}}\\) for both polarization states for the enhancing eigenstate (top row). The output intensity is here spread over a combination of spatial modes for both the horizontal and the vertical polarization. As a contrast with digital optical phase conjugation of the TRTM [11] and previous spatio-temporal control through scattering systems in the literature, the output intensity is not spatially focused to a mode or to a speckle grain, as all the output spatial modes are enhanced at that specific delay. As a comparison, we also show the reconstructed intensity of a random combination of input modes of both polarization at \\({\\tau }_{\\text{max}}\\) , with the same color-bar (bottom row). Similarly to spatial focusing experiments [4] , we define an enhancement ratio \\(\\eta ({\\tau }_{\\text{max}})\\) when enhancing the output intensity at delay time \\({\\tau }_{\\text{max}}\\) . Based on the impulse responses plotted in Fig. 2 a., \\(\\eta ({\\tau }_{\\text{max}})\\) is defined as follows: $$\\eta ({\\tau }_{\\text{max}})=\\frac{{I}_{{\\tau }_{\\text{max}}}({\\tau }_{\\text{max}})}{{I}_{\\text{rand}}({\\tau }_{\\text{max}})}$$ (2) with \\({I}_{{\\tau }_{\\text{max}}}({\\tau }_{\\text{max}})\\) and \\({I}_{\\text{rand}}({\\tau }_{\\text{max}})\\) the intensity of the impulse response of the enhancing eigenstate and the time of flight at the delay time \\({\\tau }_{\\text{max}}\\) . In Fig. 2 b, we compare the enhancement ratio \\(\\eta (\\tau )\\) calculated with the experimentally measured impulse responses (red markers) with the expected enhancement \\({\\eta }^{\\text{numerical}}(\\tau )\\) (green markers) for a set of different arrival times spread between \\(\\tau =0\\) ps and \\(\\tau =45\\) ps. The expected enhancement can be predicted with the TRTM. However, we cannot use the random matrix theory to predict the eigenvalue distribution as the MMF is not in the strong mode coupling regime [35] . The expected enhancement \\({\\eta }^{\\text{numerical}}(\\tau )\\) is calculated with Eq. 2 , where \\({I}_{{\\tau }_{\\text{max}}}^{\\text{numerical}\\,}\\) replaces \\({I}_{{\\tau }_{\\text{max}}}\\) . As the TRTM contains the full spatiotemporal relationship between the input and the output fields, \\(\\eta (\\tau )\\) follows, as expected, a similar trend as \\({\\eta }^{\\text{numerical}}(\\tau )\\) . Modes arriving at very long delays are higher-order modes, that are near cut-off. These higher-order modes experience less mode coupling and hence sharper temporal features [45] , which tends to increase the enhancement ratio for the eigenmodes at long delays. Additional data processing, such all the individual impulse responses used to calculate \\(\\eta\\) and the spatial reconstruction of the beam at the different delays, are presented in the Supplementary Note 4 . Finally, in our study we do not launch an input pulse as we measure the data with swept-wavelength digital holography. A simulation of the propagation of an input pulse, and its effect on the impulse response and the enhancement is presented in the Supplementary Note 4 . Attenuating the intensity at an arbitrary delay time The time reversal operator \\({U}_{{\\tau }_{s}}^{\\dagger }{U}_{{\\tau }_{s}}\\) enables to adjust the transmitted intensity over all spatial and polarization modes at the delay \\({\\tau }_{s}\\) . Exploiting the eigenvector associated with its maximum eigenvalue enables enhancement of intensity at \\({\\tau }_{s}\\) . Instead of enhancing the light delivery, we can also attenuate the impulse response at the delay time \\({\\tau }_{s}\\) with the eigenvector \\({E}_{\\,\\text{input}}^{\\text{min}\\,}\\) associated with the minimum eigenvalue of \\({U}_{{\\tau }_{s}}^{\\dagger }{U}_{{\\tau }_{s}}\\) . We illustrate in Fig. 3 a the experimental impulse response associated with the minimum eigenstate at a chosen delay time \\({\\tau }_{\\text{min}}=19\\) ps. The expected impulse response shows a clear minimum intensity at \\({\\tau }_{\\text{min}}\\) . Experimentally, the impulse response reveals as well a sharp minimum at \\({\\tau }_{\\text{min}}\\) , whose intensity is much smaller than the time of flight distribution at the same arrival time. Fig. 3 Attenuating light intensity at a chosen delay time. a Attenuating light intensity at \\({\\tau }_{\\text{min}}=19\\) ps. Blue: time of flight distribution of the multimode fiber. Red: experimental impulse response of the eigenstate minimizing the light intensity at \\({\\tau }_{\\text{min}}\\) . Green: numerical propagation of this eigenstate with the experimentally measured MSTM. Inset: zoom around \\({\\tau }_{\\text{min}}\\) ( b ) Attenuation ratio \\(\\rho\\) for different delay times in logarithm scale. Red: attenuation ratios calculated with the experimentally measured impulse responses. Green: attenuation ratios calculated with the numerically propagated impulse responses Full size image Similarly to the enhancement ratio defined in Eq. 2 , we also define an attenuation ratio \\(\\rho ({\\tau }_{\\text{min}})\\) for such minimum eigenstates: $$\\rho ({\\tau }_{\\text{min}})=\\frac{{I}_{\\text{rand}}({\\tau }_{\\text{min}})}{{I}_{{\\tau }_{\\text{max}}}({\\tau }_{\\text{min}})}$$ (3) We plot the attenuation ratio in logarithm scale for a set of different delay time between 0 ps and 45 ps in Fig. 3 b. The numerical extinction ratio \\(< {\\!\\!}-20\\) dB implies we should be able to completely attenuate the beam at every delay time, as the inset of Fig. 3 a illustrates for instance at specific delay \\({\\tau }_{\\text{min}}\\) . However, the experimental conditions are not allowing a pure zero-intensity in the experimentally measured impulse response. Indeed, the MSTM of the MMF has slightly decorrelated between its measurement and when the eigenstates are being measured. Besides, the superposition of the \\({N}_{\\text{input}}=254\\) input modes on the SLM might not be as accurate as expected in the experiment. Finally, the experimental stability level, whether on the detection as well as on the illumination, add some noise which prevents the observation of a pure zero-intensity in the impulse response. Ultimately, this non-zero experimental intensity tends to decrease the extinction ratio at larger delays where the time of flight gets a smaller intensity. Polarization control of the generated output state The propagating modes in the multimode fiber are polarization-resolved. In order to address all the modes, it is essential to control both the spatial and the polarization degrees of freedom. Wavefront shaping experiments in disordered materials, including multimode fiber, are often sensing only a quarter of the transmission matrix, by neglecting a polarization state both at the input and at the output [7] , [24] . Neglecting the polarization is equivalent to ignoring half of the propagating modes in the fiber. Full control of all the modes requires then the measurement of the polarization-resolved transmission matrix of the fiber. The experimental setup presented in Fig. 1 enables the measurement of the polarization-resolved MSTM of the MMF. Indeed, we can generate and detect independently the two polarization states for all the spectral components. Therefore the MSTM \\(U(\\lambda )\\) could be written as follows: $$U(\\lambda )=\\left[\\begin{array}{cc}{U}_{HH}(\\lambda )&{U}_{HV}(\\lambda )\\\\ {U}_{VH}(\\lambda )&{U}_{VV}(\\lambda )\\end{array}\\right]$$ (4) with \\({U}_{ij}(\\lambda )\\) the transmission matrix relating the input spatio-temporal field on the \\(i-th\\) polarization state to the output spatio-temporal field on the \\(j-th\\) polarization state. The dimension of each sub-matrix is \\({N}_{\\text{input}}/2\\times {N}_{\\text{output}}/2\\) . Let’s assume we want to generate an arbitrary state in a chosen polarization state \\(j\\) at the output of the fiber at a wavelength \\(\\lambda\\) , whether \\(j\\) is H or V. For this purpose we thus use \\({U}_{j}(\\lambda )=[{U}_{Hj}(\\lambda )\\,{U}_{Vj}(\\lambda )]\\) [29] . In the delay domain, the calculated TRTM from the experimentally measured MSTM is also polarization-resolved: \\({U}_{j,{\\tau }_{s}}\\) is the TRTM at the delay time \\({\\tau }_{s}\\) for the output polarization state \\(j\\) . The time reversal operator \\({U}_{j,{\\tau }_{s}}^{\\dagger }{U}_{j,{\\tau }_{s}}\\) enables then a temporal control of the overall combination of spatial modes for a specific arrival time \\({\\tau }_{s}\\) and a specific polarization state at the output of the MMF. Similarly to Fig. 2 , the output intensity can be enhanced over all the spatial modes for only the \\(j\\) -polarization state at the output. The input field \\({E}_{\\,\\text{input}}^{\\text{max,j}\\,}\\) is the eigenvector associated to the maximum eigenvalue of \\({U}_{j,{\\tau }_{s}}^{\\dagger }{U}_{j,{\\tau }_{s}}\\) . In Fig. 4 a, b, we plot the polarization-resolved impulse responses associated to such eigenstates with \\({\\tau }_{s}=8.5\\) ps. The polarization-resolved impulse responses are calculated with Eq. 1 , using only the output modes on the chosen polarization state. Figure 4 a, b clearly evidence an enhancement of the intensity overall spatial modes at \\({\\tau }_{s}\\) for either only H or only V. The reconstructed intensity at the output for H and V at the delay time \\({\\tau }_{s}\\) confirms that the transmitted light at \\({\\tau }_{s}\\) is polarized in the chosen state. An incomplete control of the input polarization state would lead to similar results, but with a lower energy delivery at \\({\\tau }_{s}\\) as half of the input power would not be controlled. Such an incomplete polarization control is shown in Supplementary Fig. 8 . The MMF in conjunction with the SLM can then be used as an accurate deterministic polarization control device in the time domain. This experiment could also be used to study polarization recovery in scattering media [46] . Fig. 4 Polarization control of the engineered state at an arbitrary delay time \\({\\tau }_{s}\\) . We choose here \\({\\tau }_{s}=8.5\\) ps. a Enhancing only the H component of the output field. b Enhancing only the V component of the output field. c Attenuating only the H component of the output field. d Attenuating only the V component of the output field. Inset: reconstructed intensity at the output at delay time \\({\\tau }_{s}\\) for both H and V. Color code: (red) total impulse response of H and V; (black) impulse response of the H component; (orange) impulse response of the V component; (blue) time of flight distribution of H and V Full size image We also demonstrate an attenuation of a chosen polarization state at the output of the fiber at the delay time \\({\\tau }_{s}\\) . Instead of launching \\({E}_{\\,\\text{input}}^{\\text{max,j}\\,}\\) , we calculate \\({E}_{\\,\\text{input}}^{\\text{min,j}\\,}\\) which is the input eigenvector associated to the minimum eigenvalue of \\({U}_{j,{\\tau }_{s}}^{\\dagger }{U}_{j,{\\tau }_{s}}\\) . Figure 4 c, d present the experimental polarization-resolved impulse responses associated to such eigenstates for attenuating either only H or only V. Figure 4 c, d unambiguously reveal a strong attenuation at \\({\\tau }_{s}\\) for the controlled output polarization state. In principle, any arbitrary polarization state can be achieved at a given delay, similarly to previous work in the monochromatic regime [47] , based on the superposition of the four sub time gated transmission matrices of Eq. 4 . Some examples are shown in Supplementary Fig. 9 . Multiple delay control In Fig. 2 , we have demonstrated the enhancement of light intensity at a chosen arbitrary time \\({\\tau }_{s}\\) over all the spatial and polarization modes. The TRTM could also be used to shape more complex impulse responses. Spatio-temporal focusing over multiple space-time positions have been demonstrated with digital optical phase conjugation of the TRTM [11] . Here we demonstrate an enhancement of the transmitted light intensity at multiple delay times with the experimental setup of Fig. 1 . The time reversal operator \\({U}_{{\\tau }_{s}}^{\\dagger }{U}_{{\\tau }_{s}}\\) enables to adjust the transmitted energy at the delay time \\({\\tau }_{s}\\) . Figs. 2 and 3 have shown this control for either enhancing or attenuating light intensity at a chosen delay time \\({\\tau }_{s}\\) . The spectral diversity of the multimode fiber ensures that this control is only affecting delays around \\({\\tau }_{s}\\) , as the transmission matrices as a function of wavelength/delay are uncorrelated over a sufficient bandwidth/time interval [21] , [23] , [24] . In order to enhance light intensity at two different delays \\({\\tau }_{1}\\) and \\({\\tau }_{2}\\) , we use the time reversal operators \\({U}_{{\\tau }_{1}}^{\\dagger }{U}_{{\\tau }_{1}}\\) and \\({U}_{{\\tau }_{2}}^{\\dagger }{U}_{{\\tau }_{2}}\\) . We calculate the input eigenvectors \\({E}_{\\,\\text{input}}^{\\text{max}\\,,{\\tau }_{1}}\\) and \\({E}_{\\,\\text{input}}^{\\text{max}\\,,{\\tau }_{2}}\\) associated to the highest eigenvalue of the two single operators. The linear superposition \\({E}_{\\,\\text{input}}^{\\text{max},{\\tau }_{1},{\\tau }_{2}}={E}_{\\text{input}}^{\\text{max},{\\tau }_{1}}+{E}_{\\text{input}}^{\\text{max}\\,,{\\tau }_{2}}\\) enables a control of the impulse response at the two delay time \\({\\tau }_{1}\\) and \\({\\tau }_{2}\\) . The phase of the solution would then be displayed on the SLM, similarly to spatio-temporal focusing at multiple delays [11] , [21] . This control can be extended to a superposition of \\({N}_{\\text{superpos.}}\\) delay times. The input field would then read: $${E}_{\\,\\text{input}}^{\\text{max},{N}_{\\text{superpos.}}} = \\sum _{i=1}^{{N}_{\\text{superpos.}}}{E}_{\\text{input}}^{\\text{max}\\,,{\\tau }_{i}}$$ (5) with \\({E}_{\\,\\text{input}}^{\\text{max}\\,,{\\tau }_{i}}\\) the eigenvector associated with the highest eigenvalue of \\({U}_{{\\tau }_{i}}^{\\dagger }{U}_{{\\tau }_{i}}\\) . In practice, \\({N}_{\\text{superpos.}}\\) would be limited by the temporal width of the impulse response of each single mode. The impulse responses may interfere for different times, especially if the \\({\\tau }_{i}\\) are close in the time domain. We plot in Supplementary Fig. 3 the temporal correlation between the different Time Resolved Transmission Matrices. The quality of the superposition of the modes on the SLM would also limit the efficiency of a high \\({N}_{\\text{superpos.}}\\) . In Fig. 5 we demonstrate multiple delay control of the transmitted intensity with the experimental setup of Fig. 1 . In Fig. 5 a, \\({N}_{\\text{superpos.}}=6\\) delay times are controlled, evenly spread between 4 ps and 44 ps. Figure 5 b illustrates a similar control over a narrower time interval. The intensity of \\({N}_{\\text{superpos.}}=7\\) delay times is enhanced, evenly spread between 4 ps and 19 ps. We clearly identify \\({N}_{\\text{superpos.}}\\) peaks in Fig. 5 a and Fig. 5 b. The enhancement ratios at the \\({N}_{\\text{superpos.}}\\) delays are much lower than the individual delay enhancement ratios presented in Fig. 2 b. However the total transmitted power remains similar for either a single delay enhancement or a multiple delay control. The amplitude ratio between these peaks could also be adjusted using an amplitude coefficient in the superposition in Eq. 5 . A control of different \\({N}_{\\text{superpos.}}\\) within the same time interval is presented in the Supplementary Figs. 10 and 11 . This multiple delay control could also be exploited in conjunction with Fig. 4 , where the two polarization states could be manipulated at different arrival times. Fig. 5 Enhancement of transmitted light intensity at multiple delay times. a Enhancing the light intensity at 6 different delays \\({\\tau }_{i}\\) evenly spread from 4 ps to 44 ps. b Enhancing the light intensity at 7 different delays \\({\\tau }_{i}\\) evenly spread from 4 ps to 19 ps Full size image The ability to manipulate multiple delay times within long delay and dispersion provided by a very compact optical system such as a multimode fiber opens interesting perspectives for compact pulse shaping experiments. We believe this multiple delay control might also be useful for coherent control and pump probe experiments for imaging purposes or for complex control of light-matter interactions."},{"section":"Discussion","content":"Over the last decade, wavefront shaping with spatial light modulators have revolutionized the study of light propagation in disordered systems, where scattering was considered as an insurmountable obstacle. Most of the previous works studied spatial control of the output field at the expense of the temporal behavior. We have demonstrated here the temporal control of light intensity averaged over all the spatial and polarization modes after propagation through a multimode fiber, at the expense of the spatial pattern. More precisely, we have shown how to enhance or attenuate the transmitted intensity at any arbitrary delay times, arbitrary polarization state, and arbitrary combinations of these. This work opens interesting perspectives in applications that require the measurement or control of spatio-temporal properties of light in scattering media. A pulse with a large number of transverse modes could be useful for the following non exhaustive list of examples: the study of correlations in disordered systems [35] , [48] , [49] , [50] telecommunications [51] , non-linear imaging [31] , [52] and endoscopy [53] , [54] , [55] through complex systems, time-gating imaging [56] , [57] , temporal focusing microscopy [58] , [59] , [60] and high power fiber lasers [61] . Our polarization control of the beam in the temporal domain could find applications in polarization sensitive optical coherence tomography [62] , [63] , [64]"},{"section":"Acknowledgements","content":"We thank Martin Plöschner for careful reading of our manuscript. This work is supported by the Australian Research Council (ARC) projects DE180100009 and DP170101400."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"Light propagation in disordered systems has been extensively studied over the past 40 years [1] . Transmitted light through such materials forms a complex pattern after propagation, known as speckle, that is the signature of the interference between a large number of modes within the sample. Although this speckle looks random, this mixing of light is nonetheless linear and thus deterministic. Wavefront shaping has revolutionized the spatial control of coherent light beams thanks to the use of spatial light modulators (SLM) in disordered systems [2] , [3] , such as biological tissue, white paint or multimode fibers. Different approaches have been developed to control the propagation of coherent light that has experienced scattering, such as iterative optimization [4] , [5] , optical phase conjugation [6] , and the measurement of the optical transmission matrix [7] . These methods have been further extended to the control of polarization [8] , spectral [9] , [10] , and temporal properties of light [11] , and widely used for imaging [12] , [13] , [14] , [15] and optical manipulation [16] . The spatial and temporal distortions of scattered light are coupled in disordered systems. Therefore, the temporal properties of a spatio-temporal speckle field can be adjusted with spatial-only control. In previous works, manipulating the temporal properties of scattered light was achieved with spatio-temporal focusing with a single SLM. The output pulse is focused on a single speckle grain, and the spectral properties of the focus are controlled to ensure a short duration of the output pulse. Spatio-temporal focusing of the output pulse can be reached for instance with optimization algorithms [17] , [18] , digital phase conjugation [19] , pulse shaping methods [20] , or via the knowledge of the transmission matrix of the sample [11] , [21] . However, the output pulse has only a short duration at this specified focus position, while the background speckle remains temporally elongated [22] . The temporal control of the spatio-temporal field is then limited to a specific spatial mode (speckle position or a specific spatial pattern), at the expense of undefined temporal behavior for the other spatial modes. Equivalently, control of the impulse response of only a small portion of the total power means the remaining field is effectively a loss and potentially a source of noise. Most of the total input power does not arrive at a controlled delay, it is scattered into the temporal background. A control of the full speckle pattern over a certain spectral bandwidth has been proposed based on the transmission matrix, with the time delay operator [23] , [24] , [25] . Nonetheless, the limited controlled spectral bandwidth does not allow a full temporal control of the spatio-temporal speckle. In this paper, we propose a novel method to adjust the temporal properties of the total output intensity at any arbitrary delay over all the spatial and polarization modes propagating through a multimode fiber. After experimentally measuring the Multi-Spectral Transmission Matrix (MSTM) of the fiber, we calculate the Time Resolved Transmission Matrix (TRTM) with a Fourier Transform. Exploiting the TRTM, we demonstrate for the first time both temporal enhancement and attenuation of the full beam at specific delay times at the expense of undefined behaviour in the spatial domain. In contrast with previous spatio-temporal focusing experiments where the temporal properties are adjusted only in the spatial focus position, all the output spatial modes are controlled at that specific delay. 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Lett.117, 053901 (2016).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.117.053901\"\n },\n {\n \"idx\": \"25\",\n \"title\": \"Ambichl, P. et al. Super- and anti-principal-modes in multimode waveguides.Phys. Rev.7, 041053 (2017).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevX.7.041053\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Experimental setup At a given wavelength \\\\(\\\\lambda\\\\) , the transmission matrix \\\\(U\\\\) of a disordered system, such as a multimode fiber, linearly links the input field (adjustable with a SLM) to the output field (detectable with a camera). \\\\(U\\\\) is a \\\\({N}_{\\\\text{input}}\\\\,\\\\times \\\\,{N}_{\\\\text{output}}\\\\) matrix, with \\\\({N}_{\\\\text{input}}\\\\) and \\\\({N}_{\\\\text{output}}\\\\) the number of spatial and polarization modes at the input and at the output of the system. Note that in a multimode fiber, all the \\\\({N}_{\\\\text{input}}\\\\) modes (on the order of 100–1000 modes depending on its geometrical parameters) could be measured in contrast with highly disordered materials [26] , [27] . The transmission matrix has been widely used to manipulate light that has suffered from scattering or mode mixing, for focusing [7] or imaging purposes [28] . Recently, the transmission matrix has been extended to the spectral domain. The Multi Spectral Transmission Matrix [9] , [21] , also known as the optical transfer function [29] , is a stack of transmission matrices \\\\(U(\\\\lambda )\\\\) for a set of different input wavelength, that can be measured by sweeping the wavelength or using hyper-spectral imaging [30] . The MSTM enables a spectral control of the output field exploiting the spectral diversity of the medium, and could be used for focusing a pattern at a given wavelength [9] , [29] , pulse shaping the output focus [21] and imaging through scattering samples [31] . We exploit the setup presented in Fig. 1 to measure the Multi Spectral Transmission Matrix of a multimode fiber (MMF). The light source is a tunable continuous wave (CW) laser (New focus TLM-8700) operating between 1510 nm and 1620 nm (bandwidth \\\\(\\\\Delta \\\\lambda =110\\\\) nm \\\\(\\\\simeq 13.5\\\\) THz). We use a phase-only spatial light modulator (Meadowlark P1920) to generate any spatial and polarization input state in amplitude and phase [32] . The SLM is located in the Fourier plane of a multimode fiber. The fiber is a 1-m length of a 50 \\\\(\\\\mu\\\\) m core radius step-index multimode fiber (Thorlabs M42L01), which theoretically supports ~127 spatial modes per polarization state [33] . A step-index multimode fiber has more mode coupling and a more complex and spread impulse response than a graded index fiber with similar dimensions. The multimode fiber we use in this experiment is not in the strong mode coupling regime [34] (see Supplementary Note 2 for more details on the mode coupling in the fiber). The output field is measured with digital off-axis holography, using an external path-length matched reference beam and an InGaAs camera (Xenics Xeva-1.7-320). The two polarization states are measured on the left and the right side of the camera using a polarizer beam displacer. Fig. 1 Apparatus of the experimental setup. SMF: single mode fiber; SLM: spatial light modulator; PBS: polarizing beam splitter; MMF: multimode fiber; NPBS: non-polarizing beam splitter; (L1, L2, L3, and L4): optical lenses. Focal length: L1: 25 mm; L2: 10 mm; L3: 4 mm; and L4: 60 mm Full size image Measurement of the Transmission Matrix of a multimode fiber We use a swept-wavelength interferometer (SWI) to measure the multi spectral transmission matrix of the MMF [29] , [32] . We measure the complete transmission matrix by generating all the spatial and polarization modes (a total of \\\\({N}_{\\\\text{input}}=254\\\\) modes) on the input facet of the MMF using the SLM. We launch the modes of the step-index fiber in a linearly polarized basis with helical wavefronts. For each individual input mode, we sweep the laser and record the output field for all the wavelengths with the digital holography beam analysis. Through the sweep, the camera is triggered at 10.6 GHz through the sweep with an external Mach-Zehnder interferometer that acts as a k-clock. This triggering method enables the measurement of \\\\({N}_{\\\\lambda }=1273\\\\) monochromatic transmission matrices per sweep. Each output field is then digitally decomposed on the fly in the Laguerre-Gaussian basis ( \\\\({N}_{\\\\text{output}}=650\\\\) modes). For a specific input mode, the corresponding 2D slice of dimension \\\\({N}_{\\\\text{output}}\\\\times {N}_{\\\\lambda }\\\\) of the MSTM is measured in ~16 seconds: the full MSTM is then measured in a bit more than an hour. Additional details on the experimental setup are provided in Supplementary Note 1 . Enhancing the intensity at an arbitrary delay time With a sufficient spectral sampling, the MSTM can be Fourier transformed to access the transmission matrices as a function of delay \\\\(U(\\\\tau )\\\\) . The obtained Time Resolved Transmission matrix (TRTM) contains the full relationship between the spatiotemporal input field to the spatiotemporal output field. The TRTM has been exploited so far to focus light in space and time [11] and to study spatio-temporal correlations [35] . We show in Supplementary Fig. 1 a set of transmission matrices at different delay times of the multimode fiber, that we calculated from our measured MSTM. We demonstrate here how to exploit the TRTM to control the light intensity integrated over all spatial and polarization modes at an arbitrary delay \\\\({\\\\tau }_{s}\\\\) . The transmission matrix of the MMF at \\\\({\\\\tau }_{s}\\\\) , that we write \\\\({U}_{{\\\\tau }_{s}}\\\\) , is extracted from the TRTM. Initially developed in acoustics [36] , [37] , the Time Reversal Operator \\\\({U}^{\\\\dagger }U\\\\) , where the \\\\(\\\\dagger\\\\) operator stands for the conjugate transpose, has been extensively used in the monochromatic regime for selective focusing [38] , and the study of open channels [39] , [40] , [41] , [42] . Indeed, its eigenvalues are directly related to the total transmitted energy at the output of the disordered system [26] , [43] . In the time domain, the eigenvectors of \\\\({U}_{{\\\\tau }_{s}}^{\\\\dagger }{U}_{{\\\\tau }_{s}}\\\\) have been used to study optimal transmission through scattering systems [44] . For instance, in order to maximize the transmitted energy at the delay time \\\\({\\\\tau }_{s}\\\\) , the input state \\\\({E}_{\\\\,\\\\text{input}}^{\\\\text{max}\\\\,}\\\\) is the eigenvector of \\\\({U}_{{\\\\tau }_{s}}^{\\\\dagger }{U}_{{\\\\tau }_{s}}\\\\) associated to the highest eigenvalue [35] . Our experimental setup enables us to measure all spatial modes in both polarizations at both ends of the fiber within \\\\(U(\\\\lambda )\\\\) , and thus within \\\\({U}_{{\\\\tau }_{s}}\\\\) . Therefore launching \\\\({E}_{\\\\,\\\\text{input}}^{\\\\text{max}\\\\,}\\\\) is thus enhancing both the horizontal and the vertical polarization states at the output of the MMF at the delay time \\\\({\\\\tau }_{s}\\\\) . In contrast with digital phase conjugation of \\\\({U}_{{\\\\tau }_{s}}\\\\) [11] , this eigenstate is not focusing light in a certain spatial mode or in a spatial position: it is enhancing the light intensity at \\\\({\\\\tau }_{s}\\\\) for all the spatial and polarization modes. We experimentally demonstrate this enhancement of light intensity at an arbitrary delay time in Fig. 2 . The delay time \\\\(\\\\tau =0\\\\) ps corresponds to the arrival time of the first principal mode of the MMF [23] . As an illustrative example, we maximize the transmitted intensity at \\\\({\\\\tau }_{\\\\text{max}}=\\\\) 8.5 ps in Fig. 2 a. Once we display the eigenvector associated to the maximum eigenvalue of \\\\({U}_{{\\\\tau }_{\\\\text{max}}}^{\\\\dagger }{U}_{{\\\\tau }_{\\\\text{max}}}\\\\) onto the SLM, we measure the impulse response \\\\(I(\\\\tau )\\\\) of the output state. For this purpose, we measure the spectrally resolved output field \\\\({E}_{\\\\text{output}}(\\\\lambda )\\\\) with the SWI, while the fixed input state is being displayed on the SLM. The impulse response is calculated from the superposition over all the \\\\({N}_{\\\\text{output}}\\\\) output modes: $$I(\\\\tau ) = \\\\sum _{j=1}^{{N}_{\\\\text{output}}}| {\\\\hat{E}}_{\\\\text{output}}(\\\\tau ,j){| }^{2}$$ (1) with \\\\({\\\\hat{E}}_{\\\\text{output}}(\\\\tau ,j)\\\\) the Fourier transform of \\\\({E}_{\\\\text{output}}(\\\\lambda )\\\\) decomposed on the j-th output mode. The impulse response is then normalized. The calculated impulse response with Eq. 1 is then equivalent to what would be measured by a large photodetector at the end of the fiber. Fig. 2 Enhancing light intensity at a chosen delay time over all the spatial and polarization modes of a multimode fiber. a Enhancing light intensity at \\\\({\\\\tau }_{\\\\text{max}}=8.7\\\\) ps. Blue: time of flight distribution of the multimode fiber. Red: experimental impulse response of the eigenstate maximizing the light intensity at \\\\({\\\\tau }_{\\\\text{max}}\\\\) . Green: numerical propagation of this eigenstate with the experimentally measured MSTM. Inset: zoom around \\\\({\\\\tau }_{\\\\text{max}}\\\\) ; Reconstructed output intensity at \\\\({\\\\tau }_{\\\\text{max}}\\\\) for the two polarization states for the eigenstate (top row) and a random combination of modes (bottom row). b Enhancement ratio \\\\(\\\\eta\\\\) for different delay times. Red: enhancement ratios calculated with the experimentally measured impulse responses. Green: enhancement ratios calculated with the numerically propagated impulse responses Full size image In Fig. 2 a, we plot the experimentally measured time of flight distribution of the multimode fiber in blue color as a reference impulse response. It corresponds to the average impulse response \\\\({I}_{\\\\text{rand}}\\\\) over a set of \\\\({N}_{\\\\text{rand}}\\\\) = 20 random combinations over all spatial and polarization input states. The experimentally measured impulse response \\\\({I}_{{\\\\tau }_{\\\\text{max}}}\\\\) of the maximum eigenstate at \\\\({\\\\tau }_{\\\\text{max}}\\\\) is plotted in red in Fig. 2 a, which clearly shows a sharp peak at \\\\({\\\\tau }_{\\\\text{max}}\\\\) . This impulse response could be numerically predicted with the MSTM. For every wavelength \\\\(\\\\lambda\\\\) , the expected output field \\\\({E}_{\\\\,\\\\text{output}}^{\\\\text{numerical}\\\\,}(\\\\lambda )\\\\) reads \\\\({E}_{\\\\,\\\\text{output}}^{\\\\text{numerical}\\\\,}(\\\\lambda )\\\\,=\\\\,U(\\\\lambda )\\\\,{E}_{\\\\,\\\\text{input}}^{\\\\text{max}\\\\,}\\\\) . From the set of \\\\({E}_{\\\\,\\\\text{output}}^{\\\\text{numerical}\\\\,}(\\\\lambda )\\\\) , we can use Eq. 1 to calculate the expected impulse response \\\\({I}_{{\\\\tau }_{\\\\text{max}}}^{\\\\text{numerical}\\\\,}\\\\) . We plot \\\\({I}_{{\\\\tau }_{\\\\text{max}}}^{\\\\text{numerical}\\\\,}\\\\) in green in Fig. 2 a, which is, as expected, very similar to the experimentally measured impulse response. From the spectrally resolved field measurements, we can reconstruct the output intensity at the delay time \\\\({\\\\tau }_{\\\\text{max}}\\\\) with a Fourier transform. In the inset of Fig. 2 a, we show the reconstructed intensity at \\\\({\\\\tau }_{\\\\text{max}}\\\\) for both polarization states for the enhancing eigenstate (top row). The output intensity is here spread over a combination of spatial modes for both the horizontal and the vertical polarization. As a contrast with digital optical phase conjugation of the TRTM [11] and previous spatio-temporal control through scattering systems in the literature, the output intensity is not spatially focused to a mode or to a speckle grain, as all the output spatial modes are enhanced at that specific delay. As a comparison, we also show the reconstructed intensity of a random combination of input modes of both polarization at \\\\({\\\\tau }_{\\\\text{max}}\\\\) , with the same color-bar (bottom row). Similarly to spatial focusing experiments [4] , we define an enhancement ratio \\\\(\\\\eta ({\\\\tau }_{\\\\text{max}})\\\\) when enhancing the output intensity at delay time \\\\({\\\\tau }_{\\\\text{max}}\\\\) . Based on the impulse responses plotted in Fig. 2 a., \\\\(\\\\eta ({\\\\tau }_{\\\\text{max}})\\\\) is defined as follows: $$\\\\eta ({\\\\tau }_{\\\\text{max}})=\\\\frac{{I}_{{\\\\tau }_{\\\\text{max}}}({\\\\tau }_{\\\\text{max}})}{{I}_{\\\\text{rand}}({\\\\tau }_{\\\\text{max}})}$$ (2) with \\\\({I}_{{\\\\tau }_{\\\\text{max}}}({\\\\tau }_{\\\\text{max}})\\\\) and \\\\({I}_{\\\\text{rand}}({\\\\tau }_{\\\\text{max}})\\\\) the intensity of the impulse response of the enhancing eigenstate and the time of flight at the delay time \\\\({\\\\tau }_{\\\\text{max}}\\\\) . In Fig. 2 b, we compare the enhancement ratio \\\\(\\\\eta (\\\\tau )\\\\) calculated with the experimentally measured impulse responses (red markers) with the expected enhancement \\\\({\\\\eta }^{\\\\text{numerical}}(\\\\tau )\\\\) (green markers) for a set of different arrival times spread between \\\\(\\\\tau =0\\\\) ps and \\\\(\\\\tau =45\\\\) ps. The expected enhancement can be predicted with the TRTM. However, we cannot use the random matrix theory to predict the eigenvalue distribution as the MMF is not in the strong mode coupling regime [35] . The expected enhancement \\\\({\\\\eta }^{\\\\text{numerical}}(\\\\tau )\\\\) is calculated with Eq. 2 , where \\\\({I}_{{\\\\tau }_{\\\\text{max}}}^{\\\\text{numerical}\\\\,}\\\\) replaces \\\\({I}_{{\\\\tau }_{\\\\text{max}}}\\\\) . As the TRTM contains the full spatiotemporal relationship between the input and the output fields, \\\\(\\\\eta (\\\\tau )\\\\) follows, as expected, a similar trend as \\\\({\\\\eta }^{\\\\text{numerical}}(\\\\tau )\\\\) . Modes arriving at very long delays are higher-order modes, that are near cut-off. These higher-order modes experience less mode coupling and hence sharper temporal features [45] , which tends to increase the enhancement ratio for the eigenmodes at long delays. Additional data processing, such all the individual impulse responses used to calculate \\\\(\\\\eta\\\\) and the spatial reconstruction of the beam at the different delays, are presented in the Supplementary Note 4 . Finally, in our study we do not launch an input pulse as we measure the data with swept-wavelength digital holography. A simulation of the propagation of an input pulse, and its effect on the impulse response and the enhancement is presented in the Supplementary Note 4 . Attenuating the intensity at an arbitrary delay time The time reversal operator \\\\({U}_{{\\\\tau }_{s}}^{\\\\dagger }{U}_{{\\\\tau }_{s}}\\\\) enables to adjust the transmitted intensity over all spatial and polarization modes at the delay \\\\({\\\\tau }_{s}\\\\) . Exploiting the eigenvector associated with its maximum eigenvalue enables enhancement of intensity at \\\\({\\\\tau }_{s}\\\\) . Instead of enhancing the light delivery, we can also attenuate the impulse response at the delay time \\\\({\\\\tau }_{s}\\\\) with the eigenvector \\\\({E}_{\\\\,\\\\text{input}}^{\\\\text{min}\\\\,}\\\\) associated with the minimum eigenvalue of \\\\({U}_{{\\\\tau }_{s}}^{\\\\dagger }{U}_{{\\\\tau }_{s}}\\\\) . We illustrate in Fig. 3 a the experimental impulse response associated with the minimum eigenstate at a chosen delay time \\\\({\\\\tau }_{\\\\text{min}}=19\\\\) ps. The expected impulse response shows a clear minimum intensity at \\\\({\\\\tau }_{\\\\text{min}}\\\\) . Experimentally, the impulse response reveals as well a sharp minimum at \\\\({\\\\tau }_{\\\\text{min}}\\\\) , whose intensity is much smaller than the time of flight distribution at the same arrival time. Fig. 3 Attenuating light intensity at a chosen delay time. a Attenuating light intensity at \\\\({\\\\tau }_{\\\\text{min}}=19\\\\) ps. Blue: time of flight distribution of the multimode fiber. Red: experimental impulse response of the eigenstate minimizing the light intensity at \\\\({\\\\tau }_{\\\\text{min}}\\\\) . Green: numerical propagation of this eigenstate with the experimentally measured MSTM. Inset: zoom around \\\\({\\\\tau }_{\\\\text{min}}\\\\) ( b ) Attenuation ratio \\\\(\\\\rho\\\\) for different delay times in logarithm scale. Red: attenuation ratios calculated with the experimentally measured impulse responses. Green: attenuation ratios calculated with the numerically propagated impulse responses Full size image Similarly to the enhancement ratio defined in Eq. 2 , we also define an attenuation ratio \\\\(\\\\rho ({\\\\tau }_{\\\\text{min}})\\\\) for such minimum eigenstates: $$\\\\rho ({\\\\tau }_{\\\\text{min}})=\\\\frac{{I}_{\\\\text{rand}}({\\\\tau }_{\\\\text{min}})}{{I}_{{\\\\tau }_{\\\\text{max}}}({\\\\tau }_{\\\\text{min}})}$$ (3) We plot the attenuation ratio in logarithm scale for a set of different delay time between 0 ps and 45 ps in Fig. 3 b. The numerical extinction ratio \\\\(< {\\\\!\\\\!}-20\\\\) dB implies we should be able to completely attenuate the beam at every delay time, as the inset of Fig. 3 a illustrates for instance at specific delay \\\\({\\\\tau }_{\\\\text{min}}\\\\) . However, the experimental conditions are not allowing a pure zero-intensity in the experimentally measured impulse response. Indeed, the MSTM of the MMF has slightly decorrelated between its measurement and when the eigenstates are being measured. Besides, the superposition of the \\\\({N}_{\\\\text{input}}=254\\\\) input modes on the SLM might not be as accurate as expected in the experiment. Finally, the experimental stability level, whether on the detection as well as on the illumination, add some noise which prevents the observation of a pure zero-intensity in the impulse response. Ultimately, this non-zero experimental intensity tends to decrease the extinction ratio at larger delays where the time of flight gets a smaller intensity. Polarization control of the generated output state The propagating modes in the multimode fiber are polarization-resolved. In order to address all the modes, it is essential to control both the spatial and the polarization degrees of freedom. Wavefront shaping experiments in disordered materials, including multimode fiber, are often sensing only a quarter of the transmission matrix, by neglecting a polarization state both at the input and at the output [7] , [24] . Neglecting the polarization is equivalent to ignoring half of the propagating modes in the fiber. Full control of all the modes requires then the measurement of the polarization-resolved transmission matrix of the fiber. The experimental setup presented in Fig. 1 enables the measurement of the polarization-resolved MSTM of the MMF. Indeed, we can generate and detect independently the two polarization states for all the spectral components. Therefore the MSTM \\\\(U(\\\\lambda )\\\\) could be written as follows: $$U(\\\\lambda )=\\\\left[\\\\begin{array}{cc}{U}_{HH}(\\\\lambda )&{U}_{HV}(\\\\lambda )\\\\\\\\ {U}_{VH}(\\\\lambda )&{U}_{VV}(\\\\lambda )\\\\end{array}\\\\right]$$ (4) with \\\\({U}_{ij}(\\\\lambda )\\\\) the transmission matrix relating the input spatio-temporal field on the \\\\(i-th\\\\) polarization state to the output spatio-temporal field on the \\\\(j-th\\\\) polarization state. The dimension of each sub-matrix is \\\\({N}_{\\\\text{input}}/2\\\\times {N}_{\\\\text{output}}/2\\\\) . Let’s assume we want to generate an arbitrary state in a chosen polarization state \\\\(j\\\\) at the output of the fiber at a wavelength \\\\(\\\\lambda\\\\) , whether \\\\(j\\\\) is H or V. For this purpose we thus use \\\\({U}_{j}(\\\\lambda )=[{U}_{Hj}(\\\\lambda )\\\\,{U}_{Vj}(\\\\lambda )]\\\\) [29] . In the delay domain, the calculated TRTM from the experimentally measured MSTM is also polarization-resolved: \\\\({U}_{j,{\\\\tau }_{s}}\\\\) is the TRTM at the delay time \\\\({\\\\tau }_{s}\\\\) for the output polarization state \\\\(j\\\\) . The time reversal operator \\\\({U}_{j,{\\\\tau }_{s}}^{\\\\dagger }{U}_{j,{\\\\tau }_{s}}\\\\) enables then a temporal control of the overall combination of spatial modes for a specific arrival time \\\\({\\\\tau }_{s}\\\\) and a specific polarization state at the output of the MMF. Similarly to Fig. 2 , the output intensity can be enhanced over all the spatial modes for only the \\\\(j\\\\) -polarization state at the output. The input field \\\\({E}_{\\\\,\\\\text{input}}^{\\\\text{max,j}\\\\,}\\\\) is the eigenvector associated to the maximum eigenvalue of \\\\({U}_{j,{\\\\tau }_{s}}^{\\\\dagger }{U}_{j,{\\\\tau }_{s}}\\\\) . In Fig. 4 a, b, we plot the polarization-resolved impulse responses associated to such eigenstates with \\\\({\\\\tau }_{s}=8.5\\\\) ps. The polarization-resolved impulse responses are calculated with Eq. 1 , using only the output modes on the chosen polarization state. Figure 4 a, b clearly evidence an enhancement of the intensity overall spatial modes at \\\\({\\\\tau }_{s}\\\\) for either only H or only V. The reconstructed intensity at the output for H and V at the delay time \\\\({\\\\tau }_{s}\\\\) confirms that the transmitted light at \\\\({\\\\tau }_{s}\\\\) is polarized in the chosen state. An incomplete control of the input polarization state would lead to similar results, but with a lower energy delivery at \\\\({\\\\tau }_{s}\\\\) as half of the input power would not be controlled. Such an incomplete polarization control is shown in Supplementary Fig. 8 . The MMF in conjunction with the SLM can then be used as an accurate deterministic polarization control device in the time domain. This experiment could also be used to study polarization recovery in scattering media [46] . Fig. 4 Polarization control of the engineered state at an arbitrary delay time \\\\({\\\\tau }_{s}\\\\) . We choose here \\\\({\\\\tau }_{s}=8.5\\\\) ps. a Enhancing only the H component of the output field. b Enhancing only the V component of the output field. c Attenuating only the H component of the output field. d Attenuating only the V component of the output field. Inset: reconstructed intensity at the output at delay time \\\\({\\\\tau }_{s}\\\\) for both H and V. Color code: (red) total impulse response of H and V; (black) impulse response of the H component; (orange) impulse response of the V component; (blue) time of flight distribution of H and V Full size image We also demonstrate an attenuation of a chosen polarization state at the output of the fiber at the delay time \\\\({\\\\tau }_{s}\\\\) . Instead of launching \\\\({E}_{\\\\,\\\\text{input}}^{\\\\text{max,j}\\\\,}\\\\) , we calculate \\\\({E}_{\\\\,\\\\text{input}}^{\\\\text{min,j}\\\\,}\\\\) which is the input eigenvector associated to the minimum eigenvalue of \\\\({U}_{j,{\\\\tau }_{s}}^{\\\\dagger }{U}_{j,{\\\\tau }_{s}}\\\\) . Figure 4 c, d present the experimental polarization-resolved impulse responses associated to such eigenstates for attenuating either only H or only V. Figure 4 c, d unambiguously reveal a strong attenuation at \\\\({\\\\tau }_{s}\\\\) for the controlled output polarization state. In principle, any arbitrary polarization state can be achieved at a given delay, similarly to previous work in the monochromatic regime [47] , based on the superposition of the four sub time gated transmission matrices of Eq. 4 . Some examples are shown in Supplementary Fig. 9 . Multiple delay control In Fig. 2 , we have demonstrated the enhancement of light intensity at a chosen arbitrary time \\\\({\\\\tau }_{s}\\\\) over all the spatial and polarization modes. The TRTM could also be used to shape more complex impulse responses. Spatio-temporal focusing over multiple space-time positions have been demonstrated with digital optical phase conjugation of the TRTM [11] . Here we demonstrate an enhancement of the transmitted light intensity at multiple delay times with the experimental setup of Fig. 1 . The time reversal operator \\\\({U}_{{\\\\tau }_{s}}^{\\\\dagger }{U}_{{\\\\tau }_{s}}\\\\) enables to adjust the transmitted energy at the delay time \\\\({\\\\tau }_{s}\\\\) . Figs. 2 and 3 have shown this control for either enhancing or attenuating light intensity at a chosen delay time \\\\({\\\\tau }_{s}\\\\) . The spectral diversity of the multimode fiber ensures that this control is only affecting delays around \\\\({\\\\tau }_{s}\\\\) , as the transmission matrices as a function of wavelength/delay are uncorrelated over a sufficient bandwidth/time interval [21] , [23] , [24] . In order to enhance light intensity at two different delays \\\\({\\\\tau }_{1}\\\\) and \\\\({\\\\tau }_{2}\\\\) , we use the time reversal operators \\\\({U}_{{\\\\tau }_{1}}^{\\\\dagger }{U}_{{\\\\tau }_{1}}\\\\) and \\\\({U}_{{\\\\tau }_{2}}^{\\\\dagger }{U}_{{\\\\tau }_{2}}\\\\) . We calculate the input eigenvectors \\\\({E}_{\\\\,\\\\text{input}}^{\\\\text{max}\\\\,,{\\\\tau }_{1}}\\\\) and \\\\({E}_{\\\\,\\\\text{input}}^{\\\\text{max}\\\\,,{\\\\tau }_{2}}\\\\) associated to the highest eigenvalue of the two single operators. The linear superposition \\\\({E}_{\\\\,\\\\text{input}}^{\\\\text{max},{\\\\tau }_{1},{\\\\tau }_{2}}={E}_{\\\\text{input}}^{\\\\text{max},{\\\\tau }_{1}}+{E}_{\\\\text{input}}^{\\\\text{max}\\\\,,{\\\\tau }_{2}}\\\\) enables a control of the impulse response at the two delay time \\\\({\\\\tau }_{1}\\\\) and \\\\({\\\\tau }_{2}\\\\) . The phase of the solution would then be displayed on the SLM, similarly to spatio-temporal focusing at multiple delays [11] , [21] . This control can be extended to a superposition of \\\\({N}_{\\\\text{superpos.}}\\\\) delay times. The input field would then read: $${E}_{\\\\,\\\\text{input}}^{\\\\text{max},{N}_{\\\\text{superpos.}}} = \\\\sum _{i=1}^{{N}_{\\\\text{superpos.}}}{E}_{\\\\text{input}}^{\\\\text{max}\\\\,,{\\\\tau }_{i}}$$ (5) with \\\\({E}_{\\\\,\\\\text{input}}^{\\\\text{max}\\\\,,{\\\\tau }_{i}}\\\\) the eigenvector associated with the highest eigenvalue of \\\\({U}_{{\\\\tau }_{i}}^{\\\\dagger }{U}_{{\\\\tau }_{i}}\\\\) . In practice, \\\\({N}_{\\\\text{superpos.}}\\\\) would be limited by the temporal width of the impulse response of each single mode. The impulse responses may interfere for different times, especially if the \\\\({\\\\tau }_{i}\\\\) are close in the time domain. We plot in Supplementary Fig. 3 the temporal correlation between the different Time Resolved Transmission Matrices. The quality of the superposition of the modes on the SLM would also limit the efficiency of a high \\\\({N}_{\\\\text{superpos.}}\\\\) . In Fig. 5 we demonstrate multiple delay control of the transmitted intensity with the experimental setup of Fig. 1 . In Fig. 5 a, \\\\({N}_{\\\\text{superpos.}}=6\\\\) delay times are controlled, evenly spread between 4 ps and 44 ps. Figure 5 b illustrates a similar control over a narrower time interval. The intensity of \\\\({N}_{\\\\text{superpos.}}=7\\\\) delay times is enhanced, evenly spread between 4 ps and 19 ps. We clearly identify \\\\({N}_{\\\\text{superpos.}}\\\\) peaks in Fig. 5 a and Fig. 5 b. The enhancement ratios at the \\\\({N}_{\\\\text{superpos.}}\\\\) delays are much lower than the individual delay enhancement ratios presented in Fig. 2 b. However the total transmitted power remains similar for either a single delay enhancement or a multiple delay control. The amplitude ratio between these peaks could also be adjusted using an amplitude coefficient in the superposition in Eq. 5 . A control of different \\\\({N}_{\\\\text{superpos.}}\\\\) within the same time interval is presented in the Supplementary Figs. 10 and 11 . This multiple delay control could also be exploited in conjunction with Fig. 4 , where the two polarization states could be manipulated at different arrival times. Fig. 5 Enhancement of transmitted light intensity at multiple delay times. a Enhancing the light intensity at 6 different delays \\\\({\\\\tau }_{i}\\\\) evenly spread from 4 ps to 44 ps. b Enhancing the light intensity at 7 different delays \\\\({\\\\tau }_{i}\\\\) evenly spread from 4 ps to 19 ps Full size image The ability to manipulate multiple delay times within long delay and dispersion provided by a very compact optical system such as a multimode fiber opens interesting perspectives for compact pulse shaping experiments. We believe this multiple delay control might also be useful for coherent control and pump probe experiments for imaging purposes or for complex control of light-matter interactions.\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"Over the last decade, wavefront shaping with spatial light modulators have revolutionized the study of light propagation in disordered systems, where scattering was considered as an insurmountable obstacle. Most of the previous works studied spatial control of the output field at the expense of the temporal behavior. We have demonstrated here the temporal control of light intensity averaged over all the spatial and polarization modes after propagation through a multimode fiber, at the expense of the spatial pattern. More precisely, we have shown how to enhance or attenuate the transmitted intensity at any arbitrary delay times, arbitrary polarization state, and arbitrary combinations of these. This work opens interesting perspectives in applications that require the measurement or control of spatio-temporal properties of light in scattering media. A pulse with a large number of transverse modes could be useful for the following non exhaustive list of examples: the study of correlations in disordered systems [35] , [48] , [49] , [50] telecommunications [51] , non-linear imaging [31] , [52] and endoscopy [53] , [54] , [55] through complex systems, time-gating imaging [56] , [57] , temporal focusing microscopy [58] , [59] , [60] and high power fiber lasers [61] . Our polarization control of the beam in the temporal domain could find applications in polarization sensitive optical coherence tomography [62] , [63] , [64]\"\n },\n {\n \"section\": \"Acknowledgements\",\n \"content\": \"We thank Martin Plöschner for careful reading of our manuscript. This work is supported by the Australian Research Council (ARC) projects DE180100009 and DP170101400.\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"This paper proposes a novel method to achieve full spatio-temporal control of light propagation through a multimode fiber by experimentally measuring the Multi-Spectral Transmission Matrix (MSTM), calculating the Time Resolved Transmission Matrix (TRTM) via Fourier Transform, and using it to control the temporal properties—both enhancement and attenuation—of the entire output beam at specific delay times across all spatial and polarization modes, thereby enabling unprecedented control over the spatio-temporal output field over a large bandwidth."},"entities":{"kind":"list like","value":["novel method","full spatio-temporal control","light propagation","multimode fiber","Multi-Spectral Transmission Matrix","MSTM","Time Resolved Transmission Matrix","TRTM","Fourier Transform","temporal properties","enhancement","attenuation","output beam","delay times","spatial modes","polarization modes","spatio-temporal output field","large bandwidth"],"string":"[\n \"novel method\",\n \"full spatio-temporal control\",\n \"light propagation\",\n \"multimode fiber\",\n \"Multi-Spectral Transmission Matrix\",\n \"MSTM\",\n \"Time Resolved Transmission Matrix\",\n \"TRTM\",\n \"Fourier Transform\",\n \"temporal properties\",\n \"enhancement\",\n \"attenuation\",\n \"output beam\",\n \"delay times\",\n \"spatial modes\",\n \"polarization modes\",\n \"spatio-temporal output field\",\n \"large bandwidth\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1.","title":"Goodman, J. 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ISSN 1094-4087.","link":"https://doi.org/10.1364%2FOPEX.13.001468"},{"idx":"59.","title":"Zhu, G., Howe, Jv, Durst, M., Zipfel, W. & Xu, C. Simultaneous spatial and temporal focusing of femtosecond pulses.Opt. Express13, 2153–2159 (2005). ISSN 1094-4087.","link":"https://doi.org/10.1364%2FOPEX.13.002153"},{"idx":"60.","title":"Escobet-Montalbán, A. et al. Wide-field multiphoton imaging through scattering media without correction.Sci. Adv.4, eaau1338 (2018). ISSN 2375-2548.","link":"https://doi.org/10.1126%2Fsciadv.aau1338"},{"idx":"61.","title":"Jackson, S. D. Towards high-power mid-infrared emission from a fibre laser.Nat. Photonics6, 423–431 (2012). ISSN 1749-4893.","link":"https://doi.org/10.1038%2Fnphoton.2012.149"},{"idx":"62.","title":"Li, J. et al. Polarization sensitive optical frequency domain imaging system for endobronchial imaging.Opt. Express23, 3390–3402 (2015). ISSN1094-4087.","link":"https://doi.org/10.1364%2FOE.23.003390"},{"idx":"63.","title":"Boer, J. Fd, Hitzenberger, C. K. & Yasuno, Y. Polarization sensitive optical coherence tomography - a review [Invited].Biomed. Opt. Express8, 1838–1873 (2017). ISSN2156-7085.","link":"https://doi.org/10.1364%2FBOE.8.001838"},{"idx":"64.","title":"Walther, J. et al. Depth-resolved birefringence imaging of collagen fiber organization in the human oral mucosa in vivo.Biomed. Opt. Express10, 1942–1956 (2019). ISSN 2156-7085.","link":"https://doi.org/10.1364%2FBOE.10.001942"}],"string":"[\n {\n \"idx\": \"1.\",\n \"title\": \"Goodman, J. W. Some fundamental properties of speckle.J Opt. Soc. Am.66, 1145–1150 (1976).\",\n \"link\": \"https://doi.org/10.1364%2FJOSA.66.001145\"\n },\n {\n \"idx\": \"2.\",\n \"title\": \"Rotter, S. & Gigan, S. Light fields in complex media: mesoscopic scattering meets wave control.Rev. Mod. Phys.89, 015005 (2017).\",\n \"link\": \"https://doi.org/10.1103%2FRevModPhys.89.015005\"\n },\n {\n \"idx\": \"3.\",\n \"title\": \"Mosk, A. 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N., Farahi, S., Moser, C. & Psaltis, D. High-resolution, lensless endoscope based on digital scanning through a multimode optical fiber.Biomed. Opt. Express4, 260–270 (2013). ISSN 2156-7085.\",\n \"link\": \"https://doi.org/10.1364%2FBOE.4.000260\"\n },\n {\n \"idx\": \"54.\",\n \"title\": \"Caravaca-Aguirre, A. M. & Piestun, R. Single multimode fiber endoscope.Opt. Express25, 1656–1665 (2017). ISSN 1094-4087.\",\n \"link\": \"https://doi.org/10.1364%2FOE.25.001656\"\n },\n {\n \"idx\": \"55.\",\n \"title\": \"Turtaev, S. et al. High-fidelity multimode fibre-based endoscopy for deep brain in vivo imaging.Light Sci. Appl.7, 92 (2018). ISSN 2047-7538.\",\n \"link\": \"https://doi.org/10.1038%2Fs41377-018-0094-x\"\n },\n {\n \"idx\": \"56.\",\n \"title\": \"Badon, A. et al. Smart optical coherence tomography for ultra-deep imaging through highly scattering media.Sci. Adv.2, e1600370 (2016). ISSN 2375-2548.\",\n \"link\": \"https://doi.org/10.1126%2Fsciadv.1600370\"\n },\n {\n \"idx\": \"57.\",\n \"title\": \"Jeong, S. et al. Focusing of light energy inside a scattering medium by controlling the time-gated multiple light scattering.Nat. Photonics12, 277 (2018). ISSN 1749-4893.\",\n \"link\": \"https://doi.org/10.1038%2Fs41566-018-0120-9\"\n },\n {\n \"idx\": \"58.\",\n \"title\": \"Oron, D., Tal, E. & Silberberg, Y. Scanningless depth-resolved microscopy.Opt. Express13, 1468–1476 (2005). ISSN 1094-4087.\",\n \"link\": \"https://doi.org/10.1364%2FOPEX.13.001468\"\n },\n {\n \"idx\": \"59.\",\n \"title\": \"Zhu, G., Howe, Jv, Durst, M., Zipfel, W. & Xu, C. Simultaneous spatial and temporal focusing of femtosecond pulses.Opt. Express13, 2153–2159 (2005). ISSN 1094-4087.\",\n \"link\": \"https://doi.org/10.1364%2FOPEX.13.002153\"\n },\n {\n \"idx\": \"60.\",\n \"title\": \"Escobet-Montalbán, A. et al. 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ISSN2156-7085.\",\n \"link\": \"https://doi.org/10.1364%2FBOE.8.001838\"\n },\n {\n \"idx\": \"64.\",\n \"title\": \"Walther, J. et al. Depth-resolved birefringence imaging of collagen fiber organization in the human oral mucosa in vivo.Biomed. Opt. Express10, 1942–1956 (2019). ISSN 2156-7085.\",\n \"link\": \"https://doi.org/10.1364%2FBOE.10.001942\"\n }\n]","isConversational":false}}},{"rowIdx":11193,"cells":{"unique_id":{"kind":"string","value":"s41467-018-06297-9"},"subfield":{"kind":"string","value":"Energy science and technology"},"title":{"kind":"string","value":"Enhanced voltage generation through electrolyte flow on liquid-filled surfaces"},"abstract":{"kind":"string","value":"The generation of electrical voltage through the flow of an electrolyte over a charged surface may be used for energy transduction. Here, we show that enhanced electrical potential differences (i.e., streaming potential) may be obtained through the flow of salt water on liquid-filled surfaces that are infiltrated with a lower dielectric constant liquid, such as oil, to harness electrolyte slip and associated surface charge. A record-high figure of merit, in terms of the voltage generated per unit applied pressure, of 0.043 mV Pa −1 is obtained through the use of the liquid-filled surfaces. In comparison with air-filled surfaces, the figure of merit associated with the liquid-filled surface increases by a factor of 1.4. These results lay the basis for innovative surface charge engineering methodology for the study of electrokinetic phenomena at the microscale, with possible application in new electrical power sources."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"The motion of an aqueous electrolyte, such as salt water, over a surface may be harnessed for the generation of electrical voltage [1] . Indeed, the exploration of such electrokinetic phenomena and their possible use for energy conversion have a long history, extending over the past two centuries (from Morrison & Osterle [2] and cited references). The relevant potential difference, termed as streaming potential ( V s ), arises due to the relative motion of charged [3] species in the electrolyte with respect to the fluid channel substrate with residual charges [4] . While superhydrophobic (SH) surfaces [5] have been posited to increase the ion velocity and the resultant potential difference, such enhancement has not been observed to date due to the inability of the air in the SH surfaces to hold charge [6] . Here, we monitor the flow and measure the V s of a NaCl-based electrolyte (consisting of Na + and Cl − dissolved in water) with a higher dielectric constant ( ε ) over a lower ε solid substrate/surface with an induced negative charge [7] , [8] in a microfluidic channel [9] . The electrically compensating positive counterions reside within the adjacent electrical double layer (EDL) [10] , which consists of an inner layer with fixed counterions adsorbed onto the surface and a diffuse layer with mobile counterions, with a thickness of the order of the Debye length ( λ D ). On application of a pressure difference (Δ P ) across the two ends of the channel, say, through a mechanical pump, Fig. 1 a–c, the electrolyte flow would be mainly constituted by the mobile counterions in the diffuse layer with the consequent charge separation yielding a streaming potential ( V s ) that is proportional to the ε ( = ε ο ε r - with ε ο = 8.854. 10 −12 C 2 N −1 m −2 as the free-space permittivity, and ε r as the relative permittivity of electrolyte e.g., ~ 80 for 0.1 mM L −1 NaCl solution) and the zeta potential ( ζ ) at the edge of the shear plane where the mobile ion motion occurs and varies inversely with the dynamic viscosity of the electrolyte ( η ) and bulk electrolyte conductivity ( κ ), as described through the Helmholtz–Smoluchowski model [7] , [11] : $$V_{\\mathrm{s}} = \\frac{{{{\\epsilon }}\\zeta }}{{\\eta \\kappa }}\\Delta P$$ (1) Fig. 1 Electrokinetics over patterned and fluid-filled surfaces. The related phenomena were investigated a through monitoring the streaming potential ( V s ) of salt water, under pressure driven flow, in a microchannel (of length 11.8 cm, width of 0.9 cm and height of ~250 μm). The upper surface of the channel was poly-dimethyl siloxane (PDMS), while the lower fluid–solid surface was engineered with the fluid (f), such as oil or air. The velocity ( u ) profile with fluid slip due to the pressure difference ( P 2 – P 1 ) applied by syringe pump is indicated. b The schematic of the flow arrangement in the microchannel and its c experimental realization, wherein the top and the bottom surfaces were separated by a silicone rubber spacer to adjust the microchannel height, and Ag/AgCl electrodes were inserted in reservoirs at either end to measure the V s . While d liquid flow over an air-filled surface (AFS) yields a finite slip velocity ( u s ) and slip length ( b ), the e flow of an electrolyte such as salt water yields a finite V s due to surface charge (indicated as negative, here)-induced influence on the electrolyte ions. f The V s can be made significantly larger when the air is replaced by a water immiscible liquid, such as oil yielding a liquid-filled surface (LFS) Full size image The ζ is considered close to the substrate–electrolyte liquid interface and is significant, in that it determines the electrical potential that may be utilized [7] , [8] for the V s . In the flowing electrolyte itself, there is a close to exponential decay of the surface potential over [10] λ D . It should be noted that Eq. ( 1 ) is based [12] on the assumptions of negligible surface (/substrate) conductivity and a very small EDL thickness [13] , with Poiseuille flow of the electrolyte. Moreover, there is an implicit assumption of the no-slip boundary condition [14] , where the liquid adjacent to the substrate wall has a zero velocity, with a finite flow velocity only at a certain distance (corresponding to the shear plane) into the fluid. Consequently, ζ would be related to the electrical potential at the edge of the shear plane in electrokinetic flows over smooth surfaces [7] , [12] . Most work on harnessing the V s through electrokinetic effects, to date, has been concerned with fluid flow over smooth surfaces (where the scale of roughness is smaller than λ D ), and consequently very small potential differences of the order of 18 mV may be predicted and obtained [11] , in correspondence to Eq. ( 1 ), i.e., for 0.1 mM L −1 NaCl, with ε r ~ 80, η ~ 10 −3 Pa·s, ζ ~ 25 mV, κ ~ 10 −3 S m −1 , with Δ P ~ 1000 Pa. With the objective of obtaining significantly larger V s , we indicate briefly the principles of our approach, which first involved the modulation of the effective ζ through introducing fluid slip via groove-patterned surfaces (in air-/liquid-filled surfaces) and modifying surface charges at the substrate surface. It has previously been considered, based on molecular dynamics simulations [15] , that slip may mobilize the Stern layer, significantly enhancing the ζ . The broad concept is that the shear plane/surface of shear (at which the electrolyte flow velocity is zero) is to be moved as close as possible to the substrate. As the surface electrical potential decays away from the substrate, the closer the shear plane is to the substrate boundary, the larger the ζ with concomitantly increased V s . Here, the zero velocity boundary condition at the surface ( y = 0) would be replaced with a Navier slip condition, \\(u_{\\mathrm{s}}\\,\\left( {y = 0} \\right) = b\\frac{{\\partial u\\left( {y = 0} \\right)}}{{\\partial y}}\\) with u s as the slip velocity and b as the slip length [16] , [17] , Fig. 1d . The larger the b , the greater the fluid velocity at the surface by a factor of \\(\\left( {1 + \\frac{b}{{\\lambda _{\\mathrm{D}}}}} \\right)\\) . A related apparent increase [18] in the ζ and the V s may then be consequently obtained [19] . As much relatively large values of b of the order of 10 μm were thought to be obtained through the use of SH surfaces [20] , and given a λ D of 9.6 nm with 1 mM NaCl, a b/λ D of ~1000 is estimated, yielding large V s . While non-uniform surface conduction [21] , e.g., as related to slip to no-slip transitions, may reduce such an enhancement, a finite b would yet contribute to V s and may be aided by surface charge induced forces. To further understand the issues related to electrokinetics over surfaces with slip, we conducted experiments (see Figs. 1b, c ) where the electrolyte flow was driven in a microfluidic channel (of length: 11.8 cm, width: 0.9 cm, and height: 250 μm) by a syringe pump, with Δ P varied in the range of 0–1200 Pa (see Methods) to measure the V s over ridged pattern surface morphology, termed as AFS (air-filled surface), Fig. 1e . The substrate material of AFS is silicon, coated by 600-nm hydrophobic parylene-C. The patterns were characterized through an interstitial/air fraction: ϕ air , through the ratio [ = d/ ( d + w )], with an average ridge width ( w ) and spacing ( d ), Fig. 2a . We report results on samples with ϕ air = 0.5, d = w = 18 μm for ϕ air = 0.75, d = 27 μm, and w = 9 μm (see Methods for channel design and fabrication-related considerations, as well as the optimization and choice of the parameters in Supplementary Note 1 ). Fig. 2 Streaming potential of air-filled surfaces. a A scanning electron microscopy (SEM) micrograph of the groove-patterned surfaces with silicon as the substrate covered by parylene-C (thickness ≈ 600 nm) parameterized on the basis of the solid surface width ( w = 18 μm), groove width ( d = 18 μm), pattern height ( h ≈ 95 μm), and the air fraction: ϕ air [ = d/ ( d + w ) = 0.5] and a solid fraction: ϕ s = 1 – ϕ air . The flow direction is perpendicular to the grooves and indicated by the arrow. Scale bar is 100 μm. b The typical response of streaming potential ( V s ) was measured, for a ϕ s = 0.5 air-filled surface (AFS), through a syringe pump-driven pressure, e.g., with an applied pressure of 440 Pa, 0.1 mM L −1 NaCl here. The red dotted line indicated the average flow velocity in the channel. The measured voltage difference between a baseline value ( V o ) and a final value ( V f ) was considered as V s . c The measured V s (0.1 mM NaCl solution) on an AFS (i.e., Air 0.5 , the subscript indicating the ϕ air ) scales linearly with the applied pressure (in the range of 200–1200 Pa), in accord with the Helmholtz–Smoluchowski model (Eq. 1 ) with a lower value for V s obtained for a smaller (/larger) ϕ s (/ ϕ air ) and rationalized as due to the absence of surface charges and contributing zeta potential ζ , in the air regions. d The V s scales with the salt water electrolyte conductivity and related concentration ( I c ) through a log [ I c ]/ [ I c ] variation for both AFS (Air 0.5 ) and parylene-coated flat surface (Flat) (concentration varying from 0.1 mM L −1 to 5 mM L −1 at 200 Pa for d ) Full size image We show that a record large figure of merit, in terms of the voltage generated per unit applied pressure, is obtained by using a liquid to fill the grooves in comparison to air-filled grooves.","references":{"indices":[2,3,5,6,7,8,9,15,16,18,19,20,21],"infos":[{"idx":"2","title":"Morrison, F. A. & Osterle, J. F. Electrokinetic energy conversion in ultrafine capillaries.J. Chem. Phys.43, 2111–2115 (1965).","link":"https://doi.org/10.1063%2F1.1697081"},{"idx":"3","title":"Bahga, S. S., Vinogradova, O. I. & Bazant, M. Z. Anisotropic electro-osmotic flow over super-hydrophobic surfaces.J. Fluid. Mech.644, 245–255 (2010).","link":"https://doi.org/10.1017%2FS0022112009992771"},{"idx":"5","title":"Rothstein, J. P. Slip on superhydrophobic surfaces.Annu. Rev. Fluid. Mech.42, 89–109 (2010).","link":"https://doi.org/10.1146%2Fannurev-fluid-121108-145558"},{"idx":"6","title":"Squires, T. M. Electrokinetic flows over inhomogeneously slipping surfaces.Phys. Fluids20, 092105 (2008).","link":"https://doi.org/10.1063%2F1.2978954"},{"idx":"7","title":"Kirby, B. J. & Hasselbrink, E. F. Zeta potential of microfluidic substrates: 1. Theory, experimental techniques, and effects on separations.Electrophoresis25, 187–202 (2004).","link":"https://doi.org/10.1002%2Felps.200305754"},{"idx":"8","title":"Kirby, B. J. & Hasselbrink, E. F. Zeta potential of microfluidic substrates: 2.Data Polym. Electrophor.25, 203–213 (2004).","link":"https://doi.org/10.1002%2Felps.200305755"},{"idx":"9","title":"Whitesides, G. M. The origins and the future of microfluidics.Nature442, 368–373 (2006).","link":"https://doi.org/10.1038%2Fnature05058"},{"idx":"15","title":"Joly, L., Ybert, C., Trizac, E. & Bocquet, L. Hydrodynamics within the electric double layer on slipping surfaces.Phys. Rev. Lett.93, 257805 (2004).","link":"https://doi.org/10.1103%2FPhysRevLett.93.257805"},{"idx":"16","title":"Bocquet, L. & Lauga, E. A smooth future?Nat. Mater.10, 334–337 (2011).","link":"https://doi.org/10.1038%2Fnmat2994"},{"idx":"18","title":"Bouzigues, C. I., Tabeling, P. & Bocquet, L. Nanofluidics in the Debye layer at hydrophilic and hydrophobic surfaces.Phys. Rev. Lett.101, 114503 (2008).","link":"https://doi.org/10.1103%2FPhysRevLett.101.114503"},{"idx":"19","title":"Khair, A. S. & Squires, T. M. Surprising consequences of ion conservation in electro-osmosis over a surface charge discontinuity.J. Fluid. Mech.615, 323–334 (2008).","link":"https://doi.org/10.1017%2FS002211200800390X"},{"idx":"20","title":"Park, S.-H. et al. Bioinspired superhydrophobic surfaces, fabricated through simple and scalable roll-to-roll processing.Sci. Rep.5, 15430 (2015).","link":"https://doi.org/10.1038%2Fsrep15430"},{"idx":"21","title":"Khair, A. S. & Squires, T. M. The influence of hydrodynamic slip on the electrophoretic mobility of a spherical colloidal particle.Phys. Fluids21, 042001 (2009).","link":"https://doi.org/10.1063%2F1.3116664"}]}},{"section":"Results","content":"Voltage generation on air-filled surfaces Considering the flow of electrolyte over air-filled surfaces, the consequent voltage–time trace, Fig. 2b , was used to determine the V s , which was monitored through a high-resistance (> 10 GΩ) voltmeter (Keithley 2700) connected across the Ag/AgCl electrodes placed at either ends of the channel, Fig. 1c . However, a significant enhancement was not obtained for AFS, with an estimated [5] b of ~10 μm. It should be noted that a lower V s was obtained for a larger ϕ air (i.e., smaller ϕ s ), for which a larger b would be expected, Fig. 2c . Moreover, very little increase over a flat unpatterned surface was obtained comparing Air 0.5 and Flat, Fig. 2d . The lower V s of AFS may be rationalized as due to the absence of a surface charge in the air regions [13] , which overwhelms the contribution of the slip. While the data here seems to indicate the general validity of Eq. ( 1 ) in terms of a linear V s -Δ P plot, the slope of the obtained experimental curves would then be directly connected to ε , ζ , η , and κ . Such parameters are then intimately connected, i.e., while independent variation would be difficult to discern experimentally, theoretical or computational input (which is outside the purview of the present experimental work) may be able to yield guidance for explicit dependence. The net V s has been construed to be related to a weighted summation of ζ in the no-slip region (over the solid fraction of the surface, i.e., ζ NS ) and the slip region (i.e., over the air of the AFS, i.e., ζ S ) [6] . The very small ζ S would adversely affect the V s and as seen through our experiments overwhelms the slip length-related contribution. Related correspondence of the obtained V s to ζ as well as the electrolyte conductivity, which is proportional to I c (the counter-ion concentration), was determined through a log [ I c ]/ [ I c ] dependence, Fig. 2d . Increased voltage on liquid-filled surfaces Subsequently, we deployed patterned surfaces where the air pockets were infiltrated by a water-immiscible liquid, e.g., oil, of a higher density compared with the NaCl electrolyte, Fig. 3a . Such liquid-filled surfaces (LFS) were hypothesized to yield a finite ζ S at the oil interfaces, in addition to a non-zero b . The LFS were fabricated through replacing the air in groove-patterned arrays by water-immiscible liquid, such as oil, and may be designed to exhibit various degrees of fluid slip [22] , [23] , [24] , as well as electrical voltages that are significantly larger than those on air-filled or SH surfaces. Oil filling was found to be reliable and stable (see Methods). Fig. 3 Enhanced streaming potential at liquid-filled surfaces. Enhanced streaming potential ( V s ) was obtained by a filling the grooves with salt water immiscible oil, e.g., Krytox GPL 104, (scale bar is 100 μm); b a significantly larger V s was obtained for an GPL 104 oil-filled surface (i.e., GPL 0.5 , with the subscript now indicating the interstitial fraction of oil), compared to one with air (Air 0.5 ) and even compared to a Flat solid substrate (all measured using 0.1 mM L −1 NaCl solution with applied pressure as 1200 Pa). c The V s could be larger or smaller compared with the control air-filled surfaces (AFS), depending on the choice of the liquid in the LFS, e.g., GPL oil and castor oil (Castor 0.5 ), respectively Full size image Additionally, the use of oil with a lower ε r compared with the flowing electrolyte would mean that the electric field from the wall is effectively propagated. The details related to the (i) choice of liquid for the LFS along with the estimation of the (ii) ζ , and the (iii) b are given in Methods, Supplementary Note 2 and Supplementary Note 3 , respectively. Using Krytox GPL 104 oil ( ε r ~ 2.1, η ~ 340 cP), a significant enhancement of the V s by 50% to around ~52 mV, compared with that of Air 0.5 , was observed, Fig. 3b , attesting to the utility of the LFS. However, it was noted that the obtained V s critically depends on the choice of the oil in the LFS. Broadly, a larger ε oil seems to yield a lower V s . For instance, a factor of two smaller value was obtained for a castor oil ( ε r ~ 4.7, η ~ 312 cP)-filled LFS, Fig. 3c . The enhanced Van der Waals energy of interaction [25] between the oil and electrolyte, proportional to the dielectric constants, and the related mutual electric polarizabilities causes a larger friction. The consequent movement of the shear plane away from the solid substrate would result in a lower zeta potential and V s . The electrolyte flow over the LFS, as well as the adjacent solid surface were parameterized [26] through the use of a material-dependent surface charge density ( σ ) related to \\({\\zeta } = \\frac{{\\lambda _{\\mathrm{D}}\\sigma }}{{\\it{\\epsilon }}}\\) and fluid slip at the respective interfaces. From literature, for the top surface of the channel (constituted from poly-dimethyl siloxane, PDMS), σ PDMS [27] ~18 mC m −2 , while for the bottom-patterned surfaces, we use σ oil [28] ~1.8 mC m −2 and σ parylene [29] ~3.6 mC m −2 . The σ and the velocity slip is inhomogeneous along the length of the channel, but constant along the channel width. The no-slip hydrodynamic boundary condition was assumed to hold true at all solid–electrolyte interfaces, while a Navier slip boundary condition over the flowing electrolyte–oil interface was assumed, with a finite slip velocity: u s and slip length: b . For a hydrophobic surface, the b may be approximated [5] to be of the order of d , i.e., ~ 18 μm. A correspondence to a Cassie state-like channel flow (with no electrolyte penetration into the grooves of the patterned surface) for LFS surface may be seen. It should be noted that simulations considering the continuity of tangential shear stress across the electrolyte–oil interface also obtained the V s very close (within 1%) to the results obtained through an assumption of a slip length. Modeling of electrolyte flow over the liquid-filled surfaces We partition [12] the pressure gradient driven electrolyte flow, following experiment, over the LFS as the superposition of (i) hydrodynamic flow, with fluid slip over electrically neutral surfaces, and (ii) an electrokinetic flow, with no-slip fluid flow over flat electrically charged surfaces, Fig. 4a–c . A multiphysics model coupling the Nernst–Planck–Poisson (N-P-P) equation with the Stokes Equation was deployed to determine the volumetric charge density profiles (the details of the governing equations, assumptions, and the boundary conditions are given in Supplementary Note 4 and Supplementary Note 5 ). The velocity profile combined with the simulated volumetric charge density profile along the channel, as indicated in Fig. 4d , arises due to the difference of ionic concentrations between the counterions and co-ions. The resultant horizontal electric field is responsible for the observed V s . The calculated average electrolyte velocity of 6 cm s −1 is in agreement with that expected from Poiseuille flow along the channel, with the estimated u s at the oil–electrolyte interface of ~2 cm s −1 at Δ P ~1200 Pa. Additionally and given the relatively large groove widths, the influence of the surface conductivity on the obtained results would be relatively small [19] . Fig. 4 Parametrization and modeling of charged fluid–solid surfaces. a The surface charge density ( σ ) of an air-filled surface (AFS) varies between zero (at the air) and a finite value at the solid, while a nominally planar liquid-filled surface (LFS) maintains a larger value of the σ . The electrokinetics on the LFS was considered in terms of a superposition of b hydrodynamic flow, with fluid slip over electrically neutral surfaces, and c an electrokinetic flow, with no-slip fluid flow over electrically charged surfaces, and solving the coupled Nernst–Planck–Poisson (NPP) equation along with the Stokes Equation to determine the d velocity profile, as well as the volumetric charge density ( ρ ) profiles. The insets indicate the computed variation of the ρ at the top poly-dimethyl siloxane (PDMS) surface and the bottom LFS Full size image"},{"section":"Discussion","content":"The values of the streaming potential arising from the simulations are in excellent agreement with those observed experimentally through considering a reduced ε r ~28 over a distance ~3 λ D proximate to the LFS and an ε r ~78 and beyond, correspondent to the bulk solution. Such aspect is in accord with the use of different values of the dielectric constant for the inner and outer regions of a solid–solution interface [12] . The LFS, consisting of both oil as well as solid surfaces, may yield an uneven shear plane with fascinating implications for novel electrokinetic phenomena [30] , [31] , such as localized concentration polarization [19] , etc. Our work also indicates that a heterogeneous surface could be patterned through the use of discrete surface charge density or applied potentials, mimicking a SH surface and making a connection with electro-wetting applications, as well as with significant literature on the modeling and parameterization of liquid flows on striped surfaces that exhibit alternate regions of slip and no-slip, with [32] , [33] and without [34] surface charge. A more detailed investigation of several issues related to the interplay of enhanced surface charge as well as increased slip velocity at the flowing electrolyte–LFS interface would then be warranted. It has been indicated, for instance, that an increased V s would result due to the larger ion convection currents [35] . Additionally, viscous effects may be playing a major role [36] , [37] . As the coefficient in Eq. 1 intimately connects the ε r , and the η , a decrease in the former may not be readily deconvoluted from an increase in the latter experimentally. In our numerical simulations, an increase in the η by a factor of ~2.5 over the equivalent length scale (from a bulk value of 0.89 cP to 2.19 cP) was necessary to obtain the experimentally observed streaming potentials, if the ε r is ~78. Our experiments have deep scientific implications underlying fluid flow interfacing with both air and liquid, as well as fluid shear at a surface, and electrokinetic phenomena as related to the localized variation of the zeta potential and nonlinearity. We anticipate that our work would revitalize research related to the fabrication of alternate electrical voltage/power sources from liquid flow over charged surfaces. Other anticipated applications extend to electrophoretic applications in biological separations (cell transport, manipulation [38] , and interactions [39] ), as well as voltage sources for lab-on-a-chip applications [40] . In summary, our work has experimentally demonstrated the largest figure of merit thus far [41] , [42] , [43] , [44] , to the best of our knowledge, with primary focus on methodologies related to enhance the streaming potential ( V s ) per unit pressure difference (Δ P ) through the use of LFS (also see Supplementary Note 6 for considerations related to overall electrokinetic efficiency). The use of the LFS yields a figure of merit increase by a factor of 1.4 in comparison to that obtained using the AFS. It has been shown that larger voltages, through a measured streaming potential, may be achieved through careful engineering of the coupled electric field and fluid flow. The application of the related increase in the electrokinetic energy conversion efficiency would need further optimization of the fluidic and electrical impedances, in concert with the streaming conductance, as matched to an appropriate load [45] . Concomitantly, unipolar transport (where for example, either Na + or Cl − ions are transported [46] ) through EDL overlap [26] , [47] in nanoscale channels may be coupled with LFS to yield much larger voltages, comparable to that of batteries [48] ."},{"section":"Methods","content":"Design considerations and fabrication of air-filled surfaces and liquid-filled surfaces The critical parameters in our channel surface design are (a) structural, i.e., the interstitial fraction of AFS/LFS: ϕ -the ratio [ = d/ ( d + w )], with an average ridge width ( w ) and spacing ( d ), as indicated in Fig. 2a, (b) dimension, the channel length ( L) and channel width ( W ) should be much larger than channel height ( H ), then the channel can be treated as two infinite parallel plates and the flow is Poiseuille flow, (c) pressure gradients, i.e., over a range of 0–1000 Pa, with higher pressures yielding greater streaming potentials ( V s ), and (d) the nature of the filling fluid, easier for oil with low surface tension to penetrate into the grooves. For instance, we have used in our electrokinetic flow experiments, ϕ ~0.5, 0.9 cm wide, 11.8 cm long, and 250 μm high channel, with a pressure difference of ~1200 Pa, using GPL oil as a filling liquid, to generate maximal V s . The LFS were fabricated through filling oil into the grooves of a patterned surface, fabricated through photolithography, and dry etching. A Si wafer (n-type, < 110 > , thickness 500 µm) was cleaned with acetone and IPA, rinsed by DI water, and subsequently baked at 180 °C for 5 min. Negative photoresist (NR9-3000) was coated on the Si wafer (3000 rpm for 40 s) and baked at 120 °C for 60 s. The pattern was defined through a mask (using the EVG 620) and developed (RD6) for 1 min. Then the resist-patterned wafer was subject to dry etching (using Oxford Plasmalab 100 RIE/ICP) to yield a trench depth ( h ) ~ 95 µm. The photoresist was then removed (using RR2 photoresist remover) over ~12 h and any residues were further removed (using PVA TePla PS100 at 120 sccm, 200 W for 90 s). The grooved surface was coated with Parylene-C (using PDS 2010 Parylene Coater) to a thickness of ~600 nm. For LFS, oils, e.g., Dupont Krytox GPL 104, Castor oil, etc. was filled into the grooves of the patterned surface. The air-/liquid-patterned surfaces were imaged using environmental scanning electron microscopy (ESEM): FEI/Phillips XL ESEM and the FEI Quanta FEG 250 ESEM. We have indicated, e.g., in Fig. 2c that a lower ϕ -the ratio of the average groove width to the overall period yields a larger V s . Choice of applied pressure and related flow rate The electrolyte flow rate in the channel, constituted from rectangular plate geometry [26] , was modulated through a syringe pump, Fig. 1c , and was transduced to an applied pressure (Δ P ) based on the Poiseuille relation: \\(\\Delta P = \\frac{{12\\eta LQ}}{{wh^3}}\\) ( Q is flow rate, L is channel length of ~11.8 cm, and h the channel height of ~250 μm). The Δ P over the channel length (where the other end was left open to atmosphere) motivated the pressure gradient driven flow, and for the experiment was in the range of 200 to1200 Pa, as measured through a manometer (UEI EM152 Dual Differential Input Manometer). It was observed that when the Δ P was larger (/smaller) than 1200 Pa (/200 Pa), that the V s was not stable. Choice of liquid to fill the interstices in the liquid-filled surfaces For the fabrication of the LFS, the filling liquid (e.g., Krytox ® GPL 104 and castor oil) was chosen primarily on the basis of immiscibility with water/electrolyte. Additional criteria would be a low surface tension—for better penetration into the interstices, low viscosity, and a smaller dielectric constant—to reduce electrolyte–oil interfacial shear stress. We observe (see Supplementary Figure 1 ) that the parylene-coated Si surface was oleophilic (/hydrophobic), and that the chosen liquid or oil could penetrate the interstices easily, Fig. 3a ."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"The motion of an aqueous electrolyte, such as salt water, over a surface may be harnessed for the generation of electrical voltage [1] . Indeed, the exploration of such electrokinetic phenomena and their possible use for energy conversion have a long history, extending over the past two centuries (from Morrison & Osterle [2] and cited references). The relevant potential difference, termed as streaming potential ( V s ), arises due to the relative motion of charged [3] species in the electrolyte with respect to the fluid channel substrate with residual charges [4] . While superhydrophobic (SH) surfaces [5] have been posited to increase the ion velocity and the resultant potential difference, such enhancement has not been observed to date due to the inability of the air in the SH surfaces to hold charge [6] . Here, we monitor the flow and measure the V s of a NaCl-based electrolyte (consisting of Na + and Cl − dissolved in water) with a higher dielectric constant ( ε ) over a lower ε solid substrate/surface with an induced negative charge [7] , [8] in a microfluidic channel [9] . The electrically compensating positive counterions reside within the adjacent electrical double layer (EDL) [10] , which consists of an inner layer with fixed counterions adsorbed onto the surface and a diffuse layer with mobile counterions, with a thickness of the order of the Debye length ( λ D ). On application of a pressure difference (Δ P ) across the two ends of the channel, say, through a mechanical pump, Fig. 1 a–c, the electrolyte flow would be mainly constituted by the mobile counterions in the diffuse layer with the consequent charge separation yielding a streaming potential ( V s ) that is proportional to the ε ( = ε ο ε r - with ε ο = 8.854. 10 −12 C 2 N −1 m −2 as the free-space permittivity, and ε r as the relative permittivity of electrolyte e.g., ~ 80 for 0.1 mM L −1 NaCl solution) and the zeta potential ( ζ ) at the edge of the shear plane where the mobile ion motion occurs and varies inversely with the dynamic viscosity of the electrolyte ( η ) and bulk electrolyte conductivity ( κ ), as described through the Helmholtz–Smoluchowski model [7] , [11] : $$V_{\\\\mathrm{s}} = \\\\frac{{{{\\\\epsilon }}\\\\zeta }}{{\\\\eta \\\\kappa }}\\\\Delta P$$ (1) Fig. 1 Electrokinetics over patterned and fluid-filled surfaces. The related phenomena were investigated a through monitoring the streaming potential ( V s ) of salt water, under pressure driven flow, in a microchannel (of length 11.8 cm, width of 0.9 cm and height of ~250 μm). The upper surface of the channel was poly-dimethyl siloxane (PDMS), while the lower fluid–solid surface was engineered with the fluid (f), such as oil or air. The velocity ( u ) profile with fluid slip due to the pressure difference ( P 2 – P 1 ) applied by syringe pump is indicated. b The schematic of the flow arrangement in the microchannel and its c experimental realization, wherein the top and the bottom surfaces were separated by a silicone rubber spacer to adjust the microchannel height, and Ag/AgCl electrodes were inserted in reservoirs at either end to measure the V s . While d liquid flow over an air-filled surface (AFS) yields a finite slip velocity ( u s ) and slip length ( b ), the e flow of an electrolyte such as salt water yields a finite V s due to surface charge (indicated as negative, here)-induced influence on the electrolyte ions. f The V s can be made significantly larger when the air is replaced by a water immiscible liquid, such as oil yielding a liquid-filled surface (LFS) Full size image The ζ is considered close to the substrate–electrolyte liquid interface and is significant, in that it determines the electrical potential that may be utilized [7] , [8] for the V s . In the flowing electrolyte itself, there is a close to exponential decay of the surface potential over [10] λ D . It should be noted that Eq. ( 1 ) is based [12] on the assumptions of negligible surface (/substrate) conductivity and a very small EDL thickness [13] , with Poiseuille flow of the electrolyte. Moreover, there is an implicit assumption of the no-slip boundary condition [14] , where the liquid adjacent to the substrate wall has a zero velocity, with a finite flow velocity only at a certain distance (corresponding to the shear plane) into the fluid. Consequently, ζ would be related to the electrical potential at the edge of the shear plane in electrokinetic flows over smooth surfaces [7] , [12] . Most work on harnessing the V s through electrokinetic effects, to date, has been concerned with fluid flow over smooth surfaces (where the scale of roughness is smaller than λ D ), and consequently very small potential differences of the order of 18 mV may be predicted and obtained [11] , in correspondence to Eq. ( 1 ), i.e., for 0.1 mM L −1 NaCl, with ε r ~ 80, η ~ 10 −3 Pa·s, ζ ~ 25 mV, κ ~ 10 −3 S m −1 , with Δ P ~ 1000 Pa. With the objective of obtaining significantly larger V s , we indicate briefly the principles of our approach, which first involved the modulation of the effective ζ through introducing fluid slip via groove-patterned surfaces (in air-/liquid-filled surfaces) and modifying surface charges at the substrate surface. It has previously been considered, based on molecular dynamics simulations [15] , that slip may mobilize the Stern layer, significantly enhancing the ζ . The broad concept is that the shear plane/surface of shear (at which the electrolyte flow velocity is zero) is to be moved as close as possible to the substrate. As the surface electrical potential decays away from the substrate, the closer the shear plane is to the substrate boundary, the larger the ζ with concomitantly increased V s . Here, the zero velocity boundary condition at the surface ( y = 0) would be replaced with a Navier slip condition, \\\\(u_{\\\\mathrm{s}}\\\\,\\\\left( {y = 0} \\\\right) = b\\\\frac{{\\\\partial u\\\\left( {y = 0} \\\\right)}}{{\\\\partial y}}\\\\) with u s as the slip velocity and b as the slip length [16] , [17] , Fig. 1d . The larger the b , the greater the fluid velocity at the surface by a factor of \\\\(\\\\left( {1 + \\\\frac{b}{{\\\\lambda _{\\\\mathrm{D}}}}} \\\\right)\\\\) . A related apparent increase [18] in the ζ and the V s may then be consequently obtained [19] . As much relatively large values of b of the order of 10 μm were thought to be obtained through the use of SH surfaces [20] , and given a λ D of 9.6 nm with 1 mM NaCl, a b/λ D of ~1000 is estimated, yielding large V s . While non-uniform surface conduction [21] , e.g., as related to slip to no-slip transitions, may reduce such an enhancement, a finite b would yet contribute to V s and may be aided by surface charge induced forces. To further understand the issues related to electrokinetics over surfaces with slip, we conducted experiments (see Figs. 1b, c ) where the electrolyte flow was driven in a microfluidic channel (of length: 11.8 cm, width: 0.9 cm, and height: 250 μm) by a syringe pump, with Δ P varied in the range of 0–1200 Pa (see Methods) to measure the V s over ridged pattern surface morphology, termed as AFS (air-filled surface), Fig. 1e . The substrate material of AFS is silicon, coated by 600-nm hydrophobic parylene-C. The patterns were characterized through an interstitial/air fraction: ϕ air , through the ratio [ = d/ ( d + w )], with an average ridge width ( w ) and spacing ( d ), Fig. 2a . We report results on samples with ϕ air = 0.5, d = w = 18 μm for ϕ air = 0.75, d = 27 μm, and w = 9 μm (see Methods for channel design and fabrication-related considerations, as well as the optimization and choice of the parameters in Supplementary Note 1 ). Fig. 2 Streaming potential of air-filled surfaces. a A scanning electron microscopy (SEM) micrograph of the groove-patterned surfaces with silicon as the substrate covered by parylene-C (thickness ≈ 600 nm) parameterized on the basis of the solid surface width ( w = 18 μm), groove width ( d = 18 μm), pattern height ( h ≈ 95 μm), and the air fraction: ϕ air [ = d/ ( d + w ) = 0.5] and a solid fraction: ϕ s = 1 – ϕ air . The flow direction is perpendicular to the grooves and indicated by the arrow. Scale bar is 100 μm. b The typical response of streaming potential ( V s ) was measured, for a ϕ s = 0.5 air-filled surface (AFS), through a syringe pump-driven pressure, e.g., with an applied pressure of 440 Pa, 0.1 mM L −1 NaCl here. The red dotted line indicated the average flow velocity in the channel. The measured voltage difference between a baseline value ( V o ) and a final value ( V f ) was considered as V s . c The measured V s (0.1 mM NaCl solution) on an AFS (i.e., Air 0.5 , the subscript indicating the ϕ air ) scales linearly with the applied pressure (in the range of 200–1200 Pa), in accord with the Helmholtz–Smoluchowski model (Eq. 1 ) with a lower value for V s obtained for a smaller (/larger) ϕ s (/ ϕ air ) and rationalized as due to the absence of surface charges and contributing zeta potential ζ , in the air regions. d The V s scales with the salt water electrolyte conductivity and related concentration ( I c ) through a log [ I c ]/ [ I c ] variation for both AFS (Air 0.5 ) and parylene-coated flat surface (Flat) (concentration varying from 0.1 mM L −1 to 5 mM L −1 at 200 Pa for d ) Full size image We show that a record large figure of merit, in terms of the voltage generated per unit applied pressure, is obtained by using a liquid to fill the grooves in comparison to air-filled grooves.\",\n \"references\": {\n \"indices\": [\n 2,\n 3,\n 5,\n 6,\n 7,\n 8,\n 9,\n 15,\n 16,\n 18,\n 19,\n 20,\n 21\n ],\n \"infos\": [\n {\n \"idx\": \"2\",\n \"title\": \"Morrison, F. A. & Osterle, J. F. Electrokinetic energy conversion in ultrafine capillaries.J. Chem. Phys.43, 2111–2115 (1965).\",\n \"link\": \"https://doi.org/10.1063%2F1.1697081\"\n },\n {\n \"idx\": \"3\",\n \"title\": \"Bahga, S. S., Vinogradova, O. I. & Bazant, M. Z. Anisotropic electro-osmotic flow over super-hydrophobic surfaces.J. Fluid. Mech.644, 245–255 (2010).\",\n \"link\": \"https://doi.org/10.1017%2FS0022112009992771\"\n },\n {\n \"idx\": \"5\",\n \"title\": \"Rothstein, J. P. Slip on superhydrophobic surfaces.Annu. Rev. Fluid. Mech.42, 89–109 (2010).\",\n \"link\": \"https://doi.org/10.1146%2Fannurev-fluid-121108-145558\"\n },\n {\n \"idx\": \"6\",\n \"title\": \"Squires, T. M. Electrokinetic flows over inhomogeneously slipping surfaces.Phys. Fluids20, 092105 (2008).\",\n \"link\": \"https://doi.org/10.1063%2F1.2978954\"\n },\n {\n \"idx\": \"7\",\n \"title\": \"Kirby, B. J. & Hasselbrink, E. F. Zeta potential of microfluidic substrates: 1. Theory, experimental techniques, and effects on separations.Electrophoresis25, 187–202 (2004).\",\n \"link\": \"https://doi.org/10.1002%2Felps.200305754\"\n },\n {\n \"idx\": \"8\",\n \"title\": \"Kirby, B. J. & Hasselbrink, E. F. Zeta potential of microfluidic substrates: 2.Data Polym. Electrophor.25, 203–213 (2004).\",\n \"link\": \"https://doi.org/10.1002%2Felps.200305755\"\n },\n {\n \"idx\": \"9\",\n \"title\": \"Whitesides, G. M. The origins and the future of microfluidics.Nature442, 368–373 (2006).\",\n \"link\": \"https://doi.org/10.1038%2Fnature05058\"\n },\n {\n \"idx\": \"15\",\n \"title\": \"Joly, L., Ybert, C., Trizac, E. & Bocquet, L. Hydrodynamics within the electric double layer on slipping surfaces.Phys. Rev. Lett.93, 257805 (2004).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.93.257805\"\n },\n {\n \"idx\": \"16\",\n \"title\": \"Bocquet, L. & Lauga, E. A smooth future?Nat. Mater.10, 334–337 (2011).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat2994\"\n },\n {\n \"idx\": \"18\",\n \"title\": \"Bouzigues, C. I., Tabeling, P. & Bocquet, L. Nanofluidics in the Debye layer at hydrophilic and hydrophobic surfaces.Phys. Rev. Lett.101, 114503 (2008).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.101.114503\"\n },\n {\n \"idx\": \"19\",\n \"title\": \"Khair, A. S. & Squires, T. M. Surprising consequences of ion conservation in electro-osmosis over a surface charge discontinuity.J. Fluid. Mech.615, 323–334 (2008).\",\n \"link\": \"https://doi.org/10.1017%2FS002211200800390X\"\n },\n {\n \"idx\": \"20\",\n \"title\": \"Park, S.-H. et al. Bioinspired superhydrophobic surfaces, fabricated through simple and scalable roll-to-roll processing.Sci. Rep.5, 15430 (2015).\",\n \"link\": \"https://doi.org/10.1038%2Fsrep15430\"\n },\n {\n \"idx\": \"21\",\n \"title\": \"Khair, A. S. & Squires, T. M. The influence of hydrodynamic slip on the electrophoretic mobility of a spherical colloidal particle.Phys. Fluids21, 042001 (2009).\",\n \"link\": \"https://doi.org/10.1063%2F1.3116664\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Voltage generation on air-filled surfaces Considering the flow of electrolyte over air-filled surfaces, the consequent voltage–time trace, Fig. 2b , was used to determine the V s , which was monitored through a high-resistance (> 10 GΩ) voltmeter (Keithley 2700) connected across the Ag/AgCl electrodes placed at either ends of the channel, Fig. 1c . However, a significant enhancement was not obtained for AFS, with an estimated [5] b of ~10 μm. It should be noted that a lower V s was obtained for a larger ϕ air (i.e., smaller ϕ s ), for which a larger b would be expected, Fig. 2c . Moreover, very little increase over a flat unpatterned surface was obtained comparing Air 0.5 and Flat, Fig. 2d . The lower V s of AFS may be rationalized as due to the absence of a surface charge in the air regions [13] , which overwhelms the contribution of the slip. While the data here seems to indicate the general validity of Eq. ( 1 ) in terms of a linear V s -Δ P plot, the slope of the obtained experimental curves would then be directly connected to ε , ζ , η , and κ . Such parameters are then intimately connected, i.e., while independent variation would be difficult to discern experimentally, theoretical or computational input (which is outside the purview of the present experimental work) may be able to yield guidance for explicit dependence. The net V s has been construed to be related to a weighted summation of ζ in the no-slip region (over the solid fraction of the surface, i.e., ζ NS ) and the slip region (i.e., over the air of the AFS, i.e., ζ S ) [6] . The very small ζ S would adversely affect the V s and as seen through our experiments overwhelms the slip length-related contribution. Related correspondence of the obtained V s to ζ as well as the electrolyte conductivity, which is proportional to I c (the counter-ion concentration), was determined through a log [ I c ]/ [ I c ] dependence, Fig. 2d . Increased voltage on liquid-filled surfaces Subsequently, we deployed patterned surfaces where the air pockets were infiltrated by a water-immiscible liquid, e.g., oil, of a higher density compared with the NaCl electrolyte, Fig. 3a . Such liquid-filled surfaces (LFS) were hypothesized to yield a finite ζ S at the oil interfaces, in addition to a non-zero b . The LFS were fabricated through replacing the air in groove-patterned arrays by water-immiscible liquid, such as oil, and may be designed to exhibit various degrees of fluid slip [22] , [23] , [24] , as well as electrical voltages that are significantly larger than those on air-filled or SH surfaces. Oil filling was found to be reliable and stable (see Methods). Fig. 3 Enhanced streaming potential at liquid-filled surfaces. Enhanced streaming potential ( V s ) was obtained by a filling the grooves with salt water immiscible oil, e.g., Krytox GPL 104, (scale bar is 100 μm); b a significantly larger V s was obtained for an GPL 104 oil-filled surface (i.e., GPL 0.5 , with the subscript now indicating the interstitial fraction of oil), compared to one with air (Air 0.5 ) and even compared to a Flat solid substrate (all measured using 0.1 mM L −1 NaCl solution with applied pressure as 1200 Pa). c The V s could be larger or smaller compared with the control air-filled surfaces (AFS), depending on the choice of the liquid in the LFS, e.g., GPL oil and castor oil (Castor 0.5 ), respectively Full size image Additionally, the use of oil with a lower ε r compared with the flowing electrolyte would mean that the electric field from the wall is effectively propagated. The details related to the (i) choice of liquid for the LFS along with the estimation of the (ii) ζ , and the (iii) b are given in Methods, Supplementary Note 2 and Supplementary Note 3 , respectively. Using Krytox GPL 104 oil ( ε r ~ 2.1, η ~ 340 cP), a significant enhancement of the V s by 50% to around ~52 mV, compared with that of Air 0.5 , was observed, Fig. 3b , attesting to the utility of the LFS. However, it was noted that the obtained V s critically depends on the choice of the oil in the LFS. Broadly, a larger ε oil seems to yield a lower V s . For instance, a factor of two smaller value was obtained for a castor oil ( ε r ~ 4.7, η ~ 312 cP)-filled LFS, Fig. 3c . The enhanced Van der Waals energy of interaction [25] between the oil and electrolyte, proportional to the dielectric constants, and the related mutual electric polarizabilities causes a larger friction. The consequent movement of the shear plane away from the solid substrate would result in a lower zeta potential and V s . The electrolyte flow over the LFS, as well as the adjacent solid surface were parameterized [26] through the use of a material-dependent surface charge density ( σ ) related to \\\\({\\\\zeta } = \\\\frac{{\\\\lambda _{\\\\mathrm{D}}\\\\sigma }}{{\\\\it{\\\\epsilon }}}\\\\) and fluid slip at the respective interfaces. From literature, for the top surface of the channel (constituted from poly-dimethyl siloxane, PDMS), σ PDMS [27] ~18 mC m −2 , while for the bottom-patterned surfaces, we use σ oil [28] ~1.8 mC m −2 and σ parylene [29] ~3.6 mC m −2 . The σ and the velocity slip is inhomogeneous along the length of the channel, but constant along the channel width. The no-slip hydrodynamic boundary condition was assumed to hold true at all solid–electrolyte interfaces, while a Navier slip boundary condition over the flowing electrolyte–oil interface was assumed, with a finite slip velocity: u s and slip length: b . For a hydrophobic surface, the b may be approximated [5] to be of the order of d , i.e., ~ 18 μm. A correspondence to a Cassie state-like channel flow (with no electrolyte penetration into the grooves of the patterned surface) for LFS surface may be seen. It should be noted that simulations considering the continuity of tangential shear stress across the electrolyte–oil interface also obtained the V s very close (within 1%) to the results obtained through an assumption of a slip length. Modeling of electrolyte flow over the liquid-filled surfaces We partition [12] the pressure gradient driven electrolyte flow, following experiment, over the LFS as the superposition of (i) hydrodynamic flow, with fluid slip over electrically neutral surfaces, and (ii) an electrokinetic flow, with no-slip fluid flow over flat electrically charged surfaces, Fig. 4a–c . A multiphysics model coupling the Nernst–Planck–Poisson (N-P-P) equation with the Stokes Equation was deployed to determine the volumetric charge density profiles (the details of the governing equations, assumptions, and the boundary conditions are given in Supplementary Note 4 and Supplementary Note 5 ). The velocity profile combined with the simulated volumetric charge density profile along the channel, as indicated in Fig. 4d , arises due to the difference of ionic concentrations between the counterions and co-ions. The resultant horizontal electric field is responsible for the observed V s . The calculated average electrolyte velocity of 6 cm s −1 is in agreement with that expected from Poiseuille flow along the channel, with the estimated u s at the oil–electrolyte interface of ~2 cm s −1 at Δ P ~1200 Pa. Additionally and given the relatively large groove widths, the influence of the surface conductivity on the obtained results would be relatively small [19] . Fig. 4 Parametrization and modeling of charged fluid–solid surfaces. a The surface charge density ( σ ) of an air-filled surface (AFS) varies between zero (at the air) and a finite value at the solid, while a nominally planar liquid-filled surface (LFS) maintains a larger value of the σ . The electrokinetics on the LFS was considered in terms of a superposition of b hydrodynamic flow, with fluid slip over electrically neutral surfaces, and c an electrokinetic flow, with no-slip fluid flow over electrically charged surfaces, and solving the coupled Nernst–Planck–Poisson (NPP) equation along with the Stokes Equation to determine the d velocity profile, as well as the volumetric charge density ( ρ ) profiles. The insets indicate the computed variation of the ρ at the top poly-dimethyl siloxane (PDMS) surface and the bottom LFS Full size image\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"The values of the streaming potential arising from the simulations are in excellent agreement with those observed experimentally through considering a reduced ε r ~28 over a distance ~3 λ D proximate to the LFS and an ε r ~78 and beyond, correspondent to the bulk solution. Such aspect is in accord with the use of different values of the dielectric constant for the inner and outer regions of a solid–solution interface [12] . The LFS, consisting of both oil as well as solid surfaces, may yield an uneven shear plane with fascinating implications for novel electrokinetic phenomena [30] , [31] , such as localized concentration polarization [19] , etc. Our work also indicates that a heterogeneous surface could be patterned through the use of discrete surface charge density or applied potentials, mimicking a SH surface and making a connection with electro-wetting applications, as well as with significant literature on the modeling and parameterization of liquid flows on striped surfaces that exhibit alternate regions of slip and no-slip, with [32] , [33] and without [34] surface charge. A more detailed investigation of several issues related to the interplay of enhanced surface charge as well as increased slip velocity at the flowing electrolyte–LFS interface would then be warranted. It has been indicated, for instance, that an increased V s would result due to the larger ion convection currents [35] . Additionally, viscous effects may be playing a major role [36] , [37] . As the coefficient in Eq. 1 intimately connects the ε r , and the η , a decrease in the former may not be readily deconvoluted from an increase in the latter experimentally. In our numerical simulations, an increase in the η by a factor of ~2.5 over the equivalent length scale (from a bulk value of 0.89 cP to 2.19 cP) was necessary to obtain the experimentally observed streaming potentials, if the ε r is ~78. Our experiments have deep scientific implications underlying fluid flow interfacing with both air and liquid, as well as fluid shear at a surface, and electrokinetic phenomena as related to the localized variation of the zeta potential and nonlinearity. We anticipate that our work would revitalize research related to the fabrication of alternate electrical voltage/power sources from liquid flow over charged surfaces. Other anticipated applications extend to electrophoretic applications in biological separations (cell transport, manipulation [38] , and interactions [39] ), as well as voltage sources for lab-on-a-chip applications [40] . In summary, our work has experimentally demonstrated the largest figure of merit thus far [41] , [42] , [43] , [44] , to the best of our knowledge, with primary focus on methodologies related to enhance the streaming potential ( V s ) per unit pressure difference (Δ P ) through the use of LFS (also see Supplementary Note 6 for considerations related to overall electrokinetic efficiency). The use of the LFS yields a figure of merit increase by a factor of 1.4 in comparison to that obtained using the AFS. It has been shown that larger voltages, through a measured streaming potential, may be achieved through careful engineering of the coupled electric field and fluid flow. The application of the related increase in the electrokinetic energy conversion efficiency would need further optimization of the fluidic and electrical impedances, in concert with the streaming conductance, as matched to an appropriate load [45] . Concomitantly, unipolar transport (where for example, either Na + or Cl − ions are transported [46] ) through EDL overlap [26] , [47] in nanoscale channels may be coupled with LFS to yield much larger voltages, comparable to that of batteries [48] .\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Design considerations and fabrication of air-filled surfaces and liquid-filled surfaces The critical parameters in our channel surface design are (a) structural, i.e., the interstitial fraction of AFS/LFS: ϕ -the ratio [ = d/ ( d + w )], with an average ridge width ( w ) and spacing ( d ), as indicated in Fig. 2a, (b) dimension, the channel length ( L) and channel width ( W ) should be much larger than channel height ( H ), then the channel can be treated as two infinite parallel plates and the flow is Poiseuille flow, (c) pressure gradients, i.e., over a range of 0–1000 Pa, with higher pressures yielding greater streaming potentials ( V s ), and (d) the nature of the filling fluid, easier for oil with low surface tension to penetrate into the grooves. For instance, we have used in our electrokinetic flow experiments, ϕ ~0.5, 0.9 cm wide, 11.8 cm long, and 250 μm high channel, with a pressure difference of ~1200 Pa, using GPL oil as a filling liquid, to generate maximal V s . The LFS were fabricated through filling oil into the grooves of a patterned surface, fabricated through photolithography, and dry etching. A Si wafer (n-type, < 110 > , thickness 500 µm) was cleaned with acetone and IPA, rinsed by DI water, and subsequently baked at 180 °C for 5 min. Negative photoresist (NR9-3000) was coated on the Si wafer (3000 rpm for 40 s) and baked at 120 °C for 60 s. The pattern was defined through a mask (using the EVG 620) and developed (RD6) for 1 min. Then the resist-patterned wafer was subject to dry etching (using Oxford Plasmalab 100 RIE/ICP) to yield a trench depth ( h ) ~ 95 µm. The photoresist was then removed (using RR2 photoresist remover) over ~12 h and any residues were further removed (using PVA TePla PS100 at 120 sccm, 200 W for 90 s). The grooved surface was coated with Parylene-C (using PDS 2010 Parylene Coater) to a thickness of ~600 nm. For LFS, oils, e.g., Dupont Krytox GPL 104, Castor oil, etc. was filled into the grooves of the patterned surface. The air-/liquid-patterned surfaces were imaged using environmental scanning electron microscopy (ESEM): FEI/Phillips XL ESEM and the FEI Quanta FEG 250 ESEM. We have indicated, e.g., in Fig. 2c that a lower ϕ -the ratio of the average groove width to the overall period yields a larger V s . Choice of applied pressure and related flow rate The electrolyte flow rate in the channel, constituted from rectangular plate geometry [26] , was modulated through a syringe pump, Fig. 1c , and was transduced to an applied pressure (Δ P ) based on the Poiseuille relation: \\\\(\\\\Delta P = \\\\frac{{12\\\\eta LQ}}{{wh^3}}\\\\) ( Q is flow rate, L is channel length of ~11.8 cm, and h the channel height of ~250 μm). The Δ P over the channel length (where the other end was left open to atmosphere) motivated the pressure gradient driven flow, and for the experiment was in the range of 200 to1200 Pa, as measured through a manometer (UEI EM152 Dual Differential Input Manometer). It was observed that when the Δ P was larger (/smaller) than 1200 Pa (/200 Pa), that the V s was not stable. Choice of liquid to fill the interstices in the liquid-filled surfaces For the fabrication of the LFS, the filling liquid (e.g., Krytox ® GPL 104 and castor oil) was chosen primarily on the basis of immiscibility with water/electrolyte. Additional criteria would be a low surface tension—for better penetration into the interstices, low viscosity, and a smaller dielectric constant—to reduce electrolyte–oil interfacial shear stress. We observe (see Supplementary Figure 1 ) that the parylene-coated Si surface was oleophilic (/hydrophobic), and that the chosen liquid or oil could penetrate the interstices easily, Fig. 3a .\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"The core research idea is to enhance the streaming potential (voltage generation per unit applied pressure) in microfluidic channels by modulating the effective zeta potential through fluid slip induced by patterned liquid-filled surfaces, as opposed to air-filled superhydrophobic surfaces, thereby overcoming limitations in surface charge and slip length that have previously restricted the electrokinetic energy conversion efficiency."},"entities":{"kind":"list like","value":["Streaming potential","microfluidic channels","effective zeta potential","fluid slip","patterned liquid-filled surfaces","air-filled superhydrophobic surfaces","surface charge","slip length","electrokinetic energy conversion efficiency"],"string":"[\n \"Streaming potential\",\n \"microfluidic channels\",\n \"effective zeta potential\",\n \"fluid slip\",\n \"patterned liquid-filled surfaces\",\n \"air-filled superhydrophobic surfaces\",\n \"surface charge\",\n \"slip length\",\n \"electrokinetic energy conversion efficiency\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"2.","title":"Morrison, F. A. & Osterle, J. F. Electrokinetic energy conversion in ultrafine capillaries.J. Chem. Phys.43, 2111–2115 (1965).","link":"https://doi.org/10.1063%2F1.1697081"},{"idx":"3.","title":"Bahga, S. S., Vinogradova, O. I. & Bazant, M. Z. Anisotropic electro-osmotic flow over super-hydrophobic surfaces.J. Fluid. Mech.644, 245–255 (2010).","link":"https://doi.org/10.1017%2FS0022112009992771"},{"idx":"5.","title":"Rothstein, J. P. Slip on superhydrophobic surfaces.Annu. Rev. Fluid. Mech.42, 89–109 (2010).","link":"https://doi.org/10.1146%2Fannurev-fluid-121108-145558"},{"idx":"6.","title":"Squires, T. M. Electrokinetic flows over inhomogeneously slipping surfaces.Phys. Fluids20, 092105 (2008).","link":"https://doi.org/10.1063%2F1.2978954"},{"idx":"7.","title":"Kirby, B. J. & Hasselbrink, E. F. Zeta potential of microfluidic substrates: 1. Theory, experimental techniques, and effects on separations.Electrophoresis25, 187–202 (2004).","link":"https://doi.org/10.1002%2Felps.200305754"},{"idx":"8.","title":"Kirby, B. J. & Hasselbrink, E. F. Zeta potential of microfluidic substrates: 2.Data Polym. Electrophor.25, 203–213 (2004).","link":"https://doi.org/10.1002%2Felps.200305755"},{"idx":"9.","title":"Whitesides, G. M. The origins and the future of microfluidics.Nature442, 368–373 (2006).","link":"https://doi.org/10.1038%2Fnature05058"},{"idx":"15.","title":"Joly, L., Ybert, C., Trizac, E. & Bocquet, L. Hydrodynamics within the electric double layer on slipping surfaces.Phys. Rev. Lett.93, 257805 (2004).","link":"https://doi.org/10.1103%2FPhysRevLett.93.257805"},{"idx":"16.","title":"Bocquet, L. & Lauga, E. A smooth future?Nat. Mater.10, 334–337 (2011).","link":"https://doi.org/10.1038%2Fnmat2994"},{"idx":"18.","title":"Bouzigues, C. I., Tabeling, P. & Bocquet, L. Nanofluidics in the Debye layer at hydrophilic and hydrophobic surfaces.Phys. Rev. Lett.101, 114503 (2008).","link":"https://doi.org/10.1103%2FPhysRevLett.101.114503"},{"idx":"19.","title":"Khair, A. S. & Squires, T. M. Surprising consequences of ion conservation in electro-osmosis over a surface charge discontinuity.J. Fluid. Mech.615, 323–334 (2008).","link":"https://doi.org/10.1017%2FS002211200800390X"},{"idx":"20.","title":"Park, S.-H. et al. Bioinspired superhydrophobic surfaces, fabricated through simple and scalable roll-to-roll processing.Sci. Rep.5, 15430 (2015).","link":"https://doi.org/10.1038%2Fsrep15430"},{"idx":"21.","title":"Khair, A. S. & Squires, T. M. The influence of hydrodynamic slip on the electrophoretic mobility of a spherical colloidal particle.Phys. Fluids21, 042001 (2009).","link":"https://doi.org/10.1063%2F1.3116664"},{"idx":"22.","title":"Yao, X. et al. Adaptive fluid-infused porous films with tunable transparency and wettability.Nat. Mater.12, 529–534 (2013).","link":"https://doi.org/10.1038%2Fnmat3598"},{"idx":"23.","title":"Wong, T.-S. et al. Bioinspired self-repairing slippery surfaces with pressure-stable omniphobicity.Nature477, 443–447 (2011).","link":"https://doi.org/10.1038%2Fnature10447"},{"idx":"26.","title":"Van Der Heyden, F. H. J., Bonthuis, D. J., Stein, D., Meyer, C. & Dekker, C. Electrokinetic energy conversion efficiency in nanofluidic channels.Nano. Lett.6, 2232–2237 (2006).","link":"https://doi.org/10.1021%2Fnl061524l"},{"idx":"27.","title":"Schrott, W. et al. Study on surface properties of PDMS microfluidic chips treated with albumin.Biomicrofluidics3, 044101 (2009).","link":"https://doi.org/10.1063%2F1.3243913"},{"idx":"28.","title":"Marinova, K. G. et al. Charging of oil−water interfaces due to spontaneous adsorption of hydroxyl ions.Langmuir12, 2045–2051 (1996).","link":"https://doi.org/10.1021%2Fla950928i"},{"idx":"29.","title":"Lo, H. W. & Tai, Y. C. Parylene-based electret power generators.J. Micromechanics Microengineering18, 104006 (2008).","link":"https://doi.org/10.1088%2F0960-1317%2F18%2F10%2F104006"},{"idx":"30.","title":"Thamida, S. K. & Chang, H.-C. Nonlinear electrokinetic ejection and entrainment due to polarization at nearly insulated edges.Phys. Fluids14, 4315 (2002).","link":"https://doi.org/10.1063%2F1.1519530"},{"idx":"31.","title":"Yossifon, G., Frankel, I. & Miloh, T. On electro-osmotic flows through microchannel junctions.Phys. Fluids18, 117108 (2006).","link":"https://doi.org/10.1063%2F1.2391701"},{"idx":"32.","title":"Zhao, H. Streaming potential generated by a pressure-driven flow over superhydrophobic stripes.Phys. Fluids23, 022003 (2011).","link":"https://doi.org/10.1063%2F1.3551616"},{"idx":"33.","title":"Ng, C. O. & Chu, H. C. W. Electrokinetic flows through a parallel-plate channel with slipping stripes on walls.Phys. Fluids23, 102002 (2011).","link":"https://doi.org/10.1063%2F1.3647582"},{"idx":"34.","title":"Lauga, E. & Stone, H. A. Effective slip in pressure-driven Stokes flow.J. Fluid. Mech.489, 55–77 (2003).","link":"https://doi.org/10.1017%2FS0022112003004695"},{"idx":"35.","title":"Yang, D. & Kwok, D. Y. Analytical tretament of flow in infinitely extended circular microchannels and the effect of slippage to increase flow efficiency.J. Micromech. Microeng.13, 115 (2003).","link":"https://doi.org/10.1088%2F0960-1317%2F13%2F1%2F316"},{"idx":"36.","title":"Lyklema, J. & Overbeek, J. T. G. On the interpretation of electrokinetic potentials.J. Colloid Sci.16, 501–512 (1961).","link":"https://doi.org/10.1016%2F0095-8522%2861%2990029-0"},{"idx":"37.","title":"Wu, P. & Qio, R. Physical origins of apparently enhanced viscosity of interfacial fluids in electrokinetic transport.Phys. Fluids23, 072005 (2011).","link":"https://doi.org/10.1063%2F1.3614534"},{"idx":"38.","title":"Li, P. C. & Harrison, D. J. Transport, manipulation, and reaction of biological cells on-chip using electrokinetic effects.Anal. Chem.69, 1564–1568 (1997).","link":"https://doi.org/10.1021%2Fac9606564"},{"idx":"39.","title":"Jacobson, S. C., McKnight, T. E. & Ramsey, J. M. Microfluidic devices for electrokinetically driven parallel and serial mixing.Anal. Chem.71, 4455–4459 (1999).","link":"https://doi.org/10.1021%2Fac990576a"},{"idx":"40.","title":"Zhang, R. et al. A streaming potential/current-based microfluidic direct current generator for self-powered nanosystems.Adv. Mater.27, 6482–6487 (2015).","link":"https://doi.org/10.1002%2Fadma.201502477"},{"idx":"41.","title":"Van Der Heyden, F. H. J., Bonthuis, D. J., Stein, D., Meyer, C. & Dekker, C. Power generation by pressure-driven transport of ions in nanofluidic channels.Nano. Lett.7, 1022–1025 (2007).","link":"https://doi.org/10.1021%2Fnl070194h"},{"idx":"42.","title":"Xie, Y., Sherwood, J. D., Shui, L., van den Berg, A. & Eijkel, J. C. T. Strong enhancement of streaming current power by application of two phase flow.Lab. Chip.11, 4006 (2011).","link":"https://doi.org/10.1039%2Fc1lc20423h"},{"idx":"43.","title":"Xie, Y. et al. Electric energy generation in single track-etched nanopores.Appl. Phys. Lett.93, 163116 (2008).","link":"https://doi.org/10.1063%2F1.3001590"},{"idx":"44.","title":"Li, Z., Liu, Y., Zheng, Y. & Xu, R. Zeta potential at the root surfaces of rice characterized by streaming potential measurements.Plant Soil386, 237–250 (2015).","link":"https://link.springer.com/doi/10.1007/s11104-014-2259-6"},{"idx":"45.","title":"Ren, Y. & Stein, D. Slip enhanced electrokinetic energy conversion in nanofluidic channels.Nanotechnology19, 195707 (2008).","link":"https://doi.org/10.1088%2F0957-4484%2F19%2F19%2F195707"},{"idx":"46.","title":"Yamada, H. & Bandaru, P. R. Enhanced electrical current densities in electrochemical systems through the use of nanostructured electrodes.Appl. Phys. Lett.104, 213901 (2014).","link":"https://doi.org/10.1063%2F1.4879837"},{"idx":"47.","title":"Van Der Heyden, F. H. J., Stein, D. & Dekker, C. Streaming currents in a single nanofluidic channel.Phys. Rev. Lett.95, 116104 (2005).","link":"https://doi.org/10.1103%2FPhysRevLett.95.116104"}],"string":"[\n {\n \"idx\": \"2.\",\n \"title\": \"Morrison, F. A. & Osterle, J. F. Electrokinetic energy conversion in ultrafine capillaries.J. Chem. Phys.43, 2111–2115 (1965).\",\n \"link\": \"https://doi.org/10.1063%2F1.1697081\"\n },\n {\n \"idx\": \"3.\",\n \"title\": \"Bahga, S. S., Vinogradova, O. I. & Bazant, M. Z. Anisotropic electro-osmotic flow over super-hydrophobic surfaces.J. Fluid. Mech.644, 245–255 (2010).\",\n \"link\": \"https://doi.org/10.1017%2FS0022112009992771\"\n },\n {\n \"idx\": \"5.\",\n \"title\": \"Rothstein, J. P. Slip on superhydrophobic surfaces.Annu. Rev. Fluid. Mech.42, 89–109 (2010).\",\n \"link\": \"https://doi.org/10.1146%2Fannurev-fluid-121108-145558\"\n },\n {\n \"idx\": \"6.\",\n \"title\": \"Squires, T. M. Electrokinetic flows over inhomogeneously slipping surfaces.Phys. Fluids20, 092105 (2008).\",\n \"link\": \"https://doi.org/10.1063%2F1.2978954\"\n },\n {\n \"idx\": \"7.\",\n \"title\": \"Kirby, B. J. & Hasselbrink, E. F. Zeta potential of microfluidic substrates: 1. Theory, experimental techniques, and effects on separations.Electrophoresis25, 187–202 (2004).\",\n \"link\": \"https://doi.org/10.1002%2Felps.200305754\"\n },\n {\n \"idx\": \"8.\",\n \"title\": \"Kirby, B. J. & Hasselbrink, E. F. Zeta potential of microfluidic substrates: 2.Data Polym. Electrophor.25, 203–213 (2004).\",\n \"link\": \"https://doi.org/10.1002%2Felps.200305755\"\n },\n {\n \"idx\": \"9.\",\n \"title\": \"Whitesides, G. M. The origins and the future of microfluidics.Nature442, 368–373 (2006).\",\n \"link\": \"https://doi.org/10.1038%2Fnature05058\"\n },\n {\n \"idx\": \"15.\",\n \"title\": \"Joly, L., Ybert, C., Trizac, E. & Bocquet, L. Hydrodynamics within the electric double layer on slipping surfaces.Phys. Rev. Lett.93, 257805 (2004).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.93.257805\"\n },\n {\n \"idx\": \"16.\",\n \"title\": \"Bocquet, L. & Lauga, E. A smooth future?Nat. Mater.10, 334–337 (2011).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat2994\"\n },\n {\n \"idx\": \"18.\",\n \"title\": \"Bouzigues, C. I., Tabeling, P. & Bocquet, L. Nanofluidics in the Debye layer at hydrophilic and hydrophobic surfaces.Phys. Rev. Lett.101, 114503 (2008).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.101.114503\"\n },\n {\n \"idx\": \"19.\",\n \"title\": \"Khair, A. S. & Squires, T. M. Surprising consequences of ion conservation in electro-osmosis over a surface charge discontinuity.J. Fluid. Mech.615, 323–334 (2008).\",\n \"link\": \"https://doi.org/10.1017%2FS002211200800390X\"\n },\n {\n \"idx\": \"20.\",\n \"title\": \"Park, S.-H. et al. Bioinspired superhydrophobic surfaces, fabricated through simple and scalable roll-to-roll processing.Sci. Rep.5, 15430 (2015).\",\n \"link\": \"https://doi.org/10.1038%2Fsrep15430\"\n },\n {\n \"idx\": \"21.\",\n \"title\": \"Khair, A. S. & Squires, T. M. The influence of hydrodynamic slip on the electrophoretic mobility of a spherical colloidal particle.Phys. Fluids21, 042001 (2009).\",\n \"link\": \"https://doi.org/10.1063%2F1.3116664\"\n },\n {\n \"idx\": \"22.\",\n \"title\": \"Yao, X. et al. Adaptive fluid-infused porous films with tunable transparency and wettability.Nat. Mater.12, 529–534 (2013).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat3598\"\n },\n {\n \"idx\": \"23.\",\n \"title\": \"Wong, T.-S. et al. Bioinspired self-repairing slippery surfaces with pressure-stable omniphobicity.Nature477, 443–447 (2011).\",\n \"link\": \"https://doi.org/10.1038%2Fnature10447\"\n },\n {\n \"idx\": \"26.\",\n \"title\": \"Van Der Heyden, F. H. J., Bonthuis, D. J., Stein, D., Meyer, C. & Dekker, C. Electrokinetic energy conversion efficiency in nanofluidic channels.Nano. Lett.6, 2232–2237 (2006).\",\n \"link\": \"https://doi.org/10.1021%2Fnl061524l\"\n },\n {\n \"idx\": \"27.\",\n \"title\": \"Schrott, W. et al. Study on surface properties of PDMS microfluidic chips treated with albumin.Biomicrofluidics3, 044101 (2009).\",\n \"link\": \"https://doi.org/10.1063%2F1.3243913\"\n },\n {\n \"idx\": \"28.\",\n \"title\": \"Marinova, K. G. et al. Charging of oil−water interfaces due to spontaneous adsorption of hydroxyl ions.Langmuir12, 2045–2051 (1996).\",\n \"link\": \"https://doi.org/10.1021%2Fla950928i\"\n },\n {\n \"idx\": \"29.\",\n \"title\": \"Lo, H. W. & Tai, Y. C. Parylene-based electret power generators.J. Micromechanics Microengineering18, 104006 (2008).\",\n \"link\": \"https://doi.org/10.1088%2F0960-1317%2F18%2F10%2F104006\"\n },\n {\n \"idx\": \"30.\",\n \"title\": \"Thamida, S. K. & Chang, H.-C. Nonlinear electrokinetic ejection and entrainment due to polarization at nearly insulated edges.Phys. Fluids14, 4315 (2002).\",\n \"link\": \"https://doi.org/10.1063%2F1.1519530\"\n },\n {\n \"idx\": \"31.\",\n \"title\": \"Yossifon, G., Frankel, I. & Miloh, T. On electro-osmotic flows through microchannel junctions.Phys. Fluids18, 117108 (2006).\",\n \"link\": \"https://doi.org/10.1063%2F1.2391701\"\n },\n {\n \"idx\": \"32.\",\n \"title\": \"Zhao, H. Streaming potential generated by a pressure-driven flow over superhydrophobic stripes.Phys. Fluids23, 022003 (2011).\",\n \"link\": \"https://doi.org/10.1063%2F1.3551616\"\n },\n {\n \"idx\": \"33.\",\n \"title\": \"Ng, C. O. & Chu, H. C. W. Electrokinetic flows through a parallel-plate channel with slipping stripes on walls.Phys. Fluids23, 102002 (2011).\",\n \"link\": \"https://doi.org/10.1063%2F1.3647582\"\n },\n {\n \"idx\": \"34.\",\n \"title\": \"Lauga, E. & Stone, H. A. Effective slip in pressure-driven Stokes flow.J. Fluid. Mech.489, 55–77 (2003).\",\n \"link\": \"https://doi.org/10.1017%2FS0022112003004695\"\n },\n {\n \"idx\": \"35.\",\n \"title\": \"Yang, D. & Kwok, D. Y. Analytical tretament of flow in infinitely extended circular microchannels and the effect of slippage to increase flow efficiency.J. Micromech. Microeng.13, 115 (2003).\",\n \"link\": \"https://doi.org/10.1088%2F0960-1317%2F13%2F1%2F316\"\n },\n {\n \"idx\": \"36.\",\n \"title\": \"Lyklema, J. & Overbeek, J. T. G. On the interpretation of electrokinetic potentials.J. Colloid Sci.16, 501–512 (1961).\",\n \"link\": \"https://doi.org/10.1016%2F0095-8522%2861%2990029-0\"\n },\n {\n \"idx\": \"37.\",\n \"title\": \"Wu, P. & Qio, R. Physical origins of apparently enhanced viscosity of interfacial fluids in electrokinetic transport.Phys. Fluids23, 072005 (2011).\",\n \"link\": \"https://doi.org/10.1063%2F1.3614534\"\n },\n {\n \"idx\": \"38.\",\n \"title\": \"Li, P. C. & Harrison, D. J. Transport, manipulation, and reaction of biological cells on-chip using electrokinetic effects.Anal. Chem.69, 1564–1568 (1997).\",\n \"link\": \"https://doi.org/10.1021%2Fac9606564\"\n },\n {\n \"idx\": \"39.\",\n \"title\": \"Jacobson, S. C., McKnight, T. E. & Ramsey, J. M. Microfluidic devices for electrokinetically driven parallel and serial mixing.Anal. Chem.71, 4455–4459 (1999).\",\n \"link\": \"https://doi.org/10.1021%2Fac990576a\"\n },\n {\n \"idx\": \"40.\",\n \"title\": \"Zhang, R. et al. A streaming potential/current-based microfluidic direct current generator for self-powered nanosystems.Adv. Mater.27, 6482–6487 (2015).\",\n \"link\": \"https://doi.org/10.1002%2Fadma.201502477\"\n },\n {\n \"idx\": \"41.\",\n \"title\": \"Van Der Heyden, F. H. J., Bonthuis, D. J., Stein, D., Meyer, C. & Dekker, C. Power generation by pressure-driven transport of ions in nanofluidic channels.Nano. Lett.7, 1022–1025 (2007).\",\n \"link\": \"https://doi.org/10.1021%2Fnl070194h\"\n },\n {\n \"idx\": \"42.\",\n \"title\": \"Xie, Y., Sherwood, J. D., Shui, L., van den Berg, A. & Eijkel, J. C. T. Strong enhancement of streaming current power by application of two phase flow.Lab. Chip.11, 4006 (2011).\",\n \"link\": \"https://doi.org/10.1039%2Fc1lc20423h\"\n },\n {\n \"idx\": \"43.\",\n \"title\": \"Xie, Y. et al. Electric energy generation in single track-etched nanopores.Appl. Phys. Lett.93, 163116 (2008).\",\n \"link\": \"https://doi.org/10.1063%2F1.3001590\"\n },\n {\n \"idx\": \"44.\",\n \"title\": \"Li, Z., Liu, Y., Zheng, Y. & Xu, R. Zeta potential at the root surfaces of rice characterized by streaming potential measurements.Plant Soil386, 237–250 (2015).\",\n \"link\": \"https://link.springer.com/doi/10.1007/s11104-014-2259-6\"\n },\n {\n \"idx\": \"45.\",\n \"title\": \"Ren, Y. & Stein, D. Slip enhanced electrokinetic energy conversion in nanofluidic channels.Nanotechnology19, 195707 (2008).\",\n \"link\": \"https://doi.org/10.1088%2F0957-4484%2F19%2F19%2F195707\"\n },\n {\n \"idx\": \"46.\",\n \"title\": \"Yamada, H. & Bandaru, P. R. Enhanced electrical current densities in electrochemical systems through the use of nanostructured electrodes.Appl. Phys. Lett.104, 213901 (2014).\",\n \"link\": \"https://doi.org/10.1063%2F1.4879837\"\n },\n {\n \"idx\": \"47.\",\n \"title\": \"Van Der Heyden, F. H. J., Stein, D. & Dekker, C. Streaming currents in a single nanofluidic channel.Phys. Rev. Lett.95, 116104 (2005).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.95.116104\"\n }\n]","isConversational":false}}},{"rowIdx":11194,"cells":{"unique_id":{"kind":"string","value":"s41467-023-43891-y"},"subfield":{"kind":"string","value":"Mathematics and computing"},"title":{"kind":"string","value":"On-chip phonon-magnon reservoir for neuromorphic computing"},"abstract":{"kind":"string","value":"Reservoir computing is a concept involving mapping signals onto a high-dimensional phase space of a dynamical system called “reservoir” for subsequent recognition by an artificial neural network. We implement this concept in a nanodevice consisting of a sandwich of a semiconductor phonon waveguide and a patterned ferromagnetic layer. A pulsed write-laser encodes input signals into propagating phonon wavepackets, interacting with ferromagnetic magnons. The second laser reads the output signal reflecting a phase-sensitive mix of phonon and magnon modes, whose content is highly sensitive to the write- and read-laser positions. The reservoir efficiently separates the visual shapes drawn by the write-laser beam on the nanodevice surface in an area with a size comparable to a single pixel of a modern digital camera. Our finding suggests the phonon-magnon interaction as a promising hardware basis for realizing on-chip reservoir computing in future neuromorphic architectures."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"The flourishing idea of using artificial neural networks (ANNs) for the solution of practical tasks shifts the computing paradigm towards heavy usage of data instead of creating lengthy instructions and logical chains forming algorithms. Being inspired by biological neural systems, an ANN constitutes a set of many interconnected non-linear functional units called ‘neurons’. To solve a task with the desired accuracy, an ANN requires training by data processing to fit the weights of the connections between neurons. Over the decades, ANNs have demonstrated their high efficiency in a wide range of important practical applications [1] including voice, image, and pattern recognition, data mining, prediction of complex dynamics, 3D reconstruction, and medical diagnostics. The more sophisticated the problem an ANN has to solve, the larger is the network required to achieve the desirable accuracy of the solution. Simulation of large networks using conventional digital computers demands enormous CPU/GPU power, memory, and energy supply [2] . A promising approach to resolve the hardware issues is neuromorphic computing, which implements ANNs in circuits [3] . Such hardware architecture should dramatically improve the transfer of data across the circuit during data processing, thus enhancing both the speed and energy efficiency of calculations. Another common drawback of ANNs is the significant time required for their training to optimize performance [4] . ANNs are most efficient if the type of task (‘the goal’) is fixed, and there is enough data for training. However, they experience significant difficulties if the amount of data is limited, or the goal varies in time [5] . Smaller networks would partly resolve this problem since fewer neurons and connections need less data and time for training, but this approach suffers from poor accuracy. A promising way to resolve this problem is the concept of ‘reservoir computing’ (RC). The ‘reservoir’ is a recurrent ANN with non-linear dynamics and fixed connections, which typically does not require any training, attached to a small readout ANN, which is trained to recognize predefined characteristics of the reservoir output. In such an architecture, the reservoir serves to ‘highlight’ important features in data sets and suppress irrelevant ones, thereby enormously improving the performance of computing. As one of many natural prototypes of RC, one could consider the human vision schematically illustrated in Fig. 1a . The visual information passed through the cornea and focused by the lens is converted by the retina photoreceptors into electrical signals (the neural pulses) that are transmitted by the optic nerve and mixed in a high-dimensional space of the visual cortex for the subsequent recognition. The complex photochemical transformation (phototransduction) by the 10 8 photoreceptors allows our brain to recognize objects, distinguish tiny differences between them, and detect thereby their motion. Such a resemblance between biological sensory systems and reservoir ANNs has invigorated research on RC-based artificial vision [6] , and other RC-systems for detection and recognition [5] , [7] . Fig. 1: Nanodevice architecture and working principle. a Schematic of human vision: an image focused by the lens onto the retina is transformed into electrical pulses that are transferred by the optic nerve to the brain for the subsequent recognition. b Design of the examined nanostructure. c Scanning electron microscopy (SEM) images (top and bottom panels) of the nanodevice surface with the nanograting (NG): complete NG with a size of 25 × 100 µm 2 (top panel), and its close-up with the visible pattern (bottom panel). The inset in the right-top corner is a microscopy image of the write (large spot) and read (small spot) laser spots on the NG surface. The size of the framed area is 10 × 10 µm 2 . d Working principle of the phonon reservoir with magnon readout. A multimode phonon wavepacket is generated by ultrafast laser excitation of the patterned ferromagnetic nanolayer. It propagates well protected from external perturbations beneath the surface in the GaAs/AlAs phonon waveguide and its continuously transformed waveform is imprinted in the magnon modes of the ferromagnetic layer. The superposition of the magnon modes driven by the phonon wavepacket at a specific distance from the write spot is detected using the read laser pulse. Full size image From the viewpoint of information theory, the reservoir maps input signals into higher dimensional information spaces and can be realized by continuous in space and/or in time nonlinear dynamical systems with storage ability. Therefore, various physical systems have been suggested for the reservoir [5] , [8] , [9] , [10] . One of the first realizations was based on waves, namely gravity waves in a vessel with water to recognize spoken numbers [11] . Nowadays, wave ‘reservoirs’ have been implemented using different physical waves such as electromagnetic waves (photons) [12] , [13] , [14] , [15] , [16] , [17] , [18] , elastic waves (phonons) [19] , [20] , and spin waves (magnons) [21] , [22] , [23] , [24] . The key factors that make waves reliable for reservoir architectures are the possibility to achieve wave packets with a vast information density; weighted summation by interference; nonlinearity resulting in wave mixing and generation of higher harmonics; hybridization of waves of different types, e.g., the formation of polarons. Recently, wave-based ‘reservoirs’ have demonstrated the ability for deep learning [25] , [26] , [27] . Despite rapid progress in the development of RC-systems, their implementation in chips suitable for practical technological applications has remained an unsolved problem. Among the major challenges with existing neuromorphic platforms [3] are power efficiency, miniaturization, separability, and robustness [5] , [8] , [10] . On-chip wave-based reservoirs have potential to address these challenges, since they allow smaller devices, good scalability, great state numbers, and fine control of interacting wave modes. However, their development calls for novel materials and designs allowing the generation of waves for the reservoir function.","references":{"indices":[1,2,3,4,5,6,7,8,9,10,12,13,14,16,17,18,19,20,21,22,23,24,25,26,27],"infos":[{"idx":"1","title":"Le Cun, Y., Bengio, Y. & Hinton, G. Deep learning.Nature521, 436–444 (2015).","link":"https://doi.org/10.1038%2Fnature14539"},{"idx":"2","title":"Strubell, E., Ganesh, A. & McCallum, A. Energy and policy considerations for modern deep learning research.AAAI34, 13693–13696 (2019).","link":"https://doi.org/10.1609%2Faaai.v34i09.7123"},{"idx":"3","title":"Christensen, D. 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Deep physical neural networks trained with backpropagation.Nature601, 549–555 (2022).","link":"https://doi.org/10.1038%2Fs41586-021-04223-6"}]}},{"section":"Results","content":"Hybrid nanostructure for phonon–magnon reservoir Here, we propose and demonstrate experimentally a new type of wave reservoir based on high-frequency acoustic and spin waves in a physical system that combines ultimate compactness, robustness, and separability. For this aim, we fabricate a nanodevice with an architecture implemented with available nanotechnology. The device, shown schematically in Fig. 1b , consists of a sandwich of GaAs/AlAs semiconductor layers, which form a waveguide for high-frequency acoustic waves, i.e., phonons [28] . The waveguide is capped by a nanograting patterned in a metal ferromagnetic Fe 0.81 Ga 0.19 layer hosting spin waves, i.e. magnons [29] . Figure 1c shows the scanning electron microscopy image of the patterned surface. We examine the performance of this device by recognition of the “visual shapes” drawn on the surface by a pulsed ‘write’ laser. The laser pulses are converted by the nanograting to a coherent multimode phonon wavepacket, which is guided along the surface and experiences continuous transformation due to the interference of the phonon modes. Phonons interact with magnons and this multimode magnon-phonon mixture forms the reservoir multidimensional space. The reservoir output is read through magnons by measuring the reflectivity of another ‘read’ laser beam. The output waveform is integrated over the drawn trajectory corresponding to a specific “shape” for the following recognition by an ANN programmed in a personal computer. We may compare this procedure with handwriting and conceptualize the reservoir operations as recognition of a handwritten character. Thus, in analogy with the human vision (Fig. 1d ), we may consider the patterned surface as the retina, which converts optical input to neural pulses (coherent phonons), processed by a multidimensional reservoir space (phonon-magnon mixture) to generate a readout recognized by the brain, namely the ANN. The sub-µm wavelength of phonons and magnons allows achieving a large number of “receptors” per area (‘’receptor” density) and make the readout sensitive to a tiny change of the optical input. It enables recognition of the symbols drawn on the scale of just several wavelengths of the ‘write’ laser. The physical processes involved in the operation of the fabricated nanodevice are demonstrated in Fig. 2 . The 100-fs pulse of the write laser, focused on the metallic ferromagnetic layer to a spot of 2-µm diameter instantaneously induces thermal stress leading to the generation of the coherent acoustic wavepacket of hypersound (up to ~100 GHz) frequencies [30] , [31] . Due to the periodic surface patterning and the corresponding spatial modulation of the optically induced stress along the \\(x\\) -axis, coherent phonons acquire the wavevectors: k x = n 2 π/d , where \\(d=200\\) nm is the nanograting period and n = 1,2,3… is the harmonic order [32] . The phonon wavepacket propagates along the \\(x\\) -axis away from the optically excited area. There are two groups of propagating phonon modes with alternative localizations. The surface modes of two polarizations, so-called Rayleigh and Sezawa waves, with the frequencies of 12.2 and 13.8 GHz ( \\(n=1\\) ), respectively, are localized in the near-surface region. Their propagation is suppressed due to the scattering at the corrugated surface, and their amplitudes drastically decrease within the propagation distances of several microns. The second group of modes, so-called W-modes, are localized in the GaAs/AlAs sandwich, which has a higher acoustic impedance than the impedances of the substrate and the ferromagnetic cap. The W-modes are analogue to the Lamb modes in plates and behave as waveguide modes, which can propagate on large distances protected from scattering [28] , [33] . Detailed information about the spectrum of the optically generated coherent phonons is presented in Supplementary Fig. 1 . Here we focus on the W-modes, which are the main information carriers in the proposed device. The number of W-modes for a specific wavevector is determined by its ratio to the total waveguide thickness. Numerical modelling (Methods) shows that the first-order W-band ( \\(n=1\\) ) consists of 22 W-modes with frequencies between 16.5 and 18 GHz as shown by the red dashed rectangle in Fig. 2a . The spectral bands with higher order n (not shown) contain a continuously increasing number of phonon modes. Fig. 2: Physical principle of the phonon-magnon reservoir. a Calculated dependences of the frequencies of the nanograting magnon modes on magnetic field. Blue solid lines show the 6 lowest modes (forming the S-band). The external magnetic field, B , is applied in the layer plane at 45 o angle relative to the NG edge. The spectral range of the 1st order W-band is shown by the pink dashed rectangle. The vertical dashed line shows the magnetic field strength at which the spectral centres of the S- and W-bands coincide. b Calculated spatial profiles of the 22 W-modes (normalized absolute value of displacement vector, exaggerated for clarity) and 6 lowest S-modes in the ferromagnetic layer ( z -projection of the normalized magnetization). c Coordinate system and relative positions of the write and read spots in the experiments. d Noise-free readout signals measured for three values of B at 20-µm distance between the write and read spots. e Colour map showing the field dependence of the spectral amplitude obtained by a fast Fourier transform of the readout signals measured at X 0 = 20 µm. The vertical dashed line shows the magnetic field strength, at which the magnon readout demonstrates the most complex waveform. The horizontal dashed lines indicate the calculated spectral position of the guided phonon W-band. f , g Typical magnon noise-free readouts measured at various horizontal ( f ) and vertical ( g ) relative shifts of the write and read spots. Left panels show the transient signals, right panels the respective fast Fourier transforms. Time t = 0 corresponds to the moment when the input laser pulse hits the ferromagnetic layer. Full size image Due to the magneto-elastic coupling the propagating phonon wavepacket interacts with magnons in the ferromagnetic layer [34] , [35] . The magnon spectrum includes a number of spectrally close S-modes formed by spin waves spatially modulated along the \\(x\\) -axis and quantized along the normal to the layer plane ( \\(z\\) -axis) [29] , [36] . The frequencies of the magnon modes depend on the strength of the external magnetic field, \\(B\\) , applied in our experiments in the layer plane at an angle of 45 degrees relative to the direction of the nanograting grooves. Figure 2a shows the calculated frequencies of the magnon modes (blue and dashed black lines) as a function of B . For any pair of phonon and magnon modes, the efficiency of their interaction depends on their spatial and spectral matching [37] , [38] . The spatial matching is determined by the mode profiles set by the device structural design. Figure 2b shows the calculated spatial profiles of the 22 first-order W-modes and the six lowest S- modes in the ferromagnetic layer. Their interaction occurs through the uniaxial ( \\({xx}\\) ) and shear ( \\({xz}\\) ) strain components of the W-modes [37] and for any pair of the illustrated W- and S-modes the overlap integral is nonzero. The spectral matching is set by the external magnetic field. The highest interaction efficiency takes place at the resonance condition when the frequencies of the interacting modes are equal. However, due to the finite spectral widths of the S- and W- modes, their interaction is still quite efficient when the modes are detuned. Moreover, magnon-phonon interaction is intrinsically nonlinear and is accompanied by the parametric effects, which result in frequency mixing of the phonon modes [39] . As a result, the magnon readout signal, M ( t ), is characterized by a complex waveform, which is the phase-sensitive superposition of all S- and W-modes in the detection area, converted to an electric signal exploiting the polar magneto-optical Kerr effect (Methods). The noise-free waveforms (see Methods) of the magnon readouts, \\(M(t)\\) , measured at three values of B at the distance \\({X}_{0}=20\\) µm between the write and read laser spots are shown in Fig. 2d . The most complex readout is observed at \\(B=170\\) mT, which corresponds to the intersection of the W-band and the six lowest S-modes, which profiles are shown in Fig. 2b . The readout complexity manifests in the maximal broadening of its fast Fourier transform spectrum, the field dependence of which is shown in Fig. 2e . The experimental results demonstrated further correspond to these experimental conditions. Figure 2f, g demonstrate how the magnon readout \\(M(t)\\) depends on the position of the read point relative to the write point where the phonon wavepacket is generated (for the scheme, see Fig. 2c ). A tiny change in the relative positions of the write and read points, separated by a distance of more than 20 µm, results in a noticeable modification of the readout waveform and its spectrum. We have checked that the device is homogeneous, and the readout signal \\(M(t)\\) does not depend on the position at the device surface when the relative position of the input and output spots is preserved. A 125-nm step in the horizontal direction is smaller than the nanograting period (the W- and S-modes’ wavelength), and much smaller than the write and read laser wavelengths of 1050 and 780 nm as well as spots diameters of 2 and 0.6 µm, respectively. The readout waveform is sensitive to both horizontal and vertical shifts, \\(\\Delta x\\) and \\(\\Delta y\\) . The strong sensitivity of \\(M(t)\\) to the relative position is governed by the properties of the propagating phonon wavepacket. Each W-mode is characterized by its individual spatial profile, frequency, velocity, and decay rate and excited by the write laser pulse with individual initial amplitude and phase. As a result of the modes’ interference, the phonon wavepacket experiences continuous transformation during its propagation along the \\(x\\) -axis as illustrated in the Supplementary Video. Moreover, the superposition of the phonon modes varies also along the wavefront ( \\(y\\) -axis) due to the finite sizes of the write and read laser spots. This spatial-temporal transformation imprinted into the magnon S-modes results in a unique spatial, temporal, and spectral variation of the readout. Notably, it is reminiscent of the interaction of neural waves in the visual cortex, which was recognized as one of the computation mechanisms in the brain [40] . Signal encoding and recognition The sensitivity of the wave packet characteristics to the write position and the resulting high selectivity of the readout infer great efficiency of the proposed device as a neuromorphic reservoir, which processes the incoming signals that are encoded in the spatial-temporal distribution of the write laser intensity. We illustrate the reservoir function by recognizing “visual shapes” drawn by the write laser on the patterned surface. Symbols are drawn by sequential step-like changing of the relative position of the write laser spot along a selected trajectory formed from the set of the discrete positions as shown in Fig. 3 . The resulting single waveform for a specific shape is integrated over the whole trajectory (Methods). For our system, we have arbitrarily selected the following six symbols for recognition: ‘L’, ‘−’, ‘O’, ‘+’, ‘T’ and ‘Z’. The corresponding write laser trajectories and the idealized images of the symbols formed by the write laser spot are shown in Fig. 3b . Ten sets of symbols were drawn using this procedure. Figure 3c shows the exemplary readout waveforms. Fig. 3: Laser “drawing” of the “visual shapes” and their sorting. a Encoding the visual shapes by sequential relative shift of the write and read spots and the following characterization of the integrated waveform. The waveform \\({M}_{{{{{{\\rm{shape}}}}}}}\\left(t\\right)\\) is the sum of the waveforms \\({M}_{i,j}(t)\\) measured in the corresponding discrete positions forming the selected trajectory normalized to their number, \\(l\\) . The trajectory illustrated on the left corresponds to the symbol ‘L’. The respective exemplary waveform, \\({M}_{{{{{{\\rm{L}}}}}}}(t),\\) (as measured) is shown in the central frame. b Selected writing trajectories (left column) and the idealized artificially constructed images of the drawn symbols (right column). The micrographs are obtained by capturing images of the write laser spot on the nanograting surface sequentially shifted along the respective trajectory. The images with the spots forming the corresponding visual shape are overlapped in a graphic redactor. c Exemplary noise-free readout waveforms integrated over the respective trajectories. d Distribution of symbols in the parameter space spanned by the Shannon entropy, the skewness, and the variance, obtained from unfiltered magnon readouts for ten drawn trajectories for each symbol. Full size image In order to demonstrate the efficiency of our reservoir we minimize the number of characteristics used for recognition by the readout ANN. To attribute the information and statistical properties of the readout signals, we utilize only three characteristics: the Shannon entropy, \\(h\\) , the variance, σ 2 , and the skewness, \\(\\mu\\) (see Methods). Figure 3d presents the distributions of the chosen parameters in the three-dimensional parameter space ( \\(h\\) , σ 2 , \\(\\mu\\) ). Each write laser trajectory is shown by a different colour. The plot reveals the formation of clouds corresponding to the specific symbols. The distribution of the parameters within each cloud is due to the uncertainty of the read laser spot position, which is of ±200 nm along all the axes and exceeds the spatial sensitivity of the readout (Fig. 2 ). As a result, each symbol shape is randomly distorted (see Supplementary Fig. 2 for details) and, thus, some clouds partially overlap (e.g. for the symbols ‘T’ and ‘Z’). Nevertheless, the clear distinction of the parameters’ clouds confirms the reservoir efficiency in filtering and sorting the encoded “visual shapes”. To produce a large set of randomly distorted “visual shapes” for the follow training of the ANN, we use the technique of data augmentation (mixing) [41] . We have assembled the corresponding “visual shapes” from randomly selected “pieces” (i.e., the waveforms of the respective discrete positions of the corresponding trajectories) from the ten pre-measured sets (Methods). 2000 readout waveforms for each symbol have been produced. The distributions of the chosen statistical parameters for the augmented symbols are shown in Supplementary Fig. 3 . The randomized “visual shapes” form larger clouds with more significant overlap, but remain well separated. To exclude the artificial character of the parameters’ distinction, we have checked that the waveforms, which do not correspond to the selected trajectories but consist of the same number of discrete positions, do not form separate clouds (see Supplementary Fig. 4 ). Since the reservoir demonstrates high selectivity in filtering symbols, that are well separated in the parameter space, various algorithms for the symbol classification can be realized. We have chosen the conventional physical reservoir approach [5] , [10] with training the readouts. To accelerate the training process, we use a simple ANN forming a feedforward Multi-Layered Perceptron with a back-propagation algorithm [42] . The readout ANN, which is schematically shown in the inset of Fig. 4a , consisted of only 11 nodes in three layers, and the single output layer. As output, the ANN returns an encoded integer to identify the letter present: 1 = ’L’, 2 = ’-’, 3 = ’O’, 4 = ’+’, 5 = ’T’, and 6 = ’Z’. Before recognition, the ANN was trained with all six letters during, in total, 2000 epochs with a learning rate of 0.001. The ANN learning rate is the largest magnitude the parameters of the network can change within one epoch. It and the other ANN hyperparameters (the number of epochs, the early stopping patience, the random seed, the number of learning layers, and the number of nodes in each of those layers) were adjusted using a grid search approach to achieve the fastest training, but their absolute values can be varied without affecting the recognition. The evolution of the training and validation errors during training epochs are shown in Fig. 4a by blue and red colour, respectively. After ~400 epochs both training and validation errors saturate at values around 0.003, implying an accuracy of 99.7%. The recognitions of the individual symbols are summarized in the confusion matrix in Fig. 4b . The matrix shows that the symbols ’L’,’-’, and ’O’ are recognised with 100% accuracy. The symbol ‘+’ is recognised in 84.5% cases, in 13.7% cases it is confused with the symbol ‘T’, and with a probability 1.8% is recognised as the symbol ‘Z’. The symbol ‘T’ is recognized with a probability 83.6%, although in 4.2% of cases it could be confused with the symbol ‘+’ and in 12.2% cases with the symbol ‘Z’. Finally, the symbol ‘Z’ is recognised with 86.1% accuracy, while it could be confused with the symbol ‘T’ with probability 13.6%, and in one case this symbol was unclassified (reflected as symbol ‘Null’ in Fig. 4b ). The main reason for confusion relates to the similarity in coding of the symbols, see Methods, and could apparently be resolved by changing the coding protocol, i.e., the way in which the laser “writes” a symbol on the device surface. For example, if we exclude symbol ‘Z’ from the set of symbols, the ANN recognizes the remaining symbols with 100% accuracy (see Supplementary Fig. 5 ). The random waveforms, i.e. those not corresponding to the six selected “visual shapes”, are randomly attributed to one of the shapes or not recognized. Fig. 4: Recognition of the “visual shapes”. a Dependence of the validation (red curve) and training (blue curve) errors on the number of epochs for the set of symbols. The inset shows a scheme of the readout ANN. b Confusion matrices for the set of symbols. Full size image To verify the key role of the phonon-magnon interaction in the reservoir functionality, we apply the above recognition procedure to the readout signals passed through different band-stop filters (see Supplemental Fig. 5 ). If the stopband of the filter was set to 15–20 GHz, which excludes the signal components within the frequency range corresponding to the first W-band, the ANN was unable to reliably distinguish the symbols from the set. The recognition becomes completely impossible if the stop band of the filter is 0–20 GHz, which also suppresses the contribution of bulk and surface phonon modes. However, cutting the frequencies above 20 GHz with preserving the main information-capable range (0–20 GHz) significantly reduces the recognition accuracy. It indicates the role of high-frequency components of the readout waveforms such as the high-order W-band harmonics, parametrically mixed first-order W-modes and broadband noise, which could play a constructive role by promoting mode mixing and enhance nonlinearity effects [43] . The separation and visualization of the specific high-frequency contributions and noise, however, are out of the scope of the present study."},{"section":"Discussion","content":"Robustness, miniaturization, and energy costs Despite the simplified sequential encoding procedure, the reservoir demonstrates all the main properties required for efficient reservoir computing. The high sensitivity of the information and statistical characteristics of the generated readout to the relative position of the write and read laser spots allows us to realize RC without an input ANN. The high selectivity of the reservoir with respect to the superposition of phonon wavepackets generated at different ‘write’ laser positions allows recognition and classification of distorted and noisy “visual shapes”. Despite unavoidable errors in the positions of the laser spots (up to 5% of the largest linear size of the symbol) and in the measurements performed on different days, the readout signals generated by the reservoir are reliably recognized by an ANN, which evidences its high robustness. The weak temperature dependences of the elastic and magnetic parameters of the used materials around room temperature [44] , [45] , [46] makes the reservoir functionality also resistant to the unavoidable temperature variations due to background heating. The demonstrated reservoir functionality relies on the volatility of the phonon wavepacket provided by the interference of guided phonon modes. The working principle resembles the prototype concept with water waves in a bucket [11] but is realized at much higher carrier frequencies of ~10 GHz and on a drastically reduced spatial scale. The total area of the NG is 25 × 100 µm 2 , of which only 10 × 30 µm 2 are used. The area used for the symbol encoding is 3.5 × 6 µm 2 , which is comparable in size with that of a single pixel in the CMOS-sensor of a modern digital camera. The readout spot diameter is below 1 µm and the confirmed spatial resolution is ~100 nm. Such compactness is achieved in a nanodevice fabricated by state-of-the-art technology, which enables the implementation of the proposed reservoir on a chip. Moreover, the compactness is combined with large operational distances. In ambient conditions at room temperature, the information encoded into a multimode phonon wavepacket can be delivered on distances of tens of microns with protection from scattering at the surface and dephasing [28] . This facilitates a complex chip architecture and multithread reservoir computing with high information density. The demonstrated magnon-phonon information processing also has very low energy costs. The phonon wavepackets are generated by the ‘write’ laser with 50 mW averaged power at 80 MHz repetition rate. The transmission of optical excitation to coherent phonons guided along the surface has low energy efficiency, because a vast amount of the absorbed optical energy is converted to non-coherent phonons and bulk phonon modes escaping to the substrate. Each optical pulse with an energy of 6 × 10 −10 J is converted to a phonon wavepacket carrying only ~10 −16 J (see Methods for the details of estimations). This value is 5-orders of magnitude smaller than a single neural spike in the human brain, which estimated energy is ~10 pJ 8 . The averaged acoustic power involved into the signal processing is just about 10 nW. Due to these extremely low numbers, the signals are averaged over 10 10 ‘write’ pulses/phonon wavepackets at every discrete point of the selected trajectory to achieve reasonable signal-to-noise ratio. It takes about 2 min for every step of the “visual shape” drawing and, thus, limits the operational speed. However, the total energy costs remain extremely low: the single symbol drawing costs about 5 µJ and the total acoustic energy, which would be spent for the drawing and processing of the symbols in the whole training process, is less than 5 mJ. For comparison, it is two-order of magnitude less than the acoustics energy transmitted by a RF acoustic filter during 1-s operation of a modern wireless communication device [47] . This estimated value does not include the energy costs of the reservoir peripherals, which, in our experiments, is based on non-optimized research laboratory equipment. Perspectives The reservoir functionality in our experiment has been demonstrated by recognition of pseudo-visual information, but the proposed reservoir can process any type of signals converted to guided multimode phonon wavepackets. Thus, further development of the phonon-magnon reservoir should exploit an advancement of the write and read methods. One prospective direction here would be to implement the optical input as a two-dimensional, spatially distributed profile of laser intensity, enabling “instantaneous imprinting” of the information on the surface. This approach requires larger operational area than the sub-wavelength “drawing” used in our study but can be combined with parallel spatially distributed reading, which will improve the operational speed. Semiconductor mode-locked lasers whose efficiency in generation of coherent hypersonic phonon wavepackets has been recently proven [48] can provide the required miniaturisation to such an approach. Another direction is real-time operation with changing information patterns. Nowadays reservoir computing is considered as one of the best machine learning frameworks for temporal or sequential data processing [49] . The ability of the presented device to operate in the GHz frequency range promises high speed of neuromorphic operations. It requires, however, input methods with significantly higher energy efficiency. We consider a piezoelectric technique of generation of acoustic waves to be the most prospective in this context. Modern RF acoustic filters, which operate in billions of wireless communication devices, possess high (up to 90%) efficiency of electric to acoustic signal conversion [47] . The carrier frequency is currently limited by several GHz but has to be extended due to the requirements of the next (6 G) communication standards. Prototype devices already demonstrate an energy conversion efficiency of several per cent at frequencies above 10 GHz [50] , [51] . It will allow real-time operations using the proposed reservoir architecture keeping the total energy costs very low. Moreover, a two-order of magnitude increase of the strain amplitudes will drastically enhance the contribution of magnon-phonon nonlinearity [39] , which is manifested only indirectly in our study. Recent proposals indicate that wave phenomena in the brain could constitute alternative information processing, complementary to the commonly accepted mechanism of learning via developing connections between neurons [52] . For example, it has been shown that the brain can achieve selectivity function as well as stimulus detection via interference of neural waves, even if the tissue is fixed and does not accumulate information via learning [40] . In this case, the brain tissue operates as wave reservoir supporting the interaction of waves with two components—excitatory and inhibitory, reminiscent of the interacting phonons and magnons in our reservoir. With this mechanism, information processing in the visual cortex is realised via excitatory-inhibitory wave interference or mixing, even further deepening the analogy with the proposed on-chip phonon-magnon reservoir. More generally, cognitive functions could be understood in terms of spatio-temporal pattern formation [53] . Therefore, the role of standing and travelling waves in brain activities attracts significant attention [54] . The understanding of neuromorphic functions in terms of waves interaction become increasingly more important with development of quantum AI and quantum reservoir computing [55] , [56] , [57] , [58] , where waves phenomena play the prime role. All this indicates potential and possibility for a new type of neuromorphic technology—wavetronics— where elements utilise travelling and standing waves for realisation of neuromorphic functions. Our results contribute to the development of this novel technology. Summary We have designed and fabricated a hybrid semiconductor–ferromagnetic nanodevice for reservoir computing. The reservoir has optical input and output and encodes the input signals into multimode phonon wavepackets propagating in a semiconductor phonon waveguide. The neuromorphic function is realised through the interaction of the multimode input-sensitive phonon wavepackets with multiple magnon modes, which provides a high separability of the reservoir. The efficiency of the reservoir is demonstrated by reliable recognition of symbols “drawn” by the laser on the nanodevice surface in an area of several square microns, which is comparable with the size of the write laser spot. Our work paves the way for developing novel ultra-compact wave-based neuromorphic architectures on chips."},{"section":"Methods","content":"Device production The examined nanodevice was epitaxially grown on a commercial GaAs substrate [(100)-semi-insulating GaAs]. First, the phonon waveguide consisting of 10 pairs of AlAs/GaAs layers of 51-nm and 72-nm thickness, respectively, was fabricated by molecular beam epitaxy. Then, the ferromagnetic Fe 0.81 Ga 0.19 layer was deposited by magnetron sputtering. The nanogratings were fabricated through focused ion beam milling (Raith VELION) with 100-nm wide grooves and ridges over an area of 25 µm × 100 µm. To ensure high beam resolution during the patterning of the gratings a low Ga + beam current of 22 pA was applied at 35 keV beam energy. The milling dose was set to 0.3 nC/µm 2 , resulting in the groove depth of 25 nm. Optical writing and reading The scheme of signal encoding and read-out exploits two mode-locked Erbium-doped ring fibre laser oscillators (TOPTICA FemtoFiber Ultra 1050 and FemtoFiber Ultra 780). The lasers generate pulses of 150-fs duration with a repetition rate of 80 MHz at wavelengths of 1050 nm (write) and 780 nm (read). The write beam was focused by a microscope objective (×20 magnification; N.A. = 0.4) to the backside of the ferromagnetic layer, through the GaAs substrate and GaAs/AlAs phonon waveguide, which are transparent at the wavelength of 1050 nm. The focused pump spot had a Gaussian intensity distribution ~ \\(\\exp \\left(-\\frac{{r}^{2}}{2{R}^{2}}\\right)\\) , where r is the distance from the centre of the spot and R = 1 µm is the spot radius at the \\(1/\\sqrt{e}\\) level. The maximal used pulsed excitation density was 2 mJ/cm 2 . The linearly polarized readout beam was focused by another microscope objective (100× magnification; N.A. = 0.8) to the front side of the nanodevice. The focused readout spot also had a Gaussian distribution with R = 0.3 µm and an energy density of 1 mJ/cm 2 . Measurement of the read-out signals was based on the polar magneto-optical Kerr effect, i.e., the rotation of the readout pulse polarization plane by an angle proportional to the normal ( z -axis) projection of the net magnetization. The oscillations in the Kerr rotation signal reflect the precession of the magnetization [59] , which is a coherent superposition of the magnon modes driven by the phonon wavepacket [60] . The polarization rotation was detected in a differential scheme by a balanced photoreceiver with a 10-MHz bandwidth. Time resolution was achieved employing an asynchronous optical sampling (ASOPS) technique [61] . The write and read oscillators were locked with a frequency offset of 1600 Hz, which changed gradually the delay between the write and read pulses. In combination with the 80-MHz repetition rate and 10-MHz photoreceiver bandwidth, it allows measuring time-resolved signals in a time window of 12.5 ns with a time resolution of ~1 ps. The relative position of the write and read spots is controlled by two independent piezo-translators with a precision of 0.05 μm. The distance X 0 ≥ 20 µm between the write and read spots was chosen to minimize the contribution of the first-order surface phonon modes, which are intensively scattered by the patterned surface [28] . For visualization of the laser spots and control of their relative positions and focusing, a magnified (×75) image of the nanodevice surface was focused on crossed micrometre slits mounted at the intermediate focal plane between the nanodevice and a photoreceiver, where it was captured by a microscope (×6.5 magnification) with CMOS camera. To show more explicitly the volatility of the readout waveform, the readout signals presented in Figs. 2 d, f, g, and 3c were passed through a 15–20 GHz band pass filter, which eliminates the broadband noise. Sequential drawing of the visual shapes We used sequential “drawing” of visual shapes by step-by-step change of the relative position of the write and read laser spots in accordance with the selected trajectory as illustrated in Fig. 3 . In the XY coordinate frame centred at the apex of the readout spot as shown in Fig. 2 , the relative position of the write spot form a 3×3 coordinate matrix as shown in Fig. 3a . The positioning of the write and read spots for each point of the selected trajectory was automatized with the following algorithm: (i) randomized positioning of the read spot on the device surface (ii) adjusting the focusing of the read beam, (iii) adjusting the focusing of the write beam and the spatial overlap of the write and read spots, (iv) shift of the read spot to the required position. The experimentally estimated error of the automized positioning was 0.2 µm for all three-coordinate axis. The signal in every point of every trajectory was averaged over 10,000 measurements. Each trajectory was measured 20 times with averaging of the resulting waveform. 10 sets with six selected trajectories in each were measured. Drawing of randomized visual shapes For augmentation of the selected visual shapes with randomized distortion for the following recognition we measured 10 sets with nine signals in each set. Each of the nine signals corresponds to the specific relative positions of the centres of the write and read spots as described above. The algorithm of the automated positioning of the write and read spots was the same as for the sequential drawing with the same positioning errors. The waveform for each point was averaged over 200,000 measurements. The 10 sets were measured on three experimental days (4 sets in the first day, and 3 sets each on the second and third days) with a complete restart of the experimental setup. The waveforms for recognition were obtained by random summation of the corresponding signals from all 10 sets. The recognized shapes were obtained by the following normalized summations: $${}^{\\prime\\prime} {{{{{\\rm{L}}}}}}^{\\prime\\prime} \\,=\t\\, ({{M}}_{1,1}+{{M}}_{2,1}+{{M}}_{3,1}+{M}_{3,2}+{M}_{3,3})/5;\\\\ {}^{\\prime\\prime} -^{\\prime\\prime}=\t\\, ({{M}}_{2,1}+{M}_{2,2}+{M}_{2,3})/3\\\\ {}^{\\prime\\prime} {{{{{\\rm{O}}}}}}^{\\prime\\prime} \\,=\t\\, ({{M}}_{1,1}+{{M}}_{1,2}+{{M}}_{1,3}+{M}_{2,1}+{M}_{2,3}+{M}_{3,1}+{M}_{3,2}+{M}_{3,3})/8;\\\\ {}^{\\prime\\prime}+^{\\prime\\prime} \\,=\t\\, ({{M}}_{1,2}+{{M}}_{2,1}+{{M}}_{2,2}+{M}_{2,3}+{M}_{3,2})/5;\\\\ {}^{\\prime\\prime} {{{{{\\rm{T}}}}}}^{\\prime\\prime} \\,=\t\\, ({{M}}_{1,1}+{{M}}_{1,2}+{{M}}_{1,3}+{M}_{2,2}+{M}_{3,2})/5;\\\\ {}^{\\prime\\prime} {{{{{\\rm{Z}}}}}}^{\\prime\\prime} \\,=\t\\, ({{M}}_{1,1}+{{M}}_{1,2}+{{M}}_{1,3}+{M}_{2,2}+{M}_{3,1}+{M}_{3,2}+{M}_{3,3})/7.$$ (1) 2000 waveforms for each symbol were randomly generated (the randomized set for “-”, which consists of only 1000 unique waveforms, included 1000 additional randomly repeated waveforms). Only the integrated waveforms were processed. The separate waveforms \\({M}_{i,j}(t)\\) measured in the corresponding positions of the write laser beam were neither processed nor used to support the following recognition. Waveform characterization For characterisation of the readout signals we use the following quantities: the Shannon entropy, \\(h\\) , the variance, σ 2 , and the skewness, \\(\\mu\\) . The signals were processed as measured, without filtering and subtracting the dc-background of the photoreceiver. To calculate \\(h\\) , we split the interval between the minimal and maximal voltage of the readout signals (−4.02 mV and −3.05 mV, respectively) into J = 10 5 bins, and then apply the formula $$h=-\\mathop{\\sum }\\limits_{j=1}^{J}{p}_{j}{{{{\\mathrm{ln}}}}}{p}_{j},$$ (2) where \\({p}_{j}\\) is the probability that the readout voltage has a value within the j th bin. For calculating the statistical characteristics, we utilised the formulas below: $${\\sigma }^{2}=\\frac{1}{N}\\mathop{\\sum }\\limits_{i=1}^{N}{({V}_{i}-\\bar{V})}^{2},$$ (3) $$\\bar{V}=\\frac{1}{N}\\mathop{\\sum }\\limits_{i=1}^{N}{V}_{i},$$ (4) $$\\mu=\\frac{1}{N{\\sigma }^{3}}\\mathop{\\sum }\\limits_{i=1}^{N}{({V}_{i}-\\bar{V})}^{3},$$ (5) where \\({V}_{i}\\) is the value of the discretised in time readout voltage at the i th time step, and \\(N\\) is the total number of time steps. Artificial neural network For processing the properties of the signals to classify the letters from the device, we employ a simple Artificial Neural Network (ANN), illustrated in the inset of Fig. 4a . The network is made of the input layer (where the properties are fed into the network) with 3 nodes (one for each property of the signal), then we have the first learning layer made up of 3 nodes, the second learning layer with 5 nodes, and the final learning layer of 3 nodes, with one final output layer. The output layer gives the classification of the input signal. Each of our layers is connected with the Rectified Linear Unit (ReLU) activation function, apart from the final learning layer connected to the output layer, which is a linear activation function. The network is coded within the Python module PyTorch [62] . We use the Adam optimiser [63] to train the network, with a Root Mean Squared Error (RMSE) loss. The network optimises its weights and biases by backpropagating the error through its connections and adjusting the values of the network’s parameters. The adjustments are made on the order of the learning rate, here taken to be 0.001. We allow the network to train for 2000 epochs. However, employing an early stopping procedure, we stop the network from overfitting the training dataset. This is further achieved by splitting the dataset of letters into non-overlapping training, validating, and testing sets, with a standard 60, 20, 20% split [64] . Numerical modelling The S- and W-modes’ profiles and the excitation and propagation of phonon wavepackets were modelled using COMSOL Multiphysics® software. The spatial profiles and dispersions of the S-modes were obtained by solution of the Landau-Lifshitz and Maxwell equations in the frequency domain [36] . The following parameters of Fe 0.81 Ga 0.19 were used for the calculations: saturation magnetization, \\({{\\mu }_{0}M}_{s}\\) = 1.76 T, exchange stiffness, D = 2 \\({{{{{\\rm{\\cdot }}}}}}\\) 10 −17 T m 2 , cubic and uniaxial anisotropy coefficients of 23 mT and 11 mT, respectively. For the calculation of the W-modes spatial profiles and spatial-temporal evolution [28] , we used the following stiffness tensor components, \\({C}_{{kl}}\\) , and the mass density, \\(\\rho\\) : for Fe 0.81 Ga 0.19 \\({C}_{11}=209\\) GPa, \\({C}_{12}=156\\) GPa, \\({C}_{44}=122\\) GPa, \\(\\rho=7800\\) kg/m 3 ; for GaAs \\({C}_{11}=119\\) GPa, \\({C}_{12}=53.8\\) GPa, \\({C}_{44}=59.5\\) GPa, \\(\\rho=5316\\) kg/m 3 ; for AlAs \\({C}_{11}=119.9\\) GPa, \\({C}_{12}=57.5\\) GPa, \\({C}_{44}=56.6\\) GPa, \\(\\rho=3760\\) kg/m 3 . We used a model nanograting structure with 150 periods (i.e. 30 µm long in \\(x\\) direction). The structure was limited in \\(z\\) direction by the size of 1.9 µm (500 nm below the phonon waveguide), and the low-reflecting boundary condition was applied to the bottom boundary to minimize unwanted reflections. Free boundary conditions were applied to the top and side boundaries of the structure. The structure was considered uniform in \\({y}\\) direction. The excitation of the structure by the optical pulse creates thermal stress which was simulated using a two-temperature model for the lattice temperatures of the ferromagnetic cap and phonon waveguide [65] . Then, the calculated thermal stress was used as the input data for the acoustic modelling. The calculations were carried out in the time domain up to 6 ns. The results of the simulation are presented in the Supplemental Video and in Fig. 2 . The energy of a single W-mode wavepacket was estimated as \\(U=\\frac{1}{2}v\\left\\langle {c}_{11}\\right\\rangle \\left\\langle {\\eta }^{2}\\right\\rangle \\approx {5\\cdot 10}^{-17}\\) J, where \\(v=\\pi {R}^{2}h\\) is the total volume of the wavepacket ( \\(R=1\\) µm is the write laser spot radius and \\(h=1.3\\) µm is the phonon waveguide thickness, \\(\\left\\langle {c}_{11}\\right\\rangle=119.5\\) GPa is the averaged stiffness constant of the GaAs/AlAs waveguide and \\(\\left\\langle {\\eta }^{2}\\right\\rangle \\approx 2\\times {10}^{-10}\\) is the averaged square of the strain amplitude."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"The flourishing idea of using artificial neural networks (ANNs) for the solution of practical tasks shifts the computing paradigm towards heavy usage of data instead of creating lengthy instructions and logical chains forming algorithms. Being inspired by biological neural systems, an ANN constitutes a set of many interconnected non-linear functional units called ‘neurons’. To solve a task with the desired accuracy, an ANN requires training by data processing to fit the weights of the connections between neurons. Over the decades, ANNs have demonstrated their high efficiency in a wide range of important practical applications [1] including voice, image, and pattern recognition, data mining, prediction of complex dynamics, 3D reconstruction, and medical diagnostics. The more sophisticated the problem an ANN has to solve, the larger is the network required to achieve the desirable accuracy of the solution. Simulation of large networks using conventional digital computers demands enormous CPU/GPU power, memory, and energy supply [2] . A promising approach to resolve the hardware issues is neuromorphic computing, which implements ANNs in circuits [3] . Such hardware architecture should dramatically improve the transfer of data across the circuit during data processing, thus enhancing both the speed and energy efficiency of calculations. Another common drawback of ANNs is the significant time required for their training to optimize performance [4] . ANNs are most efficient if the type of task (‘the goal’) is fixed, and there is enough data for training. However, they experience significant difficulties if the amount of data is limited, or the goal varies in time [5] . Smaller networks would partly resolve this problem since fewer neurons and connections need less data and time for training, but this approach suffers from poor accuracy. A promising way to resolve this problem is the concept of ‘reservoir computing’ (RC). The ‘reservoir’ is a recurrent ANN with non-linear dynamics and fixed connections, which typically does not require any training, attached to a small readout ANN, which is trained to recognize predefined characteristics of the reservoir output. In such an architecture, the reservoir serves to ‘highlight’ important features in data sets and suppress irrelevant ones, thereby enormously improving the performance of computing. As one of many natural prototypes of RC, one could consider the human vision schematically illustrated in Fig. 1a . The visual information passed through the cornea and focused by the lens is converted by the retina photoreceptors into electrical signals (the neural pulses) that are transmitted by the optic nerve and mixed in a high-dimensional space of the visual cortex for the subsequent recognition. The complex photochemical transformation (phototransduction) by the 10 8 photoreceptors allows our brain to recognize objects, distinguish tiny differences between them, and detect thereby their motion. Such a resemblance between biological sensory systems and reservoir ANNs has invigorated research on RC-based artificial vision [6] , and other RC-systems for detection and recognition [5] , [7] . Fig. 1: Nanodevice architecture and working principle. a Schematic of human vision: an image focused by the lens onto the retina is transformed into electrical pulses that are transferred by the optic nerve to the brain for the subsequent recognition. b Design of the examined nanostructure. c Scanning electron microscopy (SEM) images (top and bottom panels) of the nanodevice surface with the nanograting (NG): complete NG with a size of 25 × 100 µm 2 (top panel), and its close-up with the visible pattern (bottom panel). The inset in the right-top corner is a microscopy image of the write (large spot) and read (small spot) laser spots on the NG surface. The size of the framed area is 10 × 10 µm 2 . d Working principle of the phonon reservoir with magnon readout. A multimode phonon wavepacket is generated by ultrafast laser excitation of the patterned ferromagnetic nanolayer. It propagates well protected from external perturbations beneath the surface in the GaAs/AlAs phonon waveguide and its continuously transformed waveform is imprinted in the magnon modes of the ferromagnetic layer. The superposition of the magnon modes driven by the phonon wavepacket at a specific distance from the write spot is detected using the read laser pulse. Full size image From the viewpoint of information theory, the reservoir maps input signals into higher dimensional information spaces and can be realized by continuous in space and/or in time nonlinear dynamical systems with storage ability. Therefore, various physical systems have been suggested for the reservoir [5] , [8] , [9] , [10] . One of the first realizations was based on waves, namely gravity waves in a vessel with water to recognize spoken numbers [11] . Nowadays, wave ‘reservoirs’ have been implemented using different physical waves such as electromagnetic waves (photons) [12] , [13] , [14] , [15] , [16] , [17] , [18] , elastic waves (phonons) [19] , [20] , and spin waves (magnons) [21] , [22] , [23] , [24] . The key factors that make waves reliable for reservoir architectures are the possibility to achieve wave packets with a vast information density; weighted summation by interference; nonlinearity resulting in wave mixing and generation of higher harmonics; hybridization of waves of different types, e.g., the formation of polarons. Recently, wave-based ‘reservoirs’ have demonstrated the ability for deep learning [25] , [26] , [27] . Despite rapid progress in the development of RC-systems, their implementation in chips suitable for practical technological applications has remained an unsolved problem. Among the major challenges with existing neuromorphic platforms [3] are power efficiency, miniaturization, separability, and robustness [5] , [8] , [10] . On-chip wave-based reservoirs have potential to address these challenges, since they allow smaller devices, good scalability, great state numbers, and fine control of interacting wave modes. However, their development calls for novel materials and designs allowing the generation of waves for the reservoir function.\",\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 16,\n 17,\n 18,\n 19,\n 20,\n 21,\n 22,\n 23,\n 24,\n 25,\n 26,\n 27\n ],\n \"infos\": [\n {\n \"idx\": \"1\",\n \"title\": \"Le Cun, Y., Bengio, Y. & Hinton, G. 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Deep learning with coherent nanophotonic circuits.Nat. Photon11, 441–446 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fnphoton.2017.93\"\n },\n {\n \"idx\": \"26\",\n \"title\": \"Hughes, T. W., Williamson, I. A., Minkov, M. & Fan, S. Wave physics as an analog recurrent neural network.Sci. Adv.5, eaay6946 (2019).\",\n \"link\": \"https://doi.org/10.1126%2Fsciadv.aay6946\"\n },\n {\n \"idx\": \"27\",\n \"title\": \"Wright, L. G. et al. Deep physical neural networks trained with backpropagation.Nature601, 549–555 (2022).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-021-04223-6\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Hybrid nanostructure for phonon–magnon reservoir Here, we propose and demonstrate experimentally a new type of wave reservoir based on high-frequency acoustic and spin waves in a physical system that combines ultimate compactness, robustness, and separability. For this aim, we fabricate a nanodevice with an architecture implemented with available nanotechnology. The device, shown schematically in Fig. 1b , consists of a sandwich of GaAs/AlAs semiconductor layers, which form a waveguide for high-frequency acoustic waves, i.e., phonons [28] . The waveguide is capped by a nanograting patterned in a metal ferromagnetic Fe 0.81 Ga 0.19 layer hosting spin waves, i.e. magnons [29] . Figure 1c shows the scanning electron microscopy image of the patterned surface. We examine the performance of this device by recognition of the “visual shapes” drawn on the surface by a pulsed ‘write’ laser. The laser pulses are converted by the nanograting to a coherent multimode phonon wavepacket, which is guided along the surface and experiences continuous transformation due to the interference of the phonon modes. Phonons interact with magnons and this multimode magnon-phonon mixture forms the reservoir multidimensional space. The reservoir output is read through magnons by measuring the reflectivity of another ‘read’ laser beam. The output waveform is integrated over the drawn trajectory corresponding to a specific “shape” for the following recognition by an ANN programmed in a personal computer. We may compare this procedure with handwriting and conceptualize the reservoir operations as recognition of a handwritten character. Thus, in analogy with the human vision (Fig. 1d ), we may consider the patterned surface as the retina, which converts optical input to neural pulses (coherent phonons), processed by a multidimensional reservoir space (phonon-magnon mixture) to generate a readout recognized by the brain, namely the ANN. The sub-µm wavelength of phonons and magnons allows achieving a large number of “receptors” per area (‘’receptor” density) and make the readout sensitive to a tiny change of the optical input. It enables recognition of the symbols drawn on the scale of just several wavelengths of the ‘write’ laser. The physical processes involved in the operation of the fabricated nanodevice are demonstrated in Fig. 2 . The 100-fs pulse of the write laser, focused on the metallic ferromagnetic layer to a spot of 2-µm diameter instantaneously induces thermal stress leading to the generation of the coherent acoustic wavepacket of hypersound (up to ~100 GHz) frequencies [30] , [31] . Due to the periodic surface patterning and the corresponding spatial modulation of the optically induced stress along the \\\\(x\\\\) -axis, coherent phonons acquire the wavevectors: k x = n 2 π/d , where \\\\(d=200\\\\) nm is the nanograting period and n = 1,2,3… is the harmonic order [32] . The phonon wavepacket propagates along the \\\\(x\\\\) -axis away from the optically excited area. There are two groups of propagating phonon modes with alternative localizations. The surface modes of two polarizations, so-called Rayleigh and Sezawa waves, with the frequencies of 12.2 and 13.8 GHz ( \\\\(n=1\\\\) ), respectively, are localized in the near-surface region. Their propagation is suppressed due to the scattering at the corrugated surface, and their amplitudes drastically decrease within the propagation distances of several microns. The second group of modes, so-called W-modes, are localized in the GaAs/AlAs sandwich, which has a higher acoustic impedance than the impedances of the substrate and the ferromagnetic cap. The W-modes are analogue to the Lamb modes in plates and behave as waveguide modes, which can propagate on large distances protected from scattering [28] , [33] . Detailed information about the spectrum of the optically generated coherent phonons is presented in Supplementary Fig. 1 . Here we focus on the W-modes, which are the main information carriers in the proposed device. The number of W-modes for a specific wavevector is determined by its ratio to the total waveguide thickness. Numerical modelling (Methods) shows that the first-order W-band ( \\\\(n=1\\\\) ) consists of 22 W-modes with frequencies between 16.5 and 18 GHz as shown by the red dashed rectangle in Fig. 2a . The spectral bands with higher order n (not shown) contain a continuously increasing number of phonon modes. Fig. 2: Physical principle of the phonon-magnon reservoir. a Calculated dependences of the frequencies of the nanograting magnon modes on magnetic field. Blue solid lines show the 6 lowest modes (forming the S-band). The external magnetic field, B , is applied in the layer plane at 45 o angle relative to the NG edge. The spectral range of the 1st order W-band is shown by the pink dashed rectangle. The vertical dashed line shows the magnetic field strength at which the spectral centres of the S- and W-bands coincide. b Calculated spatial profiles of the 22 W-modes (normalized absolute value of displacement vector, exaggerated for clarity) and 6 lowest S-modes in the ferromagnetic layer ( z -projection of the normalized magnetization). c Coordinate system and relative positions of the write and read spots in the experiments. d Noise-free readout signals measured for three values of B at 20-µm distance between the write and read spots. e Colour map showing the field dependence of the spectral amplitude obtained by a fast Fourier transform of the readout signals measured at X 0 = 20 µm. The vertical dashed line shows the magnetic field strength, at which the magnon readout demonstrates the most complex waveform. The horizontal dashed lines indicate the calculated spectral position of the guided phonon W-band. f , g Typical magnon noise-free readouts measured at various horizontal ( f ) and vertical ( g ) relative shifts of the write and read spots. Left panels show the transient signals, right panels the respective fast Fourier transforms. Time t = 0 corresponds to the moment when the input laser pulse hits the ferromagnetic layer. Full size image Due to the magneto-elastic coupling the propagating phonon wavepacket interacts with magnons in the ferromagnetic layer [34] , [35] . The magnon spectrum includes a number of spectrally close S-modes formed by spin waves spatially modulated along the \\\\(x\\\\) -axis and quantized along the normal to the layer plane ( \\\\(z\\\\) -axis) [29] , [36] . The frequencies of the magnon modes depend on the strength of the external magnetic field, \\\\(B\\\\) , applied in our experiments in the layer plane at an angle of 45 degrees relative to the direction of the nanograting grooves. Figure 2a shows the calculated frequencies of the magnon modes (blue and dashed black lines) as a function of B . For any pair of phonon and magnon modes, the efficiency of their interaction depends on their spatial and spectral matching [37] , [38] . The spatial matching is determined by the mode profiles set by the device structural design. Figure 2b shows the calculated spatial profiles of the 22 first-order W-modes and the six lowest S- modes in the ferromagnetic layer. Their interaction occurs through the uniaxial ( \\\\({xx}\\\\) ) and shear ( \\\\({xz}\\\\) ) strain components of the W-modes [37] and for any pair of the illustrated W- and S-modes the overlap integral is nonzero. The spectral matching is set by the external magnetic field. The highest interaction efficiency takes place at the resonance condition when the frequencies of the interacting modes are equal. However, due to the finite spectral widths of the S- and W- modes, their interaction is still quite efficient when the modes are detuned. Moreover, magnon-phonon interaction is intrinsically nonlinear and is accompanied by the parametric effects, which result in frequency mixing of the phonon modes [39] . As a result, the magnon readout signal, M ( t ), is characterized by a complex waveform, which is the phase-sensitive superposition of all S- and W-modes in the detection area, converted to an electric signal exploiting the polar magneto-optical Kerr effect (Methods). The noise-free waveforms (see Methods) of the magnon readouts, \\\\(M(t)\\\\) , measured at three values of B at the distance \\\\({X}_{0}=20\\\\) µm between the write and read laser spots are shown in Fig. 2d . The most complex readout is observed at \\\\(B=170\\\\) mT, which corresponds to the intersection of the W-band and the six lowest S-modes, which profiles are shown in Fig. 2b . The readout complexity manifests in the maximal broadening of its fast Fourier transform spectrum, the field dependence of which is shown in Fig. 2e . The experimental results demonstrated further correspond to these experimental conditions. Figure 2f, g demonstrate how the magnon readout \\\\(M(t)\\\\) depends on the position of the read point relative to the write point where the phonon wavepacket is generated (for the scheme, see Fig. 2c ). A tiny change in the relative positions of the write and read points, separated by a distance of more than 20 µm, results in a noticeable modification of the readout waveform and its spectrum. We have checked that the device is homogeneous, and the readout signal \\\\(M(t)\\\\) does not depend on the position at the device surface when the relative position of the input and output spots is preserved. A 125-nm step in the horizontal direction is smaller than the nanograting period (the W- and S-modes’ wavelength), and much smaller than the write and read laser wavelengths of 1050 and 780 nm as well as spots diameters of 2 and 0.6 µm, respectively. The readout waveform is sensitive to both horizontal and vertical shifts, \\\\(\\\\Delta x\\\\) and \\\\(\\\\Delta y\\\\) . The strong sensitivity of \\\\(M(t)\\\\) to the relative position is governed by the properties of the propagating phonon wavepacket. Each W-mode is characterized by its individual spatial profile, frequency, velocity, and decay rate and excited by the write laser pulse with individual initial amplitude and phase. As a result of the modes’ interference, the phonon wavepacket experiences continuous transformation during its propagation along the \\\\(x\\\\) -axis as illustrated in the Supplementary Video. Moreover, the superposition of the phonon modes varies also along the wavefront ( \\\\(y\\\\) -axis) due to the finite sizes of the write and read laser spots. This spatial-temporal transformation imprinted into the magnon S-modes results in a unique spatial, temporal, and spectral variation of the readout. Notably, it is reminiscent of the interaction of neural waves in the visual cortex, which was recognized as one of the computation mechanisms in the brain [40] . Signal encoding and recognition The sensitivity of the wave packet characteristics to the write position and the resulting high selectivity of the readout infer great efficiency of the proposed device as a neuromorphic reservoir, which processes the incoming signals that are encoded in the spatial-temporal distribution of the write laser intensity. We illustrate the reservoir function by recognizing “visual shapes” drawn by the write laser on the patterned surface. Symbols are drawn by sequential step-like changing of the relative position of the write laser spot along a selected trajectory formed from the set of the discrete positions as shown in Fig. 3 . The resulting single waveform for a specific shape is integrated over the whole trajectory (Methods). For our system, we have arbitrarily selected the following six symbols for recognition: ‘L’, ‘−’, ‘O’, ‘+’, ‘T’ and ‘Z’. The corresponding write laser trajectories and the idealized images of the symbols formed by the write laser spot are shown in Fig. 3b . Ten sets of symbols were drawn using this procedure. Figure 3c shows the exemplary readout waveforms. Fig. 3: Laser “drawing” of the “visual shapes” and their sorting. a Encoding the visual shapes by sequential relative shift of the write and read spots and the following characterization of the integrated waveform. The waveform \\\\({M}_{{{{{{\\\\rm{shape}}}}}}}\\\\left(t\\\\right)\\\\) is the sum of the waveforms \\\\({M}_{i,j}(t)\\\\) measured in the corresponding discrete positions forming the selected trajectory normalized to their number, \\\\(l\\\\) . The trajectory illustrated on the left corresponds to the symbol ‘L’. The respective exemplary waveform, \\\\({M}_{{{{{{\\\\rm{L}}}}}}}(t),\\\\) (as measured) is shown in the central frame. b Selected writing trajectories (left column) and the idealized artificially constructed images of the drawn symbols (right column). The micrographs are obtained by capturing images of the write laser spot on the nanograting surface sequentially shifted along the respective trajectory. The images with the spots forming the corresponding visual shape are overlapped in a graphic redactor. c Exemplary noise-free readout waveforms integrated over the respective trajectories. d Distribution of symbols in the parameter space spanned by the Shannon entropy, the skewness, and the variance, obtained from unfiltered magnon readouts for ten drawn trajectories for each symbol. Full size image In order to demonstrate the efficiency of our reservoir we minimize the number of characteristics used for recognition by the readout ANN. To attribute the information and statistical properties of the readout signals, we utilize only three characteristics: the Shannon entropy, \\\\(h\\\\) , the variance, σ 2 , and the skewness, \\\\(\\\\mu\\\\) (see Methods). Figure 3d presents the distributions of the chosen parameters in the three-dimensional parameter space ( \\\\(h\\\\) , σ 2 , \\\\(\\\\mu\\\\) ). Each write laser trajectory is shown by a different colour. The plot reveals the formation of clouds corresponding to the specific symbols. The distribution of the parameters within each cloud is due to the uncertainty of the read laser spot position, which is of ±200 nm along all the axes and exceeds the spatial sensitivity of the readout (Fig. 2 ). As a result, each symbol shape is randomly distorted (see Supplementary Fig. 2 for details) and, thus, some clouds partially overlap (e.g. for the symbols ‘T’ and ‘Z’). Nevertheless, the clear distinction of the parameters’ clouds confirms the reservoir efficiency in filtering and sorting the encoded “visual shapes”. To produce a large set of randomly distorted “visual shapes” for the follow training of the ANN, we use the technique of data augmentation (mixing) [41] . We have assembled the corresponding “visual shapes” from randomly selected “pieces” (i.e., the waveforms of the respective discrete positions of the corresponding trajectories) from the ten pre-measured sets (Methods). 2000 readout waveforms for each symbol have been produced. The distributions of the chosen statistical parameters for the augmented symbols are shown in Supplementary Fig. 3 . The randomized “visual shapes” form larger clouds with more significant overlap, but remain well separated. To exclude the artificial character of the parameters’ distinction, we have checked that the waveforms, which do not correspond to the selected trajectories but consist of the same number of discrete positions, do not form separate clouds (see Supplementary Fig. 4 ). Since the reservoir demonstrates high selectivity in filtering symbols, that are well separated in the parameter space, various algorithms for the symbol classification can be realized. We have chosen the conventional physical reservoir approach [5] , [10] with training the readouts. To accelerate the training process, we use a simple ANN forming a feedforward Multi-Layered Perceptron with a back-propagation algorithm [42] . The readout ANN, which is schematically shown in the inset of Fig. 4a , consisted of only 11 nodes in three layers, and the single output layer. As output, the ANN returns an encoded integer to identify the letter present: 1 = ’L’, 2 = ’-’, 3 = ’O’, 4 = ’+’, 5 = ’T’, and 6 = ’Z’. Before recognition, the ANN was trained with all six letters during, in total, 2000 epochs with a learning rate of 0.001. The ANN learning rate is the largest magnitude the parameters of the network can change within one epoch. It and the other ANN hyperparameters (the number of epochs, the early stopping patience, the random seed, the number of learning layers, and the number of nodes in each of those layers) were adjusted using a grid search approach to achieve the fastest training, but their absolute values can be varied without affecting the recognition. The evolution of the training and validation errors during training epochs are shown in Fig. 4a by blue and red colour, respectively. After ~400 epochs both training and validation errors saturate at values around 0.003, implying an accuracy of 99.7%. The recognitions of the individual symbols are summarized in the confusion matrix in Fig. 4b . The matrix shows that the symbols ’L’,’-’, and ’O’ are recognised with 100% accuracy. The symbol ‘+’ is recognised in 84.5% cases, in 13.7% cases it is confused with the symbol ‘T’, and with a probability 1.8% is recognised as the symbol ‘Z’. The symbol ‘T’ is recognized with a probability 83.6%, although in 4.2% of cases it could be confused with the symbol ‘+’ and in 12.2% cases with the symbol ‘Z’. Finally, the symbol ‘Z’ is recognised with 86.1% accuracy, while it could be confused with the symbol ‘T’ with probability 13.6%, and in one case this symbol was unclassified (reflected as symbol ‘Null’ in Fig. 4b ). The main reason for confusion relates to the similarity in coding of the symbols, see Methods, and could apparently be resolved by changing the coding protocol, i.e., the way in which the laser “writes” a symbol on the device surface. For example, if we exclude symbol ‘Z’ from the set of symbols, the ANN recognizes the remaining symbols with 100% accuracy (see Supplementary Fig. 5 ). The random waveforms, i.e. those not corresponding to the six selected “visual shapes”, are randomly attributed to one of the shapes or not recognized. Fig. 4: Recognition of the “visual shapes”. a Dependence of the validation (red curve) and training (blue curve) errors on the number of epochs for the set of symbols. The inset shows a scheme of the readout ANN. b Confusion matrices for the set of symbols. Full size image To verify the key role of the phonon-magnon interaction in the reservoir functionality, we apply the above recognition procedure to the readout signals passed through different band-stop filters (see Supplemental Fig. 5 ). If the stopband of the filter was set to 15–20 GHz, which excludes the signal components within the frequency range corresponding to the first W-band, the ANN was unable to reliably distinguish the symbols from the set. The recognition becomes completely impossible if the stop band of the filter is 0–20 GHz, which also suppresses the contribution of bulk and surface phonon modes. However, cutting the frequencies above 20 GHz with preserving the main information-capable range (0–20 GHz) significantly reduces the recognition accuracy. It indicates the role of high-frequency components of the readout waveforms such as the high-order W-band harmonics, parametrically mixed first-order W-modes and broadband noise, which could play a constructive role by promoting mode mixing and enhance nonlinearity effects [43] . The separation and visualization of the specific high-frequency contributions and noise, however, are out of the scope of the present study.\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"Robustness, miniaturization, and energy costs Despite the simplified sequential encoding procedure, the reservoir demonstrates all the main properties required for efficient reservoir computing. The high sensitivity of the information and statistical characteristics of the generated readout to the relative position of the write and read laser spots allows us to realize RC without an input ANN. The high selectivity of the reservoir with respect to the superposition of phonon wavepackets generated at different ‘write’ laser positions allows recognition and classification of distorted and noisy “visual shapes”. Despite unavoidable errors in the positions of the laser spots (up to 5% of the largest linear size of the symbol) and in the measurements performed on different days, the readout signals generated by the reservoir are reliably recognized by an ANN, which evidences its high robustness. The weak temperature dependences of the elastic and magnetic parameters of the used materials around room temperature [44] , [45] , [46] makes the reservoir functionality also resistant to the unavoidable temperature variations due to background heating. The demonstrated reservoir functionality relies on the volatility of the phonon wavepacket provided by the interference of guided phonon modes. The working principle resembles the prototype concept with water waves in a bucket [11] but is realized at much higher carrier frequencies of ~10 GHz and on a drastically reduced spatial scale. The total area of the NG is 25 × 100 µm 2 , of which only 10 × 30 µm 2 are used. The area used for the symbol encoding is 3.5 × 6 µm 2 , which is comparable in size with that of a single pixel in the CMOS-sensor of a modern digital camera. The readout spot diameter is below 1 µm and the confirmed spatial resolution is ~100 nm. Such compactness is achieved in a nanodevice fabricated by state-of-the-art technology, which enables the implementation of the proposed reservoir on a chip. Moreover, the compactness is combined with large operational distances. In ambient conditions at room temperature, the information encoded into a multimode phonon wavepacket can be delivered on distances of tens of microns with protection from scattering at the surface and dephasing [28] . This facilitates a complex chip architecture and multithread reservoir computing with high information density. The demonstrated magnon-phonon information processing also has very low energy costs. The phonon wavepackets are generated by the ‘write’ laser with 50 mW averaged power at 80 MHz repetition rate. The transmission of optical excitation to coherent phonons guided along the surface has low energy efficiency, because a vast amount of the absorbed optical energy is converted to non-coherent phonons and bulk phonon modes escaping to the substrate. Each optical pulse with an energy of 6 × 10 −10 J is converted to a phonon wavepacket carrying only ~10 −16 J (see Methods for the details of estimations). This value is 5-orders of magnitude smaller than a single neural spike in the human brain, which estimated energy is ~10 pJ 8 . The averaged acoustic power involved into the signal processing is just about 10 nW. Due to these extremely low numbers, the signals are averaged over 10 10 ‘write’ pulses/phonon wavepackets at every discrete point of the selected trajectory to achieve reasonable signal-to-noise ratio. It takes about 2 min for every step of the “visual shape” drawing and, thus, limits the operational speed. However, the total energy costs remain extremely low: the single symbol drawing costs about 5 µJ and the total acoustic energy, which would be spent for the drawing and processing of the symbols in the whole training process, is less than 5 mJ. For comparison, it is two-order of magnitude less than the acoustics energy transmitted by a RF acoustic filter during 1-s operation of a modern wireless communication device [47] . This estimated value does not include the energy costs of the reservoir peripherals, which, in our experiments, is based on non-optimized research laboratory equipment. Perspectives The reservoir functionality in our experiment has been demonstrated by recognition of pseudo-visual information, but the proposed reservoir can process any type of signals converted to guided multimode phonon wavepackets. Thus, further development of the phonon-magnon reservoir should exploit an advancement of the write and read methods. One prospective direction here would be to implement the optical input as a two-dimensional, spatially distributed profile of laser intensity, enabling “instantaneous imprinting” of the information on the surface. This approach requires larger operational area than the sub-wavelength “drawing” used in our study but can be combined with parallel spatially distributed reading, which will improve the operational speed. Semiconductor mode-locked lasers whose efficiency in generation of coherent hypersonic phonon wavepackets has been recently proven [48] can provide the required miniaturisation to such an approach. Another direction is real-time operation with changing information patterns. Nowadays reservoir computing is considered as one of the best machine learning frameworks for temporal or sequential data processing [49] . The ability of the presented device to operate in the GHz frequency range promises high speed of neuromorphic operations. It requires, however, input methods with significantly higher energy efficiency. We consider a piezoelectric technique of generation of acoustic waves to be the most prospective in this context. Modern RF acoustic filters, which operate in billions of wireless communication devices, possess high (up to 90%) efficiency of electric to acoustic signal conversion [47] . The carrier frequency is currently limited by several GHz but has to be extended due to the requirements of the next (6 G) communication standards. Prototype devices already demonstrate an energy conversion efficiency of several per cent at frequencies above 10 GHz [50] , [51] . It will allow real-time operations using the proposed reservoir architecture keeping the total energy costs very low. Moreover, a two-order of magnitude increase of the strain amplitudes will drastically enhance the contribution of magnon-phonon nonlinearity [39] , which is manifested only indirectly in our study. Recent proposals indicate that wave phenomena in the brain could constitute alternative information processing, complementary to the commonly accepted mechanism of learning via developing connections between neurons [52] . For example, it has been shown that the brain can achieve selectivity function as well as stimulus detection via interference of neural waves, even if the tissue is fixed and does not accumulate information via learning [40] . In this case, the brain tissue operates as wave reservoir supporting the interaction of waves with two components—excitatory and inhibitory, reminiscent of the interacting phonons and magnons in our reservoir. With this mechanism, information processing in the visual cortex is realised via excitatory-inhibitory wave interference or mixing, even further deepening the analogy with the proposed on-chip phonon-magnon reservoir. More generally, cognitive functions could be understood in terms of spatio-temporal pattern formation [53] . Therefore, the role of standing and travelling waves in brain activities attracts significant attention [54] . The understanding of neuromorphic functions in terms of waves interaction become increasingly more important with development of quantum AI and quantum reservoir computing [55] , [56] , [57] , [58] , where waves phenomena play the prime role. All this indicates potential and possibility for a new type of neuromorphic technology—wavetronics— where elements utilise travelling and standing waves for realisation of neuromorphic functions. Our results contribute to the development of this novel technology. Summary We have designed and fabricated a hybrid semiconductor–ferromagnetic nanodevice for reservoir computing. The reservoir has optical input and output and encodes the input signals into multimode phonon wavepackets propagating in a semiconductor phonon waveguide. The neuromorphic function is realised through the interaction of the multimode input-sensitive phonon wavepackets with multiple magnon modes, which provides a high separability of the reservoir. The efficiency of the reservoir is demonstrated by reliable recognition of symbols “drawn” by the laser on the nanodevice surface in an area of several square microns, which is comparable with the size of the write laser spot. Our work paves the way for developing novel ultra-compact wave-based neuromorphic architectures on chips.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Device production The examined nanodevice was epitaxially grown on a commercial GaAs substrate [(100)-semi-insulating GaAs]. First, the phonon waveguide consisting of 10 pairs of AlAs/GaAs layers of 51-nm and 72-nm thickness, respectively, was fabricated by molecular beam epitaxy. Then, the ferromagnetic Fe 0.81 Ga 0.19 layer was deposited by magnetron sputtering. The nanogratings were fabricated through focused ion beam milling (Raith VELION) with 100-nm wide grooves and ridges over an area of 25 µm × 100 µm. To ensure high beam resolution during the patterning of the gratings a low Ga + beam current of 22 pA was applied at 35 keV beam energy. The milling dose was set to 0.3 nC/µm 2 , resulting in the groove depth of 25 nm. Optical writing and reading The scheme of signal encoding and read-out exploits two mode-locked Erbium-doped ring fibre laser oscillators (TOPTICA FemtoFiber Ultra 1050 and FemtoFiber Ultra 780). The lasers generate pulses of 150-fs duration with a repetition rate of 80 MHz at wavelengths of 1050 nm (write) and 780 nm (read). The write beam was focused by a microscope objective (×20 magnification; N.A. = 0.4) to the backside of the ferromagnetic layer, through the GaAs substrate and GaAs/AlAs phonon waveguide, which are transparent at the wavelength of 1050 nm. The focused pump spot had a Gaussian intensity distribution ~ \\\\(\\\\exp \\\\left(-\\\\frac{{r}^{2}}{2{R}^{2}}\\\\right)\\\\) , where r is the distance from the centre of the spot and R = 1 µm is the spot radius at the \\\\(1/\\\\sqrt{e}\\\\) level. The maximal used pulsed excitation density was 2 mJ/cm 2 . The linearly polarized readout beam was focused by another microscope objective (100× magnification; N.A. = 0.8) to the front side of the nanodevice. The focused readout spot also had a Gaussian distribution with R = 0.3 µm and an energy density of 1 mJ/cm 2 . Measurement of the read-out signals was based on the polar magneto-optical Kerr effect, i.e., the rotation of the readout pulse polarization plane by an angle proportional to the normal ( z -axis) projection of the net magnetization. The oscillations in the Kerr rotation signal reflect the precession of the magnetization [59] , which is a coherent superposition of the magnon modes driven by the phonon wavepacket [60] . The polarization rotation was detected in a differential scheme by a balanced photoreceiver with a 10-MHz bandwidth. Time resolution was achieved employing an asynchronous optical sampling (ASOPS) technique [61] . The write and read oscillators were locked with a frequency offset of 1600 Hz, which changed gradually the delay between the write and read pulses. In combination with the 80-MHz repetition rate and 10-MHz photoreceiver bandwidth, it allows measuring time-resolved signals in a time window of 12.5 ns with a time resolution of ~1 ps. The relative position of the write and read spots is controlled by two independent piezo-translators with a precision of 0.05 μm. The distance X 0 ≥ 20 µm between the write and read spots was chosen to minimize the contribution of the first-order surface phonon modes, which are intensively scattered by the patterned surface [28] . For visualization of the laser spots and control of their relative positions and focusing, a magnified (×75) image of the nanodevice surface was focused on crossed micrometre slits mounted at the intermediate focal plane between the nanodevice and a photoreceiver, where it was captured by a microscope (×6.5 magnification) with CMOS camera. To show more explicitly the volatility of the readout waveform, the readout signals presented in Figs. 2 d, f, g, and 3c were passed through a 15–20 GHz band pass filter, which eliminates the broadband noise. Sequential drawing of the visual shapes We used sequential “drawing” of visual shapes by step-by-step change of the relative position of the write and read laser spots in accordance with the selected trajectory as illustrated in Fig. 3 . In the XY coordinate frame centred at the apex of the readout spot as shown in Fig. 2 , the relative position of the write spot form a 3×3 coordinate matrix as shown in Fig. 3a . The positioning of the write and read spots for each point of the selected trajectory was automatized with the following algorithm: (i) randomized positioning of the read spot on the device surface (ii) adjusting the focusing of the read beam, (iii) adjusting the focusing of the write beam and the spatial overlap of the write and read spots, (iv) shift of the read spot to the required position. The experimentally estimated error of the automized positioning was 0.2 µm for all three-coordinate axis. The signal in every point of every trajectory was averaged over 10,000 measurements. Each trajectory was measured 20 times with averaging of the resulting waveform. 10 sets with six selected trajectories in each were measured. Drawing of randomized visual shapes For augmentation of the selected visual shapes with randomized distortion for the following recognition we measured 10 sets with nine signals in each set. Each of the nine signals corresponds to the specific relative positions of the centres of the write and read spots as described above. The algorithm of the automated positioning of the write and read spots was the same as for the sequential drawing with the same positioning errors. The waveform for each point was averaged over 200,000 measurements. The 10 sets were measured on three experimental days (4 sets in the first day, and 3 sets each on the second and third days) with a complete restart of the experimental setup. The waveforms for recognition were obtained by random summation of the corresponding signals from all 10 sets. The recognized shapes were obtained by the following normalized summations: $${}^{\\\\prime\\\\prime} {{{{{\\\\rm{L}}}}}}^{\\\\prime\\\\prime} \\\\,=\\t\\\\, ({{M}}_{1,1}+{{M}}_{2,1}+{{M}}_{3,1}+{M}_{3,2}+{M}_{3,3})/5;\\\\\\\\ {}^{\\\\prime\\\\prime} -^{\\\\prime\\\\prime}=\\t\\\\, ({{M}}_{2,1}+{M}_{2,2}+{M}_{2,3})/3\\\\\\\\ {}^{\\\\prime\\\\prime} {{{{{\\\\rm{O}}}}}}^{\\\\prime\\\\prime} \\\\,=\\t\\\\, ({{M}}_{1,1}+{{M}}_{1,2}+{{M}}_{1,3}+{M}_{2,1}+{M}_{2,3}+{M}_{3,1}+{M}_{3,2}+{M}_{3,3})/8;\\\\\\\\ {}^{\\\\prime\\\\prime}+^{\\\\prime\\\\prime} \\\\,=\\t\\\\, ({{M}}_{1,2}+{{M}}_{2,1}+{{M}}_{2,2}+{M}_{2,3}+{M}_{3,2})/5;\\\\\\\\ {}^{\\\\prime\\\\prime} {{{{{\\\\rm{T}}}}}}^{\\\\prime\\\\prime} \\\\,=\\t\\\\, ({{M}}_{1,1}+{{M}}_{1,2}+{{M}}_{1,3}+{M}_{2,2}+{M}_{3,2})/5;\\\\\\\\ {}^{\\\\prime\\\\prime} {{{{{\\\\rm{Z}}}}}}^{\\\\prime\\\\prime} \\\\,=\\t\\\\, ({{M}}_{1,1}+{{M}}_{1,2}+{{M}}_{1,3}+{M}_{2,2}+{M}_{3,1}+{M}_{3,2}+{M}_{3,3})/7.$$ (1) 2000 waveforms for each symbol were randomly generated (the randomized set for “-”, which consists of only 1000 unique waveforms, included 1000 additional randomly repeated waveforms). Only the integrated waveforms were processed. The separate waveforms \\\\({M}_{i,j}(t)\\\\) measured in the corresponding positions of the write laser beam were neither processed nor used to support the following recognition. Waveform characterization For characterisation of the readout signals we use the following quantities: the Shannon entropy, \\\\(h\\\\) , the variance, σ 2 , and the skewness, \\\\(\\\\mu\\\\) . The signals were processed as measured, without filtering and subtracting the dc-background of the photoreceiver. To calculate \\\\(h\\\\) , we split the interval between the minimal and maximal voltage of the readout signals (−4.02 mV and −3.05 mV, respectively) into J = 10 5 bins, and then apply the formula $$h=-\\\\mathop{\\\\sum }\\\\limits_{j=1}^{J}{p}_{j}{{{{\\\\mathrm{ln}}}}}{p}_{j},$$ (2) where \\\\({p}_{j}\\\\) is the probability that the readout voltage has a value within the j th bin. For calculating the statistical characteristics, we utilised the formulas below: $${\\\\sigma }^{2}=\\\\frac{1}{N}\\\\mathop{\\\\sum }\\\\limits_{i=1}^{N}{({V}_{i}-\\\\bar{V})}^{2},$$ (3) $$\\\\bar{V}=\\\\frac{1}{N}\\\\mathop{\\\\sum }\\\\limits_{i=1}^{N}{V}_{i},$$ (4) $$\\\\mu=\\\\frac{1}{N{\\\\sigma }^{3}}\\\\mathop{\\\\sum }\\\\limits_{i=1}^{N}{({V}_{i}-\\\\bar{V})}^{3},$$ (5) where \\\\({V}_{i}\\\\) is the value of the discretised in time readout voltage at the i th time step, and \\\\(N\\\\) is the total number of time steps. Artificial neural network For processing the properties of the signals to classify the letters from the device, we employ a simple Artificial Neural Network (ANN), illustrated in the inset of Fig. 4a . The network is made of the input layer (where the properties are fed into the network) with 3 nodes (one for each property of the signal), then we have the first learning layer made up of 3 nodes, the second learning layer with 5 nodes, and the final learning layer of 3 nodes, with one final output layer. The output layer gives the classification of the input signal. Each of our layers is connected with the Rectified Linear Unit (ReLU) activation function, apart from the final learning layer connected to the output layer, which is a linear activation function. The network is coded within the Python module PyTorch [62] . We use the Adam optimiser [63] to train the network, with a Root Mean Squared Error (RMSE) loss. The network optimises its weights and biases by backpropagating the error through its connections and adjusting the values of the network’s parameters. The adjustments are made on the order of the learning rate, here taken to be 0.001. We allow the network to train for 2000 epochs. However, employing an early stopping procedure, we stop the network from overfitting the training dataset. This is further achieved by splitting the dataset of letters into non-overlapping training, validating, and testing sets, with a standard 60, 20, 20% split [64] . Numerical modelling The S- and W-modes’ profiles and the excitation and propagation of phonon wavepackets were modelled using COMSOL Multiphysics® software. The spatial profiles and dispersions of the S-modes were obtained by solution of the Landau-Lifshitz and Maxwell equations in the frequency domain [36] . The following parameters of Fe 0.81 Ga 0.19 were used for the calculations: saturation magnetization, \\\\({{\\\\mu }_{0}M}_{s}\\\\) = 1.76 T, exchange stiffness, D = 2 \\\\({{{{{\\\\rm{\\\\cdot }}}}}}\\\\) 10 −17 T m 2 , cubic and uniaxial anisotropy coefficients of 23 mT and 11 mT, respectively. For the calculation of the W-modes spatial profiles and spatial-temporal evolution [28] , we used the following stiffness tensor components, \\\\({C}_{{kl}}\\\\) , and the mass density, \\\\(\\\\rho\\\\) : for Fe 0.81 Ga 0.19 \\\\({C}_{11}=209\\\\) GPa, \\\\({C}_{12}=156\\\\) GPa, \\\\({C}_{44}=122\\\\) GPa, \\\\(\\\\rho=7800\\\\) kg/m 3 ; for GaAs \\\\({C}_{11}=119\\\\) GPa, \\\\({C}_{12}=53.8\\\\) GPa, \\\\({C}_{44}=59.5\\\\) GPa, \\\\(\\\\rho=5316\\\\) kg/m 3 ; for AlAs \\\\({C}_{11}=119.9\\\\) GPa, \\\\({C}_{12}=57.5\\\\) GPa, \\\\({C}_{44}=56.6\\\\) GPa, \\\\(\\\\rho=3760\\\\) kg/m 3 . We used a model nanograting structure with 150 periods (i.e. 30 µm long in \\\\(x\\\\) direction). The structure was limited in \\\\(z\\\\) direction by the size of 1.9 µm (500 nm below the phonon waveguide), and the low-reflecting boundary condition was applied to the bottom boundary to minimize unwanted reflections. Free boundary conditions were applied to the top and side boundaries of the structure. The structure was considered uniform in \\\\({y}\\\\) direction. The excitation of the structure by the optical pulse creates thermal stress which was simulated using a two-temperature model for the lattice temperatures of the ferromagnetic cap and phonon waveguide [65] . Then, the calculated thermal stress was used as the input data for the acoustic modelling. The calculations were carried out in the time domain up to 6 ns. The results of the simulation are presented in the Supplemental Video and in Fig. 2 . The energy of a single W-mode wavepacket was estimated as \\\\(U=\\\\frac{1}{2}v\\\\left\\\\langle {c}_{11}\\\\right\\\\rangle \\\\left\\\\langle {\\\\eta }^{2}\\\\right\\\\rangle \\\\approx {5\\\\cdot 10}^{-17}\\\\) J, where \\\\(v=\\\\pi {R}^{2}h\\\\) is the total volume of the wavepacket ( \\\\(R=1\\\\) µm is the write laser spot radius and \\\\(h=1.3\\\\) µm is the phonon waveguide thickness, \\\\(\\\\left\\\\langle {c}_{11}\\\\right\\\\rangle=119.5\\\\) GPa is the averaged stiffness constant of the GaAs/AlAs waveguide and \\\\(\\\\left\\\\langle {\\\\eta }^{2}\\\\right\\\\rangle \\\\approx 2\\\\times {10}^{-10}\\\\) is the averaged square of the strain amplitude.\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"The core research idea is to develop on-chip wave-based reservoir computing systems using phonon and magnon wave dynamics in nanostructured materials to overcome the limitations of conventional artificial neural networks in terms of power efficiency, miniaturization, and adaptability for practical neuromorphic applications."},"entities":{"kind":"list like","value":["on-chip wave-based reservoir computing systems","phonon wave dynamics","magnon wave dynamics","nanostructured materials","conventional artificial neural networks","power efficiency","miniaturization","adaptability","neuromorphic applications"],"string":"[\n \"on-chip wave-based reservoir computing systems\",\n \"phonon wave dynamics\",\n \"magnon wave dynamics\",\n \"nanostructured materials\",\n \"conventional artificial neural networks\",\n \"power efficiency\",\n \"miniaturization\",\n \"adaptability\",\n \"neuromorphic applications\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1.","title":"Le Cun, Y., Bengio, Y. & Hinton, G. 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V. et al. 2022 roadmap on neuromorphic computing and engineering.Neuromorph. Comput. Eng.2, 022501 (2022).\",\n \"link\": \"https://doi.org/10.1088%2F2634-4386%2Fac4a83\"\n },\n {\n \"idx\": \"4.\",\n \"title\": \"Edwards, C. Growing pains for deep learning.Commun. ACM58, 14–16 (2015).\",\n \"link\": \"https://doi.org/10.1145%2F2771283\"\n },\n {\n \"idx\": \"5.\",\n \"title\": \"Tanaka, G. et al. Recent advances in physical reservoir computing: A review.Neural Netw.115, 100–123 (2019).\",\n \"link\": \"https://doi.org/10.1016%2Fj.neunet.2019.03.005\"\n },\n {\n \"idx\": \"6.\",\n \"title\": \"Zhang, Z. et al. In-sensor reservoir computing system for latent fingerprint recognition with deep ultraviolet photo-synapses and memristor array.Nat. Commun.13, 6590 (2022).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-022-34230-8\"\n },\n {\n \"idx\": \"7.\",\n \"title\": \"Lee, D. et al. In-sensor image memorization and encoding via optical neurons for bio-stimulus domain reduction toward visual cognitive processing.Nat. Commun.13, 5223 (2022).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-022-32790-3\"\n },\n {\n \"idx\": \"8.\",\n \"title\": \"Marković, D., Mizrahi, A., Querlioz, D. & Grollier, J. Physics for neuromorphic computing.Nat. Rev. Phys.2, 499–510 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs42254-020-0208-2\"\n },\n {\n \"idx\": \"9.\",\n \"title\": \"Nakajima, K. Physical reservoir computing–an introductory perspective.Jpn J. Appl. Phys.59, 060501 (2020).\",\n \"link\": \"https://doi.org/10.35848%2F1347-4065%2Fab8d4f\"\n },\n {\n \"idx\": \"10.\",\n \"title\": \"Cucchi, M., Abreu, S., Ciccone, G., Brunner, D. & Kleemann, H. Hands-on reservoir computing: a tutorial for practical implementation.Neuromorphic comput. eng.2, 032002 (2022).\",\n \"link\": \"https://doi.org/10.1088%2F2634-4386%2Fac7db7\"\n },\n {\n \"idx\": \"12.\",\n \"title\": \"Duport, F., Schneider, B., Smerieri, A., Haelterman, M. & Massar, S. All-optical reservoir computing.Opt. Express20, 22783–22795 (2012).\",\n \"link\": \"https://doi.org/10.1364%2FOE.20.022783\"\n },\n {\n \"idx\": \"13.\",\n \"title\": \"Larger, L. et al. Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing.Opt. Express20, 3241–3249 (2012).\",\n \"link\": \"https://doi.org/10.1364%2FOE.20.003241\"\n },\n {\n \"idx\": \"14.\",\n \"title\": \"Vandoorne, K. et al. Experimental demonstration of reservoir computing on a silicon photonics chip.Nat. Commun.5, 3541 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms4541\"\n },\n {\n \"idx\": \"16.\",\n \"title\": \"Antonik, P., Haelterman, M. & Massar, S. Brain-inspired photonic signal processor for generating periodic patterns and emulating chaotic systems.Phys. Rev. Appl.7, 054014 (2017).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevApplied.7.054014\"\n },\n {\n \"idx\": \"17.\",\n \"title\": \"Dong, J., Rafayelyan, M., Krzakala, F. & Gigan, S. Optical reservoir computing using multiple light scattering for chaotic systems prediction.IEEE J. Sel. Top. Quantum Electron26, 1–12 (2020).\",\n \"link\": \"https://doi.org/10.1109%2FJSTQE.2019.2936281\"\n },\n {\n \"idx\": \"18.\",\n \"title\": \"Sunada, S. & Uchida, A. Photonic neural field on a silicon chip: large-scale, high-speed neuro-inspired computing and sensing.Optica8, 1388–1396 (2021).\",\n \"link\": \"https://doi.org/10.1364%2FOPTICA.434918\"\n },\n {\n \"idx\": \"19.\",\n \"title\": \"Dion, J., Mejaouri, S. & Sylvestre, J. Reservoir computing with a single delay-coupled non-linear mechanical oscillator.J. Appl. Phys.124, 152132 (2018).\",\n \"link\": \"https://doi.org/10.1063%2F1.5038038\"\n },\n {\n \"idx\": \"20.\",\n \"title\": \"Meffan, R. C., Ijima, T., Banerjee, A., Hirotani, J. & Tsuchiya, T. Non-linear processing with a surface acoustic wave reservoir computer.Microsyst. Technol.29, 1197–1206 (2023).\",\n \"link\": \"https://link.springer.com/doi/10.1007/s00542-023-05463-4\"\n },\n {\n \"idx\": \"21.\",\n \"title\": \"Nakane, R., Tanaka, G. & Hirose, A. Reservoir computing with spin waves excited in a garnet film.IEEE Access6, 4462–4469 (2018).\",\n \"link\": \"https://doi.org/10.1109%2FACCESS.2018.2794584\"\n },\n {\n \"idx\": \"22.\",\n \"title\": \"Papp, Á., Porod, W. & Csaba, G. Nanoscale neural network using non-linear spin-wave interference.Nat. Commun.12, 6422 (2021).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-021-26711-z\"\n },\n {\n \"idx\": \"23.\",\n \"title\": \"Gartside, J. C. et al. Reconfigurable training and reservoir computing in an artificial spin-vortex ice via spin-wave fingerprinting.Nat. Nanotechnol.17, 460–469 (2022).\",\n \"link\": \"https://doi.org/10.1038%2Fs41565-022-01091-7\"\n },\n {\n \"idx\": \"24.\",\n \"title\": \"Körber, L. et al. Pattern recognition in reciprocal space with a magnon-scattering reservoir.Nat. Commun.14, 3954 (2023).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-023-39452-y\"\n },\n {\n \"idx\": \"25.\",\n \"title\": \"Shen, Y. et al. Deep learning with coherent nanophotonic circuits.Nat. Photon11, 441–446 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fnphoton.2017.93\"\n },\n {\n \"idx\": \"26.\",\n \"title\": \"Hughes, T. W., Williamson, I. A., Minkov, M. & Fan, S. Wave physics as an analog recurrent neural network.Sci. Adv.5, eaay6946 (2019).\",\n \"link\": \"https://doi.org/10.1126%2Fsciadv.aay6946\"\n },\n {\n \"idx\": \"27.\",\n \"title\": \"Wright, L. G. et al. Deep physical neural networks trained with backpropagation.Nature601, 549–555 (2022).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-021-04223-6\"\n },\n {\n \"idx\": \"28.\",\n \"title\": \"Yaremkevich, D. D. et al. Protected long-distance guiding of hypersound underneath a nanocorrugated surface.ACS Nano15, 4802–4810 (2021).\",\n \"link\": \"https://doi.org/10.1021%2Facsnano.0c09475\"\n },\n {\n \"idx\": \"29.\",\n \"title\": \"Scherbakov, A. V. et al. Optical excitation of single- and multimode magnetization precession in Fe-Ga nanolayers.Phys. Rev. Appl.11, 031003 (2019).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevApplied.11.031003\"\n },\n {\n \"idx\": \"30.\",\n \"title\": \"Thomsen, C., Grahn, H. T., Maris, H. J. & Tauc, J. Surface generation and detection of phonons by picosecond light pulses.Phys. Rev. B34, 4129–4138 (1986).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.34.4129\"\n },\n {\n \"idx\": \"31.\",\n \"title\": \"Matsuda, O., Larciprete, M. 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Commun.11, 4130 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-020-17635-1\"\n }\n]","isConversational":false}}},{"rowIdx":11195,"cells":{"unique_id":{"kind":"string","value":"s41467-021-21507-7"},"subfield":{"kind":"string","value":"Nanoscience and technology"},"title":{"kind":"string","value":"Towards explicit regulating-ion-transport: nanochannels with only function-elements at outer-surface"},"abstract":{"kind":"string","value":"Function elements (FE) are vital components of nanochannel-systems for artificially regulating ion transport. Conventionally, the FE at inner wall (FE IW ) of nanochannel − systems are of concern owing to their recognized effect on the compression of ionic passageways. However, their properties are inexplicit or generally presumed from the properties of the FE at outer surface (FE OS ), which will bring potential errors. Here, we show that the FE OS independently regulate ion transport in a nanochannel − system without FE IW . The numerical simulations, assigned the measured parameters of FE OS to the Poisson and Nernst-Planck (PNP) equations, are well fitted with the experiments, indicating the generally explicit regulating-ion-transport accomplished by FE OS without FE IW . Meanwhile, the FE OS fulfill the key features of the pervious nanochannel systems on regulating-ion-transport in osmotic energy conversion devices and biosensors, and show advantages to (1) promote power density through concentrating FE at outer surface, bringing increase of ionic selectivity but no obvious change in internal resistance; (2) accommodate probes or targets with size beyond the diameter of nanochannels. Nanochannel-systems with only FE OS of explicit properties provide a quantitative platform for studying substrate transport phenomena through nanoconfined space, including nanopores, nanochannels, nanopipettes, porous membranes and two-dimensional channels."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"Nanochannel-systems are artificial passages of ions and molecules with unique controllable performances [1] , [2] . They have been widely used in sensing [3] , [4] , [5] , drug release [6] , [7] , separation [8] , [9] , [10] , [11] , nanofluidic [12] , nanoelectrochemistry [13] , [14] , and energy conversion [15] , [16] , [17] , owing to their adjustable geometries at the nanoscale, versatile chemical compositions, and strong mechanical strength. The nanochannel-systems usually consist of three components: (1) nanochannels, (2) function elements at outer surface (FE OS ), and (3) function elements at inner wall (FE IW ) [18] , [19] . However, in traditional nanochannel-systems, there are two troublesome “black boxes” which are not well addressed: one is the role of FE OS on ion transport, which has been long-termed neglected; and the other is inexplicit chemical and physical properties of FE located deep inside nanochannel which is subject to that few test techniques with test tips or testing liquids, that can sufficiently contact with FE in the confined space at the nanoscale [2] . Currently, both theoretical [20] , [21] and experimental investigation [22] , [23] , [24] , [25] showed the synergistic effects of FE OS on regulating-ion-transport in the presence of FE IW . Compared with the confined space in nanochannel, relatively more free-spaces of OS endow FE OS with advanced characteristics, such as easy to immobilize, available for precise characterizations, receptive for foreign substrates, and potential application in new scenarios. However, till now the properties of most FE are still inexplicit or generally presumed from the measurable FE OS [26] , [27] , which would bring potential errors. Here, we confined FE at the outer surface (OS) and the edge of IW with a minimum depth as 7.5 nm (~0.01% of total IW) in nanochannels through the threshold effect of reducing-diameter down to 11 ± 3 nm, which is detectable for a host of techniques and termed as FE OS . The FE OS have been certified to regulate ion transport independently and their mechanism can be well demonstrated through the Poisson and Nernst–Planck equations assigned by measured properties from atomic force microscope (AFM), time of flight secondary ion mass spectrometry (ToF-SIMS), and solid-surface zeta potential analyzer (SSZPA). The FE OS fulfill the key capabilities of nanochannels in osmotic energy conversion and biosensing and bring new features: (1) increase of ionic selectivity but no obvious change in resistance and (2) accommodating probes or targets with size beyond the diameter of nanochannels.","references":{"indices":[1,2,3,4,5,6,7,8,9,10,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],"infos":[{"idx":"1","title":"Kowalczyk, S. W., Blosser, T. R. & Dekker, C. Biomimetic nanopores: learning from and about nature.Trends Biotechnol.29, 607–614 (2011).","link":"https://doi.org/10.1016%2Fj.tibtech.2011.07.006"},{"idx":"2","title":"Bocquet, L. Nanofluidics coming of age.Nat. Mater.19, 254–256 (2020).","link":"https://doi.org/10.1038%2Fs41563-020-0625-8"},{"idx":"3","title":"Martin, C. R. & Siwy, Z. S. Learning nature’s way: biosensing with synthetic nanopores.Science317, 331–332 (2007).","link":"https://doi.org/10.1126%2Fscience.1146126"},{"idx":"4","title":"Keyser, U. F. 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While, using the nanochannel-system only consisting of independent FE OS (Fig. 1c ) will avoid addressing the two problems: one is the role of FE OS on regulating-ion-transport and the other is unclear physicochemical properties of FE IW . Hence, the explicit relationship between the physicochemical properties of FE and function of nanochannel-systems could be realized. Fig. 1: Designed nanochannel-systems attached with FE OS and FE IW . a Stage 1, FE OS and FE IW immobilized as a whole, in which the role of FE OS on ion transport and the properties of FE IW are inexplicit (two “black boxes” exist). b Stage 2, FE OS and FE IW as distinct part, in which the role of FE OS on ion transport began to be paid attention and investigated, but the properties of FE IW are still inexplicit (start to open the 1st “black box”). c Stage 3, in this work, independent FE OS without FE IW in nanochannel-system for regulating-ion-transport (further reveal the 1st “black box”), in which the properties of both FE OS and nanochannels are measurable, making the properties of the whole nanochannel-system explicit to a great extent (avoid the trouble from the 2nd “black box”). Full size image Fabrications of nanochannel-system with only FE OS We built a nanochannel-system using an anodic aluminum oxide (AAO) membrane deposited by Au at the one side as nanochannels (named as none@OS) (Fig. 2a–e , Fig. S 1 ) [22] , [23] . For the present nanochannels, their surfaces could be divided into two parts: (1) OS refers to the outermost surface of Au and AAO at the opposite side. Because FE didn’t attach to the outermost layer of AAO, the OS only refers to the outermost layer of Au in the present work (Fig. S 2 ). (2) Inner wall (IW) refers to the residual surface of the nanochannels except for the OS (Fig. 2a ). We reduced the diameter of nanochannels from 25 ± 5 nm to 11 ± 3 nm by prolonging deposition time (0.1 nm/s for 2000 s) on purpose of restraining FE from entering the IW of nanochannels through the threshold effect (meaning that the stacking FE initially at the opening of nanochannels excluded the subsequent FE from entering IW to a great extent) (Fig. 2d, e and S 3 –S 6 ), differing from the relatively free diffusion in nanochannel within larger channel diameter [23] , [26] , [27] , [28] , [29] . Then, the three different FE, including polyacrylic acid (PAA, Mw ~5000) through Van der Waals’ force, poly(ethylene imine) (PEI, Mw ~10,000) through Van der Waals’ force [30] and DNA (Mw ~11,000) through Au-thiol interactions [22] , [23] respectively, were attached to the OS of none@OS (Figs. S 7 and S 8 ). The as-obtained nanochannel-systems were named as PAA@OS, PEI@OS, and DNA@OS, respectively. Fig. 2: The characterization of none@OS and the FE OS . a SEM image of none@OS from sectional view. The thickness of nanochannels is 65 μm. b Zoom-in version of Au coating side. c The corresponding energy dispersive X-ray spectroscopy of b . d , e SEM images of the OS coated with Au ( d ) and without Au ( e ) of none@OS from top view. f A scheme showing the present nanochannel-system and the FE distribution near the opening of nanochannel-system. The exposed surface to FE in none@OS includes OS and IW, of which the whole OS and a tiny fraction of IW are attached with FE herein. The FE OS in this work consists of all FE OS and a very small amount of FE IW . g Comparison of the distribution percentage of FE at the IW in the (i) 1st, (ii) 2nd, and (iii) 3rd stage. In the 1st stage, the FE occupy the total depth of IW (≈100%) through the random indraft of FE (the inset) [26] , [27] , [29] . In the 2nd stage, the distribution percentage of FE at IW decrease down to 5–30% through the Au–S interaction between thiol-modified FE and IW (the inset) [23] . In the 3rd stage, the distribution percentage of FE at IW sharply decline near zero. In the (iii) zoom-in version (iv), the distribution percentage of FE decrease with their molar mass, which demonstrates the threshold effects in the 3rd stage. Full size image Definitions of FE OS in nanochannel systems We further defined FE OS in the present nanochannel-system. Frankly, limited by current technologies, we cannot attach FE purely at the OS but not at the IW (Fig. 2f ). Therefore, the depth of FE at the IW of the nanochannels was measured to be as much as 180 nm (PAA@OS), 75 nm (PEI@OS), and 23 nm (DNA@OS), using the ToF-SIMS and the scanning electron microscope (SEM) [23] , which occupied a small part of the total depth of IW (65 μm) as 0.28%, 0.12%, and 0.04%, respectively. (Fig. 2g and S 9 ). Compared to the distribution percentage of FE in the 1st stage through random indraft (≈100%) [26] , [27] , [29] and in the 2nd stage through Au–S interaction (5–30%) [23] , the depth of FE occupied a tiny percentage (<0.3%) of the total depth of IW in this work, which could be attributed to the threshold effect during the penetration process of FE. We found that the FE depth at IW decreased with their molar mass that manifested the threshold effect (right figure in Fig. 2g, iv ). We, therefore, defined the FE OS in this work (Fig. 2f ), consisting of the FE at the OS of Au side and the FE at the IW existing near the opening of the nanochannels (Fig. 2g and S 10 ). Explicit role of FE OS in regulating-ion-transport We then investigated the role of FE OS in the regulating-ion-transport in the nanochannel-system using a two-electrode cell with two symmetric Ag/AgCl electrodes and 0.1 M KCl solution (Fig. 3a ). The current–voltage (I–V) plots of none@OS is asymmetric (Fig. 3b ) owing to the asymmetric structure and surface properties (Figs. S 5 and S 6 ), classified as ion-current rectification (ICR) behavior, in which I +2V (current at +2 V) / I −2V (current at −2 V) is defined as ICR ratio ( f rec ) [31] . After attached FE OS , the f rec increased (for PAA@OS and DNA@OS) and the f rec decrease lower than 1 with an opposite polarity (for PEI@OS) (Fig. 3f ). The variation of ion current above shows a similar trend as the previous reports (FE with distributions in the 1st stage described in Fig. 1 ) [30] . This effect was ascribed to the enhancement of negative charge at outer surface from highly negatived PAA rich in hydroxyl in PAA@OS or DNA with phosphodiester skeleton in DNA@OS (Fig. S 11 ), or charge reversal of surface charge for the highly positive PEI@OS rich in amino, leading to the enhancement of ion accumulation and depletion (Fig. S 16 ). In addition, the ion transport of nanochannels with independent FE OS was affected by ion strength (Fig. S 13 ). Therefore, we estimated that the FE OS influenced the ion currents through nanochannel-system above independently. Fig. 3: Explicit role of FE OS in regulating-ion-transport. a Scheme of a two-electrode cell. b I–V curves characterizing the ion transport through nanochannel-system. c The PNP equation was used for numerical simulations of the effect from FE OS on the ion transport through nanochannel, where two variables remain: ∇ c i and ∇ φ valued by the surface charge density ( σ ) (measured by SSZPA, Fig. 3d ) (1), the diameter of nanochannels (measured by AFM, Fig. 3e ) (2) and the depth of FE at IW (measured by ToF-SIMS, Fig. 2g ) (3) (More details in the section “Numerical simulation” in “Method” section). Thus, unless otherwise stated, the I is available from numerical simulations using the three parameters (1), (2) and (3) measurable in the nanochannel-system with FE OS only but without FE IW . f Comparison of the rectification ratio ( f rec ) measured from I–V tests (experimental value) with the f rec from numerical simulations based on classical equations using measured parameters (theoretical value), but not estimated values like the previous works in 1st and 2nd stage. g Fraction of the measurable surface, which is the area proportion of the OS of measurable properties using mostly current test technologies, in the total surface occupied by FE (Details in Table S1 ). Data from the present work and the previous works using functional solid-state nanochannels [27] , [29] , [56] , [57] , [58] . For I–V tests, five chips were used to obtain each error bar. Statistics of diameters have been done by counting 100 nanochannels for each kind of nanochannels. Full size image In the 1st and 2nd stage, the numerical simulations coupled the Poisson–Nernst–Planck (PNP) equations [32] among the classical equations with steady state continuity equations were performed for FE IW [16] , [17] , [21] , [28] , [29] , [30] , [33] . However, the three parameters of FE IW used in equation were unmeasurable, as (1) the depth of FE at IW, (2) the surface charge, and (3) the diameter of the nanochannels after attaching FE IW . Hence, the hypotheses of the above three parameters were generated in the 1st and 2nd stage, which is unavoidable: (1) FE may not completely cover IW, but leaving the blank area of the deep IW hard for FE to reach [34] . (2) The surface charge of FE IW is substituted by the measurable surface charge of FE OS . Surface charge density is usually different between FE OS and FE IW , due to their different grafting densities [22] . Sometimes the charge of FE OS and FE IW inversed caused by the local polarization [35] . (3) The decrement of nanochannel diameter is roughly estimated by subtracting the straighten length of FE ( D − 2 L SM , where D represent the diameter of nanochannels and L SM represent the straighten length of FE) [36] . However, the FE are mostly not straight, i.e., single-strand DNA [25] . The results from the numerical simulations, therefore, deviate from experiments and even have randomness, which indicates that the above hypotheses bring the deviations or even sometimes errors. In our nanochannel-system, we measured (1) the depth of the part of FE OS , (2) the surface charge, and (3) the diameter of nanochannels after attaching FE OS using ToF-SIMS, SSZPA, and AFM [37] , respectively (Figs. 2 g, 3 d, 3e and Figs. S 11 , S 12 ). Both the qualitative and the quantitative variation of ICR behavior from numerical simulations fitted well with the experimental results, which indicated the explicit regulating-ion-transport accomplished in 3rd stage nanochannel-system (Fig. 3f, g ). One of the important features of FE is of versatile physical and chemical properties, nanochannel-system adapt to a broad range of applications spanning from osmotic energy conversion devices [15] , [16] , [17] , [33] , [38] to biosensors [26] , [27] , [39] , [40] , [41] . Meanwhile, a nanochannel system combining with electrochemistry in a confined space is now a crucial promising field [42] , [43] , which is utilized to dynamically monitor the single molecule [44] , understand the chemical reaction [45] , characterize the single particle [46] , and probe single living cell [47] , etc. Here, we investigated whether the nanochannel-system in 3rd stage with FE OS explicit regulating-ion-transport could fulfill the applications above or even with special performances to the 1st and 2nd. Impact from FE OS on osmotic energy conversion devices The osmotic energy conversion devices were fabricated using nanochannels with only FE OS without FE IW (Fig. 4a–d and Figs. S 16 –S 19 ) [15] , [16] , [17] . The output power density of nanochannel system with independent FE OS was estimated according to the equation P L = I 2 / R L , where I is the current across the circuit and R L is the external load resistance (Fig. S 21 ). It was found that (1) output max power density increased with PAA concentration owing to the enhanced ion selectivity (Fig. 4e and Fig. S 21 ); (2) R channel was almost unchanged with increase of concentration PAA (Fig. 4e and Fig. S 21 ), where R channel was the internal resistance of the nanochannels. For (1), it is easy to understand the selectivity of the cell increased with the negative-charged FE OS (PAA) [20] . For (2), we speculated the R channel did not obviously change with the FE OS . In order to verify our assumptions, two FE OS layer by layer (LbL) assembled on the OS of nanochannels (PAA and PEI sequentially in Fig. 4f–h and Figs. S 22 –S 24 ; and PEI and PAA sequentially in Figs. S 25 and S 26 ) [30] , [48] . No distinctly increase of R channel was observed during two FE OS sequential assembly at the OS in the 3rd stage (Fig. 4i ) owing to the ignored resistance induced by FE OS , while R channel increased with the LbL assembly of PAA and PEI at the IW in the 1st and 2nd stages (Fig. S 27 ). The above results demonstrated that the increase of FE OS (with selectivity) enhanced the “output max power density” of osmotic energy conversion devices (3rd stage), without obviously raising R channel , which indicated a new conceptual route to design large power density nanochannel-system [27] , [49] , [50] , [51] . Fig. 4: Effect from FE OS on osmotic energy conversion devices. a A scheme showing the working mechanism of the osmotic energy conversion devices using nanochannel-system. The electricity by reverse electrodialysis is generated under salt gradient using a nanochannel system containing only FE OS . b Fabrication of the nanochannel-system attached with PAA as FE OS (PAA@OS) using precursor with different PAA concentration (1 or 10 mM). c AFM of PAA@OS using 1 and 10 mM PAA in precursor. d Depth distribution of the part of FE OS in nanochannel-system using precursor solution with different PAA concentration from ToF-SIMS. e Output max power density, ion selectivity and R channel of PAA@OS with different PAA concentration in precursor. f Scheme showing LbL assembly of PAA and PEI at OS sequentially. g AFM of the OS after LbL assembly. h Depth distribution of the part of FE OS (PAA and PEI) in nanochannel-system after LbL assembly. i Reversal f rec through the LbL assembly of PAA and PEI and corresponding R channel . The numerical simulations of two samples above, PAA 4 PEI 3 @OS and PAA 4 PEI 4 @OS, using their measured parameters were performed, which well fit the experimental results (Fig. S 24 ). For the osmotic energy conversion devices, five devices using FE@OS were assembled to obtain each error bar of power density and R channel . Full size image Impact from FE OS on biosensors We further demonstrated a sensing strategy employing single-strand DNA probes as FE OS in the nanochannel-systems (3rd stage) (Fig. 5a ), for the detection of a broad range of targets including inorganic ions (Hg 2+ with 1 pM limit of detection, LoD), small molecules (ATP with 1 pM LoD), proteins (lysozyme with 1 pM LoD), and cancer cells (MCF-7 cells with 400 cells mL −1 LoD) (Fig. 5b, c and Fig. S 28 ). The targets were specifically captured by the designed FE OS as probe and tailored the surface charge of the OS locally, which affect the asymmetry of surface potential in between OS and IW and change the ion transport in form of f rec signal. To confirm the sensing mechanism above, we took ATP detection using ssw-DNA as an example. In ssw-DNA structure, one kind of repeating units in ssw-DNA was designed as ATP aptamer, which specifically bonded with ATP and caused the disassembly of ssw-DNA (Fig. 5d–g and Fig. S 29 ). The variation of surface potential under the disassembly of ssw-DNA triggered by different concentration ATP was quantitatively characterized through the electrochemical approaches (Fig. 5h and Fig. S 30 ). The mechanism was also confirmed by the change grafting density of DNA at OS (Fig. S 31 ). The selective detection for ATP was also realized based on surface-charge-response sensing mechanism (Fig. 5h, i ). In the 1st and 2nd stage, because probes (as FE IW ) were immobilized at the IW, a confined space usually with diameter <100 nm, the targets with a size beyond the diameter of nanochannels can’t sufficiently contact with probe and efficiently recognized. In the 3rd stage, the OS possess the receptive characteristic for probes or targets with the size beyond the diameter of nanochannels. The recognition between probes and targets took place at the OS, which is relatively more free-spaces compared with IW in nanochannels. Here, we realized a nearly “universal” biosensor approach according to the two successful sensing processes: one is the DNA amplifications (ssw-DNA) as the probes, whose diameter is larger than the diameter of the nanochannel-system (some probes is up to 30-fold larger in Fig. 5e–g ), and the other is the MCF-7 cells as the targets, whose diameters is about than 2 magnitudes larger than the diameters of the nanochannel-system. Fig. 5: Sensing performances of FE OS as probes. a A scheme showing the working mechanism of the sensor using FE OS as probes. b Capture process of multi-scale targets through designed single-stand DNA. c Sensitivity and selectivity of the DNA@OS (DNA is a designed sequence specifically bonding with targets) for the recognition of ions (Hg 2+ ), small molecules (ATP), protein (Lysozyme) and cells (MCF7). The selective detection of multiscale targets using FE OS was realized based on the change of f rec signal output induced by the surface charge at outer surface. LoD is defined as the limitation of detection for the targets. d Formation of “supersandwich” DNA structure (ssw-DNA) with long concatamers through the successive hybridization of alternating DNA unit. And gradual disassembly of ssw-DNA based on interaction between ATP and the repeat DNA units through increasing ATP concentration. e Agarose gel electrophoresis characterizing of the ssw-DNA: 1) DNA marker; 2) p1; 3) p2 (ATP ampter); 4) p1 + p2; 5) target+p1 + p2. f Depth distribution of the “supersandwich” DNA in nanochannel-system using ToF-SIMS. g Laser scanning confocal microscopy of the OS after the assembly (top) and the disassembly of ssw-DNA (bottom). The scale bar is 20 μm. h The LoD of ATP using ssw-DNA as probe based on nanochannel method and electrochemical method (Fig. S 30 ). i Specificity of ssw-DNA@OS for ATP, in contrast with other NTPs. For the sensing performances part, five sensors using FE@OS were established to obtain each error bar of sensitivity and specificity. Full size image"},{"section":"Discussion","content":"In conclusion, we have built a de novo designed “minimalist” nanochannel-system with an explicit regulating-ion-transport feature, which has been achieved using FE OS independently without FE IW . The troubles from the “black box” for the properties of FE IW in the 1st and 2nd stage are well avoided without using FE IW , meanwhile the other “black box”, the role of FE OS on regulating-ion-transport, are further demonstrated from the “coadjutant” of FE IW in the 2nd stage to the “monodrama” in the 3rd stage. The use of independent FE OS is a new attempt to separate the enrichment, screening, and recognition process taking place at OS from the ion transport process taking place at IW within a confined space at the nanoscale. Furthermore, utilization of independent FE OS under a less restricted environment than FE IW will endow nanochannel-systems with (1) more measurable properties, such as hydrophobicity (the contact angle measurement is available for FE OS , which is hardly achieved for FE IW in nanoscale confined space) (Fig. S 32 ), (2) more characteristic technologies, such as X-ray photoelectron spectroscopy (XPS), ion microprobe mass analysis (IMMA) (the detection on FE IW requiring for destructive treatment, such as ion thinning, but not necessary for FE OS ), (3) more new performances limited to diffusions in nanochannels previously, such as reusage of FE (Fig. S 33 ), rapid responses to targets (Fig. S 34 ). The FE OS in 3rd stage like “ignition system” trigger the regulatable substrate transports through nanochannels and the abroad application space of nanochannel-systems."},{"section":"Methods","content":"Materials Poly (acrylic acid) (PAA, MW ~5000) was purchased from Ryon Biological Technology (Shanghai, China). Polyethyleneimine (PEI, MW ~10,000) was purchased from Damas-beta. Tris (hydroxymethyl) aminomethane (Tris) was purchased from Alfa Aesar. KCl, NaCl, and MgCl 2 were obtained from Aladdin reagent (Shanghai, China). Adenosine 5′-triphosphate (ATP) disodium salt solution, uridine Triphosphate (UTP), cytidine triphosphate (CTP) and guanosine triphosphate (GTP) were purchased from Sigma-Aldrich. HeLa and PC3 cells were obtained from Chinese Center for Typical Culture Collection (Wuhan, China) and cultured in DMEM (Gibco) supplemented with 10% (v/v) fetal bovine serum (FBS), 2 mg/mL NaHCO 3 , and 100 U/mL antibiotics 15 (penicillin-streptomycin). MCF-7 cells were purchased from KeyGEN Biotech Co. Ltd. (Nanjing, China) and cultured in RPMI-1640 (Gibco) supplemented with 10% FBS, 2 mg/mL NaHCO 3 , and 100 U/mL antibiotics (penicillin–streptomycin) at 37 °C in a humidified 5% CO 2 atmosphere. AAO membranes were purchased from Pu-Yuan Nano Technology Co. Ltd. (Hefei, China). The thickness of AAO membrane is 65 μm. All solutions were prepared using Millipore Milli-Q water (18 MΩ cm). All oligonucleotides are synthesized and purified by Sango Biotech Co. Ltd, (Shanghai, China). The sequences are shown in Tables S2 and S3 . Fabrication of nanochannel systems Preparation of none@OS. A physical vapor deposition (PVD) method was employed using AE Nexdap PVD platform (Angstrom Engineering Inc.) to prepare nanochannels with independent OS [22] , [23] . Two kinds of depositing materials as Au and Cr were used. To enhance the stability of Au, one layer of Cr with a thickness of 10 nm was first deposited at the one side of AAO. The Au was followed deposited at the same side of Cr layer. The circular targets were approximately parallel to the AAO membranes, ensuring the deposition direction perpendicular to the membranes. The deposition was taken by the successive deposition without replacing the target materials or releasing vacuum. The successive deposition ensured no secondary pollution at the first deposited layer. The low depositing speed of Au was applied as 0.1 nm s −1 and the time was 2000 s. The depositing speed was calibrated by the deposited thickness on the surface of the flat silicon wafer at nanometer level. The as-synthesized sample was named none@OS. Preparation of PAA@OS, PEI@OS, and DNA@OS. Aqueous solution including PAA or PEI with different concentrations was prepared. Before attaching PAA or PEI, the Au-coated OS in none@OS was treated by Ar plasma for 60 s to remove impurities. Then, the PAA solution or the PEI solution was dip-coated at the OS. The dip-coating continued for 2 min. Then the sample was washed throughout by water, and dried under N 2 gas. The sample was named PAA@OS and PEI@OS, respectively. Thiol-modified DNA was attached at the OS of none@OS using similar dip-coating method. The reaction time continued for 60 min and the ample was named DNA@OS. The as-prepared samples were then applied for further measurement and characterizations. LbL self-assembly of charged polyelectrolyte. For LbL self-assembly of PAA and PEI on the outer surface of AAO/Au nanochannels, the process was described as follows: an aqueous solution containing PAA (1 mM) was firstly spread onto at the OS of none@OS for the adsorption of PAA polyelectrolyte, and the modification time was 2 min. After the adsorption of first layer of PAA, the solution of PEI was then spread on the same side of none@OS. Subsequently, other alternative layers of PAA/PEI were also electrostatically deposited on the OS of none@OS using the same procedure. Assembly and disassembly of ssw-DNA. The DNA supersandwich structure (ssw-DNA) was prepared by mixing the thiol-modified cp-DNA solution (1 μM) into the mixture of equimolar P1 and P2 (1 μM) for 1 h. Then the solution containing ssw-DNA was dropped onto the OS of none@OS, and the reaction time was 60 min, named ssw-DNA@OS. The disassembly of ssw-DNA was achieved by treating the ssw-DNA@OS with different concentrations of ATP (1 nM–1 μM) for 60 min. Detection of Hg 2+ , ATP, lysozyme, and MCF-7 cells The aptamer solution (1 μM) for different targets were dropped onto the OS of none@OS at room temperature for 60 min. After that, 6-mercapto-1-hexanol (MCH, 100 nM) was added onto the OS for 60 min to prevent nonspecific adsorption. Subsequently, the aptamer modified nanochannels was treatment by the suspension containing various targets. The treatment time was 60 min. After the interaction, the sample was washed by buffer to remove the nonspecifically adsorbed targets. And then the as-treated samples were then applied for characterizations. Cells were first cultured in flasks with Dubelcco’s Modified Eagle’s Medium (DMEM) at 37 °C under 5% CO 2 in the cell incubator followed by centrifuging at 600 × g for 5 min, and re-dispersed in PBS (10 mM, pH 7.4) with a density of 5 × 10 5 cells mL −1 . A certain amount of cells suspension reacted with aptamer modified nanochannels for 60 min. After cells capture, the sample was washed by PBS buffer to remove the nonspecifically adsorbed cells. And then the as-prepared membranes were then applied for characterizations. Characterization SEM images and EDX analyses were taken with a field-emission scanning electron microscope (SU8010, Hitachi, Japan) equipped with Energy Dispersive Xray spectroscopy (EDS, BRUKER AXS, Germany). All samples were coated with carbon (5 nm) prior to SEM examinations. Secondary ion mass spectra of as-prepared samples were characterized by ToF-SIMS V (IONTOF, GmbH). A Bi liquid metal primary ion source was applied with an angle of 45° relative to the sample surface with a pulsed Bi 3++ primary ion beam of 30 keV and shave off fresh 60 µm × 60 µm areas for each analysis. The ToF analyzer was installed at an angle of 90° to the sample surface. Negative secondary ion spectra were collected. Mass calibration was carried out using standard procedures (mass resolving power >5000). The properties of OS of nanochannel system were characterized by an atomic force microscope (Multimode 8, Burke). The determination of OS’s surface zeta potential of nanochannel system were carried out by using SSZPA (SurPASS, Anton Paar Ltd., Austria). The pH dependence of the zeta potential and the isoelectric point (IEP) were obtained in a 0.1 M potassium chloride at the same electrical conductivity. The fluorescent measurement of nanochannel functionalized with fluorescent-dyed DNA at OS were performed by using laser scanning confocal microscope (LSM 880 confocal microscopy, Carl Zeiss) equipped with a FemtoSecond Laser (Coherent Inc.) The fluorescent-dyed membranes were placed at cover glass filling with water (around 1 cm 2 membrane with 20 μL water). For the analysis of assembly and dissembly process of ssw-DNA, the mixture was analyzed and observed using native PAGE gel. The gel was run in 16% acrylamide (containing 19/1 acrylamide/bisacrylamide) solution with 1× TBE buffer, at 100 V constant voltage for 1.5 h. The gel was directly imaged or stained with GelRed (Biotium) for 20 min to image the DNA position by using Tanon imaging system (Tanon 5200 Multi). The water contact angle measurements were operated on a DSA100 contact angle analyzer at ambient temperature and around 30% humidity. A drop of water (5.0 µL) was dropped onto the surfaces of nanochannel membranes. The surface charge density for the OS with ssw-DNA grafting. The surface charge density with ssw-DNA grafting were tested using the system described in Fig. S 28 and calculated based on the following equation [52] : $$Q = \\frac{{2n{{FAD}}_0^{1/2}C_0}}{{\\pi ^{1/2}}}t^{1/2} + Q_{{\\mathrm{dl}}} + n{{FA}}{\\it{\\Gamma }}_0,$$ (1) where n is electron transfer number, F is the Faraday constant, A is the electrode area (cm 2 ), D 0 is the diffusion coefficient (cm 2 s −1 ), C 0 is the bulk concentration (mol cm −2 ), Q dl is the capacitive charge ( C ), and n FAΓ 0 is the charge from the reduction of Γ 0 (mol cm −2 ) of adsorbed redox marker. The samples were mounted in between the two halves of a homemade electrochemical cell, which contains 1.0 mL KCl solution in each cell. I–V plots were recorded by an electrochemical workstation (CHI, ShangHai) and Keithley 6487 picoammeter (Keithley Instruments, Cleveland, OH). Two Ag/AgCl electrodes were used to apply the potential. The working electrode was placed on the Au side and the reference electrode on the AAO side if not particularly mentioned. The effective area of the membrane for ion conduction test is about 7 mm 2 . All measurements were performed at room temperature, and each test was repeated for five times. Five membranes at least were used to obtain average values. To study the ionic rectification properties, a scanning voltage from −2.0 to 2.0 V was applied across the membrane. The ion concentration on the Au side and the AAO side was kept equal (0.1 M KCl, pH 7.0), if not specially mentioned. To study the energy conversion properties [53] a salt gradient was designed, where the concentration of the solution on the Au side is higher than that on the AAO side if not particularly mentioned. Numerical simulations The ionic rectification phenomenon was theoretically investigated using a commercial finite-element software package COMSOL Multiphysics [54] , [55] . $$J_{\\rm{i}} = D_{\\rm{i}}\\left( {\\nabla c_{\\rm{i}} + \\frac{{z_{\\rm{i}}Fc_{\\rm{i}}}}{{RT}}\\nabla \\varphi } \\right),$$ (2) $$\\nabla ^2\\varphi = - \\frac{F}{\\varepsilon }{\\sum} {z_{\\rm{i}}c_{\\rm{i}}},$$ (3) $$\\nabla \\cdot J_{\\rm{i}} = 0.$$ (4) Here, Eq. ( 2 ) is the Nernst–Planck equation that descripts the transport properties of a charge nanochannel and Eq. ( 3 ) is the Poisson equation that descripts the relationship between the electric potential and ion concentration in the nanochannels. The model is generally simplified using steady-state conditions (Eq. ( 4 )). The electroosmotic flow was neglected in this work. The coupled Eqs. ( 2 – 4 ) can be solved utilizing appropriate boundary conditions (Eqs. 4 , 6 and Table S4 ). The solution yields the concentration field c i for all species and the potential φ distributions in the nanochannels. $${\\mathbf{n}} \\cdot \\nabla \\varphi = - \\frac{\\sigma }{\\varepsilon },$$ (5) $${\\mathbf{n}} \\cdot J_{\\mathrm{i}} = 0.$$ (6) The total ionic current through the nanochannel can be calculated: $$I = {\\int\\!\\!\\!\\!\\!\\int} {J_{\\rm{i}}{\\rm{d}}s} = - {\\int\\!\\!\\!\\!\\!\\int} {D\\left( {\\nabla c_{\\rm{i}} + z_{\\rm{i}}c_{\\rm{i}}\\frac{F}{{RT}}\\nabla \\varphi } \\right){\\rm{d}}s}.$$ (7) The formula above was used for numerical simulations of the effect from FE OS on the ion transport through nanochannel, where i refer to cation (K + ) and anion (Cl − ) and J i refers to ionic flux [16] . When the properties of electrolyte are set, the invariants include: D i as the diffusion coefficient, c i as the concentration of K + and Cl − , z i as the valence of K + and Cl − , T as the thermodynamic temperature. Meanwhile, in this formula, the constants include: F as the Faraday’s constant and R as the ideal gas constant. As a result, two variables remain: ∇ c i describes the concentration gradient of ion along nanochannels under external electric field and ∇ φ describes the electrical potential gradient along nanochannels, in which the φ is valued by the surface charge density ( σ ) of the FE OS and the nanochannels (measured by SSZPA, Fig. 3d ) (1). The ∇ c i and the ∇ φ are calculated by integrating the c i of electrolyte and φ over the cross-section area (d s ). The integration of d s is valued by the diameter of nanochannels (measured by AFM, Fig. 3e ) (2), and the depth of FE at IW (measured by ToF-SIMS, Fig. 2g ) (3)."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"Nanochannel-systems are artificial passages of ions and molecules with unique controllable performances [1] , [2] . They have been widely used in sensing [3] , [4] , [5] , drug release [6] , [7] , separation [8] , [9] , [10] , [11] , nanofluidic [12] , nanoelectrochemistry [13] , [14] , and energy conversion [15] , [16] , [17] , owing to their adjustable geometries at the nanoscale, versatile chemical compositions, and strong mechanical strength. The nanochannel-systems usually consist of three components: (1) nanochannels, (2) function elements at outer surface (FE OS ), and (3) function elements at inner wall (FE IW ) [18] , [19] . However, in traditional nanochannel-systems, there are two troublesome “black boxes” which are not well addressed: one is the role of FE OS on ion transport, which has been long-termed neglected; and the other is inexplicit chemical and physical properties of FE located deep inside nanochannel which is subject to that few test techniques with test tips or testing liquids, that can sufficiently contact with FE in the confined space at the nanoscale [2] . Currently, both theoretical [20] , [21] and experimental investigation [22] , [23] , [24] , [25] showed the synergistic effects of FE OS on regulating-ion-transport in the presence of FE IW . Compared with the confined space in nanochannel, relatively more free-spaces of OS endow FE OS with advanced characteristics, such as easy to immobilize, available for precise characterizations, receptive for foreign substrates, and potential application in new scenarios. However, till now the properties of most FE are still inexplicit or generally presumed from the measurable FE OS [26] , [27] , which would bring potential errors. Here, we confined FE at the outer surface (OS) and the edge of IW with a minimum depth as 7.5 nm (~0.01% of total IW) in nanochannels through the threshold effect of reducing-diameter down to 11 ± 3 nm, which is detectable for a host of techniques and termed as FE OS . The FE OS have been certified to regulate ion transport independently and their mechanism can be well demonstrated through the Poisson and Nernst–Planck equations assigned by measured properties from atomic force microscope (AFM), time of flight secondary ion mass spectrometry (ToF-SIMS), and solid-surface zeta potential analyzer (SSZPA). The FE OS fulfill the key capabilities of nanochannels in osmotic energy conversion and biosensing and bring new features: (1) increase of ionic selectivity but no obvious change in resistance and (2) accommodating probes or targets with size beyond the diameter of nanochannels.\",\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 16,\n 17,\n 18,\n 19,\n 20,\n 21,\n 22,\n 23,\n 24,\n 25,\n 26,\n 27\n ],\n \"infos\": [\n {\n \"idx\": \"1\",\n \"title\": \"Kowalczyk, S. W., Blosser, T. R. & Dekker, C. 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Soc.134, 15395–15401 (2012).\",\n \"link\": \"https://doi.org/10.1021%2Fja3053333\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Designed nanochannel-systems Different from previous nanochannel-systems (Fig. 1a, b ), we designed a new nanochannel-system (Fig. 1c ). In the 1st stage in Fig. 1a and 2nd stage in Fig. 1b , both the physicochemical properties and function on regulating-ion-transport of nanochannel system are partially unclear due to the limitation for the characterization of FE IW . While, using the nanochannel-system only consisting of independent FE OS (Fig. 1c ) will avoid addressing the two problems: one is the role of FE OS on regulating-ion-transport and the other is unclear physicochemical properties of FE IW . Hence, the explicit relationship between the physicochemical properties of FE and function of nanochannel-systems could be realized. Fig. 1: Designed nanochannel-systems attached with FE OS and FE IW . a Stage 1, FE OS and FE IW immobilized as a whole, in which the role of FE OS on ion transport and the properties of FE IW are inexplicit (two “black boxes” exist). b Stage 2, FE OS and FE IW as distinct part, in which the role of FE OS on ion transport began to be paid attention and investigated, but the properties of FE IW are still inexplicit (start to open the 1st “black box”). c Stage 3, in this work, independent FE OS without FE IW in nanochannel-system for regulating-ion-transport (further reveal the 1st “black box”), in which the properties of both FE OS and nanochannels are measurable, making the properties of the whole nanochannel-system explicit to a great extent (avoid the trouble from the 2nd “black box”). Full size image Fabrications of nanochannel-system with only FE OS We built a nanochannel-system using an anodic aluminum oxide (AAO) membrane deposited by Au at the one side as nanochannels (named as none@OS) (Fig. 2a–e , Fig. S 1 ) [22] , [23] . For the present nanochannels, their surfaces could be divided into two parts: (1) OS refers to the outermost surface of Au and AAO at the opposite side. Because FE didn’t attach to the outermost layer of AAO, the OS only refers to the outermost layer of Au in the present work (Fig. S 2 ). (2) Inner wall (IW) refers to the residual surface of the nanochannels except for the OS (Fig. 2a ). We reduced the diameter of nanochannels from 25 ± 5 nm to 11 ± 3 nm by prolonging deposition time (0.1 nm/s for 2000 s) on purpose of restraining FE from entering the IW of nanochannels through the threshold effect (meaning that the stacking FE initially at the opening of nanochannels excluded the subsequent FE from entering IW to a great extent) (Fig. 2d, e and S 3 –S 6 ), differing from the relatively free diffusion in nanochannel within larger channel diameter [23] , [26] , [27] , [28] , [29] . Then, the three different FE, including polyacrylic acid (PAA, Mw ~5000) through Van der Waals’ force, poly(ethylene imine) (PEI, Mw ~10,000) through Van der Waals’ force [30] and DNA (Mw ~11,000) through Au-thiol interactions [22] , [23] respectively, were attached to the OS of none@OS (Figs. S 7 and S 8 ). The as-obtained nanochannel-systems were named as PAA@OS, PEI@OS, and DNA@OS, respectively. Fig. 2: The characterization of none@OS and the FE OS . a SEM image of none@OS from sectional view. The thickness of nanochannels is 65 μm. b Zoom-in version of Au coating side. c The corresponding energy dispersive X-ray spectroscopy of b . d , e SEM images of the OS coated with Au ( d ) and without Au ( e ) of none@OS from top view. f A scheme showing the present nanochannel-system and the FE distribution near the opening of nanochannel-system. The exposed surface to FE in none@OS includes OS and IW, of which the whole OS and a tiny fraction of IW are attached with FE herein. The FE OS in this work consists of all FE OS and a very small amount of FE IW . g Comparison of the distribution percentage of FE at the IW in the (i) 1st, (ii) 2nd, and (iii) 3rd stage. In the 1st stage, the FE occupy the total depth of IW (≈100%) through the random indraft of FE (the inset) [26] , [27] , [29] . In the 2nd stage, the distribution percentage of FE at IW decrease down to 5–30% through the Au–S interaction between thiol-modified FE and IW (the inset) [23] . In the 3rd stage, the distribution percentage of FE at IW sharply decline near zero. In the (iii) zoom-in version (iv), the distribution percentage of FE decrease with their molar mass, which demonstrates the threshold effects in the 3rd stage. Full size image Definitions of FE OS in nanochannel systems We further defined FE OS in the present nanochannel-system. Frankly, limited by current technologies, we cannot attach FE purely at the OS but not at the IW (Fig. 2f ). Therefore, the depth of FE at the IW of the nanochannels was measured to be as much as 180 nm (PAA@OS), 75 nm (PEI@OS), and 23 nm (DNA@OS), using the ToF-SIMS and the scanning electron microscope (SEM) [23] , which occupied a small part of the total depth of IW (65 μm) as 0.28%, 0.12%, and 0.04%, respectively. (Fig. 2g and S 9 ). Compared to the distribution percentage of FE in the 1st stage through random indraft (≈100%) [26] , [27] , [29] and in the 2nd stage through Au–S interaction (5–30%) [23] , the depth of FE occupied a tiny percentage (<0.3%) of the total depth of IW in this work, which could be attributed to the threshold effect during the penetration process of FE. We found that the FE depth at IW decreased with their molar mass that manifested the threshold effect (right figure in Fig. 2g, iv ). We, therefore, defined the FE OS in this work (Fig. 2f ), consisting of the FE at the OS of Au side and the FE at the IW existing near the opening of the nanochannels (Fig. 2g and S 10 ). Explicit role of FE OS in regulating-ion-transport We then investigated the role of FE OS in the regulating-ion-transport in the nanochannel-system using a two-electrode cell with two symmetric Ag/AgCl electrodes and 0.1 M KCl solution (Fig. 3a ). The current–voltage (I–V) plots of none@OS is asymmetric (Fig. 3b ) owing to the asymmetric structure and surface properties (Figs. S 5 and S 6 ), classified as ion-current rectification (ICR) behavior, in which I +2V (current at +2 V) / I −2V (current at −2 V) is defined as ICR ratio ( f rec ) [31] . After attached FE OS , the f rec increased (for PAA@OS and DNA@OS) and the f rec decrease lower than 1 with an opposite polarity (for PEI@OS) (Fig. 3f ). The variation of ion current above shows a similar trend as the previous reports (FE with distributions in the 1st stage described in Fig. 1 ) [30] . This effect was ascribed to the enhancement of negative charge at outer surface from highly negatived PAA rich in hydroxyl in PAA@OS or DNA with phosphodiester skeleton in DNA@OS (Fig. S 11 ), or charge reversal of surface charge for the highly positive PEI@OS rich in amino, leading to the enhancement of ion accumulation and depletion (Fig. S 16 ). In addition, the ion transport of nanochannels with independent FE OS was affected by ion strength (Fig. S 13 ). Therefore, we estimated that the FE OS influenced the ion currents through nanochannel-system above independently. Fig. 3: Explicit role of FE OS in regulating-ion-transport. a Scheme of a two-electrode cell. b I–V curves characterizing the ion transport through nanochannel-system. c The PNP equation was used for numerical simulations of the effect from FE OS on the ion transport through nanochannel, where two variables remain: ∇ c i and ∇ φ valued by the surface charge density ( σ ) (measured by SSZPA, Fig. 3d ) (1), the diameter of nanochannels (measured by AFM, Fig. 3e ) (2) and the depth of FE at IW (measured by ToF-SIMS, Fig. 2g ) (3) (More details in the section “Numerical simulation” in “Method” section). Thus, unless otherwise stated, the I is available from numerical simulations using the three parameters (1), (2) and (3) measurable in the nanochannel-system with FE OS only but without FE IW . f Comparison of the rectification ratio ( f rec ) measured from I–V tests (experimental value) with the f rec from numerical simulations based on classical equations using measured parameters (theoretical value), but not estimated values like the previous works in 1st and 2nd stage. g Fraction of the measurable surface, which is the area proportion of the OS of measurable properties using mostly current test technologies, in the total surface occupied by FE (Details in Table S1 ). Data from the present work and the previous works using functional solid-state nanochannels [27] , [29] , [56] , [57] , [58] . For I–V tests, five chips were used to obtain each error bar. Statistics of diameters have been done by counting 100 nanochannels for each kind of nanochannels. Full size image In the 1st and 2nd stage, the numerical simulations coupled the Poisson–Nernst–Planck (PNP) equations [32] among the classical equations with steady state continuity equations were performed for FE IW [16] , [17] , [21] , [28] , [29] , [30] , [33] . However, the three parameters of FE IW used in equation were unmeasurable, as (1) the depth of FE at IW, (2) the surface charge, and (3) the diameter of the nanochannels after attaching FE IW . Hence, the hypotheses of the above three parameters were generated in the 1st and 2nd stage, which is unavoidable: (1) FE may not completely cover IW, but leaving the blank area of the deep IW hard for FE to reach [34] . (2) The surface charge of FE IW is substituted by the measurable surface charge of FE OS . Surface charge density is usually different between FE OS and FE IW , due to their different grafting densities [22] . Sometimes the charge of FE OS and FE IW inversed caused by the local polarization [35] . (3) The decrement of nanochannel diameter is roughly estimated by subtracting the straighten length of FE ( D − 2 L SM , where D represent the diameter of nanochannels and L SM represent the straighten length of FE) [36] . However, the FE are mostly not straight, i.e., single-strand DNA [25] . The results from the numerical simulations, therefore, deviate from experiments and even have randomness, which indicates that the above hypotheses bring the deviations or even sometimes errors. In our nanochannel-system, we measured (1) the depth of the part of FE OS , (2) the surface charge, and (3) the diameter of nanochannels after attaching FE OS using ToF-SIMS, SSZPA, and AFM [37] , respectively (Figs. 2 g, 3 d, 3e and Figs. S 11 , S 12 ). Both the qualitative and the quantitative variation of ICR behavior from numerical simulations fitted well with the experimental results, which indicated the explicit regulating-ion-transport accomplished in 3rd stage nanochannel-system (Fig. 3f, g ). One of the important features of FE is of versatile physical and chemical properties, nanochannel-system adapt to a broad range of applications spanning from osmotic energy conversion devices [15] , [16] , [17] , [33] , [38] to biosensors [26] , [27] , [39] , [40] , [41] . Meanwhile, a nanochannel system combining with electrochemistry in a confined space is now a crucial promising field [42] , [43] , which is utilized to dynamically monitor the single molecule [44] , understand the chemical reaction [45] , characterize the single particle [46] , and probe single living cell [47] , etc. Here, we investigated whether the nanochannel-system in 3rd stage with FE OS explicit regulating-ion-transport could fulfill the applications above or even with special performances to the 1st and 2nd. Impact from FE OS on osmotic energy conversion devices The osmotic energy conversion devices were fabricated using nanochannels with only FE OS without FE IW (Fig. 4a–d and Figs. S 16 –S 19 ) [15] , [16] , [17] . The output power density of nanochannel system with independent FE OS was estimated according to the equation P L = I 2 / R L , where I is the current across the circuit and R L is the external load resistance (Fig. S 21 ). It was found that (1) output max power density increased with PAA concentration owing to the enhanced ion selectivity (Fig. 4e and Fig. S 21 ); (2) R channel was almost unchanged with increase of concentration PAA (Fig. 4e and Fig. S 21 ), where R channel was the internal resistance of the nanochannels. For (1), it is easy to understand the selectivity of the cell increased with the negative-charged FE OS (PAA) [20] . For (2), we speculated the R channel did not obviously change with the FE OS . In order to verify our assumptions, two FE OS layer by layer (LbL) assembled on the OS of nanochannels (PAA and PEI sequentially in Fig. 4f–h and Figs. S 22 –S 24 ; and PEI and PAA sequentially in Figs. S 25 and S 26 ) [30] , [48] . No distinctly increase of R channel was observed during two FE OS sequential assembly at the OS in the 3rd stage (Fig. 4i ) owing to the ignored resistance induced by FE OS , while R channel increased with the LbL assembly of PAA and PEI at the IW in the 1st and 2nd stages (Fig. S 27 ). The above results demonstrated that the increase of FE OS (with selectivity) enhanced the “output max power density” of osmotic energy conversion devices (3rd stage), without obviously raising R channel , which indicated a new conceptual route to design large power density nanochannel-system [27] , [49] , [50] , [51] . Fig. 4: Effect from FE OS on osmotic energy conversion devices. a A scheme showing the working mechanism of the osmotic energy conversion devices using nanochannel-system. The electricity by reverse electrodialysis is generated under salt gradient using a nanochannel system containing only FE OS . b Fabrication of the nanochannel-system attached with PAA as FE OS (PAA@OS) using precursor with different PAA concentration (1 or 10 mM). c AFM of PAA@OS using 1 and 10 mM PAA in precursor. d Depth distribution of the part of FE OS in nanochannel-system using precursor solution with different PAA concentration from ToF-SIMS. e Output max power density, ion selectivity and R channel of PAA@OS with different PAA concentration in precursor. f Scheme showing LbL assembly of PAA and PEI at OS sequentially. g AFM of the OS after LbL assembly. h Depth distribution of the part of FE OS (PAA and PEI) in nanochannel-system after LbL assembly. i Reversal f rec through the LbL assembly of PAA and PEI and corresponding R channel . The numerical simulations of two samples above, PAA 4 PEI 3 @OS and PAA 4 PEI 4 @OS, using their measured parameters were performed, which well fit the experimental results (Fig. S 24 ). For the osmotic energy conversion devices, five devices using FE@OS were assembled to obtain each error bar of power density and R channel . Full size image Impact from FE OS on biosensors We further demonstrated a sensing strategy employing single-strand DNA probes as FE OS in the nanochannel-systems (3rd stage) (Fig. 5a ), for the detection of a broad range of targets including inorganic ions (Hg 2+ with 1 pM limit of detection, LoD), small molecules (ATP with 1 pM LoD), proteins (lysozyme with 1 pM LoD), and cancer cells (MCF-7 cells with 400 cells mL −1 LoD) (Fig. 5b, c and Fig. S 28 ). The targets were specifically captured by the designed FE OS as probe and tailored the surface charge of the OS locally, which affect the asymmetry of surface potential in between OS and IW and change the ion transport in form of f rec signal. To confirm the sensing mechanism above, we took ATP detection using ssw-DNA as an example. In ssw-DNA structure, one kind of repeating units in ssw-DNA was designed as ATP aptamer, which specifically bonded with ATP and caused the disassembly of ssw-DNA (Fig. 5d–g and Fig. S 29 ). The variation of surface potential under the disassembly of ssw-DNA triggered by different concentration ATP was quantitatively characterized through the electrochemical approaches (Fig. 5h and Fig. S 30 ). The mechanism was also confirmed by the change grafting density of DNA at OS (Fig. S 31 ). The selective detection for ATP was also realized based on surface-charge-response sensing mechanism (Fig. 5h, i ). In the 1st and 2nd stage, because probes (as FE IW ) were immobilized at the IW, a confined space usually with diameter <100 nm, the targets with a size beyond the diameter of nanochannels can’t sufficiently contact with probe and efficiently recognized. In the 3rd stage, the OS possess the receptive characteristic for probes or targets with the size beyond the diameter of nanochannels. The recognition between probes and targets took place at the OS, which is relatively more free-spaces compared with IW in nanochannels. Here, we realized a nearly “universal” biosensor approach according to the two successful sensing processes: one is the DNA amplifications (ssw-DNA) as the probes, whose diameter is larger than the diameter of the nanochannel-system (some probes is up to 30-fold larger in Fig. 5e–g ), and the other is the MCF-7 cells as the targets, whose diameters is about than 2 magnitudes larger than the diameters of the nanochannel-system. Fig. 5: Sensing performances of FE OS as probes. a A scheme showing the working mechanism of the sensor using FE OS as probes. b Capture process of multi-scale targets through designed single-stand DNA. c Sensitivity and selectivity of the DNA@OS (DNA is a designed sequence specifically bonding with targets) for the recognition of ions (Hg 2+ ), small molecules (ATP), protein (Lysozyme) and cells (MCF7). The selective detection of multiscale targets using FE OS was realized based on the change of f rec signal output induced by the surface charge at outer surface. LoD is defined as the limitation of detection for the targets. d Formation of “supersandwich” DNA structure (ssw-DNA) with long concatamers through the successive hybridization of alternating DNA unit. And gradual disassembly of ssw-DNA based on interaction between ATP and the repeat DNA units through increasing ATP concentration. e Agarose gel electrophoresis characterizing of the ssw-DNA: 1) DNA marker; 2) p1; 3) p2 (ATP ampter); 4) p1 + p2; 5) target+p1 + p2. f Depth distribution of the “supersandwich” DNA in nanochannel-system using ToF-SIMS. g Laser scanning confocal microscopy of the OS after the assembly (top) and the disassembly of ssw-DNA (bottom). The scale bar is 20 μm. h The LoD of ATP using ssw-DNA as probe based on nanochannel method and electrochemical method (Fig. S 30 ). i Specificity of ssw-DNA@OS for ATP, in contrast with other NTPs. For the sensing performances part, five sensors using FE@OS were established to obtain each error bar of sensitivity and specificity. Full size image\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"In conclusion, we have built a de novo designed “minimalist” nanochannel-system with an explicit regulating-ion-transport feature, which has been achieved using FE OS independently without FE IW . The troubles from the “black box” for the properties of FE IW in the 1st and 2nd stage are well avoided without using FE IW , meanwhile the other “black box”, the role of FE OS on regulating-ion-transport, are further demonstrated from the “coadjutant” of FE IW in the 2nd stage to the “monodrama” in the 3rd stage. The use of independent FE OS is a new attempt to separate the enrichment, screening, and recognition process taking place at OS from the ion transport process taking place at IW within a confined space at the nanoscale. Furthermore, utilization of independent FE OS under a less restricted environment than FE IW will endow nanochannel-systems with (1) more measurable properties, such as hydrophobicity (the contact angle measurement is available for FE OS , which is hardly achieved for FE IW in nanoscale confined space) (Fig. S 32 ), (2) more characteristic technologies, such as X-ray photoelectron spectroscopy (XPS), ion microprobe mass analysis (IMMA) (the detection on FE IW requiring for destructive treatment, such as ion thinning, but not necessary for FE OS ), (3) more new performances limited to diffusions in nanochannels previously, such as reusage of FE (Fig. S 33 ), rapid responses to targets (Fig. S 34 ). The FE OS in 3rd stage like “ignition system” trigger the regulatable substrate transports through nanochannels and the abroad application space of nanochannel-systems.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Materials Poly (acrylic acid) (PAA, MW ~5000) was purchased from Ryon Biological Technology (Shanghai, China). Polyethyleneimine (PEI, MW ~10,000) was purchased from Damas-beta. Tris (hydroxymethyl) aminomethane (Tris) was purchased from Alfa Aesar. KCl, NaCl, and MgCl 2 were obtained from Aladdin reagent (Shanghai, China). Adenosine 5′-triphosphate (ATP) disodium salt solution, uridine Triphosphate (UTP), cytidine triphosphate (CTP) and guanosine triphosphate (GTP) were purchased from Sigma-Aldrich. HeLa and PC3 cells were obtained from Chinese Center for Typical Culture Collection (Wuhan, China) and cultured in DMEM (Gibco) supplemented with 10% (v/v) fetal bovine serum (FBS), 2 mg/mL NaHCO 3 , and 100 U/mL antibiotics 15 (penicillin-streptomycin). MCF-7 cells were purchased from KeyGEN Biotech Co. Ltd. (Nanjing, China) and cultured in RPMI-1640 (Gibco) supplemented with 10% FBS, 2 mg/mL NaHCO 3 , and 100 U/mL antibiotics (penicillin–streptomycin) at 37 °C in a humidified 5% CO 2 atmosphere. AAO membranes were purchased from Pu-Yuan Nano Technology Co. Ltd. (Hefei, China). The thickness of AAO membrane is 65 μm. All solutions were prepared using Millipore Milli-Q water (18 MΩ cm). All oligonucleotides are synthesized and purified by Sango Biotech Co. Ltd, (Shanghai, China). The sequences are shown in Tables S2 and S3 . Fabrication of nanochannel systems Preparation of none@OS. A physical vapor deposition (PVD) method was employed using AE Nexdap PVD platform (Angstrom Engineering Inc.) to prepare nanochannels with independent OS [22] , [23] . Two kinds of depositing materials as Au and Cr were used. To enhance the stability of Au, one layer of Cr with a thickness of 10 nm was first deposited at the one side of AAO. The Au was followed deposited at the same side of Cr layer. The circular targets were approximately parallel to the AAO membranes, ensuring the deposition direction perpendicular to the membranes. The deposition was taken by the successive deposition without replacing the target materials or releasing vacuum. The successive deposition ensured no secondary pollution at the first deposited layer. The low depositing speed of Au was applied as 0.1 nm s −1 and the time was 2000 s. The depositing speed was calibrated by the deposited thickness on the surface of the flat silicon wafer at nanometer level. The as-synthesized sample was named none@OS. Preparation of PAA@OS, PEI@OS, and DNA@OS. Aqueous solution including PAA or PEI with different concentrations was prepared. Before attaching PAA or PEI, the Au-coated OS in none@OS was treated by Ar plasma for 60 s to remove impurities. Then, the PAA solution or the PEI solution was dip-coated at the OS. The dip-coating continued for 2 min. Then the sample was washed throughout by water, and dried under N 2 gas. The sample was named PAA@OS and PEI@OS, respectively. Thiol-modified DNA was attached at the OS of none@OS using similar dip-coating method. The reaction time continued for 60 min and the ample was named DNA@OS. The as-prepared samples were then applied for further measurement and characterizations. LbL self-assembly of charged polyelectrolyte. For LbL self-assembly of PAA and PEI on the outer surface of AAO/Au nanochannels, the process was described as follows: an aqueous solution containing PAA (1 mM) was firstly spread onto at the OS of none@OS for the adsorption of PAA polyelectrolyte, and the modification time was 2 min. After the adsorption of first layer of PAA, the solution of PEI was then spread on the same side of none@OS. Subsequently, other alternative layers of PAA/PEI were also electrostatically deposited on the OS of none@OS using the same procedure. Assembly and disassembly of ssw-DNA. The DNA supersandwich structure (ssw-DNA) was prepared by mixing the thiol-modified cp-DNA solution (1 μM) into the mixture of equimolar P1 and P2 (1 μM) for 1 h. Then the solution containing ssw-DNA was dropped onto the OS of none@OS, and the reaction time was 60 min, named ssw-DNA@OS. The disassembly of ssw-DNA was achieved by treating the ssw-DNA@OS with different concentrations of ATP (1 nM–1 μM) for 60 min. Detection of Hg 2+ , ATP, lysozyme, and MCF-7 cells The aptamer solution (1 μM) for different targets were dropped onto the OS of none@OS at room temperature for 60 min. After that, 6-mercapto-1-hexanol (MCH, 100 nM) was added onto the OS for 60 min to prevent nonspecific adsorption. Subsequently, the aptamer modified nanochannels was treatment by the suspension containing various targets. The treatment time was 60 min. After the interaction, the sample was washed by buffer to remove the nonspecifically adsorbed targets. And then the as-treated samples were then applied for characterizations. Cells were first cultured in flasks with Dubelcco’s Modified Eagle’s Medium (DMEM) at 37 °C under 5% CO 2 in the cell incubator followed by centrifuging at 600 × g for 5 min, and re-dispersed in PBS (10 mM, pH 7.4) with a density of 5 × 10 5 cells mL −1 . A certain amount of cells suspension reacted with aptamer modified nanochannels for 60 min. After cells capture, the sample was washed by PBS buffer to remove the nonspecifically adsorbed cells. And then the as-prepared membranes were then applied for characterizations. Characterization SEM images and EDX analyses were taken with a field-emission scanning electron microscope (SU8010, Hitachi, Japan) equipped with Energy Dispersive Xray spectroscopy (EDS, BRUKER AXS, Germany). All samples were coated with carbon (5 nm) prior to SEM examinations. Secondary ion mass spectra of as-prepared samples were characterized by ToF-SIMS V (IONTOF, GmbH). A Bi liquid metal primary ion source was applied with an angle of 45° relative to the sample surface with a pulsed Bi 3++ primary ion beam of 30 keV and shave off fresh 60 µm × 60 µm areas for each analysis. The ToF analyzer was installed at an angle of 90° to the sample surface. Negative secondary ion spectra were collected. Mass calibration was carried out using standard procedures (mass resolving power >5000). The properties of OS of nanochannel system were characterized by an atomic force microscope (Multimode 8, Burke). The determination of OS’s surface zeta potential of nanochannel system were carried out by using SSZPA (SurPASS, Anton Paar Ltd., Austria). The pH dependence of the zeta potential and the isoelectric point (IEP) were obtained in a 0.1 M potassium chloride at the same electrical conductivity. The fluorescent measurement of nanochannel functionalized with fluorescent-dyed DNA at OS were performed by using laser scanning confocal microscope (LSM 880 confocal microscopy, Carl Zeiss) equipped with a FemtoSecond Laser (Coherent Inc.) The fluorescent-dyed membranes were placed at cover glass filling with water (around 1 cm 2 membrane with 20 μL water). For the analysis of assembly and dissembly process of ssw-DNA, the mixture was analyzed and observed using native PAGE gel. The gel was run in 16% acrylamide (containing 19/1 acrylamide/bisacrylamide) solution with 1× TBE buffer, at 100 V constant voltage for 1.5 h. The gel was directly imaged or stained with GelRed (Biotium) for 20 min to image the DNA position by using Tanon imaging system (Tanon 5200 Multi). The water contact angle measurements were operated on a DSA100 contact angle analyzer at ambient temperature and around 30% humidity. A drop of water (5.0 µL) was dropped onto the surfaces of nanochannel membranes. The surface charge density for the OS with ssw-DNA grafting. The surface charge density with ssw-DNA grafting were tested using the system described in Fig. S 28 and calculated based on the following equation [52] : $$Q = \\\\frac{{2n{{FAD}}_0^{1/2}C_0}}{{\\\\pi ^{1/2}}}t^{1/2} + Q_{{\\\\mathrm{dl}}} + n{{FA}}{\\\\it{\\\\Gamma }}_0,$$ (1) where n is electron transfer number, F is the Faraday constant, A is the electrode area (cm 2 ), D 0 is the diffusion coefficient (cm 2 s −1 ), C 0 is the bulk concentration (mol cm −2 ), Q dl is the capacitive charge ( C ), and n FAΓ 0 is the charge from the reduction of Γ 0 (mol cm −2 ) of adsorbed redox marker. The samples were mounted in between the two halves of a homemade electrochemical cell, which contains 1.0 mL KCl solution in each cell. I–V plots were recorded by an electrochemical workstation (CHI, ShangHai) and Keithley 6487 picoammeter (Keithley Instruments, Cleveland, OH). Two Ag/AgCl electrodes were used to apply the potential. The working electrode was placed on the Au side and the reference electrode on the AAO side if not particularly mentioned. The effective area of the membrane for ion conduction test is about 7 mm 2 . All measurements were performed at room temperature, and each test was repeated for five times. Five membranes at least were used to obtain average values. To study the ionic rectification properties, a scanning voltage from −2.0 to 2.0 V was applied across the membrane. The ion concentration on the Au side and the AAO side was kept equal (0.1 M KCl, pH 7.0), if not specially mentioned. To study the energy conversion properties [53] a salt gradient was designed, where the concentration of the solution on the Au side is higher than that on the AAO side if not particularly mentioned. Numerical simulations The ionic rectification phenomenon was theoretically investigated using a commercial finite-element software package COMSOL Multiphysics [54] , [55] . $$J_{\\\\rm{i}} = D_{\\\\rm{i}}\\\\left( {\\\\nabla c_{\\\\rm{i}} + \\\\frac{{z_{\\\\rm{i}}Fc_{\\\\rm{i}}}}{{RT}}\\\\nabla \\\\varphi } \\\\right),$$ (2) $$\\\\nabla ^2\\\\varphi = - \\\\frac{F}{\\\\varepsilon }{\\\\sum} {z_{\\\\rm{i}}c_{\\\\rm{i}}},$$ (3) $$\\\\nabla \\\\cdot J_{\\\\rm{i}} = 0.$$ (4) Here, Eq. ( 2 ) is the Nernst–Planck equation that descripts the transport properties of a charge nanochannel and Eq. ( 3 ) is the Poisson equation that descripts the relationship between the electric potential and ion concentration in the nanochannels. The model is generally simplified using steady-state conditions (Eq. ( 4 )). The electroosmotic flow was neglected in this work. The coupled Eqs. ( 2 – 4 ) can be solved utilizing appropriate boundary conditions (Eqs. 4 , 6 and Table S4 ). The solution yields the concentration field c i for all species and the potential φ distributions in the nanochannels. $${\\\\mathbf{n}} \\\\cdot \\\\nabla \\\\varphi = - \\\\frac{\\\\sigma }{\\\\varepsilon },$$ (5) $${\\\\mathbf{n}} \\\\cdot J_{\\\\mathrm{i}} = 0.$$ (6) The total ionic current through the nanochannel can be calculated: $$I = {\\\\int\\\\!\\\\!\\\\!\\\\!\\\\!\\\\int} {J_{\\\\rm{i}}{\\\\rm{d}}s} = - {\\\\int\\\\!\\\\!\\\\!\\\\!\\\\!\\\\int} {D\\\\left( {\\\\nabla c_{\\\\rm{i}} + z_{\\\\rm{i}}c_{\\\\rm{i}}\\\\frac{F}{{RT}}\\\\nabla \\\\varphi } \\\\right){\\\\rm{d}}s}.$$ (7) The formula above was used for numerical simulations of the effect from FE OS on the ion transport through nanochannel, where i refer to cation (K + ) and anion (Cl − ) and J i refers to ionic flux [16] . When the properties of electrolyte are set, the invariants include: D i as the diffusion coefficient, c i as the concentration of K + and Cl − , z i as the valence of K + and Cl − , T as the thermodynamic temperature. Meanwhile, in this formula, the constants include: F as the Faraday’s constant and R as the ideal gas constant. As a result, two variables remain: ∇ c i describes the concentration gradient of ion along nanochannels under external electric field and ∇ φ describes the electrical potential gradient along nanochannels, in which the φ is valued by the surface charge density ( σ ) of the FE OS and the nanochannels (measured by SSZPA, Fig. 3d ) (1). The ∇ c i and the ∇ φ are calculated by integrating the c i of electrolyte and φ over the cross-section area (d s ). The integration of d s is valued by the diameter of nanochannels (measured by AFM, Fig. 3e ) (2), and the depth of FE at IW (measured by ToF-SIMS, Fig. 2g ) (3).\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"The core research idea is to investigate the previously overlooked role of function elements at the outer surface (FE OS) in regulating ion transport within nanochannel-systems by precisely confining and characterizing these elements at the edge of the inner wall with a minimum depth of 7.5 nm, demonstrating their independent influence on ionic selectivity, osmotic energy conversion, and biosensing capabilities through a combination of theoretical modeling and advanced experimental techniques."},"entities":{"kind":"list like","value":["function elements at the outer surface","FE OS","nanochannel-systems","ion transport","inner wall","7.5 nm depth","ionic selectivity","osmotic energy conversion","biosensing capabilities","theoretical modeling","advanced experimental techniques"],"string":"[\n \"function elements at the outer surface\",\n \"FE OS\",\n \"nanochannel-systems\",\n \"ion transport\",\n \"inner wall\",\n \"7.5 nm depth\",\n \"ionic selectivity\",\n \"osmotic energy conversion\",\n \"biosensing capabilities\",\n \"theoretical modeling\",\n \"advanced experimental techniques\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1.","title":"Kowalczyk, S. 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Mater.24, 962–967 (2012).","link":"https://doi.org/10.1002%2Fadma.201104814"}],"string":"[\n {\n \"idx\": \"1.\",\n \"title\": \"Kowalczyk, S. W., Blosser, T. R. & Dekker, C. Biomimetic nanopores: learning from and about nature.Trends Biotechnol.29, 607–614 (2011).\",\n \"link\": \"https://doi.org/10.1016%2Fj.tibtech.2011.07.006\"\n },\n {\n \"idx\": \"2.\",\n \"title\": \"Bocquet, L. Nanofluidics coming of age.Nat. Mater.19, 254–256 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41563-020-0625-8\"\n },\n {\n \"idx\": \"3.\",\n \"title\": \"Martin, C. R. & Siwy, Z. S. Learning nature’s way: biosensing with synthetic nanopores.Science317, 331–332 (2007).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1146126\"\n },\n {\n \"idx\": \"4.\",\n \"title\": \"Keyser, U. F. Enhancing nanopore sensing with DNA nanotechnology.Nat. Nanotechnol.11, 106–108 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fnnano.2016.2\"\n },\n {\n \"idx\": \"5.\",\n \"title\": \"Liu, L., Yang, C., Zhao, K., Li, J. & Wu, H. Ultrashort single-walled carbon nanotubes in a lipid bilayer as a new nanopore sensor.Nat. Commun.4, 2989 (2013).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms3989\"\n },\n {\n \"idx\": \"6.\",\n \"title\": \"Burns, J. R., Seifert, A., Fertig, N. & Howorka, S. A biomimetic DNA-based channel for the ligand-controlled transport of charged molecular cargo across a biological membrane.Nat. Nanotechnol.11, 152–156 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fnnano.2015.279\"\n },\n {\n \"idx\": \"7.\",\n \"title\": \"Ying, Y. L. et al. Manipulating and visualizing the dynamic aggregation-induced emission within a confined quartz nanopore.Nat. Commun.9, 3657 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-018-05832-y\"\n },\n {\n \"idx\": \"8.\",\n \"title\": \"Liang, B. et al. Microporous membranes comprising conjugated polymers with rigid backbones enable ultrafast organic-solvent nanofiltration.Nat. Chem.10, 961–967 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41557-018-0093-9\"\n },\n {\n \"idx\": \"9.\",\n \"title\": \"Hou, X., Hu, Y., Grinthal, A., Khan, M. & Aizenberg, J. Liquid-based gating mechanism with tunable multiphase selectivity and antifouling behaviour.Nature519, 70–73 (2015).\",\n \"link\": \"https://doi.org/10.1038%2Fnature14253\"\n },\n {\n \"idx\": \"10.\",\n \"title\": \"Lozada-Hidalgo, M. et al. Sieving hydrogen isotopes through two-dimensional crystals.Science351, 68–70 (2016).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.aac9726\"\n },\n {\n \"idx\": \"12.\",\n \"title\": \"Xie, Q. et al. Fast water transport in graphene nanofluidic channels.Nat. Nanotechnol.13, 238–245 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41565-017-0031-9\"\n },\n {\n \"idx\": \"13.\",\n \"title\": \"Simon, P., Gogotsi, Y. & Dunn, B. Where do batteries end and supercapacitors begin?Science6176, 1210–1211 (2014).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1249625\"\n },\n {\n \"idx\": \"14.\",\n \"title\": \"Liu, C. et al. An all-in-one nanopore battery array.Nat. Nanotechnol.9, 1031–1039 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fnnano.2014.247\"\n },\n {\n \"idx\": \"15.\",\n \"title\": \"Siria, A. et al. Giant osmotic energy conversion measured in a single transmembrane boron nitride nanotube.Nature494, 455–458 (2013).\",\n \"link\": \"https://doi.org/10.1038%2Fnature11876\"\n },\n {\n \"idx\": \"16.\",\n \"title\": \"Feng, J. et al. Single-layer MoS2nanopores as nanopower generators.Nature536, 197–200 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fnature18593\"\n },\n {\n \"idx\": \"17.\",\n \"title\": \"Gao, J. et al. High-performance ionic diode membrane for salinity gradient power generation.J. Am. Chem. 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This need is particularly critical for cervical cancer, where molecular detection of human papillomavirus (HPV) has emerged as an accurate screening method. However, implementing this transition in low- and middle-income countries has been challenging due to the high costs and centralized facilities required for current NA tests. Here, we present CreDiT (CRISPR Enhanced Digital Testing) for on-site NA detection. The CreDiT platform integrates i) a one-pot CRISPR strategy that simultaneously amplifies both target NAs and analytical signals and ii) a robust fluorescent detection based on digital communication (encoding/decoding) technology. These features enable a rapid assay (<35 minutes) in a single streamlined workflow. We demonstrate the sensitive detection of cell-derived HPV DNA targets down to single copies and accurate identification of HPV types in clinical cervical brushing specimens ( n = 121)."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"The global incidence of human papillomavirus (HPV)-driven cancers, such as cervical, anal, and head and neck cancer, is on the rise, especially in underserved regions with limited healthcare infrastructure. Cervical cancer alone accounts for over 600,000 new cases a year, with more than 80% of them occurring in low- and middle-income countries (LMICs). Prompt and reliable triaging of high-risk HPV (hrHPV) is essential to offset severe pathology bottlenecks in resource-limited regions and circumvent geographical and/or socioeconomic barriers to effective cervical cancer screening [1] , [2] , [3] . Molecular diagnostics for hrHPV have demonstrated superior sensitivity (96–100%) and specificity (90–100%) when compared to standard cytology (e.g., Pap smear) [4] . Consequently, developed countries are increasingly shifting towards centralized nucleic acid (NA) testing as the sole screening method for identifying high-risk cases. However, implementing such tests in LMICs still faces technical challenges. The primary assay method, polymerase chain reaction (PCR), incurs high fixed and operational costs, straining public health budgets [5] , [6] . Centralized tests are also incompatible with the “screen-and-treat” strategy employed in LMICs—the one-time window of intervention with cervical ablative therapy. Visual inspection with acetic acid (VIA) is a rapid, inexpensive alternative but suffers from poor accuracy (i.e., high rates of false results) [7] , [8] . These realities underline the unmet need for fast, accurate hrHPV testing, ideally administered at point-of-care (POC) in LMIC settings. Clustered regularly interspaced short palindromic repeats (CRISPR) technologies are emerging as a powerful tool in nucleic acid (NA) detection [9] , [10] , [11] . One key advantage of this approach is its capability for sequence-specific signal amplification. When a CRISPR-associated (Cas) protein binds to its target NA through guide RNA (gRNA), the resulting complex becomes an active endonuclease capable of non-selectively degrading single-stranded NAs. This property has been harnessed to amplify analytical signals, with activated Cas proteins cleaving abundant non-targeted signaling probes such as single-stranded NAs tagged with a fluorescent dye (F) and quencher (Q) pair. The rapid and isothermal nature of Cas reactions has also facilitated on-site operations (Supplementary Table 1 ). We reasoned to adopt this Cas mechanism for fast, accurate NA detection in resource-limited settings. Here, we report a robust POC technology, termed CreDiT (for CRISPR Enhanced Digital Testing), devised for decentralized HPV testing and HPV-related cancer screening. CreDiT integrates two distinct strategies: (i) a Cas reaction that concurrently amplifies and detects target NAs and (ii) an efficient signal processing inspired by digital radio communications [12] , [13] . This combination confers notable advantages: (i) CreDiT assays are fast (<20 min), isothermal, and conveniently performed in a single tube; (ii) signal detection is resilient against extraneous noise and interference, enhancing assay reliability and precision; and iii) the detection system is compact for field use without requiring complex optics. Exploiting these advantages, we implement a CreDiT platform positioned for cancer diagnostics. Specifically, we design CreDiT probes for hrHPV genes (HPV16, HPV18, HPV45, HPV31, HPV33, and HPV58) and oncoprotein mRNAs ( E6 , E7 , and p16 INK4a ), unify a workflow for NA extraction and CreDIT assay, and build a portable CreDiT system to process multiple (12) samples. The system detects cell-derived HPV DNA targets down to single copies in 35 min (including DNA extraction) and accurately classifies hrHPV types in clinical cervical brushings ( n = 121). We further expand CreDiT’s utility and clinical impact by detecting hrHPV in anal swab samples from separate patients ( n = 48) [14] , [15] . The results underscore CreDiT’s versatility and potential for HPV-related clinical assessments.","references":{"indices":[1,2,3,4,5,6,7,8,9,10,11,12,14,15],"infos":[{"idx":"1","title":"Randall, T. C. & Ghebre, R. Challenges in prevention and care delivery for women with cervical cancer in sub-Saharan Africa.Front. Oncol.6, 160 (2016).","link":"https://doi.org/10.3389%2Ffonc.2016.00160"},{"idx":"2","title":"Montealegre, J. R. et al. Racial/ethnic differences in HPV 16/18 genotypes and integration status among women with a history of cytological abnormalities.Gynecol. Oncol.148, 357–362 (2018).","link":"https://doi.org/10.1016%2Fj.ygyno.2017.12.014"},{"idx":"3","title":"Islami, F., Fedewa, S. A. & Jemal, A. Trends in cervical cancer incidence rates by age, race/ethnicity, histological subtype, and stage at diagnosis in the United States.Prev. Med.123, 316–323 (2019).","link":"https://doi.org/10.1016%2Fj.ypmed.2019.04.010"},{"idx":"4","title":"Sankaranarayanan, R. et al. HPV screening for cervical cancer in rural India.N. Engl. J. Med.360, 1385–1394 (2009).","link":"https://doi.org/10.1056%2FNEJMoa0808516"},{"idx":"5","title":"Castle, P. E. et al. Reliability of the Xpert HPV assay to detect high-risk human papillomavirus DNA in a colposcopy referral population.Am. J. Clin. Pathol143, 126–133 (2015).","link":"https://doi.org/10.1309%2FAJCP4Q0NSDHWIZGU"},{"idx":"6","title":"Zhang, J. et al. Effectiveness of high-risk human papillomavirus testing for cervical cancer screening in China: a multicenter, open-label, randomized clinical trial.JAMA Oncol.7, 263–270 (2021).","link":"https://doi.org/10.1001%2Fjamaoncol.2020.6575"},{"idx":"7","title":"Goel, A. et al. Visual inspection of the cervix with acetic acid for cervical intraepithelial lesions.Int. J. Gynaecol. Obstet.88, 25–30 (2005).","link":"https://doi.org/10.1016%2Fj.ijgo.2004.09.018"},{"idx":"8","title":"Hegde, D., Shetty, H., Shetty, P. K. & Rai, S. Diagnostic value of acetic acid comparing with conventional Pap smear in the detection of colposcopic biopsy-proved CIN.J. Cancer Res. Ther.7, 454–458 (2011).","link":"https://doi.org/10.4103%2F0973-1482.92019"},{"idx":"9","title":"Lee, C. Y. et al. Development of integrated systems for on-site infection detection.Acc. Chem. Res.54, 3991–4000 (2021).","link":"https://doi.org/10.1021%2Facs.accounts.1c00498"},{"idx":"10","title":"Aman, R., Mahas, A. & Mahfouz, M. Nucleic acid detection using CRISPR/Cas biosensing technologies.ACS Synth. Biol.9, 1226–1233 (2020).","link":"https://doi.org/10.1021%2Facssynbio.9b00507"},{"idx":"11","title":"Kaminski, M. M., Abudayyeh, O. O., Gootenberg, J. S., Zhang, F. & Collins, J. J. CRISPR-based diagnostics.Nat. Biomed. Eng.5, 643–656 (2021).","link":"https://doi.org/10.1038%2Fs41551-021-00760-7"},{"idx":"12","title":"Grove, A. C. An introduction to Walsh functions and their applications.Int. J. Math. Educ. Sci. Technol.14, 43–53 (1983).","link":"https://doi.org/10.1080%2F0020739830140107"},{"idx":"14","title":"Wei, F. et al. Epidemiology of anal human papillomavirus infection and high-grade squamous intraepithelial lesions in 29,900 men according to HIV status, sexuality, and age: a collaborative pooled analysis of 64 studies.Lancet HIV8, e531–e543 (2021).","link":"https://doi.org/10.1016%2FS2352-3018%2821%2900108-9"},{"idx":"15","title":"Marra, E., Lin, C. & Clifford, G. M. Type-specific anal human papillomavirus prevalence among men, according to sexual preference and HIV status: a systematic literature review and meta-analysis.J. Infect. Dis.219, 590–598 (2019).","link":"https://doi.org/10.1093%2Finfdis%2Fjiy556"}]}},{"section":"Results","content":"CreDiT platform development We devised the CreDiT assay to minimize user intervention, enabling process automation and effectively achieving a sample-in-answer-out test (Fig. 1a ). Our NA assay thus integrated NA amplification and Cas12a-based NA recognition (Fig. 1b ). During CreDiT’s initial phase, the polymerase enzyme separates the double-stranded target DNA, creating a single-stranded region for the gRNA molecule to recognize. This recognition triggers the Cas12a enzyme within the Cas12a/gRNA complex to act as a nuclease and cleave the F-Q DNA probes, generating a fluorescent signal. However, activated Cas12a can also interfere with RPA by cleaving the target DNA itself ( cis -cleavage) and breaking down the RPA primers ( trans -cleavage). Reassuringly, our kinetic analysis indicates that RPA is faster than these competing reactions (see Supplementary Note for details). This dominance of RPA ensures sufficient amplification of the target DNA within the CreDiT format. Fig. 1: CreDiT assay for HPV detection. a Streamlined onsite workflow. The assay executes the nucleic acid (NA) extraction and the CreDiT reaction in a single pot, minimizing the hands-on times. The total assay time is about 36 min. b CreDiT assay principle. Two reactions concurrently occur: recombinase polymerase amplification (RPA) and Cas12a-based signal generation. RPA is the major reaction that replicates the target DNA. During RPA, the polymerase displaces the DNA duplex, exposing single-stranded regions recognizable by Cas12a/gRNA complexes. This recognition event activates the complexes to cleave dye-quencher (F-Q) probes, generating fluorescent signals. As the RPA cycles proceed, more Cas12a/gRNA complexes are activated, amplifying the overall signal. All reactions are carried out at a constant temperature (42 °C) without requiring external interruptions. c A compact, integrated assay system was developed. The device had a turret-shaped heating block accommodating 12 samples and an optical module for fluorescent detection (LED light-emitting diode, PD photodiode). The heating block, mounted on a motor, rotated to position each sample for fluorescent measurements. The device had a small form factor (14 × 11 × 6.5 cm 3 ) and connected to a computer. Custom-designed software running on a computer presented a graphical user interface for device control and real-time data analysis. d Two microcontroller units (MCUs) automated the CreDiT assay processes. MCU1 controlled heating elements to maintain the desired temperatures and rotated the heating block during optical detection. MCU2 generated waveforms for fluorescent excitation and detected the resulting signal. Full size image All assay steps are performed at a fixed temperature (42 °C, isothermal) without requiring external interruptions. Furthermore, the assay is fast and sensitive, as it concurrently amplifies target DNA (RPA) and analytical signals (Cas12a/gRNA). Similar one-pot approaches have been described [16] , [17] ; our analysis may be applicable to understand their underlying reaction mechanisms and inform optimizations. To translate CreDiT’s practical advantages into on-site testing, we constructed an automated assay system (Fig. 1c and Supplementary Movie 1 ). The system hardware integrated a rotating sample holder, heating components, an optical detection module, and microcontroller units (detailed schematic in Supplementary Fig. 1 ). The sample holder, machined in aluminum, had embedded resistive heaters and temperature sensors and was turret-shaped to accommodate 12 standard PCR tubes (6 mm in diameter). The optical module was configured for fluorescent measurements, including a light-emitting diode (LED) for excitation and a photodiode for detection. We controlled the system operation using two onboard microcontrollers (Fig. 1d ), one for sample heating and the other for signal detection. The heating controller kept the sample holder at a desired temperature for a given assay step (e.g., NA extraction, CreDiT reaction). It executed a feedback loop control, adjusting input powers to heaters and a fan based on the temperature sensor readouts. When the reaction was over, this controller also rotated the sample holder using a stepping motor and positioned each sample for fluorescent detection. The signal detection controller directed optical measurements. It generated waveforms to modulate fluorescent excitation and processed the resulting light signals for demodulation. The approach enabled multi-sample measurement with an identical optical module, ensuring consistent signal detection and simplifying the system design. The developed device was compact with a form factor of 14 × 11 × 6.5 cm [3] . We also implemented user-interface software for device control and real-time data analysis on a computer (Supplementary Fig. 2 ). Robust optical detection for POC operation Ideal point-of-care (POC) devices should have fewer components to minimize design complexity and reduce cost, but this requirement may limit system reliability and analytical capacity. Fluorescent devices are in such a category. Simple optical setups may be used, although they are susceptible to environmental and system noises. Conversely, reliable fluorescent measurements use calibrated, high-end systems, challenging POC operations. To address this issue, we designed the CreDiT optical system to generate high-quality signals in a compact device. We specifically adopted the Walsh–Hadamard transform (WHT) [18] , a digital signal processing technique used in wireless communications [19] . The WHT is similar to the Fourier transform (FT) but uses non-sinusoidal, rectangular waves (Walsh functions) as an orthogonal basis (Supplementary Fig. 3a ). Any time-domain signals can be decomposed into a weighted sum of Walsh functions [12] , [20] . Moreover, Walsh functions are uniquely defined by a sequence index, which is analogous to the frequency in FT. With these attributes, the WHT can compress time-domain signals into a simple collection of coefficients for Walsh sequence values (Supplementary Fig. 3b ). The WHT method offers further advantages [21] . (i) Walsh functions, inherently binary in nature, are readily compatible with digital electronics. (ii) The signal processing algorithm, referred to as the fast Walsh–Hadamard transform (FWHT), is lightweight and efficient for portable computing, requiring only additions and subtractions of real numbers (Supplementary Fig. 4 ). This contrasts with the fast Fourier transform (FFT) algorithm, which involves multiplying complex numbers. (iii) More importantly, the WHT is more robust to external interference than other signal processing techniques. The WHT spreads the original signal over a wide frequency band [13] . Such spreading effectively reduces the relative influence of interfering signals, particularly those with narrow bandwidths [22] . To apply the WHT in fluorescent detection, we first generated a signal for fluorescent excitation (Fig. 2a , left). We specified values for the Walsh sequency indices and converted these values into a time-domain waveform using the FWHT. This waveform drove an LED, exciting fluorescent dyes in a sample. In the detection path (Fig. 2a , right), we obtained the emitting light using a photodiode (PD) and decomposed the signal via inverse FWHT. This process revealed the Walsh sequency indices used to modulate the fluorescent excitation, and the peak values at these indices were proportional to the fluorescent intensity (i.e., the CreDiT signal). The measurement was completed in 2.5 ms, minimizing the risk of photobleaching fluorophores. Fig. 2: CreDiT system engineering. a CreDiT approach to fluorescent detection. The digital input, represented by a Walsh–Hadamard (WH) sequency, is transformed into a fluorescent excitation waveform using a fast WH transform (FWHT). The resulting signal from a sample is processed via inverse FWHT (iFWHT) to recover WH sequency, whose amplitude is proportional to the fluorescent intensity. b Digital CreDiT detection outperformed analog lock-in (Fourier) and direct fluorescent detection, exhibiting a higher signal-to-noise ratio (SNR) than the others. Data were displayed as mean intensity ± s.d. from technical replicates ( n = 10). c Samples with varying fluorescein concentrations were measured. With its high SNR, digital CreDiT detection achieved >2200-fold lower detection limit (0.4 pM) than the analog lock-in method (Fourier; 1 nM). Data were displayed as mean ± s.d. from triplicate measurements. d The digital CreDiT results from ( c ) also showed an excellent correlation ( R 2 = 0.99) with those measured by a conventional plate reader. e Sample heating setup. The sample holder accommodated 12 samples for heating and was embedded with four pairs of a heater and a temperature sensor. An external fan was used to expedite sample cooling. f Sample temperature profile. The system reached different temperature targets for the CreDiT assay (e.g., NA extraction, CreDiT reaction). Thermal imaging (inset) confirmed uniform heating of all 12 samples. g The heating system maintained the target temperature with relative variations <0.6%. Data were shown as mean ± s.d. from repeated temperature measurements ( n = 100). a.u. arbitrary unit. Source data are provided as a Source Data file. Full size image We determined the optimal Walsh sequency by assessing the system’s performance under the excitation of different Walsh waveforms. Specifically, we measured the fluorescent signals from a standard sample ([fluorophore] = 100 µM) while sweeping Walsh sequency (Supplementary Fig. 5a ). The system achieved a high signal-to-noise ratio (SNR) within the range of sequences from 97 to 130 (Supplementary Fig. 5b ). We selected sequency 107 for its peak SNR in our current prototype. For different systems, the sequency number can be readily adjusted to maximize their SNR. We further compared the WHT-based detection (CreDiT) with other frequently used techniques: constant illumination and analog lock-in (Fourier) methods. For this task, we programmed the onboard microcontroller to generate either a direct current (DC) or a sinusoidal waveform for LED excitation. Acquired fluorescent signals from a sample ([fluorophore] = 100 µM) were then numerically processed to extract fluorescent intensities (Methods). This approach allowed for an unbiased comparison among signal processing algorithms by keeping the optical setup and electronics identical. The WHT method outperformed the other two, displaying the highest SNR (Fig. 2b ). In fluorophore-titration measurements, the WHT detection achieved a superior sensitivity, with its limit of detection (0.4 pM) about 2200-fold lower than that (1 nM) of the lock-in method (Fig. 2c ). The results by the WHT detection also showed excellent linear correlation ( R 2 = 0.99) with a commercial plate reader (Fig. 2d ), which confirmed the reliable analytical performance of the developed signal-processing approach. Temperature control for CreDiT reaction To enable consistent CreDiT reactions, we engineered precise temperature control for the assay device. We first optimized the sample holder design to facilitate rapid and even heating for all 12 samples. Through numerical simulations, we determined that a holder featuring hollow pockets would heat up more rapidly than a solid one, primarily due to the reduced thermal mass (Supplementary Fig. 6 ). Based on this design insight, we proceeded to fabricate a holder in aluminum. The holder had four quadrants (Fig. 2e ); each quadrant accommodated three samples and was embedded with a pair of a heater and a temperature sensor. To regulate the holder temperature, we programmed a microcontroller for feedback control: it read temperature sensors every 100 msec, averaged their outputs, and generated a control signal for heating elements or a cooling fan. Applying the method, we could create reliable temperature profiles suited for different assay steps (Fig. 2f ). On average, the accuracy was within ±0.3 °C of the target temperature, and the precision was 0.6% as measured by the coefficient of variation (Fig. 2g ). The temperature was also uniform across the 12 samples as evidenced by the overall heat distribution recorded throughout the assay (Supplementary Movie 1 ). Assay optimization for POC operation We proceeded to fine-tune the entire CreDiT assay process, aiming to establish a streamlined workflow. For NA extraction from cells, we chose to apply enzymatic cell digestion, a method that does not require external instruments (e.g., centrifuge and spin-columns) and can be controlled by adjusting the temperature (Fig. 3a ). We used a lysis buffer containing proteinase K (Pro K) to release cellular NAs and protect them from nucleases. During the lysis, the sample was heated to 56 °C to enhance Pro K activity. Using cervical cancer cells (Ca Ski), we optimized the lysis time (15 min) and the buffer composition ([Pro K] = 0.5 mg/mL) for the maximal NA yield (Fig. 3b ). Following the NA extraction, we deactivated Pro K by elevating the buffer temperature; this process was necessary to prevent Pro K from affecting the downstream CreDiT reaction. We found that a brief incubation (1 min) at 95 °C was sufficient (Supplementary Fig. 7 ). Fig. 3: Assay optimization for POC applications. a NA extraction protocol. Cells are lysed by proteinase K (Pro K) to release NAs. Subsequently, Pro K is heat-deactivated (95 °C), and the CreDiT reaction is initiated. b The lysis time and Pro K concentration ([Pro K]) were optimized to maximize the NA extraction yield. Samples containing Ca Ski cells (2.5 × 10 4 cells/mL) were analyzed for HPV16. (Left panel) The extraction yields reached a plateau (red arrow) within 15-min of lysis. HPV16 signals from 20-min lysis were used for normalization. (Right panel) The extraction yield was maximal (blue arrow) at [Pro K] = 0.5 mg/mL and comparable to that of a spin column (gold standard). Data were displayed as mean ± s.d. from technical triplicates. c Extracted NAs were PCR-amplified for the HPV16 gene. Gel electrophoresis showed comparable bands between the CreDiT and the spin-column extraction methods. bp, base pairs. d Both extraction methods (CreDiT, spin column) yielded statistically identical results (unpaired, two-sided t -test) in analytical assays. Data were displayed as mean ± s.d. from technical quadruplicates. e CreDiT reagents were lyophilized and stored under ambient conditions. The lyophilized reagents demonstrated consistent activity for at least two weeks. Ca Ski cells (2.5 × 10 4 cells/mL) were analyzed for HPV16. Data were displayed as mean ± s.d. from technical triplicates. a.u. arbitrary unit. Source data are provided as a Source Data file. Full size image The performance of our extraction protocol was similar to that of a standard method (commercial spin-column). For this evaluation, we extracted NAs from cells (Ca Ski) and subjected them to PCR and CreDiT analysis. NAs from both extraction methods had similar quality (Supplementary Fig. 8a ) and generated strong PCR bands of comparable size and intensity (Fig. 3c ). The extraction yields were also comparable between the two extraction methods, as confirmed by PCR and CreDiT assays producing statistically indistinguishable (unpaired, two-sided t -test) signals (Fig. 3d ). When compared to the method using only thermal lysis (95 °C) for NA release, the CreDiT extraction yielded six times more NA (Supplementary Fig. 8b ); this improvement was likely due to the additional benefits of Pro K, notably the degradation of nucleases (i.e., RNases and DNases) that break down NA. We further assessed the feasibility of storing samples and reagents under ambient conditions; this capability is desirable for promoting CreDiT applications in LMIC sites. From the perspective of sample storage, we tested the stability of cells in our extraction buffer. We spiked cells (Ca Ski) in the buffer and stored sample aliquots either in a refrigerator (4 °C) or under ambient conditions before NA extraction. For both storage conditions, we observed no notable decrease in NA amounts for at least two weeks (Supplementary Fig. 9 ). Regarding CreDiT reagents, we premixed the components (RPA primers, reporter probe, Cas12a gRNAs, and Cas12a protein) and lyophilized the mixture (see Methods) to facilitate its transport and storage. When evaluated via the CreDiT assay, the lyophilized premix retained its efficacy for at least two weeks in storage at ambient conditions (Fig. 3e ). Overall, these features could mitigate logistic challenges and promote CreDiT usability in resource-limited settings even without cold chain solutions. CreDiT probe design and validation We prepared a library of CreDiT probes for hrHPV detection. For a given target DNA, CreDiT used a pair of probes, one forward and one reverse, to span both ends of the target sequence (Fig. 4a ). At each end of the sequence, we designed an RPA primer and a Cas12a gRNA against a target region. The primer, typically 30–33 nucleotides in length, served to amplify the target region, and the gRNA featured a spacer sequence allowing Cas12a to bind to a region positioned a few bases (2–10 nucleotides) downstream from the primer binding site (detailed design flow in Supplementary Fig. 10 ). Because Cas12a was activated by a single-strand DNA, the gRNAs can be designed without the constraint of a protospacer adjacent motif (PAM) sequence, offering enhanced flexibility. Capitalizing on this flexibility, we designed CreDiT probes for a diverse range of targets (see Supplementary Tables 2 , 3 ): (i) genomic DNA of hrHPV subtypes prevalent in Africa (HPV16, HPV18, HPV31, HPV33, HPV45, and HPV58); (ii) oncogenic E6 / E7 mRNA whose presence may indicate progressive HPV16 infections; and (iii) p16 INK4a ( CDKN2A ) mRNA that is upregulated in transformed cells as a consequence of E6/E7 expression. Fig. 4: CreDiT probe design. a For a given target DNA (gray backbone), a pair of CreDiT probes (forward and reverse) were designed, with each probe consisting of an RPA primer (green backbone) and a Cas12a/gRNA complex (orange backbone). Upon binding to the target DNA, the primer exposes a downstream sequence complementary to gRNA. The Cas12a/gRNA complex then recognizes its target and becomes an active endonuclease to cleave F-Q reporters. HPV16 probes are shown. b Assay validation with varying reaction conditions. HPV16 DNA was used as a model target ([DNA] = 4 × 10 7 copies/mL). A high signal was only observed when all assay components were present. Data were displayed as mean ± s.d. from technical triplicates. c CreDiT (20 min) and qPCR (1 h) assays were used to analyze samples with different target DNA concentrations. The CReDiT assay exhibited a lower detection limit (40 copies/mL) and a wider dynamic range (4 orders of magnitude) than qPCR (detection limit, 400 copies/mL; dynamic range, 2.5 orders of magnitude). ∆CreDiT is the background-subtracted signal. Data were displayed as mean ± s.d. from technical triplicates. d Probe selectivity. A panel of probes was designed for hrHPV targets (HPV16, HPV18, HPV31, HPV33, HPV45, and HPV58) and GAPDH . These probes detected their intended targets with negligible off-target signals. [Target DNA] = 4 × 10 7 copies/mL. The heatmap displays mean values from technical triplicates. a.u. arbitrary unit. Source data are provided as a Source Data file. Full size image We validated the designed probes in the CreDiT assay. We first varied the composition of the assay reagents and monitored the resulting CreDiT signals (Fig. 4b and Supplementary Fig. 11 ). As expected, the fluorescent signal was only observed when all CreDiT reagents and the target DNA were present. We further observed the synergistic benefit of recognizing both the forward and reverse regions in a target DNA. The combination of forward and reverse gRNAs produced stronger analytical signals than individual gRNAs (Supplementary Fig. 12 ). We also designed an additional gRNA that recognized a middle region of the target DNA (Supplementary Table 2 ). However, this gRNA displayed the weakest signal generation, and incorporating it into the assay lowered the overall signal, likely due to competition with the more efficient forward and reverse gRNAs for Cas12a binding. Next, we compared CreDiT’s performance with the established gold standard, qPCR, and a two-step assay, where RPA (20 min) and Cas12a detection (30 min) were carried out sequentially. We used synthetic HPV16 DNA as a model target. CreDiT demonstrated superior sensitivity compared to qPCR (Fig. 4c ) and the two-step assay (Supplementary Fig. 13 ). CreDiT’s limit of detection (LOD) was down to 40 copies/mL, whereas LODs for qPCR and the two-step assay were around 400 copies/mL. CreDiT also offered an advantage in assay time, requiring only 20 min compared to 1 h for qPCR and 50 min for the two-step assay. Furthermore, the designed CreDiT probes exhibited excellent selectivity with minimal cross-reactivity, as confirmed by the CreDiT assay (Fig. 4d and Supplementary Fig. 14 ) and gel electrophoresis (Supplementary Fig. 15 ). CreDiT assay characterization With the detection system and probes ready, we characterized the entire CreDiT assay flow, beginning with NA extraction from cervical cancer cells. To assess the assay kinetics, we extracted DNA from varying numbers of Ca Ski cells (HPV16+) and performed CreDiT assays, monitoring the signal evolution over time. Even at low cell concentrations, the signal rose rapidly above the background (Fig. 5a ); this can be attributed to the dual enzymatic activities in CreDiT, namely, RPA for NA duplication and Cas12a reaction for signal amplification. From the kinetics data, we observed that the limit of quantification (LOQ) reached below a single cell cutoff (in 25 µL reaction volume) in about 20 min (Fig. 5b ); we thus set the detection time to 20 min. In cell-titration experiments (Fig. 5c ), the 20-min CreDiT assay achieved the LOQ of 0.8 cells (33 cells/mL) and a dynamic range spanning five orders of magnitude. Fig. 5: CreDiT assay characterization with cellular samples. a Temporal evolution of CreDiT signal. A varying number of Ca Ski cells were analyzed for HPV16. The signal started rising above the background about 10 min after the CreDiT reaction started. ∆CreDiT is the net signal with background subtracted. b Limits of detection (LOD) and quantification (LOQ) were estimated at different assay times from ( a ). The LOQ reached below a single cell cutoff (in 25 µL reaction volume) after 20 min of the assay start. The detection time was set to this moment. c The 20-min CreDiT assay achieved an LOQ of 0.8 cells (33 cells/mL) and a dynamic range over five orders of magnitude. Data were displayed as mean ± s.d. from technical triplicates. d A panel of CreDiT probes was tested on cervical cancer cells (1000 cells per sample). The CreDiT assay accurately detected HPV genotypes in different cervical cancer cells. The GAPDH probe was used to ascertain cellularity in samples. Data were displayed as mean ± s.d. from technical triplicates. e The CreDiT assay was compatible with mRNA detection. CreDiT probes were designed for mRNA counterparts of E6 and E7 oncoproteins of HPV16 and p16 INK4a . The assay detected high expression of E6 and E7 in HPV16-positive cells (Ca Ski, SiHa). The expression of p16 INK4a was elevated in all four cell lines. Samples contained about 1000 cells. Data were displayed as mean ± s.d. from technical triplicates. f The CreDiT results in ( e ) qualitatively matched gel electrophoresis results of RT-PCR products of the same mRNA targets. a.u. arbitrary unit, bp base pairs. Source data are provided as a Source Data file. Full size image We expanded our evaluation to other hrHPV probes, using a panel of cervical cancer cell lines of different HPV subtypes: Ca Ski (HPV16+), SiHa (HPV16+), HeLa (HPV18+), and C-33A (HPV–). DNAs were extracted from these cells (~10 5 cells) and subjected to multi-sample CreDiT assays. We used the GAPDH signal as a positive control to confirm the presence of sufficient cells. The hrHPV profiling results matched with known cellular genotypes [23] , [24] . Only on-target signals were positive, while off-target signals were all close to the background (Fig. 5d ). We further applied CreDiT to detect mRNA targets ( E6 , E7 , and p16 INK4a ), which are overexpressed in invasive, high-grade dysplasia, a precursor to malignant cervical cancer [25] . Detecting these targets can help identify high-risk cases, minimizing overtreatment. To adapt CreDiT for mRNA assays, we simply introduced reverse transcriptase into the CreDiT reagent mix, which enabled the reverse transcription to proceed concurrently with the CreDiT reaction (see Methods). The CreDiT results (Fig. 5e ) were consistent with those from gold standard RT-PCR. They showed a qualitative match with PCR-product electrophoresis (Fig. 5f and Supplementary Fig. 16 ) and a good correlation ( r = 0.88) with quantitative PCR results (Supplementary Fig. 17 ). CreDiT clinical testing We finally applied CreDiT to analyze human clinical samples. We used cervical or vaginal brushing specimens ( n = 121) collected during routine gynecologic evaluation (see Table 1 ). To encompass diverse demographics, we collected liquid-based samples from patients at a tertiary-care US hospital (Massachusetts General Hospital) as well as its affiliated primary care clinics serving multiple low-resource neighborhoods across the city (Boston, Massachusetts, USA) and adjacent areas. The samples were then assessed for HPV status in a clinical pathology laboratory (BD Onclarity HPV Assay system). These qPCR-based results served as our gold standard (see Supplementary Data 1 for the results). Residual aliquots of brushing specimens were subjected to CreDiT, including NA extraction and target DNA detection (HPV16, HPV18, HPV31, HPV33, HPV45, HPV58, and GAPDH ). All samples generated high GAPDH signals, passing the cellularity check (Supplementary Fig. 18 ). Table 1 Demographic and clinical information of the cervical brushing samples Full size table Figure 6a displays the HPV-CreDiT profile of the brushing samples and their pathology diagnoses. CreDiT accurately distinguished HPV-positive from negative samples and identified three major hrHPV types (HPV16, HPV18, and HPV45), agreeing with clinical diagnostics. For other HPV targets (HPV31, HPV33, and HPV58), CreDiT results were a subset of the clinical “others” category; the clinical diagnostics grouped ten targets (HPV31, HPV33, HPV35, HPV39, HPV51, HPV52, HPV56, HPV58, HPV66, and HPV68) in a single class. Fig. 6: Analysis of clinical samples. a Cervical brushing samples ( n = 121) were analyzed via CreDiT. When compared with clinical diagnoses (bottom row), the CreDiT results showed a good concordance. The CreDiT heatmap displays mean values from triplicate measurements. ∆CreDiT is a background-subtracted signal. b Samples were categorized according to the clinical diagnoses of three major hrHPV targets (HPV16, HPV18, and HPV45). CreDiT signals were significantly higher in clinically positive samples. The p values were <10 –6 for all three targets from the unpaired two-sided t -test. Da t a were displayed as mean ± s.d. c Receiver operating characteristic (ROC) curves were generated for HPV16, HPV18, and HPV45. The predictor was a marker-specific CreDiT signal scaled to the GAPDH signal, considering varying cellularity in a sample. Overall, CreDiT achieved high diagnostic accuracy, with the area under the curve (AUC) values >0.99 for all three HPV types. d CreDiT was used to analyze anal swap samples ( n = 48) for anal cancer screening. The results matched the clinical diagnoses (bottom row). The CreDiT heatmap displays mean values from triplicate measurements. e ROC curves were constructed using the GAPDH -scaled marker expression. The AUC values were >0.98. a.u. arbitrary unit. Source data are provided as a Source Data file. Full size image For three major hrHPV targets (HPV16, HPV18, HPV45), we further assessed CreDiT’s diagnostic performance. For a given HPV target, we dichotomized samples according to the clinical HPV status (Fig. 6b ). As a diagnostic metric, we used the marker signal scaled to GAPDH (e.g., ∆CreDiT HPV16 /∆CreDiT GAPDH for HPV16), accounting for the varying cellularity in the samples. Using this metric, we constructed a receiver operating characteristic (ROC) curve (Fig. 6c ) and then determined the cutoff for positivity that maximized sensitivity and specificity. With these cutoffs applied, the observed diagnostic accuracies were 0.975 (HPV16), 0.992 (HPV18), and 0.967 (HPV45; Supplementary Table 4 ). To demonstrate CreDiT’s extended clinical utility in other cancer screening efforts, we analyzed anal Pap test samples in a similar manner ( n = 48; Supplementary Table 5 ). HPV infection accounts for 90% of anal cancer cases [26] , with HPV16 being the most prevalent [27] . The CreDiT results (Fig. 6d ) again showed excellent agreement with clinical diagnostics, achieving high diagnostic accuracy (Fig. 6e )."},{"section":"Discussion","content":"We have developed and validated CreDiT as a robust and affordable diagnostic technology for NA detection. The assay had excellent analytical sensitivity by coupling DNA replication (i.e., RPA) with self-signal amplification of CRISPR/Cas (i.e., collateral cleavage of reporter probes). CreDiT also achieves high specificity; the analytical signal is generated when RPA primers and CRISPR/Cas recognize target sequences. The integrated assay system performed multiple functions following single sample loading, including NA extraction, NA amplification, and optical detection. This system-level engineering markedly simplified the assay procedure while improving overall assay speed (35 min) and minimizing manual sample handling (<3 min hands-on time; see Supplementary Fig. 19 for the detailed assay flow). Furthermore, the system was capable of up to 12 parallel measurements, improving the assay throughput. These technical strengths (sensitivity, throughput, integration, and ease of use) may differentiate CreDiT from other Cas-based point-of-care systems that require microfluidic components [28] , [29] , [30] , rely on microscopy [31] , or produce semi-quantitative results [32] (see Supplementary Table 1 for comparison). We envision CreDiT to empower LMIC providers with low-cost, decentralized, automated, and rapid readouts to guide “screen and treat” efforts. The same visit turnaround and fractional costs compared to local gold standards will also increase CreDiT’s prospects for adoption into clinical workflows. Molecular HPV testing has emerged as a superior alternative to conventional Pap smear and VIA tests for cervical cancer screening. It offers enhanced convenience and improved detection of high-risk precancerous lesions, particularly cervical intraepithelial neoplasia grade 3 (CIN3). This capability enables timely intervention to prevent the progression of cervical cancer [33] , [34] , [35] . As a primary screening tool, molecular HPV testing further provides a wide range of services, including (i) patient triaging and risk stratification of cytological abnormalities based on HPV subtypes, (ii) follow-up monitoring before or after treatment of CIN, and (iii) surveillance of the HPV epidemic at regional or country levels [36] , [37] , [38] . Encouraged by these clinical benefits, developed countries are increasingly shifting towards centralized HPV DNA testing as the sole screening method for identifying high-risk cases. Unfortunately, similar implementation has been challenged in LMICs due to limited access to test equipment and cost, incurring poor clinical outcomes. For instance, cervical cancer remains the second leading cause of cancer-related deaths in African women. We developed CreDiT to address such disparity by increasing access to molecular tests. Several technical features elevated CreDiT’s practical utility for onsite HPV screening. First, we seamlessly combined NA duplication (RPA) and signal amplification (Cas reaction) into a unified assay scheme (Fig. 1 ). This strategy allowed CreDiT to achieve higher (~100-fold) sensitivity than conventional PCR while conveniently carrying out the reaction in a single tube and at a constant temperature (42 °C). Second, we enhanced flexibility in the probe design. CreDiT eliminates the requirement for a PAM sequence in the target DNA; Cas12a can still be activated with crRNA binding to the target DNA’s single-strand segment displaced during the RPA reaction (Fig. 4 ). This merit facilitated the creation of a probe library capable of detecting a wide range of hrHPV targets. Third, we implemented a signal detection algorithm akin to those employed in wireless communication (Fig. 2 ). Specifically, the algorithm utilizes digitally encoded light to excite samples and then numerically decodes measured fluorescent signals to recover the original waveform. This digital approach offers superior computational efficiency and signal-to-noise ratio compared to conventional analog methods, enabling us to develop a compact CreDiT device suitable for POC applications. We propose extending the current study to enhance CreDiT’s analytical capability and accelerate its clinical translation. An immediate task is to expand CreDiT probes to detect a broader array of hrHPV targets, encompassing HPV35, HPV39, HPV51, HPV52, HPV56, HPV59, HPV66, and HPV68. Such expansion will yield comprehensive insights into the overall prevalence of distinct subtypes, which would be instrumental in devising region-specific or racially tailored countermeasures to combat HPV-associated health challenges (Supplementary Table 3 ) [3] , [39] , [40] . Another improvement would be equipping the assay to detect multiple molecular targets within a single reaction tube. Such multiplexing will adopt orthogonal Cas proteins (e.g., LwaCas13a, LbaCas13a, and PsmCas13b), with each protein type, upon target recognition, selectively cleaving reporter probes of distinct fluorescent colors [41] , [42] . In signal detection, we could explore applying our digital algorithm for multiplexing. For instance, each fluorescent-detection channel will use its own WH code to generate distinct excitation and response patterns. The total fluorescent signal is then acquired by a single photodetector and numerically decoded into individual channels. This approach would enable multi-fluorescent measurements with minimal optical components (i.e., a single fluorescent detector), substantially simplifying the detection system. We could further streamline the assay flow. In its current format, CreDiT requires a manual step to add the CreDiT mix into a sample after NA extraction. This step can be simplified with a new sample tube design. This tube would feature two compartments separated by a breakable seal. The top compartment will be pre-filled with CreDiT reagents, while the bottom will house the NA extraction buffer. This dual-chamber design will protect the CreDiT reagents throughout the NA extraction process. Following NA extraction, the CreDiT mix can be effortlessly introduced into the reaction chamber by breaking the seal—an action readily amenable to automation or manual operation. Finally, we need to deploy CreDiT to LMICs for evaluation by local end users. This real-world assessment will help identify potential oversights or challenges within the overall CreDiT workflow, encompassing logistics, pre-analytical handling, and readouts. The deployment will also foster collaboration with LMIC partners to refine the platform, ensuring its adaptability to specific clinical, economic, and environmental factors. We indeed plan to conduct field trials of CreDiT prototypes in Uganda and Ghana; the insights from these trials will guide further improvements in CreDiT’s field performance. Through these advancements, CreDiT will serve as a practical, field-deployable HPV testing solution, enabling the implementation of state-of-the-art HPV guidelines in regions where the impact is the most significant."},{"section":"Methods","content":"Ethical statement The research protocol received approval from the Partners Healthcare Institutional Review Board (IRB protocol 2022P002938). The investigators adhered to all relevant ethical regulations and institutional guidelines throughout the study. Participants provided written informed consent for the reporting and sharing of individual-level data. The gender of participants was determined based on self-reported information provided on the study consent form. No financial compensation was provided to study participants. Materials We had all oligonucleotides used in this study synthesized by Integrated DNA Technologies, Inc. (IDT; USA). The sequences of the oligonucleotides are listed in Supplementary Table 2 . TwistAmp® Basic kit and RevertAid reverse transcriptase (RTase) were purchased from TwistDx TM and Thermo Fisher Scientific (USA), respectively. EnGen® Lba Cas12a (Cas12a), RNase inhibitor (Murine), Pro K, and Luna Universal qPCR Master Mix were purchased from New England BioLabs (USA). All other chemicals were of analytical grade and used without further purification. CreDiT device construction We designed the CreDiT device using computer-aided design software (Solidworks, 2019) and fabricated the main device body through 3D printing (Form2, Formlabs). The device housed a rotating sample holder, a heating unit, an optical module for fluorescent detection, and two onboard computers. (i) The sample holder was cut in a metal block (aluminum 6061-T651, Protolabs) via a computer numerical control (CNC) machine. Into the holder, we inserted four ceramic heaters (HT15W, Thorlabs) for heating and four thermistors (10 kΩ) for temperature reading. The holder was mounted on a stepping motor (NEMA-8, Adafruit) to position each sample for optical detection. We also installed a 60 mm fan (OD6010-12HB, Orion Fans) underneath the holder for cooling purposes. (ii) The optical module had a pair of light a source and a fluorescent detector. The light source consisted of a 470-nm LED (SSL80, OSRAM), an aspheric condenser lens (ACL12708U, Thorlabs), and biconvex focusing lenses (LB1092-A, Thorlabs). The LED was attached to a metal heat sink and driven by a single-supply, low-noise LED current source driver (IRLMS2002TRPBF, Infineon). The fluorescent detector consisted of a filter set (GFP/FITC/Cy2 filter cube, Nikon), a convex lens (LB1761-A, Thorlabs), and a photodiode (S1223, Hamamatsu). The signal from the photodiode was amplified by an operational amplifier with a –3 dB bandwidth of 17 MHz (Analog Devices AD823AN) and digitized by a microcontroller. CreDiT device control We used two onboard computers for the system control. The master unit (Arduino Uno) executed a feedback loop to achieve the desired temperature profile in the sample holder, controlled the stepping motor, and communicated with an external computer. The signal-processing unit (ARM Cortex M4 microcontroller; MK20DX256) generated the waveform LED excitation, digitized the signal from the photodiode, and performed all necessary signal processing, including the Walsh–Hadamard transform. (iv) As a graphical user interface (GUI), we programmed a real-time dashboard using the Qt GUI framework (Supplementary Fig. 2 ), which communicated with the master unit over a universal serial bus (USB) connection. The firmware of the master unit was written in C++. CreDiT signal processing for fluorescent measurements For fluorescent measurements, we used Walsh functions of size 512. To generate the excitation waveform E ( t ), we specified an input vector D of length 512. The i th element of D represented the value for the i th sequency in the WH domain. Applying the Fast Walsh–Hadamard transform (FWHT) to D yielded the waveform E ( t ). We adjusted the excitation waveform by scaling it and adding a positive voltage offset to drive an LED in the full voltage swing (0–6 V). The resulting input voltage was E ( t ) = 3· W ( t ) + 3 (V), where W ( t ) is the Walsh function. We subsequently measured a Walsh-modulated fluorescence signal using a photodiode. This signal S ( t ) was directly digitized (200 kHz) by a 12-bit analog-to-digital converter built into the MCU (MK20DX256). The MCU then performed the inverse FWHT, converting S ( t ) into the sequency vector T . We finally obtained the signal intensity by taking an inner vector product of D·T . In the current CreDiT prototype, we used an input vector D 107 , whose 107th element was the only non-zero element. This sequency was close to the operating frequency of 20.9 kHz. A similar approach was applied for the Fourier modulation, with the input voltage E ( t ) = 3·sin(2πƒ· t ) + 3 (V), where ƒ is the waveform frequency. The average input power to LED had a 4:3 ratio between Walsh and Fourier modulations. Cell culture and NA extraction We purchased Ca Ski (CRL-1550), SiHa (HTB-35), HeLa (CCL2), and C-33A (HTB-31) cells from the American Type Culture Collection and grew them (37 °C, 5% CO 2 ) in the vendor-recommended growth media (RPMI-1640 for Ca Ski and DMEM for SiHa, HeLa, and C-33A) supplemented with 10% fetal bovine serum, 100 U/mL penicillin, and 100 µg/mL streptomycin. All cell lines were regularly tested for mycoplasma contamination using a mycoplasma detection kit (MycoAlert TM , Lonza). We extracted NA from these cells following the Pro K-based CreDiT extraction protocol. The lysis buffer had 0.5 mg/mL Pro K in 1× Tris-EDTA (TE; pH 8.0). As a gold standard for NA extraction, we purchased Quick-DNA Miniprep (Zymo Research) and used it according to the manufacturer’s protocol. CreDiT assay We prepared the CreDiT assay mix (21.75 µL) by combining target-specific RPA primers (5 µM; 1.2 µL for each primer), reporter probe (5 µM; 1 µL), gene-specific Cas12a gRNAs (25 µM; 0.16 µL for each target region), murine RNase inhibitor (40 U/µL; 0.5 µL), Cas12a (100 µM; 0.16 µL), 10× NEBuffer 2.1 (10 mM Tris-HCl, 50 mM NaCl, 10 mM MgCl 2 , 100 µg/mL BSA, pH 7.9 at 1× concentration; 0.5 µL), and reconstituted RPA mix (14.75 µL). After adding NA extract (2 µL) and MgOAc (280 mM, 1.25 µL), we incubated the mix at 42 °C for 20 min, followed by fluorescence measurement (see Supplementary Fig. 19 for the workflow). To analyze mRNA targets, we included 1 U/µL RTase in the CreDiT assay mix. The limit of detection (LOD) was determined based on S 0 + 3σ, where S 0 and σ are the signal and the standard deviation of a blank sample, respectively. The limit of quantification (LOQ) was determined based on S 0 + 10σ. Gel electrophoresis was performed using the Mupid One Electrophoresis System (Eurogentec) for analysis of CreDiT assay products, followed by imaging with the ChemiDoc XRS+ System (Bio-Rad). Whole gel images are shown in Supplementary Figs. 20 , 21 . Lyophilization of CreDiT reagents We prepared the CreDiT assay premix and snap-froze the mixture by immersing it in liquid nitrogen. The frozen mixture was then lyophilized overnight at room temperature on a VirTis FreezeMobile 25 L freeze dryer (SP Scientific). The lyophilized mix was stored at room temperature until use. Quantitative PCR (qPCR) and RT-qPCR We used PCR for quantitative comparison with CreDiT. For DNA targets, qPCR was employed. We prepared the PCR solution (20 µL) by mixing 250 nM IVT primers, 1× Luna Universal qPCR Master Mix, and extracted DNA. We then processed the mixture on a CFX Opus 96 real-time PCR instrument (Bio-Rad). The temperature profile was 95 °C (1 min), followed by cycles of 95 °C (15 s) and 60 °C (30 s). For mRNA targets, we performed RT-qPCR. The sample mixture (20 µL) contains 250 nM gene-specific primers, SuperScript™ III Platinum™ SYBR™ Green One-Step qRT-PCR Kit (Thermo Fisher Scientific), and target RNA (from 8000 cells). The reaction proceeded with the following protocol: 42 °C, 10 min (RT) and 94 °C, 5 min (RTase inactivation), followed by thermal cycling of 94 °C. (5 s), 56 °C (15 s), and 72 °C (15 s). The threshold cycle ( C t ) was automatically determined by the system software. Gel electrophoresis of the PCR products was run in 2% E-Gel™ EX Agarose Gels on the E-Gel™ Power Snap Electrophoresis Device, followed by imaging with Azure 280 (Azure Biosystems). Clinical sample analysis We collected cervical or anal Pap test samples (BD SurePath™ liquid-based Pap test, Becton, Dickinson and Company) during gynecological or anal exams. Clinical pathologic diagnostics were performed using the BD Onclarity HPV Assay system. Blind to the clinical diagnostic results, we analyzed aliquots (0.5 mL) of the residual clinical materials. We processed samples in the extraction buffer (0.5 mg/mL Pro K in 1× TE) following the heat-based extraction protocol. The extracted NA was then subjected to the CreDiT assay. For each NA target, we used 2 µL of NA extract. Statistics and reproducibility We used GraphPad Prism version 9.5 (GraphPad Software Inc.) or R version 4.2.2 for statistical analyses. For two-group comparisons, we used an unpaired, two-sided t -test. For all statistical tests, p values less than 0.05 were considered significant. Details on data presentation and the sample size are included in figure legends. In the study of clinical samples, no statistical method was used to predetermine the sample size, and the experiments were not randomized. No data were excluded from the analyses. The investigators analyzed the clinical samples using the CreDiT assay while blinded to the clinical diagnostic results (gold standard). 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\": \"The global incidence of human papillomavirus (HPV)-driven cancers, such as cervical, anal, and head and neck cancer, is on the rise, especially in underserved regions with limited healthcare infrastructure. Cervical cancer alone accounts for over 600,000 new cases a year, with more than 80% of them occurring in low- and middle-income countries (LMICs). Prompt and reliable triaging of high-risk HPV (hrHPV) is essential to offset severe pathology bottlenecks in resource-limited regions and circumvent geographical and/or socioeconomic barriers to effective cervical cancer screening [1] , [2] , [3] . Molecular diagnostics for hrHPV have demonstrated superior sensitivity (96–100%) and specificity (90–100%) when compared to standard cytology (e.g., Pap smear) [4] . Consequently, developed countries are increasingly shifting towards centralized nucleic acid (NA) testing as the sole screening method for identifying high-risk cases. However, implementing such tests in LMICs still faces technical challenges. The primary assay method, polymerase chain reaction (PCR), incurs high fixed and operational costs, straining public health budgets [5] , [6] . Centralized tests are also incompatible with the “screen-and-treat” strategy employed in LMICs—the one-time window of intervention with cervical ablative therapy. Visual inspection with acetic acid (VIA) is a rapid, inexpensive alternative but suffers from poor accuracy (i.e., high rates of false results) [7] , [8] . These realities underline the unmet need for fast, accurate hrHPV testing, ideally administered at point-of-care (POC) in LMIC settings. Clustered regularly interspaced short palindromic repeats (CRISPR) technologies are emerging as a powerful tool in nucleic acid (NA) detection [9] , [10] , [11] . One key advantage of this approach is its capability for sequence-specific signal amplification. When a CRISPR-associated (Cas) protein binds to its target NA through guide RNA (gRNA), the resulting complex becomes an active endonuclease capable of non-selectively degrading single-stranded NAs. This property has been harnessed to amplify analytical signals, with activated Cas proteins cleaving abundant non-targeted signaling probes such as single-stranded NAs tagged with a fluorescent dye (F) and quencher (Q) pair. The rapid and isothermal nature of Cas reactions has also facilitated on-site operations (Supplementary Table 1 ). We reasoned to adopt this Cas mechanism for fast, accurate NA detection in resource-limited settings. Here, we report a robust POC technology, termed CreDiT (for CRISPR Enhanced Digital Testing), devised for decentralized HPV testing and HPV-related cancer screening. CreDiT integrates two distinct strategies: (i) a Cas reaction that concurrently amplifies and detects target NAs and (ii) an efficient signal processing inspired by digital radio communications [12] , [13] . This combination confers notable advantages: (i) CreDiT assays are fast (<20 min), isothermal, and conveniently performed in a single tube; (ii) signal detection is resilient against extraneous noise and interference, enhancing assay reliability and precision; and iii) the detection system is compact for field use without requiring complex optics. Exploiting these advantages, we implement a CreDiT platform positioned for cancer diagnostics. Specifically, we design CreDiT probes for hrHPV genes (HPV16, HPV18, HPV45, HPV31, HPV33, and HPV58) and oncoprotein mRNAs ( E6 , E7 , and p16 INK4a ), unify a workflow for NA extraction and CreDIT assay, and build a portable CreDiT system to process multiple (12) samples. The system detects cell-derived HPV DNA targets down to single copies in 35 min (including DNA extraction) and accurately classifies hrHPV types in clinical cervical brushings ( n = 121). We further expand CreDiT’s utility and clinical impact by detecting hrHPV in anal swab samples from separate patients ( n = 48) [14] , [15] . The results underscore CreDiT’s versatility and potential for HPV-related clinical assessments.\",\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 14,\n 15\n ],\n \"infos\": [\n {\n \"idx\": \"1\",\n \"title\": \"Randall, T. C. & Ghebre, R. Challenges in prevention and care delivery for women with cervical cancer in sub-Saharan Africa.Front. Oncol.6, 160 (2016).\",\n \"link\": \"https://doi.org/10.3389%2Ffonc.2016.00160\"\n },\n {\n \"idx\": \"2\",\n \"title\": \"Montealegre, J. R. et al. Racial/ethnic differences in HPV 16/18 genotypes and integration status among women with a history of cytological abnormalities.Gynecol. Oncol.148, 357–362 (2018).\",\n \"link\": \"https://doi.org/10.1016%2Fj.ygyno.2017.12.014\"\n },\n {\n \"idx\": \"3\",\n \"title\": \"Islami, F., Fedewa, S. A. & Jemal, A. Trends in cervical cancer incidence rates by age, race/ethnicity, histological subtype, and stage at diagnosis in the United States.Prev. Med.123, 316–323 (2019).\",\n \"link\": \"https://doi.org/10.1016%2Fj.ypmed.2019.04.010\"\n },\n {\n \"idx\": \"4\",\n \"title\": \"Sankaranarayanan, R. et al. HPV screening for cervical cancer in rural India.N. Engl. J. Med.360, 1385–1394 (2009).\",\n \"link\": \"https://doi.org/10.1056%2FNEJMoa0808516\"\n },\n {\n \"idx\": \"5\",\n \"title\": \"Castle, P. E. et al. Reliability of the Xpert HPV assay to detect high-risk human papillomavirus DNA in a colposcopy referral population.Am. J. Clin. Pathol143, 126–133 (2015).\",\n \"link\": \"https://doi.org/10.1309%2FAJCP4Q0NSDHWIZGU\"\n },\n {\n \"idx\": \"6\",\n \"title\": \"Zhang, J. et al. Effectiveness of high-risk human papillomavirus testing for cervical cancer screening in China: a multicenter, open-label, randomized clinical trial.JAMA Oncol.7, 263–270 (2021).\",\n \"link\": \"https://doi.org/10.1001%2Fjamaoncol.2020.6575\"\n },\n {\n \"idx\": \"7\",\n \"title\": \"Goel, A. et al. Visual inspection of the cervix with acetic acid for cervical intraepithelial lesions.Int. J. Gynaecol. Obstet.88, 25–30 (2005).\",\n \"link\": \"https://doi.org/10.1016%2Fj.ijgo.2004.09.018\"\n },\n {\n \"idx\": \"8\",\n \"title\": \"Hegde, D., Shetty, H., Shetty, P. K. & Rai, S. Diagnostic value of acetic acid comparing with conventional Pap smear in the detection of colposcopic biopsy-proved CIN.J. Cancer Res. Ther.7, 454–458 (2011).\",\n \"link\": \"https://doi.org/10.4103%2F0973-1482.92019\"\n },\n {\n \"idx\": \"9\",\n \"title\": \"Lee, C. Y. et al. Development of integrated systems for on-site infection detection.Acc. Chem. Res.54, 3991–4000 (2021).\",\n \"link\": \"https://doi.org/10.1021%2Facs.accounts.1c00498\"\n },\n {\n \"idx\": \"10\",\n \"title\": \"Aman, R., Mahas, A. & Mahfouz, M. Nucleic acid detection using CRISPR/Cas biosensing technologies.ACS Synth. Biol.9, 1226–1233 (2020).\",\n \"link\": \"https://doi.org/10.1021%2Facssynbio.9b00507\"\n },\n {\n \"idx\": \"11\",\n \"title\": \"Kaminski, M. M., Abudayyeh, O. O., Gootenberg, J. S., Zhang, F. & Collins, J. J. CRISPR-based diagnostics.Nat. Biomed. Eng.5, 643–656 (2021).\",\n \"link\": \"https://doi.org/10.1038%2Fs41551-021-00760-7\"\n },\n {\n \"idx\": \"12\",\n \"title\": \"Grove, A. C. An introduction to Walsh functions and their applications.Int. J. Math. Educ. Sci. Technol.14, 43–53 (1983).\",\n \"link\": \"https://doi.org/10.1080%2F0020739830140107\"\n },\n {\n \"idx\": \"14\",\n \"title\": \"Wei, F. et al. Epidemiology of anal human papillomavirus infection and high-grade squamous intraepithelial lesions in 29,900 men according to HIV status, sexuality, and age: a collaborative pooled analysis of 64 studies.Lancet HIV8, e531–e543 (2021).\",\n \"link\": \"https://doi.org/10.1016%2FS2352-3018%2821%2900108-9\"\n },\n {\n \"idx\": \"15\",\n \"title\": \"Marra, E., Lin, C. & Clifford, G. M. Type-specific anal human papillomavirus prevalence among men, according to sexual preference and HIV status: a systematic literature review and meta-analysis.J. Infect. Dis.219, 590–598 (2019).\",\n \"link\": \"https://doi.org/10.1093%2Finfdis%2Fjiy556\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"CreDiT platform development We devised the CreDiT assay to minimize user intervention, enabling process automation and effectively achieving a sample-in-answer-out test (Fig. 1a ). Our NA assay thus integrated NA amplification and Cas12a-based NA recognition (Fig. 1b ). During CreDiT’s initial phase, the polymerase enzyme separates the double-stranded target DNA, creating a single-stranded region for the gRNA molecule to recognize. This recognition triggers the Cas12a enzyme within the Cas12a/gRNA complex to act as a nuclease and cleave the F-Q DNA probes, generating a fluorescent signal. However, activated Cas12a can also interfere with RPA by cleaving the target DNA itself ( cis -cleavage) and breaking down the RPA primers ( trans -cleavage). Reassuringly, our kinetic analysis indicates that RPA is faster than these competing reactions (see Supplementary Note for details). This dominance of RPA ensures sufficient amplification of the target DNA within the CreDiT format. Fig. 1: CreDiT assay for HPV detection. a Streamlined onsite workflow. The assay executes the nucleic acid (NA) extraction and the CreDiT reaction in a single pot, minimizing the hands-on times. The total assay time is about 36 min. b CreDiT assay principle. Two reactions concurrently occur: recombinase polymerase amplification (RPA) and Cas12a-based signal generation. RPA is the major reaction that replicates the target DNA. During RPA, the polymerase displaces the DNA duplex, exposing single-stranded regions recognizable by Cas12a/gRNA complexes. This recognition event activates the complexes to cleave dye-quencher (F-Q) probes, generating fluorescent signals. As the RPA cycles proceed, more Cas12a/gRNA complexes are activated, amplifying the overall signal. All reactions are carried out at a constant temperature (42 °C) without requiring external interruptions. c A compact, integrated assay system was developed. The device had a turret-shaped heating block accommodating 12 samples and an optical module for fluorescent detection (LED light-emitting diode, PD photodiode). The heating block, mounted on a motor, rotated to position each sample for fluorescent measurements. The device had a small form factor (14 × 11 × 6.5 cm 3 ) and connected to a computer. Custom-designed software running on a computer presented a graphical user interface for device control and real-time data analysis. d Two microcontroller units (MCUs) automated the CreDiT assay processes. MCU1 controlled heating elements to maintain the desired temperatures and rotated the heating block during optical detection. MCU2 generated waveforms for fluorescent excitation and detected the resulting signal. Full size image All assay steps are performed at a fixed temperature (42 °C, isothermal) without requiring external interruptions. Furthermore, the assay is fast and sensitive, as it concurrently amplifies target DNA (RPA) and analytical signals (Cas12a/gRNA). Similar one-pot approaches have been described [16] , [17] ; our analysis may be applicable to understand their underlying reaction mechanisms and inform optimizations. To translate CreDiT’s practical advantages into on-site testing, we constructed an automated assay system (Fig. 1c and Supplementary Movie 1 ). The system hardware integrated a rotating sample holder, heating components, an optical detection module, and microcontroller units (detailed schematic in Supplementary Fig. 1 ). The sample holder, machined in aluminum, had embedded resistive heaters and temperature sensors and was turret-shaped to accommodate 12 standard PCR tubes (6 mm in diameter). The optical module was configured for fluorescent measurements, including a light-emitting diode (LED) for excitation and a photodiode for detection. We controlled the system operation using two onboard microcontrollers (Fig. 1d ), one for sample heating and the other for signal detection. The heating controller kept the sample holder at a desired temperature for a given assay step (e.g., NA extraction, CreDiT reaction). It executed a feedback loop control, adjusting input powers to heaters and a fan based on the temperature sensor readouts. When the reaction was over, this controller also rotated the sample holder using a stepping motor and positioned each sample for fluorescent detection. The signal detection controller directed optical measurements. It generated waveforms to modulate fluorescent excitation and processed the resulting light signals for demodulation. The approach enabled multi-sample measurement with an identical optical module, ensuring consistent signal detection and simplifying the system design. The developed device was compact with a form factor of 14 × 11 × 6.5 cm [3] . We also implemented user-interface software for device control and real-time data analysis on a computer (Supplementary Fig. 2 ). Robust optical detection for POC operation Ideal point-of-care (POC) devices should have fewer components to minimize design complexity and reduce cost, but this requirement may limit system reliability and analytical capacity. Fluorescent devices are in such a category. Simple optical setups may be used, although they are susceptible to environmental and system noises. Conversely, reliable fluorescent measurements use calibrated, high-end systems, challenging POC operations. To address this issue, we designed the CreDiT optical system to generate high-quality signals in a compact device. We specifically adopted the Walsh–Hadamard transform (WHT) [18] , a digital signal processing technique used in wireless communications [19] . The WHT is similar to the Fourier transform (FT) but uses non-sinusoidal, rectangular waves (Walsh functions) as an orthogonal basis (Supplementary Fig. 3a ). Any time-domain signals can be decomposed into a weighted sum of Walsh functions [12] , [20] . Moreover, Walsh functions are uniquely defined by a sequence index, which is analogous to the frequency in FT. With these attributes, the WHT can compress time-domain signals into a simple collection of coefficients for Walsh sequence values (Supplementary Fig. 3b ). The WHT method offers further advantages [21] . (i) Walsh functions, inherently binary in nature, are readily compatible with digital electronics. (ii) The signal processing algorithm, referred to as the fast Walsh–Hadamard transform (FWHT), is lightweight and efficient for portable computing, requiring only additions and subtractions of real numbers (Supplementary Fig. 4 ). This contrasts with the fast Fourier transform (FFT) algorithm, which involves multiplying complex numbers. (iii) More importantly, the WHT is more robust to external interference than other signal processing techniques. The WHT spreads the original signal over a wide frequency band [13] . Such spreading effectively reduces the relative influence of interfering signals, particularly those with narrow bandwidths [22] . To apply the WHT in fluorescent detection, we first generated a signal for fluorescent excitation (Fig. 2a , left). We specified values for the Walsh sequency indices and converted these values into a time-domain waveform using the FWHT. This waveform drove an LED, exciting fluorescent dyes in a sample. In the detection path (Fig. 2a , right), we obtained the emitting light using a photodiode (PD) and decomposed the signal via inverse FWHT. This process revealed the Walsh sequency indices used to modulate the fluorescent excitation, and the peak values at these indices were proportional to the fluorescent intensity (i.e., the CreDiT signal). The measurement was completed in 2.5 ms, minimizing the risk of photobleaching fluorophores. Fig. 2: CreDiT system engineering. a CreDiT approach to fluorescent detection. The digital input, represented by a Walsh–Hadamard (WH) sequency, is transformed into a fluorescent excitation waveform using a fast WH transform (FWHT). The resulting signal from a sample is processed via inverse FWHT (iFWHT) to recover WH sequency, whose amplitude is proportional to the fluorescent intensity. b Digital CreDiT detection outperformed analog lock-in (Fourier) and direct fluorescent detection, exhibiting a higher signal-to-noise ratio (SNR) than the others. Data were displayed as mean intensity ± s.d. from technical replicates ( n = 10). c Samples with varying fluorescein concentrations were measured. With its high SNR, digital CreDiT detection achieved >2200-fold lower detection limit (0.4 pM) than the analog lock-in method (Fourier; 1 nM). Data were displayed as mean ± s.d. from triplicate measurements. d The digital CreDiT results from ( c ) also showed an excellent correlation ( R 2 = 0.99) with those measured by a conventional plate reader. e Sample heating setup. The sample holder accommodated 12 samples for heating and was embedded with four pairs of a heater and a temperature sensor. An external fan was used to expedite sample cooling. f Sample temperature profile. The system reached different temperature targets for the CreDiT assay (e.g., NA extraction, CreDiT reaction). Thermal imaging (inset) confirmed uniform heating of all 12 samples. g The heating system maintained the target temperature with relative variations <0.6%. Data were shown as mean ± s.d. from repeated temperature measurements ( n = 100). a.u. arbitrary unit. Source data are provided as a Source Data file. Full size image We determined the optimal Walsh sequency by assessing the system’s performance under the excitation of different Walsh waveforms. Specifically, we measured the fluorescent signals from a standard sample ([fluorophore] = 100 µM) while sweeping Walsh sequency (Supplementary Fig. 5a ). The system achieved a high signal-to-noise ratio (SNR) within the range of sequences from 97 to 130 (Supplementary Fig. 5b ). We selected sequency 107 for its peak SNR in our current prototype. For different systems, the sequency number can be readily adjusted to maximize their SNR. We further compared the WHT-based detection (CreDiT) with other frequently used techniques: constant illumination and analog lock-in (Fourier) methods. For this task, we programmed the onboard microcontroller to generate either a direct current (DC) or a sinusoidal waveform for LED excitation. Acquired fluorescent signals from a sample ([fluorophore] = 100 µM) were then numerically processed to extract fluorescent intensities (Methods). This approach allowed for an unbiased comparison among signal processing algorithms by keeping the optical setup and electronics identical. The WHT method outperformed the other two, displaying the highest SNR (Fig. 2b ). In fluorophore-titration measurements, the WHT detection achieved a superior sensitivity, with its limit of detection (0.4 pM) about 2200-fold lower than that (1 nM) of the lock-in method (Fig. 2c ). The results by the WHT detection also showed excellent linear correlation ( R 2 = 0.99) with a commercial plate reader (Fig. 2d ), which confirmed the reliable analytical performance of the developed signal-processing approach. Temperature control for CreDiT reaction To enable consistent CreDiT reactions, we engineered precise temperature control for the assay device. We first optimized the sample holder design to facilitate rapid and even heating for all 12 samples. Through numerical simulations, we determined that a holder featuring hollow pockets would heat up more rapidly than a solid one, primarily due to the reduced thermal mass (Supplementary Fig. 6 ). Based on this design insight, we proceeded to fabricate a holder in aluminum. The holder had four quadrants (Fig. 2e ); each quadrant accommodated three samples and was embedded with a pair of a heater and a temperature sensor. To regulate the holder temperature, we programmed a microcontroller for feedback control: it read temperature sensors every 100 msec, averaged their outputs, and generated a control signal for heating elements or a cooling fan. Applying the method, we could create reliable temperature profiles suited for different assay steps (Fig. 2f ). On average, the accuracy was within ±0.3 °C of the target temperature, and the precision was 0.6% as measured by the coefficient of variation (Fig. 2g ). The temperature was also uniform across the 12 samples as evidenced by the overall heat distribution recorded throughout the assay (Supplementary Movie 1 ). Assay optimization for POC operation We proceeded to fine-tune the entire CreDiT assay process, aiming to establish a streamlined workflow. For NA extraction from cells, we chose to apply enzymatic cell digestion, a method that does not require external instruments (e.g., centrifuge and spin-columns) and can be controlled by adjusting the temperature (Fig. 3a ). We used a lysis buffer containing proteinase K (Pro K) to release cellular NAs and protect them from nucleases. During the lysis, the sample was heated to 56 °C to enhance Pro K activity. Using cervical cancer cells (Ca Ski), we optimized the lysis time (15 min) and the buffer composition ([Pro K] = 0.5 mg/mL) for the maximal NA yield (Fig. 3b ). Following the NA extraction, we deactivated Pro K by elevating the buffer temperature; this process was necessary to prevent Pro K from affecting the downstream CreDiT reaction. We found that a brief incubation (1 min) at 95 °C was sufficient (Supplementary Fig. 7 ). Fig. 3: Assay optimization for POC applications. a NA extraction protocol. Cells are lysed by proteinase K (Pro K) to release NAs. Subsequently, Pro K is heat-deactivated (95 °C), and the CreDiT reaction is initiated. b The lysis time and Pro K concentration ([Pro K]) were optimized to maximize the NA extraction yield. Samples containing Ca Ski cells (2.5 × 10 4 cells/mL) were analyzed for HPV16. (Left panel) The extraction yields reached a plateau (red arrow) within 15-min of lysis. HPV16 signals from 20-min lysis were used for normalization. (Right panel) The extraction yield was maximal (blue arrow) at [Pro K] = 0.5 mg/mL and comparable to that of a spin column (gold standard). Data were displayed as mean ± s.d. from technical triplicates. c Extracted NAs were PCR-amplified for the HPV16 gene. Gel electrophoresis showed comparable bands between the CreDiT and the spin-column extraction methods. bp, base pairs. d Both extraction methods (CreDiT, spin column) yielded statistically identical results (unpaired, two-sided t -test) in analytical assays. Data were displayed as mean ± s.d. from technical quadruplicates. e CreDiT reagents were lyophilized and stored under ambient conditions. The lyophilized reagents demonstrated consistent activity for at least two weeks. Ca Ski cells (2.5 × 10 4 cells/mL) were analyzed for HPV16. Data were displayed as mean ± s.d. from technical triplicates. a.u. arbitrary unit. Source data are provided as a Source Data file. Full size image The performance of our extraction protocol was similar to that of a standard method (commercial spin-column). For this evaluation, we extracted NAs from cells (Ca Ski) and subjected them to PCR and CreDiT analysis. NAs from both extraction methods had similar quality (Supplementary Fig. 8a ) and generated strong PCR bands of comparable size and intensity (Fig. 3c ). The extraction yields were also comparable between the two extraction methods, as confirmed by PCR and CreDiT assays producing statistically indistinguishable (unpaired, two-sided t -test) signals (Fig. 3d ). When compared to the method using only thermal lysis (95 °C) for NA release, the CreDiT extraction yielded six times more NA (Supplementary Fig. 8b ); this improvement was likely due to the additional benefits of Pro K, notably the degradation of nucleases (i.e., RNases and DNases) that break down NA. We further assessed the feasibility of storing samples and reagents under ambient conditions; this capability is desirable for promoting CreDiT applications in LMIC sites. From the perspective of sample storage, we tested the stability of cells in our extraction buffer. We spiked cells (Ca Ski) in the buffer and stored sample aliquots either in a refrigerator (4 °C) or under ambient conditions before NA extraction. For both storage conditions, we observed no notable decrease in NA amounts for at least two weeks (Supplementary Fig. 9 ). Regarding CreDiT reagents, we premixed the components (RPA primers, reporter probe, Cas12a gRNAs, and Cas12a protein) and lyophilized the mixture (see Methods) to facilitate its transport and storage. When evaluated via the CreDiT assay, the lyophilized premix retained its efficacy for at least two weeks in storage at ambient conditions (Fig. 3e ). Overall, these features could mitigate logistic challenges and promote CreDiT usability in resource-limited settings even without cold chain solutions. CreDiT probe design and validation We prepared a library of CreDiT probes for hrHPV detection. For a given target DNA, CreDiT used a pair of probes, one forward and one reverse, to span both ends of the target sequence (Fig. 4a ). At each end of the sequence, we designed an RPA primer and a Cas12a gRNA against a target region. The primer, typically 30–33 nucleotides in length, served to amplify the target region, and the gRNA featured a spacer sequence allowing Cas12a to bind to a region positioned a few bases (2–10 nucleotides) downstream from the primer binding site (detailed design flow in Supplementary Fig. 10 ). Because Cas12a was activated by a single-strand DNA, the gRNAs can be designed without the constraint of a protospacer adjacent motif (PAM) sequence, offering enhanced flexibility. Capitalizing on this flexibility, we designed CreDiT probes for a diverse range of targets (see Supplementary Tables 2 , 3 ): (i) genomic DNA of hrHPV subtypes prevalent in Africa (HPV16, HPV18, HPV31, HPV33, HPV45, and HPV58); (ii) oncogenic E6 / E7 mRNA whose presence may indicate progressive HPV16 infections; and (iii) p16 INK4a ( CDKN2A ) mRNA that is upregulated in transformed cells as a consequence of E6/E7 expression. Fig. 4: CreDiT probe design. a For a given target DNA (gray backbone), a pair of CreDiT probes (forward and reverse) were designed, with each probe consisting of an RPA primer (green backbone) and a Cas12a/gRNA complex (orange backbone). Upon binding to the target DNA, the primer exposes a downstream sequence complementary to gRNA. The Cas12a/gRNA complex then recognizes its target and becomes an active endonuclease to cleave F-Q reporters. HPV16 probes are shown. b Assay validation with varying reaction conditions. HPV16 DNA was used as a model target ([DNA] = 4 × 10 7 copies/mL). A high signal was only observed when all assay components were present. Data were displayed as mean ± s.d. from technical triplicates. c CreDiT (20 min) and qPCR (1 h) assays were used to analyze samples with different target DNA concentrations. The CReDiT assay exhibited a lower detection limit (40 copies/mL) and a wider dynamic range (4 orders of magnitude) than qPCR (detection limit, 400 copies/mL; dynamic range, 2.5 orders of magnitude). ∆CreDiT is the background-subtracted signal. Data were displayed as mean ± s.d. from technical triplicates. d Probe selectivity. A panel of probes was designed for hrHPV targets (HPV16, HPV18, HPV31, HPV33, HPV45, and HPV58) and GAPDH . These probes detected their intended targets with negligible off-target signals. [Target DNA] = 4 × 10 7 copies/mL. The heatmap displays mean values from technical triplicates. a.u. arbitrary unit. Source data are provided as a Source Data file. Full size image We validated the designed probes in the CreDiT assay. We first varied the composition of the assay reagents and monitored the resulting CreDiT signals (Fig. 4b and Supplementary Fig. 11 ). As expected, the fluorescent signal was only observed when all CreDiT reagents and the target DNA were present. We further observed the synergistic benefit of recognizing both the forward and reverse regions in a target DNA. The combination of forward and reverse gRNAs produced stronger analytical signals than individual gRNAs (Supplementary Fig. 12 ). We also designed an additional gRNA that recognized a middle region of the target DNA (Supplementary Table 2 ). However, this gRNA displayed the weakest signal generation, and incorporating it into the assay lowered the overall signal, likely due to competition with the more efficient forward and reverse gRNAs for Cas12a binding. Next, we compared CreDiT’s performance with the established gold standard, qPCR, and a two-step assay, where RPA (20 min) and Cas12a detection (30 min) were carried out sequentially. We used synthetic HPV16 DNA as a model target. CreDiT demonstrated superior sensitivity compared to qPCR (Fig. 4c ) and the two-step assay (Supplementary Fig. 13 ). CreDiT’s limit of detection (LOD) was down to 40 copies/mL, whereas LODs for qPCR and the two-step assay were around 400 copies/mL. CreDiT also offered an advantage in assay time, requiring only 20 min compared to 1 h for qPCR and 50 min for the two-step assay. Furthermore, the designed CreDiT probes exhibited excellent selectivity with minimal cross-reactivity, as confirmed by the CreDiT assay (Fig. 4d and Supplementary Fig. 14 ) and gel electrophoresis (Supplementary Fig. 15 ). CreDiT assay characterization With the detection system and probes ready, we characterized the entire CreDiT assay flow, beginning with NA extraction from cervical cancer cells. To assess the assay kinetics, we extracted DNA from varying numbers of Ca Ski cells (HPV16+) and performed CreDiT assays, monitoring the signal evolution over time. Even at low cell concentrations, the signal rose rapidly above the background (Fig. 5a ); this can be attributed to the dual enzymatic activities in CreDiT, namely, RPA for NA duplication and Cas12a reaction for signal amplification. From the kinetics data, we observed that the limit of quantification (LOQ) reached below a single cell cutoff (in 25 µL reaction volume) in about 20 min (Fig. 5b ); we thus set the detection time to 20 min. In cell-titration experiments (Fig. 5c ), the 20-min CreDiT assay achieved the LOQ of 0.8 cells (33 cells/mL) and a dynamic range spanning five orders of magnitude. Fig. 5: CreDiT assay characterization with cellular samples. a Temporal evolution of CreDiT signal. A varying number of Ca Ski cells were analyzed for HPV16. The signal started rising above the background about 10 min after the CreDiT reaction started. ∆CreDiT is the net signal with background subtracted. b Limits of detection (LOD) and quantification (LOQ) were estimated at different assay times from ( a ). The LOQ reached below a single cell cutoff (in 25 µL reaction volume) after 20 min of the assay start. The detection time was set to this moment. c The 20-min CreDiT assay achieved an LOQ of 0.8 cells (33 cells/mL) and a dynamic range over five orders of magnitude. Data were displayed as mean ± s.d. from technical triplicates. d A panel of CreDiT probes was tested on cervical cancer cells (1000 cells per sample). The CreDiT assay accurately detected HPV genotypes in different cervical cancer cells. The GAPDH probe was used to ascertain cellularity in samples. Data were displayed as mean ± s.d. from technical triplicates. e The CreDiT assay was compatible with mRNA detection. CreDiT probes were designed for mRNA counterparts of E6 and E7 oncoproteins of HPV16 and p16 INK4a . The assay detected high expression of E6 and E7 in HPV16-positive cells (Ca Ski, SiHa). The expression of p16 INK4a was elevated in all four cell lines. Samples contained about 1000 cells. Data were displayed as mean ± s.d. from technical triplicates. f The CreDiT results in ( e ) qualitatively matched gel electrophoresis results of RT-PCR products of the same mRNA targets. a.u. arbitrary unit, bp base pairs. Source data are provided as a Source Data file. Full size image We expanded our evaluation to other hrHPV probes, using a panel of cervical cancer cell lines of different HPV subtypes: Ca Ski (HPV16+), SiHa (HPV16+), HeLa (HPV18+), and C-33A (HPV–). DNAs were extracted from these cells (~10 5 cells) and subjected to multi-sample CreDiT assays. We used the GAPDH signal as a positive control to confirm the presence of sufficient cells. The hrHPV profiling results matched with known cellular genotypes [23] , [24] . Only on-target signals were positive, while off-target signals were all close to the background (Fig. 5d ). We further applied CreDiT to detect mRNA targets ( E6 , E7 , and p16 INK4a ), which are overexpressed in invasive, high-grade dysplasia, a precursor to malignant cervical cancer [25] . Detecting these targets can help identify high-risk cases, minimizing overtreatment. To adapt CreDiT for mRNA assays, we simply introduced reverse transcriptase into the CreDiT reagent mix, which enabled the reverse transcription to proceed concurrently with the CreDiT reaction (see Methods). The CreDiT results (Fig. 5e ) were consistent with those from gold standard RT-PCR. They showed a qualitative match with PCR-product electrophoresis (Fig. 5f and Supplementary Fig. 16 ) and a good correlation ( r = 0.88) with quantitative PCR results (Supplementary Fig. 17 ). CreDiT clinical testing We finally applied CreDiT to analyze human clinical samples. We used cervical or vaginal brushing specimens ( n = 121) collected during routine gynecologic evaluation (see Table 1 ). To encompass diverse demographics, we collected liquid-based samples from patients at a tertiary-care US hospital (Massachusetts General Hospital) as well as its affiliated primary care clinics serving multiple low-resource neighborhoods across the city (Boston, Massachusetts, USA) and adjacent areas. The samples were then assessed for HPV status in a clinical pathology laboratory (BD Onclarity HPV Assay system). These qPCR-based results served as our gold standard (see Supplementary Data 1 for the results). Residual aliquots of brushing specimens were subjected to CreDiT, including NA extraction and target DNA detection (HPV16, HPV18, HPV31, HPV33, HPV45, HPV58, and GAPDH ). All samples generated high GAPDH signals, passing the cellularity check (Supplementary Fig. 18 ). Table 1 Demographic and clinical information of the cervical brushing samples Full size table Figure 6a displays the HPV-CreDiT profile of the brushing samples and their pathology diagnoses. CreDiT accurately distinguished HPV-positive from negative samples and identified three major hrHPV types (HPV16, HPV18, and HPV45), agreeing with clinical diagnostics. For other HPV targets (HPV31, HPV33, and HPV58), CreDiT results were a subset of the clinical “others” category; the clinical diagnostics grouped ten targets (HPV31, HPV33, HPV35, HPV39, HPV51, HPV52, HPV56, HPV58, HPV66, and HPV68) in a single class. Fig. 6: Analysis of clinical samples. a Cervical brushing samples ( n = 121) were analyzed via CreDiT. When compared with clinical diagnoses (bottom row), the CreDiT results showed a good concordance. The CreDiT heatmap displays mean values from triplicate measurements. ∆CreDiT is a background-subtracted signal. b Samples were categorized according to the clinical diagnoses of three major hrHPV targets (HPV16, HPV18, and HPV45). CreDiT signals were significantly higher in clinically positive samples. The p values were <10 –6 for all three targets from the unpaired two-sided t -test. Da t a were displayed as mean ± s.d. c Receiver operating characteristic (ROC) curves were generated for HPV16, HPV18, and HPV45. The predictor was a marker-specific CreDiT signal scaled to the GAPDH signal, considering varying cellularity in a sample. Overall, CreDiT achieved high diagnostic accuracy, with the area under the curve (AUC) values >0.99 for all three HPV types. d CreDiT was used to analyze anal swap samples ( n = 48) for anal cancer screening. The results matched the clinical diagnoses (bottom row). The CreDiT heatmap displays mean values from triplicate measurements. e ROC curves were constructed using the GAPDH -scaled marker expression. The AUC values were >0.98. a.u. arbitrary unit. Source data are provided as a Source Data file. Full size image For three major hrHPV targets (HPV16, HPV18, HPV45), we further assessed CreDiT’s diagnostic performance. For a given HPV target, we dichotomized samples according to the clinical HPV status (Fig. 6b ). As a diagnostic metric, we used the marker signal scaled to GAPDH (e.g., ∆CreDiT HPV16 /∆CreDiT GAPDH for HPV16), accounting for the varying cellularity in the samples. Using this metric, we constructed a receiver operating characteristic (ROC) curve (Fig. 6c ) and then determined the cutoff for positivity that maximized sensitivity and specificity. With these cutoffs applied, the observed diagnostic accuracies were 0.975 (HPV16), 0.992 (HPV18), and 0.967 (HPV45; Supplementary Table 4 ). To demonstrate CreDiT’s extended clinical utility in other cancer screening efforts, we analyzed anal Pap test samples in a similar manner ( n = 48; Supplementary Table 5 ). HPV infection accounts for 90% of anal cancer cases [26] , with HPV16 being the most prevalent [27] . The CreDiT results (Fig. 6d ) again showed excellent agreement with clinical diagnostics, achieving high diagnostic accuracy (Fig. 6e ).\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"We have developed and validated CreDiT as a robust and affordable diagnostic technology for NA detection. The assay had excellent analytical sensitivity by coupling DNA replication (i.e., RPA) with self-signal amplification of CRISPR/Cas (i.e., collateral cleavage of reporter probes). CreDiT also achieves high specificity; the analytical signal is generated when RPA primers and CRISPR/Cas recognize target sequences. The integrated assay system performed multiple functions following single sample loading, including NA extraction, NA amplification, and optical detection. This system-level engineering markedly simplified the assay procedure while improving overall assay speed (35 min) and minimizing manual sample handling (<3 min hands-on time; see Supplementary Fig. 19 for the detailed assay flow). Furthermore, the system was capable of up to 12 parallel measurements, improving the assay throughput. These technical strengths (sensitivity, throughput, integration, and ease of use) may differentiate CreDiT from other Cas-based point-of-care systems that require microfluidic components [28] , [29] , [30] , rely on microscopy [31] , or produce semi-quantitative results [32] (see Supplementary Table 1 for comparison). We envision CreDiT to empower LMIC providers with low-cost, decentralized, automated, and rapid readouts to guide “screen and treat” efforts. The same visit turnaround and fractional costs compared to local gold standards will also increase CreDiT’s prospects for adoption into clinical workflows. Molecular HPV testing has emerged as a superior alternative to conventional Pap smear and VIA tests for cervical cancer screening. It offers enhanced convenience and improved detection of high-risk precancerous lesions, particularly cervical intraepithelial neoplasia grade 3 (CIN3). This capability enables timely intervention to prevent the progression of cervical cancer [33] , [34] , [35] . As a primary screening tool, molecular HPV testing further provides a wide range of services, including (i) patient triaging and risk stratification of cytological abnormalities based on HPV subtypes, (ii) follow-up monitoring before or after treatment of CIN, and (iii) surveillance of the HPV epidemic at regional or country levels [36] , [37] , [38] . Encouraged by these clinical benefits, developed countries are increasingly shifting towards centralized HPV DNA testing as the sole screening method for identifying high-risk cases. Unfortunately, similar implementation has been challenged in LMICs due to limited access to test equipment and cost, incurring poor clinical outcomes. For instance, cervical cancer remains the second leading cause of cancer-related deaths in African women. We developed CreDiT to address such disparity by increasing access to molecular tests. Several technical features elevated CreDiT’s practical utility for onsite HPV screening. First, we seamlessly combined NA duplication (RPA) and signal amplification (Cas reaction) into a unified assay scheme (Fig. 1 ). This strategy allowed CreDiT to achieve higher (~100-fold) sensitivity than conventional PCR while conveniently carrying out the reaction in a single tube and at a constant temperature (42 °C). Second, we enhanced flexibility in the probe design. CreDiT eliminates the requirement for a PAM sequence in the target DNA; Cas12a can still be activated with crRNA binding to the target DNA’s single-strand segment displaced during the RPA reaction (Fig. 4 ). This merit facilitated the creation of a probe library capable of detecting a wide range of hrHPV targets. Third, we implemented a signal detection algorithm akin to those employed in wireless communication (Fig. 2 ). Specifically, the algorithm utilizes digitally encoded light to excite samples and then numerically decodes measured fluorescent signals to recover the original waveform. This digital approach offers superior computational efficiency and signal-to-noise ratio compared to conventional analog methods, enabling us to develop a compact CreDiT device suitable for POC applications. We propose extending the current study to enhance CreDiT’s analytical capability and accelerate its clinical translation. An immediate task is to expand CreDiT probes to detect a broader array of hrHPV targets, encompassing HPV35, HPV39, HPV51, HPV52, HPV56, HPV59, HPV66, and HPV68. Such expansion will yield comprehensive insights into the overall prevalence of distinct subtypes, which would be instrumental in devising region-specific or racially tailored countermeasures to combat HPV-associated health challenges (Supplementary Table 3 ) [3] , [39] , [40] . Another improvement would be equipping the assay to detect multiple molecular targets within a single reaction tube. Such multiplexing will adopt orthogonal Cas proteins (e.g., LwaCas13a, LbaCas13a, and PsmCas13b), with each protein type, upon target recognition, selectively cleaving reporter probes of distinct fluorescent colors [41] , [42] . In signal detection, we could explore applying our digital algorithm for multiplexing. For instance, each fluorescent-detection channel will use its own WH code to generate distinct excitation and response patterns. The total fluorescent signal is then acquired by a single photodetector and numerically decoded into individual channels. This approach would enable multi-fluorescent measurements with minimal optical components (i.e., a single fluorescent detector), substantially simplifying the detection system. We could further streamline the assay flow. In its current format, CreDiT requires a manual step to add the CreDiT mix into a sample after NA extraction. This step can be simplified with a new sample tube design. This tube would feature two compartments separated by a breakable seal. The top compartment will be pre-filled with CreDiT reagents, while the bottom will house the NA extraction buffer. This dual-chamber design will protect the CreDiT reagents throughout the NA extraction process. Following NA extraction, the CreDiT mix can be effortlessly introduced into the reaction chamber by breaking the seal—an action readily amenable to automation or manual operation. Finally, we need to deploy CreDiT to LMICs for evaluation by local end users. This real-world assessment will help identify potential oversights or challenges within the overall CreDiT workflow, encompassing logistics, pre-analytical handling, and readouts. The deployment will also foster collaboration with LMIC partners to refine the platform, ensuring its adaptability to specific clinical, economic, and environmental factors. We indeed plan to conduct field trials of CreDiT prototypes in Uganda and Ghana; the insights from these trials will guide further improvements in CreDiT’s field performance. Through these advancements, CreDiT will serve as a practical, field-deployable HPV testing solution, enabling the implementation of state-of-the-art HPV guidelines in regions where the impact is the most significant.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Ethical statement The research protocol received approval from the Partners Healthcare Institutional Review Board (IRB protocol 2022P002938). The investigators adhered to all relevant ethical regulations and institutional guidelines throughout the study. Participants provided written informed consent for the reporting and sharing of individual-level data. The gender of participants was determined based on self-reported information provided on the study consent form. No financial compensation was provided to study participants. Materials We had all oligonucleotides used in this study synthesized by Integrated DNA Technologies, Inc. (IDT; USA). The sequences of the oligonucleotides are listed in Supplementary Table 2 . TwistAmp® Basic kit and RevertAid reverse transcriptase (RTase) were purchased from TwistDx TM and Thermo Fisher Scientific (USA), respectively. EnGen® Lba Cas12a (Cas12a), RNase inhibitor (Murine), Pro K, and Luna Universal qPCR Master Mix were purchased from New England BioLabs (USA). All other chemicals were of analytical grade and used without further purification. CreDiT device construction We designed the CreDiT device using computer-aided design software (Solidworks, 2019) and fabricated the main device body through 3D printing (Form2, Formlabs). The device housed a rotating sample holder, a heating unit, an optical module for fluorescent detection, and two onboard computers. (i) The sample holder was cut in a metal block (aluminum 6061-T651, Protolabs) via a computer numerical control (CNC) machine. Into the holder, we inserted four ceramic heaters (HT15W, Thorlabs) for heating and four thermistors (10 kΩ) for temperature reading. The holder was mounted on a stepping motor (NEMA-8, Adafruit) to position each sample for optical detection. We also installed a 60 mm fan (OD6010-12HB, Orion Fans) underneath the holder for cooling purposes. (ii) The optical module had a pair of light a source and a fluorescent detector. The light source consisted of a 470-nm LED (SSL80, OSRAM), an aspheric condenser lens (ACL12708U, Thorlabs), and biconvex focusing lenses (LB1092-A, Thorlabs). The LED was attached to a metal heat sink and driven by a single-supply, low-noise LED current source driver (IRLMS2002TRPBF, Infineon). The fluorescent detector consisted of a filter set (GFP/FITC/Cy2 filter cube, Nikon), a convex lens (LB1761-A, Thorlabs), and a photodiode (S1223, Hamamatsu). The signal from the photodiode was amplified by an operational amplifier with a –3 dB bandwidth of 17 MHz (Analog Devices AD823AN) and digitized by a microcontroller. CreDiT device control We used two onboard computers for the system control. The master unit (Arduino Uno) executed a feedback loop to achieve the desired temperature profile in the sample holder, controlled the stepping motor, and communicated with an external computer. The signal-processing unit (ARM Cortex M4 microcontroller; MK20DX256) generated the waveform LED excitation, digitized the signal from the photodiode, and performed all necessary signal processing, including the Walsh–Hadamard transform. (iv) As a graphical user interface (GUI), we programmed a real-time dashboard using the Qt GUI framework (Supplementary Fig. 2 ), which communicated with the master unit over a universal serial bus (USB) connection. The firmware of the master unit was written in C++. CreDiT signal processing for fluorescent measurements For fluorescent measurements, we used Walsh functions of size 512. To generate the excitation waveform E ( t ), we specified an input vector D of length 512. The i th element of D represented the value for the i th sequency in the WH domain. Applying the Fast Walsh–Hadamard transform (FWHT) to D yielded the waveform E ( t ). We adjusted the excitation waveform by scaling it and adding a positive voltage offset to drive an LED in the full voltage swing (0–6 V). The resulting input voltage was E ( t ) = 3· W ( t ) + 3 (V), where W ( t ) is the Walsh function. We subsequently measured a Walsh-modulated fluorescence signal using a photodiode. This signal S ( t ) was directly digitized (200 kHz) by a 12-bit analog-to-digital converter built into the MCU (MK20DX256). The MCU then performed the inverse FWHT, converting S ( t ) into the sequency vector T . We finally obtained the signal intensity by taking an inner vector product of D·T . In the current CreDiT prototype, we used an input vector D 107 , whose 107th element was the only non-zero element. This sequency was close to the operating frequency of 20.9 kHz. A similar approach was applied for the Fourier modulation, with the input voltage E ( t ) = 3·sin(2πƒ· t ) + 3 (V), where ƒ is the waveform frequency. The average input power to LED had a 4:3 ratio between Walsh and Fourier modulations. Cell culture and NA extraction We purchased Ca Ski (CRL-1550), SiHa (HTB-35), HeLa (CCL2), and C-33A (HTB-31) cells from the American Type Culture Collection and grew them (37 °C, 5% CO 2 ) in the vendor-recommended growth media (RPMI-1640 for Ca Ski and DMEM for SiHa, HeLa, and C-33A) supplemented with 10% fetal bovine serum, 100 U/mL penicillin, and 100 µg/mL streptomycin. All cell lines were regularly tested for mycoplasma contamination using a mycoplasma detection kit (MycoAlert TM , Lonza). We extracted NA from these cells following the Pro K-based CreDiT extraction protocol. The lysis buffer had 0.5 mg/mL Pro K in 1× Tris-EDTA (TE; pH 8.0). As a gold standard for NA extraction, we purchased Quick-DNA Miniprep (Zymo Research) and used it according to the manufacturer’s protocol. CreDiT assay We prepared the CreDiT assay mix (21.75 µL) by combining target-specific RPA primers (5 µM; 1.2 µL for each primer), reporter probe (5 µM; 1 µL), gene-specific Cas12a gRNAs (25 µM; 0.16 µL for each target region), murine RNase inhibitor (40 U/µL; 0.5 µL), Cas12a (100 µM; 0.16 µL), 10× NEBuffer 2.1 (10 mM Tris-HCl, 50 mM NaCl, 10 mM MgCl 2 , 100 µg/mL BSA, pH 7.9 at 1× concentration; 0.5 µL), and reconstituted RPA mix (14.75 µL). After adding NA extract (2 µL) and MgOAc (280 mM, 1.25 µL), we incubated the mix at 42 °C for 20 min, followed by fluorescence measurement (see Supplementary Fig. 19 for the workflow). To analyze mRNA targets, we included 1 U/µL RTase in the CreDiT assay mix. The limit of detection (LOD) was determined based on S 0 + 3σ, where S 0 and σ are the signal and the standard deviation of a blank sample, respectively. The limit of quantification (LOQ) was determined based on S 0 + 10σ. Gel electrophoresis was performed using the Mupid One Electrophoresis System (Eurogentec) for analysis of CreDiT assay products, followed by imaging with the ChemiDoc XRS+ System (Bio-Rad). Whole gel images are shown in Supplementary Figs. 20 , 21 . Lyophilization of CreDiT reagents We prepared the CreDiT assay premix and snap-froze the mixture by immersing it in liquid nitrogen. The frozen mixture was then lyophilized overnight at room temperature on a VirTis FreezeMobile 25 L freeze dryer (SP Scientific). The lyophilized mix was stored at room temperature until use. Quantitative PCR (qPCR) and RT-qPCR We used PCR for quantitative comparison with CreDiT. For DNA targets, qPCR was employed. We prepared the PCR solution (20 µL) by mixing 250 nM IVT primers, 1× Luna Universal qPCR Master Mix, and extracted DNA. We then processed the mixture on a CFX Opus 96 real-time PCR instrument (Bio-Rad). The temperature profile was 95 °C (1 min), followed by cycles of 95 °C (15 s) and 60 °C (30 s). For mRNA targets, we performed RT-qPCR. The sample mixture (20 µL) contains 250 nM gene-specific primers, SuperScript™ III Platinum™ SYBR™ Green One-Step qRT-PCR Kit (Thermo Fisher Scientific), and target RNA (from 8000 cells). The reaction proceeded with the following protocol: 42 °C, 10 min (RT) and 94 °C, 5 min (RTase inactivation), followed by thermal cycling of 94 °C. (5 s), 56 °C (15 s), and 72 °C (15 s). The threshold cycle ( C t ) was automatically determined by the system software. Gel electrophoresis of the PCR products was run in 2% E-Gel™ EX Agarose Gels on the E-Gel™ Power Snap Electrophoresis Device, followed by imaging with Azure 280 (Azure Biosystems). Clinical sample analysis We collected cervical or anal Pap test samples (BD SurePath™ liquid-based Pap test, Becton, Dickinson and Company) during gynecological or anal exams. Clinical pathologic diagnostics were performed using the BD Onclarity HPV Assay system. Blind to the clinical diagnostic results, we analyzed aliquots (0.5 mL) of the residual clinical materials. We processed samples in the extraction buffer (0.5 mg/mL Pro K in 1× TE) following the heat-based extraction protocol. The extracted NA was then subjected to the CreDiT assay. For each NA target, we used 2 µL of NA extract. Statistics and reproducibility We used GraphPad Prism version 9.5 (GraphPad Software Inc.) or R version 4.2.2 for statistical analyses. For two-group comparisons, we used an unpaired, two-sided t -test. For all statistical tests, p values less than 0.05 were considered significant. Details on data presentation and the sample size are included in figure legends. In the study of clinical samples, no statistical method was used to predetermine the sample size, and the experiments were not randomized. No data were excluded from the analyses. The investigators analyzed the clinical samples using the CreDiT assay while blinded to the clinical diagnostic results (gold standard). 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 and implement a novel point-of-care diagnostic platform called CreDiT (CRISPR Enhanced Digital Testing) for rapid, accurate, and decentralized detection of high-risk human papillomavirus (hrHPV) types and oncoprotein mRNAs in resource-limited settings, enabling efficient cervical and anal cancer screening through a combination of CRISPR-based signal amplification and digital signal processing."},"entities":{"kind":"list like","value":["CreDiT","CRISPR Enhanced Digital Testing","point-of-care diagnostic platform","high-risk human papillomavirus","hrHPV","oncoprotein mRNAs","resource-limited settings","cervical cancer screening","anal cancer screening","CRISPR-based signal amplification","digital signal processing"],"string":"[\n \"CreDiT\",\n \"CRISPR Enhanced Digital Testing\",\n \"point-of-care diagnostic platform\",\n \"high-risk human papillomavirus\",\n \"hrHPV\",\n \"oncoprotein mRNAs\",\n \"resource-limited settings\",\n \"cervical cancer screening\",\n \"anal cancer screening\",\n \"CRISPR-based signal amplification\",\n \"digital signal processing\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1.","title":"Randall, T. 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Reliability of the Xpert HPV assay to detect high-risk human papillomavirus DNA in a colposcopy referral population.Am. J. Clin. Pathol143, 126–133 (2015).","link":"https://doi.org/10.1309%2FAJCP4Q0NSDHWIZGU"},{"idx":"6.","title":"Zhang, J. et al. Effectiveness of high-risk human papillomavirus testing for cervical cancer screening in China: a multicenter, open-label, randomized clinical trial.JAMA Oncol.7, 263–270 (2021).","link":"https://doi.org/10.1001%2Fjamaoncol.2020.6575"},{"idx":"7.","title":"Goel, A. et al. Visual inspection of the cervix with acetic acid for cervical intraepithelial lesions.Int. J. Gynaecol. Obstet.88, 25–30 (2005).","link":"https://doi.org/10.1016%2Fj.ijgo.2004.09.018"},{"idx":"8.","title":"Hegde, D., Shetty, H., Shetty, P. K. & Rai, S. Diagnostic value of acetic acid comparing with conventional Pap smear in the detection of colposcopic biopsy-proved CIN.J. Cancer Res. Ther.7, 454–458 (2011).","link":"https://doi.org/10.4103%2F0973-1482.92019"},{"idx":"9.","title":"Lee, C. Y. et al. Development of integrated systems for on-site infection detection.Acc. Chem. Res.54, 3991–4000 (2021).","link":"https://doi.org/10.1021%2Facs.accounts.1c00498"},{"idx":"10.","title":"Aman, R., Mahas, A. & Mahfouz, M. Nucleic acid detection using CRISPR/Cas biosensing technologies.ACS Synth. Biol.9, 1226–1233 (2020).","link":"https://doi.org/10.1021%2Facssynbio.9b00507"},{"idx":"11.","title":"Kaminski, M. M., Abudayyeh, O. O., Gootenberg, J. S., Zhang, F. & Collins, J. J. CRISPR-based diagnostics.Nat. Biomed. Eng.5, 643–656 (2021).","link":"https://doi.org/10.1038%2Fs41551-021-00760-7"},{"idx":"12.","title":"Grove, A. C. An introduction to Walsh functions and their applications.Int. J. Math. Educ. Sci. Technol.14, 43–53 (1983).","link":"https://doi.org/10.1080%2F0020739830140107"},{"idx":"14.","title":"Wei, F. et al. Epidemiology of anal human papillomavirus infection and high-grade squamous intraepithelial lesions in 29,900 men according to HIV status, sexuality, and age: a collaborative pooled analysis of 64 studies.Lancet HIV8, e531–e543 (2021).","link":"https://doi.org/10.1016%2FS2352-3018%2821%2900108-9"},{"idx":"15.","title":"Marra, E., Lin, C. & Clifford, G. M. Type-specific anal human papillomavirus prevalence among men, according to sexual preference and HIV status: a systematic literature review and meta-analysis.J. Infect. Dis.219, 590–598 (2019).","link":"https://doi.org/10.1093%2Finfdis%2Fjiy556"},{"idx":"16.","title":"Ding, X. et al. Ultrasensitive and visual detection of SARS-CoV-2 using all-in-one dual CRISPR-Cas12a assay.Nat. Commun.11, 4711 (2020).","link":"https://doi.org/10.1038%2Fs41467-020-18575-6"},{"idx":"17.","title":"Lee, C. Y. et al. Fluorescence polarization system for rapid COVID-19 diagnosis.Biosens. Bioelectron.178, 113049 (2021).","link":"https://doi.org/10.1016%2Fj.bios.2021.113049"},{"idx":"19.","title":"Deb, A., Sen, S. K. & Datta, A. K. Walsh functions and their applications: a review.IETE Tech. Rev.9, 238–252 (1992).","link":"https://doi.org/10.1080%2F02564602.1992.11438882"},{"idx":"21.","title":"Beer, T. Walsh transforms.Am. J. Phys.49, 466–472 (1981).","link":"https://doi.org/10.1119%2F1.12714"},{"idx":"23.","title":"Ho, N. R. Y. et al. Visual and modular detection of pathogen nucleic acids with enzyme-DNA molecular complexes.Nat. Commun.9, 3238 (2018).","link":"https://doi.org/10.1038%2Fs41467-018-05733-0"},{"idx":"24.","title":"Pathania, D. et al. Point-of-care cervical cancer screening using deep learning-based microholography.Theranostics9, 8438–8447 (2019).","link":"https://doi.org/10.7150%2Fthno.37187"},{"idx":"25.","title":"Gustinucci, D. et al. Use of cytology, E6/E7 mRNA, and p16INK4a-Ki-67 to define the management of human papillomavirus (HPV)-positive women in cervical cancer screening.Am. J. Clin. Pathol.145, 35–45 (2016).","link":"https://doi.org/10.1093%2Fajcp%2Faqv019"},{"idx":"26.","title":"de Martel, C., Plummer, M., Vignat, J. & Franceschi, S. Worldwide burden of cancer attributable to HPV by site, country and HPV type.Int. J. Cancer141, 664–670 (2017).","link":"https://doi.org/10.1002%2Fijc.30716"},{"idx":"27.","title":"Lin, C., Franceschi, S. & Clifford, G. M. Human papillomavirus types from infection to cancer in the anus, according to sex and HIV status: a systematic review and meta-analysis.Lancet Infect. Dis.18, 198–206 (2018).","link":"https://doi.org/10.1016%2FS1473-3099%2817%2930653-9"},{"idx":"28.","title":"Zhao, Y. et al. Integrating CRISPR-Cas12a into a microfluidic dual-droplet device enables simultaneous detection of HPV16 and HPV18.Anal. Chem.95, 3476–3485 (2023).","link":"https://doi.org/10.1021%2Facs.analchem.2c05320"},{"idx":"29.","title":"Xu, Z. et al. Microfluidic space coding for multiplexed nucleic acid detection via CRISPR-Cas12a and recombinase polymerase amplification.Nat. Commun.13, 6480 (2022).","link":"https://doi.org/10.1038%2Fs41467-022-34086-y"},{"idx":"30.","title":"Zhou, H. et al. Coupling CRISPR/Cas12a and recombinase polymerase amplification on a stand-alone microfluidics platform for fast and parallel nucleic acid detection.Anal. Chem.95, 3379–3389 (2023).","link":"https://doi.org/10.1021%2Facs.analchem.2c04713"},{"idx":"32.","title":"Tang, Y. et al. CLIPON: a CRISPR-enabled strategy that turns commercial pregnancy test strips into general point-of-need test devices.Angew. Chem. Int. Ed. Engl.61, e202115907 (2022).","link":"https://doi.org/10.1002%2Fanie.202115907"},{"idx":"33.","title":"McCredie, M. R. et al. Natural history of cervical neoplasia and risk of invasive cancer in women with cervical intraepithelial neoplasia 3: a retrospective cohort study.Lancet Oncol.9, 425–434 (2008).","link":"https://doi.org/10.1016%2FS1470-2045%2808%2970103-7"},{"idx":"34.","title":"Ogilvie, G. S. et al. HPV for cervical cancer screening (HPV FOCAL): complete round 1 results of a randomized trial comparing HPV-based primary screening to liquid-based cytology for cervical cancer.Int. J. Cancer140, 440–448 (2017).","link":"https://doi.org/10.1002%2Fijc.30454"},{"idx":"35.","title":"Ogilvie, G. S. et al. Effect of screening with primary cervical HPV testing vs cytology testing on high-grade cervical intraepithelial neoplasia at 48 months: the HPV FOCAL randomized clinical trial.JAMA320, 43–52 (2018).","link":"https://doi.org/10.1001%2Fjama.2018.7464"},{"idx":"36.","title":"Abreu, A. L., Souza, R. P., Gimenes, F. & Consolaro, M. E. A review of methods for detect human Papillomavirus infection.Virol. J.9, 262 (2012).","link":"https://link.springer.com/doi/10.1186/1743-422X-9-262"},{"idx":"37.","title":"Arbyn, M. et al. Accuracy of human papillomavirus testing on self-collected versus clinician-collected samples: a meta-analysis.Lancet Oncol.15, 172–183 (2014).","link":"https://doi.org/10.1016%2FS1470-2045%2813%2970570-9"},{"idx":"38.","title":"Kelly, R. S. et al. HPV testing as a triage for borderline or mild dyskaryosis on cervical cytology: results from the Sentinel Sites study.Br. J. Cancer105, 983–988 (2011).","link":"https://doi.org/10.1038%2Fbjc.2011.326"},{"idx":"39.","title":"Ragin, C. et al. Prevalence of HPV infection in racial-ethnic subgroups of head and neck cancer patients.Carcinogenesis38, 218–229 (2017).","link":"https://doi.org/10.1093%2Fcarcin%2Fbgw203"},{"idx":"40.","title":"Vidal, A. C. et al. HPV genotypes and cervical intraepithelial neoplasia in a multiethnic cohort in the southeastern USA.Cancer Causes Control25, 1055–1062 (2014).","link":"https://link.springer.com/doi/10.1007/s10552-014-0406-2"},{"idx":"41.","title":"Gootenberg, J. S. et al. Multiplexed and portable nucleic acid detection platform with Cas13, Cas12a, and Csm6.Science360, 439–444 (2018).","link":"https://doi.org/10.1126%2Fscience.aaq0179"},{"idx":"42.","title":"Jiao, C. et al. Noncanonical crRNAs derived from host transcripts enable multiplexable RNA detection by Cas9.Science372, 941–948 (2021).","link":"https://doi.org/10.1126%2Fscience.abe7106"}],"string":"[\n {\n \"idx\": \"1.\",\n \"title\": \"Randall, T. C. & Ghebre, R. Challenges in prevention and care delivery for women with cervical cancer in sub-Saharan Africa.Front. Oncol.6, 160 (2016).\",\n \"link\": \"https://doi.org/10.3389%2Ffonc.2016.00160\"\n },\n {\n \"idx\": \"2.\",\n \"title\": \"Montealegre, J. R. et al. Racial/ethnic differences in HPV 16/18 genotypes and integration status among women with a history of cytological abnormalities.Gynecol. Oncol.148, 357–362 (2018).\",\n \"link\": \"https://doi.org/10.1016%2Fj.ygyno.2017.12.014\"\n },\n {\n \"idx\": \"3.\",\n \"title\": \"Islami, F., Fedewa, S. A. & Jemal, A. Trends in cervical cancer incidence rates by age, race/ethnicity, histological subtype, and stage at diagnosis in the United States.Prev. Med.123, 316–323 (2019).\",\n \"link\": \"https://doi.org/10.1016%2Fj.ypmed.2019.04.010\"\n },\n {\n \"idx\": \"4.\",\n \"title\": \"Sankaranarayanan, R. et al. HPV screening for cervical cancer in rural India.N. Engl. J. Med.360, 1385–1394 (2009).\",\n \"link\": \"https://doi.org/10.1056%2FNEJMoa0808516\"\n },\n {\n \"idx\": \"5.\",\n \"title\": \"Castle, P. E. et al. Reliability of the Xpert HPV assay to detect high-risk human papillomavirus DNA in a colposcopy referral population.Am. J. Clin. Pathol143, 126–133 (2015).\",\n \"link\": \"https://doi.org/10.1309%2FAJCP4Q0NSDHWIZGU\"\n },\n {\n \"idx\": \"6.\",\n \"title\": \"Zhang, J. et al. Effectiveness of high-risk human papillomavirus testing for cervical cancer screening in China: a multicenter, open-label, randomized clinical trial.JAMA Oncol.7, 263–270 (2021).\",\n \"link\": \"https://doi.org/10.1001%2Fjamaoncol.2020.6575\"\n },\n {\n \"idx\": \"7.\",\n \"title\": \"Goel, A. et al. Visual inspection of the cervix with acetic acid for cervical intraepithelial lesions.Int. J. Gynaecol. Obstet.88, 25–30 (2005).\",\n \"link\": \"https://doi.org/10.1016%2Fj.ijgo.2004.09.018\"\n },\n {\n \"idx\": \"8.\",\n \"title\": \"Hegde, D., Shetty, H., Shetty, P. K. & Rai, S. Diagnostic value of acetic acid comparing with conventional Pap smear in the detection of colposcopic biopsy-proved CIN.J. Cancer Res. Ther.7, 454–458 (2011).\",\n \"link\": \"https://doi.org/10.4103%2F0973-1482.92019\"\n },\n {\n \"idx\": \"9.\",\n \"title\": \"Lee, C. Y. et al. Development of integrated systems for on-site infection detection.Acc. Chem. Res.54, 3991–4000 (2021).\",\n \"link\": \"https://doi.org/10.1021%2Facs.accounts.1c00498\"\n },\n {\n \"idx\": \"10.\",\n \"title\": \"Aman, R., Mahas, A. & Mahfouz, M. Nucleic acid detection using CRISPR/Cas biosensing technologies.ACS Synth. Biol.9, 1226–1233 (2020).\",\n \"link\": \"https://doi.org/10.1021%2Facssynbio.9b00507\"\n },\n {\n \"idx\": \"11.\",\n \"title\": \"Kaminski, M. M., Abudayyeh, O. O., Gootenberg, J. S., Zhang, F. & Collins, J. J. CRISPR-based diagnostics.Nat. Biomed. Eng.5, 643–656 (2021).\",\n \"link\": \"https://doi.org/10.1038%2Fs41551-021-00760-7\"\n },\n {\n \"idx\": \"12.\",\n \"title\": \"Grove, A. C. An introduction to Walsh functions and their applications.Int. J. Math. Educ. Sci. Technol.14, 43–53 (1983).\",\n \"link\": \"https://doi.org/10.1080%2F0020739830140107\"\n },\n {\n \"idx\": \"14.\",\n \"title\": \"Wei, F. et al. Epidemiology of anal human papillomavirus infection and high-grade squamous intraepithelial lesions in 29,900 men according to HIV status, sexuality, and age: a collaborative pooled analysis of 64 studies.Lancet HIV8, e531–e543 (2021).\",\n \"link\": \"https://doi.org/10.1016%2FS2352-3018%2821%2900108-9\"\n },\n {\n \"idx\": \"15.\",\n \"title\": \"Marra, E., Lin, C. & Clifford, G. M. Type-specific anal human papillomavirus prevalence among men, according to sexual preference and HIV status: a systematic literature review and meta-analysis.J. Infect. Dis.219, 590–598 (2019).\",\n \"link\": \"https://doi.org/10.1093%2Finfdis%2Fjiy556\"\n },\n {\n \"idx\": \"16.\",\n \"title\": \"Ding, X. et al. Ultrasensitive and visual detection of SARS-CoV-2 using all-in-one dual CRISPR-Cas12a assay.Nat. Commun.11, 4711 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-020-18575-6\"\n },\n {\n \"idx\": \"17.\",\n \"title\": \"Lee, C. Y. et al. Fluorescence polarization system for rapid COVID-19 diagnosis.Biosens. Bioelectron.178, 113049 (2021).\",\n \"link\": \"https://doi.org/10.1016%2Fj.bios.2021.113049\"\n },\n {\n \"idx\": \"19.\",\n \"title\": \"Deb, A., Sen, S. K. & Datta, A. K. Walsh functions and their applications: a review.IETE Tech. Rev.9, 238–252 (1992).\",\n \"link\": \"https://doi.org/10.1080%2F02564602.1992.11438882\"\n },\n {\n \"idx\": \"21.\",\n \"title\": \"Beer, T. Walsh transforms.Am. J. Phys.49, 466–472 (1981).\",\n \"link\": \"https://doi.org/10.1119%2F1.12714\"\n },\n {\n \"idx\": \"23.\",\n \"title\": \"Ho, N. R. Y. et al. Visual and modular detection of pathogen nucleic acids with enzyme-DNA molecular complexes.Nat. Commun.9, 3238 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-018-05733-0\"\n },\n {\n \"idx\": \"24.\",\n \"title\": \"Pathania, D. et al. Point-of-care cervical cancer screening using deep learning-based microholography.Theranostics9, 8438–8447 (2019).\",\n \"link\": \"https://doi.org/10.7150%2Fthno.37187\"\n },\n {\n \"idx\": \"25.\",\n \"title\": \"Gustinucci, D. et al. Use of cytology, E6/E7 mRNA, and p16INK4a-Ki-67 to define the management of human papillomavirus (HPV)-positive women in cervical cancer screening.Am. J. Clin. Pathol.145, 35–45 (2016).\",\n \"link\": \"https://doi.org/10.1093%2Fajcp%2Faqv019\"\n },\n {\n \"idx\": \"26.\",\n \"title\": \"de Martel, C., Plummer, M., Vignat, J. & Franceschi, S. Worldwide burden of cancer attributable to HPV by site, country and HPV type.Int. J. Cancer141, 664–670 (2017).\",\n \"link\": \"https://doi.org/10.1002%2Fijc.30716\"\n },\n {\n \"idx\": \"27.\",\n \"title\": \"Lin, C., Franceschi, S. & Clifford, G. M. Human papillomavirus types from infection to cancer in the anus, according to sex and HIV status: a systematic review and meta-analysis.Lancet Infect. Dis.18, 198–206 (2018).\",\n \"link\": \"https://doi.org/10.1016%2FS1473-3099%2817%2930653-9\"\n },\n {\n \"idx\": \"28.\",\n \"title\": \"Zhao, Y. et al. Integrating CRISPR-Cas12a into a microfluidic dual-droplet device enables simultaneous detection of HPV16 and HPV18.Anal. Chem.95, 3476–3485 (2023).\",\n \"link\": \"https://doi.org/10.1021%2Facs.analchem.2c05320\"\n },\n {\n \"idx\": \"29.\",\n \"title\": \"Xu, Z. et al. Microfluidic space coding for multiplexed nucleic acid detection via CRISPR-Cas12a and recombinase polymerase amplification.Nat. Commun.13, 6480 (2022).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-022-34086-y\"\n },\n {\n \"idx\": \"30.\",\n \"title\": \"Zhou, H. et al. Coupling CRISPR/Cas12a and recombinase polymerase amplification on a stand-alone microfluidics platform for fast and parallel nucleic acid detection.Anal. Chem.95, 3379–3389 (2023).\",\n \"link\": \"https://doi.org/10.1021%2Facs.analchem.2c04713\"\n },\n {\n \"idx\": \"32.\",\n \"title\": \"Tang, Y. et al. CLIPON: a CRISPR-enabled strategy that turns commercial pregnancy test strips into general point-of-need test devices.Angew. Chem. Int. Ed. Engl.61, e202115907 (2022).\",\n \"link\": \"https://doi.org/10.1002%2Fanie.202115907\"\n },\n {\n \"idx\": \"33.\",\n \"title\": \"McCredie, M. R. et al. Natural history of cervical neoplasia and risk of invasive cancer in women with cervical intraepithelial neoplasia 3: a retrospective cohort study.Lancet Oncol.9, 425–434 (2008).\",\n \"link\": \"https://doi.org/10.1016%2FS1470-2045%2808%2970103-7\"\n },\n {\n \"idx\": \"34.\",\n \"title\": \"Ogilvie, G. S. et al. HPV for cervical cancer screening (HPV FOCAL): complete round 1 results of a randomized trial comparing HPV-based primary screening to liquid-based cytology for cervical cancer.Int. J. Cancer140, 440–448 (2017).\",\n \"link\": \"https://doi.org/10.1002%2Fijc.30454\"\n },\n {\n \"idx\": \"35.\",\n \"title\": \"Ogilvie, G. S. et al. Effect of screening with primary cervical HPV testing vs cytology testing on high-grade cervical intraepithelial neoplasia at 48 months: the HPV FOCAL randomized clinical trial.JAMA320, 43–52 (2018).\",\n \"link\": \"https://doi.org/10.1001%2Fjama.2018.7464\"\n },\n {\n \"idx\": \"36.\",\n \"title\": \"Abreu, A. L., Souza, R. P., Gimenes, F. & Consolaro, M. E. A review of methods for detect human Papillomavirus infection.Virol. J.9, 262 (2012).\",\n \"link\": \"https://link.springer.com/doi/10.1186/1743-422X-9-262\"\n },\n {\n \"idx\": \"37.\",\n \"title\": \"Arbyn, M. et al. Accuracy of human papillomavirus testing on self-collected versus clinician-collected samples: a meta-analysis.Lancet Oncol.15, 172–183 (2014).\",\n \"link\": \"https://doi.org/10.1016%2FS1470-2045%2813%2970570-9\"\n },\n {\n \"idx\": \"38.\",\n \"title\": \"Kelly, R. S. et al. HPV testing as a triage for borderline or mild dyskaryosis on cervical cytology: results from the Sentinel Sites study.Br. J. Cancer105, 983–988 (2011).\",\n \"link\": \"https://doi.org/10.1038%2Fbjc.2011.326\"\n },\n {\n \"idx\": \"39.\",\n \"title\": \"Ragin, C. et al. Prevalence of HPV infection in racial-ethnic subgroups of head and neck cancer patients.Carcinogenesis38, 218–229 (2017).\",\n \"link\": \"https://doi.org/10.1093%2Fcarcin%2Fbgw203\"\n },\n {\n \"idx\": \"40.\",\n \"title\": \"Vidal, A. C. et al. HPV genotypes and cervical intraepithelial neoplasia in a multiethnic cohort in the southeastern USA.Cancer Causes Control25, 1055–1062 (2014).\",\n \"link\": \"https://link.springer.com/doi/10.1007/s10552-014-0406-2\"\n },\n {\n \"idx\": \"41.\",\n \"title\": \"Gootenberg, J. S. et al. Multiplexed and portable nucleic acid detection platform with Cas13, Cas12a, and Csm6.Science360, 439–444 (2018).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.aaq0179\"\n },\n {\n \"idx\": \"42.\",\n \"title\": \"Jiao, C. et al. Noncanonical crRNAs derived from host transcripts enable multiplexable RNA detection by Cas9.Science372, 941–948 (2021).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.abe7106\"\n }\n]","isConversational":false}}},{"rowIdx":11197,"cells":{"unique_id":{"kind":"string","value":"s41467-022-29771-x"},"subfield":{"kind":"string","value":"Materials science"},"title":{"kind":"string","value":"Reaching the highest efficiency of spin Hall effect of light in the near-infrared using all-dielectric metasurfaces"},"abstract":{"kind":"string","value":"The spin Hall effect of light refers to a spin-dependent transverse splitting of light at a planar interface. Previous demonstrations to enhance the splitting have suffered from exceedingly low efficiency. Achievements of the large splitting with high efficiency have been reported in the microwave, but those in the optical regime remain elusive. Here, an approach to attain the large splitting with high efficiency in the near-infrared is proposed and experimentally demonstrated at 800 nm by using a dielectric metasurface. Modulation of the complex transmission of the metasurface leads to the shifts that reach 10 λ along with efficiencies over 70% under two linear polarizations. Our work extends the recent attempts to achieve the large and efficient spin Hall effect of light, which have been limited only to the microwave, to the optical regime."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"Metasurfaces, nanostructure arrays that are designed to exhibit desired optical responses, have emerged as powerful tools to modulate various features of electromagnetic waves such as amplitudes, phases, and polarizations [1] . Whereas materials in nature have limited optical properties, the metasurfaces have shown unprecedented light manipulation capabilities despite their deep-subwavelength thickness. To maximize the light–matter interaction in a short spatial path length, the nanostructures that compose the metasurfaces have been constructed from plasmonic materials [2] , [3] . At optical wavelengths, however, plasmonic metasurfaces generally suffer from low efficiency due to Ohmic loss, thereby hindering their practical implementations. As alternatives, all-dielectric metasurfaces, constituents of which are all made of dielectric materials, have been suggested to manipulate the optical responses efficiently [4] , [5] , [6] . The dielectric metasurfaces have reproduced many exotic optical responses of the plasmonic metasurfaces such as polarization control [5] , [7] , ultrathin lenses [8] , [9] , [10] , [11] , and holographic display [12] , [13] , [14] , [15] with even higher efficiencies than the plasmonic metasurfaces. Furthermore, the metasurfaces’ compactness, compatibility, and ease of integration have made them a viable replacement for bulky optical components [16] , [17] , [18] , [19] . During the past decade, a variety of optical analogies of the spin Hall effect [20] , [21] such as spin Hall effect of light (SHEL) [22] , [23] , optical Magnus effect [24] , spin-dependent angular splitting [25] , plasmonic spin Hall effect [26] , [27] , and optical spin Hall effect [28] have regained extensive scientific interest when combined with structured media [29] , [30] , [31] , [32] , [33] such as metasurfaces and metamaterials. Among them, the SHEL is a transverse and spin-dependent spatial displacement of light at refraction or reflection [34] , [35] , [36] . At an interface between naturally occurring materials, the SHEL exhibits in general a shift that is much smaller than the wavelength [23] . However, specifically designed metasurfaces and metamaterials can enhance the shift by several orders of magnitude [37] , [38] , [39] , [40] , [41] , [42] , [43] , [44] from a few micrometers [45] to a few hundreds of micrometers [46] , [47] . Unfortunately, the drastic enhancement of the SHEL generally entails extremely low efficiency, as a result of a vanishing Fresnel coefficient [37] , [38] , [40] . A simultaneous enhancement of the shift and efficiency has been achieved very recently in the microwave regime using hyperbolic [48] and index-near-zero [49] metamaterials. However, these demonstrations cannot be readily adapted to the optical regime that encompasses ultraviolet, visible, and infrared wavelengths (10 nm < λ < 1 mm) because the operating principles rely on effective medium properties that require deep-subwavelength structures going down from hundreds of micrometers to a few tens of nanometers scale for optical operations. The lack of fabrication methods for such complicated structures and high optical losses of metals in the optical regime hinder the realization of the large and efficient SHEL. Thus, a large SHEL with high efficiency has not been achieved in the optical regime. Here we propose an all-dielectric metasurface to enhance both the shift and efficiency of the SHEL simultaneously in the near-infrared regime. In contrast to the previous approaches that have demanded a three-dimensional or stacked bulky structure to achieve the desired effective medium properties, here, a radically different approach is used by exploiting a single-layer metasurface as a compact and versatile route to modulate the transmission coefficients. To achieve high efficiency, the metasurface is designed to yield high transmission amplitudes for both s and p polarizations, with the phase difference between the two linear polarizations being far from zero. Modulation of the complex transmission results in a simultaneous enhancement of the shift and efficiency under both horizontal and vertical polarizations at 800 nm. More specifically, we experimentally demonstrate a shift and efficiency that reaches 10.0 λ and 71%, respectively, under horizontally polarized incidence and 9.9 λ and 79%, respectively, under vertically polarized incidence. Our work bridges the gap between metasurface research and their implementation towards efficient spin-sensitive devices. The large SHEL with high efficiency based on a dielectric metasurface will find wide applications in ultrahigh precision measurement and compact, spin-sensitive optical devices in nanoscale.","references":{"indices":[1,2,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,22,23,25,26,27,28,29,30,32,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49],"infos":[{"idx":"1","title":"Yu, N. & Capasso, F. Flat optics with designer metasurfaces.Nat. 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Total reflection-induced efficiency enhancement of the spin Hall effect of light.ACS Photonics8, 2705–2712 (2021).","link":"https://doi.org/10.1021%2Facsphotonics.1c00727"}]}},{"section":"Results","content":"Principle When transmitted at an optical interface, a linearly polarized Gaussian beam undergoes a transverse spatial shift by an amount [23] $$\\begin{array}{r}{\\delta }_{H}^{\\pm }/\\lambda =\\pm \\frac{\\cot {\\theta }_{i}}{2\\pi }\\left(1-\\frac{| {t}_{s}| }{| {t}_{p}| }\\cos \\phi \\right),\\\\ {\\delta }_{V}^{\\pm }/\\lambda =\\pm \\frac{\\cot {\\theta }_{i}}{2\\pi }\\left(1-\\frac{| {t}_{p}| }{| {t}_{s}| }\\cos \\phi \\right),\\end{array}$$ (1) where superscripts + and − represent left- and right-circularly polarized components of the transmitted beam, respectively, subscripts H and V correspond to the horizontal and vertical incident polarization respectively, λ is the wavelength, θ i is the incident angle, t s and t p denote Fresnel coefficients of the s - and p -polarized incidence respectively, and ϕ = arg( t s ) − arg( t p ) is the phase difference between the two Fresnel coefficients. Note that the Fresnel coefficients and θ i are those of the central wave vector. Equation ( 1 ) is valid under condition of \\({(2\\pi w/\\lambda )}^{2}\\gg {\\cot }^{2}{\\theta }_{i}\\) for a beam waist w , which is true except for a tightly confined beam under near-normal incidence ( θ i ≪ 1°). This narrow angular regime is where a transition from the SHEL to vortex generation occurs [50] and is out of our consideration. The shift is generally much smaller than the wavelength at an interface between naturally occurring media (Fig. 1a ). An intuitive approach to enhance the shift is to reduce the Fresnel coefficients in the denominator ( ∣ t p ∣ for horizontal and ∣ t s ∣ for vertical case); however, given that the efficiency is directly related to the Fresnel coefficients [48] ( ϵ H = ∣ t p ∣ 2 and ϵ V = ∣ t s ∣ 2 , Fig. 1b ), the method has the intrinsic limitation that it also degrades the efficiency significantly. Recently, two approaches to achieve large SHEL and high efficiency have been proposed [48] , [49] (Fig. 1c ). The key idea is to fix the Fresnel coefficient in the denominator to unity to ensure unity efficiency while enhancing the shift by reducing θ i . At an interface between two isotropic media, because the two Fresnel coefficients are degenerate under normal incidence, both \\(\\cos \\phi | {t}_{p}| /| {t}_{s}|\\) and \\(\\cos \\phi | {t}_{s}| /| {t}_{p}|\\) are close to unity at a small θ i , thereby giving rise to a subwavelength shift. To attain a large shift, an anisotropic medium [48] and near-index-zero metamaterial [49] have been proposed to differentiate the amplitude and phase of the remaining Fresnel coefficients, respectively. These previous methods have been, however, demonstrated in the microwave regime where metals behave as perfect electrical conductors. Realization of the large SHEL with high efficiency in the optical frequencies using this method is not trivial due to the high optical losses of metals and challenging fabrications and measurements. Fig. 1: Schematic of the spin Hall effect of light (SHEL) and a dielectric metasurface to enhance the shift δ and efficiency ϵ of the SHEL simultaneously. a , b Illustration of the SHEL in previous demonstrations. The width of the transmitted beams denotes their intensities. a The efficient SHEL is limited by the small shift, b whereas the enhancement of the shift entails a low efficiency. c Large SHEL with high efficiency. d A metasurface that transmits x - and y -polarized incidence with unity amplitudes ( ∣ t x ∣ = ∣ t y ∣ = 1) but with the opposite phase (arg( t x ) − arg( t y ) = ± π ). Full size image A dielectric metasurface can be used to overcome these difficulties, because of its low loss and the ability to control its complex transmission. Nanostructure arrays that have dimensions carefully chosen to modulate the amplitudes and phase of the transmitted beam can result in a large SHEL and high efficiency simultaneously. Ideally, a metasurface that transmits both x - and y -polarized incidence with unity amplitudes but with a phase difference of π or − π yields a SHEL that reaches \\(\\delta^\\pm / \\lambda = \\pm\\!\\!\\cot {\\theta }_{i}/\\pi\\) with near-unity efficiency at a small θ i ( ≪ π /2) (Fig. 1 d). In short, the design rule is to build a dielectric metasurface operating as a half-wave plate ( ∣ t s ∣ = ∣ t p ∣ = 1, ϕ = ± π ) under normal incidence and to examine the SHEL at small θ i . Whereas the capabilities of metasurfaces to manipulate the transmission or reflection have been well known [5] , [7] , [51] , no attempt has been made to use these qualities to produce a large and efficient SHEL. Furthermore, in comparison to the commercially available half-wave plates, this metasurface-based approach has advantages of compactness and compatibility. Design and fabrication To ensure the phase difference between t s and t p , a unit design of the metasurface has an anisotropic geometry of a rectangular-shaped nanorod on a fused silica substrate (Fig. 2a ). This rectangular nanorod is known to induce a Pancharatnam–Berry phase [25] , [52] under normal incidence. However, the phase contributes negligibly to the SHEL because the Pancharatnam-Berry phase is more dominant than the spin-redirection phase only under normal incidence and at a close vicinity of it [50] ( θ i ≪ 1°), which is not where we are interested in. Hydrogenated amorphous silicon (a-Si:H) [53] is used for its high refractive index and low optical losses (Supplementary Note 7) . This dielectric nanorod can modulate amplitudes and phases of the incoming light with substantially higher efficiency compared to plasmonic resonators. To find an optimal design for ∣ t s ∣ = ∣ t p ∣ = 1 and ϕ = ± π , five geometrical parameters, height ( H ), length ( a ), and width ( b ) of the nanorod and periodicities ( p x and p y ), are determined using a particle swarm optimization [54] to minimize 2 − ∣ t s − t p ∣ at 800 nm wavelength for a taper angle α = 5° (see “Methods” section for details). Then the metasurface is fabricated using electron beam lithography (see “Methods” section). Fig. 2: Design of a dielectric metasurface. a Schematic of a unit cell. Geometric parameters are given as: height H = 400 nm, length a = 265 nm, width b = 158 nm, periodicities p x = 378 nm and p y = 263 nm, and taper angle α . b , c Scanning electron microscopy images of the fabricated sample. b Top view, c perspective view. Inset shows a magnified image. Full size image The transmittance of the metasurface calculated by using full-wave simulation (see “Methods” section for detailed simulation) and measured by Fourier-transform infrared spectroscopy are shown in Fig. 3a . The simulated and measured transmittance agrees well with each other. A low transmittance at 726 nm under x -polarized incidence (Fig. 3a , blue) is attributed to the magnetic dipole resonance manifested by the displacement currents circulating the dielectric nanorods (see Supplementary Note 3) . In contrast, the y -polarized incidence induces the magnetic dipole resonance at a smaller wavelength near 675 nm because of the smaller rod length and periodicity and thus has a high transmittance at λ > 700 nm (Fig. 3a , red). The phase difference ϕ varies rapidly near the magnetic dipole resonances and has a value of −0.87 π at 800 nm (Fig. 3b ). To summarize, the metasurface has a high transmittance under both x - and y -polarized incidences and large ϕ at 800 nm, which is advantageous in realizing a large SHEL with high efficiency. Simulations indicate that a metasurface composed of nontapered cells with different geometrical parameters can have ϕ = ± π , which produces further enhanced SHEL (see Supplementary Note 1) . Fig. 3: Transmittance of the dielectric metasurface under normal incidence. a Simulated (markers) and measured (solid lines) transmittance under normal incidence. b Phase difference ϕ between the two polarizations obtained by numerical simulation. Full size image Experimental demonstration In general, the shift can be experimentally obtained via either weak measurement [23] or Stokes polarimetry measurement [55] , [56] . The former amplifies the shift by several orders of magnitudes where the amplification factor is determined by the setup parameters and the latter measures the Fresnel coefficients and their phase difference under oblique incidence so that the shift can be deduced from Eq. ( 1 ). Here the large SHEL with high efficiency at 800 nm is experimentally verified using a Stokes polarimetry setup (Fig. 4a , see “Methods” section for measurement details). The measurements are performed in two ways: first by setting the incident polarization as 45°-linearly polarized one to measure the phase difference ϕ and then by setting the incident polarization as horizontal and vertical to measure the amplitudes ∣ t p ∣ and ∣ t s ∣ respectively. Then the shifts can be obtained by substituting the measured ϕ , ∣ t s ∣ , and ∣ t p ∣ into Eq. ( 1 ). In all measurements, the metasurface is placed tilted by θ i ranging from 1° to 10° to examine the SHEL at various θ i . Fig. 4: Measurement of the SHEL at 800 nm. a Schematic of Stokes polarimetry set-up. HWP: half-wave plate, L: lens, QWP: quarter-wave plate, LP: linear polarizer, CCD: charge-coupled device camera. b Phase difference between t s and t p , c \\({\\delta }_{H}^{+}/\\lambda\\) , d \\({\\delta }_{V}^{+}/\\lambda\\) , e ϵ H , and f ϵ V . Curve: simulated, markers: measured. Insets in c , d show the magnified view for near-normal incidence where the analytic formula (solid) given by Eq. ( 1 ) deviates from the exact shifts (dashed) due to the breakdown of the assumption, i.e., \\({(2\\pi w/\\lambda )}^{2}\\gg {\\cot }^{2}{\\theta }_{i}\\) . Error bars of the measured data are not shown because they are all smaller than the markers. Full size image In the first measurement, we obtain the second and third Stokes parameters, $$\\begin{array}{rc}{s}_{\\!2}&=I({0}^{\\circ },4{5}^{\\circ })-I({0}^{\\circ },13{5}^{\\circ }),\\\\ {s}_{3}&=I(9{0}^{\\circ },4{5}^{\\circ })-I(9{0}^{\\circ },13{5}^{\\circ })\\end{array}$$ (2) where I ( β , γ ) is the spatial integration of the intensity when retardation angle of the quarter-wave plate is β and rotation angle of the linear polarizer is γ (Fig. 4a ). Then the measured Stokes parameters are used to calculate [56] ϕ by using a relation of \\(\\tan \\phi ={s}_{3}/{s}_{2}\\) . The measured ϕ deviates slightly from the simulated values because of the fabrication imperfections but exhibits a similar tendency (Fig. 4b ). In the second measurement, ∣ t p ∣ and ∣ t s ∣ under oblique incidence is obtained by taking the square roots of the intensity of transmitted beam under horizontally (Fig. 4e ) and vertically (Fig. 4f ) polarized incidence respectively. Note that the intensity is normalized by that measured without the sample. The shifts calculated by substituting the measured ϕ , ∣ t s ∣ , and ∣ t p ∣ in Eq. ( 1 ) and their efficiencies are shown in Fig. 4c–f . The shift is enhanced remarkably up to several wavelengths at a small θ i (≤ 5°) under both horizontal (Fig. 4 c) and vertical (Fig. 4d ) polarizations. We experimentally observe the shift of 10.0 λ along with 71% efficiency under horizontal polarization and the shift of 9.9 λ along with 79% efficiency under vertical polarization at θ i = 1°. The shifts can be further increased by reducing θ i except for an extremely narrow region near θ i = 0°. When θ i is sufficiently small (≤ 0.01°), Equation ( 1 ) deviates from the exact shifts and converges to zero rapidly (Fig. 4c, d , inset). The metasurface exhibits high transmittance for both polarizations even under oblique incidence. Despite the large SHEL, high efficiencies are observed at the wide range of θ i (Fig. 4e, f ). For completeness, the shift is also obtained using the weak measurement technique. The shift is 1.31 λ under vertically polarized light at θ i = 10°, which agrees well with the result from the Stokes polarimetry method (Supplementary Note 6) . The shift and efficiencies in our work in comparison with previous studies are summarized in Supplementary Note 4 ."},{"section":"Discussion","content":"We experimentally demonstrate simultaneous enhancement of the shift and efficiency of the spin Hall effect of light in the optical regime for the first time by using a dielectric metasurface. Whereas previous approaches to enhance the shift have suffered from low efficiencies and the achievements of a large shift with high efficiency have been limited in the microwave regime, we present the large shift and high efficiency at 800 nm by modulating the complex transmission of the metasurface. Shifts that reach 10 λ with efficiencies >70% are confirmed experimentally under two linearly polarized incidences. Furthermore, the shift can be further enhanced by reducing the taper angle through an optimization of the etching condition [57] . The simultaneous enhancement of the shift and efficiency can be also achieved in a reflective metasurface by changing geometrical parameters (Supplementary Note 5) . This approach towards the large spin Hall effect of light with high efficiency has two distinct merits. First, it facilitates ultrahigh precision measurements by increasing an amplification factor of a weak measurement. The amplification factor is directly linked to the precision, but its increase is practically bounded by the detectable signal strength. The n -fold enhancement of the SHEL efficiency provides n times stronger signal, which, in turn, enhances the precision by n times. Secondly, the large spin Hall effect of light with high efficiency in the optical regime enables realizing compact photonic devices such as spin-sensitive filters. With these clear advantages, the metasurface-driven simultaneous enhancement of the shift and efficiency will find wide applications in spin-dependent optics."},{"section":"Methods","content":"Numerical simulation and optimization Transmission coefficients of the dielectric metasurface were calculated using commercial software (COMSOL Multiphysics). For transmission under normal incidence, a unit cell with periodic boundary conditions along the x - and y -axes and perfectly matched layers and ports along the z -axis was simulated. For transmission under oblique incidence, simulation conditions were similar to these, except that the periodic boundary conditions were replaced by the Bloch boundary conditions. The dielectric metasurface was optimized to minimize 2 − ∣ t s − t p ∣ at 800 nm by using particle swarm optimization. Starting from a rectangular nanorod that has height H , length a , and width b and is periodically arranged with periodicities p x and p y , the five parameters were searched during 100 iterations. Sample fabrication The all-dielectric metasurface was fabricated using standard plasma-enhanced chemical vapor deposition (PECVD), electron-beam lithography (EBL), and reactive-ion etching. The fabrication process begins with the deposition of hydrogenated amorphous silicon (a-Si:H) on a fused silica substrate using PECVD, then spin-coating the a-Si:H with a positive photoresist (Microchem, PMMA A6) at 5000 r.p.m. for 1 min and baking at 180 °C for 5 min. Then a conductive polymer (Showa Denko, Espacer Z300) was spin-coated at 2000 r.p.m. for 1 min to prevent charge accumulation during the subsequent process. The sample was exposed using EBL at an acceleration voltage of 80 kV and developed using a commercial developer (Microchem, MIBK:IPA = 1:3) at 0 °C for 9 min. A Chromium (Cr) mask pattern was deposited using electron beam deposition and transferred to a-Si:H by using reactive-ion etching. Finally, the all-dielectric metasurface was fabricated after Cr mask etching by immersing it in Cr7 etchant for 1 min. Stokes polarimetry measurement A Stokes polarimetric technique was used to measure the shifts of the beam transmitted through the dielectric metasurface. A horizontally polarized source with a specific wavelength of 800 nm and a beam waist of 4 mm was filtered using an acousto-optic tunable filter (AOTF) from a supercontinuum laser (YSL SC-Pro). The polarization state of the incident beam was rotated using a half-wave plate. The incident beam was focused to have a beam size less than the sample size (~500 μm × 500 μm) by a lens and illuminated to the dielectric metasurface at various tilt angles (from 1° to 10°, step: 1°). The beam passed through a quarter-wave plate (Thorlabs, AQWP10M-980) and a Glan–Tompson polarizer (Thorlabs, GL15) in sequence, then was captured by a charge-coupled device camera (Thorlabs, DCC1545M). To minimize errors due to laser fluctuation, ten images were taken sequentially, then averaged to produce one image. Five consecutive measurements were performed in all experiments."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"Metasurfaces, nanostructure arrays that are designed to exhibit desired optical responses, have emerged as powerful tools to modulate various features of electromagnetic waves such as amplitudes, phases, and polarizations [1] . Whereas materials in nature have limited optical properties, the metasurfaces have shown unprecedented light manipulation capabilities despite their deep-subwavelength thickness. To maximize the light–matter interaction in a short spatial path length, the nanostructures that compose the metasurfaces have been constructed from plasmonic materials [2] , [3] . At optical wavelengths, however, plasmonic metasurfaces generally suffer from low efficiency due to Ohmic loss, thereby hindering their practical implementations. As alternatives, all-dielectric metasurfaces, constituents of which are all made of dielectric materials, have been suggested to manipulate the optical responses efficiently [4] , [5] , [6] . The dielectric metasurfaces have reproduced many exotic optical responses of the plasmonic metasurfaces such as polarization control [5] , [7] , ultrathin lenses [8] , [9] , [10] , [11] , and holographic display [12] , [13] , [14] , [15] with even higher efficiencies than the plasmonic metasurfaces. Furthermore, the metasurfaces’ compactness, compatibility, and ease of integration have made them a viable replacement for bulky optical components [16] , [17] , [18] , [19] . During the past decade, a variety of optical analogies of the spin Hall effect [20] , [21] such as spin Hall effect of light (SHEL) [22] , [23] , optical Magnus effect [24] , spin-dependent angular splitting [25] , plasmonic spin Hall effect [26] , [27] , and optical spin Hall effect [28] have regained extensive scientific interest when combined with structured media [29] , [30] , [31] , [32] , [33] such as metasurfaces and metamaterials. Among them, the SHEL is a transverse and spin-dependent spatial displacement of light at refraction or reflection [34] , [35] , [36] . At an interface between naturally occurring materials, the SHEL exhibits in general a shift that is much smaller than the wavelength [23] . However, specifically designed metasurfaces and metamaterials can enhance the shift by several orders of magnitude [37] , [38] , [39] , [40] , [41] , [42] , [43] , [44] from a few micrometers [45] to a few hundreds of micrometers [46] , [47] . Unfortunately, the drastic enhancement of the SHEL generally entails extremely low efficiency, as a result of a vanishing Fresnel coefficient [37] , [38] , [40] . A simultaneous enhancement of the shift and efficiency has been achieved very recently in the microwave regime using hyperbolic [48] and index-near-zero [49] metamaterials. However, these demonstrations cannot be readily adapted to the optical regime that encompasses ultraviolet, visible, and infrared wavelengths (10 nm < λ < 1 mm) because the operating principles rely on effective medium properties that require deep-subwavelength structures going down from hundreds of micrometers to a few tens of nanometers scale for optical operations. The lack of fabrication methods for such complicated structures and high optical losses of metals in the optical regime hinder the realization of the large and efficient SHEL. Thus, a large SHEL with high efficiency has not been achieved in the optical regime. Here we propose an all-dielectric metasurface to enhance both the shift and efficiency of the SHEL simultaneously in the near-infrared regime. In contrast to the previous approaches that have demanded a three-dimensional or stacked bulky structure to achieve the desired effective medium properties, here, a radically different approach is used by exploiting a single-layer metasurface as a compact and versatile route to modulate the transmission coefficients. To achieve high efficiency, the metasurface is designed to yield high transmission amplitudes for both s and p polarizations, with the phase difference between the two linear polarizations being far from zero. Modulation of the complex transmission results in a simultaneous enhancement of the shift and efficiency under both horizontal and vertical polarizations at 800 nm. More specifically, we experimentally demonstrate a shift and efficiency that reaches 10.0 λ and 71%, respectively, under horizontally polarized incidence and 9.9 λ and 79%, respectively, under vertically polarized incidence. Our work bridges the gap between metasurface research and their implementation towards efficient spin-sensitive devices. 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Direct visualizing the spin Hall effect of light via ultrahigh-order modes.Phys. Rev. Lett.124, 053902 (2020).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.124.053902\"\n },\n {\n \"idx\": \"48\",\n \"title\": \"Kim, M., Lee, D., Cho, H., Min, B. & Rho, J. Spin Hall effect of light with near-unity efficiency in the microwave.Laser Photonics Rev.15, 2000393 (2021).\",\n \"link\": \"https://doi.org/10.1002%2Flpor.202000393\"\n },\n {\n \"idx\": \"49\",\n \"title\": \"Kim, M. et al. Total reflection-induced efficiency enhancement of the spin Hall effect of light.ACS Photonics8, 2705–2712 (2021).\",\n \"link\": \"https://doi.org/10.1021%2Facsphotonics.1c00727\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Principle When transmitted at an optical interface, a linearly polarized Gaussian beam undergoes a transverse spatial shift by an amount [23] $$\\\\begin{array}{r}{\\\\delta }_{H}^{\\\\pm }/\\\\lambda =\\\\pm \\\\frac{\\\\cot {\\\\theta }_{i}}{2\\\\pi }\\\\left(1-\\\\frac{| {t}_{s}| }{| {t}_{p}| }\\\\cos \\\\phi \\\\right),\\\\\\\\ {\\\\delta }_{V}^{\\\\pm }/\\\\lambda =\\\\pm \\\\frac{\\\\cot {\\\\theta }_{i}}{2\\\\pi }\\\\left(1-\\\\frac{| {t}_{p}| }{| {t}_{s}| }\\\\cos \\\\phi \\\\right),\\\\end{array}$$ (1) where superscripts + and − represent left- and right-circularly polarized components of the transmitted beam, respectively, subscripts H and V correspond to the horizontal and vertical incident polarization respectively, λ is the wavelength, θ i is the incident angle, t s and t p denote Fresnel coefficients of the s - and p -polarized incidence respectively, and ϕ = arg( t s ) − arg( t p ) is the phase difference between the two Fresnel coefficients. Note that the Fresnel coefficients and θ i are those of the central wave vector. Equation ( 1 ) is valid under condition of \\\\({(2\\\\pi w/\\\\lambda )}^{2}\\\\gg {\\\\cot }^{2}{\\\\theta }_{i}\\\\) for a beam waist w , which is true except for a tightly confined beam under near-normal incidence ( θ i ≪ 1°). This narrow angular regime is where a transition from the SHEL to vortex generation occurs [50] and is out of our consideration. The shift is generally much smaller than the wavelength at an interface between naturally occurring media (Fig. 1a ). An intuitive approach to enhance the shift is to reduce the Fresnel coefficients in the denominator ( ∣ t p ∣ for horizontal and ∣ t s ∣ for vertical case); however, given that the efficiency is directly related to the Fresnel coefficients [48] ( ϵ H = ∣ t p ∣ 2 and ϵ V = ∣ t s ∣ 2 , Fig. 1b ), the method has the intrinsic limitation that it also degrades the efficiency significantly. Recently, two approaches to achieve large SHEL and high efficiency have been proposed [48] , [49] (Fig. 1c ). The key idea is to fix the Fresnel coefficient in the denominator to unity to ensure unity efficiency while enhancing the shift by reducing θ i . At an interface between two isotropic media, because the two Fresnel coefficients are degenerate under normal incidence, both \\\\(\\\\cos \\\\phi | {t}_{p}| /| {t}_{s}|\\\\) and \\\\(\\\\cos \\\\phi | {t}_{s}| /| {t}_{p}|\\\\) are close to unity at a small θ i , thereby giving rise to a subwavelength shift. To attain a large shift, an anisotropic medium [48] and near-index-zero metamaterial [49] have been proposed to differentiate the amplitude and phase of the remaining Fresnel coefficients, respectively. These previous methods have been, however, demonstrated in the microwave regime where metals behave as perfect electrical conductors. Realization of the large SHEL with high efficiency in the optical frequencies using this method is not trivial due to the high optical losses of metals and challenging fabrications and measurements. Fig. 1: Schematic of the spin Hall effect of light (SHEL) and a dielectric metasurface to enhance the shift δ and efficiency ϵ of the SHEL simultaneously. a , b Illustration of the SHEL in previous demonstrations. The width of the transmitted beams denotes their intensities. a The efficient SHEL is limited by the small shift, b whereas the enhancement of the shift entails a low efficiency. c Large SHEL with high efficiency. d A metasurface that transmits x - and y -polarized incidence with unity amplitudes ( ∣ t x ∣ = ∣ t y ∣ = 1) but with the opposite phase (arg( t x ) − arg( t y ) = ± π ). Full size image A dielectric metasurface can be used to overcome these difficulties, because of its low loss and the ability to control its complex transmission. Nanostructure arrays that have dimensions carefully chosen to modulate the amplitudes and phase of the transmitted beam can result in a large SHEL and high efficiency simultaneously. Ideally, a metasurface that transmits both x - and y -polarized incidence with unity amplitudes but with a phase difference of π or − π yields a SHEL that reaches \\\\(\\\\delta^\\\\pm / \\\\lambda = \\\\pm\\\\!\\\\!\\\\cot {\\\\theta }_{i}/\\\\pi\\\\) with near-unity efficiency at a small θ i ( ≪ π /2) (Fig. 1 d). In short, the design rule is to build a dielectric metasurface operating as a half-wave plate ( ∣ t s ∣ = ∣ t p ∣ = 1, ϕ = ± π ) under normal incidence and to examine the SHEL at small θ i . Whereas the capabilities of metasurfaces to manipulate the transmission or reflection have been well known [5] , [7] , [51] , no attempt has been made to use these qualities to produce a large and efficient SHEL. Furthermore, in comparison to the commercially available half-wave plates, this metasurface-based approach has advantages of compactness and compatibility. Design and fabrication To ensure the phase difference between t s and t p , a unit design of the metasurface has an anisotropic geometry of a rectangular-shaped nanorod on a fused silica substrate (Fig. 2a ). This rectangular nanorod is known to induce a Pancharatnam–Berry phase [25] , [52] under normal incidence. However, the phase contributes negligibly to the SHEL because the Pancharatnam-Berry phase is more dominant than the spin-redirection phase only under normal incidence and at a close vicinity of it [50] ( θ i ≪ 1°), which is not where we are interested in. Hydrogenated amorphous silicon (a-Si:H) [53] is used for its high refractive index and low optical losses (Supplementary Note 7) . This dielectric nanorod can modulate amplitudes and phases of the incoming light with substantially higher efficiency compared to plasmonic resonators. To find an optimal design for ∣ t s ∣ = ∣ t p ∣ = 1 and ϕ = ± π , five geometrical parameters, height ( H ), length ( a ), and width ( b ) of the nanorod and periodicities ( p x and p y ), are determined using a particle swarm optimization [54] to minimize 2 − ∣ t s − t p ∣ at 800 nm wavelength for a taper angle α = 5° (see “Methods” section for details). Then the metasurface is fabricated using electron beam lithography (see “Methods” section). Fig. 2: Design of a dielectric metasurface. a Schematic of a unit cell. Geometric parameters are given as: height H = 400 nm, length a = 265 nm, width b = 158 nm, periodicities p x = 378 nm and p y = 263 nm, and taper angle α . b , c Scanning electron microscopy images of the fabricated sample. b Top view, c perspective view. Inset shows a magnified image. Full size image The transmittance of the metasurface calculated by using full-wave simulation (see “Methods” section for detailed simulation) and measured by Fourier-transform infrared spectroscopy are shown in Fig. 3a . The simulated and measured transmittance agrees well with each other. A low transmittance at 726 nm under x -polarized incidence (Fig. 3a , blue) is attributed to the magnetic dipole resonance manifested by the displacement currents circulating the dielectric nanorods (see Supplementary Note 3) . In contrast, the y -polarized incidence induces the magnetic dipole resonance at a smaller wavelength near 675 nm because of the smaller rod length and periodicity and thus has a high transmittance at λ > 700 nm (Fig. 3a , red). The phase difference ϕ varies rapidly near the magnetic dipole resonances and has a value of −0.87 π at 800 nm (Fig. 3b ). To summarize, the metasurface has a high transmittance under both x - and y -polarized incidences and large ϕ at 800 nm, which is advantageous in realizing a large SHEL with high efficiency. Simulations indicate that a metasurface composed of nontapered cells with different geometrical parameters can have ϕ = ± π , which produces further enhanced SHEL (see Supplementary Note 1) . Fig. 3: Transmittance of the dielectric metasurface under normal incidence. a Simulated (markers) and measured (solid lines) transmittance under normal incidence. b Phase difference ϕ between the two polarizations obtained by numerical simulation. Full size image Experimental demonstration In general, the shift can be experimentally obtained via either weak measurement [23] or Stokes polarimetry measurement [55] , [56] . The former amplifies the shift by several orders of magnitudes where the amplification factor is determined by the setup parameters and the latter measures the Fresnel coefficients and their phase difference under oblique incidence so that the shift can be deduced from Eq. ( 1 ). Here the large SHEL with high efficiency at 800 nm is experimentally verified using a Stokes polarimetry setup (Fig. 4a , see “Methods” section for measurement details). The measurements are performed in two ways: first by setting the incident polarization as 45°-linearly polarized one to measure the phase difference ϕ and then by setting the incident polarization as horizontal and vertical to measure the amplitudes ∣ t p ∣ and ∣ t s ∣ respectively. Then the shifts can be obtained by substituting the measured ϕ , ∣ t s ∣ , and ∣ t p ∣ into Eq. ( 1 ). In all measurements, the metasurface is placed tilted by θ i ranging from 1° to 10° to examine the SHEL at various θ i . Fig. 4: Measurement of the SHEL at 800 nm. a Schematic of Stokes polarimetry set-up. HWP: half-wave plate, L: lens, QWP: quarter-wave plate, LP: linear polarizer, CCD: charge-coupled device camera. b Phase difference between t s and t p , c \\\\({\\\\delta }_{H}^{+}/\\\\lambda\\\\) , d \\\\({\\\\delta }_{V}^{+}/\\\\lambda\\\\) , e ϵ H , and f ϵ V . Curve: simulated, markers: measured. Insets in c , d show the magnified view for near-normal incidence where the analytic formula (solid) given by Eq. ( 1 ) deviates from the exact shifts (dashed) due to the breakdown of the assumption, i.e., \\\\({(2\\\\pi w/\\\\lambda )}^{2}\\\\gg {\\\\cot }^{2}{\\\\theta }_{i}\\\\) . Error bars of the measured data are not shown because they are all smaller than the markers. Full size image In the first measurement, we obtain the second and third Stokes parameters, $$\\\\begin{array}{rc}{s}_{\\\\!2}&=I({0}^{\\\\circ },4{5}^{\\\\circ })-I({0}^{\\\\circ },13{5}^{\\\\circ }),\\\\\\\\ {s}_{3}&=I(9{0}^{\\\\circ },4{5}^{\\\\circ })-I(9{0}^{\\\\circ },13{5}^{\\\\circ })\\\\end{array}$$ (2) where I ( β , γ ) is the spatial integration of the intensity when retardation angle of the quarter-wave plate is β and rotation angle of the linear polarizer is γ (Fig. 4a ). Then the measured Stokes parameters are used to calculate [56] ϕ by using a relation of \\\\(\\\\tan \\\\phi ={s}_{3}/{s}_{2}\\\\) . The measured ϕ deviates slightly from the simulated values because of the fabrication imperfections but exhibits a similar tendency (Fig. 4b ). In the second measurement, ∣ t p ∣ and ∣ t s ∣ under oblique incidence is obtained by taking the square roots of the intensity of transmitted beam under horizontally (Fig. 4e ) and vertically (Fig. 4f ) polarized incidence respectively. Note that the intensity is normalized by that measured without the sample. The shifts calculated by substituting the measured ϕ , ∣ t s ∣ , and ∣ t p ∣ in Eq. ( 1 ) and their efficiencies are shown in Fig. 4c–f . The shift is enhanced remarkably up to several wavelengths at a small θ i (≤ 5°) under both horizontal (Fig. 4 c) and vertical (Fig. 4d ) polarizations. We experimentally observe the shift of 10.0 λ along with 71% efficiency under horizontal polarization and the shift of 9.9 λ along with 79% efficiency under vertical polarization at θ i = 1°. The shifts can be further increased by reducing θ i except for an extremely narrow region near θ i = 0°. When θ i is sufficiently small (≤ 0.01°), Equation ( 1 ) deviates from the exact shifts and converges to zero rapidly (Fig. 4c, d , inset). The metasurface exhibits high transmittance for both polarizations even under oblique incidence. Despite the large SHEL, high efficiencies are observed at the wide range of θ i (Fig. 4e, f ). For completeness, the shift is also obtained using the weak measurement technique. The shift is 1.31 λ under vertically polarized light at θ i = 10°, which agrees well with the result from the Stokes polarimetry method (Supplementary Note 6) . The shift and efficiencies in our work in comparison with previous studies are summarized in Supplementary Note 4 .\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"We experimentally demonstrate simultaneous enhancement of the shift and efficiency of the spin Hall effect of light in the optical regime for the first time by using a dielectric metasurface. Whereas previous approaches to enhance the shift have suffered from low efficiencies and the achievements of a large shift with high efficiency have been limited in the microwave regime, we present the large shift and high efficiency at 800 nm by modulating the complex transmission of the metasurface. Shifts that reach 10 λ with efficiencies >70% are confirmed experimentally under two linearly polarized incidences. Furthermore, the shift can be further enhanced by reducing the taper angle through an optimization of the etching condition [57] . The simultaneous enhancement of the shift and efficiency can be also achieved in a reflective metasurface by changing geometrical parameters (Supplementary Note 5) . This approach towards the large spin Hall effect of light with high efficiency has two distinct merits. First, it facilitates ultrahigh precision measurements by increasing an amplification factor of a weak measurement. The amplification factor is directly linked to the precision, but its increase is practically bounded by the detectable signal strength. The n -fold enhancement of the SHEL efficiency provides n times stronger signal, which, in turn, enhances the precision by n times. Secondly, the large spin Hall effect of light with high efficiency in the optical regime enables realizing compact photonic devices such as spin-sensitive filters. With these clear advantages, the metasurface-driven simultaneous enhancement of the shift and efficiency will find wide applications in spin-dependent optics.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Numerical simulation and optimization Transmission coefficients of the dielectric metasurface were calculated using commercial software (COMSOL Multiphysics). For transmission under normal incidence, a unit cell with periodic boundary conditions along the x - and y -axes and perfectly matched layers and ports along the z -axis was simulated. For transmission under oblique incidence, simulation conditions were similar to these, except that the periodic boundary conditions were replaced by the Bloch boundary conditions. The dielectric metasurface was optimized to minimize 2 − ∣ t s − t p ∣ at 800 nm by using particle swarm optimization. Starting from a rectangular nanorod that has height H , length a , and width b and is periodically arranged with periodicities p x and p y , the five parameters were searched during 100 iterations. Sample fabrication The all-dielectric metasurface was fabricated using standard plasma-enhanced chemical vapor deposition (PECVD), electron-beam lithography (EBL), and reactive-ion etching. The fabrication process begins with the deposition of hydrogenated amorphous silicon (a-Si:H) on a fused silica substrate using PECVD, then spin-coating the a-Si:H with a positive photoresist (Microchem, PMMA A6) at 5000 r.p.m. for 1 min and baking at 180 °C for 5 min. Then a conductive polymer (Showa Denko, Espacer Z300) was spin-coated at 2000 r.p.m. for 1 min to prevent charge accumulation during the subsequent process. The sample was exposed using EBL at an acceleration voltage of 80 kV and developed using a commercial developer (Microchem, MIBK:IPA = 1:3) at 0 °C for 9 min. A Chromium (Cr) mask pattern was deposited using electron beam deposition and transferred to a-Si:H by using reactive-ion etching. Finally, the all-dielectric metasurface was fabricated after Cr mask etching by immersing it in Cr7 etchant for 1 min. Stokes polarimetry measurement A Stokes polarimetric technique was used to measure the shifts of the beam transmitted through the dielectric metasurface. A horizontally polarized source with a specific wavelength of 800 nm and a beam waist of 4 mm was filtered using an acousto-optic tunable filter (AOTF) from a supercontinuum laser (YSL SC-Pro). The polarization state of the incident beam was rotated using a half-wave plate. The incident beam was focused to have a beam size less than the sample size (~500 μm × 500 μm) by a lens and illuminated to the dielectric metasurface at various tilt angles (from 1° to 10°, step: 1°). The beam passed through a quarter-wave plate (Thorlabs, AQWP10M-980) and a Glan–Tompson polarizer (Thorlabs, GL15) in sequence, then was captured by a charge-coupled device camera (Thorlabs, DCC1545M). To minimize errors due to laser fluctuation, ten images were taken sequentially, then averaged to produce one image. Five consecutive measurements were performed in all experiments.\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"The core research idea is to propose an all-dielectric metasurface that simultaneously enhances both the shift and efficiency of the spin Hall effect of light (SHEL) in the near-infrared regime by utilizing a single-layer design with high transmission amplitudes and controlled phase differences between s and p polarizations, thereby achieving experimentally verified shifts of 10.0 λ and 9.9 λ with efficiencies of 71% and 79% under horizontal and vertical polarizations, respectively, at 800 nm."},"entities":{"kind":"list like","value":["all-dielectric metasurface","spin Hall effect of light","near-infrared regime","single-layer design","high transmission amplitudes","controlled phase differences","s polarization","p polarization","horizontal polarization","vertical polarization","800 nm","10.0 λ shift","9.9 λ shift","71% efficiency","79% efficiency"],"string":"[\n \"all-dielectric metasurface\",\n \"spin Hall effect of light\",\n \"near-infrared regime\",\n \"single-layer design\",\n \"high transmission amplitudes\",\n \"controlled phase differences\",\n \"s polarization\",\n \"p polarization\",\n \"horizontal polarization\",\n \"vertical polarization\",\n \"800 nm\",\n \"10.0 λ shift\",\n \"9.9 λ shift\",\n \"71% efficiency\",\n \"79% efficiency\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1.","title":"Yu, N. & Capasso, F. 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Mater.13, 139–150 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat3839\"\n },\n {\n \"idx\": \"2.\",\n \"title\": \"Schuller, J. A. et al. Plasmonics for extreme light concentration and manipulation.Nat. Mater.9, 193–204 (2010).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat2630\"\n },\n {\n \"idx\": \"4.\",\n \"title\": \"Staude, I. et al. Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks.ACS Nano7, 7824–7832 (2013).\",\n \"link\": \"https://doi.org/10.1021%2Fnn402736f\"\n },\n {\n \"idx\": \"5.\",\n \"title\": \"Arbabi, A., Horie, Y., Bagheri, M. & Faraon, A. Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission.Nat. Nanotechnol.10, 937–943 (2015).\",\n \"link\": \"https://doi.org/10.1038%2Fnnano.2015.186\"\n },\n {\n \"idx\": \"6.\",\n \"title\": \"Koshelev, K. & Kivshar, Y. Dielectric resonant metaphotonics.ACS Photonics8, 102–112 (2020).\",\n \"link\": \"https://doi.org/10.1021%2Facsphotonics.0c01315\"\n },\n {\n \"idx\": \"7.\",\n \"title\": \"Kruk, S. et al. Invited article: Broadband highly efficient dielectric metadevices for polarization control.APL Photonics1, 030801 (2016).\",\n \"link\": \"https://doi.org/10.1063%2F1.4949007\"\n },\n {\n \"idx\": \"8.\",\n \"title\": \"West, P. R. et al. All-dielectric subwavelength metasurface focusing lens.Opt. Express22, 26212–26221 (2014).\",\n \"link\": \"https://doi.org/10.1364%2FOE.22.026212\"\n },\n {\n \"idx\": \"9.\",\n \"title\": \"Arbabi, A., Horie, Y., Ball, A. J., Bagheri, M. & Faraon, A. Subwavelength-thick lenses with high numerical apertures and large efficiency based on high-contrast transmitarrays.Nat. Commun.6, 7069 (2015).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms8069\"\n },\n {\n \"idx\": \"10.\",\n \"title\": \"Kamali, S. M., Arbabi, E., Arbabi, A., Horie, Y. & Faraon, A. Highly tunable elastic dielectric metasurface lenses.Laser Photonics Rev.10, 1002–1008 (2016).\",\n \"link\": \"https://doi.org/10.1002%2Flpor.201600144\"\n },\n {\n \"idx\": \"11.\",\n \"title\": \"Yoon, G., Kim, K., Huh, D., Lee, H. & Rho, J. Single-step manufacturing of hierarchical dielectric metalens in the visible.Nat. Commun.11, 2268 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-020-16136-5\"\n },\n {\n \"idx\": \"12.\",\n \"title\": \"Hasman, E., Kleiner, V., Biener, G. & Niv, A. Polarization dependent focusing lens by use of quantized pancharatnam–berry phase diffractive optics.Appl. Phys. Lett.82, 328–330 (2003).\",\n \"link\": \"https://doi.org/10.1063%2F1.1539300\"\n },\n {\n \"idx\": \"13.\",\n \"title\": \"Zhao, W. et al. Dielectric huygens’ metasurface for high-efficiency hologram operating in transmission mode.Sci. Rep.6, 30613 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fsrep30613\"\n },\n {\n \"idx\": \"14.\",\n \"title\": \"Khorasaninejad, M., Ambrosio, A., Kanhaiya, P. & Capasso, F. Broadband and chiral binary dielectric meta-holograms.Sci. Adv.2, e1501258 (2016).\",\n \"link\": \"https://doi.org/10.1126%2Fsciadv.1501258\"\n },\n {\n \"idx\": \"15.\",\n \"title\": \"Yoon, G., Lee, D., Nam, K. T. & Rho, J. Pragmatic metasurface hologram at visible wavelength: the balance between diffraction efficiency and fabrication compatibility.ACS Photonics5, 1643–1647 (2017).\",\n \"link\": \"https://doi.org/10.1021%2Facsphotonics.7b01044\"\n },\n {\n \"idx\": \"16.\",\n \"title\": \"Lee, G.-Y. et al. Metasurface eyepiece for augmented reality.Nat. Commun.9, 4562 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-018-07011-5\"\n },\n {\n \"idx\": \"17.\",\n \"title\": \"Tittl, A. et al. Imaging-based molecular barcoding with pixelated dielectric metasurfaces.Science360, 1105–1109 (2018).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.aas9768\"\n },\n {\n \"idx\": \"18.\",\n \"title\": \"Kim, I. et al. Holographic metasurface gas sensors for instantaneous visual alarms.Sci. Adv.7, eabe9943 (2021).\",\n \"link\": \"https://doi.org/10.1126%2Fsciadv.abe9943\"\n },\n {\n \"idx\": \"19.\",\n \"title\": \"Pahlevaninezhad, H. et al. Nano-optic endoscope for high-resolution optical coherence tomography in vivo.Nat. Photon.12, 540–547 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41566-018-0224-2\"\n },\n {\n \"idx\": \"20.\",\n \"title\": \"Dyakonov, M. I. & Perel, V. Current-induced spin orientation of electrons in semiconductors.Phys. Lett. A35, 459–460 (1971).\",\n \"link\": \"https://doi.org/10.1016%2F0375-9601%2871%2990196-4\"\n },\n {\n \"idx\": \"22.\",\n \"title\": \"Onoda, M., Murakami, S. & Nagaosa, N. Hall effect of light.Phys. Rev. Lett.93, 083901 (2004).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.93.083901\"\n },\n {\n \"idx\": \"23.\",\n \"title\": \"Hosten, O. & Kwiat, P. Observation of the spin Hall effect of light via weak measurements.Science319, 787–790 (2008).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1152697\"\n },\n {\n \"idx\": \"25.\",\n \"title\": \"Chen, X. et al. Dual-polarity plasmonic metalens for visible light.Nat. Commun.3, 1198 (2012).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms2207\"\n },\n {\n \"idx\": \"26.\",\n \"title\": \"Huang, L. et al. Helicity dependent directional surface plasmon polariton excitation using a metasurface with interfacial phase discontinuity.Light Sci. Appl.2, e70 (2013).\",\n \"link\": \"https://doi.org/10.1038%2Flsa.2013.26\"\n },\n {\n \"idx\": \"27.\",\n \"title\": \"Xiao, S., Zhong, F., Liu, H., Zhu, S. & Li, J. Flexible coherent control of plasmonic spin-Hall effect.Nat. Commun.6, 8360 (2015).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms9360\"\n },\n {\n \"idx\": \"28.\",\n \"title\": \"Kavokin, A., Malpuech, G. & Glazov, M. Optical spin Hall effect.Phys. Rev. Lett.95, 136601 (2005).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.95.136601\"\n },\n {\n \"idx\": \"29.\",\n \"title\": \"Yin, X., Ye, Z., Rho, J., Wang, Y. & Zhang, X. Photonic spin Hall effect at metasurfaces.Science339, 1405–1407 (2013).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1231758\"\n },\n {\n \"idx\": \"30.\",\n \"title\": \"Kim, M., Lee, D. & Rho, J. Spin Hall effect under arbitrarily polarized or unpolarized light.Laser Photonics Rev.15, 2100138 (2021).\",\n \"link\": \"https://doi.org/10.1002%2Flpor.202100138\"\n },\n {\n \"idx\": \"32.\",\n \"title\": \"Yu, X. et al. Spin Hall effect of light based on a surface plasmonic platform.Nanophotonics10, 3031–3048 (2021).\",\n \"link\": \"https://doi.org/10.1515%2Fnanoph-2021-0217\"\n },\n {\n \"idx\": \"34.\",\n \"title\": \"Aiello, A. Goos–Hänchen and Imbert–Fedorov shifts: a novel perspective.New J. Phys.14, 013058 (2012).\",\n \"link\": \"https://doi.org/10.1088%2F1367-2630%2F14%2F1%2F013058\"\n },\n {\n \"idx\": \"35.\",\n \"title\": \"Töppel, F., Ornigotti, M. & Aiello, A. Goos–Hänchen and Imbert–Fedorov shifts from a quantum-mechanical perspective.New J. Phys.15, 113059 (2013).\",\n \"link\": \"https://doi.org/10.1088%2F1367-2630%2F15%2F11%2F113059\"\n },\n {\n \"idx\": \"36.\",\n \"title\": \"Bliokh, K. Y. & Aiello, A. Goos–Hänchen and Imbert–Fedorov beam shifts: an overview.J. Opt.15, 014001 (2013).\",\n \"link\": \"https://doi.org/10.1088%2F2040-8978%2F15%2F1%2F014001\"\n },\n {\n \"idx\": \"37.\",\n \"title\": \"Zhu, W. & She, W. 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Reactive ion etching of silica structures for integrated optics applications.J. Vac. Sci. Technol.14, 2994–3003 (1996).\",\n \"link\": \"https://doi.org/10.1116%2F1.580258\"\n }\n]","isConversational":false}}},{"rowIdx":11198,"cells":{"unique_id":{"kind":"string","value":"s41467-021-22043-0"},"subfield":{"kind":"string","value":"Optics and photonics"},"title":{"kind":"string","value":"Simultaneous readout of multiple FRET pairs using photochromism"},"abstract":{"kind":"string","value":"Förster resonant energy transfer (FRET) is a powerful mechanism to probe associations in situ. Simultaneously performing more than one FRET measurement can be challenging due to the spectral bandwidth required for the donor and acceptor fluorophores. We present an approach to distinguish overlapping FRET pairs based on the photochromism of the donor fluorophores, even if the involved fluorophores display essentially identical absorption and emission spectra. We develop the theory underlying this method and validate our approach using numerical simulations. To apply our system, we develop rsAKARev, a photochromic biosensor for cAMP-dependent protein kinase (PKA), and combine it with the spectrally-identical biosensor EKARev, a reporter for extracellular signal-regulated kinase (ERK) activity, to deliver simultaneous readout of both activities in the same cell. We further perform multiplexed PKA, ERK, and calcium measurements by including a third, spectrally-shifted biosensor. Our work demonstrates that exploiting donor photochromism in FRET can be a powerful approach to simultaneously read out multiple associations within living cells."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"Förster resonance energy transfer (FRET) is a powerful mechanism to probe associations in situ. FRET experiments are performed by labeling the system of interest with donor and acceptor fluorophores, and then using the extent of the energy transfer to probe the distances between and relative orientations of both fluorophores [1] . FRET-based approaches have gained tremendous popularity for association studies, and can be performed using both genetically encoded fluorophores and organic dyes [2] . It is also the key mechanism underlying a class of genetically encoded biosensors that convert the presence of a particular chemical stimulus into a change in FRET [3] , [4] , [5] , [6] . Most FRET experiments use a fluorescent donor and acceptor, which has the advantage that the FRET efficiency can be estimated based on the ratio between the donor-excited emission from both fluorophores [7] , [8] , [9] , [10] . The accuracy of this estimation depends on the ease with which the donor and acceptor can be spectrally separated from one another. The choice of specific FRET pairs is usually driven by a compromise between achieving a clearly observable FRET efficiency, which requires sufficient spectral overlap, and maintaining a low level of cross-talk between channels, which is made easier when the probes are more spectrally separated. As a result, the donor and acceptor tend to occupy broad regions of the visual spectrum, complicating the combination of FRET measurements with the use of other probes. Measuring multiple FRET-based constructs is further complicated if these constructs share fluorophores emitting in the same spectral band. For example, many FRET biosensors consist of a cyan and a yellow fluorescent protein, making it more difficult to combine multiple biosensors to follow, e.g., cross-talk between signaling pathways. Using more than one of these probes typically requires that separate measurements are performed for each probe, possibly in a high-throughput manner [11] . A number of strategies can facilitate FRET multiplexing [12] . The conceptually easiest approach is to select two or more FRET pairs with minimal spectral overlap, though this is limited by the available probes and the usable spectral range [13] . It may be possible to share a single fluorophore among two FRET pairs, reducing the total number of fluorophores from four to three [14] . Other approaches include the use of non-fluorescent acceptors combined with fluorescence lifetime imaging [15] , [16] , [17] , the use of homo-FRET combined with alternative readouts such as fluorescence anisotropy [18] , and the separation of spectrally overlapping pairs based on orthogonal information, such as the spatial localization of the probes within the cell [19] . Several strategies for probe multiplexing have relied on exploiting the photochemistry of the fluorophores. Photochromic labels, for example, display light-induced and reversible “on-off” switching of their fluorescence emission [20] , [21] . Upon excitation, these probes display a bright fluorescence emission that disappears in time, though the original fluorescence can be rapidly recovered by irradiating with the light of a shorter wavelength. These dynamics can be used to separate the signal of a photochromic label from that of a non-photochromic label or from the autofluorescence, or can be used to separate two or more photochromic labels that switch with different kinetics [22] , [23] , [24] , [25] , [26] . However, there has been comparatively little work on the evaluation of such strategies for the simultaneous measurement of multiple FRET pairs. In contrast, photochromism of the acceptor has been used as a way to provide a more accurate FRET quantification [27] , [28] or to measure multiple single-molecule donor-acceptor distances within the same complex [29] . Previous work has also investigated the use of a photochromic donor, where the FRET efficiency can be estimated by determining the rate of the photochromism [30] , or to facilitate the anisotropy-based readout of homo-FRET [31] . Association-induced fluorescence dynamics have also been used to provide diffraction-unlimited imaging of biosensors [32] . In this contribution, we describe the use of donor photochromism to simultaneously measure more than one FRET pair, even when the donors and acceptors display complete spectral overlap. We commence by developing the theory underlying our method and validate its performance using numerical simulations. We then develop a photochromic biosensor for cAMP-dependent protein kinase (PKA), which we call rsAKARev, and use it to demonstrate our approach through the simultaneous measurement of PKA and extracellular signal-regulated kinase (ERK) activity in live cells. Finally, we demonstrate the versatility of our approach by combining these sensors with a spectrally separated calcium biosensor to perform simultaneous biosensing of three different signaling activities in live cells.","references":{"indices":[1,2,3,4,5,7,8,9,10,11,12,13,14,15,16,17,18,19,20,22,23,24,25,26,27,28,29,30,32],"infos":[{"idx":"1","title":"Piston, D. W. & Kremers, G. J. Fluorescent protein FRET: the good, the bad and the ugly.Trends Biochem. Sci.32, 407–414 (2007).","link":"https://doi.org/10.1016%2Fj.tibs.2007.08.003"},{"idx":"2","title":"Sahoo, H. Förster resonance energy transfer—a spectroscopic nanoruler: principle and applications.J. Photochem. Photobiol. C12, 20–30 (2011).","link":"https://doi.org/10.1016%2Fj.jphotochemrev.2011.05.001"},{"idx":"3","title":"Newman, R. H., Fosbrink, M. D. & Zhang, J. Genetically encodable fluorescent biosensors for tracking signaling dynamics in living cells.Chem. 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As a control, and to estimate the effects of photodestruction, one usually also measures the acceptor emission upon direct excitation ( S AA ). A full FRET acquisition, therefore, consists of three measurements { S DD , S DA , S AA } at every timepoint and at every pixel of the detector. The S DA signal is usually contaminated by donor bleed-through and direct acceptor cross-excitation. These contributions can be quantified by using explicit correction factors α and δ [9] : $$\\alpha ={S}_{\\mathrm{DA}}/{S}_{\\mathrm{DD}}\\ \\ \\ \\,{\\mathrm{in}}\\; {\\mathrm{presence}}\\; {\\mathrm{of}}\\; {\\mathrm{only}}\\; {\\mathrm{donor}}\\,$$ (1) $$\\delta ={S}_{\\mathrm{DA}}/{S}_{\\mathrm{AA}}\\ \\ \\ \\,{\\mathrm{in}}\\; {\\mathrm{presence}}\\; {\\mathrm{of}}\\; {\\mathrm{only}}\\; {\\mathrm{acceptor}}$$ (2) These correction factors can be determined using control measurements or can be estimated given knowledge of the fluorophore spectra and the spectral sensitivity of the instrument. In general, direct control measurements are preferable since these will then be tailored to the specifics of the instrument. We assume that direct excitation of the acceptor does not lead to appreciable emission of the donor fluorophore in the acceptor channel. Not all experimenters choose to determine these correction factors, instead quantifying the energy transfer using the “raw” emission ratio R $$R=\\frac{{S}_{\\mathrm{DA}}}{{S}_{\\mathrm{DD}}}$$ (3) which will already allow the detection of relative changes in the FRET efficiency. If available, however, we can use the correction factors to calculate the sensitized emission F c , which is the donor-excited emission from the acceptor corrected for cross-talk: $${F}_{\\mathrm{c}}={S}_{\\mathrm{DA}}-\\alpha {S}_{\\mathrm{DD}}-\\delta {S}_{\\mathrm{AA}}$$ (4) Given F c , the FRET response can be estimated using the sensitized emission ratio R c : $${R}_{\\mathrm{c}}=\\frac{{F}_{\\mathrm{c}}}{{S}_{\\mathrm{DD}}}$$ (5) R c can be compared across different instruments and is more sensitive to changes in FRET than the emission ratio R . However, it is not unambiguously related to the absolute FRET efficiency (Θ) since it fails to take into account the different brightnesses of the donor and acceptor fluorophores and/or the different collection efficiencies of the instrument. The introduction of an additional correction factor γ [9] makes it possible to calculate the FRET efficiency using $${{\\Theta }}=\\frac{1}{\\gamma {S}_{\\mathrm{DD}}/{F}_{\\mathrm{c}}+1}$$ (6) as is derived in Supplementary Note 1 . γ can be estimated from model spectra, or more reliably from additional control experiments [10] . Introducing a photochromic donor The core idea of our method is that two different FRET pairs can be distinguished based on the photochromism of the donors, conceptually shown in Fig. 1 a. We therefore extend our model to the case where the donor is a photochromic fluorophore. To probe this photochromism, we expand our measurement scheme with two irradiation periods that serve to switch the donor fluorophore between fluorescent and non-fluorescent states (Fig. 1 b). This consists of a short-wavelength light pulse and fluorescence acquisition to capture the fluorescent state, followed by a longer-wavelength light pulse and another fluorescence acquisition to capture the extent of the off-switching. The net result is that every FRET acquisition is now replaced with two acquisitions, performed at the end of each irradiation period, for a total of six measurements for each FRET timepoint (two measurements each for S DD , S DA , and S AA ). S AA usually varies slowly because it is affected only by photodestruction or large structural changes in the sample, and can therefore be measured at a reduced frequency. Fig. 1: Conceptual overview. a Two FRET pairs can be distinguished by probing the photochromism of the donors since only photochromic donors react to the off- and on-switching light. b Example irradiation scheme for a cyan photochromic protein, showing the donor on-state population (dotted line) during violet and blue light irradiation (colored bars). The red dots show the fluorescence acquisitions that are performed. The inset shows representative fluorescence images of a HeLa cell, acquired in all three measurement channels ( S D D , S D A , S A A ). Scale bar, 20 μm. In practice, the light-induced off-switching does not need to be as complete as shown here. Full size image Off-switching of the donor fluorophore results in a decrease in S DD , S DA , and F c . We quantify this using the photoswitching ratios ρ D and ρ A , given by $${\\rho }_{\\mathrm{D}}=\\frac{{S}_{{\\mathrm{DD}},\\text{off}}}{{S}_{{\\mathrm{DD}},\\text{on}}}$$ (7) $${\\rho }_{\\mathrm{A}}=\\left\\{\\begin{array}{ll}\\frac{{F}_{{\\mathrm{C}},\\text{off}}}{{F}_{{\\mathrm{C}},\\text{on}}}&\\;{\\text{if}}\\; {\\text{correction}} \\;{\\text{factors}}\\; {\\text{are}} \\;{\\text{available}}\\hfill\\\\ \\frac{{S}_{{\\mathrm{DA}},\\text{off}}}{{S}_{{\\mathrm{DA}},{\\text{on}}}}&\\,\\text{otherwise}\\,\\hfill\\end{array}\\right.$$ (8) where the “on” and “off” acquisitions are those shown in Fig. 1 b. We provide two alternative definitions of ρ A in Eq. ( 8 ) depending on whether correction factors for cross-talk are available. Our methodology remains valid for both approaches as long as these definitions are used consistently. The photoswitching ratios ρ D and ρ A are thus the fractions of the original fluorescence that are left after the off-switching irradiation has been applied, with smaller values implying more pronounced off-switching. They depend on the specifics of the donor fluorophore and on the total light dose that is delivered, but also on the FRET efficiency since this process competes with photochromism. This aspect was leveraged in the development of psFRET [30] , and likewise must be taken into account when separating two FRET pairs based on photochromism. In the general case, the FRET-dependence of the photoswitching ratios ρ D and ρ A must be determined experimentally by measuring these for varying (sensitized) emission ratios or FRET efficiencies. This can be done simply by performing an experiment where the FRET pair is expressed and ρ D and ρ A are followed as a function of emission ratio or FRET efficiency. This approach will also work in the more general case where the emission contrast arises not just due to energy transfer, but also through other contributions such as a change in brightness of the donor. This process can be simplified when suitable assumptions can be made about the system. For a simple FRET process with negligible or perfectly corrected cross-talk in which only a single species is present, $${\\rho }_{\\mathrm{{D}}}={\\rho }_{\\mathrm{{A}}}$$ (9) since the acceptor emission is proportional to the amount of the donor in the fluorescent state. If we further assume that the off-switching can be described by an intrinsic quantum yield then the photoswitching ratio is given by $$\\rho ={({\\rho }_{0})}^{(1-{{\\Theta }})}$$ (10) as is derived in Supplementary Note 2 . Equation ( 10 ) shows that the full dependence is known if one simply measures ρ 0 , the photoswitching ratio in the absence of FRET, using the same settings for the instrument. As we show in the Supplementary Information, this consideration can be expanded to photochromic molecules that do not display complete off-switching but instead reach an equilibrium or plateau level. While these results can simplify the characterization of the FRET pairs, our method will still work if these assumptions cannot be made. Residual cross-talk of direct acceptor excitation into the S D A channel, for example, will increase ρ A since this contribution shows no fluorescence modulation. The derived models also do not apply to FRET-based biosensors, which exist as two interconverting species (active and inactive forms of the sensor). If no simplifying assumptions can be made, the dependence of ρ D and ρ A on the (sensitized) emission ratio or FRET efficiency will need to be determined over the full range of values, for each FRET pair and off-switching light dose used in the experiment. Separating photochromic FRET pairs We now assume that the sample is labeled with two FRET pairs, where the absorption and emission spectra are highly similar yet the donors differ in their photochromism. This situation can also arise when two different interactions are probed using two donors and a single shared acceptor. The recorded fluorescence signals are simply the sum of the signals from the individual components $${S}_{{\\mathrm{DD}},\\text{on}}={S}_{{\\mathrm{DD}},\\text{on},1}+{S}_{{\\mathrm{DD}},\\text{on},2}$$ (11) $${S}_{{\\mathrm{DA}},\\text{on}}={S}_{{\\mathrm{DA}},\\text{on},1}+{S}_{{\\mathrm{DA}},\\text{on},2}$$ (12) where “1” and “2” indicate the first and second FRET pair. After irradiation with off-switching light, we also obtain $${S}_{{\\mathrm{DD}},\\text{off}}={\\rho }_{{\\mathrm{D}},1}\\,{S}_{{\\mathrm{DD}},\\text{on},1}+{\\rho }_{D,2}\\ {S}_{{\\mathrm{DD}},\\text{on},2}$$ (13) $${S}_{{\\mathrm{DA}},\\text{off}}={\\rho }_{{\\mathrm{A}},1}\\,{S}_{{\\mathrm{DA}},\\text{on},1}+{\\rho }_{{\\mathrm{A}},2}\\ {S}_{{\\mathrm{DA}},\\text{on},2}$$ (14) Our measurement provides the S D D and S D A signals before and after off-switching. If we assume that all four photoswitching ratios ρ D and ρ A are known, then these equations constitute a system of four equations and four unknowns, which can be solved to obtain S D D ,on and S D A ,on for both species. These values can then be readily used to obtain the (sensitized) emission ratios for each pair using Eqs. ( 3 ) or ( 5 ), or the FRET efficiency using Eq. ( 6 ). In principle, the same approach could also be used to separate two FRET pairs where at least one donor is photochromic and one pair contains a dark (non-fluorescent) acceptor. In practice, we do not know the four ρ D and ρ A values as these depend on the sought-after (sensitized) emission ratio or FRET efficiency. However, the previous section considered how to determine the relationship between these parameters. Given that information, we recommend the use of an iterative procedure, in which the (sensitized) emission ratio or FRET efficiency is calculated starting from initial guesses for the photoswitching ratios. This value is then used to obtain better guesses for the photoswitching ratios, and this procedure is repeated until these values converge, which occurs rapidly in practice. A more detailed discussion of the algorithm used in this work can be found in Supplementary Note 3 . Numerical simulations We devised a series of numerical simulations to test the validity of our approach. We assumed a sample labeled with two spectrally indistinguishable FRET pairs, where one of the donors is photochromic and can be described using Eq. ( 10 ), and the other does not show appreciable photochromism. We further assumed different brightness levels, reflected in the number of photons that would be acquired in the “on” acquisition if no FRET were to occur. Noise was included in our measurement by simulating the Poisson noise intrinsic to the photon detection process. For each condition, we ran 5000 simulations where we selected random ground-truth FRET efficiencies between 0 and 0.6 for each FRET pair, and generated two simulated fluorescence acquisitions, before and after off-switching (cfr. the measurement scheme shown in Fig. 1 b). These data were then analyzed using our methodology, resulting in estimates for the FRET efficiency of each of the pairs. The accuracy of these results was then quantified using the mean absolute deviation, given by $$\\,\\text{MAD}\\,=\\frac{\\left|{{{\\Theta }}}_{1}-{\\widehat{{{\\Theta }}}}_{1}\\right|+\\left|{{{\\Theta }}}_{2}-{\\widehat{{{\\Theta }}}}_{2}\\right|}{2}$$ (15) where Θ i are the ground-truth FRET efficiencies and \\({\\hat{{{\\Theta }}}}_{i}\\) are the FRET efficiencies estimated by our method. As a negative control, we also included the result of an analysis that simply generated random FRET efficiencies. Our simulations show that our analysis works well already at low light levels, providing accurate results when 1000 or more photons are detected, and less accurate but useful results already at signal levels as low as 250 photons (Fig. 2 a). We also investigated to what extent the difference in photoswitching ratio of the FRET donors (Δ ρ ) determines the accuracy of the analysis (Fig. 2 b and Supplementary Fig. 1 ). We find that this depends on the expected FRET efficiency and the donor photoswitching ratio in the absence of FRET ( ρ 0 ). However, a ρ 0 of 0.3 or lower, readily achievable with, e.g., photochromic fluorescent proteins, works well over a broad range of FRET efficiencies. Furthermore, ρ 0 can be increased or decreased by adjusting the duration or intensity of the off-switching illumination, allowing the conditions to be tailored to the measurement. Finally, we note that further increases in accuracy are possible by including more fluorescence acquisitions as part of the off-switching (adding additional fluorescence acquisitions to each of the measurement cycles shown in Fig. 1 b). Fig. 2: Evaluation of the method on simulated data. a Combined violin and box-plot of the mean absolute deviation (MAD) of our analysis approach for different numbers of detected photons. Data were generated by performing 5000 independent simulations for each category. Violin plot shows the underlying data distribution. Box-plot middle line indicates the median, lower and upper hinges represent first and third quartile, whiskers extend to the full width of the data. b Scatter plot showing the analysis performance (MAD) for different FRET efficiencies and photoswitching ratios (in the absence of FRET) of the photochromic ( ρ 0, p ) and non-photochromic ( ρ 0, n p ) donor. Δ ρ is the difference in photoswitching ratio. Two-dimensional projections of this data are shown in Supplementary Fig. 1 . Full size image In principle, our methodology can be used to separate more than two overlapping FRET pairs, by adding additional terms to Eqs. ( 11 )–( 14 ). Each additional pair will then require one additional fluorescence acquisition per switching cycle. rsAKARev: a photochromic biosensor for PKA We decided to experimentally validate our method by developing a photochromic FRET-based biosensor for cAMP-dependent protein kinase (PKA) activity, called rsAKARev (reversibly switchable A-kinase activity reporter with EV-linker). We obtained this biosensor by replacing the donor and acceptor in AKAR3ev [33] with the cyan fluorescent protein mTFP0.7 and the yellow fluorescent protein cpVenus172. mTFP0.7 is a variant of the bright and photostable mTFP1 [34] that shows efficient photochromism with little fatigue [35] . cpVenus172 is a circular permutant of Venus that has been reported to result in an improved FRET contrast [36] , [37] , [38] . To evaluate our unmixing strategy, we chose to combine rsAKARev with the non-photochromic biosensor EKARev [33] , which reports on extracellular signal-regulated kinase (ERK) activity and consists of the cyan donor ECFP and yellow acceptor YPet. Our choice of sensors reflects the fundamental interest in the interactions between the conserved cAMP/PKA and MAPK/ERK signaling pathways, which regulate various physiological processes including cell proliferation and growth [39] , [40] . To characterize the performance of our biosensors, we performed measurements on cells expressing either rsAKARev or EKARev, and submitted them to the measurement scheme shown in Fig. 1 b. As expected, we found that the fluorescent state of the rsAKARev donor fluorophore, mTFP0.7, readily depopulates upon irradiation with blue light and can be efficiently recovered through weak irradiation with violet light (Fig. 3 a). This donor photochromism leads to a large and reversible fluorescence contrast in the donor and acceptor emission with minimal loss of fluorescence intensity (Fig. 3 b). In contrast, the EKARev donor, ECFP, displays a much more limited modulation of the fluorescence signal (Fig. 3 c). Fig. 3: Photochromism of rsAKARev and EKARev expressed in HeLa cells. a Representative fluorescence images of the donor ( S D D ) and acceptor ( S D A ) fluorescence of rsAKARev before and after off-switching. Scale bar, 40 μm. b , c Time-course of the donor and sensitized acceptor emission of rsAKARev ( b ) and EKARev ( c ) during multiple irradiation cycles. d , e Representative responses of a HeLa cell expressing rsAKARev ( d ) or EKARev ( e ) during stimulation of PKA or ERK activity. Time-traces are colored as indicated. PKA activity was stimulated with forskolin (Fsk, 50 μM) and IBMX (100 μM), ERK activity with PMA (1 μM). Full size image We next evaluated how the activation state of each biosensor affected its fluorescence emission. Cells were treated with pharmacological compounds to evoke either PKA or ERK kinase activity, and the FRET response of the respective biosensor was followed over time (Fig. 3 d, e). This response was determined via conventional FRET analysis using only the ‘on’ fluorescence acquired after every violet irradiation step in Fig. 1 b. Forskolin-(Fsk) and 3-isobutyl-1-methylxanthine (IBMX)-mediated elevation of intracellular cAMP strongly increased the FRET response of rsAKARev. Likewise, stimulation of cells with phorbol-12-myristate-13-acetate (PMA), which acts on ERK via upstream kinases [41] , [42] , elicited a strong EKARev response. As expected, our data additionally revealed that changes in PKA activity also lead to changes in the photoswitching ratios ρ D and ρ A (Fig. 3 d). Conversely, Fig. 3 e shows that the irradiation-induced fluorescence dynamics of EKARev are small and largely independent of the activation state of the sensor, consistent with the photostatic nature of ECFP. As we discussed in the theory, separating overlapping FRET pairs based on their photochromism requires knowledge of the relationship between the photoswitching ratio and the FRET efficiency. For well-behaved FRET processes, this can be done using only a single reference measurement and the analytical expression of the dependence (e.g., Eq. ( 10 )). However, systems that consist of an equilibrium between multiple discrete states, such as the active and inactive state of a biosensor, cannot be described using this relationship. Furthermore, cyan fluorescent proteins commonly show complicated spectroscopic behavior such as multi-exponential excited state lifetime decays [43] . We therefore resorted to direct measurement of the full dependence in cells expressing either rsAKARev or EKARev (Fig. 4 ). We found that the sought-after relationship could be empirically described using double-exponential (rsAKARev) or straight line (EKARev) fits, providing a model that was used in the downstream analysis. Figure 4 b does show that the photoswitching ratio increases with decreasing FRET efficiency, which is contrary to what one would expect for a two-state FRET system. This is likely due to a suboptimal value for the acceptor bleed-through correction, though this does not interfere in the separation analysis which follows since its effects are automatically taken into account via this in situ calibration. Fig. 4: Relation between FRET efficiency and photoswitching ratio. Plots for donor emission ( a , c ) and donor-excited acceptor emission ( b , d ) as determined by single-sensor experiments on COS-7 cells expressing rsAKARev (orange) or HeLa cells expressing EKARev (blue). n = 99 (rsAKARev) or 41 (EKARev) cells from 1 experiment each. Solid lines represent the empirical fitting functions for rsAKARev (double-exponential) or EKARev (straight line) that were used for downstream analysis. Full size image Simultaneous measurement of two overlapping FRET biosensors We then focused on the simultaneous measurement of rsAKARev and EKARev expressed within the same cells. Since the emission spectra of both biosensors are essentially indistinguishable, their responses would be difficult to separate based on classical FRET imaging. In order to generate predictable response profiles for rsAKARev and EKARev, we treated the cells sequentially with PMA, Fsk/IBMX, U0126, and H-89. U0126 and H-89 are inhibitors that reverse the responses of the MAPK/ERK and PKA signaling pathways, respectively [17] . Ideally, this sequence of manipulations should result in distinguishable response profiles for rsAKARev and EKARev. We initially performed these experiments on cells expressing a single biosensor, either rsAKARev, EKARev, or inactive mutants in which the kinase target threonine was mutated to alanine, denoted using a “(T/A)” suffix. We subjected these cells to the measurement scheme shown in Fig. 1 b, but analyzed the resulting responses in two different ways: either via a conventional intensity-based analysis, using only the “on” fluorescence images acquired after every violet irradiation step (Fig. 5 a, d, g), or via our full photochromism-based approach (Fig. 5 b, e, h). Applying both analysis methodologies to the same data makes it possible to detect any discrepancies between the obtained results. Details of the analysis procedure are provided in Supplementary Note 3 . The conventional analysis shows that the biosensors respond as expected to pharmacological treatment (Fig. 5 a). We did observe inhibition of the EKARev signal upon addition of Fsk/IBMX, reminiscent of previous reports where it was found that forskolin interferes with EGF-mediated ERK activation [17] . This suggests that a similar mechanism is also at play when stimulating ERK activity with PMA. As expected, the biosensor response of the T/A mutants remained insensitive to the applied sequence of manipulations. The addition of Fsk/IBMX did induce a decrease in the EKARev(T/A) response similar to that observed for EKARev, demonstrating that this response arises from a process other than phosphorylation of the target residue. We also observed a weaker than expected inhibition of the rsAKARev signal upon addition of H-89. However, additional experiments using a fresh sample of H-89 showed the expected reversibility of the biosensor response (Supplementary Fig. 2 ), demonstrating that the limited reversibility found here was due to a poorly performing batch of H-89. Fig. 5: Performance of our method in living cells expressing two overlapping FRET biosensors. PKA and/or ERK signaling responses in HeLa cells expressing one (left, middle panels) or two (right panels) FRET biosensors. Responses of rsAKARev, rsAKARev(T/A), EKARev, and EKARev(T/A) are depicted per column according to the method of analysis. Dashed traces indicate results obtained using conventional analysis based on sensitized emission, while solid traces indicate results obtained using the photoswitching-based analysis presented here. All traces represent average FRET efficiencies plus standard deviation (shaded area). n = 75 (rsAKARev), 132 (EKARev), 12 (rsAKARev(T/A)), 42 (EKARev(T/A)) cells for single-sensor experiments and 80 (rsAKARev+EKARev), 50 (rsAKARev(T/A)+EKARev), 22 (rsAKARev+EKARev(T/A)) cells for two-sensor experiments, from 3, 2, 1, 1 and 2, 1, 1 independent experiments, respectively. Cells were stimulated with PMA (1 μM), forskolin (Fsk, 50 μM) and IBMX (100 μM), U0126 (20 μM), and H-89 (20 μM). Full size image The results from the conventional analysis and our unmixing strategy are in excellent agreement for the single-sensor experiments, reproducing the same features and responses. Taken together, the differences between the conventional and the photochromism-based analysis are small and unlikely to be important for all but the most demanding quantitative experiments. Having validated our strategy at the level of single biosensors, we then moved on to cells expressing both rsAKARev and EKARev simultaneously. Our analysis reproduces essentially the same trends as observed for the single-sensor case (Fig. 5 c, f, i), demonstrating that we can indeed separate two FRET pairs that display essentially complete spectral overlap. We do find that the simultaneous imaging of two biosensors leads to a reduction in the signal-to-noise ratio of the separated responses (Supplementary Figs. 3 and 4 ), consistent with the extra processing required to separate the signals, though the cellular responses can still be readily verified. Multiplexed detection of three signaling activities Our separation strategy can be straightforwardly combined with other spectrally distinct fluorophores or sensors. As a proof-of-concept, we simultaneously expressed rsAKARev and EKARev together with the red-shifted calcium sensor RCaMP [44] . Calcium is known to modulate the signaling activities of proteins within both the cAMP/PKA [45] , [46] and the MAPK/ERK [47] signaling pathways. Additionally, it has been shown that the PKA-mediated activation of the CREB transcription factor occurs via MAPK/ERK and is mediated by calcium release from intracellular stores [48] . Such inter-pathway dependencies illustrate the potential of multi-parametric biosensing. We sequentially stimulated the cells with Fsk/IBMX, epidermal growth factor (EGF), ionomycin, and CaCl 2 in order to provoke distinguishable response profiles. Ionomycin is an ionophore that transports Ca 2+ ions over lipid bilayers, leading to both calcium release from intracellular stores and calcium uptake from or leakage to the external medium. CaCl 2 was added as an internal control for the RCaMP response. A schematic overview of the stimulated signaling pathways can be found in Supplementary Fig. 5 . Using our strategy, rsAKARev and EKARev responses could be distinguished based on their photochromism, while the red-shifted RCaMP response could be separated based on its different excitation and emission wavelengths. Figure 6 a shows the responses of the three biosensors for all 129 analyzed cells. Plots for each individual biosensor can be found in Supplementary Fig. 6 , together with images of representative cells (Supplementary Fig. 7 ) and their corresponding responses (Supplementary Fig. 8 ). To simplify the following discussion, we also visually indicate three different time intervals during our measurement at the top of Fig. 6 a. Fig. 6: Three-biosensor unmixing reveals heterogeneity in cellular responses. HEK293T cells simultaneously expressing rsAKARev, EKARev, and RCaMP biosensors. a RGB color-coded responses of RCaMP (red), EKARev (green), and rsAKARev (blue). b 3D scatter plot of the maximal response, colored by the clustering assignment of each cell. c – e Box-plots of the maximal response observed for rsAKARev ( c ), RCaMP ( d ), and EKARev ( e ) during indicated time intervals. The middle line is the median, lower and upper hinges represent first and third quartile, whiskers extend to the full width of the data, outliers (further than 1.5× the interquartile range from the hinges) are represented by filled circles. n = 129 cells from 1 experiment with 37, 51, 41 cells in classes 1, 2, 3. Two-tailed unpaired Student’s t -test ( c , e {1,3}), Welch’s unequal variance test ( e {2,3}), Mann–Whitney U test ( d ); values between brackets indicate the compared classes. “*” indicates significant differences ( p < 0.0001); all other differences are non-significant ( p > 0.05). Cells were stimulated with forskolin (Fsk, 50 μM) and IBMX (100 μM), EGF (1 μg/ml), ionomycin (Iono, 10 μM), and CaCl 2 (2 mM). Full size image As expected, stimulation with Fsk/IBMX (interval 1) strongly increases the PKA activity in all cells, additionally showing that cAMP elevation also mediates a transient calcium response. This is in line with previous work describing cross-talk occurring via PKA-mediated release of calcium from ER-stores [49] , [50] . Next, EGF stimulation results in the second release of calcium (interval 2) [51] , [52] , and in delayed ERK activity (interval 3), with varying responses across cells. The onset of the ERK response overlaps somewhat with the addition of ionomycin, though this response is already initiated before addition in most cells, and has reached a plateau by the time at which the ionomycin-induced release of calcium becomes evident (Supplementary Figs. 6 and 8 ). Taken together, this suggests that ionomycin has a small or no effect on ERK activity. Visual inspection of the data suggested heterogeneity of the cellular responses (Fig. 6 a). We quantified the maximal response and timing of the biosensor responses within each interval (see “Methods” section, Supplementary Fig. 9 ), and found that k-means clustering could divide the cellular responses into three classes based on their maximal responses, visually shown in Fig. 6 b and confirmed using partial least squares discriminant analysis (Supplementary Fig. 10 ). The maximal response of the different sensors showed significant variations between the classes (Fig. 6 c–e). Fsk/IBMX stimulation appears to have a slightly stronger effect on the PKA activity in cells of class 2, though this does not appear to translate into a higher calcium transient, which was overall very variable. Most striking are the inter-class discrepancies in calcium (interval 2) and ERK (interval 3) signaling upon EGF stimulation. More specifically, cells within classes 1 and 2 respond strongly to EGF-mediated ERK activation, whereas the effect on class 3 cells is rather weak and occurs with an additional delay (Supplementary Fig. 9 h). Additionally, the sizes of the ERK responses do not appear to be linked to the increase of the preceding calcium release. These results show the potential of classifying cells based on the heterogeneity in their responses, although it is not immediately clear from the data where this heterogeneity originates from, or if the differences in the signaling activities of the three pathways are biologically correlated in the context of this experiment. Poor expression of EGF receptors [53] , [54] or Fsk/IBMX-induced cAMP-mediated effects [55] could render some cells less sensitive to EGF stimulation, while further heterogeneity may also be contributed by the high levels of variability associated with HEK293 cells [56] . Taken together, our data further support the notion that the simultaneous imaging of multiple biosensors can readily reveal heterogeneities even in well-established systems, while demonstrating that our scheme can be readily combined with spectrally-orthogonal biosensors to further expand the range of responses that can be measured. In conclusion, in this work, we set out to separate the contributions of two or more spectrally overlapping FRET pairs based on the photochromism of the donor fluorophores, by monitoring their response to irradiation with off-switching light. We developed a framework to perform the analysis and validated its applicability using numerical simulations. We find that our method can deliver accurate results already at low light levels, though the competition between the FRET process and the photochromism must be characterized for each of the FRET pairs. We then applied our methodology to FRET-based biosensors, many of which make use of cyan and yellow fluorescent proteins. We showed that the cyan fluorescent protein mTFP0.7 could be used to make a photochromic biosensor for PKA activity, which we named rsAKARev. We combined this sensor with the spectrally overlapping biosensor EKARev, a reporter for ERK kinase activity, and showed that the contributions of both sensors could be readily separated when expressed simultaneously in live cells. Finally, we expanded these experiments to include an additional reporter for Ca 2+ , and showed that this approach could detect cellular heterogeneity in the response to pharmacological stimulation. Our method further illustrates the potential of simultaneously acquiring multiple signaling responses to investigate complex biological networks. Our method is readily compatible with existing measurement or analysis pipelines, which may make use of the raw emission ratio, sensitized emission ratio, or the calculated FRET efficiency, provided that one characterizes the dependence between this metric and the photoswitching ratio. In practice, this will likely require a separate measurement for each FRET pair where the photoswitching ratio is monitored over varying emission ratios or FRET efficiencies. Such a measurement must only be performed once for each FRET pair and off-switching light dose. The method can be readily integrated with established instrumentation, requiring only the possibility of inserting an off- and on-switching irradiation in between two FRET acquisitions. In principle, our procedure can be used to separate the contributions of more than two FRET pairs, provided that the donors are sufficiently separated in terms of their photochromism. Our use of two biosensors also reflects the limited availability of photochromic cyan fluorescent proteins, though many more opportunities are available when considering also photochromic green and red fluorescent proteins, which would be directly compatible with our method. In summary, we have shown that photochromism of the donor can be leveraged to achieve FRET multiplexing using a simple linear unmixing scheme, on the condition that the responses of the FRET pairs are well characterized. Our strategy can be readily used as part of experiments that require multiplexed interaction sensing and biosensing."},{"section":"Methods","content":"Biosensor construction The expression plasmids were generated using a combination of polymerase chain reaction (PCR), restriction enzyme digestion, and Gibson Assembly molecular cloning techniques. For Gibson Assembly reactions, vector DNA was digested with restriction enzymes cutting the multiple cloning site. All other DNA fragments were generated by PCR with Q5-polymerase using primer sequences with an optimal annealing temperature of 72 °C, calculated using the NEB Tm calculator ( https://tmcalculator.neb.com ), and with minimal overlap of 20-basepairs between neighboring fragments (Supplementary Table 2 ). All fragments were then purified using gel extraction and subsequently combined in the Gibson Assembly reaction according to manufacturer instructions. rsAKARev was generated by replacing the cDNA for YPet and ECFP in AKAR3EV (pcDNA3.1+ vector) with those for mTFP0.7 and cpVenus172, respectively. For rsAKARev(T/A), PCR primers were modified to introduce a T/A mutation in the PKA substrate region. For the generation of EKARev(T/A), PCR primers were designed to introduce a T/A mutation in the ERK substrate region. Cell culture and transfection HeLa, COS-7 and HEK293T cells were cultured in Dulbecco’s modified Eagle medium (DMEM; Gibco, #31053028) supplemented with 10% (v/v) fetal bovine serum (FBS; Gibco, #70011036), 1× Glutamax (Gibco, #35050061) and 50 μg/ml gentamycin (GM; Sigma-Aldrich, #15750060) and maintained in a humidified incubator (37 °C, 5% CO 2 ). 24 h before transfection, cells were plated onto sterile 35-mm glass-bottom dishes (MatTek) and grown to 30–70% confluence. Cells were subsequently transfected 24–48 h before imaging using FuGENE6 transfection reagent (Promega) and plasmid DNA, according to the manufacturer’s protocol. For optimal transfection results, 90 μl DMEM containing 3 μl FuGENE6 was mixed into 10 μl ultrapure water containing a total of 1 μg DNA by pipetting thoroughly, and each mixture was left for at least 20 min before addition to each dish in dropwise fashion. For multi-biosensor experiments, the relative amounts of plasmid DNA were optimized to obtain comparable expression of all constructs. Cells were then grown for another 24h (HeLa) or 48h (HEK293T, COS-7) before imaging. Cells were finally switched to serum-reduced medium comprising DMEM (HEK293T, COS-7) or FluoroBrite DMEM (Gibco, #A1896702) (HeLa), supplemented with 0.5%(v/v) FBS, 1× Glutamax and 50 μg/ml GM 4-6h (HEK293T) or 16–20h (COS-7, HeLa) before imaging. Time-lapse fluorescence imaging Cell treatment Before imaging, 35-mm glass-bottom dishes were washed twice gently with phosphate-buffered saline (PBS; Gibco, #70011036) and then imaged in the dark in either calcium-free Hank’s balanced salt solution (HBSS; Gibco, #14185052) supplemented with 0.2% (w/v) bovine serum albumin (BSA; Gibco, #10270106) (HEK293T, COS-7) at room temperature, or in FluoroBrite DMEM supplemented with 0.5% (v/v) FBS and 1× Glutamax (HeLa) at 37 °C, 5% CO 2 . Cells were treated at indicated timepoints using a final concentration of 50 μM forskolin (Fsk) (Sigma-Aldrich, #93049), 100 μM 3-isobutyl-1-methylxanthine (IBMX) (Sigma-Aldrich, #410957), 1 μg/ml epidermal growth factor (EGF) (R&D Systems, #236-EG10), 10 μM ionomycin (Iono) (Sigma-Aldrich, #407950), 2 mM CaCl 2 (Chem-Lab Analytical, #CL05.0371), 1 μM phorbol-12-myristate-13-acetate (PMA) (Sigma-Aldrich, #P1585), 20 μM U0126 (Sigma-Aldrich, #662005) and 20 μM H-89 (Sigma-Aldrich, #B1427), by mixing a 5× concentrated solution, prepared in imaging buffer from freshly thawed DMSO (Fsk, IBMX, PMA, U0126, H-89) or HBSS (CaCl 2 ) stocks, into the imaging dish. Equipment Images were acquired on an Olympus IX-71 inverted microscope coupled to a Spectra X Light Engine (Lumencor), 10× UplanSApo objective (Olympus), ORCA-Flash4.0 LT camera (Hamamatsu), IX2-RFACA motorized fluorescence cube turret (Olympus), Lambda 10-B Optical Filter Changer (Sutter) and H117 high precision motorized stage (Prior). All hardware was controlled using in-house software. Experiments performed at 37 °C, 5% CO 2 were done using a stage-top incubator (H301-NIKON-TI-S-ER, OKOLAB) coupled to a digital CO2 controller unit (CO2-UNIT-BL, OKOLAB). Image acquisition Following the notation from the theory, the S D D , S D A , and S A A detection channels were acquired with a T455LP ( S D D , S D A ) or ZT514RDC ( S A A ) dichroic mirror and an ET480/40m ( S D D ) or ET545/40m ( S D A , S A A ) emission filter (all Chroma). Each sample was excited with blue light (438/29 nm, 35.6mW; S D D , S D A ) or teal light (513/22 nm, 2.45 mW; S A A ). RCaMP intensity was acquired using a ZT561rdc dichroic and HQ572lp emission filter (all Chroma) with excitation by green light (542/33 nm, 39.0 mW). All images were acquired during 50 ms camera exposure. Each acquisition was performed during constant time intervals every 10–30 s on multiple positions of the sample. Irradiation settings for photochromism were determined as the minimum light dose required to reach steady-state fluorescence during ON and OFF switching of mTFP0.7, which was achieved by irradiating 0.2 s with violet light (390/22 nm, 26.7 mW) and 0.2 s with blue light (438/29 nm, 78.35 mW), respectively. S D D and S D A fluorescence was acquired after each irradiation step; S A A fluorescence was acquired for only a subset of timepoints to minimize acceptor photobleaching. FRET analysis Analysis of the fluorescence time-series was performed using in-house analysis software running on Igor Pro 8 (WaveMetrics). For each sample position, multiple regions of interest (ROI) corresponding to single cells were defined, and the average intensity over the shape of the cell was determined at each timepoint. Subsequently, the intensity was corrected for background by subtracting the average signal of a region outside the cell boundaries. This correction can be improved by subtracting the signal observed for non-transfected cells instead, though we did not pursue this here as the emission from non-transfected cells was much smaller than both the background and the emission from transfected cells. The resulting average intensities were used for downstream analysis. For the conventional FRET analysis of single-sensor experiments, FRET efficiencies were subsequently calculated according to Eq. ( 6 ) using only the background-corrected fluorescence detected after each violet irradiation step in the measurement scheme shown in Fig. 1 b. The photoswitching analysis of single- and two-biosensor experiments was performed as detailed in the theory using Eqs. ( 11 )–( 14 ). The analysis procedure is described in more detail in Supplementary Note 3 . From single-sensor experiments, the dependencies ρ (Θ) were determined empirically by fitting a double-exponential function (rsAKARev) or a straight line (EKARev) on the pooled cell data represented in Fig. 4 , using orthogonal distance regression. Coefficients α , δ , and γ (Supplementary Table 1 ) were calculated based on published values for fluorescence quantum yields, instrument parameters, and the available spectral data on the fluorescent protein database FPbase [57] , as described in Supplementary Note 4 . Three-biosensor unmixing experiment The FRET efficiencies for rsAKARev and EKARev were obtained as described above. The raw RCaMP intensities were obtained by averaging the corresponding emission over the same regions of interest, and also included background removal by subtracting the signal from a region of interest that did not include any cells. We found that both the FRET and the RCaMP traces showed gradual decreases in amplitude, which we attribute to photobleaching of the fluorophores. We correct this using a baseline that is determined for every trace. For FRET measurements, the baseline is determined by fitting the trace with the function $${{\\Theta }}(t)={e}^{-t/\\tau }\\cdot \\left(b+\\frac{A}{1+\\exp \\left(\\frac{{t}_{0}-t}{r}\\right)}\\right)$$ (16) where the factor between parentheses is a standard sigmoidal function. The baseline is then given by b e − t / τ . τ is typically much larger than the duration of the experiment. For RCaMP measurements, we estimated the baseline used by fitting a bi-exponential function to the parts of the trace that do not show an elevated calcium signal. We then calculated the responses as $$\\% \\,\\,{\\text{Increase}}\\,(t)=\\frac{\\,{\\text{signal}}\\,(t)-\\,{\\text{baseline}}\\,(t)}{\\,{\\text{baseline}}\\,(t)}\\cdot 100 \\%$$ (17) where “signal” is the FRET efficiency for rsAKARev and EKARev, and the fluorescence intensity for RCaMP. The resulting responses are shown in Fig. 6 a, where we manually optimized the color scaling to highlight the cellular responses. We additionally show example traces, their corresponding baselines and the calculated responses in Supplementary Figs. 11 – 13 . The cell responses to pharmacological stimulation were quantified by taking the value of the maximum response observed within the time interval following stimulation. The onset time of the FRET biosensor responses was determined as the time for which a sigmoidal fit of the response increases to 7.6% of its maximum value, while the onset time of the RCaMP data was defined as the timepoint at which the second derivative of the response reached its maximum value. Partitional k -means clustering was performed in Matlab R2019b based on the responses determined for rsAKARev over time interval 1, for RCaMP over time interval 2, and for EKARev over time interval 3. To validate the classification results of the K -means clustering, partial least squares discriminant analysis (PLS-DA) was performed on the full-time trace data sets (% increase) assuming the three classes obtained by the K -means clustering. Subsequently, clustering results were cross-validated with the PLS-DA model using a 95% confidence interval. Additional information is available in Supplementary Note 5 . Statistics and reproducibility All statistical tests were performed in GraphPad Prism 5 (GraphPad Software). Data were checked for normality using a D’Agostino & Pearson omnibus test. Subsequently, pairwise comparison was performed using the F -test for equal variances, followed by a two-tailed unpaired Student’s t -test (if F -test, p > 0.05) or Welch’s unequal variance test (if F -test, p < 0.05) for Gaussian data, or using the Mann Whitney U test for non-gaussian data. The features shown in the images of Figs. 1 b, 3 a, and Supplementary Fig. 9 could be consistently reproduced across multiple experiments. 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\": \"Förster resonance energy transfer (FRET) is a powerful mechanism to probe associations in situ. FRET experiments are performed by labeling the system of interest with donor and acceptor fluorophores, and then using the extent of the energy transfer to probe the distances between and relative orientations of both fluorophores [1] . FRET-based approaches have gained tremendous popularity for association studies, and can be performed using both genetically encoded fluorophores and organic dyes [2] . It is also the key mechanism underlying a class of genetically encoded biosensors that convert the presence of a particular chemical stimulus into a change in FRET [3] , [4] , [5] , [6] . Most FRET experiments use a fluorescent donor and acceptor, which has the advantage that the FRET efficiency can be estimated based on the ratio between the donor-excited emission from both fluorophores [7] , [8] , [9] , [10] . The accuracy of this estimation depends on the ease with which the donor and acceptor can be spectrally separated from one another. The choice of specific FRET pairs is usually driven by a compromise between achieving a clearly observable FRET efficiency, which requires sufficient spectral overlap, and maintaining a low level of cross-talk between channels, which is made easier when the probes are more spectrally separated. As a result, the donor and acceptor tend to occupy broad regions of the visual spectrum, complicating the combination of FRET measurements with the use of other probes. Measuring multiple FRET-based constructs is further complicated if these constructs share fluorophores emitting in the same spectral band. For example, many FRET biosensors consist of a cyan and a yellow fluorescent protein, making it more difficult to combine multiple biosensors to follow, e.g., cross-talk between signaling pathways. Using more than one of these probes typically requires that separate measurements are performed for each probe, possibly in a high-throughput manner [11] . A number of strategies can facilitate FRET multiplexing [12] . The conceptually easiest approach is to select two or more FRET pairs with minimal spectral overlap, though this is limited by the available probes and the usable spectral range [13] . It may be possible to share a single fluorophore among two FRET pairs, reducing the total number of fluorophores from four to three [14] . Other approaches include the use of non-fluorescent acceptors combined with fluorescence lifetime imaging [15] , [16] , [17] , the use of homo-FRET combined with alternative readouts such as fluorescence anisotropy [18] , and the separation of spectrally overlapping pairs based on orthogonal information, such as the spatial localization of the probes within the cell [19] . Several strategies for probe multiplexing have relied on exploiting the photochemistry of the fluorophores. Photochromic labels, for example, display light-induced and reversible “on-off” switching of their fluorescence emission [20] , [21] . Upon excitation, these probes display a bright fluorescence emission that disappears in time, though the original fluorescence can be rapidly recovered by irradiating with the light of a shorter wavelength. These dynamics can be used to separate the signal of a photochromic label from that of a non-photochromic label or from the autofluorescence, or can be used to separate two or more photochromic labels that switch with different kinetics [22] , [23] , [24] , [25] , [26] . However, there has been comparatively little work on the evaluation of such strategies for the simultaneous measurement of multiple FRET pairs. In contrast, photochromism of the acceptor has been used as a way to provide a more accurate FRET quantification [27] , [28] or to measure multiple single-molecule donor-acceptor distances within the same complex [29] . Previous work has also investigated the use of a photochromic donor, where the FRET efficiency can be estimated by determining the rate of the photochromism [30] , or to facilitate the anisotropy-based readout of homo-FRET [31] . Association-induced fluorescence dynamics have also been used to provide diffraction-unlimited imaging of biosensors [32] . In this contribution, we describe the use of donor photochromism to simultaneously measure more than one FRET pair, even when the donors and acceptors display complete spectral overlap. We commence by developing the theory underlying our method and validate its performance using numerical simulations. We then develop a photochromic biosensor for cAMP-dependent protein kinase (PKA), which we call rsAKARev, and use it to demonstrate our approach through the simultaneous measurement of PKA and extracellular signal-regulated kinase (ERK) activity in live cells. 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Monitoring multiple distances within a single molecule using switchable FRET.Nat. Methods7, 831–836 (2010).\",\n \"link\": \"https://doi.org/10.1038%2Fnmeth.1502\"\n },\n {\n \"idx\": \"30\",\n \"title\": \"Rainey, K. H. & Patterson, G. H. Photoswitching FRET to monitor protein–protein interactions.Proc. Natl Acad. Sci. USA116, 864–873 (2019).\",\n \"link\": \"https://doi.org/10.1073%2Fpnas.1805333116\"\n },\n {\n \"idx\": \"32\",\n \"title\": \"Mo, G. C. H. et al. Genetically encoded biosensors for visualizing live-cell biochemical activity at super-resolution.Nat. Methods14, 427–434 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fnmeth.4221\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Quantification of the FRET signal The goal of a FRET measurement is to quantify the energy transfer between the donor and acceptor fluorophore. We assume that this measurement is performed using sensitized imaging, where the donor is excited and the emission from the donor ( S DD ) and acceptor ( S DA ) are measured simultaneously or in rapid succession. As a control, and to estimate the effects of photodestruction, one usually also measures the acceptor emission upon direct excitation ( S AA ). A full FRET acquisition, therefore, consists of three measurements { S DD , S DA , S AA } at every timepoint and at every pixel of the detector. The S DA signal is usually contaminated by donor bleed-through and direct acceptor cross-excitation. These contributions can be quantified by using explicit correction factors α and δ [9] : $$\\\\alpha ={S}_{\\\\mathrm{DA}}/{S}_{\\\\mathrm{DD}}\\\\ \\\\ \\\\ \\\\,{\\\\mathrm{in}}\\\\; {\\\\mathrm{presence}}\\\\; {\\\\mathrm{of}}\\\\; {\\\\mathrm{only}}\\\\; {\\\\mathrm{donor}}\\\\,$$ (1) $$\\\\delta ={S}_{\\\\mathrm{DA}}/{S}_{\\\\mathrm{AA}}\\\\ \\\\ \\\\ \\\\,{\\\\mathrm{in}}\\\\; {\\\\mathrm{presence}}\\\\; {\\\\mathrm{of}}\\\\; {\\\\mathrm{only}}\\\\; {\\\\mathrm{acceptor}}$$ (2) These correction factors can be determined using control measurements or can be estimated given knowledge of the fluorophore spectra and the spectral sensitivity of the instrument. In general, direct control measurements are preferable since these will then be tailored to the specifics of the instrument. We assume that direct excitation of the acceptor does not lead to appreciable emission of the donor fluorophore in the acceptor channel. Not all experimenters choose to determine these correction factors, instead quantifying the energy transfer using the “raw” emission ratio R $$R=\\\\frac{{S}_{\\\\mathrm{DA}}}{{S}_{\\\\mathrm{DD}}}$$ (3) which will already allow the detection of relative changes in the FRET efficiency. If available, however, we can use the correction factors to calculate the sensitized emission F c , which is the donor-excited emission from the acceptor corrected for cross-talk: $${F}_{\\\\mathrm{c}}={S}_{\\\\mathrm{DA}}-\\\\alpha {S}_{\\\\mathrm{DD}}-\\\\delta {S}_{\\\\mathrm{AA}}$$ (4) Given F c , the FRET response can be estimated using the sensitized emission ratio R c : $${R}_{\\\\mathrm{c}}=\\\\frac{{F}_{\\\\mathrm{c}}}{{S}_{\\\\mathrm{DD}}}$$ (5) R c can be compared across different instruments and is more sensitive to changes in FRET than the emission ratio R . However, it is not unambiguously related to the absolute FRET efficiency (Θ) since it fails to take into account the different brightnesses of the donor and acceptor fluorophores and/or the different collection efficiencies of the instrument. The introduction of an additional correction factor γ [9] makes it possible to calculate the FRET efficiency using $${{\\\\Theta }}=\\\\frac{1}{\\\\gamma {S}_{\\\\mathrm{DD}}/{F}_{\\\\mathrm{c}}+1}$$ (6) as is derived in Supplementary Note 1 . γ can be estimated from model spectra, or more reliably from additional control experiments [10] . Introducing a photochromic donor The core idea of our method is that two different FRET pairs can be distinguished based on the photochromism of the donors, conceptually shown in Fig. 1 a. We therefore extend our model to the case where the donor is a photochromic fluorophore. To probe this photochromism, we expand our measurement scheme with two irradiation periods that serve to switch the donor fluorophore between fluorescent and non-fluorescent states (Fig. 1 b). This consists of a short-wavelength light pulse and fluorescence acquisition to capture the fluorescent state, followed by a longer-wavelength light pulse and another fluorescence acquisition to capture the extent of the off-switching. The net result is that every FRET acquisition is now replaced with two acquisitions, performed at the end of each irradiation period, for a total of six measurements for each FRET timepoint (two measurements each for S DD , S DA , and S AA ). S AA usually varies slowly because it is affected only by photodestruction or large structural changes in the sample, and can therefore be measured at a reduced frequency. Fig. 1: Conceptual overview. a Two FRET pairs can be distinguished by probing the photochromism of the donors since only photochromic donors react to the off- and on-switching light. b Example irradiation scheme for a cyan photochromic protein, showing the donor on-state population (dotted line) during violet and blue light irradiation (colored bars). The red dots show the fluorescence acquisitions that are performed. The inset shows representative fluorescence images of a HeLa cell, acquired in all three measurement channels ( S D D , S D A , S A A ). Scale bar, 20 μm. In practice, the light-induced off-switching does not need to be as complete as shown here. Full size image Off-switching of the donor fluorophore results in a decrease in S DD , S DA , and F c . We quantify this using the photoswitching ratios ρ D and ρ A , given by $${\\\\rho }_{\\\\mathrm{D}}=\\\\frac{{S}_{{\\\\mathrm{DD}},\\\\text{off}}}{{S}_{{\\\\mathrm{DD}},\\\\text{on}}}$$ (7) $${\\\\rho }_{\\\\mathrm{A}}=\\\\left\\\\{\\\\begin{array}{ll}\\\\frac{{F}_{{\\\\mathrm{C}},\\\\text{off}}}{{F}_{{\\\\mathrm{C}},\\\\text{on}}}&\\\\;{\\\\text{if}}\\\\; {\\\\text{correction}} \\\\;{\\\\text{factors}}\\\\; {\\\\text{are}} \\\\;{\\\\text{available}}\\\\hfill\\\\\\\\ \\\\frac{{S}_{{\\\\mathrm{DA}},\\\\text{off}}}{{S}_{{\\\\mathrm{DA}},{\\\\text{on}}}}&\\\\,\\\\text{otherwise}\\\\,\\\\hfill\\\\end{array}\\\\right.$$ (8) where the “on” and “off” acquisitions are those shown in Fig. 1 b. We provide two alternative definitions of ρ A in Eq. ( 8 ) depending on whether correction factors for cross-talk are available. Our methodology remains valid for both approaches as long as these definitions are used consistently. The photoswitching ratios ρ D and ρ A are thus the fractions of the original fluorescence that are left after the off-switching irradiation has been applied, with smaller values implying more pronounced off-switching. They depend on the specifics of the donor fluorophore and on the total light dose that is delivered, but also on the FRET efficiency since this process competes with photochromism. This aspect was leveraged in the development of psFRET [30] , and likewise must be taken into account when separating two FRET pairs based on photochromism. In the general case, the FRET-dependence of the photoswitching ratios ρ D and ρ A must be determined experimentally by measuring these for varying (sensitized) emission ratios or FRET efficiencies. This can be done simply by performing an experiment where the FRET pair is expressed and ρ D and ρ A are followed as a function of emission ratio or FRET efficiency. This approach will also work in the more general case where the emission contrast arises not just due to energy transfer, but also through other contributions such as a change in brightness of the donor. This process can be simplified when suitable assumptions can be made about the system. For a simple FRET process with negligible or perfectly corrected cross-talk in which only a single species is present, $${\\\\rho }_{\\\\mathrm{{D}}}={\\\\rho }_{\\\\mathrm{{A}}}$$ (9) since the acceptor emission is proportional to the amount of the donor in the fluorescent state. If we further assume that the off-switching can be described by an intrinsic quantum yield then the photoswitching ratio is given by $$\\\\rho ={({\\\\rho }_{0})}^{(1-{{\\\\Theta }})}$$ (10) as is derived in Supplementary Note 2 . Equation ( 10 ) shows that the full dependence is known if one simply measures ρ 0 , the photoswitching ratio in the absence of FRET, using the same settings for the instrument. As we show in the Supplementary Information, this consideration can be expanded to photochromic molecules that do not display complete off-switching but instead reach an equilibrium or plateau level. While these results can simplify the characterization of the FRET pairs, our method will still work if these assumptions cannot be made. Residual cross-talk of direct acceptor excitation into the S D A channel, for example, will increase ρ A since this contribution shows no fluorescence modulation. The derived models also do not apply to FRET-based biosensors, which exist as two interconverting species (active and inactive forms of the sensor). If no simplifying assumptions can be made, the dependence of ρ D and ρ A on the (sensitized) emission ratio or FRET efficiency will need to be determined over the full range of values, for each FRET pair and off-switching light dose used in the experiment. Separating photochromic FRET pairs We now assume that the sample is labeled with two FRET pairs, where the absorption and emission spectra are highly similar yet the donors differ in their photochromism. This situation can also arise when two different interactions are probed using two donors and a single shared acceptor. The recorded fluorescence signals are simply the sum of the signals from the individual components $${S}_{{\\\\mathrm{DD}},\\\\text{on}}={S}_{{\\\\mathrm{DD}},\\\\text{on},1}+{S}_{{\\\\mathrm{DD}},\\\\text{on},2}$$ (11) $${S}_{{\\\\mathrm{DA}},\\\\text{on}}={S}_{{\\\\mathrm{DA}},\\\\text{on},1}+{S}_{{\\\\mathrm{DA}},\\\\text{on},2}$$ (12) where “1” and “2” indicate the first and second FRET pair. After irradiation with off-switching light, we also obtain $${S}_{{\\\\mathrm{DD}},\\\\text{off}}={\\\\rho }_{{\\\\mathrm{D}},1}\\\\,{S}_{{\\\\mathrm{DD}},\\\\text{on},1}+{\\\\rho }_{D,2}\\\\ {S}_{{\\\\mathrm{DD}},\\\\text{on},2}$$ (13) $${S}_{{\\\\mathrm{DA}},\\\\text{off}}={\\\\rho }_{{\\\\mathrm{A}},1}\\\\,{S}_{{\\\\mathrm{DA}},\\\\text{on},1}+{\\\\rho }_{{\\\\mathrm{A}},2}\\\\ {S}_{{\\\\mathrm{DA}},\\\\text{on},2}$$ (14) Our measurement provides the S D D and S D A signals before and after off-switching. If we assume that all four photoswitching ratios ρ D and ρ A are known, then these equations constitute a system of four equations and four unknowns, which can be solved to obtain S D D ,on and S D A ,on for both species. These values can then be readily used to obtain the (sensitized) emission ratios for each pair using Eqs. ( 3 ) or ( 5 ), or the FRET efficiency using Eq. ( 6 ). In principle, the same approach could also be used to separate two FRET pairs where at least one donor is photochromic and one pair contains a dark (non-fluorescent) acceptor. In practice, we do not know the four ρ D and ρ A values as these depend on the sought-after (sensitized) emission ratio or FRET efficiency. However, the previous section considered how to determine the relationship between these parameters. Given that information, we recommend the use of an iterative procedure, in which the (sensitized) emission ratio or FRET efficiency is calculated starting from initial guesses for the photoswitching ratios. This value is then used to obtain better guesses for the photoswitching ratios, and this procedure is repeated until these values converge, which occurs rapidly in practice. A more detailed discussion of the algorithm used in this work can be found in Supplementary Note 3 . Numerical simulations We devised a series of numerical simulations to test the validity of our approach. We assumed a sample labeled with two spectrally indistinguishable FRET pairs, where one of the donors is photochromic and can be described using Eq. ( 10 ), and the other does not show appreciable photochromism. We further assumed different brightness levels, reflected in the number of photons that would be acquired in the “on” acquisition if no FRET were to occur. Noise was included in our measurement by simulating the Poisson noise intrinsic to the photon detection process. For each condition, we ran 5000 simulations where we selected random ground-truth FRET efficiencies between 0 and 0.6 for each FRET pair, and generated two simulated fluorescence acquisitions, before and after off-switching (cfr. the measurement scheme shown in Fig. 1 b). These data were then analyzed using our methodology, resulting in estimates for the FRET efficiency of each of the pairs. The accuracy of these results was then quantified using the mean absolute deviation, given by $$\\\\,\\\\text{MAD}\\\\,=\\\\frac{\\\\left|{{{\\\\Theta }}}_{1}-{\\\\widehat{{{\\\\Theta }}}}_{1}\\\\right|+\\\\left|{{{\\\\Theta }}}_{2}-{\\\\widehat{{{\\\\Theta }}}}_{2}\\\\right|}{2}$$ (15) where Θ i are the ground-truth FRET efficiencies and \\\\({\\\\hat{{{\\\\Theta }}}}_{i}\\\\) are the FRET efficiencies estimated by our method. As a negative control, we also included the result of an analysis that simply generated random FRET efficiencies. Our simulations show that our analysis works well already at low light levels, providing accurate results when 1000 or more photons are detected, and less accurate but useful results already at signal levels as low as 250 photons (Fig. 2 a). We also investigated to what extent the difference in photoswitching ratio of the FRET donors (Δ ρ ) determines the accuracy of the analysis (Fig. 2 b and Supplementary Fig. 1 ). We find that this depends on the expected FRET efficiency and the donor photoswitching ratio in the absence of FRET ( ρ 0 ). However, a ρ 0 of 0.3 or lower, readily achievable with, e.g., photochromic fluorescent proteins, works well over a broad range of FRET efficiencies. Furthermore, ρ 0 can be increased or decreased by adjusting the duration or intensity of the off-switching illumination, allowing the conditions to be tailored to the measurement. Finally, we note that further increases in accuracy are possible by including more fluorescence acquisitions as part of the off-switching (adding additional fluorescence acquisitions to each of the measurement cycles shown in Fig. 1 b). Fig. 2: Evaluation of the method on simulated data. a Combined violin and box-plot of the mean absolute deviation (MAD) of our analysis approach for different numbers of detected photons. Data were generated by performing 5000 independent simulations for each category. Violin plot shows the underlying data distribution. Box-plot middle line indicates the median, lower and upper hinges represent first and third quartile, whiskers extend to the full width of the data. b Scatter plot showing the analysis performance (MAD) for different FRET efficiencies and photoswitching ratios (in the absence of FRET) of the photochromic ( ρ 0, p ) and non-photochromic ( ρ 0, n p ) donor. Δ ρ is the difference in photoswitching ratio. Two-dimensional projections of this data are shown in Supplementary Fig. 1 . Full size image In principle, our methodology can be used to separate more than two overlapping FRET pairs, by adding additional terms to Eqs. ( 11 )–( 14 ). Each additional pair will then require one additional fluorescence acquisition per switching cycle. rsAKARev: a photochromic biosensor for PKA We decided to experimentally validate our method by developing a photochromic FRET-based biosensor for cAMP-dependent protein kinase (PKA) activity, called rsAKARev (reversibly switchable A-kinase activity reporter with EV-linker). We obtained this biosensor by replacing the donor and acceptor in AKAR3ev [33] with the cyan fluorescent protein mTFP0.7 and the yellow fluorescent protein cpVenus172. mTFP0.7 is a variant of the bright and photostable mTFP1 [34] that shows efficient photochromism with little fatigue [35] . cpVenus172 is a circular permutant of Venus that has been reported to result in an improved FRET contrast [36] , [37] , [38] . To evaluate our unmixing strategy, we chose to combine rsAKARev with the non-photochromic biosensor EKARev [33] , which reports on extracellular signal-regulated kinase (ERK) activity and consists of the cyan donor ECFP and yellow acceptor YPet. Our choice of sensors reflects the fundamental interest in the interactions between the conserved cAMP/PKA and MAPK/ERK signaling pathways, which regulate various physiological processes including cell proliferation and growth [39] , [40] . To characterize the performance of our biosensors, we performed measurements on cells expressing either rsAKARev or EKARev, and submitted them to the measurement scheme shown in Fig. 1 b. As expected, we found that the fluorescent state of the rsAKARev donor fluorophore, mTFP0.7, readily depopulates upon irradiation with blue light and can be efficiently recovered through weak irradiation with violet light (Fig. 3 a). This donor photochromism leads to a large and reversible fluorescence contrast in the donor and acceptor emission with minimal loss of fluorescence intensity (Fig. 3 b). In contrast, the EKARev donor, ECFP, displays a much more limited modulation of the fluorescence signal (Fig. 3 c). Fig. 3: Photochromism of rsAKARev and EKARev expressed in HeLa cells. a Representative fluorescence images of the donor ( S D D ) and acceptor ( S D A ) fluorescence of rsAKARev before and after off-switching. Scale bar, 40 μm. b , c Time-course of the donor and sensitized acceptor emission of rsAKARev ( b ) and EKARev ( c ) during multiple irradiation cycles. d , e Representative responses of a HeLa cell expressing rsAKARev ( d ) or EKARev ( e ) during stimulation of PKA or ERK activity. Time-traces are colored as indicated. PKA activity was stimulated with forskolin (Fsk, 50 μM) and IBMX (100 μM), ERK activity with PMA (1 μM). Full size image We next evaluated how the activation state of each biosensor affected its fluorescence emission. Cells were treated with pharmacological compounds to evoke either PKA or ERK kinase activity, and the FRET response of the respective biosensor was followed over time (Fig. 3 d, e). This response was determined via conventional FRET analysis using only the ‘on’ fluorescence acquired after every violet irradiation step in Fig. 1 b. Forskolin-(Fsk) and 3-isobutyl-1-methylxanthine (IBMX)-mediated elevation of intracellular cAMP strongly increased the FRET response of rsAKARev. Likewise, stimulation of cells with phorbol-12-myristate-13-acetate (PMA), which acts on ERK via upstream kinases [41] , [42] , elicited a strong EKARev response. As expected, our data additionally revealed that changes in PKA activity also lead to changes in the photoswitching ratios ρ D and ρ A (Fig. 3 d). Conversely, Fig. 3 e shows that the irradiation-induced fluorescence dynamics of EKARev are small and largely independent of the activation state of the sensor, consistent with the photostatic nature of ECFP. As we discussed in the theory, separating overlapping FRET pairs based on their photochromism requires knowledge of the relationship between the photoswitching ratio and the FRET efficiency. For well-behaved FRET processes, this can be done using only a single reference measurement and the analytical expression of the dependence (e.g., Eq. ( 10 )). However, systems that consist of an equilibrium between multiple discrete states, such as the active and inactive state of a biosensor, cannot be described using this relationship. Furthermore, cyan fluorescent proteins commonly show complicated spectroscopic behavior such as multi-exponential excited state lifetime decays [43] . We therefore resorted to direct measurement of the full dependence in cells expressing either rsAKARev or EKARev (Fig. 4 ). We found that the sought-after relationship could be empirically described using double-exponential (rsAKARev) or straight line (EKARev) fits, providing a model that was used in the downstream analysis. Figure 4 b does show that the photoswitching ratio increases with decreasing FRET efficiency, which is contrary to what one would expect for a two-state FRET system. This is likely due to a suboptimal value for the acceptor bleed-through correction, though this does not interfere in the separation analysis which follows since its effects are automatically taken into account via this in situ calibration. Fig. 4: Relation between FRET efficiency and photoswitching ratio. Plots for donor emission ( a , c ) and donor-excited acceptor emission ( b , d ) as determined by single-sensor experiments on COS-7 cells expressing rsAKARev (orange) or HeLa cells expressing EKARev (blue). n = 99 (rsAKARev) or 41 (EKARev) cells from 1 experiment each. Solid lines represent the empirical fitting functions for rsAKARev (double-exponential) or EKARev (straight line) that were used for downstream analysis. Full size image Simultaneous measurement of two overlapping FRET biosensors We then focused on the simultaneous measurement of rsAKARev and EKARev expressed within the same cells. Since the emission spectra of both biosensors are essentially indistinguishable, their responses would be difficult to separate based on classical FRET imaging. In order to generate predictable response profiles for rsAKARev and EKARev, we treated the cells sequentially with PMA, Fsk/IBMX, U0126, and H-89. U0126 and H-89 are inhibitors that reverse the responses of the MAPK/ERK and PKA signaling pathways, respectively [17] . Ideally, this sequence of manipulations should result in distinguishable response profiles for rsAKARev and EKARev. We initially performed these experiments on cells expressing a single biosensor, either rsAKARev, EKARev, or inactive mutants in which the kinase target threonine was mutated to alanine, denoted using a “(T/A)” suffix. We subjected these cells to the measurement scheme shown in Fig. 1 b, but analyzed the resulting responses in two different ways: either via a conventional intensity-based analysis, using only the “on” fluorescence images acquired after every violet irradiation step (Fig. 5 a, d, g), or via our full photochromism-based approach (Fig. 5 b, e, h). Applying both analysis methodologies to the same data makes it possible to detect any discrepancies between the obtained results. Details of the analysis procedure are provided in Supplementary Note 3 . The conventional analysis shows that the biosensors respond as expected to pharmacological treatment (Fig. 5 a). We did observe inhibition of the EKARev signal upon addition of Fsk/IBMX, reminiscent of previous reports where it was found that forskolin interferes with EGF-mediated ERK activation [17] . This suggests that a similar mechanism is also at play when stimulating ERK activity with PMA. As expected, the biosensor response of the T/A mutants remained insensitive to the applied sequence of manipulations. The addition of Fsk/IBMX did induce a decrease in the EKARev(T/A) response similar to that observed for EKARev, demonstrating that this response arises from a process other than phosphorylation of the target residue. We also observed a weaker than expected inhibition of the rsAKARev signal upon addition of H-89. However, additional experiments using a fresh sample of H-89 showed the expected reversibility of the biosensor response (Supplementary Fig. 2 ), demonstrating that the limited reversibility found here was due to a poorly performing batch of H-89. Fig. 5: Performance of our method in living cells expressing two overlapping FRET biosensors. PKA and/or ERK signaling responses in HeLa cells expressing one (left, middle panels) or two (right panels) FRET biosensors. Responses of rsAKARev, rsAKARev(T/A), EKARev, and EKARev(T/A) are depicted per column according to the method of analysis. Dashed traces indicate results obtained using conventional analysis based on sensitized emission, while solid traces indicate results obtained using the photoswitching-based analysis presented here. All traces represent average FRET efficiencies plus standard deviation (shaded area). n = 75 (rsAKARev), 132 (EKARev), 12 (rsAKARev(T/A)), 42 (EKARev(T/A)) cells for single-sensor experiments and 80 (rsAKARev+EKARev), 50 (rsAKARev(T/A)+EKARev), 22 (rsAKARev+EKARev(T/A)) cells for two-sensor experiments, from 3, 2, 1, 1 and 2, 1, 1 independent experiments, respectively. Cells were stimulated with PMA (1 μM), forskolin (Fsk, 50 μM) and IBMX (100 μM), U0126 (20 μM), and H-89 (20 μM). Full size image The results from the conventional analysis and our unmixing strategy are in excellent agreement for the single-sensor experiments, reproducing the same features and responses. Taken together, the differences between the conventional and the photochromism-based analysis are small and unlikely to be important for all but the most demanding quantitative experiments. Having validated our strategy at the level of single biosensors, we then moved on to cells expressing both rsAKARev and EKARev simultaneously. Our analysis reproduces essentially the same trends as observed for the single-sensor case (Fig. 5 c, f, i), demonstrating that we can indeed separate two FRET pairs that display essentially complete spectral overlap. We do find that the simultaneous imaging of two biosensors leads to a reduction in the signal-to-noise ratio of the separated responses (Supplementary Figs. 3 and 4 ), consistent with the extra processing required to separate the signals, though the cellular responses can still be readily verified. Multiplexed detection of three signaling activities Our separation strategy can be straightforwardly combined with other spectrally distinct fluorophores or sensors. As a proof-of-concept, we simultaneously expressed rsAKARev and EKARev together with the red-shifted calcium sensor RCaMP [44] . Calcium is known to modulate the signaling activities of proteins within both the cAMP/PKA [45] , [46] and the MAPK/ERK [47] signaling pathways. Additionally, it has been shown that the PKA-mediated activation of the CREB transcription factor occurs via MAPK/ERK and is mediated by calcium release from intracellular stores [48] . Such inter-pathway dependencies illustrate the potential of multi-parametric biosensing. We sequentially stimulated the cells with Fsk/IBMX, epidermal growth factor (EGF), ionomycin, and CaCl 2 in order to provoke distinguishable response profiles. Ionomycin is an ionophore that transports Ca 2+ ions over lipid bilayers, leading to both calcium release from intracellular stores and calcium uptake from or leakage to the external medium. CaCl 2 was added as an internal control for the RCaMP response. A schematic overview of the stimulated signaling pathways can be found in Supplementary Fig. 5 . Using our strategy, rsAKARev and EKARev responses could be distinguished based on their photochromism, while the red-shifted RCaMP response could be separated based on its different excitation and emission wavelengths. Figure 6 a shows the responses of the three biosensors for all 129 analyzed cells. Plots for each individual biosensor can be found in Supplementary Fig. 6 , together with images of representative cells (Supplementary Fig. 7 ) and their corresponding responses (Supplementary Fig. 8 ). To simplify the following discussion, we also visually indicate three different time intervals during our measurement at the top of Fig. 6 a. Fig. 6: Three-biosensor unmixing reveals heterogeneity in cellular responses. HEK293T cells simultaneously expressing rsAKARev, EKARev, and RCaMP biosensors. a RGB color-coded responses of RCaMP (red), EKARev (green), and rsAKARev (blue). b 3D scatter plot of the maximal response, colored by the clustering assignment of each cell. c – e Box-plots of the maximal response observed for rsAKARev ( c ), RCaMP ( d ), and EKARev ( e ) during indicated time intervals. The middle line is the median, lower and upper hinges represent first and third quartile, whiskers extend to the full width of the data, outliers (further than 1.5× the interquartile range from the hinges) are represented by filled circles. n = 129 cells from 1 experiment with 37, 51, 41 cells in classes 1, 2, 3. Two-tailed unpaired Student’s t -test ( c , e {1,3}), Welch’s unequal variance test ( e {2,3}), Mann–Whitney U test ( d ); values between brackets indicate the compared classes. “*” indicates significant differences ( p < 0.0001); all other differences are non-significant ( p > 0.05). Cells were stimulated with forskolin (Fsk, 50 μM) and IBMX (100 μM), EGF (1 μg/ml), ionomycin (Iono, 10 μM), and CaCl 2 (2 mM). Full size image As expected, stimulation with Fsk/IBMX (interval 1) strongly increases the PKA activity in all cells, additionally showing that cAMP elevation also mediates a transient calcium response. This is in line with previous work describing cross-talk occurring via PKA-mediated release of calcium from ER-stores [49] , [50] . Next, EGF stimulation results in the second release of calcium (interval 2) [51] , [52] , and in delayed ERK activity (interval 3), with varying responses across cells. The onset of the ERK response overlaps somewhat with the addition of ionomycin, though this response is already initiated before addition in most cells, and has reached a plateau by the time at which the ionomycin-induced release of calcium becomes evident (Supplementary Figs. 6 and 8 ). Taken together, this suggests that ionomycin has a small or no effect on ERK activity. Visual inspection of the data suggested heterogeneity of the cellular responses (Fig. 6 a). We quantified the maximal response and timing of the biosensor responses within each interval (see “Methods” section, Supplementary Fig. 9 ), and found that k-means clustering could divide the cellular responses into three classes based on their maximal responses, visually shown in Fig. 6 b and confirmed using partial least squares discriminant analysis (Supplementary Fig. 10 ). The maximal response of the different sensors showed significant variations between the classes (Fig. 6 c–e). Fsk/IBMX stimulation appears to have a slightly stronger effect on the PKA activity in cells of class 2, though this does not appear to translate into a higher calcium transient, which was overall very variable. Most striking are the inter-class discrepancies in calcium (interval 2) and ERK (interval 3) signaling upon EGF stimulation. More specifically, cells within classes 1 and 2 respond strongly to EGF-mediated ERK activation, whereas the effect on class 3 cells is rather weak and occurs with an additional delay (Supplementary Fig. 9 h). Additionally, the sizes of the ERK responses do not appear to be linked to the increase of the preceding calcium release. These results show the potential of classifying cells based on the heterogeneity in their responses, although it is not immediately clear from the data where this heterogeneity originates from, or if the differences in the signaling activities of the three pathways are biologically correlated in the context of this experiment. Poor expression of EGF receptors [53] , [54] or Fsk/IBMX-induced cAMP-mediated effects [55] could render some cells less sensitive to EGF stimulation, while further heterogeneity may also be contributed by the high levels of variability associated with HEK293 cells [56] . Taken together, our data further support the notion that the simultaneous imaging of multiple biosensors can readily reveal heterogeneities even in well-established systems, while demonstrating that our scheme can be readily combined with spectrally-orthogonal biosensors to further expand the range of responses that can be measured. In conclusion, in this work, we set out to separate the contributions of two or more spectrally overlapping FRET pairs based on the photochromism of the donor fluorophores, by monitoring their response to irradiation with off-switching light. We developed a framework to perform the analysis and validated its applicability using numerical simulations. We find that our method can deliver accurate results already at low light levels, though the competition between the FRET process and the photochromism must be characterized for each of the FRET pairs. We then applied our methodology to FRET-based biosensors, many of which make use of cyan and yellow fluorescent proteins. We showed that the cyan fluorescent protein mTFP0.7 could be used to make a photochromic biosensor for PKA activity, which we named rsAKARev. We combined this sensor with the spectrally overlapping biosensor EKARev, a reporter for ERK kinase activity, and showed that the contributions of both sensors could be readily separated when expressed simultaneously in live cells. Finally, we expanded these experiments to include an additional reporter for Ca 2+ , and showed that this approach could detect cellular heterogeneity in the response to pharmacological stimulation. Our method further illustrates the potential of simultaneously acquiring multiple signaling responses to investigate complex biological networks. Our method is readily compatible with existing measurement or analysis pipelines, which may make use of the raw emission ratio, sensitized emission ratio, or the calculated FRET efficiency, provided that one characterizes the dependence between this metric and the photoswitching ratio. In practice, this will likely require a separate measurement for each FRET pair where the photoswitching ratio is monitored over varying emission ratios or FRET efficiencies. Such a measurement must only be performed once for each FRET pair and off-switching light dose. The method can be readily integrated with established instrumentation, requiring only the possibility of inserting an off- and on-switching irradiation in between two FRET acquisitions. In principle, our procedure can be used to separate the contributions of more than two FRET pairs, provided that the donors are sufficiently separated in terms of their photochromism. Our use of two biosensors also reflects the limited availability of photochromic cyan fluorescent proteins, though many more opportunities are available when considering also photochromic green and red fluorescent proteins, which would be directly compatible with our method. In summary, we have shown that photochromism of the donor can be leveraged to achieve FRET multiplexing using a simple linear unmixing scheme, on the condition that the responses of the FRET pairs are well characterized. Our strategy can be readily used as part of experiments that require multiplexed interaction sensing and biosensing.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Biosensor construction The expression plasmids were generated using a combination of polymerase chain reaction (PCR), restriction enzyme digestion, and Gibson Assembly molecular cloning techniques. For Gibson Assembly reactions, vector DNA was digested with restriction enzymes cutting the multiple cloning site. All other DNA fragments were generated by PCR with Q5-polymerase using primer sequences with an optimal annealing temperature of 72 °C, calculated using the NEB Tm calculator ( https://tmcalculator.neb.com ), and with minimal overlap of 20-basepairs between neighboring fragments (Supplementary Table 2 ). All fragments were then purified using gel extraction and subsequently combined in the Gibson Assembly reaction according to manufacturer instructions. rsAKARev was generated by replacing the cDNA for YPet and ECFP in AKAR3EV (pcDNA3.1+ vector) with those for mTFP0.7 and cpVenus172, respectively. For rsAKARev(T/A), PCR primers were modified to introduce a T/A mutation in the PKA substrate region. For the generation of EKARev(T/A), PCR primers were designed to introduce a T/A mutation in the ERK substrate region. Cell culture and transfection HeLa, COS-7 and HEK293T cells were cultured in Dulbecco’s modified Eagle medium (DMEM; Gibco, #31053028) supplemented with 10% (v/v) fetal bovine serum (FBS; Gibco, #70011036), 1× Glutamax (Gibco, #35050061) and 50 μg/ml gentamycin (GM; Sigma-Aldrich, #15750060) and maintained in a humidified incubator (37 °C, 5% CO 2 ). 24 h before transfection, cells were plated onto sterile 35-mm glass-bottom dishes (MatTek) and grown to 30–70% confluence. Cells were subsequently transfected 24–48 h before imaging using FuGENE6 transfection reagent (Promega) and plasmid DNA, according to the manufacturer’s protocol. For optimal transfection results, 90 μl DMEM containing 3 μl FuGENE6 was mixed into 10 μl ultrapure water containing a total of 1 μg DNA by pipetting thoroughly, and each mixture was left for at least 20 min before addition to each dish in dropwise fashion. For multi-biosensor experiments, the relative amounts of plasmid DNA were optimized to obtain comparable expression of all constructs. Cells were then grown for another 24h (HeLa) or 48h (HEK293T, COS-7) before imaging. Cells were finally switched to serum-reduced medium comprising DMEM (HEK293T, COS-7) or FluoroBrite DMEM (Gibco, #A1896702) (HeLa), supplemented with 0.5%(v/v) FBS, 1× Glutamax and 50 μg/ml GM 4-6h (HEK293T) or 16–20h (COS-7, HeLa) before imaging. Time-lapse fluorescence imaging Cell treatment Before imaging, 35-mm glass-bottom dishes were washed twice gently with phosphate-buffered saline (PBS; Gibco, #70011036) and then imaged in the dark in either calcium-free Hank’s balanced salt solution (HBSS; Gibco, #14185052) supplemented with 0.2% (w/v) bovine serum albumin (BSA; Gibco, #10270106) (HEK293T, COS-7) at room temperature, or in FluoroBrite DMEM supplemented with 0.5% (v/v) FBS and 1× Glutamax (HeLa) at 37 °C, 5% CO 2 . Cells were treated at indicated timepoints using a final concentration of 50 μM forskolin (Fsk) (Sigma-Aldrich, #93049), 100 μM 3-isobutyl-1-methylxanthine (IBMX) (Sigma-Aldrich, #410957), 1 μg/ml epidermal growth factor (EGF) (R&D Systems, #236-EG10), 10 μM ionomycin (Iono) (Sigma-Aldrich, #407950), 2 mM CaCl 2 (Chem-Lab Analytical, #CL05.0371), 1 μM phorbol-12-myristate-13-acetate (PMA) (Sigma-Aldrich, #P1585), 20 μM U0126 (Sigma-Aldrich, #662005) and 20 μM H-89 (Sigma-Aldrich, #B1427), by mixing a 5× concentrated solution, prepared in imaging buffer from freshly thawed DMSO (Fsk, IBMX, PMA, U0126, H-89) or HBSS (CaCl 2 ) stocks, into the imaging dish. Equipment Images were acquired on an Olympus IX-71 inverted microscope coupled to a Spectra X Light Engine (Lumencor), 10× UplanSApo objective (Olympus), ORCA-Flash4.0 LT camera (Hamamatsu), IX2-RFACA motorized fluorescence cube turret (Olympus), Lambda 10-B Optical Filter Changer (Sutter) and H117 high precision motorized stage (Prior). All hardware was controlled using in-house software. Experiments performed at 37 °C, 5% CO 2 were done using a stage-top incubator (H301-NIKON-TI-S-ER, OKOLAB) coupled to a digital CO2 controller unit (CO2-UNIT-BL, OKOLAB). Image acquisition Following the notation from the theory, the S D D , S D A , and S A A detection channels were acquired with a T455LP ( S D D , S D A ) or ZT514RDC ( S A A ) dichroic mirror and an ET480/40m ( S D D ) or ET545/40m ( S D A , S A A ) emission filter (all Chroma). Each sample was excited with blue light (438/29 nm, 35.6mW; S D D , S D A ) or teal light (513/22 nm, 2.45 mW; S A A ). RCaMP intensity was acquired using a ZT561rdc dichroic and HQ572lp emission filter (all Chroma) with excitation by green light (542/33 nm, 39.0 mW). All images were acquired during 50 ms camera exposure. Each acquisition was performed during constant time intervals every 10–30 s on multiple positions of the sample. Irradiation settings for photochromism were determined as the minimum light dose required to reach steady-state fluorescence during ON and OFF switching of mTFP0.7, which was achieved by irradiating 0.2 s with violet light (390/22 nm, 26.7 mW) and 0.2 s with blue light (438/29 nm, 78.35 mW), respectively. S D D and S D A fluorescence was acquired after each irradiation step; S A A fluorescence was acquired for only a subset of timepoints to minimize acceptor photobleaching. FRET analysis Analysis of the fluorescence time-series was performed using in-house analysis software running on Igor Pro 8 (WaveMetrics). For each sample position, multiple regions of interest (ROI) corresponding to single cells were defined, and the average intensity over the shape of the cell was determined at each timepoint. Subsequently, the intensity was corrected for background by subtracting the average signal of a region outside the cell boundaries. This correction can be improved by subtracting the signal observed for non-transfected cells instead, though we did not pursue this here as the emission from non-transfected cells was much smaller than both the background and the emission from transfected cells. The resulting average intensities were used for downstream analysis. For the conventional FRET analysis of single-sensor experiments, FRET efficiencies were subsequently calculated according to Eq. ( 6 ) using only the background-corrected fluorescence detected after each violet irradiation step in the measurement scheme shown in Fig. 1 b. The photoswitching analysis of single- and two-biosensor experiments was performed as detailed in the theory using Eqs. ( 11 )–( 14 ). The analysis procedure is described in more detail in Supplementary Note 3 . From single-sensor experiments, the dependencies ρ (Θ) were determined empirically by fitting a double-exponential function (rsAKARev) or a straight line (EKARev) on the pooled cell data represented in Fig. 4 , using orthogonal distance regression. Coefficients α , δ , and γ (Supplementary Table 1 ) were calculated based on published values for fluorescence quantum yields, instrument parameters, and the available spectral data on the fluorescent protein database FPbase [57] , as described in Supplementary Note 4 . Three-biosensor unmixing experiment The FRET efficiencies for rsAKARev and EKARev were obtained as described above. The raw RCaMP intensities were obtained by averaging the corresponding emission over the same regions of interest, and also included background removal by subtracting the signal from a region of interest that did not include any cells. We found that both the FRET and the RCaMP traces showed gradual decreases in amplitude, which we attribute to photobleaching of the fluorophores. We correct this using a baseline that is determined for every trace. For FRET measurements, the baseline is determined by fitting the trace with the function $${{\\\\Theta }}(t)={e}^{-t/\\\\tau }\\\\cdot \\\\left(b+\\\\frac{A}{1+\\\\exp \\\\left(\\\\frac{{t}_{0}-t}{r}\\\\right)}\\\\right)$$ (16) where the factor between parentheses is a standard sigmoidal function. The baseline is then given by b e − t / τ . τ is typically much larger than the duration of the experiment. For RCaMP measurements, we estimated the baseline used by fitting a bi-exponential function to the parts of the trace that do not show an elevated calcium signal. We then calculated the responses as $$\\\\% \\\\,\\\\,{\\\\text{Increase}}\\\\,(t)=\\\\frac{\\\\,{\\\\text{signal}}\\\\,(t)-\\\\,{\\\\text{baseline}}\\\\,(t)}{\\\\,{\\\\text{baseline}}\\\\,(t)}\\\\cdot 100 \\\\%$$ (17) where “signal” is the FRET efficiency for rsAKARev and EKARev, and the fluorescence intensity for RCaMP. The resulting responses are shown in Fig. 6 a, where we manually optimized the color scaling to highlight the cellular responses. We additionally show example traces, their corresponding baselines and the calculated responses in Supplementary Figs. 11 – 13 . The cell responses to pharmacological stimulation were quantified by taking the value of the maximum response observed within the time interval following stimulation. The onset time of the FRET biosensor responses was determined as the time for which a sigmoidal fit of the response increases to 7.6% of its maximum value, while the onset time of the RCaMP data was defined as the timepoint at which the second derivative of the response reached its maximum value. Partitional k -means clustering was performed in Matlab R2019b based on the responses determined for rsAKARev over time interval 1, for RCaMP over time interval 2, and for EKARev over time interval 3. To validate the classification results of the K -means clustering, partial least squares discriminant analysis (PLS-DA) was performed on the full-time trace data sets (% increase) assuming the three classes obtained by the K -means clustering. Subsequently, clustering results were cross-validated with the PLS-DA model using a 95% confidence interval. Additional information is available in Supplementary Note 5 . Statistics and reproducibility All statistical tests were performed in GraphPad Prism 5 (GraphPad Software). Data were checked for normality using a D’Agostino & Pearson omnibus test. Subsequently, pairwise comparison was performed using the F -test for equal variances, followed by a two-tailed unpaired Student’s t -test (if F -test, p > 0.05) or Welch’s unequal variance test (if F -test, p < 0.05) for Gaussian data, or using the Mann Whitney U test for non-gaussian data. The features shown in the images of Figs. 1 b, 3 a, and Supplementary Fig. 9 could be consistently reproduced across multiple experiments. 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 is to develop and validate a novel FRET-based method that exploits donor photochromism to enable the simultaneous measurement of multiple FRET pairs, even when the fluorophores exhibit complete spectral overlap, thereby overcoming limitations in current multiplexing strategies and allowing for the concurrent monitoring of multiple signaling activities in live cells."},"entities":{"kind":"list like","value":["FRET-based method","donor photochromism","multiple FRET pairs","fluorophores","spectral overlap","multiplexing strategies","signaling activities","live cells","method validation","photochromic donors"],"string":"[\n \"FRET-based method\",\n \"donor photochromism\",\n \"multiple FRET pairs\",\n \"fluorophores\",\n \"spectral overlap\",\n \"multiplexing strategies\",\n \"signaling activities\",\n \"live cells\",\n \"method validation\",\n \"photochromic donors\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1.","title":"Piston, D. W. & Kremers, G. J. Fluorescent protein FRET: the good, the bad and the ugly.Trends Biochem. Sci.32, 407–414 (2007).","link":"https://doi.org/10.1016%2Fj.tibs.2007.08.003"},{"idx":"2.","title":"Sahoo, H. Förster resonance energy transfer—a spectroscopic nanoruler: principle and applications.J. Photochem. Photobiol. C12, 20–30 (2011).","link":"https://doi.org/10.1016%2Fj.jphotochemrev.2011.05.001"},{"idx":"3.","title":"Newman, R. H., Fosbrink, M. D. & Zhang, J. Genetically encodable fluorescent biosensors for tracking signaling dynamics in living cells.Chem. Rev.111, 3614–3666 (2011).","link":"https://doi.org/10.1021%2Fcr100002u"},{"idx":"4.","title":"Hamers, D., van Voorst Vader, L., Borst, J. W. & Goedhart, J. Development of FRET biosensors for mammalian and plant systems.Protoplasma251, 333–347 (2014).","link":"https://link.springer.com/doi/10.1007/s00709-013-0590-z"},{"idx":"5.","title":"Greenwald, E. C., Mehta, S. & Zhang, J. Genetically encoded fluorescent biosensors illuminate the spatiotemporal regulation of signaling networks.Chem. Rev.118, 11707–11794 (2018).","link":"https://doi.org/10.1021%2Facs.chemrev.8b00333"},{"idx":"7.","title":"Gordon, G. W., Berry, G., Liang, X. H., Levine, B. & Herman, B. Quantitative fluorescence resonance energy transfer measurements using fluorescence microscopy.Biophys. J.74, 2702–2713 (1998).","link":"https://doi.org/10.1016%2FS0006-3495%2898%2977976-7"},{"idx":"8.","title":"Zeug, A., Woehler, A., Neher, E. & Ponimaskin, E. G. Quantitative intensity-based FRET approaches—a comparative snapshot.Biophys. J.103, 1821–1827 (2012).","link":"https://doi.org/10.1016%2Fj.bpj.2012.09.031"},{"idx":"9.","title":"Hellenkamp, B. et al. Precision and accuracy of single-molecule FRET measurements—a multi-laboratory benchmark study.Nat. Methods15, 669–676 (2018).","link":"https://doi.org/10.1038%2Fs41592-018-0085-0"},{"idx":"10.","title":"Coullomb, A. et al. QuanTI-FRET: a framework for quantitative FRET measurements in living cells.Sci. Rep.10, 1–11 (2020).","link":"https://doi.org/10.1038%2Fs41598-020-62924-w"},{"idx":"11.","title":"Kuchenov, D. et al. High-content imaging platform for profiling intracellular signaling network activity in living cells.Cell Chem. Biol.23, 1550–1559 (2016).","link":"https://doi.org/10.1016%2Fj.chembiol.2016.11.008"},{"idx":"12.","title":"Bunt, G. & Wouters, F. S. FRET from single to multiplexed signaling events.Biophys. Rev.9, 119–129 (2017).","link":"https://link.springer.com/doi/10.1007/s12551-017-0252-z"},{"idx":"13.","title":"Depry, C., Mehta, S. & Zhang, J. Multiplexed visualization of dynamic signaling networks using genetically encoded fluorescent protein-based biosensors.Pflügers Arch.465, 373–381 (2013).","link":"https://link.springer.com/doi/10.1007/s00424-012-1175-y"},{"idx":"14.","title":"Watrob, H. M., Pan, C.-P. & Barkley, M. D. Two-step FRET as a structural tool.J. Am. Chem. Soc.125, 7336–7343 (2003).","link":"https://doi.org/10.1021%2Fja034564p"},{"idx":"15.","title":"Ganesan, S., Ameer-beg, S. M., Ng, T. T. C., Vojnovic, B. & Wouters, F. S. A dark yellow fluorescent protein (YFP)-based resonance energy-accepting chromoprotein (REACh) for Forster resonance energy transfer with GFP.Proc. Natl Acad. Sci. USA103, 4089–4094 (2006).","link":"https://doi.org/10.1073%2Fpnas.0509922103"},{"idx":"16.","title":"Grant, D. M. et al. Multiplexed FRET to image multiple signaling events in live cells.Biophys. J.95, L69–L71 (2008).","link":"https://doi.org/10.1529%2Fbiophysj.108.139204"},{"idx":"17.","title":"Demeautis, C. et al. Multiplexing PKA and ERK1&2 kinases FRET biosensors in living cells using single excitation wavelength dual colour FLIM.Sci. Rep.7, 1–14 (2017).","link":"https://doi.org/10.1038%2Fsrep41026"},{"idx":"18.","title":"Ross, B. L. et al. Single-color, ratiometric biosensors for detecting signaling activities in live cells.eLife7, e35458 (2018).","link":"https://doi.org/10.7554%2FeLife.35458"},{"idx":"19.","title":"Piljic, A. & Schultz, C. Simultaneous recording of multiple cellular events by FRET.ACS Chem. Biol.3, 156–160 (2008).","link":"https://doi.org/10.1021%2Fcb700247q"},{"idx":"20.","title":"Dedecker, P., De Schryver, F. C. & Hofkens, J. Fluorescent proteins: shine on, you crazy diamond.J. Am. Chem. Soc.135, 2387–2402 (2013).","link":"https://doi.org/10.1021%2Fja309768d"},{"idx":"22.","title":"Yan, Y., Marriott, M. E., Petchprayoon, C. & Marriott, G. Optical switch probes and optical lock-in detection (OLID) imaging microscopy: high-contrast fluorescence imaging within living systems.Biochem. J.433, 411–422 (2011).","link":"https://doi.org/10.1042%2FBJ20100992"},{"idx":"23.","title":"Querard, J. et al. Photoswitching kinetics and phase-sensitive detection add discriminative dimensions for selective fluorescence imaging.Angew. Chem.54, 2633–2637 (2015).","link":"https://doi.org/10.1002%2Fanie.201408985"},{"idx":"24.","title":"Abbandonato, G., Storti, B., Signore, G., Beltram, F. & Bizzarri, R. Quantitative optical lock-in detection for quantitative imaging of switchable and non-switchable components: quantitative optical lock-in detection.Microsc. Res. Tech.79, 929–937 (2016).","link":"https://doi.org/10.1002%2Fjemt.22724"},{"idx":"25.","title":"Quérard, J. et al. Resonant out-of-phase fluorescence microscopy and remote imaging overcome spectral limitations.Nat. Commun.8, 969 (2017).","link":"https://doi.org/10.1038%2Fs41467-017-00847-3"},{"idx":"26.","title":"Duwé, S., Vandenberg, W. & Dedecker, P. Live-cell monochromatic dual-label sub-diffraction microscopy by mt-pcSOFI.Chem. 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Sci.32, 407–414 (2007).\",\n \"link\": \"https://doi.org/10.1016%2Fj.tibs.2007.08.003\"\n },\n {\n \"idx\": \"2.\",\n \"title\": \"Sahoo, H. Förster resonance energy transfer—a spectroscopic nanoruler: principle and applications.J. Photochem. Photobiol. C12, 20–30 (2011).\",\n \"link\": \"https://doi.org/10.1016%2Fj.jphotochemrev.2011.05.001\"\n },\n {\n \"idx\": \"3.\",\n \"title\": \"Newman, R. H., Fosbrink, M. D. & Zhang, J. Genetically encodable fluorescent biosensors for tracking signaling dynamics in living cells.Chem. Rev.111, 3614–3666 (2011).\",\n \"link\": \"https://doi.org/10.1021%2Fcr100002u\"\n },\n {\n \"idx\": \"4.\",\n \"title\": \"Hamers, D., van Voorst Vader, L., Borst, J. W. & Goedhart, J. Development of FRET biosensors for mammalian and plant systems.Protoplasma251, 333–347 (2014).\",\n \"link\": \"https://link.springer.com/doi/10.1007/s00709-013-0590-z\"\n },\n {\n \"idx\": \"5.\",\n \"title\": \"Greenwald, E. C., Mehta, S. & Zhang, J. Genetically encoded fluorescent biosensors illuminate the spatiotemporal regulation of signaling networks.Chem. Rev.118, 11707–11794 (2018).\",\n \"link\": \"https://doi.org/10.1021%2Facs.chemrev.8b00333\"\n },\n {\n \"idx\": \"7.\",\n \"title\": \"Gordon, G. W., Berry, G., Liang, X. H., Levine, B. & Herman, B. Quantitative fluorescence resonance energy transfer measurements using fluorescence microscopy.Biophys. J.74, 2702–2713 (1998).\",\n \"link\": \"https://doi.org/10.1016%2FS0006-3495%2898%2977976-7\"\n },\n {\n \"idx\": \"8.\",\n \"title\": \"Zeug, A., Woehler, A., Neher, E. & Ponimaskin, E. G. Quantitative intensity-based FRET approaches—a comparative snapshot.Biophys. J.103, 1821–1827 (2012).\",\n \"link\": \"https://doi.org/10.1016%2Fj.bpj.2012.09.031\"\n },\n {\n \"idx\": \"9.\",\n \"title\": \"Hellenkamp, B. et al. Precision and accuracy of single-molecule FRET measurements—a multi-laboratory benchmark study.Nat. Methods15, 669–676 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41592-018-0085-0\"\n },\n {\n \"idx\": \"10.\",\n \"title\": \"Coullomb, A. et al. QuanTI-FRET: a framework for quantitative FRET measurements in living cells.Sci. Rep.10, 1–11 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41598-020-62924-w\"\n },\n {\n \"idx\": \"11.\",\n \"title\": \"Kuchenov, D. et al. High-content imaging platform for profiling intracellular signaling network activity in living cells.Cell Chem. Biol.23, 1550–1559 (2016).\",\n \"link\": \"https://doi.org/10.1016%2Fj.chembiol.2016.11.008\"\n },\n {\n \"idx\": \"12.\",\n \"title\": \"Bunt, G. & Wouters, F. S. FRET from single to multiplexed signaling events.Biophys. Rev.9, 119–129 (2017).\",\n \"link\": \"https://link.springer.com/doi/10.1007/s12551-017-0252-z\"\n },\n {\n \"idx\": \"13.\",\n \"title\": \"Depry, C., Mehta, S. & Zhang, J. Multiplexed visualization of dynamic signaling networks using genetically encoded fluorescent protein-based biosensors.Pflügers Arch.465, 373–381 (2013).\",\n \"link\": \"https://link.springer.com/doi/10.1007/s00424-012-1175-y\"\n },\n {\n \"idx\": \"14.\",\n \"title\": \"Watrob, H. M., Pan, C.-P. & Barkley, M. D. Two-step FRET as a structural tool.J. Am. Chem. Soc.125, 7336–7343 (2003).\",\n \"link\": \"https://doi.org/10.1021%2Fja034564p\"\n },\n {\n \"idx\": \"15.\",\n \"title\": \"Ganesan, S., Ameer-beg, S. M., Ng, T. T. C., Vojnovic, B. & Wouters, F. S. A dark yellow fluorescent protein (YFP)-based resonance energy-accepting chromoprotein (REACh) for Forster resonance energy transfer with GFP.Proc. Natl Acad. Sci. USA103, 4089–4094 (2006).\",\n \"link\": \"https://doi.org/10.1073%2Fpnas.0509922103\"\n },\n {\n \"idx\": \"16.\",\n \"title\": \"Grant, D. M. et al. Multiplexed FRET to image multiple signaling events in live cells.Biophys. J.95, L69–L71 (2008).\",\n \"link\": \"https://doi.org/10.1529%2Fbiophysj.108.139204\"\n },\n {\n \"idx\": \"17.\",\n \"title\": \"Demeautis, C. et al. Multiplexing PKA and ERK1&2 kinases FRET biosensors in living cells using single excitation wavelength dual colour FLIM.Sci. Rep.7, 1–14 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fsrep41026\"\n },\n {\n \"idx\": \"18.\",\n \"title\": \"Ross, B. L. et al. Single-color, ratiometric biosensors for detecting signaling activities in live cells.eLife7, e35458 (2018).\",\n \"link\": \"https://doi.org/10.7554%2FeLife.35458\"\n },\n {\n \"idx\": \"19.\",\n \"title\": \"Piljic, A. & Schultz, C. Simultaneous recording of multiple cellular events by FRET.ACS Chem. Biol.3, 156–160 (2008).\",\n \"link\": \"https://doi.org/10.1021%2Fcb700247q\"\n },\n {\n \"idx\": \"20.\",\n \"title\": \"Dedecker, P., De Schryver, F. C. & Hofkens, J. Fluorescent proteins: shine on, you crazy diamond.J. Am. Chem. Soc.135, 2387–2402 (2013).\",\n \"link\": \"https://doi.org/10.1021%2Fja309768d\"\n },\n {\n \"idx\": \"22.\",\n \"title\": \"Yan, Y., Marriott, M. E., Petchprayoon, C. & Marriott, G. Optical switch probes and optical lock-in detection (OLID) imaging microscopy: high-contrast fluorescence imaging within living systems.Biochem. J.433, 411–422 (2011).\",\n \"link\": \"https://doi.org/10.1042%2FBJ20100992\"\n },\n {\n \"idx\": \"23.\",\n \"title\": \"Querard, J. et al. Photoswitching kinetics and phase-sensitive detection add discriminative dimensions for selective fluorescence imaging.Angew. Chem.54, 2633–2637 (2015).\",\n \"link\": \"https://doi.org/10.1002%2Fanie.201408985\"\n },\n {\n \"idx\": \"24.\",\n \"title\": \"Abbandonato, G., Storti, B., Signore, G., Beltram, F. & Bizzarri, R. Quantitative optical lock-in detection for quantitative imaging of switchable and non-switchable components: quantitative optical lock-in detection.Microsc. Res. Tech.79, 929–937 (2016).\",\n \"link\": \"https://doi.org/10.1002%2Fjemt.22724\"\n },\n {\n \"idx\": \"25.\",\n \"title\": \"Quérard, J. et al. Resonant out-of-phase fluorescence microscopy and remote imaging overcome spectral limitations.Nat. Commun.8, 969 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-017-00847-3\"\n },\n {\n \"idx\": \"26.\",\n \"title\": \"Duwé, S., Vandenberg, W. & Dedecker, P. Live-cell monochromatic dual-label sub-diffraction microscopy by mt-pcSOFI.Chem. Commun.53, 7242–7245 (2017).\",\n \"link\": \"https://doi.org/10.1039%2FC7CC02344H\"\n },\n {\n \"idx\": \"27.\",\n \"title\": \"Jares-Eruman, E. A., Song, L. & Jovin, T. M. Photochromism-Fret (phFRET): modulation of fluorescence resonance energy transfer by a photochromic acceptor.Mol. Cryst. Liquid Cryst. Sci. Technol. A298, 151–159 (1997).\",\n \"link\": \"https://doi.org/10.1080%2F10587259708036155\"\n },\n {\n \"idx\": \"28.\",\n \"title\": \"Mao, S. et al. Optical lock-in detection of FRET using synthetic and genetically encoded optical switches.Biophys. J.94, 4515–4524 (2008).\",\n \"link\": \"https://doi.org/10.1529%2Fbiophysj.107.124859\"\n },\n {\n \"idx\": \"29.\",\n \"title\": \"Uphoff, S. et al. Monitoring multiple distances within a single molecule using switchable FRET.Nat. Methods7, 831–836 (2010).\",\n \"link\": \"https://doi.org/10.1038%2Fnmeth.1502\"\n },\n {\n \"idx\": \"30.\",\n \"title\": \"Rainey, K. H. & Patterson, G. H. Photoswitching FRET to monitor protein–protein interactions.Proc. Natl Acad. Sci. USA116, 864–873 (2019).\",\n \"link\": \"https://doi.org/10.1073%2Fpnas.1805333116\"\n },\n {\n \"idx\": \"32.\",\n \"title\": \"Mo, G. C. H. et al. Genetically encoded biosensors for visualizing live-cell biochemical activity at super-resolution.Nat. Methods14, 427–434 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fnmeth.4221\"\n },\n {\n \"idx\": \"33.\",\n \"title\": \"Komatsu, N. et al. Development of an optimized backbone of FRET biosensors for kinases and GTPases.Mol. Biol. Cell22, 4647–56 (2011).\",\n \"link\": \"https://doi.org/10.1091%2Fmbc.e11-01-0072\"\n },\n {\n \"idx\": \"34.\",\n \"title\": \"Ai, H. W., Henderson, J. N., Remington, S. J. & Campbell, R. E. Directed evolution of a monomeric, bright and photostable version of Clavularia cyan fluorescent protein: structural characterization and applications in fluorescence imaging.Biochem. J.400, 531–540 (2006).\",\n \"link\": \"https://doi.org/10.1042%2FBJ20060874\"\n },\n {\n \"idx\": \"35.\",\n \"title\": \"Henderson, J. N., Ai, H. W., Campbell, R. E. & Remington, S. J. Structural basis for reversible photobleaching of a green fluorescent protein homologue.Proc. Natl Acad. Sci. USA104, 6672–6677 (2007).\",\n \"link\": \"https://doi.org/10.1073%2Fpnas.0700059104\"\n },\n {\n \"idx\": \"36.\",\n \"title\": \"Nagai, T., Sawano, A., Park, E. S. & Miyawaki, A. 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Recently, electrons were found to become spin polarized after transmitting through chiral molecules, crystals, and their hybrids. This phenomenon, called chirality-induced spin selectivity (CISS), presents broad application potentials and far-reaching fundamental implications involving intricate interplays among structural chirality, topological states, and electronic spin and orbitals. However, the microscopic picture of how chiral geometry influences electronic spin remains elusive, given the negligible spin-orbit coupling (SOC) in organic molecules. In this work, we address this issue via a direct comparison of magnetoconductance (MC) measurements on magnetic semiconductor-based chiral molecular spin valves with normal metal electrodes of contrasting SOC strengths. The experiment reveals that a heavy-metal electrode provides SOC to convert the orbital polarization induced by the chiral molecular structure to spin polarization. Our results illustrate the essential role of SOC in the metal electrode for the CISS spin valve effect. A tunneling model with a magnetochiral modulation of the potential barrier is shown to quantitatively account for the unusual transport behavior."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"Helical textures and monopole-like chirality in electronic structures of topological materials have given rise to a plethora of emergent phenomena characterized by unusual interplays between electronic charge, spin, and orbital [1] , [2] , [3] , [4] . More recently, a parallel phenomenon in real space, in which structural chirality induces electron spin polarization in the direction of their momentum, has received increasing attention [5] , [6] , [7] . The effect, termed chirality-induced spin selectivity (CISS), was first evidenced by Mott polarimetry of photoelectrons from a nonmagnetic (NM) Au electrode through a self-assembled monolayer (SAM) of short synthetic molecules of dsDNA [8] . Since then, CISS has been observed in a variety of chiral molecular systems including macro [8] , [9] , [10] , [11] , [12] , [13] and small [14] , [15] molecules, supramolecular polymers [16] , metal-organic frameworks [17] , and hybrid organic-inorganic perovskites [18] , [19] and artificial superlattices [20] , [21] , via a host of electrical, optical, and electrochemical probes [22] , [23] , [24] , [25] . More broadly, CISS is shown to effect enantio-selective chemical reactions [26] and facilitate enantiomer separation [27] , and the adsorption of chiral molecules on the surface of a conventional superconductor was reported to induce unconventional superconductivity [28] , [29] . All these experiments suggest a highly consequential interaction between molecular structural chirality and electronic spin, which carries profound and broad implications. Despite increasing preponderance of experimental results and a great deal of theoretical efforts, the microscopic origin and physical mechanisms behind CISS remain open questions [30] , [31] . A central unsettled issue is the role of spin-orbit coupling (SOC) in the chiral media. SOC is a necessary element in the emergence of spin polarization in NM materials in general. Specifically, it is an essential ingredient in most theoretical models of CISS, whereas the SOC in the molecular materials is too weak to account for the experimentally significant spin selectivity at room temperature [32] . In order to overcome this difficulty, a number of theoretical approaches were proposed, based primarily on spin dependent scattering and tight-binding models [33] , [34] , [35] , [36] , [37] , [38] , [39] , [40] , [41] . The approaches have targeted at amplification of SOC, either its value , to account for the experimentally observed CISS spin polarization by introducing various factors such as density of scattering centers [36] , dephasing [37] , [42] , and environmental nonunitary effects [41] , or its effect , through electron-electron correlation [43] , exchange interactions [44] , vibrational and polaronic effects [45] , [46] , frictional dissipation [47] , and Berry force [48] . An alternative approach rids of reliance on SOC in chiral molecules altogether [39] , [49] , [50] . Gersten et al. [39] introduced the concept of “induced spin filtering”: A selectivity in the transmission of the electron orbital angular momentum can induce spin selectivity in the transmission process, provided that there is strong SOC in the substrate supporting the chiral SAM. This proposal, however, was questioned because CISS was observed in photoemission experiments in which the substrates have negligible SOC [11] , [15] . Liu et al. [49] noted an important difference between the manifestations of the CISS effect in photoemission setups [8] , [15] and transport in molecular junctions: [51] , [52] The former measures the “global orbital angular momentum” that includes both the orbital and spin angular momenta, whereas the latter probes spin polarization exclusively. Physically, the model suggests that chiral molecules act as an orbital filter rather than a spin filter, and the SOC in the metal electrode converts the orbital polarization into spin polarization, thus producing CISS without the need for any SOC in the molecules (see the illustration in Fig. 1a ). The orbital polarization effect, which is caused by the orbital-momentum locking - an intrinsic topological property of electronic states in a chiral material [49] , [53] , has much broader relevance beyond CISS; in particular, it presents a new pathway for spin manipulation through atomic structure engineering. So far, however, definitive experimental evidence of the effect is lacking. Fig. 1: The chiral molecular spin valve and orbital to spin conversion mechanism. a Schematic depiction of the mechanism for orbital to spin polarization conversion in a (Ga,Mn)As/chiral molecule/NM spin valve. Chiral molecules induce orbital polarization in passing electrons and subsequently, the electrode SOC converts the orbital to spin (represented by the black vertical arrows on the top electrodes) [49] . b Schematic diagram of the device structure along with the junction measurement setup. c A top-view scanning electron microscopy image of the junction region (black circle in ( b )) in a molecular spin-valve device. Full size image One of the most widely used device platforms to detect and utilize spin polarization are spin valves. In a CISS spin valve, the spin polarization of a charge current from the NM electrode through the chiral SAM is analyzed by the magnetic electrode, and the junction conductance is expected to depend on the magnetization direction of the magnetic electrode, resulting in a MC. For CISS studies, a scanning probe rendition of the spin valve, magnetic conductive atomic force microscopy (mc-AFM), has been frequently used [51] . Although its implementation is relatively straightforward, mc-AFM relies on large number of averaging to mitigate the fluctuation and instability. In contrast, thin film-based molecular junctions, in which a chiral SAM is sandwiched between two conducting electrodes, are more conducive to stable and reproducible current-voltage (I-V) and MC measurements. Such devices, however, present significant technical challenges of their own: Pinholes are almost always present in SAMs at device scales (~ µm), and in the cases of two metal electrodes, any direct contact will short out the device. We recently demonstrated that these complications can be effectively mitigated by using a ferromagnetic semiconductor, (Ga,Mn)As, as the magnetic spin analyzer [52] . The use of the magnetic semiconductor was found to alleviate the shorting problem due to the presence of a Schottky barrier at direct contact with the Au electrode. Moreover, the (Ga,Mn)As was grown by molecular beam epitaxy (MBE) on an (In,Ga)As buffer layer, and the resulting strain from the lattice mismatch produces perpendicular magnetic anisotropy (PMA) [54] , namely an out-of-plane magnetization that is collinear with the spin polarization from CISS. These two unique device characteristics enabled observation of spin-valve MC distinctly associated with CISS, and the inherent stability of the platform facilitated a first rigorous determination of the bias current dependence of the MC from CISS [52] . Leveraging this proven device platform, we fabricated and characterized a deliberately chosen set of (Ga,Mn)As/SAM/NM hetero-junctions. The experiments yielded quantitative differentiation of the magnitude and bias-dependence of the spin valve conductance in junctions with NM electrodes of contrastingly different SOC strengths (Au versus Al) and SAMs of chiral and achiral molecules. The results revealed a definitive correlation between the magnitude of the CISS spin valve conductance and the SOC strength in the NM electrode: The molecular junctions with Au electrodes exhibit significant MC whose magnitudes depend distinctly on the chirality or length of the molecules; in contrast, in otherwise identical devices with Al electrodes, regardless of the molecule involved the MC are essentially indistinguishable from those of the control samples without any molecules. A model based on magnetochiral modulation of the tunneling barrier potential [55] from orbital polarization is shown to provide quantitative account for both the magnitude and bias dependence of the MC of the two types of junctions. The work unambiguously evidenced the essential role of the contact SOC in generating observable CISS effect in chiral molecular spin valves.","references":{"indices":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,54],"infos":[{"idx":"1","title":"Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators.Rev. Mod. Phys.82, 3045 (2010).","link":"https://doi.org/10.1103%2FRevModPhys.82.3045"},{"idx":"2","title":"Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors.Rev. Mod. Phys.83, 1057 (2011).","link":"https://doi.org/10.1103%2FRevModPhys.83.1057"},{"idx":"3","title":"Yan, B. & Felser, C. Topological materials: Weyl semimetals.Annu. Rev. Condens. Matter Phys.8, 337 (2017).","link":"https://doi.org/10.1146%2Fannurev-conmatphys-031016-025458"},{"idx":"4","title":"Armitage, N. P., Mele, E. J. & Vishwanath, A. Weyl and dirac semimetals in three-dimensional solids.Rev. Mod. 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B73, 195331 (2006).","link":"https://doi.org/10.1103%2FPhysRevB.73.195331"}]}},{"section":"Results and discussion","content":"Chiral molecular spin valve and orbital to spin conversion We detect the MC in chiral molecular spin valve devices with a (Ga,Mn)As magnetic electrode and Au or Al normal metal electrode. Figure 1a shows a schematic diagram depicting the molecular junction structure and the physical mechanism for the chirality-induced orbital polarization and subsequent orbital to spin polarization conversion due to the SOC in the NM electrode. Figure 1b is schematics of the device heterostructure and setups for the quasi-four-terminal I-V and conductance measurements. Figure 1c is an SEM image of a junction; the junctions were squares of sizes ranging from 5 \\(\\times\\) 5 µm 2 to 15 \\(\\times\\) 15 µm 2 . Molecular assembly, the formation of the SAM on (Ga,Mn)As, is a critical step in the device fabrication process. For this work, alpha-helix L -polyalanine (AHPA-L) and L -cysteine served to compare chiral molecules of different molecular lengths, whereas 16-mercaptohexadecanoic acid (MHA) and 1-octadecanethiol (ODT) were used as achiral molecules of similar length but with different terminal groups. The total molecular length of AHPA-L is 5.4 nm, whereas L -cysteine is around 4 Å, and those of MHA and ODT are 2.4 nm and 2.7 nm, respectively. All four molecules contain a thiol end group, which facilitates formation of high-quality SAMs on the (Ga,Mn)As [52] , [56] , [57] . The experimental details are described in “Methods” section. As described previously [52] , despite the probable presence of direct contacts between the NM and (Ga,Mn)As (parallel conduction) through defects in the SAM, the spin valve conductance due to CISS in these junctions can be identified from the difference in junction conductance, ΔG, under opposite saturation magnetization for the (Ga,Mn)As. Figure 2 a, b shows representative sets of MC measurements with varying perpendicular magnetic field for (Ga,Mn)As/AHPA-L/NM junctions with NM electrode of Au and Al, respectively, measured at various constant bias currents at T = 4.8 K. Each MC curve shows two distinct conducting states, coinciding with the well-defined square magnetic hysteresis of the (Ga,Mn)As due to its PMA, hence a ΔG can be precisely determined. As expected, the square hystereses of the MC resemble those of the anomalous Hall effect (AHE) of the (Ga,Mn)As which is directly proportional to M. Any significant contribution of AHE of the (Ga,Mn)As in the junction MC was ruled out in our previous work [52] . Here we present an updated list of justifications in Supplementary Information (Sec. 5). Fig. 2: CISS spin valve conductance: effect of the NM electrode. Representative MC curves measured at different bias currents for (Ga,Mn)As/AHPA-L/NM junctions, with NM of ( a ) Au, ( b ) Al. c The bias dependence of \\(\\Delta G\\) for the junctions with Au and Al contact. The solid squares are measured from I-V curves, whereas empty circles are measured from MC measurements at different bias currents. The left panels illustrate how the values of \\(\\Delta G\\) are extracted from MC and I-V measurements. Full size image The square hysteresis thus facilitates a straightforward and reliable determination of detailed bias current dependence of ΔG, from I-V curves measured under the opposite magnetization states of the (Ga,Mn)As, as shown in Fig. 2c . Specifically, the \\(\\Delta G\\) from the MC measurements can be obtained from and corroborated by the I-V’s as [52] $$\\varDelta G=I(\\frac{1}{{V}_{-M}}-\\frac{1}{{V}_{+M}}),$$ (1) where \\({V}_{+M}\\) and \\({V}_{-M}\\) indicate the corresponding bias voltage upon switching the magnetization in the (Ga,Mn)As from \\(+M\\) to \\(-M\\) at the same current \\(I\\) . Figure 2c shows ΔG as functions of bias current for the junction; as expected, the two types of measurements produced consistent results. The I-V data are presented in Supplementary Fig. 1 . The same measurement and analysis procedures were applied to all devices with different molecules and NM electrodes in this study to obtain ΔG and their bias current dependences. We note that the total junction conductance (G) for the Au junction is also much greater than that of the Al junction. Nevertheless, measurements on the large number of (Ga,Mn)As/AHPA-L/Au devices clearly demonstrated that although the I-V and total conductance may vary greatly in the molecular junctions of similar structures depending on the degree of parallel conduction (quality of the SAM assembly), both the magnitude and bias current dependence of the CISS spin valve conductance ( \\(\\Delta G\\) ) were found to be consistently similar. A detailed discussion and comparison with a Au junction shown in our previous work [52] are presented in Supplementary Information (Sec. 2). We conclude that the total G is spurious (depending the molecular coverage of the SAM) and has no bearing on CISS spin-valve conductance; it is the \\(\\Delta G\\) that accurately reflects the CISS effect. The fact that different Au junctions show large variations of the I-V and total G, but exhibit similar bias-dependent \\(\\Delta G\\) , lends further credence to our model and the associated analyses and conclusion. Effect of the NM electrode Figure 2c shows CISS spin valve conductance for two (Ga,Mn)As/AHPA-L/NM junctions with Au and Al as the NM electrode. The experiment constitutes a direct comparison of the magnitude and bias dependence of \\(\\Delta G\\) for two NM electrodes of contrasting SOC strengths. The most notably result here is the pronounced differences between the junctions with Au and Al electrodes. Figure 3 shows the results from a comparative experiment with AHPA-L replaced by the much shorter chiral molecule of L -cysteine. As expected, with the same Au electrode, \\(\\Delta G\\) for the L -cysteine junctions are significantly smaller than those of the AHPA-L counterparts [51] , [58] . Remarkably, the large difference between \\(\\Delta G\\) is also present for the L -cysteine junctions with Au and Al electrodes. We emphasize that for each combination of chiral SAM and NM electrode, the entire set of measurements was repeated in multiple samples (2 to 4), each with 4 junctions, and the results were consistent. The experiments, therefore, strongly indicate that the observed significant impact of the NM electrodes on the CISS spin valve effect originates from inherent differences of Au and Al, and is independent of the specific chiral molecule used. Fig. 3: CISS spin valve conductance: chiral molecules of different lengths. Bias current dependences of \\(\\Delta G\\) for four junctions of (Ga,Mn)As/AHPA-L ( L -cysteine)/Au (Al). Full size image We have also fabricated and measured large numbers of devices with Cu and Ag electrodes, two NM materials of intermediate SOC strengths between Au and Al. However, despite the repeated attempts, for either material, we were unable to obtain results with the degree of consistency achieved in Au and Al devices. Most Ag and Cu junctions yielded very small \\(\\Delta G\\) without the regular bias dependences. We surmise that this was due to poor interfaces or even damages to the molecular SAM by Cu and Ag. As shown in Supplementary Fig. 3 , one Cu junction yielded \\(\\Delta G\\) that fits well between those of the Au and Al junctions, however, it does not exhibit the bias-current dependence consistently seen in Au and Al junctions. For these reasons, we are unable to make a definitive statement regarding the CISS spin valve conductance and SOC strengths in Cu and Ag at this point. Chiral versus achiral molecules We further examined the spin valve effect in the molecular junctions with achiral molecules. Figure 4 shows the variation of the spin valve signal from chiral to achiral molecules in the molecular junctions with the same NM electrode of Au (Fig. 4a ) and Al (Fig. 4b ). In the Au junctions, there is a clear difference in the magnitude of the current-dependent \\(\\Delta G\\) between the chiral molecule (AHPA-L) and achiral molecules (MHA and ODT). It is important to note that despite the much-diminished magnitude, \\(\\Delta G\\) in the achiral molecular junctions are not trivial. Their magnitudes are clearly above the background in the control junction; in fact, they are comparable to \\(\\Delta G\\) in the short chiral molecule ( L -cysteine) junctions. Moreover, \\(\\Delta G\\) in the MHA and ODT junctions exhibit distinct bias dependences resembling those in the chiral molecular junctions. These observations are consistent with a prediction from the orbital polarization model that nontrivial spin transport can materialize even in non-helical and even achiral systems, because in the presence of time-reversal symmetry, the emergence of orbital texture requires only inversion symmetry breaking [49] , not necessarily chirality. Both MHA and ODT molecules have a thiol end and different terminal groups, thus are polar and possess inversion asymmetry. Here, the Al junctions again provide an illuminating comparison (Fig. 4b ): There are no discernible differences in the \\(\\Delta G\\) for the AHPA-L and achiral molecular junctions. In fact, as shown in Supplementary Fig. 5 , regardless of the molecule involved, the MC of all the Al junctions are comparable to that of the control junction without any molecules. In previous [52] and current work, we have conducted extensive measurements on large number of control samples; a representative set of results are shown in Supplementary Information (Sec. 4). It is clear that the MC in the control samples were close to the noise level and lacking any well-defined bias dependence, thus constituting a viable reference for meaningful comparison of the MC in various molecular junctions. Fig. 4: CISS spin valve conductance: chiral versus achiral molecules. Bias current dependences of \\(\\Delta G\\) for the molecular junctions of different chiral (AHPA-L and L -cysteine) and achiral (MHA and ODT) molecules with ( a ) Au, and ( b ) Al contact. The pink and blue squares are the chiral molecular junctions (AHPA-L and L -cysteine), whereas green and orange squares are from the molecular junctions with achiral molecules (MHA and ODT). Full size image Tunneling model and magnetochiral modulation of potential barrier Summarizing the key experimental observations, several robust features have been conclusively identified from our experiments: An optimal two-terminal chiral molecular spin valve, as exemplified by the (Ga,Mn)As/AHPA-L/Au junction, consistently exhibits significant MC, which increases linearly with the bias current at high biases but approaches a finite value at zero-bias. Contrary to expectation from the Onsager reciprocal relation [49] , [59] , [60] , the MC is symmetric (i.e. the sign of \\(\\Delta G\\) remains the same), rather than anti-symmetric, upon reversal of the current direction. Moreover, the magnitude of the MC decreases precipitously when AHPA-L is replaced by a much shorter chiral molecule or achiral molecules. Most importantly, replacing the Au electrode by Al results in even greater reduction of the MC, to levels where the differences between junctions of the different molecules are no longer discernible. Taken together, these results have revealed valuable new insights and placed several important constraints on a viable mechanism of CISS. The orbital polarization model [49] offers a natural account for the observed qualitative differences between the molecular junctions with Au and Al electrodes. More recently, incorporating orbital polarization, a tunneling model was proposed to describe the electronic transport in chiral molecular junctions [55] . The essential ideas are depicted in Fig. 5a : The molecular chirality and electrode magnetism modulates tunneling barrier through the insulating molecular junction, termed magnetochiral modulation, which originates from the magnetochiral anisotropy [55] . Physically, a current flow induces charge accumulation, which changes the tunneling barrier in a way that depends on the magnetization and molecular chirality. We demonstrate that this model [55] can provide a semi-quantitative self-consistent description of all the key observations in the following. Fig. 5: Barrier tunneling analysis of the CISS spin valve conductance. The bias dependent ΔG for the chiral molecular junctions can be fitted to a model of magnetochiral modulation of the tunneling barrier. a Schematic depiction of the tunneling model and barrier modulation mechanism. Red and black curves illustrate the modified tunneling barriers by \\(\\downarrow\\) and \\(\\uparrow\\) magnetizations, respectively, in the (Ga,Mn)As electrode. The original barrier is represented by the gray dashed curve. The bias voltage is indicated by V. b Fitting results of the spin valve conductance for two (Ga,Mn)As/AHPA-L/NM junctions with Au and Al contact. Full size image We incorporate the magnetochiral modulation into the Simmons model [61] , [62] of metal/insulator/metal tunneling junctions. The problem of an arbitrarily shaped potential barrier is modeled into that of a rectangular barrier, which results in an explicit expression for the \\(I(V)\\) . The Simmons model and its variants have been widely applied to the modeling of electron transport in molecular junctions [63] , [64] . Based on the Simmons expression and assuming a small magnetochiral modulation of the potential barrier, in the intermediate bias range, we approximate the magnetization-dependent differential conductance through a chiral molecular junction in the form of a simple exponential relation: $$\\frac{{dI}}{{dV}}=({\\alpha }_{o}+{\\alpha }_{M}){e}^{\\beta {V}_{M}}$$ (2) where \\({\\alpha }_{o}\\) and \\(\\beta\\) are magnetization-independent coefficients reflecting the tunneling current and probability across the unmodified potential barrier, and \\({\\alpha }_{M}\\) is the coefficient that quantifies the effect from the change of the potential barrier height upon switching of the magnetization of the (Ga,Mn)As, namely the CISS spin valve conductance \\(\\Delta G\\) . The modulation of the potential barrier is small and considered a perturbation, \\({\\alpha }_{M} \\, \\ll \\, {\\alpha }_{o}\\) . \\(\\Delta G\\) as a function of the bias current (Eq. ( 1 )) can then be evaluated as: $$\\Delta G\\left(I\\right)=\\beta I\\left[\\frac{1}{{{{{{\\rm{ln}}}}}}\\left(1+\\frac{\\beta I}{{\\alpha }_{0}+{\\alpha }_{-M}}\\right)}-\\frac{1}{{{{{{\\rm{ln}}}}}}\\left(1+\\frac{\\beta I}{{\\alpha }_{0}+{\\alpha }_{+M}}\\right)}\\right]$$ (3) Figure 5b shows the best fits of the \\(\\Delta G\\) for the two (Ga,Mn)As/AHPA-L/NM junctions (NM = Au, Al) to Eq. ( 3 ). More details of the fitting procedure are presented in Sec. 6 of the Supplementary Information. In brief, the fitting is performed separately for positive and negative currents. In a typical fitting process, an optimal value of \\(\\beta\\) is first identified. With β fixed, best fit to the data to Eq. 3 is then performed with \\({\\alpha }_{o},\\,{\\alpha }_{-M}\\,\\) and \\({\\alpha }_{+M}\\) as the fitting parameters. Table 1 lists the resulting values for the parameters from the best fits. Table 1 Fitting parameters for the tunneling model Full size table Two notable features are evident in Table 1 . First, \\({\\alpha }_{o} \\, \\gg \\, {\\alpha }_{\\mp M}\\) , consistent with our assumption that the magnetochiral modulation of tunneling barrier is small, but the CISS-induced spin valve conductance is large due to the exponential dependence of the junction conductance on the barrier height. Furthermore, the magnitudes of \\({\\alpha }_{\\mp M}\\) in the junction with Au contact is an order of magnitude greater than those with Al electrode, while the values of \\({\\alpha }_{o}\\) are similar in both junctions. The result constitutes quantitative support for the hypothesis that \\({\\alpha }_{M}\\) is magnetization dependent and its strength depends on the SOC of the NM electrode. In addition, it is evident that the fittings provide excellent description of the observed bias current dependences of \\(\\Delta G\\) , and naturally account for the fact that \\(\\Delta G\\) is essentially independent of current direction. The higher order asymmetry in the data is reflected in the different values of \\({\\alpha }_{o}\\) for positive and negative current. We note that slight asymmetries in the tunneling conductance are commonly observed and expected in the Simmons model [62] for junctions with dissimilar metallic electrodes and/or asymmetric potential barriers. The fitting results for other junctions, including those of achiral molecules, are described in Sec. 7 of the Supplementary Information. The results provide a quantitative measure of the different effects of the normal metal electrode (Au versus Al) consistent with the qualitative trends of \\(\\Delta G\\) apparent in Figs. 2 – 4 and Supplementary Fig. 5 : For the Au junctions \\({\\alpha }_{M}\\) are large and show a consistent decrease from AHPA-L to the achiral molecules; in contrast, all Al junctions show much smaller \\({\\alpha }_{M}\\) without any systematic difference depending on the chirality and length of the molecules. The observation of CISS MC in the spin valves of the achiral molecules and their magnitudes are well accounted for in our model. The molecular polarity (asymmetry) factors directly into the magnetochiral modulation of the tunneling barrier [55] . Physically, the magnetochiral modulation of the tunneling barrier of the insulating molecules can be understood as an analog of the electrical magnetochiral anisotropy (eMChA) in conductors [65] . It is therefore important to note that eMChA has been observed in both chiral and (nonchiral) polar conductors (see Fig. 3 in ref. [66] for a comprehensive list). Quantitatively, the eMChA is stronger in a chiral crystal with well-defined structural chirality [67] than in a nonchiral polar solid. Therefore, finite spin-valve conductance in junctions of achiral (polar) molecules are expected in our model and their magnitudes should be smaller relative to that in chiral molecular junctions. Further theoretical work is needed to relate the quantitative differences with the molecular structures and physical parameters of Au and Al. Finally, we comment on the presence of low-bias MC, most notably in the (Ga,Mn)As/AHPA-L/Au junctions. Experimentally, finite low-bias MC has been observed in a variety of two-terminal junctions of different chiral molecules and device structures [20] , [21] , [52] , [68] , [69] . On the other hand, theoretically, there is continuing debate as to whether and how this can be reconciled with the Onsager relation, which precludes a linear-response MC in the two-terminal chiral spin valves [59] , [60] , [70] , [71] . This topic was a focal point of our previous work [52] , here we describe a possible pathway out of the dilemma within the proposed model above. Because the model relies on bias current induced magnetochiral modulation of the tunneling barrier, any hysteresis in the charging/discharging process would yield a finite remnant (“linear-response”) spin valve conductance even at zero bias. We note that bias-induced conductance switching, and I-V hysteresis have been reported in a variety of molecular junctions of different configurations and molecules [72] , [73] , [74] , [75] , [76] . Since the conductance is exponentially sensitive to the tunneling barrier, the induced spin valve conductance can be considerable even for a small remnant modulation (splitting) of the potential barrier. In one of the devices we reported on previously (Supporting Fig. 5 , ref. [52] ), where particular care was taken to examine the effect of bias history, pronounced bias-induced changes in I-V and MC were observed. The results are shown in the current Supporting Information (Sec. 8). In this junction, for applied bias currents up to 800 μA, the junction showed negligible low-bias (linear-response) MC, whereas the application of a 1000 μA bias current induced substantial increases in MC over the full bias range; most notably, a significant zero-bias MC was now present (Supplementary Fig. 8 ). Concurrently, the I-V characteristics of the junction also saw significant changes due to the applied large bias (Supplementary Fig. 8b ). An interesting contrasting example was reported in ref. [77] , where CISS MC was measured in a long lateral two-terminal device. In such a device, the electron transport is likely to be diffusive instead of tunneling, and the zero-bias MC was vanishingly small. In summary, utilizing a robust device platform of magnetic semiconductor-based molecular junctions proven effective for CISS studies, we have obtained direct experimental evidence that the SOC in the NM electrode is essential to the emergence of the CISS spin valve effect. With a Au electrode, the precipitous decrease of the spin valve conductance from AHPA-L junctions to those of shorter chiral molecule and achiral molecules is readily discerned. Replacing the Au electrode with Al results in pronounced drops of the spin valve conductance for all molecules, to the degree that the differences between the molecules and with the control junctions are no longer discernible. A model based on orbital polarization from inversion-symmetry breaking and the resulting magnetochiral modulation of the tunneling barrier potential is shown to not only consistently account for all key aspects of the experimental results, but also provide resolution to several long-standing open issues in the field, including finite low-bias magnetoresistance due to CISS in two-terminal molecular spin valves, the spin valve conductance being symmetric upon current reversal, and its magnitude often being much greater than what is expected from the spin polarization of the ferromagnet. Our work reveals a close relation between chirality and electronic properties, in which structural chirality information is transferred and transformed from molecular geometry to electronic orbital and eventually to the electronic spin via SOC. The results thus provide useful guidelines for detecting chirality-induced phenomena and designing CISS devices."},{"section":"Methods","content":"Materials The AHPA-L in the experiments was obtained from RS Synthesis, LLC. L -cysteine, MHA, and ODT were acquired from Sigma-Aldrich, Inc. Molecules, except L -cysteine, were dissolved in pure ethanol at 1 mM concentration. L -cysteine was dissolved in deionized water at a concentration of 2.5 mM. The AHPA-L solution was kept at -18 o C for storage whereas L-cysteine, MHA and ODT solution were stored in ambient conditions. The (Ga,Mn)As with perpendicular magnetic anisotropy was grown by low-temperature molecular-beam epitaxy (LT-MBE). 500 nm thick (In,Ga)As buffer layer was first grown at 450 °C on semi-insulating (001) GaAs substrates. (Ga,Mn)As films of 40 nm thickness with perpendicular magnetic anisotropy were later grown at substrate temperature of 270 °C. The (Ga,Mn)As thin film has a Curie temperature of 80 K and coercive field of 174 Oe. Device fabrication process The junction devices are fabricated in the following steps: a. Junctions Defined by Electron-Beam Lithography (EBL) : The sample was spin-coated with 2% PMMA at 4 krpm for 30 s. It was prebaked at 180 °C for 10 min on a hot plate. The EBL was performed to draw the junction patterns of size ranging from 5 \\(\\times\\) 5 µm 2 to 15 \\(\\times\\) 15 µm 2 . The sample was developed in methyl isobutyl ketone (MIBK) diluted with isopropanol (1:3) for 40 s and then in pure isopropanol for 30 s at room temperature. b. Removal of oxide layer on (Ga,Mn)As and molecular assembly: The sample was cleaned with O 2 plasma to remove any organic residue. It was set with medium power at 200 mTorr oxygen pressure for 1 min. The sample was then baked at 180 °C for 20 min on the hot plate to harden the PMMA. To remove the native oxide layer on the (Ga,Mn)As, the sample was etched with an ion mill for 1 min 30 sec. It was immersed in ethanol immediately after being taken out from the ion mill chamber before the molecular assembly. For the assembly of molecular monolayer on the (Ga,Mn)As, the sample was left in the molecular solution at room temperature for 24 h. After the assembly, the sample was rinsed with ethanol and dried with nitrogen gas. The molecular assembly process is similar for both chiral and achiral molecules. c. Top NM Electrodes Deposition : A metallic shadow mask was used to draw the electrodes patterns which was aligned with the molecular junctions in a sample under optical microscope. Thermal evaporation was done to deposit NM electrodes. During the deposition of the top magnetic electrode, the substrate is cooled with liquid nitrogen and the temperature is maintained below –50 o C. 35 nm of Au (without Cr) and 50–70 nm of Al was deposited through a shadow mask. Electrical measurements The sample was fixed on a socket with a copper base and wired with silver paint and Pt wire. All the measurements were performed in an Oxford 3 He cryostat at the temperature range of 4.2–5.5 K. The measurement procedure is similar to previously reported [52] . Magnetic field perpendicular to the sample plane was applied up to 700 Oe. The (Ga,Mn)As was first magnetized at 700 Oe, and then the magnetic field was swept at a constant rate of 350 Oe/min for measurements. DC measurements were done with Keithley 2400 as the current source and HP 3458 as the voltmeter. AC measurements were performed with SR2124 dual-phase analog lock-in amplifiers."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"Helical textures and monopole-like chirality in electronic structures of topological materials have given rise to a plethora of emergent phenomena characterized by unusual interplays between electronic charge, spin, and orbital [1] , [2] , [3] , [4] . More recently, a parallel phenomenon in real space, in which structural chirality induces electron spin polarization in the direction of their momentum, has received increasing attention [5] , [6] , [7] . The effect, termed chirality-induced spin selectivity (CISS), was first evidenced by Mott polarimetry of photoelectrons from a nonmagnetic (NM) Au electrode through a self-assembled monolayer (SAM) of short synthetic molecules of dsDNA [8] . Since then, CISS has been observed in a variety of chiral molecular systems including macro [8] , [9] , [10] , [11] , [12] , [13] and small [14] , [15] molecules, supramolecular polymers [16] , metal-organic frameworks [17] , and hybrid organic-inorganic perovskites [18] , [19] and artificial superlattices [20] , [21] , via a host of electrical, optical, and electrochemical probes [22] , [23] , [24] , [25] . More broadly, CISS is shown to effect enantio-selective chemical reactions [26] and facilitate enantiomer separation [27] , and the adsorption of chiral molecules on the surface of a conventional superconductor was reported to induce unconventional superconductivity [28] , [29] . All these experiments suggest a highly consequential interaction between molecular structural chirality and electronic spin, which carries profound and broad implications. Despite increasing preponderance of experimental results and a great deal of theoretical efforts, the microscopic origin and physical mechanisms behind CISS remain open questions [30] , [31] . A central unsettled issue is the role of spin-orbit coupling (SOC) in the chiral media. SOC is a necessary element in the emergence of spin polarization in NM materials in general. Specifically, it is an essential ingredient in most theoretical models of CISS, whereas the SOC in the molecular materials is too weak to account for the experimentally significant spin selectivity at room temperature [32] . In order to overcome this difficulty, a number of theoretical approaches were proposed, based primarily on spin dependent scattering and tight-binding models [33] , [34] , [35] , [36] , [37] , [38] , [39] , [40] , [41] . The approaches have targeted at amplification of SOC, either its value , to account for the experimentally observed CISS spin polarization by introducing various factors such as density of scattering centers [36] , dephasing [37] , [42] , and environmental nonunitary effects [41] , or its effect , through electron-electron correlation [43] , exchange interactions [44] , vibrational and polaronic effects [45] , [46] , frictional dissipation [47] , and Berry force [48] . An alternative approach rids of reliance on SOC in chiral molecules altogether [39] , [49] , [50] . Gersten et al. [39] introduced the concept of “induced spin filtering”: A selectivity in the transmission of the electron orbital angular momentum can induce spin selectivity in the transmission process, provided that there is strong SOC in the substrate supporting the chiral SAM. This proposal, however, was questioned because CISS was observed in photoemission experiments in which the substrates have negligible SOC [11] , [15] . Liu et al. [49] noted an important difference between the manifestations of the CISS effect in photoemission setups [8] , [15] and transport in molecular junctions: [51] , [52] The former measures the “global orbital angular momentum” that includes both the orbital and spin angular momenta, whereas the latter probes spin polarization exclusively. Physically, the model suggests that chiral molecules act as an orbital filter rather than a spin filter, and the SOC in the metal electrode converts the orbital polarization into spin polarization, thus producing CISS without the need for any SOC in the molecules (see the illustration in Fig. 1a ). The orbital polarization effect, which is caused by the orbital-momentum locking - an intrinsic topological property of electronic states in a chiral material [49] , [53] , has much broader relevance beyond CISS; in particular, it presents a new pathway for spin manipulation through atomic structure engineering. So far, however, definitive experimental evidence of the effect is lacking. Fig. 1: The chiral molecular spin valve and orbital to spin conversion mechanism. a Schematic depiction of the mechanism for orbital to spin polarization conversion in a (Ga,Mn)As/chiral molecule/NM spin valve. Chiral molecules induce orbital polarization in passing electrons and subsequently, the electrode SOC converts the orbital to spin (represented by the black vertical arrows on the top electrodes) [49] . b Schematic diagram of the device structure along with the junction measurement setup. c A top-view scanning electron microscopy image of the junction region (black circle in ( b )) in a molecular spin-valve device. Full size image One of the most widely used device platforms to detect and utilize spin polarization are spin valves. In a CISS spin valve, the spin polarization of a charge current from the NM electrode through the chiral SAM is analyzed by the magnetic electrode, and the junction conductance is expected to depend on the magnetization direction of the magnetic electrode, resulting in a MC. For CISS studies, a scanning probe rendition of the spin valve, magnetic conductive atomic force microscopy (mc-AFM), has been frequently used [51] . Although its implementation is relatively straightforward, mc-AFM relies on large number of averaging to mitigate the fluctuation and instability. In contrast, thin film-based molecular junctions, in which a chiral SAM is sandwiched between two conducting electrodes, are more conducive to stable and reproducible current-voltage (I-V) and MC measurements. Such devices, however, present significant technical challenges of their own: Pinholes are almost always present in SAMs at device scales (~ µm), and in the cases of two metal electrodes, any direct contact will short out the device. We recently demonstrated that these complications can be effectively mitigated by using a ferromagnetic semiconductor, (Ga,Mn)As, as the magnetic spin analyzer [52] . The use of the magnetic semiconductor was found to alleviate the shorting problem due to the presence of a Schottky barrier at direct contact with the Au electrode. Moreover, the (Ga,Mn)As was grown by molecular beam epitaxy (MBE) on an (In,Ga)As buffer layer, and the resulting strain from the lattice mismatch produces perpendicular magnetic anisotropy (PMA) [54] , namely an out-of-plane magnetization that is collinear with the spin polarization from CISS. These two unique device characteristics enabled observation of spin-valve MC distinctly associated with CISS, and the inherent stability of the platform facilitated a first rigorous determination of the bias current dependence of the MC from CISS [52] . Leveraging this proven device platform, we fabricated and characterized a deliberately chosen set of (Ga,Mn)As/SAM/NM hetero-junctions. The experiments yielded quantitative differentiation of the magnitude and bias-dependence of the spin valve conductance in junctions with NM electrodes of contrastingly different SOC strengths (Au versus Al) and SAMs of chiral and achiral molecules. The results revealed a definitive correlation between the magnitude of the CISS spin valve conductance and the SOC strength in the NM electrode: The molecular junctions with Au electrodes exhibit significant MC whose magnitudes depend distinctly on the chirality or length of the molecules; in contrast, in otherwise identical devices with Al electrodes, regardless of the molecule involved the MC are essentially indistinguishable from those of the control samples without any molecules. A model based on magnetochiral modulation of the tunneling barrier potential [55] from orbital polarization is shown to provide quantitative account for both the magnitude and bias dependence of the MC of the two types of junctions. 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Figure 1a shows a schematic diagram depicting the molecular junction structure and the physical mechanism for the chirality-induced orbital polarization and subsequent orbital to spin polarization conversion due to the SOC in the NM electrode. Figure 1b is schematics of the device heterostructure and setups for the quasi-four-terminal I-V and conductance measurements. Figure 1c is an SEM image of a junction; the junctions were squares of sizes ranging from 5 \\\\(\\\\times\\\\) 5 µm 2 to 15 \\\\(\\\\times\\\\) 15 µm 2 . Molecular assembly, the formation of the SAM on (Ga,Mn)As, is a critical step in the device fabrication process. For this work, alpha-helix L -polyalanine (AHPA-L) and L -cysteine served to compare chiral molecules of different molecular lengths, whereas 16-mercaptohexadecanoic acid (MHA) and 1-octadecanethiol (ODT) were used as achiral molecules of similar length but with different terminal groups. The total molecular length of AHPA-L is 5.4 nm, whereas L -cysteine is around 4 Å, and those of MHA and ODT are 2.4 nm and 2.7 nm, respectively. All four molecules contain a thiol end group, which facilitates formation of high-quality SAMs on the (Ga,Mn)As [52] , [56] , [57] . The experimental details are described in “Methods” section. As described previously [52] , despite the probable presence of direct contacts between the NM and (Ga,Mn)As (parallel conduction) through defects in the SAM, the spin valve conductance due to CISS in these junctions can be identified from the difference in junction conductance, ΔG, under opposite saturation magnetization for the (Ga,Mn)As. Figure 2 a, b shows representative sets of MC measurements with varying perpendicular magnetic field for (Ga,Mn)As/AHPA-L/NM junctions with NM electrode of Au and Al, respectively, measured at various constant bias currents at T = 4.8 K. Each MC curve shows two distinct conducting states, coinciding with the well-defined square magnetic hysteresis of the (Ga,Mn)As due to its PMA, hence a ΔG can be precisely determined. As expected, the square hystereses of the MC resemble those of the anomalous Hall effect (AHE) of the (Ga,Mn)As which is directly proportional to M. Any significant contribution of AHE of the (Ga,Mn)As in the junction MC was ruled out in our previous work [52] . Here we present an updated list of justifications in Supplementary Information (Sec. 5). Fig. 2: CISS spin valve conductance: effect of the NM electrode. Representative MC curves measured at different bias currents for (Ga,Mn)As/AHPA-L/NM junctions, with NM of ( a ) Au, ( b ) Al. c The bias dependence of \\\\(\\\\Delta G\\\\) for the junctions with Au and Al contact. The solid squares are measured from I-V curves, whereas empty circles are measured from MC measurements at different bias currents. The left panels illustrate how the values of \\\\(\\\\Delta G\\\\) are extracted from MC and I-V measurements. Full size image The square hysteresis thus facilitates a straightforward and reliable determination of detailed bias current dependence of ΔG, from I-V curves measured under the opposite magnetization states of the (Ga,Mn)As, as shown in Fig. 2c . Specifically, the \\\\(\\\\Delta G\\\\) from the MC measurements can be obtained from and corroborated by the I-V’s as [52] $$\\\\varDelta G=I(\\\\frac{1}{{V}_{-M}}-\\\\frac{1}{{V}_{+M}}),$$ (1) where \\\\({V}_{+M}\\\\) and \\\\({V}_{-M}\\\\) indicate the corresponding bias voltage upon switching the magnetization in the (Ga,Mn)As from \\\\(+M\\\\) to \\\\(-M\\\\) at the same current \\\\(I\\\\) . Figure 2c shows ΔG as functions of bias current for the junction; as expected, the two types of measurements produced consistent results. The I-V data are presented in Supplementary Fig. 1 . The same measurement and analysis procedures were applied to all devices with different molecules and NM electrodes in this study to obtain ΔG and their bias current dependences. We note that the total junction conductance (G) for the Au junction is also much greater than that of the Al junction. Nevertheless, measurements on the large number of (Ga,Mn)As/AHPA-L/Au devices clearly demonstrated that although the I-V and total conductance may vary greatly in the molecular junctions of similar structures depending on the degree of parallel conduction (quality of the SAM assembly), both the magnitude and bias current dependence of the CISS spin valve conductance ( \\\\(\\\\Delta G\\\\) ) were found to be consistently similar. A detailed discussion and comparison with a Au junction shown in our previous work [52] are presented in Supplementary Information (Sec. 2). We conclude that the total G is spurious (depending the molecular coverage of the SAM) and has no bearing on CISS spin-valve conductance; it is the \\\\(\\\\Delta G\\\\) that accurately reflects the CISS effect. The fact that different Au junctions show large variations of the I-V and total G, but exhibit similar bias-dependent \\\\(\\\\Delta G\\\\) , lends further credence to our model and the associated analyses and conclusion. Effect of the NM electrode Figure 2c shows CISS spin valve conductance for two (Ga,Mn)As/AHPA-L/NM junctions with Au and Al as the NM electrode. The experiment constitutes a direct comparison of the magnitude and bias dependence of \\\\(\\\\Delta G\\\\) for two NM electrodes of contrasting SOC strengths. The most notably result here is the pronounced differences between the junctions with Au and Al electrodes. Figure 3 shows the results from a comparative experiment with AHPA-L replaced by the much shorter chiral molecule of L -cysteine. As expected, with the same Au electrode, \\\\(\\\\Delta G\\\\) for the L -cysteine junctions are significantly smaller than those of the AHPA-L counterparts [51] , [58] . Remarkably, the large difference between \\\\(\\\\Delta G\\\\) is also present for the L -cysteine junctions with Au and Al electrodes. We emphasize that for each combination of chiral SAM and NM electrode, the entire set of measurements was repeated in multiple samples (2 to 4), each with 4 junctions, and the results were consistent. The experiments, therefore, strongly indicate that the observed significant impact of the NM electrodes on the CISS spin valve effect originates from inherent differences of Au and Al, and is independent of the specific chiral molecule used. Fig. 3: CISS spin valve conductance: chiral molecules of different lengths. Bias current dependences of \\\\(\\\\Delta G\\\\) for four junctions of (Ga,Mn)As/AHPA-L ( L -cysteine)/Au (Al). Full size image We have also fabricated and measured large numbers of devices with Cu and Ag electrodes, two NM materials of intermediate SOC strengths between Au and Al. However, despite the repeated attempts, for either material, we were unable to obtain results with the degree of consistency achieved in Au and Al devices. Most Ag and Cu junctions yielded very small \\\\(\\\\Delta G\\\\) without the regular bias dependences. We surmise that this was due to poor interfaces or even damages to the molecular SAM by Cu and Ag. As shown in Supplementary Fig. 3 , one Cu junction yielded \\\\(\\\\Delta G\\\\) that fits well between those of the Au and Al junctions, however, it does not exhibit the bias-current dependence consistently seen in Au and Al junctions. For these reasons, we are unable to make a definitive statement regarding the CISS spin valve conductance and SOC strengths in Cu and Ag at this point. Chiral versus achiral molecules We further examined the spin valve effect in the molecular junctions with achiral molecules. Figure 4 shows the variation of the spin valve signal from chiral to achiral molecules in the molecular junctions with the same NM electrode of Au (Fig. 4a ) and Al (Fig. 4b ). In the Au junctions, there is a clear difference in the magnitude of the current-dependent \\\\(\\\\Delta G\\\\) between the chiral molecule (AHPA-L) and achiral molecules (MHA and ODT). It is important to note that despite the much-diminished magnitude, \\\\(\\\\Delta G\\\\) in the achiral molecular junctions are not trivial. Their magnitudes are clearly above the background in the control junction; in fact, they are comparable to \\\\(\\\\Delta G\\\\) in the short chiral molecule ( L -cysteine) junctions. Moreover, \\\\(\\\\Delta G\\\\) in the MHA and ODT junctions exhibit distinct bias dependences resembling those in the chiral molecular junctions. These observations are consistent with a prediction from the orbital polarization model that nontrivial spin transport can materialize even in non-helical and even achiral systems, because in the presence of time-reversal symmetry, the emergence of orbital texture requires only inversion symmetry breaking [49] , not necessarily chirality. Both MHA and ODT molecules have a thiol end and different terminal groups, thus are polar and possess inversion asymmetry. Here, the Al junctions again provide an illuminating comparison (Fig. 4b ): There are no discernible differences in the \\\\(\\\\Delta G\\\\) for the AHPA-L and achiral molecular junctions. In fact, as shown in Supplementary Fig. 5 , regardless of the molecule involved, the MC of all the Al junctions are comparable to that of the control junction without any molecules. In previous [52] and current work, we have conducted extensive measurements on large number of control samples; a representative set of results are shown in Supplementary Information (Sec. 4). It is clear that the MC in the control samples were close to the noise level and lacking any well-defined bias dependence, thus constituting a viable reference for meaningful comparison of the MC in various molecular junctions. Fig. 4: CISS spin valve conductance: chiral versus achiral molecules. Bias current dependences of \\\\(\\\\Delta G\\\\) for the molecular junctions of different chiral (AHPA-L and L -cysteine) and achiral (MHA and ODT) molecules with ( a ) Au, and ( b ) Al contact. The pink and blue squares are the chiral molecular junctions (AHPA-L and L -cysteine), whereas green and orange squares are from the molecular junctions with achiral molecules (MHA and ODT). Full size image Tunneling model and magnetochiral modulation of potential barrier Summarizing the key experimental observations, several robust features have been conclusively identified from our experiments: An optimal two-terminal chiral molecular spin valve, as exemplified by the (Ga,Mn)As/AHPA-L/Au junction, consistently exhibits significant MC, which increases linearly with the bias current at high biases but approaches a finite value at zero-bias. Contrary to expectation from the Onsager reciprocal relation [49] , [59] , [60] , the MC is symmetric (i.e. the sign of \\\\(\\\\Delta G\\\\) remains the same), rather than anti-symmetric, upon reversal of the current direction. Moreover, the magnitude of the MC decreases precipitously when AHPA-L is replaced by a much shorter chiral molecule or achiral molecules. Most importantly, replacing the Au electrode by Al results in even greater reduction of the MC, to levels where the differences between junctions of the different molecules are no longer discernible. Taken together, these results have revealed valuable new insights and placed several important constraints on a viable mechanism of CISS. The orbital polarization model [49] offers a natural account for the observed qualitative differences between the molecular junctions with Au and Al electrodes. More recently, incorporating orbital polarization, a tunneling model was proposed to describe the electronic transport in chiral molecular junctions [55] . The essential ideas are depicted in Fig. 5a : The molecular chirality and electrode magnetism modulates tunneling barrier through the insulating molecular junction, termed magnetochiral modulation, which originates from the magnetochiral anisotropy [55] . Physically, a current flow induces charge accumulation, which changes the tunneling barrier in a way that depends on the magnetization and molecular chirality. We demonstrate that this model [55] can provide a semi-quantitative self-consistent description of all the key observations in the following. Fig. 5: Barrier tunneling analysis of the CISS spin valve conductance. The bias dependent ΔG for the chiral molecular junctions can be fitted to a model of magnetochiral modulation of the tunneling barrier. a Schematic depiction of the tunneling model and barrier modulation mechanism. Red and black curves illustrate the modified tunneling barriers by \\\\(\\\\downarrow\\\\) and \\\\(\\\\uparrow\\\\) magnetizations, respectively, in the (Ga,Mn)As electrode. The original barrier is represented by the gray dashed curve. The bias voltage is indicated by V. b Fitting results of the spin valve conductance for two (Ga,Mn)As/AHPA-L/NM junctions with Au and Al contact. Full size image We incorporate the magnetochiral modulation into the Simmons model [61] , [62] of metal/insulator/metal tunneling junctions. The problem of an arbitrarily shaped potential barrier is modeled into that of a rectangular barrier, which results in an explicit expression for the \\\\(I(V)\\\\) . The Simmons model and its variants have been widely applied to the modeling of electron transport in molecular junctions [63] , [64] . Based on the Simmons expression and assuming a small magnetochiral modulation of the potential barrier, in the intermediate bias range, we approximate the magnetization-dependent differential conductance through a chiral molecular junction in the form of a simple exponential relation: $$\\\\frac{{dI}}{{dV}}=({\\\\alpha }_{o}+{\\\\alpha }_{M}){e}^{\\\\beta {V}_{M}}$$ (2) where \\\\({\\\\alpha }_{o}\\\\) and \\\\(\\\\beta\\\\) are magnetization-independent coefficients reflecting the tunneling current and probability across the unmodified potential barrier, and \\\\({\\\\alpha }_{M}\\\\) is the coefficient that quantifies the effect from the change of the potential barrier height upon switching of the magnetization of the (Ga,Mn)As, namely the CISS spin valve conductance \\\\(\\\\Delta G\\\\) . The modulation of the potential barrier is small and considered a perturbation, \\\\({\\\\alpha }_{M} \\\\, \\\\ll \\\\, {\\\\alpha }_{o}\\\\) . \\\\(\\\\Delta G\\\\) as a function of the bias current (Eq. ( 1 )) can then be evaluated as: $$\\\\Delta G\\\\left(I\\\\right)=\\\\beta I\\\\left[\\\\frac{1}{{{{{{\\\\rm{ln}}}}}}\\\\left(1+\\\\frac{\\\\beta I}{{\\\\alpha }_{0}+{\\\\alpha }_{-M}}\\\\right)}-\\\\frac{1}{{{{{{\\\\rm{ln}}}}}}\\\\left(1+\\\\frac{\\\\beta I}{{\\\\alpha }_{0}+{\\\\alpha }_{+M}}\\\\right)}\\\\right]$$ (3) Figure 5b shows the best fits of the \\\\(\\\\Delta G\\\\) for the two (Ga,Mn)As/AHPA-L/NM junctions (NM = Au, Al) to Eq. ( 3 ). More details of the fitting procedure are presented in Sec. 6 of the Supplementary Information. In brief, the fitting is performed separately for positive and negative currents. In a typical fitting process, an optimal value of \\\\(\\\\beta\\\\) is first identified. With β fixed, best fit to the data to Eq. 3 is then performed with \\\\({\\\\alpha }_{o},\\\\,{\\\\alpha }_{-M}\\\\,\\\\) and \\\\({\\\\alpha }_{+M}\\\\) as the fitting parameters. Table 1 lists the resulting values for the parameters from the best fits. Table 1 Fitting parameters for the tunneling model Full size table Two notable features are evident in Table 1 . First, \\\\({\\\\alpha }_{o} \\\\, \\\\gg \\\\, {\\\\alpha }_{\\\\mp M}\\\\) , consistent with our assumption that the magnetochiral modulation of tunneling barrier is small, but the CISS-induced spin valve conductance is large due to the exponential dependence of the junction conductance on the barrier height. Furthermore, the magnitudes of \\\\({\\\\alpha }_{\\\\mp M}\\\\) in the junction with Au contact is an order of magnitude greater than those with Al electrode, while the values of \\\\({\\\\alpha }_{o}\\\\) are similar in both junctions. The result constitutes quantitative support for the hypothesis that \\\\({\\\\alpha }_{M}\\\\) is magnetization dependent and its strength depends on the SOC of the NM electrode. In addition, it is evident that the fittings provide excellent description of the observed bias current dependences of \\\\(\\\\Delta G\\\\) , and naturally account for the fact that \\\\(\\\\Delta G\\\\) is essentially independent of current direction. The higher order asymmetry in the data is reflected in the different values of \\\\({\\\\alpha }_{o}\\\\) for positive and negative current. We note that slight asymmetries in the tunneling conductance are commonly observed and expected in the Simmons model [62] for junctions with dissimilar metallic electrodes and/or asymmetric potential barriers. The fitting results for other junctions, including those of achiral molecules, are described in Sec. 7 of the Supplementary Information. The results provide a quantitative measure of the different effects of the normal metal electrode (Au versus Al) consistent with the qualitative trends of \\\\(\\\\Delta G\\\\) apparent in Figs. 2 – 4 and Supplementary Fig. 5 : For the Au junctions \\\\({\\\\alpha }_{M}\\\\) are large and show a consistent decrease from AHPA-L to the achiral molecules; in contrast, all Al junctions show much smaller \\\\({\\\\alpha }_{M}\\\\) without any systematic difference depending on the chirality and length of the molecules. The observation of CISS MC in the spin valves of the achiral molecules and their magnitudes are well accounted for in our model. The molecular polarity (asymmetry) factors directly into the magnetochiral modulation of the tunneling barrier [55] . Physically, the magnetochiral modulation of the tunneling barrier of the insulating molecules can be understood as an analog of the electrical magnetochiral anisotropy (eMChA) in conductors [65] . It is therefore important to note that eMChA has been observed in both chiral and (nonchiral) polar conductors (see Fig. 3 in ref. [66] for a comprehensive list). Quantitatively, the eMChA is stronger in a chiral crystal with well-defined structural chirality [67] than in a nonchiral polar solid. Therefore, finite spin-valve conductance in junctions of achiral (polar) molecules are expected in our model and their magnitudes should be smaller relative to that in chiral molecular junctions. Further theoretical work is needed to relate the quantitative differences with the molecular structures and physical parameters of Au and Al. Finally, we comment on the presence of low-bias MC, most notably in the (Ga,Mn)As/AHPA-L/Au junctions. Experimentally, finite low-bias MC has been observed in a variety of two-terminal junctions of different chiral molecules and device structures [20] , [21] , [52] , [68] , [69] . On the other hand, theoretically, there is continuing debate as to whether and how this can be reconciled with the Onsager relation, which precludes a linear-response MC in the two-terminal chiral spin valves [59] , [60] , [70] , [71] . This topic was a focal point of our previous work [52] , here we describe a possible pathway out of the dilemma within the proposed model above. Because the model relies on bias current induced magnetochiral modulation of the tunneling barrier, any hysteresis in the charging/discharging process would yield a finite remnant (“linear-response”) spin valve conductance even at zero bias. We note that bias-induced conductance switching, and I-V hysteresis have been reported in a variety of molecular junctions of different configurations and molecules [72] , [73] , [74] , [75] , [76] . Since the conductance is exponentially sensitive to the tunneling barrier, the induced spin valve conductance can be considerable even for a small remnant modulation (splitting) of the potential barrier. In one of the devices we reported on previously (Supporting Fig. 5 , ref. [52] ), where particular care was taken to examine the effect of bias history, pronounced bias-induced changes in I-V and MC were observed. The results are shown in the current Supporting Information (Sec. 8). In this junction, for applied bias currents up to 800 μA, the junction showed negligible low-bias (linear-response) MC, whereas the application of a 1000 μA bias current induced substantial increases in MC over the full bias range; most notably, a significant zero-bias MC was now present (Supplementary Fig. 8 ). Concurrently, the I-V characteristics of the junction also saw significant changes due to the applied large bias (Supplementary Fig. 8b ). An interesting contrasting example was reported in ref. [77] , where CISS MC was measured in a long lateral two-terminal device. In such a device, the electron transport is likely to be diffusive instead of tunneling, and the zero-bias MC was vanishingly small. In summary, utilizing a robust device platform of magnetic semiconductor-based molecular junctions proven effective for CISS studies, we have obtained direct experimental evidence that the SOC in the NM electrode is essential to the emergence of the CISS spin valve effect. With a Au electrode, the precipitous decrease of the spin valve conductance from AHPA-L junctions to those of shorter chiral molecule and achiral molecules is readily discerned. Replacing the Au electrode with Al results in pronounced drops of the spin valve conductance for all molecules, to the degree that the differences between the molecules and with the control junctions are no longer discernible. A model based on orbital polarization from inversion-symmetry breaking and the resulting magnetochiral modulation of the tunneling barrier potential is shown to not only consistently account for all key aspects of the experimental results, but also provide resolution to several long-standing open issues in the field, including finite low-bias magnetoresistance due to CISS in two-terminal molecular spin valves, the spin valve conductance being symmetric upon current reversal, and its magnitude often being much greater than what is expected from the spin polarization of the ferromagnet. Our work reveals a close relation between chirality and electronic properties, in which structural chirality information is transferred and transformed from molecular geometry to electronic orbital and eventually to the electronic spin via SOC. The results thus provide useful guidelines for detecting chirality-induced phenomena and designing CISS devices.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Materials The AHPA-L in the experiments was obtained from RS Synthesis, LLC. L -cysteine, MHA, and ODT were acquired from Sigma-Aldrich, Inc. Molecules, except L -cysteine, were dissolved in pure ethanol at 1 mM concentration. L -cysteine was dissolved in deionized water at a concentration of 2.5 mM. The AHPA-L solution was kept at -18 o C for storage whereas L-cysteine, MHA and ODT solution were stored in ambient conditions. The (Ga,Mn)As with perpendicular magnetic anisotropy was grown by low-temperature molecular-beam epitaxy (LT-MBE). 500 nm thick (In,Ga)As buffer layer was first grown at 450 °C on semi-insulating (001) GaAs substrates. (Ga,Mn)As films of 40 nm thickness with perpendicular magnetic anisotropy were later grown at substrate temperature of 270 °C. The (Ga,Mn)As thin film has a Curie temperature of 80 K and coercive field of 174 Oe. Device fabrication process The junction devices are fabricated in the following steps: a. Junctions Defined by Electron-Beam Lithography (EBL) : The sample was spin-coated with 2% PMMA at 4 krpm for 30 s. It was prebaked at 180 °C for 10 min on a hot plate. The EBL was performed to draw the junction patterns of size ranging from 5 \\\\(\\\\times\\\\) 5 µm 2 to 15 \\\\(\\\\times\\\\) 15 µm 2 . The sample was developed in methyl isobutyl ketone (MIBK) diluted with isopropanol (1:3) for 40 s and then in pure isopropanol for 30 s at room temperature. b. Removal of oxide layer on (Ga,Mn)As and molecular assembly: The sample was cleaned with O 2 plasma to remove any organic residue. It was set with medium power at 200 mTorr oxygen pressure for 1 min. The sample was then baked at 180 °C for 20 min on the hot plate to harden the PMMA. To remove the native oxide layer on the (Ga,Mn)As, the sample was etched with an ion mill for 1 min 30 sec. It was immersed in ethanol immediately after being taken out from the ion mill chamber before the molecular assembly. For the assembly of molecular monolayer on the (Ga,Mn)As, the sample was left in the molecular solution at room temperature for 24 h. After the assembly, the sample was rinsed with ethanol and dried with nitrogen gas. The molecular assembly process is similar for both chiral and achiral molecules. c. Top NM Electrodes Deposition : A metallic shadow mask was used to draw the electrodes patterns which was aligned with the molecular junctions in a sample under optical microscope. Thermal evaporation was done to deposit NM electrodes. During the deposition of the top magnetic electrode, the substrate is cooled with liquid nitrogen and the temperature is maintained below –50 o C. 35 nm of Au (without Cr) and 50–70 nm of Al was deposited through a shadow mask. Electrical measurements The sample was fixed on a socket with a copper base and wired with silver paint and Pt wire. All the measurements were performed in an Oxford 3 He cryostat at the temperature range of 4.2–5.5 K. The measurement procedure is similar to previously reported [52] . Magnetic field perpendicular to the sample plane was applied up to 700 Oe. The (Ga,Mn)As was first magnetized at 700 Oe, and then the magnetic field was swept at a constant rate of 350 Oe/min for measurements. DC measurements were done with Keithley 2400 as the current source and HP 3458 as the voltmeter. AC measurements were performed with SR2124 dual-phase analog lock-in amplifiers.\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"The core research idea is to investigate the mechanism of chirality-induced spin selectivity (CISS) in chiral molecular spin valves by demonstrating that orbital polarization, induced by the chiral molecular structure and converted into spin polarization via the spin-orbit coupling (SOC) of the nonmagnetic electrode, is responsible for the observed magnetoresistance, with the magnitude of the effect being directly correlated with the SOC strength of the electrode material."},"entities":{"kind":"list like","value":["chirality-induced spin selectivity","chiral molecular spin valves","orbital polarization","chiral molecular structure","spin polarization","spin-orbit coupling","nonmagnetic electrode","magnetoresistance","SOC strength","electrode material"],"string":"[\n \"chirality-induced spin selectivity\",\n \"chiral molecular spin valves\",\n \"orbital polarization\",\n \"chiral molecular structure\",\n \"spin polarization\",\n \"spin-orbit coupling\",\n \"nonmagnetic electrode\",\n \"magnetoresistance\",\n \"SOC strength\",\n \"electrode material\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1.","title":"Hasan, M. 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Vibration-enhanced spin-selective transport of electrons in the DNA double helix.Phys. Rev. B102, 035431 (2020).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.102.035431\"\n },\n {\n \"idx\": \"46.\",\n \"title\": \"Zhang, L., Hao, Y., Qin, W., Xie, S. & Qu, F. Chiral-induced spin selectivity: A polaron transport model.Phys. Rev. B102, 214303 (2020).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.102.214303\"\n },\n {\n \"idx\": \"47.\",\n \"title\": \"Volosniev, A. G. et al. Interplay between friction and spin-orbit coupling as a source of spin polarization.Phys. Rev. B104, 024430 (2021).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.104.024430\"\n },\n {\n \"idx\": \"48.\",\n \"title\": \"Wu, Y. & Subotnik, J. E. Electronic spin separation induced by nuclear motion near conical intersections.Nat. Commun.12, 700 (2021).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-020-20831-8\"\n },\n {\n \"idx\": \"49.\",\n \"title\": \"Liu, Y., Xiao, J., Koo, J. & Yan, B. 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B73, 195331 (2006).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.73.195331\"\n },\n {\n \"idx\": \"56.\",\n \"title\": \"Wang, X. et al. Robust manipulation of magnetism in dilute magnetic semiconductor (Ga,Mn)As by organic molecules.Adv. Mater.27, 8043 (2015).\",\n \"link\": \"https://doi.org/10.1002%2Fadma.201503547\"\n },\n {\n \"idx\": \"57.\",\n \"title\": \"Liu, T. et al. Molecular patterning and directed self-assembly of gold nanoparticles on GaAs.ACS Appl. Mater. Interfaces9, 43363 (2017).\",\n \"link\": \"https://doi.org/10.1021%2Facsami.7b14113\"\n },\n {\n \"idx\": \"58.\",\n \"title\": \"Mishra, S. et al. Length-dependent electron spin polarization in oligopeptides and DNA.J. Phys. Chem. C.124, 10776 (2020).\",\n \"link\": \"https://doi.org/10.1021%2Facs.jpcc.0c02291\"\n },\n {\n \"idx\": \"59.\",\n \"title\": \"Dalum, S. & Hedegård, P. Theory of chiral induced spin selectivity.Nano Lett.19, 5253 (2019).\",\n \"link\": \"https://doi.org/10.1021%2Facs.nanolett.9b01707\"\n },\n {\n \"idx\": \"60.\",\n \"title\": \"Yang, X., van der Wal, C. H. & van Wees, B. J. Detecting chirality in two-terminal electronic nanodevices.Nano Lett.20, 6148 (2020).\",\n \"link\": \"https://doi.org/10.1021%2Facs.nanolett.0c02417\"\n },\n {\n \"idx\": \"61.\",\n \"title\": \"Simmons, J. G. Generalized formula for the electric tunnel effect between similar electrodes separated by a thin insulating film.J. Appl. Phys.34, 1793 (1963).\",\n \"link\": \"https://doi.org/10.1063%2F1.1702682\"\n },\n {\n \"idx\": \"62.\",\n \"title\": \"Simmons, J. G. Electric tunnel effect between dissimilar electrodes separated by a thin insulating film.J. Appl. Phys.34, 2581 (1963).\",\n \"link\": \"https://doi.org/10.1063%2F1.1729774\"\n },\n {\n \"idx\": \"63.\",\n \"title\": \"Vilan, A. Analyzing molecular current-voltage characteristics with the Simmons tunneling model: Scaling and linearization.J. Phys. Chem. C.111, 4431 (2007).\",\n \"link\": \"https://doi.org/10.1021%2Fjp066846s\"\n },\n {\n \"idx\": \"64.\",\n \"title\": \"Akkerman, H. B. et al. Electron tunneling through alkanedithiol self-assembled monolayers in large-area molecular junctions.Proc. Natl Acad. Sci.104, 11161 (2007).\",\n \"link\": \"https://doi.org/10.1073%2Fpnas.0701472104\"\n },\n {\n \"idx\": \"65.\",\n \"title\": \"Rikken, G. L. J. A., Fölling, J. & Wyder, P. Electrical magnetochiral anisotropy.Phys. Rev. Lett.87, 236602 (2001).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.87.236602\"\n },\n {\n \"idx\": \"66.\",\n \"title\": \"Tokura, Y. & Nagaosa, N. Nonreciprocal responses from non-centrosymmetric quantum materials.Nat. Commun.9, 3740 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-018-05759-4\"\n },\n {\n \"idx\": \"67.\",\n \"title\": \"Rikken, G. L. J. A. & Avarvari, N. Strong electrical magnetochiral anisotropy in tellurium.Phys. Rev. B99, 245153 (2019).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.99.245153\"\n },\n {\n \"idx\": \"68.\",\n \"title\": \"Kulkarni, C. et al. Highly efficient and tunable filtering of electrons’ spin by supramolecular chirality of nanofiber-based materials.Adv. Mater.32, 1904965 (2020).\",\n \"link\": \"https://doi.org/10.1002%2Fadma.201904965\"\n },\n {\n \"idx\": \"69.\",\n \"title\": \"Yang, C. et al. Real-time monitoring of reaction stereochemistry through single-molecule observations of chirality-induced spin selectivity.Nat. Chem.15, 972–979 (2023).\",\n \"link\": \"https://doi.org/10.1038%2Fs41557-023-01212-2\"\n },\n {\n \"idx\": \"70.\",\n \"title\": \"Yang, X., van der Wal, C. H. & van Wees, B. J. Spin-dependent electron transmission model for chiral molecules in mesoscopic devices.Phys. Rev. B99, 024418 (2019).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.99.024418\"\n },\n {\n \"idx\": \"71.\",\n \"title\": \"Naaman, R. & Waldeck, D. H. Comment on “Spin-Dependent Electron Transmission Model for Chiral Molecules in Mesoscopic Devices,”.Phys. Rev. B101, 026403 (2020).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.101.026403\"\n },\n {\n \"idx\": \"72.\",\n \"title\": \"He, J. et al. Metal-free silicon-molecule-nanotube testbed and memory device.Nat. Mater.5, 63 (2006).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat1526\"\n },\n {\n \"idx\": \"73.\",\n \"title\": \"Nijhuis, C. A., Reus, W. F. & Whitesides, G. M. Molecular rectification in Metal-SAM-metal oxide-metal junctions.J. Am. Chem. Soc.131, 17814 (2009).\",\n \"link\": \"https://doi.org/10.1021%2Fja9048898\"\n },\n {\n \"idx\": \"74.\",\n \"title\": \"Schwarz, F. et al. Field-induced conductance switching by charge-state alternation in organometallic single-molecule junctions.Nat. Nanotechnol.11, 170 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fnnano.2015.255\"\n },\n {\n \"idx\": \"75.\",\n \"title\": \"Park, J. et al. Controlled hysteresis of conductance in molecular tunneling junctions.ACS Nano16, 4206 (2022).\",\n \"link\": \"https://doi.org/10.1021%2Facsnano.1c10155\"\n },\n {\n \"idx\": \"76.\",\n \"title\": \"Kastlunger, G. & Stadler, R. Bias-induced conductance switching in single molecule junctions containing a redox-active transition metal complex.Monatshefte F.ür. Chem. - Chem. Monthly147, 1675 (2016).\",\n \"link\": \"https://link.springer.com/doi/10.1007/s00706-016-1795-6\"\n },\n {\n \"idx\": \"77.\",\n \"title\": \"Hossain, M. A. et al. Transverse magnetoconductance in two-terminal chiral spin-selective devices.Nanoscale Horiz.8, 320 (2023).\",\n \"link\": \"https://doi.org/10.1039%2FD2NH00502F\"\n }\n]","isConversational":false}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":111,"numItemsPerPage":100,"numTotalItems":11341,"offset":11100,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6IjVHZDBvd0g5MTM2eDZjc1FvbE1zcktNWUZoRVFoUm5PVVVybEpjOUhaUEEifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc4NTc1NjM1NiwianRpIjoiMDM2ZDM0YTgtYjA4Mi00ZTdiLWI4OWUtNmUyZTlmMGNhZDYwIiwic3ViIjoiL2RhdGFzZXRzL1hpYW8tWW91dGgvTkNfUGh5c2ljcyIsImV4cCI6MTc4NTc1OTk1NiwiaXNzIjoiaHR0cHM6Ly9odWdnaW5nZmFjZS5jbyJ9.7nLxUt2sUi8EY3ocbY2KsuxX3BEPqh2p8ZVCUWLN_JRkRzffCU-DEOliHEpikTC1rDKixzltoKEAaETAuzD6Bg","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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s41467-020-16934-x | Materials science | Phonon-exciton Interactions in WSe2under a quantizing magnetic field | Strong many-body interaction in two-dimensional transitional metal dichalcogenides provides a unique platform to study the interplay between different quasiparticles, such as prominent phonon replica emission and modified valley-selection rules. A large out-of-plane magnetic field is expected to modify the exciton-phon... | [
{
"section": "Introduction",
"content": "Phonons, quasiparticles describing the collective vibrations of lattice, can strongly interact with excitons in semiconductor nanostructures [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] and significantly influence the light emission and energy relaxation [2] ,... | This research explores the phonon–exciton interaction in monolayer WSe₂ under Landau quantization, aiming to probe the unique selection rules of inter-Landau level transitions, directly observe the Landau quantization of individual charge carriers through phonon replica photoluminescence, and extract the distinct elect... | [
"phonon–exciton interaction",
"monolayer WSe₂",
"Landau quantization",
"inter-Landau level transitions",
"phonon replica photoluminescence",
"electron mass",
"hole mass",
"valley-Zeeman splitting",
"out-of-plane magnetic field",
"selection rules",
"charge carriers",
"individual charge carriers... | [
{
"idx": "1.",
"title": "Li, Z. et al. Momentum-dark intervalley exciton in monolayer tungsten diselenide brightened via chiral phonon.ACS Nano13, 14107–14113 (2019).",
"link": "https://doi.org/10.1021%2Facsnano.9b06682"
},
{
"idx": "2.",
"title": "Zhu, H. et al. Observation of chiral phonon... |
s41467-024-48544-2 | Optics and photonics | Electrically empowered microcomb laser | Optical microcomb underpins a wide range of applications from communication, metrology, to sensing. Although extensively explored in recent years, challenges remain in key aspects of microcomb such as complex soliton initialization, low power efficiency, and limited comb reconfigurability. Here we present an on-chip mi... | [
{
"section": "Introduction",
"content": "Optical frequency comb is a coherent light source that consists of many highly coherent single-frequency laser lines equally spaced in the frequency domain. Its development has revolutionized many fields including metrology, spectroscopy, and clock [1] . In recent ye... | The core research idea is to develop a fundamentally new approach to on-chip optical frequency comb generation by integrating a III-V gain element with a thin-film lithium-niobate photonic integrated circuit, leveraging resonantly enhanced electro-optic modulation, Kerr nonlinearity, and embedded optical gain within a ... | [
"on-chip optical frequency comb generation",
"III-V gain element",
"thin-film lithium-niobate photonic integrated circuit",
"resonantly enhanced electro-optic modulation",
"Kerr nonlinearity",
"embedded optical gain",
"single laser cavity",
"self-starting soliton microcomb generation",
"highly coher... | [
{
"idx": "1.",
"title": "Diddams, S. A., Vahala, K. & Udem, T. Optical frequency combs: coherently uniting the electromagnetic spectrum.Science369, eaay3676 (2020).",
"link": "https://doi.org/10.1126%2Fscience.aay3676"
},
{
"idx": "2.",
"title": "Kippenberg, T. J., Gaeta, A. L., Lipson, M. &... |
s41467-021-20965-3 | Nanoscience and technology | Surface-bound reactive oxygen species generating nanozymes for selective antibacterial action | Acting by producing reactive oxygen species (ROS) in situ, nanozymes are promising as antimicrobials. ROS’ intrinsic inability to distinguish bacteria from mammalian cells, however, deprives nanozymes of the selectivity necessary for an ideal antimicrobial. Here we report that nanozymes that generate surface-bound ROS ... | [
{
"section": "Introduction",
"content": "Antimicrobial resistance (AMR) poses a global threat to public health [1] . Depending on whether the resistance is genetically encoded or purely phenotypic, AMR can be inherited or non-inherited [2] . Inherited AMR results from mutations in existing genes or acquisit... | The core research idea is to develop nanozymes that generate surface-bound reactive oxygen species (ROS) to selectively kill bacteria over mammalian cells, leveraging the surface-bound nature of ROS and the endocytosis process in mammalian cells as an antidote mechanism, thereby enabling efficient elimination of antibi... | [
"nanozymes",
"surface-bound reactive oxygen species",
"ROS",
"endocytosis process",
"mammalian cells",
"antibiotic-resistant bacteria",
"biofilm formation",
"host toxicity",
"genetically encoded antimicrobial resistance",
"phenotypic antimicrobial resistance"
] | [
{
"idx": "2.",
"title": "Levin, B. R. & Rozen, D. E. Opinion—non-inherited antibiotic resistance.Nat. Rev. Microbiol.4, 556–562 (2006).",
"link": "https://doi.org/10.1038%2Fnrmicro1445"
},
{
"idx": "3.",
"title": "Willyard, C. Drug-resistant bacteria ranked.Nature543, 15–15 (2017).",
"li... |
ncomms2188 | Nanoscience and technology | Giant Raman gain in silicon nanocrystals | Nanostructured silicon has generated a lot of interest in the past decades as a key material for silicon-based photonics. The low absorption coefficient makes silicon nanocrystals attractive as an active medium in waveguide structures, and their third-order nonlinear optical properties are crucial for the development o... | [
{
"section": "Introduction",
"content": "Nonlinear silicon photonics is a technology for implementing various optical functionalities [1] . Silicon offers a variety of nonlinear effects that can be used to process optical signals at speeds of 100 Gbit s −1 and beyond [2] , [3] and to enable broadband electr... | The core research idea is to investigate stimulated Raman scattering in silicon nanocrystals embedded in a silica matrix for use in a Raman amplifier that achieves significantly enhanced Raman gain—up to four orders of magnitude greater than crystalline silicon—by leveraging the nonlinear optical properties of low-dime... | [
"stimulated Raman scattering",
"silicon nanocrystals",
"silica matrix",
"Raman amplifier",
"Raman gain",
"crystalline silicon",
"nonlinear optical properties",
"low-dimensional silicon",
"high-performance optical signal processing",
"compact optical signal processing",
"CMOS technology"
] | [
{
"idx": "1",
"title": "Leuthold J., Koos C., Freude W. Nonlinear silicon photonics.Nature Photon.4, 535–544 (2010).",
"link": "https://doi.org/10.1038%2Fnphoton.2010.185"
},
{
"idx": "2",
"title": "Koos C. et al. All-optical high-speed signal processing with silicon–organic hybrid slot wave... |
s41467-024-48634-1 | Materials science | Tunnel junctions based on interfacial two dimensional ferroelectrics | Van der Waals heterostructures have opened new opportunities to develop atomically thin (opto)electronic devices with a wide range of functionalities. The recent focus on manipulating the interlayer twist angle has led to the observation of out-of-plane room temperature ferroelectricity in twisted rhombohedral bilayers... | [
{
"section": "Introduction",
"content": "Ferroelectrics (FE) are a class of materials with the ability to maintain a spontaneous electric polarisation that can be switched by an external electric field. Due to this, ferroelectric materials have become the key element in devices for a broad range of electron... | The core research idea is to investigate the polarization-dependent tunneling electro-resistance in ferroelectric tunnel junctions incorporating transition metal dichalcogenide-based sliding ferroelectric interfaces, aiming to demonstrate ambipolar switching behavior and elucidate its strong dependence on the local dom... | [
"polarization-dependent tunneling electro-resistance",
"ferroelectric tunnel junctions",
"transition metal dichalcogenide",
"sliding ferroelectric interfaces",
"ambipolar switching behavior",
"local domain structure",
"reconstructed domains",
"complex network",
"domain structure",
"tunneling elect... | [
{
"idx": "1.",
"title": "Martin, L. W. & Rappe, A. M. Thin-film ferroelectric materials and their applications.Nat. Rev. Mater.2, 1–14 (2016).",
"link": "https://doi.org/10.1038%2Fnatrevmats.2016.87"
},
{
"idx": "3.",
"title": "Soni, R. et al. Giant electrode effect on tunnelling electroresi... |
s41467-018-03485-5 | Physics | Magnon detection using a ferroic collinear multilayer spin valve | Information transport and processing by pure magnonic spin currents in insulators is a promising alternative to conventional charge-current-driven spintronic devices. The absence of Joule heating and reduced spin wave damping in insulating ferromagnets have been suggested for implementing efficient logic devices. After... | [
{
"section": "Introduction",
"content": "Almost three decades ago, the discovery of the giant magnetoresistance effect (GMR) [1] , [2] marked a key moment for the research field of spintronics based on spin-polarized charge currents. The inclusion of the spin degree of freedom in information technology prom... | The core research idea is to demonstrate the magnetization orientation-dependent detection of magnonic spin currents in YIG|CoO|Co spin valve structures, where a pure magnonic spin current generated via ferromagnetic resonance spin pumping in YIG is transmitted through CoO and detected via the inverse spin Hall effect ... | [
"magnetization orientation-dependent detection",
"magnonic spin currents",
"YIG|CoO|Co spin valve structures",
"ferromagnetic resonance spin pumping",
"inverse spin Hall effect",
"spin-dependent inverse spin Hall effect",
"anomalous Hall effect-induced spin rectification",
"magnonic contribution",
"... | [
{
"idx": "1.",
"title": "Baibich, M. et al. Giant magnetoresistance of (001) Fe/(001) Cr magnetic superlattices.Phys. Rev. Lett.61, 2472 (1988).",
"link": "https://doi.org/10.1103%2FPhysRevLett.61.2472"
},
{
"idx": "2.",
"title": "Binasch, G., Grünberg, P., Saurenbach, F. & Zinn, W. Enhanced... |
s41467-021-22350-6 | Materials science | Discovery of\({\hat{\boldsymbol{C}}}_2\)rotation anomaly in topological crystalline insulator SrPb | Topological crystalline insulators (TCIs) are insulating electronic states with nontrivial topology protected by crystalline symmetries. Recently, theory has proposed new classes of TCIs protected by rotation symmetries \(\hat C_n\) , which have surface rotation anomaly evading the fermion doubling theorem, i.e., n ins... | [
{
"section": "Introduction",
"content": "Since the advent of time-reversal symmetry \\(\\hat T\\) protected topological insulators characterized by topological \\({\\Bbb Z}_2\\) invariants [1] , [2] , a generalization of the topology concept to the band insulators is topological crystalline insulators (TCIs... | The core research idea is to investigate and experimentally confirm the existence of $\hat C_2$ rotation anomaly in the binary compound SrPb by identifying two Dirac-cone surface states on its (010) surface, which are protected by the combination of time-reversal symmetry $\hat T$ and $\hat C_{2y}$ rotational symmetry. | [
"$\\hat C_2$ rotation anomaly",
"binary compound SrPb",
"Dirac-cone surface states",
"(010) surface",
"time-reversal symmetry $\\hat T$",
"$\\hat C_{2y}$ rotational symmetry",
"experimental confirmation",
"surface state identification",
"symmetry protection",
"topological materials",
"rotational... | [
{
"idx": "1.",
"title": "Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators.Rev. Mod. Phys.82, 3045 (2010).",
"link": "https://doi.org/10.1103%2FRevModPhys.82.3045"
},
{
"idx": "2.",
"title": "Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors.Rev. Mod. Phys.83, 1... |
s41467-025-56182-5 | Energy science and technology | Molecular ferroelectric self-assembled interlayer for efficient perovskite solar cells | The interfacial molecular dipole enhances the photovoltaic performance of perovskite solar cells (PSCs) by facilitating improved charge extraction. However, conventional self-assembled monolayers (SAMs) face challenges like inadequate interface coverage and weak dipole interactions. Herein, we develop a strategy using ... | [
{
"section": "Introduction",
"content": "Recently, perovskite solar cells (PSCs) have made an efficiency breakthrough beyond 26%, as is comparable to that of the silicon-based counterparts [1] , [2] , [3] . Despite the rapid evolution in efficiencies, further improvement in device performance is still requi... | The core research idea is to enhance the photovoltaic performance and operational stability of both normal and inverted perovskite solar cells by introducing a ferroelectric self-assembled monolayer (SAM), specifically 1-adamantanamine hydroiodide (ADAI), at the perovskite/hole transport layer interface, which forms a ... | [
"photovoltaic performance",
"operational stability",
"normal perovskite solar cells",
"inverted perovskite solar cells",
"ferroelectric self-assembled monolayer",
"1-adamantanamine hydroiodide",
"ADAI",
"perovskite/hole transport layer interface",
"dipole layer",
"interfacial energy level alignmen... | [
{
"idx": "1.",
"title": "Park, J. et al. Controlled growth of perovskite layers with volatile alkylammonium chlorides.Nature616, 724–730 (2023).",
"link": "https://doi.org/10.1038%2Fs41586-023-05825-y"
},
{
"idx": "2.",
"title": "Zhao, Y. et al. Inactive (PbI2)2RbCl stabilizes perovskite fil... |
s41467-023-37902-1 | Physics | Neural-network decoders for measurement induced phase transitions | Open quantum systems have been shown to host a plethora of exotic dynamical phases. Measurement-induced entanglement phase transitions in monitored quantum systems are a striking example of this phenomena. However, naive realizations of such phase transitions requires an exponential number of repetitions of the experim... | [
{
"section": "Introduction",
"content": "Entanglement entropy in closed quantum systems that thermalize generally tends to increase until reaching a volume-law behavior with entanglement spread throughout the system [1] , [2] . Coupling to a bath profoundly changes the internal evolution of the system [3] ,... | This research proposes a generic neural network approach to efficiently find decoders that map measurement outcomes to the density matrix of a reference qubit in random quantum circuits, enabling the experimental probing of measurement-induced phase transitions through the purification dynamics of the reference qubit, ... | [
"neural network approach",
"decoders",
"measurement outcomes",
"density matrix",
"reference qubit",
"random quantum circuits",
"measurement-induced phase transitions",
"purification dynamics",
"scalability",
"applicability",
"Haar-random circuits"
] | [
{
"idx": "1.",
"title": "Kim, H. & Huse, D. A. Ballistic spreading of entanglement in a diffusive nonintegrable system.Phys. Rev. Lett.111, 127205 (2013).",
"link": "https://doi.org/10.1103%2FPhysRevLett.111.127205"
},
{
"idx": "2.",
"title": "Nandkishore, R. & Huse, D. A. Many-body localiza... |
ncomms5922 | Nanoscience and technology | Angular momentum-induced circular dichroism in non-chiral nanostructures | Circular dichroism, that is, the differential absorption of a system to left and right circularly polarized light, is one of the only techniques capable of providing morphological information of certain samples. In biology, for instance, circular dichroism spectroscopy is widely used to study the structure of proteins.... | [
{
"section": "Introduction",
"content": "Since its discovery in the nineteenthth century, circular dichroism (CD) has been widely used in science. Defined as the differential absorption of left and right circular polarization (LCP or RCP) [1] , its uses are as diverse as protein spectroscopy, DNA studies an... | The core research idea is: We demonstrate that circular dichroism can be induced in a non-chiral sample under normal incidence by using vortex beams with angular momentum as the input light, breaking the mirror symmetry of the system through the internal degree of freedom of the beam rather than through structural chir... | [
"circular dichroism",
"non-chiral sample",
"normal incidence",
"vortex beams",
"angular momentum",
"mirror symmetry",
"internal degree of freedom",
"structural chirality",
"external media"
] | [
{
"idx": "2",
"title": "Kelly, S., Jess, T. J. & Price, N. C. How to study proteins by circular dichroism.Biochim. Biophys. Acta1751, 119–139 (2005).",
"link": "https://doi.org/10.1016%2Fj.bbapap.2005.06.005"
},
{
"idx": "3",
"title": "Stephens, P. J. Magnetic circular dichroism.Annu. Rev. P... |
s41467-024-55654-4 | Energy science and technology | Performance and stability analysis of all-perovskite tandem photovoltaics in light-driven electrochemical water splitting | All-perovskite tandem photovoltaics are a potentially cost-effective technology to power chemical fuel production, such as green hydrogen. However, their application is limited by deficits in open-circuit voltage and, more challengingly, poor operational stability of the photovoltaic cell. Here we report a laboratory-s... | [
{
"section": "Introduction",
"content": "Monolithic tandem solar cells offer the ability to simultaneously possess high photocurrent and photovoltage for converting sunlight into electricity for photovoltaic (PV) power and for direct generation of solar fuels [1] . Among the options for tandem solar cells, ... | This research investigates the development and degradation mechanisms of efficient all-perovskite tandem solar cells used in a light-driven water-splitting system, with a focus on enhancing voltage output through interfacial engineering and identifying stability-limiting factors—particularly at the narrow-bandgap perov... | [
"all-perovskite tandem solar cells",
"light-driven water-splitting system",
"voltage output",
"interfacial engineering",
"stability-limiting factors",
"narrow-bandgap perovskite",
"hole transport layer interface",
"solar-to-hydrogen conversion systems",
"degradation mechanisms",
"long-term perform... | [
{
"idx": "1.",
"title": "Young, J. L. et al. Direct solar-to-hydrogen conversion via inverted metamorphic multi-junction semiconductor architectures.Nat. Energy2, 17028 (2017).",
"link": "https://doi.org/10.1038%2Fnenergy.2017.28"
},
{
"idx": "2.",
"title": "Green, M. A. et al. Solar cell ef... |
s41467-025-57440-2 | Optics and photonics | Fiber-based angular demultiplexer using nanoprinted periodic structures on single-mode multicore fibers | Precise analysis of light beams is critical for modern applications, especially in integrated photonics, with traditional methods often struggling with efficient angular demultiplexing in compact environments. Here, we present a novel fiber-based approach that achieves angular demultiplexing through angle-sensitive cou... | [
{
"section": "Introduction",
"content": "Optical fibers have become an integral part of everyday life, finding applications in areas such as optical networks, infrastructure monitoring, and life sciences. Beyond telecommunications, fibers represent an ideal platform for efficiently generating and observing ... | This work proposes a fiber-based concept for angular demultiplexing through angle-sensitive coupling in nanostructure-enhanced multicore fibers, where algorithmically optimized, 3D nanoprinted axially symmetric structures on the fiber facets enable efficient and simultaneous distribution of incident light beams with di... | [
"fiber-based concept",
"angular demultiplexing",
"angle-sensitive coupling",
"nanostructure-enhanced multicore fibers",
"algorithmically optimized 3D nanoprinted axially symmetric structures",
"fiber facets",
"incident light beams",
"angles of incidence",
"single-mode cores",
"power collection",
... | [
{
"idx": "1.",
"title": "Lee, B. Review of the present status of optical fiber sensors.Opt. Fiber Technol.9, 57–79 (2003).",
"link": "https://doi.org/10.1016%2FS1068-5200%2802%2900527-8"
},
{
"idx": "2.",
"title": "Elahi, S. F. & Wang, T. D. Future and advances in endoscopy.J. Biophotonics4,... |
s41467-021-26087-0 | Physics | Modified Bose-Einstein condensation in an optical quantum gas | Open quantum systems can be systematically controlled by making changes to their environment. A well-known example is the spontaneous radiative decay of an electronically excited emitter, such as an atom or a molecule, which is significantly influenced by the feedback from the emitter’s environment, for example, by the... | [
{
"section": "Introduction",
"content": "When a Bose gas exceeds the critical phase space density, a condensation process is triggered [1] . This process can be considered as a competition between various possible condensate wavefunctions that experience different growth rates under the given experimental c... | The core research idea is to investigate how non-equilibrium conditions, created by spatially inhomogeneous optical pump geometries and controlled dissipation in photonic microcavity systems, influence the formation and dynamics of Bose–Einstein condensates, particularly in terms of their tendency to condense into stat... | [
"non-equilibrium conditions",
"spatially inhomogeneous optical pump geometries",
"controlled dissipation",
"photonic microcavity systems",
"Bose–Einstein condensates",
"lowest energy level",
"quantum simulation",
"interferometric applications"
] | [
{
"idx": "2.",
"title": "Berloff, N. G. et al. Realizing the classical XY Hamiltonian in polariton simulators.Nat. Mater.16, 1120–1126 (2017).",
"link": "https://doi.org/10.1038%2Fnmat4971"
},
{
"idx": "3.",
"title": "Vretenar, M., Kassenberg, B., Bissesar, S., Toebes, C. & Klaers, J. Contro... |
s41467-020-16435-x | Energy science and technology | Understanding the interface interaction between U3Si2fuel and SiC cladding | Triuranium disilicide (U 3 Si 2 ) fuel with silicon carbide (SiC) composite cladding is being considered as an advanced concept/accident tolerant fuel for light water reactors thus, understanding their chemical compatibility under operational and accident conditions is paramount. Here we provide a comprehensive view of... | [
{
"section": "Introduction",
"content": "Maintaining the existing high standard of living in many countries, and allowing for improvement in the standard of living for others, will require a significant reliable supply of energy into the future. With the threat of global climate change, providing that energ... | This study proposes to investigate the compatibility and interfacial interactions between U₃Si₂ nuclear fuel and SiC/SiC composite cladding as a potential accident-tolerant fuel (ATF) system through a combination of computational methods, including density functional theory (DFT) calculations of reaction energies and d... | [
"U₃Si₂ nuclear fuel",
"SiC/SiC composite cladding",
"accident-tolerant fuel",
"density functional theory",
"reaction energies",
"defect formation energies",
"high-temperature diffusion couples",
"phase characterization",
"atomic-scale mechanisms",
"interface behavior",
"computational methods",
... | [
{
"idx": "1.",
"title": "Dresselhaus, M. S. & Thomas, I. L. Alternative energy technologies.Nature414, 332–337 (2001).",
"link": "https://doi.org/10.1038%2F35104599"
},
{
"idx": "2.",
"title": "Tsouris, C. Uranium extraction: fuel from seawater.Nat. Energy2, 17022 (2017).",
"link": "http... |
s41467-021-22516-2 | Physics | Incommensurate smectic phase in close proximity to the high-Tcsuperconductor FeSe/SrTiO3 | Superconductivity is significantly enhanced in monolayer FeSe grown on SrTiO 3 , but not for multilayer films, in which large strength of nematicity develops. However, the link between the high-transition temperature superconductivity in monolayer and the correlation related nematicity in multilayer FeSe films is not w... | [
{
"section": "Introduction",
"content": "Electronic liquid crystal phases are correlation-induced intermediate states between the weak coupling Fermi liquid and localized electronic crystals [1] , [2] . The symmetries of electronic structures in these states are lower than those of the lattice. For example,... | The core research idea is to investigate the existence and implications of a unique smectic electronic phase observed in two-unit-cell-thick FeSe films grown on SrTiO3, which exhibits long-range incommensurate stripe patterns detected via scanning tunneling microscopy, and to understand how this phase, absent in both m... | [
"smectic electronic phase",
"two-unit-cell-thick FeSe films",
"SrTiO3",
"long-range incommensurate stripe patterns",
"scanning tunneling microscopy",
"superconductivity enhancement",
"nematicity",
"electronic correlations",
"substrate effects",
"iron-based superconductors"
] | [
{
"idx": "1.",
"title": "Kivelson, S. A., Fradkin, E. & Emery, V. J. Electronic liquid-crystal phases of a doped Mott insulator.Nature393, 550–553 (1998).",
"link": "https://doi.org/10.1038%2F31177"
},
{
"idx": "2.",
"title": "Fradkin, E., Kivelson, S. A., Lawler, M. J., Eisenstein, J. P. & ... |
s41467-022-34394-3 | Optics and photonics | Two-dimensional Thouless pumping of light in photonic moiré lattices | Continuous and quantized transports are profoundly different. The latter is determined by the global rather than local properties of a system, it exhibits unique topological features, and its ubiquitous nature causes its occurrence in many areas of science. Here we report the first observation of fully-two-dimensional ... | [
{
"section": "Introduction",
"content": "Transfer of matter, charge, or spin, carried by quantum or classical extended wave packets depends on the global characteristics of the system in which it occurs. Topological charge pumping that transfers current in a quantized manner was discovered by Thouless [1] i... | The core research idea is to achieve genuinely two-dimensional topological pumping of light using a novel photonic moiré lattice with enhanced flat bands, created by superimposing two sublattices that are both twisted and tilted to form a dynamically evolving, periodically restoring three-dimensional structure, thereby... | [
"two-dimensional topological pumping",
"photonic moiré lattice",
"enhanced flat bands",
"sublattices",
"twisted sublattices",
"tilted sublattices",
"dynamically evolving structure",
"periodically restoring structure",
"three-dimensional structure",
"directional transport",
"bulk modes",
"trans... | [
{
"idx": "1.",
"title": "Thouless, D. J. Quantization of particle transport.Phys. Rev. B27, 6083 (1983).",
"link": "https://doi.org/10.1103%2FPhysRevB.27.6083"
},
{
"idx": "2.",
"title": "Xiao, D., Chang, M.-C. & Niu, Q. Berry phase effects on electronic properties.Rev. Mod. Phys.82, 1959 (2... |
s41467-017-02286-6 | Nanoscience and technology | Diversity of trion states and substrate effects in the optical properties of an MoS2monolayer | Almost all experiments and future applications of transition metal dichalcogenide monolayers rely on a substrate for mechanical stability, which can significantly modify the optical spectra of the monolayer. Doping from the substrate might lead to the domination of the spectra by trions. Here we show by ab initio many-... | [
{
"section": "Introduction",
"content": "Semiconducting transition metal dichalcogenides (TMDCs) have attracted much attention as candidates for future opto-electronic applications due to their particularly rich physical properties: the indirect-to-direct band-gap transition from bulk to monolayer [1] , [2]... | The core research idea of this work is to develop a comprehensive theoretical framework using ab initio many-body perturbation theory to quantitatively understand the interplay between a transition metal dichalcogenide (TMDC) monolayer and its substrate, focusing on the formation, properties, and spectral characteristi... | [
"ab initio many-body perturbation theory",
"transition metal dichalcogenide monolayer",
"substrate",
"negative trions",
"positive trions",
"substrate-induced screening effects",
"excitonic transition energies",
"trionic transition energies",
"redshifts",
"theoretical framework",
"spectral charac... | [
{
"idx": "1.",
"title": "Mak, K. F., Lee, C., Hone, J., Shan, J. & Heinz, T. F. Atomically thinMoS2: a new direct-gap semiconductor.Phys. Rev. Lett.105, 136805 (2010).",
"link": "https://doi.org/10.1103%2FPhysRevLett.105.136805"
},
{
"idx": "2.",
"title": "Splendiani, A. et al. Emerging phot... |
s41467-020-20441-4 | Mathematics and computing | Coherence resonance in influencer networks | Complex networks are abundant in nature and many share an important structural property: they contain a few nodes that are abnormally highly connected (hubs). Some of these hubs are called influencers because they couple strongly to the network and play fundamental dynamical and structural roles. Strikingly, despite th... | [
{
"section": "Introduction",
"content": "A central discovery in network science is that a small group of highly connected hubs can couple to the network more strongly than their peers and greatly influence the network behavior [1] , [2] , [3] , [4] , [5] , [6] . Examples of network influencers can be found ... | The core research idea is to investigate how stochastic forcing of influencers in heterogeneous networks can synergize with network structure to induce optimal collective oscillations through a two-step process involving synchronization of connected nodes and macroscopic interaction mediated by a hyper-graph, revealing... | [
"stochastic forcing",
"influencers",
"heterogeneous networks",
"optimal collective oscillations",
"two-step process",
"synchronization",
"connected nodes",
"macroscopic interaction",
"hyper-graph",
"noise-induced coherence resonance effect",
"global network synchronization",
"deterministic set... | [
{
"idx": "2.",
"title": "Xu, X.-K., Zhang, J. & Small, M. Rich-club connectivity dominates assortativity and transitivity of complex networks.Phys. Rev. E82, 046117 (2010).",
"link": "https://doi.org/10.1103%2FPhysRevE.82.046117"
},
{
"idx": "3.",
"title": "Watts, D. J. & Strogatz, S. H. Col... |
s41467-024-47904-2 | Optics and photonics | Parallel wavelength-division-multiplexed signal transmission and dispersion compensation enabled by soliton microcombs and microrings | The proliferation of computation-intensive technologies has led to a significant rise in the number of datacenters, posing challenges for high-speed and power-efficient datacenter interconnects (DCIs). Although inter-DCIs based on intensity modulation and direct detection (IM-DD) along with wavelength-division multiple... | [
{
"section": "Introduction",
"content": "According to the International Data Corporation (IDC), the exponential growth of digital data generation is projected to reach 175 zettabytes by 2025 [1] due to the surgent growth of computation-intensive technologies such as artificial intelligence, the Internet of ... | This paper proposes a short-reach inter-datacenter interconnect (inter-DCI) system based on an intensity modulation and direct detection (IM-DD) optical communications architecture that utilizes a single-soliton microcomb source generated by a silicon nitride microring resonator (Si3N4 MRR) as a wavelength-division mul... | [
"short-reach inter-datacenter interconnect system",
"intensity modulation and direct detection optical communications architecture",
"single-soliton microcomb source",
"silicon nitride microring resonator",
"wavelength-division multiplexing light source",
"fully reconfigurable optical chromatic dispersion... | [
{
"idx": "3.",
"title": "Zhou, X., Urata, R. & Liu, H. Beyond 1 Tb/s intra-data center interconnect technology: IM-DD OR coherent?J. Lightwave Technol.38, 475–484 (2020).",
"link": "https://doi.org/10.1109%2FJLT.2019.2956779"
},
{
"idx": "4.",
"title": "Berikaa, E. et al. TFLN MZMs and next-... |
s41467-024-51345-2 | Energy science and technology | A binary 2D perovskite passivation for efficient and stable perovskite/silicon tandem solar cells | To achieve high power conversion efficiency in perovskite/silicon tandem solar cells, it is necessary to develop a promising wide-bandgap perovskite absorber and processing techniques in relevance. To date, the performance of devices based on wide-bandgap perovskite is still limited mainly by carrier recombination at t... | [
{
"section": "Introduction",
"content": "Metal halide perovskite solar cells (PSCs) have shown their superior optoelectronic properties to achieve a record efficiency of 26.1% for single junction devices [1] . The tunability of this material opens up its applications for light-emitting diodes, detectors, la... | The core research idea is to develop a binary two-dimensional (B-2D) perovskite structure composed of alternating-cation-interlayer (ACI) and Ruddlesden–Popper (RP) phases for surface passivation of a wide-bandgap (1.68 eV) mixed-halide perovskite in order to simultaneously reduce non-radiative recombination and enhanc... | [
"binary two-dimensional perovskite structure",
"alternating-cation-interlayer phase",
"Ruddlesden–Popper phase",
"surface passivation",
"wide-bandgap mixed-halide perovskite",
"non-radiative recombination",
"charge extraction",
"single-junction solar cells",
"monolithic perovskite/silicon tandem sol... | [
{
"idx": "2.",
"title": "Stranks, S. D. & Snaith, H. J. Metal-halide perovskites for photovoltaic and light-emitting devices.Nat. Nanotechnol.10, 391–402 (2015).",
"link": "https://doi.org/10.1038%2Fnnano.2015.90"
},
{
"idx": "3.",
"title": "Ahmadi, M., Wu, T. & Hu, B. A review on organic–in... |
ncomms7229 | Materials science | A real-time study of the benefits of co-solvents in polymer solar cell processing | The photoactive layer of organic solar cells consists of a nanoscale blend of electron-donating and electron-accepting organic semiconductors. Controlling the degree of phase separation between these components is crucial to reach efficient solar cells. In solution-processed polymer–fullerene solar cells, small amounts... | [
{
"section": "Introduction",
"content": "Focussed research efforts on organic semiconductors have pushed the certified efficiency of single junction organic solar cells to over 10% (ref. 1 ). To become a viable technology, research is now focussing on stability and lifetime [2] , [3] as well as all-solution... | This work proposes to investigate in real time during spin coating the formation of the photoactive layer in polymer solar cells processed with co-solvents, with the hypothesis that co-solvents enable control over morphology by shifting the onset of polymer aggregation to occur before large-scale phase separation, ther... | [
"spin coating",
"photoactive layer",
"polymer solar cells",
"co-solvents",
"polymer aggregation",
"large-scale phase separation",
"fullerene-rich domains",
"intimately mixed morphology",
"photovoltaic performance",
"real time investigation",
"morphology control",
"onset of polymer aggregation"... | [
{
"idx": "1",
"title": "Liu, Y. et al. Aggregation and morphology control enables multiple cases of high-efficiency polymer solar cells.Nat. Commun.5, 5293 (2014).",
"link": "https://doi.org/10.1038%2Fncomms6293"
},
{
"idx": "2",
"title": "Grossiord, N., Kroon, J. M., Andriessen, R. & Blom, ... |
ncomms5783 | Physics | Broadband optical cooling of molecular rotors from room temperature to the ground state | Laser cycling of resonances can remove entropy from a system via spontaneously emitted photons, with electronic resonances providing the fastest cooling timescales because of their rapid spontaneous relaxation. Although atoms are routinely laser-cooled, even simple molecules pose two interrelated challenges for cooling... | [
{
"section": "Introduction",
"content": "Laser cooling of atomic translational and internal energies has made possible a wide range of applications including creation of novel phases of matter [1] , atom-wave accelerometry [2] and ultraprecise timekeeping [3] . Because of their additional internal structure... | The core research idea is to demonstrate broadband rotational cooling of trapped AlH⁺ molecules using a broadband laser tuned to an electronic resonance, enabling rapid cooling from room temperature to the ground rotational state (~100 ms) by simultaneously addressing all initially excited rotational states, leveraging... | [
"Broadband rotational cooling",
"Trapped AlH⁺ molecules",
"Broadband laser",
"Electronic resonance",
"Ground rotational state",
"Vibrational–electronic decoupling",
"Unwanted vibrational excitation",
"Quantum state preparation",
"Repeated state resets",
"Molecular beam experiments",
"Trapped-mol... | [
{
"idx": "2",
"title": "Dickerson, S. M., Hogan, J. M., Sugarbaker, A., Johnson, D. M. S. & Kasevich, M. A. Multiaxis inertial sensing with long-time point source atom interferometry.Phys. Rev. Lett.111, 083001 (2013).",
"link": "https://doi.org/10.1103%2FPhysRevLett.111.083001"
},
{
"idx": "3",... |
ncomms2760 | Physics | Polariton condensation in solitonic gap states in a one-dimensional periodic potential | Manipulation of nonlinear waves in artificial periodic structures leads to spectacular spatial features, such as generation of gap solitons or onset of the Mott insulator phase transition. Cavity exciton–polaritons are strongly interacting quasiparticles offering large possibilities for potential optical technologies. ... | [
{
"section": "Introduction",
"content": "Nonlinear wave propagation in a periodic medium is fundamentally different from the propagation in a homogeneous medium [1] . Such nonlinear waves can be photons, subject to Kerr nonlinearity ( χ 3 ), or the macroscopic wave-function of an interacting Bose–Einstein c... | The core research idea is to investigate how polaritons in a one-dimensional periodic potential, under specific excitation conditions, can spontaneously condense into highly localized gap soliton states that can be either bound to or free from the pump-induced excitonic reservoir, and how these states can be manipulate... | [
"polaritons",
"one-dimensional periodic potential",
"excitation conditions",
"gap soliton states",
"pump-induced excitonic reservoir",
"hybridized symmetric configurations",
"hybridized anti-symmetric configurations",
"Josephson oscillations",
"unbinding propagation",
"coupled dynamical model"
] | [
{
"idx": "2",
"title": "Painter, O. et al. Two-dimensional photonic band-gap defect mode laser.Science284, 1819–1821 (1999) .",
"link": "https://doi.org/10.1126%2Fscience.284.5421.1819"
},
{
"idx": "4",
"title": "Notomi, M. & Taniyama, H. On-demand ultrahigh-Q cavity formation and photon pin... |
s41467-018-07230-w | Nanoscience and technology | Graphene mechanical pixels for Interferometric Modulator Displays | Electro-optic modulators based on micro-electromechanical systems have found success as elements for optical projectors, for simplified optical spectrometers, and as reflective-type screens that make use of light interference (Interferometric Modulator Display technology). The latter concept offers an exciting avenue f... | [
{
"section": "Introduction",
"content": "Graphene, the carbon monolayer and two-dimensional allotrope of graphite, has the potential to impact technology with a wide range of applications such as optical modulators for high-speed communications [1] , [2] , [3] , [4] . In contrast to modulation devices that ... | The core research idea is to develop a reflective-type display technology based on electro-optically tunable double-layer graphene membranes, leveraging their interference properties and mechanical response to electrical gating for achieving high-resolution, low-power, full-spectrum pixel performance with reduced flick... | [
"Reflective-type display technology",
"Electro-optically tunable double-layer graphene membranes",
"Interference properties",
"Mechanical response",
"Electrical gating",
"High-resolution pixel performance",
"Low-power pixel performance",
"Full-spectrum pixel performance",
"Reduced flicker effect",
... | [
{
"idx": "1.",
"title": "Bonaccorso, F., Sun, Z., Hasan, T. & Ferrari, A. Graphene photonics and optoelectronics.Nat. Photonics4, 611–622 (2010).",
"link": "https://doi.org/10.1038%2Fnphoton.2010.186"
},
{
"idx": "2.",
"title": "Yu, R., Pruneri, V. & Garcia de Abajo F. G. Resonant visible li... |
s41467-017-00280-6 | Physics | Lorentz-violating type-II Dirac fermions in transition metal dichalcogenide PtTe2 | Topological semimetals have recently attracted extensive research interests as host materials to condensed matter physics counterparts of Dirac and Weyl fermions originally proposed in high energy physics. Although Lorentz invariance is required in high energy physics, it is not necessarily obeyed in condensed matter p... | [
{
"section": "Introduction",
"content": "In three dimensional topological semimetals, the low energy excitations—Dirac or Weyl fermions—are described by the massless Dirac equation [1] . Dirac fermions are protected by certain crystal symmetries [2] , [3] , [4] . When time-reversal or inversion symmetry is ... | The core research idea is to experimentally identify and characterize type-II Dirac fermions in bulk PtTe₂ single crystals by combining angle-resolved photoemission spectroscopy (ARPES) measurements with first-principles calculations, thereby filling the last missing element in the classification of Dirac and Weyl semi... | [
"type-II Dirac fermions",
"bulk PtTe₂ single crystals",
"angle-resolved photoemission spectroscopy",
"ARPES measurements",
"first-principles calculations",
"Dirac semimetals",
"Weyl semimetals",
"strongly tilted Dirac cones",
"critical point",
"topological phase transitions"
] | [
{
"idx": "1.",
"title": "Young, S. M. et al. Dirac semimetal in three dimensions.Phys. Rev. Lett.108, 140405 (2012).",
"link": "https://doi.org/10.1103%2FPhysRevLett.108.140405"
},
{
"idx": "2.",
"title": "Yang, B.-J. & Nagaosa, N. Classification of stable three-dimensional Dirac semimetals ... |
ncomms2075 | Materials science | Controllable chirality-induced geometrical Hall effect in a frustrated highly correlated metal | A current of electrons traversing a landscape of localized spins possessing non-coplanar magnetic order gains a geometrical (Berry) phase, which can lead to a Hall voltage independent of the spin–orbit coupling within the material—a geometrical Hall effect. Here we show that the highly correlated metal UCu 5 possesses ... | [
{
"section": "Introduction",
"content": "Frustration is prevalent in nature, appearing in substances as simple as water ice to systems as complex as neural networks [1] , [2] . In general, frustration is the result of competing interactions of comparable strengths that can prevent a system from entering a s... | The core research idea is to investigate how geometrical frustration in the heavy-fermion metal UCu5 leads to a non-coplanar 4–q antiferromagnetic order below 1.2 K, which induces a chirality-driven Berry phase in itinerant electrons and results in a large, tunable geometrical Hall effect that can be controlled by smal... | [
"geometrical frustration",
"heavy-fermion metal",
"UCu5",
"non-coplanar 4–q antiferromagnetic order",
"chirality-driven Berry phase",
"itinerant electrons",
"large geometrical Hall effect",
"tunable geometrical Hall effect",
"small applied magnetic fields"
] | [
{
"idx": "3",
"title": "Moessner, R. & Chalker, J. T. Properties of a classical spin liquid: the Heisenberg pyrochlore antiferromagnet.Phys. Rev. Lett.80, 2929–2932 (1998).",
"link": "https://doi.org/10.1103%2FPhysRevLett.80.2929"
},
{
"idx": "4",
"title": "Cheong, S.- W. & Mostovoy, M. Mult... |
ncomms5766 | Materials science | Visualizing domain wall and reverse domain superconductivity | In magnetically coupled, planar ferromagnet-superconductor (F/S) hybrid structures, magnetic domain walls can be used to spatially confine the superconductivity. In contrast to a superconductor in a uniform applied magnetic field, the nucleation of the superconducting order parameter in F/S structures is governed by th... | [
{
"section": "Introduction",
"content": "Recent experimental advances in fabrication and in the measurement of mesoscopic superconductivity have renewed theoretical interest in ferromagnet-superconductor (F/S) hybrid structures. The interaction between superconductivity and magnetism, with Cooper pairs tran... | The core research idea is to investigate how the inhomogeneous magnetic stray fields produced by ferromagnetic domains in F/S heterostructures with an insulating interlayer can spatially confine and manipulate mesoscopic superconductivity at the nanoscale, leading to the nucleation of superconductivity above magnetic d... | [
"Ferromagnetic domains",
"F/S heterostructures",
"Insulating interlayer",
"Inhomogeneous magnetic stray fields",
"Mesoscopic superconductivity",
"Nanoscale",
"Superconductivity nucleation",
"Magnetic domain walls",
"Reverse domain superconductivity",
"Applied magnetic fields",
"Spontaneous vorte... | [
{
"idx": "1",
"title": "Buzdin, A. I. Proximity effects in superconductor-ferromagnet heterostructures.Rev. Mod. Phys.77, 935–976 (2005).",
"link": "https://doi.org/10.1103%2FRevModPhys.77.935"
},
{
"idx": "2",
"title": "Buzdin, A. I. & Melnikov, A. S. Domain wall superconductivity in ferrom... |
s41467-019-11804-7 | Materials science | Solvent-non-solvent rapid-injection for preparing nanostructured materials from micelles to hydrogels | Due to their distinctive molecular architecture, ABA triblock copolymers will undergo specific self-assembly processes into various nanostructures upon introduction into a B-block selective solvent. Although much of the focus in ABA triblock copolymer self-assembly has been on equilibrium nanostructures, little attenti... | [
{
"section": "Introduction",
"content": "Block polymers consist of two or more covalently attached incompatible polymer sequences, and as a result, will self-assemble into nanostructured materials with tunable morphologies and features on the 10–100 nm length scale [1] . The resulting nanoscale morphologies... | The core research idea is to investigate the experimental parameters governing the rapid-injection-induced self-assembly of ABA triblock copolymers in a B-selective solvent, with the aim of establishing a quantitative and universally applicable method to produce nanostructured hydrogels, microgels, and micelles by cont... | [
"rapid-injection-induced self-assembly",
"ABA triblock copolymers",
"B-selective solvent",
"nanostructured hydrogels",
"microgels",
"micelles",
"initial polymer concentration",
"overlap concentration (c*)",
"hierarchical structures",
"tunable properties",
"drug delivery",
"tissue engineering",... | [
{
"idx": "1.",
"title": "Bates, F. S. & Fredrickson, G. Block copolymers-designer soft materials.Phys. Today52, 32–38 (1999).",
"link": "https://doi.org/10.1063%2F1.882522"
},
{
"idx": "2.",
"title": "Bates, C. M. & Bates, F. S. 50th anniversary perspective: block polymers pure potential.Mac... |
s41467-019-08573-8 | Materials science | Enhanced photoelectrical response of thermodynamically epitaxial organic crystals at the two-dimensional limit | Owing to strong light-matter interaction, two-dimensional (2D) organic crystal is regarded as promising materials for ultrasensitive photodetectors, however it still received limited success due to degraded photoelectrical response and problems in controllable growth. Here, we find the growth of 2D organic crystal obey... | [
{
"section": "Introduction",
"content": "Two-dimensional (2D) materials are normally regarded as a promising material in future ultrasensitive photodetectors, owing to their strong light–matter interaction at the 2D limit [1] , [2] . Besides well-known 2D atomic crystals (2DACs), 2D organic crystals (2DOCs)... | The core research idea is to investigate the epitaxial growth of large-sized, thickness-controlled two-dimensional organic crystals (2DOCs) via a seed-epitaxial drop-casting method based on the Gibbs–Curie–Wulff law, and to demonstrate that the 2D limit effect in these crystals—induced by compact molecular stacking and... | [
"epitaxial growth",
"large-sized",
"thickness-controlled",
"two-dimensional organic crystals",
"2DOCs",
"seed-epitaxial drop-casting method",
"Gibbs–Curie–Wulff law",
"2D limit effect",
"compact molecular stacking",
"strong intra-layer π-stacking",
"SiO₂/Si substrates",
"photoelectrical perfor... | [
{
"idx": "1.",
"title": "Liu, X. et al. Strong light-matter coupling in two-dimensional atomic crystals.Nat. Photon9, 30–34 (2015).",
"link": "https://doi.org/10.1038%2Fnphoton.2014.304"
},
{
"idx": "2.",
"title": "Britnell, L. et al. Strong light-matter interactions in heterostructures of a... |
s41467-020-19442-0 | Astronomy and planetary science | Self-consistent kinetic model of nested electron- and ion-scale magnetic cavities in space plasmas | NASA’s Magnetospheric Multi-Scale (MMS) mission is designed to explore the proton- and electron-gyroscale kinetics of plasma turbulence where the bulk of particle acceleration and heating takes place. Understanding the nature of cross-scale structures ubiquitous as magnetic cavities is important to assess the energy pa... | [
{
"section": "Introduction",
"content": "In the plasma universe, unstable velocity distributions of ions and electrons are formed around a variety of acceleration regions such as shock waves and magnetic reconnection sites [1] , [2] , [3] , [4] , which can excite various types of plasma waves, either compre... | The core research idea is to demonstrate the existence of a kinetic equilibrium model that accounts for the survival and cross-scale properties of electron- and proton-scale nested magnetic cavities in turbulent plasmas, by incorporating adiabatic electron behavior and validating it through multi-spacecraft observation... | [
"kinetic equilibrium model",
"survival",
"cross-scale properties",
"electron-scale nested magnetic cavities",
"proton-scale nested magnetic cavities",
"turbulent plasmas",
"adiabatic electron behavior",
"multi-spacecraft observations",
"MMS constellation"
] | [
{
"idx": "1.",
"title": "Drake, J. F., Swisdak, M., Che, H. & Shay, M. A. Electron acceleration from contracting magnetic islands during reconnection.Nature443, 553–556 (2006).",
"link": "https://doi.org/10.1038%2Fnature05116"
},
{
"idx": "2.",
"title": "Matsumoto, Y., Amano, T., Kato, T. N.... |
s41467-018-05799-w | Optics and photonics | Enhanced photon collection enables four dimensional fluorescence nanoscopy of living systems | The theoretically unlimited spatial resolution of fluorescence nanoscopy often comes at the expense of time, contrast and increased dose of energy for recording. Here, we developed MoNaLISA, for Molecular Nanoscale Live Imaging with Sectioning Ability, a nanoscope capable of imaging structures at a scale of 45–65 nm wi... | [
{
"section": "Introduction",
"content": "Observing the interplay of organelles and macromolecular complexes inside living cells and tissues calls for the continuous development of minimally invasive optical systems performing at high spatio-temporal resolution. Nowadays, the spatial resolution of fluorescen... | The core research idea is to develop a novel nanoscopy technique called Molecular Nanoscale Live Imaging with Sectioning Ability (MoNaLISA), which utilizes light patterns with optimized and distinct periodicities to switch and read out reversibly switchable fluorescent proteins (rsFPs) in a way that minimizes light dos... | [
"nanoscopy technique",
"Molecular Nanoscale Live Imaging with Sectioning Ability",
"MoNaLISA",
"light patterns",
"optimized periodicities",
"distinct periodicities",
"reversibly switchable fluorescent proteins",
"rsFPs",
"light dose minimization",
"crosstalk reduction",
"photon collection enhanc... | [
{
"idx": "1.",
"title": "Hell, S. W. Far-field optical nanoscopy.Science316, 1153–1158 (2007).",
"link": "https://doi.org/10.1126%2Fscience.1137395"
},
{
"idx": "2.",
"title": "Betzig, E. et al. Imaging intracellular fluorescent proteins at nanometer resolution.Science313, 1642–1645 (2006)."... |
ncomms3933 | Materials science | Co-operativity in a nanocrystalline solid-state transition | Co-operativity is a remarkable phenomenon mostly seen in biology, where initial reaction events significantly alter the propensity of subsequent reaction events, giving rise to a nonlinear tightly regulated synergistic response. Here we have found unique evidence of atomic level co-operativity in an inorganic material.... | [
{
"section": "Introduction",
"content": "Since the first observation in 1910 of the co-operative binding of oxygen to haemoglobin [1] , [2] , where binding at any of the four sites of haemoglobin increases the oxygen affinity of the other sites, co-operative phenomena have been displayed in a host of macrom... | The core research idea is: We propose that cation exchange in nanocrystals of cadmium selenide (CdSe) exhibits positive co-operative behaviour, where initial Cu+ doping enhances the likelihood of further Cu+ doping within the same nanocrystal, leading to a sharp transition from CdSe to Cu2Se rather than a random, diffu... | [
"cation exchange",
"nanocrystals",
"cadmium selenide",
"CdSe",
"Cu+ doping",
"positive co-operative behaviour",
"sharp transition",
"Cu2Se",
"random process",
"diffusion-limited process"
] | [
{
"idx": "2",
"title": "Akers, G. K., Doyle, M. L., Myers, D. & Daugherty, M. A. Molecular code for cooperativity in hemoglobin.Science255, 54–63 (1992).",
"link": "https://doi.org/10.1126%2Fscience.1553532"
},
{
"idx": "3",
"title": "Whitty, A. Cooperativity and biological complexity.Nat. C... |
ncomms11648 | Nanoscience and technology | Shaping nanoscale magnetic domain memory in exchange-coupled ferromagnets by field cooling | The advance of magnetic nanotechnologies relies on detailed understanding of nanoscale magnetic mechanisms in materials. Magnetic domain memory (MDM), that is, the tendency for magnetic domains to repeat the same pattern during field cycling, is important for magnetic recording technologies. Here we demonstrate MDM in ... | [
{
"section": "Introduction",
"content": "Understanding the mechanisms driving nanoscale magnetic reversal and memory phenomena in layered magnetic films is essential for the development of magnetic recording and spintronic nanotechnologies [1] , [2] , [3] , [4] . Thin ferromagnetic (F) films with perpendicu... | The core research idea is to investigate how exchange coupling with an antiferromagnetic template can induce and spatially control 100% magnetic domain memory in smooth perpendicular magnetic anisotropy ferromagnetic films, enabling the on/off switching of memory properties through temperature and field-cooling conditi... | [
"exchange coupling",
"antiferromagnetic template",
"magnetic domain memory",
"perpendicular magnetic anisotropy",
"ferromagnetic films",
"temperature control",
"field-cooling conditions",
"memory properties",
"spatial control",
"100% magnetic domain memory",
"smooth ferromagnetic films"
] | [
{
"idx": "1",
"title": "Ackermann, J. Toward a universal memory.Science308, 508–510 (2005).",
"link": "https://doi.org/10.1126%2Fscience.1110549"
},
{
"idx": "2",
"title": "Chappert, C., Fert, A. & Nguyen Van Dau, F. The emergence of spin electronics in data storage.Nat. Mater.6, 813–823 (20... |
ncomms3083 | Physics | Liquid–liquid transition in a strong bulk metallic glass-forming liquid | Polymorphic phase transitions are common in crystalline solids. Recent studies suggest that phase transitions may also exist between two liquid forms with different entropy and structure. Such a liquid–liquid transition has been investigated in various substances including water, Al 2 O 3 -Y 2 O 3 and network glass for... | [
{
"section": "Introduction",
"content": "There is a growing interest in density- or entropy-driven liquid–liquid phase transitions (LLTs) [1] , [2] , [3] , [4] , [5] . The LLT of two distinct forms of liquid water has been intensely studied and debated [1] , [2] , [3] , [4] , [5] , [6] , [7] . Not limited t... | The core research idea is to investigate the hypothesis that a liquid–liquid phase transition (LLT) underlies the strong-fragile crossover in the bulk metallic glass-forming liquid Zr41.2Ti13.8Cu12.5Ni10Be22.5 (Vit.1), by experimentally observing a lambda-like heat capacity peak and corresponding structural changes in ... | [
"liquid–liquid phase transition",
"LLT",
"strong-fragile crossover",
"bulk metallic glass-forming liquid",
"Zr41.2Ti13.8Cu12.5Ni10Be22.5",
"Vit.1",
"lambda-like heat capacity peak",
"structural changes",
"stable liquid state",
"melting point",
"thermodynamic evidence",
"structural evidence",
... | [
{
"idx": "1",
"title": "Tanaka, H., Kurita, R. & Mataki, H. Liquid-liquid transition in the molecular liquid triphenyl phosphite.Phys. Rev. Lett.92, 025701 (2004).",
"link": "https://doi.org/10.1103%2FPhysRevLett.92.025701"
},
{
"idx": "2",
"title": "Kurita, R. & Tanaka, H. Critical-like phe... |
s41467-021-26436-z | Astronomy and planetary science | A GeV-TeV particle component and the barrier of cosmic-ray sea in the Central Molecular Zone | Cosmic rays are important probe of a number of fundamental physical problems such as the acceleration of high and very high energy particles in extreme astrophysical environments. The Galactic center is widely anticipated to be an important cosmic-ray source and the observations of some Imaging Atmospheric Cherenkov Te... | [
{
"section": "Introduction",
"content": "It is believed that in the Milky Way the cosmic rays (CRs) could be accelerated by shock waves in supernova remnants (e.g., [1] ) or stellar winds of massive stars [2] . Those charged relativistic particles would then propagate diffusively in the Galactic magnetic fi... | This study proposes to critically address the nature of the cosmic ray population in the Galactic Center region by conducting a dedicated morphological and spectral analysis of Fermi-LAT data in the Central Molecular Zone, incorporating the Galactic Center GeV excess component, in order to determine whether a low-energ... | [
"cosmic ray population",
"Galactic Center",
"morphological analysis",
"spectral analysis",
"Fermi-LAT data",
"Central Molecular Zone",
"Galactic Center GeV excess component",
"TeV-PeV cosmic rays",
"Imaging Atmospheric Cherenkov Telescopes",
"low-energy counterpart",
"magnetic fields",
"cosmic... | [
{
"idx": "1.",
"title": "Blasi, P. The origin of galactic cosmic rays.Astron. Astrophys. Rev.21, 70 (2013).",
"link": "https://link.springer.com/doi/10.1007/s00159-013-0070-7"
},
{
"idx": "2.",
"title": "Aharonian, F., Yang, R. & de Oña Wilhelmi, E. Massive stars as major factories of Galact... |
ncomms12477 | Mathematics and computing | Prediction of allosteric sites and mediating interactions through bond-to-bond propensities | Allostery is a fundamental mechanism of biological regulation, in which binding of a molecule at a distant location affects the active site of a protein. Allosteric sites provide targets to fine-tune protein activity, yet we lack computational methodologies to predict them. Here we present an efficient graph-theoretica... | [
{
"section": "Introduction",
"content": "Allostery is a key molecular mechanism underpinning control and modulation in a variety of cellular processes [1] , [2] . Allosteric effects are those induced on the main functional site of a biomolecule by the binding of an effector at a distant site, for example, t... | The core research idea is to develop and apply a graph-theoretic, atomistic approach that computes bond propensities through an edge-to-edge transfer function in order to identify allosteric sites and the specific bonds involved in signal propagation within proteins, thereby providing a detailed, efficient, and physico... | [
"graph-theoretic approach",
"atomistic approach",
"bond propensities",
"edge-to-edge transfer function",
"allosteric sites",
"signal propagation",
"proteins",
"physico-chemically rich method",
"allosteric pathways",
"structure-based allosteric drug discovery"
] | [
{
"idx": "1",
"title": "Monod, J., Changeux, J.-P. & Jacob., F. Allosteric proteins and cellular control systems.J. Mol. Biol.6, 306–329 (1963).",
"link": "https://doi.org/10.1016%2FS0022-2836%2863%2980091-1"
},
{
"idx": "2",
"title": "Perutz, M. F. Mechanisms of cooperativity and allosteric... |
s41467-025-57122-z | Physics | Preclinical tumor control with a laser-accelerated high-energy electron radiotherapy prototype | Radiotherapy using very-high-energy electron (VHEE) beams (50-300 MeV) has attracted considerable attention due to its advantageous dose deposition characteristics, enabling deep penetration and easy manipulation by magnetic components. One promising approach to compactly delivering these high energy electron beams in ... | [
{
"section": "Introduction",
"content": "Nowadays, cancer curability remains a significant global challenge despite constant advances in early diagnosis and treatment options. Radiotherapy, applied to more than half of the patients, is an essential part of cancer treatment that relies on the use of ionizing... | The core research idea is to develop a compact, stable, and conformal laser-wakefield acceleration (LWFA)-based very high-energy electron (VHEE) machine for clinical radiotherapy, which addresses the limitations of conventional accelerators by leveraging the high acceleration gradient of LWFA to produce a cost-effectiv... | [
"compact laser-wakefield acceleration",
"stable laser-wakefield acceleration",
"conformal laser-wakefield acceleration",
"very high-energy electron machine",
"clinical radiotherapy",
"conventional accelerators",
"high acceleration gradient",
"cost-effective electron beam",
"space-efficient electron ... | [
{
"idx": "1.",
"title": "Bentzen, S. M. Radiation therapy: intensity-modulated, image-guided, biologically optimized and evidence-based.Radiother. Oncol.77, 227–230 (2005).",
"link": "https://doi.org/10.1016%2Fj.radonc.2005.11.001"
},
{
"idx": "2.",
"title": "Otto, K. Volumetric modulated ar... |
s41467-021-24155-z | Materials science | Room-temperature oxygen vacancy migration induced reversible phase transformation during the anelastic deformation in CuO | From the mechanical perspectives, the influence of point defects is generally considered at high temperature, especially when the creep deformation dominates. Here, we show the stress-induced reversible oxygen vacancy migration in CuO nanowires at room temperature, causing the unanticipated anelastic deformation. The a... | [
{
"section": "Introduction",
"content": "The structural stability of material subjected to intense mechanical duress is the corner stone for ensuring the robust performance in micro-electro-mechanical systems and devices, whereas there is a surging demand for the effective damping system to minimize mechani... | The core research idea is to investigate the oxygen vacancy migration-mediated anelasticity in single crystalline CuO nanowires, where bending stress induces vacancy clustering and the nucleation of a metastable CuO x phase that transforms back to CuO upon load removal, providing direct atomistic evidence of point defe... | [
"oxygen vacancy migration",
"anelasticity",
"single crystalline CuO nanowires",
"bending stress",
"vacancy clustering",
"metastable CuO x phase",
"load removal",
"atomistic evidence",
"point defect-driven anelastic behavior",
"nanosized materials",
"room temperature"
] | [
{
"idx": "1.",
"title": "Lai, A., Du, Z. H., Gan, C. L. & Schuh, C. A. Shape memory and superelastic ceramics at small scales.Science341, 1505–1508 (2013).",
"link": "https://doi.org/10.1126%2Fscience.1239745"
},
{
"idx": "2.",
"title": "Gianola, D. S. & Shin, J. H. Full recovery takes time.... |
s41467-022-35206-4 | Optics and photonics | Ultrahigh-responsivity waveguide-coupled optical power monitor for Si photonic circuits operating at near-infrared wavelengths | A phototransistor is a promising candidate as an optical power monitor in Si photonic circuits since the internal gain of photocurrent enables high responsivity. However, state-of-the-art waveguide-coupled phototransistors suffer from a responsivity of lower than 10 3 A/W, which is insufficient for detecting very low p... | [
{
"section": "Introduction",
"content": "The rapid progress in Si photonics over the past decade has made it possible to fabricate large-scale photonic circuits on a Si wafer, which are being widely used as optical interconnection devices in datacenters [1] . In addition to optical fiber communication, many... | This work proposes an ultrahigh-responsivity waveguide-coupled phototransistor using an InGaAs ultrathin membrane bonded onto a Si waveguide, where the Si waveguide serves as a gate electrode to enable effective current control without inducing optical loss, thereby addressing the limitations of existing waveguide-coup... | [
"ultrahigh-responsivity waveguide-coupled phototransistor",
"InGaAs ultrathin membrane",
"Si waveguide",
"gate electrode",
"current control",
"optical loss",
"waveguide-coupled photoFETs",
"high responsivity"
] | [
{
"idx": "1.",
"title": "Siew, S. Y. et al. Review of silicon photonics technology and platform development.J. Lightwave Technol.39, 4374–4389 (2021).",
"link": "https://doi.org/10.1109%2FJLT.2021.3066203"
},
{
"idx": "2.",
"title": "Shen, Y. et al. Deep learning with coherent nanophotonic c... |
s41467-025-55993-w | Engineering | Gypsum heterogenous nucleation pathways regulated by surface functional groups and hydrophobicity | Gypsum (CaSO 4 ·2H 2 O) plays a critical role in numerous natural and industrial processes. Nevertheless, the underlying mechanisms governing the formation of gypsum crystals on surfaces with diverse chemical properties remain poorly understood due to a lack of sufficient temporal-spatial resolution. Herein, we use in ... | [
{
"section": "Introduction",
"content": "Gypsum (CaSO 4 ·2H 2 O) scale formation is important across diverse fields, impacting industrial processes and environmental systems [1] , [2] , [3] , [4] , [5] , [6] , [7] . One example is seawater desalination, where gypsum formation on reverse osmosis (RO) membran... | This research proposes that combining classical nucleation theory with molecular dynamics simulations can provide mechanistic insights into gypsum scale formation on engineered surfaces with varying chemical properties, particularly by elucidating how surface functional groups, hydrophilicity, and ion interactions infl... | [
"classical nucleation theory",
"molecular dynamics simulations",
"gypsum scale formation",
"engineered surfaces",
"surface functional groups",
"hydrophilicity",
"ion interactions",
"heterogeneous nucleation",
"growth mechanisms",
"hydrophilic surfaces",
"hydrophobic surfaces",
"mechanistic ins... | [
{
"idx": "1.",
"title": "Huang, X., Li, C., Zuo, K. & Li, Q. Predominant effect of material surface hydrophobicity on gypsum scale formation.Environ. Sci. Technol.54, 15395–15404 (2020).",
"link": "https://doi.org/10.1021%2Facs.est.0c03826"
},
{
"idx": "2.",
"title": "Yin, Y. et al. Which su... |
s41467-021-21780-6 | Nanoscience and technology | Proximity control of interlayer exciton-phonon hybridization in van der Waals heterostructures | Van der Waals stacking has provided unprecedented flexibility in shaping many-body interactions by controlling electronic quantum confinement and orbital overlap. Theory has predicted that also electron-phonon coupling critically influences the quantum ground state of low-dimensional systems. Here we introduce proximit... | [
{
"section": "Introduction",
"content": "Heterostructures of atomically thin materials provide a unique laboratory to explore novel quantum states of matter [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] . By van der Waals stacking, band structures and electronic correlations have been ta... | The core research idea is to investigate proximity-induced hybridization between phonons and electrically neutral excitons in atomically thin WSe₂/gypsum heterostructures, aiming to create and resolve new hybrid excitations called Lyman polarons through strong exciton–phonon coupling, which is tuned by aligning the int... | [
"proximity-induced hybridization",
"phonons",
"electrically neutral excitons",
"atomically thin WSe₂/gypsum heterostructures",
"Lyman polarons",
"strong exciton–phonon coupling",
"internal 1s–2p Lyman transition",
"polar phonon modes",
"phase-locked few-cycle mid-infrared probe pulses"
] | [
{
"idx": "1.",
"title": "Shimazaki, Y. et al. Strongly correlated electrons and hybrid excitons in a moiré heterostructure.Nature580, 472–477 (2020).",
"link": "https://doi.org/10.1038%2Fs41586-020-2191-2"
},
{
"idx": "2.",
"title": "Jin, C. et al. Observation of moiré excitons in WSe2/WS2he... |
s41467-024-45172-8 | Physics | Learning nonequilibrium statistical mechanics and dynamical phase transitions | Nonequilibrium statistical mechanics exhibit a variety of complex phenomena far from equilibrium. It inherits challenges of equilibrium, including accurately describing the joint distribution of a large number of configurations, and also poses new challenges as the distribution evolves over time. Characterizing dynamic... | [
{
"section": "Introduction",
"content": "Tracking time evolution and characterizing phase transitions are fundamental tasks in nonequilibrium statistical mechanics [1] , having implications to a wide range of fields including quantum transport [2] , molecular machines [3] , quantitative biology [4] , [5] , ... | The core research idea is to develop a general framework using variational autoregressive networks (VAN) to track the time evolution of nonequilibrium systems and efficiently compute the dynamical partition function, enabling the characterization of dynamical phase transitions over time by overcoming the challenges of ... | [
"variational autoregressive networks",
"nonequilibrium systems",
"dynamical partition function",
"dynamical phase transitions",
"high-dimensional probability distributions",
"non-normalized tilted generators",
"full-time spectrum estimation"
] | [
{
"idx": "2.",
"title": "Esposito, M., Harbola, U. & Mukamel, S. Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems.Rev. Mod. Phys.81, 1665 (2009).",
"link": "https://doi.org/10.1103%2FRevModPhys.81.1665"
},
{
"idx": "3.",
"title": "Seifert, U. Stoc... |
s41467-018-04608-8 | Mathematics and computing | Exploring patterns enriched in a dataset with contrastive principal component analysis | Visualization and exploration of high-dimensional data is a ubiquitous challenge across disciplines. Widely used techniques such as principal component analysis (PCA) aim to identify dominant trends in one dataset. However, in many settings we have datasets collected under different conditions, e.g., a treatment and a ... | [
{
"section": "Introduction",
"content": "Principal component analysis (PCA) is one of the most widely used methods for data exploration and visualization [1] . PCA projects the data onto a low-dimensional space and is especially powerful as an approach to visualize patterns, such as clusters, clines, and ou... | The core research idea of this work is to introduce Contrastive Principal Component Analysis (cPCA), a novel unsupervised dimensionality reduction technique that leverages a background dataset to automatically identify and visualize low-dimensional subspaces in a target dataset where the variance is enriched relative t... | [
"Contrastive Principal Component Analysis",
"cPCA",
"unsupervised dimensionality reduction technique",
"background dataset",
"target dataset",
"low-dimensional subspaces",
"variance",
"standard PCA",
"patterns",
"visualization",
"automatic identification"
] | [
{
"idx": "1.",
"title": "Hotelling, H. Analysis of a complex of statistical variables into principal components.J. Educ. Psychol.24, 417 (1933).",
"link": "https://doi.org/10.1037%2Fh0071325"
},
{
"idx": "5.",
"title": "Chen, W., Ma, H., Yu, D. & Zhang, H. SVD-based technique for interferenc... |
ncomms11424 | Materials science | All-optical design for inherently energy-conserving reversible gates and circuits | As energy efficiency becomes a paramount issue in this day and age, reversible computing may serve as a critical step towards energy conservation in information technology. The inputs of reversible computing elements define the outputs and vice versa. Some reversible gates such as the Fredkin gate are also universal; t... | [
{
"section": "Introduction",
"content": "One of the greatest challenges in computing is to reduce the energy consumption, not only for the sake of reducing the tremendous amount of electric power used by the computer-based industry (for example, Google, Amazon or Facebook) but also to enable computing devic... | The core research idea is to develop an all-optical, energy-conserving implementation of a universal reversible Fredkin gate using linear and nonlinear unitary transformations in conjunction with optical components such as directional couplers and graphene-induced silicon waveguides, with the aim of enabling efficient,... | [
"all-optical implementation",
"energy-conserving implementation",
"universal reversible Fredkin gate",
"linear unitary transformations",
"nonlinear unitary transformations",
"directional couplers",
"graphene-induced silicon waveguides",
"reversible logic operations",
"large-scale circuits",
"optic... | [
{
"idx": "1",
"title": "Caulfield, H. J. & Dolev, S. Why future supercomputing requires optics.Nat. Photon.4, 261–263 (2010).",
"link": "https://doi.org/10.1038%2Fnphoton.2010.94"
},
{
"idx": "2",
"title": "Fredkin, E. & Toffoli, T. Conservative logic.Int. J. Theor. Phys.21, 219–253 (1982)."... |
s41467-024-44754-w | Optics and photonics | Antireflective vertical-cavity surface-emitting laser for LiDAR | Multijunction vertical-cavity surface-emitting lasers (VCSELs) have gained popularity in automotive LiDARs, yet achieving a divergence of less than 16° (D86) is difficult for conventional extended cavity designs due to multiple-longitudinal-mode lasing. Our innovation, the antireflective vertical-cavity surface-emittin... | [
{
"section": "Introduction",
"content": "Compactness, fast response, and high energy conversion efficiency have made vertical-cavity surface-emitting lasers (VCSELs) a major light source for high-speed data communication [1] , [2] , [3] and sensing [4] , [5] , [6] over the past three decades. Particularly i... | This research proposes the development and analysis of antireflective vertical-cavity surface-emitting laser (AR-VCSEL) arrays with multijunction structures to address the challenges of high beam divergence and multi-longitudinal mode operation in conventional extended cavity VCSELs, aiming to achieve ultrasmall diverg... | [
"antireflective vertical-cavity surface-emitting laser arrays",
"multijunction structures",
"conventional extended cavity VCSELs",
"high beam divergence",
"multi-longitudinal mode operation",
"ultrasmall divergence",
"single longitudinal mode operation",
"LiDAR applications",
"vertical-cavity surfac... | [
{
"idx": "2.",
"title": "Choquette, K. D. & Hou, H. Q. Vertical-cavity surface emitting lasers: moving from research to manufacturing.Proc. IEEE85, 1730–1739 (1997).",
"link": "https://doi.org/10.1109%2F5.649649"
},
{
"idx": "3.",
"title": "Iga, K. Forty years of vertical-cavity surface-emit... |
ncomms9347 | Physics | Sub-10-fs population inversion in N2+in air lasing through multiple state coupling | Laser filamentation generated when intense laser pulses propagate in air has been an attractive phenomenon having a variety of potential applications such as detection and spectroscopy of gases at far distant places. It was discovered recently that the filamentation in air induces ‘lasing’, showing that electronically ... | [
{
"section": "Introduction",
"content": "Remote generation of population-inverted gain media in air is a promising step towards the realization of bright and coherent atmospheric lasers for various applications such as standoff spectroscopy [1] , sensing [2] , [3] , [4] and remote material processing [5] . ... | The core research idea is to demonstrate that the population inversion responsible for air lasing in nitrogen molecular ions (N₂⁺) at 391 nm is achieved through post-ionization optical coupling in a three-level system involving the electronic ground state X²Σg⁺ and the excited states A²Πu and B²Σu⁺, where the A²Πu stat... | [
"population inversion",
"air lasing",
"nitrogen molecular ions",
"N₂⁺",
"391 nm",
"post-ionization optical coupling",
"three-level system",
"electronic ground state X²Σg⁺",
"excited states A²Πu",
"excited states B²Σu⁺",
"population trap",
"dipole coupling strength",
"ultrafast timescale",
... | [
{
"idx": "1",
"title": "Hemmer, P. R. et al. Standoff spectroscopy via remote generation of a backward-propagating laser beam.Proc. Natl Acad. Sci. USA108, 3130–3134 (2011) .",
"link": "https://doi.org/10.1073%2Fpnas.1014401107"
},
{
"idx": "2",
"title": "Luo, Q., Liu, W. & Chin, S. L. Lasin... |
ncomms14963 | Physics | Transient birefringence of liquids induced by terahertz electric-field torque on permanent molecular dipoles | Collective low-frequency molecular motions have large impact on chemical reactions and structural relaxation in liquids. So far, these modes have mostly been accessed indirectly by off-resonant optical pulses. Here, we provide evidence that intense terahertz (THz) pulses can resonantly excite reorientational-librationa... | [
{
"section": "Introduction",
"content": "Low-frequency structural dynamics of liquids in the range from 0.1 to 10 THz (3 to 330 cm −1 ) are believed to strongly contribute to the outcome of chemical processes [1] , [2] , [3] , [4] , [5] . The underlying molecular motions can be complex and include reorienta... | The core research idea is to investigate the resonant nonlinear excitation of liquids using intense terahertz (THz) pulses in order to selectively exert ultrafast torque on both permanent and induced molecular electric dipoles, thereby enhancing transient optical birefringence signals and enabling systematic studies of... | [
"resonant nonlinear excitation",
"liquids",
"intense terahertz pulses",
"ultrafast torque",
"permanent molecular electric dipoles",
"induced molecular electric dipoles",
"transient optical birefringence signals",
"librational molecular modes",
"reorientational molecular modes",
"vibrational molecu... | [
{
"idx": "3",
"title": "Maroncelli, M. The dynamics of solvation in polar liquids.J. Mol. Liq.57, 1–37 (1993).",
"link": "https://doi.org/10.1016%2F0167-7322%2893%2980045-W"
},
{
"idx": "4",
"title": "Zusman, L. D. Outer-sphere electron-transfer in polar-solvents.Chem. Phys.49, 295–304 (1980... |
s41467-023-36134-7 | Mathematics and computing | Topological identification and interpretation for single-cell gene regulation elucidation across multiple platforms using scMGCA | Single-cell RNA sequencing provides high-throughput gene expression information to explore cellular heterogeneity at the individual cell level. A major challenge in characterizing high-throughput gene expression data arises from challenges related to dimensionality, and the prevalence of dropout events. To address thes... | [
{
"section": "Introduction",
"content": "Single-cell RNA sequencing (scRNA-Seq) [1] allows the investigation of transcriptomic landscapes; it is an increasingly popular platform for characterizing cellular heterogeneity [2] , discovering complex tissues and diseases [3] , [4] , and inferring cell trajectori... | This research proposes a graph convolutional autoencoder framework called scMGCA that integrates cell topological information and gene expression data through a novel cell-PPMI graph construction and self-optimized embedded clustering, aiming to improve the accuracy of cell clustering, visualization, and batch effect c... | [
"graph convolutional autoencoder",
"scMGCA",
"cell topological information",
"gene expression data",
"cell-PPMI graph construction",
"self-optimized embedded clustering",
"cell clustering",
"visualization",
"batch effect correction",
"single-cell RNA-seq data",
"clustering loss",
"multinomial-... | [
{
"idx": "1.",
"title": "Kolodziejczyk, A. A., Kim, J. K., Svensson, V., Marioni, J. C. & Teichmann, S. A. The technology and biology of single-cell rna sequencing.Mol. Cell58, 610–620 (2015).",
"link": "https://doi.org/10.1016%2Fj.molcel.2015.04.005"
},
{
"idx": "2.",
"title": "Saliba, A.-E... |
ncomms11341 | Physics | Laser spectroscopic probing of coexisting superfluid and insulating states of an atomic Bose–Hubbard system | A system of ultracold atoms in an optical lattice has been regarded as an ideal quantum simulator for a Hubbard model with extremely high controllability of the system parameters. While making use of the controllability, a comprehensive measurement across the weakly to strongly interacting regimes in the Hubbard model ... | [
{
"section": "Introduction",
"content": "Ultracold atoms confined in an optical lattice potential offer a novel way to study quantum many-body physics [1] , [2] . One of the most interesting problems is the quantum phase transition of ultracold bosonic atoms in a three-dimensional (3D) optical lattice from ... | The core research idea is to investigate the quantum phase transition of ultracold bosonic atoms in a three-dimensional optical lattice from a superfluid to a Mott insulating state by employing high-resolution laser spectroscopy to probe on-site interactions across weakly and strongly interacting regimes, supported by ... | [
"quantum phase transition",
"ultracold bosonic atoms",
"three-dimensional optical lattice",
"superfluid",
"Mott insulating state",
"high-resolution laser spectroscopy",
"on-site interactions",
"finite temperature Gutzwiller approximation",
"sum rule relations",
"excitation spectra",
"many-body s... | [
{
"idx": "1",
"title": "Jaksch, D. & Zoller, P. The cold atom Hubbard toolbox.Ann. Phys.315, 52–79 (2005).",
"link": "https://doi.org/10.1016%2Fj.aop.2004.09.010"
},
{
"idx": "2",
"title": "Bloch, I., Dalibard, J. & Zwerger, W. Many-body physics with ultracold gases.Rev. Mod. Phys.80, 885–96... |
s41467-024-54893-9 | Materials science | Insights into the anomalous hardness of the tantalum carbides from dislocation mobility | The tantalum carbides, TaC x , have been repeatedly shown to harden dramatically with some loss of carbon content, then soften with further decarburization. First observed in 1963, this anomalous hardness behavior has been reproduced for decades without satisfactory explanation. Prior attempts to characterize this phen... | [
{
"section": "Introduction",
"content": "Indentation hardness, or a material’s resistance to local plastic deformation, remains something of a challenge to researchers seeking to understand the inelastic deformation behavior of crystalline materials. Though not a material property per se, hardness provides ... | The core research idea is to investigate the anomalous hardness behavior in tantalum carbide (TaC x )—wherein hardness initially increases with carbon deficiency before declining—by examining how changes in dislocation slip plane dominance from {111} B1 to {110} B1 with decreasing carbon content influence plastic defor... | [
"anomalous hardness behavior",
"tantalum carbide",
"carbon deficiency",
"dislocation slip plane dominance",
"{111} B1",
"{110} B1",
"plastic deformation mechanisms",
"titanium carbide",
"hardness decrease",
"slip plane transition"
] | [
{
"idx": "1.",
"title": "Wuchina, E., Opila, E., Opeka, M., Fahrenholtz, B. & Talmy, I. UHTCs: ultra-high temperature ceramic materials for extreme environment applications.Electrochem. Soc. Interface16, 30–36 (2007).",
"link": "https://doi.org/10.1149%2F2.F04074IF"
},
{
"idx": "2.",
"title"... |
s41467-023-40877-8 | Physics | Uncovering spin-orbit coupling-independent hidden spin polarization of energy bands in antiferromagnets | Many textbook physical effects in crystals are enabled by some specific symmetries. In contrast to such ‘apparent effects’, ‘hidden effect X’ refers to the general condition where the nominal global system symmetry would disallow the effect X, whereas the symmetry of local sectors within the crystal would enable effect... | [
{
"section": "Introduction",
"content": "Many traditional textbook physical effects in crystals are enabled by some specific symmetries, encoded in the crystal space group. Such are the symmetry conditions for the apparent electric polarization which defines various order parameters such as in ferroelectric... | The core research idea is to investigate a novel, SOC-independent hidden spin polarization effect in collinear antiferromagnetic materials, which manifests locally despite the absence of the global symmetry typically required for such effects, revealing intrinsic properties of perfect crystals and expanding the range o... | [
"SOC-independent hidden spin polarization effect",
"collinear antiferromagnetic materials",
"global symmetry",
"intrinsic properties",
"perfect crystals",
"spintronics",
"novel effect",
"local manifestation",
"symmetry requirements",
"material applications",
"investigation",
"hidden spin polar... | [
{
"idx": "1.",
"title": "Litvin, D. B. Ferroelectric space groups.Acta Crystallogr. Sect. A42, 44–47,https://doi.org/10.1107/S0108767386099920(1986).",
"link": "https://doi.org/10.1107%2FS0108767386099920"
},
{
"idx": "2.",
"title": "Johnson, W. C. Jr. Protein secondary structure and circula... |
ncomms3064 | Physics | Quantum simulation of low-temperature metallic liquid hydrogen | The melting temperature of solid hydrogen drops with pressure above ~65 GPa, suggesting that a liquid state might exist at low temperatures. It has also been suggested that this low-temperature liquid state might be non-molecular and metallic, although evidence for such behaviour is lacking. Here we report results for ... | [
{
"section": "Introduction",
"content": "Ever since Wigner and Huntington’s prediction in 1935 that solid molecular hydrogen would dissociate and form an atomic and metallic phase at high pressures [1] , the phase diagram of hydrogen has been the focus of intense experimental and theoretical studies [2] , [... | The core research idea is to use ab initio path-integral molecular dynamics simulations to investigate the low-temperature phase diagram of hydrogen at ultra-high pressures (500–1,200 GPa) in order to determine the existence and properties of a predicted low-temperature metallic atomic liquid phase, which arises due to... | [
"ab initio path-integral molecular dynamics simulations",
"low-temperature phase diagram",
"hydrogen",
"ultra-high pressures",
"metallic atomic liquid phase",
"negative slope",
"melting curve",
"atomic liquid",
"solid phases",
"quantum nuclear effects"
] | [
{
"idx": "1",
"title": "Wigner, E. & Huntington, H. B. On the possibility of a metallic modification of hydrogen.J. Chem. Phys.3, 764–770 (1935).",
"link": "https://doi.org/10.1063%2F1.1749590"
},
{
"idx": "2",
"title": "Mao, H. K. & Hemley, R. J. Ultrahigh-pressure transitions in solid hydr... |
s41467-024-51110-5 | Mathematics and computing | High-performance deep spiking neural networks with 0.3 spikes per neuron | Communication by rare, binary spikes is a key factor for the energy efficiency of biological brains. However, it is harder to train biologically-inspired spiking neural networks than artificial neural networks. This is puzzling given that theoretical results provide exact mapping algorithms from artificial to spiking n... | [
{
"section": "Introduction",
"content": "Similar to the brain, neurons in spiking neural networks (SNNs) communicate via short pulses called spikes that arrive in continuous time—in striking contrast to artificial neural networks (ANNs) where neurons communicate by the exchange of real-valued signals in dis... | The core research idea is to develop a theoretically grounded training methodology for deep spiking neural networks using exact backpropagation and an exact revertible mapping between ReLU networks and time-to-first-spike (TTFS) networks, enabling stable gradient updates and achieving performance parity with artificial... | [
"training methodology",
"deep spiking neural networks",
"exact backpropagation",
"exact revertible mapping",
"ReLU networks",
"time-to-first-spike networks",
"gradient updates",
"performance parity",
"artificial neural networks",
"energy efficiency",
"sparse spiking"
] | [
{
"idx": "3.",
"title": "Yu, G. et al. Accurate recognition of colorectal cancer with semi-supervised deep learning on pathological images.Nat. Commun.12, 6311 (2021).",
"link": "https://doi.org/10.1038%2Fs41467-021-26643-8"
},
{
"idx": "5.",
"title": "Patterson, D. et al. The carbon footpri... |
s41467-019-12038-3 | Nanoscience and technology | Modeling and observation of mid-infrared nonlocality in effective epsilon-near-zero ultranarrow coaxial apertures | With advances in nanofabrication techniques, extreme-scale nanophotonic devices with critical gap dimensions of just 1–2 nm have been realized. Plasmons in such ultranarrow gaps can exhibit nonlocal response, which was previously shown to limit the field enhancement and cause optical properties to deviate from the loca... | [
{
"section": "Introduction",
"content": "The ability of metal nanostructures to localize light below the diffraction limit and enhance the optical field via surface plasmons—collective oscillations of free charge carriers—forms the basis of nano-optics and plasmonics [1] , [2] , [3] . Ultraconfined light in... | This research proposes to investigate and quantify nonlocal electrodynamics in mid-infrared nanophotonic structures—specifically, in coaxial apertures with sub-10 nm gaps—by combining a hydrodynamic model with a hybridizable discontinuous Galerkin computational method and atomic layer engineering, in order to understan... | [
"nonlocal electrodynamics",
"mid-infrared nanophotonic structures",
"coaxial apertures",
"sub-10 nm gaps",
"hydrodynamic model",
"hybridizable discontinuous Galerkin computational method",
"atomic layer engineering",
"spectral resonance",
"transmission intensity",
"transmission efficiency",
"gap... | [
{
"idx": "1.",
"title": "Barnes, W. L., Dereux, A. & Ebbesen, T. W. Surface plasmon subwavelength optics.Nature424, 824–830 (2003).",
"link": "https://doi.org/10.1038%2Fnature01937"
},
{
"idx": "2.",
"title": "Maier, S. A. et al. Local detection of electromagnetic energy transport below the ... |
s41467-024-55329-0 | Energy science and technology | Ultrathin polymer membrane for improved hole extraction and ion blocking in perovskite solar cells | Highly efficient perovskite solar cells (PSCs) in the n-i-p structure have demonstrated limited operational lifetimes, primarily due to the layer-to-layer ion diffusion in the perovskite/doped hole-transport layer (HTL) heterojunction, leading to conductivity drop in HTL and component loss in perovskite. Herein, we int... | [
{
"section": "Introduction",
"content": "Certified power conversion efficiencies (PCEs) exceeding 26% have been reported for both normal (n-i-p) [1] and inverted (p-i-n) [2] structure perovskite solar cells (PSCs). However, the long-term stability of the highly efficient n-i-p PSCs under light and heat stre... | The core research idea is to improve the operational stability of high-efficiency n-i-p perovskite solar cells by inserting an ultrathin (~7 nm) p-type polymeric interlayer (D18) that effectively blocks ion diffusion between the perovskite and Spiro-OMeTAD layers while maintaining efficient hole transport, thereby addr... | [
"high-efficiency n-i-p perovskite solar cells",
"ultrathin p-type polymeric interlayer",
"D18",
"ion diffusion",
"perovskite",
"Spiro-OMeTAD",
"operational stability",
"long-term stability",
"device efficiency",
"hole transport"
] | [
{
"idx": "1.",
"title": "Zhou, J. J. et al. Highly efficient and stable perovskite solar cells via a multifunctional hole transporting material.Joule8, 1691–1706 (2024).",
"link": "https://doi.org/10.1016%2Fj.joule.2024.02.019"
},
{
"idx": "2.",
"title": "Chen, H. et al. Improved charge extr... |
ncomms1286 | Nanoscience and technology | Plasmonic beaming and active control over fluorescent emission | Nanometallic optical antennas are rapidly gaining popularity in applications that require exquisite control over light concentration and emission processes. The search is on for high-performance antennas that offer facile integration on chips. Here we demonstrate a new, easily fabricated optical antenna design that ach... | [
{
"section": "Introduction",
"content": "Optical antennas based on metal nanostructures have recently gained significant popularity for their unique ability to concentrate light into a deep-subwavelength volume [1] , [2] . They hold great promise for many emerging applications requiring tight confinement of... | The core research idea is to develop a new type of active optical antenna that combines radiative decay engineering and beam shaping to enable highly collimated fluorescent emission from a large number of quantum emitters in a nanoscale region, with electrical control over their emission properties, surpassing the limi... | [
"active optical antenna",
"radiative decay engineering",
"beam shaping",
"fluorescent emission",
"quantum emitters",
"nanoscale region",
"electrical control",
"emission properties",
"antenna designs",
"emitter coupling",
"fabrication tolerance",
"functionality"
] | [
{
"idx": "1",
"title": "Mühlschlegel, P., Eisler, H. J., Martin, O. J. F., Hecht, B. & Pohl, D. W. Resonant optical antennas.Science308, 1607–1609 (2005).",
"link": "https://doi.org/10.1126%2Fscience.1111886"
},
{
"idx": "2",
"title": "Bharadwaj, P., Deutsch, B. & Novotny, L. Optical antenna... |
s41467-018-08227-1 | Nanoscience and technology | Planar and van der Waals heterostructures for vertical tunnelling single electron transistors | Despite a rich choice of two-dimensional materials, which exists these days, heterostructures, both vertical (van der Waals) and in-plane, offer an unprecedented control over the properties and functionalities of the resulted structures. Thus, planar heterostructures allow p-n junctions between different two-dimensiona... | [
{
"section": "Introduction",
"content": "Graphene quantum dots (GQDs) and graphene nanowires have been attracting attention because of the linear spectrum obeyed by the quasiparticles [1] , [2] (zero mass allows one to reach large quantisation energy, comparable with the room temperature, for relatively lar... | The core research idea is to develop single-electron tunnelling transistors by forming graphene quantum dots with precisely controlled size and edge passivation within a hexagonal boron nitride matrix using catalytic-assisted carbon substitution, and by creating reproducible contacts using hBN tunnelling barriers and g... | [
"single-electron tunnelling transistors",
"graphene quantum dots",
"hexagonal boron nitride matrix",
"catalytic-assisted carbon substitution",
"hBN tunnelling barriers",
"graphene electrodes",
"localised edge states",
"transport properties",
"edge passivation",
"precise size control",
"reproduci... | [
{
"idx": "1.",
"title": "Stampfer, C. et al. Tunable Coulomb blockade in nanostructured graphene.Appl. Phys. Lett.92, 012102 (2008).",
"link": "https://doi.org/10.1063%2F1.2827188"
},
{
"idx": "2.",
"title": "Guttinger, J. et al. Electron-hole crossover in graphene quantum dots.Phys. Rev. Le... |
ncomms10295 | Materials science | Magnetoelectric effect and phase transitions in CuO in external magnetic fields | Apart from being so far the only known binary multiferroic compound, CuO has a much higher transition temperature into the multiferroic state, 230 K, than any other known material in which the electric polarization is induced by spontaneous magnetic order, typically lower than 100 K. Although the magnetically induced f... | [
{
"section": "Introduction",
"content": "As a building block of high-temperature superconductors, CuO (tenorite) has been extensively studied following their discovery. Unlike other 3 d -metal monoxides, CuO has a monoclinic crystal structure [1] described by the centrosymmetric space group C 2 /c . Magneti... | The core research idea is to investigate the magnetoelectric phase diagram of CuO, a high-temperature magnetically induced multiferroic, in order to reveal how high magnetic fields can suppress the helical magnetic modulation in the multiferroic phase, thereby dramatically affecting the electric polarization, and to de... | [
"magnetoelectric phase diagram",
"CuO",
"high-temperature magnetically induced multiferroic",
"helical magnetic modulation",
"multiferroic phase",
"high magnetic fields",
"electric polarization",
"incommensurate magnetic structure",
"213 K transition",
"polarization changes"
] | [
{
"idx": "1",
"title": "Åsbrink, S. & Norrby, L.-J. A Refinement of the crystal structure of copper(II) oxide with a discussion of some exceptional E.s.fl.'s.Acta Crystallogr. B26, 8–15 (1970).",
"link": "https://doi.org/10.1107%2FS0567740870001838"
},
{
"idx": "2",
"title": "Forsyth, J. B.,... |
s41467-020-15280-2 | Physics | Matter-wave interference of a native polypeptide | The de Broglie wave nature of matter is a paradigmatic example of quantum physics and it has been exploited in precision measurements of forces and fundamental constants. However, matter-wave interferometry has remained an outstanding challenge for natural polypeptides, building blocks of life, which are fragile and di... | [
{
"section": "Introduction",
"content": "The wave nature of massive particles is a central aspect of quantum physics. The free evolution of particles is no longer described by classical trajectories, but instead by a wave-like propagation in multiple directions. Recombining the wavefronts leads to interfere... | The core research idea is to realize matter-wave interferometry with gramicidin A1, a neutral polypeptide antibiotic composed of 15 amino acids, using a combination of direct laser desorption, cold noble gas jet entrainment, and VUV photoionization to overcome the challenges of creating stable, intense molecular beams ... | [
"matter-wave interferometry",
"gramicidin A1",
"neutral polypeptide antibiotic",
"direct laser desorption",
"cold noble gas jet entrainment",
"VUV photoionization",
"molecular beams",
"fragile biomolecules",
"quantum coherence",
"interference experiments",
"wave nature",
"electronic properties... | [
{
"idx": "1.",
"title": "Cronin, A. D., Schmiedmayer, J. & Pritchard, D. E. Optics and interferometry with atoms and molecules.Rev. Mod. Phys.81, 1051–1129 (2009).",
"link": "https://doi.org/10.1103%2FRevModPhys.81.1051"
},
{
"idx": "3.",
"title": "Kovachy, T. et al. Quantum superposition at... |
s41467-018-03029-x | Energy science and technology | Ultra-high thermal effusivity materials for resonant ambient thermal energy harvesting | Materials science has made progress in maximizing or minimizing the thermal conductivity of materials; however, the thermal effusivity—related to the product of conductivity and capacity—has received limited attention, despite its importance in the coupling of thermal energy to the environment. Herein, we design materi... | [
{
"section": "Introduction",
"content": "Thermal effusivity or thermal inertia is the measure of a material’s ability to transfer thermal energy with its surroundings and is equal to the square root of the product of thermal conductivity and heat capacity [1] , [2] . The importance of high thermal conductiv... | This research proposes the development and experimental demonstration of ultra-high thermal effusivity materials and their application in a novel thermal resonator device, designed to optimize electrical power generation from ambient, broadband temperature fluctuations by leveraging the dynamic thermal energy exchange ... | [
"Ultra-high thermal effusivity materials",
"thermal resonator device",
"electrical power generation",
"ambient temperature fluctuations",
"broadband temperature fluctuations",
"dynamic thermal energy exchange",
"thermal effusivity",
"experimental demonstration",
"material development",
"thermal en... | [
{
"idx": "1.",
"title": "Balderas-Lopez, J. A. et al. Photoacoustic measurements of transparent liquid samples: thermal effusivity.Meas. Sci. Technol.6, 1163–1168 (1999).",
"link": "https://doi.org/10.1088%2F0957-0233%2F6%2F8%2F011"
},
{
"idx": "3.",
"title": "Chen, Z., Gao, D. & Shi, J. Exp... |
s41467-019-12679-4 | Materials science | On-demand orbital maneuver of multiple soft robots via hierarchical magnetomotility | Magnetic soft robots facilitate the battery-free remote control of soft robots. However, parallel control of multiple magnetic robots is challenging due to interference between robots and difficult maneuvers. Here we present the orbital maneuvering of manifold magnetic soft robots. Magneto-induced motion (magnetomotili... | [
{
"section": "Introduction",
"content": "The smooth-jointed motility of soft robots can be powered by various external stimuli such as thermal [1] , [2] , [3] , [4] , photonic [5] , [6] , pneumatic [7] , [8] , [9] , [10] , hydraulic [11] , hygroscopic [12] , [13] , [14] , [15] , combustive [16] , electrical... | This research proposes a novel approach for the untethered, battery-free manipulation of multiple soft robots through hierarchical magneto-induced orbital motion, inspired by celestial body dynamics, enabling coordinated multi-body actuation with independent velocity control and environmental adaptability in both land ... | [
"novel approach",
"untethered manipulation",
"battery-free manipulation",
"soft robots",
"hierarchical magneto-induced orbital motion",
"celestial body dynamics",
"coordinated multi-body actuation",
"independent velocity control",
"environmental adaptability",
"land settings",
"aquatic settings"... | [
{
"idx": "1.",
"title": "López-Valdeolivas, M., Liu, D., Broer, D. J. & Sánchez-Somolinos, C. 4D printed actuators with soft-robotic functions.Macromol. Rapid Commun.39, 1700710 (2018).",
"link": "https://doi.org/10.1002%2Fmarc.201700710"
},
{
"idx": "2.",
"title": "Ware, T. H., McConney, M.... |
s41467-021-22454-z | Nanoscience and technology | Targeted apoptosis of macrophages and osteoclasts in arthritic joints is effective against advanced inflammatory arthritis | Insufficient apoptosis of inflammatory macrophages and osteoclasts (OCs) in rheumatoid arthritis (RA) joints contributes toward the persistent progression of joint inflammation and destruction. Here, we deliver celastrol (CEL) to selectively induce apoptosis of OCs and macrophages in arthritic joints, with enzyme-respo... | [
{
"section": "Introduction",
"content": "Rheumatoid arthritis (RA) is a chronic autoimmune inflammatory disease characterized by synovial inflammation and joint destruction [1] . Synovial inflammation is the dominant feature in the early stage of RA [2] , [3] , [4] . In addition to the inflammatory symptoms... | The core research idea is to develop MMP9-cleavable, PEG- and RGD peptide-modified PLGA nanoparticles (PRNPs) for the targeted delivery of celastrol (CEL) to both macrophages and osteoclasts in arthritic joints, with the aim of inducing apoptosis in these cells to reduce synovial inflammation and reverse bone erosion i... | [
"MMP9-cleavable",
"PEG",
"RGD peptide",
"PLGA nanoparticles",
"PRNPs",
"celastrol",
"CEL",
"macrophages",
"osteoclasts",
"arthritic joints",
"apoptosis",
"synovial inflammation",
"bone erosion",
"advanced rheumatoid arthritis",
"targeted delivery"
] | [
{
"idx": "1.",
"title": "Smolen, J. S. et al. Rheumatoid arthritis.Nat. Rev. Dis. Prim.4, 18001 (2018).",
"link": "https://doi.org/10.1038%2Fnrdp.2018.1"
},
{
"idx": "2.",
"title": "Yeo, L. et al. Expression of chemokines CXCL4 and CXCL7 by synovial macrophages defines an early stage of rheu... |
s41467-023-38452-2 | Optics and photonics | Single-frame deep-learning super-resolution microscopy for intracellular dynamics imaging | Single-molecule localization microscopy (SMLM) can be used to resolve subcellular structures and achieve a tenfold improvement in spatial resolution compared to that obtained by conventional fluorescence microscopy. However, the separation of single-molecule fluorescence events that requires thousands of frames dramati... | [
{
"section": "Introduction",
"content": "Live-cell fluorescence imaging, requiring both low phototoxic illumination and a high imaging speed, is usually performed with a wide-field (WF) fluorescence microscope [1] . The spatial resolution of a conventional fluorescence microscope is limited by diffraction a... | This research proposes a single-frame super-resolution microscopy (SFSRM) approach that utilizes a neural network-based method to directly transform a single diffraction-limited image into a super-resolution image with up to a 10-fold resolution improvement, enabling high-speed, noninvasive, time-resolved imaging of li... | [
"single-frame super-resolution microscopy",
"SFSRM",
"neural network-based method",
"diffraction-limited image",
"super-resolution image",
"resolution improvement",
"high-speed imaging",
"noninvasive imaging",
"time-resolved imaging",
"live-cell dynamics",
"vesicle transport",
"endosome fusion... | [
{
"idx": "1.",
"title": "Stephens, D. J. & Allan, V. J. Light microscopy techniques for live cell imaging.Science300, 82–86 (2003).",
"link": "https://doi.org/10.1126%2Fscience.1082160"
},
{
"idx": "2.",
"title": "Gustafsson, M. G. Surpassing the lateral resolution limit by a factor of two u... |
s41467-023-38600-8 | Energy science and technology | Electric double layer-mediated polarization field for optimizing photogenerated carrier dynamics and thermodynamics | Photocatalytic hydrogen evolution efficiency is limited due to unfavorable carrier dynamics and thermodynamic performance. Here, we propose to introduce electronegative molecules to build an electric double layer (EDL) to generate a polarization field instead of the traditional built-in electric field to improve carrie... | [
{
"section": "Introduction",
"content": "Photocatalytic hydrogen evolution by water splitting is considered to be an important strategy to solve the energy and environmental crisis [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] . However, the solar-to-hydrogen (STH) efficiency is quite low due t... | The core research idea is to improve photocatalytic hydrogen evolution by constructing an electrical double layer (EDL) with a negative electrode outside on the surface of CuNi through ester group modification, thereby enhancing carrier dynamics and reducing the Gibbs free energy of hydrogen adsorption for more efficie... | [
"photocatalytic hydrogen evolution",
"electrical double layer",
"negative electrode",
"CuNi",
"ester group modification",
"carrier dynamics",
"Gibbs free energy",
"hydrogen adsorption",
"solar-to-hydrogen conversion"
] | [
{
"idx": "1.",
"title": "Ma, F. K. et al. 0D/2D nanocomposite visible light photocatalyst for highly stable and efficient hydrogen generation via recrystallization of CdS on MoS2nanosheets.Nano Energy27, 466–474 (2016).",
"link": "https://doi.org/10.1016%2Fj.nanoen.2016.07.014"
},
{
"idx": "2.",... |
ncomms14913 | Engineering | Molecular level detection and localization of mechanical damage in collagen enabled by collagen hybridizing peptides | Mechanical injury to connective tissue causes changes in collagen structure and material behaviour, but the role and mechanisms of molecular damage have not been established. In the case of mechanical subfailure damage, no apparent macroscale damage can be detected, yet this damage initiates and potentiates in patholog... | [
{
"section": "Introduction",
"content": "Many painful and physically debilitating conditions involve subfailure tissue damage, or mechanical damage to seemingly intact connective tissues. Examples include rotator cuff disease [1] , [2] , tendinosis [3] , [4] , whiplash [5] , [6] and low back pain [6] , [7] ... | The core research idea is to investigate whether carboxyfluorescein-labeled collagen hybridizing peptide (CF-CHP) can specifically detect, localize, and quantify molecular-level damage to collagen caused by mechanical loading in tendon tissues, and to use this technique alongside steered molecular dynamics simulations ... | [
"carboxyfluorescein-labeled collagen hybridizing peptide",
"CF-CHP",
"collagen",
"mechanical loading",
"tendon tissues",
"molecular-level damage",
"steered molecular dynamics simulations",
"subfailure loading",
"molecular mechanisms",
"detection",
"localization",
"quantification",
"investiga... | [
{
"idx": "1",
"title": "Tytherleigh-Strong, G., Hirahara, A. & Miniaci, A. Rotator cuff disease.Curr. Opin. Rheumatol.13, 135–145 (2001).",
"link": "https://doi.org/10.1097%2F00002281-200103000-00007"
},
{
"idx": "2",
"title": "Nho, S. J., Yadav, H., Shindle, M. K. & Macgillivray, J. D. Rota... |
s41467-022-30761-2 | Mathematics and computing | Towards artificial general intelligence via a multimodal foundation model | The fundamental goal of artificial intelligence (AI) is to mimic the core cognitive activities of human. Despite tremendous success in the AI research, most of existing methods have only single-cognitive ability. To overcome this limitation and take a solid step towards artificial general intelligence (AGI), we develop... | [
{
"section": "Introduction",
"content": "Science fictions and sci-fi films, that describe highly intelligent computer minds, robots, or even human-shaped ones, can be said to understand or have primitive cognitive abilities analogous to human intelligence. Since this form of human-level artificial intellige... | The core research idea is to develop a large-scale multimodal foundation model named BriVL, pre-trained through self-supervised learning on 650 million weakly correlated image-text pairs crawled from the web, using a two-tower architecture and cross-modal contrastive learning with a momentum-based negative sample queue... | [
"large-scale multimodal foundation model",
"BriVL",
"self-supervised learning",
"650 million weakly correlated image-text pairs",
"two-tower architecture",
"cross-modal contrastive learning",
"momentum-based negative sample queue",
"cross-modal understanding",
"generalization",
"imagination abilit... | [
{
"idx": "1.",
"title": "Goertzel, B. Artificial general intelligence: Concept, state of the art, and future prospects.J. Artif. Gen. Intell.5, 1–48 (2014).",
"link": "https://doi.org/10.2478%2Fjagi-2014-0001"
},
{
"idx": "2.",
"title": "LeCun, Y., Bengio, Y. & Hinton, G. Deep learning.Natur... |
s41467-019-13406-9 | Physics | Detection of the 5p– 4forbital crossing and its optical clock transition in Pr9+ | Recent theoretical works have proposed atomic clocks based on narrow optical transitions in highly charged ions. The most interesting candidates for searches of physics beyond the Standard Model are those which occur at rare orbital crossings where the shell structure of the periodic table is reordered. There are only ... | [
{
"section": "Introduction",
"content": "Current single ion clocks reach fractional frequency uncertainties \\(\\delta \\nu /\\nu\\) around \\(1{0}^{-18}\\) , and enable sensitive tests of relativity and searches for potential variations in fundamental constants [1] , [2] , [3] , [4] , [5] . These experimen... | The core research idea is to develop an ultra-precise atomic clock and probe physics beyond the Standard Model by exploiting the highly sensitive 5p² ³P₀ – 5p4f ³G₃ magnetic octupole (M3) transition in Sn-like Pr⁹⁺, a highly charged ion with reduced sensitivity to external perturbations, where the clock transition freq... | [
"ultra-precise atomic clock",
"Standard Model",
"5p² ³P₀ – 5p4f ³G₃ magnetic octupole transition",
"Sn-like Pr⁹⁺",
"highly charged ion",
"magnetic-dipole transition",
"Rydberg-Ritz combination principle",
"fine-structure constant",
"local Lorentz invariance",
"external perturbations",
"clock tra... | [
{
"idx": "1.",
"title": "Rosenband, T. et al. Frequency ratio of Al\\({}^{+}\\)and Hg\\({}^{+}\\)single-ion optical clocks; metrology at the 17th decimal place.Science319, 1808–1812 (2008).",
"link": "https://doi.org/10.1126%2Fscience.1154622"
},
{
"idx": "2.",
"title": "Chou, C. W., Hume, D... |
s41467-018-06503-8 | Materials science | Tailoring manganese oxide with atomic precision to increase surface site availability for oxygen reduction catalysis | Controlling the structure of catalysts at the atomic level provides an opportunity to establish detailed understanding of the catalytic form-to-function and realize new, non-equilibrium catalytic structures. Here, advanced thin-film deposition is used to control the atomic structure of La 2/3 Sr 1/3 MnO 3 , a well-know... | [
{
"section": "Introduction",
"content": "Understanding the structure–activity relationship of a catalyst at the atomic level has been a long-standing challenge in the pursuit of a designer catalyst [1] , [2] , [3] , [4] , [5] . Traditionally, researchers identify this relationship by varying the bulk struct... | The core research idea is to develop a method for independently tuning the surface and sub-surface structure and composition of catalysts through advanced thin-film deposition, specifically using molecular beam epitaxy to layer-by-layer grow La2/3Sr1/3MnO3 films with controlled placement of SrMnO3 in either the surface... | [
"method",
"advanced thin-film deposition",
"molecular beam epitaxy",
"La2/3Sr1/3MnO3 films",
"SrMnO3",
"surface structure",
"sub-surface structure",
"catalytic activity",
"oxygen reduction reaction",
"electronic effects",
"structural effects",
"surface reactions",
"layer-by-layer growth",
... | [
{
"idx": "1.",
"title": "Stamenkovic, V. R., Strmcnik, D., Lopes, P. P. & Markovic, N. M. Energy and fuels from electrochemical interfaces.Nat. Mater.16, 57–69 (2017).",
"link": "https://doi.org/10.1038%2Fnmat4738"
},
{
"idx": "2.",
"title": "Bruce, P. G., Freunberger, Sa, Hardwick, L. J. & ... |
s41467-020-18545-y | Astronomy and planetary science | Very-Low-Frequency transmitters bifurcate energetic electron belt in near-earth space | Very-Low-Frequency (VLF) transmitters operate worldwide mostly at frequencies of 10–30 kilohertz for submarine communications. While it has been of intense scientific interest and practical importance to understand whether VLF transmitters can affect the natural environment of charged energetic particles, for decades t... | [
{
"section": "Introduction",
"content": "Ground-based Very-Low-Frequency (VLF) transmitters radiate emissions at particular frequencies mostly over the range of 10–30 kHz with transmitted power ranging from 20 kW to two megawatts [1] , [2] , [3] , [4] . While propagating mostly within the Earth-ionosphere w... | This study proposes that ground-based VLF transmitters, through their wave emissions that penetrate into the Earth's magnetosphere, can effectively scatter and remove energetic electrons—leading to the formation of a bifurcated electron belt at L-shells ~1.8–2.5—as demonstrated by Fokker-Planck simulations and direct o... | [
"ground-based VLF transmitters",
"wave emissions",
"Earth's magnetosphere",
"energetic electrons",
"bifurcated electron belt",
"L-shells",
"Fokker-Planck simulations",
"Van Allen Probes",
"near-Earth space environment",
"hazardous electron fluxes",
"Van Allen radiation belts"
] | [
{
"idx": "1.",
"title": "Diesendorf, M. O. Amplitude variations in artificially stimulated VLF emissions.Nat. Phys. Sci.236, 56–58 (1972).",
"link": "https://doi.org/10.1038%2Fphysci236056a0"
},
{
"idx": "2.",
"title": "Cohen, M. B., Inan, U. S. & Paschal, E. Sensitive broadband ELF/VLF radi... |
s41467-017-01887-5 | Physics | Round complexity in the local transformations of quantum and classical states | In distributed quantum and classical information processing, spatially separated parties operate locally on their respective subsystems, but coordinate their actions through multiple exchanges of public communication. With interaction, the parties can perform more tasks. But how the exact number and order of exchanges ... | [
{
"section": "Introduction",
"content": "One of the most fascinating aspects of quantum information is how classical communication can enhance quantum information processing. For instance, “mixed” entanglement shared between two or more parties can be “purified” when the parties are allowed to perform Local... | The core research idea is to demonstrate that for any integer $ r \geq 1 $, there exist families of quantum (resp. classical) states for which a minimum of $ r $ communication rounds is both necessary and sufficient to achieve pure-state entanglement (resp. secret shared randomness) manipulation under LOCC/LOPC protoco... | [
"integer",
"quantum state",
"classical state",
"communication round",
"pure-state entanglement",
"secret shared randomness",
"LOCC protocol",
"LOPC protocol",
"state dimension",
"alphabet size",
"universal upper bound",
"entanglement manipulation",
"secret randomness manipulation"
] | [
{
"idx": "1.",
"title": "Bennett, C. H., DiVincenzo, D. P., Smolin, J. A. & Wootters, W. K. Mixed-state entanglement and quantum error correction.Phys. Rev. A54, 3824–3851 (1996).",
"link": "https://doi.org/10.1103%2FPhysRevA.54.3824"
},
{
"idx": "3.",
"title": "Maurer, U. M. Secret key agre... |
s41467-023-38788-9 | Optics and photonics | Asynchronous locking in metamaterials of fluids of light and sound | Lattices of exciton-polariton condensates represent an attractive platform for the study and implementation of non-Hermitian bosonic quantum systems with strong non-linear interactions. The possibility to actuate on them with a time dependent drive could provide for example the means to induce resonant inter-level tran... | [
{
"section": "Introduction",
"content": "Microcavity exciton-polaritons (simply polaritons, the quantum states formed by strongly coupled excitons and photons) constitute a hybrid system [1] that displays a plethora of striking properties. These include Bose-Einstein condensation (BEC) [2] , superfluidity [... | This work proposes the development of polaromechanical metamaterials—periodic arrays of micrometer-scale intra-cavity polariton traps that co-localize polaritons and acoustic vibrations—where phonon-mediated time-dependent inter-site coupling between polariton condensates leads to asynchronous energy locking at integer... | [
"polaromechanical metamaterials",
"micrometer-scale intra-cavity polariton traps",
"polaritons",
"acoustic vibrations",
"phonon-mediated time-dependent inter-site coupling",
"polariton condensates",
"asynchronous energy locking",
"confined phonon energy",
"driven-dissipative light fluids",
"cohere... | [
{
"idx": "1.",
"title": "Kurizkia, G. et al. Quantum technologies with hybrid systems.Proc. Natl. Acad. Sci. USA112, 3866 (2015).",
"link": "https://doi.org/10.1073%2Fpnas.1419326112"
},
{
"idx": "2.",
"title": "Kasprzak, J. et al. Bose-Einstein condensation of exciton polaritons.Nature443, ... |
s41467-021-24626-3 | Astronomy and planetary science | The search for lunar mantle rocks exposed on the surface of the Moon | The lunar surface is ancient and well-preserved, recording Solar System history and planetary evolution processes. Ancient basin-scale impacts excavated lunar mantle rocks, which are expected to remain present on the surface. Sampling these rocks would provide insight into fundamental planetary processes, including dif... | [
{
"section": "Introduction",
"content": "Earth’s Moon is a keystone for understanding the formation and evolution of silicate bodies throughout the solar system and beyond. The lunar mantle is central to unraveling the history of the Moon, but our understanding of it is restricted by the limited scope of av... | The core research idea is to identify and characterize candidate mantle rock types on the lunar surface, particularly within impact basins that may have exposed materials from the Moon's interior, using a multidisciplinary approach informed by sample analyses, laboratory experiments, and numerical models to advance our... | [
"mantle rock types",
"lunar surface",
"impact basins",
"Moon's interior",
"multidisciplinary approach",
"sample analyses",
"laboratory experiments",
"numerical models",
"lunar evolution",
"crustal processes",
"interior processes"
] | [
{
"idx": "27.",
"title": "Maurice, M., Tosi, N., Schwinger, S., Breuer, D. & Kleine, T. A long-lived magma ocean on a young Moon.Sci. Adv.6, eaba8949 (2020).",
"link": "https://doi.org/10.1126%2Fsciadv.aba8949"
},
{
"idx": "29.",
"title": "Li, Y., Deschamps, F. & Tackley, P. J. The stability... |
s41467-022-31606-8 | Optics and photonics | High output mode-locked laser empowered by defect regulation in 2D Bi2O2Se saturable absorber | Atomically thin Bi 2 O 2 Se has emerged as a novel two-dimensional (2D) material with an ultrabroadband nonlinear optical response, high carrier mobility and excellent air stability, showing great potential for the realization of optical modulators. Here, we demonstrate a femtosecond solid-state laser at 1.0 µm with Bi... | [
{
"section": "Introduction",
"content": "Ultrashort pulsed lasers with high output power have been proven indispensable in many applications, such as biomedicine [1] , spectroscopy [2] and precision micro/nanoprocessing [3] . Saturable absorber (SA) is the crucial optical element to trigger pulsed operation... | This research proposes that defect engineering in 2D bismuth oxyselenide (Bi₂O₂Se) saturable absorbers can significantly enhance their saturation intensity and ultrafast carrier dynamics, enabling high-performance mode-locked all-solid-state lasers with both high output power and ultrashort pulse duration. | [
"defect engineering",
"2D bismuth oxyselenide",
"Bi₂O₂Se saturable absorbers",
"saturation intensity",
"ultrafast carrier dynamics",
"high-performance mode-locked all-solid-state lasers",
"high output power",
"ultrashort pulse duration"
] | [
{
"idx": "1.",
"title": "Santos, H. D. A. et al. Ultrafast photochemistry produces superbright short-wave infrared dots for low-dose in vivo imaging.Nat. Commun.11, 2933 (2020).",
"link": "https://doi.org/10.1038%2Fs41467-020-16333-2"
},
{
"idx": "2.",
"title": "Ding, T. et al. Measuring the... |
s41467-023-37138-z | Nanoscience and technology | Enhanced nonlinear optomechanics in a coupled-mode photonic crystal device | The nonlinear component of the optomechanical interaction between light and mechanical vibration promises many exciting classical and quantum mechanical applications, but is generally weak. Here we demonstrate enhancement of nonlinear optomechanical measurement of mechanical motion by using pairs of coupled optical and... | [
{
"section": "Introduction",
"content": "The field of cavity optomechanics studies the interaction between a light field and mechanical vibration. On the one hand, the optomechanical interaction imprints the mechanical motion onto the light field, enabling extremely precise optical detection of position and... | The core research idea is to develop and demonstrate a coupled-mode optomechanical crystal device that fulfills the conditions for enhanced nonlinear optomechanical coupling and improved linear transduction efficiency, enabling a direct experimental comparison between single-mode and coupled-mode configurations to vali... | [
"coupled-mode optomechanical crystal device",
"enhanced nonlinear optomechanical coupling",
"improved linear transduction efficiency",
"single-mode configuration",
"coupled-mode configuration",
"multimode systems",
"stronger nonlinear quantum effects",
"more efficient optomechanical interactions",
"... | [
{
"idx": "1.",
"title": "Mason, D., Chen, J., Rossi, M., Tsaturyan, Y. & Schliesser, A. Continuous force and displacement measurement below the standard quantum limit.Nat. Phys.15, 745–749 (2019).",
"link": "https://doi.org/10.1038%2Fs41567-019-0533-5"
},
{
"idx": "2.",
"title": "Muhonen, J.... |
ncomms3556 | Optics and photonics | Demonstration of a novel focusing small-angle neutron scattering instrument equipped with axisymmetric mirrors | Small-angle neutron scattering (SANS) is the most significant neutron technique in terms of impact on science and engineering. However, the basic design of SANS facilities has not changed since the technique’s inception about 40 years ago, as all SANS instruments, save a few, are still designed as pinhole cameras. Here... | [
{
"section": "Introduction",
"content": "Small-angle neutron scattering (SANS) instruments are found at nearly every neutron-research facility, where they serve a sizable and diverse community of users. The basic instrument design remains the same, as original small-angle-scattering instruments were describ... | The core research idea is to develop and demonstrate a novel small-angle neutron scattering (SANS) instrument based on axisymmetric focusing mirrors called Wolter mirrors, which function as achromatic lenses with short focal lengths and high throughput, enabling a transformative improvement in signal rate and resolutio... | [
"small-angle neutron scattering",
"SANS instrument",
"axisymmetric focusing mirrors",
"Wolter mirrors",
"achromatic lenses",
"short focal lengths",
"high throughput",
"signal rate",
"resolution",
"pinhole-camera designs",
"microscope-like focusing systems"
] | [
{
"idx": "2",
"title": "Littrell, K. C. A comparison of different methods for improving flux and resolution on SANS instruments.Nucl. Instr. Meth. A529, 22–27 (2004).",
"link": "https://doi.org/10.1016%2Fj.nima.2004.04.151"
},
{
"idx": "3",
"title": "Kearney, P., Klein, A., Opat, G. & Gähler... |
s41467-017-02804-6 | Physics | Room temperature magneto-optic effect in silicon light-emitting diodes | In weakly spin–orbit coupled materials, the spin-selective nature of recombination can give rise to large magnetic-field effects, e.g. on the electro-luminescence of molecular semiconductors. Although silicon has weak spin–orbit coupling, observing spin-dependent recombination through magneto-electroluminescence is cha... | [
{
"section": "Introduction",
"content": "Spintronic effects in systems with weak spin–orbit coupling have attracted considerable attention due to their rich fundamental physics and potential for device applications [1] , [2] , [3] , [4] . A class of these effects can be measured optically [5] , [6] , [7] , ... | The core research idea is to investigate spin-dependent recombination in silicon light-emitting diodes (SiLEDs) fabricated using gas immersion laser doping (GILD), a technique that enables efficient emission and planar geometry control, in order to probe the spin statistics of carriers in silicon through magneto-electr... | [
"spin-dependent recombination",
"silicon light-emitting diodes",
"SiLEDs",
"gas immersion laser doping",
"GILD",
"efficient emission",
"planar geometry control",
"spin statistics",
"carriers",
"silicon",
"magneto-electroluminescence effects",
"spintronic applications",
"optoelectronic applic... | [
{
"idx": "2.",
"title": "Jansen, R. Silicon spintronics.Nat. Mater.11, 400–408 (2012).",
"link": "https://doi.org/10.1038%2Fnmat3293"
},
{
"idx": "3.",
"title": "Sun, Y. et al. Management of singlet and triplet excitons for efficient white organic light-emitting devices.Nature440, 908 (2006)... |
s41467-020-20547-9 | Materials science | Nanohole-boosted electron transport between nanomaterials and bacteria as a concept for nano–bio interactions | Biofilms contribute to bacterial infection and drug resistance and are a serious threat to global human health. Antibacterial nanomaterials have attracted considerable attention, but the inhibition of biofilms remains a major challenge. Herein, we propose a nanohole-boosted electron transport (NBET) antibiofilm concept... | [
{
"section": "Introduction",
"content": "Bacterial resistance caused by the overuse of antibiotics causes more serious harm than cancer [1] . The World Health Organization (WHO) has reported that approximately 700,000 people die each year due to drug-resistant bacterial infections [2] , and 10 million peopl... | The core research idea is to develop nanohole-enriched two-dimensional transition metal disulfide (MoS₂) nanosheets that utilize nanohole-boosted electron transport to effectively destroy bacterial extracellular polymeric substances and biofilms, thereby inhibiting Staphylococcus aureus infection with enhanced antibact... | [
"nanohole-enriched two-dimensional transition metal disulfide nanosheets",
"MoS₂ nanosheets",
"nanohole-boosted electron transport",
"bacterial extracellular polymeric substances",
"biofilms",
"Staphylococcus aureus infection",
"enhanced antibacterial efficacy",
"biocompatibility",
"resistance preve... | [
{
"idx": "1.",
"title": "Bow, E. There should be no ESKAPE for febrile neutropenic cancer patients: the dearth of effective antibacterial drugs threatens anticancer efficacy.J. Antimicrob. Chemother.68, 492–495 (2013).",
"link": "https://doi.org/10.1093%2Fjac%2Fdks512"
},
{
"idx": "2.",
"tit... |
s41467-017-01996-1 | Physics | Partially native intermediates mediate misfolding of SOD1 in single-molecule folding trajectories | Prion-like misfolding of superoxide dismutase 1 (SOD1) is associated with the disease ALS, but the mechanism of misfolding remains unclear, partly because misfolding is difficult to observe directly. Here we study the most misfolding-prone form of SOD1, reduced un-metallated monomers, using optical tweezers to measure ... | [
{
"section": "Introduction",
"content": "Cu/Zn-superoxide dismutase 1 (SOD1) is a ubiquitous cytosolic antioxidant that protects cells against damage from superoxide [1] . How SOD1 folds is of particular interest because its misfolding has been linked to amyotrophic lateral sclerosis (ALS), a fatal neurodeg... | Using single-molecule force spectroscopy, this study investigates the folding and misfolding pathways of monomeric, reduced apo-SOD1 to determine how native folding and misfolding are related, with the hypothesis that misfolded states branch from the native pathway at specific intermediate states following the formatio... | [
"single-molecule force spectroscopy",
"monomeric",
"reduced apo-SOD1",
"folding pathways",
"misfolding pathways",
"native folding",
"misfolded states",
"intermediate states",
"stable native-like core"
] | [
{
"idx": "1.",
"title": "Crapo, J. D., Oury, T., Rabouille, C., Slot, J. W. & Chang, L. Y. Copper,zinc superoxide dismutase is primarily a cytosolic protein in human cells.Proc. Natl Acad. Sci. USA89, 10405–10409 (1992).",
"link": "https://doi.org/10.1073%2Fpnas.89.21.10405"
},
{
"idx": "3.",
... |
s41467-024-52206-8 | Materials science | Quantum geometry quadrupole-induced third-order nonlinear transport in antiferromagnetic topological insulator MnBi2Te4 | The study of quantum geometry effects in materials has been one of the most important research directions in recent decades. The quantum geometry of a material is characterized by the quantum geometric tensor of the Bloch states. The imaginary part of the quantum geometry tensor gives rise to the Berry curvature while ... | [
{
"section": "Introduction",
"content": "The study of the intricate relationship between quantum geometric properties and high-order nonlinear transport properties in topological materials is at the frontier of condensed matter physics research [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [1... | The core research idea is to investigate how the quantum geometric properties—specifically the Berry curvature and quantum metric quadrupoles—induce third-order nonlinear transport phenomena in bulk antiferromagnetic topological insulator MnBi₂Te₄, where inversion symmetry suppresses second-order effects, thereby revea... | [
"quantum geometric properties",
"Berry curvature",
"quantum metric quadrupoles",
"third-order nonlinear transport phenomena",
"bulk antiferromagnetic topological insulator",
"MnBi₂Te₄",
"inversion symmetry",
"second-order effects",
"magnetic order",
"crystal symmetry",
"high-order nonlinear anom... | [
{
"idx": "1.",
"title": "Provost, J. P. & Vallee, G. Riemannian structure on manifolds of quantum states.Commun. Math. Phys.76, 289–301 (1980).",
"link": "https://link.springer.com/doi/10.1007/BF02193559"
},
{
"idx": "2.",
"title": "Gao, Y., Yang, S. A. & Niu, Q. Field induced positional shi... |
s41467-023-44114-0 | Nanoscience and technology | Parity-conserving Cooper-pair transport and ideal superconducting diode in planar germanium | Superconductor/semiconductor hybrid devices have attracted increasing interest in the past years. Superconducting electronics aims to complement semiconductor technology, while hybrid architectures are at the forefront of new ideas such as topological superconductivity and protected qubits. In this work, we engineer th... | [
{
"section": "Introduction",
"content": "III–V semiconductors have become the materials of choice for realizing high-quality hybrid devices, due to the possibility of growing epitaxial Al on top of them [1] . Gate-tunable superconducting and Andreev spin qubits [2] , [3] , [4] , [5] , [6] , parametric ampli... | The core research idea is to establish Ge/SiGe heterostructures as an alternative platform to III–V materials for hybrid devices and microwave experiments by using a shallow quantum well, enabling the coexistence of semiconductor and superconducting qubits while overcoming limitations imposed by nuclear spins in III–V ... | [
"Ge/SiGe heterostructures",
"III–V materials",
"hybrid devices",
"microwave experiments",
"shallow quantum well",
"semiconductor qubits",
"superconducting qubits",
"nuclear spins"
] | [
{
"idx": "1.",
"title": "Krogstrup, P. et al. Epitaxy of semiconductor–superconductor nanowires.Nat. Mater.14, 400–406 (2015).",
"link": "https://doi.org/10.1038%2Fnmat4176"
},
{
"idx": "2.",
"title": "Larsen, T. W. et al. Semiconductor-nanowire-based superconducting qubit.Phys. Rev. Lett.11... |
s41467-023-36675-x | Energy science and technology | Metallic glass-based triboelectric nanogenerators | Surface wear is a major hindrance in the solid/solid interface of triboelectric nanogenerators (TENG), severely affecting their output performance and stability. To reduce the mechanical input and surface wear, solid/liquid-interface alternatives have been investigated; however, charge generation capability is still lo... | [
{
"section": "Introduction",
"content": "To address the growing energy demands, various sustainable energy harvesting technologies, such as solar cell [1] , [2] , piezoelectric nanogenerators [3] , [4] , thermoelectric nanogenerators [5] , [6] and triboelectric nanogenerators (TENGs) [7] , [8] , etc., have ... | The core research idea is to enhance the triboelectrification efficiency and durability of triboelectric nanogenerators (TENGs) by employing metallic glass (MG) with disordered atomic structures as the triboelectric interface, leveraging its low atomic density, low friction coefficient, and excellent wear- and corrosio... | [
"triboelectrification efficiency",
"triboelectric nanogenerators",
"metallic glass",
"disordered atomic structures",
"low atomic density",
"low friction coefficient",
"wear-resistance",
"corrosion-resistance",
"surface charge density",
"humidity resistance",
"power output",
"theoretical limit"... | [
{
"idx": "1.",
"title": "Al-Rousan, N., Isa, N. A. M. & Desa, M. K. M. Advances in solar photovoltaic tracking systems: a review.Renew. Sustain. Energy Rev.82, 2548–2569 (2018).",
"link": "https://doi.org/10.1016%2Fj.rser.2017.09.077"
},
{
"idx": "3.",
"title": "Briscoe, J. & Dunn, S. Piezoe... |
s41467-021-23398-0 | Materials science | Laser-excited elastic guided waves reveal the complex mechanics of nanoporous silicon | Nanoporosity in silicon leads to completely new functionalities of this mainstream semiconductor. A difficult to assess mechanics has however significantly limited its application in fields ranging from nanofluidics and biosensorics to drug delivery, energy storage and photonics. Here, we present a study on laser-excit... | [
{
"section": "Introduction",
"content": "Exploitation of new “nano”-physical effects of matter confined in nanoporous media, like novel phase behaviour and transport properties, is already beginning to revolutionise technologies ranging from water filtering [1] and drug delivery [2] to energy harvesting, tr... | The core research idea is to propose a thorough assessment of the effective mechanical behaviour of nanoporous silicon by exploiting the characteristics of elastic guided waves and zero group velocity modes combined with waveguide modelling and inverse identification, aiming to evaluate the anisotropic stiffness tensor... | [
"nanoporous silicon",
"elastic guided waves",
"zero group velocity modes",
"waveguide modelling",
"inverse identification",
"anisotropic stiffness tensor",
"pore conicity",
"mechanical behaviour",
"liquid-infused states",
"functionalisation-induced changes",
"nanoporous systems"
] | [
{
"idx": "1.",
"title": "Bocquet, L. Nanofluidics coming of age.Nat. Mater.19, 254–256 (2020).",
"link": "https://doi.org/10.1038%2Fs41563-020-0625-8"
},
{
"idx": "2.",
"title": "Tzur-Balter, A., Shatsberg, Z., Beckerman, M., Segal, E. & Artzi, N. Mechanism of erosion of nanostructured porou... |
s41467-018-04718-3 | Physics | Finding influential nodes for integration in brain networks using optimal percolation theory | Global integration of information in the brain results from complex interactions of segregated brain networks. Identifying the most influential neuronal populations that efficiently bind these networks is a fundamental problem of systems neuroscience. Here, we apply optimal percolation theory and pharmacogenetic interv... | [
{
"section": "Introduction",
"content": "A fundamental question in systems neuroscience is how the brain integrates distributed and specialized networks into a coherent information processing system [1] , [2] . Brain networks are considered integrated when they exhibit long-range correlated activity over di... | The core research idea is to identify essential nodes in a memory-related brain network using different centrality measures, particularly focusing on whether "weak nodes" identified by integrative centralities such as collective influence and betweenness centrality, rather than traditional hub-centric measures, are cri... | [
"memory-related brain network",
"centrality measures",
"integrative centralities",
"collective influence",
"betweenness centrality",
"traditional hub-centric measures",
"long-range functional interactions",
"memory encoding",
"memory consolidation",
"in vivo pharmacogenetic inactivation",
"roden... | [
{
"idx": "1.",
"title": "Dehaene, S. & Naccache, L. Towards a cognitive neuroscience of consciousness: basic evidence and a workspace framework.Cognition79, 1–37 (2001).",
"link": "https://doi.org/10.1016%2FS0010-0277%2800%2900123-2"
},
{
"idx": "2.",
"title": "Sporns, O. Network attributes ... |
s41467-023-37060-4 | Materials science | Tailoring high-energy storage NaNbO3-based materials from antiferroelectric to relaxor states | Reversible field-induced phase transitions define antiferroelectric perovskite oxides and lay the foundation for high-energy storage density materials, required for future green technologies. However, promising new antiferroelectrics are hampered by transition´s irreversibility and low electrical resistivity. Here, we ... | [
{
"section": "Introduction",
"content": "The global goal to achieve net-zero CO 2 emissions by the middle of the century requires the development of efficient and environmentally benign technologies that can convert energy from renewable and sustainable resources such as sunlight and wind. Among the key com... | The core research idea is to design a series of NaNbO 3 -based materials with tailorable functional properties ranging from antiferroelectric to relaxor through compositional modification and defect chemistry engineering, aiming to achieve well-defined double polarization hysteresis loops, significantly enhanced energy... | [
"NaNbO3-based materials",
"compositional modification",
"defect chemistry engineering",
"double polarization hysteresis loops",
"energy-storage density",
"energy-storage efficiency",
"AFE–FE phase transition mechanisms",
"structural changes",
"functional properties",
"antiferroelectric",
"relaxo... | [
{
"idx": "1.",
"title": "Pan, H. et al. Ultrahigh energy storage in superparaelectric relaxor ferroelectrics.Science374, 100–104 (2021).",
"link": "https://doi.org/10.1126%2Fscience.abi7687"
},
{
"idx": "2.",
"title": "Li, J. et al. Grain-orientation-engineered multilayer ceramic capacitors ... |
s41467-017-02778-5 | Nanoscience and technology | DNA-assisted swarm control in a biomolecular motor system | In nature, swarming behavior has evolved repeatedly among motile organisms because it confers a variety of beneficial emergent properties. These include improved information gathering, protection from predators, and resource utilization. Some organisms, e.g., locusts, switch between solitary and swarm behavior in respo... | [
{
"section": "Introduction",
"content": "Nature assembles its wide variety of structures through self-assembly and self-organization where local interactions among the individual components play pivotal roles [1] . A striking example is the swarming of living organisms in which fascinating, large scale and ... | The core research idea is to demonstrate that DNA can serve as a universal, programmable interface to control the swarming behavior of kinesin-propelled microtubules by using single-stranded DNA tethered to microtubules and DNA cross-linkers as input signals for information processing, enabling reversible control of sw... | [
"DNA",
"microtubules",
"kinesin-propelled microtubules",
"single-stranded DNA",
"DNA cross-linkers",
"swarming behavior",
"programmable interface",
"information processing",
"photoresponsive elements",
"azobenzene",
"reversible control",
"swarming modes",
"tethered DNA"
] | [
{
"idx": "1.",
"title": "Whitesides, G. M. & Grzybowski, B. Self-assembly at all scales.Science295, 2418–2421 (2002).",
"link": "https://doi.org/10.1126%2Fscience.1070821"
},
{
"idx": "3.",
"title": "Beshers, S. N. & Fewell, J. H. Models of division of labor in social insects.Annu. Rev. Ento... |
s41467-022-29287-4 | Physics | Improving qubit coherence using closed-loop feedback | Superconducting qubits are a promising platform for building a larger-scale quantum processor capable of solving otherwise intractable problems. In order for the processor to reach practical viability, the gate errors need to be further suppressed and remain stable for extended periods of time. With recent advances in ... | [
{
"section": "Introduction",
"content": "High-fidelity single- and two-qubit gates are a prerequisite for high-depth circuits and quantum error correction. For many qubit modalities, including superconducting qubits, the qubit frequencies and their controls are subject to temporally correlated noise—most no... | The core research idea is to use active feedback control to stabilize the frequency drift of a flux-tunable superconducting transmon qubit and thereby suppress its temporal frequency fluctuations, resulting in improved coherence times and gate fidelities that remain stable over extended periods, even when operating far... | [
"active feedback control",
"frequency drift",
"flux-tunable superconducting transmon qubit",
"temporal frequency fluctuations",
"coherence times",
"gate fidelities",
"flux sweet spot"
] | [
{
"idx": "1.",
"title": "Paladino, E., Galperin, Y. M., Falci, G. & Altshuler, B. L. 1/f noise: Implications for solid-state quantum information.Rev. Mod. Phys.86, 361–418 (2014).",
"link": "https://doi.org/10.1103%2FRevModPhys.86.361"
},
{
"idx": "2.",
"title": "Steck, D. A., Jacobs, K., Ma... |
s41467-018-05558-x | Materials science | Coulomb-bound four- and five-particle intervalley states in an atomically-thin semiconductor | As hosts for tightly-bound electron-hole pairs carrying quantized angular momentum, atomically-thin semiconductors of transition metal dichalcogenides (TMDCs) provide an appealing platform for optically addressing the valley degree of freedom. In particular, the valleytronic properties of neutral and charged excitons i... | [
{
"section": "Introduction",
"content": "Many-body correlation is a fascinating topic that has attracted decades of experimental and theoretical investigation. In atomic physics, the positronium molecule, consisting of two electrons and two positrons bound together, took more than half a century to confirm ... | The core research idea is to investigate the experimental observation and unique spin-valley configurations of biexcitons and a five-particle bound state (composed of a bright trion and a dark exciton) in monolayer tungsten diselenide (WSe₂), a two-dimensional transition metal dichalcogenide, and to demonstrate that th... | [
"monolayer tungsten diselenide",
"two-dimensional transition metal dichalcogenide",
"biexcitons",
"five-particle bound state",
"bright trion",
"dark exciton",
"spin-valley configurations",
"valley polarization properties",
"experimental observation",
"many-body physics",
"valleytronics",
"spin... | [
{
"idx": "1.",
"title": "Cassidy, D. B. & Mills, A. P. The production of molecular positronium.Nature449, 195–197 (2007).",
"link": "https://doi.org/10.1038%2Fnature06094"
},
{
"idx": "2.",
"title": "Drüppel, M., Deilmann, T., Krüger, P. & Rohlfing, M. Diversity of trion states and substrate... |
s41467-022-29798-0 | Optics and photonics | Full-range birefringence control with piezoelectric MEMS-based metasurfaces | Dynamic polarization control is crucial for emerging highly integrated photonic systems with diverse metasurfaces being explored for its realization, but efficient, fast, and broadband operation remains a cumbersome challenge. While efficient optical metasurfaces (OMSs) involving liquid crystals suffer from inherently ... | [
{
"section": "Introduction",
"content": "Recent years brought explosive growth of OMS applications in terms of both diversity and sophistication [1] , especially with OMS configurations becoming dynamic, i.e., offering not only spatial but also temporal control over transmitted and reflected optical fields ... | The core research idea is to demonstrate electrically controlled full-range birefringence through the integration of a plasmonic optical metasurface with anisotropic meta-atoms and a thin-film piezoelectric MEMS mirror, enabling dynamic polarization control in reflection with continuously variable anisotropy and comple... | [
"electrically controlled full-range birefringence",
"plasmonic optical metasurface",
"anisotropic meta-atoms",
"thin-film piezoelectric MEMS mirror",
"dynamic polarization control",
"reflection",
"continuously variable anisotropy",
"complete encircling",
"Poincaré sphere"
] | [
{
"idx": "1.",
"title": "Ding, F., Pors, A. & Bozhevolnyi, S. I. Gradient metasurfaces: a review of fundamentals and applications.Rep. Prog. Phys.81, 026401 (2018).",
"link": "https://doi.org/10.1088%2F1361-6633%2Faa8732"
},
{
"idx": "2.",
"title": "Shaltout, A. M., Shalaev, V. M. & Brongers... |
ncomms13682 | Nanoscience and technology | Miniature optical planar camera based on a wide-angle metasurface doublet corrected for monochromatic aberrations | Optical metasurfaces are two-dimensional arrays of nano-scatterers that modify optical wavefronts at subwavelength spatial resolution. They are poised to revolutionize optics by enabling complex low-cost systems where multiple metasurfaces are lithographically stacked and integrated with electronics. For imaging applic... | [
{
"section": "Introduction",
"content": "Optical systems such as cameras, spectrometers and microscopes are conventionally made by assembling discrete bulk optical components like lenses, gratings and filters. The optical components are manufactured separately using cutting, polishing and grinding, and have... | The core research idea is to correct monochromatic aberrations in metasurface lenses by cascading two metasurfaces to form a doublet lens, enabling a wide field of view and facilitating the development of ultra-slim, low f-number cameras through vertical integration with image sensors. | [
"monochromatic aberrations",
"metasurface lenses",
"cascading",
"doublet lens",
"wide field of view",
"ultra-slim cameras",
"low f-number cameras",
"vertical integration",
"image sensors"
] | [
{
"idx": "1",
"title": "Kildishev, A. V., Boltasseva, A. & Shalaev, V. M. Planar photonics with metasurfaces.Science339, 1232009 (2013).",
"link": "https://doi.org/10.1126%2Fscience.1232009"
},
{
"idx": "2",
"title": "Yu, N. & Capasso, F. Flat optics with designer metasurfaces.Nat. Mater.13,... |
s41467-020-16108-9 | Mathematics and computing | Accurate deep neural network inference using computational phase-change memory | In-memory computing using resistive memory devices is a promising non-von Neumann approach for making energy-efficient deep learning inference hardware. However, due to device variability and noise, the network needs to be trained in a specific way so that transferring the digitally trained weights to the analog resist... | [
{
"section": "Introduction",
"content": "Deep neural networks (DNNs) have revolutionized the field of artificial intelligence and have achieved unprecedented success in cognitive tasks such as image and speech recognition. Platforms for deploying the trained model of such networks and performing inference i... | The core research idea is to improve the resilience of deep neural networks to hardware non-idealities in analog in-memory computing platforms by injecting noise into synaptic weights during software-based training, enabling accurate and robust deployment on a variety of chips with minimal hardware-specific characteriz... | [
"deep neural networks",
"hardware non-idealities",
"analog in-memory computing platforms",
"noise injection",
"synaptic weights",
"software-based training",
"accurate deployment",
"robust deployment",
"chip-specific retraining",
"hardware-specific characterization",
"resilient neural networks",
... | [
{
"idx": "5.",
"title": "Wang, Z. et al. Resistive switching materials for information processing.Nat. Rev. Mater.5, 173–195 (2020).",
"link": "https://doi.org/10.1038%2Fs41578-019-0159-3"
},
{
"idx": "6.",
"title": "Merrikh-Bayat, F. et al. High-performance mixed-signal neurocomputing with ... |
s41467-020-15506-3 | Astronomy and planetary science | Plasmapause surface wave oscillates the magnetosphere and diffuse aurora | Energy circulation in geospace lies at the heart of space weather research. In the inner magnetosphere, the steep plasmapause boundary separates the cold dense plasmasphere, which corotates with the planet, from the hot ring current/plasma sheet outside. Theoretical studies suggested that plasmapause surface waves rela... | [
{
"section": "Introduction",
"content": "The solar wind and embedded interplanetary magnetic field (IMF) play a crucial role in driving terrestrial magnetospheric energy dissipation. When the IMF has a southward component, the mass and energy in solar winds enter into and convect in the magnetosphere and ar... | This study proposes that plasmapause surface waves (PSW), generated during geomagnetic storms, act as a systematic ultra-low frequency (ULF) wave driver in the magnetosphere, and that these waves are physically connected to the formation of sawtooth-shaped auroral undulations (SA) on the equatorward edge of the diffuse... | [
"plasmapause surface waves",
"geomagnetic storms",
"ultra-low frequency wave driver",
"magnetosphere",
"sawtooth-shaped auroral undulations",
"equatorward edge",
"diffuse aurora",
"energy dissipation",
"ionosphere-magnetosphere coupling system"
] | [
{
"idx": "1.",
"title": "Lyon, J. G. The solar wind-magnetosphere-ionosphere system.Science288, 1987–1991 (2000).",
"link": "https://doi.org/10.1126%2Fscience.288.5473.1987"
},
{
"idx": "2.",
"title": "Angelopoulos, V. et al. Tail reconnection triggering substorm onset.Science321, 931–935 (2... |
s41467-023-41236-3 | Physics | An electronic origin of charge order in infinite-layer nickelates | A charge order (CO) with a wavevector \({{{{{{{\bf{q}}}}}}}}\simeq \left(\frac{1}{3},\, 0,\, 0\right)\) is observed in infinite-layer nickelates. Here we use first-principles calculations to demonstrate a charge-transfer-driven CO mechanism in infinite-layer nickelates, which leads to a characteristic Ni 1+ -Ni 2+ -Ni ... | [
{
"section": "Introduction",
"content": "Motivated by cuprates, superconductivity was proposed [1] , [2] and has been recently discovered in hole-doped infinite-layer nickelates that contain Ni 1+ ions of the similar d 9 electronic configuration [3] , [4] , [5] , [6] , [7] , [8] , [9] . While this seems to ... | core researchThe idea is to investigate whether the unique charge order observed in infinite-layer nickelates, characterized by a commensurate stripe pattern with wavevector ${\bf{q}}\simeq \left(\frac{1}{3},\, 0,\, 0\right)$, arises from a charge-transfer-driven mechanism involving multi-band physics and the interplay... | [
"infinite-layer nickelates",
"commensurate stripe pattern",
"wavevector",
"Hubbard interactions",
"Ni-$d$ orbitals",
"charge transfer energy",
"conduction bands",
"DFT+DMFT calculations",
"effective model",
"NdNiO$_2$",
"charge order",
"multi-band physics",
"charge-transfer-driven mechanism"... | [] |
s41467-019-13059-8 | Optics and photonics | Control of the temporal and polarization response of a multimode fiber | Control of the spatial and temporal properties of light propagating in disordered media have been demonstrated over the last decade using spatial light modulators. Most of the previous studies demonstrated spatial focusing to the speckle grain size, and manipulation of the temporal properties of the achieved focus. In ... | [
{
"section": "Introduction",
"content": "Light propagation in disordered systems has been extensively studied over the past 40 years [1] . Transmitted light through such materials forms a complex pattern after propagation, known as speckle, that is the signature of the interference between a large number of... | This paper proposes a novel method to achieve full spatio-temporal control of light propagation through a multimode fiber by experimentally measuring the Multi-Spectral Transmission Matrix (MSTM), calculating the Time Resolved Transmission Matrix (TRTM) via Fourier Transform, and using it to control the temporal proper... | [
"novel method",
"full spatio-temporal control",
"light propagation",
"multimode fiber",
"Multi-Spectral Transmission Matrix",
"MSTM",
"Time Resolved Transmission Matrix",
"TRTM",
"Fourier Transform",
"temporal properties",
"enhancement",
"attenuation",
"output beam",
"delay times",
"spat... | [
{
"idx": "1.",
"title": "Goodman, J. W. Some fundamental properties of speckle.J Opt. Soc. Am.66, 1145–1150 (1976).",
"link": "https://doi.org/10.1364%2FJOSA.66.001145"
},
{
"idx": "2.",
"title": "Rotter, S. & Gigan, S. Light fields in complex media: mesoscopic scattering meets wave control.... |
s41467-018-06297-9 | Energy science and technology | Enhanced voltage generation through electrolyte flow on liquid-filled surfaces | The generation of electrical voltage through the flow of an electrolyte over a charged surface may be used for energy transduction. Here, we show that enhanced electrical potential differences (i.e., streaming potential) may be obtained through the flow of salt water on liquid-filled surfaces that are infiltrated with ... | [
{
"section": "Introduction",
"content": "The motion of an aqueous electrolyte, such as salt water, over a surface may be harnessed for the generation of electrical voltage [1] . Indeed, the exploration of such electrokinetic phenomena and their possible use for energy conversion have a long history, extendi... | The core research idea is to enhance the streaming potential (voltage generation per unit applied pressure) in microfluidic channels by modulating the effective zeta potential through fluid slip induced by patterned liquid-filled surfaces, as opposed to air-filled superhydrophobic surfaces, thereby overcoming limitatio... | [
"Streaming potential",
"microfluidic channels",
"effective zeta potential",
"fluid slip",
"patterned liquid-filled surfaces",
"air-filled superhydrophobic surfaces",
"surface charge",
"slip length",
"electrokinetic energy conversion efficiency"
] | [
{
"idx": "2.",
"title": "Morrison, F. A. & Osterle, J. F. Electrokinetic energy conversion in ultrafine capillaries.J. Chem. Phys.43, 2111–2115 (1965).",
"link": "https://doi.org/10.1063%2F1.1697081"
},
{
"idx": "3.",
"title": "Bahga, S. S., Vinogradova, O. I. & Bazant, M. Z. Anisotropic ele... |
s41467-023-43891-y | Mathematics and computing | On-chip phonon-magnon reservoir for neuromorphic computing | Reservoir computing is a concept involving mapping signals onto a high-dimensional phase space of a dynamical system called “reservoir” for subsequent recognition by an artificial neural network. We implement this concept in a nanodevice consisting of a sandwich of a semiconductor phonon waveguide and a patterned ferro... | [
{
"section": "Introduction",
"content": "The flourishing idea of using artificial neural networks (ANNs) for the solution of practical tasks shifts the computing paradigm towards heavy usage of data instead of creating lengthy instructions and logical chains forming algorithms. Being inspired by biological ... | The core research idea is to develop on-chip wave-based reservoir computing systems using phonon and magnon wave dynamics in nanostructured materials to overcome the limitations of conventional artificial neural networks in terms of power efficiency, miniaturization, and adaptability for practical neuromorphic applicat... | [
"on-chip wave-based reservoir computing systems",
"phonon wave dynamics",
"magnon wave dynamics",
"nanostructured materials",
"conventional artificial neural networks",
"power efficiency",
"miniaturization",
"adaptability",
"neuromorphic applications"
] | [
{
"idx": "1.",
"title": "Le Cun, Y., Bengio, Y. & Hinton, G. Deep learning.Nature521, 436–444 (2015).",
"link": "https://doi.org/10.1038%2Fnature14539"
},
{
"idx": "2.",
"title": "Strubell, E., Ganesh, A. & McCallum, A. Energy and policy considerations for modern deep learning research.AAAI3... |
s41467-021-21507-7 | Nanoscience and technology | Towards explicit regulating-ion-transport: nanochannels with only function-elements at outer-surface | Function elements (FE) are vital components of nanochannel-systems for artificially regulating ion transport. Conventionally, the FE at inner wall (FE IW ) of nanochannel − systems are of concern owing to their recognized effect on the compression of ionic passageways. However, their properties are inexplicit or genera... | [
{
"section": "Introduction",
"content": "Nanochannel-systems are artificial passages of ions and molecules with unique controllable performances [1] , [2] . They have been widely used in sensing [3] , [4] , [5] , drug release [6] , [7] , separation [8] , [9] , [10] , [11] , nanofluidic [12] , nanoelectroche... | The core research idea is to investigate the previously overlooked role of function elements at the outer surface (FE OS) in regulating ion transport within nanochannel-systems by precisely confining and characterizing these elements at the edge of the inner wall with a minimum depth of 7.5 nm, demonstrating their inde... | [
"function elements at the outer surface",
"FE OS",
"nanochannel-systems",
"ion transport",
"inner wall",
"7.5 nm depth",
"ionic selectivity",
"osmotic energy conversion",
"biosensing capabilities",
"theoretical modeling",
"advanced experimental techniques"
] | [
{
"idx": "1.",
"title": "Kowalczyk, S. W., Blosser, T. R. & Dekker, C. Biomimetic nanopores: learning from and about nature.Trends Biotechnol.29, 607–614 (2011).",
"link": "https://doi.org/10.1016%2Fj.tibtech.2011.07.006"
},
{
"idx": "2.",
"title": "Bocquet, L. Nanofluidics coming of age.Nat... |
s41467-024-50588-3 | Engineering | Empowering the on-site detection of nucleic acids by integrating CRISPR and digital signal processing | Addressing the global disparity in cancer care necessitates the development of rapid and affordable nucleic acid (NA) testing technologies. This need is particularly critical for cervical cancer, where molecular detection of human papillomavirus (HPV) has emerged as an accurate screening method. However, implementing t... | [
{
"section": "Introduction",
"content": "The global incidence of human papillomavirus (HPV)-driven cancers, such as cervical, anal, and head and neck cancer, is on the rise, especially in underserved regions with limited healthcare infrastructure. Cervical cancer alone accounts for over 600,000 new cases a ... | The core research idea is to develop and implement a novel point-of-care diagnostic platform called CreDiT (CRISPR Enhanced Digital Testing) for rapid, accurate, and decentralized detection of high-risk human papillomavirus (hrHPV) types and oncoprotein mRNAs in resource-limited settings, enabling efficient cervical an... | [
"CreDiT",
"CRISPR Enhanced Digital Testing",
"point-of-care diagnostic platform",
"high-risk human papillomavirus",
"hrHPV",
"oncoprotein mRNAs",
"resource-limited settings",
"cervical cancer screening",
"anal cancer screening",
"CRISPR-based signal amplification",
"digital signal processing"
] | [
{
"idx": "1.",
"title": "Randall, T. C. & Ghebre, R. Challenges in prevention and care delivery for women with cervical cancer in sub-Saharan Africa.Front. Oncol.6, 160 (2016).",
"link": "https://doi.org/10.3389%2Ffonc.2016.00160"
},
{
"idx": "2.",
"title": "Montealegre, J. R. et al. Racial/... |
s41467-022-29771-x | Materials science | Reaching the highest efficiency of spin Hall effect of light in the near-infrared using all-dielectric metasurfaces | The spin Hall effect of light refers to a spin-dependent transverse splitting of light at a planar interface. Previous demonstrations to enhance the splitting have suffered from exceedingly low efficiency. Achievements of the large splitting with high efficiency have been reported in the microwave, but those in the opt... | [
{
"section": "Introduction",
"content": "Metasurfaces, nanostructure arrays that are designed to exhibit desired optical responses, have emerged as powerful tools to modulate various features of electromagnetic waves such as amplitudes, phases, and polarizations [1] . Whereas materials in nature have limite... | The core research idea is to propose an all-dielectric metasurface that simultaneously enhances both the shift and efficiency of the spin Hall effect of light (SHEL) in the near-infrared regime by utilizing a single-layer design with high transmission amplitudes and controlled phase differences between s and p polariza... | [
"all-dielectric metasurface",
"spin Hall effect of light",
"near-infrared regime",
"single-layer design",
"high transmission amplitudes",
"controlled phase differences",
"s polarization",
"p polarization",
"horizontal polarization",
"vertical polarization",
"800 nm",
"10.0 λ shift",
"9.9 λ s... | [
{
"idx": "1.",
"title": "Yu, N. & Capasso, F. Flat optics with designer metasurfaces.Nat. Mater.13, 139–150 (2014).",
"link": "https://doi.org/10.1038%2Fnmat3839"
},
{
"idx": "2.",
"title": "Schuller, J. A. et al. Plasmonics for extreme light concentration and manipulation.Nat. Mater.9, 193–... |
s41467-021-22043-0 | Optics and photonics | Simultaneous readout of multiple FRET pairs using photochromism | Förster resonant energy transfer (FRET) is a powerful mechanism to probe associations in situ. Simultaneously performing more than one FRET measurement can be challenging due to the spectral bandwidth required for the donor and acceptor fluorophores. We present an approach to distinguish overlapping FRET pairs based on... | [
{
"section": "Introduction",
"content": "Förster resonance energy transfer (FRET) is a powerful mechanism to probe associations in situ. FRET experiments are performed by labeling the system of interest with donor and acceptor fluorophores, and then using the extent of the energy transfer to probe the dista... | The core research idea is to develop and validate a novel FRET-based method that exploits donor photochromism to enable the simultaneous measurement of multiple FRET pairs, even when the fluorophores exhibit complete spectral overlap, thereby overcoming limitations in current multiplexing strategies and allowing for th... | [
"FRET-based method",
"donor photochromism",
"multiple FRET pairs",
"fluorophores",
"spectral overlap",
"multiplexing strategies",
"signaling activities",
"live cells",
"method validation",
"photochromic donors"
] | [
{
"idx": "1.",
"title": "Piston, D. W. & Kremers, G. J. Fluorescent protein FRET: the good, the bad and the ugly.Trends Biochem. Sci.32, 407–414 (2007).",
"link": "https://doi.org/10.1016%2Fj.tibs.2007.08.003"
},
{
"idx": "2.",
"title": "Sahoo, H. Förster resonance energy transfer—a spectros... |
s41467-023-40884-9 | Materials science | Interplay of structural chirality, electron spin and topological orbital in chiral molecular spin valves | Chirality has been a property of central importance in physics, chemistry and biology for more than a century. Recently, electrons were found to become spin polarized after transmitting through chiral molecules, crystals, and their hybrids. This phenomenon, called chirality-induced spin selectivity (CISS), presents bro... | [
{
"section": "Introduction",
"content": "Helical textures and monopole-like chirality in electronic structures of topological materials have given rise to a plethora of emergent phenomena characterized by unusual interplays between electronic charge, spin, and orbital [1] , [2] , [3] , [4] . More recently, ... | The core research idea is to investigate the mechanism of chirality-induced spin selectivity (CISS) in chiral molecular spin valves by demonstrating that orbital polarization, induced by the chiral molecular structure and converted into spin polarization via the spin-orbit coupling (SOC) of the nonmagnetic electrode, i... | [
"chirality-induced spin selectivity",
"chiral molecular spin valves",
"orbital polarization",
"chiral molecular structure",
"spin polarization",
"spin-orbit coupling",
"nonmagnetic electrode",
"magnetoresistance",
"SOC strength",
"electrode material"
] | [
{
"idx": "1.",
"title": "Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators.Rev. Mod. Phys.82, 3045 (2010).",
"link": "https://doi.org/10.1103%2FRevModPhys.82.3045"
},
{
"idx": "2.",
"title": "Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors.Rev. Mod. Phys.83, 1... |
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