MgO substrate, and capped with a 12-nm-thick MgO layer as shown in Fig. 1b (see “Methods”). The sample was optically excited at normal incidence, using linearly polarized 100 fs transform-limited pulses of 800 nm central wavelength from an amplified Ti:Sapphire laser. The laser pump beam was collimated with the excitation laser spot on the sample of 19.6 mm 2 (1/e 2 width), and the corresponding excitation fluence was ranging from 0.25 to 1.02 mJ cm −2 . The sample was placed in a static in-plane magnetic field of B = 63.4 mT, as shown in Fig. 1a . The forward-propagating THz emission from the sample was measured in the far-field at a distance of 10.3 cm from the sample, without any intermediate optical components, using free-space electro-optic sampling (FEOS) in a 1-mm-thick <110> ZnTe crystal gated by 100 fs, 800 nm laser pulses [26] . This detector configuration provided the required sensitivity and acceptance of spectral bandwidth (see “Methods”). A THz-transparent laser beam block was placed behind the sample to stop the propagation of the remainder of the pump laser beam in the setup (see Fig. 1a ). Fig. 1: Schematic of the experiment and the sample. a Sketch of the THz emission experiment: the sample is placed in the static in-plane magnetic field of 63.4 mT, and excited with a 100 fs, 800 nm laser pulse at normal incidence. The THz emission from the sample in the forward-propagating direction is detected using free-space electro-optic sampling in a 1 mm ZnTe crystal, without any intermediate focusing or additional guiding. b Schematic of the sample: 10-nm-thick bcc Fe film, grown on a 0.5-mm-thick MgO substrate, and capped with 12-nm-thick MgO cap. The excitation gradient of the electrons in the Fe film in such a configuration is negligible, and hot-electron diffusion from the Fe film is blocked by MgO substrate and cap. This eliminates the possibility of any ISHE-type mechanism, which relies on the motion of spin-polarized electrons in the sample, in the laser-induced magnetization dynamics M ( t ). The only possible contributions to M ( t ) in the sample remain hot-electron-driven transient demagnetization causing incoherent hot-magnon excitation (primary dynamics), and modulation of exchange interaction in Fe due to generated coherent acoustic phonons S ( t ) propagating in the structure (secondary dynamics). Full size image Challenges in implementation of the method Before we proceed with the results of our measurements and their interpretation, we outline here the challenges in the successful implementation of ultrafast THz magnetometry. The detailed solutions to these challenges are presented in Methods. First of all, THz magnetic dipole radiation E M ( t ) due to magnetization dynamics M ( t ) inside the sample must be uniquely identified as such, and properly isolated. Yet, under certain conditions, the THz emission from the photoexcited magnetic samples can, in fact, be strongly dominated by electric dipole emission E P ( t ), driven by a time-varying electric current J ( t ) = d P /d t in the sample (see “Methods”). This emission originates from the ultrafast inverse spin-Hall effect (ISHE) [17] , [19] , converting the non-radiating laser-driven hot spin currents in the sample into THz-emitting charge currents. While originating from the ultrafast superdiffusion of hot spin-polarized electrons, these THz-emitting conduction currents are in fact an intricate convolution of ultrafast spin currents [27] with the complex process of natural ultrafast conductive response in a material, involving, e.g., energy-, density-, spin- and time-dependent electron mobility (see, e.g. refs. [25] , [28] , [29] , [30] ). Therefore, this ISHE-mediated THz emission does not carry direct, but only indirect information about the laser-induced dynamics of the magnetic state in the material M ( t ). Yet, as we show below, the ISHE-driven emission easily obscures the much weaker THz emission driven purely by magnetization dynamics M ( t ), which remains the quantity of fundamental interest. We overcome this challenge by a proper sample design, where the electronic motion is precluded, thus eliminating the ISHE mechanism; and by performing control measurements, fully supported by the simulations, that unambiguously confirm the magnetic dipole origin of the detected THz fields (see “Methods”). It should be noted here, that all THz emission experiments on metallic magnetic systems known to us [17] , [18] , [19] , [20] , [21] , are, in fact, dealing with the situation of non-local spin dynamics, and hence with predominantly ISHE-type electric dipole THz emission. This includes the pioneering observation of THz emission from an optically pumped Ni/Cr nanostructure made by one of us, Beaurepaire et al. [15] , which was initially assigned to the process of laser-driven transient demagnetization M ( t ) in Ni. Our present understanding [17] , [19] , [31] , however, clearly points to the electric dipole nature of this THz emission [15] via the ISHE mechanism, enabled by the superdiffusion of laser-excited spin-polarized electrons from Ni to Cr, and resulting in the THz-emitting in-plane charge current in the Cr layer. As such, even after more than two decades since the first experiments on ultrafast magnetism [6] , and after more than a decade since the pioneering observations of THz emission from magnetic materials [15] , [16] , the fundamental process of laser-driven transient demagnetization in ferromagnetic metals has not been identified so far in a clear, unambiguous and calibrated manner, to the best of our knowledge. Another challenge is a rigorous and fully calibrated reconstruction of the source of magnetic dipole radiation, i.e., the magnetization dynamics in the sample M ( t ), from the observable, which is the THz electro-optic signal (EOS) detected in the far-field, many centimeters away from the sample. As the propagation and detection of the THz fields significantly modify their temporal structure and spectral content, such a reconstruction becomes another critical step in the successful implementation of ultrafast THz magnetometry: minor discrepancies in the signal reconstruction will lead to a significant error in the recovered M ( t ). We address this challenge by introducing an experimental calibration procedure to a known reference THz emitter, confirmed by a numerical simulation of the THz propagation function in the spectrometer (see “Methods”). The thickness of the Fe layer used in this work was chosen to be 10 nm, which ensures a virtually uniform pump intensity distribution within the film (see “Methods”). As a result, a gradient of superdiffusive motion of photoexcited electrons in the film is precluded. The MgO substrate and 12-nm-thick MgO cap layer (see Fig. 1b ) serve as diffusion barriers, inhibiting the motion of laser-excited electrons out of Fe. As a result, the motion of spin-polarized electrons in our MgO/Fe/MgO sample is precluded, thus eliminating the possibility for ISHE-type mechanisms and leaving the process of pure transient demagnetization M ( t ) as the only possible source of THz electromagnetic emission from this sample. As a control sample, we have grown a nominally identical Fe film on an MgO substrate, but now capped with a metallic Pd film of 5 nm thickness. In such a sample, the superdiffusion of laser-excited spin-polarized electrons from Fe to Pd is possible, thus leading to THz emission via the ISHE mechanism [17] , [19] . The polarization of the electric field component of the measured THz emission from both MgO/Fe/MgO and MgO/Fe/Pd samples was always perpendicular to the direction of the applied magnetic field. Rotating the polarization of the incident laser pulse about the incidence axis had no effect on the THz emission from both samples. Exemplary measured THz EOS from the MgO/Fe/MgO and MgO/Fe/Pd samples, obtained under identical excitation conditions, are shown in the inset of Fig. 2a . The THz emission from the Pd-capped sample exceeds that from the MgO-capped one by about a factor of 10 in amplitude. The dominant mechanisms of THz emission from the MgO/Fe/MgO and MgO/Fe/Pd samples can be differentiated, and uniquely identified as transient demagnetization and ISHE, respectively, using the symmetry analysis of the polarity of measured THz signals (see “Methods” for details). The order of magnitude difference in strength between the THz emission from MgO/Fe/MgO and MgO/Fe/Pd samples shown in the inset of Fig. 2a emphasizes the importance of creating the dielectric diffusion barriers in the sample, blocking the superdiffusion of hot spin-polarized electrons, and thereby suppressing the ISHE THz emission mechanism, for the observation of the pure transient demagnetization processes. We note here that any broken inversion symmetry of the film would also lead to the parasitic electric dipole emission, which was, however, observed to be negligible in our samples (see “Methods”). Fig. 2: Observable quantities: measured electro-optic signal of the THz emission, the reconstructed magnetization dynamics M ( t ), and its theoretical simulation. a Measured FEOS signal of the THz emission from MgO/Fe/MgO sample, under the excitation of 0.51 mJ cm −2 . Inset: comparison of FEOS signal obtained from MgO/Fe/MgO sample via pure magnetic dipole emission due to M ( t ), red line; and MgO/Fe/Pd sample, dominated by the electric dipole emission via ISHE, green line. b Reconstructed magnetization from experimental EOS (solid line) and simulated magnetization dynamics (dashed line). Simulation conditions are identical to the experiment. μ 0 M 0 = 2.15 T—equilibrium magnetization in Fe (see “Methods”). c Hot electrons excite hot magnons that drive primary demagnetization dynamics. d Dynamics of elongation of the Fe film due to the generated coherent optical phonon pulse. Complete magnetization dynamics M ( t ) comprises both incoherent hot-magnon and coherent acoustic phonon contributions. Full size image Reconstruction of magnetization dynamics from THz emission Having proven that the THz emission from the MgO-capped Fe nanofilm is exclusively related to the local dynamics of magnetic order in laser-excited Fe, the measured THz EOS can be rigorously reconstructed to its source, the ultrafast magnetization dynamics M ( t ) in Fe. Our reconstruction protocol is based on the rigorous experimental and theoretical characterization of the complex propagation function, which connects the THz radiation source, i.e., magnetization dynamics M ( t ) in the sample, to the observable, the THz EOS of the magnetic dipole emission E M ( t ). Once the complex propagation function is established, its deconvolution allows the quantitative extraction of the quantity of interest, magnetization dynamics M ( t ). Our reconstruction protocol is described in detail in the “Methods” section. In Fig. 2a , we show the exemplary as-measured THz EOS, and in Fig. 2b the reconstructed magnetization dynamics M ( t ), in our MgO/Fe/MgO sample (excitation fluence of 0.51 mJ cm −2 , experimental configuration as shown in Fig. 1a ). The reconstructed M ( t ) signal contains two clearly visible components: (i) a fast transient demagnetization signal with the amplitude on the order of μ 0 Δ M = 16.6 mT, with an initial drop on the timescale of about 500 fs, and recovery on the timescale of about 1.7 ps; and (ii) a time-delayed contribution of a rather complex shape, with an amplitude of about 2.5 mT, and a total duration of several picoseconds. We attribute the initial fast component to the pure hot-magnon-driven transient demagnetization process in laser-excited Fe: the laser-excited hot electrons in Fe excite nonequilibrium magnons, which in turn (incoherently) drive the demagnetization process. The dynamics of this demagnetization is reasonably well-understood, and has by now been observed by a variety of other experimental methods (see, e.g. refs. [11] , [13] , [32] , [33] ). We assign the second, time-delayed component in the reconstructed magnetization dynamics M ( t ) to the magnetoelastic effect driven by the laser-excited coherent acoustic phonon propagation in the MgO/Fe/MgO structure. The mechanism of the magnetoelastic effect is as follows. Subsequent to the laser excitation, a lattice strain is impulsively generated in Fe via electron-phonon coupling [34] , thus producing a coherent acoustic phonon pulse propagating away from the Fe film surface. This acoustic pulse will undergo reflections on the acoustic discontinuities within the structure (see e.g. ref. [35] ): partial reflections on the Fe/MgO interfaces, and total reflection on the outside cap layer MgO/air interface, as shown in Fig. 1b , thus returning back to the Fe film. The propagation of such a multiply reflected coherent acoustic phonon pulse through Fe leads to the transient modification of interatomic distances in Fe, modulating the exchange interaction in the material, and hence leading to an additional, strain-induced demagnetization channel in Fe (see, e.g. refs. [36] , [37] , [38] and references therein). We note that, to the best of our knowledge, the direct measurement of such an acoustically driven magnetization dynamics on the ultrafast (sub-)picosecond timescale, and its co-existence with the direct laser-driven demagnetization via hot magnons, has not been reported before. Comparison of experiment and theory Here, we have theoretically modeled the entire magnetization dynamics in the laser-excited MgO/Fe/MgO structure employing a methodology that uses as building blocks a microscopic multi-temperature model (MMTM), lattice strain dynamics and first-principles calculations (see “Methods” for details). In Fig. 2b , we show the exemplary measured and simulated M ( t ) signals, corresponding to identical excitation conditions (excitation fluence of 0.51 mJ cm −2 ), and demonstrating very good quantitative agreement. Figure 2c shows the calculation of the isolated hot-magnon driven transient demagnetization dynamics, which occurs on a ~500 fs timescale and recovers on a timescale of ~1.7 ps. In Fig. 2d the calculated dynamics of elongation of laser-excited Fe film is presented, corresponding to the experimental conditions in Fig. 2b : the initial strain in the 10-nm-thick Fe film builds up on a ~2 ps timescale, followed by slow (>100 ps) relaxation, convoluted with the multiple reflections of the strain waves within the Fe film, as well as the return of the strain wave reflected from the outside MgO/air interface of the sample back into the Fe film. The theoretically simulated magnetization dynamics curve M ( t ) in Fig. 2b includes the contributions of the hot-magnon driven transient demagnetization (Fig. 2c ), and the coherent-phonon-driven magnetoelastic effect (Fig. 2d ). Note that the ~2.6 mT amplitude of the acoustically driven demagnetization signal in Fig. 2b is only about 6.5 times smaller than that from the direct laser-driven transient demagnetization of 16.7 mT. In Fig. 3a, b we show the entirety of the measured magnetization dynamics signals M ( t ) in this work under the excitation fluence in the range 0.25–1.02 mJ cm −2 , and the results of the first-principles calculations as described above, fully corresponding to the experimental conditions. The maximum observed hot-magnon driven demagnetization reaches 28.8 mT (corresponding to the change in magnetic moment density of 2.26 × 10 4 J T −1 m −3 ), while the strain-driven demagnetization reaches 3.5 mT (0.18 × 10 4 J T −1 m −3 ) at the strongest excitation. Hence, the maximum measured hot-magnon – and coherent phonon-driven demagnetizations of μ 0 Δ M = 28.8 mT and μ 0 Δ M = 3.5 mT, respectively, correspond to about 1.3% and 0.16% of the equilibrium magnetization in Fe of μ 0 M 0 = 2.15 T. In this respect, we would like to emphasize the high sensitivity provided by our ultrafast magnetometry method, which allows one to reliably observe the ultrafast magnetization dynamics on the order of 10 −4 M 0 (see Figs. 2 , 3 ). Generally, the ratio of the amplitudes of these two components of transient demagnetization signal is maintained as \\(\\Delta M_{{\\mathrm{hot - magnon}}}{\\mathrm{/}}\\Delta M_{{\\mathrm{phonon}}}\\, {\\approx} \\,7\\) for the whole range of excitation fluences used in the experiment. Such a high relative efficiency of ultrafast magnetization control by strain, as compared to direct laser excitation of electrons, is quite remarkable, and points to the promise of efficient ultrafast magnetic technologies controlled by acoustic signals, as opposed to traditional optical, electric, or magnetic stimuli. Notwithstanding, the ratio \\(\\Delta M_{{\\mathrm{hot - magnon}}}{\\mathrm{/}}\\Delta M_{{\\mathrm{phonon}}} \\,{\\approx} \\,7\\) points to the dominating role of ultrafast magnon excitation for the process of laser-driven femtosecond demagnetization. Fig. 3: Comparison of measured and simulated magnetization dynamics M ( t ) in laser-excited MgO/Fe/MgO sample in this work. a Ultrafast magnetization dynamics in the Fe film at various laser pump fluences in the range 0.25–1.02 mJ cm −2 with errors shown by dashed lines, and b results of simulation corresponding to the experimental scenario in ( a ). The error bars in ( a ) were determined from the standard errors in the measured electro-optic signals (from 200 individual measurements). The experimental error was then propagated in the reconstruction protocol, leading to the intervals as indicated. Full size image Again, the experimental data and the results of modeling are in good agreement: the calculation captures both the temporal features and the magnitudes of the measured transient demagnetization signals. The small discrepancy in the shapes of measured and modeled magnetoelastic contribution might be caused by minor thickness fluctuations in the sample within the laser excitation spot, affecting the local arrival times of the acoustic phonon pulse, and thus leading to a smearing of the overall magnetoelastic signal. Further, the influence of phenomena such as the Elliott-Yafet electron-phonon spin-flip scattering process [39] , [40] , phonon-magnon coupling [41] or heat diffusion [42] , which have not been considered in our theoretical modeling, and whose contribution have been estimated only phenomenologically, might also play a certain role. For example, in our theoretical modeling a linear dependence of the electron-magnon coupling parameter (see “Methods”) on the absorbed optical pump fluence was assumed, and its value was established phenomenologically from the experimental data taken at the intermediate pump fluence of 0.51 mJ cm −2 (see Fig. 2 ). The reason for such an assumption is the current lack of solid experimental and theoretical knowledge regarding this important physical parameter. Yet, as can be seen in the comparison of the entirety of our experimental and theoretical data presented in Fig. 3 , the maximum calculated hot-magnon driven demagnetization at the maximum pump fluence of 1.02 mJ cm −2 exceeds that observed in the experiment by about 25%. While such a discrepancy is not very large, it only emphasizes the need to understand more in-depth the processes governing the magnetic response of materials to ultrafast external stimuli. Our experimental method can significantly contribute to such development."},{"section":"Discussion","content":"To summarize, we have presented a rigorous method for ultrafast magnetometry based on THz emission spectroscopy. In our proof-of-principle experiment, we observe the ultrafast demagnetization dynamics in laser-excited Fe, and uniquely identify and disentangle two contributions to this process: (i) a hot-electron-driven emission of incoherent hot magnons leading to ultrafast transient demagnetization, and (ii) a coherent acoustic-phonon-driven magnetoelastic effect. Our observations are fully supported by the results of theoretical modeling. To the best of our knowledge, this is the first simultaneous observation of both hot-magnon and coherent-phonon-driven ultrafast magnetization dynamics in a magnetic system, with hot magnons being the dominating effect. Our observations demonstrate the potential of our highly sensitive method for both fundamental research and industrial applications of magnetism. The magnetization dynamics driven by coherent acoustic phonon propagation in iron corresponds to the elongation of the Fe film on the order of only tens of fm (see Fig. 2 ). It is even more so remarkable, that the effects related to such tiny elongations could be clearly revealed in the experiment using THz radiation, i.e., sub-mm electromagnetic waves having 10 orders of magnitude longer wavelength. The surprisingly strong magnetoelastic contribution to ultrafast magnetization dynamics, leading to a clearly measurable electromagnetic emission in the THz range, holds promise for spintronic technologies based on a purely acoustic and, where possible, heat-free ultrafast control of magnetism."},{"section":"Methods","content":"Pump intensity distribution within the iron film The refractive index of iron at the pump wavelength around 800 nm is n = 2.94 + 3.39 i [43] , yielding the optical penetration depth on the order of 20 nm under bulk conditions (Beer-Lambert Law). However, multiple interferences of the pump light in the film result in a virtually constant intensity distribution profile, with the intensity variation on the order of only 1%, with the intensity minimum around the middle of the film (Supplementary Fig. 1 ). Spin current simulations Ferromagnetic 3 d metals such as Fe are known to have a small, yet non-vanishing, inverse spin Hall effect (see e.g. ref. [44] ). To exclude that any spin current along the depth of the sample would lead to an electric THz emission via ISHE mechanism, we have performed rigorous simulations. Specifically, to make our simulations parameter-free, we only use ab initio computed quantities as input. We have therefore computed ab initio the SHE and ISHE coefficients of Fe and Pd; the computed spin Hall conductivities of bulk bcc Fe and fcc Pd are 14.96 and 700.93 ( ħ / e ) S cm −1 , respectively, which implies that the computed spin Hall angle of Pd is 46.85 times larger than that of Fe (Salemi, L. & Oppeneer, P. M. unpublished). In addition we have used spin-dependent hot-electron lifetimes and velocities computed previously [45] . With these input quantities and using the ultrafast superdiffusion code [27] we have simulated the charge and spin currents in MgO/Fe/MgO and MgO/Fe/Pd structures under the same excitation condition as in our experiment (800 nm, 100 fs pulse). The results (Supplementary Fig. 2 ) demonstrate that the electric dipole contribution from the MgO/Fe/MgO structure is extremely small and can be safely neglected, leaving the transient demagnetization as the only source of its THz emission. Magnetic and electric dipole radiation The far-field electric field components of magnetic E M ( t ) and electric E P ( t ) dipole radiation from a point source [46] read $${{\\mathbf{E}}_{\\mathrm{M}}\\left( t \\right) = iFT\\left\\{ { - \\frac{1}{{4\\pi }}\\sqrt {\\frac{{\\mu _0}}{{{\\it{\\epsilon }}_0}}} k^2\\left( {{\\mathbf{n}} \\times {\\mathbf{M}}} \\right)\\frac{{e^{ikr}}}{r}} \\right\\},}$$ (1) $${{\\mathbf{E}}_{\\mathrm{P}}\\left( t \\right) = iFT\\left\\{ {\\frac{1}{{4\\pi {\\it{\\epsilon }}_0}}k^2\\left( {{\\mathbf{n}} \\times {\\mathbf{P}}} \\right) \\times {\\mathbf{n}}\\frac{{e^{ikr}}}{r}} \\right\\}.} $$ (2) Here iFT stands for inverse Fourier transform, μ 0 is the vacuum permeability, \\({\\it{\\epsilon }}_0\\) the vacuum permittivity, k is the wavevector, n the unit vector pointing towards the detecting position from the emitting dipole, M and P are, respectively, the frequency-dependent magnetization and polarization [20] (i.e., Fourier transforms of M ( t ) and P ( t )), and r is the distance from the emitting dipole to the position of detection. Symmetry distinction of THz emission Two types of samples were prepared. Both of them were 10 nm ferromagnetic iron (Fe) films deposited on an insulating MgO substrate. One was capped with a 12 nm insulating MgO layer (MgO/Fe/MgO), and the other was capped with a 5 nm metallic Pd layer (MgO/Fe/Pd). The samples were fully magnetized in-plane along the x -axis (see Fig. 1 ) by an applied external magnetic field of 50 mT, and were excited by femtosecond laser pulses with a fluence of 1.1 mJ cm −2 . First, both samples were pumped from the cap layer side, and the emitted THz waves were measured by FEOS in the far field, in the direction of the pump beam propagation. Second, we flipped the two samples while keeping the external magnetic field and the THz detection geometry unchanged, thus now pumping the samples from the substrate side. Finally, both samples were pumped at the cap layer, but the external magnetic field was reversed. The experimental results (Supplementary Fig. 4 ) show that the polarity of THz emitted from MgO/Fe/MgO only depends on the external magnetic field, while both the external magnetic field and the geometry of the sample can affect the polarity of THz emitted from MgO/Fe/Pd. We conclude that the THz emission from MgO/Fe/MgO is due to the ultrafast magnetization dynamics, whereas the electric dipole radiation caused by the spin-charge conversion dominates the THz emission in MgO/Fe/Pd. Reconstruction of magnetization dynamics We performed direct experimental characterization, fully supported by the theoretical modeling, of the THz field propagation function in our experiments, rigorously relating the THz EOS measured in the far-field, to its source, the magnetization dynamics in the sample M ( t ). For this we used a reference electric-dipole THz emitter, a 0.5-mm-thick ZnTe crystal, in place of MgO/Fe/MgO sample. As a result, we achieve precise reconstruction of the fully calibrated transfer function of our experiment, relating the dynamics of polarization P ( t ) or magnetization M ( t ) at the surface of the laser-excited THz emitter to our observable, the detected THz EOS. The basis for the reconstruction protocol is the accurate establishment of the complete transfer function of the experiment in the frequency domain, which includes (i) conversion of time-dependent magnetization (polarization) in the sample into electromagnetic radiation f rad,M ( ω ) and f rad,P ( ω ), (ii) propagation of this electromagnetic radiation in the spectrometer towards detector crystal f prop ( ω ), and (iii) detection by FEOS of the electromagnetic field in the detector crystal f d ( ω ). While the radiative functions f rad,M ( ω ) and f rad,P ( ω ) are different for magnetic dipole (MgO/Fe/MgO) and electric dipole (ZnTe) emitters, the propagation f prop ( ω ) and detection f d ( ω ) functions remain identical for both cases. The total detected EOS for magnetic and electric dipole emission cases, S M ( ω ) and S P ( ω ) respectively, are $${S_{\\mathrm{M}}\\left( \\omega \\right) = M\\left( \\omega \\right)f_{{\\mathrm{rad}},{\\mathrm{M}}}\\left( \\omega \\right)f_{{\\mathrm{prop}}}\\left( \\omega \\right)f_{\\mathrm{d}}\\left( \\omega \\right),} $$ (3) $${S_{\\mathrm{P}}\\left( \\omega \\right) = P\\left( \\omega \\right)f_{{\\mathrm{rad}},{\\mathrm{P}}}\\left( \\omega \\right)f_{{\\mathrm{prop}}}\\left( \\omega \\right)f_{\\mathrm{d}}\\left( \\omega \\right),} $$ (4) where M ( ω ) and P ( ω ) are the frequency-dependent magnetization and polarization, respectively (i.e., the Fourier trasnsforms of M ( t ) and P ( t )). The reconstruction of the quantity of interest M ( ω ) from the observable, EOS S M ( ω ) is therefore $$M\\left( \\omega \\right) = \\frac{{S_{\\mathrm{M}}\\left( \\omega \\right)}}{{f_{{\\mathrm{rad}},{\\mathrm{M}}}\\left( \\omega \\right)f_{{\\mathrm{prop}}}\\left( \\omega \\right)f_{\\mathrm{d}}\\left( \\omega \\right)}}.$$ (5) The radiative functions f rad,M ( ω ) = FT ( E M ( t )) and f rad,P ( ω ) = FT ( E P ( t )) are given by Eqs. ( 1 ) and ( 2 ). In order to experimentally establish the spectrometer function, i.e., the propagation and detection function of our experiment f prop ( ω ) f d ( ω ), we use a reference electric dipole emitter, a 0.5-mm-thick <110> ZnTe crystal. The process of THz generation by optical rectification of femtosecond laser pulses in transparent nonlinear crystals is by now very well understood, and can be modeled quantitatively very accurately from the first principles, using as an input the temporal intensity profile of the laser pulse, the value for electro-optic coefficient of the crystal, and the THz dispersion of its complex-valued refractive index (see e.g. ref. [47] for the comprehensive description of the calculation). This allows one to very accurately calculate the polarization at the outcoupling crystal surface, i.e., the parameter P ( ω ) in Eq. ( 4 ). With the knowledge of P ( ω ), it is therefore straightforward to recover the complete propagation and detection function of our experiment f prop ( ω ) f d ( ω ), from the measured EOS of the THz emission of a reference electric dipole emitter S P ( ω ) [48] . Now that the spectrometer function f prop ( ω ) f d ( ω ) is established, it can be directly applied to the reconstruction of the magnetization dynamics M ( t ) in our magnetic dipole emitter MgO/Fe/MgO, from the correspondingly measured EOS S M ( t ), using the inverse Fourier transform of the Eq. ( 5 ). This yields the recovered, and fully calibrated, magnetization dynamics M ( t ) in our experiment, shown in Figs. 2 and 3 of the main text. Incoherent excitation of magnons Although experimentally we cannot uniquely resolve which microscopic mechanism leads to the observed magnetization dynamics, in this section we show that incoherent excitation of magnons is the only plausible mechanism responsible for the ultrafast magnetization dynamics in the MgO/Fe/MgO structure. On the one hand, it has been recently demonstrated [11] , [32] , [33] that the incoherent excitation of magnon modes is a dominant mechanism contributing to ultrafast demagnetization. Moreover, Mlynczak et al. [49] have recently shown that the electron-magnon scattering process is a very relevant scattering mechanism in Fe. On the other hand, transport mechanisms, such as spin superdiffusion or heat transport, can be disregarded as shown by the simulations. Coherent processes stemming from the direct light-spin interaction can also be neglected due to the linear light polarization. Another relevant mechanism which could potentially explain the transient magnetization dynamics is the Elliott-Yafet mechanism, in which the electrons transfer angular momentum to the phonons. However, it has been theoretically shown that it only provides a small contribution to the ultrafast demagnetization in ferromagnetic 3 d metals [40] . Another plausible mechanism that has gained attention in the last years, is the spin-flip Coulomb scattering model, which is based on the ultrafast transfer of angular momentum from the electronic spin to the electronic orbital angular momentum, and subsequently to the lattice [50] . However, the lack of a theoretical model to explain the ultrafast (tens of femtoseconds) transfer of angular momentum from the electronic orbitals to the lattice, and the relatively slow demagnetization time found in our experiments (around 500 fs) suggest that this theoretically proposal is also not relevant in our experiment. As we show below, excitation of incoherent magnons by electron-magnon scattering can perfectly well explain our experimental observations. Microscopic multi-temperature model To simulate the transient magnetization dynamics M ( t ) in Fe we use a MMTM, which describes the magnetization dynamics as an incoherent excitation of magnon modes through a complex interplay between electrons, spin waves and the lattice. Our model goes well beyond the usual three-temperature model (3TM) and provides an out-of-equilibrium description of electron and lattice dynamics with an explicit inclusion of phonon-phonon scattering through an anharmonic phonon interaction term [51] . Mathematically, the temporal evolution of the three different subsystems is modeled by the following set of coupled differential equations $$C_{\\mathrm{e}}\\frac{{\\partial T_{\\mathrm{e}}}}{{\\partial t}} \t= G\\left( {T_{\\mathrm{s}} - T_{\\mathrm{e}}} \\right) + \\mathop {\\sum }\\limits_Q C_Q\\gamma _QI\\left( {T_{\\mathrm{e}}} \\right)\\left( {T_{\\mathrm{l}}^Q - T_{\\mathrm{e}}} \\right)\\left[ {1 + J\\left( {\\omega _Q,T_{\\mathrm{l}}^Q} \\right)\\left( {T_{\\mathrm{l}}^Q - T_{\\mathrm{e}}} \\right)} \\right]\\\\ \t\\quad+ \\frac{{\\partial U_{{\\mathrm{e}} - {\\mathrm{e}}}}}{{\\partial t}},$$ (6) $$C_Q\\frac{{\\partial T_{\\mathrm{l}}^Q}}{{\\partial t}} \t= - C_Q\\gamma _QI\\left( {T_{\\mathrm{e}}} \\right)\\left( {T_{\\mathrm{l}}^Q - T_{\\mathrm{e}}} \\right)\\left[ {1 + J\\left( {\\omega _Q,T_{\\mathrm{l}}^Q} \\right)\\left( {T_{\\mathrm{l}}^Q - T_{\\mathrm{e}}} \\right)} \\right]\\\\ \t\\quad - \\mathop {\\sum}\\limits_k {C_Q} \\Gamma _{Qk}\\left( {T_{\\mathrm{l}}^Q - T_{\\mathrm{l}}^k} \\right)\\left[ {1 + J\\left( {\\omega _Q,T_{\\mathrm{l}}^Q} \\right)\\left( {T_{\\mathrm{l}}^Q - T_{\\mathrm{l}}^k} \\right)} \\right]\\\\ \t\\quad+ \\frac{{\\partial U_{{\\mathrm{e}} - {\\mathrm{ph}}}}}{{\\partial t}},\\,\\,for\\,Q = Q_1, \\cdots ,Q_N,$$ (7) $$ C_{\\mathrm{s}}\\frac{{\\partial T_{\\mathrm{s}}}}{{\\partial t}} = - G\\left( {T_{\\mathrm{s}} - T_{\\mathrm{e}}} \\right).\\qquad$$ (8) Here T e , T s , and \\(T_{\\mathrm{l}}^Q\\) are the electron, spin and phonon mode-dependent temperatures. Q accounts for the N distinct and independent phonon subsystems, each of them corresponding to a specific branch, υ , and momentum, q , having different temperatures \\(T_{\\mathrm{l}}^Q\\) . C e , C s , and C Q are the temperature-dependent electronic, spin and phonon mode-dependent heat capacities, respectively . γ Q and Γ Qk are the mode-dependent linewidth due to electron-phonon and phonon-phonon interactions, respectively. G is the electron-magnon coupling constant. I ( T e ) accounts for the dependence of the electron-phonon coupling on the electronic temperature. \\(\\frac{{\\partial U_{{\\mathrm{e}} - {\\mathrm{ph}}}}}{{\\partial t}}\\) and \\(\\frac{{\\partial U_{{\\mathrm{e}} - {\\mathrm{e}}}}}{{\\partial t}}\\) describe the energy transfer rate from the laser-induced nonequlibrium electrons to the lattice through electron-phonon interaction and into thermal electrons via electron-electron interactions, respectively. \\(J\\left( {\\omega _Q,T_{\\mathrm{l}}^Q} \\right)\\) is the second-order Taylor expansion of the out-of-equilibrium phonon populations around the mode-dependent phonon temperatures (see ref. [51] for full forms). The effective spin temperature, T s , is in our model equivalent to the magnon temperature, which characterizes the incoherent excitation of magnon modes in the 3TM. It is worth mentioning that, while the model for the out-of-equilibrium description of the electron and lattice dynamics has been derived from a microscopic theory and is therefore fit-free (the parameters are ab initio computed), the coupling of these two subsystems with the spin system is introduced phenomenologically, and the electron-magnon coupling constant is chosen to reproduce the experimental demagnetization time. In addition, a correct treatment of the spin system would need to include effective mode- and q -dependent spin temperatures to account for the distinct q -dependent electron-magnon scatterings. However, currently there does not exist a reliable theoretical model to provide such q -dependent electron-magnon coupling, and therefore such description is beyond our present capabilities. Notwithstanding, we can effectively treat the global effect of heating the magnon system by considering the spin system to be individually thermalized and work with an effective electron-magnon coupling, which is then chosen to reproduce the experimental demagnetization time. From the MMTM model we determine the time evolution of the spin temperature which then is used in the analytic form of the temperature-dependent magnetization for bcc Fe $$ {M\\left( {T_{\\mathrm{s}}} \\right) = M\\left( 0 \\right)\\left( {1 - \\frac{{T_{\\mathrm{s}}}}{{T_{\\mathrm{c}}}}} \\right)^{0.39},} $$ (9) where T c = 1044 K is the Curie temperature of Fe. In addition, the transient phonon-mode dependent lattice temperatures are used to model the strain dynamics, as outlined in the following. Lattice strain dynamics We consider three different mechanisms that contribute to the creation of strain in the lattice. These mechanisms are: first, the force induced by the optically excited electrons; second, the force induced by the ultrafast demagnetization (a new electronic surface potential is generated following the magnon emission); and third, the thermal strain due to heating of the lattice. The first and second processes originate from the deformation potential mechanism [52] and happen at a time scale below 100 fs, where the strain caused by acoustic phonons depends on the deformation potential and the absorbed laser energy. The third process is induced by the changing temperature distribution in a thermoelastic model wherein a linear relation is established between the stress pulse and the varying temperature distribution. Although in the latter process the changing electronic and lattice temperatures contribute to the stress pulse, the regime of small laser intensities used in the experiment along with the smallness of the electron heat capacity as compared with the lattice one, allows one to neglect the electronic contribution. The transient lattice strain along the direction perpendicular to the film ( z ) can then be written as $$\\eta \\left( {z,t} \\right) = - \\frac{1}{{2\\rho v^{2}}}\\sum_{Q} \\int_{ - \\infty }^{ + \\infty } g_{Q} C_{Q} {\\mathrm{sgn}}\\left(z - v\\left(t - t^{\\prime} \\right) \\right)\\left. {\\frac{{\\partial T_{\\mathrm{l}}^Q}}{{\\partial t}}} \\right|_{\\left( {\\left| {z - v\\left( {t - t^{\\prime} } \\right)} \\right|,t^{\\prime} } \\right)}{{\\mathrm{d}}t}^{\\prime},$$ (10) where ρ is the density and v the sound velocity of Fe, and g Q is the mode-dependent Grüneisen parameter, which are computed from first principles. The term \\(\\frac{{\\partial T_{\\mathrm{l}}^Q}}{{\\partial t}}\\) is determined by the solution of the MMTM (see above). Thus, Eq. ( 10 ) is a modified out-of-equilibrium version of the expression provided by Wright [53] . Although it may provide the same total expansion of the lattice, it is reached through different strain dynamics. Notably, the solution of the MMTM yields that the time evolution of the strain produced by heating covers a range of several picoseconds. The different time scales of the three different processes that induce the strain pulses enable a simplification in the modeling. Thus, strain pulses generated by the deformation potential mechanisms are assumed to happen instantaneously following laser excitation with an amplitude proportional to the laser excitation, which adds to the heating induced strain. Schematically, the process is as follows: At each of the two Fe/MgO interfaces a tensile strain is generated following laser excitation. These pulses travel within the Fe layer toward the other interface at the Fe sound velocity (5060.05 m s −1 ). Correspondingly, a compressive strain is generated at the Fe/MgO interface which travels within the MgO system at the MgO sound velocity (9100 m s −1 ). When the strains traveling within the Fe film reach one of the Fe/MgO interfaces, the pulses become partially reflected (92.5% transmission, due to the different impedances between Fe and MgO) inducing a change of polarization of the strain (tensile waves become compressive waves and vice versa), while the rest goes into the MgO films. On the other hand, the strain waves are completely reflected on the MgO/air interfaces. At the end of the simulation, a finite strain remains in the system due to the heating effect that induces the thermal expansion of the system. To recover the normal volume expansion (in our case normal surface expansion, u ( t )) we integrate over the whole Fe film thickness the time-dependent strain pulses, getting $$u\\left( t \\right) = \\int_0^{z_{{\\mathrm{max}}}} \\eta \\left( {z,t} \\right){{\\mathrm{d}}z},$$ (11) where z max = 10 nm. The calculated time evolution of the system’s expansion is shown in Fig. 2d for a laser excitation of 1.02 mJ cm −2 . Magnetization dynamics The final magnetization dynamics is calculated by combining the MMTM and the strain dynamics. Thus, we can write $$ {\\frac{{M\\left( {T_{\\mathrm{s}}} \\right)}}{{M\\left( {T_0} \\right)}} = \\left( {\\frac{{T_{\\mathrm{c}} - T_{\\mathrm{s}}}}{{T_{\\mathrm{c}} - T_0}}} \\right)^{0.39} + \\Delta M\\left( {u\\left( t \\right)} \\right) + f_{{\\mathrm{recovery}}},} $$ (12) where f recovery is a phenomenological function linear in time, introduced to recover the long-time behavior of the magnetization dynamics (relaxation back to equilibrium), and which accounts for the physical processes that are not included in our model, such as phonon-magnon coupling or heat diffusion, which are relevant at long-time scales. Δ M ( u ( t )) is a function that represents the change of magnetization due to the change of the system’s volume. It is driven by the modulation of exchange interaction in iron, and phenomenologically, can be written as: $$ {\\Delta M\\left( {u\\left( t \\right)} \\right) = \\Delta J\\left( {u\\left( t \\right)} \\right)\\frac{1}{2}\\frac{{M_0}}{{J_0}},} $$ (13) where the factor 1/2 comes from the fact that around equilibrium position a 2% change of J produces a 1% change of magnetization [54] , [55] We also assume that $$ {\\Delta J\\left( {u\\left( t \\right)} \\right) = \\frac{{\\mathrm{d}}J}{{\\mathrm{d}}u}u\\left( t \\right),} $$ (14) where u ( t ) is the system elongation in z -direction (see Eq. ( 11 )). By combining Eqs. ( 13 ) and ( 14 ) we obtain $$ {\\Delta M\\left( {u\\left( t \\right)} \\right) = \\frac{{\\mathrm{d}}J}{{\\mathrm{d}}u}u\\left( t \\right)\\frac{1}{2}\\frac{{M_0}}{{J_0}},} $$ (15) where the value of the gradient of the exchange constant is fitted to provide the experimentally observed transient magnetization dynamics. For the specific case of Fe we obtain $${\\Delta M\\left( {u\\left( t \\right)} \\right) = 80.88\\frac{{\\mu _{\\mathrm{B}}}}{{\\mathrm{{\\AA}}}}u\\left( t \\right).} $$ (16) This expression is used along with the elongation values provided by Eq. ( 11 ) to obtain the transient magnetization driven by the acoustic waves in Eq. ( 12 ). As example, a system’s elongation of 40 fm would induce an increase of magnetization of the order of 0.16% of the equilibrium magnetization, as can be seen in Fig. 2b, d for a delay time between 3 and 4.5 ps. On the other hand, lattice strains of the order of 10 −3 would induce magnetization changes of the order of 37% of the equilibrium magnetization. However, it is important to mention that although those large lattice strains are experimentally accessible at short time scales, their effect on the transient magnetization might be largely shadowed by the more dominant magnetization dynamics induced by hot magnons. In addition, large elongations would introduce nonlinear effects in the above relation, and also significant changes in the magnetic anisotropy, leading to a distinct magnetization dynamics. The value of equilibrium magnetization in iron μ 0 M 0 = 2.15 T [56] , [57] , [58] was used in the calculations of relative magnetization dynamics. Sample preparation The iron films were deposited on double polished MgO <100> substrates and capped with 12 nm of MgO and 5 nm of Pd at room temperature, respectively. The molecular beam epitaxy was performed to deposit 10 nm of Fe at room temperature with subsequent 90 minutes heat treatment at 800 K, necessary to obtain highly ordered monocrystalline Fe layer. The deposition rate of Fe layer was 0.05 nm min −1 at standard 10 −10 Torr (10 −8 Pa) pressure under reflection high energy electron diffraction control to verify the crystal quality. The thicknesses were controlled in situ by quartz balance sensing."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"Magnetism [1] has traditionally been a major topic of fundamental research. It also underpins numerous modern technologies ranging from high-capacity memories [2] , [3] to novel concepts in computing and data transfer [4] , [5] . Since the discovery of sub-picosecond spin momentum removal in a magnetic system two decades ago [6] , the field of ultrafast magnetism has become a rapidly growing area in modern condensed matter physics (see e.g. refs. [7] , [8] , [9] and references therein). We owe our present understanding of ultrafast magnetization dynamics to a range of advanced laser-based experimental methods, all of which, however, feature certain drawbacks in terms of limited precision, ambiguity of measurement interpretation, or limited applicability under realistic device conditions. Spin-resolved surface science methods such as time-resolved photoemission [10] , [11] require ultra-high vacuum conditions, pristine sample surfaces, and technically challenging detection schemes. These constraints preclude the measurements on encapsulated materials or devices, as well as experimentation under typical device working conditions. Optical probes such as the magneto-optical Kerr effect (MOKE) [12] , [13] or optical second-harmonic generation [14] are versatile but are limited to the optical probe penetration depth. Further, the interpretation of MOKE signals on sub-picosecond timescales is not straightforward: pump-induced change in the magnetooptical constant of a material can make a significant contribution to the MOKE signal itself, which currently remains the subject of intense research [12] , [13] . Over a decade ago, THz electromagnetic emission from optically excited magnetic metallic structures [15] , [16] was reported for the first time, and was assigned to the presence [16] or temporal modification [15] of magnetization in laser-excited samples. A number of studies followed (see, e.g. refs. [17] , [18] , [19] , [20] , [21] , [22] , [23] , [24] ), reporting on the THz emission from laser- or THz-excited metallic and dielectric magnetic structures, and tracing this emission back to ultrafast spin- and electron-dynamics of various origins. THz emission spectroscopy relies on a fundamental principle of electrodynamics: a time-varying magnetization M ( t ) or polarization P ( t ) in the sample acts as a source of electromagnetic radiation, known as magnetic E M ( t ) or electric E P ( t ) dipole emission, respectively (see “Methods”). When the temporal variation of M ( t ) or P ( t ) occurs on the ultrafast, (sub-)picosecond timescale, the resulting radiation contains THz frequencies. The THz electromagnetic waves, in turn, experience considerably weaker absorption in metals, semiconductors, and dielectrics in comparison to ultraviolet, visible, and infrared lightwaves. Such a high penetration power becomes a natural advantage of the THz waves, making them especially suitable for probing relatively thick, multilayer structures and devices (see, e.g. ref. [25] ), where optical probes can fail. Unlike other methods, THz emission spectroscopy can be easily implemented under ambient conditions, and can be applied to encapsulated samples or devices. Importantly, such a contact-free measurement reveals the ultrafast magnetization dynamics in the sample as a whole, and not only within the probed surface region. Most crucially, however, the principles of THz emission spectroscopy must allow for direct and fully calibrated determination of the quantity of fundamental interest, the ultrafast magnetization dynamics in the sample M ( t ), from the measurable quantity—magnetic dipole emission E M ( t ). It must be emphasized here that the magnetic dipole emission E M ( t ) is connected to its source, instantaneous magnetic moment M ( t ), in the most straightforward way, not involving any potentially time-dependent coupling constants. Nonetheless, such measurements have so far remained a challenge. For example, in a dedicated comparative study [18] the discrepancy was identified between the timescales of M ( t ) estimated via THz emission and MOKE methods. Therefore, in spite of all its promise, the full potential of THz emission spectroscopy for studies of ultrafast magnetism still remains to be shown. Here, we demonstrate an accurate and practical ultrafast magnetometry method, based on THz emission spectroscopy under ambient conditions. As a proof-of-principle, we rigorously determine the ultrafast magnetization dynamics M ( t ) in a laser-excited encapsulated iron film, based on the measured electric field component E M ( t ) of the laser-driven magnetic dipole THz emission from the sample. This measurement is sufficiently sensitive to clearly resolve and disentangle various contributions to the complex magnetization dynamics in the sample, occurring on sub-picosecond and picosecond timescales. Our measurements are in quantitative agreement with the results of theoretical modeling.\",\n \"references\": {\n \"indices\": [],\n \"infos\": []\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Sample and experiment The schematic of our experiment is shown in Fig. 1a . Our sample was a 10-nm-thick single-crystalline Fe film epitaxially grown on <100> MgO substrate, and capped with a 12-nm-thick MgO layer as shown in Fig. 1b (see “Methods”). The sample was optically excited at normal incidence, using linearly polarized 100 fs transform-limited pulses of 800 nm central wavelength from an amplified Ti:Sapphire laser. The laser pump beam was collimated with the excitation laser spot on the sample of 19.6 mm 2 (1/e 2 width), and the corresponding excitation fluence was ranging from 0.25 to 1.02 mJ cm −2 . The sample was placed in a static in-plane magnetic field of B = 63.4 mT, as shown in Fig. 1a . The forward-propagating THz emission from the sample was measured in the far-field at a distance of 10.3 cm from the sample, without any intermediate optical components, using free-space electro-optic sampling (FEOS) in a 1-mm-thick <110> ZnTe crystal gated by 100 fs, 800 nm laser pulses [26] . This detector configuration provided the required sensitivity and acceptance of spectral bandwidth (see “Methods”). A THz-transparent laser beam block was placed behind the sample to stop the propagation of the remainder of the pump laser beam in the setup (see Fig. 1a ). Fig. 1: Schematic of the experiment and the sample. a Sketch of the THz emission experiment: the sample is placed in the static in-plane magnetic field of 63.4 mT, and excited with a 100 fs, 800 nm laser pulse at normal incidence. The THz emission from the sample in the forward-propagating direction is detected using free-space electro-optic sampling in a 1 mm ZnTe crystal, without any intermediate focusing or additional guiding. b Schematic of the sample: 10-nm-thick bcc Fe film, grown on a 0.5-mm-thick MgO substrate, and capped with 12-nm-thick MgO cap. The excitation gradient of the electrons in the Fe film in such a configuration is negligible, and hot-electron diffusion from the Fe film is blocked by MgO substrate and cap. This eliminates the possibility of any ISHE-type mechanism, which relies on the motion of spin-polarized electrons in the sample, in the laser-induced magnetization dynamics M ( t ). The only possible contributions to M ( t ) in the sample remain hot-electron-driven transient demagnetization causing incoherent hot-magnon excitation (primary dynamics), and modulation of exchange interaction in Fe due to generated coherent acoustic phonons S ( t ) propagating in the structure (secondary dynamics). Full size image Challenges in implementation of the method Before we proceed with the results of our measurements and their interpretation, we outline here the challenges in the successful implementation of ultrafast THz magnetometry. The detailed solutions to these challenges are presented in Methods. First of all, THz magnetic dipole radiation E M ( t ) due to magnetization dynamics M ( t ) inside the sample must be uniquely identified as such, and properly isolated. Yet, under certain conditions, the THz emission from the photoexcited magnetic samples can, in fact, be strongly dominated by electric dipole emission E P ( t ), driven by a time-varying electric current J ( t ) = d P /d t in the sample (see “Methods”). This emission originates from the ultrafast inverse spin-Hall effect (ISHE) [17] , [19] , converting the non-radiating laser-driven hot spin currents in the sample into THz-emitting charge currents. While originating from the ultrafast superdiffusion of hot spin-polarized electrons, these THz-emitting conduction currents are in fact an intricate convolution of ultrafast spin currents [27] with the complex process of natural ultrafast conductive response in a material, involving, e.g., energy-, density-, spin- and time-dependent electron mobility (see, e.g. refs. [25] , [28] , [29] , [30] ). Therefore, this ISHE-mediated THz emission does not carry direct, but only indirect information about the laser-induced dynamics of the magnetic state in the material M ( t ). Yet, as we show below, the ISHE-driven emission easily obscures the much weaker THz emission driven purely by magnetization dynamics M ( t ), which remains the quantity of fundamental interest. We overcome this challenge by a proper sample design, where the electronic motion is precluded, thus eliminating the ISHE mechanism; and by performing control measurements, fully supported by the simulations, that unambiguously confirm the magnetic dipole origin of the detected THz fields (see “Methods”). It should be noted here, that all THz emission experiments on metallic magnetic systems known to us [17] , [18] , [19] , [20] , [21] , are, in fact, dealing with the situation of non-local spin dynamics, and hence with predominantly ISHE-type electric dipole THz emission. This includes the pioneering observation of THz emission from an optically pumped Ni/Cr nanostructure made by one of us, Beaurepaire et al. [15] , which was initially assigned to the process of laser-driven transient demagnetization M ( t ) in Ni. Our present understanding [17] , [19] , [31] , however, clearly points to the electric dipole nature of this THz emission [15] via the ISHE mechanism, enabled by the superdiffusion of laser-excited spin-polarized electrons from Ni to Cr, and resulting in the THz-emitting in-plane charge current in the Cr layer. As such, even after more than two decades since the first experiments on ultrafast magnetism [6] , and after more than a decade since the pioneering observations of THz emission from magnetic materials [15] , [16] , the fundamental process of laser-driven transient demagnetization in ferromagnetic metals has not been identified so far in a clear, unambiguous and calibrated manner, to the best of our knowledge. Another challenge is a rigorous and fully calibrated reconstruction of the source of magnetic dipole radiation, i.e., the magnetization dynamics in the sample M ( t ), from the observable, which is the THz electro-optic signal (EOS) detected in the far-field, many centimeters away from the sample. As the propagation and detection of the THz fields significantly modify their temporal structure and spectral content, such a reconstruction becomes another critical step in the successful implementation of ultrafast THz magnetometry: minor discrepancies in the signal reconstruction will lead to a significant error in the recovered M ( t ). We address this challenge by introducing an experimental calibration procedure to a known reference THz emitter, confirmed by a numerical simulation of the THz propagation function in the spectrometer (see “Methods”). The thickness of the Fe layer used in this work was chosen to be 10 nm, which ensures a virtually uniform pump intensity distribution within the film (see “Methods”). As a result, a gradient of superdiffusive motion of photoexcited electrons in the film is precluded. The MgO substrate and 12-nm-thick MgO cap layer (see Fig. 1b ) serve as diffusion barriers, inhibiting the motion of laser-excited electrons out of Fe. As a result, the motion of spin-polarized electrons in our MgO/Fe/MgO sample is precluded, thus eliminating the possibility for ISHE-type mechanisms and leaving the process of pure transient demagnetization M ( t ) as the only possible source of THz electromagnetic emission from this sample. As a control sample, we have grown a nominally identical Fe film on an MgO substrate, but now capped with a metallic Pd film of 5 nm thickness. In such a sample, the superdiffusion of laser-excited spin-polarized electrons from Fe to Pd is possible, thus leading to THz emission via the ISHE mechanism [17] , [19] . The polarization of the electric field component of the measured THz emission from both MgO/Fe/MgO and MgO/Fe/Pd samples was always perpendicular to the direction of the applied magnetic field. Rotating the polarization of the incident laser pulse about the incidence axis had no effect on the THz emission from both samples. Exemplary measured THz EOS from the MgO/Fe/MgO and MgO/Fe/Pd samples, obtained under identical excitation conditions, are shown in the inset of Fig. 2a . The THz emission from the Pd-capped sample exceeds that from the MgO-capped one by about a factor of 10 in amplitude. The dominant mechanisms of THz emission from the MgO/Fe/MgO and MgO/Fe/Pd samples can be differentiated, and uniquely identified as transient demagnetization and ISHE, respectively, using the symmetry analysis of the polarity of measured THz signals (see “Methods” for details). The order of magnitude difference in strength between the THz emission from MgO/Fe/MgO and MgO/Fe/Pd samples shown in the inset of Fig. 2a emphasizes the importance of creating the dielectric diffusion barriers in the sample, blocking the superdiffusion of hot spin-polarized electrons, and thereby suppressing the ISHE THz emission mechanism, for the observation of the pure transient demagnetization processes. We note here that any broken inversion symmetry of the film would also lead to the parasitic electric dipole emission, which was, however, observed to be negligible in our samples (see “Methods”). Fig. 2: Observable quantities: measured electro-optic signal of the THz emission, the reconstructed magnetization dynamics M ( t ), and its theoretical simulation. a Measured FEOS signal of the THz emission from MgO/Fe/MgO sample, under the excitation of 0.51 mJ cm −2 . Inset: comparison of FEOS signal obtained from MgO/Fe/MgO sample via pure magnetic dipole emission due to M ( t ), red line; and MgO/Fe/Pd sample, dominated by the electric dipole emission via ISHE, green line. b Reconstructed magnetization from experimental EOS (solid line) and simulated magnetization dynamics (dashed line). Simulation conditions are identical to the experiment. μ 0 M 0 = 2.15 T—equilibrium magnetization in Fe (see “Methods”). c Hot electrons excite hot magnons that drive primary demagnetization dynamics. d Dynamics of elongation of the Fe film due to the generated coherent optical phonon pulse. Complete magnetization dynamics M ( t ) comprises both incoherent hot-magnon and coherent acoustic phonon contributions. Full size image Reconstruction of magnetization dynamics from THz emission Having proven that the THz emission from the MgO-capped Fe nanofilm is exclusively related to the local dynamics of magnetic order in laser-excited Fe, the measured THz EOS can be rigorously reconstructed to its source, the ultrafast magnetization dynamics M ( t ) in Fe. Our reconstruction protocol is based on the rigorous experimental and theoretical characterization of the complex propagation function, which connects the THz radiation source, i.e., magnetization dynamics M ( t ) in the sample, to the observable, the THz EOS of the magnetic dipole emission E M ( t ). Once the complex propagation function is established, its deconvolution allows the quantitative extraction of the quantity of interest, magnetization dynamics M ( t ). Our reconstruction protocol is described in detail in the “Methods” section. In Fig. 2a , we show the exemplary as-measured THz EOS, and in Fig. 2b the reconstructed magnetization dynamics M ( t ), in our MgO/Fe/MgO sample (excitation fluence of 0.51 mJ cm −2 , experimental configuration as shown in Fig. 1a ). The reconstructed M ( t ) signal contains two clearly visible components: (i) a fast transient demagnetization signal with the amplitude on the order of μ 0 Δ M = 16.6 mT, with an initial drop on the timescale of about 500 fs, and recovery on the timescale of about 1.7 ps; and (ii) a time-delayed contribution of a rather complex shape, with an amplitude of about 2.5 mT, and a total duration of several picoseconds. We attribute the initial fast component to the pure hot-magnon-driven transient demagnetization process in laser-excited Fe: the laser-excited hot electrons in Fe excite nonequilibrium magnons, which in turn (incoherently) drive the demagnetization process. The dynamics of this demagnetization is reasonably well-understood, and has by now been observed by a variety of other experimental methods (see, e.g. refs. [11] , [13] , [32] , [33] ). We assign the second, time-delayed component in the reconstructed magnetization dynamics M ( t ) to the magnetoelastic effect driven by the laser-excited coherent acoustic phonon propagation in the MgO/Fe/MgO structure. The mechanism of the magnetoelastic effect is as follows. Subsequent to the laser excitation, a lattice strain is impulsively generated in Fe via electron-phonon coupling [34] , thus producing a coherent acoustic phonon pulse propagating away from the Fe film surface. This acoustic pulse will undergo reflections on the acoustic discontinuities within the structure (see e.g. ref. [35] ): partial reflections on the Fe/MgO interfaces, and total reflection on the outside cap layer MgO/air interface, as shown in Fig. 1b , thus returning back to the Fe film. The propagation of such a multiply reflected coherent acoustic phonon pulse through Fe leads to the transient modification of interatomic distances in Fe, modulating the exchange interaction in the material, and hence leading to an additional, strain-induced demagnetization channel in Fe (see, e.g. refs. [36] , [37] , [38] and references therein). We note that, to the best of our knowledge, the direct measurement of such an acoustically driven magnetization dynamics on the ultrafast (sub-)picosecond timescale, and its co-existence with the direct laser-driven demagnetization via hot magnons, has not been reported before. Comparison of experiment and theory Here, we have theoretically modeled the entire magnetization dynamics in the laser-excited MgO/Fe/MgO structure employing a methodology that uses as building blocks a microscopic multi-temperature model (MMTM), lattice strain dynamics and first-principles calculations (see “Methods” for details). In Fig. 2b , we show the exemplary measured and simulated M ( t ) signals, corresponding to identical excitation conditions (excitation fluence of 0.51 mJ cm −2 ), and demonstrating very good quantitative agreement. Figure 2c shows the calculation of the isolated hot-magnon driven transient demagnetization dynamics, which occurs on a ~500 fs timescale and recovers on a timescale of ~1.7 ps. In Fig. 2d the calculated dynamics of elongation of laser-excited Fe film is presented, corresponding to the experimental conditions in Fig. 2b : the initial strain in the 10-nm-thick Fe film builds up on a ~2 ps timescale, followed by slow (>100 ps) relaxation, convoluted with the multiple reflections of the strain waves within the Fe film, as well as the return of the strain wave reflected from the outside MgO/air interface of the sample back into the Fe film. The theoretically simulated magnetization dynamics curve M ( t ) in Fig. 2b includes the contributions of the hot-magnon driven transient demagnetization (Fig. 2c ), and the coherent-phonon-driven magnetoelastic effect (Fig. 2d ). Note that the ~2.6 mT amplitude of the acoustically driven demagnetization signal in Fig. 2b is only about 6.5 times smaller than that from the direct laser-driven transient demagnetization of 16.7 mT. In Fig. 3a, b we show the entirety of the measured magnetization dynamics signals M ( t ) in this work under the excitation fluence in the range 0.25–1.02 mJ cm −2 , and the results of the first-principles calculations as described above, fully corresponding to the experimental conditions. The maximum observed hot-magnon driven demagnetization reaches 28.8 mT (corresponding to the change in magnetic moment density of 2.26 × 10 4 J T −1 m −3 ), while the strain-driven demagnetization reaches 3.5 mT (0.18 × 10 4 J T −1 m −3 ) at the strongest excitation. Hence, the maximum measured hot-magnon – and coherent phonon-driven demagnetizations of μ 0 Δ M = 28.8 mT and μ 0 Δ M = 3.5 mT, respectively, correspond to about 1.3% and 0.16% of the equilibrium magnetization in Fe of μ 0 M 0 = 2.15 T. In this respect, we would like to emphasize the high sensitivity provided by our ultrafast magnetometry method, which allows one to reliably observe the ultrafast magnetization dynamics on the order of 10 −4 M 0 (see Figs. 2 , 3 ). Generally, the ratio of the amplitudes of these two components of transient demagnetization signal is maintained as \\\\(\\\\Delta M_{{\\\\mathrm{hot - magnon}}}{\\\\mathrm{/}}\\\\Delta M_{{\\\\mathrm{phonon}}}\\\\, {\\\\approx} \\\\,7\\\\) for the whole range of excitation fluences used in the experiment. Such a high relative efficiency of ultrafast magnetization control by strain, as compared to direct laser excitation of electrons, is quite remarkable, and points to the promise of efficient ultrafast magnetic technologies controlled by acoustic signals, as opposed to traditional optical, electric, or magnetic stimuli. Notwithstanding, the ratio \\\\(\\\\Delta M_{{\\\\mathrm{hot - magnon}}}{\\\\mathrm{/}}\\\\Delta M_{{\\\\mathrm{phonon}}} \\\\,{\\\\approx} \\\\,7\\\\) points to the dominating role of ultrafast magnon excitation for the process of laser-driven femtosecond demagnetization. Fig. 3: Comparison of measured and simulated magnetization dynamics M ( t ) in laser-excited MgO/Fe/MgO sample in this work. a Ultrafast magnetization dynamics in the Fe film at various laser pump fluences in the range 0.25–1.02 mJ cm −2 with errors shown by dashed lines, and b results of simulation corresponding to the experimental scenario in ( a ). The error bars in ( a ) were determined from the standard errors in the measured electro-optic signals (from 200 individual measurements). The experimental error was then propagated in the reconstruction protocol, leading to the intervals as indicated. Full size image Again, the experimental data and the results of modeling are in good agreement: the calculation captures both the temporal features and the magnitudes of the measured transient demagnetization signals. The small discrepancy in the shapes of measured and modeled magnetoelastic contribution might be caused by minor thickness fluctuations in the sample within the laser excitation spot, affecting the local arrival times of the acoustic phonon pulse, and thus leading to a smearing of the overall magnetoelastic signal. Further, the influence of phenomena such as the Elliott-Yafet electron-phonon spin-flip scattering process [39] , [40] , phonon-magnon coupling [41] or heat diffusion [42] , which have not been considered in our theoretical modeling, and whose contribution have been estimated only phenomenologically, might also play a certain role. For example, in our theoretical modeling a linear dependence of the electron-magnon coupling parameter (see “Methods”) on the absorbed optical pump fluence was assumed, and its value was established phenomenologically from the experimental data taken at the intermediate pump fluence of 0.51 mJ cm −2 (see Fig. 2 ). The reason for such an assumption is the current lack of solid experimental and theoretical knowledge regarding this important physical parameter. Yet, as can be seen in the comparison of the entirety of our experimental and theoretical data presented in Fig. 3 , the maximum calculated hot-magnon driven demagnetization at the maximum pump fluence of 1.02 mJ cm −2 exceeds that observed in the experiment by about 25%. While such a discrepancy is not very large, it only emphasizes the need to understand more in-depth the processes governing the magnetic response of materials to ultrafast external stimuli. Our experimental method can significantly contribute to such development.\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"To summarize, we have presented a rigorous method for ultrafast magnetometry based on THz emission spectroscopy. In our proof-of-principle experiment, we observe the ultrafast demagnetization dynamics in laser-excited Fe, and uniquely identify and disentangle two contributions to this process: (i) a hot-electron-driven emission of incoherent hot magnons leading to ultrafast transient demagnetization, and (ii) a coherent acoustic-phonon-driven magnetoelastic effect. Our observations are fully supported by the results of theoretical modeling. To the best of our knowledge, this is the first simultaneous observation of both hot-magnon and coherent-phonon-driven ultrafast magnetization dynamics in a magnetic system, with hot magnons being the dominating effect. Our observations demonstrate the potential of our highly sensitive method for both fundamental research and industrial applications of magnetism. The magnetization dynamics driven by coherent acoustic phonon propagation in iron corresponds to the elongation of the Fe film on the order of only tens of fm (see Fig. 2 ). It is even more so remarkable, that the effects related to such tiny elongations could be clearly revealed in the experiment using THz radiation, i.e., sub-mm electromagnetic waves having 10 orders of magnitude longer wavelength. The surprisingly strong magnetoelastic contribution to ultrafast magnetization dynamics, leading to a clearly measurable electromagnetic emission in the THz range, holds promise for spintronic technologies based on a purely acoustic and, where possible, heat-free ultrafast control of magnetism.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Pump intensity distribution within the iron film The refractive index of iron at the pump wavelength around 800 nm is n = 2.94 + 3.39 i [43] , yielding the optical penetration depth on the order of 20 nm under bulk conditions (Beer-Lambert Law). However, multiple interferences of the pump light in the film result in a virtually constant intensity distribution profile, with the intensity variation on the order of only 1%, with the intensity minimum around the middle of the film (Supplementary Fig. 1 ). Spin current simulations Ferromagnetic 3 d metals such as Fe are known to have a small, yet non-vanishing, inverse spin Hall effect (see e.g. ref. [44] ). To exclude that any spin current along the depth of the sample would lead to an electric THz emission via ISHE mechanism, we have performed rigorous simulations. Specifically, to make our simulations parameter-free, we only use ab initio computed quantities as input. We have therefore computed ab initio the SHE and ISHE coefficients of Fe and Pd; the computed spin Hall conductivities of bulk bcc Fe and fcc Pd are 14.96 and 700.93 ( ħ / e ) S cm −1 , respectively, which implies that the computed spin Hall angle of Pd is 46.85 times larger than that of Fe (Salemi, L. & Oppeneer, P. M. unpublished). In addition we have used spin-dependent hot-electron lifetimes and velocities computed previously [45] . With these input quantities and using the ultrafast superdiffusion code [27] we have simulated the charge and spin currents in MgO/Fe/MgO and MgO/Fe/Pd structures under the same excitation condition as in our experiment (800 nm, 100 fs pulse). The results (Supplementary Fig. 2 ) demonstrate that the electric dipole contribution from the MgO/Fe/MgO structure is extremely small and can be safely neglected, leaving the transient demagnetization as the only source of its THz emission. Magnetic and electric dipole radiation The far-field electric field components of magnetic E M ( t ) and electric E P ( t ) dipole radiation from a point source [46] read $${{\\\\mathbf{E}}_{\\\\mathrm{M}}\\\\left( t \\\\right) = iFT\\\\left\\\\{ { - \\\\frac{1}{{4\\\\pi }}\\\\sqrt {\\\\frac{{\\\\mu _0}}{{{\\\\it{\\\\epsilon }}_0}}} k^2\\\\left( {{\\\\mathbf{n}} \\\\times {\\\\mathbf{M}}} \\\\right)\\\\frac{{e^{ikr}}}{r}} \\\\right\\\\},}$$ (1) $${{\\\\mathbf{E}}_{\\\\mathrm{P}}\\\\left( t \\\\right) = iFT\\\\left\\\\{ {\\\\frac{1}{{4\\\\pi {\\\\it{\\\\epsilon }}_0}}k^2\\\\left( {{\\\\mathbf{n}} \\\\times {\\\\mathbf{P}}} \\\\right) \\\\times {\\\\mathbf{n}}\\\\frac{{e^{ikr}}}{r}} \\\\right\\\\}.} $$ (2) Here iFT stands for inverse Fourier transform, μ 0 is the vacuum permeability, \\\\({\\\\it{\\\\epsilon }}_0\\\\) the vacuum permittivity, k is the wavevector, n the unit vector pointing towards the detecting position from the emitting dipole, M and P are, respectively, the frequency-dependent magnetization and polarization [20] (i.e., Fourier transforms of M ( t ) and P ( t )), and r is the distance from the emitting dipole to the position of detection. Symmetry distinction of THz emission Two types of samples were prepared. Both of them were 10 nm ferromagnetic iron (Fe) films deposited on an insulating MgO substrate. One was capped with a 12 nm insulating MgO layer (MgO/Fe/MgO), and the other was capped with a 5 nm metallic Pd layer (MgO/Fe/Pd). The samples were fully magnetized in-plane along the x -axis (see Fig. 1 ) by an applied external magnetic field of 50 mT, and were excited by femtosecond laser pulses with a fluence of 1.1 mJ cm −2 . First, both samples were pumped from the cap layer side, and the emitted THz waves were measured by FEOS in the far field, in the direction of the pump beam propagation. Second, we flipped the two samples while keeping the external magnetic field and the THz detection geometry unchanged, thus now pumping the samples from the substrate side. Finally, both samples were pumped at the cap layer, but the external magnetic field was reversed. The experimental results (Supplementary Fig. 4 ) show that the polarity of THz emitted from MgO/Fe/MgO only depends on the external magnetic field, while both the external magnetic field and the geometry of the sample can affect the polarity of THz emitted from MgO/Fe/Pd. We conclude that the THz emission from MgO/Fe/MgO is due to the ultrafast magnetization dynamics, whereas the electric dipole radiation caused by the spin-charge conversion dominates the THz emission in MgO/Fe/Pd. Reconstruction of magnetization dynamics We performed direct experimental characterization, fully supported by the theoretical modeling, of the THz field propagation function in our experiments, rigorously relating the THz EOS measured in the far-field, to its source, the magnetization dynamics in the sample M ( t ). For this we used a reference electric-dipole THz emitter, a 0.5-mm-thick ZnTe crystal, in place of MgO/Fe/MgO sample. As a result, we achieve precise reconstruction of the fully calibrated transfer function of our experiment, relating the dynamics of polarization P ( t ) or magnetization M ( t ) at the surface of the laser-excited THz emitter to our observable, the detected THz EOS. The basis for the reconstruction protocol is the accurate establishment of the complete transfer function of the experiment in the frequency domain, which includes (i) conversion of time-dependent magnetization (polarization) in the sample into electromagnetic radiation f rad,M ( ω ) and f rad,P ( ω ), (ii) propagation of this electromagnetic radiation in the spectrometer towards detector crystal f prop ( ω ), and (iii) detection by FEOS of the electromagnetic field in the detector crystal f d ( ω ). While the radiative functions f rad,M ( ω ) and f rad,P ( ω ) are different for magnetic dipole (MgO/Fe/MgO) and electric dipole (ZnTe) emitters, the propagation f prop ( ω ) and detection f d ( ω ) functions remain identical for both cases. The total detected EOS for magnetic and electric dipole emission cases, S M ( ω ) and S P ( ω ) respectively, are $${S_{\\\\mathrm{M}}\\\\left( \\\\omega \\\\right) = M\\\\left( \\\\omega \\\\right)f_{{\\\\mathrm{rad}},{\\\\mathrm{M}}}\\\\left( \\\\omega \\\\right)f_{{\\\\mathrm{prop}}}\\\\left( \\\\omega \\\\right)f_{\\\\mathrm{d}}\\\\left( \\\\omega \\\\right),} $$ (3) $${S_{\\\\mathrm{P}}\\\\left( \\\\omega \\\\right) = P\\\\left( \\\\omega \\\\right)f_{{\\\\mathrm{rad}},{\\\\mathrm{P}}}\\\\left( \\\\omega \\\\right)f_{{\\\\mathrm{prop}}}\\\\left( \\\\omega \\\\right)f_{\\\\mathrm{d}}\\\\left( \\\\omega \\\\right),} $$ (4) where M ( ω ) and P ( ω ) are the frequency-dependent magnetization and polarization, respectively (i.e., the Fourier trasnsforms of M ( t ) and P ( t )). The reconstruction of the quantity of interest M ( ω ) from the observable, EOS S M ( ω ) is therefore $$M\\\\left( \\\\omega \\\\right) = \\\\frac{{S_{\\\\mathrm{M}}\\\\left( \\\\omega \\\\right)}}{{f_{{\\\\mathrm{rad}},{\\\\mathrm{M}}}\\\\left( \\\\omega \\\\right)f_{{\\\\mathrm{prop}}}\\\\left( \\\\omega \\\\right)f_{\\\\mathrm{d}}\\\\left( \\\\omega \\\\right)}}.$$ (5) The radiative functions f rad,M ( ω ) = FT ( E M ( t )) and f rad,P ( ω ) = FT ( E P ( t )) are given by Eqs. ( 1 ) and ( 2 ). In order to experimentally establish the spectrometer function, i.e., the propagation and detection function of our experiment f prop ( ω ) f d ( ω ), we use a reference electric dipole emitter, a 0.5-mm-thick <110> ZnTe crystal. The process of THz generation by optical rectification of femtosecond laser pulses in transparent nonlinear crystals is by now very well understood, and can be modeled quantitatively very accurately from the first principles, using as an input the temporal intensity profile of the laser pulse, the value for electro-optic coefficient of the crystal, and the THz dispersion of its complex-valued refractive index (see e.g. ref. [47] for the comprehensive description of the calculation). This allows one to very accurately calculate the polarization at the outcoupling crystal surface, i.e., the parameter P ( ω ) in Eq. ( 4 ). With the knowledge of P ( ω ), it is therefore straightforward to recover the complete propagation and detection function of our experiment f prop ( ω ) f d ( ω ), from the measured EOS of the THz emission of a reference electric dipole emitter S P ( ω ) [48] . Now that the spectrometer function f prop ( ω ) f d ( ω ) is established, it can be directly applied to the reconstruction of the magnetization dynamics M ( t ) in our magnetic dipole emitter MgO/Fe/MgO, from the correspondingly measured EOS S M ( t ), using the inverse Fourier transform of the Eq. ( 5 ). This yields the recovered, and fully calibrated, magnetization dynamics M ( t ) in our experiment, shown in Figs. 2 and 3 of the main text. Incoherent excitation of magnons Although experimentally we cannot uniquely resolve which microscopic mechanism leads to the observed magnetization dynamics, in this section we show that incoherent excitation of magnons is the only plausible mechanism responsible for the ultrafast magnetization dynamics in the MgO/Fe/MgO structure. On the one hand, it has been recently demonstrated [11] , [32] , [33] that the incoherent excitation of magnon modes is a dominant mechanism contributing to ultrafast demagnetization. Moreover, Mlynczak et al. [49] have recently shown that the electron-magnon scattering process is a very relevant scattering mechanism in Fe. On the other hand, transport mechanisms, such as spin superdiffusion or heat transport, can be disregarded as shown by the simulations. Coherent processes stemming from the direct light-spin interaction can also be neglected due to the linear light polarization. Another relevant mechanism which could potentially explain the transient magnetization dynamics is the Elliott-Yafet mechanism, in which the electrons transfer angular momentum to the phonons. However, it has been theoretically shown that it only provides a small contribution to the ultrafast demagnetization in ferromagnetic 3 d metals [40] . Another plausible mechanism that has gained attention in the last years, is the spin-flip Coulomb scattering model, which is based on the ultrafast transfer of angular momentum from the electronic spin to the electronic orbital angular momentum, and subsequently to the lattice [50] . However, the lack of a theoretical model to explain the ultrafast (tens of femtoseconds) transfer of angular momentum from the electronic orbitals to the lattice, and the relatively slow demagnetization time found in our experiments (around 500 fs) suggest that this theoretically proposal is also not relevant in our experiment. As we show below, excitation of incoherent magnons by electron-magnon scattering can perfectly well explain our experimental observations. Microscopic multi-temperature model To simulate the transient magnetization dynamics M ( t ) in Fe we use a MMTM, which describes the magnetization dynamics as an incoherent excitation of magnon modes through a complex interplay between electrons, spin waves and the lattice. Our model goes well beyond the usual three-temperature model (3TM) and provides an out-of-equilibrium description of electron and lattice dynamics with an explicit inclusion of phonon-phonon scattering through an anharmonic phonon interaction term [51] . Mathematically, the temporal evolution of the three different subsystems is modeled by the following set of coupled differential equations $$C_{\\\\mathrm{e}}\\\\frac{{\\\\partial T_{\\\\mathrm{e}}}}{{\\\\partial t}} \\t= G\\\\left( {T_{\\\\mathrm{s}} - T_{\\\\mathrm{e}}} \\\\right) + \\\\mathop {\\\\sum }\\\\limits_Q C_Q\\\\gamma _QI\\\\left( {T_{\\\\mathrm{e}}} \\\\right)\\\\left( {T_{\\\\mathrm{l}}^Q - T_{\\\\mathrm{e}}} \\\\right)\\\\left[ {1 + J\\\\left( {\\\\omega _Q,T_{\\\\mathrm{l}}^Q} \\\\right)\\\\left( {T_{\\\\mathrm{l}}^Q - T_{\\\\mathrm{e}}} \\\\right)} \\\\right]\\\\\\\\ \\t\\\\quad+ \\\\frac{{\\\\partial U_{{\\\\mathrm{e}} - {\\\\mathrm{e}}}}}{{\\\\partial t}},$$ (6) $$C_Q\\\\frac{{\\\\partial T_{\\\\mathrm{l}}^Q}}{{\\\\partial t}} \\t= - C_Q\\\\gamma _QI\\\\left( {T_{\\\\mathrm{e}}} \\\\right)\\\\left( {T_{\\\\mathrm{l}}^Q - T_{\\\\mathrm{e}}} \\\\right)\\\\left[ {1 + J\\\\left( {\\\\omega _Q,T_{\\\\mathrm{l}}^Q} \\\\right)\\\\left( {T_{\\\\mathrm{l}}^Q - T_{\\\\mathrm{e}}} \\\\right)} \\\\right]\\\\\\\\ \\t\\\\quad - \\\\mathop {\\\\sum}\\\\limits_k {C_Q} \\\\Gamma _{Qk}\\\\left( {T_{\\\\mathrm{l}}^Q - T_{\\\\mathrm{l}}^k} \\\\right)\\\\left[ {1 + J\\\\left( {\\\\omega _Q,T_{\\\\mathrm{l}}^Q} \\\\right)\\\\left( {T_{\\\\mathrm{l}}^Q - T_{\\\\mathrm{l}}^k} \\\\right)} \\\\right]\\\\\\\\ \\t\\\\quad+ \\\\frac{{\\\\partial U_{{\\\\mathrm{e}} - {\\\\mathrm{ph}}}}}{{\\\\partial t}},\\\\,\\\\,for\\\\,Q = Q_1, \\\\cdots ,Q_N,$$ (7) $$ C_{\\\\mathrm{s}}\\\\frac{{\\\\partial T_{\\\\mathrm{s}}}}{{\\\\partial t}} = - G\\\\left( {T_{\\\\mathrm{s}} - T_{\\\\mathrm{e}}} \\\\right).\\\\qquad$$ (8) Here T e , T s , and \\\\(T_{\\\\mathrm{l}}^Q\\\\) are the electron, spin and phonon mode-dependent temperatures. Q accounts for the N distinct and independent phonon subsystems, each of them corresponding to a specific branch, υ , and momentum, q , having different temperatures \\\\(T_{\\\\mathrm{l}}^Q\\\\) . C e , C s , and C Q are the temperature-dependent electronic, spin and phonon mode-dependent heat capacities, respectively . γ Q and Γ Qk are the mode-dependent linewidth due to electron-phonon and phonon-phonon interactions, respectively. G is the electron-magnon coupling constant. I ( T e ) accounts for the dependence of the electron-phonon coupling on the electronic temperature. \\\\(\\\\frac{{\\\\partial U_{{\\\\mathrm{e}} - {\\\\mathrm{ph}}}}}{{\\\\partial t}}\\\\) and \\\\(\\\\frac{{\\\\partial U_{{\\\\mathrm{e}} - {\\\\mathrm{e}}}}}{{\\\\partial t}}\\\\) describe the energy transfer rate from the laser-induced nonequlibrium electrons to the lattice through electron-phonon interaction and into thermal electrons via electron-electron interactions, respectively. \\\\(J\\\\left( {\\\\omega _Q,T_{\\\\mathrm{l}}^Q} \\\\right)\\\\) is the second-order Taylor expansion of the out-of-equilibrium phonon populations around the mode-dependent phonon temperatures (see ref. [51] for full forms). The effective spin temperature, T s , is in our model equivalent to the magnon temperature, which characterizes the incoherent excitation of magnon modes in the 3TM. It is worth mentioning that, while the model for the out-of-equilibrium description of the electron and lattice dynamics has been derived from a microscopic theory and is therefore fit-free (the parameters are ab initio computed), the coupling of these two subsystems with the spin system is introduced phenomenologically, and the electron-magnon coupling constant is chosen to reproduce the experimental demagnetization time. In addition, a correct treatment of the spin system would need to include effective mode- and q -dependent spin temperatures to account for the distinct q -dependent electron-magnon scatterings. However, currently there does not exist a reliable theoretical model to provide such q -dependent electron-magnon coupling, and therefore such description is beyond our present capabilities. Notwithstanding, we can effectively treat the global effect of heating the magnon system by considering the spin system to be individually thermalized and work with an effective electron-magnon coupling, which is then chosen to reproduce the experimental demagnetization time. From the MMTM model we determine the time evolution of the spin temperature which then is used in the analytic form of the temperature-dependent magnetization for bcc Fe $$ {M\\\\left( {T_{\\\\mathrm{s}}} \\\\right) = M\\\\left( 0 \\\\right)\\\\left( {1 - \\\\frac{{T_{\\\\mathrm{s}}}}{{T_{\\\\mathrm{c}}}}} \\\\right)^{0.39},} $$ (9) where T c = 1044 K is the Curie temperature of Fe. In addition, the transient phonon-mode dependent lattice temperatures are used to model the strain dynamics, as outlined in the following. Lattice strain dynamics We consider three different mechanisms that contribute to the creation of strain in the lattice. These mechanisms are: first, the force induced by the optically excited electrons; second, the force induced by the ultrafast demagnetization (a new electronic surface potential is generated following the magnon emission); and third, the thermal strain due to heating of the lattice. The first and second processes originate from the deformation potential mechanism [52] and happen at a time scale below 100 fs, where the strain caused by acoustic phonons depends on the deformation potential and the absorbed laser energy. The third process is induced by the changing temperature distribution in a thermoelastic model wherein a linear relation is established between the stress pulse and the varying temperature distribution. Although in the latter process the changing electronic and lattice temperatures contribute to the stress pulse, the regime of small laser intensities used in the experiment along with the smallness of the electron heat capacity as compared with the lattice one, allows one to neglect the electronic contribution. The transient lattice strain along the direction perpendicular to the film ( z ) can then be written as $$\\\\eta \\\\left( {z,t} \\\\right) = - \\\\frac{1}{{2\\\\rho v^{2}}}\\\\sum_{Q} \\\\int_{ - \\\\infty }^{ + \\\\infty } g_{Q} C_{Q} {\\\\mathrm{sgn}}\\\\left(z - v\\\\left(t - t^{\\\\prime} \\\\right) \\\\right)\\\\left. {\\\\frac{{\\\\partial T_{\\\\mathrm{l}}^Q}}{{\\\\partial t}}} \\\\right|_{\\\\left( {\\\\left| {z - v\\\\left( {t - t^{\\\\prime} } \\\\right)} \\\\right|,t^{\\\\prime} } \\\\right)}{{\\\\mathrm{d}}t}^{\\\\prime},$$ (10) where ρ is the density and v the sound velocity of Fe, and g Q is the mode-dependent Grüneisen parameter, which are computed from first principles. The term \\\\(\\\\frac{{\\\\partial T_{\\\\mathrm{l}}^Q}}{{\\\\partial t}}\\\\) is determined by the solution of the MMTM (see above). Thus, Eq. ( 10 ) is a modified out-of-equilibrium version of the expression provided by Wright [53] . Although it may provide the same total expansion of the lattice, it is reached through different strain dynamics. Notably, the solution of the MMTM yields that the time evolution of the strain produced by heating covers a range of several picoseconds. The different time scales of the three different processes that induce the strain pulses enable a simplification in the modeling. Thus, strain pulses generated by the deformation potential mechanisms are assumed to happen instantaneously following laser excitation with an amplitude proportional to the laser excitation, which adds to the heating induced strain. Schematically, the process is as follows: At each of the two Fe/MgO interfaces a tensile strain is generated following laser excitation. These pulses travel within the Fe layer toward the other interface at the Fe sound velocity (5060.05 m s −1 ). Correspondingly, a compressive strain is generated at the Fe/MgO interface which travels within the MgO system at the MgO sound velocity (9100 m s −1 ). When the strains traveling within the Fe film reach one of the Fe/MgO interfaces, the pulses become partially reflected (92.5% transmission, due to the different impedances between Fe and MgO) inducing a change of polarization of the strain (tensile waves become compressive waves and vice versa), while the rest goes into the MgO films. On the other hand, the strain waves are completely reflected on the MgO/air interfaces. At the end of the simulation, a finite strain remains in the system due to the heating effect that induces the thermal expansion of the system. To recover the normal volume expansion (in our case normal surface expansion, u ( t )) we integrate over the whole Fe film thickness the time-dependent strain pulses, getting $$u\\\\left( t \\\\right) = \\\\int_0^{z_{{\\\\mathrm{max}}}} \\\\eta \\\\left( {z,t} \\\\right){{\\\\mathrm{d}}z},$$ (11) where z max = 10 nm. The calculated time evolution of the system’s expansion is shown in Fig. 2d for a laser excitation of 1.02 mJ cm −2 . Magnetization dynamics The final magnetization dynamics is calculated by combining the MMTM and the strain dynamics. Thus, we can write $$ {\\\\frac{{M\\\\left( {T_{\\\\mathrm{s}}} \\\\right)}}{{M\\\\left( {T_0} \\\\right)}} = \\\\left( {\\\\frac{{T_{\\\\mathrm{c}} - T_{\\\\mathrm{s}}}}{{T_{\\\\mathrm{c}} - T_0}}} \\\\right)^{0.39} + \\\\Delta M\\\\left( {u\\\\left( t \\\\right)} \\\\right) + f_{{\\\\mathrm{recovery}}},} $$ (12) where f recovery is a phenomenological function linear in time, introduced to recover the long-time behavior of the magnetization dynamics (relaxation back to equilibrium), and which accounts for the physical processes that are not included in our model, such as phonon-magnon coupling or heat diffusion, which are relevant at long-time scales. Δ M ( u ( t )) is a function that represents the change of magnetization due to the change of the system’s volume. It is driven by the modulation of exchange interaction in iron, and phenomenologically, can be written as: $$ {\\\\Delta M\\\\left( {u\\\\left( t \\\\right)} \\\\right) = \\\\Delta J\\\\left( {u\\\\left( t \\\\right)} \\\\right)\\\\frac{1}{2}\\\\frac{{M_0}}{{J_0}},} $$ (13) where the factor 1/2 comes from the fact that around equilibrium position a 2% change of J produces a 1% change of magnetization [54] , [55] We also assume that $$ {\\\\Delta J\\\\left( {u\\\\left( t \\\\right)} \\\\right) = \\\\frac{{\\\\mathrm{d}}J}{{\\\\mathrm{d}}u}u\\\\left( t \\\\right),} $$ (14) where u ( t ) is the system elongation in z -direction (see Eq. ( 11 )). By combining Eqs. ( 13 ) and ( 14 ) we obtain $$ {\\\\Delta M\\\\left( {u\\\\left( t \\\\right)} \\\\right) = \\\\frac{{\\\\mathrm{d}}J}{{\\\\mathrm{d}}u}u\\\\left( t \\\\right)\\\\frac{1}{2}\\\\frac{{M_0}}{{J_0}},} $$ (15) where the value of the gradient of the exchange constant is fitted to provide the experimentally observed transient magnetization dynamics. For the specific case of Fe we obtain $${\\\\Delta M\\\\left( {u\\\\left( t \\\\right)} \\\\right) = 80.88\\\\frac{{\\\\mu _{\\\\mathrm{B}}}}{{\\\\mathrm{{\\\\AA}}}}u\\\\left( t \\\\right).} $$ (16) This expression is used along with the elongation values provided by Eq. ( 11 ) to obtain the transient magnetization driven by the acoustic waves in Eq. ( 12 ). As example, a system’s elongation of 40 fm would induce an increase of magnetization of the order of 0.16% of the equilibrium magnetization, as can be seen in Fig. 2b, d for a delay time between 3 and 4.5 ps. On the other hand, lattice strains of the order of 10 −3 would induce magnetization changes of the order of 37% of the equilibrium magnetization. However, it is important to mention that although those large lattice strains are experimentally accessible at short time scales, their effect on the transient magnetization might be largely shadowed by the more dominant magnetization dynamics induced by hot magnons. In addition, large elongations would introduce nonlinear effects in the above relation, and also significant changes in the magnetic anisotropy, leading to a distinct magnetization dynamics. The value of equilibrium magnetization in iron μ 0 M 0 = 2.15 T [56] , [57] , [58] was used in the calculations of relative magnetization dynamics. Sample preparation The iron films were deposited on double polished MgO <100> substrates and capped with 12 nm of MgO and 5 nm of Pd at room temperature, respectively. The molecular beam epitaxy was performed to deposit 10 nm of Fe at room temperature with subsequent 90 minutes heat treatment at 800 K, necessary to obtain highly ordered monocrystalline Fe layer. The deposition rate of Fe layer was 0.05 nm min −1 at standard 10 −10 Torr (10 −8 Pa) pressure under reflection high energy electron diffraction control to verify the crystal quality. The thicknesses were controlled in situ by quartz balance sensing.\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"The core research idea is to develop an accurate and practical ultrafast magnetometry method based on THz emission spectroscopy under ambient conditions, enabling the direct and fully calibrated determination of ultrafast magnetization dynamics $ M(t) $ in encapsulated magnetic materials, such as laser-excited iron films, by measuring the electric field component $ E_M(t) $ of the magnetic dipole THz emission, thereby overcoming limitations of existing techniques and revealing the full potential of THz emission spectroscopy for studying ultrafast magnetism."},"entities":{"kind":"list like","value":["ultrafast magnetometry","THz emission spectroscopy","ambient conditions","ultrafast magnetization dynamics","encapsulated magnetic materials","laser-excited iron films","electric field component","magnetic dipole THz emission"],"string":"[\n \"ultrafast magnetometry\",\n \"THz emission spectroscopy\",\n \"ambient conditions\",\n \"ultrafast magnetization dynamics\",\n \"encapsulated magnetic materials\",\n \"laser-excited iron films\",\n \"electric field component\",\n \"magnetic dipole THz emission\"\n]","isConversational":false},"references":{"kind":"list like","value":[],"string":"[]","isConversational":false}}},{"rowIdx":85,"cells":{"unique_id":{"kind":"string","value":"ncomms13752"},"subfield":{"kind":"string","value":"Optics and photonics"},"title":{"kind":"string","value":"Multiple signal classification algorithm for super-resolution fluorescence microscopy"},"abstract":{"kind":"string","value":"Single-molecule localization techniques are restricted by long acquisition and computational times, or the need of special fluorophores or biologically toxic photochemical environments. Here we propose a statistical super-resolution technique of wide-field fluorescence microscopy we call the multiple signal classification algorithm which has several advantages. It provides resolution down to at least 50 nm, requires fewer frames and lower excitation power and works even at high fluorophore concentrations. Further, it works with any fluorophore that exhibits blinking on the timescale of the recording. The multiple signal classification algorithm shows comparable or better performance in comparison with single-molecule localization techniques and four contemporary statistical super-resolution methods for experiments of in vitro actin filaments and other independently acquired experimental data sets. We also demonstrate super-resolution at timescales of 245 ms (using 49 frames acquired at 200 frames per second) in samples of live-cell microtubules and live-cell actin filaments imaged without imaging buffers."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"Super-resolution fluorescence microscopy techniques aim at resolving details smaller than the Abbe diffraction limit of , where λ is the wavelength of the fluorescence emission and NA is the numerical aperture of the microscope objective. Most of these techniques use the blinking phenomenon, where fluorophores switch between a bright (fluorescent) state and a long-lived dark state. A series of images is recorded over time. Each image has different intensity distribution because a different set of fluorophores were in the bright state during each image acquisition. The temporal information contained in the series is, then, used to construct a final image with improved spatial resolution. Single-molecule localization microscopy (SMLM) techniques such as stochastic optical reconstruction microscopy (STORM) or photo-activated localization microscopy (PALM) are popular super-resolution techniques owing to their simplicity, few (if any) special requirements on instrumentation, and impressive resolution of ∼ 20 nm (refs 1 , 2 , 3 ). However, they require that the fluorophores exhibit long dark states, so that only a small subset of optically separable fluorophores are in the bright state in each frame of the image stack. This translates into requirements of long acquisition times and of photochemical environment promoting long dark states and impeding bleaching, which is toxic to live cells [4] . The limitations of SMLM have motivated development of techniques that rely on statistical independence of blinking of individual fluorophores rather than on long dark states [5] . Such techniques include super-resolution optical fluctuations imaging (SOFI [6] ), Bayesian analysis of blinking and bleaching (3B (ref. 7 )) and entropy-based super-resolution imaging (ESI [8] ). Although they relax the requirements of SMLM, they do not reach resolution achievable by SMLM ( ∼ 110 nm for SOFI [9] , 80 nm for ESI [8] and 50 nm for 3B (ref. 7 )) and they possess limitations of their own. For example, SOFI uses cumulants of the fluorescence blinking to enhance resolution; since cumulants of orders higher than six are prone to shot noise and do not have good approximations, the practically achievable resolution improvement is limited to the factor of (ref. 6 ). 3B uses a Markov process for modelling the blinking and bleaching of the fluorophores and an expectation maximization approach to determine the likelihood of an emitter being present at a given location. This approach is computationally intensive and its convergence to global minimum is not guaranteed. In the following, we propose a novel algorithm utilizing fluorescence blinking to enhance spatial resolution. The algorithm, called MUltiple SIgnal Classification ALgorithm (MUSICAL), achieves super-resolution by exploiting the eigenimages of the image stack, which statistically represent its prominent structures and, then, applying the knowledge of the point spread function (PSF) of the imaging system to localize the structures to super-resolution scales. Like SOFI or 3B and other related techniques, MUSICAL requires neither special instrumentation nor special fluorophores. The sole requirement is statistically independent blinking of individual emitters. We tested MUSICAL on images of actin filaments and compared it with STORM, showing that both techniques give comparable resolution enhancements. We also compared MUSICAL with 3B, SOFI, ESI and deconSTORM [10] on experimental data sets independently acquired by other super-resolution research groups [11] , [12] and show comparable or superior performance of MUSICAL. We also demonstrate that MUSICAL performs well in situations where STORM fails due to high density of fluorophores. Further, we show that MUSICAL can be used for live-cell fast imaging ( ∼ 49 frames amounting to a total acquisition time of less than 250 ms) of live cells expressing standard green fluorescent protein (GFP) imaged in physiologically conducive buffer devoid of chemicals that influence blinking.","references":{"indices":[4,5,6,8,9,10],"infos":[{"idx":"4","title":"Ettinger, A. & Wittmann, T. Fluorescence live cell imaging.Methods Cell Biol.123, 77–94 (2014).","link":"https://doi.org/10.1016%2FB978-0-12-420138-5.00005-7"},{"idx":"5","title":"Small, A. & Stahlheber, S. Fluorophore localization algorithms for super-resolution microscopy.Nat. Methods11, 267–279 (2014).","link":"https://doi.org/10.1038%2Fnmeth.2844"},{"idx":"6","title":"Dertinger, T., Colyer, R., Iyer, G., Weiss, S. & Enderlein, J. Fast, background-free, 3d super-resolution optical fluctuation imaging (SOFI).Proc. Natl Acad. Sci. USA106, 22287–22292 (2009).","link":"https://doi.org/10.1073%2Fpnas.0907866106"},{"idx":"8","title":"Yahiatene, I., Hennig, S., Müller, M. & Huser, T. Entropy-based super-resolution imaging (ESI): From disorder to fine detail.ACS Photonics2, 1049–1056 (2015).","link":"https://doi.org/10.1021%2Facsphotonics.5b00307"},{"idx":"9","title":"Geissbuehler, S. et al. Live-cell multiplane three-dimensional super-resolution optical fluctuation imaging.Nat. Commun.5, 5830 (2014).","link":"https://doi.org/10.1038%2Fncomms6830"},{"idx":"10","title":"Mukamel, E. A., Babcock, H. & Zhuang, X. Statistical deconvolution for superresolution fluorescence microscopy.Biophys. J.102, 2391–2400 (2012).","link":"https://doi.org/10.1016%2Fj.bpj.2012.03.070"}]}},{"section":"Results","content":"Multiple signal classification algorithm The idea of MUSICAL is inspired from MUltiple SIgnal Classification (MUSIC) used in acoustics [13] , radar signal processing [14] and electromagnetic imaging [15] for finding the contrast sources created due to scattering and contributing to the measured signal. However, MUSICAL differs from MUSIC because the emitters in fluorescence microscopy behave differently from the contrast sources encountered in scattering. Firstly, the fluorophores exhibit intermittent emission when exposed to continuous excitation, the intermittence patterns of any two fluorophores being uncorrelated. Secondly, the information of a molecule is concentrated in a small region defined by the PSF in the image plane. Thus, there is a region of confidence for the likely position of each molecule; pixels beyond that region contribute only additional noise and almost no usable information in the context of localizing molecules in the given region. Thirdly, MUSIC in its traditional form needs a higher number of receivers (pixels in the camera) and measurements (frames) than the number of contrast sources (fluorescent emitters). Thus, when imaging an area with a high density of fluorophores, the required number of frames may easily become impractically large when using the traditional form of MUSIC. MUSICAL overcomes those issues of MUSIC by adopting a sliding soft window and stitching approach. MUSIC is applied only on a small part of the image (the soft window) at a time. The soft window is, then, scanned over the whole image and the individual MUSIC reconstructions are stitched together to form the MUSICAL image. See Supplementary Fig. 1 for the flowchart of MUSICAL. Since the area corresponding to the soft window contains only a fraction of the total number of fluorophores in the imaged area, the required number of frames in the stack is considerably reduced. Besides, this approach also suppresses the problem of noise contribution from pixels further away from a given emitter location. The concept, algorithm, and implementation of MUSICAL is detailed in Supplementary Notes 1 and 2 , and Supplementary Methods , respectively. Several synthetic examples have been used to demonstrate various aspects of MUSICAL. Their details are given in Supplementary Methods . We discuss here the algorithm of MUSICAL briefly. Using the selected soft window of the image stack, MUSIC first computes the eigenimages of the soft window through singular value decomposition. The effect of the soft window is illustrated in Supplementary Fig. 2 . Each eigenimage of the selected soft window is associated to a singular value and represents a particular pattern found in the image stack. Statistically, large singular value indicates that the pattern of the corresponding eigenimage is a prominent pattern in the image stack. On the other hand, statistically less likely patterns, which include noise patterns, are characterized by small singular values. Thus, the eigenimages can be separated into two sets (or subspaces) using a threshold value σ 0 , where eigenimages with singular values more than or equal to σ 0 form the signal subspace (or the range) and the remaining eigenimages form the null subspace. The concepts of the range and the null space are discussed in Supplementary Note 3 . The choice of the value of σ 0 is discussed in Supplementary Figs 3 and 4 , and Supplementary Note 4 . Then, the projections of the PSF (an example is given in Supplementary Fig. 5 ) at a given test point on the range and the null subspace are determined, which we denote as d PR ( ) and d PN ( ) respectively. The projection d PR ( ) indicates whether the PSF at the test point is related to the patterns represented by the range, similarly for d PN ( ). If the test point indeed belongs to either the fluorophore locations or the structure represented by them, then d PN ( ) is close to zero, it is non-zero otherwise. This property of d PN ( ) is used to compute an indicator function as follows such that the value of the indicator function is very large at the point a fluorophore is present and small at the point where no fluorophore is present. An example of eigenimages and projection of PSF on them is given in Supplementary Fig. 6 and Supplementary Note 5 . Further, the sensitivity of MUSICAL to the estimate of PSF is given in Supplementary Fig. 7 . In the original form of MUSIC, as used in acoustic and electromagnetic imaging, only the distance from the null space d PN ( ) is used and α is often set as 1. However, the indicator function of MUSIC is not suitable for the sliding window approach, as shown in Supplementary Fig. 8 . The modified indicator function of MUSICAL is amenable to the sliding window approach since the inclusion of d PR ( ) in the numerator automatically weighs the result of each window, so that stitching of the results for sliding windows can be simplified as sum of the indicator functions of all the sliding windows covering a test point. At the same time, the use of α allows for better resolution by non-linearly scaling the indicator function at closely located test points. See Supplementary Note 6 for a discussion on the role of the parameter α and Supplementary Fig. 3 for an example. In vitro experiments Here, we present results of MUSICAL for in vitro experiments and in vitro test data available online [11] , [12] . In vitro experiments were performed on actin filaments tagged with Phalloidin-Atto 565 dye and serve to characterize the performance of MUSICAL. Results of samples 1 and 2 are included in Figs 1 and 2 , respectively. Figure 1: Results for regions A–C of in vitro sample 1. The mean image of 10,000 frames is shown in a . ( b , c ) The mean image and MUSICAL image for region A. ( d ) Shows the profile of section A1 shown in c . ( e – g ) The mean image, STORM image, and MUSICAL image for region B. ( h – j ) show the mean image, STORM image and MUSICAL image for region C. ( k ) shows the profile of section C1 shown in i , j . ( l ) shows the average PSFs derived from groups AA and BB of line segments shown in c , g , respectively. Scale bars: 1 μm ( a ); 250 nm ( b , c , h – j ); and 500 nm ( e – g ). Full size image Figure 2: MUSICAL results for in vitro sample 2. The first, second and third columns correspond to the image stacks acquired using laser powers 10.3 W cm −2 ( a , e , i , m , q ), 40.2 W cm −2 ( b , f , j , n , r ) and 205.6 W cm −2 ( c , g , k , o , s ), respectively. The fourth column ( d , h , l , p , t ) shows result of MUSICAL using 49 frames only from the image stack acquired with 205.6 W cm −2 excitation power. ( q – s ) Compares the intensity profiles of mean image and MUSICAL at the sections E1, E2 and E3, respectively. ( t ) Compares MUSICAL result at E3 using entire stack of 10,000 images with MUSICAL result at F3 using 49 frames only. Scale bars: 2 μm (first and second rows); and 1 μm (third and fourth rows). Full size image Experimental data provided as single-molecule localization test data set (referred to as Data-SMLM) [11] are used for comparison with other contemporary methods. Details about this data set can be found in its website. It comprises of three data, namely tubulins long sequence (15,000 frames), tubulins high density (500 frames) and tubulinAF647 (9,990 frames). The results for Data-SMLM are included in Fig. 3 . Tubulins high-density data is further used to demonstrate resolution of two microtubules using MUSICAL. Figure 3: Results on Data-SMLM data sets and synthetic example SynSTORM for comparison of MUSICAL with SMLM techniques. ( a ) Tubulins high density (500 frames), ( b ) tubulins long sequence (15,000 frames) and ( c ) TubulinAF647 (9,990 frames). PALM and STORM results in b are taken from [11] . In b , MUSICAL image (in green) is overlaid with PALM image (in magenta) and STORM image (in magenta), respectively, for comparison. Similarly, MUSICAL image (in magenta) is overlaid with STORM image (in green) in c . Full size image We breakdown the in vitro results into the following studies: characterization of the PSF and resolution of MUSICAL, performance of MUSICAL for non-sparse blinking, minimum number of frames required for MUSICAL, excitation power and MUSICAL, comparison of MUSICAL with SMLM techniques and comparison of MUSICAL with other super-resolution techniques. These studies are presented below. Point spread function and resolution of MUSICAL In vitro sample 1 is used for characterization of the experimental PSF of MUSICAL. In Fig. 1c , group of line segments AA is used to obtain the experimental PSF of MUSICAL. The lines are separated by 26 nm. The maxima of the profiles of all the lines are aligned and then their average profile is computed as representative of the PSF of MUSICAL, which is shown in Fig. 1l . The full-width at half-maximum (FWHM) of the average profile is 47.4 nm. We also determined FWHMs of all the individual profiles; they were 47.3±7.4 nm (mean±s.d.). We repeated the same procedure for another group of line segments BB shown in Fig. 1g . The lines are separated by 65 nm here. The FWHM of the average profile for group BB is 46.6 nm. The FWHMs of the individual profiles were 48.5±8.7 nm (mean±s.d.). Characterization of the PSF of MUSICAL as a function of signal to background (SBR) ratio using a synthetic example SynEx3 is given in Supplementary Fig. 9a . We observed periodicity along the actin filaments in the MUSICAL images and the minimum distance between the two peaks on actin filament to be 27.7 nm, as seen in Fig. 1d . Further, we consider the tubulins high-density data from Data-SMLM data set which has high density of emitters such that blinking is non-sparse. MUSICAL clearly illustrates the ability to resolve microtubules separated by 125 nm with a contrast value of 0.39, as seen for the section A in Fig. 3a . Characterization of structural resolution of MUSICAL is given in Supplementary Fig. 9b–d using synthetic examples SynPairDelX. To study further the observed periodicity in the actin filaments, we plot the intensity of MUSICAL image across the sections A1 and C1 of two actin filaments in Fig. 1d,k , respectively. Fourier spectra of these sections and a synthetic example with periodically placed emitters are provided in Supplementary Fig. 10 . It reveals a prominent peak at the spatial frequency corresponding to a period of 62.5 nm for A1. Another prominent peak at similar length scale occurs at sampling frequency corresponding to a period of 91.0 nm (which is ∼ 1.5 times 62.5 nm). Similarly, for the section C1, the Fourier analysis revealed a prominent peak at the spatial frequency corresponding to a period of 65.5 nm. Chiu et al . [16] reported the period of the actin double helix to be 37 nm, which corresponds to a period of 74 nm for a single helix. However, actin is known to have structural polymorphs [17] and disorders of sub-helical dimensions [18] . In addition, poly-lysine, used for stabilizing the actin filaments on the glass chamber, is known to introduce polymeric variations [19] . Thus, we suggest that this periodicity is associated with the periodicity of a single helix of the double helical structure. The performance of MUSICAL for non-sparse blinking As noted before, most localization approaches rely on long dark states of fluorophores (sparse blinking) such that the few bright fluorophores are optically resolvable. MUSICAL does not have this restriction. To illustrate this, we consider in vitro sample 2 which exhibits high density of fluorophores (see Supplementary Fig. 11 ). While STORM is not suitable for imaging such sample (see Supplementary Fig. 12 ), MUSICAL can image these samples efficiently as seen in the second and fourth rows of Fig. 2 ). We further demonstrate the ability of MUSICAL to image dense fluorophore structures through the tubulins high-density data of Data-SMLM. The results are shown in Fig. 3a . The zoom-in of the cyan marked rectangles in the MUSICAL results show that closely lying microtubules with separation smaller than the optical diffraction limit ( ∼ 265 nm for this data) can be resolved consistently using MUSICAL. Minimum number of frames required for MUSICAL If blinking is sparse, large number of frames have to be acquired to be able to collect fluorescent signal from each fluorophore. However, since sample 2 does not demonstrate sparse blinking, it is interesting to consider if MUSICAL can be applied to such sample using only a few frames. Owing to the sliding window of size N pixels, the rank of the matrix on which eigenimages are computed is less than or equal to min( N , K ), where K is the number of frames; see Supplementary Note 1 . The number of pixels in the sliding window N is determined by the PSF of the system and thus fixed. Thus, it is desirable to use K ≥ N . We have used a sliding window of size N w =7 pixels, which implies that N = =49. We applied MUSICAL on the first 49 frames of the data acquired for in vitro sample 2 under excitation power 205.6 W cm −2 . The results shown in the fourth column of Fig. 2 clearly illustrate the capability of MUSICAL to obtain super-resolution in non-sparse blinking using very few frames. This also indicates the utility of MUSICAL in the study of dynamic biological processes. We note that emitters that do not emit significant number of photons in the frames used for MUSICAL will not be imaged by MUSICAL. As a further comment on the desirable condition K ≥ N , we note that soft window of size larger than N w in Supplementary Fig. 2 does not add much value while increasing the minimum number of frames and the computation time. Thus, the sufficient number of frames is limited by the blinking rate and emitter density such that all emitters in the sample can blink in order to represent the entire structure and there are enough fluctuations in the intensity over the frames. Excitation power and MUSICAL Whereas the image stack for sample 1 was acquired under 205.6 W cm −2 , we acquired three image stacks for sample 2, each with different excitation powers, that is, 10.3, 40.2 and 205.6 W cm −2 . On one hand, the excitation power directly impacts the achievable resolution through the signal strength and SBR ratio. On the other hand, it indirectly influences the reconstruction by affecting the blinking rates of fluorophores, where high excitation powers generally result in longer dark states [20] . The MUSICAL results for these image stacks are shown in Fig. 2 . It is seen that at each excitation power, MUSICAL can reconstruct more details than the mean image. The FWHM of the peaks indicated by arrows in Fig. 2r,s are 49 and 61 nm, respectively. The result implies that the FWHM does not deteriorates significantly when the laser power is decreased from 205.6 to 40.2 W cm −2 . It also helps to identify the minimum power at which sufficient resolution is achieved while minimizing photo-bleaching. We provide an additional example using in vitro sample 3 and an empirical plot of MUSICAL FWHM as a function of excitation power in Supplementary Fig. 13 and Supplementary Note 7 . We discuss the presence of side lobes in experimental results of MUSICAL and provide an analysis of the side lobes of MUSICAL as a function of SBR in Supplementary Fig. 14 . Effect of camera noise that becomes prominent at very poor SBR is discussed in Supplementary Fig. 15 and Supplementary Note 8 . Compensation for camera noise and side lobes is discussed in Supplementary Fig. 16 and Supplementary Note 9 . Compensation of side lobes of Fig. 2m is shown in Supplementary Fig. 17 . Comparison between MUSICAL and SMLM techniques Comparison of MUSICAL and STORM results for regions B and C of sample 1 are shown in Fig. 1f,i . In addition, we compare the results of MUSICAL with SMLM techniques for the Data-SMLM data set in Fig. 3a–c . Implementation and parameters used for SMLM techniques are given in Supplementary Methods . Overlay of MUSICAL images with STORM or PALM images are also provided for tubulins long sequence and tubulinAF647. We chose these data for overlay and comparison because they demonstrate sparse blinking, and thus are favourable for SMLM. The results show good agreement between MUSICAL and SMLM techniques. Now we discuss the difference between MUSICAL and SMLM results, such as noted in Fig. 1f,g,i,j . These differences are attributed to multiple simultaneously bright fluorophores in the small volume surrounding the junctions. If a couple of closely placed emitters are blinking in a frame in a very small volume, STORM may conclude the presence of one very bright fluorophore at the centre of the small volume instead of the actual distribution of the closely placed emitters, unless a multiple-fluorophore fitting approach is incorporated [5] . The differences are discussed more explicitly in the context of cross-section lines B1-B4 and C2, C3 in Fig. 1f,g,i,j in Supplementary Fig. 18 , synthetic example SynFork in Supplementary Fig. 19 emulating the fork structure in Fig. 1f,g , and synthetic example SynSTORM in Supplementary Fig. 20 . Discussion on these results is given in Supplementary Note 10 . Comparison with other super-resolution techniques Here, we compare MUSICAL with other super-resolution techniques, namely, SOFI [6] , deconSTORM [10] , 3B (ref. 7 ) and ESI [8] , the implementation details of which are given in Supplementary Methods . All these methods use statistical analysis of the fluorescence intensity in the image stack instead of single-molecule localization. DeconSTORM combines expectation maximization and deconvolution to form a super-resolved image which converges to the maximum likelihood estimate of emitter locations. It can be used for non-sparse blinking as well. We present comparison of these techniques for region A of in vitro sample 1, tubulins high-density data in Data-SMLM, synthetic example SynEx2 and synthetic example with Poisson noise SynEx2SBR3 in Fig. 4a–d , respectively. Further, comparison of MUSICAL with these techniques on the 3B test data [12] is given in Supplementary Fig. 21 . Heavy computational requirements and convergence issue of 3B were found to be limiting in generating the comparison results. Thus, instead of using the entire image stacks, we have used lesser number of frames and smaller region for generating the results for all the methods. We have used only 300 frames from sample 1 and only 200 frames from the tubulins high-density data of Data-SMLM. SynEx2 and SynEx2SBR3 have 49 frames each. Overlaid images of results of two methods in magenta and green colours are included in Fig. 4a for the ease of comparison. Coloured triangles are used to indicate the respective colours used for the methods in the overlaid images. Figure 4: Comparison of MUSICAL with other super-resolution methods that perform statistical analysis of blinking statistics rather than SMLM. ( a ) shows comparison for region A of in vitro sample 1. Overlay of MUSICAL image with 3B and deconSTORM is given for comparison in addition to their individual results. ( b ) Comparison of MUSICAL results with 3B and deconSTORM for the tubulins high-density data of Data-SMLM. MUSICAL can resolve two closely lying microtubules (cyan coloured arrow) which are unresolvable by 3B and deconSTORM. ( c ) Comparison results for synthetic example SynEx2 which is noise-free. ( d ) Shows comparison results for synthetic example SynEx2 with Poisson noise statistics and SBR ratio 3. Includes the computation time for each method below its result. Full size image For region A of sample 1 in Fig. 4a , MUSICAL and 3B perform similar and provide better result than deconSTORM, ESI and SOFI. For Data-SMLM, MUSICAL results in sharper images than 3B and deconSTROM, as seen in Fig. 4b . In Fig. 4b , the cyan arrows indicate an instance where MUSICAL is able to resolve two microtubules that are unresolved by 3B and deconSTORM. Lastly, we discuss the comparison of MUSICAL with other methods for synthetic examples SynEx2 and SynEx2SBR3 shown in Fig. 4c,d , respectively. The details of these synthetic examples are presented in Supplementary Methods . It is seen that MUSICAL gives the sharpest result for the noise-free image stack, followed by deconSTORM and 3B. However, deconSTORM generates some artifacts inside the rings, which seem to be related to side lobes. For SynEx2SBR3, deconSTORM, 3B and MUSICAL provide reasonable reconstructions, although 3B reconstructs finer rings than MUSICAL. However, 3B generates point-like artefacts along 45° and 135° directions. This might be the case because of the very low number of frames. DeconSTORM also generates artefacts in the regions where the rings are close to each other. ESI provides good reconstruction, although visible in the logarithmic scale only. We present a summary and comparison of features of various super-resolution techniques in Supplementary Table 1 and Supplementary Note 11 . SynEx2SBR3 is used for comparison of the computation time also. The time taken by each method is reported below its result in Fig. 4d . While MUSICAL takes significantly larger computation time than SOFI and ESI, it compares well with deconSTORM and is orders of magnitudes faster than 3B. We note that no parallelization has been used for reporting the computation time of MUSICAL, although there is a scope for parallelization and thus improvement in the computation time of MUSICAL. Live-cell experiments Two sets of live-cell experiments were conducted on Chinese hamster ovary cells (CHO-K). The cells were imaged using total internal reflection fluorescence (TIRF) microscope under physiologically relevant conditions. In the first set of experiments performed on live-cell microtubules sample 1 and 2 (details in methods), we imaged microtubules labelled by transiently expressed GFP-tubulin. MUSICAL results for sample 1 are presented here. In the experiment with live-cell microtubules, we used regular GFP and standard live-cell imaging medium with no additive chemicals to influence blinking. Thus, the cells were in physiologically relevant conditions optimally suited for biological experiments. A 488-nm laser excitation at the intensity of 25 W cm −2 was used. In live cells, microtubules are highly dynamic structures and there is a continuous exchange of tubulin molecules between the microtubules and cytosol [21] . Therefore, there is a significant population of labelled tubulin molecules freely diffusing in the cytosol in addition to the labelled microtubules. To suppress the large background introduced by these freely diffusing molecules, we have heuristically used a lower bound of 0.4 times the MUSICAL intensity that corresponds to 99.9% of the histogram of intensities. This choice is analogous to the heuristic choice of the upper and lower bounds used in 3B (ref. 7 ). Further, only a part of the population of tubulin molecules in the cell is labelled. Thus, the microtubules are not uniformly labelled and fluorescent in parts only. The MUSICAL result for live-cell microtubules sample 1 in Fig. 5 is generated using the first 49 frames, each with 5 ms exposure, which amounts to a total time 245 ms for a MUSICAL image. The zoom-ins shown in Fig. 5c,e indicate that MUSICAL provides resolution of less than 100 nm in the live cells too. We note that the overlay of the microtubules as seen in the zoom-in has been observed before [22] , [23] and not considered particularly significant. We use this structure to illustrate sub-100 nm resolution and the reconstruction of details by MUSICAL. This is further validated quantitatively in Fig. 5g , which plots the distance of the closest maximum along the direction n 1 (or n 2 ) from the cyan coloured line shown in Fig. 5d . It shows that the three microtubules can be identified separately even though the distance between them is quite small. Figure 5: Result of MUSICAL for live-cell microtubules sample 1. Mean image of 49 frames is shown in a . Mean image and MUSICAL result for region A identified in a are shown in b , c . Mean image and MUSICAL result for region B identified in a are shown in d , e . Zoom-in of region C identified in e is shown in f . Scale bar, 200 nm. Sub-figure g shows the closest maxima from points along the cyan line in f in the directions n 1 (blue) and n 2 (red). Full size image The effect and benefit of using lower number of frames in live-cell experiment is demonstrated in Supplementary Fig. 22 , which shows results for region B of live-cell microtubules sample 1 with different frame numbers taken from different parts of the whole-image stack. Further, we show using the results of live-cell microtubules sample 2 in Supplementary Fig. 23 that the timescales of MUSICAL may potentially be reduced to less than 50 ms. In the second set of live-cell experiments, we imaged GFP-labelled actin in cortical cytoskeleton of a CHO-K1 cell stably expressing Lifeact-GFP. The results are given in Supplementary Fig. 24 . It shows the ability of MUSICAL to image dense nanoscale structures."},{"section":"Discussion","content":"Although the reported resolution of STORM at 20 nm is significantly better than MUSICAL, MUSICAL conclusively demonstrates less stringent requirements on sparsity of blinking, density of emitters, number of frames, excitation power and chemical environment. Further comparison of MUSICAL with other statistical super-resolution techniques, such as SOFI, deconSTORM, 3B and ESI indicates competitive and often superior performance of MUSICAL. Here, we mention that MUSICAL performs better than SOFI and ESI in terms of resolution and image quality, however at the cost of increased computational time and using the knowledge of the PSF. The analysis of sensitivity of MUSICAL to the accurate knowledge of the optical PSF in Supplementary Fig. 7 indicates that MUSICAL is not too sensitive to PSF. Although 3B shows better robustness to noise than MUSICAL, it has significantly higher computational demands unless faster implementation of 3B using cloud computation [24] is used. Lastly, to the best of our knowledge, none of the currently known super-resolution algorithms so far has demonstrated the capability to produce super-resolved images of live cells in their natural environment with total acquisition times of the order of 50–250 milliseconds using regular non-sparse blinking dyes. We do note that Huang et al . [25] have reported super-resolution at video acquisition rate of 32 frames per second, although using photoswitchable proteins or Alexa Fluor 647 in glucose oxidase and catalase-based imaging medium, as opposed to the live-cell results reported here obtained using regular GFP and without oxygen scavenging and carbon dioxide-deficient imaging medium considered toxic to live-cells [4] . We believe that MUSICAL is a step forward towards the goal of fast super-resolved imaging of biological phenomena in their natural state."},{"section":"Methods","content":"In vitro actin sample preparation We used the protocol suggested in ref. 26 for forming the in vitro actin samples. Preformed actin filaments from Cytoskeleton Inc., extracted from rabbit skeletal muscle, were used for forming the sample. In one chamber of an 8-chambered cover glass of ∼ 750 μl volume, 200 μl of 0.01% poly- L -lysine solution was incubated for 10 min. Remaining liquid was drained out using a pipette. A total of 90 μl of general actin buffer (reconstituted as suggested by Cytoskeleton’s product datasheet of general actin buffer) was introduced in the chamber. This was followed by adding 10 μl of 10 μM preformed actin filaments and 10 μl of Phalloidin Atto-565 solution (stock solution prepared as recommended by the manufacturer, Sigma-Aldrich). The contents of chamber were mixed by gently pipetting up and down. An incubation time of ∼ 45 min was allowed. Then the liquid in the chamber was removed by pipetting. For sample 1, the contents of the chamber were washed 3 times using Buffer B (see details of imaging buffer below) by a two pipette system, where 1 pipette let in the buffer and the other pipette sucked out the solution. Sample 2 was washed five times. Sample 3, with results in Supplementary Fig. 13 , was not washed at all. In sample 2, we introduced tetraspeck beads in the imaging solution and allowed the solution to settle for 3 hours. In samples 2 and 3, where the same samples are imaged with different excitation powers, the image stacks have been appropriately shifted to compensate for the drift of sample between different acquisitions. See Supplementary Table 2 for an overview of the in vitro samples. Preparation and introduction of imaging buffer We used the imaging buffer composition suggested in ref. 27 . Buffer A composed of 10 mM of TRIS (pH 8.0) and 50 mM NaCl. Buffer B composed of 50 mM TRIS (pH 8.0), 10 mM NaCl and 10% Glucose (weight per volume). GLOX solution (1 ml) was formed by vortex mixing a solution of 56 mg of Glucose oxidase, 200 μl of Catalase (17 mg ml −1 ) and 800 μl of Buffer A. MEA solution (1 M, 1 ml) was prepared using 77 mg of MEA and 1 ml of 0.25 N HCl. For one chamber of an eight-chambered cover glass of about 750 μl volume, 700 μl of imaging buffer was prepared by mixing 7 μl of GLOX solution, 70 μl of MEA solution and 620 μl of Buffer B on ice. This imaging buffer was used as the medium for in vitro actin samples. The chamber was filled completely and covered immediately to avoid replenishing of oxygen in solution. Live-cell sample preparation CHO-K1 cells were obtained from ATCC (Manassas, VA). Lifeact cells (CHO-K1 cells stably expressing Lifeact-GFP) were provided by Prof. Rachel S. Kraut (NTU, Singapore). CHO-K1 and Lifeact cells were cultivated in DMEM medium (Dulbecco’s Modified Eagle Medium, Invitrogen; Singapore) supplemented with 1% penicillin G and streptomycin (PS, PAA, Austria), and 10% fetal bovine serum (Invitrogen; Singapore) at 37 °C in 5% (v/v) CO 2 environment. GFP-tubulin plasmid was a gift from Dr Pakorn T. Kanchanawong (MBI, NUS, Singapore). Electroporation was used for transfection of the cells, during which, 90% confluent cells in a 75 cm 2 flask were washed twice with 1 × PBS, trypsinized with 0.25% trypsin-0.03% EDTA solution for ∼ 1 min at 37 °C, and then re-suspended in culture medium. Cells were precipitated by centrifugation and re-suspended in small amount of resuspension R buffer (NeonTM Transfection System, Life Technologies, Singapore) and transferred half into one electroporation cuvette (2 mm wide, Bio-Rad; Hercules, CA) for one transfection. Between 300 and 500 ng ml −1 of the plasmid were added. After electroporation pulse, cells were seeded back to prewashed cover glass (30 mm in diameter; Lakeside, Monee, IL) in a 35 mm culture dish. Transfected cells grew in the culture medium for 24–36 h before measurement. All cells were imaged in Phenol red free DMEM medium containing 10% fetal bovine serum. See Supplementary Table 2 for an overview of the live-cell samples. Imaging system and image acquisition The setup consisted of an inverted epifluorescence microscope (IX83, Olympus, Japan) equipped with a motorized TIRF illumination combiner (IX3-MITICO, Olympus, Japan) and a scientific complementary metal oxide (sCMOS) camera with 6.5 μm pixels (Orca-Flash4.0, Hamamtsu Photonics, Japan). A 488 nm laser (LAS/488/100/D) or 561 nm laser (LAS/561/100, Olympus, Germany) was connected to the TIRF illumination combiner in which the incidence angle was adjusted to give 110 nm penetration depth of the evanescent field. The 488 and 561 nm lasers were used for imaging live-cell samples or in vitro actin filaments, respectively. A × 100, numerical aperture 1.49 oil immersion objective (UAPON, Olympus, Japan) was used to illuminate the sample and collect the fluorescence image. The fluorescence light then passed through a major dichroic (ZT405/488/561/647rpc, Chroma Technology, Bellows Falls, VT) and a band-pass filter (ZET405/488/561/647m, Chroma Technology, Bellows Falls, VT). The intensity of the excitation light was determined as the power of the laser light exiting the objective divided by the illuminated area; the power of the laser light was regulated by the proprietary laser control software and by an additional OD 1 neutral density filter. The camera was controlled by Micro-Manager 1.4 (ref. 28 ). For multiple power measurements, the first measurement is done using the lowest power and the power is subsequently increased. The optical PSF computed as an airy disk has FWHM of ∼ 198 nm for these parameters and emission wavelength 593 nm of the phalloidin Atto-565 dye and about 176 nm for emission wavelength 512 nm of the lifeact-GFP. For in vitro samples 1 and 2, 10,000 frames were acquired with exposure time of 5 ms (200 frames per second) For in vitro sample 3, 20,000 frames were acquired with exposure time of 10 ms (100 frames per second). For live-cell microtubule sample 1, 1,000 frames were acquired with exposure time of 5 ms (200 frames per second). For live-cell microtubule sample 2, 49 frames were acquired with exposure time of 1 ms (1,000 frames per second). For live-cell F-actin sample, 100 frames were acquired with exposure time of 1 ms (1,000 frames per second). Data availability The data that support the findings of this study and the source codes of MUSICAL are available at https://sites.google.com/site/uthkrishth/musical ."},{"section":"Additional information","content":"How to cite this article: Agarwal, K. & Machan, R. Multiple signal classification algorithm for super-resolution fluorescence microscopy. Nat. Commun. 7, 13752 doi: 10.1038/ncomms13752 (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\": \"Super-resolution fluorescence microscopy techniques aim at resolving details smaller than the Abbe diffraction limit of , where λ is the wavelength of the fluorescence emission and NA is the numerical aperture of the microscope objective. Most of these techniques use the blinking phenomenon, where fluorophores switch between a bright (fluorescent) state and a long-lived dark state. A series of images is recorded over time. Each image has different intensity distribution because a different set of fluorophores were in the bright state during each image acquisition. The temporal information contained in the series is, then, used to construct a final image with improved spatial resolution. Single-molecule localization microscopy (SMLM) techniques such as stochastic optical reconstruction microscopy (STORM) or photo-activated localization microscopy (PALM) are popular super-resolution techniques owing to their simplicity, few (if any) special requirements on instrumentation, and impressive resolution of ∼ 20 nm (refs 1 , 2 , 3 ). However, they require that the fluorophores exhibit long dark states, so that only a small subset of optically separable fluorophores are in the bright state in each frame of the image stack. This translates into requirements of long acquisition times and of photochemical environment promoting long dark states and impeding bleaching, which is toxic to live cells [4] . The limitations of SMLM have motivated development of techniques that rely on statistical independence of blinking of individual fluorophores rather than on long dark states [5] . Such techniques include super-resolution optical fluctuations imaging (SOFI [6] ), Bayesian analysis of blinking and bleaching (3B (ref. 7 )) and entropy-based super-resolution imaging (ESI [8] ). Although they relax the requirements of SMLM, they do not reach resolution achievable by SMLM ( ∼ 110 nm for SOFI [9] , 80 nm for ESI [8] and 50 nm for 3B (ref. 7 )) and they possess limitations of their own. For example, SOFI uses cumulants of the fluorescence blinking to enhance resolution; since cumulants of orders higher than six are prone to shot noise and do not have good approximations, the practically achievable resolution improvement is limited to the factor of (ref. 6 ). 3B uses a Markov process for modelling the blinking and bleaching of the fluorophores and an expectation maximization approach to determine the likelihood of an emitter being present at a given location. This approach is computationally intensive and its convergence to global minimum is not guaranteed. In the following, we propose a novel algorithm utilizing fluorescence blinking to enhance spatial resolution. The algorithm, called MUltiple SIgnal Classification ALgorithm (MUSICAL), achieves super-resolution by exploiting the eigenimages of the image stack, which statistically represent its prominent structures and, then, applying the knowledge of the point spread function (PSF) of the imaging system to localize the structures to super-resolution scales. Like SOFI or 3B and other related techniques, MUSICAL requires neither special instrumentation nor special fluorophores. The sole requirement is statistically independent blinking of individual emitters. We tested MUSICAL on images of actin filaments and compared it with STORM, showing that both techniques give comparable resolution enhancements. We also compared MUSICAL with 3B, SOFI, ESI and deconSTORM [10] on experimental data sets independently acquired by other super-resolution research groups [11] , [12] and show comparable or superior performance of MUSICAL. We also demonstrate that MUSICAL performs well in situations where STORM fails due to high density of fluorophores. Further, we show that MUSICAL can be used for live-cell fast imaging ( ∼ 49 frames amounting to a total acquisition time of less than 250 ms) of live cells expressing standard green fluorescent protein (GFP) imaged in physiologically conducive buffer devoid of chemicals that influence blinking.\",\n \"references\": {\n \"indices\": [\n 4,\n 5,\n 6,\n 8,\n 9,\n 10\n ],\n \"infos\": [\n {\n \"idx\": \"4\",\n \"title\": \"Ettinger, A. & Wittmann, T. Fluorescence live cell imaging.Methods Cell Biol.123, 77–94 (2014).\",\n \"link\": \"https://doi.org/10.1016%2FB978-0-12-420138-5.00005-7\"\n },\n {\n \"idx\": \"5\",\n \"title\": \"Small, A. & Stahlheber, S. Fluorophore localization algorithms for super-resolution microscopy.Nat. Methods11, 267–279 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fnmeth.2844\"\n },\n {\n \"idx\": \"6\",\n \"title\": \"Dertinger, T., Colyer, R., Iyer, G., Weiss, S. & Enderlein, J. Fast, background-free, 3d super-resolution optical fluctuation imaging (SOFI).Proc. Natl Acad. Sci. USA106, 22287–22292 (2009).\",\n \"link\": \"https://doi.org/10.1073%2Fpnas.0907866106\"\n },\n {\n \"idx\": \"8\",\n \"title\": \"Yahiatene, I., Hennig, S., Müller, M. & Huser, T. Entropy-based super-resolution imaging (ESI): From disorder to fine detail.ACS Photonics2, 1049–1056 (2015).\",\n \"link\": \"https://doi.org/10.1021%2Facsphotonics.5b00307\"\n },\n {\n \"idx\": \"9\",\n \"title\": \"Geissbuehler, S. et al. Live-cell multiplane three-dimensional super-resolution optical fluctuation imaging.Nat. Commun.5, 5830 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms6830\"\n },\n {\n \"idx\": \"10\",\n \"title\": \"Mukamel, E. A., Babcock, H. & Zhuang, X. Statistical deconvolution for superresolution fluorescence microscopy.Biophys. J.102, 2391–2400 (2012).\",\n \"link\": \"https://doi.org/10.1016%2Fj.bpj.2012.03.070\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Multiple signal classification algorithm The idea of MUSICAL is inspired from MUltiple SIgnal Classification (MUSIC) used in acoustics [13] , radar signal processing [14] and electromagnetic imaging [15] for finding the contrast sources created due to scattering and contributing to the measured signal. However, MUSICAL differs from MUSIC because the emitters in fluorescence microscopy behave differently from the contrast sources encountered in scattering. Firstly, the fluorophores exhibit intermittent emission when exposed to continuous excitation, the intermittence patterns of any two fluorophores being uncorrelated. Secondly, the information of a molecule is concentrated in a small region defined by the PSF in the image plane. Thus, there is a region of confidence for the likely position of each molecule; pixels beyond that region contribute only additional noise and almost no usable information in the context of localizing molecules in the given region. Thirdly, MUSIC in its traditional form needs a higher number of receivers (pixels in the camera) and measurements (frames) than the number of contrast sources (fluorescent emitters). Thus, when imaging an area with a high density of fluorophores, the required number of frames may easily become impractically large when using the traditional form of MUSIC. MUSICAL overcomes those issues of MUSIC by adopting a sliding soft window and stitching approach. MUSIC is applied only on a small part of the image (the soft window) at a time. The soft window is, then, scanned over the whole image and the individual MUSIC reconstructions are stitched together to form the MUSICAL image. See Supplementary Fig. 1 for the flowchart of MUSICAL. Since the area corresponding to the soft window contains only a fraction of the total number of fluorophores in the imaged area, the required number of frames in the stack is considerably reduced. Besides, this approach also suppresses the problem of noise contribution from pixels further away from a given emitter location. The concept, algorithm, and implementation of MUSICAL is detailed in Supplementary Notes 1 and 2 , and Supplementary Methods , respectively. Several synthetic examples have been used to demonstrate various aspects of MUSICAL. Their details are given in Supplementary Methods . We discuss here the algorithm of MUSICAL briefly. Using the selected soft window of the image stack, MUSIC first computes the eigenimages of the soft window through singular value decomposition. The effect of the soft window is illustrated in Supplementary Fig. 2 . Each eigenimage of the selected soft window is associated to a singular value and represents a particular pattern found in the image stack. Statistically, large singular value indicates that the pattern of the corresponding eigenimage is a prominent pattern in the image stack. On the other hand, statistically less likely patterns, which include noise patterns, are characterized by small singular values. Thus, the eigenimages can be separated into two sets (or subspaces) using a threshold value σ 0 , where eigenimages with singular values more than or equal to σ 0 form the signal subspace (or the range) and the remaining eigenimages form the null subspace. The concepts of the range and the null space are discussed in Supplementary Note 3 . The choice of the value of σ 0 is discussed in Supplementary Figs 3 and 4 , and Supplementary Note 4 . Then, the projections of the PSF (an example is given in Supplementary Fig. 5 ) at a given test point on the range and the null subspace are determined, which we denote as d PR ( ) and d PN ( ) respectively. The projection d PR ( ) indicates whether the PSF at the test point is related to the patterns represented by the range, similarly for d PN ( ). If the test point indeed belongs to either the fluorophore locations or the structure represented by them, then d PN ( ) is close to zero, it is non-zero otherwise. This property of d PN ( ) is used to compute an indicator function as follows such that the value of the indicator function is very large at the point a fluorophore is present and small at the point where no fluorophore is present. An example of eigenimages and projection of PSF on them is given in Supplementary Fig. 6 and Supplementary Note 5 . Further, the sensitivity of MUSICAL to the estimate of PSF is given in Supplementary Fig. 7 . In the original form of MUSIC, as used in acoustic and electromagnetic imaging, only the distance from the null space d PN ( ) is used and α is often set as 1. However, the indicator function of MUSIC is not suitable for the sliding window approach, as shown in Supplementary Fig. 8 . The modified indicator function of MUSICAL is amenable to the sliding window approach since the inclusion of d PR ( ) in the numerator automatically weighs the result of each window, so that stitching of the results for sliding windows can be simplified as sum of the indicator functions of all the sliding windows covering a test point. At the same time, the use of α allows for better resolution by non-linearly scaling the indicator function at closely located test points. See Supplementary Note 6 for a discussion on the role of the parameter α and Supplementary Fig. 3 for an example. In vitro experiments Here, we present results of MUSICAL for in vitro experiments and in vitro test data available online [11] , [12] . In vitro experiments were performed on actin filaments tagged with Phalloidin-Atto 565 dye and serve to characterize the performance of MUSICAL. Results of samples 1 and 2 are included in Figs 1 and 2 , respectively. Figure 1: Results for regions A–C of in vitro sample 1. The mean image of 10,000 frames is shown in a . ( b , c ) The mean image and MUSICAL image for region A. ( d ) Shows the profile of section A1 shown in c . ( e – g ) The mean image, STORM image, and MUSICAL image for region B. ( h – j ) show the mean image, STORM image and MUSICAL image for region C. ( k ) shows the profile of section C1 shown in i , j . ( l ) shows the average PSFs derived from groups AA and BB of line segments shown in c , g , respectively. Scale bars: 1 μm ( a ); 250 nm ( b , c , h – j ); and 500 nm ( e – g ). Full size image Figure 2: MUSICAL results for in vitro sample 2. The first, second and third columns correspond to the image stacks acquired using laser powers 10.3 W cm −2 ( a , e , i , m , q ), 40.2 W cm −2 ( b , f , j , n , r ) and 205.6 W cm −2 ( c , g , k , o , s ), respectively. The fourth column ( d , h , l , p , t ) shows result of MUSICAL using 49 frames only from the image stack acquired with 205.6 W cm −2 excitation power. ( q – s ) Compares the intensity profiles of mean image and MUSICAL at the sections E1, E2 and E3, respectively. ( t ) Compares MUSICAL result at E3 using entire stack of 10,000 images with MUSICAL result at F3 using 49 frames only. Scale bars: 2 μm (first and second rows); and 1 μm (third and fourth rows). Full size image Experimental data provided as single-molecule localization test data set (referred to as Data-SMLM) [11] are used for comparison with other contemporary methods. Details about this data set can be found in its website. It comprises of three data, namely tubulins long sequence (15,000 frames), tubulins high density (500 frames) and tubulinAF647 (9,990 frames). The results for Data-SMLM are included in Fig. 3 . Tubulins high-density data is further used to demonstrate resolution of two microtubules using MUSICAL. Figure 3: Results on Data-SMLM data sets and synthetic example SynSTORM for comparison of MUSICAL with SMLM techniques. ( a ) Tubulins high density (500 frames), ( b ) tubulins long sequence (15,000 frames) and ( c ) TubulinAF647 (9,990 frames). PALM and STORM results in b are taken from [11] . In b , MUSICAL image (in green) is overlaid with PALM image (in magenta) and STORM image (in magenta), respectively, for comparison. Similarly, MUSICAL image (in magenta) is overlaid with STORM image (in green) in c . Full size image We breakdown the in vitro results into the following studies: characterization of the PSF and resolution of MUSICAL, performance of MUSICAL for non-sparse blinking, minimum number of frames required for MUSICAL, excitation power and MUSICAL, comparison of MUSICAL with SMLM techniques and comparison of MUSICAL with other super-resolution techniques. These studies are presented below. Point spread function and resolution of MUSICAL In vitro sample 1 is used for characterization of the experimental PSF of MUSICAL. In Fig. 1c , group of line segments AA is used to obtain the experimental PSF of MUSICAL. The lines are separated by 26 nm. The maxima of the profiles of all the lines are aligned and then their average profile is computed as representative of the PSF of MUSICAL, which is shown in Fig. 1l . The full-width at half-maximum (FWHM) of the average profile is 47.4 nm. We also determined FWHMs of all the individual profiles; they were 47.3±7.4 nm (mean±s.d.). We repeated the same procedure for another group of line segments BB shown in Fig. 1g . The lines are separated by 65 nm here. The FWHM of the average profile for group BB is 46.6 nm. The FWHMs of the individual profiles were 48.5±8.7 nm (mean±s.d.). Characterization of the PSF of MUSICAL as a function of signal to background (SBR) ratio using a synthetic example SynEx3 is given in Supplementary Fig. 9a . We observed periodicity along the actin filaments in the MUSICAL images and the minimum distance between the two peaks on actin filament to be 27.7 nm, as seen in Fig. 1d . Further, we consider the tubulins high-density data from Data-SMLM data set which has high density of emitters such that blinking is non-sparse. MUSICAL clearly illustrates the ability to resolve microtubules separated by 125 nm with a contrast value of 0.39, as seen for the section A in Fig. 3a . Characterization of structural resolution of MUSICAL is given in Supplementary Fig. 9b–d using synthetic examples SynPairDelX. To study further the observed periodicity in the actin filaments, we plot the intensity of MUSICAL image across the sections A1 and C1 of two actin filaments in Fig. 1d,k , respectively. Fourier spectra of these sections and a synthetic example with periodically placed emitters are provided in Supplementary Fig. 10 . It reveals a prominent peak at the spatial frequency corresponding to a period of 62.5 nm for A1. Another prominent peak at similar length scale occurs at sampling frequency corresponding to a period of 91.0 nm (which is ∼ 1.5 times 62.5 nm). Similarly, for the section C1, the Fourier analysis revealed a prominent peak at the spatial frequency corresponding to a period of 65.5 nm. Chiu et al . [16] reported the period of the actin double helix to be 37 nm, which corresponds to a period of 74 nm for a single helix. However, actin is known to have structural polymorphs [17] and disorders of sub-helical dimensions [18] . In addition, poly-lysine, used for stabilizing the actin filaments on the glass chamber, is known to introduce polymeric variations [19] . Thus, we suggest that this periodicity is associated with the periodicity of a single helix of the double helical structure. The performance of MUSICAL for non-sparse blinking As noted before, most localization approaches rely on long dark states of fluorophores (sparse blinking) such that the few bright fluorophores are optically resolvable. MUSICAL does not have this restriction. To illustrate this, we consider in vitro sample 2 which exhibits high density of fluorophores (see Supplementary Fig. 11 ). While STORM is not suitable for imaging such sample (see Supplementary Fig. 12 ), MUSICAL can image these samples efficiently as seen in the second and fourth rows of Fig. 2 ). We further demonstrate the ability of MUSICAL to image dense fluorophore structures through the tubulins high-density data of Data-SMLM. The results are shown in Fig. 3a . The zoom-in of the cyan marked rectangles in the MUSICAL results show that closely lying microtubules with separation smaller than the optical diffraction limit ( ∼ 265 nm for this data) can be resolved consistently using MUSICAL. Minimum number of frames required for MUSICAL If blinking is sparse, large number of frames have to be acquired to be able to collect fluorescent signal from each fluorophore. However, since sample 2 does not demonstrate sparse blinking, it is interesting to consider if MUSICAL can be applied to such sample using only a few frames. Owing to the sliding window of size N pixels, the rank of the matrix on which eigenimages are computed is less than or equal to min( N , K ), where K is the number of frames; see Supplementary Note 1 . The number of pixels in the sliding window N is determined by the PSF of the system and thus fixed. Thus, it is desirable to use K ≥ N . We have used a sliding window of size N w =7 pixels, which implies that N = =49. We applied MUSICAL on the first 49 frames of the data acquired for in vitro sample 2 under excitation power 205.6 W cm −2 . The results shown in the fourth column of Fig. 2 clearly illustrate the capability of MUSICAL to obtain super-resolution in non-sparse blinking using very few frames. This also indicates the utility of MUSICAL in the study of dynamic biological processes. We note that emitters that do not emit significant number of photons in the frames used for MUSICAL will not be imaged by MUSICAL. As a further comment on the desirable condition K ≥ N , we note that soft window of size larger than N w in Supplementary Fig. 2 does not add much value while increasing the minimum number of frames and the computation time. Thus, the sufficient number of frames is limited by the blinking rate and emitter density such that all emitters in the sample can blink in order to represent the entire structure and there are enough fluctuations in the intensity over the frames. Excitation power and MUSICAL Whereas the image stack for sample 1 was acquired under 205.6 W cm −2 , we acquired three image stacks for sample 2, each with different excitation powers, that is, 10.3, 40.2 and 205.6 W cm −2 . On one hand, the excitation power directly impacts the achievable resolution through the signal strength and SBR ratio. On the other hand, it indirectly influences the reconstruction by affecting the blinking rates of fluorophores, where high excitation powers generally result in longer dark states [20] . The MUSICAL results for these image stacks are shown in Fig. 2 . It is seen that at each excitation power, MUSICAL can reconstruct more details than the mean image. The FWHM of the peaks indicated by arrows in Fig. 2r,s are 49 and 61 nm, respectively. The result implies that the FWHM does not deteriorates significantly when the laser power is decreased from 205.6 to 40.2 W cm −2 . It also helps to identify the minimum power at which sufficient resolution is achieved while minimizing photo-bleaching. We provide an additional example using in vitro sample 3 and an empirical plot of MUSICAL FWHM as a function of excitation power in Supplementary Fig. 13 and Supplementary Note 7 . We discuss the presence of side lobes in experimental results of MUSICAL and provide an analysis of the side lobes of MUSICAL as a function of SBR in Supplementary Fig. 14 . Effect of camera noise that becomes prominent at very poor SBR is discussed in Supplementary Fig. 15 and Supplementary Note 8 . Compensation for camera noise and side lobes is discussed in Supplementary Fig. 16 and Supplementary Note 9 . Compensation of side lobes of Fig. 2m is shown in Supplementary Fig. 17 . Comparison between MUSICAL and SMLM techniques Comparison of MUSICAL and STORM results for regions B and C of sample 1 are shown in Fig. 1f,i . In addition, we compare the results of MUSICAL with SMLM techniques for the Data-SMLM data set in Fig. 3a–c . Implementation and parameters used for SMLM techniques are given in Supplementary Methods . Overlay of MUSICAL images with STORM or PALM images are also provided for tubulins long sequence and tubulinAF647. We chose these data for overlay and comparison because they demonstrate sparse blinking, and thus are favourable for SMLM. The results show good agreement between MUSICAL and SMLM techniques. Now we discuss the difference between MUSICAL and SMLM results, such as noted in Fig. 1f,g,i,j . These differences are attributed to multiple simultaneously bright fluorophores in the small volume surrounding the junctions. If a couple of closely placed emitters are blinking in a frame in a very small volume, STORM may conclude the presence of one very bright fluorophore at the centre of the small volume instead of the actual distribution of the closely placed emitters, unless a multiple-fluorophore fitting approach is incorporated [5] . The differences are discussed more explicitly in the context of cross-section lines B1-B4 and C2, C3 in Fig. 1f,g,i,j in Supplementary Fig. 18 , synthetic example SynFork in Supplementary Fig. 19 emulating the fork structure in Fig. 1f,g , and synthetic example SynSTORM in Supplementary Fig. 20 . Discussion on these results is given in Supplementary Note 10 . Comparison with other super-resolution techniques Here, we compare MUSICAL with other super-resolution techniques, namely, SOFI [6] , deconSTORM [10] , 3B (ref. 7 ) and ESI [8] , the implementation details of which are given in Supplementary Methods . All these methods use statistical analysis of the fluorescence intensity in the image stack instead of single-molecule localization. DeconSTORM combines expectation maximization and deconvolution to form a super-resolved image which converges to the maximum likelihood estimate of emitter locations. It can be used for non-sparse blinking as well. We present comparison of these techniques for region A of in vitro sample 1, tubulins high-density data in Data-SMLM, synthetic example SynEx2 and synthetic example with Poisson noise SynEx2SBR3 in Fig. 4a–d , respectively. Further, comparison of MUSICAL with these techniques on the 3B test data [12] is given in Supplementary Fig. 21 . Heavy computational requirements and convergence issue of 3B were found to be limiting in generating the comparison results. Thus, instead of using the entire image stacks, we have used lesser number of frames and smaller region for generating the results for all the methods. We have used only 300 frames from sample 1 and only 200 frames from the tubulins high-density data of Data-SMLM. SynEx2 and SynEx2SBR3 have 49 frames each. Overlaid images of results of two methods in magenta and green colours are included in Fig. 4a for the ease of comparison. Coloured triangles are used to indicate the respective colours used for the methods in the overlaid images. Figure 4: Comparison of MUSICAL with other super-resolution methods that perform statistical analysis of blinking statistics rather than SMLM. ( a ) shows comparison for region A of in vitro sample 1. Overlay of MUSICAL image with 3B and deconSTORM is given for comparison in addition to their individual results. ( b ) Comparison of MUSICAL results with 3B and deconSTORM for the tubulins high-density data of Data-SMLM. MUSICAL can resolve two closely lying microtubules (cyan coloured arrow) which are unresolvable by 3B and deconSTORM. ( c ) Comparison results for synthetic example SynEx2 which is noise-free. ( d ) Shows comparison results for synthetic example SynEx2 with Poisson noise statistics and SBR ratio 3. Includes the computation time for each method below its result. Full size image For region A of sample 1 in Fig. 4a , MUSICAL and 3B perform similar and provide better result than deconSTORM, ESI and SOFI. For Data-SMLM, MUSICAL results in sharper images than 3B and deconSTROM, as seen in Fig. 4b . In Fig. 4b , the cyan arrows indicate an instance where MUSICAL is able to resolve two microtubules that are unresolved by 3B and deconSTORM. Lastly, we discuss the comparison of MUSICAL with other methods for synthetic examples SynEx2 and SynEx2SBR3 shown in Fig. 4c,d , respectively. The details of these synthetic examples are presented in Supplementary Methods . It is seen that MUSICAL gives the sharpest result for the noise-free image stack, followed by deconSTORM and 3B. However, deconSTORM generates some artifacts inside the rings, which seem to be related to side lobes. For SynEx2SBR3, deconSTORM, 3B and MUSICAL provide reasonable reconstructions, although 3B reconstructs finer rings than MUSICAL. However, 3B generates point-like artefacts along 45° and 135° directions. This might be the case because of the very low number of frames. DeconSTORM also generates artefacts in the regions where the rings are close to each other. ESI provides good reconstruction, although visible in the logarithmic scale only. We present a summary and comparison of features of various super-resolution techniques in Supplementary Table 1 and Supplementary Note 11 . SynEx2SBR3 is used for comparison of the computation time also. The time taken by each method is reported below its result in Fig. 4d . While MUSICAL takes significantly larger computation time than SOFI and ESI, it compares well with deconSTORM and is orders of magnitudes faster than 3B. We note that no parallelization has been used for reporting the computation time of MUSICAL, although there is a scope for parallelization and thus improvement in the computation time of MUSICAL. Live-cell experiments Two sets of live-cell experiments were conducted on Chinese hamster ovary cells (CHO-K). The cells were imaged using total internal reflection fluorescence (TIRF) microscope under physiologically relevant conditions. In the first set of experiments performed on live-cell microtubules sample 1 and 2 (details in methods), we imaged microtubules labelled by transiently expressed GFP-tubulin. MUSICAL results for sample 1 are presented here. In the experiment with live-cell microtubules, we used regular GFP and standard live-cell imaging medium with no additive chemicals to influence blinking. Thus, the cells were in physiologically relevant conditions optimally suited for biological experiments. A 488-nm laser excitation at the intensity of 25 W cm −2 was used. In live cells, microtubules are highly dynamic structures and there is a continuous exchange of tubulin molecules between the microtubules and cytosol [21] . Therefore, there is a significant population of labelled tubulin molecules freely diffusing in the cytosol in addition to the labelled microtubules. To suppress the large background introduced by these freely diffusing molecules, we have heuristically used a lower bound of 0.4 times the MUSICAL intensity that corresponds to 99.9% of the histogram of intensities. This choice is analogous to the heuristic choice of the upper and lower bounds used in 3B (ref. 7 ). Further, only a part of the population of tubulin molecules in the cell is labelled. Thus, the microtubules are not uniformly labelled and fluorescent in parts only. The MUSICAL result for live-cell microtubules sample 1 in Fig. 5 is generated using the first 49 frames, each with 5 ms exposure, which amounts to a total time 245 ms for a MUSICAL image. The zoom-ins shown in Fig. 5c,e indicate that MUSICAL provides resolution of less than 100 nm in the live cells too. We note that the overlay of the microtubules as seen in the zoom-in has been observed before [22] , [23] and not considered particularly significant. We use this structure to illustrate sub-100 nm resolution and the reconstruction of details by MUSICAL. This is further validated quantitatively in Fig. 5g , which plots the distance of the closest maximum along the direction n 1 (or n 2 ) from the cyan coloured line shown in Fig. 5d . It shows that the three microtubules can be identified separately even though the distance between them is quite small. Figure 5: Result of MUSICAL for live-cell microtubules sample 1. Mean image of 49 frames is shown in a . Mean image and MUSICAL result for region A identified in a are shown in b , c . Mean image and MUSICAL result for region B identified in a are shown in d , e . Zoom-in of region C identified in e is shown in f . Scale bar, 200 nm. Sub-figure g shows the closest maxima from points along the cyan line in f in the directions n 1 (blue) and n 2 (red). Full size image The effect and benefit of using lower number of frames in live-cell experiment is demonstrated in Supplementary Fig. 22 , which shows results for region B of live-cell microtubules sample 1 with different frame numbers taken from different parts of the whole-image stack. Further, we show using the results of live-cell microtubules sample 2 in Supplementary Fig. 23 that the timescales of MUSICAL may potentially be reduced to less than 50 ms. In the second set of live-cell experiments, we imaged GFP-labelled actin in cortical cytoskeleton of a CHO-K1 cell stably expressing Lifeact-GFP. The results are given in Supplementary Fig. 24 . It shows the ability of MUSICAL to image dense nanoscale structures.\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"Although the reported resolution of STORM at 20 nm is significantly better than MUSICAL, MUSICAL conclusively demonstrates less stringent requirements on sparsity of blinking, density of emitters, number of frames, excitation power and chemical environment. Further comparison of MUSICAL with other statistical super-resolution techniques, such as SOFI, deconSTORM, 3B and ESI indicates competitive and often superior performance of MUSICAL. Here, we mention that MUSICAL performs better than SOFI and ESI in terms of resolution and image quality, however at the cost of increased computational time and using the knowledge of the PSF. The analysis of sensitivity of MUSICAL to the accurate knowledge of the optical PSF in Supplementary Fig. 7 indicates that MUSICAL is not too sensitive to PSF. Although 3B shows better robustness to noise than MUSICAL, it has significantly higher computational demands unless faster implementation of 3B using cloud computation [24] is used. Lastly, to the best of our knowledge, none of the currently known super-resolution algorithms so far has demonstrated the capability to produce super-resolved images of live cells in their natural environment with total acquisition times of the order of 50–250 milliseconds using regular non-sparse blinking dyes. We do note that Huang et al . [25] have reported super-resolution at video acquisition rate of 32 frames per second, although using photoswitchable proteins or Alexa Fluor 647 in glucose oxidase and catalase-based imaging medium, as opposed to the live-cell results reported here obtained using regular GFP and without oxygen scavenging and carbon dioxide-deficient imaging medium considered toxic to live-cells [4] . We believe that MUSICAL is a step forward towards the goal of fast super-resolved imaging of biological phenomena in their natural state.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"In vitro actin sample preparation We used the protocol suggested in ref. 26 for forming the in vitro actin samples. Preformed actin filaments from Cytoskeleton Inc., extracted from rabbit skeletal muscle, were used for forming the sample. In one chamber of an 8-chambered cover glass of ∼ 750 μl volume, 200 μl of 0.01% poly- L -lysine solution was incubated for 10 min. Remaining liquid was drained out using a pipette. A total of 90 μl of general actin buffer (reconstituted as suggested by Cytoskeleton’s product datasheet of general actin buffer) was introduced in the chamber. This was followed by adding 10 μl of 10 μM preformed actin filaments and 10 μl of Phalloidin Atto-565 solution (stock solution prepared as recommended by the manufacturer, Sigma-Aldrich). The contents of chamber were mixed by gently pipetting up and down. An incubation time of ∼ 45 min was allowed. Then the liquid in the chamber was removed by pipetting. For sample 1, the contents of the chamber were washed 3 times using Buffer B (see details of imaging buffer below) by a two pipette system, where 1 pipette let in the buffer and the other pipette sucked out the solution. Sample 2 was washed five times. Sample 3, with results in Supplementary Fig. 13 , was not washed at all. In sample 2, we introduced tetraspeck beads in the imaging solution and allowed the solution to settle for 3 hours. In samples 2 and 3, where the same samples are imaged with different excitation powers, the image stacks have been appropriately shifted to compensate for the drift of sample between different acquisitions. See Supplementary Table 2 for an overview of the in vitro samples. Preparation and introduction of imaging buffer We used the imaging buffer composition suggested in ref. 27 . Buffer A composed of 10 mM of TRIS (pH 8.0) and 50 mM NaCl. Buffer B composed of 50 mM TRIS (pH 8.0), 10 mM NaCl and 10% Glucose (weight per volume). GLOX solution (1 ml) was formed by vortex mixing a solution of 56 mg of Glucose oxidase, 200 μl of Catalase (17 mg ml −1 ) and 800 μl of Buffer A. MEA solution (1 M, 1 ml) was prepared using 77 mg of MEA and 1 ml of 0.25 N HCl. For one chamber of an eight-chambered cover glass of about 750 μl volume, 700 μl of imaging buffer was prepared by mixing 7 μl of GLOX solution, 70 μl of MEA solution and 620 μl of Buffer B on ice. This imaging buffer was used as the medium for in vitro actin samples. The chamber was filled completely and covered immediately to avoid replenishing of oxygen in solution. Live-cell sample preparation CHO-K1 cells were obtained from ATCC (Manassas, VA). Lifeact cells (CHO-K1 cells stably expressing Lifeact-GFP) were provided by Prof. Rachel S. Kraut (NTU, Singapore). CHO-K1 and Lifeact cells were cultivated in DMEM medium (Dulbecco’s Modified Eagle Medium, Invitrogen; Singapore) supplemented with 1% penicillin G and streptomycin (PS, PAA, Austria), and 10% fetal bovine serum (Invitrogen; Singapore) at 37 °C in 5% (v/v) CO 2 environment. GFP-tubulin plasmid was a gift from Dr Pakorn T. Kanchanawong (MBI, NUS, Singapore). Electroporation was used for transfection of the cells, during which, 90% confluent cells in a 75 cm 2 flask were washed twice with 1 × PBS, trypsinized with 0.25% trypsin-0.03% EDTA solution for ∼ 1 min at 37 °C, and then re-suspended in culture medium. Cells were precipitated by centrifugation and re-suspended in small amount of resuspension R buffer (NeonTM Transfection System, Life Technologies, Singapore) and transferred half into one electroporation cuvette (2 mm wide, Bio-Rad; Hercules, CA) for one transfection. Between 300 and 500 ng ml −1 of the plasmid were added. After electroporation pulse, cells were seeded back to prewashed cover glass (30 mm in diameter; Lakeside, Monee, IL) in a 35 mm culture dish. Transfected cells grew in the culture medium for 24–36 h before measurement. All cells were imaged in Phenol red free DMEM medium containing 10% fetal bovine serum. See Supplementary Table 2 for an overview of the live-cell samples. Imaging system and image acquisition The setup consisted of an inverted epifluorescence microscope (IX83, Olympus, Japan) equipped with a motorized TIRF illumination combiner (IX3-MITICO, Olympus, Japan) and a scientific complementary metal oxide (sCMOS) camera with 6.5 μm pixels (Orca-Flash4.0, Hamamtsu Photonics, Japan). A 488 nm laser (LAS/488/100/D) or 561 nm laser (LAS/561/100, Olympus, Germany) was connected to the TIRF illumination combiner in which the incidence angle was adjusted to give 110 nm penetration depth of the evanescent field. The 488 and 561 nm lasers were used for imaging live-cell samples or in vitro actin filaments, respectively. A × 100, numerical aperture 1.49 oil immersion objective (UAPON, Olympus, Japan) was used to illuminate the sample and collect the fluorescence image. The fluorescence light then passed through a major dichroic (ZT405/488/561/647rpc, Chroma Technology, Bellows Falls, VT) and a band-pass filter (ZET405/488/561/647m, Chroma Technology, Bellows Falls, VT). The intensity of the excitation light was determined as the power of the laser light exiting the objective divided by the illuminated area; the power of the laser light was regulated by the proprietary laser control software and by an additional OD 1 neutral density filter. The camera was controlled by Micro-Manager 1.4 (ref. 28 ). For multiple power measurements, the first measurement is done using the lowest power and the power is subsequently increased. The optical PSF computed as an airy disk has FWHM of ∼ 198 nm for these parameters and emission wavelength 593 nm of the phalloidin Atto-565 dye and about 176 nm for emission wavelength 512 nm of the lifeact-GFP. For in vitro samples 1 and 2, 10,000 frames were acquired with exposure time of 5 ms (200 frames per second) For in vitro sample 3, 20,000 frames were acquired with exposure time of 10 ms (100 frames per second). For live-cell microtubule sample 1, 1,000 frames were acquired with exposure time of 5 ms (200 frames per second). For live-cell microtubule sample 2, 49 frames were acquired with exposure time of 1 ms (1,000 frames per second). For live-cell F-actin sample, 100 frames were acquired with exposure time of 1 ms (1,000 frames per second). Data availability The data that support the findings of this study and the source codes of MUSICAL are available at https://sites.google.com/site/uthkrishth/musical .\"\n },\n {\n \"section\": \"Additional information\",\n \"content\": \"How to cite this article: Agarwal, K. & Machan, R. Multiple signal classification algorithm for super-resolution fluorescence microscopy. Nat. Commun. 7, 13752 doi: 10.1038/ncomms13752 (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 propose a novel super-resolution fluorescence microscopy algorithm called MUltiple SIgnal Classification ALgorithm (MUSICAL), which exploits the eigenimages of a temporally varying fluorescence image stack—reflecting the statistical structure of blinking emitters—combined with knowledge of the point spread function (PSF), to achieve high-resolution imaging comparable to SMLM techniques but without the need for special instrumentation, specialized fluorophores, or long dark states, and applicable to live-cell imaging with standard fluorescent proteins and short acquisition times."},"entities":{"kind":"list like","value":["super-resolution fluorescence microscopy algorithm","MUltiple SIgnal Classification ALgorithm","MUSICAL","eigenimages","temporally varying fluorescence image stack","statistical structure","blinking emitters","point spread function","PSF","high-resolution imaging","SMLM techniques","special instrumentation","specialized fluorophores","long dark states","live-cell imaging","standard fluorescent proteins","short acquisition times"],"string":"[\n \"super-resolution fluorescence microscopy algorithm\",\n \"MUltiple SIgnal Classification ALgorithm\",\n \"MUSICAL\",\n \"eigenimages\",\n \"temporally varying fluorescence image stack\",\n \"statistical structure\",\n \"blinking emitters\",\n \"point spread function\",\n \"PSF\",\n \"high-resolution imaging\",\n \"SMLM techniques\",\n \"special instrumentation\",\n \"specialized fluorophores\",\n \"long dark states\",\n \"live-cell imaging\",\n \"standard fluorescent proteins\",\n \"short acquisition times\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1","title":"Betzig, E. et al. Imaging intracellular fluorescent proteins at nanometer resolution.Science313, 1642–1645 (2006).","link":"https://doi.org/10.1126%2Fscience.1127344"},{"idx":"2","title":"Hess, S. T., Girirajan, T. P. & Mason, M. D. Ultra-high resolution imaging by fluorescence photoactivation localization microscopy.Biophys. J.91, 4258–4272 (2006).","link":"https://doi.org/10.1529%2Fbiophysj.106.091116"},{"idx":"3","title":"Rust, M. J., Bates, M. & Zhuang, X. Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM).Nat. Methods3, 793–796 (2006).","link":"https://doi.org/10.1038%2Fnmeth929"},{"idx":"4","title":"Ettinger, A. & Wittmann, T. Fluorescence live cell imaging.Methods Cell Biol.123, 77–94 (2014).","link":"https://doi.org/10.1016%2FB978-0-12-420138-5.00005-7"},{"idx":"5","title":"Small, A. & Stahlheber, S. Fluorophore localization algorithms for super-resolution microscopy.Nat. Methods11, 267–279 (2014).","link":"https://doi.org/10.1038%2Fnmeth.2844"},{"idx":"6","title":"Dertinger, T., Colyer, R., Iyer, G., Weiss, S. & Enderlein, J. Fast, background-free, 3d super-resolution optical fluctuation imaging (SOFI).Proc. Natl Acad. Sci. USA106, 22287–22292 (2009).","link":"https://doi.org/10.1073%2Fpnas.0907866106"},{"idx":"7","title":"Cox, S. et al. Bayesian localization microscopy reveals nanoscale podosome dynamics.Nat. Methods9, 195–200 (2012).","link":"https://doi.org/10.1038%2Fnmeth.1812"},{"idx":"8","title":"Yahiatene, I., Hennig, S., Müller, M. & Huser, T. Entropy-based super-resolution imaging (ESI): From disorder to fine detail.ACS Photonics2, 1049–1056 (2015).","link":"https://doi.org/10.1021%2Facsphotonics.5b00307"},{"idx":"9","title":"Geissbuehler, S. et al. Live-cell multiplane three-dimensional super-resolution optical fluctuation imaging.Nat. Commun.5, 5830 (2014).","link":"https://doi.org/10.1038%2Fncomms6830"},{"idx":"10","title":"Mukamel, E. A., Babcock, H. & Zhuang, X. Statistical deconvolution for superresolution fluorescence microscopy.Biophys. J.102, 2391–2400 (2012).","link":"https://doi.org/10.1016%2Fj.bpj.2012.03.070"},{"idx":"13","title":"Gruber, F. K., Marengo, E. A. & Devaney, A. J. Time-reversal imaging with multiple signal classification considering multiple scattering between the targets.J. Acoust. Soc. Am.115, 3042–3047 (2004).","link":"https://doi.org/10.1121%2F1.1738451"},{"idx":"14","title":"Schmidt, R. O. Multiple emitter location and signal parameter estimation.IEEE Trans. Antennas Propag.34, 276–280 (1986).","link":"https://doi.org/10.1109%2FTAP.1986.1143830"},{"idx":"15","title":"Chen, X. & Agarwal, K. MUSIC algorithm for two-dimensional inverse problems with special characteristics of cylinders.IEEE Trans. Antennas Propag.56, 1808–1812 (2008).","link":"https://doi.org/10.1109%2FTAP.2008.923333"},{"idx":"16","title":"Chiu, W., McGough, A., Sherman, M. B. & Schmid, M. F. High-resolution electron cryomicroscopy of macromolecular assemblies.Trends Cell Biol.9, 154–159 (1999).","link":"https://doi.org/10.1016%2FS0962-8924%2899%2901511-1"},{"idx":"17","title":"Galkin, V. E., Orlova, A., Schröder, G. F. & Egelman, E. H. Structural polymorphism in F-actin.Nat. Struct. Mol. Biol.17, 1318–1323 (2010).","link":"https://doi.org/10.1038%2Fnsmb.1930"},{"idx":"18","title":"Egelman, E., Francis, N. & DeRosier, D. F-actin is a helix with a random variable twist.Nature298, 131–135 (1982).","link":"https://doi.org/10.1038%2F298131a0"},{"idx":"19","title":"Fowler, W. E. & Aebi, U. Polymorphism of actin paracrystals induced by polylysine.J. Cell Biol.93, 452–458 (1982).","link":"https://doi.org/10.1083%2Fjcb.93.2.452"},{"idx":"20","title":"Dempsey, G. T., Vaughan, J. C., Chen, K. H., Bates, M. & Zhuang, X. Evaluation of fluorophores for optimal performance in localization-based super-resolution imaging.Nat. Methods8, 1027–1036 (2011).","link":"https://doi.org/10.1038%2Fnmeth.1768"},{"idx":"21","title":"Nogales, E. Structural insights into microtubule function.Annu. Rev. Biophys. Biomol. Struct.30, 397–420 (2001).","link":"https://doi.org/10.1146%2Fannurev.biophys.30.1.397"},{"idx":"24","title":"Hu, Y. S., Nan, X., Sengupta, P., Lippincott-Schwartz, J. & Cang, H. Accelerating 3B single-molecule super-resolution microscopy with cloud computing.Nat. Methods10, 96–97 (2013).","link":"https://doi.org/10.1038%2Fnmeth.2335"},{"idx":"25","title":"Huang, F. et al. Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms.Nat. Methods10, 653–658 (2013).","link":"https://doi.org/10.1038%2Fnmeth.2488"},{"idx":"28","title":"Edelstein, A. D. et al. Advanced methods of microscope control using μManager software.J. Biol. Methods1, e10 (2014).","link":"https://doi.org/10.14440%2Fjbm.2014.36"}],"string":"[\n {\n \"idx\": \"1\",\n \"title\": \"Betzig, E. et al. Imaging intracellular fluorescent proteins at nanometer resolution.Science313, 1642–1645 (2006).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1127344\"\n },\n {\n \"idx\": \"2\",\n \"title\": \"Hess, S. T., Girirajan, T. P. & Mason, M. D. Ultra-high resolution imaging by fluorescence photoactivation localization microscopy.Biophys. J.91, 4258–4272 (2006).\",\n \"link\": \"https://doi.org/10.1529%2Fbiophysj.106.091116\"\n },\n {\n \"idx\": \"3\",\n \"title\": \"Rust, M. J., Bates, M. & Zhuang, X. Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM).Nat. Methods3, 793–796 (2006).\",\n \"link\": \"https://doi.org/10.1038%2Fnmeth929\"\n },\n {\n \"idx\": \"4\",\n \"title\": \"Ettinger, A. & Wittmann, T. Fluorescence live cell imaging.Methods Cell Biol.123, 77–94 (2014).\",\n \"link\": \"https://doi.org/10.1016%2FB978-0-12-420138-5.00005-7\"\n },\n {\n \"idx\": \"5\",\n \"title\": \"Small, A. & Stahlheber, S. Fluorophore localization algorithms for super-resolution microscopy.Nat. Methods11, 267–279 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fnmeth.2844\"\n },\n {\n \"idx\": \"6\",\n \"title\": \"Dertinger, T., Colyer, R., Iyer, G., Weiss, S. & Enderlein, J. Fast, background-free, 3d super-resolution optical fluctuation imaging (SOFI).Proc. Natl Acad. Sci. USA106, 22287–22292 (2009).\",\n \"link\": \"https://doi.org/10.1073%2Fpnas.0907866106\"\n },\n {\n \"idx\": \"7\",\n \"title\": \"Cox, S. et al. Bayesian localization microscopy reveals nanoscale podosome dynamics.Nat. Methods9, 195–200 (2012).\",\n \"link\": \"https://doi.org/10.1038%2Fnmeth.1812\"\n },\n {\n \"idx\": \"8\",\n \"title\": \"Yahiatene, I., Hennig, S., Müller, M. & Huser, T. Entropy-based super-resolution imaging (ESI): From disorder to fine detail.ACS Photonics2, 1049–1056 (2015).\",\n \"link\": \"https://doi.org/10.1021%2Facsphotonics.5b00307\"\n },\n {\n \"idx\": \"9\",\n \"title\": \"Geissbuehler, S. et al. Live-cell multiplane three-dimensional super-resolution optical fluctuation imaging.Nat. Commun.5, 5830 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms6830\"\n },\n {\n \"idx\": \"10\",\n \"title\": \"Mukamel, E. A., Babcock, H. & Zhuang, X. Statistical deconvolution for superresolution fluorescence microscopy.Biophys. J.102, 2391–2400 (2012).\",\n \"link\": \"https://doi.org/10.1016%2Fj.bpj.2012.03.070\"\n },\n {\n \"idx\": \"13\",\n \"title\": \"Gruber, F. K., Marengo, E. A. & Devaney, A. J. Time-reversal imaging with multiple signal classification considering multiple scattering between the targets.J. Acoust. Soc. Am.115, 3042–3047 (2004).\",\n \"link\": \"https://doi.org/10.1121%2F1.1738451\"\n },\n {\n \"idx\": \"14\",\n \"title\": \"Schmidt, R. O. Multiple emitter location and signal parameter estimation.IEEE Trans. Antennas Propag.34, 276–280 (1986).\",\n \"link\": \"https://doi.org/10.1109%2FTAP.1986.1143830\"\n },\n {\n \"idx\": \"15\",\n \"title\": \"Chen, X. & Agarwal, K. MUSIC algorithm for two-dimensional inverse problems with special characteristics of cylinders.IEEE Trans. Antennas Propag.56, 1808–1812 (2008).\",\n \"link\": \"https://doi.org/10.1109%2FTAP.2008.923333\"\n },\n {\n \"idx\": \"16\",\n \"title\": \"Chiu, W., McGough, A., Sherman, M. B. & Schmid, M. F. High-resolution electron cryomicroscopy of macromolecular assemblies.Trends Cell Biol.9, 154–159 (1999).\",\n \"link\": \"https://doi.org/10.1016%2FS0962-8924%2899%2901511-1\"\n },\n {\n \"idx\": \"17\",\n \"title\": \"Galkin, V. E., Orlova, A., Schröder, G. F. & Egelman, E. H. Structural polymorphism in F-actin.Nat. Struct. Mol. Biol.17, 1318–1323 (2010).\",\n \"link\": \"https://doi.org/10.1038%2Fnsmb.1930\"\n },\n {\n \"idx\": \"18\",\n \"title\": \"Egelman, E., Francis, N. & DeRosier, D. F-actin is a helix with a random variable twist.Nature298, 131–135 (1982).\",\n \"link\": \"https://doi.org/10.1038%2F298131a0\"\n },\n {\n \"idx\": \"19\",\n \"title\": \"Fowler, W. E. & Aebi, U. Polymorphism of actin paracrystals induced by polylysine.J. Cell Biol.93, 452–458 (1982).\",\n \"link\": \"https://doi.org/10.1083%2Fjcb.93.2.452\"\n },\n {\n \"idx\": \"20\",\n \"title\": \"Dempsey, G. T., Vaughan, J. C., Chen, K. H., Bates, M. & Zhuang, X. Evaluation of fluorophores for optimal performance in localization-based super-resolution imaging.Nat. Methods8, 1027–1036 (2011).\",\n \"link\": \"https://doi.org/10.1038%2Fnmeth.1768\"\n },\n {\n \"idx\": \"21\",\n \"title\": \"Nogales, E. Structural insights into microtubule function.Annu. Rev. Biophys. Biomol. Struct.30, 397–420 (2001).\",\n \"link\": \"https://doi.org/10.1146%2Fannurev.biophys.30.1.397\"\n },\n {\n \"idx\": \"24\",\n \"title\": \"Hu, Y. S., Nan, X., Sengupta, P., Lippincott-Schwartz, J. & Cang, H. Accelerating 3B single-molecule super-resolution microscopy with cloud computing.Nat. Methods10, 96–97 (2013).\",\n \"link\": \"https://doi.org/10.1038%2Fnmeth.2335\"\n },\n {\n \"idx\": \"25\",\n \"title\": \"Huang, F. et al. Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms.Nat. Methods10, 653–658 (2013).\",\n \"link\": \"https://doi.org/10.1038%2Fnmeth.2488\"\n },\n {\n \"idx\": \"28\",\n \"title\": \"Edelstein, A. D. et al. Advanced methods of microscope control using μManager software.J. Biol. Methods1, e10 (2014).\",\n \"link\": \"https://doi.org/10.14440%2Fjbm.2014.36\"\n }\n]","isConversational":false}}},{"rowIdx":86,"cells":{"unique_id":{"kind":"string","value":"ncomms11442"},"subfield":{"kind":"string","value":"Physics"},"title":{"kind":"string","value":"Fast electronic resistance switching involving hidden charge density wave states"},"abstract":{"kind":"string","value":"The functionality of computer memory elements is currently based on multi-stability, driven either by locally manipulating the density of electrons in transistors or by switching magnetic or ferroelectric order. Another possibility is switching between metallic and insulating phases by the motion of ions, but their speed is limited by slow nucleation and inhomogeneous percolative growth. Here we demonstrate fast resistance switching in a charge density wave system caused by pulsed current injection. As a charge pulse travels through the material, it converts a commensurately ordered polaronic Mott insulating state in 1T–TaS 2 to a metastable electronic state with textured domain walls, accompanied with a conversion of polarons to band states, and concurrent rapid switching from an insulator to a metal. The large resistance change, high switching speed (30 ps) and ultralow energy per bit opens the way to new concepts in non-volatile memory devices manipulating all-electronic states."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"Systems with first-order transitions between competing states in the phase diagram are obvious candidates for resistive memories. In the last few years, pulsed laser experiments have led to an improved understanding of non-equilibrium phase transitions [1] , [2] , [3] , [4] and metastable hidden (H) states [1] , [2] , [3] , [5] . Optical memory, even though it can be switched with short pulses (35 fs) in charge density wave (CDW) systems [2] would be severely limited by associated disk-based technology. Electrical switching of CDW systems on the other hand has until now met the fundamental problem that injected charge does not directly couple to the order parameter. Instead, an applied electric field usually causes an incommensurate (IC) CDW to slide [6] , and in experiments so far the system invariably reverts back to the original state when the external field is switched off [7] , [8] . In this study we describe ultrafast non-volatile resistance switching with short pulsed current injection in 1T–TaS 2 , where injected charges create domain walls (DWs) via coupling to gradients of the CDW order parameter, converting the material from an insulator to a metal at low temperatures. From a technological viewpoint, this opens up the possibility of low-temperature ultrafast memory devices, whose absence has so far seriously impeded progress of ultrafast energy-efficient superconducting computing, for example [9] .","references":{"indices":[1,2,4,5,6,7,8,9],"infos":[{"idx":"1","title":"Ichikawa, H. et al. Transient photoinduced ‘hidden’ phase in a manganite.Nat. Mater.10, 101–105 (2011).","link":"https://doi.org/10.1038%2Fnmat2929"},{"idx":"2","title":"Stojchevska, L. et al. Ultrafast switching to a stable hidden quantum state in an electronic crystal.Science344, 177–180 (2014).","link":"https://doi.org/10.1126%2Fscience.1241591"},{"idx":"4","title":"Tokura, Y. Photoinduced phase transition: a tool for generating a hidden state of matter.J. Phys. Soc. Jpn75, 011001 (2006).","link":"https://doi.org/10.1143%2FJPSJ.75.011001"},{"idx":"5","title":"Vaskivskyi, I. et al. Controlling the metal to insulator relaxation of the metastable hidden quantum state in 1T-TaS2 .Sci. Advances1, e1500168 (2015).","link":"https://doi.org/10.1126%2Fsciadv.1500168"},{"idx":"6","title":"Monceau, P. Electronic crystals: an experimental overview.Adv. Phys.61, 325–581 (2012).","link":"https://doi.org/10.1080%2F00018732.2012.719674"},{"idx":"7","title":"Hollander, M. J. et al. Electrically driven reversible insulator-metal phase transition in 1T-TaS2 .Nano Lett.15, 1861–1866 (2015).","link":"https://doi.org/10.1021%2Fnl504662b"},{"idx":"8","title":"Brazovskii, S. et al. Plastic sliding of charge density waves: X-ray space resolved-studies versus theory of current conversion.Phys. Rev. B61, 10640–10650 (2000).","link":"https://doi.org/10.1103%2FPhysRevB.61.10640"},{"idx":"9","title":"Holmes, D. S., Ripple, A. L. & Manheimer, M. A. Energy-efficient superconducting computing; power budgets and requirements.IEEE Trans. Appl. Supercond.23, 1701610–1701610 (2015).","link":"https://doi.org/10.1109%2FTASC.2013.2244634"}]}},{"section":"Results","content":"Thermal phases of 1T–TaS 2 The experiments were performed on 1T–TaS 2 , which at high temperatures is a simple metal with a single Ta d electron band crossing the Fermi level. At T c0 =543 K, it forms an IC CDW. On cooling below T c1 =350 K, in trying to conform with the underlying lattice, the IC structure forms a nearly commensurate (NC) state in which electrons on each 13th Ta atom localize, leading to a regular hexagonal array of NC polaron clusters separated by DWs (illustrated in Fig. 1a ). Below T c2 =183 K, the DWs vanish and a fully commensurate (C) hexagonal polaronic superlattice forms, which is believed to be a Mott insulator with an energy gap Δ Mott 0.1 eV (refs 10 , 11 ). Figure 1: The equilibrium phases of 1T–TaS 2 and demonstration of switching behaviour. ( a ) An illustration of the different CDW states in 1T–TaS 2 at different temperatures. ( b ) Temporal behaviour of switching from HI to LO resistance in response to a pulse from a current source. The extrapolated voltage in the absence of switching is shown by the dashed lines. The time–response is determined by τ RC = RC where R is the two-terminal circuit resistance and changes from 53 to 44 ns. Fits of the relaxation times show that the difference between τ after switching and τ RC at the end of the pulse is<0.3 ns, setting a limit on the intrinsic speed of the device. The insert shows the measuring circuit. The small superimposed oscillations are due to ringing of the circuit. Scale bar, 2 μm. Full size image Switching by current pulse The switching experiments were performed in the Mott state on thin (50–100 nm) single crystal flakes of 1T–TaS 2 on sapphire substrates, with electrical contacts deposited over the crystal using laser direct lithography, typically 1–8 μm apart. Electrical measurements were made in the C state with pulse lengths between 1 s and 40 ps (see Methods section, Supplementary Note 1 and Supplementary Fig. 1 for details). In Fig. 1b , we show the voltage across the sample at 150 K as a function of time after the application of a 1-μs pulse from a current source for different magnitudes of pulse current. At low pulsed currents, the voltage across the sample follows the source, with rise and fall times defined by the RC constant of the circuit τ RC = R HI C =53 ns, where R HI is the intial (high) value of circuit resistance. With increasing current, above a certain threshold current, the voltage across the sample drops sharply to ∼ 60% of its expected value (dashed) and remains constant until the end of the pulse. After switching, τ RC decreases to 44 ns, consistent with the change of sample resistance (to a value designated as R LO ) after switching. Comparing exponential fits with the fall time τ s after switching and the fall time after the end of the pulse we find that they are indistinguishable from each other to within ±0.3 ns, defining an upper limit on the intrinsic switching time. The recovery of R to the initial value R HI occurs within τ R ∼ 10 ms, consistent with relaxation time data at 150 K, which changes with temperature T as , with E A =280 ∼ 2,300 K, depending on the sample substrate [5] , k B is the Boltzmann constant. To investigate the switching speed, we apply current pulses of different duration and measure the initial and final resistance with a low current after each pulse. For τ pulse >25 ns, we used a standard electrical pulse generator, whereas for τ pulse <25 ns we used a 35-fs optical pulse-triggered 30 ps risetime ( ∼ 40 ps full width at half maximum) metal–semiconductor–metal pulse source with the 1T–TaS 2 sample mounted in a 50-Ω transmission line circuit (shown in Fig. 2c ). Remarkably, provided that I > I T , we observe switching between R HI and R LO , irrespective of the duration of the applied pulse over the interval 40 ps ≲ τ pulse <0.1 s (summarized in Fig. 2a ). For τ pulse >0.1 s, switching is incomplete and does not occur at all for τ pulse >1 s. Repeating the same experiment, but starting in the low resistance state R LO , switching from R LO to R HI occurs for τ pulse >0.1 s ( Fig. 2b ). The implication of these measurements is that the R HI → R LO switching is caused by the leading edge of the pulse, in a time τ s <30 ps. On the other hand, the reverse switching R LO → R HI with τ pulse ⩾ 0.1 s is consistent with Joule heating above 100 K, where the lifetime is sufficiently short to relax the H state [5] . We label the two cycles described above as write (W) and erase (E). Figure 2: Switching speed and threshold behaviour. ( a ) Switching resistance ratio measured with a single pulse above threshold I T , each time starting from the HI resistance state (red circles). Black circles show resistivity after current pulse. ( b ) Erasing with 1-s pulses, each time starting from the LO resistance state (red circles). Black circles show resistivity after applying erase pulse. The data were obtained by alternating write/erase cycles and increasing the pulse length. ( c ) The circuit used for the 30-ps measurement with an metal–semiconductor–metal (MSM) device as the current source. ( d ) The voltage V measured during each pulse, first incrementally increasing and then decreasing the pulse current I , shown by paths (1) and (2), respectively. Above the threshold current I T , V abruptly drops between two consecutive current values. ( e ) Same as in d , but starting from the low resistance state increasing I pulse along path (1) and then reducing I pulse along (2). In all cases T =20 K and the pulse length was 10 μs. Scale bar, 2 μm. Full size image To systematically measure threshold current I T , we incrementally increase and then decrease the current pulse magnitude, while simultaneously recording the voltage across the sample. The resulting hysteretic I–V curve measured at 14 K for a 80-nm-thick crystal with 8 μm between inner contacts and τ pulse =50 μs is shown in Fig. 2d . Before switching, the I–V curve shows an unusual exponentially increasing I with V : Increasing I beyond a critical value I T , we observe a sharp drop in V . Thereafter, the I–V curve becomes ohmic over a large range of I . On the return cycle (and subsequent cycles as long as I <2 I T ), the I–V curve is linear, with a slope at the origin that is two to three orders of magnitude smaller than R HI in the C state ( Fig. 2a ). Repeating the measurement cycle, but starting with the sample in the low resistance state R LO , the response departs from linearity beyond I ∼ I T /2 ( Fig. 2e ). Thereafter, V increases very slowly with I , becoming nearly independent of current above 4 mA. In the return part of the cycle, the nearly I -independent behaviour remains until ∼ 1 mA, whereupon the system reverts to the original C state with R HI below 1 mA. More details are presented in Supplementary Fig. 2 . Temperature dependence of the switching The temperature dependence of the switching is shown in Fig. 3a for 20 K< T <205 K. For 150 K and below, each curve is measured on a freshly ordered CDW state, prepared by slowly cooling the sample from 150 K to the indicated base temperature before each measurement. Above 150 K, the temperature was gradually increased. For T up to ∼ 55 K, the behaviour is similar to that described above for T =14 ∼ 20 K ( Fig. 2d,e ). In the range 55 ∼ 165 K, an instability is observed above I T , where the voltage fluctuates with each pulse between R HI and R LO . Above 165 K, these fluctuations disappear and the switching ratio R HI /R LO eventually disappears completely above 195 K. The T -dependence of I T and V 0 obtained from a fit to the data using equation (1) is shown in Fig. 3b . Remarkably, fits to the I–V curves below threshold show that I T is almost completely independent of T , whereas V 0 gradually drops with T . (The details of the fitting are given in the Supplementary Note 2 and Supplementary Fig. 3 .) The latter behaviour is probably not entirely intrinsic, as at high temperatures the measurement time becomes comparable with the relaxation lifetime τ R of the low resistance state [5] . We also note the rather remarkable behaviour in Fig. 3a at high currents, displaying a cross-over from negative differential resistance (DR) through zero DR to a slight positive DR with increasing temperature up to 200 K. Figure 3: The T -dependence of switching. ( a ) The pulsed V–I characteristic as in Fig. 2 , measured at different temperatures and at 10 K intervals with τ pulse =50 μs. The V–I curve is measured in pulsed mode, by increasing the current incrementally and measuring the voltage at each point. The resistance drops sharply in a few consecutive measurements at the critical value of I T . The sample was reset in between each V–I curve by heating it above 310 K and then cooling down to the indicated temperature for the next measurement. ( b ) I T and V 0 as a function of temperature. Error bars are obtained from fitting the data with the exponential function. Full size image"},{"section":"Discussion","content":"Before discussing the switching mechanism, we first highlight the unusual observations. The switching is remarkably fast and works as a latch with a very sharp current threshold, which cannot be associated with dielectric breakdown into filamentary paths involving ionic motion, as in chalcogenide glasses [7] , [12] , manganite Mott insulators [13] or band insulator memristors such as TiO x (ref. 14 ). Zener tunnelling and avalanche breakdown can also be eliminated, because they do not exhibit metastability. The pulsed I–V curve below switching threshold is also qualitatively different from the observed behaviour for sliding and de-pinning of a CDW [6] , [15] , [16] . On the other hand, well above I T , the negative DR behaviour gradually crosses over to a small positive DR with increasing temperature, which is reminiscent of dissipationless CDW sliding conductivity (Frölich conductivity) observed in blue bronze at similar current densities [17] . In previous optical switching experiments, electrons (e) and holes (h) are homogeneously photoexcited and spatially overlapping [2] . The combination of narrow spectral width and unchanged spectral intensity of the collective amplitude mode in the photoexcited H state compared with the C state indicate that the two states are comparably homogeneous. Presently, the e and h are spatially separated at either electrode so the optical mechanism does not apply and we expect the structure to be less homogeneous. Scanning tunneling microscope (STM) images of the sample surface after current injection by an STM tip confirm this: they show an irregular meander of DWs rather than a regular array of DWs (see Supplementary Figs 4 and 5 ) [18] , [19] . In our experiments we find that different R LO with different relaxation properties can be reached with different pulse voltages near threshold (see Supplementary Note 3 for a few examples), indicating that different DW configurations can be created under different conditions. The free energy landscape of H states in 1T–TaS 2 clearly has multiple minima—consistent with the notion of an electronic glass. From a Landau theory viewpoint, charge cannot couple directly to the order parameter ψ of the CDW, but can couple through Lifshitz-invariant spatial gradient terms in the free energy , which are present at DWs. We propose that as the pulse of charges propagates through the sample, the energy of the system is lowered by charge carriers dynamically creating DWs. As the DWs are charged (with respect to the CDW background), they repel each other, creating a periodically textured stripe structure, an effect that is well known in correlated electron systems. The free energy F for such a system includes terms from the IC CDW formation and the DW repulsion, and has a number of minima, corresponding to states with different CDW and DW order ( Fig. 4b ) already in the absence of additional charge [20] . In the Supplementary Notes 4–5 and Supplementary Figs 6–7 , we consider the situation in detail, showing that on charge injection the C state is unstable towards the formation of a periodic DW array, IC with the underlying lattice structure. The origin of this instability lies in the negative curvature of the inverse compressibility k =d 2 F /d n 2 <0 arising from the shape of F ( n ), which in turn is a consequence of the Coulomb repulsion between DWs. The periodicity of the DWs is directly related to the injected charge density n i = n e − n h : the resulting structure is commensurate if n i =0 and IC otherwise, with a wavevector q H = n i / π corresponding to the minimum in the free energy ( Fig. 4b ). The DW structure in the current-injection-induced H state is thus similar to the NC state, but with a periodicity that depends on n i . Inhomogeneity of charge injection will naturally lead to spatially inhomogeneous DW structures observed in experiments. Nevertheless, the state conversion from polarons to band states means that it is not only the DWs, but the entire state that is metallic. Figure 4: An illustration of carrier trapping and DW formation after carrier injection. ( a ) The injected charges are trapped in the C structure causing the formation of a textured glassy electronic state. The charge density ρ is depleted at the leading edge of the pulse as it travels through the sample. ( b ) The free energy as a function of CDW wavevector based on the model of Nakanishi and Shiba [20] . q C and q I are the q -vectors of the undoped commensurate and IC states, respectively. The q -vector in the H state q H is determined by the density of carriers forming the DWs. ( c ) A depiction of the charge injection at the electrodes in the C state and ( d ) the band structure after the Mott-to-band state conversion process. Full size image Microscopically, a simple Mott–Hubbard picture [21] is inadequate for a description of the DW formation [18] . However, if we also take into account the effect of long-range Coulomb repulsion and consider pulsed carrier propagation through a commensurate polaronic crystal ( Fig. 4a ), we can qualitatively understand the origin of the textured state also from the microscopic viewpoint. The carriers launched at the electrodes first lose their excess energy e V = E e − E h =1 ∼ 10 eV ( Fig. 4c,d ) within <1 ps. Taking into account scattering and energy loss to phonons of these energetic carriers gives an avalanche multiplication factor . By the time the e and h reach the upper Hubbard band (UHB) and lower Hubbard band (LHB), respectively, the number of carriers injected by a 40-ps pulse is ∼ . Occupancy of the Hubbard bands is unstable, so as they propagate in the applied electric field, the holes annihilate with the localized electrons at the centre of each polaron, leading to the creation of voids in the polaron lattice, accompanied by a dynamic conversion of polaronic (Mott–Hubbard) states to conducting band states as the wave travels through the sample ( Fig. 4a ). (STM studies suggest that electrons are not expected to show the same effect as holes [18] .) From the STM images (see Supplementary Note 6 for details), the number of holes n h that can be accommodated in the DWs is very roughly between ∼ 1/4 and 1/3 of all the available polaron sites; thus, for our typical sample of volume 3 μm 3 we estimate n h ≈10 6 ∼ 10 7 . This is consistent with the number of injected charges in the pulse n i . However, rather than forming a homogeneous patch of voids at the electrode, as a result of the Coulomb interaction, a textured state forms ( Fig. 4a ). The time for such an DW state to form after melting is known to be τ form ∼ 1 ps (ref. 22 ), which is consistent with the observed speed of the switching process. The observed independence on pulse length can thus be understood: the DWs are formed by the carriers in the leading edge of the pulse. Above a certain threshold, the entire sample is converted and further added charges have no additional effect; thus, the trailing edge of the pulse has no further effect. Importantly, the unusual exponential I–V curves below threshold described by equation (1) can be understood within this picture in terms of carrier tunnelling through barriers resulting from the transient DW structure. We show in the SM that the tunnelling rate within such a state leading to equation (1) is R ∼ exp( V / V 0 ) β where β ∼ 1, where the barrier energy V 0 is related to the C state order parameter, giving rise to its T -dependence shown in Fig. 3b . The experiments introduce a paradigm of all-electronic switching between CDW states for data storage applications beyond current concepts [12] , [14] , [23] , [24] , [25] , [26] , [27] , [28] , [29] . Devices based on CDW state switching in 1T–TaS 2 could be scaled and integrated into commonly used cross-bar configurations [30] . The demonstrated write speed ∼ 30 ps is externally limited, but is an order of magnitude faster than the previous record for non-volatile memory [14] , [25] . The intrinsic speed appears to be <1 ps. With τ W =30 ps, the expended energy per bit is estimated to be pJ for a device with an area 2 × 2 μm 2 (shown in Supplementary Fig. 8 ), which is extremely small, comparable with current magnetic random access memory (MRAM) devices [31] . As both V T and R HI scale linearly with distance between contacts (see Supplementary Fig. 9 ), E B scales linearly with size along the contact direction and can be reduced by making the devices smaller. From a practical device viewpoint, it functions as a latch at low T . The reverse M – I transition is caused thermally, which requires that the device is heated momentarily above T c2 —which can be done on a sub-nanosecond timescale with appropriate design [25] . Moreover, in practical devices bulk Erase is used where entire blocks are erased simultaneously; thus, E speed is not a limiting device operation speed. The device operating temperature appears to be limited by T c2 ( Fig. 3a ). It is known that this can be increased significantly by Se substitution: 1T–TaS 1.5 Se 0.5 has a T c2 >300 K (ref. 32 ), introducing the possibility of room temperature operation, whereas the relaxation time τ R has been recently shown to be controllable by compressive strain [5] tuning the data retention time of the device."},{"section":"Methods","content":"Sample preparation 1T–TaS 2 crystals were grown by chemical transport method with iodine as a transport agent with the average lateral dimensions 1–5 mm and thicknesses ∼ 100 μm. Thin flakes were obtained from single crystal by exfoliating with sticky tape. The obtained 50- to 100-nm-thick flakes were deposited on sapphire substrate. Four gold electrodes were deposited on top of the flake by laser-direct lithography with initial 10-nm-thick Au/Pd layer. Electrical measurements DC transport measurements were performed using standard four-point technique. All switching experiments were performed starting from the C Mott phase by applying single electrical pulse. The pulses in the range 100 ns–1 s were produced by a standard pulse generator. For 40-ps full width at half maximum pulse we used metal–semiconductor–metal current source triggered by a 35-fs laser pulse."},{"section":"Additional information","content":"How to cite this article : Vaskivskyi, I. et al . Fast electronic resistance switching involving hidden charge density wave states. Nat. Commun. 7:11442 doi: 10.1038/ncomms11442 (2016)."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"Systems with first-order transitions between competing states in the phase diagram are obvious candidates for resistive memories. In the last few years, pulsed laser experiments have led to an improved understanding of non-equilibrium phase transitions [1] , [2] , [3] , [4] and metastable hidden (H) states [1] , [2] , [3] , [5] . Optical memory, even though it can be switched with short pulses (35 fs) in charge density wave (CDW) systems [2] would be severely limited by associated disk-based technology. Electrical switching of CDW systems on the other hand has until now met the fundamental problem that injected charge does not directly couple to the order parameter. Instead, an applied electric field usually causes an incommensurate (IC) CDW to slide [6] , and in experiments so far the system invariably reverts back to the original state when the external field is switched off [7] , [8] . In this study we describe ultrafast non-volatile resistance switching with short pulsed current injection in 1T–TaS 2 , where injected charges create domain walls (DWs) via coupling to gradients of the CDW order parameter, converting the material from an insulator to a metal at low temperatures. From a technological viewpoint, this opens up the possibility of low-temperature ultrafast memory devices, whose absence has so far seriously impeded progress of ultrafast energy-efficient superconducting computing, for example [9] .\",\n \"references\": {\n \"indices\": [\n 1,\n 2,\n 4,\n 5,\n 6,\n 7,\n 8,\n 9\n ],\n \"infos\": [\n {\n \"idx\": \"1\",\n \"title\": \"Ichikawa, H. et al. Transient photoinduced ‘hidden’ phase in a manganite.Nat. Mater.10, 101–105 (2011).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat2929\"\n },\n {\n \"idx\": \"2\",\n \"title\": \"Stojchevska, L. et al. Ultrafast switching to a stable hidden quantum state in an electronic crystal.Science344, 177–180 (2014).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1241591\"\n },\n {\n \"idx\": \"4\",\n \"title\": \"Tokura, Y. Photoinduced phase transition: a tool for generating a hidden state of matter.J. Phys. Soc. Jpn75, 011001 (2006).\",\n \"link\": \"https://doi.org/10.1143%2FJPSJ.75.011001\"\n },\n {\n \"idx\": \"5\",\n \"title\": \"Vaskivskyi, I. et al. Controlling the metal to insulator relaxation of the metastable hidden quantum state in 1T-TaS2 .Sci. Advances1, e1500168 (2015).\",\n \"link\": \"https://doi.org/10.1126%2Fsciadv.1500168\"\n },\n {\n \"idx\": \"6\",\n \"title\": \"Monceau, P. Electronic crystals: an experimental overview.Adv. Phys.61, 325–581 (2012).\",\n \"link\": \"https://doi.org/10.1080%2F00018732.2012.719674\"\n },\n {\n \"idx\": \"7\",\n \"title\": \"Hollander, M. J. et al. Electrically driven reversible insulator-metal phase transition in 1T-TaS2 .Nano Lett.15, 1861–1866 (2015).\",\n \"link\": \"https://doi.org/10.1021%2Fnl504662b\"\n },\n {\n \"idx\": \"8\",\n \"title\": \"Brazovskii, S. et al. Plastic sliding of charge density waves: X-ray space resolved-studies versus theory of current conversion.Phys. Rev. B61, 10640–10650 (2000).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.61.10640\"\n },\n {\n \"idx\": \"9\",\n \"title\": \"Holmes, D. S., Ripple, A. L. & Manheimer, M. A. Energy-efficient superconducting computing; power budgets and requirements.IEEE Trans. Appl. Supercond.23, 1701610–1701610 (2015).\",\n \"link\": \"https://doi.org/10.1109%2FTASC.2013.2244634\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Thermal phases of 1T–TaS 2 The experiments were performed on 1T–TaS 2 , which at high temperatures is a simple metal with a single Ta d electron band crossing the Fermi level. At T c0 =543 K, it forms an IC CDW. On cooling below T c1 =350 K, in trying to conform with the underlying lattice, the IC structure forms a nearly commensurate (NC) state in which electrons on each 13th Ta atom localize, leading to a regular hexagonal array of NC polaron clusters separated by DWs (illustrated in Fig. 1a ). Below T c2 =183 K, the DWs vanish and a fully commensurate (C) hexagonal polaronic superlattice forms, which is believed to be a Mott insulator with an energy gap Δ Mott 0.1 eV (refs 10 , 11 ). Figure 1: The equilibrium phases of 1T–TaS 2 and demonstration of switching behaviour. ( a ) An illustration of the different CDW states in 1T–TaS 2 at different temperatures. ( b ) Temporal behaviour of switching from HI to LO resistance in response to a pulse from a current source. The extrapolated voltage in the absence of switching is shown by the dashed lines. The time–response is determined by τ RC = RC where R is the two-terminal circuit resistance and changes from 53 to 44 ns. Fits of the relaxation times show that the difference between τ after switching and τ RC at the end of the pulse is<0.3 ns, setting a limit on the intrinsic speed of the device. The insert shows the measuring circuit. The small superimposed oscillations are due to ringing of the circuit. Scale bar, 2 μm. Full size image Switching by current pulse The switching experiments were performed in the Mott state on thin (50–100 nm) single crystal flakes of 1T–TaS 2 on sapphire substrates, with electrical contacts deposited over the crystal using laser direct lithography, typically 1–8 μm apart. Electrical measurements were made in the C state with pulse lengths between 1 s and 40 ps (see Methods section, Supplementary Note 1 and Supplementary Fig. 1 for details). In Fig. 1b , we show the voltage across the sample at 150 K as a function of time after the application of a 1-μs pulse from a current source for different magnitudes of pulse current. At low pulsed currents, the voltage across the sample follows the source, with rise and fall times defined by the RC constant of the circuit τ RC = R HI C =53 ns, where R HI is the intial (high) value of circuit resistance. With increasing current, above a certain threshold current, the voltage across the sample drops sharply to ∼ 60% of its expected value (dashed) and remains constant until the end of the pulse. After switching, τ RC decreases to 44 ns, consistent with the change of sample resistance (to a value designated as R LO ) after switching. Comparing exponential fits with the fall time τ s after switching and the fall time after the end of the pulse we find that they are indistinguishable from each other to within ±0.3 ns, defining an upper limit on the intrinsic switching time. The recovery of R to the initial value R HI occurs within τ R ∼ 10 ms, consistent with relaxation time data at 150 K, which changes with temperature T as , with E A =280 ∼ 2,300 K, depending on the sample substrate [5] , k B is the Boltzmann constant. To investigate the switching speed, we apply current pulses of different duration and measure the initial and final resistance with a low current after each pulse. For τ pulse >25 ns, we used a standard electrical pulse generator, whereas for τ pulse <25 ns we used a 35-fs optical pulse-triggered 30 ps risetime ( ∼ 40 ps full width at half maximum) metal–semiconductor–metal pulse source with the 1T–TaS 2 sample mounted in a 50-Ω transmission line circuit (shown in Fig. 2c ). Remarkably, provided that I > I T , we observe switching between R HI and R LO , irrespective of the duration of the applied pulse over the interval 40 ps ≲ τ pulse <0.1 s (summarized in Fig. 2a ). For τ pulse >0.1 s, switching is incomplete and does not occur at all for τ pulse >1 s. Repeating the same experiment, but starting in the low resistance state R LO , switching from R LO to R HI occurs for τ pulse >0.1 s ( Fig. 2b ). The implication of these measurements is that the R HI → R LO switching is caused by the leading edge of the pulse, in a time τ s <30 ps. On the other hand, the reverse switching R LO → R HI with τ pulse ⩾ 0.1 s is consistent with Joule heating above 100 K, where the lifetime is sufficiently short to relax the H state [5] . We label the two cycles described above as write (W) and erase (E). Figure 2: Switching speed and threshold behaviour. ( a ) Switching resistance ratio measured with a single pulse above threshold I T , each time starting from the HI resistance state (red circles). Black circles show resistivity after current pulse. ( b ) Erasing with 1-s pulses, each time starting from the LO resistance state (red circles). Black circles show resistivity after applying erase pulse. The data were obtained by alternating write/erase cycles and increasing the pulse length. ( c ) The circuit used for the 30-ps measurement with an metal–semiconductor–metal (MSM) device as the current source. ( d ) The voltage V measured during each pulse, first incrementally increasing and then decreasing the pulse current I , shown by paths (1) and (2), respectively. Above the threshold current I T , V abruptly drops between two consecutive current values. ( e ) Same as in d , but starting from the low resistance state increasing I pulse along path (1) and then reducing I pulse along (2). In all cases T =20 K and the pulse length was 10 μs. Scale bar, 2 μm. Full size image To systematically measure threshold current I T , we incrementally increase and then decrease the current pulse magnitude, while simultaneously recording the voltage across the sample. The resulting hysteretic I–V curve measured at 14 K for a 80-nm-thick crystal with 8 μm between inner contacts and τ pulse =50 μs is shown in Fig. 2d . Before switching, the I–V curve shows an unusual exponentially increasing I with V : Increasing I beyond a critical value I T , we observe a sharp drop in V . Thereafter, the I–V curve becomes ohmic over a large range of I . On the return cycle (and subsequent cycles as long as I <2 I T ), the I–V curve is linear, with a slope at the origin that is two to three orders of magnitude smaller than R HI in the C state ( Fig. 2a ). Repeating the measurement cycle, but starting with the sample in the low resistance state R LO , the response departs from linearity beyond I ∼ I T /2 ( Fig. 2e ). Thereafter, V increases very slowly with I , becoming nearly independent of current above 4 mA. In the return part of the cycle, the nearly I -independent behaviour remains until ∼ 1 mA, whereupon the system reverts to the original C state with R HI below 1 mA. More details are presented in Supplementary Fig. 2 . Temperature dependence of the switching The temperature dependence of the switching is shown in Fig. 3a for 20 K< T <205 K. For 150 K and below, each curve is measured on a freshly ordered CDW state, prepared by slowly cooling the sample from 150 K to the indicated base temperature before each measurement. Above 150 K, the temperature was gradually increased. For T up to ∼ 55 K, the behaviour is similar to that described above for T =14 ∼ 20 K ( Fig. 2d,e ). In the range 55 ∼ 165 K, an instability is observed above I T , where the voltage fluctuates with each pulse between R HI and R LO . Above 165 K, these fluctuations disappear and the switching ratio R HI /R LO eventually disappears completely above 195 K. The T -dependence of I T and V 0 obtained from a fit to the data using equation (1) is shown in Fig. 3b . Remarkably, fits to the I–V curves below threshold show that I T is almost completely independent of T , whereas V 0 gradually drops with T . (The details of the fitting are given in the Supplementary Note 2 and Supplementary Fig. 3 .) The latter behaviour is probably not entirely intrinsic, as at high temperatures the measurement time becomes comparable with the relaxation lifetime τ R of the low resistance state [5] . We also note the rather remarkable behaviour in Fig. 3a at high currents, displaying a cross-over from negative differential resistance (DR) through zero DR to a slight positive DR with increasing temperature up to 200 K. Figure 3: The T -dependence of switching. ( a ) The pulsed V–I characteristic as in Fig. 2 , measured at different temperatures and at 10 K intervals with τ pulse =50 μs. The V–I curve is measured in pulsed mode, by increasing the current incrementally and measuring the voltage at each point. The resistance drops sharply in a few consecutive measurements at the critical value of I T . The sample was reset in between each V–I curve by heating it above 310 K and then cooling down to the indicated temperature for the next measurement. ( b ) I T and V 0 as a function of temperature. Error bars are obtained from fitting the data with the exponential function. Full size image\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"Before discussing the switching mechanism, we first highlight the unusual observations. The switching is remarkably fast and works as a latch with a very sharp current threshold, which cannot be associated with dielectric breakdown into filamentary paths involving ionic motion, as in chalcogenide glasses [7] , [12] , manganite Mott insulators [13] or band insulator memristors such as TiO x (ref. 14 ). Zener tunnelling and avalanche breakdown can also be eliminated, because they do not exhibit metastability. The pulsed I–V curve below switching threshold is also qualitatively different from the observed behaviour for sliding and de-pinning of a CDW [6] , [15] , [16] . On the other hand, well above I T , the negative DR behaviour gradually crosses over to a small positive DR with increasing temperature, which is reminiscent of dissipationless CDW sliding conductivity (Frölich conductivity) observed in blue bronze at similar current densities [17] . In previous optical switching experiments, electrons (e) and holes (h) are homogeneously photoexcited and spatially overlapping [2] . The combination of narrow spectral width and unchanged spectral intensity of the collective amplitude mode in the photoexcited H state compared with the C state indicate that the two states are comparably homogeneous. Presently, the e and h are spatially separated at either electrode so the optical mechanism does not apply and we expect the structure to be less homogeneous. Scanning tunneling microscope (STM) images of the sample surface after current injection by an STM tip confirm this: they show an irregular meander of DWs rather than a regular array of DWs (see Supplementary Figs 4 and 5 ) [18] , [19] . In our experiments we find that different R LO with different relaxation properties can be reached with different pulse voltages near threshold (see Supplementary Note 3 for a few examples), indicating that different DW configurations can be created under different conditions. The free energy landscape of H states in 1T–TaS 2 clearly has multiple minima—consistent with the notion of an electronic glass. From a Landau theory viewpoint, charge cannot couple directly to the order parameter ψ of the CDW, but can couple through Lifshitz-invariant spatial gradient terms in the free energy , which are present at DWs. We propose that as the pulse of charges propagates through the sample, the energy of the system is lowered by charge carriers dynamically creating DWs. As the DWs are charged (with respect to the CDW background), they repel each other, creating a periodically textured stripe structure, an effect that is well known in correlated electron systems. The free energy F for such a system includes terms from the IC CDW formation and the DW repulsion, and has a number of minima, corresponding to states with different CDW and DW order ( Fig. 4b ) already in the absence of additional charge [20] . In the Supplementary Notes 4–5 and Supplementary Figs 6–7 , we consider the situation in detail, showing that on charge injection the C state is unstable towards the formation of a periodic DW array, IC with the underlying lattice structure. The origin of this instability lies in the negative curvature of the inverse compressibility k =d 2 F /d n 2 <0 arising from the shape of F ( n ), which in turn is a consequence of the Coulomb repulsion between DWs. The periodicity of the DWs is directly related to the injected charge density n i = n e − n h : the resulting structure is commensurate if n i =0 and IC otherwise, with a wavevector q H = n i / π corresponding to the minimum in the free energy ( Fig. 4b ). The DW structure in the current-injection-induced H state is thus similar to the NC state, but with a periodicity that depends on n i . Inhomogeneity of charge injection will naturally lead to spatially inhomogeneous DW structures observed in experiments. Nevertheless, the state conversion from polarons to band states means that it is not only the DWs, but the entire state that is metallic. Figure 4: An illustration of carrier trapping and DW formation after carrier injection. ( a ) The injected charges are trapped in the C structure causing the formation of a textured glassy electronic state. The charge density ρ is depleted at the leading edge of the pulse as it travels through the sample. ( b ) The free energy as a function of CDW wavevector based on the model of Nakanishi and Shiba [20] . q C and q I are the q -vectors of the undoped commensurate and IC states, respectively. The q -vector in the H state q H is determined by the density of carriers forming the DWs. ( c ) A depiction of the charge injection at the electrodes in the C state and ( d ) the band structure after the Mott-to-band state conversion process. Full size image Microscopically, a simple Mott–Hubbard picture [21] is inadequate for a description of the DW formation [18] . However, if we also take into account the effect of long-range Coulomb repulsion and consider pulsed carrier propagation through a commensurate polaronic crystal ( Fig. 4a ), we can qualitatively understand the origin of the textured state also from the microscopic viewpoint. The carriers launched at the electrodes first lose their excess energy e V = E e − E h =1 ∼ 10 eV ( Fig. 4c,d ) within <1 ps. Taking into account scattering and energy loss to phonons of these energetic carriers gives an avalanche multiplication factor . By the time the e and h reach the upper Hubbard band (UHB) and lower Hubbard band (LHB), respectively, the number of carriers injected by a 40-ps pulse is ∼ . Occupancy of the Hubbard bands is unstable, so as they propagate in the applied electric field, the holes annihilate with the localized electrons at the centre of each polaron, leading to the creation of voids in the polaron lattice, accompanied by a dynamic conversion of polaronic (Mott–Hubbard) states to conducting band states as the wave travels through the sample ( Fig. 4a ). (STM studies suggest that electrons are not expected to show the same effect as holes [18] .) From the STM images (see Supplementary Note 6 for details), the number of holes n h that can be accommodated in the DWs is very roughly between ∼ 1/4 and 1/3 of all the available polaron sites; thus, for our typical sample of volume 3 μm 3 we estimate n h ≈10 6 ∼ 10 7 . This is consistent with the number of injected charges in the pulse n i . However, rather than forming a homogeneous patch of voids at the electrode, as a result of the Coulomb interaction, a textured state forms ( Fig. 4a ). The time for such an DW state to form after melting is known to be τ form ∼ 1 ps (ref. 22 ), which is consistent with the observed speed of the switching process. The observed independence on pulse length can thus be understood: the DWs are formed by the carriers in the leading edge of the pulse. Above a certain threshold, the entire sample is converted and further added charges have no additional effect; thus, the trailing edge of the pulse has no further effect. Importantly, the unusual exponential I–V curves below threshold described by equation (1) can be understood within this picture in terms of carrier tunnelling through barriers resulting from the transient DW structure. We show in the SM that the tunnelling rate within such a state leading to equation (1) is R ∼ exp( V / V 0 ) β where β ∼ 1, where the barrier energy V 0 is related to the C state order parameter, giving rise to its T -dependence shown in Fig. 3b . The experiments introduce a paradigm of all-electronic switching between CDW states for data storage applications beyond current concepts [12] , [14] , [23] , [24] , [25] , [26] , [27] , [28] , [29] . Devices based on CDW state switching in 1T–TaS 2 could be scaled and integrated into commonly used cross-bar configurations [30] . The demonstrated write speed ∼ 30 ps is externally limited, but is an order of magnitude faster than the previous record for non-volatile memory [14] , [25] . The intrinsic speed appears to be <1 ps. With τ W =30 ps, the expended energy per bit is estimated to be pJ for a device with an area 2 × 2 μm 2 (shown in Supplementary Fig. 8 ), which is extremely small, comparable with current magnetic random access memory (MRAM) devices [31] . As both V T and R HI scale linearly with distance between contacts (see Supplementary Fig. 9 ), E B scales linearly with size along the contact direction and can be reduced by making the devices smaller. From a practical device viewpoint, it functions as a latch at low T . The reverse M – I transition is caused thermally, which requires that the device is heated momentarily above T c2 —which can be done on a sub-nanosecond timescale with appropriate design [25] . Moreover, in practical devices bulk Erase is used where entire blocks are erased simultaneously; thus, E speed is not a limiting device operation speed. The device operating temperature appears to be limited by T c2 ( Fig. 3a ). It is known that this can be increased significantly by Se substitution: 1T–TaS 1.5 Se 0.5 has a T c2 >300 K (ref. 32 ), introducing the possibility of room temperature operation, whereas the relaxation time τ R has been recently shown to be controllable by compressive strain [5] tuning the data retention time of the device.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Sample preparation 1T–TaS 2 crystals were grown by chemical transport method with iodine as a transport agent with the average lateral dimensions 1–5 mm and thicknesses ∼ 100 μm. Thin flakes were obtained from single crystal by exfoliating with sticky tape. The obtained 50- to 100-nm-thick flakes were deposited on sapphire substrate. Four gold electrodes were deposited on top of the flake by laser-direct lithography with initial 10-nm-thick Au/Pd layer. Electrical measurements DC transport measurements were performed using standard four-point technique. All switching experiments were performed starting from the C Mott phase by applying single electrical pulse. The pulses in the range 100 ns–1 s were produced by a standard pulse generator. For 40-ps full width at half maximum pulse we used metal–semiconductor–metal current source triggered by a 35-fs laser pulse.\"\n },\n {\n \"section\": \"Additional information\",\n \"content\": \"How to cite this article : Vaskivskyi, I. et al . Fast electronic resistance switching involving hidden charge density wave states. Nat. Commun. 7:11442 doi: 10.1038/ncomms11442 (2016).\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"The core research idea is to achieve ultrafast non-volatile resistance switching in 1T–TaS₂ through short pulsed current injection that creates domain walls via coupling to gradients of the charge density wave order parameter, enabling low-temperature ultrafast memory devices for energy-efficient superconducting computing."},"entities":{"kind":"list like","value":["ultrafast non-volatile resistance switching","1T–TaS₂","short pulsed current injection","domain walls","charge density wave order parameter","low-temperature ultrafast memory devices","energy-efficient superconducting computing"],"string":"[\n \"ultrafast non-volatile resistance switching\",\n \"1T–TaS₂\",\n \"short pulsed current injection\",\n \"domain walls\",\n \"charge density wave order parameter\",\n \"low-temperature ultrafast memory devices\",\n \"energy-efficient superconducting computing\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1","title":"Ichikawa, H. et al. Transient photoinduced ‘hidden’ phase in a manganite.Nat. Mater.10, 101–105 (2011).","link":"https://doi.org/10.1038%2Fnmat2929"},{"idx":"2","title":"Stojchevska, L. et al. Ultrafast switching to a stable hidden quantum state in an electronic crystal.Science344, 177–180 (2014).","link":"https://doi.org/10.1126%2Fscience.1241591"},{"idx":"4","title":"Tokura, Y. Photoinduced phase transition: a tool for generating a hidden state of matter.J. Phys. Soc. Jpn75, 011001 (2006).","link":"https://doi.org/10.1143%2FJPSJ.75.011001"},{"idx":"5","title":"Vaskivskyi, I. et al. Controlling the metal to insulator relaxation of the metastable hidden quantum state in 1T-TaS2 .Sci. Advances1, e1500168 (2015).","link":"https://doi.org/10.1126%2Fsciadv.1500168"},{"idx":"6","title":"Monceau, P. Electronic crystals: an experimental overview.Adv. Phys.61, 325–581 (2012).","link":"https://doi.org/10.1080%2F00018732.2012.719674"},{"idx":"7","title":"Hollander, M. J. et al. Electrically driven reversible insulator-metal phase transition in 1T-TaS2 .Nano Lett.15, 1861–1866 (2015).","link":"https://doi.org/10.1021%2Fnl504662b"},{"idx":"8","title":"Brazovskii, S. et al. Plastic sliding of charge density waves: X-ray space resolved-studies versus theory of current conversion.Phys. Rev. B61, 10640–10650 (2000).","link":"https://doi.org/10.1103%2FPhysRevB.61.10640"},{"idx":"9","title":"Holmes, D. S., Ripple, A. L. & Manheimer, M. A. Energy-efficient superconducting computing; power budgets and requirements.IEEE Trans. Appl. Supercond.23, 1701610–1701610 (2015).","link":"https://doi.org/10.1109%2FTASC.2013.2244634"},{"idx":"10","title":"Wilson, J., DiSalvo, F. & Mahajan, S. Charge-density waves and superlattices in metallic layered transition-metal dichalcogenides.Adv. Phys.24, 117–201 (1975).","link":"https://doi.org/10.1080%2F00018737500101391"},{"idx":"11","title":"Rossnagel, K. On the origin of charge-density waves in select layered transition-metal dichalcogenides.J. Phys. Condens. Matter23, 213001 (2011).","link":"https://doi.org/10.1088%2F0953-8984%2F23%2F21%2F213001"},{"idx":"12","title":"Cario, L., Corraze, B., Meerschaut, A. & Chauvet, O. Dielectric breakdown and current switching effect in the incommensurate layered compound (LaS)1.196VS2.Phys. Rev. B73, 155116 (2006).","link":"https://doi.org/10.1103%2FPhysRevB.73.155116"},{"idx":"13","title":"Asamitsu, A., Tomioka, Y., Kuwahara, H. & Tokura, Y. Current switching of resistive states in magnetoresistive manganites.Nature388, 50–52 (1997).","link":"https://doi.org/10.1038%2F40363"},{"idx":"14","title":"Yang, J. J., Strukov, D. B. & Stewart, D. R. Memristive devices for computing.Nat. Nanotechnol.8, 13–24 (2013).","link":"https://doi.org/10.1038%2Fnnano.2012.240"},{"idx":"15","title":"Lee, P. A. & Rice, T. M. Electric field depinning of charge density waves.Phys. Rev. B19, 3970–3980 (1979).","link":"https://doi.org/10.1103%2FPhysRevB.19.3970"},{"idx":"16","title":"Gruner, G., Zawadowski, A. & Chaikin, P. M. Nonlinear conductivity and noise due to charge-density-wave depinning in Nb Se3.Phys. Rev. Lett.46, 511–515 (1981).","link":"https://doi.org/10.1103%2FPhysRevLett.46.511"},{"idx":"17","title":"Mihaly, G. & Beauchene, P. Sliding charge density waves without damping: possible Fröhlich superconductivity in blue bronze.Solid State Commun.63, 911–914 (1987).","link":"https://doi.org/10.1016%2F0038-1098%2887%2990338-3"},{"idx":"18","title":"Cho, D. et al. Nanoscale manipulation of the Mott insulating state coupled to charge order in 1T-TaS2 .Nat Commun.7, 10453 (2016).","link":"https://doi.org/10.1038%2Fncomms10453"},{"idx":"19","title":"Ma, L. et al. A metallic mosaic phase and the origin of Mott-insulating state in 1T-TaS2 .Nat Commun.7, 10956 (2016).","link":"https://doi.org/10.1038%2Fncomms10956"},{"idx":"20","title":"Nakanishi, K. & Shiba, H. Domain-like incommensurate charge-density-wave states and the first-order incommensurate-commensurate transitions in layered tantalum dichalcogenides. I. 1T-polytype.J. Phys. Soc. Jpn43, 1839 (1977).","link":"https://doi.org/10.1143%2FJPSJ.43.1839"},{"idx":"21","title":"Fazekas, P. & Tosatti, E. Charge carrier localization in pure and doped 1T-TaS2 .Phys. B & C99, 183–187 (1980).","link":"https://doi.org/10.1016%2F0378-4363%2880%2990229-6"},{"idx":"22","title":"Eichberger, M. et al. Snapshots of cooperative atomic motions in the optical suppression of charge density waves.Nature468, 799–802 (2010).","link":"https://doi.org/10.1038%2Fnature09539"},{"idx":"23","title":"Yoshida, M., Suzuki, R., Zhang, Y., Nakano, M. & Iwasa, Y. Memristive phase switching in two-dimensional 1T-TaS2 crystals.Science Adv.1, e1500606 (2015).","link":"https://doi.org/10.1126%2Fsciadv.1500606"},{"idx":"24","title":"Kent, A. D. & Worledge, D. C. A new spin on magnetic memories.Nat. Nanotechnol.10, 187–191 (2015).","link":"https://doi.org/10.1038%2Fnnano.2015.24"},{"idx":"25","title":"Loke, D. et al. Breaking the speed limits of phase-change memory.Science336, 1566–1569 (2012).","link":"https://doi.org/10.1126%2Fscience.1221561"},{"idx":"26","title":"Sawa, A. Resistive switching in transition metal oxides.Mater. Today11, 28–36 (2008).","link":"https://doi.org/10.1016%2FS1369-7021%2808%2970119-6"},{"idx":"27","title":"Waser, R. & Aono, M. Nanoionics-based resistive switching memories.Nat. Mater.6, 833–840 (2007).","link":"https://doi.org/10.1038%2Fnmat2023"},{"idx":"28","title":"Welnic, W. Unravelling the interplay of local structure and physical properties in phase-change materials.Nat. Mater.5, 56–62 (2005).","link":"https://doi.org/10.1038%2Fnmat1539"},{"idx":"29","title":"Yoshida, M. et al. Controlling charge-density-wave states in nano-thick crystals of 1T-TaS2 .Sci. Rep.4, 7302–7307 (2014).","link":"https://doi.org/10.1038%2Fsrep07302"},{"idx":"30","title":"Wong, H. S. P. et al. Phase change memory.Proc. IEEE98, 2201–2227 (2010).","link":"https://doi.org/10.1109%2FJPROC.2010.2070050"},{"idx":"31","title":"Wong, H. S. P. & Salahuddin, S. Memory leads the way to better computing.Nat. Nanotechnol.10, 191–194 (2015).","link":"https://doi.org/10.1038%2Fnnano.2015.29"},{"idx":"32","title":"DiSalvo, F., Wilson, J., Bagley, B. & Waszczak, J. Effects of doping on charge-density waves in layer compounds.Phys. Rev. B12, 2220–2235 (1975).","link":"https://doi.org/10.1103%2FPhysRevB.12.2220"}],"string":"[\n {\n \"idx\": \"1\",\n \"title\": \"Ichikawa, H. et al. Transient photoinduced ‘hidden’ phase in a manganite.Nat. Mater.10, 101–105 (2011).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat2929\"\n },\n {\n \"idx\": \"2\",\n \"title\": \"Stojchevska, L. et al. Ultrafast switching to a stable hidden quantum state in an electronic crystal.Science344, 177–180 (2014).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1241591\"\n },\n {\n \"idx\": \"4\",\n \"title\": \"Tokura, Y. Photoinduced phase transition: a tool for generating a hidden state of matter.J. Phys. Soc. Jpn75, 011001 (2006).\",\n \"link\": \"https://doi.org/10.1143%2FJPSJ.75.011001\"\n },\n {\n \"idx\": \"5\",\n \"title\": \"Vaskivskyi, I. et al. Controlling the metal to insulator relaxation of the metastable hidden quantum state in 1T-TaS2 .Sci. Advances1, e1500168 (2015).\",\n \"link\": \"https://doi.org/10.1126%2Fsciadv.1500168\"\n },\n {\n \"idx\": \"6\",\n \"title\": \"Monceau, P. Electronic crystals: an experimental overview.Adv. Phys.61, 325–581 (2012).\",\n \"link\": \"https://doi.org/10.1080%2F00018732.2012.719674\"\n },\n {\n \"idx\": \"7\",\n \"title\": \"Hollander, M. J. et al. Electrically driven reversible insulator-metal phase transition in 1T-TaS2 .Nano Lett.15, 1861–1866 (2015).\",\n \"link\": \"https://doi.org/10.1021%2Fnl504662b\"\n },\n {\n \"idx\": \"8\",\n \"title\": \"Brazovskii, S. et al. Plastic sliding of charge density waves: X-ray space resolved-studies versus theory of current conversion.Phys. Rev. B61, 10640–10650 (2000).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.61.10640\"\n },\n {\n \"idx\": \"9\",\n \"title\": \"Holmes, D. S., Ripple, A. L. & Manheimer, M. A. Energy-efficient superconducting computing; power budgets and requirements.IEEE Trans. Appl. Supercond.23, 1701610–1701610 (2015).\",\n \"link\": \"https://doi.org/10.1109%2FTASC.2013.2244634\"\n },\n {\n \"idx\": \"10\",\n \"title\": \"Wilson, J., DiSalvo, F. & Mahajan, S. Charge-density waves and superlattices in metallic layered transition-metal dichalcogenides.Adv. Phys.24, 117–201 (1975).\",\n \"link\": \"https://doi.org/10.1080%2F00018737500101391\"\n },\n {\n \"idx\": \"11\",\n \"title\": \"Rossnagel, K. On the origin of charge-density waves in select layered transition-metal dichalcogenides.J. Phys. Condens. Matter23, 213001 (2011).\",\n \"link\": \"https://doi.org/10.1088%2F0953-8984%2F23%2F21%2F213001\"\n },\n {\n \"idx\": \"12\",\n \"title\": \"Cario, L., Corraze, B., Meerschaut, A. & Chauvet, O. Dielectric breakdown and current switching effect in the incommensurate layered compound (LaS)1.196VS2.Phys. Rev. B73, 155116 (2006).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.73.155116\"\n },\n {\n \"idx\": \"13\",\n \"title\": \"Asamitsu, A., Tomioka, Y., Kuwahara, H. & Tokura, Y. Current switching of resistive states in magnetoresistive manganites.Nature388, 50–52 (1997).\",\n \"link\": \"https://doi.org/10.1038%2F40363\"\n },\n {\n \"idx\": \"14\",\n \"title\": \"Yang, J. J., Strukov, D. B. & Stewart, D. R. Memristive devices for computing.Nat. Nanotechnol.8, 13–24 (2013).\",\n \"link\": \"https://doi.org/10.1038%2Fnnano.2012.240\"\n },\n {\n \"idx\": \"15\",\n \"title\": \"Lee, P. A. & Rice, T. M. Electric field depinning of charge density waves.Phys. Rev. B19, 3970–3980 (1979).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.19.3970\"\n },\n {\n \"idx\": \"16\",\n \"title\": \"Gruner, G., Zawadowski, A. & Chaikin, P. M. Nonlinear conductivity and noise due to charge-density-wave depinning in Nb Se3.Phys. Rev. Lett.46, 511–515 (1981).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.46.511\"\n },\n {\n \"idx\": \"17\",\n \"title\": \"Mihaly, G. & Beauchene, P. Sliding charge density waves without damping: possible Fröhlich superconductivity in blue bronze.Solid State Commun.63, 911–914 (1987).\",\n \"link\": \"https://doi.org/10.1016%2F0038-1098%2887%2990338-3\"\n },\n {\n \"idx\": \"18\",\n \"title\": \"Cho, D. et al. Nanoscale manipulation of the Mott insulating state coupled to charge order in 1T-TaS2 .Nat Commun.7, 10453 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms10453\"\n },\n {\n \"idx\": \"19\",\n \"title\": \"Ma, L. et al. A metallic mosaic phase and the origin of Mott-insulating state in 1T-TaS2 .Nat Commun.7, 10956 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms10956\"\n },\n {\n \"idx\": \"20\",\n \"title\": \"Nakanishi, K. & Shiba, H. Domain-like incommensurate charge-density-wave states and the first-order incommensurate-commensurate transitions in layered tantalum dichalcogenides. I. 1T-polytype.J. Phys. Soc. Jpn43, 1839 (1977).\",\n \"link\": \"https://doi.org/10.1143%2FJPSJ.43.1839\"\n },\n {\n \"idx\": \"21\",\n \"title\": \"Fazekas, P. & Tosatti, E. Charge carrier localization in pure and doped 1T-TaS2 .Phys. B & C99, 183–187 (1980).\",\n \"link\": \"https://doi.org/10.1016%2F0378-4363%2880%2990229-6\"\n },\n {\n \"idx\": \"22\",\n \"title\": \"Eichberger, M. et al. Snapshots of cooperative atomic motions in the optical suppression of charge density waves.Nature468, 799–802 (2010).\",\n \"link\": \"https://doi.org/10.1038%2Fnature09539\"\n },\n {\n \"idx\": \"23\",\n \"title\": \"Yoshida, M., Suzuki, R., Zhang, Y., Nakano, M. & Iwasa, Y. Memristive phase switching in two-dimensional 1T-TaS2 crystals.Science Adv.1, e1500606 (2015).\",\n \"link\": \"https://doi.org/10.1126%2Fsciadv.1500606\"\n },\n {\n \"idx\": \"24\",\n \"title\": \"Kent, A. D. & Worledge, D. C. A new spin on magnetic memories.Nat. Nanotechnol.10, 187–191 (2015).\",\n \"link\": \"https://doi.org/10.1038%2Fnnano.2015.24\"\n },\n {\n \"idx\": \"25\",\n \"title\": \"Loke, D. et al. Breaking the speed limits of phase-change memory.Science336, 1566–1569 (2012).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1221561\"\n },\n {\n \"idx\": \"26\",\n \"title\": \"Sawa, A. Resistive switching in transition metal oxides.Mater. Today11, 28–36 (2008).\",\n \"link\": \"https://doi.org/10.1016%2FS1369-7021%2808%2970119-6\"\n },\n {\n \"idx\": \"27\",\n \"title\": \"Waser, R. & Aono, M. Nanoionics-based resistive switching memories.Nat. Mater.6, 833–840 (2007).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat2023\"\n },\n {\n \"idx\": \"28\",\n \"title\": \"Welnic, W. Unravelling the interplay of local structure and physical properties in phase-change materials.Nat. Mater.5, 56–62 (2005).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat1539\"\n },\n {\n \"idx\": \"29\",\n \"title\": \"Yoshida, M. et al. Controlling charge-density-wave states in nano-thick crystals of 1T-TaS2 .Sci. Rep.4, 7302–7307 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fsrep07302\"\n },\n {\n \"idx\": \"30\",\n \"title\": \"Wong, H. S. P. et al. Phase change memory.Proc. IEEE98, 2201–2227 (2010).\",\n \"link\": \"https://doi.org/10.1109%2FJPROC.2010.2070050\"\n },\n {\n \"idx\": \"31\",\n \"title\": \"Wong, H. S. P. & Salahuddin, S. Memory leads the way to better computing.Nat. Nanotechnol.10, 191–194 (2015).\",\n \"link\": \"https://doi.org/10.1038%2Fnnano.2015.29\"\n },\n {\n \"idx\": \"32\",\n \"title\": \"DiSalvo, F., Wilson, J., Bagley, B. & Waszczak, J. Effects of doping on charge-density waves in layer compounds.Phys. Rev. B12, 2220–2235 (1975).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.12.2220\"\n }\n]","isConversational":false}}},{"rowIdx":87,"cells":{"unique_id":{"kind":"string","value":"s41467-023-41285-8"},"subfield":{"kind":"string","value":"Nanoscience and technology"},"title":{"kind":"string","value":"A magneto-activated nanoscale cytometry platform for molecular profiling of small extracellular vesicles"},"abstract":{"kind":"string","value":"Exosomal PD-L1 (exoPD-L1) has recently received significant attention as a biomarker predicting immunotherapeutic responses involving the PD1/PD-L1 pathway. However, current technologies for exosomal analysis rely primarily on bulk measurements that do not consider the heterogeneity found within exosomal subpopulations. Here, we present a nanoscale cytometry platform NanoEPIC, enabling phenotypic sorting and exoPD-L1 profiling from blood plasma. We highlight the efficacy of NanoEPIC in monitoring anti-PD-1 immunotherapy through the interrogation of exoPD-L1. NanoEPIC generates signature exoPD-L1 patterns in responders and non-responders. In mice treated with PD1-targeted immunotherapy, exoPD-L1 is correlated with tumor growth, PD-L1 burden in tumors, and the immune suppression of CD8+ tumor-infiltrating lymphocytes. Small extracellular vesicles (sEVs) with different PD-L1 expression levels display distinctive inhibitory effects on CD8 + T cells. NanoEPIC offers robust, high-throughput profiling of exosomal markers, enabling sEV subpopulation analysis. This platform holds the potential for enhanced cancer screening, personalized treatment, and therapeutic response monitoring."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"The discovery of the immune checkpoint PD-L1 has revolutionized cancer therapy. Interventions disrupting the PD-L1/PD-1 axis have improved clinical outcomes in numerous cancers including kidney, lung, breast, colon, and melanoma. However, due to the heterogeneous nature of tumors and the complexity of immunoregulatory mechanisms, only a subset of patients respond to immunotherapy [1] , [2] . Therefore, diagnostic tools that can stratify patients and enable the early classification of responders and non-responders would allow for earlier decision-making and maximize the success of cancer treatments [3] . While cell surface PD-L1 (cPD-L1) is one of the most validated predictive biomarkers for immunotherapy, it has only been shown to be predictive in less than 30% of patients [3] , [4] . Recent studies have shown that small extracellular vesicles (sEVs) can present PD-L1 on their surfaces and act as improved informative biomarkers compared to cPD-L1 [5] , [6] , [7] , [8] . sEVs are a subset of extracellular vesicles (EV) which are lipid-bilayer enclosed structures secreted by cells, and exosomes are the most studied sEVs. With a typical size range of 30–160 nm, sEVs can carry molecular constituents such as proteins, metabolites, lipids, and nucleic acids from parent cells and act as communication cargos in the tumor microenvironment [9] , [10] . Due to their high stability and high abundance in bodily fluids such as blood, urine, and saliva, sEVs are promising cancer biomarkers [10] , [11] , [12] , [13] . Exosomal PD-L1 (exoPD-L1) in particular has been shown to be a robust indicator of tumor progression and predictor of immunotherapeutic response [5] , [6] , [7] , [8] , [14] , [15] , [16] , [17] , [18] , [19] . Despite the importance of analyzing exosomal markers, there are many technical limitations in this area due to the small size and heterogeneity of tumor-derived sEVs [20] , [21] , [22] . Recent studies indicate that PD-L1 expression among EVs can be heterogeneous and that these EVs show PD-L1-dependent inhibition of T cell activation [23] . Conventional approaches for isolating sEVs such as ultracentrifugation, filtration, and polymer precipitation are time-consuming and yield low recovery and purity [24] , [25] . Additionally, standard methods for sEV analysis, such as Western blot and enzyme-linked immunosorbent assays (ELISA), are limited by their relatively low sensitivity and large sample requirements [26] . Several studies have attempted exosomal protein screening using flow cytometry [27] , [28] . In fact, with optimized high-end cytometers, PD-L1 positive EVs have been detected in clinical samples and shown relevance to immunotherapy outcome [18] , [19] . However, the detection of EVs with a size smaller than 100 nm is still a challenge and the throughput needs to be improved. More recent studies have applied novel nanomaterials [29] , [30] or microfluidic systems to detect sEVs and profile their protein contents [31] , [32] , [33] , [34] , [35] , [36] , [37] , [38] . These efforts have led to several platforms that are capable of multiparametric analysis of sEVs in clinical specimens, such as nanoplasmonic sensors [39] , [40] , electrochemical sensors [41] , and immunomagnetic approaches [42] . Despite their excellent analytical performance and broad diagnostic utilities, these methods also lack the throughput needed to handle large volumes of plasma samples. Although our understanding of the translational potential of sEVs is quickly advancing, our ability to further characterize their biological roles in diseases and exploit them as clinical biomarkers is greatly hindered without techniques to efficiently separate sEVs based on their heterogeneous marker expression. In recent years, our group has developed a number of immunoaffinity-based microfluidic technologies for the rapid, high-throughput, and inexpensive targeting of cellular biomarkers to address the heterogeneous nature of multiple diseases [43] , [44] , [45] , [46] , [47] , [48] . Surface marker profiling (e.g., EpCAM) of circulating tumor cells (CTCs) have shown clinical relevance with cancer progression [49] , [50] . We also found that cell subpopulations with different surface markers showed distinctive behaviors. For example, tumor-infiltrating lymphocytes (TILs) with medium CD39 expression are more potent in killing cancer cells compared with TILs with high or low CD39 expression [51] . We therefore hypothesized that biomarker-based profiling of sEVs could predict therapeutic outcomes. Herein, we describe a microfluidic approach for the molecular profiling of sEVs through nanoscale exosomal protein-based sorting using immunomagnetic-activated cytometry (NanoEPIC). The NanoEPIC platform can conduct high-throughput and high-resolution sorting of individual sEVs based on surface marker expression. Furthermore, NanoEPIC performs single-step processing of sEVs directly from cell-free biological fluids such as plasma or cell culture medium. Using an mouse model, we validate the reliability of NanoEPIC for predicting immunotherapeutic responses and demonstrate the strong associations between exoPD-L1 profiling with tumor PD-L1 burden and T cell suppression. Additionally, sEVs can be efficiently recovered for downstream analysis, enabling the testing of suppression of CD8+ T cells. The activation of CD8+ T cells are inhibited differently after treatment with sEVs displaying varied levels of PD-L1. Overall, this approach not only allows for the phenotypic profiling of exoPD-L1 which is a potential strategy to monitor therapeutic responses but also enables the isolation of heterogeneous sEVs for further downstream investigations, offering capabilities for sEV-focused research.","references":{"indices":[],"infos":[]}},{"section":"Results","content":"Development of the NanoEPIC platform The NanoEPIC platform is designed to enable one-step sorting and molecular profiling of sEV that are pre-labeled with antibody-functionalized magnetic nanoparticles (MNPs) (Fig. 1 ). The working principle of the NanoEPIC system relies on the differential deflection of sEVs based on their varying biomarker expression (Fig. 1a ). sEVs with higher expression of the target biomarker (e.g., PD-L1) bind to more MNPs and subsequently gain higher magnetic susceptibility. Separation of phenotypically distinct vesicles is facilitated through the directed movement of sEVs along ferromagnetic guides located below the microfluidic flow channels (Fig. 2a and Supplementary Fig. 1a ). The magnetic guides were designed to branch out laterally from the inlet to outlet to generate a magnetic field gradient when magnetized by an external magnetic field, allowing for the separation of particles with varying magnetic susceptibility (Fig. 2b, c and Supplementary Fig. 1b–d ). Made from the ultra-high permeability metal alloy Metglas 2714 A, the magnetic guides were precisely tuned to ensure an optimal balance between the Stokes’ drag force \\(({F}_{d})\\) , produced by the fluid flow, and the magnetic force ( \\({F}_{m}\\) ), which acts on the MNP-labeled sEVs (see Supplementary Text). sEVs with higher MNP loading typically experience more lateral deflection toward the edges of the device. Based on the final lateral displacements, it is possible to collect sEVs bearing four different expression levels of the target biomarker: negative, low (exo-L), medium (exo-M), and high (exo-H). To establish a uniform laminar flow and minimize non-specific deflections, samples were flow-focused with an additional buffer stream (Supplementary Fig. 2a ). Equipped with a customized 3D-printed magnetic stage, the NanoEPIC system allows for the simultaneous processing of 6 samples (Supplementary Fig. 2b ). As this approach permits marker-based profiling of sEVs, the NanoEPIC can be easily applied for the screening of exoPD-L1 to predict immunotherapeutic outcomes (Fig. 1b ). Additionally, the ability of NanoEPIC to perform high-resolution separation of individual sEVs provides opportunities for downstream analyses of specific sEV subpopulations (Fig. 1c ). To optimize the performance of the NanoEPIC platform for exoPD-L1 sorting, we evaluated three cell lines with distinct levels of PD-L1 expression: PC9, H460, and H1975. Through flow cytometry and PD-L1 ELISA, we confirmed both the cPD-L1 and exoPD-L1 levels to be lowest in PC9 and highest in H1975 (Supplementary Fig. 3a, b ). The magnetic loading on sEVs from different cell lines was inspected through transmission electron microscopy (TEM) and was found proportional to exoPD-L1 expression (Fig. 2d and Supplementary Fig. 4 ). Since H1975 sEVs displayed the broadest magnetic loading, we used H1975 as a model for optimizing the NanoEPIC platform for exoPD-L1 profiling. To alleviate the interference of larger EVs such as microvesicles (MVs) and apoptotic bodies (APOs) with larger sizes, samples were centrifuged at 10,000 g and then filtered with 0.22 µm filters during sample preparation. Therefore, while processing samples in NanoEPIC, only sEVs were introduced to the device. Also, larger vesicles require more PD-L1 to achieve magnetic deflection as the Stoke’s force is bigger. Therefore, the NanoEPIC is optimal for sEV profiling. Fig. 1: Overview of the NanoEPIC Platform. The NanoEPIC system relies on magnetic deflection to achieve phenotypic profiling and nanoscale sorting of sEVs. a Plasma extracted from whole blood was first treated with anti-PD-L1 conjugated MNPs. The MNPs-treated plasma sample is directly processed through NanoEPIC for exoPD-L1 profiling. The NanoEPIC device consists of a sample inlet and a buffer inlet. The sample is focused to the middle of the microchannel by the buffer during processing. Magnetic guides are embedded under the microchannel for exosomal sorting. Magnetically labeled sEVs experience a strong magnetic force in the direction normal to the magnetic guides due to the intensified magnetic gradient found at the edges of the guides. If the magnetic force is large enough to balance the Stokes’ drag force in the direction normal to the magnetic guide, the magnetically labeled sEVs will divert from the initial flow path and follow the magnetic guide. Unlabeled particles are focused on the negative outlet. Deflected sEVs are collected in the low/medium/high outlets based on PD-L1 expression. b ExoPD-L1 is used as a predictive marker for cancer immunotherapeutic outcomes through exoPD-L1 profiling. To showcase the utility of the platform for monitoring the therapeutic response to immunotherapy, mouse models under anti-PD-1 therapy were tracked for exoPD-L1 profiling using the NanoEPIC platform. The exoPD-L1 profiles were assessed in light of treatment outcomes. c Sorted sEVs can be used for downstream analysis. As illustrated, sEVs from different outlets were used to interact with cytotoxic T cells. Treatment with different sEV subpopulations resulted in variations in the level of T cell inhibition. Full size image Fig. 2: Design of the NanoEPIC device. a NanoEPIC includes a glass substrate with magnetic guides patterned under a fluidic channel and a PDMS cover. A permanent magnet is positioned below the NanoEPIC to facilitate magnetic deflection. b Visualization of magnetic guides through (top) (scale bar = 100 µm) and the corresponding simulation of the magnetization of magnetic guides under an external magnetic field (bottom). c Simulated magnetic gradient across the magnetic guide. d The number of anti-PD-L1 conjugated MNPs bound to sEVs from different cell lines. (upper) Representative figures of sEVs bound to anti-PD-L1 conjugated MNPs obtained by TEM. Scale bar is 100 nm. In general, the number of MNPs bound to sEVs is proportional to exoPD-L1 expression. The average number of MNPs bound to sEVs collected from H1975 cells is the highest among the three tested cell lines. Furthermore, H1975 sEVs displayed the broadest distribution of the number of MNPs bound to each sEV. e Deflection efficiency of anti-PD-L1 MNP bonded sEVs in NanoEPIC with different minimum deflection angles. Smaller deflection angles led to higher deflection efficiency. ( p = 0.0479; p = 0.0003; p < 0.0001). f Deflection efficiency of anti-PD-L1 MNP bonded sEVs in NanoEPIC with different channel heights. Smaller channel height gives higher deflection efficiency. ( p = 0.0003; p < 0.0001). g Deflection efficiency of sEVs bound with anti-CD9 MNPs. ( p = 0.0363; p = 0.0150) h Specificity of the NanoEPIC assessed through the difference in deflection efficiency between H1975 wildtype (WT) sEVs and H1975 PD-L1 knockout (KO) sEVs, which were sorted based on exoPD-L1. Isotype antibodies were used to further confirm the specificity of magnetic labeling of sEVs based on exoPD-L1. ( p < 0.0001) i , ExoPD-L1 profile distribution i n sEVs from H1975 WT cells and PD-L1 KO H1975 cells. ExoPD-L1 profile of sEVs treated with isotype antibody-MNPs is also given. n = 5 biologically independent samples for e-i . All bar plots represent mean ± s.d. * P < 0.05, ** P < 0.01, *** P < 0.001, **** P < 0.0001, one-sided unpaired t-test. Source data are provided as a Source Data file. Full size image There are two main components in the NanoEPIC device that are crucial for its performance: (1) the deflection angle of magnetic guides, and (2) flow channel height. The deflection angles allow for tuning the equilibrium between the fluid drag force and magnetic force which consequently dictates the degree of separation between the sorted sEV subpopulations. To determine the angular thresholds of our magnetic guides, we modeled 4, 10, and 20 as the approximate number of MNPs conjugated to sEVs in the low, medium, and high outlets respectively (Supplementary Fig. 5a ). Through experimentation, we observed near-saturation deflection efficiency at a minimum deflection angle of 3° (Fig. 2e ) which represents an acceptable starting point for exoPD-L1 segregation. To sort sEVs with distinct phenotypic profiles, we selected 3° for exo-L, 5° for exo-M, and 10° for exo-H subpopulations. As for the channel height, increased height allows for higher throughput but can also diminish the magnetic influence of the magnetic guides. Given that the magnetic guides are positioned below the device, sEVs migrate toward the bottom of the device to fully experience their designed magnetic trajectories. Both simulations and experimental outcomes suggest 30 µm to be the optimal height for exoPD-L1 sorting (Fig. 2f and Supplementary Fig. 5b–f ). It is worth noting that an overly strong magnetic gradient might result in trapping the MNP-labeled sEVs at the bottom of the microchannel. This problem was addressed by adjusting the distance between the bottom of the channel and the magnetic guides (Supplementary Fig. 6a ). Next, we optimized the sample flow rate to maximize the capture efficiency. Instead of anti-PD-L1, we used anti-CD9 as a sorting marker to determine the maximal sorting efficiency of sEVs. We demonstrated that a flow rate of 200 µL/h has led to the highest deflection efficiency (86.6 ± 2.9%) (Fig. 2g ). To gauge the specificity of the NanoEPIC device, we generated mutagenic PD-L1 knockouts of H1975 (PD-L1 KO) (Supplementary Fig. 6b ) and compared the exoPD-L1 profile of PD-L1 KO and wildtype H1975 (WT). As depicted in Fig. 2h , the NanoEPIC platform maintained high specificity as indicated by the low deflection efficiency of PD-L1 KO (5.4 ± 1.9%) and isotype antibody control (8.6 ± 3.4%) compared to 64.2 ± 2.2% obtained using WT sEVs. Additionally, we found that most WT sEVs were collected in the exo-H outlet, whereas most of the PD-L1 KO sEVs or sEVs treated with isotype antibody-linked MNPs were collected in the exo-L outlet (Fig. 2i ), which suggests that unlabeled sEVs might have escaped to the exo-L outlet with minimal nonspecific deflections into the exo-M and exo-H outlets. ExoPD-L1 profiling of tumor-derived sEVs with NanoEPIC Having established the high sensitivity and specificity of NanoEPIC for sorting sEVs, we further assessed the performance of exoPD-L1 profiling using the NanoEPIC platform. First, we assessed exoPD-L1 profiling in H1975 sEVs at different flow rates. While flow rates of 200 µL/h and 300 µL/h displayed nonsignificant differences in deflection efficiency (Fig. 3a ), the distributions of sEVs between the deflected outlets were much more pronounced (Fig. 3b ). More importantly, increasing the flow rate not only decreased the deflection efficiency but also shifted the distribution of sEVs towards the low-expression outlet which corroborated our simulations (Supplementary Fig. 7 ). Next, we utilized NanoEPIC for profiling exoPD-L1 in other cancer cell lines. The deflection efficiency and exoPD-L1 profiles were significantly different among all three cell lines which agreed with our previous findings (Supplementary Fig. 8a, b ). Since the expression of exoPD-L1 has previously been shown to be upregulated after treatment with IFN-γ [6] , [52] , we also compared the exoPD-L1 profiles between IFN-γ treated and untreated cells (Supplementary Fig. 8c ). Notably, the differences in exoPD-L1 profiles between IFN-γ treated and untreated cell lines were also detectable using the NanoEPIC platform (Fig. 3c–f ), which highlights the high-resolution profiling efficiency of the NanoEPIC platform. Fig. 3: Performance of the NanoEPIC platform. a Deflection of anti-PD-L1 MNPs labeled sEVs from H1975 at different flow rates. ( p = 0.0651; p < 0.0001; p = 0.0031). b ExoPD-L1 profiling of sEVs from H1975 at different flow rates. c Deflection efficiency of sEVs from different cell lines (H1975, H460, and PC9). The deflection efficiency of sEVs from corresponding cells treated by IFNγ is shown. ( p = 0.0713; p = 0.0250; p = .0029). d ExoPD-L1 profiling of sEVs collected from WT PC9 cells and IFNγ treated PC9 cells. e ExoPD-L1 profiling of sEVs collected from WT H460 cells and IFN-γ treated H460 cells. f ExoPD-L1 profiling of sEVs collected from WT H1975 cells and IFN-γ treated H1975 cells. g TEM images of sEV subpopulations collected from Negative, Low, Medium, and High outlets of the NanoEPIC. The scale bar is 50 nm. h Deflection efficiency of anti-PDL1 MNPs bounded sEVs from H1975 cells after treatment with antibody elution buffer compared to control (i.e., sEVs treated with PBS). i ExoPD-L1 profiling of H1975 sEVs after treatment with elution buffer. PBS-treated sEVs were mainly collected in the exo-H outlet, whereas elution buffer-treated sEVs were mainly collected in the exo-L and exo-M outlets. j exoPD-L1 expression of H1975 sEV subpopulations collected from low/medium/high outlets, measured by PD-L1 ELISA. ( p < 0.0001; p < 0.0001; p = 0.0004). k Western blot (WB) analysis of different sEV subpopulations compared to cell lysate (CL). n = 5 biologically independent samples for a , b . n = 3 biologically independent samples for c–f . n = 4 biologically independent samples for h – j . All samples were harvested from H1975 cells. WB assay was independently repeated 3 times for k . All bar plots represent mean ± sd. * P < 0.05, ** P < 0.01, *** P < 0.001, **** P < 0.0001, one-sided unpaired t-test. Source data are provided as a Source Data file. Full size image We next determined if the sorted sEVs could be recovered for subsequent downstream analysis. TEM imaging confirmed the number of MNPs bound to sEVs is correlated to their designated outlets (Fig. 3g ); however, to enable the utility of sorted sEVs, the MNPs must be stripped off. Unlike cells, sEVs lack innate processes that can degrade, internalize, or detach MNPs. Using antibody elution buffer, we were able to remove most MNPs while maintaining the integrity of PD-L1 proteins on sEVs (Supplementary Fig. 9a–c ). MNP release was validated through exoPD-L1 profiling using NanoEPIC (Fig. 3h, i ). The eluted MNPs can be separated from sEVs through sucrose-gradient ultracentrifugation (Supplementary Fig. 9d, e ). Using the purified sEVs, we further confirmed the difference in PD-L1 expression in sEVs collected from each of the outlets using PD-L1 ELISA and western blotting (Fig. 3j, k ). In vivo analysis of exoPD-L1 for PD-1 immunotherapy To examine the potential clinical utility of profiling exoPD-L1, we established a PD-1 immunotherapy mouse model through the injection of MC38 cells into C57BL/6 mice followed by a semiweekly administration of anti-PD-1 antibodies (Fig. 4a ). We assessed whether exoPD-L1 profiling can be used to differentiate between the heterogeneous responses to anti-PD-1 immunotherapy (Supplementary Fig. 10 ). At the endpoint of the study (23 days post-inoculation), terminal blood collection was performed and the plasma was separated. The plasma was then used for exosomal analysis using the NanoEPIC platform. The status of complete responders, partial responders, and non-responders were determined through tumor volumetric measurements at day 23. Figure 4b represents a summary of the exoPD-L1 profiles from anti-PD1 treated mice (Supplementary Fig. 11a ). The NanoEPIC analysis revealed a lower proportion of exo-L and a considerably higher proportion of exo-H sEVs found in non-responders compared to responders (Supplementary Fig. 11b ) which suggests that elevated exoPD-L1 expression led to greater immunotherapeutic resistance. It was also observed that without any immunotherapeutic pressure, exoPD-L1 profiles were unpredictable and inconclusive (Supplementary Fig. 11c, d ). We then investigated whether plasma levels of sEVs are correlated with outcome. The concentration of sEVs from all the mice were measured using NTA. No significant differences were observed between complete responders, partial responders and non-responders (Supplementary Fig. 10d ). To refine the quantitation of exoPD-L1 expression measured with the NanoEPIC platform, we generated a numerical protein expression index called the NanoEPIC score. This score reports the overall exoPD-L1 expression while incorporating the deflection efficiency and exoPD-L1 profiling from the NanoEPIC’s high-throughput and high-resolution nanoscale sorting of heterogenous sEVs (see supplementary text for the calculation of the NanoEPIC score). Using this approach, we observed a direct correlation between tumor volume and NanoEPIC score (Fig. 4c ), indicating that the NanoEPIC platform could potentially be used to monitor and predict tumor growth. Fig. 4: Analysis of exoPD-L1 in PD-1 immunotherapy mouse model. a Workflow of sEV and tumor analysis in PD-1 immunotherapeutic mouse model. Briefly, mice injected with MC38 cells were treated with either anti-PD-1 antibodies or saline (control) twice a week. 23 days post-inoculation, mice were sacrificed for plasma and tumor collection. Collected plasma was subjected to exoPD-L1 analysis using NanoEPIC while solid tumors underwent PD-L1 sorting or TIL analysis through flow cytometry. b Distribution of sEVs sorted in the low, medium, and high outlets of the NanoEPIC chip are represented as a heat map and arranged in ascending order with respect to tumor volume. c Scatterplot of tumor volume against NanoEPIC score ( n = 3). An exponential plateau model was used for the regression analysis which resulted in the following fit: y = 2213 – (2213 + 1945)e -0.008393x . d Distribution of tumor cells sorted through each outlet of the prismatic deflection chip with PD-L1 as the sorting marker. Data arranged in ascending order with respect to NanoEPIC score. sEVs from PD-L1 sorted MC38 cells were collected and processed through the NanoEPIC device and the deflection efficiency ( p = 0.0085) e along with the exoPD-L1 profiles for the low ( p = 0.0106) f , medium g , and high ( p < 0.0001) h of cPD-L1 are shown. n = 3 biologically independent samples for e – h . All data represent mean ± s.d. * P < 0.05, ** P < 0.01, *** P < 0.001, **** P < 0.0001, one-way ANOVA. Source data are provided as a Source Data file. Full size image Tumor cPD-L1 levels have traditionally been used for the indication of PD-L1-associated tumor immune evasion; however, the acquisition of PD-L1+ cells from solid tumor tissues is highly invasive. On the other hand, cPD-L1-based liquid biopsy which relies on the analysis of circulating tumor cells (CTCs) can be challenging due to the scarcity of CTCs in early cancer stages. Since sEVs are highly abundant in the plasma and simple to obtain, we were interested in determining whether the levels of plasma exoPD-L1 would reflect the levels of cPD-L1 in tumors. Using our previously established prismatic deflection cell-sorting device (PRISM) [45] , we profiled dissociated tumor cells from our C57BL/6 mice based on cPD-L1 expression (Fig. 4d and Supplementary Fig. 12a ). Based on our findings, the distribution of exoPD-L1 closely reflected the distribution of PD-L1+ cells (Supplementary Fig. 12b–d ), suggesting that exoPD-L1 expression is positively correlated with cPD-L1 expression. To further ascertain the relationship between exoPD-L1 and cPD-L1, we evaluated an MC38 colorectal cancer model. For this investigation, we first segregated MC38 cells using the PRISM device into three subpopulations of varying cPD-L1 expression, which were promptly validated through flow cytometry (Supplementary Fig. 12e ). We then collected sEVs from each of the sorted MC38 subpopulations and profiled their exoPD-L1 using the NanoEPIC device (Fig. 4e–h ). The total fraction of sEVs containing PD-L1 was found to increase with higher cPD-L1 expression in MC38 cells (Fig. 4e ). Upon examining the exoPD-L1 profiles, a similar trend was seen with a larger proportion of exo-L found in low cPD-L1 expressing MC38 and a higher proportion of exo-H in high cPD-L1 (Fig. 4f–h ). Overall, NanoEPIC is a compelling prognostic platform for PD-1 immunotherapy as exoPD-L1 is highly abundant, easily obtainable, and correlates with PD-L1 burden in tumors to a similar degree as traditional prognostic markers such as tumor cPD-L1. We next explored the relationship between exoPD-L1 and their immune effectors, particularly with CD8+ tumor-infiltrating lymphocytes (TILs). Flow cytometric analysis of tumors from partial responder, non-responder, and control (no anti-PD-1 immunotherapy) mice revealed a uniform decrease in CD45 + CD8+ TILs with higher NanoEPIC score (Fig. 5a and Supplementary Fig. 13a ). It was also found that TILs with higher NanoEPIC score had reduced proliferation, T cell activation/differentiation, cytokine secretion and T cell cytotoxicity as demonstrated by the decrease in expression of ki67, TCF7, CD69, CD137, IFN-γ, and granzyme B (GzB) (Fig. 5b–g and Supplementary Fig. 13b–g ). These findings are consistent with previous investigations, indicating a decline in TIL activity is strongly associated with higher exoPD-L1 levels [6] , [7] , [23] . While we expectedly detected a reduction in PD-1 expression with higher exoPD-L1, we also observed a striking reduction in CD152 (CTLA-4) expression (Fig. 5h, i and Supplementary Fig. 13h, i ). The CD152 pathway is involved in the inhibition of T cell activation which typically occurs in lymph nodes [53] , suggesting that sEVs can elicit longer range immune-suppression beyond the tumor site. In summary, our results demonstrated the strong potential of using exoPD-L1 profiling for predicting T cell response to immunotherapy, further supporting NanoEPIC as a robust prognostic platform. Fig. 5: Determination of NanoEPIC scores of TILs from murine PD-1 immunotherapeutic model. a Proportion of CD45 + CD8 + T cells arranged with increasing NanoEPIC scores from left to right. Flow cytometric analysis of proliferation markers ki67 b , T cell differentiation and activation markers (TCF7, CD69, and CD137) c – e , cytotoxicity (IFNγ, Granzyme B) f , g and immune-checkpoint markers (PD-1 and CD152) h , i shown as scatterplots of median fluorescence intensities and NanoEPIC score. All scatterplots were fitted with linear regression analysis. Source data are provided as a Source Data file. Full size image Interaction of T cells and sorted sEV subpopulations Having demonstrated the varying extent of TIL inactivation in vivo relative to exoPD-L1 levels, we then explored whether NanoEPIC-sorted sEVs would exhibit distinct biological activities. The prevailing model for PD-L1-mediated immunosuppression relies on the interaction between PD-L1 with PD-1 generally expressed on CD8 cytotoxic T cells. To determine the direct link between exoPD-L1 and T cell suppression, we used an in vitro model in which isolated CD8 + T cells were treated with PD-L1-sorted sEVs from the NanoEPIC device. First, we examined the relative binding of sEVs onto CD8 + T cells using SEM (Supplementary Fig. 14a ). As seen in Fig. 6a , there were significantly more exo-H sEVs bound to CD8 T cells when compared with exo-L sEVs. To confirm this observation, we performed immunofluorescent staining of sEV-bound T cells using CD9 as an indicator of sEVs (Fig. 6b and Supplementary Fig. 14b ). Through measurement of CD9 fluorescent intensities, it was confirmed that there were statistically significant differences between the binding affinity of exo-L, exo-M, and exo-H sEVs onto T cells, with the lowest binding affinity exhibited by exo-L and the highest binding affinity in exo-H (Supplementary Fig. 14c ). These results suggest exoPD-L1 levels are positively correlated to the binding affinity of sEVs to CD8 T cells. Fig. 6: Characterization of T cell properties after treatment with sorted PD-L1 + sEVs. CD8 + T cells bound to different sEV subpopulations were imaged through SEM a and confocal microscopy b . Scale bar represents 1 µm for SEM and 4 µm for confocal microscopy. Proportion of T cells positive for granzyme B c ( p = 0.0030; p = 0.0058), ki67 d ( p = 0.0101; p = 0.0350), and CD69 e ( p = 0.0044; p = 0.0119) was determined through flow cytometry after treatment with sEVs from exo-L, exo-M, exo-H, unsorted (WT), activated CD8 + T cells with no exosomal treatment (Pos), and exosomes acquired from PD-L1 knockouts (PD-L1-KO). Secretion of IFN-γ f ( p = 0.0002; p = 0.0140) and IL-2 g ( p < 0.0001; p < 0.0001) by T cells after treatment of different sEV subpopulations was determined using ELISA. h Schematic of CD8 + T cell cytotoxicity assay after treatment with varying exoPD-L1 subpopulations of sEVs. i T cell cytotoxicity assay from CD8 + T cells cocultured with either H1975 (left) ( p = 0.1872; p = 0.0418; p < 0.0001) or MC38 (right) ( p = 0.015; p = 0.0710; p = 0.0024) after treatments from exo-L, exo-M, exo-H, or no sEVs (control). n = 4 biological independent samples for c – g . n = 5 for i . All bar plots represent mean ± s.d. * P < 0.05, ** P < 0.01, *** P < 0.001, **** P < 0.0001, one-way ANOVA. Source data are provided as a Source Data file. Full size image Next, we sought to investigate the phenotypic variations of T-cells after sEV treatment. Pre-activated CD8 + T cells were treated with sEVs from each outlet of the NanoEPIC device (exo-L, exo-M, exo-H) separately. Compared with the positive control (Pos) where pre-activated CD8 T cells were not treated with sEVs, a decrease in activation, proliferation, and cytotoxicity of T cells was observed through flow cytometry after treatment with exo-L, exo-M, and exo-H as indicated by the decreasing proportions of CD69, ki67, and GzB positive CD8 T-cells (Fig. 6c–e and Supplementary Fig. 15 ). Exo-H caused stronger inhibition of T cells compared with Exo-L. As comparison, treatment with unsorted sEVs (WT) was consistently found to be within the margins of exo-L to exo-H for all analyses. Through ELISA, it was shown that the cytokines, namely IFN-γ and IL-2 were similarly reduced in T cells when treated with sEVs with higher exoPD-L1 expression (Fig. 6f, g ). For the cytokine secretion assay, fresh purified inactivated CD8 T cells were used as a negative control (Neg). To further evaluate the effects of exoPD-L1 on the cytotoxic activity of CD8 T cells, the cells treated with distinct sEV subpopulations were co-cultured with tumor cell lines (Fig. 6h ). Through LDH assay, we observed a significant reduction in CD8 T cell cytotoxicity when treated with sEVs with higher exoPD-L1 levels (Fig. 6i ). Overall, our findings demonstrate that the NanoEPIC is capable of sorting sEVs with biologically distinct activities and allow for the separation of unique sEV subpopulations. Validation of NanoEPIC platform with clinical specimens To establish that the NanoEPIC platform could be used with clinical specimens, we analyzed a small set of samples from patients undergoing anti-PD-1/anti-PD-L1 immunotherapy. Plasma samples were gathered from 10 de-identified cancer patients (multiple visits) and 5 healthy donors to determine whether exoPD-L1 levels could be measured (Supplementary Table 4 ). Comparing patient samples to healthy donor samples, we observed a significant elevation in exoPD-L1 levels (Fig. 7a ). We also analyzed changes in NanoEPIC scores before and after treatment. Notably, responders exhibited a consistent decrease in exoPD-L1 expressions (Fold change<1), while non-responders showed a general increase (Fig. 7b and Supplementary Fig. 17 ). Additionally, we observed a positive correlation between changes in NanoEPIC scores and tumor size (Fig. 7c ). These findings indicate that NanoEPIC could be applied in the future to predict immunotherapy responses and aid in treatment selection. Moreover, the ability of the NanoEPIC platform to distinguish between healthy and clinical samples highlights its potential diagnostic capabilities. Fig. 7: Assessment of clinical specimens with the NanoEPIC Platform. To evaluate the performance of the NanoEPIC platform on specimens collected from patients receiving anti PD-1/PD-L1-based immunotherapy, a total of 10 de-identified plasma samples were processed using the NanoEPIC device for exoPD-L1 profiling. a ExoPD-L1 profiling of patient samples before immunotherapy was compared with that of healthy donors. ( n = 10) ( p = 0.0002). b Changes in NanoEPIC scores were compared between responders and non-responders to immunotherapy. The fold change in NanoEPIC score represents the ratio of NanoEPIC score after immunotherapy over NanoEPIC score before immunotherapy. A fold change of 1 signifies no change in NanoEPIC score. The correlation between fold change in NanoEPIC score and c tumor size obtained through CT scans. n = 3 technical replicates for a – c . All bar plots represent mean ± s.d. *** P < 0.001, one-sided unpaired t-test. Source data are provided as a Source Data file. Full size image"},{"section":"Discussion","content":"Here, we have presented a platform that utilizes magnetic-activated ranking for nanoscale cytometry of sEVs. We conceptually validated the feasibility of magnetic labeling of sEVs with MNPs and demonstrated nanoscale deflection of individual sEVs. The optimized NanoEPIC can perform phenotypic profiling with high efficiency, specificity, and throughput. As a microfluidics-based platform, the NanoEPIC is advantageous as it is simple to fabricate, low-cost, and user-friendly. The NanoEPIC platform can be adapted to a variety of applications. Since the magnetic deflection of sEVs relies on the binding of antibody-decorated MNPs, NanoEPIC permits sorting sEVs based on the level of any surface marker. The NanoEPIC’s capability to segregate sEVs based on surface marker expression allows for precise analysis of specific sEV subpopulations that were previously not feasible. While sEVs have been used to examine the performance of NanoEPIC, this nanoscale cytometry system can be widely adapted for the sorting and phenotypic profiling of other submicron/nanoscale particles such as certain bacteria or subcellular organelles. To our knowledge, few other technologies can perform high-resolution molecular sorting and profiling at that throughput. Overall, we demonstrated how NanoEPIC could potentially be used as a cancer diagnostic tool. The NanoEPIC was used in mouse models to monitor the response to anti-PD1 immunotherapy and produce signature exoPD-L1 profiles that can be used to distinguish between responders and non-responders to immunotherapy. Our the numerical NanoEPIC scores not only reflected tumor progression but also demonstrated a high correlation with the suppression of TILs. The downstream analysis showed that different sEV subpopulations with varying PD-L1 expression exhibited different inhibitory behaviors on T cells. Given that different exosomal biomarkers such as CTLA4, TIM3, CD47 are being studied for the prediction of immunotherapy response, the NanoEPIC can be used to profile panels of exosomal biomarkers and ultimately be applied for further clinical investigations to guide clinical decisions and improve therapeutic outcomes. In addition to the diagnostic utility of NanoEPIC, therapeutic sEVs are emerging as promising drug delivery agents due to their excellent biocompatibility, low immunogenicity, persistence in the blood, and ability to cross various biological barriers [54] . Several reports have demonstrated the utility of sEVs as drug delivery vehicles for drug-resistant lung cancer [55] , [56] , [57] . Hence, the ability to isolate distinct subpopulations of sEVs with favorable phenotypes and superior delivery properties would offer many opportunities for the treatment of lung cancer [54] . This necessitates high-throughput methods capable of accurately sorting sEVs from large volumes of patients’ blood, which suggests a future utility for the platform. Given that drug loading to sEVs can also be heterogeneous [58] , the NanoEPIC system can be used to sort out sEV subpopulations with high drug loading, which can potentially reduce the dose of sEVs required for effective treatment. Another application of NanoEPIC can be sorting of cargo-loaded sEVs based on their specificity to recipient cells. sEVs engineered with antibodies specific to recipient cells have been used for cargo delivery [59] . Sorting of high marker expressed sEVs can increase the uptake of cargo-loaded sEVs by the recipient cells. Also, we have shown that the NanoEPIC system was able to sort magnetically labeled concentrated sEVs (up to 10 12 particles/mL) with high deflection efficiency (more than 80%) at a flow rate of 200 µL/h. The NanoEPIC was amenable for simultaneous processing of up to 6 microfluidic chips, which further increases its total throughput. While the NanoEPIC exhibits merits for biomarker profiling and sorting, as a magneto-activated system, it does lack the flexibility of simultaneously detecting multiple markers like FACS. More work must be done to multiplex biomarker profiling and sorting. One potential way to overcome this problem is through serial sorting with multiple magnetic labeling steps [59] , [60] . Another way is to use MNPs with distinctive magnetic properties such as size and magnetic susceptibility [61] , [62] , [63] . Continued efforts to assess novel magnetic sorting methodologies for facilitating multiplex analyses on the NanoEPIC platform hold tremendous promise in expanding the scope of extracellular vesicle investigations. Such advancements have the potential to uncover broader clinical implications on a larger scale."},{"section":"Methods","content":"Ethical statement This research complies with all relevant ethical regulations and the related protocols were approved by the University of Toronto Research Ethics Board (REB). All animal experiments were operated following the protocols approved by the University of Toronto Animal Care Committee. Human plasma samples were obtained from PrecisionMed LLC upon the approval by the Western Institutional Review Board® (Puyallup, WA, USA) and after informed consents were signed. Whole blood samples were collected from 10 patients under anti-PD1/anti-PD-L1 immunotherapy ( n = 8 female donors and n = 2 male donors aged 41–76) (Fig. 7 , Supplementary Fig 17 and Supplementary Table 4 ). Plasma samples were extracted from whole blood immediately after blood collection and stored at −80 °C. In all cases, participants or legal representatives were informed in detail about the research purpose and informed consent was signed. Fabrication of NanoEPIC The design of NanoEPIC was sketched using a CAD software (AutoCAD). Chrome masks containing the designed patterns were printed using a mask maker (Heidelberg µPG 501). The fabrication of NanoEPIC includes three main procedures: (i) patterning of magnetic guides, (ii) generation of microchannels, and (iii) binding of the devices (Supplementary Fig. 16 ). For patterning the magnetic guides, a sheet of Metglas 2714 A was glued onto borosilicate glass wafers (452, UniversityWafer) with a thin spun-coated layer of epoxy (7370A38, McMaster-Carr) and left to dry overnight. Excessive epoxies were removed with acetone and isopropanol (IPA). The magnetic guides were photolithographically patterned onto a layer of 2 µm thick S1811 photoresist that was spun-coated onto Metglas. After developing the S1811 with MF-CD-26, Metglas was wet-etched using an etchant consisting of 3.6% HCl and 14.3% H 2 O 2 in water. For the generation of microchannels, the S1811 photoresist was stripped with AZ300T (Integrated Micromaterials) and coated with Omnicoat (Kayaku Advanced Materials) to improve SU-8 photoresist adhesion. SU-8 3035 was used to develop a 40µm-thick SU-8 layer to encapsulate the magnetic guides and a second 30 µm-thick SU-8 layer was developed on top to generate the microchannels. For device binding, PDMS cover pieces that were hole punched at the inlet and outlet sites were first treated with plasma for 1 min, then treated with 10% APTES in water for 30 min. The pieces were then rinsed with deionized water and bonded onto the SU-8 channels. The bonded chips were incubated at 70 °C overnight with vertical pressure to enhance the binding strength. Cell culture Human NSCLC cell lines H1975 (CRL-5908, ATCC), H460 (HTB-177, ATCC), and PC9 (90071810, Sigma) were cultured in RPMI medium (350-007-CL, Wisent) while mouse colon cancer cell line MC38 (ENH204-FP, Kerafast) were cultured in DMEM (D5796, Sigma). All culture media were supplemented with 10% FBS (Wisent), 1% penicillin-streptomycin (Sigma), and incubated at 37 °C with 5% CO 2 . For the stimulation of cells with IFN-γ, cells were incubated in medium supplemented with 100 ng/mL of either human or mouse IFN-γ (Peprotech) for 48 h before the collection of cell and sEV samples. Human CD8 T cells were isolated from human blood mononuclear cells (70025, Stemcell Technologies) using Human CD8 + T cell isolation kit (130-096-495, Miltenyi Biotec) while mice T cells were collected from the spleen of C57BL/6 mice (Jackson Laboratory) and isolated with mouse CD8 + T cell isolation kit (130-096-543, Miltenyi Biotec). Human T cells were cultured in T cell expansion medium (10981, STEMCELL Technologies) with 10 ng/mL of human IL-2 (78036.3, STEMCELL Technologies). Mouse T cells were cultured in RPMI medium supplemented with 10% FBS and 10 ng/mL of mouse IL-2 (130-120-334, Miltenyi Biotec). T cells were activated with Human CD3/CD28/CD2 activation kit (10990, STEMCELL Technologies) at 25 µL/mL or Mouse CD3/CD28 beads (11456D, Thermofisher) at 25 µL/mL at every second passage. Genetic modifications To generate the PD-L1 knockout (KO) H1975 cell lines, H1975 wildtype (WT) cells were first infected with lenti-Cas9 (52962LV, Addgene) followed by blasticidin selection to establish Cas9 expressing cell lines (H1975-cas9). The H1975-cas9 cells were then simultaneously infected with two different guide RNA targeting PD-L1 (target sequence: GGTTCCCAAGGACCTATATG and CGCTGCATGATCAGCTATGG) that were previously packaged into lentivirus by Thermofisher (A32042). Monoclonal expansion was performed following puromycin selection and the knockout clones were identified with flow cytometry (cytoFLEX S, Beckman Coulter) for surface PD-L1. For the modification of MC38 cell lines, gRNA used to knockout PD-L1 (target sequences: GTTTACTATCACGGCTCCAA and GGGGAGAGCCTCGCTGCCAA) were purchased from Integrated DNA Technologies and lentiviral plasmids were obtained from Genscript (pLentiCRISPRv2 backbone) respectively. Lentiviral packaging was performed through co-transfection of HEK293T cells with psPAX2, pMD2.G, and the transfer plasmid. The medium was changed after 24 h and then collected at 24 h intervals. MC38 cells were infected with the lentivirus and selected with 10 µg/mL of puromycin. Fluorescent-activated cell sorting (BD FACSAria IIIu cell sorter) was performed to select cells negative for PD-L1 from the knockout population. Flow cytometry was used to confirm the stable maintenance of PD-L1 expression levels. Purification of sEVs For sEVs collected from cultured cells, the base culture medium was replaced with medium supplemented with 10% sEV-depleted FBS (Gibco) when cells reached ~70% confluency. Cells were then cultured for 48 hours then the culture medium was collected for sEV isolation. The culture medium was centrifuged at 2000 g for 30 minutes to remove dead cells and cellular debris followed by another centrifugation at 10,000 g for 1 h to remove large microvesicles. The supernatant were filtered with 0.22 µm filter and then mixed with an sEV precipitation kit (4478359, Thermofisher) at a 2:1 volume ratio, incubated overnight, and then centrifuged at 10,000 g for 1 h to pellet the sEVs. SEVs were resuspended with 1% bovine serum albumin (BSA, Sigma) with protease inhibitor (11836170001, Roche) and phosphatase inhibitor (4906845001, Roche) in PBS. All buffers were filtered with 0.22 µm filters (SLGV033R, Sigma) prior to mixing with sEVs. For blood samples, plasma was extracted by collecting the supernatant after 2-step centrifugation at 2000 g for 20 minutes followed by 10,000 g for 20 min. Samples were directly processed through the NanoEPIC device as discussed later. Magnetic labeling of sEVs 10 nm diameter protein-G cross-linked iron oxide magnetic nanoparticles at 1 mg/mL (IPG-10-01) were purchased from Ocean NanoTech and mixed with either human anti-CD9 antibody (10626D, Thermofisher), human anti-PD-L1 (14-5983-82, Thermofisher), or mouse anti-PD-L1 (14-5982-82, Thermofisher) in a 15:1 volume ratio and chilled overnight at 4 °C. The antibody-bead conjugates were then pelleted through centrifugation at 100,000 g for 45 minutes and then resuspended in 0.1% BSA to reach a final concentration of 0.5 mg/mL and stored at 4 °C until later use. For the labeling of sEVs, 10 9 sEVs in 100 µL of 1% BSA were mixed with 5 µL of the functionalized magnetic nanoparticles (MNPs) and agitated overnight at 4 °C prior to processing through the NanoEPIC device. To ensure the maximum interaction between EVs and MNPs and maximized magnetic labeling efficiency, an excessive amount of functionalized MNPs were added to the sEV sample. For the preparation of sEV-spiked plasma samples, 100 µL of sEVs in 1% BSA were mixed with the same volume of plasma and treated with 10 µL of anti-PD-L1 conjugated MNPs at 4 °C overnight. As a control, 100 µL of 1% BSA was mixed with the same volume of plasma and treated with 10 µL of anti-PD-L1 conjugated MNPs at 4 °C overnight. For mouse plasma samples, 100 µL of 2-fold diluted plasma was treated with 5 µL of mouse anti-PD-L1 conjugated MNPs at 4 °C overnight. Processing sEV samples through NanoEPIC Before chip processing, all buffers were filtered through a 0.22 µm syringe filter and degassed under a vacuum for 15 minutes. NanoEPIC chips were first treated with 0.1% Pluronic F-68 (Thermofisher) in deionized water for 15 min then washed with running buffer (1% BSA in PBS) to minimize non-specific adhesion. The device was then positioned on 4 laterally stacked N52-Neodymium magnets (BY042-N52, K&J Magnetics) for all further processing. 100 µL of magnetically labeled sEV samples were mixed in a 1:1 volume ratio with running buffer and loaded into the NanoEPIC device. Fluidic processing was performed through a syringe pump (Fusion 100, Chemyx) that was set to withdraw at a flow rate of 200 µL/h. Subsequent to sample processing, the devices were washed with PBS at a flow rate of 400 µL/h. Characterization of sEVs The size distribution and concentration of sEVs were determined through nanoparticle tracking analysis (NTA) with NanoSight NS300 (Malvern). Transmission electron microscopy (TEM) was performed for the visualization of sEVs binding to beads. Samples for TEM were fixed at a 1:1 volume ratio of 4% paraformaldehyde (PFA, F8775-4X25ML, Sigma) diluted in PBS. Specimen preparations were then performed by the Nanoscale Biomedical Imaging Facility (NBIF) in Peter Gilgan Centre for Research and Learning at The Hospital for Sick Children (Toronto, CA). Briefly, the sample was dropped on a glow-discharged formvar/carbon-supported copper grid. After staining with 2% uranyl acetate, the grids were air-dried and visualized using a Hitachi HT7800 transmission electron microscope. Separation of sEVs from antibody-functionalized MNPs Two elution buffers were tested, including glycine-HCl (Sigma) and Pierce IgG elution buffer (Thermofisher). For glycine-HCl, 200 mM and 300 mM were tested. SEVs collected from the outlets of the NanoEPIC device were treated with an equal volume of elution buffer and immediately incubated at 37 °C for 30 min. The sEV samples were then separated from eluted MNPs through sucrose gradient centrifugation (0.5 M and 1.5 M sucrose concentration) running at 120,000 g for 1.5 h. sEVs were collected from the interface between the 0.5 M and 1.5 M sucrose while the released MNPs were pelleted at the bottom of the tube. To remove the excessive sucrose, the collected sEVs were washed with 5 mL of PBS and pelleted through centrifugation at 100,000 g for 2 h. Western blot analysis Cell lysates were prepared by adding RIPA buffer (PI89900, Thermofisher) supplemented with protease inhibitor and phosphatase inhibitor to cell samples followed by a 14,000 g centrifugation for 15 minutes to collect the supernatant. sEV lysates were prepared by adding an equal volume of RIPA buffer and incubated at room temperature for 30 min. Lysates of different sEV subpopulations were normalized based on their total protein weight measured by BCA protein assay (Thermofisher). Lysates were separated using a 4–15% SDS-PAGE and transferred onto a nitrocellulose membrane. Blots were blocked with 3% BSA in Tris-buffered saline with Tween 20 (TBST) and incubated with primary antibodies at 4 °C overnight at the recommended supplier concentrations. For secondary staining, HRP-conjugated antibodies were added and left at room temperature for 1 h. The blots were developed with ECL substrate (Pierce). Information on the primary and secondary antibodies can be found in Supplementary Table 2 . ELISA All lysate samples were prepared as described above. The quantifications of protein through ELISA were performed by following the protocols from the supplier. Information on the ELISA kits can be found in Supplementary Table 3 . Flow cytometry Flow cytometric analysis was used to determine the protein expression in cells. Samples were typically fixed with 4% PFA followed by 20 minutes of blocking in 2% BSA. After primary antibody staining, samples were then washed and processed through cytoFLEX S flow cytometer (Beckman Coulter). For intracellular staining, samples were permeabilized with 0.2% Triton-X-100 (Sigma) after fixation. Information on the antibodies can be found in Supplementary Table 2 . The collected data were analyzed using FlowJo. Immunotherapeutic mouse model All animal experiments were operated in accordance with the protocols approved by the University of Toronto Animal Care Committee, with no tumors above the maximal limit of 2000 mm 3 and tumor size did not exceed 20 mm in any directions. For our syngeneic mouse model, female C57BL/6 mice (6–8 weeks old) purchased from Jackson Laboratory were subcutaneously injected with 6 × 10 6 MC38 cells in 100 µL of PBS. Starting one week after inoculation, 200 µg of anti-PD-1 antibodies (ICH1132UL, Ichor. Bio) were administered intraperitoneally twice a week (control mice were injected with saline). Tumors were measured twice a week with a digital caliper and volume was calculated using the following formula: (width) 2 × length/2. Mice were euthanized when the tumors exceeded 20 mm in one dimension or 23 days after inoculation. Blood was immediately collected through cardiac puncture and tumors and spleen were surgically removed. Plasma was isolated from blood samples for exosomal sorting through the NanoEPIC device. For mouse CD8+ T cell isolation, spleens were homogenized with a cell scraper (83.3950, Sarstedt), filtered through a 70 µm mesh cell strainer (352350, Falcon), and treated with RBC lysis buffer (00-4333-57, Thermofisher) before magnetic separation. Tumor cells were dissociated with Tumor Dissociation Kit (130-096-730) from Miltenyi Biotec and were either analyzed for tumor-infiltrating lymphocytes (TILs) through flow cytometry or sorted for PD-L1 through our previously reported prismatic deflection chip (PRISM). For data analysis, mice were categorized into 3 groups based on their differential responses to immunotherapy: “complete responders” referred to mice that had a significant reduction in tumor volume, “partial responders” displayed no significant changes in tumor volume, and “non-responders” displayed a significant development in tumor volume from the initiation of immunotherapy to the endpoint of the study (23 days post inoculation) (Supplementary Fig. 10a, b ). T cell-sEV binding assay To facilitate the binding of sEVs to CD8 T cells, 100 µL of CD8+ T cells in fresh medium were dispensed into 96-well plates at a concentration of 2 × 10 6 cells/mL. Cells were activated for 24 h with the addition of CD3/CD28/CD2 for human lines or CD3/CD28 beads for mice lines as mentioned earlier. Inactivated CD8+ T cells used as controls were supplemented with PBS instead of the activation cocktail. Preactivated CD8+ T cells were treated with 10 µL of sEV subpopulations (1 mg/mL) for 48 h at 37 °C and 5% CO 2 . After sEV treatment, CD8+ T cells under different conditions were harvested and used for flow cytometry to measure T cell activation, proliferation, and cytotoxicity. T cell culture media were collected to measure cytokine secretion with ELISA. Visualization of sEV-bound T cells To visualize the binding of sEVs to T cells, scanning electron microscopy (SEM) was used. 100 µL of preactivated CD8+ T cells (10 6 cells/mL) were treated with 10 µL of sEVs (1 mg/mL) for 2 h at 37 °C. For SEM, we first immobilized the T cells onto gold electrodes functionalized with anti-CD8. Primary fixation was achieved through overnight incubation of samples in 0.1 M phosphate buffer containing 4% paraformaldehyde and 1% glutaraldehyde. Samples were then additionally fixed with 1% osmium tetroxide in 0.1 M phosphate buffer for 30 minutes. Samples were dehydrated using a gradually increasing ethanol series, followed by critical point drying. Dehydrated samples were then spin-coated in gold and imaged using the Prisma E scanning electron microscope. Confocal microscopy was additionally used to verify the relative binding of different sEV subpopulations to T cells. The T cell-sEV binding protocol is modified to improve the fluorescence intensity. 100 µL of sEVs (1 mg/mL) of each subpopulation were incubated with 5 µL of Anti-CD9-PE (Thermofisher) overnight at 4 °C. SEV samples were then washed with PBS at 100,000 g for 2 h. The pelleted sEVs were then resuspended in 100 µL of PBS. For cell staining, 100 µL of activated T cells (10 6 cells/mL) were incubated with 1 µL of DiD lipid dye (V22887, Thermofisher) for 15 minutes at 37 °C. Cells were washed 4 times with PBS and then blocked in 400 µL of 1% BSA for 15 min. For each condition, 10 µL of PE-stained sEVs were added to the 100 µL of T cell samples and incubated at 37 °C for 2 hours. Samples were then washed with 1 mL of PBS to remove unbound sEVs, then fixed with 4% PFA for 15 minutes. After another PBS wash, 1 drop of nuclear stain (R37606, Thermofisher) in 100 µL of PBS was added and samples were incubated for 10 minutes at room temperature. After a final wash with PBS, samples were then imaged using the Zeiss LSM 880 microscope. Cytotoxic CD8 T cell-mediated tumor killing assay To measure the effect of exoPD-L1 on T cell-induced tumor cytotoxicity, we co-cultured sEV-treated CD8+ T cells with tumor cells and performed LDH-cytotoxicity assay (ab197004, Abcam). Briefly, sEVs from H1975 WT and MC38 WT were sorted through the NanoEPIC device and subsequently used to treat T cells using the aforementioned protocols. PD-L1 KO H1975 and PD-L1 KO MC38 cells were seeded into 96-well plates at 5,000 cells/well and incubated at 37 °C for 24 h. For co-culture, 100 µL of sEV-treated T cells (1.5 × 105 cells/mL) were added to the tumor cells and cultured for 48 h. LDH assay was subsequently performed following the supplier’s protocol (Abcam). Clinical sample processing Plasma samples were obtained from de-identified cancer patients who were undergoing anti-PD-1 or anti-PD-L1 immunotherapy, procured from PrecisionMed LLC (Supplementary Table 4 ). These samples were voluntarily collected by PrecisionMed LLC after approval from Western Institutional Review Board® (WIRB®) (Puyallup, WA, USA) (Protocol No.: 6054 WIRB® Protocol #20161289). Samples were obtained from n = 8 females and n = 2 males aged 47–76. All plasma samples were pre-filtered with 0.22 µm filters. The concentration of sEV in the plasma was measured to be between 0.6 and 1.74 × 10 11 particles/mL. For sample processing, 50 µL of plasma sample was first diluted 4 times with 1% filtered BSA. The diluted sample was then treated with 2.5 µL of anti-PD-L1 conjugated MNPs and incubated at 4 °C overnight. The mixture was then processed through the NanoEPIC system at 200 µL/h. sEVs collected from different outlets were analyzed with NTA as mentioned above. Gender analysis was not performed due to the limited number of available samples. Statistical analysis and reproducibility All statistical analyses were performed with GraphPad Prism version 9.0.0. All graphical data are represented as mean ± s.d. unless stated otherwise. Each dot represents a biological replicate unless specified otherwise. Unpaired student t-test was used to compare two different groups while two-way ANOVA was used to compare multiple groups. P value < 0.05 was considered statistically significant. No statistical method was used to predetermine sample size. No data were excluded from the data analyses. The investigators were blinded to allocation of treatment during mice experiments but not blinded when assessing outcome. 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 discovery of the immune checkpoint PD-L1 has revolutionized cancer therapy. Interventions disrupting the PD-L1/PD-1 axis have improved clinical outcomes in numerous cancers including kidney, lung, breast, colon, and melanoma. However, due to the heterogeneous nature of tumors and the complexity of immunoregulatory mechanisms, only a subset of patients respond to immunotherapy [1] , [2] . Therefore, diagnostic tools that can stratify patients and enable the early classification of responders and non-responders would allow for earlier decision-making and maximize the success of cancer treatments [3] . While cell surface PD-L1 (cPD-L1) is one of the most validated predictive biomarkers for immunotherapy, it has only been shown to be predictive in less than 30% of patients [3] , [4] . Recent studies have shown that small extracellular vesicles (sEVs) can present PD-L1 on their surfaces and act as improved informative biomarkers compared to cPD-L1 [5] , [6] , [7] , [8] . sEVs are a subset of extracellular vesicles (EV) which are lipid-bilayer enclosed structures secreted by cells, and exosomes are the most studied sEVs. With a typical size range of 30–160 nm, sEVs can carry molecular constituents such as proteins, metabolites, lipids, and nucleic acids from parent cells and act as communication cargos in the tumor microenvironment [9] , [10] . Due to their high stability and high abundance in bodily fluids such as blood, urine, and saliva, sEVs are promising cancer biomarkers [10] , [11] , [12] , [13] . Exosomal PD-L1 (exoPD-L1) in particular has been shown to be a robust indicator of tumor progression and predictor of immunotherapeutic response [5] , [6] , [7] , [8] , [14] , [15] , [16] , [17] , [18] , [19] . Despite the importance of analyzing exosomal markers, there are many technical limitations in this area due to the small size and heterogeneity of tumor-derived sEVs [20] , [21] , [22] . Recent studies indicate that PD-L1 expression among EVs can be heterogeneous and that these EVs show PD-L1-dependent inhibition of T cell activation [23] . Conventional approaches for isolating sEVs such as ultracentrifugation, filtration, and polymer precipitation are time-consuming and yield low recovery and purity [24] , [25] . Additionally, standard methods for sEV analysis, such as Western blot and enzyme-linked immunosorbent assays (ELISA), are limited by their relatively low sensitivity and large sample requirements [26] . Several studies have attempted exosomal protein screening using flow cytometry [27] , [28] . In fact, with optimized high-end cytometers, PD-L1 positive EVs have been detected in clinical samples and shown relevance to immunotherapy outcome [18] , [19] . However, the detection of EVs with a size smaller than 100 nm is still a challenge and the throughput needs to be improved. More recent studies have applied novel nanomaterials [29] , [30] or microfluidic systems to detect sEVs and profile their protein contents [31] , [32] , [33] , [34] , [35] , [36] , [37] , [38] . These efforts have led to several platforms that are capable of multiparametric analysis of sEVs in clinical specimens, such as nanoplasmonic sensors [39] , [40] , electrochemical sensors [41] , and immunomagnetic approaches [42] . Despite their excellent analytical performance and broad diagnostic utilities, these methods also lack the throughput needed to handle large volumes of plasma samples. Although our understanding of the translational potential of sEVs is quickly advancing, our ability to further characterize their biological roles in diseases and exploit them as clinical biomarkers is greatly hindered without techniques to efficiently separate sEVs based on their heterogeneous marker expression. In recent years, our group has developed a number of immunoaffinity-based microfluidic technologies for the rapid, high-throughput, and inexpensive targeting of cellular biomarkers to address the heterogeneous nature of multiple diseases [43] , [44] , [45] , [46] , [47] , [48] . Surface marker profiling (e.g., EpCAM) of circulating tumor cells (CTCs) have shown clinical relevance with cancer progression [49] , [50] . We also found that cell subpopulations with different surface markers showed distinctive behaviors. For example, tumor-infiltrating lymphocytes (TILs) with medium CD39 expression are more potent in killing cancer cells compared with TILs with high or low CD39 expression [51] . We therefore hypothesized that biomarker-based profiling of sEVs could predict therapeutic outcomes. Herein, we describe a microfluidic approach for the molecular profiling of sEVs through nanoscale exosomal protein-based sorting using immunomagnetic-activated cytometry (NanoEPIC). The NanoEPIC platform can conduct high-throughput and high-resolution sorting of individual sEVs based on surface marker expression. Furthermore, NanoEPIC performs single-step processing of sEVs directly from cell-free biological fluids such as plasma or cell culture medium. Using an mouse model, we validate the reliability of NanoEPIC for predicting immunotherapeutic responses and demonstrate the strong associations between exoPD-L1 profiling with tumor PD-L1 burden and T cell suppression. Additionally, sEVs can be efficiently recovered for downstream analysis, enabling the testing of suppression of CD8+ T cells. The activation of CD8+ T cells are inhibited differently after treatment with sEVs displaying varied levels of PD-L1. Overall, this approach not only allows for the phenotypic profiling of exoPD-L1 which is a potential strategy to monitor therapeutic responses but also enables the isolation of heterogeneous sEVs for further downstream investigations, offering capabilities for sEV-focused research.\",\n \"references\": {\n \"indices\": [],\n \"infos\": []\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Development of the NanoEPIC platform The NanoEPIC platform is designed to enable one-step sorting and molecular profiling of sEV that are pre-labeled with antibody-functionalized magnetic nanoparticles (MNPs) (Fig. 1 ). The working principle of the NanoEPIC system relies on the differential deflection of sEVs based on their varying biomarker expression (Fig. 1a ). sEVs with higher expression of the target biomarker (e.g., PD-L1) bind to more MNPs and subsequently gain higher magnetic susceptibility. Separation of phenotypically distinct vesicles is facilitated through the directed movement of sEVs along ferromagnetic guides located below the microfluidic flow channels (Fig. 2a and Supplementary Fig. 1a ). The magnetic guides were designed to branch out laterally from the inlet to outlet to generate a magnetic field gradient when magnetized by an external magnetic field, allowing for the separation of particles with varying magnetic susceptibility (Fig. 2b, c and Supplementary Fig. 1b–d ). Made from the ultra-high permeability metal alloy Metglas 2714 A, the magnetic guides were precisely tuned to ensure an optimal balance between the Stokes’ drag force \\\\(({F}_{d})\\\\) , produced by the fluid flow, and the magnetic force ( \\\\({F}_{m}\\\\) ), which acts on the MNP-labeled sEVs (see Supplementary Text). sEVs with higher MNP loading typically experience more lateral deflection toward the edges of the device. Based on the final lateral displacements, it is possible to collect sEVs bearing four different expression levels of the target biomarker: negative, low (exo-L), medium (exo-M), and high (exo-H). To establish a uniform laminar flow and minimize non-specific deflections, samples were flow-focused with an additional buffer stream (Supplementary Fig. 2a ). Equipped with a customized 3D-printed magnetic stage, the NanoEPIC system allows for the simultaneous processing of 6 samples (Supplementary Fig. 2b ). As this approach permits marker-based profiling of sEVs, the NanoEPIC can be easily applied for the screening of exoPD-L1 to predict immunotherapeutic outcomes (Fig. 1b ). Additionally, the ability of NanoEPIC to perform high-resolution separation of individual sEVs provides opportunities for downstream analyses of specific sEV subpopulations (Fig. 1c ). To optimize the performance of the NanoEPIC platform for exoPD-L1 sorting, we evaluated three cell lines with distinct levels of PD-L1 expression: PC9, H460, and H1975. Through flow cytometry and PD-L1 ELISA, we confirmed both the cPD-L1 and exoPD-L1 levels to be lowest in PC9 and highest in H1975 (Supplementary Fig. 3a, b ). The magnetic loading on sEVs from different cell lines was inspected through transmission electron microscopy (TEM) and was found proportional to exoPD-L1 expression (Fig. 2d and Supplementary Fig. 4 ). Since H1975 sEVs displayed the broadest magnetic loading, we used H1975 as a model for optimizing the NanoEPIC platform for exoPD-L1 profiling. To alleviate the interference of larger EVs such as microvesicles (MVs) and apoptotic bodies (APOs) with larger sizes, samples were centrifuged at 10,000 g and then filtered with 0.22 µm filters during sample preparation. Therefore, while processing samples in NanoEPIC, only sEVs were introduced to the device. Also, larger vesicles require more PD-L1 to achieve magnetic deflection as the Stoke’s force is bigger. Therefore, the NanoEPIC is optimal for sEV profiling. Fig. 1: Overview of the NanoEPIC Platform. The NanoEPIC system relies on magnetic deflection to achieve phenotypic profiling and nanoscale sorting of sEVs. a Plasma extracted from whole blood was first treated with anti-PD-L1 conjugated MNPs. The MNPs-treated plasma sample is directly processed through NanoEPIC for exoPD-L1 profiling. The NanoEPIC device consists of a sample inlet and a buffer inlet. The sample is focused to the middle of the microchannel by the buffer during processing. Magnetic guides are embedded under the microchannel for exosomal sorting. Magnetically labeled sEVs experience a strong magnetic force in the direction normal to the magnetic guides due to the intensified magnetic gradient found at the edges of the guides. If the magnetic force is large enough to balance the Stokes’ drag force in the direction normal to the magnetic guide, the magnetically labeled sEVs will divert from the initial flow path and follow the magnetic guide. Unlabeled particles are focused on the negative outlet. Deflected sEVs are collected in the low/medium/high outlets based on PD-L1 expression. b ExoPD-L1 is used as a predictive marker for cancer immunotherapeutic outcomes through exoPD-L1 profiling. To showcase the utility of the platform for monitoring the therapeutic response to immunotherapy, mouse models under anti-PD-1 therapy were tracked for exoPD-L1 profiling using the NanoEPIC platform. The exoPD-L1 profiles were assessed in light of treatment outcomes. c Sorted sEVs can be used for downstream analysis. As illustrated, sEVs from different outlets were used to interact with cytotoxic T cells. Treatment with different sEV subpopulations resulted in variations in the level of T cell inhibition. Full size image Fig. 2: Design of the NanoEPIC device. a NanoEPIC includes a glass substrate with magnetic guides patterned under a fluidic channel and a PDMS cover. A permanent magnet is positioned below the NanoEPIC to facilitate magnetic deflection. b Visualization of magnetic guides through (top) (scale bar = 100 µm) and the corresponding simulation of the magnetization of magnetic guides under an external magnetic field (bottom). c Simulated magnetic gradient across the magnetic guide. d The number of anti-PD-L1 conjugated MNPs bound to sEVs from different cell lines. (upper) Representative figures of sEVs bound to anti-PD-L1 conjugated MNPs obtained by TEM. Scale bar is 100 nm. In general, the number of MNPs bound to sEVs is proportional to exoPD-L1 expression. The average number of MNPs bound to sEVs collected from H1975 cells is the highest among the three tested cell lines. Furthermore, H1975 sEVs displayed the broadest distribution of the number of MNPs bound to each sEV. e Deflection efficiency of anti-PD-L1 MNP bonded sEVs in NanoEPIC with different minimum deflection angles. Smaller deflection angles led to higher deflection efficiency. ( p = 0.0479; p = 0.0003; p < 0.0001). f Deflection efficiency of anti-PD-L1 MNP bonded sEVs in NanoEPIC with different channel heights. Smaller channel height gives higher deflection efficiency. ( p = 0.0003; p < 0.0001). g Deflection efficiency of sEVs bound with anti-CD9 MNPs. ( p = 0.0363; p = 0.0150) h Specificity of the NanoEPIC assessed through the difference in deflection efficiency between H1975 wildtype (WT) sEVs and H1975 PD-L1 knockout (KO) sEVs, which were sorted based on exoPD-L1. Isotype antibodies were used to further confirm the specificity of magnetic labeling of sEVs based on exoPD-L1. ( p < 0.0001) i , ExoPD-L1 profile distribution i n sEVs from H1975 WT cells and PD-L1 KO H1975 cells. ExoPD-L1 profile of sEVs treated with isotype antibody-MNPs is also given. n = 5 biologically independent samples for e-i . All bar plots represent mean ± s.d. * P < 0.05, ** P < 0.01, *** P < 0.001, **** P < 0.0001, one-sided unpaired t-test. Source data are provided as a Source Data file. Full size image There are two main components in the NanoEPIC device that are crucial for its performance: (1) the deflection angle of magnetic guides, and (2) flow channel height. The deflection angles allow for tuning the equilibrium between the fluid drag force and magnetic force which consequently dictates the degree of separation between the sorted sEV subpopulations. To determine the angular thresholds of our magnetic guides, we modeled 4, 10, and 20 as the approximate number of MNPs conjugated to sEVs in the low, medium, and high outlets respectively (Supplementary Fig. 5a ). Through experimentation, we observed near-saturation deflection efficiency at a minimum deflection angle of 3° (Fig. 2e ) which represents an acceptable starting point for exoPD-L1 segregation. To sort sEVs with distinct phenotypic profiles, we selected 3° for exo-L, 5° for exo-M, and 10° for exo-H subpopulations. As for the channel height, increased height allows for higher throughput but can also diminish the magnetic influence of the magnetic guides. Given that the magnetic guides are positioned below the device, sEVs migrate toward the bottom of the device to fully experience their designed magnetic trajectories. Both simulations and experimental outcomes suggest 30 µm to be the optimal height for exoPD-L1 sorting (Fig. 2f and Supplementary Fig. 5b–f ). It is worth noting that an overly strong magnetic gradient might result in trapping the MNP-labeled sEVs at the bottom of the microchannel. This problem was addressed by adjusting the distance between the bottom of the channel and the magnetic guides (Supplementary Fig. 6a ). Next, we optimized the sample flow rate to maximize the capture efficiency. Instead of anti-PD-L1, we used anti-CD9 as a sorting marker to determine the maximal sorting efficiency of sEVs. We demonstrated that a flow rate of 200 µL/h has led to the highest deflection efficiency (86.6 ± 2.9%) (Fig. 2g ). To gauge the specificity of the NanoEPIC device, we generated mutagenic PD-L1 knockouts of H1975 (PD-L1 KO) (Supplementary Fig. 6b ) and compared the exoPD-L1 profile of PD-L1 KO and wildtype H1975 (WT). As depicted in Fig. 2h , the NanoEPIC platform maintained high specificity as indicated by the low deflection efficiency of PD-L1 KO (5.4 ± 1.9%) and isotype antibody control (8.6 ± 3.4%) compared to 64.2 ± 2.2% obtained using WT sEVs. Additionally, we found that most WT sEVs were collected in the exo-H outlet, whereas most of the PD-L1 KO sEVs or sEVs treated with isotype antibody-linked MNPs were collected in the exo-L outlet (Fig. 2i ), which suggests that unlabeled sEVs might have escaped to the exo-L outlet with minimal nonspecific deflections into the exo-M and exo-H outlets. ExoPD-L1 profiling of tumor-derived sEVs with NanoEPIC Having established the high sensitivity and specificity of NanoEPIC for sorting sEVs, we further assessed the performance of exoPD-L1 profiling using the NanoEPIC platform. First, we assessed exoPD-L1 profiling in H1975 sEVs at different flow rates. While flow rates of 200 µL/h and 300 µL/h displayed nonsignificant differences in deflection efficiency (Fig. 3a ), the distributions of sEVs between the deflected outlets were much more pronounced (Fig. 3b ). More importantly, increasing the flow rate not only decreased the deflection efficiency but also shifted the distribution of sEVs towards the low-expression outlet which corroborated our simulations (Supplementary Fig. 7 ). Next, we utilized NanoEPIC for profiling exoPD-L1 in other cancer cell lines. The deflection efficiency and exoPD-L1 profiles were significantly different among all three cell lines which agreed with our previous findings (Supplementary Fig. 8a, b ). Since the expression of exoPD-L1 has previously been shown to be upregulated after treatment with IFN-γ [6] , [52] , we also compared the exoPD-L1 profiles between IFN-γ treated and untreated cells (Supplementary Fig. 8c ). Notably, the differences in exoPD-L1 profiles between IFN-γ treated and untreated cell lines were also detectable using the NanoEPIC platform (Fig. 3c–f ), which highlights the high-resolution profiling efficiency of the NanoEPIC platform. Fig. 3: Performance of the NanoEPIC platform. a Deflection of anti-PD-L1 MNPs labeled sEVs from H1975 at different flow rates. ( p = 0.0651; p < 0.0001; p = 0.0031). b ExoPD-L1 profiling of sEVs from H1975 at different flow rates. c Deflection efficiency of sEVs from different cell lines (H1975, H460, and PC9). The deflection efficiency of sEVs from corresponding cells treated by IFNγ is shown. ( p = 0.0713; p = 0.0250; p = .0029). d ExoPD-L1 profiling of sEVs collected from WT PC9 cells and IFNγ treated PC9 cells. e ExoPD-L1 profiling of sEVs collected from WT H460 cells and IFN-γ treated H460 cells. f ExoPD-L1 profiling of sEVs collected from WT H1975 cells and IFN-γ treated H1975 cells. g TEM images of sEV subpopulations collected from Negative, Low, Medium, and High outlets of the NanoEPIC. The scale bar is 50 nm. h Deflection efficiency of anti-PDL1 MNPs bounded sEVs from H1975 cells after treatment with antibody elution buffer compared to control (i.e., sEVs treated with PBS). i ExoPD-L1 profiling of H1975 sEVs after treatment with elution buffer. PBS-treated sEVs were mainly collected in the exo-H outlet, whereas elution buffer-treated sEVs were mainly collected in the exo-L and exo-M outlets. j exoPD-L1 expression of H1975 sEV subpopulations collected from low/medium/high outlets, measured by PD-L1 ELISA. ( p < 0.0001; p < 0.0001; p = 0.0004). k Western blot (WB) analysis of different sEV subpopulations compared to cell lysate (CL). n = 5 biologically independent samples for a , b . n = 3 biologically independent samples for c–f . n = 4 biologically independent samples for h – j . All samples were harvested from H1975 cells. WB assay was independently repeated 3 times for k . All bar plots represent mean ± sd. * P < 0.05, ** P < 0.01, *** P < 0.001, **** P < 0.0001, one-sided unpaired t-test. Source data are provided as a Source Data file. Full size image We next determined if the sorted sEVs could be recovered for subsequent downstream analysis. TEM imaging confirmed the number of MNPs bound to sEVs is correlated to their designated outlets (Fig. 3g ); however, to enable the utility of sorted sEVs, the MNPs must be stripped off. Unlike cells, sEVs lack innate processes that can degrade, internalize, or detach MNPs. Using antibody elution buffer, we were able to remove most MNPs while maintaining the integrity of PD-L1 proteins on sEVs (Supplementary Fig. 9a–c ). MNP release was validated through exoPD-L1 profiling using NanoEPIC (Fig. 3h, i ). The eluted MNPs can be separated from sEVs through sucrose-gradient ultracentrifugation (Supplementary Fig. 9d, e ). Using the purified sEVs, we further confirmed the difference in PD-L1 expression in sEVs collected from each of the outlets using PD-L1 ELISA and western blotting (Fig. 3j, k ). In vivo analysis of exoPD-L1 for PD-1 immunotherapy To examine the potential clinical utility of profiling exoPD-L1, we established a PD-1 immunotherapy mouse model through the injection of MC38 cells into C57BL/6 mice followed by a semiweekly administration of anti-PD-1 antibodies (Fig. 4a ). We assessed whether exoPD-L1 profiling can be used to differentiate between the heterogeneous responses to anti-PD-1 immunotherapy (Supplementary Fig. 10 ). At the endpoint of the study (23 days post-inoculation), terminal blood collection was performed and the plasma was separated. The plasma was then used for exosomal analysis using the NanoEPIC platform. The status of complete responders, partial responders, and non-responders were determined through tumor volumetric measurements at day 23. Figure 4b represents a summary of the exoPD-L1 profiles from anti-PD1 treated mice (Supplementary Fig. 11a ). The NanoEPIC analysis revealed a lower proportion of exo-L and a considerably higher proportion of exo-H sEVs found in non-responders compared to responders (Supplementary Fig. 11b ) which suggests that elevated exoPD-L1 expression led to greater immunotherapeutic resistance. It was also observed that without any immunotherapeutic pressure, exoPD-L1 profiles were unpredictable and inconclusive (Supplementary Fig. 11c, d ). We then investigated whether plasma levels of sEVs are correlated with outcome. The concentration of sEVs from all the mice were measured using NTA. No significant differences were observed between complete responders, partial responders and non-responders (Supplementary Fig. 10d ). To refine the quantitation of exoPD-L1 expression measured with the NanoEPIC platform, we generated a numerical protein expression index called the NanoEPIC score. This score reports the overall exoPD-L1 expression while incorporating the deflection efficiency and exoPD-L1 profiling from the NanoEPIC’s high-throughput and high-resolution nanoscale sorting of heterogenous sEVs (see supplementary text for the calculation of the NanoEPIC score). Using this approach, we observed a direct correlation between tumor volume and NanoEPIC score (Fig. 4c ), indicating that the NanoEPIC platform could potentially be used to monitor and predict tumor growth. Fig. 4: Analysis of exoPD-L1 in PD-1 immunotherapy mouse model. a Workflow of sEV and tumor analysis in PD-1 immunotherapeutic mouse model. Briefly, mice injected with MC38 cells were treated with either anti-PD-1 antibodies or saline (control) twice a week. 23 days post-inoculation, mice were sacrificed for plasma and tumor collection. Collected plasma was subjected to exoPD-L1 analysis using NanoEPIC while solid tumors underwent PD-L1 sorting or TIL analysis through flow cytometry. b Distribution of sEVs sorted in the low, medium, and high outlets of the NanoEPIC chip are represented as a heat map and arranged in ascending order with respect to tumor volume. c Scatterplot of tumor volume against NanoEPIC score ( n = 3). An exponential plateau model was used for the regression analysis which resulted in the following fit: y = 2213 – (2213 + 1945)e -0.008393x . d Distribution of tumor cells sorted through each outlet of the prismatic deflection chip with PD-L1 as the sorting marker. Data arranged in ascending order with respect to NanoEPIC score. sEVs from PD-L1 sorted MC38 cells were collected and processed through the NanoEPIC device and the deflection efficiency ( p = 0.0085) e along with the exoPD-L1 profiles for the low ( p = 0.0106) f , medium g , and high ( p < 0.0001) h of cPD-L1 are shown. n = 3 biologically independent samples for e – h . All data represent mean ± s.d. * P < 0.05, ** P < 0.01, *** P < 0.001, **** P < 0.0001, one-way ANOVA. Source data are provided as a Source Data file. Full size image Tumor cPD-L1 levels have traditionally been used for the indication of PD-L1-associated tumor immune evasion; however, the acquisition of PD-L1+ cells from solid tumor tissues is highly invasive. On the other hand, cPD-L1-based liquid biopsy which relies on the analysis of circulating tumor cells (CTCs) can be challenging due to the scarcity of CTCs in early cancer stages. Since sEVs are highly abundant in the plasma and simple to obtain, we were interested in determining whether the levels of plasma exoPD-L1 would reflect the levels of cPD-L1 in tumors. Using our previously established prismatic deflection cell-sorting device (PRISM) [45] , we profiled dissociated tumor cells from our C57BL/6 mice based on cPD-L1 expression (Fig. 4d and Supplementary Fig. 12a ). Based on our findings, the distribution of exoPD-L1 closely reflected the distribution of PD-L1+ cells (Supplementary Fig. 12b–d ), suggesting that exoPD-L1 expression is positively correlated with cPD-L1 expression. To further ascertain the relationship between exoPD-L1 and cPD-L1, we evaluated an MC38 colorectal cancer model. For this investigation, we first segregated MC38 cells using the PRISM device into three subpopulations of varying cPD-L1 expression, which were promptly validated through flow cytometry (Supplementary Fig. 12e ). We then collected sEVs from each of the sorted MC38 subpopulations and profiled their exoPD-L1 using the NanoEPIC device (Fig. 4e–h ). The total fraction of sEVs containing PD-L1 was found to increase with higher cPD-L1 expression in MC38 cells (Fig. 4e ). Upon examining the exoPD-L1 profiles, a similar trend was seen with a larger proportion of exo-L found in low cPD-L1 expressing MC38 and a higher proportion of exo-H in high cPD-L1 (Fig. 4f–h ). Overall, NanoEPIC is a compelling prognostic platform for PD-1 immunotherapy as exoPD-L1 is highly abundant, easily obtainable, and correlates with PD-L1 burden in tumors to a similar degree as traditional prognostic markers such as tumor cPD-L1. We next explored the relationship between exoPD-L1 and their immune effectors, particularly with CD8+ tumor-infiltrating lymphocytes (TILs). Flow cytometric analysis of tumors from partial responder, non-responder, and control (no anti-PD-1 immunotherapy) mice revealed a uniform decrease in CD45 + CD8+ TILs with higher NanoEPIC score (Fig. 5a and Supplementary Fig. 13a ). It was also found that TILs with higher NanoEPIC score had reduced proliferation, T cell activation/differentiation, cytokine secretion and T cell cytotoxicity as demonstrated by the decrease in expression of ki67, TCF7, CD69, CD137, IFN-γ, and granzyme B (GzB) (Fig. 5b–g and Supplementary Fig. 13b–g ). These findings are consistent with previous investigations, indicating a decline in TIL activity is strongly associated with higher exoPD-L1 levels [6] , [7] , [23] . While we expectedly detected a reduction in PD-1 expression with higher exoPD-L1, we also observed a striking reduction in CD152 (CTLA-4) expression (Fig. 5h, i and Supplementary Fig. 13h, i ). The CD152 pathway is involved in the inhibition of T cell activation which typically occurs in lymph nodes [53] , suggesting that sEVs can elicit longer range immune-suppression beyond the tumor site. In summary, our results demonstrated the strong potential of using exoPD-L1 profiling for predicting T cell response to immunotherapy, further supporting NanoEPIC as a robust prognostic platform. Fig. 5: Determination of NanoEPIC scores of TILs from murine PD-1 immunotherapeutic model. a Proportion of CD45 + CD8 + T cells arranged with increasing NanoEPIC scores from left to right. Flow cytometric analysis of proliferation markers ki67 b , T cell differentiation and activation markers (TCF7, CD69, and CD137) c – e , cytotoxicity (IFNγ, Granzyme B) f , g and immune-checkpoint markers (PD-1 and CD152) h , i shown as scatterplots of median fluorescence intensities and NanoEPIC score. All scatterplots were fitted with linear regression analysis. Source data are provided as a Source Data file. Full size image Interaction of T cells and sorted sEV subpopulations Having demonstrated the varying extent of TIL inactivation in vivo relative to exoPD-L1 levels, we then explored whether NanoEPIC-sorted sEVs would exhibit distinct biological activities. The prevailing model for PD-L1-mediated immunosuppression relies on the interaction between PD-L1 with PD-1 generally expressed on CD8 cytotoxic T cells. To determine the direct link between exoPD-L1 and T cell suppression, we used an in vitro model in which isolated CD8 + T cells were treated with PD-L1-sorted sEVs from the NanoEPIC device. First, we examined the relative binding of sEVs onto CD8 + T cells using SEM (Supplementary Fig. 14a ). As seen in Fig. 6a , there were significantly more exo-H sEVs bound to CD8 T cells when compared with exo-L sEVs. To confirm this observation, we performed immunofluorescent staining of sEV-bound T cells using CD9 as an indicator of sEVs (Fig. 6b and Supplementary Fig. 14b ). Through measurement of CD9 fluorescent intensities, it was confirmed that there were statistically significant differences between the binding affinity of exo-L, exo-M, and exo-H sEVs onto T cells, with the lowest binding affinity exhibited by exo-L and the highest binding affinity in exo-H (Supplementary Fig. 14c ). These results suggest exoPD-L1 levels are positively correlated to the binding affinity of sEVs to CD8 T cells. Fig. 6: Characterization of T cell properties after treatment with sorted PD-L1 + sEVs. CD8 + T cells bound to different sEV subpopulations were imaged through SEM a and confocal microscopy b . Scale bar represents 1 µm for SEM and 4 µm for confocal microscopy. Proportion of T cells positive for granzyme B c ( p = 0.0030; p = 0.0058), ki67 d ( p = 0.0101; p = 0.0350), and CD69 e ( p = 0.0044; p = 0.0119) was determined through flow cytometry after treatment with sEVs from exo-L, exo-M, exo-H, unsorted (WT), activated CD8 + T cells with no exosomal treatment (Pos), and exosomes acquired from PD-L1 knockouts (PD-L1-KO). Secretion of IFN-γ f ( p = 0.0002; p = 0.0140) and IL-2 g ( p < 0.0001; p < 0.0001) by T cells after treatment of different sEV subpopulations was determined using ELISA. h Schematic of CD8 + T cell cytotoxicity assay after treatment with varying exoPD-L1 subpopulations of sEVs. i T cell cytotoxicity assay from CD8 + T cells cocultured with either H1975 (left) ( p = 0.1872; p = 0.0418; p < 0.0001) or MC38 (right) ( p = 0.015; p = 0.0710; p = 0.0024) after treatments from exo-L, exo-M, exo-H, or no sEVs (control). n = 4 biological independent samples for c – g . n = 5 for i . All bar plots represent mean ± s.d. * P < 0.05, ** P < 0.01, *** P < 0.001, **** P < 0.0001, one-way ANOVA. Source data are provided as a Source Data file. Full size image Next, we sought to investigate the phenotypic variations of T-cells after sEV treatment. Pre-activated CD8 + T cells were treated with sEVs from each outlet of the NanoEPIC device (exo-L, exo-M, exo-H) separately. Compared with the positive control (Pos) where pre-activated CD8 T cells were not treated with sEVs, a decrease in activation, proliferation, and cytotoxicity of T cells was observed through flow cytometry after treatment with exo-L, exo-M, and exo-H as indicated by the decreasing proportions of CD69, ki67, and GzB positive CD8 T-cells (Fig. 6c–e and Supplementary Fig. 15 ). Exo-H caused stronger inhibition of T cells compared with Exo-L. As comparison, treatment with unsorted sEVs (WT) was consistently found to be within the margins of exo-L to exo-H for all analyses. Through ELISA, it was shown that the cytokines, namely IFN-γ and IL-2 were similarly reduced in T cells when treated with sEVs with higher exoPD-L1 expression (Fig. 6f, g ). For the cytokine secretion assay, fresh purified inactivated CD8 T cells were used as a negative control (Neg). To further evaluate the effects of exoPD-L1 on the cytotoxic activity of CD8 T cells, the cells treated with distinct sEV subpopulations were co-cultured with tumor cell lines (Fig. 6h ). Through LDH assay, we observed a significant reduction in CD8 T cell cytotoxicity when treated with sEVs with higher exoPD-L1 levels (Fig. 6i ). Overall, our findings demonstrate that the NanoEPIC is capable of sorting sEVs with biologically distinct activities and allow for the separation of unique sEV subpopulations. Validation of NanoEPIC platform with clinical specimens To establish that the NanoEPIC platform could be used with clinical specimens, we analyzed a small set of samples from patients undergoing anti-PD-1/anti-PD-L1 immunotherapy. Plasma samples were gathered from 10 de-identified cancer patients (multiple visits) and 5 healthy donors to determine whether exoPD-L1 levels could be measured (Supplementary Table 4 ). Comparing patient samples to healthy donor samples, we observed a significant elevation in exoPD-L1 levels (Fig. 7a ). We also analyzed changes in NanoEPIC scores before and after treatment. Notably, responders exhibited a consistent decrease in exoPD-L1 expressions (Fold change<1), while non-responders showed a general increase (Fig. 7b and Supplementary Fig. 17 ). Additionally, we observed a positive correlation between changes in NanoEPIC scores and tumor size (Fig. 7c ). These findings indicate that NanoEPIC could be applied in the future to predict immunotherapy responses and aid in treatment selection. Moreover, the ability of the NanoEPIC platform to distinguish between healthy and clinical samples highlights its potential diagnostic capabilities. Fig. 7: Assessment of clinical specimens with the NanoEPIC Platform. To evaluate the performance of the NanoEPIC platform on specimens collected from patients receiving anti PD-1/PD-L1-based immunotherapy, a total of 10 de-identified plasma samples were processed using the NanoEPIC device for exoPD-L1 profiling. a ExoPD-L1 profiling of patient samples before immunotherapy was compared with that of healthy donors. ( n = 10) ( p = 0.0002). b Changes in NanoEPIC scores were compared between responders and non-responders to immunotherapy. The fold change in NanoEPIC score represents the ratio of NanoEPIC score after immunotherapy over NanoEPIC score before immunotherapy. A fold change of 1 signifies no change in NanoEPIC score. The correlation between fold change in NanoEPIC score and c tumor size obtained through CT scans. n = 3 technical replicates for a – c . All bar plots represent mean ± s.d. *** P < 0.001, one-sided unpaired t-test. Source data are provided as a Source Data file. Full size image\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"Here, we have presented a platform that utilizes magnetic-activated ranking for nanoscale cytometry of sEVs. We conceptually validated the feasibility of magnetic labeling of sEVs with MNPs and demonstrated nanoscale deflection of individual sEVs. The optimized NanoEPIC can perform phenotypic profiling with high efficiency, specificity, and throughput. As a microfluidics-based platform, the NanoEPIC is advantageous as it is simple to fabricate, low-cost, and user-friendly. The NanoEPIC platform can be adapted to a variety of applications. Since the magnetic deflection of sEVs relies on the binding of antibody-decorated MNPs, NanoEPIC permits sorting sEVs based on the level of any surface marker. The NanoEPIC’s capability to segregate sEVs based on surface marker expression allows for precise analysis of specific sEV subpopulations that were previously not feasible. While sEVs have been used to examine the performance of NanoEPIC, this nanoscale cytometry system can be widely adapted for the sorting and phenotypic profiling of other submicron/nanoscale particles such as certain bacteria or subcellular organelles. To our knowledge, few other technologies can perform high-resolution molecular sorting and profiling at that throughput. Overall, we demonstrated how NanoEPIC could potentially be used as a cancer diagnostic tool. The NanoEPIC was used in mouse models to monitor the response to anti-PD1 immunotherapy and produce signature exoPD-L1 profiles that can be used to distinguish between responders and non-responders to immunotherapy. Our the numerical NanoEPIC scores not only reflected tumor progression but also demonstrated a high correlation with the suppression of TILs. The downstream analysis showed that different sEV subpopulations with varying PD-L1 expression exhibited different inhibitory behaviors on T cells. Given that different exosomal biomarkers such as CTLA4, TIM3, CD47 are being studied for the prediction of immunotherapy response, the NanoEPIC can be used to profile panels of exosomal biomarkers and ultimately be applied for further clinical investigations to guide clinical decisions and improve therapeutic outcomes. In addition to the diagnostic utility of NanoEPIC, therapeutic sEVs are emerging as promising drug delivery agents due to their excellent biocompatibility, low immunogenicity, persistence in the blood, and ability to cross various biological barriers [54] . Several reports have demonstrated the utility of sEVs as drug delivery vehicles for drug-resistant lung cancer [55] , [56] , [57] . Hence, the ability to isolate distinct subpopulations of sEVs with favorable phenotypes and superior delivery properties would offer many opportunities for the treatment of lung cancer [54] . This necessitates high-throughput methods capable of accurately sorting sEVs from large volumes of patients’ blood, which suggests a future utility for the platform. Given that drug loading to sEVs can also be heterogeneous [58] , the NanoEPIC system can be used to sort out sEV subpopulations with high drug loading, which can potentially reduce the dose of sEVs required for effective treatment. Another application of NanoEPIC can be sorting of cargo-loaded sEVs based on their specificity to recipient cells. sEVs engineered with antibodies specific to recipient cells have been used for cargo delivery [59] . Sorting of high marker expressed sEVs can increase the uptake of cargo-loaded sEVs by the recipient cells. Also, we have shown that the NanoEPIC system was able to sort magnetically labeled concentrated sEVs (up to 10 12 particles/mL) with high deflection efficiency (more than 80%) at a flow rate of 200 µL/h. The NanoEPIC was amenable for simultaneous processing of up to 6 microfluidic chips, which further increases its total throughput. While the NanoEPIC exhibits merits for biomarker profiling and sorting, as a magneto-activated system, it does lack the flexibility of simultaneously detecting multiple markers like FACS. More work must be done to multiplex biomarker profiling and sorting. One potential way to overcome this problem is through serial sorting with multiple magnetic labeling steps [59] , [60] . Another way is to use MNPs with distinctive magnetic properties such as size and magnetic susceptibility [61] , [62] , [63] . Continued efforts to assess novel magnetic sorting methodologies for facilitating multiplex analyses on the NanoEPIC platform hold tremendous promise in expanding the scope of extracellular vesicle investigations. Such advancements have the potential to uncover broader clinical implications on a larger scale.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Ethical statement This research complies with all relevant ethical regulations and the related protocols were approved by the University of Toronto Research Ethics Board (REB). All animal experiments were operated following the protocols approved by the University of Toronto Animal Care Committee. Human plasma samples were obtained from PrecisionMed LLC upon the approval by the Western Institutional Review Board® (Puyallup, WA, USA) and after informed consents were signed. Whole blood samples were collected from 10 patients under anti-PD1/anti-PD-L1 immunotherapy ( n = 8 female donors and n = 2 male donors aged 41–76) (Fig. 7 , Supplementary Fig 17 and Supplementary Table 4 ). Plasma samples were extracted from whole blood immediately after blood collection and stored at −80 °C. In all cases, participants or legal representatives were informed in detail about the research purpose and informed consent was signed. Fabrication of NanoEPIC The design of NanoEPIC was sketched using a CAD software (AutoCAD). Chrome masks containing the designed patterns were printed using a mask maker (Heidelberg µPG 501). The fabrication of NanoEPIC includes three main procedures: (i) patterning of magnetic guides, (ii) generation of microchannels, and (iii) binding of the devices (Supplementary Fig. 16 ). For patterning the magnetic guides, a sheet of Metglas 2714 A was glued onto borosilicate glass wafers (452, UniversityWafer) with a thin spun-coated layer of epoxy (7370A38, McMaster-Carr) and left to dry overnight. Excessive epoxies were removed with acetone and isopropanol (IPA). The magnetic guides were photolithographically patterned onto a layer of 2 µm thick S1811 photoresist that was spun-coated onto Metglas. After developing the S1811 with MF-CD-26, Metglas was wet-etched using an etchant consisting of 3.6% HCl and 14.3% H 2 O 2 in water. For the generation of microchannels, the S1811 photoresist was stripped with AZ300T (Integrated Micromaterials) and coated with Omnicoat (Kayaku Advanced Materials) to improve SU-8 photoresist adhesion. SU-8 3035 was used to develop a 40µm-thick SU-8 layer to encapsulate the magnetic guides and a second 30 µm-thick SU-8 layer was developed on top to generate the microchannels. For device binding, PDMS cover pieces that were hole punched at the inlet and outlet sites were first treated with plasma for 1 min, then treated with 10% APTES in water for 30 min. The pieces were then rinsed with deionized water and bonded onto the SU-8 channels. The bonded chips were incubated at 70 °C overnight with vertical pressure to enhance the binding strength. Cell culture Human NSCLC cell lines H1975 (CRL-5908, ATCC), H460 (HTB-177, ATCC), and PC9 (90071810, Sigma) were cultured in RPMI medium (350-007-CL, Wisent) while mouse colon cancer cell line MC38 (ENH204-FP, Kerafast) were cultured in DMEM (D5796, Sigma). All culture media were supplemented with 10% FBS (Wisent), 1% penicillin-streptomycin (Sigma), and incubated at 37 °C with 5% CO 2 . For the stimulation of cells with IFN-γ, cells were incubated in medium supplemented with 100 ng/mL of either human or mouse IFN-γ (Peprotech) for 48 h before the collection of cell and sEV samples. Human CD8 T cells were isolated from human blood mononuclear cells (70025, Stemcell Technologies) using Human CD8 + T cell isolation kit (130-096-495, Miltenyi Biotec) while mice T cells were collected from the spleen of C57BL/6 mice (Jackson Laboratory) and isolated with mouse CD8 + T cell isolation kit (130-096-543, Miltenyi Biotec). Human T cells were cultured in T cell expansion medium (10981, STEMCELL Technologies) with 10 ng/mL of human IL-2 (78036.3, STEMCELL Technologies). Mouse T cells were cultured in RPMI medium supplemented with 10% FBS and 10 ng/mL of mouse IL-2 (130-120-334, Miltenyi Biotec). T cells were activated with Human CD3/CD28/CD2 activation kit (10990, STEMCELL Technologies) at 25 µL/mL or Mouse CD3/CD28 beads (11456D, Thermofisher) at 25 µL/mL at every second passage. Genetic modifications To generate the PD-L1 knockout (KO) H1975 cell lines, H1975 wildtype (WT) cells were first infected with lenti-Cas9 (52962LV, Addgene) followed by blasticidin selection to establish Cas9 expressing cell lines (H1975-cas9). The H1975-cas9 cells were then simultaneously infected with two different guide RNA targeting PD-L1 (target sequence: GGTTCCCAAGGACCTATATG and CGCTGCATGATCAGCTATGG) that were previously packaged into lentivirus by Thermofisher (A32042). Monoclonal expansion was performed following puromycin selection and the knockout clones were identified with flow cytometry (cytoFLEX S, Beckman Coulter) for surface PD-L1. For the modification of MC38 cell lines, gRNA used to knockout PD-L1 (target sequences: GTTTACTATCACGGCTCCAA and GGGGAGAGCCTCGCTGCCAA) were purchased from Integrated DNA Technologies and lentiviral plasmids were obtained from Genscript (pLentiCRISPRv2 backbone) respectively. Lentiviral packaging was performed through co-transfection of HEK293T cells with psPAX2, pMD2.G, and the transfer plasmid. The medium was changed after 24 h and then collected at 24 h intervals. MC38 cells were infected with the lentivirus and selected with 10 µg/mL of puromycin. Fluorescent-activated cell sorting (BD FACSAria IIIu cell sorter) was performed to select cells negative for PD-L1 from the knockout population. Flow cytometry was used to confirm the stable maintenance of PD-L1 expression levels. Purification of sEVs For sEVs collected from cultured cells, the base culture medium was replaced with medium supplemented with 10% sEV-depleted FBS (Gibco) when cells reached ~70% confluency. Cells were then cultured for 48 hours then the culture medium was collected for sEV isolation. The culture medium was centrifuged at 2000 g for 30 minutes to remove dead cells and cellular debris followed by another centrifugation at 10,000 g for 1 h to remove large microvesicles. The supernatant were filtered with 0.22 µm filter and then mixed with an sEV precipitation kit (4478359, Thermofisher) at a 2:1 volume ratio, incubated overnight, and then centrifuged at 10,000 g for 1 h to pellet the sEVs. SEVs were resuspended with 1% bovine serum albumin (BSA, Sigma) with protease inhibitor (11836170001, Roche) and phosphatase inhibitor (4906845001, Roche) in PBS. All buffers were filtered with 0.22 µm filters (SLGV033R, Sigma) prior to mixing with sEVs. For blood samples, plasma was extracted by collecting the supernatant after 2-step centrifugation at 2000 g for 20 minutes followed by 10,000 g for 20 min. Samples were directly processed through the NanoEPIC device as discussed later. Magnetic labeling of sEVs 10 nm diameter protein-G cross-linked iron oxide magnetic nanoparticles at 1 mg/mL (IPG-10-01) were purchased from Ocean NanoTech and mixed with either human anti-CD9 antibody (10626D, Thermofisher), human anti-PD-L1 (14-5983-82, Thermofisher), or mouse anti-PD-L1 (14-5982-82, Thermofisher) in a 15:1 volume ratio and chilled overnight at 4 °C. The antibody-bead conjugates were then pelleted through centrifugation at 100,000 g for 45 minutes and then resuspended in 0.1% BSA to reach a final concentration of 0.5 mg/mL and stored at 4 °C until later use. For the labeling of sEVs, 10 9 sEVs in 100 µL of 1% BSA were mixed with 5 µL of the functionalized magnetic nanoparticles (MNPs) and agitated overnight at 4 °C prior to processing through the NanoEPIC device. To ensure the maximum interaction between EVs and MNPs and maximized magnetic labeling efficiency, an excessive amount of functionalized MNPs were added to the sEV sample. For the preparation of sEV-spiked plasma samples, 100 µL of sEVs in 1% BSA were mixed with the same volume of plasma and treated with 10 µL of anti-PD-L1 conjugated MNPs at 4 °C overnight. As a control, 100 µL of 1% BSA was mixed with the same volume of plasma and treated with 10 µL of anti-PD-L1 conjugated MNPs at 4 °C overnight. For mouse plasma samples, 100 µL of 2-fold diluted plasma was treated with 5 µL of mouse anti-PD-L1 conjugated MNPs at 4 °C overnight. Processing sEV samples through NanoEPIC Before chip processing, all buffers were filtered through a 0.22 µm syringe filter and degassed under a vacuum for 15 minutes. NanoEPIC chips were first treated with 0.1% Pluronic F-68 (Thermofisher) in deionized water for 15 min then washed with running buffer (1% BSA in PBS) to minimize non-specific adhesion. The device was then positioned on 4 laterally stacked N52-Neodymium magnets (BY042-N52, K&J Magnetics) for all further processing. 100 µL of magnetically labeled sEV samples were mixed in a 1:1 volume ratio with running buffer and loaded into the NanoEPIC device. Fluidic processing was performed through a syringe pump (Fusion 100, Chemyx) that was set to withdraw at a flow rate of 200 µL/h. Subsequent to sample processing, the devices were washed with PBS at a flow rate of 400 µL/h. Characterization of sEVs The size distribution and concentration of sEVs were determined through nanoparticle tracking analysis (NTA) with NanoSight NS300 (Malvern). Transmission electron microscopy (TEM) was performed for the visualization of sEVs binding to beads. Samples for TEM were fixed at a 1:1 volume ratio of 4% paraformaldehyde (PFA, F8775-4X25ML, Sigma) diluted in PBS. Specimen preparations were then performed by the Nanoscale Biomedical Imaging Facility (NBIF) in Peter Gilgan Centre for Research and Learning at The Hospital for Sick Children (Toronto, CA). Briefly, the sample was dropped on a glow-discharged formvar/carbon-supported copper grid. After staining with 2% uranyl acetate, the grids were air-dried and visualized using a Hitachi HT7800 transmission electron microscope. Separation of sEVs from antibody-functionalized MNPs Two elution buffers were tested, including glycine-HCl (Sigma) and Pierce IgG elution buffer (Thermofisher). For glycine-HCl, 200 mM and 300 mM were tested. SEVs collected from the outlets of the NanoEPIC device were treated with an equal volume of elution buffer and immediately incubated at 37 °C for 30 min. The sEV samples were then separated from eluted MNPs through sucrose gradient centrifugation (0.5 M and 1.5 M sucrose concentration) running at 120,000 g for 1.5 h. sEVs were collected from the interface between the 0.5 M and 1.5 M sucrose while the released MNPs were pelleted at the bottom of the tube. To remove the excessive sucrose, the collected sEVs were washed with 5 mL of PBS and pelleted through centrifugation at 100,000 g for 2 h. Western blot analysis Cell lysates were prepared by adding RIPA buffer (PI89900, Thermofisher) supplemented with protease inhibitor and phosphatase inhibitor to cell samples followed by a 14,000 g centrifugation for 15 minutes to collect the supernatant. sEV lysates were prepared by adding an equal volume of RIPA buffer and incubated at room temperature for 30 min. Lysates of different sEV subpopulations were normalized based on their total protein weight measured by BCA protein assay (Thermofisher). Lysates were separated using a 4–15% SDS-PAGE and transferred onto a nitrocellulose membrane. Blots were blocked with 3% BSA in Tris-buffered saline with Tween 20 (TBST) and incubated with primary antibodies at 4 °C overnight at the recommended supplier concentrations. For secondary staining, HRP-conjugated antibodies were added and left at room temperature for 1 h. The blots were developed with ECL substrate (Pierce). Information on the primary and secondary antibodies can be found in Supplementary Table 2 . ELISA All lysate samples were prepared as described above. The quantifications of protein through ELISA were performed by following the protocols from the supplier. Information on the ELISA kits can be found in Supplementary Table 3 . Flow cytometry Flow cytometric analysis was used to determine the protein expression in cells. Samples were typically fixed with 4% PFA followed by 20 minutes of blocking in 2% BSA. After primary antibody staining, samples were then washed and processed through cytoFLEX S flow cytometer (Beckman Coulter). For intracellular staining, samples were permeabilized with 0.2% Triton-X-100 (Sigma) after fixation. Information on the antibodies can be found in Supplementary Table 2 . The collected data were analyzed using FlowJo. Immunotherapeutic mouse model All animal experiments were operated in accordance with the protocols approved by the University of Toronto Animal Care Committee, with no tumors above the maximal limit of 2000 mm 3 and tumor size did not exceed 20 mm in any directions. For our syngeneic mouse model, female C57BL/6 mice (6–8 weeks old) purchased from Jackson Laboratory were subcutaneously injected with 6 × 10 6 MC38 cells in 100 µL of PBS. Starting one week after inoculation, 200 µg of anti-PD-1 antibodies (ICH1132UL, Ichor. Bio) were administered intraperitoneally twice a week (control mice were injected with saline). Tumors were measured twice a week with a digital caliper and volume was calculated using the following formula: (width) 2 × length/2. Mice were euthanized when the tumors exceeded 20 mm in one dimension or 23 days after inoculation. Blood was immediately collected through cardiac puncture and tumors and spleen were surgically removed. Plasma was isolated from blood samples for exosomal sorting through the NanoEPIC device. For mouse CD8+ T cell isolation, spleens were homogenized with a cell scraper (83.3950, Sarstedt), filtered through a 70 µm mesh cell strainer (352350, Falcon), and treated with RBC lysis buffer (00-4333-57, Thermofisher) before magnetic separation. Tumor cells were dissociated with Tumor Dissociation Kit (130-096-730) from Miltenyi Biotec and were either analyzed for tumor-infiltrating lymphocytes (TILs) through flow cytometry or sorted for PD-L1 through our previously reported prismatic deflection chip (PRISM). For data analysis, mice were categorized into 3 groups based on their differential responses to immunotherapy: “complete responders” referred to mice that had a significant reduction in tumor volume, “partial responders” displayed no significant changes in tumor volume, and “non-responders” displayed a significant development in tumor volume from the initiation of immunotherapy to the endpoint of the study (23 days post inoculation) (Supplementary Fig. 10a, b ). T cell-sEV binding assay To facilitate the binding of sEVs to CD8 T cells, 100 µL of CD8+ T cells in fresh medium were dispensed into 96-well plates at a concentration of 2 × 10 6 cells/mL. Cells were activated for 24 h with the addition of CD3/CD28/CD2 for human lines or CD3/CD28 beads for mice lines as mentioned earlier. Inactivated CD8+ T cells used as controls were supplemented with PBS instead of the activation cocktail. Preactivated CD8+ T cells were treated with 10 µL of sEV subpopulations (1 mg/mL) for 48 h at 37 °C and 5% CO 2 . After sEV treatment, CD8+ T cells under different conditions were harvested and used for flow cytometry to measure T cell activation, proliferation, and cytotoxicity. T cell culture media were collected to measure cytokine secretion with ELISA. Visualization of sEV-bound T cells To visualize the binding of sEVs to T cells, scanning electron microscopy (SEM) was used. 100 µL of preactivated CD8+ T cells (10 6 cells/mL) were treated with 10 µL of sEVs (1 mg/mL) for 2 h at 37 °C. For SEM, we first immobilized the T cells onto gold electrodes functionalized with anti-CD8. Primary fixation was achieved through overnight incubation of samples in 0.1 M phosphate buffer containing 4% paraformaldehyde and 1% glutaraldehyde. Samples were then additionally fixed with 1% osmium tetroxide in 0.1 M phosphate buffer for 30 minutes. Samples were dehydrated using a gradually increasing ethanol series, followed by critical point drying. Dehydrated samples were then spin-coated in gold and imaged using the Prisma E scanning electron microscope. Confocal microscopy was additionally used to verify the relative binding of different sEV subpopulations to T cells. The T cell-sEV binding protocol is modified to improve the fluorescence intensity. 100 µL of sEVs (1 mg/mL) of each subpopulation were incubated with 5 µL of Anti-CD9-PE (Thermofisher) overnight at 4 °C. SEV samples were then washed with PBS at 100,000 g for 2 h. The pelleted sEVs were then resuspended in 100 µL of PBS. For cell staining, 100 µL of activated T cells (10 6 cells/mL) were incubated with 1 µL of DiD lipid dye (V22887, Thermofisher) for 15 minutes at 37 °C. Cells were washed 4 times with PBS and then blocked in 400 µL of 1% BSA for 15 min. For each condition, 10 µL of PE-stained sEVs were added to the 100 µL of T cell samples and incubated at 37 °C for 2 hours. Samples were then washed with 1 mL of PBS to remove unbound sEVs, then fixed with 4% PFA for 15 minutes. After another PBS wash, 1 drop of nuclear stain (R37606, Thermofisher) in 100 µL of PBS was added and samples were incubated for 10 minutes at room temperature. After a final wash with PBS, samples were then imaged using the Zeiss LSM 880 microscope. Cytotoxic CD8 T cell-mediated tumor killing assay To measure the effect of exoPD-L1 on T cell-induced tumor cytotoxicity, we co-cultured sEV-treated CD8+ T cells with tumor cells and performed LDH-cytotoxicity assay (ab197004, Abcam). Briefly, sEVs from H1975 WT and MC38 WT were sorted through the NanoEPIC device and subsequently used to treat T cells using the aforementioned protocols. PD-L1 KO H1975 and PD-L1 KO MC38 cells were seeded into 96-well plates at 5,000 cells/well and incubated at 37 °C for 24 h. For co-culture, 100 µL of sEV-treated T cells (1.5 × 105 cells/mL) were added to the tumor cells and cultured for 48 h. LDH assay was subsequently performed following the supplier’s protocol (Abcam). Clinical sample processing Plasma samples were obtained from de-identified cancer patients who were undergoing anti-PD-1 or anti-PD-L1 immunotherapy, procured from PrecisionMed LLC (Supplementary Table 4 ). These samples were voluntarily collected by PrecisionMed LLC after approval from Western Institutional Review Board® (WIRB®) (Puyallup, WA, USA) (Protocol No.: 6054 WIRB® Protocol #20161289). Samples were obtained from n = 8 females and n = 2 males aged 47–76. All plasma samples were pre-filtered with 0.22 µm filters. The concentration of sEV in the plasma was measured to be between 0.6 and 1.74 × 10 11 particles/mL. For sample processing, 50 µL of plasma sample was first diluted 4 times with 1% filtered BSA. The diluted sample was then treated with 2.5 µL of anti-PD-L1 conjugated MNPs and incubated at 4 °C overnight. The mixture was then processed through the NanoEPIC system at 200 µL/h. sEVs collected from different outlets were analyzed with NTA as mentioned above. Gender analysis was not performed due to the limited number of available samples. Statistical analysis and reproducibility All statistical analyses were performed with GraphPad Prism version 9.0.0. All graphical data are represented as mean ± s.d. unless stated otherwise. Each dot represents a biological replicate unless specified otherwise. Unpaired student t-test was used to compare two different groups while two-way ANOVA was used to compare multiple groups. P value < 0.05 was considered statistically significant. No statistical method was used to predetermine sample size. No data were excluded from the data analyses. The investigators were blinded to allocation of treatment during mice experiments but not blinded when assessing outcome. 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 validate a high-throughput microfluidic platform called NanoEPIC for the molecular profiling and efficient isolation of heterogeneous small extracellular vesicles (sEVs) based on surface marker expression, particularly exosomal PD-L1 (exoPD-L1), to predict immunotherapeutic responses and enable downstream functional analysis of sEV-mediated T cell suppression in cancer."},"entities":{"kind":"list like","value":["NanoEPIC","microfluidic platform","molecular profiling","heterogeneous small extracellular vesicles","surface marker expression","exosomal PD-L1","exoPD-L1","immunotherapeutic responses","downstream functional analysis","sEV-mediated T cell suppression","cancer","high-throughput","validation"],"string":"[\n \"NanoEPIC\",\n \"microfluidic platform\",\n \"molecular profiling\",\n \"heterogeneous small extracellular vesicles\",\n \"surface marker expression\",\n \"exosomal PD-L1\",\n \"exoPD-L1\",\n \"immunotherapeutic responses\",\n \"downstream functional analysis\",\n \"sEV-mediated T cell suppression\",\n \"cancer\",\n \"high-throughput\",\n \"validation\"\n]","isConversational":false},"references":{"kind":"list like","value":[],"string":"[]","isConversational":false}}},{"rowIdx":88,"cells":{"unique_id":{"kind":"string","value":"s41467-023-42678-5"},"subfield":{"kind":"string","value":"Optics and photonics"},"title":{"kind":"string","value":"Polarization structured light 3D depth image sensor for scenes with reflective surfaces"},"abstract":{"kind":"string","value":"Highly reflective surfaces are notorious in the field of depth sensing and three-dimensional (3D) imaging because they can cause severe errors in perception of the depth. Despite recent progress in addressing this challenge, there are still no robust and error-free solutions. Here, we devise a polarization structured light 3D sensor for solving these problems, in which high-contrast-grating (HCG) vertical-cavity surface-emitting lasers (VCSELs) are used to exploit the polarization property. We demonstrate accurate depth measurements of the reflective surfaces and objects behind them in various imaging situations. In addition, the absolute error and effective measurement range are measured to prove the applicability for a wide range of 3D applications. Our work innovatively combines polarization and depth information, opening the way for fully understanding and applying polarization properties in the 3D domain."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"Highly reflective surfaces, such as glass, mirror, and water surface, are common scenes in 3D imaging. They often cause irrevocable errors in depth sensing and 3D imaging, such as mirages leading to wrong depth measurements or specular reflection blinding 3D sensors. However, the correct perception of the surrounding environment is critical for many applications, such as robotics, scene reconstruction, and virtual reality. Hence, it is extremely important to manage the problem of reflective surfaces. In recent decades, there are plenty of techniques managing to detect the reflective surfaces, including polarization imaging, active projectors and its fusion with other sensors, and deep learning. However, due to the effects of reflective or multipath noise, most of these techniques will result in erroneous measurements of the reflective surfaces. Polarization imaging is able to obtain the 3D shape of an object by analyzing several images of reflected intensity at different polarized angles. Its first use to determine the 3D orientations of reflective surfaces dated back to the 1990s [1] . Since then, polarization imaging has been used to reconstruct the shape of transparent [2] , [3] , [4] and specular [5] , [6] , [7] , [8] objects. Furthermore, the integration with cues from various depth sensors [9] , [10] , [11] , [12] , [13] , [14] enables polarization imaging to obtain better shape recoveries of the targets. However, polarization imaging needs to capture multiple polarized images and can only estimate relative depths. In addition, most of them requires knowledge of the refractive index of the object, while some rely on other factors to ease this need, such as various light conditions [15] , [16] or multiwavelength [17] . In [11] , Berger combined polarization and stereo vision and further extended the application to general environments with reflective surfaces. However, due to the inability to differentiate mirages from real objects, stereo vision will result in a virtual and wrong distance. Some methods use multiple polarized images [18] , [19] , [20] , [21] , [22] or local reflection cues [23] , [24] to separate the reflection. In some cases, an additional infrared sensor is used to provide depth information and to remove the mirage images [25] , [26] . One major drawback of polarization-stereo-vision methods is that they do not result in correct depth measurements of the reflective surfaces (e.g., the depth of the glass). Active projectors, such as laser range finders and structured light projector, are widely used in the field of depth sensing. In the case of laser range finder, in order to deal with the reflective surfaces, some researchers use the reflected intensity profile to determine the glass area [27] , [28] , [29] . The fusion of laser range finder and sonar [30] , [31] is also developed for navigation in the glass environment. In addition, the method used in polarization imaging is applied to the laser range finder, in which the degree of polarization is calculated and serves as a detection standard of glass areas [32] . However, these methods require scanning around to determine the depth and glass points, thus they are still in the stage of 2D route mapping. As for structured light projectors, a method [33] employs the fusion of Kinect and sonar to obtain the depth images of glass scenes. But, due to the sparse data and narrow angle of the sonar, it needs to sweep multiple times to obtain enough information. Besides, the fusion with sonar will make the system costly, huge and complex. Deep learning is also a good candidate for the detection of reflective surfaces [34] , [35] , [36] , [37] , where a network model trained on abundant images of reflective surfaces is employed to mark out the target area. But we find that even the recent works [36] , [37] might still misjudge the areas with frames or borders, such as an empty frame without glass, as the reflective surfaces. Therefore, a compact and robust method is still needed for applications in environments full of highly reflective surfaces. Here, we develop a polarization structured light (PSL) 3D sensor [38] with polarization properties on both the transmitter (TX) and receiver (RX). In TX, high-contrast-grating (HCG) vertical-cavity surface-emitting lasers (VCSELs) are specially designed to provide structured light with a strong polarization selection ratio. In RX, a polarization-selection CMOS camera is designed to receive the signal selectively. According to Fresnel’s theory [39] , the specular reflection from a reflective surface maintains the same polarization as the incident polarized light. However, diffused reflection from objects without smooth surfaces does not exhibit any polarization even incident by a strongly polarized light. Hence, using a polarization-selection CMOS camera can differentiate reflection from a reflective surface. Thus, the depth information of a reflective surface or objects behind it can be obtained based on the choice of the polarizer direction. Here, we report three experiments to demonstrate how PSL 3D sensors can be used to see as well as to see through highly reflective surfaces.","references":{"indices":[2,3,4,7,13,15,18,20,22,25,26,28,33],"infos":[{"idx":"2","title":"Miyazaki, D., Kagesawa, M. & Ikeuchi, K. Transparent surface modeling from a pair of polarization images.IEEE Trans. Pattern Anal. Mach. Intell.26, 73–82 (2004).","link":"https://doi.org/10.1109%2FTPAMI.2004.1261080"},{"idx":"3","title":"Miyazaki, D. & Ikeuchi, K. Shape estimation of transparent objects by using inverse polarization ray tracing.IEEE Trans. Pattern Anal. Mach. Intell.29, 2018–2030 (2007).","link":"https://doi.org/10.1109%2FTPAMI.2007.1117"},{"idx":"4","title":"Xu, X., Qiao, Y. & Qiu, B. Reconstructing the surface of transparent objects by polarized light measurements.Opt. Express25, 26296–26309 (2017).","link":"https://doi.org/10.1364%2FOE.25.026296"},{"idx":"7","title":"Morel, O., Stolz, C., Meriaudeau, F. & Gorria, P. Active lighting applied to three-dimensional reconstruction of specular metallic surfaces by polarization imaging.Appl. Opt.45, 4062–4068 (2006).","link":"https://doi.org/10.1364%2FAO.45.004062"},{"idx":"13","title":"Tian, X., Liu, R., Wang, Z. & Ma, J. High quality 3d reconstruction based on fusion of polarization imaging and binocular stereo vision.Inf. Fusion77, 19–28 (2022).","link":"https://doi.org/10.1016%2Fj.inffus.2021.07.002"},{"idx":"15","title":"Thilak, V., Voelz, D. G. & Creusere, C. D. Polarization-based index of refraction and reflection angle estimation for remote sensing applications.Appl. Opt.46, 7527–7536 (2007).","link":"https://doi.org/10.1364%2FAO.46.007527"},{"idx":"18","title":"Kong, N., Tai, Y.-W. & Shin, S. Y. High-quality reflection separation using polarized images.IEEE Trans. Image Process.20, 3393–3405 (2011).","link":"https://doi.org/10.1109%2FTIP.2011.2155080"},{"idx":"20","title":"Kong, N., Tai, Y.-W. & Shin, S. Y. A physically-based approach to reflection separation: from physical modeling to constrained optimization.IEEE Trans. Pattern Anal. Mach. Intell.36, 209–221 (2014).","link":"https://doi.org/10.1109%2FTPAMI.2013.45"},{"idx":"22","title":"Wen, S., Zheng, Y. & Lu, F. Polarization guided specular reflection separation.IEEE Trans. Image Process.30, 7280–7291 (2021).","link":"https://doi.org/10.1109%2FTIP.2021.3104188"},{"idx":"25","title":"Sun, J., Chang, Y., Jung, C. & Feng, J. Multi-modal reflection removal using convolutional neural networks.IEEE Signal Process. Lett.26, 1011–1015 (2019).","link":"https://doi.org/10.1109%2FLSP.2019.2915560"},{"idx":"26","title":"Chang, Y., Jung, C. & Sun, J. Joint reflection removal and depth estimation from a single image.IEEE Trans. Cybern.51, 5836–5849 (2021).","link":"https://doi.org/10.1109%2FTCYB.2019.2959381"},{"idx":"28","title":"Wang, X. & Wang, J. Detecting glass in simultaneous localisation and mapping.Robot. Autonomous Syst.88, 97–103 (2017).","link":"https://doi.org/10.1016%2Fj.robot.2016.11.003"},{"idx":"33","title":"Zhang, Y., Ye, M., Manocha, D. & Yang, R. 3D reconstruction in the presence of glass and mirrors by acoustic and visual fusion.IEEE Trans. Pattern Anal. Mach. Intell.40, 1785–1798 (2018).","link":"https://doi.org/10.1109%2FTPAMI.2017.2723883"}]}},{"section":"Results","content":"Polarization structured light 3D sensor The PSL 3D sensor is illustrated in Fig. 1 a. It consists of a transmitter TX, a receiver RX and an RGB camera. In TX, we have embedded an HCG-VCSEL array inside, whose working wavelength is 940 nm. VCSELs are critical sources for dot projectors used in structured light cameras [40] . Typical VCSEL uses distributed Bragg reflectors (DBRs) as its top and bottom mirrors. The DBRs are many 10s of layers of planar heteroepitaxial layers and do not provide or maintain a fixed polarization to the VCSELs. HCG is a thin-film subwavelength metastructure that is effective in providing high reflection with a fixed polarization. Hence, HCG VCSELs have exhibited a very high polarization selection ratio independent of operating temperature or drive conditions [41] . With this specially designed HCG-VCSEL array, the TX of PSL 3D sensor can project dot-array structured light (see Supplementary Fig. S1 and Methods) in either transverse electric (TE) or transverse magnetic (TM) polarization. Fig. 1: Principle of the polarization structured light 3D sensor. a The model of PSL 3D sensor. In TX, polarized structured light is produced by an HCG-VCSEL array inside. In RX, signals are received selectively by a rotatable polarizer mounted on the receiving infrared CMOS camera. In addition to TX and RX, PSL 3D sensor contains a RGB camera in the middle as well. b The scheme of seeing through the reflective surface. On the upper row, it shows the 3D sensor can obtain depth information of both the reflective surface and the objects behind. On the lower row, the 3D sensor can remove the depth of the reflective surface. The specific steps are shown in the flow diagram. c The scheme of seeing against reflective noise. On the left side, it shows the 3D sensor can obtain the correct depth of the scene by eliminating the effect of reflective noise. The flow of this mission is shown on the right. Note that the plane of the PSL 3D sensor is set to be x - y plane and the light propagation direction is in z direction. Full size image The polarization direction of the structured light emitted from TX is denoted by a double-head blue arrow in Fig. 1 a and its projection range is 75. 5° × 65. 7°, which is indicated by a pink area bounded by gray dot arrow lines. Here, we set the plane of the PSL 3D sensor to be x - y plane and the light propagation direction to be z , then define the horizontal ( x direction) polarization, as pointed by the blue arrow in Fig. 1 a, to be TM polarization. In RX end, a rotatable polarizer is mounted on the infrared CMOS camera. Its polarization direction is denoted by a double-head green arrow and its receiving angle is 74° × 50. 5°, which is indicated by a pink area bounded by orange dot arrow lines. The polarizer can be rotated to receive the dot patterns selectively. By calculating the spatial displacements of known dot patterns, depth information can be obtained for an entire image at high frame rates. Seeing through the reflective surface According to the polarization feature of the HCG-VCSEL-based PSL 3D sensor, we can fully exploit its merits for 3D imaging in scenarios with large reflective surfaces. Based on Fresnel equations [39] , when S- or P- polarized light is incident on a highly reflective surface, the reflection will maintain the same polarized direction. As shown in the upper picture of Fig. 1 b, TX of the PSL 3D sensor projects TM-polarization (P-polarization, P in1 ) structured light onto the scene. The reflected light from the reflective surface ( P out1 ), which is denoted by the black solid line, will bounce back with the same polarization as the TX, while the light reflected from the object behind the reflective surface ( S out2 , P out2 ), whose light path is denoted by the black dot line, is polarized in various directions due to diffuse reflection. Then RX can receive the signals ( P out1,2 ) of both the reflective surface and the object behind with its polarizer set in the same polarized direction, i.e., x direction. As shown in the lower picture of Fig. 1 b, if we change the direction of the polarizer to be orthogonal, i.e., y direction, the depth information of the reflective surface is eliminated, leaving that ( S out2 ) from the diffuse object behind. Thus, for seeing through the reflective surface, the process can be divided into three steps, as illustrated in the flow of Fig. 1 b. Step 1 is the projection of polarized structured light with a specific polarization direction ( P in1 ). Step 2 is the setting of RX polarization direction to be orthogonal to TX. Step 3 is obtaining the depth information of the scene ( S out2 ), where the part of the reflective surface ( P out1 ) is removed. In addition, the distinction between two polarization settings of RX can be further used to complete the reflective surface. Seeing against the reflective noise In Fig. 1 c, we also show the principle of working against reflective noise. In this sketch, the influence of natural light ( S in3 , P in3 ) is indicated by the gray dash line, whose incident plane is in the x – z plane. Its reflected component is stronger in S-polarization ( y direction, S out3 ) according to Fresnel equations [39] . Hence it can be eliminated when a P-polarized TX and RX are used to enhance the signal-to-noise ratio (SNR), enabling clear 3D image acquisition of the whole scene where there is strong reflective noise from the natural light. The process for this mission can be concluded into four steps, as illustrated in the flow of Fig. 1 c. Step 1 is the projection of polarized structured light with specific polarization direction ( P in1 ). Step 2 is the determination of polarization direction of noise ( S out3 ). Step 3 is the setting of RX polarization direction to be orthogonal to the reflective noise. Step 4 is obtaining the correct depth information of the scene ( P out1 ). Completing the reflective surface In addition to seeing through the reflective surface and seeing against the reflective noise, we further present the PSL 3D sensor’s capability to detect and complete the reflective surfaces. The two-step working principle is illustrated in Fig. 2 . In step 1, we extract the glass region by the collaboration of the depth and color channels. In the depth channel, we first obtain the depth image of the scene in polarization 0°. Polarization 0° means that both the TX and RX of the PSL 3D sensor are in the same polarization, here we set them in TM polarization. Next, we rotate the RX to the orthogonal polarization, obtaining the depth image in polarization 90°. As illustrated in the principle of seeing through the reflective surface, the glass part can be eliminated because its reflected polarization is orthogonal to the RX. Then, subtraction is applied between these two depth images to capture the area with sharp contrast. Meanwhile, in the color channel, we adopt the deep learning method [36] to get the glass boundary from the RGB image of the scene. For a glass pixel, it is predicted to be 1 (white) and for a non-glass pixel, it is 0 (black). Combining the subtraction result in the depth channel and the boundary result in the color channel, the depth points belonging to the glass can be determined and extracted, which is highlighted in red in the depth image. Note that, for scene without glass, there is no sharp contrast between two polarization settings in the depth channel (see Supplementary Fig. S3 and the results of completing the reflective surface). Fig. 2: Principle of completing the reflective surface. In step 1, the glass region is extracted through a combination of depth and color channels, where the glass points are highlighted in red. In depth channel, two depth images are obtained in polarization 0 ∘ and 90 ∘ respectively. Then, subtraction is employed between these two depth images for extraction of the glass points. In color channel, the glass boundary is predicted from the color image of the scene using deep learning method. In step 2, the extracted glass points are transformed to the world coordinate for fitting. After fitting, the results are mapped to the color channel. In this channel, the reflective surface can be interpolated and completed inside the glass boundary. At last, the completed point cloud can be converted back to the world coordinate, as shown in the before and after comparison on the right side. Full size image After getting the glass region, we move forward to step 2. The extracted glass depth points P d = [ u , v , z ] ⊤ are transformed to the world coordinate of the infrared camera P ir = [ x , y , z ] ⊤ using the following equation: $$\\begin{array}{rcl}x&=&\\frac{z(u-{c}_{x})}{{f}_{x}}\\\\ y&=&\\frac{z(v-{c}_{y})}{{f}_{y}},\\end{array}$$ (1) where { f x , f y , c x , c y } are the internal parameters of the infrared camera. Then the glass is fitted to a large enough area in this world coordinate (Supplementary Fig. S4 explains why it needs to fit in the world coordinate). The fitted glass points are next transformed to the world coordinate of the RGB camera \\({P}_{{{{{{{{\\rm{rgb}}}}}}}}}={[{x}^{{\\prime} },{y}^{{\\prime} },{z}^{{\\prime} }]}^{\\top }\\) with the external parameters, i.e. the rotational matrix R and translational matrix T : $${P}_{{{{{{{{\\rm{rgb}}}}}}}}}=R{P}_{{{{{{{{\\rm{ir}}}}}}}}}+T.$$ (2) These points P rgb are mapped to the color channel \\({P}_{c}={[{u}^{{\\prime} },{v}^{{\\prime} },{z}^{{\\prime} }]}^{\\top }\\) using equation ( 1 ) but with the internal parameters of the RGB camera \\(\\{{f}_{x}^{{\\prime} },{f}_{y}^{{\\prime} },{c}_{x}^{{\\prime} },{c}_{y}^{{\\prime} }\\}\\) . In the color channel, we can interpolate inside the glass boundary according to the fitted glass points P c and finish the completion of the glass. At last, the completed point cloud can be converted back to the world coordinate. As shown in the right part of step 2, the original point cloud is plotted in the first row, where only the middle region of the glass is detected. After applying step 1 and 2, the whole glass area can be completed, which is highlighted in the second row with red color. With this proposed method, not only the glass can be seen and completed correctly, but also the objects behind the glass can be reconstructed, enabling seeing and seeing through the reflective surfaces. Analysis of 3D imaging in scenes with reflective surfaces Before showing the results of three experiments, we first analyse 3D imaging in scenes with reflective surfaces. Among the reflective surfaces, the indoor glass wall is a typically challenging situation (Fig. 3 a, b). In this scene, we place the sensor in front of a glass wall, with a reflecting image of two persons and the sensor on the same side of the glass at about 1.2 m distance in front. In this situation, the depth image obtained with a stereo vision camera (Intel RealSense D455) is shown in Fig. 3 c, which is unable to see the depth of the glass. In fact, the stereo vision camera provides erroneous depth measurements based upon the image of the persons reflected by the glass with a measured distance of about 2.4 m behind the glass wall, whereas the actual should be 1.2 m in front of the glass wall. In contrast, when TM polarized structured light patterns produced from the TX of PSL 3D sensor hit the glass, depth information of part of the glass can be reconstructed with TM polarization in RX, as shown by the area encircled by the dashed line in Fig. 3 d. These measured values agree well with the actual distance and this correct depth information can be further used to demonstrate and complete the glass wall. Fig. 3: Analysis of 3D imaging in scenes with reflective surfaces. ( a ) and ( b ) are scheme and picture of the indoor glass wall scene. The glass wall is about 1.2 m in front of the sensor, with a reflecting image of two persons and the sensor. Note that two persons are on the same side as the sensor. c Depth image from the stereo vision camera (Intel RealSense D455). d Depth image from the PSL 3D sensor. The glass part is denoted by the dashed line. ( e ) and ( g ) are testing scenes of the deep learning method. Their corresponding prediction results are shown in ( f ) and ( h ). The jet color bar applies to ( c ) and ( d ), the gray color bar applies to ( f ) and ( h ). Full size image Deep learning is an effective tool to detect glass [36] , [37] , where a trained deep neural network is used to predict the glass part. However, such RGB-image-based deep learning method will cause misjudgement in some scenes. As shown in Fig. 3 e, a cardboard box with its front surface removed is placed in front of the sensor. Inside the box, there is a box of gloves and a metal breadboard. The RGB image of this scene is sent to the network for prediction and its result is shown in Fig. 3 f. The prediction result is the probability of being considered as glass, whose range is 0 to 1. For a glass pixel, it is evaluated as 1 (white) and for a non-glass pixel, it is 0 (black). From the result, we can see that almost the whole front surface area is considered to be the glass area. This is because the context of having box boundaries and objects inside will lead to errors in prediction. Moreover, we test another glass door scene. As shown in Fig. 3 g, we place the sensor in front of a soundproof room with its door open. That is, there is glass on the left and no glass on the right. Inside the soundproof room, a cardboard box with a basketball on its top is placed on the left, and another box and a book are on the right. Again, the result in Fig. 3 h shows that the network predicts both parts as glass areas. Due to these disadvantages, a much more robust method for the detection of reflective surfaces is required to develop. On the one hand, it needs to obtain the correct 3D information on reflective surfaces. On the other hand, it should be able to determine whether the depth information belongs to them. In the following, we will illustrate how the PSL 3D sensor and its corresponding imaging methods facilitate in such a mission. Results of seeing through the reflective surface We start with a typical situation, i.e. glass door with objects behind, to illustrate PSL 3D sensor’s ability to see through the reflective surface. We analyze different factors that can affect this mission, including the distance from the 3D sensor to the glass, the 3D sensor incident angle, the distance of objects behind glass and the density of objects behind glass. Their specific conditions are listed in Table 1 . In Fig. 4 , we show the influence of the first factor. The color pictures of the scene are shown in the first column. In this situation, the PSL 3D sensor is facing the glass door with a distance changing from 0.4 m to 1.2 m. Behind the glass door, there is a movable cabinet with a basketball on the top. Their distances to the glass are fixed to 0.5 m. The TX of the PSL 3D sensor projects TM-polarized structured light to the scene. Then the structured light is reflected back to the RX, containing depth information of both the glass door and objects behind. As mentioned in the principle, the reflected light of the glass door maintains the same polarization as the TX, while those of the cabinet and basketball are in various polarized directions due to diffuse reflection. Table 1 Influence factors for seeing through the reflective surfaces Full size table Fig. 4: Seeing through the reflective surface. ( a )–( c ) are the results of different distances from 3D sensor to glass. Polarization 0 ∘ is the depth map when TX and RX are in the same polarization. Polarization 90 ∘ is the depth map when TX and RX are in the orthogonal polarization. The yellow-black color bars apply to both second and third columns. Full size image Hence, when we set the RX to be TM polarization, depth information of the glass door, cabinet, and basketball can all be reconstructed, which is shown in Polarization 0° of Fig. 4 a. In this depth map, bright yellow represents the near points and black represents the far or missing points. The middle bright yellow area accurately reveals the depth of the glass, while the farther cabinet and basketball are darker. Unlike Fig. 3 d, the central part shows the depth of the basketball rather than the glass because reflection from the basketball is stronger. Next, if we rotate the RX to the orthogonal direction, i.e. TE polarization, the glass part can be filtered, leaving clear depth information of the cabinet and basketball (Polarization 90° in Fig. 4 a). As we change the distance to 0.8 m, the middle ball area becomes smaller because the glass depth begins to dominate (dot line in Polarization 0° of Fig. 4 b). As for 1.2 m (Fig. 4 c), the middle is almost replaced by glass depth, while the whole proportion of glass depth is smaller as compared to that of 0.4 m. In Polarization 90° for both cases, the glass part can still be eliminated. More details about the incident angle of 3D sensor, distance and density of objects behind glass are displayed in Supplementary Fig. S5 – S8 . Regardless of different incidence angles, or different distances and object densities, the 3D sensor can eliminate the depth of the glass and achieve the function of seeing through the reflective surface. Supplementary Movie 1 clearly shows this phenomenon by rotating the RX polarizer 360 degrees. The high contrast between Polarization 0° and Polarization 90° can further serve as a determination method of the reflective surface, which will be demonstrated in the section after next. Results of seeing against the reflective noise We also set up outdoor and indoor scenes to demonstrate the advantage of detecting against noise (Fig. 5 ). We first show a case of outdoor glass scene (Fig. 5 a). In this situation, a book is placed behind a tilted glass, where there is strong specular reflection from the natural light. We can see that the reflected noise incapacitates the stereo vision camera for obtaining the depth of the scene, leading to black-irregular-patch errors in the depth map (encircled by white and red dot lines while the book is indicated in red). However, as mentioned in the Fig. 1 c, this noise has a larger component of S polarization. Note that in this setting, S polarization is in the horizontal direction ( x direction) when facing the glass. Therefore, we can utilize P-polarized (i.e. y -direction polarization) TX and RX to filter this noise, enhancing the SNR. As shown in Fig. 5 c, the book behind the glass (encircled by the red dot line) can be measured clearly with the PSL 3D sensor in this P-polarized setting. Fig. 5: Seeing against the reflective noise. ( a )–( c ) are comparison results of an outdoor glass scene, where ( a ) is the picture of the scene, ( b ) is the depth map from stereo vision camera (Intel RealSense D455) and ( c ) is the depth map from PSL 3D sensor. ( d ) – ( l ) are comparison results of an indoor corner scene. The experiments are done in different height, whose schemes are shown in ( d ), ( g ), and ( j ). The corresponding depth map and point cloud from stereo vision camera are displayed in ( e ), ( h ) and ( k ) respectively, while those from PSL 3D sensor are in ( f ), ( i ), and ( l ). The jet color bar with range (0.48 m, 1.1 m) applies to ( b ), ( c ), and the one with range (0.3 m, 1 m) applies to the depth map column. Full size image Besides this sunlight noise, the shadow of objects may also severely blind the depth camera based on stereo vision. As illustrated before in Fig. 3 c, the reflection noise of two persons makes the stereo vision camera provide erroneous measurement. Another outdoor scene with reflection overlapped with the detecting object is also illustrated in Supplementary Fig. S9 , where the stereo vision camera is still affected by the reflection of outside objects while the PSL 3D sensor can get the depth information correctly. In the next case, we analyze the ability to detect against noise in the situation of wall corner, which is a common scene for indoor serving robot. The results are displayed in Fig. 5 d–l. As shown in Fig. 5 d, the comparison is first done at a height of 0.24 m and a distance of 0.7 m away from the corner, with the sensor 45° downward. In this situation, due to the influence of reflection and multipath noise, the stereo vision camera will make the wrong depth judgment. As indicated by the red dot line in Fig. 5 e, the depth has gone deep below the floor plane. Only a small part (orange dot line) has accurate floor information. However, for the PSL 3D sensor (Fig. 5 f), a complete and correct shape of the corner can be obtained by setting TX and RX to be P polarization ( y direction). Then we change to the height of 0.4 m (Fig. 5 g), with a distance 0.7 m and an incline angle 45° unchanged. It means that the incident angle decreases. In this condition, the correct floor area obtained from the stereo vision camera starts to increase (orange dot line in Fig. 5 h), but the depth area encircled by the red dot line still penetrates into the floor. As we change to the height of 0.56 m (Fig. 5 j), the floor area of the stereo vision camera goes back to the correct plane (Fig. 5 k), but there are still some black speckles in the depth map because of noise. However, for the PSL 3D sensor, the intact depth map and point cloud are maintained in these two heights (Fig. 5 i and l) with this P-polarized setting. For these common scenes of reflective surface, PSL 3D sensor can remove noise through specific polarization combinations, improve the signal-to-noise ratio, and obtain complete depth information of the scene. More polarization combination experiments and comparison with time of flight (ToF) camera, presented in Supplementary Fig. S10– - S13 , further demonstrate the superiority of PSL 3D sensor in these scenarios. Results of completing the reflective surface In the following, we prove the feasibility of completing the reflective surfaces. The results of eight common scenes are selected to illustrate in Fig. 6 and Supplementary Fig. S14 . In Fig. 6 , we display four of these scenes, including balcony glass, soundproof room glass, spherical glass and office door glass. In the balcony glass scene, a yellow square box is placed outside the glass door which has reflections of the interior furniture. With polarization 0° and polarization 90° settings, we obtain two depth images of the scene. In polarization 0°, partial of the glass and the box outside can both be reconstructed while the glass part is eliminated in polarization 90°. Compared to the stereo vision camera, which fails to detect the glass depth and misjudges the reflections as real objects, such a change in depth channel can serve as a reliable cue for determining the reflective surfaces. Thus, using the subtraction of two depth images and the glass boundary predicted from the RGB image of the scene, we are able to extract the glass region, which is highlighted in red in the fifth row. The extracted glass points are then used to fit and complete the reflective surface. The completed depth image is shown in the seventh row, where the partially empty glass has now been filled in. The final comparison between the original point cloud and the completed point cloud is shown in the last two rows. In the completed point cloud, we can see that the new glass plane, which is colored in red, matches well with the glass frame and the 3D information of the box outside the glass can be acquired simultaneously. Fig. 6: Detection and completion results. The results of four scenes are presented column-wise in this figure. The first row is the RGB image of each scene. The second and third rows are the depth map from polarization 0 ∘ and polarization 90 ∘ settings respectively. The fourth row is the predicted glass boundary using the deep learning method. The fifth row is the extracted glass region, which is obtained from the combination of depth map subtraction and glass boundary. The sixth row shows the fitting of the extracted glass points in the world coordinate. The seventh row is the completed depth map. The original point cloud and the completed point cloud are shown in the last two rows. In the completed point cloud, the completed glass is highlighted in red. The parula color bar applies to the last two rows and the yellow-black color bar applies to the rows of polarization 0 ∘ , polarization 90 ∘ , and depth completion. Full size image As for the soundproof room glass scene, we execute the same process. In this scene, the right door of the soundproof room is open, that is, only the left side has glass. Inside the room, a cardboard box with a basketball on its top is placed on the left, and another box and a book are on the right. As seen in the depth images, only the left side with glass has a large change while the right side shows no significant change. If simply relying on the predicted glass boundary from deep learning, we will mistakenly consider the right side as glass. However, our method can determine the regions that actually have glass and extract effective depth information belonging to the glass. Similarly, this accurately extracted depth information is used for the following fitting and completion. In the completed results, the left glass door and objects inside the room can be reconstructed, as if exploring the inside and outside of the soundproof room from God’s perspective. Besides planar glass, our method can also be applied to curved glass, which is displayed in the third column. In this situation, we use a spherical exhibition stand with two 3D-printed models inside as the detecting object. Likewise, two depth images are obtained from two polarization directions. We can see that the front face of the glass ball is eliminated in polarization 90°. With the ball-shaped glass boundary, we acquire the glass region in the fifth row. These points are from the spherical glass so the fitting result is a quadric surface. Then we complete the glass inside the glass boundary and get the completion result. As seen from the point cloud in the last row, it completes well the front surface of the spherical glass. Note that, there are still some misjudged points in the predicted glass boundary, such as the unsmoothness and the false positive outliers at the ball edge. These minor flaws can be solved by optimizing the predicting network in future work. At last, we show an office door glass scene with a larger tilted angle than the first column. In like manner, there is a cabinet with a ball on its top behind the glass. Due to different imaging angles and distances, the position of the glass depth is different in polarization 0°, whereas it will not affect the elimination of glass in polarization 90°. With the same subtraction method, the glass points are extracted and shown in the fifth row. Then the glass door is fitted and completed. From the completion results in the seventh row and the last row, we can see that the glass can be completed well and the box and ball behind the glass can also be reconstructed, reinforcing the idea of seeing and seeing through the reflective surfaces. It can be seen from the above experiments that the position and proportion of effective glass information are variational for different experiment situations. This is because of the combined effect of the illumination angle and the cleanliness of the reflective surface. In Supplementary Fig. S15 , we analyze the influence of different cleanliness of the reflective surface on the detection area. In fact, if the reflective surface perfectly follows the specular reflection model, only an area with the size of the emitter will be fully received. But in common scenes, this kind of situation rarely exists on the reflective surface. Actually, it will be affected by the randomly distributed particles on the surface, expanding the reflection range to a certain extent, which is the so-called off-specular reflection [42] , [43] . As our experiments show, the PSL 3D sensor can extract enough effective glass information in various scenes. We also describe the effective measurement angles in Supplementary Fig. S16 . We fix the position of the sensor facing the glass and then measure the working range by rotating the sensor horizontally and vertically to measure the proportion of the glass points. We use the number at 0° as a base and the ratio down to 2% as the limit, and the measured working range is ± 45° horizontally and ± 30° vertically. Moreover, the rest of the eight scenes are also illustrated in Supplementary Fig. S14 , demonstrating the general applicability of our method. As for the measurement accuracy, we use the absolute error to evaluate, and the results are shown in Fig. 7 . Among them, Fig. 7 a-h are the error maps of eight scenes, where Fig. 7 a-d correspond to the four scenes in Fig. 6 and Fig. 7 e-h correspond to the four scenes in Supplementary Fig. S14 . Specifically, we cover the reflective surfaces with thin stickers and conduct depth detection on the scene again, the fitted depth of which is used as a benchmark. We then calculate the absolute error between the depth map of the completed reflective surface and this benchmark. Then we perform boxplot statistics on these eight error maps, and the results are shown in Fig. 7 i. The green box includes the median and the upper and lower quartiles of each error map, the gray points are outliers exceeding 1.5 times the interquartile range, and the yellow circle dashed line is the mean value of each error map. From the error analysis, we can see that the errors are maintained at the level of 1mm, demonstrating the precise reconstruction of our method. Note that there are outliers in the error maps. These errors mainly come from the fluctuation of the data, such as the influence of the interreflection, which can be reduced to a much lower level if we include the model to suppress it in the future. Fig. 7: Error analysis. ( a )–( h ) are the error maps of eight scenes of reflective surfaces, which are the absolute errors between the completed reflective surface and the fitted benchmark. i The boxplot graph of these eight error maps. It includes the box, mean value and outliers of each map, showing their statistical properties. Full size image The above experiments of completion are from a single view. If we want to reconstruct the whole scene, we need to employ our method in different views. In Supplementary Fig. S17 , we show the reconstruction of a fish tank by employing our method in four sides of the fish tank. In addition, a more complicated scene with a mirror at the back of glass is used to prove the feasibility of our method in Supplementary Fig. S18 . The ToF camera is also compared in Supplementary Fig. S19 , which once again demonstrates the distinction of the PSL 3D sensor."},{"section":"Discussion","content":"In summary, we have invented an HCG-VCSEL-based polarization structured light (PSL) 3D sensor and proposed the corresponding imaging methods for 3D reconstruction in a wide type of scenes with highly reflective surfaces. We first chose a glass scene to demonstrate the ability of the PSL 3D sensor to obtain 3D information of objects behind reflective surfaces. Next, we compare with the stereo vision camera and ToF camera, and show that in the scene with strong reflective noise, through a specific polarization setting, our sensor can eliminate the reflective noise and accurately obtain the 3D image of the scene. Then, we select 8 different scenes to prove that our method can be robustly applied to the detection and completion of the reflective surfaces. Moreover, we show that the effective working range of incident angle is as large as 90° × 60°. In contrast, stereo vision cameras, ToF cameras, and deep learning methods will cause erroneous measurements in these scenarios. At last, we evaluate the absolute errors of our detecting results, and the mean error of each scene maintains at the level as low as 1mm, equivalent to 0.1%. Through these different experiments, we demonstrate that the PSL 3D sensor can realize seeing and seeing through the reflective surfaces. Currently, the proposed methods are conducted in post-processing, but in fact they can be integrated into the module. We can also modify and optimize the design of the deep learning network, so that they can be transferred to the processing chip of the sensor for real-time detection and completion. On the receiving end, we can also design a unique polarizer array like that in [44] , which can be coated on the CMOS camera and receive the signal from different polarizations simultaneously, making our system more compact. Serving as the eyes of a robot, PSL 3D sensors will find many applications such as service robots and logistics robots. These robots will inevitably encounter the problem of reflective surfaces such as the glass door and floor corner shown in the main text. The PSL 3D sensor can make them handle these cases correctly. Besides, object or scene reconstruction of reflective surfaces, such as office and exhibition hall, will also benefit from this method. Thus, with the special polarization characteristic, the PSL 3D sensor can be further extended to a wide range of indoor and outdoor applications."},{"section":"Methods","content":"Generation of dot-array structured light The polarized dot-array structured light is generated directly from the TX. Inside the TX, there is a 940 nm HCG-VCSEL array, a collimating lens with 4.8 mm focal length and a diffractive optical element (DOE). The HCG-VCSEL array and the DOE are mounted at the focal points of both sides of the lens. By designing the bandwidth and airgap of the HCG, the VCSEL sources produce polarized Gaussian beams. These beams propagate to the lens and are collimated to compensate the divergence. Then the collimated light travels to the DOE. Passing through the DOE, one beam can diffract to 11 × 9 orders uniformly. If we properly position the VCSEL on the array, then the far-field distribution of the laser array can be duplicated in 11 × 9 copies, enabling the production of dot-array structured light over 30,000 points. The far-field dot-array distribution is shown in Supplementary Fig. S2 . Postprocessing of the point cloud data and RGB images The reading and drawing of point clouds are done by the Point Cloud Processing of Computer Vision Toolbox in MATLAB. The fitting uses the fit function of Curve Fitting Toolbox with its robust option set to bisquare. The other processing of the point clouds is also based on MATLAB. The prediction of the reflective surface boundary is based on PyTorch. In addition, the visualization of the point clouds in Fig. 5 , Supplementary Fig. S10 – S13 and Supplementary Fig. S17 is based on CloudCompare [45] . Calculation of the depth of reflective surfaces By calculating the spatial displacements of known dot patterns, depth information of reflective surfaces can be obtained according to triangulation. The received dot patterns on the reflective surfaces are shown in Supplementary Fig. S20 and a model of triangulation is also shown to illustrate how depth is calculated."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"Highly reflective surfaces, such as glass, mirror, and water surface, are common scenes in 3D imaging. They often cause irrevocable errors in depth sensing and 3D imaging, such as mirages leading to wrong depth measurements or specular reflection blinding 3D sensors. However, the correct perception of the surrounding environment is critical for many applications, such as robotics, scene reconstruction, and virtual reality. Hence, it is extremely important to manage the problem of reflective surfaces. In recent decades, there are plenty of techniques managing to detect the reflective surfaces, including polarization imaging, active projectors and its fusion with other sensors, and deep learning. However, due to the effects of reflective or multipath noise, most of these techniques will result in erroneous measurements of the reflective surfaces. Polarization imaging is able to obtain the 3D shape of an object by analyzing several images of reflected intensity at different polarized angles. Its first use to determine the 3D orientations of reflective surfaces dated back to the 1990s [1] . Since then, polarization imaging has been used to reconstruct the shape of transparent [2] , [3] , [4] and specular [5] , [6] , [7] , [8] objects. Furthermore, the integration with cues from various depth sensors [9] , [10] , [11] , [12] , [13] , [14] enables polarization imaging to obtain better shape recoveries of the targets. However, polarization imaging needs to capture multiple polarized images and can only estimate relative depths. In addition, most of them requires knowledge of the refractive index of the object, while some rely on other factors to ease this need, such as various light conditions [15] , [16] or multiwavelength [17] . In [11] , Berger combined polarization and stereo vision and further extended the application to general environments with reflective surfaces. However, due to the inability to differentiate mirages from real objects, stereo vision will result in a virtual and wrong distance. Some methods use multiple polarized images [18] , [19] , [20] , [21] , [22] or local reflection cues [23] , [24] to separate the reflection. In some cases, an additional infrared sensor is used to provide depth information and to remove the mirage images [25] , [26] . One major drawback of polarization-stereo-vision methods is that they do not result in correct depth measurements of the reflective surfaces (e.g., the depth of the glass). Active projectors, such as laser range finders and structured light projector, are widely used in the field of depth sensing. In the case of laser range finder, in order to deal with the reflective surfaces, some researchers use the reflected intensity profile to determine the glass area [27] , [28] , [29] . The fusion of laser range finder and sonar [30] , [31] is also developed for navigation in the glass environment. In addition, the method used in polarization imaging is applied to the laser range finder, in which the degree of polarization is calculated and serves as a detection standard of glass areas [32] . However, these methods require scanning around to determine the depth and glass points, thus they are still in the stage of 2D route mapping. As for structured light projectors, a method [33] employs the fusion of Kinect and sonar to obtain the depth images of glass scenes. But, due to the sparse data and narrow angle of the sonar, it needs to sweep multiple times to obtain enough information. Besides, the fusion with sonar will make the system costly, huge and complex. Deep learning is also a good candidate for the detection of reflective surfaces [34] , [35] , [36] , [37] , where a network model trained on abundant images of reflective surfaces is employed to mark out the target area. But we find that even the recent works [36] , [37] might still misjudge the areas with frames or borders, such as an empty frame without glass, as the reflective surfaces. Therefore, a compact and robust method is still needed for applications in environments full of highly reflective surfaces. Here, we develop a polarization structured light (PSL) 3D sensor [38] with polarization properties on both the transmitter (TX) and receiver (RX). In TX, high-contrast-grating (HCG) vertical-cavity surface-emitting lasers (VCSELs) are specially designed to provide structured light with a strong polarization selection ratio. In RX, a polarization-selection CMOS camera is designed to receive the signal selectively. According to Fresnel’s theory [39] , the specular reflection from a reflective surface maintains the same polarization as the incident polarized light. However, diffused reflection from objects without smooth surfaces does not exhibit any polarization even incident by a strongly polarized light. Hence, using a polarization-selection CMOS camera can differentiate reflection from a reflective surface. Thus, the depth information of a reflective surface or objects behind it can be obtained based on the choice of the polarizer direction. Here, we report three experiments to demonstrate how PSL 3D sensors can be used to see as well as to see through highly reflective surfaces.\",\n \"references\": {\n \"indices\": [\n 2,\n 3,\n 4,\n 7,\n 13,\n 15,\n 18,\n 20,\n 22,\n 25,\n 26,\n 28,\n 33\n ],\n \"infos\": [\n {\n \"idx\": \"2\",\n \"title\": \"Miyazaki, D., Kagesawa, M. & Ikeuchi, K. Transparent surface modeling from a pair of polarization images.IEEE Trans. Pattern Anal. Mach. Intell.26, 73–82 (2004).\",\n \"link\": \"https://doi.org/10.1109%2FTPAMI.2004.1261080\"\n },\n {\n \"idx\": \"3\",\n \"title\": \"Miyazaki, D. & Ikeuchi, K. Shape estimation of transparent objects by using inverse polarization ray tracing.IEEE Trans. Pattern Anal. Mach. Intell.29, 2018–2030 (2007).\",\n \"link\": \"https://doi.org/10.1109%2FTPAMI.2007.1117\"\n },\n {\n \"idx\": \"4\",\n \"title\": \"Xu, X., Qiao, Y. & Qiu, B. Reconstructing the surface of transparent objects by polarized light measurements.Opt. Express25, 26296–26309 (2017).\",\n \"link\": \"https://doi.org/10.1364%2FOE.25.026296\"\n },\n {\n \"idx\": \"7\",\n \"title\": \"Morel, O., Stolz, C., Meriaudeau, F. & Gorria, P. Active lighting applied to three-dimensional reconstruction of specular metallic surfaces by polarization imaging.Appl. Opt.45, 4062–4068 (2006).\",\n \"link\": \"https://doi.org/10.1364%2FAO.45.004062\"\n },\n {\n \"idx\": \"13\",\n \"title\": \"Tian, X., Liu, R., Wang, Z. & Ma, J. High quality 3d reconstruction based on fusion of polarization imaging and binocular stereo vision.Inf. Fusion77, 19–28 (2022).\",\n \"link\": \"https://doi.org/10.1016%2Fj.inffus.2021.07.002\"\n },\n {\n \"idx\": \"15\",\n \"title\": \"Thilak, V., Voelz, D. G. & Creusere, C. D. Polarization-based index of refraction and reflection angle estimation for remote sensing applications.Appl. Opt.46, 7527–7536 (2007).\",\n \"link\": \"https://doi.org/10.1364%2FAO.46.007527\"\n },\n {\n \"idx\": \"18\",\n \"title\": \"Kong, N., Tai, Y.-W. & Shin, S. Y. High-quality reflection separation using polarized images.IEEE Trans. Image Process.20, 3393–3405 (2011).\",\n \"link\": \"https://doi.org/10.1109%2FTIP.2011.2155080\"\n },\n {\n \"idx\": \"20\",\n \"title\": \"Kong, N., Tai, Y.-W. & Shin, S. Y. A physically-based approach to reflection separation: from physical modeling to constrained optimization.IEEE Trans. Pattern Anal. Mach. Intell.36, 209–221 (2014).\",\n \"link\": \"https://doi.org/10.1109%2FTPAMI.2013.45\"\n },\n {\n \"idx\": \"22\",\n \"title\": \"Wen, S., Zheng, Y. & Lu, F. Polarization guided specular reflection separation.IEEE Trans. Image Process.30, 7280–7291 (2021).\",\n \"link\": \"https://doi.org/10.1109%2FTIP.2021.3104188\"\n },\n {\n \"idx\": \"25\",\n \"title\": \"Sun, J., Chang, Y., Jung, C. & Feng, J. Multi-modal reflection removal using convolutional neural networks.IEEE Signal Process. Lett.26, 1011–1015 (2019).\",\n \"link\": \"https://doi.org/10.1109%2FLSP.2019.2915560\"\n },\n {\n \"idx\": \"26\",\n \"title\": \"Chang, Y., Jung, C. & Sun, J. Joint reflection removal and depth estimation from a single image.IEEE Trans. Cybern.51, 5836–5849 (2021).\",\n \"link\": \"https://doi.org/10.1109%2FTCYB.2019.2959381\"\n },\n {\n \"idx\": \"28\",\n \"title\": \"Wang, X. & Wang, J. Detecting glass in simultaneous localisation and mapping.Robot. Autonomous Syst.88, 97–103 (2017).\",\n \"link\": \"https://doi.org/10.1016%2Fj.robot.2016.11.003\"\n },\n {\n \"idx\": \"33\",\n \"title\": \"Zhang, Y., Ye, M., Manocha, D. & Yang, R. 3D reconstruction in the presence of glass and mirrors by acoustic and visual fusion.IEEE Trans. Pattern Anal. Mach. Intell.40, 1785–1798 (2018).\",\n \"link\": \"https://doi.org/10.1109%2FTPAMI.2017.2723883\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Polarization structured light 3D sensor The PSL 3D sensor is illustrated in Fig. 1 a. It consists of a transmitter TX, a receiver RX and an RGB camera. In TX, we have embedded an HCG-VCSEL array inside, whose working wavelength is 940 nm. VCSELs are critical sources for dot projectors used in structured light cameras [40] . Typical VCSEL uses distributed Bragg reflectors (DBRs) as its top and bottom mirrors. The DBRs are many 10s of layers of planar heteroepitaxial layers and do not provide or maintain a fixed polarization to the VCSELs. HCG is a thin-film subwavelength metastructure that is effective in providing high reflection with a fixed polarization. Hence, HCG VCSELs have exhibited a very high polarization selection ratio independent of operating temperature or drive conditions [41] . With this specially designed HCG-VCSEL array, the TX of PSL 3D sensor can project dot-array structured light (see Supplementary Fig. S1 and Methods) in either transverse electric (TE) or transverse magnetic (TM) polarization. Fig. 1: Principle of the polarization structured light 3D sensor. a The model of PSL 3D sensor. In TX, polarized structured light is produced by an HCG-VCSEL array inside. In RX, signals are received selectively by a rotatable polarizer mounted on the receiving infrared CMOS camera. In addition to TX and RX, PSL 3D sensor contains a RGB camera in the middle as well. b The scheme of seeing through the reflective surface. On the upper row, it shows the 3D sensor can obtain depth information of both the reflective surface and the objects behind. On the lower row, the 3D sensor can remove the depth of the reflective surface. The specific steps are shown in the flow diagram. c The scheme of seeing against reflective noise. On the left side, it shows the 3D sensor can obtain the correct depth of the scene by eliminating the effect of reflective noise. The flow of this mission is shown on the right. Note that the plane of the PSL 3D sensor is set to be x - y plane and the light propagation direction is in z direction. Full size image The polarization direction of the structured light emitted from TX is denoted by a double-head blue arrow in Fig. 1 a and its projection range is 75. 5° × 65. 7°, which is indicated by a pink area bounded by gray dot arrow lines. Here, we set the plane of the PSL 3D sensor to be x - y plane and the light propagation direction to be z , then define the horizontal ( x direction) polarization, as pointed by the blue arrow in Fig. 1 a, to be TM polarization. In RX end, a rotatable polarizer is mounted on the infrared CMOS camera. Its polarization direction is denoted by a double-head green arrow and its receiving angle is 74° × 50. 5°, which is indicated by a pink area bounded by orange dot arrow lines. The polarizer can be rotated to receive the dot patterns selectively. By calculating the spatial displacements of known dot patterns, depth information can be obtained for an entire image at high frame rates. Seeing through the reflective surface According to the polarization feature of the HCG-VCSEL-based PSL 3D sensor, we can fully exploit its merits for 3D imaging in scenarios with large reflective surfaces. Based on Fresnel equations [39] , when S- or P- polarized light is incident on a highly reflective surface, the reflection will maintain the same polarized direction. As shown in the upper picture of Fig. 1 b, TX of the PSL 3D sensor projects TM-polarization (P-polarization, P in1 ) structured light onto the scene. The reflected light from the reflective surface ( P out1 ), which is denoted by the black solid line, will bounce back with the same polarization as the TX, while the light reflected from the object behind the reflective surface ( S out2 , P out2 ), whose light path is denoted by the black dot line, is polarized in various directions due to diffuse reflection. Then RX can receive the signals ( P out1,2 ) of both the reflective surface and the object behind with its polarizer set in the same polarized direction, i.e., x direction. As shown in the lower picture of Fig. 1 b, if we change the direction of the polarizer to be orthogonal, i.e., y direction, the depth information of the reflective surface is eliminated, leaving that ( S out2 ) from the diffuse object behind. Thus, for seeing through the reflective surface, the process can be divided into three steps, as illustrated in the flow of Fig. 1 b. Step 1 is the projection of polarized structured light with a specific polarization direction ( P in1 ). Step 2 is the setting of RX polarization direction to be orthogonal to TX. Step 3 is obtaining the depth information of the scene ( S out2 ), where the part of the reflective surface ( P out1 ) is removed. In addition, the distinction between two polarization settings of RX can be further used to complete the reflective surface. Seeing against the reflective noise In Fig. 1 c, we also show the principle of working against reflective noise. In this sketch, the influence of natural light ( S in3 , P in3 ) is indicated by the gray dash line, whose incident plane is in the x – z plane. Its reflected component is stronger in S-polarization ( y direction, S out3 ) according to Fresnel equations [39] . Hence it can be eliminated when a P-polarized TX and RX are used to enhance the signal-to-noise ratio (SNR), enabling clear 3D image acquisition of the whole scene where there is strong reflective noise from the natural light. The process for this mission can be concluded into four steps, as illustrated in the flow of Fig. 1 c. Step 1 is the projection of polarized structured light with specific polarization direction ( P in1 ). Step 2 is the determination of polarization direction of noise ( S out3 ). Step 3 is the setting of RX polarization direction to be orthogonal to the reflective noise. Step 4 is obtaining the correct depth information of the scene ( P out1 ). Completing the reflective surface In addition to seeing through the reflective surface and seeing against the reflective noise, we further present the PSL 3D sensor’s capability to detect and complete the reflective surfaces. The two-step working principle is illustrated in Fig. 2 . In step 1, we extract the glass region by the collaboration of the depth and color channels. In the depth channel, we first obtain the depth image of the scene in polarization 0°. Polarization 0° means that both the TX and RX of the PSL 3D sensor are in the same polarization, here we set them in TM polarization. Next, we rotate the RX to the orthogonal polarization, obtaining the depth image in polarization 90°. As illustrated in the principle of seeing through the reflective surface, the glass part can be eliminated because its reflected polarization is orthogonal to the RX. Then, subtraction is applied between these two depth images to capture the area with sharp contrast. Meanwhile, in the color channel, we adopt the deep learning method [36] to get the glass boundary from the RGB image of the scene. For a glass pixel, it is predicted to be 1 (white) and for a non-glass pixel, it is 0 (black). Combining the subtraction result in the depth channel and the boundary result in the color channel, the depth points belonging to the glass can be determined and extracted, which is highlighted in red in the depth image. Note that, for scene without glass, there is no sharp contrast between two polarization settings in the depth channel (see Supplementary Fig. S3 and the results of completing the reflective surface). Fig. 2: Principle of completing the reflective surface. In step 1, the glass region is extracted through a combination of depth and color channels, where the glass points are highlighted in red. In depth channel, two depth images are obtained in polarization 0 ∘ and 90 ∘ respectively. Then, subtraction is employed between these two depth images for extraction of the glass points. In color channel, the glass boundary is predicted from the color image of the scene using deep learning method. In step 2, the extracted glass points are transformed to the world coordinate for fitting. After fitting, the results are mapped to the color channel. In this channel, the reflective surface can be interpolated and completed inside the glass boundary. At last, the completed point cloud can be converted back to the world coordinate, as shown in the before and after comparison on the right side. Full size image After getting the glass region, we move forward to step 2. The extracted glass depth points P d = [ u , v , z ] ⊤ are transformed to the world coordinate of the infrared camera P ir = [ x , y , z ] ⊤ using the following equation: $$\\\\begin{array}{rcl}x&=&\\\\frac{z(u-{c}_{x})}{{f}_{x}}\\\\\\\\ y&=&\\\\frac{z(v-{c}_{y})}{{f}_{y}},\\\\end{array}$$ (1) where { f x , f y , c x , c y } are the internal parameters of the infrared camera. Then the glass is fitted to a large enough area in this world coordinate (Supplementary Fig. S4 explains why it needs to fit in the world coordinate). The fitted glass points are next transformed to the world coordinate of the RGB camera \\\\({P}_{{{{{{{{\\\\rm{rgb}}}}}}}}}={[{x}^{{\\\\prime} },{y}^{{\\\\prime} },{z}^{{\\\\prime} }]}^{\\\\top }\\\\) with the external parameters, i.e. the rotational matrix R and translational matrix T : $${P}_{{{{{{{{\\\\rm{rgb}}}}}}}}}=R{P}_{{{{{{{{\\\\rm{ir}}}}}}}}}+T.$$ (2) These points P rgb are mapped to the color channel \\\\({P}_{c}={[{u}^{{\\\\prime} },{v}^{{\\\\prime} },{z}^{{\\\\prime} }]}^{\\\\top }\\\\) using equation ( 1 ) but with the internal parameters of the RGB camera \\\\(\\\\{{f}_{x}^{{\\\\prime} },{f}_{y}^{{\\\\prime} },{c}_{x}^{{\\\\prime} },{c}_{y}^{{\\\\prime} }\\\\}\\\\) . In the color channel, we can interpolate inside the glass boundary according to the fitted glass points P c and finish the completion of the glass. At last, the completed point cloud can be converted back to the world coordinate. As shown in the right part of step 2, the original point cloud is plotted in the first row, where only the middle region of the glass is detected. After applying step 1 and 2, the whole glass area can be completed, which is highlighted in the second row with red color. With this proposed method, not only the glass can be seen and completed correctly, but also the objects behind the glass can be reconstructed, enabling seeing and seeing through the reflective surfaces. Analysis of 3D imaging in scenes with reflective surfaces Before showing the results of three experiments, we first analyse 3D imaging in scenes with reflective surfaces. Among the reflective surfaces, the indoor glass wall is a typically challenging situation (Fig. 3 a, b). In this scene, we place the sensor in front of a glass wall, with a reflecting image of two persons and the sensor on the same side of the glass at about 1.2 m distance in front. In this situation, the depth image obtained with a stereo vision camera (Intel RealSense D455) is shown in Fig. 3 c, which is unable to see the depth of the glass. In fact, the stereo vision camera provides erroneous depth measurements based upon the image of the persons reflected by the glass with a measured distance of about 2.4 m behind the glass wall, whereas the actual should be 1.2 m in front of the glass wall. In contrast, when TM polarized structured light patterns produced from the TX of PSL 3D sensor hit the glass, depth information of part of the glass can be reconstructed with TM polarization in RX, as shown by the area encircled by the dashed line in Fig. 3 d. These measured values agree well with the actual distance and this correct depth information can be further used to demonstrate and complete the glass wall. Fig. 3: Analysis of 3D imaging in scenes with reflective surfaces. ( a ) and ( b ) are scheme and picture of the indoor glass wall scene. The glass wall is about 1.2 m in front of the sensor, with a reflecting image of two persons and the sensor. Note that two persons are on the same side as the sensor. c Depth image from the stereo vision camera (Intel RealSense D455). d Depth image from the PSL 3D sensor. The glass part is denoted by the dashed line. ( e ) and ( g ) are testing scenes of the deep learning method. Their corresponding prediction results are shown in ( f ) and ( h ). The jet color bar applies to ( c ) and ( d ), the gray color bar applies to ( f ) and ( h ). Full size image Deep learning is an effective tool to detect glass [36] , [37] , where a trained deep neural network is used to predict the glass part. However, such RGB-image-based deep learning method will cause misjudgement in some scenes. As shown in Fig. 3 e, a cardboard box with its front surface removed is placed in front of the sensor. Inside the box, there is a box of gloves and a metal breadboard. The RGB image of this scene is sent to the network for prediction and its result is shown in Fig. 3 f. The prediction result is the probability of being considered as glass, whose range is 0 to 1. For a glass pixel, it is evaluated as 1 (white) and for a non-glass pixel, it is 0 (black). From the result, we can see that almost the whole front surface area is considered to be the glass area. This is because the context of having box boundaries and objects inside will lead to errors in prediction. Moreover, we test another glass door scene. As shown in Fig. 3 g, we place the sensor in front of a soundproof room with its door open. That is, there is glass on the left and no glass on the right. Inside the soundproof room, a cardboard box with a basketball on its top is placed on the left, and another box and a book are on the right. Again, the result in Fig. 3 h shows that the network predicts both parts as glass areas. Due to these disadvantages, a much more robust method for the detection of reflective surfaces is required to develop. On the one hand, it needs to obtain the correct 3D information on reflective surfaces. On the other hand, it should be able to determine whether the depth information belongs to them. In the following, we will illustrate how the PSL 3D sensor and its corresponding imaging methods facilitate in such a mission. Results of seeing through the reflective surface We start with a typical situation, i.e. glass door with objects behind, to illustrate PSL 3D sensor’s ability to see through the reflective surface. We analyze different factors that can affect this mission, including the distance from the 3D sensor to the glass, the 3D sensor incident angle, the distance of objects behind glass and the density of objects behind glass. Their specific conditions are listed in Table 1 . In Fig. 4 , we show the influence of the first factor. The color pictures of the scene are shown in the first column. In this situation, the PSL 3D sensor is facing the glass door with a distance changing from 0.4 m to 1.2 m. Behind the glass door, there is a movable cabinet with a basketball on the top. Their distances to the glass are fixed to 0.5 m. The TX of the PSL 3D sensor projects TM-polarized structured light to the scene. Then the structured light is reflected back to the RX, containing depth information of both the glass door and objects behind. As mentioned in the principle, the reflected light of the glass door maintains the same polarization as the TX, while those of the cabinet and basketball are in various polarized directions due to diffuse reflection. Table 1 Influence factors for seeing through the reflective surfaces Full size table Fig. 4: Seeing through the reflective surface. ( a )–( c ) are the results of different distances from 3D sensor to glass. Polarization 0 ∘ is the depth map when TX and RX are in the same polarization. Polarization 90 ∘ is the depth map when TX and RX are in the orthogonal polarization. The yellow-black color bars apply to both second and third columns. Full size image Hence, when we set the RX to be TM polarization, depth information of the glass door, cabinet, and basketball can all be reconstructed, which is shown in Polarization 0° of Fig. 4 a. In this depth map, bright yellow represents the near points and black represents the far or missing points. The middle bright yellow area accurately reveals the depth of the glass, while the farther cabinet and basketball are darker. Unlike Fig. 3 d, the central part shows the depth of the basketball rather than the glass because reflection from the basketball is stronger. Next, if we rotate the RX to the orthogonal direction, i.e. TE polarization, the glass part can be filtered, leaving clear depth information of the cabinet and basketball (Polarization 90° in Fig. 4 a). As we change the distance to 0.8 m, the middle ball area becomes smaller because the glass depth begins to dominate (dot line in Polarization 0° of Fig. 4 b). As for 1.2 m (Fig. 4 c), the middle is almost replaced by glass depth, while the whole proportion of glass depth is smaller as compared to that of 0.4 m. In Polarization 90° for both cases, the glass part can still be eliminated. More details about the incident angle of 3D sensor, distance and density of objects behind glass are displayed in Supplementary Fig. S5 – S8 . Regardless of different incidence angles, or different distances and object densities, the 3D sensor can eliminate the depth of the glass and achieve the function of seeing through the reflective surface. Supplementary Movie 1 clearly shows this phenomenon by rotating the RX polarizer 360 degrees. The high contrast between Polarization 0° and Polarization 90° can further serve as a determination method of the reflective surface, which will be demonstrated in the section after next. Results of seeing against the reflective noise We also set up outdoor and indoor scenes to demonstrate the advantage of detecting against noise (Fig. 5 ). We first show a case of outdoor glass scene (Fig. 5 a). In this situation, a book is placed behind a tilted glass, where there is strong specular reflection from the natural light. We can see that the reflected noise incapacitates the stereo vision camera for obtaining the depth of the scene, leading to black-irregular-patch errors in the depth map (encircled by white and red dot lines while the book is indicated in red). However, as mentioned in the Fig. 1 c, this noise has a larger component of S polarization. Note that in this setting, S polarization is in the horizontal direction ( x direction) when facing the glass. Therefore, we can utilize P-polarized (i.e. y -direction polarization) TX and RX to filter this noise, enhancing the SNR. As shown in Fig. 5 c, the book behind the glass (encircled by the red dot line) can be measured clearly with the PSL 3D sensor in this P-polarized setting. Fig. 5: Seeing against the reflective noise. ( a )–( c ) are comparison results of an outdoor glass scene, where ( a ) is the picture of the scene, ( b ) is the depth map from stereo vision camera (Intel RealSense D455) and ( c ) is the depth map from PSL 3D sensor. ( d ) – ( l ) are comparison results of an indoor corner scene. The experiments are done in different height, whose schemes are shown in ( d ), ( g ), and ( j ). The corresponding depth map and point cloud from stereo vision camera are displayed in ( e ), ( h ) and ( k ) respectively, while those from PSL 3D sensor are in ( f ), ( i ), and ( l ). The jet color bar with range (0.48 m, 1.1 m) applies to ( b ), ( c ), and the one with range (0.3 m, 1 m) applies to the depth map column. Full size image Besides this sunlight noise, the shadow of objects may also severely blind the depth camera based on stereo vision. As illustrated before in Fig. 3 c, the reflection noise of two persons makes the stereo vision camera provide erroneous measurement. Another outdoor scene with reflection overlapped with the detecting object is also illustrated in Supplementary Fig. S9 , where the stereo vision camera is still affected by the reflection of outside objects while the PSL 3D sensor can get the depth information correctly. In the next case, we analyze the ability to detect against noise in the situation of wall corner, which is a common scene for indoor serving robot. The results are displayed in Fig. 5 d–l. As shown in Fig. 5 d, the comparison is first done at a height of 0.24 m and a distance of 0.7 m away from the corner, with the sensor 45° downward. In this situation, due to the influence of reflection and multipath noise, the stereo vision camera will make the wrong depth judgment. As indicated by the red dot line in Fig. 5 e, the depth has gone deep below the floor plane. Only a small part (orange dot line) has accurate floor information. However, for the PSL 3D sensor (Fig. 5 f), a complete and correct shape of the corner can be obtained by setting TX and RX to be P polarization ( y direction). Then we change to the height of 0.4 m (Fig. 5 g), with a distance 0.7 m and an incline angle 45° unchanged. It means that the incident angle decreases. In this condition, the correct floor area obtained from the stereo vision camera starts to increase (orange dot line in Fig. 5 h), but the depth area encircled by the red dot line still penetrates into the floor. As we change to the height of 0.56 m (Fig. 5 j), the floor area of the stereo vision camera goes back to the correct plane (Fig. 5 k), but there are still some black speckles in the depth map because of noise. However, for the PSL 3D sensor, the intact depth map and point cloud are maintained in these two heights (Fig. 5 i and l) with this P-polarized setting. For these common scenes of reflective surface, PSL 3D sensor can remove noise through specific polarization combinations, improve the signal-to-noise ratio, and obtain complete depth information of the scene. More polarization combination experiments and comparison with time of flight (ToF) camera, presented in Supplementary Fig. S10– - S13 , further demonstrate the superiority of PSL 3D sensor in these scenarios. Results of completing the reflective surface In the following, we prove the feasibility of completing the reflective surfaces. The results of eight common scenes are selected to illustrate in Fig. 6 and Supplementary Fig. S14 . In Fig. 6 , we display four of these scenes, including balcony glass, soundproof room glass, spherical glass and office door glass. In the balcony glass scene, a yellow square box is placed outside the glass door which has reflections of the interior furniture. With polarization 0° and polarization 90° settings, we obtain two depth images of the scene. In polarization 0°, partial of the glass and the box outside can both be reconstructed while the glass part is eliminated in polarization 90°. Compared to the stereo vision camera, which fails to detect the glass depth and misjudges the reflections as real objects, such a change in depth channel can serve as a reliable cue for determining the reflective surfaces. Thus, using the subtraction of two depth images and the glass boundary predicted from the RGB image of the scene, we are able to extract the glass region, which is highlighted in red in the fifth row. The extracted glass points are then used to fit and complete the reflective surface. The completed depth image is shown in the seventh row, where the partially empty glass has now been filled in. The final comparison between the original point cloud and the completed point cloud is shown in the last two rows. In the completed point cloud, we can see that the new glass plane, which is colored in red, matches well with the glass frame and the 3D information of the box outside the glass can be acquired simultaneously. Fig. 6: Detection and completion results. The results of four scenes are presented column-wise in this figure. The first row is the RGB image of each scene. The second and third rows are the depth map from polarization 0 ∘ and polarization 90 ∘ settings respectively. The fourth row is the predicted glass boundary using the deep learning method. The fifth row is the extracted glass region, which is obtained from the combination of depth map subtraction and glass boundary. The sixth row shows the fitting of the extracted glass points in the world coordinate. The seventh row is the completed depth map. The original point cloud and the completed point cloud are shown in the last two rows. In the completed point cloud, the completed glass is highlighted in red. The parula color bar applies to the last two rows and the yellow-black color bar applies to the rows of polarization 0 ∘ , polarization 90 ∘ , and depth completion. Full size image As for the soundproof room glass scene, we execute the same process. In this scene, the right door of the soundproof room is open, that is, only the left side has glass. Inside the room, a cardboard box with a basketball on its top is placed on the left, and another box and a book are on the right. As seen in the depth images, only the left side with glass has a large change while the right side shows no significant change. If simply relying on the predicted glass boundary from deep learning, we will mistakenly consider the right side as glass. However, our method can determine the regions that actually have glass and extract effective depth information belonging to the glass. Similarly, this accurately extracted depth information is used for the following fitting and completion. In the completed results, the left glass door and objects inside the room can be reconstructed, as if exploring the inside and outside of the soundproof room from God’s perspective. Besides planar glass, our method can also be applied to curved glass, which is displayed in the third column. In this situation, we use a spherical exhibition stand with two 3D-printed models inside as the detecting object. Likewise, two depth images are obtained from two polarization directions. We can see that the front face of the glass ball is eliminated in polarization 90°. With the ball-shaped glass boundary, we acquire the glass region in the fifth row. These points are from the spherical glass so the fitting result is a quadric surface. Then we complete the glass inside the glass boundary and get the completion result. As seen from the point cloud in the last row, it completes well the front surface of the spherical glass. Note that, there are still some misjudged points in the predicted glass boundary, such as the unsmoothness and the false positive outliers at the ball edge. These minor flaws can be solved by optimizing the predicting network in future work. At last, we show an office door glass scene with a larger tilted angle than the first column. In like manner, there is a cabinet with a ball on its top behind the glass. Due to different imaging angles and distances, the position of the glass depth is different in polarization 0°, whereas it will not affect the elimination of glass in polarization 90°. With the same subtraction method, the glass points are extracted and shown in the fifth row. Then the glass door is fitted and completed. From the completion results in the seventh row and the last row, we can see that the glass can be completed well and the box and ball behind the glass can also be reconstructed, reinforcing the idea of seeing and seeing through the reflective surfaces. It can be seen from the above experiments that the position and proportion of effective glass information are variational for different experiment situations. This is because of the combined effect of the illumination angle and the cleanliness of the reflective surface. In Supplementary Fig. S15 , we analyze the influence of different cleanliness of the reflective surface on the detection area. In fact, if the reflective surface perfectly follows the specular reflection model, only an area with the size of the emitter will be fully received. But in common scenes, this kind of situation rarely exists on the reflective surface. Actually, it will be affected by the randomly distributed particles on the surface, expanding the reflection range to a certain extent, which is the so-called off-specular reflection [42] , [43] . As our experiments show, the PSL 3D sensor can extract enough effective glass information in various scenes. We also describe the effective measurement angles in Supplementary Fig. S16 . We fix the position of the sensor facing the glass and then measure the working range by rotating the sensor horizontally and vertically to measure the proportion of the glass points. We use the number at 0° as a base and the ratio down to 2% as the limit, and the measured working range is ± 45° horizontally and ± 30° vertically. Moreover, the rest of the eight scenes are also illustrated in Supplementary Fig. S14 , demonstrating the general applicability of our method. As for the measurement accuracy, we use the absolute error to evaluate, and the results are shown in Fig. 7 . Among them, Fig. 7 a-h are the error maps of eight scenes, where Fig. 7 a-d correspond to the four scenes in Fig. 6 and Fig. 7 e-h correspond to the four scenes in Supplementary Fig. S14 . Specifically, we cover the reflective surfaces with thin stickers and conduct depth detection on the scene again, the fitted depth of which is used as a benchmark. We then calculate the absolute error between the depth map of the completed reflective surface and this benchmark. Then we perform boxplot statistics on these eight error maps, and the results are shown in Fig. 7 i. The green box includes the median and the upper and lower quartiles of each error map, the gray points are outliers exceeding 1.5 times the interquartile range, and the yellow circle dashed line is the mean value of each error map. From the error analysis, we can see that the errors are maintained at the level of 1mm, demonstrating the precise reconstruction of our method. Note that there are outliers in the error maps. These errors mainly come from the fluctuation of the data, such as the influence of the interreflection, which can be reduced to a much lower level if we include the model to suppress it in the future. Fig. 7: Error analysis. ( a )–( h ) are the error maps of eight scenes of reflective surfaces, which are the absolute errors between the completed reflective surface and the fitted benchmark. i The boxplot graph of these eight error maps. It includes the box, mean value and outliers of each map, showing their statistical properties. Full size image The above experiments of completion are from a single view. If we want to reconstruct the whole scene, we need to employ our method in different views. In Supplementary Fig. S17 , we show the reconstruction of a fish tank by employing our method in four sides of the fish tank. In addition, a more complicated scene with a mirror at the back of glass is used to prove the feasibility of our method in Supplementary Fig. S18 . The ToF camera is also compared in Supplementary Fig. S19 , which once again demonstrates the distinction of the PSL 3D sensor.\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"In summary, we have invented an HCG-VCSEL-based polarization structured light (PSL) 3D sensor and proposed the corresponding imaging methods for 3D reconstruction in a wide type of scenes with highly reflective surfaces. We first chose a glass scene to demonstrate the ability of the PSL 3D sensor to obtain 3D information of objects behind reflective surfaces. Next, we compare with the stereo vision camera and ToF camera, and show that in the scene with strong reflective noise, through a specific polarization setting, our sensor can eliminate the reflective noise and accurately obtain the 3D image of the scene. Then, we select 8 different scenes to prove that our method can be robustly applied to the detection and completion of the reflective surfaces. Moreover, we show that the effective working range of incident angle is as large as 90° × 60°. In contrast, stereo vision cameras, ToF cameras, and deep learning methods will cause erroneous measurements in these scenarios. At last, we evaluate the absolute errors of our detecting results, and the mean error of each scene maintains at the level as low as 1mm, equivalent to 0.1%. Through these different experiments, we demonstrate that the PSL 3D sensor can realize seeing and seeing through the reflective surfaces. Currently, the proposed methods are conducted in post-processing, but in fact they can be integrated into the module. We can also modify and optimize the design of the deep learning network, so that they can be transferred to the processing chip of the sensor for real-time detection and completion. On the receiving end, we can also design a unique polarizer array like that in [44] , which can be coated on the CMOS camera and receive the signal from different polarizations simultaneously, making our system more compact. Serving as the eyes of a robot, PSL 3D sensors will find many applications such as service robots and logistics robots. These robots will inevitably encounter the problem of reflective surfaces such as the glass door and floor corner shown in the main text. The PSL 3D sensor can make them handle these cases correctly. Besides, object or scene reconstruction of reflective surfaces, such as office and exhibition hall, will also benefit from this method. Thus, with the special polarization characteristic, the PSL 3D sensor can be further extended to a wide range of indoor and outdoor applications.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Generation of dot-array structured light The polarized dot-array structured light is generated directly from the TX. Inside the TX, there is a 940 nm HCG-VCSEL array, a collimating lens with 4.8 mm focal length and a diffractive optical element (DOE). The HCG-VCSEL array and the DOE are mounted at the focal points of both sides of the lens. By designing the bandwidth and airgap of the HCG, the VCSEL sources produce polarized Gaussian beams. These beams propagate to the lens and are collimated to compensate the divergence. Then the collimated light travels to the DOE. Passing through the DOE, one beam can diffract to 11 × 9 orders uniformly. If we properly position the VCSEL on the array, then the far-field distribution of the laser array can be duplicated in 11 × 9 copies, enabling the production of dot-array structured light over 30,000 points. The far-field dot-array distribution is shown in Supplementary Fig. S2 . Postprocessing of the point cloud data and RGB images The reading and drawing of point clouds are done by the Point Cloud Processing of Computer Vision Toolbox in MATLAB. The fitting uses the fit function of Curve Fitting Toolbox with its robust option set to bisquare. The other processing of the point clouds is also based on MATLAB. The prediction of the reflective surface boundary is based on PyTorch. In addition, the visualization of the point clouds in Fig. 5 , Supplementary Fig. S10 – S13 and Supplementary Fig. S17 is based on CloudCompare [45] . Calculation of the depth of reflective surfaces By calculating the spatial displacements of known dot patterns, depth information of reflective surfaces can be obtained according to triangulation. The received dot patterns on the reflective surfaces are shown in Supplementary Fig. S20 and a model of triangulation is also shown to illustrate how depth is calculated.\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"This research proposes a polarization structured light (PSL) 3D sensor, which utilizes polarization properties on both the transmitter and receiver sides—employing high-contrast-grating vertical-cavity surface-emitting lasers (HCG VCSELs) for structured light with strong polarization selection and a polarization-selection CMOS camera—to accurately capture depth information of highly reflective surfaces and objects behind them by leveraging Fresnel’s theory to differentiate specular reflections from diffuse ones."},"entities":{"kind":"list like","value":["polarization structured light","3D sensor","polarization properties","transmitter","receiver","high-contrast-grating vertical-cavity surface-emitting lasers","HCG VCSELs","structured light","polarization-selection CMOS camera","depth information","highly reflective surfaces","objects","Fresnel’s theory","specular reflections","diffuse reflections"],"string":"[\n \"polarization structured light\",\n \"3D sensor\",\n \"polarization properties\",\n \"transmitter\",\n \"receiver\",\n \"high-contrast-grating vertical-cavity surface-emitting lasers\",\n \"HCG VCSELs\",\n \"structured light\",\n \"polarization-selection CMOS camera\",\n \"depth information\",\n \"highly reflective surfaces\",\n \"objects\",\n \"Fresnel’s theory\",\n \"specular reflections\",\n \"diffuse reflections\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"2.","title":"Miyazaki, D., Kagesawa, M. & Ikeuchi, K. Transparent surface modeling from a pair of polarization images.IEEE Trans. Pattern Anal. Mach. Intell.26, 73–82 (2004).","link":"https://doi.org/10.1109%2FTPAMI.2004.1261080"},{"idx":"3.","title":"Miyazaki, D. & Ikeuchi, K. Shape estimation of transparent objects by using inverse polarization ray tracing.IEEE Trans. Pattern Anal. Mach. Intell.29, 2018–2030 (2007).","link":"https://doi.org/10.1109%2FTPAMI.2007.1117"},{"idx":"4.","title":"Xu, X., Qiao, Y. & Qiu, B. Reconstructing the surface of transparent objects by polarized light measurements.Opt. Express25, 26296–26309 (2017).","link":"https://doi.org/10.1364%2FOE.25.026296"},{"idx":"7.","title":"Morel, O., Stolz, C., Meriaudeau, F. & Gorria, P. Active lighting applied to three-dimensional reconstruction of specular metallic surfaces by polarization imaging.Appl. Opt.45, 4062–4068 (2006).","link":"https://doi.org/10.1364%2FAO.45.004062"},{"idx":"13.","title":"Tian, X., Liu, R., Wang, Z. & Ma, J. High quality 3d reconstruction based on fusion of polarization imaging and binocular stereo vision.Inf. Fusion77, 19–28 (2022).","link":"https://doi.org/10.1016%2Fj.inffus.2021.07.002"},{"idx":"15.","title":"Thilak, V., Voelz, D. G. & Creusere, C. D. Polarization-based index of refraction and reflection angle estimation for remote sensing applications.Appl. Opt.46, 7527–7536 (2007).","link":"https://doi.org/10.1364%2FAO.46.007527"},{"idx":"18.","title":"Kong, N., Tai, Y.-W. & Shin, S. Y. High-quality reflection separation using polarized images.IEEE Trans. Image Process.20, 3393–3405 (2011).","link":"https://doi.org/10.1109%2FTIP.2011.2155080"},{"idx":"20.","title":"Kong, N., Tai, Y.-W. & Shin, S. Y. A physically-based approach to reflection separation: from physical modeling to constrained optimization.IEEE Trans. Pattern Anal. Mach. Intell.36, 209–221 (2014).","link":"https://doi.org/10.1109%2FTPAMI.2013.45"},{"idx":"22.","title":"Wen, S., Zheng, Y. & Lu, F. Polarization guided specular reflection separation.IEEE Trans. Image Process.30, 7280–7291 (2021).","link":"https://doi.org/10.1109%2FTIP.2021.3104188"},{"idx":"25.","title":"Sun, J., Chang, Y., Jung, C. & Feng, J. Multi-modal reflection removal using convolutional neural networks.IEEE Signal Process. Lett.26, 1011–1015 (2019).","link":"https://doi.org/10.1109%2FLSP.2019.2915560"},{"idx":"26.","title":"Chang, Y., Jung, C. & Sun, J. Joint reflection removal and depth estimation from a single image.IEEE Trans. Cybern.51, 5836–5849 (2021).","link":"https://doi.org/10.1109%2FTCYB.2019.2959381"},{"idx":"28.","title":"Wang, X. & Wang, J. Detecting glass in simultaneous localisation and mapping.Robot. Autonomous Syst.88, 97–103 (2017).","link":"https://doi.org/10.1016%2Fj.robot.2016.11.003"},{"idx":"33.","title":"Zhang, Y., Ye, M., Manocha, D. & Yang, R. 3D reconstruction in the presence of glass and mirrors by acoustic and visual fusion.IEEE Trans. Pattern Anal. Mach. Intell.40, 1785–1798 (2018).","link":"https://doi.org/10.1109%2FTPAMI.2017.2723883"},{"idx":"40.","title":"Huang, M. C., Ye, Z. & Chang-Hasnain, C. J. A surface-emitting laser incorporating a high-index-contrast subwavelength grating.Nat. Photonics1, 119–122 (2007).","link":"https://doi.org/10.1038%2Fnphoton.2006.80"},{"idx":"42.","title":"Torrance, K. E. & Sparrow, E. M. Theory for off-specular reflection from roughened surfaces.J. Opt. Soc. Am.57, 1105–1114 (1967).","link":"https://doi.org/10.1364%2FJOSA.57.001105"},{"idx":"43.","title":"Wolff, L. B., Nayar, S. K. & Oren, M. Improved diffuse reflection models for computer vision.Int. J. Comput. Vis.30, 55–71 (1998).","link":"https://doi.org/10.1023%2FA%3A1008017513536"}],"string":"[\n {\n \"idx\": \"2.\",\n \"title\": \"Miyazaki, D., Kagesawa, M. & Ikeuchi, K. Transparent surface modeling from a pair of polarization images.IEEE Trans. Pattern Anal. Mach. Intell.26, 73–82 (2004).\",\n \"link\": \"https://doi.org/10.1109%2FTPAMI.2004.1261080\"\n },\n {\n \"idx\": \"3.\",\n \"title\": \"Miyazaki, D. & Ikeuchi, K. Shape estimation of transparent objects by using inverse polarization ray tracing.IEEE Trans. Pattern Anal. Mach. Intell.29, 2018–2030 (2007).\",\n \"link\": \"https://doi.org/10.1109%2FTPAMI.2007.1117\"\n },\n {\n \"idx\": \"4.\",\n \"title\": \"Xu, X., Qiao, Y. & Qiu, B. Reconstructing the surface of transparent objects by polarized light measurements.Opt. Express25, 26296–26309 (2017).\",\n \"link\": \"https://doi.org/10.1364%2FOE.25.026296\"\n },\n {\n \"idx\": \"7.\",\n \"title\": \"Morel, O., Stolz, C., Meriaudeau, F. & Gorria, P. Active lighting applied to three-dimensional reconstruction of specular metallic surfaces by polarization imaging.Appl. Opt.45, 4062–4068 (2006).\",\n \"link\": \"https://doi.org/10.1364%2FAO.45.004062\"\n },\n {\n \"idx\": \"13.\",\n \"title\": \"Tian, X., Liu, R., Wang, Z. & Ma, J. High quality 3d reconstruction based on fusion of polarization imaging and binocular stereo vision.Inf. Fusion77, 19–28 (2022).\",\n \"link\": \"https://doi.org/10.1016%2Fj.inffus.2021.07.002\"\n },\n {\n \"idx\": \"15.\",\n \"title\": \"Thilak, V., Voelz, D. G. & Creusere, C. D. Polarization-based index of refraction and reflection angle estimation for remote sensing applications.Appl. Opt.46, 7527–7536 (2007).\",\n \"link\": \"https://doi.org/10.1364%2FAO.46.007527\"\n },\n {\n \"idx\": \"18.\",\n \"title\": \"Kong, N., Tai, Y.-W. & Shin, S. Y. High-quality reflection separation using polarized images.IEEE Trans. Image Process.20, 3393–3405 (2011).\",\n \"link\": \"https://doi.org/10.1109%2FTIP.2011.2155080\"\n },\n {\n \"idx\": \"20.\",\n \"title\": \"Kong, N., Tai, Y.-W. & Shin, S. Y. A physically-based approach to reflection separation: from physical modeling to constrained optimization.IEEE Trans. Pattern Anal. Mach. Intell.36, 209–221 (2014).\",\n \"link\": \"https://doi.org/10.1109%2FTPAMI.2013.45\"\n },\n {\n \"idx\": \"22.\",\n \"title\": \"Wen, S., Zheng, Y. & Lu, F. Polarization guided specular reflection separation.IEEE Trans. Image Process.30, 7280–7291 (2021).\",\n \"link\": \"https://doi.org/10.1109%2FTIP.2021.3104188\"\n },\n {\n \"idx\": \"25.\",\n \"title\": \"Sun, J., Chang, Y., Jung, C. & Feng, J. Multi-modal reflection removal using convolutional neural networks.IEEE Signal Process. Lett.26, 1011–1015 (2019).\",\n \"link\": \"https://doi.org/10.1109%2FLSP.2019.2915560\"\n },\n {\n \"idx\": \"26.\",\n \"title\": \"Chang, Y., Jung, C. & Sun, J. Joint reflection removal and depth estimation from a single image.IEEE Trans. Cybern.51, 5836–5849 (2021).\",\n \"link\": \"https://doi.org/10.1109%2FTCYB.2019.2959381\"\n },\n {\n \"idx\": \"28.\",\n \"title\": \"Wang, X. & Wang, J. Detecting glass in simultaneous localisation and mapping.Robot. Autonomous Syst.88, 97–103 (2017).\",\n \"link\": \"https://doi.org/10.1016%2Fj.robot.2016.11.003\"\n },\n {\n \"idx\": \"33.\",\n \"title\": \"Zhang, Y., Ye, M., Manocha, D. & Yang, R. 3D reconstruction in the presence of glass and mirrors by acoustic and visual fusion.IEEE Trans. Pattern Anal. Mach. Intell.40, 1785–1798 (2018).\",\n \"link\": \"https://doi.org/10.1109%2FTPAMI.2017.2723883\"\n },\n {\n \"idx\": \"40.\",\n \"title\": \"Huang, M. C., Ye, Z. & Chang-Hasnain, C. J. A surface-emitting laser incorporating a high-index-contrast subwavelength grating.Nat. Photonics1, 119–122 (2007).\",\n \"link\": \"https://doi.org/10.1038%2Fnphoton.2006.80\"\n },\n {\n \"idx\": \"42.\",\n \"title\": \"Torrance, K. E. & Sparrow, E. M. Theory for off-specular reflection from roughened surfaces.J. Opt. Soc. Am.57, 1105–1114 (1967).\",\n \"link\": \"https://doi.org/10.1364%2FJOSA.57.001105\"\n },\n {\n \"idx\": \"43.\",\n \"title\": \"Wolff, L. B., Nayar, S. K. & Oren, M. Improved diffuse reflection models for computer vision.Int. J. Comput. Vis.30, 55–71 (1998).\",\n \"link\": \"https://doi.org/10.1023%2FA%3A1008017513536\"\n }\n]","isConversational":false}}},{"rowIdx":89,"cells":{"unique_id":{"kind":"string","value":"ncomms3791"},"subfield":{"kind":"string","value":"Materials science"},"title":{"kind":"string","value":"Atomic-scale mechanisms of ferroelastic domain-wall-mediated ferroelectric switching"},"abstract":{"kind":"string","value":"Polarization switching in ferroelectric thin films occurs via nucleation and growth of 180° domains through a highly inhomogeneous process in which the kinetics are largely controlled by defects, interfaces and pre-existing domain walls. Here we present the first real-time, atomic-scale observations and phase-field simulations of domain switching dominated by pre-existing, but immobile, ferroelastic domains in Pb(Zr 0.2 Ti 0.8 )O 3 thin films. Our observations reveal a novel hindering effect, which occurs via the formation of a transient layer with a thickness of several unit cells at an otherwise charged interface between a ferroelastic domain and a switched domain. This transient layer possesses a low-magnitude polarization, with a dipole glass structure, resembling the dead layer. The present study provides an atomic level explanation of the hindering of ferroelectric domain motion by ferroelastic domains. Hindering can be overcome either by applying a higher bias or by removing the as-grown ferroelastic domains in fabricated nanostructures."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"Ferroelectric thin films usually possess polydomain structures determined by electromechanical boundary conditions. For example, ferroelastic twinning structures with polarization rotations of 71° or 109° in rhombohedral perovskites (for example, BiFeO 3 ) or of 90° in tetragonal perovskites (for example, PbZr 0.2 Ti 0.8 O 3 (PZT)) provide an energetically favourable means of reducing the strain energy in epitaxial thin films. Many properties of ferroelectric thin films are closely correlated to the structure and dynamic behaviour of domains, and thus can be manipulated by domain engineering. For example, the piezoelectric response in epitaxial PZT thin films is particularly sensitive to embedded ferroelastic domains and can be dramatically enhanced by manipulating the domain structure to either remove these domains or increase their mobility [1] . Above a critical film thickness of about 100 nm in PZT [1] , [2] , stress-relieving ferroelastic domains generally form a dense square network of crossing domains, which, in epitaxial thin films, are usually nearly immobile because of the strong coupling between charge and strain in the system. Given the high density and poor mobility of such ferroelastic domains in epitaxial films, interactions between the switched 180° domains and these pre-existing ferroelastic domains dominate the overall polarization-switching kinetics. In particular, the presence of ferroelastic domains reduces the responsiveness of the thin film by lowering the spontaneous out-of-film polarization and reducing the electromechanical piezoelectric response, d 33 , in piezoforce microscopy-switching experiments [1] , [3] . During switching, these domains are often observed to undergo 180° switching in place rather than 90° switching. Although this removes charged domain boundaries between the switched domains and the embedded ferroelastic domains, the polarization is not completely aligned with the applied electric field, leading to the observed decrease in the measured spontaneous polarization. These ferroelastic domains also lead to a decrease in the relative stability of a written domain structure and have been seen to be associated with back switching of the thin film to its initial state after the removal of the applied electric field [3] , [4] . This behaviour is especially important in non-volatile devices as their ability to retain the current state is determined by the stability of newly written domains [5] . Macroscopic ferroelectric-switching studies naturally tend to aggregate interactions between domains into the total system behaviour [6] . An understanding of the local response and underlying dynamic behaviour of individual domain walls and defects throughout the thickness of the film during the switching process is necessary, however, to engineer reliable ferroelectric devices [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] . This is especially important in ferroelectric memory devices scaled down to the nanometre scale to achieve a high data storage density where a single defect may dominate the total switching process. At this feature size, it is critical to understand how specific interactions between isolated immobile ferroelastic domains and switched ferroelectric domains influence local switching dynamics. Several studies have attempted to elucidate the specific interactions around ferroelastic domain walls that lead to their strong influence on macroscopic switching kinetics in tetragonal ferroelectrics [2] , [3] , [4] , [15] , [16] . For example, the preferential nucleation of 180° domains at 90° domain walls was reported in barium titanate using optical techniques [15] . Polarization switching through preferential paths was also observed in PZT thin films using piezoresponse force microscopy [3] , whereas numerical modelling of the polarization switching around embedded 90° domains showed that the reduction in the nucleation potential near the domain boundary was the result of a large built-in electrostatic field at the surface of the film lowering the energy barrier to nucleation [17] . The numerical study, however, did not investigate polarization switching beyond the initial nucleation. Ferroelastic domain walls have also been found to be active in switching kinetics beyond facilitating domain nucleation [2] , [4] . Little [15] found, for instance, that in bulk crystals, 180° domain wall motion was abruptly arrested at 90° domain walls. On the basis of electrical measurements, Li and Alexe [2] proposed that 90° domain walls inhibit the motion of 180° domain walls in the plane of the film, slowing the switching kinetics and reducing the overall response of the ferroelectric to an applied electric field. Electrical measurements, however, yield only statistically averaged results, and surface probe techniques are unable to observe subsurface switching paths. Using either method, detailed atomic-scale mechanisms governing the interactions between 90° and 180° domain boundaries are difficult to determine. Understanding local switching pathways, however, is critical as ferroelectrics find use in miniaturized electronic devices such as capacitors and memory devices [9] where the switching kinetics of individual domains dominate the device’s response. To overcome the resolution limits of optical and electrical measurement techniques, we use transmission electron microscopy (TEM) with in situ switching along with numerical phase-field simulations of the full switching process to determine the interaction mechanisms between 90° and 180° domain walls during switching. In comparison with general measurement techniques, in situ TEM can be used to directly observe atomic-scale domain interactions in real time during switching through the thickness of the film, enabling switching dynamics to be correlated with local defect microstructures in real time [13] , [18] , [19] . Employing spherical aberration-corrected TEM with sub-angstrom resolution [20] , [21] , we are able to directly map the local polarization at the atomic scale [9] , [10] . We combine this with mesoscale phase-field modelling of the dynamic polarization-switching processes to corroborate our observations with current models of ferroelectric thin films. Our key finding is the identification of a dipole glass structure formed at the immobile ferroelastic domain boundary during the switching process as the primary atomic-scale mechanism for hindering of the domain-switching process by immobile ferroelastic domains. We find that formation of this layer could occur to reduce the system’s electrostatic energy and allow formation of stable head-to-head domain boundaries in the thin films. The impact of this layer on the switching process and strategies to overcome this effect are discussed.","references":{"indices":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],"infos":[{"idx":"1","title":"Nagarajan, V. et al. Dynamics of ferroelastic domains in ferroelectric thin films.Nat. Mater.2, 43–47 (2003).","link":"https://doi.org/10.1038%2Fnmat800"},{"idx":"2","title":"Li, W. & Alexe, M. Investigation on switching kinetics in epitaxial Pb(Zr0.2Ti0.8)O3 ferroelectric thin films: role of the 90 degrees domain walls.Appl. Phys. Lett.91, 262903 (2007).","link":"https://doi.org/10.1063%2F1.2825414"},{"idx":"3","title":"Ganpule, C. S. et al. Role of 90 degrees domains in lead zirconate titanate thin films.Appl. Phys. Lett.77, 292–294 (2000).","link":"https://doi.org/10.1063%2F1.126954"},{"idx":"4","title":"Roelofs, A. et al. Depolarizing-field-mediated 180 degrees switching in ferroelectric thin films with 90 degrees domains.Appl. Phys. Lett.80, 1424–1426 (2002).","link":"https://doi.org/10.1063%2F1.1448653"},{"idx":"5","title":"Gao, P. et al. Direct observations of retention failure in ferroelectric memories.Adv. Mater.24, 1106–1110 (2012).","link":"https://doi.org/10.1002%2Fadma.201103983"},{"idx":"6","title":"Dawber, M., Rabe, K. M. & Scott, J. F. Physics of thin-film ferroelectric oxides.Rev. Mod. Phys.77, 1083–1130 (2005).","link":"https://doi.org/10.1103%2FRevModPhys.77.1083"},{"idx":"7","title":"Merz, W. J. Domain formation and domain wall motions in ferroelectric BaTiO3 single crystals.Phys. Rev.95, 690–698 (1954).","link":"https://doi.org/10.1103%2FPhysRev.95.690"},{"idx":"8","title":"Jesse, S. et al. Direct imaging of the spatial and energy distribution of nucleation centres in ferroelectric materials.Nat. Mater.7, 209–215 (2008).","link":"https://doi.org/10.1038%2Fnmat2114"},{"idx":"9","title":"Nelson, C. T. et al. Domain dynamics during ferroelectric switching.Science334, 968–971 (2011).","link":"https://doi.org/10.1126%2Fscience.1206980"},{"idx":"10","title":"Gao, P. et al. Revealing the role of defects in ferroelectric switching with atomic resolution.Nat. Commun.2, 591 (2011).","link":"https://doi.org/10.1038%2Fncomms1600"},{"idx":"11","title":"Chu, M. W. et al. Impact of misfit dislocations on the polarization instability of epitaxial nanostructured ferroelectric perovskites.Nat. Mater.3, 87–90 (2004).","link":"https://doi.org/10.1038%2Fnmat1057"},{"idx":"12","title":"Wang, R. V. et al. Reversible chemical switching of a ferroelectric film.Phys. Rev. Lett.102, 047601 (2009).","link":"https://doi.org/10.1103%2FPhysRevLett.102.047601"},{"idx":"13","title":"Chang, H. J. et al. Watching domains grow:In-situstudies of polarization switching by combined scanning probe and scanning transmission electron microscopy.J. Appl. Phys.110, 052014 (2011).","link":"https://doi.org/10.1063%2F1.3623779"},{"idx":"14","title":"Dearaujo, C. A. P., Cuchiaro, J. D., McMillan, L. D., Scott, M. C. & Scott, J. F. Fatigue-free ferroelectric capacitors with platinum-electrodes.Nature374, 627–629 (1995).","link":"https://doi.org/10.1038%2F374627a0"},{"idx":"15","title":"Little, E. A. Dynamic behavior of domain walls in barium titanate.Phys. Rev.98, 978–984 (1955).","link":"https://doi.org/10.1103%2FPhysRev.98.978"},{"idx":"16","title":"Kohli, M., Muralt, P. & Setter, N. Removal of 90 degrees domain pinning in (100) Pb(Zr0.15Ti0.85)O3 thin films by pulsed operation.Appl. Phys. Lett.72, 3217–3219 (1998).","link":"https://doi.org/10.1063%2F1.121554"},{"idx":"17","title":"Choudhury, S. et al. Effect of ferroelastic twin walls on local polarization switching: phase-field modeling.Appl. Phys. Lett.93, 162901 (2008).","link":"https://doi.org/10.1063%2F1.2993330"},{"idx":"18","title":"Sato, Y., Hirayama, T. & Ikuhara, Y. Real-time direct observations of polarization reversal in a piezoelectric crystal: Pb(Mg1/3Nb2/3)O3-PbTiO3 studied viain situelectrical biasing transmission electron microscopy.Phys. Rev. Lett.107, 187601 (2011).","link":"https://doi.org/10.1103%2FPhysRevLett.107.187601"},{"idx":"19","title":"Winkler, C. R. et al. Accessing intermediate ferroelectric switching regimes with time-resolved transmission electron microscopy.J. Appl. Phys.112, 052013–052019 (2012).","link":"https://doi.org/10.1063%2F1.4746082"},{"idx":"20","title":"Jia, C. L. et al. Atomic-scale study of electric dipoles near charged and uncharged domain walls in ferroelectric films.Nat. Mater.7, 57–61 (2008).","link":"https://doi.org/10.1038%2Fnmat2080"},{"idx":"21","title":"Catalan, G. et al. Flexoelectric rotation of polarization in ferroelectric thin films.Nat. Mater.10, 963–967 (2011).","link":"https://doi.org/10.1038%2Fnmat3141"}]}},{"section":"Results","content":"Phase-field modelling Phase-field modelling [22] , [23] , [24] of an isolated 90° domain embedded in a c -poled PZT thin film was used to understand the starting electromechanical state of the film. The results ( Fig. 1a–d ) show that the ferroelastic domain walls develop an electrical charge and are associated with significant strains despite lying on nominally charge and strain neutral {101} planes. In fact, an electrical potential step exists across a ferroelastic domain wall in bulk tetragonal PbTiO 3 according to first-principles calculations [25] and phase-field simulations [26] , indicating 90° domain walls may be inherently charged. In addition, the effect of the substrate constraint may lead to additional polarization charges at the domain wall because of different polarization magnitudes for the c- and a -domains in an epitaxial thin film ( c refers to domains with the polarization oriented along the film normal and a refers to those with the polarization oriented in-plane). The existence of this bound charge can have a significant impact on the local switching dynamics. Because of the presence of charged 90° domain walls, the externally applied bias changes less quickly at the surface of the film, reducing the electric field there ( Fig. 1b,c ). As a result, a larger applied bias is required to generate the critical electric field for switching in the vicinity of the 90° domain wall. In addition, clamping of the ferroelectric film by the epitaxial substrate leads to local inhomogeneous strain distributions ( Fig. 1d ), which may influence local switching kinetics in a more complicated manner. Figure 1: Ferroelastic domains in a tetragonal PZT thin film. ( a ) The calculated potential distribution around an isolated a -domain showing the domain walls are moderately charged. ( b ) Calculated potential distribution at 8.4 V without and ( c ) with an embedded a -domain. ( d ) Calculated elastic strain distribution in the absence of external bias showing large compressive strain near the a -domain/substrate interface. ( e ) TEM image of the in situ biasing experimental set-up. A sharp mobile tungsten tip acts as an electrode, whereas the bottom SRO electrode is grounded. Scale bar, 100 nm. The schematic below the micrograph illustrates the observed ferroelastic domains. ( f ) Selected area electron diffraction pattern acquired from a region including both a -domain and c -domain showing splitting of the diffraction spots. The red rectangle indicates the c -domain diffraction pattern, whereas the yellow rectangle corresponds to the a -domain diffraction pattern. ( g ) Atomic resolution HAADF image of an embedded a -domain overlaid with its lattice rotation angle mapping generated by GPA. The colour boundaries correspond to the 90°-domain walls. Scale bar, 5 nm. ( h ) Plot of polarization vectors for a 90°-domain wall showing head-to-tail configuration. Scale bar, 2 nm. Full size image Ferroelastic domain wall structures Isolated ferroelastic domains were investigated in c -oriented tetragonal PZT films, with a thickness of ~200 nm, grown on {001} SrTiO 3 (STO) substrates buffered with an ~20 nm thick SrRuO 3 (SRO) conducting electrode. Cross-sectional TEM specimens were prepared by conventional mechanical polishing and argon ion milling. A TEM image ( Fig. 1e ) shows a mixture of a / c -domains necessary to accommodate the residual epitaxial stain. A selected area electron diffraction pattern acquired from a region that includes both a -domains (with the polarization oriented towards right) and c -domains (with the polarization oriented upwards) shows the splitting of reflection spots ( Fig. 1f ). There is a lattice tilt, α , of ~2.7° between lattice planes across the 90°-domain wall, corresponding to a lattice constant ratio, c/a = tan (45°+ α /2), of ~1.048, consistent with previous X-ray diffraction results [27] . This lattice tilt enables the 90°-domain walls to be easily distinguished by geometric phase analysis (GPA) [28] of atomic resolution high angle annular dark field (HAADF) images ( Fig. 1g ). Figure 1h shows a map of the polarization vectors around a 90°-domain wall obtained by calculating the offsets of the Pb sublattice [29] , demonstrating that the 90°-domain walls are on {101} planes. This head-to-tail configuration of polarization vectors minimizes the bound charge at the domain wall [7] . This analysis also shows that the width of 90° domain walls is approximately one unit cell according to the polarization vector map in Fig. 1h . In situ ferroelectric switching In situ observations of polarization switching were carried out by application of an electric field between a scanning tungsten tip electrode and the grounded SRO electrode in the TEM ( Fig. 1e ). Figure 2 shows sequential TEM images during switching of a large c -domain adjacent to an a -domain. In Fig. 2b , a new − c -domain (where the negative sign indicates downward polarization) nucleates and grows at the top interface at a bias of 8.2 V, leading to the contrast change from bright to dark. Although previous studies proposed that the nucleation of c -domains preferentially begins from 90° domain walls [3] , [8] , [15] , our in situ TEM observations revealed that nucleation occurs only at the top interface with a negative bias [5] , [10] . The switched − c -domain grew with increasing voltage ( Fig. 2b,c ). Under a constant voltage ( Fig. 2c–g ), the domain grew slowly, which may result from a relatively high density of dislocations acting as weak pinning centres in the film [10] . Dislocations also account for the sawtooth shape of the domain ( Fig. 2c ). As a single needle-like domain was observed in dislocation-free PZT thin films grown on more closely lattice matched DyScO 3 substrates [10] , local pinning causes non-uniform growth of the switched domain front. Figure 2: Hindering effect of a ferroelastic domain. A chronological TEM bright field image series shows the evolution of a − c -domain at biases of 0→(+10)→0 V. Original polarization of c -domain in PZT film is upward ( a ). Scale bar, 100 nm. Nucleation of – c -domain with downward polarization occurs at the top interface of tungsten and PZT at 8.2 V ( b ). It grows forward slowly with the increase in bias and/or duration ( c ), but its motion is hindered by the 90°-domain boundary ( d – g ). After the voltage is removed, the domain is stable with a slight relaxation of the 180°-domain wall ( h ). Thin arrows indicate possible dislocation pinning sites. Below each image a schematic shows the observed domains. In each image contrast, thermodynamic analysis and high-resolution polarization mapping were used to determine the domain polarization. Full size image Once the created − c -domain reached the edge of the a -domain ( Fig. 2d ), the 180°-domain wall motion ceased ( Fig. 2e–g ), showing the ferroelastic domain wall acted as an obstacle to the propagation of the domain wall. When a 180°-domain wall impinges on the ferroelastic domain, the out-of-plane component of the electric field favours the 90°-ferroelastic switching of the a -domain (in which polarization rotates from pointing right to downwards). Such switching, however, is unlikely to happen at moderate field strengths because of the large energy cost associated with the lattice distortion and accompanying elastic strain incurred when switching the ferroelastic domain [30] . In addition, a -domains are often stabilized by misfit dislocations at the film interfaces and consequently are even more difficult to switch [31] , [32] ."},{"section":"Discussion","content":"The pinning observed here differs considerably from that seen in ferroelastic switching in BiFeO 3 by Wrinkler et al. [33] where they observed that growing domains were repelled by pre-existing ferroelastic domain boundaries and that many of these metastable domains disappeared when the bias was removed. In our observations, the switched ferroelectric domain was impeded by the ferroelastic domain and was stable after the removal of applied voltage. Atomically resolved HAADF images were used to characterize the structure of 90° domain walls around the ferroelastic domain after switching ( Fig. 3 ). Mapping the ferroelastic variants around the domain wall using both GPA ( Fig. 3a ) and lattice plane spacing determined from the Pb atom positions ( Fig. 3b ) shows that the lattice orientation change remains sharp across both domain walls during the switching process, and indicates no ferroelastic switching or crystal structure change occurred around the 90°-domain wall during this process. In addition, strain distributions determined from GPA calculations are not obviously affected by the ferroelectric polarization switching ( Supplementary Fig. S1 ). The corresponding polarization angle map calculated by the Pb sublattice offset in Fig. 3c , however, shows that the upper wall is diffuse. There is a transition zone with thickness of ~10 unit cells in which the polarization vectors are randomly oriented and have substantially reduced magnitudes ( Fig. 3e ). As the contrast in a HAADF image comes from the average of many unit cells along the beam direction, it seems that the transition zone consists of many tiny domains polarized in different orientations, resulting in a glassy polarization configuration or dipole glass. Unlike traditional glasses, this region remains crystalline; only the long-range order of the polarization collapses. The uncharged 90° domain wall in the bottom-left corner of Fig. 3c , however, is unaffected and the polarization angle map shows a sharply defined boundary ( Fig. 3f ), similar to that seen before switching ( Fig. 1h ). Figure 3: Charged ferroelastic domain boundary between − c -domain and a -domain. ( a ) A map of lattice tilt angle based on GPA calculations overlaid on its atomic resolution HAADF image. The region is the enlarged view of the rectangle in Fig. 2h , including three domains: − c -domain (upper right), a -domain (middle) and c -domain (lower left). Scale bar, 2 nm. ( b ) The map of lattice ratios of out-of-plane to in-plane calculated from the Pb sublattice of the same region. Scale bar, 2 nm. ( c ) Polarization angle map showing the transition zone, highlighted by the white-dashed line, between − c -domain and a -domain. Scale bar, 2 nm. ( d ) Polarization vector map of the switched a -domain corresponding to the rectangle number 1 in c . Scale bar, 1 nm. ( e ) Polarization vector map from the transition zone corresponding to the rectangle number 2 in c . Scale bar, 1 nm. ( f ) Polarization vector map at the a -domain/ c -domain boundary corresponding to the rectangle number 3 in c . Scale bar, 1 nm. Full size image In contrast to previous studies that observed transition layers with suppressed polarization at the substrate interface induced by dislocations [34] , [35] , non-ferroelectric layers at grain boundaries [36] or charge compensation caused irregularly shaped domain walls near the interface [37] ; here the generation of the dipole glass at the charged 90° domain walls may result from the high energy of the head-to-head polarization configuration. As 90° domain walls are on {101} planes, the bound charge density from the termination of the polarization at the domain wall, 2 P S cos θ , is 1.42 P S , where P S is the spontaneous polarization. Given an approximate value for P S of 70 μC cm −2 (ref. 30 ), this corresponds to a charge density of 99 μC cm −2 . Formation of such an energetic, charged domain wall is not favourable, and it is reasonable that a spontaneous relaxation may occur to form a transition zone of mixed polarization to reduce the charge density, increasing the stability of the domain wall. This also reduces the strength of in-plane electric fields at the domain wall that favour 180° ferroelectric switching within the ferroelastic domain such that they are insufficient to cause switching. Therefore, the front of the newly switched − c -domain remains stable and pinned at the a -domain, preventing complete switching at small-to-moderate biases [2] . Such a transition zone only occurs at the charged 90° boundary, whereas the uncharged domain walls remain sharp ( Supplementary Fig. S2 ). With a further increase in applied bias, the transition zone becomes thinner as dipoles in the dipole glass layer are increasingly aligned along the {00-1} direction, leading to an increase in the bound charge density at the boundary. This also increases the in-plane electric field at the boundary and can induce 180° switching within the a -domain, overcoming the hindering effect. Figure 4a–c shows that at +15 V, the a -domain is switched into a − a -domain (with the polarization pointing left), allowing the 180°-domain wall to pass through the a -domain and continue to grow forward ( Fig. 4b,c ). Although the 180°-domain wall in the a -domain is not shown clearly in Fig. 4c , we did observe it in other cases, as shown in Supplementary Fig. S3 . In addition, if the polarization within the a -domain had not switched, the upper and lower 90° domain walls would become oppositely charged, inducing a large electric field within the a -domain that would favour 180° switching [4] . Figure 4: One hundred and eighty degree polarization switching within a ferroelastic domain. ( a ) In situ TEM observation of the switching process through an embedded a -domain. A − c -domain with downward polarization nucleates from the top interface ( a , b ) at a bias of +8 V, it passes through a ferroelastic domain with the polarization pointing towards right ( c ) at a bias of approximately +15 V and keeps growing at +15 V. ( c ) Polarization rotation (180°) has occurred within the ferroelastic domain. Below each TEM image a schematic shows the domain orientations determined using a combination of image contrast, thermodynamic analysis and high-resolution polarization mapping. Scale bar, 50 nm. ( d ) Phase-field simulation of the switching process around an embedded a -domain. The switched − c -domain ( e ) nucleates directly underneath the electrode and extends into the film. Near the charged 90°-domain wall, large in-plane electric fields cause the nucleation of a switched − a -domain with the polarization pointing towards left ( f ). The a -domain is replaced by an − a -domain, creating another charged 90° domain wall, which facilitates the switching of far c -domain with upward polarization ( g ). ( h ) Simulation of the switched − c -domain impeded by an a -domain at an averaged field of ~229 kV cm −1 . ( i ) The local distribution of electric potential of the rectangle region in h . ( j ) The potential profile along the red arrow in h . The thin black arrow indicates the location of the charged domain wall. ( k ) The local distribution of the in-plane electric field of the rectangular region in h indicating large in-plane fields across the charged boundary. ( l ) The in-plane electric field profile along the red arrow in h . The thin black arrow indicates the charged domain wall. Full size image As the switching process within the a -domain is usually too fast and local to observe with in situ TEM, the phase-field modelling technique [22] , [23] , [24] was employed to simulate the switching around an embedded a -domain and reveal the detailed dynamics. In agreement with the experimental results, the simulation reproduced the hindering effect and 180° switching of the embedded 90° domain boundary during the switching process. Figure 4d–g shows that the progression of domain growth during the simulation is similar to the experimental observations. Initially, the switched − c -domain nucleates directly underneath the electrode and extends into the thin film ( Fig. 4d,e ). The simulation results in Fig. 4h–l show that the interaction between the bound charges at the a -domain wall and the switched − c -domain wall generates a large depolarization potential and in-plane electric field. These fields subsequently lead to the nucleation of 180° switching within a -domains to reduce the bound charge in the region between the a -domain and the switched − c -domain ( Fig. 4f ). Under continued applied potential, this switched − a -domain replaced the previous a -domain, forming another, charged, head-to-head 90° domain wall with the far c -domain ( Fig. 4g ), which facilitates the 180° switching in the far c -domain. Incomplete switching caused by pre-existing ferroelastic domain wall pinning may account for the degraded performance in ferroelectric devices operating at low fields [1] , [2] . Our in situ TEM studies demonstrate that the hindering effect of 90° ferroelastic domains on 180° ferroelectric switching can be overcome by applying higher electric bias ( Fig. 4 ). This is consistent with previous reports on the enhancement of ferroelectric response by improving the mobility of the as-grown a -domains with high-electric field pulses [16] or using nanofabricated island structures [1] . Reversing the applied voltage, the recently created domain can be switched back, as shown in Fig. 5 . The back switching is typically realized by shrinking of the c -domains due to the bound charges at the domain walls, essentially exactly reversing the domain growth process ( Fig. 5a–d ). Eventually, the film returns back to the original domain structure ( Fig. 5d ). The negative voltage required for back switching is smaller than the positive switching voltage because of the built-in electrical field induced by the Schottky barrier between PZT and SRO, which favours the upward polarization and hence facilitates back switching [10] . Figure 5: Back switching. Reversing the voltage, the switched domain shrinks ( a – c ), passing through the ferroelastic domain ( d ) as a result of back switching. Simultaneously, the whole ferroelastic domain is switched back to original state in d . Below each TEM image a schematic shows the domain orientations determined using a combination of image contrast, thermodynamic analysis and high-resolution polarization mapping. Scale bar, 50 nm. Full size image In conclusion, our results clearly show that ferroelastic a -domains in tetragonal PZT thin films hinder motion of ferroelectric 180° domain walls by forming charged ferroelastic domain boundaries, leading to incomplete switching. To the best of our knowledge, the atomic-scale structure of these charged 90° domain wall is observed for the first time in this study. Charged 90° domain walls exhibit ‘smearing’ of the local polarization at the interface and a ‘dipole glass’ transition layer formed by introducing nanodomains consisting of reduced polarization and disorder to reduce the charge density and depolarization energy, while the charged and uncharged 180° domain wall ( Supplementary Fig. S3 ) and uncharged 90° domain walls ( Fig. 1h ) remain sharp. We propose that the formation of this layer provides the fundamental atomic-scale mechanism for hindering at ferroelastic walls during switching, preventing a simpler, lower voltage-switching process through ferroelastic domains in PZT. This mechanism also probably explains similar hindering at ferroelastic domain walls observed in other ferroelectrics such as rhombohedral BiFeO 3 (ref. 13 ). This result also indicates that a charged 90° domain wall may have an even larger energy density than that of a charged 180° domain wall because of the charge–strain coupling."},{"section":"Methods","content":"Material synthesis and characterization Epitaxial PZT and SRO films were grown on single crystal {001} STO substrates by 90 o off-axis sputtering [38] . SRO was grown at 560 °C and 200 mTorr with 3:2 ratio of Ar to O 2 and PZT was grown at 530 °C and 200 mTorr with a 1:1 ratio of Ar to O 2 gasses. Cross-sectional TEM samples were made by conventional mechanical polishing and argon ion milling (Gatan). Atomic resolution scanning TEM images used in this manuscript were acquired using the FEI Titan 80-300 (TEAM 0.5) at Lawrence Berkeley National Lab. In situ TEM In situ TEM experiments were carried out with a JEOL 2010F and a spherical aberration ( C s )-corrected JEOL2100F using a Nanofactory TEM-scanning tunnelling microscopy platform. Electrical bias was applied between a home-made tungsten tip, acting as a movable electrode, and the conductive SRO bottom electrode, which was connected to ground. External voltages were applied using a HP 6614c for direct current voltages and a HP 3324a function generator for triangle waveforms. The domain evolution was recorded by a (Gatan) charge-coupled device or a (Gatan) television-rate camera at 29 frames using diffraction contrast imaging with diffraction g =202 p . Phase-field modelling Phase-field modelling is a mesoscale-modelling approach that evolves the distribution of the three polarization components, P i , in time towards a minimum energy state by solving the time-dependent Ginzburg–Landau equation [22] , [23] , [39] , [40] , [41] In equation (1), F is the volume integral of the total free energy of the system expressed in terms of the polarization components and L is the kinetic coefficient. The phase-field model considers contributions to the total free energy from the bulk free energy, f bulk ; the electrical dipole interaction energy, f electric ; the elastic interaction energy, f elastic ; and the polarization gradient energy, f gradient . Altogether, the total free energy expressed in terms of the polarization parameters has the form [39] , [40] where f i are the local contributions of each type of energy. In equation (2), the strain distribution is introduced as a second-order parameter to model the elastic response of the system [23] . Bulk crystal free energy was modelled using a sixth-order Landau polynomial of the form: where α 1 , α 11 , α 12 , α 111 , α 112 and α 123 are the empirical fitting parameters for the Landau polynomial to describe phase transitions in Pb(Zr 0.2 Ti 0.8 )O 3 under stress-free conditions. For this simulation, coefficients measured by Haun, et al. [42] , [43] , [44] were used. Electrical dipole interactions between bound polarization charges and the electric field distribution, E i , were considered through the electrostatic interaction energy [24] , [45] In equation (4), ε 0 is the permittivity of free space and κ ij is the relative background dielectric constant tensor. Using a specified polarization distribution, the electric field distribution was found by first finding the potential distribution, ϕ , through the Poisson equation [24] and using this result to determine the electric field via E i =− ϕ ,i . For the thin film geometry with an electrically conductive bottom electrode under an applied bias, fixed electrical potential boundary conditions at the film surfaces were used to solve equation (5). For the planar bottom electrode in the system, the potential was held uniformly constant at zero bias throughout the simulation. In contrast, the electrode used at the top surface for switching was modelled using a piezoresponse force microscope (PFM)-like potential simulated using the Lorentz-like distribution of width γ (ref. 17 ) In equation (6), the centre of the electrode potential distribution was assumed to be located at position ( x 0 , y 0 ) and to apply a potential, ϕ 0 , at the centre of the distribution. For these simulations, the applied potential distribution width parameter γ was assumed to have a value of 10 nm. Ferroelectric PZT undergoes a spontaneous deformation, ε ij 0 , as a result of the electrostrictive effect. Elastic energy associated with the resulting total strain in the system, ε ij , is (ref. 23 ) where C ijkl is the stiffness tensor of PZT. The spontaneous strain associated with the ferroelectric transformation in equation (6) is related to the polarization via the experimentally measured electrostrictive tensor, Q ijkl Similar to the potential, the strain distribution was determined for a given polarization distribution by solving the equations for mechanical stability in the thin film: Details about solving equations (7, 8, 9) for the film/substrate system and the required boundary conditions are presented in depth elsewhere [23] , [39] . Domain wall energy in the system was included through the polarization gradient energy. For the sake of simplicity, isotropic gradient energy was assumed, so the total gradient energy was expressed as (ref. 23 ) In equation (10), G 1111 is the gradient energy coefficient. For this simulation, its value was chosen to be G 1111 / G 110 =0.6 in reduced units. G 110 was related to the voxel size in the simulation Δ, through . For an assumed grid spacing of Δ=1 nm, G 110 =1.5 × 10 −10 C −2 m 4 N. Using equations (3, 4, 5, 6, 7, 8, 9, 10), the temporal evolution of the thin film under an applied potential was obtained by using a linear approximation to the time derivative in equation (1) and solving the resulting equation using the semi-implicit Fourier spectral method [22] . This method applies periodic boundary conditions in the x 1 and x 2 dimensions while applying the specified boundary conditions in the out-of-plane x 3 dimension. For this simulation, a regular grid of 256Δ × 48Δ × 96Δ voxels was used with periodic boundary conditions along the first two dimensions. The thickness of the film was taken to be 56Δ while the substrate was assumed to have a thickness of 14Δ (ref. 39 ). Each cubic voxel was assumed to have an edge length of 1 nm. Further, the substrate was assumed to apply a biaxial substrate strain to the film of –0.7% as a result of the lattice mismatch between the STO substrate and Pb(Zr 0.2 Ti 0.8 )O 3 film. This strain represents a partially relaxed biaxial strain in the system and was applied by assuming a constant average background strain when solving equation (9) above. Several layers of substrate were included to allow some small relaxation of the substrate in response to deformation in the film. However, for solving equation (9), the displacement at the bottom of the substrate was assumed to be zero. The time step used was Δ t =0.01 where Δ t =1/( L | α 1 | T =25 °C ) (ref. 23 ). In Fig. 1a–c , the potential distribution is normalized by a characteristic potential defined in the simulation as ( HP 0 | α 11 | T =25 °C ) where H is the voxel edge length and P 0 is the spontaneous polarization assumed to be around 0.757 Cm −2 . In all, the characteristic potential is 0.123 V."},{"section":"Additional information","content":"How to cite this article: Gao, P. et al. Atomic-scale mechanisms of ferroelastic domain-wall-mediated ferroelectric switching. Nat. Commun. 4:2791 doi: 10.1038/ncomms3791 (2013)."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"Ferroelectric thin films usually possess polydomain structures determined by electromechanical boundary conditions. For example, ferroelastic twinning structures with polarization rotations of 71° or 109° in rhombohedral perovskites (for example, BiFeO 3 ) or of 90° in tetragonal perovskites (for example, PbZr 0.2 Ti 0.8 O 3 (PZT)) provide an energetically favourable means of reducing the strain energy in epitaxial thin films. Many properties of ferroelectric thin films are closely correlated to the structure and dynamic behaviour of domains, and thus can be manipulated by domain engineering. For example, the piezoelectric response in epitaxial PZT thin films is particularly sensitive to embedded ferroelastic domains and can be dramatically enhanced by manipulating the domain structure to either remove these domains or increase their mobility [1] . Above a critical film thickness of about 100 nm in PZT [1] , [2] , stress-relieving ferroelastic domains generally form a dense square network of crossing domains, which, in epitaxial thin films, are usually nearly immobile because of the strong coupling between charge and strain in the system. Given the high density and poor mobility of such ferroelastic domains in epitaxial films, interactions between the switched 180° domains and these pre-existing ferroelastic domains dominate the overall polarization-switching kinetics. In particular, the presence of ferroelastic domains reduces the responsiveness of the thin film by lowering the spontaneous out-of-film polarization and reducing the electromechanical piezoelectric response, d 33 , in piezoforce microscopy-switching experiments [1] , [3] . During switching, these domains are often observed to undergo 180° switching in place rather than 90° switching. Although this removes charged domain boundaries between the switched domains and the embedded ferroelastic domains, the polarization is not completely aligned with the applied electric field, leading to the observed decrease in the measured spontaneous polarization. These ferroelastic domains also lead to a decrease in the relative stability of a written domain structure and have been seen to be associated with back switching of the thin film to its initial state after the removal of the applied electric field [3] , [4] . This behaviour is especially important in non-volatile devices as their ability to retain the current state is determined by the stability of newly written domains [5] . Macroscopic ferroelectric-switching studies naturally tend to aggregate interactions between domains into the total system behaviour [6] . An understanding of the local response and underlying dynamic behaviour of individual domain walls and defects throughout the thickness of the film during the switching process is necessary, however, to engineer reliable ferroelectric devices [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] . This is especially important in ferroelectric memory devices scaled down to the nanometre scale to achieve a high data storage density where a single defect may dominate the total switching process. At this feature size, it is critical to understand how specific interactions between isolated immobile ferroelastic domains and switched ferroelectric domains influence local switching dynamics. Several studies have attempted to elucidate the specific interactions around ferroelastic domain walls that lead to their strong influence on macroscopic switching kinetics in tetragonal ferroelectrics [2] , [3] , [4] , [15] , [16] . For example, the preferential nucleation of 180° domains at 90° domain walls was reported in barium titanate using optical techniques [15] . Polarization switching through preferential paths was also observed in PZT thin films using piezoresponse force microscopy [3] , whereas numerical modelling of the polarization switching around embedded 90° domains showed that the reduction in the nucleation potential near the domain boundary was the result of a large built-in electrostatic field at the surface of the film lowering the energy barrier to nucleation [17] . The numerical study, however, did not investigate polarization switching beyond the initial nucleation. Ferroelastic domain walls have also been found to be active in switching kinetics beyond facilitating domain nucleation [2] , [4] . Little [15] found, for instance, that in bulk crystals, 180° domain wall motion was abruptly arrested at 90° domain walls. On the basis of electrical measurements, Li and Alexe [2] proposed that 90° domain walls inhibit the motion of 180° domain walls in the plane of the film, slowing the switching kinetics and reducing the overall response of the ferroelectric to an applied electric field. Electrical measurements, however, yield only statistically averaged results, and surface probe techniques are unable to observe subsurface switching paths. Using either method, detailed atomic-scale mechanisms governing the interactions between 90° and 180° domain boundaries are difficult to determine. Understanding local switching pathways, however, is critical as ferroelectrics find use in miniaturized electronic devices such as capacitors and memory devices [9] where the switching kinetics of individual domains dominate the device’s response. To overcome the resolution limits of optical and electrical measurement techniques, we use transmission electron microscopy (TEM) with in situ switching along with numerical phase-field simulations of the full switching process to determine the interaction mechanisms between 90° and 180° domain walls during switching. In comparison with general measurement techniques, in situ TEM can be used to directly observe atomic-scale domain interactions in real time during switching through the thickness of the film, enabling switching dynamics to be correlated with local defect microstructures in real time [13] , [18] , [19] . Employing spherical aberration-corrected TEM with sub-angstrom resolution [20] , [21] , we are able to directly map the local polarization at the atomic scale [9] , [10] . We combine this with mesoscale phase-field modelling of the dynamic polarization-switching processes to corroborate our observations with current models of ferroelectric thin films. Our key finding is the identification of a dipole glass structure formed at the immobile ferroelastic domain boundary during the switching process as the primary atomic-scale mechanism for hindering of the domain-switching process by immobile ferroelastic domains. We find that formation of this layer could occur to reduce the system’s electrostatic energy and allow formation of stable head-to-head domain boundaries in the thin films. The impact of this layer on the switching process and strategies to overcome this effect are discussed.\",\n \"references\": {\n \"indices\": [\n 1,\n 2,\n 3,\n 4,\n 5,\n 6,\n 7,\n 8,\n 9,\n 10,\n 11,\n 12,\n 13,\n 14,\n 15,\n 16,\n 17,\n 18,\n 19,\n 20,\n 21\n ],\n \"infos\": [\n {\n \"idx\": \"1\",\n \"title\": \"Nagarajan, V. et al. Dynamics of ferroelastic domains in ferroelectric thin films.Nat. Mater.2, 43–47 (2003).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat800\"\n },\n {\n \"idx\": \"2\",\n \"title\": \"Li, W. & Alexe, M. Investigation on switching kinetics in epitaxial Pb(Zr0.2Ti0.8)O3 ferroelectric thin films: role of the 90 degrees domain walls.Appl. Phys. Lett.91, 262903 (2007).\",\n \"link\": \"https://doi.org/10.1063%2F1.2825414\"\n },\n {\n \"idx\": \"3\",\n \"title\": \"Ganpule, C. S. et al. Role of 90 degrees domains in lead zirconate titanate thin films.Appl. Phys. Lett.77, 292–294 (2000).\",\n \"link\": \"https://doi.org/10.1063%2F1.126954\"\n },\n {\n \"idx\": \"4\",\n \"title\": \"Roelofs, A. et al. Depolarizing-field-mediated 180 degrees switching in ferroelectric thin films with 90 degrees domains.Appl. Phys. Lett.80, 1424–1426 (2002).\",\n \"link\": \"https://doi.org/10.1063%2F1.1448653\"\n },\n {\n \"idx\": \"5\",\n \"title\": \"Gao, P. et al. Direct observations of retention failure in ferroelectric memories.Adv. Mater.24, 1106–1110 (2012).\",\n \"link\": \"https://doi.org/10.1002%2Fadma.201103983\"\n },\n {\n \"idx\": \"6\",\n \"title\": \"Dawber, M., Rabe, K. M. & Scott, J. F. Physics of thin-film ferroelectric oxides.Rev. Mod. Phys.77, 1083–1130 (2005).\",\n \"link\": \"https://doi.org/10.1103%2FRevModPhys.77.1083\"\n },\n {\n \"idx\": \"7\",\n \"title\": \"Merz, W. J. Domain formation and domain wall motions in ferroelectric BaTiO3 single crystals.Phys. Rev.95, 690–698 (1954).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRev.95.690\"\n },\n {\n \"idx\": \"8\",\n \"title\": \"Jesse, S. et al. Direct imaging of the spatial and energy distribution of nucleation centres in ferroelectric materials.Nat. Mater.7, 209–215 (2008).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat2114\"\n },\n {\n \"idx\": \"9\",\n \"title\": \"Nelson, C. T. et al. Domain dynamics during ferroelectric switching.Science334, 968–971 (2011).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1206980\"\n },\n {\n \"idx\": \"10\",\n \"title\": \"Gao, P. et al. Revealing the role of defects in ferroelectric switching with atomic resolution.Nat. Commun.2, 591 (2011).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms1600\"\n },\n {\n \"idx\": \"11\",\n \"title\": \"Chu, M. W. et al. Impact of misfit dislocations on the polarization instability of epitaxial nanostructured ferroelectric perovskites.Nat. Mater.3, 87–90 (2004).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat1057\"\n },\n {\n \"idx\": \"12\",\n \"title\": \"Wang, R. V. et al. Reversible chemical switching of a ferroelectric film.Phys. Rev. Lett.102, 047601 (2009).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.102.047601\"\n },\n {\n \"idx\": \"13\",\n \"title\": \"Chang, H. J. et al. Watching domains grow:In-situstudies of polarization switching by combined scanning probe and scanning transmission electron microscopy.J. Appl. Phys.110, 052014 (2011).\",\n \"link\": \"https://doi.org/10.1063%2F1.3623779\"\n },\n {\n \"idx\": \"14\",\n \"title\": \"Dearaujo, C. A. P., Cuchiaro, J. D., McMillan, L. D., Scott, M. C. & Scott, J. F. Fatigue-free ferroelectric capacitors with platinum-electrodes.Nature374, 627–629 (1995).\",\n \"link\": \"https://doi.org/10.1038%2F374627a0\"\n },\n {\n \"idx\": \"15\",\n \"title\": \"Little, E. A. Dynamic behavior of domain walls in barium titanate.Phys. Rev.98, 978–984 (1955).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRev.98.978\"\n },\n {\n \"idx\": \"16\",\n \"title\": \"Kohli, M., Muralt, P. & Setter, N. Removal of 90 degrees domain pinning in (100) Pb(Zr0.15Ti0.85)O3 thin films by pulsed operation.Appl. Phys. Lett.72, 3217–3219 (1998).\",\n \"link\": \"https://doi.org/10.1063%2F1.121554\"\n },\n {\n \"idx\": \"17\",\n \"title\": \"Choudhury, S. et al. Effect of ferroelastic twin walls on local polarization switching: phase-field modeling.Appl. Phys. Lett.93, 162901 (2008).\",\n \"link\": \"https://doi.org/10.1063%2F1.2993330\"\n },\n {\n \"idx\": \"18\",\n \"title\": \"Sato, Y., Hirayama, T. & Ikuhara, Y. Real-time direct observations of polarization reversal in a piezoelectric crystal: Pb(Mg1/3Nb2/3)O3-PbTiO3 studied viain situelectrical biasing transmission electron microscopy.Phys. Rev. Lett.107, 187601 (2011).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.107.187601\"\n },\n {\n \"idx\": \"19\",\n \"title\": \"Winkler, C. R. et al. Accessing intermediate ferroelectric switching regimes with time-resolved transmission electron microscopy.J. Appl. Phys.112, 052013–052019 (2012).\",\n \"link\": \"https://doi.org/10.1063%2F1.4746082\"\n },\n {\n \"idx\": \"20\",\n \"title\": \"Jia, C. L. et al. Atomic-scale study of electric dipoles near charged and uncharged domain walls in ferroelectric films.Nat. Mater.7, 57–61 (2008).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat2080\"\n },\n {\n \"idx\": \"21\",\n \"title\": \"Catalan, G. et al. Flexoelectric rotation of polarization in ferroelectric thin films.Nat. Mater.10, 963–967 (2011).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat3141\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Phase-field modelling Phase-field modelling [22] , [23] , [24] of an isolated 90° domain embedded in a c -poled PZT thin film was used to understand the starting electromechanical state of the film. The results ( Fig. 1a–d ) show that the ferroelastic domain walls develop an electrical charge and are associated with significant strains despite lying on nominally charge and strain neutral {101} planes. In fact, an electrical potential step exists across a ferroelastic domain wall in bulk tetragonal PbTiO 3 according to first-principles calculations [25] and phase-field simulations [26] , indicating 90° domain walls may be inherently charged. In addition, the effect of the substrate constraint may lead to additional polarization charges at the domain wall because of different polarization magnitudes for the c- and a -domains in an epitaxial thin film ( c refers to domains with the polarization oriented along the film normal and a refers to those with the polarization oriented in-plane). The existence of this bound charge can have a significant impact on the local switching dynamics. Because of the presence of charged 90° domain walls, the externally applied bias changes less quickly at the surface of the film, reducing the electric field there ( Fig. 1b,c ). As a result, a larger applied bias is required to generate the critical electric field for switching in the vicinity of the 90° domain wall. In addition, clamping of the ferroelectric film by the epitaxial substrate leads to local inhomogeneous strain distributions ( Fig. 1d ), which may influence local switching kinetics in a more complicated manner. Figure 1: Ferroelastic domains in a tetragonal PZT thin film. ( a ) The calculated potential distribution around an isolated a -domain showing the domain walls are moderately charged. ( b ) Calculated potential distribution at 8.4 V without and ( c ) with an embedded a -domain. ( d ) Calculated elastic strain distribution in the absence of external bias showing large compressive strain near the a -domain/substrate interface. ( e ) TEM image of the in situ biasing experimental set-up. A sharp mobile tungsten tip acts as an electrode, whereas the bottom SRO electrode is grounded. Scale bar, 100 nm. The schematic below the micrograph illustrates the observed ferroelastic domains. ( f ) Selected area electron diffraction pattern acquired from a region including both a -domain and c -domain showing splitting of the diffraction spots. The red rectangle indicates the c -domain diffraction pattern, whereas the yellow rectangle corresponds to the a -domain diffraction pattern. ( g ) Atomic resolution HAADF image of an embedded a -domain overlaid with its lattice rotation angle mapping generated by GPA. The colour boundaries correspond to the 90°-domain walls. Scale bar, 5 nm. ( h ) Plot of polarization vectors for a 90°-domain wall showing head-to-tail configuration. Scale bar, 2 nm. Full size image Ferroelastic domain wall structures Isolated ferroelastic domains were investigated in c -oriented tetragonal PZT films, with a thickness of ~200 nm, grown on {001} SrTiO 3 (STO) substrates buffered with an ~20 nm thick SrRuO 3 (SRO) conducting electrode. Cross-sectional TEM specimens were prepared by conventional mechanical polishing and argon ion milling. A TEM image ( Fig. 1e ) shows a mixture of a / c -domains necessary to accommodate the residual epitaxial stain. A selected area electron diffraction pattern acquired from a region that includes both a -domains (with the polarization oriented towards right) and c -domains (with the polarization oriented upwards) shows the splitting of reflection spots ( Fig. 1f ). There is a lattice tilt, α , of ~2.7° between lattice planes across the 90°-domain wall, corresponding to a lattice constant ratio, c/a = tan (45°+ α /2), of ~1.048, consistent with previous X-ray diffraction results [27] . This lattice tilt enables the 90°-domain walls to be easily distinguished by geometric phase analysis (GPA) [28] of atomic resolution high angle annular dark field (HAADF) images ( Fig. 1g ). Figure 1h shows a map of the polarization vectors around a 90°-domain wall obtained by calculating the offsets of the Pb sublattice [29] , demonstrating that the 90°-domain walls are on {101} planes. This head-to-tail configuration of polarization vectors minimizes the bound charge at the domain wall [7] . This analysis also shows that the width of 90° domain walls is approximately one unit cell according to the polarization vector map in Fig. 1h . In situ ferroelectric switching In situ observations of polarization switching were carried out by application of an electric field between a scanning tungsten tip electrode and the grounded SRO electrode in the TEM ( Fig. 1e ). Figure 2 shows sequential TEM images during switching of a large c -domain adjacent to an a -domain. In Fig. 2b , a new − c -domain (where the negative sign indicates downward polarization) nucleates and grows at the top interface at a bias of 8.2 V, leading to the contrast change from bright to dark. Although previous studies proposed that the nucleation of c -domains preferentially begins from 90° domain walls [3] , [8] , [15] , our in situ TEM observations revealed that nucleation occurs only at the top interface with a negative bias [5] , [10] . The switched − c -domain grew with increasing voltage ( Fig. 2b,c ). Under a constant voltage ( Fig. 2c–g ), the domain grew slowly, which may result from a relatively high density of dislocations acting as weak pinning centres in the film [10] . Dislocations also account for the sawtooth shape of the domain ( Fig. 2c ). As a single needle-like domain was observed in dislocation-free PZT thin films grown on more closely lattice matched DyScO 3 substrates [10] , local pinning causes non-uniform growth of the switched domain front. Figure 2: Hindering effect of a ferroelastic domain. A chronological TEM bright field image series shows the evolution of a − c -domain at biases of 0→(+10)→0 V. Original polarization of c -domain in PZT film is upward ( a ). Scale bar, 100 nm. Nucleation of – c -domain with downward polarization occurs at the top interface of tungsten and PZT at 8.2 V ( b ). It grows forward slowly with the increase in bias and/or duration ( c ), but its motion is hindered by the 90°-domain boundary ( d – g ). After the voltage is removed, the domain is stable with a slight relaxation of the 180°-domain wall ( h ). Thin arrows indicate possible dislocation pinning sites. Below each image a schematic shows the observed domains. In each image contrast, thermodynamic analysis and high-resolution polarization mapping were used to determine the domain polarization. Full size image Once the created − c -domain reached the edge of the a -domain ( Fig. 2d ), the 180°-domain wall motion ceased ( Fig. 2e–g ), showing the ferroelastic domain wall acted as an obstacle to the propagation of the domain wall. When a 180°-domain wall impinges on the ferroelastic domain, the out-of-plane component of the electric field favours the 90°-ferroelastic switching of the a -domain (in which polarization rotates from pointing right to downwards). Such switching, however, is unlikely to happen at moderate field strengths because of the large energy cost associated with the lattice distortion and accompanying elastic strain incurred when switching the ferroelastic domain [30] . In addition, a -domains are often stabilized by misfit dislocations at the film interfaces and consequently are even more difficult to switch [31] , [32] .\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"The pinning observed here differs considerably from that seen in ferroelastic switching in BiFeO 3 by Wrinkler et al. [33] where they observed that growing domains were repelled by pre-existing ferroelastic domain boundaries and that many of these metastable domains disappeared when the bias was removed. In our observations, the switched ferroelectric domain was impeded by the ferroelastic domain and was stable after the removal of applied voltage. Atomically resolved HAADF images were used to characterize the structure of 90° domain walls around the ferroelastic domain after switching ( Fig. 3 ). Mapping the ferroelastic variants around the domain wall using both GPA ( Fig. 3a ) and lattice plane spacing determined from the Pb atom positions ( Fig. 3b ) shows that the lattice orientation change remains sharp across both domain walls during the switching process, and indicates no ferroelastic switching or crystal structure change occurred around the 90°-domain wall during this process. In addition, strain distributions determined from GPA calculations are not obviously affected by the ferroelectric polarization switching ( Supplementary Fig. S1 ). The corresponding polarization angle map calculated by the Pb sublattice offset in Fig. 3c , however, shows that the upper wall is diffuse. There is a transition zone with thickness of ~10 unit cells in which the polarization vectors are randomly oriented and have substantially reduced magnitudes ( Fig. 3e ). As the contrast in a HAADF image comes from the average of many unit cells along the beam direction, it seems that the transition zone consists of many tiny domains polarized in different orientations, resulting in a glassy polarization configuration or dipole glass. Unlike traditional glasses, this region remains crystalline; only the long-range order of the polarization collapses. The uncharged 90° domain wall in the bottom-left corner of Fig. 3c , however, is unaffected and the polarization angle map shows a sharply defined boundary ( Fig. 3f ), similar to that seen before switching ( Fig. 1h ). Figure 3: Charged ferroelastic domain boundary between − c -domain and a -domain. ( a ) A map of lattice tilt angle based on GPA calculations overlaid on its atomic resolution HAADF image. The region is the enlarged view of the rectangle in Fig. 2h , including three domains: − c -domain (upper right), a -domain (middle) and c -domain (lower left). Scale bar, 2 nm. ( b ) The map of lattice ratios of out-of-plane to in-plane calculated from the Pb sublattice of the same region. Scale bar, 2 nm. ( c ) Polarization angle map showing the transition zone, highlighted by the white-dashed line, between − c -domain and a -domain. Scale bar, 2 nm. ( d ) Polarization vector map of the switched a -domain corresponding to the rectangle number 1 in c . Scale bar, 1 nm. ( e ) Polarization vector map from the transition zone corresponding to the rectangle number 2 in c . Scale bar, 1 nm. ( f ) Polarization vector map at the a -domain/ c -domain boundary corresponding to the rectangle number 3 in c . Scale bar, 1 nm. Full size image In contrast to previous studies that observed transition layers with suppressed polarization at the substrate interface induced by dislocations [34] , [35] , non-ferroelectric layers at grain boundaries [36] or charge compensation caused irregularly shaped domain walls near the interface [37] ; here the generation of the dipole glass at the charged 90° domain walls may result from the high energy of the head-to-head polarization configuration. As 90° domain walls are on {101} planes, the bound charge density from the termination of the polarization at the domain wall, 2 P S cos θ , is 1.42 P S , where P S is the spontaneous polarization. Given an approximate value for P S of 70 μC cm −2 (ref. 30 ), this corresponds to a charge density of 99 μC cm −2 . Formation of such an energetic, charged domain wall is not favourable, and it is reasonable that a spontaneous relaxation may occur to form a transition zone of mixed polarization to reduce the charge density, increasing the stability of the domain wall. This also reduces the strength of in-plane electric fields at the domain wall that favour 180° ferroelectric switching within the ferroelastic domain such that they are insufficient to cause switching. Therefore, the front of the newly switched − c -domain remains stable and pinned at the a -domain, preventing complete switching at small-to-moderate biases [2] . Such a transition zone only occurs at the charged 90° boundary, whereas the uncharged domain walls remain sharp ( Supplementary Fig. S2 ). With a further increase in applied bias, the transition zone becomes thinner as dipoles in the dipole glass layer are increasingly aligned along the {00-1} direction, leading to an increase in the bound charge density at the boundary. This also increases the in-plane electric field at the boundary and can induce 180° switching within the a -domain, overcoming the hindering effect. Figure 4a–c shows that at +15 V, the a -domain is switched into a − a -domain (with the polarization pointing left), allowing the 180°-domain wall to pass through the a -domain and continue to grow forward ( Fig. 4b,c ). Although the 180°-domain wall in the a -domain is not shown clearly in Fig. 4c , we did observe it in other cases, as shown in Supplementary Fig. S3 . In addition, if the polarization within the a -domain had not switched, the upper and lower 90° domain walls would become oppositely charged, inducing a large electric field within the a -domain that would favour 180° switching [4] . Figure 4: One hundred and eighty degree polarization switching within a ferroelastic domain. ( a ) In situ TEM observation of the switching process through an embedded a -domain. A − c -domain with downward polarization nucleates from the top interface ( a , b ) at a bias of +8 V, it passes through a ferroelastic domain with the polarization pointing towards right ( c ) at a bias of approximately +15 V and keeps growing at +15 V. ( c ) Polarization rotation (180°) has occurred within the ferroelastic domain. Below each TEM image a schematic shows the domain orientations determined using a combination of image contrast, thermodynamic analysis and high-resolution polarization mapping. Scale bar, 50 nm. ( d ) Phase-field simulation of the switching process around an embedded a -domain. The switched − c -domain ( e ) nucleates directly underneath the electrode and extends into the film. Near the charged 90°-domain wall, large in-plane electric fields cause the nucleation of a switched − a -domain with the polarization pointing towards left ( f ). The a -domain is replaced by an − a -domain, creating another charged 90° domain wall, which facilitates the switching of far c -domain with upward polarization ( g ). ( h ) Simulation of the switched − c -domain impeded by an a -domain at an averaged field of ~229 kV cm −1 . ( i ) The local distribution of electric potential of the rectangle region in h . ( j ) The potential profile along the red arrow in h . The thin black arrow indicates the location of the charged domain wall. ( k ) The local distribution of the in-plane electric field of the rectangular region in h indicating large in-plane fields across the charged boundary. ( l ) The in-plane electric field profile along the red arrow in h . The thin black arrow indicates the charged domain wall. Full size image As the switching process within the a -domain is usually too fast and local to observe with in situ TEM, the phase-field modelling technique [22] , [23] , [24] was employed to simulate the switching around an embedded a -domain and reveal the detailed dynamics. In agreement with the experimental results, the simulation reproduced the hindering effect and 180° switching of the embedded 90° domain boundary during the switching process. Figure 4d–g shows that the progression of domain growth during the simulation is similar to the experimental observations. Initially, the switched − c -domain nucleates directly underneath the electrode and extends into the thin film ( Fig. 4d,e ). The simulation results in Fig. 4h–l show that the interaction between the bound charges at the a -domain wall and the switched − c -domain wall generates a large depolarization potential and in-plane electric field. These fields subsequently lead to the nucleation of 180° switching within a -domains to reduce the bound charge in the region between the a -domain and the switched − c -domain ( Fig. 4f ). Under continued applied potential, this switched − a -domain replaced the previous a -domain, forming another, charged, head-to-head 90° domain wall with the far c -domain ( Fig. 4g ), which facilitates the 180° switching in the far c -domain. Incomplete switching caused by pre-existing ferroelastic domain wall pinning may account for the degraded performance in ferroelectric devices operating at low fields [1] , [2] . Our in situ TEM studies demonstrate that the hindering effect of 90° ferroelastic domains on 180° ferroelectric switching can be overcome by applying higher electric bias ( Fig. 4 ). This is consistent with previous reports on the enhancement of ferroelectric response by improving the mobility of the as-grown a -domains with high-electric field pulses [16] or using nanofabricated island structures [1] . Reversing the applied voltage, the recently created domain can be switched back, as shown in Fig. 5 . The back switching is typically realized by shrinking of the c -domains due to the bound charges at the domain walls, essentially exactly reversing the domain growth process ( Fig. 5a–d ). Eventually, the film returns back to the original domain structure ( Fig. 5d ). The negative voltage required for back switching is smaller than the positive switching voltage because of the built-in electrical field induced by the Schottky barrier between PZT and SRO, which favours the upward polarization and hence facilitates back switching [10] . Figure 5: Back switching. Reversing the voltage, the switched domain shrinks ( a – c ), passing through the ferroelastic domain ( d ) as a result of back switching. Simultaneously, the whole ferroelastic domain is switched back to original state in d . Below each TEM image a schematic shows the domain orientations determined using a combination of image contrast, thermodynamic analysis and high-resolution polarization mapping. Scale bar, 50 nm. Full size image In conclusion, our results clearly show that ferroelastic a -domains in tetragonal PZT thin films hinder motion of ferroelectric 180° domain walls by forming charged ferroelastic domain boundaries, leading to incomplete switching. To the best of our knowledge, the atomic-scale structure of these charged 90° domain wall is observed for the first time in this study. Charged 90° domain walls exhibit ‘smearing’ of the local polarization at the interface and a ‘dipole glass’ transition layer formed by introducing nanodomains consisting of reduced polarization and disorder to reduce the charge density and depolarization energy, while the charged and uncharged 180° domain wall ( Supplementary Fig. S3 ) and uncharged 90° domain walls ( Fig. 1h ) remain sharp. We propose that the formation of this layer provides the fundamental atomic-scale mechanism for hindering at ferroelastic walls during switching, preventing a simpler, lower voltage-switching process through ferroelastic domains in PZT. This mechanism also probably explains similar hindering at ferroelastic domain walls observed in other ferroelectrics such as rhombohedral BiFeO 3 (ref. 13 ). This result also indicates that a charged 90° domain wall may have an even larger energy density than that of a charged 180° domain wall because of the charge–strain coupling.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Material synthesis and characterization Epitaxial PZT and SRO films were grown on single crystal {001} STO substrates by 90 o off-axis sputtering [38] . SRO was grown at 560 °C and 200 mTorr with 3:2 ratio of Ar to O 2 and PZT was grown at 530 °C and 200 mTorr with a 1:1 ratio of Ar to O 2 gasses. Cross-sectional TEM samples were made by conventional mechanical polishing and argon ion milling (Gatan). Atomic resolution scanning TEM images used in this manuscript were acquired using the FEI Titan 80-300 (TEAM 0.5) at Lawrence Berkeley National Lab. In situ TEM In situ TEM experiments were carried out with a JEOL 2010F and a spherical aberration ( C s )-corrected JEOL2100F using a Nanofactory TEM-scanning tunnelling microscopy platform. Electrical bias was applied between a home-made tungsten tip, acting as a movable electrode, and the conductive SRO bottom electrode, which was connected to ground. External voltages were applied using a HP 6614c for direct current voltages and a HP 3324a function generator for triangle waveforms. The domain evolution was recorded by a (Gatan) charge-coupled device or a (Gatan) television-rate camera at 29 frames using diffraction contrast imaging with diffraction g =202 p . Phase-field modelling Phase-field modelling is a mesoscale-modelling approach that evolves the distribution of the three polarization components, P i , in time towards a minimum energy state by solving the time-dependent Ginzburg–Landau equation [22] , [23] , [39] , [40] , [41] In equation (1), F is the volume integral of the total free energy of the system expressed in terms of the polarization components and L is the kinetic coefficient. The phase-field model considers contributions to the total free energy from the bulk free energy, f bulk ; the electrical dipole interaction energy, f electric ; the elastic interaction energy, f elastic ; and the polarization gradient energy, f gradient . Altogether, the total free energy expressed in terms of the polarization parameters has the form [39] , [40] where f i are the local contributions of each type of energy. In equation (2), the strain distribution is introduced as a second-order parameter to model the elastic response of the system [23] . Bulk crystal free energy was modelled using a sixth-order Landau polynomial of the form: where α 1 , α 11 , α 12 , α 111 , α 112 and α 123 are the empirical fitting parameters for the Landau polynomial to describe phase transitions in Pb(Zr 0.2 Ti 0.8 )O 3 under stress-free conditions. For this simulation, coefficients measured by Haun, et al. [42] , [43] , [44] were used. Electrical dipole interactions between bound polarization charges and the electric field distribution, E i , were considered through the electrostatic interaction energy [24] , [45] In equation (4), ε 0 is the permittivity of free space and κ ij is the relative background dielectric constant tensor. Using a specified polarization distribution, the electric field distribution was found by first finding the potential distribution, ϕ , through the Poisson equation [24] and using this result to determine the electric field via E i =− ϕ ,i . For the thin film geometry with an electrically conductive bottom electrode under an applied bias, fixed electrical potential boundary conditions at the film surfaces were used to solve equation (5). For the planar bottom electrode in the system, the potential was held uniformly constant at zero bias throughout the simulation. In contrast, the electrode used at the top surface for switching was modelled using a piezoresponse force microscope (PFM)-like potential simulated using the Lorentz-like distribution of width γ (ref. 17 ) In equation (6), the centre of the electrode potential distribution was assumed to be located at position ( x 0 , y 0 ) and to apply a potential, ϕ 0 , at the centre of the distribution. For these simulations, the applied potential distribution width parameter γ was assumed to have a value of 10 nm. Ferroelectric PZT undergoes a spontaneous deformation, ε ij 0 , as a result of the electrostrictive effect. Elastic energy associated with the resulting total strain in the system, ε ij , is (ref. 23 ) where C ijkl is the stiffness tensor of PZT. The spontaneous strain associated with the ferroelectric transformation in equation (6) is related to the polarization via the experimentally measured electrostrictive tensor, Q ijkl Similar to the potential, the strain distribution was determined for a given polarization distribution by solving the equations for mechanical stability in the thin film: Details about solving equations (7, 8, 9) for the film/substrate system and the required boundary conditions are presented in depth elsewhere [23] , [39] . Domain wall energy in the system was included through the polarization gradient energy. For the sake of simplicity, isotropic gradient energy was assumed, so the total gradient energy was expressed as (ref. 23 ) In equation (10), G 1111 is the gradient energy coefficient. For this simulation, its value was chosen to be G 1111 / G 110 =0.6 in reduced units. G 110 was related to the voxel size in the simulation Δ, through . For an assumed grid spacing of Δ=1 nm, G 110 =1.5 × 10 −10 C −2 m 4 N. Using equations (3, 4, 5, 6, 7, 8, 9, 10), the temporal evolution of the thin film under an applied potential was obtained by using a linear approximation to the time derivative in equation (1) and solving the resulting equation using the semi-implicit Fourier spectral method [22] . This method applies periodic boundary conditions in the x 1 and x 2 dimensions while applying the specified boundary conditions in the out-of-plane x 3 dimension. For this simulation, a regular grid of 256Δ × 48Δ × 96Δ voxels was used with periodic boundary conditions along the first two dimensions. The thickness of the film was taken to be 56Δ while the substrate was assumed to have a thickness of 14Δ (ref. 39 ). Each cubic voxel was assumed to have an edge length of 1 nm. Further, the substrate was assumed to apply a biaxial substrate strain to the film of –0.7% as a result of the lattice mismatch between the STO substrate and Pb(Zr 0.2 Ti 0.8 )O 3 film. This strain represents a partially relaxed biaxial strain in the system and was applied by assuming a constant average background strain when solving equation (9) above. Several layers of substrate were included to allow some small relaxation of the substrate in response to deformation in the film. However, for solving equation (9), the displacement at the bottom of the substrate was assumed to be zero. The time step used was Δ t =0.01 where Δ t =1/( L | α 1 | T =25 °C ) (ref. 23 ). In Fig. 1a–c , the potential distribution is normalized by a characteristic potential defined in the simulation as ( HP 0 | α 11 | T =25 °C ) where H is the voxel edge length and P 0 is the spontaneous polarization assumed to be around 0.757 Cm −2 . In all, the characteristic potential is 0.123 V.\"\n },\n {\n \"section\": \"Additional information\",\n \"content\": \"How to cite this article: Gao, P. et al. Atomic-scale mechanisms of ferroelastic domain-wall-mediated ferroelectric switching. Nat. Commun. 4:2791 doi: 10.1038/ncomms3791 (2013).\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"The core research idea is to investigate the atomic-scale mechanisms by which immobile ferroelastic domains hinder polarization switching in ferroelectric thin films, particularly through the formation of a dipole glass structure at domain boundaries, using a combination of in situ transmission electron microscopy and mesoscale phase-field simulations to directly observe and model the interaction dynamics between 90° and 180° domain walls during switching."},"entities":{"kind":"list like","value":["atomic-scale mechanisms","immobile ferroelastic domains","polarization switching","ferroelectric thin films","dipole glass structure","domain boundaries","in situ transmission electron microscopy","mesoscale phase-field simulations","90° domain walls","180° domain walls","interaction dynamics","switching","direct observation","modeling","domain wall interactions"],"string":"[\n \"atomic-scale mechanisms\",\n \"immobile ferroelastic domains\",\n \"polarization switching\",\n \"ferroelectric thin films\",\n \"dipole glass structure\",\n \"domain boundaries\",\n \"in situ transmission electron microscopy\",\n \"mesoscale phase-field simulations\",\n \"90° domain walls\",\n \"180° domain walls\",\n \"interaction dynamics\",\n \"switching\",\n \"direct observation\",\n \"modeling\",\n \"domain wall interactions\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1","title":"Nagarajan, V. et al. Dynamics of ferroelastic domains in ferroelectric thin films.Nat. Mater.2, 43–47 (2003).","link":"https://doi.org/10.1038%2Fnmat800"},{"idx":"2","title":"Li, W. & Alexe, M. Investigation on switching kinetics in epitaxial Pb(Zr0.2Ti0.8)O3 ferroelectric thin films: role of the 90 degrees domain walls.Appl. Phys. Lett.91, 262903 (2007).","link":"https://doi.org/10.1063%2F1.2825414"},{"idx":"3","title":"Ganpule, C. S. et al. Role of 90 degrees domains in lead zirconate titanate thin films.Appl. Phys. Lett.77, 292–294 (2000).","link":"https://doi.org/10.1063%2F1.126954"},{"idx":"4","title":"Roelofs, A. et al. Depolarizing-field-mediated 180 degrees switching in ferroelectric thin films with 90 degrees domains.Appl. Phys. Lett.80, 1424–1426 (2002).","link":"https://doi.org/10.1063%2F1.1448653"},{"idx":"5","title":"Gao, P. et al. Direct observations of retention failure in ferroelectric memories.Adv. Mater.24, 1106–1110 (2012).","link":"https://doi.org/10.1002%2Fadma.201103983"},{"idx":"6","title":"Dawber, M., Rabe, K. M. & Scott, J. F. Physics of thin-film ferroelectric oxides.Rev. Mod. Phys.77, 1083–1130 (2005).","link":"https://doi.org/10.1103%2FRevModPhys.77.1083"},{"idx":"7","title":"Merz, W. J. Domain formation and domain wall motions in ferroelectric BaTiO3 single crystals.Phys. Rev.95, 690–698 (1954).","link":"https://doi.org/10.1103%2FPhysRev.95.690"},{"idx":"8","title":"Jesse, S. et al. Direct imaging of the spatial and energy distribution of nucleation centres in ferroelectric materials.Nat. Mater.7, 209–215 (2008).","link":"https://doi.org/10.1038%2Fnmat2114"},{"idx":"9","title":"Nelson, C. T. et al. Domain dynamics during ferroelectric switching.Science334, 968–971 (2011).","link":"https://doi.org/10.1126%2Fscience.1206980"},{"idx":"10","title":"Gao, P. et al. Revealing the role of defects in ferroelectric switching with atomic resolution.Nat. Commun.2, 591 (2011).","link":"https://doi.org/10.1038%2Fncomms1600"},{"idx":"11","title":"Chu, M. W. et al. Impact of misfit dislocations on the polarization instability of epitaxial nanostructured ferroelectric perovskites.Nat. Mater.3, 87–90 (2004).","link":"https://doi.org/10.1038%2Fnmat1057"},{"idx":"12","title":"Wang, R. V. et al. Reversible chemical switching of a ferroelectric film.Phys. Rev. Lett.102, 047601 (2009).","link":"https://doi.org/10.1103%2FPhysRevLett.102.047601"},{"idx":"13","title":"Chang, H. J. et al. Watching domains grow:In-situstudies of polarization switching by combined scanning probe and scanning transmission electron microscopy.J. Appl. Phys.110, 052014 (2011).","link":"https://doi.org/10.1063%2F1.3623779"},{"idx":"14","title":"Dearaujo, C. A. P., Cuchiaro, J. D., McMillan, L. D., Scott, M. C. & Scott, J. F. Fatigue-free ferroelectric capacitors with platinum-electrodes.Nature374, 627–629 (1995).","link":"https://doi.org/10.1038%2F374627a0"},{"idx":"15","title":"Little, E. A. Dynamic behavior of domain walls in barium titanate.Phys. Rev.98, 978–984 (1955).","link":"https://doi.org/10.1103%2FPhysRev.98.978"},{"idx":"16","title":"Kohli, M., Muralt, P. & Setter, N. Removal of 90 degrees domain pinning in (100) Pb(Zr0.15Ti0.85)O3 thin films by pulsed operation.Appl. Phys. Lett.72, 3217–3219 (1998).","link":"https://doi.org/10.1063%2F1.121554"},{"idx":"17","title":"Choudhury, S. et al. Effect of ferroelastic twin walls on local polarization switching: phase-field modeling.Appl. Phys. Lett.93, 162901 (2008).","link":"https://doi.org/10.1063%2F1.2993330"},{"idx":"18","title":"Sato, Y., Hirayama, T. & Ikuhara, Y. Real-time direct observations of polarization reversal in a piezoelectric crystal: Pb(Mg1/3Nb2/3)O3-PbTiO3 studied viain situelectrical biasing transmission electron microscopy.Phys. Rev. Lett.107, 187601 (2011).","link":"https://doi.org/10.1103%2FPhysRevLett.107.187601"},{"idx":"19","title":"Winkler, C. R. et al. Accessing intermediate ferroelectric switching regimes with time-resolved transmission electron microscopy.J. Appl. Phys.112, 052013–052019 (2012).","link":"https://doi.org/10.1063%2F1.4746082"},{"idx":"20","title":"Jia, C. L. et al. Atomic-scale study of electric dipoles near charged and uncharged domain walls in ferroelectric films.Nat. Mater.7, 57–61 (2008).","link":"https://doi.org/10.1038%2Fnmat2080"},{"idx":"21","title":"Catalan, G. et al. Flexoelectric rotation of polarization in ferroelectric thin films.Nat. Mater.10, 963–967 (2011).","link":"https://doi.org/10.1038%2Fnmat3141"},{"idx":"22","title":"Chen, L. Q. & Shen, J. Applications of semi-implicit Fourier-spectral method to phase field equations.Comput. Phys. Commun.108, 147–158 (1998).","link":"https://doi.org/10.1016%2FS0010-4655%2897%2900115-X"},{"idx":"23","title":"Li, Y. L., Hu, S. Y., Liu, Z. K. & Chen, L. Q. Effect of substrate constraint on the stability and evolution of ferroelectric domain structures in thin films.Acta Mater.50, 395–411 (2002).","link":"https://doi.org/10.1016%2FS1359-6454%2801%2900360-3"},{"idx":"24","title":"Li, Y. L. et al. Ferroelectric domain structures in SrBi2Nb2O9 epitaxial thin films: electron microscopy and phase-field simulations.J. Appl. Phys.95, 6332–6340 (2004).","link":"https://doi.org/10.1063%2F1.1707211"},{"idx":"25","title":"Meyer, B. & Vanderbilt, D. Ab initio study of ferroelectric domain walls in PbTiO3 .Phys. Rev. B65, 104111 (2002).","link":"https://doi.org/10.1103%2FPhysRevB.65.104111"},{"idx":"26","title":"Ishibashi, Y. The 90-degrees-wall in the tetragonal phase of BaTiO3-type ferroelectrics.J. Phys. Soc. Jpn62, 1044–1047 (1993).","link":"https://doi.org/10.1143%2FJPSJ.62.1044"},{"idx":"27","title":"Klee, M., Eusemann, R., Waser, R., Brand, W. & Vanhal, H. Processing and electrical-properties of Pb(ZrxTi1-x)O3 (X=0.2-0.75) films- compasion of metallooranic decomposition and sol-gel processes.J. Appl. Phys.72, 1566–1576 (1992).","link":"https://doi.org/10.1063%2F1.351726"},{"idx":"28","title":"Hytch, M. J., Snoeck, E. & Kilaas, R. Quantitative measurement of displacement and strain fields from HREM micrographs.Ultramicroscopy74, 131–146 (1998).","link":"https://doi.org/10.1016%2FS0304-3991%2898%2900035-7"},{"idx":"29","title":"Nelson, C. T. et al. Spontaneous vortex nanodomain arrays at ferroelectric heterointerfaces.Nano Lett.11, 828–834 (2011).","link":"https://doi.org/10.1021%2Fnl1041808"},{"idx":"30","title":"Damjanovic, D. Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics.Rep. Prog. Phys.61, 1267–1324 (1998).","link":"https://doi.org/10.1088%2F0034-4885%2F61%2F9%2F002"},{"idx":"31","title":"Su, D. et al. Origin of 90 degrees domain wall pinning in Pb(Zr0.2Ti0.8)O3 heteroepitaxial thin films.Appl. Phys. Lett.99, 102902 (2011).","link":"https://doi.org/10.1063%2F1.3634028"},{"idx":"32","title":"Kiguchi, T. et al. Configuration and local elastic interaction of ferroelectric domains and misfit dislocation in PbTiO3/SrTiO3 epitaxial thin films.Sci. Technol. Adv. Mater.12, 034413–034421 (2011).","link":"https://doi.org/10.1088%2F1468-6996%2F12%2F3%2F034413"},{"idx":"33","title":"Winkler, C. R., Damodaran, A. R., Karthik, J., Martin, L. W. & Taheri, M. L. Direct observation of ferroelectric domain switching in varying electric field regimes usingin situTEM.Micron43, 1121–1126 (2012).","link":"https://doi.org/10.1016%2Fj.micron.2012.02.009"},{"idx":"34","title":"Alpay, S. P., Misirlioglu, I. B., Nagarajan, V. & Ramesh, R. Can interface dislocations degrade ferroelectric properties?Appl. Phys. Lett.85, 2044 (2004).","link":"https://doi.org/10.1063%2F1.1788894"},{"idx":"35","title":"Misirlioglu, I. B., Vasiliev, A. L., Aindow, M. & Alpay, S. P. Strong degradation of physical properties and formation of a dead layer in ferroelectric films due to interfacial dislocations.Integr. Ferroelectr.71, 67–80 (2005).","link":"https://doi.org/10.1080%2F10584580590964709"},{"idx":"36","title":"Zhao, Z. et al. Grain-size effects on the ferroelectric behavior of dense nanocrystalline BaTiO3 ceramics.Phys. Rev. B70, 024107 (2004).","link":"https://doi.org/10.1103%2FPhysRevB.70.024107"},{"idx":"37","title":"Kim, Y. M. et al. Interplay of octahedral tilts and polar order in BiFeO3 films.Adv. Mater.25, 2497–2504 (2013).","link":"https://doi.org/10.1002%2Fadma.201204584"},{"idx":"38","title":"Eom, C. B. et al.In situgrown YNa2Cu3O7-d thin-films from single-target magnetron sputtering.Appl. Phys. Lett.55, 595–597 (1989).","link":"https://doi.org/10.1063%2F1.102436"},{"idx":"39","title":"Chen, L. Q. Phase-field method of phase transitions/domain structures in ferroelectric thin films: a review.J. Am. Ceram. Soc.91, 1835–1844 (2008).","link":"https://doi.org/10.1111%2Fj.1551-2916.2008.02413.x"},{"idx":"40","title":"Li, Y. L., Hu, S. Y., Liu, Z. K. & Chen, L. Q. Phase-field model of domain structures in ferroelectric thin films.Appl. Phys. Lett.78, 3878–3880 (2001).","link":"https://doi.org/10.1063%2F1.1377855"},{"idx":"41","title":"Zhang, J. X. et al. Computer simulation of ferroelectric domain structures in epitaxial BiFeO3 thin films.J. Appl. Phys.103, 094111 (2008).","link":"https://doi.org/10.1063%2F1.2927385"},{"idx":"42","title":"Haun, M. J., Zhuang, Z. Q., Furman, E., Jang, S. J. & Cross, L. E. Thermodynamic theory of the lead zirconate-titanate solid-solution system,.3. curie constant and 6th order polarization interaction dielectric stiffness coefficients.Ferroelectrics99, 45–54 (1989).","link":"https://doi.org/10.1080%2F00150198908221438"},{"idx":"43","title":"Haun, M. J., Furman, E., McKinstry, H. A. & Cross, L. E. Thermodynamic theory of the lead zirconate-titanate solid-solution system,.2. tricritical behavior.Ferroelectrics99, 27–44 (1989).","link":"https://doi.org/10.1080%2F00150198908221437"},{"idx":"44","title":"Haun, M. J., Furman, E., Jang, S. J. & Cross, L. E. Thermodynamic theory of the lead zirconate-titanate solid-solution system,. 1. phenomenology.Ferroelectrics99, 13–25 (1989).","link":"https://doi.org/10.1080%2F00150198908221436"},{"idx":"45","title":"Li, Y. L., Hu, S. Y., Liu, Z. K. & Chen, L. Q. Effect of electrical boundary conditions on ferroelectric domain structures in thin films.Appl. Phys. Lett.81, 427–429 (2002).","link":"https://doi.org/10.1063%2F1.1492025"}],"string":"[\n {\n \"idx\": \"1\",\n \"title\": \"Nagarajan, V. et al. Dynamics of ferroelastic domains in ferroelectric thin films.Nat. Mater.2, 43–47 (2003).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat800\"\n },\n {\n \"idx\": \"2\",\n \"title\": \"Li, W. & Alexe, M. Investigation on switching kinetics in epitaxial Pb(Zr0.2Ti0.8)O3 ferroelectric thin films: role of the 90 degrees domain walls.Appl. Phys. Lett.91, 262903 (2007).\",\n \"link\": \"https://doi.org/10.1063%2F1.2825414\"\n },\n {\n \"idx\": \"3\",\n \"title\": \"Ganpule, C. S. et al. Role of 90 degrees domains in lead zirconate titanate thin films.Appl. Phys. Lett.77, 292–294 (2000).\",\n \"link\": \"https://doi.org/10.1063%2F1.126954\"\n },\n {\n \"idx\": \"4\",\n \"title\": \"Roelofs, A. et al. Depolarizing-field-mediated 180 degrees switching in ferroelectric thin films with 90 degrees domains.Appl. Phys. Lett.80, 1424–1426 (2002).\",\n \"link\": \"https://doi.org/10.1063%2F1.1448653\"\n },\n {\n \"idx\": \"5\",\n \"title\": \"Gao, P. et al. Direct observations of retention failure in ferroelectric memories.Adv. Mater.24, 1106–1110 (2012).\",\n \"link\": \"https://doi.org/10.1002%2Fadma.201103983\"\n },\n {\n \"idx\": \"6\",\n \"title\": \"Dawber, M., Rabe, K. M. & Scott, J. F. Physics of thin-film ferroelectric oxides.Rev. Mod. Phys.77, 1083–1130 (2005).\",\n \"link\": \"https://doi.org/10.1103%2FRevModPhys.77.1083\"\n },\n {\n \"idx\": \"7\",\n \"title\": \"Merz, W. J. Domain formation and domain wall motions in ferroelectric BaTiO3 single crystals.Phys. Rev.95, 690–698 (1954).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRev.95.690\"\n },\n {\n \"idx\": \"8\",\n \"title\": \"Jesse, S. et al. Direct imaging of the spatial and energy distribution of nucleation centres in ferroelectric materials.Nat. Mater.7, 209–215 (2008).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat2114\"\n },\n {\n \"idx\": \"9\",\n \"title\": \"Nelson, C. T. et al. Domain dynamics during ferroelectric switching.Science334, 968–971 (2011).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1206980\"\n },\n {\n \"idx\": \"10\",\n \"title\": \"Gao, P. et al. Revealing the role of defects in ferroelectric switching with atomic resolution.Nat. Commun.2, 591 (2011).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms1600\"\n },\n {\n \"idx\": \"11\",\n \"title\": \"Chu, M. W. et al. Impact of misfit dislocations on the polarization instability of epitaxial nanostructured ferroelectric perovskites.Nat. Mater.3, 87–90 (2004).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat1057\"\n },\n {\n \"idx\": \"12\",\n \"title\": \"Wang, R. V. et al. Reversible chemical switching of a ferroelectric film.Phys. Rev. Lett.102, 047601 (2009).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.102.047601\"\n },\n {\n \"idx\": \"13\",\n \"title\": \"Chang, H. J. et al. Watching domains grow:In-situstudies of polarization switching by combined scanning probe and scanning transmission electron microscopy.J. Appl. Phys.110, 052014 (2011).\",\n \"link\": \"https://doi.org/10.1063%2F1.3623779\"\n },\n {\n \"idx\": \"14\",\n \"title\": \"Dearaujo, C. A. P., Cuchiaro, J. D., McMillan, L. D., Scott, M. C. & Scott, J. F. Fatigue-free ferroelectric capacitors with platinum-electrodes.Nature374, 627–629 (1995).\",\n \"link\": \"https://doi.org/10.1038%2F374627a0\"\n },\n {\n \"idx\": \"15\",\n \"title\": \"Little, E. A. Dynamic behavior of domain walls in barium titanate.Phys. Rev.98, 978–984 (1955).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRev.98.978\"\n },\n {\n \"idx\": \"16\",\n \"title\": \"Kohli, M., Muralt, P. & Setter, N. Removal of 90 degrees domain pinning in (100) Pb(Zr0.15Ti0.85)O3 thin films by pulsed operation.Appl. Phys. Lett.72, 3217–3219 (1998).\",\n \"link\": \"https://doi.org/10.1063%2F1.121554\"\n },\n {\n \"idx\": \"17\",\n \"title\": \"Choudhury, S. et al. Effect of ferroelastic twin walls on local polarization switching: phase-field modeling.Appl. Phys. Lett.93, 162901 (2008).\",\n \"link\": \"https://doi.org/10.1063%2F1.2993330\"\n },\n {\n \"idx\": \"18\",\n \"title\": \"Sato, Y., Hirayama, T. & Ikuhara, Y. Real-time direct observations of polarization reversal in a piezoelectric crystal: Pb(Mg1/3Nb2/3)O3-PbTiO3 studied viain situelectrical biasing transmission electron microscopy.Phys. Rev. Lett.107, 187601 (2011).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.107.187601\"\n },\n {\n \"idx\": \"19\",\n \"title\": \"Winkler, C. R. et al. Accessing intermediate ferroelectric switching regimes with time-resolved transmission electron microscopy.J. Appl. Phys.112, 052013–052019 (2012).\",\n \"link\": \"https://doi.org/10.1063%2F1.4746082\"\n },\n {\n \"idx\": \"20\",\n \"title\": \"Jia, C. L. et al. Atomic-scale study of electric dipoles near charged and uncharged domain walls in ferroelectric films.Nat. Mater.7, 57–61 (2008).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat2080\"\n },\n {\n \"idx\": \"21\",\n \"title\": \"Catalan, G. et al. Flexoelectric rotation of polarization in ferroelectric thin films.Nat. Mater.10, 963–967 (2011).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat3141\"\n },\n {\n \"idx\": \"22\",\n \"title\": \"Chen, L. Q. & Shen, J. Applications of semi-implicit Fourier-spectral method to phase field equations.Comput. Phys. Commun.108, 147–158 (1998).\",\n \"link\": \"https://doi.org/10.1016%2FS0010-4655%2897%2900115-X\"\n },\n {\n \"idx\": \"23\",\n \"title\": \"Li, Y. L., Hu, S. Y., Liu, Z. K. & Chen, L. Q. Effect of substrate constraint on the stability and evolution of ferroelectric domain structures in thin films.Acta Mater.50, 395–411 (2002).\",\n \"link\": \"https://doi.org/10.1016%2FS1359-6454%2801%2900360-3\"\n },\n {\n \"idx\": \"24\",\n \"title\": \"Li, Y. L. et al. Ferroelectric domain structures in SrBi2Nb2O9 epitaxial thin films: electron microscopy and phase-field simulations.J. Appl. Phys.95, 6332–6340 (2004).\",\n \"link\": \"https://doi.org/10.1063%2F1.1707211\"\n },\n {\n \"idx\": \"25\",\n \"title\": \"Meyer, B. & Vanderbilt, D. Ab initio study of ferroelectric domain walls in PbTiO3 .Phys. Rev. B65, 104111 (2002).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.65.104111\"\n },\n {\n \"idx\": \"26\",\n \"title\": \"Ishibashi, Y. The 90-degrees-wall in the tetragonal phase of BaTiO3-type ferroelectrics.J. Phys. Soc. Jpn62, 1044–1047 (1993).\",\n \"link\": \"https://doi.org/10.1143%2FJPSJ.62.1044\"\n },\n {\n \"idx\": \"27\",\n \"title\": \"Klee, M., Eusemann, R., Waser, R., Brand, W. & Vanhal, H. Processing and electrical-properties of Pb(ZrxTi1-x)O3 (X=0.2-0.75) films- compasion of metallooranic decomposition and sol-gel processes.J. Appl. Phys.72, 1566–1576 (1992).\",\n \"link\": \"https://doi.org/10.1063%2F1.351726\"\n },\n {\n \"idx\": \"28\",\n \"title\": \"Hytch, M. J., Snoeck, E. & Kilaas, R. Quantitative measurement of displacement and strain fields from HREM micrographs.Ultramicroscopy74, 131–146 (1998).\",\n \"link\": \"https://doi.org/10.1016%2FS0304-3991%2898%2900035-7\"\n },\n {\n \"idx\": \"29\",\n \"title\": \"Nelson, C. T. et al. Spontaneous vortex nanodomain arrays at ferroelectric heterointerfaces.Nano Lett.11, 828–834 (2011).\",\n \"link\": \"https://doi.org/10.1021%2Fnl1041808\"\n },\n {\n \"idx\": \"30\",\n \"title\": \"Damjanovic, D. Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics.Rep. Prog. Phys.61, 1267–1324 (1998).\",\n \"link\": \"https://doi.org/10.1088%2F0034-4885%2F61%2F9%2F002\"\n },\n {\n \"idx\": \"31\",\n \"title\": \"Su, D. et al. Origin of 90 degrees domain wall pinning in Pb(Zr0.2Ti0.8)O3 heteroepitaxial thin films.Appl. Phys. Lett.99, 102902 (2011).\",\n \"link\": \"https://doi.org/10.1063%2F1.3634028\"\n },\n {\n \"idx\": \"32\",\n \"title\": \"Kiguchi, T. et al. Configuration and local elastic interaction of ferroelectric domains and misfit dislocation in PbTiO3/SrTiO3 epitaxial thin films.Sci. Technol. Adv. Mater.12, 034413–034421 (2011).\",\n \"link\": \"https://doi.org/10.1088%2F1468-6996%2F12%2F3%2F034413\"\n },\n {\n \"idx\": \"33\",\n \"title\": \"Winkler, C. R., Damodaran, A. R., Karthik, J., Martin, L. W. & Taheri, M. L. Direct observation of ferroelectric domain switching in varying electric field regimes usingin situTEM.Micron43, 1121–1126 (2012).\",\n \"link\": \"https://doi.org/10.1016%2Fj.micron.2012.02.009\"\n },\n {\n \"idx\": \"34\",\n \"title\": \"Alpay, S. P., Misirlioglu, I. B., Nagarajan, V. & Ramesh, R. Can interface dislocations degrade ferroelectric properties?Appl. Phys. Lett.85, 2044 (2004).\",\n \"link\": \"https://doi.org/10.1063%2F1.1788894\"\n },\n {\n \"idx\": \"35\",\n \"title\": \"Misirlioglu, I. B., Vasiliev, A. L., Aindow, M. & Alpay, S. P. Strong degradation of physical properties and formation of a dead layer in ferroelectric films due to interfacial dislocations.Integr. Ferroelectr.71, 67–80 (2005).\",\n \"link\": \"https://doi.org/10.1080%2F10584580590964709\"\n },\n {\n \"idx\": \"36\",\n \"title\": \"Zhao, Z. et al. Grain-size effects on the ferroelectric behavior of dense nanocrystalline BaTiO3 ceramics.Phys. Rev. B70, 024107 (2004).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.70.024107\"\n },\n {\n \"idx\": \"37\",\n \"title\": \"Kim, Y. M. et al. Interplay of octahedral tilts and polar order in BiFeO3 films.Adv. Mater.25, 2497–2504 (2013).\",\n \"link\": \"https://doi.org/10.1002%2Fadma.201204584\"\n },\n {\n \"idx\": \"38\",\n \"title\": \"Eom, C. B. et al.In situgrown YNa2Cu3O7-d thin-films from single-target magnetron sputtering.Appl. Phys. Lett.55, 595–597 (1989).\",\n \"link\": \"https://doi.org/10.1063%2F1.102436\"\n },\n {\n \"idx\": \"39\",\n \"title\": \"Chen, L. Q. Phase-field method of phase transitions/domain structures in ferroelectric thin films: a review.J. Am. Ceram. Soc.91, 1835–1844 (2008).\",\n \"link\": \"https://doi.org/10.1111%2Fj.1551-2916.2008.02413.x\"\n },\n {\n \"idx\": \"40\",\n \"title\": \"Li, Y. L., Hu, S. Y., Liu, Z. K. & Chen, L. Q. Phase-field model of domain structures in ferroelectric thin films.Appl. Phys. Lett.78, 3878–3880 (2001).\",\n \"link\": \"https://doi.org/10.1063%2F1.1377855\"\n },\n {\n \"idx\": \"41\",\n \"title\": \"Zhang, J. X. et al. Computer simulation of ferroelectric domain structures in epitaxial BiFeO3 thin films.J. Appl. Phys.103, 094111 (2008).\",\n \"link\": \"https://doi.org/10.1063%2F1.2927385\"\n },\n {\n \"idx\": \"42\",\n \"title\": \"Haun, M. J., Zhuang, Z. Q., Furman, E., Jang, S. J. & Cross, L. E. Thermodynamic theory of the lead zirconate-titanate solid-solution system,.3. curie constant and 6th order polarization interaction dielectric stiffness coefficients.Ferroelectrics99, 45–54 (1989).\",\n \"link\": \"https://doi.org/10.1080%2F00150198908221438\"\n },\n {\n \"idx\": \"43\",\n \"title\": \"Haun, M. J., Furman, E., McKinstry, H. A. & Cross, L. E. Thermodynamic theory of the lead zirconate-titanate solid-solution system,.2. tricritical behavior.Ferroelectrics99, 27–44 (1989).\",\n \"link\": \"https://doi.org/10.1080%2F00150198908221437\"\n },\n {\n \"idx\": \"44\",\n \"title\": \"Haun, M. J., Furman, E., Jang, S. J. & Cross, L. E. Thermodynamic theory of the lead zirconate-titanate solid-solution system,. 1. phenomenology.Ferroelectrics99, 13–25 (1989).\",\n \"link\": \"https://doi.org/10.1080%2F00150198908221436\"\n },\n {\n \"idx\": \"45\",\n \"title\": \"Li, Y. L., Hu, S. Y., Liu, Z. K. & Chen, L. Q. Effect of electrical boundary conditions on ferroelectric domain structures in thin films.Appl. Phys. Lett.81, 427–429 (2002).\",\n \"link\": \"https://doi.org/10.1063%2F1.1492025\"\n }\n]","isConversational":false}}},{"rowIdx":90,"cells":{"unique_id":{"kind":"string","value":"s41467-024-47353-x"},"subfield":{"kind":"string","value":"Nanoscience and technology"},"title":{"kind":"string","value":"Single-photon detection using large-scale high-temperature MgB2sensors at 20 K"},"abstract":{"kind":"string","value":"Ultra-fast single-photon detectors with high current density and operating temperature can benefit space and ground applications, including quantum optical communication systems, lightweight cryogenics for space crafts, and medical use. Here we demonstrate magnesium diboride (MgB 2 ) thin-film superconducting microwires capable of single-photon detection at 1.55 μ m optical wavelength. We used helium ions to alter the properties of MgB 2 , resulting in microwire-based detectors exhibiting single-photon sensitivity across a broad temperature range of up to 20 K, and detection efficiency saturation for 1 μ m wide microwires at 3.7 K. Linearity of detection rate vs incident power was preserved up to at least 100 Mcps. Despite the large active area of up to 400 × 400 μ m 2 , the reset time was found to be as low as ~ 1 ns. Our research provides possibilities for breaking the operating temperature limit and maximum single-pixel count rate, expanding the detector area, and raises inquiries about the fundamental mechanisms of single-photon detection in high-critical-temperature superconductors."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"Superconducting Nanowire Single-Photon Detectors (SNSPDs) [1] have become crucial for a variety of applications, including quantum optics [2] , [3] , [4] , [5] and security [2] , [6] , [7] , [8] , deep-space communication [9] , biomedical imaging [10] , [11] , [12] , and light detection and ranging (LIDAR) [13] , [14] , [15] . SNSPDs have achieved record-breaking efficiencies: a broadband high-efficiency detection from the soft X-ray [16] to mid-infrared spectral range [17] ; close to 100 % system detection efficiency [18] , [19] , [20] ; sub-3 ps temporal resolution (timing jitter) [21] ; and low noise (referred to as dark count rate) of 6 × 10 −6 cps [22] . Despite this significant progress, the detection system requires cooling to at least 4.2 K (and often lower) due to the low critical temperature, T C < 10 K, of utilized superconductors (NbN, NbTiN, WSi, MoSi) [23] . These temperatures require costly and bulky cryocoolers that limit practical application. As a consequence, there has been a significant interest in superconductors with higher critical temperatures that would allow for operation at elevated temperatures [24] . In addition to the critical temperature, another material-related limitation arises due to the kinetic inductance, L k , of superconducting devices [25] . In order to achieve a high photon-detection efficiency, one typically uses superconducting films of thicknesses ≤ 10 nm [26] and high-resistivity (disordered) superconductors [17] . As a result, the films exhibit large values of L k (10 2 -10 3 pH/square) thereby limiting the detector reset time [25] , particularly problematic for large-area SNSPDs. The high critical temperature ( T C ) superconductors with low kinetic inductance may be regarded as viable alternatives for detectors. In an attempt to increase the operational temperature, higher T C materials have been widely explored. Early studies of thin films have suffered from serious challenges [27] in the development of quantum detectors based on high- T C superconductors. The lateral proximity effect [28] with internal constrictions and grain boundaries [29] , non-uniform distribution of the superconducting order parameters across the structure [30] , instability of superconducting parameters [31] , sensitivity to baking, and chemical interaction during the fabrication all impose limitations on the capacity to replicate detectors in a consistent manner. Moreover, the reduction of sensor dimensions, especially crucial for the detection of infrared photons with low energy, may lead to a suppression of superconductivity in low- and high- T C materials [32] . This phenomenon could limit both the sensitivity and operating temperature of potential devices. Very recently, attempts to realize SNSPDs using two-dimensional high- T C materials revealed single-photon response in superconducting nanowires made of thin flakes of Bi 2 Sr 2 CaCu 2 O 8+ δ , with single-photon response up to 25 K [33] , [34] . One major drawback of exfoliated flakes, however, is the inability to scale up the detector’s active area. Magnesium diboride (MgB 2 ) is a promising candidate for single-photon detectors that has been pursued previously for SNSPDs with mixed success [35] , [36] , [37] . The low kinetic inductance ( L k ~ 2 pH/square [36] ) and critical temperature of 39 K make MgB 2 attractive from practical viewpoints. In SNSPDs made of MgB 2 thin films with T C = 30 K, single-photon response in the optical range has been demonstrated at T ≈ 10 K [35] , [37] and at 5 K for the near-infrared [36] , [38] . However, the non-uniformity of thin films over large areas [39] , relatively low critical temperature [40] , and low switching current in MgB 2 thin films remain essential unsolved challenges. The latest progress in Hybrid Physical-Chemical Vapor Deposition (HPCVD) has illuminated a route to grow high-quality superconducting films of MgB 2 for single-photon detectors. In this work, we demonstrate SNSPDs fabricated from high-quality MgB 2 films with an active area of hundreds of square micrometers, critical temperature approaching the bulk value, and single-photon sensitivity up to 20 K (Fig. 1 ). To achieve this result, we used a helium-ion-beam-based irradiation process [41] that additionally enabled saturated detection of 1.55 μ m single photons at 3.7 K. In contrast to previous MgB 2 detector attempts, we chose the micro-scale SNSPD geometry (width of the wire, W ≈ 1 μ m) recently demonstrated in a number of reports using low- T C superconductors [42] , [43] , [44] , [45] . In comparison to nanowires ( W ≈ 100 nm), superconducting microwires have far larger switching currents and a lower total kinetic inductance, making them attractive for the fabrication of large-area detectors. We observed a reset time of ~ 1 ns enabling a count rate of 100 Mcps for a device with an active area up to 400 × 400 μ m 2 . Fig. 1: Superconducting MgB 2 microwire-based single-photon detectors. a Scanning-electron microscopy (SEM) image of a 1 μ m-wide meander-shaped microwire device. The smooth transitions of the microwires to the electrodes prevent current crowding. The scale bar is 100 μ m. b Example of the R ( T ) dependence of a 5 μ m-wide meander-shaped microwire device. The measurement was done in a two-terminal configuration. The residual resistance ratio ( R R R ) was 2.25. c The voltage pulse from a photon event measured in the 1 μ m MgB 2 microwire single-photon detector at 20 K and λ = 1.55 μ m. The device was biased to 98% of its switching current. Full size image","references":{"indices":[1,2,3,4,5,6,7,8,10,11,12,13,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,38,39,40,41,42,43,44,45],"infos":[{"idx":"1","title":"Semenov, A. D., Gol’tsman, G. N. & Korneev, A. A. Quantum detection by current carrying superconducting film.Physica C: Superconduct.351, 4 (2001).","link":"https://doi.org/10.1016%2FS0921-4534%2800%2901637-3"},{"idx":"2","title":"Hadfield, R. H. Single-photon detectors for optical quantum information applications.Nat. Photonics3, 696–705 (2009).","link":"https://doi.org/10.1038%2Fnphoton.2009.230"},{"idx":"3","title":"Shalm, L. K. et al. Strong Loophole-Free Test of Local Realism.Phys. Rev. Lett.115, 250402 (2015).","link":"https://doi.org/10.1103%2FPhysRevLett.115.250402"},{"idx":"4","title":"Joch, D. J. et al. Certified random-number generation from quantum steering.Phys. Rev. A106, L050401 (2022).","link":"https://doi.org/10.1103%2FPhysRevA.106.L050401"},{"idx":"5","title":"Guccione, G. et al. Connecting heterogeneous quantum networks by hybrid entanglement swapping.Sci. Adv.6, eaba4508 (2020).","link":"https://doi.org/10.1126%2Fsciadv.aba4508"},{"idx":"6","title":"Valivarthi, R. et al. Quantum teleportation across a metropolitan fibre network.Nat. Photonics10, 676–680 (2016).","link":"https://doi.org/10.1038%2Fnphoton.2016.180"},{"idx":"7","title":"Gisin, N. & Thew, R. Quantum communication.Nat. Photonics3, 165–171 (2007).","link":"https://doi.org/10.1038%2Fnphoton.2007.22"},{"idx":"8","title":"Liao, S.-K. et al. Satellite-relayed intercontinental quantum network.Phys. Rev. Lett.120, 030501 (2018).","link":"https://doi.org/10.1103%2FPhysRevLett.120.030501"},{"idx":"10","title":"Zhao, Q.-Y. et al. Single-photon imager based on a superconducting nanowire delay line.Nat. Photonics11, 247–251 (2017).","link":"https://doi.org/10.1038%2Fnphoton.2017.35"},{"idx":"11","title":"Xia, F. et al. Short-wave infrared confocal fluorescence imaging of deep mouse brain with a superconducting nanowire single-photon detector.ACS Photonics8, 2800–2810 (2021).","link":"https://doi.org/10.1021%2Facsphotonics.1c01018"},{"idx":"12","title":"Ozana, N. et al. Superconducting nanowire single-photon sensing of cerebral blood flow.Neurophotonics8, 035006 (2021).","link":"https://doi.org/10.1117%2F1.NPh.8.3.035006"},{"idx":"13","title":"Guan, Y. et al. Lidar with superconducting nanowire single-photon detectors: Recent advances and developments.Optics Lasers Eng.156, 107102 (2022).","link":"https://doi.org/10.1016%2Fj.optlaseng.2022.107102"},{"idx":"15","title":"Salvoni, D. et al. Lidar techniques for a SNSPD-based measurement.J. Phys. Conf. Ser.1182, 012014 (2019).","link":"https://doi.org/10.1088%2F1742-6596%2F1182%2F1%2F012014"},{"idx":"16","title":"Inderbitzin, K., Engel, A., Schilling, A., Il’in, K. & Siegel, M. An ultra-fast superconducting Nb nanowire single-photon detector for soft x-rays.Appl. Phys. Lett.101, 162601 (2012).","link":"https://doi.org/10.1063%2F1.4759046"},{"idx":"17","title":"Verma, V. B. et al. Single-photon detection in the mid-infrared up to 10μm wavelength using tungsten silicide superconducting nanowire detectors.APL Photonics6, 056101 (2021).","link":"https://doi.org/10.1063%2F5.0048049"},{"idx":"18","title":"Reddy, D. V., Nerem, R. R., Nam, S. W., Mirin, R. P. & Verma, V. B. Superconducting nanowire single-photon detectors with 98% system detection efficiency at 1550 nm.Optica7, 1649–1653 (2020).","link":"https://doi.org/10.1364%2FOPTICA.400751"},{"idx":"19","title":"Hu, P. et al. Detecting single infrared photons toward optimal system detection efficiency.Opt. Express28, 36884–36891 (2020).","link":"https://doi.org/10.1364%2FOE.410025"},{"idx":"20","title":"Chang, J. et al. Detecting telecom single photons with 99.5% system detection efficiency and high time resolution.APL Photonics6, 036114 (2021).","link":"https://doi.org/10.1063%2F5.0039772"},{"idx":"21","title":"Korzh, B. et al. Demonstration of sub-3 ps temporal resolution with a superconducting nanowire single-photon detector.Nat. Photonics14, 250–255 (2020).","link":"https://doi.org/10.1038%2Fs41566-020-0589-x"},{"idx":"22","title":"Hochberg, Y. et al. Detecting sub-GeV dark matter with superconducting nanowires.Phys. Rev. Lett.123, 151802 (2019).","link":"https://doi.org/10.1103%2FPhysRevLett.123.151802"},{"idx":"23","title":"Natarajan, C. M., Tanner, M. G. & Hadfield, R. H. Superconducting nanowire single-photon detectors: physics and applications.Supercond. Sci. Technol.25, 063001 (2012).","link":"https://doi.org/10.1088%2F0953-2048%2F25%2F6%2F063001"},{"idx":"24","title":"Shibata, H. Review of superconducting nanostrip photon detectors using various superconductors.IEICE Trans. Electron.104, 429–434 (2021).","link":"https://doi.org/10.1587%2Ftransele.2020SUI0001"},{"idx":"25","title":"Kerman, A. J. et al. Kinetic-inductance-limited reset time of superconducting nanowire photon counters.Appl. Phys. Lett.88, 111116 (2006).","link":"https://doi.org/10.1063%2F1.2183810"},{"idx":"26","title":"Hofherr, M. et al. Intrinsic detection efficiency of superconducting nanowire single-photon detectors with different thicknesses.J. Appl. Phys.108, 014507 (2010).","link":"https://doi.org/10.1063%2F1.3437043"},{"idx":"27","title":"Ejrnaes, M. et al. Observation of dark pulses in 10 nm thick YBCO nanostrips presenting hysteretic current voltage characteristics.Supercond. Sci. Technol.30, 12LT02 (2017).","link":"https://doi.org/10.1088%2F1361-6668%2Faa94b9"},{"idx":"28","title":"Charaev, I. et al. Proximity effect model of ultranarrow NbN strips.Phys. Rev. B96, 184517 (2017).","link":"https://doi.org/10.1103%2FPhysRevB.96.184517"},{"idx":"29","title":"Dane, A. E. et al. Bias sputtered NbN and superconducting nanowire devices.Appl. Phys. Lett.111, 122601 (2017).","link":"https://doi.org/10.1063%2F1.4990066"},{"idx":"30","title":"Kimmel, G. J., Glatz, A., Vinokur, V. M. & Sadovskyy, I. A. Edge effect pinning in mesoscopic superconducting strips with non-uniform distribution of defects.Sci. Rep.9, 211 (2019).","link":"https://doi.org/10.1038%2Fs41598-018-36285-4"},{"idx":"31","title":"Singh, U. R. et al. Spatial inhomogeneity of the superconducting gap and order parameter in FeSe0.4Te0.6.Phys. Rev. B88, 155124 (2013).","link":"https://doi.org/10.1103%2FPhysRevB.88.155124"},{"idx":"32","title":"Arpaia, R. et al. Transport properties of ultrathin YBa2Cu3O7−δnanowires: A route to single-photon detection.Phys. Rev. B96, 064525 (2017).","link":"https://doi.org/10.1103%2FPhysRevB.96.064525"},{"idx":"33","title":"Charaev, I. et al. Single-photon detection using high-temperature superconductors.Nat. Nanotechnol.18, 343–349 (2023).","link":"https://doi.org/10.1038%2Fs41565-023-01325-2"},{"idx":"34","title":"Merino, R. et al. Two-dimensional cuprate nanodetector with single telecom photon sensitivity at T = 20K˙.2D Mater.10, 021001 (2023).","link":"https://doi.org/10.1088%2F2053-1583%2Facb4a8"},{"idx":"35","title":"Shibata, H., Akazaki, T. & Tokura, Y. Fabrication of MgB2nanowire single-photon detector with meander structure.Appl. Phys. Express6, 023101 (2013).","link":"https://doi.org/10.7567%2FAPEX.6.023101"},{"idx":"36","title":"Cherednichenko, S., Acharya, N., Novoselov, E. & Drakinskiy, V. Low kinetic inductance superconducting MgB2nanowires with a 130 ps relaxation time for single-photon detection applications.Supercond. Sci. Technol.34, 044001 (2021).","link":"https://doi.org/10.1088%2F1361-6668%2Fabdeda"},{"idx":"38","title":"Shibata, H., Takesue, H., Honjo, T., Akazaki, T. & Tokura, Y. Single-photon detection using magnesium diboride superconducting nanowires.Appl. Phys. Lett.97, 212504 (2010).","link":"https://doi.org/10.1063%2F1.3518723"},{"idx":"39","title":"Wolak, M. et al. Fabrication and characterization of ultrathin MgB2films for hot-electron bolometer applications.IEEE Trans. Appl. Supercond.25, 1–5 (2015).","link":"https://doi.org/10.1109%2FTASC.2015.2390415"},{"idx":"40","title":"Shibata, H., Akazaki, T. & Tokura, Y. Ultrathin MgB2films fabricated by molecular beam epitaxy and rapid annealing.Supercond. Sci. Technol.26, 035005 (2013).","link":"https://doi.org/10.1088%2F0953-2048%2F26%2F3%2F035005"},{"idx":"41","title":"Zhang, W. et al. Saturating intrinsic detection efficiency of superconducting nanowire single-photon detectors via defect engineering.Phys. Rev. Appl.12, 044040 (2019).","link":"https://doi.org/10.1103%2FPhysRevApplied.12.044040"},{"idx":"42","title":"Korneeva, Y. et al. Optical single-photon detection in micrometer-scale NbN bridges.Phys. Rev. Appl.9, 064037 (2018).","link":"https://doi.org/10.1103%2FPhysRevApplied.9.064037"},{"idx":"43","title":"Korneeva, Y. et al. Different single-photon response of wide and narrow superconducting MoxSi1−xstrips.Phys. Rev. Appl.13, 024011 (2020).","link":"https://doi.org/10.1103%2FPhysRevApplied.13.024011"},{"idx":"44","title":"Chiles, J. et al. Superconducting microwire detectors based on WSi with single-photon sensitivity in the near-infrared.Appl. Phys. Lett.116, 242602 (2020).","link":"https://doi.org/10.1063%2F5.0006221"},{"idx":"45","title":"Charaev, I. et al. Large-area microwire MoSi single-photon detectors at 1550 nm wavelength.Appl. Phys. Lett.116, 242603 (2020).","link":"https://doi.org/10.1063%2F5.0005439"}]}},{"section":"Results","content":"Design and characterization of the MgB 2 detectors To fabricate detectors out of MgB 2 films, we systematically optimized the Hybrid Physical-Chemical Vapor Deposition (HPCVD) process to produce relatively thin films with low surface roughness and high critical temperature. We concentrated on 5–10 nm thin films (Fig. 2 b), where a high critical-current density can be achieved [36] . This thickness range also facilitates stronger absorption of incident light in the infrared range [46] . Thicknesses were verified using x-ray reflectivity (XRR) (See Supplementary Note 6) and scanning transmission electron microscopy (STEM). We also used cross-sectional energy dispersive x-ray spectroscopy (EDS) to identify and map elemental composition. Figure 2 c shows the results of qualitative analyses of a superconducting MgB 2 film deposited on the 6H-SiC substrate (See Methods and Supplementary Note 1 for details). While the total thickness of the studied sample was ≈ 12 nm, the superconducting core of MgB 2 was measured to be only ≈ 7 nm, and observed to be sandwiched between two oxide layers, one on the top of the film (2.5 nm) and the other one at the substrate interface (1.5 nm). A thin layer (assumed to be boron) was observed between superconducting MgB 2 and the top oxide layer. Boron is difficult to detect due to its light atomic mass but has been successfully identified in this region in similar samples (See Supplementary Fig. 9) . These samples exhibited surprising resilience to degradation during standard lithography, etching, and contact deposition processes as well as storage in nitrogen over several months. Other films (unpatterned) lasted over three years in a vacuum desiccator. However, similar films have shown degradation under similar storage and processing conditions, indicating that the problem of the reliability and robustness of such films remains unresolved. Fig. 2: Material analyses of superconducting MgB 2 films. a Atomic-force microscope (AFM) image of the surface topography of a MgB 2 film (sample F180, Supplementary Fig. 1) . The root-mean-square (RMS) surface roughness over several micrometers was less than 1 nm. b Cross-section of thin MgB 2 film taken via scanning transmission electron microscopy (STEM). Using focused-ion beam (FIB) preparation, a cross-section of the material was transferred onto a suitable TEM grid. The presence of carbon on top of the film is a result of the sample preparation. c Energy-Dispersive X-ray Spectroscopy (EDS) for elemental characterization of Si, Mg, and O. The EDS spectrum provides the compositional information for the deposited MgB 2 films. Notably, the superconducting MgB 2 core is sandwiched between two oxides. While the MgB 2 films readily pick up oxygen from the ambient atmosphere forming oxides on the top, the presence of O 2 at the interface is likely from residual SiO x on the SiC substrate surface. Full size image We analyzed the sample surface using Atomic Force Microscopy (AFM), observing a surface micro-roughness (RMS) of less than 1 nm over several square micrometers of film area (Fig. 2 a). This low roughness suggests a minimal variation of the superconducting energy gap across the structure which is crucial for avoiding constrictions and thus maximizing the switching current [47] . We used electron-beam lithography (EBL) to define a meander structure for SNSPDs with various dimensions, 1200 − 10200 μ m in length and 1 − 5 μ m in width, with a filling factor of 0.28 (Fig. 1 a). For fabrication details and device geometries, see Methods and Supplementary Note 3 . We used 12 magnesium diboride films with various thicknesses for our experiment. Out of these, six films underwent material analyses, while the remaining six were employed in the nanofabrication process to create detectors. These devices, as initially fabricated, exhibited only dark counts. Motivated by our desire for single-photon detection we deliberately added defects by using 30-keV helium ion (He + ) irradiation [41] . In contrast to patterning with heavier ions (e.g. gallium or xenon), the exposed regions were not observably etched by the He + beam irradiation. The detector area was entirely exposed with He + ions of 5 × 10 15 ions/cm 2 (See Supplementary Note 5) . Such exposure had only a mild effect on switching current I sw , T C , and the normal state resistance R s (all changed by less than 5%). After fabrication, we characterized the transport properties of our high- T C superconducting MgB 2 microwires. Figure 1 b shows the temperature dependence of the resistance, R ( T ), of a typical MgB 2 fabricated device, revealing the critical temperature of 37.6 ± 0.3 K as determined from the maximum of the d R(T) / d T . The obtained value is somewhat lower than those obtained for the parent MgB 2 film (38.7 K). This decrease indicates a mild degradation of the material’s superconducting properties during the fabrication and ion beam irradiation. Additional information on superconducting and electrical properties can be found in Supplementary Note 4 . An important characteristic of single-photon detectors, enabling the generation of a voltage pulse upon single-photon absorption, is their metastable state that emerges under current biasing [48] , [49] , [50] . This metastable state is essential for detector operation and appears due to competition between the current-induced Joule self-heating of the microwire in the resistive state and electron cooling processes. This state manifests as a hysteresis in the I - V characteristics. Figure 3 a shows an example of an I - V curve measured in a 5 μ m-wide MgB 2 microwire device at a temperature (20 K) well below T C when the microwire is current-biased. A slight hysteretic behavior, characterized by the switching current I s w = 673 μ A and the retrapping current I r = 490 μ A, is observed. Fig. 3: Single-photon detection by superconducting MgB 2 microwires. a I - V curve for 5 μ m MgB 2 microwire detector measured at T = 20 K in the two-terminal configuration. b Photon count rate vs attenuation factor, at different powers of incident lights for 1 μ m-wide MgB 2 device at given I b ≈ 0.98 I s w and λ = 1.55 μ m. Measurement shows linearity over five orders of magnitude in count rates. c Normalized PCR vs relative bias current for different wire widths measured in the microscale MgB 2 devices at 20 K. d Photon and dark-count rate ( D C R ) of 1- μ m wide detector as a function of the absolute bias current ( I b ) at 20 K. e The PCR and DCR , normalized to its maximum value, as a function of the relative bias current, I b / I s w , measured in 1- μ m MgB 2 detector at given temperatures T and 1550-nm wavelength ( λ ). Notably, the count rate at 3.7 K is approaching a saturation plateau that suggests the internal detection efficiency of absorbed photons is approaching 100%. f PCR vs I b for different λ measured in 5- μ m wide MgB 2 device at T = 20 K. Full size image We characterized the switching current of an array of devices by performing I - V sweeps for wire widths from 1 μ m to 5 μ m at different temperatures; the switching current results are collected in Supplementary Note 4 . They do not follow linear fits to the wire width, as one would expect the approaching zero, likely due to exceeding the width of the wire in relation to the Pearl length [51] (this value is ~1.3 μ m for the penetration depth of 90 nm [36] ) and non-uniform current distribution in the superconducting portion [51] . Notably, the deviation from the linear trend becomes more pronounced as the temperature decreases, aligning with the temperature-dependent penetration depth. Photoresponse measurements To perform the photoresponse measurements, we mounted the chip with MgB 2 detectors in a variable-temperature cryostat equipped with RF coax cables and an optical fiber. The latter was held approximately 1 cm away from the device so that a defocused continuous wave laser beam covered the whole chip area. The simplified circuit diagrams that were used for the photoresponse measurements can be found in [see Fig. 3 b [33] ]. Detectors were measured in a conventional SNSPD configuration in which the device was biased through a DC input of the bias tee using an isolated voltage source with a bias resistor ranging from 1 to 10 kΩ. The AC output was connected to the low-noise amplifier whose output was fed to an oscilloscope or a photon counter. To mitigate latching effects, we used a low-bandwidth reset loop formed by a shunt inductor, L s h , and resistor, R s h [52] in parallel with the device. Figure 1 c shows an example of a generated photovoltage, measured across the current-biased MgB 2 microwires when the device was exposed to the laser beam radiation of wavelength λ = 1.55 μ m. The traces of these devices shared common features with the photoresponse of conventional NbN SNSPDs. After reaching the maximum value within ~ 1 ns, the voltage exhibited a slower decay with the characteristic time τ , often referred to as dead or recovery time, which depends on the total kinetic inductance, L k , of the superconducting circuit and the load resistance [53] . The measured values (in particular, an example in Fig. 1 c), τ ≈ 1.3 ns (determined as the time when the signal dropped to 30% of its maximum value) are in agreement with previous measurements of the kinetic inductance in MgB 2 films for different detector lengths at various temperatures [36] . The voltage spikes in MgB 2 devices were observed below and above liquid-helium temperature and could be detected up to T = 20 K. At higher temperatures, detectors did not exhibit a hysteresis, and thus no voltage pulses were observed upon illuminating the detectors with low-intensity laser light. Single-photon sensitivity of MgB 2 detectors The MgB 2 devices exhibited single-photon sensitivity in the technologically-important 1550-nm telecommunications wavelength at the different powers of incident lights (Fig. 3 b) and rates up to 100 Mcps. To obtain further insight into the performance of detectors, we recorded the photon count rate, PCR (the number of pulses per unit time), as a function of the bias current, I b . In these measurements, the light underwent an additional 40 dB of attenuation. Figure 3 c shows the PCR normalized to its maximum value, measured in MgB 2 devices with widths of 1-5 μ m upon exposing it to the λ = 1.55 μ m laser light at 20 K. Despite the biasing near the transition point at 99% of the switching current, the operational bias range remains in the microampere range, making it similar to low-temperature detectors. In the dark, spontaneous voltage pulses emerge when the device is biased close to the switching current (97-99%) (Fig. 3 d), similar to low- T detectors. The absolute value of these dark counts did not exceed 2 × 10 3 s −1 , comparable to the values in conventional NbN SNSPDs. Upon illumination, the counts appeared at onset current I b = 0.96 I c whereas the PCR of the 1 μ m wide device showed some tendency to saturation upon approaching I s w at 3.7 K (Fig. 3 e). This saturation means high internal detector efficiency [54] . With increasing T to 20 K, the onset current decreased together with I sw , as expected for superconducting devices (Fig. 3 e). Furthermore, we found that the PCR for a given I b differed for photon energies corresponding to photons with λ = 780 nm and λ = 1.55 μ m (Fig. 3 f). Single-photon operation was established through the observation of linear scaling of the photon count rate on the radiation power. In the Supplementary Note 9 , we provide estimates for the MgB 2 detector efficiency, D E . To this end, we used estimated absorption of thin films by determination of the impedance function [46] . We found that at 20 K, at which P C R is × 10 3 s −1 for the 40 dB attenuation, the D E is of the order of 7.6%. At 10 and 3.7 K, we find D E exceeding 10%, to be 11.3%, and 14.1% respectively. Note, that these estimates are approximate as they do not account for the polarization dependence of meander geometry that can lead to both enhancement and reduction of the apparent efficiency. Nevertheless, we envision that D E can be boosted by integrating the MgB 2 detector into photonic cavities providing enhanced light-matter interaction. We also found that the timing jitter of MgB 2 detectors is similar to that of conventional SNSPDs ( ~ 50 ps) (see Supplementary Note 8) with an active area at least one order of magnitude smaller than our devices. We used the 1 μ m wide detector with an active area of 200 by 200 μ m 2 . This experiment was performed using the apparatus as described in the Supplementary Note 8 ."},{"section":"Discussion","content":"While the experimental results show single-photon detection in micro-scale wire MgB 2 detectors at elevated temperatures several features of the data require discussion. The key aspects to discuss are (1) the details of the I - V characteristics after He + exposure; (2) the mechanism of defect formation with He + ion exposure; (3) the impact of the two-band character of MgB 2 ; and (4) the mechanism of photodetection in microwires; (5) the enhancement of the detection efficiency. In Fig. 3 , the IV-curve depicts multiple transitions occurring during the distinct jump from the superconducting to the resistive state. This observation on IV curves appeared after irradiation with He + ions and was unexpected. This data supports the hypothesis of non-uniform post-exposure with He + ions. The instability of the beam along with stitching error [55] is expected to result in variation of the superconducting gap over the detector area. We observed, at various temperatures and low bias currents, a noticeable increase in photon count rate with bias current which supports this hypothesis, suggesting localized areas with a lower gap and thus relatively higher internal detection efficiency. We furthermore expect that non-uniformity might have been introduced in the patterning and processing, therefore different regions of the wire could be participating in the detection process at different T and j bias . With process development including He + irradiation dose distribution correction, improved results may thus be expected. The role of He + irradiation at various doses on MgB 2 films remains not fully explored. Drawing from the data reported for low- T C SNSPDs composed of NbTiN [56] and NbN [41] , it’s observed that even at He + dosages considerably lower than those necessary to induce noticeable effects on transport properties like critical current density or normal state conductivity, there is an enhancement in single-photon detection efficiency. In our study, we employed a dosage of 5 × 10 15 ions/cm 2 (equivalent to 50 ions/nm 2 ), a dose that aligns with the reported dosage required for 8-to-12 nm-thick NbTiN films to begin exhibiting single-photon response. This dose resulted in only modest variations, up to 10%, in parameters such as critical current density, critical temperature, and sheet resistance. These results closely approximate our findings. Nevertheless, comprehensive investigations in this area are still pending. Apart from affecting MgB 2 film itself, the process of helium ion implantation causes defects and amorphization in the SiC substrate [57] . As a result, the efficiency of energy transfer from the superconducting structure to the substrate is likely to be affected. This effect may prolong the lifespan of the normal domain, thereby increasing the probability of detection before the normal domain collapses. In devices subjected to a relatively low dose of irradiation (5 × 10 15 ions/cm 2 ), we noticed a decrease in the retrapping current. While the switching current remained unchanged, the retrapping current reduced to 90% of its nominal value compared to non-irradiated devices (Supplementary Fig. 7) . Although magnesium diboride has two bands with different gaps: π -band with a gap of 0.6 k B T C and σ -band with a gap of 2.2 k B T C , the electron doping by atoms may initiate the interband scattering in MgB 2 . The filling band effect increases the value of the small gap and, together with band filling, leads to the merging of the two gaps [58] . Our work points out that it is the uniformity of the superconducting gap, rather than the absolute gap, that is crucial. The large bias current is not entirely unexpected in the case of high- T C microwire devices. The magnitude of the bias current is contingent upon the cross-sectional area of the wires and the critical temperature of the materials involved. In contrast to materials with lower critical temperatures typically employed in the production of superconducting nanowire single-photon detectors (SNSPDs), such as NbN, WSi, and MoSi, magnesium diboride has a critical temperature at least three times higher. That means that to achieve the same sensitivity, the gap must be suppressed by some means other than temperature, and elevating current density is one way to achieve a sufficiently small gap for detection. It is also worth noting that, the width of wires surpassed the Pearl length in MgB 2 . Wires wider than the Pearl length are expected to display non-uniform current distribution across the wire. The calculated Pearl length for MgB 2 films, approximately 1.3 μ m, is four times lower than the widest wire used in our experiment. As a result, the ratio of the experimentally observed switching current density to the current density for spontaneous vortex entry is expected to degrade, proportionally to the following: \\(\\sim 1/\\sqrt{1+W/2\\pi {\\Lambda }_{{{{{{{{\\rm{P}}}}}}}}}}\\) [59] , where Λ P is the Pearl Length. Surprisingly, contrary to theoretical predictions [59] , we observed single-photon detection in devices with wire widths of up to 5 μ m. This observation challenges the previously known limit in wire width from a practical standpoint, even though the mechanism of detection under these conditions remains unclear. While the detection efficiency approaches the limit for a single-layer detector, primarily due to absorption in thin films, there exists an opportunity for further enhancement by integrating the detector into an optical waveguide. The integration of MgB 2 detectors onto a chip with a waveguide faces limitations imposed by the material that is used for the waveguide. The polycrystalline structure of magnesium diboride introduces lattice mismatch that impacts on superconducting and transport properties of films. On silicon carbide (SiC), it is feasible to produce MgB 2 films as thin as 3 nm, maintaining a critical temperature higher than 30 K with a narrow transition. Epitaxial growth, a crucial factor for achieving a high critical temperature in thin films, is employed in this process. Although sapphire (Al 2 O 3 ) has also proven to be a suitable substrate for MgB 2 films, the demonstration of MgB 2 films as thin as those on SiC remains unreported. Alternatively, the approach that involves light coupled through the substrate with an optical cavity positioned atop the detector can enhance the absorption efficiency in MgB 2 detectors."},{"section":"Methods","content":"Sputtering and fab To fabricate detectors out of MgB 2 films we started with film deposition. Ultra-thin MgB 2 films were made using Hybrid Physical-Chemical Vapor Deposition (HPCVD) in a custom-built system [60] , [61] . Magnesium melts at 650 degrees Celsius and interacts with boron. The latter is produced via the thermal decomposition of gaseous diborane (B 2 H 6 ) supplied in a mixture with hydrogen. Films were made on (0001) SiC substrates which provide excellent lattice matching to the hexagonal lattice of c-oriented MgB 2 films. Devices were fabricated out of MgB 2 films on the 6H-SiC substrate with 5 nm titanium/ 50 nm gold contact pads using standard e-beam lithography, followed by metal deposition and lift-off in 45 °C N-Methylpyrrolidone (NMP). In order to produce the wires out of MgB 2 films, another round of e-beam lithography was performed, using 330-nm-thick high-resolution positive e-beam resist (ZEP520A). The patterns were then transferred onto the MgB 2 film by Ar + milling at a beam voltage of 300 V and Ar flow of 9 sccm for 12 minutes in a series of one-minute etches with three-minute-long intervals between each etch to avoid overheating of the sample. The residual resist was removed by immersing the substrate in the 45 °C NMP. After fabrication and initial testing, we irradiated our samples with He + using a Zeiss Orion Microscope equipped with a Raith pattern generator. The irradiation was realized by sweeping the beam across the detector area. The exposed area was a rectangle with dimensions of active detector areas (to cover the whole meander area). The beam-limiting aperture was set to the largest diameter to maximize the beam current. The dose was varied in the range from 10 15 to 10 20 ions/cm 2 (Supplementary Fig. 5) . Scanning Transmission Electron Microscopy (STEM) STEM imaging was performed with a probe-corrected Thermo Fisher Scientific Themis Z G3 60–300 kV S/TEM operated at 200 kV with a beam current of 30-90 pA and 25 mrad convergence angle. Elemental analysis was accomplished by using EDS (Energy Dispersive X-ray Spectroscopy). For cross-sectional STEM characterization, samples were prepared using a Raith VELION focused ion beam-scanning electron microscope (FIB-SEM) system with carbon and platinum protection layers on top of the sample."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"Superconducting Nanowire Single-Photon Detectors (SNSPDs) [1] have become crucial for a variety of applications, including quantum optics [2] , [3] , [4] , [5] and security [2] , [6] , [7] , [8] , deep-space communication [9] , biomedical imaging [10] , [11] , [12] , and light detection and ranging (LIDAR) [13] , [14] , [15] . SNSPDs have achieved record-breaking efficiencies: a broadband high-efficiency detection from the soft X-ray [16] to mid-infrared spectral range [17] ; close to 100 % system detection efficiency [18] , [19] , [20] ; sub-3 ps temporal resolution (timing jitter) [21] ; and low noise (referred to as dark count rate) of 6 × 10 −6 cps [22] . Despite this significant progress, the detection system requires cooling to at least 4.2 K (and often lower) due to the low critical temperature, T C < 10 K, of utilized superconductors (NbN, NbTiN, WSi, MoSi) [23] . These temperatures require costly and bulky cryocoolers that limit practical application. As a consequence, there has been a significant interest in superconductors with higher critical temperatures that would allow for operation at elevated temperatures [24] . In addition to the critical temperature, another material-related limitation arises due to the kinetic inductance, L k , of superconducting devices [25] . In order to achieve a high photon-detection efficiency, one typically uses superconducting films of thicknesses ≤ 10 nm [26] and high-resistivity (disordered) superconductors [17] . As a result, the films exhibit large values of L k (10 2 -10 3 pH/square) thereby limiting the detector reset time [25] , particularly problematic for large-area SNSPDs. The high critical temperature ( T C ) superconductors with low kinetic inductance may be regarded as viable alternatives for detectors. In an attempt to increase the operational temperature, higher T C materials have been widely explored. Early studies of thin films have suffered from serious challenges [27] in the development of quantum detectors based on high- T C superconductors. The lateral proximity effect [28] with internal constrictions and grain boundaries [29] , non-uniform distribution of the superconducting order parameters across the structure [30] , instability of superconducting parameters [31] , sensitivity to baking, and chemical interaction during the fabrication all impose limitations on the capacity to replicate detectors in a consistent manner. Moreover, the reduction of sensor dimensions, especially crucial for the detection of infrared photons with low energy, may lead to a suppression of superconductivity in low- and high- T C materials [32] . This phenomenon could limit both the sensitivity and operating temperature of potential devices. Very recently, attempts to realize SNSPDs using two-dimensional high- T C materials revealed single-photon response in superconducting nanowires made of thin flakes of Bi 2 Sr 2 CaCu 2 O 8+ δ , with single-photon response up to 25 K [33] , [34] . One major drawback of exfoliated flakes, however, is the inability to scale up the detector’s active area. Magnesium diboride (MgB 2 ) is a promising candidate for single-photon detectors that has been pursued previously for SNSPDs with mixed success [35] , [36] , [37] . The low kinetic inductance ( L k ~ 2 pH/square [36] ) and critical temperature of 39 K make MgB 2 attractive from practical viewpoints. In SNSPDs made of MgB 2 thin films with T C = 30 K, single-photon response in the optical range has been demonstrated at T ≈ 10 K [35] , [37] and at 5 K for the near-infrared [36] , [38] . However, the non-uniformity of thin films over large areas [39] , relatively low critical temperature [40] , and low switching current in MgB 2 thin films remain essential unsolved challenges. The latest progress in Hybrid Physical-Chemical Vapor Deposition (HPCVD) has illuminated a route to grow high-quality superconducting films of MgB 2 for single-photon detectors. In this work, we demonstrate SNSPDs fabricated from high-quality MgB 2 films with an active area of hundreds of square micrometers, critical temperature approaching the bulk value, and single-photon sensitivity up to 20 K (Fig. 1 ). To achieve this result, we used a helium-ion-beam-based irradiation process [41] that additionally enabled saturated detection of 1.55 μ m single photons at 3.7 K. In contrast to previous MgB 2 detector attempts, we chose the micro-scale SNSPD geometry (width of the wire, W ≈ 1 μ m) recently demonstrated in a number of reports using low- T C superconductors [42] , [43] , [44] , [45] . In comparison to nanowires ( W ≈ 100 nm), superconducting microwires have far larger switching currents and a lower total kinetic inductance, making them attractive for the fabrication of large-area detectors. We observed a reset time of ~ 1 ns enabling a count rate of 100 Mcps for a device with an active area up to 400 × 400 μ m 2 . Fig. 1: Superconducting MgB 2 microwire-based single-photon detectors. a Scanning-electron microscopy (SEM) image of a 1 μ m-wide meander-shaped microwire device. The smooth transitions of the microwires to the electrodes prevent current crowding. The scale bar is 100 μ m. b Example of the R ( T ) dependence of a 5 μ m-wide meander-shaped microwire device. The measurement was done in a two-terminal configuration. The residual resistance ratio ( R R R ) was 2.25. c The voltage pulse from a photon event measured in the 1 μ m MgB 2 microwire single-photon detector at 20 K and λ = 1.55 μ m. The device was biased to 98% of its switching current. Full size image\",\n \"references\": {\n \"indices\": [\n 1,\n 2,\n 3,\n 4,\n 5,\n 6,\n 7,\n 8,\n 10,\n 11,\n 12,\n 13,\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 28,\n 29,\n 30,\n 31,\n 32,\n 33,\n 34,\n 35,\n 36,\n 38,\n 39,\n 40,\n 41,\n 42,\n 43,\n 44,\n 45\n ],\n \"infos\": [\n {\n \"idx\": \"1\",\n \"title\": \"Semenov, A. D., Gol’tsman, G. N. & Korneev, A. A. Quantum detection by current carrying superconducting film.Physica C: Superconduct.351, 4 (2001).\",\n \"link\": \"https://doi.org/10.1016%2FS0921-4534%2800%2901637-3\"\n },\n {\n \"idx\": \"2\",\n \"title\": \"Hadfield, R. H. Single-photon detectors for optical quantum information applications.Nat. Photonics3, 696–705 (2009).\",\n \"link\": \"https://doi.org/10.1038%2Fnphoton.2009.230\"\n },\n {\n \"idx\": \"3\",\n \"title\": \"Shalm, L. K. et al. Strong Loophole-Free Test of Local Realism.Phys. Rev. Lett.115, 250402 (2015).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.115.250402\"\n },\n {\n \"idx\": \"4\",\n \"title\": \"Joch, D. J. et al. Certified random-number generation from quantum steering.Phys. Rev. A106, L050401 (2022).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevA.106.L050401\"\n },\n {\n \"idx\": \"5\",\n \"title\": \"Guccione, G. et al. Connecting heterogeneous quantum networks by hybrid entanglement swapping.Sci. Adv.6, eaba4508 (2020).\",\n \"link\": \"https://doi.org/10.1126%2Fsciadv.aba4508\"\n },\n {\n \"idx\": \"6\",\n \"title\": \"Valivarthi, R. et al. Quantum teleportation across a metropolitan fibre network.Nat. Photonics10, 676–680 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fnphoton.2016.180\"\n },\n {\n \"idx\": \"7\",\n \"title\": \"Gisin, N. & Thew, R. Quantum communication.Nat. Photonics3, 165–171 (2007).\",\n \"link\": \"https://doi.org/10.1038%2Fnphoton.2007.22\"\n },\n {\n \"idx\": \"8\",\n \"title\": \"Liao, S.-K. et al. Satellite-relayed intercontinental quantum network.Phys. Rev. Lett.120, 030501 (2018).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.120.030501\"\n },\n {\n \"idx\": \"10\",\n \"title\": \"Zhao, Q.-Y. et al. Single-photon imager based on a superconducting nanowire delay line.Nat. Photonics11, 247–251 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fnphoton.2017.35\"\n },\n {\n \"idx\": \"11\",\n \"title\": \"Xia, F. et al. Short-wave infrared confocal fluorescence imaging of deep mouse brain with a superconducting nanowire single-photon detector.ACS Photonics8, 2800–2810 (2021).\",\n \"link\": \"https://doi.org/10.1021%2Facsphotonics.1c01018\"\n },\n {\n \"idx\": \"12\",\n \"title\": \"Ozana, N. et al. Superconducting nanowire single-photon sensing of cerebral blood flow.Neurophotonics8, 035006 (2021).\",\n \"link\": \"https://doi.org/10.1117%2F1.NPh.8.3.035006\"\n },\n {\n \"idx\": \"13\",\n \"title\": \"Guan, Y. et al. Lidar with superconducting nanowire single-photon detectors: Recent advances and developments.Optics Lasers Eng.156, 107102 (2022).\",\n \"link\": \"https://doi.org/10.1016%2Fj.optlaseng.2022.107102\"\n },\n {\n \"idx\": \"15\",\n \"title\": \"Salvoni, D. et al. Lidar techniques for a SNSPD-based measurement.J. Phys. Conf. Ser.1182, 012014 (2019).\",\n \"link\": \"https://doi.org/10.1088%2F1742-6596%2F1182%2F1%2F012014\"\n },\n {\n \"idx\": \"16\",\n \"title\": \"Inderbitzin, K., Engel, A., Schilling, A., Il’in, K. & Siegel, M. An ultra-fast superconducting Nb nanowire single-photon detector for soft x-rays.Appl. Phys. Lett.101, 162601 (2012).\",\n \"link\": \"https://doi.org/10.1063%2F1.4759046\"\n },\n {\n \"idx\": \"17\",\n \"title\": \"Verma, V. B. et al. Single-photon detection in the mid-infrared up to 10μm wavelength using tungsten silicide superconducting nanowire detectors.APL Photonics6, 056101 (2021).\",\n \"link\": \"https://doi.org/10.1063%2F5.0048049\"\n },\n {\n \"idx\": \"18\",\n \"title\": \"Reddy, D. V., Nerem, R. R., Nam, S. W., Mirin, R. P. & Verma, V. B. Superconducting nanowire single-photon detectors with 98% system detection efficiency at 1550 nm.Optica7, 1649–1653 (2020).\",\n \"link\": \"https://doi.org/10.1364%2FOPTICA.400751\"\n },\n {\n \"idx\": \"19\",\n \"title\": \"Hu, P. et al. Detecting single infrared photons toward optimal system detection efficiency.Opt. Express28, 36884–36891 (2020).\",\n \"link\": \"https://doi.org/10.1364%2FOE.410025\"\n },\n {\n \"idx\": \"20\",\n \"title\": \"Chang, J. et al. Detecting telecom single photons with 99.5% system detection efficiency and high time resolution.APL Photonics6, 036114 (2021).\",\n \"link\": \"https://doi.org/10.1063%2F5.0039772\"\n },\n {\n \"idx\": \"21\",\n \"title\": \"Korzh, B. et al. Demonstration of sub-3 ps temporal resolution with a superconducting nanowire single-photon detector.Nat. Photonics14, 250–255 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41566-020-0589-x\"\n },\n {\n \"idx\": \"22\",\n \"title\": \"Hochberg, Y. et al. Detecting sub-GeV dark matter with superconducting nanowires.Phys. Rev. Lett.123, 151802 (2019).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.123.151802\"\n },\n {\n \"idx\": \"23\",\n \"title\": \"Natarajan, C. M., Tanner, M. G. & Hadfield, R. H. Superconducting nanowire single-photon detectors: physics and applications.Supercond. Sci. Technol.25, 063001 (2012).\",\n \"link\": \"https://doi.org/10.1088%2F0953-2048%2F25%2F6%2F063001\"\n },\n {\n \"idx\": \"24\",\n \"title\": \"Shibata, H. Review of superconducting nanostrip photon detectors using various superconductors.IEICE Trans. Electron.104, 429–434 (2021).\",\n \"link\": \"https://doi.org/10.1587%2Ftransele.2020SUI0001\"\n },\n {\n \"idx\": \"25\",\n \"title\": \"Kerman, A. J. et al. Kinetic-inductance-limited reset time of superconducting nanowire photon counters.Appl. Phys. Lett.88, 111116 (2006).\",\n \"link\": \"https://doi.org/10.1063%2F1.2183810\"\n },\n {\n \"idx\": \"26\",\n \"title\": \"Hofherr, M. et al. Intrinsic detection efficiency of superconducting nanowire single-photon detectors with different thicknesses.J. Appl. Phys.108, 014507 (2010).\",\n \"link\": \"https://doi.org/10.1063%2F1.3437043\"\n },\n {\n \"idx\": \"27\",\n \"title\": \"Ejrnaes, M. et al. Observation of dark pulses in 10 nm thick YBCO nanostrips presenting hysteretic current voltage characteristics.Supercond. Sci. Technol.30, 12LT02 (2017).\",\n \"link\": \"https://doi.org/10.1088%2F1361-6668%2Faa94b9\"\n },\n {\n \"idx\": \"28\",\n \"title\": \"Charaev, I. et al. Proximity effect model of ultranarrow NbN strips.Phys. Rev. B96, 184517 (2017).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.96.184517\"\n },\n {\n \"idx\": \"29\",\n \"title\": \"Dane, A. E. et al. Bias sputtered NbN and superconducting nanowire devices.Appl. Phys. Lett.111, 122601 (2017).\",\n \"link\": \"https://doi.org/10.1063%2F1.4990066\"\n },\n {\n \"idx\": \"30\",\n \"title\": \"Kimmel, G. J., Glatz, A., Vinokur, V. M. & Sadovskyy, I. A. Edge effect pinning in mesoscopic superconducting strips with non-uniform distribution of defects.Sci. Rep.9, 211 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41598-018-36285-4\"\n },\n {\n \"idx\": \"31\",\n \"title\": \"Singh, U. R. et al. Spatial inhomogeneity of the superconducting gap and order parameter in FeSe0.4Te0.6.Phys. Rev. B88, 155124 (2013).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.88.155124\"\n },\n {\n \"idx\": \"32\",\n \"title\": \"Arpaia, R. et al. Transport properties of ultrathin YBa2Cu3O7−δnanowires: A route to single-photon detection.Phys. Rev. B96, 064525 (2017).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.96.064525\"\n },\n {\n \"idx\": \"33\",\n \"title\": \"Charaev, I. et al. Single-photon detection using high-temperature superconductors.Nat. Nanotechnol.18, 343–349 (2023).\",\n \"link\": \"https://doi.org/10.1038%2Fs41565-023-01325-2\"\n },\n {\n \"idx\": \"34\",\n \"title\": \"Merino, R. et al. Two-dimensional cuprate nanodetector with single telecom photon sensitivity at T = 20K˙.2D Mater.10, 021001 (2023).\",\n \"link\": \"https://doi.org/10.1088%2F2053-1583%2Facb4a8\"\n },\n {\n \"idx\": \"35\",\n \"title\": \"Shibata, H., Akazaki, T. & Tokura, Y. Fabrication of MgB2nanowire single-photon detector with meander structure.Appl. Phys. Express6, 023101 (2013).\",\n \"link\": \"https://doi.org/10.7567%2FAPEX.6.023101\"\n },\n {\n \"idx\": \"36\",\n \"title\": \"Cherednichenko, S., Acharya, N., Novoselov, E. & Drakinskiy, V. Low kinetic inductance superconducting MgB2nanowires with a 130 ps relaxation time for single-photon detection applications.Supercond. Sci. Technol.34, 044001 (2021).\",\n \"link\": \"https://doi.org/10.1088%2F1361-6668%2Fabdeda\"\n },\n {\n \"idx\": \"38\",\n \"title\": \"Shibata, H., Takesue, H., Honjo, T., Akazaki, T. & Tokura, Y. Single-photon detection using magnesium diboride superconducting nanowires.Appl. Phys. Lett.97, 212504 (2010).\",\n \"link\": \"https://doi.org/10.1063%2F1.3518723\"\n },\n {\n \"idx\": \"39\",\n \"title\": \"Wolak, M. et al. Fabrication and characterization of ultrathin MgB2films for hot-electron bolometer applications.IEEE Trans. Appl. Supercond.25, 1–5 (2015).\",\n \"link\": \"https://doi.org/10.1109%2FTASC.2015.2390415\"\n },\n {\n \"idx\": \"40\",\n \"title\": \"Shibata, H., Akazaki, T. & Tokura, Y. Ultrathin MgB2films fabricated by molecular beam epitaxy and rapid annealing.Supercond. Sci. Technol.26, 035005 (2013).\",\n \"link\": \"https://doi.org/10.1088%2F0953-2048%2F26%2F3%2F035005\"\n },\n {\n \"idx\": \"41\",\n \"title\": \"Zhang, W. et al. Saturating intrinsic detection efficiency of superconducting nanowire single-photon detectors via defect engineering.Phys. Rev. Appl.12, 044040 (2019).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevApplied.12.044040\"\n },\n {\n \"idx\": \"42\",\n \"title\": \"Korneeva, Y. et al. Optical single-photon detection in micrometer-scale NbN bridges.Phys. Rev. Appl.9, 064037 (2018).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevApplied.9.064037\"\n },\n {\n \"idx\": \"43\",\n \"title\": \"Korneeva, Y. et al. Different single-photon response of wide and narrow superconducting MoxSi1−xstrips.Phys. Rev. Appl.13, 024011 (2020).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevApplied.13.024011\"\n },\n {\n \"idx\": \"44\",\n \"title\": \"Chiles, J. et al. Superconducting microwire detectors based on WSi with single-photon sensitivity in the near-infrared.Appl. Phys. Lett.116, 242602 (2020).\",\n \"link\": \"https://doi.org/10.1063%2F5.0006221\"\n },\n {\n \"idx\": \"45\",\n \"title\": \"Charaev, I. et al. Large-area microwire MoSi single-photon detectors at 1550 nm wavelength.Appl. Phys. Lett.116, 242603 (2020).\",\n \"link\": \"https://doi.org/10.1063%2F5.0005439\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Design and characterization of the MgB 2 detectors To fabricate detectors out of MgB 2 films, we systematically optimized the Hybrid Physical-Chemical Vapor Deposition (HPCVD) process to produce relatively thin films with low surface roughness and high critical temperature. We concentrated on 5–10 nm thin films (Fig. 2 b), where a high critical-current density can be achieved [36] . This thickness range also facilitates stronger absorption of incident light in the infrared range [46] . Thicknesses were verified using x-ray reflectivity (XRR) (See Supplementary Note 6) and scanning transmission electron microscopy (STEM). We also used cross-sectional energy dispersive x-ray spectroscopy (EDS) to identify and map elemental composition. Figure 2 c shows the results of qualitative analyses of a superconducting MgB 2 film deposited on the 6H-SiC substrate (See Methods and Supplementary Note 1 for details). While the total thickness of the studied sample was ≈ 12 nm, the superconducting core of MgB 2 was measured to be only ≈ 7 nm, and observed to be sandwiched between two oxide layers, one on the top of the film (2.5 nm) and the other one at the substrate interface (1.5 nm). A thin layer (assumed to be boron) was observed between superconducting MgB 2 and the top oxide layer. Boron is difficult to detect due to its light atomic mass but has been successfully identified in this region in similar samples (See Supplementary Fig. 9) . These samples exhibited surprising resilience to degradation during standard lithography, etching, and contact deposition processes as well as storage in nitrogen over several months. Other films (unpatterned) lasted over three years in a vacuum desiccator. However, similar films have shown degradation under similar storage and processing conditions, indicating that the problem of the reliability and robustness of such films remains unresolved. Fig. 2: Material analyses of superconducting MgB 2 films. a Atomic-force microscope (AFM) image of the surface topography of a MgB 2 film (sample F180, Supplementary Fig. 1) . The root-mean-square (RMS) surface roughness over several micrometers was less than 1 nm. b Cross-section of thin MgB 2 film taken via scanning transmission electron microscopy (STEM). Using focused-ion beam (FIB) preparation, a cross-section of the material was transferred onto a suitable TEM grid. The presence of carbon on top of the film is a result of the sample preparation. c Energy-Dispersive X-ray Spectroscopy (EDS) for elemental characterization of Si, Mg, and O. The EDS spectrum provides the compositional information for the deposited MgB 2 films. Notably, the superconducting MgB 2 core is sandwiched between two oxides. While the MgB 2 films readily pick up oxygen from the ambient atmosphere forming oxides on the top, the presence of O 2 at the interface is likely from residual SiO x on the SiC substrate surface. Full size image We analyzed the sample surface using Atomic Force Microscopy (AFM), observing a surface micro-roughness (RMS) of less than 1 nm over several square micrometers of film area (Fig. 2 a). This low roughness suggests a minimal variation of the superconducting energy gap across the structure which is crucial for avoiding constrictions and thus maximizing the switching current [47] . We used electron-beam lithography (EBL) to define a meander structure for SNSPDs with various dimensions, 1200 − 10200 μ m in length and 1 − 5 μ m in width, with a filling factor of 0.28 (Fig. 1 a). For fabrication details and device geometries, see Methods and Supplementary Note 3 . We used 12 magnesium diboride films with various thicknesses for our experiment. Out of these, six films underwent material analyses, while the remaining six were employed in the nanofabrication process to create detectors. These devices, as initially fabricated, exhibited only dark counts. Motivated by our desire for single-photon detection we deliberately added defects by using 30-keV helium ion (He + ) irradiation [41] . In contrast to patterning with heavier ions (e.g. gallium or xenon), the exposed regions were not observably etched by the He + beam irradiation. The detector area was entirely exposed with He + ions of 5 × 10 15 ions/cm 2 (See Supplementary Note 5) . Such exposure had only a mild effect on switching current I sw , T C , and the normal state resistance R s (all changed by less than 5%). After fabrication, we characterized the transport properties of our high- T C superconducting MgB 2 microwires. Figure 1 b shows the temperature dependence of the resistance, R ( T ), of a typical MgB 2 fabricated device, revealing the critical temperature of 37.6 ± 0.3 K as determined from the maximum of the d R(T) / d T . The obtained value is somewhat lower than those obtained for the parent MgB 2 film (38.7 K). This decrease indicates a mild degradation of the material’s superconducting properties during the fabrication and ion beam irradiation. Additional information on superconducting and electrical properties can be found in Supplementary Note 4 . An important characteristic of single-photon detectors, enabling the generation of a voltage pulse upon single-photon absorption, is their metastable state that emerges under current biasing [48] , [49] , [50] . This metastable state is essential for detector operation and appears due to competition between the current-induced Joule self-heating of the microwire in the resistive state and electron cooling processes. This state manifests as a hysteresis in the I - V characteristics. Figure 3 a shows an example of an I - V curve measured in a 5 μ m-wide MgB 2 microwire device at a temperature (20 K) well below T C when the microwire is current-biased. A slight hysteretic behavior, characterized by the switching current I s w = 673 μ A and the retrapping current I r = 490 μ A, is observed. Fig. 3: Single-photon detection by superconducting MgB 2 microwires. a I - V curve for 5 μ m MgB 2 microwire detector measured at T = 20 K in the two-terminal configuration. b Photon count rate vs attenuation factor, at different powers of incident lights for 1 μ m-wide MgB 2 device at given I b ≈ 0.98 I s w and λ = 1.55 μ m. Measurement shows linearity over five orders of magnitude in count rates. c Normalized PCR vs relative bias current for different wire widths measured in the microscale MgB 2 devices at 20 K. d Photon and dark-count rate ( D C R ) of 1- μ m wide detector as a function of the absolute bias current ( I b ) at 20 K. e The PCR and DCR , normalized to its maximum value, as a function of the relative bias current, I b / I s w , measured in 1- μ m MgB 2 detector at given temperatures T and 1550-nm wavelength ( λ ). Notably, the count rate at 3.7 K is approaching a saturation plateau that suggests the internal detection efficiency of absorbed photons is approaching 100%. f PCR vs I b for different λ measured in 5- μ m wide MgB 2 device at T = 20 K. Full size image We characterized the switching current of an array of devices by performing I - V sweeps for wire widths from 1 μ m to 5 μ m at different temperatures; the switching current results are collected in Supplementary Note 4 . They do not follow linear fits to the wire width, as one would expect the approaching zero, likely due to exceeding the width of the wire in relation to the Pearl length [51] (this value is ~1.3 μ m for the penetration depth of 90 nm [36] ) and non-uniform current distribution in the superconducting portion [51] . Notably, the deviation from the linear trend becomes more pronounced as the temperature decreases, aligning with the temperature-dependent penetration depth. Photoresponse measurements To perform the photoresponse measurements, we mounted the chip with MgB 2 detectors in a variable-temperature cryostat equipped with RF coax cables and an optical fiber. The latter was held approximately 1 cm away from the device so that a defocused continuous wave laser beam covered the whole chip area. The simplified circuit diagrams that were used for the photoresponse measurements can be found in [see Fig. 3 b [33] ]. Detectors were measured in a conventional SNSPD configuration in which the device was biased through a DC input of the bias tee using an isolated voltage source with a bias resistor ranging from 1 to 10 kΩ. The AC output was connected to the low-noise amplifier whose output was fed to an oscilloscope or a photon counter. To mitigate latching effects, we used a low-bandwidth reset loop formed by a shunt inductor, L s h , and resistor, R s h [52] in parallel with the device. Figure 1 c shows an example of a generated photovoltage, measured across the current-biased MgB 2 microwires when the device was exposed to the laser beam radiation of wavelength λ = 1.55 μ m. The traces of these devices shared common features with the photoresponse of conventional NbN SNSPDs. After reaching the maximum value within ~ 1 ns, the voltage exhibited a slower decay with the characteristic time τ , often referred to as dead or recovery time, which depends on the total kinetic inductance, L k , of the superconducting circuit and the load resistance [53] . The measured values (in particular, an example in Fig. 1 c), τ ≈ 1.3 ns (determined as the time when the signal dropped to 30% of its maximum value) are in agreement with previous measurements of the kinetic inductance in MgB 2 films for different detector lengths at various temperatures [36] . The voltage spikes in MgB 2 devices were observed below and above liquid-helium temperature and could be detected up to T = 20 K. At higher temperatures, detectors did not exhibit a hysteresis, and thus no voltage pulses were observed upon illuminating the detectors with low-intensity laser light. Single-photon sensitivity of MgB 2 detectors The MgB 2 devices exhibited single-photon sensitivity in the technologically-important 1550-nm telecommunications wavelength at the different powers of incident lights (Fig. 3 b) and rates up to 100 Mcps. To obtain further insight into the performance of detectors, we recorded the photon count rate, PCR (the number of pulses per unit time), as a function of the bias current, I b . In these measurements, the light underwent an additional 40 dB of attenuation. Figure 3 c shows the PCR normalized to its maximum value, measured in MgB 2 devices with widths of 1-5 μ m upon exposing it to the λ = 1.55 μ m laser light at 20 K. Despite the biasing near the transition point at 99% of the switching current, the operational bias range remains in the microampere range, making it similar to low-temperature detectors. In the dark, spontaneous voltage pulses emerge when the device is biased close to the switching current (97-99%) (Fig. 3 d), similar to low- T detectors. The absolute value of these dark counts did not exceed 2 × 10 3 s −1 , comparable to the values in conventional NbN SNSPDs. Upon illumination, the counts appeared at onset current I b = 0.96 I c whereas the PCR of the 1 μ m wide device showed some tendency to saturation upon approaching I s w at 3.7 K (Fig. 3 e). This saturation means high internal detector efficiency [54] . With increasing T to 20 K, the onset current decreased together with I sw , as expected for superconducting devices (Fig. 3 e). Furthermore, we found that the PCR for a given I b differed for photon energies corresponding to photons with λ = 780 nm and λ = 1.55 μ m (Fig. 3 f). Single-photon operation was established through the observation of linear scaling of the photon count rate on the radiation power. In the Supplementary Note 9 , we provide estimates for the MgB 2 detector efficiency, D E . To this end, we used estimated absorption of thin films by determination of the impedance function [46] . We found that at 20 K, at which P C R is × 10 3 s −1 for the 40 dB attenuation, the D E is of the order of 7.6%. At 10 and 3.7 K, we find D E exceeding 10%, to be 11.3%, and 14.1% respectively. Note, that these estimates are approximate as they do not account for the polarization dependence of meander geometry that can lead to both enhancement and reduction of the apparent efficiency. Nevertheless, we envision that D E can be boosted by integrating the MgB 2 detector into photonic cavities providing enhanced light-matter interaction. We also found that the timing jitter of MgB 2 detectors is similar to that of conventional SNSPDs ( ~ 50 ps) (see Supplementary Note 8) with an active area at least one order of magnitude smaller than our devices. We used the 1 μ m wide detector with an active area of 200 by 200 μ m 2 . This experiment was performed using the apparatus as described in the Supplementary Note 8 .\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"While the experimental results show single-photon detection in micro-scale wire MgB 2 detectors at elevated temperatures several features of the data require discussion. The key aspects to discuss are (1) the details of the I - V characteristics after He + exposure; (2) the mechanism of defect formation with He + ion exposure; (3) the impact of the two-band character of MgB 2 ; and (4) the mechanism of photodetection in microwires; (5) the enhancement of the detection efficiency. In Fig. 3 , the IV-curve depicts multiple transitions occurring during the distinct jump from the superconducting to the resistive state. This observation on IV curves appeared after irradiation with He + ions and was unexpected. This data supports the hypothesis of non-uniform post-exposure with He + ions. The instability of the beam along with stitching error [55] is expected to result in variation of the superconducting gap over the detector area. We observed, at various temperatures and low bias currents, a noticeable increase in photon count rate with bias current which supports this hypothesis, suggesting localized areas with a lower gap and thus relatively higher internal detection efficiency. We furthermore expect that non-uniformity might have been introduced in the patterning and processing, therefore different regions of the wire could be participating in the detection process at different T and j bias . With process development including He + irradiation dose distribution correction, improved results may thus be expected. The role of He + irradiation at various doses on MgB 2 films remains not fully explored. Drawing from the data reported for low- T C SNSPDs composed of NbTiN [56] and NbN [41] , it’s observed that even at He + dosages considerably lower than those necessary to induce noticeable effects on transport properties like critical current density or normal state conductivity, there is an enhancement in single-photon detection efficiency. In our study, we employed a dosage of 5 × 10 15 ions/cm 2 (equivalent to 50 ions/nm 2 ), a dose that aligns with the reported dosage required for 8-to-12 nm-thick NbTiN films to begin exhibiting single-photon response. This dose resulted in only modest variations, up to 10%, in parameters such as critical current density, critical temperature, and sheet resistance. These results closely approximate our findings. Nevertheless, comprehensive investigations in this area are still pending. Apart from affecting MgB 2 film itself, the process of helium ion implantation causes defects and amorphization in the SiC substrate [57] . As a result, the efficiency of energy transfer from the superconducting structure to the substrate is likely to be affected. This effect may prolong the lifespan of the normal domain, thereby increasing the probability of detection before the normal domain collapses. In devices subjected to a relatively low dose of irradiation (5 × 10 15 ions/cm 2 ), we noticed a decrease in the retrapping current. While the switching current remained unchanged, the retrapping current reduced to 90% of its nominal value compared to non-irradiated devices (Supplementary Fig. 7) . Although magnesium diboride has two bands with different gaps: π -band with a gap of 0.6 k B T C and σ -band with a gap of 2.2 k B T C , the electron doping by atoms may initiate the interband scattering in MgB 2 . The filling band effect increases the value of the small gap and, together with band filling, leads to the merging of the two gaps [58] . Our work points out that it is the uniformity of the superconducting gap, rather than the absolute gap, that is crucial. The large bias current is not entirely unexpected in the case of high- T C microwire devices. The magnitude of the bias current is contingent upon the cross-sectional area of the wires and the critical temperature of the materials involved. In contrast to materials with lower critical temperatures typically employed in the production of superconducting nanowire single-photon detectors (SNSPDs), such as NbN, WSi, and MoSi, magnesium diboride has a critical temperature at least three times higher. That means that to achieve the same sensitivity, the gap must be suppressed by some means other than temperature, and elevating current density is one way to achieve a sufficiently small gap for detection. It is also worth noting that, the width of wires surpassed the Pearl length in MgB 2 . Wires wider than the Pearl length are expected to display non-uniform current distribution across the wire. The calculated Pearl length for MgB 2 films, approximately 1.3 μ m, is four times lower than the widest wire used in our experiment. As a result, the ratio of the experimentally observed switching current density to the current density for spontaneous vortex entry is expected to degrade, proportionally to the following: \\\\(\\\\sim 1/\\\\sqrt{1+W/2\\\\pi {\\\\Lambda }_{{{{{{{{\\\\rm{P}}}}}}}}}}\\\\) [59] , where Λ P is the Pearl Length. Surprisingly, contrary to theoretical predictions [59] , we observed single-photon detection in devices with wire widths of up to 5 μ m. This observation challenges the previously known limit in wire width from a practical standpoint, even though the mechanism of detection under these conditions remains unclear. While the detection efficiency approaches the limit for a single-layer detector, primarily due to absorption in thin films, there exists an opportunity for further enhancement by integrating the detector into an optical waveguide. The integration of MgB 2 detectors onto a chip with a waveguide faces limitations imposed by the material that is used for the waveguide. The polycrystalline structure of magnesium diboride introduces lattice mismatch that impacts on superconducting and transport properties of films. On silicon carbide (SiC), it is feasible to produce MgB 2 films as thin as 3 nm, maintaining a critical temperature higher than 30 K with a narrow transition. Epitaxial growth, a crucial factor for achieving a high critical temperature in thin films, is employed in this process. Although sapphire (Al 2 O 3 ) has also proven to be a suitable substrate for MgB 2 films, the demonstration of MgB 2 films as thin as those on SiC remains unreported. Alternatively, the approach that involves light coupled through the substrate with an optical cavity positioned atop the detector can enhance the absorption efficiency in MgB 2 detectors.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Sputtering and fab To fabricate detectors out of MgB 2 films we started with film deposition. Ultra-thin MgB 2 films were made using Hybrid Physical-Chemical Vapor Deposition (HPCVD) in a custom-built system [60] , [61] . Magnesium melts at 650 degrees Celsius and interacts with boron. The latter is produced via the thermal decomposition of gaseous diborane (B 2 H 6 ) supplied in a mixture with hydrogen. Films were made on (0001) SiC substrates which provide excellent lattice matching to the hexagonal lattice of c-oriented MgB 2 films. Devices were fabricated out of MgB 2 films on the 6H-SiC substrate with 5 nm titanium/ 50 nm gold contact pads using standard e-beam lithography, followed by metal deposition and lift-off in 45 °C N-Methylpyrrolidone (NMP). In order to produce the wires out of MgB 2 films, another round of e-beam lithography was performed, using 330-nm-thick high-resolution positive e-beam resist (ZEP520A). The patterns were then transferred onto the MgB 2 film by Ar + milling at a beam voltage of 300 V and Ar flow of 9 sccm for 12 minutes in a series of one-minute etches with three-minute-long intervals between each etch to avoid overheating of the sample. The residual resist was removed by immersing the substrate in the 45 °C NMP. After fabrication and initial testing, we irradiated our samples with He + using a Zeiss Orion Microscope equipped with a Raith pattern generator. The irradiation was realized by sweeping the beam across the detector area. The exposed area was a rectangle with dimensions of active detector areas (to cover the whole meander area). The beam-limiting aperture was set to the largest diameter to maximize the beam current. The dose was varied in the range from 10 15 to 10 20 ions/cm 2 (Supplementary Fig. 5) . Scanning Transmission Electron Microscopy (STEM) STEM imaging was performed with a probe-corrected Thermo Fisher Scientific Themis Z G3 60–300 kV S/TEM operated at 200 kV with a beam current of 30-90 pA and 25 mrad convergence angle. Elemental analysis was accomplished by using EDS (Energy Dispersive X-ray Spectroscopy). For cross-sectional STEM characterization, samples were prepared using a Raith VELION focused ion beam-scanning electron microscope (FIB-SEM) system with carbon and platinum protection layers on top of the sample.\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"This research proposes that high-quality magnesium diboride (MgB₂) microwire-based superconducting nanowire single-photon detectors (SNSPDs), fabricated using hybrid physical-chemical vapor deposition and helium-ion-beam irradiation, can achieve single-photon sensitivity up to 20 K with large active areas and improved performance metrics such as fast reset times and high count rates, overcoming limitations of previous high-temperature superconducting materials and detector geometries."},"entities":{"kind":"list like","value":["high-quality magnesium diboride microwire","superconducting nanowire single-photon detectors","hybrid physical-chemical vapor deposition","helium-ion-beam irradiation","single-photon sensitivity","active areas","reset times","count rates","high-temperature superconducting materials","detector geometries"],"string":"[\n \"high-quality magnesium diboride microwire\",\n \"superconducting nanowire single-photon detectors\",\n \"hybrid physical-chemical vapor deposition\",\n \"helium-ion-beam irradiation\",\n \"single-photon sensitivity\",\n \"active areas\",\n \"reset times\",\n \"count rates\",\n \"high-temperature superconducting materials\",\n \"detector geometries\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1.","title":"Semenov, A. D., Gol’tsman, G. N. & Korneev, A. A. Quantum detection by current carrying superconducting film.Physica C: Superconduct.351, 4 (2001).","link":"https://doi.org/10.1016%2FS0921-4534%2800%2901637-3"},{"idx":"2.","title":"Hadfield, R. H. Single-photon detectors for optical quantum information applications.Nat. Photonics3, 696–705 (2009).","link":"https://doi.org/10.1038%2Fnphoton.2009.230"},{"idx":"3.","title":"Shalm, L. K. et al. Strong Loophole-Free Test of Local Realism.Phys. Rev. Lett.115, 250402 (2015).","link":"https://doi.org/10.1103%2FPhysRevLett.115.250402"},{"idx":"4.","title":"Joch, D. J. et al. Certified random-number generation from quantum steering.Phys. Rev. A106, L050401 (2022).","link":"https://doi.org/10.1103%2FPhysRevA.106.L050401"},{"idx":"5.","title":"Guccione, G. et al. Connecting heterogeneous quantum networks by hybrid entanglement swapping.Sci. Adv.6, eaba4508 (2020).","link":"https://doi.org/10.1126%2Fsciadv.aba4508"},{"idx":"6.","title":"Valivarthi, R. et al. Quantum teleportation across a metropolitan fibre network.Nat. Photonics10, 676–680 (2016).","link":"https://doi.org/10.1038%2Fnphoton.2016.180"},{"idx":"7.","title":"Gisin, N. & Thew, R. Quantum communication.Nat. Photonics3, 165–171 (2007).","link":"https://doi.org/10.1038%2Fnphoton.2007.22"},{"idx":"8.","title":"Liao, S.-K. et al. Satellite-relayed intercontinental quantum network.Phys. Rev. Lett.120, 030501 (2018).","link":"https://doi.org/10.1103%2FPhysRevLett.120.030501"},{"idx":"10.","title":"Zhao, Q.-Y. et al. Single-photon imager based on a superconducting nanowire delay line.Nat. Photonics11, 247–251 (2017).","link":"https://doi.org/10.1038%2Fnphoton.2017.35"},{"idx":"11.","title":"Xia, F. et al. Short-wave infrared confocal fluorescence imaging of deep mouse brain with a superconducting nanowire single-photon detector.ACS Photonics8, 2800–2810 (2021).","link":"https://doi.org/10.1021%2Facsphotonics.1c01018"},{"idx":"12.","title":"Ozana, N. et al. Superconducting nanowire single-photon sensing of cerebral blood flow.Neurophotonics8, 035006 (2021).","link":"https://doi.org/10.1117%2F1.NPh.8.3.035006"},{"idx":"13.","title":"Guan, Y. et al. Lidar with superconducting nanowire single-photon detectors: Recent advances and developments.Optics Lasers Eng.156, 107102 (2022).","link":"https://doi.org/10.1016%2Fj.optlaseng.2022.107102"},{"idx":"15.","title":"Salvoni, D. et al. Lidar techniques for a SNSPD-based measurement.J. Phys. Conf. Ser.1182, 012014 (2019).","link":"https://doi.org/10.1088%2F1742-6596%2F1182%2F1%2F012014"},{"idx":"16.","title":"Inderbitzin, K., Engel, A., Schilling, A., Il’in, K. & Siegel, M. An ultra-fast superconducting Nb nanowire single-photon detector for soft x-rays.Appl. Phys. Lett.101, 162601 (2012).","link":"https://doi.org/10.1063%2F1.4759046"},{"idx":"17.","title":"Verma, V. B. et al. Single-photon detection in the mid-infrared up to 10μm wavelength using tungsten silicide superconducting nanowire detectors.APL Photonics6, 056101 (2021).","link":"https://doi.org/10.1063%2F5.0048049"},{"idx":"18.","title":"Reddy, D. V., Nerem, R. R., Nam, S. W., Mirin, R. P. & Verma, V. B. Superconducting nanowire single-photon detectors with 98% system detection efficiency at 1550 nm.Optica7, 1649–1653 (2020).","link":"https://doi.org/10.1364%2FOPTICA.400751"},{"idx":"19.","title":"Hu, P. et al. Detecting single infrared photons toward optimal system detection efficiency.Opt. Express28, 36884–36891 (2020).","link":"https://doi.org/10.1364%2FOE.410025"},{"idx":"20.","title":"Chang, J. et al. Detecting telecom single photons with 99.5% system detection efficiency and high time resolution.APL Photonics6, 036114 (2021).","link":"https://doi.org/10.1063%2F5.0039772"},{"idx":"21.","title":"Korzh, B. et al. Demonstration of sub-3 ps temporal resolution with a superconducting nanowire single-photon detector.Nat. Photonics14, 250–255 (2020).","link":"https://doi.org/10.1038%2Fs41566-020-0589-x"},{"idx":"22.","title":"Hochberg, Y. et al. Detecting sub-GeV dark matter with superconducting nanowires.Phys. Rev. Lett.123, 151802 (2019).","link":"https://doi.org/10.1103%2FPhysRevLett.123.151802"},{"idx":"23.","title":"Natarajan, C. M., Tanner, M. G. & Hadfield, R. H. Superconducting nanowire single-photon detectors: physics and applications.Supercond. Sci. Technol.25, 063001 (2012).","link":"https://doi.org/10.1088%2F0953-2048%2F25%2F6%2F063001"},{"idx":"24.","title":"Shibata, H. Review of superconducting nanostrip photon detectors using various superconductors.IEICE Trans. Electron.104, 429–434 (2021).","link":"https://doi.org/10.1587%2Ftransele.2020SUI0001"},{"idx":"25.","title":"Kerman, A. J. et al. Kinetic-inductance-limited reset time of superconducting nanowire photon counters.Appl. Phys. Lett.88, 111116 (2006).","link":"https://doi.org/10.1063%2F1.2183810"},{"idx":"26.","title":"Hofherr, M. et al. Intrinsic detection efficiency of superconducting nanowire single-photon detectors with different thicknesses.J. Appl. Phys.108, 014507 (2010).","link":"https://doi.org/10.1063%2F1.3437043"},{"idx":"27.","title":"Ejrnaes, M. et al. Observation of dark pulses in 10 nm thick YBCO nanostrips presenting hysteretic current voltage characteristics.Supercond. Sci. Technol.30, 12LT02 (2017).","link":"https://doi.org/10.1088%2F1361-6668%2Faa94b9"},{"idx":"28.","title":"Charaev, I. et al. Proximity effect model of ultranarrow NbN strips.Phys. Rev. B96, 184517 (2017).","link":"https://doi.org/10.1103%2FPhysRevB.96.184517"},{"idx":"29.","title":"Dane, A. E. et al. Bias sputtered NbN and superconducting nanowire devices.Appl. Phys. Lett.111, 122601 (2017).","link":"https://doi.org/10.1063%2F1.4990066"},{"idx":"30.","title":"Kimmel, G. J., Glatz, A., Vinokur, V. M. & Sadovskyy, I. A. Edge effect pinning in mesoscopic superconducting strips with non-uniform distribution of defects.Sci. Rep.9, 211 (2019).","link":"https://doi.org/10.1038%2Fs41598-018-36285-4"},{"idx":"31.","title":"Singh, U. R. et al. Spatial inhomogeneity of the superconducting gap and order parameter in FeSe0.4Te0.6.Phys. Rev. B88, 155124 (2013).","link":"https://doi.org/10.1103%2FPhysRevB.88.155124"},{"idx":"32.","title":"Arpaia, R. et al. Transport properties of ultrathin YBa2Cu3O7−δnanowires: A route to single-photon detection.Phys. Rev. B96, 064525 (2017).","link":"https://doi.org/10.1103%2FPhysRevB.96.064525"},{"idx":"33.","title":"Charaev, I. et al. Single-photon detection using high-temperature superconductors.Nat. Nanotechnol.18, 343–349 (2023).","link":"https://doi.org/10.1038%2Fs41565-023-01325-2"},{"idx":"34.","title":"Merino, R. et al. Two-dimensional cuprate nanodetector with single telecom photon sensitivity at T = 20K˙.2D Mater.10, 021001 (2023).","link":"https://doi.org/10.1088%2F2053-1583%2Facb4a8"},{"idx":"35.","title":"Shibata, H., Akazaki, T. & Tokura, Y. Fabrication of MgB2nanowire single-photon detector with meander structure.Appl. Phys. Express6, 023101 (2013).","link":"https://doi.org/10.7567%2FAPEX.6.023101"},{"idx":"36.","title":"Cherednichenko, S., Acharya, N., Novoselov, E. & Drakinskiy, V. Low kinetic inductance superconducting MgB2nanowires with a 130 ps relaxation time for single-photon detection applications.Supercond. Sci. Technol.34, 044001 (2021).","link":"https://doi.org/10.1088%2F1361-6668%2Fabdeda"},{"idx":"38.","title":"Shibata, H., Takesue, H., Honjo, T., Akazaki, T. & Tokura, Y. Single-photon detection using magnesium diboride superconducting nanowires.Appl. Phys. Lett.97, 212504 (2010).","link":"https://doi.org/10.1063%2F1.3518723"},{"idx":"39.","title":"Wolak, M. et al. Fabrication and characterization of ultrathin MgB2films for hot-electron bolometer applications.IEEE Trans. Appl. Supercond.25, 1–5 (2015).","link":"https://doi.org/10.1109%2FTASC.2015.2390415"},{"idx":"40.","title":"Shibata, H., Akazaki, T. & Tokura, Y. Ultrathin MgB2films fabricated by molecular beam epitaxy and rapid annealing.Supercond. Sci. Technol.26, 035005 (2013).","link":"https://doi.org/10.1088%2F0953-2048%2F26%2F3%2F035005"},{"idx":"41.","title":"Zhang, W. et al. Saturating intrinsic detection efficiency of superconducting nanowire single-photon detectors via defect engineering.Phys. Rev. Appl.12, 044040 (2019).","link":"https://doi.org/10.1103%2FPhysRevApplied.12.044040"},{"idx":"42.","title":"Korneeva, Y. et al. Optical single-photon detection in micrometer-scale NbN bridges.Phys. Rev. Appl.9, 064037 (2018).","link":"https://doi.org/10.1103%2FPhysRevApplied.9.064037"},{"idx":"43.","title":"Korneeva, Y. et al. Different single-photon response of wide and narrow superconducting MoxSi1−xstrips.Phys. Rev. Appl.13, 024011 (2020).","link":"https://doi.org/10.1103%2FPhysRevApplied.13.024011"},{"idx":"44.","title":"Chiles, J. et al. Superconducting microwire detectors based on WSi with single-photon sensitivity in the near-infrared.Appl. Phys. Lett.116, 242602 (2020).","link":"https://doi.org/10.1063%2F5.0006221"},{"idx":"45.","title":"Charaev, I. et al. Large-area microwire MoSi single-photon detectors at 1550 nm wavelength.Appl. Phys. Lett.116, 242603 (2020).","link":"https://doi.org/10.1063%2F5.0005439"},{"idx":"46.","title":"Semenov, A. et al. Optical and transport properties of ultrathin NbN films and nanostructures.Phys. Rev. B80, 054510 (2009).","link":"https://doi.org/10.1103%2FPhysRevB.80.054510"},{"idx":"47.","title":"Cooper, L. Superconductivity in the neighborhood of metallic contacts.Phys. Rev. Lett.6, 689 (1961).","link":"https://doi.org/10.1103%2FPhysRevLett.6.689"},{"idx":"48.","title":"Skocpol, W., Beasley, M. & Tinkham, M. Self-heating hotspots in superconducting thin-film microbridges.J. Appl. Phys.45, 4054–4066 (1974).","link":"https://doi.org/10.1063%2F1.1663912"},{"idx":"49.","title":"Yang, J. et al. Modeling the electrical and thermal response of superconducting nanowire single-photon detectors.IEEE Trans. Appl. Supercond.17, 581–585 (2007).","link":"https://doi.org/10.1109%2FTASC.2007.898660"},{"idx":"50.","title":"Stockhausen, A. et al. Adjustment of self-heating in long superconducting thin film NbN microbridges.Supercond. Sci. Technol.25, 035012 (2012).","link":"https://doi.org/10.1088%2F0953-2048%2F25%2F3%2F035012"},{"idx":"51.","title":"Pearl, J. Current distribution in superconducting films carrying quantized fluxoids.Appl. Phys. Lett.5, 65–66 (1964).","link":"https://doi.org/10.1063%2F1.1754056"},{"idx":"52.","title":"Cheng, R. et al. Unveiling photon statistics with a 100-pixel photon-number-resolving detector.Nat. Photon.17, 112–119 (2023).","link":"https://doi.org/10.1038%2Fs41566-022-01119-3"},{"idx":"53.","title":"Kerman, A., Yang, J., Molnar, R., Dauler, E. & Berggren, K. Electrothermal feedback in superconducting nanowire single-photon detectors.Phys. Rev. B79, 100509 (2009).","link":"https://doi.org/10.1103%2FPhysRevB.79.100509"},{"idx":"54.","title":"Marsili, F. et al. Detecting single infrared photons with 93% system efficiency.Nat. Photonics7, 210–214 (2013).","link":"https://doi.org/10.1038%2Fnphoton.2013.13"},{"idx":"55.","title":"He, S., Tian, R., Wu, W., Li, W. & Wang, D. Helium-ion-beam nanofabrication: extreme processes and applications.Int. J. Extrem. Manuf.3, 012001 (2020).","link":"https://doi.org/10.1088%2F2631-7990%2Fabc673"},{"idx":"56.","title":"Strohauer, S. et al. Site-selective enhancement of superconducting nanowire single-photon detectors via local helium ion irradiation.Adv Quantum Technol.6, 2300139 (2023).","link":"https://doi.org/10.1002%2Fqute.202300139"},{"idx":"57.","title":"Fan, Y. et al. Xenon ion implantation induced defects and amorphization in 4H-SiC: Insights from MD simulation and Raman spectroscopy characterization.Ceramics International49, 26654 (2023).","link":"https://doi.org/10.1016%2Fj.ceramint.2023.05.200"},{"idx":"58.","title":"Kortus, J., Dolgov, O., Kremer, R. & Golubov, A. Band filling and interband scattering effects in MgB2: carbon versus aluminum doping.Phys. Rev. Lett.94, 027002 (2005).","link":"https://doi.org/10.1103%2FPhysRevLett.94.027002"},{"idx":"59.","title":"Vodolazov, D. Single-photon detection by a dirty current-carrying superconducting strip based on the kinetic-equation approach.Phys. Rev. Appl.7, 034014 (2017).","link":"https://doi.org/10.1103%2FPhysRevApplied.7.034014"},{"idx":"60.","title":"Zeng, X. et al. Superconducting MgB2thin films on silicon carbide substrates by hybrid physical-chemical vapor deposition.Appl. Phys. Lett.82, 2097–2099 (2003).","link":"https://doi.org/10.1063%2F1.1563840"}],"string":"[\n {\n \"idx\": \"1.\",\n \"title\": \"Semenov, A. D., Gol’tsman, G. N. & Korneev, A. A. Quantum detection by current carrying superconducting film.Physica C: Superconduct.351, 4 (2001).\",\n \"link\": \"https://doi.org/10.1016%2FS0921-4534%2800%2901637-3\"\n },\n {\n \"idx\": \"2.\",\n \"title\": \"Hadfield, R. H. Single-photon detectors for optical quantum information applications.Nat. Photonics3, 696–705 (2009).\",\n \"link\": \"https://doi.org/10.1038%2Fnphoton.2009.230\"\n },\n {\n \"idx\": \"3.\",\n \"title\": \"Shalm, L. K. et al. Strong Loophole-Free Test of Local Realism.Phys. Rev. Lett.115, 250402 (2015).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.115.250402\"\n },\n {\n \"idx\": \"4.\",\n \"title\": \"Joch, D. J. et al. Certified random-number generation from quantum steering.Phys. Rev. A106, L050401 (2022).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevA.106.L050401\"\n },\n {\n \"idx\": \"5.\",\n \"title\": \"Guccione, G. et al. Connecting heterogeneous quantum networks by hybrid entanglement swapping.Sci. Adv.6, eaba4508 (2020).\",\n \"link\": \"https://doi.org/10.1126%2Fsciadv.aba4508\"\n },\n {\n \"idx\": \"6.\",\n \"title\": \"Valivarthi, R. et al. Quantum teleportation across a metropolitan fibre network.Nat. Photonics10, 676–680 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fnphoton.2016.180\"\n },\n {\n \"idx\": \"7.\",\n \"title\": \"Gisin, N. & Thew, R. Quantum communication.Nat. Photonics3, 165–171 (2007).\",\n \"link\": \"https://doi.org/10.1038%2Fnphoton.2007.22\"\n },\n {\n \"idx\": \"8.\",\n \"title\": \"Liao, S.-K. et al. Satellite-relayed intercontinental quantum network.Phys. Rev. Lett.120, 030501 (2018).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.120.030501\"\n },\n {\n \"idx\": \"10.\",\n \"title\": \"Zhao, Q.-Y. et al. Single-photon imager based on a superconducting nanowire delay line.Nat. Photonics11, 247–251 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fnphoton.2017.35\"\n },\n {\n \"idx\": \"11.\",\n \"title\": \"Xia, F. et al. Short-wave infrared confocal fluorescence imaging of deep mouse brain with a superconducting nanowire single-photon detector.ACS Photonics8, 2800–2810 (2021).\",\n \"link\": \"https://doi.org/10.1021%2Facsphotonics.1c01018\"\n },\n {\n \"idx\": \"12.\",\n \"title\": \"Ozana, N. et al. Superconducting nanowire single-photon sensing of cerebral blood flow.Neurophotonics8, 035006 (2021).\",\n \"link\": \"https://doi.org/10.1117%2F1.NPh.8.3.035006\"\n },\n {\n \"idx\": \"13.\",\n \"title\": \"Guan, Y. et al. Lidar with superconducting nanowire single-photon detectors: Recent advances and developments.Optics Lasers Eng.156, 107102 (2022).\",\n \"link\": \"https://doi.org/10.1016%2Fj.optlaseng.2022.107102\"\n },\n {\n \"idx\": \"15.\",\n \"title\": \"Salvoni, D. et al. Lidar techniques for a SNSPD-based measurement.J. Phys. Conf. Ser.1182, 012014 (2019).\",\n \"link\": \"https://doi.org/10.1088%2F1742-6596%2F1182%2F1%2F012014\"\n },\n {\n \"idx\": \"16.\",\n \"title\": \"Inderbitzin, K., Engel, A., Schilling, A., Il’in, K. & Siegel, M. An ultra-fast superconducting Nb nanowire single-photon detector for soft x-rays.Appl. Phys. Lett.101, 162601 (2012).\",\n \"link\": \"https://doi.org/10.1063%2F1.4759046\"\n },\n {\n \"idx\": \"17.\",\n \"title\": \"Verma, V. B. et al. Single-photon detection in the mid-infrared up to 10μm wavelength using tungsten silicide superconducting nanowire detectors.APL Photonics6, 056101 (2021).\",\n \"link\": \"https://doi.org/10.1063%2F5.0048049\"\n },\n {\n \"idx\": \"18.\",\n \"title\": \"Reddy, D. V., Nerem, R. R., Nam, S. W., Mirin, R. P. & Verma, V. B. Superconducting nanowire single-photon detectors with 98% system detection efficiency at 1550 nm.Optica7, 1649–1653 (2020).\",\n \"link\": \"https://doi.org/10.1364%2FOPTICA.400751\"\n },\n {\n \"idx\": \"19.\",\n \"title\": \"Hu, P. et al. Detecting single infrared photons toward optimal system detection efficiency.Opt. Express28, 36884–36891 (2020).\",\n \"link\": \"https://doi.org/10.1364%2FOE.410025\"\n },\n {\n \"idx\": \"20.\",\n \"title\": \"Chang, J. et al. Detecting telecom single photons with 99.5% system detection efficiency and high time resolution.APL Photonics6, 036114 (2021).\",\n \"link\": \"https://doi.org/10.1063%2F5.0039772\"\n },\n {\n \"idx\": \"21.\",\n \"title\": \"Korzh, B. et al. Demonstration of sub-3 ps temporal resolution with a superconducting nanowire single-photon detector.Nat. Photonics14, 250–255 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41566-020-0589-x\"\n },\n {\n \"idx\": \"22.\",\n \"title\": \"Hochberg, Y. et al. Detecting sub-GeV dark matter with superconducting nanowires.Phys. Rev. Lett.123, 151802 (2019).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.123.151802\"\n },\n {\n \"idx\": \"23.\",\n \"title\": \"Natarajan, C. M., Tanner, M. G. & Hadfield, R. H. Superconducting nanowire single-photon detectors: physics and applications.Supercond. Sci. Technol.25, 063001 (2012).\",\n \"link\": \"https://doi.org/10.1088%2F0953-2048%2F25%2F6%2F063001\"\n },\n {\n \"idx\": \"24.\",\n \"title\": \"Shibata, H. Review of superconducting nanostrip photon detectors using various superconductors.IEICE Trans. Electron.104, 429–434 (2021).\",\n \"link\": \"https://doi.org/10.1587%2Ftransele.2020SUI0001\"\n },\n {\n \"idx\": \"25.\",\n \"title\": \"Kerman, A. J. et al. Kinetic-inductance-limited reset time of superconducting nanowire photon counters.Appl. Phys. Lett.88, 111116 (2006).\",\n \"link\": \"https://doi.org/10.1063%2F1.2183810\"\n },\n {\n \"idx\": \"26.\",\n \"title\": \"Hofherr, M. et al. Intrinsic detection efficiency of superconducting nanowire single-photon detectors with different thicknesses.J. Appl. Phys.108, 014507 (2010).\",\n \"link\": \"https://doi.org/10.1063%2F1.3437043\"\n },\n {\n \"idx\": \"27.\",\n \"title\": \"Ejrnaes, M. et al. Observation of dark pulses in 10 nm thick YBCO nanostrips presenting hysteretic current voltage characteristics.Supercond. Sci. Technol.30, 12LT02 (2017).\",\n \"link\": \"https://doi.org/10.1088%2F1361-6668%2Faa94b9\"\n },\n {\n \"idx\": \"28.\",\n \"title\": \"Charaev, I. et al. Proximity effect model of ultranarrow NbN strips.Phys. Rev. B96, 184517 (2017).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.96.184517\"\n },\n {\n \"idx\": \"29.\",\n \"title\": \"Dane, A. E. et al. Bias sputtered NbN and superconducting nanowire devices.Appl. Phys. Lett.111, 122601 (2017).\",\n \"link\": \"https://doi.org/10.1063%2F1.4990066\"\n },\n {\n \"idx\": \"30.\",\n \"title\": \"Kimmel, G. J., Glatz, A., Vinokur, V. M. & Sadovskyy, I. A. Edge effect pinning in mesoscopic superconducting strips with non-uniform distribution of defects.Sci. Rep.9, 211 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41598-018-36285-4\"\n },\n {\n \"idx\": \"31.\",\n \"title\": \"Singh, U. R. et al. Spatial inhomogeneity of the superconducting gap and order parameter in FeSe0.4Te0.6.Phys. Rev. B88, 155124 (2013).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.88.155124\"\n },\n {\n \"idx\": \"32.\",\n \"title\": \"Arpaia, R. et al. Transport properties of ultrathin YBa2Cu3O7−δnanowires: A route to single-photon detection.Phys. Rev. B96, 064525 (2017).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.96.064525\"\n },\n {\n \"idx\": \"33.\",\n \"title\": \"Charaev, I. et al. Single-photon detection using high-temperature superconductors.Nat. Nanotechnol.18, 343–349 (2023).\",\n \"link\": \"https://doi.org/10.1038%2Fs41565-023-01325-2\"\n },\n {\n \"idx\": \"34.\",\n \"title\": \"Merino, R. et al. Two-dimensional cuprate nanodetector with single telecom photon sensitivity at T = 20K˙.2D Mater.10, 021001 (2023).\",\n \"link\": \"https://doi.org/10.1088%2F2053-1583%2Facb4a8\"\n },\n {\n \"idx\": \"35.\",\n \"title\": \"Shibata, H., Akazaki, T. & Tokura, Y. Fabrication of MgB2nanowire single-photon detector with meander structure.Appl. Phys. Express6, 023101 (2013).\",\n \"link\": \"https://doi.org/10.7567%2FAPEX.6.023101\"\n },\n {\n \"idx\": \"36.\",\n \"title\": \"Cherednichenko, S., Acharya, N., Novoselov, E. & Drakinskiy, V. Low kinetic inductance superconducting MgB2nanowires with a 130 ps relaxation time for single-photon detection applications.Supercond. Sci. Technol.34, 044001 (2021).\",\n \"link\": \"https://doi.org/10.1088%2F1361-6668%2Fabdeda\"\n },\n {\n \"idx\": \"38.\",\n \"title\": \"Shibata, H., Takesue, H., Honjo, T., Akazaki, T. & Tokura, Y. Single-photon detection using magnesium diboride superconducting nanowires.Appl. Phys. Lett.97, 212504 (2010).\",\n \"link\": \"https://doi.org/10.1063%2F1.3518723\"\n },\n {\n \"idx\": \"39.\",\n \"title\": \"Wolak, M. et al. Fabrication and characterization of ultrathin MgB2films for hot-electron bolometer applications.IEEE Trans. Appl. Supercond.25, 1–5 (2015).\",\n \"link\": \"https://doi.org/10.1109%2FTASC.2015.2390415\"\n },\n {\n \"idx\": \"40.\",\n \"title\": \"Shibata, H., Akazaki, T. & Tokura, Y. Ultrathin MgB2films fabricated by molecular beam epitaxy and rapid annealing.Supercond. Sci. Technol.26, 035005 (2013).\",\n \"link\": \"https://doi.org/10.1088%2F0953-2048%2F26%2F3%2F035005\"\n },\n {\n \"idx\": \"41.\",\n \"title\": \"Zhang, W. et al. Saturating intrinsic detection efficiency of superconducting nanowire single-photon detectors via defect engineering.Phys. Rev. Appl.12, 044040 (2019).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevApplied.12.044040\"\n },\n {\n \"idx\": \"42.\",\n \"title\": \"Korneeva, Y. et al. Optical single-photon detection in micrometer-scale NbN bridges.Phys. Rev. Appl.9, 064037 (2018).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevApplied.9.064037\"\n },\n {\n \"idx\": \"43.\",\n \"title\": \"Korneeva, Y. et al. Different single-photon response of wide and narrow superconducting MoxSi1−xstrips.Phys. Rev. Appl.13, 024011 (2020).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevApplied.13.024011\"\n },\n {\n \"idx\": \"44.\",\n \"title\": \"Chiles, J. et al. Superconducting microwire detectors based on WSi with single-photon sensitivity in the near-infrared.Appl. Phys. Lett.116, 242602 (2020).\",\n \"link\": \"https://doi.org/10.1063%2F5.0006221\"\n },\n {\n \"idx\": \"45.\",\n \"title\": \"Charaev, I. et al. Large-area microwire MoSi single-photon detectors at 1550 nm wavelength.Appl. Phys. Lett.116, 242603 (2020).\",\n \"link\": \"https://doi.org/10.1063%2F5.0005439\"\n },\n {\n \"idx\": \"46.\",\n \"title\": \"Semenov, A. et al. Optical and transport properties of ultrathin NbN films and nanostructures.Phys. Rev. B80, 054510 (2009).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.80.054510\"\n },\n {\n \"idx\": \"47.\",\n \"title\": \"Cooper, L. Superconductivity in the neighborhood of metallic contacts.Phys. Rev. Lett.6, 689 (1961).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.6.689\"\n },\n {\n \"idx\": \"48.\",\n \"title\": \"Skocpol, W., Beasley, M. & Tinkham, M. Self-heating hotspots in superconducting thin-film microbridges.J. Appl. Phys.45, 4054–4066 (1974).\",\n \"link\": \"https://doi.org/10.1063%2F1.1663912\"\n },\n {\n \"idx\": \"49.\",\n \"title\": \"Yang, J. et al. Modeling the electrical and thermal response of superconducting nanowire single-photon detectors.IEEE Trans. Appl. Supercond.17, 581–585 (2007).\",\n \"link\": \"https://doi.org/10.1109%2FTASC.2007.898660\"\n },\n {\n \"idx\": \"50.\",\n \"title\": \"Stockhausen, A. et al. Adjustment of self-heating in long superconducting thin film NbN microbridges.Supercond. Sci. Technol.25, 035012 (2012).\",\n \"link\": \"https://doi.org/10.1088%2F0953-2048%2F25%2F3%2F035012\"\n },\n {\n \"idx\": \"51.\",\n \"title\": \"Pearl, J. Current distribution in superconducting films carrying quantized fluxoids.Appl. Phys. Lett.5, 65–66 (1964).\",\n \"link\": \"https://doi.org/10.1063%2F1.1754056\"\n },\n {\n \"idx\": \"52.\",\n \"title\": \"Cheng, R. et al. Unveiling photon statistics with a 100-pixel photon-number-resolving detector.Nat. Photon.17, 112–119 (2023).\",\n \"link\": \"https://doi.org/10.1038%2Fs41566-022-01119-3\"\n },\n {\n \"idx\": \"53.\",\n \"title\": \"Kerman, A., Yang, J., Molnar, R., Dauler, E. & Berggren, K. Electrothermal feedback in superconducting nanowire single-photon detectors.Phys. Rev. B79, 100509 (2009).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.79.100509\"\n },\n {\n \"idx\": \"54.\",\n \"title\": \"Marsili, F. et al. Detecting single infrared photons with 93% system efficiency.Nat. Photonics7, 210–214 (2013).\",\n \"link\": \"https://doi.org/10.1038%2Fnphoton.2013.13\"\n },\n {\n \"idx\": \"55.\",\n \"title\": \"He, S., Tian, R., Wu, W., Li, W. & Wang, D. Helium-ion-beam nanofabrication: extreme processes and applications.Int. J. Extrem. Manuf.3, 012001 (2020).\",\n \"link\": \"https://doi.org/10.1088%2F2631-7990%2Fabc673\"\n },\n {\n \"idx\": \"56.\",\n \"title\": \"Strohauer, S. et al. Site-selective enhancement of superconducting nanowire single-photon detectors via local helium ion irradiation.Adv Quantum Technol.6, 2300139 (2023).\",\n \"link\": \"https://doi.org/10.1002%2Fqute.202300139\"\n },\n {\n \"idx\": \"57.\",\n \"title\": \"Fan, Y. et al. Xenon ion implantation induced defects and amorphization in 4H-SiC: Insights from MD simulation and Raman spectroscopy characterization.Ceramics International49, 26654 (2023).\",\n \"link\": \"https://doi.org/10.1016%2Fj.ceramint.2023.05.200\"\n },\n {\n \"idx\": \"58.\",\n \"title\": \"Kortus, J., Dolgov, O., Kremer, R. & Golubov, A. Band filling and interband scattering effects in MgB2: carbon versus aluminum doping.Phys. Rev. Lett.94, 027002 (2005).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.94.027002\"\n },\n {\n \"idx\": \"59.\",\n \"title\": \"Vodolazov, D. Single-photon detection by a dirty current-carrying superconducting strip based on the kinetic-equation approach.Phys. Rev. Appl.7, 034014 (2017).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevApplied.7.034014\"\n },\n {\n \"idx\": \"60.\",\n \"title\": \"Zeng, X. et al. Superconducting MgB2thin films on silicon carbide substrates by hybrid physical-chemical vapor deposition.Appl. Phys. Lett.82, 2097–2099 (2003).\",\n \"link\": \"https://doi.org/10.1063%2F1.1563840\"\n }\n]","isConversational":false}}},{"rowIdx":91,"cells":{"unique_id":{"kind":"string","value":"s41467-023-38939-y"},"subfield":{"kind":"string","value":"Optics and photonics"},"title":{"kind":"string","value":"Collective excitations of a bound-in-the-continuum condensate"},"abstract":{"kind":"string","value":"Spectra of low-lying elementary excitations are critical to characterize properties of bosonic quantum fluids. Usually these spectra are difficult to observe, due to low occupation of non-condensate states compared to the ground state. Recently, low-threshold Bose-Einstein condensation was realised in a symmetry-protected bound state in the continuum, at a saddle point, thanks to coupling of this electromagnetic resonance to semiconductor excitons. While it has opened the door to long-living polariton condensates, their intrinsic collective properties are still unexplored. Here we unveil the peculiar features of the Bogoliubov spectrum of excitations in this system. Thanks to the dark nature of the bound-in-the-continuum state, collective excitations lying directly above the condensate become observable in enhanced detail. We reveal interesting aspects, such as energy-flat parts of the dispersion characterized by two parallel stripes in photoluminescence pattern, pronounced linearization at non-zero momenta in one of the directions, and a strongly anisotropic velocity of sound."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"Bound states in the continuum (BICs) have been originally proposed as a mathematical feature of the Schrödinger equation in presence of specially prepared potentials [1] , [2] , [3] . Similar states have been later found in a wide range of systems like graphene, topological insulators, atomic superlattices, dielectric photonic crystals, metasurfaces and patterned optical waveguides [4] . In the latter cases, coupling in both the real and the imaginary parts of the two counter-propagating electromagnetic modes leads to the opening of an energy gap, with one of the new eigenmodes being bright and the other exhibiting vanishing radiative losses despite the non-Hermiticity of the underlying Hamiltonian [5] , [6] . Symmetry-protected optical BICs occuring from two interfering resonances enable light confinement and are routinely used in grating surface-emitting and distributed feedback lasers [7] , [8] , [9] , where emission from the BIC mode manifests itself in a specific two-lobe far field pattern. Interestingly, these quasi-infinite-lifetime photonic states can couple to matter excitations, such as surface plasmons [6] or excitons in monolayer transition-metal dichalcogenides [10] , [11] and quantum wells [12] and form polaritons with interesting features, such as greatly enhanced tunable lifetimes and strong nonlinearities. Polaritons are hybrid half-light half-matter quasi-particles in semiconductor heterostructures, typically embedded in a Fabry-Pérot microcavity to enhance the coupling of electronic excitations (e.g. excitons) with photons and to reduce the losses [13] . Such low-dimensional bosonic systems have been shown to provide conditions for observation of collective behaviors like Bose-Einstein condensation [14] , [15] , [16] and superfluidity [17] , [18] , [19] , at the same time allowing for a direct measurement of their momentum and spatial distributions via the emitted light. The onset of Bose-Einstein condensation is accompanied with a narrowed enhanced emission from the ground state at k = 0 of the polariton energy dispersion, which consequently blueshifts due to particle-particle interactions. At the same time, the momentum-energy dispersion on top of the condensate becomes linearized in accordance with the textbook Bogoliubov prediction [20] , therefore giving origin to superfluidity as follows from the Landau criterion. Multiple experiments with cavity exciton-polaritons have provided evidence of such linearization [21] , [22] , [23] , [24] , [25] despite the driven-dissipative nature of the system suggesting the diffusive character of the Bogoliubov dispersion [26] , [27] with a zero real part in the region of small momenta. Precise observation of the shape of the excitations spectrum due to the thermal and quantum depletion of the macroscopically occupied ground state is also possible, but requires substantial momentum-space filtering covering a much brighter signal from the condensate [28] , [29] or using refined interferometric techniques [25] . In this work, we study the elementary excitation spectrum of a polariton Bose condensate arising from a BIC state in a planar nanostructured waveguide. It has been recently reported [12] that due to the band folding effect and the consequent coupling with the quantum well exciton mode, the condensate which is formed on the lowest polariton branch appears, counter-intuitively, at a saddle point of the dispersion rather than in a global energy minimum. Since the particles accumulate in the quasi-BIC state with a very long lifetime, much longer than the carriers relaxation time, the polariton condensate in such a state appears to be a paradigmatic system for studying the excitation spectrum of a condensate in thermal equilibrium [30] . In such state the condensate cannot directly radiate and, as we demonstrate in the following, measurements of the shape of the spectrum of elementary excitations are possible in great detail and even without any momentum filtering. We derive theoretically the finite-temperature Bogoliubov dispersion for polaritons accumulating in the BIC and show that a local energy minimum starts to appear around the saddle point k = 0 in the spectrum of excitations which, possibly, could help in the formation of a long-living Bose-Einstein condensate, with the real part of the spectrum being non-zero despite the negative effective mass in one of the directions and losses present in the system. We experimentally observe various and very distinct characteristics of collective excitations dispersion strongly dependent on the direction in k -space. The detailed knowledge of the Bogoliubov spectrum allows to investigate the angular profile of the sound velocity in such highly anisotropic system. We believe that this study could open a way of controlling the condensate properties by engineering its excitation spectrum.","references":{"indices":[2,3,4,5,6,7,9,10,11,12,14,15,16,17,18,19,21,22,23,24,25,26,27,28,29,30],"infos":[{"idx":"2","title":"Friedrich, H. & Wintgen, D. Interfering resonances and bound states in the continuum.Phys. Rev. A32, 3231 (1985).","link":"https://doi.org/10.1103%2FPhysRevA.32.3231"},{"idx":"3","title":"Capasso, F. et al. Observation of an electronic bound state above a potential well.Nature358, 565 (1992).","link":"https://doi.org/10.1038%2F358565a0"},{"idx":"4","title":"Hsu, C. W., Zhen, B., Stone, A. D., Joannopoulos, J. D. & Soljačic, M. Bound states in the continuum.Nat. Rev. Mater.1, 16048 (2016).","link":"https://doi.org/10.1038%2Fnatrevmats.2016.48"},{"idx":"5","title":"Zhen, B. et al. Spawning rings of exceptional points out of Dirac cones.Nature525, 354 (2015).","link":"https://doi.org/10.1038%2Fnature14889"},{"idx":"6","title":"Lu, L. et al. Engineering a light-matter strong coupling regime in perovskite-based plasmonic metasurface: quasi-bound state in the continuum and exceptional points.Photon. Res.8, A91–A100 (2020).","link":"https://doi.org/10.1364%2FPRJ.404743"},{"idx":"7","title":"Noll, R. J. & Macomber, S. H. Analysis of grating surface emitting lasers.IEEE J. Quant. Electron.26, 456 (1990).","link":"https://doi.org/10.1109%2F3.52121"},{"idx":"9","title":"Wang, Y. et al. Highly controllable etchless perovskite microlasers based on bound states in the continuum.ACS Nano15, 7386–7391 (2021).","link":"https://doi.org/10.1021%2Facsnano.1c00673"},{"idx":"10","title":"Koshelev, K. L., Sychev, S. K., Sadrieva, Z. F., Bogdanov, A. A. & Iorsh, I. V. Strong coupling between excitons in transition metal dichalcogenides and optical bound states in the continuum.Phys. Rev. B98, 161113(R) (2018).","link":"https://doi.org/10.1103%2FPhysRevB.98.161113"},{"idx":"11","title":"Kravtsov, V. et al. Nonlinear polaritons in a monolayer semiconductor coupled to optical bound states in the continuum.Light. Sci. Appl.9, 56 (2020).","link":"https://doi.org/10.1038%2Fs41377-020-0286-z"},{"idx":"12","title":"Ardizzone, V. et al. Polariton Bose-Einstein condensate from a Bound State in the Continuum.Nature605, 447–452 (2022).","link":"https://doi.org/10.1038%2Fs41586-022-04583-7"},{"idx":"14","title":"Kasprzak, J. et al. Bose-Einstein condensation of exciton polaritons.Nature443, 409–414 (2006).","link":"https://doi.org/10.1038%2Fnature05131"},{"idx":"15","title":"Balili, R., Hartwell, V., Snoke, D., Pfeiffer, L. & West, K. Bose-Einstein condensation of microcavity polaritons in a trap.Science316, 1007–1010 (2007).","link":"https://doi.org/10.1126%2Fscience.1140990"},{"idx":"16","title":"Byrnes, T., Kim, N. Y. & Yamamoto, Y. Exciton-polariton condensates.Nat. Phys.10, 803–813 (2014).","link":"https://doi.org/10.1038%2Fnphys3143"},{"idx":"17","title":"Cristofolini, P. et al. Optical superfluid phase transitions and trapping of polariton condensates.Phys. Rev. Lett.110, 186403 (2013).","link":"https://doi.org/10.1103%2FPhysRevLett.110.186403"},{"idx":"18","title":"Amo, A. et al. Superfluidity of polaritons in semiconductor microcavities.Nat. Phys.5, 805–810 (2009).","link":"https://doi.org/10.1038%2Fnphys1364"},{"idx":"19","title":"Lerario, G. et al. Room-temperature superfluidity in a polariton condensate.Nat. Phys.13, 837–841 (2017).","link":"https://doi.org/10.1038%2Fnphys4147"},{"idx":"21","title":"Utsunomiya, S. et al. Observation of Bogoliubov excitations in exciton-polariton condensates.Nat. Phys.4, 700 (2008).","link":"https://doi.org/10.1038%2Fnphys1034"},{"idx":"22","title":"Kohnle, V. et al. From single particle to superfluid excitations in a dissipative polariton gas.Phys. Rev. Lett.106, 255302 (2011).","link":"https://doi.org/10.1103%2FPhysRevLett.106.255302"},{"idx":"23","title":"Pieczarka, M. et al. Ghost branch photoluminescence from a polariton fluid under nonresonant excitation.Phys. Rev. Lett.115, 186401 (2015).","link":"https://doi.org/10.1103%2FPhysRevLett.115.186401"},{"idx":"24","title":"Stepanov, P. et al. Dispersion relation of the collective excitations in a resonantly driven polariton fluid.Nat. Commun.10, 3869 (2019).","link":"https://doi.org/10.1038%2Fs41467-019-11886-3"},{"idx":"25","title":"Ballarini, D. et al. Directional Goldstone waves in polariton condensates close to equilibrium.Nat. Commun.11, 217 (2020).","link":"https://doi.org/10.1038%2Fs41467-019-13733-x"},{"idx":"26","title":"Wouters, M. & Carusotto, I. Excitations in a nonequilibrium bose-einstein condensate of exciton polaritons.Phys. Rev. Lett.99, 140402 (2007).","link":"https://doi.org/10.1103%2FPhysRevLett.99.140402"},{"idx":"27","title":"Byrnes, T., Horikiri, T., Ishida, N., Fraser, M. & Yamamoto, Y. Negative Bogoliubov dispersion in exciton-polariton condensates.Phys. Rev. B85, 075130 (2012).","link":"https://doi.org/10.1103%2FPhysRevB.85.075130"},{"idx":"28","title":"Pieczarka, M. et al. Observation of quantum depletion in a non-equilibrium exciton-polariton condensate.Nat. Commun.11, 429 (2020).","link":"https://doi.org/10.1038%2Fs41467-019-14243-6"},{"idx":"29","title":"Pieczarka, M. et al. Bogoliubov excitations of a polariton condensate in dynamical equilibrium with an incoherent reservoir.Phys. Rev. B105, 224515 (2022).","link":"https://doi.org/10.1103%2FPhysRevB.105.224515"},{"idx":"30","title":"Grudinina, A. M., Kurbakov, I. L., Lozovik, Y. E. & Voronova, N. S. Finite-temperature Hartree-Fock-Bogoliubov theory for exciton-polaritons.Phys. Rev. B104, 125301 (2021).","link":"https://doi.org/10.1103%2FPhysRevB.104.125301"}]}},{"section":"Results","content":"Saddle-point polariton condensate The sample studied here, sketched in Fig. 1 a, is a GaAs/AlGaAs waveguide hosting 12 20-nm-thick GaAs quantum wells embedded in a Al 0.4 Ga 0.6 As core. A set of linear diffraction gratings is etched on the surface of the waveguide to allow for the observation, within the lightcone, of the propagating guided modes. By changing the grating period a and the filling factor it is possible to tune the energy of the BIC (i.e. its excitonic fraction) and the width of the energy gap separating the lower-polariton BIC, with ultra-long lifetimes, and the leaky polaritonic modes. The coupling of the counter-propagating TE-modes of electromagnetic waves to the exciton results in the four branches of polariton dispersion, the analytical derivation of which are provided in the Supplementary Information ( SI) . Of interest, in this work, are the two anisotropic lowest branches which are shown in Fig. 1 b. It is clearly seen that the shape of the upper of the two lower-polariton modes (ULP) resembles the regular lower cavity exciton-polariton, albeit with differing effective masses along and perpendicular to the grating, whereas the lower (LLP) mode is distinct from the standard dispersion showing different behaviors along k x – and k y –directions. In particular, we note the appearance of a negative effective mass in the x –direction, with an absolute value much lower than the positive mass in the y –direction. It is also important to note that, when expanded at k → 0 in Taylor series up to the second order, the LLP dispersion appears to have an imaginary contribution in the x –direction (for more details, see the SI) . This imaginary term provides the k x –dependent loss rate of polaritons on the lowest branch, with the zero radiative loss at k = 0, as was experimentally verified in ref. [12] . Fig. 1: Polariton dispersions around the BIC state. a Sketch of the etched waveguide structure. The waveguide is made of 12 GaAs/Al 0.4 Ga 0.6 As (20-nm-thick each) layer pairs on a substrate, with the top 90 nm textured with a periodic 1D grating with a period a = 243 nm. The grating sizes are 50 μm along y and 300 μm along x -direction. b Three-dimensional dispersions of the two lower exciton-polariton modes (see SI) , with the energy reference taken at the saddle point of the lowest mode, calculated for Δ 0 = − 0.6 meV, ℏ Ω R = 15 meV, U = − 1.3 meV, ℏ γ = 0.51 meV, n g = 4.3, m X = 0.22 m 0 . c , d Cross-cuts of the energy-momentum experimental dispersion along k y = 0 ( c ) and k x = 0.025 μm −1 ( d ) showing two orders of magnitude difference in the lower polariton effective mass along x and y directions, \\(1/{m}_{{{{{{{{\\rm{LP}}}}}}}}}^{x(y)}\\) (see text). The yellow dashed lines show theoretical fits of the two lower polariton branches with the parameters as in b . e , f Comparison of experimental far field emission below (on the left-hand side) and above condensation threshold (on the right-hand side) showing energy versus k x and k y respectively; panel f is a cut along k x = 0.02 μm −1 , as for panel d . The colorscales for intensity are linear, given in arb. units and normalized separately for panels c , d and e , f . Full size image The polariton dispersions along k x and k y can be directly measured by energy and momentum resolved photoluminescence (PL). Figure 1 c and d show two different cuts (along k x and k y respectively) of the PL of the single-particle lower polariton dispersions. While along k x (Fig. 1 c) the ULP and LLP have the masses of opposite sign, this is not the case along k y where the mass is positive for both (Fig. 1d) . The dispersion in Fig. 1 d has been taken at a slightly finite wavevector in the x –direction ( k x ~ 0.02 μm −1 ) to avoid the completely dark stripe of the BIC state at k x = 0. These experimental dispersions are in very good agreement with the analytical model which is shown by the overlaid yellow dashed lines in both panels. We note that the range of horizontal axis along k y is four times larger than that of k x and that the effective masses along the two directions differ by about two orders of magnitude, being in quantitative agreement with the theory developed in the SI. Both images in Fig. 1 c and d show the PL emission below threshold, when the upper of the two lower-polariton branches is emitting more light than the lowest one, as it is much more lossy around k = 0 (see SI) , while their populations are comparable. When increasing the excitation power, polaritons condense into the BIC at the saddle point of the LLP branch [12] . Figure 1 e (right-hand side) shows the cross-cut of the dispersion along k x with the time-integrated PL measured slightly above the condensation threshold. We only show one of the two bright spots forming to either side of the saddle-point condensate, blueshifted compared to the below-threshold image shown on the left-hand side of the panel. Figure 1 f (right-hand side) shows the corresponding dispersion along k y for k x ~ 0.02 μm −1 , i.e. close to the inner edge of the bright spot of Fig. 1 e above threshold. The latter has an interesting form, with an energy-flattened emission roughly extending from −0.5 μm −1 (not visible) to +0.5 μm −1 continuing into linear tails going up in energy. It is important to note that, as we will show in the following, the flat region in this case is not related to the diffusive character of the condensate. On the contrary, it is a consequence of both the very long polariton lifetime and the saddle shape of the dispersion. Hartree–Fock–Bogoliubov theory for elementary excitations To explain these features observed in Fig. 1 e, f, we derive the Bogoliubov spectrum of excitations on top of the condensate forming in the BIC state, i.e. in the vicinity of the k = 0 saddle point of the LLP branch for low finite temperatures in the presence of the (dark) exciton reservoir [30] . The Hamiltonian of lower polaritons is expressed in the second quantization as $$\\hat{H}=\t\\int\\,d{{{{{{{\\bf{r}}}}}}}}{\\hat{P}}_{-}^{{{{\\dagger}}} }({{{{{{{\\bf{r}}}}}}}})[{\\varepsilon }_{-}(\\hat{{{{{{{{\\bf{p}}}}}}}}})-{\\mu }_{-}]{\\hat{P}}_{-}({{{{{{{\\bf{r}}}}}}}}) \\\\ \t+\\int\\,d{{{{{{{\\bf{r}}}}}}}}{\\hat{P}}_{+}^{{{{\\dagger}}} }({{{{{{{\\bf{r}}}}}}}})[{\\varepsilon }_{+}(\\hat{{{{{{{{\\bf{p}}}}}}}}})-{\\mu }_{+}]{\\hat{P}}_{+}({{{{{{{\\bf{r}}}}}}}})\\\\ \t+\\frac{1}{2}\\int\\,d{{{{{{{\\bf{r}}}}}}}}d{{{{{{{{\\bf{r}}}}}}}}}^{{\\prime} }{\\hat{Q}}^{{{{\\dagger}}} }({{{{{{{\\bf{r}}}}}}}}){\\hat{Q}}^{{{{\\dagger}}} }({{{{{{{{\\bf{r}}}}}}}}}^{{\\prime} })U({{{{{{{\\bf{r}}}}}}}}-{{{{{{{{\\bf{r}}}}}}}}}^{{\\prime} })\\hat{Q}({{{{{{{{\\bf{r}}}}}}}}}^{{\\prime} })\\hat{Q}({{{{{{{\\bf{r}}}}}}}}) \\\\ \t+\\int\\,d{{{{{{{\\bf{r}}}}}}}}d{{{{{{{{\\bf{r}}}}}}}}}^{{\\prime} }{\\hat{\\tilde{Q}}}^{{{{\\dagger}}} }({{{{{{{\\bf{r}}}}}}}}){\\hat{Q}}^{{{{\\dagger}}} }({{{{{{{{\\bf{r}}}}}}}}}^{{\\prime} })U({{{{{{{\\bf{r}}}}}}}}-{{{{{{{{\\bf{r}}}}}}}}}^{{\\prime} })\\hat{Q}({{{{{{{{\\bf{r}}}}}}}}}^{{\\prime} })\\hat{\\tilde{Q}}({{{{{{{\\bf{r}}}}}}}}),$$ (1) where \\({\\hat{P}}_{\\pm }({{{{{{{\\bf{r}}}}}}}})\\) are the ULP and LLP field operators, respectively, \\(\\hat{{{{{{{{\\bf{p}}}}}}}}}=-i\\hslash \\nabla,{\\varepsilon }_{\\pm }({{{{{{{\\bf{p}}}}}}}})={E}_{\\pm }^{{{{{{{{\\rm{LP}}}}}}}}}({{{{{{{\\bf{p}}}}}}}})-{{{{{{{\\rm{Re}}}}}}}}\\,{E}_{-}^{{{{{{{{\\rm{LP}}}}}}}}}(0)\\) are the corresponding bare particle dispersions (the shape of \\({E}_{\\pm }^{{{{{{{{\\rm{LP}}}}}}}}}\\,({{{{{{{\\bf{p}}}}}}}})\\) is given in the SI) counted from the saddle point of the LLP branch, μ ± is the chemical potential, \\(\\hat{Q}({{{{{{{\\bf{r}}}}}}}})=\\int\\,d{{{{{{{{\\bf{r}}}}}}}}}^{{\\prime} }[X({{{{{{{{\\bf{r}}}}}}}}}^{{\\prime} }-{{{{{{{\\bf{r}}}}}}}}){\\hat{P}}_{-}({{{{{{{{\\bf{r}}}}}}}}}^{{\\prime} })+C({{{{{{{{\\bf{r}}}}}}}}}^{{\\prime} }-{{{{{{{\\bf{r}}}}}}}}){\\hat{P}}_{+}({{{{{{{{\\bf{r}}}}}}}}}^{{\\prime} })]\\) is the exciton field operator, and \\(\\hat{\\tilde{Q}}({{{{{{{\\bf{r}}}}}}}})\\) describes the field of background excitons that do not directly convert into polaritons (e.g. dark excitons). In the above, \\(X({{{{{{{{\\bf{r}}}}}}}}}^{{\\prime} }-{{{{{{{\\bf{r}}}}}}}})=(1/S){\\sum }_{{{{{{{{\\bf{p}}}}}}}}}{X}_{{{{{{{{\\bf{p}}}}}}}}}\\exp \\{i{{{{{{{\\bf{p}}}}}}}}\\cdot ({{{{{{{{\\bf{r}}}}}}}}}^{{\\prime} }-{{{{{{{\\bf{r}}}}}}}})/\\hslash \\},C({{{{{{{{\\bf{r}}}}}}}}}^{{\\prime} }-{{{{{{{\\bf{r}}}}}}}})=(1/S){\\sum }_{{{{{{{{\\bf{p}}}}}}}}}{C}_{{{{{{{{\\bf{p}}}}}}}}}\\exp \\{i{{{{{{{\\bf{p}}}}}}}}\\cdot ({{{{{{{{\\bf{r}}}}}}}}}^{{\\prime} }-{{{{{{{\\bf{r}}}}}}}})/\\hslash \\},{X}_{{{{{{{{\\bf{p}}}}}}}}}\\) and C p are the exciton and photon Hopfield coefficients, S is the quantization area. For simplicity, the exciton-exciton pair interaction potential is assumed to be contact: U ( r ) = g δ ( r ). The Hamiltonian ( 1 ) allows to write the Heisenberg equations for the evolution of both lower polariton fields \\({\\hat{P}}_{\\pm }({{{{{{{\\bf{r}}}}}}}},t)\\) . To obtain the excitation spectrum on top of the Bose condensate of BIC polaritons, we will assume that above threshold the LLP is macroscopically populated, which is supported by the experimental observation (see Fig. 1 e). Since the polaritons occupying the ULP branch do not convert into LLP polaritons because of the different symmetry of the underlying photon and exciton modes, we exclude the ULP operators from consideration when describing the macroscopically occupied saddle point. In this case, one can separate the condensate in the regular Bogoliubov fashion and, by averaging the Heisenberg equation for \\({\\hat{P}}_{-}\\) in the Hartree-Fock mean-field approximation (see “Methods” section), obtain the expression for the chemical potential of the LLP polaritons: $${\\mu }_{-}=g|{X}_{0}{|}^{2}\\left({n}_{0}|{X}_{0}{|}^{2}+2{n}_{Q}^{{\\prime} }+\\tilde{n}\\right),$$ (2) where n 0 is the condensate density, \\({n}_{Q}^{{\\prime} }\\equiv \\langle {\\hat{Q}}^{{\\prime} {{{\\dagger}}} }({{{{{{{\\bf{r}}}}}}}}){\\hat{Q}}^{{\\prime} }({{{{{{{\\bf{r}}}}}}}})\\rangle\\) is the non-condensate exciton density due to finite temperature, and \\(\\tilde{n}\\equiv \\langle {\\hat{\\tilde{Q}}}^{{{{\\dagger}}} }({{{{{{{\\bf{r}}}}}}}})\\hat{\\tilde{Q}}({{{{{{{\\bf{r}}}}}}}})\\rangle\\) is the background reservoir density. We note that μ − contains the thermal contribution \\({n}_{Q}^{{\\prime} }\\) and the dark reservoir contribution \\(\\tilde{n}\\) compared to the regularly used expression for the lower-polariton chemical potential μ LP = g n 0 ∣ X 0 ∣ 4 . Importantly, Eq. ( 2 ) corresponds to the experimentally-observed condensate blueshift, which will be used together with the excitation spectrum below to define n 0 and \\(\\tilde{n}\\) at a given pump power (note that at low temperatures considered here, \\({n}_{Q}^{{\\prime} }\\, \\ll \\,{n}_{0}\\) ). Diagonalizing the Hamiltonian ( 1 ) within the Hartree-Fock-Bogoliubov framework, one obtains the excitation spectrum: $${E}_{{{{{{{{\\bf{p}}}}}}}}}=\\sqrt{{{{{{{{\\mathcal{E}}}}}}}}({{{{{{{\\mathcal{E}}}}}}}}+2g{n}_{0}|{X}_{0}{|}^{2}|{X}_{{{{{{{{\\bf{p}}}}}}}}}{|}^{2})}\\quad \\,{{\\mbox{with}}}\\,\\quad {{{{{{{\\mathcal{E}}}}}}}}={\\varepsilon }_{-}({{{{{{{\\bf{p}}}}}}}})+g(|{X}_{{{{{{{{\\bf{p}}}}}}}}}{|}^{2}-|{X}_{0}{|}^{2})({n}_{0}|{X}_{0}{|}^{2}+2{n}_{Q}^{{\\prime} }+\\tilde{n}).$$ (3) It needs to be underlined that while Eq. ( 3 ) has the form looking similar to the textbook Bogoliubov prediction, it is, in fact, substantially different. Besides noticing that \\({{{{{{{\\mathcal{E}}}}}}}}\\) contains the reservoir contributions \\({n}_{Q}^{{\\prime} },\\tilde{n}\\) and the renormalization due to the dependence of the Hopfield coefficient X p on momentum, the bare polariton dispersion ε − ( p ) itself is very unusual: in the k x –direction it features, at the same time, the negative effective mass and momentum-dependent photon losses (see Supplementary Fig. 1 and Eq. ( 8) in the SI). Due to the latter, despite the negatively-defined \\({{{{{{{\\rm{Re}}}}}}}}{E}_{{{{{{{{\\bf{p}}}}}}}}}^{2}\\) in the vicinity of k x = 0, the real part of Eq. ( 3 ) is nowhere zero except the special points of the dispersion where \\({{{{{{{\\rm{Im}}}}}}}}{E}_{{{{{{{{\\bf{p}}}}}}}}}^{2}\\) changes sign. The excitation spectrum does not acquire a diffusive character predicted for systems with dissipation [26] , [27] . In fact, here the non-zero imaginary part of ε − ( p ) ensures the existence of low-lying states just above the condensate with \\({{{{{{{\\rm{Re}}}}}}}}{E}_{{{{{{{{\\bf{p}}}}}}}}} > 0\\) . The analysis of the imaginary part of the spectrum is given below. Figure 2a shows the real part of the Bogoliubov dispersion of excitations (Eq. 3 ) on top of the condensate formed in the saddle point of the LLP branch. The linearization at small momenta in the y –direction is clearly seen. Figure 2 b, c show the cuts of \\({{{{{{{\\rm{Re}}}}}}}}{E}_{{{{{{{{\\bf{p}}}}}}}}}\\) in Fig. 2 a along different near-zero k x (b) and k y = 0 (c), displaying the flattening of the dispersion in the k y –direction. As can be anticipated from ( 3 ), the extent of the energy-flat part along k y depends on the chosen near-zero k x , while the slope of the linear tails of the excitation spectrum at larger k y (for each k x ) is unambiguously defined by the polariton condensate density n 0 . Moreover, the zoom-in on the small momenta region displays a local minimum in the k x –direction (see the inset of Fig. 2 c). At the values of k x outside this local minimum, the Bogoliubov spectrum recovers the negative slopes characteristic to the single-particle LLP dispersion. Fig. 2: Dispersion of elementary excitations of the saddle-point polariton condensate. a The Bogoliubov dispersion of elementary excitations (real part of ( 3 )) with added blueshift of the saddle point. b Cross-cut of a at different k x from 0.005 to 0.045 μm −1 with the step 0.01 μm −1 , showing the flattening of the dispersion along the k y –direction close to the BIC state at k x = 0, calculated for condensate density n 0 = 4 × 10 10 cm −2 , reservoir density \\(\\tilde{n}=3\\times 1{0}^{10}\\) cm −2 . The gray dashed lines show the bare single-particle dispersions ε ± ( k x = 0, k y ). c Cross-cut of a at k y = 0, showing the real part of the Bogoliubov excitations spectrum E p given by ( 3 ). The gray dashed lines show the bare single-particle dispersions ε ± ( k x , k y = 0). Inset shows a close-up of the local minimum of the excitations spectrum, for varying n 0 from 1 to 6 × 10 10 cm −2 . Parameters as on Fig. 1 , g = 2.5 μeVμm 2 , T = 10 K. d , e Experimental time-resolved photoluminescence images above threshold displaying the energy-momentum dispersion of excitations vs. k y ; the excitation spectrum is evident in both the flat part and the linearized tails. f Experimental PL image of the dispersion of excitations vs. k x . In d – f the red lines show theoretical fitting of the Bogoliubov dispersion according to the real part of ( 3 ), with n 0 = 4 × 10 10 cm \\({}^{-2},\\tilde{n}=3\\times 1{0}^{10}\\) cm −2 ( d ) and n 0 = 5 × 10 10 cm \\({}^{-2},\\tilde{n}=4\\times 1{0}^{10}\\) cm −2 ( e , f ). The light-blue dashed lines show the single-particle LLP dispersion. The gray dotted lines in d , e mark the blueshift at different powers. The colorscales are in arb. units and normalized separately for d , e and f . Full size image Excitations luminescence dynamics in the pulsed regime The experimental PL images of collective excitations obtained in the regime of pulsed excitation are shown in Fig. 2 d–f. While the features very close to k x ~ 0 are below the experimental resolution and correspond to the dark region due to the long BIC lifetime, the dispersion along k y shows a striking correspondence to the theory derived above. To perform a better comparison, we have measured the temporal dynamics of the dispersion of excitations in PL along k y . This allows us to rule out any temporal smearing of the dispersion and, at the same time, track the change of the dispersion for different condensate and reservoir densities. Figure 2 d-e display two different snapshots of the emission along k y for k x ~ 0.02 μm −1 . These snapshots corresponding to different times clearly exhibit an energy-flat region with less-pronounced linear tails at higher values of k y . Thanks to the intrinsic dark nature of the condensate from a BIC state, it is possible to directly measure these features without any filtering in the momentum space. The red solid lines in Fig. 2 d, e show the analytical cross-cuts of the dispersion (Eq. 3 ) along k x ~ 0.02 μm −1 , for two values of densities calculated from the fitting of the measured linear tails together with the blueshift (as compared to the single-particle dispersion shown by the light-blue dashed lines). Full temporal dynamics of the PL vs. k y is provided in the Supplementary Movie 1 , while additional snapshots corresponding to different densities are shown in the SI. Experimentally following this dynamics allows to observe in time the delayed formation of the dark condensate after the pulse arrival, accompanied by the theoretically predicted energy-flat parts of the excitation spectrum, its shifting down with time due to the decreasing blueshift and, finally, disappearance of the flat parts and recovering of the single-particle near-parabolic LLP dispersion as the system goes below threshold. Finally, Fig. 2 f shows the energy-resolved PL image along the k x –direction above threshold. While the BIC polariton condensate in the local minimum of the modified dispersion stays dark, the excitations on top of the condensate occupying the very narrow region around k x = 0 are emitting light. Here, since under pulsed non-resonant excitation the blueshift is strongly time dependent, the resulting time-integrated measurement in Fig. 2 f looks apparently broadened in energy (see the corresponding enlarged curves for different densities in the inset of Fig. 2 c, getting smeared over time) which appears as a two-lobe image. Dispersion of collective excitations at different k x under continuous-wave excitation In order to see a more pronounced linearisation of the dispersion tails, we turn to continuous-wave (c.w.) excitation that allows to reach a density state while avoiding the energy blur induced by the dynamical change of energy of the condensate under time-integrated measurements. For fixed excitation conditions, we compare the PL from energy dispersion versus k y resolved at different values of k x as shown in Fig. 3 : one cut closer to k x = 0 (a), another directly within the bright spot with maximal emission that is seen above threshold in Fig. 2 f (b), and a final one when k x is further away from the center (c). As one can observe, while the PL from the non-condensate (out-of-the-BIC) lower polaritons in panels (a) and (c) is weak enough to allow registering the emission from the upper branch, in panel (b) the PL from the flat part of the dispersion is very bright, making the ULP invisible on the same (normalized) scale. Matching the observed PL with the theoretical expressions for the dispersion of excitations Eq. ( 3 ) and blueshift Eq. ( 2 ) for all three cases and using the condensate and reservoir densities as fitting parameters allows us to define exactly the values of k x at which those cuts were measured. Fig. 3: Continuous-wave excitation study. a – c Experimentally measured (left-hand sides) and theoretically calculated (right-hand sides) photoluminescence from the system vs. k y at three different values of k x above threshold: a k x = 0.019 μm −1 , b k x = 0.0228 μm −1 , c k x = 0.16 μm −1 . The overlaid red dotted lines represent the real part of the excitation spectrum \\({{{{{{{\\rm{Re}}}}}}}}{E}_{{{{{{{{\\bf{p}}}}}}}}}({k}_{y})\\) at corresponding values of k x , white dotted lines in a , c show the ULP and LLP single-particle dispersions. In b , the blueshifted lower polariton dispersion is shown for comparison. Intensity colorscales are in arb. units and normalized for each panel separately. d Real part (dashed line), counted from the blueshifted saddle point of the dispersion, and imaginary part (solid line) of E p vs. k x in the vicinity of k x ~ 0. The red arrows indicate the values of k x at which the panels a and b are measured. e Imaginary part of the spectrum vs. k y corresponding to panels a (yellow), b (blue), and c (green). For all panels, calculation parameters are the same as in Fig. 2 and densities estimated from fitting are n 0 = 5 × 10 10 cm \\({}^{-2},\\tilde{n}=11\\times 1{0}^{10}\\) cm −2 . Full size image The first straightforward outcome of this study is the clearly increased slope of the linear part of the Bogoliubov dispersion for the states lying above the flat stripe, as shown in Fig. 3 a, b. While the difference in energy at which the energy-flat part is seen in (a) and (b) lies within the linewidth (see the red arrows in Fig. 3 d or the inset of Fig. 2 c), the length of the stripe is changing with k x and is different for (a) and (b), in agreement with the theoretical Fig. 2 b. The tails of the dispersion outside the flat part in both (a) and (b) are clearly more visible than in the case of pulsed excitation studied above, and are linearized in accordance with Eq. ( 3 ). The shifted LLP dispersion is plotted in panel (b) together with the Bogoliubov spectrum. When moving away from the condensate, the dispersion of excitations recovers the shape of the lower polariton branch, while still staying blueshifted, as shown in Fig. 3 c. For comparison, in the right-hand sides of panels Fig. 3 a–c we plot the theoretically calculated PL emission, using the excitation spectrum derived above (note that since we neglected the upper polaritons in the theory, we do not reproduce the ULP luminescence). In this calculation, we assume that at finite temperature the normal branch of the Bogoliubov spectrum is predominantly populated (due to thermal depletion of the condensate). As a more subtle and more meaningful insight from fine resolving the PL in k x close to the BIC, we reveal that the maximal emission (panel b), i.e. the flat line in the Energy vs. k y dispersion, corresponds to the point in k x where the real part of the excitation spectrum E (counted from the blueshifted saddle point) goes to zero. This analysis is presented in Fig. 3 d, where we indicate by arrows the two values of k x at which panels (a) and (b) were measured ( k x corresponding to panel (c) lies out of the range of values plotted in (d) and is not marked). As discussed above, the only values of k x where \\({{{{{{{\\rm{Re}}}}}}}}{E}_{{{{{{{{\\bf{p}}}}}}}}}=0\\) correspond to the points in which the imaginary part of \\({E}_{{{{{{{{\\bf{p}}}}}}}}}^{2}\\) changes sign, i.e. the points that lie at the same energy as the dark condensate in the BIC, thus making the scattering from the condensate towards these states energetically effortless. Looking at the imaginary part of the dispersion, plotted as well in Fig. 3 d, allows us to conclude that since \\({{{{{{{\\rm{Im}}}}}}}}{E}_{{{{{{{{\\bf{p}}}}}}}}}\\) is negative for all k x > 0, the condensate with \\({{{{{{{\\rm{Im}}}}}}}}{E}_{{{{{{{{\\bf{p}}}}}}}}}=0\\) stays nevertheless stable and there is no macroscopic gain in the states indicated by the arrow ‘b’. However, since they correspond to the largest \\(|{{{{{{{\\rm{Im}}}}}}}}{E}_{{{{{{{{\\bf{p}}}}}}}}}|\\) , these states correspond to the maximum PL intensity observed in the experiments. We conclude that the particles residing in the dark condensate at k x = 0 at any scattering event may jump to the states ‘b’ where they leak out of the system due to maximal loss. For completeness of the analysis, we plot \\({{{{{{{\\rm{Im}}}}}}}}{E}_{{{{{{{{\\bf{p}}}}}}}}}\\) vs. k y in Fig. 3 e. One can see that the large imaginary part and hence short lifetimes are characteristic only for the states along the energy-flat part of the dispersion E p ( k y ), whereas the linear tails, corresponding to low-lying excitations above the saddle-point condensate, feature narrow linewidth and can be described adequately within the equilibrium theory. Anisotropy in the momentum space To study the anisotropic dispersion of excitations discussed above, we measure the photoluminescence emission under pulsed excitation in the plane ( k x , k y ) which is shown in Fig. 4 for different energies close to the BIC state. For a clearer visualization, we suggest to watch the Supplementary Movie 2 showing the experimental cross-cut of the PL emission in the far field at different energies. The exotic shape with two parallel stripes in the middle at a given energy can be directly related to the populated states of the excitation spectrum where \\({{{{{{{\\rm{Re}}}}}}}}{E}_{{{{{{{{\\bf{p}}}}}}}}}=0\\) and the absolute value of the negative \\({{{{{{{\\rm{Im}}}}}}}}{E}_{{{{{{{{\\bf{p}}}}}}}}}\\) is large, as discussed above and confirmed by the calculated \\({{{{{{{\\rm{Im}}}}}}}}{E}_{{{{{{{{\\bf{p}}}}}}}}}\\) dependence on k x and k y in Fig. 3 d, e. It is important to note that the distribution of the PL in two parallel stripes appears only above threshold: without the macroscopic population of the saddle-point, there are no states lying at the same energy with the condensate and hence no population of the energy-flat states along k y that exhibit strong emission. For comparison, we provide in the SI and the Supplementary Movie 3 the corresponding data taken below threshold, where the energy corresponding to the saddle point stays fully dark. In the above-threshold case of Fig. 4 , due to time averaging in the pulsed excitation setting, the two-stripe pattern which lies at the energy of the condensate is visible in more than one panel, as the condensate shifts down over time (due to decreasing blueshift). We note that the observed peculiar far field PL distribution in this case is very different from the two-lobe pattern of grating-based lasers [8] , [9] , where, despite the similar dark line along k x = 0, the lasing happens exclusively at the photonic BIC energy and the intensity distribution along k y is a Gaussian centered at k y = 0. In our case, the PL emission along both directions and in energy follows the calculated dispersion Eq. ( 3 ), with the length of the parallel lines shorter when the condensate density is lower, and the curved tails at nonzero momenta appearing outside the two-stripe region (whose emission is weaker since \\(|{{{{{{{\\rm{Im}}}}}}}}{E}_{{{{{{{{\\bf{p}}}}}}}}}|\\) is much smaller). In all panels of Fig. 4 , the theoretically calculated cross-cuts of the real part of E p at the corresponding energies above the BIC state are overlayed as dashed lines on top of the experimental images. The maximum of the experimental emission appears at those states where the real part of the dispersion E p crosses the given energy (the brighter spots move away from the center in the panels (c) and (d) that correspond to higher energies). Fig. 4: Momentum-space scan. a – d Experimental PL maps in the ( k x , k y ) domain taken at different energies just above the blueshifted BIC polariton condensate (as denoted on the panels). The line k x = 0 above threshold stays fully dark, making the PL profile in the vicinity of the condensate look like two parallel stripes corresponding to the flat parts of \\({{{{{{{\\rm{Re}}}}}}}}{E}_{{{{{{{{\\bf{p}}}}}}}}}({k}_{y})\\) appearing at small non-zero k x . These maps are time-integrated in the pulsed excitation setting. Intensity colorscale is in arb. units and normalized the same way for all panels. Condensate and reservoir densities n 0 = 5 × 10 10 cm \\({}^{-2},\\tilde{n}=4\\times 1{0}^{10}\\) cm −2 , other parameters as in Fig. 2 . Full size image From the point of view of the anisotropy of collective excitations which is illustrated by the above study of the spectrum, it is also interesting to address the limit of very small momenta. Experimentally, since the region of k x corresponding to the positive slope of \\({{{{{{{\\rm{Re}}}}}}}}{E}_{{{{{{{{\\bf{p}}}}}}}}}\\) is very small, the shift in energy which is acquired within this narrow Δ k x lies within the linewidth. At the same time, it is not possible to follow the change in energy with k y along the positive slope of the saddle (from Fig. 2 a one sees that at k x → 0 the energy-flat part along k y disappears and the dispersion is linear) due to the darkness of the states corresponding to k x = 0 (at any k y ). Theoretically, however, taking the limit p → 0 in Eq. ( 3 ) allows to obtain the sound velocity which appears to be strongly dependent on direction (on the angle φ in the polar coordinate system): $${c}_{{{{{{{{\\rm{s}}}}}}}}}(\\varphi )={{{{{{{\\rm{Re}}}}}}}}\\left\\{\\sqrt{{\\mu }_{{{{{{{{\\rm{LP}}}}}}}}}\\left[{\\cos }^{2}\\varphi \\left(\\frac{1}{{m}_{{{{{{{{\\rm{LP}}}}}}}}}^{x}}+\\frac{i}{{s}_{x}}\\right)+\\frac{{\\sin }^{2}\\varphi }{{m}_{{{{{{{{\\rm{LP}}}}}}}}}^{y}}\\right]\\left[1+\\frac{2\\mu }{\\sqrt{{(\\hslash \\Omega )}^{2}+{\\Delta }^{2}}}\\right]}\\right\\},$$ (4) where \\({m}_{{{{{{{{\\rm{LP}}}}}}}}}^{x(y)}\\) are the effective masses of LLP in the x ( y )–direction, and s x is the mass-like parameter defined by the radiative losses rate γ and the radiative coupling of the modes U (for more details, see the SI) . In a standard case of real positive masses in both directions, for experimentally-relevant parameters (resulting in \\(1/|{m}_{{{{{{{{\\rm{LP}}}}}}}}}^{x}|\\gg 1/{m}_{{{{{{{{\\rm{LP}}}}}}}}}^{y}\\) ) the velocity given by Eq. ( 4 ) would significantly increase when approaching the x –axis (the direction corresponding to \\(\\varphi=\\pi m,m\\in {\\mathbb{Z}}\\) ). However, here the expression for the sound velocity is modified by the negative \\({m}_{{{{{{{{\\rm{LP}}}}}}}}}^{x}\\) and by the imaginary contribution s x , which makes the definition of the sound velocity more complicated. In Fig. 5 b we plot the sound velocity dependence on the angle φ revealing a very anisotropic profile. The increase of c s in the x –direction is clearly seen from the difference of slopes in the linearized parts of the Bogoliubov dispersion (see the zoom-in at the low-momenta region in Fig. 5 a): while the slope along k y at k x = 0 corresponds to very low \\({c}_{{{{{{{{\\rm{s}}}}}}}}}^{y}\\) , the shallow parts of the spectrum at finite but small k x result in the sound velocity in the directions slightly deviating from the y –axis being very close to zero (see Fig. 5 c). Much steeper slopes along k x at k y = 0 provide maxima of c s at φ = π m , as shown in Fig. 5 b for various condensate densities. It is important to note that taking into account the non-radiative exciton losses would additionally affect the behavior of the excitation spectrum along k x in a very narrow vicinity of the saddle point. A more detailed discussion of this matter is provided in the SI . Fig. 5: Anisotropic Bogoliubov sound velocity. a A close-up view of the local potential well created in the real part of the Bogoliubov spectrum of elementary excitations around the saddle point, with essentially different slopes \\({c}_{{{{{{{{\\rm{s}}}}}}}}}^{x}\\gg {c}_{{{{{{{{\\rm{s}}}}}}}}}^{y}\\) (where \\({c}_{{{{{{{{\\rm{s}}}}}}}}}^{x}={c}_{{{{{{{{\\rm{s}}}}}}}}}(\\varphi=\\pi m),{c}_{{{{{{{{\\rm{s}}}}}}}}}^{y}={c}_{{{{{{{{\\rm{s}}}}}}}}}(\\varphi=\\pi [m+1]/2),m\\in {\\mathbb{Z}}\\) ). The colored dashed lines are guide to the eye showing different directions along the anisotropic dispersion slopes linearized in the vicinity of k = 0. b The Bogoliubov sound velocity profile according to Eq. ( 4 ). The colored dashed lines mark different angles corresponding to the slopes marked in a , see discussion in the text. c Close-up of the sound velocity profile at the vicinity of π /2 (corresponding to the black rectangle in b ). The condensate density n 0 in b , c is varied from 1 to 6 × 10 10 cm \\({}^{-2},\\tilde{n}=4\\times 1{0}^{10}\\) cm −2 , other parameters as in Fig. 2 . Full size image"},{"section":"Discussion","content":"In this work, we studied theoretically and experimentally the spectrum of elementary excitations of the BIC polariton condensate arising from the coupling of excitons to the photonic modes in a patterned semiconductor waveguide. The studied system is exotic and very different from the well-studied case of microcavity polaritons. We show that despite the saddle-like shape of the single-particle dispersion of lower polaritons, which exhibits a maximum in one of the directions in k –space, with the accumulation of particles in the saddle point, a local minimum is created with the positive slopes of the Bogoliubov dispersion, which may help in further accumulation of polaritons. At the same time, two parallel stripes in momentum space, which correspond to the states at the same energy as the condensate—but being bright—appear. It is, in fact, the interplay of the negative mass and the mometum-dependent losses that results in the existence of such unique, extended in momentum, states outside the condensate. This unusual anisotropic shape of the excitation spectrum yields in the k y –direction to two energy-flat zones (near k x = 0) separated by the dark line at k x = 0. Thanks to the quasi-infinite lifetime of the BIC state, the condensate itself does not emit light hence allowing to neatly observe the populated states around and above the saddle point. With both the slopes of the linear parts of the dispersion that follow the energy-flat regions and the interaction-induced blueshift being directly experimentally accessible, our theory allows to precisely define the macroscopic density of the dark polariton condensate. Thus, while the observation of the excitation spectrum in such a system is striking on its own, it also allows to obtain the information about the saddle-point polariton condensate which is otherwise invisible. Remarkably, despite the negative mass in the x –direction and the presence of losses, the real part of the spectrum is positive everywhere at k → 0, allowing to extract the velocity of sound in every direction. The latter appears to be highly anisotropic, dropping to almost zero in the directions next to the y -axis, i.e. slightly misaligned with the grating principal axis, and growing in the direction of propagation along the waveguide. It needs to be noted that the anisotropic sound velocity was reported previously for atomic BECs with anisotropic dipolar interactions [31] , [32] . In case of dipolar BECs, however, transport measurements reveal the anisotropy of the critical Landau velocity rather than the velocity of sound [33] , since the dispersion of excitations is affected by direction-dependent roton softening. The situation realized in our work can be better compared with anisotropic superfluidity reported for excitons [34] and cavity photon BEC [35] in periodically modulated planar structures. At the same time, we stress that here we deal with a more exotic case of anisotropy: even though there is a well-defined finite sound velocity, superfluidity in the sense of Landau criterion cannot be reached in all directions of propagation. Indeed, no matter how small the velocity would be below c s in the x –direction, due to the negative slopes of the dispersion, it could always elastically create an excitation out of the condensate. However if an obstacle is propagating strictly along the y –axis, the velocity \\({c}_{{{{{{{{\\rm{s}}}}}}}}}^{y}\\) would be a proper Landau sound velocity below which superfluidity can be observed. This illustrates the richness of saddle-point condensates and their anisotropic behaviors. Along with the previously proposed topological-dispersion engineering [36] , we believe that this work demonstrates the high interest in engineering the excitation spectrum of the condensate e.g. via implementing different grating symmetries—as a tool to impart new properties to the condensate itself—and underlines once again the astonishing richness of polariton systems, as well as multiple possible avenues for further investigations."},{"section":"Methods","content":"Theoretical methods Starting from the Hamiltonian ( 1 ) in the neglection of the ULP fields, one can separate the macroscopically occupied condensate state as $${\\hat{P}}_{-}({{{{{{{\\bf{r}}}}}}}},t)=\\sqrt{{n}_{0}}+{\\hat{P}}_{-}^{{\\prime} }({{{{{{{\\bf{r}}}}}}}},t),\\quad \\hat{Q}({{{{{{{\\bf{r}}}}}}}},t)={X}_{0}\\sqrt{{n}_{0}}+{\\hat{Q}}^{{\\prime} }({{{{{{{\\bf{r}}}}}}}},t),$$ (5) where n 0 is the condensate density, \\({\\hat{Q}}^{{\\prime} }({{{{{{{\\bf{r}}}}}}}},t)=\\int\\,X({{{{{{{{\\bf{r}}}}}}}}}^{{\\prime} }-{{{{{{{\\bf{r}}}}}}}}){\\hat{P}}_{-}^{{\\prime} }({{{{{{{{\\bf{r}}}}}}}}}^{{\\prime} },t)d{{{{{{{{\\bf{r}}}}}}}}}^{{\\prime} }\\) , and the average \\(\\langle {\\hat{P}}_{-}^{{\\prime} }({{{{{{{\\bf{r}}}}}}}},t)\\rangle=\\langle {\\hat{Q}}^{{\\prime} }({{{{{{{\\bf{r}}}}}}}},t)\\rangle=0\\) . The substitution Eq. ( 5 ) allows to rewrite the triple products of the field operators in Eq. ( 1 ) in the Hartree-Fock mean-field approximation as $${\\hat{Q}}^{{{{\\dagger}}} }({{{{{{{\\bf{r}}}}}}}},t)\\hat{Q}({{{{{{{\\bf{r}}}}}}}},t)\\hat{Q}({{{{{{{\\bf{r}}}}}}}},t)=\t{X}_{0}\\sqrt{{n}_{0}}(|{X}_{0}{|}^{2}{n}_{0}+2{n}_{Q}^{{\\prime} }) \\\\ \t+2{\\hat{Q}}^{{\\prime} }({{{{{{{\\bf{r}}}}}}}},t)(|{X}_{0}{|}^{2}{n}_{0}+{n}_{Q}^{{\\prime} })+{\\hat{Q}}^{{\\prime} {{{\\dagger}}} }({{{{{{{\\bf{r}}}}}}}},t){X}_{0}^{2}{n}_{0},$$ (6) $${\\hat{\\tilde{Q}}}^{{{{\\dagger}}} }({{{{{{{\\bf{r}}}}}}}},t)\\hat{\\tilde{Q}}({{{{{{{\\bf{r}}}}}}}},t)\\hat{Q}({{{{{{{\\bf{r}}}}}}}},t)=\\left[{X}_{0}\\sqrt{{n}_{0}}+{\\hat{Q}}^{{\\prime} }({{{{{{{\\bf{r}}}}}}}},t)\\right]\\tilde{n}$$ (7) with notations \\({n}_{Q}^{{\\prime} }=\\langle {\\hat{Q}}^{{\\prime} {{{\\dagger}}} }({{{{{{{\\bf{r}}}}}}}},t){\\hat{Q}}^{{\\prime} }({{{{{{{\\bf{r}}}}}}}},t)\\rangle,\\tilde{n}=\\langle {\\hat{\\tilde{Q}}}^{{{{\\dagger}}} }({{{{{{{\\bf{r}}}}}}}},t)\\hat{\\tilde{Q}}({{{{{{{\\bf{r}}}}}}}},t)\\rangle\\) . Using the Hamiltonian Eq. ( 1 ) with the substitutions Eqs. ( 5 ) and ( 6 ), ( 7 ), the Heisenberg equation for the non-condensed part of the polariton field \\({\\hat{P}}_{-}^{{\\prime} }({{{{{{{\\bf{r}}}}}}}},t)\\) can be written as $$i\\hslash {\\partial }_{t}{\\hat{P}}_{-}^{{\\prime} }({{{{{{{\\bf{r}}}}}}}},t)=\t\\left[{\\varepsilon }_{-}(-i\\hslash \\nabla )-{\\mu }_{-}\\right]{\\hat{P}}_{-}^{{\\prime} }({{{{{{{\\bf{r}}}}}}}},t)\\\\ \t+g\\int\\,d{{{{{{{{\\bf{r}}}}}}}}}^{{\\prime} }{X}^{*}({{{{{{{{\\bf{r}}}}}}}}}^{{\\prime} }-{{{{{{{\\bf{r}}}}}}}})\\left[(2{n}_{Q}^{{\\prime} }+2{n}_{0}|{X}_{0}{|}^{2}+\\tilde{n}){\\hat{Q}}^{{\\prime} }({{{{{{{{\\bf{r}}}}}}}}}^{{\\prime} },t) \\right. \\\\ \t\\left.+{X}_{0}^{2}{n}_{0}{\\hat{Q}}^{{\\prime} {{{\\dagger}}} }({{{{{{{{\\bf{r}}}}}}}}}^{{\\prime} },t)\\right].$$ (8) In Fourier space, ( 8 ) takes the form \\(i\\hslash {\\partial }_{t}{\\hat{P}}_{{{{{{{{\\bf{p}}}}}}}}}(t)=[{\\varepsilon }_{-}({{{{{{{\\bf{p}}}}}}}})-{\\mu }_{-}+g(2{n}_{0}|{X}_{0}{|}^{2}+2{n}_{Q}^{{\\prime} }+\\tilde{n})]|{X}_{{{{{{{{\\bf{p}}}}}}}}}{|}^{2}{\\hat{P}}_{{{{{{{{\\bf{p}}}}}}}}}(t)+g{n}_{0}{X}_{0}^{2}{X}_{{{{{{{{\\bf{p}}}}}}}}}^{*2}{\\hat{P}}_{-{{{{{{{\\bf{p}}}}}}}}}^{{{{\\dagger}}} }(t)\\) that allows to perform the Bogoliubov transformation $${\\hat{P}}_{{{{{{{{\\bf{p}}}}}}}}}(t)={u}_{{{{{{{{\\bf{p}}}}}}}}}{\\hat{\\alpha }}_{{{{{{{{\\bf{p}}}}}}}}}{e}^{-i{E}_{{{{{{{{\\bf{p}}}}}}}}}t/\\hslash }-{v}_{-{{{{{{{\\bf{p}}}}}}}}}^{*}{\\hat{\\alpha }}_{-{{{{{{{\\bf{p}}}}}}}}}^{{{{\\dagger}}} }{e}^{i{E}_{{{{{{{{\\bf{p}}}}}}}}}^{*}t/\\hslash }\\,\\,\\,\\,{{\\mbox{with}}}\\,\\,\\,|{u}_{{{{{{{{\\bf{p}}}}}}}}}{|}^{2}-|{v}_{-{{{{{{{\\bf{p}}}}}}}}}{|}^{2}=1,$$ (9) where \\({\\hat{\\alpha }}_{{{{{{{{\\bf{p}}}}}}}}}\\) is the annihilation operator of the Bogoliubov excitation with momentum p . Eq. ( 9 ) yields the excitation spectrum Eq. ( 3 ). We note that due to the underlying photon dispersion whose imaginary part is strongly momentum-dependent, the Hopfield coefficients X p , C p as well as the Bogoliubov amplitudes u p , v − p are complex, and careful treatment of complex conjugations throughout the theory is required. Moreover, choosing the Bogoliubov transformation in the specific shape given by Eq. ( 9 ) defines the so-called ghost branch (GB) of excitations as \\({E}_{{{{{{{{\\bf{p}}}}}}}}}^{{{{{{{{\\rm{GB}}}}}}}}}=-{E}_{{{{{{{{\\bf{p}}}}}}}}}^{*}\\) (this definition of the GB is consistent with that used in ref. [27] ). Theoretical calculations of the intensity of photoluminescence from the excitations around the BIC state that are presented in Fig. 4 are performed using the standard expression [37] , [38] : $${{{{{{{\\rm{PL}}}}}}}}({k}_{x},{k}_{y};\\omega )\\propto \\left(1-{\\left|{X}_{{{{{{{{\\bf{p}}}}}}}}}\\right|}^{2}\\right){{{{{{{\\rm{Re}}}}}}}}\\int\\nolimits_{0}^{\\infty }dt\\,{e}^{-it(\\omega -i{0}^{+})} < {P}_{{{{{{{{\\bf{p}}}}}}}}}^{{{{\\dagger}}} }(t){P}_{{{{{{{{\\bf{p}}}}}}}}}(0) > .$$ (10) Experiments All the PL data reported in this work has been acquired using a confocal setup. The objective back focal plane is imaged on the entrance slit of the spectrometer allowing to obtain the energy versus k dispersion. Two different magnification of the objective focal plane are used in order to properly image the dispersions along k x and k y which have very different effective masses and hence very different extension in the k -space. The laser used in the experiments is a fs-pulsed laser having a repetition rate of 80 MHz and a pulse duration of ~100 fs. Excitation wavelength is set at 780 nm. The time resolved measurements are obtained by scanning the far-field emission and acquiring for each k y an energy versus time temporal trace with a streak camera. The set of temporal traces is then used to reconstruct the dynamics in the Energy vs k y space. Supplementary Movie 1 shows an example of temporal dynamics reconstructed in this way. The emission in the ( k x , k y )–plane at a given energy like the images shown in Fig. 4 are obtained by scanning the far-field emission and acquiring a set of Energy vs k x dispersions, each one corresponding to a different k y . The data are then merged and cut at a given energy in the plane k x , k y . Supplementary Movies 2 and 3 show a sequence of such cuts for different energies above and below threshold, respectively."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"Bound states in the continuum (BICs) have been originally proposed as a mathematical feature of the Schrödinger equation in presence of specially prepared potentials [1] , [2] , [3] . Similar states have been later found in a wide range of systems like graphene, topological insulators, atomic superlattices, dielectric photonic crystals, metasurfaces and patterned optical waveguides [4] . In the latter cases, coupling in both the real and the imaginary parts of the two counter-propagating electromagnetic modes leads to the opening of an energy gap, with one of the new eigenmodes being bright and the other exhibiting vanishing radiative losses despite the non-Hermiticity of the underlying Hamiltonian [5] , [6] . Symmetry-protected optical BICs occuring from two interfering resonances enable light confinement and are routinely used in grating surface-emitting and distributed feedback lasers [7] , [8] , [9] , where emission from the BIC mode manifests itself in a specific two-lobe far field pattern. Interestingly, these quasi-infinite-lifetime photonic states can couple to matter excitations, such as surface plasmons [6] or excitons in monolayer transition-metal dichalcogenides [10] , [11] and quantum wells [12] and form polaritons with interesting features, such as greatly enhanced tunable lifetimes and strong nonlinearities. Polaritons are hybrid half-light half-matter quasi-particles in semiconductor heterostructures, typically embedded in a Fabry-Pérot microcavity to enhance the coupling of electronic excitations (e.g. excitons) with photons and to reduce the losses [13] . Such low-dimensional bosonic systems have been shown to provide conditions for observation of collective behaviors like Bose-Einstein condensation [14] , [15] , [16] and superfluidity [17] , [18] , [19] , at the same time allowing for a direct measurement of their momentum and spatial distributions via the emitted light. The onset of Bose-Einstein condensation is accompanied with a narrowed enhanced emission from the ground state at k = 0 of the polariton energy dispersion, which consequently blueshifts due to particle-particle interactions. At the same time, the momentum-energy dispersion on top of the condensate becomes linearized in accordance with the textbook Bogoliubov prediction [20] , therefore giving origin to superfluidity as follows from the Landau criterion. Multiple experiments with cavity exciton-polaritons have provided evidence of such linearization [21] , [22] , [23] , [24] , [25] despite the driven-dissipative nature of the system suggesting the diffusive character of the Bogoliubov dispersion [26] , [27] with a zero real part in the region of small momenta. Precise observation of the shape of the excitations spectrum due to the thermal and quantum depletion of the macroscopically occupied ground state is also possible, but requires substantial momentum-space filtering covering a much brighter signal from the condensate [28] , [29] or using refined interferometric techniques [25] . In this work, we study the elementary excitation spectrum of a polariton Bose condensate arising from a BIC state in a planar nanostructured waveguide. It has been recently reported [12] that due to the band folding effect and the consequent coupling with the quantum well exciton mode, the condensate which is formed on the lowest polariton branch appears, counter-intuitively, at a saddle point of the dispersion rather than in a global energy minimum. Since the particles accumulate in the quasi-BIC state with a very long lifetime, much longer than the carriers relaxation time, the polariton condensate in such a state appears to be a paradigmatic system for studying the excitation spectrum of a condensate in thermal equilibrium [30] . In such state the condensate cannot directly radiate and, as we demonstrate in the following, measurements of the shape of the spectrum of elementary excitations are possible in great detail and even without any momentum filtering. We derive theoretically the finite-temperature Bogoliubov dispersion for polaritons accumulating in the BIC and show that a local energy minimum starts to appear around the saddle point k = 0 in the spectrum of excitations which, possibly, could help in the formation of a long-living Bose-Einstein condensate, with the real part of the spectrum being non-zero despite the negative effective mass in one of the directions and losses present in the system. We experimentally observe various and very distinct characteristics of collective excitations dispersion strongly dependent on the direction in k -space. The detailed knowledge of the Bogoliubov spectrum allows to investigate the angular profile of the sound velocity in such highly anisotropic system. We believe that this study could open a way of controlling the condensate properties by engineering its excitation spectrum.\",\n \"references\": {\n \"indices\": [\n 2,\n 3,\n 4,\n 5,\n 6,\n 7,\n 9,\n 10,\n 11,\n 12,\n 14,\n 15,\n 16,\n 17,\n 18,\n 19,\n 21,\n 22,\n 23,\n 24,\n 25,\n 26,\n 27,\n 28,\n 29,\n 30\n ],\n \"infos\": [\n {\n \"idx\": \"2\",\n \"title\": \"Friedrich, H. & Wintgen, D. Interfering resonances and bound states in the continuum.Phys. Rev. A32, 3231 (1985).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevA.32.3231\"\n },\n {\n \"idx\": \"3\",\n \"title\": \"Capasso, F. et al. Observation of an electronic bound state above a potential well.Nature358, 565 (1992).\",\n \"link\": \"https://doi.org/10.1038%2F358565a0\"\n },\n {\n \"idx\": \"4\",\n \"title\": \"Hsu, C. W., Zhen, B., Stone, A. D., Joannopoulos, J. D. & Soljačic, M. Bound states in the continuum.Nat. Rev. Mater.1, 16048 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fnatrevmats.2016.48\"\n },\n {\n \"idx\": \"5\",\n \"title\": \"Zhen, B. et al. Spawning rings of exceptional points out of Dirac cones.Nature525, 354 (2015).\",\n \"link\": \"https://doi.org/10.1038%2Fnature14889\"\n },\n {\n \"idx\": \"6\",\n \"title\": \"Lu, L. et al. Engineering a light-matter strong coupling regime in perovskite-based plasmonic metasurface: quasi-bound state in the continuum and exceptional points.Photon. Res.8, A91–A100 (2020).\",\n \"link\": \"https://doi.org/10.1364%2FPRJ.404743\"\n },\n {\n \"idx\": \"7\",\n \"title\": \"Noll, R. J. & Macomber, S. H. Analysis of grating surface emitting lasers.IEEE J. Quant. Electron.26, 456 (1990).\",\n \"link\": \"https://doi.org/10.1109%2F3.52121\"\n },\n {\n \"idx\": \"9\",\n \"title\": \"Wang, Y. et al. Highly controllable etchless perovskite microlasers based on bound states in the continuum.ACS Nano15, 7386–7391 (2021).\",\n \"link\": \"https://doi.org/10.1021%2Facsnano.1c00673\"\n },\n {\n \"idx\": \"10\",\n \"title\": \"Koshelev, K. L., Sychev, S. K., Sadrieva, Z. F., Bogdanov, A. A. & Iorsh, I. V. Strong coupling between excitons in transition metal dichalcogenides and optical bound states in the continuum.Phys. Rev. B98, 161113(R) (2018).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.98.161113\"\n },\n {\n \"idx\": \"11\",\n \"title\": \"Kravtsov, V. et al. Nonlinear polaritons in a monolayer semiconductor coupled to optical bound states in the continuum.Light. Sci. Appl.9, 56 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41377-020-0286-z\"\n },\n {\n \"idx\": \"12\",\n \"title\": \"Ardizzone, V. et al. Polariton Bose-Einstein condensate from a Bound State in the Continuum.Nature605, 447–452 (2022).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-022-04583-7\"\n },\n {\n \"idx\": \"14\",\n \"title\": \"Kasprzak, J. et al. Bose-Einstein condensation of exciton polaritons.Nature443, 409–414 (2006).\",\n \"link\": \"https://doi.org/10.1038%2Fnature05131\"\n },\n {\n \"idx\": \"15\",\n \"title\": \"Balili, R., Hartwell, V., Snoke, D., Pfeiffer, L. & West, K. Bose-Einstein condensation of microcavity polaritons in a trap.Science316, 1007–1010 (2007).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1140990\"\n },\n {\n \"idx\": \"16\",\n \"title\": \"Byrnes, T., Kim, N. Y. & Yamamoto, Y. Exciton-polariton condensates.Nat. Phys.10, 803–813 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys3143\"\n },\n {\n \"idx\": \"17\",\n \"title\": \"Cristofolini, P. et al. Optical superfluid phase transitions and trapping of polariton condensates.Phys. Rev. Lett.110, 186403 (2013).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.110.186403\"\n },\n {\n \"idx\": \"18\",\n \"title\": \"Amo, A. et al. Superfluidity of polaritons in semiconductor microcavities.Nat. Phys.5, 805–810 (2009).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys1364\"\n },\n {\n \"idx\": \"19\",\n \"title\": \"Lerario, G. et al. Room-temperature superfluidity in a polariton condensate.Nat. Phys.13, 837–841 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys4147\"\n },\n {\n \"idx\": \"21\",\n \"title\": \"Utsunomiya, S. et al. Observation of Bogoliubov excitations in exciton-polariton condensates.Nat. Phys.4, 700 (2008).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys1034\"\n },\n {\n \"idx\": \"22\",\n \"title\": \"Kohnle, V. et al. From single particle to superfluid excitations in a dissipative polariton gas.Phys. Rev. Lett.106, 255302 (2011).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.106.255302\"\n },\n {\n \"idx\": \"23\",\n \"title\": \"Pieczarka, M. et al. Ghost branch photoluminescence from a polariton fluid under nonresonant excitation.Phys. Rev. Lett.115, 186401 (2015).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.115.186401\"\n },\n {\n \"idx\": \"24\",\n \"title\": \"Stepanov, P. et al. Dispersion relation of the collective excitations in a resonantly driven polariton fluid.Nat. Commun.10, 3869 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-019-11886-3\"\n },\n {\n \"idx\": \"25\",\n \"title\": \"Ballarini, D. et al. Directional Goldstone waves in polariton condensates close to equilibrium.Nat. Commun.11, 217 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-019-13733-x\"\n },\n {\n \"idx\": \"26\",\n \"title\": \"Wouters, M. & Carusotto, I. Excitations in a nonequilibrium bose-einstein condensate of exciton polaritons.Phys. Rev. Lett.99, 140402 (2007).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.99.140402\"\n },\n {\n \"idx\": \"27\",\n \"title\": \"Byrnes, T., Horikiri, T., Ishida, N., Fraser, M. & Yamamoto, Y. Negative Bogoliubov dispersion in exciton-polariton condensates.Phys. Rev. B85, 075130 (2012).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.85.075130\"\n },\n {\n \"idx\": \"28\",\n \"title\": \"Pieczarka, M. et al. Observation of quantum depletion in a non-equilibrium exciton-polariton condensate.Nat. Commun.11, 429 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-019-14243-6\"\n },\n {\n \"idx\": \"29\",\n \"title\": \"Pieczarka, M. et al. Bogoliubov excitations of a polariton condensate in dynamical equilibrium with an incoherent reservoir.Phys. Rev. B105, 224515 (2022).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.105.224515\"\n },\n {\n \"idx\": \"30\",\n \"title\": \"Grudinina, A. M., Kurbakov, I. L., Lozovik, Y. E. & Voronova, N. S. Finite-temperature Hartree-Fock-Bogoliubov theory for exciton-polaritons.Phys. Rev. B104, 125301 (2021).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.104.125301\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Saddle-point polariton condensate The sample studied here, sketched in Fig. 1 a, is a GaAs/AlGaAs waveguide hosting 12 20-nm-thick GaAs quantum wells embedded in a Al 0.4 Ga 0.6 As core. A set of linear diffraction gratings is etched on the surface of the waveguide to allow for the observation, within the lightcone, of the propagating guided modes. By changing the grating period a and the filling factor it is possible to tune the energy of the BIC (i.e. its excitonic fraction) and the width of the energy gap separating the lower-polariton BIC, with ultra-long lifetimes, and the leaky polaritonic modes. The coupling of the counter-propagating TE-modes of electromagnetic waves to the exciton results in the four branches of polariton dispersion, the analytical derivation of which are provided in the Supplementary Information ( SI) . Of interest, in this work, are the two anisotropic lowest branches which are shown in Fig. 1 b. It is clearly seen that the shape of the upper of the two lower-polariton modes (ULP) resembles the regular lower cavity exciton-polariton, albeit with differing effective masses along and perpendicular to the grating, whereas the lower (LLP) mode is distinct from the standard dispersion showing different behaviors along k x – and k y –directions. In particular, we note the appearance of a negative effective mass in the x –direction, with an absolute value much lower than the positive mass in the y –direction. It is also important to note that, when expanded at k → 0 in Taylor series up to the second order, the LLP dispersion appears to have an imaginary contribution in the x –direction (for more details, see the SI) . This imaginary term provides the k x –dependent loss rate of polaritons on the lowest branch, with the zero radiative loss at k = 0, as was experimentally verified in ref. [12] . Fig. 1: Polariton dispersions around the BIC state. a Sketch of the etched waveguide structure. The waveguide is made of 12 GaAs/Al 0.4 Ga 0.6 As (20-nm-thick each) layer pairs on a substrate, with the top 90 nm textured with a periodic 1D grating with a period a = 243 nm. The grating sizes are 50 μm along y and 300 μm along x -direction. b Three-dimensional dispersions of the two lower exciton-polariton modes (see SI) , with the energy reference taken at the saddle point of the lowest mode, calculated for Δ 0 = − 0.6 meV, ℏ Ω R = 15 meV, U = − 1.3 meV, ℏ γ = 0.51 meV, n g = 4.3, m X = 0.22 m 0 . c , d Cross-cuts of the energy-momentum experimental dispersion along k y = 0 ( c ) and k x = 0.025 μm −1 ( d ) showing two orders of magnitude difference in the lower polariton effective mass along x and y directions, \\\\(1/{m}_{{{{{{{{\\\\rm{LP}}}}}}}}}^{x(y)}\\\\) (see text). The yellow dashed lines show theoretical fits of the two lower polariton branches with the parameters as in b . e , f Comparison of experimental far field emission below (on the left-hand side) and above condensation threshold (on the right-hand side) showing energy versus k x and k y respectively; panel f is a cut along k x = 0.02 μm −1 , as for panel d . The colorscales for intensity are linear, given in arb. units and normalized separately for panels c , d and e , f . Full size image The polariton dispersions along k x and k y can be directly measured by energy and momentum resolved photoluminescence (PL). Figure 1 c and d show two different cuts (along k x and k y respectively) of the PL of the single-particle lower polariton dispersions. While along k x (Fig. 1 c) the ULP and LLP have the masses of opposite sign, this is not the case along k y where the mass is positive for both (Fig. 1d) . The dispersion in Fig. 1 d has been taken at a slightly finite wavevector in the x –direction ( k x ~ 0.02 μm −1 ) to avoid the completely dark stripe of the BIC state at k x = 0. These experimental dispersions are in very good agreement with the analytical model which is shown by the overlaid yellow dashed lines in both panels. We note that the range of horizontal axis along k y is four times larger than that of k x and that the effective masses along the two directions differ by about two orders of magnitude, being in quantitative agreement with the theory developed in the SI. Both images in Fig. 1 c and d show the PL emission below threshold, when the upper of the two lower-polariton branches is emitting more light than the lowest one, as it is much more lossy around k = 0 (see SI) , while their populations are comparable. When increasing the excitation power, polaritons condense into the BIC at the saddle point of the LLP branch [12] . Figure 1 e (right-hand side) shows the cross-cut of the dispersion along k x with the time-integrated PL measured slightly above the condensation threshold. We only show one of the two bright spots forming to either side of the saddle-point condensate, blueshifted compared to the below-threshold image shown on the left-hand side of the panel. Figure 1 f (right-hand side) shows the corresponding dispersion along k y for k x ~ 0.02 μm −1 , i.e. close to the inner edge of the bright spot of Fig. 1 e above threshold. The latter has an interesting form, with an energy-flattened emission roughly extending from −0.5 μm −1 (not visible) to +0.5 μm −1 continuing into linear tails going up in energy. It is important to note that, as we will show in the following, the flat region in this case is not related to the diffusive character of the condensate. On the contrary, it is a consequence of both the very long polariton lifetime and the saddle shape of the dispersion. Hartree–Fock–Bogoliubov theory for elementary excitations To explain these features observed in Fig. 1 e, f, we derive the Bogoliubov spectrum of excitations on top of the condensate forming in the BIC state, i.e. in the vicinity of the k = 0 saddle point of the LLP branch for low finite temperatures in the presence of the (dark) exciton reservoir [30] . The Hamiltonian of lower polaritons is expressed in the second quantization as $$\\\\hat{H}=\\t\\\\int\\\\,d{{{{{{{\\\\bf{r}}}}}}}}{\\\\hat{P}}_{-}^{{{{\\\\dagger}}} }({{{{{{{\\\\bf{r}}}}}}}})[{\\\\varepsilon }_{-}(\\\\hat{{{{{{{{\\\\bf{p}}}}}}}}})-{\\\\mu }_{-}]{\\\\hat{P}}_{-}({{{{{{{\\\\bf{r}}}}}}}}) \\\\\\\\ \\t+\\\\int\\\\,d{{{{{{{\\\\bf{r}}}}}}}}{\\\\hat{P}}_{+}^{{{{\\\\dagger}}} }({{{{{{{\\\\bf{r}}}}}}}})[{\\\\varepsilon }_{+}(\\\\hat{{{{{{{{\\\\bf{p}}}}}}}}})-{\\\\mu }_{+}]{\\\\hat{P}}_{+}({{{{{{{\\\\bf{r}}}}}}}})\\\\\\\\ \\t+\\\\frac{1}{2}\\\\int\\\\,d{{{{{{{\\\\bf{r}}}}}}}}d{{{{{{{{\\\\bf{r}}}}}}}}}^{{\\\\prime} }{\\\\hat{Q}}^{{{{\\\\dagger}}} }({{{{{{{\\\\bf{r}}}}}}}}){\\\\hat{Q}}^{{{{\\\\dagger}}} }({{{{{{{{\\\\bf{r}}}}}}}}}^{{\\\\prime} })U({{{{{{{\\\\bf{r}}}}}}}}-{{{{{{{{\\\\bf{r}}}}}}}}}^{{\\\\prime} })\\\\hat{Q}({{{{{{{{\\\\bf{r}}}}}}}}}^{{\\\\prime} })\\\\hat{Q}({{{{{{{\\\\bf{r}}}}}}}}) \\\\\\\\ \\t+\\\\int\\\\,d{{{{{{{\\\\bf{r}}}}}}}}d{{{{{{{{\\\\bf{r}}}}}}}}}^{{\\\\prime} }{\\\\hat{\\\\tilde{Q}}}^{{{{\\\\dagger}}} }({{{{{{{\\\\bf{r}}}}}}}}){\\\\hat{Q}}^{{{{\\\\dagger}}} }({{{{{{{{\\\\bf{r}}}}}}}}}^{{\\\\prime} })U({{{{{{{\\\\bf{r}}}}}}}}-{{{{{{{{\\\\bf{r}}}}}}}}}^{{\\\\prime} })\\\\hat{Q}({{{{{{{{\\\\bf{r}}}}}}}}}^{{\\\\prime} })\\\\hat{\\\\tilde{Q}}({{{{{{{\\\\bf{r}}}}}}}}),$$ (1) where \\\\({\\\\hat{P}}_{\\\\pm }({{{{{{{\\\\bf{r}}}}}}}})\\\\) are the ULP and LLP field operators, respectively, \\\\(\\\\hat{{{{{{{{\\\\bf{p}}}}}}}}}=-i\\\\hslash \\\\nabla,{\\\\varepsilon }_{\\\\pm }({{{{{{{\\\\bf{p}}}}}}}})={E}_{\\\\pm }^{{{{{{{{\\\\rm{LP}}}}}}}}}({{{{{{{\\\\bf{p}}}}}}}})-{{{{{{{\\\\rm{Re}}}}}}}}\\\\,{E}_{-}^{{{{{{{{\\\\rm{LP}}}}}}}}}(0)\\\\) are the corresponding bare particle dispersions (the shape of \\\\({E}_{\\\\pm }^{{{{{{{{\\\\rm{LP}}}}}}}}}\\\\,({{{{{{{\\\\bf{p}}}}}}}})\\\\) is given in the SI) counted from the saddle point of the LLP branch, μ ± is the chemical potential, \\\\(\\\\hat{Q}({{{{{{{\\\\bf{r}}}}}}}})=\\\\int\\\\,d{{{{{{{{\\\\bf{r}}}}}}}}}^{{\\\\prime} }[X({{{{{{{{\\\\bf{r}}}}}}}}}^{{\\\\prime} }-{{{{{{{\\\\bf{r}}}}}}}}){\\\\hat{P}}_{-}({{{{{{{{\\\\bf{r}}}}}}}}}^{{\\\\prime} })+C({{{{{{{{\\\\bf{r}}}}}}}}}^{{\\\\prime} }-{{{{{{{\\\\bf{r}}}}}}}}){\\\\hat{P}}_{+}({{{{{{{{\\\\bf{r}}}}}}}}}^{{\\\\prime} })]\\\\) is the exciton field operator, and \\\\(\\\\hat{\\\\tilde{Q}}({{{{{{{\\\\bf{r}}}}}}}})\\\\) describes the field of background excitons that do not directly convert into polaritons (e.g. dark excitons). In the above, \\\\(X({{{{{{{{\\\\bf{r}}}}}}}}}^{{\\\\prime} }-{{{{{{{\\\\bf{r}}}}}}}})=(1/S){\\\\sum }_{{{{{{{{\\\\bf{p}}}}}}}}}{X}_{{{{{{{{\\\\bf{p}}}}}}}}}\\\\exp \\\\{i{{{{{{{\\\\bf{p}}}}}}}}\\\\cdot ({{{{{{{{\\\\bf{r}}}}}}}}}^{{\\\\prime} }-{{{{{{{\\\\bf{r}}}}}}}})/\\\\hslash \\\\},C({{{{{{{{\\\\bf{r}}}}}}}}}^{{\\\\prime} }-{{{{{{{\\\\bf{r}}}}}}}})=(1/S){\\\\sum }_{{{{{{{{\\\\bf{p}}}}}}}}}{C}_{{{{{{{{\\\\bf{p}}}}}}}}}\\\\exp \\\\{i{{{{{{{\\\\bf{p}}}}}}}}\\\\cdot ({{{{{{{{\\\\bf{r}}}}}}}}}^{{\\\\prime} }-{{{{{{{\\\\bf{r}}}}}}}})/\\\\hslash \\\\},{X}_{{{{{{{{\\\\bf{p}}}}}}}}}\\\\) and C p are the exciton and photon Hopfield coefficients, S is the quantization area. For simplicity, the exciton-exciton pair interaction potential is assumed to be contact: U ( r ) = g δ ( r ). The Hamiltonian ( 1 ) allows to write the Heisenberg equations for the evolution of both lower polariton fields \\\\({\\\\hat{P}}_{\\\\pm }({{{{{{{\\\\bf{r}}}}}}}},t)\\\\) . To obtain the excitation spectrum on top of the Bose condensate of BIC polaritons, we will assume that above threshold the LLP is macroscopically populated, which is supported by the experimental observation (see Fig. 1 e). Since the polaritons occupying the ULP branch do not convert into LLP polaritons because of the different symmetry of the underlying photon and exciton modes, we exclude the ULP operators from consideration when describing the macroscopically occupied saddle point. In this case, one can separate the condensate in the regular Bogoliubov fashion and, by averaging the Heisenberg equation for \\\\({\\\\hat{P}}_{-}\\\\) in the Hartree-Fock mean-field approximation (see “Methods” section), obtain the expression for the chemical potential of the LLP polaritons: $${\\\\mu }_{-}=g|{X}_{0}{|}^{2}\\\\left({n}_{0}|{X}_{0}{|}^{2}+2{n}_{Q}^{{\\\\prime} }+\\\\tilde{n}\\\\right),$$ (2) where n 0 is the condensate density, \\\\({n}_{Q}^{{\\\\prime} }\\\\equiv \\\\langle {\\\\hat{Q}}^{{\\\\prime} {{{\\\\dagger}}} }({{{{{{{\\\\bf{r}}}}}}}}){\\\\hat{Q}}^{{\\\\prime} }({{{{{{{\\\\bf{r}}}}}}}})\\\\rangle\\\\) is the non-condensate exciton density due to finite temperature, and \\\\(\\\\tilde{n}\\\\equiv \\\\langle {\\\\hat{\\\\tilde{Q}}}^{{{{\\\\dagger}}} }({{{{{{{\\\\bf{r}}}}}}}})\\\\hat{\\\\tilde{Q}}({{{{{{{\\\\bf{r}}}}}}}})\\\\rangle\\\\) is the background reservoir density. We note that μ − contains the thermal contribution \\\\({n}_{Q}^{{\\\\prime} }\\\\) and the dark reservoir contribution \\\\(\\\\tilde{n}\\\\) compared to the regularly used expression for the lower-polariton chemical potential μ LP = g n 0 ∣ X 0 ∣ 4 . Importantly, Eq. ( 2 ) corresponds to the experimentally-observed condensate blueshift, which will be used together with the excitation spectrum below to define n 0 and \\\\(\\\\tilde{n}\\\\) at a given pump power (note that at low temperatures considered here, \\\\({n}_{Q}^{{\\\\prime} }\\\\, \\\\ll \\\\,{n}_{0}\\\\) ). Diagonalizing the Hamiltonian ( 1 ) within the Hartree-Fock-Bogoliubov framework, one obtains the excitation spectrum: $${E}_{{{{{{{{\\\\bf{p}}}}}}}}}=\\\\sqrt{{{{{{{{\\\\mathcal{E}}}}}}}}({{{{{{{\\\\mathcal{E}}}}}}}}+2g{n}_{0}|{X}_{0}{|}^{2}|{X}_{{{{{{{{\\\\bf{p}}}}}}}}}{|}^{2})}\\\\quad \\\\,{{\\\\mbox{with}}}\\\\,\\\\quad {{{{{{{\\\\mathcal{E}}}}}}}}={\\\\varepsilon }_{-}({{{{{{{\\\\bf{p}}}}}}}})+g(|{X}_{{{{{{{{\\\\bf{p}}}}}}}}}{|}^{2}-|{X}_{0}{|}^{2})({n}_{0}|{X}_{0}{|}^{2}+2{n}_{Q}^{{\\\\prime} }+\\\\tilde{n}).$$ (3) It needs to be underlined that while Eq. ( 3 ) has the form looking similar to the textbook Bogoliubov prediction, it is, in fact, substantially different. Besides noticing that \\\\({{{{{{{\\\\mathcal{E}}}}}}}}\\\\) contains the reservoir contributions \\\\({n}_{Q}^{{\\\\prime} },\\\\tilde{n}\\\\) and the renormalization due to the dependence of the Hopfield coefficient X p on momentum, the bare polariton dispersion ε − ( p ) itself is very unusual: in the k x –direction it features, at the same time, the negative effective mass and momentum-dependent photon losses (see Supplementary Fig. 1 and Eq. ( 8) in the SI). Due to the latter, despite the negatively-defined \\\\({{{{{{{\\\\rm{Re}}}}}}}}{E}_{{{{{{{{\\\\bf{p}}}}}}}}}^{2}\\\\) in the vicinity of k x = 0, the real part of Eq. ( 3 ) is nowhere zero except the special points of the dispersion where \\\\({{{{{{{\\\\rm{Im}}}}}}}}{E}_{{{{{{{{\\\\bf{p}}}}}}}}}^{2}\\\\) changes sign. The excitation spectrum does not acquire a diffusive character predicted for systems with dissipation [26] , [27] . In fact, here the non-zero imaginary part of ε − ( p ) ensures the existence of low-lying states just above the condensate with \\\\({{{{{{{\\\\rm{Re}}}}}}}}{E}_{{{{{{{{\\\\bf{p}}}}}}}}} > 0\\\\) . The analysis of the imaginary part of the spectrum is given below. Figure 2a shows the real part of the Bogoliubov dispersion of excitations (Eq. 3 ) on top of the condensate formed in the saddle point of the LLP branch. The linearization at small momenta in the y –direction is clearly seen. Figure 2 b, c show the cuts of \\\\({{{{{{{\\\\rm{Re}}}}}}}}{E}_{{{{{{{{\\\\bf{p}}}}}}}}}\\\\) in Fig. 2 a along different near-zero k x (b) and k y = 0 (c), displaying the flattening of the dispersion in the k y –direction. As can be anticipated from ( 3 ), the extent of the energy-flat part along k y depends on the chosen near-zero k x , while the slope of the linear tails of the excitation spectrum at larger k y (for each k x ) is unambiguously defined by the polariton condensate density n 0 . Moreover, the zoom-in on the small momenta region displays a local minimum in the k x –direction (see the inset of Fig. 2 c). At the values of k x outside this local minimum, the Bogoliubov spectrum recovers the negative slopes characteristic to the single-particle LLP dispersion. Fig. 2: Dispersion of elementary excitations of the saddle-point polariton condensate. a The Bogoliubov dispersion of elementary excitations (real part of ( 3 )) with added blueshift of the saddle point. b Cross-cut of a at different k x from 0.005 to 0.045 μm −1 with the step 0.01 μm −1 , showing the flattening of the dispersion along the k y –direction close to the BIC state at k x = 0, calculated for condensate density n 0 = 4 × 10 10 cm −2 , reservoir density \\\\(\\\\tilde{n}=3\\\\times 1{0}^{10}\\\\) cm −2 . The gray dashed lines show the bare single-particle dispersions ε ± ( k x = 0, k y ). c Cross-cut of a at k y = 0, showing the real part of the Bogoliubov excitations spectrum E p given by ( 3 ). The gray dashed lines show the bare single-particle dispersions ε ± ( k x , k y = 0). Inset shows a close-up of the local minimum of the excitations spectrum, for varying n 0 from 1 to 6 × 10 10 cm −2 . Parameters as on Fig. 1 , g = 2.5 μeVμm 2 , T = 10 K. d , e Experimental time-resolved photoluminescence images above threshold displaying the energy-momentum dispersion of excitations vs. k y ; the excitation spectrum is evident in both the flat part and the linearized tails. f Experimental PL image of the dispersion of excitations vs. k x . In d – f the red lines show theoretical fitting of the Bogoliubov dispersion according to the real part of ( 3 ), with n 0 = 4 × 10 10 cm \\\\({}^{-2},\\\\tilde{n}=3\\\\times 1{0}^{10}\\\\) cm −2 ( d ) and n 0 = 5 × 10 10 cm \\\\({}^{-2},\\\\tilde{n}=4\\\\times 1{0}^{10}\\\\) cm −2 ( e , f ). The light-blue dashed lines show the single-particle LLP dispersion. The gray dotted lines in d , e mark the blueshift at different powers. The colorscales are in arb. units and normalized separately for d , e and f . Full size image Excitations luminescence dynamics in the pulsed regime The experimental PL images of collective excitations obtained in the regime of pulsed excitation are shown in Fig. 2 d–f. While the features very close to k x ~ 0 are below the experimental resolution and correspond to the dark region due to the long BIC lifetime, the dispersion along k y shows a striking correspondence to the theory derived above. To perform a better comparison, we have measured the temporal dynamics of the dispersion of excitations in PL along k y . This allows us to rule out any temporal smearing of the dispersion and, at the same time, track the change of the dispersion for different condensate and reservoir densities. Figure 2 d-e display two different snapshots of the emission along k y for k x ~ 0.02 μm −1 . These snapshots corresponding to different times clearly exhibit an energy-flat region with less-pronounced linear tails at higher values of k y . Thanks to the intrinsic dark nature of the condensate from a BIC state, it is possible to directly measure these features without any filtering in the momentum space. The red solid lines in Fig. 2 d, e show the analytical cross-cuts of the dispersion (Eq. 3 ) along k x ~ 0.02 μm −1 , for two values of densities calculated from the fitting of the measured linear tails together with the blueshift (as compared to the single-particle dispersion shown by the light-blue dashed lines). Full temporal dynamics of the PL vs. k y is provided in the Supplementary Movie 1 , while additional snapshots corresponding to different densities are shown in the SI. Experimentally following this dynamics allows to observe in time the delayed formation of the dark condensate after the pulse arrival, accompanied by the theoretically predicted energy-flat parts of the excitation spectrum, its shifting down with time due to the decreasing blueshift and, finally, disappearance of the flat parts and recovering of the single-particle near-parabolic LLP dispersion as the system goes below threshold. Finally, Fig. 2 f shows the energy-resolved PL image along the k x –direction above threshold. While the BIC polariton condensate in the local minimum of the modified dispersion stays dark, the excitations on top of the condensate occupying the very narrow region around k x = 0 are emitting light. Here, since under pulsed non-resonant excitation the blueshift is strongly time dependent, the resulting time-integrated measurement in Fig. 2 f looks apparently broadened in energy (see the corresponding enlarged curves for different densities in the inset of Fig. 2 c, getting smeared over time) which appears as a two-lobe image. Dispersion of collective excitations at different k x under continuous-wave excitation In order to see a more pronounced linearisation of the dispersion tails, we turn to continuous-wave (c.w.) excitation that allows to reach a density state while avoiding the energy blur induced by the dynamical change of energy of the condensate under time-integrated measurements. For fixed excitation conditions, we compare the PL from energy dispersion versus k y resolved at different values of k x as shown in Fig. 3 : one cut closer to k x = 0 (a), another directly within the bright spot with maximal emission that is seen above threshold in Fig. 2 f (b), and a final one when k x is further away from the center (c). As one can observe, while the PL from the non-condensate (out-of-the-BIC) lower polaritons in panels (a) and (c) is weak enough to allow registering the emission from the upper branch, in panel (b) the PL from the flat part of the dispersion is very bright, making the ULP invisible on the same (normalized) scale. Matching the observed PL with the theoretical expressions for the dispersion of excitations Eq. ( 3 ) and blueshift Eq. ( 2 ) for all three cases and using the condensate and reservoir densities as fitting parameters allows us to define exactly the values of k x at which those cuts were measured. Fig. 3: Continuous-wave excitation study. a – c Experimentally measured (left-hand sides) and theoretically calculated (right-hand sides) photoluminescence from the system vs. k y at three different values of k x above threshold: a k x = 0.019 μm −1 , b k x = 0.0228 μm −1 , c k x = 0.16 μm −1 . The overlaid red dotted lines represent the real part of the excitation spectrum \\\\({{{{{{{\\\\rm{Re}}}}}}}}{E}_{{{{{{{{\\\\bf{p}}}}}}}}}({k}_{y})\\\\) at corresponding values of k x , white dotted lines in a , c show the ULP and LLP single-particle dispersions. In b , the blueshifted lower polariton dispersion is shown for comparison. Intensity colorscales are in arb. units and normalized for each panel separately. d Real part (dashed line), counted from the blueshifted saddle point of the dispersion, and imaginary part (solid line) of E p vs. k x in the vicinity of k x ~ 0. The red arrows indicate the values of k x at which the panels a and b are measured. e Imaginary part of the spectrum vs. k y corresponding to panels a (yellow), b (blue), and c (green). For all panels, calculation parameters are the same as in Fig. 2 and densities estimated from fitting are n 0 = 5 × 10 10 cm \\\\({}^{-2},\\\\tilde{n}=11\\\\times 1{0}^{10}\\\\) cm −2 . Full size image The first straightforward outcome of this study is the clearly increased slope of the linear part of the Bogoliubov dispersion for the states lying above the flat stripe, as shown in Fig. 3 a, b. While the difference in energy at which the energy-flat part is seen in (a) and (b) lies within the linewidth (see the red arrows in Fig. 3 d or the inset of Fig. 2 c), the length of the stripe is changing with k x and is different for (a) and (b), in agreement with the theoretical Fig. 2 b. The tails of the dispersion outside the flat part in both (a) and (b) are clearly more visible than in the case of pulsed excitation studied above, and are linearized in accordance with Eq. ( 3 ). The shifted LLP dispersion is plotted in panel (b) together with the Bogoliubov spectrum. When moving away from the condensate, the dispersion of excitations recovers the shape of the lower polariton branch, while still staying blueshifted, as shown in Fig. 3 c. For comparison, in the right-hand sides of panels Fig. 3 a–c we plot the theoretically calculated PL emission, using the excitation spectrum derived above (note that since we neglected the upper polaritons in the theory, we do not reproduce the ULP luminescence). In this calculation, we assume that at finite temperature the normal branch of the Bogoliubov spectrum is predominantly populated (due to thermal depletion of the condensate). As a more subtle and more meaningful insight from fine resolving the PL in k x close to the BIC, we reveal that the maximal emission (panel b), i.e. the flat line in the Energy vs. k y dispersion, corresponds to the point in k x where the real part of the excitation spectrum E (counted from the blueshifted saddle point) goes to zero. This analysis is presented in Fig. 3 d, where we indicate by arrows the two values of k x at which panels (a) and (b) were measured ( k x corresponding to panel (c) lies out of the range of values plotted in (d) and is not marked). As discussed above, the only values of k x where \\\\({{{{{{{\\\\rm{Re}}}}}}}}{E}_{{{{{{{{\\\\bf{p}}}}}}}}}=0\\\\) correspond to the points in which the imaginary part of \\\\({E}_{{{{{{{{\\\\bf{p}}}}}}}}}^{2}\\\\) changes sign, i.e. the points that lie at the same energy as the dark condensate in the BIC, thus making the scattering from the condensate towards these states energetically effortless. Looking at the imaginary part of the dispersion, plotted as well in Fig. 3 d, allows us to conclude that since \\\\({{{{{{{\\\\rm{Im}}}}}}}}{E}_{{{{{{{{\\\\bf{p}}}}}}}}}\\\\) is negative for all k x > 0, the condensate with \\\\({{{{{{{\\\\rm{Im}}}}}}}}{E}_{{{{{{{{\\\\bf{p}}}}}}}}}=0\\\\) stays nevertheless stable and there is no macroscopic gain in the states indicated by the arrow ‘b’. However, since they correspond to the largest \\\\(|{{{{{{{\\\\rm{Im}}}}}}}}{E}_{{{{{{{{\\\\bf{p}}}}}}}}}|\\\\) , these states correspond to the maximum PL intensity observed in the experiments. We conclude that the particles residing in the dark condensate at k x = 0 at any scattering event may jump to the states ‘b’ where they leak out of the system due to maximal loss. For completeness of the analysis, we plot \\\\({{{{{{{\\\\rm{Im}}}}}}}}{E}_{{{{{{{{\\\\bf{p}}}}}}}}}\\\\) vs. k y in Fig. 3 e. One can see that the large imaginary part and hence short lifetimes are characteristic only for the states along the energy-flat part of the dispersion E p ( k y ), whereas the linear tails, corresponding to low-lying excitations above the saddle-point condensate, feature narrow linewidth and can be described adequately within the equilibrium theory. Anisotropy in the momentum space To study the anisotropic dispersion of excitations discussed above, we measure the photoluminescence emission under pulsed excitation in the plane ( k x , k y ) which is shown in Fig. 4 for different energies close to the BIC state. For a clearer visualization, we suggest to watch the Supplementary Movie 2 showing the experimental cross-cut of the PL emission in the far field at different energies. The exotic shape with two parallel stripes in the middle at a given energy can be directly related to the populated states of the excitation spectrum where \\\\({{{{{{{\\\\rm{Re}}}}}}}}{E}_{{{{{{{{\\\\bf{p}}}}}}}}}=0\\\\) and the absolute value of the negative \\\\({{{{{{{\\\\rm{Im}}}}}}}}{E}_{{{{{{{{\\\\bf{p}}}}}}}}}\\\\) is large, as discussed above and confirmed by the calculated \\\\({{{{{{{\\\\rm{Im}}}}}}}}{E}_{{{{{{{{\\\\bf{p}}}}}}}}}\\\\) dependence on k x and k y in Fig. 3 d, e. It is important to note that the distribution of the PL in two parallel stripes appears only above threshold: without the macroscopic population of the saddle-point, there are no states lying at the same energy with the condensate and hence no population of the energy-flat states along k y that exhibit strong emission. For comparison, we provide in the SI and the Supplementary Movie 3 the corresponding data taken below threshold, where the energy corresponding to the saddle point stays fully dark. In the above-threshold case of Fig. 4 , due to time averaging in the pulsed excitation setting, the two-stripe pattern which lies at the energy of the condensate is visible in more than one panel, as the condensate shifts down over time (due to decreasing blueshift). We note that the observed peculiar far field PL distribution in this case is very different from the two-lobe pattern of grating-based lasers [8] , [9] , where, despite the similar dark line along k x = 0, the lasing happens exclusively at the photonic BIC energy and the intensity distribution along k y is a Gaussian centered at k y = 0. In our case, the PL emission along both directions and in energy follows the calculated dispersion Eq. ( 3 ), with the length of the parallel lines shorter when the condensate density is lower, and the curved tails at nonzero momenta appearing outside the two-stripe region (whose emission is weaker since \\\\(|{{{{{{{\\\\rm{Im}}}}}}}}{E}_{{{{{{{{\\\\bf{p}}}}}}}}}|\\\\) is much smaller). In all panels of Fig. 4 , the theoretically calculated cross-cuts of the real part of E p at the corresponding energies above the BIC state are overlayed as dashed lines on top of the experimental images. The maximum of the experimental emission appears at those states where the real part of the dispersion E p crosses the given energy (the brighter spots move away from the center in the panels (c) and (d) that correspond to higher energies). Fig. 4: Momentum-space scan. a – d Experimental PL maps in the ( k x , k y ) domain taken at different energies just above the blueshifted BIC polariton condensate (as denoted on the panels). The line k x = 0 above threshold stays fully dark, making the PL profile in the vicinity of the condensate look like two parallel stripes corresponding to the flat parts of \\\\({{{{{{{\\\\rm{Re}}}}}}}}{E}_{{{{{{{{\\\\bf{p}}}}}}}}}({k}_{y})\\\\) appearing at small non-zero k x . These maps are time-integrated in the pulsed excitation setting. Intensity colorscale is in arb. units and normalized the same way for all panels. Condensate and reservoir densities n 0 = 5 × 10 10 cm \\\\({}^{-2},\\\\tilde{n}=4\\\\times 1{0}^{10}\\\\) cm −2 , other parameters as in Fig. 2 . Full size image From the point of view of the anisotropy of collective excitations which is illustrated by the above study of the spectrum, it is also interesting to address the limit of very small momenta. Experimentally, since the region of k x corresponding to the positive slope of \\\\({{{{{{{\\\\rm{Re}}}}}}}}{E}_{{{{{{{{\\\\bf{p}}}}}}}}}\\\\) is very small, the shift in energy which is acquired within this narrow Δ k x lies within the linewidth. At the same time, it is not possible to follow the change in energy with k y along the positive slope of the saddle (from Fig. 2 a one sees that at k x → 0 the energy-flat part along k y disappears and the dispersion is linear) due to the darkness of the states corresponding to k x = 0 (at any k y ). Theoretically, however, taking the limit p → 0 in Eq. ( 3 ) allows to obtain the sound velocity which appears to be strongly dependent on direction (on the angle φ in the polar coordinate system): $${c}_{{{{{{{{\\\\rm{s}}}}}}}}}(\\\\varphi )={{{{{{{\\\\rm{Re}}}}}}}}\\\\left\\\\{\\\\sqrt{{\\\\mu }_{{{{{{{{\\\\rm{LP}}}}}}}}}\\\\left[{\\\\cos }^{2}\\\\varphi \\\\left(\\\\frac{1}{{m}_{{{{{{{{\\\\rm{LP}}}}}}}}}^{x}}+\\\\frac{i}{{s}_{x}}\\\\right)+\\\\frac{{\\\\sin }^{2}\\\\varphi }{{m}_{{{{{{{{\\\\rm{LP}}}}}}}}}^{y}}\\\\right]\\\\left[1+\\\\frac{2\\\\mu }{\\\\sqrt{{(\\\\hslash \\\\Omega )}^{2}+{\\\\Delta }^{2}}}\\\\right]}\\\\right\\\\},$$ (4) where \\\\({m}_{{{{{{{{\\\\rm{LP}}}}}}}}}^{x(y)}\\\\) are the effective masses of LLP in the x ( y )–direction, and s x is the mass-like parameter defined by the radiative losses rate γ and the radiative coupling of the modes U (for more details, see the SI) . In a standard case of real positive masses in both directions, for experimentally-relevant parameters (resulting in \\\\(1/|{m}_{{{{{{{{\\\\rm{LP}}}}}}}}}^{x}|\\\\gg 1/{m}_{{{{{{{{\\\\rm{LP}}}}}}}}}^{y}\\\\) ) the velocity given by Eq. ( 4 ) would significantly increase when approaching the x –axis (the direction corresponding to \\\\(\\\\varphi=\\\\pi m,m\\\\in {\\\\mathbb{Z}}\\\\) ). However, here the expression for the sound velocity is modified by the negative \\\\({m}_{{{{{{{{\\\\rm{LP}}}}}}}}}^{x}\\\\) and by the imaginary contribution s x , which makes the definition of the sound velocity more complicated. In Fig. 5 b we plot the sound velocity dependence on the angle φ revealing a very anisotropic profile. The increase of c s in the x –direction is clearly seen from the difference of slopes in the linearized parts of the Bogoliubov dispersion (see the zoom-in at the low-momenta region in Fig. 5 a): while the slope along k y at k x = 0 corresponds to very low \\\\({c}_{{{{{{{{\\\\rm{s}}}}}}}}}^{y}\\\\) , the shallow parts of the spectrum at finite but small k x result in the sound velocity in the directions slightly deviating from the y –axis being very close to zero (see Fig. 5 c). Much steeper slopes along k x at k y = 0 provide maxima of c s at φ = π m , as shown in Fig. 5 b for various condensate densities. It is important to note that taking into account the non-radiative exciton losses would additionally affect the behavior of the excitation spectrum along k x in a very narrow vicinity of the saddle point. A more detailed discussion of this matter is provided in the SI . Fig. 5: Anisotropic Bogoliubov sound velocity. a A close-up view of the local potential well created in the real part of the Bogoliubov spectrum of elementary excitations around the saddle point, with essentially different slopes \\\\({c}_{{{{{{{{\\\\rm{s}}}}}}}}}^{x}\\\\gg {c}_{{{{{{{{\\\\rm{s}}}}}}}}}^{y}\\\\) (where \\\\({c}_{{{{{{{{\\\\rm{s}}}}}}}}}^{x}={c}_{{{{{{{{\\\\rm{s}}}}}}}}}(\\\\varphi=\\\\pi m),{c}_{{{{{{{{\\\\rm{s}}}}}}}}}^{y}={c}_{{{{{{{{\\\\rm{s}}}}}}}}}(\\\\varphi=\\\\pi [m+1]/2),m\\\\in {\\\\mathbb{Z}}\\\\) ). The colored dashed lines are guide to the eye showing different directions along the anisotropic dispersion slopes linearized in the vicinity of k = 0. b The Bogoliubov sound velocity profile according to Eq. ( 4 ). The colored dashed lines mark different angles corresponding to the slopes marked in a , see discussion in the text. c Close-up of the sound velocity profile at the vicinity of π /2 (corresponding to the black rectangle in b ). The condensate density n 0 in b , c is varied from 1 to 6 × 10 10 cm \\\\({}^{-2},\\\\tilde{n}=4\\\\times 1{0}^{10}\\\\) cm −2 , other parameters as in Fig. 2 . Full size image\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"In this work, we studied theoretically and experimentally the spectrum of elementary excitations of the BIC polariton condensate arising from the coupling of excitons to the photonic modes in a patterned semiconductor waveguide. The studied system is exotic and very different from the well-studied case of microcavity polaritons. We show that despite the saddle-like shape of the single-particle dispersion of lower polaritons, which exhibits a maximum in one of the directions in k –space, with the accumulation of particles in the saddle point, a local minimum is created with the positive slopes of the Bogoliubov dispersion, which may help in further accumulation of polaritons. At the same time, two parallel stripes in momentum space, which correspond to the states at the same energy as the condensate—but being bright—appear. It is, in fact, the interplay of the negative mass and the mometum-dependent losses that results in the existence of such unique, extended in momentum, states outside the condensate. This unusual anisotropic shape of the excitation spectrum yields in the k y –direction to two energy-flat zones (near k x = 0) separated by the dark line at k x = 0. Thanks to the quasi-infinite lifetime of the BIC state, the condensate itself does not emit light hence allowing to neatly observe the populated states around and above the saddle point. With both the slopes of the linear parts of the dispersion that follow the energy-flat regions and the interaction-induced blueshift being directly experimentally accessible, our theory allows to precisely define the macroscopic density of the dark polariton condensate. Thus, while the observation of the excitation spectrum in such a system is striking on its own, it also allows to obtain the information about the saddle-point polariton condensate which is otherwise invisible. Remarkably, despite the negative mass in the x –direction and the presence of losses, the real part of the spectrum is positive everywhere at k → 0, allowing to extract the velocity of sound in every direction. The latter appears to be highly anisotropic, dropping to almost zero in the directions next to the y -axis, i.e. slightly misaligned with the grating principal axis, and growing in the direction of propagation along the waveguide. It needs to be noted that the anisotropic sound velocity was reported previously for atomic BECs with anisotropic dipolar interactions [31] , [32] . In case of dipolar BECs, however, transport measurements reveal the anisotropy of the critical Landau velocity rather than the velocity of sound [33] , since the dispersion of excitations is affected by direction-dependent roton softening. The situation realized in our work can be better compared with anisotropic superfluidity reported for excitons [34] and cavity photon BEC [35] in periodically modulated planar structures. At the same time, we stress that here we deal with a more exotic case of anisotropy: even though there is a well-defined finite sound velocity, superfluidity in the sense of Landau criterion cannot be reached in all directions of propagation. Indeed, no matter how small the velocity would be below c s in the x –direction, due to the negative slopes of the dispersion, it could always elastically create an excitation out of the condensate. However if an obstacle is propagating strictly along the y –axis, the velocity \\\\({c}_{{{{{{{{\\\\rm{s}}}}}}}}}^{y}\\\\) would be a proper Landau sound velocity below which superfluidity can be observed. This illustrates the richness of saddle-point condensates and their anisotropic behaviors. Along with the previously proposed topological-dispersion engineering [36] , we believe that this work demonstrates the high interest in engineering the excitation spectrum of the condensate e.g. via implementing different grating symmetries—as a tool to impart new properties to the condensate itself—and underlines once again the astonishing richness of polariton systems, as well as multiple possible avenues for further investigations.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Theoretical methods Starting from the Hamiltonian ( 1 ) in the neglection of the ULP fields, one can separate the macroscopically occupied condensate state as $${\\\\hat{P}}_{-}({{{{{{{\\\\bf{r}}}}}}}},t)=\\\\sqrt{{n}_{0}}+{\\\\hat{P}}_{-}^{{\\\\prime} }({{{{{{{\\\\bf{r}}}}}}}},t),\\\\quad \\\\hat{Q}({{{{{{{\\\\bf{r}}}}}}}},t)={X}_{0}\\\\sqrt{{n}_{0}}+{\\\\hat{Q}}^{{\\\\prime} }({{{{{{{\\\\bf{r}}}}}}}},t),$$ (5) where n 0 is the condensate density, \\\\({\\\\hat{Q}}^{{\\\\prime} }({{{{{{{\\\\bf{r}}}}}}}},t)=\\\\int\\\\,X({{{{{{{{\\\\bf{r}}}}}}}}}^{{\\\\prime} }-{{{{{{{\\\\bf{r}}}}}}}}){\\\\hat{P}}_{-}^{{\\\\prime} }({{{{{{{{\\\\bf{r}}}}}}}}}^{{\\\\prime} },t)d{{{{{{{{\\\\bf{r}}}}}}}}}^{{\\\\prime} }\\\\) , and the average \\\\(\\\\langle {\\\\hat{P}}_{-}^{{\\\\prime} }({{{{{{{\\\\bf{r}}}}}}}},t)\\\\rangle=\\\\langle {\\\\hat{Q}}^{{\\\\prime} }({{{{{{{\\\\bf{r}}}}}}}},t)\\\\rangle=0\\\\) . The substitution Eq. ( 5 ) allows to rewrite the triple products of the field operators in Eq. ( 1 ) in the Hartree-Fock mean-field approximation as $${\\\\hat{Q}}^{{{{\\\\dagger}}} }({{{{{{{\\\\bf{r}}}}}}}},t)\\\\hat{Q}({{{{{{{\\\\bf{r}}}}}}}},t)\\\\hat{Q}({{{{{{{\\\\bf{r}}}}}}}},t)=\\t{X}_{0}\\\\sqrt{{n}_{0}}(|{X}_{0}{|}^{2}{n}_{0}+2{n}_{Q}^{{\\\\prime} }) \\\\\\\\ \\t+2{\\\\hat{Q}}^{{\\\\prime} }({{{{{{{\\\\bf{r}}}}}}}},t)(|{X}_{0}{|}^{2}{n}_{0}+{n}_{Q}^{{\\\\prime} })+{\\\\hat{Q}}^{{\\\\prime} {{{\\\\dagger}}} }({{{{{{{\\\\bf{r}}}}}}}},t){X}_{0}^{2}{n}_{0},$$ (6) $${\\\\hat{\\\\tilde{Q}}}^{{{{\\\\dagger}}} }({{{{{{{\\\\bf{r}}}}}}}},t)\\\\hat{\\\\tilde{Q}}({{{{{{{\\\\bf{r}}}}}}}},t)\\\\hat{Q}({{{{{{{\\\\bf{r}}}}}}}},t)=\\\\left[{X}_{0}\\\\sqrt{{n}_{0}}+{\\\\hat{Q}}^{{\\\\prime} }({{{{{{{\\\\bf{r}}}}}}}},t)\\\\right]\\\\tilde{n}$$ (7) with notations \\\\({n}_{Q}^{{\\\\prime} }=\\\\langle {\\\\hat{Q}}^{{\\\\prime} {{{\\\\dagger}}} }({{{{{{{\\\\bf{r}}}}}}}},t){\\\\hat{Q}}^{{\\\\prime} }({{{{{{{\\\\bf{r}}}}}}}},t)\\\\rangle,\\\\tilde{n}=\\\\langle {\\\\hat{\\\\tilde{Q}}}^{{{{\\\\dagger}}} }({{{{{{{\\\\bf{r}}}}}}}},t)\\\\hat{\\\\tilde{Q}}({{{{{{{\\\\bf{r}}}}}}}},t)\\\\rangle\\\\) . Using the Hamiltonian Eq. ( 1 ) with the substitutions Eqs. ( 5 ) and ( 6 ), ( 7 ), the Heisenberg equation for the non-condensed part of the polariton field \\\\({\\\\hat{P}}_{-}^{{\\\\prime} }({{{{{{{\\\\bf{r}}}}}}}},t)\\\\) can be written as $$i\\\\hslash {\\\\partial }_{t}{\\\\hat{P}}_{-}^{{\\\\prime} }({{{{{{{\\\\bf{r}}}}}}}},t)=\\t\\\\left[{\\\\varepsilon }_{-}(-i\\\\hslash \\\\nabla )-{\\\\mu }_{-}\\\\right]{\\\\hat{P}}_{-}^{{\\\\prime} }({{{{{{{\\\\bf{r}}}}}}}},t)\\\\\\\\ \\t+g\\\\int\\\\,d{{{{{{{{\\\\bf{r}}}}}}}}}^{{\\\\prime} }{X}^{*}({{{{{{{{\\\\bf{r}}}}}}}}}^{{\\\\prime} }-{{{{{{{\\\\bf{r}}}}}}}})\\\\left[(2{n}_{Q}^{{\\\\prime} }+2{n}_{0}|{X}_{0}{|}^{2}+\\\\tilde{n}){\\\\hat{Q}}^{{\\\\prime} }({{{{{{{{\\\\bf{r}}}}}}}}}^{{\\\\prime} },t) \\\\right. \\\\\\\\ \\t\\\\left.+{X}_{0}^{2}{n}_{0}{\\\\hat{Q}}^{{\\\\prime} {{{\\\\dagger}}} }({{{{{{{{\\\\bf{r}}}}}}}}}^{{\\\\prime} },t)\\\\right].$$ (8) In Fourier space, ( 8 ) takes the form \\\\(i\\\\hslash {\\\\partial }_{t}{\\\\hat{P}}_{{{{{{{{\\\\bf{p}}}}}}}}}(t)=[{\\\\varepsilon }_{-}({{{{{{{\\\\bf{p}}}}}}}})-{\\\\mu }_{-}+g(2{n}_{0}|{X}_{0}{|}^{2}+2{n}_{Q}^{{\\\\prime} }+\\\\tilde{n})]|{X}_{{{{{{{{\\\\bf{p}}}}}}}}}{|}^{2}{\\\\hat{P}}_{{{{{{{{\\\\bf{p}}}}}}}}}(t)+g{n}_{0}{X}_{0}^{2}{X}_{{{{{{{{\\\\bf{p}}}}}}}}}^{*2}{\\\\hat{P}}_{-{{{{{{{\\\\bf{p}}}}}}}}}^{{{{\\\\dagger}}} }(t)\\\\) that allows to perform the Bogoliubov transformation $${\\\\hat{P}}_{{{{{{{{\\\\bf{p}}}}}}}}}(t)={u}_{{{{{{{{\\\\bf{p}}}}}}}}}{\\\\hat{\\\\alpha }}_{{{{{{{{\\\\bf{p}}}}}}}}}{e}^{-i{E}_{{{{{{{{\\\\bf{p}}}}}}}}}t/\\\\hslash }-{v}_{-{{{{{{{\\\\bf{p}}}}}}}}}^{*}{\\\\hat{\\\\alpha }}_{-{{{{{{{\\\\bf{p}}}}}}}}}^{{{{\\\\dagger}}} }{e}^{i{E}_{{{{{{{{\\\\bf{p}}}}}}}}}^{*}t/\\\\hslash }\\\\,\\\\,\\\\,\\\\,{{\\\\mbox{with}}}\\\\,\\\\,\\\\,|{u}_{{{{{{{{\\\\bf{p}}}}}}}}}{|}^{2}-|{v}_{-{{{{{{{\\\\bf{p}}}}}}}}}{|}^{2}=1,$$ (9) where \\\\({\\\\hat{\\\\alpha }}_{{{{{{{{\\\\bf{p}}}}}}}}}\\\\) is the annihilation operator of the Bogoliubov excitation with momentum p . Eq. ( 9 ) yields the excitation spectrum Eq. ( 3 ). We note that due to the underlying photon dispersion whose imaginary part is strongly momentum-dependent, the Hopfield coefficients X p , C p as well as the Bogoliubov amplitudes u p , v − p are complex, and careful treatment of complex conjugations throughout the theory is required. Moreover, choosing the Bogoliubov transformation in the specific shape given by Eq. ( 9 ) defines the so-called ghost branch (GB) of excitations as \\\\({E}_{{{{{{{{\\\\bf{p}}}}}}}}}^{{{{{{{{\\\\rm{GB}}}}}}}}}=-{E}_{{{{{{{{\\\\bf{p}}}}}}}}}^{*}\\\\) (this definition of the GB is consistent with that used in ref. [27] ). Theoretical calculations of the intensity of photoluminescence from the excitations around the BIC state that are presented in Fig. 4 are performed using the standard expression [37] , [38] : $${{{{{{{\\\\rm{PL}}}}}}}}({k}_{x},{k}_{y};\\\\omega )\\\\propto \\\\left(1-{\\\\left|{X}_{{{{{{{{\\\\bf{p}}}}}}}}}\\\\right|}^{2}\\\\right){{{{{{{\\\\rm{Re}}}}}}}}\\\\int\\\\nolimits_{0}^{\\\\infty }dt\\\\,{e}^{-it(\\\\omega -i{0}^{+})} < {P}_{{{{{{{{\\\\bf{p}}}}}}}}}^{{{{\\\\dagger}}} }(t){P}_{{{{{{{{\\\\bf{p}}}}}}}}}(0) > .$$ (10) Experiments All the PL data reported in this work has been acquired using a confocal setup. The objective back focal plane is imaged on the entrance slit of the spectrometer allowing to obtain the energy versus k dispersion. Two different magnification of the objective focal plane are used in order to properly image the dispersions along k x and k y which have very different effective masses and hence very different extension in the k -space. The laser used in the experiments is a fs-pulsed laser having a repetition rate of 80 MHz and a pulse duration of ~100 fs. Excitation wavelength is set at 780 nm. The time resolved measurements are obtained by scanning the far-field emission and acquiring for each k y an energy versus time temporal trace with a streak camera. The set of temporal traces is then used to reconstruct the dynamics in the Energy vs k y space. Supplementary Movie 1 shows an example of temporal dynamics reconstructed in this way. The emission in the ( k x , k y )–plane at a given energy like the images shown in Fig. 4 are obtained by scanning the far-field emission and acquiring a set of Energy vs k x dispersions, each one corresponding to a different k y . The data are then merged and cut at a given energy in the plane k x , k y . Supplementary Movies 2 and 3 show a sequence of such cuts for different energies above and below threshold, respectively.\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"This work proposes that the elementary excitation spectrum of a polariton Bose-Einstein condensate formed in a bound state in the continuum (BIC) within a planar nanostructured waveguide exhibits a unique Bogoliubov dispersion characterized by a local energy minimum around the saddle point k = 0, non-zero real part despite negative effective mass and system losses, and strong anisotropy in the sound velocity, offering a novel platform for detailed study and control of condensate properties through excitation spectrum engineering."},"entities":{"kind":"list like","value":["polariton Bose-Einstein condensate","bound state in the continuum","planar nanostructured waveguide","Bogoliubov dispersion","local energy minimum","saddle point","effective mass","system losses","strong anisotropy","sound velocity","excitation spectrum engineering"],"string":"[\n \"polariton Bose-Einstein condensate\",\n \"bound state in the continuum\",\n \"planar nanostructured waveguide\",\n \"Bogoliubov dispersion\",\n \"local energy minimum\",\n \"saddle point\",\n \"effective mass\",\n \"system losses\",\n \"strong anisotropy\",\n \"sound velocity\",\n \"excitation spectrum engineering\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"2.","title":"Friedrich, H. & Wintgen, D. Interfering resonances and bound states in the continuum.Phys. Rev. A32, 3231 (1985).","link":"https://doi.org/10.1103%2FPhysRevA.32.3231"},{"idx":"3.","title":"Capasso, F. et al. Observation of an electronic bound state above a potential well.Nature358, 565 (1992).","link":"https://doi.org/10.1038%2F358565a0"},{"idx":"4.","title":"Hsu, C. W., Zhen, B., Stone, A. D., Joannopoulos, J. D. & Soljačic, M. Bound states in the continuum.Nat. Rev. Mater.1, 16048 (2016).","link":"https://doi.org/10.1038%2Fnatrevmats.2016.48"},{"idx":"5.","title":"Zhen, B. et al. Spawning rings of exceptional points out of Dirac cones.Nature525, 354 (2015).","link":"https://doi.org/10.1038%2Fnature14889"},{"idx":"6.","title":"Lu, L. et al. Engineering a light-matter strong coupling regime in perovskite-based plasmonic metasurface: quasi-bound state in the continuum and exceptional points.Photon. Res.8, A91–A100 (2020).","link":"https://doi.org/10.1364%2FPRJ.404743"},{"idx":"7.","title":"Noll, R. J. & Macomber, S. H. Analysis of grating surface emitting lasers.IEEE J. Quant. Electron.26, 456 (1990).","link":"https://doi.org/10.1109%2F3.52121"},{"idx":"9.","title":"Wang, Y. et al. Highly controllable etchless perovskite microlasers based on bound states in the continuum.ACS Nano15, 7386–7391 (2021).","link":"https://doi.org/10.1021%2Facsnano.1c00673"},{"idx":"10.","title":"Koshelev, K. L., Sychev, S. K., Sadrieva, Z. F., Bogdanov, A. A. & Iorsh, I. V. Strong coupling between excitons in transition metal dichalcogenides and optical bound states in the continuum.Phys. Rev. B98, 161113(R) (2018).","link":"https://doi.org/10.1103%2FPhysRevB.98.161113"},{"idx":"11.","title":"Kravtsov, V. et al. Nonlinear polaritons in a monolayer semiconductor coupled to optical bound states in the continuum.Light. Sci. Appl.9, 56 (2020).","link":"https://doi.org/10.1038%2Fs41377-020-0286-z"},{"idx":"12.","title":"Ardizzone, V. et al. Polariton Bose-Einstein condensate from a Bound State in the Continuum.Nature605, 447–452 (2022).","link":"https://doi.org/10.1038%2Fs41586-022-04583-7"},{"idx":"14.","title":"Kasprzak, J. et al. Bose-Einstein condensation of exciton polaritons.Nature443, 409–414 (2006).","link":"https://doi.org/10.1038%2Fnature05131"},{"idx":"15.","title":"Balili, R., Hartwell, V., Snoke, D., Pfeiffer, L. & West, K. Bose-Einstein condensation of microcavity polaritons in a trap.Science316, 1007–1010 (2007).","link":"https://doi.org/10.1126%2Fscience.1140990"},{"idx":"16.","title":"Byrnes, T., Kim, N. Y. & Yamamoto, Y. Exciton-polariton condensates.Nat. Phys.10, 803–813 (2014).","link":"https://doi.org/10.1038%2Fnphys3143"},{"idx":"17.","title":"Cristofolini, P. et al. Optical superfluid phase transitions and trapping of polariton condensates.Phys. Rev. Lett.110, 186403 (2013).","link":"https://doi.org/10.1103%2FPhysRevLett.110.186403"},{"idx":"18.","title":"Amo, A. et al. Superfluidity of polaritons in semiconductor microcavities.Nat. Phys.5, 805–810 (2009).","link":"https://doi.org/10.1038%2Fnphys1364"},{"idx":"19.","title":"Lerario, G. et al. Room-temperature superfluidity in a polariton condensate.Nat. Phys.13, 837–841 (2017).","link":"https://doi.org/10.1038%2Fnphys4147"},{"idx":"21.","title":"Utsunomiya, S. et al. Observation of Bogoliubov excitations in exciton-polariton condensates.Nat. Phys.4, 700 (2008).","link":"https://doi.org/10.1038%2Fnphys1034"},{"idx":"22.","title":"Kohnle, V. et al. From single particle to superfluid excitations in a dissipative polariton gas.Phys. Rev. Lett.106, 255302 (2011).","link":"https://doi.org/10.1103%2FPhysRevLett.106.255302"},{"idx":"23.","title":"Pieczarka, M. et al. Ghost branch photoluminescence from a polariton fluid under nonresonant excitation.Phys. Rev. Lett.115, 186401 (2015).","link":"https://doi.org/10.1103%2FPhysRevLett.115.186401"},{"idx":"24.","title":"Stepanov, P. et al. Dispersion relation of the collective excitations in a resonantly driven polariton fluid.Nat. Commun.10, 3869 (2019).","link":"https://doi.org/10.1038%2Fs41467-019-11886-3"},{"idx":"25.","title":"Ballarini, D. et al. Directional Goldstone waves in polariton condensates close to equilibrium.Nat. Commun.11, 217 (2020).","link":"https://doi.org/10.1038%2Fs41467-019-13733-x"},{"idx":"26.","title":"Wouters, M. & Carusotto, I. Excitations in a nonequilibrium bose-einstein condensate of exciton polaritons.Phys. Rev. Lett.99, 140402 (2007).","link":"https://doi.org/10.1103%2FPhysRevLett.99.140402"},{"idx":"27.","title":"Byrnes, T., Horikiri, T., Ishida, N., Fraser, M. & Yamamoto, Y. Negative Bogoliubov dispersion in exciton-polariton condensates.Phys. Rev. B85, 075130 (2012).","link":"https://doi.org/10.1103%2FPhysRevB.85.075130"},{"idx":"28.","title":"Pieczarka, M. et al. Observation of quantum depletion in a non-equilibrium exciton-polariton condensate.Nat. Commun.11, 429 (2020).","link":"https://doi.org/10.1038%2Fs41467-019-14243-6"},{"idx":"29.","title":"Pieczarka, M. et al. Bogoliubov excitations of a polariton condensate in dynamical equilibrium with an incoherent reservoir.Phys. Rev. B105, 224515 (2022).","link":"https://doi.org/10.1103%2FPhysRevB.105.224515"},{"idx":"30.","title":"Grudinina, A. M., Kurbakov, I. L., Lozovik, Y. E. & Voronova, N. S. Finite-temperature Hartree-Fock-Bogoliubov theory for exciton-polaritons.Phys. Rev. B104, 125301 (2021).","link":"https://doi.org/10.1103%2FPhysRevB.104.125301"},{"idx":"31.","title":"Ticknor, C., Wilson, R. M. & Bohn, J. L. Anisotropic superfluidity in a dipolar bose gas.Phys. Rev. Lett.106, 065301 (2011).","link":"https://doi.org/10.1103%2FPhysRevLett.106.065301"},{"idx":"32.","title":"Bismut, G. et al. Anisotropic excitation spectrum of a dipolar quantum bose gas.Phys. Rev. Lett.109, 155302 (2012).","link":"https://doi.org/10.1103%2FPhysRevLett.109.155302"},{"idx":"33.","title":"Wenzel, M. et al. Anisotropic superfluid behavior of a dipolar Bose-Einstein condensate.Phys. Rev. Lett.121, 030401 (2018).","link":"https://doi.org/10.1103%2FPhysRevLett.121.030401"},{"idx":"34.","title":"Lozovik, Y. E., Kurbakov, I. L. & Volkov, P. A. Anisotropic superfluidity of two-dimensional excitons in a periodic potential.Phys. Rev. B95, 245430 (2017).","link":"https://doi.org/10.1103%2FPhysRevB.95.245430"},{"idx":"36.","title":"Sigurdsson, H., Li, G. & Liew, T. C. H. Spontaneous and superfluid chiral edge states in exciton-polariton condensates.Phys. Rev. B96, 115453 (2017).","link":"https://doi.org/10.1103%2FPhysRevB.96.115453"},{"idx":"37.","title":"Ciuti, C., Schwendimann, P. & Quattropani, A. Parametric luminescence of microcavity polaritons.Phys. Rev. B63, 041303(R) (2001).","link":"https://doi.org/10.1103%2FPhysRevB.63.041303"}],"string":"[\n {\n \"idx\": \"2.\",\n \"title\": \"Friedrich, H. & Wintgen, D. Interfering resonances and bound states in the continuum.Phys. Rev. A32, 3231 (1985).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevA.32.3231\"\n },\n {\n \"idx\": \"3.\",\n \"title\": \"Capasso, F. et al. Observation of an electronic bound state above a potential well.Nature358, 565 (1992).\",\n \"link\": \"https://doi.org/10.1038%2F358565a0\"\n },\n {\n \"idx\": \"4.\",\n \"title\": \"Hsu, C. W., Zhen, B., Stone, A. D., Joannopoulos, J. D. & Soljačic, M. Bound states in the continuum.Nat. Rev. Mater.1, 16048 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fnatrevmats.2016.48\"\n },\n {\n \"idx\": \"5.\",\n \"title\": \"Zhen, B. et al. Spawning rings of exceptional points out of Dirac cones.Nature525, 354 (2015).\",\n \"link\": \"https://doi.org/10.1038%2Fnature14889\"\n },\n {\n \"idx\": \"6.\",\n \"title\": \"Lu, L. et al. Engineering a light-matter strong coupling regime in perovskite-based plasmonic metasurface: quasi-bound state in the continuum and exceptional points.Photon. Res.8, A91–A100 (2020).\",\n \"link\": \"https://doi.org/10.1364%2FPRJ.404743\"\n },\n {\n \"idx\": \"7.\",\n \"title\": \"Noll, R. J. & Macomber, S. H. Analysis of grating surface emitting lasers.IEEE J. Quant. Electron.26, 456 (1990).\",\n \"link\": \"https://doi.org/10.1109%2F3.52121\"\n },\n {\n \"idx\": \"9.\",\n \"title\": \"Wang, Y. et al. Highly controllable etchless perovskite microlasers based on bound states in the continuum.ACS Nano15, 7386–7391 (2021).\",\n \"link\": \"https://doi.org/10.1021%2Facsnano.1c00673\"\n },\n {\n \"idx\": \"10.\",\n \"title\": \"Koshelev, K. L., Sychev, S. K., Sadrieva, Z. F., Bogdanov, A. A. & Iorsh, I. V. Strong coupling between excitons in transition metal dichalcogenides and optical bound states in the continuum.Phys. Rev. B98, 161113(R) (2018).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.98.161113\"\n },\n {\n \"idx\": \"11.\",\n \"title\": \"Kravtsov, V. et al. Nonlinear polaritons in a monolayer semiconductor coupled to optical bound states in the continuum.Light. Sci. Appl.9, 56 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41377-020-0286-z\"\n },\n {\n \"idx\": \"12.\",\n \"title\": \"Ardizzone, V. et al. Polariton Bose-Einstein condensate from a Bound State in the Continuum.Nature605, 447–452 (2022).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-022-04583-7\"\n },\n {\n \"idx\": \"14.\",\n \"title\": \"Kasprzak, J. et al. Bose-Einstein condensation of exciton polaritons.Nature443, 409–414 (2006).\",\n \"link\": \"https://doi.org/10.1038%2Fnature05131\"\n },\n {\n \"idx\": \"15.\",\n \"title\": \"Balili, R., Hartwell, V., Snoke, D., Pfeiffer, L. & West, K. Bose-Einstein condensation of microcavity polaritons in a trap.Science316, 1007–1010 (2007).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1140990\"\n },\n {\n \"idx\": \"16.\",\n \"title\": \"Byrnes, T., Kim, N. Y. & Yamamoto, Y. Exciton-polariton condensates.Nat. Phys.10, 803–813 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys3143\"\n },\n {\n \"idx\": \"17.\",\n \"title\": \"Cristofolini, P. et al. Optical superfluid phase transitions and trapping of polariton condensates.Phys. Rev. Lett.110, 186403 (2013).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.110.186403\"\n },\n {\n \"idx\": \"18.\",\n \"title\": \"Amo, A. et al. Superfluidity of polaritons in semiconductor microcavities.Nat. Phys.5, 805–810 (2009).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys1364\"\n },\n {\n \"idx\": \"19.\",\n \"title\": \"Lerario, G. et al. Room-temperature superfluidity in a polariton condensate.Nat. Phys.13, 837–841 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys4147\"\n },\n {\n \"idx\": \"21.\",\n \"title\": \"Utsunomiya, S. et al. Observation of Bogoliubov excitations in exciton-polariton condensates.Nat. Phys.4, 700 (2008).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys1034\"\n },\n {\n \"idx\": \"22.\",\n \"title\": \"Kohnle, V. et al. From single particle to superfluid excitations in a dissipative polariton gas.Phys. Rev. Lett.106, 255302 (2011).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.106.255302\"\n },\n {\n \"idx\": \"23.\",\n \"title\": \"Pieczarka, M. et al. Ghost branch photoluminescence from a polariton fluid under nonresonant excitation.Phys. Rev. Lett.115, 186401 (2015).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.115.186401\"\n },\n {\n \"idx\": \"24.\",\n \"title\": \"Stepanov, P. et al. Dispersion relation of the collective excitations in a resonantly driven polariton fluid.Nat. Commun.10, 3869 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-019-11886-3\"\n },\n {\n \"idx\": \"25.\",\n \"title\": \"Ballarini, D. et al. Directional Goldstone waves in polariton condensates close to equilibrium.Nat. Commun.11, 217 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-019-13733-x\"\n },\n {\n \"idx\": \"26.\",\n \"title\": \"Wouters, M. & Carusotto, I. Excitations in a nonequilibrium bose-einstein condensate of exciton polaritons.Phys. Rev. Lett.99, 140402 (2007).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.99.140402\"\n },\n {\n \"idx\": \"27.\",\n \"title\": \"Byrnes, T., Horikiri, T., Ishida, N., Fraser, M. & Yamamoto, Y. Negative Bogoliubov dispersion in exciton-polariton condensates.Phys. Rev. B85, 075130 (2012).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.85.075130\"\n },\n {\n \"idx\": \"28.\",\n \"title\": \"Pieczarka, M. et al. Observation of quantum depletion in a non-equilibrium exciton-polariton condensate.Nat. Commun.11, 429 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-019-14243-6\"\n },\n {\n \"idx\": \"29.\",\n \"title\": \"Pieczarka, M. et al. Bogoliubov excitations of a polariton condensate in dynamical equilibrium with an incoherent reservoir.Phys. Rev. B105, 224515 (2022).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.105.224515\"\n },\n {\n \"idx\": \"30.\",\n \"title\": \"Grudinina, A. M., Kurbakov, I. L., Lozovik, Y. E. & Voronova, N. S. Finite-temperature Hartree-Fock-Bogoliubov theory for exciton-polaritons.Phys. Rev. B104, 125301 (2021).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.104.125301\"\n },\n {\n \"idx\": \"31.\",\n \"title\": \"Ticknor, C., Wilson, R. M. & Bohn, J. L. Anisotropic superfluidity in a dipolar bose gas.Phys. Rev. Lett.106, 065301 (2011).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.106.065301\"\n },\n {\n \"idx\": \"32.\",\n \"title\": \"Bismut, G. et al. Anisotropic excitation spectrum of a dipolar quantum bose gas.Phys. Rev. Lett.109, 155302 (2012).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.109.155302\"\n },\n {\n \"idx\": \"33.\",\n \"title\": \"Wenzel, M. et al. Anisotropic superfluid behavior of a dipolar Bose-Einstein condensate.Phys. Rev. Lett.121, 030401 (2018).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.121.030401\"\n },\n {\n \"idx\": \"34.\",\n \"title\": \"Lozovik, Y. E., Kurbakov, I. L. & Volkov, P. A. Anisotropic superfluidity of two-dimensional excitons in a periodic potential.Phys. Rev. B95, 245430 (2017).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.95.245430\"\n },\n {\n \"idx\": \"36.\",\n \"title\": \"Sigurdsson, H., Li, G. & Liew, T. C. H. Spontaneous and superfluid chiral edge states in exciton-polariton condensates.Phys. Rev. B96, 115453 (2017).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.96.115453\"\n },\n {\n \"idx\": \"37.\",\n \"title\": \"Ciuti, C., Schwendimann, P. & Quattropani, A. Parametric luminescence of microcavity polaritons.Phys. Rev. B63, 041303(R) (2001).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.63.041303\"\n }\n]","isConversational":false}}},{"rowIdx":92,"cells":{"unique_id":{"kind":"string","value":"s41467-020-16069-z"},"subfield":{"kind":"string","value":"Materials science"},"title":{"kind":"string","value":"Twist-tailoring Coulomb correlations in van der Waals homobilayers"},"abstract":{"kind":"string","value":"The recent discovery of artificial phase transitions induced by stacking monolayer materials at magic twist angles represents a paradigm shift for solid state physics. Twist-induced changes of the single-particle band structure have been studied extensively, yet a precise understanding of the underlying Coulomb correlations has remained challenging. Here we reveal in experiment and theory, how the twist angle alone affects the Coulomb-induced internal structure and mutual interactions of excitons. In homobilayers of WSe 2 , we trace the internal 1 s –2 p resonance of excitons with phase-locked mid-infrared pulses as a function of the twist angle. Remarkably, the exciton binding energy is renormalized by up to a factor of two, their lifetime exhibits an enhancement by more than an order of magnitude, and the exciton-exciton interaction is widely tunable. Our work opens the possibility of tailoring quasiparticles in search of unexplored phases of matter in a broad range of van der Waals heterostructures."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"To tackle the formidable many-body scenario of Coulomb correlations in solids, Lev Landau introduced the concept of quasiparticles. These fictitious entities, which consist of particles dressed by their many-body interactions, are usually characteristic of material composition. Yet van der Waals layered materials provide an efficient tuning knob: the twist angle between adjacent layers can turn the semimetal graphene into a Mott insulator [1] , superconductor [2] , or ferromagnet [3] ; topological phases [4] , [5] , [6] , [7] , spin-pseudospin coupling [8] , and shear solitons [9] have been discussed for twisted bilayers of transition metal dichalcogenides (TMDs). TMDs form a unique laboratory to explore such effects because quantum confinement in two dimensions and reduced dielectric screening give rise to exceptionally strong Coulomb interactions. In the monolayer limit, this situation allows hydrogen-like electron-hole pairs—excitons—to bind with energies on the order of hundreds of meV (refs. [10] , [11] , [12] ). In van der Waals heterostructures, the relative orientation between adjacent layers strongly influences the electronic structure because of a subatomic variation of the orbital overlap of adjacent layers [1] , [2] , [3] , [4] , [5] , [6] , [9] , [13] , [14] . Hybridization effects between the electronic states of the constituent layers allow the band structure to be externally controlled by the twist angle θ (refs. [13] , [15] , [16] , [17] , [18] , [19] , [20] ). Interband photoluminescence combined with theoretical calculations have provided valuable insight into these phenomena [16] , [21] , [22] , but cannot resolve the internal excitonic structure which reveals the Coulomb correlations within a bilayer (BL). Interband spectroscopy can only probe optically bright excitons, whereas the formation and decay dynamics of excitons in prototypical monolayer TMDs such as WSe 2 —and especially in twisted BLs—are often dominated by dark states [23] , [24] . Mid-infrared (MIR) photons, in contrast, may directly interrogate Lyman-like 1 s –2 p transitions of excitonic species, irrespective of interband dipole moments, center-of-mass momenta, spin, or spatial separation of electron and hole. This concept [25] has been employed to probe Coulomb correlations [26] in quantum wells [27] , TMD monolayers [28] , [29] and heterostructures [30] , where electronic correlations are dominated by chemical composition and quantum confinement. By exploiting the internal 1 s –2 p resonance of excitons in homobilayers of WSe 2 , we demonstrate how the twist angle renormalizes the exciton binding energy by a factor of up to two and enhances their lifetime by more than an order of magnitude, whereas the exciton–exciton interaction is widely tunable at a certain angle. This fascinating scenario can be quantitatively explained with our microscopic many-body theory, which identifies the wavefunction overlap, interlayer hopping, and hybridization effects at the atomic interface as effective means to synthesize custom-tailored hybrid exciton species.","references":{"indices":[1,2,3,4,5,6,7,8,9,10,11,12,13,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],"infos":[{"idx":"1","title":"Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices.Nature556, 80–84 (2018).","link":"https://doi.org/10.1038%2Fnature26154"},{"idx":"2","title":"Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices.Nature556, 43–50 (2018).","link":"https://doi.org/10.1038%2Fnature26160"},{"idx":"3","title":"Sharpe, A. L. et al. Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene.Science365, 605–608 (2019).","link":"https://doi.org/10.1126%2Fscience.aaw3780"},{"idx":"4","title":"Tong, Q. et al. Topological mosaics in moiré superlattices of van der Waals heterobilayers.Nat. Phys.13, 356–362 (2016).","link":"https://doi.org/10.1038%2Fnphys3968"},{"idx":"5","title":"Wu, F., Lovorn, T. & MacDonald, A. H. Topological exciton bands in Moiré Heterojunctions.Phys. Rev. Lett.118, 147401 (2017).","link":"https://doi.org/10.1103%2FPhysRevLett.118.147401"},{"idx":"6","title":"Yu, H., Liu, G.-B., Tang, J., Xu, X. & Yao, W. Moiré excitons: from programmable quantum emitter arrays to spin-orbit–coupled artificial lattices.Sci. Adv.3, e1701696 (2017).","link":"https://doi.org/10.1126%2Fsciadv.1701696"},{"idx":"7","title":"Wu, F., Lovorn, T., Tutuc, E., Martin, I. & MacDonald, A. H. Topological insulators in twisted transition metal dichalcogenide homobilayers.Phys. Rev. Lett.122, 086402 (2019).","link":"https://doi.org/10.1103%2FPhysRevLett.122.086402"},{"idx":"8","title":"Jin, C. et al. Identification of spin, valley and moiré quasi-angular momentum of interlayer excitons.Nat. Phys.15, 1140–1144 (2019).","link":"https://doi.org/10.1038%2Fs41567-019-0631-4"},{"idx":"9","title":"Naik, M. H. & Jain, M. Ultraflatbands and shear solitons in Moiré patterns of twisted bilayer transition metal dichalcogenides.Phys. Rev. Lett.121, 266401 (2018).","link":"https://doi.org/10.1103%2FPhysRevLett.121.266401"},{"idx":"10","title":"Novoselov, K. S., Mishchenko, A., Carvalho, A. & Castro Neto, A. H. 2D materials and van der Waals heterostructures.Science353, aac9439 (2016).","link":"https://doi.org/10.1126%2Fscience.aac9439"},{"idx":"11","title":"Ugeda, M. M. et al. Giant bandgap renormalization and excitonic effects in a monolayer transition metal dichalcogenide semiconductor.Nat. Mater.13, 1091–1095 (2014).","link":"https://doi.org/10.1038%2Fnmat4061"},{"idx":"12","title":"Xu, X., Yao, W., Xiao, D. & Heinz, T. F. Spin and pseudospins in layered transition metal dichalcogenides.Nat. Phys.10, 343–350 (2014).","link":"https://doi.org/10.1038%2Fnphys2942"},{"idx":"13","title":"Alexeev, E. M. et al. Resonantly hybridized excitons in moiré superlattices in van der Waals heterostructures.Nature567, 81–86 (2019).","link":"https://doi.org/10.1038%2Fs41586-019-0986-9"},{"idx":"15","title":"Kunstmann, J. et al. Momentum-space indirect interlayer excitons in transition-metal dichalcogenide van der Waals heterostructures.Nat. Phys.14, 801–805 (2018).","link":"https://doi.org/10.1038%2Fs41567-018-0123-y"},{"idx":"16","title":"Van Der Zande, A. M. et al. Tailoring the electronic structure in bilayer molybdenum disulfide via interlayer twist.Nano Lett.14, 3869–3875 (2014).","link":"https://doi.org/10.1021%2Fnl501077m"},{"idx":"17","title":"Jiang, T. et al. Valley and band structure engineering of folded MoS2bilayers.Nat. Nanotechnol.9, 825–829 (2014).","link":"https://doi.org/10.1038%2Fnnano.2014.176"},{"idx":"18","title":"Horng, J. et al. Observation of interlayer excitons in MoSe2single crystals.Phys. Rev. B97, 241404 (2018).","link":"https://doi.org/10.1103%2FPhysRevB.97.241404"},{"idx":"19","title":"Hsu, W.-T. et al. Tailoring excitonic states of van der Waals bilayers through stacking configuration, band alignment, and valley spin.Sci. Adv.5, eaax7407 (2019).","link":"https://doi.org/10.1126%2Fsciadv.aax7407"},{"idx":"20","title":"Gerber, I. C. et al. Interlayer excitons in bilayer MoS2with strong oscillator strength up to room temperature.Phys. Rev. B99, 035443 (2019).","link":"https://doi.org/10.1103%2FPhysRevB.99.035443"},{"idx":"21","title":"Liu, K. et al. Evolution of interlayer coupling in twisted molybdenum disulfide bilayers.Nat. Commun.5, 4966 (2014).","link":"https://doi.org/10.1038%2Fncomms5966"},{"idx":"22","title":"Wang, Z., Chiu, Y.-H., Honz, K., Mak, K. F. & Shan, J. Electrical tuning of interlayer exciton gases in WSe2bilayers.Nano Lett.18, 137–143 (2018).","link":"https://doi.org/10.1021%2Facs.nanolett.7b03667"},{"idx":"23","title":"Nayak, P. K. et al. Probing evolution of twist-angle-dependent Interlayer excitons in MoSe2/WSe2van der Waals heterostructures.ACS Nano11, 4041–4050 (2017).","link":"https://doi.org/10.1021%2Facsnano.7b00640"},{"idx":"24","title":"Zhang, X.-X., You, Y., Zhao, S. Y. F. & Heinz, T. F. Experimental evidence for dark excitons in monolayer WSe2.Phys. Rev. Lett.115, 257403 (2015).","link":"https://doi.org/10.1103%2FPhysRevLett.115.257403"},{"idx":"25","title":"Ulbricht, R., Hendry, E., Shan, J., Heinz, T. F. & Bonn, M. Carrier dynamics in semiconductors studied with time-resolved terahertz spectroscopy.Rev. Mod. Phys.83, 543–586 (2011).","link":"https://doi.org/10.1103%2FRevModPhys.83.543"},{"idx":"26","title":"Huber, R. et al. How many-particle interactions develop after ultrafast excitation of an electron–hole plasma.Nature414, 286–289 (2001).","link":"https://doi.org/10.1038%2F35104522"},{"idx":"27","title":"Kaindl, R. A., Carnahan, M. A., Hägele, D., Lövenich, R. & Chemla, D. S. Ultrafast terahertz probes of transient conducting and insulating phases in an electron–hole gas.Nature423, 734–738 (2003).","link":"https://doi.org/10.1038%2Fnature01676"},{"idx":"28","title":"Poellmann, C. et al. Resonant internal quantum transitions and femtosecond radiative decay of excitons in monolayer WSe2.Nat. Mater.14, 889–893 (2015).","link":"https://doi.org/10.1038%2Fnmat4356"},{"idx":"29","title":"Yong, C.-K. et al. Valley-dependent exciton fine structure and Autler–Townes doublets from Berry phases in monolayer MoSe2.Nat. Mater.18, 1065–1070 (2019).","link":"https://doi.org/10.1038%2Fs41563-019-0447-8"},{"idx":"30","title":"Merkl, P. et al. Ultrafast transition between exciton phases in van der Waals heterostructures.Nat. Mater.18, 691–696 (2019).","link":"https://doi.org/10.1038%2Fs41563-019-0337-0"}]}},{"section":"Results","content":"Rydberg spectroscopy of excitons in twisted WSe 2 BLs The WSe 2 BL samples are produced by mechanical exfoliation and subsequent deterministic transfer onto a diamond substrate (see Methods). Monolayer flakes with diameters exceeding 70 µm (Fig. 1a ) are stamped on top of each other. Before the transfer, the crystallographic orientations of the layers are determined by polarization-resolved second-harmonic generation [15] , [31] . The component of the second harmonic I 2ω,|| polarized parallel to the near-infrared pump laser indicates the armchair directions of each monolayer (inset Fig. 1a , blue/orange flower patterns), revealing the relative twist angle θ between both layers (see Methods). We fabricated a series of BLs with θ ranging from 0° to 60° (see Supplementary Figs. 1 and 2 ). Fig. 1: NIR pump–MIR probe spectroscopy of twisted WSe 2 bilayers. a Optical microscope image of a representative twisted homobilayer (BL) on a diamond substrate. The blue frame indicates the lower WSe 2 monolayer, which is covered by a second WSe 2 monolayer (orange frame) forming a twisted BL in the overlap region (black frame). Scale bar: 50 µm. Inset: component of the second harmonic I 2ω,|| polarized parallel to the pump laser light. The characteristic patterns of sixfold symmetry indicate the armchair directions of the bottom (blue dots) and top (orange dots) monolayers, enclosing a twist angle, θ = 53°, close to the natural 2H configuration. b Normalized photoluminescence spectra for samples with different twist angles θ , recorded at a temperature of 5 K (vertically offset for clarity). The black dotted line serves as a guide to the eye highlighting the evolution of the K-Λ transition. c Sketch of the experiment, where a near-infrared (NIR) pump pulse (orange intensity envelope) optically injects 1 s A excitons in the WSe 2 BLs. The excited sample is probed by a mid-infrared waveform (MIR, red curve). The exciton dispersion for a given orbital quantum number is depicted as a function of the center-of-mass momentum q (gray paraboloids). Red arrows indicate the 1 s –2 p transitions of the exciton, interrogated by the MIR photons at an energy ħω MIR . Full size image The samples were pre-characterized using photoluminescence spectroscopy with an excitation wavelength of 488 nm (see Methods). The resulting spectra (Fig. 1b ) feature a single emission peak at an energy of ~1.54 eV for the pristine BL with θ = 60° (2H stacking), which we attribute to the phonon-assisted recombination of K-Λ excitons [32] . For θ = 59°, this peak blue shifts by 20 meV, whereas a second maximum emerges at an energy of ~1.65 eV, corresponding to the direct K-K transition. This trend continues for θ = 53° until the two spectral features merge for θ = 30°. When the twist angle is further reduced to θ = 5° and θ = 0° (i.e., the 3R stacking) the two peaks are discernible again and move towards energies of 1.52 eV and 1.66 eV, respectively. Similar photoluminescence characteristics of BLs have been reported in the literature [16] , [17] , [21] . To explore the actual microscopic origin and the role of Coulomb correlations in the photoexcited twisted BLs, we study the internal structure and binding energy of the excitons. To this end, we generate 1 s A excitons in the WSe 2 BLs with a 100-fs near-infrared (NIR) laser pulse centered at an energy of 1.68 eV (Fig. 1c , orange intensity envelope). After a variable pump-probe delay time t pp , the exciton ensemble is probed by a phase-locked MIR pulse (Fig. 1c , red wave), the electric field of which is monitored by electro-optic sampling (see Methods and Supplementary Fig. 3 ). The intraexcitonic 1 s –2 p transition (Fig. 1c , red arrows) imprints a characteristic change on the MIR waveform, from which information about the excitonic correlations of the non-equilibrium system can be extracted. A Fourier domain Fresnel analysis [26] , [27] , [28] , [30] directly reveals the pump-induced change of the full dielectric response, including the change of the real part of the optical conductivity, Δ σ 1 , and the dielectric function, Δ ε 1 , at variable delays t pp . Fig. 2 summarizes the changes in dielectric function measured for seven WSe 2 BLs with varying twist angles θ , at t pp = 5.1 ps, where the photoexcited state has thermalized (see Supplementary Fig. 4 ). For all twist angles, Δ σ 1 (Fig. 2a ) exhibits a distinct maximum with a corresponding dispersive feature in Δ ε 1 (Fig. 2b ), signifying a resonant response of the photoexcited sample. As we demonstrate below, these resonances relate to the internal 1 s –2 p transition of BL excitons. Between θ = 60° and 30°, the resonance blue shifts with decreasing angle (Fig. 2a dashed line, blue arrow). Strikingly, for θ = 0° the resonance exhibits the opposite behavior: it shifts substantially to the red and gains oscillator strength (red arrow). A phenomenological analysis of the data with a two-component model (see Methods) confirms a systematic blue shift of the resonance (Fig. 2c ) from 100 meV for θ = 60° to 121 meV for θ = 30°. Importantly, the resonance at θ = 0° is red shifted to 67 meV—corresponding to an energy reduction by a factor of two. This strong asymmetry of the 1 s –2 p transition energy—a direct measure of the exciton binding energy [28] , [30] (see Methods)—for θ = 0° and θ = 60° is in stark contrast to the almost degenerate interband photoluminescence (see Fig. 1b ). Fig. 2: Evolution of the 1 s –2 p resonance with interlayer twist angle. a , b Pump-induced changes of the real parts of the optical conductivity Δ σ 1 ( a ) and the dielectric function Δ ε 1 ( b ) for a fixed pump-probe delay time t pp = 5.1 ps as a function of the photon energy for samples with different twist angles θ . The blue spheres represent the dielectric response measured for the photoexcited WSe 2 BLs (pump fluence Φ = 27 µJ cm −2 ; sample temperature, 5 K). The blue shaded areas indicate the fits to the experimental data by a phenomenological model (see Methods). The dashed line and the blue/red arrow indicate the blue/red shift of the intraexcitonic 1 s –2 p resonance energy for increasing misalignment angle with respect to the 2H stacking. c Intraexcitonic 1 s –2 p transition energy ħω res (black spheres) extracted from the data in a , b and derived from the microscopic theory (blue circles) as function of the twist angle θ . For θ = 21° and θ = 30°, microscopic calculations of hybrid excitons were not feasible. In the case of θ = 0°, the intraexcitonic resonance exhibits a strong dependence on the exciton density (see Fig. 4c and Supplementary Note 4 ). The gray sphere indicates the resonance energy as obtained for a pump fluence of Φ = 7 µJ cm −2 , yielding the best agreement with the microscopic theory. The error bars represent the 95% confidence interval of the fitting procedure. Full size image Microscopic many-body theory To explain these surprising observations, we have to go beyond single-particle band structure effects and include Coulomb correlations, by a combination of ab initio density functional theory (DFT) calculations with density matrix theory (see Methods). In WSe 2 BLs, the energetic minimum of the conduction band lies at the Λ and Λ′ points. Thus, following creation of excitons at the Κ point (Fig. 3a , red arrow), electrons scatter via phonons into the global energetic minima [33] (Fig. 3a , blue arrow), corroborated by a transient shift of the excitonic 1 s –2 p resonance (see Supplementary Fig. 4 ). The conduction band states in the Λ and Λ′ valleys are composed of selenium p orbitals, whose overlap with the corresponding wavefunctions of the neighboring layer leads to efficient interlayer hopping [34] and hybridization of the excitonic wavefunctions (X hyb ). We study this scenario by tracing excitons with their hole fixed in one layer—without loss of generality assigned to the top layer—whereas the electrons are located in Λ-like valleys of either the top (Λ top ) or the bottom (Λ bot ) layer (see Fig. 3b and Methods), forming intra- (X intra ) or interlayer excitons (X inter ). The energy of both exciton species is influenced by the electron-hole Coulomb attraction. The interlayer exciton energy depends further on the spin-orbit splitting of the conduction band in the bottom layer. The parabolic dispersion relations of the excitons, E intra and E inter , are mutually offset in momentum space by θ (see Fig. 3b ). Depending on the wavefunction overlap of X intra and X inter in momentum space (Fig. 3c , blue/orange grid), these states can hybridize effectively. The overlap (Fig. 3c , violet surfaces) depends on θ and the orbital quantum number of the excitonic states, which defines the momentum spread of the wavefunctions. Upon hybridization, bonding and anti-bonding exciton states are formed. As the energy splitting between bonding and anti-bonding 1 s states is as large as ~500 meV, the anti-bonding exciton state is pushed outside of our MIR probe spectrum. Therefore, we only consider the energetically lower 1 s - and 2 p -hybrid states (Fig. 3d , magenta lines). As the energy reduction differs for these two states, the 1 s –2 p transition energy varies with θ (Fig. 3d ). Fig. 3: Hybridization between intra- and interlayer excitons. a Single-particle band structure of the WSe 2 BLs for θ = 60°, calculated by density functional theory (see Methods). After resonant excitation (red arrow) at the K point, electrons scatter via phonons (blue arrow) into the energetically favorable Λ valley to form hybrid excitons (X hyb ). b Within the microscopic model, the electron is located in a Λ-like valley of the top (Λ top ) or bottom (Λ bot ) layer for a finite crystallographic misalignment. The parabolic dispersion of intra- ( E intra ) and interlayer ( E inter ) excitons is shifted in momentum space by twisting the layers, owing to the momentum mismatch Δ k between Λ top and Λ bot . Strong interlayer wavefunction overlap promotes electron hopping and exciton hybridization (pink glow). c The overlap (violet surface) of excitonic wavefunctions (blue and orange grids) in momentum space depends on Δ k . For clarity, we only show the real part of the exciton wavefunctions. d Hybridization of 1 s and 2 p excitons lowers the energy levels (magenta lines) with respect to the corresponding non-hybridized states X intra and X inter (blue/orange lines). The situation is sketched for θ = 60°, θ = 50°, and θ = 0°. Zero energy is set to the non-hybridized 1 s state at θ = 60°. The vertical black arrows mark respective 1 s –2 p transition energies. Full size image For θ = 60°, X intra and X inter feature very different energies (Fig. 3d ). This situation is characteristic of the 2H order, where the momentum overlap is strongest for intra- and interlayer exciton states containing electrons from the Λ and Λ′ valley, respectively. As hybridization can only occur for spin-like electron bands and spin-orbit splitting is of opposite sign for the two valleys, the non-hybridized excitons are offset by the strong spin-orbit splitting. Despite this energy mismatch, the energy of X hyb is clearly reduced. For increasing misalignment ( θ < 60°) with respect to 2H stacking, the 1 s states, which are spread out over a broad range of momenta, continue to effectively hybridize. Conversely, the 2 p wavefunctions are more strongly localized in momentum space. Hence misalignment towards θ = 30° rapidly quenches hybridization in the 2 p state and therefore reduces the energy gain. The combination of these mechanisms results in a distinct blue shift of the 1 s –2 p resonance, in excellent agreement with the experiment (Fig. 2c , blue circles). For 3R stacking, X intra and X inter are almost degenerate because both states contain electrons from the same valley and are therefore not offset by spin-orbit splitting. In particular, the 2 p states are almost perfectly degenerate, which enhances the excitonic hybridization as compared to the pristine bilayer ( θ = 60°) and leads to an even smaller 1 s –2 p transition energy (see Fig. 3d ). Tailoring the excitonic lifetime The variable degree of interlayer hybridization of excitons allows us to tune the exciton dynamics as well as the exciton–exciton interaction (Fig. 4 ). We trace the temporal evolution of the exciton population by mapping out the pump-induced change of the MIR electric field Δ E at a fixed electro-optic sampling time, t eos = 0 fs (see ref. [28] and Methods), for a variable pump-probe delay time t pp . Fig. 4a shows a rapid appearance of exciton density at t pp = 0 ps, for all twist angles, whereas the decay dynamics depend sensitively on θ . An exponential fit (Fig. 4a , dashed lines) shows a systematic reduction of the lifetime of the hybrid excitons (Fig. 4b ) from τ = 54 ps to 8 ps as θ is reduced from 60° to 30°. Intriguingly, for θ = 0°, the lifetime rises dramatically to a value of 148 ps. This behavior can be understood by considering the hybridized wavefunctions in real space. Fig. 4: Ultrafast dynamics and density-dependent energy renormalization of hybridized excitons. a Pump-induced change of the mid-infrared electric field Δ E as a function of the pump-probe delay time t pp recorded at fixed electro-optic sampling time t eos = 0 fs for different twist angles θ (spheres). Curves are vertically offset for clarity. The dashed lines represent fits to the data with an exponential decay. b Exciton decay time τ as a function of the twist angle θ , determined by the decay constant of the fit in a . The error bars indicate one standard deviation. c 1 s –2 p resonance energy ħω res for different pump fluences Φ at a pump-probe delay time t pp = 5.1 ps recorded for twist angles θ = 0°, 5°, and 60°, respectively (spheres). The resonance energies were extracted from the dielectric response by fits to a phenomenological model (see Supplementary Fig. 5 ). The blue circle marks the resonance energy derived from the microscopic theory shown in Fig. 2c . The sample was kept at a temperature of 5 K. The error bars represent the 95% confidence interval of the fitting procedure. d Calculated probability density of the electron within the two layers for a hybrid exciton for twist angles θ = 60°, 50°, and 0°, where the hole is chosen to remain at a fixed position in the top layer. Full size image To do this, we keep the position of the hole fixed in the upper TMD layer and plot the probability density of the electron (Fig. 4d , blue peaks). Because of strong interlayer hopping between the Λ and Λ′ points of the adjacent layers, the electron delocalizes within the BL for θ = 60°. A distinct probability density in the bottom layer (23%) signifies the interlayer character of the hybrid exciton. By changing the twist angle to θ = 50°, the orbital overlap of the conduction band states shrinks, reducing the probability density of the electron in the bottom layer (18%). With both electron and hole wavefunctions overlapping strongly in the top layer, the lifetime of the exciton drops, as processes such as phonon-assisted radiative decay [32] , [35] or Auger recombination [28] likely become more efficient. For θ = 30°, the interlayer hopping becomes even weaker, which further reduces the exciton lifetime. In contrast, for θ = 0° interlayer hopping is facilitated by the energetic degeneracy of the Λ-valley conduction band states discussed above. In this case, the strong delocalization of the electron over the top (58%) and bottom (42%) layers distinctly extends the exciton lifetime as the hybrid state becomes more interlayer-like. At the same time, the delocalized wavefunctions provide a larger transition dipole moment, enhancing the amplitude of Δ σ 1 for θ = 0° as compared with θ = 60° (Fig. 2a ). Shaping the interaction between excitons The twist angle controlled interlayer delocalization can also be employed to tune the interaction between excitons. To this end, we record the position of the 1 s –2 p resonance as a function of the pump fluence (Fig. 4c ). For θ = 0°, a dramatic red shift is observed, whereas the effect is less pronounced for θ = 5° and completely absent for θ = 60° (see Supplementary Fig. 5 , for the full dielectric response). We expect that an effective renormalization of the exciton binding energy should ultimately occur for all twist angles, when the exciton densities are increased beyond the values accessible in our experiment. Microscopically, the critical exciton density at which screening effects become important is determined by the exciton wavefunction. The larger the respective Bohr radius, the lower the critical density at which the orbitals start to overlap. Hence, strong interlayer delocalization accompanied by a larger lateral extent of the wavefunction ( θ = 0°, see Fig. 4d ) enables excitons to effectively screen the mutual Coulomb attraction between electron and hole. In addition, the delocalization of the excitons across the two layers for θ = 0° increases their polarizability and enables an out-of-plane dipole-dipole type interaction [36] . Consequently, hybrid excitons experience a stronger renormalization of their internal transitions with increasing exciton density as compared to previous reports on intralayer excitons [28] , [37] . Thus, for increasing pump fluences the exciton binding energy is further reduced beyond the value of the hybrid exciton with negligible many-body interactions. Therefore, the theoretically predicted transition energy (see Figs. 2 c and 4c , blue circle) is in very good agreement with the experimental results only for low fluences (compare grey sphere in Fig. 2c )."},{"section":"Discussion","content":"In conclusion, we have tailored Coulomb correlations by interlayer hybridization in twisted WSe 2 BLs. By accessing internal 1 s –2 p transitions of the K-Λ excitons, we reveal systematic changes of their binding energy and lifetime with twist angle. A microscopic model based on density functional and density matrix theory quantitatively describes the underlying hybridization effects. Furthermore, we reveal how the density-dependent exciton–exciton interaction is renormalized by the twist angle. These findings underpin a qualitatively different way of fine-tuning the internal structure of excitons in TMDs and controlling the electronic and optical properties in van der Waals structures based on a single material. Tailoring many-body correlations over a large energy range can now be systematically extended to the full library of two-dimensional crystals, which should guide the search for man-made electronic phases in van der Waals heterostructures."},{"section":"Methods","content":"Sample fabrication and photoluminescence spectroscopy The TMD monolayers were exfoliated mechanically from a bulk single crystal using the viscoelastic transfer method [38] . Both constituent WSe 2 monolayers were inspected under an optical microscope and subsequently stacked on top of each other on a diamond substrate. During the transfer process, the lateral position and rotation of the layers can be controlled accurately using micro-positioning stages. To remove any adsorbates trapped between the monolayers, the BLs were annealed at a temperature of 150 °C and a pressure of 1 × 10 −5 mbar for 5 hours. Subsequently, we recorded the microphotoluminescence intensity of the BLs with a continuous-wave laser (wavelength, 488 nm) focused onto the sample using a microscope objective (numerical aperture, 0.6). The radiation emitted by the BLs is then detected in backscattering geometry, dispersed in a spectrometer, and read out by a charge-coupled device camera. Second-harmonic generation The twist angle θ was determined for each BL by polarization-resolved second-harmonic generation (SHG) under rotation of the plane of polarization [31] . Isolated monolayer regions of the structure were excited by a linearly-polarized femtosecond mode-locked Ti:sapphire laser and the intensity of the SHG was recorded. Characteristic sixfold patterns, whose maxima indicate the armchair directions of the monolayers, are then obtained. For the representative structure in Fig. 1a , the intensity maxima span an angle ~7°. As the method is not sensitive to the electric field of the second harmonic but only to its intensity, an ambiguity remains between twist angles of θ = 7° and θ = 60°-7°. This ambiguity can be resolved by recording the sixfold pattern of the SHG polarization anisotropy of the bilayer area of the sample (see Supplementary Fig. 2 ). The strong suppression of the SHG emission of the bilayer region attests to the fact that the two layers are almost perfectly anti-aligned [39] , [40] corresponding to an angle close to the natural 2H stacking. Thus, the twist angle of the sample presented in Fig. 1a can be determined as θ = 53°. For the other BLs, an identical procedure is performed (see Supplementary Note 1 ). Ultrafast NIR pump–MIR probe spectroscopy A schematic of the experimental principle is depicted in Fig. 1c . 1 s A excitons are resonantly created in the WSe 2 BL by photoexcitation with 100-fs pump pulses centered at an energy of 1.69 eV (Supplementary Fig. 3a , inset) from a home-built Ti:sapphire laser amplifier (repetition rate, 400 kHz). The phase-locked MIR probe pulses are generated by optical rectification in a GaSe crystal (thickness, 10 µm) using a fraction of the laser output. After a variable delay time t pp , the probe pulses are transmitted through the photoexcited sample and the MIR waveform is recorded by electro-optic sampling in a second 10 µm-thick GaSe crystal as a function of the electro-optic sampling time t eos (Supplementary Fig. 3a ). The MIR probe pulses are centered at a frequency of 28 THz with a full width at half maximum of 14 THz and a spectral phase, which is approximately flat between 11 and 44 THz (Supplementary Fig. 3b ). Any changes to the electric field, Δ E ( t eos ), of the waveform induced by the transient dielectric response of the sample are resolved in absolute amplitude and phase by serial lock-in detection as a function of the electro-optic sampling time t eos . The full complex-valued dielectric response, characterized by Δ σ 1 and Δ ε 1 , can be retrieved from these data by applying a transfer matrix formalism [26] , [28] , [30] . This technique is sensitive to the total population of bound and unbound electron-hole pairs, irrespective of interband selection rules, and can, thus, address optically dark and bright excitons [41] . Microscopic many-particle theory To calculate the internal quantum transitions of excitons in the twisted BL, a many-particle Hamiltonian was diagonalized accounting for the Coulomb interaction on a Hartree-Fock level. For WSe 2 BLs, ab initio calculations suggest that momentum-dark K-Λ excitons represent the lowest states. Therefore, the Hamiltonian was restricted to electrons and holes at these high-symmetry points, taking into account the energetically most favorable dipole-allowed spin configuration. The BL was then treated as a bipartite system, using the monolayer eigenstates as basis functions of the two subsystems. Here, the layer-localized basis functions were treated in an effective mass approximation with parameters extracted from ab initio simulations for monolayers. The Coulomb interaction strength was implemented on a semiclassical level assuming anisotropic homogeneous slabs to account for the complex dielectric environment of the BL system. Ab initio parameters for the anisotropic dielectric tensor of WSe 2 were used and the influence of the substrate was included. The overlap of electronic wavefunctions of the two-layer subsystem gives rise to interlayer hybridization effects. Only the hybridization of electrons at the Λ point was considered, which is known to be orders of magnitude stronger than at the K point owing to different orbital compositions. The interlayer interaction strength was extracted from ab initio calculations of the BL band structure for different stacking angles. The Hamiltonian was diagonalized by first transforming it into an exciton frame to account for the Coulomb interaction. Subsequently, eigenenergies and wavefunctions were represented in terms of intra- and interlayer excitons, resulting from the Wannier equation for the BL. The overlap of electronic wavefunctions between the two layers results in an interaction of intra- and interlayer excitons. Finally, the excitonic Hamiltonian was diagonalized yielding intralayer-interlayer exciton hybrids. For the numerical calculation of the bilayer eigenstates we consider the hybridization of a large number of both intra- and interlayer excitons including states with angular momentum of s- , p- , d-, and f -type. The measured transition energies were then compared with the energetically lowest dipole-allowed transition of the calculated hybrid exciton Rydberg series where we neglect higher-order corrections such as a Berry curvature-induced splitting of the otherwise degenerate p states. For additional details see Supplementary Note 5 . Ab initio DFT calculations The electronic band structure was calculated on the basis of density functional theory in the Perdew-Burke-Ernzerhof approximation, as implemented in the Quantum ESPRESSO package. The atomic positions and in-plane lattice constants of twisted BL WSe 2 were optimized with different relative twist angles and the resulting atomic geometries were used to derive an average equilibrium distance between the two tungsten sublayers. Based on the optimized geometry of a 2H stacked BL, the evolution of the electronic band structure for varying interlayer distance was simulated. The required interlayer interaction strength to calculate the exciton hybridization was extracted from the splitting of the electronic bands at the Λ point. To obtain a continuous twist angle dependence of the interaction parameter, the interlayer distances obtained for computationally feasible angles were linearly interpolated. Spin-orbit interactions and quasiparticle energy corrections from the G 0 W 0 approximation were explicitly included to calculate the full band structure for 2H stacked WSe 2 . Here, the YAMBO code was used to calculate the quasiparticle corrections with spin-orbit effects. Subsequently, GW calculations with the BerkeleyGW code were performed, combined with a non-uniform Neck Subsampling method to derive an additional scissor-shift correction for the electronic band gaps. For additional details, see Supplementary Note 6 . Two-component model A phenomenological two-component model was used to extract the quantitative properties of the 1 s –2 p transition from the transient dielectric response. To this end, the pump-induced change of the dielectric function Δ ε ( ω ) = Δ ε 1 ( ω ) + Δ σ 1 ( ω )/( ε 0 ω ) is described by the Drude–Lorentz model: $$\\Delta \\varepsilon \\left( \\omega \\right) = \\frac{{n_{\\mathrm{X}}e^2}}{{d\\varepsilon _0\\mu }}\\frac{{f_{1s,2p}}}{{\\omega _{{\\mathrm{res}}}^2 - \\omega ^2 - {\\mathrm{i}}\\omega {\\mathrm{\\Delta }}}} - \\frac{{n_{{\\mathrm{fc}}}e^2}}{{d\\varepsilon _0{\\upmu}}}\\frac{1}{{\\omega ^2 + {\\mathrm{i}}\\omega \\Gamma }}$$ (1) where e is the electron charge and ε 0 denotes the vacuum permittivity. The Lo-rentzian resonance with oscillator strength f 1 s ,2 p describes the internal 1 s –2 p transition of the excitons within the WSe 2 BL as determined by the density matrix theory (see Methods). Owing to the twist angle-dependent hybridization, the oscillator strength equals f 1 s ,2 p = 0.34, f 1 s ,2 p = 0.33, f 1 s ,2 p = 0.23, f 1 s ,2 p = 0.18, f 1s,2p = 0.19, f 1 s ,2 p = 0.26, and f 1 s ,2 p = 0.42 for θ = 60°, 59°, 53°, 30°, 21°, 5°, and 0°, respectively. The resonance is further characterized by the exciton density n X , the resonance energy ħω res and the linewidth Δ, which are free fit parameters. The effective mass of the electron-hole pairs μ = 0.23 m 0 is determined with the electron and hole effective masses within the WSe 2 BL (ref. [42] ), where m 0 is the free-electron mass. The sample is treated as a thin slab in this model with a thickness d = 0.7 nm (ref. [28] ) of each layer. The second term in Δ ε ( ω ) is a Drude model describing unbound electron-hole pairs or excitonic transitions energetically below the spectral window of the MIR probe pulses [30] . Here, the density of free carriers n fc and their scattering rate Γ govern the resulting dielectric response. Strict limits on the possible values of the free parameters are set by the fact that both independently retrieved spectra Δ σ 1 and Δ ε 1 have to be reproduced simultaneously. Without any further restrictions, the numerical adaption of the measured spectra yields an overall good fit quality."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"To tackle the formidable many-body scenario of Coulomb correlations in solids, Lev Landau introduced the concept of quasiparticles. These fictitious entities, which consist of particles dressed by their many-body interactions, are usually characteristic of material composition. Yet van der Waals layered materials provide an efficient tuning knob: the twist angle between adjacent layers can turn the semimetal graphene into a Mott insulator [1] , superconductor [2] , or ferromagnet [3] ; topological phases [4] , [5] , [6] , [7] , spin-pseudospin coupling [8] , and shear solitons [9] have been discussed for twisted bilayers of transition metal dichalcogenides (TMDs). TMDs form a unique laboratory to explore such effects because quantum confinement in two dimensions and reduced dielectric screening give rise to exceptionally strong Coulomb interactions. In the monolayer limit, this situation allows hydrogen-like electron-hole pairs—excitons—to bind with energies on the order of hundreds of meV (refs. [10] , [11] , [12] ). In van der Waals heterostructures, the relative orientation between adjacent layers strongly influences the electronic structure because of a subatomic variation of the orbital overlap of adjacent layers [1] , [2] , [3] , [4] , [5] , [6] , [9] , [13] , [14] . Hybridization effects between the electronic states of the constituent layers allow the band structure to be externally controlled by the twist angle θ (refs. [13] , [15] , [16] , [17] , [18] , [19] , [20] ). Interband photoluminescence combined with theoretical calculations have provided valuable insight into these phenomena [16] , [21] , [22] , but cannot resolve the internal excitonic structure which reveals the Coulomb correlations within a bilayer (BL). Interband spectroscopy can only probe optically bright excitons, whereas the formation and decay dynamics of excitons in prototypical monolayer TMDs such as WSe 2 —and especially in twisted BLs—are often dominated by dark states [23] , [24] . Mid-infrared (MIR) photons, in contrast, may directly interrogate Lyman-like 1 s –2 p transitions of excitonic species, irrespective of interband dipole moments, center-of-mass momenta, spin, or spatial separation of electron and hole. This concept [25] has been employed to probe Coulomb correlations [26] in quantum wells [27] , TMD monolayers [28] , [29] and heterostructures [30] , where electronic correlations are dominated by chemical composition and quantum confinement. By exploiting the internal 1 s –2 p resonance of excitons in homobilayers of WSe 2 , we demonstrate how the twist angle renormalizes the exciton binding energy by a factor of up to two and enhances their lifetime by more than an order of magnitude, whereas the exciton–exciton interaction is widely tunable at a certain angle. This fascinating scenario can be quantitatively explained with our microscopic many-body theory, which identifies the wavefunction overlap, interlayer hopping, and hybridization effects at the atomic interface as effective means to synthesize custom-tailored hybrid exciton species.\",\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 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 28,\n 29,\n 30\n ],\n \"infos\": [\n {\n \"idx\": \"1\",\n \"title\": \"Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices.Nature556, 80–84 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fnature26154\"\n },\n {\n \"idx\": \"2\",\n \"title\": \"Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices.Nature556, 43–50 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fnature26160\"\n },\n {\n \"idx\": \"3\",\n \"title\": \"Sharpe, A. L. et al. Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene.Science365, 605–608 (2019).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.aaw3780\"\n },\n {\n \"idx\": \"4\",\n \"title\": \"Tong, Q. et al. Topological mosaics in moiré superlattices of van der Waals heterobilayers.Nat. Phys.13, 356–362 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys3968\"\n },\n {\n \"idx\": \"5\",\n \"title\": \"Wu, F., Lovorn, T. & MacDonald, A. H. Topological exciton bands in Moiré Heterojunctions.Phys. Rev. Lett.118, 147401 (2017).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.118.147401\"\n },\n {\n \"idx\": \"6\",\n \"title\": \"Yu, H., Liu, G.-B., Tang, J., Xu, X. & Yao, W. Moiré excitons: from programmable quantum emitter arrays to spin-orbit–coupled artificial lattices.Sci. Adv.3, e1701696 (2017).\",\n \"link\": \"https://doi.org/10.1126%2Fsciadv.1701696\"\n },\n {\n \"idx\": \"7\",\n \"title\": \"Wu, F., Lovorn, T., Tutuc, E., Martin, I. & MacDonald, A. H. Topological insulators in twisted transition metal dichalcogenide homobilayers.Phys. Rev. Lett.122, 086402 (2019).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.122.086402\"\n },\n {\n \"idx\": \"8\",\n \"title\": \"Jin, C. et al. Identification of spin, valley and moiré quasi-angular momentum of interlayer excitons.Nat. Phys.15, 1140–1144 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41567-019-0631-4\"\n },\n {\n \"idx\": \"9\",\n \"title\": \"Naik, M. H. & Jain, M. Ultraflatbands and shear solitons in Moiré patterns of twisted bilayer transition metal dichalcogenides.Phys. Rev. Lett.121, 266401 (2018).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.121.266401\"\n },\n {\n \"idx\": \"10\",\n \"title\": \"Novoselov, K. S., Mishchenko, A., Carvalho, A. & Castro Neto, A. H. 2D materials and van der Waals heterostructures.Science353, aac9439 (2016).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.aac9439\"\n },\n {\n \"idx\": \"11\",\n \"title\": \"Ugeda, M. M. et al. Giant bandgap renormalization and excitonic effects in a monolayer transition metal dichalcogenide semiconductor.Nat. Mater.13, 1091–1095 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat4061\"\n },\n {\n \"idx\": \"12\",\n \"title\": \"Xu, X., Yao, W., Xiao, D. & Heinz, T. F. Spin and pseudospins in layered transition metal dichalcogenides.Nat. Phys.10, 343–350 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys2942\"\n },\n {\n \"idx\": \"13\",\n \"title\": \"Alexeev, E. M. et al. Resonantly hybridized excitons in moiré superlattices in van der Waals heterostructures.Nature567, 81–86 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-019-0986-9\"\n },\n {\n \"idx\": \"15\",\n \"title\": \"Kunstmann, J. et al. Momentum-space indirect interlayer excitons in transition-metal dichalcogenide van der Waals heterostructures.Nat. Phys.14, 801–805 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41567-018-0123-y\"\n },\n {\n \"idx\": \"16\",\n \"title\": \"Van Der Zande, A. M. et al. Tailoring the electronic structure in bilayer molybdenum disulfide via interlayer twist.Nano Lett.14, 3869–3875 (2014).\",\n \"link\": \"https://doi.org/10.1021%2Fnl501077m\"\n },\n {\n \"idx\": \"17\",\n \"title\": \"Jiang, T. et al. Valley and band structure engineering of folded MoS2bilayers.Nat. Nanotechnol.9, 825–829 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fnnano.2014.176\"\n },\n {\n \"idx\": \"18\",\n \"title\": \"Horng, J. et al. Observation of interlayer excitons in MoSe2single crystals.Phys. Rev. B97, 241404 (2018).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.97.241404\"\n },\n {\n \"idx\": \"19\",\n \"title\": \"Hsu, W.-T. et al. Tailoring excitonic states of van der Waals bilayers through stacking configuration, band alignment, and valley spin.Sci. Adv.5, eaax7407 (2019).\",\n \"link\": \"https://doi.org/10.1126%2Fsciadv.aax7407\"\n },\n {\n \"idx\": \"20\",\n \"title\": \"Gerber, I. C. et al. Interlayer excitons in bilayer MoS2with strong oscillator strength up to room temperature.Phys. Rev. B99, 035443 (2019).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.99.035443\"\n },\n {\n \"idx\": \"21\",\n \"title\": \"Liu, K. et al. Evolution of interlayer coupling in twisted molybdenum disulfide bilayers.Nat. Commun.5, 4966 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms5966\"\n },\n {\n \"idx\": \"22\",\n \"title\": \"Wang, Z., Chiu, Y.-H., Honz, K., Mak, K. F. & Shan, J. Electrical tuning of interlayer exciton gases in WSe2bilayers.Nano Lett.18, 137–143 (2018).\",\n \"link\": \"https://doi.org/10.1021%2Facs.nanolett.7b03667\"\n },\n {\n \"idx\": \"23\",\n \"title\": \"Nayak, P. K. et al. Probing evolution of twist-angle-dependent Interlayer excitons in MoSe2/WSe2van der Waals heterostructures.ACS Nano11, 4041–4050 (2017).\",\n \"link\": \"https://doi.org/10.1021%2Facsnano.7b00640\"\n },\n {\n \"idx\": \"24\",\n \"title\": \"Zhang, X.-X., You, Y., Zhao, S. Y. F. & Heinz, T. F. Experimental evidence for dark excitons in monolayer WSe2.Phys. Rev. Lett.115, 257403 (2015).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.115.257403\"\n },\n {\n \"idx\": \"25\",\n \"title\": \"Ulbricht, R., Hendry, E., Shan, J., Heinz, T. F. & Bonn, M. Carrier dynamics in semiconductors studied with time-resolved terahertz spectroscopy.Rev. Mod. Phys.83, 543–586 (2011).\",\n \"link\": \"https://doi.org/10.1103%2FRevModPhys.83.543\"\n },\n {\n \"idx\": \"26\",\n \"title\": \"Huber, R. et al. How many-particle interactions develop after ultrafast excitation of an electron–hole plasma.Nature414, 286–289 (2001).\",\n \"link\": \"https://doi.org/10.1038%2F35104522\"\n },\n {\n \"idx\": \"27\",\n \"title\": \"Kaindl, R. A., Carnahan, M. A., Hägele, D., Lövenich, R. & Chemla, D. S. Ultrafast terahertz probes of transient conducting and insulating phases in an electron–hole gas.Nature423, 734–738 (2003).\",\n \"link\": \"https://doi.org/10.1038%2Fnature01676\"\n },\n {\n \"idx\": \"28\",\n \"title\": \"Poellmann, C. et al. Resonant internal quantum transitions and femtosecond radiative decay of excitons in monolayer WSe2.Nat. Mater.14, 889–893 (2015).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat4356\"\n },\n {\n \"idx\": \"29\",\n \"title\": \"Yong, C.-K. et al. Valley-dependent exciton fine structure and Autler–Townes doublets from Berry phases in monolayer MoSe2.Nat. Mater.18, 1065–1070 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41563-019-0447-8\"\n },\n {\n \"idx\": \"30\",\n \"title\": \"Merkl, P. et al. Ultrafast transition between exciton phases in van der Waals heterostructures.Nat. Mater.18, 691–696 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41563-019-0337-0\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Rydberg spectroscopy of excitons in twisted WSe 2 BLs The WSe 2 BL samples are produced by mechanical exfoliation and subsequent deterministic transfer onto a diamond substrate (see Methods). Monolayer flakes with diameters exceeding 70 µm (Fig. 1a ) are stamped on top of each other. Before the transfer, the crystallographic orientations of the layers are determined by polarization-resolved second-harmonic generation [15] , [31] . The component of the second harmonic I 2ω,|| polarized parallel to the near-infrared pump laser indicates the armchair directions of each monolayer (inset Fig. 1a , blue/orange flower patterns), revealing the relative twist angle θ between both layers (see Methods). We fabricated a series of BLs with θ ranging from 0° to 60° (see Supplementary Figs. 1 and 2 ). Fig. 1: NIR pump–MIR probe spectroscopy of twisted WSe 2 bilayers. a Optical microscope image of a representative twisted homobilayer (BL) on a diamond substrate. The blue frame indicates the lower WSe 2 monolayer, which is covered by a second WSe 2 monolayer (orange frame) forming a twisted BL in the overlap region (black frame). Scale bar: 50 µm. Inset: component of the second harmonic I 2ω,|| polarized parallel to the pump laser light. The characteristic patterns of sixfold symmetry indicate the armchair directions of the bottom (blue dots) and top (orange dots) monolayers, enclosing a twist angle, θ = 53°, close to the natural 2H configuration. b Normalized photoluminescence spectra for samples with different twist angles θ , recorded at a temperature of 5 K (vertically offset for clarity). The black dotted line serves as a guide to the eye highlighting the evolution of the K-Λ transition. c Sketch of the experiment, where a near-infrared (NIR) pump pulse (orange intensity envelope) optically injects 1 s A excitons in the WSe 2 BLs. The excited sample is probed by a mid-infrared waveform (MIR, red curve). The exciton dispersion for a given orbital quantum number is depicted as a function of the center-of-mass momentum q (gray paraboloids). Red arrows indicate the 1 s –2 p transitions of the exciton, interrogated by the MIR photons at an energy ħω MIR . Full size image The samples were pre-characterized using photoluminescence spectroscopy with an excitation wavelength of 488 nm (see Methods). The resulting spectra (Fig. 1b ) feature a single emission peak at an energy of ~1.54 eV for the pristine BL with θ = 60° (2H stacking), which we attribute to the phonon-assisted recombination of K-Λ excitons [32] . For θ = 59°, this peak blue shifts by 20 meV, whereas a second maximum emerges at an energy of ~1.65 eV, corresponding to the direct K-K transition. This trend continues for θ = 53° until the two spectral features merge for θ = 30°. When the twist angle is further reduced to θ = 5° and θ = 0° (i.e., the 3R stacking) the two peaks are discernible again and move towards energies of 1.52 eV and 1.66 eV, respectively. Similar photoluminescence characteristics of BLs have been reported in the literature [16] , [17] , [21] . To explore the actual microscopic origin and the role of Coulomb correlations in the photoexcited twisted BLs, we study the internal structure and binding energy of the excitons. To this end, we generate 1 s A excitons in the WSe 2 BLs with a 100-fs near-infrared (NIR) laser pulse centered at an energy of 1.68 eV (Fig. 1c , orange intensity envelope). After a variable pump-probe delay time t pp , the exciton ensemble is probed by a phase-locked MIR pulse (Fig. 1c , red wave), the electric field of which is monitored by electro-optic sampling (see Methods and Supplementary Fig. 3 ). The intraexcitonic 1 s –2 p transition (Fig. 1c , red arrows) imprints a characteristic change on the MIR waveform, from which information about the excitonic correlations of the non-equilibrium system can be extracted. A Fourier domain Fresnel analysis [26] , [27] , [28] , [30] directly reveals the pump-induced change of the full dielectric response, including the change of the real part of the optical conductivity, Δ σ 1 , and the dielectric function, Δ ε 1 , at variable delays t pp . Fig. 2 summarizes the changes in dielectric function measured for seven WSe 2 BLs with varying twist angles θ , at t pp = 5.1 ps, where the photoexcited state has thermalized (see Supplementary Fig. 4 ). For all twist angles, Δ σ 1 (Fig. 2a ) exhibits a distinct maximum with a corresponding dispersive feature in Δ ε 1 (Fig. 2b ), signifying a resonant response of the photoexcited sample. As we demonstrate below, these resonances relate to the internal 1 s –2 p transition of BL excitons. Between θ = 60° and 30°, the resonance blue shifts with decreasing angle (Fig. 2a dashed line, blue arrow). Strikingly, for θ = 0° the resonance exhibits the opposite behavior: it shifts substantially to the red and gains oscillator strength (red arrow). A phenomenological analysis of the data with a two-component model (see Methods) confirms a systematic blue shift of the resonance (Fig. 2c ) from 100 meV for θ = 60° to 121 meV for θ = 30°. Importantly, the resonance at θ = 0° is red shifted to 67 meV—corresponding to an energy reduction by a factor of two. This strong asymmetry of the 1 s –2 p transition energy—a direct measure of the exciton binding energy [28] , [30] (see Methods)—for θ = 0° and θ = 60° is in stark contrast to the almost degenerate interband photoluminescence (see Fig. 1b ). Fig. 2: Evolution of the 1 s –2 p resonance with interlayer twist angle. a , b Pump-induced changes of the real parts of the optical conductivity Δ σ 1 ( a ) and the dielectric function Δ ε 1 ( b ) for a fixed pump-probe delay time t pp = 5.1 ps as a function of the photon energy for samples with different twist angles θ . The blue spheres represent the dielectric response measured for the photoexcited WSe 2 BLs (pump fluence Φ = 27 µJ cm −2 ; sample temperature, 5 K). The blue shaded areas indicate the fits to the experimental data by a phenomenological model (see Methods). The dashed line and the blue/red arrow indicate the blue/red shift of the intraexcitonic 1 s –2 p resonance energy for increasing misalignment angle with respect to the 2H stacking. c Intraexcitonic 1 s –2 p transition energy ħω res (black spheres) extracted from the data in a , b and derived from the microscopic theory (blue circles) as function of the twist angle θ . For θ = 21° and θ = 30°, microscopic calculations of hybrid excitons were not feasible. In the case of θ = 0°, the intraexcitonic resonance exhibits a strong dependence on the exciton density (see Fig. 4c and Supplementary Note 4 ). The gray sphere indicates the resonance energy as obtained for a pump fluence of Φ = 7 µJ cm −2 , yielding the best agreement with the microscopic theory. The error bars represent the 95% confidence interval of the fitting procedure. Full size image Microscopic many-body theory To explain these surprising observations, we have to go beyond single-particle band structure effects and include Coulomb correlations, by a combination of ab initio density functional theory (DFT) calculations with density matrix theory (see Methods). In WSe 2 BLs, the energetic minimum of the conduction band lies at the Λ and Λ′ points. Thus, following creation of excitons at the Κ point (Fig. 3a , red arrow), electrons scatter via phonons into the global energetic minima [33] (Fig. 3a , blue arrow), corroborated by a transient shift of the excitonic 1 s –2 p resonance (see Supplementary Fig. 4 ). The conduction band states in the Λ and Λ′ valleys are composed of selenium p orbitals, whose overlap with the corresponding wavefunctions of the neighboring layer leads to efficient interlayer hopping [34] and hybridization of the excitonic wavefunctions (X hyb ). We study this scenario by tracing excitons with their hole fixed in one layer—without loss of generality assigned to the top layer—whereas the electrons are located in Λ-like valleys of either the top (Λ top ) or the bottom (Λ bot ) layer (see Fig. 3b and Methods), forming intra- (X intra ) or interlayer excitons (X inter ). The energy of both exciton species is influenced by the electron-hole Coulomb attraction. The interlayer exciton energy depends further on the spin-orbit splitting of the conduction band in the bottom layer. The parabolic dispersion relations of the excitons, E intra and E inter , are mutually offset in momentum space by θ (see Fig. 3b ). Depending on the wavefunction overlap of X intra and X inter in momentum space (Fig. 3c , blue/orange grid), these states can hybridize effectively. The overlap (Fig. 3c , violet surfaces) depends on θ and the orbital quantum number of the excitonic states, which defines the momentum spread of the wavefunctions. Upon hybridization, bonding and anti-bonding exciton states are formed. As the energy splitting between bonding and anti-bonding 1 s states is as large as ~500 meV, the anti-bonding exciton state is pushed outside of our MIR probe spectrum. Therefore, we only consider the energetically lower 1 s - and 2 p -hybrid states (Fig. 3d , magenta lines). As the energy reduction differs for these two states, the 1 s –2 p transition energy varies with θ (Fig. 3d ). Fig. 3: Hybridization between intra- and interlayer excitons. a Single-particle band structure of the WSe 2 BLs for θ = 60°, calculated by density functional theory (see Methods). After resonant excitation (red arrow) at the K point, electrons scatter via phonons (blue arrow) into the energetically favorable Λ valley to form hybrid excitons (X hyb ). b Within the microscopic model, the electron is located in a Λ-like valley of the top (Λ top ) or bottom (Λ bot ) layer for a finite crystallographic misalignment. The parabolic dispersion of intra- ( E intra ) and interlayer ( E inter ) excitons is shifted in momentum space by twisting the layers, owing to the momentum mismatch Δ k between Λ top and Λ bot . Strong interlayer wavefunction overlap promotes electron hopping and exciton hybridization (pink glow). c The overlap (violet surface) of excitonic wavefunctions (blue and orange grids) in momentum space depends on Δ k . For clarity, we only show the real part of the exciton wavefunctions. d Hybridization of 1 s and 2 p excitons lowers the energy levels (magenta lines) with respect to the corresponding non-hybridized states X intra and X inter (blue/orange lines). The situation is sketched for θ = 60°, θ = 50°, and θ = 0°. Zero energy is set to the non-hybridized 1 s state at θ = 60°. The vertical black arrows mark respective 1 s –2 p transition energies. Full size image For θ = 60°, X intra and X inter feature very different energies (Fig. 3d ). This situation is characteristic of the 2H order, where the momentum overlap is strongest for intra- and interlayer exciton states containing electrons from the Λ and Λ′ valley, respectively. As hybridization can only occur for spin-like electron bands and spin-orbit splitting is of opposite sign for the two valleys, the non-hybridized excitons are offset by the strong spin-orbit splitting. Despite this energy mismatch, the energy of X hyb is clearly reduced. For increasing misalignment ( θ < 60°) with respect to 2H stacking, the 1 s states, which are spread out over a broad range of momenta, continue to effectively hybridize. Conversely, the 2 p wavefunctions are more strongly localized in momentum space. Hence misalignment towards θ = 30° rapidly quenches hybridization in the 2 p state and therefore reduces the energy gain. The combination of these mechanisms results in a distinct blue shift of the 1 s –2 p resonance, in excellent agreement with the experiment (Fig. 2c , blue circles). For 3R stacking, X intra and X inter are almost degenerate because both states contain electrons from the same valley and are therefore not offset by spin-orbit splitting. In particular, the 2 p states are almost perfectly degenerate, which enhances the excitonic hybridization as compared to the pristine bilayer ( θ = 60°) and leads to an even smaller 1 s –2 p transition energy (see Fig. 3d ). Tailoring the excitonic lifetime The variable degree of interlayer hybridization of excitons allows us to tune the exciton dynamics as well as the exciton–exciton interaction (Fig. 4 ). We trace the temporal evolution of the exciton population by mapping out the pump-induced change of the MIR electric field Δ E at a fixed electro-optic sampling time, t eos = 0 fs (see ref. [28] and Methods), for a variable pump-probe delay time t pp . Fig. 4a shows a rapid appearance of exciton density at t pp = 0 ps, for all twist angles, whereas the decay dynamics depend sensitively on θ . An exponential fit (Fig. 4a , dashed lines) shows a systematic reduction of the lifetime of the hybrid excitons (Fig. 4b ) from τ = 54 ps to 8 ps as θ is reduced from 60° to 30°. Intriguingly, for θ = 0°, the lifetime rises dramatically to a value of 148 ps. This behavior can be understood by considering the hybridized wavefunctions in real space. Fig. 4: Ultrafast dynamics and density-dependent energy renormalization of hybridized excitons. a Pump-induced change of the mid-infrared electric field Δ E as a function of the pump-probe delay time t pp recorded at fixed electro-optic sampling time t eos = 0 fs for different twist angles θ (spheres). Curves are vertically offset for clarity. The dashed lines represent fits to the data with an exponential decay. b Exciton decay time τ as a function of the twist angle θ , determined by the decay constant of the fit in a . The error bars indicate one standard deviation. c 1 s –2 p resonance energy ħω res for different pump fluences Φ at a pump-probe delay time t pp = 5.1 ps recorded for twist angles θ = 0°, 5°, and 60°, respectively (spheres). The resonance energies were extracted from the dielectric response by fits to a phenomenological model (see Supplementary Fig. 5 ). The blue circle marks the resonance energy derived from the microscopic theory shown in Fig. 2c . The sample was kept at a temperature of 5 K. The error bars represent the 95% confidence interval of the fitting procedure. d Calculated probability density of the electron within the two layers for a hybrid exciton for twist angles θ = 60°, 50°, and 0°, where the hole is chosen to remain at a fixed position in the top layer. Full size image To do this, we keep the position of the hole fixed in the upper TMD layer and plot the probability density of the electron (Fig. 4d , blue peaks). Because of strong interlayer hopping between the Λ and Λ′ points of the adjacent layers, the electron delocalizes within the BL for θ = 60°. A distinct probability density in the bottom layer (23%) signifies the interlayer character of the hybrid exciton. By changing the twist angle to θ = 50°, the orbital overlap of the conduction band states shrinks, reducing the probability density of the electron in the bottom layer (18%). With both electron and hole wavefunctions overlapping strongly in the top layer, the lifetime of the exciton drops, as processes such as phonon-assisted radiative decay [32] , [35] or Auger recombination [28] likely become more efficient. For θ = 30°, the interlayer hopping becomes even weaker, which further reduces the exciton lifetime. In contrast, for θ = 0° interlayer hopping is facilitated by the energetic degeneracy of the Λ-valley conduction band states discussed above. In this case, the strong delocalization of the electron over the top (58%) and bottom (42%) layers distinctly extends the exciton lifetime as the hybrid state becomes more interlayer-like. At the same time, the delocalized wavefunctions provide a larger transition dipole moment, enhancing the amplitude of Δ σ 1 for θ = 0° as compared with θ = 60° (Fig. 2a ). Shaping the interaction between excitons The twist angle controlled interlayer delocalization can also be employed to tune the interaction between excitons. To this end, we record the position of the 1 s –2 p resonance as a function of the pump fluence (Fig. 4c ). For θ = 0°, a dramatic red shift is observed, whereas the effect is less pronounced for θ = 5° and completely absent for θ = 60° (see Supplementary Fig. 5 , for the full dielectric response). We expect that an effective renormalization of the exciton binding energy should ultimately occur for all twist angles, when the exciton densities are increased beyond the values accessible in our experiment. Microscopically, the critical exciton density at which screening effects become important is determined by the exciton wavefunction. The larger the respective Bohr radius, the lower the critical density at which the orbitals start to overlap. Hence, strong interlayer delocalization accompanied by a larger lateral extent of the wavefunction ( θ = 0°, see Fig. 4d ) enables excitons to effectively screen the mutual Coulomb attraction between electron and hole. In addition, the delocalization of the excitons across the two layers for θ = 0° increases their polarizability and enables an out-of-plane dipole-dipole type interaction [36] . Consequently, hybrid excitons experience a stronger renormalization of their internal transitions with increasing exciton density as compared to previous reports on intralayer excitons [28] , [37] . Thus, for increasing pump fluences the exciton binding energy is further reduced beyond the value of the hybrid exciton with negligible many-body interactions. Therefore, the theoretically predicted transition energy (see Figs. 2 c and 4c , blue circle) is in very good agreement with the experimental results only for low fluences (compare grey sphere in Fig. 2c ).\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"In conclusion, we have tailored Coulomb correlations by interlayer hybridization in twisted WSe 2 BLs. By accessing internal 1 s –2 p transitions of the K-Λ excitons, we reveal systematic changes of their binding energy and lifetime with twist angle. A microscopic model based on density functional and density matrix theory quantitatively describes the underlying hybridization effects. Furthermore, we reveal how the density-dependent exciton–exciton interaction is renormalized by the twist angle. These findings underpin a qualitatively different way of fine-tuning the internal structure of excitons in TMDs and controlling the electronic and optical properties in van der Waals structures based on a single material. Tailoring many-body correlations over a large energy range can now be systematically extended to the full library of two-dimensional crystals, which should guide the search for man-made electronic phases in van der Waals heterostructures.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Sample fabrication and photoluminescence spectroscopy The TMD monolayers were exfoliated mechanically from a bulk single crystal using the viscoelastic transfer method [38] . Both constituent WSe 2 monolayers were inspected under an optical microscope and subsequently stacked on top of each other on a diamond substrate. During the transfer process, the lateral position and rotation of the layers can be controlled accurately using micro-positioning stages. To remove any adsorbates trapped between the monolayers, the BLs were annealed at a temperature of 150 °C and a pressure of 1 × 10 −5 mbar for 5 hours. Subsequently, we recorded the microphotoluminescence intensity of the BLs with a continuous-wave laser (wavelength, 488 nm) focused onto the sample using a microscope objective (numerical aperture, 0.6). The radiation emitted by the BLs is then detected in backscattering geometry, dispersed in a spectrometer, and read out by a charge-coupled device camera. Second-harmonic generation The twist angle θ was determined for each BL by polarization-resolved second-harmonic generation (SHG) under rotation of the plane of polarization [31] . Isolated monolayer regions of the structure were excited by a linearly-polarized femtosecond mode-locked Ti:sapphire laser and the intensity of the SHG was recorded. Characteristic sixfold patterns, whose maxima indicate the armchair directions of the monolayers, are then obtained. For the representative structure in Fig. 1a , the intensity maxima span an angle ~7°. As the method is not sensitive to the electric field of the second harmonic but only to its intensity, an ambiguity remains between twist angles of θ = 7° and θ = 60°-7°. This ambiguity can be resolved by recording the sixfold pattern of the SHG polarization anisotropy of the bilayer area of the sample (see Supplementary Fig. 2 ). The strong suppression of the SHG emission of the bilayer region attests to the fact that the two layers are almost perfectly anti-aligned [39] , [40] corresponding to an angle close to the natural 2H stacking. Thus, the twist angle of the sample presented in Fig. 1a can be determined as θ = 53°. For the other BLs, an identical procedure is performed (see Supplementary Note 1 ). Ultrafast NIR pump–MIR probe spectroscopy A schematic of the experimental principle is depicted in Fig. 1c . 1 s A excitons are resonantly created in the WSe 2 BL by photoexcitation with 100-fs pump pulses centered at an energy of 1.69 eV (Supplementary Fig. 3a , inset) from a home-built Ti:sapphire laser amplifier (repetition rate, 400 kHz). The phase-locked MIR probe pulses are generated by optical rectification in a GaSe crystal (thickness, 10 µm) using a fraction of the laser output. After a variable delay time t pp , the probe pulses are transmitted through the photoexcited sample and the MIR waveform is recorded by electro-optic sampling in a second 10 µm-thick GaSe crystal as a function of the electro-optic sampling time t eos (Supplementary Fig. 3a ). The MIR probe pulses are centered at a frequency of 28 THz with a full width at half maximum of 14 THz and a spectral phase, which is approximately flat between 11 and 44 THz (Supplementary Fig. 3b ). Any changes to the electric field, Δ E ( t eos ), of the waveform induced by the transient dielectric response of the sample are resolved in absolute amplitude and phase by serial lock-in detection as a function of the electro-optic sampling time t eos . The full complex-valued dielectric response, characterized by Δ σ 1 and Δ ε 1 , can be retrieved from these data by applying a transfer matrix formalism [26] , [28] , [30] . This technique is sensitive to the total population of bound and unbound electron-hole pairs, irrespective of interband selection rules, and can, thus, address optically dark and bright excitons [41] . Microscopic many-particle theory To calculate the internal quantum transitions of excitons in the twisted BL, a many-particle Hamiltonian was diagonalized accounting for the Coulomb interaction on a Hartree-Fock level. For WSe 2 BLs, ab initio calculations suggest that momentum-dark K-Λ excitons represent the lowest states. Therefore, the Hamiltonian was restricted to electrons and holes at these high-symmetry points, taking into account the energetically most favorable dipole-allowed spin configuration. The BL was then treated as a bipartite system, using the monolayer eigenstates as basis functions of the two subsystems. Here, the layer-localized basis functions were treated in an effective mass approximation with parameters extracted from ab initio simulations for monolayers. The Coulomb interaction strength was implemented on a semiclassical level assuming anisotropic homogeneous slabs to account for the complex dielectric environment of the BL system. Ab initio parameters for the anisotropic dielectric tensor of WSe 2 were used and the influence of the substrate was included. The overlap of electronic wavefunctions of the two-layer subsystem gives rise to interlayer hybridization effects. Only the hybridization of electrons at the Λ point was considered, which is known to be orders of magnitude stronger than at the K point owing to different orbital compositions. The interlayer interaction strength was extracted from ab initio calculations of the BL band structure for different stacking angles. The Hamiltonian was diagonalized by first transforming it into an exciton frame to account for the Coulomb interaction. Subsequently, eigenenergies and wavefunctions were represented in terms of intra- and interlayer excitons, resulting from the Wannier equation for the BL. The overlap of electronic wavefunctions between the two layers results in an interaction of intra- and interlayer excitons. Finally, the excitonic Hamiltonian was diagonalized yielding intralayer-interlayer exciton hybrids. For the numerical calculation of the bilayer eigenstates we consider the hybridization of a large number of both intra- and interlayer excitons including states with angular momentum of s- , p- , d-, and f -type. The measured transition energies were then compared with the energetically lowest dipole-allowed transition of the calculated hybrid exciton Rydberg series where we neglect higher-order corrections such as a Berry curvature-induced splitting of the otherwise degenerate p states. For additional details see Supplementary Note 5 . Ab initio DFT calculations The electronic band structure was calculated on the basis of density functional theory in the Perdew-Burke-Ernzerhof approximation, as implemented in the Quantum ESPRESSO package. The atomic positions and in-plane lattice constants of twisted BL WSe 2 were optimized with different relative twist angles and the resulting atomic geometries were used to derive an average equilibrium distance between the two tungsten sublayers. Based on the optimized geometry of a 2H stacked BL, the evolution of the electronic band structure for varying interlayer distance was simulated. The required interlayer interaction strength to calculate the exciton hybridization was extracted from the splitting of the electronic bands at the Λ point. To obtain a continuous twist angle dependence of the interaction parameter, the interlayer distances obtained for computationally feasible angles were linearly interpolated. Spin-orbit interactions and quasiparticle energy corrections from the G 0 W 0 approximation were explicitly included to calculate the full band structure for 2H stacked WSe 2 . Here, the YAMBO code was used to calculate the quasiparticle corrections with spin-orbit effects. Subsequently, GW calculations with the BerkeleyGW code were performed, combined with a non-uniform Neck Subsampling method to derive an additional scissor-shift correction for the electronic band gaps. For additional details, see Supplementary Note 6 . Two-component model A phenomenological two-component model was used to extract the quantitative properties of the 1 s –2 p transition from the transient dielectric response. To this end, the pump-induced change of the dielectric function Δ ε ( ω ) = Δ ε 1 ( ω ) + Δ σ 1 ( ω )/( ε 0 ω ) is described by the Drude–Lorentz model: $$\\\\Delta \\\\varepsilon \\\\left( \\\\omega \\\\right) = \\\\frac{{n_{\\\\mathrm{X}}e^2}}{{d\\\\varepsilon _0\\\\mu }}\\\\frac{{f_{1s,2p}}}{{\\\\omega _{{\\\\mathrm{res}}}^2 - \\\\omega ^2 - {\\\\mathrm{i}}\\\\omega {\\\\mathrm{\\\\Delta }}}} - \\\\frac{{n_{{\\\\mathrm{fc}}}e^2}}{{d\\\\varepsilon _0{\\\\upmu}}}\\\\frac{1}{{\\\\omega ^2 + {\\\\mathrm{i}}\\\\omega \\\\Gamma }}$$ (1) where e is the electron charge and ε 0 denotes the vacuum permittivity. The Lo-rentzian resonance with oscillator strength f 1 s ,2 p describes the internal 1 s –2 p transition of the excitons within the WSe 2 BL as determined by the density matrix theory (see Methods). Owing to the twist angle-dependent hybridization, the oscillator strength equals f 1 s ,2 p = 0.34, f 1 s ,2 p = 0.33, f 1 s ,2 p = 0.23, f 1 s ,2 p = 0.18, f 1s,2p = 0.19, f 1 s ,2 p = 0.26, and f 1 s ,2 p = 0.42 for θ = 60°, 59°, 53°, 30°, 21°, 5°, and 0°, respectively. The resonance is further characterized by the exciton density n X , the resonance energy ħω res and the linewidth Δ, which are free fit parameters. The effective mass of the electron-hole pairs μ = 0.23 m 0 is determined with the electron and hole effective masses within the WSe 2 BL (ref. [42] ), where m 0 is the free-electron mass. The sample is treated as a thin slab in this model with a thickness d = 0.7 nm (ref. [28] ) of each layer. The second term in Δ ε ( ω ) is a Drude model describing unbound electron-hole pairs or excitonic transitions energetically below the spectral window of the MIR probe pulses [30] . Here, the density of free carriers n fc and their scattering rate Γ govern the resulting dielectric response. Strict limits on the possible values of the free parameters are set by the fact that both independently retrieved spectra Δ σ 1 and Δ ε 1 have to be reproduced simultaneously. Without any further restrictions, the numerical adaption of the measured spectra yields an overall good fit quality.\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"By exploiting the internal 1s–2p resonance of excitons in homobilayers of WSe₂, we demonstrate that the twist angle between layers renormalizes the exciton binding energy by up to a factor of two, enhances their lifetime by more than an order of magnitude, and allows wide tunability of exciton–exciton interactions, which can be quantitatively explained by a microscopic many-body theory identifying wavefunction overlap, interlayer hopping, and hybridization effects as key mechanisms for engineering custom-tailored hybrid exciton species."},"entities":{"kind":"list like","value":["internal 1s–2p resonance","excitons","homobilayers","WSe₂","twist angle","exciton binding energy","exciton lifetime","exciton–exciton interactions","microscopic many-body theory","wavefunction overlap","interlayer hopping","hybridization effects","hybrid exciton species"],"string":"[\n \"internal 1s–2p resonance\",\n \"excitons\",\n \"homobilayers\",\n \"WSe₂\",\n \"twist angle\",\n \"exciton binding energy\",\n \"exciton lifetime\",\n \"exciton–exciton interactions\",\n \"microscopic many-body theory\",\n \"wavefunction overlap\",\n \"interlayer hopping\",\n \"hybridization effects\",\n \"hybrid exciton species\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1.","title":"Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices.Nature556, 80–84 (2018).","link":"https://doi.org/10.1038%2Fnature26154"},{"idx":"2.","title":"Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices.Nature556, 43–50 (2018).","link":"https://doi.org/10.1038%2Fnature26160"},{"idx":"3.","title":"Sharpe, A. L. et al. Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene.Science365, 605–608 (2019).","link":"https://doi.org/10.1126%2Fscience.aaw3780"},{"idx":"4.","title":"Tong, Q. et al. Topological mosaics in moiré superlattices of van der Waals heterobilayers.Nat. Phys.13, 356–362 (2016).","link":"https://doi.org/10.1038%2Fnphys3968"},{"idx":"5.","title":"Wu, F., Lovorn, T. & MacDonald, A. H. Topological exciton bands in Moiré Heterojunctions.Phys. Rev. Lett.118, 147401 (2017).","link":"https://doi.org/10.1103%2FPhysRevLett.118.147401"},{"idx":"6.","title":"Yu, H., Liu, G.-B., Tang, J., Xu, X. & Yao, W. Moiré excitons: from programmable quantum emitter arrays to spin-orbit–coupled artificial lattices.Sci. Adv.3, e1701696 (2017).","link":"https://doi.org/10.1126%2Fsciadv.1701696"},{"idx":"7.","title":"Wu, F., Lovorn, T., Tutuc, E., Martin, I. & MacDonald, A. H. Topological insulators in twisted transition metal dichalcogenide homobilayers.Phys. Rev. Lett.122, 086402 (2019).","link":"https://doi.org/10.1103%2FPhysRevLett.122.086402"},{"idx":"8.","title":"Jin, C. et al. Identification of spin, valley and moiré quasi-angular momentum of interlayer excitons.Nat. Phys.15, 1140–1144 (2019).","link":"https://doi.org/10.1038%2Fs41567-019-0631-4"},{"idx":"9.","title":"Naik, M. H. & Jain, M. Ultraflatbands and shear solitons in Moiré patterns of twisted bilayer transition metal dichalcogenides.Phys. Rev. Lett.121, 266401 (2018).","link":"https://doi.org/10.1103%2FPhysRevLett.121.266401"},{"idx":"10.","title":"Novoselov, K. S., Mishchenko, A., Carvalho, A. & Castro Neto, A. H. 2D materials and van der Waals heterostructures.Science353, aac9439 (2016).","link":"https://doi.org/10.1126%2Fscience.aac9439"},{"idx":"11.","title":"Ugeda, M. M. et al. Giant bandgap renormalization and excitonic effects in a monolayer transition metal dichalcogenide semiconductor.Nat. Mater.13, 1091–1095 (2014).","link":"https://doi.org/10.1038%2Fnmat4061"},{"idx":"12.","title":"Xu, X., Yao, W., Xiao, D. & Heinz, T. F. Spin and pseudospins in layered transition metal dichalcogenides.Nat. Phys.10, 343–350 (2014).","link":"https://doi.org/10.1038%2Fnphys2942"},{"idx":"13.","title":"Alexeev, E. M. et al. Resonantly hybridized excitons in moiré superlattices in van der Waals heterostructures.Nature567, 81–86 (2019).","link":"https://doi.org/10.1038%2Fs41586-019-0986-9"},{"idx":"15.","title":"Kunstmann, J. et al. Momentum-space indirect interlayer excitons in transition-metal dichalcogenide van der Waals heterostructures.Nat. Phys.14, 801–805 (2018).","link":"https://doi.org/10.1038%2Fs41567-018-0123-y"},{"idx":"16.","title":"Van Der Zande, A. M. et al. Tailoring the electronic structure in bilayer molybdenum disulfide via interlayer twist.Nano Lett.14, 3869–3875 (2014).","link":"https://doi.org/10.1021%2Fnl501077m"},{"idx":"17.","title":"Jiang, T. et al. Valley and band structure engineering of folded MoS2bilayers.Nat. Nanotechnol.9, 825–829 (2014).","link":"https://doi.org/10.1038%2Fnnano.2014.176"},{"idx":"18.","title":"Horng, J. et al. Observation of interlayer excitons in MoSe2single crystals.Phys. Rev. B97, 241404 (2018).","link":"https://doi.org/10.1103%2FPhysRevB.97.241404"},{"idx":"19.","title":"Hsu, W.-T. et al. Tailoring excitonic states of van der Waals bilayers through stacking configuration, band alignment, and valley spin.Sci. Adv.5, eaax7407 (2019).","link":"https://doi.org/10.1126%2Fsciadv.aax7407"},{"idx":"20.","title":"Gerber, I. C. et al. Interlayer excitons in bilayer MoS2with strong oscillator strength up to room temperature.Phys. Rev. B99, 035443 (2019).","link":"https://doi.org/10.1103%2FPhysRevB.99.035443"},{"idx":"21.","title":"Liu, K. et al. Evolution of interlayer coupling in twisted molybdenum disulfide bilayers.Nat. Commun.5, 4966 (2014).","link":"https://doi.org/10.1038%2Fncomms5966"},{"idx":"22.","title":"Wang, Z., Chiu, Y.-H., Honz, K., Mak, K. F. & Shan, J. Electrical tuning of interlayer exciton gases in WSe2bilayers.Nano Lett.18, 137–143 (2018).","link":"https://doi.org/10.1021%2Facs.nanolett.7b03667"},{"idx":"23.","title":"Nayak, P. K. et al. Probing evolution of twist-angle-dependent Interlayer excitons in MoSe2/WSe2van der Waals heterostructures.ACS Nano11, 4041–4050 (2017).","link":"https://doi.org/10.1021%2Facsnano.7b00640"},{"idx":"24.","title":"Zhang, X.-X., You, Y., Zhao, S. Y. F. & Heinz, T. F. Experimental evidence for dark excitons in monolayer WSe2.Phys. Rev. Lett.115, 257403 (2015).","link":"https://doi.org/10.1103%2FPhysRevLett.115.257403"},{"idx":"25.","title":"Ulbricht, R., Hendry, E., Shan, J., Heinz, T. F. & Bonn, M. Carrier dynamics in semiconductors studied with time-resolved terahertz spectroscopy.Rev. Mod. Phys.83, 543–586 (2011).","link":"https://doi.org/10.1103%2FRevModPhys.83.543"},{"idx":"26.","title":"Huber, R. et al. How many-particle interactions develop after ultrafast excitation of an electron–hole plasma.Nature414, 286–289 (2001).","link":"https://doi.org/10.1038%2F35104522"},{"idx":"27.","title":"Kaindl, R. A., Carnahan, M. A., Hägele, D., Lövenich, R. & Chemla, D. S. Ultrafast terahertz probes of transient conducting and insulating phases in an electron–hole gas.Nature423, 734–738 (2003).","link":"https://doi.org/10.1038%2Fnature01676"},{"idx":"28.","title":"Poellmann, C. et al. Resonant internal quantum transitions and femtosecond radiative decay of excitons in monolayer WSe2.Nat. Mater.14, 889–893 (2015).","link":"https://doi.org/10.1038%2Fnmat4356"},{"idx":"29.","title":"Yong, C.-K. et al. Valley-dependent exciton fine structure and Autler–Townes doublets from Berry phases in monolayer MoSe2.Nat. Mater.18, 1065–1070 (2019).","link":"https://doi.org/10.1038%2Fs41563-019-0447-8"},{"idx":"30.","title":"Merkl, P. et al. Ultrafast transition between exciton phases in van der Waals heterostructures.Nat. Mater.18, 691–696 (2019).","link":"https://doi.org/10.1038%2Fs41563-019-0337-0"},{"idx":"31.","title":"Lin, K.-Q., Bange, S. & Lupton, J. M. Quantum interference in second-harmonic generation from monolayer WSe2.Nat. Phys.15, 242–246 (2019).","link":"https://doi.org/10.1038%2Fs41567-018-0384-5"},{"idx":"32.","title":"Lindlau, J. et al. The role of momentum-dark excitons in the elementary optical response of bilayer WSe2.Nat. Commun.9, 2586 (2018).","link":"https://doi.org/10.1038%2Fs41467-018-04877-3"},{"idx":"33.","title":"Waldecker, L. et al. Momentum-resolved view of electron-phonon coupling in multilayer WSe2.Phys. Rev. Lett.119, 036803 (2017).","link":"https://doi.org/10.1103%2FPhysRevLett.119.036803"},{"idx":"34.","title":"Liu, G.-B., Xiao, D., Yao, Y., Xu, X. & Yao, W. Electronic structures and theoretical modelling of two-dimensional group-VIB transition metal dichalcogenides.Chem. Soc. Rev.44, 2643–2663 (2015).","link":"https://doi.org/10.1039%2FC4CS00301B"},{"idx":"35.","title":"Brem, S. et al. Phonon-assisted photoluminescence from indirect excitons in monolayers of transition-metal dichalcogenides.Nano Lett.4, 2849–2856 (2020).","link":"https://doi.org/10.1021%2Facs.nanolett.0c00633"},{"idx":"36.","title":"Miller, B. et al. Long-lived direct and indirect interlayer excitons in van der waals heterostructures.Nano Lett.17, 5229–5237 (2017).","link":"https://doi.org/10.1021%2Facs.nanolett.7b01304"},{"idx":"37.","title":"Cunningham, P. D., Hanbicki, A. T., McCreary, K. M. & Jonker, B. T. Photoinduced bandgap renormalization and exciton binding energy reduction in WS2.ACS Nano11, 12601–12608 (2017).","link":"https://doi.org/10.1021%2Facsnano.7b06885"},{"idx":"38.","title":"Castellanos-Gomez, A. et al. Deterministic transfer of two-dimensional materials by all-dry viscoelastic stamping.2D Mater.1, 011002 (2014).","link":"https://doi.org/10.1088%2F2053-1583%2F1%2F1%2F011002"},{"idx":"40.","title":"Hsu, W. et al. Second harmonic generation from artificially stacked transition metal dichalcogenide twisted Bilayers.ACS Nano8, 2951–2958 (2014).","link":"https://doi.org/10.1021%2Fnn500228r"},{"idx":"41.","title":"Berghäuser, G. et al. Mapping of the dark exciton landscape in transition metal dichalcogenides.Phys. Rev. B98, 020301(R) (2018).","link":"https://doi.org/10.1103%2FPhysRevB.98.020301"},{"idx":"42.","title":"Kormányos, A. et al. k·p theory for two-dimensional transition metal dichalcogenide semiconductors.2D Mater.2, 022001 (2015).","link":"https://doi.org/10.1088%2F2053-1583%2F2%2F2%2F022001"}],"string":"[\n {\n \"idx\": \"1.\",\n \"title\": \"Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices.Nature556, 80–84 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fnature26154\"\n },\n {\n \"idx\": \"2.\",\n \"title\": \"Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices.Nature556, 43–50 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fnature26160\"\n },\n {\n \"idx\": \"3.\",\n \"title\": \"Sharpe, A. L. et al. Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene.Science365, 605–608 (2019).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.aaw3780\"\n },\n {\n \"idx\": \"4.\",\n \"title\": \"Tong, Q. et al. Topological mosaics in moiré superlattices of van der Waals heterobilayers.Nat. Phys.13, 356–362 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys3968\"\n },\n {\n \"idx\": \"5.\",\n \"title\": \"Wu, F., Lovorn, T. & MacDonald, A. H. Topological exciton bands in Moiré Heterojunctions.Phys. Rev. Lett.118, 147401 (2017).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.118.147401\"\n },\n {\n \"idx\": \"6.\",\n \"title\": \"Yu, H., Liu, G.-B., Tang, J., Xu, X. & Yao, W. Moiré excitons: from programmable quantum emitter arrays to spin-orbit–coupled artificial lattices.Sci. Adv.3, e1701696 (2017).\",\n \"link\": \"https://doi.org/10.1126%2Fsciadv.1701696\"\n },\n {\n \"idx\": \"7.\",\n \"title\": \"Wu, F., Lovorn, T., Tutuc, E., Martin, I. & MacDonald, A. H. Topological insulators in twisted transition metal dichalcogenide homobilayers.Phys. Rev. Lett.122, 086402 (2019).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.122.086402\"\n },\n {\n \"idx\": \"8.\",\n \"title\": \"Jin, C. et al. Identification of spin, valley and moiré quasi-angular momentum of interlayer excitons.Nat. Phys.15, 1140–1144 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41567-019-0631-4\"\n },\n {\n \"idx\": \"9.\",\n \"title\": \"Naik, M. H. & Jain, M. Ultraflatbands and shear solitons in Moiré patterns of twisted bilayer transition metal dichalcogenides.Phys. Rev. Lett.121, 266401 (2018).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.121.266401\"\n },\n {\n \"idx\": \"10.\",\n \"title\": \"Novoselov, K. S., Mishchenko, A., Carvalho, A. & Castro Neto, A. H. 2D materials and van der Waals heterostructures.Science353, aac9439 (2016).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.aac9439\"\n },\n {\n \"idx\": \"11.\",\n \"title\": \"Ugeda, M. M. et al. Giant bandgap renormalization and excitonic effects in a monolayer transition metal dichalcogenide semiconductor.Nat. Mater.13, 1091–1095 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat4061\"\n },\n {\n \"idx\": \"12.\",\n \"title\": \"Xu, X., Yao, W., Xiao, D. & Heinz, T. F. Spin and pseudospins in layered transition metal dichalcogenides.Nat. Phys.10, 343–350 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys2942\"\n },\n {\n \"idx\": \"13.\",\n \"title\": \"Alexeev, E. M. et al. Resonantly hybridized excitons in moiré superlattices in van der Waals heterostructures.Nature567, 81–86 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-019-0986-9\"\n },\n {\n \"idx\": \"15.\",\n \"title\": \"Kunstmann, J. et al. Momentum-space indirect interlayer excitons in transition-metal dichalcogenide van der Waals heterostructures.Nat. Phys.14, 801–805 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41567-018-0123-y\"\n },\n {\n \"idx\": \"16.\",\n \"title\": \"Van Der Zande, A. M. et al. Tailoring the electronic structure in bilayer molybdenum disulfide via interlayer twist.Nano Lett.14, 3869–3875 (2014).\",\n \"link\": \"https://doi.org/10.1021%2Fnl501077m\"\n },\n {\n \"idx\": \"17.\",\n \"title\": \"Jiang, T. et al. Valley and band structure engineering of folded MoS2bilayers.Nat. Nanotechnol.9, 825–829 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fnnano.2014.176\"\n },\n {\n \"idx\": \"18.\",\n \"title\": \"Horng, J. et al. Observation of interlayer excitons in MoSe2single crystals.Phys. Rev. B97, 241404 (2018).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.97.241404\"\n },\n {\n \"idx\": \"19.\",\n \"title\": \"Hsu, W.-T. et al. Tailoring excitonic states of van der Waals bilayers through stacking configuration, band alignment, and valley spin.Sci. Adv.5, eaax7407 (2019).\",\n \"link\": \"https://doi.org/10.1126%2Fsciadv.aax7407\"\n },\n {\n \"idx\": \"20.\",\n \"title\": \"Gerber, I. C. et al. Interlayer excitons in bilayer MoS2with strong oscillator strength up to room temperature.Phys. Rev. B99, 035443 (2019).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.99.035443\"\n },\n {\n \"idx\": \"21.\",\n \"title\": \"Liu, K. et al. Evolution of interlayer coupling in twisted molybdenum disulfide bilayers.Nat. Commun.5, 4966 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms5966\"\n },\n {\n \"idx\": \"22.\",\n \"title\": \"Wang, Z., Chiu, Y.-H., Honz, K., Mak, K. F. & Shan, J. Electrical tuning of interlayer exciton gases in WSe2bilayers.Nano Lett.18, 137–143 (2018).\",\n \"link\": \"https://doi.org/10.1021%2Facs.nanolett.7b03667\"\n },\n {\n \"idx\": \"23.\",\n \"title\": \"Nayak, P. K. et al. Probing evolution of twist-angle-dependent Interlayer excitons in MoSe2/WSe2van der Waals heterostructures.ACS Nano11, 4041–4050 (2017).\",\n \"link\": \"https://doi.org/10.1021%2Facsnano.7b00640\"\n },\n {\n \"idx\": \"24.\",\n \"title\": \"Zhang, X.-X., You, Y., Zhao, S. Y. F. & Heinz, T. F. Experimental evidence for dark excitons in monolayer WSe2.Phys. Rev. Lett.115, 257403 (2015).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.115.257403\"\n },\n {\n \"idx\": \"25.\",\n \"title\": \"Ulbricht, R., Hendry, E., Shan, J., Heinz, T. F. & Bonn, M. Carrier dynamics in semiconductors studied with time-resolved terahertz spectroscopy.Rev. Mod. Phys.83, 543–586 (2011).\",\n \"link\": \"https://doi.org/10.1103%2FRevModPhys.83.543\"\n },\n {\n \"idx\": \"26.\",\n \"title\": \"Huber, R. et al. How many-particle interactions develop after ultrafast excitation of an electron–hole plasma.Nature414, 286–289 (2001).\",\n \"link\": \"https://doi.org/10.1038%2F35104522\"\n },\n {\n \"idx\": \"27.\",\n \"title\": \"Kaindl, R. A., Carnahan, M. A., Hägele, D., Lövenich, R. & Chemla, D. S. Ultrafast terahertz probes of transient conducting and insulating phases in an electron–hole gas.Nature423, 734–738 (2003).\",\n \"link\": \"https://doi.org/10.1038%2Fnature01676\"\n },\n {\n \"idx\": \"28.\",\n \"title\": \"Poellmann, C. et al. Resonant internal quantum transitions and femtosecond radiative decay of excitons in monolayer WSe2.Nat. Mater.14, 889–893 (2015).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat4356\"\n },\n {\n \"idx\": \"29.\",\n \"title\": \"Yong, C.-K. et al. Valley-dependent exciton fine structure and Autler–Townes doublets from Berry phases in monolayer MoSe2.Nat. Mater.18, 1065–1070 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41563-019-0447-8\"\n },\n {\n \"idx\": \"30.\",\n \"title\": \"Merkl, P. et al. Ultrafast transition between exciton phases in van der Waals heterostructures.Nat. Mater.18, 691–696 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41563-019-0337-0\"\n },\n {\n \"idx\": \"31.\",\n \"title\": \"Lin, K.-Q., Bange, S. & Lupton, J. M. Quantum interference in second-harmonic generation from monolayer WSe2.Nat. Phys.15, 242–246 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41567-018-0384-5\"\n },\n {\n \"idx\": \"32.\",\n \"title\": \"Lindlau, J. et al. The role of momentum-dark excitons in the elementary optical response of bilayer WSe2.Nat. Commun.9, 2586 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-018-04877-3\"\n },\n {\n \"idx\": \"33.\",\n \"title\": \"Waldecker, L. et al. Momentum-resolved view of electron-phonon coupling in multilayer WSe2.Phys. Rev. Lett.119, 036803 (2017).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.119.036803\"\n },\n {\n \"idx\": \"34.\",\n \"title\": \"Liu, G.-B., Xiao, D., Yao, Y., Xu, X. & Yao, W. Electronic structures and theoretical modelling of two-dimensional group-VIB transition metal dichalcogenides.Chem. Soc. Rev.44, 2643–2663 (2015).\",\n \"link\": \"https://doi.org/10.1039%2FC4CS00301B\"\n },\n {\n \"idx\": \"35.\",\n \"title\": \"Brem, S. et al. Phonon-assisted photoluminescence from indirect excitons in monolayers of transition-metal dichalcogenides.Nano Lett.4, 2849–2856 (2020).\",\n \"link\": \"https://doi.org/10.1021%2Facs.nanolett.0c00633\"\n },\n {\n \"idx\": \"36.\",\n \"title\": \"Miller, B. et al. Long-lived direct and indirect interlayer excitons in van der waals heterostructures.Nano Lett.17, 5229–5237 (2017).\",\n \"link\": \"https://doi.org/10.1021%2Facs.nanolett.7b01304\"\n },\n {\n \"idx\": \"37.\",\n \"title\": \"Cunningham, P. D., Hanbicki, A. T., McCreary, K. M. & Jonker, B. T. Photoinduced bandgap renormalization and exciton binding energy reduction in WS2.ACS Nano11, 12601–12608 (2017).\",\n \"link\": \"https://doi.org/10.1021%2Facsnano.7b06885\"\n },\n {\n \"idx\": \"38.\",\n \"title\": \"Castellanos-Gomez, A. et al. Deterministic transfer of two-dimensional materials by all-dry viscoelastic stamping.2D Mater.1, 011002 (2014).\",\n \"link\": \"https://doi.org/10.1088%2F2053-1583%2F1%2F1%2F011002\"\n },\n {\n \"idx\": \"40.\",\n \"title\": \"Hsu, W. et al. Second harmonic generation from artificially stacked transition metal dichalcogenide twisted Bilayers.ACS Nano8, 2951–2958 (2014).\",\n \"link\": \"https://doi.org/10.1021%2Fnn500228r\"\n },\n {\n \"idx\": \"41.\",\n \"title\": \"Berghäuser, G. et al. Mapping of the dark exciton landscape in transition metal dichalcogenides.Phys. Rev. B98, 020301(R) (2018).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.98.020301\"\n },\n {\n \"idx\": \"42.\",\n \"title\": \"Kormányos, A. et al. k·p theory for two-dimensional transition metal dichalcogenide semiconductors.2D Mater.2, 022001 (2015).\",\n \"link\": \"https://doi.org/10.1088%2F2053-1583%2F2%2F2%2F022001\"\n }\n]","isConversational":false}}},{"rowIdx":93,"cells":{"unique_id":{"kind":"string","value":"ncomms15961"},"subfield":{"kind":"string","value":"Nanoscience and technology"},"title":{"kind":"string","value":"Imaging quasiperiodic electronic states in a synthetic Penrose tiling"},"abstract":{"kind":"string","value":"Quasicrystals possess long-range order but lack the translational symmetry of crystalline solids. In solid state physics, periodicity is one of the fundamental properties that prescribes the electronic band structure in crystals. In the absence of periodicity and the presence of quasicrystalline order, the ways that electronic states change remain a mystery. Scanning tunnelling microscopy and atomic manipulation can be used to assemble a two-dimensional quasicrystalline structure mapped upon the Penrose tiling. Here, carbon monoxide molecules are arranged on the surface of Cu(111) one at a time to form the potential landscape that mimics the ionic potential of atoms in natural materials by constraining the electrons in the two-dimensional surface state of Cu(111). The real-space images reveal the presence of the quasiperiodic order in the electronic wave functions and the Fourier analysis of our results links the energy of the resonant states to the local vertex structure of the quasicrystal."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"The unique long range order of quasicrystals is evidenced by scattering [1] . Dan Shechtman first observed the 10-fold symmetric electron diffraction patterns from a rapidly cooled Al–Mn alloy [2] . Shechtman discovered a new type of material with long-range orientational and quasiperiodic order, but with rotational symmetries that cannot coexist with the translational symmetry found in classically defined crystals [3] . The sharp scattering peaks are a result of this quasiperiodic order, which can be visualized by the formation of moiré patterns. This order arises from the repetition of the same tiling patterns over and over again, with well-defined rules albeit in a non-periodic manner [3] . The electronic behaviour in periodic systems is dominated by extended wavefunctions while, in contrast, completely aperiodic or disordered systems are dominated by localized wavefunctions [4] . Quasicrystals tread right in the middle between these two conflicting limits, combining the lack of periodicity with the repetitions over long range, leading some studies to suggest critical wave functions which are neither localized nor extended [5] . The Penrose tiling is a classic example of quasiperiodic order, and it has been studied by mathematicians even before the discovery of quasicrystals [6] . Tight-binding studies for the Penrose vertex model quasicrystal, with the nearest neighbour edge hopping showed a largely degenerate state in the middle of the band containing almost 10% of all states in the band [7] , [8] . This degeneracy is typical for quasiperiodic tilings, and is due to families of strictly localized states [9] supported on a finite number of vertices, which reappear at various places throughout the tiling [10] . These localized states may be caused not by a disordered local potential of the sites, but instead by the local vertex structure that differentiates different sites [11] . The density of states of Penrose tiling quasicrystals is often described as spiky and presenting pseudogaps [12] , [13] . Photoemission measurements of natural quasicrystals have observed the existence of a pseudogap, but they are not able to resolve the fine structure of the density of states [14] , [15] . Tunnelling spectroscopic studies have also measured a pseudogap and they observe a spatially inhomogeneous local density of states, with signatures of electronic localization [16] , [17] . Even though high-quality imaging of quasicrystals has been achieved [18] , [19] , many of the most pivotal theoretical predictions for the electronic properties in quasicrystals is yet to be to supported experimentally. The use of synthetic quantum matter has been an extremely fruitful path to explore the physics behind natural systems. Synthetic quasicrystals have been modelled for cold atoms [20] , [21] and photonics [22] . In this experiment we realize a form of quantum simulation through the creation of artificial lattices [23] , [24] . To create a quasicrystal where we had unsurpassed control over the structure and imaging capabilities, we use the atomic manipulation capabilities of a scanning tunnelling microscope (STM) [25] to assemble the quasicrystal one molecule at a time and spectroscopic imaging to visualize the electronic density of states.","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":"Baake, M. & Grimm, U. Mathematical diffraction of aperiodic structures.Chem. Soc. Rev.41, 6821–6824 (2012).","link":"https://doi.org/10.1039%2Fc2cs35120j"},{"idx":"2","title":"Shechtman, D., Blech, I., Gratias, D. & Cahn, J. W. Metallic phase with long-range orientational order and no translational symmetry.Phys. Rev. Lett.53, 1951–1953 (1984).","link":"https://doi.org/10.1103%2FPhysRevLett.53.1951"},{"idx":"3","title":"Levine, D. & Steinhardt, P. J. Quasicrystals. I. Definition and structure.Phys. Rev. B34, 596–616 (1986).","link":"https://doi.org/10.1103%2FPhysRevB.34.596"},{"idx":"4","title":"Anderson, P. W. Absence of diffusion in certain random lattices.Phys. Rev.109, 1492–1505 (1958).","link":"https://doi.org/10.1103%2FPhysRev.109.1492"},{"idx":"5","title":"Schreiber, M. & Grussbach, H. Multifractal wave functions at the Anderson transition.Phys. Rev. Lett.67, 607–610 (1991).","link":"https://doi.org/10.1103%2FPhysRevLett.67.607"},{"idx":"6","title":"de Bruijn, N. G. Algebraic theory of Penrose's non-periodic tilings of the plane. I.Indag. Math.84, 39–52 (1981).","link":"https://doi.org/10.1016%2F1385-7258%2881%2990016-0"},{"idx":"7","title":"Odagaki, T. Properties of tight-binding electrons in a two-dimensional quasi-lattice.Solid State Commun.60, 693–696 (1986).","link":"https://doi.org/10.1016%2F0038-1098%2886%2990423-0"},{"idx":"8","title":"Kohmoto, M. & Sutherland, B. Electronic and vibrational-modes on a Penrose lattice – localized states and band-structure.Phys. Rev. B34, 3849–3853 (1986).","link":"https://doi.org/10.1103%2FPhysRevB.34.3849"},{"idx":"9","title":"Arai, M., Tokihiro, T., Fujiwara, T. & Kohmoto, M. Strictly localized states on a two-dimensional Penrose lattice.Phys. Rev. B38, 1621–1626 (1988).","link":"https://doi.org/10.1103%2FPhysRevB.38.1621"},{"idx":"10","title":"Rieth, T. & Schreiber, M. Identification of spatially confined states in two-dimensional quasiperiodic lattices.Phys. Rev. B51, 15827–15832 (1995).","link":"https://doi.org/10.1103%2FPhysRevB.51.15827"},{"idx":"11","title":"Sutherland, B. Localization of electronic wave functions due to local topology.Phys. Rev. B34, 5208–5211 (1986).","link":"https://doi.org/10.1103%2FPhysRevB.34.5208"},{"idx":"12","title":"Hafner, J. & Krajčí, M. Electronic structure and stability of quasicrystals: Quasiperiodic dispersion relations and pseudogaps.Phys. Rev. Lett.68, 2321–2324 (1992).","link":"https://doi.org/10.1103%2FPhysRevLett.68.2321"},{"idx":"13","title":"Zijlstra, E. S. & Janssen, T. Density of states and localization of electrons in a tight-binding model on the Penrose tiling.Phys. Rev. B61, 3377–3383 (2000).","link":"https://doi.org/10.1103%2FPhysRevB.61.3377"},{"idx":"14","title":"Stadnik, Z. M., Purdie, D., Baer, Y. & Lograsso, T. A. Absence of fine structure in the photoemission spectrum of the icosahedral Al-Pd-Mn quasicrystal.Phys. Rev. B64, 214202 (2001).","link":"https://doi.org/10.1103%2FPhysRevB.64.214202"},{"idx":"15","title":"Rotenberg, E., Theis, W. & Horn, K. Electronic structure investigations of quasicrystals.Prog. Surf. Sci.75, 237–253 (2004).","link":"https://doi.org/10.1016%2Fj.progsurf.2004.05.002"},{"idx":"16","title":"Widmer, R., Gröning, P., Feuerbacher, M. & Gröning, O. Experimental signatures of spiky local density of states in quasicrystals.Phys. Rev. B79, 104202 (2009).","link":"https://doi.org/10.1103%2FPhysRevB.79.104202"},{"idx":"17","title":"Mäder, R., Widmer, R., Gröning, P., Steurer, W. & Gröning, O. Correlating scanning tunneling spectroscopy with the electrical resistivity of Al-based quasicrystals and approximants.Phys. Rev. B87, 075425 (2013).","link":"https://doi.org/10.1103%2FPhysRevB.87.075425"},{"idx":"18","title":"Wasio, N. A. et al. Self-assembly of hydrogen-bonded two-dimensional quasicrystals.Nature507, 86–89 (2014).","link":"https://doi.org/10.1038%2Fnature12993"},{"idx":"19","title":"Urgel, J. I. et al. Quasicrystallinity expressed in two-dimensional coordination networks.Nat. Chem.8, 657–662 (2016).","link":"https://doi.org/10.1038%2Fnchem.2507"},{"idx":"20","title":"Gopalakrishnan, S., Martin, I. & Demler, E. A. Quantum quasicrystals of spin-orbit-coupled dipolar bosons.Phys. Rev. Lett.111, 185304 (2013).","link":"https://doi.org/10.1103%2FPhysRevLett.111.185304"},{"idx":"21","title":"Jagannathan, A. & Duneau, M. An eightfold optical quasicrystal with cold atoms.Europhys. Lett.104, 66003 (2013).","link":"https://doi.org/10.1209%2F0295-5075%2F104%2F66003"},{"idx":"22","title":"Vardeny, Z. V., Nahata, A. & Agrawal, A. Optics of photonic quasicrystals.Nat. Photon.7, 177–187 (2013).","link":"https://doi.org/10.1038%2Fnphoton.2012.343"},{"idx":"23","title":"Gomes, K. K., Mar, W., Ko, W., Guinea, F. & Manoharan, H. C. Designer Dirac fermions and topological phases in molecular graphene.Nature483, 306–310 (2012).","link":"https://doi.org/10.1038%2Fnature10941"},{"idx":"24","title":"Polini, M., Guinea, F., Lewenstein, M., Manoharan, H. C. & Pellegrini, V. Artificial honeycomb lattices for electrons, atoms and photons.Nat. Nano8, 625–633 (2013).","link":"https://doi.org/10.1038%2Fnnano.2013.161"},{"idx":"25","title":"Stroscio, J. A. & Eigler, D. M. Atomic and molecular manipulation with the scanning tunneling microscope.Science254, 1319–1326 (1991).","link":"https://doi.org/10.1126%2Fscience.254.5036.1319"}]}},{"section":"Results","content":"Assembly and design of our synthetic quasicrystal Our experiment starts with a perfectly flat and clean Cu (111) surface, where we adsorbed carbon monoxide (CO) molecules. We used the STM tip to move each CO molecule into their correct positions prescribed by the rhombic Penrose tiling. This Penrose tiling is formed by the arrangement of two kinds of rhombi with the same side length but different vertex angles. One CO molecule is placed at the centre of each rhombus of the tiling, repelling the electrons to the vertices of the tiles and allowing flow along their edges. This kind of arrangement of electrons at the vertices of the tiles and hopping along their edges is known as the Penrose vertex model quasicrystal, and it has been extensively studied by tight-binding model calculations [26] . The STM topograph of the final assembled quasicrystal is shown in Fig. 1a , with the overlay showing the Penrose tiling. The completed quasicrystal is made up of 460 CO molecules, spaced such that the length of the side of each rhombus measures a 0 =1.6 nm. Since the CO molecules register to the position of Cu atoms on the surfaces, their placement is not exact for this non-periodic arrangement, but the imprecision is always smaller than 0.15 nm. Figure 1: Synthetic Penrose tiling quasicrystal. ( a ) STM topograph of assembled quasicrystal composed of 460 CO molecules measured at a bias voltage V =10 mV and setpoint current I =1 nA. The CO molecules are located at the centre of each dark spot in the topograph. The overlay on the right side is the Penrose tiling composed of rhombi with side length a 0 =1.6 nm and vertices angles 72°/108° (blue) and 36°/144° (green). Scale bar, 4 nm. ( b ) Atlas of the eight types of vertex sites encountered in the Penrose vertex model tiling. Full size image The Penrose tiling forms eight structurally different sites [27] , associated with each type of the first order vertex structure in the tiling, classified by the star combination of rhombi vertices. Interestingly, not every combination of rhombi is found in the Penrose tiling, as some configurations lead to spaces that cannot be extended. The eight types of sites allowed are shown in Fig. 1b , where the vertex at the centre of each image is the vertex site. In our experiment, we aim to weight the role played by the vertex types prescribed by the quasicrystalline structure in determining the local density of states. Because each vertex site is formed by a different arrangement of CO molecules, we may end up with slightly different local potentials for each vertex site. However, two of the sites shown in Fig. 1b , site-E and site-F, are formed by the same union of five ‘fat’ rhombi, but are classified as structurally different [6] . Even though they are locally identical at the first order, those sites are formed by different matching rules imposed by the global quasicrystalline order, and as a result they should exhibit different electronic behaviour. The complete quasicrystal is displayed in Fig. 2a . The visualization of the electronic states is done through the measurement of the differential tunnelling conductance at different bias voltages, normalized by the spatially averaged tunnelling conductance measured on the bare Cu surface (see Methods for details). The normalized conductance map taken at the Fermi energy of the Cu surface states (bias voltage V =0 mV) reveals an intriguing pattern ( Fig. 2b ), where electrons form a standing wave resonance but only at particular sites. The histogram of the normalized conductance sorted by the site type reveals that the electrons are predominantly localized on the B-sites ( Fig. 2c ). The diagram in Fig. 2d highlights the position of the B-sites and makes the observed pattern even clearer upon a direct comparison with the conductance map. At this particular energy, electrons resonate at a particular local vertex structure of the quasicrystal sites but the existence of this unique state right at the Fermi energy is not accidental. Figure 2: Visualizing a quasicrystalline electronic state. ( a ) Topograph of the assembled quasicrystal composed of 460 CO molecules (dark spots in the image) measured with a bias voltage V =500 mV and setpoint current I =500 pA over a 42 nm by 42 nm field of view. Scale bar, 5 nm. ( b ) Normalized differential conductance map of the same field of view as a measured at bias voltage V =0 mV. ( c ) Histogram of the normalized differential conductance at bias voltage V =0 mV averaged over each type of vertex site. ( d ) Diagram of all vertex sites of the quasicrystal displaying the same field of view as a with site-B locations highlighted in cyan. ( e ) Fourier transform of the conductance map displayed in b with the arrow representing a wave-number value equal to twice the Fermi wave-number of the bare copper surface states ( k F ). Scale bar, 4 nm −1 . ( f ) Fourier transform of quasicrystal model structure built with the same proportions as the one used in the experiment but with tens of thousands of sites to improve sharpness of Fourier peaks. The arrow represents the same wavenumber (2 k F ) as in e . Full size image Electronic behaviour of our synthetic quasicrystal We have designed the tile size such that the position of the brightest peak in the Fourier transform of the Penrose tiling is equal to twice the Fermi wave-number ( k F =2.1 nm −1 ) of the electrons in the surface state of copper. In periodic lattices, this match corresponds to when the wave-vector first touches the edge of the Brillouin zone, transforming the band structure from electron-like to hole-like. This change in the curvature of the energy dispersion relation creates a nesting at the Fermi surface, which leads to a resonance in the density of states, which can be measured by tunnelling spectroscopy. Figure 2e is the Fourier transform of the conductance map shown in Fig. 2b . The outer ring originates from scattering of the electronic waves outside of the lattice and it has a radius of 2 k F . Notice that it overlaps with the brightest peak of the Fourier transform of the model Penrose tiling ( Fig. 2f ). Quasicrystals do not have a well-defined Brillouin zone or dispersion relation but the Bragg peaks in the Fourier transform still correlate to the quasicrystalline order. The brightest peak is associated with the repetition of B-sites (the most abundant of all site types) even though this repetition is non-periodic, and seems to result in our observed standing wave resonance at those sites. We show additional normalized conductance maps and their Fourier transforms in Supplementary Note 1 ( Supplementary Fig. 1 ). To further clarify the connection between the quasiperiodic repetition of the B-sites and the brightest peak in the Fourier transform, we have taken the Fourier transform of the model Penrose tiling with only B-sites ( Supplementary Fig. 2b ) to examine the effects of their quasiperiodic arrangement. We see the same brightest peak as the Fourier transform with the full model Penrose tiling ( Fig. 2f ), which indicates that the chosen tile size for our Penrose tiling indeed leads to a resonant state due to the quasiperiodic nature of the tiling. A more detailed explanation can be found in Supplementary Note 2 . While the structural differences between these sites and the connection to the quasicrystalline order is clear, tight-binding model calculations on the Penrose tiling have found that sites that are structurally similar can present different local density of states [9] . The electronic behaviour in our synthetic Penrose tiling is dictated by the Cu surface state electrons and their limited lifetime due to scattering to bulk states upon hitting the CO molecules. It is reasonable to assume that the electronic behaviour of a site is due to its first order vertex structure. To prove this nearest-neighbour limit in our system, we performed further analysis of the B-sites separated according to their second order vertex structure [27] ( Supplementary Fig. 3 ) and there was a very small variance between the behaviour of the second order vertex structures. This analysis, along with a more detailed explanation can be found in Supplementary Note 2 . Unlike tight-binding predictions [9] which predict the presence of forbidden B-sites at the Fermi energy, we see the same electronic behaviour from each B-site in our synthetic quasicrystal system. We must emphasize though that the electronic states are not being determined simply by the local potential created by the CO molecules. We can demonstrate this by showing how the E and F-sites have distinct electronic behaviour. We measured conductance maps at different energies to further explore the dependence of resonant states in our synthetic Penrose tiling on the quasiperiodic arrangement of vertex sites. Conductance maps measured at bias voltage V =−200 mV ( Fig. 3a ) and V =−100 mV ( Fig. 3b ) show distinct patterns dominated by different site types. The map at V =−200 mV is dominated by the E, G and H-sites ( Fig. 3c ), while the map at V =−100 mV is dominated by the F-sites ( Fig. 3d ). The histograms of the conductance maps sorted by site type further demonstrate that pattern ( Fig. 3e ). On the basis of our comparison of the second order vertex structures of the B-sites ( Supplementary Fig. 3 ) and the limited lifetime of the Cu surface state electrons, we assume the local potential of a site in our system is solely dependent on the arrangement of the nearest neighbours, so the E and F-sites have different local vertex structures but the same local potential since they are formed by the same pattern of five fat rhombi. We emphasize that even though the E and F-sites have the same local potential, the local density of states of these sites displays different energy dependence. The local density of states therefore depends on the arrangement of CO molecules beyond the first order vertex structure, which could reflect confinement effects from the second order vertex structure or the quasiperiodic arrangement of sites. Figure 3: Topologically distinct sites. ( a ) Normalized differential conductance map of a 42 nm by 42 nm field of view measured at bias voltage V =−200 mV. Scale bar, 5 nm. ( b ) Conductance map of the same field of view but at bias voltage V =−100 mV. ( c ) Diagram of all vertex sites of the quasicrystal displaying the same field of view as above and with site-E, site-G and site-H highlighted in cyan. ( d ) Diagram of all vertex sites of the quasicrystal displaying the same field of view as above and with site-F highlighted in cyan. ( e ) Histogram of the normalized differential conductance at bias voltage V =−200 mV (left) and V =−100 mV (right) averaged over each type of vertex site. Full size image Our next step to corroborate the dependence of the density of states on the local vertex structure was to measure the differential conductance spectra over a large sampling of sites. We measured 1/5 of the vertex sites up to a radius of 12 nm ( Fig. 4a ). We avoided the edges of the sample to minimize the finite size effects. We normalized each differential conductance spectrum in the same manner as we did with the maps, by dividing each spectrum by the spatially averaged spectrum of the bare copper surface ( Fig. 4b ). We compare this normalization method to a few others in Supplementary Note 3 ( Supplementary Fig. 4 ). The total normalized conductance spectrum ( Fig. 4c ) can be obtained by averaging all spectra and it displays a spiky morphology, as expected for quasicrystals [28] . The abundance of peaks in the Fourier transform results in an abundance of gaps and resonances in the density of states. We note that our synthetic system has strong broadening effects in the conductance spectra because of the short lifetime of our surface state electrons due to scattering to the bulk states of Cu. Figure 4: Tunnelling spectroscopy survey. ( a ) The grey diagram illustrates the Penrose tiling assembled in our experiment. The colour dots mark the 61 electron sites where we measured the differential conductance spectra, with each colour corresponding to a site type. ( b ) The red line is the average of all 61 differential conductance spectra measure in the quasicrystal and the grey line a differential conductance spectra of the bare Cu (111) surface, spatially averaged over 100 distinct points. ( c ) The total normalized conductance spectra, calculated by the ratio of the two spectra presented above. ( d ) Normalized conductance spectra averaged by site type for all eight sites in the Penrose tiling. The y -scale refers to the bottom spectra taken at the A-sites. Other spectra have been offset for clarity by 0.3 units in the y -scale above the previous spectrum. Full size image When comparing spectra taken at the same site types we noticed that they presented a very small variance between different locations. On the other hand, we noticed that spectra taken at different site types were consistently different, with a variance one hundred times larger than the variance of spectra taken at the same site types. In Fig. 4d , we present the spectra averaged by each site type. Notice that the sites with smaller areas (less distance between them and the nearest CO molecules in our system) show resonant states at higher energies than those with larger areas. These energy differences could be partially due to confinement effects in our synthetic Penrose tiling. However, since the E and F-sites have the same confinement at first order as each other and exhibit different resonant states ( Fig. 3 ), both the local vertex structure and quasiperiodic order of the system are important in determining the electronic states."},{"section":"Discussion","content":"The creation of a synthetic Penrose tiling allowed for the first time to visualize the local density of state of a two-dimensional quasicrystal with sub-nanometer resolution. Our measurements have identified the existence of resonant states associated with Fourier peaks in reciprocal space and made a clear connection to the local vertex structure of each site. The extension of this experimental approach to include measurements of the changes in the electronic states due to scattering from defects could also solve other puzzles in the understanding of quasicrystals. Those measurements along with Fourier analysis should be able to reveal whether the electronic states are extended and dispersive or localized."},{"section":"Methods","content":"Sample preparation A single crystal Cu (111) sample was prepared by repeated cycles of sputtering (Ar ions, 1 keV) and annealing at a temperature of 1000 K in an ultra-high vacuum chamber. To deposit the CO molecules on the surface, we briefly exposed it to 1 × 10 −8 torr partial pressure of CO gas, while keeping the sample temperature below 50 K. Normalized differential conductance spectroscopy All differential conductance spectra and maps were measure with a lock-in amplifier technique, with set point voltage V =−500 mV, set point current I =0.5 nA and bias voltage modulation with root-mean-square average of dV =3.5 mV. All energies are measured with the feedback loop open but the feedback loop is closed at the set point conditions to move between each point. The differential conductance measured by a scanning tunnelling microscope is proportional to the density of states of the assembled structure, but is also proportional to the density of states of the STM tip, as well as the density of states of the bulk copper crystal. To isolate the density of states of the assembled lattice, we divided our measured conductance maps and tunnelling spectroscopy by the spatially averaged differential conductance of the bare surface of copper. The bare copper differential conductance was measured with the same tip conditions as the other measurements, and it is averaged over 100 distinct sites. This normalization minimized the contribution of varying tip conditions and the copper background and ensured that we mainly considered electronic contributions from our assembled quasicrystal. We also find that the normalized spectra are a much better fit to theoretical models of tunnelling spectroscopy, such as scattering theory. Data availability The data that support the findings of this study are available from the corresponding author on request."},{"section":"Additional information","content":"How to cite this article: Collins, L. C. et al . Imaging quasiperiodic electronic states in a synthetic Penrose tiling. Nat. Commun. 8, 15961 doi: 10.1038/ncomms15961 (2017). 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\": \"The unique long range order of quasicrystals is evidenced by scattering [1] . Dan Shechtman first observed the 10-fold symmetric electron diffraction patterns from a rapidly cooled Al–Mn alloy [2] . Shechtman discovered a new type of material with long-range orientational and quasiperiodic order, but with rotational symmetries that cannot coexist with the translational symmetry found in classically defined crystals [3] . The sharp scattering peaks are a result of this quasiperiodic order, which can be visualized by the formation of moiré patterns. This order arises from the repetition of the same tiling patterns over and over again, with well-defined rules albeit in a non-periodic manner [3] . The electronic behaviour in periodic systems is dominated by extended wavefunctions while, in contrast, completely aperiodic or disordered systems are dominated by localized wavefunctions [4] . Quasicrystals tread right in the middle between these two conflicting limits, combining the lack of periodicity with the repetitions over long range, leading some studies to suggest critical wave functions which are neither localized nor extended [5] . The Penrose tiling is a classic example of quasiperiodic order, and it has been studied by mathematicians even before the discovery of quasicrystals [6] . Tight-binding studies for the Penrose vertex model quasicrystal, with the nearest neighbour edge hopping showed a largely degenerate state in the middle of the band containing almost 10% of all states in the band [7] , [8] . This degeneracy is typical for quasiperiodic tilings, and is due to families of strictly localized states [9] supported on a finite number of vertices, which reappear at various places throughout the tiling [10] . These localized states may be caused not by a disordered local potential of the sites, but instead by the local vertex structure that differentiates different sites [11] . The density of states of Penrose tiling quasicrystals is often described as spiky and presenting pseudogaps [12] , [13] . Photoemission measurements of natural quasicrystals have observed the existence of a pseudogap, but they are not able to resolve the fine structure of the density of states [14] , [15] . Tunnelling spectroscopic studies have also measured a pseudogap and they observe a spatially inhomogeneous local density of states, with signatures of electronic localization [16] , [17] . Even though high-quality imaging of quasicrystals has been achieved [18] , [19] , many of the most pivotal theoretical predictions for the electronic properties in quasicrystals is yet to be to supported experimentally. The use of synthetic quantum matter has been an extremely fruitful path to explore the physics behind natural systems. Synthetic quasicrystals have been modelled for cold atoms [20] , [21] and photonics [22] . In this experiment we realize a form of quantum simulation through the creation of artificial lattices [23] , [24] . To create a quasicrystal where we had unsurpassed control over the structure and imaging capabilities, we use the atomic manipulation capabilities of a scanning tunnelling microscope (STM) [25] to assemble the quasicrystal one molecule at a time and spectroscopic imaging to visualize the electronic density of states.\",\n \"references\": {\n \"indices\": [\n 1,\n 2,\n 3,\n 4,\n 5,\n 6,\n 7,\n 8,\n 9,\n 10,\n 11,\n 12,\n 13,\n 14,\n 15,\n 16,\n 17,\n 18,\n 19,\n 20,\n 21,\n 22,\n 23,\n 24,\n 25\n ],\n \"infos\": [\n {\n \"idx\": \"1\",\n \"title\": \"Baake, M. & Grimm, U. Mathematical diffraction of aperiodic structures.Chem. Soc. Rev.41, 6821–6824 (2012).\",\n \"link\": \"https://doi.org/10.1039%2Fc2cs35120j\"\n },\n {\n \"idx\": \"2\",\n \"title\": \"Shechtman, D., Blech, I., Gratias, D. & Cahn, J. W. Metallic phase with long-range orientational order and no translational symmetry.Phys. Rev. Lett.53, 1951–1953 (1984).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.53.1951\"\n },\n {\n \"idx\": \"3\",\n \"title\": \"Levine, D. & Steinhardt, P. J. Quasicrystals. I. Definition and structure.Phys. Rev. B34, 596–616 (1986).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.34.596\"\n },\n {\n \"idx\": \"4\",\n \"title\": \"Anderson, P. W. Absence of diffusion in certain random lattices.Phys. Rev.109, 1492–1505 (1958).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRev.109.1492\"\n },\n {\n \"idx\": \"5\",\n \"title\": \"Schreiber, M. & Grussbach, H. Multifractal wave functions at the Anderson transition.Phys. Rev. Lett.67, 607–610 (1991).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.67.607\"\n },\n {\n \"idx\": \"6\",\n \"title\": \"de Bruijn, N. G. Algebraic theory of Penrose's non-periodic tilings of the plane. I.Indag. Math.84, 39–52 (1981).\",\n \"link\": \"https://doi.org/10.1016%2F1385-7258%2881%2990016-0\"\n },\n {\n \"idx\": \"7\",\n \"title\": \"Odagaki, T. Properties of tight-binding electrons in a two-dimensional quasi-lattice.Solid State Commun.60, 693–696 (1986).\",\n \"link\": \"https://doi.org/10.1016%2F0038-1098%2886%2990423-0\"\n },\n {\n \"idx\": \"8\",\n \"title\": \"Kohmoto, M. & Sutherland, B. Electronic and vibrational-modes on a Penrose lattice – localized states and band-structure.Phys. Rev. B34, 3849–3853 (1986).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.34.3849\"\n },\n {\n \"idx\": \"9\",\n \"title\": \"Arai, M., Tokihiro, T., Fujiwara, T. & Kohmoto, M. Strictly localized states on a two-dimensional Penrose lattice.Phys. Rev. B38, 1621–1626 (1988).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.38.1621\"\n },\n {\n \"idx\": \"10\",\n \"title\": \"Rieth, T. & Schreiber, M. Identification of spatially confined states in two-dimensional quasiperiodic lattices.Phys. Rev. B51, 15827–15832 (1995).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.51.15827\"\n },\n {\n \"idx\": \"11\",\n \"title\": \"Sutherland, B. Localization of electronic wave functions due to local topology.Phys. Rev. B34, 5208–5211 (1986).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.34.5208\"\n },\n {\n \"idx\": \"12\",\n \"title\": \"Hafner, J. & Krajčí, M. Electronic structure and stability of quasicrystals: Quasiperiodic dispersion relations and pseudogaps.Phys. Rev. Lett.68, 2321–2324 (1992).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.68.2321\"\n },\n {\n \"idx\": \"13\",\n \"title\": \"Zijlstra, E. S. & Janssen, T. Density of states and localization of electrons in a tight-binding model on the Penrose tiling.Phys. Rev. B61, 3377–3383 (2000).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.61.3377\"\n },\n {\n \"idx\": \"14\",\n \"title\": \"Stadnik, Z. M., Purdie, D., Baer, Y. & Lograsso, T. A. Absence of fine structure in the photoemission spectrum of the icosahedral Al-Pd-Mn quasicrystal.Phys. Rev. B64, 214202 (2001).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.64.214202\"\n },\n {\n \"idx\": \"15\",\n \"title\": \"Rotenberg, E., Theis, W. & Horn, K. Electronic structure investigations of quasicrystals.Prog. Surf. Sci.75, 237–253 (2004).\",\n \"link\": \"https://doi.org/10.1016%2Fj.progsurf.2004.05.002\"\n },\n {\n \"idx\": \"16\",\n \"title\": \"Widmer, R., Gröning, P., Feuerbacher, M. & Gröning, O. Experimental signatures of spiky local density of states in quasicrystals.Phys. Rev. B79, 104202 (2009).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.79.104202\"\n },\n {\n \"idx\": \"17\",\n \"title\": \"Mäder, R., Widmer, R., Gröning, P., Steurer, W. & Gröning, O. Correlating scanning tunneling spectroscopy with the electrical resistivity of Al-based quasicrystals and approximants.Phys. Rev. B87, 075425 (2013).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.87.075425\"\n },\n {\n \"idx\": \"18\",\n \"title\": \"Wasio, N. A. et al. Self-assembly of hydrogen-bonded two-dimensional quasicrystals.Nature507, 86–89 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fnature12993\"\n },\n {\n \"idx\": \"19\",\n \"title\": \"Urgel, J. I. et al. Quasicrystallinity expressed in two-dimensional coordination networks.Nat. Chem.8, 657–662 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fnchem.2507\"\n },\n {\n \"idx\": \"20\",\n \"title\": \"Gopalakrishnan, S., Martin, I. & Demler, E. A. Quantum quasicrystals of spin-orbit-coupled dipolar bosons.Phys. Rev. Lett.111, 185304 (2013).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.111.185304\"\n },\n {\n \"idx\": \"21\",\n \"title\": \"Jagannathan, A. & Duneau, M. An eightfold optical quasicrystal with cold atoms.Europhys. Lett.104, 66003 (2013).\",\n \"link\": \"https://doi.org/10.1209%2F0295-5075%2F104%2F66003\"\n },\n {\n \"idx\": \"22\",\n \"title\": \"Vardeny, Z. V., Nahata, A. & Agrawal, A. Optics of photonic quasicrystals.Nat. Photon.7, 177–187 (2013).\",\n \"link\": \"https://doi.org/10.1038%2Fnphoton.2012.343\"\n },\n {\n \"idx\": \"23\",\n \"title\": \"Gomes, K. K., Mar, W., Ko, W., Guinea, F. & Manoharan, H. C. Designer Dirac fermions and topological phases in molecular graphene.Nature483, 306–310 (2012).\",\n \"link\": \"https://doi.org/10.1038%2Fnature10941\"\n },\n {\n \"idx\": \"24\",\n \"title\": \"Polini, M., Guinea, F., Lewenstein, M., Manoharan, H. C. & Pellegrini, V. Artificial honeycomb lattices for electrons, atoms and photons.Nat. Nano8, 625–633 (2013).\",\n \"link\": \"https://doi.org/10.1038%2Fnnano.2013.161\"\n },\n {\n \"idx\": \"25\",\n \"title\": \"Stroscio, J. A. & Eigler, D. M. Atomic and molecular manipulation with the scanning tunneling microscope.Science254, 1319–1326 (1991).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.254.5036.1319\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Assembly and design of our synthetic quasicrystal Our experiment starts with a perfectly flat and clean Cu (111) surface, where we adsorbed carbon monoxide (CO) molecules. We used the STM tip to move each CO molecule into their correct positions prescribed by the rhombic Penrose tiling. This Penrose tiling is formed by the arrangement of two kinds of rhombi with the same side length but different vertex angles. One CO molecule is placed at the centre of each rhombus of the tiling, repelling the electrons to the vertices of the tiles and allowing flow along their edges. This kind of arrangement of electrons at the vertices of the tiles and hopping along their edges is known as the Penrose vertex model quasicrystal, and it has been extensively studied by tight-binding model calculations [26] . The STM topograph of the final assembled quasicrystal is shown in Fig. 1a , with the overlay showing the Penrose tiling. The completed quasicrystal is made up of 460 CO molecules, spaced such that the length of the side of each rhombus measures a 0 =1.6 nm. Since the CO molecules register to the position of Cu atoms on the surfaces, their placement is not exact for this non-periodic arrangement, but the imprecision is always smaller than 0.15 nm. Figure 1: Synthetic Penrose tiling quasicrystal. ( a ) STM topograph of assembled quasicrystal composed of 460 CO molecules measured at a bias voltage V =10 mV and setpoint current I =1 nA. The CO molecules are located at the centre of each dark spot in the topograph. The overlay on the right side is the Penrose tiling composed of rhombi with side length a 0 =1.6 nm and vertices angles 72°/108° (blue) and 36°/144° (green). Scale bar, 4 nm. ( b ) Atlas of the eight types of vertex sites encountered in the Penrose vertex model tiling. Full size image The Penrose tiling forms eight structurally different sites [27] , associated with each type of the first order vertex structure in the tiling, classified by the star combination of rhombi vertices. Interestingly, not every combination of rhombi is found in the Penrose tiling, as some configurations lead to spaces that cannot be extended. The eight types of sites allowed are shown in Fig. 1b , where the vertex at the centre of each image is the vertex site. In our experiment, we aim to weight the role played by the vertex types prescribed by the quasicrystalline structure in determining the local density of states. Because each vertex site is formed by a different arrangement of CO molecules, we may end up with slightly different local potentials for each vertex site. However, two of the sites shown in Fig. 1b , site-E and site-F, are formed by the same union of five ‘fat’ rhombi, but are classified as structurally different [6] . Even though they are locally identical at the first order, those sites are formed by different matching rules imposed by the global quasicrystalline order, and as a result they should exhibit different electronic behaviour. The complete quasicrystal is displayed in Fig. 2a . The visualization of the electronic states is done through the measurement of the differential tunnelling conductance at different bias voltages, normalized by the spatially averaged tunnelling conductance measured on the bare Cu surface (see Methods for details). The normalized conductance map taken at the Fermi energy of the Cu surface states (bias voltage V =0 mV) reveals an intriguing pattern ( Fig. 2b ), where electrons form a standing wave resonance but only at particular sites. The histogram of the normalized conductance sorted by the site type reveals that the electrons are predominantly localized on the B-sites ( Fig. 2c ). The diagram in Fig. 2d highlights the position of the B-sites and makes the observed pattern even clearer upon a direct comparison with the conductance map. At this particular energy, electrons resonate at a particular local vertex structure of the quasicrystal sites but the existence of this unique state right at the Fermi energy is not accidental. Figure 2: Visualizing a quasicrystalline electronic state. ( a ) Topograph of the assembled quasicrystal composed of 460 CO molecules (dark spots in the image) measured with a bias voltage V =500 mV and setpoint current I =500 pA over a 42 nm by 42 nm field of view. Scale bar, 5 nm. ( b ) Normalized differential conductance map of the same field of view as a measured at bias voltage V =0 mV. ( c ) Histogram of the normalized differential conductance at bias voltage V =0 mV averaged over each type of vertex site. ( d ) Diagram of all vertex sites of the quasicrystal displaying the same field of view as a with site-B locations highlighted in cyan. ( e ) Fourier transform of the conductance map displayed in b with the arrow representing a wave-number value equal to twice the Fermi wave-number of the bare copper surface states ( k F ). Scale bar, 4 nm −1 . ( f ) Fourier transform of quasicrystal model structure built with the same proportions as the one used in the experiment but with tens of thousands of sites to improve sharpness of Fourier peaks. The arrow represents the same wavenumber (2 k F ) as in e . Full size image Electronic behaviour of our synthetic quasicrystal We have designed the tile size such that the position of the brightest peak in the Fourier transform of the Penrose tiling is equal to twice the Fermi wave-number ( k F =2.1 nm −1 ) of the electrons in the surface state of copper. In periodic lattices, this match corresponds to when the wave-vector first touches the edge of the Brillouin zone, transforming the band structure from electron-like to hole-like. This change in the curvature of the energy dispersion relation creates a nesting at the Fermi surface, which leads to a resonance in the density of states, which can be measured by tunnelling spectroscopy. Figure 2e is the Fourier transform of the conductance map shown in Fig. 2b . The outer ring originates from scattering of the electronic waves outside of the lattice and it has a radius of 2 k F . Notice that it overlaps with the brightest peak of the Fourier transform of the model Penrose tiling ( Fig. 2f ). Quasicrystals do not have a well-defined Brillouin zone or dispersion relation but the Bragg peaks in the Fourier transform still correlate to the quasicrystalline order. The brightest peak is associated with the repetition of B-sites (the most abundant of all site types) even though this repetition is non-periodic, and seems to result in our observed standing wave resonance at those sites. We show additional normalized conductance maps and their Fourier transforms in Supplementary Note 1 ( Supplementary Fig. 1 ). To further clarify the connection between the quasiperiodic repetition of the B-sites and the brightest peak in the Fourier transform, we have taken the Fourier transform of the model Penrose tiling with only B-sites ( Supplementary Fig. 2b ) to examine the effects of their quasiperiodic arrangement. We see the same brightest peak as the Fourier transform with the full model Penrose tiling ( Fig. 2f ), which indicates that the chosen tile size for our Penrose tiling indeed leads to a resonant state due to the quasiperiodic nature of the tiling. A more detailed explanation can be found in Supplementary Note 2 . While the structural differences between these sites and the connection to the quasicrystalline order is clear, tight-binding model calculations on the Penrose tiling have found that sites that are structurally similar can present different local density of states [9] . The electronic behaviour in our synthetic Penrose tiling is dictated by the Cu surface state electrons and their limited lifetime due to scattering to bulk states upon hitting the CO molecules. It is reasonable to assume that the electronic behaviour of a site is due to its first order vertex structure. To prove this nearest-neighbour limit in our system, we performed further analysis of the B-sites separated according to their second order vertex structure [27] ( Supplementary Fig. 3 ) and there was a very small variance between the behaviour of the second order vertex structures. This analysis, along with a more detailed explanation can be found in Supplementary Note 2 . Unlike tight-binding predictions [9] which predict the presence of forbidden B-sites at the Fermi energy, we see the same electronic behaviour from each B-site in our synthetic quasicrystal system. We must emphasize though that the electronic states are not being determined simply by the local potential created by the CO molecules. We can demonstrate this by showing how the E and F-sites have distinct electronic behaviour. We measured conductance maps at different energies to further explore the dependence of resonant states in our synthetic Penrose tiling on the quasiperiodic arrangement of vertex sites. Conductance maps measured at bias voltage V =−200 mV ( Fig. 3a ) and V =−100 mV ( Fig. 3b ) show distinct patterns dominated by different site types. The map at V =−200 mV is dominated by the E, G and H-sites ( Fig. 3c ), while the map at V =−100 mV is dominated by the F-sites ( Fig. 3d ). The histograms of the conductance maps sorted by site type further demonstrate that pattern ( Fig. 3e ). On the basis of our comparison of the second order vertex structures of the B-sites ( Supplementary Fig. 3 ) and the limited lifetime of the Cu surface state electrons, we assume the local potential of a site in our system is solely dependent on the arrangement of the nearest neighbours, so the E and F-sites have different local vertex structures but the same local potential since they are formed by the same pattern of five fat rhombi. We emphasize that even though the E and F-sites have the same local potential, the local density of states of these sites displays different energy dependence. The local density of states therefore depends on the arrangement of CO molecules beyond the first order vertex structure, which could reflect confinement effects from the second order vertex structure or the quasiperiodic arrangement of sites. Figure 3: Topologically distinct sites. ( a ) Normalized differential conductance map of a 42 nm by 42 nm field of view measured at bias voltage V =−200 mV. Scale bar, 5 nm. ( b ) Conductance map of the same field of view but at bias voltage V =−100 mV. ( c ) Diagram of all vertex sites of the quasicrystal displaying the same field of view as above and with site-E, site-G and site-H highlighted in cyan. ( d ) Diagram of all vertex sites of the quasicrystal displaying the same field of view as above and with site-F highlighted in cyan. ( e ) Histogram of the normalized differential conductance at bias voltage V =−200 mV (left) and V =−100 mV (right) averaged over each type of vertex site. Full size image Our next step to corroborate the dependence of the density of states on the local vertex structure was to measure the differential conductance spectra over a large sampling of sites. We measured 1/5 of the vertex sites up to a radius of 12 nm ( Fig. 4a ). We avoided the edges of the sample to minimize the finite size effects. We normalized each differential conductance spectrum in the same manner as we did with the maps, by dividing each spectrum by the spatially averaged spectrum of the bare copper surface ( Fig. 4b ). We compare this normalization method to a few others in Supplementary Note 3 ( Supplementary Fig. 4 ). The total normalized conductance spectrum ( Fig. 4c ) can be obtained by averaging all spectra and it displays a spiky morphology, as expected for quasicrystals [28] . The abundance of peaks in the Fourier transform results in an abundance of gaps and resonances in the density of states. We note that our synthetic system has strong broadening effects in the conductance spectra because of the short lifetime of our surface state electrons due to scattering to the bulk states of Cu. Figure 4: Tunnelling spectroscopy survey. ( a ) The grey diagram illustrates the Penrose tiling assembled in our experiment. The colour dots mark the 61 electron sites where we measured the differential conductance spectra, with each colour corresponding to a site type. ( b ) The red line is the average of all 61 differential conductance spectra measure in the quasicrystal and the grey line a differential conductance spectra of the bare Cu (111) surface, spatially averaged over 100 distinct points. ( c ) The total normalized conductance spectra, calculated by the ratio of the two spectra presented above. ( d ) Normalized conductance spectra averaged by site type for all eight sites in the Penrose tiling. The y -scale refers to the bottom spectra taken at the A-sites. Other spectra have been offset for clarity by 0.3 units in the y -scale above the previous spectrum. Full size image When comparing spectra taken at the same site types we noticed that they presented a very small variance between different locations. On the other hand, we noticed that spectra taken at different site types were consistently different, with a variance one hundred times larger than the variance of spectra taken at the same site types. In Fig. 4d , we present the spectra averaged by each site type. Notice that the sites with smaller areas (less distance between them and the nearest CO molecules in our system) show resonant states at higher energies than those with larger areas. These energy differences could be partially due to confinement effects in our synthetic Penrose tiling. However, since the E and F-sites have the same confinement at first order as each other and exhibit different resonant states ( Fig. 3 ), both the local vertex structure and quasiperiodic order of the system are important in determining the electronic states.\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"The creation of a synthetic Penrose tiling allowed for the first time to visualize the local density of state of a two-dimensional quasicrystal with sub-nanometer resolution. Our measurements have identified the existence of resonant states associated with Fourier peaks in reciprocal space and made a clear connection to the local vertex structure of each site. The extension of this experimental approach to include measurements of the changes in the electronic states due to scattering from defects could also solve other puzzles in the understanding of quasicrystals. Those measurements along with Fourier analysis should be able to reveal whether the electronic states are extended and dispersive or localized.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Sample preparation A single crystal Cu (111) sample was prepared by repeated cycles of sputtering (Ar ions, 1 keV) and annealing at a temperature of 1000 K in an ultra-high vacuum chamber. To deposit the CO molecules on the surface, we briefly exposed it to 1 × 10 −8 torr partial pressure of CO gas, while keeping the sample temperature below 50 K. Normalized differential conductance spectroscopy All differential conductance spectra and maps were measure with a lock-in amplifier technique, with set point voltage V =−500 mV, set point current I =0.5 nA and bias voltage modulation with root-mean-square average of dV =3.5 mV. All energies are measured with the feedback loop open but the feedback loop is closed at the set point conditions to move between each point. The differential conductance measured by a scanning tunnelling microscope is proportional to the density of states of the assembled structure, but is also proportional to the density of states of the STM tip, as well as the density of states of the bulk copper crystal. To isolate the density of states of the assembled lattice, we divided our measured conductance maps and tunnelling spectroscopy by the spatially averaged differential conductance of the bare surface of copper. The bare copper differential conductance was measured with the same tip conditions as the other measurements, and it is averaged over 100 distinct sites. This normalization minimized the contribution of varying tip conditions and the copper background and ensured that we mainly considered electronic contributions from our assembled quasicrystal. We also find that the normalized spectra are a much better fit to theoretical models of tunnelling spectroscopy, such as scattering theory. Data availability The data that support the findings of this study are available from the corresponding author on request.\"\n },\n {\n \"section\": \"Additional information\",\n \"content\": \"How to cite this article: Collins, L. C. et al . Imaging quasiperiodic electronic states in a synthetic Penrose tiling. Nat. Commun. 8, 15961 doi: 10.1038/ncomms15961 (2017). 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 experimentally realize and investigate the electronic properties of a synthetic quasicrystal, specifically a Penrose tiling-based structure, using atomic-scale manipulation with a scanning tunnelling microscope to assemble the lattice and spectroscopic imaging to visualize its electronic density of states, with the aim of testing theoretical predictions about critical wavefunctions, localized states, and pseudogaps in quasiperiodic systems."},"entities":{"kind":"list like","value":["synthetic quasicrystal","Penrose tiling-based structure","atomic-scale manipulation","scanning tunnelling microscope","spectroscopic imaging","electronic density of states","critical wavefunctions","localized states","pseudogaps","quasiperiodic systems","experimental realization","electronic properties","lattice assembly"],"string":"[\n \"synthetic quasicrystal\",\n \"Penrose tiling-based structure\",\n \"atomic-scale manipulation\",\n \"scanning tunnelling microscope\",\n \"spectroscopic imaging\",\n \"electronic density of states\",\n \"critical wavefunctions\",\n \"localized states\",\n \"pseudogaps\",\n \"quasiperiodic systems\",\n \"experimental realization\",\n \"electronic properties\",\n \"lattice assembly\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1","title":"Baake, M. & Grimm, U. Mathematical diffraction of aperiodic structures.Chem. Soc. Rev.41, 6821–6824 (2012).","link":"https://doi.org/10.1039%2Fc2cs35120j"},{"idx":"2","title":"Shechtman, D., Blech, I., Gratias, D. & Cahn, J. W. Metallic phase with long-range orientational order and no translational symmetry.Phys. Rev. Lett.53, 1951–1953 (1984).","link":"https://doi.org/10.1103%2FPhysRevLett.53.1951"},{"idx":"3","title":"Levine, D. & Steinhardt, P. J. Quasicrystals. I. Definition and structure.Phys. Rev. B34, 596–616 (1986).","link":"https://doi.org/10.1103%2FPhysRevB.34.596"},{"idx":"4","title":"Anderson, P. W. Absence of diffusion in certain random lattices.Phys. Rev.109, 1492–1505 (1958).","link":"https://doi.org/10.1103%2FPhysRev.109.1492"},{"idx":"5","title":"Schreiber, M. & Grussbach, H. Multifractal wave functions at the Anderson transition.Phys. Rev. Lett.67, 607–610 (1991).","link":"https://doi.org/10.1103%2FPhysRevLett.67.607"},{"idx":"6","title":"de Bruijn, N. G. Algebraic theory of Penrose's non-periodic tilings of the plane. I.Indag. Math.84, 39–52 (1981).","link":"https://doi.org/10.1016%2F1385-7258%2881%2990016-0"},{"idx":"7","title":"Odagaki, T. Properties of tight-binding electrons in a two-dimensional quasi-lattice.Solid State Commun.60, 693–696 (1986).","link":"https://doi.org/10.1016%2F0038-1098%2886%2990423-0"},{"idx":"8","title":"Kohmoto, M. & Sutherland, B. Electronic and vibrational-modes on a Penrose lattice – localized states and band-structure.Phys. Rev. B34, 3849–3853 (1986).","link":"https://doi.org/10.1103%2FPhysRevB.34.3849"},{"idx":"9","title":"Arai, M., Tokihiro, T., Fujiwara, T. & Kohmoto, M. Strictly localized states on a two-dimensional Penrose lattice.Phys. Rev. B38, 1621–1626 (1988).","link":"https://doi.org/10.1103%2FPhysRevB.38.1621"},{"idx":"10","title":"Rieth, T. & Schreiber, M. Identification of spatially confined states in two-dimensional quasiperiodic lattices.Phys. Rev. B51, 15827–15832 (1995).","link":"https://doi.org/10.1103%2FPhysRevB.51.15827"},{"idx":"11","title":"Sutherland, B. Localization of electronic wave functions due to local topology.Phys. Rev. B34, 5208–5211 (1986).","link":"https://doi.org/10.1103%2FPhysRevB.34.5208"},{"idx":"12","title":"Hafner, J. & Krajčí, M. Electronic structure and stability of quasicrystals: Quasiperiodic dispersion relations and pseudogaps.Phys. Rev. Lett.68, 2321–2324 (1992).","link":"https://doi.org/10.1103%2FPhysRevLett.68.2321"},{"idx":"13","title":"Zijlstra, E. S. & Janssen, T. Density of states and localization of electrons in a tight-binding model on the Penrose tiling.Phys. Rev. B61, 3377–3383 (2000).","link":"https://doi.org/10.1103%2FPhysRevB.61.3377"},{"idx":"14","title":"Stadnik, Z. M., Purdie, D., Baer, Y. & Lograsso, T. A. Absence of fine structure in the photoemission spectrum of the icosahedral Al-Pd-Mn quasicrystal.Phys. Rev. B64, 214202 (2001).","link":"https://doi.org/10.1103%2FPhysRevB.64.214202"},{"idx":"15","title":"Rotenberg, E., Theis, W. & Horn, K. Electronic structure investigations of quasicrystals.Prog. Surf. Sci.75, 237–253 (2004).","link":"https://doi.org/10.1016%2Fj.progsurf.2004.05.002"},{"idx":"16","title":"Widmer, R., Gröning, P., Feuerbacher, M. & Gröning, O. Experimental signatures of spiky local density of states in quasicrystals.Phys. Rev. B79, 104202 (2009).","link":"https://doi.org/10.1103%2FPhysRevB.79.104202"},{"idx":"17","title":"Mäder, R., Widmer, R., Gröning, P., Steurer, W. & Gröning, O. Correlating scanning tunneling spectroscopy with the electrical resistivity of Al-based quasicrystals and approximants.Phys. Rev. B87, 075425 (2013).","link":"https://doi.org/10.1103%2FPhysRevB.87.075425"},{"idx":"18","title":"Wasio, N. A. et al. Self-assembly of hydrogen-bonded two-dimensional quasicrystals.Nature507, 86–89 (2014).","link":"https://doi.org/10.1038%2Fnature12993"},{"idx":"19","title":"Urgel, J. I. et al. Quasicrystallinity expressed in two-dimensional coordination networks.Nat. Chem.8, 657–662 (2016).","link":"https://doi.org/10.1038%2Fnchem.2507"},{"idx":"20","title":"Gopalakrishnan, S., Martin, I. & Demler, E. A. Quantum quasicrystals of spin-orbit-coupled dipolar bosons.Phys. Rev. Lett.111, 185304 (2013).","link":"https://doi.org/10.1103%2FPhysRevLett.111.185304"},{"idx":"21","title":"Jagannathan, A. & Duneau, M. An eightfold optical quasicrystal with cold atoms.Europhys. Lett.104, 66003 (2013).","link":"https://doi.org/10.1209%2F0295-5075%2F104%2F66003"},{"idx":"22","title":"Vardeny, Z. V., Nahata, A. & Agrawal, A. Optics of photonic quasicrystals.Nat. Photon.7, 177–187 (2013).","link":"https://doi.org/10.1038%2Fnphoton.2012.343"},{"idx":"23","title":"Gomes, K. K., Mar, W., Ko, W., Guinea, F. & Manoharan, H. C. Designer Dirac fermions and topological phases in molecular graphene.Nature483, 306–310 (2012).","link":"https://doi.org/10.1038%2Fnature10941"},{"idx":"24","title":"Polini, M., Guinea, F., Lewenstein, M., Manoharan, H. C. & Pellegrini, V. Artificial honeycomb lattices for electrons, atoms and photons.Nat. Nano8, 625–633 (2013).","link":"https://doi.org/10.1038%2Fnnano.2013.161"},{"idx":"25","title":"Stroscio, J. A. & Eigler, D. M. Atomic and molecular manipulation with the scanning tunneling microscope.Science254, 1319–1326 (1991).","link":"https://doi.org/10.1126%2Fscience.254.5036.1319"},{"idx":"27","title":"Repetowicz, P., Grimm, U. & Schreiber, M. Exact eigenstates of tight-binding Hamiltonians on the Penrose tiling.Phys. Rev. B58, 13482–13490 (1998).","link":"https://doi.org/10.1103%2FPhysRevB.58.13482"}],"string":"[\n {\n \"idx\": \"1\",\n \"title\": \"Baake, M. & Grimm, U. Mathematical diffraction of aperiodic structures.Chem. Soc. Rev.41, 6821–6824 (2012).\",\n \"link\": \"https://doi.org/10.1039%2Fc2cs35120j\"\n },\n {\n \"idx\": \"2\",\n \"title\": \"Shechtman, D., Blech, I., Gratias, D. & Cahn, J. W. Metallic phase with long-range orientational order and no translational symmetry.Phys. Rev. Lett.53, 1951–1953 (1984).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.53.1951\"\n },\n {\n \"idx\": \"3\",\n \"title\": \"Levine, D. & Steinhardt, P. J. Quasicrystals. I. Definition and structure.Phys. Rev. B34, 596–616 (1986).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.34.596\"\n },\n {\n \"idx\": \"4\",\n \"title\": \"Anderson, P. W. Absence of diffusion in certain random lattices.Phys. Rev.109, 1492–1505 (1958).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRev.109.1492\"\n },\n {\n \"idx\": \"5\",\n \"title\": \"Schreiber, M. & Grussbach, H. Multifractal wave functions at the Anderson transition.Phys. Rev. Lett.67, 607–610 (1991).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.67.607\"\n },\n {\n \"idx\": \"6\",\n \"title\": \"de Bruijn, N. G. Algebraic theory of Penrose's non-periodic tilings of the plane. I.Indag. Math.84, 39–52 (1981).\",\n \"link\": \"https://doi.org/10.1016%2F1385-7258%2881%2990016-0\"\n },\n {\n \"idx\": \"7\",\n \"title\": \"Odagaki, T. Properties of tight-binding electrons in a two-dimensional quasi-lattice.Solid State Commun.60, 693–696 (1986).\",\n \"link\": \"https://doi.org/10.1016%2F0038-1098%2886%2990423-0\"\n },\n {\n \"idx\": \"8\",\n \"title\": \"Kohmoto, M. & Sutherland, B. Electronic and vibrational-modes on a Penrose lattice – localized states and band-structure.Phys. Rev. B34, 3849–3853 (1986).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.34.3849\"\n },\n {\n \"idx\": \"9\",\n \"title\": \"Arai, M., Tokihiro, T., Fujiwara, T. & Kohmoto, M. Strictly localized states on a two-dimensional Penrose lattice.Phys. Rev. B38, 1621–1626 (1988).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.38.1621\"\n },\n {\n \"idx\": \"10\",\n \"title\": \"Rieth, T. & Schreiber, M. Identification of spatially confined states in two-dimensional quasiperiodic lattices.Phys. Rev. B51, 15827–15832 (1995).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.51.15827\"\n },\n {\n \"idx\": \"11\",\n \"title\": \"Sutherland, B. Localization of electronic wave functions due to local topology.Phys. Rev. B34, 5208–5211 (1986).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.34.5208\"\n },\n {\n \"idx\": \"12\",\n \"title\": \"Hafner, J. & Krajčí, M. Electronic structure and stability of quasicrystals: Quasiperiodic dispersion relations and pseudogaps.Phys. Rev. Lett.68, 2321–2324 (1992).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.68.2321\"\n },\n {\n \"idx\": \"13\",\n \"title\": \"Zijlstra, E. S. & Janssen, T. Density of states and localization of electrons in a tight-binding model on the Penrose tiling.Phys. Rev. B61, 3377–3383 (2000).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.61.3377\"\n },\n {\n \"idx\": \"14\",\n \"title\": \"Stadnik, Z. M., Purdie, D., Baer, Y. & Lograsso, T. A. Absence of fine structure in the photoemission spectrum of the icosahedral Al-Pd-Mn quasicrystal.Phys. Rev. B64, 214202 (2001).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.64.214202\"\n },\n {\n \"idx\": \"15\",\n \"title\": \"Rotenberg, E., Theis, W. & Horn, K. Electronic structure investigations of quasicrystals.Prog. Surf. Sci.75, 237–253 (2004).\",\n \"link\": \"https://doi.org/10.1016%2Fj.progsurf.2004.05.002\"\n },\n {\n \"idx\": \"16\",\n \"title\": \"Widmer, R., Gröning, P., Feuerbacher, M. & Gröning, O. Experimental signatures of spiky local density of states in quasicrystals.Phys. Rev. B79, 104202 (2009).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.79.104202\"\n },\n {\n \"idx\": \"17\",\n \"title\": \"Mäder, R., Widmer, R., Gröning, P., Steurer, W. & Gröning, O. Correlating scanning tunneling spectroscopy with the electrical resistivity of Al-based quasicrystals and approximants.Phys. Rev. B87, 075425 (2013).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.87.075425\"\n },\n {\n \"idx\": \"18\",\n \"title\": \"Wasio, N. A. et al. Self-assembly of hydrogen-bonded two-dimensional quasicrystals.Nature507, 86–89 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fnature12993\"\n },\n {\n \"idx\": \"19\",\n \"title\": \"Urgel, J. I. et al. Quasicrystallinity expressed in two-dimensional coordination networks.Nat. Chem.8, 657–662 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fnchem.2507\"\n },\n {\n \"idx\": \"20\",\n \"title\": \"Gopalakrishnan, S., Martin, I. & Demler, E. A. Quantum quasicrystals of spin-orbit-coupled dipolar bosons.Phys. Rev. Lett.111, 185304 (2013).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.111.185304\"\n },\n {\n \"idx\": \"21\",\n \"title\": \"Jagannathan, A. & Duneau, M. An eightfold optical quasicrystal with cold atoms.Europhys. Lett.104, 66003 (2013).\",\n \"link\": \"https://doi.org/10.1209%2F0295-5075%2F104%2F66003\"\n },\n {\n \"idx\": \"22\",\n \"title\": \"Vardeny, Z. V., Nahata, A. & Agrawal, A. Optics of photonic quasicrystals.Nat. Photon.7, 177–187 (2013).\",\n \"link\": \"https://doi.org/10.1038%2Fnphoton.2012.343\"\n },\n {\n \"idx\": \"23\",\n \"title\": \"Gomes, K. K., Mar, W., Ko, W., Guinea, F. & Manoharan, H. C. Designer Dirac fermions and topological phases in molecular graphene.Nature483, 306–310 (2012).\",\n \"link\": \"https://doi.org/10.1038%2Fnature10941\"\n },\n {\n \"idx\": \"24\",\n \"title\": \"Polini, M., Guinea, F., Lewenstein, M., Manoharan, H. C. & Pellegrini, V. Artificial honeycomb lattices for electrons, atoms and photons.Nat. Nano8, 625–633 (2013).\",\n \"link\": \"https://doi.org/10.1038%2Fnnano.2013.161\"\n },\n {\n \"idx\": \"25\",\n \"title\": \"Stroscio, J. A. & Eigler, D. M. Atomic and molecular manipulation with the scanning tunneling microscope.Science254, 1319–1326 (1991).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.254.5036.1319\"\n },\n {\n \"idx\": \"27\",\n \"title\": \"Repetowicz, P., Grimm, U. & Schreiber, M. Exact eigenstates of tight-binding Hamiltonians on the Penrose tiling.Phys. Rev. B58, 13482–13490 (1998).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.58.13482\"\n }\n]","isConversational":false}}},{"rowIdx":94,"cells":{"unique_id":{"kind":"string","value":"s41467-019-13077-6"},"subfield":{"kind":"string","value":"Materials science"},"title":{"kind":"string","value":"Light-fuelled freestyle self-oscillators"},"abstract":{"kind":"string","value":"Self-oscillation is a phenomenon where an object sustains periodic motion upon non-periodic stimulus. It occurs commonly in nature, a few examples being heartbeat, sea waves and fluttering of leaves. Stimuli-responsive materials allow creating synthetic self-oscillators fuelled by different forms of energy, e.g. heat, light and chemicals, showing great potential for applications in power generation, autonomous mass transport, and self-propelled micro-robotics. However, most of the self-oscillators are based on bending deformation, thereby limiting their possibilities of being implemented in practical applications. Here, we report light-fuelled self-oscillators based on liquid crystal network actuators that can exhibit three basic oscillation modes: bending, twisting and contraction-expansion. We show that a time delay in material response dictates the self-oscillation dynamics, and realize a freestyle self-oscillator that combines numerous oscillation modes simultaneously by adjusting the excitation beam position. The results provide new insights into understanding of self-oscillation phenomenon and offer new designs for future self-propelling micro-robots."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"Leaves oscillate in response to wind flow. Intriguingly, the frequency of such oscillation can be almost constant, even if driven by a gust of wind that lacks the corresponding periodicity. Leaf oscillation, together with a broad range of phenomena including heartbeat, bridge swaying, sea waves, etc., exemplify naturally occurring self-oscillatory processes [1] . The cause of such self-oscillation is the interplay between the oscillator motion and the force that triggers the oscillation. In the case of leaves, torsional galloping of the leaf shades the aerodynamic vortex generated behind the surface, thus providing positive feedback to the motion that sustains the vibration with a defined periodicity [2] . Generally, the onset of self-oscillation needs to meet critical conditions [3] . This is evident from daily observations: only one or few leaves with specific orientation with respect to the wind flow direction oscillate under a breeze, while the neighbouring leaves with different orientation remain stationary. Passive oscillating elements in nature rely on an external driving force, such as wind. Stimuli-responsive soft materials allow realizing externally fuelled, man-made self-oscillators, the motion of which is sustained by inner forces that arise due to stimuli-induced changes in material properties. Self-oscillators fuelled by, e.g. light [4] , [5] , heat [6] , and chemical reactions [7] have been generated, offering a possibility towards self-sustained motions without the need of human control. Towards this goal, several device functions are demonstrated, including electric power generation [6] , [8] , mass transport [9] , [10] , mill [11] , and self-propelling locomotion [5] , [7] , [12] . Among the different stimuli, light is particularly promising, due to its sustainability, precise controllability, and omnipresence, the Sun being the ultimate energy source. However, most of the light-driven self-oscillators reported to date are using bending deformation as the main degree of freedom of movement or, in few cases, the combination of bending and twisting [13] , [14] . The broader the range of available oscillation modes, the more sophisticated autonomous devices one can potentially construct. Therefore, there is a need for self-sustained actuators with versatile motion and complex oscillation modes. Liquid crystal elastomers and polymer networks (LCNs) are an important class of soft actuators that have been widely used in micro-robotics [15] , [16] , [17] , [18] , [19] . Their utility is based on coupling between elasticity due to the crosslinked network and anisotropic molecular orientation arising from the LC character. As a result of this combination, LCNs may exhibit pronounced macroscopic shape changes in response to external stimuli. By embedding light-sensitive dyes into the LCN, the material can reversibly and rapidly deform upon light excitation [20] . The photoactuation can be triggered either photothermally [21] or photochemically [22] (or by a combination of these), both leading to light-induced control over the degree of molecular alignment. Programming the molecular alignment in LCNs leads to diverse forms of shape changes [23] , enabling the desired deformation mode, and in some cases self-oscillation [4] , [5] , [13] , [14] , to be realized. Typically, light-fuelled self-oscillators rely on a bending LCN strip, in which the oscillation is triggered by alternate activation of the two surfaces [4] , or an optical configuration that provides self-shadowing within the motion cycles [24] . Being based on cantilever-type or strip-like geometries, the oscillation has been explained with a model connecting the self-oscillation frequency with their harmonic resonances [24] , [25] . However, there are some complications in this approach. First, the photomechanical oscillators experience significant damping during the fast motion because of the small mass/inertia compared to the friction/drag. Such damping causes large energy loss and requires compensation from the external stimulus, different to conventional harmonic oscillation. Second, the energy flow into the material by light absorption is oscillating during the cyclic movement, which in turn gives rise to oscillating temperature in the LCN. The change in intensity/temperature leads to change in modulus [26] and thus to a non-constant resonance frequency. Owing to the above considerations, several important issues need to be addressed, such as the possibility of obtaining versatile oscillating modes, understanding the feedback mechanism that actually fuels the motion and compensates for the damping loss, and the differences between LCN self-oscillation and conventional cantilever oscillation. Here we try to address the above questions by exploring the mechanism of self-oscillation in LCN photoactuators. We first theoretically point out that the time delay in the material response provides a positive feedback to the motion, serving as the key to self-sustained oscillation. Then we devise three cantilever-type photoactuators capable of exhibiting the three basic oscillation modes: bending, twisting, and contraction–expansion. We show that the oscillation frequencies of the latter two diverge from the expected natural frequencies. Finally, we demonstrate a freestyle self-oscillator by hanging an LCN photoactuator on a thread, in which all the above degrees of freedom can be combined. The freestyle oscillator has numerous stabilized oscillation modes upon light excitation at different sample positions and is able to evolve spontaneously between different modes upon irradiation with a constant light beam. We believe that the generalized model provided, together with the freestyle self-oscillator demonstrated, can give new insights into soft matter mechanics.","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],"infos":[{"idx":"1","title":"Jenkins, A. Self-oscillation.Phys. Rep.525, 167–222 (2013).","link":"https://doi.org/10.1016%2Fj.physrep.2012.10.007"},{"idx":"2","title":"Leclercq, T., Peake, N. & de Langre, E. Does flutter prevent drag reduction by reconfiguration?Proc. R. Soc. A474, 20170678 (2018).","link":"https://doi.org/10.1098%2Frspa.2017.0678"},{"idx":"3","title":"Tadrist, L., Julio, K., Saudreau, M. & de Langre, E. Leaf flutter by torsional galloping: experiments and model.J. Fluids Struct.56, 1–10 (2015).","link":"https://doi.org/10.1016%2Fj.jfluidstructs.2015.04.001"},{"idx":"4","title":"White, T. J. et al. A high frequency photodriven polymer oscillator.Soft Matter4, 1796–1798 (2008).","link":"https://doi.org/10.1039%2Fb805434g"},{"idx":"5","title":"Gelebart, A. H. et al. Making waves in a photoactive polymer film.Nature546, 632–636 (2017).","link":"https://doi.org/10.1038%2Fnature22987"},{"idx":"6","title":"Wang, X. Q. et al. In-built thermo-mechanical cooperative feedback mechanism for self-propelled multimodal locomotion and electricity generation.Nat. Commun.9, 3438 (2018).","link":"https://doi.org/10.1038%2Fs41467-018-06011-9"},{"idx":"7","title":"Maeda, S., Hara, Y., Sakai, T., Yoshida, R. & Hashimoto, S. Self-walking gel.Adv. Mater.19, 3480–3484 (2007).","link":"https://doi.org/10.1002%2Fadma.200700625"},{"idx":"8","title":"Tang, R. et al. Optical pendulum generator based on photomechanical liquid-crystalline actuators.ACS Appl. Mater. Interfaces7, 8393–8397 (2015).","link":"https://doi.org/10.1021%2Facsami.5b01732"},{"idx":"9","title":"Baumann, A. et al. Motorizing fibres with geometric zero-energy modes.Nat. Mater.17, 523–527 (2018).","link":"https://doi.org/10.1038%2Fs41563-018-0062-0"},{"idx":"10","title":"Ahn, C., Li, K. & Cai, S. Light or thermally-powered autonomous rolling of an elastomer rod.ACS Appl. Mater. Interfaces10, 25689–25696 (2018).","link":"https://doi.org/10.1021%2Facsami.8b07563"},{"idx":"11","title":"Vantomme, G., Gelebart, A. H., Broer, D. J. & Meijer, E. W. A four-blade light-driven plastic mill based on hydrazone liquid-crystal networks.Tetrahedron73, 4963–4967 (2017).","link":"https://doi.org/10.1016%2Fj.tet.2017.06.041"},{"idx":"12","title":"Wie, J. J., Shankar, M. R. & White, T. J. Photomotility of polymers.Nat. Commun.7, 13260 (2016).","link":"https://doi.org/10.1038%2Fncomms13260"},{"idx":"13","title":"Lee, K. M. et al. Photodriven, flexural–torsional oscillation of glassy azobenzene liquid crystal polymer networks.Adv. Funct. Mater.21, 2913–2918 (2011).","link":"https://doi.org/10.1002%2Fadfm.201100333"},{"idx":"14","title":"Ge, F., Yang, R., Tong, X., Camerel, F. & Zhao, Y. A multifunctional dye-doped liquid crystal polymer actuator: light-guided transportation, turning in locomotion, and autonomous motion.Angew. Chem. Int. Ed.57, 11758–11763 (2018).","link":"https://doi.org/10.1002%2Fanie.201807495"},{"idx":"15","title":"Zeng, H., Wasylczyk, P., Wiersma, D. S. & Priimagi, A. Light robots: bridging the gap between microrobotics and photomechanics in soft materials.Adv. Mater.30, 1703554 (2018).","link":"https://doi.org/10.1002%2Fadma.201703554"},{"idx":"16","title":"Ohm, C., Brehmer, M. & Zentel, R. Liquid crystalline elastomers as actuators and sensors.Adv. Mater.22, 3366–3387 (2010).","link":"https://doi.org/10.1002%2Fadma.200904059"},{"idx":"17","title":"Nocentini, S., Parmeggiani, C., Martella, D. & Wiersma, D. S. Optically driven soft micro robotics.Adv. Opt. Mater.6, 1800207 (2018).","link":"https://doi.org/10.1002%2Fadom.201800207"},{"idx":"18","title":"Ambulo, C. P. et al. Four-dimensional printing of liquid crystal elastomers.ACS Appl. Mater. Interfaces9, 37332–37339 (2017).","link":"https://doi.org/10.1021%2Facsami.7b11851"},{"idx":"19","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":"20","title":"Ikeda, T., Mamiya, J.-I. & Yu, Y. Photomechanics of liquid-crystalline elastomers and other polymers.Angew. Chem. Int. Ed.46, 506–528 (2007).","link":"https://doi.org/10.1002%2Fanie.200602372"},{"idx":"21","title":"Yang, Y., Pei, Z., Li, Z., Wei, Y. & Ji, Y. Making and remaking dynamic 3d structures by shining light on flat liquid crystalline vitrimer films without a mold.J. Am. Chem. Soc.138, 2118–2121 (2016).","link":"https://doi.org/10.1021%2Fjacs.5b12531"},{"idx":"22","title":"Li, M. H., Keller, P., Li, B., Wang, X. & Brunet, M. Light‐driven side-on nematic elastomer actuators.Adv. Mater.15, 569–572 (2003).","link":"https://doi.org/10.1002%2Fadma.200304552"},{"idx":"23","title":"White, T. J. & Broer, D. J. Programmable and adaptive mechanics with liquid crystal polymer networks and elastomers.Nat. Mater.14, 1087–1098 (2015).","link":"https://doi.org/10.1038%2Fnmat4433"},{"idx":"24","title":"Gelebart, A. H., Vantomme, G., Meijer, B. E. W. & Broer, D. J. Mastering the photothermal effect in liquid crystal networks: a general approach for self-sustained mechanical oscillators.Adv. Mater.29, 1606712 (2017).","link":"https://doi.org/10.1002%2Fadma.201606712"},{"idx":"25","title":"Serak, S. et al. Liquid crystalline polymer cantilever oscillators fueled by light.Soft Matter6, 779–783 (2010).","link":"https://doi.org/10.1039%2FB916831A"},{"idx":"26","title":"Kumar, K., Schenning, A. P. H. J., Broer, D. J. & Liu, D. Regulating the modulus of a chiral liquid crystal polymer network by light.Soft Matter12, 3196–3201 (2016).","link":"https://doi.org/10.1039%2FC6SM00114A"}]}},{"section":"Results","content":"Origin of light-fuelled self-oscillation While tree leaves oscillate in response to wind flow (Fig. 1a, b ), the photoresponse of a soft actuator results from its mechanical deformability upon light illumination. Taking a bending LCN actuator as an example (Fig. 1c ), the bending angle increases when increasing the light intensity I (Fig. 1d ), which can be qualitatively described as a linear dependence between the deformation ( D ) and the energy absorbed ( E ) by the actuator (Fig. 1f, g ). Note that the deformation does not have to be bending as exemplified here but any combination of twisting/bending/contraction, as will be illustrated later. Owing to the directionality of the incident light field, the effective light-absorbing area A depends on the deformed geometry. This can be noticed from Fig. 1e , illustrating that the higher the bending angle, the smaller the light-absorbing area. The relation between A and D can be qualitatively described as a linear curve (Fig. 1g ), in which the axis of A can be linked to E through a simple relationship: E = A · I . Imposing the two curves gives an intersection, determined by the light intensity, at which the equilibrium occurs (Fig. 1h ). Fig. 1 The origin of self-oscillation in natural and in artificial light-fuelled systems. a , b Photographs of a leaf self-oscillating under a breeze. Inset of b shows time-dependent displacement of the centre of mass of the leaf, indicating periodic movement with well-defined frequency. Scale bar is 5 cm. A free-standing LCN actuator ( c ) bends slightly upon low-intensity irradiation (50 mW cm −2 , d ) and more pronouncedly when the intensity is increased (200 mW cm −2 , e ). f – h Schematic drawings showing the qualitative dependence between absorbed energy E , deformation D , effective light-absorbing area A , and equilibrium position upon illumination. i Harmonic oscillator (solid line, numerical solution to equation, \\(\\ddot x + \\omega _{\\mathrm{o}}^2x = 0\\) ) and an oscillator experiencing damping (dashed line, numerical solution to equation, \\(\\ddot x + \\varsigma \\dot x + \\omega _{\\mathrm{o}}^2x = 0\\) ), where ω 0 = 2π, \\(\\dot x\\left( 0 \\right) = 0,x\\left( 0 \\right) = 1\\) , \\(\\varsigma = 0.1\\) . Self-oscillation induced by j a minor time delay σ and k a large σ . Numerical solutions to equation \\(\\ddot x - \\left( {\\sigma \\omega _{\\mathrm{o}}^2 - \\varsigma - \\eta x^2} \\right)\\dot x + \\omega _{\\mathrm{o}}^2x = 0\\) , where \\(\\varsigma = 0.1,\\omega _0 = 2{\\mathrm{\\pi }}\\) , \\(\\dot x\\left( 0 \\right) = 1,x\\left( 0 \\right) = 0\\) , and \\(\\sigma \\omega _{\\mathrm{o}}^2 = 1,\\eta = 3\\) in j and \\(\\sigma \\omega _{\\mathrm{o}}^2 = 20,\\eta = 60\\) in k Full size image Around this equilibrium, the most familiar case is the harmonic oscillator, where an object experiences a restoring force proportional to its displacement x . The equation of motion can be written as: $$\\ddot x + \\omega _{\\mathrm{o}}^2x = 0,$$ (1) where ω o is the angular frequency of the system. Solving Eq. ( 1 ) yields harmonic oscillation with a constant amplitude and periodicity 2π/ ω o , defined by its natural frequency, as shown by the solid line in Fig. 1i . However, in most cases, the oscillator experiences damping from its environment, a restoring force proportional to the velocity of its motion: $$\\ddot x + \\varsigma \\dot x + \\omega _{\\mathrm{o}}^2x = 0,$$ (2) where ϛ is the damping ratio. Accounting for the damping, the amplitude gradually decreases, due to energy dissipation from the system to the environment (the dashed line in Fig. 1i ). For a light-responsive mechanical cantilever, the restoring force arises from a sequence of processes: light absorption by the photoactive molecules, conversion of light energy into heat, heat transport across the sample area, and finally, build-up of the inner stress needed for deformation. All these contribute to a time delay σ in the actuation. Thus the equation of motion can be written as $$\\ddot x\\left( t \\right) + \\varsigma \\dot x(t) + \\omega _{\\mathrm{o}}^2x\\left( {t - \\sigma } \\right) = 0.$$ (3) The value of σ may vary significantly depending on the material system, and it may also depend on the exact position of the LCN within the oscillation cycle. In order to reveal the contribution of this delay into the motion kinetics, we have simplified the model and consider only an invariant σ in the motion equation. We use Taylor expansion of Eq. ( 3 ), yielding $$\\ddot x + \\varsigma \\dot x + \\omega _o^2\\left( {x - \\sigma \\frac{{\\dot x}}{{1!}} + \\sigma ^2\\frac{{\\ddot x}}{{2!}} - \\sigma ^2\\frac{{\\mathop {x}\\limits^{...} }}{{3!}} \\cdots } \\right) = 0$$ (4) In this equation, the damping term \\(- \\sigma \\omega _{\\mathrm{o}}^2\\dot x\\) has a negative value, indicating that the oscillation amplitude increases to a certain extent, until the response becomes nonlinear. For example, air drag would significantly increase at large oscillation amplitudes, leading to amplitude saturation. Herein we adopt a nonlinear positive damping term \\(\\eta x^2\\dot x\\) (where η is a positive constant) based on van der Pol model [27] and simplify Eq. ( 4 ) using first-order approximation. The equation of motion for a light-responsive actuator at the equilibrium then appears as $$\\ddot x - \\left( {\\sigma \\omega _{\\mathrm{o}}^2 - \\varsigma - \\eta x^2} \\right)\\dot x + \\omega _{\\mathrm{o}}^2x = 0.$$ (5) Numerical solution to Eq. ( 5 ) when σ is small is given in Fig. 1j , showing an exponential growth of the oscillation amplitude until reaching saturation. Comparing Fig. 1i, j one may notice that all these oscillations exhibit very similar periodicity, matching the conventional cantilever natural frequency, ω o /2π. However, when σ increases, the periodicity increases and the form of oscillation becomes non-sinusoidal, as shown in Fig. 1k . Numerical calculations on the oscillation frequency and the waveform upon increasing σ are given in Supplementary Fig. 1 . The key to self-oscillation is the time delay in the material response [1] : while an oscillator passes through the equilibrium ( x = 0), the delayed force remains in the system showing the same direction as the velocity, thus pushing the object out of the equilibrium and providing positive feedback to the motion. Such feedback extracts the energy from the external energy source (light), fuelling the oscillator to sustain the motion. In the following, we explore the relation between the deformation degrees of freedom and the equilibrium positions in different light-fuelled self-oscillators and connect the observed oscillation modes to the material delay. Basic oscillation modes The three basic deformation modes in a cantilever—bending, contraction–expansion, and twisting—are schematically shown in Fig. 2a . Their amplitudes can be quantified as out-of-plane bending angle α , displacement d , and twisting angle Ф around the cantilever axis, respectively. To realize the basic deformations, we used planar-aligned LCN cantilevers, with a director along the cantilever for bending and contraction and perpendicular to it for twisting (Fig. 2b ). The bending/twisting and contracting actuators are based on different LCN materials, as further elaborated in “Methods” and the Supplementary Methods. Fig. 2 Light-fuelled self-oscillators based on three basic deformation modes. a Schematic drawing of the three basic deformation modes in a cantilever: bending, contraction, and twisting. The deformation can be quantified through out-of-plane bending angle α , displacement d , and twisting angle Ф for each mode. b Molecular alignment and irradiation geometry to initiate self-oscillation in each deformation mode. c Photographs of bending actuator with size 5.5 × 1.5 × 0.05 mm 3 upon 180 mW laser beam (spot size 2.2 mm) and the corresponding oscillation dynamics. d Photographs of cylindrical actuator (100 μm diameter) upon 100 mW focused laser excitation (spot size 20 μm) and the contracting–expanding oscillation dynamics. e Twisting self-oscillator (size: 3.5 × 3.5 × 0.05 mm 3 , laser: 180 mW, spot 2.2 mm) and the corresponding oscillation dynamics. The scale bars are 2 mm in c , e , 100 μm in d Full size image The photoactuation is triggered with a 488 nm continuous-wave laser beam, using different irradiation geometries for the different modes as depicted in Fig. 2b . Bending is obtained by light propagation along the cantilever axis, in which case the laser alternately activates both surfaces during the cyclic motion. The photoactuation results from light absorption gradient across the thickness, which in turn gives rise to photoinduced stress on the irradiated surface, and bending towards the illuminating light source. Combination of photochemical and photothermal effects is likely responsible for this type of oscillation [13] . Contraction occurs when the beam is focused on the tip of the cantilever, heating up the material, followed by subsequent cooling and expansion once the cantilever has moved away from the laser spot. Thus photothermal effects are responsible for this actuation mode. Twisting/torsional deformation takes place when the strip is illuminated from the side, causing an angular displacement around the cantilever axis and alternate exposure of the two cantilever surfaces, driven by similar mechanism as in the case of bending. Further details on photoactuation and the experimental arrangements are given in “Methods” and Supplementary Methods, respectively. Equilibrium positions, where α = d = Ф = 0, can be found for each mode. When driven out of equilibrium (initiated by air fluctuation or a mechanical trigger), the cantilever returns to the equilibrium position due to elastic force, and the delay in the material response causes the self-oscillation as explained earlier. The frequency of the self-oscillation depends on the desired mode. The bending oscillation shown in Fig. 2c has the highest frequency of 80 Hz among the three basic modes (see the slow-motion video, Supplementary Movie 1 ), well matching with the calculated natural resonance frequency of the cantilever (about 90 Hz, see natural frequency calculations in Supplementary Methods for further details). This indicates a minimal time delay σ in such mode, yielding self-oscillation dominated by ω o (Eq. ( 5 ), Fig. 1j ). The natural resonance frequency is determined by the cantilever geometry and the material rigidity, as has been systematically studied in several LCN self-oscillators [24] , [25] . In contrast, the contraction–expansion oscillator as well as the twisting one exhibit much longer periodicities, and resonance frequencies in the range of 0.5–18 Hz for contraction and about 1 Hz for twisting can be observed (Fig. 2d, e ; Supplementary Movies 2 and 3 ). By comparing the observed frequencies to the natural resonances, estimated to be >20 kHz for the longitudinal vibration mode and ca. 170 Hz for the torsional one (see natural frequency calculations in Supplementary Methods), the frequencies of light-fuelled self-oscillations are 2–4 orders of magnitude lower. We ascribe this to significant increase in σ in those modes. During contraction–expansion, the light energy is absorbed within the focused spot area located at the tip of the cantilever. After the light absorption, heat has to be conducted along the cantilever axis to trigger deformation, thus posing a huge time delay. This delay strongly depends on the amount of energy being absorbed by the material. A set of stable oscillations with a broad range of frequencies, from 0.5 to 18 Hz, are observed by slightly tuning the laser spot position (same power) on the sample (Supplementary Fig. 2 ). We attribute this to differences in light scattering due to inhomogeneity and hence different amount of light being absorbed, which in turn yields different contraction length and time delay in the material response. In the twisting-mode oscillator, the time delay arises from the stress accumulation required for the material to switch its angular orientation. In this case, there seem to be two stable states, as indicated by the rectangular waveform shown in Fig. 2e . More detailed analysis on the contraction–expansion and twisting self-oscillation modes have been conducted, with results shown in Supplementary Figs. 3 and 4 . By maintaining the same excitation beam position but changing the input power, the frequency of the contraction–expansion mode varies between 16 and 20 Hz (see the Fourier transform in Supplementary Fig. 3b ). However, no direct relation between the oscillation frequency and the excitation power can be deduced. The twisting actuator, instead, maintains the same oscillation amplitude irrespective of the laser power, but its oscillation frequency increases from 0.7 to 2 Hz (Supplementary Fig. 4b, c ). Such findings seem to contradict conventional conceptions about cantilever mechanics, where the structural rigidity determines the natural resonance frequency, while the input power may affect only the oscillation amplitude. These observations, together with the non-sinusoidal waveforms and the considerably low oscillation frequencies compared to the natural resonance frequencies, indicate that the time delay becomes the dominating factor in the contracting and twisting self-oscillators. Note that, for large time delays, the simplified model given by Eq. ( 5 ) will no longer be accurate and more elaborate numerical calculation methods would be needed for solving the corresponding kinetic equations. A freestyle self-oscillator The actuators described above exhibit only one selected oscillation mode. In order to combine the different modes into a single actuator, we use a main-chain LCN with high degree of deformability [28] (>50% contraction upon order–disorder phase transition; thermal actuation and mechanical properties between different LCNs given in Supplementary Figs 5 and 6 ). Material details are given in “Methods”; for further information about sample preparation, see the Supplementary Methods. Upon excitation with a laser spot, the localized strain builds up stress inside the material, causing additional degrees of freedom for the entire deformed structure, as schematized in Fig. 3a and experimentally demonstrated in Fig. 3b–e . The strip (Fig. 3b ) bends when illuminated with a laser spot (100 mW) at the centre of the cantilever (Fig. 3c ). Moving the laser spot horizontally, close to the edge of the cantilever, triggers combined twisting and bending deformation (Fig. 3d ). By increasing the light intensity from 100 to 140 mW, a larger extent of deformation can be triggered, and bending and contraction occur simultaneously (Fig. 3e ). To better visualize the twisting, we attached a weight to the end of the actuator (1.4 g, 250 times the weight of the actuator) to suppress bending (Fig. 2f ). When the laser spot deviates from the centre of the actuator, contraction and twisting can be observed (Fig. 2g ). Fig. 3 A freestyle self-oscillator. a Schematic drawing of a soft actuator combining the three basic deformation modes (twisting, bending, and contraction). An LCN actuator ( b ) that bends ( c ), bends and twists ( d ), bends and contracts ( e ), upon changing the irradiation conditions. The circle marks the position of the laser spot. With an external load that prevents bending ( f ), the actuator contracts and twists simultaneously when excited from one side as indicated in g . h Schematics of a light-driven self-oscillator swinging on a thread. i – k Snapshots of the freestyle oscillator with different oscillation modes upon laser excitation (488 nm, 160 mW) at different positions. LCN strip size: 3 × 25 × 0.1 mm 3 . Inset of j shows the microscopic image of the hole punctured in the LCN actuator (scale bar 200 μm). Inset of k shows the displacement of the end of the strip during the oscillation process. All scale bars: 2 mm Full size image To trigger self-oscillation, the excitation beam orientation and spot location need to meet critical conditions. We realize this by puncturing a hole into the LCN (inset of Fig. 3j ) and hanging it vertically on a thread (human hair), thereby providing freedom to swing around the thread axis, as schematically illustrated in Fig. 3h . Under such configuration, any kind of light-induced deformation (bending and/or contraction and/or twisting) shifts the centre of mass of the actuator (Fig. 3h ; 1 → 2). In return, gravity imposes a torque to rotate the deformed structure around the thread axis (2 → 3), and the strip moves out of the illuminating area, cools down, and relaxes back to the original position (3 → 1′), and the cycle restarts. Triggered by subsequent sequences of the above-described cycles, the strip oscillates continuously upon a constant laser beam excitation. By slightly changing the position of the laser spot, different oscillation modes can be obtained (Fig. 3i–k and Supplementary Movie 4 ). Each mode has a well-defined periodicity, as exemplified in the inset of Fig. 3k . During the distinct stable oscillation modes, the laser beam is not spatially or temporally modified. However, by slightly adjusting the laser spot position along the LCN strip, the actuator may exhibit fast-varying geometry, oscillating like a dancer on a thread (Supplementary Movie 5 ), which we coin as light-fuelled freestyle self-oscillation. Stabilized oscillations and evolution between the modes For more detailed characterization of the oscillating modes, we labelled the free end of the LCN strip with fluorescent particles (Fig. 4a , and further details are given in data collection in Supplementary Methods). With the help of the fluorescent markers, we can track the trajectory of oscillation in the x – y plane, offering a distinct fingerprint for each oscillation mode. Each mode prevails for a relatively long time, typically at least 5 s, maintaining the same amplitude and frequency (Supplementary Fig. 7 ). However, the oscillation frequency may significantly vary for the different modes, e.g. from 0.4 to 6.7 Hz between examples shown in Fig. 4b–g , and no connection between the oscillation frequency and amplitude seems to exist. The trajectories with zero area correspond to reciprocal oscillation (Fig. 4e–g ) while those with non-zero area indicate non-reciprocal motion during the oscillation cycle (Fig. 4b–d ). Fig. 4 Stabilized modes and evolution between the modes. a Schematic drawing of the arrangement for tracking the oscillator position. b – g Examples of different self-oscillation modes upon excitation with a laser spot at different positions. Data shows the tracking trajectory on x – y plane for one oscillation cycle. h Evolution between modes upon a constant laser excitation. The x – y trajectory spontaneously changes upon prolonged excitation ( i , j ), while maintaining the similar oscillating periodicity ( k ) until 190 s when the oscillation vanishes Full size image Figure 4h presents an intriguing oscillation behaviour under constant irradiation conditions: spontaneous evolution between different oscillation modes. In short term, the periodicity of each mode is constant (Fig. 4k ) but varies somewhat over extended time periods, e.g. from 0.35 to 0.45 s within 3 min (Supplementary Fig. 8 ). The spontaneous evolution between the modes (Fig. 4i, j ) without changing the excitation beam position is ascribed to irreversibility of the actuating material, which is quantified by monitoring a freestanding LCN upon cyclic excitation (Supplementary Fig. 9 ). The cyclic actuation shows that the material maintains its deformability, i.e. the maximum strain obtained upon constant irradiation, while the relaxation process shows certain degree of reduction with a decay constant of about 80 cycles. The reason for such photomechanical irreversibility may be attributed to kinetic deformation driven by elastic entropy, where cyclic stretching–coiling gives rise to friction between the polymeric chains, and photobleaching of the dyes. During the oscillation, the actuator undergoes hundreds of deformation–relaxation cycles. Hence, even slight changes in material response may shift the deformation mode and lead to spontaneous evolution between the oscillation modes. At about 190 s, the oscillator stops at a stable position, where the negative damping condition is no longer met ( \\(\\sigma \\omega _{\\mathrm{o}}^2 \\, > \\, \\varsigma\\) , Eq. 5 ), possibility due to the irreversibility of the material response. The prediction and control of the evolution between different modes during the freestyle oscillation is difficult to attain, due to the multiple degrees of freedom in deformation and complexity of light–matter interactions in responsive soft materials, resulting in different time delay in material response at different stages of the oscillation cycle. Despite the lack of control, the intriguing freestyle self-oscillation phenomenon provides insight into the process that, we believe, can be generalized into different kinds of responsive materials with arbitrary geometry, deformability, and stimulus responsivity, as long as a feedback mechanism has been established for sustaining the periodic motion. From general material concept towards practical applications The above light-driven system combines the basic oscillation modes—bending, contraction, and twisting—into a single freestyle oscillator. Although the movements are complex, the freestyle oscillator is governed by the same physics as conventional oscillators with simpler movements. We believe that similar principles can be applied also to other types of stimuli-responsive systems. For example, thermoresponsive nylon fibres [9] , carbon-nanotube-based bi-layer actuators [29] , thermoactive elastomer rods [10] , hydrogels [30] , and azobenzene crystals [31] have all shown self-sustained cyclic movements. Different from approaches in systems chemistry [32] , [33] , which aim at programming positive/negative feedback between reversible chemical reactions to induce oscillating behaviours, the approach demonstrated here focuses on photomechanical response in macroscopic samples, driven by time delay in material response and not restricted in few specific materials. Also note that the self-oscillation discussed in this study is different from the phenomenon of fluctuation, which, however, in some cases has been coined as chaotic self-oscillation [34] . A well-defined frequency component in the Fourier transform can serve as a good indicator for the occurrence of self-oscillation. To combine self-oscillators into locomotive devices to achieve self-sustained micro-robots is challenging. First, most of the actuators used in self-oscillation studies are free-standing strips lacking external loading. Upon devising a linkage between the motor (the oscillator) and the robot body (the moving element), the mechanical conditions change. Upon locomotion, the oscillator’s position with respect to the illumination source, as well as its geometry and mechanical properties, change, affecting the equilibrium conditions crucial for sustaining the oscillation. Thus a conventional mechanical design, e.g. biking the wheel [35] , may lead to a failure in long-time conversion of oscillating motion into mechanical work. Second, efficient translocation often requires a nonreciprocal deformation, which in turn requires temporal coordination of a series of self-oscillation events or within different sequences of a self-oscillation cycle with certain phase delays [36] , [37] . However, coupling between different self-oscillators sets extra hurdles. Third, light-responsive materials may not provide enough force to overcome the friction/drag experienced during the robotic motion, hence other soft mechanic properties like snapping and instabilities [37] need to be further investigated in self-oscillating systems, in order to reach their full potential. Although challenges exist, pioneering examples have shed light on device integration, e.g. mass transportation [9] , [10] and self-propelled walking [5] . We hope more self-sustained devices will be reported in the future, and that self-sustained robots can be integrated with more sophisticated functions by the joint scientific development in this field. A useful application can be foreseen in realizing self-oscillator within fluidic environment, which may eventually give birth to self-sustained pumping and cilia-like collective movement for efficient microfluidic manipulation. However, the dragging force (large ς) may pose great hurdle to meet the required negative damping condition \\((\\sigma \\omega _{\\mathrm{o}}^2 \\, > \\, \\varsigma )\\) . A possible solution could be to investigate stimuli-responsive materials with a large delay σ in material response, such as light/heat-responsive hydrogels [38] . Beyond the above-mentioned robotic applications that aim to transfer input light energy into mechanical work output, there is a frontier of optical applications to be investigated. A self-oscillator is essentially an optical chopper. However, compared to the conventional electronic motorized chopping system, it has many advantages, such as compact size, light weight, and, being remotely powered, possibility towards integration into micro-devices. An oscillating strip can modulate a signal beam (635 nm) periodically in its transmission, as shown in Fig. 5a : during the down-bending stage, the beam is transmitted, while the up-bending stage blocks the beam. In other words, the signal beam transmission is controlled and modulated by the self-oscillator (Fig. 5b ). Furthermore, one single strip can even control two signal beams simultaneously. As shown in Fig. 5c , the two signal beams can be in phase, when propagating along the same side of the strip, or 180° out-of-phase, when propagating from the opposite sides of the strip (Fig. 5d ). Fig. 5 Self-oscillator-based beam modulator. a A self-oscillating strip (bending mode, driven by a continuous 488 nm, 190 mW) can modulate the transmission of a signal beam (635 nm). The signal beam is focused on the upper part of the oscillator as shown by the schematic drawing, while the transmitted beam profiles are shown in the photos. Scale bar: 1 cm. b Transmission measurement of the beam being modulated by the self-oscillator. c Two signal beams are propagating along the same side of the oscillator and the measured transmission intensities being modulated in phase. d Two signal beams are propagating along the opposite sides of the oscillator and the measured transmission intensities being modulated 180° out-of-phase. Blue curves in c , d show the scattered 488 nm light intensity for phase reference. Strip size: 1 × 10 × 0.05 mm 3 Full size image"},{"section":"Discussion","content":"We study the mechanism of light-driven self-oscillation and show that a time delay due to the material response is the key to sustain the periodicity of oscillation. By interplay between the equilibrium position upon irradiation and specific form of light-induced deformation pre-programmed into the LCN by molecular-alignment control, three basic oscillation modes—bending, contraction, and twisting—are demonstrated. A main-chain LCN actuator combines the basic oscillation modes, yielding freestyle oscillation when hung on a thread. Such freestyle oscillator exhibits different stabilized modes upon continuous laser excitation at different sample positions and evolution between the modes across a relatively long period under a constant light field. The results provide insight into light-driven self-oscillation in soft materials and may in longer term open up new approaches to realize light-driven motors for self-sustained soft micro-robotics."},{"section":"Methods","content":"Materials in brief The LCN photoactuators demonstrated here are photopolymerized from monomer mixtures containing liquid crystalline mesogens, crosslinkers, light-sensitive dyes, and photoinitiators, via one-step or two-step fabrication strategies. Different materials were used for different deformation/oscillation modes. For bending and twisting, a side-chain LCN with 28 mol% crosslinker (of which 6 mol% are azobenzene crosslinkers) is used [39] ; for contraction–expansion, we used a side-chain LCN polymerized from different LC mesogens and reduced crosslinker concentration (10 mol%) [40] to render the LCN softer; for freestyle actuator, we used a main-chain LCN synthesized through two-step fabrication process [28] . All details about material preparation procedures are given in Supplementary Methods. Photoactuation in brief An unfocused, continuous-wave 488 nm laser beam was propagated along the axis of the LCN cantilever in case of bending-mode oscillation and perpendicular to the axis for the twisting mode. Upon laser excitation, the azobenzene moieties in the LCN undergo repeated trans - cis - trans isomerization. Owing to limited light-penetration depth, isomerization gradient across the film thickness is formed, which, together with photothermal heating, gives rise to cantilever deformation towards the light source. As a result, the laser beam illuminates alternately both cantilever surfaces, yielding periodic bending/twisting. Further details on the optical configuration are given in Supplementary Methods. For contraction–expansion mode oscillation, we used a purely photothermally driven actuator, where light was absorbed at a tip of an LCN fibre and the energy was transferred into heat and conducted along the cantilever to the rest of the actuator body. The temperature difference between illuminated and non-illuminated surfaces is negligible due to fast thermal transport occurring at microscopic scale. Bending and twisting actuators are film like, with length-to-thickness ratio about 100. For contraction–expansion mode, a microscopic fibre with 100 µm diameter was used, with length-to-thickness ratio around 1. Because of the low length-to-thickness ratio, the bending of the fibre was suppressed. The LCN used for freestyle oscillation was also a photothermal actuator. Because of its large deformability, multiple degrees of freedom of deformation can be activated by impinging the light beam at different sample positions and/or by changing the excitation power. In all cases, the excitation beam position needs to be finely tuned in order to reach the stable self-oscillating movement. Once it self-oscillates, the duration can last for typically >30 min for bending and twisting modes, >50 min for contacting–expanding mode, and >30 s for the freestyle mode. More details about photomechanics and optical configurations are given in Supplementary Methods."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"Leaves oscillate in response to wind flow. Intriguingly, the frequency of such oscillation can be almost constant, even if driven by a gust of wind that lacks the corresponding periodicity. Leaf oscillation, together with a broad range of phenomena including heartbeat, bridge swaying, sea waves, etc., exemplify naturally occurring self-oscillatory processes [1] . The cause of such self-oscillation is the interplay between the oscillator motion and the force that triggers the oscillation. In the case of leaves, torsional galloping of the leaf shades the aerodynamic vortex generated behind the surface, thus providing positive feedback to the motion that sustains the vibration with a defined periodicity [2] . Generally, the onset of self-oscillation needs to meet critical conditions [3] . This is evident from daily observations: only one or few leaves with specific orientation with respect to the wind flow direction oscillate under a breeze, while the neighbouring leaves with different orientation remain stationary. Passive oscillating elements in nature rely on an external driving force, such as wind. Stimuli-responsive soft materials allow realizing externally fuelled, man-made self-oscillators, the motion of which is sustained by inner forces that arise due to stimuli-induced changes in material properties. Self-oscillators fuelled by, e.g. light [4] , [5] , heat [6] , and chemical reactions [7] have been generated, offering a possibility towards self-sustained motions without the need of human control. Towards this goal, several device functions are demonstrated, including electric power generation [6] , [8] , mass transport [9] , [10] , mill [11] , and self-propelling locomotion [5] , [7] , [12] . Among the different stimuli, light is particularly promising, due to its sustainability, precise controllability, and omnipresence, the Sun being the ultimate energy source. However, most of the light-driven self-oscillators reported to date are using bending deformation as the main degree of freedom of movement or, in few cases, the combination of bending and twisting [13] , [14] . The broader the range of available oscillation modes, the more sophisticated autonomous devices one can potentially construct. Therefore, there is a need for self-sustained actuators with versatile motion and complex oscillation modes. Liquid crystal elastomers and polymer networks (LCNs) are an important class of soft actuators that have been widely used in micro-robotics [15] , [16] , [17] , [18] , [19] . Their utility is based on coupling between elasticity due to the crosslinked network and anisotropic molecular orientation arising from the LC character. As a result of this combination, LCNs may exhibit pronounced macroscopic shape changes in response to external stimuli. By embedding light-sensitive dyes into the LCN, the material can reversibly and rapidly deform upon light excitation [20] . The photoactuation can be triggered either photothermally [21] or photochemically [22] (or by a combination of these), both leading to light-induced control over the degree of molecular alignment. Programming the molecular alignment in LCNs leads to diverse forms of shape changes [23] , enabling the desired deformation mode, and in some cases self-oscillation [4] , [5] , [13] , [14] , to be realized. Typically, light-fuelled self-oscillators rely on a bending LCN strip, in which the oscillation is triggered by alternate activation of the two surfaces [4] , or an optical configuration that provides self-shadowing within the motion cycles [24] . Being based on cantilever-type or strip-like geometries, the oscillation has been explained with a model connecting the self-oscillation frequency with their harmonic resonances [24] , [25] . However, there are some complications in this approach. First, the photomechanical oscillators experience significant damping during the fast motion because of the small mass/inertia compared to the friction/drag. Such damping causes large energy loss and requires compensation from the external stimulus, different to conventional harmonic oscillation. Second, the energy flow into the material by light absorption is oscillating during the cyclic movement, which in turn gives rise to oscillating temperature in the LCN. The change in intensity/temperature leads to change in modulus [26] and thus to a non-constant resonance frequency. Owing to the above considerations, several important issues need to be addressed, such as the possibility of obtaining versatile oscillating modes, understanding the feedback mechanism that actually fuels the motion and compensates for the damping loss, and the differences between LCN self-oscillation and conventional cantilever oscillation. Here we try to address the above questions by exploring the mechanism of self-oscillation in LCN photoactuators. We first theoretically point out that the time delay in the material response provides a positive feedback to the motion, serving as the key to self-sustained oscillation. Then we devise three cantilever-type photoactuators capable of exhibiting the three basic oscillation modes: bending, twisting, and contraction–expansion. We show that the oscillation frequencies of the latter two diverge from the expected natural frequencies. Finally, we demonstrate a freestyle self-oscillator by hanging an LCN photoactuator on a thread, in which all the above degrees of freedom can be combined. The freestyle oscillator has numerous stabilized oscillation modes upon light excitation at different sample positions and is able to evolve spontaneously between different modes upon irradiation with a constant light beam. We believe that the generalized model provided, together with the freestyle self-oscillator demonstrated, can give new insights into soft matter mechanics.\",\n \"references\": {\n \"indices\": [\n 1,\n 2,\n 3,\n 4,\n 5,\n 6,\n 7,\n 8,\n 9,\n 10,\n 11,\n 12,\n 13,\n 14,\n 15,\n 16,\n 17,\n 18,\n 19,\n 20,\n 21,\n 22,\n 23,\n 24,\n 25,\n 26\n ],\n \"infos\": [\n {\n \"idx\": \"1\",\n \"title\": \"Jenkins, A. Self-oscillation.Phys. Rep.525, 167–222 (2013).\",\n \"link\": \"https://doi.org/10.1016%2Fj.physrep.2012.10.007\"\n },\n {\n \"idx\": \"2\",\n \"title\": \"Leclercq, T., Peake, N. & de Langre, E. Does flutter prevent drag reduction by reconfiguration?Proc. R. Soc. A474, 20170678 (2018).\",\n \"link\": \"https://doi.org/10.1098%2Frspa.2017.0678\"\n },\n {\n \"idx\": \"3\",\n \"title\": \"Tadrist, L., Julio, K., Saudreau, M. & de Langre, E. Leaf flutter by torsional galloping: experiments and model.J. Fluids Struct.56, 1–10 (2015).\",\n \"link\": \"https://doi.org/10.1016%2Fj.jfluidstructs.2015.04.001\"\n },\n {\n \"idx\": \"4\",\n \"title\": \"White, T. J. et al. A high frequency photodriven polymer oscillator.Soft Matter4, 1796–1798 (2008).\",\n \"link\": \"https://doi.org/10.1039%2Fb805434g\"\n },\n {\n \"idx\": \"5\",\n \"title\": \"Gelebart, A. H. et al. Making waves in a photoactive polymer film.Nature546, 632–636 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fnature22987\"\n },\n {\n \"idx\": \"6\",\n \"title\": \"Wang, X. Q. et al. In-built thermo-mechanical cooperative feedback mechanism for self-propelled multimodal locomotion and electricity generation.Nat. Commun.9, 3438 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-018-06011-9\"\n },\n {\n \"idx\": \"7\",\n \"title\": \"Maeda, S., Hara, Y., Sakai, T., Yoshida, R. & Hashimoto, S. Self-walking gel.Adv. Mater.19, 3480–3484 (2007).\",\n \"link\": \"https://doi.org/10.1002%2Fadma.200700625\"\n },\n {\n \"idx\": \"8\",\n \"title\": \"Tang, R. et al. Optical pendulum generator based on photomechanical liquid-crystalline actuators.ACS Appl. Mater. Interfaces7, 8393–8397 (2015).\",\n \"link\": \"https://doi.org/10.1021%2Facsami.5b01732\"\n },\n {\n \"idx\": \"9\",\n \"title\": \"Baumann, A. et al. Motorizing fibres with geometric zero-energy modes.Nat. Mater.17, 523–527 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41563-018-0062-0\"\n },\n {\n \"idx\": \"10\",\n \"title\": \"Ahn, C., Li, K. & Cai, S. Light or thermally-powered autonomous rolling of an elastomer rod.ACS Appl. Mater. Interfaces10, 25689–25696 (2018).\",\n \"link\": \"https://doi.org/10.1021%2Facsami.8b07563\"\n },\n {\n \"idx\": \"11\",\n \"title\": \"Vantomme, G., Gelebart, A. H., Broer, D. J. & Meijer, E. W. A four-blade light-driven plastic mill based on hydrazone liquid-crystal networks.Tetrahedron73, 4963–4967 (2017).\",\n \"link\": \"https://doi.org/10.1016%2Fj.tet.2017.06.041\"\n },\n {\n \"idx\": \"12\",\n \"title\": \"Wie, J. J., Shankar, M. R. & White, T. J. Photomotility of polymers.Nat. Commun.7, 13260 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms13260\"\n },\n {\n \"idx\": \"13\",\n \"title\": \"Lee, K. M. et al. Photodriven, flexural–torsional oscillation of glassy azobenzene liquid crystal polymer networks.Adv. Funct. Mater.21, 2913–2918 (2011).\",\n \"link\": \"https://doi.org/10.1002%2Fadfm.201100333\"\n },\n {\n \"idx\": \"14\",\n \"title\": \"Ge, F., Yang, R., Tong, X., Camerel, F. & Zhao, Y. A multifunctional dye-doped liquid crystal polymer actuator: light-guided transportation, turning in locomotion, and autonomous motion.Angew. Chem. Int. Ed.57, 11758–11763 (2018).\",\n \"link\": \"https://doi.org/10.1002%2Fanie.201807495\"\n },\n {\n \"idx\": \"15\",\n \"title\": \"Zeng, H., Wasylczyk, P., Wiersma, D. S. & Priimagi, A. Light robots: bridging the gap between microrobotics and photomechanics in soft materials.Adv. Mater.30, 1703554 (2018).\",\n \"link\": \"https://doi.org/10.1002%2Fadma.201703554\"\n },\n {\n \"idx\": \"16\",\n \"title\": \"Ohm, C., Brehmer, M. & Zentel, R. Liquid crystalline elastomers as actuators and sensors.Adv. Mater.22, 3366–3387 (2010).\",\n \"link\": \"https://doi.org/10.1002%2Fadma.200904059\"\n },\n {\n \"idx\": \"17\",\n \"title\": \"Nocentini, S., Parmeggiani, C., Martella, D. & Wiersma, D. S. Optically driven soft micro robotics.Adv. Opt. Mater.6, 1800207 (2018).\",\n \"link\": \"https://doi.org/10.1002%2Fadom.201800207\"\n },\n {\n \"idx\": \"18\",\n \"title\": \"Ambulo, C. P. et al. Four-dimensional printing of liquid crystal elastomers.ACS Appl. Mater. Interfaces9, 37332–37339 (2017).\",\n \"link\": \"https://doi.org/10.1021%2Facsami.7b11851\"\n },\n {\n \"idx\": \"19\",\n \"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).\",\n \"link\": \"https://doi.org/10.1002%2Fmarc.201700710\"\n },\n {\n \"idx\": \"20\",\n \"title\": \"Ikeda, T., Mamiya, J.-I. & Yu, Y. Photomechanics of liquid-crystalline elastomers and other polymers.Angew. Chem. Int. Ed.46, 506–528 (2007).\",\n \"link\": \"https://doi.org/10.1002%2Fanie.200602372\"\n },\n {\n \"idx\": \"21\",\n \"title\": \"Yang, Y., Pei, Z., Li, Z., Wei, Y. & Ji, Y. Making and remaking dynamic 3d structures by shining light on flat liquid crystalline vitrimer films without a mold.J. Am. Chem. Soc.138, 2118–2121 (2016).\",\n \"link\": \"https://doi.org/10.1021%2Fjacs.5b12531\"\n },\n {\n \"idx\": \"22\",\n \"title\": \"Li, M. H., Keller, P., Li, B., Wang, X. & Brunet, M. Light‐driven side-on nematic elastomer actuators.Adv. Mater.15, 569–572 (2003).\",\n \"link\": \"https://doi.org/10.1002%2Fadma.200304552\"\n },\n {\n \"idx\": \"23\",\n \"title\": \"White, T. J. & Broer, D. J. Programmable and adaptive mechanics with liquid crystal polymer networks and elastomers.Nat. Mater.14, 1087–1098 (2015).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat4433\"\n },\n {\n \"idx\": \"24\",\n \"title\": \"Gelebart, A. H., Vantomme, G., Meijer, B. E. W. & Broer, D. J. Mastering the photothermal effect in liquid crystal networks: a general approach for self-sustained mechanical oscillators.Adv. Mater.29, 1606712 (2017).\",\n \"link\": \"https://doi.org/10.1002%2Fadma.201606712\"\n },\n {\n \"idx\": \"25\",\n \"title\": \"Serak, S. et al. Liquid crystalline polymer cantilever oscillators fueled by light.Soft Matter6, 779–783 (2010).\",\n \"link\": \"https://doi.org/10.1039%2FB916831A\"\n },\n {\n \"idx\": \"26\",\n \"title\": \"Kumar, K., Schenning, A. P. H. J., Broer, D. J. & Liu, D. Regulating the modulus of a chiral liquid crystal polymer network by light.Soft Matter12, 3196–3201 (2016).\",\n \"link\": \"https://doi.org/10.1039%2FC6SM00114A\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Origin of light-fuelled self-oscillation While tree leaves oscillate in response to wind flow (Fig. 1a, b ), the photoresponse of a soft actuator results from its mechanical deformability upon light illumination. Taking a bending LCN actuator as an example (Fig. 1c ), the bending angle increases when increasing the light intensity I (Fig. 1d ), which can be qualitatively described as a linear dependence between the deformation ( D ) and the energy absorbed ( E ) by the actuator (Fig. 1f, g ). Note that the deformation does not have to be bending as exemplified here but any combination of twisting/bending/contraction, as will be illustrated later. Owing to the directionality of the incident light field, the effective light-absorbing area A depends on the deformed geometry. This can be noticed from Fig. 1e , illustrating that the higher the bending angle, the smaller the light-absorbing area. The relation between A and D can be qualitatively described as a linear curve (Fig. 1g ), in which the axis of A can be linked to E through a simple relationship: E = A · I . Imposing the two curves gives an intersection, determined by the light intensity, at which the equilibrium occurs (Fig. 1h ). Fig. 1 The origin of self-oscillation in natural and in artificial light-fuelled systems. a , b Photographs of a leaf self-oscillating under a breeze. Inset of b shows time-dependent displacement of the centre of mass of the leaf, indicating periodic movement with well-defined frequency. Scale bar is 5 cm. A free-standing LCN actuator ( c ) bends slightly upon low-intensity irradiation (50 mW cm −2 , d ) and more pronouncedly when the intensity is increased (200 mW cm −2 , e ). f – h Schematic drawings showing the qualitative dependence between absorbed energy E , deformation D , effective light-absorbing area A , and equilibrium position upon illumination. i Harmonic oscillator (solid line, numerical solution to equation, \\\\(\\\\ddot x + \\\\omega _{\\\\mathrm{o}}^2x = 0\\\\) ) and an oscillator experiencing damping (dashed line, numerical solution to equation, \\\\(\\\\ddot x + \\\\varsigma \\\\dot x + \\\\omega _{\\\\mathrm{o}}^2x = 0\\\\) ), where ω 0 = 2π, \\\\(\\\\dot x\\\\left( 0 \\\\right) = 0,x\\\\left( 0 \\\\right) = 1\\\\) , \\\\(\\\\varsigma = 0.1\\\\) . Self-oscillation induced by j a minor time delay σ and k a large σ . Numerical solutions to equation \\\\(\\\\ddot x - \\\\left( {\\\\sigma \\\\omega _{\\\\mathrm{o}}^2 - \\\\varsigma - \\\\eta x^2} \\\\right)\\\\dot x + \\\\omega _{\\\\mathrm{o}}^2x = 0\\\\) , where \\\\(\\\\varsigma = 0.1,\\\\omega _0 = 2{\\\\mathrm{\\\\pi }}\\\\) , \\\\(\\\\dot x\\\\left( 0 \\\\right) = 1,x\\\\left( 0 \\\\right) = 0\\\\) , and \\\\(\\\\sigma \\\\omega _{\\\\mathrm{o}}^2 = 1,\\\\eta = 3\\\\) in j and \\\\(\\\\sigma \\\\omega _{\\\\mathrm{o}}^2 = 20,\\\\eta = 60\\\\) in k Full size image Around this equilibrium, the most familiar case is the harmonic oscillator, where an object experiences a restoring force proportional to its displacement x . The equation of motion can be written as: $$\\\\ddot x + \\\\omega _{\\\\mathrm{o}}^2x = 0,$$ (1) where ω o is the angular frequency of the system. Solving Eq. ( 1 ) yields harmonic oscillation with a constant amplitude and periodicity 2π/ ω o , defined by its natural frequency, as shown by the solid line in Fig. 1i . However, in most cases, the oscillator experiences damping from its environment, a restoring force proportional to the velocity of its motion: $$\\\\ddot x + \\\\varsigma \\\\dot x + \\\\omega _{\\\\mathrm{o}}^2x = 0,$$ (2) where ϛ is the damping ratio. Accounting for the damping, the amplitude gradually decreases, due to energy dissipation from the system to the environment (the dashed line in Fig. 1i ). For a light-responsive mechanical cantilever, the restoring force arises from a sequence of processes: light absorption by the photoactive molecules, conversion of light energy into heat, heat transport across the sample area, and finally, build-up of the inner stress needed for deformation. All these contribute to a time delay σ in the actuation. Thus the equation of motion can be written as $$\\\\ddot x\\\\left( t \\\\right) + \\\\varsigma \\\\dot x(t) + \\\\omega _{\\\\mathrm{o}}^2x\\\\left( {t - \\\\sigma } \\\\right) = 0.$$ (3) The value of σ may vary significantly depending on the material system, and it may also depend on the exact position of the LCN within the oscillation cycle. In order to reveal the contribution of this delay into the motion kinetics, we have simplified the model and consider only an invariant σ in the motion equation. We use Taylor expansion of Eq. ( 3 ), yielding $$\\\\ddot x + \\\\varsigma \\\\dot x + \\\\omega _o^2\\\\left( {x - \\\\sigma \\\\frac{{\\\\dot x}}{{1!}} + \\\\sigma ^2\\\\frac{{\\\\ddot x}}{{2!}} - \\\\sigma ^2\\\\frac{{\\\\mathop {x}\\\\limits^{...} }}{{3!}} \\\\cdots } \\\\right) = 0$$ (4) In this equation, the damping term \\\\(- \\\\sigma \\\\omega _{\\\\mathrm{o}}^2\\\\dot x\\\\) has a negative value, indicating that the oscillation amplitude increases to a certain extent, until the response becomes nonlinear. For example, air drag would significantly increase at large oscillation amplitudes, leading to amplitude saturation. Herein we adopt a nonlinear positive damping term \\\\(\\\\eta x^2\\\\dot x\\\\) (where η is a positive constant) based on van der Pol model [27] and simplify Eq. ( 4 ) using first-order approximation. The equation of motion for a light-responsive actuator at the equilibrium then appears as $$\\\\ddot x - \\\\left( {\\\\sigma \\\\omega _{\\\\mathrm{o}}^2 - \\\\varsigma - \\\\eta x^2} \\\\right)\\\\dot x + \\\\omega _{\\\\mathrm{o}}^2x = 0.$$ (5) Numerical solution to Eq. ( 5 ) when σ is small is given in Fig. 1j , showing an exponential growth of the oscillation amplitude until reaching saturation. Comparing Fig. 1i, j one may notice that all these oscillations exhibit very similar periodicity, matching the conventional cantilever natural frequency, ω o /2π. However, when σ increases, the periodicity increases and the form of oscillation becomes non-sinusoidal, as shown in Fig. 1k . Numerical calculations on the oscillation frequency and the waveform upon increasing σ are given in Supplementary Fig. 1 . The key to self-oscillation is the time delay in the material response [1] : while an oscillator passes through the equilibrium ( x = 0), the delayed force remains in the system showing the same direction as the velocity, thus pushing the object out of the equilibrium and providing positive feedback to the motion. Such feedback extracts the energy from the external energy source (light), fuelling the oscillator to sustain the motion. In the following, we explore the relation between the deformation degrees of freedom and the equilibrium positions in different light-fuelled self-oscillators and connect the observed oscillation modes to the material delay. Basic oscillation modes The three basic deformation modes in a cantilever—bending, contraction–expansion, and twisting—are schematically shown in Fig. 2a . Their amplitudes can be quantified as out-of-plane bending angle α , displacement d , and twisting angle Ф around the cantilever axis, respectively. To realize the basic deformations, we used planar-aligned LCN cantilevers, with a director along the cantilever for bending and contraction and perpendicular to it for twisting (Fig. 2b ). The bending/twisting and contracting actuators are based on different LCN materials, as further elaborated in “Methods” and the Supplementary Methods. Fig. 2 Light-fuelled self-oscillators based on three basic deformation modes. a Schematic drawing of the three basic deformation modes in a cantilever: bending, contraction, and twisting. The deformation can be quantified through out-of-plane bending angle α , displacement d , and twisting angle Ф for each mode. b Molecular alignment and irradiation geometry to initiate self-oscillation in each deformation mode. c Photographs of bending actuator with size 5.5 × 1.5 × 0.05 mm 3 upon 180 mW laser beam (spot size 2.2 mm) and the corresponding oscillation dynamics. d Photographs of cylindrical actuator (100 μm diameter) upon 100 mW focused laser excitation (spot size 20 μm) and the contracting–expanding oscillation dynamics. e Twisting self-oscillator (size: 3.5 × 3.5 × 0.05 mm 3 , laser: 180 mW, spot 2.2 mm) and the corresponding oscillation dynamics. The scale bars are 2 mm in c , e , 100 μm in d Full size image The photoactuation is triggered with a 488 nm continuous-wave laser beam, using different irradiation geometries for the different modes as depicted in Fig. 2b . Bending is obtained by light propagation along the cantilever axis, in which case the laser alternately activates both surfaces during the cyclic motion. The photoactuation results from light absorption gradient across the thickness, which in turn gives rise to photoinduced stress on the irradiated surface, and bending towards the illuminating light source. Combination of photochemical and photothermal effects is likely responsible for this type of oscillation [13] . Contraction occurs when the beam is focused on the tip of the cantilever, heating up the material, followed by subsequent cooling and expansion once the cantilever has moved away from the laser spot. Thus photothermal effects are responsible for this actuation mode. Twisting/torsional deformation takes place when the strip is illuminated from the side, causing an angular displacement around the cantilever axis and alternate exposure of the two cantilever surfaces, driven by similar mechanism as in the case of bending. Further details on photoactuation and the experimental arrangements are given in “Methods” and Supplementary Methods, respectively. Equilibrium positions, where α = d = Ф = 0, can be found for each mode. When driven out of equilibrium (initiated by air fluctuation or a mechanical trigger), the cantilever returns to the equilibrium position due to elastic force, and the delay in the material response causes the self-oscillation as explained earlier. The frequency of the self-oscillation depends on the desired mode. The bending oscillation shown in Fig. 2c has the highest frequency of 80 Hz among the three basic modes (see the slow-motion video, Supplementary Movie 1 ), well matching with the calculated natural resonance frequency of the cantilever (about 90 Hz, see natural frequency calculations in Supplementary Methods for further details). This indicates a minimal time delay σ in such mode, yielding self-oscillation dominated by ω o (Eq. ( 5 ), Fig. 1j ). The natural resonance frequency is determined by the cantilever geometry and the material rigidity, as has been systematically studied in several LCN self-oscillators [24] , [25] . In contrast, the contraction–expansion oscillator as well as the twisting one exhibit much longer periodicities, and resonance frequencies in the range of 0.5–18 Hz for contraction and about 1 Hz for twisting can be observed (Fig. 2d, e ; Supplementary Movies 2 and 3 ). By comparing the observed frequencies to the natural resonances, estimated to be >20 kHz for the longitudinal vibration mode and ca. 170 Hz for the torsional one (see natural frequency calculations in Supplementary Methods), the frequencies of light-fuelled self-oscillations are 2–4 orders of magnitude lower. We ascribe this to significant increase in σ in those modes. During contraction–expansion, the light energy is absorbed within the focused spot area located at the tip of the cantilever. After the light absorption, heat has to be conducted along the cantilever axis to trigger deformation, thus posing a huge time delay. This delay strongly depends on the amount of energy being absorbed by the material. A set of stable oscillations with a broad range of frequencies, from 0.5 to 18 Hz, are observed by slightly tuning the laser spot position (same power) on the sample (Supplementary Fig. 2 ). We attribute this to differences in light scattering due to inhomogeneity and hence different amount of light being absorbed, which in turn yields different contraction length and time delay in the material response. In the twisting-mode oscillator, the time delay arises from the stress accumulation required for the material to switch its angular orientation. In this case, there seem to be two stable states, as indicated by the rectangular waveform shown in Fig. 2e . More detailed analysis on the contraction–expansion and twisting self-oscillation modes have been conducted, with results shown in Supplementary Figs. 3 and 4 . By maintaining the same excitation beam position but changing the input power, the frequency of the contraction–expansion mode varies between 16 and 20 Hz (see the Fourier transform in Supplementary Fig. 3b ). However, no direct relation between the oscillation frequency and the excitation power can be deduced. The twisting actuator, instead, maintains the same oscillation amplitude irrespective of the laser power, but its oscillation frequency increases from 0.7 to 2 Hz (Supplementary Fig. 4b, c ). Such findings seem to contradict conventional conceptions about cantilever mechanics, where the structural rigidity determines the natural resonance frequency, while the input power may affect only the oscillation amplitude. These observations, together with the non-sinusoidal waveforms and the considerably low oscillation frequencies compared to the natural resonance frequencies, indicate that the time delay becomes the dominating factor in the contracting and twisting self-oscillators. Note that, for large time delays, the simplified model given by Eq. ( 5 ) will no longer be accurate and more elaborate numerical calculation methods would be needed for solving the corresponding kinetic equations. A freestyle self-oscillator The actuators described above exhibit only one selected oscillation mode. In order to combine the different modes into a single actuator, we use a main-chain LCN with high degree of deformability [28] (>50% contraction upon order–disorder phase transition; thermal actuation and mechanical properties between different LCNs given in Supplementary Figs 5 and 6 ). Material details are given in “Methods”; for further information about sample preparation, see the Supplementary Methods. Upon excitation with a laser spot, the localized strain builds up stress inside the material, causing additional degrees of freedom for the entire deformed structure, as schematized in Fig. 3a and experimentally demonstrated in Fig. 3b–e . The strip (Fig. 3b ) bends when illuminated with a laser spot (100 mW) at the centre of the cantilever (Fig. 3c ). Moving the laser spot horizontally, close to the edge of the cantilever, triggers combined twisting and bending deformation (Fig. 3d ). By increasing the light intensity from 100 to 140 mW, a larger extent of deformation can be triggered, and bending and contraction occur simultaneously (Fig. 3e ). To better visualize the twisting, we attached a weight to the end of the actuator (1.4 g, 250 times the weight of the actuator) to suppress bending (Fig. 2f ). When the laser spot deviates from the centre of the actuator, contraction and twisting can be observed (Fig. 2g ). Fig. 3 A freestyle self-oscillator. a Schematic drawing of a soft actuator combining the three basic deformation modes (twisting, bending, and contraction). An LCN actuator ( b ) that bends ( c ), bends and twists ( d ), bends and contracts ( e ), upon changing the irradiation conditions. The circle marks the position of the laser spot. With an external load that prevents bending ( f ), the actuator contracts and twists simultaneously when excited from one side as indicated in g . h Schematics of a light-driven self-oscillator swinging on a thread. i – k Snapshots of the freestyle oscillator with different oscillation modes upon laser excitation (488 nm, 160 mW) at different positions. LCN strip size: 3 × 25 × 0.1 mm 3 . Inset of j shows the microscopic image of the hole punctured in the LCN actuator (scale bar 200 μm). Inset of k shows the displacement of the end of the strip during the oscillation process. All scale bars: 2 mm Full size image To trigger self-oscillation, the excitation beam orientation and spot location need to meet critical conditions. We realize this by puncturing a hole into the LCN (inset of Fig. 3j ) and hanging it vertically on a thread (human hair), thereby providing freedom to swing around the thread axis, as schematically illustrated in Fig. 3h . Under such configuration, any kind of light-induced deformation (bending and/or contraction and/or twisting) shifts the centre of mass of the actuator (Fig. 3h ; 1 → 2). In return, gravity imposes a torque to rotate the deformed structure around the thread axis (2 → 3), and the strip moves out of the illuminating area, cools down, and relaxes back to the original position (3 → 1′), and the cycle restarts. Triggered by subsequent sequences of the above-described cycles, the strip oscillates continuously upon a constant laser beam excitation. By slightly changing the position of the laser spot, different oscillation modes can be obtained (Fig. 3i–k and Supplementary Movie 4 ). Each mode has a well-defined periodicity, as exemplified in the inset of Fig. 3k . During the distinct stable oscillation modes, the laser beam is not spatially or temporally modified. However, by slightly adjusting the laser spot position along the LCN strip, the actuator may exhibit fast-varying geometry, oscillating like a dancer on a thread (Supplementary Movie 5 ), which we coin as light-fuelled freestyle self-oscillation. Stabilized oscillations and evolution between the modes For more detailed characterization of the oscillating modes, we labelled the free end of the LCN strip with fluorescent particles (Fig. 4a , and further details are given in data collection in Supplementary Methods). With the help of the fluorescent markers, we can track the trajectory of oscillation in the x – y plane, offering a distinct fingerprint for each oscillation mode. Each mode prevails for a relatively long time, typically at least 5 s, maintaining the same amplitude and frequency (Supplementary Fig. 7 ). However, the oscillation frequency may significantly vary for the different modes, e.g. from 0.4 to 6.7 Hz between examples shown in Fig. 4b–g , and no connection between the oscillation frequency and amplitude seems to exist. The trajectories with zero area correspond to reciprocal oscillation (Fig. 4e–g ) while those with non-zero area indicate non-reciprocal motion during the oscillation cycle (Fig. 4b–d ). Fig. 4 Stabilized modes and evolution between the modes. a Schematic drawing of the arrangement for tracking the oscillator position. b – g Examples of different self-oscillation modes upon excitation with a laser spot at different positions. Data shows the tracking trajectory on x – y plane for one oscillation cycle. h Evolution between modes upon a constant laser excitation. The x – y trajectory spontaneously changes upon prolonged excitation ( i , j ), while maintaining the similar oscillating periodicity ( k ) until 190 s when the oscillation vanishes Full size image Figure 4h presents an intriguing oscillation behaviour under constant irradiation conditions: spontaneous evolution between different oscillation modes. In short term, the periodicity of each mode is constant (Fig. 4k ) but varies somewhat over extended time periods, e.g. from 0.35 to 0.45 s within 3 min (Supplementary Fig. 8 ). The spontaneous evolution between the modes (Fig. 4i, j ) without changing the excitation beam position is ascribed to irreversibility of the actuating material, which is quantified by monitoring a freestanding LCN upon cyclic excitation (Supplementary Fig. 9 ). The cyclic actuation shows that the material maintains its deformability, i.e. the maximum strain obtained upon constant irradiation, while the relaxation process shows certain degree of reduction with a decay constant of about 80 cycles. The reason for such photomechanical irreversibility may be attributed to kinetic deformation driven by elastic entropy, where cyclic stretching–coiling gives rise to friction between the polymeric chains, and photobleaching of the dyes. During the oscillation, the actuator undergoes hundreds of deformation–relaxation cycles. Hence, even slight changes in material response may shift the deformation mode and lead to spontaneous evolution between the oscillation modes. At about 190 s, the oscillator stops at a stable position, where the negative damping condition is no longer met ( \\\\(\\\\sigma \\\\omega _{\\\\mathrm{o}}^2 \\\\, > \\\\, \\\\varsigma\\\\) , Eq. 5 ), possibility due to the irreversibility of the material response. The prediction and control of the evolution between different modes during the freestyle oscillation is difficult to attain, due to the multiple degrees of freedom in deformation and complexity of light–matter interactions in responsive soft materials, resulting in different time delay in material response at different stages of the oscillation cycle. Despite the lack of control, the intriguing freestyle self-oscillation phenomenon provides insight into the process that, we believe, can be generalized into different kinds of responsive materials with arbitrary geometry, deformability, and stimulus responsivity, as long as a feedback mechanism has been established for sustaining the periodic motion. From general material concept towards practical applications The above light-driven system combines the basic oscillation modes—bending, contraction, and twisting—into a single freestyle oscillator. Although the movements are complex, the freestyle oscillator is governed by the same physics as conventional oscillators with simpler movements. We believe that similar principles can be applied also to other types of stimuli-responsive systems. For example, thermoresponsive nylon fibres [9] , carbon-nanotube-based bi-layer actuators [29] , thermoactive elastomer rods [10] , hydrogels [30] , and azobenzene crystals [31] have all shown self-sustained cyclic movements. Different from approaches in systems chemistry [32] , [33] , which aim at programming positive/negative feedback between reversible chemical reactions to induce oscillating behaviours, the approach demonstrated here focuses on photomechanical response in macroscopic samples, driven by time delay in material response and not restricted in few specific materials. Also note that the self-oscillation discussed in this study is different from the phenomenon of fluctuation, which, however, in some cases has been coined as chaotic self-oscillation [34] . A well-defined frequency component in the Fourier transform can serve as a good indicator for the occurrence of self-oscillation. To combine self-oscillators into locomotive devices to achieve self-sustained micro-robots is challenging. First, most of the actuators used in self-oscillation studies are free-standing strips lacking external loading. Upon devising a linkage between the motor (the oscillator) and the robot body (the moving element), the mechanical conditions change. Upon locomotion, the oscillator’s position with respect to the illumination source, as well as its geometry and mechanical properties, change, affecting the equilibrium conditions crucial for sustaining the oscillation. Thus a conventional mechanical design, e.g. biking the wheel [35] , may lead to a failure in long-time conversion of oscillating motion into mechanical work. Second, efficient translocation often requires a nonreciprocal deformation, which in turn requires temporal coordination of a series of self-oscillation events or within different sequences of a self-oscillation cycle with certain phase delays [36] , [37] . However, coupling between different self-oscillators sets extra hurdles. Third, light-responsive materials may not provide enough force to overcome the friction/drag experienced during the robotic motion, hence other soft mechanic properties like snapping and instabilities [37] need to be further investigated in self-oscillating systems, in order to reach their full potential. Although challenges exist, pioneering examples have shed light on device integration, e.g. mass transportation [9] , [10] and self-propelled walking [5] . We hope more self-sustained devices will be reported in the future, and that self-sustained robots can be integrated with more sophisticated functions by the joint scientific development in this field. A useful application can be foreseen in realizing self-oscillator within fluidic environment, which may eventually give birth to self-sustained pumping and cilia-like collective movement for efficient microfluidic manipulation. However, the dragging force (large ς) may pose great hurdle to meet the required negative damping condition \\\\((\\\\sigma \\\\omega _{\\\\mathrm{o}}^2 \\\\, > \\\\, \\\\varsigma )\\\\) . A possible solution could be to investigate stimuli-responsive materials with a large delay σ in material response, such as light/heat-responsive hydrogels [38] . Beyond the above-mentioned robotic applications that aim to transfer input light energy into mechanical work output, there is a frontier of optical applications to be investigated. A self-oscillator is essentially an optical chopper. However, compared to the conventional electronic motorized chopping system, it has many advantages, such as compact size, light weight, and, being remotely powered, possibility towards integration into micro-devices. An oscillating strip can modulate a signal beam (635 nm) periodically in its transmission, as shown in Fig. 5a : during the down-bending stage, the beam is transmitted, while the up-bending stage blocks the beam. In other words, the signal beam transmission is controlled and modulated by the self-oscillator (Fig. 5b ). Furthermore, one single strip can even control two signal beams simultaneously. As shown in Fig. 5c , the two signal beams can be in phase, when propagating along the same side of the strip, or 180° out-of-phase, when propagating from the opposite sides of the strip (Fig. 5d ). Fig. 5 Self-oscillator-based beam modulator. a A self-oscillating strip (bending mode, driven by a continuous 488 nm, 190 mW) can modulate the transmission of a signal beam (635 nm). The signal beam is focused on the upper part of the oscillator as shown by the schematic drawing, while the transmitted beam profiles are shown in the photos. Scale bar: 1 cm. b Transmission measurement of the beam being modulated by the self-oscillator. c Two signal beams are propagating along the same side of the oscillator and the measured transmission intensities being modulated in phase. d Two signal beams are propagating along the opposite sides of the oscillator and the measured transmission intensities being modulated 180° out-of-phase. Blue curves in c , d show the scattered 488 nm light intensity for phase reference. Strip size: 1 × 10 × 0.05 mm 3 Full size image\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"We study the mechanism of light-driven self-oscillation and show that a time delay due to the material response is the key to sustain the periodicity of oscillation. By interplay between the equilibrium position upon irradiation and specific form of light-induced deformation pre-programmed into the LCN by molecular-alignment control, three basic oscillation modes—bending, contraction, and twisting—are demonstrated. A main-chain LCN actuator combines the basic oscillation modes, yielding freestyle oscillation when hung on a thread. Such freestyle oscillator exhibits different stabilized modes upon continuous laser excitation at different sample positions and evolution between the modes across a relatively long period under a constant light field. The results provide insight into light-driven self-oscillation in soft materials and may in longer term open up new approaches to realize light-driven motors for self-sustained soft micro-robotics.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Materials in brief The LCN photoactuators demonstrated here are photopolymerized from monomer mixtures containing liquid crystalline mesogens, crosslinkers, light-sensitive dyes, and photoinitiators, via one-step or two-step fabrication strategies. Different materials were used for different deformation/oscillation modes. For bending and twisting, a side-chain LCN with 28 mol% crosslinker (of which 6 mol% are azobenzene crosslinkers) is used [39] ; for contraction–expansion, we used a side-chain LCN polymerized from different LC mesogens and reduced crosslinker concentration (10 mol%) [40] to render the LCN softer; for freestyle actuator, we used a main-chain LCN synthesized through two-step fabrication process [28] . All details about material preparation procedures are given in Supplementary Methods. Photoactuation in brief An unfocused, continuous-wave 488 nm laser beam was propagated along the axis of the LCN cantilever in case of bending-mode oscillation and perpendicular to the axis for the twisting mode. Upon laser excitation, the azobenzene moieties in the LCN undergo repeated trans - cis - trans isomerization. Owing to limited light-penetration depth, isomerization gradient across the film thickness is formed, which, together with photothermal heating, gives rise to cantilever deformation towards the light source. As a result, the laser beam illuminates alternately both cantilever surfaces, yielding periodic bending/twisting. Further details on the optical configuration are given in Supplementary Methods. For contraction–expansion mode oscillation, we used a purely photothermally driven actuator, where light was absorbed at a tip of an LCN fibre and the energy was transferred into heat and conducted along the cantilever to the rest of the actuator body. The temperature difference between illuminated and non-illuminated surfaces is negligible due to fast thermal transport occurring at microscopic scale. Bending and twisting actuators are film like, with length-to-thickness ratio about 100. For contraction–expansion mode, a microscopic fibre with 100 µm diameter was used, with length-to-thickness ratio around 1. Because of the low length-to-thickness ratio, the bending of the fibre was suppressed. The LCN used for freestyle oscillation was also a photothermal actuator. Because of its large deformability, multiple degrees of freedom of deformation can be activated by impinging the light beam at different sample positions and/or by changing the excitation power. In all cases, the excitation beam position needs to be finely tuned in order to reach the stable self-oscillating movement. Once it self-oscillates, the duration can last for typically >30 min for bending and twisting modes, >50 min for contacting–expanding mode, and >30 s for the freestyle mode. More details about photomechanics and optical configurations are given in Supplementary Methods.\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"The core research idea is to explore the mechanism of self-oscillation in liquid crystal network (LCN) photoactuators, particularly focusing on how time-delayed material response provides positive feedback to sustain motion, and to demonstrate versatile self-oscillation modes—such as bending, twisting, and contraction–expansion—in both standard cantilever-type and novel freestyle LCN actuators that can combine multiple degrees of freedom and spontaneously transition between oscillation modes under constant light excitation."},"entities":{"kind":"list like","value":["self-oscillation","liquid crystal network","LCN photoactuators","time-delayed material response","positive feedback","bending","twisting","contraction–expansion","standard cantilever-type actuators","novel freestyle LCN actuators","multiple degrees of freedom","spontaneous transition","oscillation modes","constant light excitation"],"string":"[\n \"self-oscillation\",\n \"liquid crystal network\",\n \"LCN photoactuators\",\n \"time-delayed material response\",\n \"positive feedback\",\n \"bending\",\n \"twisting\",\n \"contraction–expansion\",\n \"standard cantilever-type actuators\",\n \"novel freestyle LCN actuators\",\n \"multiple degrees of freedom\",\n \"spontaneous transition\",\n \"oscillation modes\",\n \"constant light excitation\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1.","title":"Jenkins, A. Self-oscillation.Phys. Rep.525, 167–222 (2013).","link":"https://doi.org/10.1016%2Fj.physrep.2012.10.007"},{"idx":"2.","title":"Leclercq, T., Peake, N. & de Langre, E. Does flutter prevent drag reduction by reconfiguration?Proc. R. Soc. A474, 20170678 (2018).","link":"https://doi.org/10.1098%2Frspa.2017.0678"},{"idx":"3.","title":"Tadrist, L., Julio, K., Saudreau, M. & de Langre, E. Leaf flutter by torsional galloping: experiments and model.J. Fluids Struct.56, 1–10 (2015).","link":"https://doi.org/10.1016%2Fj.jfluidstructs.2015.04.001"},{"idx":"4.","title":"White, T. J. et al. A high frequency photodriven polymer oscillator.Soft Matter4, 1796–1798 (2008).","link":"https://doi.org/10.1039%2Fb805434g"},{"idx":"5.","title":"Gelebart, A. H. et al. Making waves in a photoactive polymer film.Nature546, 632–636 (2017).","link":"https://doi.org/10.1038%2Fnature22987"},{"idx":"6.","title":"Wang, X. Q. et al. In-built thermo-mechanical cooperative feedback mechanism for self-propelled multimodal locomotion and electricity generation.Nat. Commun.9, 3438 (2018).","link":"https://doi.org/10.1038%2Fs41467-018-06011-9"},{"idx":"7.","title":"Maeda, S., Hara, Y., Sakai, T., Yoshida, R. & Hashimoto, S. Self-walking gel.Adv. Mater.19, 3480–3484 (2007).","link":"https://doi.org/10.1002%2Fadma.200700625"},{"idx":"8.","title":"Tang, R. et al. Optical pendulum generator based on photomechanical liquid-crystalline actuators.ACS Appl. Mater. Interfaces7, 8393–8397 (2015).","link":"https://doi.org/10.1021%2Facsami.5b01732"},{"idx":"9.","title":"Baumann, A. et al. Motorizing fibres with geometric zero-energy modes.Nat. Mater.17, 523–527 (2018).","link":"https://doi.org/10.1038%2Fs41563-018-0062-0"},{"idx":"10.","title":"Ahn, C., Li, K. & Cai, S. Light or thermally-powered autonomous rolling of an elastomer rod.ACS Appl. Mater. Interfaces10, 25689–25696 (2018).","link":"https://doi.org/10.1021%2Facsami.8b07563"},{"idx":"11.","title":"Vantomme, G., Gelebart, A. H., Broer, D. J. & Meijer, E. W. A four-blade light-driven plastic mill based on hydrazone liquid-crystal networks.Tetrahedron73, 4963–4967 (2017).","link":"https://doi.org/10.1016%2Fj.tet.2017.06.041"},{"idx":"12.","title":"Wie, J. J., Shankar, M. R. & White, T. J. Photomotility of polymers.Nat. Commun.7, 13260 (2016).","link":"https://doi.org/10.1038%2Fncomms13260"},{"idx":"13.","title":"Lee, K. M. et al. Photodriven, flexural–torsional oscillation of glassy azobenzene liquid crystal polymer networks.Adv. Funct. Mater.21, 2913–2918 (2011).","link":"https://doi.org/10.1002%2Fadfm.201100333"},{"idx":"14.","title":"Ge, F., Yang, R., Tong, X., Camerel, F. & Zhao, Y. A multifunctional dye-doped liquid crystal polymer actuator: light-guided transportation, turning in locomotion, and autonomous motion.Angew. Chem. Int. Ed.57, 11758–11763 (2018).","link":"https://doi.org/10.1002%2Fanie.201807495"},{"idx":"15.","title":"Zeng, H., Wasylczyk, P., Wiersma, D. S. & Priimagi, A. Light robots: bridging the gap between microrobotics and photomechanics in soft materials.Adv. Mater.30, 1703554 (2018).","link":"https://doi.org/10.1002%2Fadma.201703554"},{"idx":"16.","title":"Ohm, C., Brehmer, M. & Zentel, R. Liquid crystalline elastomers as actuators and sensors.Adv. Mater.22, 3366–3387 (2010).","link":"https://doi.org/10.1002%2Fadma.200904059"},{"idx":"17.","title":"Nocentini, S., Parmeggiani, C., Martella, D. & Wiersma, D. S. Optically driven soft micro robotics.Adv. Opt. Mater.6, 1800207 (2018).","link":"https://doi.org/10.1002%2Fadom.201800207"},{"idx":"18.","title":"Ambulo, C. P. et al. Four-dimensional printing of liquid crystal elastomers.ACS Appl. Mater. Interfaces9, 37332–37339 (2017).","link":"https://doi.org/10.1021%2Facsami.7b11851"},{"idx":"19.","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":"20.","title":"Ikeda, T., Mamiya, J.-I. & Yu, Y. Photomechanics of liquid-crystalline elastomers and other polymers.Angew. Chem. Int. Ed.46, 506–528 (2007).","link":"https://doi.org/10.1002%2Fanie.200602372"},{"idx":"21.","title":"Yang, Y., Pei, Z., Li, Z., Wei, Y. & Ji, Y. Making and remaking dynamic 3d structures by shining light on flat liquid crystalline vitrimer films without a mold.J. Am. Chem. Soc.138, 2118–2121 (2016).","link":"https://doi.org/10.1021%2Fjacs.5b12531"},{"idx":"22.","title":"Li, M. H., Keller, P., Li, B., Wang, X. & Brunet, M. Light‐driven side-on nematic elastomer actuators.Adv. Mater.15, 569–572 (2003).","link":"https://doi.org/10.1002%2Fadma.200304552"},{"idx":"23.","title":"White, T. J. & Broer, D. J. Programmable and adaptive mechanics with liquid crystal polymer networks and elastomers.Nat. Mater.14, 1087–1098 (2015).","link":"https://doi.org/10.1038%2Fnmat4433"},{"idx":"24.","title":"Gelebart, A. H., Vantomme, G., Meijer, B. E. W. & Broer, D. J. Mastering the photothermal effect in liquid crystal networks: a general approach for self-sustained mechanical oscillators.Adv. Mater.29, 1606712 (2017).","link":"https://doi.org/10.1002%2Fadma.201606712"},{"idx":"25.","title":"Serak, S. et al. Liquid crystalline polymer cantilever oscillators fueled by light.Soft Matter6, 779–783 (2010).","link":"https://doi.org/10.1039%2FB916831A"},{"idx":"26.","title":"Kumar, K., Schenning, A. P. H. J., Broer, D. J. & Liu, D. Regulating the modulus of a chiral liquid crystal polymer network by light.Soft Matter12, 3196–3201 (2016).","link":"https://doi.org/10.1039%2FC6SM00114A"},{"idx":"27.","title":"van der Pol, B. On relaxation-oscillations.Philos. Mag.2, 978–992 (1926).","link":"https://doi.org/10.1080%2F14786442608564127"},{"idx":"28.","title":"Liu, L., Liu, M. H., Deng, L. L., Lin, B. P. & Yang, H. Near-infrared chromophore functionalized soft actuator with ultrafast photoresponsive speed and superior mechanical property.J. Am. Chem. Soc.139, 11333–11336 (2017).","link":"https://doi.org/10.1021%2Fjacs.7b06410"},{"idx":"29.","title":"Zhang, X. et al. Photoactuators and motors based on carbon nanotubes with selective chirality distributions.Nat. Commun.5, 2983 (2014).","link":"https://doi.org/10.1038%2Fncomms3983"},{"idx":"30.","title":"He, X. et al. Synthetic homeostatic materials with chemo-mechano-chemical self-regulation.Nature487, 214–218 (2012).","link":"https://doi.org/10.1038%2Fnature11223"},{"idx":"31.","title":"Uchida, E., Azumi, R. & Norikane, Y. Light-induced crawling of crystals on a glass surface.Nat. Commun.6, 7310 (2015).","link":"https://doi.org/10.1038%2Fncomms8310"},{"idx":"32.","title":"Leira-Iglesias, J., Tassoni, A., Adachi, T., Stich, M. & Hermans, T. M. Oscillations, travelling fronts and patterns in a supramolecular system.Nat. Nanotechnol.13, 1021–1027 (2018).","link":"https://doi.org/10.1038%2Fs41565-018-0270-4"},{"idx":"33.","title":"Novák, B. & Tyson, J. J. Design principles of biochemical oscillators.Nat. Rev. Mol. Cell Biol.9, 981–991 (2008).","link":"https://doi.org/10.1038%2Fnrm2530"},{"idx":"34.","title":"Kumar, K. et al. A chaotic self-oscillating sunlight-driven polymer actuator.Nat. Commun.7, 11975 (2016).","link":"https://doi.org/10.1038%2Fncomms11975"},{"idx":"35.","title":"Ge, F. & Zhao, Y. New function for thermal phase transition-based polymer actuators: autonomous motion on a surface of constant temperature.Chem. Sci.8, 6307–6312 (2017).","link":"https://doi.org/10.1039%2FC7SC01792H"},{"idx":"36.","title":"Duncan, P. N., Nguyen, T. V. & Hui, E. E. Pneumatic oscillator circuits for timing and control of integrated microfluidics.Proc. Natl Acad. Sci.110, 18104–18109 (2013).","link":"https://doi.org/10.1073%2Fpnas.1310254110"},{"idx":"37.","title":"Preston, D. J. et al. A soft ring oscillator.Sci. Robot.4, eaaw5496 (2019).","link":"https://doi.org/10.1126%2Fscirobotics.aaw5496"},{"idx":"38.","title":"Zhao, Y. et al. Soft phototactic swimmer based on self-sustained hydrogel oscillator.Sci. Robot.4, eaax7112 (2019).","link":"https://doi.org/10.1126%2Fscirobotics.aax7112"},{"idx":"39.","title":"Lahikainen, M., Zeng, H. & Priimagi, A. Reconfigurable photoactuator through synergistic use of photochemical and photothermal effects.Nat. Commun.9, 4148 (2018).","link":"https://doi.org/10.1038%2Fs41467-018-06647-7"},{"idx":"40.","title":"Nocentini, S., Martella, D., Wiersma, D. S. & Parmeggiani, C. Beam steering by liquid crystal elastomer fibres.Soft Matter13, 8590–8596 (2017).","link":"https://doi.org/10.1039%2FC7SM02063E"}],"string":"[\n {\n \"idx\": \"1.\",\n \"title\": \"Jenkins, A. Self-oscillation.Phys. Rep.525, 167–222 (2013).\",\n \"link\": \"https://doi.org/10.1016%2Fj.physrep.2012.10.007\"\n },\n {\n \"idx\": \"2.\",\n \"title\": \"Leclercq, T., Peake, N. & de Langre, E. Does flutter prevent drag reduction by reconfiguration?Proc. R. Soc. A474, 20170678 (2018).\",\n \"link\": \"https://doi.org/10.1098%2Frspa.2017.0678\"\n },\n {\n \"idx\": \"3.\",\n \"title\": \"Tadrist, L., Julio, K., Saudreau, M. & de Langre, E. Leaf flutter by torsional galloping: experiments and model.J. Fluids Struct.56, 1–10 (2015).\",\n \"link\": \"https://doi.org/10.1016%2Fj.jfluidstructs.2015.04.001\"\n },\n {\n \"idx\": \"4.\",\n \"title\": \"White, T. J. et al. A high frequency photodriven polymer oscillator.Soft Matter4, 1796–1798 (2008).\",\n \"link\": \"https://doi.org/10.1039%2Fb805434g\"\n },\n {\n \"idx\": \"5.\",\n \"title\": \"Gelebart, A. H. et al. Making waves in a photoactive polymer film.Nature546, 632–636 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fnature22987\"\n },\n {\n \"idx\": \"6.\",\n \"title\": \"Wang, X. Q. et al. In-built thermo-mechanical cooperative feedback mechanism for self-propelled multimodal locomotion and electricity generation.Nat. Commun.9, 3438 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-018-06011-9\"\n },\n {\n \"idx\": \"7.\",\n \"title\": \"Maeda, S., Hara, Y., Sakai, T., Yoshida, R. & Hashimoto, S. Self-walking gel.Adv. Mater.19, 3480–3484 (2007).\",\n \"link\": \"https://doi.org/10.1002%2Fadma.200700625\"\n },\n {\n \"idx\": \"8.\",\n \"title\": \"Tang, R. et al. Optical pendulum generator based on photomechanical liquid-crystalline actuators.ACS Appl. Mater. Interfaces7, 8393–8397 (2015).\",\n \"link\": \"https://doi.org/10.1021%2Facsami.5b01732\"\n },\n {\n \"idx\": \"9.\",\n \"title\": \"Baumann, A. et al. Motorizing fibres with geometric zero-energy modes.Nat. Mater.17, 523–527 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41563-018-0062-0\"\n },\n {\n \"idx\": \"10.\",\n \"title\": \"Ahn, C., Li, K. & Cai, S. Light or thermally-powered autonomous rolling of an elastomer rod.ACS Appl. Mater. Interfaces10, 25689–25696 (2018).\",\n \"link\": \"https://doi.org/10.1021%2Facsami.8b07563\"\n },\n {\n \"idx\": \"11.\",\n \"title\": \"Vantomme, G., Gelebart, A. H., Broer, D. J. & Meijer, E. W. A four-blade light-driven plastic mill based on hydrazone liquid-crystal networks.Tetrahedron73, 4963–4967 (2017).\",\n \"link\": \"https://doi.org/10.1016%2Fj.tet.2017.06.041\"\n },\n {\n \"idx\": \"12.\",\n \"title\": \"Wie, J. J., Shankar, M. R. & White, T. J. Photomotility of polymers.Nat. Commun.7, 13260 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms13260\"\n },\n {\n \"idx\": \"13.\",\n \"title\": \"Lee, K. M. et al. Photodriven, flexural–torsional oscillation of glassy azobenzene liquid crystal polymer networks.Adv. Funct. Mater.21, 2913–2918 (2011).\",\n \"link\": \"https://doi.org/10.1002%2Fadfm.201100333\"\n },\n {\n \"idx\": \"14.\",\n \"title\": \"Ge, F., Yang, R., Tong, X., Camerel, F. & Zhao, Y. A multifunctional dye-doped liquid crystal polymer actuator: light-guided transportation, turning in locomotion, and autonomous motion.Angew. Chem. Int. Ed.57, 11758–11763 (2018).\",\n \"link\": \"https://doi.org/10.1002%2Fanie.201807495\"\n },\n {\n \"idx\": \"15.\",\n \"title\": \"Zeng, H., Wasylczyk, P., Wiersma, D. S. & Priimagi, A. Light robots: bridging the gap between microrobotics and photomechanics in soft materials.Adv. Mater.30, 1703554 (2018).\",\n \"link\": \"https://doi.org/10.1002%2Fadma.201703554\"\n },\n {\n \"idx\": \"16.\",\n \"title\": \"Ohm, C., Brehmer, M. & Zentel, R. Liquid crystalline elastomers as actuators and sensors.Adv. Mater.22, 3366–3387 (2010).\",\n \"link\": \"https://doi.org/10.1002%2Fadma.200904059\"\n },\n {\n \"idx\": \"17.\",\n \"title\": \"Nocentini, S., Parmeggiani, C., Martella, D. & Wiersma, D. S. Optically driven soft micro robotics.Adv. Opt. Mater.6, 1800207 (2018).\",\n \"link\": \"https://doi.org/10.1002%2Fadom.201800207\"\n },\n {\n \"idx\": \"18.\",\n \"title\": \"Ambulo, C. P. et al. Four-dimensional printing of liquid crystal elastomers.ACS Appl. Mater. Interfaces9, 37332–37339 (2017).\",\n \"link\": \"https://doi.org/10.1021%2Facsami.7b11851\"\n },\n {\n \"idx\": \"19.\",\n \"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).\",\n \"link\": \"https://doi.org/10.1002%2Fmarc.201700710\"\n },\n {\n \"idx\": \"20.\",\n \"title\": \"Ikeda, T., Mamiya, J.-I. & Yu, Y. Photomechanics of liquid-crystalline elastomers and other polymers.Angew. Chem. Int. Ed.46, 506–528 (2007).\",\n \"link\": \"https://doi.org/10.1002%2Fanie.200602372\"\n },\n {\n \"idx\": \"21.\",\n \"title\": \"Yang, Y., Pei, Z., Li, Z., Wei, Y. & Ji, Y. Making and remaking dynamic 3d structures by shining light on flat liquid crystalline vitrimer films without a mold.J. Am. Chem. Soc.138, 2118–2121 (2016).\",\n \"link\": \"https://doi.org/10.1021%2Fjacs.5b12531\"\n },\n {\n \"idx\": \"22.\",\n \"title\": \"Li, M. H., Keller, P., Li, B., Wang, X. & Brunet, M. Light‐driven side-on nematic elastomer actuators.Adv. Mater.15, 569–572 (2003).\",\n \"link\": \"https://doi.org/10.1002%2Fadma.200304552\"\n },\n {\n \"idx\": \"23.\",\n \"title\": \"White, T. J. & Broer, D. J. Programmable and adaptive mechanics with liquid crystal polymer networks and elastomers.Nat. Mater.14, 1087–1098 (2015).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat4433\"\n },\n {\n \"idx\": \"24.\",\n \"title\": \"Gelebart, A. H., Vantomme, G., Meijer, B. E. W. & Broer, D. J. Mastering the photothermal effect in liquid crystal networks: a general approach for self-sustained mechanical oscillators.Adv. Mater.29, 1606712 (2017).\",\n \"link\": \"https://doi.org/10.1002%2Fadma.201606712\"\n },\n {\n \"idx\": \"25.\",\n \"title\": \"Serak, S. et al. Liquid crystalline polymer cantilever oscillators fueled by light.Soft Matter6, 779–783 (2010).\",\n \"link\": \"https://doi.org/10.1039%2FB916831A\"\n },\n {\n \"idx\": \"26.\",\n \"title\": \"Kumar, K., Schenning, A. P. H. J., Broer, D. J. & Liu, D. Regulating the modulus of a chiral liquid crystal polymer network by light.Soft Matter12, 3196–3201 (2016).\",\n \"link\": \"https://doi.org/10.1039%2FC6SM00114A\"\n },\n {\n \"idx\": \"27.\",\n \"title\": \"van der Pol, B. On relaxation-oscillations.Philos. Mag.2, 978–992 (1926).\",\n \"link\": \"https://doi.org/10.1080%2F14786442608564127\"\n },\n {\n \"idx\": \"28.\",\n \"title\": \"Liu, L., Liu, M. H., Deng, L. L., Lin, B. P. & Yang, H. Near-infrared chromophore functionalized soft actuator with ultrafast photoresponsive speed and superior mechanical property.J. Am. Chem. Soc.139, 11333–11336 (2017).\",\n \"link\": \"https://doi.org/10.1021%2Fjacs.7b06410\"\n },\n {\n \"idx\": \"29.\",\n \"title\": \"Zhang, X. et al. Photoactuators and motors based on carbon nanotubes with selective chirality distributions.Nat. Commun.5, 2983 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms3983\"\n },\n {\n \"idx\": \"30.\",\n \"title\": \"He, X. et al. Synthetic homeostatic materials with chemo-mechano-chemical self-regulation.Nature487, 214–218 (2012).\",\n \"link\": \"https://doi.org/10.1038%2Fnature11223\"\n },\n {\n \"idx\": \"31.\",\n \"title\": \"Uchida, E., Azumi, R. & Norikane, Y. Light-induced crawling of crystals on a glass surface.Nat. Commun.6, 7310 (2015).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms8310\"\n },\n {\n \"idx\": \"32.\",\n \"title\": \"Leira-Iglesias, J., Tassoni, A., Adachi, T., Stich, M. & Hermans, T. M. Oscillations, travelling fronts and patterns in a supramolecular system.Nat. Nanotechnol.13, 1021–1027 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41565-018-0270-4\"\n },\n {\n \"idx\": \"33.\",\n \"title\": \"Novák, B. & Tyson, J. J. Design principles of biochemical oscillators.Nat. Rev. Mol. Cell Biol.9, 981–991 (2008).\",\n \"link\": \"https://doi.org/10.1038%2Fnrm2530\"\n },\n {\n \"idx\": \"34.\",\n \"title\": \"Kumar, K. et al. A chaotic self-oscillating sunlight-driven polymer actuator.Nat. Commun.7, 11975 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms11975\"\n },\n {\n \"idx\": \"35.\",\n \"title\": \"Ge, F. & Zhao, Y. New function for thermal phase transition-based polymer actuators: autonomous motion on a surface of constant temperature.Chem. Sci.8, 6307–6312 (2017).\",\n \"link\": \"https://doi.org/10.1039%2FC7SC01792H\"\n },\n {\n \"idx\": \"36.\",\n \"title\": \"Duncan, P. N., Nguyen, T. V. & Hui, E. E. Pneumatic oscillator circuits for timing and control of integrated microfluidics.Proc. Natl Acad. Sci.110, 18104–18109 (2013).\",\n \"link\": \"https://doi.org/10.1073%2Fpnas.1310254110\"\n },\n {\n \"idx\": \"37.\",\n \"title\": \"Preston, D. J. et al. A soft ring oscillator.Sci. Robot.4, eaaw5496 (2019).\",\n \"link\": \"https://doi.org/10.1126%2Fscirobotics.aaw5496\"\n },\n {\n \"idx\": \"38.\",\n \"title\": \"Zhao, Y. et al. Soft phototactic swimmer based on self-sustained hydrogel oscillator.Sci. Robot.4, eaax7112 (2019).\",\n \"link\": \"https://doi.org/10.1126%2Fscirobotics.aax7112\"\n },\n {\n \"idx\": \"39.\",\n \"title\": \"Lahikainen, M., Zeng, H. & Priimagi, A. Reconfigurable photoactuator through synergistic use of photochemical and photothermal effects.Nat. Commun.9, 4148 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-018-06647-7\"\n },\n {\n \"idx\": \"40.\",\n \"title\": \"Nocentini, S., Martella, D., Wiersma, D. S. & Parmeggiani, C. Beam steering by liquid crystal elastomer fibres.Soft Matter13, 8590–8596 (2017).\",\n \"link\": \"https://doi.org/10.1039%2FC7SM02063E\"\n }\n]","isConversational":false}}},{"rowIdx":95,"cells":{"unique_id":{"kind":"string","value":"s41467-025-56960-1"},"subfield":{"kind":"string","value":"Engineering"},"title":{"kind":"string","value":"Adaptative two-phase thermal circulation system for complex-shaped electronic device cooling"},"abstract":{"kind":"string","value":"Thermal management using a vapor-liquid two-phase circulation system is challenging in compact and complex-shaped electronic devices. In this study, we design and fabricate a heat pipe that can adapt to various shapes, regardless of space constraints. The heat pipe is capable of bending or twisting in three dimensions, making it suitable for electronic devices of arbitrary shapes. It effectively transfers heat from in-plane chips to out-of-plane spaces through flexible circulation pathways. This two-phase heat cycle system achieves an ultra-high thermal conductivity of up to 11,363 W/m·K. The flexible and adaptive design strategy enables efficient heat transfer in complex and compact environments."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"The tech industry has consistently evolved towards miniaturization, with chip circuit fabrication becoming more refined and electronic device designs increasingly compact [1] , [2] , [3] . Conversely, the heat flux density of electronic components is rapidly increasing [4] , which significantly impacting device performance [5] , [6] . Research shows that the lifespan of an electronic device is directly linked to its operating temperature, with every 10 °C increase potentially doubling the failure rate [7] , [8] . Therefore, thermal management is a crucial aspect of electronic design [9] , [10] . To address this issue, various technologies, such as forced air cooling and microchannel cooling, have been developed as effective heat dissipation methods. Unfortunately, the compact size, high integration, and non-standard shapes of most emerging electronic devices [11] significantly limit direct application of many thermal management components due to space constraints. Passive thermal management technologies utilize the inherent high thermal conductivity of materials to spread heat concentrated at high heat flux density points over a larger surface area. With their small spatial requirements, high thermal conductivity, and low application costs, they have become a general solution for dissipating hot spots in compact electronic devices [12] , [13] . Flexible thermal film is a technological approach with high design freedom [14] , [15] . It can adapt flexibly to the complex surfaces of electronic devices [16] , [17] and offers a wide range of conformable curvatures and adjustable cooling areas [18] , [19] , demonstrating high level of design freedom. However, single-phase solid-state conductive materials, such as graphene films or graphene paper, exhibit significant attenuation over long-distance heat transfer, limiting their effective transmission range in high heat flux density applications. Phase-change heat transfer systems, such as micro heat pipes and vapor chambers, utilize the phase change of the working fluid to transfer heat, achieving extremely high thermal conductivity [20] , [21] . This is another major approach in passive thermal management. To ensure the stable operation of the phase change cycle within the heat pipe, it is necessary to maintain a high vacuum inside the system. This imposes stringent requirements on the sealing properties of the micro heat pipe. Traditional heat pipe fabrication processes typically employ welding or flattening techniques for shaping, but these methods make it difficult to construct complex three-dimensional phase-change pathways, thus limiting the design freedom of heat pipe configurations. This results in the majority of two-phase circulation systems maintaining simple 1D or 2D configuration [22] , [23] . Although these systems are widely applied in devices with standard shapes such as smartphones and laptops [24] , [25] , thermal management in electronics with more complex shapes, like Augmented Reality (AR) and Virtual Reality (VR) headsets or cameras, is still largely confined to the plane of the chips [26] (Fig. 1a ), leaving a vast untapped space available for heat distribution. Fig. 1: Conceptual illustration and customized design strategy for the adaptative heat pipe (AHP) in complex 3D electronic devices. a In-plane thermal management of traditional phase change system in electronic device. b Out-plane thermal management of 3D two-phase circulation by AHP. c Working mechanism and the structure of AHP, consisting of a self-supporting wick and aluminum-plastic film casing to form a vapor-liquid co-planar two-phase circulation structure. d Design freedom of two-phase heat cycle pathways inside the AHP. e Comparison of design freedom and thermal conductivity among AHP, single-phase heat conduction, and flexible two-phase heat cycle technologies. f Reverse design of AHP (Illustrated with the mixed reality headset example). g Custom conformal AHP for typical complex 3D electronic devices. Full size image Flexible heat pipes provide a solution for transferring heat out-of-plane. Some studies use corrugated tubes as the heat pipe casing, allowing for normal and radial bending, while demonstrating good thermal conductivity [27] . However, the tubular design limits integration into most electronic devices, imposing constraints on design freedom. Alternatively, flexible composite films can be used as casings to enhance the flexibility of flat heat pipes [28] , [29] , providing another approach for extending two-phase heat cycle pathways out-of-plane. However, this flexibility often comes at the expense of some heat transfer efficiency in structural design, and flexible deformation may cause distortion of the internal two-phase flow channels. As a result, most flexible flat heat pipes experience a reduction in heat transfer performance in three-dimensional configurations [30] , [31] . The deformation of most flexible flat heat pipes is typically limited to simple normal bending, significantly constraining design flexibility for more complex forms of electronic devices. Phase-change flexible filaments are another emerging technology [32] . For example, squid-inspired pulsating heat pipes [33] can achieve a thermal conductivity of up to 6750 W/m·K while offering high deformation flexibility, making them effective for spatial thermal management. However, to maximize the heat dissipation area within limited assembly spaces in electronic devices, developing phase-change cycle technologies with greater design freedom is necessary. This work integrates customized reverse engineering with data-driven three-dimensional thermal bonding technology to design and fabricate a three-dimensional two-phase thermal circulation system that can adopt arbitrary shapes, free from assembly space constraints. This thermal circulation system is termed the adaptative heat pipe (AHP). In the AHP system, the working fluid absorbs heat under vacuum conditions, evaporating into vapor at the evaporator, which then travels along a three-dimensional vapor path, conforming to the electronic device to the condenser. At the condenser, the vapor condenses which then returns to the evaporator through the three-dimensional capillary path of the wick, establishing a continuous, two-phase, three-dimensional thermal circulation. (Fig. 1c ). The AHP possesses four degrees of design freedom: normal bending, radial bending, twisting, and lateral adjustment (Fig. 1d ). It can establish stable three-dimensional phase-change pathways for electronic devices with any curved surfaces and complex configurations (as Supplementary Movie 1 ), achieving a maximum thermal conductivity of 11363 W/m·K. The AHP technology was compared with various passive thermal management technologies in terms of thermal conductivity and design freedom. The AHP maintains the ultra-high thermal conductivity advantage of the gas-liquid phase-change cycle while offering design freedom in any shape (Fig. 1e ). Therefore, in practical applications, the AHP thermal circulation system can be considered a material with ultra-high thermal conductivity, featuring integrated structure-function characteristics. In this work, the AHP serves not only as a thermal management component integrated into the electronic device but also as internal support, or even as part of the device’s shell or other structural components. The special three-dimensional shape alters the characteristics of the vapor-liquid two-phase cycle, endowing the phase change cycling system with multifunctional and intelligent features, providing application patterns for the next generation of phase change heat transfer systems.","references":{"indices":[1,2,3,4,5,6,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,27,28,29,30,31,32,33],"infos":[{"idx":"1","title":"Van Erp, R., Soleimanzadeh, R., Nela, L., Kampitsis, G. & Matioli, E. Co-designing electronics with microfluidics for more sustainable cooling.Nature585, 211–216 (2020).","link":"https://doi.org/10.1038%2Fs41586-020-2666-1"},{"idx":"2","title":"Lukatskaya, M. R., Dunn, B. & Gogotsi, Y. Multidimensional materials and device architectures for future hybrid energy storage.Nat. Commun.7, 12647 (2016).","link":"https://doi.org/10.1038%2Fncomms12647"},{"idx":"3","title":"Yang, Z., Yang, T., Cai, W. & Cai, T. Miniaturization of optical spectrometers.Science371, eabe0722 (2021).","link":"https://doi.org/10.1126%2Fscience.abe0722"},{"idx":"4","title":"Song, H. et al. Two-dimensional materials for thermal management applications.Joule2, 442–463 (2018).","link":"https://doi.org/10.1016%2Fj.joule.2018.01.006"},{"idx":"5","title":"Wang, C. et al. A thermal management strategy for electronic devices based on moisture sorption-desorption processes.Joule4, 435–447 (2020).","link":"https://doi.org/10.1016%2Fj.joule.2019.12.005"},{"idx":"6","title":"Li, M. D. et al. Thermal management of chips by a device prototype using synergistic effects of 3D heat-conductive network and electrocaloric refrigeration.Nat. Commun.13, 5849 (2022).","link":"https://doi.org/10.1038%2Fs41467-022-33596-z"},{"idx":"8","title":"Unsal, O. S. et al. Impact of parameter variations on circuits and microarchitecture.IEEE Micro26, 30–39 (2006).","link":"https://doi.org/10.1109%2FMM.2006.122"},{"idx":"9","title":"Aftab, W. et al. Phase change material-integrated latent heat storage systems for sustainable energy solutions.Energy Environ. Sci.14, 4268–4291 (2021).","link":"https://doi.org/10.1039%2FD1EE00527H"},{"idx":"10","title":"Li, W. M., Yang, S. Y., Chen, Y. P., Li, C. & Wang, Z. K. Tesla valves and capillary structures-activated thermal regulator.Nat. Commun.14, 3996 (2023).","link":"https://doi.org/10.1038%2Fs41467-023-39289-5"},{"idx":"11","title":"Yoo, S. W. et al. Responsive materials and mechanisms as thermal safety systems for skin-interfaced electronic devices.Nat. Commun.14, 1024 (2023).","link":"https://doi.org/10.1038%2Fs41467-023-36690-y"},{"idx":"12","title":"Wei, X. H., Liu, L., Zhang, J. X., Shi, J. L. & Guo, Q. G. Mechanical, electrical, thermal performances and structure characteristics of flexible graphite sheets.J. Mater. Sci.45, 2449–2455 (2010).","link":"https://link.springer.com/doi/10.1007/s10853-010-4216-y"},{"idx":"13","title":"Vu, M. C. et al. Scalable ultrarobust thermoconductive nonflammable bioinspired papers of graphene nanoplatelet crosslinked aramid nanofibers for thermal management and electromagnetic Shielding.J. Mater. Chem.9, 8527 (2021).","link":"https://doi.org/10.1039%2FD0TA12306D"},{"idx":"14","title":"Bartlett, M. D. et al. High thermal conductivity in soft elastomers with elongated liquid metal inclusions.Proc. Natl Acad. Sci. USA28, 2143–2148 (2017).","link":"https://doi.org/10.1073%2Fpnas.1616377114"},{"idx":"15","title":"Ren, N. et al. Shear-induced fabrication of cellulose nanofibril/liquid metal nanocomposite films for flexible electromagnetic interference shielding and thermal management.ACS Appl. Mater. Interfaces16, 17904–17917 (2024).","link":"https://doi.org/10.1021%2Facsami.4c01220"},{"idx":"16","title":"Shen, B., Zhai, W. & Zheng, W. Ultrathin flexible graphene film: an excellent thermal conducting material with efficient EMI shielding.Adv. Funct. Mater.24, 4542–4548 (2014).","link":"https://doi.org/10.1002%2Fadfm.201400079"},{"idx":"17","title":"Liang, G. et al. Phase change material filled hybrid 2D/3D graphene structure with ultra-high thermal effusivity for effective thermal managemen.Carbon176, 11–20 (2021).","link":"https://doi.org/10.1016%2Fj.carbon.2020.12.046"},{"idx":"18","title":"Kong, Q. Q. et al. Hierarchical graphene–carbon fiber composite paper as a flexible lateral heat spreader.Adv. Funct. Mater.24, 4222–4228 (2014).","link":"https://doi.org/10.1002%2Fadfm.201304144"},{"idx":"19","title":"Xin, G. et al. Highly thermally conductive and mechanically strong graphene fibers.Science349, 1083–1087 (2015).","link":"https://doi.org/10.1126%2Fscience.aaa6502"},{"idx":"20","title":"Jaipurkar, T., Kant, P., Khandekar, S., Bhattacharya, B. & Paralikar, S. Thermo-mechanical design and characterization of flexible heat pipes.Appl Therm. Eng.126, 1199–1208 (2017).","link":"https://doi.org/10.1016%2Fj.applthermaleng.2017.01.036"},{"idx":"21","title":"Yang, C. et al. Fabrication and performance evaluation of flexible heat pipes for potential thermal control of foldable electronics.Appl Therm. Eng.96, 445–453 (2016).","link":"https://doi.org/10.1016%2Fj.applthermaleng.2015.11.078"},{"idx":"22","title":"Gibbons, M. J., Marengo, M. & Marengo, T. A review of heat pipe technology for foldable electronic devices.Appl. Therm. Eng.194, 117087 (2021).","link":"https://doi.org/10.1016%2Fj.applthermaleng.2021.117087"},{"idx":"23","title":"Xin, F., Ma, T. & Wang, Q. Thermal performance analysis of flat heat pipe with graded mini-grooves wick.Appl. Energy228, 2129–2139 (2018).","link":"https://doi.org/10.1016%2Fj.apenergy.2018.07.053"},{"idx":"24","title":"Tang, H. et al. Review of applications and developments of ultra-thin micro heat pipes for electronic cooling.Appl. Energy223, 383–400 (2018).","link":"https://doi.org/10.1016%2Fj.apenergy.2018.04.072"},{"idx":"25","title":"Luo, J. L., Mo, D. C., Wang, Y. Q. & Lyu, S. S. Biomimetic copper forest wick enables high thermal conductivity ultrathin heat pipe.ACS Nano15, 6614–6621 (2021).","link":"https://doi.org/10.1021%2Facsnano.0c09961"},{"idx":"27","title":"Lee, D. & Byon, C. Fabrication and characterization of pure-metal-based submillimeter-thick flexible flat heat pipe with innovative wick structures.Int J. Heat. Mass Transf.122, 306–314 (2018).","link":"https://doi.org/10.1016%2Fj.ijheatmasstransfer.2018.01.135"},{"idx":"28","title":"Liu, C., Li, Q. & Fan, D. Fabrication and performance evaluation of flexible flat heat pipes for the thermal control of deployable structure.Int. J. Heat. Mass Transf.144, 118661 (2019).","link":"https://doi.org/10.1016%2Fj.ijheatmasstransfer.2019.118661"},{"idx":"29","title":"Lim, J. & Kim, J. Fabrication and experimental evaluation of a polymer-based flexible pulsating heat pipe.Energ. Convers. Manag.126, 358–364 (2018).","link":"https://doi.org/10.1016%2Fj.enconman.2017.11.022"},{"idx":"30","title":"Jung, C., Lim, J. & Kim, S. J. Fabrication and evaluation of a high-performance flexible pulsating heat pipe hermetically sealed with metal.Int J. Heat. Mass Transf.149, 119180 (2020).","link":"https://doi.org/10.1016%2Fj.ijheatmasstransfer.2019.119180"},{"idx":"31","title":"Huang, J. et al. Development of novel flexible heat pipe with multistage design inspired by structure of human spine.Appl. Therm. Eng.175, 115392 (2020).","link":"https://doi.org/10.1016%2Fj.applthermaleng.2020.115392"},{"idx":"32","title":"Kang, Z., Jiang, S., Hong, Y. & Fan, J. Squid‐like soft heat pipe for multiple heat transport.Droplet1, 182–191 (2022).","link":"https://doi.org/10.1002%2Fdro2.25"},{"idx":"33","title":"Kang, Z., Hong, Y., Jiang, S. & Fan, J. Composite filament with super high effective thermal conductivity.Mater. Today Phys.34, 101067 (2023).","link":"https://doi.org/10.1016%2Fj.mtphys.2023.101067"}]}},{"section":"Results","content":"The freeform construction method of AHP To address the challenges of producing complex-shaped adaptative heat pipes, we present a rapid and cost-effective solution for customized fabrication. Instead of directly printing heat pipes using selective laser melting (SLM) technology, this work utilizes digital light processing (DLP) 3D printing technology for the indirect fabrication of complex-shaped AHP. The designed AHP structure, shown in Fig. 1c consists of two components: the heat pipe casing and the self-supporting wick. The casing is composed of two layers of aluminum-plastic film, with the composition detailed in Supplementary Fig. 2 . Due to the thermoplastic properties of the inner polypropylene layer, applying pressure at a specific temperature allows for thermal bonding of the aluminum-plastic film casing. The internal structure includes a self-supporting wick, created from three-dimensionally structured multi-layer copper mesh. The capillary force of the wick is enhanced by chemically deposited hydrophilic nanosheets (as detailed in Supplementary Fig. 3 ). The self-supporting wick and the aluminum-plastic film casing collaborate to form a vapor-liquid co-planar two-phase circulation structure. The wick functions as the liquid pathway, while an independent vapor pathway is created between the wick and the casing. By using the cross-section as the design unit for AHP enables the topology of two-phase circulation along any path of the three-dimensional coordinate system, which forms the foundation for the freeform design of AHP. During the design process, a customized 3D configuration of the AHP must be determined based on the complex shape of the target electronic device. Inspired by reverse engineering techniques in architecture, we have developed a reverse design strategy for customized AHP, as shown in Fig. 1f . Taking Mixed Reality (MR) headset as an example, the design process starts by converting the target model into spatial point cloud data (Fig. 1fI ). Trajectories are planned along the edges and areas of abrupt curvature of the target, and critical point coordinates on these trajectories are extracted (Fig. 1fII–III ). As the aluminum-plastic film shell needs to be transformed from an initial 2D plane into a 3D curved surface, the 2D shape of the aluminum-plastic film must be defined according to the 3D surface to prevent sealing defects. Triangular meshes are generated between points on adjacent trajectories to form the 3D surface (Fig. 1fVI ). Then, using a topological equivalence mapping algorithm, the corresponding 2D plane shape of the 3D surface is generated (Fig. 1fV ). The detailed description of the algorithm can be found in the method section. In the fabrication process, a major challenge is the thermal bonding of the Al/plastic film casing with a 3D curved surface, which directly impacts the sealing capability of the two-phase circulation system. Two key factors are identified in the thermal bonding process: temperature and pressure. In this work, the fastening force of bolts provides three-dimensional normal pressure, while Joule heat generated by flexible carbon fabric as an internal heat source. Customized resin molds are created based on the extracted trajectory points by DLP photopolymerization 3D printing technology (Supplementary Fig. 4a ). Carbon fabric is used as a flexible heater and is assembled with the heat pipe components in a customized 3D-printed resin mold. The thermal edge sealing of the complex-shaped casing is achieved by the combined effects of three-dimensional pressure and internal heat (Supplementary Fig. 4c ). The detailed description of the fabrication method is provided in the method section and Supplementary Movie 2 . The AHP customized for MR headset efficiently performs two-phase thermal cycling under complex three-dimensional configurations (Fig. 1fVI ). Its spatial thermal management performance under the effect of a point heat source is illustrated by thermal imaging, as shown in Fig. 1fVII . Due to the characteristics of two-phase cycling heat transfer, a relatively uniform temperature distribution is maintained over the entire annular curved surface. We selected typical 3D complex electronic devices such as cameras, unmanned aerial vehicles, and lamps as design targets. The trajectory point clouds of their casings were extracted, and AHPs with customized shapes were fabricated. The customized AHP conforms to the freeform surface of target electronic devices, establishing a three-dimensional vapor-liquid two-phase circulation path and performing out-plane 3D thermal management in these devices, as shown in Fig. 1g . The influence of 3D shape on the entrainment effect in two-phase counterflow The operating limit of a heat pipe is the maximum heat capacity it can transfer. When the heat source exceeds this limit, it disrupts the two-phase cycle within the heat pipe, leading to operating failure. Based on heat pipe theory, there are five critical limits for heat pipes [34] . To investigate the impact of the 3D segment on heat pipe performance, we initially conducted theoretical calculations on the two-phase thermal cycle limit under the basic structural parameters of AHP, as shown in Fig. 2a (The theoretical derivation process can be found in Supplementary Note 1 ). Calculations show that within the typical operating temperature range of micro heat pipes using water as the working fluid, the AHP is primarily limited by the capillary limit and the entrainment limit. The capillary limit occurs when the pressure drop during the two-phase cycle exceeds the capillary force provided by the wick [35] , failing to satisfy Eq. ( 1 ). As a result, the working fluid cannot circulate back quickly enough, causing the evaporator to dry out. The entrainment limit occurs at the vapor-liquid interface, where excessively high relative speeds of vapor and liquid counterflow, lead to the vapor entraining the working liquid back towards the condenser [36] , [37] , as illustrated in Fig. 2b . $$\\varDelta {P}_{cp}\\ge \\varDelta {P}_{v}+\\varDelta {P}_{l}+\\varDelta {P}_{g}$$ (1) Fig. 2: Mitigation effects of 3D segment on two-phase counterflow entrainment. a Theoretical operating limits of adaptative heat pipe (AHP). b Entrainment effects in 2D two-phase counterflow. c Schematic of mitigated two-phase counterflow entrainment in 3D AHP. d Operating performance of AHP without 3D segment. e Operating performance of AHP with 90° local 3D segment. f Operating performance of AHP with 180° local 3D segment. Thermal response time and testing illustration for normal bending angles of ( g , h ) 0°, ( i , j ) 45°, ( k , l ) 90°, ( m , n ) 135°, and ( o , p ) 180° local 3D segments. q Theoretical prediction of entrainment limit for AHP without 3D sections. r Modified theoretical prediction of the entrainment limit for AHP with a 90° local 3D segment. s Modified theoretical prediction of the entrainment limit for AHP with a 180° local 3D segment. Full size image To characterize the 3D shape of AHP parametrically, the normal bending state of the heat pipe is used as the fundamental 3D configuration, as illustrated in Fig. 2c . When the AHP is straight, i.e., with a bending angle of 0°, its operating performance is as shown in Fig. 2d . Thermocouples are placed at both the hot and cold ends of the AHP, and two additional thermocouples are positioned equidistantly at the middle section. The straight AHP shows the smallest temperature difference between the cold and hot ends ( \\(\\varDelta T1-4=T1-T4\\) ) at 1 W. As the heating power increases, \\(\\varDelta T1-4\\) gradually increases, indicating that the straight AHP operates more efficient two-phase circulation at low power input. When the heating power reaches 9.3 W, the temperature at the hot end ( T 1 ) rises rapidly, whereas the other temperature points ( T 2 , T 3 , T 4 ) fail to follow. This indicates that the evaporator of the AHP cannot absorb the increased power, thus reaching its operating limit. By introducing a 3D segment with a 90° bending angle in the AHP (Fig. 2e ), the minimum \\(\\varDelta T1-4\\) is observed at 4 W, and the heat transfer limit extends to 11.5 W, an increase of 23.7%. When the normal bending angle is increased to 180° (Fig. 2f ), the minimum \\(\\varDelta T1-4\\) occurs at 10 W, and the operating limit is further increased to 13 W (an increase of 44.4%). Supplementary Fig. 6a–c further illustrated the heat transfer performance of AHP with normal bending angles of 45° and 135°. The results indicate that the operating limit of the AHP occurs within the power range of the entrainment limit. Increasing the normal bending angle improves the AHP’s operating limit. This suggests that the 3D configuration of the heat pipe potentially mitigates the occurrence of the entrainment limit. By combining theoretical formulas, we further investigated the mechanism of the mitigation effect of 3D configurations on the entrainment limitation. The entrainment limit for the vapor-liquid interface can be theoretically calculated using flooding correlations in a counterflow vapor-liquid system. The criteria for entrainment instabilities can be determined using Eq. ( 2 ) [37] . $${({k}_{v})}^{\\frac{1}{2}}+{({k}_{l})}^{\\frac{1}{2}}=\\sqrt{3.2}\\,\\tanh (0.5B{o}^{\\frac{1}{2}})\\equiv {C}_{k}$$ (2) The Bond number ( Bo ) is a dimensionless parameter characterizing the interface fluid properties (Eq. ( 3 )), primarily determined by the surface tension \\({\\sigma }\\) of working fluid and characteristics parameter D for the interface. k v and k l respectively represent the dimensionless momentum fluxes of the vapor and liquid phase (Detailed calculation formulas are provided in Supplementary Note 2 ). $$Bo=D{[g({{\\rho }}_{l}+{{\\rho }}_{v})/{\\sigma }]}^{\\frac{1}{2}}$$ (3) We theoretically calculated the entrainment limit at different input powers, as shown in Fig. 2q . The results reveal that as power increases, the most noticeably changing variable is the dimensionless momentum flux of the vapor k v , suggesting that the drastic variation in vapor velocity as power changes is crucial in influencing the entrainment effect. Due to the high sealing requirement, directly measuring vapor flow velocity inside the heat pipes at micro-scales is challenging. Therefore, we indirectly assessed vapor velocity across different 3D configurations by measuring the thermal response time and revising entrainment limit criteria for 3D configurations. The normal bending angles tested were 0°(Fig. 2g, h ), 45°(Fig. 2i, j ), 90°(Fig. 2k, l ), 135°(Fig. 2m, n ), and 180°(Fig. 2o, p ), respectively. Four thermocouples were placed at the same positions on the AHP for different 3D configurations, and the thermal response time intervals between \\(T1\\) and the other three points were measured upon contact of the heating plate at the evaporator, with input powers of 1 W and 9 W. \\(\\varDelta t1-2\\) , \\(\\varDelta t2-3\\) , and \\(\\varDelta t3-4\\) represents the thermal response time of the vapor flow before the 3D segment, through the 3D segment, and from the 3D segment to the cold end, respectively. The results show that as the bending angle increases, there is negligible change in \\(\\varDelta t1-2\\) , while \\(\\varDelta t2-3\\) significantly increases, and \\(\\varDelta t3-4\\) shows a modest increase. The extended thermal response time due to the 3D sections primarily results from changes in vapor flow, while liquid flow is more influenced by the wick’s capillary structure. As illustrated in Supplementary Figs. 7 – 12 , the introduction of curved sections primarily affects the vapor flow pressure drop, and changes in liquid pressure drop are negligible. In addition to momentum exchange, the vapor-liquid interface of the wick undergoes evaporation and condensation of the working fluid, resulting in complex flow conditions [38] , [39] . With an increasing bending angle, the flow behavior at the interface becomes more complex as high-speed vapor passes through the bending section. Consequently, the thermal response time for vapor to pass through the 3D segment is greatly extended, while the straight segment after the 3D segment shows a minor reduction in thermal flow velocity due to reduced momentum. By modifying the momentum flux based on the increments of \\(\\varDelta t3-4\\) at different bending angles, as illustrated in Fig. 2r, s , we found that the predictions for the entrainment limit closely align with the increases in the power limit. Since the condensation mainly occurs at the condenser of the heat pipe, leading to a higher concentration of working fluid at this segment, entrainment is more likely to occur. Therefore, the construction of the 3D segment able to mitigate entrainment occurrence in the condenser segment. Additionally, we constructed local 3D segments at different locations along the two-phase cycle path of the AHP and evaluated heat transfer performance, as shown in Supplementary Fig. 13a . Results showed that constructing a local 3D segment at the condenser reduces the maximum power of the AHP by 1 W compared to other locations. This reduction could be attributed to significant interactions at the vapor-liquid interface within the 3D pathway, which adversely affects the liquid return at the condenser, where the liquid concentration is higher. A detailed description can be seen in Supplementary Fig. 13 . The robustness of AHP to spatial state changes Wearable and mobile electronic devices frequently change spatial state based on user demands, which often prevents their cooling components from operating in the most advantageous state. In two-phase thermal circulation systems, spatial state changes affect the gravitational pressure drop, represented by \\(\\varDelta Pg\\) in Eq. ( 1 ). To ensure the stable operation of the AHP, it is essential to minimize gravitational pressure drop changes across various spatial states. We theoretically calculated the capillary driving force, vapor pressure drop, and liquid pressure drop of the AHP at different heating powers and identified the maximum allowable gravitational pressure drop, as shown in Supplementary Fig. 14 . The results show that as power increases, the theoretically permissible range of gravitational pressure drop gradually decreases. By discretizing the AHP and defining the spatial state change parameters α and β , and calculating the weighted average of the gravitational pressure drops across the discrete segments, we derived the formula for the 3D gravitational pressure drop of the AHP in various spatial states, as Eq. ( 4 ). The complete derivation is provided in Supplementary Note 3 . $${P}_{g}={\\sum }_{k=1}^{n}\\frac{{l}_{k}}{L}{\\rho }g\\,\\sin ({{\\theta }}_{k}({\\alpha },{\\beta }))$$ (4) We selected three different heat pipe configurations: a straight flat heat pipe (2D), a fully deformed 3D AHP (full 3D), and an AHP with 3D deformation in the middle section while the remaining section remains flat (partial 3D). All three heat pipes maintain a complete two-phase heat cycle pathway, effectively eliminating the impact caused by structural collapse in flexible heat pipes under deformation. The primary difference lies in the gravitational pressure drop of the liquid working fluid across different spatial orientations. Using the gravitational pressure drop formula for AHP, we evaluated the theoretical changes in gravitational pressure drops for 2D, partial 3D, and full 3D configurations of AHP across six spatial states, as shown in Fig. 3b . The results indicate that the maximum change in gravitational pressure drop for the 2D straight heat pipe across different spatial states is 2560 Pa. For AHP with partial 3D and full 3D configurations, the maximum changes are 870 Pa and 960 Pa, respectively, equating to 34% and 37.5% of that for the straight 2D heat pipe. A smaller change in gravitational pressure drop indicates that spatial state variations have less impact on the heat pipe’s operational performance, enhancing the robustness of electronic devices against spatial changes in practical applications. Fig. 3: Spatial robustness and mechanical strength testing of the adaptative heat pipe (AHP). a Discrete weighted calculation of gravitational pressure drop in AHP. b Theoretical values of gravitational pressure drop in different spatial states. c Heat transfer performance of the straight 2D heat pipe across different spatial states. d Heat transfer performance of the AHP with full 3D segment across different spatial states. e Schematic diagram of the AHP tensile test. f An AHP in the shape of tensile specimens. g A copper heat dissipation sheet in the shape of tensile specimens. h Heat transfer performance of the tensile-shaped AHP and copper sheet under 4 W heating power. i Tensile stress-strain curve of the AHP. j Tensile test of the AHP. k Schematic diagram of the AHP flexural test. l An AHP in the shape of flexural specimens. m A copper heat dissipation sheet in the shape of flexural specimens. n Heat transfer performance of the flexural-shaped AHP and copper sheet under 4 W heating power. o Flexural stress-strain curve of the AHP. n Flexural test of the AHP. Full size image The operating performance of AHP was evaluated across different configurations. After stabilizing at a heating power of 8 W, the 2D heat pipe and the 3D AHP underwent sequential spatial state changes, with temperatures at the hot and cold ends recorded. The results indicate that with changes in spatial state, the straight 2D heat pipe exhibits significant temperature fluctuations between the hot and cold ends, which increase the temperature gradient, severely disrupting the vapor-liquid two-phase cycle (Fig. 3c ). In contrast, both the partial 3D (Supplementary Fig. 15 ) and full 3D (Fig. 3d ) configurations of AHP maintain relatively stable temperatures at the hot and cold ends, resulting in minimal changes in the temperature difference. Theoretical calculations indicate that the permissible gravitational pressure drop for the straight 2D heat pipe across six spatial states is 1281 Pa, with a maximum resistance to gravitational pressure drop of 1203.2 Pa, which is close to the permissible limit. This results in considerable variations in capillary backflow resistance for the straight 2D heat pipe across different spatial states, indicating weaker robustness to spatial changes. The 3D configured AHP, with smaller variations in gravitational pressure drop, has maximum gravitational pressure drops of 435.7 Pa and 609.2 Pa across different spatial states, significantly below the allowable limits. This indicates good spatial robustness, making it suitable for the spatial state changes required by compact electronic devices. Mechanical strength of AHP With its freeform design characteristics, the AHP can integrate into the mechanical assembly of electronic devices in various forms, similar to two-dimensional thermal materials, while maintaining efficient phase-change heat transfer performance. It exhibits integrated structural and functional application characteristics. To demonstrate the structural support properties of the AHP as a casing or support component for electronic devices, we conducted tensile and flexural tests and compared the results with those of commercial copper heat dissipation plates. Given the thin-plate structure of the AHP, we designed a dumbbell-shaped tensile specimen (tensile section width of 10 mm) and conducted tensile tests on both the AHP and copper plates (Fig. 3e–g ). The tensile strength of the AHP ranges between 30 and 40 MPa (Fig. 3h ), similar to that of rigid plastics like epoxy resin (20.67–40 MPa) [40] , [41] , [42] . Although the tensile strength of AHP shows a significant difference compared to copper (>200 MPa), it is sufficient to meet the mechanical requirements of most mobile and wearable device enclosures. Moreover, the internal two-phase cycle enables the AHP to exhibit extremely high thermal conductivity as a device-level application material, as shown in Fig. 3i . The AHP, composed of a composite structure with an aluminum-plastic film casing and a copper wire mesh wick, undergoes multiple mechanical stages during the tensile test. In the initial tension stage, the aluminum-plastic film casing and copper wire mesh wick transition from elastic deformation to plastic deformation (Supplementary Fig. 16aI–II ). When the tensile stress reaches 36.18 MPa, a slight decrease in tensile stress is observed. The images indicate that at this point, while the heat pipe casing remains intact, the AHP elongates slightly, primarily due to the fracture of the internal wick. Subsequently, the aluminum-plastic film continues to stretch (Supplementary Fig. 16aIII ), and the tensile stress-displacement curve shows characteristics of plastic deformation, resulting in complete fracture when the tensile stress reaches 40.01 MPa (Supplementary Fig. 16aIV ). Additionally, due to the thin-plate structure of the AHP, a flexural performance test was conducted (Fig. 3k ). The AHP and copper plate specimens were prepared as rectangles with a length of 150 mm and a width of 30 mm (Fig. 3l–m ). Under a heating power of 4 W, the AHP displayed a more uniform temperature gradient, while the copper plate exhibited uneven temperature distribution (Fig. 3n ), with significant heat accumulation at the hot end. The resulting pressure-displacement curve is shown in Fig. 3o . Compared to the tensile performance, the difference in flexural performance between the two thin-plate materials is smaller: the maximum pressure for the copper plate is 54.7 N, while for the AHP it is 37.68 N, approximately 68.9% of the copper plate’s value. Out-plane 3D thermal management of customized AHP in a VR headset The assembly of thermal management components is often a key point of contention between thermal engineers and structural designers in electronic device design. The AHP design utilizes the available space after other components are installed, free from the spatial constraints, to achieve out-plane 3D thermal management. Moreover, AHP possesses integrated structural-functional characteristics. Beyond its two-phase thermal circulation function, it can also serve as a casing or support component, further enhancing the compactness of electronic devices. As representative products of compact and complex-shaped devices, virtual reality (VR) headsets are widely adopted in both industrial and home settings [43] , [44] . Research indicates that the face is the one of the most heat-sensitive parts of human body [45] , [46] . VR headset panels typically feature an enclosed design, leading to significant heat accumulation during operation [47] . However, compact and complex device configurations hinder the effective integration of thermal management components [48] . Here, we designed a customized AHP specifically tailored to the special construction of VR headsets, deviating from the conventional design of regular-shaped cooling components. The customized AHP functions not only as a thermal management element but also as part of the head-mounted component, allowing the heat from the simulated chip to be directly transferred through the AHP for natural convection with the external air and maximizing the cooling area within limited assembly space. A schematic of the VR device is shown in Fig. 4a . To ensure alignment with the VR device’s structural characteristics, the AHP contacts the simulated chip at the center of the panel, distributing the heat along three separate paths. This heat is then transferred outside through gaps between the face pad and the control panel, extending along three head straps to form a closed-loop structure. The installation of the customized AHP in the VR headset, as illustrated in Fig. 4b , demonstrates that the AHP configuration conforms to the structural characteristics of the VR headset. Fig. 4: Customized adaptative heat pipe (AHP) for VR headset. a Reverse design of the AHP for the VR headset, extracting the head-mount component to create a customized AHP that provides both cooling capability and mounting functionality. b VR headset after co-integration of the AHP. c Infrared thermal imaging results of the commercial flat heat pipe under a heating power of 4 W. d Infrared thermal imaging of the commercial flexible graphene film under simulated chip heating power of 4 W. e Infrared thermal imaging of the customized AHP under simulated chip heating power of 4 W. f Comparison of the temperatures at the hot end for the customized AHP, conventional flat micro heat pipe, and commercial flexible graphene film across different power settings. g User testing of the structural-functional integrated AHP. Full size image In commercial applications, flat microheat pipes are used as cooling components in VR devices due to their two-phase thermal cycling characteristics. For instance, the Quest Pro VR headset by Meta Inc. integrates flat heat pipes within the panel for chip cooling [26] . Additionally, graphene thermal films, an emerging electronic cooling strategy, are used in some electronic products due to their high flexibility and thermal conductivity [49] , [50] , [51] . To assess the 3D two-phase thermal management efficiency of AHP, it was compared with commercial flat heat pipes and graphene films in a VR headset (as Supplementary Movie 3 ). Detailed structural and assembly information is provided in Supplementary Figs. 17 – 20 . With a 4 W heating power applied to the simulated chip, thermal management performance is shown in Fig. 4c–e . Due to the heat transfer characteristics of two-phase cycling, 2D flat heat pipes exhibit uniform temperature distribution. However, constrained by the 2D form, they can only manage heat within the plane of the panel, limiting the cooling area and resulting in the hot end temperature exceeding 100 °C (Fig. 4c ). As a flexible cooling material, graphene film can manage heat in three dimensions according to device shape, but it is limited by the properties of single-phase thermal conduction. The graphene film exhibits a significant temperature gradient, with hot end reaching 84.7 °C (Fig. 4d ). A customized AHP combines the phase-change cooling capabilities of 2D flat heat pipes with the design freedom of graphene film, achieving comprehensive 3D two-phase circulation cooling for VR devices. As shown in Fig. 4e , AHP maintains excellent temperature uniformity, reducing the temperature to 47.8 °C under the same power input. We compared the chip temperature under different cooling conditions and input powers, as shown in Fig. 4f . Without cooling measures, the simulated chip heat diffuses throughout the panel via direct contact, reaching 100 °C at 2 W and exceeding 120 °C at 3 W. When a 2D micro heat pipe is installed on the panel, the heating power can be stabilized at 3 W, with the hot end temperature reaching 83 °C. At 4 W, the temperature at the hot end exceeds 100 °C. Installing graphene film along the head strap significantly increases the heat dissipation area and enhances cooling performance compared to the 2D heat pipe, allowing the heating power to increase further to 5 W. With the integration of AHP, a significant reduction in temperature is observed. At the input power of 3 W, the average temperature of the simulated chip decreased by 48.59% (from 85°C to 43.7 °C) and 38.62% (from 71.2 °C to 43.7 °C) compared to the 2D heat pipe and graphene film, respectively. Additionally, combined with the 3D configuration’s mitigation effects of two-phase counterflow entrainment, the customized AHP shows a pronounced advantage in the improved heat transfer limit. Taking 100 °C as the limit, the simulated chip temperature exceeds 100 °C at 4 W with the 2D heat pipe and at 6 W with graphene film. With AHP cooling, however, the simulated chip can be increased to 19 W, an increment of 4.75 and 3.17 times compared to the 2D heat pipe and graphene film, respectively. The AHP demonstrates structural-functional integration. Specifically designed for head-mounting components, the customized AHP not only performs two-phase thermal circulation but also provides structural support for the user’s head. As shown in Fig. 4g , user testing was conducted with a simulated chip heating power of 6 W, during which the highest temperature maintained by the customized AHP was 43 °C. Under the same power settings, temperatures for commercial cooling strategies exceed 100 °C, posing burn risks to users. Detailed information on the user testing is provided in Supplementary Movie 4 . Stretchable two-phase circulation system In electronic devices, thermal management components like heat pipes will operate two-phase circulation under extreme environments [52] . Intense vibrations experienced during spacecraft launches or while driving automobiles can significantly impact electronic devices [53] , [54] . In addition, potential heat transfer requirements between cross-axis planes of aerospace robotic arms (Supplementary Fig. 21 ) pose a severe challenge to traditional heat pipe technology. Special 3D structural design can impart exceptional mechanical properties and advanced functions to the device [55] . By utilizing the freeform design characteristics of AHP, we have developed a stretchable AHP with a spiral three-dimensional shape, targeting the spring components in shock absorbers (Fig. 5a ). This design allows the heat pipe to extend over a long-range, functioning similarly to a spring. It adapts to dynamic spatial relationships between heat transfer planes, ensuring stable heat exchange. Fig. 5: Development of stretchable vapor-liquid phase-change circulation system. a Extraction of spring element curves from the shock absorber enables the designed adaptative heat pipe (AHP) to offer not only heat exchange capabilities but also damping and stretchability. b A stretchable AHP. c Characteristics of the stretchable AHP (L represents the minimum pitch of the stretchable heat pipe). d Dynamic temperature distribution curves of the stretchable heat pipe under different heating powers. e Heat transfer performance of the stretchable heat pipe under different stretching conditions. f Operation of the stretchable AHP under simulated extreme conditions, including compression, stretching, mild vibration, and intense vibration. g Cycle test of compression and stretching for the stretchable AHP. Full size image The developed stretchable AHP and its dynamic operating state are shown in Fig. 5b , with its two-phase circulation during dynamic stretching illustrated in Fig. 5c . The axial length can be compressed from 2L to L and extended to 3 L, achieving a threefold extension range. Heat transfer performance tests were conducted on the stretchable heat pipe with a pitch of 2 L under different heating powers. As shown in Fig. 5d , temperatures were measured at three locations: the hot end (15 mm from the edge), the cold end (20 mm from the edge), and the middle section, with temperature changes recorded as heating power increased from 1 W to 4 W. Results indicate that as heating power increased from 1 W to 4 W, the temperature difference between the hot and cold ends decreased from 11 °C to 9 °C, with a thermal resistance of 2.34 K/W and a thermal conductivity of 5335 W/m·K. In a helical configuration, the AHP has minimal curvature changes without smooth transition sections. This results in less effective curvature coordination between the membrane layers on both sides of the heat pipe during fabrication, causing some compression of the internal wick and affecting the stretchable AHP’s heat transfer efficiency. However, in the constructed 3D helical two-phase heat cycle pathway, the macroscopic deformation from mechanical stretching has minimal impact on the stretchable heat pipe. As shown in Fig. 5e , thermal conductivity of the stretchable heat pipe was measured under various stretching states. When the pitch of the stretchable heat pipe is L, 2 L, or 3 L, its heat transfer performance shows minimal variation, with thermal resistance fluctuating within ±0.15 K/W. Heat transfer operation between different planes of the AHP was simulated, as shown in Fig. 5f . After thermal stabilization, the AHP underwent compression, stretching, mild vibration, and intense vibration (complete process available in Supplementary Movie 5 ). Thermal imaging revealed no severe performance loss in the AHP. After thermal stabilization, a test of 200 compression and extension cycles was conducted (Fig. 5g ). Fluctuations were observed in the temperature at the heat pipe’s hot end. These fluctuations may be attributed to changes in vapor pathways during compression and extension and to slight compression of the two-phase structure during stretching. Additionally, the hot end temperature remained relatively stable within 65.5 °C to 68.5 °C, while a gradual decrease in cold end temperature was observed. This suggests that while the stretching cycles affected the heat pipe performance, they did not cause edge leakage or significant blockage in the two-phase cycle. Instead, the fluctuations likely result from deformation of the two-phase structure due to repeated stretching and compression. Magnetically-controlled two-phase circulation heat exchange robot The fourth industrial revolution, marked by advancements in information technology and artificial intelligence, has ushered in the era of intelligence, making device smartification an inevitable path for future development [56] . In thermal management, a key challenge is the cooling random hot spots, specifically dynamic heat dissipation when point heat source coordinates change over time. Leveraging the freeform design property of the adaptative heat pipe, we introduce the concept of vapor-liquid two-phase circulation heat exchange robots, developing intelligent application modes for conventional phase change systems. The AHP was designed as an enclosed cube (Fig. 6a ), with the entire cube serving as a unified vapor-liquid two-phase circulation path (Fig. 6b ). Its two-dimensional mapping form is shown in Fig. 6c , with the two-phase cycling path marked in the diagram. As shown in Fig. 6d–e , the heat transfer performance of the heat exchange robot achieves a maximum thermal conductivity of 4699 W/m·K with the heat source on surface 3. The thermal conductivity between surface 3 and surface 1 is 3720 W/m·K, and between surface 3 and surface 6 is 3524 W/m·K. After fabricating the cubic AHP, magnets with different polarizations were embedded inside each of the six faces, forming the AHP heat exchange robot. Leveraging its cubic 3D form, the AHP heat exchange robot can be maneuvered by flipping through the magnetic field at the base, allowing it to approach coordinates of randomly generated hot spots (Fig. 6f ). When the AHP heat exchange robot covers a hot spot, heat is quickly conducted away through the robot’s two-phase circulation walls, preventing further heat accumulation (Fig. 6g ). To illustrate the design concept of the AHP heat exchange robot, we simulated its operation under random heat sources, as shown in Fig. 6h (for a detailed view, see Supplementary Movie 6 ). When a random hot spot emerges within the plane (Fig. 6hI ), the AHP robot quickly flips and moves closer to the hot spot (Fig. 6hII–III ). By contacting the hot spot with its bottom wall, the robot uses the fast startup and efficient heat transfer of its two-phase circulation walls to rapidly disperse heat throughout the entire shell of the robot, expanding the originally localized air convection area to all six faces of the cube (Fig. 6hIV ). After the first hot spot dissipates, a second one forms (Fig. 6hV ). The AHP heat exchange robot continues to flip, contact, and dissipate the hot spot (Fig. 6hVI–VII ) and then moves to the next hot spot (Fig. 6hVIII ). Fig. 6: Dynamic cooling of two-phase circulation heat exchange robot system. a Extraction of external curves of the cubic model to plan the two-phase circulation path of the adaptative heat pipe (AHP) and construct a fully enclosed cubic AHP. b Structural schematic of a cubic AHP. c Schematic of magnetic pole arrangement for heat exchange robot. d Temperature distribution at the center points of the six faces under a bottom point heat source. e Thermal resistance and thermal conductivity at a heating power of 5 W. f Magnetic actuation flipping method for the AHP heat exchange robot. g Heat transfer process demonstrated by the cubic AHP when positioned above a hot spot. h The AHP heat exchange robot performing dynamic heat dissipation in response to random hot spots (The upper and lower sections of the diagram respectively represent the workflow under an optical camera and an infrared thermal imaging camera). Full size image"},{"section":"Discussion","content":"In summary, the AHP developed in this work leverages reverse engineering and data-driven three-dimensional thermal bonding technology, functioning as a phase-change circulation system with unprecedented shape design freedom. The three-dimensional two-phase thermal cycling path is customized to the configuration and thermal requirements of target electronic devices, facilitating direct heat exchange between the chip and the external environment, offering a solution to thermal management challenges in small, compact, and complex-shaped electronic devices. The three-dimensional pathway improves the heat transfer limit of the vapor-liquid phase-change system and potentially mitigates entrainment effects at the vapor-liquid interface in counterflow. Theoretical calculations based on flooding correlations reveal vapor flow rate as a key factor under varying input powers. Adjusting the correlation equations with velocity attenuation aligns calculated power increases closely with experimental measurements. Moreover, compared to 2D phase-change systems, 3D two-phase systems exhibit enhanced robustness to spatial state changes under high power conditions. Under spatial state changes, the maximum theoretical gravitational pressure drop for the 3D-configured AHP reduced to 34% of that in the 2D configuration. This diminished influence of gravity on two-phase circulation meets the spatial state change requirements of wearable and mobile electronic devices. This study developed a customized AHP tailored for VR headset structures. Compared to commercial flat 2D heat pipes and graphene films, the customized AHP reduced the simulated chip temperature by 48.59% (41.3 °C) and 38.62% (27.5 °C), respectively, at the same input power of 3 W. It also increased the power limit by 4.75 times (from 4 W to 19 W) and 3.17 times (from 6 W to 19 W), respectively. The three-dimensional configuration endows the AHP with integrated structural-functional characteristics, enabling it to serve as both a thermal management component and a head-mounted structure, providing structural support. The AHP’s spiral and cubic designs impart features such as stretchability and dynamic heat dissipation, paving the way for multifunctional and intelligent development of two-phase thermal circulation systems."},{"section":"Methods","content":"Fabrication of freeform AHP In the fabrication process, DLP 3D printing technology is first used to create molds for shaping the copper wire mesh and pre-encapsulating the AHP (Supplementary Fig. 4a ). The mold for the copper mesh is designed to serve as the self-supporting structure for the wick, while the heat pipe sealing mold is designed with space reserved exclusively for the edges of the AHP, ensuring contact only at the these points. As shown in Supplementary Fig. 4b , the mold shapes the copper mesh into a three-dimensional form by assembling multiple layers of flat copper mesh within it. Thanks to the high flexibility and ductility of the copper mesh, upon releasing the bolts, the multi-layer copper mesh retains the customized shape of the AHP at the macro level and forms a self-supporting structure of the wick at the micro level. The formed multi-layer copper mesh then undergoes sintering and chemical deposition processes to form stable capillary pores and enhance hydrophilicity. The aluminum-plastic film, wick, flexible carbon fabric, and mold are assembled together and secured with bolts to apply pressure. To generate heat for the thermal bonding process, apply voltage to both sides of the carbon fabric for Joule heating. This step completes the thermal bonding of the aluminum-plastic film casing, forming a complex three-dimensional shape (Supplementary Fig. 4c ). After releasing the bolts, a pre-encapsulated AHP (three sides sealed, one side open) is obtained. Next, venting treatment is performed through a charging port on the pre-encapsulated AHP, followed by vacuuming and filling with working fluid (deionized water). The final side is then sealed to complete the fabrication of a customized AHP with two-phase thermal cycling capabilities. Detailed information about the fabrication process is provided in Supplementary Fig. 5 and Supplementary Movie 2 , offering a more intuitive understanding of the fabrication method. Materials Aluminum-plastic film (D-EL40H(3)) as heat pipe casing was purchased from Dai Nippon Printing Co., Japan. Copper wire mesh (200 mesh copper) used as wick was purchased from Anping County Bolin Metal Wire Mesh Co., China. The carbon fabric (WOS1009, CeTech, USA) after cutting, was utilized as a flexible heater. Chemicals including sodium persulfate (Na 2 S 2 O 8 ), sodium hydroxide (NaOH), deionized water, and ethanol were purchased from Shanghai Aladdin Biochemical Technology Co. 3D printing of shaping and pre-encapsulating molds A customized configuration of the three-dimensional resin mold is created using Computer-Aided Design (CAD) software and exported in STL format. This STL file is then imported into Voxeldance Tango slicing software to add support structures and slice the model. The sliced file is loaded into a Digital Light Processing (DLP) 3D printing machine (Reflect2, DONZY, China). The printing parameters are set as follows: light source intensity at 5 mW/cm², exposure time at 2.5 s, and layer thickness at 50μm. Begin the printing process. After printing, the mold is removed and immersed in an ultrasonic cleaner (SK5210LHC, KUDOS, China) filled with 95% alcohol, with the cleaning parameters set to 53 kHz and 150 W for 60 s. After ultrasonic cleaning, the mold is dried in a 60 °C oven (DZF−6020, Shanghai Jinghong, China). Topological equivalence mapping algorithm The implementation of the 2D topological mapping algorithm for the AHP 3D triangular sheet is conducted using commercial software MATLAB. Trajectory planning of the three-dimensional point cloud is performed in CloudCompare software to obtain point coordinates along the trajectory, forming 3D triangular meshes between points on two closely adjacent trajectories. This forms the basis for mapping the surface from 3D triangular meshes to a corresponding 2D plane, guiding the planar shape design of the aluminum-plastic film casing. The specific process for implementing the topological mapping algorithm is described as follows: Step 1: Identified the central triangle mesh within the triangulated surface model to serve as the base triangle for mapping. Step 2: Map the base triangle onto the plane while preserving its spatial characteristics unchanged. Mark the corresponding vertices, edges, and triangle as unmapped. Step 3: Identified the adjacent 3D triangular mesh, \\(\\varDelta {1}_{i}\\) , that currently processing. Among the triangular meshes that are not yet marked as unmapped, select one that shares an edge with a triangle already marked as unmapped to be the current triangle under processing. Step 4: Confirm the position of the undetermined vertex of the currently processed triangle in the 2D plane. Calculate the normal vector passing through this point in the 3D surface, and determine the normal plane \\(\\overrightarrow{n}\\) corresponding to this normal vector \\(\\overrightarrow{{P}_{n}}\\) . Select a vertex adjacent to the vertex currently being processed, and determine the projection of this adjacent vertex on the normal plane \\(\\overrightarrow{{P}_{n}}\\) of the 3D surface. Then, calculate the curvature \\(K{1}_{i}\\) at the processed vertex. In the 2D plane, ensure that the curvature \\(K{2}_{i}\\) of the undetermined point is the same as \\(K{1}_{i}\\) . This approach can allow determine the direction between the undetermined point and its adjacent point. To determine the 2D position of the currently processed point, ensure that the area \\(S{2}_{i}\\) of the triangle in the 2D plane matches the area \\(S{1}_{i}\\) of its counterpart in 3D space. Step 5: Mark the currently processed point and the triangular surface as unmapped, record the number of unmapped triangles, and repeat steps 3 and 4. Hydrophilic treatment of wick The capillary force of the wick significantly influences the performance of heat pipes [57] . To efficiently fabricate wicks with intricate three-dimensional shapes, flexible copper wire mesh is used. The capillary pores are created by sintering multi-layered copper wire mesh, and hydrophilic capillary structures on the copper mesh surface are formed via chemical deposition [31] , [58] , with detailed procedures depicted in Supplementary Fig. 3 . The process begins with precise cutting of copper mesh using a pulsed laser. The layered copper mesh is then assembled within a mold to achieve complex three-dimensional forms. To enhance the bonding strength between layers and stabilize the capillary pores, the shaped multi-layer copper wire mesh undergoes a sintering process. Subsequently, the samples are cleaned with ultrasonic cleaning and treated chemically, with the reaction formula presented in Eqs. ( 5 )–( 7 ) [59] , [60] . After hydrophilic treatment, the surface of wick features CuO micro/nano hydrophilic structures (Supplementary Figs. 23 – 24 ), forming the liquid pathway for two-phase circulation of AHP. The contrast in hydrophilicity of the wick before and after the hydrophilic treatment is demonstrated in Supplementary Fig. 25 . $$Cu+2NaOH+N{a}_{2}{S}_{2}{O}_{8}\\to Cu{(OH)}_{2}+N{a}_{2}S{O}_{4}$$ (5) $$Cu{(OH)}_{2}+2O{H}^{-}\\to {[Cu{(OH)}_{4}]}^{2-}$$ (6) $${[Cu{(OH)}_{4}]}^{2-}\\to CuO\\downarrow+{H}_{2}O+2O{H}^{-}$$ (7) Characterization The morphology of hydrophilic wick in Supplementary Fig. 23 was captured by Scanning Electron Microscopy (SUPRA55 SAPPHIRE, ZEISS, German), as well as the energy dispersive X-ray spectrometer (EDS) in Supplementary Fig. 24 . The structure of wick was observed by optical microscope (AO-HK830, Aosvi Co. Ltd, China). The thermal imaging of the AHP during startup and stable operation were recorded by an infrared thermal imaging camera (FLIR T440, FLIR Systems Company, USA). Experimental setup The testing setup for evaluating the performance of the AHP includes a DC power supply (WPS3010B, Shenzhen WANPTEK Electronic Technology Co., China), a data acquisition system (34901 A, KEYSIGHT, USA), Type K thermocouples (Omega, USA), and a heating plate (10 cm × 10 cm). The heating plate is mounted 10 mm from the edge on the hot end of the AHP. Thermal conductive silicone grease (6.5 W/m·K) is applied between the heating plate and AHP to minimize contact thermal resistance. Two thermocouples are positioned at each end of the AHP: one at the hot end, 10 mm from the heat pipe’s edge, and the other at the cold end, 15 mm from the edge. Additional thermocouples are evenly placed in the middle, conforming to the customized shape of the AHP, to assess thermal distribution throughout the device. The temperatures recorded by the thermocouples are captured by the data acquisition system and transmitted to a computer. The temperatures measured at the hot and cold ends are used to compute the heat pipe’s thermal resistance [31] , [61] and thermal conductivity [21] , [62] , with the calculation formulas detailed in Supplementary Note 4 . A parametric study on the heat transfer performance of the AHP structure was conducted, with variations in parameters such as thickness, length, and width. Test results and analysis are provided in Supplementary Figs. 26 – 27 . Numerical methodology To streamline the computational model and focus on the impact of the heat pipe’s geometry on the two-phase cycle, the simulation employs a design featuring distinct vapor and liquid fields, encapsulated by a metal film layer that serves as the heat pipe’s shell, as depicted in Supplementary Fig. 28 . Cuboids with an area of 10 × 10 mm² and a thickness of 0.3 mm are used as both the heating plate and condenser. Their centers are located 15 mm from the heat pipe’s ends, positioned near the wall on the wick side (Supplementary Fig. 28b, c ). To maintain a uniform total length for the two-phase circulation across all models, the central line length of the various 3D heat pipes is kept consistent with that of a straight heat pipe. The operation of the two-phase cycle in the heat pipe is simulated using COMSOL Multiphysics 6.2. This simulation employs the Laminar Flow module to analyze vapor flow within the vapor chamber, the Brinkman Equations module for describing liquid flow within the wick, and the Porous Media Heat Transfer module for addressing heat transfer across different sections of the heat pipe. Non-isothermal flow multiphysics coupling integrates the Laminar Flow and Brinkman Equations modules. The effect of gravity may vary and become complex under different deformation conditions. To eliminate the influence of gravity, the gravity option is disabled in both the Laminar Flow and Brinkman Equations modules. The vapor chamber boundary is set to the vapor pressure of water. To maintain equal water mass flux between the vapor chamber and wick field, velocity boundary conditions are set at their interface. $${{{\\bf{u}}}}_{2}=\\frac{{{{\\bf{u}}}}_{1}\\cdot {{\\rho }}_{2}}{{{\\rho }}_{1}}$$ (8) Where u represents the fluid velocity vector, and ρ denotes density. The subscript 1 refers to the vapor chamber domain, and subscript 2 refers to the wick domain. Heat exchange during the evaporation-condensation process is simulated by adding a boundary heat source at the interface between the vapor chamber and the wick. $${Q}_{b}={{{\\bf{u}}}}_{n}\\cdot {h}_{fg}\\cdot {\\rho }$$ (9) Where u n represents the normal vector of velocity u , and h fg is the latent heat of vaporization of water. Simulations are conducted on heat pipes of four different shapes: straight, normal bending, radial bending, and twisting, as shown in Supplementary Fig. 28d–g . In the results, probes were placed 10 mm from the cold and hot end boundaries in both the vapor chamber and the wick to calculate the pressure drop for both the gas and liquid phases. Simulations are conducted at different angles (Supplementary Figs. 7 – 9 ) and different radius (Supplementary Figs. 10 – 12 ). A validity test for the numerical simulation of the heat pipe is conducted following methods from reported studies [63] , [64] . Using the modeling setting and simulation condition provided in the referenced report [65] , a numerical simulation is performed on a straight heat pipe with multiple heat sources at low-temperature operation, as depicted in Supplementary Fig. 29 . Four evaporators are positioned at specific locations on one end of the straight heat pipe, with a condenser is placed at a defined coordinate on the other end (Supplementary Fig. 29a ). Different power inputs are applied to each of the four evaporators, and the temperatures along the heat pipe wall are measured and compared. Since explicit cooling conditions for the condenser are not specified in the literature, various parameters are adjusted for comparison, as shown in Supplementary Fig. 29b–e ."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"The tech industry has consistently evolved towards miniaturization, with chip circuit fabrication becoming more refined and electronic device designs increasingly compact [1] , [2] , [3] . Conversely, the heat flux density of electronic components is rapidly increasing [4] , which significantly impacting device performance [5] , [6] . Research shows that the lifespan of an electronic device is directly linked to its operating temperature, with every 10 °C increase potentially doubling the failure rate [7] , [8] . Therefore, thermal management is a crucial aspect of electronic design [9] , [10] . To address this issue, various technologies, such as forced air cooling and microchannel cooling, have been developed as effective heat dissipation methods. Unfortunately, the compact size, high integration, and non-standard shapes of most emerging electronic devices [11] significantly limit direct application of many thermal management components due to space constraints. Passive thermal management technologies utilize the inherent high thermal conductivity of materials to spread heat concentrated at high heat flux density points over a larger surface area. With their small spatial requirements, high thermal conductivity, and low application costs, they have become a general solution for dissipating hot spots in compact electronic devices [12] , [13] . Flexible thermal film is a technological approach with high design freedom [14] , [15] . It can adapt flexibly to the complex surfaces of electronic devices [16] , [17] and offers a wide range of conformable curvatures and adjustable cooling areas [18] , [19] , demonstrating high level of design freedom. However, single-phase solid-state conductive materials, such as graphene films or graphene paper, exhibit significant attenuation over long-distance heat transfer, limiting their effective transmission range in high heat flux density applications. Phase-change heat transfer systems, such as micro heat pipes and vapor chambers, utilize the phase change of the working fluid to transfer heat, achieving extremely high thermal conductivity [20] , [21] . This is another major approach in passive thermal management. To ensure the stable operation of the phase change cycle within the heat pipe, it is necessary to maintain a high vacuum inside the system. This imposes stringent requirements on the sealing properties of the micro heat pipe. Traditional heat pipe fabrication processes typically employ welding or flattening techniques for shaping, but these methods make it difficult to construct complex three-dimensional phase-change pathways, thus limiting the design freedom of heat pipe configurations. This results in the majority of two-phase circulation systems maintaining simple 1D or 2D configuration [22] , [23] . Although these systems are widely applied in devices with standard shapes such as smartphones and laptops [24] , [25] , thermal management in electronics with more complex shapes, like Augmented Reality (AR) and Virtual Reality (VR) headsets or cameras, is still largely confined to the plane of the chips [26] (Fig. 1a ), leaving a vast untapped space available for heat distribution. Fig. 1: Conceptual illustration and customized design strategy for the adaptative heat pipe (AHP) in complex 3D electronic devices. a In-plane thermal management of traditional phase change system in electronic device. b Out-plane thermal management of 3D two-phase circulation by AHP. c Working mechanism and the structure of AHP, consisting of a self-supporting wick and aluminum-plastic film casing to form a vapor-liquid co-planar two-phase circulation structure. d Design freedom of two-phase heat cycle pathways inside the AHP. e Comparison of design freedom and thermal conductivity among AHP, single-phase heat conduction, and flexible two-phase heat cycle technologies. f Reverse design of AHP (Illustrated with the mixed reality headset example). g Custom conformal AHP for typical complex 3D electronic devices. Full size image Flexible heat pipes provide a solution for transferring heat out-of-plane. Some studies use corrugated tubes as the heat pipe casing, allowing for normal and radial bending, while demonstrating good thermal conductivity [27] . However, the tubular design limits integration into most electronic devices, imposing constraints on design freedom. Alternatively, flexible composite films can be used as casings to enhance the flexibility of flat heat pipes [28] , [29] , providing another approach for extending two-phase heat cycle pathways out-of-plane. However, this flexibility often comes at the expense of some heat transfer efficiency in structural design, and flexible deformation may cause distortion of the internal two-phase flow channels. As a result, most flexible flat heat pipes experience a reduction in heat transfer performance in three-dimensional configurations [30] , [31] . The deformation of most flexible flat heat pipes is typically limited to simple normal bending, significantly constraining design flexibility for more complex forms of electronic devices. Phase-change flexible filaments are another emerging technology [32] . For example, squid-inspired pulsating heat pipes [33] can achieve a thermal conductivity of up to 6750 W/m·K while offering high deformation flexibility, making them effective for spatial thermal management. However, to maximize the heat dissipation area within limited assembly spaces in electronic devices, developing phase-change cycle technologies with greater design freedom is necessary. This work integrates customized reverse engineering with data-driven three-dimensional thermal bonding technology to design and fabricate a three-dimensional two-phase thermal circulation system that can adopt arbitrary shapes, free from assembly space constraints. This thermal circulation system is termed the adaptative heat pipe (AHP). In the AHP system, the working fluid absorbs heat under vacuum conditions, evaporating into vapor at the evaporator, which then travels along a three-dimensional vapor path, conforming to the electronic device to the condenser. At the condenser, the vapor condenses which then returns to the evaporator through the three-dimensional capillary path of the wick, establishing a continuous, two-phase, three-dimensional thermal circulation. (Fig. 1c ). The AHP possesses four degrees of design freedom: normal bending, radial bending, twisting, and lateral adjustment (Fig. 1d ). It can establish stable three-dimensional phase-change pathways for electronic devices with any curved surfaces and complex configurations (as Supplementary Movie 1 ), achieving a maximum thermal conductivity of 11363 W/m·K. The AHP technology was compared with various passive thermal management technologies in terms of thermal conductivity and design freedom. The AHP maintains the ultra-high thermal conductivity advantage of the gas-liquid phase-change cycle while offering design freedom in any shape (Fig. 1e ). Therefore, in practical applications, the AHP thermal circulation system can be considered a material with ultra-high thermal conductivity, featuring integrated structure-function characteristics. In this work, the AHP serves not only as a thermal management component integrated into the electronic device but also as internal support, or even as part of the device’s shell or other structural components. The special three-dimensional shape alters the characteristics of the vapor-liquid two-phase cycle, endowing the phase change cycling system with multifunctional and intelligent features, providing application patterns for the next generation of phase change heat transfer systems.\",\n \"references\": {\n \"indices\": [\n 1,\n 2,\n 3,\n 4,\n 5,\n 6,\n 8,\n 9,\n 10,\n 11,\n 12,\n 13,\n 14,\n 15,\n 16,\n 17,\n 18,\n 19,\n 20,\n 21,\n 22,\n 23,\n 24,\n 25,\n 27,\n 28,\n 29,\n 30,\n 31,\n 32,\n 33\n ],\n \"infos\": [\n {\n \"idx\": \"1\",\n \"title\": \"Van Erp, R., Soleimanzadeh, R., Nela, L., Kampitsis, G. & Matioli, E. Co-designing electronics with microfluidics for more sustainable cooling.Nature585, 211–216 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-020-2666-1\"\n },\n {\n \"idx\": \"2\",\n \"title\": \"Lukatskaya, M. R., Dunn, B. & Gogotsi, Y. Multidimensional materials and device architectures for future hybrid energy storage.Nat. Commun.7, 12647 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms12647\"\n },\n {\n \"idx\": \"3\",\n \"title\": \"Yang, Z., Yang, T., Cai, W. & Cai, T. Miniaturization of optical spectrometers.Science371, eabe0722 (2021).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.abe0722\"\n },\n {\n \"idx\": \"4\",\n \"title\": \"Song, H. et al. Two-dimensional materials for thermal management applications.Joule2, 442–463 (2018).\",\n \"link\": \"https://doi.org/10.1016%2Fj.joule.2018.01.006\"\n },\n {\n \"idx\": \"5\",\n \"title\": \"Wang, C. et al. A thermal management strategy for electronic devices based on moisture sorption-desorption processes.Joule4, 435–447 (2020).\",\n \"link\": \"https://doi.org/10.1016%2Fj.joule.2019.12.005\"\n },\n {\n \"idx\": \"6\",\n \"title\": \"Li, M. D. et al. Thermal management of chips by a device prototype using synergistic effects of 3D heat-conductive network and electrocaloric refrigeration.Nat. Commun.13, 5849 (2022).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-022-33596-z\"\n },\n {\n \"idx\": \"8\",\n \"title\": \"Unsal, O. S. et al. Impact of parameter variations on circuits and microarchitecture.IEEE Micro26, 30–39 (2006).\",\n \"link\": \"https://doi.org/10.1109%2FMM.2006.122\"\n },\n {\n \"idx\": \"9\",\n \"title\": \"Aftab, W. et al. Phase change material-integrated latent heat storage systems for sustainable energy solutions.Energy Environ. Sci.14, 4268–4291 (2021).\",\n \"link\": \"https://doi.org/10.1039%2FD1EE00527H\"\n },\n {\n \"idx\": \"10\",\n \"title\": \"Li, W. M., Yang, S. Y., Chen, Y. P., Li, C. & Wang, Z. K. Tesla valves and capillary structures-activated thermal regulator.Nat. Commun.14, 3996 (2023).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-023-39289-5\"\n },\n {\n \"idx\": \"11\",\n \"title\": \"Yoo, S. W. et al. Responsive materials and mechanisms as thermal safety systems for skin-interfaced electronic devices.Nat. Commun.14, 1024 (2023).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-023-36690-y\"\n },\n {\n \"idx\": \"12\",\n \"title\": \"Wei, X. H., Liu, L., Zhang, J. X., Shi, J. L. & Guo, Q. G. Mechanical, electrical, thermal performances and structure characteristics of flexible graphite sheets.J. Mater. Sci.45, 2449–2455 (2010).\",\n \"link\": \"https://link.springer.com/doi/10.1007/s10853-010-4216-y\"\n },\n {\n \"idx\": \"13\",\n \"title\": \"Vu, M. C. et al. Scalable ultrarobust thermoconductive nonflammable bioinspired papers of graphene nanoplatelet crosslinked aramid nanofibers for thermal management and electromagnetic Shielding.J. Mater. Chem.9, 8527 (2021).\",\n \"link\": \"https://doi.org/10.1039%2FD0TA12306D\"\n },\n {\n \"idx\": \"14\",\n \"title\": \"Bartlett, M. D. et al. High thermal conductivity in soft elastomers with elongated liquid metal inclusions.Proc. Natl Acad. Sci. USA28, 2143–2148 (2017).\",\n \"link\": \"https://doi.org/10.1073%2Fpnas.1616377114\"\n },\n {\n \"idx\": \"15\",\n \"title\": \"Ren, N. et al. Shear-induced fabrication of cellulose nanofibril/liquid metal nanocomposite films for flexible electromagnetic interference shielding and thermal management.ACS Appl. Mater. Interfaces16, 17904–17917 (2024).\",\n \"link\": \"https://doi.org/10.1021%2Facsami.4c01220\"\n },\n {\n \"idx\": \"16\",\n \"title\": \"Shen, B., Zhai, W. & Zheng, W. Ultrathin flexible graphene film: an excellent thermal conducting material with efficient EMI shielding.Adv. Funct. Mater.24, 4542–4548 (2014).\",\n \"link\": \"https://doi.org/10.1002%2Fadfm.201400079\"\n },\n {\n \"idx\": \"17\",\n \"title\": \"Liang, G. et al. Phase change material filled hybrid 2D/3D graphene structure with ultra-high thermal effusivity for effective thermal managemen.Carbon176, 11–20 (2021).\",\n \"link\": \"https://doi.org/10.1016%2Fj.carbon.2020.12.046\"\n },\n {\n \"idx\": \"18\",\n \"title\": \"Kong, Q. Q. et al. Hierarchical graphene–carbon fiber composite paper as a flexible lateral heat spreader.Adv. Funct. Mater.24, 4222–4228 (2014).\",\n \"link\": \"https://doi.org/10.1002%2Fadfm.201304144\"\n },\n {\n \"idx\": \"19\",\n \"title\": \"Xin, G. et al. Highly thermally conductive and mechanically strong graphene fibers.Science349, 1083–1087 (2015).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.aaa6502\"\n },\n {\n \"idx\": \"20\",\n \"title\": \"Jaipurkar, T., Kant, P., Khandekar, S., Bhattacharya, B. & Paralikar, S. Thermo-mechanical design and characterization of flexible heat pipes.Appl Therm. Eng.126, 1199–1208 (2017).\",\n \"link\": \"https://doi.org/10.1016%2Fj.applthermaleng.2017.01.036\"\n },\n {\n \"idx\": \"21\",\n \"title\": \"Yang, C. et al. Fabrication and performance evaluation of flexible heat pipes for potential thermal control of foldable electronics.Appl Therm. Eng.96, 445–453 (2016).\",\n \"link\": \"https://doi.org/10.1016%2Fj.applthermaleng.2015.11.078\"\n },\n {\n \"idx\": \"22\",\n \"title\": \"Gibbons, M. J., Marengo, M. & Marengo, T. A review of heat pipe technology for foldable electronic devices.Appl. Therm. Eng.194, 117087 (2021).\",\n \"link\": \"https://doi.org/10.1016%2Fj.applthermaleng.2021.117087\"\n },\n {\n \"idx\": \"23\",\n \"title\": \"Xin, F., Ma, T. & Wang, Q. Thermal performance analysis of flat heat pipe with graded mini-grooves wick.Appl. Energy228, 2129–2139 (2018).\",\n \"link\": \"https://doi.org/10.1016%2Fj.apenergy.2018.07.053\"\n },\n {\n \"idx\": \"24\",\n \"title\": \"Tang, H. et al. Review of applications and developments of ultra-thin micro heat pipes for electronic cooling.Appl. Energy223, 383–400 (2018).\",\n \"link\": \"https://doi.org/10.1016%2Fj.apenergy.2018.04.072\"\n },\n {\n \"idx\": \"25\",\n \"title\": \"Luo, J. L., Mo, D. C., Wang, Y. Q. & Lyu, S. S. Biomimetic copper forest wick enables high thermal conductivity ultrathin heat pipe.ACS Nano15, 6614–6621 (2021).\",\n \"link\": \"https://doi.org/10.1021%2Facsnano.0c09961\"\n },\n {\n \"idx\": \"27\",\n \"title\": \"Lee, D. & Byon, C. Fabrication and characterization of pure-metal-based submillimeter-thick flexible flat heat pipe with innovative wick structures.Int J. Heat. Mass Transf.122, 306–314 (2018).\",\n \"link\": \"https://doi.org/10.1016%2Fj.ijheatmasstransfer.2018.01.135\"\n },\n {\n \"idx\": \"28\",\n \"title\": \"Liu, C., Li, Q. & Fan, D. Fabrication and performance evaluation of flexible flat heat pipes for the thermal control of deployable structure.Int. J. Heat. Mass Transf.144, 118661 (2019).\",\n \"link\": \"https://doi.org/10.1016%2Fj.ijheatmasstransfer.2019.118661\"\n },\n {\n \"idx\": \"29\",\n \"title\": \"Lim, J. & Kim, J. Fabrication and experimental evaluation of a polymer-based flexible pulsating heat pipe.Energ. Convers. Manag.126, 358–364 (2018).\",\n \"link\": \"https://doi.org/10.1016%2Fj.enconman.2017.11.022\"\n },\n {\n \"idx\": \"30\",\n \"title\": \"Jung, C., Lim, J. & Kim, S. J. Fabrication and evaluation of a high-performance flexible pulsating heat pipe hermetically sealed with metal.Int J. Heat. Mass Transf.149, 119180 (2020).\",\n \"link\": \"https://doi.org/10.1016%2Fj.ijheatmasstransfer.2019.119180\"\n },\n {\n \"idx\": \"31\",\n \"title\": \"Huang, J. et al. Development of novel flexible heat pipe with multistage design inspired by structure of human spine.Appl. Therm. Eng.175, 115392 (2020).\",\n \"link\": \"https://doi.org/10.1016%2Fj.applthermaleng.2020.115392\"\n },\n {\n \"idx\": \"32\",\n \"title\": \"Kang, Z., Jiang, S., Hong, Y. & Fan, J. Squid‐like soft heat pipe for multiple heat transport.Droplet1, 182–191 (2022).\",\n \"link\": \"https://doi.org/10.1002%2Fdro2.25\"\n },\n {\n \"idx\": \"33\",\n \"title\": \"Kang, Z., Hong, Y., Jiang, S. & Fan, J. Composite filament with super high effective thermal conductivity.Mater. Today Phys.34, 101067 (2023).\",\n \"link\": \"https://doi.org/10.1016%2Fj.mtphys.2023.101067\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"The freeform construction method of AHP To address the challenges of producing complex-shaped adaptative heat pipes, we present a rapid and cost-effective solution for customized fabrication. Instead of directly printing heat pipes using selective laser melting (SLM) technology, this work utilizes digital light processing (DLP) 3D printing technology for the indirect fabrication of complex-shaped AHP. The designed AHP structure, shown in Fig. 1c consists of two components: the heat pipe casing and the self-supporting wick. The casing is composed of two layers of aluminum-plastic film, with the composition detailed in Supplementary Fig. 2 . Due to the thermoplastic properties of the inner polypropylene layer, applying pressure at a specific temperature allows for thermal bonding of the aluminum-plastic film casing. The internal structure includes a self-supporting wick, created from three-dimensionally structured multi-layer copper mesh. The capillary force of the wick is enhanced by chemically deposited hydrophilic nanosheets (as detailed in Supplementary Fig. 3 ). The self-supporting wick and the aluminum-plastic film casing collaborate to form a vapor-liquid co-planar two-phase circulation structure. The wick functions as the liquid pathway, while an independent vapor pathway is created between the wick and the casing. By using the cross-section as the design unit for AHP enables the topology of two-phase circulation along any path of the three-dimensional coordinate system, which forms the foundation for the freeform design of AHP. During the design process, a customized 3D configuration of the AHP must be determined based on the complex shape of the target electronic device. Inspired by reverse engineering techniques in architecture, we have developed a reverse design strategy for customized AHP, as shown in Fig. 1f . Taking Mixed Reality (MR) headset as an example, the design process starts by converting the target model into spatial point cloud data (Fig. 1fI ). Trajectories are planned along the edges and areas of abrupt curvature of the target, and critical point coordinates on these trajectories are extracted (Fig. 1fII–III ). As the aluminum-plastic film shell needs to be transformed from an initial 2D plane into a 3D curved surface, the 2D shape of the aluminum-plastic film must be defined according to the 3D surface to prevent sealing defects. Triangular meshes are generated between points on adjacent trajectories to form the 3D surface (Fig. 1fVI ). Then, using a topological equivalence mapping algorithm, the corresponding 2D plane shape of the 3D surface is generated (Fig. 1fV ). The detailed description of the algorithm can be found in the method section. In the fabrication process, a major challenge is the thermal bonding of the Al/plastic film casing with a 3D curved surface, which directly impacts the sealing capability of the two-phase circulation system. Two key factors are identified in the thermal bonding process: temperature and pressure. In this work, the fastening force of bolts provides three-dimensional normal pressure, while Joule heat generated by flexible carbon fabric as an internal heat source. Customized resin molds are created based on the extracted trajectory points by DLP photopolymerization 3D printing technology (Supplementary Fig. 4a ). Carbon fabric is used as a flexible heater and is assembled with the heat pipe components in a customized 3D-printed resin mold. The thermal edge sealing of the complex-shaped casing is achieved by the combined effects of three-dimensional pressure and internal heat (Supplementary Fig. 4c ). The detailed description of the fabrication method is provided in the method section and Supplementary Movie 2 . The AHP customized for MR headset efficiently performs two-phase thermal cycling under complex three-dimensional configurations (Fig. 1fVI ). Its spatial thermal management performance under the effect of a point heat source is illustrated by thermal imaging, as shown in Fig. 1fVII . Due to the characteristics of two-phase cycling heat transfer, a relatively uniform temperature distribution is maintained over the entire annular curved surface. We selected typical 3D complex electronic devices such as cameras, unmanned aerial vehicles, and lamps as design targets. The trajectory point clouds of their casings were extracted, and AHPs with customized shapes were fabricated. The customized AHP conforms to the freeform surface of target electronic devices, establishing a three-dimensional vapor-liquid two-phase circulation path and performing out-plane 3D thermal management in these devices, as shown in Fig. 1g . The influence of 3D shape on the entrainment effect in two-phase counterflow The operating limit of a heat pipe is the maximum heat capacity it can transfer. When the heat source exceeds this limit, it disrupts the two-phase cycle within the heat pipe, leading to operating failure. Based on heat pipe theory, there are five critical limits for heat pipes [34] . To investigate the impact of the 3D segment on heat pipe performance, we initially conducted theoretical calculations on the two-phase thermal cycle limit under the basic structural parameters of AHP, as shown in Fig. 2a (The theoretical derivation process can be found in Supplementary Note 1 ). Calculations show that within the typical operating temperature range of micro heat pipes using water as the working fluid, the AHP is primarily limited by the capillary limit and the entrainment limit. The capillary limit occurs when the pressure drop during the two-phase cycle exceeds the capillary force provided by the wick [35] , failing to satisfy Eq. ( 1 ). As a result, the working fluid cannot circulate back quickly enough, causing the evaporator to dry out. The entrainment limit occurs at the vapor-liquid interface, where excessively high relative speeds of vapor and liquid counterflow, lead to the vapor entraining the working liquid back towards the condenser [36] , [37] , as illustrated in Fig. 2b . $$\\\\varDelta {P}_{cp}\\\\ge \\\\varDelta {P}_{v}+\\\\varDelta {P}_{l}+\\\\varDelta {P}_{g}$$ (1) Fig. 2: Mitigation effects of 3D segment on two-phase counterflow entrainment. a Theoretical operating limits of adaptative heat pipe (AHP). b Entrainment effects in 2D two-phase counterflow. c Schematic of mitigated two-phase counterflow entrainment in 3D AHP. d Operating performance of AHP without 3D segment. e Operating performance of AHP with 90° local 3D segment. f Operating performance of AHP with 180° local 3D segment. Thermal response time and testing illustration for normal bending angles of ( g , h ) 0°, ( i , j ) 45°, ( k , l ) 90°, ( m , n ) 135°, and ( o , p ) 180° local 3D segments. q Theoretical prediction of entrainment limit for AHP without 3D sections. r Modified theoretical prediction of the entrainment limit for AHP with a 90° local 3D segment. s Modified theoretical prediction of the entrainment limit for AHP with a 180° local 3D segment. Full size image To characterize the 3D shape of AHP parametrically, the normal bending state of the heat pipe is used as the fundamental 3D configuration, as illustrated in Fig. 2c . When the AHP is straight, i.e., with a bending angle of 0°, its operating performance is as shown in Fig. 2d . Thermocouples are placed at both the hot and cold ends of the AHP, and two additional thermocouples are positioned equidistantly at the middle section. The straight AHP shows the smallest temperature difference between the cold and hot ends ( \\\\(\\\\varDelta T1-4=T1-T4\\\\) ) at 1 W. As the heating power increases, \\\\(\\\\varDelta T1-4\\\\) gradually increases, indicating that the straight AHP operates more efficient two-phase circulation at low power input. When the heating power reaches 9.3 W, the temperature at the hot end ( T 1 ) rises rapidly, whereas the other temperature points ( T 2 , T 3 , T 4 ) fail to follow. This indicates that the evaporator of the AHP cannot absorb the increased power, thus reaching its operating limit. By introducing a 3D segment with a 90° bending angle in the AHP (Fig. 2e ), the minimum \\\\(\\\\varDelta T1-4\\\\) is observed at 4 W, and the heat transfer limit extends to 11.5 W, an increase of 23.7%. When the normal bending angle is increased to 180° (Fig. 2f ), the minimum \\\\(\\\\varDelta T1-4\\\\) occurs at 10 W, and the operating limit is further increased to 13 W (an increase of 44.4%). Supplementary Fig. 6a–c further illustrated the heat transfer performance of AHP with normal bending angles of 45° and 135°. The results indicate that the operating limit of the AHP occurs within the power range of the entrainment limit. Increasing the normal bending angle improves the AHP’s operating limit. This suggests that the 3D configuration of the heat pipe potentially mitigates the occurrence of the entrainment limit. By combining theoretical formulas, we further investigated the mechanism of the mitigation effect of 3D configurations on the entrainment limitation. The entrainment limit for the vapor-liquid interface can be theoretically calculated using flooding correlations in a counterflow vapor-liquid system. The criteria for entrainment instabilities can be determined using Eq. ( 2 ) [37] . $${({k}_{v})}^{\\\\frac{1}{2}}+{({k}_{l})}^{\\\\frac{1}{2}}=\\\\sqrt{3.2}\\\\,\\\\tanh (0.5B{o}^{\\\\frac{1}{2}})\\\\equiv {C}_{k}$$ (2) The Bond number ( Bo ) is a dimensionless parameter characterizing the interface fluid properties (Eq. ( 3 )), primarily determined by the surface tension \\\\({\\\\sigma }\\\\) of working fluid and characteristics parameter D for the interface. k v and k l respectively represent the dimensionless momentum fluxes of the vapor and liquid phase (Detailed calculation formulas are provided in Supplementary Note 2 ). $$Bo=D{[g({{\\\\rho }}_{l}+{{\\\\rho }}_{v})/{\\\\sigma }]}^{\\\\frac{1}{2}}$$ (3) We theoretically calculated the entrainment limit at different input powers, as shown in Fig. 2q . The results reveal that as power increases, the most noticeably changing variable is the dimensionless momentum flux of the vapor k v , suggesting that the drastic variation in vapor velocity as power changes is crucial in influencing the entrainment effect. Due to the high sealing requirement, directly measuring vapor flow velocity inside the heat pipes at micro-scales is challenging. Therefore, we indirectly assessed vapor velocity across different 3D configurations by measuring the thermal response time and revising entrainment limit criteria for 3D configurations. The normal bending angles tested were 0°(Fig. 2g, h ), 45°(Fig. 2i, j ), 90°(Fig. 2k, l ), 135°(Fig. 2m, n ), and 180°(Fig. 2o, p ), respectively. Four thermocouples were placed at the same positions on the AHP for different 3D configurations, and the thermal response time intervals between \\\\(T1\\\\) and the other three points were measured upon contact of the heating plate at the evaporator, with input powers of 1 W and 9 W. \\\\(\\\\varDelta t1-2\\\\) , \\\\(\\\\varDelta t2-3\\\\) , and \\\\(\\\\varDelta t3-4\\\\) represents the thermal response time of the vapor flow before the 3D segment, through the 3D segment, and from the 3D segment to the cold end, respectively. The results show that as the bending angle increases, there is negligible change in \\\\(\\\\varDelta t1-2\\\\) , while \\\\(\\\\varDelta t2-3\\\\) significantly increases, and \\\\(\\\\varDelta t3-4\\\\) shows a modest increase. The extended thermal response time due to the 3D sections primarily results from changes in vapor flow, while liquid flow is more influenced by the wick’s capillary structure. As illustrated in Supplementary Figs. 7 – 12 , the introduction of curved sections primarily affects the vapor flow pressure drop, and changes in liquid pressure drop are negligible. In addition to momentum exchange, the vapor-liquid interface of the wick undergoes evaporation and condensation of the working fluid, resulting in complex flow conditions [38] , [39] . With an increasing bending angle, the flow behavior at the interface becomes more complex as high-speed vapor passes through the bending section. Consequently, the thermal response time for vapor to pass through the 3D segment is greatly extended, while the straight segment after the 3D segment shows a minor reduction in thermal flow velocity due to reduced momentum. By modifying the momentum flux based on the increments of \\\\(\\\\varDelta t3-4\\\\) at different bending angles, as illustrated in Fig. 2r, s , we found that the predictions for the entrainment limit closely align with the increases in the power limit. Since the condensation mainly occurs at the condenser of the heat pipe, leading to a higher concentration of working fluid at this segment, entrainment is more likely to occur. Therefore, the construction of the 3D segment able to mitigate entrainment occurrence in the condenser segment. Additionally, we constructed local 3D segments at different locations along the two-phase cycle path of the AHP and evaluated heat transfer performance, as shown in Supplementary Fig. 13a . Results showed that constructing a local 3D segment at the condenser reduces the maximum power of the AHP by 1 W compared to other locations. This reduction could be attributed to significant interactions at the vapor-liquid interface within the 3D pathway, which adversely affects the liquid return at the condenser, where the liquid concentration is higher. A detailed description can be seen in Supplementary Fig. 13 . The robustness of AHP to spatial state changes Wearable and mobile electronic devices frequently change spatial state based on user demands, which often prevents their cooling components from operating in the most advantageous state. In two-phase thermal circulation systems, spatial state changes affect the gravitational pressure drop, represented by \\\\(\\\\varDelta Pg\\\\) in Eq. ( 1 ). To ensure the stable operation of the AHP, it is essential to minimize gravitational pressure drop changes across various spatial states. We theoretically calculated the capillary driving force, vapor pressure drop, and liquid pressure drop of the AHP at different heating powers and identified the maximum allowable gravitational pressure drop, as shown in Supplementary Fig. 14 . The results show that as power increases, the theoretically permissible range of gravitational pressure drop gradually decreases. By discretizing the AHP and defining the spatial state change parameters α and β , and calculating the weighted average of the gravitational pressure drops across the discrete segments, we derived the formula for the 3D gravitational pressure drop of the AHP in various spatial states, as Eq. ( 4 ). The complete derivation is provided in Supplementary Note 3 . $${P}_{g}={\\\\sum }_{k=1}^{n}\\\\frac{{l}_{k}}{L}{\\\\rho }g\\\\,\\\\sin ({{\\\\theta }}_{k}({\\\\alpha },{\\\\beta }))$$ (4) We selected three different heat pipe configurations: a straight flat heat pipe (2D), a fully deformed 3D AHP (full 3D), and an AHP with 3D deformation in the middle section while the remaining section remains flat (partial 3D). All three heat pipes maintain a complete two-phase heat cycle pathway, effectively eliminating the impact caused by structural collapse in flexible heat pipes under deformation. The primary difference lies in the gravitational pressure drop of the liquid working fluid across different spatial orientations. Using the gravitational pressure drop formula for AHP, we evaluated the theoretical changes in gravitational pressure drops for 2D, partial 3D, and full 3D configurations of AHP across six spatial states, as shown in Fig. 3b . The results indicate that the maximum change in gravitational pressure drop for the 2D straight heat pipe across different spatial states is 2560 Pa. For AHP with partial 3D and full 3D configurations, the maximum changes are 870 Pa and 960 Pa, respectively, equating to 34% and 37.5% of that for the straight 2D heat pipe. A smaller change in gravitational pressure drop indicates that spatial state variations have less impact on the heat pipe’s operational performance, enhancing the robustness of electronic devices against spatial changes in practical applications. Fig. 3: Spatial robustness and mechanical strength testing of the adaptative heat pipe (AHP). a Discrete weighted calculation of gravitational pressure drop in AHP. b Theoretical values of gravitational pressure drop in different spatial states. c Heat transfer performance of the straight 2D heat pipe across different spatial states. d Heat transfer performance of the AHP with full 3D segment across different spatial states. e Schematic diagram of the AHP tensile test. f An AHP in the shape of tensile specimens. g A copper heat dissipation sheet in the shape of tensile specimens. h Heat transfer performance of the tensile-shaped AHP and copper sheet under 4 W heating power. i Tensile stress-strain curve of the AHP. j Tensile test of the AHP. k Schematic diagram of the AHP flexural test. l An AHP in the shape of flexural specimens. m A copper heat dissipation sheet in the shape of flexural specimens. n Heat transfer performance of the flexural-shaped AHP and copper sheet under 4 W heating power. o Flexural stress-strain curve of the AHP. n Flexural test of the AHP. Full size image The operating performance of AHP was evaluated across different configurations. After stabilizing at a heating power of 8 W, the 2D heat pipe and the 3D AHP underwent sequential spatial state changes, with temperatures at the hot and cold ends recorded. The results indicate that with changes in spatial state, the straight 2D heat pipe exhibits significant temperature fluctuations between the hot and cold ends, which increase the temperature gradient, severely disrupting the vapor-liquid two-phase cycle (Fig. 3c ). In contrast, both the partial 3D (Supplementary Fig. 15 ) and full 3D (Fig. 3d ) configurations of AHP maintain relatively stable temperatures at the hot and cold ends, resulting in minimal changes in the temperature difference. Theoretical calculations indicate that the permissible gravitational pressure drop for the straight 2D heat pipe across six spatial states is 1281 Pa, with a maximum resistance to gravitational pressure drop of 1203.2 Pa, which is close to the permissible limit. This results in considerable variations in capillary backflow resistance for the straight 2D heat pipe across different spatial states, indicating weaker robustness to spatial changes. The 3D configured AHP, with smaller variations in gravitational pressure drop, has maximum gravitational pressure drops of 435.7 Pa and 609.2 Pa across different spatial states, significantly below the allowable limits. This indicates good spatial robustness, making it suitable for the spatial state changes required by compact electronic devices. Mechanical strength of AHP With its freeform design characteristics, the AHP can integrate into the mechanical assembly of electronic devices in various forms, similar to two-dimensional thermal materials, while maintaining efficient phase-change heat transfer performance. It exhibits integrated structural and functional application characteristics. To demonstrate the structural support properties of the AHP as a casing or support component for electronic devices, we conducted tensile and flexural tests and compared the results with those of commercial copper heat dissipation plates. Given the thin-plate structure of the AHP, we designed a dumbbell-shaped tensile specimen (tensile section width of 10 mm) and conducted tensile tests on both the AHP and copper plates (Fig. 3e–g ). The tensile strength of the AHP ranges between 30 and 40 MPa (Fig. 3h ), similar to that of rigid plastics like epoxy resin (20.67–40 MPa) [40] , [41] , [42] . Although the tensile strength of AHP shows a significant difference compared to copper (>200 MPa), it is sufficient to meet the mechanical requirements of most mobile and wearable device enclosures. Moreover, the internal two-phase cycle enables the AHP to exhibit extremely high thermal conductivity as a device-level application material, as shown in Fig. 3i . The AHP, composed of a composite structure with an aluminum-plastic film casing and a copper wire mesh wick, undergoes multiple mechanical stages during the tensile test. In the initial tension stage, the aluminum-plastic film casing and copper wire mesh wick transition from elastic deformation to plastic deformation (Supplementary Fig. 16aI–II ). When the tensile stress reaches 36.18 MPa, a slight decrease in tensile stress is observed. The images indicate that at this point, while the heat pipe casing remains intact, the AHP elongates slightly, primarily due to the fracture of the internal wick. Subsequently, the aluminum-plastic film continues to stretch (Supplementary Fig. 16aIII ), and the tensile stress-displacement curve shows characteristics of plastic deformation, resulting in complete fracture when the tensile stress reaches 40.01 MPa (Supplementary Fig. 16aIV ). Additionally, due to the thin-plate structure of the AHP, a flexural performance test was conducted (Fig. 3k ). The AHP and copper plate specimens were prepared as rectangles with a length of 150 mm and a width of 30 mm (Fig. 3l–m ). Under a heating power of 4 W, the AHP displayed a more uniform temperature gradient, while the copper plate exhibited uneven temperature distribution (Fig. 3n ), with significant heat accumulation at the hot end. The resulting pressure-displacement curve is shown in Fig. 3o . Compared to the tensile performance, the difference in flexural performance between the two thin-plate materials is smaller: the maximum pressure for the copper plate is 54.7 N, while for the AHP it is 37.68 N, approximately 68.9% of the copper plate’s value. Out-plane 3D thermal management of customized AHP in a VR headset The assembly of thermal management components is often a key point of contention between thermal engineers and structural designers in electronic device design. The AHP design utilizes the available space after other components are installed, free from the spatial constraints, to achieve out-plane 3D thermal management. Moreover, AHP possesses integrated structural-functional characteristics. Beyond its two-phase thermal circulation function, it can also serve as a casing or support component, further enhancing the compactness of electronic devices. As representative products of compact and complex-shaped devices, virtual reality (VR) headsets are widely adopted in both industrial and home settings [43] , [44] . Research indicates that the face is the one of the most heat-sensitive parts of human body [45] , [46] . VR headset panels typically feature an enclosed design, leading to significant heat accumulation during operation [47] . However, compact and complex device configurations hinder the effective integration of thermal management components [48] . Here, we designed a customized AHP specifically tailored to the special construction of VR headsets, deviating from the conventional design of regular-shaped cooling components. The customized AHP functions not only as a thermal management element but also as part of the head-mounted component, allowing the heat from the simulated chip to be directly transferred through the AHP for natural convection with the external air and maximizing the cooling area within limited assembly space. A schematic of the VR device is shown in Fig. 4a . To ensure alignment with the VR device’s structural characteristics, the AHP contacts the simulated chip at the center of the panel, distributing the heat along three separate paths. This heat is then transferred outside through gaps between the face pad and the control panel, extending along three head straps to form a closed-loop structure. The installation of the customized AHP in the VR headset, as illustrated in Fig. 4b , demonstrates that the AHP configuration conforms to the structural characteristics of the VR headset. Fig. 4: Customized adaptative heat pipe (AHP) for VR headset. a Reverse design of the AHP for the VR headset, extracting the head-mount component to create a customized AHP that provides both cooling capability and mounting functionality. b VR headset after co-integration of the AHP. c Infrared thermal imaging results of the commercial flat heat pipe under a heating power of 4 W. d Infrared thermal imaging of the commercial flexible graphene film under simulated chip heating power of 4 W. e Infrared thermal imaging of the customized AHP under simulated chip heating power of 4 W. f Comparison of the temperatures at the hot end for the customized AHP, conventional flat micro heat pipe, and commercial flexible graphene film across different power settings. g User testing of the structural-functional integrated AHP. Full size image In commercial applications, flat microheat pipes are used as cooling components in VR devices due to their two-phase thermal cycling characteristics. For instance, the Quest Pro VR headset by Meta Inc. integrates flat heat pipes within the panel for chip cooling [26] . Additionally, graphene thermal films, an emerging electronic cooling strategy, are used in some electronic products due to their high flexibility and thermal conductivity [49] , [50] , [51] . To assess the 3D two-phase thermal management efficiency of AHP, it was compared with commercial flat heat pipes and graphene films in a VR headset (as Supplementary Movie 3 ). Detailed structural and assembly information is provided in Supplementary Figs. 17 – 20 . With a 4 W heating power applied to the simulated chip, thermal management performance is shown in Fig. 4c–e . Due to the heat transfer characteristics of two-phase cycling, 2D flat heat pipes exhibit uniform temperature distribution. However, constrained by the 2D form, they can only manage heat within the plane of the panel, limiting the cooling area and resulting in the hot end temperature exceeding 100 °C (Fig. 4c ). As a flexible cooling material, graphene film can manage heat in three dimensions according to device shape, but it is limited by the properties of single-phase thermal conduction. The graphene film exhibits a significant temperature gradient, with hot end reaching 84.7 °C (Fig. 4d ). A customized AHP combines the phase-change cooling capabilities of 2D flat heat pipes with the design freedom of graphene film, achieving comprehensive 3D two-phase circulation cooling for VR devices. As shown in Fig. 4e , AHP maintains excellent temperature uniformity, reducing the temperature to 47.8 °C under the same power input. We compared the chip temperature under different cooling conditions and input powers, as shown in Fig. 4f . Without cooling measures, the simulated chip heat diffuses throughout the panel via direct contact, reaching 100 °C at 2 W and exceeding 120 °C at 3 W. When a 2D micro heat pipe is installed on the panel, the heating power can be stabilized at 3 W, with the hot end temperature reaching 83 °C. At 4 W, the temperature at the hot end exceeds 100 °C. Installing graphene film along the head strap significantly increases the heat dissipation area and enhances cooling performance compared to the 2D heat pipe, allowing the heating power to increase further to 5 W. With the integration of AHP, a significant reduction in temperature is observed. At the input power of 3 W, the average temperature of the simulated chip decreased by 48.59% (from 85°C to 43.7 °C) and 38.62% (from 71.2 °C to 43.7 °C) compared to the 2D heat pipe and graphene film, respectively. Additionally, combined with the 3D configuration’s mitigation effects of two-phase counterflow entrainment, the customized AHP shows a pronounced advantage in the improved heat transfer limit. Taking 100 °C as the limit, the simulated chip temperature exceeds 100 °C at 4 W with the 2D heat pipe and at 6 W with graphene film. With AHP cooling, however, the simulated chip can be increased to 19 W, an increment of 4.75 and 3.17 times compared to the 2D heat pipe and graphene film, respectively. The AHP demonstrates structural-functional integration. Specifically designed for head-mounting components, the customized AHP not only performs two-phase thermal circulation but also provides structural support for the user’s head. As shown in Fig. 4g , user testing was conducted with a simulated chip heating power of 6 W, during which the highest temperature maintained by the customized AHP was 43 °C. Under the same power settings, temperatures for commercial cooling strategies exceed 100 °C, posing burn risks to users. Detailed information on the user testing is provided in Supplementary Movie 4 . Stretchable two-phase circulation system In electronic devices, thermal management components like heat pipes will operate two-phase circulation under extreme environments [52] . Intense vibrations experienced during spacecraft launches or while driving automobiles can significantly impact electronic devices [53] , [54] . In addition, potential heat transfer requirements between cross-axis planes of aerospace robotic arms (Supplementary Fig. 21 ) pose a severe challenge to traditional heat pipe technology. Special 3D structural design can impart exceptional mechanical properties and advanced functions to the device [55] . By utilizing the freeform design characteristics of AHP, we have developed a stretchable AHP with a spiral three-dimensional shape, targeting the spring components in shock absorbers (Fig. 5a ). This design allows the heat pipe to extend over a long-range, functioning similarly to a spring. It adapts to dynamic spatial relationships between heat transfer planes, ensuring stable heat exchange. Fig. 5: Development of stretchable vapor-liquid phase-change circulation system. a Extraction of spring element curves from the shock absorber enables the designed adaptative heat pipe (AHP) to offer not only heat exchange capabilities but also damping and stretchability. b A stretchable AHP. c Characteristics of the stretchable AHP (L represents the minimum pitch of the stretchable heat pipe). d Dynamic temperature distribution curves of the stretchable heat pipe under different heating powers. e Heat transfer performance of the stretchable heat pipe under different stretching conditions. f Operation of the stretchable AHP under simulated extreme conditions, including compression, stretching, mild vibration, and intense vibration. g Cycle test of compression and stretching for the stretchable AHP. Full size image The developed stretchable AHP and its dynamic operating state are shown in Fig. 5b , with its two-phase circulation during dynamic stretching illustrated in Fig. 5c . The axial length can be compressed from 2L to L and extended to 3 L, achieving a threefold extension range. Heat transfer performance tests were conducted on the stretchable heat pipe with a pitch of 2 L under different heating powers. As shown in Fig. 5d , temperatures were measured at three locations: the hot end (15 mm from the edge), the cold end (20 mm from the edge), and the middle section, with temperature changes recorded as heating power increased from 1 W to 4 W. Results indicate that as heating power increased from 1 W to 4 W, the temperature difference between the hot and cold ends decreased from 11 °C to 9 °C, with a thermal resistance of 2.34 K/W and a thermal conductivity of 5335 W/m·K. In a helical configuration, the AHP has minimal curvature changes without smooth transition sections. This results in less effective curvature coordination between the membrane layers on both sides of the heat pipe during fabrication, causing some compression of the internal wick and affecting the stretchable AHP’s heat transfer efficiency. However, in the constructed 3D helical two-phase heat cycle pathway, the macroscopic deformation from mechanical stretching has minimal impact on the stretchable heat pipe. As shown in Fig. 5e , thermal conductivity of the stretchable heat pipe was measured under various stretching states. When the pitch of the stretchable heat pipe is L, 2 L, or 3 L, its heat transfer performance shows minimal variation, with thermal resistance fluctuating within ±0.15 K/W. Heat transfer operation between different planes of the AHP was simulated, as shown in Fig. 5f . After thermal stabilization, the AHP underwent compression, stretching, mild vibration, and intense vibration (complete process available in Supplementary Movie 5 ). Thermal imaging revealed no severe performance loss in the AHP. After thermal stabilization, a test of 200 compression and extension cycles was conducted (Fig. 5g ). Fluctuations were observed in the temperature at the heat pipe’s hot end. These fluctuations may be attributed to changes in vapor pathways during compression and extension and to slight compression of the two-phase structure during stretching. Additionally, the hot end temperature remained relatively stable within 65.5 °C to 68.5 °C, while a gradual decrease in cold end temperature was observed. This suggests that while the stretching cycles affected the heat pipe performance, they did not cause edge leakage or significant blockage in the two-phase cycle. Instead, the fluctuations likely result from deformation of the two-phase structure due to repeated stretching and compression. Magnetically-controlled two-phase circulation heat exchange robot The fourth industrial revolution, marked by advancements in information technology and artificial intelligence, has ushered in the era of intelligence, making device smartification an inevitable path for future development [56] . In thermal management, a key challenge is the cooling random hot spots, specifically dynamic heat dissipation when point heat source coordinates change over time. Leveraging the freeform design property of the adaptative heat pipe, we introduce the concept of vapor-liquid two-phase circulation heat exchange robots, developing intelligent application modes for conventional phase change systems. The AHP was designed as an enclosed cube (Fig. 6a ), with the entire cube serving as a unified vapor-liquid two-phase circulation path (Fig. 6b ). Its two-dimensional mapping form is shown in Fig. 6c , with the two-phase cycling path marked in the diagram. As shown in Fig. 6d–e , the heat transfer performance of the heat exchange robot achieves a maximum thermal conductivity of 4699 W/m·K with the heat source on surface 3. The thermal conductivity between surface 3 and surface 1 is 3720 W/m·K, and between surface 3 and surface 6 is 3524 W/m·K. After fabricating the cubic AHP, magnets with different polarizations were embedded inside each of the six faces, forming the AHP heat exchange robot. Leveraging its cubic 3D form, the AHP heat exchange robot can be maneuvered by flipping through the magnetic field at the base, allowing it to approach coordinates of randomly generated hot spots (Fig. 6f ). When the AHP heat exchange robot covers a hot spot, heat is quickly conducted away through the robot’s two-phase circulation walls, preventing further heat accumulation (Fig. 6g ). To illustrate the design concept of the AHP heat exchange robot, we simulated its operation under random heat sources, as shown in Fig. 6h (for a detailed view, see Supplementary Movie 6 ). When a random hot spot emerges within the plane (Fig. 6hI ), the AHP robot quickly flips and moves closer to the hot spot (Fig. 6hII–III ). By contacting the hot spot with its bottom wall, the robot uses the fast startup and efficient heat transfer of its two-phase circulation walls to rapidly disperse heat throughout the entire shell of the robot, expanding the originally localized air convection area to all six faces of the cube (Fig. 6hIV ). After the first hot spot dissipates, a second one forms (Fig. 6hV ). The AHP heat exchange robot continues to flip, contact, and dissipate the hot spot (Fig. 6hVI–VII ) and then moves to the next hot spot (Fig. 6hVIII ). Fig. 6: Dynamic cooling of two-phase circulation heat exchange robot system. a Extraction of external curves of the cubic model to plan the two-phase circulation path of the adaptative heat pipe (AHP) and construct a fully enclosed cubic AHP. b Structural schematic of a cubic AHP. c Schematic of magnetic pole arrangement for heat exchange robot. d Temperature distribution at the center points of the six faces under a bottom point heat source. e Thermal resistance and thermal conductivity at a heating power of 5 W. f Magnetic actuation flipping method for the AHP heat exchange robot. g Heat transfer process demonstrated by the cubic AHP when positioned above a hot spot. h The AHP heat exchange robot performing dynamic heat dissipation in response to random hot spots (The upper and lower sections of the diagram respectively represent the workflow under an optical camera and an infrared thermal imaging camera). Full size image\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"In summary, the AHP developed in this work leverages reverse engineering and data-driven three-dimensional thermal bonding technology, functioning as a phase-change circulation system with unprecedented shape design freedom. The three-dimensional two-phase thermal cycling path is customized to the configuration and thermal requirements of target electronic devices, facilitating direct heat exchange between the chip and the external environment, offering a solution to thermal management challenges in small, compact, and complex-shaped electronic devices. The three-dimensional pathway improves the heat transfer limit of the vapor-liquid phase-change system and potentially mitigates entrainment effects at the vapor-liquid interface in counterflow. Theoretical calculations based on flooding correlations reveal vapor flow rate as a key factor under varying input powers. Adjusting the correlation equations with velocity attenuation aligns calculated power increases closely with experimental measurements. Moreover, compared to 2D phase-change systems, 3D two-phase systems exhibit enhanced robustness to spatial state changes under high power conditions. Under spatial state changes, the maximum theoretical gravitational pressure drop for the 3D-configured AHP reduced to 34% of that in the 2D configuration. This diminished influence of gravity on two-phase circulation meets the spatial state change requirements of wearable and mobile electronic devices. This study developed a customized AHP tailored for VR headset structures. Compared to commercial flat 2D heat pipes and graphene films, the customized AHP reduced the simulated chip temperature by 48.59% (41.3 °C) and 38.62% (27.5 °C), respectively, at the same input power of 3 W. It also increased the power limit by 4.75 times (from 4 W to 19 W) and 3.17 times (from 6 W to 19 W), respectively. The three-dimensional configuration endows the AHP with integrated structural-functional characteristics, enabling it to serve as both a thermal management component and a head-mounted structure, providing structural support. The AHP’s spiral and cubic designs impart features such as stretchability and dynamic heat dissipation, paving the way for multifunctional and intelligent development of two-phase thermal circulation systems.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Fabrication of freeform AHP In the fabrication process, DLP 3D printing technology is first used to create molds for shaping the copper wire mesh and pre-encapsulating the AHP (Supplementary Fig. 4a ). The mold for the copper mesh is designed to serve as the self-supporting structure for the wick, while the heat pipe sealing mold is designed with space reserved exclusively for the edges of the AHP, ensuring contact only at the these points. As shown in Supplementary Fig. 4b , the mold shapes the copper mesh into a three-dimensional form by assembling multiple layers of flat copper mesh within it. Thanks to the high flexibility and ductility of the copper mesh, upon releasing the bolts, the multi-layer copper mesh retains the customized shape of the AHP at the macro level and forms a self-supporting structure of the wick at the micro level. The formed multi-layer copper mesh then undergoes sintering and chemical deposition processes to form stable capillary pores and enhance hydrophilicity. The aluminum-plastic film, wick, flexible carbon fabric, and mold are assembled together and secured with bolts to apply pressure. To generate heat for the thermal bonding process, apply voltage to both sides of the carbon fabric for Joule heating. This step completes the thermal bonding of the aluminum-plastic film casing, forming a complex three-dimensional shape (Supplementary Fig. 4c ). After releasing the bolts, a pre-encapsulated AHP (three sides sealed, one side open) is obtained. Next, venting treatment is performed through a charging port on the pre-encapsulated AHP, followed by vacuuming and filling with working fluid (deionized water). The final side is then sealed to complete the fabrication of a customized AHP with two-phase thermal cycling capabilities. Detailed information about the fabrication process is provided in Supplementary Fig. 5 and Supplementary Movie 2 , offering a more intuitive understanding of the fabrication method. Materials Aluminum-plastic film (D-EL40H(3)) as heat pipe casing was purchased from Dai Nippon Printing Co., Japan. Copper wire mesh (200 mesh copper) used as wick was purchased from Anping County Bolin Metal Wire Mesh Co., China. The carbon fabric (WOS1009, CeTech, USA) after cutting, was utilized as a flexible heater. Chemicals including sodium persulfate (Na 2 S 2 O 8 ), sodium hydroxide (NaOH), deionized water, and ethanol were purchased from Shanghai Aladdin Biochemical Technology Co. 3D printing of shaping and pre-encapsulating molds A customized configuration of the three-dimensional resin mold is created using Computer-Aided Design (CAD) software and exported in STL format. This STL file is then imported into Voxeldance Tango slicing software to add support structures and slice the model. The sliced file is loaded into a Digital Light Processing (DLP) 3D printing machine (Reflect2, DONZY, China). The printing parameters are set as follows: light source intensity at 5 mW/cm², exposure time at 2.5 s, and layer thickness at 50μm. Begin the printing process. After printing, the mold is removed and immersed in an ultrasonic cleaner (SK5210LHC, KUDOS, China) filled with 95% alcohol, with the cleaning parameters set to 53 kHz and 150 W for 60 s. After ultrasonic cleaning, the mold is dried in a 60 °C oven (DZF−6020, Shanghai Jinghong, China). Topological equivalence mapping algorithm The implementation of the 2D topological mapping algorithm for the AHP 3D triangular sheet is conducted using commercial software MATLAB. Trajectory planning of the three-dimensional point cloud is performed in CloudCompare software to obtain point coordinates along the trajectory, forming 3D triangular meshes between points on two closely adjacent trajectories. This forms the basis for mapping the surface from 3D triangular meshes to a corresponding 2D plane, guiding the planar shape design of the aluminum-plastic film casing. The specific process for implementing the topological mapping algorithm is described as follows: Step 1: Identified the central triangle mesh within the triangulated surface model to serve as the base triangle for mapping. Step 2: Map the base triangle onto the plane while preserving its spatial characteristics unchanged. Mark the corresponding vertices, edges, and triangle as unmapped. Step 3: Identified the adjacent 3D triangular mesh, \\\\(\\\\varDelta {1}_{i}\\\\) , that currently processing. Among the triangular meshes that are not yet marked as unmapped, select one that shares an edge with a triangle already marked as unmapped to be the current triangle under processing. Step 4: Confirm the position of the undetermined vertex of the currently processed triangle in the 2D plane. Calculate the normal vector passing through this point in the 3D surface, and determine the normal plane \\\\(\\\\overrightarrow{n}\\\\) corresponding to this normal vector \\\\(\\\\overrightarrow{{P}_{n}}\\\\) . Select a vertex adjacent to the vertex currently being processed, and determine the projection of this adjacent vertex on the normal plane \\\\(\\\\overrightarrow{{P}_{n}}\\\\) of the 3D surface. Then, calculate the curvature \\\\(K{1}_{i}\\\\) at the processed vertex. In the 2D plane, ensure that the curvature \\\\(K{2}_{i}\\\\) of the undetermined point is the same as \\\\(K{1}_{i}\\\\) . This approach can allow determine the direction between the undetermined point and its adjacent point. To determine the 2D position of the currently processed point, ensure that the area \\\\(S{2}_{i}\\\\) of the triangle in the 2D plane matches the area \\\\(S{1}_{i}\\\\) of its counterpart in 3D space. Step 5: Mark the currently processed point and the triangular surface as unmapped, record the number of unmapped triangles, and repeat steps 3 and 4. Hydrophilic treatment of wick The capillary force of the wick significantly influences the performance of heat pipes [57] . To efficiently fabricate wicks with intricate three-dimensional shapes, flexible copper wire mesh is used. The capillary pores are created by sintering multi-layered copper wire mesh, and hydrophilic capillary structures on the copper mesh surface are formed via chemical deposition [31] , [58] , with detailed procedures depicted in Supplementary Fig. 3 . The process begins with precise cutting of copper mesh using a pulsed laser. The layered copper mesh is then assembled within a mold to achieve complex three-dimensional forms. To enhance the bonding strength between layers and stabilize the capillary pores, the shaped multi-layer copper wire mesh undergoes a sintering process. Subsequently, the samples are cleaned with ultrasonic cleaning and treated chemically, with the reaction formula presented in Eqs. ( 5 )–( 7 ) [59] , [60] . After hydrophilic treatment, the surface of wick features CuO micro/nano hydrophilic structures (Supplementary Figs. 23 – 24 ), forming the liquid pathway for two-phase circulation of AHP. The contrast in hydrophilicity of the wick before and after the hydrophilic treatment is demonstrated in Supplementary Fig. 25 . $$Cu+2NaOH+N{a}_{2}{S}_{2}{O}_{8}\\\\to Cu{(OH)}_{2}+N{a}_{2}S{O}_{4}$$ (5) $$Cu{(OH)}_{2}+2O{H}^{-}\\\\to {[Cu{(OH)}_{4}]}^{2-}$$ (6) $${[Cu{(OH)}_{4}]}^{2-}\\\\to CuO\\\\downarrow+{H}_{2}O+2O{H}^{-}$$ (7) Characterization The morphology of hydrophilic wick in Supplementary Fig. 23 was captured by Scanning Electron Microscopy (SUPRA55 SAPPHIRE, ZEISS, German), as well as the energy dispersive X-ray spectrometer (EDS) in Supplementary Fig. 24 . The structure of wick was observed by optical microscope (AO-HK830, Aosvi Co. Ltd, China). The thermal imaging of the AHP during startup and stable operation were recorded by an infrared thermal imaging camera (FLIR T440, FLIR Systems Company, USA). Experimental setup The testing setup for evaluating the performance of the AHP includes a DC power supply (WPS3010B, Shenzhen WANPTEK Electronic Technology Co., China), a data acquisition system (34901 A, KEYSIGHT, USA), Type K thermocouples (Omega, USA), and a heating plate (10 cm × 10 cm). The heating plate is mounted 10 mm from the edge on the hot end of the AHP. Thermal conductive silicone grease (6.5 W/m·K) is applied between the heating plate and AHP to minimize contact thermal resistance. Two thermocouples are positioned at each end of the AHP: one at the hot end, 10 mm from the heat pipe’s edge, and the other at the cold end, 15 mm from the edge. Additional thermocouples are evenly placed in the middle, conforming to the customized shape of the AHP, to assess thermal distribution throughout the device. The temperatures recorded by the thermocouples are captured by the data acquisition system and transmitted to a computer. The temperatures measured at the hot and cold ends are used to compute the heat pipe’s thermal resistance [31] , [61] and thermal conductivity [21] , [62] , with the calculation formulas detailed in Supplementary Note 4 . A parametric study on the heat transfer performance of the AHP structure was conducted, with variations in parameters such as thickness, length, and width. Test results and analysis are provided in Supplementary Figs. 26 – 27 . Numerical methodology To streamline the computational model and focus on the impact of the heat pipe’s geometry on the two-phase cycle, the simulation employs a design featuring distinct vapor and liquid fields, encapsulated by a metal film layer that serves as the heat pipe’s shell, as depicted in Supplementary Fig. 28 . Cuboids with an area of 10 × 10 mm² and a thickness of 0.3 mm are used as both the heating plate and condenser. Their centers are located 15 mm from the heat pipe’s ends, positioned near the wall on the wick side (Supplementary Fig. 28b, c ). To maintain a uniform total length for the two-phase circulation across all models, the central line length of the various 3D heat pipes is kept consistent with that of a straight heat pipe. The operation of the two-phase cycle in the heat pipe is simulated using COMSOL Multiphysics 6.2. This simulation employs the Laminar Flow module to analyze vapor flow within the vapor chamber, the Brinkman Equations module for describing liquid flow within the wick, and the Porous Media Heat Transfer module for addressing heat transfer across different sections of the heat pipe. Non-isothermal flow multiphysics coupling integrates the Laminar Flow and Brinkman Equations modules. The effect of gravity may vary and become complex under different deformation conditions. To eliminate the influence of gravity, the gravity option is disabled in both the Laminar Flow and Brinkman Equations modules. The vapor chamber boundary is set to the vapor pressure of water. To maintain equal water mass flux between the vapor chamber and wick field, velocity boundary conditions are set at their interface. $${{{\\\\bf{u}}}}_{2}=\\\\frac{{{{\\\\bf{u}}}}_{1}\\\\cdot {{\\\\rho }}_{2}}{{{\\\\rho }}_{1}}$$ (8) Where u represents the fluid velocity vector, and ρ denotes density. The subscript 1 refers to the vapor chamber domain, and subscript 2 refers to the wick domain. Heat exchange during the evaporation-condensation process is simulated by adding a boundary heat source at the interface between the vapor chamber and the wick. $${Q}_{b}={{{\\\\bf{u}}}}_{n}\\\\cdot {h}_{fg}\\\\cdot {\\\\rho }$$ (9) Where u n represents the normal vector of velocity u , and h fg is the latent heat of vaporization of water. Simulations are conducted on heat pipes of four different shapes: straight, normal bending, radial bending, and twisting, as shown in Supplementary Fig. 28d–g . In the results, probes were placed 10 mm from the cold and hot end boundaries in both the vapor chamber and the wick to calculate the pressure drop for both the gas and liquid phases. Simulations are conducted at different angles (Supplementary Figs. 7 – 9 ) and different radius (Supplementary Figs. 10 – 12 ). A validity test for the numerical simulation of the heat pipe is conducted following methods from reported studies [63] , [64] . Using the modeling setting and simulation condition provided in the referenced report [65] , a numerical simulation is performed on a straight heat pipe with multiple heat sources at low-temperature operation, as depicted in Supplementary Fig. 29 . Four evaporators are positioned at specific locations on one end of the straight heat pipe, with a condenser is placed at a defined coordinate on the other end (Supplementary Fig. 29a ). Different power inputs are applied to each of the four evaporators, and the temperatures along the heat pipe wall are measured and compared. Since explicit cooling conditions for the condenser are not specified in the literature, various parameters are adjusted for comparison, as shown in Supplementary Fig. 29b–e .\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"This research proposes the development and integration of a three-dimensional, shape-adaptive phase-change thermal circulation system called the adaptative heat pipe (AHP), which combines customized reverse engineering with data-driven thermal bonding to enable ultra-high thermal conductivity and full design freedom for efficient thermal management in complex, non-standard electronic devices."},"entities":{"kind":"list like","value":["three-dimensional phase-change thermal circulation system","adaptative heat pipe","customized reverse engineering","data-driven thermal bonding","ultra-high thermal conductivity","full design freedom","complex non-standard electronic devices","thermal management","shape-adaptive system","phase-change material"],"string":"[\n \"three-dimensional phase-change thermal circulation system\",\n \"adaptative heat pipe\",\n \"customized reverse engineering\",\n \"data-driven thermal bonding\",\n \"ultra-high thermal conductivity\",\n \"full design freedom\",\n \"complex non-standard electronic devices\",\n \"thermal management\",\n \"shape-adaptive system\",\n \"phase-change material\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1.","title":"Van Erp, R., Soleimanzadeh, R., Nela, L., Kampitsis, G. & Matioli, E. Co-designing electronics with microfluidics for more sustainable cooling.Nature585, 211–216 (2020).","link":"https://doi.org/10.1038%2Fs41586-020-2666-1"},{"idx":"2.","title":"Lukatskaya, M. R., Dunn, B. & Gogotsi, Y. Multidimensional materials and device architectures for future hybrid energy storage.Nat. Commun.7, 12647 (2016).","link":"https://doi.org/10.1038%2Fncomms12647"},{"idx":"3.","title":"Yang, Z., Yang, T., Cai, W. & Cai, T. Miniaturization of optical spectrometers.Science371, eabe0722 (2021).","link":"https://doi.org/10.1126%2Fscience.abe0722"},{"idx":"4.","title":"Song, H. et al. Two-dimensional materials for thermal management applications.Joule2, 442–463 (2018).","link":"https://doi.org/10.1016%2Fj.joule.2018.01.006"},{"idx":"5.","title":"Wang, C. et al. A thermal management strategy for electronic devices based on moisture sorption-desorption processes.Joule4, 435–447 (2020).","link":"https://doi.org/10.1016%2Fj.joule.2019.12.005"},{"idx":"6.","title":"Li, M. D. et al. Thermal management of chips by a device prototype using synergistic effects of 3D heat-conductive network and electrocaloric refrigeration.Nat. Commun.13, 5849 (2022).","link":"https://doi.org/10.1038%2Fs41467-022-33596-z"},{"idx":"8.","title":"Unsal, O. S. et al. Impact of parameter variations on circuits and microarchitecture.IEEE Micro26, 30–39 (2006).","link":"https://doi.org/10.1109%2FMM.2006.122"},{"idx":"9.","title":"Aftab, W. et al. Phase change material-integrated latent heat storage systems for sustainable energy solutions.Energy Environ. Sci.14, 4268–4291 (2021).","link":"https://doi.org/10.1039%2FD1EE00527H"},{"idx":"10.","title":"Li, W. M., Yang, S. Y., Chen, Y. P., Li, C. & Wang, Z. K. Tesla valves and capillary structures-activated thermal regulator.Nat. Commun.14, 3996 (2023).","link":"https://doi.org/10.1038%2Fs41467-023-39289-5"},{"idx":"11.","title":"Yoo, S. W. et al. Responsive materials and mechanisms as thermal safety systems for skin-interfaced electronic devices.Nat. Commun.14, 1024 (2023).","link":"https://doi.org/10.1038%2Fs41467-023-36690-y"},{"idx":"12.","title":"Wei, X. H., Liu, L., Zhang, J. X., Shi, J. L. & Guo, Q. G. Mechanical, electrical, thermal performances and structure characteristics of flexible graphite sheets.J. Mater. Sci.45, 2449–2455 (2010).","link":"https://link.springer.com/doi/10.1007/s10853-010-4216-y"},{"idx":"13.","title":"Vu, M. C. et al. Scalable ultrarobust thermoconductive nonflammable bioinspired papers of graphene nanoplatelet crosslinked aramid nanofibers for thermal management and electromagnetic Shielding.J. Mater. Chem.9, 8527 (2021).","link":"https://doi.org/10.1039%2FD0TA12306D"},{"idx":"14.","title":"Bartlett, M. D. et al. High thermal conductivity in soft elastomers with elongated liquid metal inclusions.Proc. Natl Acad. Sci. USA28, 2143–2148 (2017).","link":"https://doi.org/10.1073%2Fpnas.1616377114"},{"idx":"15.","title":"Ren, N. et al. Shear-induced fabrication of cellulose nanofibril/liquid metal nanocomposite films for flexible electromagnetic interference shielding and thermal management.ACS Appl. Mater. Interfaces16, 17904–17917 (2024).","link":"https://doi.org/10.1021%2Facsami.4c01220"},{"idx":"16.","title":"Shen, B., Zhai, W. & Zheng, W. Ultrathin flexible graphene film: an excellent thermal conducting material with efficient EMI shielding.Adv. Funct. Mater.24, 4542–4548 (2014).","link":"https://doi.org/10.1002%2Fadfm.201400079"},{"idx":"17.","title":"Liang, G. et al. Phase change material filled hybrid 2D/3D graphene structure with ultra-high thermal effusivity for effective thermal managemen.Carbon176, 11–20 (2021).","link":"https://doi.org/10.1016%2Fj.carbon.2020.12.046"},{"idx":"18.","title":"Kong, Q. Q. et al. Hierarchical graphene–carbon fiber composite paper as a flexible lateral heat spreader.Adv. Funct. Mater.24, 4222–4228 (2014).","link":"https://doi.org/10.1002%2Fadfm.201304144"},{"idx":"19.","title":"Xin, G. et al. Highly thermally conductive and mechanically strong graphene fibers.Science349, 1083–1087 (2015).","link":"https://doi.org/10.1126%2Fscience.aaa6502"},{"idx":"20.","title":"Jaipurkar, T., Kant, P., Khandekar, S., Bhattacharya, B. & Paralikar, S. Thermo-mechanical design and characterization of flexible heat pipes.Appl Therm. Eng.126, 1199–1208 (2017).","link":"https://doi.org/10.1016%2Fj.applthermaleng.2017.01.036"},{"idx":"21.","title":"Yang, C. et al. Fabrication and performance evaluation of flexible heat pipes for potential thermal control of foldable electronics.Appl Therm. Eng.96, 445–453 (2016).","link":"https://doi.org/10.1016%2Fj.applthermaleng.2015.11.078"},{"idx":"22.","title":"Gibbons, M. J., Marengo, M. & Marengo, T. A review of heat pipe technology for foldable electronic devices.Appl. Therm. Eng.194, 117087 (2021).","link":"https://doi.org/10.1016%2Fj.applthermaleng.2021.117087"},{"idx":"23.","title":"Xin, F., Ma, T. & Wang, Q. Thermal performance analysis of flat heat pipe with graded mini-grooves wick.Appl. Energy228, 2129–2139 (2018).","link":"https://doi.org/10.1016%2Fj.apenergy.2018.07.053"},{"idx":"24.","title":"Tang, H. et al. Review of applications and developments of ultra-thin micro heat pipes for electronic cooling.Appl. Energy223, 383–400 (2018).","link":"https://doi.org/10.1016%2Fj.apenergy.2018.04.072"},{"idx":"25.","title":"Luo, J. L., Mo, D. C., Wang, Y. Q. & Lyu, S. S. Biomimetic copper forest wick enables high thermal conductivity ultrathin heat pipe.ACS Nano15, 6614–6621 (2021).","link":"https://doi.org/10.1021%2Facsnano.0c09961"},{"idx":"27.","title":"Lee, D. & Byon, C. Fabrication and characterization of pure-metal-based submillimeter-thick flexible flat heat pipe with innovative wick structures.Int J. Heat. Mass Transf.122, 306–314 (2018).","link":"https://doi.org/10.1016%2Fj.ijheatmasstransfer.2018.01.135"},{"idx":"28.","title":"Liu, C., Li, Q. & Fan, D. Fabrication and performance evaluation of flexible flat heat pipes for the thermal control of deployable structure.Int. J. Heat. Mass Transf.144, 118661 (2019).","link":"https://doi.org/10.1016%2Fj.ijheatmasstransfer.2019.118661"},{"idx":"29.","title":"Lim, J. & Kim, J. Fabrication and experimental evaluation of a polymer-based flexible pulsating heat pipe.Energ. Convers. Manag.126, 358–364 (2018).","link":"https://doi.org/10.1016%2Fj.enconman.2017.11.022"},{"idx":"30.","title":"Jung, C., Lim, J. & Kim, S. J. Fabrication and evaluation of a high-performance flexible pulsating heat pipe hermetically sealed with metal.Int J. Heat. Mass Transf.149, 119180 (2020).","link":"https://doi.org/10.1016%2Fj.ijheatmasstransfer.2019.119180"},{"idx":"31.","title":"Huang, J. et al. Development of novel flexible heat pipe with multistage design inspired by structure of human spine.Appl. Therm. Eng.175, 115392 (2020).","link":"https://doi.org/10.1016%2Fj.applthermaleng.2020.115392"},{"idx":"32.","title":"Kang, Z., Jiang, S., Hong, Y. & Fan, J. Squid‐like soft heat pipe for multiple heat transport.Droplet1, 182–191 (2022).","link":"https://doi.org/10.1002%2Fdro2.25"},{"idx":"33.","title":"Kang, Z., Hong, Y., Jiang, S. & Fan, J. Composite filament with super high effective thermal conductivity.Mater. Today Phys.34, 101067 (2023).","link":"https://doi.org/10.1016%2Fj.mtphys.2023.101067"},{"idx":"35.","title":"Ma, Y. G. et al. Experimental study on sodium screen-wick heat pipe capillary limit.Appl. Therm. Eng.227, 120397 (2023).","link":"https://doi.org/10.1016%2Fj.applthermaleng.2023.120397"},{"idx":"36.","title":"Kim, B. H. & Kim, G. P. A microscale model for thin-film evaporation in capillary wick structures.Int. J. Heat. Mass Transf.38, 1427–1442 (1995).","link":"https://doi.org/10.1016%2F0017-9310%2894%2900249-U"},{"idx":"38.","title":"Ranjan, R., Tien, J. Y. & Tien, S. V. Analysis of the critical weber number at the onset of liquid entrainment in capillary-driven heat pipes.Int. J. Heat. Mass Transf.54, 169–179 (2011).","link":"https://doi.org/10.1016%2Fj.ijheatmasstransfer.2010.09.037"},{"idx":"39.","title":"Ranjan, R., Murthy, J. Y. & Garimella, S. V. Analysis of the wicking and thin-film evaporation characteristics of wick microstructures.ASME.J. Heat. Transf.131, 101001 (2009).","link":"https://doi.org/10.1115%2F1.3160538"},{"idx":"40.","title":"Tesfaw, S., Bogale, T. M. & Fatoba, O. Evaluation of tensile and flexural strength properties of virgin and recycled high-density polyethylene (HDPE) for pipe fitting application.Mater. Today Proc.62, 3103–3113 (2022).","link":"https://doi.org/10.1016%2Fj.matpr.2022.03.385"},{"idx":"41.","title":"Cofaru, N. F., Pascu, A., Oleksik, M. & Petruse, R. Tensile properties of 3D-printed continuous-fiber-reinforced plastics.Mater. Plast.58, 271–282 (2021).","link":"https://doi.org/10.37358%2FMP.21.4.5552"},{"idx":"42.","title":"Tuohedi, N. & Wang, Q. Preparation and evaluation of epoxy resin prepared from the liquefied product of cotton stalk.Processes9, 1417 (2021).","link":"https://doi.org/10.3390%2Fpr9081417"},{"idx":"43.","title":"Li, X., Yi, W., Chi, H. L., Wang, X. & Wang, A. P. C. A critical review of virtual and augmented reality (VR/AR) applications in construction safety.Autom. Constr.86, 150–162 (2018).","link":"https://doi.org/10.1016%2Fj.autcon.2017.11.003"},{"idx":"44.","title":"Pot-Kolder, R. M. C. A. et al. Virtual-reality-based cognitive behavioural therapy versus waiting list control for paranoid ideation and social avoidance in patients with psychotic disorders: a single-blind randomised controlled trial.Lancet Psychiatry5, 217–226 (2018).","link":"https://doi.org/10.1016%2FS2215-0366%2818%2930053-1"},{"idx":"45.","title":"Cotter, J. D. & Taylor, N. A. S. The distribution of cutaneous sudomotor and alliesthesial thermosensitivity in mildly heat‐stressed humans: an open‐loop approach.J. Physiol.565, 335–345 (2005).","link":"https://doi.org/10.1113%2Fjphysiol.2004.081562"},{"idx":"46.","title":"Buyan, M. P. et al. Facial warming and tinted helmet visors.Int. J. Ind. Ergon.36, 11–16 (2006).","link":"https://doi.org/10.1016%2Fj.ergon.2005.06.005"},{"idx":"49.","title":"Moon, J. Y. et al. Layer-engineered large-area exfoliation of graphene.Sci. Adv.6, eabc6601 (2020).","link":"https://doi.org/10.1126%2Fsciadv.abc6601"},{"idx":"50.","title":"Fu, Y. F. et al. Graphene related materials for thermal management.2D Mater.7, 012001 (2020).","link":"https://doi.org/10.1088%2F2053-1583%2Fab48d9"},{"idx":"51.","title":"Li, H. L., Xiao, S. N., Yu, H. L., Xue, Y. H. & Yang, J. H. A review of graphene-based films for heat dissipation.N. Carbon Mater.36, 897–910 (2021).","link":"https://doi.org/10.1016%2FS1872-5805%2821%2960092-6"},{"idx":"52.","title":"Wang, G. et al. Recent advances in the mechanics of 2D materials.Int. J. Extrem. Manuf.5, 032002 (2023).","link":"https://doi.org/10.1088%2F2631-7990%2Faccda2"},{"idx":"53.","title":"Tang, J., Tang, D., Ren, F. & Li, H. Design and experimental study of a VCM-based whole-spacecraft vibration isolation system.J. Aerosp. Eng.31, 04018045 (2018).","link":"https://doi.org/10.1061%2F%28ASCE%29AS.1943-5525.0000871"},{"idx":"54.","title":"Tang, Y. et al. Noise and vibration suppression in hybrid electric vehicles: State of the art and challenges.Renew. Sustain. Energy Rev.124, 109782 (2020).","link":"https://doi.org/10.1016%2Fj.rser.2020.109782"},{"idx":"55.","title":"Cao, Z. et al. Inversely engineered biomimetic flexible network scaffolds for soft tissue regeneration.Sci. Adv.9, adi8606 (2023).","link":"https://doi.org/10.1126%2Fsciadv.adi8606"},{"idx":"56.","title":"Meng, Q., Huang, Y., Li, L., Wu, F. & Chen, R. Smart batteries for powering the future.Joule8, 344–373 (2024).","link":"https://doi.org/10.1016%2Fj.joule.2024.01.011"},{"idx":"57.","title":"Huang, X. & Franchi, G. Design and fabrication of hybrid bi-modal wick structure for heat pipe application.J. Porous Mater.15, 635–642 (2008).","link":"https://link.springer.com/doi/10.1007/s10934-007-9143-1"},{"idx":"58.","title":"Tang, Y. et al. Experimental investigation of capillary force in a novel sintered copper mesh wick for ultra-thin heat pipes.Appl. Therm. Eng.115, 1020–1030 (2017).","link":"https://doi.org/10.1016%2Fj.applthermaleng.2016.12.056"},{"idx":"59.","title":"Chen, X. et al. Synthesis and characterization of superhydrophobic functionalized Cu(OH)2 nanotube arrays on copper foil.Appl. Surf. Sci.255, 4015–4019 (2009).","link":"https://doi.org/10.1016%2Fj.apsusc.2008.10.104"},{"idx":"60.","title":"Zhang, Y., Yu, X., Zhou, Q., Chen, F. & Li, K. Fabrication of superhydrophobic copper surface with ultra-low water roll angle.Appl. Surf. Sci.256, 1883–1887 (2010).","link":"https://doi.org/10.1016%2Fj.apsusc.2009.10.024"},{"idx":"61.","title":"Min, C., Gao, X., Gao, F. & Wang, K. Thermal performance analyses of pulsating heat pipe for application in proton exchange member fuel cell.Energy Convers. Manag.259, 115566 (2022).","link":"https://doi.org/10.1016%2Fj.enconman.2022.115566"},{"idx":"62.","title":"Mehrali, M. et al. Effect of nitrogen-doped graphene nanofluid on the thermal performance of the grooved copper heat pipe.Energy Convers. Manag.118, 459–473 (2016).","link":"https://doi.org/10.1016%2Fj.enconman.2016.04.028"},{"idx":"64.","title":"Rice, J. & Faghri, A. Analysis of screen wick heat pipes, including capillary dry-out limitations.J. Thermophys. Heat. Transf.21, 475–486 (2007).","link":"https://doi.org/10.2514%2F1.24809"},{"idx":"65.","title":"Naphon, P. & Wongwises, S. A review of flow and heat transfer characteristics in curved tubes.Renew. Sust. Energy Rev.10, 463–490 (2006).","link":"https://doi.org/10.1016%2Fj.rser.2004.09.014"}],"string":"[\n {\n \"idx\": \"1.\",\n \"title\": \"Van Erp, R., Soleimanzadeh, R., Nela, L., Kampitsis, G. & Matioli, E. Co-designing electronics with microfluidics for more sustainable cooling.Nature585, 211–216 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-020-2666-1\"\n },\n {\n \"idx\": \"2.\",\n \"title\": \"Lukatskaya, M. R., Dunn, B. & Gogotsi, Y. Multidimensional materials and device architectures for future hybrid energy storage.Nat. Commun.7, 12647 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms12647\"\n },\n {\n \"idx\": \"3.\",\n \"title\": \"Yang, Z., Yang, T., Cai, W. & Cai, T. Miniaturization of optical spectrometers.Science371, eabe0722 (2021).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.abe0722\"\n },\n {\n \"idx\": \"4.\",\n \"title\": \"Song, H. et al. Two-dimensional materials for thermal management applications.Joule2, 442–463 (2018).\",\n \"link\": \"https://doi.org/10.1016%2Fj.joule.2018.01.006\"\n },\n {\n \"idx\": \"5.\",\n \"title\": \"Wang, C. et al. A thermal management strategy for electronic devices based on moisture sorption-desorption processes.Joule4, 435–447 (2020).\",\n \"link\": \"https://doi.org/10.1016%2Fj.joule.2019.12.005\"\n },\n {\n \"idx\": \"6.\",\n \"title\": \"Li, M. D. et al. Thermal management of chips by a device prototype using synergistic effects of 3D heat-conductive network and electrocaloric refrigeration.Nat. Commun.13, 5849 (2022).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-022-33596-z\"\n },\n {\n \"idx\": \"8.\",\n \"title\": \"Unsal, O. S. et al. Impact of parameter variations on circuits and microarchitecture.IEEE Micro26, 30–39 (2006).\",\n \"link\": \"https://doi.org/10.1109%2FMM.2006.122\"\n },\n {\n \"idx\": \"9.\",\n \"title\": \"Aftab, W. et al. Phase change material-integrated latent heat storage systems for sustainable energy solutions.Energy Environ. Sci.14, 4268–4291 (2021).\",\n \"link\": \"https://doi.org/10.1039%2FD1EE00527H\"\n },\n {\n \"idx\": \"10.\",\n \"title\": \"Li, W. M., Yang, S. Y., Chen, Y. P., Li, C. & Wang, Z. K. Tesla valves and capillary structures-activated thermal regulator.Nat. Commun.14, 3996 (2023).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-023-39289-5\"\n },\n {\n \"idx\": \"11.\",\n \"title\": \"Yoo, S. W. et al. Responsive materials and mechanisms as thermal safety systems for skin-interfaced electronic devices.Nat. Commun.14, 1024 (2023).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-023-36690-y\"\n },\n {\n \"idx\": \"12.\",\n \"title\": \"Wei, X. H., Liu, L., Zhang, J. X., Shi, J. L. & Guo, Q. G. Mechanical, electrical, thermal performances and structure characteristics of flexible graphite sheets.J. Mater. Sci.45, 2449–2455 (2010).\",\n \"link\": \"https://link.springer.com/doi/10.1007/s10853-010-4216-y\"\n },\n {\n \"idx\": \"13.\",\n \"title\": \"Vu, M. C. et al. Scalable ultrarobust thermoconductive nonflammable bioinspired papers of graphene nanoplatelet crosslinked aramid nanofibers for thermal management and electromagnetic Shielding.J. Mater. Chem.9, 8527 (2021).\",\n \"link\": \"https://doi.org/10.1039%2FD0TA12306D\"\n },\n {\n \"idx\": \"14.\",\n \"title\": \"Bartlett, M. D. et al. High thermal conductivity in soft elastomers with elongated liquid metal inclusions.Proc. Natl Acad. Sci. USA28, 2143–2148 (2017).\",\n \"link\": \"https://doi.org/10.1073%2Fpnas.1616377114\"\n },\n {\n \"idx\": \"15.\",\n \"title\": \"Ren, N. et al. Shear-induced fabrication of cellulose nanofibril/liquid metal nanocomposite films for flexible electromagnetic interference shielding and thermal management.ACS Appl. Mater. Interfaces16, 17904–17917 (2024).\",\n \"link\": \"https://doi.org/10.1021%2Facsami.4c01220\"\n },\n {\n \"idx\": \"16.\",\n \"title\": \"Shen, B., Zhai, W. & Zheng, W. Ultrathin flexible graphene film: an excellent thermal conducting material with efficient EMI shielding.Adv. Funct. Mater.24, 4542–4548 (2014).\",\n \"link\": \"https://doi.org/10.1002%2Fadfm.201400079\"\n },\n {\n \"idx\": \"17.\",\n \"title\": \"Liang, G. et al. Phase change material filled hybrid 2D/3D graphene structure with ultra-high thermal effusivity for effective thermal managemen.Carbon176, 11–20 (2021).\",\n \"link\": \"https://doi.org/10.1016%2Fj.carbon.2020.12.046\"\n },\n {\n \"idx\": \"18.\",\n \"title\": \"Kong, Q. Q. et al. Hierarchical graphene–carbon fiber composite paper as a flexible lateral heat spreader.Adv. Funct. Mater.24, 4222–4228 (2014).\",\n \"link\": \"https://doi.org/10.1002%2Fadfm.201304144\"\n },\n {\n \"idx\": \"19.\",\n \"title\": \"Xin, G. et al. Highly thermally conductive and mechanically strong graphene fibers.Science349, 1083–1087 (2015).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.aaa6502\"\n },\n {\n \"idx\": \"20.\",\n \"title\": \"Jaipurkar, T., Kant, P., Khandekar, S., Bhattacharya, B. & Paralikar, S. Thermo-mechanical design and characterization of flexible heat pipes.Appl Therm. Eng.126, 1199–1208 (2017).\",\n \"link\": \"https://doi.org/10.1016%2Fj.applthermaleng.2017.01.036\"\n },\n {\n \"idx\": \"21.\",\n \"title\": \"Yang, C. et al. Fabrication and performance evaluation of flexible heat pipes for potential thermal control of foldable electronics.Appl Therm. Eng.96, 445–453 (2016).\",\n \"link\": \"https://doi.org/10.1016%2Fj.applthermaleng.2015.11.078\"\n },\n {\n \"idx\": \"22.\",\n \"title\": \"Gibbons, M. J., Marengo, M. & Marengo, T. A review of heat pipe technology for foldable electronic devices.Appl. Therm. Eng.194, 117087 (2021).\",\n \"link\": \"https://doi.org/10.1016%2Fj.applthermaleng.2021.117087\"\n },\n {\n \"idx\": \"23.\",\n \"title\": \"Xin, F., Ma, T. & Wang, Q. Thermal performance analysis of flat heat pipe with graded mini-grooves wick.Appl. Energy228, 2129–2139 (2018).\",\n \"link\": \"https://doi.org/10.1016%2Fj.apenergy.2018.07.053\"\n },\n {\n \"idx\": \"24.\",\n \"title\": \"Tang, H. et al. Review of applications and developments of ultra-thin micro heat pipes for electronic cooling.Appl. Energy223, 383–400 (2018).\",\n \"link\": \"https://doi.org/10.1016%2Fj.apenergy.2018.04.072\"\n },\n {\n \"idx\": \"25.\",\n \"title\": \"Luo, J. L., Mo, D. C., Wang, Y. Q. & Lyu, S. S. Biomimetic copper forest wick enables high thermal conductivity ultrathin heat pipe.ACS Nano15, 6614–6621 (2021).\",\n \"link\": \"https://doi.org/10.1021%2Facsnano.0c09961\"\n },\n {\n \"idx\": \"27.\",\n \"title\": \"Lee, D. & Byon, C. Fabrication and characterization of pure-metal-based submillimeter-thick flexible flat heat pipe with innovative wick structures.Int J. Heat. Mass Transf.122, 306–314 (2018).\",\n \"link\": \"https://doi.org/10.1016%2Fj.ijheatmasstransfer.2018.01.135\"\n },\n {\n \"idx\": \"28.\",\n \"title\": \"Liu, C., Li, Q. & Fan, D. Fabrication and performance evaluation of flexible flat heat pipes for the thermal control of deployable structure.Int. J. Heat. Mass Transf.144, 118661 (2019).\",\n \"link\": \"https://doi.org/10.1016%2Fj.ijheatmasstransfer.2019.118661\"\n },\n {\n \"idx\": \"29.\",\n \"title\": \"Lim, J. & Kim, J. Fabrication and experimental evaluation of a polymer-based flexible pulsating heat pipe.Energ. Convers. Manag.126, 358–364 (2018).\",\n \"link\": \"https://doi.org/10.1016%2Fj.enconman.2017.11.022\"\n },\n {\n \"idx\": \"30.\",\n \"title\": \"Jung, C., Lim, J. & Kim, S. J. Fabrication and evaluation of a high-performance flexible pulsating heat pipe hermetically sealed with metal.Int J. Heat. Mass Transf.149, 119180 (2020).\",\n \"link\": \"https://doi.org/10.1016%2Fj.ijheatmasstransfer.2019.119180\"\n },\n {\n \"idx\": \"31.\",\n \"title\": \"Huang, J. et al. Development of novel flexible heat pipe with multistage design inspired by structure of human spine.Appl. Therm. Eng.175, 115392 (2020).\",\n \"link\": \"https://doi.org/10.1016%2Fj.applthermaleng.2020.115392\"\n },\n {\n \"idx\": \"32.\",\n \"title\": \"Kang, Z., Jiang, S., Hong, Y. & Fan, J. Squid‐like soft heat pipe for multiple heat transport.Droplet1, 182–191 (2022).\",\n \"link\": \"https://doi.org/10.1002%2Fdro2.25\"\n },\n {\n \"idx\": \"33.\",\n \"title\": \"Kang, Z., Hong, Y., Jiang, S. & Fan, J. Composite filament with super high effective thermal conductivity.Mater. Today Phys.34, 101067 (2023).\",\n \"link\": \"https://doi.org/10.1016%2Fj.mtphys.2023.101067\"\n },\n {\n \"idx\": \"35.\",\n \"title\": \"Ma, Y. G. et al. Experimental study on sodium screen-wick heat pipe capillary limit.Appl. Therm. Eng.227, 120397 (2023).\",\n \"link\": \"https://doi.org/10.1016%2Fj.applthermaleng.2023.120397\"\n },\n {\n \"idx\": \"36.\",\n \"title\": \"Kim, B. H. & Kim, G. P. A microscale model for thin-film evaporation in capillary wick structures.Int. J. Heat. Mass Transf.38, 1427–1442 (1995).\",\n \"link\": \"https://doi.org/10.1016%2F0017-9310%2894%2900249-U\"\n },\n {\n \"idx\": \"38.\",\n \"title\": \"Ranjan, R., Tien, J. Y. & Tien, S. V. Analysis of the critical weber number at the onset of liquid entrainment in capillary-driven heat pipes.Int. J. Heat. Mass Transf.54, 169–179 (2011).\",\n \"link\": \"https://doi.org/10.1016%2Fj.ijheatmasstransfer.2010.09.037\"\n },\n {\n \"idx\": \"39.\",\n \"title\": \"Ranjan, R., Murthy, J. Y. & Garimella, S. V. Analysis of the wicking and thin-film evaporation characteristics of wick microstructures.ASME.J. Heat. Transf.131, 101001 (2009).\",\n \"link\": \"https://doi.org/10.1115%2F1.3160538\"\n },\n {\n \"idx\": \"40.\",\n \"title\": \"Tesfaw, S., Bogale, T. M. & Fatoba, O. Evaluation of tensile and flexural strength properties of virgin and recycled high-density polyethylene (HDPE) for pipe fitting application.Mater. Today Proc.62, 3103–3113 (2022).\",\n \"link\": \"https://doi.org/10.1016%2Fj.matpr.2022.03.385\"\n },\n {\n \"idx\": \"41.\",\n \"title\": \"Cofaru, N. F., Pascu, A., Oleksik, M. & Petruse, R. Tensile properties of 3D-printed continuous-fiber-reinforced plastics.Mater. Plast.58, 271–282 (2021).\",\n \"link\": \"https://doi.org/10.37358%2FMP.21.4.5552\"\n },\n {\n \"idx\": \"42.\",\n \"title\": \"Tuohedi, N. & Wang, Q. Preparation and evaluation of epoxy resin prepared from the liquefied product of cotton stalk.Processes9, 1417 (2021).\",\n \"link\": \"https://doi.org/10.3390%2Fpr9081417\"\n },\n {\n \"idx\": \"43.\",\n \"title\": \"Li, X., Yi, W., Chi, H. L., Wang, X. & Wang, A. P. C. A critical review of virtual and augmented reality (VR/AR) applications in construction safety.Autom. Constr.86, 150–162 (2018).\",\n \"link\": \"https://doi.org/10.1016%2Fj.autcon.2017.11.003\"\n },\n {\n \"idx\": \"44.\",\n \"title\": \"Pot-Kolder, R. M. C. A. et al. Virtual-reality-based cognitive behavioural therapy versus waiting list control for paranoid ideation and social avoidance in patients with psychotic disorders: a single-blind randomised controlled trial.Lancet Psychiatry5, 217–226 (2018).\",\n \"link\": \"https://doi.org/10.1016%2FS2215-0366%2818%2930053-1\"\n },\n {\n \"idx\": \"45.\",\n \"title\": \"Cotter, J. D. & Taylor, N. A. S. The distribution of cutaneous sudomotor and alliesthesial thermosensitivity in mildly heat‐stressed humans: an open‐loop approach.J. Physiol.565, 335–345 (2005).\",\n \"link\": \"https://doi.org/10.1113%2Fjphysiol.2004.081562\"\n },\n {\n \"idx\": \"46.\",\n \"title\": \"Buyan, M. P. et al. Facial warming and tinted helmet visors.Int. J. Ind. Ergon.36, 11–16 (2006).\",\n \"link\": \"https://doi.org/10.1016%2Fj.ergon.2005.06.005\"\n },\n {\n \"idx\": \"49.\",\n \"title\": \"Moon, J. Y. et al. Layer-engineered large-area exfoliation of graphene.Sci. Adv.6, eabc6601 (2020).\",\n \"link\": \"https://doi.org/10.1126%2Fsciadv.abc6601\"\n },\n {\n \"idx\": \"50.\",\n \"title\": \"Fu, Y. F. et al. Graphene related materials for thermal management.2D Mater.7, 012001 (2020).\",\n \"link\": \"https://doi.org/10.1088%2F2053-1583%2Fab48d9\"\n },\n {\n \"idx\": \"51.\",\n \"title\": \"Li, H. L., Xiao, S. N., Yu, H. L., Xue, Y. H. & Yang, J. H. A review of graphene-based films for heat dissipation.N. Carbon Mater.36, 897–910 (2021).\",\n \"link\": \"https://doi.org/10.1016%2FS1872-5805%2821%2960092-6\"\n },\n {\n \"idx\": \"52.\",\n \"title\": \"Wang, G. et al. Recent advances in the mechanics of 2D materials.Int. J. Extrem. Manuf.5, 032002 (2023).\",\n \"link\": \"https://doi.org/10.1088%2F2631-7990%2Faccda2\"\n },\n {\n \"idx\": \"53.\",\n \"title\": \"Tang, J., Tang, D., Ren, F. & Li, H. Design and experimental study of a VCM-based whole-spacecraft vibration isolation system.J. Aerosp. Eng.31, 04018045 (2018).\",\n \"link\": \"https://doi.org/10.1061%2F%28ASCE%29AS.1943-5525.0000871\"\n },\n {\n \"idx\": \"54.\",\n \"title\": \"Tang, Y. et al. Noise and vibration suppression in hybrid electric vehicles: State of the art and challenges.Renew. Sustain. Energy Rev.124, 109782 (2020).\",\n \"link\": \"https://doi.org/10.1016%2Fj.rser.2020.109782\"\n },\n {\n \"idx\": \"55.\",\n \"title\": \"Cao, Z. et al. Inversely engineered biomimetic flexible network scaffolds for soft tissue regeneration.Sci. Adv.9, adi8606 (2023).\",\n \"link\": \"https://doi.org/10.1126%2Fsciadv.adi8606\"\n },\n {\n \"idx\": \"56.\",\n \"title\": \"Meng, Q., Huang, Y., Li, L., Wu, F. & Chen, R. Smart batteries for powering the future.Joule8, 344–373 (2024).\",\n \"link\": \"https://doi.org/10.1016%2Fj.joule.2024.01.011\"\n },\n {\n \"idx\": \"57.\",\n \"title\": \"Huang, X. & Franchi, G. Design and fabrication of hybrid bi-modal wick structure for heat pipe application.J. Porous Mater.15, 635–642 (2008).\",\n \"link\": \"https://link.springer.com/doi/10.1007/s10934-007-9143-1\"\n },\n {\n \"idx\": \"58.\",\n \"title\": \"Tang, Y. et al. Experimental investigation of capillary force in a novel sintered copper mesh wick for ultra-thin heat pipes.Appl. Therm. Eng.115, 1020–1030 (2017).\",\n \"link\": \"https://doi.org/10.1016%2Fj.applthermaleng.2016.12.056\"\n },\n {\n \"idx\": \"59.\",\n \"title\": \"Chen, X. et al. Synthesis and characterization of superhydrophobic functionalized Cu(OH)2 nanotube arrays on copper foil.Appl. Surf. Sci.255, 4015–4019 (2009).\",\n \"link\": \"https://doi.org/10.1016%2Fj.apsusc.2008.10.104\"\n },\n {\n \"idx\": \"60.\",\n \"title\": \"Zhang, Y., Yu, X., Zhou, Q., Chen, F. & Li, K. Fabrication of superhydrophobic copper surface with ultra-low water roll angle.Appl. Surf. Sci.256, 1883–1887 (2010).\",\n \"link\": \"https://doi.org/10.1016%2Fj.apsusc.2009.10.024\"\n },\n {\n \"idx\": \"61.\",\n \"title\": \"Min, C., Gao, X., Gao, F. & Wang, K. Thermal performance analyses of pulsating heat pipe for application in proton exchange member fuel cell.Energy Convers. Manag.259, 115566 (2022).\",\n \"link\": \"https://doi.org/10.1016%2Fj.enconman.2022.115566\"\n },\n {\n \"idx\": \"62.\",\n \"title\": \"Mehrali, M. et al. Effect of nitrogen-doped graphene nanofluid on the thermal performance of the grooved copper heat pipe.Energy Convers. Manag.118, 459–473 (2016).\",\n \"link\": \"https://doi.org/10.1016%2Fj.enconman.2016.04.028\"\n },\n {\n \"idx\": \"64.\",\n \"title\": \"Rice, J. & Faghri, A. Analysis of screen wick heat pipes, including capillary dry-out limitations.J. Thermophys. Heat. Transf.21, 475–486 (2007).\",\n \"link\": \"https://doi.org/10.2514%2F1.24809\"\n },\n {\n \"idx\": \"65.\",\n \"title\": \"Naphon, P. & Wongwises, S. A review of flow and heat transfer characteristics in curved tubes.Renew. Sust. Energy Rev.10, 463–490 (2006).\",\n \"link\": \"https://doi.org/10.1016%2Fj.rser.2004.09.014\"\n }\n]","isConversational":false}}},{"rowIdx":96,"cells":{"unique_id":{"kind":"string","value":"ncomms4860"},"subfield":{"kind":"string","value":"Materials science"},"title":{"kind":"string","value":"Electron spin resonance and spin–valley physics in a silicon double quantum dot"},"abstract":{"kind":"string","value":"Silicon quantum dots are a leading approach for solid-state quantum bits. However, developing this technology is complicated by the multi-valley nature of silicon. Here we observe transport of individual electrons in a silicon CMOS-based double quantum dot under electron spin resonance. An anticrossing of the driven dot energy levels is observed when the Zeeman and valley splittings coincide. A detected anticrossing splitting of 60 MHz is interpreted as a direct measure of spin and valley mixing, facilitated by spin–orbit interaction in the presence of non-ideal interfaces. A lower bound of spin dephasing time of 63 ns is extracted. We also describe a possible experimental evidence of an unconventional spin–valley blockade, despite the assumption of non-ideal interfaces. This understanding of silicon spin–valley physics should enable better control and read-out techniques for the spin qubits in an all CMOS silicon approach."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"It has long been speculated that qubits based on individual electron spins in Si quantum dots (QDs) have considerable potential for quantum information processing. Attractive features are the extremely long coherence time of spins in Si bulk materials and the possibility to approach zero hyperfine interaction to nuclear spins in isotopically purified structures. Furthermore, the extensive collection of Complementary Metal-Oxide-Semiconductor (CMOS)-based techniques, accumulated over decades, is expected to be very important for fabricating many qubits. Electric and magnetic fields along with charge detection enable qubit gates and read-out. A long coherence time was recently confirmed for the singlet and triplet states in a Si/SiGe double quantum dot (DQD) qubit [1] . Electron spin resonance (ESR) is a direct means to drive rotations of a spin qubit. For electron spins bound in Si, an ensemble of spins in various structured materials [2] , single electrons in a single defect [3] and in a single donor [4] have been explored with ESR, using various detection schemes. Physical implementations of ESR on individual bound electronic spins have proven to be successful in GaAs-based QD transport experiments [5] , [6] , where the essential role of the spin (Pauli) blockade and the nuclear spin bath in that systems were established. However, spin detection via electronic transport in gate-defined Si QDs has remained challenging. Here we report the detection of microwave-driven ESR transport of individual electrons in a silicon metal-oxide-semiconductor (MOS)-based DQD. The lifting of the blockade via ESR is detectable only in a narrow region where Zeeman split spin states of different valley content anticross where the Zeeman splitting E Z equals the valley splitting of E V ≃ 86.2 μ eV. We show that the anticrossing is due to spin–orbit coupling (SOC) in the heterostructure, in the presence of interface roughness, that mixes spin and valley states (similar mixing mechanism was first established in the electron spin relaxation in a small single Si QD [7] ). The gap at anticrossing of Δ f anticross ≃ 60 MHz is a measure of this spin–valley mixing and also provides a means to access higher valley states via ESR. Analysis of the ESR spectrum provides a lower-bound estimation of an inhomogeneous decoherence time, , which is much longer than that in GaAs [8] ; it also compares well with the direct measurement of for a Si qubit encoded by singlet–triplet states [1] . The nature of the experimental blockade regime, and its dependence on the applied magnetic field and interdot energy detuning are discussed. Since the observed blockade region includes detunings larger than the valley splitting, we can conclude that a spin–valley blockade takes place, related to the impossibility for an inelastic (via phonons) electron tunnelling to happen. This blockade survives even in the presence of a non-ideal interface. The observations made in this paper encourage further development of Si MOS-based spin qubits and further suggest that the additional valley degree of freedom [1] , [7] , [9] , [10] , [11] , [12] is critical to understanding silicon qubits.","references":{"indices":[1,2,3,4,5,6,7,8,9,10,11,12],"infos":[{"idx":"1","title":"Maune, B. M. et al. Coherent singlet-triplet oscillations in a silicon-based double quantum dot.Nature481, 344–347 (2012).","link":"https://doi.org/10.1038%2Fnature10707"},{"idx":"2","title":"Morton, J. J. L., McCamey, D. R., Eriksson, M. A. & Lyon, S. A. Embracing the quantum limit in silicon computing.Nature479, 345–353 (2011).","link":"https://doi.org/10.1038%2Fnature10681"},{"idx":"3","title":"Xiao, M., Martin, I., Yablonovitch, E. & Jiang, H. W. Electrical detection of the spin resonance of a single electron in a silicon field-effect transistor.Nature430, 435–439 (2004).","link":"https://doi.org/10.1038%2Fnature02727"},{"idx":"4","title":"Pla, J. J. et al. A single-atom electron spin qubit in silicon.Nature489, 541–545 (2012).","link":"https://doi.org/10.1038%2Fnature11449"},{"idx":"5","title":"Koppens, F. H. L. et al. Driven coherent oscillations of a single electron spin in a quantum dot.Nature442, 766–771 (2006).","link":"https://doi.org/10.1038%2Fnature05065"},{"idx":"6","title":"Pioro-Ladrière, M. et al. Electrically driven single-electron spin resonance in a slanting Zeeman field.Nat. Phys.4, 776–779 (2008).","link":"https://doi.org/10.1038%2Fnphys1053"},{"idx":"7","title":"Yang, C. H. et al. Spin-valley lifetimes in a silicon quantum dot with tunable valley splitting.Nat. Commun.4, 2069 (2013).","link":"https://doi.org/10.1038%2Fncomms3069"},{"idx":"8","title":"Petta, J. R. et al. Coherent manipulation of coupled electron spins in semiconductor quantum dots.Science309, 2180–2184 (2005).","link":"https://doi.org/10.1126%2Fscience.1116955"},{"idx":"9","title":"Goswami, S. et al. Controllable valley splitting in silicon quantum devices.Nat. Phys.3, 41–45 (2007).","link":"https://doi.org/10.1038%2Fnphys475"},{"idx":"10","title":"Culcer, D., Saraiva, A. L., Koiller, B., Hu, X. & Das Sarma, S. Valley-based noise-resistant quantum computation using si quantum dots.Phys. Rev. Lett.108, 126804 (2012).","link":"https://doi.org/10.1103%2FPhysRevLett.108.126804"},{"idx":"11","title":"Culcer, D., Hu, X. & Das Sarma, S. Interface roughness, valley-orbit coupling, and valley manipulation in quantum dots.Phys. Rev. B82, 205315 (2010).","link":"https://doi.org/10.1103%2FPhysRevB.82.205315"},{"idx":"12","title":"Zhang, L., Luo, J. W., Saraiva, A., Koiller, B. & Zunger, A. Genetic design of enhanced valley splitting towards a spin qubit in silicon.Nat. Commun.4, 2396 (2013).","link":"https://doi.org/10.1038%2Fncomms3396"}]}},{"section":"Results","content":"DQD device The cross-sectional view of the Si MOS QD device is shown in Fig. 1a . A scanning electron microscope image of the essential part of a similar device is shown in Fig. 1b . A DQD is defined by six confinement gates, which are labelled as ‘T’, ‘L’, ‘PL’, ‘M’, ‘PR’ and ‘R’. A coplanar strip (CPS) loop (see Methods for device details), situated about 1.5 μm away, is used to deliver an oscillating (AC) magnetic field B ac , perpendicular to the DQD interface. The oscillation frequency f ac is scanned to resonance with the electron spin precession oscillations in an in-plane external magnetic field B , Fig. 1b . The DQD is characterized by the DC transport current. Figure 1c shows a typical charge stability diagram of the device with source–drain bias voltage of V sd =−1 mV, in which the transport current is recorded while the plunger gates V pl and V pr are scanned [13] . The device does not contain a charge-sensing channel and the identified electron numbers are the approximate ones. The estimated electron occupation numbers in the left and right dots are labelled by ( N L , N R ). At lower electron numbers (more negative voltage at the plunger gates ‘PL’, ‘PR’) the tunnelling from (out of) source (drain) is suppressed, so higher biasing triangles will be examined on electronic transport. Electron transitions into and out of the left (right) dot are labelled by white dashed (blue dash-dotted) lines in Fig. 1c . The honeycomb structure and the biasing triangles here show the characteristic features of a well-defined DQD [13] . Figure 1: Silicon MOS DQD. ( a ) The cross-sectional view of the device. ( b ) The scanning electron microscope image of a similar device (scale bar, 1 μm). DQD (blue circles) is defined at the Si/SiO 2 interface by the confinement gates (labelled as ‘T’, ‘L’, ‘PL’, ‘M’, ‘PR’ and ‘R’). Microwave applied to CPS loop generates an AC magnetic field at the QD. ( c ) Stability diagram at few-electron region (log scale), with additional labels for estimated electron numbers and guidelines for transitions between different electron configurations. Full size image Spin blockade Spin blockade of the electronic transport is the well-known method for sensing and manipulation of confined electron spins in semiconductor heterostructures [1] , [5] , [8] , [14] . For a DQD confining two electrons, the standard statement is that an electron cannot flip spin under tunnelling, and so a transition from a (1,1) charge configuration to a (2,0) configuration is only possible between the corresponding singlet or triplet spin states: S(1,1)→S(2,0), T(1,1)→T(2,0), respecting the Pauli exclusion principle (see Fig. 2a,b insets). In a typically biased DQD (with detuning much larger than tunnelling, ε>> t c ), the delocalized states S(1,1), T(1,1) are only slightly shifted by an exchange energy , while the localized states S(2,0), T(2,0) are split by large Δ ST >> J (given a higher orbital excitation of a few hundred μ eV). Thus, the energy of the state T(2,0) is much higher, and transition of T(1,1) to T(2,0) is energetically forbidden, while transition of S(1,1) to S(2,0) is still allowed. Figure 2: Spin blockade at biasing triangle labelled as ‘M’ in Fig. 1c. ( a , b ) Transport data at forward- and reverse-bias voltage applied to source and drain (external magnetic field B =0 T). The low-current region in b is a signature of spin blockade. Insets demonstrate that at forward-bias voltage, only spin up can tunnel into the left dot and then tunnel out, which contributes to the transport current, while at reverse-bias voltage, the transport is prohibited by Pauli exclusion principle once a spin down electron tunnels into the right dot. ( c ) Transport data at reverse-bias voltage with external magnetic field on ( B =1 T). Valence electron numbers are labelled. The leakage current in the spin blockade region is well suppressed by the magnetic field. ( d ) Leakage current in spin blockade region measured at different magnetic fields. Full size image In Si, the conduction electrons may belong to different valley configurations v i , ( i =1, 2 at the two-dimensional interface): the electron state acquires a valley index, v 1 or v 2 , denoting two states that have the same spin and orbital content, but are split by a valley energy E V ; we further refer to these states as valley states, although they are certain superposition of the original valleys [15] , (see Supplementary Note 1 ). Thus, the simple spin blockade can be applied to the lowest [v1, v1] valley states. In what follows, we first make an attempt to describe the experimental blockade and ESR following the simple spin blockade picture. The implications of higher valley states, [v1, v2], [v2, v2], will be considered later (also see Supplementary Notes 2–4 ). We focus on the biasing triangle labelled as ‘M’ in Fig. 1c , which occurs at the transition between the electron states (4,2) and (3,3). Assuming only ‘valence’ electron configurations take place in the transport [16] , [17] , we use hereafter the effective electron occupancy (2,0) and (1,1), as labelled in Fig. 2c . As a key signature of the spin blockade, the forward-bias ( Fig. 2a ) transport is allowed within the whole detuning region, while the reverse-bias ( Fig. 2b ) transport shows a low-current region [18] . As illustrated, at forward bias (inset of Fig. 2a ), only spin singlet S(2,0) can be formed when the second electron tunnels into the left dot (DQD is in the (1,0) configuration before tunnelling), and then the S(2,0) state can make transition to (1,0) state through a S(1,1) state. Therefore, a continuous flowing transport current will be observed. However, at reverse bias (inset of Fig. 2b ), once a triplet state T(1,1) is formed, it cannot make transition to a (2,0) charge state, and thus blocks the current [14] , [18] . With a finite magnetic field applied (specifically, in this experiment we used an in-plane field, parallel to the DQD, and oriented along the [110] crystallographic direction at the Si interface, Fig. 1a,b ), the triplet T(1,1) state splits into three states: T + (1,1), T 0 (1,1) and T − (1,1), and the current can be blocked by loading electrons into any of these three states. The low-current (blocked) region, traced by the white trapezoid in Fig. 2b , implies a (2,0) singlet–triplet splitting of (see, however, the discussion of spin–valley blockade below). The current in the spin blockade region in Fig. 2b is not completely suppressed; in real systems, the electron spins can be mixed or flipped by the nuclear field hyperfine interaction [19] , [20] , SOC [20] , [21] or co-tunnelling [17] , [21] , [22] , [23] , which generates a finite leakage current and lifts the spin blockade. This leakage can be strongly suppressed, in Fig. 2c , by applying an external magnetic field [17] , [22] , which is parallel to the Si/SiO 2 interface. Almost one order of magnitude suppression of the leakage current, as shown in Fig. 2d , allows us to probe the electron spin at higher field ( B ≳ 0.5 T) with a good sensitivity. Detection of ESR and phase coherence time The single ESR is observed by setting the DQD in the spin blockade region ( Fig. 3a , green dot in the inset of Fig. 3d , corresponding to an interdot detuning ε≈100 μ eV, in this case), and applying an oscillating magnetic field via the CPS loop [5] . At a frequency where the microwave energy matches the Zeeman splitting of a single electron spin ( hf ac = gμ B B , where μ B is Bohr magneton and g is the electron g -factor), the spins can flip. Since, however, the AC field rotates the spins in the two dots simultaneously, the two electrons will remain within the triplet subspace and the spin blockade will not be lifted (see also Supplementary Note 3 ). Thus, a spin mixing mechanism is required to mix the triplet (1,1) subspace with the singlet (1,1) ( Fig. 3b ), that can subsequently inelastically tunnel to a (2,0) state, lifting the blockade ( Fig. 3c ). Similar to a GaAs DQD system [5] , an inhomogeneous nuclear hyperfine (HF) field σ N could mix the singlet S(1,1) and triplets T(1,1), assuming the HF energy, E N ≡ gμ B σ N , is larger than the singlet–triplet exchange splitting, (see, for example, refs 24 , 25 ). While, at a finite external magnetic field B , the polarized triplets T + (1,1), T − (1,1) are spin blocked since their HF mixing with the singlet is energy suppressed, an AC field resonant to the Zeeman energy splitting brings them in resonance with the T 0 (1,1) state that can mix to the singlet S(1,1). In a Si-based system, however, the inhomogeneous nuclear field is one to two orders of magnitude smaller than that in GaAs system, and one would expect a much weaker ESR signal. Figure 3: Single ESR scheme and data. ( a – c ) Single ESR detection scheme. ( a ) An additional electron tunnels into the right dot but transport is blocked due to Pauli exclusion principle. ( b ) Microwave drives electron spin in resonance hence opens a transport channel. ( c ) Second electron with flipped spin tunnels out, contributing to the transport current. ( d ) A typical ESR signal obtained by fixing microwave frequency at f ac =20.77 GHz (microwave power P =−28 dBm) while scanning magnetic field (averaged over 20 curves). Inset: green dot in the biasing triangle (at B =0.7425 T, frequency f ac =20.71 GHz and microwave power P =−20 dBm) shows where the ESR is detected. ( e ) Normalized leakage current (see text) as a function of magnetic field and microwave frequency. ESR peak position shows linear relation between magnetic field and microwave frequency. Full size image The ESR peak is observed within a relatively narrow microwave frequency range, Δ f ac ≤1 GHz, by measuring the leakage current in the spin blockade region as a function of the external magnetic field. In Fig. 3d it is shown for a fixed microwave frequency ( f ac =20.77 GHz). We have verified that the ESR signal can only be detected in the spin blockade region of the reverse-biasing triangle in Fig. 2b , and persists up to 500 mK. By measuring the leakage current as a function of the magnetic field and microwave frequency, the ESR signal is presented as a sloped straight line in the two-dimensional space ( Fig. 3e ). For better contrast, the leakage current in Fig. 3e is normalized by the average current at each frequency. The slope of the ESR line gives an effective g -factor of g =1.97±0.07, compatible to Si. The relatively large error is because the ESR signal is only visible in a narrow frequency window. Surprisingly, the ESR signal is just as strong as that in a GaAs system [5] , where there is a much larger nuclear HF field. The line width (Δ B ESR ≡FWHM) of the ESR peak in Fig. 3d is Δ B ESR =0.2 mT, which is one order of magnitude smaller compared with the HF field value of ~2 mT measured in GaAs QD [5] . The power dependence of the ESR line width is plotted in Fig. 4a , showing weak dependence at low power, and power broadening by the applied AC magnetic field, and finally a saturation [26] , [27] . The saturation effect is also seen in the power dependence of the ESR peak height as shown in Fig. 4b : the ESR peak first increases linearly as the power increases [5] , [28] , and finally decreases at higher power due to the disturbance of additional electric field or photon-assisted tunnelling. Figure 4: Power dependence of ESR peak. ESR signal ( a ) line width and ( b ) peak height (with error bars) as a function of applied microwave power. Both of them show power saturation. The microwave power is measured at the output of the microwave generator. Full size image At low microwave power, the line width of the ESR peak would be determined by the nuclear field fluctuation, assuming that this mechanism dominates. The narrowest ESR line below the saturation that we observed is around Δ B ESR =0.15 mT, giving an estimated nuclear HF field (in z -direction) in a single dot . This value gives a lower bound for the inhomogeneously broadened spin dephasing time (assuming a singlet–triplet qubit) ( ħ is the reduced Planck constant), which is significantly longer than that measured in GaAs DQD system via free induction decay [8] . The value of the estimated hyperfine field, however, is six times larger than that expected [29] in Si (with natural abundance of 29 Si nuclei), and also confirmed experimentally for SiGe QDs [1] . The discrepancy may be explained by noting that in our Si DQD with transport measurement setup, the inelastic interdot tunnelling rate may happen to be large: Γ in ≳ gμ B σ N ,z / ħ ~10 7 s −1 , so that the ESR line may be broadened by the inelastic transition itself (see Supplementary Note 5 ). Assuming only this broadening mechanism, the experimental FWHM-ESR, Δ B ESR , allows to estimate Γ in ≈2.6 × 10 7 s −1 , comparable to recent measurements in Si/SiO 2 DQDs [30] . Anticrossing feature in the ESR spectrum By exploring the two-dimensional ESR spectroscopy (see Fig. 3e ) at higher magnetic field and microwave frequencies, we notice that the straight ESR line splits and forms an anticrossing feature at a frequency f anticross ≃ 20.84 GHz; also notice a ‘remnant’ of the straight ESR line in between the anticrossing ( Fig. 5a ). The anticrossing position (corresponding to an energy difference of hf anticross ≃ 86.2 μ eV) is determined to be independent of the interdot detuning ε (for example, in the range of ε =50–250 μ eV the anticrossing frequency does not shift at different detunings within an error bar of 20 MHz). The size of the anticrossing gap can be readily obtained from the f-B diagram to be about Δ f anticross ≃ 60±10 MHz. Similar ESR anticrossing features were observed for a different biasing triangle than that mentioned in Fig. 1c (see Supplementary Note 3 ). Since the experiment involves two coupled QDs, at first sight, one would relate the anticrossing feature with the level crossing of, for example, the T − (1,1) and S(2,0) states at finite magnetic field, if a spin–orbit interaction would dominate the anticrossing [31] . However, the independence on detuning rules out this possibility, as such crossing would be a strong function of detuning [32] . Therefore, to explain the anticrossing one should only include the states with the same charge configuration. Figure 5: ESR spectroscopy in a Si MOS DQD. ( a ) Normalized leakage current as a function of the external magnetic field and microwave frequency. ESR lines show anticrossing feature with a frequency gap Δ f anticross =60±10 MHz (black arrows), at a position B V ≃ 0.746 T (Methods). ( b ) Left panel: field-dependent energy diagram of electrons in the (1,1) charge states in case of small exchange energy: J <| a R | (see text). Dotted (blue) rectangles label the spin-blocked states that are similar to GaAs, but with valley index added; dotted (red) circles label the spin–valley-blocked states: , , eventually surviving for a non-ideal interface (see text). The other (non-blocked) states are: , and , . Right panels: zoom in of the level anticrossing of states with different valley content due to SOC; the size of the energy gaps is given by only two dipole matrix elements, a R , b L (see text). ( c ) Simulated anticrossing features in the ESR spectroscopy, using the measured valley energy splitting E V =86.2 μeV and the anticrossing splitting of equation (3) (second splitting is invisible for | b L |<<| a R |). Full size image Interpretation of the anticrossing feature Because the independency on detuning of the anticrossing, we will identify it as due to anticrossing of singlet and triplet (1,1) states of different valley content. The two-electron states now acquire an additional valley index [v i , v j ], i , j =1, 2. For zero magnetic field one has three groups of degenerate levels denoted as [v1, v1], [v1, v2], [v2, v2] (16 states in total [33] , see Supplementary Note 1 ), that are split off each other by the valley splitting energy of E V = hf anticross ≃ 86.2 μ eV measured in the experiment (we have neglected for the moment the finite exchange splitting J , as well as the small nuclear HF field in Si, see below). The polarized triplet states from each group (anti)cross the singlet(s)/triplet(s) of the other group at a magnetic field B V = E V /( gμ B ) ≃ 0.746 T (see Supplementary Note 2 ). For example, the triplet state will anticross the upper singlets S v1v2 , S v2v1 and the unpolarized triplets , (see Fig. 5b and Supplementary Figs 1–3 for other situations). Simultaneously, for B ~ B V , the triplets , anticross the lower levels S v1v1 , and the triplets , anticross the upper levels S v2v2 , . Thus, three anticrossing ‘spots’ are formed at a Zeeman splitting E Z = E V , Fig. 5b . Since the source–drain voltage is large, eV sd ≃ 1 meV>> E V , all (1,1) states are loaded, and the ESR resonance transport takes place for the different groups of transitions. The mixing mechanism of different spin–valley states is due to the SOC in the presence of non-ideal Si/SiO 2 interface [7] ; here it can be parameterized by only two dipole matrix elements where β D ( α R ) is the Dresselhaus (Rashba) SOC parameter (also, valley splitting may be different in the right/left dot), m t =0.198 m e is the transverse effective mass for conduction electrons, and, for example, is the wave function of an electron confined in the right dot [34] . We note that for an ideally flat interface the above matrix elements are exactly zero. However, in the presence of disorder/roughness the valley envelop functions of both valleys are perturbed [7] , providing non-zero dipole matrix elements; the latter just parameterize the presence of a kind of roughness, that is a non-flat interface, atomic steps, or defects. The splitting at anticrossing is directly observable via ESR, and (for | a R |>>| b L |) is given by providing a dipole matrix element, , of the order of x 12 ~15–55 nm for the measured gap (see Supplementary Note 2 ). We note that the same matrix elements were recently shown to be responsible for spin–valley mixing and fast (phonon) relaxation of spin states at the so-called ‘hot spot’ in a single Si/SiO 2 QD [7] (see also recent calculations of such matrix element that confirm their order of magnitude value, by modelling the interface roughness with single atomic steps at the QD interface [35] ). The description of the ESR and the spin blockade, in general, are more involved since (1,1) states of various valley content are loaded, starting from a (1,0) state. For a fixed external magnetic field there are several transitions in resonance with the AC oscillating magnetic field, Fig. 5b . For the direction of the magnetic field chosen in the experiment ( Fig. 1b ) the AC Hamiltonian is: , and couples triplet states within each valley subspace (both the AC and nuclear HF interaction do not mix different valley subspaces). Therefore, the AC coupling between each pair of resonant eigenstates will be dependent on the projection of these states to the corresponding triplets, T ± , T 0 , making some of the transitions suppressed. The contribution to the ESR signal of the valley subspaces [v1, v1], [v2, v2] ( Fig. 5b ) is similar to that in a GaAs DQD [5] , [8] while in a regime where nuclear HF mixing is suppressed. Even in the absence of (random) nuclear HF field mixing of S–T 0 states, the spin–valley mixing mechanism allows a finite ESR signal to be observed. For instance, for the spin-blocked state , a resonant ESR transition to the upper state is possible, since it is a coherent mixture of the states and S v1v1 at the anticrossing spot I ( Fig. 5b ). Thus, the ESR leakage current will increase at anticrossing due to relative increase of a singlet (or triplet, see below) state, that can tunnel inelastically to a (2,0) state. Since the size of the splitting of 60 MHz is equivalent to an ‘effective magnetic field’ of ~2.1 mT, this explains qualitatively that the observed ESR signal is as strong as for GaAs DQDs, even though the nuclear HF field in Si is two orders of magnitude smaller than that in GaAs. Out of anticrossing the ESR signal decreases as well as the S–T 0 mixing; numerically, for example, for the lowest hybridized state ( Fig. 5b ) a ten times smaller admixing of non-blocked states (S v1v1 , in this case) corresponds to the Zeeman detuning | E Z − E V |≈7 Δ anticross , explaining a ‘bright ESR range’ of ~2| E Z − E V |≈0.8 GHz (or ~0.03 mT), comparable to the experimentally observed range, Fig. 5a . Another mechanism of ESR signal suppression (especially of the sloped ESR line) is due to the finite exchange energy splitting. A finite exchange splitting, , lifts the singlet–triplet degeneracy for each group of valley states, far from the anticrossing region, and forms eigenstates ( Fig. 5b ), where some of them will be blocked. For a coherent tunnelling of t c ≈5–10 μ eV the estimated exchange splitting is in the range of J ≈0.2–0.8 μ eV, so it is much larger than the nuclear HF energy in Si, E N ≈3 neV. (Even though we do not measure J directly, a situation when J < E N is unlikely, since in this case the ESR suppression out of anticrossing cannot be explained, see Supplementary Note 3 .) Thus, the standard mechanism of a S–T 0 mixing via the HF field will be energetically suppressed far from anticrossing for the [v1, v1] and [v2, v2] valley states, and so its corresponding contribution to the observed ESR signal. Since in the ESR transport experiment upper valley states are loaded, one needs to consider one more mechanism of ESR signal suppression. We assume that within the [v1, v2] valley subspace, the polarized state is spin–valley blocked (see the discussion of spin–valley blockade below). Since out of anticrossing it is equally coupled via Ĥ ac to two degenerate states , Fig. 5b , this creates a coherent superposition of these states, in which the singlet part, , is cancelled, while the triplet is spin–valley blocked. Further HF coupling of to the unblocked singlet is suppressed by the finite exchange splitting J ( Fig. 5b , left and right panels), and so is the ESR signal. The above arguments complete the explanation of suppression of the ESR signal out of anticrossing, observed experimentally ( Fig. 5a ). At anticrossing the spin–valley mixing and the AC driving lift the blockade, making the observation of ESR possible. In Fig. 5c we plotted the energy difference for each pair of states, with an intensity given by the absolute value of the AC coupling that qualitatively reconstructs the anticrossing picture observed experimentally. Despite the many different transitions that give rise to multiple ESR lines/crossings, the AC coupling filters out many of them (still involving all three anticrossing spots, see Fig. 5b ), that leaves us with only one anticrossing and a straight line in between, Fig. 5c . Actually, just this picture requires to have the inequality | a R |>>| b L |, mentioned above. Spin–valley blockade It is worth now to consider the observed blockade in the absence of AC driving and for a finite magnetic field, where the leakage current is suppressed for the whole region of interdot detuning, (see Fig. 2c,d ), including that which is 2–3 times larger than the valley splitting E V ≃ 86.2 μ eV. This means that a type of spin–valley blockade is experimentally observed. Since the blockade is magnetic field independent for B ≳ 0.5 T, Fig. 2d , and particularly for B ~ B V , one needs to consider several possible scenarios of spin–valley blockade. In all scenarios the blockade means impossibility for an inelastic transition (via phonons) to happen. For the usual spin blockade (1) it is since phonon emission cannot flip spin, and so a triplet T(1,1) cannot decay to the localized singlet S(2,0). The different type of spin–valley blockade, if it happened, is since phonon emission cannot change the valley content of the state [36] (2), or it cannot change the ‘valley parity’ of the state (3) (related to specific cancellation of phonon decay amplitudes, see below and Supplementary Note 3 ). It is essential for our argument that all spin–valley states, Fig. 6 , are loaded continuously, for any fixed magnetic field B . Assuming the dominance of a phonon inelastic relaxation (like in a single Si/SiO 2 QD [7] ), we introduce the phonon decay amplitudes between various spin–valley states [37] , (see Supplementary Note 3 ), and consider the scenarios: (1) The spin blockade. For ε < E V and B ≶ B V the polarized triplet states , are spin blocked similar to a GaAs system [5] , [8] (analogously, the higher valley states are spin blocked as well, Fig. 6 ). In the region of B ~ B V , the states , still remain blocked. (2) Spin–valley blockade I. For larger detuning, ε > E V , the spin blockade remains, Fig. 6 ; however, the state T v1v1 may decay to the triplet two-electron state in the left dot, since it is energetically allowed; the blockade will depend on the off-diagonal-in-valley phonon decay amplitude : for the ideal case (since umklapp transitions are suppressed, while the envelope functions of the states |v 1 ›, |v 2 › are identical for ideal interface, see Supplementary Note 1 ). The matrix element could be non-zero for a non-ideal interface, similar to the matrix element equations (1) and (2). Even if this type of valley blockade is lifted (for ), there could be a second type of spin–valley blockade, Fig. 6 . (3) Spin–valley blockade II. The overall blockade could not yet be lifted, since the spin–valley triplet state (as well as its polarized counterparts) cannot decay to the corresponding state, which we call a spin–valley phonon selection rule. Indeed, the corresponding diagonal-in-valley phonon decay amplitudes cancel if the equality holds: . The equality is exact for an ideal interface, (for identical v 1 , v 2 envelope functions) and is likely to hold at least approximately even for a non-ideal interface, see Supplementary Note 2 . (4) Spin–valley blockade III. Since the blockade is observed at B ~ B V , one more candidate for a blocked state is the eigenstate (as an alternative of the blocked state ) , Figs 5b and 6 . To be blocked, this requires the above equality, , and also . As to the blockade alternatives (2–4) considered here, we notice that, if the v 1 , v 2 were the lowest orbital states in each dot, none of the above equalities would hold, and the blockade would be lifted just at ε > E V . Figure 6: Spin blockade versus spin–valley blockade. Schematics of spin blockade and spin–valley blockade in the case of , and for interdot detuning ε > E V (see text). For the energy levels of the localized (2,0) states (shown on the left), the states and are degenerate; the ‘true’ singlet–triplet energy splitting is to the higher orbital (triplet) state . The energy levels of the delocalized (1,1) states (shown on the right) are fine split by the exchange coupling J (see text and Supplementary Note 3 ). The polarized (Zeeman split)±triplets of the (1,1), (2,0) states are not shown, and for the [v 1 , v 2 ] group of (1,1) states only the relevant blocked states are shown: namely, the two polarized triplets (shown schematically as one level) are referred to the spin–valley blockade (3); the unpolarized state is referred to the spin–valley blockade (4) (see also Fig. 5b ). The spin blockade (1) and the three types of possible spin–valley blockade (2–4) are explained in the text. The blockade from the upper valley (1,1) state T v2v2 shown as (1′) and (2′) is similar to (1), (2) (see Supplementary Note 3 ). Full size image The spin–valley blockade may be lifted for the above alternatives either at detuning ε > E V and/or at . The latter is possible when the state T v1v2 +T v2v1 matches in energy the usual orbital state in the left dot, (see Fig. 6 , Supplementary Fig. 4 and Supplementary Note 4 ). Since in the experiment the inequality holds: , the states match at an energy larger than E V , namely at the experimentally observed , which implies a true S–T splitting The measured experimental blockade cannot distinguish which of the above alternatives has happened. However, the single fact that we have observed a blockade of the leakage current for detunings ε > E V allows us to state that we have observed the spin–valley blockade associated with one of the alternatives (2–4)."},{"section":"Discussion","content":"In conclusion, we have observed the ESR using spin or spin–valley blockade in a gate-defined Si MOS DQD. The ESR signal is significantly enhanced where Zeeman levels of different valley content (anti)cross, due to the spin–valley mixing arising from spin–orbit interaction at non-ideal QD interfaces. From the ESR line width, the spin dephasing time is estimated as , which is significantly longer than that in a GaAs system. The discovery of the anomalous anticrossing demonstrates the possibility to characterize and manipulate spin–valley states using ESR, for individual qubits. In a long run, with a better understanding of the device physics of Si QDs, one can choose, design and operate qubits in regimes that are better suited for robust quantum computation (examples could include making valley splitting large enough across devices or improving surface interfaces). Our results improve the outlook of Si MOS QDs as a platform for high-coherence spin qubits, in the now leading microelectronics material."},{"section":"Methods","content":"Device fabrication The sample used in this experiment was fabricated on undoped commercial Si wafer with a 50-nm thermal SiO 2 (refs 38 , 39 ). First, the ohmic contacts were made by phosphorous ion implantation followed by a high-temperature annealing. Then, the confinement gates as well as the CPS loop were defined by electron beam lithography. Before putting on the global accumulation gate (Cr/Au), a 120-nm Al 2 O 3 , which serves as an insulating layer between confinement gates and accumulation gates, was grown by atomic layer deposition. See Fig. 1a for the cross-section layout of the DQD device. We use aluminium as the material for CPS loop and obtain a few Ω loop resistance to maximize the transmission of microwave signal. Electrical measurement The device is mounted on the cold finger of a dilution refrigerator with a base temperature of 80 mK. An in-plane magnetic field is created via a superconducting magnet; possible trapping of a residual magnetic flux may cause an overall shift of the magnetic field read-off by a few mT. The electron temperature is about 200 mK. A semirigid cable delivers the microwave, which is generated by HP/Agilent Signal Generator 8673B, to the coplanar loop. The cable has an attenuation of 20 dB at the frequency of 20 GHz. A low-noise current amplifier is used to measure the QD transport current."},{"section":"Additional information","content":"How to cite this article: Hao, X. et al. Electron spin resonance and spin–valley physics in a silicon double quantum dot. Nat. Commun. 5:3860 doi: 10.1038/ncomms4860 (2014)."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"It has long been speculated that qubits based on individual electron spins in Si quantum dots (QDs) have considerable potential for quantum information processing. Attractive features are the extremely long coherence time of spins in Si bulk materials and the possibility to approach zero hyperfine interaction to nuclear spins in isotopically purified structures. Furthermore, the extensive collection of Complementary Metal-Oxide-Semiconductor (CMOS)-based techniques, accumulated over decades, is expected to be very important for fabricating many qubits. Electric and magnetic fields along with charge detection enable qubit gates and read-out. A long coherence time was recently confirmed for the singlet and triplet states in a Si/SiGe double quantum dot (DQD) qubit [1] . Electron spin resonance (ESR) is a direct means to drive rotations of a spin qubit. For electron spins bound in Si, an ensemble of spins in various structured materials [2] , single electrons in a single defect [3] and in a single donor [4] have been explored with ESR, using various detection schemes. Physical implementations of ESR on individual bound electronic spins have proven to be successful in GaAs-based QD transport experiments [5] , [6] , where the essential role of the spin (Pauli) blockade and the nuclear spin bath in that systems were established. However, spin detection via electronic transport in gate-defined Si QDs has remained challenging. Here we report the detection of microwave-driven ESR transport of individual electrons in a silicon metal-oxide-semiconductor (MOS)-based DQD. The lifting of the blockade via ESR is detectable only in a narrow region where Zeeman split spin states of different valley content anticross where the Zeeman splitting E Z equals the valley splitting of E V ≃ 86.2 μ eV. We show that the anticrossing is due to spin–orbit coupling (SOC) in the heterostructure, in the presence of interface roughness, that mixes spin and valley states (similar mixing mechanism was first established in the electron spin relaxation in a small single Si QD [7] ). The gap at anticrossing of Δ f anticross ≃ 60 MHz is a measure of this spin–valley mixing and also provides a means to access higher valley states via ESR. Analysis of the ESR spectrum provides a lower-bound estimation of an inhomogeneous decoherence time, , which is much longer than that in GaAs [8] ; it also compares well with the direct measurement of for a Si qubit encoded by singlet–triplet states [1] . The nature of the experimental blockade regime, and its dependence on the applied magnetic field and interdot energy detuning are discussed. Since the observed blockade region includes detunings larger than the valley splitting, we can conclude that a spin–valley blockade takes place, related to the impossibility for an inelastic (via phonons) electron tunnelling to happen. This blockade survives even in the presence of a non-ideal interface. The observations made in this paper encourage further development of Si MOS-based spin qubits and further suggest that the additional valley degree of freedom [1] , [7] , [9] , [10] , [11] , [12] is critical to understanding silicon qubits.\",\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 ],\n \"infos\": [\n {\n \"idx\": \"1\",\n \"title\": \"Maune, B. M. et al. Coherent singlet-triplet oscillations in a silicon-based double quantum dot.Nature481, 344–347 (2012).\",\n \"link\": \"https://doi.org/10.1038%2Fnature10707\"\n },\n {\n \"idx\": \"2\",\n \"title\": \"Morton, J. J. L., McCamey, D. R., Eriksson, M. A. & Lyon, S. A. Embracing the quantum limit in silicon computing.Nature479, 345–353 (2011).\",\n \"link\": \"https://doi.org/10.1038%2Fnature10681\"\n },\n {\n \"idx\": \"3\",\n \"title\": \"Xiao, M., Martin, I., Yablonovitch, E. & Jiang, H. W. Electrical detection of the spin resonance of a single electron in a silicon field-effect transistor.Nature430, 435–439 (2004).\",\n \"link\": \"https://doi.org/10.1038%2Fnature02727\"\n },\n {\n \"idx\": \"4\",\n \"title\": \"Pla, J. J. et al. A single-atom electron spin qubit in silicon.Nature489, 541–545 (2012).\",\n \"link\": \"https://doi.org/10.1038%2Fnature11449\"\n },\n {\n \"idx\": \"5\",\n \"title\": \"Koppens, F. H. L. et al. Driven coherent oscillations of a single electron spin in a quantum dot.Nature442, 766–771 (2006).\",\n \"link\": \"https://doi.org/10.1038%2Fnature05065\"\n },\n {\n \"idx\": \"6\",\n \"title\": \"Pioro-Ladrière, M. et al. Electrically driven single-electron spin resonance in a slanting Zeeman field.Nat. Phys.4, 776–779 (2008).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys1053\"\n },\n {\n \"idx\": \"7\",\n \"title\": \"Yang, C. H. et al. Spin-valley lifetimes in a silicon quantum dot with tunable valley splitting.Nat. Commun.4, 2069 (2013).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms3069\"\n },\n {\n \"idx\": \"8\",\n \"title\": \"Petta, J. R. et al. Coherent manipulation of coupled electron spins in semiconductor quantum dots.Science309, 2180–2184 (2005).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1116955\"\n },\n {\n \"idx\": \"9\",\n \"title\": \"Goswami, S. et al. Controllable valley splitting in silicon quantum devices.Nat. Phys.3, 41–45 (2007).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys475\"\n },\n {\n \"idx\": \"10\",\n \"title\": \"Culcer, D., Saraiva, A. L., Koiller, B., Hu, X. & Das Sarma, S. Valley-based noise-resistant quantum computation using si quantum dots.Phys. Rev. Lett.108, 126804 (2012).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.108.126804\"\n },\n {\n \"idx\": \"11\",\n \"title\": \"Culcer, D., Hu, X. & Das Sarma, S. Interface roughness, valley-orbit coupling, and valley manipulation in quantum dots.Phys. Rev. B82, 205315 (2010).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.82.205315\"\n },\n {\n \"idx\": \"12\",\n \"title\": \"Zhang, L., Luo, J. W., Saraiva, A., Koiller, B. & Zunger, A. Genetic design of enhanced valley splitting towards a spin qubit in silicon.Nat. Commun.4, 2396 (2013).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms3396\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"DQD device The cross-sectional view of the Si MOS QD device is shown in Fig. 1a . A scanning electron microscope image of the essential part of a similar device is shown in Fig. 1b . A DQD is defined by six confinement gates, which are labelled as ‘T’, ‘L’, ‘PL’, ‘M’, ‘PR’ and ‘R’. A coplanar strip (CPS) loop (see Methods for device details), situated about 1.5 μm away, is used to deliver an oscillating (AC) magnetic field B ac , perpendicular to the DQD interface. The oscillation frequency f ac is scanned to resonance with the electron spin precession oscillations in an in-plane external magnetic field B , Fig. 1b . The DQD is characterized by the DC transport current. Figure 1c shows a typical charge stability diagram of the device with source–drain bias voltage of V sd =−1 mV, in which the transport current is recorded while the plunger gates V pl and V pr are scanned [13] . The device does not contain a charge-sensing channel and the identified electron numbers are the approximate ones. The estimated electron occupation numbers in the left and right dots are labelled by ( N L , N R ). At lower electron numbers (more negative voltage at the plunger gates ‘PL’, ‘PR’) the tunnelling from (out of) source (drain) is suppressed, so higher biasing triangles will be examined on electronic transport. Electron transitions into and out of the left (right) dot are labelled by white dashed (blue dash-dotted) lines in Fig. 1c . The honeycomb structure and the biasing triangles here show the characteristic features of a well-defined DQD [13] . Figure 1: Silicon MOS DQD. ( a ) The cross-sectional view of the device. ( b ) The scanning electron microscope image of a similar device (scale bar, 1 μm). DQD (blue circles) is defined at the Si/SiO 2 interface by the confinement gates (labelled as ‘T’, ‘L’, ‘PL’, ‘M’, ‘PR’ and ‘R’). Microwave applied to CPS loop generates an AC magnetic field at the QD. ( c ) Stability diagram at few-electron region (log scale), with additional labels for estimated electron numbers and guidelines for transitions between different electron configurations. Full size image Spin blockade Spin blockade of the electronic transport is the well-known method for sensing and manipulation of confined electron spins in semiconductor heterostructures [1] , [5] , [8] , [14] . For a DQD confining two electrons, the standard statement is that an electron cannot flip spin under tunnelling, and so a transition from a (1,1) charge configuration to a (2,0) configuration is only possible between the corresponding singlet or triplet spin states: S(1,1)→S(2,0), T(1,1)→T(2,0), respecting the Pauli exclusion principle (see Fig. 2a,b insets). In a typically biased DQD (with detuning much larger than tunnelling, ε>> t c ), the delocalized states S(1,1), T(1,1) are only slightly shifted by an exchange energy , while the localized states S(2,0), T(2,0) are split by large Δ ST >> J (given a higher orbital excitation of a few hundred μ eV). Thus, the energy of the state T(2,0) is much higher, and transition of T(1,1) to T(2,0) is energetically forbidden, while transition of S(1,1) to S(2,0) is still allowed. Figure 2: Spin blockade at biasing triangle labelled as ‘M’ in Fig. 1c. ( a , b ) Transport data at forward- and reverse-bias voltage applied to source and drain (external magnetic field B =0 T). The low-current region in b is a signature of spin blockade. Insets demonstrate that at forward-bias voltage, only spin up can tunnel into the left dot and then tunnel out, which contributes to the transport current, while at reverse-bias voltage, the transport is prohibited by Pauli exclusion principle once a spin down electron tunnels into the right dot. ( c ) Transport data at reverse-bias voltage with external magnetic field on ( B =1 T). Valence electron numbers are labelled. The leakage current in the spin blockade region is well suppressed by the magnetic field. ( d ) Leakage current in spin blockade region measured at different magnetic fields. Full size image In Si, the conduction electrons may belong to different valley configurations v i , ( i =1, 2 at the two-dimensional interface): the electron state acquires a valley index, v 1 or v 2 , denoting two states that have the same spin and orbital content, but are split by a valley energy E V ; we further refer to these states as valley states, although they are certain superposition of the original valleys [15] , (see Supplementary Note 1 ). Thus, the simple spin blockade can be applied to the lowest [v1, v1] valley states. In what follows, we first make an attempt to describe the experimental blockade and ESR following the simple spin blockade picture. The implications of higher valley states, [v1, v2], [v2, v2], will be considered later (also see Supplementary Notes 2–4 ). We focus on the biasing triangle labelled as ‘M’ in Fig. 1c , which occurs at the transition between the electron states (4,2) and (3,3). Assuming only ‘valence’ electron configurations take place in the transport [16] , [17] , we use hereafter the effective electron occupancy (2,0) and (1,1), as labelled in Fig. 2c . As a key signature of the spin blockade, the forward-bias ( Fig. 2a ) transport is allowed within the whole detuning region, while the reverse-bias ( Fig. 2b ) transport shows a low-current region [18] . As illustrated, at forward bias (inset of Fig. 2a ), only spin singlet S(2,0) can be formed when the second electron tunnels into the left dot (DQD is in the (1,0) configuration before tunnelling), and then the S(2,0) state can make transition to (1,0) state through a S(1,1) state. Therefore, a continuous flowing transport current will be observed. However, at reverse bias (inset of Fig. 2b ), once a triplet state T(1,1) is formed, it cannot make transition to a (2,0) charge state, and thus blocks the current [14] , [18] . With a finite magnetic field applied (specifically, in this experiment we used an in-plane field, parallel to the DQD, and oriented along the [110] crystallographic direction at the Si interface, Fig. 1a,b ), the triplet T(1,1) state splits into three states: T + (1,1), T 0 (1,1) and T − (1,1), and the current can be blocked by loading electrons into any of these three states. The low-current (blocked) region, traced by the white trapezoid in Fig. 2b , implies a (2,0) singlet–triplet splitting of (see, however, the discussion of spin–valley blockade below). The current in the spin blockade region in Fig. 2b is not completely suppressed; in real systems, the electron spins can be mixed or flipped by the nuclear field hyperfine interaction [19] , [20] , SOC [20] , [21] or co-tunnelling [17] , [21] , [22] , [23] , which generates a finite leakage current and lifts the spin blockade. This leakage can be strongly suppressed, in Fig. 2c , by applying an external magnetic field [17] , [22] , which is parallel to the Si/SiO 2 interface. Almost one order of magnitude suppression of the leakage current, as shown in Fig. 2d , allows us to probe the electron spin at higher field ( B ≳ 0.5 T) with a good sensitivity. Detection of ESR and phase coherence time The single ESR is observed by setting the DQD in the spin blockade region ( Fig. 3a , green dot in the inset of Fig. 3d , corresponding to an interdot detuning ε≈100 μ eV, in this case), and applying an oscillating magnetic field via the CPS loop [5] . At a frequency where the microwave energy matches the Zeeman splitting of a single electron spin ( hf ac = gμ B B , where μ B is Bohr magneton and g is the electron g -factor), the spins can flip. Since, however, the AC field rotates the spins in the two dots simultaneously, the two electrons will remain within the triplet subspace and the spin blockade will not be lifted (see also Supplementary Note 3 ). Thus, a spin mixing mechanism is required to mix the triplet (1,1) subspace with the singlet (1,1) ( Fig. 3b ), that can subsequently inelastically tunnel to a (2,0) state, lifting the blockade ( Fig. 3c ). Similar to a GaAs DQD system [5] , an inhomogeneous nuclear hyperfine (HF) field σ N could mix the singlet S(1,1) and triplets T(1,1), assuming the HF energy, E N ≡ gμ B σ N , is larger than the singlet–triplet exchange splitting, (see, for example, refs 24 , 25 ). While, at a finite external magnetic field B , the polarized triplets T + (1,1), T − (1,1) are spin blocked since their HF mixing with the singlet is energy suppressed, an AC field resonant to the Zeeman energy splitting brings them in resonance with the T 0 (1,1) state that can mix to the singlet S(1,1). In a Si-based system, however, the inhomogeneous nuclear field is one to two orders of magnitude smaller than that in GaAs system, and one would expect a much weaker ESR signal. Figure 3: Single ESR scheme and data. ( a – c ) Single ESR detection scheme. ( a ) An additional electron tunnels into the right dot but transport is blocked due to Pauli exclusion principle. ( b ) Microwave drives electron spin in resonance hence opens a transport channel. ( c ) Second electron with flipped spin tunnels out, contributing to the transport current. ( d ) A typical ESR signal obtained by fixing microwave frequency at f ac =20.77 GHz (microwave power P =−28 dBm) while scanning magnetic field (averaged over 20 curves). Inset: green dot in the biasing triangle (at B =0.7425 T, frequency f ac =20.71 GHz and microwave power P =−20 dBm) shows where the ESR is detected. ( e ) Normalized leakage current (see text) as a function of magnetic field and microwave frequency. ESR peak position shows linear relation between magnetic field and microwave frequency. Full size image The ESR peak is observed within a relatively narrow microwave frequency range, Δ f ac ≤1 GHz, by measuring the leakage current in the spin blockade region as a function of the external magnetic field. In Fig. 3d it is shown for a fixed microwave frequency ( f ac =20.77 GHz). We have verified that the ESR signal can only be detected in the spin blockade region of the reverse-biasing triangle in Fig. 2b , and persists up to 500 mK. By measuring the leakage current as a function of the magnetic field and microwave frequency, the ESR signal is presented as a sloped straight line in the two-dimensional space ( Fig. 3e ). For better contrast, the leakage current in Fig. 3e is normalized by the average current at each frequency. The slope of the ESR line gives an effective g -factor of g =1.97±0.07, compatible to Si. The relatively large error is because the ESR signal is only visible in a narrow frequency window. Surprisingly, the ESR signal is just as strong as that in a GaAs system [5] , where there is a much larger nuclear HF field. The line width (Δ B ESR ≡FWHM) of the ESR peak in Fig. 3d is Δ B ESR =0.2 mT, which is one order of magnitude smaller compared with the HF field value of ~2 mT measured in GaAs QD [5] . The power dependence of the ESR line width is plotted in Fig. 4a , showing weak dependence at low power, and power broadening by the applied AC magnetic field, and finally a saturation [26] , [27] . The saturation effect is also seen in the power dependence of the ESR peak height as shown in Fig. 4b : the ESR peak first increases linearly as the power increases [5] , [28] , and finally decreases at higher power due to the disturbance of additional electric field or photon-assisted tunnelling. Figure 4: Power dependence of ESR peak. ESR signal ( a ) line width and ( b ) peak height (with error bars) as a function of applied microwave power. Both of them show power saturation. The microwave power is measured at the output of the microwave generator. Full size image At low microwave power, the line width of the ESR peak would be determined by the nuclear field fluctuation, assuming that this mechanism dominates. The narrowest ESR line below the saturation that we observed is around Δ B ESR =0.15 mT, giving an estimated nuclear HF field (in z -direction) in a single dot . This value gives a lower bound for the inhomogeneously broadened spin dephasing time (assuming a singlet–triplet qubit) ( ħ is the reduced Planck constant), which is significantly longer than that measured in GaAs DQD system via free induction decay [8] . The value of the estimated hyperfine field, however, is six times larger than that expected [29] in Si (with natural abundance of 29 Si nuclei), and also confirmed experimentally for SiGe QDs [1] . The discrepancy may be explained by noting that in our Si DQD with transport measurement setup, the inelastic interdot tunnelling rate may happen to be large: Γ in ≳ gμ B σ N ,z / ħ ~10 7 s −1 , so that the ESR line may be broadened by the inelastic transition itself (see Supplementary Note 5 ). Assuming only this broadening mechanism, the experimental FWHM-ESR, Δ B ESR , allows to estimate Γ in ≈2.6 × 10 7 s −1 , comparable to recent measurements in Si/SiO 2 DQDs [30] . Anticrossing feature in the ESR spectrum By exploring the two-dimensional ESR spectroscopy (see Fig. 3e ) at higher magnetic field and microwave frequencies, we notice that the straight ESR line splits and forms an anticrossing feature at a frequency f anticross ≃ 20.84 GHz; also notice a ‘remnant’ of the straight ESR line in between the anticrossing ( Fig. 5a ). The anticrossing position (corresponding to an energy difference of hf anticross ≃ 86.2 μ eV) is determined to be independent of the interdot detuning ε (for example, in the range of ε =50–250 μ eV the anticrossing frequency does not shift at different detunings within an error bar of 20 MHz). The size of the anticrossing gap can be readily obtained from the f-B diagram to be about Δ f anticross ≃ 60±10 MHz. Similar ESR anticrossing features were observed for a different biasing triangle than that mentioned in Fig. 1c (see Supplementary Note 3 ). Since the experiment involves two coupled QDs, at first sight, one would relate the anticrossing feature with the level crossing of, for example, the T − (1,1) and S(2,0) states at finite magnetic field, if a spin–orbit interaction would dominate the anticrossing [31] . However, the independence on detuning rules out this possibility, as such crossing would be a strong function of detuning [32] . Therefore, to explain the anticrossing one should only include the states with the same charge configuration. Figure 5: ESR spectroscopy in a Si MOS DQD. ( a ) Normalized leakage current as a function of the external magnetic field and microwave frequency. ESR lines show anticrossing feature with a frequency gap Δ f anticross =60±10 MHz (black arrows), at a position B V ≃ 0.746 T (Methods). ( b ) Left panel: field-dependent energy diagram of electrons in the (1,1) charge states in case of small exchange energy: J <| a R | (see text). Dotted (blue) rectangles label the spin-blocked states that are similar to GaAs, but with valley index added; dotted (red) circles label the spin–valley-blocked states: , , eventually surviving for a non-ideal interface (see text). The other (non-blocked) states are: , and , . Right panels: zoom in of the level anticrossing of states with different valley content due to SOC; the size of the energy gaps is given by only two dipole matrix elements, a R , b L (see text). ( c ) Simulated anticrossing features in the ESR spectroscopy, using the measured valley energy splitting E V =86.2 μeV and the anticrossing splitting of equation (3) (second splitting is invisible for | b L |<<| a R |). Full size image Interpretation of the anticrossing feature Because the independency on detuning of the anticrossing, we will identify it as due to anticrossing of singlet and triplet (1,1) states of different valley content. The two-electron states now acquire an additional valley index [v i , v j ], i , j =1, 2. For zero magnetic field one has three groups of degenerate levels denoted as [v1, v1], [v1, v2], [v2, v2] (16 states in total [33] , see Supplementary Note 1 ), that are split off each other by the valley splitting energy of E V = hf anticross ≃ 86.2 μ eV measured in the experiment (we have neglected for the moment the finite exchange splitting J , as well as the small nuclear HF field in Si, see below). The polarized triplet states from each group (anti)cross the singlet(s)/triplet(s) of the other group at a magnetic field B V = E V /( gμ B ) ≃ 0.746 T (see Supplementary Note 2 ). For example, the triplet state will anticross the upper singlets S v1v2 , S v2v1 and the unpolarized triplets , (see Fig. 5b and Supplementary Figs 1–3 for other situations). Simultaneously, for B ~ B V , the triplets , anticross the lower levels S v1v1 , and the triplets , anticross the upper levels S v2v2 , . Thus, three anticrossing ‘spots’ are formed at a Zeeman splitting E Z = E V , Fig. 5b . Since the source–drain voltage is large, eV sd ≃ 1 meV>> E V , all (1,1) states are loaded, and the ESR resonance transport takes place for the different groups of transitions. The mixing mechanism of different spin–valley states is due to the SOC in the presence of non-ideal Si/SiO 2 interface [7] ; here it can be parameterized by only two dipole matrix elements where β D ( α R ) is the Dresselhaus (Rashba) SOC parameter (also, valley splitting may be different in the right/left dot), m t =0.198 m e is the transverse effective mass for conduction electrons, and, for example, is the wave function of an electron confined in the right dot [34] . We note that for an ideally flat interface the above matrix elements are exactly zero. However, in the presence of disorder/roughness the valley envelop functions of both valleys are perturbed [7] , providing non-zero dipole matrix elements; the latter just parameterize the presence of a kind of roughness, that is a non-flat interface, atomic steps, or defects. The splitting at anticrossing is directly observable via ESR, and (for | a R |>>| b L |) is given by providing a dipole matrix element, , of the order of x 12 ~15–55 nm for the measured gap (see Supplementary Note 2 ). We note that the same matrix elements were recently shown to be responsible for spin–valley mixing and fast (phonon) relaxation of spin states at the so-called ‘hot spot’ in a single Si/SiO 2 QD [7] (see also recent calculations of such matrix element that confirm their order of magnitude value, by modelling the interface roughness with single atomic steps at the QD interface [35] ). The description of the ESR and the spin blockade, in general, are more involved since (1,1) states of various valley content are loaded, starting from a (1,0) state. For a fixed external magnetic field there are several transitions in resonance with the AC oscillating magnetic field, Fig. 5b . For the direction of the magnetic field chosen in the experiment ( Fig. 1b ) the AC Hamiltonian is: , and couples triplet states within each valley subspace (both the AC and nuclear HF interaction do not mix different valley subspaces). Therefore, the AC coupling between each pair of resonant eigenstates will be dependent on the projection of these states to the corresponding triplets, T ± , T 0 , making some of the transitions suppressed. The contribution to the ESR signal of the valley subspaces [v1, v1], [v2, v2] ( Fig. 5b ) is similar to that in a GaAs DQD [5] , [8] while in a regime where nuclear HF mixing is suppressed. Even in the absence of (random) nuclear HF field mixing of S–T 0 states, the spin–valley mixing mechanism allows a finite ESR signal to be observed. For instance, for the spin-blocked state , a resonant ESR transition to the upper state is possible, since it is a coherent mixture of the states and S v1v1 at the anticrossing spot I ( Fig. 5b ). Thus, the ESR leakage current will increase at anticrossing due to relative increase of a singlet (or triplet, see below) state, that can tunnel inelastically to a (2,0) state. Since the size of the splitting of 60 MHz is equivalent to an ‘effective magnetic field’ of ~2.1 mT, this explains qualitatively that the observed ESR signal is as strong as for GaAs DQDs, even though the nuclear HF field in Si is two orders of magnitude smaller than that in GaAs. Out of anticrossing the ESR signal decreases as well as the S–T 0 mixing; numerically, for example, for the lowest hybridized state ( Fig. 5b ) a ten times smaller admixing of non-blocked states (S v1v1 , in this case) corresponds to the Zeeman detuning | E Z − E V |≈7 Δ anticross , explaining a ‘bright ESR range’ of ~2| E Z − E V |≈0.8 GHz (or ~0.03 mT), comparable to the experimentally observed range, Fig. 5a . Another mechanism of ESR signal suppression (especially of the sloped ESR line) is due to the finite exchange energy splitting. A finite exchange splitting, , lifts the singlet–triplet degeneracy for each group of valley states, far from the anticrossing region, and forms eigenstates ( Fig. 5b ), where some of them will be blocked. For a coherent tunnelling of t c ≈5–10 μ eV the estimated exchange splitting is in the range of J ≈0.2–0.8 μ eV, so it is much larger than the nuclear HF energy in Si, E N ≈3 neV. (Even though we do not measure J directly, a situation when J < E N is unlikely, since in this case the ESR suppression out of anticrossing cannot be explained, see Supplementary Note 3 .) Thus, the standard mechanism of a S–T 0 mixing via the HF field will be energetically suppressed far from anticrossing for the [v1, v1] and [v2, v2] valley states, and so its corresponding contribution to the observed ESR signal. Since in the ESR transport experiment upper valley states are loaded, one needs to consider one more mechanism of ESR signal suppression. We assume that within the [v1, v2] valley subspace, the polarized state is spin–valley blocked (see the discussion of spin–valley blockade below). Since out of anticrossing it is equally coupled via Ĥ ac to two degenerate states , Fig. 5b , this creates a coherent superposition of these states, in which the singlet part, , is cancelled, while the triplet is spin–valley blocked. Further HF coupling of to the unblocked singlet is suppressed by the finite exchange splitting J ( Fig. 5b , left and right panels), and so is the ESR signal. The above arguments complete the explanation of suppression of the ESR signal out of anticrossing, observed experimentally ( Fig. 5a ). At anticrossing the spin–valley mixing and the AC driving lift the blockade, making the observation of ESR possible. In Fig. 5c we plotted the energy difference for each pair of states, with an intensity given by the absolute value of the AC coupling that qualitatively reconstructs the anticrossing picture observed experimentally. Despite the many different transitions that give rise to multiple ESR lines/crossings, the AC coupling filters out many of them (still involving all three anticrossing spots, see Fig. 5b ), that leaves us with only one anticrossing and a straight line in between, Fig. 5c . Actually, just this picture requires to have the inequality | a R |>>| b L |, mentioned above. Spin–valley blockade It is worth now to consider the observed blockade in the absence of AC driving and for a finite magnetic field, where the leakage current is suppressed for the whole region of interdot detuning, (see Fig. 2c,d ), including that which is 2–3 times larger than the valley splitting E V ≃ 86.2 μ eV. This means that a type of spin–valley blockade is experimentally observed. Since the blockade is magnetic field independent for B ≳ 0.5 T, Fig. 2d , and particularly for B ~ B V , one needs to consider several possible scenarios of spin–valley blockade. In all scenarios the blockade means impossibility for an inelastic transition (via phonons) to happen. For the usual spin blockade (1) it is since phonon emission cannot flip spin, and so a triplet T(1,1) cannot decay to the localized singlet S(2,0). The different type of spin–valley blockade, if it happened, is since phonon emission cannot change the valley content of the state [36] (2), or it cannot change the ‘valley parity’ of the state (3) (related to specific cancellation of phonon decay amplitudes, see below and Supplementary Note 3 ). It is essential for our argument that all spin–valley states, Fig. 6 , are loaded continuously, for any fixed magnetic field B . Assuming the dominance of a phonon inelastic relaxation (like in a single Si/SiO 2 QD [7] ), we introduce the phonon decay amplitudes between various spin–valley states [37] , (see Supplementary Note 3 ), and consider the scenarios: (1) The spin blockade. For ε < E V and B ≶ B V the polarized triplet states , are spin blocked similar to a GaAs system [5] , [8] (analogously, the higher valley states are spin blocked as well, Fig. 6 ). In the region of B ~ B V , the states , still remain blocked. (2) Spin–valley blockade I. For larger detuning, ε > E V , the spin blockade remains, Fig. 6 ; however, the state T v1v1 may decay to the triplet two-electron state in the left dot, since it is energetically allowed; the blockade will depend on the off-diagonal-in-valley phonon decay amplitude : for the ideal case (since umklapp transitions are suppressed, while the envelope functions of the states |v 1 ›, |v 2 › are identical for ideal interface, see Supplementary Note 1 ). The matrix element could be non-zero for a non-ideal interface, similar to the matrix element equations (1) and (2). Even if this type of valley blockade is lifted (for ), there could be a second type of spin–valley blockade, Fig. 6 . (3) Spin–valley blockade II. The overall blockade could not yet be lifted, since the spin–valley triplet state (as well as its polarized counterparts) cannot decay to the corresponding state, which we call a spin–valley phonon selection rule. Indeed, the corresponding diagonal-in-valley phonon decay amplitudes cancel if the equality holds: . The equality is exact for an ideal interface, (for identical v 1 , v 2 envelope functions) and is likely to hold at least approximately even for a non-ideal interface, see Supplementary Note 2 . (4) Spin–valley blockade III. Since the blockade is observed at B ~ B V , one more candidate for a blocked state is the eigenstate (as an alternative of the blocked state ) , Figs 5b and 6 . To be blocked, this requires the above equality, , and also . As to the blockade alternatives (2–4) considered here, we notice that, if the v 1 , v 2 were the lowest orbital states in each dot, none of the above equalities would hold, and the blockade would be lifted just at ε > E V . Figure 6: Spin blockade versus spin–valley blockade. Schematics of spin blockade and spin–valley blockade in the case of , and for interdot detuning ε > E V (see text). For the energy levels of the localized (2,0) states (shown on the left), the states and are degenerate; the ‘true’ singlet–triplet energy splitting is to the higher orbital (triplet) state . The energy levels of the delocalized (1,1) states (shown on the right) are fine split by the exchange coupling J (see text and Supplementary Note 3 ). The polarized (Zeeman split)±triplets of the (1,1), (2,0) states are not shown, and for the [v 1 , v 2 ] group of (1,1) states only the relevant blocked states are shown: namely, the two polarized triplets (shown schematically as one level) are referred to the spin–valley blockade (3); the unpolarized state is referred to the spin–valley blockade (4) (see also Fig. 5b ). The spin blockade (1) and the three types of possible spin–valley blockade (2–4) are explained in the text. The blockade from the upper valley (1,1) state T v2v2 shown as (1′) and (2′) is similar to (1), (2) (see Supplementary Note 3 ). Full size image The spin–valley blockade may be lifted for the above alternatives either at detuning ε > E V and/or at . The latter is possible when the state T v1v2 +T v2v1 matches in energy the usual orbital state in the left dot, (see Fig. 6 , Supplementary Fig. 4 and Supplementary Note 4 ). Since in the experiment the inequality holds: , the states match at an energy larger than E V , namely at the experimentally observed , which implies a true S–T splitting The measured experimental blockade cannot distinguish which of the above alternatives has happened. However, the single fact that we have observed a blockade of the leakage current for detunings ε > E V allows us to state that we have observed the spin–valley blockade associated with one of the alternatives (2–4).\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"In conclusion, we have observed the ESR using spin or spin–valley blockade in a gate-defined Si MOS DQD. The ESR signal is significantly enhanced where Zeeman levels of different valley content (anti)cross, due to the spin–valley mixing arising from spin–orbit interaction at non-ideal QD interfaces. From the ESR line width, the spin dephasing time is estimated as , which is significantly longer than that in a GaAs system. The discovery of the anomalous anticrossing demonstrates the possibility to characterize and manipulate spin–valley states using ESR, for individual qubits. In a long run, with a better understanding of the device physics of Si QDs, one can choose, design and operate qubits in regimes that are better suited for robust quantum computation (examples could include making valley splitting large enough across devices or improving surface interfaces). Our results improve the outlook of Si MOS QDs as a platform for high-coherence spin qubits, in the now leading microelectronics material.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Device fabrication The sample used in this experiment was fabricated on undoped commercial Si wafer with a 50-nm thermal SiO 2 (refs 38 , 39 ). First, the ohmic contacts were made by phosphorous ion implantation followed by a high-temperature annealing. Then, the confinement gates as well as the CPS loop were defined by electron beam lithography. Before putting on the global accumulation gate (Cr/Au), a 120-nm Al 2 O 3 , which serves as an insulating layer between confinement gates and accumulation gates, was grown by atomic layer deposition. See Fig. 1a for the cross-section layout of the DQD device. We use aluminium as the material for CPS loop and obtain a few Ω loop resistance to maximize the transmission of microwave signal. Electrical measurement The device is mounted on the cold finger of a dilution refrigerator with a base temperature of 80 mK. An in-plane magnetic field is created via a superconducting magnet; possible trapping of a residual magnetic flux may cause an overall shift of the magnetic field read-off by a few mT. The electron temperature is about 200 mK. A semirigid cable delivers the microwave, which is generated by HP/Agilent Signal Generator 8673B, to the coplanar loop. The cable has an attenuation of 20 dB at the frequency of 20 GHz. A low-noise current amplifier is used to measure the QD transport current.\"\n },\n {\n \"section\": \"Additional information\",\n \"content\": \"How to cite this article: Hao, X. et al. Electron spin resonance and spin–valley physics in a silicon double quantum dot. Nat. Commun. 5:3860 doi: 10.1038/ncomms4860 (2014).\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"The core research idea is to investigate and demonstrate the detection of microwave-driven electron spin resonance (ESR) in a silicon metal-oxide-semiconductor (MOS)-based double quantum dot (DQD) to enable the manipulation and read-out of individual electron spin qubits, leveraging spin–valley mixing induced by spin–orbit coupling at an anticrossing of Zeeman and valley-split states, and to show that a spin–valley blockade regime allows for the observation of long coherence times and access to higher valley states, thereby supporting the development of scalable Si-based spin qubits with potential advantages over GaAs systems."},"entities":{"kind":"list like","value":["microwave-driven electron spin resonance","silicon metal-oxide-semiconductor","double quantum dot","electron spin qubits","spin–valley mixing","spin–orbit coupling","anticrossing","Zeeman states","valley-split states","spin–valley blockade regime","long coherence times","higher valley states","scalable Si-based spin qubits","GaAs systems"],"string":"[\n \"microwave-driven electron spin resonance\",\n \"silicon metal-oxide-semiconductor\",\n \"double quantum dot\",\n \"electron spin qubits\",\n \"spin–valley mixing\",\n \"spin–orbit coupling\",\n \"anticrossing\",\n \"Zeeman states\",\n \"valley-split states\",\n \"spin–valley blockade regime\",\n \"long coherence times\",\n \"higher valley states\",\n \"scalable Si-based spin qubits\",\n \"GaAs systems\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1","title":"Maune, B. M. et al. Coherent singlet-triplet oscillations in a silicon-based double quantum dot.Nature481, 344–347 (2012).","link":"https://doi.org/10.1038%2Fnature10707"},{"idx":"2","title":"Morton, J. J. L., McCamey, D. R., Eriksson, M. A. & Lyon, S. A. Embracing the quantum limit in silicon computing.Nature479, 345–353 (2011).","link":"https://doi.org/10.1038%2Fnature10681"},{"idx":"3","title":"Xiao, M., Martin, I., Yablonovitch, E. & Jiang, H. W. Electrical detection of the spin resonance of a single electron in a silicon field-effect transistor.Nature430, 435–439 (2004).","link":"https://doi.org/10.1038%2Fnature02727"},{"idx":"4","title":"Pla, J. J. et al. A single-atom electron spin qubit in silicon.Nature489, 541–545 (2012).","link":"https://doi.org/10.1038%2Fnature11449"},{"idx":"5","title":"Koppens, F. H. L. et al. Driven coherent oscillations of a single electron spin in a quantum dot.Nature442, 766–771 (2006).","link":"https://doi.org/10.1038%2Fnature05065"},{"idx":"6","title":"Pioro-Ladrière, M. et al. Electrically driven single-electron spin resonance in a slanting Zeeman field.Nat. Phys.4, 776–779 (2008).","link":"https://doi.org/10.1038%2Fnphys1053"},{"idx":"7","title":"Yang, C. H. et al. Spin-valley lifetimes in a silicon quantum dot with tunable valley splitting.Nat. Commun.4, 2069 (2013).","link":"https://doi.org/10.1038%2Fncomms3069"},{"idx":"8","title":"Petta, J. R. et al. Coherent manipulation of coupled electron spins in semiconductor quantum dots.Science309, 2180–2184 (2005).","link":"https://doi.org/10.1126%2Fscience.1116955"},{"idx":"9","title":"Goswami, S. et al. Controllable valley splitting in silicon quantum devices.Nat. Phys.3, 41–45 (2007).","link":"https://doi.org/10.1038%2Fnphys475"},{"idx":"10","title":"Culcer, D., Saraiva, A. L., Koiller, B., Hu, X. & Das Sarma, S. Valley-based noise-resistant quantum computation using si quantum dots.Phys. Rev. Lett.108, 126804 (2012).","link":"https://doi.org/10.1103%2FPhysRevLett.108.126804"},{"idx":"11","title":"Culcer, D., Hu, X. & Das Sarma, S. Interface roughness, valley-orbit coupling, and valley manipulation in quantum dots.Phys. Rev. B82, 205315 (2010).","link":"https://doi.org/10.1103%2FPhysRevB.82.205315"},{"idx":"12","title":"Zhang, L., Luo, J. W., Saraiva, A., Koiller, B. & Zunger, A. Genetic design of enhanced valley splitting towards a spin qubit in silicon.Nat. Commun.4, 2396 (2013).","link":"https://doi.org/10.1038%2Fncomms3396"},{"idx":"13","title":"van der Wiel, W. G. et al. Electron transport through double quantum dots.Rev. Mod. Phys.75, 1–22 (2002).","link":"https://doi.org/10.1103%2FRevModPhys.75.1"},{"idx":"14","title":"Ono, K., Austing, D. G., Tokura, Y. & Tarucha, S. Current rectification by Pauli exclusion in a weakly coupled double quantum dot system.Science297, 1313–1317 (2002).","link":"https://doi.org/10.1126%2Fscience.1070958"},{"idx":"15","title":"Tahan, C., Friesen, M. & Joint, R. Decoherence of electron spin qubits in Si-based quantum computers.Phys. Rev. B66, 035314 (2002).","link":"https://doi.org/10.1103%2FPhysRevB.66.035314"},{"idx":"16","title":"Shaji, N. et al. Spin blockade and lifetime-enhanced transport in a few-electron Si/SiGe double quantum dot.Nat. Phys.4, 540–544 (2008).","link":"https://doi.org/10.1038%2Fnphys988"},{"idx":"17","title":"Lai, N. S. et al. Pauli spin blockade in a highly tunable silicon double quantum dot.Sci. Rep.1, 110 (2011).","link":"https://doi.org/10.1038%2Fsrep00110"},{"idx":"18","title":"Johnson, A. C., Petta, J. R., Marcus, C. M., Hanson, M. P. & Gossard, A. C. Singlet-triplet spin blockade and charge sensing in a few-electron double quantum dot.Phys. Rev. B72, 165308 (2005).","link":"https://doi.org/10.1103%2FPhysRevB.72.165308"},{"idx":"19","title":"Koppens, F. H. L. et al. Control and detection of singlet-triplet mixing in a random nuclear field.Science309, 1346–1350 (2005).","link":"https://doi.org/10.1126%2Fscience.1113719"},{"idx":"20","title":"Nadj-Perge, S. et al. Disentangling the effects of spin-orbit and hyperfine interactions on spin blockade.Phys. Rev. B81, 201305 (2010).","link":"https://doi.org/10.1103%2FPhysRevB.81.201305"},{"idx":"21","title":"Yamahata, G. et al. Magnetic field dependence of Pauli spin blockade: a window into the sources of spin relaxation in silicon quantum dots.Phys. Rev. B86, 115322 (2012).","link":"https://doi.org/10.1103%2FPhysRevB.86.115322"},{"idx":"22","title":"Coish, W. A. & Qassemi, F. Leakage-current line shapes from inelastic cotunneling in the Pauli spin blockade regime.Phys. Rev. B84, 245407 (2011).","link":"https://doi.org/10.1103%2FPhysRevB.84.245407"},{"idx":"23","title":"Koh, T. S., Simmons, C. B., Eriksson, M. A., Coppersmith, S. N. & Friesen, M. Unconventional transport in the ‘hole’ regime of a Si double quantum dot.Phys. Rev. Lett.106, 186801 (2011).","link":"https://doi.org/10.1103%2FPhysRevLett.106.186801"},{"idx":"24","title":"Jouravlev, O. N. & Nazarov, Y. V. Electron transport in a double quantum dot governed by a nuclear magnetic field.Phys. Rev. Lett.96, 176804 (2006).","link":"https://doi.org/10.1103%2FPhysRevLett.96.176804"},{"idx":"25","title":"Taylor, J. M. et al. Relaxation, dephasing, and quantum control of electron spins in double quantum dots.Phys. Rev. B76, 035315 (2007).","link":"https://doi.org/10.1103%2FPhysRevB.76.035315"},{"idx":"26","title":"Koppens, F. H. L. et al. Detection of single electron spin resonance in a double quantum dot.J. Appl. Phys.101, 081706 (2007).","link":"https://doi.org/10.1063%2F1.2722734"},{"idx":"28","title":"Engel, H.-A. & Loss, D. Detection of single spin decoherence in a quantum dot via charge currents.Phys. Rev. Lett.86, 4648–4651 (2001).","link":"https://doi.org/10.1103%2FPhysRevLett.86.4648"},{"idx":"29","title":"Assali, L. V. C. et al. Hyperfine interactions in silicon quantum dots.Phys. Rev. B83, 165301 (2011).","link":"https://doi.org/10.1103%2FPhysRevB.83.165301"},{"idx":"30","title":"Wang, K., Payette, C., Dovzhenko, Y., Deelman, P. W. & Petta, J. R. Charge relaxation in a single-electron Si/SiGe double quantum dot.Phys. Rev. Lett.111, 046801 (2013).","link":"https://doi.org/10.1103%2FPhysRevLett.111.046801"},{"idx":"31","title":"Danon, J. & Nazarov, Y. V. Pauli spin blockade in the presence of strong spin-orbit coupling.Phys. Rev. B80, 041301 (2009).","link":"https://doi.org/10.1103%2FPhysRevB.80.041301"},{"idx":"32","title":"Nadj-Perge, S. et al. Spectroscopy of spin-orbit quantum bits in indium antimonide nanowires.Phys. Rev. Lett.108, 166801 (2012).","link":"https://doi.org/10.1103%2FPhysRevLett.108.166801"},{"idx":"33","title":"Culcer, D., Cywiński, L., Li, Q., Hu, X. & Das Sarma, S. Quantum dot spin qubits in silicon: multivalley physics.Phys. Rev. B82, 155312 (2010).","link":"https://doi.org/10.1103%2FPhysRevB.82.155312"},{"idx":"34","title":"Burkard, G., Loss, D. & DiVincenzo, D. P. Coupled quantum dots as quantum gates.Phys. Rev. B59, 2070–2078 (1999).","link":"https://doi.org/10.1103%2FPhysRevB.59.2070"},{"idx":"35","title":"Gamble, J. K., Eriksson, M. A., Coppersmith, S. N. & Friesen, M. Disorder-induced valley-orbit hybrid states in Si quantum dots.Phys. Rev. B88, 035310 (2013).","link":"https://doi.org/10.1103%2FPhysRevB.88.035310"},{"idx":"36","title":"Pályi, A. & Burkard, G. Hyperfine-induced valley mixing and the spin-valley blockade in carbon-based quantum dots.Phys. Rev. B80, 201404(R) (2009).","link":"https://doi.org/10.1103%2FPhysRevB.80.201404"},{"idx":"38","title":"Xiao, M., House, M. G. & Jiang, H. W. Measurement of the spin relaxation time of single electrons in a silicon metal-oxide-semiconductor-based quantum dot.Phys. Rev. Lett.104, 096801 (2010).","link":"https://doi.org/10.1103%2FPhysRevLett.104.096801"},{"idx":"39","title":"Xiao, M., House, M. G. & Jiang, H. W. Parallel spin filling and energy spectroscopy in few-electron Si metal-on-semiconductor-based quantum dots.Appl. Phys. Lett.97, 032103 (2010).","link":"https://doi.org/10.1063%2F1.3464324"}],"string":"[\n {\n \"idx\": \"1\",\n \"title\": \"Maune, B. M. et al. Coherent singlet-triplet oscillations in a silicon-based double quantum dot.Nature481, 344–347 (2012).\",\n \"link\": \"https://doi.org/10.1038%2Fnature10707\"\n },\n {\n \"idx\": \"2\",\n \"title\": \"Morton, J. J. L., McCamey, D. R., Eriksson, M. A. & Lyon, S. A. Embracing the quantum limit in silicon computing.Nature479, 345–353 (2011).\",\n \"link\": \"https://doi.org/10.1038%2Fnature10681\"\n },\n {\n \"idx\": \"3\",\n \"title\": \"Xiao, M., Martin, I., Yablonovitch, E. & Jiang, H. W. Electrical detection of the spin resonance of a single electron in a silicon field-effect transistor.Nature430, 435–439 (2004).\",\n \"link\": \"https://doi.org/10.1038%2Fnature02727\"\n },\n {\n \"idx\": \"4\",\n \"title\": \"Pla, J. J. et al. A single-atom electron spin qubit in silicon.Nature489, 541–545 (2012).\",\n \"link\": \"https://doi.org/10.1038%2Fnature11449\"\n },\n {\n \"idx\": \"5\",\n \"title\": \"Koppens, F. H. L. et al. Driven coherent oscillations of a single electron spin in a quantum dot.Nature442, 766–771 (2006).\",\n \"link\": \"https://doi.org/10.1038%2Fnature05065\"\n },\n {\n \"idx\": \"6\",\n \"title\": \"Pioro-Ladrière, M. et al. Electrically driven single-electron spin resonance in a slanting Zeeman field.Nat. Phys.4, 776–779 (2008).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys1053\"\n },\n {\n \"idx\": \"7\",\n \"title\": \"Yang, C. H. et al. Spin-valley lifetimes in a silicon quantum dot with tunable valley splitting.Nat. Commun.4, 2069 (2013).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms3069\"\n },\n {\n \"idx\": \"8\",\n \"title\": \"Petta, J. R. et al. Coherent manipulation of coupled electron spins in semiconductor quantum dots.Science309, 2180–2184 (2005).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1116955\"\n },\n {\n \"idx\": \"9\",\n \"title\": \"Goswami, S. et al. Controllable valley splitting in silicon quantum devices.Nat. Phys.3, 41–45 (2007).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys475\"\n },\n {\n \"idx\": \"10\",\n \"title\": \"Culcer, D., Saraiva, A. L., Koiller, B., Hu, X. & Das Sarma, S. Valley-based noise-resistant quantum computation using si quantum dots.Phys. Rev. Lett.108, 126804 (2012).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.108.126804\"\n },\n {\n \"idx\": \"11\",\n \"title\": \"Culcer, D., Hu, X. & Das Sarma, S. Interface roughness, valley-orbit coupling, and valley manipulation in quantum dots.Phys. Rev. B82, 205315 (2010).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.82.205315\"\n },\n {\n \"idx\": \"12\",\n \"title\": \"Zhang, L., Luo, J. W., Saraiva, A., Koiller, B. & Zunger, A. Genetic design of enhanced valley splitting towards a spin qubit in silicon.Nat. Commun.4, 2396 (2013).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms3396\"\n },\n {\n \"idx\": \"13\",\n \"title\": \"van der Wiel, W. G. et al. Electron transport through double quantum dots.Rev. Mod. Phys.75, 1–22 (2002).\",\n \"link\": \"https://doi.org/10.1103%2FRevModPhys.75.1\"\n },\n {\n \"idx\": \"14\",\n \"title\": \"Ono, K., Austing, D. G., Tokura, Y. & Tarucha, S. Current rectification by Pauli exclusion in a weakly coupled double quantum dot system.Science297, 1313–1317 (2002).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1070958\"\n },\n {\n \"idx\": \"15\",\n \"title\": \"Tahan, C., Friesen, M. & Joint, R. Decoherence of electron spin qubits in Si-based quantum computers.Phys. Rev. B66, 035314 (2002).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.66.035314\"\n },\n {\n \"idx\": \"16\",\n \"title\": \"Shaji, N. et al. Spin blockade and lifetime-enhanced transport in a few-electron Si/SiGe double quantum dot.Nat. Phys.4, 540–544 (2008).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys988\"\n },\n {\n \"idx\": \"17\",\n \"title\": \"Lai, N. S. et al. Pauli spin blockade in a highly tunable silicon double quantum dot.Sci. Rep.1, 110 (2011).\",\n \"link\": \"https://doi.org/10.1038%2Fsrep00110\"\n },\n {\n \"idx\": \"18\",\n \"title\": \"Johnson, A. C., Petta, J. R., Marcus, C. M., Hanson, M. P. & Gossard, A. C. Singlet-triplet spin blockade and charge sensing in a few-electron double quantum dot.Phys. Rev. B72, 165308 (2005).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.72.165308\"\n },\n {\n \"idx\": \"19\",\n \"title\": \"Koppens, F. H. L. et al. Control and detection of singlet-triplet mixing in a random nuclear field.Science309, 1346–1350 (2005).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1113719\"\n },\n {\n \"idx\": \"20\",\n \"title\": \"Nadj-Perge, S. et al. Disentangling the effects of spin-orbit and hyperfine interactions on spin blockade.Phys. Rev. B81, 201305 (2010).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.81.201305\"\n },\n {\n \"idx\": \"21\",\n \"title\": \"Yamahata, G. et al. Magnetic field dependence of Pauli spin blockade: a window into the sources of spin relaxation in silicon quantum dots.Phys. Rev. B86, 115322 (2012).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.86.115322\"\n },\n {\n \"idx\": \"22\",\n \"title\": \"Coish, W. A. & Qassemi, F. Leakage-current line shapes from inelastic cotunneling in the Pauli spin blockade regime.Phys. Rev. B84, 245407 (2011).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.84.245407\"\n },\n {\n \"idx\": \"23\",\n \"title\": \"Koh, T. S., Simmons, C. B., Eriksson, M. A., Coppersmith, S. N. & Friesen, M. Unconventional transport in the ‘hole’ regime of a Si double quantum dot.Phys. Rev. Lett.106, 186801 (2011).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.106.186801\"\n },\n {\n \"idx\": \"24\",\n \"title\": \"Jouravlev, O. N. & Nazarov, Y. V. Electron transport in a double quantum dot governed by a nuclear magnetic field.Phys. Rev. Lett.96, 176804 (2006).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.96.176804\"\n },\n {\n \"idx\": \"25\",\n \"title\": \"Taylor, J. M. et al. Relaxation, dephasing, and quantum control of electron spins in double quantum dots.Phys. Rev. B76, 035315 (2007).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.76.035315\"\n },\n {\n \"idx\": \"26\",\n \"title\": \"Koppens, F. H. L. et al. Detection of single electron spin resonance in a double quantum dot.J. Appl. Phys.101, 081706 (2007).\",\n \"link\": \"https://doi.org/10.1063%2F1.2722734\"\n },\n {\n \"idx\": \"28\",\n \"title\": \"Engel, H.-A. & Loss, D. Detection of single spin decoherence in a quantum dot via charge currents.Phys. Rev. Lett.86, 4648–4651 (2001).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.86.4648\"\n },\n {\n \"idx\": \"29\",\n \"title\": \"Assali, L. V. C. et al. Hyperfine interactions in silicon quantum dots.Phys. Rev. B83, 165301 (2011).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.83.165301\"\n },\n {\n \"idx\": \"30\",\n \"title\": \"Wang, K., Payette, C., Dovzhenko, Y., Deelman, P. W. & Petta, J. R. Charge relaxation in a single-electron Si/SiGe double quantum dot.Phys. Rev. Lett.111, 046801 (2013).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.111.046801\"\n },\n {\n \"idx\": \"31\",\n \"title\": \"Danon, J. & Nazarov, Y. V. Pauli spin blockade in the presence of strong spin-orbit coupling.Phys. Rev. B80, 041301 (2009).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.80.041301\"\n },\n {\n \"idx\": \"32\",\n \"title\": \"Nadj-Perge, S. et al. Spectroscopy of spin-orbit quantum bits in indium antimonide nanowires.Phys. Rev. Lett.108, 166801 (2012).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.108.166801\"\n },\n {\n \"idx\": \"33\",\n \"title\": \"Culcer, D., Cywiński, L., Li, Q., Hu, X. & Das Sarma, S. Quantum dot spin qubits in silicon: multivalley physics.Phys. Rev. B82, 155312 (2010).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.82.155312\"\n },\n {\n \"idx\": \"34\",\n \"title\": \"Burkard, G., Loss, D. & DiVincenzo, D. P. Coupled quantum dots as quantum gates.Phys. Rev. B59, 2070–2078 (1999).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.59.2070\"\n },\n {\n \"idx\": \"35\",\n \"title\": \"Gamble, J. K., Eriksson, M. A., Coppersmith, S. N. & Friesen, M. Disorder-induced valley-orbit hybrid states in Si quantum dots.Phys. Rev. B88, 035310 (2013).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.88.035310\"\n },\n {\n \"idx\": \"36\",\n \"title\": \"Pályi, A. & Burkard, G. Hyperfine-induced valley mixing and the spin-valley blockade in carbon-based quantum dots.Phys. Rev. B80, 201404(R) (2009).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.80.201404\"\n },\n {\n \"idx\": \"38\",\n \"title\": \"Xiao, M., House, M. G. & Jiang, H. W. Measurement of the spin relaxation time of single electrons in a silicon metal-oxide-semiconductor-based quantum dot.Phys. Rev. Lett.104, 096801 (2010).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.104.096801\"\n },\n {\n \"idx\": \"39\",\n \"title\": \"Xiao, M., House, M. G. & Jiang, H. W. Parallel spin filling and energy spectroscopy in few-electron Si metal-on-semiconductor-based quantum dots.Appl. Phys. Lett.97, 032103 (2010).\",\n \"link\": \"https://doi.org/10.1063%2F1.3464324\"\n }\n]","isConversational":false}}},{"rowIdx":97,"cells":{"unique_id":{"kind":"string","value":"s41467-023-39358-9"},"subfield":{"kind":"string","value":"Materials science"},"title":{"kind":"string","value":"Interspecies exciton interactions lead to enhanced nonlinearity of dipolar excitons and polaritons in MoS2homobilayers"},"abstract":{"kind":"string","value":"Nonlinear interactions between excitons strongly coupled to light are key for accessing quantum many-body phenomena in polariton systems. Atomically-thin two-dimensional semiconductors provide an attractive platform for strong light-matter coupling owing to many controllable excitonic degrees of freedom. Among these, the recently emerged exciton hybridization opens access to unexplored excitonic species, with a promise of enhanced interactions. Here, we employ hybridized interlayer excitons (hIX) in bilayer MoS 2 to achieve highly nonlinear excitonic and polaritonic effects. Such interlayer excitons possess an out-of-plane electric dipole as well as an unusually large oscillator strength allowing observation of dipolar polaritons (dipolaritons) in bilayers in optical microcavities. Compared to excitons and polaritons in MoS 2 monolayers, both hIX and dipolaritons exhibit ≈ 8 times higher nonlinearity, which is further strongly enhanced when hIX and intralayer excitons, sharing the same valence band, are excited simultaneously. This provides access to an unusual nonlinear regime which we describe theoretically as a mixed effect of Pauli exclusion and exciton-exciton interactions enabled through charge tunnelling. The presented insight into many-body interactions provides new tools for accessing few-polariton quantum correlations."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"Excitons in two-dimensional transition metal dichalcogenides (TMDs) have large oscillator strengths and binding energies [1] , making them attractive as a platform for studies of strong light-matter coupling in optical microcavities [2] , [3] , [4] , [5] . A variety of polaritonic states have been realised using monolayers of MX 2 (M=Mo, W; X=S, Se) embedded in tunable [3] , [5] , [6] , [7] and monolithic microcavities [8] , [9] , [10] , [11] , [12] . One of the central research themes in polaritonics is the study of nonlinear interactions leading to extremely rich phenomena such as Bose-Einstein condensation [13] , [14] , polariton lasing [15] , [16] or optical parametric amplification [17] . Polaritons formed from tightly bound neutral intralayer excitons in TMDs are not expected to show strong nonlinearity. However, pronounced nonlinear behaviour was observed for trion polaritons [7] , [18] and Rydberg polaritons [19] . Enhanced nonlinearity can be achieved by employing excitonic states with a physically separated electron and hole, e.g. in adjacent atomic layers [20] or quantum wells [21] , [22] , [23] , [24] , [25] . Such interlayer excitons have a large out-of-plane electric dipole moment, and thus can strongly mutually interact [26] . Typically, however, interlayer or ‘spatially indirect’ excitons possess low oscillator strength [20] , [27] . Thus, in order to strongly couple to cavity photons, hybridization with high-oscillator-strength intralayer excitons is required [11] , [22] , [24] , [25] , [28] . An attractive approach for realization of dipolar excitons and polaritons is to employ the recently discovered exciton hybridization in MoS 2 bilayers [29] , [30] . This approach allows realization of uniform samples suitable for the observation of macroscopic many-body phenomena [31] . Interlayer excitons unique to bilayer MoS 2 possess a large oscillator strength, comparable to that of the intralayer exciton, arising from interlayer hybridization of valence band states, aided by a favourable orbital overlap and a relatively small spin-orbit splitting among semiconducting TMDs [29] . Such hybridized interlayer excitons (hIX) are highly tunable using out-of-plane electric field [32] , [33] and their valley degree of freedom persists up to room temperature [34] . Here we use hIXs in bilayer MoS 2 to realize highly nonlinear excitonic and dipolaritonic effects. We unravel a previously unexplored interaction regime involving intra- and interlayer excitons stemming from the fermionic nature of the charge carriers in a valence band shared between different excitonic species. This regime, accessible using broadband excitation resonant with both hIX and intralayer exciton transitions, provides strong (up to 10 times) enhancement of the exciton nonlinearity, already enhanced by up to 8 times in MoS 2 bilayers compared with monolayers. We support our experimental findings with a theoretical discussion of nonlinear contributions to energy shifts and exciton broadening, which become enabled in the presence of charge tunnelling.","references":{"indices":[1,2,3,5,6,7,9,10,11,13,14,15,16,17,18,20,22,24,25,26,27,28,29,30,31,32,33,34],"infos":[{"idx":"1","title":"Wang, G. et al. Colloquium: Excitons in atomically thin transition metal dichalcogenides.Rev. Mod. Phys.90, 21001 (2018).","link":"https://doi.org/10.1103%2FRevModPhys.90.021001"},{"idx":"2","title":"Liu, X. et al. Strong light-matter coupling in two-dimensional atomic crystals.Nat. Photonics9, 30 (2014).","link":"https://doi.org/10.1038%2Fnphoton.2014.304"},{"idx":"3","title":"Dufferwiel, S. et al. Exciton-polaritons in van der Waals heterostructures embedded in tunable microcavities.Nat. Commun.6, 8579 (2015).","link":"https://doi.org/10.1038%2Fncomms9579"},{"idx":"5","title":"Sidler, M. et al. Fermi polaron-polaritons in charge-tunable atomically thin semiconductors.Nat. Phys.13, 255 (2017).","link":"https://doi.org/10.1038%2Fnphys3949"},{"idx":"6","title":"Dufferwiel, S. et al. Valley-addressable polaritons in atomically thin semiconductors.Nat. Photonics11, 497 (2017).","link":"https://doi.org/10.1038%2Fnphoton.2017.125"},{"idx":"7","title":"Lyons, T. et al. Giant effective zeeman splitting in a monolayer semiconductor realized by spin-selective strong light–matter coupling.Nat. Photonics16, 632 (2022).","link":"https://doi.org/10.1038%2Fs41566-022-01025-8"},{"idx":"9","title":"Gu, J., Chakraborty, B., Khatoniar, M. & Menon, V. M. A room-temperature polariton light-emitting diode based on monolayer WS2.Nat. Nanotechnol.14, 1024 (2019).","link":"https://doi.org/10.1038%2Fs41565-019-0543-6"},{"idx":"10","title":"Lundt, N. et al. Optical valley hall effect for highly valley-coherent exciton-polaritons in an atomically thin semiconductor.Nat. Nanotechnol.14, 770 (2019).","link":"https://doi.org/10.1038%2Fs41565-019-0492-0"},{"idx":"11","title":"Zhang, L. et al. Van der waals heterostructure polaritons with moiré-induced nonlinearity.Nature591, 61 (2021).","link":"https://doi.org/10.1038%2Fs41586-021-03228-5"},{"idx":"13","title":"Deng, H., Haug, H. & Yamamoto, Y. Exciton-polariton Bose-Einstein condensation.Rev. Mod. Phys.82, 1489 (2010).","link":"https://doi.org/10.1103%2FRevModPhys.82.1489"},{"idx":"14","title":"Kasprzak, J. et al. Bose-Einstein condensation of exciton polaritons.Nature443, 409 (2006).","link":"https://doi.org/10.1038%2Fnature05131"},{"idx":"15","title":"Christopoulos, S. et al. Room-temperature polariton lasing in semiconductor microcavities.Phys. Rev. Lett.98, 126405 (2007).","link":"https://doi.org/10.1103%2FPhysRevLett.98.126405"},{"idx":"16","title":"Bhattacharya, P. et al. Room temperature electrically injected polariton laser.Phys. Rev. Lett.112, 236802 (2014).","link":"https://doi.org/10.1103%2FPhysRevLett.112.236802"},{"idx":"17","title":"Amo, A. et al. Collective fluid dynamics of a polariton condensate in a semiconductor microcavity.Nature457, 291 (2009).","link":"https://doi.org/10.1038%2Fnature07640"},{"idx":"18","title":"Emmanuele, R. et al. Highly nonlinear trion-polaritons in a monolayer semiconductor.Nat. Commun.11, 3589 (2020).","link":"https://doi.org/10.1038%2Fs41467-020-17340-z"},{"idx":"20","title":"Rivera, P. et al. Interlayer valley excitons in heterobilayers of transition metal dichalcogenides.Nat. Nanotechnol.13, 1004 (2018).","link":"https://doi.org/10.1038%2Fs41565-018-0193-0"},{"idx":"22","title":"Kyriienko, O., Magnusson, E. B. & Shelykh, I. A. Spin dynamics of cold exciton condensates.Phys. Rev. B86, 115324 (2012).","link":"https://doi.org/10.1103%2FPhysRevB.86.115324"},{"idx":"24","title":"Cristofolini, P. et al. Coupling quantum tunneling with cavity photons.Science336, 704 (2012).","link":"https://doi.org/10.1126%2Fscience.1219010"},{"idx":"25","title":"Togan, E., Lim, H.-T., Faelt, S., Wegscheider, W. & Imamoglu, A. Enhanced interactions between dipolar polaritons.Phys. Rev. Lett.121, 227402 (2018).","link":"https://doi.org/10.1103%2FPhysRevLett.121.227402"},{"idx":"26","title":"Butov, L., Gossard, A. & Chemla, D. Macroscopically ordered state in an exciton system.Nature418, 751 (2002).","link":"https://doi.org/10.1038%2Fnature00943"},{"idx":"27","title":"Fox, A., Miller, D., Livescu, G., Cunningham, J. & Jan, W. Excitonic effects in coupled quantum wells.Phys. Rev. B44, 6231 (1991).","link":"https://doi.org/10.1103%2FPhysRevB.44.6231"},{"idx":"28","title":"Alexeev, E. M. et al. Resonantly hybridized excitons in moiré superlattices in van der waals heterostructures.Nature567, 81 (2019).","link":"https://doi.org/10.1038%2Fs41586-019-0986-9"},{"idx":"29","title":"Gerber, I. C. et al. Interlayer excitons in bilayer MoS2with strong oscillator strength up to room temperature.Phys. Rev. B99, 035443 (2019).","link":"https://doi.org/10.1103%2FPhysRevB.99.035443"},{"idx":"30","title":"Paradisanos, I. et al. Controlling interlayer excitons in MoS2layers grown by chemical vapor deposition.Nat. Commun.11, 2391 (2020).","link":"https://doi.org/10.1038%2Fs41467-020-16023-z"},{"idx":"31","title":"Amo, A. et al. Collective fluid dynamics of a polariton condensate in a semiconductor microcavity.Nature457, 291 (2009).","link":"https://doi.org/10.1038%2Fnature07640"},{"idx":"32","title":"Leisgang, N. et al. Giant Stark splitting of an exciton in bilayer MoS2.Nat. Nanotechnol.15, 901 (2020).","link":"https://doi.org/10.1038%2Fs41565-020-0750-1"},{"idx":"33","title":"Lorchat, E. et al. Excitons in bilayer mos 2 displaying a colossal electric field splitting and tunable magnetic response.Phys. Rev. Lett.126, 037401 (2021).","link":"https://doi.org/10.1103%2FPhysRevLett.126.037401"},{"idx":"34","title":"Peimyoo, N. et al. Electrical tuning of optically active interlayer excitons in bilayer MoS2.Nat. Nanotechnol.16, 888 (2021).","link":"https://doi.org/10.1038%2Fs41565-021-00916-1"}]}},{"section":"Results","content":"Inter- and intra-layer exciton hybridization in bilayer MoS 2 Our heterostructure samples consists of a MoS 2 bilayer (BL) sandwiched between hBN and placed on a distributed Bragg reflector (DBR). Figure 1 a shows a bright field microscope image of the encapsulated BL MoS 2 . A sketch of the side view of the device is displayed in Fig. 1 b. The reflectance contrast (RC) spectrum of the studied MoS 2 bilayer, displayed in Fig. 1 c, shows three peaks: the intralayer neutral excitons X A at at 1.937 eV (see Fig. 1 d), hybridized interlayer exciton hIX at 2.004 eV and hybridized B-exciton at 2.113 eV. Due to the quantum tunnelling of holes, B-excitons hybridize with an interlayer exciton (IX) (Fig. 1 d), which is a direct transition in the bilayer momentum space [29] . The ratio of the integrated intensities of X A and hIX is 4.5. Based on these data, we estimate the electron-hole separation d = 0.55 nm (see details in Supplementary Note S1) in agreement with previous studies [34] . We further confirm the nature of the hIX states by placing the BL MoS 2 in magnetic field where the valley degeneracy is lifted (Fig. 1 e). In agreement with recent studies [33] , [35] , we measure a Zeeman splitting with an opposite sign and larger magnitude in hIX compared with X A (−3.5 versus 1.5 meV at 8 Tesla). Fig. 1: Homobilayer MoS 2 and its optical response. a Bright field microscope image of an encapsulated BL MoS 2 transferred on top of a DBR. b Schematic side-view of the fabricated heterostructure comprising a BL MoS 2 sandwiched between few-layer hBN. c Reflectance contrast (RC) spectrum of the sample measured at low temperature (4 K) showing three distinct absorption features at 1.937 eV, 2.004 eV and 2.113 eV for X A , hIX and hX B , respectively. The measured linewidths for X A , hIX, and hX B are 20, 23 and 64 meV, respectively. RC is calculated using the formula in the top-right corner of the graph. d Sketch of the conduction and valence bands in two adjacent layers of MoS 2 , displaying the allowed optical transitions of A and B direct intralayer excitons (X A and X B ) and interlayer excitons (IX) for spin-up states (black lines) at the K point in the bilayer momentum space. IX hybridizes with X B through the hole tunnelling between the two layers (red dashed arrow). At the K' point of the bilayer Brillouin zone, the same configuration applies for the states with the opposite spins. e RC spectra of excitons in BL MoS 2 detected in two circular polarizations in an out-of-plane magnetic field of 8 T at T =4 K. Zeeman shifts of opposite signs are observed for X A and hIX. The absorption peak of the charged intralayer exciton (X \\({}_{{{{{{{{\\rm{A}}}}}}}}}^{*}\\) ) shows near unity circular polarization. Full size image Hybridised interlayer excitons in the strong-coupling regime We study the strong coupling regime in a tunable planar microcavity (Fig. 2 a) formed by a silver mirror and a planar DBR [3] . RC scans as a function of the cavity mode detuning Δ = E c a v − E e x c , where E c a v and E e x c are the cavity mode and the corresponding exciton energy, respectively, are shown in Fig. 2 b, c. Characteristic anticrossings of the cavity mode with X A and hIX are observed, resulting in lower, middle and upper polariton branches (LPB, MPB, and UPB, respectively). The extracted Rabi splittings are \\({\\Omega }_{{{{{{{{{\\rm{X}}}}}}}}}_{{{{{{{{\\rm{A}}}}}}}}}}=38\\) meV for X A and Ω hIX = 19 meV for hIX (Supplementary Note S2) . Figure 2 d shows the RC spectra in the vicinity of the anticrossing with hIX, providing a more detailed view of the formation of the MPB and UPB. The intensity of the polariton peaks is relatively low for the states with a high exciton fraction at positive (negative) cavity detunings for the MPB (UPB). As the Rabi splitting scales as a square root of the oscillator strength, the ratio \\({\\Omega }_{{{{{{{{{\\rm{X}}}}}}}}}_{{{{{{{{\\rm{A}}}}}}}}}}/{\\Omega }_{{{{{{{{\\rm{hIX}}}}}}}}}=2\\) is in a good agreement with the RC data for integrated intensities of X A and hIX. From the Rabi splitting ratio we can estimate the tunnelling constant J leading to the B-exciton hybridization. The corresponding coefficient is J = 48 meV (see Supplementary Note S1 for details), matching the density functional theory predictions [29] . In polarization-resolved cavity scans in an out-of-plane magnetic field (Fig. 2 e, f), similarly to hIX behaviour, we observe opposite and larger Zeeman splitting for dipolaritons relative to the intralayer polaritons (see Supplementary Fig. S4) . Chiral dipolariton states are observed distinguished by their opposite circular polarization (Fig. 2 f). Fig. 2: Strong exciton-photon coupling in MoS 2 bilayers. a Schematics of the tunable open microcavity composed of a bottom DBR and a top semi-transparent silver mirror. b , c Low temperature (4K) RC spectra measured as a function of the cavity-exciton detuning (Δ = E cav − E exc ) for cavity scans across b X A and c hIX energies. White dotted lines show the fitting obtained using the coupled oscillator model providing the Rabi splittings Ω hIX = 19 meV and \\({\\Omega }_{{{{{{{{{\\rm{X}}}}}}}}}_{{{{{{{{\\rm{A}}}}}}}}}}=38\\) meV. d RC spectra measured for the cavity-exciton detunings in the vicinity of the anticrossing between hIX and the cavity mode. e Dipolariton dispersion measured with circularly polarized detection for 8 T magnetic field. The orange and black solid curves are the coupled oscillator model fits for σ + and σ − detection, respectively. The positions of the Zeeman-split hIX peaks are shown by dashed lines. f σ + (orange) and σ − (black) RC spectra measured at 8 T at the hIX-cavity anticrossing. Fitting with two Lorentzians (solid lines) is shown. Full size image Nonlinear properties of hybridized interlayer excitons We investigate the nonlinear response of X A and hIX in the bare BL flake performing fluence-dependent reflectivity experiments, illuminating the sample with ultrashort (≈150 fs) pump pulses (in a single beam experiment) in both narrow band (NB, full-width at half maximum, FWHM = 28 nm) and broad-band (BB, FWHM = 50 nm) configurations (see “Methods”). Our separate resonant pump-probe experiments have confirmed that the lifetimes of the hIX and X A states are considerably longer than the pulse duration of ≈ 150 fs (Supplementary Note S3) . Measured RC spectra are shown in Fig. 3 a, b for the NB and in Fig. 3 c for BB excitation. In the NB case, the excitation was tuned to excite either X A or hIX independently, while in the BB case, both resonances were excited simultaneously. Fig. 3: Exciton nonlinearity in MoS 2 bilayers. RC spectra measured with the NB (FWHM=28 nm) excitation for the X A ( a ) and hIX ( b ), and with the BB (FWHM = 50 nm) excitation ( c ) at different fluences. The dashed curves are guide for the eye. d The energy shift Δ E (top) and normalized integrated RC intensity (bottom) as a function of the exciton density for the hIX (left) and X A (right). Solid (open) symbols show the results for the BB (NB) excitation. For the normalized RC we divide the spectrally integrated RC at each laser fluence by its maximum value measured for the whole power dependence. Schematic diagram showing exciton generation under the NB ( e ) and BB ( f ) excitation. In ( e ) only generation of hIX is shown. In ( f ), the holes of the two excitonic species share the same valence band. Full size image As seen in Fig. 3 a, b both X A and hIX spectra behave similarly upon increasing the power of the NB excitation: a blueshift of several meV is observed, accompanied by the peak broadening and bleaching. For the BB excitation, however, a different nonlinear behaviour is observed as shown in Fig. 3 c: the broadening and complete suppression of the hIX peak is observed at much lower powers, accompanied by a redshift. This is in contrast to X A , whose behaviour is similar under the two excitation regimes. The resulting energy shifts and intensities are shown in Fig. 3 d as a function of the exciton density (see details in Supplementary Note S4 and S5) . Figure 3 d quantifies the trends observed in Fig. 3 a–c showing for the BB excitation an abrupt bleaching of the hIX peak above the hIX density 5 × 10 3 μm −2 accompanied by a redshift of ≈ 4 meV and a 12 meV broadening (see Supplementary Note S6) . For the NB case, a similar decrease in peak intensity is observed only around 4 × 10 4 μm −2 , accompanied with a peak blueshift of ≈ 7 meV and a broadening exceeding 15 meV (see Supplementary Figure S10) . However, it is apparent that the observed behaviour under the two excitation regimes is similar for X A . A similar blueshift, broadening and saturation are observed at slightly higher densities compared to the hIX under the NB excitation (Supplementary Note S7) . We also find that due to the increased excitonic Bohr radius, the onset of the nonlinear behaviour for X A in bilayers occurs at a lower exciton density than for X A in monolayers (Supplementary Note S8) . We develop a microscopic model to describe the contrasting phenomena under the NB and BB excitation. Under the NB excitation, either X A or hIX excitons are created as sketched in Fig. 3 e. In this case, nonlinearity arises from Coulomb exciton-exciton interactions causing the blueshift and dephasing [36] . For simplicity, in the main text we will use a Coulomb potential V Coul combining the exchange and direct terms further detailed in Supplementary Note S9 . We confirm that for the intralayer exciton-exciton interaction (X A -X A ) the dominant nonlinear contribution comes from the Coulomb exchange processes, as in the monolayer case [36] , [37] , while for the hIX-hIX scattering the dominant contribution is from the direct Coulomb (dipole-dipole) interaction terms [22] . We find that for the modest electron-hole separation d = 0.55 nm in the bilayer, V Coul is overall 2.3 times stronger for hIX compared with X A . For X A and hIX of the same dipole orientation the Coulomb interaction is repulsive, corresponding to the experimentally observed blueshifts. At the same time, opposite dipole orientation leads to the negative energy shifts, and the total contribution of interlayer scattering depends on possible asymmetry in the system (Supplementary Note 11) . Analysing the shapes of the reflectance spectra in the NB case, we note that they depend on the rates of radiative (Γ R ) and non-radiative (Γ NR ) processes. The area under RC curves is described by the ratio Γ R /(Γ R + Γ NR ). This ratio changes under the increased excitation if the rates depend on the exciton densities. Specifically, we account for the scattering-induced non-radiative processes that microscopically scale as Γ NR ∝ ∣ V Coul ∣ 2 n , i.e. depend on the absolute value of the combined matrix elements for the Coulomb interactions and the exciton density n [36] . This process allows reproducing the RC behaviour and bleaching at increasing pump intensity. Moreover, it explains stronger nonlinearity for X A in bilayers compared to monolayers. Namely, the scattering scales with the exciton Bohr radius, V Coul ∝ α , which is larger in the bilayers due to the enhanced screening (Supplementary Note S9) . We note that the full analysis of the observed behaviour should take into account the inhomogeneous broadening, the effect of which nonetheless can be neglected for the qualitative analysis presented in the paper. In the BB case, both X A and hIX excitons are generated simultaneously, and together with intraspecies scattering (X A -X A and hIX-hIX), interspecies scattering (X A -hIX) occurs, similarly to the direct-indirect exciton Coulomb scattering in double quantum wells [38] . Since X A and hIX are formed by the holes from the same valence band (Fig. 3 f), an additional contribution arises from the phase space filling, i.e. the commutation relations for the excitons (composite bosons) start to deviate from the ideal weak-density limit once more particles are created [39] . For particles of the same flavour, the phase space filling enables nonlinear saturation effects in the strong coupling regime, similar to polariton saturation observed in Ref. [18] . However, in the presence of several exciton species, we reveal a distinct mixture of the phase space filling and inter-exciton interactions which we term the tunnelling-enabled nonlinearity . Specifically, we note that the commutator of the X A annihilation operator ( \\(\\hat{X}\\) ) and hIX creation operator ( \\({\\hat{I}}^{{{{\\dagger}}} }\\) ) is non-zero, \\([\\hat{X}({{{{{{{\\bf{p}}}}}}}}),{\\hat{I}}^{{{{\\dagger}}} }({{{{{{{\\bf{q}}}}}}}})]=-{\\hat{B}}_{{{{{{{{\\bf{p}}}}}}}},{{{{{{{\\bf{q}}}}}}}}}\\) , meaning that modes are not independent. Here p , q are exciton momenta and \\({\\hat{B}}_{{{{{{{{\\bf{p}}}}}}}},{{{{{{{\\bf{q}}}}}}}}}\\) is an operator denoting the deviation from the ideal commuting case ( \\({\\hat{B}}_{{{{{{{{\\bf{p}}}}}}}},{{{{{{{\\bf{q}}}}}}}}}=0\\) ) of distinct bosons where holes do not compete for the valence band space. This statistical property of modes that share a hole, and experience phase space filling, has consequences for the nonlinear response. Namely, the total energy is evaluated as an expectation value over a many-body state with both X A and hIX excitons, \\(\\left|{N}_{{{{{{{{\\rm{X}}}}}}}}},{N}_{{{{{{{{\\rm{hIX}}}}}}}}}\\right\\rangle :=(\\mathop{\\prod }\\nolimits_{{{{{{{{\\bf{p}}}}}}}}}^{{N}_{{{{{{{{\\rm{X}}}}}}}}}}{\\hat{X}}^{{{{\\dagger}}} })(\\mathop{\\prod }\\nolimits_{{{{{{{{\\bf{q}}}}}}}}}^{{N}_{{{{{{{{\\rm{hIX}}}}}}}}}}{\\hat{I}}^{{{{\\dagger}}} })\\left|{\\Omega }_{0}\\right\\rangle\\) , where N X and N hIX particles are created from the ground state \\(\\left|{\\Omega }_{0}\\right\\rangle\\) . If the excitonic modes are independent, the contributions from X A and hIX simply add up. However, the hole coexistence in the valence band induces the excitonic interspecies scattering. The phase space filling then affects both Coulomb-related processes, radiative lifetime of excitonic modes, and scattering amplitudes, which can lead to redshifts for hIX (see Supplementary Notes S9 – S11) . The interplay of these effects is unique to bilayers as tunnelling is absent in monolayers. Moreover, in contrast to standard epitaxially-grown double quantum wells where the layers are separated up to 10 nm, much smaller interlayer separation in TMD bilayers (0.5 nm) allows for stronger tunnelling-enabled nonlinearity. We note that the magnitude of shifts depends on the mode populations as well as interaction constants. For instance, the redshift for X A scales as \\(\\Delta {E}_{{{{{{{{{\\rm{X}}}}}}}}}_{{{{{{{{\\rm{A}}}}}}}}}}={g}_{{{{{{{{\\rm{D}}}}}}}}-{{{{{{{\\rm{I}}}}}}}}}{n}_{{{{{{{{\\rm{hIX}}}}}}}}}\\) . The population of hIX is much smaller than X A for a given power and the imbalance grows further as the fluence is increased (see Supplementary Note S12) . This can explain that the redshift induced by tunnelling-enabled nonlinear processes for X A is less apparent than for hIX under BB illumination. At the same time, the shifts for hIX are proportional to the density of intralayer excitons, \\({n}_{{{{{{{{{\\rm{X}}}}}}}}}_{{{{{{{{\\rm{A}}}}}}}}}}\\) , and thus the effect of the BB excitation shall be most pronounced for the interlayer mode. In addition, hIX-hIX scattering plays a significant role for increasing nonradiative rates. In this case, interactions of both repulsive and attractive sign lead to enhanced scattering channels (irrespective of dipole orientation), and thus results into RC spectra bleaching at lower hIX exciton densities (Supplementary Note S9) . Using the estimated tunnelling-enabled nonlinearity coefficients, we model the RC in the BB regime and qualitatively reproduce the strong bleaching and redshift for hIX at the increased density (see Supplementary Fig. S13) . We observe the same BB-enhanced tunnelling-enabled nonlinearity in samples with lower inhomogeneous broadening (Supplementary Note S13) , confirming that our findings are not masked by disorder. In our analysis on the excitonic nonlinearities, we neglect the effect of trions and biexcitons [40] , [41] , since the natural doping is relatively small in our sample (see Fig. 1 c, e) and the excitation densities required to generate a substantial population of biexcitons are considerably higher than those used in our experiments [42] , [43] , [44] . Moreover, we observe a redshift only for interlayer excitons and specifically under broadband excitation, while an excess of charges would have produced the same effect on both intra and interlayer excitons. In fact, the interspecies interactions in our study are mediated through charge tunnelling and cannot be explained as a consequence of the presence of many-body complexes. Nonlinear behaviour of dipolar polaritons We investigate nonlinear properties of dipolar polaritons in a monolithic (fixed-length) cavity created by a silver mirror on top of a PMMA spacer (245 nm thick) covering the hBN-encapsulated MoS 2 homobilayer placed on the DBR. The cavity mode energy can be tuned by varying the angle of observation (0 degrees corresponds to normal incidence). We use a microscopy setup optimized for Fourier-plane imaging, thus allowing simultaneous detection of reflectivity spectra in a range of angles as shown in (Fig. 4 a) displaying the measured polariton dispersion. In this experiment, the cavity mode is tuned around hIX and only two polariton branches LPB and UPB are observed at low fluence of 0.6 μ J cm −2 with a characteristic Rabi splitting of 17.5 meV. In Fig. 4 b, at an increased fluence of 58.5 μJ cm −2 , only a weakly coupled cavity mode is visible. Figure 4 c shows RC spectra taken at ~6.5 ∘ around the anticrossing at different laser fluences. The collapse of the two polariton peaks into one peak signifying the transition to the weak coupling regime is observed above 25 μJ cm −2 . The LPB and UPB energies extracted using the coupled oscillator model (Supplementary Fig. S5) are shown in (Fig. 4 d). As the polariton density is increased, the LPB and UPB approach each other almost symmetrically, converging to the exciton energy. The corresponding normalized Rabi splitting (Ω/Ω m a x , where Ω m a x is measured at low fluence) is shown in Fig. 4 d, e as a function of the total polariton density. Fig. 4: Nonlinear behaviour of dipolaritons. a , b Reflectance contrast spectra measured at different laser fluences for the MoS 2 bilayer placed in a monolithic cavity. a The low fluence case (0.6 μJ cm −2 ). A clear anticrossing at 6.5 ∘ is observed. Dashed red lines show the results of the fitting using a coupled oscillator model, with two polariton branches LPB and UPB formed. White and orange lines show the energies of the uncoupled cavity mode and hIX state, respectively. The vertical line marks the anticrossing angle. b The high fluence case (58.5 μJ cm −2 ). A complete collapse of the strong coupling regime is observed, with the disappearance of the anticrossing and transition into the weak coupling regime. c RC spectra measured at the anticrossing at 6.5 ∘ as a function of the laser fluence. d Measured UPB and LPB peak energies at 6.5 ∘ as a function of the laser fluence (see top axis) and the corresponding polariton density (bottom axis). e Symbols show the Rabi splittings normalized by the Rabi splitting measured at the lowest power (Ω/Ω m a x ) as deduced from ( d ). The line shows the fitting using our theoretical model (Supplementary Note S9) . Full size image In this experiment, the cavity mode is considerably above the X A energy, which therefore is not coupled to the cavity. Hence, the extracted Rabi splittings are fitted with a theoretically predicted trend of Ω for the NB excitation regime (Supplementary Note S9) . A nonlinear polariton coefficient β = 0.86 μ eV μ m 2 is extracted by differentiating the fitted function with respect to the polariton density. Comparing our results to X A intralayer-exciton-polaritons in monolayers in similar cavities [11] , we observe that the nonlinearity coefficient for dipolar interlayer polaritons is about an order of magnitude larger. This is in a good agreement with the theoretically predicted intrinsic nonlinearity of hybridized interlayer polaritons (Supplementary Note S9) , and with our experimental data for X A taken on a monolayer embedded in a microcavity (Supplementary Note S14) ."},{"section":"Discussion","content":"In summary, we report the nonlinear exciton and exciton-polariton behaviour in MoS 2 homobilayers, a unique system where hybridized interlayer exciton states can be realized having a large oscillator strength. We find that nonlinearity in MoS 2 bilayers can be enhanced when both the intralayer and interlayer states are excited simultaneously, the regime that qualitatively changes the exciton-exciton interaction through the tunnelling-enabled nonlinearity. This approach enriches the exciton nonlinear behaviour in atomically thin semiconductors, leading to unique effects compared to any other semiconducting system. In this broad-band excitation regime, the bleaching of the hIX absorption occurs at 8 times lower hIX densities compared to the narrow-band excitation case when the interlayer excitons are generated on their own. In addition to this, we find that the dipolar nature of hIX states in MoS 2 homobilayers already results in 10 times stronger nonlinearity compared with the intralayer excitons in MoS 2 monolayers. Thus, we report on an overall enhancement of the exciton nonlinearity by nearly two orders of magnitude. Thanks to the large oscillator strength, hIX can enter the strong coupling regime in MoS 2 bilayers placed in microcavities, as realized in our work. Similarly to hIX states themselves, dipolar polaritons also show 10 times stronger nonlinearity compared with exciton-polaritons in MoS 2 monolayers. We conclude that dipolaritons in MoS 2 bilayers uniquely combine strong nonlinearity, large Rabi splitting and fabrication reproducibility among other TMD systems studied so far [9] , [11] , [18] (see Supplementary Note 14) . We note that in the monolayer samples, the nonlinearity arising from Pauli blocking is only pronounced in the strong light-matter coupling regime [18] , [45] , and is not significant in the weak coupling. On the contrary, for bilayers we discover other nonlinear processes that crucially arise due to the presence of interlayer tunnelling. This is specific to the broad band (BB) illumination regime, where the simultaneous occupation of intra- and interlayer exciton states effectively activates this channel of nonlinearity. We expect that in microcavities where the cavity mode is coupled to both hIX and X A in MoS 2 bilayers, and the excitation similar to the broad-band regime can thus be realized, the nonlinear polariton coefficient will be dramatically enhanced owing to the tunnelling-enabled nonlinearity effect, allowing highly nonlinear polariton system to be realized. We thus predict that MoS 2 bilayers will be an attractive platform for realization of quantum-correlated polaritons with applications in polariton logic networks [46] and polariton blockade [47] , [48] . After the submission of this manuscript another article reporting strong coupling and nonlinear effects in MoS 2 bilayers was published [49] ."},{"section":"Methods","content":"Sample fabrication The hBN/MoS 2 /hBN heterostructures were assembled using a PDMS polymer stamp method. The PMMA spacer for the monolithic cavity was deposited using a spin-coating technique, while a silver mirror of 45 nm was thermally evaporated on top of it. Optical studies Broad-band excitation was used to measure the reflectance contrast (RC) spectra of the devices at cryogenic temperatures, defined as RC = ( R sub − R BL )/ R sub , where R sub and R BL are the substrate and MoS 2 bilayer reflectivity, respectively. For the magnetic field studies the same RC measurements were performed using unpolarized light in excitation and with λ /4 and λ /2 waveplates and a linear polarizer in collection to resolve σ + and σ − polarized light. The low temperature measurements using the tunable cavity were carried out in a liquid helium bath cryostat (T = 4.2K) equipped with a superconducting magnet and free beam optical access. We used a white light LED as a source. RC spectra were measured at each cavity length and were integrated over the angles within 5 degrees from normal incidence. The RC spectra measured in the cavity are fitted using Lorentzians. The peak positions are then used to fit to a coupled oscillator model, producing the Rabi splitting and the exciton and cavity mode energies. The measurements on the monolithic cavity were performed in a helium flow cryostat (T = 6K). For the power-dependent RC experiments, we used supercontinuum radiation produced by 100 fs Ti:Sapphire laser pulses at 2 kHz repetition rate at 1.55 eV propagating through a thin sapphire crystal. The supercontinuun radiation was then filtered to produce the desired narrow-band excitation. Density calculations All the exciton and polariton densities were calculated following the procedure introduced by L. Zhang et al. [11] , taking into account the spectral overlap of the spectrum of the excitation laser and the investigated exciton peak (see further details in Supplementary Note S4) ."}],"string":"[\n {\n \"section\": \"Introduction\",\n \"content\": \"Excitons in two-dimensional transition metal dichalcogenides (TMDs) have large oscillator strengths and binding energies [1] , making them attractive as a platform for studies of strong light-matter coupling in optical microcavities [2] , [3] , [4] , [5] . A variety of polaritonic states have been realised using monolayers of MX 2 (M=Mo, W; X=S, Se) embedded in tunable [3] , [5] , [6] , [7] and monolithic microcavities [8] , [9] , [10] , [11] , [12] . One of the central research themes in polaritonics is the study of nonlinear interactions leading to extremely rich phenomena such as Bose-Einstein condensation [13] , [14] , polariton lasing [15] , [16] or optical parametric amplification [17] . Polaritons formed from tightly bound neutral intralayer excitons in TMDs are not expected to show strong nonlinearity. However, pronounced nonlinear behaviour was observed for trion polaritons [7] , [18] and Rydberg polaritons [19] . Enhanced nonlinearity can be achieved by employing excitonic states with a physically separated electron and hole, e.g. in adjacent atomic layers [20] or quantum wells [21] , [22] , [23] , [24] , [25] . Such interlayer excitons have a large out-of-plane electric dipole moment, and thus can strongly mutually interact [26] . Typically, however, interlayer or ‘spatially indirect’ excitons possess low oscillator strength [20] , [27] . Thus, in order to strongly couple to cavity photons, hybridization with high-oscillator-strength intralayer excitons is required [11] , [22] , [24] , [25] , [28] . An attractive approach for realization of dipolar excitons and polaritons is to employ the recently discovered exciton hybridization in MoS 2 bilayers [29] , [30] . This approach allows realization of uniform samples suitable for the observation of macroscopic many-body phenomena [31] . Interlayer excitons unique to bilayer MoS 2 possess a large oscillator strength, comparable to that of the intralayer exciton, arising from interlayer hybridization of valence band states, aided by a favourable orbital overlap and a relatively small spin-orbit splitting among semiconducting TMDs [29] . Such hybridized interlayer excitons (hIX) are highly tunable using out-of-plane electric field [32] , [33] and their valley degree of freedom persists up to room temperature [34] . Here we use hIXs in bilayer MoS 2 to realize highly nonlinear excitonic and dipolaritonic effects. We unravel a previously unexplored interaction regime involving intra- and interlayer excitons stemming from the fermionic nature of the charge carriers in a valence band shared between different excitonic species. This regime, accessible using broadband excitation resonant with both hIX and intralayer exciton transitions, provides strong (up to 10 times) enhancement of the exciton nonlinearity, already enhanced by up to 8 times in MoS 2 bilayers compared with monolayers. We support our experimental findings with a theoretical discussion of nonlinear contributions to energy shifts and exciton broadening, which become enabled in the presence of charge tunnelling.\",\n \"references\": {\n \"indices\": [\n 1,\n 2,\n 3,\n 5,\n 6,\n 7,\n 9,\n 10,\n 11,\n 13,\n 14,\n 15,\n 16,\n 17,\n 18,\n 20,\n 22,\n 24,\n 25,\n 26,\n 27,\n 28,\n 29,\n 30,\n 31,\n 32,\n 33,\n 34\n ],\n \"infos\": [\n {\n \"idx\": \"1\",\n \"title\": \"Wang, G. et al. Colloquium: Excitons in atomically thin transition metal dichalcogenides.Rev. Mod. Phys.90, 21001 (2018).\",\n \"link\": \"https://doi.org/10.1103%2FRevModPhys.90.021001\"\n },\n {\n \"idx\": \"2\",\n \"title\": \"Liu, X. et al. Strong light-matter coupling in two-dimensional atomic crystals.Nat. Photonics9, 30 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fnphoton.2014.304\"\n },\n {\n \"idx\": \"3\",\n \"title\": \"Dufferwiel, S. et al. Exciton-polaritons in van der Waals heterostructures embedded in tunable microcavities.Nat. Commun.6, 8579 (2015).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms9579\"\n },\n {\n \"idx\": \"5\",\n \"title\": \"Sidler, M. et al. Fermi polaron-polaritons in charge-tunable atomically thin semiconductors.Nat. Phys.13, 255 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys3949\"\n },\n {\n \"idx\": \"6\",\n \"title\": \"Dufferwiel, S. et al. Valley-addressable polaritons in atomically thin semiconductors.Nat. Photonics11, 497 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fnphoton.2017.125\"\n },\n {\n \"idx\": \"7\",\n \"title\": \"Lyons, T. et al. Giant effective zeeman splitting in a monolayer semiconductor realized by spin-selective strong light–matter coupling.Nat. Photonics16, 632 (2022).\",\n \"link\": \"https://doi.org/10.1038%2Fs41566-022-01025-8\"\n },\n {\n \"idx\": \"9\",\n \"title\": \"Gu, J., Chakraborty, B., Khatoniar, M. & Menon, V. M. A room-temperature polariton light-emitting diode based on monolayer WS2.Nat. Nanotechnol.14, 1024 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41565-019-0543-6\"\n },\n {\n \"idx\": \"10\",\n \"title\": \"Lundt, N. et al. Optical valley hall effect for highly valley-coherent exciton-polaritons in an atomically thin semiconductor.Nat. Nanotechnol.14, 770 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41565-019-0492-0\"\n },\n {\n \"idx\": \"11\",\n \"title\": \"Zhang, L. et al. Van der waals heterostructure polaritons with moiré-induced nonlinearity.Nature591, 61 (2021).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-021-03228-5\"\n },\n {\n \"idx\": \"13\",\n \"title\": \"Deng, H., Haug, H. & Yamamoto, Y. Exciton-polariton Bose-Einstein condensation.Rev. Mod. Phys.82, 1489 (2010).\",\n \"link\": \"https://doi.org/10.1103%2FRevModPhys.82.1489\"\n },\n {\n \"idx\": \"14\",\n \"title\": \"Kasprzak, J. et al. Bose-Einstein condensation of exciton polaritons.Nature443, 409 (2006).\",\n \"link\": \"https://doi.org/10.1038%2Fnature05131\"\n },\n {\n \"idx\": \"15\",\n \"title\": \"Christopoulos, S. et al. Room-temperature polariton lasing in semiconductor microcavities.Phys. Rev. Lett.98, 126405 (2007).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.98.126405\"\n },\n {\n \"idx\": \"16\",\n \"title\": \"Bhattacharya, P. et al. Room temperature electrically injected polariton laser.Phys. Rev. Lett.112, 236802 (2014).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.112.236802\"\n },\n {\n \"idx\": \"17\",\n \"title\": \"Amo, A. et al. Collective fluid dynamics of a polariton condensate in a semiconductor microcavity.Nature457, 291 (2009).\",\n \"link\": \"https://doi.org/10.1038%2Fnature07640\"\n },\n {\n \"idx\": \"18\",\n \"title\": \"Emmanuele, R. et al. Highly nonlinear trion-polaritons in a monolayer semiconductor.Nat. Commun.11, 3589 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-020-17340-z\"\n },\n {\n \"idx\": \"20\",\n \"title\": \"Rivera, P. et al. Interlayer valley excitons in heterobilayers of transition metal dichalcogenides.Nat. Nanotechnol.13, 1004 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41565-018-0193-0\"\n },\n {\n \"idx\": \"22\",\n \"title\": \"Kyriienko, O., Magnusson, E. B. & Shelykh, I. A. Spin dynamics of cold exciton condensates.Phys. Rev. B86, 115324 (2012).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.86.115324\"\n },\n {\n \"idx\": \"24\",\n \"title\": \"Cristofolini, P. et al. Coupling quantum tunneling with cavity photons.Science336, 704 (2012).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1219010\"\n },\n {\n \"idx\": \"25\",\n \"title\": \"Togan, E., Lim, H.-T., Faelt, S., Wegscheider, W. & Imamoglu, A. Enhanced interactions between dipolar polaritons.Phys. Rev. Lett.121, 227402 (2018).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.121.227402\"\n },\n {\n \"idx\": \"26\",\n \"title\": \"Butov, L., Gossard, A. & Chemla, D. Macroscopically ordered state in an exciton system.Nature418, 751 (2002).\",\n \"link\": \"https://doi.org/10.1038%2Fnature00943\"\n },\n {\n \"idx\": \"27\",\n \"title\": \"Fox, A., Miller, D., Livescu, G., Cunningham, J. & Jan, W. Excitonic effects in coupled quantum wells.Phys. Rev. B44, 6231 (1991).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.44.6231\"\n },\n {\n \"idx\": \"28\",\n \"title\": \"Alexeev, E. M. et al. Resonantly hybridized excitons in moiré superlattices in van der waals heterostructures.Nature567, 81 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-019-0986-9\"\n },\n {\n \"idx\": \"29\",\n \"title\": \"Gerber, I. C. et al. Interlayer excitons in bilayer MoS2with strong oscillator strength up to room temperature.Phys. Rev. B99, 035443 (2019).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.99.035443\"\n },\n {\n \"idx\": \"30\",\n \"title\": \"Paradisanos, I. et al. Controlling interlayer excitons in MoS2layers grown by chemical vapor deposition.Nat. Commun.11, 2391 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-020-16023-z\"\n },\n {\n \"idx\": \"31\",\n \"title\": \"Amo, A. et al. Collective fluid dynamics of a polariton condensate in a semiconductor microcavity.Nature457, 291 (2009).\",\n \"link\": \"https://doi.org/10.1038%2Fnature07640\"\n },\n {\n \"idx\": \"32\",\n \"title\": \"Leisgang, N. et al. Giant Stark splitting of an exciton in bilayer MoS2.Nat. Nanotechnol.15, 901 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41565-020-0750-1\"\n },\n {\n \"idx\": \"33\",\n \"title\": \"Lorchat, E. et al. Excitons in bilayer mos 2 displaying a colossal electric field splitting and tunable magnetic response.Phys. Rev. Lett.126, 037401 (2021).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.126.037401\"\n },\n {\n \"idx\": \"34\",\n \"title\": \"Peimyoo, N. et al. Electrical tuning of optically active interlayer excitons in bilayer MoS2.Nat. Nanotechnol.16, 888 (2021).\",\n \"link\": \"https://doi.org/10.1038%2Fs41565-021-00916-1\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Inter- and intra-layer exciton hybridization in bilayer MoS 2 Our heterostructure samples consists of a MoS 2 bilayer (BL) sandwiched between hBN and placed on a distributed Bragg reflector (DBR). Figure 1 a shows a bright field microscope image of the encapsulated BL MoS 2 . A sketch of the side view of the device is displayed in Fig. 1 b. The reflectance contrast (RC) spectrum of the studied MoS 2 bilayer, displayed in Fig. 1 c, shows three peaks: the intralayer neutral excitons X A at at 1.937 eV (see Fig. 1 d), hybridized interlayer exciton hIX at 2.004 eV and hybridized B-exciton at 2.113 eV. Due to the quantum tunnelling of holes, B-excitons hybridize with an interlayer exciton (IX) (Fig. 1 d), which is a direct transition in the bilayer momentum space [29] . The ratio of the integrated intensities of X A and hIX is 4.5. Based on these data, we estimate the electron-hole separation d = 0.55 nm (see details in Supplementary Note S1) in agreement with previous studies [34] . We further confirm the nature of the hIX states by placing the BL MoS 2 in magnetic field where the valley degeneracy is lifted (Fig. 1 e). In agreement with recent studies [33] , [35] , we measure a Zeeman splitting with an opposite sign and larger magnitude in hIX compared with X A (−3.5 versus 1.5 meV at 8 Tesla). Fig. 1: Homobilayer MoS 2 and its optical response. a Bright field microscope image of an encapsulated BL MoS 2 transferred on top of a DBR. b Schematic side-view of the fabricated heterostructure comprising a BL MoS 2 sandwiched between few-layer hBN. c Reflectance contrast (RC) spectrum of the sample measured at low temperature (4 K) showing three distinct absorption features at 1.937 eV, 2.004 eV and 2.113 eV for X A , hIX and hX B , respectively. The measured linewidths for X A , hIX, and hX B are 20, 23 and 64 meV, respectively. RC is calculated using the formula in the top-right corner of the graph. d Sketch of the conduction and valence bands in two adjacent layers of MoS 2 , displaying the allowed optical transitions of A and B direct intralayer excitons (X A and X B ) and interlayer excitons (IX) for spin-up states (black lines) at the K point in the bilayer momentum space. IX hybridizes with X B through the hole tunnelling between the two layers (red dashed arrow). At the K' point of the bilayer Brillouin zone, the same configuration applies for the states with the opposite spins. e RC spectra of excitons in BL MoS 2 detected in two circular polarizations in an out-of-plane magnetic field of 8 T at T =4 K. Zeeman shifts of opposite signs are observed for X A and hIX. The absorption peak of the charged intralayer exciton (X \\\\({}_{{{{{{{{\\\\rm{A}}}}}}}}}^{*}\\\\) ) shows near unity circular polarization. Full size image Hybridised interlayer excitons in the strong-coupling regime We study the strong coupling regime in a tunable planar microcavity (Fig. 2 a) formed by a silver mirror and a planar DBR [3] . RC scans as a function of the cavity mode detuning Δ = E c a v − E e x c , where E c a v and E e x c are the cavity mode and the corresponding exciton energy, respectively, are shown in Fig. 2 b, c. Characteristic anticrossings of the cavity mode with X A and hIX are observed, resulting in lower, middle and upper polariton branches (LPB, MPB, and UPB, respectively). The extracted Rabi splittings are \\\\({\\\\Omega }_{{{{{{{{{\\\\rm{X}}}}}}}}}_{{{{{{{{\\\\rm{A}}}}}}}}}}=38\\\\) meV for X A and Ω hIX = 19 meV for hIX (Supplementary Note S2) . Figure 2 d shows the RC spectra in the vicinity of the anticrossing with hIX, providing a more detailed view of the formation of the MPB and UPB. The intensity of the polariton peaks is relatively low for the states with a high exciton fraction at positive (negative) cavity detunings for the MPB (UPB). As the Rabi splitting scales as a square root of the oscillator strength, the ratio \\\\({\\\\Omega }_{{{{{{{{{\\\\rm{X}}}}}}}}}_{{{{{{{{\\\\rm{A}}}}}}}}}}/{\\\\Omega }_{{{{{{{{\\\\rm{hIX}}}}}}}}}=2\\\\) is in a good agreement with the RC data for integrated intensities of X A and hIX. From the Rabi splitting ratio we can estimate the tunnelling constant J leading to the B-exciton hybridization. The corresponding coefficient is J = 48 meV (see Supplementary Note S1 for details), matching the density functional theory predictions [29] . In polarization-resolved cavity scans in an out-of-plane magnetic field (Fig. 2 e, f), similarly to hIX behaviour, we observe opposite and larger Zeeman splitting for dipolaritons relative to the intralayer polaritons (see Supplementary Fig. S4) . Chiral dipolariton states are observed distinguished by their opposite circular polarization (Fig. 2 f). Fig. 2: Strong exciton-photon coupling in MoS 2 bilayers. a Schematics of the tunable open microcavity composed of a bottom DBR and a top semi-transparent silver mirror. b , c Low temperature (4K) RC spectra measured as a function of the cavity-exciton detuning (Δ = E cav − E exc ) for cavity scans across b X A and c hIX energies. White dotted lines show the fitting obtained using the coupled oscillator model providing the Rabi splittings Ω hIX = 19 meV and \\\\({\\\\Omega }_{{{{{{{{{\\\\rm{X}}}}}}}}}_{{{{{{{{\\\\rm{A}}}}}}}}}}=38\\\\) meV. d RC spectra measured for the cavity-exciton detunings in the vicinity of the anticrossing between hIX and the cavity mode. e Dipolariton dispersion measured with circularly polarized detection for 8 T magnetic field. The orange and black solid curves are the coupled oscillator model fits for σ + and σ − detection, respectively. The positions of the Zeeman-split hIX peaks are shown by dashed lines. f σ + (orange) and σ − (black) RC spectra measured at 8 T at the hIX-cavity anticrossing. Fitting with two Lorentzians (solid lines) is shown. Full size image Nonlinear properties of hybridized interlayer excitons We investigate the nonlinear response of X A and hIX in the bare BL flake performing fluence-dependent reflectivity experiments, illuminating the sample with ultrashort (≈150 fs) pump pulses (in a single beam experiment) in both narrow band (NB, full-width at half maximum, FWHM = 28 nm) and broad-band (BB, FWHM = 50 nm) configurations (see “Methods”). Our separate resonant pump-probe experiments have confirmed that the lifetimes of the hIX and X A states are considerably longer than the pulse duration of ≈ 150 fs (Supplementary Note S3) . Measured RC spectra are shown in Fig. 3 a, b for the NB and in Fig. 3 c for BB excitation. In the NB case, the excitation was tuned to excite either X A or hIX independently, while in the BB case, both resonances were excited simultaneously. Fig. 3: Exciton nonlinearity in MoS 2 bilayers. RC spectra measured with the NB (FWHM=28 nm) excitation for the X A ( a ) and hIX ( b ), and with the BB (FWHM = 50 nm) excitation ( c ) at different fluences. The dashed curves are guide for the eye. d The energy shift Δ E (top) and normalized integrated RC intensity (bottom) as a function of the exciton density for the hIX (left) and X A (right). Solid (open) symbols show the results for the BB (NB) excitation. For the normalized RC we divide the spectrally integrated RC at each laser fluence by its maximum value measured for the whole power dependence. Schematic diagram showing exciton generation under the NB ( e ) and BB ( f ) excitation. In ( e ) only generation of hIX is shown. In ( f ), the holes of the two excitonic species share the same valence band. Full size image As seen in Fig. 3 a, b both X A and hIX spectra behave similarly upon increasing the power of the NB excitation: a blueshift of several meV is observed, accompanied by the peak broadening and bleaching. For the BB excitation, however, a different nonlinear behaviour is observed as shown in Fig. 3 c: the broadening and complete suppression of the hIX peak is observed at much lower powers, accompanied by a redshift. This is in contrast to X A , whose behaviour is similar under the two excitation regimes. The resulting energy shifts and intensities are shown in Fig. 3 d as a function of the exciton density (see details in Supplementary Note S4 and S5) . Figure 3 d quantifies the trends observed in Fig. 3 a–c showing for the BB excitation an abrupt bleaching of the hIX peak above the hIX density 5 × 10 3 μm −2 accompanied by a redshift of ≈ 4 meV and a 12 meV broadening (see Supplementary Note S6) . For the NB case, a similar decrease in peak intensity is observed only around 4 × 10 4 μm −2 , accompanied with a peak blueshift of ≈ 7 meV and a broadening exceeding 15 meV (see Supplementary Figure S10) . However, it is apparent that the observed behaviour under the two excitation regimes is similar for X A . A similar blueshift, broadening and saturation are observed at slightly higher densities compared to the hIX under the NB excitation (Supplementary Note S7) . We also find that due to the increased excitonic Bohr radius, the onset of the nonlinear behaviour for X A in bilayers occurs at a lower exciton density than for X A in monolayers (Supplementary Note S8) . We develop a microscopic model to describe the contrasting phenomena under the NB and BB excitation. Under the NB excitation, either X A or hIX excitons are created as sketched in Fig. 3 e. In this case, nonlinearity arises from Coulomb exciton-exciton interactions causing the blueshift and dephasing [36] . For simplicity, in the main text we will use a Coulomb potential V Coul combining the exchange and direct terms further detailed in Supplementary Note S9 . We confirm that for the intralayer exciton-exciton interaction (X A -X A ) the dominant nonlinear contribution comes from the Coulomb exchange processes, as in the monolayer case [36] , [37] , while for the hIX-hIX scattering the dominant contribution is from the direct Coulomb (dipole-dipole) interaction terms [22] . We find that for the modest electron-hole separation d = 0.55 nm in the bilayer, V Coul is overall 2.3 times stronger for hIX compared with X A . For X A and hIX of the same dipole orientation the Coulomb interaction is repulsive, corresponding to the experimentally observed blueshifts. At the same time, opposite dipole orientation leads to the negative energy shifts, and the total contribution of interlayer scattering depends on possible asymmetry in the system (Supplementary Note 11) . Analysing the shapes of the reflectance spectra in the NB case, we note that they depend on the rates of radiative (Γ R ) and non-radiative (Γ NR ) processes. The area under RC curves is described by the ratio Γ R /(Γ R + Γ NR ). This ratio changes under the increased excitation if the rates depend on the exciton densities. Specifically, we account for the scattering-induced non-radiative processes that microscopically scale as Γ NR ∝ ∣ V Coul ∣ 2 n , i.e. depend on the absolute value of the combined matrix elements for the Coulomb interactions and the exciton density n [36] . This process allows reproducing the RC behaviour and bleaching at increasing pump intensity. Moreover, it explains stronger nonlinearity for X A in bilayers compared to monolayers. Namely, the scattering scales with the exciton Bohr radius, V Coul ∝ α , which is larger in the bilayers due to the enhanced screening (Supplementary Note S9) . We note that the full analysis of the observed behaviour should take into account the inhomogeneous broadening, the effect of which nonetheless can be neglected for the qualitative analysis presented in the paper. In the BB case, both X A and hIX excitons are generated simultaneously, and together with intraspecies scattering (X A -X A and hIX-hIX), interspecies scattering (X A -hIX) occurs, similarly to the direct-indirect exciton Coulomb scattering in double quantum wells [38] . Since X A and hIX are formed by the holes from the same valence band (Fig. 3 f), an additional contribution arises from the phase space filling, i.e. the commutation relations for the excitons (composite bosons) start to deviate from the ideal weak-density limit once more particles are created [39] . For particles of the same flavour, the phase space filling enables nonlinear saturation effects in the strong coupling regime, similar to polariton saturation observed in Ref. [18] . However, in the presence of several exciton species, we reveal a distinct mixture of the phase space filling and inter-exciton interactions which we term the tunnelling-enabled nonlinearity . Specifically, we note that the commutator of the X A annihilation operator ( \\\\(\\\\hat{X}\\\\) ) and hIX creation operator ( \\\\({\\\\hat{I}}^{{{{\\\\dagger}}} }\\\\) ) is non-zero, \\\\([\\\\hat{X}({{{{{{{\\\\bf{p}}}}}}}}),{\\\\hat{I}}^{{{{\\\\dagger}}} }({{{{{{{\\\\bf{q}}}}}}}})]=-{\\\\hat{B}}_{{{{{{{{\\\\bf{p}}}}}}}},{{{{{{{\\\\bf{q}}}}}}}}}\\\\) , meaning that modes are not independent. Here p , q are exciton momenta and \\\\({\\\\hat{B}}_{{{{{{{{\\\\bf{p}}}}}}}},{{{{{{{\\\\bf{q}}}}}}}}}\\\\) is an operator denoting the deviation from the ideal commuting case ( \\\\({\\\\hat{B}}_{{{{{{{{\\\\bf{p}}}}}}}},{{{{{{{\\\\bf{q}}}}}}}}}=0\\\\) ) of distinct bosons where holes do not compete for the valence band space. This statistical property of modes that share a hole, and experience phase space filling, has consequences for the nonlinear response. Namely, the total energy is evaluated as an expectation value over a many-body state with both X A and hIX excitons, \\\\(\\\\left|{N}_{{{{{{{{\\\\rm{X}}}}}}}}},{N}_{{{{{{{{\\\\rm{hIX}}}}}}}}}\\\\right\\\\rangle :=(\\\\mathop{\\\\prod }\\\\nolimits_{{{{{{{{\\\\bf{p}}}}}}}}}^{{N}_{{{{{{{{\\\\rm{X}}}}}}}}}}{\\\\hat{X}}^{{{{\\\\dagger}}} })(\\\\mathop{\\\\prod }\\\\nolimits_{{{{{{{{\\\\bf{q}}}}}}}}}^{{N}_{{{{{{{{\\\\rm{hIX}}}}}}}}}}{\\\\hat{I}}^{{{{\\\\dagger}}} })\\\\left|{\\\\Omega }_{0}\\\\right\\\\rangle\\\\) , where N X and N hIX particles are created from the ground state \\\\(\\\\left|{\\\\Omega }_{0}\\\\right\\\\rangle\\\\) . If the excitonic modes are independent, the contributions from X A and hIX simply add up. However, the hole coexistence in the valence band induces the excitonic interspecies scattering. The phase space filling then affects both Coulomb-related processes, radiative lifetime of excitonic modes, and scattering amplitudes, which can lead to redshifts for hIX (see Supplementary Notes S9 – S11) . The interplay of these effects is unique to bilayers as tunnelling is absent in monolayers. Moreover, in contrast to standard epitaxially-grown double quantum wells where the layers are separated up to 10 nm, much smaller interlayer separation in TMD bilayers (0.5 nm) allows for stronger tunnelling-enabled nonlinearity. We note that the magnitude of shifts depends on the mode populations as well as interaction constants. For instance, the redshift for X A scales as \\\\(\\\\Delta {E}_{{{{{{{{{\\\\rm{X}}}}}}}}}_{{{{{{{{\\\\rm{A}}}}}}}}}}={g}_{{{{{{{{\\\\rm{D}}}}}}}}-{{{{{{{\\\\rm{I}}}}}}}}}{n}_{{{{{{{{\\\\rm{hIX}}}}}}}}}\\\\) . The population of hIX is much smaller than X A for a given power and the imbalance grows further as the fluence is increased (see Supplementary Note S12) . This can explain that the redshift induced by tunnelling-enabled nonlinear processes for X A is less apparent than for hIX under BB illumination. At the same time, the shifts for hIX are proportional to the density of intralayer excitons, \\\\({n}_{{{{{{{{{\\\\rm{X}}}}}}}}}_{{{{{{{{\\\\rm{A}}}}}}}}}}\\\\) , and thus the effect of the BB excitation shall be most pronounced for the interlayer mode. In addition, hIX-hIX scattering plays a significant role for increasing nonradiative rates. In this case, interactions of both repulsive and attractive sign lead to enhanced scattering channels (irrespective of dipole orientation), and thus results into RC spectra bleaching at lower hIX exciton densities (Supplementary Note S9) . Using the estimated tunnelling-enabled nonlinearity coefficients, we model the RC in the BB regime and qualitatively reproduce the strong bleaching and redshift for hIX at the increased density (see Supplementary Fig. S13) . We observe the same BB-enhanced tunnelling-enabled nonlinearity in samples with lower inhomogeneous broadening (Supplementary Note S13) , confirming that our findings are not masked by disorder. In our analysis on the excitonic nonlinearities, we neglect the effect of trions and biexcitons [40] , [41] , since the natural doping is relatively small in our sample (see Fig. 1 c, e) and the excitation densities required to generate a substantial population of biexcitons are considerably higher than those used in our experiments [42] , [43] , [44] . Moreover, we observe a redshift only for interlayer excitons and specifically under broadband excitation, while an excess of charges would have produced the same effect on both intra and interlayer excitons. In fact, the interspecies interactions in our study are mediated through charge tunnelling and cannot be explained as a consequence of the presence of many-body complexes. Nonlinear behaviour of dipolar polaritons We investigate nonlinear properties of dipolar polaritons in a monolithic (fixed-length) cavity created by a silver mirror on top of a PMMA spacer (245 nm thick) covering the hBN-encapsulated MoS 2 homobilayer placed on the DBR. The cavity mode energy can be tuned by varying the angle of observation (0 degrees corresponds to normal incidence). We use a microscopy setup optimized for Fourier-plane imaging, thus allowing simultaneous detection of reflectivity spectra in a range of angles as shown in (Fig. 4 a) displaying the measured polariton dispersion. In this experiment, the cavity mode is tuned around hIX and only two polariton branches LPB and UPB are observed at low fluence of 0.6 μ J cm −2 with a characteristic Rabi splitting of 17.5 meV. In Fig. 4 b, at an increased fluence of 58.5 μJ cm −2 , only a weakly coupled cavity mode is visible. Figure 4 c shows RC spectra taken at ~6.5 ∘ around the anticrossing at different laser fluences. The collapse of the two polariton peaks into one peak signifying the transition to the weak coupling regime is observed above 25 μJ cm −2 . The LPB and UPB energies extracted using the coupled oscillator model (Supplementary Fig. S5) are shown in (Fig. 4 d). As the polariton density is increased, the LPB and UPB approach each other almost symmetrically, converging to the exciton energy. The corresponding normalized Rabi splitting (Ω/Ω m a x , where Ω m a x is measured at low fluence) is shown in Fig. 4 d, e as a function of the total polariton density. Fig. 4: Nonlinear behaviour of dipolaritons. a , b Reflectance contrast spectra measured at different laser fluences for the MoS 2 bilayer placed in a monolithic cavity. a The low fluence case (0.6 μJ cm −2 ). A clear anticrossing at 6.5 ∘ is observed. Dashed red lines show the results of the fitting using a coupled oscillator model, with two polariton branches LPB and UPB formed. White and orange lines show the energies of the uncoupled cavity mode and hIX state, respectively. The vertical line marks the anticrossing angle. b The high fluence case (58.5 μJ cm −2 ). A complete collapse of the strong coupling regime is observed, with the disappearance of the anticrossing and transition into the weak coupling regime. c RC spectra measured at the anticrossing at 6.5 ∘ as a function of the laser fluence. d Measured UPB and LPB peak energies at 6.5 ∘ as a function of the laser fluence (see top axis) and the corresponding polariton density (bottom axis). e Symbols show the Rabi splittings normalized by the Rabi splitting measured at the lowest power (Ω/Ω m a x ) as deduced from ( d ). The line shows the fitting using our theoretical model (Supplementary Note S9) . Full size image In this experiment, the cavity mode is considerably above the X A energy, which therefore is not coupled to the cavity. Hence, the extracted Rabi splittings are fitted with a theoretically predicted trend of Ω for the NB excitation regime (Supplementary Note S9) . A nonlinear polariton coefficient β = 0.86 μ eV μ m 2 is extracted by differentiating the fitted function with respect to the polariton density. Comparing our results to X A intralayer-exciton-polaritons in monolayers in similar cavities [11] , we observe that the nonlinearity coefficient for dipolar interlayer polaritons is about an order of magnitude larger. This is in a good agreement with the theoretically predicted intrinsic nonlinearity of hybridized interlayer polaritons (Supplementary Note S9) , and with our experimental data for X A taken on a monolayer embedded in a microcavity (Supplementary Note S14) .\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"In summary, we report the nonlinear exciton and exciton-polariton behaviour in MoS 2 homobilayers, a unique system where hybridized interlayer exciton states can be realized having a large oscillator strength. We find that nonlinearity in MoS 2 bilayers can be enhanced when both the intralayer and interlayer states are excited simultaneously, the regime that qualitatively changes the exciton-exciton interaction through the tunnelling-enabled nonlinearity. This approach enriches the exciton nonlinear behaviour in atomically thin semiconductors, leading to unique effects compared to any other semiconducting system. In this broad-band excitation regime, the bleaching of the hIX absorption occurs at 8 times lower hIX densities compared to the narrow-band excitation case when the interlayer excitons are generated on their own. In addition to this, we find that the dipolar nature of hIX states in MoS 2 homobilayers already results in 10 times stronger nonlinearity compared with the intralayer excitons in MoS 2 monolayers. Thus, we report on an overall enhancement of the exciton nonlinearity by nearly two orders of magnitude. Thanks to the large oscillator strength, hIX can enter the strong coupling regime in MoS 2 bilayers placed in microcavities, as realized in our work. Similarly to hIX states themselves, dipolar polaritons also show 10 times stronger nonlinearity compared with exciton-polaritons in MoS 2 monolayers. We conclude that dipolaritons in MoS 2 bilayers uniquely combine strong nonlinearity, large Rabi splitting and fabrication reproducibility among other TMD systems studied so far [9] , [11] , [18] (see Supplementary Note 14) . We note that in the monolayer samples, the nonlinearity arising from Pauli blocking is only pronounced in the strong light-matter coupling regime [18] , [45] , and is not significant in the weak coupling. On the contrary, for bilayers we discover other nonlinear processes that crucially arise due to the presence of interlayer tunnelling. This is specific to the broad band (BB) illumination regime, where the simultaneous occupation of intra- and interlayer exciton states effectively activates this channel of nonlinearity. We expect that in microcavities where the cavity mode is coupled to both hIX and X A in MoS 2 bilayers, and the excitation similar to the broad-band regime can thus be realized, the nonlinear polariton coefficient will be dramatically enhanced owing to the tunnelling-enabled nonlinearity effect, allowing highly nonlinear polariton system to be realized. We thus predict that MoS 2 bilayers will be an attractive platform for realization of quantum-correlated polaritons with applications in polariton logic networks [46] and polariton blockade [47] , [48] . After the submission of this manuscript another article reporting strong coupling and nonlinear effects in MoS 2 bilayers was published [49] .\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Sample fabrication The hBN/MoS 2 /hBN heterostructures were assembled using a PDMS polymer stamp method. The PMMA spacer for the monolithic cavity was deposited using a spin-coating technique, while a silver mirror of 45 nm was thermally evaporated on top of it. Optical studies Broad-band excitation was used to measure the reflectance contrast (RC) spectra of the devices at cryogenic temperatures, defined as RC = ( R sub − R BL )/ R sub , where R sub and R BL are the substrate and MoS 2 bilayer reflectivity, respectively. For the magnetic field studies the same RC measurements were performed using unpolarized light in excitation and with λ /4 and λ /2 waveplates and a linear polarizer in collection to resolve σ + and σ − polarized light. The low temperature measurements using the tunable cavity were carried out in a liquid helium bath cryostat (T = 4.2K) equipped with a superconducting magnet and free beam optical access. We used a white light LED as a source. RC spectra were measured at each cavity length and were integrated over the angles within 5 degrees from normal incidence. The RC spectra measured in the cavity are fitted using Lorentzians. The peak positions are then used to fit to a coupled oscillator model, producing the Rabi splitting and the exciton and cavity mode energies. The measurements on the monolithic cavity were performed in a helium flow cryostat (T = 6K). For the power-dependent RC experiments, we used supercontinuum radiation produced by 100 fs Ti:Sapphire laser pulses at 2 kHz repetition rate at 1.55 eV propagating through a thin sapphire crystal. The supercontinuun radiation was then filtered to produce the desired narrow-band excitation. Density calculations All the exciton and polariton densities were calculated following the procedure introduced by L. Zhang et al. [11] , taking into account the spectral overlap of the spectrum of the excitation laser and the investigated exciton peak (see further details in Supplementary Note S4) .\"\n }\n]","isConversational":false},"core_idea":{"kind":"string","value":"The core research idea is to utilize hybridized interlayer excitons in bilayer MoS₂ to achieve highly nonlinear excitonic and dipolaritonic effects by exploiting the fermionic nature of charge carriers in a shared valence band, leading to a strong enhancement of exciton nonlinearity through broadband excitation resonant with both inter- and intralayer exciton transitions."},"entities":{"kind":"list like","value":["hybridized interlayer excitons","bilayer MoS₂","nonlinear excitonic effects","dipolaritonic effects","fermionic nature","charge carriers","shared valence band","exciton nonlinearity","broadband excitation","interlayer exciton transitions","intralayer exciton transitions"],"string":"[\n \"hybridized interlayer excitons\",\n \"bilayer MoS₂\",\n \"nonlinear excitonic effects\",\n \"dipolaritonic effects\",\n \"fermionic nature\",\n \"charge carriers\",\n \"shared valence band\",\n \"exciton nonlinearity\",\n \"broadband excitation\",\n \"interlayer exciton transitions\",\n \"intralayer exciton transitions\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1.","title":"Wang, G. et al. Colloquium: Excitons in atomically thin transition metal dichalcogenides.Rev. Mod. Phys.90, 21001 (2018).","link":"https://doi.org/10.1103%2FRevModPhys.90.021001"},{"idx":"2.","title":"Liu, X. et al. Strong light-matter coupling in two-dimensional atomic crystals.Nat. Photonics9, 30 (2014).","link":"https://doi.org/10.1038%2Fnphoton.2014.304"},{"idx":"3.","title":"Dufferwiel, S. et al. Exciton-polaritons in van der Waals heterostructures embedded in tunable microcavities.Nat. Commun.6, 8579 (2015).","link":"https://doi.org/10.1038%2Fncomms9579"},{"idx":"5.","title":"Sidler, M. et al. Fermi polaron-polaritons in charge-tunable atomically thin semiconductors.Nat. Phys.13, 255 (2017).","link":"https://doi.org/10.1038%2Fnphys3949"},{"idx":"6.","title":"Dufferwiel, S. et al. Valley-addressable polaritons in atomically thin semiconductors.Nat. Photonics11, 497 (2017).","link":"https://doi.org/10.1038%2Fnphoton.2017.125"},{"idx":"7.","title":"Lyons, T. et al. Giant effective zeeman splitting in a monolayer semiconductor realized by spin-selective strong light–matter coupling.Nat. Photonics16, 632 (2022).","link":"https://doi.org/10.1038%2Fs41566-022-01025-8"},{"idx":"9.","title":"Gu, J., Chakraborty, B., Khatoniar, M. & Menon, V. M. A room-temperature polariton light-emitting diode based on monolayer WS2.Nat. Nanotechnol.14, 1024 (2019).","link":"https://doi.org/10.1038%2Fs41565-019-0543-6"},{"idx":"10.","title":"Lundt, N. et al. Optical valley hall effect for highly valley-coherent exciton-polaritons in an atomically thin semiconductor.Nat. Nanotechnol.14, 770 (2019).","link":"https://doi.org/10.1038%2Fs41565-019-0492-0"},{"idx":"11.","title":"Zhang, L. et al. Van der waals heterostructure polaritons with moiré-induced nonlinearity.Nature591, 61 (2021).","link":"https://doi.org/10.1038%2Fs41586-021-03228-5"},{"idx":"13.","title":"Deng, H., Haug, H. & Yamamoto, Y. Exciton-polariton Bose-Einstein condensation.Rev. Mod. Phys.82, 1489 (2010).","link":"https://doi.org/10.1103%2FRevModPhys.82.1489"},{"idx":"14.","title":"Kasprzak, J. et al. Bose-Einstein condensation of exciton polaritons.Nature443, 409 (2006).","link":"https://doi.org/10.1038%2Fnature05131"},{"idx":"15.","title":"Christopoulos, S. et al. Room-temperature polariton lasing in semiconductor microcavities.Phys. Rev. Lett.98, 126405 (2007).","link":"https://doi.org/10.1103%2FPhysRevLett.98.126405"},{"idx":"16.","title":"Bhattacharya, P. et al. Room temperature electrically injected polariton laser.Phys. Rev. Lett.112, 236802 (2014).","link":"https://doi.org/10.1103%2FPhysRevLett.112.236802"},{"idx":"17.","title":"Amo, A. et al. Collective fluid dynamics of a polariton condensate in a semiconductor microcavity.Nature457, 291 (2009).","link":"https://doi.org/10.1038%2Fnature07640"},{"idx":"18.","title":"Emmanuele, R. et al. Highly nonlinear trion-polaritons in a monolayer semiconductor.Nat. Commun.11, 3589 (2020).","link":"https://doi.org/10.1038%2Fs41467-020-17340-z"},{"idx":"20.","title":"Rivera, P. et al. Interlayer valley excitons in heterobilayers of transition metal dichalcogenides.Nat. Nanotechnol.13, 1004 (2018).","link":"https://doi.org/10.1038%2Fs41565-018-0193-0"},{"idx":"22.","title":"Kyriienko, O., Magnusson, E. B. & Shelykh, I. A. Spin dynamics of cold exciton condensates.Phys. Rev. B86, 115324 (2012).","link":"https://doi.org/10.1103%2FPhysRevB.86.115324"},{"idx":"24.","title":"Cristofolini, P. et al. Coupling quantum tunneling with cavity photons.Science336, 704 (2012).","link":"https://doi.org/10.1126%2Fscience.1219010"},{"idx":"25.","title":"Togan, E., Lim, H.-T., Faelt, S., Wegscheider, W. & Imamoglu, A. Enhanced interactions between dipolar polaritons.Phys. Rev. Lett.121, 227402 (2018).","link":"https://doi.org/10.1103%2FPhysRevLett.121.227402"},{"idx":"26.","title":"Butov, L., Gossard, A. & Chemla, D. Macroscopically ordered state in an exciton system.Nature418, 751 (2002).","link":"https://doi.org/10.1038%2Fnature00943"},{"idx":"27.","title":"Fox, A., Miller, D., Livescu, G., Cunningham, J. & Jan, W. Excitonic effects in coupled quantum wells.Phys. Rev. B44, 6231 (1991).","link":"https://doi.org/10.1103%2FPhysRevB.44.6231"},{"idx":"28.","title":"Alexeev, E. M. et al. Resonantly hybridized excitons in moiré superlattices in van der waals heterostructures.Nature567, 81 (2019).","link":"https://doi.org/10.1038%2Fs41586-019-0986-9"},{"idx":"29.","title":"Gerber, I. C. et al. Interlayer excitons in bilayer MoS2with strong oscillator strength up to room temperature.Phys. Rev. B99, 035443 (2019).","link":"https://doi.org/10.1103%2FPhysRevB.99.035443"},{"idx":"30.","title":"Paradisanos, I. et al. Controlling interlayer excitons in MoS2layers grown by chemical vapor deposition.Nat. Commun.11, 2391 (2020).","link":"https://doi.org/10.1038%2Fs41467-020-16023-z"},{"idx":"31.","title":"Amo, A. et al. Collective fluid dynamics of a polariton condensate in a semiconductor microcavity.Nature457, 291 (2009).","link":"https://doi.org/10.1038%2Fnature07640"},{"idx":"32.","title":"Leisgang, N. et al. Giant Stark splitting of an exciton in bilayer MoS2.Nat. Nanotechnol.15, 901 (2020).","link":"https://doi.org/10.1038%2Fs41565-020-0750-1"},{"idx":"33.","title":"Lorchat, E. et al. Excitons in bilayer mos 2 displaying a colossal electric field splitting and tunable magnetic response.Phys. Rev. Lett.126, 037401 (2021).","link":"https://doi.org/10.1103%2FPhysRevLett.126.037401"},{"idx":"34.","title":"Peimyoo, N. et al. Electrical tuning of optically active interlayer excitons in bilayer MoS2.Nat. Nanotechnol.16, 888 (2021).","link":"https://doi.org/10.1038%2Fs41565-021-00916-1"},{"idx":"35.","title":"Gong, Z. et al. Magnetoelectric effects and valley-controlled spin quantum gates in transition metal dichalcogenide bilayers.Nat. Commun.4, 2053 (2013).","link":"https://doi.org/10.1038%2Fncomms3053"},{"idx":"36.","title":"Erkensten, D., Brem, S. & Malic, E. Exciton-exciton interaction in transition metal dichalcogenide monolayers and van der waals heterostructures.Phys. Rev. B103, 045426 (2021).","link":"https://doi.org/10.1103%2FPhysRevB.103.045426"},{"idx":"37.","title":"Shahnazaryan, V., Iorsh, I., Shelykh, I. A. & Kyriienko, O. Exciton-exciton interaction in transition-metal dichalcogenide monolayers.Phys. Rev. B96, 115409 (2017).","link":"https://doi.org/10.1103%2FPhysRevB.96.115409"},{"idx":"38.","title":"Kristinsson, K., Kyriienko, O., Liew, T. C. H. & Shelykh, I. A. Continuous terahertz emission from dipolaritons.Phys. Rev. B88, 245303 (2013).","link":"https://doi.org/10.1103%2FPhysRevB.88.245303"},{"idx":"39.","title":"Combescot, M., Betbeder-Matibet, O. & Dubin, F. The many-body physics of composite bosons.Phys. Rep.463, 215 (2008).","link":"https://doi.org/10.1016%2Fj.physrep.2007.11.003"},{"idx":"40.","title":"Drüppel, M., Deilmann, T., Krüger, P. & Rohlfing, M. Diversity of trion states and substrate effects in the optical properties of an MoS2monolayer.Nat. Commun.8, 2117 (2017).","link":"https://doi.org/10.1038%2Fs41467-017-02286-6"},{"idx":"41.","title":"Zhumagulov, Y. V. et al. Microscopic theory of exciton and trion polaritons in doped monolayers of transition metal dichalcogenides.npj Comput. Mater.8, 92 (2022).","link":"https://doi.org/10.1038%2Fs41524-022-00775-x"},{"idx":"42.","title":"Sie, E. J., Frenzel, A. J., Lee, Y.-H., Kong, J. & Gedik, N. Intervalley biexcitons and many-body effects in monolayer mos 2.Phys. Rev. B92, 125417 (2015).","link":"https://doi.org/10.1103%2FPhysRevB.92.125417"},{"idx":"43.","title":"Mai, C. et al. Many-body effects in valleytronics: direct measurement of valley lifetimes in single-layer MoS2.Nano Lett.14, 202 (2014).","link":"https://doi.org/10.1021%2Fnl403742j"},{"idx":"44.","title":"Wang, W. et al. Studying of the biexciton characteristics in monolayer mos2.J. Phys. Chem. C.124, 1749 (2019).","link":"https://doi.org/10.1021%2Facs.jpcc.9b11582"},{"idx":"45.","title":"Tassone, F. & Yamamoto, Y. Exciton-exciton scattering dynamics in a semiconductor microcavity and stimulated scattering into polaritons.Phys. Rev. B59, 10830 (1999).","link":"https://doi.org/10.1103%2FPhysRevB.59.10830"},{"idx":"46.","title":"Berloff, N. G. et al. Realizing the classical xy hamiltonian in polariton simulators.Nat. Mater.16, 1120 (2017).","link":"https://doi.org/10.1038%2Fnmat4971"},{"idx":"47.","title":"Delteil, A. et al. Towards polariton blockade of confined exciton–polaritons.Nat. Mater.18, 219 (2019).","link":"https://doi.org/10.1038%2Fs41563-019-0282-y"},{"idx":"48.","title":"Kyriienko, O., Krizhanovskii, D. & Shelykh, I. Nonlinear quantum optics with trion polaritons in 2d monolayers: conventional and unconventional photon blockade.Phys. Rev. Lett.125, 197402 (2020).","link":"https://doi.org/10.1103%2FPhysRevLett.125.197402"}],"string":"[\n {\n \"idx\": \"1.\",\n \"title\": \"Wang, G. et al. Colloquium: Excitons in atomically thin transition metal dichalcogenides.Rev. Mod. Phys.90, 21001 (2018).\",\n \"link\": \"https://doi.org/10.1103%2FRevModPhys.90.021001\"\n },\n {\n \"idx\": \"2.\",\n \"title\": \"Liu, X. et al. Strong light-matter coupling in two-dimensional atomic crystals.Nat. Photonics9, 30 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fnphoton.2014.304\"\n },\n {\n \"idx\": \"3.\",\n \"title\": \"Dufferwiel, S. et al. Exciton-polaritons in van der Waals heterostructures embedded in tunable microcavities.Nat. Commun.6, 8579 (2015).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms9579\"\n },\n {\n \"idx\": \"5.\",\n \"title\": \"Sidler, M. et al. Fermi polaron-polaritons in charge-tunable atomically thin semiconductors.Nat. Phys.13, 255 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fnphys3949\"\n },\n {\n \"idx\": \"6.\",\n \"title\": \"Dufferwiel, S. et al. Valley-addressable polaritons in atomically thin semiconductors.Nat. Photonics11, 497 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fnphoton.2017.125\"\n },\n {\n \"idx\": \"7.\",\n \"title\": \"Lyons, T. et al. Giant effective zeeman splitting in a monolayer semiconductor realized by spin-selective strong light–matter coupling.Nat. Photonics16, 632 (2022).\",\n \"link\": \"https://doi.org/10.1038%2Fs41566-022-01025-8\"\n },\n {\n \"idx\": \"9.\",\n \"title\": \"Gu, J., Chakraborty, B., Khatoniar, M. & Menon, V. M. A room-temperature polariton light-emitting diode based on monolayer WS2.Nat. Nanotechnol.14, 1024 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41565-019-0543-6\"\n },\n {\n \"idx\": \"10.\",\n \"title\": \"Lundt, N. et al. Optical valley hall effect for highly valley-coherent exciton-polaritons in an atomically thin semiconductor.Nat. Nanotechnol.14, 770 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41565-019-0492-0\"\n },\n {\n \"idx\": \"11.\",\n \"title\": \"Zhang, L. et al. Van der waals heterostructure polaritons with moiré-induced nonlinearity.Nature591, 61 (2021).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-021-03228-5\"\n },\n {\n \"idx\": \"13.\",\n \"title\": \"Deng, H., Haug, H. & Yamamoto, Y. Exciton-polariton Bose-Einstein condensation.Rev. Mod. Phys.82, 1489 (2010).\",\n \"link\": \"https://doi.org/10.1103%2FRevModPhys.82.1489\"\n },\n {\n \"idx\": \"14.\",\n \"title\": \"Kasprzak, J. et al. Bose-Einstein condensation of exciton polaritons.Nature443, 409 (2006).\",\n \"link\": \"https://doi.org/10.1038%2Fnature05131\"\n },\n {\n \"idx\": \"15.\",\n \"title\": \"Christopoulos, S. et al. Room-temperature polariton lasing in semiconductor microcavities.Phys. Rev. Lett.98, 126405 (2007).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.98.126405\"\n },\n {\n \"idx\": \"16.\",\n \"title\": \"Bhattacharya, P. et al. Room temperature electrically injected polariton laser.Phys. Rev. Lett.112, 236802 (2014).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.112.236802\"\n },\n {\n \"idx\": \"17.\",\n \"title\": \"Amo, A. et al. Collective fluid dynamics of a polariton condensate in a semiconductor microcavity.Nature457, 291 (2009).\",\n \"link\": \"https://doi.org/10.1038%2Fnature07640\"\n },\n {\n \"idx\": \"18.\",\n \"title\": \"Emmanuele, R. et al. Highly nonlinear trion-polaritons in a monolayer semiconductor.Nat. Commun.11, 3589 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-020-17340-z\"\n },\n {\n \"idx\": \"20.\",\n \"title\": \"Rivera, P. et al. Interlayer valley excitons in heterobilayers of transition metal dichalcogenides.Nat. Nanotechnol.13, 1004 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41565-018-0193-0\"\n },\n {\n \"idx\": \"22.\",\n \"title\": \"Kyriienko, O., Magnusson, E. B. & Shelykh, I. A. Spin dynamics of cold exciton condensates.Phys. Rev. B86, 115324 (2012).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.86.115324\"\n },\n {\n \"idx\": \"24.\",\n \"title\": \"Cristofolini, P. et al. Coupling quantum tunneling with cavity photons.Science336, 704 (2012).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1219010\"\n },\n {\n \"idx\": \"25.\",\n \"title\": \"Togan, E., Lim, H.-T., Faelt, S., Wegscheider, W. & Imamoglu, A. Enhanced interactions between dipolar polaritons.Phys. Rev. Lett.121, 227402 (2018).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.121.227402\"\n },\n {\n \"idx\": \"26.\",\n \"title\": \"Butov, L., Gossard, A. & Chemla, D. Macroscopically ordered state in an exciton system.Nature418, 751 (2002).\",\n \"link\": \"https://doi.org/10.1038%2Fnature00943\"\n },\n {\n \"idx\": \"27.\",\n \"title\": \"Fox, A., Miller, D., Livescu, G., Cunningham, J. & Jan, W. Excitonic effects in coupled quantum wells.Phys. Rev. B44, 6231 (1991).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.44.6231\"\n },\n {\n \"idx\": \"28.\",\n \"title\": \"Alexeev, E. M. et al. Resonantly hybridized excitons in moiré superlattices in van der waals heterostructures.Nature567, 81 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-019-0986-9\"\n },\n {\n \"idx\": \"29.\",\n \"title\": \"Gerber, I. C. et al. Interlayer excitons in bilayer MoS2with strong oscillator strength up to room temperature.Phys. Rev. B99, 035443 (2019).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.99.035443\"\n },\n {\n \"idx\": \"30.\",\n \"title\": \"Paradisanos, I. et al. Controlling interlayer excitons in MoS2layers grown by chemical vapor deposition.Nat. Commun.11, 2391 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-020-16023-z\"\n },\n {\n \"idx\": \"31.\",\n \"title\": \"Amo, A. et al. Collective fluid dynamics of a polariton condensate in a semiconductor microcavity.Nature457, 291 (2009).\",\n \"link\": \"https://doi.org/10.1038%2Fnature07640\"\n },\n {\n \"idx\": \"32.\",\n \"title\": \"Leisgang, N. et al. Giant Stark splitting of an exciton in bilayer MoS2.Nat. Nanotechnol.15, 901 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41565-020-0750-1\"\n },\n {\n \"idx\": \"33.\",\n \"title\": \"Lorchat, E. et al. Excitons in bilayer mos 2 displaying a colossal electric field splitting and tunable magnetic response.Phys. Rev. Lett.126, 037401 (2021).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.126.037401\"\n },\n {\n \"idx\": \"34.\",\n \"title\": \"Peimyoo, N. et al. Electrical tuning of optically active interlayer excitons in bilayer MoS2.Nat. Nanotechnol.16, 888 (2021).\",\n \"link\": \"https://doi.org/10.1038%2Fs41565-021-00916-1\"\n },\n {\n \"idx\": \"35.\",\n \"title\": \"Gong, Z. et al. Magnetoelectric effects and valley-controlled spin quantum gates in transition metal dichalcogenide bilayers.Nat. Commun.4, 2053 (2013).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms3053\"\n },\n {\n \"idx\": \"36.\",\n \"title\": \"Erkensten, D., Brem, S. & Malic, E. Exciton-exciton interaction in transition metal dichalcogenide monolayers and van der waals heterostructures.Phys. Rev. B103, 045426 (2021).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.103.045426\"\n },\n {\n \"idx\": \"37.\",\n \"title\": \"Shahnazaryan, V., Iorsh, I., Shelykh, I. A. & Kyriienko, O. Exciton-exciton interaction in transition-metal dichalcogenide monolayers.Phys. Rev. B96, 115409 (2017).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.96.115409\"\n },\n {\n \"idx\": \"38.\",\n \"title\": \"Kristinsson, K., Kyriienko, O., Liew, T. C. H. & Shelykh, I. A. Continuous terahertz emission from dipolaritons.Phys. Rev. B88, 245303 (2013).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.88.245303\"\n },\n {\n \"idx\": \"39.\",\n \"title\": \"Combescot, M., Betbeder-Matibet, O. & Dubin, F. The many-body physics of composite bosons.Phys. Rep.463, 215 (2008).\",\n \"link\": \"https://doi.org/10.1016%2Fj.physrep.2007.11.003\"\n },\n {\n \"idx\": \"40.\",\n \"title\": \"Drüppel, M., Deilmann, T., Krüger, P. & Rohlfing, M. Diversity of trion states and substrate effects in the optical properties of an MoS2monolayer.Nat. Commun.8, 2117 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fs41467-017-02286-6\"\n },\n {\n \"idx\": \"41.\",\n \"title\": \"Zhumagulov, Y. V. et al. Microscopic theory of exciton and trion polaritons in doped monolayers of transition metal dichalcogenides.npj Comput. Mater.8, 92 (2022).\",\n \"link\": \"https://doi.org/10.1038%2Fs41524-022-00775-x\"\n },\n {\n \"idx\": \"42.\",\n \"title\": \"Sie, E. J., Frenzel, A. J., Lee, Y.-H., Kong, J. & Gedik, N. Intervalley biexcitons and many-body effects in monolayer mos 2.Phys. Rev. B92, 125417 (2015).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.92.125417\"\n },\n {\n \"idx\": \"43.\",\n \"title\": \"Mai, C. et al. Many-body effects in valleytronics: direct measurement of valley lifetimes in single-layer MoS2.Nano Lett.14, 202 (2014).\",\n \"link\": \"https://doi.org/10.1021%2Fnl403742j\"\n },\n {\n \"idx\": \"44.\",\n \"title\": \"Wang, W. et al. Studying of the biexciton characteristics in monolayer mos2.J. Phys. Chem. C.124, 1749 (2019).\",\n \"link\": \"https://doi.org/10.1021%2Facs.jpcc.9b11582\"\n },\n {\n \"idx\": \"45.\",\n \"title\": \"Tassone, F. & Yamamoto, Y. Exciton-exciton scattering dynamics in a semiconductor microcavity and stimulated scattering into polaritons.Phys. Rev. B59, 10830 (1999).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevB.59.10830\"\n },\n {\n \"idx\": \"46.\",\n \"title\": \"Berloff, N. G. et al. Realizing the classical xy hamiltonian in polariton simulators.Nat. Mater.16, 1120 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat4971\"\n },\n {\n \"idx\": \"47.\",\n \"title\": \"Delteil, A. et al. Towards polariton blockade of confined exciton–polaritons.Nat. Mater.18, 219 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41563-019-0282-y\"\n },\n {\n \"idx\": \"48.\",\n \"title\": \"Kyriienko, O., Krizhanovskii, D. & Shelykh, I. Nonlinear quantum optics with trion polaritons in 2d monolayers: conventional and unconventional photon blockade.Phys. Rev. Lett.125, 197402 (2020).\",\n \"link\": \"https://doi.org/10.1103%2FPhysRevLett.125.197402\"\n }\n]","isConversational":false}}},{"rowIdx":98,"cells":{"unique_id":{"kind":"string","value":"ncomms15729"},"subfield":{"kind":"string","value":"Physics"},"title":{"kind":"string","value":"Reconstruction of stochastic temporal networks through diffusive arrival times"},"abstract":{"kind":"string","value":"Temporal networks have opened a new dimension in defining and quantification of complex interacting systems. Our ability to identify and reproduce time-resolved interaction patterns is, however, limited by the restricted access to empirical individual-level data. Here we propose an inverse modelling method based on first-arrival observations of the diffusion process taking place on temporal networks. We describe an efficient coordinate-ascent implementation for inferring stochastic temporal networks that builds in particular but not exclusively on the null model assumption of mutually independent interaction sequences at the dyadic level. The results of benchmark tests applied on both synthesized and empirical network data sets confirm the validity of our algorithm, showing the feasibility of statistically accurate inference of temporal networks only from moderate-sized samples of diffusion cascades. Our approach provides an effective and flexible scheme for the temporally augmented inverse problems of network reconstruction and has potential in a broad variety of applications."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"The collective functionality of complex networks emerges as a consequence of the interactions among their constituents. Recently, it has been empirically observed that, besides the spatially or topologically structured organization, the temporal constraints imposed on many complex interacting systems add a further dimension crucial for understanding their generic structure and dynamics. Internal links in these so-called temporal networks [1] evolve over time and are inherently registered by a series of rhythmically activated events among interacting actors at discrete time stamps. The causal sequence of the time-ordered links strikingly affects accessibility [2] , an essential and revealing characteristic, especially in social and communication networks related to human activities. Such a profound shift from static to temporally intermittent interactivity properties has expanded, challenged or even redefined many fundamental concepts of networks, including path length [3] , [4] , clustering correlation [3] , [4] , node centrality [3] , [5] , communicability [6] , structural controllability [7] , motif [8] and community structure [9] . Of another interest are the anomalous patterns (for example, bursts and heavy tails) embedded in the interevent time (IET) distribution of temporal interactions [10] , [11] . This marked departure of human activities from Poissonian behaviour substantially alters the dynamical process that takes place on networks, such as epidemic spreading [10] , [11] , [12] , [13] , [14] , random walks [15] , [16] , synchronization [17] , cooperative evolution [18] , consensus and coordination processes [19] , [20] . Undoubtedly, identifying the temporal interaction pattern is a first step in understanding and controlling collective dynamics of empirical temporal networked systems. The vast majority of generative models for temporal networks—aimed at reproducing the empirical sequences of the time-stamped interactions—by and large require a priori knowledge of the raw data, or relevant statistics on the underlying interaction patterns [21] , [22] , [23] , [24] . However, strong limitations often arise regarding the availability of time-resolved interaction data at the individual level. Apart from increased technological expenditures, data collection is also hindered by small observation windows or samples reflecting individual activities only at coarse time granularity. Some data sets potentially suffer from other statistical deficiencies resulting from, for example, the participants with highly correlated behaviour from a skewed population. Another, even more problematic, restriction is imposed by data observability issues. The specifics of temporal interactions, particularly in social and financial networks, are in general obscured due to the privacy concerns of the participants, making their time-extended network structure unobservable, at least in principle. Extracting temporal networks from measurable data at the collective level has hence become a very desirable task. In viral marketing, for example, mining social networks is of critical value to identify highly influential customers who directly affect other consumers’ decision making [25] . However, the interpersonal interactions (word-of-mouth recommendations) are often rendered private and thus indirectly accessible. In such circumstances, the recorded product dissemination history (for example, when customers purchase the products) provides an observable data source for the social network mining. Other candidate data applicable to similar inferential tasks include observed information cascades, such as propagating memes through blog posts [26] and tweets [27] on social media. In this paper, we focus on exploiting such time-of-arrival data collected from the diffusion process taking place on networks, partially because diverse temporal interactions serve as the local propagation mechanisms for material or information exchange across a population in a variety of realistic scenarios, ranging from the spread of infectious diseases to the diffusion of cultural fads and the proliferation of innovative ideas (see, for example, a recent review [28] ). We show later on that it is fundamentally possible to learn latent networks by discovering both structural and temporal regularities in diffusion process data. Here we restrict on reconstructing a class of stochastic temporal networks (STNs)—which can be generally taken as a null model preserving temporal statistics for all dyads of interacting individuals but ignoring higher-order correlations across them [15] . Specifically, an STN builds, on the basis of a time-aggregated static network, an extra temporal dimension by assigning to each link a mutually independent random IET, denoting the interval of activation of the events occurring on the link in a renewal manner [29] . This simplification was at first made for analytical convenience [15] , and a number of exact methods have in recent years been developed for quantifying diffusion dynamics of complex networked systems when temporal characteristics of pairwise interactions are incorporated [16] , [28] , [30] , [31] . Compared to purely phenomenological (for example, regression-based) models [32] , [33] , [34] , the STN is endowed with better predictive power in both theoretical and applied domains (see Supplementary Note 1 for a brief review on the related literature). Towards an effective null modelling procedure for temporal networks in a data-driven fashion, we conceive the STN model as a convenient descriptive device for explaining time-course data of observed diffusion processes thereon, and we carry out extensive benchmark tests on a variety of simulated and empirical temporal networks to validate the reconstruction efficacy of our approach. We further discuss the inferential complexity of temporal networks in terms of entropy of underlying diffusion pathways from an information-theoretic viewpoint.","references":{"indices":[],"infos":[]}},{"section":"Results","content":"Overall sketch of the reconstruction method The null modelling method for temporal networks we develop is to some extent intuitive and pragmatic. As sketched in Fig. 1 , the topological structure of underlying temporal networks can be directly recovered using the superposed spreading paths encoded in the observable arrival order in the diffusion processes. On the other hand, the statistical temporal properties of dyadic interactions can be also exacted from the time differences of arrivals in time courses of diffusion, an incomplete observation of waiting times associated with possible diffusion routes. We show that in both cases, a soft (namely, probabilistic) censoring indicator whether a link lies in actual diffusion routes, called branching coefficient, plays a pivotal role in our inferential framework. In the following, we first introduce the forward model for information diffusion on temporal networks, and describe the construction of first-order STNs by decorrelating the dyadic interaction sequences of an empirical network. We then derive the likelihood of observing a specific diffusion cascade as well as the corresponding branching coefficients. Based on them, we exhibit a coordinate-ascent scheme which alternates between estimating the latent time-aggregated network by Markov chain Monte Carlo (MCMC) techniques and determining the dyad-level WTDs from self-consistency conditions. Additional details on mathematical proofs, performance assessments of algorithms, empirical validation results, as well as further discussions are deferred in Supplementary Notes 2–12 . Figure 1: Schematic representation of our inferential framework. ( a ) The STN tuple to be inferred which confines the underlying diffusion process, both spatially and temporally. Specifically, a diffusion cascade originated by any source node (red circle) can only proceed on the time-aggregated graph , and relay delay of diffusion along any network link (green arrow) is a random variable τ (called waiting time) with the corresponding distribution denoted by WTD ρ ( τ ). ( b ) A simulated diffusion cascade using our generative model with realized waiting times attached to network links. Here, DAT data are obtained from first-arrival observations of the diffusion cascade, with each entry t v equal to the length of the shortest diffusion path (blue dashed arrow) connecting nodes s * and v . The dashed arrows highlight that the waiting times are unobservable in our problem settings. ( c ) An illustrative result of topology inference using multiple diffusion cascades. Intuitively, the network topology can be recovered by uniting the realized (or inferred) diffusion paths, ( ). Here, denotes the underlying diffusion tree associated with the observed cascade, which is inferable from diffusion data as a spanning tree with maximum link weights β uv (called branching coefficient, see definition below). ( d ) Kernel density estimation for WTD ρ ( τ ) using as incomplete observations of underlying waiting times. For brevity, we adopt Dirac delta kernel function K h ( τ )= δ ( τ ). The sector located at each node v represents its associated branching coefficients, each β uv defining the probability that an upstream neighbouring node spreads the information to node v in a specific cascade. Thus the self-consistent estimator simply sums over all observed cascades (only one is shown here) Dirac deltas of amplitude β uv placed at the TDOAs, d uv = t v − t u , as well as itself truncated above d uv and then renormalized to . The first term corresponds to the contribution from link ( u , v ) as a diffusion branch leading to the tight constraint τ uv = d uv with probability β uv , and the second from link ( u , v ) as a chord having the censored waiting time τ uv > d uv with the complementary probability . ( e ) The coordinate-ascent implementation of our inferential method. The overall procedure is initialized with a first guess for the WTD, and alternates until convergence between the topology inference and density estimation steps using observed diffusion data and the most recent estimates. Additional algorithmic details can be found in Supplementary Note 7 . Full size image Forward generative model We first introduce the forward model used to describe data-generating processes. The temporal network on which diffusion takes place is represented in terms of a set of nodes as well as a set of events observed within a time window [0, T w ]. Each event is depicted by a temporal link from node u to node v activated during a time interval [ t , t + δt ). Here, we assume for simplicity that the duration of events is infinitesimally small ( δt→ 0) in order to avoid several links to be present simultaneously. Accordingly, the STN model built on consists of two key ingredients. First, we obtain the time-aggregated graph by projecting the temporal events onto topological links , each having as its pre-image a list of events, , in order of their activation times. Second, to generate statistically accurate temporal links, we substitute random sequences of synthesized events with IETs drawn i.i.d. from the probability density function fitted on the empirical data for each link separately. It is noteworthy here that the STN model defines ensembles of random temporal networks with the realistic IET distributions at the dyadic level, while being simplified by the automatic elimination of inter-link correlations. Consider now a node introduced as the information source that initializes the diffusion process on , during which each node has one of two mutually exclusive states: (i) informed, if it has already received the information through any incident link; or (ii) ignorant, if it has not been informed so far. In a broadcast manner, the information is transmitted along any time-respecting path which consists of consecutive events of increasing times [that is, with t 1 <…< t l ]. The diffusive arrival time (DAT) at node v is defined as the instant at which node v enters the informed state, and the path through which the information arrives first is termed the fastest time-respecting path (FTP). Here, we assume without loss of generality that t v <∞ for any node ; in other words, the underlying network is connected and contains at least one time-respecting path between any source and receiver pair. Likelihood functional for the observed cascade We next calculate the likelihood of observing a single diffusion cascade of DATs , attempting to search the STN model that fits to the diffusion data. To this end, it is conventional to introduce the concept of the waiting time τ uv occurring on each link . It is defined as the relay interval for which node u has to wait since it becomes informed until it activates a next link incident to node v for transmitting the information. In the case of uncorrelated IETs, τ uv is randomly drawn according to a length-biased probability law which takes the form [35] where is the mean of the IET distribution , and is the Heaviside unit-step function. Denote by the set of such WTDs assigned to respective links of , and the STN model is thus also fully described by , as illustrated in Fig. 1a . Hence the likelihood of observing a given cascade considers all possible FTPs weighted by their conditional probabilities of occurrence and is written where is the likelihood that the cascade is observed along a set of underlying FTPs, and represents the time differences of arrival between nodes u and v . Here denotes any possible union of FTPs from source s * to respective other nodes compatible with the diffusion cascade . Note that constitutes an acyclic directed subgraph of with likelihood , and ideally, we call an ( s *-rooted) diffusion tree. Obviously, the condition of coinciding with actual diffusion pathways requires each branch to produce an exact waiting time, , as well as each chord to produce instead a right-censored (namely, bounded from below) waiting time, , in order to guarantee the minimum-time optimality of the diffusion that occurs along , as shown in Fig. 1b . After some algebra (see Supplementary Note 2 for details), we have the following logarithmic likelihood functional: where and are the survival and hazard functions of WTDs [36] , respectively. Route-specific branching coefficient We are now in the position to introduce the branching coefficient , the conditional probability of link ( u , v ) acting as a diffusion branch (that is, u is the first to inform v ) given a specific cascade on STN . By uncorrelated properties of dyadic network interactions, it is simply given by Equation (4) has a clear interpretation in terms of superposition of inhomogeneous point processes [37] . The simultaneous exposure of v to any informed neighbour u constitutes a set of competing risk events, leading to a cumulative hazard rate at which diffusion will occur at time t . Thus equation (4) immediately follows from Superposition Theorem by equating t to the realized value of v ’s DAT, t v . More detailed derivation of is presented in Supplementary Note 2 . Note that to recover the underlying STN requires observation of multiple diffusion cascades triggered from different regions of the network. Suppose we collect a sample of independent cascades and aim at finding the optimal tuple as the maximizer of the following log-likelihood where C denotes the number of observed diffusion cascades. To be concrete, we resort to an iterative coordinate-ascent strategy to search and alternately. In what follows we specify the two steps outlined above and present numerical results obtained by applying to a variety of benchmark STNs and empirical temporal networks. Time-aggregated topology inference This step takes as input observed DATs D as well as WTDs ρ that are known or estimated. However, finding the maximum likelihood (ML) time-aggregated graph belongs to a wide class of submodular function optimization problems, which is in general computationally hard (see Supplementary Note 3 ). Alternatively, rather than picking out a single ML estimate, we apply MCMC to integrate over all possible configurations using weights proportional to their likelihood values . Here we employ a Gibbs sampler [38] which starts from an initial graph and iteratively flips any link, say ( u , v ), with acceptance probability one by one, where represents the marginal gain of the link flipping operation, that is, where are the branching coefficients defined in equation (4) using the current configuration . Here we adopt the shorthand notation ( ) for the network obtained by adding (removing) link from . As schematically shown in Fig. 1c , the proposed procedure is inclined to sample links with large aggregate branching coefficients. This is roughly equivalent to reconstruction of the network topology using as building blocks the diffusion trees inferred from independent cascades (see Supplementary Note 2 for discussion on the role of branching coefficients in our inferential scheme). Next we quantify the performance of our inference procedure. In view of the structural sparsity of many real-world networked systems, we preferably select a measurement index, called break-even point (BEP), which strikes the optimal balance between precision and recall on true-positive links, as illustrated in Fig. 2 . To test our approach, we carry out extensive numerical simulations for various types of benchmark time-aggregated networks and WTDs. We evaluate the attained BEP, as well as two other standard indices, the area under the receiver operating characteristic curve and the area under the precision-recall curve (AUPR) (see Supplementary Note 4 ). Because high inference accuracy can always be achieved, we report in Table 1 the minimum sample size for assuring at least 0.95 area under the receiver operating characteristic, area under the precision-recall curve and BEP, respectively, showing that universal high inference accuracy can be achieved from only a moderate sample size of diffusion observations. More detailed descriptions and complete results of benchmark tests for our inference algorithm can be found in Supplementary Table 5 . Figure 2: Break-even point of topology inference using diffusion cascades. ( a – e ) The time-aggregated graph configuration with BEP optimality is attained by equating the number of false-positive links (red solid lines) to the number of false-negative links (black dashed lines) in estimated configurations. Percentages in parentheses correspond to the estimated BEPs. ( f ) Real and estimated PR curves with respect to increasing sample sizes of C =5,10,15,20,25,30,40, respectively. The BEPs correspond to intersections of the PR curves with the diagonal of the unit square (dotted line). The time-aggregated network is an Erdös–Rényi random graph of size N =100 and average degree , and the real WTD is Gaussian of mean μ =2 and variance , which is assumed here to be explicit for topology inference. Full size image Table 1 Summary of critical relative sample sizes for STN reconstruction. Full size table Waiting-time distribution estimation This step takes as input observed DATs D as well as an underlying time-aggregated network . To implement nonparametric estimation of WTD , the major difficulty stems from the indeterminacy of the diffusion trees and the censored nature of waiting times occurring on chord links during the diffusion cascades. Here, we adopt the ‘Redistribute-to-the-Right’ formulation [39] as an imputation scheme to tackle this censored data problem, by which we obtain the following self-consistent equations for estimating the WTDs: where is a kernel smoother with bandwidth h , is the Heaviside unit-step function, and denotes the convolution operator. The first term in the braces corresponds to the probability of link ( u , v ) acting as a diffusion branch, and the second term corresponds to that of link ( u , v ) acting as a chord which contributes a truncated WTD of above , as shown in Fig. 1d . Particularly, in the case of an underlying tree , reduces to standard kernel density estimation [40] as . In the most general case of an arbitrary network , the consistency of the WTD estimator is in essence guaranteed by our choice of weights [ ] in soft assignments of respective links to diffusion branches in an expectation-maximization manner [41] . We numerically verify the consistency of the estimator for non-identically distributed WTDs attached to a small-size network, as illustrated in Fig. 3 . A rigorous proof is presented in Supplementary Note 5 . Figure 3: Asymptotic consistency of the iterative WTD estimator. ( a – h ) Numerical verification is illustrated through large samples of diffusion cascades synthesized on a small-size network, with several benchmark WTDs (black lines) assigned to departure links of respective nodes. The iteratively estimated WTDs (gradient-coloured lines) ultimately converge to the correct distributions. ( i ) The illustrative network consisting of N =8 nodes and L =16 (double-directed) links. The sample size of diffusion cascades used for the WTD estimation is C =10 4 , the kernel bandwidth is chosen as h =10 −3 , and the a priori guess as exponentials of mean μ =1 (see Supplementary Note 6 for practical selection criteria of both and h ). Full size image When only limited amount of diffusion data are available for inference, the reconstruction accuracy undergoes a phase transition as the sample increases (see Supplementary Fig. 6 ). Hence we explore the minimum relative sample size for simultaneously successfully reconstructing both underlying networks and associated WTDs only from observed diffusion cascades, as displayed in Table 1 . Somewhat surprisingly, the same reconstruction accuracy can generally be achieved using even less diffusion cascades in a considerable fraction of benchmark tests, comparing to the case where the underlying WTDs are explicitly known. This counter-intuition can be in part explained by the fact that our self-consistent WTD estimator provides an adaptive mechanism for the fitting of observed diffusion data. Specifically, through the process of inference, the estimator acquires subtle features of the empirical WTD in accordance with the particular realizations of waiting times, and hence outperforms the result given the true WTD ρ which, even if observed data truly follow it, differs more or less from , especially when the sample size is small. We design a parametric bootstrap procedure to provide confidence bands to assess the variability in the estimated WTDs. Detailed implementation and illustrative numerical results are present in Supplementary Note 7 . We also apply our method to several realistic temporal networks to test its validity under more practical situations. Relative empirical validation results are reported in Table 2 . Keeping in mind the gap between our STN model from reality, we introduce three statistical measures to quantify the deviation of empirical data sets from the null model assumption of i.i.d. distributed WTDs: distributional standard deviation, Pearson correlation coefficient and normalized mutual information. On one hand, we find that the WTDs underlying these real-world temporal networks have relatively small distributional standard deviation, which partially justifies our null model assumption. This considerable homogeneity in temporal interactivity of individuals can be interpreted by the typical distribution of human response time which is shown to be heavy-tailed with power exponent between 1 and 2. On the other hand, large Pearson correlation coefficient or normalized mutual information of empirical WTDs implies the salience of inter-link correlations, either (negatively) linear or non-linear, in many realistic temporal networks. This leads to substantially decreased reconstruction accuracy (see BEP 1 and BEP 2 in Table 2 ), suggesting correction of the self-consistent relation of WTDs to incorporate such polyadic correlations (see Supplementary Note 9 for discussion on the extended higher-order STN models). In addition to the discrepancy between empirical temporal networks and our STN model, another bottleneck causing the reduced BEP 1 comes from the iterative coordinate-ascent procedure. The reconstruction accuracy often collapses due to the positive feedback loop between topology inference and density estimation steps that amplifies the estimation error caused by the unrealistic null model assumptions. We further compare the reconstruction results using empirical WTDs as prior input to break such unexpected feedback loops, showing an acceptable efficacy of our topology inference method even when applied to realistic temporal networks (see BEP 3 and BEP 4 in Table 2 ). We also find that the reconstruction of can be sensibly improved given correct prior WTD ρ , and on the contrary, the estimation density for ρ does not benefit from the addition of topological knowledge (see Supplementary Figs 20–21 ). A possible reason lies in the extremely large supports of heavy-tailed WTDs, in which case our equidistant binning-based WTD estimator has a large number of parameters to be estimated from self-consistent iterations and hence demonstrates poor convergence properties (see Supplementary Note 10 for further discussion on the effect of binning and the Fourier-domain density estimation). Table 2 Empirical validation results using realistic temporal network data. Full size table Inferential complexity of temporal networks Numerical results of our benchmark tests reveal a complicated picture of how structural and temporal properties of the underlying network synergically affect the critical amount of diffusion data needed to attain a given reconstruction accuracy. To more comprehensively quantify the inferential complexity of diffusion structure embodied in temporal networks, we introduce a single entropic measure as follows: Note that reflects the entropy of possible diffusion trees associated with a given cascade on , thus extending the concept of network complexity which counts the (logarithmic) number of spanning trees contained in a static graph [42] . Additional details on the derivation of equation (8) and related discussion are presented in Supplementary Note 11 . Next we examine the inferential complexity of diffusion structure in various types of benchmark WTDs, finding a natural positive correlation between the intrinsic complexity of the underlying diffusion processes and the critical relative sample size for reconstruction goals, as shown in Fig. 4 . A more systematic analysis requires parametric generation of network ensembles that are tunable from both structural or temporal aspects to decouple the underlying influences. Here we first focus on three important structural properties of time-aggregated networks: clustering coefficient, average path length and connectivity heterogeneity. Numerical results show a nonlinear dependence of network complexity on the topological indices. Particularly in the practically relevant regime of salient small-world effects (characterized by large clustering coefficient and small average path length), the inferential complexity is positively (negatively) correlated with clustering coefficient (average path length) of the time-aggregated network. For most specific distributional shapes used as our benchmark WTDs, the heterogeneity in degree distribution also increases the inferential complexity of networks. Intuitively, diffusion on small-world and/or highly heterogeneous networks is inclined to produce, due to frequent occurrences of competing-risk censoring, indiscernible ‘downstream’ DATs that are close to each other, thus complicating the task of network reconstruction. We further examine the roles of the first three (normalized central) moments of WTDs, finding that the inferential complexity is positively (negatively) correlated with variance and skewness (mean, or lower bound of the support) of underlying WTDs. Detailed numerical results can be found in Supplementary Figs 24–30 . Figure 4: Inferential complexity of benchmark STNs. ( a – d ) Scatter plot of relative inferential complexity [ ] versus critical relative sample size for ensuring at least 0.95 AUROC, AUPR and BEP, respectively, as well as that for simultaneously recovering both the underlying networks and WTDs. Here, the angular brackets denote the average over ensembles of diffusion cascades. Each point represents an average over at least ten independent realizations with respect to a specific combination from benchmark networks and WTDs, and the critical values of respective relative sample sizes are listed in Supplementary Table 5 . Numerical results show that there is positive correlation between the inferential complexity and the critical relative sample size for achieving network reconstruction at a given accuracy, implying that the more complex (or indeterminate) the underlying diffusion structure, the more difficult the inference of temporal networks from data. Full size image"},{"section":"Discussion","content":"In summary, we have developed a general framework to reconstruct STNs as a null model of temporal networks by fitting time-course observations of the diffusion process taking place on them. To alleviate the ill-posedness of this time-extended inverse problem, we have decomposed the task of network reconstruction into structural and temporal aspects, that is, an unsupervised topology inference for the time-aggregated networks using MCMC sampling, and a nonparametric density estimation for the associated WTDs via self-consistent iterations, respectively. We have given a rigorous consistency proof for the proposed WTD estimator, and have numerically shown that the iterative algorithm frequently possesses good convergence properties given properly selected, data-driven initial guesses. We have also applied our method to various types of benchmark STNs and empirical temporal networks, showing that it is statistically possible to discover latent temporal networks with high accuracy only from a moderate amount of diffusion data. Despite the inability to recover actual serial snapshots of networks, the reconstructed STN builds up both structural and temporal statistics sufficient to enable reliable prediction of the population-level diffusion behaviour, which can thus be of practical interest in a broad range of applications, particularly in time-critical and privacy-sensitive circumstances related to human activities. Our reconstruction method is not only restricted to the data generative process considered in the paper, but is also adapted to many variants of the original inference task. For example, recent work [27] has demonstrated the predictive power of the Hawkes process method for quantifying and tracing information cascades on social media, provided a well-defined memory kernel (a mathematical equivalent to the WTD in our model) fitted to the empirical mechanism for diffusion. Our method then provides an alternative operative route for estimating the memory kernel from early time courses of diffusion, as well as for updating the posterior WTD self-consistently with new available process data as diffusion proceeds, while not necessitating access to possibly privacy-sensitive information at the individual level. Another important extension is to use observed time courses of disease outbreaks to deduce the underlying population structure and temporal transmission pattern in epidemiological studies. Here, the essential difference between a generic epidemic process and the information broadcasting considered here lies in that the recovery process (for example, self-healing) prevents to a large extent the possible transmission of diseases from infected individuals to the remainder of the population. Restated in the language of survival analysis, the observation of successful transmissions is potentially ‘censored’ by the recovery events occurred among the infected population. Under this circumstance, one can directly make use of subdistributed WTDs (that is, ) to encapsulate arbitrary distributions of transmission and recovery times [30] , and apply the same inferential scheme with few modifications. We note here that the residual probability mass located at the infinity, , meaning the probability of endemic transmissions being interrupted by node u ’s recovery, causes a decrease in the effective number of diffusion cascades that are helpful for predicting if link ( u , v ) is present—as a consequence, the larger r uv , the less observably succeeded transmission routes, the more diffusion data required to extract the latent network. To exclude this nuisance factor we therefore have focused on the information diffusion model that mimics susceptible-infected epidemics (with r uv =0) to capture the effective sample sizes for specific reconstruction goals. In addition to the simplified forward diffusion model, we have made several assumptions in this paper, of which the most important is that the STNs are assumed to have no cross-dyad interdependence. This is the key property that enables the localized computation of likelihoods and branching coefficients using only neighbourhood information, thus ensuring an efficient implementation with polynomial complexity. We have discussed the second-order STN model that assumes a joint distribution of waiting times occurring on any successive dyads. Such polyadic correlations destroy the applicability of Matrix-Tree Theorems, leading to the necessity of a tree-enumeration procedure with exponential complexity. Leaving aside temporarily the issue of implementation expenditure, our inferential scheme can be naturally extended to the higher-order correlated case (see Supplementary Note 9 ). We have furthermore devised a pairwise approximation of branching coefficients in order to speed up our algorithms. Another significant one, mainly adopted in our benchmark tests, is the homogeneous population assumption, stating that the underlying WTDs ρ uv ( τ ) are identical, despite the individual heterogeneity or frailty widely prevalent in many real-life populations. Again, it should be pointed out that although our framework is applicable to non-identical WTDs, as well as higher-order, polyadically correlated cases, the established STN model to some extent sacrifices simplicity and runs the risk of overfitting. The ultimate goal of null modelling is to reproduce expected patterns that capture predictive information in data, while not being overly complicated. Let us further inspect the complexity and scalability of our method. While it is generally difficult to quantify the complexity of the iterative estimator even for the first-order STNs, benchmark test results numerically illustrate that the proposed procedure, when initialized with appropriate first guesses (see Supplementary Note 6 ), has quite good iterative convergence, irrespective of either structural properties of the time-aggregated topologies or distributional shapes of the dyad-specific WTDs. Thus our algorithm can practically be implemented with complexity O ( CN 2 M ), where C , N and M are the numbers of diffusion cascades, network nodes and MCMC iterations, respectively. Because M only exerts influence on the precision of a posteriori prediction of links and is independent of the problem size, the actual complexity of our procedure is then O ( CN 2 ), which reaches the theoretically optimal time complexity. Put another way, the network reconstruction problem can be regarded as a sequence of O ( N 2 ) likelihood-ratio hypothesis tests, in which the marginal gain of each possible link calculated using O ( C ) diffusion cascades is utilized to decide if the corresponding dyad of nodes is truly connected. Despite the proposed method with the (near-)inherent, quadratic complexity, there is a widespread need for more scalable and efficient, hopefully linear-time, inverse modelling approaches for empirical complex systems, especially in the era of Big Data. The potential solution to this impasse is to shift from predicting individual links to zooming out to the coarse-grained or mesoscopic topological scales using more sophisticated inference techniques borrowed from, for example, large-scale hypothesis testing [43] and community-based time series analysis [44] . There are also numerous conceivable approaches for our topology inference step. A closely related alternative one, which is formulated in a Bayesian fashion, is to introduce prior information about the underlying diffusion network into the estimation scheme to encompass a broader spectrum of expected structural features. Here we have exemplified this with the standard -norm sparsity priors, turning the maximum likelihood into a maximum a posteriori (MAP) estimation by virtue of slight differences in calculating the marginal gains of network links (see Supplementary Note 12 ). Numerical results have demonstrated the significantly improved reconstruction performance for STNs. Furthermore, parametric network models (such as exponential random graphs [45] ) and advanced optimization techniques for the MAP problem, as well as practically available prior information other than collective diffusion data (for example, partially observed topology structure, relevant metadata of network agents, and so on) can also be incorporated into our reconstruction scheme with little technical effort. Another contribution of the proposed inferential framework is the density estimation for underlying WTDs. Because of the context in which we are interested, an ideal null model of temporal networks would be reproductive of time variability in empirical interactions, as well as predictive of their collective dynamical properties at the desired level, in particular with the help of the WTD extracted from true latent (usually anomalous) diffusion patterns. When the goal is to solely create temporality information with prior network structures, rather than to reconstruct the topology of diffusion substrates, the density estimation can be pursued by a radically sharpened diffusion cascades. Under the homogeneous population assumption, meaning a population-wide fitting of the WTD commonly shared by all underlying dyadic interactions, one can expect a reliable WTD estimation even from a single diffusion cascade, unless either network size or observation window is very small. Consequently, our method has potential applications in epidemiological studies, especially for inverse modelling of general epidemics with non-Poissonian behaviour [30] , which range from estimating parameters such as the effective reproductive number using subdistributed WTDs to identifying the invasion routes of epidemics using branching coefficients. Finally, the central role played by branching coefficients—in the universal imputation scheme for treating the censored data problem arising from competing risk events of networked diffusion—has implications for definition and refinement of the dynamics-based node or link centrality and information-theoretic complexity measures for temporal networks."},{"section":"Methods","content":"Benchmark test data To test the performance of our inference algorithm, we have carried out extensive numerical experiments using three categories of time-aggregated benchmark topologies (deterministic, stochastic and empirical static networks) in combination with several different types of benchmark WTDs, as listed in Supplementary Tables 1 and 3 . To provide further empirical validation, we also apply our method to several realistic temporal contact networks, as listed in Supplementary Table 2 . Specifically, we reconstruct the STN model from synthesized diffusion cascades respectively using empirical contact lists and the STN tuple that is fitted to the original data (see Supplementary Table 4 for detailed experimental settings). Network data or their generative models can be found in relevant references. Gibbs sampler for time-aggregated graphs We apply MCMC method with Gibbs sampling to explore the configuration space of the underlying time-aggregated network. The configuration transition is based on the link-flipping operation with success probability according to the likelihood ratio between the new configuration and the old configuration, that is, the marginal gain of the flipping operation (equation (6)). In our benchmark test, the burn-in period and the maximum lag are set to 10, and the number M =200 of MCMC samples are drawn. See Supplementary Note 7 for detailed algorithm implementation. Iterative procedure for the WTD estimator To solve the self-consistent equations for , the algorithm carries out an iterative procedure as follows: Start with a priori guess for the underlying WTDs and repeat the following synchronous updates for all entries of until convergence The convergence is judged by a small change in the estimated between successive iterations. Here we adopt the Kolmogorov–Smirnov divergence, and terminate the iteration whenever , where is survival function of WTD , and ɛ is a predetermined error threshold. The detailed self-consistent iterative scheme is presented in Supplementary Note 7 . In our numerical validation, we assume for brevity that internal temporal interactions among a population satisfy the homogeneity condition, corresponding to first-order the STN with i.i.d. dyad-level WTDs. To initialize the iterative procedure, we use exponential WTD as the first guess, and set smoothing kernel bandwidth h =0.05. Practical parameter selection criteria, as well as their effects on convergence properties and performance of the density estimator are discussed in Supplementary Note 6 . Notations We summarize the notations used throughout this paper in Supplementary Table 6 . Data availability All relevant data are available from the authors on request."},{"section":"Additional information","content":"How to cite this article: Li, Xun & Li, Xiang et al . Reconstruction of stochastic temporal networks through diffusive arrival times. Nat. Commun. 8, 15729 doi: 10.1038/ncomms15729 (2017). 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\": \"The collective functionality of complex networks emerges as a consequence of the interactions among their constituents. Recently, it has been empirically observed that, besides the spatially or topologically structured organization, the temporal constraints imposed on many complex interacting systems add a further dimension crucial for understanding their generic structure and dynamics. Internal links in these so-called temporal networks [1] evolve over time and are inherently registered by a series of rhythmically activated events among interacting actors at discrete time stamps. The causal sequence of the time-ordered links strikingly affects accessibility [2] , an essential and revealing characteristic, especially in social and communication networks related to human activities. Such a profound shift from static to temporally intermittent interactivity properties has expanded, challenged or even redefined many fundamental concepts of networks, including path length [3] , [4] , clustering correlation [3] , [4] , node centrality [3] , [5] , communicability [6] , structural controllability [7] , motif [8] and community structure [9] . Of another interest are the anomalous patterns (for example, bursts and heavy tails) embedded in the interevent time (IET) distribution of temporal interactions [10] , [11] . This marked departure of human activities from Poissonian behaviour substantially alters the dynamical process that takes place on networks, such as epidemic spreading [10] , [11] , [12] , [13] , [14] , random walks [15] , [16] , synchronization [17] , cooperative evolution [18] , consensus and coordination processes [19] , [20] . Undoubtedly, identifying the temporal interaction pattern is a first step in understanding and controlling collective dynamics of empirical temporal networked systems. The vast majority of generative models for temporal networks—aimed at reproducing the empirical sequences of the time-stamped interactions—by and large require a priori knowledge of the raw data, or relevant statistics on the underlying interaction patterns [21] , [22] , [23] , [24] . However, strong limitations often arise regarding the availability of time-resolved interaction data at the individual level. Apart from increased technological expenditures, data collection is also hindered by small observation windows or samples reflecting individual activities only at coarse time granularity. Some data sets potentially suffer from other statistical deficiencies resulting from, for example, the participants with highly correlated behaviour from a skewed population. Another, even more problematic, restriction is imposed by data observability issues. The specifics of temporal interactions, particularly in social and financial networks, are in general obscured due to the privacy concerns of the participants, making their time-extended network structure unobservable, at least in principle. Extracting temporal networks from measurable data at the collective level has hence become a very desirable task. In viral marketing, for example, mining social networks is of critical value to identify highly influential customers who directly affect other consumers’ decision making [25] . However, the interpersonal interactions (word-of-mouth recommendations) are often rendered private and thus indirectly accessible. In such circumstances, the recorded product dissemination history (for example, when customers purchase the products) provides an observable data source for the social network mining. Other candidate data applicable to similar inferential tasks include observed information cascades, such as propagating memes through blog posts [26] and tweets [27] on social media. In this paper, we focus on exploiting such time-of-arrival data collected from the diffusion process taking place on networks, partially because diverse temporal interactions serve as the local propagation mechanisms for material or information exchange across a population in a variety of realistic scenarios, ranging from the spread of infectious diseases to the diffusion of cultural fads and the proliferation of innovative ideas (see, for example, a recent review [28] ). We show later on that it is fundamentally possible to learn latent networks by discovering both structural and temporal regularities in diffusion process data. Here we restrict on reconstructing a class of stochastic temporal networks (STNs)—which can be generally taken as a null model preserving temporal statistics for all dyads of interacting individuals but ignoring higher-order correlations across them [15] . Specifically, an STN builds, on the basis of a time-aggregated static network, an extra temporal dimension by assigning to each link a mutually independent random IET, denoting the interval of activation of the events occurring on the link in a renewal manner [29] . This simplification was at first made for analytical convenience [15] , and a number of exact methods have in recent years been developed for quantifying diffusion dynamics of complex networked systems when temporal characteristics of pairwise interactions are incorporated [16] , [28] , [30] , [31] . Compared to purely phenomenological (for example, regression-based) models [32] , [33] , [34] , the STN is endowed with better predictive power in both theoretical and applied domains (see Supplementary Note 1 for a brief review on the related literature). Towards an effective null modelling procedure for temporal networks in a data-driven fashion, we conceive the STN model as a convenient descriptive device for explaining time-course data of observed diffusion processes thereon, and we carry out extensive benchmark tests on a variety of simulated and empirical temporal networks to validate the reconstruction efficacy of our approach. We further discuss the inferential complexity of temporal networks in terms of entropy of underlying diffusion pathways from an information-theoretic viewpoint.\",\n \"references\": {\n \"indices\": [],\n \"infos\": []\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Overall sketch of the reconstruction method The null modelling method for temporal networks we develop is to some extent intuitive and pragmatic. As sketched in Fig. 1 , the topological structure of underlying temporal networks can be directly recovered using the superposed spreading paths encoded in the observable arrival order in the diffusion processes. On the other hand, the statistical temporal properties of dyadic interactions can be also exacted from the time differences of arrivals in time courses of diffusion, an incomplete observation of waiting times associated with possible diffusion routes. We show that in both cases, a soft (namely, probabilistic) censoring indicator whether a link lies in actual diffusion routes, called branching coefficient, plays a pivotal role in our inferential framework. In the following, we first introduce the forward model for information diffusion on temporal networks, and describe the construction of first-order STNs by decorrelating the dyadic interaction sequences of an empirical network. We then derive the likelihood of observing a specific diffusion cascade as well as the corresponding branching coefficients. Based on them, we exhibit a coordinate-ascent scheme which alternates between estimating the latent time-aggregated network by Markov chain Monte Carlo (MCMC) techniques and determining the dyad-level WTDs from self-consistency conditions. Additional details on mathematical proofs, performance assessments of algorithms, empirical validation results, as well as further discussions are deferred in Supplementary Notes 2–12 . Figure 1: Schematic representation of our inferential framework. ( a ) The STN tuple to be inferred which confines the underlying diffusion process, both spatially and temporally. Specifically, a diffusion cascade originated by any source node (red circle) can only proceed on the time-aggregated graph , and relay delay of diffusion along any network link (green arrow) is a random variable τ (called waiting time) with the corresponding distribution denoted by WTD ρ ( τ ). ( b ) A simulated diffusion cascade using our generative model with realized waiting times attached to network links. Here, DAT data are obtained from first-arrival observations of the diffusion cascade, with each entry t v equal to the length of the shortest diffusion path (blue dashed arrow) connecting nodes s * and v . The dashed arrows highlight that the waiting times are unobservable in our problem settings. ( c ) An illustrative result of topology inference using multiple diffusion cascades. Intuitively, the network topology can be recovered by uniting the realized (or inferred) diffusion paths, ( ). Here, denotes the underlying diffusion tree associated with the observed cascade, which is inferable from diffusion data as a spanning tree with maximum link weights β uv (called branching coefficient, see definition below). ( d ) Kernel density estimation for WTD ρ ( τ ) using as incomplete observations of underlying waiting times. For brevity, we adopt Dirac delta kernel function K h ( τ )= δ ( τ ). The sector located at each node v represents its associated branching coefficients, each β uv defining the probability that an upstream neighbouring node spreads the information to node v in a specific cascade. Thus the self-consistent estimator simply sums over all observed cascades (only one is shown here) Dirac deltas of amplitude β uv placed at the TDOAs, d uv = t v − t u , as well as itself truncated above d uv and then renormalized to . The first term corresponds to the contribution from link ( u , v ) as a diffusion branch leading to the tight constraint τ uv = d uv with probability β uv , and the second from link ( u , v ) as a chord having the censored waiting time τ uv > d uv with the complementary probability . ( e ) The coordinate-ascent implementation of our inferential method. The overall procedure is initialized with a first guess for the WTD, and alternates until convergence between the topology inference and density estimation steps using observed diffusion data and the most recent estimates. Additional algorithmic details can be found in Supplementary Note 7 . Full size image Forward generative model We first introduce the forward model used to describe data-generating processes. The temporal network on which diffusion takes place is represented in terms of a set of nodes as well as a set of events observed within a time window [0, T w ]. Each event is depicted by a temporal link from node u to node v activated during a time interval [ t , t + δt ). Here, we assume for simplicity that the duration of events is infinitesimally small ( δt→ 0) in order to avoid several links to be present simultaneously. Accordingly, the STN model built on consists of two key ingredients. First, we obtain the time-aggregated graph by projecting the temporal events onto topological links , each having as its pre-image a list of events, , in order of their activation times. Second, to generate statistically accurate temporal links, we substitute random sequences of synthesized events with IETs drawn i.i.d. from the probability density function fitted on the empirical data for each link separately. It is noteworthy here that the STN model defines ensembles of random temporal networks with the realistic IET distributions at the dyadic level, while being simplified by the automatic elimination of inter-link correlations. Consider now a node introduced as the information source that initializes the diffusion process on , during which each node has one of two mutually exclusive states: (i) informed, if it has already received the information through any incident link; or (ii) ignorant, if it has not been informed so far. In a broadcast manner, the information is transmitted along any time-respecting path which consists of consecutive events of increasing times [that is, with t 1 <…< t l ]. The diffusive arrival time (DAT) at node v is defined as the instant at which node v enters the informed state, and the path through which the information arrives first is termed the fastest time-respecting path (FTP). Here, we assume without loss of generality that t v <∞ for any node ; in other words, the underlying network is connected and contains at least one time-respecting path between any source and receiver pair. Likelihood functional for the observed cascade We next calculate the likelihood of observing a single diffusion cascade of DATs , attempting to search the STN model that fits to the diffusion data. To this end, it is conventional to introduce the concept of the waiting time τ uv occurring on each link . It is defined as the relay interval for which node u has to wait since it becomes informed until it activates a next link incident to node v for transmitting the information. In the case of uncorrelated IETs, τ uv is randomly drawn according to a length-biased probability law which takes the form [35] where is the mean of the IET distribution , and is the Heaviside unit-step function. Denote by the set of such WTDs assigned to respective links of , and the STN model is thus also fully described by , as illustrated in Fig. 1a . Hence the likelihood of observing a given cascade considers all possible FTPs weighted by their conditional probabilities of occurrence and is written where is the likelihood that the cascade is observed along a set of underlying FTPs, and represents the time differences of arrival between nodes u and v . Here denotes any possible union of FTPs from source s * to respective other nodes compatible with the diffusion cascade . Note that constitutes an acyclic directed subgraph of with likelihood , and ideally, we call an ( s *-rooted) diffusion tree. Obviously, the condition of coinciding with actual diffusion pathways requires each branch to produce an exact waiting time, , as well as each chord to produce instead a right-censored (namely, bounded from below) waiting time, , in order to guarantee the minimum-time optimality of the diffusion that occurs along , as shown in Fig. 1b . After some algebra (see Supplementary Note 2 for details), we have the following logarithmic likelihood functional: where and are the survival and hazard functions of WTDs [36] , respectively. Route-specific branching coefficient We are now in the position to introduce the branching coefficient , the conditional probability of link ( u , v ) acting as a diffusion branch (that is, u is the first to inform v ) given a specific cascade on STN . By uncorrelated properties of dyadic network interactions, it is simply given by Equation (4) has a clear interpretation in terms of superposition of inhomogeneous point processes [37] . The simultaneous exposure of v to any informed neighbour u constitutes a set of competing risk events, leading to a cumulative hazard rate at which diffusion will occur at time t . Thus equation (4) immediately follows from Superposition Theorem by equating t to the realized value of v ’s DAT, t v . More detailed derivation of is presented in Supplementary Note 2 . Note that to recover the underlying STN requires observation of multiple diffusion cascades triggered from different regions of the network. Suppose we collect a sample of independent cascades and aim at finding the optimal tuple as the maximizer of the following log-likelihood where C denotes the number of observed diffusion cascades. To be concrete, we resort to an iterative coordinate-ascent strategy to search and alternately. In what follows we specify the two steps outlined above and present numerical results obtained by applying to a variety of benchmark STNs and empirical temporal networks. Time-aggregated topology inference This step takes as input observed DATs D as well as WTDs ρ that are known or estimated. However, finding the maximum likelihood (ML) time-aggregated graph belongs to a wide class of submodular function optimization problems, which is in general computationally hard (see Supplementary Note 3 ). Alternatively, rather than picking out a single ML estimate, we apply MCMC to integrate over all possible configurations using weights proportional to their likelihood values . Here we employ a Gibbs sampler [38] which starts from an initial graph and iteratively flips any link, say ( u , v ), with acceptance probability one by one, where represents the marginal gain of the link flipping operation, that is, where are the branching coefficients defined in equation (4) using the current configuration . Here we adopt the shorthand notation ( ) for the network obtained by adding (removing) link from . As schematically shown in Fig. 1c , the proposed procedure is inclined to sample links with large aggregate branching coefficients. This is roughly equivalent to reconstruction of the network topology using as building blocks the diffusion trees inferred from independent cascades (see Supplementary Note 2 for discussion on the role of branching coefficients in our inferential scheme). Next we quantify the performance of our inference procedure. In view of the structural sparsity of many real-world networked systems, we preferably select a measurement index, called break-even point (BEP), which strikes the optimal balance between precision and recall on true-positive links, as illustrated in Fig. 2 . To test our approach, we carry out extensive numerical simulations for various types of benchmark time-aggregated networks and WTDs. We evaluate the attained BEP, as well as two other standard indices, the area under the receiver operating characteristic curve and the area under the precision-recall curve (AUPR) (see Supplementary Note 4 ). Because high inference accuracy can always be achieved, we report in Table 1 the minimum sample size for assuring at least 0.95 area under the receiver operating characteristic, area under the precision-recall curve and BEP, respectively, showing that universal high inference accuracy can be achieved from only a moderate sample size of diffusion observations. More detailed descriptions and complete results of benchmark tests for our inference algorithm can be found in Supplementary Table 5 . Figure 2: Break-even point of topology inference using diffusion cascades. ( a – e ) The time-aggregated graph configuration with BEP optimality is attained by equating the number of false-positive links (red solid lines) to the number of false-negative links (black dashed lines) in estimated configurations. Percentages in parentheses correspond to the estimated BEPs. ( f ) Real and estimated PR curves with respect to increasing sample sizes of C =5,10,15,20,25,30,40, respectively. The BEPs correspond to intersections of the PR curves with the diagonal of the unit square (dotted line). The time-aggregated network is an Erdös–Rényi random graph of size N =100 and average degree , and the real WTD is Gaussian of mean μ =2 and variance , which is assumed here to be explicit for topology inference. Full size image Table 1 Summary of critical relative sample sizes for STN reconstruction. Full size table Waiting-time distribution estimation This step takes as input observed DATs D as well as an underlying time-aggregated network . To implement nonparametric estimation of WTD , the major difficulty stems from the indeterminacy of the diffusion trees and the censored nature of waiting times occurring on chord links during the diffusion cascades. Here, we adopt the ‘Redistribute-to-the-Right’ formulation [39] as an imputation scheme to tackle this censored data problem, by which we obtain the following self-consistent equations for estimating the WTDs: where is a kernel smoother with bandwidth h , is the Heaviside unit-step function, and denotes the convolution operator. The first term in the braces corresponds to the probability of link ( u , v ) acting as a diffusion branch, and the second term corresponds to that of link ( u , v ) acting as a chord which contributes a truncated WTD of above , as shown in Fig. 1d . Particularly, in the case of an underlying tree , reduces to standard kernel density estimation [40] as . In the most general case of an arbitrary network , the consistency of the WTD estimator is in essence guaranteed by our choice of weights [ ] in soft assignments of respective links to diffusion branches in an expectation-maximization manner [41] . We numerically verify the consistency of the estimator for non-identically distributed WTDs attached to a small-size network, as illustrated in Fig. 3 . A rigorous proof is presented in Supplementary Note 5 . Figure 3: Asymptotic consistency of the iterative WTD estimator. ( a – h ) Numerical verification is illustrated through large samples of diffusion cascades synthesized on a small-size network, with several benchmark WTDs (black lines) assigned to departure links of respective nodes. The iteratively estimated WTDs (gradient-coloured lines) ultimately converge to the correct distributions. ( i ) The illustrative network consisting of N =8 nodes and L =16 (double-directed) links. The sample size of diffusion cascades used for the WTD estimation is C =10 4 , the kernel bandwidth is chosen as h =10 −3 , and the a priori guess as exponentials of mean μ =1 (see Supplementary Note 6 for practical selection criteria of both and h ). Full size image When only limited amount of diffusion data are available for inference, the reconstruction accuracy undergoes a phase transition as the sample increases (see Supplementary Fig. 6 ). Hence we explore the minimum relative sample size for simultaneously successfully reconstructing both underlying networks and associated WTDs only from observed diffusion cascades, as displayed in Table 1 . Somewhat surprisingly, the same reconstruction accuracy can generally be achieved using even less diffusion cascades in a considerable fraction of benchmark tests, comparing to the case where the underlying WTDs are explicitly known. This counter-intuition can be in part explained by the fact that our self-consistent WTD estimator provides an adaptive mechanism for the fitting of observed diffusion data. Specifically, through the process of inference, the estimator acquires subtle features of the empirical WTD in accordance with the particular realizations of waiting times, and hence outperforms the result given the true WTD ρ which, even if observed data truly follow it, differs more or less from , especially when the sample size is small. We design a parametric bootstrap procedure to provide confidence bands to assess the variability in the estimated WTDs. Detailed implementation and illustrative numerical results are present in Supplementary Note 7 . We also apply our method to several realistic temporal networks to test its validity under more practical situations. Relative empirical validation results are reported in Table 2 . Keeping in mind the gap between our STN model from reality, we introduce three statistical measures to quantify the deviation of empirical data sets from the null model assumption of i.i.d. distributed WTDs: distributional standard deviation, Pearson correlation coefficient and normalized mutual information. On one hand, we find that the WTDs underlying these real-world temporal networks have relatively small distributional standard deviation, which partially justifies our null model assumption. This considerable homogeneity in temporal interactivity of individuals can be interpreted by the typical distribution of human response time which is shown to be heavy-tailed with power exponent between 1 and 2. On the other hand, large Pearson correlation coefficient or normalized mutual information of empirical WTDs implies the salience of inter-link correlations, either (negatively) linear or non-linear, in many realistic temporal networks. This leads to substantially decreased reconstruction accuracy (see BEP 1 and BEP 2 in Table 2 ), suggesting correction of the self-consistent relation of WTDs to incorporate such polyadic correlations (see Supplementary Note 9 for discussion on the extended higher-order STN models). In addition to the discrepancy between empirical temporal networks and our STN model, another bottleneck causing the reduced BEP 1 comes from the iterative coordinate-ascent procedure. The reconstruction accuracy often collapses due to the positive feedback loop between topology inference and density estimation steps that amplifies the estimation error caused by the unrealistic null model assumptions. We further compare the reconstruction results using empirical WTDs as prior input to break such unexpected feedback loops, showing an acceptable efficacy of our topology inference method even when applied to realistic temporal networks (see BEP 3 and BEP 4 in Table 2 ). We also find that the reconstruction of can be sensibly improved given correct prior WTD ρ , and on the contrary, the estimation density for ρ does not benefit from the addition of topological knowledge (see Supplementary Figs 20–21 ). A possible reason lies in the extremely large supports of heavy-tailed WTDs, in which case our equidistant binning-based WTD estimator has a large number of parameters to be estimated from self-consistent iterations and hence demonstrates poor convergence properties (see Supplementary Note 10 for further discussion on the effect of binning and the Fourier-domain density estimation). Table 2 Empirical validation results using realistic temporal network data. Full size table Inferential complexity of temporal networks Numerical results of our benchmark tests reveal a complicated picture of how structural and temporal properties of the underlying network synergically affect the critical amount of diffusion data needed to attain a given reconstruction accuracy. To more comprehensively quantify the inferential complexity of diffusion structure embodied in temporal networks, we introduce a single entropic measure as follows: Note that reflects the entropy of possible diffusion trees associated with a given cascade on , thus extending the concept of network complexity which counts the (logarithmic) number of spanning trees contained in a static graph [42] . Additional details on the derivation of equation (8) and related discussion are presented in Supplementary Note 11 . Next we examine the inferential complexity of diffusion structure in various types of benchmark WTDs, finding a natural positive correlation between the intrinsic complexity of the underlying diffusion processes and the critical relative sample size for reconstruction goals, as shown in Fig. 4 . A more systematic analysis requires parametric generation of network ensembles that are tunable from both structural or temporal aspects to decouple the underlying influences. Here we first focus on three important structural properties of time-aggregated networks: clustering coefficient, average path length and connectivity heterogeneity. Numerical results show a nonlinear dependence of network complexity on the topological indices. Particularly in the practically relevant regime of salient small-world effects (characterized by large clustering coefficient and small average path length), the inferential complexity is positively (negatively) correlated with clustering coefficient (average path length) of the time-aggregated network. For most specific distributional shapes used as our benchmark WTDs, the heterogeneity in degree distribution also increases the inferential complexity of networks. Intuitively, diffusion on small-world and/or highly heterogeneous networks is inclined to produce, due to frequent occurrences of competing-risk censoring, indiscernible ‘downstream’ DATs that are close to each other, thus complicating the task of network reconstruction. We further examine the roles of the first three (normalized central) moments of WTDs, finding that the inferential complexity is positively (negatively) correlated with variance and skewness (mean, or lower bound of the support) of underlying WTDs. Detailed numerical results can be found in Supplementary Figs 24–30 . Figure 4: Inferential complexity of benchmark STNs. ( a – d ) Scatter plot of relative inferential complexity [ ] versus critical relative sample size for ensuring at least 0.95 AUROC, AUPR and BEP, respectively, as well as that for simultaneously recovering both the underlying networks and WTDs. Here, the angular brackets denote the average over ensembles of diffusion cascades. Each point represents an average over at least ten independent realizations with respect to a specific combination from benchmark networks and WTDs, and the critical values of respective relative sample sizes are listed in Supplementary Table 5 . Numerical results show that there is positive correlation between the inferential complexity and the critical relative sample size for achieving network reconstruction at a given accuracy, implying that the more complex (or indeterminate) the underlying diffusion structure, the more difficult the inference of temporal networks from data. Full size image\"\n },\n {\n \"section\": \"Discussion\",\n \"content\": \"In summary, we have developed a general framework to reconstruct STNs as a null model of temporal networks by fitting time-course observations of the diffusion process taking place on them. To alleviate the ill-posedness of this time-extended inverse problem, we have decomposed the task of network reconstruction into structural and temporal aspects, that is, an unsupervised topology inference for the time-aggregated networks using MCMC sampling, and a nonparametric density estimation for the associated WTDs via self-consistent iterations, respectively. We have given a rigorous consistency proof for the proposed WTD estimator, and have numerically shown that the iterative algorithm frequently possesses good convergence properties given properly selected, data-driven initial guesses. We have also applied our method to various types of benchmark STNs and empirical temporal networks, showing that it is statistically possible to discover latent temporal networks with high accuracy only from a moderate amount of diffusion data. Despite the inability to recover actual serial snapshots of networks, the reconstructed STN builds up both structural and temporal statistics sufficient to enable reliable prediction of the population-level diffusion behaviour, which can thus be of practical interest in a broad range of applications, particularly in time-critical and privacy-sensitive circumstances related to human activities. Our reconstruction method is not only restricted to the data generative process considered in the paper, but is also adapted to many variants of the original inference task. For example, recent work [27] has demonstrated the predictive power of the Hawkes process method for quantifying and tracing information cascades on social media, provided a well-defined memory kernel (a mathematical equivalent to the WTD in our model) fitted to the empirical mechanism for diffusion. Our method then provides an alternative operative route for estimating the memory kernel from early time courses of diffusion, as well as for updating the posterior WTD self-consistently with new available process data as diffusion proceeds, while not necessitating access to possibly privacy-sensitive information at the individual level. Another important extension is to use observed time courses of disease outbreaks to deduce the underlying population structure and temporal transmission pattern in epidemiological studies. Here, the essential difference between a generic epidemic process and the information broadcasting considered here lies in that the recovery process (for example, self-healing) prevents to a large extent the possible transmission of diseases from infected individuals to the remainder of the population. Restated in the language of survival analysis, the observation of successful transmissions is potentially ‘censored’ by the recovery events occurred among the infected population. Under this circumstance, one can directly make use of subdistributed WTDs (that is, ) to encapsulate arbitrary distributions of transmission and recovery times [30] , and apply the same inferential scheme with few modifications. We note here that the residual probability mass located at the infinity, , meaning the probability of endemic transmissions being interrupted by node u ’s recovery, causes a decrease in the effective number of diffusion cascades that are helpful for predicting if link ( u , v ) is present—as a consequence, the larger r uv , the less observably succeeded transmission routes, the more diffusion data required to extract the latent network. To exclude this nuisance factor we therefore have focused on the information diffusion model that mimics susceptible-infected epidemics (with r uv =0) to capture the effective sample sizes for specific reconstruction goals. In addition to the simplified forward diffusion model, we have made several assumptions in this paper, of which the most important is that the STNs are assumed to have no cross-dyad interdependence. This is the key property that enables the localized computation of likelihoods and branching coefficients using only neighbourhood information, thus ensuring an efficient implementation with polynomial complexity. We have discussed the second-order STN model that assumes a joint distribution of waiting times occurring on any successive dyads. Such polyadic correlations destroy the applicability of Matrix-Tree Theorems, leading to the necessity of a tree-enumeration procedure with exponential complexity. Leaving aside temporarily the issue of implementation expenditure, our inferential scheme can be naturally extended to the higher-order correlated case (see Supplementary Note 9 ). We have furthermore devised a pairwise approximation of branching coefficients in order to speed up our algorithms. Another significant one, mainly adopted in our benchmark tests, is the homogeneous population assumption, stating that the underlying WTDs ρ uv ( τ ) are identical, despite the individual heterogeneity or frailty widely prevalent in many real-life populations. Again, it should be pointed out that although our framework is applicable to non-identical WTDs, as well as higher-order, polyadically correlated cases, the established STN model to some extent sacrifices simplicity and runs the risk of overfitting. The ultimate goal of null modelling is to reproduce expected patterns that capture predictive information in data, while not being overly complicated. Let us further inspect the complexity and scalability of our method. While it is generally difficult to quantify the complexity of the iterative estimator even for the first-order STNs, benchmark test results numerically illustrate that the proposed procedure, when initialized with appropriate first guesses (see Supplementary Note 6 ), has quite good iterative convergence, irrespective of either structural properties of the time-aggregated topologies or distributional shapes of the dyad-specific WTDs. Thus our algorithm can practically be implemented with complexity O ( CN 2 M ), where C , N and M are the numbers of diffusion cascades, network nodes and MCMC iterations, respectively. Because M only exerts influence on the precision of a posteriori prediction of links and is independent of the problem size, the actual complexity of our procedure is then O ( CN 2 ), which reaches the theoretically optimal time complexity. Put another way, the network reconstruction problem can be regarded as a sequence of O ( N 2 ) likelihood-ratio hypothesis tests, in which the marginal gain of each possible link calculated using O ( C ) diffusion cascades is utilized to decide if the corresponding dyad of nodes is truly connected. Despite the proposed method with the (near-)inherent, quadratic complexity, there is a widespread need for more scalable and efficient, hopefully linear-time, inverse modelling approaches for empirical complex systems, especially in the era of Big Data. The potential solution to this impasse is to shift from predicting individual links to zooming out to the coarse-grained or mesoscopic topological scales using more sophisticated inference techniques borrowed from, for example, large-scale hypothesis testing [43] and community-based time series analysis [44] . There are also numerous conceivable approaches for our topology inference step. A closely related alternative one, which is formulated in a Bayesian fashion, is to introduce prior information about the underlying diffusion network into the estimation scheme to encompass a broader spectrum of expected structural features. Here we have exemplified this with the standard -norm sparsity priors, turning the maximum likelihood into a maximum a posteriori (MAP) estimation by virtue of slight differences in calculating the marginal gains of network links (see Supplementary Note 12 ). Numerical results have demonstrated the significantly improved reconstruction performance for STNs. Furthermore, parametric network models (such as exponential random graphs [45] ) and advanced optimization techniques for the MAP problem, as well as practically available prior information other than collective diffusion data (for example, partially observed topology structure, relevant metadata of network agents, and so on) can also be incorporated into our reconstruction scheme with little technical effort. Another contribution of the proposed inferential framework is the density estimation for underlying WTDs. Because of the context in which we are interested, an ideal null model of temporal networks would be reproductive of time variability in empirical interactions, as well as predictive of their collective dynamical properties at the desired level, in particular with the help of the WTD extracted from true latent (usually anomalous) diffusion patterns. When the goal is to solely create temporality information with prior network structures, rather than to reconstruct the topology of diffusion substrates, the density estimation can be pursued by a radically sharpened diffusion cascades. Under the homogeneous population assumption, meaning a population-wide fitting of the WTD commonly shared by all underlying dyadic interactions, one can expect a reliable WTD estimation even from a single diffusion cascade, unless either network size or observation window is very small. Consequently, our method has potential applications in epidemiological studies, especially for inverse modelling of general epidemics with non-Poissonian behaviour [30] , which range from estimating parameters such as the effective reproductive number using subdistributed WTDs to identifying the invasion routes of epidemics using branching coefficients. Finally, the central role played by branching coefficients—in the universal imputation scheme for treating the censored data problem arising from competing risk events of networked diffusion—has implications for definition and refinement of the dynamics-based node or link centrality and information-theoretic complexity measures for temporal networks.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Benchmark test data To test the performance of our inference algorithm, we have carried out extensive numerical experiments using three categories of time-aggregated benchmark topologies (deterministic, stochastic and empirical static networks) in combination with several different types of benchmark WTDs, as listed in Supplementary Tables 1 and 3 . To provide further empirical validation, we also apply our method to several realistic temporal contact networks, as listed in Supplementary Table 2 . Specifically, we reconstruct the STN model from synthesized diffusion cascades respectively using empirical contact lists and the STN tuple that is fitted to the original data (see Supplementary Table 4 for detailed experimental settings). Network data or their generative models can be found in relevant references. Gibbs sampler for time-aggregated graphs We apply MCMC method with Gibbs sampling to explore the configuration space of the underlying time-aggregated network. The configuration transition is based on the link-flipping operation with success probability according to the likelihood ratio between the new configuration and the old configuration, that is, the marginal gain of the flipping operation (equation (6)). In our benchmark test, the burn-in period and the maximum lag are set to 10, and the number M =200 of MCMC samples are drawn. See Supplementary Note 7 for detailed algorithm implementation. Iterative procedure for the WTD estimator To solve the self-consistent equations for , the algorithm carries out an iterative procedure as follows: Start with a priori guess for the underlying WTDs and repeat the following synchronous updates for all entries of until convergence The convergence is judged by a small change in the estimated between successive iterations. Here we adopt the Kolmogorov–Smirnov divergence, and terminate the iteration whenever , where is survival function of WTD , and ɛ is a predetermined error threshold. The detailed self-consistent iterative scheme is presented in Supplementary Note 7 . In our numerical validation, we assume for brevity that internal temporal interactions among a population satisfy the homogeneity condition, corresponding to first-order the STN with i.i.d. dyad-level WTDs. To initialize the iterative procedure, we use exponential WTD as the first guess, and set smoothing kernel bandwidth h =0.05. Practical parameter selection criteria, as well as their effects on convergence properties and performance of the density estimator are discussed in Supplementary Note 6 . Notations We summarize the notations used throughout this paper in Supplementary Table 6 . Data availability All relevant data are available from the authors on request.\"\n },\n {\n \"section\": \"Additional information\",\n \"content\": \"How to cite this article: Li, Xun & Li, Xiang et al . Reconstruction of stochastic temporal networks through diffusive arrival times. Nat. Commun. 8, 15729 doi: 10.1038/ncomms15729 (2017). 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 develop a data-driven null model of stochastic temporal networks that can reconstruct latent network structures and temporal interaction patterns from observed diffusion process data, such as time-of-arrival records, by capturing both structural and temporal regularities inherent in the dynamics of complex systems."},"entities":{"kind":"list like","value":["data-driven null model","stochastic temporal networks","latent network structures","temporal interaction patterns","observed diffusion process data","time-of-arrival records","structural regularities","temporal regularities","dynamics of complex systems"],"string":"[\n \"data-driven null model\",\n \"stochastic temporal networks\",\n \"latent network structures\",\n \"temporal interaction patterns\",\n \"observed diffusion process data\",\n \"time-of-arrival records\",\n \"structural regularities\",\n \"temporal regularities\",\n \"dynamics of complex systems\"\n]","isConversational":false},"references":{"kind":"list like","value":[],"string":"[]","isConversational":false}}},{"rowIdx":99,"cells":{"unique_id":{"kind":"string","value":"s41467-023-38237-7"},"subfield":{"kind":"string","value":"Materials science"},"title":{"kind":"string","value":"Epitaxial growth of highly symmetrical branched noble metal-semiconductor heterostructures with efficient plasmon-induced hot-electron transfer"},"abstract":{"kind":"string","value":"Epitaxial growth is one of the most commonly used strategies to precisely tailor heterostructures with well-defined compositions, morphologies, crystal phases, and interfaces for various applications. However, as epitaxial growth requires a small interfacial lattice mismatch between the components, it remains a challenge for the epitaxial synthesis of heterostructures constructed by materials with large lattice mismatch and/or different chemical bonding, especially the noble metal-semiconductor heterostructures. Here, we develop a noble metal-seeded epitaxial growth strategy to prepare highly symmetrical noble metal-semiconductor branched heterostructures with desired spatial configurations, i.e., twenty CdS (or CdSe) nanorods epitaxially grown on twenty exposed (111) facets of Ag icosahedral nanocrystal, albeit a large lattice mismatch (more than 40%). Importantly, a high quantum yield (QY) of plasmon-induced hot-electron transferred from Ag to CdS was observed in epitaxial Ag-CdS icosapods (18.1%). This work demonstrates that epitaxial growth can be achieved in heterostructures composed of materials with large lattice mismatches. The constructed epitaxial noble metal-semiconductor interfaces could be an ideal platform for investigating the role of interfaces in various physicochemical processes."},"sections":{"kind":"list like","value":[{"section":"Introduction","content":"Heterostructures have attracted tremendous attention due to their physicochemical properties arising from their morphologies [1] , [2] , [3] , interfaces [4] , [5] , [6] , and spatial arrangements of different components [7] , [8] . The precise control over their nanoscale structures is essential to understand the structure-property correlation and enhance their performance in various applications, such as electronics [9] , [10] , catalysis [11] , [12] , solar energy conversion [13] , [14] , etc. To date, the seeded/templated epitaxial growth is the most commonly used strategy to precisely tailor the hierarchical heterostructures [15] , [16] , [17] , [18] , [19] . Epitaxial growth of a secondary material on a specific facet of the seed/template allows it to follow the crystallographic orientation of the seed/template [17] , [20] . Combined with delicate control over the exposed facet and crystal phase of seeds/templates, this strategy has been successfully used to prepare various hierarchical heterostructures with well-defined spatial structures [21] , [22] , interfaces [23] , [24] , crystal phases [25] , [26] , [27] , and programmable components [28] , [29] . However, the aforementioned seeded/templated epitaxial growth is mainly used to prepare heterostructures composed of components with similar lattice structures and/or chemical bonding, e.g., metal-metal [20] , [26] , [27] , metal oxide-metal oxide [30] , [31] , metal chalcogenide-metal chalcogenide [1] , [16] , [22] , [23] , [28] , perovskite-perovskite [29] , and metal-organic framework (MOF)-MOF [32] , [33] heterostructures due to the common requirement of similar structure (normally with lattice mismatch smaller than 5%) for the epitaxial growth [17] , [18] , [34] . Therefore, it remains a great challenge for the epitaxial growth of heterostructures made of distinctly different materials, especially for the noble metal-semiconductor heterostructures, in which two materials could have a large lattice mismatch (normally larger than 20%) and different chemical bonding [18] , [24] , [35] , [36] , [37] , [38] . Commonly, the preparation of noble metal-semiconductor heterostructures relies on conventional non-epitaxial strategies including chemical deposition [39] , [40] , cation-exchange-facilitated non-epitaxial growth [35] , [41] , phase transfer [42] , photochemical deposition [43] , chemical extraction [44] , [45] , sol-gel method [46] , etc. Although the obtained non-epitaxial noble metal-semiconductor heterostructures have shown great potential in diverse fields, such as catalysis [6] , [14] , [40] , [43] , [46] , photovoltaic devices [4] , [14] , [47] and sensors [36] , [44] , [48] , their wide application has been hindered due to difficulties in the precisely defined control over their structures and interfaces. Therefore, a reliable epitaxial growth method is urgently desired to rationally construct hierarchical noble metal-semiconductor heterostructures with precisely controlled architectures and well-defined interfaces. In this work, we report a controlled synthesis of highly symmetrical branched noble metal-semiconductor heterostructures via a noble-metal-seeded epitaxial growth. The 1D II-VI semiconductor nanorods can be uniformly grown on the exposed (111) facets of zero-dimensional platonic Ag nanocrystals. The obtained heterostructure possesses the same crystallographic symmetry as the noble metal seed, i.e., twenty CdS (or CdSe) nanorods radially grown on twenty exposed (111) facets of the five-fold multi-twinned Ag icosahedron, respectively, denoted as Ag-CdS (or Ag-CdSe) icosapods. Importantly, the epitaxial growth of CdS (or CdSe) nanorods on Ag icosahedral nanocrystals has been achieved albeit a large lattice mismatch between Ag and CdS (or CdSe) (more than 40%). Impressively, the pump-probe transient absorption (TA) spectroscopy reveals the ultrafast plasmon hot-electron transfer process in the epitaxial heterostructures with high QY of the plasmon-induced hot-electron transferred from noble metal to semiconductor, i.e., 18.1% and 17.6% for the Ag-CdS and Ag-CdSe icosapods, respectively.","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],"infos":[{"idx":"1","title":"Borys, N. J., Walter, M. J., Huang, J., Talapin, D. V. & Lupton, J. M. The role of particle morphology in interfacial energy transfer in CdSe/CdS heterostructure nanocrystals.Science330, 1371–1374 (2010).","link":"https://doi.org/10.1126%2Fscience.1198070"},{"idx":"2","title":"Gu, J., Zhang, Y. W. & Tao, F. Shape control of bimetallic nanocatalysts through well-designed colloidal chemistry approaches.Chem. Soc. Rev.41, 8050–8065 (2012).","link":"https://doi.org/10.1039%2Fc2cs35184f"},{"idx":"3","title":"Jiang, L. Z. et al. Facet engineering accelerates spillover hydrogenation on highly diluted metal nanocatalysts.Nat. Nanotechnol.15, 848–853 (2020).","link":"https://doi.org/10.1038%2Fs41565-020-0746-x"},{"idx":"4","title":"Clavero, C. Plasmon-induced hot-electron generation at nanoparticle/metal-oxide interfaces for photovoltaic and photocatalytic devices.Nat. Photon.8, 95–103 (2014).","link":"https://doi.org/10.1038%2Fnphoton.2013.238"},{"idx":"5","title":"Liu, M. X. et al. Lattice anchoring stabilizes solution-processed semiconductors.Nature570, 96–101 (2019).","link":"https://doi.org/10.1038%2Fs41586-019-1239-7"},{"idx":"6","title":"Yuan, W. T. et al. In situ manipulation of the active Au-TiO2interface with atomic precision during CO oxidation.Science371, 517–521 (2021).","link":"https://doi.org/10.1126%2Fscience.abe3558"},{"idx":"7","title":"Zhao, M. T. et al. Metal-organic frameworks as selectivity regulators for hydrogenation reactions.Nature539, 76–80 (2016).","link":"https://doi.org/10.1038%2Fnature19763"},{"idx":"8","title":"Cheng, K. et al. Maximizing noble metal utilization in solid catalysts by control of nanoparticle location.Science377, 204–208 (2022).","link":"https://doi.org/10.1126%2Fscience.abn8289"},{"idx":"9","title":"Kagan, C. R., Lifshitz, E., Sargent, E. H. & Talapin, D. V. Building devices from colloidal quantum dots.Science353, aac5523 (2016).","link":"https://doi.org/10.1126%2Fscience.aac5523"},{"idx":"10","title":"Liu, Y. et al. Van der Waals heterostructures and devices.Nat. Rev. Mater.1, 16042 (2016).","link":"https://doi.org/10.1038%2Fnatrevmats.2016.42"},{"idx":"11","title":"Losch, P. et al. Colloidal nanocrystals for heterogeneous catalysis.Nano Today24, 15–47 (2019).","link":"https://doi.org/10.1016%2Fj.nantod.2018.12.002"},{"idx":"12","title":"Vogt, C. & Weckhuysen, B. M. The concept of active site in heterogeneous catalysis.Nat. Rev. Chem.6, 89–111 (2022).","link":"https://doi.org/10.1038%2Fs41570-021-00340-y"},{"idx":"13","title":"Lu, Q. P., Yu, Y. F., Ma, Q. L., Chen, B. & Zhang, H. 2D transition-metal-dichalcogenide-nanosheet-based composites for photocatalytic and electrocatalytic hydrogen evolution reactions.Adv. Mater.28, 1917–1933 (2016).","link":"https://doi.org/10.1002%2Fadma.201503270"},{"idx":"14","title":"Linic, S., Chavez, S. & Elias, R. Flow and extraction of energy and charge carriers in hybrid plasmonic nanostructures.Nat. Mater.20, 916–924 (2021).","link":"https://doi.org/10.1038%2Fs41563-020-00858-4"},{"idx":"15","title":"Lauhon, L. J., Gudiksen, M. S., Wang, C. L. & Lieber, C. M. Epitaxial core-shell and core-multishell nanowire heterostructures.Nature420, 57–61 (2002).","link":"https://doi.org/10.1038%2Fnature01141"},{"idx":"16","title":"Talapin, D. V. et al. Seeded growth of highly luminescent CdSe/CdS nanoheterostructures with rod and tetrapod morphologies.Nano Lett.7, 2951–2959 (2007).","link":"https://doi.org/10.1021%2Fnl072003g"},{"idx":"17","title":"Tan, C. L., Chen, J. Z., Wu, X.-J. & Zhang, H. Epitaxial growth of hybrid nanostructures.Nat. Rev. Mater.3, 17089 (2018).","link":"https://doi.org/10.1038%2Fnatrevmats.2017.89"},{"idx":"18","title":"Kum, H. et al. Epitaxial growth and layer-transfer techniques for heterogeneous integration of materials for electronic and photonic devices.Nat. Electro2, 439–450 (2019).","link":"https://doi.org/10.1038%2Fs41928-019-0314-2"},{"idx":"19","title":"Kelso, M. V., Mahenderkar, N. K., Chen, Q. Z., Tubbesing, J. Z. & Switzer, J. A. Spin coating epitaxial films.Science364, 166–169 (2019).","link":"https://doi.org/10.1126%2Fscience.aaw6184"},{"idx":"20","title":"Habas, S. E., Lee, H., Radmilovic, V., Somorjai, G. A. & Yang, P. Shaping binary metal nanocrystals through epitaxial seeded growth.Nat. Mater.6, 692–697 (2007).","link":"https://doi.org/10.1038%2Fnmat1957"},{"idx":"21","title":"Miszta, K. et al. Hierarchical self-assembly of suspended branched colloidal nanocrystals into superlattice structures.Nat. Mater.10, 872–876 (2011).","link":"https://doi.org/10.1038%2Fnmat3121"},{"idx":"22","title":"Wu, X.-J. et al. Controlled growth of high-density CdS and CdSe nanorod arrays on selective facets of two-dimensional semiconductor nanoplates.Nat. Chem.8, 470–475 (2016).","link":"https://doi.org/10.1038%2Fnchem.2473"},{"idx":"23","title":"Li, M. Y. et al. Epitaxial growth of a monolayer WSe2-MoS2lateral p-n junction with an atomically sharp interface.Science349, 524–528 (2015).","link":"https://doi.org/10.1126%2Fscience.aab4097"},{"idx":"24","title":"Kanne, T. et al. Epitaxial Pb on InAs nanowires for quantum devices.Nat. Nanotechnol.16, 776–781 (2021).","link":"https://doi.org/10.1038%2Fs41565-021-00900-9"},{"idx":"25","title":"Chen, Y. et al. Phase engineering of nanomaterials.Nat. Rev. Chem.4, 243–256 (2020).","link":"https://doi.org/10.1038%2Fs41570-020-0173-4"},{"idx":"26","title":"Lu, Q. P. et al. Crystal phase-based epitaxial growth of hybrid noble metal nanostructures on 4H/fcc Au nanowires.Nat. Chem.10, 456–461 (2018).","link":"https://doi.org/10.1038%2Fs41557-018-0012-0"},{"idx":"27","title":"Liu, J. W. et al. Selective epitaxial growth of Rh nanorods on 2H/fccheterophase Au nanosheets to form 1D/2D Rh-Au heterostructures for highly efficient hydrogen evolution.J. Am. Chem. Soc.143, 4387–4396 (2021).","link":"https://doi.org/10.1021%2Fjacs.1c00612"},{"idx":"28","title":"Zhang, Z. W. et al. Robust epitaxial growth of two-dimensional heterostructures, multiheterostructures, and superlattices.Science357, 788–792 (2017).","link":"https://doi.org/10.1126%2Fscience.aan6814"},{"idx":"29","title":"Shi, E. Z. et al. Two-dimensional halide perovskite lateral epitaxial heterostructures.Nature580, 614–620 (2020).","link":"https://doi.org/10.1038%2Fs41586-020-2219-7"},{"idx":"30","title":"Oh, M. H. et al. Design and synthesis of multigrain nanocrystals via geometric misfit strain.Nature577, 359–363 (2020).","link":"https://doi.org/10.1038%2Fs41586-019-1899-3"},{"idx":"31","title":"Forticaux, A., Hacialioglu, S., DeGrave, J. P., Dziedzic, R. & Jin, S. Three-dimensional mesoscale heterostructures of ZnO nanowire arrays epitaxially grown on CuGaO2nanoplates as individual diodes.ACS Nano7, 8224–8232 (2013).","link":"https://doi.org/10.1021%2Fnn4037078"},{"idx":"32","title":"Yao, Y. et al. Heteroepitaxial growth of multiblock Ln-MOF microrods for photonic barcodes.Angew. Chem. Int. Ed.58, 13803–13807 (2019).","link":"https://doi.org/10.1002%2Fanie.201907433"},{"idx":"33","title":"Zhao, M. T. et al. Selective epitaxial growth of oriented hierarchical metal-organic framework heterostructures.J. Am. Chem. Soc.142, 8953–8961 (2020).","link":"https://doi.org/10.1021%2Fjacs.0c02489"},{"idx":"34","title":"Grundmann, M. Formation of epitaxial domains: unified theory and survey of experimental results.Phys. Status Solidi B248, 805–824 (2011).","link":"https://doi.org/10.1002%2Fpssb.201046530"},{"idx":"35","title":"Zhang, J. T., Tang, Y., Lee, K. & Min, O. Y. Nonepitaxial growth of hybrid core-shell nanostructures with large lattice mismatches.Science327, 1634–1638 (2010).","link":"https://doi.org/10.1126%2Fscience.1184769"},{"idx":"36","title":"Liu, J. & Zhang, J. T. Nanointerface chemistry: lattice-mismatch-directed synthesis and application of hybrid nanocrystals.Chem. Rev.120, 2123–2170 (2020).","link":"https://doi.org/10.1021%2Facs.chemrev.9b00443"},{"idx":"37","title":"Palmstrom, C. J. Epitaxy of dissimilar materials.Annu. Rev. Mater. Sci.25, 389–415 (1995).","link":"https://doi.org/10.1146%2Fannurev.ms.25.080195.002133"},{"idx":"38","title":"Pendharkar, M. et al. Parity-preserving and magnetic field-resilient superconductivity in InSb nanowires with Sn shells.Science372, 508–511 (2021).","link":"https://doi.org/10.1126%2Fscience.aba5211"},{"idx":"39","title":"Mokari, T., Rothenberg, E., Popov, I., Costi, R. & Banin, U. Selective growth of metal tips onto semiconductor quantum rods and tetrapods.Science304, 1787–1790 (2004).","link":"https://doi.org/10.1126%2Fscience.1097830"},{"idx":"40","title":"Wolff, C. M. et al. All-in-one visible-light-driven water splitting by combining nanoparticulate and molecular co-catalysts on CdS nanorods.Nat. Energy3, 862–869 (2018).","link":"https://doi.org/10.1038%2Fs41560-018-0229-6"},{"idx":"41","title":"Fan, Z. X. et al. Synthesis of 4H/fcc-Au@metal sulfide core shell nanoribbons.J. Am. Chem. Soc.137, 10910–10913 (2015).","link":"https://doi.org/10.1021%2Fjacs.5b06405"},{"idx":"42","title":"Yang, J. & Ying, J. Y. A general phase-transfer protocol for metal ions and its application in nanocrystal synthesis.Nat. Mater.8, 683–689 (2009).","link":"https://doi.org/10.1038%2Fnmat2490"},{"idx":"43","title":"Ben-Shahar, Y. et al. Optimal metal domain size for photocatalysis with hybrid semiconductor-metal nanorods.Nat. Commun.7, 10413 (2016).","link":"https://doi.org/10.1038%2Fncomms10413"},{"idx":"44","title":"Chen, J. Z. et al. Ag@MoS2core-shell heterostructure as SERS platform to reveal the hydrogen evolution active sites of single-layer MoS2.J. Am. Chem. Soc.142, 7161–7167 (2020).","link":"https://doi.org/10.1021%2Fjacs.0c01649"},{"idx":"45","title":"Han, S. K., Gu, C., Gong, M. & Yu, S. H. A Trialkylphosphine-driven chemical transformation route to Ag- and Bi-based chalcogenides.J. Am. Chem. Soc.137, 5390–5396 (2015).","link":"https://doi.org/10.1021%2Fjacs.5b00041"},{"idx":"46","title":"Atta, S., Pennington, A. M., Celik, F. E. & Fabris, L. TiO2on gold nanostars enhances photocatalytic water reduction in the near-infrared regime.Chem4, 2140–2153 (2018).","link":"https://doi.org/10.1016%2Fj.chempr.2018.06.004"},{"idx":"47","title":"Jang, Y. H. et al. Plasmonic solar cells: from rational design to mechanism overview.Chem. Rev.116, 14982–15034 (2016).","link":"https://doi.org/10.1021%2Facs.chemrev.6b00302"},{"idx":"48","title":"Knight, M. W., Sobhani, H., Nordlander, P. & Halas, N. J. Photodetection with active optical antennas.Science332, 702–704 (2011).","link":"https://doi.org/10.1126%2Fscience.1203056"}]}},{"section":"Results","content":"Material synthesis and characterizations Figure 1a schematically illustrates the synthesis of Ag-CdS icosapods. Briefly, five-fold multi-twinned Ag icosahedral nanocrystals with a size of 13.3 ± 0.6 nm were first synthesized (Supplementary Fig. 1 ), which were then used as seeds for the epitaxial growth of CdS nanorods via a hot injection method (see the Methods section). Scanning electron microscope (SEM) (Fig. 1b ), transmission electron microscopy (TEM) (Fig. 1c ) and high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM, Fig. 1d ) images show the high purity and uniformity of the synthesized Ag-CdS icosapods with a size of 108.2 ± 5.5 nm, and the length and diameter of CdS nanorods are around 50 and 7 nm, respectively (Supplementary Fig. 2 ). Fig. 1: Synthesis and characterization of Ag-CdS icosapods. a Schematic illustration of the synthesis of Ag-CdS icosapods by using the pre-synthesized Ag nanocrystals, i.e., Ag icosahedral nanocrystals, as seeds for the epitaxial growth of CdS nanorods. C 2 , C 3 , and C 5 indicate the observed rotational axes of the schematically drawn Ag icosahedral nanocrystals (left) and Ag-CdS icosapods (right). b – d SEM ( b ), TEM ( c ), and HAADF-STEM ( d ) images of Ag-CdS icosapods. e – g HAADF-STEM images of a representative Ag-CdS icosapod observed along its C 2 ( e ), C 3 ( f ), and C 5 ( g ) rotational axes, respectively. h HAADF-STEM and the corresponding elemental mapping images of a representative Ag-CdS icosapod. Full size image To reveal the detailed three-dimensional (3D) architecture of the obtained Ag-CdS icosapods, a series of HAADF-STEM images of a representative Ag-CdS icosapod tilted with different angles were collected. The 3D reconstruction shows that twenty CdS nanorods radially extend from twenty exposed facets of the Ag icosahedron seed (Supplementary Movie 1 ), confirming that the Ag-CdS icosapod has the same rotational symmetry as the Ag icosahedral nanocrystal. As shown in Supplementary Fig. 3a–f , the Ag icosahedral nanocrystal exhibits twofold (C 2 ), threefold (C 3 ), and five-fold (C 5 ) symmetrical Moiré patterns, which are consistent with the previously reported five-fold multi-twinned icosahedral nanocrystals [49] . After the growth of CdS nanorods, the obtained Ag-CdS icosapods with C 2 , C 3 , C 5 symmetrical patterns (Supplementary Fig. 3g–i ) have also been observed under HAADF-STEM (Fig. 1e–g ), which are consistent with the HRTEM images (Supplementary Fig. 3j–l ). In particular, when projected along its C 3 rotational axis (Fig. 1f ), the Ag-CdS icosapod presents a sophisticated star-like pattern with the observed eighteen branches radially extended from the bright core, while the other two branches are unobserved since they are perpendicular to the projected plane. Moreover, a five-fold pattern of the Ag-CdS icosapod with the C 5 symmetry exhibits different contrast along the ten extending directions of branches, due to the overlapped images of two nanorods in the same direction when viewed along its C 5 rotational axis (Fig. 1g ). The HAADF-STEM and the corresponding energy-dispersive X-ray spectroscopy (EDS) elemental mapping images (Fig. 1h ) of a representative Ag-CdS icosapod reveal that the Ag element is located in the core, while the Cd and S elements are homogeneously distributed over the twenty nanorods. X-ray photoelectron spectroscopy (XPS) was used to determine the valence state of components in Ag-CdS icosapods (Supplementary Fig. 4 ). In particular, the peaks located at 373.7 and 367.7 eV can be ascribed to Ag 3d 3/2 and Ag 3d 5/2 of zerovalent Ag, respectively (Supplementary Fig. 4b ). Combined XPS results with the Ag K-edge XANES spectra (Supplementary Fig. 5 ), it can be concluded that the Ag nanocrystals have not been sulfurized during the synthesis of Ag-CdS icosapods, which might arise from the trioctylphosphine (TOP) used in the synthesis [44] , [45] , [50] . Furthermore, the X-ray diffraction (XRD) pattern of the synthesized Ag-CdS icosapods can be well indexed to the wurtzite CdS and fcc -Ag (Supplementary Fig. 6 ). The aforementioned results clearly demonstrate the successful growth of CdS nanorods on the Ag icosahedral nanocrystals, and the obtained Ag-CdS icosapods show the uniform size as well as well-defined morphology with sophisticated symmetrical branched architecture. Ultraviolet photoelectron spectroscopy (UPS) was used to reveal the interfacial electronic structure of Ag-CdS icosapods. As shown in Supplementary Fig. 7a , the cut-off energies (E cut-off ) of Ag nanocrystals and Ag-CdS icosapods are determined to be 16.95 eV and 17.20 eV, respectively. The Fermi levels of Ag nanocrystals and Ag-CdS icosapods are estimated to be −4.27 eV and −4.02 eV relative to the vacuum level, respectively. Compared with the Ag nanocrystals, the shift (≈0.25 eV) of the Fermi level of Ag-CdS icosapods demonstrates a Schottky barrier was formed between Ag and n-type semiconductor CdS in the Ag-CdS icosapods [51] . Since the Fermi level, conduction band (CB), and valence band (VB) of CdS are measured to be −3.92 eV, −3.61 eV, and −6.22 eV, respectively (Supplementary Fig. 7b, c ), the interfacial electronic structure of Ag-CdS icosapods can be schematically shown in Supplementary Fig. 7d . Importantly, the synthesis of noble metal-semiconductor heterostructures with tunable components can be achieved. By exchanging the trioctylphosphine sulfide (TOPS) with trioctylphosphine selenium (TOPSe) as the chalcogen precursor in our experiment (see the “Methods” section), branched Ag-CdSe icosapods have been obtained. The TEM and HAADF-STEM images of Ag-CdSe icosapods (Supplementary Fig. 8 ) and the EDS elemental mapping result (Supplementary Fig. 9 ) of a representative Ag-CdSe icosapod demonstrate the successful growth of CdSe nanorods on the icosahedral Ag seeds. The XPS spectra of Ag-CdSe icosapods confirm the Ag core in Ag-CdSe icosapods maintained zerovalent (Supplementary Fig. 10 ), and the XRD spectrum matches well with the wurtzite CdSe and fcc -Ag (Supplementary Fig. 11 ). Epitaxial noble metal-semiconductor interfaces The interfacial structures between the icosahedral Ag seed and the grown CdS and CdSe nanorods are very important for the formation of such sophisticated noble metal-semiconductor heterostructures. As shown in Supplementary Fig. 12a , although it is very difficult to use spherical aberration-corrected high-angle annular dark-field STEM (Cs-HAADF-STEM) to clearly observe the detailed interfacial structure in the synthesized Ag-CdS icosapod due to its sophisticated architecture with large size of 108.2 ± 5.5 nm (Fig. 1b–d and Supplementary Fig. 2 ), the epitaxial growth of CdS rods on Ag with the epitaxial interface between the Ag (111) and CdS (002) planes, referred to as Ag (111) ∥ CdS (002), can be confirmed by the fast Fourier transformation (FFT) pattern (Supplementary Fig. 12b ). To identify the Ag-CdS interface, Ag-CdS icosapods extracted from the reactor at ~60 s were characterized by the Cs-HAADF-STEM (Supplementary Fig. 13a ). As shown in Fig. 2a , the measured lattice distances of 0.14, 0.24, 0.34 and 0.36 nm are well-matched with the interplanar distances of Ag (220), Ag (111), CdS (002), and CdS (100) planes, respectively. Besides the sharp Ag-CdS interface observed, it is also found that the Ag (111) planes are parallel to the CdS (002) planes, and Ag (220) planes are parallel to CdS (100) planes. Together with the aligned spots in the FFT pattern (Fig. 2b ), it can be concluded that the CdS are epitaxially grown on Ag icosahedral nanocrystals with an epitaxial relationship of Ag (111) ∥ CdS (002) and Ag (220) ∥ CdS (100). Moreover, as shown in the Cs-HAADF-STEM image (Fig. 2a ) and inverse FFT image (Fig. 2c ), the nearest neighbor atoms on the Ag surface are determined to be S of CdS ( \\(00\\) \\(\\overline{1}\\) ) plane since the S atoms have lower contrast than the Cd atoms under the STEM mode, thus CdS is grown along the [001] direction [52] (Supplementary Fig. 14 ). Similarly, as shown in Fig. 2d–f , when CdSe was grown on the icosahedral Ag seeds at a short reaction time of 1 min (Supplementary Fig. 13b ), instead of 3 min used for the preparation of aforementioned Ag-CdSe icosapods (Supplementary Figs. 8 and 9 ), an epitaxial relationship can also be found between Ag and CdSe. The epitaxial Ag-CdSe interface was determined to be Ag (111) ∥ CdSe (002) and Ag (220) ∥ CdSe (100), and CdSe grows along the [001] direction with the nearest neighbor Se atoms on the Ag surface (Fig. 2f ). Based on the aforementioned results, the epitaxial interfaces of Ag-CdS and Ag-CdSe can be schematically illustrated in Supplementary Fig. 15 and Supplementary Data 1 . Importantly, despite the large lattice mismatch (more than 40%) between Ag and CdS (or CdSe) (Supplementary Table 1 ), which is much greater than the previously reported value for the epitaxial growth (normally <5%) [17] , [18] , [30] , [36] , the epitaxial interfaces can still be achieved in our synthesized Ag-CdS and Ag-CdSe heterostructures. Fig. 2: Characterization of epitaxial interfaces in Ag-CdS and Ag-CdSe heterostructures obtained at a short reaction time of 1 min. a Cs-HAADF-STEM image of the Ag-CdS interface. b , c The FFT pattern ( b ) and inverse FFT image ( c ) of the Ag-CdS interface. The silver square and silver triangle in ( b ) are applied to mark the FFT spots indexed to Ag (111) and Ag (220), respectively. The yellow circle and yellow rhombus are applied to mark the FFT spots indexed to CdS (002) and CdS (100), respectively. d Cs-HAADF-STEM image of the Ag-CdSe interface. e , f The FFT pattern ( e ) and inverse FFT image ( f ) of the Ag-CdSe interface. The silver square and silver triangle in ( e ) are applied to mark the FFT spots indexed to Ag (111) and Ag (220), respectively. The red circle and red rhombus are applied to mark the FFT spots indexed to CdSe (002) and CdSe (100), respectively. The silver, blue, yellow, red balls in ( a ), ( c ), ( d ), ( f ) represent the scheme of Ag, Cd, S, Se atoms, respectively. Full size image Ultrafast plasmon-induced charge transfer The successful preparation of noble metal-semiconductor with well-defined morphologies and epitaxial interfaces provides a possibility to explore and understand their structure-property relationships. Here, we systematically studied the plasmon-induced charge transfer process in the obtained Ag-CdS and Ag-CdSe icosapods using pump-probe TA spectroscopy. As schematically shown in Fig. 3a , the pump laser-excited plasmon in Ag, which decays into electron-hole pairs, is followed by hot-electron transfer to semiconductor and/or direct interfacial charge transfer transition. The UV-visible absorption spectra of Ag icosahedral seeds, CdS nanorods (Supplementary Fig. 16 ), and the obtained Ag-CdS icosapods are presented in Fig. 3b . The strong peak for the blue curve located at 408 nm can be assigned to the localized surface plasmon resonance (LSPR) absorption peak of the Ag icosahedral nanocrystals. After the growth of CdS on Ag, two peaks located at 462 and 548 nm appear (red curve in Fig. 3b ). The peak at 462 nm, which is similar to the absorption peak of CdS nanorods (465 nm, black curve in Fig. 3b ), is assigned to the 1Σ exciton transition of CdS between the CB and valence band VB levels [53] , and the peak at 548 nm should be attributed to the redshifted LSPR of Ag nanocrystals. We then performed pump-probe TA spectroscopy using the 550 nm laser to excite Ag-CdS icosapods. As the photon energy of 550 nm laser is below the band gap of the CdS nanorods, the Ag LSPR bands in Ag-CdS icosapods are selectively excited (Supplementary Fig. 17 ). The TA spectrum of Ag-CdS icosapods, displayed as a two-dimensional pseudo-color plot in Fig. 3c , shows the distinct bleaching bands centered at ~480 and ~550 nm that can be attributed to the bleach of the CdS 1Σ band and Ag LSPR band, respectively. As the CdS was not excited, the bleach signal of the CdS 1Σ band can be attributed to the plasmon-induced hot-electron transfer from the Ag core to CdS nanorods. Fig. 3: Plasmon-induced charge separation in Ag-CdS icosapods. a Schematic illustration of plasmon-induced hot-electron transfer in Ag-CdS icosapods after femtosecond laser pumping. The dashed ellipsis schematically illustrates the plasmon-induced interfacial charge-transfer transition in Ag-CdS icosapods. b UV-visible absorption spectra of the colloidal solutions of Ag icosahedral nanocrystals, CdS nanorods, and Ag-CdS icosapods in toluene. Insets: photographs of the corresponding colloid solutions. c Two-dimensional pseudo-color plots of pump-visible probe TA spectra of the colloid solutions of Ag-CdS in toluene. X-axis: probe wavelength; Y-axis: pump-probe delay; Color: change in absorbance (ΔAbs), shown in milli-optical density unit (mOD). Gray shading is used to avoid the disturbance of the 550 nm pump laser to the probed signal. d Averaged mid-IR TA spectra of Ag-CdS icosapods from 1 to 2 ps (red curve), 20 to 40 ps (orange curve), 100 to 200 ps (blue curve), and 600 to 900 ps (black curve) after 550 nm laser excitation. e Time-resolved mid-IR absorption spectrum of Ag-CdS icosapods. Gray curves represent the multiexponential fits as described in Eq. 6 . A negligible intraband absorption signal (black squares) is observed in a control experiment conducted on CdS nanorods. Inset in e : zoom-in kinetics spectrum for the Ag-CdS icosapods. Source data are provided as a Source data file. Full size image Moreover, as shown in Fig. 3a , when the plasmon-induced hot-electron is transferred from the Ag to CdS or CdSe, it can generate a positive mid-IR TA signal due to the 1Σ-to-1П intraband transition, which enables us to directly observe the plasmon-induced hot-electron transfer process and quantify its QY from Ag to CdS or CdSe [53] , [54] , [55] . As shown in the averaged mid-IR TA spectra of Ag-CdS icosapods (Fig. 3d ), positive mid-IR TA signals were observed after the excitation of the Ag LSPR band, further confirming the presence of plasmon-induced hot-electron in the CB of CdS. The QY of plasmon-induced hot-electron transferred from Ag to CdS can be determined by the peak amplitude of the CdS intraband absorption signal according to Eq. 4 , which is calculated to be 18.1% (see “Methods”, Supplementary Figs. 18 , 19 and Supplementary Table 2 for details). This QY value is much higher than that previously reported in nonepitaxial noble metal-semiconductor heterostructures [53] , [55] , [56] , [57] , [58] , for instance, about 2.75% for Au-tipped CdS nanorods [53] . Fitting the time-resolved mid-IR absorption curve of Ag-CdS icosapods (Fig. 3e ) based on Eq. 6 yielded a formation time of 18.0 ± 0.3 fs and a multi-exponential decay with a half-life of 1.9 ± 0.16 ps, which corresponds to plasmon-induced hot-electron transfer time and charge-recombination time, respectively (Supplementary Table 3 ). A similar result was obtained in the epitaxial Ag-CdSe icosapods. The measured QY and the time of plasmon-induced hot-electron transfer were determined to be 17.6% and 7.5 ± 7.2 fs when epitaxial Ag-CdSe icosapods were pumped by a 700 nm laser, respectively (Supplementary Figs. 19 – 22 ; Supplementary Tables 3 and 4 ). The plasmon-induced hot-electron transfer processes in the epitaxial Ag-CdS and Ag-CdSe icosapods are much faster than the conventional indirect plasmon-induced hot-electron transfer process (commonly more than 100 fs [53] , [55] ). On such a short timescale (<20 fs), the plasmon directly decays into the electrons in the CB of semiconductors via the interfacial charge-transfer transition before rapid cooling of hot electrons induced by electron-electron scattering [54] , [55] , [58] , [59] , [60] . Unfortunately, because the transient IR selectively probes the electrons injected into the CB of CdS, similar to the previous reports [54] , our study was not able to provide direct information about the effect of the metal-semiconductor interfacial states on the plasmon hot electron transfer and recombination processes. In summary, we have demonstrated a noble-metal-seeded epitaxial growth method to achieve the controlled synthesis of highly symmetrical noble metal-semiconductor heterostructures, i.e., Ag-CdS and Ag-CdSe icosapods. Abrupt epitaxial Ag-CdS/CdSe interfaces are observed in the Ag-CdS and Ag-CdSe icosapods, which could be concluded as Ag (111) ∥ CdS/CdSe (002) and Ag (220) ∥ CdS/CdSe (100), albeit a large lattice mismatch (more than 40%). Detailed pump-probe TA-IR spectroscopy studies reveal high QYs of plasmon-induced hot-electron transferred from noble metal to semiconductor in the Ag-CdS (18.1%) and Ag-CdSe (17.6%) icosapods. This work demonstrates a strategy to rationally design and controllably synthesize sophisticated hierarchical noble metal-semiconductor heterostructures. The constructed epitaxial interface provides an ideal platform for investigating the role of the metal-semiconductor interface in various physicochemical processes."},{"section":"Methods","content":"Materials Silver nitrate (AgNO 3 , 99.9999% trace metals basis), cadmium oxide (CdO, ≥99.99% trace metals basis), cadmium chloride (CdCl 2 , 99.99% trace metals basis), sulfur powder (S, ≥99.98% trace metals basis), selenium powder (Se, 99.99%), oleylamine (technical grade, 70%), trioctylphosphine (97%), trioctylphosphine oxide (technical grade, 90%), oleic acid (technical grade, 90%) were purchased from Sigma-Aldrich. N-octadecylphosphonic acid (95%), n-hexylphosphonic acid (95%) were bought from Alfa Aesar. Hexane (≥99.0%), toluene (≥99.0%), and methanol (≥99.9%) were purchased from Aladdin and used as received without any further purification. Synthesis of five-fold multi-twinned Ag icosahedral nanocrystals Typically, after 85 mg of AgNO 3 were mixed with 10 mL of oleylamine in a 50 mL three-neck flask, the mixture was degassed at 60 °C for 30 min under vigorous magnetic stirring at 750 r.p.m. It was then quickly heated up (≥15 °C min −1 ) to 220 °C under N 2 atmosphere. After the temperature was maintained at 220 °C for 1 h, the solution was cooled down to 160 °C, and a mixture of 60 μL of acetate and 0.5 mL of oleylamine was injected into the flask. After the temperature was kept at 160 °C for 10 min, the reaction was stopped by removing the heating mantle. The flask was then cooled down to room temperature naturally. The obtained brownish product was precipitated by centrifuge at 5800 × g . for 2 min. The precipitate was then washed 3 times using the mixture of hexane and acetone (V hexane /V acetone = 5/1). Finally, the obtained Ag nanocrystals were re-dispersed in 2.5 mL of TOP as seeds for the synthesis of Ag-CdS icosapods and Ag-CdSe icosapods. Synthesis of Ag-CdS and Ag-CdSe icosapods TOPS and TOPSe stock solutions were first prepared by dissolving 18 mg of sulfur powder or 10 mg of selenium powder in 0.5 mL of TOP in a glove box filled with argon, respectively. For the synthesis of Ag-CdS icosapods, 115 mg of CdO, 18 mg of CdCl 2 , 580 mg of n-octadecylphosphonic acid, 160 mg of n-hexylphosphonic acid, and 6 g of trioctylphosphine oxide were added into a 50 mL three-neck flask and degassed at 120 °C for 30 min under vigorous magnetic stirring at 750 r.p.m. After it was heated to 360 °C under N 2 atmosphere, the solution becomes transparent. Then, 1 mL of TOP was injected into the flask, and the temperature was kept at 360 °C for 15 min. A mixture of 0.5 mL of the above-mentioned Ag seeds in TOP and 0.5 mL of TOPS stock was injected into the flask at 360 °C swiftly. After the injection of Ag seeds and TOPS, the temperature was kept at 350 °C for 6 min. Then the reaction was stopped by removing the heating mantle. After the solution was cooled down to 100 °C, 10 mL of toluene was injected into the reaction flask. Then, 5 mL of methanol was added to the solution, and the product was collected by centrifuge at 5800 × g for 3 min. The obtained precipitate was washed with the mixture of toluene and methanol (V toluene /V methanol = 5/1) and then dispersed into 50 mL of toluene. For the synthesis of Ag-CdSe icosapods, all the conditions are the same with the synthesis of Ag-CdS icosapods except the mixture of 0.5 mL of the above-mentioned Ag seeds in TOP and 0.5 mL of TOPSe stock was injected into the flask at 310 °C and the temperature was kept at 300 °C for 3 min. Note that the narrow size distribution of Ag seeds (Supplementary Fig. 1 ) is critical for synthesizing such delicate Ag-CdS and Ag-CdSe icosapods. Synthesis of CdS and CdSe nanorods TOPS and TOPSe stock solutions were first prepared by dissolving 32 mg of sulfur powder or 40 mg of selenium powder in 0.5 mL of TOP in a glove box filled with argon, respectively. For the synthesis of CdS nanorods, 120 mg of CdO, 580 mg of n-octadecylphosphonic acid, 160 mg of n-hexylphosphonic acid, and 6 g of trioctylphosphine oxide were added into a 50 mL three-neck flask and degassed at 120 °C for 30 min under vigorous magnetic stirring at 750 r.p.m. After it was heated to 370 °C under N 2 atmosphere, the solution becomes transparent. Then, 1 mL of TOP was injected into the flask, and the temperature was kept at 360 °C for 15 min. 0.5 mL of TOPS stock was injected into the flask at 370 °C swiftly, and the temperature was kept at 360 °C for 3 min. Then, the reaction was stopped by removing the heating mantle. After the solution was cooled down to 100 °C, 10 mL of toluene was injected into the reaction flask. Then, 5 mL of methanol was added to the solution, and the product was collected by centrifuge at 5800 × g for 3 min. The obtained precipitate was washed with the mixture of toluene and methanol (V toluene /V methanol = 5/1) and then dispersed into 50 mL of toluene. For the synthesis of CdSe nanorods, all the conditions are the same as the synthesis of CdS nanorods except 0.5 mL of TOPSe stock was injected into the flask at 380 °C and the temperature was kept at 360 °C for 1 min. Characterization Powder XRD patterns of the dried samples were recorded on Bruker D8 diffractometer at a scanning rate of 2° min −1 using Cu Kα radiation (λ = 1.5406 Å). XPS and UPS spectra were recorded on the ESCALAB 250Xi (Thermo Fisher Scientific) system. The XPS results were calibrated with the reference C 1s peak located at 284.6 eV. The X-ray absorption fine structure (XAFS) experiments under transmission mode were performed at the 7-BM of the National Synchrotron Light Source II (NSLS-II) which is using a channel-cut monochromator. The NSLS-II is rung at 400 mA electron beam current under top-off mode. Samples for SEM and TEM characterizations were prepared by dropping nanomaterial dispersions in toluene on a silicon substrate and amorphous carbon-coated copper grid, respectively. SEM images were obtained using FESEM, JEOL JSM-7800F. TEM characterization was performed with a JEOL JEM-2100 and FEI TF20 operated at 200 kV. Dark-field STEM, Cs-HAADF STEM, and elemental mapping images were obtained using FEI Titan G2 60-300 operated at 300 kV as well as a JEM-ARM200F operated at 200 kV. The 3D reconstruction of Ag-CdS icosapods was carried out using FEI Titan G2 60-300 operated at 60 kV. Visible femtosecond transient absorption The visible femtosecond transient absorption measurements were conducted in a Helios spectrometer (Ultrafast Systems LLC) with pump and probe beams derived from a regenerative amplified Ti: Sapphire laser system (Coherent Astrella, 35 fs, 4 mJ per pulse, and 1 kHz repetition rate). The 800 nm output pulse was split into two beams with a beam splitter. One beam passed through a tunable optical parametric amplifier (OperA solo, Coherent) to generate a tunable visible pump. During the measurement, the pump beam was chopped by a synchronized chopper to 500 Hz. The other beam was attenuated and focused on a CaF 2 window to generate the white light continuum with a wavelength range from 350 nm to 800 nm, referred to as the probe beam. The probe beam was focused into a 1-mm path-length quartz cuvette (Starna) containing the sample in toluene. The transmission of the probe was collected by a fiber optics-coupled multichannel spectrometer with complementary metal-oxide-semiconductor (CMOS) sensors and detected at a frequency of 1 kHz (Ultrafast systems, Helios). The delay between the pump and probe pulses was controlled by a motorized delay stage. Samples in 1-mm cuvettes were used for all spectroscopy measurements and stirred vigorously during the measurements. The data were analyzed with surface Xplorer. Mid-IR femtosecond transient absorption Mid-IR TA experiments were conducted using a commercial Ti:Sapphire regenerative amplifier (Astrella, Coherent) at 800 nm with a repetition rate of 1 kHz and pulse duration of 35 fs. An optical amplifier (OperA solo, Coherent) pumped by the regenerative amplifier was used to provide a pump beam with tunable wavelengths. Another optical amplifier (OperA solo, Coherent) pumped by the regenerative amplifier was used to generate a tunable IR probe via difference frequency generation (DFG) by signal and idler beams. The pump beam was chopped by a synchronized chopper to 500 Hz and focused on a 1.5-mm path-length Harrick cell containing the sample in toluene. The IR probe, which was also focused on a 1.5-mm path length Harrick cell containing the sample in toluene, overlapped with the temporally delayed pump beam controlled by a movable delay stage. The probe intensity with and without the pump was analyzed by a Teledyne/phasetech nitrogen-cooled 128 × 128 MCT detector. All measurements were carried out under ambient conditions. QY measurement of the plasmon-induced hot-electron transfer in Ag-CdS heterostructures The QY of the plasmon-induced hot-electrons transferred from the Ag core to the CdS under 550 nm excitation can be quantified by the mid-IR intraband absorption at around 3340 cm −1 of CB electrons in CdS. The CB electron in CdS produces a broad IR absorption which can be attributed to the transition from 1Σ to a higher level (such as 1П), as shown in Fig. 3a . To determine the plasmon-induced hot-electron transfer QY, the amplitude of IR signal per CdS CB electron was quantified, denoted as S 0 . Hence, S 0 can be calculated by Eq. 1 : $${{{{{{\\rm{S}}}}}}}_{0}=\\frac{\\Delta {{{{{\\rm{A}}}}}}\\left({{{{{\\rm{CdS}}}}}},400\\,{{{{{\\rm{nm}}}}}}\\right)}{{{{{{\\rm{N}}}}}}({{{{{\\rm{CdSCB}}}}}}{{{{{\\rm{e}}}}}}{{{{{\\rm{lectrons}}}}}})}$$ (1) where ∆A is the maximum signal amplitude obtained from the kinetics; N(CdS CB electrons) is the number of electrons in CdS CB band, which can be determined by the number of absorbed photons by CdS (every photon creates one electron in CdS CB band under 400 nm excitation). Therefore, N(CdS CB electrons) equals the pump photon flux in the pump/probe overlap region times the sample absorption \\(\\left(\\frac{{{{{{\\rm{power}}}}}}/{hv}}{{{{{{\\rm{beam}}}}}}\\; {{{{{{\\rm{size}}}}}}}}\\times (1-{10}^{-{{{{{\\rm{OD}}}}}}})\\right)\\) , where hv is photon energy. The QY is calculated by the number of hot electrons generated in Ag-CdS per absorbed photons under 550 nm excitation, as shown in Eq. 2 . $${{{{{\\rm{QY}}}}}} \t=\\frac{{{{{{\\rm{N}}}}}}({{{{{\\rm{hot}}}}}}\\,{{{{{\\rm{electrons}}}}}})}{{{{{{\\rm{N}}}}}}({{{{{\\rm{absorbed}}}}}}\\,{{{{{\\rm{photons}}}}}})} \\\\ \t=\\frac{\\frac{\\Delta {{{{{\\rm{A}}}}}}\\left({{{{{\\rm{Ag}}}}}}-{{{{{\\rm{CdS}}}}}},550\\,{{{{{\\rm{nm}}}}}}\\right)}{{{{{{{\\rm{S}}}}}}}_{0}}}{\\frac{{{{{{\\rm{power}}}}}}\\left(550\\,{{{{{\\rm{nm}}}}}}\\right)/{hv}(550\\,{{{{{\\rm{nm}}}}}})}{{{{{{\\rm{beam}}}}}}\\,{{{{{\\rm{size}}}}}}\\left(550\\,{{{{{\\rm{nm}}}}}}\\right)}\\times \\left(1-{10}^{-{{{{{\\rm{OD}}}}}}\\left({{{{{\\rm{Ag}}}}}}-{{{{{\\rm{CdS}}}}}},550\\,{{{{{\\rm{nm}}}}}}\\right)}\\right)} \\\\ \t=\\frac{{{{{{\\rm{beam}}}}}}\\,{{{{{\\rm{size}}}}}}(550\\,{{{{{\\rm{nm}}}}}})}{{{{{{\\rm{beam}}}}}}\\,{{{{{\\rm{size}}}}}}\\,(400\\,{{{{{\\rm{nm}}}}}})}\\times \\left[\\frac{\\frac{\\Delta {{{{{\\rm{A}}}}}}\\left({{{{{\\rm{Ag}}}}}}-{{{{{\\rm{CdS}}}}}},550\\,{{{{{\\rm{nm}}}}}}\\right)}{{{{{{\\rm{power}}}}}}\\left(550\\,{{{{{\\rm{nm}}}}}}\\right)/{hv}(550\\,{{{{{\\rm{nm}}}}}})}}{(1-{10}^{-{{{{{\\rm{OD}}}}}}({{{{{\\rm{Ag}}}}}}-{{{{{\\rm{CdS}}}}}},550\\,{{{{{\\rm{nm}}}}}})})}\\bigg/\\frac{\\frac{\\Delta\\, {{{{{\\rm{A}}}}}}\\left({{{{{\\rm{CdS}}}}}},400\\,{{{{{\\rm{nm}}}}}}\\right)}{{{{{{\\rm{power}}}}}}(400\\,{{{{{\\rm{nm}}}}}})/{hv}(400\\,{{{{{\\rm{nm}}}}}})}}{(1-{1\\,0}^{-{{{{{\\rm{OD}}}}}}({{{{{\\rm{CdS}}}}}},400\\,{{{{{\\rm{nm}}}}}\\,})})}\\right]$$ (2) In Eq. 2 , beam size (550 nm) and beam size (400 nm) represent the overlap areas of pump and probe beams under 550 nm, and 400 nm pump laser excitation, respectively, whose ratio can be determined in a calibration sample (Cd 3 P 2 quantum dots in this work) under the same conditions. The calibration sample is excited at both wavelengths and is expected to have the same QY for generating CB electrons. Then \\(\\frac{{{{{{\\rm{beam}}}}}}\\,{{{{{\\rm{size}}}}}}(550\\,{{{{{\\rm{nm}}}}}})}{{{{{{\\rm{beam}}}}}}\\,{{{{{\\rm{size}}}}}}(400\\,{{{{{\\rm{nm}}}}}})}\\) can be described as follows: $$\\frac{{{{{{\\rm{beam}}}}}}\\,{{{{{\\rm{size}}}}}}(550\\,{{{{{\\rm{nm}}}}}})}{{{{{{\\rm{beam}}}}}}\\,{{{{{\\rm{size}}}}}}(400\\,{{{{{\\rm{nm}}}}}})}=\\frac{\\frac{\\Delta {{{{{\\rm{A}}}}}}\\left({{{{{\\rm{Cd}}}}}}3{{{{{\\rm{P}}}}}}2,400\\,{{{{{\\rm{nm}}}}}}\\right)}{{{{{{\\rm{power}}}}}}\\left(400\\,{{{{{\\rm{nm}}}}}}\\right)/{hv}(400\\,{{{{{\\rm{nm}}}}}})}}{(1-{10}^{-{{{{{\\rm{OD}}}}}}({{{{{\\rm{Cd}}}}}}3{{{{{\\rm{P}}}}}}2,400\\,{{{{{\\rm{nm}}}}}})})}\\bigg/\\frac{\\frac{\\Delta {{{{{\\rm{A}}}}}}\\left({{{{{\\rm{Cd}}}}}}3{{{{{\\rm{P}}}}}}2,550\\,{{{{{\\rm{nm}}}}}}\\right)}{{{{{{\\rm{power}}}}}}(550\\,{{{{{\\rm{nm}}}}}})/{hv}(550\\,{{{{{\\rm{nm}}}}}})}}{(1-{10}^{-{{{{{\\rm{OD}}}}}}({{{{{\\rm{Cd}}}}}}3{{{{{\\rm{P}}}}}}2,550\\,{{{{{\\rm{nm}}}}}})})}$$ (3) Thus, the QY of plasmon-induced hot-electron in Ag-CdS icosapods is given by Eq. 4 . $${{{{{\\rm{QY}}}}}}=\\frac{\\frac{\\Delta {{{{{\\rm{S}}}}}}({{{{{\\rm{Ag}}}}}}-{{{{{\\rm{CdS}}}}}},550\\,{{{{{\\rm{nm}}}}}})}{1-{10}^{-{{{{{\\rm{OD}}}}}}({{{{{\\rm{Ag}}}}}}-{{{{{\\rm{CdS}}}}}},550\\,{{{{{\\rm{nm}}}}}})}}}{\\frac{\\Delta {{{{{\\rm{S}}}}}}({{{{{\\rm{CdS}}}}}},400\\,{{{{{\\rm{nm}}}}}})}{1-{10}^{-{{{{{\\rm{OD}}}}}}({{{{{\\rm{CdS}}}}}},400\\,{{{{{\\rm{nm}}}}}})}}}\\times \\frac{\\frac{\\Delta {{{{{\\rm{S}}}}}}({{{{{\\rm{Cd}}}}}}3{{{{{\\rm{P}}}}}}2,400\\,{{{{{\\rm{nm}}}}}})}{1-{10}^{-{{{{{\\rm{OD}}}}}}({{{{{\\rm{Cd}}}}}}3{{{{{\\rm{P}}}}}}2,400\\,{{{{{\\rm{nm}}}}}})}}}{\\frac{\\Delta {{{{{\\rm{S}}}}}}({{{{{\\rm{Cd}}}}}}3{{{{{\\rm{P}}}}}}2,550\\,{{{{{\\rm{nm}}}}}})}{1-{10}^{-{{{{{\\rm{OD}}}}}}({{{{{\\rm{Cd}}}}}}3{{{{{\\rm{P}}}}}}2,550\\,{{{{{\\rm{nm}}}}}})}}}$$ (4) Here, we define \\(\\Delta {{{{{\\rm{S}}}}}}=\\frac{\\Delta {{{{{\\rm{A}}}}}}}{{{{{{\\rm{power}}}}}}}\\) , a quantity to be obtained from the transient IR measurement. To improve the measurement accuracy and to check the linearity of the signal, ∆A as a function of excitation power was measured for all samples. As shown in Supplementary Fig. 18 , the TA signal amplitude increases linearly with the excitation power. ∆S can be obtained from the slope of the fitted linear line. For all samples, the OD is controlled at around 0.2–0.8. The values of epitaxial Ag-CdS icosapods are listed in Supplementary Table 2 . It should be noted that because the nanorods are directly grown on the Ag nanoparticles, there may be slight differences in the diameter of the CdS nanorods control sample compared to the diameter of the nanorods in the heterostructure. This could affect the QY as the optical properties of the nanorods are dependent upon their size (absorption cross section), therefore, the QYs for each sample were calculated to the best of our ability using the independently synthesized CdS nanorods controls. QY measurement of the plasmon-induced hot-electron transfer in Ag-CdSe icosapods The QY of the electrons transferred from the Ag core to the CdSe under 700 nm excitation can be quantified by the mid-IR intraband absorption at around 3340 cm −1 of CB electrons in CdSe. Similar to Ag-CdS icosapods, the QY is calculated by the number of hot electrons generated in Ag-CdSe icosapods per absorbed photons under 700 nm excitation. CdSe nanorods are excited by a 400 nm laser. Cd 3 P 2 quantum dots are excited by 400 nm laser and 700 nm laser for calibration. The QY of plasmon-induced hot-electron transfer in Ag-CdSe icosapods was shown in Eq. 5 . $${{{{{\\rm{QY}}}}}}=\\frac{\\frac{\\Delta {{{{{\\rm{S}}}}}}({{{{{\\rm{Ag}}}}}}-{{{{{\\rm{CdSe}}}}}},700\\,{{{{{\\rm{nm}}}}}})}{1-{10}^{-{{{{{\\rm{OD}}}}}}({{{{{\\rm{Ag}}}}}}-{{{{{\\rm{CdSe}}}}}},700\\,{{{{{\\rm{nm}}}}}})}}}{\\frac{\\Delta {{{{{\\rm{S}}}}}}({{{{{\\rm{CdSe}}}}}},400\\,{{{{{\\rm{nm}}}}}})}{1-{10}^{-{{{{{\\rm{OD}}}}}}({{{{{\\rm{CdSe}}}}}},400\\,{{{{{\\rm{nm}}}}}})}}}\\times \\frac{\\frac{\\Delta {{{{{\\rm{S}}}}}}({{{{{\\rm{Cd}}}}}}3{{{{{\\rm{P}}}}}}2,400\\,{{{{{\\rm{nm}}}}}})}{1-{10}^{-{{{{{\\rm{OD}}}}}}({{{{{\\rm{Cd}}}}}}3{{{{{\\rm{P}}}}}}2,400\\,{{{{{\\rm{nm}}}}}})}}}{\\frac{\\Delta {{{{{\\rm{S}}}}}}({{{{{\\rm{Cd}}}}}}3{{{{{\\rm{P}}}}}}2,700\\,{{{{{\\rm{nm}}}}}})}{1-{10}^{-{{{{{\\rm{OD}}}}}}({{{{{\\rm{Cd}}}}}}3{{{{{\\rm{P}}}}}}2,700\\,{{{{{\\rm{nm}}}}}})}}}$$ (5) For all samples, the OD is controlled at around 0.4–0.7. The values used in Eq. 4 is listed in Supplementary Table 4 . The multiexponential fit of the decay kinetics of Ag-CdS and Ag-CdSe icospads The normalized TA IR kinetics shown in Fig. 3e and Supplementary Fig. 23b are fitted by a single exponential rise and multi-exponential decay function according to Eq. 6 : $${{{{{\\rm{S}}}}}}\\left({{{{{\\rm{t}}}}}}\\right)={{{{{\\rm{IRF}}}}}}\\otimes \\left[-{{\\exp }}\\left(-\\frac{{{{{{\\rm{t}}}}}}}{{T}_{f}}\\right)+\\mathop{\\sum }\\limits_{1}^{3}{A}_{i}{{\\exp }}\\left(-\\frac{{{{{{\\rm{t}}}}}}}{{T}_{i}}\\right)\\right]$$ (6) The instrument response function (IRF) of the measurement was determined in a 1 mm silicon wafer and can be well modeled by a Gaussian function with an FWHM of ~177 ± 25 fs. T f is the time constant of the signal rise. A i and T i ( i = 1,2,3) are amplitude and time constants of the multiexponential function decay function, respectively, \\({\\sum }_{1}^{3}{A}_{i}=1\\) . The fitting parameters are listed in Supplementary Table 3 . 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\": \"Heterostructures have attracted tremendous attention due to their physicochemical properties arising from their morphologies [1] , [2] , [3] , interfaces [4] , [5] , [6] , and spatial arrangements of different components [7] , [8] . The precise control over their nanoscale structures is essential to understand the structure-property correlation and enhance their performance in various applications, such as electronics [9] , [10] , catalysis [11] , [12] , solar energy conversion [13] , [14] , etc. To date, the seeded/templated epitaxial growth is the most commonly used strategy to precisely tailor the hierarchical heterostructures [15] , [16] , [17] , [18] , [19] . Epitaxial growth of a secondary material on a specific facet of the seed/template allows it to follow the crystallographic orientation of the seed/template [17] , [20] . Combined with delicate control over the exposed facet and crystal phase of seeds/templates, this strategy has been successfully used to prepare various hierarchical heterostructures with well-defined spatial structures [21] , [22] , interfaces [23] , [24] , crystal phases [25] , [26] , [27] , and programmable components [28] , [29] . However, the aforementioned seeded/templated epitaxial growth is mainly used to prepare heterostructures composed of components with similar lattice structures and/or chemical bonding, e.g., metal-metal [20] , [26] , [27] , metal oxide-metal oxide [30] , [31] , metal chalcogenide-metal chalcogenide [1] , [16] , [22] , [23] , [28] , perovskite-perovskite [29] , and metal-organic framework (MOF)-MOF [32] , [33] heterostructures due to the common requirement of similar structure (normally with lattice mismatch smaller than 5%) for the epitaxial growth [17] , [18] , [34] . Therefore, it remains a great challenge for the epitaxial growth of heterostructures made of distinctly different materials, especially for the noble metal-semiconductor heterostructures, in which two materials could have a large lattice mismatch (normally larger than 20%) and different chemical bonding [18] , [24] , [35] , [36] , [37] , [38] . Commonly, the preparation of noble metal-semiconductor heterostructures relies on conventional non-epitaxial strategies including chemical deposition [39] , [40] , cation-exchange-facilitated non-epitaxial growth [35] , [41] , phase transfer [42] , photochemical deposition [43] , chemical extraction [44] , [45] , sol-gel method [46] , etc. Although the obtained non-epitaxial noble metal-semiconductor heterostructures have shown great potential in diverse fields, such as catalysis [6] , [14] , [40] , [43] , [46] , photovoltaic devices [4] , [14] , [47] and sensors [36] , [44] , [48] , their wide application has been hindered due to difficulties in the precisely defined control over their structures and interfaces. Therefore, a reliable epitaxial growth method is urgently desired to rationally construct hierarchical noble metal-semiconductor heterostructures with precisely controlled architectures and well-defined interfaces. In this work, we report a controlled synthesis of highly symmetrical branched noble metal-semiconductor heterostructures via a noble-metal-seeded epitaxial growth. The 1D II-VI semiconductor nanorods can be uniformly grown on the exposed (111) facets of zero-dimensional platonic Ag nanocrystals. The obtained heterostructure possesses the same crystallographic symmetry as the noble metal seed, i.e., twenty CdS (or CdSe) nanorods radially grown on twenty exposed (111) facets of the five-fold multi-twinned Ag icosahedron, respectively, denoted as Ag-CdS (or Ag-CdSe) icosapods. Importantly, the epitaxial growth of CdS (or CdSe) nanorods on Ag icosahedral nanocrystals has been achieved albeit a large lattice mismatch between Ag and CdS (or CdSe) (more than 40%). Impressively, the pump-probe transient absorption (TA) spectroscopy reveals the ultrafast plasmon hot-electron transfer process in the epitaxial heterostructures with high QY of the plasmon-induced hot-electron transferred from noble metal to semiconductor, i.e., 18.1% and 17.6% for the Ag-CdS and Ag-CdSe icosapods, respectively.\",\n \"references\": {\n \"indices\": [\n 1,\n 2,\n 3,\n 4,\n 5,\n 6,\n 7,\n 8,\n 9,\n 10,\n 11,\n 12,\n 13,\n 14,\n 15,\n 16,\n 17,\n 18,\n 19,\n 20,\n 21,\n 22,\n 23,\n 24,\n 25,\n 26,\n 27,\n 28,\n 29,\n 30,\n 31,\n 32,\n 33,\n 34,\n 35,\n 36,\n 37,\n 38,\n 39,\n 40,\n 41,\n 42,\n 43,\n 44,\n 45,\n 46,\n 47,\n 48\n ],\n \"infos\": [\n {\n \"idx\": \"1\",\n \"title\": \"Borys, N. J., Walter, M. J., Huang, J., Talapin, D. V. & Lupton, J. M. The role of particle morphology in interfacial energy transfer in CdSe/CdS heterostructure nanocrystals.Science330, 1371–1374 (2010).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1198070\"\n },\n {\n \"idx\": \"2\",\n \"title\": \"Gu, J., Zhang, Y. W. & Tao, F. Shape control of bimetallic nanocatalysts through well-designed colloidal chemistry approaches.Chem. Soc. Rev.41, 8050–8065 (2012).\",\n \"link\": \"https://doi.org/10.1039%2Fc2cs35184f\"\n },\n {\n \"idx\": \"3\",\n \"title\": \"Jiang, L. Z. et al. Facet engineering accelerates spillover hydrogenation on highly diluted metal nanocatalysts.Nat. Nanotechnol.15, 848–853 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41565-020-0746-x\"\n },\n {\n \"idx\": \"4\",\n \"title\": \"Clavero, C. Plasmon-induced hot-electron generation at nanoparticle/metal-oxide interfaces for photovoltaic and photocatalytic devices.Nat. Photon.8, 95–103 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fnphoton.2013.238\"\n },\n {\n \"idx\": \"5\",\n \"title\": \"Liu, M. X. et al. Lattice anchoring stabilizes solution-processed semiconductors.Nature570, 96–101 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-019-1239-7\"\n },\n {\n \"idx\": \"6\",\n \"title\": \"Yuan, W. T. et al. In situ manipulation of the active Au-TiO2interface with atomic precision during CO oxidation.Science371, 517–521 (2021).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.abe3558\"\n },\n {\n \"idx\": \"7\",\n \"title\": \"Zhao, M. T. et al. Metal-organic frameworks as selectivity regulators for hydrogenation reactions.Nature539, 76–80 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fnature19763\"\n },\n {\n \"idx\": \"8\",\n \"title\": \"Cheng, K. et al. Maximizing noble metal utilization in solid catalysts by control of nanoparticle location.Science377, 204–208 (2022).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.abn8289\"\n },\n {\n \"idx\": \"9\",\n \"title\": \"Kagan, C. R., Lifshitz, E., Sargent, E. H. & Talapin, D. V. Building devices from colloidal quantum dots.Science353, aac5523 (2016).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.aac5523\"\n },\n {\n \"idx\": \"10\",\n \"title\": \"Liu, Y. et al. Van der Waals heterostructures and devices.Nat. Rev. Mater.1, 16042 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fnatrevmats.2016.42\"\n },\n {\n \"idx\": \"11\",\n \"title\": \"Losch, P. et al. Colloidal nanocrystals for heterogeneous catalysis.Nano Today24, 15–47 (2019).\",\n \"link\": \"https://doi.org/10.1016%2Fj.nantod.2018.12.002\"\n },\n {\n \"idx\": \"12\",\n \"title\": \"Vogt, C. & Weckhuysen, B. M. The concept of active site in heterogeneous catalysis.Nat. Rev. Chem.6, 89–111 (2022).\",\n \"link\": \"https://doi.org/10.1038%2Fs41570-021-00340-y\"\n },\n {\n \"idx\": \"13\",\n \"title\": \"Lu, Q. P., Yu, Y. F., Ma, Q. L., Chen, B. & Zhang, H. 2D transition-metal-dichalcogenide-nanosheet-based composites for photocatalytic and electrocatalytic hydrogen evolution reactions.Adv. Mater.28, 1917–1933 (2016).\",\n \"link\": \"https://doi.org/10.1002%2Fadma.201503270\"\n },\n {\n \"idx\": \"14\",\n \"title\": \"Linic, S., Chavez, S. & Elias, R. Flow and extraction of energy and charge carriers in hybrid plasmonic nanostructures.Nat. Mater.20, 916–924 (2021).\",\n \"link\": \"https://doi.org/10.1038%2Fs41563-020-00858-4\"\n },\n {\n \"idx\": \"15\",\n \"title\": \"Lauhon, L. J., Gudiksen, M. S., Wang, C. L. & Lieber, C. M. Epitaxial core-shell and core-multishell nanowire heterostructures.Nature420, 57–61 (2002).\",\n \"link\": \"https://doi.org/10.1038%2Fnature01141\"\n },\n {\n \"idx\": \"16\",\n \"title\": \"Talapin, D. V. et al. Seeded growth of highly luminescent CdSe/CdS nanoheterostructures with rod and tetrapod morphologies.Nano Lett.7, 2951–2959 (2007).\",\n \"link\": \"https://doi.org/10.1021%2Fnl072003g\"\n },\n {\n \"idx\": \"17\",\n \"title\": \"Tan, C. L., Chen, J. Z., Wu, X.-J. & Zhang, H. Epitaxial growth of hybrid nanostructures.Nat. Rev. Mater.3, 17089 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fnatrevmats.2017.89\"\n },\n {\n \"idx\": \"18\",\n \"title\": \"Kum, H. et al. Epitaxial growth and layer-transfer techniques for heterogeneous integration of materials for electronic and photonic devices.Nat. Electro2, 439–450 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41928-019-0314-2\"\n },\n {\n \"idx\": \"19\",\n \"title\": \"Kelso, M. V., Mahenderkar, N. K., Chen, Q. Z., Tubbesing, J. Z. & Switzer, J. A. Spin coating epitaxial films.Science364, 166–169 (2019).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.aaw6184\"\n },\n {\n \"idx\": \"20\",\n \"title\": \"Habas, S. E., Lee, H., Radmilovic, V., Somorjai, G. A. & Yang, P. Shaping binary metal nanocrystals through epitaxial seeded growth.Nat. Mater.6, 692–697 (2007).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat1957\"\n },\n {\n \"idx\": \"21\",\n \"title\": \"Miszta, K. et al. Hierarchical self-assembly of suspended branched colloidal nanocrystals into superlattice structures.Nat. Mater.10, 872–876 (2011).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat3121\"\n },\n {\n \"idx\": \"22\",\n \"title\": \"Wu, X.-J. et al. Controlled growth of high-density CdS and CdSe nanorod arrays on selective facets of two-dimensional semiconductor nanoplates.Nat. Chem.8, 470–475 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fnchem.2473\"\n },\n {\n \"idx\": \"23\",\n \"title\": \"Li, M. Y. et al. Epitaxial growth of a monolayer WSe2-MoS2lateral p-n junction with an atomically sharp interface.Science349, 524–528 (2015).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.aab4097\"\n },\n {\n \"idx\": \"24\",\n \"title\": \"Kanne, T. et al. Epitaxial Pb on InAs nanowires for quantum devices.Nat. Nanotechnol.16, 776–781 (2021).\",\n \"link\": \"https://doi.org/10.1038%2Fs41565-021-00900-9\"\n },\n {\n \"idx\": \"25\",\n \"title\": \"Chen, Y. et al. Phase engineering of nanomaterials.Nat. Rev. Chem.4, 243–256 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41570-020-0173-4\"\n },\n {\n \"idx\": \"26\",\n \"title\": \"Lu, Q. P. et al. Crystal phase-based epitaxial growth of hybrid noble metal nanostructures on 4H/fcc Au nanowires.Nat. Chem.10, 456–461 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41557-018-0012-0\"\n },\n {\n \"idx\": \"27\",\n \"title\": \"Liu, J. W. et al. Selective epitaxial growth of Rh nanorods on 2H/fccheterophase Au nanosheets to form 1D/2D Rh-Au heterostructures for highly efficient hydrogen evolution.J. Am. Chem. Soc.143, 4387–4396 (2021).\",\n \"link\": \"https://doi.org/10.1021%2Fjacs.1c00612\"\n },\n {\n \"idx\": \"28\",\n \"title\": \"Zhang, Z. W. et al. Robust epitaxial growth of two-dimensional heterostructures, multiheterostructures, and superlattices.Science357, 788–792 (2017).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.aan6814\"\n },\n {\n \"idx\": \"29\",\n \"title\": \"Shi, E. Z. et al. Two-dimensional halide perovskite lateral epitaxial heterostructures.Nature580, 614–620 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-020-2219-7\"\n },\n {\n \"idx\": \"30\",\n \"title\": \"Oh, M. H. et al. Design and synthesis of multigrain nanocrystals via geometric misfit strain.Nature577, 359–363 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-019-1899-3\"\n },\n {\n \"idx\": \"31\",\n \"title\": \"Forticaux, A., Hacialioglu, S., DeGrave, J. P., Dziedzic, R. & Jin, S. Three-dimensional mesoscale heterostructures of ZnO nanowire arrays epitaxially grown on CuGaO2nanoplates as individual diodes.ACS Nano7, 8224–8232 (2013).\",\n \"link\": \"https://doi.org/10.1021%2Fnn4037078\"\n },\n {\n \"idx\": \"32\",\n \"title\": \"Yao, Y. et al. Heteroepitaxial growth of multiblock Ln-MOF microrods for photonic barcodes.Angew. Chem. Int. Ed.58, 13803–13807 (2019).\",\n \"link\": \"https://doi.org/10.1002%2Fanie.201907433\"\n },\n {\n \"idx\": \"33\",\n \"title\": \"Zhao, M. T. et al. Selective epitaxial growth of oriented hierarchical metal-organic framework heterostructures.J. Am. Chem. Soc.142, 8953–8961 (2020).\",\n \"link\": \"https://doi.org/10.1021%2Fjacs.0c02489\"\n },\n {\n \"idx\": \"34\",\n \"title\": \"Grundmann, M. Formation of epitaxial domains: unified theory and survey of experimental results.Phys. Status Solidi B248, 805–824 (2011).\",\n \"link\": \"https://doi.org/10.1002%2Fpssb.201046530\"\n },\n {\n \"idx\": \"35\",\n \"title\": \"Zhang, J. T., Tang, Y., Lee, K. & Min, O. Y. Nonepitaxial growth of hybrid core-shell nanostructures with large lattice mismatches.Science327, 1634–1638 (2010).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1184769\"\n },\n {\n \"idx\": \"36\",\n \"title\": \"Liu, J. & Zhang, J. T. Nanointerface chemistry: lattice-mismatch-directed synthesis and application of hybrid nanocrystals.Chem. Rev.120, 2123–2170 (2020).\",\n \"link\": \"https://doi.org/10.1021%2Facs.chemrev.9b00443\"\n },\n {\n \"idx\": \"37\",\n \"title\": \"Palmstrom, C. J. Epitaxy of dissimilar materials.Annu. Rev. Mater. Sci.25, 389–415 (1995).\",\n \"link\": \"https://doi.org/10.1146%2Fannurev.ms.25.080195.002133\"\n },\n {\n \"idx\": \"38\",\n \"title\": \"Pendharkar, M. et al. Parity-preserving and magnetic field-resilient superconductivity in InSb nanowires with Sn shells.Science372, 508–511 (2021).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.aba5211\"\n },\n {\n \"idx\": \"39\",\n \"title\": \"Mokari, T., Rothenberg, E., Popov, I., Costi, R. & Banin, U. Selective growth of metal tips onto semiconductor quantum rods and tetrapods.Science304, 1787–1790 (2004).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1097830\"\n },\n {\n \"idx\": \"40\",\n \"title\": \"Wolff, C. M. et al. All-in-one visible-light-driven water splitting by combining nanoparticulate and molecular co-catalysts on CdS nanorods.Nat. Energy3, 862–869 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41560-018-0229-6\"\n },\n {\n \"idx\": \"41\",\n \"title\": \"Fan, Z. X. et al. Synthesis of 4H/fcc-Au@metal sulfide core shell nanoribbons.J. Am. Chem. Soc.137, 10910–10913 (2015).\",\n \"link\": \"https://doi.org/10.1021%2Fjacs.5b06405\"\n },\n {\n \"idx\": \"42\",\n \"title\": \"Yang, J. & Ying, J. Y. A general phase-transfer protocol for metal ions and its application in nanocrystal synthesis.Nat. Mater.8, 683–689 (2009).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat2490\"\n },\n {\n \"idx\": \"43\",\n \"title\": \"Ben-Shahar, Y. et al. Optimal metal domain size for photocatalysis with hybrid semiconductor-metal nanorods.Nat. Commun.7, 10413 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms10413\"\n },\n {\n \"idx\": \"44\",\n \"title\": \"Chen, J. Z. et al. Ag@MoS2core-shell heterostructure as SERS platform to reveal the hydrogen evolution active sites of single-layer MoS2.J. Am. Chem. Soc.142, 7161–7167 (2020).\",\n \"link\": \"https://doi.org/10.1021%2Fjacs.0c01649\"\n },\n {\n \"idx\": \"45\",\n \"title\": \"Han, S. K., Gu, C., Gong, M. & Yu, S. H. A Trialkylphosphine-driven chemical transformation route to Ag- and Bi-based chalcogenides.J. Am. Chem. Soc.137, 5390–5396 (2015).\",\n \"link\": \"https://doi.org/10.1021%2Fjacs.5b00041\"\n },\n {\n \"idx\": \"46\",\n \"title\": \"Atta, S., Pennington, A. M., Celik, F. E. & Fabris, L. TiO2on gold nanostars enhances photocatalytic water reduction in the near-infrared regime.Chem4, 2140–2153 (2018).\",\n \"link\": \"https://doi.org/10.1016%2Fj.chempr.2018.06.004\"\n },\n {\n \"idx\": \"47\",\n \"title\": \"Jang, Y. H. et al. Plasmonic solar cells: from rational design to mechanism overview.Chem. Rev.116, 14982–15034 (2016).\",\n \"link\": \"https://doi.org/10.1021%2Facs.chemrev.6b00302\"\n },\n {\n \"idx\": \"48\",\n \"title\": \"Knight, M. W., Sobhani, H., Nordlander, P. & Halas, N. J. Photodetection with active optical antennas.Science332, 702–704 (2011).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1203056\"\n }\n ]\n }\n },\n {\n \"section\": \"Results\",\n \"content\": \"Material synthesis and characterizations Figure 1a schematically illustrates the synthesis of Ag-CdS icosapods. Briefly, five-fold multi-twinned Ag icosahedral nanocrystals with a size of 13.3 ± 0.6 nm were first synthesized (Supplementary Fig. 1 ), which were then used as seeds for the epitaxial growth of CdS nanorods via a hot injection method (see the Methods section). Scanning electron microscope (SEM) (Fig. 1b ), transmission electron microscopy (TEM) (Fig. 1c ) and high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM, Fig. 1d ) images show the high purity and uniformity of the synthesized Ag-CdS icosapods with a size of 108.2 ± 5.5 nm, and the length and diameter of CdS nanorods are around 50 and 7 nm, respectively (Supplementary Fig. 2 ). Fig. 1: Synthesis and characterization of Ag-CdS icosapods. a Schematic illustration of the synthesis of Ag-CdS icosapods by using the pre-synthesized Ag nanocrystals, i.e., Ag icosahedral nanocrystals, as seeds for the epitaxial growth of CdS nanorods. C 2 , C 3 , and C 5 indicate the observed rotational axes of the schematically drawn Ag icosahedral nanocrystals (left) and Ag-CdS icosapods (right). b – d SEM ( b ), TEM ( c ), and HAADF-STEM ( d ) images of Ag-CdS icosapods. e – g HAADF-STEM images of a representative Ag-CdS icosapod observed along its C 2 ( e ), C 3 ( f ), and C 5 ( g ) rotational axes, respectively. h HAADF-STEM and the corresponding elemental mapping images of a representative Ag-CdS icosapod. Full size image To reveal the detailed three-dimensional (3D) architecture of the obtained Ag-CdS icosapods, a series of HAADF-STEM images of a representative Ag-CdS icosapod tilted with different angles were collected. The 3D reconstruction shows that twenty CdS nanorods radially extend from twenty exposed facets of the Ag icosahedron seed (Supplementary Movie 1 ), confirming that the Ag-CdS icosapod has the same rotational symmetry as the Ag icosahedral nanocrystal. As shown in Supplementary Fig. 3a–f , the Ag icosahedral nanocrystal exhibits twofold (C 2 ), threefold (C 3 ), and five-fold (C 5 ) symmetrical Moiré patterns, which are consistent with the previously reported five-fold multi-twinned icosahedral nanocrystals [49] . After the growth of CdS nanorods, the obtained Ag-CdS icosapods with C 2 , C 3 , C 5 symmetrical patterns (Supplementary Fig. 3g–i ) have also been observed under HAADF-STEM (Fig. 1e–g ), which are consistent with the HRTEM images (Supplementary Fig. 3j–l ). In particular, when projected along its C 3 rotational axis (Fig. 1f ), the Ag-CdS icosapod presents a sophisticated star-like pattern with the observed eighteen branches radially extended from the bright core, while the other two branches are unobserved since they are perpendicular to the projected plane. Moreover, a five-fold pattern of the Ag-CdS icosapod with the C 5 symmetry exhibits different contrast along the ten extending directions of branches, due to the overlapped images of two nanorods in the same direction when viewed along its C 5 rotational axis (Fig. 1g ). The HAADF-STEM and the corresponding energy-dispersive X-ray spectroscopy (EDS) elemental mapping images (Fig. 1h ) of a representative Ag-CdS icosapod reveal that the Ag element is located in the core, while the Cd and S elements are homogeneously distributed over the twenty nanorods. X-ray photoelectron spectroscopy (XPS) was used to determine the valence state of components in Ag-CdS icosapods (Supplementary Fig. 4 ). In particular, the peaks located at 373.7 and 367.7 eV can be ascribed to Ag 3d 3/2 and Ag 3d 5/2 of zerovalent Ag, respectively (Supplementary Fig. 4b ). Combined XPS results with the Ag K-edge XANES spectra (Supplementary Fig. 5 ), it can be concluded that the Ag nanocrystals have not been sulfurized during the synthesis of Ag-CdS icosapods, which might arise from the trioctylphosphine (TOP) used in the synthesis [44] , [45] , [50] . Furthermore, the X-ray diffraction (XRD) pattern of the synthesized Ag-CdS icosapods can be well indexed to the wurtzite CdS and fcc -Ag (Supplementary Fig. 6 ). The aforementioned results clearly demonstrate the successful growth of CdS nanorods on the Ag icosahedral nanocrystals, and the obtained Ag-CdS icosapods show the uniform size as well as well-defined morphology with sophisticated symmetrical branched architecture. Ultraviolet photoelectron spectroscopy (UPS) was used to reveal the interfacial electronic structure of Ag-CdS icosapods. As shown in Supplementary Fig. 7a , the cut-off energies (E cut-off ) of Ag nanocrystals and Ag-CdS icosapods are determined to be 16.95 eV and 17.20 eV, respectively. The Fermi levels of Ag nanocrystals and Ag-CdS icosapods are estimated to be −4.27 eV and −4.02 eV relative to the vacuum level, respectively. Compared with the Ag nanocrystals, the shift (≈0.25 eV) of the Fermi level of Ag-CdS icosapods demonstrates a Schottky barrier was formed between Ag and n-type semiconductor CdS in the Ag-CdS icosapods [51] . Since the Fermi level, conduction band (CB), and valence band (VB) of CdS are measured to be −3.92 eV, −3.61 eV, and −6.22 eV, respectively (Supplementary Fig. 7b, c ), the interfacial electronic structure of Ag-CdS icosapods can be schematically shown in Supplementary Fig. 7d . Importantly, the synthesis of noble metal-semiconductor heterostructures with tunable components can be achieved. By exchanging the trioctylphosphine sulfide (TOPS) with trioctylphosphine selenium (TOPSe) as the chalcogen precursor in our experiment (see the “Methods” section), branched Ag-CdSe icosapods have been obtained. The TEM and HAADF-STEM images of Ag-CdSe icosapods (Supplementary Fig. 8 ) and the EDS elemental mapping result (Supplementary Fig. 9 ) of a representative Ag-CdSe icosapod demonstrate the successful growth of CdSe nanorods on the icosahedral Ag seeds. The XPS spectra of Ag-CdSe icosapods confirm the Ag core in Ag-CdSe icosapods maintained zerovalent (Supplementary Fig. 10 ), and the XRD spectrum matches well with the wurtzite CdSe and fcc -Ag (Supplementary Fig. 11 ). Epitaxial noble metal-semiconductor interfaces The interfacial structures between the icosahedral Ag seed and the grown CdS and CdSe nanorods are very important for the formation of such sophisticated noble metal-semiconductor heterostructures. As shown in Supplementary Fig. 12a , although it is very difficult to use spherical aberration-corrected high-angle annular dark-field STEM (Cs-HAADF-STEM) to clearly observe the detailed interfacial structure in the synthesized Ag-CdS icosapod due to its sophisticated architecture with large size of 108.2 ± 5.5 nm (Fig. 1b–d and Supplementary Fig. 2 ), the epitaxial growth of CdS rods on Ag with the epitaxial interface between the Ag (111) and CdS (002) planes, referred to as Ag (111) ∥ CdS (002), can be confirmed by the fast Fourier transformation (FFT) pattern (Supplementary Fig. 12b ). To identify the Ag-CdS interface, Ag-CdS icosapods extracted from the reactor at ~60 s were characterized by the Cs-HAADF-STEM (Supplementary Fig. 13a ). As shown in Fig. 2a , the measured lattice distances of 0.14, 0.24, 0.34 and 0.36 nm are well-matched with the interplanar distances of Ag (220), Ag (111), CdS (002), and CdS (100) planes, respectively. Besides the sharp Ag-CdS interface observed, it is also found that the Ag (111) planes are parallel to the CdS (002) planes, and Ag (220) planes are parallel to CdS (100) planes. Together with the aligned spots in the FFT pattern (Fig. 2b ), it can be concluded that the CdS are epitaxially grown on Ag icosahedral nanocrystals with an epitaxial relationship of Ag (111) ∥ CdS (002) and Ag (220) ∥ CdS (100). Moreover, as shown in the Cs-HAADF-STEM image (Fig. 2a ) and inverse FFT image (Fig. 2c ), the nearest neighbor atoms on the Ag surface are determined to be S of CdS ( \\\\(00\\\\) \\\\(\\\\overline{1}\\\\) ) plane since the S atoms have lower contrast than the Cd atoms under the STEM mode, thus CdS is grown along the [001] direction [52] (Supplementary Fig. 14 ). Similarly, as shown in Fig. 2d–f , when CdSe was grown on the icosahedral Ag seeds at a short reaction time of 1 min (Supplementary Fig. 13b ), instead of 3 min used for the preparation of aforementioned Ag-CdSe icosapods (Supplementary Figs. 8 and 9 ), an epitaxial relationship can also be found between Ag and CdSe. The epitaxial Ag-CdSe interface was determined to be Ag (111) ∥ CdSe (002) and Ag (220) ∥ CdSe (100), and CdSe grows along the [001] direction with the nearest neighbor Se atoms on the Ag surface (Fig. 2f ). Based on the aforementioned results, the epitaxial interfaces of Ag-CdS and Ag-CdSe can be schematically illustrated in Supplementary Fig. 15 and Supplementary Data 1 . Importantly, despite the large lattice mismatch (more than 40%) between Ag and CdS (or CdSe) (Supplementary Table 1 ), which is much greater than the previously reported value for the epitaxial growth (normally <5%) [17] , [18] , [30] , [36] , the epitaxial interfaces can still be achieved in our synthesized Ag-CdS and Ag-CdSe heterostructures. Fig. 2: Characterization of epitaxial interfaces in Ag-CdS and Ag-CdSe heterostructures obtained at a short reaction time of 1 min. a Cs-HAADF-STEM image of the Ag-CdS interface. b , c The FFT pattern ( b ) and inverse FFT image ( c ) of the Ag-CdS interface. The silver square and silver triangle in ( b ) are applied to mark the FFT spots indexed to Ag (111) and Ag (220), respectively. The yellow circle and yellow rhombus are applied to mark the FFT spots indexed to CdS (002) and CdS (100), respectively. d Cs-HAADF-STEM image of the Ag-CdSe interface. e , f The FFT pattern ( e ) and inverse FFT image ( f ) of the Ag-CdSe interface. The silver square and silver triangle in ( e ) are applied to mark the FFT spots indexed to Ag (111) and Ag (220), respectively. The red circle and red rhombus are applied to mark the FFT spots indexed to CdSe (002) and CdSe (100), respectively. The silver, blue, yellow, red balls in ( a ), ( c ), ( d ), ( f ) represent the scheme of Ag, Cd, S, Se atoms, respectively. Full size image Ultrafast plasmon-induced charge transfer The successful preparation of noble metal-semiconductor with well-defined morphologies and epitaxial interfaces provides a possibility to explore and understand their structure-property relationships. Here, we systematically studied the plasmon-induced charge transfer process in the obtained Ag-CdS and Ag-CdSe icosapods using pump-probe TA spectroscopy. As schematically shown in Fig. 3a , the pump laser-excited plasmon in Ag, which decays into electron-hole pairs, is followed by hot-electron transfer to semiconductor and/or direct interfacial charge transfer transition. The UV-visible absorption spectra of Ag icosahedral seeds, CdS nanorods (Supplementary Fig. 16 ), and the obtained Ag-CdS icosapods are presented in Fig. 3b . The strong peak for the blue curve located at 408 nm can be assigned to the localized surface plasmon resonance (LSPR) absorption peak of the Ag icosahedral nanocrystals. After the growth of CdS on Ag, two peaks located at 462 and 548 nm appear (red curve in Fig. 3b ). The peak at 462 nm, which is similar to the absorption peak of CdS nanorods (465 nm, black curve in Fig. 3b ), is assigned to the 1Σ exciton transition of CdS between the CB and valence band VB levels [53] , and the peak at 548 nm should be attributed to the redshifted LSPR of Ag nanocrystals. We then performed pump-probe TA spectroscopy using the 550 nm laser to excite Ag-CdS icosapods. As the photon energy of 550 nm laser is below the band gap of the CdS nanorods, the Ag LSPR bands in Ag-CdS icosapods are selectively excited (Supplementary Fig. 17 ). The TA spectrum of Ag-CdS icosapods, displayed as a two-dimensional pseudo-color plot in Fig. 3c , shows the distinct bleaching bands centered at ~480 and ~550 nm that can be attributed to the bleach of the CdS 1Σ band and Ag LSPR band, respectively. As the CdS was not excited, the bleach signal of the CdS 1Σ band can be attributed to the plasmon-induced hot-electron transfer from the Ag core to CdS nanorods. Fig. 3: Plasmon-induced charge separation in Ag-CdS icosapods. a Schematic illustration of plasmon-induced hot-electron transfer in Ag-CdS icosapods after femtosecond laser pumping. The dashed ellipsis schematically illustrates the plasmon-induced interfacial charge-transfer transition in Ag-CdS icosapods. b UV-visible absorption spectra of the colloidal solutions of Ag icosahedral nanocrystals, CdS nanorods, and Ag-CdS icosapods in toluene. Insets: photographs of the corresponding colloid solutions. c Two-dimensional pseudo-color plots of pump-visible probe TA spectra of the colloid solutions of Ag-CdS in toluene. X-axis: probe wavelength; Y-axis: pump-probe delay; Color: change in absorbance (ΔAbs), shown in milli-optical density unit (mOD). Gray shading is used to avoid the disturbance of the 550 nm pump laser to the probed signal. d Averaged mid-IR TA spectra of Ag-CdS icosapods from 1 to 2 ps (red curve), 20 to 40 ps (orange curve), 100 to 200 ps (blue curve), and 600 to 900 ps (black curve) after 550 nm laser excitation. e Time-resolved mid-IR absorption spectrum of Ag-CdS icosapods. Gray curves represent the multiexponential fits as described in Eq. 6 . A negligible intraband absorption signal (black squares) is observed in a control experiment conducted on CdS nanorods. Inset in e : zoom-in kinetics spectrum for the Ag-CdS icosapods. Source data are provided as a Source data file. Full size image Moreover, as shown in Fig. 3a , when the plasmon-induced hot-electron is transferred from the Ag to CdS or CdSe, it can generate a positive mid-IR TA signal due to the 1Σ-to-1П intraband transition, which enables us to directly observe the plasmon-induced hot-electron transfer process and quantify its QY from Ag to CdS or CdSe [53] , [54] , [55] . As shown in the averaged mid-IR TA spectra of Ag-CdS icosapods (Fig. 3d ), positive mid-IR TA signals were observed after the excitation of the Ag LSPR band, further confirming the presence of plasmon-induced hot-electron in the CB of CdS. The QY of plasmon-induced hot-electron transferred from Ag to CdS can be determined by the peak amplitude of the CdS intraband absorption signal according to Eq. 4 , which is calculated to be 18.1% (see “Methods”, Supplementary Figs. 18 , 19 and Supplementary Table 2 for details). This QY value is much higher than that previously reported in nonepitaxial noble metal-semiconductor heterostructures [53] , [55] , [56] , [57] , [58] , for instance, about 2.75% for Au-tipped CdS nanorods [53] . Fitting the time-resolved mid-IR absorption curve of Ag-CdS icosapods (Fig. 3e ) based on Eq. 6 yielded a formation time of 18.0 ± 0.3 fs and a multi-exponential decay with a half-life of 1.9 ± 0.16 ps, which corresponds to plasmon-induced hot-electron transfer time and charge-recombination time, respectively (Supplementary Table 3 ). A similar result was obtained in the epitaxial Ag-CdSe icosapods. The measured QY and the time of plasmon-induced hot-electron transfer were determined to be 17.6% and 7.5 ± 7.2 fs when epitaxial Ag-CdSe icosapods were pumped by a 700 nm laser, respectively (Supplementary Figs. 19 – 22 ; Supplementary Tables 3 and 4 ). The plasmon-induced hot-electron transfer processes in the epitaxial Ag-CdS and Ag-CdSe icosapods are much faster than the conventional indirect plasmon-induced hot-electron transfer process (commonly more than 100 fs [53] , [55] ). On such a short timescale (<20 fs), the plasmon directly decays into the electrons in the CB of semiconductors via the interfacial charge-transfer transition before rapid cooling of hot electrons induced by electron-electron scattering [54] , [55] , [58] , [59] , [60] . Unfortunately, because the transient IR selectively probes the electrons injected into the CB of CdS, similar to the previous reports [54] , our study was not able to provide direct information about the effect of the metal-semiconductor interfacial states on the plasmon hot electron transfer and recombination processes. In summary, we have demonstrated a noble-metal-seeded epitaxial growth method to achieve the controlled synthesis of highly symmetrical noble metal-semiconductor heterostructures, i.e., Ag-CdS and Ag-CdSe icosapods. Abrupt epitaxial Ag-CdS/CdSe interfaces are observed in the Ag-CdS and Ag-CdSe icosapods, which could be concluded as Ag (111) ∥ CdS/CdSe (002) and Ag (220) ∥ CdS/CdSe (100), albeit a large lattice mismatch (more than 40%). Detailed pump-probe TA-IR spectroscopy studies reveal high QYs of plasmon-induced hot-electron transferred from noble metal to semiconductor in the Ag-CdS (18.1%) and Ag-CdSe (17.6%) icosapods. This work demonstrates a strategy to rationally design and controllably synthesize sophisticated hierarchical noble metal-semiconductor heterostructures. The constructed epitaxial interface provides an ideal platform for investigating the role of the metal-semiconductor interface in various physicochemical processes.\"\n },\n {\n \"section\": \"Methods\",\n \"content\": \"Materials Silver nitrate (AgNO 3 , 99.9999% trace metals basis), cadmium oxide (CdO, ≥99.99% trace metals basis), cadmium chloride (CdCl 2 , 99.99% trace metals basis), sulfur powder (S, ≥99.98% trace metals basis), selenium powder (Se, 99.99%), oleylamine (technical grade, 70%), trioctylphosphine (97%), trioctylphosphine oxide (technical grade, 90%), oleic acid (technical grade, 90%) were purchased from Sigma-Aldrich. N-octadecylphosphonic acid (95%), n-hexylphosphonic acid (95%) were bought from Alfa Aesar. Hexane (≥99.0%), toluene (≥99.0%), and methanol (≥99.9%) were purchased from Aladdin and used as received without any further purification. Synthesis of five-fold multi-twinned Ag icosahedral nanocrystals Typically, after 85 mg of AgNO 3 were mixed with 10 mL of oleylamine in a 50 mL three-neck flask, the mixture was degassed at 60 °C for 30 min under vigorous magnetic stirring at 750 r.p.m. It was then quickly heated up (≥15 °C min −1 ) to 220 °C under N 2 atmosphere. After the temperature was maintained at 220 °C for 1 h, the solution was cooled down to 160 °C, and a mixture of 60 μL of acetate and 0.5 mL of oleylamine was injected into the flask. After the temperature was kept at 160 °C for 10 min, the reaction was stopped by removing the heating mantle. The flask was then cooled down to room temperature naturally. The obtained brownish product was precipitated by centrifuge at 5800 × g . for 2 min. The precipitate was then washed 3 times using the mixture of hexane and acetone (V hexane /V acetone = 5/1). Finally, the obtained Ag nanocrystals were re-dispersed in 2.5 mL of TOP as seeds for the synthesis of Ag-CdS icosapods and Ag-CdSe icosapods. Synthesis of Ag-CdS and Ag-CdSe icosapods TOPS and TOPSe stock solutions were first prepared by dissolving 18 mg of sulfur powder or 10 mg of selenium powder in 0.5 mL of TOP in a glove box filled with argon, respectively. For the synthesis of Ag-CdS icosapods, 115 mg of CdO, 18 mg of CdCl 2 , 580 mg of n-octadecylphosphonic acid, 160 mg of n-hexylphosphonic acid, and 6 g of trioctylphosphine oxide were added into a 50 mL three-neck flask and degassed at 120 °C for 30 min under vigorous magnetic stirring at 750 r.p.m. After it was heated to 360 °C under N 2 atmosphere, the solution becomes transparent. Then, 1 mL of TOP was injected into the flask, and the temperature was kept at 360 °C for 15 min. A mixture of 0.5 mL of the above-mentioned Ag seeds in TOP and 0.5 mL of TOPS stock was injected into the flask at 360 °C swiftly. After the injection of Ag seeds and TOPS, the temperature was kept at 350 °C for 6 min. Then the reaction was stopped by removing the heating mantle. After the solution was cooled down to 100 °C, 10 mL of toluene was injected into the reaction flask. Then, 5 mL of methanol was added to the solution, and the product was collected by centrifuge at 5800 × g for 3 min. The obtained precipitate was washed with the mixture of toluene and methanol (V toluene /V methanol = 5/1) and then dispersed into 50 mL of toluene. For the synthesis of Ag-CdSe icosapods, all the conditions are the same with the synthesis of Ag-CdS icosapods except the mixture of 0.5 mL of the above-mentioned Ag seeds in TOP and 0.5 mL of TOPSe stock was injected into the flask at 310 °C and the temperature was kept at 300 °C for 3 min. Note that the narrow size distribution of Ag seeds (Supplementary Fig. 1 ) is critical for synthesizing such delicate Ag-CdS and Ag-CdSe icosapods. Synthesis of CdS and CdSe nanorods TOPS and TOPSe stock solutions were first prepared by dissolving 32 mg of sulfur powder or 40 mg of selenium powder in 0.5 mL of TOP in a glove box filled with argon, respectively. For the synthesis of CdS nanorods, 120 mg of CdO, 580 mg of n-octadecylphosphonic acid, 160 mg of n-hexylphosphonic acid, and 6 g of trioctylphosphine oxide were added into a 50 mL three-neck flask and degassed at 120 °C for 30 min under vigorous magnetic stirring at 750 r.p.m. After it was heated to 370 °C under N 2 atmosphere, the solution becomes transparent. Then, 1 mL of TOP was injected into the flask, and the temperature was kept at 360 °C for 15 min. 0.5 mL of TOPS stock was injected into the flask at 370 °C swiftly, and the temperature was kept at 360 °C for 3 min. Then, the reaction was stopped by removing the heating mantle. After the solution was cooled down to 100 °C, 10 mL of toluene was injected into the reaction flask. Then, 5 mL of methanol was added to the solution, and the product was collected by centrifuge at 5800 × g for 3 min. The obtained precipitate was washed with the mixture of toluene and methanol (V toluene /V methanol = 5/1) and then dispersed into 50 mL of toluene. For the synthesis of CdSe nanorods, all the conditions are the same as the synthesis of CdS nanorods except 0.5 mL of TOPSe stock was injected into the flask at 380 °C and the temperature was kept at 360 °C for 1 min. Characterization Powder XRD patterns of the dried samples were recorded on Bruker D8 diffractometer at a scanning rate of 2° min −1 using Cu Kα radiation (λ = 1.5406 Å). XPS and UPS spectra were recorded on the ESCALAB 250Xi (Thermo Fisher Scientific) system. The XPS results were calibrated with the reference C 1s peak located at 284.6 eV. The X-ray absorption fine structure (XAFS) experiments under transmission mode were performed at the 7-BM of the National Synchrotron Light Source II (NSLS-II) which is using a channel-cut monochromator. The NSLS-II is rung at 400 mA electron beam current under top-off mode. Samples for SEM and TEM characterizations were prepared by dropping nanomaterial dispersions in toluene on a silicon substrate and amorphous carbon-coated copper grid, respectively. SEM images were obtained using FESEM, JEOL JSM-7800F. TEM characterization was performed with a JEOL JEM-2100 and FEI TF20 operated at 200 kV. Dark-field STEM, Cs-HAADF STEM, and elemental mapping images were obtained using FEI Titan G2 60-300 operated at 300 kV as well as a JEM-ARM200F operated at 200 kV. The 3D reconstruction of Ag-CdS icosapods was carried out using FEI Titan G2 60-300 operated at 60 kV. Visible femtosecond transient absorption The visible femtosecond transient absorption measurements were conducted in a Helios spectrometer (Ultrafast Systems LLC) with pump and probe beams derived from a regenerative amplified Ti: Sapphire laser system (Coherent Astrella, 35 fs, 4 mJ per pulse, and 1 kHz repetition rate). The 800 nm output pulse was split into two beams with a beam splitter. One beam passed through a tunable optical parametric amplifier (OperA solo, Coherent) to generate a tunable visible pump. During the measurement, the pump beam was chopped by a synchronized chopper to 500 Hz. The other beam was attenuated and focused on a CaF 2 window to generate the white light continuum with a wavelength range from 350 nm to 800 nm, referred to as the probe beam. The probe beam was focused into a 1-mm path-length quartz cuvette (Starna) containing the sample in toluene. The transmission of the probe was collected by a fiber optics-coupled multichannel spectrometer with complementary metal-oxide-semiconductor (CMOS) sensors and detected at a frequency of 1 kHz (Ultrafast systems, Helios). The delay between the pump and probe pulses was controlled by a motorized delay stage. Samples in 1-mm cuvettes were used for all spectroscopy measurements and stirred vigorously during the measurements. The data were analyzed with surface Xplorer. Mid-IR femtosecond transient absorption Mid-IR TA experiments were conducted using a commercial Ti:Sapphire regenerative amplifier (Astrella, Coherent) at 800 nm with a repetition rate of 1 kHz and pulse duration of 35 fs. An optical amplifier (OperA solo, Coherent) pumped by the regenerative amplifier was used to provide a pump beam with tunable wavelengths. Another optical amplifier (OperA solo, Coherent) pumped by the regenerative amplifier was used to generate a tunable IR probe via difference frequency generation (DFG) by signal and idler beams. The pump beam was chopped by a synchronized chopper to 500 Hz and focused on a 1.5-mm path-length Harrick cell containing the sample in toluene. The IR probe, which was also focused on a 1.5-mm path length Harrick cell containing the sample in toluene, overlapped with the temporally delayed pump beam controlled by a movable delay stage. The probe intensity with and without the pump was analyzed by a Teledyne/phasetech nitrogen-cooled 128 × 128 MCT detector. All measurements were carried out under ambient conditions. QY measurement of the plasmon-induced hot-electron transfer in Ag-CdS heterostructures The QY of the plasmon-induced hot-electrons transferred from the Ag core to the CdS under 550 nm excitation can be quantified by the mid-IR intraband absorption at around 3340 cm −1 of CB electrons in CdS. The CB electron in CdS produces a broad IR absorption which can be attributed to the transition from 1Σ to a higher level (such as 1П), as shown in Fig. 3a . To determine the plasmon-induced hot-electron transfer QY, the amplitude of IR signal per CdS CB electron was quantified, denoted as S 0 . Hence, S 0 can be calculated by Eq. 1 : $${{{{{{\\\\rm{S}}}}}}}_{0}=\\\\frac{\\\\Delta {{{{{\\\\rm{A}}}}}}\\\\left({{{{{\\\\rm{CdS}}}}}},400\\\\,{{{{{\\\\rm{nm}}}}}}\\\\right)}{{{{{{\\\\rm{N}}}}}}({{{{{\\\\rm{CdSCB}}}}}}{{{{{\\\\rm{e}}}}}}{{{{{\\\\rm{lectrons}}}}}})}$$ (1) where ∆A is the maximum signal amplitude obtained from the kinetics; N(CdS CB electrons) is the number of electrons in CdS CB band, which can be determined by the number of absorbed photons by CdS (every photon creates one electron in CdS CB band under 400 nm excitation). Therefore, N(CdS CB electrons) equals the pump photon flux in the pump/probe overlap region times the sample absorption \\\\(\\\\left(\\\\frac{{{{{{\\\\rm{power}}}}}}/{hv}}{{{{{{\\\\rm{beam}}}}}}\\\\; {{{{{{\\\\rm{size}}}}}}}}\\\\times (1-{10}^{-{{{{{\\\\rm{OD}}}}}}})\\\\right)\\\\) , where hv is photon energy. The QY is calculated by the number of hot electrons generated in Ag-CdS per absorbed photons under 550 nm excitation, as shown in Eq. 2 . $${{{{{\\\\rm{QY}}}}}} \\t=\\\\frac{{{{{{\\\\rm{N}}}}}}({{{{{\\\\rm{hot}}}}}}\\\\,{{{{{\\\\rm{electrons}}}}}})}{{{{{{\\\\rm{N}}}}}}({{{{{\\\\rm{absorbed}}}}}}\\\\,{{{{{\\\\rm{photons}}}}}})} \\\\\\\\ \\t=\\\\frac{\\\\frac{\\\\Delta {{{{{\\\\rm{A}}}}}}\\\\left({{{{{\\\\rm{Ag}}}}}}-{{{{{\\\\rm{CdS}}}}}},550\\\\,{{{{{\\\\rm{nm}}}}}}\\\\right)}{{{{{{{\\\\rm{S}}}}}}}_{0}}}{\\\\frac{{{{{{\\\\rm{power}}}}}}\\\\left(550\\\\,{{{{{\\\\rm{nm}}}}}}\\\\right)/{hv}(550\\\\,{{{{{\\\\rm{nm}}}}}})}{{{{{{\\\\rm{beam}}}}}}\\\\,{{{{{\\\\rm{size}}}}}}\\\\left(550\\\\,{{{{{\\\\rm{nm}}}}}}\\\\right)}\\\\times \\\\left(1-{10}^{-{{{{{\\\\rm{OD}}}}}}\\\\left({{{{{\\\\rm{Ag}}}}}}-{{{{{\\\\rm{CdS}}}}}},550\\\\,{{{{{\\\\rm{nm}}}}}}\\\\right)}\\\\right)} \\\\\\\\ \\t=\\\\frac{{{{{{\\\\rm{beam}}}}}}\\\\,{{{{{\\\\rm{size}}}}}}(550\\\\,{{{{{\\\\rm{nm}}}}}})}{{{{{{\\\\rm{beam}}}}}}\\\\,{{{{{\\\\rm{size}}}}}}\\\\,(400\\\\,{{{{{\\\\rm{nm}}}}}})}\\\\times \\\\left[\\\\frac{\\\\frac{\\\\Delta {{{{{\\\\rm{A}}}}}}\\\\left({{{{{\\\\rm{Ag}}}}}}-{{{{{\\\\rm{CdS}}}}}},550\\\\,{{{{{\\\\rm{nm}}}}}}\\\\right)}{{{{{{\\\\rm{power}}}}}}\\\\left(550\\\\,{{{{{\\\\rm{nm}}}}}}\\\\right)/{hv}(550\\\\,{{{{{\\\\rm{nm}}}}}})}}{(1-{10}^{-{{{{{\\\\rm{OD}}}}}}({{{{{\\\\rm{Ag}}}}}}-{{{{{\\\\rm{CdS}}}}}},550\\\\,{{{{{\\\\rm{nm}}}}}})})}\\\\bigg/\\\\frac{\\\\frac{\\\\Delta\\\\, {{{{{\\\\rm{A}}}}}}\\\\left({{{{{\\\\rm{CdS}}}}}},400\\\\,{{{{{\\\\rm{nm}}}}}}\\\\right)}{{{{{{\\\\rm{power}}}}}}(400\\\\,{{{{{\\\\rm{nm}}}}}})/{hv}(400\\\\,{{{{{\\\\rm{nm}}}}}})}}{(1-{1\\\\,0}^{-{{{{{\\\\rm{OD}}}}}}({{{{{\\\\rm{CdS}}}}}},400\\\\,{{{{{\\\\rm{nm}}}}}\\\\,})})}\\\\right]$$ (2) In Eq. 2 , beam size (550 nm) and beam size (400 nm) represent the overlap areas of pump and probe beams under 550 nm, and 400 nm pump laser excitation, respectively, whose ratio can be determined in a calibration sample (Cd 3 P 2 quantum dots in this work) under the same conditions. The calibration sample is excited at both wavelengths and is expected to have the same QY for generating CB electrons. Then \\\\(\\\\frac{{{{{{\\\\rm{beam}}}}}}\\\\,{{{{{\\\\rm{size}}}}}}(550\\\\,{{{{{\\\\rm{nm}}}}}})}{{{{{{\\\\rm{beam}}}}}}\\\\,{{{{{\\\\rm{size}}}}}}(400\\\\,{{{{{\\\\rm{nm}}}}}})}\\\\) can be described as follows: $$\\\\frac{{{{{{\\\\rm{beam}}}}}}\\\\,{{{{{\\\\rm{size}}}}}}(550\\\\,{{{{{\\\\rm{nm}}}}}})}{{{{{{\\\\rm{beam}}}}}}\\\\,{{{{{\\\\rm{size}}}}}}(400\\\\,{{{{{\\\\rm{nm}}}}}})}=\\\\frac{\\\\frac{\\\\Delta {{{{{\\\\rm{A}}}}}}\\\\left({{{{{\\\\rm{Cd}}}}}}3{{{{{\\\\rm{P}}}}}}2,400\\\\,{{{{{\\\\rm{nm}}}}}}\\\\right)}{{{{{{\\\\rm{power}}}}}}\\\\left(400\\\\,{{{{{\\\\rm{nm}}}}}}\\\\right)/{hv}(400\\\\,{{{{{\\\\rm{nm}}}}}})}}{(1-{10}^{-{{{{{\\\\rm{OD}}}}}}({{{{{\\\\rm{Cd}}}}}}3{{{{{\\\\rm{P}}}}}}2,400\\\\,{{{{{\\\\rm{nm}}}}}})})}\\\\bigg/\\\\frac{\\\\frac{\\\\Delta {{{{{\\\\rm{A}}}}}}\\\\left({{{{{\\\\rm{Cd}}}}}}3{{{{{\\\\rm{P}}}}}}2,550\\\\,{{{{{\\\\rm{nm}}}}}}\\\\right)}{{{{{{\\\\rm{power}}}}}}(550\\\\,{{{{{\\\\rm{nm}}}}}})/{hv}(550\\\\,{{{{{\\\\rm{nm}}}}}})}}{(1-{10}^{-{{{{{\\\\rm{OD}}}}}}({{{{{\\\\rm{Cd}}}}}}3{{{{{\\\\rm{P}}}}}}2,550\\\\,{{{{{\\\\rm{nm}}}}}})})}$$ (3) Thus, the QY of plasmon-induced hot-electron in Ag-CdS icosapods is given by Eq. 4 . $${{{{{\\\\rm{QY}}}}}}=\\\\frac{\\\\frac{\\\\Delta {{{{{\\\\rm{S}}}}}}({{{{{\\\\rm{Ag}}}}}}-{{{{{\\\\rm{CdS}}}}}},550\\\\,{{{{{\\\\rm{nm}}}}}})}{1-{10}^{-{{{{{\\\\rm{OD}}}}}}({{{{{\\\\rm{Ag}}}}}}-{{{{{\\\\rm{CdS}}}}}},550\\\\,{{{{{\\\\rm{nm}}}}}})}}}{\\\\frac{\\\\Delta {{{{{\\\\rm{S}}}}}}({{{{{\\\\rm{CdS}}}}}},400\\\\,{{{{{\\\\rm{nm}}}}}})}{1-{10}^{-{{{{{\\\\rm{OD}}}}}}({{{{{\\\\rm{CdS}}}}}},400\\\\,{{{{{\\\\rm{nm}}}}}})}}}\\\\times \\\\frac{\\\\frac{\\\\Delta {{{{{\\\\rm{S}}}}}}({{{{{\\\\rm{Cd}}}}}}3{{{{{\\\\rm{P}}}}}}2,400\\\\,{{{{{\\\\rm{nm}}}}}})}{1-{10}^{-{{{{{\\\\rm{OD}}}}}}({{{{{\\\\rm{Cd}}}}}}3{{{{{\\\\rm{P}}}}}}2,400\\\\,{{{{{\\\\rm{nm}}}}}})}}}{\\\\frac{\\\\Delta {{{{{\\\\rm{S}}}}}}({{{{{\\\\rm{Cd}}}}}}3{{{{{\\\\rm{P}}}}}}2,550\\\\,{{{{{\\\\rm{nm}}}}}})}{1-{10}^{-{{{{{\\\\rm{OD}}}}}}({{{{{\\\\rm{Cd}}}}}}3{{{{{\\\\rm{P}}}}}}2,550\\\\,{{{{{\\\\rm{nm}}}}}})}}}$$ (4) Here, we define \\\\(\\\\Delta {{{{{\\\\rm{S}}}}}}=\\\\frac{\\\\Delta {{{{{\\\\rm{A}}}}}}}{{{{{{\\\\rm{power}}}}}}}\\\\) , a quantity to be obtained from the transient IR measurement. To improve the measurement accuracy and to check the linearity of the signal, ∆A as a function of excitation power was measured for all samples. As shown in Supplementary Fig. 18 , the TA signal amplitude increases linearly with the excitation power. ∆S can be obtained from the slope of the fitted linear line. For all samples, the OD is controlled at around 0.2–0.8. The values of epitaxial Ag-CdS icosapods are listed in Supplementary Table 2 . It should be noted that because the nanorods are directly grown on the Ag nanoparticles, there may be slight differences in the diameter of the CdS nanorods control sample compared to the diameter of the nanorods in the heterostructure. This could affect the QY as the optical properties of the nanorods are dependent upon their size (absorption cross section), therefore, the QYs for each sample were calculated to the best of our ability using the independently synthesized CdS nanorods controls. QY measurement of the plasmon-induced hot-electron transfer in Ag-CdSe icosapods The QY of the electrons transferred from the Ag core to the CdSe under 700 nm excitation can be quantified by the mid-IR intraband absorption at around 3340 cm −1 of CB electrons in CdSe. Similar to Ag-CdS icosapods, the QY is calculated by the number of hot electrons generated in Ag-CdSe icosapods per absorbed photons under 700 nm excitation. CdSe nanorods are excited by a 400 nm laser. Cd 3 P 2 quantum dots are excited by 400 nm laser and 700 nm laser for calibration. The QY of plasmon-induced hot-electron transfer in Ag-CdSe icosapods was shown in Eq. 5 . $${{{{{\\\\rm{QY}}}}}}=\\\\frac{\\\\frac{\\\\Delta {{{{{\\\\rm{S}}}}}}({{{{{\\\\rm{Ag}}}}}}-{{{{{\\\\rm{CdSe}}}}}},700\\\\,{{{{{\\\\rm{nm}}}}}})}{1-{10}^{-{{{{{\\\\rm{OD}}}}}}({{{{{\\\\rm{Ag}}}}}}-{{{{{\\\\rm{CdSe}}}}}},700\\\\,{{{{{\\\\rm{nm}}}}}})}}}{\\\\frac{\\\\Delta {{{{{\\\\rm{S}}}}}}({{{{{\\\\rm{CdSe}}}}}},400\\\\,{{{{{\\\\rm{nm}}}}}})}{1-{10}^{-{{{{{\\\\rm{OD}}}}}}({{{{{\\\\rm{CdSe}}}}}},400\\\\,{{{{{\\\\rm{nm}}}}}})}}}\\\\times \\\\frac{\\\\frac{\\\\Delta {{{{{\\\\rm{S}}}}}}({{{{{\\\\rm{Cd}}}}}}3{{{{{\\\\rm{P}}}}}}2,400\\\\,{{{{{\\\\rm{nm}}}}}})}{1-{10}^{-{{{{{\\\\rm{OD}}}}}}({{{{{\\\\rm{Cd}}}}}}3{{{{{\\\\rm{P}}}}}}2,400\\\\,{{{{{\\\\rm{nm}}}}}})}}}{\\\\frac{\\\\Delta {{{{{\\\\rm{S}}}}}}({{{{{\\\\rm{Cd}}}}}}3{{{{{\\\\rm{P}}}}}}2,700\\\\,{{{{{\\\\rm{nm}}}}}})}{1-{10}^{-{{{{{\\\\rm{OD}}}}}}({{{{{\\\\rm{Cd}}}}}}3{{{{{\\\\rm{P}}}}}}2,700\\\\,{{{{{\\\\rm{nm}}}}}})}}}$$ (5) For all samples, the OD is controlled at around 0.4–0.7. The values used in Eq. 4 is listed in Supplementary Table 4 . The multiexponential fit of the decay kinetics of Ag-CdS and Ag-CdSe icospads The normalized TA IR kinetics shown in Fig. 3e and Supplementary Fig. 23b are fitted by a single exponential rise and multi-exponential decay function according to Eq. 6 : $${{{{{\\\\rm{S}}}}}}\\\\left({{{{{\\\\rm{t}}}}}}\\\\right)={{{{{\\\\rm{IRF}}}}}}\\\\otimes \\\\left[-{{\\\\exp }}\\\\left(-\\\\frac{{{{{{\\\\rm{t}}}}}}}{{T}_{f}}\\\\right)+\\\\mathop{\\\\sum }\\\\limits_{1}^{3}{A}_{i}{{\\\\exp }}\\\\left(-\\\\frac{{{{{{\\\\rm{t}}}}}}}{{T}_{i}}\\\\right)\\\\right]$$ (6) The instrument response function (IRF) of the measurement was determined in a 1 mm silicon wafer and can be well modeled by a Gaussian function with an FWHM of ~177 ± 25 fs. T f is the time constant of the signal rise. A i and T i ( i = 1,2,3) are amplitude and time constants of the multiexponential function decay function, respectively, \\\\({\\\\sum }_{1}^{3}{A}_{i}=1\\\\) . The fitting parameters are listed in Supplementary Table 3 . 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":"This research proposes a novel epitaxial growth method to synthesize highly symmetrical, branched noble metal-semiconductor heterostructures—specifically, Ag-CdS or Ag-CdSe icosapods—with precisely controlled architectures and well-defined interfaces, despite a large lattice mismatch (>40%), by leveraging the crystallographic symmetry of zero-dimensional platonic Ag nanocrystals to enable uniform growth of one-dimensional II-VI semiconductor nanorods on their (111) facets."},"entities":{"kind":"list like","value":["novel epitaxial growth method","noble metal-semiconductor heterostructures","Ag-CdS icosapods","Ag-CdSe icosapods","crystallographic symmetry","zero-dimensional platonic Ag nanocrystals","one-dimensional II-VI semiconductor nanorods","lattice mismatch","(111) facets","precisely controlled architectures","well-defined interfaces","semiconductor nanorods","Ag nanocrystals","II-VI semiconductor","icosapods"],"string":"[\n \"novel epitaxial growth method\",\n \"noble metal-semiconductor heterostructures\",\n \"Ag-CdS icosapods\",\n \"Ag-CdSe icosapods\",\n \"crystallographic symmetry\",\n \"zero-dimensional platonic Ag nanocrystals\",\n \"one-dimensional II-VI semiconductor nanorods\",\n \"lattice mismatch\",\n \"(111) facets\",\n \"precisely controlled architectures\",\n \"well-defined interfaces\",\n \"semiconductor nanorods\",\n \"Ag nanocrystals\",\n \"II-VI semiconductor\",\n \"icosapods\"\n]","isConversational":false},"references":{"kind":"list like","value":[{"idx":"1.","title":"Borys, N. J., Walter, M. J., Huang, J., Talapin, D. V. & Lupton, J. M. The role of particle morphology in interfacial energy transfer in CdSe/CdS heterostructure nanocrystals.Science330, 1371–1374 (2010).","link":"https://doi.org/10.1126%2Fscience.1198070"},{"idx":"2.","title":"Gu, J., Zhang, Y. W. & Tao, F. Shape control of bimetallic nanocatalysts through well-designed colloidal chemistry approaches.Chem. Soc. Rev.41, 8050–8065 (2012).","link":"https://doi.org/10.1039%2Fc2cs35184f"},{"idx":"3.","title":"Jiang, L. Z. et al. Facet engineering accelerates spillover hydrogenation on highly diluted metal nanocatalysts.Nat. Nanotechnol.15, 848–853 (2020).","link":"https://doi.org/10.1038%2Fs41565-020-0746-x"},{"idx":"4.","title":"Clavero, C. Plasmon-induced hot-electron generation at nanoparticle/metal-oxide interfaces for photovoltaic and photocatalytic devices.Nat. Photon.8, 95–103 (2014).","link":"https://doi.org/10.1038%2Fnphoton.2013.238"},{"idx":"5.","title":"Liu, M. X. et al. Lattice anchoring stabilizes solution-processed semiconductors.Nature570, 96–101 (2019).","link":"https://doi.org/10.1038%2Fs41586-019-1239-7"},{"idx":"6.","title":"Yuan, W. T. et al. In situ manipulation of the active Au-TiO2interface with atomic precision during CO oxidation.Science371, 517–521 (2021).","link":"https://doi.org/10.1126%2Fscience.abe3558"},{"idx":"7.","title":"Zhao, M. T. et al. Metal-organic frameworks as selectivity regulators for hydrogenation reactions.Nature539, 76–80 (2016).","link":"https://doi.org/10.1038%2Fnature19763"},{"idx":"8.","title":"Cheng, K. et al. Maximizing noble metal utilization in solid catalysts by control of nanoparticle location.Science377, 204–208 (2022).","link":"https://doi.org/10.1126%2Fscience.abn8289"},{"idx":"9.","title":"Kagan, C. R., Lifshitz, E., Sargent, E. H. & Talapin, D. V. Building devices from colloidal quantum dots.Science353, aac5523 (2016).","link":"https://doi.org/10.1126%2Fscience.aac5523"},{"idx":"10.","title":"Liu, Y. et al. Van der Waals heterostructures and devices.Nat. Rev. Mater.1, 16042 (2016).","link":"https://doi.org/10.1038%2Fnatrevmats.2016.42"},{"idx":"11.","title":"Losch, P. et al. Colloidal nanocrystals for heterogeneous catalysis.Nano Today24, 15–47 (2019).","link":"https://doi.org/10.1016%2Fj.nantod.2018.12.002"},{"idx":"12.","title":"Vogt, C. & Weckhuysen, B. M. The concept of active site in heterogeneous catalysis.Nat. Rev. Chem.6, 89–111 (2022).","link":"https://doi.org/10.1038%2Fs41570-021-00340-y"},{"idx":"13.","title":"Lu, Q. P., Yu, Y. F., Ma, Q. L., Chen, B. & Zhang, H. 2D transition-metal-dichalcogenide-nanosheet-based composites for photocatalytic and electrocatalytic hydrogen evolution reactions.Adv. Mater.28, 1917–1933 (2016).","link":"https://doi.org/10.1002%2Fadma.201503270"},{"idx":"14.","title":"Linic, S., Chavez, S. & Elias, R. Flow and extraction of energy and charge carriers in hybrid plasmonic nanostructures.Nat. Mater.20, 916–924 (2021).","link":"https://doi.org/10.1038%2Fs41563-020-00858-4"},{"idx":"15.","title":"Lauhon, L. J., Gudiksen, M. S., Wang, C. L. & Lieber, C. M. Epitaxial core-shell and core-multishell nanowire heterostructures.Nature420, 57–61 (2002).","link":"https://doi.org/10.1038%2Fnature01141"},{"idx":"16.","title":"Talapin, D. V. et al. Seeded growth of highly luminescent CdSe/CdS nanoheterostructures with rod and tetrapod morphologies.Nano Lett.7, 2951–2959 (2007).","link":"https://doi.org/10.1021%2Fnl072003g"},{"idx":"17.","title":"Tan, C. L., Chen, J. Z., Wu, X.-J. & Zhang, H. Epitaxial growth of hybrid nanostructures.Nat. Rev. Mater.3, 17089 (2018).","link":"https://doi.org/10.1038%2Fnatrevmats.2017.89"},{"idx":"18.","title":"Kum, H. et al. Epitaxial growth and layer-transfer techniques for heterogeneous integration of materials for electronic and photonic devices.Nat. Electro2, 439–450 (2019).","link":"https://doi.org/10.1038%2Fs41928-019-0314-2"},{"idx":"19.","title":"Kelso, M. V., Mahenderkar, N. K., Chen, Q. Z., Tubbesing, J. Z. & Switzer, J. A. Spin coating epitaxial films.Science364, 166–169 (2019).","link":"https://doi.org/10.1126%2Fscience.aaw6184"},{"idx":"20.","title":"Habas, S. E., Lee, H., Radmilovic, V., Somorjai, G. A. & Yang, P. Shaping binary metal nanocrystals through epitaxial seeded growth.Nat. Mater.6, 692–697 (2007).","link":"https://doi.org/10.1038%2Fnmat1957"},{"idx":"21.","title":"Miszta, K. et al. Hierarchical self-assembly of suspended branched colloidal nanocrystals into superlattice structures.Nat. Mater.10, 872–876 (2011).","link":"https://doi.org/10.1038%2Fnmat3121"},{"idx":"22.","title":"Wu, X.-J. et al. Controlled growth of high-density CdS and CdSe nanorod arrays on selective facets of two-dimensional semiconductor nanoplates.Nat. Chem.8, 470–475 (2016).","link":"https://doi.org/10.1038%2Fnchem.2473"},{"idx":"23.","title":"Li, M. Y. et al. Epitaxial growth of a monolayer WSe2-MoS2lateral p-n junction with an atomically sharp interface.Science349, 524–528 (2015).","link":"https://doi.org/10.1126%2Fscience.aab4097"},{"idx":"24.","title":"Kanne, T. et al. Epitaxial Pb on InAs nanowires for quantum devices.Nat. Nanotechnol.16, 776–781 (2021).","link":"https://doi.org/10.1038%2Fs41565-021-00900-9"},{"idx":"25.","title":"Chen, Y. et al. Phase engineering of nanomaterials.Nat. Rev. Chem.4, 243–256 (2020).","link":"https://doi.org/10.1038%2Fs41570-020-0173-4"},{"idx":"26.","title":"Lu, Q. P. et al. Crystal phase-based epitaxial growth of hybrid noble metal nanostructures on 4H/fcc Au nanowires.Nat. Chem.10, 456–461 (2018).","link":"https://doi.org/10.1038%2Fs41557-018-0012-0"},{"idx":"27.","title":"Liu, J. W. et al. Selective epitaxial growth of Rh nanorods on 2H/fccheterophase Au nanosheets to form 1D/2D Rh-Au heterostructures for highly efficient hydrogen evolution.J. Am. Chem. Soc.143, 4387–4396 (2021).","link":"https://doi.org/10.1021%2Fjacs.1c00612"},{"idx":"28.","title":"Zhang, Z. W. et al. Robust epitaxial growth of two-dimensional heterostructures, multiheterostructures, and superlattices.Science357, 788–792 (2017).","link":"https://doi.org/10.1126%2Fscience.aan6814"},{"idx":"29.","title":"Shi, E. Z. et al. Two-dimensional halide perovskite lateral epitaxial heterostructures.Nature580, 614–620 (2020).","link":"https://doi.org/10.1038%2Fs41586-020-2219-7"},{"idx":"30.","title":"Oh, M. H. et al. Design and synthesis of multigrain nanocrystals via geometric misfit strain.Nature577, 359–363 (2020).","link":"https://doi.org/10.1038%2Fs41586-019-1899-3"},{"idx":"31.","title":"Forticaux, A., Hacialioglu, S., DeGrave, J. P., Dziedzic, R. & Jin, S. Three-dimensional mesoscale heterostructures of ZnO nanowire arrays epitaxially grown on CuGaO2nanoplates as individual diodes.ACS Nano7, 8224–8232 (2013).","link":"https://doi.org/10.1021%2Fnn4037078"},{"idx":"32.","title":"Yao, Y. et al. Heteroepitaxial growth of multiblock Ln-MOF microrods for photonic barcodes.Angew. Chem. Int. Ed.58, 13803–13807 (2019).","link":"https://doi.org/10.1002%2Fanie.201907433"},{"idx":"33.","title":"Zhao, M. T. et al. Selective epitaxial growth of oriented hierarchical metal-organic framework heterostructures.J. Am. Chem. Soc.142, 8953–8961 (2020).","link":"https://doi.org/10.1021%2Fjacs.0c02489"},{"idx":"34.","title":"Grundmann, M. Formation of epitaxial domains: unified theory and survey of experimental results.Phys. Status Solidi B248, 805–824 (2011).","link":"https://doi.org/10.1002%2Fpssb.201046530"},{"idx":"35.","title":"Zhang, J. T., Tang, Y., Lee, K. & Min, O. Y. Nonepitaxial growth of hybrid core-shell nanostructures with large lattice mismatches.Science327, 1634–1638 (2010).","link":"https://doi.org/10.1126%2Fscience.1184769"},{"idx":"36.","title":"Liu, J. & Zhang, J. T. Nanointerface chemistry: lattice-mismatch-directed synthesis and application of hybrid nanocrystals.Chem. Rev.120, 2123–2170 (2020).","link":"https://doi.org/10.1021%2Facs.chemrev.9b00443"},{"idx":"37.","title":"Palmstrom, C. J. Epitaxy of dissimilar materials.Annu. Rev. Mater. Sci.25, 389–415 (1995).","link":"https://doi.org/10.1146%2Fannurev.ms.25.080195.002133"},{"idx":"38.","title":"Pendharkar, M. et al. Parity-preserving and magnetic field-resilient superconductivity in InSb nanowires with Sn shells.Science372, 508–511 (2021).","link":"https://doi.org/10.1126%2Fscience.aba5211"},{"idx":"39.","title":"Mokari, T., Rothenberg, E., Popov, I., Costi, R. & Banin, U. Selective growth of metal tips onto semiconductor quantum rods and tetrapods.Science304, 1787–1790 (2004).","link":"https://doi.org/10.1126%2Fscience.1097830"},{"idx":"40.","title":"Wolff, C. M. et al. All-in-one visible-light-driven water splitting by combining nanoparticulate and molecular co-catalysts on CdS nanorods.Nat. Energy3, 862–869 (2018).","link":"https://doi.org/10.1038%2Fs41560-018-0229-6"},{"idx":"41.","title":"Fan, Z. X. et al. Synthesis of 4H/fcc-Au@metal sulfide core shell nanoribbons.J. Am. Chem. Soc.137, 10910–10913 (2015).","link":"https://doi.org/10.1021%2Fjacs.5b06405"},{"idx":"42.","title":"Yang, J. & Ying, J. Y. A general phase-transfer protocol for metal ions and its application in nanocrystal synthesis.Nat. Mater.8, 683–689 (2009).","link":"https://doi.org/10.1038%2Fnmat2490"},{"idx":"43.","title":"Ben-Shahar, Y. et al. Optimal metal domain size for photocatalysis with hybrid semiconductor-metal nanorods.Nat. Commun.7, 10413 (2016).","link":"https://doi.org/10.1038%2Fncomms10413"},{"idx":"44.","title":"Chen, J. Z. et al. Ag@MoS2core-shell heterostructure as SERS platform to reveal the hydrogen evolution active sites of single-layer MoS2.J. Am. Chem. Soc.142, 7161–7167 (2020).","link":"https://doi.org/10.1021%2Fjacs.0c01649"},{"idx":"45.","title":"Han, S. K., Gu, C., Gong, M. & Yu, S. H. A Trialkylphosphine-driven chemical transformation route to Ag- and Bi-based chalcogenides.J. Am. Chem. Soc.137, 5390–5396 (2015).","link":"https://doi.org/10.1021%2Fjacs.5b00041"},{"idx":"46.","title":"Atta, S., Pennington, A. M., Celik, F. E. & Fabris, L. TiO2on gold nanostars enhances photocatalytic water reduction in the near-infrared regime.Chem4, 2140–2153 (2018).","link":"https://doi.org/10.1016%2Fj.chempr.2018.06.004"},{"idx":"47.","title":"Jang, Y. H. et al. Plasmonic solar cells: from rational design to mechanism overview.Chem. Rev.116, 14982–15034 (2016).","link":"https://doi.org/10.1021%2Facs.chemrev.6b00302"},{"idx":"48.","title":"Knight, M. W., Sobhani, H., Nordlander, P. & Halas, N. J. Photodetection with active optical antennas.Science332, 702–704 (2011).","link":"https://doi.org/10.1126%2Fscience.1203056"},{"idx":"49.","title":"Wang, H. L., Zhou, S., Gilroy, K. D., Cai, Z. S. & Xia, Y. N. Icosahedral nanocrystals of noble metals: synthesis and applications.Nano Today15, 121–144 (2017).","link":"https://doi.org/10.1016%2Fj.nantod.2017.06.011"},{"idx":"50.","title":"Sines, I. T. & Schaak, R. E. Phase-selective chemical extraction of selenium and sulfur from nanoscale metal chalcogenides: a general strategy for synthesis, purification, and phase targeting.J. Am. Chem. Soc.133, 1294–1297 (2011).","link":"https://doi.org/10.1021%2Fja110374d"},{"idx":"51.","title":"Wager, J. F. & Kuhn, K. Device physics modeling of surfaces and interfaces from an induced gap state perspective.Crit. Rev. Solid State Mater. Sci.42, 373–415 (2017).","link":"https://doi.org/10.1080%2F10408436.2016.1223013"},{"idx":"52.","title":"Bladt, E. et al. Atomic structure of wurtzite CdSe (core)/CdS (giant shell) nanobullets related to epitaxy and growth.J. Am. Chem. Soc.138, 14288–14293 (2016).","link":"https://doi.org/10.1021%2Fjacs.6b06443"},{"idx":"53.","title":"Wu, K. F., Rodriguez-Cordoba, W. E., Yang, Y. & Lian, T. Q. Plasmon-induced hot electron transfer from the Au tip to CdS rod in CdS-Au nanoheterostructures.Nano Lett.13, 5255–5263 (2013).","link":"https://doi.org/10.1021%2Fnl402730m"},{"idx":"54.","title":"Wu, K., Chen, J., McBride, J. R. & Lian, T. Efficient hot-electron transfer by a plasmon-induced interfacial charge-transfer transition.Science349, 632–635 (2015).","link":"https://doi.org/10.1126%2Fscience.aac5443"},{"idx":"55.","title":"Ratchford, D. C. Plasmon-induced charge transfer: challenges and outlook.ACS Nano13, 13610–13614 (2019).","link":"https://doi.org/10.1021%2Facsnano.9b08829"},{"idx":"56.","title":"Tagliabue, G. et al. Ultrafast hot-hole injection modifies hot-electron dynamics in Au/p-GaN heterostructures.Nat. Mater.19, 1312–1318 (2020).","link":"https://doi.org/10.1038%2Fs41563-020-0737-1"},{"idx":"57.","title":"Cresi, J. S. P. et al. Ultrafast dynamics of plasmon-mediated charge transfer in Ag@CeO2studied by free electron laser time-resolved X-ray absorption spectroscopy.Nano Lett.21, 1729–1734 (2021).","link":"https://doi.org/10.1021%2Facs.nanolett.0c04547"},{"idx":"58.","title":"Tan, S. et al. Plasmonic coupling at a metal/semiconductor interface.Nat. Photon.11, 806–812 (2017).","link":"https://doi.org/10.1038%2Fs41566-017-0049-4"},{"idx":"59.","title":"Long, R. & Prezhdo, O. V. Instantaneous generation of charge-separated state on TiO2surface sensitized with plasmonic nanoparticles.J. Am. Chem. Soc.136, 4343–4354 (2014).","link":"https://doi.org/10.1021%2Fja5001592"},{"idx":"60.","title":"Tan, S. et al. Ultrafast plasmon-enhanced hot electron generation at Ag nanocluster/Graphite heterojunctions.J. Am. Chem. Soc.139, 6160–6168 (2017).","link":"https://doi.org/10.1021%2Fjacs.7b01079"}],"string":"[\n {\n \"idx\": \"1.\",\n \"title\": \"Borys, N. J., Walter, M. J., Huang, J., Talapin, D. V. & Lupton, J. M. The role of particle morphology in interfacial energy transfer in CdSe/CdS heterostructure nanocrystals.Science330, 1371–1374 (2010).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1198070\"\n },\n {\n \"idx\": \"2.\",\n \"title\": \"Gu, J., Zhang, Y. W. & Tao, F. Shape control of bimetallic nanocatalysts through well-designed colloidal chemistry approaches.Chem. Soc. Rev.41, 8050–8065 (2012).\",\n \"link\": \"https://doi.org/10.1039%2Fc2cs35184f\"\n },\n {\n \"idx\": \"3.\",\n \"title\": \"Jiang, L. Z. et al. Facet engineering accelerates spillover hydrogenation on highly diluted metal nanocatalysts.Nat. Nanotechnol.15, 848–853 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41565-020-0746-x\"\n },\n {\n \"idx\": \"4.\",\n \"title\": \"Clavero, C. Plasmon-induced hot-electron generation at nanoparticle/metal-oxide interfaces for photovoltaic and photocatalytic devices.Nat. Photon.8, 95–103 (2014).\",\n \"link\": \"https://doi.org/10.1038%2Fnphoton.2013.238\"\n },\n {\n \"idx\": \"5.\",\n \"title\": \"Liu, M. X. et al. Lattice anchoring stabilizes solution-processed semiconductors.Nature570, 96–101 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-019-1239-7\"\n },\n {\n \"idx\": \"6.\",\n \"title\": \"Yuan, W. T. et al. In situ manipulation of the active Au-TiO2interface with atomic precision during CO oxidation.Science371, 517–521 (2021).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.abe3558\"\n },\n {\n \"idx\": \"7.\",\n \"title\": \"Zhao, M. T. et al. Metal-organic frameworks as selectivity regulators for hydrogenation reactions.Nature539, 76–80 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fnature19763\"\n },\n {\n \"idx\": \"8.\",\n \"title\": \"Cheng, K. et al. Maximizing noble metal utilization in solid catalysts by control of nanoparticle location.Science377, 204–208 (2022).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.abn8289\"\n },\n {\n \"idx\": \"9.\",\n \"title\": \"Kagan, C. R., Lifshitz, E., Sargent, E. H. & Talapin, D. V. Building devices from colloidal quantum dots.Science353, aac5523 (2016).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.aac5523\"\n },\n {\n \"idx\": \"10.\",\n \"title\": \"Liu, Y. et al. Van der Waals heterostructures and devices.Nat. Rev. Mater.1, 16042 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fnatrevmats.2016.42\"\n },\n {\n \"idx\": \"11.\",\n \"title\": \"Losch, P. et al. Colloidal nanocrystals for heterogeneous catalysis.Nano Today24, 15–47 (2019).\",\n \"link\": \"https://doi.org/10.1016%2Fj.nantod.2018.12.002\"\n },\n {\n \"idx\": \"12.\",\n \"title\": \"Vogt, C. & Weckhuysen, B. M. The concept of active site in heterogeneous catalysis.Nat. Rev. Chem.6, 89–111 (2022).\",\n \"link\": \"https://doi.org/10.1038%2Fs41570-021-00340-y\"\n },\n {\n \"idx\": \"13.\",\n \"title\": \"Lu, Q. P., Yu, Y. F., Ma, Q. L., Chen, B. & Zhang, H. 2D transition-metal-dichalcogenide-nanosheet-based composites for photocatalytic and electrocatalytic hydrogen evolution reactions.Adv. Mater.28, 1917–1933 (2016).\",\n \"link\": \"https://doi.org/10.1002%2Fadma.201503270\"\n },\n {\n \"idx\": \"14.\",\n \"title\": \"Linic, S., Chavez, S. & Elias, R. Flow and extraction of energy and charge carriers in hybrid plasmonic nanostructures.Nat. Mater.20, 916–924 (2021).\",\n \"link\": \"https://doi.org/10.1038%2Fs41563-020-00858-4\"\n },\n {\n \"idx\": \"15.\",\n \"title\": \"Lauhon, L. J., Gudiksen, M. S., Wang, C. L. & Lieber, C. M. Epitaxial core-shell and core-multishell nanowire heterostructures.Nature420, 57–61 (2002).\",\n \"link\": \"https://doi.org/10.1038%2Fnature01141\"\n },\n {\n \"idx\": \"16.\",\n \"title\": \"Talapin, D. V. et al. Seeded growth of highly luminescent CdSe/CdS nanoheterostructures with rod and tetrapod morphologies.Nano Lett.7, 2951–2959 (2007).\",\n \"link\": \"https://doi.org/10.1021%2Fnl072003g\"\n },\n {\n \"idx\": \"17.\",\n \"title\": \"Tan, C. L., Chen, J. Z., Wu, X.-J. & Zhang, H. Epitaxial growth of hybrid nanostructures.Nat. Rev. Mater.3, 17089 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fnatrevmats.2017.89\"\n },\n {\n \"idx\": \"18.\",\n \"title\": \"Kum, H. et al. Epitaxial growth and layer-transfer techniques for heterogeneous integration of materials for electronic and photonic devices.Nat. Electro2, 439–450 (2019).\",\n \"link\": \"https://doi.org/10.1038%2Fs41928-019-0314-2\"\n },\n {\n \"idx\": \"19.\",\n \"title\": \"Kelso, M. V., Mahenderkar, N. K., Chen, Q. Z., Tubbesing, J. Z. & Switzer, J. A. Spin coating epitaxial films.Science364, 166–169 (2019).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.aaw6184\"\n },\n {\n \"idx\": \"20.\",\n \"title\": \"Habas, S. E., Lee, H., Radmilovic, V., Somorjai, G. A. & Yang, P. Shaping binary metal nanocrystals through epitaxial seeded growth.Nat. Mater.6, 692–697 (2007).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat1957\"\n },\n {\n \"idx\": \"21.\",\n \"title\": \"Miszta, K. et al. Hierarchical self-assembly of suspended branched colloidal nanocrystals into superlattice structures.Nat. Mater.10, 872–876 (2011).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat3121\"\n },\n {\n \"idx\": \"22.\",\n \"title\": \"Wu, X.-J. et al. Controlled growth of high-density CdS and CdSe nanorod arrays on selective facets of two-dimensional semiconductor nanoplates.Nat. Chem.8, 470–475 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fnchem.2473\"\n },\n {\n \"idx\": \"23.\",\n \"title\": \"Li, M. Y. et al. Epitaxial growth of a monolayer WSe2-MoS2lateral p-n junction with an atomically sharp interface.Science349, 524–528 (2015).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.aab4097\"\n },\n {\n \"idx\": \"24.\",\n \"title\": \"Kanne, T. et al. Epitaxial Pb on InAs nanowires for quantum devices.Nat. Nanotechnol.16, 776–781 (2021).\",\n \"link\": \"https://doi.org/10.1038%2Fs41565-021-00900-9\"\n },\n {\n \"idx\": \"25.\",\n \"title\": \"Chen, Y. et al. Phase engineering of nanomaterials.Nat. Rev. Chem.4, 243–256 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41570-020-0173-4\"\n },\n {\n \"idx\": \"26.\",\n \"title\": \"Lu, Q. P. et al. Crystal phase-based epitaxial growth of hybrid noble metal nanostructures on 4H/fcc Au nanowires.Nat. Chem.10, 456–461 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41557-018-0012-0\"\n },\n {\n \"idx\": \"27.\",\n \"title\": \"Liu, J. W. et al. Selective epitaxial growth of Rh nanorods on 2H/fccheterophase Au nanosheets to form 1D/2D Rh-Au heterostructures for highly efficient hydrogen evolution.J. Am. Chem. Soc.143, 4387–4396 (2021).\",\n \"link\": \"https://doi.org/10.1021%2Fjacs.1c00612\"\n },\n {\n \"idx\": \"28.\",\n \"title\": \"Zhang, Z. W. et al. Robust epitaxial growth of two-dimensional heterostructures, multiheterostructures, and superlattices.Science357, 788–792 (2017).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.aan6814\"\n },\n {\n \"idx\": \"29.\",\n \"title\": \"Shi, E. Z. et al. Two-dimensional halide perovskite lateral epitaxial heterostructures.Nature580, 614–620 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-020-2219-7\"\n },\n {\n \"idx\": \"30.\",\n \"title\": \"Oh, M. H. et al. Design and synthesis of multigrain nanocrystals via geometric misfit strain.Nature577, 359–363 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41586-019-1899-3\"\n },\n {\n \"idx\": \"31.\",\n \"title\": \"Forticaux, A., Hacialioglu, S., DeGrave, J. P., Dziedzic, R. & Jin, S. Three-dimensional mesoscale heterostructures of ZnO nanowire arrays epitaxially grown on CuGaO2nanoplates as individual diodes.ACS Nano7, 8224–8232 (2013).\",\n \"link\": \"https://doi.org/10.1021%2Fnn4037078\"\n },\n {\n \"idx\": \"32.\",\n \"title\": \"Yao, Y. et al. Heteroepitaxial growth of multiblock Ln-MOF microrods for photonic barcodes.Angew. Chem. Int. Ed.58, 13803–13807 (2019).\",\n \"link\": \"https://doi.org/10.1002%2Fanie.201907433\"\n },\n {\n \"idx\": \"33.\",\n \"title\": \"Zhao, M. T. et al. Selective epitaxial growth of oriented hierarchical metal-organic framework heterostructures.J. Am. Chem. Soc.142, 8953–8961 (2020).\",\n \"link\": \"https://doi.org/10.1021%2Fjacs.0c02489\"\n },\n {\n \"idx\": \"34.\",\n \"title\": \"Grundmann, M. Formation of epitaxial domains: unified theory and survey of experimental results.Phys. Status Solidi B248, 805–824 (2011).\",\n \"link\": \"https://doi.org/10.1002%2Fpssb.201046530\"\n },\n {\n \"idx\": \"35.\",\n \"title\": \"Zhang, J. T., Tang, Y., Lee, K. & Min, O. Y. Nonepitaxial growth of hybrid core-shell nanostructures with large lattice mismatches.Science327, 1634–1638 (2010).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1184769\"\n },\n {\n \"idx\": \"36.\",\n \"title\": \"Liu, J. & Zhang, J. T. Nanointerface chemistry: lattice-mismatch-directed synthesis and application of hybrid nanocrystals.Chem. Rev.120, 2123–2170 (2020).\",\n \"link\": \"https://doi.org/10.1021%2Facs.chemrev.9b00443\"\n },\n {\n \"idx\": \"37.\",\n \"title\": \"Palmstrom, C. J. Epitaxy of dissimilar materials.Annu. Rev. Mater. Sci.25, 389–415 (1995).\",\n \"link\": \"https://doi.org/10.1146%2Fannurev.ms.25.080195.002133\"\n },\n {\n \"idx\": \"38.\",\n \"title\": \"Pendharkar, M. et al. Parity-preserving and magnetic field-resilient superconductivity in InSb nanowires with Sn shells.Science372, 508–511 (2021).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.aba5211\"\n },\n {\n \"idx\": \"39.\",\n \"title\": \"Mokari, T., Rothenberg, E., Popov, I., Costi, R. & Banin, U. Selective growth of metal tips onto semiconductor quantum rods and tetrapods.Science304, 1787–1790 (2004).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1097830\"\n },\n {\n \"idx\": \"40.\",\n \"title\": \"Wolff, C. M. et al. All-in-one visible-light-driven water splitting by combining nanoparticulate and molecular co-catalysts on CdS nanorods.Nat. Energy3, 862–869 (2018).\",\n \"link\": \"https://doi.org/10.1038%2Fs41560-018-0229-6\"\n },\n {\n \"idx\": \"41.\",\n \"title\": \"Fan, Z. X. et al. Synthesis of 4H/fcc-Au@metal sulfide core shell nanoribbons.J. Am. Chem. Soc.137, 10910–10913 (2015).\",\n \"link\": \"https://doi.org/10.1021%2Fjacs.5b06405\"\n },\n {\n \"idx\": \"42.\",\n \"title\": \"Yang, J. & Ying, J. Y. A general phase-transfer protocol for metal ions and its application in nanocrystal synthesis.Nat. Mater.8, 683–689 (2009).\",\n \"link\": \"https://doi.org/10.1038%2Fnmat2490\"\n },\n {\n \"idx\": \"43.\",\n \"title\": \"Ben-Shahar, Y. et al. Optimal metal domain size for photocatalysis with hybrid semiconductor-metal nanorods.Nat. Commun.7, 10413 (2016).\",\n \"link\": \"https://doi.org/10.1038%2Fncomms10413\"\n },\n {\n \"idx\": \"44.\",\n \"title\": \"Chen, J. Z. et al. Ag@MoS2core-shell heterostructure as SERS platform to reveal the hydrogen evolution active sites of single-layer MoS2.J. Am. Chem. Soc.142, 7161–7167 (2020).\",\n \"link\": \"https://doi.org/10.1021%2Fjacs.0c01649\"\n },\n {\n \"idx\": \"45.\",\n \"title\": \"Han, S. K., Gu, C., Gong, M. & Yu, S. H. A Trialkylphosphine-driven chemical transformation route to Ag- and Bi-based chalcogenides.J. Am. Chem. Soc.137, 5390–5396 (2015).\",\n \"link\": \"https://doi.org/10.1021%2Fjacs.5b00041\"\n },\n {\n \"idx\": \"46.\",\n \"title\": \"Atta, S., Pennington, A. M., Celik, F. E. & Fabris, L. TiO2on gold nanostars enhances photocatalytic water reduction in the near-infrared regime.Chem4, 2140–2153 (2018).\",\n \"link\": \"https://doi.org/10.1016%2Fj.chempr.2018.06.004\"\n },\n {\n \"idx\": \"47.\",\n \"title\": \"Jang, Y. H. et al. Plasmonic solar cells: from rational design to mechanism overview.Chem. Rev.116, 14982–15034 (2016).\",\n \"link\": \"https://doi.org/10.1021%2Facs.chemrev.6b00302\"\n },\n {\n \"idx\": \"48.\",\n \"title\": \"Knight, M. W., Sobhani, H., Nordlander, P. & Halas, N. J. Photodetection with active optical antennas.Science332, 702–704 (2011).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.1203056\"\n },\n {\n \"idx\": \"49.\",\n \"title\": \"Wang, H. L., Zhou, S., Gilroy, K. D., Cai, Z. S. & Xia, Y. N. Icosahedral nanocrystals of noble metals: synthesis and applications.Nano Today15, 121–144 (2017).\",\n \"link\": \"https://doi.org/10.1016%2Fj.nantod.2017.06.011\"\n },\n {\n \"idx\": \"50.\",\n \"title\": \"Sines, I. T. & Schaak, R. E. Phase-selective chemical extraction of selenium and sulfur from nanoscale metal chalcogenides: a general strategy for synthesis, purification, and phase targeting.J. Am. Chem. Soc.133, 1294–1297 (2011).\",\n \"link\": \"https://doi.org/10.1021%2Fja110374d\"\n },\n {\n \"idx\": \"51.\",\n \"title\": \"Wager, J. F. & Kuhn, K. Device physics modeling of surfaces and interfaces from an induced gap state perspective.Crit. Rev. Solid State Mater. Sci.42, 373–415 (2017).\",\n \"link\": \"https://doi.org/10.1080%2F10408436.2016.1223013\"\n },\n {\n \"idx\": \"52.\",\n \"title\": \"Bladt, E. et al. Atomic structure of wurtzite CdSe (core)/CdS (giant shell) nanobullets related to epitaxy and growth.J. Am. Chem. Soc.138, 14288–14293 (2016).\",\n \"link\": \"https://doi.org/10.1021%2Fjacs.6b06443\"\n },\n {\n \"idx\": \"53.\",\n \"title\": \"Wu, K. F., Rodriguez-Cordoba, W. E., Yang, Y. & Lian, T. Q. Plasmon-induced hot electron transfer from the Au tip to CdS rod in CdS-Au nanoheterostructures.Nano Lett.13, 5255–5263 (2013).\",\n \"link\": \"https://doi.org/10.1021%2Fnl402730m\"\n },\n {\n \"idx\": \"54.\",\n \"title\": \"Wu, K., Chen, J., McBride, J. R. & Lian, T. Efficient hot-electron transfer by a plasmon-induced interfacial charge-transfer transition.Science349, 632–635 (2015).\",\n \"link\": \"https://doi.org/10.1126%2Fscience.aac5443\"\n },\n {\n \"idx\": \"55.\",\n \"title\": \"Ratchford, D. C. Plasmon-induced charge transfer: challenges and outlook.ACS Nano13, 13610–13614 (2019).\",\n \"link\": \"https://doi.org/10.1021%2Facsnano.9b08829\"\n },\n {\n \"idx\": \"56.\",\n \"title\": \"Tagliabue, G. et al. Ultrafast hot-hole injection modifies hot-electron dynamics in Au/p-GaN heterostructures.Nat. Mater.19, 1312–1318 (2020).\",\n \"link\": \"https://doi.org/10.1038%2Fs41563-020-0737-1\"\n },\n {\n \"idx\": \"57.\",\n \"title\": \"Cresi, J. S. P. et al. Ultrafast dynamics of plasmon-mediated charge transfer in Ag@CeO2studied by free electron laser time-resolved X-ray absorption spectroscopy.Nano Lett.21, 1729–1734 (2021).\",\n \"link\": \"https://doi.org/10.1021%2Facs.nanolett.0c04547\"\n },\n {\n \"idx\": \"58.\",\n \"title\": \"Tan, S. et al. Plasmonic coupling at a metal/semiconductor interface.Nat. Photon.11, 806–812 (2017).\",\n \"link\": \"https://doi.org/10.1038%2Fs41566-017-0049-4\"\n },\n {\n \"idx\": \"59.\",\n \"title\": \"Long, R. & Prezhdo, O. V. Instantaneous generation of charge-separated state on TiO2surface sensitized with plasmonic nanoparticles.J. Am. Chem. Soc.136, 4343–4354 (2014).\",\n \"link\": \"https://doi.org/10.1021%2Fja5001592\"\n },\n {\n \"idx\": \"60.\",\n \"title\": \"Tan, S. et al. Ultrafast plasmon-enhanced hot electron generation at Ag nanocluster/Graphite heterojunctions.J. Am. Chem. Soc.139, 6160–6168 (2017).\",\n \"link\": \"https://doi.org/10.1021%2Fjacs.7b01079\"\n }\n]","isConversational":false}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":0,"numItemsPerPage":100,"numTotalItems":11341,"offset":0,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6IjVHZDBvd0g5MTM2eDZjc1FvbE1zcktNWUZoRVFoUm5PVVVybEpjOUhaUEEifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc4NTc1NTE4MiwianRpIjoiMTgwYTBlMDQtY2FlYy00Y2NlLTk1ZDAtNTEyZDJkN2RjMGFhIiwic3ViIjoiL2RhdGFzZXRzL1hpYW8tWW91dGgvTkNfUGh5c2ljcyIsImV4cCI6MTc4NTc1ODc4MiwiaXNzIjoiaHR0cHM6Ly9odWdnaW5nZmFjZS5jbyJ9.u2zD6VlMWG59abZjJUw3B8CfrWWt0m5Gbg2I1VqiJz7WojU-hNMUy-MeqCoKIjdtyhXhNid2ys4gw7zR1BLBBw","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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s41467-021-25455-0 | Materials science | 4K-memristor analog-grade passive crossbar circuit | The superior density of passive analog-grade memristive crossbar circuits enables storing large neural network models directly on specialized neuromorphic chips to avoid costly off-chip communication. To ensure efficient use of such circuits in neuromorphic systems, memristor variations must be substantially lower than... | [
{
"section": "Introduction",
"content": "Analog-grade nonvolatile memories, such as those based on floating-gate transistor [1] , [2] , [3] , phase-change [4] , [5] , [6] , ferroelectric [7] , [8] , magnetic [9] , solid-state electrolyte [10] , [11] , [12] , [13] , organic [14] , [15] , and metal-oxide [16]... | The main contribution of this work is to address the challenges of insufficient uniformity, density, and analog properties in existing memristive crossbar circuits by developing a uniform, CMOS-compatible fabrication technology for large-scale, passively integrated, analog-grade ReRAM crossbar arrays that enable high-p... | [
"memristive crossbar circuits",
"uniformity",
"density",
"analog properties",
"CMOS-compatible fabrication technology",
"large-scale passively integrated analog-grade ReRAM crossbar arrays",
"high-precision in-memory computing",
"back-end-of-the-line 3D integration",
"neuromorphic computing applicat... | [
{
"idx": "1.",
"title": "Chakrabartty, S. & Cauwenberghs, G. Sub-microwatt analog VLSI trainable pattern classifier.IEEE J. Solid-State Circuits42, 1169–1179 (2007).",
"link": "https://doi.org/10.1109%2FJSSC.2007.894803"
},
{
"idx": "2.",
"title": "Ramakrishnan, S. & Hasler, J. Vector-matrix... |
s41467-022-32482-y | Energy science and technology | Moisture-triggered fast crystallization enables efficient and stable perovskite solar cells | Understanding the function of moisture on perovskite is challenging since the random environmental moisture strongly disturbs the perovskite structure. Here, we develop various N 2 -protected characterization techniques to comprehensively study the effect of moisture on the efficient cesium, methylammonium, and formami... | [
{
"section": "Introduction",
"content": "Depositing high-quality perovskite films with few defects plays a crucial role in achieving efficient and stable perovskite solar cells (PSCs) [1] , [2] , [3] , [4] , [5] , [6] , [7] . Perovskite bulk doping and interface modification have been demonstrated to be eff... | The core research idea is to investigate the direct effect of moisture (H₂O) on the intermediate and final perovskite films with the composition (Cs₀.₀₅FA₀.₇₅MA₀.₂₀)Pb(I₀.₉₆Br₀.₀₄)₃ by systematically controlling humidity during film processing, in order to decouple the role of water from other environmental factors and... | [
"moisture",
"H₂O",
"perovskite films",
"Cs₀.₀₅FA₀.₇₅MA₀.₂₀Pb(I₀.₉₆Br₀.₀₄)₃",
"humidity control",
"film processing",
"environmental factors",
"perovskite crystallization",
"defects",
"efficiency",
"stability",
"perovskite solar cells"
] | [
{
"idx": "1.",
"title": "Kojima, A., Teshima, K., Shirai, Y. & Miyasaka, T. Organometal halide perovskites as visible-light sensitizers for photovoltaic cells.J. Am. Chem. Soc.131, 6050–6051 (2009).",
"link": "https://doi.org/10.1021%2Fja809598r"
},
{
"idx": "2.",
"title": "Burschka, J. et a... |
s41467-021-22601-6 | Physics | Estimating the effect of social inequalities on the mitigation of COVID-19 across communities in Santiago de Chile | We study the spatio-temporal spread of SARS-CoV-2 in Santiago de Chile using anonymized mobile phone data from 1.4 million users, 22% of the whole population in the area, characterizing the effects of non-pharmaceutical interventions (NPIs) on the epidemic dynamics. We integrate these data into a mechanistic epidemic m... | [
{
"section": "Introduction",
"content": "As of September 1, 2020, Chile has reported more than 400,000 cases and 590 SARS-CoV-2 deaths per million, becoming one of the worst COVID-19 epidemic globally [1] . Officially, the first SARS-CoV-2 case in Chile was detected on March 3, 2020 [2] . Although other cas... | This research proposes to model the spatial and temporal spread of SARS-CoV-2 in 37 comunas of Santiago, Chile, using mobility and contact data inferred from mobile devices within a stochastic, age-structured epidemic model to quantify the impact of nonpharmaceutical interventions—particularly the full lockdown—on the ... | [
"SARS-CoV-2",
"spatial and temporal spread",
"37 comunas of Santiago",
"Chile",
"mobility data",
"contact data",
"mobile devices",
"stochastic age-structured epidemic model",
"nonpharmaceutical interventions",
"full lockdown",
"epidemic progression",
"socioeconomic disparities",
"pandemic bu... | [
{
"idx": "4.",
"title": "Aleta, A. et al. Modelling the impact of testing, contact tracing and household quarantine on second waves of COVID-19.Nat. Hum. Behav.4, 964–971 (2020).",
"link": "https://doi.org/10.1038%2Fs41562-020-0931-9"
},
{
"idx": "7.",
"title": "Goscé, L., Phillips, P. A., S... |
ncomms1283 | Materials science | A monolithically integrated plasmonic infrared quantum dot camera | In the past few years, there has been increasing interest in surface plasmon-polaritons, as a result of the strong near-field enhancement of the electric fields at a metal–dielectric interface. Here we show the first demonstration of a monolithically integrated plasmonic focal plane array (FPA) in the mid-infrared regi... | [
{
"section": "Introduction",
"content": "Infrared imagers operating in the 3–12 μm range are useful for a variety of applications ranging from night vision to non-invasive medical diagnostics [1] . The past decade has witnessed dramatic advances in infrared imaging technology with the introduction of new ma... | The core research idea is to develop a monolithically integrated plasmonic infrared camera that leverages near-field spectrally resonant coupling between surface plasmon-polariton structures and quantum dot infrared detectors to enable biomimetic imaging with enhanced spectral and polarization sensitivity, while mainta... | [
"monolithically integrated plasmonic infrared camera",
"near-field spectrally resonant coupling",
"surface plasmon-polariton structures",
"quantum dot infrared detectors",
"biomimetic imaging",
"enhanced spectral sensitivity",
"enhanced polarization sensitivity",
"standard focal plane array fabricatio... | [
{
"idx": "1",
"title": "Hutchinson, L. Imaging: digital infrared breast scan shows promise for detecting cancer.Nat. Rev. Clin. Oncol.7, 483 (2010).",
"link": "https://doi.org/10.1038%2Fnrclinonc.2010.132"
},
{
"idx": "2",
"title": "Krishna, S., Gunapala, S. D., Bandara, S. V., Hill, C. & Ti... |
s41467-022-32007-7 | Mathematics and computing | ProtGPT2 is a deep unsupervised language model for protein design | Protein design aims to build novel proteins customized for specific purposes, thereby holding the potential to tackle many environmental and biomedical problems. Recent progress in Transformer-based architectures has enabled the implementation of language models capable of generating text with human-like capabilities. ... | [
{
"section": "Introduction",
"content": "Natural language processing (NLP) has seen extraordinary advances in recent years. Large pre-trained language models have drastically transformed the NLP field and with it, many of the tools we use in our daily lives, such as chatbots, smart assistants, or translatio... | The core research idea is to develop ProtGPT2, an autoregressive Transformer model with 738 million parameters, trained on approximately 50 million non-annotated protein sequences, to generate de novo protein sequences that are structurally and functionally comparable to natural proteins while exploring underrepresente... | [
"ProtGPT2",
"autoregressive Transformer model",
"738 million parameters",
"50 million non-annotated protein sequences",
"de novo protein sequences",
"structurally comparable proteins",
"functionally comparable proteins",
"underrepresented regions of the protein space",
"efficient high-throughput pro... | [
{
"idx": "1.",
"title": "Yang, K. K., Wu, Z. & Arnold, F. H. Machine-learning-guided directed evolution for protein engineering.Nat. Methods16, 687–694 (2019).",
"link": "https://doi.org/10.1038%2Fs41592-019-0496-6"
},
{
"idx": "2.",
"title": "Ferruz, N. & Höcker, B. Controllable protein des... |
s41467-021-23005-2 | Materials science | TRPM7 kinase-mediated immunomodulation in macrophage plays a central role in magnesium ion-induced bone regeneration | Despite the widespread observations on the osteogenic effects of magnesium ion (Mg 2+ ), the diverse roles of Mg 2+ during bone healing have not been systematically dissected. Here, we reveal a previously unknown, biphasic mode of action of Mg 2+ in bone repair. During the early inflammation phase, Mg 2+ contributes to... | [
{
"section": "Introduction",
"content": "Bone tissue has a substantial capacity for repair and regeneration after injury or surgical treatment. However, the natural healing of bone can be a slow process that often fails to restore the bone to its original strength and structure [1] . Thus, clinical interven... | This study systematically investigates the dose-dependent and time-dependent immunomodulatory effects of magnesium ions (Mg²⁺) on the monocyte-macrophage-osteoblast axis during bone healing, revealing that Mg²⁺ creates a pro-osteogenic immune microenvironment via TRPM7 signaling in the early phase and influences bone r... | [
"dose-dependent immunomodulatory effects",
"time-dependent immunomodulatory effects",
"magnesium ions",
"Mg²⁺",
"monocyte-macrophage-osteoblast axis",
"bone healing",
"pro-osteogenic immune microenvironment",
"TRPM7 signaling",
"NF-κB activation",
"extracellular matrix mineralization inhibition",
... | [
{
"idx": "1.",
"title": "Gong, T. et al. Nanomaterials and bone regeneration.Bone Res.3, 15029 (2015).",
"link": "https://doi.org/10.1038%2Fboneres.2015.29"
},
{
"idx": "2.",
"title": "Rude, R. K., Gruber, H. E., Wei, L. Y., Frausto, A. & Mills, B. G. Magnesium deficiency: effect on bone and... |
s41467-018-07293-9 | Energy science and technology | Passive directional sub-ambient daytime radiative cooling | Demonstrations of passive daytime radiative cooling have primarily relied on complex and costly spectrally selective nanophotonic structures with high emissivity in the transparent atmospheric spectral window and high reflectivity in the solar spectrum. Here, we show a directional approach to passive radiative cooling ... | [
{
"section": "Introduction",
"content": "Cooling technologies are essential for refrigeration and thermal management applications. Existing cooling processes primarily rely on vapor compression and fluid-cooled systems despite their complexity and high cost. Passive cooling approaches such as atmospheric ra... | This research proposes a directional approach to achieve sub-ambient passive atmospheric radiative cooling during the day by exploiting the angular confinement of solar flux—completely blocking radiative exchange in the direct-solar direction while allowing energy transfer in other directions—to enable significant cool... | [
"directional approach",
"sub-ambient passive atmospheric radiative cooling",
"angular confinement",
"solar flux",
"direct-solar direction",
"radiative exchange",
"reflective emitters",
"absorptive emitters",
"high solar irradiation",
"energy transfer"
] | [
{
"idx": "1.",
"title": "Florides, G. A., Tassou, S. A., Kalogirou, S. A. & Wrobel, L. C. Review of solar and low energy cooling technologies for buildings.Renew. Sustain. Energy Rev.6, 557–572 (2002).",
"link": "https://doi.org/10.1016%2FS1364-0321%2802%2900016-3"
},
{
"idx": "2.",
"title":... |
s41467-019-09924-1 | Energy science and technology | Fast lithium growth and short circuit induced by localized-temperature hotspots in lithium batteries | Fast-charging and high-energy-density batteries pose significant safety concerns due to high rates of heat generation. Understanding how localized high temperatures affect the battery is critical but remains challenging, mainly due to the difficulty of probing battery internal temperature with high spatial resolution. ... | [
{
"section": "Introduction",
"content": "Energy storage with batteries is critical to enable renewable energy technologies and environmental sustainability [1] , [2] . Significant progress has been made on the development of rechargeable lithium-based batteries in recent decades [3] , [4] , [5] , [6] . Howe... | This research investigates how localized temperature hotspots within lithium metal batteries influence lithium growth dynamics and proposes a temperature-induced battery shorting mechanism as a critical safety concern, utilizing a laser to create internal hotspots and a micro-Raman spectroscopy platform to measure loca... | [
"localized temperature hotspots",
"lithium metal batteries",
"lithium growth dynamics",
"temperature-induced battery shorting mechanism",
"laser",
"internal hotspots",
"micro-Raman spectroscopy platform",
"local temperature effects",
"accelerated lithium deposition rates",
"thermal management stra... | [
{
"idx": "1.",
"title": "Chu, S., Cui, Y. & Liu, N. The path towards sustainable energy.Nat. Mater.16, 16–22 (2017).",
"link": "https://doi.org/10.1038%2Fnmat4834"
},
{
"idx": "2.",
"title": "Dunn, B., Kamath, H. & Tarascon, J.-M. Electrical energy storage for the grid: a battery of choices.... |
s41467-020-20451-2 | Astronomy and planetary science | Sulfur monoxide dimer chemistry as a possible source of polysulfur in the upper atmosphere of Venus | The abundance of SO dimers (SO) 2 in the upper atmosphere of Venus and their implications for the enigmatic ultraviolet absorption has been investigated in several studies over the past few years. However, the photochemistry of sulfur species in the upper atmosphere of Venus is still not well understood and the identit... | [
{
"section": "Introduction",
"content": "The photochemistry of sulfur species in the upper atmosphere of Venus is not well understood and is poorly quantified. The known parent molecules upwelling from the deep atmosphere are SO 2 and OCS. Near and above the cloud tops, these molecules are photolyzed, leadi... | The core research idea is to investigate whether the photochemistry of sulfur monoxide (SO) dimers, particularly their photolysis and reaction pathways, can serve as a significant source of S₂ and more complex sulfur species, such as polysulfur and polysulfur oxides, which may be responsible for the unexplained ultravi... | [
"photochemistry",
"sulfur monoxide dimers",
"photolysis",
"reaction pathways",
"S₂",
"polysulfur",
"polysulfur oxides",
"ultraviolet absorption",
"Venus's atmosphere",
"photochemical model",
"laboratory studies",
"Venus Express mission"
] | [
{
"idx": "2.",
"title": "Krasnopolsky, V. A. Sulfur aerosol in the clouds of Venus.Icarus274, 33–36 (2016).",
"link": "https://doi.org/10.1016%2Fj.icarus.2016.03.010"
},
{
"idx": "3.",
"title": "Krasnopolsky, V. A. Disulfur dioxide and its near-UV absorption in the photochemical model of Ven... |
s41467-021-22520-6 | Nanoscience and technology | Nanoscale magnonic Fabry-Pérot resonator for low-loss spin-wave manipulation | Active control of propagating spin waves on the nanoscale is essential for beyond-CMOS magnonic computing. Here, we experimentally demonstrate reconfigurable spin-wave transport in a hybrid YIG-based material structure that operates as a Fabry-Pérot nanoresonator. The magnonic resonator is formed by a local frequency d... | [
{
"section": "Introduction",
"content": "A Fabry–Pérot resonator (or interferometer) is an optical cavity made of two parallel reflecting surfaces. Optical waves transmit through the resonator when incoming waves interfere constructively with circulating waves inside the cavity, whereas destructive interfer... | The core research idea is to introduce a magnonic Fabry–Pérot nanoresonator that enables versatile manipulation of low-loss spin-wave transport through local dynamic dipolar coupling between a ferromagnetic metal stripe and a nanometer-thick YIG film, producing interference effects that suppress transmission at discret... | [
"magnonic Fabry–Pérot nanoresonator",
"low-loss spin-wave transport",
"local dynamic dipolar coupling",
"ferromagnetic metal stripe",
"nanometer-thick YIG film",
"interference effects",
"transmission suppression",
"discrete frequencies",
"micrometer-long spin waves",
"subwavelength nanostripes"
] | [
{
"idx": "3.",
"title": "Khitun, A., Bao, M. & Wang, K. L. Magnonic logic circuits.J. Phys. D: Appl. Phys.43, 264005 (2010).",
"link": "https://doi.org/10.1088%2F0022-3727%2F43%2F26%2F264005"
},
{
"idx": "4.",
"title": "Chumak, A. V., Vasyuchka, V. I., Serga, A. A. & Hillebrands, B. Magnon s... |
s41467-021-24571-1 | Materials science | Large-area synthesis of nanoscopic catalyst-decorated conductive MOF film using microfluidic-based solution shearing | Conductive metal-organic framework (C-MOF) thin-films have a wide variety of potential applications in the field of electronics, sensors, and energy devices. The immobilization of various functional species within the pores of C-MOFs can further improve the performance and extend the potential applications of C-MOFs th... | [
{
"section": "Introduction",
"content": "Conductive metal–organic frameworks (C-MOFs) are emerging materials receiving a great deal of interest in recent years due to their many attractive properties, such as high porosity, narrow pore-size distribution and periodically organized pores, adjustable bandgap, ... | This research proposes the use of microfluidic channel-embedded solution shearing (MiCS) to simultaneously synthesize high-quality, nanoscale conductive metal–organic framework (C-MOF) thin films and immobilize metal nanoparticles within their pores during film formation, enabling precise thickness control, large-area ... | [
"microfluidic channel-embedded solution shearing",
"MiCS",
"high-quality nanoscale conductive metal–organic framework",
"C-MOF thin films",
"metal nanoparticles",
"film formation",
"precise thickness control",
"large-area scalability",
"enhanced catalytic performance",
"batteries",
"sensors"
] | [
{
"idx": "1.",
"title": "Xie, L. S., Skorupskii, G. & Dincă, M. Electrically conductive metal–organic frameworks.Chem. Rev.120, 8536–8580 (2020).",
"link": "https://doi.org/10.1021%2Facs.chemrev.9b00766"
},
{
"idx": "2.",
"title": "Sun, L., Campbell, M. G. & Dinca, M. Electrically conductive... |
ncomms15701 | Materials science | Universal emulsion stabilization from the arrested adsorption of rough particles at liquid-liquid interfaces | Surface heterogeneities, including roughness, significantly affect the adsorption, motion and interactions of particles at fluid interfaces. However, a systematic experimental study, linking surface roughness to particle wettability at a microscopic level, is currently missing. Here we synthesize a library of all-silic... | [
{
"section": "Introduction",
"content": "Micro- and nanoparticles adsorbed at fluid interfaces are an essential ingredient of a very broad class of materials and processes, from stabilizing emulsions and foams [1] , [2] , [3] , to the fabrication of nanostructured materials [4] , [5] . Understanding the sin... | The core research idea is to investigate how surface roughness affects the wetting behavior and adsorption dynamics of colloidal particles at oil–water interfaces, with a focus on understanding how roughness-induced contact line pinning leads to metastable particle orientations and interfacial deformations that can be ... | [
"surface roughness",
"wetting behavior",
"adsorption dynamics",
"colloidal particles",
"oil–water interfaces",
"contact line pinning",
"metastable particle orientations",
"interfacial deformations",
"Pickering emulsifiers",
"oil-in-water emulsions",
"water-in-oil emulsions"
] | [] |
s41467-024-47788-2 | Engineering | Snail-inspired robotic swarms: a hybrid connector drives collective adaptation in unstructured outdoor environments | Terrestrial self-reconfigurable robot swarms offer adaptable solutions for various tasks. However, most existing swarms are limited to controlled indoor settings, and often compromise stability due to their freeform connections. To address these issues, we present a snail robotic swarm system inspired by land snails, t... | [
{
"section": "Introduction",
"content": "Robot swarms draw inspiration from natural swarm systems observed in various species, such as fish [1] , [2] , [3] , insects [4] , [5] , [6] , and birds [7] . In nature, these organisms exhibit collective behaviors that enable them to accomplish tasks that are far be... | This research proposes the development of a 3D freeform self-reconfigurable snail-inspired robot swarm with a dual-mode connection mechanism—combining magnetic adhesion and vacuum suction—to enable efficient task execution in unstructured outdoor environments by overcoming terrain challenges and achieving both high mob... | [
"3D freeform self-reconfigurable snail-inspired robot swarm",
"dual-mode connection mechanism",
"magnetic adhesion",
"vacuum suction",
"unstructured outdoor environments",
"terrain challenges",
"high mobility",
"robust interconnectivity",
"efficient task execution"
] | [
{
"idx": "1.",
"title": "Berlinger, F., Gauci, M. & Nagpal, R. Implicit coordination for 3d underwater collective behaviors in a fish-inspired robot swarm.Sci. Robot.6, eabd8668 (2021).",
"link": "https://doi.org/10.1126%2Fscirobotics.abd8668"
},
{
"idx": "2.",
"title": "Hemelrijk, C. K. & H... |
s41467-018-04276-8 | Nanoscience and technology | Controlling evanescent waves using silicon photonic all-dielectric metamaterials for dense integration | Ultra-compact, densely integrated optical components manufactured on a CMOS-foundry platform are highly desirable for optical information processing and electronic-photonic co-integration. However, the large spatial extent of evanescent waves arising from nanoscale confinement, ubiquitous in silicon photonic devices, c... | [
{
"section": "Introduction",
"content": "One of the long-standing goals of nanophotonics is the integration of electronic and photonic circuitry on a single CMOS chip for applications ranging from information processing and data centers to massively parallel sensing [1] , [2] , [3] , [4] , [5] , [6] , [7] ,... | The core research idea is to develop a novel, scalable, and low-loss photonic integration platform compatible with CMOS technology by utilizing all-dielectric anisotropic metamaterials as a cladding to engineer photonic skin-depth, thereby enabling ultra-compact waveguides with significantly reduced cross-talk and bend... | [
"photonic integration platform",
"CMOS technology",
"all-dielectric anisotropic metamaterials",
"photonic skin-depth",
"ultra-compact waveguides",
"cross-talk reduction",
"bending loss reduction",
"light confinement mechanism",
"relaxed total internal reflection",
"evanescent wave control"
] | [
{
"idx": "1.",
"title": "Jalali, B. & Fathpour, S. Silicon photonics.J. Light Technol.24, 4600–4615 (2006).",
"link": "https://doi.org/10.1109%2FJLT.2006.885782"
},
{
"idx": "2.",
"title": "Bogaerts, W. et al. Nanophotonic waveguides in silicon-on-insulator fabricated with CMOS technology.J.... |
s41467-021-21137-z | Optics and photonics | Photo thermal effect graphene detector featuring 105 Gbit s−1NRZ and 120 Gbit s−1PAM4 direct detection | One of the main challenges of next generation optical communication is to increase the available bandwidth while reducing the size, cost and power consumption of photonic integrated circuits. Graphene has been recently proposed to be integrated with silicon photonics to meet these goals because of its high mobility, fa... | [
{
"section": "Introduction",
"content": "The global number of devices connected to the IP network is growing at a compound annual growth rate (CAGR) of 10%, and is expected to exceed 28 billion connected devices by 2023 [1] , [2] . The speed of these connections will also increase, with 5G devices expected ... | This research proposes that integrating a 50 Ω-matched photo-thermoelectric (PTE) graphene photodetector on a silicon photonic waveguide, which operates without dark current and achieves a voltage responsivity of 3.5 V W⁻¹ and a 3 dB bandwidth of 70 GHz, can enable high-speed optical data detection beyond the performan... | [
"50 Ω-matched photo-thermoelectric graphene photodetector",
"silicon photonic waveguide",
"dark current",
"voltage responsivity",
"3.5 V W⁻¹",
"3 dB bandwidth",
"70 GHz",
"germanium-based detectors",
"105 GBaud NRZ",
"120 Gb s⁻¹",
"PAM4 modulation",
"high-bandwidth photonic transceivers",
"l... | [
{
"idx": "4.",
"title": "Saad, W., Bennis, M. & Chen, M. A vision of 6G wireless systems: applications, trends, technologies, and open research problems.IEEE Netw.34, 134–142 (2020).",
"link": "https://doi.org/10.1109%2FMNET.001.1900287"
},
{
"idx": "9.",
"title": "Witzens, J. High-speed sil... |
ncomms11524 | Physics | Quantum annealing for the number-partitioning problem using a tunable spin glass of ions | Exploiting quantum properties to outperform classical ways of information processing is an outstanding goal of modern physics. A promising route is quantum simulation, which aims at implementing relevant and computationally hard problems in controllable quantum systems. Here we demonstrate that in a trapped ion setup, ... | [
{
"section": "Introduction",
"content": "Spin models are paradigms of multidisciplinary science: They find several applications in various fields of physics, from condensed matter to high-energy physics, but also beyond the physical sciences. In neuroscience, the famous Hopfield model describes brain functi... | The core research idea is to demonstrate how trapped ions, described by a time-dependent Dicke-like model and reducible to an effective Ising-type spin Hamiltonian with long-range, pattern-imprinted couplings, can be used as a quantum annealer to solve classical spin glass models and potentially address NP-hard optimiz... | [
"trapped ions",
"time-dependent Dicke-like model",
"effective Ising-type spin Hamiltonian",
"long-range couplings",
"pattern-imprinted couplings",
"quantum annealer",
"classical spin glass models",
"NP-hard optimization problems",
"number partitioning",
"adiabatic state preparation",
"quantum si... | [
{
"idx": "1",
"title": "Hopfield, J. J. Neural networks and physical systems with emergent collective computational abilities.Proc. Natl Acad. Sci. USA79, 2554–2558 (1982).",
"link": "https://doi.org/10.1073%2Fpnas.79.8.2554"
},
{
"idx": "3",
"title": "Barahona, F. On the computational compl... |
s41467-023-42516-8 | Materials science | Recastable assemblies of carbon dots into mechanically robust macroscopic materials | Assembly of nanoparticles into macroscopic materials with mechanical robustness, green processability, and recastable ability is an important and challenging task in materials science and nanotechnology. As an emerging nanoparticle with superior properties, macroscopic materials assembled from carbon dots will inherit ... | [
{
"section": "Introduction",
"content": "Self-assembly of nanoparticles into macroscopic materials is an important target of materials science and nanotechnology, since it paves an avenue towards innovative materials with functions through collective couplings of the optical, electrical, and mechanical prop... | The core research idea is to develop a recastable, self-healing, and mechanically robust macroscopic material through the self-assembly of carbon dots (CDs) modified with ureido pyrimidinone (UPy) groups, leveraging quadruple hydrogen bonding interactions to enhance interparticle associations while preserving the photo... | [
"recastable material",
"self-healing material",
"mechanically robust material",
"macroscopic material",
"self-assembly",
"carbon dots",
"ureido pyrimidinone groups",
"quadruple hydrogen bonding interactions",
"interparticle associations",
"photoluminescent properties",
"collective mechanical per... | [
{
"idx": "1.",
"title": "Begley, M. R., Gianola, D. S. & Ray, T. R. Bridging functional nanocomposites to robust macroscale devices.Science364, eaav4299 (2019).",
"link": "https://doi.org/10.1126%2Fscience.aav4299"
},
{
"idx": "2.",
"title": "Lee, M. S., Yee, D. W., Ye, M. & Macfarlane, R. J... |
ncomms9616 | Materials science | Water desalination with a single-layer MoS2nanopore | Efficient desalination of water continues to be a problem facing the society. Advances in nanotechnology have led to the development of a variety of nanoporous membranes for water purification. Here we show, by performing molecular dynamics simulations, that a nanopore in a single-layer molybdenum disulfide can effecti... | [
{
"section": "Introduction",
"content": "Producing fresh water is currently a great challenge facing the society [1] , [2] , [3] , [4] . High capital costs and low efficiency of current desalination technology motivate the need for advances in desalination technology [5] , [6] . Approximately, half of the c... | The core research idea is to investigate the effectiveness of a single-layer molybdenum disulfide (MoS₂) membrane with precisely engineered nanopores in achieving efficient water desalination by examining how pore size, chemistry, geometry, and applied hydrostatic pressure influence ion rejection and water permeation u... | [
"single-layer molybdenum disulfide membrane",
"nanopores",
"water desalination",
"pore size",
"pore chemistry",
"pore geometry",
"hydrostatic pressure",
"ion rejection",
"water permeation",
"molecular dynamics simulations"
] | [
{
"idx": "1",
"title": "Elimelech, M. & Phillip, W. A. The future of seawater desalination: energy, technology, and the environment.Science333, 712–717 (2011).",
"link": "https://doi.org/10.1126%2Fscience.1200488"
},
{
"idx": "2",
"title": "Zhao, S. F., Zou, L., Tang, C. Y. Y. & Mulcahy, D. ... |
ncomms2481 | Materials science | Aromatic porous-honeycomb electrodes for a sodium-organic energy storage device | Rechargeable batteries using organic electrodes and sodium as a charge carrier can be high-performance, affordable energy storage devices due to the abundance of both sodium and organic materials. However, only few organic materials have been found to be active in sodium battery systems. Here we report a high-performan... | [
{
"section": "Introduction",
"content": "Next-generation rechargeable batteries are key devices for the successful development of green technologies [1] , for example, a grid system combined with large-scale storage and renewable energy, and electric vehicles. Currently, rechargeable lithium batteries are t... | The core research idea is to develop bipolar porous polymeric frameworks as a new class of affordable organic cathodes for sodium-based energy storage devices, specifically an aromatic porous-honeycomb cathode, which demonstrates a long cycle life and performance comparable to lithium-ion batteries, aiming to address t... | [
"bipolar porous polymeric frameworks",
"affordable organic cathodes",
"sodium-based energy storage devices",
"aromatic porous-honeycomb cathode",
"long cycle life",
"lithium-ion batteries",
"high-specific energy",
"high-specific power",
"next-generation rechargeable batteries",
"energy storage",
... | [
{
"idx": "1",
"title": "Tarascon, J.-M. . Key challenges in future Li-battery research.Phil. Trans. R. Soc. A368, 3227–3341 (2010) .",
"link": "https://doi.org/10.1098%2Frsta.2010.0112"
},
{
"idx": "2",
"title": "Tarascon, J.-M. . Is lithium new gold?Nat. Chem.2, 510 (2010) .",
"link": "... |
s41467-022-34932-z | Mathematics and computing | GhostKnockoff inference empowers identification of putative causal variants in genome-wide association studies | Recent advances in genome sequencing and imputation technologies provide an exciting opportunity to comprehensively study the contribution of genetic variants to complex phenotypes. However, our ability to translate genetic discoveries into mechanistic insights remains limited at this point. In this paper, we propose a... | [
{
"section": "Introduction",
"content": "Recent advances in genome sequencing technologies and improvement in genotype imputation accuracy enable large-scale genetic studies with hundreds of thousands of samples and tens of millions of variants. The ultimate goal of such studies is to provide a credible set... | The core research idea is to propose a novel method called GhostKnockoff that enables efficient knockoff-based inference using freely available GWAS summary statistics—without requiring individual-level data—to enhance locus discovery, prioritize causal variants, and improve the identification of biologically and clini... | [
"GhostKnockoff",
"knockoff-based inference",
"GWAS summary statistics",
"individual-level data",
"locus discovery",
"causal variants",
"genetic associations",
"false discovery rate control",
"meta-analysis",
"sample overlap"
] | [
{
"idx": "2.",
"title": "Schaid, D. J., Chen, W. & Larson, N. B. From genome-wide associations to candidate causal variants by statistical fine-mapping.Nat. Rev. Genet.19, 491 (2018).",
"link": "https://doi.org/10.1038%2Fs41576-018-0016-z"
},
{
"idx": "3.",
"title": "Boyle, E. A., Li, Y. I. ... |
s41467-019-09170-5 | Energy science and technology | NASICON-type air-stable and all-climate cathode for sodium-ion batteries with low cost and high-power density | The development of low-cost and long-lasting all-climate cathode materials for the sodium ion battery has been one of the key issues for the success of large-scale energy storage. One option is the utilization of earth-abundant elements such as iron. Here, we synthesize a NASICON-type tuneable Na 4 Fe 3 (PO 4 ) 2 (P 2 ... | [
{
"section": "Introduction",
"content": "Cost-efficient large-scale energy storage systems (EESs) have been in high demand in recent years due to the rapid development of renewable energy resources (i.e., solar, wind, geothermal, and tidal energy) [1] , [2] . Although advanced lithium-ion battery (LIBs) tec... | This research proposes the development and investigation of a novel, low-cost, environmentally friendly NASICON-type cathode material, Na₄Fe₃(PO₄)₂(P₂O₇), with a focus on its exceptional structural stability, fast sodium ion diffusion, and outstanding electrochemical performance across a wide range of temperatures, aim... | [
"NASICON-type cathode material",
"Na₄Fe₃(PO₄)₂(P₂O₇)",
"structural stability",
"sodium ion diffusion",
"electrochemical performance",
"low-cost",
"environmentally friendly",
"all-climate operability",
"sodium-ion batteries",
"large-scale energy storage"
] | [
{
"idx": "1.",
"title": "Armand, M. & Tarascon, J.-M. Building better batteries.Nature451, 652–657 (2008).",
"link": "https://doi.org/10.1038%2F451652a"
},
{
"idx": "2.",
"title": "Yabuuchi, N., Kubota, K., Dahbi, M. & Komaba, S. Research development on sodium-ion batteries.Chem. Rev.114, 11... |
s41467-020-16062-6 | Nanoscience and technology | Solvent-mediated assembly of atom-precise gold–silver nanoclusters to semiconducting one-dimensional materials | Bottom-up design of functional device components based on nanometer-sized building blocks relies on accurate control of their self-assembly behavior. Atom-precise metal nanoclusters are well-characterizable building blocks for designing tunable nanomaterials, but it has been challenging to achieve directed assembly to ... | [
{
"section": "Introduction",
"content": "The remarkable progress in the synthesis, structural discovery, functionalization and theoretical understanding of ligand-stabilized, atom-precise metal nanoclusters has opened fascinating opportunities to use these well-defined nanometer-size building blocks for des... | The core research idea is to demonstrate a solvent-mediated assembly of 1-ethynyladamantane-protected, intermetallic 34-atom Au–Ag clusters into 1D cluster polymers connected by direct metal–metal bonds, forming macroscopic single crystals with anisotropic semiconducting properties that exhibit p-type behavior, highly ... | [
"solvent-mediated assembly",
"1-ethynyladamantane-protected",
"intermetallic 34-atom Au–Ag clusters",
"1D cluster polymers",
"direct metal–metal bonds",
"macroscopic single crystals",
"anisotropic semiconducting properties",
"p-type behavior",
"highly anisotropic electrical conductivity",
"field-e... | [] |
s41467-021-23103-1 | Mathematics and computing | Spectral bias and task-model alignment explain generalization in kernel regression and infinitely wide neural networks | A theoretical understanding of generalization remains an open problem for many machine learning models, including deep networks where overparameterization leads to better performance, contradicting the conventional wisdom from classical statistics. Here, we investigate generalization error for kernel regression, which,... | [
{
"section": "Introduction",
"content": "Learning machines aim to find statistical patterns in data that generalize to previously unseen samples [1] . How well they perform in doing so depends on factors such as the size and the nature of the training data set, the complexity of the learning task, and the i... | The core research idea of this work is to develop a theory of generalization in kernel regression—applicable to real datasets and motivated by its correspondence to deep learning—that analytically predicts generalization performance based on the interplay between the training sample size, the spectral properties of the... | [
"kernel regression",
"generalization performance",
"training sample size",
"spectral properties",
"kernel",
"target function",
"kernel's eigenfunctions",
"inductive bias",
"overparameterized models",
"spectral bias",
"non-monotonic generalization error",
"double-descent phenomenon"
] | [
{
"idx": "3.",
"title": "Belkin, M., Hsu, D., Ma, S. & Mandal, S. Reconciling modern machine-learning practice and the classical bias–variance trade-off.Proc. Natl Acad. Sci.116, 15849–15854 (2019).",
"link": "https://doi.org/10.1073%2Fpnas.1903070116"
},
{
"idx": "4.",
"title": "LeCun, Y., ... |
ncomms2965 | Nanoscience and technology | Integrating DNA strand-displacement circuitry with DNA tile self-assembly | DNA nanotechnology has emerged as a reliable and programmable way of controlling matter at the nanoscale through the specificity of Watson–Crick base pairing, allowing both complex self-assembled structures with nanometer precision and complex reaction networks implementing digital and analog behaviors. Here we show ho... | [
{
"section": "Introduction",
"content": "Biological cells are self-organized information-based molecular systems that acquire material and information, make molecules and decisions, and take actions that regulate their internal functions and their interactions with the environment [1] . The mechanisms under... | The core research idea is to integrate DNA strand-displacement circuits with DNA tile self-assembly to achieve autonomous isothermal control over the timing and composition of nanoscale structure fabrication, specifically demonstrating that an upstream entropy-driven catalyst circuit can release a signal molecule to ac... | [
"DNA strand-displacement circuits",
"DNA tile self-assembly",
"autonomous isothermal control",
"nanoscale structure fabrication",
"entropy-driven catalyst circuit",
"signal molecule",
"downstream polymerization",
"double-crossover tiles",
"DNA nanotubes",
"programmable molecular construction",
"... | [
{
"idx": "1",
"title": "Mann, S. Life as a nanoscale phenomenon.Angew. Chem. Int. Ed.47, 5306–5320 (2008).",
"link": "https://doi.org/10.1002%2Fanie.200705538"
},
{
"idx": "2",
"title": "Leiman, P. G. Kanamaru, S. Mesyanzhinov, V. V. Arsaka, F. & Rossman, M. G. Structure and morphogenesis of... |
ncomms13966 | Optics and photonics | Shifting molecular localization by plasmonic coupling in a single-molecule mirage | Over the last decade, two fields have dominated the attention of sub-diffraction photonics research: plasmonics and fluorescence nanoscopy. Nanoscopy based on single-molecule localization offers a practical way to explore plasmonic interactions with nanometre resolution. However, this seemingly straightforward techniqu... | [
{
"section": "Introduction",
"content": "Plasmonics makes use of surface plasmon polaritons confined to nanometrically structured metals in order to control optical fields and interactions at the nanoscale [1] , [2] . Locally intensified optical fields around metallic nanoparticles have enabled highly sensi... | The core research idea is to investigate how plasmonic coupling between single emitters and gold nanoparticles induces far-field emission modulations that result in significant localization inaccuracies, akin to a "single-molecule mirage," by integrating DNA origami-based nanosystem fabrication, 3D single-molecule loca... | [
"plasmonic coupling",
"single emitters",
"gold nanoparticles",
"far-field emission modulations",
"localization inaccuracies",
"single-molecule mirage",
"DNA origami-based nanosystem fabrication",
"3D single-molecule localization nanoscopy",
"FDFD full-field simulations",
"nanoscale optical imaging... | [
{
"idx": "3",
"title": "Kinkhabwala, A. A., Yu, Z., Fan, S. & Moerner, W. E. Fluorescence correlation spectroscopy at high concentrations using gold bowtie nanoantennas.Chem. Phys.406, 3–8 (2012).",
"link": "https://doi.org/10.1016%2Fj.chemphys.2012.04.011"
},
{
"idx": "4",
"title": "Acuna, ... |
s41467-024-54021-7 | Physics | Tuning the BCS-BEC crossover of electron-hole pairing with pressure | In graphite, a moderate magnetic field confines electrons and holes into their lowest Landau levels. In the extreme quantum limit, two insulating states with a dome-like field dependence of the their critical temperatures are induced by the magnetic field. Here, we study the evolution of the first dome (below 60 T) und... | [
{
"section": "Introduction",
"content": "In 1961, Mott made the observation that Coulomb attraction between electrons and holes of a semi-metal can form bound pairs known as excitons [1] . Knox then proposed that a sufficiently large exciton binding energy would lead to an insulating state, quite distinct f... | The core research idea is to investigate the evolution of the phase diagram and Fermi surface of graphite under high magnetic field and hydrostatic pressure in order to demonstrate that the insulating state observed within a dome-shaped region in the (field, temperature) plane is characterized by a BCS-BEC crossover, w... | [
"phase diagram",
"Fermi surface",
"graphite",
"high magnetic field",
"hydrostatic pressure",
"insulating state",
"dome-shaped region",
"BCS-BEC crossover",
"weak-coupling BCS behavior",
"density of states",
"BEC transition temperature"
] | [
{
"idx": "1.",
"title": "Mott, N. F. The transition to the metallic state.Philos. Mag.6, 287 (1961).",
"link": "https://doi.org/10.1080%2F14786436108243318"
},
{
"idx": "5.",
"title": "Jérome, D., Rice, T. M. & Kohn, W. Excitonic insulator.Phys. Rev.158, 462 (1967).",
"link": "https://do... |
s41467-018-03758-z | Mathematics and computing | Instability of expanding bacterial droplets | Suspensions of motile bacteria or synthetic microswimmers, termed active matter, manifest a remarkable propensity for self-organization, and formation of large-scale coherent structures. Most active matter research deals with almost homogeneous in space systems and little is known about the dynamics of strongly heterog... | [
{
"section": "Introduction",
"content": "Collective motion and self-organization of motile suspensions is active topic of research [1] , [2] . Recent studies uncovered many salient features of active matter, both living [3] , [4] , [5] , [6] , [7] and synthetic [8] , [9] , [10] . Various theoretical approac... | The core research idea is to investigate the instability dynamics of highly concentrated bacterial droplets formed by rapid rotation in a suspension, where the cessation of rotation leads to a violent explosion driven by self-propulsion and hydrodynamic interactions, and to characterize this phenomenon experimentally a... | [
"instability dynamics",
"highly concentrated bacterial droplets",
"rapid rotation",
"suspension",
"cessation of rotation",
"violent explosion",
"self-propulsion",
"hydrodynamic interactions",
"generalized swim pressure model",
"rod-like particles",
"small Péclet number",
"experimental characte... | [
{
"idx": "1.",
"title": "Ramaswamy, S. The mechanics and statistics of active matter.Ann. Rev. Condens Matter Phys.1, 323–345 (2010).",
"link": "https://doi.org/10.1146%2Fannurev-conmatphys-070909-104101"
},
{
"idx": "2.",
"title": "Vicsek, T. & Zafeiris, A. Collective motion.Phys. Rep.517, ... |
s41467-021-23366-8 | Energy science and technology | Sub-micro droplet reactors for green synthesis of Li3VO4anode materials in lithium ion batteries | The conventional solid-state reaction suffers from low diffusivity, high energy consumption, and uncontrolled morphology. These limitations are competed by the presence of water in solution route reaction. Herein, based on concept of combining above methods, we report a facile solid-state reaction conducted in water va... | [
{
"section": "Introduction",
"content": "Since the first commercial products of Sony in 1991, lithium-ion batteries (LIBs) and other potentially replaceable metal-ion storage sources such as Na + ion [1] , [2] , Mg 2+ ion [3] , and Al 3+ ion [4] batteries, have attracted much attention for developing system... | This study proposes a novel and environmentally friendly strategy that combines acid–base reaction (ABR) synthesis in a humid atmosphere with calcium (Ca²⁺) doping to overcome the limitations of conventional solid-state reaction methods, aiming to control the morphology and particle size of Li₃VO₄, enhance its electron... | [
"acid–base reaction synthesis",
"humid atmosphere",
"calcium doping",
"solid-state reaction methods",
"Li₃VO₄",
"electronic conductivity",
"electrochemical performance",
"lithium-ion batteries",
"sodium-ion batteries",
"battery electrodes",
"scalable fabrication",
"energy-efficient fabrication... | [
{
"idx": "2.",
"title": "Gong, S., Lee, J. & Kim, H. S. Development of electrode architecture using Sb–rGO composite and CMC binder for high-performance sodium-ion battery anodes.J. Korean Ceram Soc.57, 91–97 (2020).",
"link": "https://link.springer.com/doi/10.1007/s43207-019-00012-0"
},
{
"idx"... |
s41467-020-16206-8 | Optics and photonics | A self-operating broadband spectrometer on a droplet | Small-scale Fourier transform spectrometers are rapidly revolutionizing infrared spectro-chemical analysis, enabling on-site and remote sensing applications that were hardly imaginable just few years ago. While most devices reported to date rely on advanced photonic integration technologies, here we demonstrate a minia... | [
{
"section": "Introduction",
"content": "Detecting the spectral power distribution of electromagnetic radiation and how that is modified by the interaction with matter is one of the founding techniques of experimental science. Uncountable scientific and industrial applications require to identify/quantify s... | The core research idea is to demonstrate that a droplet evaporating on the tip of a single-mode optical fiber can function as a self-operating, miniaturized Fourier Transform spectrometer by utilizing the spontaneous displacement of the droplet's surface during evaporation as a mechanical drive for phase-scanning inter... | [
"droplet",
"single-mode optical fiber",
"Fourier Transform spectrometer",
"spontaneous displacement",
"droplet's surface",
"evaporation",
"mechanical drive",
"phase-scanning interferometry",
"spectral distribution",
"incident radiation",
"robustness",
"simplicity",
"disposability",
"integr... | [
{
"idx": "1.",
"title": "Yang, Z. et al. Single-nanowire spectrometers.Science365, 1017–1020 (2019).",
"link": "https://doi.org/10.1126%2Fscience.aax8814"
},
{
"idx": "2.",
"title": "Bao, J. & Bawendi, M. G. A colloidal quantum dot spectrometer.Nature523, 67–70 (2015).",
"link": "https:/... |
s41467-017-00582-9 | Energy science and technology | Ultrathin high band gap solar cells with improved efficiencies from the world’s oldest photovoltaic material | Selenium was used in the first solid state solar cell in 1883 and gave early insights into the photoelectric effect that inspired Einstein’s Nobel Prize work; however, the latest efficiency milestone of 5.0% was more than 30 years ago. The recent surge of interest towards high-band gap absorbers for tandem applications... | [
{
"section": "Introduction",
"content": "In 1873, Willoughby Smith discovered the photoconductivity of selenium (Se) [1] . Ten years later, Charles Fritts created the first solid solar cells by coating metal foils with selenium and thin layers of gold. Puzzled by the peculiar current behavior of his cell un... | This research proposes that redesigning the device structure of selenium-based solar cells—by incorporating a MoOx hole-selective layer, significantly reducing the selenium absorber thickness, and optimizing a ZnMgO buffer layer—can overcome long-standing efficiency limitations and enable a resurgence of selenium as a ... | [
"selenium-based solar cells",
"MoOx hole-selective layer",
"selenium absorber thickness",
"ZnMgO buffer layer",
"efficiency limitations",
"tandem photovoltaic devices",
"device structure redesign",
"low-cost",
"stable high-bandgap top absorber"
] | [
{
"idx": "1.",
"title": "Smith, W. Effect of light on selenium during the passage of an electric current.Nature7, 303 (1873).",
"link": "https://doi.org/10.1038%2F007303b0"
},
{
"idx": "2.",
"title": "Fritts, C. E. On a new form of selenium cell, and some electrical discoveries made by its u... |
s41467-018-02841-9 | Physics | Visualizing heavy fermion confinement and Pauli-limited superconductivity in layered CeCoIn5 | Layered material structures play a key role in enhancing electron–electron interactions to create correlated metallic phases that can transform into unconventional superconducting states. The quasi-two-dimensional electronic properties of such compounds are often inferred indirectly through examination of bulk properti... | [
{
"section": "Introduction",
"content": "A central theme of the research on unconventional superconductivity has been its strong relationship to reduced dimensionality [1] , [2] , [3] , [4] . For example, the layered crystal structure of high- T c superconductors gives rise to strongly two-dimensional (2D) ... | The core research idea is to investigate the layer-dependent electronic structure and superconducting properties of the quasi-two-dimensional heavy fermion superconductor CeCoIn5 using cross-sectional scanning tunneling microscopy, with a focus on probing the confinement of heavy quasiparticles, the spatial variation o... | [
"quasi-two-dimensional heavy fermion superconductor",
"CeCoIn5",
"cross-sectional scanning tunneling microscopy",
"electronic structure",
"superconducting properties",
"heavy quasiparticles",
"pseudogap",
"d-wave order parameter",
"defects",
"vortex phase",
"Pauli-limited transition"
] | [
{
"idx": "1.",
"title": "Norman, M. R. The challenge of unconventional superconductivity.Science332, 196–200 (2011).",
"link": "https://doi.org/10.1126%2Fscience.1200181"
},
{
"idx": "2.",
"title": "Monthoux, P., Pines, D. & Lonzarich, G. G. Superconductivity without phonons.Nature450, 1177–... |
s41467-018-08210-w | Materials science | Self-assembly of fractal liquid crystal colloids | Nematic liquid crystals are anisotropic fluids that self-assemble into vector fields, which are governed by geometrical and topological laws. Consequently, particulate or droplet inclusions self-assemble in nematic domains through a balance of topological defects. Here, we use double emulsions of water droplets inside ... | [
{
"section": "Introduction",
"content": "In a world governed by symmetry and invariance principles, complex mathematical concepts often realize themselves in seemingly simple physical systems [1] . Topology is a study of geometrical properties that are preserved by continuous deformations, which is mostly a... | The core research idea is to investigate how size differences between water droplets with radial boundary conditions inside larger radial nematic liquid crystal droplets influence the spontaneous formation of complex three-dimensional structures, including fractals, by leveraging topological and elastic properties, and... | [
"water droplets",
"radial boundary conditions",
"nematic liquid crystal droplets",
"three-dimensional structures",
"fractals",
"topological properties",
"elastic properties",
"numerical analysis",
"Thomson problem",
"topological defects"
] | [
{
"idx": "1.",
"title": "Gross, D. J. The role of symmetry in fundamental physics.Proc. Natl Acad. Sci. USA93, 14256–14259 (1996).",
"link": "https://doi.org/10.1073%2Fpnas.93.25.14256"
},
{
"idx": "3.",
"title": "Ellis, G. Topology and cosmology.Gen. Relativ. Gravit.2, 7–21 (1971).",
"l... |
s41467-020-14366-1 | Energy science and technology | Highly efficient three-dimensional solar evaporator for high salinity desalination by localized crystallization | Solar-driven water evaporation represents an environmentally benign method of water purification/desalination. However, the efficiency is limited by increased salt concentration and accumulation. Here, we propose an energy reutilizing strategy based on a bio-mimetic 3D structure. The spontaneously formed water film, wi... | [
{
"section": "Introduction",
"content": "Facing the globally occurring water scarcity situation, solar-driven water evaporation or solar steam generation is considered as a promising technology for potential applications in desalination and clean water preparation [1] , [2] , [3] , as solar energy is the on... | The core research idea is to develop a biomimetic 3D evaporator system with hierarchical water pathways and structure inhomogeneity that enables interfacial water film inhomogeneity management, thereby achieving high energy efficiency, enhanced water evaporation rates, and sustainable desalination performance under hig... | [
"biomimetic 3D evaporator system",
"hierarchical water pathways",
"structure inhomogeneity",
"interfacial water film inhomogeneity management",
"high energy efficiency",
"enhanced water evaporation rates",
"sustainable desalination performance",
"high salinity",
"position-related energy utilization"... | [
{
"idx": "1.",
"title": "Oki, T. & Kanae, S. Global hydrological cycles and world water resources.Science313, 1068–1072 (2006).",
"link": "https://doi.org/10.1126%2Fscience.1128845"
},
{
"idx": "2.",
"title": "Ni, G. et al. Steam generation under one sun enabled by a floating structure with ... |
s41467-023-38185-2 | Optics and photonics | Dispersion-engineered metasurfaces reaching broadband 90% relative diffraction efficiency | Dispersion results from the variation of index of refraction as well as electric field confinement in sub-wavelength structures. It usually results in efficiency decrease in metasurface components leading to troublesome scattering into unwanted directions. In this letter, by dispersion engineering, we report a set of e... | [
{
"section": "Introduction",
"content": "The demand of compact and high-end devices for sensing, imaging as well as virtual and augmented reality is ever increasing. Photonic metasurfaces, consisting of sub-wavelength spaced nanostructures, are an emerging platform to realize the target performances. Recent... | The core research idea is to develop polarization-insensitive and broadband metasurface components with high relative diffraction efficiency across nearly the entire visible spectrum by leveraging dispersion-engineered anisotropic nanostructures, enabling improved performance in applications such as metagratings, chrom... | [
"polarization-insensitive metasurface components",
"broadband metasurface components",
"high relative diffraction efficiency",
"visible spectrum",
"dispersion-engineered anisotropic nanostructures",
"metagratings",
"chromatic metalenses",
"beam shapers",
"unwanted diffracted orders",
"noise minimi... | [
{
"idx": "1.",
"title": "Zhang, C. et al. Low-loss metasurface optics down to the deep ultraviolet region.Light. Sci. Appl.9, 55 (2020).",
"link": "https://doi.org/10.1038%2Fs41377-020-0287-y"
},
{
"idx": "2.",
"title": "Khorasaninejad, M. et al. Metalenses at visible wavelengths: diffractio... |
s41467-024-49352-4 | Energy science and technology | Intrinsically stretchable organic photovoltaics by redistributing strain to PEDOT:PSS with enhanced stretchability and interfacial adhesion | Intrinsically stretchable organic photovoltaics have emerged as a prominent candidate for the next-generation wearable power generators regarding their structural design flexibility, omnidirectional stretchability, and in-plane deformability. However, formulating strategies to fabricate intrinsically stretchable organi... | [
{
"section": "Introduction",
"content": "Organic photovoltaics (OPVs) have garnered attention as promising advanced wearable power generators for on-skin electronics, owing to their remarkably high power conversion efficiency (PCE) and flexibility [1] , [2] , [3] . To secure the structural and functional in... | This study proposes a delocalization and redistribution strategy for intrinsically stretchable organic photovoltaics (IS-OPVs) that enhances their endurance to high tensile strains and cyclic stretching durability while maintaining a high initial power conversion efficiency (PCE), achieved through the incorporation of ... | [
"delocalization",
"redistribution",
"intrinsically stretchable organic photovoltaics",
"high tensile strains",
"cyclic stretching durability",
"power conversion efficiency",
"zwitterion additive",
"ION E",
"stretchability",
"interfacial adhesion",
"PEDOT:PSS electrode",
"terpolymer-based activ... | [
{
"idx": "1.",
"title": "Liu, R., Wang, Z. L., Fukuda, K. & Someya, T. Flexible self-charging power sources.Nat. Rev. Mater.7, 870–886 (2022).",
"link": "https://doi.org/10.1038%2Fs41578-022-00441-0"
},
{
"idx": "2.",
"title": "Oh, J. Y. & Bao, Z. Second skin enabled by advanced electronics.... |
s41467-024-49727-7 | Physics | Transcription regulates the spatio-temporal dynamics of genes through micro-compartmentalization | Although our understanding of the involvement of heterochromatin architectural factors in shaping nuclear organization is improving, there is still ongoing debate regarding the role of active genes in this process. In this study, we utilize publicly-available Micro-C data from mouse embryonic stem cells to investigate ... | [
{
"section": "Introduction",
"content": "Chromosome conformation assays like Hi-C unveiled hierarchical organization of chromosomes within eukaryotic nuclei [1] , [2] . In metazoans, Mbp-scale “checkerboard” patterns in contact maps reveal spatial segregation of chromosomes into a euchromatic “A” compartmen... | This study proposes that RNA Polymerase II (Pol II) occupancy acts as a key determinant of transcription-dependent 3D gene folding, with Pol II-mediated phase separation giving rise to transcriptionally active subcompartments and intragenic contact enrichment, and Pol II-mediated condensation coupled with transcription... | [
"RNA Polymerase II",
"Pol II occupancy",
"transcription-dependent 3D gene folding",
"Pol II-mediated phase separation",
"transcriptionally active subcompartments",
"intragenic contact enrichment",
"Pol II-mediated condensation",
"transcriptional bursting",
"reduced gene mobility",
"unifying framew... | [
{
"idx": "1.",
"title": "Eagen, K. P. Principles of chromosome architecture revealed by Hi-C.Trends Biochem. Sci.43, 469–478 (2018).",
"link": "https://doi.org/10.1016%2Fj.tibs.2018.03.006"
},
{
"idx": "2.",
"title": "Jerkovic, I. & Cavalli, G. Understanding 3D genome organization by multidi... |
s41467-020-16075-1 | Energy science and technology | Benign ferroelastic twin boundaries in halide perovskites for charge carrier transport and recombination | Grain boundaries have been established to impact charge transport, recombination and thus the power conversion efficiency of metal halide perovskite thin film solar cells. As a special category of grain boundaries, ferroelastic twin boundaries have been recently discovered to exist in both CH 3 NH 3 PbI 3 thin films an... | [
{
"section": "Introduction",
"content": "Metal halide perovskite (MHP) materials have exhibited great potential in realizing next-generation low cost and high efficiency solar cells [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , and their certified power conversion efficiency (PCE) has already... | The core research idea is to investigate whether ferroelastic twin boundaries in metal halide perovskite materials, unlike grain boundaries, do not negatively impact carrier transport or act as recombination centers, and whether they can enhance the stability and flexibility of perovskite solar cells without compromisi... | [
"ferroelastic twin boundaries",
"metal halide perovskite materials",
"grain boundaries",
"carrier transport",
"recombination centers",
"perovskite solar cells",
"stability",
"flexibility",
"efficiency"
] | [
{
"idx": "1.",
"title": "Burschka, J. et al. Sequential deposition as a route to high-performance perovskite-sensitized solar cells.Nature499, 316 (2013).",
"link": "https://doi.org/10.1038%2Fnature12340"
},
{
"idx": "2.",
"title": "Chen, W. et al. Efficient and stable large-area perovskite ... |
s41467-022-32974-x | Physics | QED radiative corrections for accelerator neutrinos | Neutrino oscillation experiments at accelerator energies aim to establish charge-parity violation in the neutrino sector by measuring the energy-dependent rate of ν e appearance and ν μ disappearance in a ν μ beam. These experiments can precisely measure ν μ cross sections at near detectors, but ν e cross sections are ... | [
{
"section": "Introduction",
"content": "Current and future accelerator neutrino oscillation experiments [1] , [2] , [3] , [4] , [5] , [6] observe primarily muon neutrinos and antineutrinos in their near detectors, but must precisely interpret electron-neutrino and antineutrino interactions in far detectors... | This work proposes a computational framework for precisely calculating differences between muon- and electron-neutrino interactions, including QED radiative corrections, to improve the interpretation of νe appearance signals in accelerator neutrino oscillation experiments and enable percent-level control necessary for ... | [
"computational framework",
"muon-neutrino interactions",
"electron-neutrino interactions",
"QED radiative corrections",
"νe appearance signals",
"accelerator neutrino oscillation experiments",
"CP violation",
"percent-level control",
"neutrino oscillation",
"interaction differences",
"radiative ... | [
{
"idx": "1.",
"title": "Abe, K. et al. The T2K experiment.Nucl. Instrum. Meth. A659, 106–135 (2011).",
"link": "https://doi.org/10.1016%2Fj.nima.2011.06.067"
},
{
"idx": "2.",
"title": "Abe, K. et al. Constraint on the matter-antimatter symmetry-violating phase in neutrino oscillations.Natu... |
s41467-023-44271-2 | Mathematics and computing | Gene-SGAN: discovering disease subtypes with imaging and genetic signatures via multi-view weakly-supervised deep clustering | Disease heterogeneity has been a critical challenge for precision diagnosis and treatment, especially in neurologic and neuropsychiatric diseases. Many diseases can display multiple distinct brain phenotypes across individuals, potentially reflecting disease subtypes that can be captured using MRI and machine learning ... | [
{
"section": "Introduction",
"content": "Neurologic and neuropsychiatric diseases are associated with pathologic processes, which lead to heterogeneous brain changes modified by underlying genetic determinants, as well as lifestyle and environmental factors. Imaging has been a cornerstone of studying the hu... | The core research idea is to develop and apply a deep learning method called Gene-SGAN that integrates imaging and genetic data to model the heterogeneity of brain diseases by identifying genetically-associated endophenotypic imaging signatures, enabling the clustering of patients into biologically interpretable and cl... | [
"Gene-SGAN",
"deep learning method",
"imaging data",
"genetic data",
"brain diseases",
"genetically-associated endophenotypic imaging signatures",
"neuroanatomical patterns",
"genetic underpinnings",
"clinical profiles",
"precision diagnostics",
"patient stratification",
"neuropathologic proce... | [
{
"idx": "1.",
"title": "Davatzikos, C. Machine learning in neuroimaging: progress and challenges.Neuroimage197, 652–6 (2019).",
"link": "https://doi.org/10.1016%2Fj.neuroimage.2018.10.003"
},
{
"idx": "2.",
"title": "Davatzikos, C., Xu, F., An, Y., Fan, Y. & Resnick, S. M. Longitudinal prog... |
s41467-022-32065-x | Physics | Pressure-induced monotonic enhancement ofTcto over 30 K in superconducting Pr0.82Sr0.18NiO2thin films | The successful synthesis of superconducting infinite-layer nickelate thin films with the highest T c ≈ 15 K has ignited great enthusiasm for this material class as potential analogs of the high- T c cuprates. Pursuing a higher T c is always an imperative task in studying a new superconducting material system. Here we r... | [
{
"section": "Introduction",
"content": "Since the discovery of high- T c superconductivity in cuprates [1] , numerous experimental and theoretical investigations have been carried out aiming at finding more superconductors with higher T c and unveiling the mysteries mechanisms. After over 30 years of endea... | The core research idea is to investigate the effect of high pressure on the superconducting properties of infinite-layer Pr 0.82 Sr 0.18 NiO 2 thin films, with the hypothesis that applying pressure can significantly enhance the superconducting transition temperature (T c), as demonstrated by the observed increase from ... | [
"high pressure",
"superconducting properties",
"infinite-layer Pr0.82Sr0.18NiO2 thin films",
"superconducting transition temperature (Tc)",
"12.1 GPa",
"ambient pressure",
"17 K",
"31 K"
] | [
{
"idx": "1.",
"title": "Bednorz, J. G. & Müller, K. A. Possible highTcsuperconductivity in the Ba-La-Cu-O system.Z. Phys. B64, 189 (1986).",
"link": "https://link.springer.com/doi/10.1007/BF01303701"
},
{
"idx": "2.",
"title": "Gao, L. et al. Study of superconductivity in the Hg-Ba-Ca-Cu-O ... |
s41467-023-41923-1 | Materials science | Microrobots powered by concentration polarization electrophoresis (CPEP) | Second-order electrokinetic flow around colloidal particles caused by concentration polarization electro-osmosis (CPEO) can result in a phoretic motion of asymmetric particle dimers in a homogeneous AC electrical field, which we refer to as concentration polarization electro-phoresis (CPEP). To demonstrate this actuati... | [
{
"section": "Introduction",
"content": "According to Smoluchowski’s century-old theory, electrophoresis of colloidal particles is shape-independent [1] . In combination with the time reversibility of hydrodynamics at low Reynolds numbers, shape-independence implies that even asymmetric particles will not d... | The core research idea is to investigate a novel propulsion mechanism for weakly polarizable, asymmetric colloidal particles based on concentration polarization electroosmosis (CPEO), referred to as concentration polarization electrophoresis (CPEP), which enables directed movement in a homogeneous AC electric field thr... | [
"novel propulsion mechanism",
"weakly polarizable asymmetric colloidal particles",
"concentration polarization electroosmosis (CPEO)",
"concentration polarization electrophoresis (CPEP)",
"homogeneous AC electric field",
"electro-orientation",
"induced hydrodynamic flows",
"precise 2D maneuverability"... | [] |
s41467-024-50045-1 | Materials science | Self-powered triboelectric-responsive microneedles with controllable release of optogenetically engineered extracellular vesicles for intervertebral disc degeneration repair | Excessive exercise is an etiological factor of intervertebral disc degeneration (IVDD). Engineered extracellular vesicles (EVs) exhibit excellent therapeutic potential for disease-modifying treatments. Herein, we fabricate an exercise self-powered triboelectric-responsive microneedle (MN) assay with the sustainable rel... | [
{
"section": "Introduction",
"content": "Low back pain (LBP) is the most prevalent musculoskeletal disorder and the leading cause of disability worldwide, posing a severe socioeconomic burden and decreasing quality of life [1] , [2] , [3] . LBP spans a spectrum of different types of pain anatomically extend... | The core research idea is to develop a self-powered triboelectric-responsive microneedle array that controllably releases optogenetically engineered extracellular vesicles (EXPLOR EVs) to target intervertebral disc degeneration by modulating inflammation through TRAM1-mediated regulation of TREX1 localization, thereby ... | [
"self-powered triboelectric-responsive microneedle array",
"optogenetically engineered extracellular vesicles",
"EXPLOR EVs",
"intervertebral disc degeneration",
"TRAM1-mediated regulation",
"TREX1 localization",
"wearable therapeutic strategy",
"exercise-responsive therapeutic strategy",
"disease-m... | [
{
"idx": "1.",
"title": "Knezevic, N. N., Candido, K. D., Vlaeyen, J. W. S., Van Zundert, J. & Cohen, S. P. Low back pain.Lancet398, 78–92 (2021).",
"link": "https://doi.org/10.1016%2FS0140-6736%2821%2900733-9"
},
{
"idx": "2.",
"title": "Cieza, A. et al. Global estimates of the need for reh... |
s41467-019-08882-y | Nanoscience and technology | Universal nuclear focusing of confined electron spins | For spin-based quantum computation in semiconductors, dephasing of electron spins by a fluctuating background of nuclear spins is a main obstacle. Here we show that this nuclear background can be precisely controlled in generic quantum dots by periodically exciting electron spins. We demonstrate this universal phenomen... | [
{
"section": "Introduction",
"content": "Semiconductor quantum dots (QDs) exhibit tunable atomic-like electronic states [1] . For this reason, QDs are referred to as artificial atoms and may serve as hosts for spin quantum bits (qubits), the main building block for spin-based quantum computation [2] . One o... | The core research idea is to investigate spin mode-locking as a universal phenomenon in many-electron GaAs/AlGaAs quantum dots and to demonstrate that it arises from dynamical nuclear polarization induced by periodic optical excitation, offering a feedback mechanism that synchronizes nuclear spin polarization to the la... | [
"spin mode-locking",
"many-electron GaAs/AlGaAs quantum dots",
"dynamical nuclear polarization",
"periodic optical excitation",
"feedback mechanism",
"nuclear spin polarization",
"laser repetition rate",
"electron spin coherence times",
"hyperfine-induced dephasing"
] | [
{
"idx": "1.",
"title": "Kastner, M. A. Artificial atoms.Phys. Today46, 24–31 (1993).",
"link": "https://doi.org/10.1063%2F1.881393"
},
{
"idx": "2.",
"title": "Loss, D. & DiVincenzo, D. P. Quantum computation with quantum dots.Phys. Rev. A57, 120–126 (1998).",
"link": "https://doi.org/1... |
s41467-018-04014-0 | Physics | Nonzero Berry phase in quantum oscillations from giant Rashba-type spin splitting in LaTiO3/SrTiO3heterostructures | The manipulation of the spin degrees of freedom in a solid has been of fundamental and technological interest recently for developing high-speed, low-power computational devices. There has been much work focused on developing highly spin-polarized materials and understanding their behavior when incorporated into so-cal... | [
{
"section": "Introduction",
"content": "In the presence of an electric field, inversion symmetry in a solid is broken which, in addition to spin–orbit coupling, lifts the spin degeneracy [1] . This effect is known as Rashba-type spin splitting, and the resulting Fermi surface consists of two pockets, an in... | The core research idea is to investigate ultra-thin LaTiO3/SrTiO3 heterostructures to demonstrate that Rashba-type spin–orbit coupling, evidenced by Shubnikov–de Haas oscillations, weak anti-localization, and large anisotropic magnetoresistance, is significantly enhanced in these systems compared to thicker films and o... | [
"Ultra-thin LaTiO3/SrTiO3 heterostructures",
"Rashba-type spin–orbit coupling",
"Shubnikov–de Haas oscillations",
"weak anti-localization",
"large anisotropic magnetoresistance",
"thicker films",
"oxide interfaces",
"coupling coefficient",
"LaAlO3/SrTiO3"
] | [
{
"idx": "2.",
"title": "Popovic, Z. S. & Satpathy, S. Wedge-shaped potential and airy-function electron localization in oxide superlattices.Phys. Rev. Lett.94, 176805 (2005).",
"link": "https://doi.org/10.1103%2FPhysRevLett.94.176805"
},
{
"idx": "3.",
"title": "Ohtomo, A., Muller, D. A., G... |
s41467-022-29359-5 | Physics | Chiral structures of electric polarization vectors quantified by X-ray resonant scattering | Resonant elastic X-ray scattering (REXS) offers a unique tool to investigate solid-state systems providing spatial knowledge from diffraction combined with electronic information through the enhanced absorption process, allowing the probing of magnetic, charge, spin, and orbital degrees of spatial order together with e... | [
{
"section": "Introduction",
"content": "Recently, novel types of polar chiral structures have been discovered in epitaxial ferroelectric PbTiO 3 /SrTiO 3 superlattice structures. Depending on the number of layer repetitions, number of unit cells in each layer, and type of substrate, various phases such as ... | The core research idea is to develop a detailed theoretical framework for quantitatively analyzing X-ray resonant scattering intensity from chiral electric-polarization structures in PbTiO₃/SrTiO₃ superlattices, including a method to determine the resonant-scattering amplitude components for arbitrary polarization dire... | [
"X-ray resonant scattering intensity",
"chiral electric-polarization structures",
"PbTiO₃/SrTiO₃ superlattices",
"resonant-scattering amplitude components",
"polarization directions",
"rotational approximation",
"anisotropic tensors",
"first-principles calculations",
"coherence length",
"correlati... | [
{
"idx": "1.",
"title": "Yadav, A. K. et al. Observation of polar vortices in oxide superlattices.Nature530, 198–201 (2016).",
"link": "https://doi.org/10.1038%2Fnature16463"
},
{
"idx": "2.",
"title": "Damodaran, A. R. et al. Phase coexistence and electric-field control of toroidal order in... |
s41467-022-34718-3 | Nanoscience and technology | Flash drug release from nanoparticles accumulated in the targeted blood vessels facilitates the tumour treatment | Tumour microenvironment hinders nanoparticle transport deep into the tissue precluding thorough treatment of solid tumours and metastatic nodes. We introduce an anticancer drug delivery concept termed FlaRE (Flash Release in Endothelium), which represents alternative to the existing approaches based on enhanced permeab... | [
{
"section": "Introduction",
"content": "Formulating drug molecules in nanoscale carriers is an attractive approach to improve the therapeutic index of anticancer pharmaceuticals [1] , [2] . The nanocarriers protect therapeutic cargo from degradation in harsh extracellular environment, mediate transportatio... | The core research idea is to develop a novel non-EPR-based drug delivery strategy called FlaRE (Flash Release in Endothelium) that enhances the therapeutic efficiency of anticancer nanomedicines by promoting rapid drug release within tumour microcapillaries, thereby enabling high local drug concentrations that drive ef... | [
"FlaRE",
"Flash Release in Endothelium",
"non-EPR-based drug delivery strategy",
"anticancer nanomedicines",
"tumour microcapillaries",
"rapid drug release",
"high local drug concentrations",
"endothelium",
"cancer cells",
"nanoparticle extravasation",
"EPR effect"
] | [
{
"idx": "1.",
"title": "Mitchell, M. J. et al. Engineering precision nanoparticles for drug delivery.Nat. Rev. Drug. Discov.20, 101–124 (2021).",
"link": "https://doi.org/10.1038%2Fs41573-020-0090-8"
},
{
"idx": "2.",
"title": "Nikitin, M. P. et al. Enhancement of the blood-circulation time... |
ncomms9860 | Physics | Compensating for population sampling in simulations of epidemic spread on temporal contact networks | Data describing human interactions often suffer from incomplete sampling of the underlying population. As a consequence, the study of contagion processes using data-driven models can lead to a severe underestimation of the epidemic risk. Here we present a systematic method to alleviate this issue and obtain a better es... | [
{
"section": "Introduction",
"content": "Human interactions play an important role in determining the potential transmission routes of infectious diseases and other contagion phenomena [1] . Their measure and characterization represent an invaluable contribution to the study of transmissible diseases [2] . ... | The core research idea is to develop and evaluate systematic methods for estimating the outcomes of epidemic spreading models on whole populations by constructing surrogate contact data for missing individuals in incompletely sampled temporal networks, using only the structural and temporal information available in res... | [
"epidemic spreading models",
"whole populations",
"surrogate contact data",
"missing individuals",
"incompletely sampled temporal networks",
"structural information",
"temporal information",
"resampled empirical data",
"epidemic risk",
"simulations",
"incomplete contact data"
] | [
{
"idx": "2",
"title": "Read, J. M., Edmunds, W. J., Riley, S., Lessler, J. & Cummings, D. A. T. Close encounters of the infectious kind: methods to measure social mixing behaviour.Epidemiol. Infect.140, 2117–2130 (2012).",
"link": "https://doi.org/10.1017%2FS0950268812000842"
},
{
"idx": "3",
... |
s41467-024-54586-3 | Physics | Element-specific X-Ray detection of electron paramagnetic resonance in thin films of quantum bits | Element-specific magnetism accessible by synchrotron-based X-ray spectroscopy has proven to be valuable to study spin and orbital moments of transition metals and lanthanides in technologically relevant thin-film and monolayer samples. The access to coherent spin superposition states relevant for emergent quantum techn... | [
{
"section": "Introduction",
"content": "Recently, interest in versatile, coherent quantum systems has grown massively due to the rise of quantum technologies [1] . Molecular qubits are of small, nanometer size, and they can be functionalized and easily organized on surfaces [2] . Reports on Rabi oscillatio... | The core research idea is to develop and demonstrate X-ray detected electron paramagnetic resonance (XDEPR) as a novel technique that combines the element-specific, high-sensitivity capabilities of X-ray magnetic circular dichroism (XMCD) with the ultrahigh spectral resolution of EPR, enabling the detection of microwav... | [
"X-ray detected electron paramagnetic resonance",
"XDEPR",
"element-specific",
"high-sensitivity capabilities",
"X-ray magnetic circular dichroism",
"XMCD",
"ultrahigh spectral resolution",
"electron paramagnetic resonance",
"EPR",
"microwave-driven changes",
"longitudinal magnetic moments",
"... | [
{
"idx": "1.",
"title": "Heinrich, A. J. et al. Quantum-coherent nanoscience.Nat. Nanotechnol.16, 1318–1329 (2021).",
"link": "https://doi.org/10.1038%2Fs41565-021-00994-1"
},
{
"idx": "2.",
"title": "Tesi, L. et al. Modular approach to creating functionalized surface arrays of molecular qub... |
ncomms15950 | Nanoscience and technology | Microwave-free nuclear magnetic resonance at molecular scales | The implementation of nuclear magnetic resonance (NMR) at the nanoscale is a major challenge, as the resolution of conventional methods is limited to mesoscopic scales. Approaches based on quantum spin probes, such as the nitrogen-vacancy (NV) centre in diamond, have achieved nano-NMR under ambient conditions. However,... | [
{
"section": "Introduction",
"content": "The discovery of nuclear magnetic resonance (NMR), and its related technologies, was one of the great scientific achievements of the twentieth century, contributing to significant advances in areas ranging from materials science to healthcare. However, the limitation... | This research proposes a microwave-free nano-NMR technique that utilizes static field tuning to exploit natural spin interactions between a nitrogen-vacancy (NV) center in diamond and external nuclear spins, enabling non-invasive, high-sensitivity detection of proton nuclear spins in nanoscale samples through all-optic... | [
"microwave-free nano-NMR technique",
"static field tuning",
"nitrogen-vacancy center",
"diamond",
"external nuclear spins",
"non-invasive detection",
"high-sensitivity detection",
"proton nuclear spins",
"nanoscale samples",
"all-optical control",
"longitudinal relaxation time",
"T1 measuremen... | [
{
"idx": "1",
"title": "Doherty, M. W. et al. The nitrogen-vacancy colour centre in diamond.Phys. Rep.528, 1–45 (2013).",
"link": "https://doi.org/10.1016%2Fj.physrep.2013.02.001"
},
{
"idx": "2",
"title": "Chernobrod, B. M. & Berman, G. P. Spin microscope based on optically detected magneti... |
s41467-024-53488-8 | Materials science | Zn/Pt dual-site single-atom driven difunctional superimposition-augmented sonosensitizer for sonodynamic therapy boosted ferroptosis of cancer | Sonodynamic therapy (SDT) as a non-invasive antitumor strategy has been widely concerned. However, the rapid electron (e - ) and hole (h + ) recombination of traditional inorganic semiconductor sonosensitizers under ultrasonic (US) stimulation greatly limits the production of reactive oxygen species (ROS). Herein, we r... | [
{
"section": "Introduction",
"content": "Sonodynamic therapy (SDT) is a non-invasive treatment method, in which sonosensitizer could produce toxic reactive oxygen species (ROS) under ultrasound (US) stimuli to achieve tumor suppression [1] , [2] , [3] , [4] . Compared with photodynamic therapy (PDT), SDT wi... | This research proposes that a Zn/Pt dual-site single-atom loaded TiO₂ sonosensitizer (Zn/Pt SATs) can significantly enhance sonodynamic therapy (SDT) by synergistically optimizing charge separation through atomic-level electronic structure regulation, thereby improving reactive oxygen species (ROS) generation, inducing... | [
"Zn/Pt dual-site single-atom loaded TiO₂ sonosensitizer",
"Zn/Pt SATs",
"sonodynamic therapy",
"SDT",
"charge separation",
"atomic-level electronic structure regulation",
"reactive oxygen species",
"ROS",
"ferroptosis",
"GSH depletion",
"lipid peroxide accumulation",
"tumor suppression",
"so... | [
{
"idx": "1.",
"title": "Gong, F. et al. Preparation of TiH1.924nanodots by liquid-phase exfoliation for enhanced sonodynamic cancer therapy.Nat. Commun.11, 3712 (2020).",
"link": "https://doi.org/10.1038%2Fs41467-020-17485-x"
},
{
"idx": "2.",
"title": "Wen, D., Li, K., Deng, R., Feng, J. &... |
ncomms1976 | Optics and photonics | Plasmonic black gold by adiabatic nanofocusing and absorption of light in ultra-sharp convex grooves | Excitation of localized and delocalized surface plasmon resonances can be used for turning excellent reflectors of visible light, such as gold and silver, into efficient absorbers, whose wavelength, polarization or angular bandwidths are however necessarily limited owing to the resonant nature of surface plasmon excita... | [
{
"section": "Introduction",
"content": "Early blackened metallic surfaces produced by the distillation of metals at relatively high pressures [1] , as well as recent blackening of metals [2] , [3] and silicon [4] caused by intense laser irradiation, suppress reflection of light through a combined effect of... | The core research idea is to achieve efficient broadband light absorption with nanostructured metal surfaces by utilizing one- and two-dimensional arrays of ultra-sharp convex metal grooves that enable adiabatic nanofocusing of gap surface plasmon modes excited by scattering off subwavelength-sized wedges, thereby maki... | [
"Nanostructured metal surfaces",
"One-dimensional arrays",
"Two-dimensional arrays",
"Ultra-sharp convex metal grooves",
"Adiabatic nanofocusing",
"Gap surface plasmon modes",
"Subwavelength-sized wedges",
"Optical energy nanofocusing",
"Tapered plasmonic waveguides",
"Plasmonic phenomenon"
] | [
{
"idx": "1",
"title": "Pfund, A. H. The optical properties of metallic and crystalline powders.J. Opt. Soc. Am.23, 375–378 (1933).",
"link": "https://doi.org/10.1364%2FJOSA.23.000375"
},
{
"idx": "2",
"title": "Vorobyev, A. Y. & Guo, C. Enhanced absorptance of gold following multipulse femt... |
s41467-020-15405-7 | Mathematics and computing | Social media usage reveals recovery of small businesses after natural hazard events | The challenge of nowcasting the effect of natural hazard events (e.g., earthquakes, floods, hurricanes) on assets, people and society is of primary importance for assessing the ability of such systems to recover from extreme events. Traditional recovery estimates, such as surveys and interviews, are usually costly, tim... | [
{
"section": "Introduction",
"content": "Post-event management is of interest to organisations that deliver aid, provide insurance, and operate in areas affected by natural hazards. The steps in disaster management are categorised into phases, with previous literature identifying anywhere from four to eight... | This paper proposes that business downtime following natural disasters can be estimated in real time using publicly available Facebook posts from local businesses, without the need for natural language processing or semantic analysis, by analyzing changes in posting activity before, during, and after three diverse natu... | [
"business downtime",
"natural disasters",
"Facebook posts",
"local businesses",
"natural language processing",
"semantic analysis",
"posting activity",
"natural hazard events",
"Nepal",
"Mexico",
"Puerto Rico"
] | [
{
"idx": "3.",
"title": "Olshansky, R. B., Hopkins, L. D. & Johnson, L. A. Disaster and recovery: processes compressed in time.Nat. Hazards Rev.13, 173–178 (2012).",
"link": "https://doi.org/10.1061%2F%28ASCE%29NH.1527-6996.0000077"
},
{
"idx": "4.",
"title": "Horney, J., Dwyer, C., Aminto, ... |
s41467-020-18741-w | Physics | Stabilizing hidden room-temperature ferroelectricity via a metastable atomic distortion pattern | Nonequilibrium atomic structures can host exotic and technologically relevant properties in otherwise conventional materials. Oxygen octahedral rotation forms a fundamental atomic distortion in perovskite oxides, but only a few patterns are predominantly present at equilibrium. This has restricted the range of possible... | [
{
"section": "Introduction",
"content": "Despite their simple structure, ABO 3 perovskites offer a wide range of functionalities including superconductivity, metal–insulator transition, ferroelectricity, and ferromagnetism. These functionalities largely originate from the rotation of corner-connected BO 6 o... | The core research idea is to stabilize a nonequilibrium oxygen octahedral rotation (OOR) pattern in CaTiO₃ through oxide heterostructure engineering, thereby enabling room-temperature ferroelectricity that is otherwise suppressed in the common equilibrium OOR configuration. | [
"nonequilibrium oxygen octahedral rotation",
"OOR pattern",
"CaTiO₃",
"oxide heterostructure engineering",
"room-temperature ferroelectricity",
"equilibrium OOR configuration"
] | [] |
s41467-021-21799-9 | Materials science | Quasi-continuous melting of model polymer monolayers prompts reinterpretation of polymer melting | Condensed matter textbooks teach us that melting cannot be continuous and indeed experience, including with polymers and other long-chain compounds, tells us that it is a strongly first-order transition. However, here we report nearly continuous melting of monolayers of ultralong n-alkane C 390 H 782 on graphite, obser... | [
{
"section": "Introduction",
"content": "Crystal melting is normally a strongly first-order transition. At the melting point ( T m ), a pure substance faces a stark choice between being a low-energy, low-entropy crystal or a high-energy, high-entropy liquid (Fig. 1a ). In some cases 3D long-range positional... | The core research idea is to investigate whether nearly continuous equilibrium melting from the surface inward, as hypothesized for long-chain molecules and polymers, can occur in monolayers of the longest exactly monodisperse alkane synthesized, n-C390H782, when supported on a graphite surface, using atomic force micr... | [
"nearly continuous equilibrium melting",
"long-chain molecules",
"polymers",
"monolayers",
"n-C390H782",
"graphite surface",
"atomic force microscopy",
"molecular dynamics simulations",
"semiquantitative analytical model"
] | [
{
"idx": "2.",
"title": "Flory, P. J. & Vrij, A. Melting points of linear-chain homologs. The normal paraffin hydrocarbons.J. Am. Chem. Soc.85, 3548–3553 (1963).",
"link": "https://doi.org/10.1021%2Fja00905a004"
},
{
"idx": "3.",
"title": "Zachmann, H. G. Berechnung der entropie von kettenmo... |
s41467-024-51383-w | Materials science | Light-regulated soliton dynamics in liquid crystals | Electrically powered solitons are particle-like field configurations in out-of-equilibrium nematics that have garnered significant interest. However, their random generation and lack of controllable motion have limited their application. Here, we present a reconfigurable optoelectronic approach capable of regulating th... | [
{
"section": "Introduction",
"content": "Solitons, or solitary waves, are ubiquitous in nature. They are self-sustained, localized wave entities that can propagate in nonlinear media while maintaining their shape and velocity constant after pairwise collisions [1] . The full significance of the solitary wav... | The core research idea is to develop an optoelectronic approach based on the photoconductive effect for reconfigurable, light-actuated control of three-dimensional dynamic solitons called directrons in nematic liquid crystals, enabling precise, real-time micromanipulation of their full lifecycle—including generation, p... | [
"optoelectronic approach",
"photoconductive effect",
"reconfigurable control",
"light-actuated control",
"three-dimensional dynamic solitons",
"directrons",
"nematic liquid crystals",
"real-time micromanipulation",
"virtual patterned electrodes",
"conventional global electric field control",
"di... | [
{
"idx": "2.",
"title": "Zabusky, N. J. & Kruskal, M. D. Interaction of “Solitons” in a collisionless plasma and the recurrence of initial states.Phys. Rev. Lett.15, 240–243 (1965).",
"link": "https://doi.org/10.1103%2FPhysRevLett.15.240"
},
{
"idx": "3.",
"title": "Blanco-Redondo, A., de St... |
s41467-024-45036-1 | Optics and photonics | Chirality manipulation of ultrafast phase switches in a correlated CDW-Weyl semimetal | Light engineering of correlated states in topological materials provides a new avenue of achieving exotic topological phases inaccessible by conventional tuning methods. Here we demonstrate a light control of correlation gaps in a model charge-density-wave (CDW) and polaron insulator (TaSe 4 ) 2 I recently predicted to... | [
{
"section": "Introduction",
"content": "Ultrafast laser excitation has emerged as an important approach to tune and realize dynamical topological phase switch of functionalities, e.g., light-induced Floquet-Weyl phase [1] , photo-quench of Weyl semimetal [2] and light-induced Dirac [3] and Weyl nodes [4] .... | The core research idea is to propose and demonstrate an experimental approach using ultrafast laser excitation and polarization-resolved THz emission spectroscopy to accurately access, tune, and characterize hidden topological phases—specifically a light-induced Weyl semimetal phase—in the correlated Weyl semimetal (Ta... | [
"ultrafast laser excitation",
"polarization-resolved THz emission spectroscopy",
"hidden topological phases",
"light-induced Weyl semimetal phase",
"correlated Weyl semimetal",
"(TaSe₄)₂I",
"strong electron-phonon coupling",
"polaron physics",
"topological axion insulator phase",
"Weyl phase"
] | [
{
"idx": "1.",
"title": "Hubener, H., Sentef, M. A., Giovannini, U. D., Kemper, A. F. & Rubio, A. Creating stable Floquet-Weyl semimetals by laser-driving of 3D Dirac materials.Nat. Commun.8, 13940 (2017).",
"link": "https://doi.org/10.1038%2Fncomms13940"
},
{
"idx": "2.",
"title": "Sie, E. ... |
ncomms2870 | Optics and photonics | Picosecond pulses from wavelength-swept continuous-wave Fourier domain mode-locked lasers | Ultrafast lasers have a crucial function in many fields of science; however, up to now, high-energy pulses directly from compact, efficient and low-power semiconductor lasers are not available. Therefore, we introduce a new approach based on temporal compression of the continuous-wave, wavelength-swept output of Fourie... | [
{
"section": "Introduction",
"content": "Compact and reliable ultrashort pulse, high-energy laser sources [1] are of great importance for many fields in science and engineering, such as material processing, biomedical optics or optical communication. Compared with other gain media, the application of fully ... | The core research idea is to introduce and investigate a fundamentally new concept of ultrashort pulse generation based on Fourier domain mode-locked (FDML) lasers, which enables high pulse energies exceeding 100 nJ at low repetition rates of several 100 kHz directly from an electrically pumped semiconductor-based osci... | [
"Ultrashort pulse generation",
"Fourier domain mode-locked lasers",
"FDML lasers",
"High pulse energies",
"100 nJ",
"Low repetition rates",
"100 kHz",
"Electrically pumped semiconductor oscillator",
"Optically storing",
"Wavelength sweep",
"Long delay fiber",
"Energy-storage limitations",
"S... | [
{
"idx": "1",
"title": "Keller U. Recent developments in compact ultrafast lasers.Nature424, 831–838 (2003).",
"link": "https://doi.org/10.1038%2Fnature01938"
},
{
"idx": "3",
"title": "Delfyett P. J. et al. High-power ultrafast laser-diodes.IEEE J. Quantum Electron.28, 2203–2219 (1992).",
... |
s41467-019-10959-7 | Materials science | Strength of carbon nanotubes depends on their chemical structures | Single-walled carbon nanotubes theoretically possess ultimate intrinsic tensile strengths in the 100–200 GPa range, among the highest in existing materials. However, all of the experimentally reported values are considerably lower and exhibit a considerable degree of scatter, with the lack of structural information inh... | [
{
"section": "Introduction",
"content": "High-strength and lightweight materials have always been highly sought after structural materials in a broad range of research fields, such as the fabrication of the safest and most fuel-efficient aircraft, or the construction of massive architectural structures. Sin... | This study proposes that the ultimate tensile strength of real, structure-defined single-walled carbon nanotubes is dependent on their chiral structure, with small-diameter, near-armchair nanotubes exhibiting the highest strengths, and that this structural dependence can be understood through intrinsic inter-atomic str... | [
"ultimate tensile strength",
"structure-defined single-walled carbon nanotubes",
"chiral structure",
"small-diameter nanotubes",
"near-armchair nanotubes",
"intrinsic inter-atomic stress distribution",
"stress concentration",
"unavoidable defects",
"empirical formula",
"macroscopic materials"
] | [
{
"idx": "1.",
"title": "Iijima, S. & Ichihashi, T. Single-shell carbon nanotubes of 1-nm diameter.Nature363, 603–605 (1993).",
"link": "https://doi.org/10.1038%2F363603a0"
},
{
"idx": "2.",
"title": "Novoselov, K. S. et al. Electric field effect in atomically thin carbon films.Science306, 6... |
s41467-023-42551-5 | Astronomy and planetary science | Evidence of an upper ionospheric electric field perturbation correlated with a gamma ray burst | Earth’s atmosphere, whose ionization stability plays a fundamental role for the evolution and endurance of life, is exposed to the effect of cosmic explosions producing high energy Gamma-ray-bursts. Being able to abruptly increase the atmospheric ionization, they might deplete stratospheric ozone on a global scale. Dur... | [
{
"section": "Introduction",
"content": "Evidence of ionospheric disturbance induced by a gamma-ray burst (GRB) was first reported in 1988 by Fishman and Inan [1] as due to the GRB occurred on 1st August 1983, the strongest ever observed at that time, with a total fluence exceeding 10 −3 e r g s / c m 2 / s... | This study investigates the hypothesis that the intense gamma-ray burst GRB221009A, which occurred on October 9th, 2022, induced significant ionospheric disturbances not only in the bottom-side ionosphere but also at higher altitudes around 500 km, as evidenced by satellite observations and a newly developed analytical... | [
"gamma-ray burst GRB221009A",
"ionospheric disturbances",
"bottom-side ionosphere",
"satellite observations",
"analytical model",
"ionospheric conductivity",
"electric field variations",
"October 9th",
"2022",
"500 km altitude"
] | [
{
"idx": "1.",
"title": "Fishman, G. & Inan, U. Observation of an ionospheric disturbance caused by a gamma-ray burst.Nature331, 418–420 (1988).",
"link": "https://doi.org/10.1038%2F331418a0"
},
{
"idx": "3.",
"title": "Tanaka, Y. et al. First very low frequency detection of short repeated b... |
s41467-022-33960-z | Nanoscience and technology | Zero energy states clustering in an elemental nanowire coupled to a superconductor | Nanoelectronic hybrid devices combining superconductors and a one-dimensional nanowire are promising platforms to realize topological superconductivity and its resulting exotic excitations. The bulk of experimental studies in this context are transport measurements where conductance peaks allow to perform a spectroscop... | [
{
"section": "Introduction",
"content": "Topological superconductivity (TS) is predicted to exist naturally in some exotic p -wave superconductors [1] . It is however actively sought for in low-dimensional nanoscale devices where it can be engineered by combining a standard BCS superconductor with 1D conduc... | The core research idea is to investigate proximity-induced superconductivity in an ultra-clean carbon nanotube with weak spin-orbit interaction in the one-dimensional regime, where the level spacing δ is less than or comparable to the induced superconducting gap Δ*, in order to study mesoscopic fluctuations of the mini... | [
"proximity-induced superconductivity",
"ultra-clean carbon nanotube",
"weak spin-orbit interaction",
"one-dimensional regime",
"level spacing",
"induced superconducting gap",
"mesoscopic fluctuations",
"mini-gap",
"nontopological zero-energy states",
"Majorana modes",
"magnetic field",
"topolo... | [
{
"idx": "1.",
"title": "Das Sarma, S., Nayak, C. & Tewari, S. Proposal to stabilize and detect half-quantum vortices in strontium ruthenate thin films: Non-abelian braiding statistics of vortices in apx+ipysuperconductor.Phys. Rev. B73, 220502 (2006).",
"link": "https://doi.org/10.1103%2FPhysRevB.73.22... |
s41467-022-29452-9 | Optics and photonics | Observation of optical gyromagnetic properties in a magneto-plasmonic metamaterial | Metamaterials with artificial optical properties have attracted significant research interest. In particular, artificial magnetic resonances with non-unity permeability tensor at optical frequencies in metamaterials have been reported. However, only non-unity diagonal elements of the permeability tensor have been demon... | [
{
"section": "Introduction",
"content": "Metamaterials have attracted considerable research interest in the past two decades because of their artificial electromagnetic properties, facilitating the realization of left-handed materials [1] , [2] , [3] , invisible cloaking [4] , [5] , and superlens [6] , [7] ... | This study proposes that hybridizing plasmonic nanostructures with magneto-optical materials can enable the realization of optical-frequency gyromagnetic metamaterials with enhanced off-diagonal magnetic permeability tensor components, thereby allowing simultaneous manipulation of both permittivity and permeability ten... | [
"plasmonic nanostructures",
"magneto-optical materials",
"optical-frequency gyromagnetic metamaterials",
"off-diagonal magnetic permeability tensor components",
"permittivity tensor",
"permeability tensor",
"electromagnetic wave control",
"hybridizing",
"magnetic permeability tensor",
"advanced el... | [
{
"idx": "1.",
"title": "Pendry, J. B., Holden, A. J., Stewart, W. J. & Youngs, I. I. Extremely low frequency plasmons in metallic mesostructures.Phys. Rev. Lett.76, 4773–4776 (1996).",
"link": "https://doi.org/10.1103%2FPhysRevLett.76.4773"
},
{
"idx": "2.",
"title": "Pendry, J. B. Negative... |
s41467-017-01504-5 | Optics and photonics | Virtual photons in the ground state of a dissipative system | Much of the novel physics predicted to be observable in the ultrastrong light–matter coupling regime rests on the hybridisation between states with different numbers of excitations, leading to a population of virtual photons in the system’s ground state. In this article, exploiting an exact diagonalisation approach, we... | [
{
"section": "Introduction",
"content": "The study of the interaction between light and matter has been one of the cornerstones in the development of quantum mechanics. In most cases the light–matter coupling is weak enough to be intuitively described in terms of the emission and absorption of photons, whil... | The core research idea is to investigate, through a non-perturbative theoretical approach, whether ultrastrong light–matter coupling phenomena—particularly the presence of virtual photons in the ground state—can still be observed in systems with large losses that prevent the achievement of the strong coupling regime, b... | [
"non-perturbative theoretical approach",
"ultrastrong light–matter coupling",
"virtual photons",
"ground state",
"strong coupling regime",
"virtual photon population",
"graphene layers",
"hybrid quantum systems",
"intrinsic limitations",
"technological limitations",
"lossy systems",
"light–mat... | [
{
"idx": "1.",
"title": "Ciuti, C., Bastard, G. & Carusotto, I. Quantum vacuum properties of the intersubband cavity polariton field.Phys. Rev. B72, 115303 (2005).",
"link": "https://doi.org/10.1103%2FPhysRevB.72.115303"
},
{
"idx": "2.",
"title": "Anappara, A. A. et al. Signatures of the ul... |
s41467-019-10451-2 | Optics and photonics | Tomographic near-eye displays | The ultimate 3D displays should provide both psychological and physiological cues for depth recognition. However, it has been challenging to satisfy the essential features without making sacrifices in the resolution, frame rate, and eye box. Here, we present a tomographic near-eye display that supports a wide depth of ... | [
{
"section": "Introduction",
"content": "An ideal three-dimensional (3D) display system provides an immersive and realistic experience. To approach the ideal 3D display system, the following points should be considered: first, the visual characteristics of a real object that make it look real; second, allev... | This research proposes a tomographic near-eye display that synchronizes focus-tunable optics with a fast spatially adjustable backlight to effectively alleviate the trade-off among spatial resolution, depth of field, and frame rate, enabling a wide depth of field, quasi-continuous accommodation, omni-directional motion... | [
"tomographic near-eye display",
"focus-tunable optics",
"fast spatially adjustable backlight",
"spatial resolution",
"depth of field",
"frame rate",
"wide depth of field",
"quasi-continuous accommodation",
"omni-directional motion parallax",
"full resolution",
"full frame rate",
"performance m... | [
{
"idx": "1.",
"title": "Hong, J., Min, S.-W. & Lee, B. Integral floating display systems for augmented reality.Appl. Opt.51, 4201–4209 (2012).",
"link": "https://doi.org/10.1364%2FAO.51.004201"
},
{
"idx": "2.",
"title": "Jang, C. et al. Retinal 3d: augmented reality near-eye display via pu... |
ncomms7661 | Physics | Polarization control at spin-driven ferroelectric domain walls | Unusual electronic states arise at ferroelectric domain walls due to the local symmetry reduction, strain gradients and electrostatics. This particularly applies to improper ferroelectrics, where the polarization is induced by a structural or magnetic order parameter. Because of the subordinate nature of the polarizati... | [
{
"section": "Introduction",
"content": "Ferroelectric domain walls represent a naturally occurring type of oxide interface that develops versatile electronic ground states. In particular, the polarity mismatch at charged walls leads to characteristics that clearly distinguish them from the interior of the ... | The core research idea is to demonstrate that the polarization state at spin-driven ferroelectric domain walls in Mn₀.₉₅Co₀.₀₅WO₄ can be dynamically and reversibly manipulated by applying magnetic fields, enabling a new method to control the charge configuration of ferroelectric domain walls without moving the walls th... | [
"polarization state",
"spin-driven ferroelectric domain walls",
"Mn₀.₉₅Co₀.₀₅WO₄",
"magnetic fields",
"charge configuration",
"ferroelectric domain walls",
"moderate voltage",
"180° domain wall",
"magnetic field-induced reorientation",
"polarization reorientation",
"side-by-side configuration",
... | [
{
"idx": "1",
"title": "Salje, E. K. H. Multiferroic domain boundaries as active memory devices: trajectories towards domain boundary engineering.Chem. Phys. Chem.11, 940–950 (2010) .",
"link": "https://doi.org/10.1002%2Fcphc.200900943"
},
{
"idx": "2",
"title": "Salje, E. K. H. & Zhang, H. ... |
ncomms1728 | Materials science | Perpendicular exchange bias in ferrimagnetic spin valves | The exchange bias effect refers to the shift of the hysteresis loop of a ferromagnet in direct contact to an antiferromagnet. For applications in spintronics a robust and tunable exchange bias is required. Here we show experimental evidence for a perpendicular exchange bias in a prototypical ferrimagnetic spin valve co... | [
{
"section": "Introduction",
"content": "After the discovery of the giant magnetoresistance [1] , [2] , the exchange bias (EB) effect has become an integral part of spintronics with implications for basic research and for numerous device applications like random access magnetic storage units and spin valves... | The core research idea is to demonstrate that perpendicular exchange bias can be set, controlled, and reversed at room temperature in exchange-coupled ferrimagnetic systems through an interlayer exchange decoupling mechanism, using a prototype artificial DyCo₅/Ta(x)/Fe₇₆Gd₂₄ spin-valve trilayer system where the hard fe... | [
"perpendicular exchange bias",
"exchange-coupled ferrimagnetic systems",
"interlayer exchange decoupling mechanism",
"artificial DyCo₅/Ta(x)/Fe₇₆Gd₂₄ spin-valve trilayer system",
"DyCo₅",
"Ta(x)",
"Fe₇₆Gd₂₄",
"hard ferrimagnet",
"single magnetic domain pinning layer",
"soft ferrimagnet",
"magnet... | [
{
"idx": "1",
"title": "Baibich, M. N. et al. Giant magnetoresistance of (001)Fe/(001)Cr magnetic superlattices.Phys. Rev. Lett.61, 2472–2475 (1988).",
"link": "https://doi.org/10.1103%2FPhysRevLett.61.2472"
},
{
"idx": "2",
"title": "Binasch, G., Grünberg, P., Saurenbach, F. & Zinn, W. Enha... |
s41467-024-47147-1 | Materials science | Low thermal contact resistance boron nitride nanosheets composites enabled by interfacial arc-like phonon bridge | Two-dimensional materials with ultrahigh in-plane thermal conductivity are ideal for heat spreader applications but cause significant thermal contact resistance in complex interfaces, limiting their use as thermal interface materials. In this study, we present an interfacial phonon bridge strategy to reduce the thermal... | [
{
"section": "Introduction",
"content": "Two-dimensional (2D) materials, such as graphene and hexagonal boron nitride nanosheets (BNNSs), have emerged as promising materials for studying nanoscale thermal transport and developing advanced thermal management solutions [1] , [2] . The atomically thin nature o... | The core research idea is to develop a novel interfacial phonon bridge strategy using a stacking-cutting method to align hexagonal boron nitride nanosheets (BNNSs) in a polymer composite, creating an arc-like microstructure that significantly reduces interfacial thermal resistance and enhances phonon transport efficien... | [
"novel interfacial phonon bridge strategy",
"stacking-cutting method",
"hexagonal boron nitride nanosheets",
"BNNSs",
"polymer composite",
"arc-like microstructure",
"interfacial thermal resistance",
"phonon transport efficiency",
"high-performance thermal interface materials",
"TIMs",
"superior... | [
{
"idx": "1.",
"title": "Song, H. et al. Two-dimensional materials for thermal management applications.Joule2, 442–463 (2018).",
"link": "https://doi.org/10.1016%2Fj.joule.2018.01.006"
},
{
"idx": "2.",
"title": "Sun, B. et al. Dislocation-induced thermal transport anisotropy in single-cryst... |
ncomms11419 | Physics | Quantum coding with finite resources | The quantum capacity of a memoryless channel determines the maximal rate at which we can communicate reliably over asymptotically many uses of the channel. Here we illustrate that this asymptotic characterization is insufficient in practical scenarios where decoherence severely limits our ability to manipulate large qu... | [
{
"section": "Introduction",
"content": "One of the quintessential topics in quantum information theory is the study of reliable quantum information transmission over noisy quantum channels. Here ‘channel’ simply refers to a description of a physical evolution. In the standard formulation, one considers com... | This work investigates the performance of quantum coding schemes in realistic scenarios with a limited number of channel uses, proposing the concept of quantum channel dispersion as a second fundamental parameter—alongside quantum capacity—to more accurately characterize the trade-off between transmission rate, error p... | [
"quantum coding schemes",
"realistic scenarios",
"limited number of channel uses",
"quantum channel dispersion",
"quantum capacity",
"transmission rate",
"error probability",
"finite blocklength",
"quantum communication",
"second fundamental parameter",
"trade-off",
"work",
"concept",
"per... | [
{
"idx": "1",
"title": "Shannon, C. A mathematical theory of communication.Bell Syst. Tech. J.27, 379–423 (1948).",
"link": "https://doi.org/10.1002%2Fj.1538-7305.1948.tb01338.x"
},
{
"idx": "2",
"title": "Barnum, H., Knill, E. & Nielsen, M. A. On quantum fidelities and channel capacities.IE... |
s41467-022-31346-9 | Astronomy and planetary science | Identification and characterization of a new ensemble of cometary organic molecules | In-situ study of comet 1P/Halley during its 1986 apparition revealed a surprising abundance of organic coma species. It remained unclear, whether or not these species originated from polymeric matter. Now, high-resolution mass-spectrometric data collected at comet 67P/Churyumov-Gerasimenko by ESA’s Rosetta mission unve... | [
{
"section": "Introduction",
"content": "Comets are thought to be composed of refractories and ices aggregated in the outer Solar System. However, the recent comet mission Rosetta, targeting the Jupiter-family comet 67P/Churyumov-Gerasimenko (hereafter 67P), delivered evidence that this traditional classifi... | The core research idea is to identify and characterize a new ensemble of complex organic molecules in the inner coma of comet 67P/Churyumov-Gerasimenko using high-resolution mass spectrometric data from the Rosetta mission, demonstrating their absence of significant polymeric content and comparing their chemical proper... | [
"complex organic molecules",
"inner coma",
"comet 67P/Churyumov-Gerasimenko",
"high-resolution mass spectrometric data",
"Rosetta mission",
"polymeric content",
"average sum formula",
"sp²:sp³ ratio",
"CH₂:CH₃ ratio",
"hydrogen deficiency index",
"Solar System reservoirs",
"interstellar medium... | [
{
"idx": "1.",
"title": "Altwegg, K., Balsiger, H. & Fuselier, S. A. Cometary chemistry and the origin of icy solar system bodies: the view after.Rosetta. Annu. Rev. Astron. Astrophys.57, 113–55 (2019).",
"link": "https://doi.org/10.1146%2Fannurev-astro-091918-104409"
},
{
"idx": "2.",
"titl... |
s41467-024-48461-4 | Engineering | Regulation of cerebral blood flow boosts precise brain targeting of vinpocetine-derived ionizable-lipidoid nanoparticles | Despite advances in active drug targeting for blood-brain barrier penetration, two key challenges persist: first, attachment of a targeting ligand to the drug or drug carrier does not enhance its brain biodistribution; and second, many brain diseases are intricately linked to microcirculation disorders that significant... | [
{
"section": "Introduction",
"content": "As the world population ages, the incidence of brain-related diseases has risen significantly, and such diseases are now the leading cause of mortality [1] . A pressing challenge lies in the dearth or absence of disease-modifying drugs for patients afflicted with the... | The core research idea is to develop a novel brain-targeting drug delivery system called vinpocetine-derived ionizable-lipidoid nanoparticles (VIP) that leverages the cerebral blood flow-enhancing properties and structural features of vinpocetine to improve the delivery of nucleic acid-based and other therapeutic agent... | [
"brain-targeting drug delivery system",
"vinpocetine-derived ionizable-lipidoid nanoparticles",
"vinpocetine",
"cerebral blood flow-enhancing properties",
"structural features",
"nucleic acid-based therapeutic agents",
"blood-brain barrier",
"brain diseases",
"cerebrovascular impairments"
] | [
{
"idx": "1.",
"title": "Roth, G. A. et al. Global Burden of Cardiovascular Diseases and Risk Factors, 1990-2019: Update From the GBD 2019 Study.J. Am. Coll. Cardiol.76, 2982–3021 (2020).",
"link": "https://doi.org/10.1016%2Fj.jacc.2020.11.010"
},
{
"idx": "2.",
"title": "Calcoen, D., Elias,... |
s41467-018-08160-3 | Physics | The structured backbone of temporal social ties | In many data sets, information on the structure and temporality of a system coexists with noise and non-essential elements. In networked systems for instance, some edges might be non-essential or exist only by chance. Filtering them out and extracting a set of relevant connections is a non-trivial task. Moreover, mehod... | [
{
"section": "Introduction",
"content": "The analysis of large-scale empirical data sets, and in particular of complex networked data, is often made difficult by the nature of the data itself: data may be noisy [1] , [2] , [3] , and contain both robust, generalizable properties and details specific to the c... | The core research idea is to develop a method for extracting an irreducible backbone of statistically significant ties and higher-order temporal structures from temporal networks by defining a temporal null model that accounts for node activity levels and enables significance testing of interactions at multiple scales,... | [
"Temporal networks",
"Irreducible backbone",
"Statistically significant ties",
"Higher-order temporal structures",
"Temporal null model",
"Node activity levels",
"Significance testing",
"Static network approaches",
"Temporal motifs",
"Temporally aggregated data",
"Multiple scales",
"Non-random... | [
{
"idx": "1.",
"title": "Butts, C. T. Network inference, error, and informant (in)accuracy: a Bayesian approach.Soc. Netw.25, 103–140 (2003).",
"link": "https://doi.org/10.1016%2FS0378-8733%2802%2900038-2"
},
{
"idx": "2.",
"title": "Newman, M. Network structure from rich but noisy data.Nat.... |
s41467-020-19128-7 | Optics and photonics | Shortwave-infrared meso-patterned imaging enables label-free mapping of tissue water and lipid content | Water and lipids are key participants in many biological processes, but there are few non-invasive methods that provide quantification of these components in vivo, and none that can isolate and quantify lipids in the blood. Here we develop a new imaging modality termed shortwave infrared meso-patterned imaging (SWIR-MP... | [
{
"section": "Introduction",
"content": "Water and lipids compose 60–80% of the human body and are inextricably linked to proper cellular function [1] . Alterations in the concentrations and spatial distribution of these components are hallmarks of many conditions including cardiovascular disease [2] , [3] ... | The core research idea is to develop and demonstrate a novel optical imaging technique called SWIR Meso-Patterned Imaging (SWIR-MPI) that enables quantitative, label-free, noninvasive imaging of tissue water and lipid content through the skin and within superficial venous circulation by exploiting the distinct optical ... | [
"SWIR Meso-Patterned Imaging",
"SWIR-MPI",
"optical imaging technique",
"tissue water",
"lipid content",
"skin",
"superficial venous circulation",
"shortwave infrared",
"SWIR spectral region",
"optical absorption characteristics",
"optical scattering",
"hemoglobin absorption",
"visible wavel... | [
{
"idx": "2.",
"title": "Fuster, J. J., Ouchi, N., Gokce, N. & Walsh, K. Obesity-induced changes in adipose tissue microenvironment and their impact on cardiovascular disease.Circ. Res.118, 1786–1807 (2016).",
"link": "https://doi.org/10.1161%2FCIRCRESAHA.115.306885"
},
{
"idx": "3.",
"title... |
s41467-018-07932-1 | Physics | On chaotic dynamics in transcription factors and the associated effects in differential gene regulation | The control of proteins by a transcription factor with periodically varying concentration exhibits intriguing dynamical behaviour. Even though it is accepted that transcription factors vary their dynamics in response to different situations, insight into how this affects downstream genes is lacking. Here, we investigat... | [
{
"section": "Introduction",
"content": "The regulation and control of protein production is a vital element in all living organisms. This process can be highly complicated, involving a large number of steps. However, despite stochastic fluctuations, life is characterised by a high level of organisation ind... | This study investigates how oscillatory and chaotic dynamics of transcription factors, such as NF-κB, influence the expression of downstream genes with varying affinities, with the hypothesis that such complex dynamics can differentially regulate gene expression, generate cellular heterogeneity, and provide selective a... | [
"oscillatory dynamics",
"chaotic dynamics",
"transcription factors",
"NF-κB",
"downstream genes",
"gene expression",
"cellular heterogeneity",
"multi-toxic environments",
"complex dynamics",
"selective advantages",
"varying affinities"
] | [
{
"idx": "1.",
"title": "Hoffmann, A., Levchenko, A., Scott, M. L. & Baltimore, D. The IκB-NF-κB signaling module: temporal control and selective gene activation.Science298, 1241–1245 (2002).",
"link": "https://doi.org/10.1126%2Fscience.1071914"
},
{
"idx": "2.",
"title": "Nelson, D. E. et a... |
s41467-024-50057-x | Energy science and technology | Capillary compression induced outstanding n-type thermoelectric power factor in CNT films towards intelligent temperature controller | One-dimensional carbon nanotubes are promising candidates for thermoelectrics because of their excellent electrical and mechanical properties. However, the large n-type power factor remains elusive in macroscopic carbon nanotubes films. Herein, we report an outstanding n-type power factor of 6.75 mW m −1 K −2 for macro... | [
{
"section": "Introduction",
"content": "Thermoelectric (TE) materials can realize the direct conversion between heat and electricity, which offers promising potential in the applications of both waste heat recovery and solid-state cooling [1] . For waste heat recovery, previous works typically focus on red... | This research proposes a novel material processing approach, combining a developed floating catalyst chemical vapor deposition (FCCVD) with capillary compression-assisted alignment and a compressing process, to fabricate macroscopic carbon nanotube (CNT) films with highly ordered CNT assemblies, aiming to achieve both ... | [
"novel material processing approach",
"floating catalyst chemical vapor deposition (FCCVD)",
"capillary compression-assisted alignment",
"compressing process",
"macroscopic carbon nanotube (CNT) films",
"highly ordered CNT assemblies",
"p-type power factors (PF)",
"n-type power factors (PF)",
"elect... | [
{
"idx": "1.",
"title": "Bell, L. E. Cooling, heating, generating power, and recovering waste heat with thermoelectric systems.Science321, 1457–1461 (2008).",
"link": "https://doi.org/10.1126%2Fscience.1158899"
},
{
"idx": "2.",
"title": "Wang, H. & Yu, C. Organic thermoelectrics: materials ... |
s41467-024-45830-x | Optics and photonics | Probing molecules in gas cells of subwavelength thickness with high frequency resolution | Miniaturizing and integrating atomic vapor cells is widely investigated for the purposes of fundamental measurements and technological applications such as quantum sensing. Extending such platforms to the realm of molecular physics is a fascinating prospect that paves the way for compact frequency metrology as well as ... | [
{
"section": "Introduction",
"content": "Interfacing atomic and molecular gases with compact photonic platforms is a long-standing goal promising for quantum sensing [1] , [2] , [3] , [4] , [5] , optical image processing [6] , gas lasers and super-continuum sources [7] , [8] , as well as time keeping [1] an... | The core research idea is to explore the potential of thin-cells for sub-Doppler molecular spectroscopy by probing low-pressure molecular gases confined in micrometric-scale cells, enabling high-resolution studies of rovibrational transitions with applications in compact frequency referencing and fundamental physics ex... | [
"thin-cells",
"sub-Doppler molecular spectroscopy",
"low-pressure molecular gases",
"micrometric-scale cells",
"high-resolution studies",
"rovibrational transitions",
"compact frequency referencing",
"fundamental physics experiments"
] | [
{
"idx": "1.",
"title": "Knappe, S. et al. A microfabricated atomic clock.Appl. Phys. Lett.85, 1460–1462 (2004).",
"link": "https://doi.org/10.1063%2F1.1787942"
},
{
"idx": "2.",
"title": "Shah, V., Knappe, S., Schwindt, P. D. D. & Kitching, J. Subpicotesla atomic magnetometry with a microfa... |
ncomms2383 | Physics | Motional averaging in a superconducting qubit | Superconducting circuits with Josephson junctions are promising candidates for developing future quantum technologies. Of particular interest is to use these circuits to study effects that typically occur in complex condensed-matter systems. Here we employ a superconducting quantum bit—a transmon—to perform an analogue... | [
{
"section": "Introduction",
"content": "The ability to resolve energy variations Δ E occurring in a time interval Δ t is fundamentally limited by the energy-time uncertainty relation . Consider for example a system of spin-1/2 particles filling a porous material ( Fig. 1a ) in an external magnetic field; t... | The core research idea is to observe motional averaging and narrowing in a single artificial atom, a quantum bit (qubit), with a simulated fast-fluctuating environment under direct experimental control, contrasting it with the typical indirect manipulation of fluctuation rates in condensed-matter systems. | [
"motional averaging",
"narrowing",
"artificial atom",
"quantum bit",
"qubit",
"simulated fast-fluctuating environment",
"direct experimental control",
"indirect manipulation",
"fluctuation rates",
"condensed-matter systems"
] | [
{
"idx": "1",
"title": "Anderson, P. W. . A mathematical model for the narrowing of spectral lines by exchange or motion.J. Phys. Soc. Jpn.9, 316–339 (1954) .",
"link": "https://doi.org/10.1143%2FJPSJ.9.316"
},
{
"idx": "3",
"title": "Jiang, L. et al. Coherence of an optically illuminated si... |
s41467-018-05932-9 | Materials science | Measurement of complex optical susceptibility for individual carbon nanotubes by elliptically polarized light excitation | The complex optical susceptibility is the most fundamental parameter characterizing light-matter interactions and determining optical applications in any material. In one-dimensional (1D) materials, all conventional techniques to measure the complex susceptibility become invalid. Here we report a methodology to measure... | [
{
"section": "Introduction",
"content": "One-dimensional (1D) materials are at the center of significant research effort of nano science and technology. For example, carbon nanotubes, a 1D material from rolled-up graphene, have shown fascinating optical properties, e.g., quantized optical transitions [1] , ... | The core research idea is to develop a methodology for directly measuring the complex optical susceptibility of individual one-dimensional materials, such as carbon nanotubes, using an elliptical-polarization-based optical homodyne detection technique that enables the simultaneous determination of both the real and ima... | [
"methodology",
"complex optical susceptibility",
"one-dimensional materials",
"carbon nanotubes",
"elliptical-polarization-based optical homodyne detection technique",
"real part of susceptibility",
"imaginary part of susceptibility",
"Kramers–Kronig relation",
"conventional methods",
"weak optica... | [
{
"idx": "1.",
"title": "Saito, R., Fujita, M., Dresselhaus, G. & Dresselhaus, M. S. Electronic-structure of chiral graphene tubules.Appl. Phys. Lett.60, 2204–2206 (1992).",
"link": "https://doi.org/10.1063%2F1.107080"
},
{
"idx": "2.",
"title": "Kataura, H. et al. Optical properties of sing... |
s41467-020-16739-y | Nanoscience and technology | Towards fully integrated photonic displacement sensors | The field of optical metrology with its high precision position, rotation and wavefront sensors represents the basis for lithography and high resolution microscopy. However, the on-chip integration—a task highly relevant for future nanotechnological devices—necessitates the reduction of the spatial footprint of sensing... | [
{
"section": "Introduction",
"content": "Optical metrology is one of the corner stones of quality control in modern nanotechnological devices, lithography and high accuracy localization. Grating based schemes and interferometers [1] , [2] , [3] , both enabling deep sub-wavelength distance sensing for a huge... | The core research idea is to implement a two-dimensional displacement sensor based on position-dependent directional scattering of a single dipolar nanoantenna, utilizing transverse Kerker scattering and Huygens dipoles to achieve nanometrological precision by measuring the relative position between an external light s... | [
"two-dimensional displacement sensor",
"position-dependent directional scattering",
"single dipolar nanoantenna",
"transverse Kerker scattering",
"Huygens dipoles",
"nanometrological precision",
"external light source",
"photonic microchip",
"position-dependent coupling efficiency",
"six-way photo... | [
{
"idx": "1.",
"title": "Teimel, A. Technology and applications of grating interferometers in high-precision measurement.Precis. Eng.14, 147–154 (1992).",
"link": "https://doi.org/10.1016%2F0141-6359%2892%2990003-F"
},
{
"idx": "3.",
"title": "Berkovic, G. & Shafir, E. Optical methods for di... |
ncomms13547 | Materials science | Electromagnon dispersion probed by inelastic X-ray scattering in LiCrO2 | Inelastic X-ray scattering with meV energy resolution (IXS) is an ideal tool to measure collective excitations in solids and liquids. In non-resonant scattering condition, the cross-section is strongly dominated by lattice vibrations (phonons). However, it is possible to probe additional degrees of freedom such as magn... | [
{
"section": "Introduction",
"content": "The coupling between magnetic and lattice degrees of freedom gives rise to many interesting effects. It can induce multiferroic order with ferroelectric polarization coupled to the magnetic structure [1] , [2] , [3] or it can generate dynamic mixed magnon–phonon exci... | The core research idea is to demonstrate that LiCrO₂ exhibits exceptionally strong magnon–phonon coupling, enabling the direct observation of electromagnon dispersion via inelastic X-ray scattering and revealing the role of non-collinear magnetic order and exchange striction in inducing linear magnon–phonon coupling in... | [
"LiCrO₂",
"magnon–phonon coupling",
"electromagnon dispersion",
"inelastic X-ray scattering",
"non-collinear magnetic order",
"exchange striction",
"linear magnon–phonon coupling",
"two-dimensional S = 3/2 Heisenberg triangular lattice antiferromagnet"
] | [
{
"idx": "1",
"title": "Eerenstein, W., Mathur, N. D. & Scott, J. F. Multiferroic and magnetoelectric materials.Nature442, 759–765 (2006).",
"link": "https://doi.org/10.1038%2Fnature05023"
},
{
"idx": "2",
"title": "Tokura, Y. Multiferroics as quantum electromagnets.Science312, 1481–1482 (20... |
s41467-023-43156-8 | Mathematics and computing | Relational visual representations underlie human social interaction recognition | Humans effortlessly recognize social interactions from visual input. Attempts to model this ability have typically relied on generative inverse planning models, which make predictions by inverting a generative model of agents’ interactions based on their inferred goals, suggesting humans use a similar process of mental... | [
{
"section": "Introduction",
"content": "Humans easily make social evaluations from visual input, such as deciding whether two agents are interacting or who is a friend versus a foe. Early work by Heider and Simmel (1944) showed that humans can recognize rich information about others’ interactions even from... | The core research idea is to develop a bottom-up visual neural network model, called SocialGNN, that incorporates relational inductive biases in the form of graph structures to capture how humans recognize social interactions from spatiotemporal visual cues without relying on explicit mental state representations or co... | [
"bottom-up visual neural network model",
"SocialGNN",
"relational inductive biases",
"graph structures",
"humans",
"social interactions",
"spatiotemporal visual cues",
"explicit mental state representations",
"computationally expensive physical simulations"
] | [
{
"idx": "1.",
"title": "Heider, F. & Simmel, M. An experimental study of apparent behavior.Am. J. Psychol.57, 243–259 (1944).",
"link": "https://doi.org/10.2307%2F1416950"
},
{
"idx": "2.",
"title": "Hamlin, J. K., Wynn, K. & Bloom, P. Social evaluation by preverbal infants.Nature450, 557–5... |
s41467-019-13258-3 | Physics | Observation of a topological nodal surface and its surface-state arcs in an artificial acoustic crystal | Three-dimensional (3D) gapless topological phases can be classified by the dimensionality of the band degeneracies, including zero-dimensional (0D) nodal points, one-dimensional (1D) nodal lines, and two-dimensional (2D) nodal surfaces. Both nodal points and nodal lines have been realized recently in photonics and acou... | [
{
"section": "Introduction",
"content": "Specially designed structures including metamaterials and artificial crystals in classical-wave systems can possess topological properties, which offer fundamentally new degrees of freedom to manipulate the classical waves in photonics, acoustics, and mechanics [1] ,... | This work proposes the experimental realization and characterization of a twofold topological nodal surface in a chiral 3D phononic crystal, stabilized by non-symmorphic lattice symmetry and time-reversal symmetry, which carries a topological charge of 2 and exhibits a topological bulk-edge correspondence through Fermi... | [
"twofold topological nodal surface",
"chiral 3D phononic crystal",
"non-symmorphic lattice symmetry",
"time-reversal symmetry",
"topological charge of 2",
"topological bulk-edge correspondence",
"Fermi-arc-like surface states",
"Weyl cones",
"higher-dimensional topologically charged band degeneracy"... | [
{
"idx": "1.",
"title": "Lu, L., Joannopoulos, J. D. & Soljačić, M. Topological photonics.Nat. Photon.8, 821–829 (2014).",
"link": "https://doi.org/10.1038%2Fnphoton.2014.248"
},
{
"idx": "2.",
"title": "Yang, Z. et al. Topological acoustics.Phys. Rev. Lett.114, 114301 (2015).",
"link": ... |
s41467-021-23603-0 | Materials science | Sub-nanometer confinement enables facile condensation of gas electrolyte for low-temperature batteries | Confining molecules in the nanoscale environment can lead to dramatic changes of their physical and chemical properties, which opens possibilities for new applications. There is a growing interest in liquefied gas electrolytes for electrochemical devices operating at low temperatures due to their low melting point. How... | [
{
"section": "Introduction",
"content": "Batteries that can sustain ultralow temperatures (<−30 °C) are essential for extending the operation capability of existing energy storage systems as well as enabling human presence to the outer space and deep ocean worlds [1] , [2] , [3] , [4] , [5] . The state-of-t... | The core research idea is to develop a mechanically robust, self-supporting MOF-polymer membrane with sub-nanometer pores that enables capillary condensation of fluoromethane gas electrolyte at reduced pressures, thereby achieving high ionic conductivity and stable low-temperature battery operation down to −60 °C witho... | [
"mechanically robust",
"self-supporting MOF-polymer membrane",
"sub-nanometer pores",
"capillary condensation",
"fluoromethane gas electrolyte",
"reduced pressures",
"high ionic conductivity",
"stable low-temperature battery operation",
"−60 °C",
"energy density",
"long-term stability"
] | [
{
"idx": "1.",
"title": "Smart, M. C. et al. The use of lithium-ion batteries for JPL’s Mars missions.Electrochim. Acta268, 27–40 (2018).",
"link": "https://doi.org/10.1016%2Fj.electacta.2018.02.020"
},
{
"idx": "2.",
"title": "Chin, K. B. et al. Energy storage technologies for small satelli... |
s41467-021-25720-2 | Energy science and technology | High resolution global spatiotemporal assessment of rooftop solar photovoltaics potential for renewable electricity generation | Rooftop solar photovoltaics currently account for 40% of the global solar photovoltaics installed capacity and one-fourth of the total renewable capacity additions in 2018. Yet, only limited information is available on its global potential and associated costs at a high spatiotemporal resolution. Here, we present a hig... | [
{
"section": "Introduction",
"content": "From powering the National Aeronautics and Space Administration (NASA’s) Vanguard satellites in 1958 to lighting homes in sub-Saharan Africa, solar photovoltaics (PV) technology has come a long way. Rooftop Solar photovoltaics (RTSPV) technology as a subset of the so... | This study proposes a hybrid framework integrating "Top-down" and "Bottom-up" methods using a machine learning model to provide a high-resolution, global assessment of rooftop solar photovoltaic (RTSPV) technical potential and associated levelized costs at a monthly temporal resolution, addressing the current lack of c... | [
"hybrid framework",
"Top-down method",
"Bottom-up method",
"machine learning model",
"high-resolution assessment",
"global assessment",
"rooftop solar photovoltaic (RTSPV)",
"technical potential",
"levelized costs",
"monthly temporal resolution",
"geo-mapped evaluations",
"decentralized RTSPV ... | [
{
"idx": "17.",
"title": "Izquierdo, S., Rodrigues, M. & Fueyo, N. A method for estimating the geographical distribution of the available roof surface area for large-scale photovoltaic energy-potential evaluations.Sol. Energy82, 929–939 (2008).",
"link": "https://doi.org/10.1016%2Fj.solener.2008.03.007"... |
s41467-018-07282-y | Optics and photonics | Free-space creation of ultralong anti-diffracting beam with multiple energy oscillations adjusted using optical pen | A light beam propagating over an infinite anti-diffracting distance requires infinite power to preserve its shape. However, the fundamental barrier of finite power in free space has made the problem of diffraction insurmountable in recent decades. To overcome this limitation, we report an approach that employs the mult... | [
{
"section": "Introduction",
"content": "Diffraction is a natural property of light beams that tends to broaden the beams during propagation in free space, and fighting against the diffraction effect is one of the most important topics in the community of optics. However, it was not until 1987 that the adve... | The core research idea is to overcome the fundamental limitation of finite anti-diffracting distances for light beams in free space by employing a multiple energy oscillation mechanism that enables energy recharge, thereby generating ultralong anti-diffracting (UAD) light beams with tunable properties through a versati... | [
"finite anti-diffracting distances",
"light beams",
"free space",
"multiple energy oscillation mechanism",
"energy recharge",
"ultralong anti-diffracting light beams",
"tunable properties",
"versatile optical pen",
"energy oscillations",
"amplitude",
"position",
"phase"
] | [
{
"idx": "1.",
"title": "Durnin, J., Miceli, J. J. & Eberly, J. H. Diffraction-free beams.Phys. Rev. Lett.58, 1499–1501 (1987).",
"link": "https://doi.org/10.1103%2FPhysRevLett.58.1499"
},
{
"idx": "2.",
"title": "Chattrapiban, N., Rogers, E. A., Cofield, D., Hill, W. T. III. & Roy, R. Gener... |
s41467-020-19261-3 | Nanoscience and technology | Dynamically-enhanced strain in atomically thin resonators | Graphene and related two-dimensional (2D) materials associate remarkable mechanical, electronic, optical and phononic properties. As such, 2D materials are promising for hybrid systems that couple their elementary excitations (excitons, phonons) to their macroscopic mechanical modes. These built-in systems may yield en... | [
{
"section": "Introduction",
"content": "Since the first demonstration of mechanical resonators made from suspended graphene layers [1] , considerable progress has been made to conceive nano-mechanical systems based on 2D materials [2] , [3] with well-characterised performances [4] , [5] , [6] , [7] , [8] ,... | Using micro-Raman scattering spectroscopy in resonators made from pristine suspended graphene monolayers, this research demonstrates efficient strain-mediated coupling between the "built-in" quantum degrees of freedom (specifically, Raman-active optical phonons) and macroscopic flexural vibrations, revealing anomalousl... | [
"micro-Raman scattering spectroscopy",
"pristine suspended graphene monolayers",
"resonators",
"quantum degrees of freedom",
"Raman-active optical phonons",
"macroscopic flexural vibrations",
"dynamically-induced strain",
"harmonic vibrations",
"graphene-based 2D resonators",
"hybrid opto-electro-... | [
{
"idx": "1.",
"title": "Bunch, J. S. et al. Electromechanical resonators from graphene sheets.Science315, 490–493 (2007).",
"link": "https://doi.org/10.1126%2Fscience.1136836"
},
{
"idx": "2.",
"title": "Castellanos-Gomez, A., Singh, V., van der Zant, H. S. J. & Steele, G. A. Mechanics of f... |
s41467-020-17935-6 | Optics and photonics | Ultrafast terahertz magnetometry | A material’s magnetic state and its dynamics are of great fundamental research interest and are also at the core of a wide plethora of modern technologies. However, reliable access to magnetization dynamics in materials and devices on the technologically relevant ultrafast timescale, and under realistic device-operatio... | [
{
"section": "Introduction",
"content": "Magnetism [1] has traditionally been a major topic of fundamental research. It also underpins numerous modern technologies ranging from high-capacity memories [2] , [3] to novel concepts in computing and data transfer [4] , [5] . Since the discovery of sub-picosecond... | The core research idea is to develop an accurate and practical ultrafast magnetometry method based on THz emission spectroscopy under ambient conditions, enabling the direct and fully calibrated determination of ultrafast magnetization dynamics $ M(t) $ in encapsulated magnetic materials, such as laser-excited iron fil... | [
"ultrafast magnetometry",
"THz emission spectroscopy",
"ambient conditions",
"ultrafast magnetization dynamics",
"encapsulated magnetic materials",
"laser-excited iron films",
"electric field component",
"magnetic dipole THz emission"
] | [] |
ncomms13752 | Optics and photonics | Multiple signal classification algorithm for super-resolution fluorescence microscopy | Single-molecule localization techniques are restricted by long acquisition and computational times, or the need of special fluorophores or biologically toxic photochemical environments. Here we propose a statistical super-resolution technique of wide-field fluorescence microscopy we call the multiple signal classificat... | [
{
"section": "Introduction",
"content": "Super-resolution fluorescence microscopy techniques aim at resolving details smaller than the Abbe diffraction limit of , where λ is the wavelength of the fluorescence emission and NA is the numerical aperture of the microscope objective. Most of these techniques use... | The core research idea is to propose a novel super-resolution fluorescence microscopy algorithm called MUltiple SIgnal Classification ALgorithm (MUSICAL), which exploits the eigenimages of a temporally varying fluorescence image stack—reflecting the statistical structure of blinking emitters—combined with knowledge of ... | [
"super-resolution fluorescence microscopy algorithm",
"MUltiple SIgnal Classification ALgorithm",
"MUSICAL",
"eigenimages",
"temporally varying fluorescence image stack",
"statistical structure",
"blinking emitters",
"point spread function",
"PSF",
"high-resolution imaging",
"SMLM techniques",
... | [
{
"idx": "1",
"title": "Betzig, E. et al. Imaging intracellular fluorescent proteins at nanometer resolution.Science313, 1642–1645 (2006).",
"link": "https://doi.org/10.1126%2Fscience.1127344"
},
{
"idx": "2",
"title": "Hess, S. T., Girirajan, T. P. & Mason, M. D. Ultra-high resolution imagi... |
ncomms11442 | Physics | Fast electronic resistance switching involving hidden charge density wave states | The functionality of computer memory elements is currently based on multi-stability, driven either by locally manipulating the density of electrons in transistors or by switching magnetic or ferroelectric order. Another possibility is switching between metallic and insulating phases by the motion of ions, but their spe... | [
{
"section": "Introduction",
"content": "Systems with first-order transitions between competing states in the phase diagram are obvious candidates for resistive memories. In the last few years, pulsed laser experiments have led to an improved understanding of non-equilibrium phase transitions [1] , [2] , [3... | The core research idea is to achieve ultrafast non-volatile resistance switching in 1T–TaS₂ through short pulsed current injection that creates domain walls via coupling to gradients of the charge density wave order parameter, enabling low-temperature ultrafast memory devices for energy-efficient superconducting comput... | [
"ultrafast non-volatile resistance switching",
"1T–TaS₂",
"short pulsed current injection",
"domain walls",
"charge density wave order parameter",
"low-temperature ultrafast memory devices",
"energy-efficient superconducting computing"
] | [
{
"idx": "1",
"title": "Ichikawa, H. et al. Transient photoinduced ‘hidden’ phase in a manganite.Nat. Mater.10, 101–105 (2011).",
"link": "https://doi.org/10.1038%2Fnmat2929"
},
{
"idx": "2",
"title": "Stojchevska, L. et al. Ultrafast switching to a stable hidden quantum state in an electron... |
s41467-023-41285-8 | Nanoscience and technology | A magneto-activated nanoscale cytometry platform for molecular profiling of small extracellular vesicles | Exosomal PD-L1 (exoPD-L1) has recently received significant attention as a biomarker predicting immunotherapeutic responses involving the PD1/PD-L1 pathway. However, current technologies for exosomal analysis rely primarily on bulk measurements that do not consider the heterogeneity found within exosomal subpopulations... | [
{
"section": "Introduction",
"content": "The discovery of the immune checkpoint PD-L1 has revolutionized cancer therapy. Interventions disrupting the PD-L1/PD-1 axis have improved clinical outcomes in numerous cancers including kidney, lung, breast, colon, and melanoma. However, due to the heterogeneous nat... | The core research idea is to develop and validate a high-throughput microfluidic platform called NanoEPIC for the molecular profiling and efficient isolation of heterogeneous small extracellular vesicles (sEVs) based on surface marker expression, particularly exosomal PD-L1 (exoPD-L1), to predict immunotherapeutic resp... | [
"NanoEPIC",
"microfluidic platform",
"molecular profiling",
"heterogeneous small extracellular vesicles",
"surface marker expression",
"exosomal PD-L1",
"exoPD-L1",
"immunotherapeutic responses",
"downstream functional analysis",
"sEV-mediated T cell suppression",
"cancer",
"high-throughput",
... | [] |
s41467-023-42678-5 | Optics and photonics | Polarization structured light 3D depth image sensor for scenes with reflective surfaces | Highly reflective surfaces are notorious in the field of depth sensing and three-dimensional (3D) imaging because they can cause severe errors in perception of the depth. Despite recent progress in addressing this challenge, there are still no robust and error-free solutions. Here, we devise a polarization structured l... | [
{
"section": "Introduction",
"content": "Highly reflective surfaces, such as glass, mirror, and water surface, are common scenes in 3D imaging. They often cause irrevocable errors in depth sensing and 3D imaging, such as mirages leading to wrong depth measurements or specular reflection blinding 3D sensors.... | This research proposes a polarization structured light (PSL) 3D sensor, which utilizes polarization properties on both the transmitter and receiver sides—employing high-contrast-grating vertical-cavity surface-emitting lasers (HCG VCSELs) for structured light with strong polarization selection and a polarization-select... | [
"polarization structured light",
"3D sensor",
"polarization properties",
"transmitter",
"receiver",
"high-contrast-grating vertical-cavity surface-emitting lasers",
"HCG VCSELs",
"structured light",
"polarization-selection CMOS camera",
"depth information",
"highly reflective surfaces",
"objec... | [
{
"idx": "2.",
"title": "Miyazaki, D., Kagesawa, M. & Ikeuchi, K. Transparent surface modeling from a pair of polarization images.IEEE Trans. Pattern Anal. Mach. Intell.26, 73–82 (2004).",
"link": "https://doi.org/10.1109%2FTPAMI.2004.1261080"
},
{
"idx": "3.",
"title": "Miyazaki, D. & Ikeuc... |
ncomms3791 | Materials science | Atomic-scale mechanisms of ferroelastic domain-wall-mediated ferroelectric switching | Polarization switching in ferroelectric thin films occurs via nucleation and growth of 180° domains through a highly inhomogeneous process in which the kinetics are largely controlled by defects, interfaces and pre-existing domain walls. Here we present the first real-time, atomic-scale observations and phase-field sim... | [
{
"section": "Introduction",
"content": "Ferroelectric thin films usually possess polydomain structures determined by electromechanical boundary conditions. For example, ferroelastic twinning structures with polarization rotations of 71° or 109° in rhombohedral perovskites (for example, BiFeO 3 ) or of 90° ... | The core research idea is to investigate the atomic-scale mechanisms by which immobile ferroelastic domains hinder polarization switching in ferroelectric thin films, particularly through the formation of a dipole glass structure at domain boundaries, using a combination of in situ transmission electron microscopy and ... | [
"atomic-scale mechanisms",
"immobile ferroelastic domains",
"polarization switching",
"ferroelectric thin films",
"dipole glass structure",
"domain boundaries",
"in situ transmission electron microscopy",
"mesoscale phase-field simulations",
"90° domain walls",
"180° domain walls",
"interaction ... | [
{
"idx": "1",
"title": "Nagarajan, V. et al. Dynamics of ferroelastic domains in ferroelectric thin films.Nat. Mater.2, 43–47 (2003).",
"link": "https://doi.org/10.1038%2Fnmat800"
},
{
"idx": "2",
"title": "Li, W. & Alexe, M. Investigation on switching kinetics in epitaxial Pb(Zr0.2Ti0.8)O3 ... |
s41467-024-47353-x | Nanoscience and technology | Single-photon detection using large-scale high-temperature MgB2sensors at 20 K | Ultra-fast single-photon detectors with high current density and operating temperature can benefit space and ground applications, including quantum optical communication systems, lightweight cryogenics for space crafts, and medical use. Here we demonstrate magnesium diboride (MgB 2 ) thin-film superconducting microwire... | [
{
"section": "Introduction",
"content": "Superconducting Nanowire Single-Photon Detectors (SNSPDs) [1] have become crucial for a variety of applications, including quantum optics [2] , [3] , [4] , [5] and security [2] , [6] , [7] , [8] , deep-space communication [9] , biomedical imaging [10] , [11] , [12] ,... | This research proposes that high-quality magnesium diboride (MgB₂) microwire-based superconducting nanowire single-photon detectors (SNSPDs), fabricated using hybrid physical-chemical vapor deposition and helium-ion-beam irradiation, can achieve single-photon sensitivity up to 20 K with large active areas and improved ... | [
"high-quality magnesium diboride microwire",
"superconducting nanowire single-photon detectors",
"hybrid physical-chemical vapor deposition",
"helium-ion-beam irradiation",
"single-photon sensitivity",
"active areas",
"reset times",
"count rates",
"high-temperature superconducting materials",
"det... | [
{
"idx": "1.",
"title": "Semenov, A. D., Gol’tsman, G. N. & Korneev, A. A. Quantum detection by current carrying superconducting film.Physica C: Superconduct.351, 4 (2001).",
"link": "https://doi.org/10.1016%2FS0921-4534%2800%2901637-3"
},
{
"idx": "2.",
"title": "Hadfield, R. H. Single-phot... |
s41467-023-38939-y | Optics and photonics | Collective excitations of a bound-in-the-continuum condensate | Spectra of low-lying elementary excitations are critical to characterize properties of bosonic quantum fluids. Usually these spectra are difficult to observe, due to low occupation of non-condensate states compared to the ground state. Recently, low-threshold Bose-Einstein condensation was realised in a symmetry-protec... | [
{
"section": "Introduction",
"content": "Bound states in the continuum (BICs) have been originally proposed as a mathematical feature of the Schrödinger equation in presence of specially prepared potentials [1] , [2] , [3] . Similar states have been later found in a wide range of systems like graphene, topo... | This work proposes that the elementary excitation spectrum of a polariton Bose-Einstein condensate formed in a bound state in the continuum (BIC) within a planar nanostructured waveguide exhibits a unique Bogoliubov dispersion characterized by a local energy minimum around the saddle point k = 0, non-zero real part des... | [
"polariton Bose-Einstein condensate",
"bound state in the continuum",
"planar nanostructured waveguide",
"Bogoliubov dispersion",
"local energy minimum",
"saddle point",
"effective mass",
"system losses",
"strong anisotropy",
"sound velocity",
"excitation spectrum engineering"
] | [
{
"idx": "2.",
"title": "Friedrich, H. & Wintgen, D. Interfering resonances and bound states in the continuum.Phys. Rev. A32, 3231 (1985).",
"link": "https://doi.org/10.1103%2FPhysRevA.32.3231"
},
{
"idx": "3.",
"title": "Capasso, F. et al. Observation of an electronic bound state above a po... |
s41467-020-16069-z | Materials science | Twist-tailoring Coulomb correlations in van der Waals homobilayers | The recent discovery of artificial phase transitions induced by stacking monolayer materials at magic twist angles represents a paradigm shift for solid state physics. Twist-induced changes of the single-particle band structure have been studied extensively, yet a precise understanding of the underlying Coulomb correla... | [
{
"section": "Introduction",
"content": "To tackle the formidable many-body scenario of Coulomb correlations in solids, Lev Landau introduced the concept of quasiparticles. These fictitious entities, which consist of particles dressed by their many-body interactions, are usually characteristic of material c... | By exploiting the internal 1s–2p resonance of excitons in homobilayers of WSe₂, we demonstrate that the twist angle between layers renormalizes the exciton binding energy by up to a factor of two, enhances their lifetime by more than an order of magnitude, and allows wide tunability of exciton–exciton interactions, whi... | [
"internal 1s–2p resonance",
"excitons",
"homobilayers",
"WSe₂",
"twist angle",
"exciton binding energy",
"exciton lifetime",
"exciton–exciton interactions",
"microscopic many-body theory",
"wavefunction overlap",
"interlayer hopping",
"hybridization effects",
"hybrid exciton species"
] | [
{
"idx": "1.",
"title": "Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices.Nature556, 80–84 (2018).",
"link": "https://doi.org/10.1038%2Fnature26154"
},
{
"idx": "2.",
"title": "Cao, Y. et al. Unconventional superconductivity in magic-angle g... |
ncomms15961 | Nanoscience and technology | Imaging quasiperiodic electronic states in a synthetic Penrose tiling | Quasicrystals possess long-range order but lack the translational symmetry of crystalline solids. In solid state physics, periodicity is one of the fundamental properties that prescribes the electronic band structure in crystals. In the absence of periodicity and the presence of quasicrystalline order, the ways that el... | [
{
"section": "Introduction",
"content": "The unique long range order of quasicrystals is evidenced by scattering [1] . Dan Shechtman first observed the 10-fold symmetric electron diffraction patterns from a rapidly cooled Al–Mn alloy [2] . Shechtman discovered a new type of material with long-range orientat... | The core research idea is to experimentally realize and investigate the electronic properties of a synthetic quasicrystal, specifically a Penrose tiling-based structure, using atomic-scale manipulation with a scanning tunnelling microscope to assemble the lattice and spectroscopic imaging to visualize its electronic de... | [
"synthetic quasicrystal",
"Penrose tiling-based structure",
"atomic-scale manipulation",
"scanning tunnelling microscope",
"spectroscopic imaging",
"electronic density of states",
"critical wavefunctions",
"localized states",
"pseudogaps",
"quasiperiodic systems",
"experimental realization",
"... | [
{
"idx": "1",
"title": "Baake, M. & Grimm, U. Mathematical diffraction of aperiodic structures.Chem. Soc. Rev.41, 6821–6824 (2012).",
"link": "https://doi.org/10.1039%2Fc2cs35120j"
},
{
"idx": "2",
"title": "Shechtman, D., Blech, I., Gratias, D. & Cahn, J. W. Metallic phase with long-range o... |
s41467-019-13077-6 | Materials science | Light-fuelled freestyle self-oscillators | Self-oscillation is a phenomenon where an object sustains periodic motion upon non-periodic stimulus. It occurs commonly in nature, a few examples being heartbeat, sea waves and fluttering of leaves. Stimuli-responsive materials allow creating synthetic self-oscillators fuelled by different forms of energy, e.g. heat, ... | [
{
"section": "Introduction",
"content": "Leaves oscillate in response to wind flow. Intriguingly, the frequency of such oscillation can be almost constant, even if driven by a gust of wind that lacks the corresponding periodicity. Leaf oscillation, together with a broad range of phenomena including heartbea... | The core research idea is to explore the mechanism of self-oscillation in liquid crystal network (LCN) photoactuators, particularly focusing on how time-delayed material response provides positive feedback to sustain motion, and to demonstrate versatile self-oscillation modes—such as bending, twisting, and contraction–... | [
"self-oscillation",
"liquid crystal network",
"LCN photoactuators",
"time-delayed material response",
"positive feedback",
"bending",
"twisting",
"contraction–expansion",
"standard cantilever-type actuators",
"novel freestyle LCN actuators",
"multiple degrees of freedom",
"spontaneous transiti... | [
{
"idx": "1.",
"title": "Jenkins, A. Self-oscillation.Phys. Rep.525, 167–222 (2013).",
"link": "https://doi.org/10.1016%2Fj.physrep.2012.10.007"
},
{
"idx": "2.",
"title": "Leclercq, T., Peake, N. & de Langre, E. Does flutter prevent drag reduction by reconfiguration?Proc. R. Soc. A474, 2017... |
s41467-025-56960-1 | Engineering | Adaptative two-phase thermal circulation system for complex-shaped electronic device cooling | Thermal management using a vapor-liquid two-phase circulation system is challenging in compact and complex-shaped electronic devices. In this study, we design and fabricate a heat pipe that can adapt to various shapes, regardless of space constraints. The heat pipe is capable of bending or twisting in three dimensions,... | [
{
"section": "Introduction",
"content": "The tech industry has consistently evolved towards miniaturization, with chip circuit fabrication becoming more refined and electronic device designs increasingly compact [1] , [2] , [3] . Conversely, the heat flux density of electronic components is rapidly increasi... | This research proposes the development and integration of a three-dimensional, shape-adaptive phase-change thermal circulation system called the adaptative heat pipe (AHP), which combines customized reverse engineering with data-driven thermal bonding to enable ultra-high thermal conductivity and full design freedom fo... | [
"three-dimensional phase-change thermal circulation system",
"adaptative heat pipe",
"customized reverse engineering",
"data-driven thermal bonding",
"ultra-high thermal conductivity",
"full design freedom",
"complex non-standard electronic devices",
"thermal management",
"shape-adaptive system",
... | [
{
"idx": "1.",
"title": "Van Erp, R., Soleimanzadeh, R., Nela, L., Kampitsis, G. & Matioli, E. Co-designing electronics with microfluidics for more sustainable cooling.Nature585, 211–216 (2020).",
"link": "https://doi.org/10.1038%2Fs41586-020-2666-1"
},
{
"idx": "2.",
"title": "Lukatskaya, M... |
ncomms4860 | Materials science | Electron spin resonance and spin–valley physics in a silicon double quantum dot | Silicon quantum dots are a leading approach for solid-state quantum bits. However, developing this technology is complicated by the multi-valley nature of silicon. Here we observe transport of individual electrons in a silicon CMOS-based double quantum dot under electron spin resonance. An anticrossing of the driven do... | [
{
"section": "Introduction",
"content": "It has long been speculated that qubits based on individual electron spins in Si quantum dots (QDs) have considerable potential for quantum information processing. Attractive features are the extremely long coherence time of spins in Si bulk materials and the possibi... | The core research idea is to investigate and demonstrate the detection of microwave-driven electron spin resonance (ESR) in a silicon metal-oxide-semiconductor (MOS)-based double quantum dot (DQD) to enable the manipulation and read-out of individual electron spin qubits, leveraging spin–valley mixing induced by spin–o... | [
"microwave-driven electron spin resonance",
"silicon metal-oxide-semiconductor",
"double quantum dot",
"electron spin qubits",
"spin–valley mixing",
"spin–orbit coupling",
"anticrossing",
"Zeeman states",
"valley-split states",
"spin–valley blockade regime",
"long coherence times",
"higher val... | [
{
"idx": "1",
"title": "Maune, B. M. et al. Coherent singlet-triplet oscillations in a silicon-based double quantum dot.Nature481, 344–347 (2012).",
"link": "https://doi.org/10.1038%2Fnature10707"
},
{
"idx": "2",
"title": "Morton, J. J. L., McCamey, D. R., Eriksson, M. A. & Lyon, S. A. Embr... |
s41467-023-39358-9 | Materials science | Interspecies exciton interactions lead to enhanced nonlinearity of dipolar excitons and polaritons in MoS2homobilayers | Nonlinear interactions between excitons strongly coupled to light are key for accessing quantum many-body phenomena in polariton systems. Atomically-thin two-dimensional semiconductors provide an attractive platform for strong light-matter coupling owing to many controllable excitonic degrees of freedom. Among these, t... | [
{
"section": "Introduction",
"content": "Excitons in two-dimensional transition metal dichalcogenides (TMDs) have large oscillator strengths and binding energies [1] , making them attractive as a platform for studies of strong light-matter coupling in optical microcavities [2] , [3] , [4] , [5] . A variety ... | The core research idea is to utilize hybridized interlayer excitons in bilayer MoS₂ to achieve highly nonlinear excitonic and dipolaritonic effects by exploiting the fermionic nature of charge carriers in a shared valence band, leading to a strong enhancement of exciton nonlinearity through broadband excitation resonan... | [
"hybridized interlayer excitons",
"bilayer MoS₂",
"nonlinear excitonic effects",
"dipolaritonic effects",
"fermionic nature",
"charge carriers",
"shared valence band",
"exciton nonlinearity",
"broadband excitation",
"interlayer exciton transitions",
"intralayer exciton transitions"
] | [
{
"idx": "1.",
"title": "Wang, G. et al. Colloquium: Excitons in atomically thin transition metal dichalcogenides.Rev. Mod. Phys.90, 21001 (2018).",
"link": "https://doi.org/10.1103%2FRevModPhys.90.021001"
},
{
"idx": "2.",
"title": "Liu, X. et al. Strong light-matter coupling in two-dimensi... |
ncomms15729 | Physics | Reconstruction of stochastic temporal networks through diffusive arrival times | Temporal networks have opened a new dimension in defining and quantification of complex interacting systems. Our ability to identify and reproduce time-resolved interaction patterns is, however, limited by the restricted access to empirical individual-level data. Here we propose an inverse modelling method based on fir... | [
{
"section": "Introduction",
"content": "The collective functionality of complex networks emerges as a consequence of the interactions among their constituents. Recently, it has been empirically observed that, besides the spatially or topologically structured organization, the temporal constraints imposed o... | The core research idea is to develop a data-driven null model of stochastic temporal networks that can reconstruct latent network structures and temporal interaction patterns from observed diffusion process data, such as time-of-arrival records, by capturing both structural and temporal regularities inherent in the dyn... | [
"data-driven null model",
"stochastic temporal networks",
"latent network structures",
"temporal interaction patterns",
"observed diffusion process data",
"time-of-arrival records",
"structural regularities",
"temporal regularities",
"dynamics of complex systems"
] | [] |
s41467-023-38237-7 | Materials science | Epitaxial growth of highly symmetrical branched noble metal-semiconductor heterostructures with efficient plasmon-induced hot-electron transfer | Epitaxial growth is one of the most commonly used strategies to precisely tailor heterostructures with well-defined compositions, morphologies, crystal phases, and interfaces for various applications. However, as epitaxial growth requires a small interfacial lattice mismatch between the components, it remains a challen... | [
{
"section": "Introduction",
"content": "Heterostructures have attracted tremendous attention due to their physicochemical properties arising from their morphologies [1] , [2] , [3] , interfaces [4] , [5] , [6] , and spatial arrangements of different components [7] , [8] . The precise control over their nan... | This research proposes a novel epitaxial growth method to synthesize highly symmetrical, branched noble metal-semiconductor heterostructures—specifically, Ag-CdS or Ag-CdSe icosapods—with precisely controlled architectures and well-defined interfaces, despite a large lattice mismatch (>40%), by leveraging the crystallo... | [
"novel epitaxial growth method",
"noble metal-semiconductor heterostructures",
"Ag-CdS icosapods",
"Ag-CdSe icosapods",
"crystallographic symmetry",
"zero-dimensional platonic Ag nanocrystals",
"one-dimensional II-VI semiconductor nanorods",
"lattice mismatch",
"(111) facets",
"precisely controlle... | [
{
"idx": "1.",
"title": "Borys, N. J., Walter, M. J., Huang, J., Talapin, D. V. & Lupton, J. M. The role of particle morphology in interfacial energy transfer in CdSe/CdS heterostructure nanocrystals.Science330, 1371–1374 (2010).",
"link": "https://doi.org/10.1126%2Fscience.1198070"
},
{
"idx": ... |
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