Abstract
Nonmagnetic Rashba systems with broken inversion symmetry are expected to exhibit nonreciprocal charge transport, a new paradigm of unidirectional magnetoresistance in the absence of ferromagnetic layer. So far, most work on nonreciprocal transport has been solely limited to cryogenic temperatures, which is a major obstacle for exploiting the roomtemperature twoterminal devices based on such a nonreciprocal response. Here, we report a nonreciprocal charge transport behavior up to room temperature in semiconductor αGeTe with coexisting the surface and bulk Rashba states. The combination of the band structure measurements and theoretical calculations strongly suggest that the nonreciprocal response is ascribed to the giant bulk Rashba spin splitting rather than the surface Rashba states. Remarkably, we find that the magnitude of the nonreciprocal response shows an unexpected nonmonotonical dependence on temperature. The extended theoretical model based on the secondorder spin–orbit coupled magnetotransport enables us to establish the correlation between the nonlinear magnetoresistance and the spin textures in the Rashba system. Our findings offer significant fundamental insight into the physics underlying the nonreciprocity and may pave a route for future rectification devices.
Introduction
The nonreciprocal transport of propagating particles or quasiparticles in noncentrosymmetric materials has opened up various avenues for research on symmetryrelated physical phenomena as well as potential applications in optical isolators, circulators, and microwave diodes over a broad range of frequencies^{1,2,3}. On the basis of symmetry arguments, a striking electrical manifestation of inversion symmetry breaking is the emergence of nonreciprocal charge transport, i.e., inequivalent rightward and leftward currents^{4}. Under further breaking time inversion symmetry via applying a magnetic field B, nonreciprocal charge transport characterized by the currentdirection Idependent nonlinear resistivity \(R({\mathbf{B}},{\mathbf{I}})\) can be expressed as^{5,6,7}
where R_{0}, β, and γ are the resistance at zero magnetic field, the coefficient of the normal magnetoresistance, and the nonreciprocal coefficient, respectively. In this context, the nonreciprocal response scales linearly with both the applied electric current and the magnetic field, which has been recently discovered in polar semiconductors^{6}, topological insulators (TIs)^{8}, and several interface/surface Rashba systems^{9} with spinmomentum locked bands. Unlike magnetoresistance in ferromagnet/heavy metals (FM/HMs) or FM/TI bilayers, in which the FM layer plays an essential role as a source of spindependent scattering, the nonreciprocal charge transport in noncentrosymmetric materials without FM layers introduces a new paradigm of unidirectional magnetoresistance (UMR) as a consequence of the secondorder response to the electric field^{10,11,12,13,14}. Such a UMR sparks a surge of interest in realizing twoterminal rectification, memory, and logic devices^{7,14,15}. To date, more efforts to hunt for the materials with larger γ values by taking the spin–orbit interaction and Fermi energy into account are being made in interface/surface Rashba systems^{9,15,16}. However, given the low Rashba spin splitting energy, e.g., 3 meV (~35 k_{B}) in LaAlO_{3}/SrTiO_{3}^{9}, 5 meV (~58 k_{B}) in Ge(111)^{15}, nonreciprocal transport can only be observed at very low temperature, and here γvalue decreases dramatically with increasing temperature due to the thermal fluctuation. To exploit the numerous possible applications in Rashba systems, the preservation of nonreciprocal charge transport at room temperature is vigorously pursued.
Although Rashba effect is commonly associated with lowdimensional systems and heterostructures, the recent discovery of sizeable Rashba splitting in bulk materials has attracted much attention^{17,18,19}. αGeTe, one of the emergent ferroelectric Rashba semiconductors, has a noncentrosymmetric crystal structure up to a high critical temperature T_{c} ~ 700 K^{20,21,22}. As theoretically predicted and experimentally verified, αGeTe forms a giant bulk Rashbatype spin splitting with the largest observed Rashba constant up to α ~ 5 eV Å and hosts electric fieldcontrolled Rashbatype spin textures as well^{23,24,25}. Furthermore, the corresponding spin splitting energy in αGeTe, proportional to α^{2}, reaches up to ~200 meV (~2300 k_{B})^{23}, one order of magnitude stronger than the thermal energy k_{B}T at room temperature, which thus makes it as a prominent platform to realize the nonreciprocal charge transport even at room temperature. In this work, we demonstrate the existence of nonreciprocal charge transport up to 300 K originating from the bulk Rashba states in αGeTe. Nonreciprocal coefficient γ exhibits a nonmonotonic dependence with increasing temperature T. The physical mechanism underlying the characteristics can be understood by combining the angleresolved photoelectron spectroscopy (ARPES) measurements and theoretical calculations.
Results and discussion
Basic characterizations
We fabricated the Teterminated αGeTe films on Al_{2}O_{3} (0001) substrates by molecular beam epitaxy (MBE). The rhombohedral crystal structure of αGeTe (space group R3m) with the displaced adjacent Ge and Te layers is schematically presented in Fig. 1a. Such a noncentrosymmetric structure manifests itself as the ferroelectric order P along c axis and the Rashbatype spin–orbit splitting bands^{26}. To verify the Rashbatype splitting, we performed in situ ARPES measurements using the photon energy of hν = 21.2 eV. Figure 1b shows the map of electronic band structure along the highsymmetry direction \(\overline {\Gamma}  \overline {\mathrm{K}}\), in which the Rashba splitting is most pronounced. The energy dispersion of bulk Rashba states presents a momentum splitting along \(\overline {\Gamma}  \overline {\mathrm{K}}\) with Δk ≈ 0.1 Å^{−1} and the giant Rashba parameter around 4.3 eV Å, in good agreement with other reports and calculations for Teterminated αGeTe^{23,24,25,26,27}. We also resolve the bandcrossing point (BCP) of bulk Rashba state at \({\bar{\mathrm{{\Gamma}}}}\), which is located ~0.14 eV below the Fermi level. Meanwhile, the surface states crossing the Fermi level at significantly higher momenta are also clearly discernable, in particular their linear dispersions in the vicinity of the \({\bar{\mathrm{{\Gamma}}}}\) point. In addition, the Fermi level position μ is nearly independent of the temperature T, as shown in Supplementary Fig. 4.
αGeTe thin film is patterned into Hall devices for transport measurements (see “Methods”). Figure 1c shows the temperature dependence of the resistivity of αGeTe. Unlike a usual semiconductor, αGeTe shows a low resistivity ρ ~ 0.12 mΩ cm at 300 K and a metallic behavior down to 3 K is observed clearly in its temperature dependence. Hall measurements display a linear dependence of the Hall resistances R_{xy} on the applied magnetic field, a typical ordinary Hall effect in a usual semiconductor with the conventional single carrier, as shown in the inset of Fig. 1d. The positive slope indicates that the dominant carriers are holes (Ptype) in αGeTe films. This is also confirmed via ARPES measurement, which demonstrates that Fermi surface lies in the valence band. As shown in Fig. 1d, the extracted carrier concentration n decreases from 3.0 × 10^{20} cm^{−3} at 300 K to a minimum 2.8 × 10^{20} cm^{−3} at 100 K, then it increases upon further lowering the temperature. Below ~15 K, the carrier concentration reaches a saturation value 2.9 × 10^{20} cm^{−3}. Although the carrier concentration shows anomalous temperature dependence from 3 to 300 K, its absolute change in the magnitude remains weak, revealing a negligible shift of Fermi level with temperature. According to previous reports, the metallic resistivity and high Ptype carrier concentrations are ascribed to the natural tendency for Ge deficiency in αGeTe films^{28,29,30,31}. However, the anomalous temperaturedependent carrier concentration remains puzzling, which will be discussed later.
Nonreciprocal transport response
To explore the existence of nonreciprocal transport response, we performed angulardependent ac harmonic measurements. The first (R_{ω}) and the secondharmonic resistance (R_{2ω}) upon injecting sinusoidal ac current \(\tilde I_\omega\) into devices are simultaneously detected during the rotation of the applied magnetic field in three different geometries, as illustrated in Fig. 2a. The rotation angles in xy, yz, and zx planes are defined with respect to +x, +z, and +z axis, respectively. Under a fixed strength of the magnetic field B = 9 T and at T = 10 K, the angledependent R_{ω} is found to be a sinusoidal curve with a period of 180° in all measurement geometries (see Supplementary Fig. 8), while R_{2ω} in the xy and yz planes displays a sinusoidal angular dependence with nearly same amplitude and a period of 360°, as shown in Fig. 2b. In contrast, R_{2ω} is vanishingly small when rotating the magneticfield angle in xz plane. These results indicate that R_{2ω} follows \({\mathbf{I}} \cdot ({\mathbf{P}} \times {\mathbf{B}})\) with the polarization P along +z direction and reaches its maximum value when B is applied along the +y axis.
As shown in Fig. 2c, even at 300 K, we still observed an angulardependent \(R_{2\omega }\), following \(R_{2\omega } = {\mathrm{{\Delta}}}R_{2\omega }{\mathrm{sin}}\varphi\) at a fixed injected current density \(j = 7.5 \times 10^5\;{\mathrm{Acm}}^{  2}\) in the xy rotation plane with different magneticfield strengths. Here, \({\mathrm{{\Delta}}}R_{2\omega }\) denotes the amplitudes of the angulardependent \(R_{2\omega }\) and φ represents the angle between the applied magnetic field and +x. Remarkably, \(R_{2\omega }\) changes sign when reversing the magnetic field, demonstrating a characteristic UMR. The values of \({\mathrm{{\Delta}}}R_{2\omega }\) were extracted at varied magnetic field B, as plotted in Fig. 2d. The \({\mathrm{{\Delta}}}R_{2\omega }\) clearly scales linearly with B with a negligible intercept. Moreover, the angulardependent \(R_{2\omega }\) is also measured by varying the current density j at a fixed magnetic field B = 9 T. The values of \({\mathrm{{\Delta}}}R_{2\omega }\) show a linear dependence of j, as shown in Fig. 2e. The UMR with a bilinear magnetoresistance characteristic, \({\mathrm{{\Delta}}}R_{2\omega }\) proportional to both the applied magnetic field B and the injected current j, was observed, which unambiguously verifies the existence of the nonreciprocal charge transport in αGeTe. Similar UMR characteristics were also observed in other nonmagnetic noncentrosymmetric systems with spinmomentum locking, such as BiTeBr^{6}, WTe_{2}^{32}, and SrTiO_{3}^{9}, which are distinct from those in FM/HM and FM/TI systems. The UMR in FM/HM or FM/TI originates from the spin accumulation at the interfaces induced by the spin Hall effect or the Rashba–Edelstein effect and complies with the chiral rule \(({\mathbf{j}} \times {\hat{\mathbf{z}}}) \cdot {\mathbf{M}}\) with the magnetization M^{10,11,12,33}. Thus, the values of \({\mathrm{{\Delta}}}R_{2\omega }\) in these heterostructures are also linear with the magnitude of the applied ac current and reach a saturated value at the saturation of M^{10,11}. According to the Rashba Hamiltonian \(\alpha ({\mathbf{k}} \times {\upsigma}) \cdot {\hat{\mathbf{z}}}\), the shift of wave vector Δk upon injecting charge current j gives rise to the term \(\alpha \left( {{\mathrm{{\Delta}}}{\mathbf{k}} \times {\upsigma}} \right) \cdot {\hat{\mathbf{z}}}\sim ({\hat{\mathbf{z}}} \times {\mathbf{j}}) \cdot \sigma\), which means that a pseudomagnetic field ~\({\hat{\mathbf{z}}} \times {\mathbf{j}}\) acts on the spins σ resulting in the asymmetric spindependent scattering (i.e., UMR)^{15,34}.
Furthermore, the nonreciprocal coefficient γ over a temperature range of 3–300 K in αGeTe can be extracted by \(\gamma = 2{\mathrm{{\Delta}}}R_{2\omega }/(R_0BI)\)^{6}. Figure 2f reports γ as a function of temperature T. We note that γ has the same order of magnitude over the entire temperature range. Upon increasing temperature, γ decreases slightly from 1.60 × 10^{−3} A^{−1} T^{−1} at 3 K to a minimum 1.55 × 10^{−3} A^{−1} T^{−1} at 50 K, and then increases to 2.34 × 10^{−3} A^{−1} T^{−1} at about 200 K. As the temperature increases from 200 to 300 K, γ monotonically decreases to 1.95 × 10^{−3} A^{−1} T^{−1}. The observed temperaturedependent γ(T) is remarkably distinct from those reported in other Rashba systems, in which γ significantly decreases to a very low value with increasing temperature to only a few tens of Kelvin^{6,9,15}. The remarkable trend of γ(T) from 3 K to the room temperature was reconfirmed in other αGeTe films, as shown in Supplementary Fig. 13. This unambiguously indicates that the feature displayed in Fig. 2f is authentic and one of the characteristic properties of αGeTe. The toy model based on the pseudomagnetic field discussed above is unable to properly reproduce the temperature dependence of γ^{15}. To date, two scenarios have been proposed to explain the temperaturedependent γ(T). One is associated with shift of Fermi level driven by temperature, which was used to interpret the temperaturedependent nonlinear magnetotransport observed in semimetal WTe_{2}^{32}. The other scenario involves the density of occupied states modulated by the temperaturedependent Fermi–Dirac distribution function^{6}. This model could be applied to αGeTe by considering the fact that the Fermi level position only slightly varies with the temperature, as confirmed via ARPES and carrier concentration measurements. Furthermore, in contrast with most systems investigated previously, αGeTe possesses both surface and bulk Rashba states; henceforth, their relative contributions to the magnetotransport and in particular to the temperature dependence of γ requires further scrutiny. In the following part, the contributions from both the surface and bulk Rashba states to the nonreciprocal charge transport in αGeTe are theoretically analyzed.
Theoretical analysis of nonreciprocal charge transport
On the basis of the fundamental symmetry principles, Onsager reciprocal theorem allows the nonreciprocal response existing in the systems without the inversion symmetry when the time reversal symmetry is also broken^{4,5}. Furthermore, the broken inversion symmetry manifests itself as the Rashbatype spin splitting of energy bands with spinmomentum locking^{35,36}. Therefore, nonreciprocal charge transport can be investigated using Boltzmann transport equation in Rashbatype bands^{6}. Here, we firstly construct the physical picture for the nonreciprocal transport of αGeTe by extending the nonlinear secondorder spin–orbit coupled magnetotransport model^{6,8,37,38}. This model was successfully employed to interpret the bilinear magnetoresistance of TI Bi_{2}Se_{3} by taking into account of the first and secondorder correction of the carrier distribution. For the sake of simplicity and without loss of generality, we only discuss a 2D Rashba system and neglect the variation of the Rashba constant α with temperature^{6}. Figure 3a shows a schematic of the valence band structures with Rashbatype spin splitting in momentum space (k_{x}, k_{y}), described by the Rashba Hamiltonian \(h = \frac{{\hbar \left( {k_x^2 + k_y^2} \right)}}{{2m^ \ast }} + \alpha ( {k_x\sigma _y  k_y\sigma _x} )\) with α = 4.3 eVÅ, the Pauli matrices σ_{x} (σ_{y}), and the effective mass of carrier m^{*}. The Fermi surfaces consist of two identical spin helicities above the BCP at zero magnetic field is defined as the zeropoint energy), whereas there are two opposite spin helicities below BCP, as illustrated in Fig. 3b, c. When applying an electric field E_{x}, the Fermi contours shift along the k_{x} direction. The carrier distribution function is expanded as \(f = f_0 + f_1 + f_2 + O\left( {k^3}\right)\). Here, f_{0} is the equilibrium Fermi–Dirac distribution function and \(f_n = \left( {\frac{{e\tau E_x}}{\hbar }\frac{\partial }{{\partial k_x}}} \right)^nf_0\) is the norder correction to f_{0} with the scattering time τ, the elementary charge e and the Planck’s constant \(\hbar\). The firstorder distribution f_{1} gives rise to the firstorder charge current \(J_x^1 = e{\int} {\frac{{d^2{\mathbf{k}}}}{{\left( {2\pi } \right)^2}}v\left( {\mathbf{k}} \right)f_1({\mathbf{k}})}\) with group velocity v(k) and Fermi contours subsequently generate an extra imbalance of spin population known as the Rashba–Edelstein Effect^{34,39}. In addition, a nonlinear spin current \(J_S^2\left( {E_x^2} \right)\) is also simultaneously triggered by the secondorder distribution f_{2} combined with the spinmomentum locking in Rashba bands^{40}. Note that the nonlinear spin current \(J_S^2(E_x^2)\) due to carriers with energy below the BCP (Fig. 3c) is partially compensated because the spin chirality of the inner Fermi contour is opposite to that of the outer one. In contrast, for carriers with energy above the BCP (Fig. 3b), \(J_{\mathrm{S}}^2\left( {E_x^2} \right)\) is enhanced due to the cooperative contributions of the identical spin helicities. When an external magnetic field is applied along the +y or −y direction, \(J_{\mathrm{S}}^2(E_x^2)\) is partially converted into a magnetic fielddirection dependent secondorder charge current \(J_x^2(E_x^2)\) (denoted by \(J_x^2\) below), giving rise to a UMR (i.e., nonreciprocal charge transport), as shown in Fig. 3d–e.
The secondorder charge current is further quantified in the following calculations. The Rashba Hamiltonian with inplane magnetic field B_{y} oriented along y axis reads as
Here, g and μ_{B} are the Landé gfactor and the Bohr magneton, respectively^{6,41}. For a given applied magnetic field, the typical band dispersion of the Rashba gas is depicted in Fig. 4a and can be parsed into three energy regions. The applied magnetic field gives rise to an energy shift with a value of \(g\mu _{\mathrm{B}} {B_y}\), corresponding to energy region I \(( {  \frac{{g\mu _{\mathrm{B}} {B_y}}}{2}  \frac{{m^ \ast \alpha ^2}}{{2\hbar ^2}} {\,}<{\,} E {\,}<{\,} \frac{{g\mu _{\mathrm{B}} {B_y}}}{2}  \frac{{m^ \ast \alpha ^2}}{{2\hbar ^2}}})\). Due to the giant pseudomagnetic field of the Rashba spin–orbit coupling up to ~120 T, the region I is narrow under our experimental magnetic field ~9 T. Furthermore, the inner and outer Rashba Fermi contours host opposite spin helicities below BCP (region III \(E < \frac{{\hbar ^2}}{{2m^ \ast \alpha ^2}}( {\frac{{g\mu _{\mathrm{B}}B_y}}{2}})^2\)), whereas they display identical spin helicities above BCP (region I and region II \(\frac{{\hbar ^2}}{{2m^ \ast \alpha ^2}}( {\frac{{g\mu _{\mathrm{B}}B_y}}{2}})^2 {\,}< {\,}E {\,}<{\,}  \frac{{g\mu _{\mathrm{B}} {B_y}}}{2}  \frac{{m^ \ast \alpha ^2}}{{2\hbar ^2}}\)). The secondorder charge current \(J_x^2\) is given by the following integral,
At a fixed E_{x}, \(J_x^2\) is proportional to the experimental \({\Delta}R_{2\omega }\) \(\left( {J_x^2 \propto {\Delta}R_{2\omega }} \right)\) (see Supplementary Information)^{32}.
Nonreciprocal charge transport manifests itself as a bilinear magnetoresistance, that is, ΔR_{2ω} scales linearly with both the injected charge current and the applied magnetic field. The former is implicit in Eq. (3), whereas the latter is numerically calculated in the following. Figure 4b summarizes the dependence of \(J_x^2\) as a function of B for various Fermi level positions μ. \(J_x^2\) increases linearly with B at μ = 0.13 eV and 0.066 eV (region II). Besides, \(J_x^2\) nonlinearly changes with B at μ = 0.18 eV (region I) and T = 0 K, and then \(J_x^2(B)\) returns to the linear relationship with increasing temperature (see Supplementary Fig. 2). In those cases, the change in sign of \(J_x^2\) occurs when reversing magnetic field B. In addition, the magnitude of \(J_x^2\) is mostly negligible at all magnetic fields with μ = −0.066 eV (region III). Therefore, as described above, the characteristics of bilinear magnetoresistances reappear in the nonlinear secondorder spin–orbit coupled magnetotransport model.
To identify the relative contributions of surface and bulk Rashba states in the nonreciprocal charge transport, we calculated the dependence of \(J_x^2\) on the Fermi level position μ based on Eq. (3), as illustrated in Fig. 4c. \(J_x^2\) exhibits a significant enhancement with a peak in region I and gradually decreases as Fermi level position shifts toward the lower energy in region II and then fades away with Fermi surface across BCP into region III. It is shown that \(J_x^2\) is quite sensitive to μ. In addition, the peak of \(J_x^2\) in region I progressively sinks upon increasing the temperature. Note that \(J_x^2\) in region III is nearly negligible in comparison with those in region I and II, demonstrating that the contributions of two opposite spin helicities to \(J_x^2\) compensate each other. These theoretical results provide a guideline to distinguish the contributions of surface and bulk Rashba states in the nonreciprocal charge transport. As shown in Fig. 1b, the band structure of αGeTe includes bulk as well as surface Rashba states. Based on ARPES measurements, the Fermi level lies below BCP for surface Rashba states, but above BCP for bulk Rashba states. As a consequence, it is reasonable to claim that the bulk Rashba rather than surface states are the dominant contribution to the nonreciprocal charge transport in αGeTe.
To gain further insight into the temperaturedependent γ(T), we calculated the dependence of \({\mathrm{{\Gamma}}}\left( T \right) = \left[ {\gamma \left( T \right)  \gamma \left( {0{\mathrm{K}}} \right)} \right]/\gamma \left( {0{\mathrm{K}}} \right)\), the fractional difference of γ at T and at 0 K, as a function of the temperature for various values of μ. To do so, we only consider the temperaturerelated broadening of the distribution functions. As plotted in Fig. 4d, Γ decreases monotonically with increasing temperature at μ = 0.18 eV and μ = 0.039 eV, which has been well documented in previous reports^{6,9,15}. In comparison, a novel temperature dependence Γ(T) is observed at μ = 0.066 eV, indicating γ increases monotonically as temperature increases, which has never been reported before. Assigning μ = 0.14 eV, Γ(T) shows a nonmonotonic evolution with temperature, indicating that γ increases upon increasing the temperature below ~200 K and then decreases as the temperature rises, which is in excellent qualitative agreement with our experimental observations (Fig. 2f). In particular, the calculated Fermi level position μ = 0.14 eV is also comparable with that of ARPES measurements, as shown in Fig. 1b. Consequently, it can be concluded that the temperature dependence of γ(T) derives from the variation of carrier occupation states with temperature and the sensitivity of \(J_x^2\) to the Fermi level position. This conclusion is further supported by the consistency between the experimental and theoretical temperature dependence of the carrier concentration, \(n = {\int} {\frac{{f_0d^3{\mathbf{k}}}}{{(2\pi )^3}}}\), as shown in Fig. 1d.
In summary, we unambiguously demonstrated the existence of nonreciprocal charge transport up to room temperature in Rashba semiconductor αGeTe, in which both the surface and bulk Rashba states exist. The nonreciprocal charge transport yields a UMR with a bilinear magnetoresistance characteristic. More interestingly, we observed an unconventional temperaturedependent nonreciprocal coefficient γ, in which γ increases with raising temperature below 200 K and monotonically decreases in the range of 200–300 K. To understand the physics underlying these observations, a secondorder spin–orbit coupled magnetotransport model considering the distinction of the spin chirality in Rashba bands has been developed. The combination of the ARPES measurements and theoretical calculations strongly suggests that the nonreciprocal response originates from the bulk rather than surface Rashba states, and that the unconventional temperature dependence of γ(T) is related to the Fermi level position and the secondorder correction of the distribution function. Our work offers valuable insight into the nonreciprocal response and provides pathways towards realizing the roomtemperature twoterminal spintronic devices.
Methods
Sample preparation
αGeTe films were fabricated on insulating Al_{2}O_{3} (0001) substrates by MBE with a base pressure of <2 × 10^{−9} mbar. The epiready substrate was annealed at 500 °C for 2 h in vacuum before the epitaxy. The deposition was performed using Ge and Te effusion cells set at T_{Ge} = 1140 °C and T_{Te} = 310 °C with the substrate temperature at 200 °C. Then, the samples were annealed at the deposition temperature for 30 min to improve the crystalline quality of samples.
ARPES measurements
The grown samples are transferred into the in situ ARPES chamber with a base pressure lower than 10^{−10} mbar. We used the He discharge lamp (HeI α, hv = 21.2 eV) as the photon source and then detected photoelectrons using Scienta DA30 analyzer with an energy resolution of 20 meV and angular resolution of 0.5°. ARPES measurements were performed at various temperatures.
Transport measurements
The films were patterned into Hall bar devices with the width of 5 ~30 μm (Hall bar with width of 5 μm is used for the harmonic measurement in the main text) by standard photolithography technique and Ar ion milling, and then Ti(10 nm)/Au(50 nm) electrical contacts were deposited via the electronbeam evaporation. The devices were bonded to the horizontal or vertical rotatable sample holders using Al wires and then installed in the Physical Property Measurement System (PPMS, Quantum Design) to perform the electrical properties measurements. Keithley 6221 current source was used to supply sinusoidal ac current with a frequency of 13 Hz. Meanwhile, the inphase first (0° phase) and outofphase (\(\frac{\pi }{2}\) degree phase) secondharmonic voltage signals were probed using two Stanford Research SR830 lockin amplifiers.
Data availability
The data that support this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank Dr. Keita Hamamoto and Dr. Toshiya Ideue for their useful discussions. We thank Dr. Aitian Chen for the technical support on preparing the devices. The work reported was funded by King Abdullah University of Science and Technology (KAUST), Office of Sponsored Research (OSR) under the Award numbers CRF2015SENSORS2708 and CRF20183717CRG7. This work is also supported by the National Key Research Program of China (grant numbers 2016YFA0300701 and 2017YFB0702702), the National Natural Sciences Foundation of China (Grant numbers 52031015, 1187411, and 51427801), and the Key Research Program of Frontier Sciences, CAS (Grant numbers QYZDJSSWJSC023, KJZDSWM01, and ZDYZ20122).
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Yan Li, Yang Li, Z.H.C., and X.X.Z. conceived and designed the experiments. Yang Li and X.Y. grew the films and performed the ARPES measurements. Yan Li, B.F., and C.H.Z. fabricated the devices. Yan Li, P.L., Y.W., D.X.Z., C.H.Z., and X.H. carried out the transport measurements. Yan Li performed the theoretical calculations with assistance from A.M. Yan Li and Yang Li wrote the manuscript. All authors discussed the results and contributed to the manuscript preparation.
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Li, Y., Li, Y., Li, P. et al. Nonreciprocal charge transport up to room temperature in bulk Rashba semiconductor αGeTe. Nat Commun 12, 540 (2021). https://doi.org/10.1038/s41467020208407
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DOI: https://doi.org/10.1038/s41467020208407
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