Nonreciprocal charge transport up to room temperature in bulk Rashba semiconductor α-GeTe

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 room-temperature two-terminal devices based on such a nonreciprocal response. Here, we report a nonreciprocal charge transport behavior up to room temperature in semiconductor α-GeTe 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 non-monotonical dependence on temperature. The extended theoretical model based on the second-order 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.


Supplementary Note 1: Method details of ac harmonic measurements
When applying an electric field Ex, the charge current density can be expanded as = 1 + 2 = 1 + 2 2 (S1) Here, 1 and 2 are first-and second-order conductance, respectively.
In the case of 2 2 ≪ 1 , the longitudinal resistivity writes as 1 , = = Therefore, the second-order harmonic resistance 2 is expressed as, The amplitude of 2 is extracted as, The combination of Eq. S3 and Eq. S9 give ∆ 2 ∝ 2 or 2 ∝ ∆ 2 at a fixed Ex.
Supplementary Note 2: Calculation of the nonreciprocal charge transport Supplementary Fig. 1 Illustrations of the Rashba-type band structure. Rashba-type band structure with in-plane magnetic field and shape of the Fermi surface at varied .
In 2D momentum space (kx, ky), the corresponding Hamiltonian with in-plane magnetic field By along y-axis can be expressed as 2 Here, ℏ , , * , =57 (assuming the same value as SnTe 3 ) and are the Planck's constant, Rashba parameter, effective mass of electrons, Landé g-factor and Bohr magneton, respectively. and are the Pauli matrices. Eigenvalues of the Hamiltonian correspond to the two spectral branches, The total charge current density is given by the following integral, Here, ( ) = ∂ ( )/ℏ ∂ , f1 and f2 are the carrier group velocity, first and second order occupation functions, respectively. According to Boltzmann equation, f1 and f2 can be described using the Fermi-Dirac distribution f0 as 1 = ℏ 0 (S13) (S14) Here, and E are the scattering time and electric field. As seen above, the first and second charge currents depend on the terms weighed by f1 and f2. Supplementary Fig. 2 shows the magnetic field dependence of the second-order charge current 2 . It is found that the second order current is odd function about the applied magnetic field, i.e., 2 (− ) = − 2 ( ). Moreover, it is noteworthy that the characteristics of the nonreciprocal charge transport with temperature and Fermi level position are also directly influenced by the types of carriers considering that the parabolic curves of the bulk Rashba-type bands point downwards or upwards. The nonreciprocal response up to room temperature strongly benefits from its giant bulk Rashba spin splitting energy and stable noncentrosymmetric structure.
Supplementary Fig. 2 The calculated second-order charge current . 2 (in unit of = 3 2 2 * ℏ 2 2 ) as functions of the applied magnetic field By at different Fermi level position at 200 K.
In the text, we discuss a 2D Rashba to state the negligible contribution of the surface Rashba states to the nonreciprocal charge transport in α-GeTe. Here, the 2D model is extended to the bulk Rashba systems. The Hamiltonian with By along y-axis in 3D Rashba system reads as 2 , Eigenvalues of the corresponding Hamiltonian writes as, We can follow the calculation in 2D model via setting The corresponding first-,3 1 and second-order charge current ,3 2 is given by the following In order to further confirm the evolution of the Fermi level and the Rashba constant with temperature, we carried out the ARPES measurements on α-GeTe at varied temperatures. It is seen that the Fermi level position and the Rashba constant in α-GeTe keep almost unchanged with varying temperature, as shown in Supplementary Fig. 4, which is consistent with the recent report 4 . This is also in consistent with the fact that the absolute change in the temperature dependent carrier concentration remains weak in the magnitude, as shown in Fig. 1(d). To identify the contribution of the thermoelectric effect (Nernst effect) to the second-order harmonic resistance, the angular dependent second-order harmonic transverse (Rxy) resistance is also measured, as shown in Supplementary Fig. 9. It is seen that 2 doesn't nearly manifest cos or sin behaviors with R 2 /R 2 ≪ / . Therefore, we can neglect the contribution of the thermoelectric effect to the measured second-order harmonic longitudinal resistance. For convenience, we refer to the α-GeTe film with a thickness of 64 nm in main text. In reality, similar experimental results can be also observed in other α-GeTe films with different thickness.
In the following, we show our observations in the α-GeTe with a thickness of 40 nm (sample 2) as an example, as shown in Supplementary Fig. 10-13. We further performed the measurements of the nonreciprocal charge transport above 300 K using the PPMS system. As shown in Supplementary Fig. 14a, the nonreciprocal transport behavior in α-GeTe is still evident at 375 K. In addition, the vs. T curve still follows the same trend at a temperature much high than room temperature, as shown in Supplementary Fig. 14b.