Odd Nonlinear Conductivity under Spatial Inversion in Chiral Tellurium

Electrical transport in noncentrosymmetric materials departs from the well-established phenomenological Ohm’s law. Instead of a linear relation between current and electric field, a nonlinear conductivity emerges along specific crystallographic directions. This nonlinear transport is fundamentally related to the lack of spatial inversion symmetry. However, the experimental implications of an inversion symmetry operation on the nonlinear conductivity remain to be explored. Here, we report on a large, nonlinear conductivity in chiral tellurium. By measuring samples with opposite handedness, we demonstrate that the nonlinear transport is odd under spatial inversion. Furthermore, by applying an electrostatic gate, we modulate the nonlinear output by a factor of 300, reaching the highest reported value excluding engineered heterostructures. Our results establish chiral tellurium as an ideal compound not just to study the fundamental interplay between crystal structure, symmetry operations and nonlinear transport; but also to develop wireless rectifiers and energy-harvesting chiral devices.

Figure 1

Basic characterization of Te. (a) False-color SEM image of a typical starlike device used in this study. The etched Te flake (green) is contacted by Pt contacts (blue). The scale bar corresponds to $2\mathrm{\mu }\mathrm{m}$. (b) Temperature dependence of the parallel first-harmonic resistance ($R_{\parallel }^{\omega }$) when $I^{\omega }$ is injected along the $x$ and $z$ axis. The data points for the latter are multiplied by 10 for clarity. (c) Parallel ($R_{\parallel }^{\omega }$) and transverse ($R_{\perp }^{\omega }$) first-harmonic resistance as a function of $\theta$ at 10 K. The dashed lines are fits to the equations introduced in the main text. (d) $\alpha$-angle-dependent magnetoresistance of $R_{zz}^{\omega }$ at 9 T and 50 K. The gray dashed line indicates the $R_{zz}^{\omega }$ value at zero field. All data were measured in device S1.

Figure 2

Nonlinear conductivity. (a),(b) Second-order parallel $V_{\parallel }^{2\omega }$ (a) and transverse $V_{\perp }^{2\omega }$ (b) voltage as a function of ${(I^{\omega })}^2$ at different $\theta$ angles. The second-order voltage depends linearly on the square of $I^{\omega }$. (c) The nonlinear conductivity [slopes of the dependences in (a) and (b)] as a function of $\theta$. The dashed lines are fits to the experimental data with Eqs. (2) and (3). All measurements were performed in device S1 at 10 K.

Figure 3

Spatial inversion operation. (a) Spatial inversion operation over trigonal Te results on a change of handedness, 180° rotation of the $x-y$ plane triangles pattern, and sign change of the nonlinear susceptibility tensor. STEM images of Te lamellas obtained by cutting perpendicular to the $z$ axis devices S1 and S2 (scale bars correspond to 2 nm). (b),(c) $\alpha$-angle-dependent magnetoresistance of $R_{zz}^{2\omega }$ at 9 T measured at 50 K in device S1 (b) and at 10 K in device S2 (c). The $R_{zz}^{2\omega }$ signal determines that devices S1 and S2 are left- and right-handed, respectively. (d),(e) Nonlinear conductivity as a function of $\theta$ at 10 K for device S1 (d) and S2 (e). The dashed lines are fits to the experimental data with Eq. (2). Data and fit in (d) are the same as in Fig. 2.

Figure 4

(a) Temperature and (b) gate dependence of the resistivity along the $x$ axis (${\rho }_{xx}$). (c) The nonlinear conductivity [taken from the slopes of $V_{\perp }^{2\omega }$ as a function of ${(I^{\omega })}^2$ at $\theta =45°$ in Figs. S7(a) and S7(b) [46] ] as a function of ${\rho }_{xx}$, tuned by either temperature (solid black circles) or gate (open red circles). The dashed lines are fits to experimental data with Eq. (4). (d) Comparison between fits to Eq. (4) with (black dashed line) and without (blue dashed line) the linear term ($\gamma$) for the temperature-tuned dataset.