Electric spin and valley Hall effects

Abstract

The electric Hall effect (EHE) is a newly identified Hall effect characterized by a perpendicular electric field inducing a transverse charge current in two-dimensional (2D) systems. Here, we propose a spin and valley version of EHE. We demonstrate that the transverse spin and valley currents can be generated in an all-in-one tunnel junction based on a buckled 2D hexagonal material in response to a perpendicular electric field, referred to as the electric spin Hall effect and electric valley Hall effect, respectively. These effects arise from the perpendicular-electric-field-induced backreflection phase of electrons in the junction spacer, independent of Berry curvature. The valley Hall conductance exhibits an odd response to the perpendicular electric field, whereas the spin Hall conductance shows an even one. The predicted effects can further enable the transverse separation of a pair of pure spin-valley-locked states with full spin-valley polarization while preserving time-reversal symmetry, as manifested by equal spin and valley Hall angles. Our findings present a new mechanism for realizing the spin and valley Hall effects and provide a novel route to the full electric-field manipulation of spin and valley degrees of freedom, with significant potential for future applications in spintronics and valleytronics.

Figures (4)

• Figure 1: Schematic of the all-in-one tunnel junction based on a buckled 2D material (top panel) and of its side view (bottom panel), where the left and right electrode regions are separated by the central spacer region controlled by the top and bottom gates. A perpendicular electric field $\mathcal{E}$ (blue arrow) is applied on the right electrode region. The electrons with both opposite spin and valley are transversely separated, denoted by the red and green arrows. The red and blue circles in the bottom panel represent the $A$ and $B$ sublattices, respectively.
• Figure 2: Contour plot of the transmission probability $T_{\eta s}$ as a function of the junction length $w$ and the transverse wave number $q_y$ for $E=10\meV$, $V=20\meV$, and $\lambda_{\mathrm{SO}}=4\meV$. (a)-(d) Spin- and valley-dependent transmission for $\mathcal{E}=22meV\per\angstrom$. The spin-valley channel labeled by $|\eta,s\rangle$ is indicated above the corresponding figure panel. (e) Transmission at $\mathcal{E}=0$.
• Figure 3: [(a), (b)] Valley Hall conductance $\sigma^{\mathrm{V}}_{yx}$ versus $w$ and $\mathcal{E}$, respectively. [(c), (d)] Spin Hall conductance $\sigma^{\mathrm{S}}_{yx}$ versus $w$ and $\mathcal{E}$, respectively. $N_0=EW/\pi\hbar v_F$ is the number of the transverse modes with $W$ being the width of the junction. The incident energy is $E=8\meV$. In the legend, the electric field is given in $meV\per\angstrom$ and the length in $nm$. The remaining parameters are taken to be the same as Fig. \ref{['fig:tran']}. The inset in (b) shows a zoom-in of the boxed region.
• Figure 4: Hall angle (a) and spin-valley polarization (b) versus $\mathcal{E}$ at $E=8meV$, $\lambda_{\mathrm{SO}}=4\meV$, and $w=198nm$.