Three-dimensional quantum anomalous Hall effect in Weyl semimetals

Abstract

The quantum anomalous Hall effect (QAHE) is a quantum phenomenon in which a two-dimensional system exhibits a quantized Hall resistance $h/e^2$ in the absence of magnetic field, where $h$ is the Planck constant and $e$ is the electron charge. In this work, we extend this novel phase to three dimensions and thus propose a three-dimensional QAHE exhibiting richer and more versatile transport behaviors. We first confirm this three-dimensional QAHE through the quantized Chern number, then establish its bulk-boundary correspondence, and finally reaffirm it via the distinctive transport properties. Remarkably, we find that the three-dimensional QAHE hosts two chiral surface states along one spatial direction while a pair of chiral hinge states along another direction, and the location of the hinge states depends sensitively on the Fermi energy. These two types of boundary states are further connected through a perpendicular chiral surface states, whose chirality is also Fermi energy dependent. Consequently, depending on the transport direction, its Hall resistance can quantize to $0$, $h/e^2$, or $\pm h/e^2$ when the Fermi energy is tuned across the charge neutral point. This three-dimensional QAHE not only fill the gap in the Hall effect family but also holds significant potentials in device applications such as in-memory computing.

Figures (2)

• Figure 1: Construction of 3D QAHE and its bulk-boundary correspondence. (a) Schematics for the band dispersions in the absence of spin-orbit coupling ($t_{\text{s}}=0$). (b) Calculated bulk band spectrum of the system along highly symmetric points in the 1st BZ with Rashba spin-orbit coupling ($t_{\text{s}}=0.5t$). (c) Two-dimensional surface band spectra of a slab and corresponding wavefunction distributions on $k_{\text{x}}$-$k_{\text{y}}$ (left), $k_{\text{z}}$-$k_{\text{x}}$ (right) planes, respectively. The surface band spectra on the $k_{\text{y}}-k_{\text{z}}$ plane are presented in Supplementary Note 2A. Here, the thickness of the slab is $L=10$. (d,e) Band spectra on a one-dimensional nanowire (left panels) and corresponding wave function distributions. The red and blue colors here represent the propagating direction identical to the group velocity. The system size is $L_{\text{y/x}}=L_{\text{z}}=10$. (f) One-dimensional band structure on a nanowire extending infinitely along $z$ direction and the average position $\langle y/L_{\text{y}}\rangle$. The system size is $L_{\text{x}}=L_{\text{y}}=8$. (g) Mechanism of the emergence of chiral surface states along $z$ direction. (h) Edge picture of the 3D QAHE at $E_{\text{f}}>0$ (top) and $E_{\text{f}}<0$ (bottom).
• Figure 2: Transport properties of the 3D QAHE. (a,b,c) Six different Hall bar configurations in three dimensions and corresponding edge states (colorful arrows) at different Fermi energy $E_{\text{f}}$. Note that, for clarity, only the edge states at $E_{\text{f}}>0$ ($E_{\text{f}}<0$) are labeled in top (bottom) panels, respectively, in (b) and (c), while both cases are allowed the same Hall bar configuration. Hall resistance and longitudinal resistance for each configuration obtained from the edge picture analysis are labeled below. (d,e,f) Hall resistances (solid lines) and longitudinal resistances (dashed lines) versus the Fermi energy $E_{\text{f}}$ for different Hall bar configurations calculated by using Green's function method. Here, the size for all six metallic leads in each Hall bar is $10\times10$, and the distance between neighboring leads is $L=20$. All data are obtained in the presence of a random disorder with strength $W=0.5t$ under 100 times ensemble average.