Three-dimensional quantum Hall effect in Weyl semimetals

We analytically study the three-dimensional (3D) quantum Hall effect (QHE) in a thin film of a Weyl semimetal from the perspective of bulk states. We derive the Chern numbers for $B\ne 0$ from the Chern numbers for $B=0$ through a topological analysis and obtain a phase diagram of Chern numbers in Weyl semimetals. We demonstrate how the relative alignment of the Weyl nodes and the thickness of the film influence the quantum Hall plateaus and predict a peculiar phase diagram for the QHE in a thin film of a Weyl semimetal. Our work reveals the nature of the 3D QHE in Weyl semimetals from the bulk states.

Figure 1

Schematic illustration of (a) the two Weyl nodes in $(k_x,k_y,k_z)$ space and (b) a thin film of a Weyl semimetal in $(x,y,z)$ space with a magnetic field along the $z$ direction and an electrical field along the $x$ direction.

Figure 2

Energy spectrum as a function of $k_z$ for (a), (c) $B=0$ and (b), (d) $B>0$. In (a) and (b), $k_{wz}<0$ is assumed, and in (d) and (c), $k_{wz}>0$ is assumed. The Chern numbers $C\left(k_z\right)$ are labeled in the figure, and $\pm$ denote the chiralities of the Weyl nodes.

Figure 3

Energy spectrum as a function of $k_z$ for (a) $B=0$ and (b) $B>0$, when the Weyl nodes are separated in the $x\text{−}y$ plane, namely, $k_{wz}=0$. The Chern numbers $C\left(k_z\right)$ are labeled in the figure.

Figure 4

(a), (c) Phase diagram for QHE on the $E_F/E_g$ vs $L_z{k}_w$ plane, and (b), (d) quantized Hall conductivity in units of $e^2/h$ as a function of $E_F/E_g$ for some different values of sample thicknesses $L_z$. In (a) and (b), the Weyl nodes are separated along the $z$ direction. In (c) and (d), the Weyl nodes are separated in the $x\text{−}y$ plane. The parameters are $k_w=\pi /10{\text{nm}}^{-1}$, and $\mathbf{B}=(0,0,5T)$.