Charge and Spin Transport in Edge Channels of a $\nu =0$ Quantum Hall System on the Surface of Topological Insulators

Three-dimensional topological insulators of finite thickness can show the quantum Hall effect (QHE) at the filling factor $\nu =0$ under an external magnetic field if there is a finite potential difference between the top and bottom surfaces. We calculate energy spectra of surface Weyl fermions in the $\nu =0$ QHE and find that gapped edge states with helical spin structure are formed from Weyl fermions on the side surfaces under certain conditions. These edge channels account for the nonlocal charge transport in the $\nu =0$ QHE which is observed in a recent experiment on $({\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x{)}_2{\mathrm{Te}}_3$ films. The edge channels also support spin transport due to the spin-momentum locking. We propose an experimental setup to observe various spintronics functions such as spin transport and spin conversion.

Figure 1

Schematic picture of the two-dimensional (2D) model for the surface Weyl fermions of a topological insulator in a magnetic field. A 2D plane in the lower panel represents the top, bottom, and side surfaces, where the strength of the magnetic field is $\pm B$ for $x>d$ and $x<-d$ and vanishing for $-d<x<d$.

Figure 2

Band structure of the surface Weyl fermions in the 2D model. (a) The energy levels of $H$ in the Landau gauge [Eq. (1)] are plotted against the momentum $k_y$ for $2d=30\mathrm{nm}$, $B=15\mathrm{T}$, $2V_0=50\mathrm{meV}$. Colors encode the expectation value of the position $⟨x⟩$; the Landau levels on the top and bottom surfaces are depicted in red and blue in the region $k_y<0$, while dispersive bands in green are edge channels on the side surface. LLs at the top (bottom) surface are labeled by indices $N_T$ ($N_B$). (b) Spin textures in the $(s_x,s_z)$ plane and (c) expectation values $⟨x⟩$ of the two bands connected to the $N_T=0$ and $N_B=0$ LLs. Edge channels ($k_y\gtrsim 0$) show the spin-momentum locking, while the $N=0$ Landau levels on the top and bottom surfaces are spin polarized.

Figure 3

A setup for the detection of charge and spin transport through edge channels. Applying a voltage between the electrodes 1 and 2 in the left induces charge and spin transport through the edge channels in the $\nu =0$ quantum Hall state. The nonlocal charge transport is measured by the voltage $V_{34}$ between the electrodes 3 and 4. The spin current is detected by the voltage $V_3$ between the two sides of the Pt electrode 3 via the inverse spin Hall effect.

Figure 4

(a) Spin conductance as a function of the Fermi energy. Spin conductance is finite in the $\nu =0$ quantum Hall state when the Fermi level is in the edge channels. (b) Spin conversion rate plotted against electric fields $E_y$. Dots represent the spin conversion rate obtained from numerical solutions of the time-dependent Schrödinger equation, while the solid curve is given by the Zener tunneling formula [Eq. (9)]. Results in both panels are obtained for the same parameters used in Fig. 2.