Surface-step defect in three-dimensional topological insulators: Electric manipulation of spin and quantum spin Hall effect

We study the influence of a step defect on surface states in three-dimensional topological insulators subject to a perpendicular magnetic field. By calculating the energy spectrum of the surface states, we find that Landau levels (LLs) can form on flat regions of the surface and are distant from the step defect, and several subbands emerge at the side surface of the step defect. The subband which connects to the two zeroth LLs is spin polarized and chiral. In particular, when the electron transports along the side surface, the electron spin direction can be manipulated arbitrarily by gate voltage. Also, no reflection occurs even if the electron spin direction is changed. This provides a fascinating avenue to control the electron spin easily and coherently. In addition, regarding the subbands with a high LL index, there exist spin-momentum locking helical states and the quantum spin Hall effect can appear.

Figure 1

(a) Schematic of upper and lower surfaces connected by a side surface (step defect) in a 3D TI under a perpendicular magnetic field. (b) Mapping of the above surfaces in the regions of $x>d,-d<x<d$, and $x<-d$ of a 2D model. The potential energies of the upper and lower surfaces can be controlled by the gate voltage and no magnetic field threads into the side surface.

Figure 2

Dispersion relation of the 2D model with (a) $V_0=0\mathrm{meV}$ and (b) $V_0=25\mathrm{meV}$. The other parameters are $2d=30\mathrm{nm}$ and $B=15\mathrm{T}$.

Figure 3

Spatial distribution ${\mathbf{s}}_{x,y,z}$ of the electron spin for the chiral state, which connects to the two zeroth LLs of the upper and lower surfaces, with the potential energy (a) $V_0=0\mathrm{meV}$, (b) $V_0=10\mathrm{meV}$, (c) $V_0=25\mathrm{meV}$, and (d) $V_0=50\mathrm{meV}$. The insets display a probability density $\rho \left(x\right)$. Here, we choose $k_y=0.01{\mathrm{nm}}^{-1},2d=30\mathrm{nm}$, and $B=15\mathrm{T}$.

Figure 4

(a) A setup for manipulation of the electron spin under nonuniform gate voltage. (b) Spin texture in the ${\mathbf{s}}_x\text{−}{\mathbf{s}}_z$ plane as a function of the gate voltage $V_0$. Tilting angle $\theta$ vs $V_0$ for (c) different magnetic fields $B$, for (d) various heights $2d$ of the side surface, and for (e) several momenta $k_y$. The unmentioned parameters are $B=15\mathrm{T},2d=30\mathrm{nm}$, and $k_y=0.01{\mathrm{nm}}^{-1}$.

Figure 5

(a) and (c) Spatial distribution ${\mathbf{s}}_{x,y,z}$ for the electronic state in the front side surface, which connects the two LLs of $N=1$, with (a) $k_y=0.1{\mathrm{nm}}^{-1}$ and (c) $k_y=-0.1{\mathrm{nm}}^{-1}$. (b) and (d) show the corresponding ${\mathbf{s}}_{x,y,z}$ for the back side surface. The parameters are $B=15\mathrm{T},2d=30\mathrm{nm}$, and $V_0=0\mathrm{meV}$. In the calculation of (b) and (d), we make a local rotation of the $x$ and $z$ coordinates around the $y$ axis by $-{90}^{\circ }$, and the region of $-d<x<d$ is still the side surface. The inset shows a schematic view of a convex platform in the surface of the 3D TI, and the front and back side surfaces form naturally.