Spin-Hall Insulator

Recent theories predict dissipationless spin current induced by an electric field in doped semiconductors. Nevertheless, the charge current is still dissipative in these systems. In this work, we theoretically predict the dissipationless spin-Hall effect, without any accompanying charge current, in some classes of band insulators, including zero-gap semiconductors such as HgTe and narrow-gap semiconductors such as PbTe. This effect is similar to the quantum-Hall effect in that all the states below the gap contribute and there occurs no dissipation. However, the spin-Hall conductance is not quantized even in two dimensions. This is the first example of a nontrivial topological structure in a band insulator without any magnetic field.

Figure 1

Spin-Hall conductivity ${\sigma }_s$ as a function of ${\gamma }_2/{\gamma }_3$, calculated from the tight-binding Hamiltonian modeling a zero-gap semiconductor without doping. $e_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}}=m{a}^2{E}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}}/{\gamma }_3$ is a dimensionless parameter representing the uniaxial compressive strain, where $2E_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}}$ is an energy splitting at $\mathbf{k}=\mathbf{0}$ due to the strain. $e_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}}\sim 0.014$ corresponds to the value of $\alpha$-Sn in a compressive stress $3.18\times {10}^9{\mathrm{d}\mathrm{y}\mathrm{n}/\mathrm{c}\mathrm{m}}^2$ [14]. Inset: a schematic picture of the band structure for the zero-gap semiconductors with uniaxial strain. The labels LH, HH, and CB corresponds to the light-hole, heavy-hole, and the conduction bands in the conventional semiconductors with cubic symmetry, respectively. $E_F$ is the Fermi energy.

Figure 2

Spin-Hall conductivity ${\sigma }_s$ as a function of $\lambda$ for various $m=Mva$, calculated from the tight-binding Hamiltonian modeling a narrow-gap semiconductor without doping.