Electromagnetic spin polarization on the surface of topological insulator

We study the spin polarization of the electrons on the surface of topological insulators under a dc electric field or a circularly polarized light by using the Keldysh Green's function formalism. When a dc electric field $E_x\sim {10}^3\mathrm{V}/\mathrm{m}$ is applied, a spin polarization $\langle {\sigma }_y\rangle \simeq 5.2\times {10}^{-8}{\AA }^{-2}$ is induced. Furthermore, we also find that a light illumination induces the out-of-plane component of the spin polarization as a result of the inverse Faraday effect. The magnitude of the spin polarization is proportional to the square of the lifetime $\tau$ and $\langle {\sigma }_z\rangle \simeq 2\times {10}^{-10}{\AA }^{-2}$ for typical parameters. Finally, we investigate the spin polarization in the presence of the warping term. By the symmetry consideration of the system, we find that the out-of-plane spin polarization is cubic of the current and that the magnitude of the induced spin polarization depends on the direction of the applied current.

Figure 1

The perturbed Green's function obtained by the self-consistent Born approximation. The single and double lines mean the unperturbed and perturbed Green's functions, respectively, and the dashed lines are scattering potentials by the impurities.

Figure 2

We have to include the diagrams shown in (a) to satisfy the Ward-Takahashi identity, which ensures the gauge invariance. (b) and (c) are the iterative method we use to get the $n$th ladder diagrams shown above.

Figure 3

The circularly polarized light propagating along the $z$ axis induces the spin polarization $\langle {\sigma }_z\rangle$.

Figure 4

(a) and (b) correspond to the terms $g_{\omega }V{g}_{\omega -\Omega }V{g}_{\omega }$ and $g_{\omega }V{g}_{\omega +\Omega }V{g}_{\omega }$, respectively. These terms give the diagonal components in the Fourier-transformed $\omega$ space.

Figure 5

The spin configuration of the eigenvectors for the $s=+$ band. The arrows in this figure indicate the in-plane spin components of the eigenstates and the solid circle (cross) indicates the out-of-plane spin components. At the cusp points the spin lies completely in the $xy$ plane just as in the case without the warping term. However, at other points the spin has nonzero ${\sigma }_z$ component, which is the largest in its magnitude at the valley points.