Resonant Faraday and Kerr effects due to in-gap states on the surface of a topological insulator

When Dirac electrons on the surface of a topological insulator are gapped, the resulting quantum anomalous Hall effect leads to universal magneto-optical Faraday and Kerr effects in the low-frequency limit. However, at higher frequencies different excitations can leave their own fingerprints on the magneto-optics and can therefore be probed. In particular, we investigate the role of localized in-gap states—which inevitably appear in the presence of charged impurities—on these higher-frequency magneto-optical effects. We have shown that these states resonantly contribute to the Hall conductivity and are magneto-optically active. These in-gap states lead to peculiar resonant signatures in the frequency dependence of the Faraday and Kerr angles, distinct in character to the contribution of in-gap excitonic states, and they can be probed in ellipsometry measurements.

Figure 1

Here we plot the optical Hall conductivity in units of half the gap $(2\Delta$ is the magnetically induced gap) taking into account both the effect of localized impurity states (the subject of this paper) and chiral excitons—which can be clearly distinguished—and we compare it to the pure, noninteracting optical conductivities (dashed line). The chemical potential is at $\mu =-\Delta$ and there is a density $N/S=0.035a_0^{-2}$ [see Eq. (8)] of Coulomb impurity states with dimensionless coupling to electrons of $\alpha =0.3$. The exciton contribution is calculated with dimensionless Coulomb coupling between electrons and holes of ${\alpha }_{\mathrm{c}}=0.18$, and it is calculated in Ref. [32]. (See Sec. 2 for a discussion of $\alpha$ and ${\alpha }_c=e^2/\epsilon \hslash v_{\mathrm{F}}$.)

Figure 2

Energies of the first six states localized on a charged impurity. The vertical line represents the $\alpha$ we consider for our numerical results.

Figure 3

Given the four positions of the chemical potential illustrated in (a), Coulomb coupling $\alpha =0.3$, and a density of $N/S=0.035a_0^2$; (b) and (c) show the longitudinal and Hall conductivities, respectively. Note that the largest features are at $2\left|\mu \right|$, when the electromagnetic waves excite electrons from the valence to the conduction band. The lower-frequency features occur when electromagnetic waves excite electrons from the valence to the bound states $\left(\mu \le -\Delta \right)$ or when electromagnetic waves excite electrons from the bound state to the conduction band $\left(\mu \ge \Delta \right)$.

Figure 4

The transmittance of the electromagnetic wave for a thin film of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ in the case of a filled valence band and an unoccupied bound state $\left(\mu =-\Delta \right)$ and an occupied bound state $\left(\mu =\Delta \right)$.

Figure 5

The measurable optical quantities of (a) the Faraday angle, (b) the ellipticity of the transmitted wave, (c) the Kerr angle, and (d) the ellipticity of the reflected wave for ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ in the case of a filled valence band with an unoccupied bound state $\left(\mu =-\Delta \right)$ and an occupied bound state $\left(\mu =\Delta \right)$. As expected, the features correspond to the features in the optical conductivities.