Surface-Quantized Anomalous Hall Current and the Magnetoelectric Effect in Magnetically Disordered Topological Insulators

We study theoretically the role of quenched magnetic disorder at the surface of topological insulators by numerical simulation and scaling analysis based on the massive Dirac fermion model. This addresses the problem of Anderson localization on chiral anomaly. It is found that all the surface states are localized, while the transverse conductivity is quantized to be $\pm \frac{e^2}{2h}$ as long as the Fermi energy is within the bulk gap. This greatly facilitates the realization of the topological magnetoelectric effect proposed by Qi et al. [Phys. Rev. B 78, 195424 (2008)] with the surface magnetization direction being controlled by the simultaneous application of magnetic and electric fields.

Figure 1

(a) Illustration of a magnetically doped topological insulator with a cylindrical geometry. (b) Surface massive Dirac dispersion. The Hall conductivity ${\sigma }_{xy}$ is quantized when the Fermi level lies in the surface gap. ${\sigma }_{xy}$ deviates from $e^2/2h$ when the Fermi level is out of the surface gap.

Figure 2

(a) Density of states, (b) Hall conductivity, and (c) diagonal conductivity of massive Dirac fermions are shown as a function of the Fermi energy divided by the mass gap $E/m$. Lines are guides for the eyes. System sizes are $L=4\hslash v_F/m$, $6\hslash v_F/m$, and $9\hslash v_F/m$.

Figure 3

Scaling flow of ${\sigma }_{xx}$ and ${\sigma }_{xy}$ as increasing system size, indicating $({\sigma }_{xx},{\sigma }_{xy})\to (0,\pm 1/2)$, in units of $e^2/h$, at $L\to \infty$. Curves are guides for the eyes. Inset: The surface Hall current is related to the bulk orbital magnetization induced by an electric field.

Figure 4

Illustration of domain structures and electromagnetic control of them. The domain wall is charged in the case of (b).