Surface quantized anomalous Hall current and magneto-electric effect in magnetically disordered topological insulators

We study theoretically the role of quenched magnetic disorder at the surface of a topological insulator by numerical simulation and scaling analysis. 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 {\it et al.} (Phys. Rev. B$\mathbf{\text{78}}$, 195424 (2008)) with the surface magnetization direction being controlled by the simultaneous application of magnetic and electric fields .