Competing effects of magnetic impurities in the anomalous Hall effect on the surface of a topological insulator

We investigate the anomalous Hall effect (AHE) on the surface of a topological insulator induced by a finite concentration of magnetic impurities, and find topologically nontrivial and trivial mechanisms simultaneously contributing to the Hall conductivity. In the topologically nontrivial mechanism, the impurities gap the surface spectrum and result in a half-integer quantized intrinsic Hall conductivity in units $e^2/h$, while in the topologically trivial mechanism, the half-integer quantized plateau is modified by impurity-induced localized states via a gap-filling process. The nonmagnetic charge potential itself, though participating in the gap-filling process, cannot induce the AHE. In the presence of a finite magnetic potential, the charge potential would destroy the symmetric distribution of the Hall conductivity by redistributing the localized levels. More interestingly, the sign of the Hall conductivity is tunable by changing the strength of the charge potential.

Figure 1

(a) The low-energy band structure, given by $-\mathrm{Im}\left\{\mathrm{Tr}\left[G(\mathbf{k},{\omega }^{+})\right]\right\}/\pi$, for $U_0=0$ and $JS=1$. (b)–(d) The DOS of the TSSs as a function of $\omega$ for some different choices of the parameters $U_0$ and $JS$. Here, $JS=J|{\mathbf{S}}_z|$, and the other parameters are taken to be $\mu =0, \mathrm{\Lambda }=300\mathrm{meV}$ (see, e.g., Ref. [8]) and $\rho =0.1$ (the experimental value is about 0.08 measured in Ref. [41]). $U_0$ and $JS$ are in units of $\mathrm{meV}$.

Figure 2

Hall conductivities ${\sigma }_{xy}^{\text{I}}$ and ${\sigma }_{xy}^{\text{II}}$ vs $E_{\mathrm{F}}$ for $JS=0.2$ in (a) and (b), and varied $JS$ in (c) and (d). Here, $U_0=0$, and the other parameters are chosen to be the same as in Fig. 1.

Figure 3

Hall conductivity ${\sigma }_{xy}={\sigma }_{xy}^{\text{I}}+{\sigma }_{xy}^{\text{II}}$ vs $E_{\mathrm{F}}$ in (a), and vs $U_0$ in (b), for $J|S_z|=2$. The other parameters are the same as in Fig. 1.