Extrinsic anomalous Hall conductivity of a topologically nontrivial conduction band

A key step towards dissipationless transport devices is the quantum anomalous Hall effect, which is characterized by an integer quantized Hall conductance in a ferromagnetic insulator with strong spin-orbit coupling. In this work, the anomalous Hall effect due to the impurity scattering, namely the extrinsic anomalous Hall effect, is studied when the Fermi energy crosses with the topologically nontrivial conduction band of a quantum anomalous Hall system. Two major extrinsic contributions, the side-jump and skew-scattering Hall conductivities, are calculated using the diagrammatic techniques in which both nonmagnetic and magnetic scattering are taken into account simultaneously. The side-jump Hall conductivity changes its sign at a critical sheet carrier density for the nontrivial phase, while it remains sign unchanged for the trivial phase. The critical sheet carrier densities estimated with realistic parameters lie in an experimentally accessible range. The results imply that a quantum anomalous Hall system could be identified in the good-metal regime.

Figure 1

$|\pm ,\mathbf{k}\rangle$ represent conduction and valence bands of a quantum anomalous Hall system, respectively. When the Fermi energy (dashed) lies between the two bands, the anomalous Hall conductivity is governed dominantly by the intrinsic mechanism (${\sigma }_{xy}^i$), which is due to the chiral edge states (red solid). When the Fermi energy crosses with the conduction (or valence) band, the extrinsic mechanisms (${\sigma }_{xy}^{sj}$ for side jump and ${\sigma }_{xy}^{sk}$ for skew scattering) also contribute to the anomalous Hall conductivity.

Figure 2

The diagrams for the extrinsic Hall conductivity can be summarized to only 5 of them. Top four: Side-jump contribution. Bottom: Skew-scattering contribution. $\pm$ are band indices. $\mathbf{k},{\mathbf{k}}^{\prime },{\mathbf{k}}^{\prime \prime }$ are wave vectors. The left-bound and right-bound arrowed lines stand for retarded ($G^R$) and advanced ($G^A$) Green's functions, respectively. $U_{\mathbf{k},{\mathbf{k}}^{\prime }}$ is the scattering matrix element. The dashed lines represent the correlation between scattering. $v^{x/y}$ are the bare velocities. The shadow area and tilde represent the vortex correction to the bare velocities.

Figure 3

The side-jump Hall conductivity as a function of sheet carrier density. For all cases $\Delta =0.1$ eV and $v=4\times {10}^5$ m/s. $B=10$ eV Å${}^2$ for the nontrivial (quantum anomalous Hall) case (a) and $B=-10$ eV Å${}^2$ for the trivial case. For better comparison, the parameters are not adopted directly from those in the experiments but of the same orders. $n_C$ is about $4\times {10}^{12}$/cm${}^2$ in this case.

Figure 4

The sum of intrinsic and side-jump Hall conductivities as functions of sheet carrier density for different $V_m/V_0$. A square device is assumed so conductivity is equivalent to conductance. Parameters: $v_F=4\times {10}^5$ m/s, (a) $\Delta =0.1$ eV, $B=10$ eV Å${}^2$; (b) $\Delta =-0.1$ eV, $B=-10$ eV Å${}^2$; (c) $\Delta =-0.1$ eV, $B=10$ eV Å${}^2$; (d) $\Delta =0.1$ eV, $B=-10$ eV Å${}^2$.