Anomalous Hall effect in a two-dimensional electron gas

The anomalous Hall effect in a magnetic two-dimensional electron gas with Rashba spin-orbit coupling is studied within the Kubo-Streda formalism in the presence of pointlike potential impurities. We find that all contributions to the anomalous Hall conductivity vanish to leading order in disorder strength when both chiral subbands are occupied. In the situation that only the majority subband is occupied, all terms are finite in the weak scattering limit and the total anomalous Hall conductivity is dominated by skew scattering. We compare our results to previous treatments and resolve some of the discrepancies present in the literature.

Figure 1

Single particle dispersion for (a) small spin-orbit interaction $\alpha k_F∕h=0.2$ and (b) large spin-orbit interaction $\alpha k_F∕h=2.0$.

Figure 2

Diagrammatic representation (a) of the bare bubble, (b) of the ladder vertex corrections, and (c) of the skew-scattering contribution.

Figure 3

Full vertex including ladder and skew-scattering diagrams.

Figure 4

(Color online) Anomalous Hall conductivity for $\alpha k_F∕h=0.2$ and an impurity concentration of $n_i=0.1$ plotted as a function of ${ϵ}_F∕h$ (from right to left) and as a function of $1∕\tau =n_{i}m{V}_0^2$ in units of $h$ (from back to front): upper left panel, total anomalous Hall conductivity [Eq. (19)]; upper right panel, skew-scattering contribution [Eq. (35b)]; lower left panel; bare bubble plus ladder vertex corrections [Eq. (25)+Eq. (31)]; lower right panel, $-{\sigma }^{\mathrm{II}}$ [Eq. (23)]. All conductivities are plotted in units of $e^2$.

Figure 5

(Color online) Anomalous Hall conductivity for $\alpha k_F∕h=10.0$ and an impurity concentration of $n_i=0.1$ plotted as a function of ${ϵ}_F∕h$ (from right to left) and as a function of $1∕\tau =n_{i}m{V}_0^2$ in units of $h$ (from back to front): upper left panel, total anomalous Hall conductivity [Eq. (19)]; upper right panel, skew-scattering contribution [Eq. (35b)]; lower left panel, bare bubble plus ladder vertex corrections [Eq. (25)+Eq. (31)]; lower right panel, $-{\sigma }^{\mathrm{II}}$ [Eq. (23)]. All conductivities are plotted in units of $e^2$.