An Extended Extrinsic Mechanism for Anomalous Hall Effect

The extrinsic mechanism for anomalous Hall effect in ferromagnets is extended to include the contributions both from spin-orbit-dependent impurity scattering and from the spin-orbit coupling induced by external electric fields. The results obtained suggest that, within the framework of the extrinsic mechanisms, the anomalous Hall current in a ferromagnet may also contain a substantial amount of dissipationless contribution independent of impurity scattering. After the contribution from the spin-orbit coupling induced by external electric fields is included, the total anomalous Hall conductivity is about two times larger than that due to spin-orbit dependent impurity scatterings.