Anomalous Hall Effect in Ferromagnetic Semiconductors in the Hopping Transport Regime

We present a theory of the anomalous Hall effect in ferromagnetic (Ga,Mn)As in the regime when conduction is due to phonon-assisted hopping of holes between localized states in the impurity band. We show that the microscopic origin of the anomalous Hall conductivity in this system can be attributed to a phase that a hole gains when hopping around closed-loop paths in the presence of spin-orbit interactions and background magnetization of the localized Mn moments. Mapping the problem to a random resistor network, we derive an analytic expression for the macroscopic anomalous Hall conductivity σ_xy^AH. We show that σ_xy^AH is proportional to the first derivative of the density of states ϱ(ϵ) and thus can be expected to change sign as a function of impurity band filling. We also show that σ_xy^AH depends on temperature as the longitudinal conductivity σ_xx within logarithmic accuracy.