Anomalous Hall Effect in Ferromagnetic Semiconductors in the Hopping Transport Regime

Abstract

We present a theory of the Anomalous Hall Effect (AHE) 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 ${\sigma }_{xy}^{AH}$. We show that ${\sigma }_{xy}^{AH}$ is proportional to the first derivative of the density of states $\varrho (\epsilon )$ and thus can be expected to change sign as a function of impurity band filling. We also show that ${\sigma }_{xy}^{AH}$ depends on temperature as the longitudinal conductivity ${\sigma }_{xx}$ within logarithmic accuracy.

arXiv comment4 pages, 1 eps figure, final version

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