Anomalous Hall Effect in Ferromagnetic Semiconductors in the Hopping Transport Regime

TL;DR

This paper develops a microscopic theory for the Anomalous Hall Effect in ferromagnetic semiconductors in the hopping regime, linking it to phase accumulation during hole hopping influenced by spin-orbit interactions and magnetization.

Contribution

It introduces an analytic expression for the anomalous Hall conductivity in the hopping regime, connecting it to the density of states and temperature dependence.

Findings

$\sigma_{xy}^{AH}$ is proportional to the derivative of the density of states.

$\sigma_{xy}^{AH}$ can change sign with impurity band filling.

$\sigma_{xy}^{AH}$ varies logarithmically with temperature, tracking $\sigma_{xx}$.

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.