Anomalous Hall Effect in Ferromagnetic Semiconductors in the Hopping Transport Regime

We present a theory of the anomalous Hall effect in ferromagnetic $(\mathrm{G}\mathrm{a},\mathrm{M}\mathrm{n})\mathrm{A}\mathrm{s}$ 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 ${\sigma }_{xy}^{\mathrm{A}\mathrm{H}}$. We show that ${\sigma }_{xy}^{\mathrm{A}\mathrm{H}}$ is proportional to the first derivative of the density of states $\varrho (ϵ)$ and thus can be expected to change sign as a function of impurity band filling. We also show that ${\sigma }_{xy}^{\mathrm{A}\mathrm{H}}$ depends on temperature as the longitudinal conductivity ${\sigma }_{xx}$ within logarithmic accuracy.

Figure 1

A backbone node with an associated triad. $I$ is the current flowing in a macrobond terminating at site 1 in response to the external electric field $E$.