Scaling relation of the anomalous Hall effect in (Ga,Mn)As

We present magnetotransport studies performed on an extended set of (Ga,Mn)As samples at 4.2 K with longitudinal conductivities ${\ensuremath{\sigma}}_{xx}$ ranging from the low-conductivity to the high-conductivity regime. The anomalous Hall conductivity ${\ensuremath{\sigma}}_{xy}^{(\text{AH})}$ is extracted from the measured longitudinal and Hall resistivities. A transition from ${\ensuremath{\sigma}}_{xy}^{(\text{AH})}=20\text{ }{\ensuremath{\Omega}}^{\ensuremath{-}1}\text{ }{\text{cm}}^{\ensuremath{-}1}$ due to the Berry phase effect in the high-conductivity regime to a scaling relation ${\ensuremath{\sigma}}_{xy}^{(\text{AH})}\ensuremath{\propto}{\ensuremath{\sigma}}_{xx}^{1.6}$ for low-conductivity samples is observed. This scaling relation is consistent with a recently developed unified theory of the anomalous Hall effect in the framework of the Keldysh formalism. It turns out to be independent of crystallographic orientation, growth conditions, Mn concentration, and strain, and can therefore be considered universal for low-conductivity (Ga,Mn)As. The relation plays a crucial role when deriving values of the hole concentration from magnetotransport measurements in low-conductivity (Ga,Mn)As. In addition, the hole diffusion constants for the high-conductivity samples are determined from the measured longitudinal conductivities.