Coherent description of the intrinsic and extrinsic anomalous Hall effect in disordered alloys on an $ab$ $initio$ level

Abstract: A coherent description of the anomalous Hall effect (AHE) is presented that is applicable to pure as well as disordered alloy systems by treating all sources of the AHE on equal footing. This is achieved by an implementation of the Kubo-St\v{r}eda equation using the fully relativistic Korringa-Kohn-Rostoker (KKR) Green's function method in combination with the Coherent Potential Approximation (CPA) alloy theory. Applications to the pure elemental ferromagnets bcc-Fe and fcc-Ni led to results in full accordance with previous work. For the alloy systems fcc-Fe$x$Pd${1-x}$ and fcc-Ni$x$Pd${1-x}$ very satisfying agreement with experiment could be achieved for the anomalous Hall conductivity (AHC) over the whole range of concentration. To interpret these results an extension of the definition for the intrinsic AHC is suggested. Plotting the corresponding extrinsic AHC versus the longitudinal conductivity a linear relation is found in the dilute regimes, that allows a detailed discussion of the role of the skew and side-jump scattering processes.