Transport theory for disordered multiple-band systems: Anomalous Hall effect and anisotropic magnetoresistance

Abstract

We present a study of transport in multiple-band noninteracting Fermi metallic systems based on the Keldysh formalism and the self-consistent $T$-matrix approximation (TMA) taking into account the effects of Berry curvature due to spin-orbit coupling. We apply this formalism to a Rashba two-dimensional electron-gas ferromagnet and calculate the anomalous Hall effect (AHE) and anisotropic magnetoresistance (AMR). The numerical calculations of the AHE reproduce analytical results in the metallic regime revealing the crossover between the skew-scattering mechanism dominating in the clean systems and intrinsic mechanism dominating in the moderately dirty systems. As we increase the disorder further, the AHE starts to diminish due to the spectral broadening of the quasiparticles. Although for certain parameters this reduction of the AHE can be approximated as ${\ensuremath{\sigma}}_{xy}\ensuremath{\sim}{\ensuremath{\sigma}}_{xx}^{\ensuremath{\varphi}}$, with $\ensuremath{\varphi}$ varying around 1.6, this is found not to be true in general as ${\ensuremath{\sigma}}_{xy}$ can go through a change in sign as a function of disorder strength in some cases. Furthermore, the disordered region consistent with the TMA is relatively narrow and a theory with a wider range of applicability in strong disorder limit is called for. By considering the higher order skew-scattering processes, we resolve some discrepancies between the AHE results obtained by using the Keldysh, Kubo, and Boltzmann approaches. We also show that similar higher order processes are important for the AMR when the nonvertex and vertex parts cancel each other. We calculate the AMR in anisotropic systems properly taking into account the anisotropy of the nonequilibrium distribution function. These calculations confirm recent findings on the unreliability of common approximations to the Boltzmann equation.