Anomalous Hall Effect in Disordered Multiband Metals

We present a microscopic theory of the anomalous Hall effect (AHE) in metallic multiband ferromagnets, which accounts for all scattering-independent contributions, i.e., both the intrinsic and the so-called side jump. For a model of Gaussian disorder, the AHE is expressed solely in terms of the material’s electronic band structure. Our theory handles systematically the interband-scattering coherence effects. We demonstrate the method in the 2D Rashba and 3D ferromagnetic (III,Mn)V semiconductor models. Our formalism is directly amenable to ab initio treatments for a wide range of ferromagnetic metals.

Figure 1

The SJC diagrams for ${\sigma }_{ij}$ expressed in the band eigenstate basis described by the indices $m$ and $n$ ($m\ne n$). The bold lines correspond to ${\widehat{G}}_d^{R(A)}$ (that can be replaced by disorder-free GF’s for calculation of the SJC [11], while dashed lines correspond to disorder strength $\mathcal{V}$. By iterating Eq. (9) and expanding as in Eq. (6), the leading-order contributions in Eq. (7) can be expressed as two components of SJC. The diagrams beginning and ending with velocity involving single (multiple) band(s) are termed as intraband (interband) diagrams.

Figure 2

SJC and IC as a function of the mean field; (a) ${\Delta }_{\mathrm{so}}\to \infty$ corresponds to four-band model and ${\Delta }_{\mathrm{so}}=341\mathrm{meV}$ corresponds to GaAs host, the hole density is $0.35{\mathrm{nm}}^{-3}$; (b) the plots correspond to the InAs (GaAs) hosts with the hole densities $0.1(0.35){\mathrm{nm}}^{-3}$. By arrows, we mark the mean fields $\Omega =25(122)\mathrm{meV}$ for the former (latter) [5, 14].