Noncommutative quantum mechanics and skew scattering in ferromagnetic metals

The anomalous Hall effect is classified into two effects based on the mechanism. The first one is the intrinsic Hall effect due to the Berry curvature in momentum space. This is a Hall effect that solely arises from the band structure of solids. On the other hand, another contribution to the Hall effect, the so-called extrinsic mechanism, comes from impurity scatterings such as skew scattering and side jump. These two mechanisms are often discussed separately; the intrinsic Hall effect is related to the Berry curvature of the band while the skew scattering is studied using the scattering theory approaches. However, we here show that, in an electronic system with finite Berry curvature, the skew scattering by nonmagnetic impurities is described by the noncommutative nature of the real-space coordinates due to the Berry curvature of the Block wave functions. The anomalous Hall effect due to this skew scattering is estimated and compared with the intrinsic contribution.