Role of band-index-dependent transport relaxation times in anomalous Hall effect

We revisit model calculations of the anomalous Hall effect (AHE) and show that, in isotropic Rashba-coupled two-dimensional electron gas with pointlike potential impurities, the full solution of the semiclassical Boltzmann equation (SBE) may differ from the widely used $1/{\ensuremath{\tau}}^{||}$ and $1/{\ensuremath{\tau}}^{\ensuremath{\perp}}$ solution [Schliemann and Loss, Phys. Rev. B 68, 165311 (2003)]. Our approach to solving the SBE is consistent with the integral equation approach [Vyborny et al., Phys. Rev. B 79, 045427 (2009)] but in the present case, we reduce the description to band-index-dependent transport relaxation times. When both Rashba bands are partially occupied, these are determined by solving a system of linear equations. Detailed calculations show that, for intrinsic and hybrid skew scatterings the difference between $1/{\ensuremath{\tau}}^{||}$ and $1/{\ensuremath{\tau}}^{\ensuremath{\perp}}$ and the full solution of SBE is notable for large Fermi energies. For coordinate-shift effects, the side-jump velocity acquired in the interband elastic-scattering process is shown to be more important for larger Rashba coupling and may even exceed the intraband one for the outer Rashba band. The coordinate-shift contribution to AHE in the considered case notably differs from that in the limit of smooth disorder potential analyzed before.