Intrinsic anomalous Hall effect in ferromagnetic metals studied by the multi d-orbital tight-binding model

To elucidate the origin of anomalous Hall effect (AHE) in ferromagnetic transition metals, we study the intrinsic AHE based on a multi-orbital (xz,yz)-tight-binding model. We find that a large anomalous velocity comes from the off-diagonal (inter-orbital) hopping. By this reason, the present model shows a large intrinsic anomalous Hall conductivity (AHC) which is compatible with typical experimental values in ferromagnets [100-1000 [1/\Omega cm]], without necessity to assume a special band structure at the Fermi level. In good metals where resistivity \rho is small, the intrinsic AHC is constant (dissipation-less) as found by Karplus and Luttinger. In bad metals, however, we find that the AHC is proportional to \rho^{-2} when \hbar/2\tau is larger than the minimum band-splitting measured from the Fermi level. This crossover behavior of the intrinsic AHE, which was first derived in J. Phys. Soc. Jpn. 63 (1994) 2627, is recently observed in various ferromagnetic metals universally by A. Asamitsu et al. We also stress that the present (xz,yz)-tight binding model shows a huge spin Hall effect in a paramagnetic state.