Theory of Intrinsic Anomalous Hall Effect and Spin Hall Effect in Transition Metals

To elucidate the origin of anomalous Hall effect (AHE) in ferromagnetic transition metals, we study the intrinsic AHE based on a multi-orbital tight-binding model. A large anomalous velocity comes from the atomic $d$-orbital degrees of freedom. We derive a general expression for the intrinsic anomalous Hall conductivity (AHC) which is valid for any damping rate $\hbar/2\tau$. This expression enables us to calculate the AHC in metals with a wide range of resistivity $\rho$. The obtained AHC is almost constant with a value of $10^2\sim10^3\Omega^{-1}{\rm cm}^{-1}$ when $\rho$ is small, as found by Karplus and Luttinger. However, this relation does not hold any more in bad metals; we show that AHC is proportional to $\rho^{-2}$ when $\hbar/2\tau$ is larger than the minimum band-splitting measured from the Fermi level, $\Delta$. This crossover behavior of the intrinsic AHE, which was first derived by H. Kontani and K. Yamada [J. Phys. Soc. Jpn. {\bf 63} (1994) 2627], is recently observed in various ferromagnetic metals universally by A. Asamitsu et al. We also present the mechanism of spin hall effect in transition metal oxides.