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

Abstract

To elucidate the origin of anomalous Hall effect (AHE) in ferromagnetic transition metals, we study the intrinsic AHE based on a multiorbital $({d}_{xz},{d}_{yz})$ tight-binding model. We find that a large anomalous velocity comes from the off-diagonal (interorbital) hopping. For this reason, the present model shows a large intrinsic anomalous Hall conductivity (AHC) which is compatible with typical experimental values in ferromagnets $({10}^{2}--{10}^{3}\phantom{\rule{0.3em}{0ex}}{\mathrm{\ensuremath{\Omega}}}^{\ensuremath{-}1}\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1})$, without the necessity to assume a special band structure at the Fermi level. In good metals where $\ensuremath{\rho}$ is small, the intrinsic AHC is constant (dissipationless) as found by Karplus and Luttinger. In bad metals, however, we find that the AHC is proportional to ${\ensuremath{\rho}}^{\ensuremath{-}2}$ when $\ensuremath{\hbar}∕2\ensuremath{\tau}$ is larger than the minimum band splitting measured from the Fermi level $\ensuremath{\Delta}$. This crossover behavior of the intrinsic AHE, which was derived in J. Phys. Soc. Jpn. 63, 2627 (1994), was recently observed in various ferromagnetic metals universally by Asamitsu et al. We also stress that the present $({d}_{xz},{d}_{yz})$ tight-binding model shows a huge spin Hall effect in a paramagnetic state.