Spin Hall effect in Sr2RuO4 and transition metals (Nb,Ta)

We study the intrinsic spin Hall conductivity (SHC) and the $d$-orbital Hall conductivity (OHC) in metallic $d$-electron systems based on the multiorbital tight-binding model. The obtained Hall conductivities are much larger than that in $p$-type semiconductors. The origin of these huge Hall effects is the "effective Aharonov-Bohm phase" induced by the signs of inter-orbital hopping integrals as well as atomic spin-orbit interaction. Huge SHC and OHC due to this mecahnism is ubiquitous in multiorbital transition metals.