We study the intrinsic spin Hall conductivity (SHC) in various $5d$ transition metals (Ta, W, Re, Os, Ir, Pt, and Au) and $4d$ transition metals (Nb, Mo, Tc, Ru, Rh, Pd, and Ag) based on the Naval Research Laboratory tight-binding model, which enables us to perform quantitatively reliable analysis. In each metal, the obtained intrinsic SHC is independent of resistivity in the low resistive regime $(\rho <50\mu \Omega \mathrm{cm})$ whereas it decreases in proportion to ${\rho }^{-2}$ in the high resistive regime. In the low resistive regime, the SHC takes a large positive value in Pt and Pd, both of which have approximately nine $d$ electrons per ion $(n_d=9)$. On the other hand, the SHC takes a large negative value in Ta, Nb, W, and Mo, where $n_d<5$. In transition metals, a conduction electron acquires the trajectory-dependent phase factor that originates from the atomic wave function. This phase factor, which is reminiscent of the Aharonov–Bohm phase, is the origin of the SHC in paramagnetic metals and that of the anomalous Hall conductivity in ferromagnetic metals. Furthermore, each transition metal shows huge and positive $d$-orbital Hall conductivity (OHC), independent of the strength of the spin-orbit interaction. Since the OHC is much larger than the SHC, it will be possible to realize an orbitronics device made of transition metals.

• Received 8 November 2007

DOI:https://doi.org/10.1103/PhysRevB.77.165117