Intrinsic spin Hall effect and orbital Hall effect in4dand5dtransition metals

Abstract

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 $(\ensuremath{\rho}<50\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\ensuremath{\Omega}\phantom{\rule{0.2em}{0ex}}\mathrm{cm})$ whereas it decreases in proportion to ${\ensuremath{\rho}}^{\ensuremath{-}2}$ 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.