Study of Intrinsic Spin Hall Effect and Orbital Hall Effect in 4d- and 5d- Transition 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 (\(\rho < 50 \mu\Omega\text{cm}\)) 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 wavefunction. 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), independently of the strength of the spin-orbit interaction (SOI). Since the OHC is much larger than the SHC, it will be possible to realize a {\it orbitronics device} made of transition metals.