Study of Intrinsic Spin Hall Effect and Orbital Hall Effect in 4d- and 5d- Transition Metals

TL;DR

This study analyzes the intrinsic spin and orbital Hall effects in 4d and 5d transition metals, revealing their dependence on resistivity, electron count, and phase factors, with implications for orbitronics devices.

Contribution

It provides a quantitative analysis of intrinsic spin and orbital Hall conductivities in transition metals using a tight-binding model, highlighting their dependence on electron count and resistivity regimes.

Findings

Large positive SHC in Pt and Pd with ~9 d-electrons

Negative SHC in Ta, Nb, W, Mo with <5 d-electrons

Huge positive orbital Hall conductivity in all studied 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.