Spin and orbital Hall effect in non-magnetic transition metals: extrinsic vs intrinsic contributions

TL;DR

This paper uses Kubo's formalism to analyze the orbital and spin Hall effects in transition metals, highlighting the roles of intrinsic and extrinsic contributions, doping, and temperature effects on conductivities.

Contribution

It provides a detailed comparison of intrinsic and extrinsic contributions to orbital and spin Hall conductivities in transition metals, emphasizing the impact of doping and temperature.

Findings

Intrinsic contributions dominate at finite temperature in undoped metals.

Extrinsic contributions depend on spin-orbit coupling and decrease with temperature.

Doping significantly alters the orbital and spin Hall conductivities.

Abstract

Kubo's linear response formalism has been used to calculate the orbital Hall conductivity (OHC) for non-magnetic undoped and doped transition metal systems, focusing on the impact of different types of disorder and the role of vertex corrections for the OHC. The doping- and temperature-dependence of the OH conductivity have been investigated and compared with corresponding results for the spin Hall conductivity (SHC). A strong difference has been found between the results for undoped and doped metallic systems. For elemental systems at finite temperature a dominating role of the intrinsic contribution to the temperature-dependent OH and SH conductivities is found. Moreover, the different temperature dependent behavior of the intrinsic SOC-independent OHC and SOC-driven SCH indicates a non-trivial relationship between these quantities. It is shown, that in contrast to the intrinsic part of the OH and SH conductivities, the extrinsic contributions in doped systems are determined by spin-orbit coupling for both of them. It is dominating at low temperature, strongly decreasing at higher temperatures due to the increasing impact of the electron-phonon scattering.