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

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.