Extrinsic Spin Hall Effect in Inhomogeneous Systems

Interconversion between spin and charge signals is one of the most important processes in spintronics. This is done most commonly by the direct and inverse spin Hall effects (SHE). Recently, An et al. observed an enhanced charge-to-spin conversion, comparable to heavy metals, in naturally oxidized Cu (nox-Cu) [1]. Okano et al. reported a non-reciprocal charge-spin interconversion in nox-Cu [2]. The so-called spin-vorticity coupling (SVC) was suggested as the origin of these results.

In this work, we take a different approach and study the problem in terms of the extrinsic SHE in a inhomogeneous system. We model the nox-Cu system by nearly free electrons subject to impurities with spin-orbit interaction and with spatially modulated distribution. We calculate spin accumulation and spin current induced by an electric field using Kubo formula and discuss the relation between them. The results are summarized by a generalized spin diffusion equation with a spin source term given by the divergence of the spin Hall current. This source term can be put in a form of SVC only if the spin Hall angle is independent of impurity concentration, as in the case of skew scattering.

[1] H. An et al., Nat. Commun. 7, 12069 (2016).
[2] G. Okano et al., Phys. Rev. Lett. 122, 217701 (2019).