Origin of the magnetic spin Hall effect: Spin current vorticity in the Fermi sea

Alexander Mook,Ingrid Mertig,Robin R Neumann,Annika Johansson,Jürgen Henk

doi:10.1103/physrevresearch.2.023065

Abstract

The interplay of spin-orbit coupling (SOC) and magnetism gives rise to a plethora of charge-to-spin conversion phenomena that harbor great potential for spintronics applications. In addition to the spin Hall effect, magnets may exhibit a magnetic spin Hall effect (MSHE), as was recently discovered [Kimata \textit{et al.}, Nature \textbf{565}, 627-630 (2019)]. To date, the MSHE is still awaiting its intuitive explanation. Here we relate the MSHE to the vorticity of spin currents in the Fermi sea, which explains pictorially the origin of the MSHE. For all magnetic Laue groups that allow for nonzero spin current vorticities the related tensor elements of the MSH conductivity are given. Minimal requirements for the occurrence of a MSHE are compatibility with either a magnetization or a magnetic toroidal quadrupole. This finding implies in particular that the MSHE is expected in all ferromagnets with sufficiently large SOC. To substantiate our symmetry analysis, we present various models, in particular a two-dimensional magnetized Rashba electron gas, that corroborate an interpretation by means of spin current vortices. Considering thermally induced spin transport and the magnetic spin Nernst effect in magnetic insulators, which are brought about by magnons, our findings for electron transport can be carried over to the realm of spincaloritronics, heat-to-spin conversion, and energy harvesting.

Highlights

While the anomalous Hall effect (AHE) in a magnet [22] produces a transverse charge current density upon applying an electric field E, the spin Hall effect (SHE) in a nonmagnet produces a transverse spin current density jγ = σ γ E (γ = x, y, z indicates the transported spin component)
Invoking the constant relaxation time approximation, we identified spin current vortices in the Fermi sea as origin of the magnetic spin Hall effect (MSHE)
It goes without saying that, due to Onsager’s reciprocity relation [105], the spin current vorticity (SCV) covers an inverse MSHE, that is a transverse charge current caused by a spin bias

Summary

Introduction

While the anomalous Hall effect (AHE) in a magnet [22] produces a transverse charge current density upon applying an electric field E, the SHE in a nonmagnet produces a transverse spin current density jγ = σ γ E (γ = x, y, z indicates the transported spin component). The intrinsic SHE [23,24] is explained by spinning electrons that experience a spin-dependent Magnus force caused by spin-orbit coupling (SOC). It appears as if “built-in” spin-dependent magnetic fields evoke spindependent Lorentz forces that result in a transverse pure. Of antisymmetric spin conductivity tensor elements traditionally associated with a SHE (applied field, current flow direction, and transported spin component are mutually orthogonal). For the rest of this paper, we refer to this SHE as “conventional SHE.”

Objectives

Findings

Discussion

Conclusion