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 et al., Nature 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.
I. FROM THE CONVENTIONAL TO THE MAGNETIC SPIN HALL EFFECTThe spin Hall effect (SHE) [1] and its inverse are without doubt important discoveries [2][3][4][5][6] in the field of spintronics [7,8]. They serve not only as 'working horses' for generating and detecting spin currents [9] but also as key ingredients in spin-orbit torque devices for electric magnetization switching [10][11][12]. Compared to spin-transfer torque devices [13-17], spin-orbit torque devices are faster, more robust, and consume less power upon operation [18][19][20]; for a recent review see Ref. 21. 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 j γ = σ γ E (γ = x, y, z indicates the transported spin component). Mathematically, the SHE is quantified by the antisymmetric part of the spin conductivity tensor σ γ . For example, the σ z xy element comprises z-polarized spin currents in x direction as a response to an electric field in y direction.In a simple picture, 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 spin-dependent Lorentz forces that result in a transverse pure spin current. The extrinsic SHE [25][26][27] is covered by Mott scattering at defects [28].Since the SHE does not rely on broken time-reversal symmetry (TRS), it is featured in nonmagnetic metals [29] or semiconductors [2]. Imposing few demands on a material's properties, a SHE can be expected in...