Thermodynamic ground and Berry-phase origin of the planar spin Hall effect

The thermodynamic ground of the spin transport is elaborated, demonstrating the Onsager's reciprocal relation. The latter is also confirmed by deriving the spin Hall effect and its inverse using the semiclassical theory and the linear response theory. Importantly, the intrinsic contribution to the spin conductivity that satisfies the Onsager's relation has only Hall-type component and hence does not dissipate heat, in analogy to the anomalous Hall effect and fulfilling the thermodynamic requirement. Based on the expression of the spin Hall conductivity, the Berry-phase origin is elaborated for the recent-discovered planar spin Hall effect which features current-induced spin polarization within the plane of the Hall deflection. We unravel a spin-repulsion vector governing the planar spin Hall effect, providing a transparent criteria for identifying intrinsic planar spin Hall materials. Finite spin-repulsion vector is found permitted in 13 crystalline point groups, revealing a big number of unexplored planar spin Hall systems. Our result can be used for the quantitative calculation of the planar spin Hall effect, facilitating further material hunting and experimental investigation.