Spin Circuit Representation for the Spin Hall Effect

Spin circuits with four component voltages and currents have been developed and used in the past to analyze various structures, which include non-collinear ferromagnets. Recent demonstrations of large spin orbit torques in heavy metals like Pt, Ta, and W open up new possibilities in spintronic applications by providing an alternative way to write information into a magnet. Here, we extend the four component (one charge and three spins) conductance matrix to include materials with spin Hall effect based on the standard diffusion equation. Our proposed spin circuit successfully reproduces standard results like spin Hall effect (SHE), inverse spin Hall effect, and spin Hall magnetoresistance. This circuit representation also makes it straightforward to analyze new configurations. We present two examples, namely, 1) the possibility of spin injection using giant SHE (GSHE) materials into semiconductors without tunneling barriers, and 2) the effect of spin ground on one surface to enhance spin current injection from the opposite surface in a thin GSHE sample. Finally, we provide an elemental conductance matrix for a small cubic structure which can be used as a building block to analyze any arbitrarily shaped GSHE material.