Influence of spin-orbit and spin-Hall effects on the spin-Seebeck current beyond linear response: A Fokker-Planck approach

We study the spin transport theoretically in heterostructures consisting of a ferromagnetic metallic thin film sandwiched between heavy-metal and oxide layers. The spin current in the heavy-metal layer is generated via the spin Hall effect, whereas the oxide layer induces at the interface with the ferromagnetic layer a spin-orbital coupling of the Rashba type. Impact of the spin-Hall effect and Rashba spin-orbit coupling on the spin-Seebeck current is explored with a particular emphasis on nonlinear effects. Technically, we employ the Fokker-Planck approach and contrast the analytical expressions with full numerical micromagnetic simulations. We show that, when an external magnetic field ${H}₀$ is aligned parallel (antiparallel) to the Rashba field, the spin-orbit coupling enhances (reduces) the spin pumping current. In turn, the spin-Hall effect and the Dzyaloshinskii-Moriya interaction are shown to increase the spin pumping current.