# Influence of spin-orbit and spin-Hall effects on the spin Seebeck current beyond linear response

## TL;DR

This paper investigates how spin-orbit and spin-Hall effects influence the spin Seebeck current in heterostructures, highlighting nonlinear effects and the roles of external magnetic fields, spin Hall effect, and Dzyaloshinskii-Moriya interaction.

## Contribution

It provides a theoretical analysis of nonlinear spin transport effects in heterostructures with Rashba spin-orbit coupling and spin Hall effect, combining analytical and numerical methods.

## Key findings

- Spin-orbit coupling enhances or reduces spin pumping depending on magnetic field alignment.
- Spin Hall effect and Dzyaloshinskii-Moriya interaction increase spin pumping current.
- Nonlinear effects significantly influence spin Seebeck current behavior.

## Abstract

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, while 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 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.