Unified Treatment of Spin Torques using a Coupled Magnetisation Dynamics and Three-Dimensional Spin Current Solver

A three-dimensional spin current solver based on a generalised spin drift-diffusion description, including the spin Hall effect, is integrated with a magnetisation dynamics solver. The resulting model is shown to simultaneously reproduce the spin-orbit torques generated using the spin Hall effect, spin pumping torques generated by magnetisation dynamics in multilayers, as well as the spin transfer torques acting on magnetisation regions with spatial gradients, whilst field-like and spin-like torques are reproduced in a spin valve geometry. Two approaches to modelling interfaces are analysed, one based on the spin mixing conductance and the other based on continuity of spin currents where the spin dephasing length governs the absorption of transverse spin components. In both cases analytical formulas are derived for the spin-orbit torques in a heavy metal / ferromagnet bilayer geometry, showing in general both field-like and damping-like torques are generated. The limitations of the analytical approach are discussed, showing that even in a simple bilayer geometry, due to the non-uniformity of the spin currents, a full three-dimensional treatment is required. Finally the model is applied to the quantitative analysis of the spin Hall angle in Pt by reproducing published experimental data on the ferromagnetic resonance linewidth in the bilayer geometry.