Unified treatment of spin torques using a coupled magnetisation dynamics and three-dimensional spin current solver

Abstract

A three-dimensional spin current solver based on a generalised spin drift-diffusion description, including the bulk and interfacial spin Hall effects, 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. The model is further applied to the analysis of the spin Hall angle in Pt by reproducing published experimental ferromagnetic resonance data in the bilayer geometry.

Boris computational spintronics—High performance multi-mesh magnetic and spin transport modeling software

“…It may be shown that under the assumption of negligible in-plane spin diffusion this expression is equivalent to Zhang-Li STTs as given in Eq. ( 2), 9,17,67 where the non-adiabaticity parameter is constant and given by β ffi λ 2 J /λ 2 sf in the limit of long spin dephasing length and long domain walls. The assumption of negligible in-plane spin diffusion breaks down for rapidly varying magnetization textures such as vortices and skyrmions, and this can lead to spatially varying and enhanced non-adiabaticity.…”

Effect of inter-layer spin diffusion on skyrmion motion in magnetic multilayers

“…Spin torques included in the magnetisation dynamics equation can be computed self-consistently using a drift-diffusion model 24,25 . Within this model the charge and spin current densities are given as:Here J S is a rank-2 tensor such that J S ij signifies the flow of the j component of spin polarisation in the direction i , J C is the charge current density, E is the electric field, S is the spin accumulation, and m is the normalised magnetisation.…”

“…For an N/F interface with current in the plane, if diffusion effects are negligible, the drift-diffusion equations may be solved analytically to obtain the resulting interfacial spin torques due to SHE 25 . These are given as a combination of damping-like and field-like spin-orbit torques as:Here p = z × e Jc , where e Jc is the charge current direction.…”

Nonresonant amplification of spin waves through interface magnetoelectric effect and spin-transfer torque

“…The coupled charge-spin-magnetization dynamics driven by the ac voltage is considered on the base of the diffusive model 28 – 31 . The free charge density current is described in the metal layers by the equation: which includes contributions from electron drift driven by electric field , diffusion driven by the gradient of free charge density and a diffusive spin polarisation representing spin conductivity contribution, respectively.…”

“…Although we implemented both spin-dependent surface screening and spin-dependent conductivity as sources of the spin accumulation in the model, the latter is neglible since interface effects play dominant role in the considered system. For the interaction between spin accumulation and magnetization at interfaces we applied the continuous approach for the spin-transfer torque 28 , 29 .…”