Models of spin torque using self-consistent solutions of the magnetisation and spin accumulation

A model of spin accumulation (m) is proposed to develop theoretical approaches to calculate the m in any arbitrary magnetic structure. The model is based on generalising the approach of Zhang, Levy and Fert (PRL 88, 236601, 2002). The calculation involves the layer-wise discretisation of the structure and the development of semi-analytical approaches to solve for the equilibrium m throughout the structure. Interestingly, the layer discretisation allows the treatment of diff�use interfaces using a gradual variation of the magnetic and transport properties across the interface. The e�ffect of the interfaces between a ferromagnet and a nonmagnet and between two ferromagnets on spin injection is investigated. The formalism for calculating the m is first generalised by taking m as the di�fference of spin-up and spin-down density of states, which is necessary for treating the interface between diff�erent ferromagnets. Then, the e�ffect of atomic species interdiffusion at the interface is included by using Ficks's law. It is shown that the discontinuity of the m at the interface depends strongly on the degree of interface mixing. Subsequently, current-induced domain wall (DW) motion in a ferromagnetic thin fi�lm driven by a spin-polarised current is investigated using an atomistic model coupled with a standard Landau-Lifshitz-Gilbert equation. The inclusion of the spin-transfer torque is represented as an additional �field. The m is calculated self-consistently and naturally includes the adiabatic and non-adiabatic contributions depending on the rate of change of magnetisation relative to the spin di�ffusion length. In this work, it is importantly found that the constants �x and �x used in the standard micromagnetic model do not provide a good description of the spin torque phenomenon due to the non-physical behaviour. Therefore, it is suggested to describe the spin-transfer torque directly from the m. Finally, the evolution of the magnetisation and m are demonstrated by introducing a spin-polarised current into a material containing a DW whose width is varied by changing the anisotropy constant. It is found that the adiabatic spin torque tends to develop in the direction of the magnetisation whereas the non-adiabatic spin torque arising from the mistracking of conduction electrons and local magnetisation results in out-of-plane magnetisation components. However, the adiabatic spin torque signifi�cantly dominates the dynamics of magnetisation. The total spin torque acting on the magnetisation increases with anisotropy constant due to the increasing magnetisation gradient. This leads to increasing DW displacement.