Layer thickness and angular dependence of spin transfer torque in ferromagnetic trilayers

Abstract

We investigate the spin transfer torque arising from a perpendicular-to-plane current in a ferromagnetic (FM)-nonmagnetic-FM trilayer. Our analysis is based on the spin drift-diffusion model, modified for the case of noncollinear magnetization, i.e., with arbitrary angle θ between the magnetization orientation of the two FM layers. By solving the electrochemical potential and spin accumulation across the trilayer, we obtain the θ dependence of the magnetoresistance and spin transfer torque in the free FM layer. The optimal magnetization orientation θmax and the corresponding maximum torque τmax are investigated as a function of the FM layer thicknesses. Based on the analysis, we propose that (i) the free (fixed) FM layer thickness be set at approximately the transverse (longitudinal) spin relaxation lengths, and (ii) the relative FM orientation be biased at some intermediate angles instead of the conventional collinear configuration, in order to maximize the current-induced magnetization switching effect.