Combined ballistic and diffusive model of spin-polarized current-induced magnetization switching in pseudo-spin-valve structure

We present a theoretical model of spin transport and spin transfer in a $\mathrm{Co}∕\mathrm{Cu}∕\mathrm{Co}$ pseudo-spin-valve (PSV) structure, which combines ballistic spin injection across the interfaces of the PSV, and diffusive spin relaxation within the free Co layer. The ballistic spin injection model considers spin-differential transmission and reflection probabilities at the two $\mathrm{Co}\text{−}\mathrm{Cu}$ interfaces, and the effect of multiple reflections at the interfaces. This ballistic process causes the incident spin current at the spacer-free Co interface to undergo spin rotation and to be reduced to a fraction of the spin current in the pinned Co layer. There are two contributions to the spin transfer to the free Co layer, i.e., (a) the fraction of the incident spin current which is “absorbed” at the interface in order to conserve spin momentum at the interface (neglected in previous purely diffusive models) and (b) spin relaxation of the transverse spin accumulation, due to a combination of spin scattering and precession. The magnitudes of these two components are calculated based on typical experimental parameters, and the switching fields due to spin torques in the in-plane and out-of-plane directions are derived by considering a modified Landau-Lifshitz-Gilbert (LLG) equation. Based on known values of switching magnetic fields of $\mathrm{Co}∕\mathrm{Cu}∕\mathrm{Co}$ PSV, the calculated critical current density ranges from $2.5\text{to}8.2\times {10}^7\mathrm{A}{\mathrm{cm}}^{-2}$, in agreement with observed values in current-induced magnetization switching experiments.