Irreversible spin-transfer and magnetization reversal under spin-injection

In the context of spin electronics, the two spin-channel model assumes that the spin carriers are composed of two distinct populations: the conduction electrons of spin up, and the conduction electrons of spin down. In order to distinguish the paramagnetic and ferromagnetic contributions in spin injection, we describe the current injection with four channels : the two spin populations of the conduction bands ($s$ or paramagnetic) and the two spin populations of the more correlated electrons ($d$ or ferromagnetic). The redistribution of the conduction electrons at the interface is described by relaxation mechanisms between the channels. Providing that the $d$ majority-spin band is frozen, $s-d$ relaxation essentially concerns the minority-spin channels. Accordingly, even in the abscence of spin-flip scattering (i.e. without standard spin-accumulation or giant magnetoresistance), the $s-d$ relaxation leads to a $d$ spin accumulation effect. The coupled diffusion equations for the two relaxation processes ($s-d$ and spin-flip) are derived. The link with the ferromagnetic order parameter $\vec{M}$ is performed by assuming that only the $d$ channel contributes to the Landau-Lifshitz-Gilbert equation. The effect of magnetization reversal induced by spin injection is explained by these relaxations under the assumption that the spins of the conduction electrons act as environmental degrees of freedom on the magnetization.