Electron transport in noncollinear magnetic multilayers

I have studied electron transport in noncollinear magnetic multilayers Co/Cu/Co. We have studied this problem by using the Boltzmann equation in the layer-by-layer approach. My study differs in two fundamental aspects from previous ones that use the Boltzmann approach: I do not a priori set the transverse part of the spinor distributions to zero in the ferromagnetic layers, and I have found the necessary condition and mechanism for the "injection" of these components in the magnetic layers. In this manner I have been able to achieve a steady state for the spin current across the noncollinear multilayer in which the spin current continuously changes its direction over a distance given by a new transport length scale. In my picture the spin angular momentum of the current does not go directly to exerting a torque on the background magnetization; rather it creates the transverse spin accumulation. It is this accumulation exerts the torque on the remainder of the Fermi sea.; My major finding is that by allowing for the injection of transverse components of the spin current we always obtain a lower resistance of noncollinear magnetic multilayer. In part due to the lower resistance and also due to the larger spin polarization of the current near the interfaces I find the spin torque due to the current is larger when I allow for transverse accumulation, e.g., when adjacent magnetic layers are 150° apart the spin torque is 50% greater for the same amount of electrical energy expended. Also, I have found the transverse spin current can go into the ferromagnetic layers about 3nm for a typical ferromagnetic 3d transition-metal such as Co. This has a direct impact on the thickness dependence of the spin torque in a thin magnetic layer. Finally I have used the time dependent spin diffusion equation, to study the time evolution of spin torque, and find it achieves its steady state in about 75 femtoseconds after undergoing damped oscillation with a period of about 5 femtoseconds.