Generalization of a circuit theory for current perpendicular to plane magnetoresistance and current-driven torque

Extensions of an existing circuit theory for current perpendicular to plane magnetoresistance and current-driven torque in noncollinear magnetic-multilayer pillar devices are presented. Our expressions for monodomain critical-current threshold Jc and giant magnetoresistance ΔR are firstly derived in terms of assumed spin-channel resistances for each of the two ferromagnets. Spinflips are thus neglected. We find a class of closed linear relationships connecting Jc -1 and ΔR. We then derive more general expressions for these quantities which take into account spin-flip relaxation. In this case, we assume analytically calculable linear 2×2 current-voltage matrices for the separate two-channel ferromagnets. These expressions again lead to a class of closed linear relationships connecting Jc -1 and ΔR. The latter generalization gives a simple theoretical framework to take into account bulk and interfacial spin flip and more complicated multilayer structures often used in experiments.