Using fully self-consistent methods, we study spin transport in fabricable spin valve systems consisting of two magnetic layers, a superconducting layer, and a spacer normal layer between the ferromagnets. Our methods ensure that the proper relations between spin current gradients and spin transfer torques are satisfied. We present results as a function of geometrical parameters, interfacial barrier values, misalignment angle between the ferromagnets, and bias voltage. Our main results are for the spin current and spin accumulation as functions of position within the spin valve structure. We see precession of the spin current about the exchange fields within the ferromagnets, and penetration of the spin current into the superconductor for biases greater than the critical bias, defined in the text. The spin accumulation exhibits oscillating behavior in the normal metal, with a strong dependence on the physical parameters both as to the structure and formation of the peaks. We also study the bias dependence of the spatially averaged spin transfer torque and spin accumulation. We examine the critical-bias effect of these quantities, and their dependence on the physical parameters. Our results are predictive of the outcome of future experiments, as they take into account imperfect interfaces and a realistic geometry.
Bibliographical noteFunding Information:
The authors thank Ilya Krivorotov (Irvine) and Chien-Te Wu (National Chiao Tung University) for many helpful discussions. This work was supported in part by US Department of Energy Grant No. DE-SC0014467.
© 2018 American Physical Society.