Full calculation of inter-conversion between charge, spin, and heat current using a common partial differential equation platform

We present a simple implementation of calculation of spin current profiles using a partial differential equation platform. By solving multiple scalar potentials, spin injection, spin/charge inter-conversion, and thermal spin injection phenomena can be well reproduced numerically. As a demonstration, we show spin current generation and detection in a composite of ferromagnetic, spin conducting, and spin-Hall-metallic materials. Furthermore, we present a model extended to three-dimensionally polarized spin current and describe the matrix for spin/charge current inter-conversion in a conductive ferromagnet, which allows for numerical reproduction of anomalous and planar Hall effects. It is found that the planar Hall voltage is in part generated by spin Hall conductivities, though its magnitude is orders smaller than that induced by the anisotropic magnetoresistance. Our method will contribute to further development of effective and feasible simulations of spin-current-mediated systems.