The unification of electric and thermal spin transfer torque in spintronics

Abstract There exist two kinds of spin transfer torque in spintronics: the electric spin transfer torque (STT) caused by spin-polarized current and the thermal spin torque (TSTT) caused by thermal spin current. In this manuscript, we will derive a unified torque which contain both of them. Based on spinor Boltzmann equation, we investigate the torques in a magnetoelectric device applied by voltage bias and temperature gradient, and find that the unified STT are contributed by two sources: the spin-polarized current induced by bias and the thermal spin current induced by the temperature gradient, moreover the TSTT can be further divided as local equilibrium TSTT and nonequilibrium TSTT. In addition to the usual electric STT and TSTT, a new term also appear in our expression, which can be interpreted by the inhomogeneity of magnetization. We illustrate them by an example of electric and thermal transport through a ferromagnet subject to some device, in which those torques originate from different sources are manifested quantitatively.