Thermal spin–orbit torque in spintronics

Based on the spinor Boltzmann equation (SBE) formalism, we present a theory of temperature-dependent thermal spin–orbit torque for a system in the presence of Rashba spin–orbit interaction. Under the local equilibrium assumption, we can expand the distribution function of spinor Boltzmann equation around local equilibrium distribution; then, the spin diffusion equation can be derived from SBE, where the spin transfer torque, spin orbit torque as well as thermal spin–orbit torque can be naturally obtained. It is shown that this thermal spin–orbit torque originates from the temperature gradient of local equilibrium distribution function, which is explicit and straightforward than previous works. Finally, we illustrate them by an example of spin-polarized transport through a ferromagnet with Rashba spin–orbit coupling, in which those torques driven whatever by temperature gradient or bias are manifested quantitatively