Spin Currents at the Interface and Spin Hall Torque

We compare two different approaches to compute the spin Hall torque at the interface between a spin Hall metallic layer and a ferromagnet. In one approach, one attributes a spin mixing conductance g(up down arrow) to the interface, while the other employs the thermodynamic theory for the magnetic moment currents. The main difference between the outcomes of the two approaches is the field-like torque term that, in the first approach, is due to the imaginary part of the spin mixing conductance Im[g(up down arrow)] and, in the second one, is proportional to the effective damping alpha' of the ferromagnet.