Spin Currents at the Interface and Spin Hall Torque

Abstract

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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$g_{\uparrow \downarrow }$ </tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text {Im}[g_{\uparrow \downarrow }]$ </tex-math></inline-formula> and, in the second one, is proportional to the effective damping <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha ^{\prime }$ </tex-math></inline-formula> of the ferromagnet.