Spin Hall effect versus Rashba torque: a Diffusive Approach

Current-driven magnetization dynamics of single ferromagnets in heavy metal/ferromagnet bilayers has been recently realized. In this work, spin torque induced by so-called Rashba spin-orbit coupling and spin Hall effect are considered within a diffusive model. The dependence of the resulting torque as a function of the thicknesses of the ferromagnet and heavy metal is analyzed. We show that (i) both torques are on the form ${\bf T}=T_{||}{\bf m}\times{\bf y}+T_\bot{\bf m}\times({\bf y}\times{\bf m})$, (ii) the ratio $T_{||}/T_\bot$ strongly depends on the thickness of the layers and (iii) the thickness dependence of the spin torque provides an indication of the origin of the (Rashba- or spin Hall effect-induced) spin torque .