First-principles study of the angular dependence of the spin-orbit torque in Pt/Co and Pd/Co bilayers

Spin-orbit torque (SOT) induced by spin Hall and interfacial effects in heavy-metal (HM)/ferromagnetic (FM) bilayers has recently been employed to switch the magnetization direction using in-plane current injection. In this paper, using the Keldysh Green's function approach and first-principles electronic structure calculations we determine the fieldlike (FL) and dampinglike (DL) components of the SOT for the HM/Co (HM = Pt,Pd) bilayers. Our approach yields the angular dependence of both the FL- and DL-SOT on the magnetization direction without assuming a priori their angular form. Decomposition of the SOT into the Fermi sea and Fermi surface contributions reveals that the SOT is dominated by the latter. Due to the large lattice mismatch between the Co and the HM we have also determined the effect of tensile biaxial strain on both the FL- and DL-SOT components. The calculated dependence of FL- and DL-SOT on the HM thickness is overall in good agreement with experiment. The dependence of the SOT with the position of the Fermi level suggests that the DL-SOT is dominated by the spin Hall effect of the bulk HM.