First-principles calculation of spin-orbit torques in ferromagnet/heavy-metal bilayers

The spin-orbit torque in Co/Pt, Co/Pd, and Co/Au bilayers is calculated using a first-principles non-equilibrium Green's function formalism with an explicit treatment of disorder. The torque is formally split in the Fermi-sea and Fermi-surface contributions. The Fermi-sea term is important at low temperatures, but the Fermi-surface term dominates at room temperature. In addition to the usual damping-like and field-like terms, the odd torque contains a sizeable planar Hall-like term (mE)m×(z×m), which contributes to damping and has been observed experimentally. While the torques that contribute to damping are largely due to spin-orbit coupling on the heavy-metal atoms, the field-like torque does not require it. The dependence of the torque on the thicknesses of the layers is also examined.