Partitioning the spin-orbit torque in heavy metal/ferromagnet heterostructures

In heterostructures composed of heavy metals and ferromagnets, the spin-orbit torque is the result of reduced symmetry and the presence of spin-orbit coupling. In order to optimize the spin-orbit torque efficiency, we provide a general scheme to identify the contributions to the intrinsic spin-orbit torque due to the wave-function change induced by an external electric field in the steady state. Using a toy model of a bilayer system with atomic p-orbitals, we find both spin current from spin Hall effect and Berry curvature-like pseudo-magnetic field contribute to the torque. We next apply our analysis to first-principles calculation of spin-orbit torque in multilayer Pt/Co and 1T’-WTe2/Co systems. The additional in-plane symmetry breaking in 1T’-WTe2 allows for an unconventional out-of-plane torque. We find that the atomic orbital current can be transferred to spin current at the interface heavy metal layer, which is the dominant spin current contribution to the torque. The contributions identified by our partitioning scheme in the ferromagnet/heavy metal heterostructure could shed light on how to improve the spin-orbit torque switch efficiency.