Field-free switching in perpendicular magnetic tunnel junctions (P-MTJs) can be achieved by combined injection of spin-transfer torque (STT) and spin-orbit torque (SOT) currents. In this paper, we derived the relationship between the STT and SOT critical current densities under combined injection. We included the damping-like torque (DLT) and field-like torque (FLT) components of both the STT and SOT. The results were derived when the ratio of the FLT to the DLT component of the SOT was positive. We observed that the relationship between the critical SOT and STT current densities depended on the damping constant and the magnitude of the FLT component of the STT and the SOT current. We also noted that, unlike the FLT component of SOT, the magnitude and sign of the FLT component of STT did not have a significant effect on the STT and SOT current densities required for switching. The derived results agreed well with micromagnetic simulations. The results of this work can serve as a guideline to model and develop spintronic devices using a combined injection of STT and SOT currents.
Keywords: combined spin-transfer torque (STT) and spin-orbit torque (SOT) switching; damping like torque; field like torque; magnetic tunnel junction.