Phase diagrams of MgO magnetic tunnel junctions including the perpendicular spin-transfer torque in different geometries

We calculate the switching voltages for MgO-based magnetic tunnel junctions taking into account both the in-plane and the fieldlike spin-torque terms. To this end, we analytically solve the Landau-Lifshitz-Gilbert equation for a generalized geometry. We assume that the in-plane spin-torque varies linearly with the applied voltage, while the fieldlike torque exhibits a quadratic voltage dependence. Specifically, we consider that the free layer has two generic, orthogonal anisotropy components, one of which is along the direction defined by the magnetization of the reference layer, which also serves as a polarizer. The resulting formalism is applied to three different, experimentally relevant geometries: tunnel junctions with both the free and the reference layers magnetized in the plane of the layers, junctions with fully perpendicular anisotropy, and perpendicular junctions with an additional in-plane easy axis, respectively. We find that for in-plane devices, the quadratic dependence of the fieldlike torque on the applied voltage can lead to back hopping, which remains possible if we insert an additional linear term for the bias dependence of the fieldlike spin-torque comparable to current experimental results. For perpendicular anisotropy junctions neither back hopping nor spin-transfer-driven steady-state precession are expected. An additional in-plane shape anisotropy component stabilizes canted states in tunnel junctions with perpendicular anisotropy for specific values of voltage and field. The results are consistent with numerical integration of the Landau-Lifshitz-Gilbert equation and in good agreement with recent experiments involving perpendicular magnetic anisotropy magnetic tunnel junctions.