Theory of ferromagnetic resonance driven by the combined action of spin-transfer torque and voltage-controlled magnetic anisotropy

An analytical study of the spectral line shape of ferromagnetic resonance (FMR) detected by spin rectification effect and driven by the combined action of spin-transfer torque (STT) and voltage-controlled magnetic anisotropy (VCMA) is developed. The system under consideration consists of a magnetic tunnel junction (MTJ). Explicit expressions for the symmetric and asymmetric components of the rectified voltage are derived, where the role of the VCMA, in-plane STT, and field-like torque is clearly identified and discussed. Typical geometrical configurations are particularly analyzed and compared with recent experimental results. The analytical findings show that the change of sign in the FMR response upon reversal of the magnetization is completely due to VCMA. By distinguishing in-plane, out-of-plane, and full magnetization reversal processes, it is shown that the VCMA induces a change of sign in the symmetric part for the in-plane and out-of-plane magnetization reversal, while the asymmetric part change its sign under a full and in-plane reversion of the magnetization. Explicit expressions of the symmetric and asymmetric contributions of the spectral line shape allow us to detect under what conditions the STT and VCMA can increase or decrease the FMR spectral line shape. The proposed theory allows access to a better understanding of the physics behind ferromagnetic resonance phenomena, promoting potential applications in STT+VCMA-based MTJs.