Nonlinear stability analysis for a spin-torque nano-oscillator with a perpendicular spin polarizer: A pendulum-like model

An analytical pendulum-like model is developed to describe the dynamics of spin torque nano-oscillators that contain a perpendicular spin current polarizer combined with an in-plane magnetized free layer. Both the in-plane localized oscillation mode and out-of-plane (OP) large angle precession mode have been observed in the absence of damping effect, which can be attributed to the competition between the in-plane uniaxial anisotropic and demagnetization energies. These two types of dynamical oscillation modes correspond to the spin-transfer torque effect acting as a conservative or non-conservative driving force, respectively. We theoretically predict the critical currents for the excitation of the OP procession mode, which are well consistent with the macrospin simulations based on the Landau-Lifshitz-Gilbert equation. Moreover, we show the phase portrait of magnetization precession dynamics at different current and magnetic fields.