Hysteretic mutual phase-locking of perpendicular-to-plane polarizer spin-torque nano-oscillator pairs analyzed by a generalized pendulum-like model

Abstract

At present, the Kuramoto model is the standard and widely accepted theoretical approach for analyzing the synchronization of spin-torque nano-oscillators (STNOs) coupled by an interaction. Nevertheless, the oscillatory decaying regime and the initial condition (IC)-dependence (hysteretic) that exist in the synchronization of many types of STNOs cannot be explained by this model. In order to more precisely elucidate the physical mechanisms behind the two phenomena, in this paper we develop a generalized pendulum-like model based on the two common features of non-linear auto-oscillators: one is the stability of the amplitude/energy of dynamic states and the other is the non-linear dynamic state energy of oscillators. In this new model, we find that the Newtonian-like particle with sufficient kinetic energy can overcome the barrier of phase-locking potential to evolve into a stable asynchronization state, leading to the IC-dependent synchronization. Furthermore, due to the presence of kinetic energy, this particle can also oscillate around the minima of the phase-locking potential, leading to the oscillatory decaying regime. Thereby, in this work, we adopt this new model to analyze the IC-dependent mutual synchronization of perpendicular-to-plane-STNO pairs, and then we suggest that the initial conditions can be controlled to avoid such a phenomenon by using magnetic dipolar coupling.