Phase locking of spin-torque nano-oscillator pairs with magnetic dipolar coupling

A spin-torque nanopillar oscillator (STNO) that combines a perpendicular-to-plane polarizer (PERP) with an in-plane magnetized free layer is a good candidate for phase locking, which opens a potential approach to enhancement of the output power of STNOs. In this paper, the magnetic dipolar coupling effect is used as the driving force to synchronize two STNOs. We develop an approximation theory for synchronizing two identical and nonidentical pairs of PERP STNOs, by which the critical current of synchronization, dipolar coupling strength, phase-locking transient time, and frequency can be analytically predicted. These predictions are further confirmed by macrospin and micromagnetic simulations. Finally, we show the phase diagrams of the phase locking as a function of applied current and separation between two STNOs.