Theory of mutual phase locking of spin-torque nanosized oscillators

A theory of mutual phase locking of a pair of generating magnetic nanocontacts driven by a spin-polarized current is developed using the general theory of coupled nonlinear oscillators. Expressions for the phase-locking band and for the frequency of phase-locked oscillations are derived and compared to result of experiments. It is shown that in typical experiments, where the separation $a$ between the nanocontacts lies in the interval $0.2<a<1.5\mu \mathrm{m}$, the coupling between the nanocontacts leading to the phase-locking effect is mainly caused by propagating spin waves radiating into the common free layer, rather than by the direct magnetodipole interaction.