Coherence of spin-torque microwave oscillators

Recently discovered effect of microwave generation in current-driven magnetic nano-structures caused by the spin-transfer torque opens a possibility for the development of a new class of tunable microwave auto-oscillators. The spin-torque oscillators (STO) are strongly nonlinear as their frequency $\omega (P)$ and total (positive plus negative) damping $\Gamma (P)$ are dependent on the oscillation power $P$ . We developed a theory of the generation linewidth of a \textit{nonlinear} auto-oscillator, and showed that the nonlinear frequency shift (characterized by the coefficient$N=d\omega /dP)$ leads to an effective increase of the phase noise. In a strongly supercritical regime, when the oscillation energy $E(P)$ is much larger than the thermal energy $k_{B}T$, the generation linewidth of a STO can be written as $\Delta \omega =\Delta \omega _0 [1+(N/\Gamma _{eff} )^2]$, where $\Delta \omega _0 =\Gamma (0)[k_B T/E(P)]$ is the oscillator linewidth without account for the nonlinear frequency shift and $\Gamma _{eff} =d\Gamma /dP$ is the effective nonlinear damping of the oscillator. Our theory explains the following features of the STO linewidth observed in experiment: (i) general linewidth narrowing with the increases in the bias current $I$ and the oscillation energy $E(P)$; (ii) presence of a minimum in the linewidth dependence on the orientation of the external bias magnetic field; (iii) linear dependence of the linewidth on the absolute temperature. Our theory also demonstrates that in the array of $n$ phase-locked STO the generation linewidth decreases linearly with the increase on the number of oscillators $n$, while the generated power $P $ increases as $n^2$.