Generation linewidth of mode-hopping spin torque oscillators

Experiments on spin torque oscillators commonly observe multimode signals. Recent theoretical works have ascribed the multimode signal generation to coupling between energy-separated spin wave modes. Here, we analyze in detail the dynamics generated by such mode coupling. We show analytically that the mode-hopping dynamics broaden the generation linewidth and makes it generally well described by a Voigt line shape. Furthermore, we show that the mode-hopping contribution to the linewidth can dominate, in which case it provides a direct measure of the mode-hopping rate. Due to the thermal drive of mode-hopping events, the mode-hopping rate also provides information on the energy barrier separating modes and temperature-dependent linewidth broadening. Our results are in good agreement with experiments, revealing the physical mechanism behind the linewidth broadening in multimode spin torque oscillators.

Figure 1

Phase spaces spanned by $\theta$ and $\psi$ for (a) $K=0$: single mode; (b) $K=0.2$: near-single modes and coexistence; and (c) $K=1$: coexistence and periodic mode transitions. For the case $K=0.2$, the basins of attraction are shown for (d) ${\varphi }_c=0$ and (e) ${\varphi }_c=\pi /2$. In the latter, the spiral feature is reminiscent of a particle with friction in a double potential well.

Figure 2

(a) Time trace of $\theta$ exhibiting mode-hopping events between $\pm \langle {\theta }_o\rangle$ (dashed lines, red online). The section in the black box is detailed in panel (b), where the underlying relaxation frequency is observed. The inset shows the exponential distribution of the time difference between mode-hopping events, in agreement with a Poisson process. (c) Phase space of the time trace (a) showing that the stable modes (fixed points indicated by black dots) are thermally driven to mode hop via saddle points (black crosses).

Figure 3

(a) Fourier transform of the autocorrelation calculated from the numerical integration of Eq. (3) (black). Mode-hooping events are included as a Poisson process with a mode-hopping rate calculated from the numerical solution of Eq. (1) (red [gray]). The best Voigt fit is shown in blue (gray). (b) Voigt linewidth as a function of the coupling strength $K$, obtained from the best fit of the autocorrelation (black) and including mode-hopping events (blue [gray]). The analytical estimate is shown by red (gray) squares. The mode-hopping events dominate the linewidth when $K>0.3$. (c) Experimentally measured linewdith (red [gray] circles) and linewidth obtained from Eq. (6) (blue [gray] line) with $\Delta E=52$ meV extracted from a single experimental linewidth data point at 303 K. This simple fit shows a remarkably good agreement with the experimental data trend as well as its magnitude.