Parametric excitation of magnetic vortex gyrations in spin-torque nano-oscillators

We experimentally demonstrate that large amplitude magnetic vortex gyrations can be parametrically excited by the injection of radio-frequency (rf) current at twice the natural frequency of the gyrotropic vortex-core motion. The mechanism of excitation is based on the parallel pumping of vortex motion by the rf orthoradial field generated by the injected current. Theoretical analysis shows that experimental results can be interpreted as the manifestation of parametric amplification when the rf current is small, and of parametric instability when the rf current is above a certain threshold. By taking into account the energy nonlinearities, we succeed to describe the amplitude saturation of vortex oscillations as well as the coexistence of stable regimes.

Figure 1

Scheme of the circuit used in the experiments.

Figure 2

(a)–(c) Colored maps of emitted power for the frequency of the gyrotropic core motion vs the external rf frequency measured (a) for a small $I_{\mathrm{rf}}=0.8$ mA (corresponding to a power of $-18$ dBm) and (c) a large $I_{\mathrm{rf}}=1.6$ mA (corresponding to $-12$ dBm) current (rms value). $I_{\mathrm{dc}}=3$ mA and $H_{\mathrm{perp}}=4.48$ kG. Both currents and field are applied perpendicular to the sample plane. (b)–(d) Power spectral density vs frequency measured with an external rf current at $f_{\mathrm{rf}}=275$ MHz.

Figure 3

Phase diagram in the $(f_{\mathrm{rf}},j_{\mathrm{rf}})$ plane with indications of the region of parametric instability (red) and parametric amplification (blue). Dashed lines represent two specific cases for $I_{\mathrm{rf}}=1.8$ mA ($-11$ dBm) and $I_{\mathrm{rf}}=0.6$ mA ($-20$ dBm). The values of the parameters used are $P=0.225$, $I_{dc}=2.5$ mA, $\alpha =0.01$ (LL damping), and $x_0=0.3$.

Figure 4

(a) Phase diagram in the $(f_{\mathrm{rf}},j_{\mathrm{rf}})$ plane with indications of the region of parametric instability (red), coexistence of stable regimes (green), and parametric amplification (blue). (b) Analytically computed amplitude response vs $f_{\mathrm{rf}}$, according to Eq. (23) with $I_{\mathrm{rf}}=1.8$ mA, which corresponds to a power value of $-11$ dBm. The values of the parameters used are $P=0.225$, $I_{\mathrm{dc}}=2.5$ mA, $\alpha =0.01$ (LL damping), and $x_0=0.3$.

Figure 5

(a) Vortex frequency $f$ vs rf frequency $f_{\mathrm{rf}}$. (b) Power density vs rf frequency $f_{\mathrm{rf}}$. All values are the result of a Lorentzian fit applied to the raw spectra in the same manner of Fig. 2. (c)–(e) Power density vs frequency $f$ for three values of $f_{\mathrm{rf}}$: 275, 295, and 305 MHz. Note that these measurements have been made with an out-of-plane field $H_{\mathrm{perp}}=4.48$ kG, a subcritical dc current $I_{\mathrm{dc}}$ $=$ 2.5 mA, and a rf current $I_{\mathrm{rf}}=1.8$ mA, corresponding to a power of $-11$ dBm.