Double resonance response in nonlinear magnetic vortex dynamics

We present experimental evidences for the dynamical bifurcation behavior of ac-driven magnetic vortex core gyration in a ferromagnetic disk. The dynamical bifurcation, i.e., appearance and disappearance of two stable dynamical states in the vortex gyration, occurring as the amplitude of the driving Oersted field increases to $B_{Oe}>B_{Oe}^{cr}$, manifests itself in a double resonance response in the dependence of homodyne the dc-voltage signal on the frequency $\omega$ of the applied microwave current. We find that the frequency range $\delta \omega$ between the two resonance features strongly increases with the excitation power. Our analysis based on the model of a low dissipative nonlinear oscillator subject to a resonant alternating force is in good agreement with the experimental results. This allows us to determine quantitatively key parameters of magnetic vortex dynamics, i.e., the critical value of the driving Oersted field $B_{Oe}^{cr}$ for nonlinear dynamics to occur, the resonant frequency, and the quality factor as well as damping of the magnetic vortex gyration.

Figure 1

(a) SEM image of the magnetic disk and electric contacts, and schematic configuration of homodyne detection. The distance between two electric contacts is about 500 nm. (b) Results of AMR measurements as a function of the external field. The blue (red) line is the result for applied in-plane magnetic field parallel to the current ($B_{\mathrm{ext}}//I$) [applied field perpendicular to current ($B_{\mathrm{ext}}\perp I$)].

Figure 2

(a) The dependence of homodyne signal ($V_{\mathit{DC}}^{\max }-V_{\mathit{DC}}^{\min }$) and the frequency region $\delta f$ on an in-plane applied external field (5 mT $<H_{\mathrm{ext}}<$ 15 mT) for two angles $\theta =0^{\circ }$ (blue squares) and 90${}^{\circ }$ (red circles). (b) The dependence of homodyne signal on the frequency of applied microwave current. The in-plane applied magnetic field $H_{\mathrm{ext}}=+15$ mT parallel to the direction of the ac current ($\theta =0^{\circ }$) was used.

Figure 3

(a) The $\delta f$ as a function of in-plane applied field (blue squares for $\theta =0^{\circ }$ and red circles for $\theta ={180}^{\circ }$). Inset shows the calculated oscillating Oe fields as a function of the position in the $y$ direction.(b) The $\delta f$ as a function of the calculated power of in-plane oscillating Oe field (blue squares for $\theta =0^{\circ }$ and red circles for $\theta ={180}^{\circ }$). Blue and red solid lines are fitted curves from the time-dependent classical nonlinear oscillator in nonequilibrium states for $\theta =0^{\circ }$ and $\theta ={180}^{\circ }$, respectively. The fitting curves are calculated by Eq. (2).

Figure 4

The three dimensional surface current distribution ($x$ component) is calculated by means of the Comsol multiphysics package. Two Au contacts (1 $\mu$m$\times$1 $\mu$m and thickness = 110 nm) are located on both sides and 1 $\mu$m Py disk (thickness = 37 nm) is located at the center of the geometry.

Figure 5

The displacements of the VC along the $x$ direction as a function of the time ($t=0\sim 50$ ns). Red, black, and blue lines indicate the $y$ displacements only Oe field contribution, only STT effect, and Oe field contribution + STT effect, respectively. The diameters of the VC trajectories are 64 nm (only STT effect), 160 nm (only Oe field contribution), and 180 nm (STT + Oe field). The phase difference between only the Oe and the combined STT and Oe field is about 15${}^{\circ }$.