Nonlinear gyrotropic vortex dynamics in ferromagnetic dots

The quasistationary and transient (nanosecond) regimes of nonlinear vortex dynamics in a soft magnetic dot driven by an oscillating external field are studied. We derive a nonlinear dynamical system of equations for the vortex core position and phase, assuming that the main source of nonlinearity comes from the magnetostatic energy. In the stationary regime, we demonstrate the occurrence of a fold-over bifurcation and calculate analytically the resonant nonlinear vortex frequencies as a function of the amplitude and frequency of the applied driving field. In the transient regime, we show that the vortex core dynamics are described by an oscillating trajectory radius. The resulting dynamics contain multiple frequencies with amplitude decaying in time. Finally, we evaluate the ranges of the system parameters leading to a vortex core instability (core polarization reversal).

Figure 1

(Color online) Amplitude of the vortex core stationary orbits vs amplitude of the external field $\left[{\omega }_0/\left(2\pi \right)=250\text{MHz}\right]$.

Figure 2

(Color online) Amplitudes of the stationary orbits vs the external field frequency for ${\omega }_0/\left(2\pi \right)=250\text{MHz}$. The hyperbolas indicate the critical values of the vortex velocity above which core reversal occurs for $L_e=L$ (solid line) and $L_e=L/2$ (dashed line), see Sec. 5.

Figure 3

(Color online) Phase space of Eq. (6) when ${\omega }_0/\left(2\pi \right)=250\text{MHz}$ and $h_0=0.012{\omega }_0$. The lines indicate the critical values of the vortex velocity above which the core reversal occurs for $L_e/L=1$ (solid line) and $L_e/L=0.5$ (dashed line), see Sec. 5. The shaded region is composed of trajectories that are beyond the reversal curve but also go to vortex core reversal.

Figure 4

(Color online) Resonance diagram for ${\omega }_0/\left(2\pi \right)=250\text{MHz}$. The solid line depicts the resonance conditions while the dashed line shows when the critical orbits reach VC reversal (estimated for $L_e=L/2$). The area between the dashed lines indicates the region of the vortex core reversal, see Sec. 5.

Figure 5

(Color online) Transient spectrum of the vortex motion with ${\omega }_0/\left(2\pi \right)=100\text{MHz}$, $\omega =1.2{\omega }_0$, $h_0=0.012{\omega }_0$, and $d=0.003$.