9.4.3 Radial Modes

Initially, an external field of 4 kA/m is applied parallel to the symmetry axis of the cylindrical nanodot (perpendicular to the circular plane on the top and bottom) and the Landau-Lifshitz-Gilbert equation is integrated with strong damping ($\alpha =1$) until equilibrium is reached. Then the external field is switched off and the free oscillation in zero field is studied. The moment, when the external field is switched off, is also the starting time ($t=0$) of our measurements.

Fig. 9.34 shows $M_z$ as a function of the radius of the nanodot for different times. It shows the uniform oscillation of $M_z$ across the whole radius. The amplitude of the oscillation is larger at the circumference of the nanodot and decreases towards the center. The vortex core remains almost undisturbed. The uniformity of the oscillation is even better visible in Fig. 9.35, where $M_z$ is plotted as a function of time for different radii. The oscillation is in phase for the chosen positions on the nanodot. The results for $M_z$ are shown in Figs. 9.35 and 9.36. The main oscillation has a frequency of about 12 GHz, which is in excellent agreement with the analytical predictions. However, the error in the Fourier transforms is quite large, because the total sampling time covers only six oscillations. Thus, one can only estimate, that the true frequency is $12 \pm 2 \mathrm{GHz}$.

Figure 9.34: Snapshots of $M_z$ as a function of the radius at different times during one oscillation period. The magnetization fluctuation outside the vortex core is very uniform, while the vortex core remains undisturbed (``rigid'').

Figure 9.35: $M_z$ as a function of time at different radii. The magnetization is perfectly in phase in all spots, which emphasizes the uniformity of the excitation mode.

Figure 9.36: Fourier spectrum of $M_z$ (of Fig. 9.35) at different radii.

Fig. 9.37 shows, that there is no phase shift or difference in amplitude between the oscillations of $M_z$ at $(R\vert)$ and $(0\vert R)$. The Fourier spectrum (Fig. 9.38) once again shows the peak at a frequency of 12 GHz.

Figure 9.37: $M_z$ as a function of time at $(R\vert)$ and $(0\vert R)$.

Figure 9.38: Fourier spectrum of $M_z$ (of Fig. 9.37) at $(R\vert)$ and $(0\vert R)$.

Even more interesting than $M_z$ is $M_r$ for comparison with analytical results. Fig. 9.39 shows $M_r$ as a function of the radius on the bottom circular plane, through the center and on the top circular plane of the nanodot.

Figure 9.39: $M_r$ as a function of the radius at different $z$ positions.

The oscillation of $M_r$ is also very uniform (Fig. 9.40). However, a small decrease in the amplitude or a phase shift is observed close to the circumference. Still, the vortex core is once again hardly influenced.

Figure 9.40: $M_r$ as a function of the radius at different times.

Fig. 9.41 shows the oscillation of $M_r$ and $M_z$. As predicted by the rigid vortex model, there is a phase shift of $90^\circ$. The Fourier spectrum (Fig. 9.42) shows another time the peak at 12 GHz.

Figure 9.41: $M_r$ and $M_z$ as a function of time at different radii.

Figure 9.42: Fourier spectrum of $M_r$ and $M_z$ (of Fig. 9.41) at different radii.

Also $M_r$ is nicely in phase at different positions on the circumference (Fig. 9.43) and oscillates at the expected 12 GHz (Fig. 9.44).

Figure 9.43: $M_r$ as a function of time at $(R\vert)$ and $(0\vert R)$.

Figure 9.44: Fourier spectrum of $M_r$ (of Fig. 9.43) at $(R\vert)$ and $(0\vert R)$.

Finally, it is also worth to have a look at $M_\varphi$. Its variation is only very small (Fig. 9.45), and oscillates at 12 GHz (Fig. 9.46).

Figure 9.45: $M_\varphi$ as a function of the radius at different times.

Figure 9.46: $M_\varphi$ as a function of time at different radii.

In order to get a more accurate Fourier transform, another simulation with a very small damping constant of $\alpha=0.0001$ has been made. With this small damping constant it takes many cycles until the oscillation is damped out. Thus, the measurement time is a lot longer, which leads to a Fourier transform with higher resolution.

Figure 9.47: Simulation results for a nanodot with $L/R=0.2$ under applied out-of-plane field $\mu_0 H_z=0.005 \mathrm{T}$.

[Oscillation of $\langle M_z \rangle$ as a function of time.]

[Fourier spectrum]

Figure: Simulation results for a nanodot with $L/R=40 \mathrm{nm}/200 \mathrm{nm}=0.2$.

[ $\langle M_z \rangle$ as a function of time.]

[Fourier spectrum]

The time dependence of the average magnetization $\langle M_z \rangle$ and the Fourier spectra for nanodots with an aspect ratio of $L/R=20 \mathrm{nm}/100 \mathrm{nm}=0.2$ and $L/R=40 \mathrm{nm}/200 \mathrm{nm}=0.2$ are given in Figs. 9.47(a) and (b) and 9.48(a) and (b), respectively.

For constant aspect ratio $L/R=0.2$ we find an eigenfrequency of approximately 12.6 GHz. However, for a nanodot with $L/R=40 \mathrm{nm}/200 \mathrm{nm}=0.2$ a very pronounced beating is observed (Fig. 9.48). This is due to the fact, that there is another eigenfrequency of 11.6 GHz very close to the 12.6 GHz oscillation. However, the main peak position depends only on the combination ratio $L/R$. This confirms the magnetostatic origin of the mode. But the physical picture is more complicated for large $L$, when the magnetization dependence on $z$ may be essential. Our numerical calculations confirmed that the eigenfunctions which correspond to low-lying part of the vortex dot excitation spectrum have radial symmetry.