9.4.1 Vortex Precession

The dynamic behavior of the magnetic nanodots has been studied by instantaneously applying an external field of 80 Oe (8 mT, 6.4 kA/m) in plane perpendicular to the dot axis. Even though the experiment was started from the equilibrium magnetization distribution in zero field, $M_x$ and $M_y$ show a quite irregular behavior during the first 0.5 ns. During this time the vortex core ``adapts'' to the applied external field and deforms while it does already start its precession towards equilibrium. A low damping constant of $\alpha = 0.05$ has been used.

Figs. 9.28 and 9.29 show $\langle M_x \rangle$ and $\langle M_y \rangle$ as a function of time. Simulation ``ad/10'' uses an inhomogeneous mesh with very small finite elements (edge length 2 nm) in the center, where the vortex core is found in equilibrium, and a smooth transition to a coarser mesh outside the core (up to an edge length of 10 nm at the circumference - cf. Tab. 9.2). Simulation ``006/07'' uses a homogeneous mesh with an average mesh size of 6 nm, simulation ``006/08'' uses the same mesh with a shorter time step and simulation ``004/02'' uses a mesh with 4 nm edge length. Obviously, simulation ``ad/10'' exhibits strong deviations from the other results, because the vortex has to move into the coarser mesh as it is pushed out of the center of the nanodot due to the external field. However, the coarser mesh leads to a bad approximation of the vortex core and an inaccurate result.

Figure: Oscillation of $\langle M_x \rangle$ as the vortex core precesses towards equilibrium.

Figure: Oscillation of $\langle M_y \rangle$ as the vortex core precesses towards equilibrium.

In contrast, the simulations using the uniform meshes give results, which are in good agreement. The precession frequency of 0.65 GHz is also confirmed by the results of Guslienko and coworkers [130]. In addition it has been found, that the magnetostatic energy oscillates in phase with $M_x$ (Fig. 9.30). This has to be ascribed to variations in the surface charge density on the circumference.

Figure 9.30: Oscillation of the magnetostatic energy.

The time evolution of $\langle M_x \rangle$ (the average of $M_x$ over the whole nanodot) for a dot with an aspect ratio of $L/R=10 \mathrm{nm}/100 \mathrm{nm}=0.1$ is given in Fig. 9.31(a). Then the damped oscillation, which is caused by the spiral motion of the vortex core towards its equilibrium position, is observed. The corresponding Fourier spectrum is given in Fig. 9.31(b) and shows a sharp peak at a frequency of 0.7 GHz.

Figure: Simulation results for vortex precession in a nanodot with $L/R=20 \mathrm{nm}/100 \mathrm{nm}=0.2$ under applied in-plane field $\mu_0 H_x=0.01 \mathrm{T}$.

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

[Fourier spectrum]

Figure 9.32: Translation mode eigenfrequencies versus aspect ratio $L/R$ for nanodots with $R=100 \mathrm{nm}$.

Fig. 9.32 shows the results of the translation mode eigenfrequencies of various nanodots with a radius $R=100 \mathrm{nm}$ and a thickness between 10 nm and 40 nm. The results are in good agreement with the results of a finite difference model and the analytical ``two-vortices'' model presented in Ref. [130].

Figure 9.33: Energy over time for a dot with $L/R=0.1$ and $\mu_0 H_x=0.01 \mathrm{T}$.

The decreasing total energy (dissipation due to damping with $\alpha = 0.05$ in the Landau-Lifshitz equation of motion) and the swapping between magnetostatic and Zeeman energy (which shifted by 180$^\circ$) are shown in Fig. 9.33. The exchange energy remains constant, because the vortex core, which accounts for most of the exchange energy, precesses without changing its shape. This confirms the analytical description of the translational mode suggested in Ref. [131].

Direct experimental observation of this mode in an isolated vortex using time-resolved Kerr microscopy has recently been reported by Park et al. [132]. There is good qualitative agreement with the analytical and numerical models, but still a few questions concerning the quantitative discrepancies and damping times remain open.