Excitations with negative dispersion in a spin vortex

Micron-sized ferromagnetic permalloy disks having an in-plane vortexlike configuration are excited by a fast-rise-time magnetic-field pulse perpendicular to the plane. The excited modes are imaged using time-resolved magneto-optic Kerr microscopy and Fourier transformation. Two types of modes are observed: modes with circular nodes and modes with diametric nodes. The frequency of the modes with circular nodes increases with the number of nodes. In contrast, the frequency of the modes with diametric nodes decreases with the number of nodes. This behavior is explained accurately by an analytical model.

Figure 1

(Color) Images of the eigenmodes for $R=3\mathrm{\mu m}$. The top row shows the absolute value of the Fourier amplitude at resonance, the bottom row the phase. The modal maps are composed from two half-images: the left from the micromagnetic simulation; the right from the experiment. (a)–(c): Axially symmetric modes showing concentric nodes $(n=1,2,3,m=0)$. (a) $2.80\mathrm{GHz}$. (b) $3.91\mathrm{GHz}$. (c) $4.49\mathrm{GHz}$. (d), (e): These modes have nodal lines going across the image. (d) $2.08\mathrm{GHz}$, one azimuthal node (1,1). (e) $1.69\mathrm{GHz}$, two azimuthal nodes (1,2). Notice that across the nodal lines the phase changes by $180°$. A Hamming cutoff window was used to obtain the fast Fourier transform (FFT) data from the $3.33\text{−}\mathrm{ns}\text{−}$long scans. The Fourier spectrum consists of a sequence of resonances along the frequency axis corresponding to the eigenmodes. Although the spacing in the frequency domain of the raw data is $df=1∕(T_{end}-T_{start})=1∕3.33\mathrm{ns}=0.30\mathrm{GHz}$, the exact location of the maximum of these resonances can be determined with a higher accuracy when using zero filling. The half-images from the time-domain simulation data were obtained using the same method.

Figure 2

Dispersion relations. (a) The frequency for modes of the type $(n,m=0)$ as a function of the radial mode number $n$: $R=1\mu \mathrm{m}(엯)$, $1.5\mu \mathrm{m}(◇)$, $2\mu \mathrm{m}(▵)$, $3\mu \mathrm{m}(▿)$. The error bars are $\pm df∕2$. (b) The frequency for modes of the type $(n=1,m)$ as a function of the azimuthal mode number $m$. The corresponding data points of the micromagnetic simulation based on the LLG code[18] are diagramed as small full symbols and dashed lines. Results from the variational model as discussed in the text are represented as solid and the dotted lines. The dotted line is calculated by the diagonal elements and the solid line from the full matrix eigenvalues.

Figure 3

Scaling. When$f(n,m){(R∕d)}^{1∕2}\doteq \tilde{f}$ is plotted vs $m$ at fixed $n$ or vs $n$ at fixed $m$, all data points for different radii fall onto one single dispersion curve $\tilde{f}\left(m\right)$ and $\tilde{f}\left(n\right)$. The symbol and line attributes are the same as in Fig. 2.