Optical detection of vortex spin-wave eigenmodes in microstructured ferromagnetic disks

We examine the excitation of spin-wave eigenmodes in the vortex state of microsized ferromagnetic circular dots made of Permalloy (${\mathrm{Ni}}_{81}{\mathrm{Fe}}_{19}$) both theoretically and experimentally using Brillouin light scattering microscopy. We report on the detection of the radial spin-wave eigenmodes of single elements with high mode number (up to $n$ = 13 for the largest disk radius 2.5 $\mu$m). Theoretically we obtain an equation for the eigenfrequencies valid for arbitrary dot aspect ratios (thickness/radius) within the magnetostatic approximation. We demonstrate the influence of the disk radius on the spatial mode profiles, in particular, changes in the pinning of the dynamical magnetization at the edges and in the center of the disks. The measured spin-wave eigenfrequencies are in good agreement with our calculations for the disks with different radii.

Figure 1

Schematic layout of the sample geometry. Individual, 40 nm thick, cylindrical ${\mathrm{Ni}}_{81}{\mathrm{Fe}}_{19}$ disks of different radii ranging from 2.5 $\mu$m down to 250 nm are positioned in the center of a microcoil having an inner and outer radius of 8 $\mu$m and 11 $\mu$m, respectively. The driving microwave magnetic field is directed perpendicularly to the disk's plane along the $z$ axis as depicted by red arrows circulating around the microcoil. The magnetic moments within the disk are aligned along the $\varphi$ direction following circular lines around the vortex core.

Figure 2

Spin-wave spectra measured in ${\mathrm{Ni}}_{81}{\mathrm{Fe}}_{19}$ disks with different radii $R$ of (a) 2.5 $\mu$m, (b) 2 $\mu$m, (c) 1 $\mu$m, (d) 750 nm, and (e) 250 nm. In the largest disk with $R=2.5\mu$m, the radial spin-wave eigenmodes with mode numbers up to $n=13$ can be clearly seen. The disk thickness is $L=40$ nm.

Figure 3

Measured two-dimensional intensity profiles of the radial spin-wave eigenmodes in a Permalloy disk with 2.5 $\mu$m radius. As can be clearly seen from the intensity distributions, the measured eigenmodes excited at the frequencies of 4.3, 6.9, and 8.5 GHz can be identified as mere radially symmetric modes having odd numbers of circular antinodes $n=1$, 3, and 5, respectively.

Figure 4

The radial spin-wave mode frequencies measured by Brillouin light scattering microscopy (filled symbols) and calculated using Eq. (3) (empty symbols) as a function of the disk aspect ratio $\beta =L/R$. The lines connecting the calculated frequency values are guides to the eye.