Engineering spin wave spectra in thick ${\mathrm{Ni}}_{80}{\mathrm{Fe}}_{20}$ rings by using competition between exchange and dipolar fields

Control of the spin wave dynamics in nanomagnetic elements is very important for the realization of a broad range of novel magnonic devices. Here we study experimentally the spin wave resonance in thick ferromagnetic rings (100 nm) using perpendicular ferromagnetic resonance spectroscopy. Different from what was observed for the continuous film of the same thickness, or from rings with similar lateral dimensions but with lower thicknesses, the spectra of thick patterned rings show a nonmonotonic dependence of the mode intensity on the resonance field for a fixed frequency. To explain this effect, the theoretical approach by considering the dependence of the mode profiles on both the radial and axial coordinates was developed. It was demonstrated that such unusual behavior is a result of the competition between exchange and dipolar fields acting at the spin excitations in the structure under study. The calculations are in a good agreement with the experimental results.

Figure 1

SEM image of the periodic array of ${\mathrm{Ni}}_{80}{\mathrm{Fe}}_{20}$ rings. Bottom left inset: SEM image of the isolated ring with indicated dimensions. Top right inset: the geometry of the experiment.

Figure 2

The microwave absorption spectra taken at 11 GHz for (a) 100-nm-thick continuous film and (b) circular rings.

Figure 3

Resonance fields extracted as a function of the excitation frequency for the continuous film: squares, dots, and triangles are experimental results; lines are calculations by Kittel formula (1) with $K_1=0.47\pi /a, K_2=1.4\pi /a, K_3=2.3\pi /a$, as explained in the text.

Figure 4

Standing spin wave profiles ${\mu }_{\mathrm{m}}\left(\rho \right)$ as functions of the radial coordinate for the first five quantization numbers, calculated by formula (3) with corresponding $C_m=-\frac{J_0\left({\beta }_m\right)}{Y_0\left({\beta }_m\right)}$ and ${\beta }_1=11.77, {\beta }_2=23.554, {\beta }_3=35.337, {\beta }_4=47.12, {\beta }_5=58.902$.

Figure 5

Resonance fields extracted as a function of the excitation frequency for the circular ring: triangles and squares are experimental results; solid lines are theoretical calculations by formula (8). The mode with indices $\left(m,n\right)=\left(1,2\right)$ is the highest (magenta) line (experimental black squares), the next (second) lower line corresponds to the mode with indices $\left(m,n\right)=\left(1,1\right)$ and has the largest intensity in the FMR experiment (red triangles). The next modes are $\left(m,n\right)=\left(3,2\right)$ and $\left(m,n\right)=\left(3,1\right)$, as indicated in the figure.

Figure 6

Сomponents of the dipole-dipole field $4\pi F_1(k_m,K)$ and $4\pi F_2(k_m,K)$ as a function of out-of-plane wave vector $K$ with fixed in-plane wave vectors $k_1={\beta }_1/R$ and $k_3={\beta }_3/R$ [calculations by the formulas (10a) and (10b)].

Figure 7

Bottom panel: calculated intensity of four spin wave modes observed in rings. Top panel: experimental data for absorption derivative at 11 GHz for comparison.