Microwave magnetization dynamics in ferromagnetic spherical nanoshells

Ferromagnetic particles of nano- and micrometer-size range are in high demand owing to their promising technological applications in magnetic memory storage, biomedical treatment, and microwave devices. New absorbing materials are required to possess the properties of low density and strong absorption in a broad frequency band. The dynamic susceptibility spectra of nanosized spherical shells supporting onion and vortex magnetization configurations are studied as a function of particle radius $40--125\phantom{\rule{0.16em}{0ex}}\mathrm{nm}$ and shell thickness $10--115\phantom{\rule{0.16em}{0ex}}\mathrm{nm}$ by means of numerical micromagnetic simulation. For the saturated shell, the frequencies of higher-order spin-wave modes are found to rapidly approach the subterahertz frequency regime ($200--300\phantom{\rule{0.16em}{0ex}}\mathrm{GHz}$) when the geometry is transformed from the single-domain sphere to the spherical shell. The frequency of each mode is proportional to ${R}_{2}^{\ensuremath{-}2}$ when the sphere radius outer ${R}_{2}$ is below 30 nm in reasonable agreement with the exchange approximation. Thickness-dependent vortex core gyrotropic modes were studied in the absence of an external bias field in the frequency range $0.1--30\phantom{\rule{0.16em}{0ex}}\mathrm{GHz}$. The size dependence of the vortex spin excitations is complicated by mixing of the modes at different particle sizes. For shells with large thickness and outer radii above ${R}_{2}\ensuremath{\ge}100\phantom{\rule{0.16em}{0ex}}\mathrm{nm}$ the amplitudes of both $n=0$ and $n=1$ vortex flexural modes are less intense than even higher-frequency resonances. However, the $n=0$ mode becomes the dominant mode in the dynamic susceptibility when either the radius or thickness of the particle becomes sufficiently small.