High frequency permeability of the composite with ferromagnetic spherical shells

The paper studies high-frequency permeability of the composite materials consisting of hollow ferromagnetic particles embedded into the non-magnetic media. We model the ferromagnetic particles in composite by spherical shells: the thickness of the ferromagnetic region $d$ compared to the particles' diameter $D$ can vary in a wide range, from $d\ll D$ to $d\sim D$. We assume that the magnetization distribution in such a particle is non-uniform, but forms a vortex-like structure: the magnetization is twisted in some plane outside two vortex cores placed at the poles of the particle. We consider two types of magnetic anisotropy, which help to stabilize such a magnetic configuration: the easy-plane magnetic anisotropy and the ``circular'' uniaxial magnetic anisotropy with easy axis rotated together with the magnetization direction outside the vortex core. The high-frequency permeability of such a composite material has been studied in the limit of non-interacting particles. We study the dependence of the permeability on the ratio $d/D$. We showed that for $d\sim D$ the composite's permeability behaves in similar manner for both types of magnetic anisotropy. It was shown also that for the second type of magnetic anisotropy and in the limit $d/D\ll1$ the frequency dependence of the particle's susceptibility is quite similar to that for the thin film. At the same time, the magnetization oscillations in the ac field are non-homogeneous for both types of anisotropy.