High frequency permeability of the composite with ferromagnetic spherical shells

TL;DR

This paper models the high-frequency permeability of composites with hollow ferromagnetic spherical shells, analyzing how the permeability depends on shell thickness and magnetic anisotropy, revealing similarities to thin film behavior.

Contribution

It introduces a model for ferromagnetic spherical shells with vortex-like magnetization and studies their high-frequency permeability considering different anisotropies and shell thicknesses.

Findings

Permeability behavior is similar for both anisotropy types when shell thickness is comparable to diameter.

For thin shells with circular anisotropy, susceptibility resembles that of thin films.

Magnetization oscillations are non-homogeneous in the composite.

Abstract

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.