Magnetoelectric Properties of Cylindrical Ferromagnetic Particles

Within the framework of the phenomenological approach, the emergence of an inhomogeneous magnetic state and the associated appearance of inhomogeneous electric polarization in the volume of small magnetic particles have been studied. The microscopic mechanism of this relation between magnetization and polarization is due to spin-orbit interaction. The specific form of magnetization and polarization is determined by the shape and size of cylindrical particles. Using the free energy expression for magnetization, a nonuniform distribution of magnetization in the form of three-dimensional magnetic vortices is obtained. A vortex state appears only for cylinders with a radius greater than a certain critical value, while for particles with a smaller radius, a uniform magnetic state arises. In the vortex state, nonuniform electric polarization emerges. The local polarization vectors have the form of rays directed towards the cylinder axis. The region of existence of such inhomogeneous states has been determined. A change in the local electric polarization of small magnetic particles in an external magnetic field is considered. An expression for the magnetoelectric susceptibility is obtained.