Electric Polarization in Small Magnetic Particles

In the framework of the phenomenological approach, we obtain an inhomogeneous vortex distribution of magnetization and associated inhomogeneous electric polarization in small magnetic particles. The microscopic mechanism of this connection between magnetization and polarization is due to the Dzyaloshinskii–Moriya interaction. Expressions of magnetization and of polarization are determined by the geometric shape of the nanoparticles. The phase transition in small spherical particles of ferromagnetics is studied in the framework of the Ginzburg–Landau phenomenological theory. Using the free energy expression for magnetization in spherical nanoparticles, we obtain a nonuniform distribution of the magnetization in the form of three-dimensional magnetic vortices. For large sphere radii, a phase transition to a vortex state is possible, while for a radius that is less than a certain critical value, the nanoparticle can be in a homogeneous state. The inhomogeneous electric polarization is calculated. In our case, the electric polarization has the form of two-dimensional hedgehogs. The area of existence of such inhomogeneous states is determined.