Electric polarization in bi-layered ferromagnetic film with combined magnetic anisotropy

Magnetoelectric phenomena become one of the most attractive fields of magnetism. One of discussable items is inhomogeneous magnetoelectricity leading to appearance of electric polarization of magnetic domain walls, improper polarization of multiferroics etc. In our article we attract attention to the modulation of electric polarization by magnetic inhomogeneity in exchange coupled ferromagnetic film whose layers differ by magnetic anisotropy. Our goal is to explore the influence of combined magnetic anisotropy (especially its cubic component) on the behavior of electric polarization of bi-layered film placed in magnetic field. We perform theoretical analysis in a frame of phenomenological modeling of spins structures considering two geometries of magnetic field (magnetic field oriented perpendicular to a film plane and magnetic field oriented in a film plane along “hard magnetization” axis). Our results show that the presence of cubic magnetic anisotropy (Kc<0) in the layers allocates the planes of magnetic inhomogeneities and correspondingly the directions of electric polarization. We demonstrate that magnetic field applied along the “hard magnetization” axis leads to the rotation of electric polarization in the 45° range and magnetic field applied along normal to a film influences the magnitude of electric polarization leading to the lowering of polarization after attaining the maximum value.