Field-induced spin-modulated transitions in epitaxial (001) BiFeO$_{3}$ films

Alterations in the ground state of thin (001) films of a BiFeO$_{3}$-type multiferroic in a magnetic field are studied theoretically, with changes in the energy of induced anisotropy taken into account. The anisotropy vs. field phase diagrams identifying the stability regions of homogeneous antiferromagnetic states and the regions of emergence of spatially modulated antiferromagnetic states are constructed for three mutually orthogonal orientations of applied magnetic field. We show that, as the magnetic field decreases, the transformation of a homogeneous phase into a spatially modulated state occurs at the instability point of the homogeneous state via gradual emergence of the conical phase that transforms into a planar cycloid with the decreasing magnetic field. A multiferroic film grown on a (001) substrate develops considerable anisotropy of the energy of spatially modulated state, depending on the modulation orientation. Meanwhile, cycloids with different orientations undergo the transitions from incommensurate phase into the homogeneous state differently: either the conical cycloid is formed followed by its collapse into the homogeneous state or an unlimitedly growing domain of the homogeneous phase is formed within the flat cycloid. Examples of field-induced changes in magnetization, with changes in spin states taken into account, are provided. These results are of value in practical applications of multiferroic strain engineering.