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Phenomenological theory of magneto-electric coupling in granular multiferroics

We study coupling between the ferroelectric polarization and magnetization of granular ferromagnetic film using a phenomenological model of combined multiferroic system consisting of granular ferromagnetic film placed above the ferroelectric (FE) layer. The coupling is due to screening of Coulomb interaction in the granular film by the FE layer. Below the FE Curie temperature the magnetization has hysteresis as a function of electric field. Below the magnetic ordering temperature the polarization has hysteresis as a function of magnetic field. We study the magneto-electric coupling for weak and strong spatial dispersion of the FE layer. The effect of mutual influence decreases with increasing the spatial dispersion of the FE layer. For weak dispersion the strongest coupling occurs in the vicinity of the ferroelectric-paraelectric phase transition. For strong dispersion the situation is the opposite. We study the magneto-electric coupling as a function of distance between the FE layer and the granular film. For large distances the coupling decays exponentially due to the exponential decrease of electric field produced by the oscillating charges in the granular ferromagnetic film.