Theory of magnetoelectric effects at microwave frequencies in a piezoelectric/magnetostrictive multilayer composite

A phenomenological theory is proposed to treat the magnetoelectric (ME) coupling at frequencies corresponding to ferromagnetic resonance in a multilayer composite consisting of alternate layers of piezoelectric and magnetostrictive phases. We discuss two models: (i) a simple two-layer (bimorph) structure and (ii) a generalized approach in which the multilayer structure is considered to be a homogeneous medium. Expressions for the stress induced shift $\ensuremath{\delta}H$ in the ferromagnetic resonance field due to an applied electric field E have been obtained for both cases. For a bimorph, $\ensuremath{\delta}H$ is directly proportional to the product of the applied electric field and the ME coupling constant. For a nickel ferrite--lead zirconate titanate (PZT) two layer structure, the theory predicts a factor of 5 stronger effect than in yttrium iron garnet-PZT. When the composite is considered to be a homogeneous medium, the corresponding shift $\ensuremath{\delta}H$ is given by $2{M}_{0}$ ${(B}_{33}\ensuremath{-}{B}_{31})$ E, where ${M}_{0}$ is the composite magnetization and B's are the ME coefficients. For this model, a method for the calculation of magnetoelectric coefficients from experimental data is presented.