Flexural deformation and bending mode of magnetoelectric nanobilayer

A model is discussed for magnetoelectric (ME) coupling in a nanobilayer of ferromagnetic and piezoelectric phases on a substrate. The theory is applied to a specific case, nickel ferrite and lead zirconate titanate bilayer on strontium titanate substrate. Both low-frequency coupling and ME interactions at frequencies corresponding to bending oscillations have been considered. The influence of lattice mismatch between the substrate and piezoelectric and piezomagnetic layers on material parameters has been estimated using the Landau–Ginsburg–Devonshire phenomenological thermodynamic theory. At low frequencies, the model predicts a maximum in the ME coupling strength when the substrate is of the same thickness as the bilayer. The ME voltage coefficient drops with further increase in substrate thickness due to clamping effects. The bending oscillations in nanobilayers are expected to occur at ultralow frequencies (<5 Hz) compared to longitudinal acoustic modes at 100–200 kHz for bilayers of nominal dimensions. The mode frequency shows an increase with increasing substrate thickness, and the resonance value of the ME coefficient is predicted to show a minimum when the substrate-to-bilayer thickness ratio is 0.3.