Theory of low-frequency magnetoelectric coupling in magnetostrictive-piezoelectric bilayers

A theoretical model is presented for low-frequency magnetoelectric (ME) effects in bilayers of magnetostrictive and piezoelectric phases. A novel approach, the introduction of an interface coupling parameter k, is proposed for the consideration of actual boundary conditions at the interface. An averaging method is used to estimate effective material parameters. Expressions for ME voltage coefficients ${\ensuremath{\alpha}}_{E}^{\ensuremath{'}}=\ensuremath{\delta}E/\ensuremath{\delta}H,$ where $\ensuremath{\delta}E$ is the induced electric field for an applied ac magnetic field $\ensuremath{\delta}H,$ are obtained by solving elastostatic and electrostatic equations. We consider both unclamped and rigidly clamped bilayers and three different field orientations of importance: (i) longitudinal fields $({\ensuremath{\alpha}}_{E,L}^{\ensuremath{'}})$ in which the poling field E, bias field H, and ac fields $\ensuremath{\delta}E$ and $\ensuremath{\delta}H$ are all parallel to each other and perpendicular to the sample plane, (ii) transverse fields $({\ensuremath{\alpha}}_{E,T}^{\ensuremath{'}})$ for in-plane H and $\ensuremath{\delta}H$ parallel to each other and perpendicular to out-of-plane E and $\ensuremath{\delta}E,$ and (iii) in-plane longitudinal fields $({\ensuremath{\alpha}}_{E,IL}^{\ensuremath{'}})$ for all the fields parallel to each other and to the sample plane. The theory predicts a giant ME coupling for bilayers with cobalt ferrite (CFO), nickel ferrite (NFO), or lanthanum strontium manganite (LSMO) for the magnetostrictive phase and barium titanate (BTO) or lead zirconate titanate (PZT) for the piezoelectric phase. Estimates of ${\ensuremath{\alpha}}_{E}^{\ensuremath{'}}$ are carried out as a function of the interface coupling k and volume fraction \ensuremath{\nu} for the piezoelectric phase. In unclamped samples, ${\ensuremath{\alpha}}_{E}^{\ensuremath{'}}$ increases with increasing k. The strongest coupling occurs for equal volume of the two phases for transverse and longitudinal cases, but a maximum occurs at $\ensuremath{\nu}=0.1$ for the in-plane longitudinal case. Upon clamping the bilayer, the ME effect is strengthened for the longitudinal case and is weakened for the transverse case. Other important results of the theory are as follows. (i) The strongest ME coupling is expected for the in-plane longitudinal fields and the weakest coupling for the (out-of-plane) longitudinal case. (ii) In ferrite-based composites, ${\ensuremath{\alpha}}_{E,T}^{\ensuremath{'}}$ and ${\ensuremath{\alpha}}_{E,IL}^{\ensuremath{'}}$ are a factor of 2--10 higher than ${\ensuremath{\alpha}}_{E,L}.$ (iii) The highest ME voltage coefficients are expected for CFO-PZT and the lowest values are for LSMO-PZT. Results of the present model are compared with available data on the volume and static magnetic field dependence of ${\ensuremath{\alpha}}_{E}^{\ensuremath{'}}.$ We infer, from the comparison, ideal interface conditions in NFO-PZT and poor interface coupling for CFO-PZT and LSMO-PZT.