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 alpha(E)(')=deltaE/deltaH, where deltaE is the induced electric field for an applied ac magnetic field deltaH, 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 (alpha(E,L)(')) in which the poling field E, bias field H, and ac fields deltaE and deltaH are all parallel to each other and perpendicular to the sample plane, (ii) transverse fields (alpha(E,T)(')) for in-plane H and deltaH parallel to each other and perpendicular to out-of-plane E and deltaE, and (iii) in-plane longitudinal fields (alpha(E,IL)(')) 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 alpha(E)(') are carried out as a function of the interface coupling k and volume fraction nu for the piezoelectric phase. In unclamped samples, alpha(E)(') 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 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, alpha(E,T)(') and alpha(E,IL)(') are a factor of 2-10 higher than 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 alpha(E)('). We infer, from the comparison, ideal interface conditions in NFO-PZT and poor interface coupling for CFO-PZT and LSMO-PZT.