Effective Piezoelectric Coefficients of Ferroelectric Thin Films on Elastic Substrates

In micro-electromechanical systems consisting of a piezoelectric thin film on a substrate, due to the clamping by the substrate, the effective piezoelectric film properties are different from the bulk material behavior. However, it is of particular difficulty to determine the transverse piezoelectric parameters for such a system. A simple theoretical model by Muralt et al. (1996) allows calculation of the transverse piezoelectric coefficients in terms of the bulk parameters of the piezoelectric material. Relying on the assumption of a rigid wafer this model is a reasonable first approximation, but on the other hand, for the high accuracy needed in technical applications, it may not always be sufficient. Therefore, a more complex theoretical model for a piezoelectric thin film on an elastic substrate was derived, which delivers more realistic results for the transverse piezoelectric coefficients. This model takes the elasticity of the substrate into account, while the PZT layer is fully covered by an electrode and the vertical displacements are suppressed at the bottom of the substrate. In this way, the model represents a system of infinite lateral extent with no overall bending. As the next step, finite element simulations were carried out to verify the simple theoretical model and the new developed model, and there was a good agreement between the theoretical models and the numerical results. Furthermore, a parametric study was performed considering the influence of various bulk material parameters. Finally, the newly proposed theoretical model was compared to more realistic models where the PZT layer possesses an isolated electrode spot instead of being covered fully by an electrode. Different options were used for the boundary conditions at the bottom of the substrate. First, the same boundary conditions as for the new theoretical model were chosen (suppression of the vertical displacements at the bottom of the substrate). Second, the bottom of the substrate was free to move such that overall bending was no longer prevented. As the main result, the comparison to the new theoretical model taking into account the elastic substrate showed only negligible differences, and, thus, it is suggested for the determination of the effective piezoelectric parameters of piezoelectric layers on elastic substrates.