Thickness mode material constants of a supported piezoelectric film

Abstract

New thick film ceramic processing techniques and microelectromechanical systems require material characterization of a piezoelectric film supported by a substrate. An analytical solution of the one-dimensional wave equation for multiple layered systems driven in the thickness mode is presented. The impedance across the piezoelectric layer is derived and expressed in terms of the material properties of the two materials. This includes the open-circuit elastic stiffness c33D, the clamped permittivity ε33S and the h33 piezoelectric constant of the piezoelectric layer and the elastic stiffness csD of the substrate. The properties are expressed as complex variables in order to account for the losses within the materials. The material parameters of the solution are extracted from experimental results using a modified Levenberg–Marquardt technique. The capabilities of this nondestructive technique are demonstrated using experimental and simulated impedance spectra of lead zirconium titanate sol gel composite coatings within the thickness range of 15–70 μm on aluminum and on platinum electroded silicon substrates and by simulating the mass loading of a quartz thickness resonator. The analytical solution allows for the extension of the Institute of Electrical and Electronics Engineers free resonator impedance technique to supported films and provides a method for determining the material parameters of a piezoelectric coating.