Intrinsic stress in thin films deposited on anisotropic substrates and its influence on the natural frequencies of piezoelectric resonators

Abstract

A description of intrinsic stress in thin films deposited on anisotropic (piezoelectric) substrates is obtained from the rotationally invariant equations of nonlinear elasticity (and electroelasticity). The accompanying residual stress state in the substrate is, of course, included in the description, as is the residual stress state in the thin film. The equations for small dynamic fields superposed on the static (intrinsic plus residual) bias are obtained, and the equation for the perturbation in eigenfrequency of a piezoelectric resonator due to the intrinsic and residual stress state is presented. The static biasing equations for both extension and flexure of the plated crystal plate, which are required for the determination of the residual stress state from the intrinsic stress state, are obtained. The important case of a thin film deposited on one side of a plate substrate is treated in detail, and the relation of the flexural curvatures of the plated plate to the components of the intrinsic stress is found. The values of the intrinsic stress components in thin titanium–gold films deposited on one side of AT-cut quartz substrates are obtained from measurements of the curvatures in three independent directions.