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.

The Love surface-acoustic wave seen as a guided non-Newtonian quasi-particle

“…If one brutally sets the thickness d equal to zero, then (17) reduces to c = c T 2 and the notion of surface waves disappears altogether as the resulting wave degenerates into a face shear wave of constant amplitude in the substrate. However, there exists a more subtle limit in considering, as done by Tiersten [11] (also Tiersten et al [12]), the approximation of a thin elastic perfectly glued film that corresponds to the approximation of d being very small compared to the wavelength of the surface wave: that is, d < < λ . This limit coincides with the a priori mechanical description considered by Murdoch [13], where the limit material surface x 2 = 0 is endowed from the start with a surface mass density ρ ^ and a shear surface elasticity of coefficient μ ^ (this is an energy per unit surface).…”

Frequency shifts in plate crystal resonators induced by electric, magnetic, or mechanical fields in surface films

“…There are other frequency effects of the surface films that are more complicated. These include the film intrinsic stresses [6], [7] resulting from manufacturing processes and the deformation of the films caused by electric or magnetic fields when the films have piezoelectric/piezomagnetic couplings [8], [9]. The intrinsic stresses and multiphysical couplings in the films manifest themselves in a crystal resonator through the stresses and strains they produce, called initial or biasing fields in resonators.…”

“…because of the complexity of these theories, the effects of biasing fields in resonators resulting from surface films are rarely studied. Film intrinsic stress and thermal expansion were treated separately in [7] and [12] for resonator applications.…”

Frequency shifts in crystal resonators due to intrinsic stresses in unequal thickness electrodes

“…We consider the case when both electrodes are under the same planar isotropic intrinsic stress [3]. Then 0 ,…”

“…Intrinsic stress in electrodes deposited on resonators is one source of stresses in resonators [1][2][3]. Recently, it has been shown that electrodes with different thickness on the major surfaces of a plate resonator cause the fundamental thickness-shear mode center (nodal plane) to shift away from the middle plane of the plate [4].…”