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Two-Dimensional Analysis of Trapped-Energy Vibrations of AT-Cut Quartz Resonators with Large Mass-Loading

We numerically studied both the mass-loading and elastic effects of Au-electrodes on the fundamental thickness-shear (TS) mode of a very thin plate. To solve the two-dimensional forced vibrations of an infinite AT-cut quartz plate with infinite strip Au-electrodes, an efficient technique that utilizes the one-dimensional finite element method was developed. Using the technique, the piezoelectrically induced admittance Yp of the plate was derived with a complex value, and then the frequency f1 and the equivalent inductance L1 of the fundamental TS resonance were calculated as a function of the Au-electrode thickness. The results show that the influence of Au-elasticity on f1 and L1 is negligible compared with mass-loading R, in the wide range of R=0.04–0.36, and that the change of L1 is proportional to R. We also find that the single wave approximation, which was considered to hold only for small R<0.1, holds well even for large mass-loading R=0.36.