Energy trapping of thickness-shear modes in inverted-mesa AT-cut quartz piezoelectric resonators

abstract We studied thickness-shear vibration frequencies and modes in an inverted-mesa AT-cut quartz plate piezoelectric resonator thinner in the center and thicker near the edges. It is electroded in the central part and unelectroded near the edges. A theoretical analysis was performed using the scalar differential equations by Stevens and Tiersten for thickness-shear vibrations of electroded and unelectroded quartz plates. Based on the variational formulation of the scalar differential equations established in a previous paper and the variation-based Ritz method with trigonometric functions as basis functions, free vibration resonant frequencies and the corresponding thickness-shear modes were obtained. It was found that the electrodes tend to trap the vibration in the central part of the resonator (energy trapping) but the curvature of the resonator surface has an opposite effect. The trapping is sensitive to the electrode thickness, the electrode length, and the curvature of the resonator surface. Hence accurate design is needed to achieve proper energy trapping in inverted-mesa resonators.