Coupling Between Thickness-Shear and Flexural Modes in AT-Cut Quartz Mesa Resonators with Beveled Edges

Beveled edges are inevitably formed in micro AT-cut quartz mesa resonators (QMRs) during the photolithography process. This local geometric variation alters the thickness distribution and edge stiffness, which in turn affects the coupling between the primary thickness-shear mode and adjacent parasitic modes. To elucidate the mechanism, we establish a vibration analysis model for the micro AT-cut QMR with beveled edge profiles based on the first-order Mindlin plate theory. The coupling between thickness-shear and flexure is the focus of the investigation. The uniform thickness regions and the beveled edge region are expressed using analytical solutions and power series expansions, respectively. The eigenfrequency equation is then derived through the interface continuity and free boundary conditions. The theoretical frequency spectra show good agreement with finite element results so that the accuracy of the proposed analytical approach is validated. A mode coupling intensity index is constructed based on the discrete Fourier spectrum of the surface displacement in the mesa region, and a modal kinetic energy ratio is introduced to characterize the modal coupling. The influence of structural parameters on the spectral characteristics and mode coupling intensity is analyzed. The results indicate that mesa parameters mainly regulate the resonance frequency and energy trapping of the primary thickness-shear mode. In contrast, due to changes in flexure stiffness of the ends and the reflection angle of elastic waves, the beveled edge parameters affect the coupling intensity between the primary thickness-shear mode and parasitic modes. The optimized parameter combinations lead to low coupling states with nearly pure thickness-shear vibration, while the mesa- and bevel-length related low coupling intervals exhibit better fabrication robustness.