An Integrated Analysis of Vibrations of Quartz Crystal Plates with the Mindlin Plate Theory for Resonator Design and Optimization

The high frequency vibrations of quartz crystal plates have been extensively studied for the design and optimization of resonators which work at the thickness-shear vibration mode with relatively high frequency in the RF range. Earlier research work started from the development of special plate equations required for the high frequency vibrations of quartz crystal plates by Mindlin, known as the Mindlin plate theory which is one of the few innovative theory and methods specifically targeted for the bulk acoustic wave (BAW) resonators. By using the Mindlin plate theory with selected vibrations modes, or carrying out analysis in the specified frequency range, we can design quartz crystal resonators, which have the thickness-shear mode that strongly couples to flexural, face-shear, and other modes, with precise knowledge on the effect of mode coupling for separation of vibration modes to enhance the dominance of thickness-shear vibrations. The Mindlin plate theory has been effective and also serves as the only tool for such analysis and design of quartz crystal resonators in this interdisciplinary subject pursued by both electrical and mechanical engineers in the development of this mechanical device used as one of essential electronic circuit components. The core development of the Mindlin plate theory has been efforts of a close group dedicated to the research of wave propagation in finite piezoelectric solids active in the electronic engineering community. Important knowledge and methods include the thermal and acceleration effects which are vital in the design of high precision quartz crystal resonators. Such theories have been utilized in many ways, like approximations in the early stage and the finite element and other numerical implementation lately, to increase the accuracy of analysis in the design evaluation. Experimental verifications of analyses have been made in connection with device development, and design tools have been created by many engineers and groups without publicizing details and functions. As we look back to the theoretical studies, we feel it is necessary to have some basic results checked and validated in conjunction with empirical design methods. In the meantime, it is also necessary to analyze the performance parameters such as the vibration frequency, mode coupling, quality factor, thermal effect, and electrical circuit parameters of quartz crystal resonators with given design parameters. As an effort to establish design method and process based on computational modeling, we draw a roadmap of analyses needed in the product design, and identified needed theoretical formulation and solution strategy. Of course, all the analyses have to be validated with experimental data by our manufacturing process. In addition to the known verification, we are also in the process to develop a computational model for circuit parameters. The final tools based our objective will have the initial design completed with predicted performance and circuit parameters for engineering realization. Such a design process based on wave propagation in finite piezoelectric structures can also be adopted in other acoustic weave devices such as surface acoustic wave (SAW) resonators.