Energy-trapping analysis for the bi-stepped mesa quartz crystal microbalance using the finite element method

Abstract

AT-cut quartz crystal plate resonators have been widely used for different frequency applications, such as the quartz crystal microbalance (QCM), owing to its high frequency-temperature stability. In this study, a finite element program based on two-dimensional Mindlin AT-quartz plate equations for thickness shearing, thickness twist and flexure vibration is written using MATLAB language. The thickness-shearing mode shape and its overtone of the rectangular AT-cut quartz resonator are computed using the program. The energy-trapping effects on a conventional resonator, a single-step-mesa resonator and a stepped bi-mesa structure resonator are analyzed. The convergence study is presented for a rectangular quartz plate without electrodes, and the frequency spectrum of the quartz plate around the fundamental TS mode is computed. A factor related to the vibrational energy of the thickness-shearing mode is defined to evaluate the energy-trapping characteristics of different structure designs. The comparisons between the conventional quartz resonator and mesa-design resonator show that the mesa design can trap the vibration energy within the electrode area more effectively. The computation of the energy-trapping factor as a function of the dimensions of the mesa depth and bi-mesa width design are performed. The results show that when the mesa depth of the single mesa reaches a certain value, further energy-trapping improvement cannot be achieved by increasing the mesa depth. A bi-mesa design can further improve the decoupling characteristics of the resonator beyond that of the single-mesa resonator.