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

Abstract

AT-cut quartz crystal plate resonators have been widely used for different frequencyapplications, such as the quartz crystal microbalance (QCM), owing to its highfrequency–temperature stability. In this study, a finite element program basedon two-dimensional Mindlin AT-quartz plate equations for thickness shearing,thickness twist and flexure vibration is written using MATLAB language. Thethickness-shearing mode shape and its overtone of the rectangular AT-cut quartz resonatorare computed using the program. The energy-trapping effects on a conventionalresonator, a single-step-mesa resonator and a stepped bi-mesa structure resonatorare analyzed. The convergence study is presented for a rectangular quartz platewithout electrodes, and the frequency spectrum of the quartz plate around thefundamental TS mode is computed. A factor related to the vibrational energy of thethickness-shearing mode is defined to evaluate the energy-trapping characteristics ofdifferent structure designs. The comparisons between the conventional quartzresonator and mesa-design resonator show that the mesa design can trap thevibration energy within the electrode area more effectively. The computation of theenergy-trapping factor as a function of the dimensions of the mesa depth and bi-mesawidth design are performed. The results show that when the mesa depth of the singlemesa reaches a certain value, further energy-trapping improvement cannot beachieved by increasing the mesa depth. A bi-mesa design can further improve thedecoupling characteristics of the resonator beyond that of the single-mesa resonator.