Energy trapping of thickness-extensional modes in thin film bulk acoustic wave resonators

We perform a theoretical analysis on a rectangular trapped-energy piezoelectric resonator operating with thickness-extensional modes of a thin zinc oxide film on a silicon layer. The driving electrode covers the central part of the resonator only. The two-dimensional scalar differential equation for such a structure derived by Tiersten and Stevens is employed which can describe in-plane mode variations. A variational formulation is developed which provides the theoretical foundation for the Ritz method used in our analysis. Free vibration frequencies and modes are obtained and examined. Modes with vibrations trapped under the driving electrode are found. An approximation used to obtain the classical result of the resonator by Tiersten and Stevens, i.e., the neglect of the four corner regions of the resonator, is removed in our variational analysis. It is shown that their approximation causes a frequency difference on the order of dozens of parts per million for the fundamental thickness-extensional operating mode, quite significant in resonator design and operation.