Analysis of thickness-extensional modes in energy-trapped thin film resonators

A theoretical analysis on a rectangular energy-trapped piezoelectric thin zinc oxide film resonator operating with thickness-extensional modes is performed in this paper. The two-dimensional scalar differential equations derived by Tiersten and Stevens are used which can describe the in-plane mode distribution. Based on the scalar equations, we construct a variational formulation which provides a theoretical foundation for the Ritz method in our analysis. Free vibration frequencies and corresponding mode shapes are obtained and discussed. Modes with vibration mainly under the electroded area are proved to exist. The results show that the classical method with an approximation of neglecting the four corner regions can cause a frequency error on the order of dozens of parts per million for the fundamental thickness-extensional modes, which is significant for the design and operation of the FBARs.