Second overtone thickness-extensional vibrations in ZnO piezoelectric film bulk acoustic resonators: modified algorithm for 2D complex transcendental equations

This article investigates the second overtone thickness-extensional (TE2) vibrations and associated mode-coupling behaviors in ZnO piezoelectric film bulk acoustic resonator (FBAR), utilizing its wave dispersion relation and the higher-order stress balance principle. By superimposing the general wave solutions of multiple eigenmodes within the frequency range of the TE2 mode, mode-coupling solutions for ZnO FBAR are constructed. The substitution of these mode-coupling solutions into the higher-order stress balance principle, as laterally weak boundary conditions, leads to the frequency spectrogram equation, determining the relationship between resonance frequency and plate length-to-thickness ratio. A modified algorithm that combines the bisection method and the complex modulus ratio method is developed to solve the dispersion equation and frequency spectrogram equation (namely a kind of 2D complex transcendental equations) accurately and efficiently. The obtained results indicate that the operational TE2 mode may couple to unwanted 3^rd thickness-shear, fundamental thickness-shear, and flexural modes. Moreover, the mode-coupling behaviors depend strongly on resonance frequencies and plate length-to-thickness ratio. The displacement distributions of total displacement components, alongside the main displacement components of all considered eigenmodes, clearly demonstrate the variety of coupling behaviors. According to the obtained frequency spectrograms, the desirable values of plate length-to-thickness ratio for a clean operating mode with very weak coupling intensity are determined. These findings are of vital importance for the understanding of the mode-coupling mechanism in overtone thickness-extensional FBARs, which will facilitate the structural design and optimization of FBAR devices.