Trapped hybrid modes in solidly mounted resonators based on c-axis oriented hexagonal crystals

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Wathen, Adam D.
Munir, Farasat
Hunt, William D.
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Assuming an idealized piezoelectric bulk acoustic wave resonator, one typically calculates the velocity of the fundamental bulk acoustic mode as the measured frequency times twice the thickness of the piezoelectric film. In c-axis 6mm hexagonal crystals of (e.g., ZnO or AlN), both the longitudinal and thickness shear modes are peizoelectrically active using thickness excitation and lateral-field excitation, respectively. Without a loss of generality, we concentrate our study on ZnO films. The theoretical velocity of the pure thickness shear mode in sputtered ZnO, based strictly on reported material properties, is calculated to be approximately 2580 m/s. However, a variety of acoustic velocities for the thickness shear mode in ZnO have been reported in the literature ranging from about 3100–3500 m/s. These reported values represent a 20%–36% increase in acoustic velocity relative to the theoretical values. In the literature, this deviation is typically attributed to ZnO film inconsistencies and other phenomena which can be difficult to quantify. We propose that the reported inconsistencies may be attributed to a hybrid acoustic mode comprised of a coupling of shear and longitudinal particle displacements. In this paper, we present a theoretical description of a hybrid mode in ZnO solidly mounted resonator (SMR) devices. We begin first with an experimental verification of a mode with a changing velocity in a ZnO SMR with the only variable being the ZnO thickness. Using the acoustic velocity through the thickness as an effective velocity with which to reference the mode, we find the effective acoustic velocity to range from 3100–3900 m/s, with increasing ZnO thickness. We then start from the first principles of piezoelectric acoustic wave propagation and derive three coupled partial differential equations describing a hybrid mode comprised of the coupling between longitudinal and shear particle displacement and the corresponding piezoelectrically generated potential in the ZnO film. The qualitative findings described by the derived equations are then further investigated with finite element simulation (COMSOL MULTIPHYSICS®). We simulate the performance of our experimental devices using the COMSOL platform, examine the eigenfrequencies of the structure, and find a hybrid mode which is trapped both vertically and laterally in the ZnO film. Calculating the effective velocity of the simulated modes, we find the simulated effective velocities to be within 1.5% of our measured results. Finally, we compare simulation results to experimentally measured results of a previously observed hybrid mode and achieve a 0.2% agreement.
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