Coupled vibrations of thickness-extensional FBARs under stress-strain biasing state

Abstract

The method of frequency spectrum quantitative prediction (FSQP) is extended to investigate high-frequency and mode-coupling vibrations of piezoelectric film bulk acoustic resonators (FBARs) subject to initial stress-strain biasing fields for the first time. In numerical examples, we explore the cases of uniaxial compressive and tensile initial stresses along the in-plane and thickness directions, respectively. Derived from the nonlinear electroelastic theory, the governing and constitutive equations for piezoelectric films under complex stress-strain biasing states are formulated. Based on these formulations, the first step of the FSQP involves obtaining exact dispersion curves of bulk waves propagating in FBARs with different stress-strain biasing fields through the classical displacement method. Subsequently, mode-coupling solutions of physical fields are constructed for the prestressed FBARs operating with the thickness-extensional mode by superimposing relevant eigenmodes in dispersion curves. The second step of the FSQP involves deriving Hamilton's principle of piezoelectric film with initial stress-strain biasing fields using the perturbation method. Finally, the frequency spectrograms describing coupling vibration intensities between the thickness-extensional mode and unwanted eigenmodes are obtained by substituting mode-coupling solutions into Hamilton's principle, which verifies the effectiveness of the extended FSQP method for addressing dynamic problems in FBARs with biasing fields. The influences of both the amplitudes and orientations of initial stresses on the frequency spectral curves are examined. Mode-shape diagrams and displacement distributions of mutually coupled eigenmodes are presented to illustrate diverse mode-coupling behaviors in thickness-extensional FBARs under complex stress-strain biasing states. Numerical results indicate that the stress-strain biasing fields significantly affect the electromechanical properties of piezoelectric films, including effective elastic, piezoelectric, and dielectric constants. Consequently, these stress-strain biasing states exert substantial changes in frequencies and propagation wavenumbers of various mode branches within frequency ranges of the thickness-extensional mode branch. Furthermore, due to changes in propagation wavenumber, frequency spectral curves experience remarkable horizontal shifts along the length-to-thickness ratio axis, significantly altering mode-coupling behaviors of FBARs. Induced initial strains can enhance shift amplitudes of frequency spectral curves caused by initial stresses. In addition, stress-strain biasing fields result in significant shifts of frequency spectral curves along the frequency axis through the strain-stiffening or -softening effect, which can be harnessed to modulate resonance frequencies of FBARs. This study offers a solid foundation for frequency tunability, mode-coupling control, and structural designs in FBAR devices with residual stresses.