An Investigation of Lateral Modes in FBAR Resonators

Using first principles and the constitutive equations for a piezoelectric, we solve the 2-D acoustic wave inside a single, infinite, piezoelectric membrane to study the dispersion of thin film bulk acoustic resonator (FBAR) lateral modes, with and without infinitely thin electrodes. The acoustic eigenfunction is a dual wave, composed of longitudinal and shear components, able to satisfy the 2-D acoustic boundary conditions at the vacuum interfaces. For the single piezoelectric slab, we obtain analytical expressions of the dispersion for frequencies near the longitudinal resonant frequency (Fs) of the resonator. These expressions are more useful for the understanding of dispersion in FBARs and more elegant than numerical methods like finite-element modeling and various matrix methods. We additionally find that the interaction between the resonator’s electrodes and the acoustic wave modifies the lateral-mode dispersion when compared to the case with no electrodes. When correctly accounting for these interactions, the dispersion zero is placed clearly at Fs, unlike what is calculated from a 2-D model without electrodes where the dispersion zero is placed at Fp. This is important since all experimental evidence of measures FBAR resonators shows that the dispersion zero is at Fs. Furthermore, we introduce an electrical current-flow model for the propagating acoustic wave inside the electroded piezoelectric, and based on this model, we can discuss an electrode-loss mechanism for FBAR lateral modes which depends on dispersion. From our model, it results that lateral modes with real $k_{x}$ have higher electrode dissipation if they are closer to the resonant frequency. This is consistent with the typical behavior of measured FBAR filters where the maximum lateral mode damage on the insertion loss takes place for frequencies immediately below Fs.