The Main Lateral Mode Approximation of a Film Bulk Acoustic Resonator With Perfect Metal Electrodes

Using first principles and the constitutive equations of a piezoelectric crystal, we solve the 2-D problem inside a three-layer film bulk acoustic resonator (FBAR) in order to study the dispersion and parasitic lateral modes' characteristic of the structure. In our main lateral mode approximation, described here in detail, we construct the acoustic wave by combining the ideal "piston" mode and the main dispersion branch lateral mode. By limiting our analysis to the practical range of frequencies near the series resonance of the stack, where the lateral component $k_{x}$ of the ${k}$ vector is small, we find analytical expressions for the FBAR acoustic wave and for the dispersion of the three-layer stack. When lateral boundary conditions are added to the acoustic problem of a laterally finite resonator, we employ our theory to estimate the amplitude and the propagation of the lateral modes and then compare the theoretical predictions with the measurements of fabricated FBARs and finite-element simulation results. We are able to distinguish between a "clamped" and a "quasi-free" lateral interface by comparing the amplitude strength of the lateral modes produced, and we discuss how optimum lateral boundaries can be engineered with perimeter frames for realistic resonators.