An analysis of thickness-extensional trapped energy resonant device structures with rectangular electrodes in the piezoelectric thin film on silicon configuration

Abstract

It has recently been shown that highly uniform thin layers can be etched in a small well-defined region of a silicon wafer and that good quality thin piezoelectric films such as zinc-oxide can be deposited along with the electrodes to form high frequency trapped energy resonant device structures. The accompanying analytical work has been for pure thickness vibrations only. In this work an analysis of essentially thickness-extensional trapped energy modes in the thin piezoelectric film on silicon composite structure is performed. It is shown that for small wavenumbers along the plate the dispersion equation is isotropic in the plane of the plate even though the silicon is severely anisotropic in that plane. From the resulting dispersion relation an asymptotic differential equation describing the mode shape along the surface of the composite plate vibrating in the vicinity of a thickness-extensional resonance is obtained along with the associated edge conditions. Since the mode is essentially thickness-extensional, trapping does not usually occur in the electroded regions for the fundamental mode of the flat composite plate. However, by appropriately thickening the silicon outside the electroded regions trapping in the electroded regions can be made to occur for the fundamental mode. Furthermore, in the case of zinc-oxide on silicon, trapping in the electroded regions can be made to occur for the fundamental mode of a flat plate simply by making the zinc-oxide sufficiently thicker than the etched silicon. The aforementioned asymptotic differential equation and edge conditions are applied in the analyses of the steady-state vibrations of trapped energy resonators and acoustically coupled two-pole resonator filters, both with rectangular electrodes. Lumped parameter representations of the admittance in the case of the single-pole device and admittance matrix in the case of the two-pole device, which are valid in the vicinity of resonances, are obtained. The analyses hold for the first few thickness-extensional modes and their accompanying lateral overtones.