A semi-analytical modeling approach for laterally-vibrating thin-film piezoelectric-on-silicon micromechanical resonators

In this paper, we propose an analytical model that more accurately captures the electro-mechanical behavior of laterally-vibrating extensional-mode thin-film piezoelectric-on-silicon resonators. The proposed modeling approach considers the actual vibration mode shape of the resonator instead of relying on simplified modal functions such as assuming that the displacements occur only within the plane of fabrication and follow a sine function in the vibration plane. Out-of-plane and planar deviations in the mode shape from an ideal sine function arise from differences in the acoustic velocities between the piezoelectric film and underlying silicon. The proposed model allows more accurate prediction of the key lumped parameters in both the mechanical (e.g. equivalent point force associated with the reverse piezoelectric effect, effective stiffness) and electrical domain (i.e. motional capacitance) as a function of electrode coverage. Compared to an existing 1D approximation model proposed in a seminal work, the lumped parameters derived from the proposed model show notably closer agreement with 3D coupled-domain finite element (FE) simulations and also the measured results from fabricated devices. The semi-analytical nature of our modeling approach relies on the details of the mode shape provided. As such, our model has the benefit of allowing increasing levels of precision as desired depending on the amount of detail that is included in the FE modal analysis.