Connecting Q to TCF for MEMS and piezoelectric resonators

TL;DR

This paper presents a phenomenological model linking the quality factor and temperature coefficient of frequency in MEMS resonators, revealing that nonlinear effects influencing TCF also limit Q, with practical applications to piezoelectric resonators.

Contribution

It introduces a new model connecting Q and TCF through anharmonic oscillator theory, extending to piezoelectric MEMS and providing a formula for estimating resonator losses.

Findings

The same nonlinear terms affect both Q and TCF.

A concise formula estimates resonator loss related to Woodruff's damping.

Model successfully applied to published data on AlN-on-diamond resonators.

Abstract

Two critical characteristics for any MEMS resonator are the quality factor ($Q$) and the temperature coefficient of frequency ($TCF$). The connection between $Q$ and $TCF$ is demonstrated here with a phenomenological anharmonic oscillator model. Specifically, it is shown that the same nonlinear terms responsible for the $TCF$ set an upper limit on the resonator's $Q$. A concise formula is found to estimate this loss and is shown to be closely related to Woodruff's formula for Akhiezer damping. The use of this formula is illustrated by extending the model to an important class of MEMS; piezoelectric resonators. Finally, the model is applied to published data for an AlN-on-diamond piezoelectric resonator. The focus here is on MEMS resonators, but the method should apply broadly to any resonance with non-zero $TCF$.