Thermoelastic Damping in Layered Microresonators: Critical Frequencies, Peak Values, and Rule of Mixture

The design of high-Q microresonators requires a careful evaluation of thermoelastic damping (TED). Here, we consider the widely used class of flexural-mode bilayered beam resonators consisting of a substrate coated with a thin film and derive an analytical expression in the form of an infinite series to compute the frequency dependence of TED. The nature and structure of the thermoelastic dissipation spectrum were explored by computing TED over several decades of frequency in a large set of materials and structures. TED exhibits a rich variety of spectral features that include multiple well-resolved peaks. An approximate rule of mixture was obtained by retaining only the leading term of the infinite series. The accuracy and utility of the one-term approximation, which can be regarded as the analog of Zener's formula for bilayered beams, are discussed. Taken together, our results can be used to optimize the design of layered microresonators and gain useful insight into a fundamental mechanism of dissipation.