Thermoelastic Damping in Hollow and Slotted Microresonators

Microresonators employed in microelectromechanical systems for sensing and communications are growing increasingly more sophisticated in terms of structural geometry and mode shapes. Accompanying this increase in sophistication is a corresponding need to develop accurate analytical models to predict the dynamic properties of such resonators. Here, we present an analytical framework to compute thermoelastic damping (TED) in the general class of microresonators characterized by structural discontinuities in the form of slots or internal channels. The temperature field within the resonators is obtained by solving the one-way coupled equation of thermoelastic heat conduction in a piecewise fashion, thereby capturing the effects of structural discontinuities interrupting heat conduction within the beam. The framework is validated by comparison with previously reported finite-element analysis and measurements of damping in slotted microresonators. The analysis leads to an expression for TED in the form of rapidly converging infinite series, and accurate closed-form expressions are obtained by retaining the leading terms. These simple formulas enable a rapid exploration of the design space over a full range of parameters, as illustrated for the case of hollow single-crystal-silicon beams containing internal channels. For constant channel volume, the peak value of TED reduces monotonically with the ratio of channel width to channel height. The analysis is used to identify designs that reduce TED to values that are less than 2 times 10^-5 at all frequencies.