Loss due to transverse thermoelastic currents in microscale resonators

In order to understand the loss mechanisms associated with microscale resonators we have studied the importance of thermoelastic (TE) damping due to transverse thermal currents. In the work presented here, we study this damping mechanism as it applies to structures involving torsional vibration, or in general possessing a non-trivial mode shape such as those associated with microelectromechanical systems (MEMS) devices. A model of thermoelastic dissipation is presented that is based on the observation that the resonant modes of elastic structures almost always contain some flexural component. We determine a flexural energy participation factor and apply this to Zener’s model for damping of a simple reed in pure bending. Predictions agree well with internal friction measurements for a macroscale single-crystal silicon double paddle oscillator (300 μm thick) at temperatures from 130 to 300 K. The approach has also been successfully applied to predict microscale oscillator (1.5 μm thick) internal friction measurements at room temperature. Our results indicate that the internal friction arising from this mechanism is strong and can be quite significant for silicon-based MEMS (Q<10⁴) and persists down to 50 nm scale structures even for nominally torsional or even slightly asymmetric compressional devices which one might conclude have no loss. The importance of the thermoelastic mechanism is examined as a function of material properties. From this perspective, diamond possesses desirable thermal expansivity and diffusivity that is examined with our model.