Finite element model of thermoelastic damping in MEMS

Abstract

Damping in MEMS resonators was studied experimentally and numerically. Quality factor measurements were performed on Draper gyroscopes made from boron doped silicon wafers with varying amount of germanium (0%, 2%, 23%, 30% ). The quality factors of gyroscopes with germanium were measured to be lower than those without germanium, due to increased anelastic damping. Specifically, the decreased thermal conductivity in the devices with germanium causes those devices to experience thermoelastic damping of a greater magnitude than the germanium-free devices. The amount of damping exhibited is found to be well explained by existing analytical expressions for thermoelastic dissipation in a beam model. The governing equations of thermo elasticity dictate that the amount of damping that a resonator undergoes is a function of both material properties as well as device geometry. Damping will become greatest at operating cycle times that are of the same scale as the thermal relaxation times of the device material. Due to the fact that analytical expressions exist for only a few simple geometries, a finite element model was developed to evaluate thermoelastic damping in more complicated geometries. The finite element model is demonstrated to be in good qualitative agreement with the analytical expressions, and is used to analyze the impact of design modifications such as the addition of fillets and anchors to a simple beam model. It is shown that depending on the size scale of the resonator (which dictates the amount of internal damping), these geometric modifications may either hinder or improve resonator damping characteristics.