Analytical modeling of n-type doped silicon elastic constants and frequency-compensation of Lamé mode microresonators

Abstract

A model for predicting n-type doped silicon elastic constants is presented in this paper. The model approach is based on an empirical exponential expression derived for the uniaxial deformation potential as a function of doping level using the experimental data available in the literature. The derived expression allows for the determination of the deformation potential constant for n-type doping level, otherwise challenging to be measured experimentally. Our model result shows that the predicted silicon elastic constants are in good agreement with previously reported experimental values with an average error for c11 as 0.58% and for c12 as 2.12% at the room temperature, while c44 does not alter with changing n-type doping. We also apply our analytical model to Lamé mode MEMS resonator to analyze temperature dependence of the resonance frequency and show that the first order temperature coefficient of frequency has zero values at doping concentration of 0.5 and 1×1019 cm−3 indicating the existence of frequency turnover at room temperature. The model presented in this work can further aid in achieving temperature stability of silicon MEMS resonators.