Temperature frequency stability study of extensional mode N-doped silicon MEMS resonator

Analytical and numerical modeling techniques are presented to predict n-type-doped silicon resonators’ temperature coefficient frequencies for the extensional mode of vibration by modeling of single-crystal silicon ([100] and [110]) directions. We utilized a previously reported empirical exponential expression for the uniaxial deformation potential that allows us to determine the deformation potential constant. It was observed that by using the empirical model of exponential expression for the uniaxial deformation potential, our result shows that the frequency variation with temperature is closely dependent on the direction and the variation of the different doping levels. We apply our analytical model to the [100] and [110] length extensional modes of a Microelectromechanical Systems (MEMS) resonator. We calculated the resonance frequency with temperature dependence and found that the first-order temperature coefficient frequency shows zero values at doping concentration levels of 0.5 and 1 × 1019 cm−3, which indicates the existence of frequency turnover at room temperature. We also extended our study by finite element analysis of MEMS resonators to investigate the mode shapes and resonant frequency for different aspect ratios in the extensional mode of vibration. It is observed that the mode shapes are more stressed with higher aspect ratios, and the frequency variation (ppm) of the resonance frequency between the [100] and [110] crystal directions of the MEMS resonator increases with increasing aspect ratios.