Design of resonators for the determination of the temperature coefficients of elastic constants of monocrystalline silicon

More accurate sets of the first and second order temperature coefficients are presented for monocrystalline silicon of a low-doped p-type silicon and high-doped n-type silicon. These values are extrapolated from the frequency-temperature curve of high-Q silicon resonators. Three types of resonators have been designed and fabricated using silicon bulk micromachining, each with a mode of vibration depending as close as possible on one of the three independent elastic constants. The electromechanical coupling and the related equivalent circuit of an electrostatic drive have been established for two of these modes. The temperature coefficients of the three independent elastic constants, as well as their estimated variances are computed through the fitting of the measured frequency-temperature curves, compared with the sensitivity of each mode to the elastic constants determined by simulations