Dependence of temperature coefficient of frequency (TCf) on crystallography and eigenmode in N-doped silicon contour mode micromechanical resonators

• Quantitative study on the temperature coefficient of frequency (TCf) of the micromechanical resonator is performed.
• The effects on n-type doping on the temperature dependence of elastic constants is calculated based on free carrier contribution on elastic constants.
• The dependence of TCf on both crystal orientation and vibration mode is discussed for single-crystal-silicon based resonator.
• Multiple samples with various dimensions are studied and the scaling effect is discussed.
• This work indicates doping is a viable approach to enhance the temperature stability of silicon based micromechanical resonator.

This paper reports the effects of crystal orientations on the temperature coefficient of frequency (TCf) of single crystal silicon square-plate micromechanical resonators vibrating in two distinct contour modes: Lamé mode and square extensional (SE). For the Lamé mode, the same TCf was found over several devices aligned to the 〈1 1 0〉 direction, while much greater variation in the TCf was observed among the devices aligned against the 〈1 0 0〉 direction. For the SE mode, the devices in both 〈1 0 0〉 and 〈1 1 0〉 orientations exhibit similar TCf values for varying doping levels. The sensitivity of TCf to doping concentration is also investigated. The TCf of Lamé 〈1 0 0〉 device is more easily influenced by n-type doping concentration than SE mode devices in both orientations while the Lamé 〈1 0 0〉 device is almost immune to doping variation. Devices with different dimensions are tested, and the TCf values are proved to be free of size scaling. Quantitative study based on free carrier contribution on elastic constants is performed and supports our observations. Close agreement among experiments, theoretical predictions and simulations is achieved.