Influence of the A-F effect on the temperature stability of silicon micromechanical resonators

Micromechanical resonators (MEMS resonators) often show a discrepancy between the frequency-temperature (ƒ-T) characteristics they have in open-loop and closed-loop measurements. We show that this discrepancy is due to the nonlinear amplitude-frequency (A-ƒ) effect: the quality factor (Q) can increase by more than a factor of 2 when the resonators go from a high temperature to a low temperature; simple oscillators thus allow the resonator amplitude to change with temperature; then, the A-ƒ effect becomes temperature-dependent, causing the discrepancy. We also show how the discrepancy adversely affects the temperature stability of single-crystal-silicon (SCS) resonators and demonstrate a new closed-loop system that improves the stability by removing the discrepancy.