Single-Crystal Silicon Thermal-Piezoresistive Resonators as High-Stability Frequency References

This paper reports on single-crystal silicon (Si) thermal-piezoresistive resonators (TPRs) achieving ~0.2ppb-level frequency stability in phase-locked loop (PLL) measurements. A pair of resonators operating in a balanced-bridge configuration is presented, with one device being driven at resonance and the other used to null the parasitic background responses. The resonance frequency of the driven TPR has been measured over 40 hours with closed-loop continuous tracking by PLL and yields an Allan deviation <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sigma _{\text {A}} \approx 2.66$ </tex-math></inline-formula>ppb at an averaging time of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tau \approx 4.95$ </tex-math></inline-formula>s which is the best reported value among all Si TPRs studied to date. Further, an external DC power feedback loop is implemented alongside the PLL to enhance the frequency stability of the TPR, to achieve <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sigma _{\text {A}} \approx 0.236$ </tex-math></inline-formula>ppb at <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tau \approx 1.2$ </tex-math></inline-formula>s, the best short-term frequency stability among all reported Si MEMS counterparts. This result suggests that such TPRs with precise DC control can potentially achieve frequency stabilities comparable to, or better than, existing state-of-the-art resonators used in oscillator circuits, with significantly reduced external thermal control requirements and power demands.[2024-0121]