Passive Thermal Compensation of Silicon Resonators Through Modal Engineering

Abstract

This paper presents an insight-driven design approach for improving the thermal stability of silicon MEMS resonators by linking temperature-dependent material properties with geometry-controlled modal behavior. Rather than treating temperature coefficients of frequency (TCFs) as aggregate fitting parameters, the approach interprets frequency drift through the relative contributions of individual elastic compliance constants, providing physical intuition for resonator design. Elastic and compliance properties were experimentally extracted from ultra-highly doped silicon (n-type, $9.33\times 10^{19}{\,}\mathrm {cm}^{-3}$ ) wafers and incorporated into simulations to guide the development of pseudo-Lamé (PL) resonators with reduced temperature sensitivity. Carefully selected geometric modifications redistribute modal participation of compliance constants, enabling passive thermal compensation without additional fabrication complexity. Fabricated devices characterized from −20°C to 70°C achieved a first-order temperature coefficient of −0.39 ppm/°C, a second-order coefficient of −7.3 ppb/°C2, and a total frequency deviation of 38 ppm. These results demonstrate that physically informed design can produce low-cost, thermally stable resonator candidates suitable for precision timing applications in next-generation MEMS oscillators while maintaining simple fabrication and compact device architecture.