Temperature coefficient of frequency modeling for CMOS-MEMS bulk mode composite resonators

CMOS-MEMS resonators, which are promising building blocks for achieving monolithic integration of MEMS structure, can be used for timing and filtering applications, and control circuitry. SiO₂ has been used to make MEMS resonators with quality factor Q > 10⁴, but temperature instability remains a major challenge. In this paper, a design that uses an embedded metal block for temperature compensation is proposed and shows sub-ppm temperature stability (-0.21 ppm/K). A comprehensive analytical model is derived and applied to analyze and optimize the temperature coefficient of frequency (TCF) of the CMOS-MEMS composite material resonator. Comparison with finite element method simulation demonstrates good accuracy. The model can also be applied to predict and analyze the TCF of MEMS resonators with arbitrary mode shape, and its integration with simulation packages enables interactive and efficient design process.