An analytical temperature-dependent design model for contour-mode MEMS resonators and oscillators verified by measurements
The importance of micro-electromechanical systems (MEMS) for radio-frequency (RF) applications is rapidly growing. In RF mobile-communication systems, MEMS-based circuits enable a compact implementation, low power consumption and high RF performance, e.g., bulk-acoustic wave filters with low insertion loss and low noise or fast and reliable MEMS switches. However, the cross-hierarchical modelling of micro-electronic and micro-electromechanical constituents together in one multi-physical design process is still not as established as the design of integrated micro-electronic circuits, such as operational amplifiers. To close the gap between micro-electronics and micro-electromechanics, this paper presents an analytical approach towards the linear top-down design of MEMS resonators, based on their electrical specification, by the solution of the mechanical wave equation. In view of the central importance of thermal effects for the performance and stability of MEMS-based RF circuits, the temperature dependence was included in the model; the aim was to study the variations of the RF parameters of the resonators and to enable a temperature dependent MEMS oscillator simulation. The variations of the resonator parameters with respect to the ambient temperature were then verified by RF measurements in a vacuum chamber at temperatures between -35 °C and 85 °C. The systematic body of data revealed temperature coefficients of the resonant frequency between -26 ppm/K and -20 ppm/K, which are in good agreement with other data from the literature. Based on the MEMS resonator model derived, a MEMS oscillator was designed, simulated, and measured in a vacuum chamber yielding a measured temperature coefficient of the oscillation frequency of -26.3 ppm/K. The difference of the temperature coefficients of frequency of oscillator and resonator turned out to be mainly influenced by the limited Q-factor of the MEMS device. In both studies, the analytical model and the measurement showed very good agreement in terms of temperature dependence and the prediction of fabrication results of the resonators designed. This analytical modelling approach serves therefore as an important step towards the design and simulation of micro-electronics and micro-electromechanics in one uniform design process. Furthermore, temperature dependences of MEMS oscillators can now be studied by simulations instead of time-consuming and complex measurements.