Modeling approach to analyze bonding stress in UHF quartz resonators

We have observed a phenomenon of high precision MEMS quartz resonators to change their frequency-temperature characteristics when they are mounted or bonded onto a substrate. This is due to the difference in thermal expansion coefficients between the quartz and substrate. When the temperature is changed, the mounting points between the quartz resonator and substrate become a source of mounting stress/strain in the resonator. We have defined a zero-stress temperature as the temperature at which the mounting stress is zero. We could determine the zero-stress temperature from the aging data of the resonator. We have derived a set of incremental equations for small vibrations superposed on mounting stress/strain that included the zero-stress temperature. The equations were employed in a COMSOL model of a UHF quartz resonator. The resonator frequency versus temperature profile was calculated by the change in eigenfrequency of the thickness shear mode as a function of the temperature. The eigenvalue problem of the resonator was modeled in COMSOL. The frequency-temperature curve of the resonator was shown to rotate counter-clockwise with the mounting stiffness and zero-stress temperature with respect to the frequency-temperature curve of the same resonator with no bonding stress. Furthermore the frequency-temperature curve of a bonded resonator will intersect the frequency-temperature curve of the same resonator without bonding stress at the zero-stress temperature.