Epitaxially-encapsulated quad mass resonator with shaped comb fingers for frequency tuning

We present an epitaxially-encapsulated 2×2mm
²
quad-mass resonator (QMR) with shaped comb fingers for frequency tuning. While shaped electrodes have been used for frequency tuning of linear resonators, the device studied here has very high quality factor (g=100,000) resulting in a very narrowband resonance which, without the shaped electrodes, results in undesirable nonlinear behavior such as amplitude-frequency dependence and instability of the oscillator loop at large amplitudes. We demonstrate that through the shaped comb finger design, both frequency tuning (over 90Hz) and large amplitude oscillation (1.25 μm amplitude, a factor of 100 compared to performance without the shaped electrodes) are possible. Furthermore, we demonstrate how critical dimension loss in the fabrication process can change the shape of the designed shaped finger and introduce electrostatic stiffness hardening.