Nonlinear response of very high frequency contour mode resonators

We explore the source of nonlinearities in Aluminum Nitride (AlN) Contour Mode Resonators (CMRs) operating in the Very High Frequency (VHF) range. We demonstrate that the red-shift of the resonance frequency found in VHF CMRs when the input RF power increases is due to nonlinear stiffness appearing from self-heating, and variable damping due to geometric nonlinearities. Moreover, we find a linear relationship between the variable damping coefficient and the resonator quality factor (Q). Such nonlinear mechanisms are modeled using a spring-mass-damper physical system and, in the electrical domain, a modified Butterworth-Van Dyke (MBVD) circuit where the nonlinear stiffness and variable damping are captured by a charge-dependent motional capacitor and a charge-dependent motional resistor, respectively. Detailed guidelines are provided to accurately analyze nonlinear CMRs using full-wave numerical simulations based on a finite-element method. Such simulations allow us to isolate the influence of each independent nonlinear mechanism and establish a relation between variable damping and geometric nonlinearities. Circuit and full-wave numerical simulations are in good agreement with measured data from fabricated 225 MHz CMRs exhibiting different Q. Finally, we exploit nonlinearities in high-Q CMRs to generate frequency combs at the MHz range opening the door to new exciting applications in telecommunication and sensing.