Multi-Time-Scale Frequency Drift in a CMOS-MEMS Oscillator Modeled by Poole-Frenkel Dielectric Sidewall Charging

CMOS-MEMS resonator oscillators exhibit multi-time-scale frequency drift arising from charge injection and transport in dielectric sidewalls. This work develops a physics-based charging model that couples energy-derived electrostatic stiffness with time-varying surface charge following Poole-Frenkel conduction with Hartke correction. In contrast to prior analytical models that focused only on the electrostatic force within the air gap, the proposed framework retains the effects of stored electrical energy in the dielectric, enabling it to capture geometry-dependent softening and hardening driven by DC polarization and evolving charge history. The proposed model quantitatively reproduces both the sign and magnitude of frequency drift from application of unipolar and bipolar voltage steps, predicts the characteristic multi-time-scale decay followed by slow recovery arising from parallel fast and slow chargetransport pathways, and matches the observed scaling of relaxation time constants in a double-sided tapered-combdrive resonator oscillator. Furthermore, the proposed model is validated against data not used in parameter fitting, demonstrating robust predictive capability.