Modeling Nonlinear MEMS Beams and the Chaotic Duffing Oscillator in SPICE

Abstract

Nonlinear MEMS beams have been modeled using SPICE. This allows for the complex dynamics of MEMS resonators to be observed parallel to their supporting electronics via circuit simulation. Silicon generally provides suitably linear parameters for use in MEMS. However, nonlinearities may arise due to issues such as amplitude-frequency (A-F) effect, large displacement of the proof mass, pull-in voltage, fatigue, material or electrical parameters, process variation, simplified beam modeling and nonlinear spring constants. By modeling these effects in SPICE, the design of electronics that automatically test, calibrate, report or even mitigate these effects is aided. Single-crystal silicon is a highly linear material up until its failure, especially type <100>. High quality factor MEMS devices may, however, be affected by even small nonlinear terms in the material's Young's modulus. Geometric deformations may also occur due to decreases in cross-sectional area of beams in reaction to stretching and loading. Specifically, by including nonlinear geometric effects of MEMS beams and nonlinear terms in the Young's modulus of <100> and <110> silicon - nonlinear and chaotic oscillations are shown to arise via SPICE simulation. Using this SPICE modeling method, electronic systems were designed to monitor the nonlinear parameters of MEMS beams that cause A-F effect and chaotic Duffing oscillations. Extracting parameters such as those from the oscillation's Poincare section may yield advantage in built-in self-test (BIST) applications. The features in these nonlinear oscillations extend parameters to monitor and potentially calibrate MEMS devices for reliability, stability and processing variation.