Introduction:

1. Introduction: Perfectly Matched Layers: • High frequency surface-micromachined MEMS resonators have many applications • Filters, frequency references, sensors • Need high quality factors • Difficult to predict analytically • Existing tools predict frequency, but not Q • Anchor loss is a major damping source • Simulate anchor loss with perfectly matched layers • Illustrate anchor loss in disk resonators • Predict surprising sensitivity to geometry • Assume waves from the anchor are not reflected (i.e. the substrate is semi-infinite). • Add damping at the boundaries to absorb waves • Implemented in standard FEA codes using a complex-valued change of coordinates • Effectively change properties smoothly for perfect matching of mechanical impedance Basic Loss Mechanism: Model of a Disk Resonator: Device micrographs (top) and schematic (bottom) Displacement and mean energy flux at resonance • Simulated and built poly-SiGe disk resonators • 31.5 and 41.5 micron radii, 1.5 micron height • Post is 1.5 micron radius, 1 microns height • Fabricated dimensions vary from nominal • Axisymmetric finite element model, bicubic elements with 0.25 micron node spacing • Dominant mode is not purely radial • Includes a small bending motion • Vertical motion at post pumps elastic waves into the substrate • More bending motion when “radial” and “bending” modes are close in frequency Conclusions: Design Sensitivity: • Anchor loss is complicated even for disks! • Surprising dips in Q from interacting modes • Poisson coupling is important: acoustic approximations are inadequate • Need CAD tools to predict damping • Simulate wafer with a perfectly matched layer • Have integrated anchor loss and thermoelastic damping models into HiQLab simulator • http://www.cs.berkeley.edu/~dbindel/hiqlab/ Simulated Q for two modes (solid lines, left) at different film thicknesses matches lab measurement (dots). The behavior is explained by the interaction of two complex frequencies near a critical geometry.