BSAC Research Review Individual Project

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.