1. ©2005 University of California Prepublication Data Spring 2005 MEMS Resonator Simulation (RTH46/JD) David S. Bindel, Emmanuel Quevy, Tsuyoshi Koyama Sanjay Govindjee, James W. Demmel, Roger T. Howe

2. ©2005 University of California Prepublication Data Spring 2005 Introduction 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 –Loss is surprisingly sensitive to geometry variations Thermoelastic damping can also be significant

3. ©2005 University of California Prepublication Data Spring 2005 Loss Model of a Disk Resonator Device micrographs (top) and schematic (bottom)

4. ©2005 University of California Prepublication Data Spring 2005 Loss Model of a Disk Resonator 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 micron height –Fabricated dimensions vary from nominal

5. ©2005 University of California Prepublication Data Spring 2005 Basic Loss Mechanism Displacement and mean energy flux at resonance Dominant mode is not purely radial!

6. ©2005 University of California Prepublication Data Spring 2005 Basic Loss Mechanism Dominant mode is not purely radial! –Includes a small bending motion –Vertical motion at post pumps elastic waves into substrate –More bending motion when “radial” and “bending” modes are close in frequency

7. ©2005 University of California Prepublication Data Spring 2005 Design Sensitivity Simulated Q (solid line, left) very sensitive to film thickness –Matches experimental data (dots) Sensitivity comes from interaction between two poles which come close at critical thicknesses (right)

8. ©2005 University of California Prepublication Data Spring 2005 Thermoelastic Damping Compute thermoelastic interactions from coupled PDEs –Matches Zener’s formula on a beam geometry –Works more generally than Zener’s formula

9. ©2005 University of California Prepublication Data Spring 2005 TED and Grain Boundaries Grain boundaries affect thermal conductivity and add another thermal length scale Working to include grain boundary effects in our thermoelastic damping simulations

10. ©2005 University of California Prepublication Data Spring 2005 Conclusions Need CAD tools to predict damping –Simulations of anchor losses using perfectly matched layers –Simulations of thermoelastic damping through coupled PDEs Illustrated usefulness of our approach on a disk resonator –Both simulation and experiment show surprising dips in Q from interactions between modes –Poisson coupling is important; acoustic approximations are inadequate to capture the behavior

11. ©2005 University of California Prepublication Data Spring 2005 References http://www.cs.berkeley.edu/~dbindel/hiqlab –Program code is freely available –Tutorial slides and relevant papers also available Papers –“Elastic PMLs for resonator anchor loss simulation.” Tech report UCB/SEMM-2005/01. Submitted to IJNME. –“Anchor Loss Simulation in Resonators.” MEMS 2005.