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CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Computer-Aided Design forMicro-Electro-Mechanical Systems
Eigenvalues, Energy Losses, and Dick Tracy Watches
D. Bindel
Computer Science DivisionDepartment of EECS
University of California, Berkeley
Sandia, 26 Jan 2006
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
The Computational Science Picture
Application modelingCheckerboard resonatorDisk resonatorShear ring resonator
Mathematical analysisPhysical modeling and finite element technologyStructured eigenproblems and reduced-order modelsParameter-dependent eigenproblems
Software engineeringHiQLabSUGARFEAPMEX
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
The Computational Science Picture
Application modelingCheckerboard resonatorDisk resonatorShear ring resonator
Mathematical analysisPhysical modeling and finite element technologyStructured eigenproblems and reduced-order modelsParameter-dependent eigenproblems
Software engineeringHiQLabSUGARFEAPMEX
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Outline
1 Resonant MEMS
2 Anchor Losses
3 Complex Symmetry
4 Disk Resonator Analysis
CAD forMEMS
ResonantMEMSMEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Outline
1 Resonant MEMS
2 Anchor Losses
3 Complex Symmetry
4 Disk Resonator Analysis
CAD forMEMS
ResonantMEMSMEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
What are MEMS?
CAD forMEMS
ResonantMEMSMEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
What are MEMS?
CAD forMEMS
ResonantMEMSMEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
MEMS Basics
Micro-Electro-Mechanical SystemsChemical, fluid, thermal, optical (MECFTOMS?)
Applications:Sensors (inertial, chemical, pressure)Ink jet printers, biolab chipsRadio devices: cell phones, inventory tags, pico radio
Use integrated circuit (IC) fabrication technologyTiny, but still classical physics
CAD forMEMS
ResonantMEMSMEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Radio-Frequency MEMS
Microguitars from Cornell University (1997 and 2003)
MHz-GHz mechanical resonatorsImpact: smaller, lower-power cell phonesOther uses:
Sensing elements (e.g. chemical sensors)Really high-pitch guitars
CAD forMEMS
ResonantMEMSMEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Micromechanical Filters
Mechanical filter
Capacitive senseCapacitive drive
Radio signal
Filtered signal
Mechanical high-frequency (high MHz-GHz) filterYour cell phone is mechanical!New MEMS filters can be integrated with circuitry=⇒ smaller and lower power
CAD forMEMS
ResonantMEMSMEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Ultimate Success
“Calling Dick Tracy!”
CAD forMEMS
ResonantMEMSMEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Designing Transfer Functions
Time domain:
Mu′′ + Cu′ + Ku = bφ(t)y(t) = pT u
Frequency domain:
−ω2Mu + iωCu + K u = bφ(ω)
y(ω) = pT u
Transfer function:
H(ω) = pT (−ω2M + iωC + K )−1by(ω) = H(ω)φ(ω)
CAD forMEMS
ResonantMEMSMEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Narrowband Filter Needs
20 log10 |H(ω)|
ω
Want “sharp” poles for narrowband filters=⇒ Want to minimize damping
CAD forMEMS
ResonantMEMSMEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Checkerboard Resonator
D+
D−
D+
D−
S+ S+
S−
S−
Anchored at outside cornersExcited at northwest cornerSensed at southeast cornerSurfaces move only a few nanometers
CAD forMEMS
ResonantMEMSMEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Checkerboard Model Reduction
Finite element model: N = 2154Expensive to solve for every H(ω) evaluation!
Build a reduced-order model to approximate behaviorReduced system of 80 to 100 vectorsEvaluate H(ω) in milliseconds instead of secondsWithout damping: standard Arnoldi projectionWith damping: Second-Order ARnoldi (SOAR)
CAD forMEMS
ResonantMEMSMEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Checkerboard Simulation
0 2 4 6 8 10
x 10−5
0
2
4
6
8
10
12
x 10
9 9.2 9.4 9.6 9.8
x 107
−200
−180
−160
−140
−120
−100
Frequency (Hz)
Am
plitu
de (
dB)
9 9.2 9.4 9.6 9.8
x 107
0
1
2
3
4
Frequency (Hz)
Pha
se (
rad)
CAD forMEMS
ResonantMEMSMEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Checkerboard Measurement
S. Bhave, MEMS 05
CAD forMEMS
ResonantMEMSMEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Damping and Q
Designers want high quality of resonance (Q)Dimensionless damping in a one-dof system
d2udt2 + Q−1 du
dt+ u = F (t)
For a resonant mode with frequency ω ∈ C:
Q :=|ω|
2 Im(ω)=
Stored energyEnergy loss per radian
CAD forMEMS
ResonantMEMSMEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Enter HiQLab
Existing codes do not compute quality factors... and awkward to prototype new solvers... and awkward to programmatically define meshesSo I wrote a new finite element code: HiQLab
CAD forMEMS
ResonantMEMSMEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
HiQLab Structure
User interfaces
(C++)Core libraries
Solver library(C, C++, Fortran, MATLAB)
Element library(C++)
Problem description(Lua)
(MATLAB, Lua)
Full scripting language for mesh inputCallbacks for boundary conditions, material propertiesMATLAB interface for quick algorithm prototypingCross-language bindings are automatically generated
CAD forMEMS
ResonantMEMSMEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Contributions
Built predictive model used to design checkerboardUsed model reduction to get thousand-fold speedup– fast enough for interactive useWrote FEAPMEX to script parameter studiesWrote a new code, HiQLab, to study damping
CAD forMEMS
ResonantMEMS
AnchorLossesSubstrate modeling
1-D Example
Finite Elements
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Outline
1 Resonant MEMS
2 Anchor Losses
3 Complex Symmetry
4 Disk Resonator Analysis
CAD forMEMS
ResonantMEMS
AnchorLossesSubstrate modeling
1-D Example
Finite Elements
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Example: Disk Resonator
SiGe disk resonators built by E. Quévy
CAD forMEMS
ResonantMEMS
AnchorLossesSubstrate modeling
1-D Example
Finite Elements
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Damping Mechanisms
Possible loss mechanisms:Fluid dampingMaterial lossesThermoelastic dampingAnchor loss
Model substrate as semi-infinite with a
Perfectly Matched Layer (PML).
CAD forMEMS
ResonantMEMS
AnchorLossesSubstrate modeling
1-D Example
Finite Elements
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Perfectly Matched Layers
Complex coordinate transformationGenerates a “perfectly matched” absorbing layerIdea works with general linear wave equations
Electromagnetics (Berengér, 1994)Quantum mechanics – exterior complex scaling(Simon, 1979)Elasticity in standard finite element framework(Basu and Chopra, 2003)
CAD forMEMS
ResonantMEMS
AnchorLossesSubstrate modeling
1-D Example
Finite Elements
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Model Problem
Domain: x ∈ [0,∞)
Governing eq:
∂2u∂x2 −
1c2
∂2u∂t2 = 0
Fourier transform:
d2udx2 + k2u = 0
Solution:u = coute−ikx + cineikx
CAD forMEMS
ResonantMEMS
AnchorLossesSubstrate modeling
1-D Example
Finite Elements
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Model with Perfectly Matched Layer
Transformed domainx
σ
Regular domain
dxdx
= λ(x) where λ(s) = 1− iσ(s)
d2udx2 + k2u = 0
u = coute−ik x + cineik x
CAD forMEMS
ResonantMEMS
AnchorLossesSubstrate modeling
1-D Example
Finite Elements
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Model with Perfectly Matched Layer
Transformed domainx
σ
Regular domain
dxdx
= λ(x) where λ(s) = 1− iσ(s),
1λ
ddx
(1λ
dudx
)+ k2u = 0
u = coute−ikx−kΣ(x) + cineikx+kΣ(x)
Σ(x) =
∫ x
0σ(s) ds
CAD forMEMS
ResonantMEMS
AnchorLossesSubstrate modeling
1-D Example
Finite Elements
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Model with Perfectly Matched Layer
Transformed domainx
σ
Regular domain
If solution clamped at x = L then
cin
cout= O(e−kγ) where γ = Σ(L) =
∫ L
0σ(s) ds
CAD forMEMS
ResonantMEMS
AnchorLossesSubstrate modeling
1-D Example
Finite Elements
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Model Problem Illustrated
Outgoing exp(−ix) Incoming exp(ix)
Transformed coordinate
Re(x)
0 2 4 6 8 10 12 14 16 18
0 5 10 15 200 5 10 15 20
-4
-2
0
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
CAD forMEMS
ResonantMEMS
AnchorLossesSubstrate modeling
1-D Example
Finite Elements
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Model Problem Illustrated
Outgoing exp(−ix) Incoming exp(ix)
Transformed coordinate
Re(x)
0 2 4 6 8 10 12 14 16 18
0 5 10 15 200 5 10 15 20
-4
-2
0
-2
-1
0
1
2
3
-1
-0.5
0
0.5
1
CAD forMEMS
ResonantMEMS
AnchorLossesSubstrate modeling
1-D Example
Finite Elements
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Model Problem Illustrated
Outgoing exp(−ix) Incoming exp(ix)
Transformed coordinate
Re(x)
0 2 4 6 8 10 12 14 16 18
0 5 10 15 200 5 10 15 20
-4
-2
0
-4
-2
0
2
4
6
-1
-0.5
0
0.5
1
CAD forMEMS
ResonantMEMS
AnchorLossesSubstrate modeling
1-D Example
Finite Elements
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Model Problem Illustrated
Outgoing exp(−ix) Incoming exp(ix)
Transformed coordinate
Re(x)
0 2 4 6 8 10 12 14 16 18
0 5 10 15 200 5 10 15 20
-4
-2
0
-10
-5
0
5
10
15
-1
-0.5
0
0.5
1
CAD forMEMS
ResonantMEMS
AnchorLossesSubstrate modeling
1-D Example
Finite Elements
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Model Problem Illustrated
Outgoing exp(−ix) Incoming exp(ix)
Transformed coordinate
Re(x)
0 2 4 6 8 10 12 14 16 18
0 5 10 15 200 5 10 15 20
-4
-2
0
-20
0
20
40
-1
-0.5
0
0.5
1
CAD forMEMS
ResonantMEMS
AnchorLossesSubstrate modeling
1-D Example
Finite Elements
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Model Problem Illustrated
Outgoing exp(−ix) Incoming exp(ix)
Transformed coordinate
Re(x)
0 2 4 6 8 10 12 14 16 18
0 5 10 15 200 5 10 15 20
-4
-2
0
-50
0
50
100
-1
-0.5
0
0.5
1
CAD forMEMS
ResonantMEMS
AnchorLossesSubstrate modeling
1-D Example
Finite Elements
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Finite Element Implementation
x(ξ)
ξ2
ξ1
x1
x2 x2
x1
Ωe Ωe
Ω
x(x)
Combine PML and isoparametric mappings
ke =
∫Ω
BT DBJ dΩ
me =
∫Ω
ρNT NJ dΩ
Matrices are complex symmetric
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Outline
1 Resonant MEMS
2 Anchor Losses
3 Complex Symmetry
4 Disk Resonator Analysis
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Eigenvalues and Model Reduction
Want to know about the transfer function H(ω):
H(ω) = pT (K − ω2M)−1b
Can eitherLocate poles of H (eigenvalues of (K , M))Plot H in a frequency range (Bode plot)
Usual tactic: subspace projectionBuild an Arnoldi basis VCompute with much smaller V ∗KV and V ∗MV
Can we do better?
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Variational Principles
Variational form for complex symmetric eigenproblems:Hermitian (Rayleigh quotient):
ρ(v) =v∗Kvv∗Mv
Complex symmetric (modified Rayleigh quotient):
θ(v) =vT KvvT Mv
First-order accurate eigenvectors =⇒Second-order accurate eigenvalues.Key: relation between left and right eigenvectors.
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Accurate Model Reduction
Build new projection basis from V :
W = orth[Re(V ), Im(V )]
span(W ) contains both Kn and Kn=⇒ double digits correct vs. projection with VW is a real-valued basis=⇒ projected system is complex symmetric
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Contributions
New formulation of perfectly matched layersEasy to apply PML to axisymmetric, 2D, or 3D modelsSame formulation applies to electromagnetics, etc.
Analysis of discretization error for perfectly matchedlayers
Results in cheap, automatic parameter optimizationStructure-preserving model reduction for complexsymmetric systems
Double accuracy for same work as standard method
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Outline
1 Resonant MEMS
2 Anchor Losses
3 Complex Symmetry
4 Disk Resonator Analysis
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Disk Resonator Simulations
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Disk Resonator Mesh
PML region
Wafer (unmodeled)
Electrode
Resonating disk
0 1 2 3 4
x 10−5
−4
−2
0
2x 10
−6
Axisymmetric model with bicubic meshAbout 10K nodal points in converged calculation
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Mesh Convergence
Mesh density
Com
pute
dQ
Cubic
LinearQuadratic
1 2 3 4 5 6 7 80
1000
2000
3000
4000
5000
6000
7000
Cubic elements converge with reasonable mesh density
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Model Reduction Accuracy
Frequency (MHz)
Tra
nsf
er(d
B)
Frequency (MHz)
Phase
(deg
rees
)
47.2 47.25 47.3
47.2 47.25 47.3
0
100
200
-80
-60
-40
-20
0
Results from ROM (solid and dotted lines) nearindistinguishable from full model (crosses)
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Model Reduction Accuracy
Frequency (MHz)
|H(ω
)−
Hreduced(ω
)|/H
(ω)|
Arnoldi ROM
Structure-preserving ROM
45 46 47 48 49 50
10−6
10−4
10−2
Preserve structure =⇒get twice the correct digits
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Response of the Disk Resonator
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Time-Averaged Energy Flux
0 0.5 1 1.5 2 2.5 3 3.5
x 10−5
−2
0
2
x 10−6
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10−6
−1
−0.5
0
0.5
1
1.5
2
2.5x 10
−6
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Variation in Quality of Resonance
Film thickness (µm)
Q
1.2 1.3 1.4 1.5 1.6 1.7 1.8100
102
104
106
108
Simulation and lab measurements vs. disk thickness
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Explanation of Q Variation
Real frequency (MHz)
Imagin
ary
freq
uen
cy(M
Hz)
ab
cdd
e
a b
cdd
e
a = 1.51 µm
b = 1.52 µm
c = 1.53 µm
d = 1.54 µm
e = 1.55 µm
46 46.5 47 47.5 480
0.05
0.1
0.15
0.2
0.25
Interaction of two nearby eigenmodes
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Contributions
Built disk model in HiQLab and verified against labmeasurementsDemonstrated dominance of anchor loss for this devicePredicted geometric sensitivity of quality factor Q(which was subsequently verified in the lab)Explained Q sensitivity in terms of mode interference
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Contributions Summary (1)
Application modelingFinite element models of several devicesDiscovery of effects of mode interferenceImportance of anchor loss vs thermoelastic damping
Mathematical analysisReformulation of perfectly-matched layersAnalysis of discretization and parameter choice in PMLsComplex symmetry-preserving model reductionPerturbation solution for thermoelastic damping
Software: HiQLab, FEAPMEX, SUGAR
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Contributions Summary (2)
HiQLab (about 33000 lines of code)Collaborations at Berkeley, Cornell, Stanford, Bosch
FEAPMEX (about 5000 lines of code)2400+ page viewsUsed for instrument models, stochastic structuralanalysis, ultrasonic nondestructive evaluation problems,material parameter identification problems
SUGAR (about 18000 lines of code)2000+ downloadsUsed in classes at Berkeley, Cornell, Johns HopkinsContinued research use for design optimization
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Other Contributions
Lowered complexity of roots from O(n3) to O(n2)
Code to go into next LAPACK releaseDeveloped first sparse subspace continuation code
Going into the next Matcont release
Developed new network tomography methodDesigned initial security model for OceanStoreServed as IEEE 754R secretaryResponsible for last CLAPACK version
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Future Work
Code developmentStructural elements and elements for different physicsDesign and implementation of parallelized version
Theoretical analysisMore damping mechanismsSensitivity analysis and variational model reduction
Application collaborationsUse of nonlinear effects (quasi-static and dynamic)New designs (e.g. internal dielectric drives)Continued experimental comparisons
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Conclusions
RF MEMS are a great source of problemsInteresting applicationsInteresting physics (and not altogether understood)Interesting numerical mathematics
http://www.cs.berkeley.edu/~dbindel
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Sources of Damping
Fluid dampingAir is a viscous fluid (Re 1)Can operate in a vacuumShown not to dominate in many RF designs
Material lossesLow intrinsic losses in silicon, diamond, germaniumTerrible material losses in metals
Thermoelastic dampingVolume changes induce temperature changeDiffusion of heat leads to mechanical loss
Anchor lossElastic waves radiate from structure
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Sources of Damping
Fluid dampingAir is a viscous fluid (Re 1)Can operate in a vacuumShown not to dominate in many RF designs
Material lossesLow intrinsic losses in silicon, diamond, germaniumTerrible material losses in metals
Thermoelastic dampingVolume changes induce temperature changeDiffusion of heat leads to mechanical loss
Anchor lossElastic waves radiate from structure
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Fabrication Outline
300 um
1 Si wafer2 Deposit 2 microns SiO23 Pattern and etch SiO2 layer4 Deposit 2 microns polycrystalline Si5 Pattern and etch Si layer6 Release etch remaining SiO2
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Fabrication Outline
300 um
2 um
1 Si wafer2 Deposit 2 microns SiO23 Pattern and etch SiO2 layer4 Deposit 2 microns polycrystalline Si5 Pattern and etch Si layer6 Release etch remaining SiO2
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Fabrication Outline
300 um
2 um
1 Si wafer2 Deposit 2 microns SiO23 Pattern and etch SiO2 layer4 Deposit 2 microns polycrystalline Si5 Pattern and etch Si layer6 Release etch remaining SiO2
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Fabrication Outline
300 um
2 um
2 um
1 Si wafer2 Deposit 2 microns SiO23 Pattern and etch SiO2 layer4 Deposit 2 microns polycrystalline Si5 Pattern and etch Si layer6 Release etch remaining SiO2
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Fabrication Outline
300 um
2 um
2 um
1 Si wafer2 Deposit 2 microns SiO23 Pattern and etch SiO2 layer4 Deposit 2 microns polycrystalline Si5 Pattern and etch Si layer6 Release etch remaining SiO2
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Fabrication Outline
300 um
2 um
2 um
1 Si wafer2 Deposit 2 microns SiO23 Pattern and etch SiO2 layer4 Deposit 2 microns polycrystalline Si5 Pattern and etch Si layer6 Release etch remaining SiO2
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Fabrication Result
300 um
2 um
2 um
(C. Nguyen, iMEMS 02)
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Role of simulation
HiQLab: Modeling RF MEMSExplore fundamental device physics
Particularly details of damping
Detailed finite element modelingReduced models eventually to go into SUGAR
SUGAR: “Be SPICE to the MEMS world”Fast enough for early design stagesSimple enough to attract usersSupport design, analysis, optimization, synthesisVerify models by comparison to measurement
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Why simulate?
“Build and break” is too expensiveWafer processing costs months, thousands of dollarsFabrication is impreciseDays or weeks to take good measurements
Good experiments need good hypothesesEven when device behavior is understood, still need tounderstand system behavior
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
From simulation to synthesis
Computer can assist at many levels:Fundamental physicsDetailed device modelsSystem-level models and macromodelsMetrologyDesign optimizationDesign synthesis
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Research thrusts
Model development (e.g. new finite elements)Numerical algorithms (e.g. model reduction)Numerical software engineering (SUGAR, HiQLab)Metrology and comparison to measurementOptimization and design synthesis
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Research group
Faculty StudentsA. Agogino (ME) D. Bindel (CS)Z. Bai (Math,CS,UCD) J.V. Clark (AS&T)J. Demmel (Math,CS) C. Cobb (ME)S. Govindjee (CEE) D. Garmire (CS)R. Howe (EE,ME) T. Koyama (CEE)K.S.J. Pister (EE) J. Nie (Math)C. Sequin (CS) H. Wei (CEE)
Y. Zhang (CEE)
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
SUGAR
Goal: “Be SPICE to the MEMS world”Fast enough for early design stagesSimple enough to attract usersSupport design, analysis, optimization, synthesisVerify models by comparison to measurement
System assembly
Models
Solvers
Matlab Web Library
Sensitivity analysis
Static analysis
Steady−state analysis
Transient analysis
Results
Netlist
Interfaces
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
SUGAR: Analysis of a micromirror
(Mirror design by M. Last)
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
SUGAR: Design synthesis
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
SUGAR: Comparison to measurement
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Elastic PMLs
∫Ω
ε(w) : C : ε(u) dΩ− ω2∫
Ωρw · u dΩ =
∫Γ
w · tndΓ
ε(u) =
(∂u∂x
)s
Start from standard weak formIntroduce transformed x with ∂x
∂x = Λ
Map back to reference system (JΛ = det(Λ))All terms are symmetric in w and u
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AnchorLosses
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Conclusions
Elastic PMLs
∫Ω
ε(w) : C : ε(u) dΩ− ω2∫
Ωρw · u dΩ =
∫Γ
w · tndΓ
ε(u) =
(∂u∂x
)s
=
(∂u∂x
Λ−1)s
Start from standard weak formIntroduce transformed x with ∂x
∂x = Λ
Map back to reference system (JΛ = det(Λ))All terms are symmetric in w and u
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Elastic PMLs
∫Ω
ε(w) : C : ε(u) JΛ dΩ− ω2∫
Ωρw · u JΛ dΩ =
∫Γ
w · tn dΓ
ε(u) =
(∂u∂x
)s
=
(∂u∂x
Λ−1)s
Start from standard weak formIntroduce transformed x with ∂x
∂x = Λ
Map back to reference system (JΛ = det(Λ))All terms are symmetric in w and u
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Elastic PMLs
∫Ω
ε(w) : C : ε(u) JΛ dΩ− ω2∫
Ωρw · u JΛ dΩ =
∫Γ
w · tn dΓ
ε(u) =
(∂u∂x
)s
=
(∂u∂x
Λ−1)s
Start from standard weak formIntroduce transformed x with ∂x
∂x = Λ
Map back to reference system (JΛ = det(Λ))All terms are symmetric in w and u
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Continuum 2D model problem
k
L
λ(x) =
1− iβ|x − L|p, x > L1 x ≤ L.
1λ
∂
∂x
(1λ
∂u∂x
)+
∂2u∂y2 + k2u = 0
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Continuum 2D model problem
k
L
λ(x) =
1− iβ|x − L|p, x > L1 x ≤ L.
1λ
∂
∂x
(1λ
∂u∂x
)− k2
y u + k2u = 0
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Continuum 2D model problem
k
L
λ(x) =
1− iβ|x − L|p, x > L1 x ≤ L.
1λ
∂
∂x
(1λ
∂u∂x
)+ k2
x u = 0
1D problem, reflection of O(e−kxγ)
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Discrete 2D model problem
k
L
Discrete Fourier transform in ySolve numerically in xProject solution onto infinite space traveling modesExtension of Collino and Monk (1998)
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AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Nondimensionalization
k
L
λ(x) =
1− iβ|x − L|p, x > L1 x ≤ L.
Rate of stretching: βhp
Elements per wave: (kxh)−1 and (kyh)−1
Elements through the PML: N
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Nondimensionalization
k
L
λ(x) =
1− iβ|x − L|p, x > L1 x ≤ L.
Rate of stretching: βhp
Elements per wave: (kxh)−1 and (kyh)−1
Elements through the PML: N
CAD forMEMS
ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Discrete reflection behavior
Number of PML elements
log10(β
h)
− log10
(r) at (kh)−1 = 10
1
1
1
2
2
2
2 2 2 2
333
3
3
3
3
444
4
4
5 10 15 20 25 30
-1.5
-1
-0.5
0
0.5
1
Quadratic elements, p = 1, (kxh)−1 = 10
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Conclusions
Discrete reflection decomposition
Model discrete reflection as two parts:Far-end reflection (clamping reflection)
Approximated well by continuum calculationGrows as (kxh)−1 grows
Interface reflectionDiscrete effect: mesh does not resolve decayDoes not depend on NGrows as (kxh)−1 shrinks
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ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Discrete reflection behavior
Number of PML elements
log10(β
h)
− log10
(r) at (kh)−1 = 10
1
1
1
2
2
2
2 2 2 2
333
3
3
3
3
444
4
4
5 10 15 20 25 30
-1.5
-1
-0.5
0
0.5
1
Number of PML elements
log10(β
h)
− log10(rinterface + rnominal) at (kh)−1 = 10
1
11
2
2
2
2 2 2 2
333
3
3
3
444
4
4
5 10 15 20 25 30
-1.5
-1
-0.5
0
0.5
1
Quadratic elements, p = 1, (kxh)−1 = 10
Model does well at predicting actual reflectionSimilar picture for other wavelengths, element types,stretch functions
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Conclusions
Choosing PML parameters
Discrete reflection dominated byInterface reflection when kx largeFar-end reflection when kx small
Heuristic for PML parameter choiceChoose an acceptable reflection levelChoose β based on interface reflection at kmax
xChoose length based on far-end reflection at kmin
x
CAD forMEMS
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AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Thermoelastic damping (TED)
u is displacement and T = T0 + θ is temperature
σ = Cε− βθ1ρutt = ∇ · σ
ρcvθt = ∇ · (κ∇θ)− βT0 tr(εt)
Volumetric strain rate drives energy transfer frommechanical to thermal domain
Irreversible diffusion =⇒ mechanical dampingNot often an important factor at the macro scaleRecognized source of damping in microresonators
Zener: semi-analytical approximation for TED in beamsWe consider the fully coupled system
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Conclusions
Nondimensionalization
σ = Cε− ξθ1utt = ∇ · σθt = η∇2θ − tr(εt)
ξ :=
(β
ρc
)2 T0
cvand η :=
κ
ρcv cL
Length ∼ LTime ∼ L/c, where c =
√E/ρ
Temperature ∼ T0β
ρcv
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AnchorLosses
ComplexSymmetry
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Conclusions
Scaling analysis
σ = Cε− ξθ1utt = ∇ · σθt = η∇2θ − tr(εt)
ξ :=
(β
ρc
)2 T0
cvand η :=
κ
ρcv cL
Micron-scale poly-Si devices: ξ and η are ∼ 10−4.Small η leads to thermal boundary layersLinearize about ξ = 0
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Conclusions
Discrete mode equations
σ = Cε− ξθ1utt = ∇ · σθt = η∇2θ − tr(εt)
σ = Cε− ξθ1−ω2u = ∇ · σ
iωθ = η∇2θ − iω tr(ε)
−ω2Muuu + Kuuu + Kutθ = 0iωDttθ + Kttθ + iωDtuu = 0
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Conclusions
Perturbation computation
−ω2Muuu + Kuuu + Kutθ = 0iωDttθ + Kttθ + iωDtuu = 0
Approximate ω by perturbation about Kut = 0:
−ω20Muuu0 + Kuuu0 = 0
iω0Dttθ0 + Kttθ0 + iω0Dtuu0 = 0
Choose v : vT u0 6= 0 and compute[(−ω2
0Muu + Kuu) −2ω0Muuu0vT 0
] [δuδω
]=
[−Kutθ0
0
]
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AnchorLosses
ComplexSymmetry
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Conclusions
Comparison to Zener’s model
105
106
107
108
109
1010
10−7
10−6
10−5
10−4
The
rmoe
last
ic D
ampi
ng Q
−1
Frequency f(Hz)
Zener’s Formula
HiQlab Results
Comparison of fully coupled simulation to Zenerapproximation over a range of frequenciesReal and imaginary parts after first-order correctionagree to about three digits with Arnoldi
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Conclusions
Thermoelastic boundary layer
0 20 40 60 80 100 1200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 20 40 60 80 100 1200
0.5
1x 10
−4
One-dimensional test problem (longitudinal mode in abar)Fixed temperature and displacement at leftFree at right
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Conclusions
Shear ring resonator
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9hw = 5.000000e−05
Value = 2.66E+07 Hz.
Ring is driven in a shearing motionCan couple ring to other resonatorsHow do we track the desired mode?
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ResonantMEMS
AnchorLosses
ComplexSymmetry
DiskResonatorAnalysis
Conclusions
Mode tracking
Find a continuous solution to(K (s)− ω(s)2M(s)
)u(s) = 0.
K and M are symmetric and M > 0Eigenvectors are M-orthogonalPerturbation theory gives good shiftsLook if u(s + h) and u(s) are on the same path bylooking at u(s + h)T M(s + h)u(s)
Many more subtleties in the nonsymmetric caseFocus of the CIS algorithm