Computer Aided Design of Micro-Electro-Mechanical …bindel/present/2008-11...Backup slides...

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ResonantMEMS andmodels

HiQLab

Anchor lossesand diskresonators

Thermoelasticlosses andbeamresonators

Conclusion

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Computer Aided Design ofMicro-Electro-Mechanical SystemsFrom Energy Losses to Dick Tracy Watches

D. Bindel

Courant Institute for Mathematical SciencesNew York University

Stanford University, 12 Nov 2008

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Conclusion

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The Computational Science Picture

Application modelingDisk resonatorBeam resonatorShear ring resonator, checkerboard, ...

Mathematical analysisPhysical modeling and finite element technologyStructured eigenproblems and reduced-order modelsParameter-dependent eigenproblems

Software engineeringHiQLabSUGARFEAPMEX / MATFEAP

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Conclusion

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The Computational Science Picture

Application modelingDisk resonatorBeam resonatorShear ring resonator, checkerboard, ...

Mathematical analysisPhysical modeling and finite element technologyStructured eigenproblems and reduced-order modelsParameter-dependent eigenproblems

Software engineeringHiQLabSUGARFEAPMEX / MATFEAP

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Conclusion

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Outline

1 Resonant MEMS and models

2 HiQLab

3 Anchor losses and disk resonators

4 Thermoelastic losses and beam resonators

5 Conclusion

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Outline

2 HiQLab

5 Conclusion

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What are MEMS?

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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

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Resonant RF MEMS

Microguitars from Cornell University (1997 and 2003)

MHz-GHz mechanical resonatorsFavorite application: radio on chipClose second: really high-pitch guitars

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The Mechanical Cell Phone

filter

Tuning

control

...RF amplifier /

preselector

Oscillator

IF amplifier /

Your cell phone has many moving parts!What if we replace them with integrated MEMS?

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Ultimate Success

“Calling Dick Tracy!”

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Disk Resonator

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Disk Resonator

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Electromechanical Model

Kirchoff’s current law and balance of linear momentum:

(C(u)V ) + GV = Iexternal

Mutt + Ku −∇u

V ∗C(u)V)

= Fexternal

Linearize about static equilibium (V0, u0):

C(u0) δVt + G δV + (∇uC(u0) · δut) V0 = δIexternal

M δutt + K δu +∇u (V ∗0 C(u0) δV ) = δFexternal

K = K − 12

∂u2 (V ∗0 C(u0)V0)

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Electromechanical Model

Assume time-harmonic steady state, no external forces:[iωC + G iωB−BT K − ω2M

] [δVδu

[δIexternal

]Eliminate the mechanical terms:

Y (ω) δV = δIexternal

Y (ω) = iωC + G + iωH(ω)

H(ω) = BT (K − ω2M)−1B

Goal: Understand electromechanical piece (iωH(ω)).As a function of geometry and operating pointPreferably as a simple circuit

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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

To understand Q, we need damping models!

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The Designer’s Dream

Ideally, would likeSimple models for behavioral simulationParameterized for design optimizationIncluding all relevant physicsWith reasonably fast and accurate set-up

We aren’t there yet.

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Outline

2 HiQLab

5 Conclusion

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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

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Heritage of HiQLab

SUGAR: SPICE for the MEMS worldSystem-level simulation using modified nodal analysisFlexible device description languageC core with MATLAB interfaces and numerical routines

FEAPMEX: MATLAB + a finite element codeMATLAB interfaces for steering, testing solvers, runningparameter studiesTime-tested finite element architectureBut old F77, brittle in places

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Other Ingredients

“Lesser artists borrow. Great artists steal.”– Picasso, Dali, Stravinsky?

Lua: www.lua.orgEvolved from simulator data languages (DEL and SOL)Pascal-like syntax fits on one page; complete languagedescription is 21 pagesFast, freely available, widely used in game design

MATLAB: www.mathworks.com“The Language of Technical Computing”OCTAVE also works well

Standard numerical libraries: ARPACK, UMFPACKMATEXPR, MWRAP, and other utilities

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HiQLab Structure

User interfaces

(C++)Core libraries

Solver library(C, C++, Fortran, MATLAB)

Element library(C++)

Problem description(Lua)

(MATLAB, Lua)

Standard finite element structures + some new ideasFull scripting language for mesh inputCallbacks for boundary conditions, material propertiesMATLAB interface for quick algorithm prototypingCross-language bindings are automatically generated

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Outline

2 HiQLab

5 Conclusion

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Damping Mechanisms

Possible loss mechanisms:Fluid dampingMaterial lossesThermoelastic dampingAnchor loss

Model substrate as semi-infinite with a

Perfectly Matched Layer (PML).

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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)

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Model Problem

Domain: x ∈ [0,∞)

Governing eq:

∂2u∂x2 −

∂2u∂t2 = 0

Fourier transform:

d2udx2 + k2u = 0

Solution:u = coute−ikx + cineikx

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Model with Perfectly Matched Layer

Transformed domainx

Regular domain

= λ(x) where λ(s) = 1− iσ(s)

d2udx2 + k2u = 0

u = coute−ik x + cineik x

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Transformed domainx

Regular domain

= λ(x) where λ(s) = 1− iσ(s),

)+ k2u = 0

u = coute−ikx−kΣ(x) + cineikx+kΣ(x)

Σ(x) =

0σ(s) ds

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Transformed domainx

Regular domain

If solution clamped at x = L then

cout= O(e−kγ) where γ = Σ(L) =

0σ(s) ds

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Model Problem Illustrated

Outgoing exp(−ix) Incoming exp(ix)

Transformed coordinate

0 2 4 6 8 10 12 14 16 18

0 5 10 15 200 5 10 15 20

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0 2 4 6 8 10 12 14 16 18

0 5 10 15 200 5 10 15 20

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0 2 4 6 8 10 12 14 16 18

0 5 10 15 200 5 10 15 20

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0 2 4 6 8 10 12 14 16 18

0 5 10 15 200 5 10 15 20

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0 2 4 6 8 10 12 14 16 18

0 5 10 15 200 5 10 15 20

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0 2 4 6 8 10 12 14 16 18

0 5 10 15 200 5 10 15 20

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Finite Element Implementation

Ωe Ωe

Combine PML and isoparametric mappings

BT DBJ dΩ

ρNT NJ dΩ

Matrices are complex symmetric

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Eigenvalues and Model Reduction

Want to know about the transfer function H(ω):

H(ω) = BT (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 V for a Krylov subspace Kn

Compute with much smaller V ∗KV and V ∗MVCan we do better?

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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.

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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

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Disk Resonator Simulations

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Disk Resonator Mesh

PML region

Wafer (unmodeled)

Electrode

Resonating disk

0 1 2 3 4

x 10−5

Axisymmetric model with bicubic meshAbout 10K nodal points in converged calculation

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Mesh Convergence

Mesh density

LinearQuadratic

1 2 3 4 5 6 7 80

Cubic elements converge with reasonable mesh density

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Model Reduction Accuracy

Frequency (MHz)

47.2 47.25 47.3

Results from ROM (solid and dotted lines) nearlyindistinguishable from full model (crosses)

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Model Reduction Accuracy

Frequency (MHz)

Hreduced(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

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Response of the Disk Resonator

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Variation in Quality of Resonance

Film thickness (µm)

1.2 1.3 1.4 1.5 1.6 1.7 1.8100

Simulation and lab measurements vs. disk thickness

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Explanation of Q Variation

Real frequency (MHz)

Imagin

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

Interaction of two nearby eigenmodes

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Outline

2 HiQLab

5 Conclusion

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Thermoelastic Damping (TED)

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Thermoelastic Damping (TED)

u is displacement and T = T0 + θ is temperature

σ = Cε− βθ1ρu = ∇ · σ

ρcv θ = ∇ · (κ∇θ)− βT0 tr(ε)

Coupling between temperature and volumetric strain:Compression and expansion =⇒ heating and coolingHeat 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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Nondimensionalized Equations

Continuum equations:

σ = Cε− ξθ1u = ∇ · σθ = η∇2θ − tr(ε)

Discrete equations:

Muuu + Kuuu = ξKuθθ + fCθθθ + ηKθθθ = −Cθuu

Micron-scale poly-Si devices: ξ and η are ∼ 10−4.Linearize about ξ = 0

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Perturbative Mode Calculation

Discretized mode equation:

(−ω2Muu + Kuu)u = ξKuθθ

(iωCθθ + ηKθθ)θ = −iωCθuu

First approximation about ξ = 0:

(−ω20Muu + Kuu)u0 = 0

(iω0Cθθ + ηKθθ)θ0 = −iω0Cθuu0

First-order correction in ξ:

−δ(ω2)Muuu0 + (−ω20Muu + Kuu)δu = ξKuθθ0

Multiply by uT0 :

δ(ω2) = −ξ

0 Kuθθ0

uT0 Muuu0

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Zener’s Model

1 Clarence Zener investigated TED in late 30s-early 40s.2 Model for beams common in MEMS literature.3 “Method of orthogonal thermodynamic potentials” ==

perturbation method + a variational method.

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Comparison to Zener’s Model

10−7

10−6

10−5

10−4

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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Outline

2 HiQLab

5 Conclusion

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Onward!

What about:Modeling more geometrically complex devices?Modeling general dependence on geometry?Modeling general dependence on operating point?Computing nonlinear dynamics?Digesting all this to help designers?

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Ongoing and Future Work

Code developmentAutomation tools (MWRAP and MATEXPR)Design and implementation of parallelized versionStructural elements and elements for different physics

Theoretical analysisError estimate and bounds for resonance calculationsSensitivity analysis and variational model reductionMore damping mechanisms

Application collaborationsUse of nonlinear effects (quasi-static and dynamic)New designs (e.g. internal dielectric drives)Continued experimental comparisons

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Conclusions

RF MEMS are a great source of problemsInteresting applicationsInteresting physics (and not altogether understood)Interesting computing challenges

http://www.cims.nyu.edu/~dbindel

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Concluding Thoughts

The difference between art and science is thatscience is what we understand well enough toexplain to a computer. Art is everything else.

Donald Knuth

The purpose of computing is insight, not numbers.Richard Hamming

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Conclusion

Backup slidesCheckerboardresonators

Nonlinear eigenvalueperturbation

Hello world!

Reflection Analysis

Checkerboard Resonator

Anchored at outside cornersExcited at northwest cornerSensed at southeast cornerSurfaces move only a few nanometers

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Conclusion

Hello world!

Reflection Analysis

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)

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Conclusion

Hello world!

Reflection Analysis

Checkerboard Simulation

0 2 4 6 8 10

x 10−5

9 9.2 9.4 9.6 9.8

−200

−180

−160

−140

−120

−100

Frequency (Hz)

9 9.2 9.4 9.6 9.8

Frequency (Hz)

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Hello world!

Reflection Analysis

Checkerboard Measurement

S. Bhave, MEMS 05

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Conclusion

Hello world!

Reflection Analysis

Contributions

Built predictive model used to design checkerboardUsed model reduction to get thousand-fold speedup– fast enough for interactive use

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Conclusion

Hello world!

Reflection Analysis

General Picture

If w∗A = 0 and Av = 0 then

δ(w∗Av) = w∗(δA)v

This impliesIf A = A(λ) and w = w(v), have

w∗(v)A(ρ(v))v = 0.

ρ stationary when (ρ(v), v) is a nonlinear eigenpair.If A(λ, ξ) and w∗

0 and v0 are null vectors for A(λ0, ξ0),

w∗0 (Aλδλ + Aξδξ)v0 = 0.

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Hello world!

Reflection Analysis

HiQLab’s Hello World

0 2 4 6 8 10x 10−6

6x 10−6

mesh = Mesh:new(2)mat = make_material(’silicon2’, ’planestrain’)mesh:blocks2d( 0, l , -w/2.0, w/2.0 ,

mesh:set_bc(function(x,y)if x == 0 then return ’uu’, 0, 0; end

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Hello world!

Reflection Analysis

HiQLab’s Hello World

>> mesh = Mesh_load(’beammesh.lua’);>> [M,K] = Mesh_assemble_mk(mesh);>> [V,D] = eigs(K,M, 5, ’sm’);>> opt.axequal = 1; opt.deform = 1;>> Mesh_scale_u(mesh, V(:,1), 2, 1e-6);>> plotfield2d(mesh, opt);

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Conclusion

Hello world!

Reflection Analysis

Continuum 2D model problem

λ(x) =

1− iβ|x − L|p, x > L1 x ≤ L.

∂u∂x

∂2u∂y2 + k2u = 0

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Conclusion

Hello world!

Reflection Analysis

λ(x) =

1− iβ|x − L|p, x > L1 x ≤ L.

∂u∂x

)− k2

y u + k2u = 0

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Conclusion

Hello world!

Reflection Analysis

λ(x) =

1− iβ|x − L|p, x > L1 x ≤ L.

∂u∂x

x u = 0

1D problem, reflection of O(e−kxγ)

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Conclusion

Hello world!

Reflection Analysis

Discrete 2D model problem

Discrete Fourier transform in ySolve numerically in xProject solution onto infinite space traveling modesExtension of Collino and Monk (1998)

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Conclusion

Hello world!

Reflection Analysis

Nondimensionalization

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

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Conclusion

Hello world!

Reflection Analysis

Nondimensionalization

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

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Conclusion

Hello world!

Reflection Analysis

Discrete reflection behavior

Number of PML elements

log10(β

− log10

(r) at (kh)−1 = 10

2 2 2 2

5 10 15 20 25 30

Quadratic elements, p = 1, (kxh)−1 = 10

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Conclusion

Hello world!

Reflection Analysis

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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Conclusion

Hello world!

Reflection Analysis

Discrete reflection behavior

log10(β

− log10

(r) at (kh)−1 = 10

2 2 2 2

5 10 15 20 25 30

log10(β

− log10(rinterface + rnominal) at (kh)−1 = 10

2 2 2 2

5 10 15 20 25 30

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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Conclusion

Hello world!

Reflection Analysis

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

Computer Aided Design of Micro-Electro-Mechanical …bindel/present/2008-11...Backup slides...

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