Damping Mechanisms in Resonant Microsystemsbindel/present/2008-03...Backup slides Electromechanical...

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ResonantMEMS andmodels

Anchor lossesand diskresonators

Thermoelasticlosses andbeamresonators

Conclusion

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Damping Mechanisms in ResonantMicrosystems

D. Bindel

Courant Institute for Mathematical SciencesNew York University

Materials Working Group, 10 Mar 2008

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Conclusion

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Outline

1 Resonant MEMS and models

2 Anchor losses and disk resonators

3 Thermoelastic losses and beam resonators

4 Conclusion

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Outline

4 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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Conclusion

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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

4 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

4 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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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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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

4 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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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

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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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Conclusion

Backup slidesHiQLab

Checkerboardresonators

Hello world!

Reflection Analysis

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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Backup slidesHiQLab

Hello world!

Reflection Analysis

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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Hello world!

Reflection Analysis

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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Hello world!

Reflection Analysis

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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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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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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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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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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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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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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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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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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Backup slidesHiQLab

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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Backup slidesHiQLab

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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Backup slidesHiQLab

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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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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Backup slidesHiQLab

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

Backup slidesHiQLab

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

Damping Mechanisms in Resonant Microsystemsbindel/present/2008-03...Backup slides Electromechanical...

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