Eigenproblems inResonant MEMS Design

David Bindel

UC Berkeley, CS Division

Eigenproblems inResonant MEMS Design – p.1/21

What are MEMS?

Eigenproblems inResonant MEMS Design – p.2/21

RF MEMS

Microguitars from Cornell University (1997 and 2003)

MHz-GHz mechanical resonators

Uses:RF signal processing (better cell phones)Sensing elements (e.g. chemical sensors)Really high-pitch guitars

Eigenproblems inResonant MEMS Design – p.3/21

Micromechanical filters

Filtered signal

Mechanical filter

Capacitive senseCapacitive drive

Radio signal

Your cell phone is already mechanical!Uses a quartz surface-acoustic wave (SAW) filter

Can do better using MEMSMEMS filters can be placed on-chipVersus SAWs: smaller, lower power

Success =⇒ “Calling Dick Tracy!”Eigenproblems inResonant MEMS Design – p.4/21

Damping

Want to minimize dampingMeasure by “quality of resonance”

Q =|ω|

Im(ω)

Electronic filters have too muchUnderstanding of damping in MEMS is lacking

Several sources of dampingAnchor lossThermoelastic dampingFluid dampingMaterial losses

Eigenproblems inResonant MEMS Design – p.5/21

Damping

Want to minimize dampingMeasure by “quality of resonance”

Q =|ω|

Im(ω)

Electronic filters have too muchUnderstanding of damping in MEMS is lacking

Several sources of dampingAnchor lossThermoelastic dampingFluid dampingMaterial losses

Eigenproblems inResonant MEMS Design – p.5/21

Example: Disk anchor loss

V+

DiskElectrode

WaferV+

V−

SiGe disk resonators built by E. Quévy

Axisymmetric model with bicubic mesh, about 10Knodal points

Eigenproblems inResonant MEMS Design – p.6/21

Perfectly matched layers

Model half-space with a perfectly matched layerComplex coordinate change x 7→ z(x;ω)

Apply a complex coordinate transformationGenerates a non-physical absorbing layer

Idea works with general linear wave equationsFirst applied to Maxwell’s equations (Berengér 95)Similar idea introduced earlier in quantummechanics (exterior complex scaling, Simon 79)

Eigenproblems inResonant MEMS Design – p.7/21

Scalar wave example−c2uzz − ω2u = 0

0 5 10 15 20−1

−0.5

0

0.5

1Outgoing wave exp(−iz)

0 5 10 15 20−1

−0.5

0

0.5

1Incoming wave exp(iz)

0 2 4 6 8 10 12 14 16 18−5

−4

−3

−2

−1

0Transformed coordinate z = x + iy

Eigenproblems inResonant MEMS Design – p.8/21

Scalar wave example−c2uzz − ω2u = 0

0 5 10 15 20−1

−0.5

0

0.5

1Outgoing wave exp(−iz)

0 5 10 15 20−2

−1

0

1

2

3Incoming wave exp(iz)

0 2 4 6 8 10 12 14 16 18−5

−4

−3

−2

−1

0Transformed coordinate z = x + iy

Eigenproblems inResonant MEMS Design – p.8/21

Scalar wave example−c2uzz − ω2u = 0

0 5 10 15 20−1

−0.5

0

0.5

1Outgoing wave exp(−iz)

0 5 10 15 20−4

−2

0

2

4

6Incoming wave exp(iz)

0 2 4 6 8 10 12 14 16 18−5

−4

−3

−2

−1

0Transformed coordinate z = x + iy

Eigenproblems inResonant MEMS Design – p.8/21

Scalar wave example−c2uzz − ω2u = 0

0 5 10 15 20−1

−0.5

0

0.5

1Outgoing wave exp(−iz)

0 5 10 15 20−10

−5

0

5

10

15Incoming wave exp(iz)

0 2 4 6 8 10 12 14 16 18−5

−4

−3

−2

−1

0Transformed coordinate z = x + iy

Eigenproblems inResonant MEMS Design – p.8/21

Scalar wave example−c2uzz − ω2u = 0

0 5 10 15 20−1

−0.5

0

0.5

1Outgoing wave exp(−iz)

0 5 10 15 20−20

−10

0

10

20

30

40Incoming wave exp(iz)

0 2 4 6 8 10 12 14 16 18−5

−4

−3

−2

−1

0Transformed coordinate z = x + iy

Eigenproblems inResonant MEMS Design – p.8/21

Scalar wave example−c2uzz − ω2u = 0

0 5 10 15 20−1

−0.5

0

0.5

1Outgoing wave exp(−iz)

0 5 10 15 20−40

−20

0

20

40

60

80

100Incoming wave exp(iz)

0 2 4 6 8 10 12 14 16 18−5

−4

−3

−2

−1

0Transformed coordinate z = x + iy

Clamp solution at transformed end to isolate outgoing wave.Eigenproblems inResonant MEMS Design – p.8/21

Choice of transformations

Generally z depends nontrivially on ω

Needed for frequency-independent attenuationCommon choice is

dz

dx= 1 − σ(x)/k

What if we use a fixed transformation?Can choose to absorb well over finite ω rangeSolve a linear eigenvalue problemAmounts to rational approx of true radiationcondition (in discrete case)

Eigenproblems inResonant MEMS Design – p.9/21

Behavior with fixed transformations

0 5 10 15 20 25 30

0

5

10

15

20

25

30

Start with (K − ω2M)u = e1

Eigenproblems inResonant MEMS Design – p.10/21

Behavior with fixed transformations

0 5 10 15 20 25 30

0

5

10

15

20

25

30

Schur complement to eliminate PML unknowns

Eigenproblems inResonant MEMS Design – p.10/21

Behavior with fixed transformations

20 30 40 50 60 70 80 90 10010

−4

10−3

10−2

10−1

Elements per wavelength

Rel

ativ

e er

ror

in b

ound

ary

coef

ficie

nt

Compare last coefficient with exact (discrete) BC

Eigenproblems inResonant MEMS Design – p.10/21

Complex symmetry

Finite element equations (forced vibration) are

−ω2Mu + Ku = F

where M and K are complex symmetric.

Row and column eigenvectors are transposes

Second-order accuracy with modified Rayleigh quotient:

θ(v) = (vTKv)/(vT Mv)

Can have vT Mv ≈ 0

Propagating modes (continuous spectrum)Not the modes of interest for resonators

Eigenproblems inResonant MEMS Design – p.11/21

Q variation

1.2 1.3 1.4 1.5 1.6 1.7 1.810

0

102

104

106

108

Film thickness (µm)

Q

Small geometry variation =⇒ large damping variation

Solid line is simulated; dots are measured

Eigenproblems inResonant MEMS Design – p.12/21

Effect of varying film thickness

46 46.5 47 47.5 480

0.05

0.1

0.15

0.2

0.25

Real frequency (MHz)

Imag

inar

y fr

eque

ncy

(MH

z)

ab

cd

e

a bc

d

e

a = 1.51µmb = 1.52µmc = 1.53µmd = 1.54µmd = 1.55µm

Sudden dip in Q comes from an interaction between a(mostly) bending mode and a (mostly) radial mode

Eigenproblems inResonant MEMS Design – p.13/21

Model reduction

Would like a reduced model which

Preserves second-order accuracy for converged eigs

Keeps at least Arnoldi’s accuracy otherwise

Is physically meaningful

Idea:

Build an Arnoldi basis V

Double the size: W = orth([Re(V ), Im(V )])

Use W as a projection basis

Resulting system is still a Galerkin approximation withreal shape functions for the continuum PML equations

Eigenproblems inResonant MEMS Design – p.14/21

Example: Disk resonator response

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

Eigenproblems inResonant MEMS Design – p.15/21

Example: Disk resonator response

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

Eigenproblems inResonant MEMS Design – p.16/21

Thermoelastic damping (TED)

u is displacement, T = T0 + θ is temperature

σ = Cε−βθ1

ρutt = ∇ · σ

ρcvθt = ∇2θ−βT0 tr(εt)

Second-order mechanical + first-order thermal equation

Eigenproblems inResonant MEMS Design – p.17/21

Thermoelastic damping (TED)

u is displacement, T = T0 + θ is temperature

σ = Cε−βθ1

ρutt = ∇ · σ

ρcvθt = ∇2θ−βT0 tr(εt)

Second-order mechanical + first-order thermal equation

Temperature change causes stress (thermal expansion)

Volumetric strain rate causes thermal fluctuations

Eigenproblems inResonant MEMS Design – p.17/21

Thermoelastic damping (TED)

Non-dimensionalized equation:

σ = Cε − ξθ1

utt = ∇ · σ

θt = η∇2θ − tr(εt)

Typical MEMS scales: ξ and η small

Perturbation about ξ = 0 is effective

Eigenproblems inResonant MEMS Design – p.18/21

Perturbation computation

Discrete time-harmonic equations:

−ω2Muuu + Kuuu + Kutθ = 0

iωDttθ + Kttθ + iωDtuu = 0

Approximate ω by perturbation about Kuθ = 0:

−ω2

0Muuu0 + Kuuu0 = 0

iω0Dθθθ0 + Kθθθ0 + iω0Dtuu0 = 0

Choose v : vT u0 6= 0 and compute[

(−ω20Muu + Kuu) −2ω0Muuu0

vT 0

] [

δu

δω

]

=

[

−Kuθθ0

0

]

Eigenproblems inResonant MEMS Design – p.19/21

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

Good match to Zener’s approximation for TED in beams

Real and imaginary parts after first-order correctionagree to about three digits with Arnoldi

Eigenproblems inResonant MEMS Design – p.20/21

Conclusions

MEMS resonator simulations give interesting problems

Damped resonators =⇒ nonlinear eigenproblemsIntroduce auxiliary variables to get exact orapproximate linear problemThere’s still useful structure in non-Hermitianproblems!

References:Bindel and Govindjee. “Elastic PMLs for ResonatorAnchor Loss Simulation.” (IJNME, to appear)HiQLab home page:www.cs.berkeley.edu/ dbindel/hiqlab/

Eigenproblems inResonant MEMS Design – p.21/21