Eigenproblems in Resonant MEMS Design David Bindel UC Berkeley, CS Division Eigenproblems inResonant MEMS Design – p.1/21
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