Eigenproblems in Resonant MEMS Design

Eigenproblems in Resonant MEMS Design
Eigenproblems in
Resonant 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
Radio signal
Mechanical filter
Filtered signal
Capacitive drive
Capacitive sense
Your cell phone is already mechanical!
Uses a quartz surface-acoustic wave (SAW) filter
Can do better using MEMS
MEMS filters can be placed on-chip
Versus SAWs: smaller, lower power
Success =⇒ “Calling Dick Tracy!”
Eigenproblems inResonant MEMS Design – p.4/21
Damping
Want to minimize damping
Measure by “quality of resonance”
|ω|
Q=
Im(ω)
Electronic filters have too much
Understanding of damping in MEMS is lacking
Several sources of damping
Anchor loss
Thermoelastic damping
Fluid damping
Material losses
Eigenproblems inResonant MEMS Design – p.5/21
Damping
Want to minimize damping
Measure by “quality of resonance”
|ω|
Q=
Im(ω)
Electronic filters have too much
Understanding of damping in MEMS is lacking
Several sources of damping
Anchor loss
Thermoelastic damping
Fluid damping
Material losses
Eigenproblems inResonant MEMS Design – p.5/21
Example: Disk anchor loss
Electrode
Disk
V−
Wafer
V+
V+
SiGe disk resonators built by E. Quévy
Axisymmetric model with bicubic mesh, about 10K
nodal points
Eigenproblems inResonant MEMS Design – p.6/21
Perfectly matched layers
Model half-space with a perfectly matched layer
Complex coordinate change x 7→ z(x; ω)
Apply a complex coordinate transformation
Generates a non-physical absorbing layer
Idea works with general linear wave equations
First applied to Maxwell’s equations (Berengér 95)
Similar idea introduced earlier in quantum
mechanics (exterior complex scaling, Simon 79)
Eigenproblems inResonant MEMS Design – p.7/21
Scalar wave example
−c2 uzz − ω 2 u = 0
Outgoing wave exp(−iz)
Incoming wave exp(iz)
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
0
5
10
15
−1
20
0
5
10
15
20
Transformed coordinate z = x + iy
0
−1
−2
−3
−4
−5
0
2
4
6
8
10
12
14
16
18
Eigenproblems inResonant MEMS Design – p.8/21
Scalar wave example
−c2 uzz − ω 2 u = 0
Outgoing wave exp(−iz)
Incoming wave exp(iz)
1
3
2
0.5
1
0
0
−0.5
−1
−1
0
5
10
15
−2
20
0
5
10
15
20
Transformed coordinate z = x + iy
0
−1
−2
−3
−4
−5
0
2
4
6
8
10
12
14
16
18
Eigenproblems inResonant MEMS Design – p.8/21
Scalar wave example
−c2 uzz − ω 2 u = 0
Outgoing wave exp(−iz)
Incoming wave exp(iz)
1
6
4
0.5
2
0
0
−0.5
−1
−2
0
5
10
15
−4
20
0
5
10
15
20
Transformed coordinate z = x + iy
0
−1
−2
−3
−4
−5
0
2
4
6
8
10
12
14
16
18
Eigenproblems inResonant MEMS Design – p.8/21
Scalar wave example
−c2 uzz − ω 2 u = 0
Outgoing wave exp(−iz)
Incoming wave exp(iz)
1
15
10
0.5
5
0
0
−0.5
−1
−5
0
5
10
15
20
−10
0
5
10
15
20
Transformed coordinate z = x + iy
0
−1
−2
−3
−4
−5
0
2
4
6
8
10
12
14
16
18
Eigenproblems inResonant MEMS Design – p.8/21
Scalar wave example
−c2 uzz − ω 2 u = 0
Outgoing wave exp(−iz)
Incoming wave exp(iz)
1
40
30
0.5
20
0
10
0
−0.5
−10
−1
0
5
10
15
20
−20
0
5
10
15
20
Transformed coordinate z = x + iy
0
−1
−2
−3
−4
−5
0
2
4
6
8
10
12
14
16
18
Eigenproblems inResonant MEMS Design – p.8/21
Scalar wave example
−c2 uzz − ω 2 u = 0
Outgoing wave exp(−iz)
Incoming wave exp(iz)
1
100
80
0.5
60
40
0
20
0
−0.5
−20
−1
0
5
10
15
20
−40
0
5
10
15
20
Transformed coordinate z = x + iy
0
−1
−2
−3
−4
−5
0
2
4
6
8
10
12
14
16
18
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 attenuation
Common choice is
dz
= 1 − σ(x)/k
dx
What if we use a fixed transformation?
Can choose to absorb well over finite ω range
Solve a linear eigenvalue problem
Amounts to rational approx of true radiation
condition (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 − ω 2 M )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
−1
Relative error in boundary coefficient
10
−2
10
−3
10
−4
10
20
30
40
50
60
70
80
Elements per wavelength
90
100
Compare last coefficient with exact (discrete) BC
Eigenproblems inResonant MEMS Design – p.10/21
Complex symmetry
Finite element equations (forced vibration) are
−ω 2 M u + Ku = F
where M and K are complex symmetric.
Row and column eigenvectors are transposes
Second-order accuracy with modified Rayleigh quotient:
θ(v) = (v T Kv)/(v T M v)
Can have v T M v ≈ 0
Propagating modes (continuous spectrum)
Not the modes of interest for resonators
Eigenproblems inResonant MEMS Design – p.11/21
Q variation
8
10
6
Q
10
4
10
2
10
0
10
1.2
1.3
1.4
1.5
1.6
Film thickness (µm)
1.7
1.8
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
0.25
e
a
b
d
Imaginary frequency (MHz)
c
0.2
0.15
0.1
0.05
0
46
a = 1.51µm
b = 1.52µm
c = 1.53µm
d = 1.54µm
d = 1.55µm
46.5
d c
e b
a
47
Real frequency (MHz)
47.5
48
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 with
real shape functions for the continuum PML equations
Eigenproblems inResonant MEMS Design – p.14/21
Example: Disk resonator response
Transfer (dB)
0
-20
-40
-60
-80
47.2
47.25
47.3
Frequency (MHz)
Phase (degrees)
200
100
0
47.2
47.25
47.3
Frequency (MHz)
Eigenproblems inResonant MEMS Design – p.15/21
Example: Disk resonator response
|H(ω) − Hreduced(ω)|/H(ω)|
10−2
10−4
10−6
Structure-preserving ROM
Arnoldi ROM
45
46
47
48
49
50
Frequency (MHz)
Eigenproblems inResonant MEMS Design – p.16/21
Thermoelastic damping (TED)
u is displacement, T = T0 + θ is temperature
σ = C−βθ1
ρutt = ∇ · σ
ρcv θt = ∇2 θ−βT0 tr(t )
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 = C−βθ1
ρutt = ∇ · σ
ρcv θt = ∇2 θ−βT0 tr(t )
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 = Ĉ − ξθ1
utt = ∇ · σ
θt = η∇2 θ − tr(t )
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:
−ω 2 Muu u + Kuu u + Kut θ = 0
iωDtt θ + Ktt θ + iωDtu u = 0
Approximate ω by perturbation about Kuθ = 0:
−ω02 Muu u0 + Kuu u0 = 0
iω0 Dθθ θ0 + Kθθ θ0 + iω0 Dtu u0 = 0
Choose v : v T u0 6= 0 and compute
"
#" # "
#
(−ω02 Muu + Kuu ) −2ω0 Muu u0 δu
−Kuθ θ0
=
T
v
0
δω
0
Eigenproblems inResonant MEMS Design – p.19/21
Comparison to Zener’s model
−4
Thermoelastic Damping Q
−1
10
−5
10
Zener’s Formula
−6
HiQlab Results
10
−7
10
5
10
6
10
7
10
8
10
9
10
10
10
Frequency f(Hz)
Good match to Zener’s approximation for TED in beams
Real and imaginary parts after first-order correction
agree to about three digits with Arnoldi
Eigenproblems inResonant MEMS Design – p.20/21
Conclusions
MEMS resonator simulations give interesting problems
Damped resonators =⇒ nonlinear eigenproblems
Introduce auxiliary variables to get exact or
approximate linear problem
There’s still useful structure in non-Hermitian
problems!
References:
Bindel and Govindjee. “Elastic PMLs for Resonator
Anchor Loss Simulation.” (IJNME, to appear)
HiQLab home page:
www.cs.berkeley.edu/d̃bindel/hiqlab/
Eigenproblems inResonant MEMS Design – p.21/21
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