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Tools for MEMS SimulationDavid Bindel
UC Berkeley, CS Division
Tools for MEMS Simulation – p.1/34
Outline
Basic MEMS applications and ideas
The many roles of computation
Detailed examples: RF elementsCheckerboard filterDisk resonator
Conclusions
Tools for MEMS Simulation – p.2/34
What are MEMS?
Tools for MEMS Simulation – p.3/34
MEMS Basics
Micro-electro-mechanical systemsChemical, fluid, thermal, optical (MECFTOMS?)
Applications:Sensors (inertial, chemical, pressure)Ink jet printers, biolab chipsRF devices: cell phones, inventory tags, pico radio“Smart dust”
Use integrated circuit (IC) fabrication technology
Large surface area / volume ratio
Still mostly classical (vs. nanosystems)
Tools for MEMS Simulation – p.4/34
Fabrication outline
300 um
1. Si wafer
Tools for MEMS Simulation – p.5/34
Fabrication outline
300 um
2 um
1. Si wafer
2. Deposit 2 microns SiO2
Tools for MEMS Simulation – p.5/34
Fabrication outline
300 um
2 um
1. Si wafer
2. Deposit 2 microns SiO2
3. Pattern and etch SiO2 layer
Tools for MEMS Simulation – p.5/34
Fabrication outline
300 um
2 um
2 um
1. Si wafer
2. Deposit 2 microns SiO2
3. Pattern and etch SiO2 layer
4. Deposit 2 microns polycrystalline Si
Tools for MEMS Simulation – p.5/34
Fabrication outline
300 um
2 um
2 um
1. Si wafer
2. Deposit 2 microns SiO2
3. Pattern and etch SiO2 layer
4. Deposit 2 microns polycrystalline Si
5. Pattern and etch Si layer
Tools for MEMS Simulation – p.5/34
Fabrication outline
300 um
2 um
2 um
1. Si wafer
2. Deposit 2 microns SiO2
3. Pattern and etch SiO2 layer
4. Deposit 2 microns polycrystalline Si
5. Pattern and etch Si layer
6. Release etch remaining SiO2
Tools for MEMS Simulation – p.5/34
Fabrication result
300 um
2 um
2 um
(C. Nguyen, iMEMS 01)
Tools for MEMS Simulation – p.6/34
Fabrication characteristics
300 um
2 um
2 um
Characteristic dimensions: microns
Geometry is “2.5” dimensional
Relatively loose fabrication tolerances
Difficult to characterize
Tools for MEMS Simulation – p.7/34
Why simulate?
“Build and break” is too expensiveWafer processing costs months, thousands of dollarsFabrication is impreciseDays or weeks to take good measurements
Good experiments need good hypotheses
Even when device behavior is understood, still need tounderstand system behavior
Tools for MEMS Simulation – p.8/34
From simulation to synthesis
Computer can assist at many levels:
Fundamental physics
Detailed device models
System-level models and macromodels
Metrology
Design optimization
Design synthesis
Tools for MEMS Simulation – p.9/34
Research thrusts
Model development (e.g. new finite elements)
Numerical algorithms (e.g. model reduction)
Numerical software engineering (SUGAR, HiQLab)
Metrology and comparison to measurement
Optimization and design synthesis
Tools for MEMS Simulation – p.10/34
Research group
Faculty Students
A. Agogino (ME) D. Bindel (CS)Z. Bai (Math,CS,UCD) J.V. Clark (AS&T)J. Demmel (Math,CS) C. Cobb (ME)S. Govindjee (CEE) D. Garmire (CS)R. Howe (EE,ME) T. Koyama (CEE)K.S.J. Pister (EE) J. Nie (Math)C. Sequin (CS) H. Wei (CEE)
Y. Zhang (CEE)
Tools for MEMS Simulation – p.11/34
SUGAR
Goal: “Be SPICE to the MEMS world”
Fast enough for early design stages
Simple enough to attract users
Support design, analysis, optimization, synthesis
Verify models by comparison to measurement
System assembly
Models
Solvers
Matlab Web Library
Sensitivity analysis
Static analysis
Steady−state analysis
Transient analysis
Results
Netlist
Interfaces
Tools for MEMS Simulation – p.12/34
SUGAR: Analysis of a micromirror
(Mirror design by M. Last)
Tools for MEMS Simulation – p.13/34
SUGAR: Design synthesis
Tools for MEMS Simulation – p.14/34
SUGAR: Comparison to measurement
Tools for MEMS Simulation – p.15/34
RF resonators
Microguitars from Cornell University (1997 and 2003)
Frequency references
Sensing elements
Filter elements
Neural networks
Really high-pitch guitars
Tools for MEMS Simulation – p.16/34
Micromechanical filters
Mechanical filter
Capacitive senseCapacitive drive
Radio signal
Filtered signal
Mechanical high-frequency (high MHz-GHz) filter
Saves power and cost over electronic filters
Advantage over piezo-actuated quartz SAW filtersIntegrated into chipLow power
Tools for MEMS Simulation – p.17/34
Governing equations
Time domain:
Mu′′ + Cu′ + Ku = Pφ
y = V T u
Frequency domain:
H(ω) = V T (−ω2M + iωC + K)−1P
y = Hφ
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.510
−4
10−2
100
102
104
106
108
Tools for MEMS Simulation – p.18/34
First proposed design: Checkerboard
� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �� � � � � � � � � � � � � � �
� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �
����
����
� �� �� �� ����
�
� �� �� �
�����
� �� � � �� �� �� �� �� �� �� �� �
� �� � � �� �
D+
D−
D+
D−
S+ S+
S−
S−
Array of loosely coupled resonators
Anchored at outside corners
Excited at northwest corner
Sensed at southeast corner
Surfaces move only a few nanometers
Tools for MEMS Simulation – p.19/34
Design questions
−8 −6 −4 −2 0 2 4 6 8
x 10−5
−8
−6
−4
−2
0
2
4
6
8x 10−5
−8 −6 −4 −2 0 2 4 6 8
x 10−5
−8
−6
−4
−2
0
2
4
6
8x 10−5
Where should drive and sense be placed?
How should the individual resonators be connected?
How should the system be anchored?
How many components? What topology?
Tools for MEMS Simulation – p.20/34
Checkerboard response
95 MHz 100 MHz
Tools for MEMS Simulation – p.21/34
Interactive visualization
0 2 4 6 8 10
x 10−5
0
2
4
6
8
10
12
x 10
9 9.2 9.4 9.6 9.8
x 107
−200
−180
−160
−140
−120
−100
Frequency (Hz)
Am
plitu
de (
dB)
9 9.2 9.4 9.6 9.8
x 107
0
1
2
3
4
Frequency (Hz)
Pha
se (
rad)
Tools for MEMS Simulation – p.22/34
Disk resonator model
� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �
� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �
WaferV−
V+V+
PML region
DiskElectrode
SiGe disk resonators built by E. Quévy
Tools for MEMS Simulation – p.23/34
Sources of damping
Fluid dampingAir is a viscous fluid (Re � 1)Can operate in a vacuumShown not to dominate in many RF designs
Anchor lossElastic waves radiate from structure
Thermoelastic dampingVolume changes induce temperature changeDiffusion of heat leads to mechanical loss
Material losses (catch-all)Low intrinsic losses in silicon, diamond, germaniumTerrible material losses in metals
Tools for MEMS Simulation – p.24/34
Sources of damping
Fluid dampingAir is a viscous fluid (Re � 1)Can operate in a vacuumShown not to dominate in many RF designs
Anchor lossElastic waves radiate from structure
Thermoelastic dampingVolume changes induce temperature changeDiffusion of heat leads to mechanical loss
Material losses (catch-all)Low intrinsic losses in silicon, diamond, germaniumTerrible material losses in metals
Tools for MEMS Simulation – p.24/34
Substrate model
Goal: Understand energy loss in this resonator
Dominant loss is elastic radiation from anchor.
Disk resonator is much smaller than substrate
Very little energy leaving the post is reflected backSubstrate is semi-infinite from disk’s perspective
Possible semi-infinite modelsMatched asymptotic modesDirichlet-to-Neumann mapsBoundary dampersPerfectly matched layers
Tools for MEMS Simulation – p.25/34
Perfectly matched layers
Apply a complex coordinate transformation
Generates a non-physical absorbing layer
No impedance mismatch between the computationaldomain and the 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)
Tools for MEMS Simulation – p.26/34
Scalar wave example
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
Tools for MEMS Simulation – p.27/34
Scalar wave example
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
Tools for MEMS Simulation – p.27/34
Scalar wave example
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
Tools for MEMS Simulation – p.27/34
Scalar wave example
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
Tools for MEMS Simulation – p.27/34
Scalar wave example
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
Tools for MEMS Simulation – p.27/34
Scalar wave example
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.
Tools for MEMS Simulation – p.27/34
Disk resonator mesh
� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �
� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �� � � � � � � � � � � � �
WaferV−
V+V+
PML region
DiskElectrode
0 1 2 3 4
x 10−5
−4
−2
0
2x 10
−6
Axisymmetric model with bicubic mesh
About 10K nodal points in converged calculation
Tools for MEMS Simulation – p.28/34
Response of the disk resonator
Tools for MEMS Simulation – p.29/34
Time-averaged energy flux
0 0.5 1 1.5 2 2.5 3 3.5
x 10−5
−2
0
2
x 10−6
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10−6
−1
−0.5
0
0.5
1
1.5
2
2.5x 10
−6
Tools for MEMS Simulation – p.30/34
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
Compute complex frequencies by shift-and-invertArnoldi with an analytically determined shift
Surprising variation – experimentally observed – in Q
as film thickness changes!
Tools for MEMS Simulation – p.31/34
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
Non-normal interaction between the modes
Tools for MEMS Simulation – p.32/34
Truth in advertising
1.2 1.3 1.4 1.5 1.6 1.7 1.810
0
102
104
106
108
1010
Film thickness (µm)
Q
Data from a set of 30µm radius disks.
Tools for MEMS Simulation – p.33/34
Conclusions
Simulation tools are good for MEMS designersReduce expensive “build-and-break” designReveal behavior that is hard to measure directly
MEMS problems are good for tool designersInteresting physics (and not altogether understood)Interesting numerical mathematics
Tools for MEMS Simulation – p.34/34
