Behavioral Modeling of Viscous
Damping in MEMS
Satish
Vemuri
2000
Advisor:
Prof.
Fedder/Mukheriee
Electrical g~ Computer
ENGINEERING
Behavioral Modeling of Viscous Damping in MEMS
M.S. Thesis Report
by
S atish
Department of Electrical
Kumar Vemuri
and Computer Engineering
Carnegie Mellon Univeristy
Committee
Prof. Gary K. Fedder
Dr. Tamal Mukherjee
Acknowledgements
I wouldlike to extend mysincere thanks to myadvisors Prof. GaryK. Fedder and Dr. Tamal
Mukherjeefor their invaluable guidanceand unlimited patience. I wouldlike to thank mycolleagues Q. Jing, Sitaraman Iyer, Vishal Gupta and other MEMS
group membersfor their valuable
discussions. Finally, I like to thank myparents and mybrother for all the support and encouragement.
This research was sponsored by Defence AdvancedResearch Projects Agency (DARPA)
Composite CADprogram, under grant number F30602-97-2-0332.
Table of contents
1 Introduction
..................................................................................................................
1
2 Squeeze
filmdamping
..................................................................................................
3
2.1Introduction
...........................................................................................................
3
2.2Squeeze
Film
Model
.............................................................................................
4
2.2.1Squeeze
filmphysics
..................................................................................
4
2.2.2Gasrarefaction
effects................................................................................
4
2.2.3 Squeezeforce: Solution to compressible,linearized Reynoldsequation....... 5
2.3Finiteelement
simulation
of squeeze
film...........................................................
6
2.3.1Trivial
boundary
conditions
.........................................................................
8
2.3.2Non-trivial
boundary
conditions
................................................................
9
2.3.3 Trivial boundary
conditionsvs. non-trivial boundary
conditions.................
9
2.4Lumped
parameter
model
...................................................................................
10
2.5Non
idealities
......................................................................................................
12
2.5.1Edge
andfinite-sizeeffects.......................................................................
12
2.5.2Edge-effect
model
.....................................................................................
13
2.5.3Extended
model
........................................................................................
16
2.6Squeeze
filmsimulation
results..........................................................................
17
2.6.1Stepfunction
response
..............................................................................
17
2.6.2Aspect
ratiodependency
...........................................................................
17
2.5.3Amplitude
ofmotion
................................................................................
18
2.6.4Frequency
dependency
.............................................................................
19
3 Lateral
viscous
damping
...........................................................................................
,23
3.1Introduction
.......................................................................................................... 23
3.2Slidefilmmodel
..................................................................................................
24
3.2.1Model
forcontinuum
conditions
..............................................................
24
3.2.1Gasrarefaction
effects.............................................................................
25
3.2.3Solutionfor the velocityprofilewithgas rarefaction................................
26
3.3Finiteelement
simulation
of slidefilms..............................................................
27
3.3.1Trivialboundary
conditions
......................................................................
28
3.3.2Non-trivial
boundary
conditions
...............................................................
28
3.3.3 Trivial boundary
conditionsvs. non-trivial boundary
conditions................
29
3.4Lumped
parameter
model
...................................................................................
29
3.4.1Edgeeffectsin the lumped
parameter
model............................................
31
3.4.2Stepfunction
response
..............................................................................
32
3.4.3Frequency
dependency
..............................................................................
33
4 System
level simulation
andexperimental
verification.............................................
35
4.1CMOS
bandpass
resonator
filter .........................................................................
35
4.2Lateral
microresonator.
.......................................................................................
37
5 Conclusions
andfuturework
.....................................................................................
39
References
.................................................................................................................... 40
Appendix
....................................................................................................................... 41
1. Introduction
MicroElectroMechanicalSystems
(MEMS)
are often characterized by structures that are a few
microns in size, separated by micron-sized gaps. At these sizes and gaps, viscous air damping
dominates other energy dissipation mechanismsat atmospheric pressure, thus, affecting the
dynamic behavior of the devices. Also, the numberof such gaps can be very large in a MEMS
device. Hence, fast and accurate behavioral simulation of the dynamicsof MEMS
systems necessitates accurate, low-order dampingmodels. Further, to be useful in the design flow, these models
need to be parameterized, so that they can be used to modeldampingfor a wide range of structure
size, gap, frequency and amplitudeof motion.
Viscous dampingdue to the fluid surrounding the movingstructures can be of two types:
squeeze film dampingand lateral damping.In squeeze film damping,there are two parallel structures as shownin Fig. 1.1. Theyare in relative motionperpendicular to the plane, thus squeezing
the air film betweenthe structures. The air counters the motionby exerting an opposingforce. Fig.
1.2 showsthe case of lateral damping.Here the motionof the plate is parallel to the plane of the
structure. Theresulting fluid flow causes energy dissipation and exerts an opposingforce whichis
a function of the velocity of the movingplate, the dimensionsof the plate, its distance from other
parallel structures, and the viscosity of the fluid.
~
Velocity
Moving Plate
ga
~ Squeeze
]
Film
Fig. 1.1: Cross section of a squeezefilm betweena movingplate and the substrate
This report presents lumpedparameter models for squeeze film and lateral damping. The
accuracy of these modelsdownto MEMS
scaled structures, and their parameterized nature makes
them useful for design and simulation of MEMS
devices. With the edge-effect contribution to the
MovingPlate
Velocity
Substrate
Fig. 1.2: Cross section of a movingplate that is laterally damped
dampingforce becomingsignificant in micron-scaled structures, emphasis is laid on including
these effects.
In chapter 2, squeeze film damping model is discussed. Finite element simulations of
squeeze film are presented. The derivation of the behavioral modelfor squeeze film from existing
analytical model[1] and its extension to include the edgeand the finite-size effects are discussed.
The significance of the contribution of these effects for MEMS
scaled devices is established. The
edge-effects enhanced behavioral model is evaluated, for varying frequency, amplitude of plate
motionand the aspect ratio of the plate by comparingit with FEMsimulations. Simulation results
for a square waveinput are presented.
In chapter 3, modelingof the fluid flow invokedby a laterally movingstructure and the derivation of the behavioral modelfor lateral dampingare discussed. The accuracy of the behavioral
modelis quantified by comparisonwith the finite element simulations. Simulation results for a
square waveinput and for varying aspect ratio and oscillation frequencyare discussed.
Chapter 4 establishes the use of the presented dampingmodelsin system level simulation and
design, by use in the nodal simulation of a bandpassfilter and a lateral resonator. Thesimulation
results are comparedwith the experimental results whichestablishes the accuracy of the damping
models.
Finally, chapter 5 concludes the overall workand discusses somefuture work that could be
donein this direction.
2. Squeeze film damping
2.1 Introduction
Whena fluid film betweentwo parallel surfaces is squeezedby the relative motionof the surfaces towards each other, a force that opposesthe motionof the structures is produced. This type
of energy dissipation mechanism,called squeeze film damping, is encountered very commonlyin
MEMS
devices. A squeeze film is characterized by physical dimensions like width, length and
thickness of the fluid film that is squeezed. Other parameters that determinethe dampingare the
ambientpressure, the temperature of device operation, viscosity coefficient of the fluid and the
velocity of the oscillating plate.
Exampleapplications include MEMS
systems that have a proof mass oscillating
vertically
above a static bottom plane or a differential combdrive. Dampingof the proof massin a z-axis
accelerometer is a fundamentalconsideration in the design of the accelerometer. In differential
combdrives, the squeezed air between the stator and the rotor fingers contributes most of the
dampingin the differential combdrives. Fig. 2.1 shows the squeeze film betweenthe stator and
rotor fingers of a differential combdrive. Thestator finger movesin the direction markedin Fig.
2.1 (b). The thickness of the structure is equal to the width of the squeezefilm (typically a
microns); the overlap distance of the stator and rotor fingers is equal to the length of the squeeze
film (tens of microns); and the equilibrium gap betweenthe stator and the rotor fingers is equal
the gap betweenthe top and the bottomplate (typically a couple of microns).
Ro.tor
Displacement
Stator
Squeeze Film
(a)
(b)
Fig. 2.1 (a): SEM
picture showingthe stator and the rotor fingers of a differential combdrive
(b): Picture showingthe squeezefilm betweenthe stator and rotor fingers
2.2 Squeeze
film
model
2.2.1 Squeeze film physics
The gas pressure distribution across the movingplate, as a result of squeezefilm effect, is
governedby the Reynolds equation. Underisothermal conditions, the pressure (that is close to
ambientpressure) is inversely proportional to the volumeoccupiedby the gas, or, directly proportional to the density of the gas. Whenthe plates are parallel and the relative displacementof the
movingplate is small as comparedto the distance betweenthe plates, the second-order, non-linear
partial differential equationsimplifies to the linearized Reynoldsequation (EQ.2.1) and to EQ.2.2
under isothermal conditions [2, 3].
EQ. 2.1
~--~)_ ,~,
,en ox----~+a-Tv-)(}-j])-~-7
~ =-a-7
EQ. 2.2
wheret9 is the density of the gas, p is the changein pressure fromthe ambientpressure, Pa is the
ambientpressure, g is the gapbetweenthe plates, ~1 is the viscosity coefficient of air andz is the
vertical displacementof the movingplate, x and y are the cartesian coordinates.
2.2.2 Gas rarefaction effects
The squeeze film sizes that are found in MEMS
devices mayresult in a condition wherethe
fluid film can no longer be considered to be within continuumlimits. As the film thickness is
reduced, the mean-free path of the molecules becomessignificant as comparedto the critical
dimension,whichis the squeeze film thickness in this case. This regime of operation can be given
by a dimensionless quantity called the Knudsennumber, Kn, which is the ratio of the meanfree
path of the gas molecules under consideration )~, to the gap between the plates, g. When
Kn < 1/100, the fluid can be considered to be in the continuumlimits and the no-slip assumption
that the plate velocity and the fluid velocity at plate surface is equal remainsvalid. Onthe other
end, if Kn > 1, then molecular regime of operation should be considered. Slip conditions begin to
occur when)~ becomesa fraction of g (>1/100). The meanfree path of air is about 70 nmat standard temperature and pressure. The gaps that are commonlyencountered in squeeze film damping
can be a couple of microns. So slip conditions have to be taken into account. At lower pressures K
n
is higher because)~ increases, and the region of operation shifts moreand moretowardsthe molecular regime.
Theeffect of slip-conditions can be modeledby using a value of effective viscosity whichis
lower than the actual viscosity of the fluid. There are manyexisting equations to modelthe viscosity of the fluid flowing in a narrowchannel as summarizedin [3]. Weuse the value pointed to by
Veijola et al. [3]. Theequationfor effective viscosity is givenby:
lie # =
rl
tse
I + 9,6381(,
EQ. 2.3
wherelieffis the effective viscosity, li is the fluid viscosity, Kn is the Knudsennumber.Fig. 2.2
plots the effective viscosity as a function of gapbetweenthe plates, the decreasein the effective
viscosity can be seen with decreasing gap.
20
19
¯ "=-
18
~o , ~7
>¢ ~;~ 16
~4
13
0
2
4
6
8
Gap (~tm)
10
12
Fig. 2.2: Effective viscosity as a function of gap
2.2.3 Squeeze force: Solution to compressible, linearized Reynolds
equation
The linearized Reynoldsequation (EQ. 2.2) can be solved under compressible conditions for
sinusoidal oscillation of the a rectangular plate. The solution consists of a spring force in phase
with the plate displacementand a dampingforce in phase with the plate velocity[Blechs]. Theinte-
gral of the gaugepressure over the area of the plate gives the following relations for the damping
andthe spring forces [1]"
EQ. 2.4
odd
EQ. 2.5
whereFd and Fs are the dampingand spring forces, I is the length, w is the widthand c is the aspect
ratio of the plate (width/length), z is the amplitudeof motion,g the gap and cr is the squeezenumber given by:
12q~weO~
~
e = pag
EQ. 2.6
where0~ is the angular frequencyof oscillation.
2.3 Finite element simulation of squeeze film
Finite element simulations of the dynamics of squeeze films are performed using a code
which solves three-dimensional Navier-Stokes equation. One such commercialcode used in this
thesis is CFD-ACE
[4, 5]. Transient simulations of a plate oscillating vertically above an infinite
substrate are performed. Thedampingforce on the oscillating plate is determinedby the integration of the gaugepressure (pressure deviation from the ambient pressure) over the surface of the
oscillating plate.
To verify the finite element simulations, a well knownand analyzed problem was solved. A
square plate of size lmmon a side, oscillating with an amplitudeof 0.1 gmand a frequency of 1
kHzwas chosen. Fig 2.3 showsthe dampingforce with time for a gap of 2 ~tm. The dampingcoefficient can be extracted from the peak force. Theanalytical equation for the dampingcoefficient
for a square plate of length I on a side and a gap g, is given by EQ.2,7 [6].
B = 0.422
4l]eyfl
whererleff is the effective viscosity coefficient of the fluid.
EQ. 2.7
The values of damping coefficient
at the plate edges and the analytical
obtained from the FEAmethod with ambient pressure set
method are 0.96 and 1.0 respectively.
They match to within
4%which verifies and quantifies the accuracy of the finite element simulations.
0.40
0.30
0.20
0.10
0,00
-0,10
-0.20
-0.30
-0.40
0
1
1
2
2
3
Time(ms)
Fig. 2.3: Dampingforce on a sinusoidally oscillating
The accuracy of the finite
3
plate
element simulations also depends on the boundary conditions used.
Twodifferent boundary conditions can be used in the simulation of the squeeze film damping: trivial boundary conditions and non-trivial
boundary conditions.
The advantages of each approach is
discussed next.
pressure
(Pa)
/ displacement~~
width
length (l)
(a):
~Symmetry
Squeeze film with TBC
(b): Pressure distribution across top plate
Fig. 2.4:20 gmsquare plate oscillating
with 0.1 gm amplitude at 1 kHz
2.3.1
Trivial
boundary conditions
In trivial boundaryconditions, the squeezefilm space is represented by a 3-D grid betweenan
oscillating plate and substrate as shownin Fig. 2.4 (a). To exploit the symmetryconditions for
reducing the numberof grid cells and the computerresources in terms of memoryand CPUtime,
only a quarter section of the squeeze film is used in simulations. Ambientpressure boundarycondition is used at the plate edges. Fig. 2.4.b showsthe pressure distribution across a the quarter plate
whena square plate of size 20 ~mis oscillating sinusoidally with an amplitudeof 0.1gin and a frequencyof 1 kHz.It can be noticed that the gaugepressure is forced to zero at the plate-edges.
Substrate
Squeeze Film
Oscillating Plate
Fig. 2.5 (a): Squeeze film with extendedboundary
Ambient Pressure
p (pa)
\ ,~ymmetry
Fig. 2.5 (b): Pressure distribution across and aroundthe top plate (20 gmX 20 ~tm)
oscillating with an amplitude of 0.1 gmat 1 kHz
2.3.2
Non-trivial
boundary conditions
The use of non-trivial boundaryconditions are morerealistic boundaryconditions by allowing the solver to arrive at the gaugepressure at the plate edges. This is done by extendingthe control volumearound the oscillating plate and setting the pressure to ambient pressure at the
extendedcontrol volumeboundary. Fig. 2.5 (a) showsthe system with non-trivial boundaryconditions with blocks of air surroundingthe plate oscillating abovean infinite substrate. Fig. 2.5 (b)
showsthe pressure distribution across the quarter plate. Notice that the pressure contours extend
beyondthe plate edges.
2.3.3 Trivial boundaryconditions vs. non-trivial boundaryconditions
In practice, the pressure at the plate edges is non-zero. Whentrivial boundaryconditions
(TBC)are used, this valt~e is forced to zero and causes underestimationof the gaugepressure distribution across the plate. This results in the underestimationof the dampingforce due to squeeze
action. This problem can be mitigated by the use of non-trivial boundary conditions (NTBC)
whichmodelsthe boundaryconditions more realistically.
The following table provides the squeeze force predicted by the use of NTBCand TBCwhen
a plate of varying size oscillates with an amplitude of 0.1 gmand frequencyof 1 kHz, 2 gmabove
the substrate. It can be observedfrom the table that the difference in force predicted by TBCand
NTBC
decreases as the plate size increases. This can be intuitively explained because the edgeeffects andthe finite-size effects decreasewith increasing plate size reachingzero for an infinitely
large plate. The dominanceof the edge and the finite-size
effects for sub-100 gm plates has
already been experimentallyverified in case of lateral viscous damping.For developingbehavioral
Plate Size
TBC
NTBC
Difference
20btm
109.1 pN
151.0 pN
27.8%
100~tm
lmm
62.5 nN
67.1 nN
7.6%
0.65 mN
0.67 mN
2.9%
models for dampingfor use in MEMS-scale
structures, all finite element simulations are done
using non-trivial boundaryconditions to take care of the edgeand the finite-size effects.
The use of NTBC
results in the increase in the numberof grid cells from 500 to 5000 as a
result of which, a typical transient simulation time increases from 5 minutesin TBCto 60 minutes
in NTBCon a 360 MHzSun Ultra-Sparc. The memoryrequirements of the simulation is 5 MBin
TBCand 45 MBin NTBC.Thoughthere is an increase in the use of computer resources in terms
of CPU-timeand memory,this provides the solver with more realistic boundaryconditions to give
accurate results.
2.4 Lumpedparameter model
A lumpedparameter model consisting of parallel branches of series connected spring and
damperelements can be derived to model the dampingforce given by EQ. 2.4 and EQ. 2.5, with
force as the throughvariable and velocity as the across variable. To derive the values of the spring
and the damperelements, we consider the following spring-damperelement ladder:
Force F
+
velocity v
Fig 2.6 A branch with series connected spring and damperelements
Thevelocity across the terminals of the ladder is the sumof the relative velocities across the
spring and the damperelements (EQ. 2.7 and EQ. 2.8). Considering the through variable and the
across variable to be vectors of angular frequencym, the final relation is describedin EQ.2.11.
V = Vspring-[- [/’damper
EQ. 2.8
EQ. 2.9
=~- ~-(,~-)
=
F
z
ekBeco
/r e + coeBe
EQ.2.1o
Bkeco
e (oeB
ke +
EQ. 2.11
The real part, in phase with the displacementvector, is the spring componentand the imaginary part is the dampingcomponent. Uponcomparingthe imaginary and real parts of EQ. 2.11
10
with the force terms in EQ. 2.4 and EQ. 2.5 respectively, the values of the spring and the damper
elementsis obtainedfor a given pair of odd integers m, n [7].
61wP~
ert
k,.,, - (mn) 4
(~ + Z)
76~([W)3qehr
EQ. 2.12
EQ. 2.13
With the elements as given in the above equations, the lumpedparameter modelcan be represented as shownin Fig. 2.7. Eachseries branch represents a pair of spring and dampingforces for
a given value of m, n. The numberof such branches dependson the required accuracy. The truncation error can be restricted to under 1%with the use of six branches.
Force
Velocity
Fig. 2.7: The spring-damper lumpedparameter model
This model has been implementedin Verilog-A [8] for use with the node-based MEMS
simulation tool developed at Carnegie MellonUniversity [9, 10]. It has also been implementedin
MAST
to be used with a similar library in SABER
environment [11]. The model accepts the displacement across the terminals of the modelas lumpedparameter inputs and uses a voltage controlled voltage source to generate velocity by a differential operator. Thesumof the force through
all the parallel branches represents the total dampingforce. The modelis parameterizedand can be
used in design. The input parameters are length and width of the plate, gap (the thickness of the
film) and the ambient pressure. The effective viscosity is calculated using Knudsennumberwhich
in turn is evaluatedfor a given input ambientpressure.
11
Someequivalent circuit,
lumpedparameter modelsfor squeeze film dampinghave been pro-
posed in the past [3, 12]. There are a couple of advantagesof representing the modelin terms of
spring and damperelementsrather than circuit elementslike resistors, capacitors and inductors [3,
12]. Firstly, the mappingof the throughvariable (force) and the across variable (position) from
mechanicaldomainto the electrical domainis not canonical. So is the case with the elements used
in the lumpedparameter model. For example, the spring element can be modeledby an inductive
or a capacitive element while a dampingelement can be modeledby a resistive or a conductive
(both dissipative) element. Secondly, in an integrated MEMS
simulation environment, there are
different domains(e.g. electrical,
mechanicaland magnetic domains), each of whichhas its own
through and across variable. If an electrical equivalent circuit modelis used to modelmechanical
dampingelements, there arises a need to separate the current which represents the force of
mechanicaldomainfrom the current that represents the electrical through variable. Hence,ease of
use, the use of spring-dampermodelis helpful.
2.5 Nonidealities
2.5.1 Edge and finite-size
effects
The force equations EQ. 2.4 and EQ. 2.5 derived from the linearized Reynold’s equation do
not consider the edgeand the finite-size effects. Theedge-effects are significant at tens of microns
scaled structure as shownin section 2.3. To study the accuracy of the spring-damper lumped
parameter model based on EQ. 2.12 and EQ. 2.13, we compare the response of the behavioral
modelwith that of finite-element simulations, for varying squeezefilm size. Theerror in the peak
dampingforce predicted by the behavioral model as comparedto FEMsimulations for a plate
oscillating with an amplitudeof 0.1 ~tm and frequencyof 1 kHzis shownin the plot in Fig. 2.8.
Thedegradation in the accuracy of the modelat plate sizes in the range of few tens of microns is
noticeable. This is because the edge and the finite-size effects becomesignificant at these MEMSscaled plate sizes.
12
8O
,9.0
70
~
60
._~
~ o~"
50
N
2O
a_
0
40
~~p
20
40
60
Plate length (pro)
g: 2 p m
frequ.ency:1 kHz
htude:
O.1 #.tin
80
1 O0
Fig 2.8: Accuracydegradation of the modelfor small plate sizes
2.5.2 Edge-effect model
There can be several waysin whichthe edge and the finite-size effects can be modeled.Here
wedescribe a couple of approaches and outline our implementation.
Onetechnique to modelthe edge effects is the effective viscosity of the modelto compensate
for the underestimationof the dampingforce. This approachquantitatively increases the estimated
dampingforce, but does not provide a direct qualitative intuition to modelthe edge-effects. As
effective viscosity is already used to model the slip conditions encountered in narrow gaps, a
modelof~leff =f(slip, edge) will makeit difficult for designers to understandwhicheffect is dominant.
Anotherapproachcan be to use an effective pressure above the ambient pressure to take care
of the non-zero pressure at the plate edges. Pressure is one of the input parameters of the model
that is used in the evaluation of the Knudsennumberand the effective viscosity. Hencethis effect
is not distinct fromthe effective viscosity approach.
There is a need to modelthe edge effects using parameters that are decoupledfrom the other
parameters. The plate width and length are such independent parameters. Hence, increasing the
plate size to account for the non-zero gauge pressure at the plate edges is one straightforward
methodfor modelingthe edge effects. This solution is also physically intuitive becausethe differential pressure becomeszero only at a finite non-zero distance from the plate edge. So extending
13
gap: 2 ~tm
Oscillation amplitude: 0.1 ~tm
Oscillation frequency: 1 kHz
x~O
x
0.75
20~mplate
p(x=O) 0.5
lOOgmplate
p(x)
0.25
\
\
0|
~;
4
6
|
: 8L= 2.2 btn~,
|
Distance from plate edge (~m)
Fig. 2.9: Pressure distribution beyondthe plate edge
the plate edge and using trivial boundaryconditions at the extendededge provide a wayof obtaining morerealistic pressure distribution across the plate. Fig. 2.9 showsthe pressure distribution
and decay beyondthe plate edges whenthe top plate is movingsinusoidally with an amplitude of
0.1 btm and frequency of lkHz.
The non-zeropressure distribution along the original plate edge is accountedfor by extending
the plate edge by an amount6L and using TBCat the extendededge [7].
The value of 6L shownin the Fig. 2.9 illustrates
the physical intuition behind the solution
adopted. The quantitative equation for the plate extension is obtained by comparingthe peak
dampingforce of the original behavioral modelwith that from FEA.The parametersthat can affect
8L are identified as the gap, plate dimensions,frequencyof oscillation and the amplitudeof oscil-
14
2.5
2.0
1.5
8L/g
1.0
0.5
o.o
0
20
40
60
80
1 O0
Plate length (~tm)
Fig. 2.10: Linear fit with least sumof squarederrors for square plates oscillating with
frequency of lkHz and an amplitude of 0.1 gm
lation. Screening experiments were performedto find the dependencyof 6L on these factors by
varying these parameters one at a time. All experiments were performedfor square plates. The
plate size wasvaried from 5 btmto 100 gin, gap size from 1 ~tmto 5 gmand the frequencyup to 1
MHz.The amplitude of oscillation
was kept under 10%of the gap. For above mentioned plate
sizes and gaps, the value of 6L that gives the closest matchbetweenthe behavioral and finite element results was determined, gL has a first-order
dependency on the gap and a second-order
dependencyon the plate size. Fig. 2.10 shows the plot of ~L/g as a function of the plate size.
Larger plates cause morepressure perturbation at the plate edgeand the pressure settles to ambient
pressure further awayfrom the edge. Hence,larger plates require a higher value of 6L.
Thebest fit modelfor 6L is given by EQ. 2.14.
5L(g, L) = 9(0.8792 + 0. IL}
EQ. 2.14
whereL is the plate dimensionin microns, g is the gap and 6L is the extension.
Whenthe value of gap is decreased, wewouldexpect the squeezeforce to increase as the continuum squeeze action is proportional to I/g 3" This can lead to the conclusion that 6L should
increase whenthe gap is decreased. Onthe contrary, the equation showsthat fiL is directly proportional to the gap. To understandthis, weneed to consider the regime of operation. Wecan consider
15
the air to be within the continuum limits when the Knudsen number is << 1 (<0.01). The molecular
meanfree path of air at standard temperature and pressure is 0.07 btm. So for gaps < 7 ~tm, the continuum limits are no longer valid. Slip conditions begin to occur. In such conditions,
the edge
effects increase with increasing gap and hence 6L is directly proportional to gap.
2.5.3
Extended model
The accuracy of the model can be extended by modeling the edge and the finite
size effects.
The width w, and the length 1 of the squeeze film are increased by ~L to give the effective
width
weffand effective length leffas in EQ.2.15 and EQ. 2.16.
Wee= I¢ + ~)L(~,w)
EQ. 2.15
/~,r =/+fiLl’if, l}
EQ. 2.16
The values of the spring and the damper elements in the extended model is given by substituting weffand lefffor
w and l respectively in EQ. 2.12 and EQ. 2.13. Uponextending the plate dimen-
sions, the improved accuracy of the model is shownin the plot in Fig. 2.11. The accuracy is within
5%for plate sizes in the range of 5 ~tm to 100 ~m.
80
70
- ~ - Error : Withoutplate extension
60
+ Error : With plate extension
50
g: 2
frequency: 1 kHz
amplitude: 0.1 ~tm
40
30
20
10
0
-10
0
20
40
60
80
100
Ratelength
Fig. 2.11: Improved accuracy of the model with extended edges of square plates
16
2.6 Squeeze film simulation results
In this section, squeeze film simulation results using the extended lumpedparameter model
described in section 2.5 are discussed. Simulationresults for the input of three dominantharmonics of a square waveare presented. Behavioral and finite element simulations are comparedfor
varying aspect ratio of the plate, amplitudeof motionand frequencyof oscillation. A systemlevel
simulation of a resonator ban@assfilter having squeeze film dampingas the dominantsource of
energydissipation is presented. Simulationresults are validated experimentally.
2.6.1 Step function response
The squeeze film model has been derived under the assumptionof sinusoidal oscillations of
the top plate. To study the accuracyof the modelfor non-sinusoidal input displacement,behavioral
simulations with the first three harmonicsof a square waveare performedand the results compared to the finite elementsimulations. Fig 2.12 showsthe displacementof the top plate whichis
the sum of the three dominantharmonicsof a square waveof amplitude 0.1 ~tm and a frequency of
10 kHz. The force on an 100 btm square plate, with a gap of 2 ktmand 4 ~tm, as a function of time
is plotted in Fig. 2.13 (a) and Fig. 2.13 (b) respectively. Thedifference in peakforce is under 1%
the case of the 2 ~tmgap and 15%in the case of the 4 ~tmgap.
2.6.2 Aspect ratio dependency
Whilederiving the finite size and the edge-effect modeland determiningthe accuracy of the
modelin the section 2.5, square plates were considered. The effect of aspect ratio on squeezefilm
dampingis studied by keeping the area constant and changing the length to width ratio of the
squeeze film. Dampingforce on a plate of area 100009m2 is determined for aspect ratios in the
range of 1-10. As the aspect ratio increases, as expected, the squeezeaction decreases becauseair
underneaththe plate with higher length to width ratio can flow out without muchsqueeze action.
Fig. 2.14 showsthat the behavioral simulations matchthe finite elementsimulations to within 7%.
This validates the edge effect modelfor higher aspect ratios, even though the modelwas derived
with square plates.
17
0.15
z 0.1
~ 0.05
~ o
~ -0.05
~. -0.1
._~
"~ -0.15
0
0.0001
0.00005
0.00015
0.0002
0.00025
0.0003
time (s)
Fig 2.12: Displacementof the top plate with three dominantharmonicsof a square wave
Behavioralsimulation
g=2~tm
FE]Vlsimulation
2
-3
0.0000
0.0001
0.0001
0.0002
0.0002
0.0003
0.0003
Time (s)
Fig. 2.13 (a): Force on a 100 btmsquare plate with gap of 2 ~tm
0.5
0.4
0.3
~ 0.2
Z
~, 0.1
~ 0
¯ -0.1
o -0.2
I,I.
-0.3
-0.4
-0.5
Behavioralsimulation
FEM
simulation
0
0.00005
0.0001
0.00015
0.0002
0.00025
g =4 ~tm
0.0003
time (s)
Fig. 2.13 (b): Force on a 100 ~tm square plate with gap of 4 ~tm
2.6.3 Amplitude of motion
The damping model consists of non-linear spring and damper elements. Thus, the damping
force is not linearly proportional to the displacement. However,for displacementsthat are small
comparedto the gap (<10%),the modelbehaves close to a linear system.
18
g=2 ~tm
frequency = 10 kHz
Behavioral amplitude= O. 1 ~tm
1.2
1
sirnula!ion.
0.8
0.6
0.4
0.2
0
0
2
4
6
length/width
8
10
Fig. 2.14 (a): Ratio of force on rectangular plate to that on square plate for a -8 m2area
-2
-4
0
2
4
6
8
10
length/width
Fig. 2.14 (b): Error in peak force whencomparedto FEMsimulations
The peak dampingforce for a 10 kHz sinusoidal input of amplitude varying from 40 nmto
200 nmis shownin Fig. 2.15 (a) - Fig. 2.15 (d) for square plates of size 20 ~tmand 100 ~tm,
gaps of 2~tm,4 ~m. The error, as comparedto the numericalsimulations is within 10%for all the
four cases. Theplot showsthe linearity of the system decreases with increasing gap.
2.6.4
Frequency dependency
Frequencyof the plate oscillations is an important parameter in determining the squeeze
force. Whenthe plate oscillates with low frequency, the air flow is nearly incompressible. In such
cases, the dampingforce that is proportional to the velocity dominatesthe spring force. The total
squeezeforce is proportional to the frequencyfor a given amplitudeof motionand for fixed phys-
19
0.25
3
2.5
0.2
2
0.15
1.5
1
0.1
Behavioralsimulation
FEMsimulation
~
0.05
0.5
0
0
0.05
0.1
0.15
0.2
0.05
0.25
0.1
0.15
0.2
0.2~
Amplitude (~m)
Amplitude (#m_)
0
0
tu -8
-10
-6
0.05
0,1
0.15
0.2
0.05
0
0.25
0.1
0.15
0.2
0.25
Amplitude (~tm)
Amplitude (~m)
Fig. 2.15 (b): Force and error for a 20
square plate and 4 gmgap
Fig. 2.15 (a): Force and error for a 20 ~tmsquare
plate and 2 gm gap
1.6
1.4
1.2
1
~ Behavioral simulation
- - -~- -, FEM
simulation
0.25
0.2
o.a
0.15
0.6
0.4
0.2
0
0.1
0.05
0
0.05 0.1
0
0.15 0.2 0.25
0
Amplitude (~m)
0.05
0.1
0.15 0.2
Amplitude (l~m)
0.25
0
-1
-2
0 0.05
0.1 0.15 0.2
Amplitude(gm)
0.25
-10
0
0.05
0.1
0.15
0.2
Amplitude (~m)
,
0.25
Fig. 2.15 (c): Force and error for 100 gmsquare Fig 2.15 (d): Force and error for 100 p.m squar~
plate and 2 gmgap
plate and 4 ~tm gap
20
~ Damping force
..... *. Spring
force
0.5
0.4
~
o.3
*
......
{o.2
~.*
....e.......
0
5
10
15
Squeezenumber
Cut-off frequency
20
Fig 2.16: Spring and damping forces per unit displacement as a function of squeeze number
00000
0000
g=2
000
~,
1 O0
g=4
lO
gm
/-
1
0.1
100
,
1000
10000
Frequency
(Hz)
~ FEN simulation
- - -m-- - Behavioral
simulation
-- -m-- - FEM
simulation
~ Behavioral simulation
100000
,
1000000
Fig. 2.17: Peak force for different frequencies for a 100 gmsquare plate
1000000
100000
10000
1000
100
10
1
100
g
=
2
gm---,,.,,,.~~~~
~.~’~-~~"
~.,~~,~,.,..,~.~,~"~’~,~
g = 4 ,Ltm~
1000
10000
Frequency
(Hz)
~ FEMsirnulation
-- ..- Behavioral simulation
-.. ~.. - FEM
simulation
~ Behavioral simulation
100000
1000000
Fig, 2.18: Peak force for a 20 gmsquare plate
ical dimensions of the squeeze film. At high frequencies, the flow invoked becomes compressible.
As a result, the inertial effects of the fluid contribute to the damping. The relative contribution of
the spring and the dampingterms is plotted as a function of squeeze number(EQ. 2.7) in Fig. 2.16.
Cut-off frequency is defined as the frequency at which the spring and the damping terms contribute equally. At frequencies comparable to and higher than the cut-off frequency, the inertial
effects
21
are significant. As a result, the gauge pressure is not negligible comparedto the ambientpressure
contradicting one of the assumptionsin the squeezefilm model[1]. Hencethe error at the high frequencies can be significant. Fig. 2.17 plots the peak force as a function of frequencyfor the case of
a 100 ~tm square plate with gaps of 2 ~mand 4 ~m. For the gap of 2 ~tm, cut-off frequencyfor an
100 ~msquare plate is 670 kHz. FromFEMsimulations, the peak gauge pressure at 1 MHzis as
high as 12 kPa. This is about 12%of the ambient pressure (0.1 MPa). The assumption of small
gaugepressure fails resulting in the high error noticeable at frequencies close to 1 MHz.Theerror
in peak dampingforce for a gap of 2 ~tm is about 50%while it is 29%for a gap of 4~tm. This can
be attributed to decrease in the squeezenumberwith the increase in gap. Fig. 2.18 plots the peak
force for a 20 ~tm square plate. Accuracyis within 10%in this case as cut-off frequencyis 16 MHz
for g =2 ~tm.
22
3. Lateral viscous damping
3.1 Introduction
Viscous lateral damping,also knownas slide film damping,occurs whentwo parallel plates
are in relative tangential motion.Therelative motionof the parallel structures invokesa fluid flow
that exerts a force opposingthe relative motion. The force exerted by this mechanismdependson
the velocity gradient of the fluid near the plane of the oscillating structure, the viscosity of the
medium
and the area of the oscillating structure. Thevelocity gradient of the fluid in turn depends
on the velocity of the oscillating plate, andits distancefromother parallel structures that are in relative motion. Hencea slide film betweentwo parallel plates is characterized by the length and the
widthof the plates, the distance betweenthem,their relative velocity and the viscosity of the film.
Viscous dampingis a dominantsource of energy dissipation in laterally-driven microstructures. A lateral resonator is a common
example. Theproofmassoscillating parallel to a substrate
experiencesviscous drag by fluid both on the top and the bottomsurfaces. In lateral combdrives,
lateral dampingcan be a dominant source of damping. There are four different lateral damping
sources in a lateral combdrive. Betweenthe stator and the rotor fingers, there are two slide films
associated with each rotor finger (one on each side of the rotor finger) as in Fig. 3.1 (a). These
films are characterizedby the overlaplength of the stator and the rotor fingers, the thickness of the
fingers and the equilibrium distance betweenthe fingers. There is additional dampingfrom the
fluid aboveand belowthe rotor finger. This dampingsource acts on the entire rotor area as shown
in Fig. 3.1 (b). Thebottomfilm is characterized by the distance fromthe substrate and the top film
can be infinitely thick if exposedto atmosphere.
.[]
~
B
[]
Slide film
Rotor finger
Rotor truss
Stator
¢~ Substrate
(a) Topview showingthe slide films
betweenthe stator and rotor fingers
(b) Cross section showingthe slide films
on the top and bottomof the rotor
Fig. 3.1: Slide films in a lateral combdrive
23
3.2 Slide
film model
3.2.1 Model for continuum conditions
The oscillations of a planar structure in a gaseous mediuminvoke flow in the surrounding
medium.This section describes the theoretical aspects and the waythe invoked flow is modeled.
Consider a plate in the X-Yplane oscillating in the X-direction. The dynamicsof the gas flow
invoked can be modeledby the one-dimensionaldiffusion equation (EQ. 3.1) [13, 14].
OV(Z) OZv(z)
-~t
~
Ot
Oz
EQ. 3.1
where~t is the kinematicviscosity of the gas and v(z) is the gas velocity component
in the x direction. Thekinematicviscosity is the ratio ~l/P wherevl is the viscosity coefficient or the dynamic
viscosity of the gas and 9 is the density of the gas..
Oscillating plate, v=vo ] Displacement
~ement
Oscillating plate,
v=v
0
~
L__~
Substrate
Substrate
Fig. 3.2: Slide film betweena planar oscillating structure and the substrate
For a fully established flow without any transient consideration, the flow can be consideredto
Couetteflow with a linear velocity profile as in Fig. 3.2 (a). Thereare no inertial effects in this
steady state (established) flow. If the continuumlimits are considered,the velocity of the gas at the
plate surface is equal to the velocity of the plate. Underthese conditions, the dampingcoefficient
is givenby [13, 14]. :
B : ~]/i
EQ. 3.2
whereAis the area of the movingplate and g is the thickness of the slide film.
Couette dampingmodel does not take inertial effects into consideration and hence can be
inaccurate at high frequencies. The frequency dependentStokes flow modeltakes into account the
inertial effects. For a plate velocity of v0cos(c0t), the velocity profile in Stokes flow decaysexpo-
24
nentially as shownin Fig. 3.2 (b) and the phase shift varies linearly as we moveawayfrom the
plate. Undercontinuumassumptions, the gas velocity profile and the dampingare given in EQ. 3.3
and EQ. 3.4.
(q9 - qz)
v(z) = o sinh
sinh (qg)
~ = ~IAqcoth(q9)
whereq is the complexfrequencyvariable given by (j03/~t) 1/2, 03 is the oscillation frequencyand g
is the kinetic viscosity and ~ is the complexdampingcoefficient. A complexdampingcoefficient
meansthat there is a dampingforce (the real part) and there is a spring force in phasewith the displacement(the imaginary part). As the frequency tends to zero, the Stokes dampingmodel
approachesthe Couette dampingmodel. The amplitudeof the velocity profile decays by a factor of
1/2.
e over a distance equal to the penetration depth 6 given by (03/(2g))
3.2.2 Gasrarefaction effects
Since the critical film dimension,the gap betweenthe two plates, is in the range of microns,
the meanfree path is comparableto the gap. This causes the gas rarefaction effect, whichcan be
modeledwith an effective viscosity. EQ.3.5 gives the value of effective kinematic viscosity, geff
and effective dynamicviscosity, ~leff [14].
#
’u~tt 1 + 2K.
r]~, -
EQ. 3.5
1+
18
15
0
2
4
6
Gap
(,u,m)
8
Fig. 3.3: Effective kinematicviscosity as a function of gap
25
whereKn is the Knudsennumberwhichis the ratio of the meanfree path of the gas molecules to
the distance betweenthe plates. Fig. 3.3 plots the effective kinematicviscosity of air at standard
pressure and temperature, as a function of the plate distance.
The gas velocity at the plate surface is not the sameas the plate velocity due to gas rarefaction. Slip conditions occur and the velocities at the plate and substrate surfaces, whichare the
boundaryconditions for EQ. 3.1, are given in EQ. 3.6 and EQ. 3.7 [14].
v{O) = v o +)~z=O
EQ. 3.6
V(g) = -L 0v(z)
Oz e=g
EQ. 3.7
wherev0 is the plate velocity, )~ is the molecularmeanfree path of air and g is the gapbetweenthe
plate and the substrate.
3.2.3 Solution for the velocity profile with gas rarefaction
For sinusoidal oscillations of the plate, the solution to EQ.3.1 with boundaryconditions as in
EQ. 3.6 and EQ. 3.7 is given as [14]:
v(z)-Vo
sinh (qg - qz) + qLcosh(qg -
EQ. 3.8
Fromthe velocity profile, the dampingcoefficient ~ can be extracted as in EQ. 3.8.
Or(Z).
COSh (qg) + q)~sinh (qg)
v o 0z z=0=~lefeAq (l+qZ)~Z)sinh(qg)+Zq)~cosh(qg)
qeffj
EQ. 3.9
where~1 is the viscosity coefficient (also called dynamicviscosity), Ais the area of the plate surface. Gasrarefaction effects are taken into accountby using geff and ~leff for g and rl respectively
as in equation3.5.
As the frequency of oscillation tends to zero, the dampingforce tends to that of Couette
dampingforce. For very high frequencies, this modeltends to Couette dampingmodel with a gap
equal to the molecularmeanfree path of air (~).
26
Amplitude: 0.1 ~tm
80.00
60.00
uency" 1 kHz
40.00
20.00
0.00
-20.00
-40.00
-60.00
-80.00
0
1
1
2
2
Time(0.1 ms)
3
3
Fig. 3.4: Lateral dampingforce on a sinusoidally oscillating plate
3.3 Finite element simulation of slide films
FEMsimulations to evaluate the lateral dampingforce on a plate oscillating laterally above
an infinite substrate were performed.To verify the finite elementsimulations being performed,the
force on a square plate of size lmmon a side, oscillating with an amplitudeof 0.1 ~tm and a frequencyof 10 ld-Iz was determined. Fig 3.4 showsthe dampingforce with time for a gap of 2 ~tm.
The dampingcoefficient can be extracted from the peak force. Since the gap is small comparedto
the penetration depth at 10 kHz, Couette modelwith linear air velocity profile betweenthe plate
and the substrate can be used to evaluate the dampingcoefficient as in EQ.3.10 [14].
B -- rl~ttA
EQ.3. i0
whereg is the gap betweenthe plates, Ais the area of the oscillating plate and "qeffis the effective
dynamicviscosity of air.
ambient pressure
displacement
gap (gJ
length (l)
width/2(w/2
symmetry
Fig. 3.5: Trivial boundaryconditions in finite elementsimulation of slide film damping
27
The values of dampingcoefficient obtained from the FEAmethodand the analytical method
are 9.5 ~tNs/mand 9.84 ~tNs/mrespectively. They matchto within 4%whichverifies and quantifies the accuracyof the finite simulations.
As in the case of squeezefilm, the type of boundaryconditions used affect the accuracyof the
finite simulations. Theuse of trivial and non-trivial boundaryconditions in the simulationsof lateral dampingand the advantages of each approachare discussed next.
3.3.1 Trivial boundary conditions
In trivial boundaryconditions (TBC),the slide film is represented by a 3-D structured block
enclosed betweentwo wall surfaces and the edges of the wall surfaces are fixed at ambient pressure. Therepresentation does not consider the finite non-zerothickness of the plate. To exploit the
symmetry
conditions for efficient simulation, one half of the plate is used as shownin Fig. 3.5.
3.3.2 Non-trivial boundary conditions
The use of non-trivial boundary conditions (NTBC)provide the solver with the freedom
evaluating the pressure and hence the air flow around the oscillating structure. The oscillating
plate is surroundedby blocks of air on all directions and the ambientpressure boundaryconditions
Substrate
Slide Film
Oscillating Plate
Sym~metry
Fig. 3.6: Use of non-trivial boundaryconditions in slide film damping
28
are set at the extended boundaries to modelmore realistic conditions. Fig. 3.6 showsthe system
used for the finite element simulations with NTBC
3.3.3 Trivial boundary conditions vs. non-trivial boundaryconditions
The following table provides the lateral dampingforce predicted by the use of NTBCand
TBCwhena plate of varying size oscillates with an amplitude of 0.1 btm and frequencyof 10 kHz,
2 gmabovethe substrate. It can be observedfromthe table that the difference in force predicted by
TBC,NTBC
and the Couette dampingmodel decreases as the plate size increases. The dominance
of the edge and the finite-size effects for sizes less than 100 ~tmon a side has beenexperimentally
observedby Zhanget al. [15].
Plate Size
Theoretical
TBC
NTBC
20.065 txN/m
20.0597 gN/m
20.0727 gN/m
100urn
20.0597 btN/m
20.0597 laN/m
lmm
20.0597 gN/m
20.0597 gN/m
20.060 ~tN/m
(Couette damping)
20um
20.064 gN/m
As in the case of squeeze film damping,the force is underestimatedwhenTBCare used. This
is mitigated by the use of NTBC,but at the expense of additional computerresources. The use of
NTBC
results in the increase in the numberof grid cells from 100 to 4000 as a result of which, a
typical transient simulation time increases from 4 minutes in TBCto 45 minutes in NTBC
on a
360 MHzSun Ultra-Sparc machine. The memoryrequirements of the simulation is 5 MBin TBC
and 45 MBin NTBC.Non-trivial boundary conditions are used for all the simulations used to
characterize behavioral model.
3.4 Lumped parameter model
The numerator and the denominatorof the complexdampingcoefficient (EQ. 3.9) can be represented by a Taylor series in variable q about the dc value of q (q=0) as in EQ.3.11. Theratio has
29
only even powersof q in both the numerator and the denominator whichmakesit a ratio of complex powerseries in angular frequency.
i~ nlq
i=O
evol7
EQ. 3.11
i=O
eveB
whereni are the evencoefficients in the numeratorand di are those in the dominatorand are functions of the gap (g) and the meanfree path 0~) of air molecules.Theequations for the coefficients
are given in the appendix.
Thecondition for the convergenceof each series can be set by selecting the ratio of the two
successive terms of the real and imaginaryparts in each series to be less than unity. Truncation
error can be controlled by changingthe numberof terms in the series and the ratio of successive
terms. The minimum
ratio of the absolute values of successive terms in the real and the imaginary
parts of the numeratorand the denominatorof EQ. 3.11 is g4~o2/24~t2.The ratio g4o32/24~t2is kept
under 0.1 to truncate the series after first three terms. As an upper boundon the operation fre-1.
quencyfor a given g and g. For examplefor g = 2 ~tmand g = 17.9/,tNms/kg,0~ < 6.9 x 106 s
vel_a
disp_a
disp_b
~
)Pos(vel_a,vel_b)
d(Pos(disp_a,disp_.b)
dt
Fig. 3.7: Lateral dampingmodelin Verilog-A
The transfer function described is implementedin Verilog-Aas a Laplace transform. The one
dimensionalslide film modelhas two pins indicating the displacementof the oscillating plate and
the substrate. This is convertedinto relative velocity using a time derivative operator. This relative
velocity is used to computethe dampingin the frequency domainand inverse Laplace transform
back to dampingforce between the plate and the substrate. The input parameters of the model
30
2.4
Amplitude: 0.1 ~tm
Frequency: 10 kHz
Gap: 2 ~tm
Width: 100
2.2
2
1.8
Displacement
1.6
1.4
1.2
1
0
20
40
60
80
1 O0
Length=2/.tm to 100gm
Length(microns)
Fig. 3.8: Ratio of peak dampingforce for varying length
include the length and the width of the plate, the gap and the ambient pressure. Fig. 3.7 shows the
circuit implementation of the model.
3.4.1
Edge effects
in the lumped parameter model
The damping force given by EQ. 3.9 does not take into account the edge and the finite
size
effects which are dominant at micron-scale devices, This observation was made experimentally by
Zhang. et al. and it was suggested that the dampingforce be multiplied by an empirical correction
factor of about 2 to 3 [15]. Since the motion of the plate is in the plane of the structure, the edge
effects can show different
dependencies on the width and the length. To study this,
we keep one
dimension constant while varying the other. Let the dimension of the edge along the direction of
motion be 1 (length) and that of the perpendicular edge be w (width). The length and the width
varied one at a time while keeping the other fixed at 100 ~m to quantify the difference in damping
force obtained from behavioral and finite elements simulations.
Fig.3.8 plots the ratio of peak forces from lumped parameter and FEMsimulations for lengths
varying from 2 gm to 100 ~tm. For/=2 gm, the force estimated by behavioral simulation using the
model represented by EQ. 3.9 is less by a factor of 2.3 than the force from FEMsimulations, given
the edge effects contribution due to finite width is negligible. This is in agreement with the empirical factor used by Zhanget al. Peak force is a sufficient metric at the frequencies in the range of
31
1.35
Amplitude: 0.1 gm
Frequency: 10 kHz
Gap: 2 gm
Length: 100 ~m
1.3
1.25
Displacement
1.2
1.15
1.1
1.05
0
Length=100ktm
20
40
60
80
1 O0
Width (gm)
Fig. 3.9: Ratio of peak dampingforce for varying width
10’s of kHzas the dampingfore dominatesover the spring force and the associated phase differ°.
enceis less than I
Fig. 3.9 showsthe ratio of peak forces for varying width. The ratio of the forces from behavioral and FEMsimulations is about 1.33 for a 2 ~tm width. This showsthat the edge effects depend
on the aspect ratio as well as the orientation of the plate.
Thecontribution of the edgeeffects is established qualitatively. For quantitative analysis and
derivation of the edge effect model, further investigation of the dependencieson gap and the coupled effect of the length and width is required and is left as future work.Further simulationsof the
lateral dampingmodelwill be presented with a fixed plate extension of 4 btm on each side. This
translates into a multiplication factor of about 1.08 for a 100 ~tmsquareplate whichis close to the
ratio shownin Fig. 3.8 and 3.9.
3.4.2 Step function response
The dampingmodel used in the derivation of the lumpedparameter model was derived for
sinusoidal input. To study the response to non-sinusoidal input, a sumof the first three dominant
harmonicsof a square waveof amplitude 0.1 gmwas used as input. Fig. 3.10 are plots of the lateral displacement and force on a 100 ktm square, oscillating plate. The plate dimensions were
32
0.15
~- 0.1
£
.__.
g 0.05
o
8 -o.o5
._~ -0.1
-0.15
0
20
40
60
80
1 oo
120
140
160
180
200
Fig. 3.10 (a): Square wave displacement of the top plate
~ FEMsimulation
- -~ o-. Behavbral
simulatbn
0
20
40
60
80
1 O0
120
140
160
180
200
extended by 4 gmfor which the damping force gets multiplied by a factor of 1.08, which is close
to the edge effect ratio in Fig. 3.8 and Fig. 3.9.
3.4.3
Frequency dependency
The lateral
damping force is highly frequency dependent. At low frequencies,
the plate surface can respond to the plate oscillations.
force and the velocity is zero. Inertial
nent. As the frequency increases, inertial
the fluid near
In this case, the phase difference between the
effects are negligible and hence there is no spring compoeffects come into play and the force on the oscillating
plate has both a damping and a spring component. The total
force can be analyzed by comparing
the amplitude and phase of the force from the behavioral and the FEMsimulations.
Fig. 3.11 (a)
showsthe plot of magnitude
as a functionof frequencyon a 100gmplate. It canbe seen that the
peak force is almost linear with frequency, at least until 1MHz,after which the amplitude increases
more rapidly with increasing frequency. The phase difference between the force and the velocity is
plotted in Fig. 3.11 (b). The phase is close to zero for frequencies up to 10 kHz. Phase increases
33
linearly
of finite
with frequency.
The phase response
element model for frequencies
100
of the behavioral
model matches to within 2 ° to that
up to 1 MHz,
Gap: 2 ~tm
Oscillation
amplitude:
0.1
gm
_~ev-
10
0.1
I’"
0.01
1000
10000
Fig. 3.11 (a):
Frequency(Hz)
"~" Behavioral
100000
1000000
Force amplitude on a 100 ~tm square plate
3O
25
~ 20
2
[- ’
-
-
"-
-’--’~-
~ehEMavioral~
~,~ 15
~ 10
5
0
1000
10000
100000
1000000
Frequency(Hz)
Fig. 3.11 (b):
Phase of damping force with respect
to velocity
34
4. System level simulation and experimental verification
This chapter provides the system-level simulation of a bandpass filter and a folded-flexure
microresonator. The simulation is done using the NODAS
cell library developed at Carnegie Mellon University [9, 10]. The bandpass filter has squeeze film as dominantdampingsource and the
lateral resonator has slide film dampingas the dominantsource.
4.1 CMOS
bandpass resonator filter
To emphasize the use of the behavioral squeeze film model in capturing the dynamicsof
dampingaccurately, results of a system level simulation of a bandpassresonator filter are presented and experimentally validated.
Fig. 4.1 shows the schematic diagram of a 550 kHz CMOS
micromechanical bandpass filter
[16]. The filter is designedusing the NODAS
parameterizedcell library. Thefilter system is composed of three resonators coupled by "O" springs and is integrated with the associated electrical
interface circuitry. Eachresonator is designed for a 550 kHzresonance frequency. The system has
three resonant peaks scattered aroundthe natural frequencyof 550 kHz. Thefilter consists of three
pairs of differential, electrostatic combdrives used for actuation and sensing. The differential
combdrives have associated squeeze film dampingwhichis the dominant source of energy dissipation in the combdrives as well as in the resonator filter. Thesqueezefilm betweenthe stator and
the rotor fingers has a length of 37 ~m,a width of 4.8 ~tm and a thickness of 1.8 ~tm. The number
of such squeezefilms in each combdrive is twice the numberof rotor fingers (one on each side of
the rotor finger). Thelateral dampingmodelused in the simulation is a Couette flow with the gap
assumedequal to the penetration depth at 550 kHz.
The output voltage obtained at the sensing combdrive is buffered by the electronics. Fig 4.2
showsthe output voltage measuredat the output of the interface circuitry. Withthe use of the
dampingmodel with edge effect enhancements, the output voltage at the resonant frequencies
matches the experimental measurementsto within 0.6 dB, whichis equivalent to 7%error in the
gain. The results obtained with the use of the behavioral modelwithout any edge extensions differ
35
~g spring
Proof mass
Differential
(a):CMOSresonator
Driving
resonator
Coupling
resonator
Coupling
spring
Sensing
resonator
(b): Bandpassresonator filter schematic
Fig. 4.1" NODAS
schematic of the bandpass resonator filter
I
I
I
I
I
I
I
l
-40
Vout
(dB)
-60
Experimental
Simulation: With plate extension
Simulation: Withoutplate extension
-80
I
460
480
500
520
540
560
Frequency (kHz)
Fig 4.2: ACtransfer characteristics of the bandpassfilter
58O
600
36
Folded flexure
Shuttle mass
Anchor points
Fig. 4.3. Layout view of the lateral folded-flexure comb-drivemicroresonator
by 8 dB, a gain error of over 150%.The
lateral dampingassociated with the filter is smaller than
the squeeze dampingby over an order of magnitude.
4.2 Lateral microresonator
In a lateral microresonator, the lateral slide film dampingcan be a dominantenergydissipation mechanism.Fig. 4.2 showsthe layout view of a folded-flexure lateral resonator [17]. Thesystem can be modeledby a mass-damper-springsystem characterized by a resonant frequency and
quality factor.
Theslide films associated with the all the movingcomponentslike the combdrive, the beams
and the shuttle masshave associated slide film dampingcontributing to energy dissipation in the
system. Different slide films in the combdrive include the films betweenthe stator and the rotor
fingers and betweenthe rotor and the substrate. A detailed description of these is providedin section 3.1. At frequencies less than 100 kHz, the dampingforce on the bottomsurface of the microstructure dominates as the distance between the substrate and the structure (2 ~m)is muchless
than the penetration depth. At frequencies close to or greater than 100 kHz, the penetration depth
of air becomesless than 10 ~m. Undersuch conditions, the dampingforce on the top surface contributes significantly to the overall damping.
37
-140
-145
-150
-155
-160
0
2000
4000
6000
Frequency(Hz)
8000
10000
Fig. 4.4: Frequencyresponse of a 7.4 kHzmicroresonator
As mentioned in chapter 3, a constant edge effect model has been used in which all the
dimensionsare extendedby 4 ~tm. Fig. 4.4 showsthe plot of resonator displacementas a function
of frequency. The dampingassociated with the top surface is not included as the penetration depth
at 10 kHzfrequency is about 26 #m. This makesthe dampingassociated with the top surface over
an order of magnitudesmaller than the dampingon the bottomsurface. This can contribute significantly for frequencies close to 1 MHz.Fig. 4.4 shows the frequency response of a 7.4 kHz
microresonator obtained from NODAS
simulation. Table 4.1 shows simulated and experimental
values of resonant frequencyand quality factor for this resonator. Simulationresults are in good
agreement with the experimental results.
The simulated resonant frequency matches to within
1.3 %and the simulated Q factor matchesto within 8.5%of the experimentalvalues. Table 4.1 also
showsthe results of microresonators designed for 30 kHzand 100 kHzresonant frequency. It can
be seen that the error in the Q value increases with increasing frequency. This can be attributed
mainly to two reasons. The constant edge extension model is an approximatemodel for the edge
effects. It’s accuracyneeds to be verified for frequencies close to 100 kHz. Thedampingon the top
surface (exposedto atmosphere)that is not taken care of in the schematiccontributes to the overall
damping.
Table 4.1" Resonantfrequency(fr) and quality factor (Q) for different microresonators
designfrequency
fr (experiment
Q(experiment)
fr (NODAS)
7.31 kHz
Q (NODAS)
10 kHz
7.4 kHz
8.3
9
30 kHz
22.7 kHz
25.6
21.9 khz
31.6
100 kHz
74.1 kHz
80
72.6
130
38
5. Conclusions and future work
This thesis describes behavioral models for squeeze film and slide film dampingin MEMS
devices. The models are parameterized and can be used in mixed domainsystem level simulation
of MEMS.
It is shownthat the edge-effects contribute significantly to the overall dampingfor
plate sizes <50gmon a side. Extensionof the plate edges, as one of the possible waysto modelthe
edge-effects, is discussed. Theedge effect modelis derived for the case of squeezefilm for gaps in
the range of 1 gmto 5 ~tm, plate sizes in the range of 5 gmto 100 gm. For slide films, it is
shownthat the edgeeffects are dependenton both the aspect ratio and the orientation of the slide
film. Gas rarefaction effects have been modeledusing an effective viscosity model.
To showthe utility of the modelsin modelingthe dampingdynamics,results of system level
simulations have been presented and experimentally validated. With these models the designer
will be able to modelthe squeezefilm and the lateral (slide film) dampingwith adequate accuracy.
There are several directions for future work. The presence of holes in the microstructure significantly alters the dampingforce on the oscillating structure. Theeffect of the hole size and the
hole density on the dampingforce needs to be studied. A constant edge extension modelhas been
used to modelthe edge-effects in lateral damping.The gap and plate-size dependentedge effect
modelderivation can extend the accuracy of the edge-effect modelto wider range of input parameters. The effect of the thickness of the oscillating plate on the squeezefilm and lateral damping
needs to be assessed. The study of the effect of the presence of other static or movingstructures
near a squeeze film or a slide film can be useful in understandingthe dampingin case of complex
geometries.
39
References
[1] J. J. Blech, "OnIsothermal SqueezeFilms", Journal of Lubrication Technology,V. 105, 1983,
pp. 615-620.
[2] J. B. Starr, "Squeeze Film Dampingin solid State Accelerometer", Tech Digest, IEEESolid
State Sensor and Actuator Workshop,Hilton HeadIsland, SC, USA,June 1990, pp.44-47.
[3] T. Veijola, H. Kuisma, J. Lahdenpera, and T. Ryhenen, "Equivalent-Circuit Modelof the
SqueezedGas Film in a Silicon Accelerator", Sensors and Actuators A, vol. 48, 1995, pp.239248.
[4[ CFD-VIEW,CFD-GEOM,
CFD-GUIUser Manuals, CFDRC,Huntsville,
AL.
[5] M. Torowski, Z. Chen, A. prezekwas, "High Fidelity and Behavioral Simulation of Air Damping in MEMS",
Prec. MSM’99, pp. 241-244.
[6] E. S. Kim, Y. H. Cho, and M. U. Kim, "Effect of holes and edges on the squeeze film damping
of perforated mico-mechancialstructures", IEEEInt. Conf MEMS,1999, pp. 206-301.
[7] S. Vemuri, G. K. Fedder, T. Mukherjee, "Low-OrderSqueeze Film Modelof Simulation of
MEMS
Devices", Prec. MSM,2000.
[8] Verilog-A reference Manual.
[9] O. E. Vandemeer,"Nodal Design of Actuators and Sensors (NODAS)",M.S. Thesis,
CarnegieMellon University, 1998.
[10] C~ K. Fedder, Q. Jing, "A Hierarchical Circuit-level Design Methodologyfor Micormechanical Systems", Trans. Circuits and Systems II, TCAS1999.
[ 11] MAST
Reference Manual, Release 4.2, AnalogyInc., Beaverton, OR, 1997.
[12] T. Veijola, T. Ryh~inen,H. Kuisma,and J. Lahdenpera,"Circuit Simulation Modelof
Gas
Dampingin Microstructures With Non-Trivial Geometries," Transducers’95 - EurosensorsIX,
Stockholm, June 1995.
[13] Y. H. Cho, B. M. Kwak,A. E Pisano, and R. T. Howe,"Slide film dampingin laterally driven
microstructures", Sensors and Actuators A, vol. 40, 1994, pp. 31-39.
[14] T. Veijola, "CompactDampingModels for Lateral Strucutres Including Gas Rarefaction
Effects", Prec. MSM
, 2000.
[15] X. Zhang, and W.C. Tang, "Viscous Air Dampingin laterally Driven Microstructures", Sensors and Materials, vol. 7, No. 6, 1995, pp. 415-430.
[16] Q. Jing, H. Hue, T. Mukherjee, L. R. Carley, and G K. Fedder, "CMOS
Micromechanical
Bandpass Filter Design using a hierarchical MEMS
Circuit Library", IEEElnt. Conf MEMS,
2000.
[17] T. Mukherjee,S. Iyer, G. K. Fedder, "Optimization-basedsynthesis of microresonators", Sensors and Actuators A 70, 1998, pp. 118-127.
4O
Appendix
The equation for slide film dampingcoefficient including the gas rarefaction effects is:
cosh(q9) + qLsinh(q9)
~ _ qeffA av(z)
v ~z z=o = qeeeAq (1 +q L )sinh(qg)+
2q)~cosh(qg)
The numerator and the denominator are odd functions in the complex frequency variable q
and hence have only odd powers in their taylor series expansion in q about q=O.
Let the numerator be N(q) and the dominator be D(q). The taylor series expansion
N(q)/
(IleffA) is:
q+@(g2/2+g~,)+q5(g4/24+g3~,/6)+q7(g6/720+gSL/120)+q9(g8/40320+g7~/5040)+q
3628800+g9~,/362880)+O[q13].
Similarly the denominator D(q) is given by:
2 ~+g~, 2)+q5 (g 5/120+g 4~,/12+g 32
q(g+2~,)+q 3(g 3/6+g
~, /6)+q 7(g7 /5040+g 6~,/360+g 52
120)+qg(g9/362880+g8~/20160+g7~,2/5040)+q
1 ~ (g ~ ~/399 ! 6800+g~ 0~,/1814400+g9~,2/
362880)+0[q13].
The ratio of the two series has only even powers ofq in the numerator and denominator as in:
i~ niq
-~_ OVOI’I
i=O
TheCoefficients ni and di are as in the abovementionedseries.
41