Nonlinear Electromechanical Behavior of an Electrostatic Microrelay
M.-A. Grktillat
3,
I),
Y .-.J Yang 2), E. S. Hung 2), V. Rabinovich 2), G. K. Ananthasuresh
N.F. de Rooij I ) and S. D. Senturia
4A3.02
3,
I ) Institute of Microtechnology, University of Neuchikl,
Jaquet-Droz 1, CH-2007 Neuchgtel, Switzerland, marc.gretillat@imt.unine.ch
Microsystems Technology Laboratories, Massachusetts Institute of Technology, Cambridge, MA, USA
Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA, USA
SUMMARY
In this paper experiments on the quasistatic and
dynamic behavior of the electrostatic polysilicon
microrelay fabricated at IMT are presented together with
new nonlinear dynamical models. The models simulate
the pull-in voltage, the resonance frequency, the quality
factor, the switching time and the electrostatic spring
softening of the microrelay.
Simulation of the pull-in voltage has been done with
MIT's MEMCAD system, the resonance frequency has
been simulated with AbaqusTM, while macro-models
developed at MIT have been used to simulate the
dynamics of the nonlinear actuator in the presence of
squeeze-film damping
Keywords: electrostatic actuator, modeling, squeeze-film
damping.
INTRODUCTION
This paper reports the simulation of the nonlinear
electromechanical behavior of an electrostatically
actuated microbeam, fabricated as the moving part of a
polysilicon microrelay. The electrical performance of the
relay has been reported elsewhere [1,2]. This paper
emphasis a comparison between the basic measured
properties (switching voltage, resonant behavior and
switching speed) and corresponding quasi-static and
dynamic models. The combination of different simulation
tools (MEMCAD [3], AbaqusTM and MIT's macromodels [4,5,6,7]) offer good insights into the behavior of
the polysilicon microrelay, including the squeeze-film
effects due to the air under the beam.
circuit consists of two polysilicon layers: the first one is
on the substrate (1" polysilicon) and the second one is
under the beam (2ndpolysilicon). Figure 2 shows a SEM
of a 350 pm long microrelay. More details about the
realisation and first characterization of these microrelays
are given in [ 1,2].
PULL-IN VOLTAGE
The pull-in voltage is measured with a reverse bias
on the p-n junction, and a steady bias between the
polysilicon beam and the p+ substrate (-33V for the 300
pm and -27 V for the 3 5 0 pm long relays). In the presence
of this bias, the potential on the counter electrode (ndoped region) is increased until the device pulls in. The
result is 48.7 V for a 300 pm long relay and 42.8 V for a
3 5 0 pm relay.
Simulation of the pull-in voltage has been done with
MIT's MEMCAD system (Figure 3) [ 3 ] . Because of the
p-n junction bias and because the polysilicon of the
working circuit under the beam is at the same potential as
the substrate and hence screens the field of the upper
polysilicon electrode in that region, pull-in was modeled
with different potentials in the different regions of the
structure.
I
p-type silicon substrate
MECHANICAL DEVICE
p-type region
The mechanical part of the microrelay consists of a
polysiliconi silicon-nitridel polysilicon microbridge
realized by sacrificial-layer technology. Figure 1 shows a
schematic cross section of the microrelay. The actuation
circuit is formed between the polysilicon layer of the
microbridge (3rdpolysilicon electrode) and the n-doped
region in the substrate (counter electrode). The working
n-type region
aluminum
0-7803-3829-4/97/$10.000 1997 IEEE
LPCVD Si-nitride
I
CVD Si-dioxide &
aLPCVD
Si-nitride
EEI
1st & 2nd P-doped
LPCVD poly-Si
3rd P-doped
LPCVD poly-si
Figure 1: Schematic cross section of the relay
1141
TRANSDUCERS '97
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Chicago, June 16-19, 1997
4A3.02
Figure 3: View of the model realized with MEMCAD
for a 350 p m long microrelay (36 V have been
applied on the counter electrode). Note the dflerent
parts under the beam which correspond to the
driving electrode in the substrate and the 2nd
polysilicon under the beam.
I
Figure 2: SEM of a 350 p m long microrelay.
An equivalent Young's modulus (Y) of 200 GPa was
used to simulate the multilayer beam. A volume average
of the different materials was used to estimate this
equivalent Young's modulus from values for the nitride
and polysilicon Young's modulus [8]. For the beam ends,
both stepped-up and fully clamped boundary conditions
(BCs) were investigated. Figure 4 shows the importance
of such investigation in the modeling of free standing
beams for the case of a residual stress 0=130 MPa. The
effect of the boundary conditions is more than 10 YOon
the pull-in voltage.
Using the stepped-up support, the residual stress
value was adjusted to 145 Mpa to match the pull-in
voltage as well as the resonance frequency of a 300 pm
long device. Figure 5 shows the simulated displacement
of two microrelays of different lengths for different
applied voltages. These curves are given by MEMCAD,
which also gives the shape of the microrelay for each
applied voltage (Figure 6). The simulated pull-in voltages
are 48.5 V for the 300 pm relays and 41 V for the 350 pm
relay which agree with experiment within 4.5%. A
summary of these simulation results is given in Table 1.
Using the same mechanical properties as in the pullin voltage simulation, a small amplitude modal analysis
was performed with AbaqusTM to get the resonance
frequencies as well as the shapes of the five first modes.
An error smaller than 5% has been obtained for the first
resonance frequency. The five first modes are shown in
Figure 7.
DYNAMIC MODELING
The dynamic behavior of these devices has also been
simulated, but because of various approximations, the
dynamic models are less accurate than the quasi-static
models used for the pull-in and resonance frequency.
TRANSDUCERS '97
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Chicago, June 16-19, 1997
Stepped-up beam
+
2:
-Fully
clamped beam1
. . . . . . . . . . . . . . . . . . . .
05c
K
n
35
45
Actuation [VI
40
50
55
Figure 4: Comparison of the effect of the boundary
conditions on the pull-in voltage (Y = 200 GPa and o
= 130 Mpa)
I -350pm
36
long beam
38
40
long beam1
-300pm
42
44
46
Actuation [VI
48
50
j
:
!
Figure 5: Simulation of the pull-in voltage. The
values are close to the measured values of 48.7 V for
the 300 ,um long relay and 42.8 V for the 350 ,um
long relay (Y = 200 GPa and U = 145 Mpa).
Y 142
4A3.02
Table I : Summary ofthe simulation results
36 V
Pull-in
11 Resonance freauencv
1
42 V
46 V
7
48 V
49 V
Figure 6: M E M A D plots of five applied voltages.
This kind of simulation allows to get the pull-in
voltage of the device, here a multilayer
polysilicon/nitride/polysilicon microrelay.
Two different models have been investigated [4, 51. They
differ only in the numerical treatment of the beam
mechanics. Both include squeeze-film air damping by
solving the Reynolds equation using finite difference
methods. The first one uses a modal analysis which
includes the step boundary conditions but not the stressstiffening in large deflection [6] while the second solves
the Euler wide beam equation with stress stiffening, but
with fully clamped beam ends [ 5 ] . Both models required
parametric adjustment to fit the pull-in voltage. The
stiffness was selected for the first model adjustment,
while the residual stress was selected for the second one.
Figure 8 shows the experimental switching time versus
actuation voltage for a 300 pm long relay, together with
several simulations. Once the pull-in voltage is matched
by adjusting respectively the stiffness or the residual
stress, the curves fit the measured variation of pull-in time
with voltage very well. It should be emphasized that the
residual stress and the stiffness adjustment used here is to
provide compensation for the unmodeled effects in the
two models, and does not mean that the physical values
have changed from the previous values.
15
Figure 7: Abaqus T'plots of the jive first modes with
their frequency. This kind of simulation is very useful
to realize a dynamic macro model (6, 71.
10"
46
48
50
52
54
Actuation Voltage [VI
56
Measurements
+
uic: wide hcarn: rcsidun; srrcss=I 161.IPa
--.--G.--
ulzr n i c k besin: residual ~.iress=l00Mi'a
Skq:~J-iipb e i ~ i i reaidxi1
:
s t n s I-iSYIi'a
Stepped-up beam: residual stress=] 45MPa.
8 "/o increased stiffiicss
Figure 8: Switching time versus voltage f o r a 300 ,um
long micvorelay. Two different models have been
used to simulate the behavior of this device, with
parametric adjustments lo match the pull-in voltage
as explained in the text.
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4A3.02
It has also been possible to simulate the effect of air
pressure on the resonance frequency and the quality
factor. Figure 9 shows the measured and simulated data
for a 300 pm long microrelay. Such simulation has also
been realized for a 350 pm long microrelay and is being
presented separately [7].
Figure 10 gives the spring softening effect for a
300 pm long microrelay. A change of about 2% has been
observed for 15V applied to the beam.
I
+
measured 300 pm --simulated
300 wml
-1.5 1
CONCLUSION
Full modeling of the nonlinear dynamic behavior of
an electrostatic microrelay has been demonstrated. The
pull-in voltage, the switching time, the resonance
frequency and the quality factor of such compliant
structures have been simulated. Squeeze-film damping
has been included for the dynamic simulations. This
modeling illustrates the potential for predicting the
behavior of such micro electro mechanical systems with
good accuracy.
ACKNOWLEDGEMENTS
The authors would like to thank Ms. Pochon,
Mr. Clerc, Mr. Jeanneret, Dr. Linder and Mr. ThiCbaud,
(of IMT) for their help in the conception and the
realization of the microrelays.
Parts of this work have been funded by the Swiss
Foundation
for
Research
in
Microtechnology
FSRM (92/06), and by DARPA (J-FBI-95-215).
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0
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600 -
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400 -
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i
365
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364
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361
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0.1
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1
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10
100
pressure
Figure 9: Small amplitude simulation of the effect of
air damping [7] Because no intrinsic mechanical
damping was included in the modeling, the quality
factor measurements and the simulation diverge for
pressures less than I mbar
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0
2
4
6
8 10 12
Substrate Voltage [VI
14
16
Figure IO: Spring softening effect: measurements
and simulation for the 300 p m long microrelay.
REFERENCES
[ 11 M.-A. Gretillat et al., “Integrated circuit compatible
electrostatic polysilicon microrelays”, Micromech.
Microeng., 5 (1995), pp. 156-160.
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[3] J. Gilbert et al., “3D Coupled Electro-mechanics for
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[4] Y.J.Yang et al., “Numerical simulation of
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[5] R. Gupta et al., “Pull-in dynamics of electrostatically
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[6] G. K. Ananthasuresh et al., “An approach to
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ASME
International
Mechanical
Engineering Congress and Exposition, Symposium on
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[7] Y.-J. Yang et al., “Effect of air damping on the
dynamics
of
non-uniform
deformations
of
microstructures”, Transducers 97, Chicago, USA (1 997).
[8] 0. Tabata et al., “Mechanical property measurements
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