Engineering, 2010, 2, 55-59
doi:10.4236/eng.2010.21007 lished Online January 2010 (http://www.SciRP.org/journal/eng/).
55
Pub
Analysis for Pull-In Voltage of a Multilayered
Micro-Bridge Driven by Electrostatic Force
Yu LIU1,2,3, Guochao WANG1,2,3, Hongyun YANG1,2,3
1Automobile College, Chongqing University of Technology, Chongqing, China
2Key Laboratory of Automobile Parts & Test Technique in Chongqing, Chongqing, China
3Chongqing Engineering Research Center for Automobile Power System and Control, Chongqing, China
Received August 8, 2009; revised September 4, 2009; accepted September 12, 2009
Abstract
A trial solution for bending deflection of a multilayered micro-bridge subject to a voltage induced load is
presented. The relation between the applied voltage and the displacements of the micro-bridge in the pull-in
state is analyzed by energy method. Furthermore, two analytical expressions about normalized displacement
and pull-in voltage are carried out. It’s proved that the value of normalized displacement is not influenced by
residual stress if axial and shear deformation is ignored. Finally, the theoretical results are compared with
that of FEM, and they show good agreement.
Keywords: MEMS, Electrostatic Actuation, Multilayered Micro-Bridge, Trial Solution, Energy Method,
Pull-In Voltage
1. Introduction
Moving part is the most frequently used one in MEMS
structures. There are various principles that can be used to
drive the moving part, including electrostatic, piezoelectric,
thermal, magnetism, etc. Compared with the others, the
method of electrostatic force drive is more attractive [1,2]
because of its larger force caused by micro-effect and
non-contact, which is beneficial to high precision. Further-
more, the process of MEMS devices driven by electro-
static force is compatible with IC process.
Micro-bridge structure driven by electrostatic force is
familiar in MEMS devices, such as changeable capaci-
tors, RF switches, micro-resonators, pressure sensors,
and so on. On one hand, it is a common structure being
used to obtain the mechanical parameters of film, just
like Young’s Modulus, residual stress, yield strength and
bending strength [3–5]. But on the other hand, MEMS
devices with micro-bridge structure are often with low
reliability and worse quality, which is induced by pull-in
phenomena. So it’s worthy to research the pull-in phe-
nomena of micro-bridge structure subject to a voltage
induced load, which is important to MEMS design and
optimization. For example, drive voltage must be less
than pull-in voltage for most structures driven by elec-
trostatic force, while it’s opposite for DMD and self-
measure unit of micro-accelerometer [6].
Although many people study the pull-in phenomena of
the monolayered micro-bridge structure driven by elec-
trostatic force, and get some valuable conclusions, they,
however, cannot be applied to the multilayered micro-
bridge structure. So some people [7] devote themselves
to research on the pull-in phenomena of the multilayered
micro-bridge structure. Whereas the analytical expres-
sion is complicated, this depends on the introduction of
the assumptions and the form of the trial solution.
This paper presents a trial solution for bending deflec-
tion of a multilayered micro-bridge subject to a voltage
induced load. The relation between the applied voltage
and the displacements of the micro-bridge in the pull-in
state is analyzed by energy method. Furthermore, two
analytical expressions about normalized displacement
and pull-in voltage are carried out. It’s proved that the
value of normalized displacement is not influenced by
residual stress if axial and shear deformation is ignored.
Finally, the theoretical results are compared with that of
FEM, and they show good agreement.
2. Model
The micro-bridge with a rectangular cross-section is
made of n layer materials, illustrated in Figure 1. Here
we assume the top of the beam is a conductor, while the
others are dielectric. The electrode under the beam is

Y. LIU ET AL.
56
Figure 1. Schematic picture of a multilayered micro-bridge
subject to a voltage induced load.
Figure 2. Deformation of an infinitesimal line element dx.
covered with the insulated material, which is fixed on the
substrate. It’s not difficult to imagine that micro-bridge
will be bended when the voltage is added to the beam
and the electrode.
Let’s explain the meanings of some symbols firstly,
where l is the length of the beam. , ,
i
hi Eri

are the
thickness, Young’s Modulus and relative permittivity of
the i-layer material respectively. ,
0
h0r

are the thick-
ness and relative permittivity of the insulated layer above
the electrode respectively. V is the voltage added to the
beam and the electrode. h is the gap distance between the
bottom of the beam and the top of the electrode when
.
0V
Set symmetric axis of beam’s cross section as z axis,
and positive direction is upwards. Set the equivalent
neutral axis as y axis, and its position is needed to be
confirmed. Then x axis is the perpendicular axis of yz
plane and through the origin. The distance from the
equivalent neutral axis to the bottom of the beam is.
It’s clear that
eff
z
1
1
2
n
eff i
i
z


As shown in Figure 2, according to the plane cross-
section assumption on pure bending, we calculate the
strain i

of the fiber bb while, which is in the
-layer and the primary length as well as the gap distance
to neutral surface are dx, respectively.
0V
i
i
z
i
i
z




where

is curvature radius of the neutral surface.
Based on the assumption on pure bending, that is,
there is no normal stress between the longitudinal fibers;
we calculate the stress i

of the fiber bb according to
Hooke’s law. Of course, the deformation is at the elastic
deformation period.
i
iii i
z
EE 


Uncountable micro internal force idA

composes a
system of parallel force in space, which can be simplified
as three internal force components. What we’re con-
cerned about is the component parallel to
x
-axis,
namely axial force. Based on the equilib-
rium of static equilibrium
i
A
N  dA
0XN


and the follow-
ing formula,
12
12
112 12
1121
112 2
1112 22
1112 2 2
1
1
n
n
effeffn eff
effeffn eff
i
A
nn
AA A
nn n
AA A
hzhh zhhh z
nnn
zhzhhhz
NdA
dA dAdA
E z dAEzdAEzdA
E bzdzEbzdzEbzdz

 



 
 

 

 










 
 
 





We get
1
11
1
()
2
j nj
j
ji
ji
eff n
ii
i
h
Eh h
z
Eh








Therefore,
0eff
zz


1
() 1,2,,
i
ijeff
j
zhzi

 
n
where are the coordinates of the micro-bridge’s
top and bottom surface along -axis, while
0n
zz,
z11n zz

,,
are that of layer-layer interface respectively, as shown in
Figure 1.
3. Solution
h due to the different
Young’s Modulus of n layer materials.
Here we choose a trial solution for bending deflection of

Y. LIU ET AL. 57
a multilayered micro-bridge subject to a voltage induced
load as following [8]:
2
2
()(1 cos)2 sin
x
x
wx AA
ll


 
where A is the undetermined coefficient and 2A is the
maximum deflection of the micro-bridge.
It’s acceptable to neglect the influence of axial and
shear deformation on bending due to the small displace-
ment assumption of the micro-bridge.
3.1. No Consideration of Residual Stress
For this situation, the strain i

of the i-layer material
only includes flexural strainbi

. That is,
22
22
42
cos
ibi
dwA x
zz l
dx l


  
So the unit volume deformation energy and the de-
formation energy of the i-layer material are described by
the following formula:
2
1
2
ii
uE
i


1
2
0
2
i
ii
lz
i
siiii
vz
Eb
Uudvdzdx 



Therefore, the total deformation energy of the mi-
cro-bridge Us is equal to the summation of the deforma-
tion energy of n layer materials. That is,
1
2
0
11
2
i
i
nn
lz
i
ssi i
z
ii
Eb
U Udzdx 





2
43
1
1
2
() (
12
n
ii i
i
blA Ezz
l




3
)
Considering the influence of the insulated layers on the
gap distance, we introduce the equivalent gap distance
[9]
h

01 1
01 (1
n
rr rn
hh h
hh
 




)
where iri h

is the equivalent air gap distance of the
i-layer insulated material. Then the model in Figure 1 is
simplified as Figure 3, where

 is the equivalent thick-
ness, which will be discussed in detail later.
So the electrostatic energy can be described as [8]
2
0
2( 2)
e
Vb l
U
hh A

 

where 0

is the permittivity of vacuum.
Therefore, the energy function is described as [10]
s
e
UU


Because the value of the electrostatic force is in in-
verse proportion to the square of distance, and the distri-
bution of electric field is influenced by structural dimen-
sion as well as position, so the issue of bending equilib-
rium stability should be considered. It exists when the
electrostatic force is equal to the elastic force. If the
voltage V exceeds the critical value, the equilibrium state
is destroyed and the micro-bridge will contact with the
fixed plane, which is called pull-in phenomena. The
minimum critical voltage is called pull-in voltage,
marked as VPI. Here,
0
A


 and
2
20
A


 (1)
Mark the maximum deflection of the micro-bridge
under VPI as 2Acr, then
2
3
)
~
2
~
(
~
2
)()
2
(
6
2
2
3
1
3
1
4



crii
n
i
i
cr Ahhh
V
zzE
l
A 
(2)
2
5
)
~
2
~
(
~
2
3
)()
2
(
6
122
2
3
1
3
1
4



crii
n
i
iAhhh
V
zzE
l

(3)
So
0.2
cr
A
h
 (4)
And the normalized displacement
20.4
cr
A
h
 

3.2. Consideration of Residual Stress
For this situation, the strain i

of the i-layer material
includes not only flexural strain bi

but also the strain
Figure 3. The equivalent model of a multilayered micro-
bridge.

Y. LIU ET AL.
58
induced by the residual stress i

of the i-layer material.
That is,
22
22
42
cos
ii
ibi
ii
dwA x
zz
EE ldx l
i
i
E

 
 
Similarly, the deformation energy of the micro-bridge
1
2
0
1
2 2
433
1
11
2
2
() ()
12 2
i
i
nlz
i
si
z
i
nn
ii
ii i
ii
i
Eb
Udzdx
blh blA Ez z
l

E










(5)
Because the second part of Formula (5) is not the function
of the undetermined coefficient A, so the results from
Formula (1) are same to that of Subsection 3.1.
It’s clear that the value of normalized displacement is
not influenced by residual stress if axial and shear de-
formation is ignored.
Based on Equations (2) and (4), we get
333
1
1
4
0
()
48.290
n
ii i
i
PI
hEzz
Vl






(6)
For the multilayered micro-bridge with a rectangular
cross-section, the area moment of inertia
)(
3
~3
0
32
0
zz
b
dzbzIn
z
z
n 
and the flexural stiffness





n
i
iii
n
i
z
zizzE
b
dzbzEIE i
i1
3
1
3
1
2)(
3
~~
1
then the equivalent Young’s Modulus
3
0
3
1
3
1
3)(
~
~~
~
zz
zzE
I
IE
E
n
n
i
iii





and the equivalent thickness
1
13
333
0
12
() 4()
n
Izz
b






Therefore, the formula (6) is simplified as
33
4
0
12.0725
PI
E
h
Vl



  
(7)
It’s clear that the multilayered micro-bridge can be
considered as a monolayered micro-bridge if only the
equivalent Young’s Modulus and the equivalent
thickness
E


 are carried out.
In the above discussion, we assume the top of the
beam is a conductor while the others are dielectric. If the
conductor is not on the top of the micro-bridge, or more
than one layer is conductor, the equivalent gap distance
will differ.
h

4. Validation
All the above deduction is based on a given trial solution
for bending deflection. Are they correct? How about the
accuracy? Now we try to prove it by simulation.
Firstly, some examples relating to the monolayered
micro-bridges are considered. For the monolayered mi-
cro-bridge given in paper [11], the value of VPI is 32.5V,
and the simulated values by ANSYS as well as Intellisuite
are 37.6V, 39.5V respectively, which are presented in
paper [11]. Whereas the analytical value based on the
Formula (7) is 34.2V, which is more close to the simu-
lated values. And for that given in paper [12], the simu-
lated value of VPI is 40V, while the analytical value based
on the Formula (7) is 40.99V. The error is only 2.48%.
Secondly, a multilayered micro-bridge with a rectan-
gular cross-section is considered. It’s made of two layer
materials. And the electrode under it is covered with
0.5 m

Si3N4.
Model 1: the top material is gold. And the other is
Si3N4.
Model 2: the top material is Si3N4. And the other is gold.
Table 1. Geometry and material parameters of the bi-lay-
ered micro-bridge.
Item Symbol (unit)
Model
1 Model 2
length l(m

) 400
width b(m

) 50
gap distance (V=0) h(m

) 2.0
thickness of the
insulated layer 0
h(m

) 0.5
1
h(m

) 2.0 0.5
thickness
2
h(m

) 0.5 2.0
relative permittiv-
ity 0r

=1r

8.0
1
E(GPa ) 210 57
Young’s Modulus
2
E
()
GPa 57 210
Table 2. PI V,
β
of the bi-layered micro-bridge.
ItemModel Analytical values Simulated valuesError
model 137.9671V 40.2187V -5.60%
VPI model 231.9798V 33.3438V -4.09%
model 10.4069 -1.70%
model 1#0.3933 1.70%
model 20.3972 0.70%

model 2#
0.4
0.4132 -3.19%
Note: # Residual stress of the gold layer is 100 MPa.

Y. LIU ET AL.
59
The geometry and material parameters of the bi-layered
micro-bridge are listed in Table 1. The micro-bridge will
be bended when subject to a voltage induced load.
[2] G. Li and N. R. Aluru, “Efficient mixed-domain analysis
of electrostatic MEMS,” ICCAD, pp. 474, 2002.
[3] T. Y. Zhang, Y. J. Su, C. F. Qian, et al. “Micro-bridge
testing of silicon nitride thin films deposited on silicon
wafers,” Acta Mater, Vol. 48, pp. 2843, 2000.
For Models 1 and 2, we get the simulated values of the
pull-in voltage VPI and the normalized displacement

by FEM. Table 2 lists the simulated performances and
the analytical values of VPI and

, as well as the error
of them. It shows good correlation. So the trial solution
of deflection is acceptable.
[4] Y. Zhou, C. S. Yang, J. A. Chen, et al. “Investigation of
Young’s modulus and residual stress of copper film micro-
bridges by MEMS technology,” Modern Scientific In-
struments, Vol. 4, pp. 45, 2003.
[5] M. J. Wang, Y. Zhou, J. A. Chen, et al. “Measurements of
elastic modulus and residual stress of nickel film by micro-
bridge testing methods,” Electronic Components & Mate-
rials, Vol. 23, No. 12, pp. 13, 2004.
5. Conclusions
This paper presents a trial solution for bending deflection
of a multilayered micro-bridge subject to a voltage induced
load. The relation between the applied voltage and the
displacements of the micro-bridge in the pull-in state is
analyzed by energy method. Furthermore, two analytical
expressions about normalized displacement and pull-in
voltage are carried out. The theoretical results are com-
pared with that of FEM, and they show good agreement.
[6] H. Yang, M. H. Bao, S. Q. Shen, et al. “The displacement
characteristics of the micromechanical structures driven
by electrostatic force,” Journal of Fudan University
(Natural Science), Vol. 38, No. 3, pp. 282, 1999.
[7] H. Rong, Q. A. Huang, M. Nie, et al. “An analytical model
for pull-in voltage of doubly-clamped multi-layer beams,”
Chinese Journal of Semiconductors, Vol. 24, No. 11, pp.
1185, 2003.
[8] D. H. Sun, Y. Q. Huang, W. Zheng, et al. “On the mod-
eling methodology of MEMS system-level simulation,”
Journal of Xiamen University (Natural Science), Vol. 40,
No. 2, pp. 297, 2001.
All results in this paper are based on the form of the
selected trial solution. The better the selected trial solu-
tion is, the higher the accuracy of solution is. Moreover,
the axial and shear deformation is ignored, which maybe
has influence on the stress and strain. Besides, in order to
get an analytical expression of VPI, it’s difficult to con-
sider the influence of some secondary effects, such as the
marginal effect of electric field. However, this paper pro-
vides an analytical model with high accuracy for a mul-
tilayered micro-bridge driven by electrostatic force, which
is beneficial to design, optimization and application of
MEMS devices with the micro-bridge structure.
[9] M. Nie, Q. A. Huang, J. H. Wang, et al. “Analysis of
deflection and pull-in voltage of a multi-layer cantilever
under an electrostatic load,” Chinese Journal of me-
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[10] Y. Y. Wang, T. I. Kamins, B. Y. Zhao, et al. Polysilicon
Film and its Application in IC (2nd Version), Science
Press, Beijing, 2001.
[11] C. S. Wang, W. B. Zhang, J. Fang, et al. “Research on
coupled electro-mechanical analysis and application for
typical components in MEMS,” Journal of Mechanical
Strength, Vol. 23, No. 4, pp. 503, 2001.
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