9-11 April 2008
?EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5
RF-MEMS beam components: FEM modelling and
experimental identification
of pull-in in presence of residual stress
Alberto Ballestra1, Eugenio Brusa2, Giorgio De Pasquale1, Mircea Gh. Munteanu3, Aurelio Som?1
1 Politecnico di Torino, Mechanical Department,
c.so Duca degli Abruzzi, 24 ? 10129 Torino, Italy;
alberto.ballestra@polito.it, giorgio.depasquale@polito.it, aurelio.soma@polito.it,
phone ++39 011 5646951; fax ++39 011 5646999
2,3 Universit? degli Studi di Udine, Dept. Electrical, Management and Mechanical Engineering
via delle Scienze 208 ? 33100 Udine, Italy; 2 eugenio.brusa@uniud.it, 3 munteanu@uniud.it
phone 2++39 0432 558299, 3++39 0432 558243, fax ++39 0432 558251
Abstract-In this paper an experimental validation of numerical
approaches aimed to predict the coupled behaviour of
microbeams for out-of-plane bending tests is performed. This
work completes a previous investigation concerning in plane
microbeams bending.
Often out-of-plane microcantilevers and clamped-clamped
microbeams suffer the presence of residual strain and stress,
which affect the value of pull-in voltage. In case of microcantilever
an accurate modelling includes the effect of the initial curvature
due to microfabrication. In double clamped microbeams a pre-
loading applied by tensile stress is considered. Geometrical
nonlinearity caused by mechanical coupling between axial and
flexural behaviour is detected and modelled.
Experimental results demonstrate a good agreement between
FEM approaches proposed and tests. A fairly fast and accurate
prediction of pull-in condition is performed, thus numerical
models can be used to identify residual stress in microbridges by
reverse analysis from the measured value of pull-in voltage.
I. INTRODUCTION
Microcantilevers and microbridges are currently widely used
in RF applications as microswitches and microresonators [1],
[2] and in experimental micromechanics, where materials
mechanical properties and strength are measured. Therefore, it
is required implementing efficient numerical models to predict
the electromechanical performance of microstructures actuated
by electric field [2], [3], [4], [5]. A wide variety of approaches
has been proposed in literature to predict static and dynamic
behaviour of microbeams [1], [6], [7], [8]. Experimental
validation is therefore aimed to verify their effectiveness in
predicting pull-in condition and frequency response. Usually
model sensitivity on the uncertainties of numerical values of
design parameters and material properties is investigated. Very
often is rather difficult to know precisely material properties
and microspecimen dimensions.
FEM original approaches developed by the authors were
proposed in [9], [10], [11], [12] and were already validated in
[13]. A preliminary experimental investigation was aimed to
predict the static behaviour of planar microcantilevers, for in-
plane bending test. Present research is devoted to complete
previous investigation activity focusing on out-of-plane
bending microbeams.
Fully coupled electromechanical problem has electrical and
mechanical coordinates, which are linked by the
electromechanical coupling effect. Pull-in condition is
responsible of a snap down of microbeam on the counter-
electrode. In present case pull-in may be affected by some
initial stress or strain present on the microsystem before the
application of electric field. Moreover, it is well known that the
problem is nonlinear because of the dependence of
electromechanical force on displacement and voltage and,
sometimes, of the so-called geometrical nonlinearity [10], [11],
[12], [13]. To include these effects, models of microcantilever
have to introduce the approximated analytical description of
the initial curvature. It is usually sufficient to predict with
enough accuracy the pull-in voltage and displacement, with an
error of 2-3% maximum. For microbridges with double clamps
axial stress is required to perform a coherent simulation of the
actual system.
II. SPECIMEN CHARACTERIZATION
A. Fabrication process and measurement methods
Specimens used for this work were realized by ITC-IRST
Research Center (Trento, Italy), by means of the so-called RF
Switch (RFS) Surface Micromachining process. Gold is used
for the suspended structures; material is deposited through
electroplating by means of a chromium-gold PVD adhesion
layer [14], [15]. Profilometric measures and pull-in tests were
performed by Fogale Zoomsurf 3D optical profiling system,
based on non-contact optical interferometry [16]. The lateral
resolution is ?0.3 ?m, while the vertical resolution reaches
?0.5?10-4 ?m [17].Tables 1 and 2 show the dimensions of
microbeams used as specimens in testing, all measures are
9-11 April 2008
?EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5
Fig. 1-Geometry 4 series cantilever beams (L=200 um)
expressed in micron. In particular, t means beam thickness, L is
length, w is width, g initial gap, y tip displacement for
cantilever. Star symbol indicates measured value instead of
nominal. Pull-in voltage (VPI) was measured by gradually
increasing voltage between suspended microbeam and ground,
until the collapse of the structure on the ground counter-
electrode.
Experimental pull-in was observed between the voltage
range reported in tables I e II.
B. Residual stress
Presence of an initial curvature in microcantilever specimens
is associated to an increase of pull-in condition. In double
clamped microbeams pull-in depends on a pre-stress tensile
loading caused by microfabrication process. Residual stress
origin is supposed to be diffusion of chrome of adhesion layer
among gold grains [15] and difference of thermal expansion
coefficient (CTE) between the gold structure and the
photoresist layer underneath [18]. Residual stress grows every
time there is a temperature difference during microfabrication
process. Stress is maximum at the interface
gold/chromium/photoresist and decreases accross the
microbeam thickness. In microcantilever, once removed the
support substrate, traction at the interface is removed and turns
into a deformation. Therefore initial curvature can be seen as
an initial strain, which bends the microstructure out-of-plane
[17]. In microbridges free bending is forbidden since
microbeam is overconstrained by clamps. Therefore residual
stress holds and strain is prevented .
III. MODELLING
According to the detailed descriptions of numerical models
available in [1], [2], [3], [4], [5], [6], this paper will focus on
the experimental validation of numerical results computed.
Few modelling issues will be here resumed. Electro-
mechanical force couples mechanical (displacements,
rotations) and electrical (voltages, charges) degrees of freedom,
and equilibrium equations in both the static and dynamic
domains appear nonlinear. Moreover, typical dimensions of
RF-MEMS introduce a second cause of nonlinearity, being
referred to as ?geometrical?. In case of microcantilever a large
displacement of the tip requires to resort to a nonlinear
mechanical solution to find the actual equilibrium condition.
This usually means to implement an iterative procedure which
applies force step by step and locally linearizes the structural
problem. In case of double clamped microbeam even in
presence of small displacement the mechanical coupling
occurring between the axial and flexural behaviour introduces
either a hardening or softening effect on the structural stiffness.
In both cases above mentioned, to solve the coupled problem a
sequential solution is performed. Computation of the electric
field distribution and of the related electromechanical force for
a given equilibrium configuration of the deflected microbeam
is separated from the mechanical analysis aimed to find the
deformed shape of the beam under the electromechanical load.
This sequence justifies implementing a computational loop. If
the geometrical nonlinearity is active, a second iterative loop
has to be implemented for each step of the electromechanical
solution to find the actual equilibrium condition. Numerical
methods are used to discretize the structure and the dielectric
material. In present case both the dielectric and mechanical
domains were meshed by FEM. A sequential approach based
on Newton Raphson iteration method was implemented. The
coupled electromechanical problem was solved by 2D and 3D
models, by means of ANSYS code. In case of dynamic analysis
Newmark Modified Algorithm was implemented and tested in
connection with the non incremental approach for
geometrically nonlinear structures. ANSYS code, MATLAB
and FORTRAN implementations were performed and
numerical results were compared to the experimental ones.
Alternately the coupled-field problem was solved by using
coupled-field elements through a direct approach available in
ANSYS as 1D transducer element TRANS126. It couples
electro-mechanical domains and consists of a reduced-order
model with structural translations and electric potential as
degrees of freedom.
Initial deformed shape of microcantilver was analytically
described by means of beam curvature ? and axial strain ?
being written as function of flexural displacement v, axial
displacement u, axial load N and bending moment M as
follows:
;2
2
EJ
M
dx
vd =?=?
EA
N
dx
du ==? (1)
Generalized force vector F was then formulated as:
{ } { } [ ] [ ] [ ]? ?? ?
== ?
?
?
?
???
?
??
?
??
?+
???
?
???
?
??
?
??
??= elel n
i l T
TT
n
i l
T dxDBdx
N
MBFF
1 0
0
1 0
0
?
? (2)
where initial values of force N0 and Moment M0 can be
computed from the initial stress ?0; ?0T is the initial thermal
strain and ?0T the curvature in case of thermal contribution.
9-11 April 2008
?EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5
TABLE 1
CANTILEVER BEAMS: NOMINAL AND MEASURED (*) DIMENSIONS, MEASURED AND CALCULATED PULL-IN VOLTAGES
TABLE 2
CLAMPED-CLAMPED BEAMS: NOMINAL AND MEASURED (*) DIMENSIONS, MEASURED AND CALCULATED PULL-IN VOLTAGES. W.O.STANDS FOR WITHOUT
-1
0
1
2
3
4
5
6
7
8
9
0 25 50 75 100 125 150 175 200
X (?m )
Z (
?m
) .
0V
20V
30V
35V
40V
44V
46V
47V
-1
0
1
2
3
4
5
6
7
8
9
0 100 200 300 400 500
X (?m)
Z (
?m
) .
0 V
26V
47 V
60 V
65 V
70 V
Fig. 2. Sequence of static equilibrium conditions measured by Fogale
Zoomsurf 3D during the pull-in tested performed on microcantilever.
Geometry 2, sample 2.
To predict the actual pull-in of microbridge layout, axial effort
had to be identified and included into the FEM models. This
investigation was performed in ANSYS by applying either a
distributed internal pressure at nodes or an initial strain as real
constant. This action allowed estimating residual stress values
by tuning the numerical pull-in tension on the experimental
result.
Fig. 3. Sequence of static equilibrium conditions measured by Fogale
Zoomsurf 3D during the pull-in tested performed on microbridge.
Geometry 5, sample 1.
IV. DISCUSSION
In Table 1 calculated values of pull-in voltage of
microcantilevers are compared to the measured ones. For
geometry 1 all methods overestimated pull-in. In particular
FEM 3D model was far from the true result. This was due to
some problems of mesh morphing in presence of a very narrow
gap. Geometry 2 shows a better agreement. Fringing effect is
Ge
om
.
Sa
mp
le
t L L* w* t* g* y* VPI* Exper.
VPI
Trans
126
VPI
Ansys
2D
VPI
Matlab
2D
VPI
Ansys
3D
1 1 3 540 531.4 33.5 2.953 2.996 6.334 10?11 16 15 15 18
1 2 3 540 535.2 32.9 2.966 2.913 4.158 10?11 12 15 15 18
1 3 3 540 534.3 33.3 3.012 2.883 6.613 10?11 16 15 15 18
2 1 1.8 200 190.5 32.4 1.842 2.971 3.845 43?44 45 49 51 46
2 2 1.8 200 190.3 32.0 1.817 3.107 4.139 46?47 48 49 51 46
2 3 1.8 200 190.3 32.1 1.820 3.170 3.932 47?48 48 49 51 46
3 1 3 200 189.7 33.0 2.594 2.897 1.130 58?59 43 82 50 70
3 2 3 200 190.1 32.6 2.578 2.939 1.270 56?57 45 82 50 70
3 3 3 200 189.7 32.8 2.614 2.968 1.342 57?58 45 82 50 70
4 1 4.8 200 189.8 33.7 4.899 3.004 0.049 81?82 84 100 100 80
4 2 4.8 200 190.2 33.3 4.875 3.002 0.044 90?91 84 100 100 80
4 3 4.8 200 190.6 33.7 4.799 3.079 0.032 88?89 84 100 100 80
Ge
om
.
Sa
mp
le
t L L* w* t* g* VPI* Exper.
VPI w.o.
stress
Trans126
Pre-Stress
(MPa)
VPI w.o.
stress Ansys
2D
5 1 3 540 541.8 32.2 2.68 2.83 57?58 27 30 29
5 2 3 540 541.0 32.3 2.7 2.81 59?60 27 32 29
5 3 3 540 544.3 32.4 2.792 2.913 59?60 29 29 29
6 1 6 375 371.4 13.9 5.627 3.110 180?190 191 0 195
7 1 4.8 650 650.0 11.9 t*+g*=9.17 88?89 72 20 70
7 2 4.8 650 653.1 11.9 6.08 3.041 88?89 72 20 70
7 3 4.8 650 655.1 12.5 6.01 3.114 88?89 72 20 70
9-11 April 2008
?EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5
Fig. 4. Analysis of the effect of the tensile load on geometry 5.
more important and has to be evaluated by a 3D analysis to fit
experimental result. In case of geometry 3 a prediction
performed by ANSYS worse than by the non incremental
approach was observed. Three dimensional effects are more
effective in geometry 4, where force inputted in 2D models has
to be tuned. Table 2 shows a large difference between pull-in
voltage estimated without including residual stress and
experimental measures.
Microbridge 5 suffers the highest mismatch. Geometry 6
shows a good agreement and thus a pre-stress almost null. In
case of geometry 7 pre-stress was present and detected. Tables
demonstrate that upward initial curvature in geometrical
nonlinear microcantilever is a very difficult configuration to be
analysed. In fact, all 2D and 3D approaches implemented had
to operate with a narrow gap and mesh morphing met some
problems about pull-in, when the tip is close to the counter-
electrode. Reduced order model based on TRANS126 gave
better results, since it did mesh dielectric region. Double
clamped microbeams strongly suffer pre-stress loading. A
sensitivity analysis concerning initial tensile stress was
performed. As Fig.4 shows for geometry 5 different pre-stress
conditions affect significantly pull-in voltage. It varies with
axial stress from VPI= 29 (0 pre-stress) to 41 (10 MPa), 49
(20 MPa) and 57 (30 MPa). All models needed to be tuned by
inputting a suitable value of tensile stress. In practice, all the
microspecimens studied exhibited geometrical nonlinearity,
thus requiring to resort to a nonlinear structural analysis based
on the iterative solution of the mechanical problem within the
sequential approach.
V. CONCLUSIONS
An experimental investigation was performed to validate
numerical approaches aimed to predict the behaviour of
microbeams, nonlinearities due to electromechanical coupling
and geometry were taken into account. The strong effect of
residual stress on the pull-in voltage was detected and included
in the models. The experimental results demonstrated a good
agreement between the FEM approaches proposed, the
methods allow a fairly fast and accurate prediction of the
microbeams behaviour.
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