9-11 April 2008
?EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5
Validation of Compact Models of Microcantilever
Actuators for RF-MEMS Application
Eugenio Brusa1, Antonio Della Gaspera, Mircea Gh. Munteanu2,
Dept. Elect., Manag. Mech. Engineering - Universit? degli Studi di Udine, via delle Scienze 208 ? 33100 Udine, Italy;
1eugenio.brusa@uniud.it, 2munteanu@uniud.it, phone 1++39-0432-558299, 2++39-0432-558243, fax ++39-0432-558251
Abstract-Microcantilever specimens for in-plane and out-of-
plane bending tests are here analyzed. Experimental validation
of 2D and 3D numerical models is performed. Main features of
in-plane and out-of-plane layouts are then discussed.
Effectiveness of plane models to predict pull-in in presence of
geometric nonlinearity due to a large tip displacement and
initial curvature of microbeam is investigated. The paper is
aimed to discuss the capability of 2D models to be used as
compact tools to substitute some model order reduction
techniques, which appear unsuitable in presence of both
electromechanical and geometric nonlinearities.
I. INTRODUCTION
Reconfigurable layout is a typical feature of circuits used for
radio-frequency application [1]. This goal is achieved by means
of cantilever and double clamped microbeams, electrostatically
actuated, being used as switches, resonators and varactors [1-
4]. Design needs for a precise prediction of pull-in condition
and frequency response. An effective modelling of these static
and dynamic behaviours is rather difficult, because of the
electromechanical coupling. Electromechanical forces
nonlinearly depend on mechanical displacement, electric
charge and voltage. In presence of large strain or displacement
a structural nonlinear solution has to be implemented [5-7].
Analytical solutions for microcantilever and double clamped
microbeams were formulated and included the effects of
stretching and large displacement, i.e. the so-called geometrical
nonlinearity [5,6]. Corrections suitable to predict fringing
effect of electric field were even proposed [1,3]. These models
assume an ideal beam geometry, which differs from the actual
structure in some details of the constraint and electrode shape.
Microcantilevers currently proposed by industry look like
specimens depicted in Figs.1 and 2. A first geometry is based
on ?in-plane? bending test, i.e. microbeam deflection occurs in
a plane parallel to the profiling system?s target. In ?out-of-
plane? bending actuation microbeam tip moves towards the
target [8]. Often microbeam is a part of a wider structure, e.g. it
connects the rigid plate of a varactor to the fixed frame [9]. All
these aspects motivate the need for a library of structural
numerical models within the simulator used to predict the
response of the whole electronic circuit. Accuracy and fastness
are main requirements for this modelling activity. Authors
demonstrated in previous papers that static pull-in of in-plane
bending specimens is accurately predicted by a numerical
solution based on sequential approach with voltage increments
[7,10,11]. Geometric nonlinearity becomes relevant for large
displacements, close to pull-in voltage. A double nonlinear
solution, including geometrical effect, has to be implemented.
In this case iterative solution is applied at each step of the
sequential approach and makes extremely heavy the
computational effort. Iteration can be avoided by using a
special finite beam element and a non-incremental formulation
[10,12,13]. Effectiveness of 2D FEM models in case of in-
plane bending specimens was a little bit surprising, because no
electric force concentration due to the finite dimensions of the
geometry was taken into account [11]. In this paper a detailed
analysis of in-plane microcantilevers is performed to complete
that investigation. Results are compared to those of out-of-
plane microcantilevers, which show an initial curvature of
microbeam. Main features of the two above mentioned layouts
are described. To define suitable criteria to proceed with a
model order reduction useful for the dynamic analysis, limits of
the coupled field analysis in 2D models are discussed, by
investigating some three dimensional effects of the electric
field. Numerical results are simultaneously compared to the
experiments done on microspecimens made of epitaxial
polysilicon and gold.
950 ?m
800 ?m
cantilever beam
connection pads
gap
(A)
(B)
counter-electrode
Fig. 1. Microcantilever built for in-plane bending test (top view) .
Fig. 2. Microcantilever built for out-of-plane bending test (front view) .
9-11 April 2008
?EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5
II. SPECIMENS FOR IN-PLANE BENDING TEST
Experimental validation of numerical models developed in
previous papers was performed on eight geometries, described
in Table I. FEM static analysis was performed in ANSYS
code, as sequential solution with mesh morphing in dielectric
region, then through the non-incremental FEM sequential
approach tested in [10,12,13] and by a combined sequential
FEM/BEM solution [11]. Last two methods were implemented
in Matlab. An additional comparison included results of the
Discrete Geometric Approach (DGA), recently proposed [14].
As Fig.3 shows in case of geometry 4, all the 2D approaches
converge to the experimental curve, provided that Young
modulus of epitaxial polysilicon was clearly identified and
geometrical nonlinearity due to the large tip displacement was
included. FEM 3D (ANSYS, SOLID122 electrostatic,
SOLID185, elastic) analysis revealed that the actual
distribution of the electrostatic force is fairly different in terms
of average and peak values (Fig.4). Nevertheless, effectiveness
of 2D models appeared surprisingly good. A deeper
investigation of three dimensional effects of electric field
allowed finding that agreement between 2D models and
experiments was assured by constraint and electrode
geometries.
TABLE I
MICROCANTILEVER SPECIMENS FOR IN-PLANE BENDING TEST
n. Length [?m] Width [?m] Thickness [?m] Gap [?m]
1 101 ? 0.1 15 1.80 ? 0.02 5.0 ? 0.3
2 101 ? 0.1 15 1.80 ? 0.02 10.0 ? 0.3
3 101 ? 0.1 15 1.80 ? 0.02 20.1 ? 0.3
4 205 ? 0.2 15 1.90 ? 0.02 10.0 ? 0.3
5 205 ? 0.2 15 1.90 ? 0.02 20.0 ? 0.3
6 805 ? 0.5 15 2.70 ? 0.04 39.6 ? 0.3
7 805 ? 0.5 15 2.70 ? 0.04 200 ? 0.5
8 805 ? 0.5 15 2.70 ? 0.04 400 ? 0.5
Legend
???? Linear (166 GPa) ? Experiments
? ? Non incremental (150 GPa) ? ? ? (166 GPa)
? ANSYS PLANE121/183 (150 GPa) + (166 GPa)
Fig. 3. Example of experimental validation on in-plane microcantilever
Length
[?m]
half-
width
[?m]
Electrostatic pressure [MPa]
Fig. 4. Actual distribution of electrostatic pressure on half-width of in-plane
microcantilever according to a 3D FEM model.
Electrode
Electrode
Cantilever
microbeam
Wafer plane
Clamp Upper gap
gap
Fig. 5. Crucial aspects of in-plane microcantilever
In case of geometry 5, where half-width is c = 7.5 ?m, ratio
between the peak values of force computed by 3D FEM model
and 2D respectively was 1.88. This result assumed that
actuation voltage was applied to the counter-electrode, equal
width for electrode and counter-electrode, rounded tip, no
surface behind the beam. All these assumptions play a
significant role in case of in-plane actuators (Fig.5). Charge
concentration at the tip and along the edges increases
electrostatic force, more largely as peak than as average value.
This concentration is localized on a small area (Fig.4).
Rectangular and rounded tips have higher force ratio 3D/2D
than sharp triangular tip. Indefinitely long counter-electrode
and zero voltage applied on the microbeam bring the above
force ratio up to 2.1. Actually, wafer surface below the lateral
edge of microcantilever decreases this ratio to 1.05, thus
allowing 2D model predicting the actual pull-in. In case of
equal width for upper and counter-electrode ratio tends to 1,
being higher when counter-electrode width is larger than the
electrode?s one.
Geometric characterization of in-plane microcantilever is
rather difficult. Profiling system offers a very high resolution
along the optical axis, while it is lower on the target plane.
Thickness and gap measurements for in-plane microcantilever
is less accurate, as Table I shows. Numerical prediction of pull-
in is consequently ineffective, if nominal values of these
parameters are inputted. For a given correspondence of
numerical and experimental values, and in case of geometry 5
following discretizations were set used: DGA (FORTRAN,
sequential method) 15000 elements, 32000 DoFs (electrical);
9-11 April 2008
?EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5
756 elements and 4000 Dofs (structural); FEM (ANSYS;
iterative, mesh morphing) 80 PLANE183 (solid beams), 3000
PLANE 121 (electrical); FEM (MATLAB; sequential; non
incremental; mesh morphing) 3036 elements, 346 nodes
(electrical); 41 nodes, 40 Timoshenko two-node beam
elements; FEM/BEM (MATLAB, sequential, non-incremental)
337 two-node boundary elements, 188 nodes (electrical); 31
nodes, 30 two-node Timoshenko beam. As it is clearly
described in [15], BEM allows reducing the number of DoFs in
dielectric region, but boundary elements assure a better
accuracy within the element field than on boundaries. To
predict accurately voltages on the microbeam a mesh
refinement is required, although DoFs are less than in FEM.
Fig. 6. Potential distribution around the microcantilever half-width in case of
null voltage applied to counter-electrode (left) or to beam surface (right).
Fig. 7. 2D and 3D predictions of potential distribution around the
microcantilever half-width in case of null voltage applied to beam surface and
in presence of wafer surface (right edge in 2D model).
Fig. 8. Effect of finite dimensions of counter-electrode
on in-plane microcantilever half-width.
III. SPECIMENS FOR OUT-OF-PLANE BENDING TEST
Four geometries of golden microcantilevers for out-of-plane
bending were built to perform a parametric analysis. Each one
included several specimens. Table II summarizes relevant
parameters. Numerical data are written by describing the range
of measured values among different specimens and the
measurement errors.
TABLE II
MICROCANTILEVER SPECIMENS FOR OUT-OF-PLANE BENDING TEST
n. Length
[?m]
Width
[?m]
Thickness
[?m]
Gap
[?m]
y
[?m]
9 531:535
? 0.3
32:33
? 0.3
2.9:3.0
? 0.5.10-4
2.88:2.99
? 0.5.10-4
4.15:6.6
? 0.5.10-4
10 190 ? 0.3 32
? 0.3
1.8
? 0.5.10-4
2.97:3.17
? 0.5.10-4
3.8:4.1
? 0.5.10-4
11 190 ? 0.3 32:33
? 0.3
2.57:2.61
? 0.5.10-4
2.89:2.97 ?
0.5.10-4
1.13:1.34
? 0.5.10-4
12 190 ? 0.3 33
? 0.3
4.79:4.89
? 0.5.10-4
3.0? 0.5.10-4 0.04
? 0.5.10-4
Precision in measuring gap and thickness is here higher than
in case of in-plane actuators. Nevertheless, specimens exhibit
some differences in length, thickness and gap. This layout has
two peculiarities. Counter-electrode only partially fills the gap,
and the anchor is a structural component with a defined
geometry. A crucial aspect was the initial curvature of
geometry 9,10 and 11 due to some differences of diffusion of
Chromium of the seed-layer among the deposited layers.
Models had to include this curvature to fit experimental pull-in
voltage. In fact, while microfabrication may induce a residual
stress gradient across the beam section in double clamped
beam, in microcantilevers stress vanishes since it imposes an
initial strain and curvature which moves the free tip. For given
initial strain ?0, curvature ?0, accidental thermal effects and
axial or flexural preloads, N0 and M0 respectively, stress-strain
relations integrated at beam?s cross section become:
??
?
??
?+
???
?
???
?
??
?
??
??
??
?
??
?
??
?
??
?=
??
?
??
?
0
0
0
0
0
0
0
0
M
N
EJ
EA
M
N
T
T
z ?
?
?
? (1)
where thermal effects are:
( ) ( )?== lTT dAyxyTxT
0
000 ,; ???? (2)
Symbols mean Young modulus, E, beam section, A, beam
second moment of area, J, axial effort, N, bending moment, M.
Temperature distribution may include a constant contribution
T0 along beam thickness (y axis), and a distribution T(x,y)
variable along beam length (x) and thickness (y).
Experimental results were similar for the four geometries
tested. Those of geometry 10 are depicted in Fig.9. FEM and
FEM/BEM approaches converge to a numerical solution which
overestimates the actual pull-in voltage. In this case benefits of
in-plane layout are absent. Fringing effect is more relevant. To
fit experiments it was required to perform a 3D FEM analysis
and compute the correction factor for the electromechanical
force. It was observed that electric field in 2D models,
including separately microbeam length and width, gave easily
this number. FEM 3D models did show some problems
because of mesh morphing operation applied to a so narrow
gap. A sensitivity analysis confirmed that electrostatic pressure
is quite uniformely distributed along the length and the width
of the microbeam, while only peak values of force strongly
9-11 April 2008
?EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5
depend on thickness and gap values. In fact, an electrical
analysis on the undeformed configuration of the microbeam
allowed computing a correction factor suitable to find
experimental pull-in.
Fig. 9. Example of experimental validation on out-of-plane microcantilever
(geometry 10). Experiments (black point) are compared to nonlinear FEM
(bold line), nonlinear FEM/BEM (dashed) an geometrically linear solution
(grey continuous line).
IV. DISCUSSION
Model order reduction of nonlinear and second order dynamic
microsystems is still a difficult task. No definitive approaches
were successfully tested, although some methods demonstrated
to be effective in some specific application [1,2,3,16,17].
Nonlinearity is a crucial aspect and microcantilevers exhibit
both electromechanical and geometric nonlinearities. Choice is
either solving an analytical formulation of the coupled
problem, by reducing DoFs involved, or resorting to a
numerical sequential solution. In this case mesh morphing
inhibits the use of MOR methods [17]. Ad hoc linearization
was already proposed, in absence of geometric nonlinearity
[19], while geometric nonlinear MEMS can be characterized by
2D static models. These allow identifying microsystem
stiffness to be used together with damping for dynamic
analysis. For in-plane configuration 3D effects of electric field
are less relevant, but a very accurate measure of design
parameters is required. In case of out-of-plane layout it is just
the opposite. Experimental validation demonstrated that force
input for 2D models can be calibrated on 3D FEM electrical
analyses. Dynamic analyses can be then performed.
V. CONCLUSION
Two dimensional models are often considered poorly effective
to predict static and dynamic behaviour of microbeam RF-
MEMS. Microcantilevers with in-plane or out-of-plane
bending can be accurately and fairly fast analysed by 2D
models, based on sequential non incremental approach
implemented in FEM, FEM/BEM or DGA. 2D model tuning
can be done by performing a FEM analysis of three
dimensional effects of electric field. Geometric nonlinearity is
relevant for all the specimens tested. In-plane microcantilevers
analysis suffers any inaccuracy in measuring the parameters
used as inputs. For out-of-plane microbenders fringing and
three dimensional effects of electric field have to be carefully
evaluated together with the initial curvature, often present.
Where model order reduction techniques fail because of the
double nonlinearity of actuation and geometrical effects, 2D
models appear suitable to extract few lumped parameters to
perform dynamic analysis. This procedure requires an
evaluation of electric field singularities to correct the
electrostatic force input.
ACKNOWLEDGMENT
This work was partially funded by the Italian Ministry of
University under grant 2005/2005091729, Operative Unit of
Udine (p.i. E.Brusa).
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