Microcantilever specimens for in-plane and out-ofplane 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.Comment: Submitted on behalf of EDA Publishing Association
  (http://irevues.inist.fr/handle/2042/16838

Brusa, Eugenio

Della-Gaspera, Antonio

Munteanu, Mircea Gh.

English

arXiv

Microcantilever specimens for in-plane and out-ofplane 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-Revues

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 Brusa
1
, Antonio Della Gaspera, Mircea Gh. Munteanu
2
,  
Dept. Elect., Manag. Mech. Engineering - Università degli Studi di Udine, via delle Scienze 208 – 33100 Udine, Italy; 
1
eugenio.brusa@uniud.it, 
2
munteanu@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 µm 
800 µm 
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 [µm] Width [µm] Thickness [µm] Gap [µm] 
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)                           ο    Experiments 
−  −    Non incremental    (150 GPa)    •    (166 GPa) 
  •  ANSYS PLANE121/183   (150 GPa)        +        (166 GPa) 
Fig. 3. Example of experimental validation on in-plane microcantilever 
Len
gth
[µm
] 
half-width [µm]
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 µm, 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  
[µm] 
Width 
 [µm] 
Thickness  
[µm] 
Gap  
[µm] 
y 
[µm] 
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 ε0, curvature κ0, 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 κ
ε
κ
ε
 (1) 
where thermal effects are:  
( ) ( )∫==
l
TT dAyxyTxT
0
000 ,; ακαε  (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). 
REFERENCES 
[1]  G.Rebeiz, RF-MEMS: Theory, design and technology, John Wiley 
InterSciences, 2003. 
[2] F.De Bona, E.Enikov ed., Microsystems Mechanical Design, CISM 
Lectures Series, Springer Verlag, ISBN: 3-211-37453-1, 2006. 
[3] S.D.Senturia, Microsystems design, Kluwer, Boston, 2001. 
[4]  F.Najar et al. “Modeling and design of variable-geometry 
electrostatic microactuators”, J.Micromech.Microeng., Vol. 15, 
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[5] B.Choi, E.G.Lovell “Improved analysis of microbeams under 
mechanical and electrostatic loads”, J.Micromech. Microeng. 7 
(1997) 24–29. 
[6]  S.Pamidighantam, R.Puer, K.Baert, H.Tilmans, “Pull-in voltage 
analysis of electrostatically actuated beam structures with fixed–
fixed and fixed–free end conditions”, J. Micromech. Microeng. 12 
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[7]  A.Somà et al. “Large deflections of microbeams under electrostatic 
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[8] C.F.Tsou., H.Yin, W.Fang “On the out-of-plane deformation of V-
shaped micromachined beams”, J. Micromech. Microeng. Vol.11 
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[10] Munteanu M., De Bona F., Collenz A., Brusa E.– “Geometrical 
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microcantilevers”, J.Microsystems Technology, in press. 
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field analysis of electrostatic microactuators dynamics”, 
Proc.ECCOMAS Thematic Conf.Coup.Prob. 2005 Santorini, 
Greece, ISBN: 84-95999-71-4. 
[13] E.Brusa, M.Gh.Munteanu “Coupled-field FEM nonlinear dynamics 
analysis of continuous microsystems by non incremental approach”, 
Analog Int.Circ. Sig. Proc., 48 (2006) pp.7-14. 
[14] P.Bettini, E.Brusa, M.Gh.Munteanu, R.Specogna, F.Trevisan, 
“Static behaviour prediction of microelectrostatic actuators by 
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[15] D.Ekeom, “Electrostatic-Mechanical Direct Coupling and Fem-bem 
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Validation of compact models of microcantilever actuators for RF-MEMS application

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

Eugenio Brusa

Antonio Della Gaspera

Mircea Gh. Munteanu

CiteSeerX

Validation of Compact Models of Microcantilever Actuators for RF-MEMS Application

Microcantilever specimens for in-plane and out-ofplane 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 nonlinearitie

Della Gaspera, A.

Gh. Munteanu, M.

PORTO Publications Open Repository TOrino

Microcantilever specimens for in-plane and out-ofplane
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

BRUSA, Eugenio

A. DELLA GASPERA

M. G.H. MUNTEANU

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

http://hal.archives-ouvertes.fr/docs/00/27/77/11/PDF/dtip08218.pdf

Validation of compact models of microcantilever actuators for RF-MEMS application

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