Stresa, Italy, 25-27 April 2007
MODELING OF T-SHAPED MICROCANTILEVER RESONATORS
Margarita Narducci, Eduard Figueras, Isabel Gr?cia, Luis Fonseca, Joaquin Santander, Carles
Can?
Centro Nacional de Microelectr?nica de Barcelona, CNM-IMB (CSIC), Campus UAB, 08193
Bellaterra, Barcelona, Spain. Phone: +34935947700, Fax: +34935801496
Email: margarita.narducci@cnm.es
ABSTRACT
The extensive research and development of
micromechanical resonators is trying to allow the use of
these devices for highly sensitive applications.
Microcantilevers are some of the simplest MEMS
structure and had been proved to be a good platform due
to its excellent mechanical properties. A cantilever
working in dynamic mode, adjust its resonance frequency
depending on changes in both, the spring constant ( k )
and mass ( m ) of the resonator. The aim of this work was
to model a cantilever structure to determine the optimal
dimensions in which the resonance frequency would be a
function dominated by mass changes and not stiffness
changes. In order to validate the model a set of
microcantilevers were fabricated and characterized.
Index Terms ? cantilever, resonance frequency,
piezoactuator, piezoresistor resonator.
1. INTRODUCTION
Silicon microcantilevers have been increasingly used for
sensor applications, due mainly to its high sensitivity.
This type of sensor can be classified into two kinds,
depending on the mode of operation, static and dynamic.
The static cantilever detects deformations and the
dynamic detects resonant frequency shift. The present
work is focused on dynamic cantilevers.
The resonance frequency on a cantilever working in
dynamic mode is given by equation (1).
effm
kf
pi2
1= (1)
Where k is the spring constant and effm is the
effective or dynamic mass. As show equation (1), any
change in effm and k will change the frequency of the
cantilever. However a varying spring constant is one
variable that is usually not taken into consideration in
studies dealing with cantilever sensors [2].
This paper proposes a model of a T-shaped cantilever
in which the change on the spring constant caused by an
extra mass has been included in the calculations. This
model allow to determine the optimal geometry
dimensions of the structure in order to make the
resonance frequency dominated by mass changes instead
of stiffness and therefore improve the sensitivity of the
cantilever resonator.
2. CANTILEVER DESIGN
The cantilevers were designed to resonate in flexural
mode perpendicular to the substrate and were driven to
reach their mechanical resonance by a ceramic-insulated
multilayer piezoactuator glued at the backside and the
resonance frequency was measured by four piezoresistors
in a Wheatstone bridge configuration. The resonators
were designed as three beams that are hold together by
means of an extra rectangular mass, as can be seen in
figure 1.
Figure 1. Diagram of a T-shaped cantilever.
Where 1l , 1w and h are, respectively, the length,
width and thickness of each beam, 2l and 2w the length
and width of each extra mass.
?EDA Publishing/DTIP 2007
ISBN: 978-2-35500-000-3
Margarita Narducci, Eduard Figueras, Isabel Gr?cia, Luis Fonseca, Joaquin Santander, Carles
Can?
Modeling of T-Shaped Microcantilever Resonators
The resonance frequency of each single cantilever
after the attachment of the extra mass is given by the next
equation [15]:
2124,02
1
mm
kkf
+
?+=
pi (2)
Where kk ?+ is the spring constant after the mass is
added, 124,0 m is the effective mass of the cantilever
beam and 2m is the added mass.
In order to calculate the spring constant after the
attachment of the mass ( kk ?+ ), the general equation of
the elastic curve was used:
)(2
2
LxEIF
dx
yd ??= (3)
Where F is the applied force to the structure, E is
the Young modulus of the material and I is the moment
of inertia. The model was simplified assuming that the
three beams oscillate in the first mode and at the same
time, therefore the equation (3) was applied to one single
beam and its attached mass. As the equation (3) depend
on x , it was necessary to split the cantilever into two
regions. The first region lay between 10 lx ?? , a mass
free cantilever, and the second one between 21 lxl ?? ,
the extra mass [5]. That is:
?
?
?
???
?
?+??=
?+??=
=
121
22
2
2
121
12
1
2
2
2
));((
));((
lxllxEIF
dx
yd
lxllxEIF
dx
yd
dx
yd (4)
Where I is the area moment of inertia of a rectangular
beam and can be expressed as:
12
3
1
1
hwI = ;
12
3
2
2
hwI = (5)
The equation (4) was integrated two times in order to
find the deflection )(xy , see equation (6):
?
?
?
???
?
?+++??=
?+++??=
=
1
2
21
3
2
2
1
2
21
3
1
1
);2)(6(
);2)(6(
)(
lxdcxxllxEIFy
lxbaxxllxEIFy
xy (6)
Subjected to:
)()();()(
0)0(;0)0(;0)0(
1
'
21
'
11211
''
1
'
11
lylylyly
yyy
==
=== (7)
Using the cantilever boundary conditions, the
integration constants were calculated:
1
22
2
12
3
112
2
11
3
1
1
2212
2
11211
2
1
6
)33(
2
)22(
0
0
w
wllwlwllwld
w
wllwlwllwlc
b
a
??+?=
++???=
=
=
(8)
Substituting (5) and (8) into (6), the equation for the
deflection )(xy was obtained. Then it is necessary to
evaluate the solution in 21 llx += to determine maxy ,
that is:
???
?
???
? +++?=
1
2
21
1
2
2
1
1
3
1
2
3
2
3max
334
w
ll
w
ll
w
l
w
l
Eh
Fy (9)
Next, to calculate kk ?+ , the general equation for the
spring constant was used:
maxy
Fk = (10)
Where F is the force acting on the spring, and maxy
is the maximum displacement of the spring. Replacing (9)
into (10) and simplifying: the new spring constant can be
shown to be:
( )22212221231132 21
3
334 wllwllwlwl
wwEhkk
+++
?=?+ (11)
If the corresponding area to the second region of the
cantilever would be eliminated ( 02 =l and 02 =w ), then
the equation (11) would become into the well known
equation of the spring constant for a simple rectangular
cantilever [2, 5]:
3
1
1
3
4 l
wEhk ?= (12)
?EDA Publishing/DTIP 2007
ISBN: 978-2-35500-000-3
Margarita Narducci, Eduard Figueras, Isabel Gr?cia, Luis Fonseca, Joaquin Santander, Carles
Can?
Modeling of T-Shaped Microcantilever Resonators
3. FABRICATION PROCESS
The fabrication process is illustrated in Figure 2. The
resonators were fabricated following a 7-mask process,
starting with an N-type SOI double side polish wafer. The
silicon layer is 15?m thick over a 2?m buried oxide and
450?m of bulk silicon. The process starts with a 180?
grown dry oxide layer and then a 1175? silicon nitride
layer is deposited (Figure 2a). The first level mask is used
to define the active zone on the frontside, and then a
10600? field oxide is grown. After that, with the second
mask the backside window is defined (Figure 2b). Next,
the silicon nitride on the frontside is removed and a
Boron implantation of 1? 1015 cm-2 and 50eV is
performed to define the resistivity of the Wheatstone
bridge resistors, subsequently trough the third mask a
new implantation fix the resistivity of contacts and
heaters. Then a 1.3?m BPTEOS oxide is deposited
(Figure 2c). The fourth mask is used to open contacts.
Aluminum is deposited and patterned using the fifth mask
(Figure 2d) to define metal connections and bonding
pads. Afterwards a 0.4?m PECVD oxide and 0.4?m
PECVD nitride are deposited (passivation layer) and
patterned with the sixth mask (Figure 2e). The seventh
mask is used to define the motif on the frontside, after
that using the nitride mask on the backside the silicon
substrate is etched using a KOH bath. After that, the
15?m membrane is etched by reactive ion etching and the
cantilever is released (Figure 2f). Finally the whole
structure is glued to a ceramic-insulated multilayer
piezoactuator. The figure 3 shows a photograph of one
fabricated cantilever.
Two chips were fabricated. One chip (chip 1) contains
structures of dimensions in the range of 400?300?m2 and
the other (chip 2) contains structures of dimensions in the
range of 200?150?m2. The dimensions of the fabricated
cantilevers are shown in the table 1.
l * (?m) 1w (?m) 2l (?m) 2w (?m)
1 400 64 350 100
2 400 64 300 100
3 400 64 200 100
4 400 64 70 100
5 200 32 175 50
6 200 32 150 50
7 200 32 100 50
8 200 32 30 50
Table 1. Dimensions of the cantilevers fabricated. For all
structures the thickness h is 15?m. *where 21 lll +=
Figure 2. Fabrication process flow.
?EDA Publishing/DTIP 2007
ISBN: 978-2-35500-000-3
Margarita Narducci, Eduard Figueras, Isabel Gr?cia, Luis Fonseca, Joaquin Santander, Carles
Can?
Modeling of T-Shaped Microcantilever Resonators
Figure 3. Photograph of a fabricated cantilever.
4. DEVICE CHARACTERIZATION
To perform the characterization of the cantilevers, first
the piezoactuator was connected to the source of the
network analyzer (HP 4195) and the detector
(Wheatstone bridge) to a voltage amplifier (G=10) which
in turn was connected to the input of the HP 4195. The
instrumental setup is shown in figure 4.
Using this experimental setup was possible to obtain
the magnitude and phase response and therefore measure
resonance frequency ( rf ) and quality factor ( Q ). The
measurements were performed with eight (8) samples of
each cantilever. The average measured values are
contained in table 2. The estimated measurement error
was calculated to be %1_ ?averf and %15?aveQ .
Figure 4. Experimental setup to perform the
measurements.
Resonance frequency (
rf ) in KHz
Quality factor ( Q )
1 98 680
2 92 870
3 89 810
4 97 820
5 354 950
6 334 1200
7 324 1050
8 400 940
Table 2. Results for the experimental measurements.
In order to compare experimental values with the
theoretical model (equation 2), the relationship between
rf and extra mass is illustrated in figure 5.
Figure 5. Plot of resonance frequency vs. extra mass length
?EDA Publishing/DTIP 2007
ISBN: 978-2-35500-000-3
Margarita Narducci, Eduard Figueras, Isabel Gr?cia, Luis Fonseca, Joaquin Santander, Carles
Can?
Modeling of T-Shaped Microcantilever Resonators
As can be seen from the theoretical trace in figure 5
there are two kinds of behaviors, the first one lay between
ll 8.00 2 ?? and it is dominated by changes on the
resonator extra mass (more mass, lower value of
resonance frequency) and the second one lll ?? 28.0
dominated by changes on the spring constant (more mass,
slightly higher value of resonance frequency). Whereas in
the trace of the experimental measurements (chip 1 and
chip 2), the two behaviors are delimited by the 60% of l .
This difference among the theoretical and the
experimental trace could be attribute to an error on
estimating the effective mass owing to an increment of it
due to the extra mass. In consequence, the effective mass
of this cantilever should be recalculated. With the
purpose of estimate a more accurate value of the effective
mass, the cantilevers were simulated using Finite Element
Analysis (FEA) with ANSYS. From the new values of
effm obtained with the FEA and the spring constant
(computed with the equation 11) the resonance frequency
was recalculate (equation 1). To compare these results
with the experimental values, the relationship between
rf and extra mass is illustrated in figure 6, which shows
clearly a better agreement with the measured data.
5. CONCLUSIONS
In conclusion, this paper focuses on the design and
modeling of a Silicon-based microcantilever resonator for
highly sensitive applications. A T-shaped cantilever
model for the resonant frequency had been calculated,
finding a polynomial function with a minimum point
marking the limit between the two types of behaviors
previously mentioned. This minimum point for the added
mass was measured to be in ~60% of the total length of
the cantilever. So, attaching a rectangular mass with
2l <0.6 l , shifts in frequency could be attribute to changes
on the resonator mass. There was good agreement
between predicted behavior for the resonant frequency
and experimental values. Featuring high sensitivity, only
to mass changes, this microcantilever resonator is
promising to become a platform for sensor applications.
Consequently, an extension of this work would be to
deposit polymer on the cantilever in order to study the
sensitivity of the resonator.
6. ACKNOWLEDGEMENTS
This work was supported by the CICYT-TIC-2002-0554-
C03-02 project.
7. REFERENCES
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Figure 6. Plot of resonance frequency vs. extra mass length
?EDA Publishing/DTIP 2007
ISBN: 978-2-35500-000-3
Margarita Narducci, Eduard Figueras, Isabel Gr?cia, Luis Fonseca, Joaquin Santander, Carles
Can?
Modeling of T-Shaped Microcantilever Resonators
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?EDA Publishing/DTIP 2007
ISBN: 978-2-35500-000-3