9-11 April 2008
?EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5
On the Determination of Poisson?s Ratio of Stressed
Monolayer and Bilayer Submicron Thick Films
P. Martins
1
, C. Malhaire
1
, S. Brida
2
and D. Barbier
1
1
Universit? de Lyon, INSA-Lyon, INL, CNRS UMR 5270, Villeurbanne, F-69621, France
(Tel : +33-4-72438795; E-mail: paolo.martins@insa-lyon.fr)
2
ESTERLINE Auxitrol SA, Esterline Sensors Group, F-18941, France
(Tel : +33-2-48667857; sbrida@auxitrol.com)
Abstract-In this paper, the bulge test is used to determine the
mechanical properties of very thin dielectric membranes.
Commonly, this experimental method permits to determine the
residual stress (?
0
) and biaxial Young?s modulus (E/(1- ?)).
Associating square and rectangular membranes with different
length to width ratios, the Poisson?s ratio (?) can also be
determined. LPCVD Si
3
N
4
monolayer and Si
3
N
4
/SiO
2
bilayer
membranes, with thicknesses down to 100 nm, have been
characterized giving results in agreement with literature for
Si
3
N
4
, E = 212 ? 14 GPa, ?
0
= 420 ? 8 and ? = 0.29.
I. INTRODUCTION
The development of Micro Electro Mechanical Systems
(MEMS) has become an economic stake since 80?s and more
recently the Nano Electro Mechanical Systems (NEMS)
began to be developed with the downscaling trend. However,
with the downscaling, accurate measurement of mechanical
properties becomes a hot challenge especially since these
properties may depend on the fabrication process. This may
have consequences on MEMS performances and reliability
[1-3]. Furthermore, architectures are more and more
complex such as multilayers which make the determination
of mechanical properties more difficult for each constituent
material.
So far, no mechanical method exists for the simultaneous
determination of the three main mechanical parameters:
Young?s modulus E, residual stress ?
0
and Poisson?s ratio ?,
except the well-known ?Bulge test? method. Indeed, the
bulge test is commonly used to determine the residual stress
?
0
and the biaxial Young?s modulus E/(1-?) on square or
circular membranes. Some authors have also shown that the
association of membranes with different shapes permits the
determination of the Poisson?s ratio [4, 5]. However, few
studies deal with the determination of the Poisson?s ratio of
very thin film membranes with significant results [5].
Indeed, J. S. Mitchell et al. [6] relate the difficulties in
determining this ratio because the bulge test method is very
sensitive to the geometrical errors.
In this study, an attempt was made to determine E, ?
0
and
? by means of bulge test on very thin (~ 100 nm) Si
3
N
4
square and rectangular membranes with different length to
width ratio (1 < b/a < 12). Few similar studies have been
made on dielectric membranes with thicknesses down to 100
nm.
In this work, we have also assessed the mixture law as a
general rule to extract the Young?s modulus, residual stress
and Poisson?s ratio of each film of submicron thick
Si
3
N
4
/SiO
2
bilayers.
II. BACKGROUND
The bulge test consists in applying a pressure P on a
membrane and in measuring its maximal deflection h at its
center (Fig. 1).
Fig. 1. Bulge test principle
Mechanical properties like Young?s modulus E, the
residual stress ?
0
and the Poisson?s ratio ? can be determine
from [8-10]:
()
3
30
1 244
2
t Et tE
PC(a,b) h hf(,a,b) h
aa112 1
?
=+ +?
?????
(1)
t represents the membrane thickness, 2a and 2b are the
membrane width and length, respectively. C
1
(b/a) and ? are
coefficients that depend on the membrane shape and f (?,
b/a) depends also on the membrane shape and on the
Poisson?s ratio. We can note that in the case of large
deflections (h/t >> 1), the second term in Eq. 1 (depending
on ?) can be neglected.
Lots of studies have been made to optimize the coefficient
values as a function of the membrane shape and in order to
take into account the particular clamping conditions of
micromachined membranes [4-5] [10-13]. Tab. I presents
different values of C
1
(b/a), f (?, b/a)) and ? found in the
literature as a function of the membrane shape.
In this study, the coefficients C
1
(b/a) and f (?, b/a) have
been recalculated with Finite Element simulations (FE) using
ANSYS software in order to verify their validity for very
thin films membranes (~ 100 nm) and to compare with the
literature values (Sees paragraph IV).
This experimental method is efficient to determine the
residual stress ?
0
and the biaxial Young?s modulus E/(1-?)
on thin film. Eq. 1 shows that E and ? are highly correlated
and to find one of these parameters, the other must be
9-11 April 2008
?EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5
assumed. To determine ? independently of E, experiments
performed on square and rectangular membranes must be
associated [4-6]. Indeed, it is possible to compare the ratio of
the cubic coefficients in Eq. 1 to the ratio of the function f
(?, b/a) for square and rectangular membranes (Eq. 2).
4
Square
Rect Rect
square Square Rect
a
Slope f ( , b / a)
Slope f ( , b / a) a
???
=
??
?
??
(2)
Moreover, the experimental results of J. J. Vlassak et al.
[5], for rectangular membranes with b/a > 4, showed that
membranes can be considered as infinite along the length
and the deflection is independent to the aspect ratio.
The other interest of this study is to apply this method in
the case of very thin multilayer films by using the simple
formula of the mixture law (Eq. 3) [14-15] which can be
applied for E, ?
0
and ?.
12 n
composite 1 2 n
total total total
tt t
MMMM= + + ??? +
(3)
M
composite
represents either the biaxial modulus or the
residual stress of the composite membrane with n layers. t
1
,
t
2
? t
n
are the thicknesses of each component layer, t
total
is
the multilayer thickness and M
1
, M
2
? M
n
represent either
E/(1-?) or ?
0
of each layer.
III. SAMPLE PREPARATION AND EXPERIMENTAL SETUP
A. Sample preparation
Dielectric membranes have been fabricated on <100> p-
type, double-side polished, 100 mm silicon substrates using
a standard micromachining process. Silicon nitride films
have been deposited at 835 ?C by LPCVD on thermally
oxidized silicon substrates (Fig. 2). Two wafers issued from
the same fabrication process have been processed: Si
3
N
4
film
(first wafer) with a thickness of 104 nm and bilayer
Si
3
N
4
/SiO
2
film (second wafer) with a thickness of 188 nm
(t(Si
3
N
4
) = 90 nm and t(SiO
2
) = 98 nm). Free standing
membranes have been obtained through silicon anisotropic
etching in a KOH solution (Fig. 2). Several samples have
been obtained with different shapes (square and rectangular).
The resulting sample characteristics are summarized in Tab.
II.
SiO
2
thermal
oxidation
Si
3
N
4
LPCVD
4? Si <100> wafer RIE etching
wet etching
Photolithography
patterning
KOH
etching
wet
etching
Fig. 2. Membranes fabrication: process steps
Pressure-deflection measurements (P (h)) have been
performed using a WYCO NT1100 white-light
interferometer microscope (Fig. 3). Pressures ranging from
10 mbar to 1 bar (depending on geometry) have been
applied. Wax has been used to fix our samples on a sample
holder.
Fig. 3. Optical interferometer setup
IV. RESULTS AND DISCUSSION
A. Finite Element simulations
Finite Element (FE) simulations, using ANSYS?
software, have been developed in order to check if the
coefficients C
1
(b/a) and f (?, b/a) that were found in the
literature were always valid for our very thin membranes.
Fig. 4 shows the evolution of these two coefficients as a
function of the b/a ratio for 100 nm thick, 1 mm width
membranes. An arbitrary Young's modulus value of 220 GPa
and a Poisson?s ratio of 0.3 have been chosen for this study.
The obtained results are in close agreement with literature
values. Moreover, as shown by J. J. Vlassak et al. [5], for
increasing b/a values from 5, C
1
(b/a) and f (?, b/a) become
quasi independent of the b/a aspect ratio (see Fig. 4).
Evolution of C
1
(b/a) and f (?, b/a)
0
1
2
3
4
0246810
b/a (length to width ratio)
C
1
(
b
/a)
an
d
f
(
0
.3, b
/
a)
C1 (b/a)
f (0.3, b/a)
Fig. 4. Evolution of C
1
(b/a) and f (?, b/a) as a function of the shape b/a for
1 mm width, 100 nm thick membranes and assuming a Young's modulus of
220 GPa and a Poisson?s ratio of 0.3
TABLE I
EXAMPLES OF COEFFICIENTS USED FOR DIFFERENT SHAPES
b/a ? [8] C
1
f (?, b/a) f (0.3, b/a)
1 1.26 ? 10
-3
3.39 [5]
3.45 [12]
3.42 [10]
3.39 (FE)
(0.8+0.062?)
-3
[5]
1.994(1-0.271?) [12]
1.91(1-0.207?) [10]
1.82
1.83
1.79
1.80 (FE)
2 2.54 ? 10
-3
2.19 [10]
2.18 (FE)
1.08(1-0.181?) [10] 1.02 [10]
1.0 (FE)
? 2.6 ? 10
-3
2 [5]
2 (FE)
8/[6(1+?)] [5] 1.02
0.9 (FE)
9-11 April 2008
?EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5
In this study, according to the simulations, the analytical
model proposed by J. J. Vlassak et al. is still valid for our
200 nm thick or less membranes and it was used to
determine the mechanical properties of the Si
3
N
4
monolayer
and the Si
3
N
4
/SiO
2
bilayer self-standing films.
B. Experimental results
In the case of 2M, 3M and 4M samples, with the same
geometrical parameters (Tab. II), experimental results have
been obtained with a good reproducibility. Moreover, for all
membranes, no hysteresis phenomenon has been observed
during load and unload cycles showing a linear behavior of
the membranes despite of the large induced deflections (> 90
?m).
A P/h as a function of h? normalized representation of the
pressure-displacement results can be made in order to extract
the y-intercept and the slope of the curves (see fig. 5 and 6).
A Poisson?s ratio of 0.3 for LPCVD Si
3
N
4
was assumed
according to the literature values [16] to calculate the
Young?s modulus E and the residual stress ?
0
for both Si
3
N
4
and Si
3
N
4
/SiO
2
membranes. These values are summarized in
Tab. II for each sample. For the monolayer Si
3
N
4
membranes, a mean Young?s modulus value of 212 ? 14
GPa and a mean residual stress of 420 ? 8 MPa have been
found. For the composite Si
3
N
4
/SiO
2
bilayer membranes, the
results were 147 ? 8 GPa and 107 ? 2 MPa for the Young?s
modulus and the residual stress, respectively.
Si
3
N
4
linearisation
0,0E+00
4,0E+08
8,0E+08
1,2E+09
0,0E+00 2,0E-09 4,0E-09 6,0E-09 8,0E-09
h? (m?)
P/
h
(
P
a
.
m
-1
)
1M
2M
3M
4M
5M
Fig. 5. Normalized pressure-displacement (P = f (h?)) curves of Si
3
N
4
membranes
Si
3
N
4
/SiO
2
linearisation
0,0E+00
2,0E+09
4,0E+09
6,0E+09
8,0E+09
0,0E+00 1,5E-09 3,0E-09 4,5E-09
h? (m?)
P/
h
(
P
a
.
m
-1
)
1B
2B
3B
4B
5B
Fig. 6. Normalized pressure-displacement (P/h = f (h?)) curves of Si
3
N
4
/SiO
2
membranes
TABLE II
RESULTS OBTAINED FOR EACH MEMBRANE USING ? = 0.3
n? 2a (mm) 2b (mm) b/a ?
0
(MPa) E (GPa)
1M 3.104 3.104 1 439?27 210?16
2M 2.131 2.131 1 400?27 217?19
3M 2.131 2.131 1 409?25 214?16
4M 2.14 2.14 1 429?29 211?18
Si
3
N
4
(t
=
1
0
4
n
m
)
5M 1.138 2.131 1.9 414?34 219?26
1B 1.89 1.89 1 104?8 150?14
2B 0.662 0.662 1 113?9 153?17
3B 0.750 0.750 1 100?8 156?17
4B 1.39 7.80 5.6 103?8 139?15Si
3
N
4
/S
iO
2
(
t
= 188 nm
)
5B 0.27 3.28 12.1 115?10 145?16
C. Determination of the Poisson?s ratio
Tab. III shows the different Poisson's ratio values obtained
for different pairs of samples and from the analytical model
proposed by Vlassak et al. and from Eq. 2.
TABLE III
CALCULATED POISSON RATIO FOR Si
3
N
4
AND Si
3
N
4
/SiO
2
MEMBRANES
? ??
5M/1M 0.22 0.05
5M/2M 0.29 0.07
5M/3M 0.27 0.06
Si
3
N
4
5M/4M 0.24 0.05
4B/1B 0,33 0.05
4B/2B 0.38 0.09
4B/3B 0.41 0.09
5B/1B 0.23 0.05
5B/2B 0.29 0.08
Si
3
N
4
/S
iO
2
5B/3B 0.33 0.09
In the case of Si
3
N
4
membranes, a Poisson?s ratio
scattering is observed between 0.22 and 0.29 for an expected
value between 0.25 and 0.3 (LPCVD Si
3
N
4
). As regards the
Si
3
N
4
/SiO
2
bilayer membranes, the composite Poisson?s ratio
results are more scattered and higher than for the Si
3
N
4
monolayer (0.23 < ? < 0.41) whereas values lower than
those of Si
3
N
4
monolayer membranes were expected (except
for 5B/1B samples). Even if the Poisson?s ratio obtained for
Si
3
N
4
monolayers are close to the expected value, it is
obvious that the determination of an accurate Poisson?s ratio
is very difficult, especially for bilayer membranes.
Moreover, the high uncertainties in Tab. III are calculated
from the lateral dimensions uncertainties showing the
importance to know accurately these geometrical parameters.
Membranes that were issued from the same wafer should
have the same mechanical properties but differences lower
than 10 % were observed on Young?s modulus results (Tab.
II). These differences could come from the presence of a
film thickness gradient or a stress gradient across each
wafer. In this study, a mean thickness value was assumed for
each wafer to calculate the mechanical properties. This may
9-11 April 2008
?EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5
also explain the scattering on the Poisson?s ratio values
calculated for Si
3
N
4
. Moreover, sometimes we have
observed underetching profiles on some Si
3
N
4
/SiO
2
membranes when lateral dimensions were lower than 1 mm
(see Fig. 7). Even for large length/width ratio, this slightly
changes the clamping conditions of the membranes and may
etching influence the experimental results.
The model used to determine the Poisson?s ratio is also
critical. For example, for mono or multilayer membranes, the
model of E. Bonnotte et al. leads to extreme Poisson?s ratio
values (>0.45) compared to that of J. J. Vlassak et al. This
last model is the most appropriate to calculate ?.
Fig. 7. Si
3
N
4
/SiO
2
Back-side membranes after anisotropic wet etching
(pictures obtained using interferometer microscope)
However, when two membranes, square and rectangular,
lead to very close values for the Young?s modulus (we can
assume that the film thickness is similar for the two samples
and that the clamping conditions are good), then a precise
determination of the Poisson?s ratio can be made. Indeed, the
1M (square) and 5M (rectangular) Si
3
N
4
membranes, lead to
very close Young?s modulus values and the calculated
Poisson?s ratio (Tab. II and III), in close agreement with
literature values for LPCVD Si
3
N
4
[5, 7, 16]. The same
observation can be made for the 1B and 5B Si
3
N
4
/SiO
2
membranes giving a composite Poisson?s ratio of 0.23.
D. Application of mixture law
Using the mixture law (Eq. 3), an attempt was made in
order to calculate the mechanical properties of thermal SiO
2
with our experimental results.
With E (Si
3
N
4
) ? 212 GPa and E (Si
3
N
4
/SiO
2
) ? 147 GPa,
we obtained E (SiO
2
) ? 87 GPa. With ?
0
(Si
3
N
4
) ? 420 MPa
and ?
0
(Si
3
N
4
/SiO
2
) ? 107 MPa, we obtained a compressive
stress ?
0
(SiO
2
) ? -180 MPa. Finally, with ? (Si
3
N
4
) ? 0.29
and ? (Si
3
N
4
/SiO
2
) ? 0.23, we obtained ? (SiO
2
) ? 0.17.
The mechanical properties calculated on thermal SiO
2
are
in close agreement with the literature values [17-19].
V. CONCLUSION
These results show that the determination of E, ?
0
and ?
by means of the bulge test method remains possible even for
deep submicron monolayer or multilayer thin films. Large
deflections can be imposed to the membranes without any
plastic deformation, which simplifies the associated
mechanical model. Finite Element simulations show that the
coefficient values found by J. J. Vlassak et al. were well
suited for our studied samples. But the accuracy of the
results depends strongly on the geometrical parameters
especially the thickness of the membranes. Young?s modulus
values and residual stress have been determined with
accuracy better than 10 %. But the accuracy on the Poisson's
ratio is about 20% in the best case. This highlights the
difference between theory and experience because achieving
well-controled free-standing submicron thick films is not
trivial. Finally, a simple mixture law has given promising
results on standard materials.
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