Full text of DOI:10.1557/JMR.2002.0266

Stress relaxation of a patterned microstructure on
a diaphragm
D.W. Zhenga)
Department of Materials Science and Engineering, University of California at Los Angeles,
Los Angeles, California 90095-1595
X.H. Wang and K. Shyu
Manifold Engineering, City of Industry, California 91748
C. Chen
Department of Materials Science and Engineering, University of California at Los Angeles,
Los Angeles, California 90095-1595
C-T. Chang
Manifold Engineering, City of Industry, California 91748
K.N. Tu
Department of Materials Science and Engineering, University of California at Los Angeles,
Los Angeles, California 90095-1595
A.K. Mal
Department of Mechanical and Aerospace Engineering, University of California at Los Angeles,
Los Angeles, California 90095-1595
Y.F. Guo
Advanced Interconnect and System Laboratory (AISL), Semiconductor Product Sector (SPS),
Motorola Corp., Tempe, Arizona 85204
(Received 16 December 2001; accepted 25 April 2002)
Stress relaxation of a patterned thin film on diaphragms of different material and
thickness was investigated through experimental study and numerical simulation. The
diaphragm deflections, caused by relaxation of the residual stress in a patterned thin
film residing on top, were measured using a Twyman–Green laser interferometer. The
first diaphragm used was a Si3N4(top)/SiO2/Si composite diaphragm and the second
a 0.5-␮m-thick Si3N4 membrane. Custom-written simulation software, which uses a
novel numerical algorithm named Nonlinear Sequential Analysis (N-LISA), was
utilized to calculate the stress distribution in the patterned thin film and the diaphragm
substrate. Agreement between the model and the experimental results was satisfactory.
Simulation of the system balance between a tensile-stressed circular Ti film and a
stress-free Si substrate of different thickness clearly shows a transition in the substrate
behavior from a pure plate to a pure membrane. Interestingly, the deflection of the
Si substrate caused by the residual stress in the Ti film reaches its maximum at a
certain substrate thickness where plate and membrane characteristics coexist. This
study addresses some basic mechanics issues involved in modern devices dealing with
thin diaphragms.
I. INTRODUCTION
urgent need to thoroughly understand such a system to
facilitate the research and development of many critical
technologies.
Stress relaxation of a patterned thin film on a substrate
is a fundamental solid mechanics problem, which finds
many important applications in modern microelectronics.
Although the interaction of a stressed thin film with a
thick substrate is well understood, the interaction of
a patterned and stressed thin film with a thin substrate has
not been carefully investigated. Currently, there is an
One important application lies in the image placement
of an x-ray mask used for deep-sub-micron lithography.
Here the system typically consists of a thin Si diaphragm
(e.g., 2 ␮m thick) or silicon nitride film, which supports
a layer of patterned high-Z absorber material. As the
supporting substrate thins down to a membrane, both
the bending and in-plane elongation of the substrate need
to be considered and the stresses in the system become
much more complicated. As pointed out by the review
a)Present address: Lightcross, Inc., 2630 Corporate Place, Mon-
terey Park, CA 91754.
e-mail: dzheng@lightcross.com
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D.W. Zheng et al.: Stress relaxation of a patterned microstructure on a diaphragm
articles,1,2 the x-ray mask itself is the most critical and
this modeling effort, we are able to understand the basic
mechanics behind the interaction of a thin patterned film
with residual stress and a thin substrate, where this thin
substrate can be multilayered and may have intrinsic
stresses.
difficult system element of x-ray lithography. The dia-
phragm as well as the absorber materials has residual
stresses. The in-plane distortion of the diaphragm caused
by stress relaxation of the absorber material has to be
corrected to improve the overall accuracy to meet the
stringent total error tolerance of 45 nm for 0.13 ␮m
complementary metal-oxide semiconductor (CMOS) de-
vices.3 One practice utilized by researchers is to fabricate
an exploratory mask, measure the movement of fiducials
distributed on the mask, and record the error.4–7 This
information is then fed back to the original mask design
to reflect the mask pattern shift due to stress. Significant
improvement has been achieved through this approach.
Nevertheless, it is obviously more economical to use
simulations to predict these numbers, and several efforts
are underway for the development of such a software.8–10
Interaction of a prestressed and patterned thin film
with a thin substrate is also quite typical in the field of
micro-electro-mechanical-system (MEMS) devices.11–14
A patterned film residing on top of a diaphragm or beam
supported either at four edges or one side is quite typical
in various MEMS devices. Knowledge of the stress in
various films, and the deflection of the supporting dia-
phragm or beam caused by this residual stess, is impor-
tant for initial device design and long-term reliability. As
an example, Fig. 1(a) displays a schematic diagram
showing the cross-sectional view of a CO chemical sen-
sor. A top view optical image of the sensor is presented
as Fig. 1(b). The deflection of the supporting Si dia-
phragm caused by the stress in various metal lines was
captured by a Twyman–Green laser interferometer15,16
and the fringe images are given in Fig. 1(c).
In summary, from the device process integration point
of view, two things are of key concern. One is how much
wafer deformation occurs due to the stress in various
layers, the other is the possible stress reduction due to
membrane response. An example of the latter is the stress
in e-beam evaporated nickel (Ni). Typically its thickness
on a regular wafer is limited to about 2500 Å. However,
if Ni is deposited onto a membrane, a significant stress
reduction might occur, resulting in more relaxed process
constraints. Stresses in a patterned microstructure could
cause local deflection in the wafer but it is typically not
an issue to lithography for generic CMOS devices. How-
ever, in MEMS devices, stress relaxation in thin dia-
phragms could cause significant local deformation in the
substrate, which could potentially pose an alignment
problem.
II. EXPERIMENTAL
A. Sample fabrication
For the sake of comparison, a 0.5-␮m-thick Ti film
was patterned to different sizes and on top of square
diaphragms, which had different thickness as well as
number of layers of diverse nature. In all cases, the dia-
phragms were formed before the Ti film was deposited
and patterned.
Fabrication of thin diaphragms with controlled thick-
ness was achieved using thin dielectrics as etch masks.
The properties of these samples are described briefly in
the following subsections.
1. Fabrication of a 22-µm-thick Si diaphragm
A 400-␮m-thick p-type (100)Si wafer was used as the
starting substrate, on top of which an n-type epitaxial
layer 22 ␮m in thickness, (100)Si was grown. Low-
temperature oxide (LTO) of 0.35 ␮m and 0.4 ␮m of
plasma-enhanced chemical vapor deposition (PECVD)
SixNyHz were deposited on both sides of the Si wafer as
etch masks. The composition and thickness of the
SixNyHz layer was carefully tuned so the overall bending
moment exerted onto the Si substrate by the SixNyHz and
LTO layers and its temperature coefficient was mini-
mized. The Si etching was done using 30 wt% KOH
solution at 85 °C. This was due to the excellent (100)
over (111) plane etch ratio (greater than 400) compared
to tetra methyl ammonia hydroxide (TMAH; less than
50) and EDP (about 35).17 A constant voltage source
reverse-biased the p-n junction formed by the n-type epi
Si and the p-type substrate. The etch-end-point was de-
tected when all p-type Si was removed as indicated by an
abrupt increase in the current. The uniformity of etching
was about 0.3 ␮m across 2750 × 2750 ␮m area.18
2. Fabrication of a 4.6-µm-thick composite
Si diaphragm
A 300-␮m-thick, (100)Si wafer with 3-␮m heavily-
boron-diffusion-doped (5 × 1019/cm3) top layer (pur-
chased from North Carolina Microelectronics Center,
NC) was used as the starting substrate. Wet oxide 0.5 ␮m
in thickness was grown onto the wafer at 1050 °C, fol-
lowed by a deposition of 0.15-␮m LPCVD stoichiomet-
ric Si3N4. The residual stress due to the wet oxide and
LPCVD nitride blanket film was measured to be
−300 MPa compressive and 1 GPa tensile, respectively,
using FleXus machine (KLA Tencor Corp, San Jose,
CA). The back opening was patterned by first etching
the nitride away using CF4:O2 ס 125:25 mTorr gas
In this paper, some experimental evidence is pre-
sented, from which a few basic concepts to be used in the
numerical modeling are extracted. Then the numerical
method is applied to a model where a fixed metal pattern
is placed on Si diaphragms of different thickness, and
which is difficult to fabricate experimentally. Through
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D.W. Zheng et al.: Stress relaxation of a patterned microstructure on a diaphragm
chemistry at 200 W for 2.5 min, followed by a wet etch
in buffered oxide etchant (BOE) to remove the oxide.
This induced an over-etching of nitride and an etching
into the oxide layer underneath of only about 1–2 kÅ.
The use of BOE wet etchant instead of dry etchant to
remove the oxide helped to minimize possible lateral
etching during Si wet etch at a later stage. Si was etched
first using 30 wt% KOH at 75 °C to a thickness of about
100 ␮m, and then switched into ethylene diamine pyro-
catechol (EDP) solution (purchased from Transene Elec-
tronics, Inc., Danvers, MA) at 90 °C with condensation
setup. Due to the significant etch rate reduction of EDP
when the doping level in Si was degenerated17 (reduction
of about 50 times, when the doping level is higher than
5 × 1019 cm−3), a timed etch was performed to reduce Si
thickness to 4.6 ␮m. The uniformity of the back surface
was better than 0.18 ␮m across a 1600 × 1600 ␮m area.
It was found that if the Si thickness were reduced to
about 2.8 ␮m, then the composite diaphragm Si3N4/
SiO2/Si would start to buckle. Even after the removal of
the Si3N4/SiO2 bilayer dielectrics, the diaphragm re-
mained buckled, which suggested that some plastic de-
formation had occurred in the Si diaphragm.19
3. Fabrication of a 0.5-µm-thick low-stress
Si3N4 diaphragm
The low-stress nitride diaphragm was among the easi-
est to fabricate because of the excellent resistance of
nitride to KOH, TMAH, and EDP etchants. In this proc-
ess, a layer of low-stress nitride was deposited onto both
sides of the (100)Si wafer, and the backside layer was
patterned using the CF4/O2 plasma mentioned previ-
ously. KOH was picked to etch Si because it has an
excellent (100) over (111) plane etch ratio and is less
expensive.
B. Analysis of diaphragm deformation
1. Contribution from thin dielectrics
To evaluate the effect from the thin dielectrics, two
groups of samples were fabricated for comparison. A
half-micron-thick Ti film was patterned into dot (circu-
lar) shapes with different diameters on top of the 22-␮m
composite Si diaphragms, as described in Sec. II. A. 1
(fabrication of a 22-␮m-thick Si diaphragm). This makes
up Group A. The comparison Group B consisted of pure
22-␮m-thick Si diaphragm with dots of patterned Ti re-
siding on top, identical to Group A. Group B samples
were simply made by removing the dielectric layers from
the Group A sample; the cross-sectional schematics are
shown in Figs. 2(a) and 2(b). After image processing, the
diaphragm deflection along a half meridian, caused by
the residual stress in the Ti film, is plotted as Fig. 2(c).
The diaphragm deflections with and without the dielec-
trics can be seen to be signficantly different.
FIG. 1. (a) Schematic view of the cross-section of a CO sensor. (b)
Optical microscope image of the top view of the sensor showing the
metal layers. (c) Interferometric images showing the deflection of
the sensor diaphragm caused by the residual stress in various thin film
layers.
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D.W. Zheng et al.: Stress relaxation of a patterned microstructure on a diaphragm
One distinction between Group A and Group B
substrate and the residual stress of the dielectric coatings
have to be considered. On the other hand, there is obvi-
ously an increase in the overall bending stiffness due to
the presence of the thin dielectrics. However, considering
that (i) the Young’s moduli of the 3 materials are in a
decreasing sequence of nitride (270 Gpa), Si (169 Gpa),
SiO2 (72.9 Gpa); (ii) the thickness of Si is much larger
than the other 2 dielectrics Si (22 ␮m), SiO2 (0.4 ␮m),
Si3N4 (0.35 ␮m), and (iii) the bending stiffness is in-
versely proportional to the flexural rigidity EI where E
and I are the Young’s modulus and area moment of in-
ertia of the cross section respectively, this effect is prob-
ably not adequate to explain the change of diaphragm
deflection alone.
samples is that the 22-␮m-thick Si diaphragm behaves
like a plate, while the composite sample behaves like a
plate bonded to a membrane (dielectrics) with residual
stress. If we recall the difference between a plate and a
membrane,19 vertical loads in a plate are balanced by
bending and shear stresses, while in a membrane, they
are balanced by the vertical components of the tensile
“membrane stresses” caused by the membrane’s local
curvature. In general, when the substrate is relatively
thick, e.g., 200 ␮m or thicker for a 4-in. wafer, the effect
from the thin dielectric is negligible. For example, sup-
pose we use a FleXus machine to measure the stresses in
multi-layered thin films on a thick wafer. When the stress
in the 2nd deposited layer is calculated, the wafer cur-
vature effect from the 2nd layer is typically considered as
linearly additive. The bending stiffness of the substrate
dominates over the residual stress in the 1st layer. How-
ever, in the present case, when the substrate is thinned
down to 20 ␮m, both the bending stiffness of the
2. Effect of the overall bending moment
For comparison purposes, two additional groups of
samples were fabricated and named Groups C and D.
Group C consists of samples based on the description of
Sec. II. A. 2. A schematic diagram of the cross-section
of this group is plotted in Fig. 3(a). Group D is different
in the sense that the pad oxide used was reduced from
0.5 to 0.1 ␮m; the schematic diagram of the cross section
of this group is shown in Fig. 4(a). Identical Ti dots were
patterned on top of these diaphragms, whose deflection
caused by the residual stress in the Ti pattern are dis-
played in Figs. 3(b) and 4(b), respectively.
As previously discussed, considering the residual
stress level in LPCVD Si3N4 and wet thermal oxide, it
appears that the bending moment exerted onto the thin Si
FIG. 2. Schematic diagram of the cross-section of (a) a composite
Si3N4 (0.16 ␮m) SiO2(0.5 ␮m)/Si (22 ␮m) Si diaphragm, and (b) a
22-␮m Si diaphragm. (c) Comparison of total diaphragm deflection
caused by the residual stress in the Ti dot pattern residing on top:
(Group A) pure Si diaphragm, (Group B) Si diaphragm with dielectric
capping layer.
FIG. 3. (a) Schematic diagram of the cross-section of a composite
Si3N4 (0.16 ␮m)/SiO2(0.5 ␮m)/Si diaphragm. (b) Deflection curve of
the diaphragm caused by the residual stress in the Ti dot pattern
residing on top.
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D.W. Zheng et al.: Stress relaxation of a patterned microstructure on a diaphragm
substrate is balanced better in the case of Group C
bending stiffness is negligible. It has a very steep slope
around the Ti pattern edge due to the residual stress in the
Ti pattern, and the bending stiffness of the membrane
does play a role in this very local area.
sample. In contrast, Group D has unbalanced bending
moment from the tensile stress in the Si3N4. We believe
this is the main reason why the substrate bending is much
smaller for the latter case, because more is needed to
rotate the unbalanced in-plane tensile stress. On the other
hand, as in the previous case (Sec. II. B. 2), there is a
difference in the overall bending stiffness due to
the Si3N4 layer. The contribution from the stresses in the
dielectric layer needs further investigation.
III. ANALYSIS OF THE THIN
DIAPHRAGM DEFLECTION
It is well known that as the thickness of the substrate
decreases, the behavior of the substrate changes from a
plate effect dominated behavior to a membrane effect
dominated behavior. Based on the observations in
Sec. II, for a relatively thick Si substrate, the residual
stress in a pattern residing on top would bend it to the
shape of a relatively low-order polynomial. On the other
hand, for an ultrathin nitride membrane, the deformation
of the substrate occurs only in a localized area immedi-
ately below the edge of the pattern. Particularly for the
latter case, modeling efforts would be very helpful in
understanding the mechanics of the deformation in that
localized area, since very steep rotation of the membrane
could result in very high stresses. It is apparent that for
certain substrate thicknesses both the plate effect (sub-
strate bending stiffness) and the membrane effect (rota-
tion of the residual stress in a membrane to balance
external force) can play a role. The mathematical formu-
lation of the associated nonlinear boundary value prob-
lem is well known21 and will not be repeated. In the
present problem, where the maximum vertical deflection
is much smaller than the radial dimensions of the plate,
the formation leads to the well known von Ka´rma´n
3. Stress relaxation on a pure membrane
Similar to the previous case, a Ti blanket film was
e-beam evaporated onto a 0.5-␮m-thick LPCVD Si3N4
diaphragm described in Sec. II. A. 3 and patterned to
dots 1000, 100, and 50 ␮m in diameter. A sche-
matic cross-sectional view of the sample is provided as
Fig. 5(a), and the deflection of the supporting diaphragm
is displayed in Fig. 5(b).
Comparing the deflection profile of the thin Si3N4 dia-
phragm with what was observed in the former cases,
the diaphragm has no deflection immediately below the
Ti pattern and far from the Ti pattern; only the area of
the diaphragm immediately below the edge of the Ti
pattern has a drastic change in shape. Recalling that in a
membrane, external vertical loads are balanced by rota-
tion of the in-plane stress, this thin nitride diaphragm
behaves primarily like a pure membrane. The membrane
does not bend outside the Ti pattern area, because its
FIG. 4. (a) Schematic diagram of the cross-section of a composite
Si3N4 (0.16 ␮m)/SiO2(0.1 ␮m)/Si diaphragm. (b) Deflection curve of
the diaphragm caused by the residual stress on the Ti dot pattern
residing on top.
FIG. 5. (a) Schematic diagram of the cross-section of a low-stress
Si3N4 (0.5 ␮m) diaphragm. (b) Deflection curve of the diaphragm
caused by the residual stress in the Ti dot pattern residing on top.
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D.W. Zheng et al.: Stress relaxation of a patterned microstructure on a diaphragm
equations21 with fixed conditions at the diaphragm
boundary. Since the equations are highly complex and
nonlinear, they need to be solved numerically.
pure plate to a pure membrane effect could be identified
clearly. In Fig. 6(a), as the Si diaphragm thickness
decreases from 10 to 3 ␮m, its deflection increases, as
can be expected. However, as the Si diaphragm thickness
decreases further, the maximum diaphragm deflection
starts to decrease, as can be seen in Fig. 6(b).
At a thickness of 0.5 ␮m, the Si diaphragm deforms
locally, and the effect from a prestressed pattern sitting
on top only propagates roughly within a distance of 2R,
where R (ס 100 ␮m) is the radius of the Ti pattern. This
An algorithm named Nonlinear Sequential Analysis
(N-LISA) has been recently developed to solve the von
Ka´rma´n equations based on the modified steepest de-
scent method and the initial stress method.22,23 The two
methods form the basic framework of N-LISA; however,
a few additional features have been introduced in the
approach utilized in this paper. The domain discretization
procedure and the formation of the basic algebraic equa-
tions in N-LISA follow that of the finite element method
(FEM). For nonlinear deformation of a thick diaphragm,
which includes the in-plane membrane stresses as well as
the bending stresses of the plate, there are usually many
stationary points in the solution space of the algebraic
equations depending on the specific problem. Thus the
trial solution can easily converge to stationary points
other than the real solution during iteration cycles. To
avoid this, the following steps were taken.
TABLE I. Summarization of the material mechanical properties used
in the simulation.
Young’s modulus Poisson’s Residual stress
Material
(GPa)
ratio
(MPa)
Ti
110
270
72.9
169
270
0.31
0.27
0.17
0.28
0.27
337
1000
−300
0
Si3N4 (stoichiometric)
SiO2
Si
Si3N4 (low stress)
28.54
First of all, the residual stress in the thin film pattern
was divided into many portions, which were unevenly
partitioned. At the beginning, a very small portion was
applied and the corresponding solution was calculated,
and this was called one sequence. Then the residual stress
value was added and obtained the solution; this was re-
peated until the deformation corresponding to the real
residual stress in the pattern was reached. The reason for
this is that the capability of the diaphragm to resist the
external load is quite sensitive to the shape of the dia-
phragm, especially for the high residual stress system.
We have observed that during the numerical experiment,
too fast of a loading at the beginning leads the iteration
solution to converge to a deformation shape, which has
many local ripples. On the other hand, within one se-
quence, the convergence of the problem with a differen-
tial loading on top of the deformation shape from the
previous sequence step was good. The details of the nu-
merical simulation will be published elsewhere.
To gain some physical insights into our problem, a
virtual case was simulated. A circular, 0.5-␮m-thick,
100-␮m-diameter Ti disc was patterned on top of a cir-
cular Si diaphragm with a radius of 1380 ␮m. Such a
geometry would allow an axisymmetric model to be util-
ized. The boundary was assumed to be fixed, and an
in-plane normal initial stress of 337 MPa was assigned
to the Ti disc, which corresponded to the real value
measured from e-beam deposited samples. The Si dia-
phragm was assumed to be stress-free; the other material
properties used in the model are listed in Table I. The
problem is clearly axisymmetric. The thickness of the Si
diaphragm was allowed to vary, and the calculated de-
flections of the Si diaphragm are plotted in Figs. 6(a) and
6(b). Since the Si diaphragms used in this simulation do
not have residual stress, the diaphragm transition from a
FIG. 6. The Si diaphragm deflection caused by a 100-␮m-diameter
pad residing on top. (a) Si diaphragm thickness ranges from 10 to
3.0 ␮m. (b) Si diaphragm thickness ranges from 3 to 0.5 ␮m.
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D.W. Zheng et al.: Stress relaxation of a patterned microstructure on a diaphragm
curve is labeled “MEMBRANE” in the plot. If we look at
as 100 MPa (tensile). Again the total deflection of the Si
diaphragm is plotted against its thickness, which is
shown by the stars in Fig. 7. It is very clear that the
presence of a tensile residual stress will decrease the total
diaphragm deflection, and it also shifts the diaphragm
thickness corresponding to the maximum deflection to a
smaller value.
the diaphragm deflection profile for the case of the Si
thickness of 3.0 ␮m, it is a combination of the “PLATE”
and “MEMBRANE” type of deformation curve, and is
therefore a combined “MEMBRANE + PLATE” case, as
marked in the figure. It is interesting to note that in this
transition region, the Si diaphragm’s total deflection
reaches it maximum. If we plot the total deflection of the
Si diaphragm versus its thickness, we obtain the open
dots shown in Fig. 7. To see the effect of the residual
stress, we fixed all other conditions except the Si dia-
phragm’s residual stress, which was arbitrarily assigned
Now it is time to look into the real and more difficult
cases mentioned in Sec. II. The 200-␮m-diameter Ti pat-
tern of Sec. II. A. 2 is selected for study. The simulation
procedure is as follows. Since we could not stimulate the
deposition of a blanket film on a diaphragm, we assume
that the film was deposited on a bulk substrate and then
the Si was etched from the bottom to form a diaphragm.
An initial stress was assigned to the oxide to achieve a
compressive stress value of 300 MPa after stress relaxa-
tion on a bulk Si substrate, which was 525 ␮m thick. This
procedure was repeated for nitride and Ti film to obtain
their corresponding residual stress in accordance with the
FleXus-machine-measured values. Then the Si, as out-
lined by the dashed lines in Fig. 8, is removed layer-by-
layer using the numerical etching algorithm.19,24 Similar
work was done to the Ti film, which was partially etched
eventually. At this stage, the composite diaphragm de-
flection could be calculated and converted to equivalent
interferometric fringes assuming that the diaphragm cen-
ter is the brightest spot. Since only an axisymmetric
model was available, the calculated fringes were
bounded within a circle. A comparison between the
experimental and numerical fringe images for this case is
presented in Fig. 9. The results obtained are qualitatively
similar. Therefore, it seems that our simulation approach
is reasonable.
FIG. 7. The maximum of Si diaphragm deflection, caused by the
stress relaxation of a pre-stressed Ti pattern residing on top, versus
the diaphragm thickness.
The thin low-stress nitride diaphragm described in
Sec. II. A 3 is another interesting case for modeling. The
nitride diaphragm is a half-micron thick, so a laser light
can penetrate through it and reflect off the Ti/nitride
interface. A very thin coating of e-beam Al of about
FIG. 8. The schematic cross-sectional view of a test sample, which
consists of a Ti pattern residing on Si3N4/SiO2/Si composite
diaphragm.
FIG. 9. A comparison between the calculated and the experimental fringe pattern for a 200-␮m-diameter Ti dot on a 4.5-␮m-thick composite Si
diaphragm: (a) experimental, (b) simulated.
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D.W. Zheng et al.: Stress relaxation of a patterned microstructure on a diaphragm
Initiative Reward (SBIR) Contract No. 9960511. The au-
thors are grateful to Jordan Neysmith and Bernard Hart
from ELOtechnologies, Inc., Torrance, CA, for the
proofreading of this paper.
REFERENCES
1. R.E. Acosta, “A review of deposition, patterning, and stress of
metal films used as absorber in masks for synchrotron X-ray
lithography”, in Proceedings of the Second International Sympo-
sium on Electrochemical Microfabrication, edited by M. Datta,
K. Sheppard, and J.D. Dukovic (Electrochemical Society, Pen-
nington, NJ, 1995), p. 88.
2. J. Silverman, J. Vac. Sci. Technol. B 15, 2117 (1997).
3. 1997 Semiconductor Industry Association Roadmap, Sematech,
Austin, TX.
FIG. 10. Calculated low-stress nitride deflection caused by a pre-
stressed Ti pattern residing on top.
4. S. Tsuboi, Y. Yamashita, T. Matsuo, T. Ohta, T. Shoki, T. Yoshihara,
T. Taguchi, S. Mitsui, S. Noda, K. Suzuki, H. Hoga, Y.I. Yamaguchi,
and K. Suzuki, Jpn. J. Appl. Phys. 35, 2845 (1996).
5. A.H. Fisher, M.F. Laudon, R.L. Engelstad, E.G. Lovell, and
F. Cerrina, J. Vac. Sci. Technol. B 15, 2249 (1997).
6. W.J. Dauksher, D.J. Resnick, K.D. Cummings, J. Baker, R.B. Gregory,
N.D. Theodore, J.A. Chan, W.A. Johnson, C.J. Mogab, M-A. Nicolet,
and J.S. Reid, J. Vac. Sci. Technol. B 13, 3103 (1995).
7. S.C. Nash and T.B. Faure, J. Vac. Sci. Technol. B 9, 3324 (1991).
8. M. Oda, S. Uchiyama, and T. Matsuda, Jpn. J. Appl. Phys. 35, part
I, 6469 (1996).
9. S. Uchiyama, M. Oda, and T. Matsuda, J. Vac. Sci. Technol. B 14,
4332 (1996).
10. M. Laudon, A. Fisher, R. Engelstad, F. Cerrina, K. Cummings,
W. Dauksher, D. Resnick, W. Johnson, and D. Puisto, Microelec-
tron. Eng. 35, 549 (1997).
200 Å does not improve this situation. Therefore, while
the overall deformation of the nitride diaphragm could be
identified unambiguously, the situation around the Ti
bump edge is not clear. This is a situation in which this
simulation could help in gaining some insight. The re-
sidual stress of the nitride film was measured using the
bulge-testing method.19,20 The mechanical properties
used for low-stress nitride and Ti are given in Table I.
The simulation result is displayed as Fig. 10. It is obvious
that the general shape of the calculated and measured
deflection curves match very well. Since the interfero-
metric image does not have sufficient resolution to dis-
tinguish between the calculated curve of Fig. 10 and the
measured curve of Fig. 5, and since our model is axisym-
metric while the real diaphragm is a square, a direct
comparison between the two cases was not attempted.
11. G.F. Cardinale, D.G. Howitt, W.M. Clift, K.F. McCarty, D.L. Medlin,
P.B. Mirkarimi, and N.R. Moody, Thin Solid Films 287, 214 (1996).
12. G.F. Cardinale, D.G. Howitt, K.F. McCarty, D.L. Medlin,
P.B. Mirkarimi, and N.R. Moody, Diamond Relat. Mater. 5, 1295
(1996).
13. E.H. Yang, S.S. Yang, and S.H. Yoo, Appl. Phys. Lett. 67, 912
(1995).
14. X. Ding, W.H. Ko, and J.M. Mansour, Sens. Actuators A 21–23,
866 (1990).
IV. CONCLUSION
15. Handbook on Experimental Mechanics, edited by Albert S. Kobayashi
(Society for Experimental Mechanics, Prentice-Hall, Englewood
Cliffs, NJ, 1987).
16. D. Zheng, R.P. Hwang, X.Y. Dou, C.P. Yeh, M. Prakash, K. Boardman,
and G. Ridsdale, in Proceedings of 47th Electronic Components
and Technology Conference (IEEE, New York, 1997), p. 1176.
17. Marc Madou, Fundamentals of Microfabrication (CRC Press,
Boca Raton, FL, 1997).
18. D.W. Zheng, Xinhua Wang, K. Shyu, C-T. Chang, Y. Guo, V. Sarihan,
W. Wen, and K.N. Tu, J. Vac. Sci. Technol. B 17, 2178 (1999).
19. D.W. Zheng, Ph.D. dissertation, University of California at Los
Angeles (1999), p. 37.
20. D.W. Zheng, Y.H. Xu, Y.P. Tsai, K.N. Tu, P. Patterson, B. Zhao,
Q-Z. Liu, and M. Brongo, Appl. Phys. Lett. 76, 2000 (2002).
21. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and
Shells (McGraw-Hill, Columbus, OH, 1959).
The deflection of a diaphragm of different thicknesses
caused by the residual stress in a patterned thin film residing
on top was studied experimentally and modeled using nu-
merical simulation. The transition of the diaphragm behav-
ior from a pure plate, to plate plus membrane, and then to
a pure membrane is clearly shown using numerical simu-
lations. The numerical model was supported by experimen-
tal measurements of the out-of-plane deformation of such a
diaphragm. We found that at certain diaphragm thickness,
the diaphragm’s total deflection reaches a maximum with
fixed residual stress in the pattern sitting on top. A tensile
stress in the diaphragm would shift this peak position and
reduce the total deflection.
22. O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method,
4th ed. (McGraw-Hill, Columbus, OH, 1991).
23. K. Feng, Numerical Computation Methods (Defense Industry
Press, Beijing, China, 1978, in Chinese).
24. X.H. Wang, K. Shyu, C-T. Chang, D.W. Zheng, and W. Wen, in
Materials Reliability in Microelectronics IX (Mater. Res. Soc.
Symp. Proc. 563, Warrendale, PA, 1999), p. 243.
ACKNOWLEDGMENTS
Manifold Engineering and the University of California
at Los Angeles acknowledge the support from the
National Science Foundation under Small Business
1802
J. Mater. Res., Vol. 17, No. 7, Jul 2002
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