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273
ISSN 1392 - 1207. MECHANIKA. 2012 Volume 18(3): 273-279
Residual stress in a thin-film microoptoelectromechanical (MOEMS)
membrane
K. Malinauskas*, V. Ostaševičius**, R. Daukševičius***, V. Grigaliūnas**** *Kaunas University of Technology, Kęstučio 27, 44312 Kaunas, Lithuania, E-mail: [email protected]
**Kaunas University of Technology, Kęstučio 27, 44312 Kaunas, Lithuania, E-mail: [email protected]
***Kaunas University of Technology, Studentų 65, 51369 Kaunas, Lithuania, E-mail: [email protected]
****Kaunas University of Technology, Savanorių 271, 50131 Kaunas, Lithuania, E-mail: [email protected]
http://dx.doi.org/10.5755/j01.mech.18.3.1880
1. Introduction
Microoptoelectromechanical systems (MOEMS)
is not some special class of microelectromechanical sys-
tems (MEMS) but in fact it is MEMS merged with micro-
optics which involves sensing or manipulating optical sig-
nals [1]. There are numerous membrane-based MOEMS
devices involved in various precise measurements such as
pressure sensors, accelerometers as well as resonators, mi-
cromotors and capacitive micromachined ultrasonic trans-
ducers (CMUTs). In MEMS devices such as CMUTs, the
width of a membrane is typically 50 - 100 μm while the
gap height reaches 0.1 μm in order to maximize device
efficiency. Hence, the aspect ratio of these microdevices is
as high as 1:1000. Only 0.01 degrees initial membrane bow
puts the membrane in contact with the bottom substrate,
making the device inoperable. During design stage it is
necessary to consider all possible initial membrane deflec-
tion contributors in order to ensure proper device opera-
tion. There is a need to emphasize that all the derived ana-
lytical formulations and simulation studies assume an ini-
tially flat membrane shape. This contributes to unexpected
device response as compared to theoretical response.
MOEMS devices frequently employ free-standing thin-
film structures to reflect or diffract light. Stress-induced
out-of-plane deformation must be small in comparison to
the optical wavelength of interest to avoid compromising
device performance. A principal source of contour errors in
micromachined structures is residual strain that results
from thin-film fabrication and structural release. Surface
micromachined films are deposited at temperatures signifi-
cantly above ambient and they are frequently doped to im-
prove their electrical conductivity. Both processes impose
residual stresses in the thin films. When sacrificial layers
of the device are dissolved, residual stresses in the elastic
structural layers are partially relieved by deformation of
the structural layers. Stress gradients through the thickness
of a micromachined film are particularly troublesome from
an optical standpoint, because they can cause significant
curvature of a free-standing thin-film structure even when
the average stress through the thickness of the film is zero.
The relationship between stress and curvature in thin-film
structures is an active area of research, both for the devel-
opment of MOEMS technology and for the fundamental
science of film growth [2]. To summarize there are three
main factors that cause a membrane-based structure to
bow:
1) residual stress developed during the deposition;
2) the effect atmospheric pressure on the membrane
(constant ~ 0.1 MPa);
3) thermal stress contribution during deposition.
2. Thin-film stress
The formation of thin films during fabrication of a
MOEMS device typically takes place at an elevated tem-
perature and the film growth process gives rise to the thin
film stress. Two main components that lead to internal or
residual stresses in thin films are thermal stresses and in-
trinsic stresses. Thermal stresses are induced due to strain
misfits as a result of differences in the temperature de-
pendent coefficient of thermal expansion between the thin
film and a substrate material such as silicon. Meanwhile,
intrinsic stresses are generated due to strain misfits en-
countered during phase transformation in the formation of
a solid layer of a thin film. Residual or internal thin film
stress therefore can be defined as the summation of the
thermal and intrinsic thin film stress components [1]
R T I (1)
where R is the residual thin film stress, T is the thermal
stress component, I is the intrinsic stress component.
3. Governing equations for stress in thin films
Between a film and substrate the stress is predom-
inantly caused by incompatibilities or misfits due to differ-
ences in thermal expansion, phase transformations with
volume changes and densification of the film [1]. Simple
solutions of mechanics of materials are therefore employed
to study the mechanical residual stress induced in thin
films. The solution that will be discussed here involves the
biaxial bending of a thin plate [2]. After a film is deposited
onto a substrate at an elevated temperature, it cools down
to a room temperature. When the film/substrate composite
is cooled, they contract with different magnitudes because
of different coefficients of thermal expansion between the
film and the substrate. The film is subsequently strained
elastically to match the substrate and remain attached,
causing the substrate to bend. This along with the intrinsic
film stress developed during film growth, gives rise to a
total residual film stress [2-6]. A relationship between the
biaxial stress in a plate and the bending moment will now
be discussed. Parts of the derivation are based on Nix’s
analysis [2]. Fig. 1 presents free body diagram illustrating
bending moment acting on a plate. From Fig. 1 the bending
moment per unit length along the edge of the plate M, is
274
related to the stresses in the plate by the following relation-
ship
322 2
2 212
h h
xxh h
hM ydy y dy
(2)
where y is the distance from the neutral axis, α is a con-
stant and xx zz y .
The stresses are given by
3
12xx zz
My
h (3)
Fig. 1 Free body diagram showing bending moment acting
on a plate
Note that the moment is defined to be positive and
will produce a positive stress in the positive y direction.
Fig. 2 below shows a picture of relationship between cur-
vature and strain.
Fig. 2 Relationship between bending strain and curvature
A negative curvature for pure bending as a result
of a tensile strain is shown in Fig. 2. The strain is given by
R y R y
y KyR R
(4)
The curvature-strain relationship is thus given by
1 yK
R y
(5)
The strain expressed in terms of the biaxial stress
is derived from Hooke’s law and is given by
1 2x x x (6)
By substitution, the curvature in terms of the biax-
ial bending moment is given by
3
1 12s
s
v MK
E h
(7)
The results from the bending moment analysis can
be extended for both the film and substrate. It is important
to note that the thin film stress equation that will be devel-
oped is applicable only for a single thin film on a flat sub-
strate. The film stress equation was first developed by
Stoney for a beam but it has since been generalized for a
thin film on a substrate. The equation is applicable if the
following conditions are satisfied:
1) the elastic properties of the substrate is known for a
specific orientation;
2) the thickness of the film is uniform and f st t ;
3) the stress in the film is equibiaxial and the film is in
a state of plane stress;
4) the out-of-plane stress and strains are zero;
5) the film adhere perfectly to the substrate [3].
Fig. 3 depicts the force per unit length and the
moment per unit length that are acting on the film (Ff and
Mf), and substrate (Fs and Ms) respectively. The thickness
of the film and the thickness of the substrate are denoted
by tf and ts.
Fig. 3 Force per unit length and bending moment per unit
length acting on thin film and substrate
If a biaxial tension stress is assumed, then
xx zz f . The force on the film and substrate are
equal and opposite and the film force per unit length is
given by f f fF t . The moment per unit length of the
substrate is thus
2
sf f
tM t (8)
The resulting curvature of the film and substrate
composite is therefore given by
3 3
1 112 12
2
s s sf f
s s s
v v tMK t
E Eh t
(9)
The stress that a single layer of thin film exerts on
a substrate is thus
2 2
1 6 1 6
s s s sf
s f s f
E t E tk
v t v t R
(10)
where Es is the Young’s modulus of the substrate, νs is the
Poisson ratio of the substrate, R is the radius of curvature
of the film and substrate composite.
This equation is the fundamental equation that
calculates the residual stress experienced by a thin film.
The equation is applicable for a single film deposited onto
a substrate, in which the film thickness is very small com-
pared to the substrate thickness.
275
4. Working principle of a MOEMS pressure sensor
Novel MOEMS pressure sensor under develop-
ment is composed of periodical diffraction grading, which
is integrated with semiconductor laser diode and photo
element matrix. The grading in the micromembrane is gen-
erated using some specific etching techniques. Working
principle of the pressure sensor can be described as fol-
lows: beam of the laser in diffraction grating is split into
exactly described positions (diffraction maximums). If
some pressure is applied, deformation of the micro-
membrane changes distance between diffraction maxi-
mums. This displacement change can be calibrated in pres-
sure units, like variation in resistance is calibrated into
pressure units in the case of a piezoresistive sensor. Chan-
ging distance between elements making optical pair, sensi-
tivity of the device can be increased remarkably. Principle
scheme of the research object with and without optical
grating is presented in Figs. 4 and 5 respectively.
a
b
Fig. 4 a) Micromembrane (P (pressure) =0, Pa); b) Micro-
membrane (P (pressure) >0, Pa)
a
b
Fig. 5 a) Micromembrane with optical grating and laser
beam (pressure is zero); b) Micromembrane with
optical grating when pressure is applied
5. Fabrication technology
For the deposition of Si3N4 layer surface-
micromaching technology was used. In order to form opti-
cal grating bulk micromaching technology was used. Du-
ring etching process the top side of the wafer is coated with
low stress transparent Si3N4, where using RIE (reactive ion
etching) techniques diffraction grating is to be formed
(transparent also for IR radiance) [7-9]. The principal of
formation of membrane is simple. Having silicon dioxide
wafer of 300 μm thickness polysilicon is deposited on a
semiconductor wafer, by pyrolyzing (decomposing ther-
mally) silane, SiH4, inside a low-pressure reactor 25-
130 Pa at a temperature of 580 to 650°C. This pyrolysis
process involves the following basic reaction: SiH4 --
> Si + 2H2. The rate of polysilicon deposition increases
rapidly with temperature, since it follows the Arrhenius
equation
aE / RTk Ae
(11)
where k is rate constant, A is prefactor, Ea is the activation
energy in electron volts, R is the universal gas constant and
T is the absolute temperature in degrees Kelvin. The acti-
vation energy for polysilicon deposition is about 1.7 eV.
Procedure of formation of micro membrane and optical
grating is presented in Fig. 6.
Fig. 6 Schematics of process for the formation of a micro-
membrane
In order to find out if fabrication process was suc-
cessful some pictures of particular micromembrane where
done using scanning electron microscope (SEM). Analyz-
ing the pictures presented below it can be observed that the
fabrication process was not successful. Fig. 7 represents
cracks of microfabricated micromembranes. Invoking the-
oretical and practical knowledge most probably reasons for
the failure and cracks of micromembranes could be:
1) the residual stresses are too big;
2) some dust during fabrication process appeared on
the surface;
3) the concentration of etchant KOH was too big leav-
ing the structure extremely thin and vulnerable.
Information is important for hot imprint microfabrication
technology and surface roughness analysis [10-11].
276
Fig. 7 Fabrication cracks of membrane
6. Eigenfrequency analysis
Eigenfrequency is one of the frequencies at which
an oscillatory system can vibrate. Micromembranes were
formed of two materials: on double polished thick silicon
substrate thin film polysilicon layer was deposited at a high
temperature. When assembly cooled down to a room tem-
perature, the film and the substrate shrunk differently and
caused strain in the film. Taking mentioned phenomenon
into account, the analysis in this section show how thermal
residual stress changes structure’s resonant frequency. As-
suming the material is isotropic, the stress is constant
through the film thickness, and the stress component in the
direction normal to the substrate is zero. The stress- strain
relationship is then
1r / E (12)
where E is Young’s modulus, ν is Poisson’s ratio, ε is
strain given by
T (13)
where Δα is the difference between thermal expansion co-
efficients, and ΔT is the difference between the deposition
temperature and the normal operating temperature.
As three different dimensions micromembranes
were fabricated, modeling also considered membranes of
different dimensions. As far as width of particular speci-
mens coincides the radius of structures used for numerical
modeling was: 0.4 mm, 1 mm, and 5 mm respectively.
Fig. 8 and Table 1 represents scheme of the micro-
membrane with exact dimensions, physical mechanical
properties and equations used for numerical modeling.
Fig. 8 Schematic representation of a micromembrane that
is used for numerical modeling
Table 1
Physical and mechanical properties of micromembrane
Description and symbol Value Unit
Radius of membrane 0.4, 1, 5 mm
Thickness t 20 m
Young’s modulus E 155 GPa
Density 2330 kg/m3
Poisson’s ratio ν 0.23 -
Room temperature T0 20 C
Deposition temperature T1 600 C
Residual stress r 50 MPa
Residual strain 1r / E -
Coefficient of thermal
expansion (1/K) 1 0/ T T -
Mechanical model of a micromembrane was cre-
ated using finite element (FE) modeling software Comsol
Multiphysics. FE model describes microstructure dynamics
by the following classic equation of motion presented in a
general matrix form [12, 13]
, ,M U C U K U Q t U U (14)
where [M], [C], [K] are mass, damping and stiffness matri-
ces respectively; U , U , U are displacement, accel-
eration and velocity vectors respectively; , ,Q t U U is
vector, representing the sum of the forces acting on the
micro-membrane.
Eigenfrequency analysis was performed for the
micromembrane of three different dimensions. The mod-
eled micromembranes were fixed in the entire perimeter
just leaving free translational movement in z direction
(Fig. 8), i.e. free translational movement was possible just
in one direction. Results are presented below
(Fig. 9 – 0.4 mm radius membrane, Fig. 10 – 1 mm radius
membrane, Fig. 11 – 5 mm radius membrane). For the
evaluation and modeling of residual thermal stresses the
temperature differences are between 600C and ambient
room temperature of 20C. The equations used for the
evaluation are presented in Table 1.
277
a
b
c
d
Fig. 9 Thermal stress influence on resonant frequency of
micro-membrane, radius is 0.4 mm: a) eigen-
frequency is 498 kHz, modeling without thermal
stresses; b) eigen-frequency is 802 kHz, modeling
with thermal stresses; c) deformed shape of geome-
try of the membrane under thermal stresses; d) von
Mises stress distribution going through the center of
membrane
a
b
c
d
Fig. 10 Thermal stress influence on resonant frequency of
micromembrane, radius is 1 mm: a) eigen-
frequency is 80.35 kHz, modeling without thermal
stresses; b) eigen-frequency is 260 kHz, modeling
with thermal stresses; c) deformed shape of geome-
try of the membrane under thermal stresses; d) von
Mises stress distribution going through the center
of membrane
278
a
b
c
d
Fig. 11 Thermal stress influence on resonant frequency of
micromembrane, radius is 5 mm: a) eigen-
frequency is 3.26 kHz, modeling without thermal
stresses; b) eigen-frequency is 48.48 kHz modeling
with thermal stresses; c) deformed shape of geome-
try of the membrane under thermal stresses; d) von
Mises stress distribution going through the center
of membrane
Table 2
Resonant frequencies with and without residual stress
Radius of membrane 0.4 mm 1 mm 5 mm
Without stress, kHz 498 80.35 3.26
With residual stress, kHz 802 260 48.48
Judging from the modeling results it can be easily
observed that having smaller radius membrane and the
same thickness of it the influence of residual stresses on
membrane decreases as the area of membrane decreases
(Table 2). Comparing resonant frequencies of smallest
radius membrane it can be noticed that solving the problem
including residual stresses resonant frequencies differs less
than two times. Thus, thermal stresses for millimeter radius
membrane even more than 3 times make a difference to
eigenmodes of structure. Resonant frequency of 5 mm
membrane including thermal stress already gives a rise
even 14 times. Von Mises stress distribution is most no-
ticeable near the fixing points of the microdevice. There-
fore, it is obvious that in order to properly fabricate opera-
ble micromembrane area and width ratio of the micro-
device needs to be as small as possible.
7. Conclusions
Micromembranes of different dimensions were
modeled and fabricated. Modeling results show, that the
smaller the area of the membrane the smaller influence of
thermal stresses will have on it. Fabrication show that
some residual stresses are left in the structure, despite the
fact that it is not desired result. Moreover, there were a lot
of problems with DLC coating, since during etching pro-
cess the film of DLC started to crumble away from the
silicon wafer.
For further analysis of a micromembrane, fluid-
structure interaction models will be developed using finite
element method. Fabrication will continue with formation
of diffraction grating on surface of micromembranes and
using different solution etchant.
Acknowledgments
This research was funded by a grant (No. MIP-
060/2012) from the Research Council of Lithuania.
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K. Malinauskas, V. Ostaševičius, R. Daukševičius,
V. Grigaliūnas
PLONASLUOKSNIŲ
MIKROOPTOELEKTROMECHANINIŲ MEMBRANŲ
LIEKAMIEJI ĮTEMPIAI
R e z i u m ė
Gamybos metu atsiradę liekamieji įtempiai gali
turėti ypač didelę reikšmę MOEMS veikimui ir patikimu-
mui. Galima drąsiai teigti, kad paviršinio mikroformavimo
būdu jokio prietaiso negalima pagaminti be liekamųjų
įtempių. Dažniausiai MOEMS gamyboje pasitaikantys
liekamieji įtempiai susidaro kaip tik dėl temperatūros po-
kyčių, kurie atsiranda užgarinant plonus sluoksnius ant
norimų bandinių esant aukštai temperatūrai ir kai naujos
struktūros bandinys atvėsta iki kambario temperatūros.
Šiame straipsnyje pateikiama mikro membranos
paviršinio formavimo technologija. Taip pat yra aprašyta
objekto principinė schema ir veikimo principas. Naudojan-
tis Comsol Multiphysics modeliavimo įrankiu yra sumode-
liuotos membranos, palyginti savieji dažniai esant tempera-
tūros poveikiui, kuris atsiranda gamybos metu, ir jo nesant.
Taip pat yra pateiktas galimas problemos sprendimas.
K. Malinauskas, V. Ostaševičius, R. Daukševičius,
V. Grigaliūnas
S u m m a r y
RESIDUAL STRESS IN A THIN-FILM
MICROOPTOELECTROMECHANICAL (MOEMS)
MEMBRANE
Residual stress from the thin film deposition pro-
cess can have extremely important effects on the function-
ality and reliability of MOEMS devices. Almost all sur-
face-micromachined thin films are subject to residual
stresses. The most common is thermal stress, which ac-
companies a change in temperature when thin-film is
evaporated on substrate and when it cools down to room
temperature. This paper presents the surface-micro-
machined micromembrane micromachining technology.
The principle scheme of the object is presented and work-
ing principle of micromembrane is described. Furthermore,
using powerful modeling software Comsol Multiphysics
comparison of eigenfrequencies of structure with thermal
stresses and without it is presented. A possible problem
solution will also be included.
Keywords: MOEMS, Residual stresses, thermal stress,
eigenfrequencies with and without residual stresses.
Received May 03, 2011
Accepted May 13, 2012