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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: karolis.malinauskas@ktu.lt

**Kaunas University of Technology, Kęstučio 27, 44312 Kaunas, Lithuania, E-mail: vytautas.ostasevicius@ktu.lt

***Kaunas University of Technology, Studentų 65, 51369 Kaunas, Lithuania, E-mail: rolanas.dauksevicius@ktu.lt

****Kaunas University of Technology, Savanorių 271, 50131 Kaunas, Lithuania, E-mail: viktoras.grigaliunas@ktu.lt

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