Residual stress in a thin-film microoptoelectromechanical (MOEMS) membrane/Plonasluoksniu mikrooptoelektromechaniniu membranu liekamieji itempiai.

Malinauskas, K. ; Ostasevicius, V. ; Dauksevicius, R. 等

1. Introduction

Microoptoelectromechanical systems (MOEMS) is not some special class of microelectromechanical systems (MEMS) but in fact it is MEMS merged with microoptics which involves sensing or manipulating optical signals [1]. There are numerous membrane-based MOEMS devices involved in various precise measurements such as pressure sensors, accelerometers as well as resonators, micromotors and capacitive micromachined ultrasonic transducers (CMUTs). In MEMS devices such as CMUTs, the width of a membrane is typically 50-100 [micro]m while the gap height reaches 0.1 [micro]m 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 deflection contributors in order to ensure proper device operation. There is a need to emphasize that all the derived analytical formulations and simulation studies assume an initially 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 significantly above ambient and they are frequently doped to improve 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 development 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 temperature 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 intrinsic stresses. Thermal stresses are induced due to strain misfits as a result of differences in the temperature dependent coefficient of thermal expansion between the thin film and a substrate material such as silicon. Meanwhile, intrinsic stresses are generated due to strain misfits encountered 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]

[[sigma].sub.R] = [[sigma].sub.T] + [[sigma].sub.1] (1)

where [[sigma].sub.R] is the residual thin film stress, [[sigma].sub.T] is the thermal stress component, [[sigma].sub.I] is the intrinsic stress component.

3. Governing equations for stress in thin films

Between a film and substrate the stress is predominantly caused by incompatibilities or misfits due to differences 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 related to the stresses in the plate by the following relationship

M = [[integral].sup.h/2.sub.-h/2][[sigma].sub.xx]ydy = [[integral].sup.h/2.sub.- h/2][alpha][y.sup.2]dy = [alpha][[h.sup.3]/12] (2)

where y is the distance from the neutral axis, [alpha] is a constant and [[sigma].sub.xx] = [[sigma].sub.zz] = [alpha]y.

The stresses are given by

[[sigma].sub.xx] = [[sigma].sub.zz] = [12M/[h.sup.3]]y (3)

[FIGURE 1 OMITTED]

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 curvature and strain.

[FIGURE 2 OMITTED]

A negative curvature for pure bending as a result of a tensile strain is shown in Fig. 2. The strain is given by

[epsilon](y) = [(R + y)[theta] - R[theta]]/R[theta] = y/R = -Ky (4)

The curvature-strain relationship is thus given by

K = -1/R = -[epsilon](y)/y (5)

The strain expressed in terms of the biaxial stress is derived from Hooke's law and is given by

[increment of x] = x1 - x2 (6)

By substitution, the curvature in terms of the biaxial bending moment is given by

K = [(1 - [v.sub.s])/[E.sub.s]][12M/[h.sup.3]] (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 developed is applicable only for a single thin film on a flat substrate. 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 [t.sub.f] < [t.sub.s];

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 ([F.sub.f] and [M.sub.f]), and substrate ([F.sub.s] and [M.sub.s]) respectively. The thickness of the film and the thickness of the substrate are denoted by [t.sub.f] and [t.sub.s].

[FIGURE 3 OMITTED]

If a biaxial tension stress is assumed, then [[sigma].sub.xx] = [[sigma].sub.zz] = [[sigma].sub.f]. The force on the film and substrate are equal and opposite and the film force per unit length is given by [F.sub.f] = [[sigma].sub.f][t.sub.f]. The moment per unit length of the substrate is thus

M = -[[sigma].sub.f][t.sub.f][[t.sub.s]/2] (8)

The resulting curvature of the film and substrate composite is therefore given by

K = [-(1-[v.sub.s])/[E.sub.s]][12M/[h.sup.3]] = [-(1 - [v.sub.s])/[E.sub.s]][12/[t.sup.3.sub.s]](- [[sigma].sub.f][t.sub.f][[t.sub.s]/2]) (9)

The stress that a single layer of thin film exerts on a substrate is thus

[[sigma].sub.f] = ([E.sub.s]/[1 - [v.sub.s]])[[t.sup.2.sub.s]/6[t.sub.f]]k = ([E.sub.s]/[1 - [v.sub.s]])[t.sup.2.sub.s]/6[t.sub.f]R (10)

where [E.sub.s] is the Young's modulus of the substrate, [v.sub.s] 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 compared to the substrate thickness.

4. Working principle of a MOEMS pressure sensor

Novel MOEMS pressure sensor under development is composed of periodical diffraction grading, which is integrated with semiconductor laser diode and photo element matrix. The grading in the micromembrane is generated using some specific etching techniques. Working principle of the pressure sensor can be described as follows: beam of the laser in diffraction grating is split into exactly described positions (diffraction maximums). If some pressure is applied, deformation of the micromembrane changes distance between diffraction maximums. This displacement change can be calibrated in pressure units, like variation in resistance is calibrated into pressure units in the case of a piezoresistive sensor. Changing distance between elements making optical pair, sensitivity 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.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

5. Fabrication technology

For the deposition of [Si.sub.3][N.sub.4] layer surface-micromaching technology was used. In order to form optical grating bulk micromaching technology was used. During etching process the top side of the wafer is coated with low stress transparent [Si.sub.3][N.sub.4], 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 ?im thickness polysilicon is deposited on a semiconductor wafer, by pyrolyzing (decomposing thermally) silane, SiH4, inside a low-pressure reactor 25-130 Pa at a temperature of 580 to 650[degrees]C. This pyrolysis process involves the following basic reaction: Si[H.sub.4] -- > Si + 2[H.sub.2]. The rate of polysilicon deposition increases rapidly with temperature, since it follows the Arrhenius equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

where k is rate constant, A is prefactor, [E.sub.a] is the activation energy in electron volts, R is the universal gas constant and T is the absolute temperature in degrees Kelvin. The activation energy for polysilicon deposition is about 1.7 eV. Procedure of formation of micro membrane and optical grating is presented in Fig. 6.

[FIGURE 6 OMITTED]

In order to find out if fabrication process was successful some pictures of particular micromembrane where done using scanning electron microscope (SEM). Analyzing the pictures presented below it can be observed that the fabrication process was not successful. Fig. 7 represents cracks of microfabricated micromembranes. Invoking theoretical 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 leaving the structure extremely thin and vulnerable.

Information is important for hot imprint microfabrication technology and surface roughness analysis [10-11].

[FIGURE 7 OMITTED]

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 temperature, 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. Assuming 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

[epsilon] = [[sigma].sub.r](1 - v)/E (12)

where E is Young's modulus, v is Poisson's ratio, [epsilon] is strain given by

[epsilon] = [DELTA][alpha][DELTA]T (13)

where [DELTA][alpha] is the difference between thermal expansion coefficients, and [increment of T] 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 specimens 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 micromembrane with exact dimensions, physical mechanical properties and equations used for numerical modeling.

[FIGURE 8 OMITTED]

Mechanical model of a micromembrane was created 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]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

where [M], [C], [K] are mass, damping and stiffness matrices respectively; {[??]}, {[??]}, {U} are displacement, acceleration and velocity vectors respectively; {Q(t,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 modeled 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 600[degrees]C and ambient room temperature of 20[degrees]C. The equations used for the evaluation are presented in Table 1.

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

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 noticeable near the fixing points of the microdevice. Therefore, it is obvious that in order to properly fabricate operable micromembrane area and width ratio of the microdevice 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 process the film of DLC started to crumble away from the silicon wafer.

For further analysis of a micromembrane, fluidstructure 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. MIP060/2012) from the Research Council of Lithuania.

References

[1.] Vinci, R.P.; Vlassak, J.J. 1996. Mechanical behavior of thin films, Annu. Rev. Mater. Sci. 26:431-462. http://dx.doi.org/10.1146/annurev.ms.26.080196.002243.

[2.] Nix, W.D. 2005. The Mechanical Properties of Thin Films, Department of Material Science and Engineering, Stanford University.

[3.] Janssen, G.C.A.M. et al. 2009. Celebrating the 100th anniversary of the Stoney equation for film stress: Developments from polycrystalline steel strips to single crystal silicon wafers, Thin Solid Films 517: 1858-1867. http://dx.doi.org/10.1016/j.tsf.2008.07.014.

[4.] Stoney, G.G. 1999. The tension of metallic films deposited by electrolysis, Proc. Royal Soc. London, A82: 172-175.

[5.] Fang, W.; Wickert, J.A. 1996. Determining mean and gradient residual stresses in thin films using micromachined cantilevers, J. Micromechan. Microeng. 6(3): 301-309. http://dx.doi.org/10.1088/0960-1317/6/3/002.

[6.] Yuan, F.; Shih, Y.; Knight, L.V.; Perkins, R.T.; Allred, D.D. 1992. Using thin films to produce precision, figured X-ray optics, Thin Solid Films 220(1-2): 284-288. http://dx.doi.org/10.1016/0040-6090(92)90586-Z.

[7.] Senturia, S.D. 2002. Microsystem design, Kluwer Academic Publishers, Norwell, MA, 30-35.

[8.] Judy, J.W. 2000. Microelectromechanical systems (MEMS): fabrication, design and applications, Smart Materials and Structures 10: 1115-1134. http://dx.doi.org/10.1088/0964-1726/10/6/301.

[9.] Comtois, J.; Bright, V.; Phipps, M. 1995. Thermal microactuators for surface micromachining processes, in Proceeding SPIE 2642, 10-21. http://dx.doi.org/10.1117/12.221154.

[10.] Augutis, V.; Saunoris, M. 2010. Estimation of surface roughness using high frequency vibrations, Journal of Vibroengineering 12: 429-436.

[11.] Narijauskaite, B.; Gaidys, R.; Palevicius, A.; Janusas, G. 2011. Simulation of hot imprint process of periodic microstructure using elasto-plastic material model, Journal of Vibroengineering 13: 232-237.

[12.] Ostasevicius, V.; Gaidys, R.; Dauksevicius, R. 2009. Numerical analysis of dynamic effects of a nonlinear vibro-impact process for enhancing the reliability of contact-type MEMS devices, Sensors 9(12): 10201-10216. http://dx.doi.org/10.3390/s91210201.

[13.] Ostasevicius, V. ; Dauksevicius, R. ; Gaidys, R. 2007. Numerical analysis of fluid-structure interaction effects on vibrations of cantilever microstructure, Journal of Sound and Vibration 308(3-5): 660-673. http://dx.doi.org/10.1016/jjsv.2007.03.072.

K. Malinauskas *, V. Ostasevicius **, R. Dauksevicius ***, V. Grigaliunas ****

* Kaunas University of Technology, Kestucio 27, 44312 Kaunas, Lithuania, E-mail: karolis.malinauskas@ktu.lt

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

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

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

doi: 10.5755/j01.mech.18.3.1880

Table 1

Physical and mechanical properties of micromembrane

Description and symbol       Value            Unit

Radius of membrane         0.4, 1, 5           mm

Thickness t                   20             [micro]m

Young's modulus E             155             GPa

Density [rho]                2330         kg/[m.sup.3]

Poisson's ratio v            0.23              -

Room temperature              20           [degrees]C
[T.sub.0]

Deposition                    600          [degrees]C
temperature [T.sub.1]

Residual stress               50              MPa
[[sigma].sub.r]

Residual strain         [[sigma].sub.r]        -
[epsilon]                  (1 - v)/E

Coefficient of thermal    [epsilon]/           -
expansion (1/K)          ([T.sub.1] -
                          [T.sub.0])

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