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Full-field measurement of nonuniform stresses of

thin films at high temperature

Xuelin Dong,1 Xue Feng,1,* Keh-Chih Hwang,1 Shaopeng Ma,2 and Qinwei Ma2

1AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China

2Department of Engineering Mechanics, Beijing Institute of Technology, Beijing 100083, China

*fengxue@tsinghua.edu.cn

Abstract: Coherent gradient sensing (CGS), a shear interferometry method,

is developed to measure the full-field curvatures of a film/substrate system

at high temperature. We obtain the relationship between an interferogram

phase and specimen topography, accounting for temperature effect. The

self-interference of CGS combined with designed setup can reduce the air

effect. The full-field phases can be extracted by fast Fourier transform. Both

nonuniform thin-film stresses and interfacial stresses are obtained by the

extended Stoney’s formula. The evolution of thermo-stresses verifies the

feasibility of the proposed interferometry method and implies the

“nonlocal” effect featured by the experimental results.

©2011 Optical Society of America

OCIS codes: (240.0310) Thin films; (120.3180) Interferometry; (120.6780) Temperature;

(310.4925) Other properties.

References and links

1. L. B. Freund and S. Suresh, Thin Film Materials; Stress, Defect Formation and Surface Evolution (Cambridge

University Press, 2003).

2. G. G. Stoney, “The tension of metallic films deposited by electrolysis,” Proc. R. Soc. Lond., A Contain. Pap.

Math. Phys. Character 82(553), 172–175 (1909).

3. L. B. Freund, J. A. Floro, and E. Chason, “Extensions of the Stoney formula for substrate curvature to

configurations with thin substrates or large deformations,” Appl. Phys. Lett. 74(14), 1987–1989 (1999).

4. T. S. Park and S. Suresh, “Effects of line and passivation geometry on curvature evolution during processing and

thermal cycling in copper interconnect lines,” Acta Mater. 48(12), 3169–3175 (2000).

5. L. B. Freund, “Substrate curvature due to thin film mismatch strain in the nonlinear deformation range,” J. Mech.

Phys. Solids 48(6–7), 1159–1174 (2000).

6. Y. Huang and A. J. Rosakis, “Extension of Stoney's formula to non-uniform temperature distributions in thin

film/substrate systems. The case of radial symmetry,” J. Mech. Phys. Solids 53(11), 2483–2500 (2005).

7. X. Feng, Y. G. Huang, H. Q. Jiang, D. Ngo, and A. J. Rosakis, “The effect of thin film/substrate radii on the

stoney formula for thin film/substrate subjected to nonuniform axisymmetric misfit strain and temperature,” J.

Mech. Mater. Struct. 1(6), 1041–1053 (2006).

8. D. Ngo, X. Feng, Y. Huang, A. J. Rosakis, and M. A. Brown, “Thin film/substrate systems featuring arbitrary

film thickness and misfit strain distributions. Part I: analysis for obtaining film stress from non-local curvature

information,” Int. J. Solids Struct. 44(6), 1745–1754 (2007).

9. M. A. Brown, A. J. Rosakis, X. Feng, Y. Huang, and E. Ustundag, “Thin film substrate systems featuring

arbitrary film thickness and misfit strain distributions. Part II: experimental validation of the non-local stress

curvature relations,” Int. J. Solids Struct. 44(6), 1755–1767 (2007).

10. X. Feng, Y. Huang, and A. J. Rosakis, “Stresses in a multilayer thin film/substrate system subjected to

nonuniform temperature,” J. Appl. Mech. 75(2), 021022 (2008).

11. P. A. Flinn, D. S. Gardner, and W. D. Nix, “Measurement and interpretation of stress in aluminum-based

metallization as a function of thermal history,” IEEE Trans. Electron. Dev. 34(3), 689–699 (1987).

12. E. Chason and B. W. Sheldon, “Monitoring stress in thin films during processing,” Surf. Eng. 19(5), 387–391

(2003).

13. H. V. Tippur, S. Krishnaswamy, and A. J. Rosakis, “A coherent gradient sensor for crack tip deformation

measurements: analysis and experimental results,” Int. J. Fract. 48(3), 193–204 (1991).

14. H. V. Tippur, “Coherent gradient sensing: a Fourier optics analysis and applications to fracture,” Appl. Opt.

31(22), 4428–4439 (1992).

15. A. J. Rosakis, R. P. Singh, Y. Tsuji, E. Kolawa, and N. R. Moore, “Full field measurements of curvature using

coherent gradient sensing: application to thin film characterization,” Thin Solid Films 325(1–2), 42–54 (1998).

16. M. A. Brown, T.-S. Park, A. Rosakis, E. Ustundag, Y. Huang, N. Tamura, and B. Valek, “A comparison of X-ray

microdiffraction and coherent gradient sensing in measuring discontinuous curvatures in thin film: substrate

systems,” J. Appl. Mech. 73(5), 723–729 (2006).

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17. J. Tao, L. H. Lee, and J. C. Bilello, “Nondestructive evaluation of residual-stresses in thin-films via x-ray-

diffraction topography methods,” J. Electron. Mater. 20(7), 819–825 (1991).

18. D. Post, B. Han, and P. Ifju, High Sensitivity Moire: Experimental Analysis for Mechanics and Materials

(Springer-Verlag, 1994).

19. Y. Y. Hung, “Shearography for non-destructive evaluation of composite structures,” Opt. Lasers Eng. 24(2–3),

161–182 (1996).

20. J. C. Murphy and L. C. Aamodt, “Photothermal spectroscopy using optical beam probing - mirage effect,” J.

Appl. Phys. 51(9), 4580–4588 (1980).

21. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based

topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982).

22. T. S. Park, S. Suresh, A. J. Rosakis, and J. Ryu, “Measurement of full-field curvature and geometrical instability

of thin film-substrate systems through CGS interferometry,” J. Mech. Phys. Solids 51(11–12), 2191–2211

(2003).

1. Introduction

Thin films deposited on various types of substrates are applied in many technologies,

including electronic circuits, integrated optical devices, microelectromechanical systems

(MEMS), systems-on-a-chip structures, as well as coatings used for thermal protection,

oxidation, and corrosion resistance. The stresses in the films induced by fabrication or diverse

processes are crucial to the performance and reliability of these devices. It is recognized that

the mismatch in thermal expansion coefficients between the film and substrate subjected to a

changing temperature environment is one of the dominant factors that cause the undesirable

stresses. For instance, the interconnect wires or other function elements in integrated circuits

(ICs) may fail because of the temperature cycling [1]. Consequently, the thin-film stresses

measurements especially under high temperature conditions are important to improve the thin-

film/substrate systems.

The most widely used method to determine the thin-film stresses at present is based on the

measurement of the substrate curvature and Stoney’s formula [2]. However, the rigid

assumptions of Stoney’s formula, such as uniform thin-film stress, uniform deformation over

the entire system, and infinitesimal strains and rotations of the system cannot be satisfied in

real situations. To infer thin-film stress by substrate curvature accurately, a number of

extensions of Stoney’s formula have been derived to relax some assumptions [3–10]. Huang

and Rosakis [6] studied the thin film/substrate system subjected to nonuniform but

axisymmetric temperature distribution; they relaxed the uniform stress assumption. Recently,

Feng and his associates [10] considered a circular multilayer thin-film/substrate system

subjected to nonuniform and nonaxisymmetrical temperature distribution and derived an

extension of Stoney’s formula that was more universal. There are a few techniques for

curvatures measurement, such as the scanning laser method [11], a multibeam optical stress

sensor (MOSS) [12], the coherent gradient sensing (CGS) method [13–16], and x-ray

diffraction [17]. Compared with other methods, CGS, one type of shear interferometry, has

distinguished advantages, including full-field measurement and vibration insensitivity.

Although Moire and shearography methods had been used for high-temperature displacement

measurement, they were not specified for thin-film/substrate systems [18,19]. This paper

presents an effective method based on extended CGS for full-field curvatures measurement in

high-temperature environment, which can be insensitive to the disturbance of air flow

resulting from the temperature. Moreover, the full-field curvatures are calculated by the fast

Fourier transform (FFT) method, and nonuniform stresses of thin films at high temperature

are obtained by the extension of Stoney’s formula. The “nonlocal” effect is also analyzed.

2. The thermal effects on shear interferometry and the experimental setup

CGS method is a full-field curvature measurement technique that is sensitive to the surface

slope of the specimen by laterally shearing the wavefront reflected from the sample. The CGS

setup for high-temperature measurement is illustrated in Fig. 1(a). A collimated laser beam

passes through a beam splitter and is then directed to the reflecting specimen surface in the

temperature chamber with a quartz window. The reflected beam from the specimen is further

reflected by the beam splitter and then passes through two Ronchi gratings, G1 and G2, with

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the same density (40 lines/mm) separated by a distance . The diffracted beams from the two

gratings are converged to interfere using a lens. Either of the ± 1 diffraction orders is filtered

by the filtering aperture to obtain the interferogram recorded by a CCD camera.

Fig. 1. The experimental setup and the thermal effect: (a) schematic of CGS setup for high

temperature measurement, (b) thermal effect on the optical path length.

CGS interferogram by

shearing in x direction

Wrapped phase map

Unwrapped phase

map φ(x)(x,y)

Substrate curvature in

Cartensian coordinates

Substrate curvature in

cylindrical coordinates

Nonuniform stresses

of thin film

CGS interferogram by

shearing in y direction

Phase unwrapping

Eq. (5)

Eqs. (6)-(10)

FFT

Wrapped phase map

Unwrapped phase

map φ(y)(x,y)

FFT

Fig. 2. The flow chart of the measurement of nonuniform film stresses.

During the heating process, the air density varies owing to the thermal effect, which

changes the refractive index of the air. Thus it is difficult to obtain the stable interferogram

fringes, which is a critical challenge for optical measurement at high temperature. To analyze

the thermal effect on CGS method at high temperature, we assume the (x,y) plane is set at the

window of the temperature chamber and z = f(x,y) represents the shape function of the

specimen in Cartesian coordinates as shown in Fig. 1(b). The refractive index of air is

nonuniform, which is expressed as n(x,y,z). With the assumption

change in optical path length, S(x,y), can be calculated by considering the thermal effect [15]

22

x

2

y

,,

1

fff

, the net

( , )

f x y

0

( , )

S x y

2( , , )d .

n x y z z

(1)

If the reflective wavefront is sheared in the y direction, partially differentiating S(x,y) with y

leads to

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( , )

f x y

0

( , )

S x y

y

( , , )

n x y z

y

( , )

f x y

y

2d2 , ( , ),

n x f x y z

.

z

(2)

The x-direction shearing will give a similar result. When the temperature becomes stable,

the refractive index of the air will distribute uniformly and can be expressed as [20]

0

1

( )1,

1

n

n t

at

(3)

where n(t) and n0 are the refractive indexes of the air at t°C and 0°C, respectively, and a is a

constant that equals to 0.00367°C1. Substituting Eq. (3) into Eq. (2) and considering both the

x- and y-direction shearing give the phase of the interferogram [13]

0

0

1

4 ( , )

f x y

x

f x y

y

( , )

x y

1

1

n

,

1

4 ( , )

( , )

x y

1

1

x

y

n

p

at

p at

(4)

where φ(x)(x,y) and φ(y)(x,y) are the phase distribution of the fringes obtained by shearing the

reflected wavefront in the x and y directions, respectively, and p is the pitch of the gratings G1

and G2. Since n01 is much smaller than 1, (n01)/(1 + at) is a higher-order term and can be

neglected. It is important to notice that the higher the temperature is, the weaker the thermal

effect is on the refractive index. Therefore, the CGS governing equation for high temperature

can be given as

2 ( )

x

2

2 ( )

y

2

2( )

y

( , )

f x y

x

f x y

y

( , )

x y

x

x y

y

4

( , )( , )

,

4

( , )

f x y

x y

( , )

x y

x

4

xx

yy

xyyx

p

p

p

(5)

where

curvature.

Accordingly, the temperature chamber is also designed deliberately in order to reduce the

air effect. During measurement, the laser beam vertically passes through a quartz window on

the side of the temperature chamber, where air convection is very weak at the stable

temperature. The thermo-isolation materials are fixed around the window in order to reduce

the temperature gradients near chamber window. CGS principle relies on the self-interference

based on Eq. (4). Therefore, the thickness and the changes of refractive index of quartz

window have only little effect on the interferometry.

The phase distribution can be calculated by FFT [21,22], such as φ(x)(x,y) =

arctan{Im[Ax(x,y)]/Re[Ax(x,y)]} and φ(y)(x,y) = arctan{Im[Ay(x,y)]/Re[Ay(x,y)]} for x and y

directions shear interferometry, respectively, where Im[A(x,y)] and Re[A(x,y)] denote the

imaginary and real parts of complex amplitude A(x,y), and the superscripts x and y represent

the shearing directions, respectively. The unwrapping algorithm is performed by MATLAB

subroutine, and then the full-field curvatures are obtained from Eq. (5).

Usually, the complicated process of wafer inevitably introduces the nonuniform

deformation or misfit, which can result in serious thermo-stress due to temperature. However,

classical Stoney’s formula considers only for the uniform situation, which cannot catch the

real stresses status. Feng and his associates [10] had derived an extension of Stoney’s formula

for a multilayer thin-film/substrate system subjected to nonuniform and nonaxisymmetrical

xx

is the curvature in x direction,

yy

is the curvature in y direction, and

xy

is the twist

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temperature distribution. In the later part, we will use the cylindrical coordinates to present the

stresses analysis. Then the nonuniform thin-film stresses from the nonuniform curvatures of

the substrate can be expressed as [10]

2

s

( )

f

rr

( )

f

(1 ) (1

)2

()

(1) (1

) (1)

(1 ) (1

)2

3

1

6(1)

2

(1) (1

) (1)

( 1)( cossin

rr

fssf

rr rr

sssff

s

fssf

s

sf

s

sssff

m

mm

E h

h

r

R

mCmSm

1

,

)

m

(6)

2

s

( )

f

rr

( )

f

2

1

(1

)

1

6(1) (1)(1)

(1)( 1),

( cossin)

ssf

sfssff

rr

mm

m

mm

E h

h

r

R

r

R

mmm

CmSm

(7)

2

s

( )

f

r

2

1

(1

)

1

6(1) (1) (1)

1

2

(1)(1)(sin cos) ,

ssf

sfs

sff

mm

rmm

m

E h

h

r

R

r

R

mmmCmSm

(8)

1

2

s

2

s

1

1

()( 1)( cossin) ,

2

6(1)

m

s

s

r rrmm

m

E h

r

R

m mCmSm

rR

(9)

1

2

s

2

s

1

1

1

r

()(1)(sincos) ,

2

6(1)

m

s

s

rrmm

m

E h

r

R

m mCmSm

R

(10)

where hs and hf are the thickness of the substrate and thin film, respectively; R is the radius of

the system;

rr

and

are the in-plane stresses of the thin film in the radial and

circumferential directions, respectively;

r

is the film shear stress; and τr and τθ are the

interfacial shear stresses between the substrate and thin film in the radial and circumferential

1

R

rr

R

1

m

C

R

νs and νf are the Poisson’s ratio of the substrate and film, respectively. αs and αf represent the

thermal expansion coefficients of the substrate and film, respectively. It should be noticed that

( )

f

( )

f

( )

f

directions, respectively.

2

2

00

() d d

r r

rr

is the average curvature of

the substrate as well as

2

2

00

()cos() d d

m

R

rr

m

R

and

2

2

00

1

R

()sin() d d

m

R

mrr

Sm

R

. Es is the Young’s modulus of the substrate.

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the thin-film nonuniform stresses are not only dependent on the local curvatures of the

substrate, but they are also related to the “nonlocal” curvatures (average curvature).

In summary, the measurement of nonuniform stresses of thin films at high temperature

contains the following steps. First, use CGS method to obtain the interferograms of the

specimen at high temperature. Second, calculate the phase distribution from the fringe pattern

by FFT. Third, use Eq. (5) to obtain the curvatures of the substrate, and then transfer them into

cylindrical coordinates. Finally, substitute the curvatures into Eqs. (6)–(10) to obtain the

nonuniform stresses of thin film. These steps are schematically illustrated in Fig. 2.

3. Experimental results and discussion

3.1 Substrate curvature measurement

The specimen consists of SiO2 film grown by thermal oxidation on Si substrate, which is the

representative wafer structure widely used in semiconductor industry. The thicknesses of the

SiO2 film and Si substrate were 500nm and 500μm, respectively; their radius was 10mm. The

geometry size agreed with the assumption hf

as shown in Fig. 1(a). The back of the specimen was supported by a stiff frame through point

contact. Moreover, the contact between the specimen and the bottom support was also point

contact because the specimen was circled shaped. Therefore, the specimen could expand

freely subjected to temperature, and there was no additional stresses induced by the boundary

condition. As the temperature was elevated from room temperature to high temperature (e.g.

~300°C), the CGS interferograms were recorded by a CCD camera. Figure 3 shows the

interferograms obtained at 300°C. The red fringes in Figs. 3(a) and 3(b) represent the contour

curves of the specimen surface slope in lateral (x direction) and vertical (y direction)

directions, respectively. The wrapped phase map is calculated by FFT method and shown in

Figs. 3(c) and 3(d), respectively. As illustrated by the process flowchart in Fig. 2, unwrapping

the phase map in Figs. 3(c) and 3(d) by using the standard MATLAB algorithm and then

substituting the results into Eq. (5) would give the curvatures distribution of the substrate in

Cartesian coordinates at 300°C. We used Zernike polynomials to fit the unwrapped phase

maps and then differentiated them by using Eq. (5). Figures 4(a) and 4(b) show the

corresponding system curvatures distribution in x and y directions, respectively, while Fig.

4(c) shows the twist curvature distribution. It is obvious that the curvature distribution is

nonuniform and thus violates the Stoney’s formula assumption. The curvatures in the vicinity

of the edge become much greater than those in the other area due to the edge effect.

hs R. The specimen was placed vertically,

Fig. 3. Interferograms at 300°C and their wrapped phase maps: (a) interferogram obtained by

shearing laterally, (b) interferogram obtained by shearing vertically, (c) wrapped phase map for

Fig. 3(a), (d) wrapped phase map for Fig. 3(b).

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Fig. 4. The substrate curvatures measured at 300°C: (a) curvature

xx

in lateral direction, (b)

curvature

yy

in vertical direction, (c) twist curvature

xy

.

3.2 Nonuniform stresses of the thin film

To calculate the film stresses at high temperature, we select the room temperature as a

reference state and use

rrrr

,

, and

Eqs. (6)–(10), where the superscripts h and r represent the curvatures obtained at high

temperature and room temperature, respectively. The physical parameters of the system are Es

= 170GPa, νs = 0.22, αs = 0.25 × 106°C1, Ef = 71GPa, νf = 0.16, and αs = 0.5 × 106°C1 [1].

The thin film stresses for 300°C are shown in Fig. 5. Figures 5(a), 5(b), and 5(c) show the film

stresses

rr

(radial direction),

(circumferential direction), and

respectively. Figures 5(d) and 5(e) show the interfacial shear stresses

(circumferential direction) between the film and the substrate, respectively. It is found that

the magnitude order of thin-film stresses is at GPa. For most areas,

and the shear stress

r

is large. The nonuniformity of the film stresses becomes more

severe owing to the nonlocal effect shown in Eqs. (6)–(10). In addition, the interfacial stresses

r and

Actually, the shear stresses on interface are more dangerous in most cases.

To investigate the thermo-stresses of thin film subjected to varied temperature, we

conducted the experiment from room temperature to 300°C with the step of 50°C. Then the

full-field stresses can be obtained at the different temperatures following the same process as

above. The film stresses of the central point in the specimen are selected to illustrate the

thermo-stress evolution, as shown in Fig. 6.

beginning room temperature then increases to 45MPa (in tension) at 200°C; however,

drops down to 190MPa at 250°C again and then reaches to 130MPa at 300°C. The

fluctuation of

rr

may result from the nonuniformity and the nonlocal effect. Meanwhile

( )

f

monotonically decreases from tension to compression with the increase of temperature.

( )

h

( )

r

( )

h

( )

r

( )

h

r

( )

r

r

to replace

rr

,

, and

r

in

( )

f

( )

f

( )

f

r

(shear stress),

r (radial direction) and

( )

f

rr

is not equal to

( )

f

,

( )

f

with the magnitude of a few MPa are rather smaller compared with the film stresses.

( )

f

rr

is 130MPa (in compression) at the

( )

f

rr

( )

f

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Fig. 5. The nonuniform stresses of the thin film measured at 300°C: (a) stress

( ) f

rr

in radial

direction, (b) stress

( ) f

in circumferential direction, (c) shear stress

( ) f

r

, (d) interfacial

shear stress

r in radial direction, (e) interfacial shear stress

in circumferential direction.

Fig. 6. The film stresses in radial and circumferential directions at the central point of the

specimen vs. temperature.

4. Conclusions

The results presented here demonstrate to use a coherent gradient sensing method to measure

the thin-film/substrate system curvature at high temperature and calculate the nonuniform

stresses of the film by the extension of Stoney’s formula. This optical technique is featured as

full-field nonuniform curvatures measurement and vibration insensitivity. A SiO2 film grown

on a Si wafer is used to verify the proposed method, which can be potentially extended to

higher temperature. These results provide a fundamental approach to understand the thin-film

stresses and the feasible measurement method for high temperature.

Acknowledgments

We gratefully acknowledge the support from the National Natural Science Foundation of

China (Grant Nos. 90816007, 10902059, 10820101048, 10832005) and the Foundation for the

Author of National Excellent Doctoral Dissertation of China (FANEDD) (No. 2007B30).

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