Page 1

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).

#144476 - $15.00 USD

(C) 2011 OSA

Received 21 Mar 2011; revised 20 May 2011; accepted 2 Jun 2011; published 23 Jun 2011

4 July 2011 / Vol. 19, No. 14 / OPTICS EXPRESS 13201

Page 2

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

#144476 - $15.00 USD

(C) 2011 OSA

Received 21 Mar 2011; revised 20 May 2011; accepted 2 Jun 2011; published 23 Jun 2011

4 July 2011 / Vol. 19, No. 14 / OPTICS EXPRESS 13202

Page 3

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

#144476 - $15.00 USD

(C) 2011 OSA

Received 21 Mar 2011; revised 20 May 2011; accepted 2 Jun 2011; published 23 Jun 2011

4 July 2011 / Vol. 19, No. 14 / OPTICS EXPRESS 13203

Page 4

( , )

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

#144476 - $15.00 USD

(C) 2011 OSA

Received 21 Mar 2011; revised 20 May 2011; accepted 2 Jun 2011; published 23 Jun 2011

4 July 2011 / Vol. 19, No. 14 / OPTICS EXPRESS 13204

Page 5

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.

#144476 - $15.00 USD

(C) 2011 OSA

Received 21 Mar 2011; revised 20 May 2011; accepted 2 Jun 2011; published 23 Jun 2011

4 July 2011 / Vol. 19, No. 14 / OPTICS EXPRESS 13205