Stresses in a Multilayer Thin Film/Substrate ... - Semantic Scholar

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Keywords: multilayer thin films, nonuniform film temperatures and stresses, nonuniform system ... received June 6, 2007; published online February 27, 2008.
X. Feng Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, P.R. China

Y. Huang1 e-mail: [email protected] Department of Civil/Environmental Engineering and Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208

A. J. Rosakis Graduate Aeronautical Laboratory, California Institute of Technology, Pasadena, CA 91125

Stresses in a Multilayer Thin Film/Substrate System Subjected to Nonuniform Temperature Current methodologies used for the inference of thin film stress through curvature measurements are strictly restricted to uniform film stress and system curvature states over the entire system of a single thin film on a substrate. By considering a circular multilayer thin film/substrate system subjected to nonuniform temperature distributions, we derive relations between the stresses in each film and temperature, and between the system curvatures and temperature. These relations featured a “local” part that involves a direct dependence of the stress or curvature components on the temperature at the same point, and a “nonlocal” part, which reflects the effect of temperature of other points on the location of scrutiny. We also derive relations between the film stresses in each film and the system curvatures, which allow for the experimental inference of such stresses from full-field curvature measurements in the presence of arbitrary nonuniformities. These relations also feature a “nonlocal” dependence on curvatures making full-field measurements of curvature a necessity for the correct inference of stress. The interfacial shear tractions between the films and between the film and substrate are proportional to the gradient of the first curvature invariant, and can also be inferred experimentally. 关DOI: 10.1115/1.2755178兴 Keywords: multilayer thin films, nonuniform film temperatures and stresses, nonuniform system curvatures, nonlocal stress-curvature relations, interfacial shears

1

Introduction

Substrates formed of suitable solid-state materials may be used as platforms to support various thin film structures. Integrated electronic circuits, integrated optical devices and optoelectronic circuits, microelectromechanical systems deposited on wafers, three-dimensional electronic circuits, systems-on-a-chip structures, lithographic reticles, and flat panel display systems are examples of such thin film structures integrated on various types of plate substrates. The stress buildup in the thin film is important to the reliability and performance of these devices and systems. Stoney 关1兴 studied a system composed of a thin film of thickness h f , deposited on a relatively thick substrate, of thickness hs, and derived a simple relation between the curvature ␬ of the system and the stress ␴共f兲 of the film as follows: Eshs2␬ ␴共 f 兲 = 6h f 共1 − ␯s兲

共1.1兲

In the above, the subscripts “f” and “s” denote the thin film and substrate, respectively, and E and ␯ are the Young’s modulus and Poisson’s ratio. Equation 共1.1兲 is called the Stoney formula, and it has been extensively used in the literature to infer film stress changes from experimental measurement of system curvature changes 关2兴. Stoney’s formula was based on a number of assumptions: 共i兲

Both the film thickness h f and the substrate thickness hs are uniform and h f Ⰶ hs Ⰶ R, where R represents the characteristic length in the lateral direction 共e.g., system radius R shown in Fig. 1兲;

1 Corresponding author. Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received April 27, 2007; final manuscript received June 6, 2007; published online February 27, 2008. Review conducted by Robert M. McMeeking.

Journal of Applied Mechanics

共ii兲 The strains and rotations of the plate system are infinitesimal; 共iii兲 Both the film and substrate are homogeneous, isotropic, and linearly elastic; 共iv兲 The film stress states are in-plane isotropic or equibiaxial 共two equal stress components in any two, mutually orthogonal in-plane directions兲 while the out-of-plane direct stress and all shear stresses vanish; 共v兲 The system’s curvature components are equibiaxial 共two equal direct curvatures兲 while the twist curvature vanishes in all directions; and 共vi兲 All surviving stress and curvature components are spatially constant over the plate system’s surface, a situation that is often violated in practice. Despite the explicitly stated assumptions of spatial stress and curvature uniformity, the Stoney formula is often, arbitrarily, applied to cases of practical interest where these assumptions are violated. This is typically done by applying Stoney’s formula pointwise, and thus extracting a local value of stress from a local measurement of the curvature of the system. This approach of inferring film stress clearly violates the uniformity assumptions of the analysis and, as such, its accuracy as an approximation is expected to deteriorate as the levels of curvature nonuniformity become more severe. Following the initial formulation by Stoney, a number of extensions have been derived to relax some assumptions. Such extensions of the initial formulation include relaxation of the assumption of equibiaxiality as well as the assumption of small deformations/deflections. A biaxial form of Stoney formula 共with different direct stress values and nonzero in-plane shear stress兲 was derived by relaxing the assumption 共v兲 of curvature equibiaxiality 关2兴. Related analyses treating discontinuous films in the form of bare periodic lines 关3兴 or composite films with periodic line structures 共e.g., bare or encapsulated periodic lines兲 have also been derived 关4–6兴. These latter analyses have removed the assumptions 共iv兲 and 共v兲 of equibiaxiality and have allowed the

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Fig. 1 A schematic diagram of a multilayer thin film/substrate system, showing the cylindrical coordinates „r , ␪ , z…

existence of three independent curvature and stress components in the form of two, nonequal, direct components and one shear or twist component. However, the uniformity assumption 共vi兲 of all of these quantities over the entire plate system was retained. In addition to the above, single, multiple, and graded films and substrates have been treated in various “large” deformation analyses 关7–10兴. These analyses have removed both the restrictions of an equibiaxial curvature state as well as the assumption 共ii兲 of infinitesimal deformations. They have allowed for the prediction of kinematically nonlinear behavior and bifurcations in curvature states that have also been observed experimentally 关11,12兴. These bifurcations are transformations from an initially equibiaxial to a subsequently biaxial curvature state that may be induced by an increase in film stress beyond a critical level. This critical level is intimately related to the system aspect ratio, i.e., the ratio of inplane to thickness dimension and the elastic stiffness. These analyses also retain the assumption 共vi兲 of spatial curvature and stress uniformity across the system. However, they allow for deformations to evolve from an initially spherical shape to an energetically favored shape 共e.g., ellipsoidal, cylindrical, or saddle shapes兲 that features three different, still spatially constant, curvature components 关6,11兴. The above-discussed extensions of Stoney’s methodology have not relaxed the most restrictive of Stoney’s original assumption 共vi兲 of spatial uniformity that does not allow film stress and system curvature components to vary in the thin film/substrate system. This crucial assumption is often violated in practice, since film stresses and the associated system curvatures are nonuniformly distributed. Recently, Huang et al. 关13兴 and Huang and Rosakis 关14兴 relaxed the assumption 共vi兲 共and also 共iv兲 and 共v兲兲 to study the thin film/substrate system subjected to nonuniform, axisymmetric misfit strain 共in thin film兲 and temperature change 共in both thin film and substrate兲, respectively, while Huang and Rosakis 关15兴 and Ngo et al. 关16兴 studied the thin film/substrate system subject to arbitrarily nonuniform 共e.g., nonaxisymmetric兲 misfit strain and temperature. The most important result is that the film stresses depend nonlocally on the system curvatures; i.e., they depend on curvatures of the entire system. The relations between film stresses and system curvatures are established for arbitrarily nonuniform misfit strain and temperature change, and such relations degenerate to Stoney’s formula for uniform, equibiaxial stresses and curvatures. Feng et al. 关17兴 relaxed part of the assumption 共i兲 to study the thin film and substrate of different radii. Ngo et al. 关18兴 com021022-2 / Vol. 75, MARCH 2008

pletely relax the assumption 共i兲 to study arbitrarily nonuniform thickness of the thin film. They derived an analytical relation between the film stresses and system curvatures that allows for the accurate experimental inference of film stress from full-field curvature measurements once the film thickness distribution is known. Brown et al. 关19兴 used two independent types of X-ray microdiffraction to measure both substrate slope and film stress across the diameter of an axisymmetric thin film/substrate specimen composed of a Si substrate on which a smaller circular W film island was deposited. The substrate slopes, measured by polychromatic 共white beam兲 X-ray microdiffraction, were used to calculate curvature fields and to, thus, infer the film stress distribution using both the “local” Stoney formula and the new, nonlocal relation. The variable film thickness, which was independently measured, was also an input to the new relation. These were then compared with the film stress measured independently through monochromatic X-ray diffraction in the sample to validate the new analytical relation 关18兴. Many thin film/substrate systems involve multiple layers of thin films. The main purpose of this paper is to extend the above analyses by Huang, Rosakis, and co-workers to a system composed of multilayer thin films on a substrate subjected to nonuniform temperature distribution. We will relate stresses in each film and system curvatures to the temperature distribution, and ultimately derive a relation between the stresses in each film and system curvatures that would allow for the accurate experimental inference of film stresses from full-field and real-time curvature measurements.

2

Axisymmetric Temperature Distribution

We first consider a system of multilayer thin films deposited on a substrate subjected to axisymmetric temperature distribution T共r兲, where r is the radial coordinate 共Fig. 1兲. The thin films and substrate are circular in the lateral direction and have a radius R. The deformation is axisymmetric and is therefore independent of the polar angle ␪, where 共r , ␪ , z兲 are cylindrical coordinates with the origin at the center of the substrate 共Fig. 1兲. 2.1 Governing Equations. Let h f i共i = 1 , . . . , n兲 denote the thickness of the ith thin film 共Fig. 1兲. The total film thickness h f n = 兺i=1 h f i of all n thin films is much less than the substrate thickness hs, and both are much less than R; i.e., h f Ⰶ hs Ⰶ R. The Young’s modulus, Poisson’s ratio, and coefficient of thermal expansion of the ith film and substrate are denoted by E f i, ␯ f i, ␣ f i, Es, ␯s, and ␣s, respectively. The substrate is modeled as a plate since it can be subjected to bending and hs Ⰶ R. The thin films are modeled as membranes that have no bending rigidities due to their small thickness h f i Ⰶ hs. Therefore, they all have the same in-plane displacement u f 共r兲 in the radial direction. The strains are ␧rr = du f / dr and ␧␪␪ = u f / r. The stresses in the ith thin film can be obtained from the linear thermo-elastic constitutive model as 共i兲 ␴rr =

共i兲 ␴␪␪ =

Efi

冋 冋

uf du f + ␯ f i − 共1 + ␯ f i兲␣ f iT r 1 − ␯2f i dr Efi 1−

␯2f i

␯fi

du f u f + − 共1 + ␯ f i兲␣ f iT dr r

册 册

共2.1兲

The membrane forces in the ith thin film are 共i兲 Nr共 f i兲 = h f i␴rr

共i兲 N␪共 f i兲 = h f i␴␪␪

共2.2兲

For a nonuniform temperature distribution T = T共r兲, the shear stress tractions at the film/substrate and film/film interfaces do not vanish, and are denoted by ␶共i兲共r兲共i = 1 , . . . , n兲 as shown in Fig. 2. The normal stress tractions ␴zz still vanish because thin films have Transactions of the ASME

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dM r M r − M ␪ hs + + Q − ␶ 共1兲 = 0 dr r 2

共2.8兲

dQ Q + =0 dr r

共2.9兲

where Q is the shear force normal to the neutral axis. Substitution of Eq. 共2.5兲 into Eq. 共2.7兲 yields dT 1 − ␯s2 共1兲 d2us 1 dus us − 2 = 共1 + ␯s兲␣s − ␶ 2 + dr dr r dr r E sh s

共2.10兲

Elimination of Q from Eqs. 共2.8兲 and 共2.9兲, in conjunction with Eq. 共2.6兲, gives Fig. 2 A schematic diagram of the nonuniform shear traction distribution at the film/film and film/substrate interfaces

d3w 1 d2w 1 dw 6共1 − ␯s2兲 共1兲 + − ␶ = dr3 r dr2 r2 dr Eshs2

共2.11兲

The continuity of displacement across the film/substrate interface requires no bending rigidities. The equilibrium equations for thin films, accounting for the effect of interface shear stress tractions, become



dNr共 f 1兲 dr dNr共 f 2兲 dr

+

Nr共 f 1兲 − N␪共 f 1兲

+

Nr共 f 2兲 − N␪共 f 2兲

+

Nr共 f n兲 − N␪共 f n兲

]

r r

− 共 ␶ 共1兲 − ␶ 共2兲兲 = 0 − 共 ␶ 共2兲 − ␶ 共3兲兲 = 0

共2.3兲

u f = us −

dr

r

−␶

共n兲

=0



共1兲

=

E f ih f i

兺 1−␯ i=1

i=1





E f ih f i d2u f 1 du f u f − + = ␶ 共1兲 + 2 dr2 r dr r2 f

兺 1−v

i

n



E f ih f i␣ f i dT

i=1

1 − v f i dr 共2.4兲

Let us denote the displacement in the radial 共r兲 direction at the neutral axis 共z = 0兲 of the substrate, and w the displacement in the normal 共z兲 direction. The forces and bending moments in the substrate are obtained from the linear thermo-elastic constitutive model as Nr共s兲

=

N␪共s兲 =

E sh s 1 − ␯s2 E sh s 1 − ␯s2

冋 冋

us dus + ␯s − 共1 + ␯s兲␣sT r dr dus us + − 共1 + ␯s兲␣sT ␯s dr r

冉 冉

d w ␯s dw + Mr = r dr 12共1 − ␯s2兲 dr2 Eshs3

M␪ =

Eshs3 12共1 −

␯s2兲

␯s

2

2

冊 冊

d w 1 dw + dr2 r dr

册 册

兺 i=1

+ ␯s兲␣s − 共1 + ␯ f i兲␣ f i兴

dT dr

共2.13兲

i

which is a remarkable result that holds regardless of boundary conditions at the edge r = R. Therefore, the interface shear stress is proportional to the gradient of temperature. For uniform temperature T = constant, the interface shear stress vanishes; i.e., ␶共1兲 = 0. Substitution of the above solution for shear stress ␶共1兲 into Eqs. 共2.11兲 and 共2.10兲 yields ordinary differential equations for displacements w and us in the substrate. Their solutions, at the limit h f  hs are n

E f ih f i 1 − ␯s2 1 dw =6 2 关共1 + ␯s兲␣s − 共1 + ␯ f i兲␣ f i兴 2 dr r Eshs i=1 1 − ␯ f i





r

␩T共␩兲d␩

0

B1 r 2

共2.14兲

us = 共1 + ␯s兲␣s

共2.6兲

␯s2

E f ih f i 1 − 4 1 − ␯2f Eshs

共2.5兲 1 r



r

␩T共␩兲d␩ +

0

B2 r 2

共2.15兲

where B1 and B2 are constants to be determined by boundary conditions. The displacement in the thin films is then obtained from Eq. 共2.12兲 as u f = 共1 + ␯s兲␣s

1 r



r

␩T共␩兲d␩ +

0





B 2 h sB 1 − r 2 4

共2.16兲

The first boundary condition at the free edge r = R requires that the net force vanish n

共2.7兲

The out-of-plane force and moment equilibrium equations are given by Journal of Applied Mechanics

1+

+

The shear stress ␶共1兲 at the film/substrate interface is equivalent to the distributed axial force ␶共1兲共r兲 and bending moment 共hs / 2兲␶共1兲共r兲 applied at the neutral axis 共z = 0兲 of the substrate. The in-plane force equilibrium equation of the substrate then becomes dNr共s兲 Nr共s兲 − N␪共s兲 + + ␶ 共1兲 = 0 dr r

2 关共1 fi n

Substitution of Eqs. 共2.1兲–共2.3兲 and the summation of its left-hand side yield n

共2.12兲

Equations 共2.4兲 and 共2.10兲–共2.12兲 constitute four ordinary differential equations for u f , us, w, and ␶共1兲. We can eliminate u f , us, andw from these four equations to obtain the shear stress at the film/substrate interface in terms of temperature as n

dNr共 f n兲

hs dw 2 dr

兺N

共 f i兲 r

+ Nr共s兲 = 0

at r = R

共2.17兲

i=1

which gives MARCH 2008, Vol. 75 / 021022-3

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B2 = 共1 − ␯s兲␣s¯T

共2.18兲

for h f Ⰶ hs, where ¯T = 共2 / R2兲兰R0 ␩T共␩兲d␩ = 兰兰TdA / ␲R2 is the average temperature over the entire system. The second boundary condition at the free edge r = R is vanishing of net moment, i.e., n



hs N 共 f i兲 = 0 2 i=1 r

Mr − which gives B1 = 6

1 − ␯s2

n

E f ih f i

兺 1−␯

Eshs2 i=1

2 fi



共2.19兲

at r = R

共1 + v f i兲共1 − vs兲 1 + vs



2.2 Stresses in Multilayer Thin Films and System Curvatures. The system curvatures are related to the out-of-plane displacement w by ␬rr = d2w / dr2 and ␬␪␪ = dw / rdr. Their sum is given by 1 − ␯s Eshs2



We extend the Stoney formula for a multilayer thin film/ substrate system by eliminating the nonuniform axisymmetric temperature in order to establish a direct relation between the stresses in the ith thin film and system curvatures. Both ␬rr − ␬␪␪ in 共f 兲 共f 兲 Eq. 共2.23兲 and ␴rri − ␴␪␪i in Eq. 共2.25兲 are proportional to T 2 r − 共2 / r 兲兰0␩T共␩兲d␩, and therefore can be directly related by 共 f i兲 共 f i兲 ␴rr − ␴␪␪ =

共␣s − ␣ f i兲 − 共vs − ␯ f i兲␣s ¯T 共2.20兲

␬rr + ␬␪␪ = 12

3 Extension of Stoney Formula for a Multilayer Thin Film/Substrate System Subjected to Axisymmetric Temperature Distribution

1 + ␯s A␣¯T + A␯␣共T − ¯T兲 2



共2.21兲

n

E f ih f i

兺 1−v i=1

n

A ␯␣ ⬅

E f ih f i

兺 1−v i=1

2 关共1 fi

fi

␬rr + ␬␪␪ =

1 ␲R2

冕冕

+ ␯s兲␣s − 共1 + ␯ f i兲␣ f i兴

␬rr − ␬␪␪ = 6

Eshs2

A ␯␣

冋 冕 2 T− 2 r

A

共 f i兲 ␴rr



Efi 1 − ␯fi

共 f i兲 ␴␪␪

r

␩T共␩兲d␩

0



=

1 + ␯fi

共1 + ␯s兲␣s

冋 冕 2 T− 2 r

␬rr + ␬␪␪ = 12

r

␩T共␩兲d␩

0



共2.24兲

␶ =

E f jh f j

兺 1−v j=i

2 关共1 + vs兲␣s − 共1 + v f j兲␣ f j兴 fj

共 f i兲 共 f i兲 ␴rr + ␴␪␪ =

␩共␬rr + ␬␪␪兲d␩

0

Eshs2

A␣¯T

共3.3兲

dT dr

Eshs2

A␯␣共T − ¯T兲

共3.4兲



共1 + vs兲␣s − 2␣ f i 共1 + vs兲A␯␣

共␬rr + ␬␪␪ − ␬rr + ␬␪␪兲

册 共3.5兲

Equations 共3.1兲 and 共3.5兲 provide direct relations between stresses in each thin film and system curvatures. Stresses at a point in each thin film depend not only on curvatures at the same point 共local dependence兲, but also on the average curvature in the entire substrate 共nonlocal dependence兲. The interface stress ␶共i兲 can also be directly related to system curvatures via

␶ =

共2.26兲

1 − ␯s2

Efi ␣s − ␣ f i Eshs2 ␬rr + ␬␪␪ 6共1 − ␯s兲 1 − v f i A␣ +

共i兲

where the summation is from the ith thin film to the last 共nth兲. 021022-4 / Vol. 75, MARCH 2008

R

Elimination of temperature deviation T − ¯T and average temperature ¯T from Eqs. 共3.3兲, 共3.4兲, and 共2.24兲 gives the sum of stresses in the ith thin film in terms of curvature as

n

They are identical to Huang and Rosakis 关14兴 for a single thin film if the Young’s modulus, Poisson’s ratio, and coefficient of thermal expansion are substituted by Ei, vi, and ␣i of the ith thin film, respectively. The shear stresses along the film/film or film/ substrate interface can be obtained from the equilibrium equation 共2.3兲. Specifically, the shear stress of ith thin film is given by n

1 − ␯s

␬rr + ␬␪␪ − ␬rr + ␬␪␪ = 6

共2.25兲

共i兲



which can be related to the average temperature ¯T by averaging both sides of Eq. 共2.21兲, i.e.,

共2.23兲

¯ + 关共1 + ␯ 兲␣ − 2␣ 兴共T − ¯T兲其 兵2共␣s − ␣ f i兲T s s fi

Efi

2 R2

共3.2兲

共2.22兲

As compared to the system curvatures for a single thin film 关14兴, Eqs. 共2.21兲 and 共2.23兲 can be obtained by replacing the single film properties by the sum of multilayer film properties in Eq. 共2.22兲. The stresses in the ith thin film can be obtained from the inplane displacement u f as 共 f i兲 共 f i兲 ␴rr + ␴␪␪ =

共␬rr + ␬␪␪兲␩d␩d␪ =

where A␣ is given in Eq. 共2.22兲. The deviation from the average curvature ␬rr + ␬␪␪ − ␬rr + ␬␪␪ can be related to the deviation from the average temperature T − ¯T from Eq. 共2.21兲 as

共 ␣ s − ␣ f i兲

The first term on the right-hand side corresponds to the 共constant兲 average temperature ¯T, while the second term gives the deviation T − ¯T from the constant temperature. The difference between two system curvatures is 1 − ␯s2

共3.1兲

where A␯␣ is given in Eq. 共2.22兲. We define the average system curvature ␬rr + ␬␪␪ as

where ¯T is the average temperature in the thin film/substrate system, and A␣ ⬅

Eshs2␣s E f i ␬rr − ␬␪␪ 6共1 − vs兲 1 + v f i A␯␣

Eshs2 6共1 −

vs2兲

E f kh f k

兺 1−v k=i

2 关共1 fk

+ vs兲␣s − 共1 + v f k兲␣ f k兴 A ␯␣

d共␬rr + ␬␪␪兲 dr 共3.6兲

This provides a remarkably simple way to estimate the interface shear stress from radial gradients of the two nonzero system curvatures.

4

Arbitrary Temperature Distribution

Similar to Huang and Rosakis 关15兴 for a single thin film on a substrate, we expand the arbitrary nonuniform temperature distribution T共r , ␪兲 to the Fourier series: Transactions of the ASME

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T共r, ␪兲 =

T共cm兲共r兲cos m␪ +

m=0



Ts共m兲共r兲sin m␪

␬rr =

共4.1兲

m=0

⳵ 2w ⳵ r2

␬␪␪ =



1 ⳵ w 1 ⳵ 2w + r ⳵ r r2 ⳵ ␪2

␬ r␪ =

冉 冊

⳵ 1 ⳵w ⳵r r ⳵␪

The sum of system curvatures is related to the temperature by

where

T共cm兲共r兲 and Ts共m兲共r兲 =

1 ␲

1 = ␲





1 2␲

T共c0兲共r兲 =



2␲

␬rr + ␬␪␪ = 12

T共r, ␪兲d␪

1 − ␯s2 Eshs2 ⬁



− A ␯␣

T共r, ␪兲cos m␪d␪

␬ r␪ = 3

1 − ␯s2 Eshs2

+ sin m␪



rm R2m+2

R

T共r, ␪兲sin m␪d␪

共m ⱖ 1兲



2 r2

A ␯␣ T −





r



␩T共c0兲d␩ −

m=1

0

冉 冕 冉 冊 册冉 冕 冋兺 冉 冕 冕 冊册 冉 冕 冊 m=1

m+1 sin m␪ rm+2

␩1−mTs共m兲d␩

r

冕 冕

␩1−mTs共m兲d␩ R

␩m+1T共cm兲d␩ − cos m␪

0



r

冊册 冊 冊



r

␩m+1Ts共m兲d␩

0

+6

共4.2兲

1 − ␯s2 Eshs2





4 A ␣ − A ␯␣ 3 + ␯s

冊兺 冋 冉 冊 ⬁

m+1 r m+2 m R R

m=1

冊兺 冋 冉 冊 ⬁

m=1

m+1 r m+2 m R R

m

共4.3兲 ⬁

␩m+1Ts共m兲d␩ +

0

4 A ␣ − A ␯␣ 3 + ␯s

0

␩m+1Ts共m兲d␩

0

r

1 − ␯s2 3 Eshs2



0

␩m+1T共cm兲d␩ + sin m␪

0

R

␩m+1T共cm兲d␩ + sin m␪

r

R

cos m␪



r

R

r

m−2

r R

冉 冕

␩1−mT共cm兲d␩ + sin m␪

␩m+1T共cm兲共␩兲d␩

共0兲 where ¯T = 共2 / R2兲兰R0 ␩Tc 共␩兲d␩ = 共1 / ␲R2兲 兰 兰AT共␩ , ␸兲dA is the average temperature over the entire area A of the thin film, dA = ␩d␩d␸, and A␣ and A␯␣ are given in Eq. 共2.22兲. The difference between two curvatures, i.e., ␬rr − ␬␪␪, and the twist ␬r␪ are given by

m+1 cos m␪ rm+2

R

cos m␪

m−2

冋 冕 册冎

R

cos m␪

␩m+1Ts共m兲共␩兲d␩

0

2␲

A ␯␣ −

− cos m␪

共m + 1兲

m=1

0

− 共m − 1兲

冊兺



1 + ␯s 4 A␣¯T + A␣ A␯␣共T − ¯T兲 + 共1 + ␯s兲 2 3 + ␯s



2␲

0

兺 共m − 1兲r

m=1

Eshs2

0

The analysis is similar to Huang and Rosakis 关15兴, except it is now for multilayer thin films on a substrate. The system curvatures are

␬rr − ␬␪␪ = 6

1 − ␯s



冉 冕 冉 冊 册冉 冕 R

␩1−mT共cm兲d␩

共m − 1兲rm−2 sin m␪

m=1

r

m

− 共m − 1兲

m−2

r R

R

sin m␪

␩m+1T共cm兲d␩

0

R

− cos m␪

␩m+1Ts共m兲d␩

共4.4兲

0

As compared to the system curvatures for a single thin film 关15兴, Eqs. 共4.2兲–共4.4兲 can be obtained by replacing the film properties by the sum of multilayer film properties in Eqs. 共2.22兲. 共f 兲 共f 兲 The sum of stresses ␴rri + ␴␪␪i in the ith thin film is related to the temperature by Efi

共 f i兲 共 f i兲 ␴rr + ␴␪␪ =

1 − ␯fi



+ sin m␪

冉 冕



¯ + 关共1 + ␯ 兲␣ − 2␣ 兴共T − ¯T兲 + 2共1 − ␯ 兲␣ 2共␣s − ␣ f i兲T s s fi s s



R

␩m+1Ts共m兲d␩

0

共f 兲

共f 兲

2 T− 2 r



冊冎



m+1 m cos m␪ 2m+2 r R m=1

R

␩m+1T共cm兲d␩

0

共4.5兲 共f 兲

The difference between stresses, ie., ␴rri − ␴␪␪i , and shear stress ␴r␪i are given by 共 f i兲 ␴rr



共 f i兲 ␴␪␪

=

Efi 1 + ␯fi − 1兲r

共1 + ␯s兲␣s

m−2



冉 冕

冉 冕

␩T共c0兲d␩



m+1 cos m␪ rm+2

r

+ sin m␪

r

r

␩m+1Ts共m兲d␩

+ sin m␪

0



m=1

m+1 r m Rm+2 R

m

− 共m − 1兲



共m

m=1

0



␩1−mTs共m兲d␩



r

␩m+1T共cm兲d␩

R

␩1−mT共cm兲d␩

R

⫻ cos m␪



m=1

0

R

cos m␪

冕 冉 冕 冊兺 冕 冊 兺 冋冉冊 冉冊 册 冕 冊冎 ⬁

r

r R

m−2

R

␩m+1T共cm兲d␩ + sin m␪

0

Journal of Applied Mechanics

␩m+1Ts共m兲d␩

共4.6兲

0

MARCH 2008, Vol. 75 / 021022-5

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␴r共 ␪f i兲 =

Efi 2共1 + ␯ f i兲 − cos m␪





兺 冉 冕 冊 兺 冋冉冊 ⬁

共1 + ␯s兲␣s −

m+1 sin m␪ rm+2

m=1



R

␩1−mTs共m兲d␩

+

m=1

r

r

冕 冊 冉 冊 册冉 冕

0

m+1 r m Rm+2 R



r

␩m+1T共cm兲d␩ − cos m␪

␩m+1Ts共m兲d␩ +

m

− 共m − 1兲

r R



R

m=1

0

R

␩m+1T共cm兲d␩ − cos m␪

sin m␪

␩1−mT共cm兲d␩

r

R

m−2

冉 冕 冕 冊冎

共m − 1兲rm−2 sin m␪

0

␩m+1Ts共m兲d␩

共4.7兲

0

Equations 共4.5兲–共4.7兲 are identical to Huang and Rosakis 关15兴 for a single thin film if the Young’s modulus, Poisson’s ratio, and coefficient of thermal expansion are substituted by Ei, ␯i, and ␣i of the ith thin film, respectively. 共i兲 共i兲 The shear stresses ␶r and ␶␪ at the film/film and film/substrate interfaces are related to the temperature by n

␶r共i兲 =

E f jh f j

兺 1−␯ j=i

+ 1兲 n

␶␪共i兲

=

兺 j=i

2 关共1 + ␯s兲␣s − 共1 + ␯ f j兲␣ f j兴 fj

冉 冕

rm−1 cos m␪ R2m+2

⳵T +2 ⳵r

再兺 n

␩m+1T共cm兲d␩ + sin m␪

0

冉 冕

rm−1 sin m␪ R2m+2



n

共␣s − ␣ f j兲 −

R

␩m+1Ts共m兲d␩

再兺 0

1 ⳵T 关共1 + ␯s兲␣s − 共1 + ␯ f j兲␣ f j兴 −2 r ⳵␪ 1 − ␯2f

+ 1兲

1 − ␯f j

j=i

R

E f jh f j

j

E f jh f j

R

␩m+1T共cm兲d␩ − cos m␪

0

n



j=i

E f jh f j 1 − ␯f j



兺 1−␯ j=i

共␣s − ␣ f j兲 −

␩m+1Ts共m兲d␩

0

where the summation is from the ith thin film to the last 共nth兲.



兺 1−␯ j=i

Sm =

1 ␲R2

冕冕

共␬rr + ␬␪␪兲

冉冊

冕冕

共␬rr + ␬␪␪兲

冉冊

A

A



m

sin m␸dA

R

共5.1兲





E f i ␣s Eshs2 ␬rr − ␬␪␪ − 共m + 1兲 6共1 − ␯s兲 1 + ␯ f i A␯␣ m=1

冋冉冊

r ⫻ m R

m

冉冊 册

r − 共m − 1兲 R

Efi Eshs2 6共1 − ␯s兲 1 − ␯ f i

共 f i兲 共 f i兲 ␴rr + ␴␪␪ =

+



冉冊 册 r R

␣s − ␣ f i A␣

共1 + vs兲A␯␣



m共m

m=1

␬rr + ␬␪␪

共␬rr + ␬␪␪ − ␬rr + ␬␪␪兲

共1 + ␯s兲␣s − 2␣ f i 3 + ␯s ␣s − ␣ f i −2 1 + vs A␣ 共1 + vs兲A␯␣

兺 共m + 1兲冉 R 冊 ⬁

r



m

共Cm cos m␪ + Sm sin m␪兲

m=1



where ␬rr + ␬␪␪ = C0 = 共1 / ␲R2兲 兰 兰A共␬rr + ␬␪␪兲dA is the average curvature over entire area A of the thin film, and A␣ and A␯␣ are given in Eq. 共2.22兲. Equations 共5.2兲–共5.4兲 provide direct relations between individual film stresses and system curvatures. It is important to note that stresses at a point in each thin film depend not only on curvatures at the same point 共local dependence兲, but also on the curvatures in the entire substrate 共nonlocal dependence兲 via the coefficients Cm and Sm. 共i兲 共i兲 The shear stresses ␶r and ␶␪ at the film/film and film/substrate interfaces can also be directly related to system curvatures via

␶r共i兲



=

Eshs2 6共1 − ␯s2兲

共5.2兲

冋冉冊

m−2

共Cm sin m␪ − Sm cos m␪兲

021022-6 / Vol. 75, MARCH 2008







E f i ␣s Eshs2 1 r ␬ r␪ + ␴r共 ␪f i兲 = 共m + 1兲 m 2 m=1 6共1 − ␯s兲 1 + ␯ f i A␯␣ R − 共m − 1兲



共1 + vs兲␣s − 2␣ f i

+

m−2

⫻共Cm cos m␪ + Sm sin m␪兲 ⬁

+ ␯s兲␣s − 共1 + ␯ f j兲␣ f j兴

2 关共1 fj

共5.4兲

cos m␸dA

where the integration is over the entire area A of the thin film, and dA = ␩d␩d␸. Elimination of temperature gives the stresses in each thin film in terms of system curvatures by 共 f i兲 共 f i兲 ␴rr − ␴␪␪ =

冎兺 ⬁

m

R



m共m

m=1

共4.9兲

We extend the Stoney formula for a multilayer thin film/ substrate system by establishing the direct relation between the stresses in each thin film and system curvatures. Similar to Huang and Rosakis 关15兴 for a single thin film, we first define the coefficients Cm and Sm, related to the system curvatures by 1 ␲R2

fj

E f jh f j

5 Extension of Stoney Formula for a Multilayer Thin Film/Substrate System Subjected to Arbitrary Temperature Distribution

Cm =

2 关共1 + ␯s兲␣s − 共1 + ␯ f j兲␣ f j兴

共4.8兲 n

R

冎兺 ⬁

E f jh f j



1 − 2R

m





n

k=i

n

4

E f kh f k

k=i

共3 + ␯s兲 −

2 关共1 fk

+ vs兲␣s − 共1 + v f k兲␣ f k兴 A ␯␣

兺 1−v

2 关共1 fk

E f kh f k

兺 1−v k=i

A␣

fk

⳵ 共␬rr + ␬␪␪兲 ⳵r

+ vs兲␣s − 共1 + v f k兲␣ f k兴 A ␯␣

n

共5.3兲

E f kh f k

兺 1−v

共 ␣ s − ␣ f k兲

冧 Transactions of the ASME

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兺 m共m + 1兲共C

m

cos m␪ + Sm sin m␪兲

m=1

␶␪共i兲 =

Eshs2 6共1 − ␯s2兲



n

1 + ␬␪␪兲 + 2R

k=i



n

4

2 关共1 fk

k=i

2 关共1 fk

,

共5.5兲

1 ⳵ 共␬rr r ⳵␪

+ vs兲␣s − 共1 + v f k兲␣ f k兴 A ␯␣

E f kh f k

兺 1−v

fk

共 ␣ s − ␣ f k兲

A␣









+ vs兲␣s − 共1 + v f k兲␣ f k兴

E f kh f k

兺 1−v

k=i



m−1

A ␯␣

n

共3 + ␯s兲

r R

References

E f kh f k

兺 1−v

冉冊

兺 m共m + 1兲共C

m

sin m␪ − Sm cos m␪兲

m=1

冉冊 r R

m−1



共5.6兲

This provides a way to determine the interface shear stresses from the system curvatures. It also displays a nonlocal dependence via the coefficients Cm and Sm.

6

delamination is a commonly encountered form of failure during wafer manufacturing, the ability to estimate the level and distribution of such stresses from wafer-level metrology might prove to be invaluable in enhancing the reliability of such systems.

Concluding Remarks

The analytical solution is obtained for a multilayer thin film/ substrate system subjected to arbitrary temperature distribution. The stresses in each thin film and system curvatures are obtained in terms of the temperature. The direct relation between the stresses in each thin film and system curvatures is also obtained. The dependence of the film stresses on curvatures is not generally “local,” i.e., the stress components at a point on the film will depend on both the local value of the curvature components 共at the same point兲 and on the value of curvatures of all other points 共nonlocal dependence兲. The presence of nonlocal contributions in such relations also has implications regarding the nature of diagnostic methods needed to perform wafer-level film stress measurements. Notably, the existence of nonlocal terms necessitates the use of full-field methods capable of measuring curvature components over the entire surface of the plate system 共or wafer兲. Furthermore, measurement of all independent components of the curvature field is necessary. This is because the stress state at a point depends on curvature contributions from the entire plate surface. The nonuniform temperature distribution also results in shear stresses along the film/film and film/substrate interfaces. The relation between the shear stresses and system curvatures provides an effective method to estimate the shear stresses. Since film

Journal of Applied Mechanics

关1兴 Stoney, G. G., 1909, “The Tension of Metallic Films Deposited by Electrolysis,” Proc. R. Soc. London, Ser. A, A82, pp. 172–175. 关2兴 Freund, L. B., and Suresh, S., 2004, Thin Film Materials; Stress, Defect Formation and Surface Evolution, Cambridge University Press, Cambridge, U.K. 关3兴 Wikstrom, A., Gudmundson, P., and Suresh, S., 1999, “Thermoelastic Analysis of Periodic Thin Lines Deposited on a Substrate,” J. Mech. Phys. Solids, 47, pp. 1113–1130. 关4兴 Shen, Y. L., Suresh, S., and Blech, I. A., 1996, “Stresses, Curvatures, and Shape Changes Arising from Patterned Lines on Silicon Wafers,” J. Appl. Phys., 80, pp. 1388–1398. 关5兴 Wikstrom, A., Gudmundson, P., and Suresh, S., 1999, “Analysis of Average Thermal Stresses in Passivated Metal Interconnects,” J. Appl. Phys., 86, pp. 6088–6095. 关6兴 Park, T.-S., and Suresh, S., 2000, “Effects of Line and Passivation Geometry on Curvature Evolution During Processing and Thermal Cycling in Copper Interconnect Lines,” Acta Mater., 48, pp. 3169–3175. 关7兴 Masters, C. B., and Salamon, N. J., 1993, “Geometrically Nonlinear Stress– Deflection Relations for Thin Film/Substrate Systems,” Int. J. Eng. Sci., 31, pp. 915–925. 关8兴 Salamon, N. J., and Masters, C. B., 1995, “Bifurcation in Isotropic Thin Film/ Substrate Plates,” Int. J. Solids Struct., 32, pp. 473–481. 关9兴 Finot, M., Blech, I. A., Suresh, S., and Fijimoto, H., 1997, “Large Deformation and Geometric Instability of Substrates with Thin-Film Deposits,” J. Appl. Phys., 81, pp. 3457–3464. 关10兴 Freund, L. B., 2000, “Substrate Curvature Due to Thin Film Mismatch Strain in the Nonlinear Deformation Range,” J. Mech. Phys. Solids, 48, pp. 1159– 1174. 关11兴 Lee, H., Rosakis, A. J., and Freund, L. B., 2001, “Full Field Optical Measurement of Curvatures in Ultra-thin Film/Substrate Systems in the Range of Geometrically Nonlinear Deformations,” J. Appl. Phys., 89, pp. 6116–6129. 关12兴 Park, T.-S., Suresh, S., Rosakis, A. J., and Ryu, J., 2003, “Measurement of Full Field Curvature and Geometrical Instability of Thin Film-Substrate Systems Through CGS Interferometry,” J. Mech. Phys. Solids, 51, pp. 2191–2211. 关13兴 Huang, Y., Ngo, D., and Rosakis, A. J., 2005, “Non-uniform, Axisymmetric Misfit Strain in Thin Films Bonded on Plate Substrates Systems: The Relation Between Non-uniform Film Stresses and System Curvatures,” Acta Mech. Sin., 21, pp. 362–370. 关14兴 Huang, Y., and Rosakis, A. J., 2005, “Extension of Stoney’s Formula to Nonuniform Temperature Distributions in Thin Film/Substrate Systems. The Case of Radial Symmetry,” J. Mech. Phys. Solids, 53, pp. 2483–2500. 关15兴 Huang, Y., and Rosakis, A. J., 2007, “Extension of Stoney’s Formula to Arbitrary Temperature Distributions in Thin Film/Substrate Systems,” ASME J. Appl. Mech., 74, pp. 1225–1233. 关16兴 Ngo, D., Huang, Y., Rosakis, A. J., and Feng, X., 2006, “Spatially Nonuniform, Isotropic Misfit Strain in Thin Films Bonded on Plate Substrates: The Relation Between Non-uniform Stresses and System Curvatures,” Thin Solid Films, 515, pp. 2220–2229. 关17兴 Feng, X., Huang, Y., Jiang, H., Ngo, D., and Rosakis, A. J., 2006, “The Effect of Thin Film/Substrate Radii on the Stoney Formula for Thin Film/Substrate Subjected to Non-uniform Axisymmetric Misfit Strain and Temperature,” J. Mech. Mater. Struct., 1, pp. 1041–1054. 关18兴 Ngo, D., Feng, X., Huang, Y., Rosakis, A. J., and Brown, M. A., 2007, “Thin Film/Substrate Systems Featuring Arbitrary Film Thickness and Misfit Strain Distributions: Part I. Analysis for Obtaining Film Stress From Nonlocal Curvature Information,” Int. J. Solids Struct., 44, pp. 1745–1754. 关19兴 Brown, M. A., Rosakis, A. J., Feng, X., Huang, Y., and Ustundag, E., 2007, “Thin Film/Substrate Systems Featuring Arbitrary Film Thickness and Misfit Strain Distributions: Part II. Experimental Validation of the Non-local StressCurvature Relations,” Int. J. Solids Struct., 44, pp. 1755–1767.

MARCH 2008, Vol. 75 / 021022-7

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