Int J Fract (2007) 148:331–342DOI 10.1007/s10704-008-9205-7

ORIGINAL PAPER

Influence of interfacial delamination on channel crackingof elastic thin films

Haixia Mei · Yaoyu Pang · Rui Huang

Received: 7 December 2007 / Accepted: 10 April 2008 / Published online: 6 May 2008© Springer Science+Business Media B.V. 2008

Abstract Channeling cracks in brittle thin �lms havebeen observed to be a key reliability issue for advancedinterconnects and other integrated structures. Most the-oretical studies to date have assumed no delaminationat the interface, while experiments have observed chan-nel cracks both with and without interfacial delam-ination. This paper analyzes the effect of interfacialdelamination on the fracture condition of brittle thin�lms on elastic substrates. It is found that, depend-ing on the elastic mismatch and interface toughness,a channel crack may grow with no delamination, witha stable delamination, or with unstable delamination.For a �lm on a relatively compliant substrate, a crit-ical interface toughness is predicted, which separatesstable and unstable delamination. For a �lm on a rel-atively stiff substrate, however, a channel crack growswith no delamination when the interface toughness isgreater than a critical value, while stable delaminationalong with the channel crack is possible only in a smallrange of interface toughness for a speci�c elastic mis-match. An effective energy release rate for the steady-state growth of a channel crack is de�ned to accountfor the in�uence of interfacial delamination on both thefracture driving force and the resistance, which can besignificantly higher than the energy release rate assum-ing no delamination.

H. Mei · Y. Pang· R. Huang (B)Department of Aerospace Engineering and EngineeringMechanics, University of Texas, Austin, TX 78712, USAe-mail: ruihuang@mail.utexas.edu

Keywords Channel cracking· Delamination·Thin �lms · Interface

1 Introduction

Integrated structures with mechanically soft compo-nents have recently been pursued over a wide range ofnovel applications, from high performance integratedcircuits in microelectronics (Ho etal. 2004) to uncon-ventional organic electronics (Dodabalapur 2006) andstretchable electronics (Wagner etal. 2004; Khang etal.2006), along with the ubiquitous integration of hard andsoft materials in biological systems (Gao 2006). In par-ticular, the integration of low dielectric constant (low k)materials in advanced interconnects of microelectronicshas posed significant challenges for reliability issuesresulting from the compromised mechanical properties.Two failure modes have been reported, one for cohesivefracture (Liu etal. 2004) and the other for interfacialdelamination (Liu etal. 2007). The former pertains tothe brittleness of the low-k materials subjected to ten-sion, and the latter manifests due to poor adhesionbetween low-k and surrounding materials (Tsui etal.2006). This paper considers concomitant cohesive frac-ture and interfacial delamination as a hybrid failuremode in integrated thin-�lm structures.

One common cohesive fracture mode for thin �lmsunder tension is channel cracking (Fig.1). Previousstudies have shown that the driving force (i.e., theenergy release rate) for the steady-state growth of a

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332 H. Mei et al.

Substrate

Film

σfhf

Gss

(a)

Substrate

Film

hf

Gdd d

(b)

Fig. 1 (a) Illustration of a channel crack with no interfacialdelamination; (b) a channel crack with symmetric interfacialdelamination of widthd on both sides far behind the channelfront

channel crack depends on the constraint effect ofsurrounding layers (Hutchinson and Suo 1992). Fora brittle thin �lm on an elastic substrate, the drivingforce increases for increasingly compliant substrates(Beuth 1992; Huang etal. 2003). The effect of constraintcan be partly lost as the substrate deforms plastically(Ambrico and Begley 2002) or viscoelastically(Huang etal. 2002; Suo etal. 2003). More recent stud-ies have focused on the effects of stacked buffer layers(Tsui etal. 2005; Cordero etal. 2007) and patterned�lm structures (Liu etal. 2004). In most of these stud-ies, the interfaces between the �lm and the substrateor the buffer layers are assumed to remain perfectlybonded as the channel crack grows in the �lm (Fig.1a).However, the stress concentration at the root of thechannel crack may drive interfacial delamination(Ye etal. 1992). While some experimental observationsclearly showed no delamination (Tsui etal. 2005;He etal. 2004), others observed delamination of the

interface (Tsui etal. 2005; Suo 2003). There are twoquestions yet to be answered:First, under what condi-tion would the growth of a channel crack be accompa-nied by interfacial delamination? Second, how wouldthe interfacial delamination (if occurring) affect thefracture condition or reliability in integrated thin filmstructures? To answer these questions, this paper con-siders steady-state channel cracking of an elastic thin�lm on an elastic substrate and theoretically exam-ines the effect of concomitant interfacial delamination.Section2 brie�y reviews the concept of steady-statedriving force for a channel crack growing withoutdelamination. Section3 analyzes interfacial delamina-tion emanating from the root of a long, straight chan-nel crack. In Sect.4, the fracture driving force for thesteady-state growth of a channel crack with interfacialdelamination is determined. A �nite element modelis used to calculate the energy release rates for boththe interfacial delamination and the steady-state chan-nel cracking. Moreover, to account for the in�uenceof interfacial delamination on the fracture resistance,we de�ne an effective energy release rate that dependson the interface toughness as well as the elastic mis-match between the �lm and the substrate. In conclu-sion, Sect.5 summarizes the �ndings and emphasizesthe impact of interfacial delamination on the reliabilityof integrated structures with mechanically softcomponents.

2 Channel cracking without delamination

As illustrated in Fig.1a, assuming no interfacial delam-ination, the energy release rate for the steady-stategrowth of a channel crack in a thin elastic �lm bondedto a thick elastic substrate is (Beuth 1992; Hutchinsonand Suo 1992):

Gss = Z(α, β)σ 2

f hf

Ef, (1)

whereσ f is the tensile stress in the �lm,hf is the �lmthickness, andEf = Ef /(1 − ν2

f ) is the plane strainmodulus of the �lm with Young’s modulusEf and Pois-son’s ratioνf . The dimensionless coef�cientZ dependson the elastic mismatch between the �lm and the sub-strate, through the Dundurs’ parameters

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In�uence of interfacial delamination on channel cracking 333

α = Ef − Es

Ef + Esand

β = Ef (1−νf )(1 − 2νs)−Es(1 − νs)(1 − 2νf )

2(1 − νf )(1 − νs)(Ef + Es). (2)

When the �lm and the substrate have identical elas-tic moduli, we haveα = β = 0 and Z = 1.976.The value ofZ decreases slightly for a compliant �lmon a relatively stiff substrate (Ef < Es andα <0). Amore compliant substrate (α > 0), on the other hand,provides less constraint against �lm cracking. Thus,Z increases asα increases. For very compliant sub-strates (e.g., organic low-k dielectrics, polymers, etc.),Z increases rapidly, withZ >30 for α >0.99 (Beuth1992; Huang etal. 2003). The effect ofβ is secondaryand often ignored.

In general, the steady-state energy release rate ofchannel cracking can be calculated from a two-dimen-sional (2D) model (Beuth 1992; Huang etal. 2003) asfollows:

Gss = 12hf

hf∫

0

σ f δ(z)dz, (3)

whereδ(z) is the opening displacement of the cracksurfaces far behind the channel front (see Fig.1a). Dueto the constraint by the substrate, the crack openingdoes not change as the channel front advances and theenergy release rate attains a steady state, independentof the channel length. Three-dimensional analyses haveshown that the steady state is reached when the lengthof a channel crack exceeds two to three times the �lmthickness for a relatively stiff substrate (Nakamura andKamath 1992), but the crack length to reach the steadystate can be significantly longer for more compliantsubstrate materials (Ambrico and Begley 2002). Thepresent study focuses on the steady state.

3 Interfacial delamination from channel root

Now consider an interfacial crack emanating from theroot of a channel crack at each side (Fig.1b). For along, straight channel crack, we assume a steady statefar behind the channel front, where the interfacial crackhas a �nite width,d. The energy release rate for theinterfacial crack can be written in a similar form asEq.1:

Gd = Zd

(d

hf, α, β

)σ2

f hf

Ef, (4)

where Zd is a dimensionless function that can bedetermined from a two-dimensional plane strain prob-lem as illustrated in Fig.2a. In the present study, a �niteelement model is constructed to calculate the interfa-cial energy release rate. By symmetry, only half of the�lm/substrate structure is modeled along with properboundary conditions (Fig.2b). The �nite element pack-age ABAQUS is employed and an example mesh nearthe interfacial crack is shown in Fig.2c. Close to thetip of the interfacial crack, a very �ne mesh is used(Fig.2d), with a set of singular elements around thecrack tip. Far away, in�nite elements are used for boththe �lm and substrate to eliminate possible size effectsof the model. The method of J-integral is adopted forthe calculation of the interfacial energy release rate. Inall calculations, we setνf = νs = 1

3 such thatβ = α/4,while the mismatch parameterα is varied.

The dimensionless coef�cientZd is determined bynormalization of the numerical results according toEq.4, which is plotted in Fig.3 as a function of the nor-malized delamination width,d/hf , for different elasticmismatch parameters. TheZd function has two limits.First, whend/hf → ∞ (long crack limit), the interfa-cial crack reaches a steady state with the energy releaserate

Gdss = σ2

f hf

2Ef, (5)

and thusZd → 0.5. The steady-state energy releaserate for the interfacial crack is independent of the cracklength as well as the elastic mismatch. On the otherhand, whend/hf → 0 (short crack limit), the interfa-cial energy release rate follows a power law (He andHutchinson 1989a):

Zd ∼(

d

hf

)1−2λ

, (6)

whereλ depends on the elastic mismatch and can bedetermined by solving the equation (Zak and Williams1963)

cosλπ = 2(α − β)

1 − β(1 − λ)2 − α − β2

1 − β2 . (7)

More details about the solution at the short crack limitas well as comparisons with the �nite element resultsare given in the Appendix. Here we discuss three sce-narios at the short crack limit, which would eventuallydetermine the condition for channel cracking with orwithout interfacial delamination. First, whenα = β =0 (no elastic mismatch), we haveλ = 0.5. In this case,

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334 H. Mei et al.

Fig. 2 (a) Schematics ofthe 2D plane strain model ofa steady-state channel crackwith interfacialdelamination; (b) geometryof the �nite element model,with uniform normaltraction(σ f ) acting onto thesurface of the channel crackand a symmetry boundarycondition for the substrate;(c) an example �niteelement mesh, with in�nityelements along the bottomand right boundaries; (d) adetailed mesh around the tipof the interfacial crack

film

substrate

hf

2d

(a) (b)

(d)

Crack

(c)

0 1 2 30.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

d/hf

Zd

β=α/4 α=�0.99

α=�0.6

α=0

α=0.2

α=0.6

Fig. 3 Normalized energy release rate of interfacial delami-nation from the root of a channel crack as a function of thenormalized delamination width for different elastic mismatchparameters

Zd approaches a constant asd/hf → 0. As shownin the Appendix, an analytical solution predicts that

Zd(0,0,0) → 0.9878, which compares well with ournumerical results (Fig.A2). Whenα > 0 (β = α/4),we haveλ > 0.5. Consequently,Zd → ∞ asd/hf →0. As shown in Fig.3, for bothα = 0 andα > 0, theinterfacial energy release rate monotonically decreasesas the delamination width increases. On the other hand,whenα < 0, we have 0< λ < 0.5, and thus,Zd → 0asd/hf → 0. Interestingly, the numerical results inFig.3 show that, instead of a monotonic variation withrespect to the crack length, the interfacial energy releaserate oscillates between the short and long crack lim-its for the cases withα <0. Such an oscillation leadsto local maxima of the interfacial energy release rate,which in some cases (e.g.,α = −0.6) can be greaterthan the steady state value at the long crack limit.

Previously,Ye etal.(1992) gave an approximate for-mula for theZd function based on their �nite elementcalculations. Although the formula has similar asymp-totic limits for long and short cracks as the analyticalsolutions, it gives inaccurate results for at least twocases. First, in the case of no elastic mismatch, the

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In�uence of interfacial delamination on channel cracking 335

formula predicts thatZd → 0.748 asd/hf →0, about25% lower than the analytical solution. Second, theinterfacial energy release rates for intermediate cracklengths by the approximate formula in general do notcompare closely with numerical results, especially forcases withα <0, where the oscillation and the max-ima are not well captured by the approximation. Aswill be discussed later, the maximum interfacial energyrelease rate forα≤ 0 is critical for determining the con-dition of interfacial delamination along side the chan-nel crack. Another previous study byYu etal. (2001)investigated interfacial delamination under two differ-ent edge conditions. While the steady-state interfacialenergy release rate is the same for all edge conditions,the short crack limit strongly depends on the edge effect.

A necessary condition for steady-state channelcracking with concomitant interfacial delamination isthat the interfacial crack arrests at a �nite width. Thedelamination width can be determined by comparingthe interfacial energy release rate in Eq.4 to the inter-face toughness. In general, the interface toughnessdepends on the phase angel of mode mix (Hutchinsonand Suo 1992), which in turn depends on the delami-nation width, as shown in Fig.4. Due to the oscillatorynature of the stress singularity at the interfacial cracktip (Rice 1988), a length scale has to be used to de�nethe phase angle. Here we take the �lm thicknesshf asthe length scale, and de�ne the mode angle as

0 1 2 3 4 540

45

50

55

60

65

d/hf

ψ (

deg)

β = α/4

α=�0.99

α=�0.6

α=0

α=0.6

Fig. 4 Phase angle of the mode mix for interfacial delaminationas a function of the normalized delamination width for differ-ent elastic mismatch parameters. The dashed line indicates thesteady-state phase angle(52◦) for the case of zero elastic mis-match(α = β = 0)

ψ = tan−1

(Im(Khiε

f )

Re(Khiεf )

), (8)

whereK = K1 + i K2 is the complex stress intensity

factor, andε = 12π ln

(1−β1+β

). The real and imaginary

parts of the complex stress intensity factor are calcu-lated by the interaction integral method in ABAQUS.Figure4shows that the phase angle quickly approachesa steady state

ψss = ω(α, β), (9)

as given bySuo and Hutchinson(1990). When the �lmand the substrate have identical elastic moduli(α =β = 0), we haveψss = ω(0,0) = 52◦. Consideringthe fact that the variation of the phase angle with respectto the delamination width is relatively small and con-�ned within a small range of short cracks(d < hf ), wetake the constant steady-state phase angle, Eq.9, in thesubsequent discussions and assume that the interfacetoughness is independent of the delamination width,i.e.,�i = �i (ψss). Then, the width of the interfacialdelamination can be determined by requiring that

Zd

(ds

hf, α, β

)= �i = Ef �i (ψss)

σ2f hf

. (10)

The right-hand side of Eq.10 is the normalized inter-face toughness, independent of the interfacial cracklength. In the following, we discuss possible solutionsto Eq.10 for different elastic mismatches.

First, whenα = β = 0 (i.e., no elastic mismatch),the Zd function has a maximum,Zd → 0.9878 asd/hf → 0, and it approaches the steady state,Zd → 0.5,for long cracks. Consequently, when�i ≥ 0.9878(strong interface), the interfacial energy release rate isalways lower than the interface toughness, and thus nodelamination would occur (i.e.,ds = 0). On the otherhand, when�i ≤ 0.5 (weak interface), the interfacialenergy release rate is always higher than the interfacetoughness. In this case, the interfacial crack would growunstably to in�nity (i.e.,ds → ∞), causing spalling ofthe �lm from the substrate, unless the interfacial crackis arrested by other features such as geometric edges ormaterial junctions. Only for an intermediate interfacetoughness with 0.9878> �i > 0.5, Eq.10 has a �nitesolution, 0< ds < ∞, in which case the channel crackgrows with concomitant interfacial delamination of thewidth ds . The stable delamination width is plotted asa function of the normalized interface toughness�i inFig.5.

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336 H. Mei et al.

0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

Interface toughness, Γi/Σ

d s/hf

α=�0.6

α=0

α=0.6

Fig. 5 The normalized stable delamination width as a functionof the normalized interface toughness�i = �i/�, where� =σ2

f hf /Ef

Next, whenα > 0 (i.e., a stiff �lm on a relativelycompliant substrate), theZd function is unboundedas d/hf → 0. Thus, for all interfaces with�i >0.5,a stable delamination widthds can be obtained fromEq.10. This indicates that interfacial delaminationwould always occur concomitantly with the channelcrack when the substrate is more compliant than the�lm. As shown in Fig.5, the delamination widthincreases as the normalized interface toughnessdecreases. When�i ≤ 0.5, the interfacial crack growsunstably and the delamination width approachesin�nity.

Whenα <0 (i.e., a compliant �lm on a relativelystiff substrate), theZd function necessarily starts fromzero atd/hf = 0, but has a local maximum (Zdm)beforeit approaches the steady state value. The valueZdm

decreases asα decreases, which is greater than 0.5when 0>α> − 0.89 and lower than 0.5 whenα ≤−0.89. Consequently, when 0>α>−0.89, no interfa-cial delamination occurs if�i ≥ Zdm , and stable delam-ination if 0.5<�i < Zdm . On the other hand, whenα≤ −0.89, stable delamination cannot occur; the chan-nel crack either has no delamination for�i ≥ 0.5 orcauses unstable delamination for�i <0.5. The stabil-ity of the interfacial delamination is dictated by thetrend of the interfacial energy release rate with respectto the delamination width (Fig.3). Although Eq.10hasa �nite solution for α <0 and�i <0.5, the interfa-cial crack is unstable because∂Zd

∂d > 0 (the minor

�1 0 1α

0.5

0.99I II�A

II�B

III�B III�A

Γi / Σ

α = �0.89

Fig. 6 A map for interfacial delamination from the root of achannel crack(β = α/4): (I) no delamination, (II) stable delam-ination, and (III) unstable delamination, where A and B denotedelamination without and with an initiation barrier, respectively

oscillation of theZd function has been ignored here).Moreover, for both the stable and unstable delamina-tion, a critical defect size is required for the initiationof the interfacial delamination, since the energy releaserate approaches zero for very short cracks(d/hf → 0).This sets a barrier for the initiation of interfacial delam-ination from the channel crack when the substrate ismechanically stiffer than the �lm.

The above discussion is summarized in Fig.6 asan interfacial delamination map for different combi-nations of �lm/substrate elastic mismatch and inter-face toughness. Three regions are identi�ed for (I) nodelamination, (II) stable delamination, and (III) unsta-ble delamination. In regions II and III, sub-regionsfor delamination without and with an initiation barrierare denoted by A and B, respectively. The boundarybetween Region I and Region II-B is determined fromthe present �nite element calculations, correspondingto the maximum interfacial energy release rate for 0>

α> − 0.89. In an experimental study byTsui etal.(2005), no interfacial delamination was observed forchannel cracking of a low k �lm directly deposited ona Si substrate, while a �nite delamination was observedwhen a polymer buffer layer was sandwiched betweenthe �lm and the substrate. These observations are con-sistent with the delamination map. In the former case,the elastic mismatch between the �lm and the sub-strate,α = −0.91, thus no delamination when thenormalized interface toughness�i ≥ 0.5 (i.e., Region

123

In�uence of interfacial delamination on channel cracking 337

I in Fig.6). With a polymer buffer layer, however, theelastic mismatch between the low k material and thepolymer is,α = 0.4. Although the polymer layer isrelatively thin, it qualitatively changes the interfacialbehavior from that forα < 0 (Region I) to that forα > 0 (Region II-A). More experimental evidenceswith different combinations of elastic mismatch, inter-face toughness, and �lm stress would be needed forfurther validation of the predicted delamination map.

4 Channel cracking with stable delamination

As the question regarding the occurrence of interfa-cial delamination from the root of a channel crack isaddressed in the previous section, the next questionis: how would the interfacial delamination in�uencethe driving force for the growth of a channel crack?Again, we consider the steady-state growth. With a sta-ble delamination along each side of the channel crack(Fig.2a), the substrate constraint on the opening ofthe channel crack is relaxed. Consequently, the steady-state energy release rate calculated from Eq.3becomesgreater than Eq.1. A dimensional consideration leadsto

G∗ss = Z∗

(d

hf, α, β

)σ2

f hf

Ef, (11)

where Z∗ is a new dimensionless coef�cient thatdepends on the width of interfacial delamination(d/hf )

in addition to the elastic mismatch parameters. From anenergetic consideration, we obtain that

G∗ss = Gss + 2

hf

d∫

0

Gd(a)da, (12)

whereGss is the steady-state energy release rate of thechannel crack with no delamination as given in Eq.1,andGd(a) is the energy release rate of the interfacialcrack of widtha as given in Eq.4. Whend/hf → 0,G∗

ss → Gss or Z∗ → Z , recovering Eq.1; whend/hf → ∞, Z∗ → ∞. Furthermore, asds/hf → ∞,since the interfacial crack approaches the steady state(Gd → Gd

ss andZd → 0.5), the increase of the energyrelease rate is simply

�G∗ss = 2

hfGd

ss�d, or �Z∗ = �d

hf, (13)

which dictates that the coef�cientZ∗ increases withthe normalized delamination widthd/hf linearly witha slope of 1 at the limit of long delamination.

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

Delamination width, d/hf

Z* �

Z

α=�0.6 (Z=1.360)

α=0 (Z=1.976)

α=0.6 (Z=3.747)

α=0.9 (Z=8.848)

Fig. 7 Increase of the driving force for steady-state channelcracking due to concomitant interfacial delamination

The same �nite element model as illustrated in Fig.2is employed to calculateZ∗, by integrating the open-ing displacement along the surface of the channel crackas Eq.3. Figure7 plots the difference,Z∗ − Z , as afunction ofd/hf for different elastic mismatch param-eters. For a compliant �lm on a relative stiff substrate(α < 0), the increase due to interfacial delaminationis almost linear for the entire range of delaminationwidth. For a stiff �lm on a relatively compliant sub-strate(α > 0), however, the increase is nonlinear forshort interfacial delamination and then approaches astraight line of slope 1 as predicted by Eq.13. Appar-ently, with interfacial delamination, the driving forcefor channel cracking can be significantly higher thanthat assuming no delamination.

As discussed in the previous section, the stabledelamination width,ds/hf , can be obtained as a func-tion of the normalized interface toughness,�i , byEq.10, as shown in Fig.5. Thus, the coef�cientZ∗in Eq.11 may also be plotted as a function of�i , asshown in Fig.8. Whenα >0, Z∗ → Z as�i → ∞, andZ∗ → ∞ as�i → 0.5; in between,Z∗ increases as�i

decreases, because the interfacial delamination widthincreases. Whenα= 0, Z∗ = Z as�i ≥ 0.9878 (i.e.,no delamination). When 0>α> − 0.89, Z∗ increasesfrom Z to in�nity within a narrow window of�i , wherestable delamination is predicted (Region II-B in Fig.6).Forα < − 0.89, eitherZ∗ = Z for no delamination orZ∗ → ∞ for unstable delamination. Therefore, Fig.8explicitly illustrates the in�uence of the interface tough-ness on the driving force of channel cracking in the �lm.

123

338 H. Mei et al.

0.5 1 1.50

2

4

6

8

10

Interface toughness, Γi/Σ

Z*

α=�0.6

α=0

α=0.6

Fig. 8 In�uence of the normalized interface toughness(� =σ2

f hf /Ef ) on the steady-state driving force for channelcracking

While the interfacial delamination, if occurring,relaxes the constraint on crack opening thus enhancesthe fracture driving force, it also requires additionalenergy to fracture the interface as the channel crackadvances. An energetic condition can thus be stated: ifthe increase in the energy release exceeds the fractureenergy needed for delamination, growth of the chan-nel crack with interfacial delamination is energeticallyfavored; otherwise, the channel crack grows with nodelamination. It can be shown that this condition isconsistent with the delamination map in Fig.6. Con-sidering the interfacial fracture energy, a fracture con-dition for steady-state growth of a channel crack canbe written as

G∗ss ≥ �f + Wd , (14)

where�f is the cohesive fracture toughness of the �lm,andWd is the energy required to delaminate the inter-face accompanying per unit area growth of the channelcrack. For stable delamination of widthd = ds at bothsides of a channel crack, the delamination energy is

Wd = 2hf

ds∫

0

�i (ψ(a))da ≈ 2�i (ψss)ds

hf. (15)

Again, the phase angle of the interfacial crack isapproximately taken as a constant independent of thecrack length. Whends = 0, Eq.14 recovers the condi-tion for cohesive fracture of the �lm, i.e.,Gss ≥ �f .

Equation14may not be convenient to apply directly,since both sides of the equation (driving force and resis-

tance, respectively) increase with the interfacial delam-ination. By movingWd to the left hand side and notingthat the stable delamination width is a function of theinterface toughness, we de�ne an effective driving forcefor the steady-state channel cracking:

Ge f fss = G∗

ss − Wd = Zef f (�i , α, β)σ2

f hf

Ef. (16)

with

Zef f = Z∗(

ds

hf, α, β

)− 2�i

ds

hf. (17)

Using the effective energy release rate, the condi-tion for the steady-state channel cracking is simply acomparison betweenGe f f

ss and�f , the latter being aconstant independent of the interface. Figure9 plotsthe ratio,Zef f /Z(α, β), as a function of�i for differ-ent elastic mismatch parameters. At the limit of highinterface toughness(�i → ∞), ds → 0 andZef f → Z ,which recovers the case of channel cracking with nodelamination. The effective driving force increases asthe normalized interface toughness deceases. Comparedto Fig.8, the in�uence of interfacial delamination on theeffective driving force is reduced after considering theinterfacial fracture energy.

5 Summary and discussions

This paper considers concomitant interfacial delami-nation and channel cracking in elastic thin �lms. Twomain conclusions are summarized as follows.

• Stable interfacial delamination along a channelcrack is predicted for certain combinations of �lm/substrate elastic mismatch, interface toughness,and �lm stress, as summarized in a delaminationmap (Fig.6), together with conditions for no delam-ination and unstable delamination.

• Interfacial delamination not only increases thefracture driving force for steady-state growth ofthe channel crack, but also adds to the fractureresistance by requiring additional energy for theinterfacial fracture. An effective energy release ratefor channel cracking is de�ned, which depends onthe interface toughness (Fig.9) in addition to theelastic mismatch and can be considerably higherthan the energy release rate assuming nodelamination.

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In�uence of interfacial delamination on channel cracking 339

0.5 1 1.50.9

1

1.1

1.2

1.3

1.4

1.5

Interface toughness, Γi/Σ

Zef

f/Z

α=�0.6

α=0

α=0.6

Fig. 9 Effective driving force for steady-state channelcracking as a function of the normalized interface toughness(�= σ2

f hf /Ef )

5.1 Implications for reliability of integrated structures

The present study predicts that channel cracking in anelastic thin �lm on a relatively compliant substrate isalways accompanied by interfacial delamination, eitherstable or unstable, depending on the interface tough-ness. This differs from the case for an elastic �lm ona relatively stiff substrate, in which channel cracks maygrow without interfacial delamination (Region I inFig.6). This difference may have important implica-tions for reliability of integrated structures. As an exam-ple, for interconnect structures in microelectronics, thelow-k dielectrics is usually more compliant comparedto the surrounding materials (Liu etal. 2004). Therefore,fracture of the low-k dielectrics by channel crackingis typically not accompanied by interfacial delamina-tion. However, when a more compliant buffer layer isadded adjacent to the low-k �lm, interfacial delami-nation can occur concomitantly with channel crackingof the low-k �lm (Tsui etal. 2005). Moreover, a rela-tively stiff cap layer (e.g., SiN) is often deposited ontop of the low-k �lm (Liu etal. 2007). Channel crack-ing of the cap layer on low-k could be significantlyenhanced by interfacial delamination. Flexible elec-tronics is another area of applications where compli-ant substrates have to be used extensively along withmechanically stiffer �lms for the functional devices andinterconnects (Wagner etal. 2004; Khang etal. 2006).Here, interfacial delamination could play a critical rolein the reliability assessment. As shown in a previous

study byLi and Suo(2007), the stretchability of metalthin-�lm interconnects on a compliant substrate can bedramatically reduced by interfacial delamination. Forbrittle thin �lms on compliant substrates, as consideredin the present study, interfacial delamination has a sim-ilar effect on the fracture and thus deformability of thedevices.

5.2 On substrate thickness effect

The present study considers only in�nitely thick sub-strates. The effect of �nite substrate thickness onchannel cracking without interface delamination hasbeen investigated previously (e.g.,Huang etal. 2003;Vlassak 2003). Depending on the boundary conditionsat the bottom and edges of the substrate, the steady-state energy release rate for channel cracking in a thin-�lm coating on the substrate may increase or decreaseas the substrate thickness decreases. A similar effectis intuitively expected for interfacial delamination, butan in-depth understanding of the substrate thicknesseffect on the coupling between channel cracking andinterfacial delamination will be left for future studies.

5.3 On other fracture modes

In addition to channel cracking, other failure modes ofthin elastic �lms have also been observed and studied(Hutchinson and Suo 1992). For example,Beuth(1992)considered partially cracked thin �lms with the chan-nel depth less than the �lm thickness and found that,when the �lm is more compliant than the substrate, thecrack may not propagate all the way to the �lm/sub-strate interface. In this case, interfacial delamination isunlikely to occur, consistent with the result from thepresent study for the cases withα <0. For a crackimpinging an interface,He and Hutchinson(1989a)examined the competition between crack de�ection intothe interface and penetration into the substrate; theyproposed a criterion based on asymptotic solutions forthe respective energy release rates along with the ratiobetween the toughness of the interface and that of thesubstrate.Ye etal.(1992) showed that, when the sub-strate is brittle with a relatively low toughness (�s <�f ),the channel crack may penetrate into the substrate andreach a stable depth. The competition among differentfracture modes may be addressed in more details forspeci�c thin-�lm material systems.

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340 H. Mei et al.

Acknowledgements This work is supported by the NationalScience Foundation through Grant No. 0547409.

Appendix: Short delamination crackfrom a channel root

This Appendix summarizes asymptotic solutionsfrom previous studies (He and Hutchinson 1989a, b;Hutchinson and Suo 1992) for short delamination cracksemanating from the root of a channel crack (i.e.,d/hf →0 in Fig.2a), and presents comparisons with numericalresults from the �nite element model shown in Fig.2.

A.1 Zero elastic mismatch (α = β = 0)

This is a case of crack kinking in a homogeneous solid.Without the interfacial delamination, the channel crackin the �lm is equivalent to a two-dimensional edgecrack, with the stress intensity factor at the root

K I = 1.1215σ f√πhf . (A1)

For a small crack segment(d hf ) kinking out of theplane of the edge crack, the stress intensity factors atthe new crack tip are linearly related to the stress inten-sity factors at the tip of the parent crack (Hutchinsonand Suo 1992), namely

K dI = c11K I + c12K II

K dII = c21K I + c22K II

, (A2)

where the coef�cients,ci j , depend on the kink angle, asgiven byHayashi and Nemat-Nasser(1981). Follow-ingHe and Hutchinson(1989b), the coef�cients can bewritten as

c11 = CR + DR, c21 = CI − DI , (A3)

whereC = CR + iCI and D = DR + i DI are twocomplex valued functions, with the subscriptsR andI denoting their real and imaginary parts, respectively.An approximation byCotterell(1965) gives that

C = 12

(e−iω/2 + e−i3ω/2

),

D = 14

(e−iω/2−ei3ω/2

), (A4)

whereω is the kink angle.Cotterell and Rice(1980)have shown that this approximation is asymptoticallycorrect for small kink angles and is reasonably accuratefor kink angles as large as 45◦ or even 90◦, dependingon the mode mix.

For the present problem with a channel crack kink-ing into the interface, the kink angle is 90◦ andK II = 0.Under the plane strain condition, the energy release rateof the interfacial crack is

Gd(d → 0) = (K dI )

2 + (K dII )

2

E

= (1.1215)2π(c211 + c2

21)σ2

f hf

E. (A5)

A comparison between Eq.A5 and Eq.4 gives thedimensionless coef�cient

Zd(d → 0) = 1.258π(c211 + c2

21). (A6)

Using the approximation (A4) withω = π/2, we obtainthat

c11 =√

24, c21 = −

√2

4. (A7)

Inserting (A7) into (A6) gives that

Zd(d → 0) = 0.9878. (A8)

This approximation agrees well with the numericalresult shown in Fig.3, whereZd = 0.9923 ford/hf =10−3.

A.2 Crack de�ection at a bimaterial interface

For an interface between two elastic materials with gen-eral elastic mismatch, an asymptotic solution byHeand Hutchinson(1989a) gives the energy release rateof the delamination crack emanating from a perpendic-ular channel crack (Fig.2a,d hf ):

Gd =(

1

Ef+ 1

Es

)K 2

1 + K 22

2 cosh2πε

=(

1

Ef+ 1

Es

)k2

1[|C |2+|D|2+2Re(C D)]2 cosh2πε

d1−2λ.

(A9)

where C and D are dimensionless, complex valuedfunctions of α and β, k1 is a real valued constantrepresenting the stress intensity at the root of the chan-nel crack, andλ is determined by Eq.7. This asymp-totic solution leads to a power-law dependence of theZd function on the interfacial crack length asd → 0,namely

Zd = Gd

σ2f hf /Ef

∼(

d

hf

)1−2λ

. (A10)

123

In�uence of interfacial delamination on channel cracking 341

0.001 0.01 0.1 1 10

0.3

0.4

0.5

0.6

d/hf

Zd

α=�0.99 (λ = 0.312)

α=�0.6 (λ = 0.388)

Fig. A1 Normalized energy release rate of interfacial delamina-tion emanating from the root of a channel crack, forα = −0.99andα = −0.6. The asymptotic power law, Eq.A10, is repre-sented by the straight lines at the short crack limit with slopes,1 − 2λ = 0.376 and 0.224, respectively

0.001 0.01 0.1 1 100.5

1

10

d/hf

Zd

α=0.6 (λ = 0.654)

α=0.2 (λ = 0.542)

α=0 (λ = 0.5)

Fig. A2 Normalized energy release rate of interfacial delami-nation emanating from the root of a channel crack, forα = 0,α = 0.2, andα = 0.6. The asymptotic power law, Eq.A10,is represented by the straight lines at the short crack limit withslopes, 1− 2λ = 0, −0.084, and−0.308, respectively

FiguresA1 andA2 plot the numerical solutions ofZd from the �nite element model (Fig.2), in compar-ison with the asymptotic solution. Whenα <0 (β =α/4), 0< λ < 0.5 and the log–log plot ofZd vs.d/hf

(Fig.A1) approaches a straight line of positive slope(1− 2λ > 0) asd/hf → 0. Whenα > 0,λ > 0.5 andthe log–log plot (Fig.A2) approaches a straight line ofnegative slope(1−2λ < 0). Whenα = β = 0 (no elas-

tic mismatch),λ = 0.5 andZd approaches a constantwith zero slope in the log–log plot (Fig.A2). The com-parisons show good agreement between the numericalresults and the asymptotic power law for short delam-ination cracks.

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