Influence of Interfacial Delamination on Channel Cracking of Elastic Thin Films. Haixia Mei, Yaoyu Pang, Rui Huang

Transcription

1 Inluence o Interacial Delamination on Channel Cracking o Elastic Thin Films Haixia Mei, Yaoyu Pang, Rui Huang Department o Aerospace Engineering an Engineering Mechanics, University o Texas, Austin, TX 7871 ( ruihuang@mail.utexas.eu) Abstract Channeling cracks in brittle thin ilms have been observe to be a key reliability issue or avance interconnects an other integrate structures. Most theoretical stuies to ate have assume no elamination at the interace, while experiments have observe channel cracks both with an without interacial elamination. This paper analyzes the eect o interacial elamination on the racture conition o brittle thin ilms on elastic substrates. It is oun that, epening on the elastic mismatch an interace toughness, a channel crack may grow with no elamination, with a stable elamination, or with unstable elamination. For a ilm on a relatively compliant substrate, a critical interace toughness is preicte, which separates stable an unstable elamination. For a ilm on a relatively sti substrate, however, a channel crack grows with no elamination when the interace toughness is greater than a critical value, while stable elamination along with the channel crack is possible only in a small range o interace toughness or a speciic elastic mismatch. An eective energy release rate or the steay-state growth o a channel crack is eine to account or the inluence o interacial elamination on both the racture riving orce an the resistance, which can be signiicantly higher than the energy release rate assuming no elamination. Key wors: channel cracking, elamination, thin ilms, interace. 1

2 1. Introuction Integrate structures with mechanically sot components have recently been pursue over a wie range o novel applications, rom high perormance integrate circuits in microelectronics (Ho et al., 004) to unconventional organic electronics (Doabalapur, 006) an stretchable electronics (Wagner et al., 004; Khang et al., 006), along with the ubiquitous integration o har an sot materials in biological systems (Gao, 006). In particular, the integration o low ielectric constant (low k) materials in avance interconnects o microelectronics has pose signiicant challenges or reliability issues resulting rom the compromise mechanical properties. Two ailure moes have been reporte, one or cohesive racture (Liu et al., 004) an the other or interacial elamination (Liu et al., 007). The ormer pertains to the brittleness o the low-k materials subjecte to tension, an the latter maniests ue to poor ahesion between low-k an surrouning materials (Tsui et al., 006). This paper consiers concomitant cohesive racture an interacial elamination as a hybri ailure moe in integrate thin-ilm structures. One common cohesive racture moe or thin ilms uner tension is channel cracking (Figure 1). Previous stuies have shown that the riving orce (i.e., the energy release rate) or the steay-state growth o a channel crack epens on the constraint eect o surrouning layers (Hutchinson an Suo, 199). For a brittle thin ilm on an elastic substrate, the riving orce increases or increasingly compliant substrates (Beuth, 199; Huang et al., 003). The eect o constraint can be partly lost as the substrate eorms plastically (Ambrico an Begley, 00) or viscoelastically (Huang et al., 00; Suo et al., 003). More recent stuies have ocuse on the eects o stacke buer layers (Tsui et al., 005; Corero et al, 007) an patterne ilm structures (Liu et al., 004). In most o these stuies, the interaces between the ilm an the substrate or the buer layers are assume to remain perectly bone as the channel crack grows

3 in the ilm (Figure 1a). However, the stress concentration at the root o the channel crack may rive interacial elamination (Ye et al., 199). While some experimental observations clearly showe no elamination (Tsui et al., 005; He, et al. 004), others observe elamination o the interace (Tsui et al., 005; Suo, 003). There are two questions yet to be answere: First, uner what conition woul the growth o a channel crack be accompanie by interacial elamination? Secon, how woul the interacial elamination (i occurring) aect the racture conition or reliability in integrate thin ilm structures? To answer these questions, this paper consiers steay-state channel cracking o an elastic thin ilm on an elastic substrate an theoretically examines the eect o concomitant interacial elamination. Section briely reviews the concept o steay-state riving orce or a channel crack growing without elamination. Section 3 analyzes interacial elamination emanating rom the root o a long, straight channel crack. In Section 4, the racture riving orce or the steay-state growth o a channel crack with interacial elamination is etermine. A inite element moel is use to calculate the energy release rates or both the interacial elamination an the steay-state channel cracking. Moreover, to account or the inluence o interacial elamination on the racture resistance, we eine an eective energy release rate that epens on the interace toughness as well as the elastic mismatch between the ilm an the substrate. In conclusion, Section 5 summarizes the inings an emphasizes the impact o interacial elamination on the reliability o integrate structures with mechanically sot components. 3

4 . Channel cracking without elamination As illustrate in Figure 1a, assuming no interacial elamination, the energy release rate or the steay-state growth o a channel crack in a thin elastic ilm bone to a thick elastic substrate is (Beuth, 199; Hutchinson an Suo, 199): G ss σ h = Z( α, β ), (1) E where σ is the tensile stress in the ilm, is the ilm thickness, an h E ( ν ) = is the E 1 plane strain moulus o the ilm with Young s moulus E an Poisson s ratio ν. The imensionless coeicient Z epens on the elastic mismatch between the ilm an the substrate, through the Dunurs parameters E Es α = an E + E s E (1 ν )(1 ν s) Es(1 ν s)(1 ν ) β =. () (1 ν )(1 ν ) s ( E + E ) s When the ilm an the substrate have ientical elastic mouli, we have α = β = 0 an Z = The value o Z ecreases slightly or a compliant ilm on a relatively sti substrate ( E < an α < 0 ). A more compliant substrate ( α > 0 ), on the other han, provies less E s constraint against ilm cracking. Thus, Z increases as α increases. For very compliant substrates (e.g., organic low-k ielectrics, polymers, etc.), Z increases rapily, with Z > 30 or α > (Beuth, 199; Huang et al., 003). The eect o β is seconary an oten ignore. In general, the steay-state energy release rate o channel cracking can be calculate rom a two-imensional (D) moel (Beuth, 199; Huang et al., 003) as ollows: G ss = 1 h h 0 σ δ ( z) z, (3) 4

5 where δ (z) is the opening isplacement o the crack suraces ar behin the channel ront (see Fig. 1a). Due to the constraint by the substrate, the crack opening oes not change as the channel ront avances an the energy release rate attains a steay state, inepenent o the channel length. Three-imensional analyses have shown that the steay state is reache when the length o a channel crack excees two to three times the ilm thickness or a relatively sti substrate (Nakamura an Kamath, 199), but the crack length to reach the steay state can be signiicantly longer or more compliant substrate materials (Ambrico an Begley, 00). The present stuy ocuses on the steay state. 3. Interacial elamination rom channel root Now consier an interacial crack emanating rom the root o a channel crack at each sie (Figure 1b). For a long, straight channel crack, we assume a steay state ar behin the channel ront, where the interacial crack has a inite with,. The energy release rate or the interacial crack can be written in a similar orm as Eq. (1): G σ h = Z, α, β, (4) h E where Z is a imensionless unction that can be etermine rom a two-imensional plane strain problem as illustrate in Figure a. In the present stuy, a inite element moel is constructe to calculate the interacial energy release rate. By symmetry, only hal o the ilm/substrate structure is moele along with proper bounary conitions (Figure b). The inite element package ABAQUS is employe an an example mesh near the interacial crack is shown in Figure c. Close to the tip o the interacial crack, a very ine mesh is use (Figure ), with a set o singular elements aroun the crack tip. Far away, ininite elements are use or both the ilm 5

6 an substrate to eliminate possible size eects o the moel. The metho o J-integral is aopte or the calculation o the interacial energy release rate. In all calculations, we set ν = ν s 1 = 3 such that β = α 4, while the mismatch parameter α is varie. The imensionless coeicient Z is etermine by normalization o the numerical results accoring to Eq. (4), which is plotte in Figure 3 as a unction o the normalize elamination with,, or ierent elastic mismatch parameters. The unction has two limits. First, h Z when / h release rate (long crack limit), the interacial crack reaches a steay state with the energy G ss σ h =, (5) E an thus Z 0.5. The steay-state energy release rate or the interacial crack is inepenent o the crack length as well as the elastic mismatch. On the other han, when / h 0 (short crack limit), the interacial energy release rate ollows a power law (He an Hutchinson, 1989a): 1 λ ~ Z, (6) h where λ epens on the elastic mismatch an can be etermine by solving the equation (Zak an Williams, 1963) ( β α ) ( ) α + β 1 λ + cosλπ =. (7) 1+ β 1 β More etails about the solution at the short crack limit as well as comparisons with the inite element results are given in the Appenix. Here we iscuss three scenarios at the short crack limit, which woul eventually etermine the conition or channel cracking with or without 6

7 interacial elamination. First, when α = β = 0 (no elastic mismatch), we have λ = In this case, Z approaches a constant as / 0. As shown in the Appenix, an analytical solution h preicts that Z ( 0,0,0) , which compares well with our numerical results (Figure A). When α > 0 ( β = α 4 ), we have λ > Consequently, Z as / 0. As shown in Figure 3, or both α = 0 an α > 0, the interacial energy release rate monotonically ecreases as the elamination with increases. On the other han, when α < 0, we have 0 < λ < 0. 5, an thus, Z 0 as / 0. Interestingly, the numerical results in Figure 3 show that, instea o h a monotonic variation with respect to the crack length, the interacial energy release rate oscillates between the short an long crack limits or the cases with α < 0. Such an oscillation leas to local maxima o the interacial energy release rate, which in some cases (e.g., α = 0. 6 ) can be greater than the steay state value at the long crack limit. h Previously, Ye et al. (199) gave an approximate ormula or the Z unction base on their inite element calculations. Although the ormula has similar asymptotic limits or long an short cracks as the analytical solutions, it gives inaccurate results or at least two cases. First, in the case o no elastic mismatch, the ormula preicts that Z as / 0, about 5% lower than the analytical solution. Secon, the interacial energy release rates or intermeiate crack lengths by the approximate ormula in general o not compare closely with numerical results, especially or cases with α < 0, where the oscillation an the maxima are not well capture by the approximation. As will be iscusse later, the maximum interacial energy release rate or α 0 is critical or etermining the conition o interacial elamination along sie the channel crack. Another previous stuy by Yu et al. (001) investigate interacial elamination uner two ierent ege conitions. While the steay-state interacial energy h 7

8 release rate is the same or all ege conitions, the short crack limit strongly epens on the ege eect. A necessary conition or steay-state channel cracking with concomitant interacial elamination is that the interacial crack arrests at a inite with. The elamination with can be etermine by comparing the interacial energy release rate in Eq. (4) to the interace toughness. In general, the interace toughness epens on the phase angel o moe mix (Hutchinson an Suo, 199), which in turn epens on the elamination with, as shown in Figure 4. Due to the oscillatory nature o the stress singularity at the interacial crack tip (Rice, 1988), a length scale has to be use to eine the phase angle. Here we take the ilm thickness h as the length scale, an eine the moe angle as iε ( Kh ) iε ( Kh ) Im ψ = tan 1, (8) Re 1 1 β where K = K 1 + ik is the complex stress intensity actor, an ε = ln. The real an π 1+ β imaginary parts o the complex stress intensity actor are calculate by the interaction integral metho in ABAQUS. Figure 4 shows that the phase angle quickly approaches a steay state ( β ) ψ ss = ω α,, (9) as given by Suo an Hutchinson (1990). When the ilm an the substrate have ientical elastic mouli ( = β = 0 α ), we have ( ) o ψ ss = ω 0,0 = 5. Consiering the act that the variation o the phase angle with respect to the elamination with is relatively small an conine within a small range o short cracks ( < h ), we take the constant steay-state phase angle, Eq. (9), in the subsequent iscussions an assume that the interace toughness is inepenent o the 8

9 elamination with, i.e., Γ etermine by requiring that i = Γi ( ψ ss ). Then, the with o the interacial elamination can be, α, β = Γ E Γ = h s i Z i h σ ( ψ ) ss. (10) The right-han sie o Eq. (10) is the normalize interace toughness, inepenent o the interacial crack length. In the ollowing, we iscuss possible solutions to Eq. (10) or ierent elastic mismatches. First, when α = β = 0 (i.e., no elastic mismatch), the unction has a maximum, Z as / 0, an it approaches the steay state, 0. 5, or long cracks. Consequently, when h Γ i (strong interace), the interacial energy release rate is always Z Z lower than the interace toughness, an thus no elamination woul occur (i.e., s = 0 ). On the other han, when Γ i 0.5 (weak interace), the interacial energy release rate is always higher than the interace toughness. In this case, the interacial crack woul grow unstably to ininity (i.e., s ), causing spalling o the ilm rom the substrate, unless the interacial crack is arreste by other eatures such as geometric eges or material junctions. Only or an intermeiate interace toughness with > Γ i > 0.5, Eq. (10) has a inite solution, 0 <, in which < s case the channel crack grows with concomitant interacial elamination o the with s. The stable elamination with is plotte as a unction o the normalize interace toughness Figure 5. Γ i in Next, when α > 0 (i.e., a sti ilm on a relatively compliant substrate), the unction is Z unboune as / h 0. Thus, or all interaces with Γ i > 0. 5, a stable elamination with s 9

10 can be obtaine rom Eq. (10). This inicates that interacial elamination woul always occur concomitantly with the channel crack when the substrate is more compliant than the ilm. As shown in Figure 5, the elamination with increases as the normalize interace toughness ecreases. When Γ i 0. 5, the interacial crack grows unstably an the elamination with approaches ininity. When α < 0 (i.e., a compliant ilm on a relatively sti substrate), the unction Z necessarily starts rom zero at Z m / h = 0, but has a local maximum ( ) beore it approaches the steay state value. The value Zm ecreases as α ecreases, which is greater than 0.5 when 0 > α > 0.89 an lower than 0.5 when α Consequently, when 0 > α > 0. 89, no interacial elamination occurs i Γ i Zm, an stable elamination i.5 < Γi < Zm 0. On the other han, when α 0. 89, stable elamination cannot occur; the channel crack either has no elamination or Γ i 0. 5 or causes unstable elamination or < 0. 5 Γ i. The stability o the interacial elamination is ictate by the tren o the interacial energy release rate with respect to the elamination with (Fig. 3). Although Eq. (10) has a inite solution or α < 0 an Z Γ i < 0.5, the interacial crack is unstable because > 0 (the minor oscillation o the Z unction has been ignore here). Moreover, or both the stable an unstable elamination, a critical eect size is require or the initiation o the interacial elamination, since the energy release rate approaches zero or very short cracks ( ). This sets a barrier or the initiation o interacial elamination rom the channel crack when the substrate is mechanically stier than the ilm. / h 0 10

11 The above iscussion is summarize in Figure 6 as an interacial elamination map or ierent combinations o ilm/substrate elastic mismatch an interace toughness. Three regions are ientiie or (I) no elamination, (II) stable elamination, an (III) unstable elamination. In regions II an III, sub-regions or elamination without an with an initiation barrier are enote by A an B, respectively. The bounary between Region I an Region II-B is etermine rom the present inite element calculations, corresponing the maximum interacial energy release rate or 0 > α > In an experimental stuy by Tsui et al. (005), no interacial elamination was observe or channel cracking o a low k ilm irectly eposite on a Si substrate, while a inite elamination was observe when a polymer buer layer was sanwiche between the ilm an the substrate. These observations are consistent with the elamination map. In the ormer case, the elastic mismatch between the ilm an the substrate, α = 0. 91, thus no elamination when the normalize interace toughness Γ i 0. 5 (i.e., Region I in Figure 6). With a polymer buer layer, however, the elastic mismatch between the low k material an the polymer is, α = Although the polymer layer is relatively thin, it qualitatively changes the interacial behavior rom that or α < 0 (Region I) to that or α > 0 (Region II-A). More experimental eviences with ierent combinations o elastic mismatch, interace toughness, an ilm stress woul be neee or urther valiation o the preicte elamination map. 4. Channel cracking with stable elamination As the question regaring the occurrence o interacial elamination rom the root o a channel crack is aresse in the previous section, the next question is: how woul the interacial elamination inluence the riving orce or the growth o a channel crack? Again, we consier the steay-state growth. With a stable elamination along each sie o the channel crack (Figure 11

12 a), the substrate constraint on the opening o the channel crack is relaxe. Consequently, the steay-state energy release rate calculate rom Eq. (3) becomes greater than Eq. (1). A imensional consieration leas to G * ss * σ h = Z, α, β, (11) h E where * Z is a new imensionless coeicient that epens on the with o interacial elamination ( h ) in aition to the elastic mismatch parameters. From an energetic consieration, we obtain that G * ss = G ss + h 0 G ( a) a, (1) where G ss is the steay-state energy release rate o the channel crack with no elamination as given in Eq. (1), an (a) G is the energy release rate o the interacial crack o with a as given h * * ss ss Z h in Eq. (4). When / 0, G G or Z, recovering Eq. (1); when /, * Z. Furthermore, as s / h, since the interacial crack approaches the steay state G ss ( G an 0. 5 ), the increase o the energy release rate is simply Z ΔG * ss = h G ss Δ, or Δ Δ Z * =, (13) h which ictates that the coeicient * Z increases with the normalize elamination with linearly with a slope o 1 at the limit o long elamination. The same inite element moel as illustrate in Figure is employe to calculate / h * Z, by integrating the opening isplacement along the surace o the channel crack as Eq. (3). Figure 7 1

13 plots the ierence, * Z Z, as a unction o / or ierent elastic mismatch parameters. For a compliant ilm on a relative sti substrate ( α < 0 ), the increase ue to interacial elamination is almost linear or the entire range o elamination with. For a sti ilm on a relatively compliant substrate ( α > 0 ), however, the increase is nonlinear or short interacial elamination an then approaches a straight line o slope 1 as preicte by Eq. (13). Apparently, with interacial elamination, the riving orce or channel cracking can be signiicantly higher than that assuming no elamination. h As iscusse in the previous section, the stable elamination with, s / h, can be obtaine as a unction o the normalize interace toughness, Γ i, by Eq. (10), as shown in Figure 5. Thus, the coeicient Z * in Eq. (11) may also be plotte as a unction o Γ i, as shown in * Figure 8. When α > 0, Z Z as * Γ i, an Z as Γ i 0. 5 ; in between, Z * increases as Γ i ecreases, because the interacial elamination with increases. When α = 0, * Z = Z as Γ i (i.e., no elamination). When 0 > α > 0. 89, * Z increases rom Z to ininity within a narrow winow o Γ i, where stable elamination is preicte (Region II-B in Figure 6). For α < 0. 89, either Z = Z * or no elamination or * Z or unstable elamination. Thereore, Figure 8 explicitly illustrates the inluence o the interace toughness on the riving orce o channel cracking in the ilm. While the interacial elamination, i occurring, relaxes the constraint on crack opening thus enhances the racture riving orce, it also requires aitional energy to racture the interace as the channel crack avances. An energetic conition can thus be state: i the increase in the energy release excees the racture energy neee or elamination, growth o the channel crack with interacial elamination is energetically avore; otherwise, the channel crack grows with 13

14 no elamination. It can be shown that this conition is consistent with the elamination map in Figure 6. Consiering the interacial racture energy, a racture conition or steay-state growth o a channel crack can be written as G Γ + W * ss, (14) where Γ is the cohesive racture toughness o the ilm, an W is the energy require to elaminate the interace accompanying per unit area growth o the channel crack. For stable elamination o with = s at both sies o a channel crack, the elamination energy is W s = Γi h 0 ( ψ ( a) ) a Γ ( ψ ) i ss h s. (15) Again, the phase angle o the interacial crack is approximately taken as a constant inepenent o the crack length. When ilm, i.e., G Γ. ss s = 0, Eq. (14) recovers the conition or cohesive racture o the Equation (14) may not be convenient to apply irectly, since both sies o the equation (riving orce an resistance, respectively) increase with the interacial elamination. By moving W to the let han sie an noting that the stable elamination with is a unction o the interace toughness, we eine an eective riving orce or the steay-state channel cracking: G e ss * σ h = Gss W = Ze ( Γi, α, β ). (16) E with Z e * s s = Z, α, β Γ i. (17) h h 14

15 Using the eective energy release rate, the conition or the steay-state channel cracking is simply a comparison between e Gss an Γ, the latter being a constant inepenent o the interace. Figure 9 plots the ratio, Z( α, β ) Z e, as a unction o Γ i or ierent elastic mismatch parameters. At the limit o high interace toughness ( Γ i ), 0 an Z, which recovers the case o channel cracking with no elamination. The eective riving orce increases as the normalize interace toughness eceases. Compare to Figure 8, the inluence o interacial elamination on the eective riving orce is reuce ater consiering the interacial racture energy. s Z e 5. Summary This paper consiers concomitant interacial elamination an channel cracking in elastic thin ilms. Two main conclusions are summarize as ollows. Stable interacial elamination along a channel crack is preicte or certain combinations o ilm/substrate elastic mismatch, interace toughness, an ilm stress, as summarize in a elamination map (Figure 6), together with conitions or no elamination an unstable elamination. Interacial elamination not only increases the racture riving orce or steay-state growth o the channel crack, but also as to the racture resistance by requiring aitional energy or the interacial racture. An eective energy release rate or channel cracking is eine, which epens on the interace toughness (Figure 9) in aition to the elastic mismatch an can be consierably higher than the energy release rate assuming no elamination. 15

16 In particular, it is preicte that channel cracking in an elastic thin ilm on a relatively compliant substrate is always accompanie by interacial elamination, either stable or unstable, epening on the interace toughness. This iers rom the case or an elastic ilm on a relatively sti substrate, in which channel cracks may grow without interacial elamination (Region I in Figure 6). This ierence may have important implications or reliability o integrate structures. As an example, or interconnect structures in microelectronics, the low-k ielectrics is usually more compliant compare to the surrouning materials (Liu et al., 004). Thereore, racture o the low-k ielectrics by channel cracking is typically not accompanie by interacial elamination. However, when a more complaint buer layer is ae ajacent to the low-k ilm, interacial elamination can occur concomitantly with channel cracking o the low-k ilm (Tsui et al., 005). Moreover, a relatively sti cap layer (e.g., SiN) is oten eposite on top o the low-k ilm (Liu et al., 007). Channel cracking o the cap layer on low-k coul be signiicantly enhance by interacial elamination. Flexible electronics is another area o applications where compliant substrates have to be use extensively along with mechanically stier ilms or the unctional evices an interconnects (Wagner et al., 004; Khang et al., 006). Here, interacial elamination coul play a critical role in the reliability assessment. As shown in a previous stuy by Li an Suo (007), the stretchability o metal thin-ilm interconnects on a compliant substrate can be ramatically reuce by interacial elamination. For brittle thin ilms on compliant substrates, as consiere in the present stuy, interacial elamination has a similar eect on the racture an thus eormability o the evices. Acknowlegments This work is supporte by the National Science Founation through Grant No

17 Appenix: Short elamination crack rom a channel root This Appenix summarizes asymptotic solutions rom previous stuies (He an Hutchinson, 1989a & 1989b; Hutchinson an Suo, 199) or short elamination cracks emanating rom the root o a channel crack (i.e., 0 in Figure a), an presents comparisons with numerical results rom the inite element moel shown in Figure. h A.1 Zero elastic mismatch ( α = β = 0) This is a case o crack kinking in a homogeneous soli. Without the interacial elamination, the channel crack in the ilm is equivalent to a two-imensional ege crack, with the stress intensity actor at the root K I = 1.115σ πh. (A1) For a small crack segment ( << h ) kinking out o the plane o the ege crack, the stress intensity actors at the new crack tip are linearly relate to the stress intensity actors at the tip o the parent crack (Hutchinson an Suo, 199), namely K K I II = c = c 11 1 K K I I + c 1 + c K K II II, (A) where the coeicients, c ij, epen on the kink angle, as given by Hayashi an Nemat-Nasser (1981). Following He an Hutchinson (1989b), the coeicients can be written as c = C R + D R 11, I c1 = C I D, (A3) where C = C R + ic I an D = D R + idi are two complex value unctions, with the subscripts R an I enoting their real an imaginary parts, respectively. An approximation by Cotterell (1965) gives that 17

18 C 1 iω / i3 = ( e + e ω ), 1 iω i3 D = ( e e ω ) 4, (A4) where ω is the kink angle. Cotterell an Rice (1980) have shown that this approximation is asymptotically correct or small kink angles an is reasonably accurate or kink angles as large as 45 or even 90, epening on the moe mix. For the present problem with a channel crack kinking into the interace, the kink angle is 90 an = 0. Uner the plane strain conition, the energy release rate o the interacial crack is K II G σ h (A5) E E ( ) ( K I ) + ( K II ) 0 = = ( 1.115) π ( c + c ) A comparison between Eq. (A5) an Eq. (4) gives the imensionless coeicient ( 0) = 1.58 ( c c ) Using the approximation (A4) with ω = π, we obtain that Z π +. (A6) 11 1 c 11 =, 4 c 1 =. (A7) 4 Inserting (A7) into (A6) gives that Z ( 0) = (A8) This approximation agrees well with the numerical result shown in Figure 3, where Z = or = h 18

19 A. Crack election at a bimaterial interace For an interace between two elastic materials with general elastic mismatch, an asymptotic solution by He an Hutchinson (1989a) gives the energy release rate o the elamination crack emanating rom a perpenicular channel crack (Figure a, << h ): G ( CD) ] 1 λ 1 1 K1 + K 1 1 k1 [ C + D + Re = + = + E E s cosh πε E E s cosh πε. (A9) where C an D are imensionless, complex value unctions o α an β, k 1 is a real value constant representing the stress intensity at the root o the channel crack, an λ is etermine by Eq. (7). This asymptotic solution leas to a power-law epenence o the Z unction on the interacial crack length as 0, namely 1 λ G ~ Z =. (A10) σ / h E h Figures A1 an A plot the numerical solutions o Z rom the inite element moel (Figure ), in comparison with the asymptotic solution. When α < 0 ( β = α / 4 ), 0 < λ < 0. 5 an the log-log plot o Z vs h (Figure A1) approaches a straight line o positive slope ( 1 λ > 0 ) as 0. When α > 0, λ > 0. 5 an the log-log plot (Figure A) approaches a h straight line o negative slope ( 1 λ < 0 ). When α = β = 0 (no elastic mismatch), λ = 0. 5 an Z approaches a constant with zero slope in the log-log plot (Figure A). The comparisons show goo agreement between the numerical results an the asymptotic power law or short elamination cracks. 19

20 Reerences Ambrico JM, Begley MR (00) The role o initial law size, elastic compliance an plasticity in channel cracking o thin ilms. Thin Soli Films 419: Beuth JL (199) Cracking o thin bone ilm in resiual tension. Int. J. Solis Struct. 9: Corero N, Yoon J, Suo Z (007) Channel cracks in hermetic coating consisting o organic an inorganic layers. Appl. Phys. Lett. 90: Cotterell B (1965) On brittle racture paths. Int. J. Fracture Mech. 1: Cotterell B, Rice JR (1980) Slightly curve or kinke cracks. Int. J. Fracture 16: Doabalapur A (006) Organic an polymer transistors or electronics. Materials Toay 9: Gao H (006) Application o racture mechanics concepts to hierarchical biomechanics o bone an bone-like materials. Int. J. Fracture 138: Hayashi K, Nemat-Nasser S (1981) Energy-release rate an crack kinking uner combine loaing. J. Appl. Mech. 48: He J, Xu G, Suo Z (004) Experimental etermination o crack riving orces in integrate structures. Proc. 7th Int. Workshop on Stress-Inuce Phenomena in Metallization (Austin, Texas, June 004), pp He MY, Hutchinson JW (1989a) Crack election at an interace between issimilar elastic materials. Int. J. Solis Struct. 5: He MY, Hutchinson JW (1989b) Kinking o a crack out o an interace. J. Appl. Mech. 56:

21 Ho PS, Wang G, Ding M, Zhao JH, Dai X (004) Reliability issues or lip-chip packages. Microelectronics Reliability 44: Huang R, Prevost JH, Huang ZY, Suo Z (003) Channel-cracking o thin ilms with the extene inite element metho. Eng. Frac. Mech. 70: Huang R, Prevost JH, Suo Z (00) Loss o constraint on racture in thin ilm structures ue to creep. Acta Mater. 50: Hutchinson JW, Suo Z (199) Mixe moe cracking in layere materials. Avances in Applie Mechanics 9: Khang DY, Jiang HQ, Huang Y, Rogers JA (006). A stretchable orm o single-crystal silicon or high-perormance electronics on rubber substrate. Science 311: Li T, Suo Z (007). Ductility o thin metal ilms on polymer substrates moulate by interacial ahesion. International Journal o Solis an Structures 44: Liu XH, Lane MW, Shaw TM, Liniger EG, Rosenberg RR, Eelstein DC (004) Low-k BEOL mechanical moeling. Proc. Avance Metallization Conerence: Liu XH, Lane MW, Shaw TM, Simonyi E (007) Delamination in patterne ilms. Int. J. Solis Struct. 44: Nakamura T, Kamath SM (199) Three-imensional eects in thin-ilm racture mechanics. Mech. Mater. 13: Rice JR (1988) Elastic racture concepts or interacial cracks. J. Appl. Mech. 55: Suo Z (003) Reliability o Interconnect Structures. In Volume 8: Interacial an Nanoscale Failure (W. Gerberich, W. Yang, Eitors) o Comprehensive Structural Integrity (I. Milne, R.O. Ritchie, B. Karihaloo, Eitors-in-Chie), Elsevier, Amsteram, pp

22 Suo Z, Hutchinson JW (1990) Interace crack between two elastic layers. Int. J. Fracture 43: Suo Z, Prevost JH, Liang J (003) Kinetics o crack initiation an growth in organic-containing integrate structures. J. Mech. Phys. Solis 51: Tsui TY, McKerrow AJ, Vlassak JJ (005) Constraint eects on thin ilm channel cracking behavior. J. Mater. Res. 0: Tsui TY, McKerrow AJ, Vlassak JJ (006) The eect o water iusion on the ahesion o organosilicate glass ilm stacks. J. Mech. Phys. Solis 54: Wagner S, Lacour SP, Jones J, Hsu PI, Sturm J, Li T, Suo Z (004) Electronic skin: architecture an components. Physica E 5: Ye T, Suo Z, Evans, AG (199) Thin ilm cracking an the roles o substrate an interace. Int. J. Solis Struct. 9: Yu HH, He MY, Hutchinson JW (001) Ege eects in thin ilm elamination. Acta Mater. 49: Zak AR, Williams ML (1963) Crack point stress singularities at a bi-material interace. J. Appl. Mech. 30:

23 FIGURE CAPTIONS Figure 1. (a) Illustration o a channel crack with no interacial elamination; (b) a channel crack with symmetric interacial elamination o with on both sies ar behin the channel ront. Figure. (a) Schematics o the D plane strain moel o a steay-state channel crack with interacial elamination; (b) geometry o the inite element moel, with uniorm normal traction ( σ ) acting onto the surace o the channel crack an a symmetry bounary conition or the substrate; (c) an example inite element mesh, with ininity elements along the bottom an right bounaries; () a etaile mesh aroun the tip o the interacial crack. Figure 3. Normalize energy release rate o interacial elamination rom the root o a channel crack as a unction o the normalize elamination with or ierent elastic mismatch parameters. Figure 4. Phase angle o the moe mix or interacial elamination as a unction o the normalize elamination with or ierent elastic mismatch parameters. The ashe line inicates o the steay-state phase angle (5 ) or the case o zero elastic mismatch ( α = β = 0 ). Figure 5. The normalize stable elamination with as a unction o the normalize interace toughness Γ i = Γi Σ, where Σ = σ h E. 3

24 Figure 6. A map or interacial elamination rom the root o a channel crack ( β = α 4 ): (I) no elamination, (II) stable elamination, an (III) unstable elamination, where A an B enote elamination without an with an initiation barrier, respectively. Figure 7. Increase o the riving orce or steay-state channel cracking ue to concomitant interacial elamination. Figure 8. Inluence o the normalize interace toughness ( Σ = σ h riving orce or channel cracking. E ) on the steay-state Figure 9. Eective riving orce or steay-state channel cracking as a unction o the normalize interace toughness ( Σ = σ h E ). Figure A1. Normalize energy release rate o interacial elamination emanating rom the root o a channel crack, or α = an α = The asymptotic power law, Eq. (A10), is represente by the straight lines at the short crack limit with slopes, 1 λ = an 0.4, respectively. Figure A. Normalize energy release rate o interacial elamination emanating rom the root o a channel crack, or α = 0, α = 0., an α = The asymptotic power law, Eq. (A10), is represente by the straight lines at the short crack limit with slopes, 1 λ = 0, , an , respectively. 4

25 G ss Film h σ (a) Substrate Film h G (b) Substrate Figure 1. (a) Illustration o a channel crack with no interacial elamination; (b) a channel crack with symmetric interacial elamination o with on both sies ar behin the channel ront. 5

26 h ilm substrate (a) (b) Crack (c) () Figure. (a) Schematics o the D plane strain moel o a steay-state channel crack with interacial elamination; (b) geometry o the inite element moel, with uniorm normal traction ( σ ) acting onto the surace o the channel crack an a symmetry bounary conition or the substrate; (c) an example inite element mesh, with ininity elements along the bottom an right bounaries; () a etaile mesh aroun the tip o the interacial crack. 6

27 Z β=α/4 α= 0.99 α= 0.6 α=0 α=0. α= /h Figure 3. Normalize energy release rate o interacial elamination rom the root o a channel crack as a unction o the normalize elamination with or ierent elastic mismatch parameters. 7

28 65 60 β = α/4 ψ (eg) α= 0.99 α= 0.6 α=0 α= /h Figure 4. Phase angle o the moe mix or interacial elamination as a unction o the normalize elamination with or ierent elastic mismatch parameters. The ashe line inicates o the steay-state phase angle (5 ) or the case o zero elastic mismatch ( α = β = 0 ). 8

29 3.5 α= 0.6 α=0 α=0.6 s /h Interace toughness, Γ i /Σ Figure 5. The normalize stable elamination with as a unction o the normalize interace toughness Γ i = Γi Σ, where Σ = σ h E. 9

30 Γ i / Σ I 0.99 II A II B 0.5 III B III A α = 0.89 α Figure 6. A map or interacial elamination rom the root o a channel crack ( β = α 4 ): (I) no elamination, (II) stable elamination, an (III) unstable elamination, where A an B enote elamination without an with an initiation barrier, respectively. 30

31 α= 0.6 (Z=1.360) α=0 (Z=1.976) α=0.6 (Z=3.747) α=0.9 (Z=8.848) Z * Z Delamination with, /h Figure 7. Increase o the riving orce or steay-state channel cracking ue to concomitant interacial elamination. 31

32 α= 0.6 α=0 α=0.6 Z * Interace toughness, Γ i /Σ Figure 8. Inluence o the normalize interace toughness ( Σ = σ h riving orce or channel cracking. E ) on the steay-state 3

33 α= 0.6 α=0 α=0.6 Z e /Z Interace toughness, Γ i /Σ Figure 9. Eective riving orce or steay-state channel cracking as a unction o the normalize interace toughness ( Σ = σ h E ). 33

34 Z α= 0.99 (λ = 0.31) α= 0.6 (λ = 0.388) /h Figure A1. Normalize energy release rate o interacial elamination emanating rom the root o a channel crack, or α = an α = The asymptotic power law, Eq. (A10), is represente by the ashe straight lines at the short crack limit with slopes, 1 λ = an 0.4, respectively. 34

35 10 α=0.6 (λ = 0.654) α=0. (λ = 0.54) α=0 (λ = 0.5) Z /h Figure A. Normalize energy release rate o interacial elamination emanating rom the root o a channel crack, or α = 0, α = 0., an α = The asymptotic power law, Eq. (A10), is represente by the ashe straight lines at the short crack limit with slopes, 1 λ = 0, , an , respectively. 35