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`INTERFACE SCIENCE
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`Volume 3-1996
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`Editorial . . .
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`. . . . . . . . . .. M. Riihle and D.J. Srolovitz
`167
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`FEATURED ARTICLE
`169
`The Mechanics and Physics of Thin Film Decohesion and its Measurement .....A. Bagchi and A. G. Evans
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`. . .. T. Kehagias,
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`Pyramidal Slip in Electron Beam Heated Deformed Titanium .
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`R Komninou, P. Grigoriadis, G.P. Dimatrakopulos, J.G. Antonopoulos and TI Karakostas 195—::—
`Evolution of Triple Junction Defect Structure at Plastic Deformation of Metallic Polycrystals . .
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`S.G. Zaichenko, A.V.Shalz'mova,A.0. Titov andA.M. Glezer. .... .... ..... 203
`Influence of Periodic and Random Two-Dimensional Nuclei Distribution on the Recrystallization Kinetics
`209
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`Wiimer
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`Grain Boundary Interdiffusion in the Case of Concentration—Dependent Grain Boundary Diffusion
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`The Atomic Structure of a 2 = 5 [O01]/(310) Grain-Boundary in an Al-5% Mg Alloy by High-Resolution
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`. . .. M. Shamsuzzoha, I. Vazquez, RA. Deymier and David J. Smith..... .. . . . .. . ..Electron Microscopy . . . 227
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`INTERFACE SCIENCE 3, 169-193 (1996)
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`NOTICE ‘This material maymbe protected
`by copyright law (Title 17 US. Code)
`Pfevideqby ihe Univ_ersity of Washington Libraries
`
`The Mechanics and Physics of Thin Film Decohesion and its Measurement
`
`A. BAGCHI AND A.G. EVANS
`Division ofApplied Sciences, Harvard University, Cambridge, Massachusetts 02138
`
`Received December 1, 1995; Accepted December 1, 1995
`
`Abstract. The intent ofthis review is to utilize the mechanics ofthin films in order to define quantitative procedures
`for predicting interface decohesion motivated by residual stress. The emphasis is on the role of the interface debond
`energy, especially methods for measuring this parameter in an accurate and reliable manner. Experimental results
`for metal films on dielectric substrates are reviewed and possible mechanisms are discussed.
`
`
`
`Keywords:
`
`Notation
`
`crack half length
`initial crack size
`section area
`Burgers vector
`constant 33
`constants related to thin film decohesion
`dislocation free zone near crack
`Young’s modulus
`plane strain value of E
`average modulus (Eq. (2.9))
`strain energy release rate
`steady-state Q
`reduced Gs; caused by bending
`film thickness
`critical superlayer thickness
`substrate thickness
`
`sectional modulus
`stress intensity factor
`characteristic length
`prescribed length to define the mode mixity
`moment
`constant related to Dundurs parameters
`(Eq. (2.33))
`edge force
`distance from crack tip
`plastic zone size
`
`\
`
`'
`
`’
`R0
`
`7'
`
`
`
`T
`U '
`Wad
`Y
`
`cx
`,5
`y
`8
`5
`n
`9
`tc
`A
`p.
`v
`3;‘
`1'1
`
`cr,-,-
`GR
`0'0
`g*
`1/r
`
`cu
`I‘
`l‘,-
`
`external stress
`stored energy
`work of adhesion
`dimensionless quantity for the
`K -calibration
`Dundurs parameter (Eq. (2.3))
`second Dundurs parameter (Eq. (2.3))
`interaction angle (Eq. (2.31))
`location of neutral axis
`strain
`thickness ratio, h/H
`polar angle
`curvature
`cracking number
`shear modulus
`Poisson’s ratio
`loading combination (Eq. (2.38)) ’
`non-dimensional Q55
`
`_
`
`stress tensor
`residual stress
`yield strength
`peak stress for the cohesive zone rupture
`mode mixity angle
`mode mixity defined at a prescribed
`length
`relative loading phase
`fracture energy
`interface fracture energy
`
`4
`
`

`
`l70
`
`Bagchi and Evans
`
`I‘,
`I‘,,
`To
`
`A
`
`6
`2
`
`substrate fracture energy
`plastic dissipation
`plastic dissipation for the cohesive zone
`rupture
`normalized location of neutral axis, 8 / H
`
`oscillation index (Eq. (2.5))
`modulus ratio, E1 /E2
`
`1.
`
`Introduction
`
`The number of applications for thin films and multilay-
`ers that take advantage of their special mechanical, ther-
`mal, electronic and optical characteristics has steadily
`increased. The associated technologies include multi-
`chip modules, thermal and oxidation protection coat-
`ings, wear and abrasion resistance coatings, etc.
`In
`general, the layers are deposited by vapor deposition
`(either physical or chemical). One of the problems, that
`has limited the more widespread use of such systems,
`has been the incidence either of interface decohesion
`or of delamination within one of the brittle constituents
`
`motivated by residual stresses [l-5]. Such stresses are
`inevitable in vapor deposited layers and are exacer-
`bated when the constituent materials have vastly dif-
`fering thermomechanical properties, such as polymers
`on metals and metals on ceramics. The stresses arise
`
`( 1) Intrinsic stresses develop during
`for two reasons.
`deposition [6]. These stresses persist, unless they are
`relaxed by plastic deformation or annealing.
`(2) The
`mismatch in thermal expansion induces stresses when
`the temperature is changed [7].
`Controlling the stress "in order to inhibit decohesion
`and delamination without compromising the functional
`characteristics of the system is not usually an option.
`Instead, thermomechanical design of multilayer sys-
`tems to resist these failure modes is required. This goal
`is crucially dependent upon the attainment of an ade-
`quate interface debond toughness, F,-. The toughness
`requirement is manifest in the fail-safe design solution,
`[1,8]
`
`1*, 2 ha;/EA
`
`(1.1)
`
`is its appropriate
`1:?
`where h is the film thickness,
`Young’s modulus (plane strain or biaxial plane stress),
`OR is the residual stress and Ais a cracking number
`(of the order unity). When Eq. (1.1) is satisfied, there
`is insufficient energy stored in the film to,permit an
`interface crack to propagate and the film must remain ,
`attached to the substrate.
`
`In order to implement this fail-safe criterion, meth-
`ods for the accurate measurement of l‘,- on the actua1
`interfaces of relevance must exist. The principal in-
`tent of the present review is to describe and analyze
`the available methods with the objective of identifying
`those capable of providing the quantitative information
`needed to apply Eq. (1.1). There have been several re-
`views on aspects of this topic. These include surveys
`of test methods, [9-12] the thermomechanical integrity
`of films and multilayers [13], the mechanics of crack
`growth along interfaces [14], residual stresses and their
`origin [15]. The present review differs from these by
`focusing on the quantitative aspects of thin film deco-
`hesion and its measurement. Most thin film adhesion
`
`tests empirically infer the adhesive strength by subject-
`ing the film to some external loading (like scratching,
`pulling or inflating) and measuring the load at which
`decohesion occurs. These tests are simple and effec-
`tive for routine ranking of bond quality. However, they
`do not measure 1",-, because the strain energy release
`rate cannot be deconvoluted from the work done by
`the external load [12]. An ideal test should duplicate
`the practical situation as closely as possible and be able
`to modulate the available strain energy.
`It must also
`explicitly incorporate the contribution to decohesion
`from the residual stress. The test methods are assessed
`
`against this ideal.
`
`2. Mechanics of Thin Film Decohesion
`
`2.1. Basic Principles
`
`Most decohesion problems of interest involve films
`subject to residual tension. This case is given the major
`emphasis in the present article. Relatively few remarks
`are made about the corresponding problem when the
`films are in compression. Films in tension are able
`to decohere from the subsuate by relaxing the resid-
`ual stress in the film above the interface crack. For
`
`the simplest case of a thin, homogeneous film sub-
`ject to uniform residual stress on a thick substrate, the
`steady-state energy release rate, Qss, for an interface
`crack is given by the strain energy in the film. The
`non-dimensional form for a film is,
`
`H = Ea./atth
`
`(2.1)
`
`where 1'1 is a non-dimensional quantity of the order
`‘ unity. The same form arises for other problems, but
`its numerical magnitude differs, as elaborated below-
`
`5
`
`

`
`The Mechanics and Physics of Thin‘ Film Decohesion and its Measurement
`
`171
`
`Decohesion takes place when 935 exceeds the interface
`debond energy, Pi,
`
`refer to the materials above and below the interface,
`respectively (Fig. 2.1). The near tip stresses ar,~j for a
`traction-free crack are [17]
`
`[Re{Kr‘€}a,§.(9, e)
`
`1 2
`
`7rr
`
`0'31’ —
`
`+ Im(Kri€}a}}(e, e)],
`
`(2.4)
`
`where i = J: and (r, 6) are the polar coordinates
`centered at the crack tip (Fig. 2.1). The dimensionless
`angular functions a,-11.0}; reduce to well-known trigono-
`metric forms in homogeneous materials [18]. The bi-
`material parameter 6 (also known as the oscillation
`index) is related to )8 by [17]
`
`__1_ H3
`
`e —2n_ln[1+fl:].
`
`(2.5)
`
`Notice that the stress intensity factor is a complex quan-
`tity for an interfacial crack, formally defined as [19,
`20]
`'
`A
`
`K=K[+iK2.
`
`(2.6)
`
`Gs; Z T’:
`
`(2-2)
`
`I‘',- may be a strong function of the mode
`However,
`mixity, manifest in a mixity angle 1/1, defined below
`(Eq. (2.11)) [14]. Hence, it is not suflicient to know
`gss; mp must also be calculated. Moreover, to design
`against decohesion, the interfacial fracture toughness
`must be measured as a function of 1,0. Both topics are
`elaborated in this article.
`
`2.1.1. Interfacial Crack Tip Fields. The determina-
`tion of Q and 1// for any interface crack problem begins
`with basic elasticity solutions. These consider a crack
`at an interface joining isotropic, linearly elastic, ma-
`terials (Fig. 2.1) with E,-, v,- and p.,- being the Young’s
`modulus, Poisson’s ratio and shear modulus. For plane
`problems with traction boundary conditions, there are
`only two nondimensional combinations of the four in-
`dependent material moduli. These are the Dundurs
`elastic mismatch parameters [16], oz and ,5, given by,
`
`_ m(Ic2 + 1) — H-2(Kl+1)
`"’ " m<«<2+ 1)+/1-2(‘Cl + 1)
`
`E I?‘ ‘€2,
`E1-l"E2
`fl __ 111062 —1)— .U«2(K1+1)
`#1062 +1)+ IJ-2(K1+1)’
`
`Its real and imaginary parts K 1 and K2, respectively, are
`similar to the conventional mode I and mode 11 intensity
`
`factors. This intensity factor can be normalized by
`suitably scaling of]. and or}; [21], such that the interface
`traction ahead of the crack tip asymptotes to
`
`(2.3)
`
`.
`Kris
`(O-yy + ldXy)9=0 = J5;-F
`
`(2.7)
`
`The energy release rate is related to the amplitude of
`K by [22]
`
`9 = <1 — fi2)|K|2/E‘
`
`(2.8)
`
`Here E* denotes an average modulus defined as
`
`1
`l
`l
`1
`— = — T T .
`
`2.9
`
`For practical purposes, K is a parameter which relates
`the external stress T and the specimen geometry to the
`near-tip stress fields. The generic form is [20, 21]
`
`K = YT«/LL-is ea”,
`
`(2.10)
`
`where Y is a dimensionless real positive quantity, L a
`characteristic in-plane length (e.g., crack length, layer
`
`where ;c,- = 3 — 412,- for plane strain and 1c; = (3 —
`v,-)/(1 + v.-) for plane stress. The subscripts 1 and 2
`
`
`
`\
`\\ .
`
`
`
`
`I’
`
`
`
`
`Interface
`/
`,
`/’ /
`/
`Crack
`
`/' / / //
`/
`’
`
`
`Figure 2.1. Crack lying along a bimaterial interface.
`
`
`
`~
`
`fl‘
`
`2.1)
`filer
`. but
`
`low’
`
`6
`
`

`
`172
`
`Bagchi and Evans
`
`thickness), and 1/r is the mode mixity angle
`
`rmi. .1.)
`
`up = 1n{K1:"€
`
`/IKI]
`
`(2.11)
`
`,-.:.'
`
`A
`
`A
`
`FKW1 I L1)
`
`Elln T"(‘lL
`
`The parameters Y and rlr can be evaluated by stress
`analysis.
`In general, they depend on the moduli, the
`geometry and loading details. Such calibrations have
`been listed elsewhere [14, 23].
`The analogs of the mode I and mode 11 stress inten-
`sity factors are not constants. Instead, they are func-
`tions of r and can be denoted K; (r) and Kn(r), such
`that [21}
`
`K1(r) E Re(Kr"‘*) -_- YT«/Zcoshl/— e ln(L/r)],
`Kn(r) E 1m(Kr"€) = YT«/fsin[1lI— e ln(L/r)].
`(2.12)
`
`Clearly, the ratio of the shear and the normal com-
`ponents of the interface traction is not constant. To
`address this complexity, 51 fixed length is introduced.
`Then, the mode mixity, 1//, can be defined unambigu-
`ously as,
`
`itif =1n[K12-‘E/1K1]
`
`(2.13)
`
`For convenience, i. may be chosen to correspond to
`some fracture process zone size. Then, the interface
`tractions can be re-expressed as
`
`a,_,. = )K|(2m)-‘/2 cos[(1?+ e1n(r/11)],
`a,,,. = 1K1(2m)-V2 sin[z/3+ e 1n(r/£)].
`
`(2.14)
`
`and the mode mixity becomes
`
`12; = tan-‘
`
`01'
`
`(Z = tan-1
`
`"yr
`
`r=i.
`
`(2.15)
`
`(2.16)
`
`It is equal to the traction phase at the prescribed length,
`r = L. The interface’ toughness also has an implicit
`dependence on 1/! and L.
`It shifts as the choice of
`L is varied, as schematically illustrated in Fig. 2.2.
`This shift is moderate for interfaces having a small
`value of e. For example, the Cu/SiO2 system has:
`6 = —— 0.045. Therefore, a change in choice of 1: by
`even a factor of 100 shifts tir by only 12°.
`
`Q2
`
`V‘
`
`Figure 2.2. Procedure for shifting the toughness function from one
`choice of the reference length to another.
`
`2.2.
`
`Interface Cracks in Bilayers
`
`2.2.1. Forces and Moments. The above results can be
`
`used to analyze a semi-infinite interface crack between
`two isotropic elastic layers under generalized edge
`loading conditions (Fig. 2.3a). The solutions provide
`
`Figure 2.3. Superimposition scheme for the bimaterial structure
`with generalized edge loading, having an interfacinl crack along th€
`negative x) axis with its tip at the origin.
`
`1.
`ll.
`
`7
`
`

`
`The Mechanics and Physics of Thin Film Decohesion and its Measurement
`
`173
`
`(he basic methodology for calculating the mode mixity.
`Force P and moment M equilibria dictate that [24],
`
`where 2 represents summation over all i layers (in
`Fig. 2.4, i = 2). Here, the distances, y,~, are measured
`from the same reference. Choosing this reference as
`
`(2.17)
`
`the bottom of layer #2, 8 becomes
`
`P1 —— P2 — P3 = 0,
`h
`
`M1—M2+P1(E+H—(3)
`+P2(5-3') —M3 =0.
`
`H
`
`(2.18)
`
`‘».:;,,:
`
`nbe ,
`teen _
`idgfi ,1-1
`vide "
`
`The quantity 6 is the height of the neutral axis of the
`bimaterial beam from the bottom surface. Only four
`among these six loading parameters are actually inde-
`pendent. These are, P1, P3, M1 and M3. The number
`of independent load parameters can be further reduced
`to only two, through superposition (Fig. 2.3b). These
`parameters are P and M, given by,
`
`M
`P = P1—C(P3-C2—3h
`M = M, — C3M3,
`
`(2-19)
`
`where the C ’s are dimensionless numbers that must be
`calculated in accordance with the following five steps:
`obtain the position of the neutral axis, evaluate the sec-
`tional modulus, obtain the section area, calculate the
`stresses, determine the C3.
`An expression for 8 is found by using the concept of
`equivalent section (Fig. 2.4) and by utilizing the defi-
`nition that the first moment of area across the neutral
`axis vanishes;
`
`such that
`
`yielding
`
`Z A.-<y.~ — 6) = 0
`i
`5 2 Ar’ = Z Azys.
`'8 _ Zi Aiyi
`2 Ai
`
`-
`
`(2.20)
`
`A
`
`E _ [1-l-2Er)+En2]
`
`2n(1 + 2n)
`
`h
`
`(2.21)
`
`'
`
`where r; = h/H and 2 = E1/E2.
`The next step is to calculate the dimensionless sec-
`tional modulus [0 with respect
`to its neutral axis
`(Fig. 2.5). The easiest approach is to divide the equiva-
`lent section into rectangles with their edges lying along
`the neutral axis, yielding
`
`10 = [ABCD + IPQRS " Ishadedareass
`
`such that,
`
`+ A3 —
`
`— Afl
`
`(2.22)
`
`The dimensionless area of the composite section A0 is
`given by
`
`1
`Ao=—+E.
`7}
`
`(2.23)
`
`The stresses in the composite beam in Fig. 2.3b are
`
`M3
`P3
`—— —-
`
`(M. +
`
`)
`
`23
`
`~
`
`+——.
`
`h3I°y~—6 <y< H-6
`
`hA0
`U22 = 012 = 0
`
`(2-24)
`
`Bimaterial Beam Cross-section
`
`Equivalent Section
`
`, h
`
`Layer#1
`
`E,
`
`
`
`
`
`H Layer #2
`
`E2
`
`<(————1—-—————->
`
`
`
`8
`
`

`
`174
`
`Bagchi and Evans
`
`Equivalent Section
`
`
`
`Figure 2.5. Procedure for finding the dimensionless sectional modulus for the bimaterial structure.
`
`which, after rearrangement, yields
`
`(2.27)
`
`12M
`P
`__ _ __ _ H
`
`
`
`
`
`
`
`h3 iy +5h 2)
`
`h
`_ _
`
`
`
`stxessin(c)
`
`-—
`
`P1
`
`h
`
`12M]
`
`k3
`
`<}’ H + 5 —'
`
`h
`
`s.:.___..:_,?.?:_»
`stress in (a)
`
`+ EP3 + EM3
`M0
`h-‘I0 3’
`stress in (b)
`
`where y is now measured from the neutral axis of
`the composite section. Also, from superposition, the
`stresses in the layers are related by
`
`U1i()’)c = 011()’)a + 011()’)b.
`
`(2-25)
`
`where the subscripts refer to the structure shown in
`Fig. 2.3. Edge loading yields the stresses in layer #1 as
`
`IZM1 *
`P1
`‘
`0110’ )u = ‘*7 — h3 3’
`P
`12M ,_
`°‘11()’*)c = '77 — 75- ’
`
`.
`
`(2-26)
`
`where y* denotes the distance measured from the neu-
`tral axis of layer #1, passing through the its mid-section.
`Since 31* = y — (6 + h/2), stress superposition in layer
`#1 gives
`
`Since Eq. (2.27) should hold for all values of y in the
`range, H — 8 < y < H — 6 + h, the coefficient on
`y as well as the constant term in the above expression
`must be zero. This requirement enables the C ’s to be
`evaluated as
`‘
`
`1
`2 l
`C==-— ———A+—,
`
`2
`
`lo (77
`
`2)
`
`‘Z
`C=—,
`
`A0
`
`E
`C = i-.
`3
`1210
`
`(2.28)
`
`The same results can be used to evaluate P and M, by
`superposing the structures in Figs. 2.3a and 2.3b:
`H-—8+k
`
`H-5
`
`P = P] ‘-f
`M ‘—‘_ Ml —/
`H-6
`
`H--.+}l
`
`<711()’)d)’.
`011(}’)[)’ - (H -5 + §)]d)’=
`(2.29)
`
`h
`
`For this purpose, the stresses are obtained from thé
`Bq. (2.26). Once P and M are found, the force and mo-
`ment for the lower layer (Fig. 2.3) can be obtained from
`equilibrium as: P* = P and M* = M + P(H + h)/'2.
`
`“-
`
`F
`36!
`
`'~l’.-3
`~
`
`.
`
`4
`
`'.
`I
`
`I
`
`V
`
`_i
`*
`‘
`
`3I
`
`i
`
`a.
`
`9
`
`

`
`The Mechanics and Physics of Thin Film Decohesion and its Measurement
`
`l75
`
`This can be re-expressed as
`
`gss __1[P2+M2+2 PM
`Ah
`U13
`
`'2E‘1
`
`\//Tihz
`
`sin 3/],
`
`(2.3%)
`
`The quantities A, I and y are now given by
`
`A
`
`1
`1
`= ~—————————, 1 = ——————,
`1+ E(4n + 6172 + 3773)
`12(1 + 2173)
`
`siny = 62n2(1+ n)~/A1.
`
`(2.31)
`
`The corresponding stress intensity factor is [l1],
`
`Figure 2.6. Energy release rate can be found by considering the
`energy far ahead and far behind the crack tip. in the volume elements
`found by translation and superposition as illustrated above, for unit
`length of crack extension.
`
`2.2.2. Energy Release Rates and Mode Mixities.
`The steady-state strain energy release rate can be com-
`puted from the difference between energy stored in the
`structure per ifnit length far ahead and far behind the
`crack tip (Fig. 2.6),
`
`
`
`[K]
`
`M1
`P2
`2 = -— —— 2
`
`[Ah + M3 +
`
`PM
`
`‘
`
`_..AIh2 s1ny]2
`
`p2
`—-
`
`where p relates to the Dundurs parameters as
`
`=
`
`p
`
`1 — at
`,/ 1 _ /52
`
`.
`
`2.32
`
`(
`
`)
`
`2.33
`
`(
`
`)
`
`To obtain the real and imaginary parts of K, linearity
`and dimensional considerations are exploited, leading
`to the following general expression
`
`K =
`
`P
`
`ia ¢Ah
`
`+1,
`
`M
`
`«/1713
`
`.
`
`p
`
`]_.h—«e,
`«/22
`
`(2.34)
`
`where a and b are dimensionless complex numbers.
`They depend only on the geometric parameter 7; and the
`Dundurs parameters, and can be found by substitution
`of Eq. (2.34) into Eq. (2.32), yielding
`
`_ia:
`a — e
`
`,
`
`b = -—ie“‘‘’‘*'’’),
`
`(2.35)
`
`such that
`
`Zsiny =&b-l-a5
`
`where w(cz, )3. n) is a real angular quantity tabulated
`by Suo and Hutchinson [24].
`Taking the reference length = h gives
`
`Re[Kh"e]=-£—[ P cosa)+ M sin(a)+y)]
`[.6 zi P
`_
`_ M
`:1
`Im[Kh ]
`‘/EL/A_h_s1na)
`‘/}I?cos(a)+)/) .
`
`\/E ~/Ah
`
`x/Ih3
`
`(2.36)
`
`
`
`10
`
`h3)
`215, < h +
`0
`+ 1
`(P2+12M*2)
`+ E77(1+ V7)2l+12F(1+ BT73)
`+ 12?-§(l + r7)PM:|.
`(2.30a)'
`
`.
`6
`S8 = —'(Ua + Ub " Uc)
`3a
`1
`
`M2
`
`P2
`
`2.29) A:
`
`1 the A
`mo- '
`.rom
`
`1)/2. ,_
`
`=
`
`2132 H
`P2
`
`1
`
`H3
`
`M2
`
`2
`
`10
`
`

`
`176
`
`Bagchi and Evans
`
`Therefore, the mode mixity, at the prescribed length
`r = h ahead of the crack tip, is given by
`
`Therefore, the equivalent load and moment can be ob.
`lainfid. using EC} (2-19). 2181
`
`=
`
`w
`
`tan
`
`§'sina)——cos(w+ )1)
`‘l ——————:————
`
`lcosa)-l-%in(a)+)’)
`
`'
`
`2.37
`
`(
`
`)
`
`where; measures the loading combination:
`
`I’:
`
`1
`1
`h1—C —C ----A+~— ,
`
`2
`
`U
`2
`1
`GR
`M = -—aRh“C3 (— — A + -).
`
`1 Ph
`
`§ =
`
`(2.38)
`
`Since h << H(77 ——> 0),
`
`1
`
`17
`
`1
`
`2
`
`(240)
`
`I _'
`l
`The
`
`awn‘
`-
`
`V
`
`p
`..
`
`,i
`sTh
`
`mg
`
`» mi
`be
`th.
`
`1 2
`
`.
`
`
`N.
`a.
`si
`
`It
`1-‘
`
`l l
`
`'
`
`’
`
`a.
`l
`
`,
`
`2.3.
`
`Thin Film Decohesion
`
`For thin films, superposition allows the residual stress
`to be simulated by the edge loaded structure in Fig. 2.7c.
`For a uniform stress 03,
`
`'
`
`and
`
`_
`
`1
`
`lint)/lo = 11113 (— + E) ='_oo;
`'7.’
`"-'
`77
`
`_
`
`,
`
`1
`
`3,£%'°=l‘.%3l‘l3(A'z) '3(A“:;>“l
`
`1
`
`2
`
`1
`
`3 A
`
`1
`
`1
`
`P1 = Ps=o«h,
`
`M1 = 0,
`
`h
`
`a)
`
`\
`
`°n
`
`#2
`
`“H
`
`b
`
`’
`
`+
`
`=
`
`#2
`
`
`
`d)
`
`Figure 2.7. The cut and paste procedure for constructing the solution to the thin film decohesion problem. Here the interfacial crack is driven
`by the residual stress present in the film. Note that the stress intensity is exactly simulated by the edge loaded structure in Fig. 2.3c.
`
`11
`
`11
`
`

`
`The Mechanics and Physics of Thin Film Decohesion and its Measurement
`
`V
`
`177
`
`the crack are not fully relieved, causing the energy re— A
`lease rate to be diminished. The redistributed stresses
`
`must be determined before obtaining Q55 and 1//. For a
`generalized multilayer with non-uniform stresses, the
`analysis is unwieldy. Here, only the general method is
`described. Explicit results are given for a bilayer film
`on a substrate.
`
`If N layers were dis-
`2.4.1. Energy Release Rate.
`connected, all of the residual strain energy would be
`available and the energy release rate 935 would be given
`by Eq. (2.1), summed over the layers. For a thick sub-
`strate,
`
`fie Zhhwi)
`i=N
`
`(2.44)
`
`The actual energy release rate is lower. It is diminished
`by A955, which is dependent on the stresses that remain
`in the layers, because they are connected,
`
`9s=9g+A9e
`
`Q49
`
`When the layers are in residual tension and the film
`bends upwards after decohesion in an attempt to relax
`the strains (Fig. 2.8), the resultant stresses in each layer
`can be related to the forces, P,», moments, M,-, and
`
`(2.42)
`
`curvature, /c, by
`
`therefore,
`
`.
`
`ti1—l3)C1
`1' C
`
`im
`27->0
`
`2
`
`2
`= . _ =0;
`
`A«1il~I>nooi:/lo]
`1'
`[EC A-l-1)]
`I0‘->00 [
`
`= 1m —- ——-
`7;->0
`I0
`7]
`)3
`_. = 0
`
`—-
`2
`
`= :
`
`o
`
`lim C3 = lim
`rt->0
`
`Hence, Eq. (2.40) can be specialized to
`
`P=mn
`
`M = 0.
`
`(2.41)
`
`Also, the limiting values of A, I and y are obtained as:
`
`1
`1‘ A=1’ ——————————— =1;
`
`n‘.’35i1+2(4n+6n2+3n3)i
`#3?)
`11
`[
`1
`J
`1
`1
`1'
`.im = m ——————-— =—;.
`n—»O
`n~>0
`l2(1 + 3n3)
`12
`lin})(sin y) = lin})[6E n1(1 + r;)«/AI] = 0.
`n—-—
`rt-+
`.
`
`In this limit, Q55 is given by Eq. (2.1), such that
`
`h
`
`K55 =-‘O'R(:-Z-)
`
`‘/2
`
`.
`
`.
`
`h"Ee”"p.
`
`The parameter 3;‘ takes the limiting value
`
`1'
`
`fills’);
`
`whence
`
`= 1'
`
`~13}: i A M]
`
`1 Ph
`——- = ,
`
`°°
`
`
`
`
`
`E~>oo
`
`5 cos w + sm(w + 3/)
`
`1;; = lim {tan" i:—-————-——-——————£smw —— c(_)S(w + y.):” E w
`
`(2.43)
`
`This indicates that, for thin film decohesion, the mode
`rnixity 1/; —> ca [24]. Moreover, since the film stress di-
`minishes as the interface decoheres, this energy release
`behavior is entirely controlled by elasticity, even when
`thefilm has yielded upon prior thermal processing [14].
`
`2.4. Multilayer Films
`
`lriven
`
`Many cases exist in which decohesion may occur at
`an interface below the surface film. Additional con-
`
`siderations are then involved in calculating the energy
`release rate and the mode mixity. The key new feature
`recognizes that the stresses in a multilayered film above .
`
`12
`
`07(2) = Pi/hi + zEzIۤ
`
`(2-46)
`
`IC = M,- E,» /1,.
`
`(2.47)
`
`where
`
`with
`
`1,
`
`.~. /1,3/12,
`
`(2.43)
`
`where z now denotes the vertical distance from the
`
`neutral axis in each separate layer, whereas I,- are the
`actual sectional moduli. The retained strain energy
`Uc is found by first integrating the stress to obtain the
`contribution from each layer and then adding, resulting
`in a diminished energy release rate,
`
`Agss E
`
`/ [a,~(z)]2 dz
`-36‘: = -2
`=“Z(%.)l%?+-“iii
`
`h,«/2
`
`-ll;/2
`
`(2.49)
`
`It is now required to provide expressions that relate P,~,
`M; and K to the stresses and the film thicknesses.
`
`12
`
`

`
`
`
`N
`
`—.A
`
`' o
`
`M1
`
`1/K
`
`
`
`,
`Decchesion,a
`C----c—-—--‘-"‘-—-...--"""*-I
`
`=>
`
`178
`
`Bagchi and Evans
`
`E1,V1
`
`*
`
` I
`
`52. V2
`
`____,_--___,__4
`
`
`
`(8)
`
`(b) -
`
`Figure 2.8. A schematic showing the behavior of a bilayer film subject to residual tension as it decoheres from the substrate. The stresses
`01. 0'2 are the misfit‘stresses, which provide the forces P; and the moments M,- in the metal bilayer above the decohesion crack. The curvature
`of the decohered bilayer film is 1:.
`
`and
`
`K =
`
`5011+ h2)(5l — $2)
`mt+EwvEm.+EmvEmz+hi+Mm+wnn
`(2.53)
`,
`,
`with c,- = a,_,- /E,-(1 — v,-). The moments are obtained
`from Eq.
`(2.47) using [C from Eq.
`(2.53). The final
`result for the strain energy release rate is determined
`by using the forces and moments in Eq.
`(2.49) with
`(2.44) and (2.45).
`.
`The following numerical example illustrates this
`method by calculating the energy release rate for a
`Cu/Cr bilayer film, described in Section 4.4, having
`the following properties,
`
`. , n), the num-
`.
`For a multilayer film (i = 1, 2, 3, .
`ber of unknowns is 2n + 1, because each layer has two
`(a force, P,-, and a moment, M,-) in addition to the cur-
`vature, K, of the film after decohesion. The solution
`requires 2n + 1 linear equations. The first two stem
`from equilibrium considerations
`
`Of
`
`and
`
`or
`
`(2.50)
`
`(2.51)
`
`2 FORCE = 0;
`‘
`
`m+m+m+m=o
`
`Z MOMENT = 0-.
`.
`ii‘/Ii = n—1 Pi[/1,--|?:h,, + ":3 M]-
`
`i=l
`
`i=l
`
`K=i+l
`
`The remaining n — 1 equations involve strain compati-
`bility at the n — 1 interfaces. For rth interface, this can
`be expressed as (r =1, 2,... , n-1)
`
`
`P
`P
`h
`r+l
`8r+ _ r +_r_E =8,’-H _ _
`Erhr
`2
`Er+lhr+l
`
`
`h
`_ r+lIC
`2
`(2.52)
`
`where the strain terms 5; should be negative for layers
`
`in residual tension before decohesion.
`For a bilayer film, solutions for P and K obtained
`from these results are [25]
`
`P =
`
`+ E2h§:‘K
`
`6011 -1- ha)
`
`13
`
`A _
`Tf2._4..
`date
`pan
`to t
`wk]-
`mo
`[26
`]
`fiat
`lib.
`pa,
`(er
`:2‘
`
`‘
`l
`_
`
`5
`
`
`g
`‘
`
`1
`.
`
`9
`_
`I
`,
`4
`
`v
`
`b
`f
`§
`i
`l
`
`j
`
`E] = 93 GPa,
`
`0;“ = 1675 MPa,
`
`v1 = 0.21,
`5, = 0.0142
`UR_2 = 50 MP3,
`U2 =
`E2 = 120 GP21,
`TheCrthickness h], is allowedto vary overanominal
`
`.52 = 0.0003
`
`range of 0-100 nm. The procedure is then repeated
`for a wide range of Cu thicknesses hg. The variation
`in the Crlayer produces a spectrum of energy release
`rates, 95, (Fig. 2.9). The effect of the Cu thickness
`is noteworthy: 9,, increases as the Cu thickness de-
`
`creases, for all Cr thicknesses. This can be explained
`as follows. The contribution of the Cu layer to 935 is
`marginal, since it ‘scales with the square of the film
`stress, and furthermore Um >> 03,2. However, during
`stress redistribution following decohesion, theCu layer
`
`13
`
`

`
`- fl,
`i ha
`
`Layer #1
`Layer #2
`
`E2 , V2
`
`
`
`D3on
`
`N
`
`—-L U1
`
`—.L
`
`0U1
`
`.°
`
`
`
`
`
`EnergyReleaseRate,635(Jm‘2)
`
`
`
`asses 2
`ature
`
`)2}
`
`2.53)
`
`.ned ,
`inal
`ned ‘ };
`vith.
`
`The Mechanics and Physics of Thin Film Decohesion and its Measurement
`
`179
`
`E, ,v,
`
`
`
`lnterfacial
`Crack
`
`
`
`
`4¢
`
`0
`
`20
`
`40
`
`60
`
`80
`
`100
`
`Cr Superlayer Thickness,
`
`(nm)
`
`(8)
`
`Figure 2.9. Variation in the strain energy release rate with
`chromium superlayer thickness.
`
`acts as a sink by storing elastic strain energy which
`increases the magnitude of Agss.
`
`To evaluate 1//, it is necessary to
`2.4.2. Mode Mixity.
`determine both the real (K 1) and the imaginary (K2)
`parts of K (Section 2.2). A general solution has yet
`to be developed. Results are presented for a bilayer in
`which the elastic moduli of the two films are identical,
`though distinct from the substrate (E1 55 E3 gé E3)
`[26].
`Referring to the generalized loading of the bimate-
`rial system (Fig. 2.10) and recalling that overall equi-
`librium provides two constraints among the six loading
`parameters, there are only three independent parame-
`ters [24]. From the equivalence of the two systems
`in Fig. 2.10, the three parameters can be expressed in
`terms of 03,; and mg; as follows,
`
`Pl =-P3 = <7R.1h1+0'R.2h2;
`h
`
`h
`
`M1= ‘7R.lhl(‘%) “°'R.2h2('%')§
`M3 =aR_,h,(H-a+h2+—2l)
`+O'R_2h2
`-' 5 + -2)
`
`/1
`
`/17
`
`2
`
`(2.54)
`
`Where 8 is now the height of the neutral axis from the
`bottom of the substrate. Once Pg, M 1 and M3 have been
`found, M and P are obtained from Eq. (2.19). Then
`S is obtained from Eqs. (2.31) and (2.38). Finally, 1/r
`is obtained from Eq.
`(2.37). Some results for Cr/Cu
`bilayers are shown in Fig. 2.11. The most interesting _
`
`14
`
`
`
`(b)
`
`Figure 2.10. A schematic illustrating the reduction of the bilayer
`film decohesion problem to the bimaterial system with an interfa-
`cial crack, when E1 2 E2 gé E,. The stresses cr1.a'2 are the misfit
`stresses present in the bilayer, whereas the forces P's and the mo-
`ments M 's stand forthe generalized loading in the bimaterial system.
`
`
`
`ModeMixity,degree
`
`h = film thickness
`
`Cu = 25 nm
`
`0
`
`20
`
`40
`
`60
`
`80
`
`100
`
`Cr Superlayer Thickness, nm
`
`Figure 2. I I .
`thickness.
`
`Variation in the phase angle with chromium superlayer
`
`14
`
`

`
`180
`
`Bagchi and Evans
`
`0 z 14; (h,/ha) z 50°
`
`M (h,/I13)
`
`
`
`
`0
`
`20
`
`40
`
`60
`
`80
`
`1 00
`
`Cr Superlayer Thickness,
`
`(nm)
`
`
`
`
`
`ModeMlxityAngle,1;;degree
`
`(a) A schematic illustrating the proposed approach of
`Figure 2.12.
`varying the mode tnixity with a trilayer test specimen configuration;
`(b) the resultant variation in the mode mixity as a function of the Cr
`superlayer thickness, at chosen values of the dimensionless thickness
`of the top Cu layer, where h3 = 500 nm.
`
`result is -that the phase angle can be quite small for
`bilayer films (1/1 —> 0°), but recovers the single layer
`value for bilayer film,
`ip C-Z 50°, as the Cr thickness
`reduces to zero [27]. A similar analysis performed for
`trilayers (Fig. 2.12) indicates that the top layer can be
`used to modulate the mode mixity over a significant
`
`range [26].
`
`3. Mechanisms of Interface Crack Growth
`
`Cracks at interfaces extend in accordance with several
`different mechanisms [28].
`In some cases, the inter-
`faces have sufficient bond strength, relative to the metal
`yield strength, that the cracks extend in the metal, by
`a ductile mechanism. Such “strong” interfaces are not
`
`considered in this discussion. The sole emphasis is on
`“brittle" interfaces, devoid of reaction layers, in which
`
`the interface crack causes complete separation of the
`constituent materials.
`
`Decohesion at such interfaces is fundamentally con-
`trolled by the bonding between the atoms across the
`interface. The associated behavior is simulated by im_
`posing a stress normal to the interface and determining
`the displacements of the atoms across it [29]. This pro-
`cedure identifies two paramete