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Simulations of Damage, Crack Initiation, and Propagation in Interlayer Dielectric Structures: Understanding Assembly-Induced Fracture in Dies

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IEEE TRANSACTIONS ON DEVICE AND MATERIALS RELIABILITY, VOL. 12, NO. 2, JUNE 2012 241
Simulations of Damage, Crack Initiation, andPropagation in Interlayer Dielectric Structures:Understanding Assembly-Induced Fracture in Dies
Abhishek Tambat, Hung-Yun Lin, Ganesh Subbarayan, Dae Young Jung, and Bahgat Sammakia
Abstract
—Performance enhancement by lowering the dielectricconstant of interlayer dielectric (ILD) materials often compro-mises the mechanical integrity of the dielectric stack. At thepresent time, fracture in the ILD stacks induced by assembly toeither an organic substrate or a die stack (3-D) is an importantreliability consideration. These interactions include what is pop-ularly referred to as the chip-package interactions. In this paper,we develop insights on the potential crack initiation site within theILD, die–substrate geometrical parameters that cause most dam-age,aswellasinsightsonthemanufacturingprocessthatiscriticalto failure. Towards this end, we utilize analytical models basedon classical elasticity theory as well as sophisticated numericaltechniques that are capable of nucleating and propagating cracksat arbitrary locations within the structure without remeshing.Speciﬁcally,weanalyticallyestimatethestrengthofsingularitiesatall the possible multimaterial corners in the ILD stack to provideinsight on the likely damage nucleation sites for various materialconﬁgurations in the ILD stack. Two novel numerical approachesareusedforfracturesimulation.Intheﬁrst,cracksaremodeledasdiscontinuous enrichments over an underlying continuous behav-ioral approximation. In the second approach, the underlying ma-terial description is enriched with a cohesive damage descriptionwhose stiffness is evolved according to a prescribed damage law.Multilevel ﬁnite-element models are used to determine the loadimposedontheILDstructurebythesubstrate.Maximumdamageinduced in the ILD stack by the above load is used as an indicatorof the reliability risk. Parametric simulations are conducted byvarying ILD material, die size, die thickness, as well as the soldermaterial. Through analytical models of bonded assemblies, weidentify groups of relevant dimensionless parameters to relate thenumerically estimated damage in ILD stacks to the die/substratematerial and geometrical parameters. We demonstrate that thedamage in the ILD stack is least when the ﬂexural rigidity of the die is matched to that of the assembled substrate. We alsodemonstrate that ILD damage is only weakly correlated to sheardeformation on the die surface due to assembly. We generalize theabove observations into mathematical ﬁts (for use as design rules)
Manuscript received October 21, 2011; revised March 18, 2012; acceptedMarch 22, 2012. Date of publication April 17, 2012; date of current versionJune 6, 2012. This study was supported by Semiconductor Research Corpo-ration under Task id 1292.061. The development of the code used in thisstudy was made possible by support from Intel Corporation. The authors arethankful for this support. The authors are very thankful to Drs. Sean King andSatish Radhakrishan of Intel and Vikas Gupta of Texas Instruments for all thesuggestions and guidance during the course of this work.A. Tambat, H.-Y. Lin, and G. Subbarayan are with the School of Mechan-ical Engineering, Purdue University, West Lafayette, IN 47907 USA (e-mail:atambat@purdue.edu; lin137@purdue.edu; ganeshs@purdue.edu).D. Y. Jung and B. Sammakia are with the Binghamton University,Binghamton, NY 13850 USA (e-mail: djung@binghamton.edu; bahgat@binghamton.edu).Color versions of one or more of the ﬁgures in this paper are available onlineat http://ieeexplore.ieee.org.Digital Object Identiﬁer 10.1109/TDMR.2012.2195006
relating damage in ILD stacks to ILD material choice, relativesubstrate ﬂexural rigidity, and die size.
Index Terms
—Chip-package interactions, crack nucleation,damage mechanics, design guidelines, ﬂip-chip packages, fracturemechanics, multiple singularities, reliability.
I. I
NTRODUCTION
T
HE TREND towards porous, lower dielectric constant in-terlayerdielectric(ILD)materialsoftenhasaconsequenceon the mechanical integrity of the ILD stack. At the presenttime, challenging dielectric and package material sets havemade ILD fracture, both at interfaces and within the dielectricmaterial, an important reliability consideration. The thermalstresses arising during fabrication, testing, or service due tocoefﬁcient of thermal expansion mismatch between packageand die is often attributed as the main driving force for fracturesin ILD stacks [1]–[3]. Thus, assessing the damage or risk of fracture in ILD stacks during the back end of line fabricationand subsequent processes such as solder reﬂow and underﬁllcuring iscriticalat thistime. Similar assembly interactions existin the emerging 3-D or vertically integrated packages [4], [5].These packages often rely on through silicon vias (TSV) toenable heterogeneous integration. The TSVs may be placed ona small pitch, as small as 10
µ
m [6], and their collective effecton die deformation is not well understood at the present time.There are both experimental and numerical challenges asso-ciated with analyzing chip-substrate interactions. Experimentalchallengesmostlyrelatetothe ability to predict bond strength of various interfaces encountered in microelectronic devices [7].The numerical modeling challenges include an ability toinitiate and propagate arbitrary cracks (interface or bulk) ina computationally efﬁcient manner. There is relatively littleresearch in modeling initiation and propagation of arbitrarycracks in structures. Thus, demonstration of such numericalprocedures to microelectronic systems in general, and dielectricstacks in particular do not appear to exist in prior literature.Perhaps because of this challenge, there are relatively fewstudies attempting to model propagation of cracks in ILDstructures. One of the few is the work by Ocana
et al.
[8], whoused ﬁnite-element models with damage at crack tip describedusing a cohesive zone model to simulate fracture propagationin ILD structures. This study required cohesive elements beinginserted
a priori along all potential crack paths
. In otherstudies, Stolarska and Chopp [9] used the extended ﬁnite el-ement method (XFEM) to model thermal fatigue cracking in
1530-4388/$31.00 © 2012 IEEE
242 IEEE TRANSACTIONS ON DEVICE AND MATERIALS RELIABILITY, VOL. 12, NO. 2, JUNE 2012
dielectric structures. Zhang
et al.
[3] have evaluated the energyrelease rates corresponding to prespeciﬁed cracks within theILD structure. In general, none of the prior studies appear tohave predicted the likely sites of crack initiation within ILDstacks or developed computational techniques to automaticallypropagate, along arbitrary paths, either a newly initiated, or apredeﬁned crack.The goal of the present paper is to investigate assembly-induced damage to ILD stacks using a novel computationalprocedure that enables one to identify locations where one ormore cracks initiate and subsequently track the progression of the cracks within the stack. Prior to numerical simulations, thelikely sites of fracture in the ILD stack are analytically identi-ﬁed through asymptotic analysis of the strength of singularitiesin multimaterial corners prevalent in the ILD stack. It is wellknown that the stress ﬁelds at the interface corners exhibit asingular behavior for linear elastic materials that is proportionalto
r
−
λ
i
[10], [11] where
λ
i
are the strengths of the singularities.Liu
et al.
[12] were among the ﬁrst to demonstrate the impor-tance of multiple singularities in micro-electronic structures.Luo and Subbarayan [10] parametrically analyzed the numberand strengths of singularities in multimaterial wedges that areof relevance to interconnect structures. Further, they developeda consistent procedure for characterizing the generalized stressintensity factors in the presence of multiple singularities. Thisanalysis relies on elasticity theory and provides a preliminaryestimate of the relative risk of fracture at various materialcorners in the absence of information about far ﬁeld loading.Next, a newly developed computational procedure (referredto here as the hierarchical partition of unity ﬁeld compositionsor HPFC, [13]–[15]) is adopted to model crack initiation andpropagation in the ILD structures. In [15], two types of en-richments were developed to model fracture. In the ﬁrst, thecrack was modeled as a behavioral discontinuity deﬁned onthe underlying continuous behavioral approximation. However,since enrichments of the continuous ﬁeld by the discontinousone presupposes the existence of a crack, the utility of behav-ioral enrichment for modeling crack initiation is limited. In thesecond approach, the underlying material modulus is enrichedwith a cohesive damage description. The damage parameterthen dictates the nucleation and the growth of the crack. Abrief description of the computational methodology ensues inthe following sections. A multilevel ﬁnite-element modelingtechnique is used in conjunction with HPFC procedure in whichthe boundary conditions from the critical location in the
global
model (package level model) are applied to the
local
model(ILD model).The factors that govern the failures in ILD stacks can bebroadly classiﬁed into die level effects and package level ef-fects. Die level effects include the size/thickness of the dies,conﬁguration/designoftheinterconnectlines,andtheparticularchoice of dielectric materials in the stack. At the package level,solder and underﬁll material properties, substrate layup as wellas substrate thickness have an impact on ILD stack reliability.The relative importance of the die-level and package-levelfactors are analyzed in the present paper through a systematic,parametric study. Using groups of geometric and material para-meters identiﬁed by Chen and Nelson [16] as controlling the re-
Fig. 1. A schematic illustration of the baseline interconnect structure with thematerial junction conﬁgurations identiﬁed by numbers 1–8.
sponse of trilayer bonded assemblies, the relative importance of shear and ﬂexural deformation of the substrate/die combinationon the damage in ILD stacks is analyzed. Finally, design rulesare developed to generalize the numerical simulations of thepresent study to other assembly conﬁgurations not consideredhere.II. A
NALYTICAL
E
STIMATION OF
S
TRENGTH OF
S
INGULARITIES
Singular stress ﬁelds arise in layered systems consisting of dissimilar materials. Cracks may initiate at these singular pointsin addition to locations where pre-existing defects may existin the material. In general, the stresses near the edge of aninterface may be described asymptotically as [17]:
σ
ij
=
N
n
=1
K
n
r
−
λ
n
f
n
ij
(
θ
)
(1)where
r
and
θ
are polar coordinates,
N
is the total number of singularities,
λ
n
are the strengths of the singularities. The an-gular function
f
n
ij
(
θ
)
can be completely described through anasymptoticanalysis.LuoandSubbarayan[10]characterizedthenumberandstrengthofsingularitiesinmultimaterialwedgesthatareofrelevancetomicroelectronicinterconnectstructures.Theyextended the solution procedure by Seweryn and Molski [18]for characterizing multimaterial corners under generalizedloading conditions. Further, they showed that considering sec-ondary singularities is essential to capturing the angular varia-tion of the stress ﬁelds around the material junctions accurately.Thebaselineinterconnectmodelconﬁgurationusedforanalysisin this work is shown in Fig. 1. The combinations of possiblebi- and trimaterial junctions are also labeled in the ﬁgure.In this paper, SiO
2
, SiCOH and ultra low-
k
(ULK) materialswere considered as the candidate ILD materials used in theinterconnect structure. The material properties used for theanalysis are tabulated in Table I.Following the procedure by Luo and Subbarayan [10], thestrengths of singularities
λ
n
of all material wedges (see Fig. 2)present in the interconnect structure were calculated. Thesevalues are listed in Table II. Corresponding to each corner,the two most dominant singularities were computed and listedin the table. Clearly, the trimaterial junction 7 identiﬁed in
TAMBAT
et al.
: SIMULATIONS OF DAMAGE, CRACK INITIATION, AND PROPAGATION IN INTERLAYER DIELECTRIC STRUCTURES 243
TABLE IP
ROPERTIES OF
M
ATERIALS
U
SED IN THE
I
NTERCONNECT
M
ODELS
Fig. 2. The eight different material junctions occurring in the structuresconsidered in the present study.TABLE IIT
HE
C
ALCULATED
S
TRENGTHS OF
S
INGULARITIES
A
SSOCIATED
W
ITHTHE
M
ATERIAL
C
ORNERS
S
HOWN IN
F
IG
. 2
the ﬁgure is the weakest as determined through asymptoticanalysis. It is also observed that the ULK material choiceincreases the strength of the dominant singularity at the criticaltrimaterial junctions. The strengths of the dominant singularityof the three most critical material junctions (5, 7, and 8) againstthe elastic modulus of the dielectric materials at the includedangle are plotted in Fig. 3. It is observed that the strength of thedominant singularity continuously increases with a decrease inmodulus value and approaches a maximum value of 0.5 (sameas for a crack) for an ideal dielectric (air). This indicates ahigher propensity to failure at material junctions that containan included region of weaker dielectrics.Strictly speaking, the strength of singularity only indicatesthe relative risk of crack initiation associated with each material
Fig. 3. In low-
k
dielectric materials, as the elastic modulus is decreased,the strength of the dominant singularity increases. For an ideal dielectric, thestrength of the singularity is 0.5, the same as that for a crack.TABLE IIIT
HE
M
ATERIAL
C
ONFIGURATIONS OF
ILD S
TACKS
E
VALUATEDIN THE
P
RESENT
S
TUDY
combination under same loading conditions. Following theabove analytical approach, in order to capture the stress ﬁeldsaccurately and to predict failure, one would need to characterizethe generalized stress intensity factors
K
n
in (1) correspondingto the speciﬁc load and boundary conditions applied on thestructure [10]. Then, the stress intensity factors may be com-pared to experimentally determined critical values to make de-cisionsoncrack initiation.Thechallenge withsuchanapproachis that the deﬁnition of the generalized stress intensity factors isoften nonunique (see discussion in [10]) and the experimentallydetermined critical values of the generalized stress intensityfactors are also often not available. Therefore, instead, in thispaper, we numerically model the fracture initiation/propagationin structures with various conﬁgurations of ILD materials thatare listed in Table III using a cohesive damage law. We relatethe observed crack initiation site to corners 5, 7, and 8 identiﬁedas being potentially risk prone in this section.III. M
ODELING
M
ETHODOLOGY
The existence and location of fracture in the interconnectstructure is greatly inﬂuenced by the assembly process thatcouples the substrate to the die thereby inducing stress, as wellas the package conﬁguration that modulates the nature/extentof stress. Therefore, in any model aimed at quantifying stress,both the process steps that induce stress as well as the pack-age conﬁguration must be faithfully captured. In the presentwork, these are captured in a “global” ﬁnite-element model of the entire assembly beginning with the ILD deposition step,followed by solder reﬂow and under ﬁll steps. While model-ing the above manufacturing process ﬂow, the critical processcondition together with the corresponding critical location inthe interconnect structure where the stress in ILD stack ismaximum are identiﬁed. The critical state of stress identiﬁed
244 IEEE TRANSACTIONS ON DEVICE AND MATERIALS RELIABILITY, VOL. 12, NO. 2, JUNE 2012
through the global ﬁnite-element models is then converted intoappropriate displacement boundary conditionsthatarenextpre-scribed on a (local) interconnect model to study potential risk of crack initiation as well as to study possible crack propagationpath. The global ﬁnite-element model is described in detail inAppendix A. The focus below is on a novel meshless procedureused to identify risk of fracture and to simulate the potentialpath of crack propagation.
A. Representation of Cracks through Enriched Field Approximation
The mathematical concept of HPFC developed in [13], [22]posits that an arbitrary approximation to a behavior or materialproperty over a domain can be hierarchically composed fromapproximations over subdomains provided such compositionssatisfy the partition of unity property [23]. Rayasam
et al.
[22]suggested a natural composition of the complex domain bymirroring the constructive solid geometry procedure of CAD,with each subdomain formed from a geometrically simple“primitive” domain. This allowed the material property as wellas the behavior combination within primitive domains (forminga “primitive” design state) to vary arbitrarily enabling oneto model complex spatially varying materials such as gradedmaterials. Further, they developed rules to construct the globaldesign state through a weighted composition of primitive de-sign states such that the weights satisfy partition of unity overthe global domain. The ideas developed in [13], [22], were
further extended by Tambat and Subbarayan [15] to enrichthe above constructed approximation with known behavior ormaterial property on surfaces (or curves or even vertices) withinthe domain. These enriched approximations are the meansadopted to describe cracks in the present study and are thereforedescribed brieﬂy below.The modeling begins by considering the geometry of thedomain
Ω
and the behavioral ﬁeld in it to be parametricallydeﬁned as the mapping
V
Ω
(
ξ,η,ζ
)
→
x
and
f
Ω
(
ξ,η,ζ
)
→
u
,respectively. Also, the geometry of the lower dimensional en-riching geometry is parametrically deﬁned as
C
Γ
e
(
s,t
)
. Usinghierarchical compositions, the global approximation at anypoint
x
is represented as (see Fig. 4):
f
(
x
) =
w
Ω
f
Ω
(
x
(
ξ,η,ζ
)) +
w
e
Ω
f
e
Γ
(
x
)
(2)where weights
w
Ω
and
w
e
Ω
are constructed such that
w
Ω
+
w
e
Ω
= 1
. Since the inﬂuence of the enriching ﬁeld must decaywith distance from the enriching geometrical entity, the weightﬁelds are required to be monotonically decreasing functions of distance. One form of weight ﬁeld is the Gaussian weight ﬁeld:
w
e
Ω
=
e
−
dd
exp
2
w
Γ
e
(3)where
d
is a monotonic measure of distance from the enrichinggeometry
Γ
e
,
d
exp
is a scaling factor, and
w
Γ
e
is the weightprescribed on the enriching geometry
Γ
e
. The weight ﬁeld thusconstructed limitstheinﬂuence ofenriching featureover aﬁnitedistance. In the present work, the approximations both overthe domain and on the enriching entities are constructed using
Fig. 4. The global approximation is constructed by composing the lower orderprimitive with the higher order primitive.
Nonuniform Rational B-Splines (NURBS), which are popularin CAD for modeling geometrical surfaces [24]. The use of NURBS for analysis of behavior was proposed by the authorsin [25], [26]. More recently, such analyses have been referred
to as isogeometric analysis [27] since these approximations arebuilt isoparametrically on the geometrical model. In general, aNURBS surface has the mathematical form:
S
(
s,t
) =
n
i
i
=0
n
j
j
=0
R
ij
(
s,t
)
P
ij
(4)where
P
ij
is the
ij
th
control point (position) vector,
n
i
arenumber of control points in the
i
th
direction (
n
j
is similarlyin
j
th
direction) and
R
ij
are the rational basis functions deﬁnedas the tensor product
R
ij
(
s,t
) =
N
i,p
(
s
)
N
j,q
(
t
)
w
ij
n
α
α
=0
n
β
β
=0
N
α,p
(
s
)
N
β,q
(
t
)
w
αβ
(5)and
N
i,p
are the
p
th
degree B-spline basis functions deﬁnedthrough the recursive relation (
N
j,q
deﬁnition is analogous):
N
i,
0
=
1
if
s
i
≤
s<s
i
+1
0
otherwise
N
i,p
=
s
−
s
i
s
i
+
p
−
s
i
N
i,p
−
1
(
s
)+
s
i
+
p
+1
−
ss
i
+
p
+1
−
s
i
+1
N
i
+1
,p
−
1
(
s
)
(6)
TAMBAT
et al.
: SIMULATIONS OF DAMAGE, CRACK INITIATION, AND PROPAGATION IN INTERLAYER DIELECTRIC STRUCTURES 245
deﬁned typically on nonperiodic and nonuniform knot interval
U
=
{
0
,...,
0
,s
p
+1
,...,s
m
−
p
−
1
,
1
,...,
1
}
(7)NURBS basis functions exhibit properties such as partition of unity, local support, domain of inﬂuence, smoothness, convexhull, and nonnegativity that are critical to ensuring convergenceof approximations to a known solution. An isogeometric ap-proximation is constructed isoparametrically on the geometricentity by replacing the control point vectors
P
ij
with the behav-ioral unknowns
ˆ
u
ij
. Thus, a behavioral approximation may beisoparametrically deﬁned analogous to (4) as:
u
(
s,t
) =
I
R
I
(
s,t
)ˆ
u
I
(8)where, the summation over
i
and
j
have been replaced with asingle summation over all grid points for convenience. In thisform,the NURBS approximations resemble the shape functions
N
I
(
s,t
)
used in the ﬁnite-element method. Henceforth, for thisreason, the basis function is denoted as
N
I
.In the present study, following [15], the enriching ﬁeld isapproximated at the control points of the
enriching entity
asshown below:
f
e
Γ
(
P
(
x
)) =
f
e
Γ
(
s,t
) =
ψ
(
x
)
I
N
I
(
s,t
)ˆ
u
I
(9)where
ˆ
u
I
are the ﬁeld unknowns at the control points deﬁningthe enriching geometry
S
Γ
e
(
s,t
)
,
N
I
(
s,t
)
are the NURBSbasis functions, and
ψ
provides the required spatial modulationof the ﬁeld
u
to together achieve the desired enrichment.
P
(
x
)
is a projection from
x
to the shortest distance point
(
s,t
)
on theenriching surface. That is,
P
:
x
→
(
s,t
)
.The choice of enrichment function depends on an
a priori
knowledge of the behavior. For example, to model the discon-tinuity in the solution ﬁeld across a crack surface, a Heavisidestep function is used as enrichment such that
ψ
= 1
above thecrack surface and
ψ
=
−
1
below it [13], [15], [28], [29]. As
before, the continuous and discontinuous ﬁelds are modeledcompletely independent of each other and composed to obeypartition of unity. The signiﬁcant computational advantage of such an enrichment strategy is that the changes are localized tocrack geometry during crack propagation.
B. Stress Intensity Factor Evaluation and Crack Propagation
The decision to propagate the crack may be based on lin-ear elastic fracture mechanics (LEFM), speciﬁcally, on thecalculated energy release rate relative to fracture toughness.Two types of failure modes are commonly observed in ILDstructures: cohesive fracture of dielectrics and interfacial de-lamination. In the case of cohesive fracture, the stress inten-sity factors may be directly obtained from the crack openingdisplacements as (replacing
ν
with
ν/
1 +
ν
under plane stressconditions) [30]:
K
I
=
µ
√
2
π
(∆
u
y
)
√
r
(2
−
2
ν
)
(10)
K
II
=
µ
√
2
π
(∆
u
x
)
√
r
(2
−
2
ν
)
(11)where
µ
is the shear modulus,
ν
is Poisson’s ratio,
r
is thedistancefromthecracktiptothecorrelationpoint,and
∆
u
x
and
∆
u
y
are the crack-opening displacements along the co-ordinatedirections at the correlation point. The crack propagation direc-tion is determined using the maximum circumferential stresscriterion:
∆
θ
c
= 2arctan
1
−
1 + 8
K
II
K
I
2
4
K
II
K
I
(12)Crack propagates when the energy release rate of the cohe-sive crack exceeds the fracture toughness of the bulk material,i.e., when
G
≥
G
c
. Energy release rate is related to the stressintensity factors as:
G
=
κ
+ 18
µ
K
2
I
+
K
2
II
(13)where, the material parameter
κ
= 3
−
4
ν
.In the case of interfacial cracks, due to asymmetry in theelastic moduli with respect to a bimaterial interface, the stressand displacement ﬁeld around an interfacial crack tip in generalcannot be decoupled into pure Mode I and Mode II ﬁelds.Interfacial crack propagation is in general under a mixed modecondition. The stress intensity factors deﬁned by Rice [31] canbe calculated using the displacement extrapolation method as:
K
I
=2
S
2
πr
[(∆
u
y
−
2
ǫ
∆
u
x
)cos
R
+(∆
u
x
+ 2
ǫ
∆
u
y
)sin
R
]
(14)
K
II
=2
S
2
πr
[
−
(∆
u
y
−
2
ǫ
∆
u
x
)sin
R
+(∆
u
x
+ 2
ǫ
∆
u
y
)cos
R
]
(15)where
S
= 2cosh(
ǫπ
)
(
κ
1
+1)
µ
1
+
(
κ
2
+1)
µ
2
(16)
R
=
ǫ
ln
rl
k
(17)Here,
κ
1
and
κ
2
are the
κ
values of the two materials atthe interface, respectively,
l
k
is a characteristic length of thesystem and
ǫ
is the material oscillation index, related to secondDundur’s parameter
β
as:
ǫ
= 12
π
ln
1
−
β
1 +
β
(18)The energy release rate and the stress intensity factors thushave the following relationship:
G
= 116cosh
2
(
ǫπ
)
κ
1
+ 1
µ
1
+
κ
2
+ 1
µ
2
K
2
I
+
K
2
II
(19)

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