Finite Element Modeling of Stress Evolution in Sn Films due to Growth of the Cu6Sn5 Intermetallic Compound

Finite Element Modeling of Stress Evolution in Sn Filmsdue to Growth of the Cu6Sn5 Intermetallic Compound

ERIC BUCHOVECKY,1,2 NITIN JADHAV,1 ALLAN F. BOWER,1

and ERIC CHASON1

1.—Division of Engineering, Brown University, Providence, RI 02666, USA. 2.—e-mail:[email protected]

We use finite element simulations to quantitatively evaluate differentmechanisms for the generation of stress in Sn films due to growth of theCu6Sn5 intermetallic phase at the Cu-Sn interface. We find that elastic andplastic behavior alone are not sufficient to reproduce the experimentallymeasured stress evolution. However, when grain boundary diffusion isincluded, the model results agree well with experimental observations.Examination of conditions necessary to produce the observed stresses providesinsight into potential strategies for minimizing stress generation and thusmitigating Sn whisker growth.

Key words: Pb-free solder, Sn whisker, finite element, grain boundarydiffusion

INTRODUCTION

The growth of long, single-crystal whiskers fromthe surface of thin Sn coatings on Cu conductorswas first observed over 50 years ago.1 Since Sncoatings are highly desirable on copper electronicinterconnections as a means to reduce oxidation ofthe conductors and improve solderability, thepotential for circuit failure due to whisker growthposes a significant threat to reliability. The problemwas largely eliminated when it was discovered thatalloying Sn with a small fraction of Pb preventedwhisker formation.2,3 However, with the recentadoption of Pb-free manufacturing practices in theelectronics industry, whisker growth from Sn-platedCu conductors has re-emerged as a significantthreat, particularly in high-reliability applications.

The importance of compressive stress in Sn filmsas a primary driving force for whisker growth is wellestablished.3,4 This stress can result from mechan-ical deformation during processing and handling ofcomponents, thermal mismatch, or in the case of Snfilms on Cu, the spontaneous growth of intermetal-lic compounds (IMCs). The latter process is of par-ticular concern as it is difficult to control and can

continue over long periods of time. A fundamentalunderstanding of the process of stress generation inSn films could potentially provide insight intostrategies to minimize stress and thereby mitigatewhisker growth. However, there is no clear con-sensus regarding the mechanisms by which local-ized growth of IMCs produces stress within the Snfilm.5–14

In order to understand how stress develops inresponse to IMC growth, it is necessary to identifythe mechanical deformation processes operating inthe surrounding Sn film. Recent experimental workprovides evidence of extensive dislocation activitywithin the Sn film, especially in the neighborhood ofIMC grains,13,14 indicating that the Sn grains yieldplastically. This conclusion is also consistent withreported stress measurements within the Sn layerwhich are very near the nominal yield stress for Sn(14.5 MPa15) and remain relatively constant overtime.7,11,13,16 In addition, self-diffusion along grainboundaries within Sn is assumed to provide a fastpathway for long-range transport.5,17 The roleplayed by these plastic deformation processes in thedevelopment of stress within Sn films, however, hasnot been quantitatively assessed.

In this paper, we present a series of finite elementsimulations that allow us to quantitatively evaluatedifferent mechanisms for the generation of stress in

(Received March 16, 2009; accepted July 23, 2009;published online August 12, 2009)

Journal of ELECTRONIC MATERIALS, Vol. 38, No. 12, 2009 Special Issue Paper

DOI: 10.1007/s11664-009-0911-3� 2009 TMS

2676

Sn films due to growth of the Cu6Sn5 intermetalliccompound at the Cu-Sn interface. Specifically, weexamine the role of elastic and plastic deformationwithin Sn grains and the role of stress-driven dif-fusion of Sn along grain boundaries. It should benoted that we do not address the nucleation ofwhiskers, nor include the effect of local stressrelaxation in the region surrounding a whisker.Also, for the purpose of this study we limit attentionto stress generation in as-deposited Sn films and donot consider the effect of solder reflow, which cansignificantly alter the microstructure, thickness,and morphology of the Sn.

Our simulations show that elastic deformationand dislocation-mediated plastic flow alone are notsufficient to reproduce the experimentally measuredrelationship between stress and IMC volume.However, when grain boundary diffusion is included,the model results agree well with experimentalobservations. Examination of conditions necessary toproduce the observed stresses provide insights intopotential strategies for minimizing stress generationand thus mitigating whisker growth.

EXPERIMENTAL BACKGROUND

The relationship between IMC growth and filmstress has been investigated through a series ofexperiments which allow simultaneous measure-ment of IMC volume and stress in the Sn for the samesamples (presented previously by Chason et al.13).Here, we briefly review those results, calling atten-tion to several key features. Figure 1 shows mea-sured IMC volume per area and the correspondingaverage stress in the Sn layer obtained for a 1.45 lmSn film electroplated over 0.6 lm of Cu. It is impor-tant to note that the measured stress representsmean biaxial stress, i.e., the average of the twoin-plane normal stress components, (r11 + r22)/2,integrated over the thickness of the Sn layer.

Examining the plot of IMC growth over time(Fig. 1a), IMC volume per area is seen to increasemonotonically. The corresponding stress in the Sn,shown in Fig. 1b, is initially tensile but quicklybecomes compressive. The stress reaches a maximumcompressive value of about �12 MPa after approxi-mately 20 h, then remains relatively constant eventhough IMCs continue to grow. The observation thatthe average stress appears to attain a ‘‘steady-state’’value indicates that further IMC growth is accom-modated by processes that produce little additionalstress within the Sn, such as plastic yield.

To understand how stress is generated by IMCgrowth, it is necessary to consider the nature of theIMC transformation reaction. At room temperature,Cu and Sn react irreversibly to form the Cu6Sn5

IMC. Scanning electron microscopy (SEM) andtransmission electron microscopy (TEM) analysesreveal that IMC particles nucleate within the Sn atthe Cu-Sn interface, typically with greater proba-bility at the triple junctions where grain boundaries

between Sn grains meet the Cu interface.13,14,18

After nucleation, growth of the particles proceeds atthe IMC/Sn interface,5,19 with IMC particles pri-marily growing up into the Sn film.13,14,16,18 TEManalysis also provides evidence of dislocation activ-ity within the Sn layer, particularly near IMC par-ticles.13,14 This suggests that IMC growth inducessufficient stress within the surrounding Sn grains tocause dislocation-mediated plastic deformation.

In the case of pure Sn on Cu, SEM and TEMexamination of cross-sections through the thicknessof the film show that IMC growth occurs only at theSn-Cu interface.13,14 There is no evidence of isolatedIMC particles away from the Sn-Cu interface, eitherwithin Sn-Sn grain boundaries or in the Sn grainsthemselves. Thus, a model for stress generationmust explain how IMC growth that is restricted tothe base of the Sn layer results in the developmentof stress throughout the thickness of the film.

Stress generation in the Sn layer is intimatelyrelated to the volume changes associated with theformation of Cu-Sn intermetallic compounds. Herewe consider only reaction of Cu and Sn to formCu6Sn5, since this is the sole IMC phase observed atroom temperature.5 As discussed above, Cu6Sn5 isobserved to grow into the Sn. This requires thetransport of Cu to the IMC growth front, where itthen reacts with the adjacent Sn.5,19 The net volumechange associated with the reaction is therefore

0.0

0.1

0.2

0.3

IMC

Vol

./Are

a [µ

m]

(a)

σ bia

xial

[MP

a]

0 20 40 60 80 100 120 140t [hr]

-15

-10

-5

0

5(b)

Fig. 1. Experimental measurements (symbols) of (a) IMC volumeper area, and (b) average stress within the Sn layer for samples with1.45 lm pure Sn plated on 0.6 lm Cu. Measurements correspond-ing to the same time were obtained from the same sample. Theempirically fit curve (solid line) in (a) was used to model IMC growthin the numerical simulations and has the form VIMC=A ¼ð0:014 lm h�0:6Þt0:6. Data previously published in Chason et al.13

Finite Element Modeling of Stress Evolution in Sn Films due to Growthof the Cu6Sn5 Intermetallic Compound

2677

partitioned into two spatially separated volumechanges: volume loss within the Cu layer and vol-ume increase within the Sn layer. Calculation of theresulting strain within the Sn layer depends on thedetails of the transformation from Sn to IMC. Whilethese details are not well understood, the conse-quences of two limiting cases can be explored.

Consider a fixed volume of Sn adjacent to agrowing IMC particle. If all of the Sn atoms in thisregion rearrange to form Cu6Sn5 as Cu atomsarrive, then the region will undergo a large volu-metric expansion. Referring to the molar volumedata summarized in Table I, the initial Sn volume ofð0:45ÞVSn

m ¼ 7:4 cm3=mol is replaced by VIMCm ¼

10:7 cm3=mol, corresponding to a 45% volumeincrease. This value is an upper bound, since someof the IMC volume may also be accommodated bymotion of the Cu-Sn interface.

At the other extreme, it is possible for the IMC tooccupy the same volume as the Sn it replaces ifsufficient Sn is expelled as point defects during thetransformation. The expelled Sn could either beaccommodated by the surrounding Sn lattice oradjacent grain boundaries. Stress generated by thediffusion of expelled Sn along grain boundaries willbe discussed in the ‘‘Results and Discussion’’ sec-tion. For the case of Sn point defects inserted intothe surrounding Sn lattice, the extremely low bulkself-diffusivity for Sn of D � Oð10�18cm2=s)15

would effectively limit the volumetric expansion toa region immediately surrounding the IMC parti-cles. The resulting localized expansion would bevery similar to that produced by expansion of theIMC itself. For the purpose of modeling stress evo-lution in the Sn film, we assume that the transfor-mation of Sn to Cu6Sn5 produces a localizedvolumetric expansion of 45%. Details of thenumerical implementation are presented in thefollowing section. The effect of volume loss withinthe Cu substrate is not considered here. Since theCu film is constrained against lateral deformationby the underlying Si wafer, stress in the Cu layerhas little influence on the stress in the Sn.

SIMULATION DETAILS

Our finite element simulations model a polycrys-talline Sn film with columnar grain structurebonded to a Cu substrate, as shown in Fig. 2. The

copper substrate has a thickness of 0.6 lm and istreated as a homogeneous solid layer. The Sn filmhas a thickness of 1.45 lm and consists of a periodicarray of identical hexagonal grains measuring0.75 lm on a side (grain size is approximately1.4 lm). The Sn grains are separated by verticallyoriented grain boundaries which support stress-driven mass transport. We assume that flux ofmaterial out of the grain boundaries onto the freesurface is prevented by the presence of a passivatingoxide layer. Mechanical behavior of the oxide layer,however, is not included in the model. Symmetryboundary conditions are applied at the lateralboundaries of the model cell to effectively model aninfinite film, and the base of the model cell is heldfixed. All dimensions and parameters used in thesimulations are listed in Table II.

The Sn grains and the Cu layer are treated asisotropic, elastic–plastic solids with Young’s modu-lus Ea, Poisson’s ratio ma, and yield stress Ya (wherea is either Sn or Cu). Plastic deformation within thesolid elements is determined by application of theisotropic von Mises yield criterion (J2 plastic con-stitutive law). We assume a yield stress value underuniaxial tension for Sn of 14.5 MPa.15 While werecognize that the constitutive behavior of Sn issignificantly more complex (exhibiting stronganisotropy in elastic and plastic deformation andstress relaxation via creep) the simplified behaviormodeled here is sufficient to establish the mecha-nisms by which stress develops and is transmittedthrough the Sn film.

In addition to elastic and plastic deformationwithin the solids, we explicitly model stress-drivenmass diffusion along the Sn-Sn grain boundaries,following the finite element formulation describedby Bower and Wininger.24 The chemical potentialdriving the flux of atoms along a grain boundary isapproximated as l = �Xrn, where X is the atomic

Table I. Density and Molar Volume Data. Note thatthe Molar Volume of Cu6Sn5 Corresponds to 1 moleof Atoms and is Based on the Composition of 5/11

(=0.45) Atomic Fraction Sn

Species Density Molar Volume

Cu 8.96 g/cm3 7.09 cm3/molSn 7.30 g/cm3 16.3 cm3/molCu6Sn5 8.28 g/cm3 10.7 cm3/mol

hSn

hCu

L

Sn

Cu

Fig. 2. The computational model used in the finite element simula-tions. The diagram on the left shows the Cu-Sn bilayer film repre-sented by the model. The Sn layer is comprised of columnarhexagonal grains separated by vertically oriented grain boundarieswhich can act as fast diffusion pathways. The highlighted regionshows the outline of the actual model cell, which is shown in greaterdetail on the right. By applying symmetry boundary conditions on thelateral boundaries, the model cell simulates a perfectly regular film ofinfinite extent.

Buchovecky, Jadhav, Bower, and Chason2678

volume of Sn and rn is the normal stress actingacross the plane of the grain boundary. The totalflux of volume along the grain boundary [volume/(lÆt)] is then given by

~jv ¼ �dDgb

kTrl � XdDgb

kTrrn; (1)

where d is the grain boundary width, Dgb is thegrain boundary diffusivity at temperature T, k isBoltzmann’s constant, and the gradient is taken inthe plane of the grain boundary.

Differences in the normal stress across the grainboundary from one location to another lead tovariations in the volume flux along the grainboundary. This nonuniform volume flux results inmass transport along the grain boundary: volume isremoved from regions of more compressive normalstress and added to regions of less compressivenormal stress. In the finite element formulation, therelative velocity of opposite faces of the grainboundary (normal to the grain boundary plane) iscalculated from the divergence of the volume flux:

½vn� ¼ r~jt �XdDgb

kTr2rn: (2)

Thus, material particles on opposite sides of thegrain boundary are displaced relative to oneanother, allowing local strain associated withstress-driven mass transport along grain bound-aries to be explicitly modeled.

Stress is generated in our model by the growth ofhemispherical IMC nodules up into the Sn from theCu-Sn interface (Fig. 3). These nodules grow radi-ally into the Sn layer by the progressive transfor-mation of adjacent Sn into IMC. In the simulations,the radius of the IMC nodules, rIMC, is increased

incrementally over a series of time steps, at a ratespecified to correspond to the experimentally mea-sured IMC growth curve (Fig. 1). During each timestep, a thin shell of Sn (�25 nm thick) surroundingthe existing IMC nodules is transformed to IMC,simultaneously increasing the volume of IMC whileconsuming some of the Sn.

The spatial extent of the IMC is specified in thesimulation by a field variable, U, which is assigned avalue between 0 and 1 at each node of the finiteelement mesh. U = 0 corresponds to Sn, while U = 1corresponds to IMC. Initially, all nodes within theSn layer are assigned U = 0. During subsequenttime steps, the value of U is updated at nodes whichare to be transformed to IMC. The value of U at eachnode is determined by its distance, r, from the cen-ter of the nearest IMC nodule via a smoothed stepfunction: U ¼ 1=2½1� tanhð2ðr� rIMCÞ=wÞ�; where wis the characteristic width of the IMC-Sn interface.Note that U varies from 0.98 to 0.02 betweenr = (rIMC � w) and r = (rIMC + w). Smoothing of theIMC-Sn interface prevents abrupt property changesas the interface passes through the finite elementmesh. In the simulations, w = 0.1 lm.

The transformation of Sn to IMC is implementedin the finite element simulations by correlatingmaterial properties (stiffness, yield strength, molarvolume) with the local value of U. The material hasthe properties of Sn where U = 0 and the propertiesof IMC where U = 1. Property values are interpo-lated for intermediate values of U. The molar vol-ume change associated with the transformation ismodeled by applying a stress-free transformationstrain25 corresponding to a 45% volumetric expan-sion. After application of the transformation strain,the material is in a state of compressive stress andwould have to undergo a 45% expansion to become

Table II. Parameters Used in the Simulations

Quantity Symbol Value Ref.

Sn film thickness hSn 1.45 lmCu film thickness hCu 0.6 lmLength of hexagonal grain edge L 0.75 lmElastic modulus of Sn ESn 50 GPa [15]Poisson’s ratio of Sn mSn 0.36 [15]Yield strength of Sn YSn 14.5 MPa [15]Elastic modulus of Cu ECu 117 GPa [15]Poisson’s ratio of Cu mCu 0.34 [15]Yield strength of Cu YCu 200 MPa [20]Elastic modulus of Cu6Sn5 EIMC 86 GPa [21]Poisson’s ratio of Cu6Sn5 mIMC 0.30 [21]Yield strength of Cu6Sn5 YIMC 2000 MPa [22]Grain boundary diffusivity of Sn Dgb 4.8 9 10�9cm2/s [23]Grain boundary width of Sn d 0.5 nm [23]Atomic volume of Sn X 0.027 nm3

Temperature T 298 K

Note: Cu6Sn5 is not expected to deform plastically, but a yield strength is required in the model to allow transformation of Sn to IMC. YIMC

is given a very large value, consistent with experimental measurements.20


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stress free. The transformation process is assumedto occur quasistatically, and stress equilibrium ismaintained throughout the simulation.

To account for the elimination of grain boundarieswithin the transformed region, grain boundary dif-fusivity is scaled logarithmically with U via therelation D�gb ¼ �UDgb, where D�gb is the scaled diffu-sivity and � is a finite constant chosen to be as smallas possible without causing numerical problems inthe solution. In the present case, � ¼ 2� 10�10:

Finite element simulations were carried out usingthe ABAQUS software package, with specializedgrain boundary diffusion elements implemented viaa user-defined subroutine. For each simulation,mean biaxial stress through the entire Sn layer iscomputed as a volume weighted average of(r11 + r22)/2 taken over all unexpanded elements.This corresponds to the stress measured experi-mentally in the Sn via the wafer curvature tech-nique described by Chason et al.13 In addition, wecompute stress profiles through the Sn film alongtwo vertical transects: one directly above a growingIMC nodule at a Sn-Sn-Sn triple junction, the otherlocated at the center of a Sn grain. At each of 22vertical positions through the film, stresses areaveraged over all elements within a 0.15 lm radiusof the respective transect lines.

RESULTS AND DISCUSSION

The development of stress due to IMC growth haspreviously been explained in terms of elastic stressfields generated by IMC particles7,9 or combinedelastic and plastic behavior within layers11 oraround IMC particles.14 In addition, models forstress relaxation via whisker growth have included

mass transport via grain boundary diffusion.6,11,17

However, the role of grain boundary diffusion in thedevelopment of stress within the Sn layer has notbeen quantitatively investigated. The primary goalof this paper is to determine which of the variousassumptions regarding constitutive behavior of theSn film best explains the experimental stress mea-surements. In the following discussion, we examinethe consequences of three different combinations ofconstitutive behaviors: purely elastic deformationwithin Sn grains; elastic and plastic deformationwithin Sn grains; and elastic and plastic deforma-tion within Sn grains combined with diffusion alonggrain boundaries. Contour plots of the mean biaxialstress, as well as stress profiles along verticaltransects, resulting from different choices of con-stitutive behavior are presented in Fig. 4. The cor-responding history of average Sn stress over time isplotted for each case in Fig. 5. We examine themeaning of the results for each case below.

We first consider the case of purely elasticbehavior. The development of stress over time,shown in Fig. 5a, appears to mirror the growth ofIMCs (curve in Fig. 1). The average stresses calcu-lated in this case are an order of magnitude morecompressive than those measured experimentally.Closer examination of the stress field in Fig. 4a and dreveals stresses of over 1000 MPa immediatelysurrounding the IMC nodules. However, the stressfield diminishes rapidly with distance from the IMC,and regions of both compressive and tensile in-planestress are observed in the overlying Sn. Within theupper third of the Sn layer, the magnitude of theelastic stresses is only on the order of a few MPa.Thus, if the Sn were capable of supporting stressesin excess of 1 GPa, a purely elastic mechanism fortransmitting stress would result in little compres-sive stress distributed through the thickness of thefilm.

We next consider the case of elastic–plasticbehavior within the Sn layer, but with no grainboundary diffusion. As discussed in the ‘‘Introduc-tion,’’ microscopy reveals significant dislocationactivity within the Sn grains, providing direct evi-dence of plastic deformation. In the finite elementanalysis, a plastically deformed zone (rmises ‡14.5 MPa in Fig. 6a) develops at the base of the Snlayer, around the growing IMC particles. Yieldwithin the Sn allows the expansion of the IMC to beaccommodated without generating the large stres-ses seen in the elastic case. Beyond the plasticallydeformed region, however, the stress field is elasticand therefore diminishes rapidly with distance fromthe IMC, as seen in Fig. 4b. In addition, regions ofboth compressive and tensile biaxial stress areobserved in the Sn overlying the plastic zone.

As shown in Fig. 5b, the biaxial stress becomessteadily more compressive with continued IMCgrowth, reaching a value of �6 MPa after 150 h.The gradual increase in compressive stress aver-aged through the Sn reflects the increasing fraction

Simulated IMC Growth

0.000.090.180.270.360.45

AppliedVolume Expansion

Fig. 3. IMC growth is simulated by applying a volumetric expansionto hemispherical regions distributed along the Sn-Sn grain bound-aries at the base of the Sn layer. The radius of the expanded regionsis increased incrementally to model the progressive growth of theIMC. In the images shown, the corresponding IMC volume/area is0.1 lm.


of the Sn layer that is at its yield stress. In contrast,the experimentally observed stress in the Snreaches a maximum compressive value of 10 MPa to12 MPa within the first 20 h, then remains rela-tively constant.

Thus, while there is experimental evidence ofdislocation-mediated plastic deformation within Sngrains,14 modeling indicates that elastic–plasticbehavior alone is insufficient to explain theobserved stress evolution associated with IMCgrowth. As in the case of purely elastic behavior,significant stresses develop in the immediateneighborhood of the IMC nodules, but die out veryrapidly upward through the thickness of the film. Asa result, very little compressive stress is transmit-ted through the thickness of the film—near thesurface of the film, both tensile and compressivebiaxial stresses are generated with magnitudes ofless than 1 MPa.

Finally, we examine the stress that developswhen IMC growth can be accommodated by masstransport through the Sn film via stress-drivengrain boundary diffusion, in addition to elastic andplastic behavior within the Sn grains. In this case,the average biaxial stress in the Sn quickly reaches

a maximum compressive value, then remains rela-tively constant as IMC growth continues (Fig. 5b).This results from the redistribution of volume alonggrain boundaries. Where compressive stresses nor-mal to the grain boundaries are largest, in andaround the plastic zone, Sn is removed from theadjacent grains and transported to less compressiveregions of the grain boundary. Since the stressgradients are largely normal to the plane of the film,this results in net transport of volume from the baseof the film toward the surface. As shown in Fig. 4cand f, the redistribution of volume via grainboundary diffusion produces relatively uniformcompressive stress through the thickness of the Snlayer.

The transport of Sn along grain boundaries alsoaffects the extent of the plastic deformation zonewithin the Sn grains. Contours of von Mises equiv-alent stress (Fig. 6a) show that, in the absence ofgrain boundary diffusion, the plastic zone (rmises ‡14.5 MPa) is restricted to a narrow band at the baseof the film, immediately surrounding the IMC par-ticles. However, when grain boundary diffusion isactive (Fig. 6b), the majority of the film is at or nearthe nominal yield stress.

(b)

xy

z

Purely Elastic Elastic-Plastic;No G.B. Diffusion

Elastic-Plastic; With G.B. Diffusion

1510

50

-5-10-15-20-25

σbiaxial[MPa]

600400200

0-200-400-600-800

-1000

σbiaxial[MPa]

1510

50

-5-10-15-20-25

σbiaxial[MPa]

(d) (e) (f)

z1 z1 z1z2z2 z2

z1 z2z1 z2

z1 z2

1.0

0.5

1.5

z [µm]

-500 0 500σbiaxial [MPa]

-30 -20 0 10σbiaxial [MPa]

z [µm]

-10

1.0

0.5

1.5

-30 -20 0 10σbiaxial [MPa]

z [µm]

-10

1.0

0.5

1.5

(a) (c)

IMC IMC

IMC IMC

IMC IMC

Fig. 4. Contours of mean biaxial stress through the film (a–c) with stress profiles plotted for the same cases (d–e). Arrows labeled ‘‘z1’’ and ‘‘z2’’on the contour plots indicate the locations of the vertical transects used to generate the stress profiles. Note that only the case including grainboundary diffusion (c,f) produces significant compressive stress through the Sn layer. In all cases, t = 40 h corresponding to totalVIMC/A = 0.13 lm.


2681

This mechanism for stress transmission helpsexplain why the measured stress in the Sn reaches asteady-state value which is relatively insensitive tofurther IMC growth. Only a small volume of IMCgrowth is required to drive sufficient transport of Snupward through the film along grain boundariesand produce a relatively uniform stress statethrough the thickness of the film. After the stress inthe Sn reaches the yield strength, additional IMCgrowth has little effect.

In our simulations, we assume that the transfor-mation of Sn to IMC is accompanied by local volumeexpansion. As discussed in the ‘‘ExperimentalBackground’’ section, another possibility is that Sncould be expelled to the grain boundaries during thereaction with little or no volume change in thetransformed region. To evaluate this case, we con-sider the amount of volume that would have to beadded to the grain boundaries to produce theexperimentally observed stress. A biaxial strain of2 9 10�4 is sufficient to bring the Sn to its nominalyield stress of 14.5 MPa (e ¼ ð1� mSnÞYSn=ESn �2� 10�4). For a columnar film with a grain size/ = 1.4 lm, this corresponds to a thickening of thegrain boundaries by an amount, Dd = e/ � 0.28 nm,roughly equivalent to the addition of a monolayer ofSn atoms. Given the exceedingly small volume ofexcess Sn that would be required in the grainboundaries to produce uniform compressive stressin the Sn, we conclude that IMC transformationinvolving Sn expulsion rather than (or in additionto) direct expansion is also a plausible mechanismfor stress generation. Regardless of the details of theIMC transformation reaction, our primary conclu-sion, that grain boundary diffusion plays a centralrole in the development of stress within the film,remains unchanged.

The grain boundary diffusion mechanism forstress transmission that we have just demonstrateddepends critically on two features of the Sn film: (1)columnar microstructure, and (2) the presence of apassivating oxide layer. Both are necessary to pre-vent relaxation of the stress—and therefore, theirdisruption may provide strategies for minimizingstress development within the Sn film.

The columnar microstructure of Sn films, with veryfew grain boundaries oriented parallel or oblique tothe plane of the film, prevents stress relaxation viaCoble creep. To summarize the argument given byBoettinger et al.,11 grain boundaries parallel to thetraction-free surface of the Sn would be subject tovery small normal stress and thus have a lower dif-fusion potential relative to vertically oriented grainboundaries. This would drive mass transport fromvertical to horizontal grain boundaries, resulting inreduced in-plane biaxial compression and slightoverall thickening of the film. This raises an inter-esting question that could be answered by additionalmodeling: What fraction of parallel or oblique grainboundaries is sufficient to allow stress relaxation byCoble creep?

0 20 40 60 80 100 120 140

0 20 40 60 80 100 120 140

t [hr]

-100

-50

0

σ bia

xial

[MP

a]

(a)Purely Elastic Sn

No G.B. diffusion

-150

-15

-10

-5

0

5

σ bia

xial

[MP

a]

t [hr]

(b)Elastic-Plastic Sn

No G.B. diffusion

G.B. diffusion with oxide

G.B. diffusion, no oxide

Fig. 5. Mean biaxial stress, averaged over the untransformed Sn.Symbols (.) are experimentally measured values; lines are simu-lation results. Results for the purely elastic case (a) are shownseparately from elastic–plastic cases (b), as the stress levels differby an order of magnitude.

0.02.55.07.5

10.012.514.5

Mises Stress[MPa]

Elastic-Plastic;No G.B. Diffusion

Elastic-Plastic; With G.B. Diffusion

(a) (b)

IMC IMC

IMC IMC

Fig. 6. Contours of von Mises equivalent stress for simulations (a)without and (b) with diffusion along grain boundaries. In both cases,elastic–plastic constitutive behavior is assumed within the Sn grains.Without grain boundary diffusion (a), plastic yield (rmises ‡ 14.5 MPa)is restricted to a band near the base of the film. However, whenmaterial can be transported via grain boundary diffusion (b) theplastically deforming region extends farther into the Sn and the entirefilm is near yield. t = 40 h and total VIMC/A = 0.13 lm in both casesshown.


The passivating oxide plays a similar constrain-ing role. Without the oxide, diffusion of Sn out of thegrain boundaries and onto the free surface of the Snwould be possible. Mass could be removed fromgrain boundaries with high compressive normalstress and transported to the traction-free surface ofthe Sn where the diffusion potential is much lower.Experimentally, in situ film stress measurementschange by an amount consistent with the relaxationof compressive stress within the Sn layer when thesurface oxide is removed by etching.13 To demon-strate the effect of the oxide in our finite elementsimulation, we eliminated the zero flux condition atthe top of the grain boundaries which allows diffu-sion of material out onto the surface of the Sn. Asseen in Figs. 5b and 7, removal of the surface oxideresults in nearly complete relaxation of stress in theSn, except in the immediate vicinity of the IMCs.

We also tested the sensitivity of the modeled stressevolution to the value assumed for grain boundarydiffusivity. Variation of the grain boundary diffusiv-ity by two orders of magnitude above and belowthe literature value (4:8� 10�11cm2=s � Dgb � 4:8�10�7cm2=s) produces a change of only about 10% inthe computed average stress levels. From theseresults we conclude that the time scale governinggrain boundary diffusion is much shorter than thatgoverning IMC growth. Thus the development ofstress in the Sn film is not limited by the rate oftransport along grain boundaries, with the possibleexception of the earliest phase of IMC formationwhen growth rates appear to be very fast.

The rapid time scale for grain boundary diffusion,together with the small excess volume within grainboundaries required to produce compressive stress,also implies that the average stress in the Sn shouldbe relatively insensitive to the thickness of the film.As seen in Fig. 5c and f, relatively uniform biax-ial compressive stress, near the yield strength,

is produced in the Sn lying above the region of IMCgrowth. In simulations in which the film thicknessis varied from 1.5 lm to 9 lm (with all otherparameters identical) the average stress in the filmsvaries by less than 10%.

CONCLUSIONS

The growth of IMC particles at the base of a Snfilm generates strain that must be accommodated bythe surrounding Sn. Finite element simulationsreveal that the details of the resulting stress field inthe Sn, as well as the average stress through the Snlayer, are strongly dependent on the deformationprocesses active within the Sn. Following is a sum-mary of our conclusions.

1. Stress-driven grain boundary diffusion, coupledwith elastic and plastic behavior within Sn grains,provides an effective mechanism for transmittingstress through the thickness of the Sn. Verticalstress gradients arising from the expansion ofIMCs at the base of the Sn film drive masstransport upward along grain boundaries. Theexcess volume in the grain boundaries producescompressive stress resulting in relatively uniformstress though the entire thickness of the Sn.

2. This mechanism can explain why the stress inthe Sn appears to quickly reach a ‘‘steady-state’’value that remains constant even as IMCs con-tinue to grow. Once sufficient volume has beendriven into the grain boundaries to bring theentire film to yield, further IMC growth isaccommodated plastically and produces no addi-tional stress. The small excess grain boundaryvolume required to bring the film to yield and thefast rate of grain boundary diffusion relative tothe rate of IMC growth helps explain how themeasured Sn stress can reach its steady-statecompressive value during the early stage of IMCgrowth.

3. The stress produced by elastic and plastic defor-mation within Sn grains, without fast transportof mass along grain boundaries, is largely con-centrated in a region immediately surroundingthe growing IMC particles, at the base of the Snlayer. Large stress gradients develop through thethickness of the film, as the stress field dimin-ishes rapidly with distance from the IMC.In-plane stresses near the surface of the filmhave both tensile and compressive character andare only a few MPa in magnitude.

4. The effectiveness of grain boundary diffusion as amechanism for transmitting compressive stressthrough the film depends on the columnarmicrostructure of the Sn, as well as the presenceof a passivating oxide layer. If either of thesecould be disrupted, strain generated by thegrowth of IMCs could be relaxed via a Coblecreep process without producing compressivestress in the Sn, thus reducing the driving forcefor whisker growth.

xy

z

1510

50

-5-10-15-20-25

σbiaxial[MPa]

-20 0 10σbiaxial [MPa]

z [µm]

-10

1.0

0.5

1.5

z1 z2

IMC IMC

z1z2

(a) (b)

Fig. 7. When material is allowed to diffuse out of grain boundariesand across the free surface of the Sn, as would be the case if therewere no passivating oxide layer present, IMC growth produces littlestress through the thickness of the Sn. (a) Contours of mean biaxialstress at t = 40 h, corresponding to total VIMC/A = 0.13 lm. (b)Stress profiles along vertical transects (marked by arrows z1 and z2)through the Sn layer.


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ACKNOWLEDGEMENTS

The authors gratefully acknowledge the supportof the NSF through the Brown MRSEC (DMR0079964) and helpful contributions from G. Barr.

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Finite Element Modeling of Stress Evolution in Sn Films due to Growth of the Cu6Sn5 Intermetallic Compound

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