Experimental and Theoretical Studies of Cu-Sn ... · according to the phase diagram shown in Fig....

Experimental and Theoretical Studies of Cu-Sn IntermetallicPhase Growth During High-Temperature Storage of EutecticSnAg Interconnects

A. MOROZOV,1,2 A.B. FREIDIN,2,3 V.A. KLINKOV,3 A.V. SEMENCHA,3

W.H. MÜLLER,1,5 and T. HAUCK4

1.—Institute of Mechanics, Technische Universität Berlin, Berlin, Germany. 2.—Institute forProblems in Mechanical Engineering of the Russian Academy of Sciences, St. Petersburg, Russia.3.—Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia. 4.—NXPSemiconductors, Schatzbogen 7, 81829 Munich, Germany. 5.—e-mail: [email protected]

In this paper, the growth of intermetallic compound (IMC) layers is consid-ered. After soldering, an IMC layer appears and establishes a mechanicalcontact between eutectic tin-silver solder bumps and Cu interconnects inmicroelectronic components. Intermetallics are relatively brittle in compar-ison with copper and tin. In addition, IMC formation is typically based onmulti-component diffusion, which may include vacancy migration leading toKirkendall voiding. Consequently, the rate of IMC growth has a strongimplication on solder joint reliability. Experiments show that the intermetalliclayers grow considerably when the structure is exposed to heat. Mechanicalstresses may also affect intermetallic growth behavior. These stresses arisenot only from external loadings but also from thermal mismatch of thematerials constituting the joint, and from the mismatch produced by thechange in shape and volume due to the chemical reactions of IMC formation.This explains why in this paper special attention is being paid to the influenceof stresses on the kinetics of the IMC growth. We develop an approach thatcouples mechanics with the chemical reactions leading to the formation ofIMC, based on the thermodynamically sound concept of the chemical affinitytensor, which was recently used in general statements and solutions ofmechanochemistry problems. We start with a report of experimental findingsregarding the IMC growth at the interface between copper pads and tin basedsolder alloys in different microchips during a high temperature storage test.Then we analyze the growth kinetics by means of a continuum model. Bycombining experiment, theory, and a comparison of experimental data andtheoretical predictions we finally find the values of the diffusion coefficientand an estimate for the chemical reaction constant. A comparison with liter-ature data is also performed.

Key words: IMC growth, chemical reaction kinetics,high temperature storage test, chemical affinity tensor

(Received March 27, 2020; accepted August 20, 2020;published online September 18, 2020)

Journal of ELECTRONIC MATERIALS, Vol. 49, No. 12, 2020

https://doi.org/10.1007/s11664-020-08433-y� 2020 The Author(s)

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OVERVIEW ON INTERMETALLICCOMPOUND GROWTH

The main technological process for creating anelectrical contact between components in a (micro-)electric circuit is soldering. Different eutectic tin-based alloys containing various metallic chemicalelements (Zn, Bi, Pb, Ag, Cu, etc.) are used for thisprocess. One of the most common lead-free solderalloys in microelectronics is eutectic SnAg3.7. Thisalloy is also used in ball-grid array (BGA) compo-nents in the microelectronic industry for solderbumps and paste.1

An intermetallic, also known as an intermetalliccompound, intermetallic alloy, ordered intermetallicalloy, and a long-range-ordered alloy is a type ofmetallic alloy that forms a solid-state compoundexhibiting defined stoichiometry and ordered crystalstructure, see, e.g., Ref.3. In electric circuits Cu isused as a conductive material for contacting thesurfaces of the electronic components. During sol-dering the solder material melts and gets in contactwith the copper substrate so that a thin layer of aparticular Inter-Metallic Compound (IMC) forms atthe interface between the Cu and the tin-basedsolder. In the absence of an IMC layer the bondbetween the solder and the substrate is weak, sincethere is almost no interaction between the metals atthe boundary. In Sn-Cu and Sn-Ag-Cu eutecticalloys the electrical and mechanical contact isestablished by one (Cu6Sn5) or two (Cu6Sn5 andCu3Sn) intermetallic phases,

4 respectively. Their

formation is determined by the temperature regime,according to the phase diagram shown in Fig. 1. Itinvolves three stages: dissolving of Cu in liquid Sn,a chemical reaction between the components, andfurther crystallization. Further growth of the IMClayer takes place in the solid state. The formation ofIMCs occurs according to the following chemicalreactions between copper and tin,

6 Cu þ 5 Sn ! Cu6Sn5;

3 Cu þ Sn ! Cu3Sn:ð1Þ

Cu6Sn5 is also referred to as the g and Cu3Sn as thee-phase.

As mentioned earlier, the existence of the IMC isa necessary condition for the electro-mechanicalcontact. However, when compared to pure copper ortin the IMC is more brittle. This may lead to adecrease of the reliability of the joint. In addition,intense diffusion of Cu (from the pad) or Sn atomsthrough the IMC may lead to void formation due tothe Kirkendall effect.5

There are many experimental studies of the IMCgrowth in solders of the aforementioned two andother phases of more complex stoichiometry,6-8 suchas microrelief mapping of the interface, the deter-mination of the chemical composition of the phases,or the influence of the soldering time on the kineticsof the IMC formation, etc. In these systems thegrowth velocities of the IMC interfaces are muchhigher than in homogeneously phased solders.

Fig. 1. Cu-Sn phase diagram reprinted from Ref. 2 under the terms of the Creative Commons license. Cu3Sn and Cu6Sn5 phases are pointed outwith red lines (Color figure online).

Experimental and Theoretical Studies of Cu-Sn Intermetallic Phase Growth During High-Temperature Storage of Eutectic SnAg Interconnects

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A typical empirical kinetic equation for the IMCphase growth is based on fitting the experimentallyobserved data and has the form9,10

h ¼ h0 þ kt1=n; ð2Þ

where h and h0 are the current and initial newphase thickness, t is the time, k and n are growthconstants. Under the assumption that the majorcontribution to IMC growth is bulk diffusion, thepower law (2) takes the form of a square rootdependency (see, e.g., Refs. 11 and 12). In otherwords, any deviation from the square root behaviorindicates that not only bulk diffusion is governingthe growth kinetics.9 In this case the growthconstant k may be expressed through interdiffusioncoefficients, D.13 The temperature dependence ofthe latter can be expressed by an Arrheniusequation,

D ¼ D0 exp �Q

RT

� �; ð3Þ

where D0 is a pre-exponential factor, Q is thereaction activation energy, R is the gas constant,and T is the absolute temperature in K, see, e.g.,Ref. 14 and the references therein. Based on theseideas a model for intermetallic growth in thin Snjoints between Cu substrates was proposed in Ref.15 with application to solder microjoints.

An understanding of the IMC formation processand predicting its kinetics in various range oftemperatures is essential for evaluating the struc-tural integrity of solder interconnects. The electricalcurrent during use can heat up a solder bump to100-150�C. This heat stimulates IMC growth and asa result reduces the lifetime of the joint.

This paper is organized as follows. In ‘‘Experi-ment Overview’’ section the experimental proce-dures are explained. ‘‘Experimental Results’’ sectioncontains the experimental results of the IMCgrowth including a chemical composition analysisof the IMC layers. A mathematical model for theIMC growth kinetics, which couples mechanicalstresses and chemical reaction rate, is presented in‘‘Theoretical Model’’ section. A simplified modelproblem is formulated and solved analytically inthe same section. A comparison of the experimentaland analytical results is also presented in ‘‘Theo-retical Model’’ section, resulting in diffusion andreaction coefficients, which can be compared withthe literature. Finally, conclusions and an outlookare presented in ‘‘Conclusions’’ section.

EXPERIMENT OVERVIEW

Specimen Preparation

Two different types of microchips (referred to asSeries I and II in Fig. 2 and in what follows) withBGA packages were available. The solder balls wereapproximately of the same diameter, 500 lm. The

frames of the microchips were made of plastic ormetal for Series I or II, respectively.

For the commercially obtained packages, theexact information about the solder material, thepackage substrate material, and its surface finishmaterials, as well the ball attachment process wasnot available. Clearly, all of this affects the initialIMC formation, as well as the diffusion processesduring its growth. Therefore, in order to obtain asomewhat clearer picture, we have carried out ourown investigations as follows. According to ourEDXS (energy-dispersive x-ray spectroscopy) anal-ysis, no other elements were found in the IMCregion apart from Cu and Sn. Hence, it is fair toassume that the solder material was eutectic tinsilver (AgSn), and the substrate had an OSP(organic surface protection) coating. Moreover, notethat our experimental work was carried out inorder to validate the analytical model of IMCgrowth but not of its formation. The thickness ofthe IMC layers was measured before the heattreatment and used as an initial condition for thetheoretical model.

In order to reduce the number of tests and stillget a reliable statistical data, our experimentalprocedure to determine the kinetics of the IMCphase growth was organized as follows. Before theexperiment each of two types of microchips wasdissected through an array of balls, as shownschematically in Fig. 3a. The cross section waspolished with 3 lm diamond polishing suspension.In order to track the interface propagation, twomicro-indenter marks (Vickers method) wereapplied to each ball by a Buehler MicroMet 5103microindentation hardness tester. The marksserved as reference for tracking the IMC growthinterfaces in the neighborhood, Fig. 3b. After theheat treatment and the EDXS analysis, the spec-imens were returned to the oven. The same crosssections of the same solder balls were examinedduring the experiment. Note that only three ballsper series were continuously monitored during theheat treatment, which was already quite time-consuming and operationally elaborate. In ourcurves below showing thickness of IMC over time,an average of the observed growth for the threeballs per series is displayed.

This procedure does not allow to fix the specimenin an epoxy casing. Indeed, the melting temperatureof the plastic is higher than melting temperature ofthe solder material. However, the epoxy curingtemperature is close to the experimental tempera-ture of 150�C. Therefore, all polishing was per-formed ‘‘by hand,’’ which resulted in a relativelyrough polished surface. Nevertheless, the quality isfair enough to obtain the chemical interface kineticsdata, which is the aim of this study. Figure 4 showsthe cross section of a solder ball. Differences incontrast correspond to different chemicalcomposition.

Morozov, Freidin, Klinkov, Semencha, Müller, and Hauck7196

Experimental Procedure

In the current study we analyze the process ofIMC growth in microprocessors with BGA packagesduring a high temperature storage test. Accordingto the JESD22-A103 specification this experimentwas performed at 150�C in a vacuum oven over1100 h. The Series I ball was examined after 0 h,120 h, 240 h, 360 h, 680 h, and 1000 h, and theSeries II ball at 0 h, 120 h, 240 h, 360 h, 480 h,800 h, and 1120 h, respectively. The chemical com-positions were obtained from EDXS analysis

performed on an Oxford Instruments INCA X-Maxdetector fitted on a MIRA 3 (TESCAN) microscopeat voltage of 20 kV. The thickness evaluation wasbased on postprocessing of micrographs with Pythonscripts.

EXPERIMENTAL RESULTS

The results for the two series are qualitativelydifferent even though the corresponding sampleswere heated and cooled under the same conditions.It is suspected that this is due to the differentcasings and the associated heat conduction proper-ties: The thickness of the microprocessor case forSeries I (plastic casing) was about 150 lm, and forSeries II (metal casing) about 360 lm. Because ofdifferences in the case materials and their thick-nesses for samples of Series I and II, the true coolingrate down to room temperature may differ. Bothseries of specimens contain a plastic circuit boardand BGA in their assembly. However, specimensfrom Series II have an additional metal framedmicrochip. Therefore, the cooling time for Series IIspecimens is longer than for specimens from SeriesI. The formation of IMC phases is a function of twocompeting processes, namely growth and dissolu-tion, which in turn depend on the cooling rate. This

Series I Series II

Fig. 2. Photos of the microprocessor boards.

Fig. 3. Cutting scheme applied to the microprocessor (a). Crosssection of a solder ball with two micro-indenter marks (b).


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may lead to the formation of various phases inmicroprocessors of various designs.

The right inset of Fig. 4 shows that an IMC layerforms at the copper-solder interface right after theattachment of solder balls on the microprocessorboard copper substrates. A spectroscopic analysisshowed that its composition corresponds to thecompound Cu6Sn5. Note that the shape of theCu6Sn5 layer has a comb-like structure (alsoreferred to as scallops in the literature16) for all ofthe studied samples. Its relief is governed by theroughness of the copper substrate, and the initialaverage height in the samples was in the range 1.9-3.3 lm. In addition, the tetragonal crystal structureof Sn results in an anisotropy of the diffusivity of theCu atoms.17,18 It was also shown that the reliefdepends on the crystalline structure of the under-lying copper pad,19–21 which can be fine-grainedpolycrystalline or even single-crystal based.

Chemical Composition of the Compounds

EDXS analysis was performed at each stage of theexperiment in order to obtain the composition of thematerials observed through the microscope. The setof examined points for specimens from Series I after1000 h heating is shown in Fig. 5 and the corre-sponding compositions are listed in Table I. Allresults are given in at.% (atom percent). One shouldnote that the composition 54.5 at.% Cu and45.5 at.% Sn corresponds to Cu6Sn5 and the com-position 75 at.% Cu and 25 at.% Sn to Cu3Sn,respectively. From the analyzed data it follows thatduring heat treatment (up to 1000 h) the chemicalcomposition (stoichiometric ratio) of the IMCremained constant and no other intermetallic com-pounds (e.g., Cu10Sn3 or Cu41Sn11) were detected.

The set of points for the EDXS analysis of thesolder material and the Cu substrate of a specimenfrom Series I after 1000 h heating is shown in Fig. 6and the results of the analysis are listed in Table II.

Fig. 4. Cross sections of a solder ball from Series I before heat treatment, general view (a) and interface zone, (b).

Fig. 5. First set of points from the interface region for EDXS analysis,a ball from Series I at t ¼ 1000 h. The corresponding data is listed inTable I.

Table I. Data obtained from the EDXS analysis forthe points in Fig. 5, values are given in at.%

Point Cu Sn

1 54.5 45.52 52.4 47.63 52.7 47.34 53.1 46.9Average 53.2 46.8


Points 1 and 2 in Fig. 6 are located in the coppersubstrate area. However, they contain 2.6 and 5.6at.% of Sn, respectively. Moreover, during separa-tion of Sn atoms from Cu6Sn5 a reaction proceedsaccording to the following scheme:

Cu6Sn5 � 3 Sn ! 2 Cu3Sn: ð4Þ

As a result of Eq. 4 and according to the phasediagram in Fig. 1, a new Cu3Sn IMC should then beformed at the Cu-Cu6Sn5 interface. However, inSeries I this compound was not observed, possiblydue to an extremely small thickness of the Cu3Snlayer, which could not be detected. We will commenton this issue again in context with our Series IImeasurements.

The set of examined points for a specimen fromSeries II after 1120 h heating is shown in Fig. 7 andthe results of the chemical composition analysis arelisted in Table III.

It can be seen from Fig. 7 that, unlike Series I,there are no ‘‘islands’’ in the IMC layer and theboundaries of the layers show less roughness (comb-like structure). During heat treatment a composi-tion of two IMC configurations was temporarilyobserved. However, at the later stages of theexperiment (after 360 h) the Cu3Sn phase vanished.The reason for this is unknown. In any case this

phase is not so easy to detect; to quote from Ref. 14‘‘However, it is well known that there is typically athin layer of Cu3Sn present between Cu andCu6Sn5, which, immediately after reflow, is gener-ally only measurable by transmission electronmicroscopy (TEM), although it can be resolved alsoby optical microscopy. Based on literature data, weestimate the thickness of the Cu3Sn layer to beabout 0.1 lm.’’

Thickness Data for the IMC Layers

In order to determine the characteristics of thegrowth kinetics of the intermetallic compounds, aset of experiments was conducted using Series I andII BGAs when exposed to a constant temperature.For each series, measurements were carried out asfollows.

Heat treatment was performed at a constanttemperature of 150�C. Measurements of the thick-ness were done at the same location of three balls,namely near the micro-indenter mark. This proce-dure allows us to track the thickness of the layers atcertain points and to average between the ballsobtained within each set of microprocessor BGAs.

As it was mentioned above, in the Series I ballsonly the Cu6Sn5 phase was detected. It is knownfrom the literature that the formation of the Cu6Sn5phase is largely determined by the diffusion

Fig. 7. Set of points from the interface region for the EDXS analysisa ball from Series II at t ¼ 1120 h. The corresponding data is listed inTable III.

Table III. Data obtained from the EDXS analysisfor the points in Fig. 7, values are given in at.%

Point Cu Sn

1 0.0 100.02 31.5 68.53 54.7 45.34 97.0 3.0

Table II. Data obtained from the EDXS analysis forthe points in Fig. 6; values are given in at.%

Point Cu Sn

1 97.4 2.62 94.4 5.63 0.0 100.04 1.0 99.05 9.5 90.56 11.8 88.2

Fig. 6. Second set of points from the interface region for the EDXSanalysis, a ball from Series I at t ¼ 1000 h. The corresponding datais listed in Table II.


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processes of Sn, and the diffusion rate determinesthe thickness of the IMC layer.1,22 Figure 8 showstypical micrographs of a sample after 120 h storageat a constant temperature of 150�C.

By comparing Fig. 4 (initial state) and Fig. 8, itfollows that the thickness of the Cu6Sn5 layersignificantly increased. The comb structure of theIMC phase is preserved. Note that in one set ofsamples the thickness of Cu6Sn5 varies along theinterface from 3.1 lm to 9.1 lm (in comparison, theinitial layer thickness was 1.9-3.3 lm, i.e., itincreased 2-3 times). A large dispersion of thick-ness, as well as a formation of IMC agglomerations,can be explained by uneven grain growth rate.23

Figure 8 shows that there are local border areas,‘‘islands’’, in which the initial composition of thesolder is preserved.

Intermetallic layers after 120 and 240 h heattreatment are shown in Fig. 9a, b, respectively. Theapproximate shape of the interfaces between thesolder, IMC and the substrate is outlined in red.One should note that the chemical composition ofthe IMC remains unchanged and the appearance ofnew crystalline phases was not detected. The inter-metallic compound grows in both directions, towardthe solder and toward the substrate. However, thegrowth rate of the latter is much less than that onthe solder side. From Fig. 9 it is also seen that the

Fig. 8. Cross section of a solder bump from Series I after t ¼ 120 h heat treatment; general view (a) and enlarged region indicated by the redrectangle (b) (Color figure online).

Fig. 9. IMC profiles in the solder bump from Series I after heat treatment: t ¼ 120 h (a) and t ¼ 240 h (b). Spheroids of Cu6Sn5 are marked withgreen (Color figure online).


area of the ‘‘island’’ regions decreases with time,which indicates that the Cu6Sn5 grains grow in alldirections and not only perpendicular to the copperinterface.

The dark ‘‘island’’ in the solder area (marked bygreen color in Fig. 9) are spheroids of Cu6Sn5 thatformed by spalling from the IMC-solderinterface.1,24

Both Sn and Cu6Sn5 islands are independentspheroids. We believe that if we polish more mate-rial from the surface of the specimen, these spher-oids will vanish. Therefore, since these areas do notreally belong to the interface, they are not consid-ered, i.e., neither added nor subtracted duringfurther evaluation of the IMC phase thickness.

With further heat treatment a consistent increasein the thickness of the IMC layer is observed.Figure 10 shows the interface profiles at treatmenttimes 360, 680 and 1000 h. It can be seen that theformation of a locally thick intermetallic layeroccurs. When comparing Fig. 10a, b, it follows thatthe areas of the solder ‘‘islands’’ slightly increasedupon heat treatment. This can be explained bysurface effects after the initial cut such that mate-rial may have moved out of plane and was thenremoved during polishing.

Microphotographs for the Series II specimen areshown in Fig. 11 for 0 h up to 120 h. One shouldnote that for this series the thickness of the IMClayer was less planar than for Series I. The initialthickness of the Cu6Sn5 phase varied from 1.7 to4.0 lm.

Fig. 10. IMC profiles in a solder bump from Series I after heat treatment: t ¼ 360 h (a), t ¼ 680 h (b) and t ¼ 1000 h (c).

Fig. 11. Cross section of a solder ball from Series II in the interface region before treatment at t ¼ 0 (a) and after treatment at t ¼ 120 h (b).

Fig. 12. Cross section of the interface region in a ball from Series IIafter 480 h heat treatment. The chemical composition of the markedarea obtained by the EDXS analysis is given in Table IV.


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As shown in Fig. 11b, two intermetallic phasesformed after treatment: Cu6Sn5 and Cu3Sn. This isdifferent from the result obtained for balls of SeriesI. As it is known from the literature the formation ofCu3Sn occurs on the interface between the coppersubstrate and the Cu6Sn5 compound. The followingchemical reaction describes its formation:25

Cu6Sn5 þ 9 Cu ! 5 Cu3Sn: ð5Þ

Despite storing in a vacuum oven, oxides and othercompounds formed on the surface of the specimencross sections. This made it hard to evaluate theexact thickness of the Cu6Sn5 layer for specimens ofSeries II at time t ¼ 480 h, since part of the layerwas covered with other materials (‘‘oxidation dirt’’),see Fig. 12.

The EDXS analysis shows the presence of 4.62%oxygen in the region where only Cu is expected (seepoints 1 and 2 in Fig. 6, or point 4 in Fig. 7). Toremove the dirt cover additional polishing wascarried out. In the following results, for this partic-ular measurement, the thickness of the only visibleIMC layer is presented. However, this measurementis excluded during the analysis for validation of thetheoretical model. The thickness of the polishedlayer can be estimated from Fig. 13, which showsthe IMC interface profiles of Series II microchips atthe same spot at time t ¼ 800 h (a) and t ¼ 1120 h(b). Therefore the thickness of the removed layer isnot greater than 10 lm.

In order to get statistical data about the thicknessof the IMC all micrographs were postprocessed withPython scripts. The positions of one interface weresubtracted from the other along the specimen. The

Fig. 13. Cross section of the IMC profile in a solder bump from Series II after heat treatment: t ¼ 800 h (a) and t ¼ 1120 h (b).

Fig. 14. Growth of the mean thickness of Cu6Sn5 phase for thespecimen from Series I and II. Experimental points are connected bydashed lines for visual clarity only.

Fig. 15. Growth of the mean thickness of Cu3Sn for the specimensfrom Series I and II. Experimental points are connected by dashedlines for visual clarity only.


mean value was chosen as the current thicknessand the standard deviation was obtained in order toobtain an error estimate.

Results for Cu6Sn5 in Series I and II specimensare shown in Fig. 14. Because of the uneven growthof the IMC layer and its comb-like shape, thestandard deviation from the mean value increasesin time.

The IMC type Cu3Sn was observed only in SeriesII specimens and even then only for a limited timeperiod. The change of its thickness is shown inFig. 15. Thus, during annealing for more than480 h, the Cu3Sn phase dissolved, but the mecha-nism of this process is not understood.

THEORETICAL MODEL

The novelty of the theoretical part of the presentpaper is that we model the effects of stresses on thereaction rate with the use of the chemical affinitytensor, which was derived in Ref. 26 and is used ingeneral statements of coupled problems of ‘‘diffu-sion–chemistry–mechanics’’ (see, e.g., Ref. 27 andreferences therein). One of the aims of this work isto validate and quantify the model of reaction frontkinetics on the basis of the experimental results forthe IMC growth.

Chemical Affinity Tensor and GeneralProblem Statement

The formation of the IMC can be considered achemical reaction of the following type:

n�B� þ n�B� ! nþBþ; ð6Þ

where n�, n� and nþ are the stoichiometric coeffi-cients, B�, B� and Bþ are the chemical formulas ofthe initial material (Sn), diffusive reactant (Cu), andthe reaction product (Cu6Sn5), respectively. Thereaction between B� and B� is localized at thereaction front C and is supported by the diffusion ofB� through Bþ as schematically shown in Fig. 16. Itis assumed that all delivered B� is consumed by thereaction.

The reaction is accompanied by the change ofvolume at the reaction front. This deformationproduces mechanical stresses at the front which in

turn may affect the front propagation. Two sourcescontribute into the volume change, deformation dueto the chemical transformation itself and deforma-tion due to the diffusion of the diffusive reactant(see, e.g., Refs. 26-28). The total deformation can beestimated as follows. A material B� is placed on oneside of the reaction front and the mix of thematerials Bþ and B� is on the other side. Becauseof the chemical reaction (6), the elementary volumedV� ¼ n�M�=q� transforms into the volumedVþ ¼ nþMþ=qþ where M� and q� are the molarmasses and reference mass densities of solid reac-tants B�. The ratio of these volumes is

Jch ¼nþMþ=qþn�M�=q�

: ð7Þ

The diffusion may produce additional volumenn�M�=q� inside material Bþ, where M� and q� arethe molar mass and reference mass density of thediffusive reactant B�. The parameter n reflects thedeformational interaction between the reactants B�and Bþ. In fact, from the point view of theory ofdeformations, B� ‘‘transforms’’ into the mix of Bþand B� at the interface C. Then the ratio of stress-free volumes of transformed materials coexistingacross the reaction front is given by

Jtr ¼nþMþ=qþ þ nn�M�=q�

n�M�=q�: ð8Þ

The case n ¼ 0 corresponds to a solid skeletonapproach, according to which the diffusion of B� isnot accompanied by volume expansion of Bþ. Thecase n ¼ 1 corresponds to adding the volumes of Bþand B�. In fact, n depends on the mechanism of thediffusion of Cu and saturations ability of Cu6Sn5with respect to diffusive Cu. The diffusion of Cuthrough Cu6Sn5 may occur by vacancy mechanismthrough two counter fluxes of Cu and vacancies, andatoms of Cu take the places of the vacancies. Thismay also lead to shrinkage of Cu6Sn5, whichcorresponds to the case n< 0. By (8), for the reaction(5) between Sn and Cu the ratio of volumes we findJtr ¼ 1:44 (44% volume expansion) if n ¼ 0 (solidskeleton approach) and Jtr ¼ 0:92 (8% volumeshrinkage) if n ¼ �1. To our knowledge the questionabout the value of the transformation strain accom-panying IMC formation remains open and estimatesof the relative volume change are in the range from�10%,13,29,30 up to þ44%.31,32 Further we fit modelparameters for n ¼ �0:5, which corresponds tovolume ratio Jtr ¼ 1:18. In particular, such a valueof deformation may be consistent with a small strainapproach used in further modeling.

To keep strain compatibility (i.e., continuity of abody) across the reaction front in the presence ofvolume change Jtr, additional strains appear andproduce internal stresses. These internal stressestogether with external loading influence the reac-tion front velocity. We model stress effects based on

Fig. 16. A solid undergoing a chemical reaction between B� and B�localized at the reaction front .


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the concept of the chemical affinity tensor. Thenotion of the chemical affinity arises from funda-mental results by Gibbs and De Donde (see, e.g.,Ref. 33). It was shown that in the case of a chemicalreaction the factor conjugate to the reaction rate xin the expression of the entropy production Pmultiplied by temperature T was equal to thecombination of the chemical potentials of the reac-tion constituents:

TP ¼ Ax; A ¼ �X

nkMklk; ð9Þ

where lk is the chemical potential of the k-thconstituent (per unit mass), Mk is the molar massof k-th constituent, the stoichiometric coefficient nkcontributes to the sum with a positive sign ‘‘þ’’ if thek-th constituent is produced in the reaction andwith a negative sign ‘‘�’’ if the k-th constituent isconsumed. For the reaction (6) the classical chem-ical affinity would be

A ¼ n�M�l� þ n�M�l� � nþMþlþ; ð10Þ

where lþ; l� and l� are the mass densities of thechemical potentials of the constituents Bþ;B� andB�, respectively. The factor A was called the chem-ical affinity of the reaction. Then a kinetic equationin a form of the dependence of the reaction rate onthe chemical affinity can be formulated, and one ofthe accepted dependencies for the reaction is theequation34

x ¼ x 1 � exp � ART

� �� ; ð11Þ

where x is the so-called partial rate of a directreaction, which is defined by the concentrations ofthe reactants, R is the universal gas constant.

The relationships (9)-(11) were formulated forreactions in systems such as gases or liquids forwhich chemical potentials and, therefore, chemicalaffinity can be presented as scalar values. Theobservation that the phase equilibrium and chem-ical reaction take place not just in a point but at anoriented area element passing through the point,led to the idea of a tensorial nature of the chemicalpotential and chemical affinity (see the discussionon the tensorial nature of a chemical potential inRef. 35 and on the tensorial nature of the chemicalaffinity in Ref. 27). One can see here an analogy tothe concept of stress. A stress state is determined bya scalar pressure acting in a point in the case ofliquids and gases. However, in the case of a solid,instead of pressure, a stress tensor determines atraction acting at the oriented area element passingthrough a point.

It was shown that the reaction rate is determinedby the normal component Ann of the chemicalaffinity tensor A, see the derivations in Refs. 26,36, and 37], and the chemical affinity tensor has thesame mathematical form as the scalar affinity (10)but with tensorial chemical potentials instead of

scalar ones. A kinetic equation similar to Eq. 11 canbe taken for the reaction rate xn at the area elementwith the normal n:

xn ¼ k�c 1 � exp �AnnRT

� �� ; ð12Þ

where it is taken into account that in the case of thereaction between solid and gaseous constituents xcan be taken in a form x ¼ k�c where c is the partialmolar concentration of the diffusive reactant.

If the reaction rate is found then the normalcomponent of the reaction front velocity Vn can beobtained from the mass balance at the reactionfront:

Vn ¼n�M�q�

xn: ð13Þ

Mechanical stresses affect the reaction via thenormal component of the affinity tensor. In order tohandle further analytical calculations and to obtainthe unknown parameters of the model, linear elasticsolid reactants are considered. In this case theconstitutive equations of B� and Bþ are

r� ¼ C� : e� � a�hIð Þ;

rþ ¼ Cþ :ðeþ � etr � aþhIÞ;ð14Þ

and the Helmholtz free energies of the solid reac-tants are represented by

f�ðe;hÞ ¼ g� þ1

2e� � a�hIð Þ :C� : e� � a�hIð Þ;

fþðe;hÞ ¼ gþ þ1

2ðeþ � etr � aþhIÞ :Cþ :ðeþ � etr � aþhIÞ;

ð15Þ

where C� are stiffness tensors and a� are coeffi-cients of thermal expansion, g� are the temperaturedependent chemical energies of the reactants B�, his the temperature change. For the sake of simplic-ity we take an isotropic transformation strain tensor

etr ¼ etrI; ð16Þ

where etr ¼ ðJ1=3tr � 1Þ is the same for all points of thedomain occupied by Bþ neglecting differencesrelated with deviations of the concentration of thediffusive reactant.

We also accept that the chemical potential of thediffusive reactant is given by

M�l� ¼ g� þ RT lnc

c�; ð17Þ

where g� is the chemical energy of the diffusiveconstituent B� and c� is a reference concentration ofB�.

It can be shown (see, e.g., Ref. 36) that from (14)-(17) it follows that


Ann ¼n�M�q�

vþ n�RT lnc

c�; ð18Þ

where the contribution of mechanical and chemicalenergies is presented by

v ¼ cþ 12r� : ðe� � a�hIÞ

� 12rþ : ðeþ � etr � aþhIÞ þ r� : ½½e��

ð19Þ

(note that due to displacement and traction conti-nuity conditions r� : ½½e�� ¼ rþ : ½½e��), where

c ¼ g� � gþ þq�

n�M�n�g� ð20Þ

is the combination of the chemical energies of thereactants.

The mechanical stresses r and the concentration cof the diffusing reactant B� can be found from thesolution of the system of equations which include:

(i) Mechanical equilibrium equations:

r � r ¼ 0 ð21Þ

with boundary and interface conditions:

ujX1¼ u0; r � njX2¼ t0;

½½u��jC¼ 0; ½½r�� njC¼ 0;ð22Þ

where u0 and t0 are the given displacement andtraction vectors at the corresponding outer bound-aries X1 and X2 of the body; square brackets denotethe jump of a value across the reaction front,½½/�� ¼ /þ � /�;

(ii) The constitutive equations (14) of the solidreactants;

(iii) The diffusion equation defined over thedomain occupied by the material Bþ

Dc ¼ 0; ð23Þ

with the boundary and interface conditions:

Dn � rc� aðc� � cÞ ¼ 0 at Xþ;

Dn � rcþ n�xn ¼ 0 at C;ð24Þ

where a is the mass transfer coefficient, or thedissolution constant38,39 at the outer boundary ofthe body, n is the normal directed outward thedomain Bþ. For the sake of simplicity we considerthe stage of steady-state diffusion with stationarydiffusion equation.

The first boundary condition in (24) defines theflux of B� through the external boundary in relationwith the solvability of B� in Bþ and means that thesupply of B� through the boundary stops if thesaturation c� is reached. Of course, one can formallyprescribe the Dirichlet boundary condition c ¼ c� atCþ. However, the Robin boundary condition (24)1

looks more reasonable to us from a physical point ofview in the considered case, because of realization.

The second boundary conditions in (24) followsfrom the mass balance between the supply andconsumption of the reactant B� at the reaction frontif additionally to take into account that the velocityof diffusive particles is much more than the velocityof the reaction front.

(iv) To close the system of equations one has toadd the kinetic equation (12) for xn which definesthe reaction front velocity via (13) with the depen-dence of Ann on the stresses and concentration givenby (18)-(20).

Analytical Solution of a Model Problem

As mentioned in the previous sections, only theCu6Sn5 intermetallic phase was detected at allstages of the experiment. Therefore, for simplicityof the further analytical analysis, we consider onlythis IMC. It was pointed out that the interfacebetween Cu6Sn5 and Sn moves much faster thanone between Cu6Sn5 and Cu. Therefore, it is alsoassumed that only the interface between the IMCand the Sn is moving. The maximal thickness of theIMC layer observed in the experiment was about15 lm (Fig. 14) which is much smaller than thediameter of the solder ball (approx. 500 lm, seeFig. 4). The above gives a reason to consider asimple boundary value problem of mechanochem-istry with a plane chemical reaction front propagat-ing in an infinite layer. Similar plane problems formoving interfaces have been previously studiedanalytically and numerically, see e.g., Refs. 40-42.

Because of the experimental setup, the crosssections of the specimens were stress-free surfacesduring the whole experiment. The chemical reactionfront propagation was observed at these stress-freesurfaces. Strictly speaking, one cannot use either aplane strain or a plane stress simplification in orderto find the stresses at the reaction front. Neverthe-less, since one can find a kinetic equation for theinterface movement for both plane stress and planestrain formulations in a closed form, we solve themodel problem in this paper in both formulations,assuming that the real behavior may be somewhat‘‘in between.’’ This gives the opportunity to fit thetheoretical prediction with the experimentalresults, as described in the ‘‘Fitting the ModelParameters’’ section. Based on the fitted data, wefurther obtain an estimate for the diffusion coeffi-cient and for the chemical reaction kinetic constant.

Consider an elastic layer with a cross section inthe xy-coordinate plane and the plane reaction frontpropagating in the y-direction from y ¼ 0 to y ¼ Hwhere H is the layer thickness and h is the currentreaction front position (Fig. 17). The storage tem-perature in the oven is T and the reference temper-ature is T0, h ¼ T � T0.

Assume that, by boundary conditions, the upperside of the layer is traction-free, then


7205

r�y jy¼H ¼ 0; ð25Þ

and the lower side is fixed, then the displacementuþjy¼0 ¼ 0. Assume that the displacement is zero inx-direction: u�x ¼ 0 and, thus,

e�x ¼ 0: ð26Þ

The displacement and traction continuity condi-tions at the reaction front will give, in particular,the continuity of y-components of the displacementvector and the stress tensor:

½½uy��y¼h ¼ 0; ½½ry��y¼h ¼ 0: ð27Þ

The plane strain or the plane stress conditions areu�z ¼ 0 or r�z ¼ 0, respectively. The equilibriumequations and boundary and interface conditionsin both plane statements will be satisfied for zeronon-diagonal components of stress and strain ten-sors, which is why these components are notmentioned above. The displacement continuity con-dition at the reaction front also demands continuityof ez, which strictly speaking is not fulfilled in theplane stress formulation, but we neglect the input ofthis formal incompatibility.

By constitutive equations (14),

e�x � a�h� etr� ¼1

E�r�x � m�ðr�y þ r�z Þ

� �; ð28Þ

where E� and m� are the Young’s moduli andPoisson’s ratio of the materials B�,etr� ¼ 0; etrþ � etr. The formulas for e�y and e�z followfrom (28) by cyclic permutations of x, y, and z.

Substituting strains and stresses found from (28)by using the conditions (25)-(27) and by taking intoaccount all constraints regarding plane strain andplane stress formulations into (19), we obtain thatthe contribution v in Ann does not depend on thefront position, and

v ¼cþ 12E�a2�h

2 � 12Eþðaþhþ etrÞ2; plane stress;

cþ E�1�m� a2�h

2 � Eþ1�mþ ðaþhþ etrÞ2; plane strain:

(ð29Þ

Then, by (12), the reaction rate becomes equal to

xn ¼ k�ðc� �c�Þ; ð30Þ

where the stoichiometric coefficient n� is renormal-ized to one with respect to all other stoichiometriccoefficients in (6) (n� ! n�=n�;nþ ! nþ=n�), and �is defined as

� ¼ exp �n�M�q�

vRT

� �: ð31Þ

Diffusive Cu atoms are supplied through thelower side y ¼ 0 and the formation of the inter-metallic occurs at the interface between Cu6Sn5 andSn at y ¼ h. The diffusion equation takes the form

d2c

dy2¼ 0; y 2 ½0;h� ð32Þ

with the boundary conditions

Ddc

dy

��y¼h

þxn ¼ 0;

Ddc

dy

��y¼0

þaðc� � cð0ÞÞ ¼ 0;ð33Þ

where xn is given by (30).The solution of the diffusion problem finally gives

the concentration at the reaction front as a functionof the reaction front position

cjy¼h ¼ c�1 þ � k�a þ

k�hD

1 þ k�a þ

k�hD

: ð34Þ

Then the reaction rate takes a form of the depen-dence of the front position

xn ¼k�c�ð1 � �Þ1 þ k�a þ

k�hD

: ð35Þ

Substitution of the expression (35) of the reactionrate into the formula (13) for the interface velocityresults in the explicit equation for the dependence ofthe front position on time:

dh

dt¼ n�M�

q�

k�c�ð1 � �Þ1 þ k�a þ

k�hD

: ð36Þ

Integration yields an explicit dependence of time onthe front position:

Fig. 17. Model description.


tðhÞ ¼12k�D ðh2 � h20Þ þ 1 þ

k�a

ðh� h0Þ

n�M�q�

k�c�ð1 � �Þ; ð37Þ

where h0 is the initial thickness of the IMC layerappeared just after the solder bump attachment.

The dependence (37) is rewritten in the standardform of the parabolic law

hðtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC1tþ C2

pþ C3; ð38Þ

where

C1 ¼ 2Dn�M�q�

c�ð1 � �Þ;

C2 ¼ h0 �D1

k�þ 1a

� �� 2;

C3 ¼ �D1

k�þ 1a

� �:

ð39Þ

Note that the parameters k� and a occur in (39) onlyin a combination of the sum of the inverse values.Note also that the influence of mechanical actions isrepresented by the parameter �, which is constantin the considered case. External mechanical loadingmay lead to the dependence � on the front position,which affects the front behavior.

The parabolic law (38) will be used in the nextsection for fitting parameters to experimental data.

Fitting the Model Parameters

The mechanical properties of Cu, Sn and corre-sponding IMCs can be found in the literature.Young’s modulus, Poisson’s ratio, and the coefficientof thermal expansion are listed in Table V.43,44

Molar masses and densities are given in Table VI.45

The solubility of Cu in the IMC is defined as themaximum achievable concentration of copper in aCu6Sn5 lattice. According to the phase diagram inFig. 1 two stable phases can appear in the IMC atT ¼ 150�C: Cu6Sn5 and Cu3Sn. By the reaction (5)

the phase Cu3Sn forms when one mole of Cu6Sn5reacts with nine moles of Cu. Therefore, we estimatethat the maximum amount of copper which can bedissolved in Cu6Sn5 is 9 mol of Cu per 1 mol of theIMC. In addition, here we assume that diffusiveatoms of Cu do not change the volume of the IMC.Hence, with the aforementioned assumptions andbased on the general definition of molar concentra-tion, c� ¼ 9qCu6Sn5=MCu6Sn5 ¼ 76 10�6 mol/mm3was computed for the reference value.

As mentioned earlier, the chemical energy param-eter c is determined by the Cu6Sn5 formationenergy. This energy is defined empirically. Thedependence of this energy on temperature in oper-ating temperatures range can be approximated byRef. 46

DG ¼ �7747:65 � 0:371T ðJ/molÞ; ð40Þ

where T is taken in K. In order to calculate c, theformation energy has to be divided by the molarvolume VCu6Sn5 ¼ 11:28 cm3=mol,

47 since c unit is anenergy density per unit volume.

Certain material and chemical parametersremain unknown, i.e., D, k�, a. In order to get anestimate for these values one can try to fit thesquare root curve, Eq. 38, constraining the param-eters such that physically acceptable values for Dand k� will result. The fitting process was performedby using weighted least squares, where the weightsof the points depend on their estimated error.

It should be noted that the equations proposed inthe ‘‘Chemical Affinity Tensor and General ProblemStatement’’ section for the chemical reaction frontkinetics represent the velocity of the interface in theinitial, undeformed configuration, while the micro-graphs track the position of the interface in thecurrent, deformed configuration. Therefore, beforefitting, the analytical curve and the experimentalpoints were transferred to the initial configuration,by multiplying by the corresponding transformationstrain value.

The fitting curves for Series I and II specimensare shown in Figs. 18 and 19, respectively.

As it was noted in the ‘‘Experiment Overview’’section, the specimens of Series II were covered withoxides and other compounds at time t ¼ 480 h sothat a polishing procedure was required. Therefore,it is not surprising that the measured thickness ofthe IMC layer at that data point departs from thegeneral pattern, Fig. 19, and contains an error thatis hard to assess. Because of that, the data point wasexcluded from the fitting curve procedure. Then theunknown material parameters D, k�, a can becalculated from (39). As it was noted, the kineticparameter k� and the mass transfer coefficient aappear only in combination, so they cannot beresolved uniquely. Found values of D and thecombination 1=k� þ 1=a are shown in Table VII.

In Ref. 48 diffusion coefficients of Cu in the IMCswere calculated based on the measured composition

Table V. Mechanical material properties used inthe model.43,44

Material E; GPa m a; K�1

Sn 50 0.36 22 10�6Cu6Sn5 118 0.31 18 10�6

Table IV. Results of the EDXS analysis of themarked area in Fig. 12

Element C O Si Cl Cu Sn

Average at % 20.57 4.62 0.33 0.47 70.62 3.39


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profiles of the diffusion zones within the tempera-ture range of 130-200�C. The authors showed thatthe diffusion coefficient depends highly on temper-ature, stating that D ¼ 0:38 10�17m2=s atT ¼ 130�C, D ¼ 9:5 10�17m2=s at T ¼ 150�C andD ¼ 60 10�17m2=s at T ¼ 170�C. Keeping this inmind, one can conclude that obtaining the result ofthe same order of magnitude can be considered as agood agreement. A comparison of the diffusionparameters, averaged from the two experimentalresults, with other works is shown in Table VIII.

Both values of diffusion coefficients obtained inthis work (basing on plane stress and plane strainformulations) correlate with the data from variouspapers and can be used as a reference for morecomplicated models based on the chemical affinitytensor concept. These models can be extended sincelarge strains are involved and non-linear materialsas well.

CONCLUSIONS

A high temperature storage test was carried outfor two groups of microchips with eutectic SnAgsolder ball grid arrays. The specimens were cut andpolished before the heat treatment. The growth ofthe intermetallic phase was examined by using thesame set of solder balls through the entire experi-ment. The proposed experimental procedureallowed to determine IMC growth kinetics analyz-ing the small set of specimens. With the use ofmicroindenter marks a relative movement of theIMC interfaces and change of layer thickness wereobtained. This in turn lead to the evaluation andquantification of the kinetics of intermetallicgrowth.

The growth of the Cu6Sn5 intermetallic compoundwas modeled analytically based on the chemicalaffinity tensor concept and by taking temperaturedependence into account. The experimental resultswere used to determine the kinetic parameters inthe chemical affinity tensor model for the first time.The simplest model of infinitely wide layers of linearelastic solids was analyzed and a theoretical pre-diction of the growth kinetics was obtained. Theinfluence of mechanical stresses on IMC growth wastaken into account. By comparison of the experi-mental data with the theoretical model the values ofthe diffusion coefficient and of the chemical reactionconstant were estimated. The obtained diffusion

Fig. 18. Fitted square root dependence (38) for Series I specimens.

Fig. 19. Fitted square root dependence (38) for Series II specimens.The point marked with asterisk was excluded from the fittingprocedure due to bad surface condition of the specimen.

Table VI. Molar masses and densities used in themodel.45

Sn Cu Cu6Sn5 Cu3Sn

q; g=cm3 7.280 8.960 8.280 9.140M, g/mol 118.7 63.55 974.8 309.3

Table VII. Estimated diffusion coefficient for two series of specimens

Parameter Formulation Series I Series II Average

D;m2=s Plane stress 2.1 1.5 1.8

10�17 Plane strain 6.4 4.8 5.61=k� þ 1=a, s/m Plane stress 89 57 7310�7 Plane strain 28 18 23


coefficients correlate with the results from works ofother researchers. The kinetic parameters of themodel can now be used as reference values for moregeneral cases with the kinetic equation based on thechemical affinity tensor.

ACKNOWLEDGMENTS

The authors appreciate the support of the RussianScience Foundation (Grant No. 19-19-00552).

FUNDING

Open Access funding provided by Projekt DEAL.

CONFLICT OF INTEREST

The authors declare that they have no conflict ofinterest.

OPEN ACCESS

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https://doi.org/10.1109/EuroSimE.2018.8369860

Experimental and Theoretical Studies of Cu-Sn Intermetallic Phase Growth During High-Temperature Storage of Eutectic SnAg InterconnectsAbstractOverview on Intermetallic Compound GrowthExperiment OverviewSpecimen PreparationExperimental Procedure

Experimental ResultsChemical Composition of the CompoundsThickness Data for the IMC Layers

Theoretical ModelChemical Affinity Tensor and General Problem StatementAnalytical Solution of a Model ProblemFitting the Model Parameters

ConclusionsFundingOpen AccessReferences

Experimental and Theoretical Studies of Cu-Sn ... · according to the phase diagram shown in Fig....

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