liquid reactions between copper or

J Mater Sci: Mater Electron DOI 10.1007/s10854-014-2221-7

Phase growth competition in solid/liquid reactions between copper or Cu3Sn compound and liquid tin-based solder Oleksii Liashenko • Andriy M. Gusak Fiqiri Hodaj

•

Received: 21 May 2014 / Accepted: 28 July 2014 Ó Springer Science+Business Media New York 2014

Abstract Interfacial reaction between solid e-Cu3Sn compound and liquid Sn at 250 °C is studied for the first time. The reaction product formed at the e-Cu3Sn/liquid Sn interface consists of the single g-Cu6Sn5 phase. The growth kinetics of the g phase formed at the incremental e/liquid Sn couple (e/g/Sn configuration) is compared to that of g phase formed at the classical Cu/liquid Sn couple (Cu/e/g/Sn configuration). The experimental method consists first in processing of intimate interfaces by dipping peaces of solid e-Cu3Sn compound and Cu in liquid Sn for 1 s at 250 °C. Afterwards, isothermal holding of such preperformed couples for 10, 30, 120 and 480 min at 250 °C are performed for both couples. A theoretical analysis of the growth kinetics of g phase and comparison of its growth in both configurations are performed.

1 Introduction Soldering is and in future will remain an important technological process in microelectronics and in other fields [1, 2]. Solders are, as a rule, eutectic tin-based alloys containing silver, copper etc. Most widespread is soldering of copper or nickel. Typical soldering of copper consists of (i) eutectic melting (reflow) of solder bump and (ii) reaction of molten solder with substrate leading to formation and growth of one or two intermetallics, g-Cu6Sn5 phase and e-Cu3Sn phase

O. Liashenko F. Hodaj (&) SIMAP, University of Grenoble Alpes, 38000 Grenoble, France e-mail: [email protected] O. Liashenko A. M. Gusak Cherkasy National University, Cherkasy, Ukraine

[3–10]. Mechanical bonding is provided mainly by g-phase, which has peculiar morphology–scallops of individual grains separated by some boundaries, nature of which is still under discussion–liquid channels of molten solder [8, 9, 11], prewetted grain boundaries [12], ordinary grain boundaries [13, 14]. Contrary to solid state aging at temperatures 200 °C and lower, when both phases grow simultaneously and demonstrate similar growth rates, in reaction with liquid solder the growth of g-phase proceeds much faster than that of e-phase. On the other hand, presence and growth of e-phase is very important since just in this phase the Kirkendall voiding takes place due to much higher diffusivity of copper than that of tin in this phase. Voiding deteriorates the solder contact and finally leads to its failure. Obviously, diffusive interaction between g-phase and e-phase during soldering should influence the kinetics of their competition and growth. Recently, we estimated the critical thickness of g-phase, necessary for the growth of e-phase, as about 0.6–1 lm [15]. It should be also useful to see, how the presence or absence of one phase may influence the growth kinetics of another phase. In this paper, below, we report the direct experimental comparison between the phase growth kinetics at copper/liquid tin and e-phase/liquid tin interfaces respectively. For this purpose we specially prepared the e-phase as a material to interact with molten tin. In this case only a single g-phase can be formed and grow. We measured the growth kinetics of g-phase in both copper/liquid tin and e-Cu3Sn/liquid tin couples, and compared the kinetics. We also made a theoretical analysis of phase growth for both cases. In all cases the amount of tin in reaction was limited (layer of tin\200 lm). Due to very fast diffusion of copper in liquid tin (about 10-9 m2 s-1) all tin is saturated with

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J Mater Sci: Mater Electron

copper in few seconds. Therefore all our experiments were actually performed with liquid tin saturated with copper. We compare experimental results and theoretical predictions and also estimate the product of channel width and diffusivity inside liquid channels formed in the g-phase.

cross-section 3 images (about 200 lm in length) were taken. In this way, the size distribution and the average value of the reaction layer thickness are obtained by a very large number of measurements.

3 Results 2 Experimental procedure The Cu3Sn compound was prepared from 99.99 % Cu and 99.9 % Sn base metals by melting in an alumina chamber furnace under vacuum. The metals were weighted to obtain the overall composition Cu-24.7 at% Sn, a total mass of about 2 g and then placed in an alumina crucible 10 mm in diameter. The melting cycle included heating to 800 °C under a vacuum of 3 9 10-6 mbar, isothermal holding for 20 min at 800 °C under Ar, cooling to 600 °C and isothermal holding for 1 h at this temperature. Thereafter the alloy was cooled naturally to the room temperature under Ar. The obtained pieces of the Cu3Sn compound were cross-sectioned, grounded, polished and ultrasonically cleaned with basic solutions. Cylindrical peaces 5–10 mm in diameter and 1–2 mm thick were thus obtained for reaction experiments. X-ray diffraction (XRD) analysis of the coupons showed the formation of the single phase Cu3Sn intermetallic (IMC). Cu coupons (99.99 % purity) with dimensions 10 9 6 9 1.5 mm3 were grounded and fine polished from one side and ultrasonically cleaned. The Cu coupons and pieces of Cu3Sn were dipped firstly in a conventional liquid RMA flux and then in the liquid melt of pure tin at 250 °C for 1 s. The thickness of tin layer over the Cu and Cu3Sn substrates varied from 20 to 200 lm. The obtained sandwich structures (Sn/substrate/Sn) were cut in small pieces of approximately 2 9 2 mm2. Each piece was then placed in a not reactive crucible, put in a heating chamber with precise temperature control and heated up to 250 °C with a heating rate of 70 °C min-1. After isothermal holding at 250 °C for the aimed reaction time, the samples were cooled down rapidly by high air flow resulting in an approximate cooling rate of 100 °C min-1. The reaction times were 10, 30, 120 and 480 min respectively. Afterwards, the samples were mounted in the resin, crosssectioned, polished and ultrasonically cleaned. Thereafter, optical microscope and Leo–S440 scanning-electron microscope (SEM) in backscatter mode at 15 kV were employed to examine the cross-section of the samples. The average thickness of the IMCs was calculated by measuring the total area of the IMCs and dividing it by the total length of the images. In fact, backscattered electron images are examined by an image analysis software (ImageJ) and the thickness of the interfacial layer is measured every 230 nm (about 4 measurements per micrometer). For each

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Figure 1 gives SEM micrographs of the reaction product formed at the Cu/alloy interface and at the e-Cu3Sn/alloy interface for the samples which were aged at 250 °C for 10, 30, 120 and 280 min. Figure 1a, c, e, g show that whatever the holding time at 250 °C, the reaction product at the Cu/ alloy interface consists of two IMC layers: a scallopshaped compound g-Cu6Sn5 on the solder side and a thin layer of e-Cu3Sn compound on the Cu side. This observation is in agreement with the experimental results reported in previous studies, see for example Refs. [1, 10]. Figure 1b, d, f, h show that whatever the holding time at 250 °C, the reaction product at the e-Cu3Sn/alloy interface consists of a single layer: a scallop-shaped compound g-Cu6Sn5 presenting a very similar morphology as that of g layer formed at Cu/alloy interface. For comparison purpose, two typical micrographs of the cross section along more than 0.5 mm of the interface are given in Fig. 2 for Cu/alloy and Cu3Sn/alloy samples after isothermal holding at 250 °C for 2 h. For each sample, the average thickness e of each layer (eg and ee for Cu/tin couple and e0 g for e-Cu3Sn/tin couple) are calculated using Image J software. The average values of e from samples used for each holding time are given in Table 1. This table shows that, for the Cu/alloy samples, the average thickness of the e-Cu3Sn layer ee is much lower than that of g-Cu6Sn5 layer eg regardless of the holding time at 250 °C and ee increases from about 0.5 lm to 4 lm when the holding time increases from 10 to 480 min. It shows also that the thickness of the g-Cu6Sn5 layer formed at Cu/alloy interface (eg) is very close to that of g formed at e-Cu3Sn/alloy interface. In Fig. 3, the average thickness of each layer (eg and ee for Cu/tin couple and e0 g for Cu3Sn/tin couple) are plotted against the square root and cubic root of time. The error bars in this figure gives the standard deviation from the average thickness of the interfacial layer calculated according to the procedure presented in Sect. 2. These presentations are in relation to the fact that the mechanism of the phase growth in Cu/Sn system is still under discussion. Indeed, the most popular candidates for the power law of g-phase growth kinetics are 1/2 and 1/3. Therefore, we represented the data of growth kinetics by two different ways–(i) as dependence of average phase thickness on the SQUARE ROOT of time (Fig. 3a), and alternatively, (ii) as dependence on the CUBIC ROOT of time (Fig. 3b).

J Mater Sci: Mater Electron Fig. 1 SEM micrograph of the Cu/Sn and e-Cu3Sn/Sn couple interfaces after 10 min (a, b), 30 min (c, d), 120 min (e, f) and 480 min (g, h) of reaction at 250 °C

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

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J Mater Sci: Mater Electron Fig. 2 Typical SEM micrographs of the cross section of the Cu/Sn couple (a) and e-Cu3Sn/Sn (b) couple along more than 0.5 mm of the interface, after isothermal holding at 250 °C for 2 h

(a)

(b)

Table 1 Average values of the e-Cu3Sn and g-Cu6Sn5 layers formed at the Cu/Sn interface and of g-Cu6Sn5 layer formed at Cu3Sn/Sn interface after different isothermal holding at 250 °C Time, min

Cu/Sn couple

10

Cu3Sn/Sn couple g-Cu6Sn5 thickness, lm

g-Cu6Sn5 thickness, lm

0.50 ± 0.27

3.31 ± 0.29

3.12 ± 0.13

30

0.91 ± 0.09

3.52 ± 0.74

5.27 ± 0.32

120

2.07 ± 0.65

7.58 ± 0.14

10.46 ± 0.86

480

3.97 ± 0.77

12.75 ± 1.00

13.81 ± 1.11

e-Cu3Sn thickness, lm

mechanisms for the layer growth. Yet, at least for g-phase, the reached average thicknesses are much larger than initial ones. So, in representation we neglected the initial thickness just after contact formation. From Fig. 3, we deduce the growth constants for g-Cu6Sn6 phase layer: (a) (b)

For k1/3 For and

the Cu/Sn couple: k1/2 = 4.87 lm h-1/2 and = 6.08 lm h-1/3, the e-Cu3Sn/Sn couple: k0 1/2 = 5.89 lm h-1/2 k0 1/3 = 7.16 lm h-1/3.

These values lead to a ratio between the experimental growth constants: Strictly speaking, more correctly would be to represent the results in the form: 2 3 ðDX Þ2 ¼ ðDX0 Þ2 þ k1=2 t or ðDX Þ3 ¼ ðDX0 Þ3 þ k1=3 t where DX0 and DX = e are the average thicknesses of a given layer i (i = e or g) just after contact formation (t = 0) and after a reaction time t respectively. k1/2 and k1/3 are growth constants of layer i depending on the operation

k01=2 k1=2

1:21 and

k01=3 k1=3

4 Theoretical analysis In this section we will discuss and compare, from a theoretical point of view, the growth kinetics of g-Cu6Sn5 layer in two configurations:

(a) Cu substrate ×

(b) k

= 4,87

Cu substrate

k

= 1,72

Cu3Sn substrate

k ´ = 7,16

k ´ = 5,89

Thickness, µm

t1/2, hours1/2

Fig. 3 a Thickness (e) of the g-phase and e-phase for the Cu/Sn diffusion couple and thickness of the g-phase for the Cu3Sn/Sn diffusion couple as a function of the square root of time. The straight lines represent the linear fit of the data for e = kt1/2. b Thickness of

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k = 6,08 ×

Thickness, µm

Cu3Sn substrate

1:18

t1/3, hours1/3

the g-phase for the Cu/Sn and Cu3Sn/Sn diffusion couples as a function of the cube root of time. The straight lines represent the linear fit of the data for e = kt1/3

J Mater Sci: Mater Electron Fig. 4 a Schematic presentation of variation of Cu concentration through the solid Cu/liquid Sn and b through the Cu3Sn/liquid system. c Schematic presentation of variation of the Gibbs freeenergy formation of (Sn,Cu) liquid phase, (Cu,Sn) solid phase and g-Cu6Sn5 and e-Cu3Sn compounds at T = 523 K indicating the stable equilibria (solid line) and the metastable liquid/e equilibrium (dash line). References states: stable states at 523 K (pure liquid Sn and pure solid Cu) [15, 20]

a.

(a)

(b)

Cu/e/g/Sn (Cu/Sn couple) and (b) e/g/Sn configuration (incremental couple) and with the following assumptions concerning the growth mechanism of g-Cu6Sn5 layer: 1. 2.

(c)

growth kinetics limited by solid state diffusion– growth in parabolic regime and growth kinetics limited by diffusion in liquid intergranular channels inside the g-Cu6Sn5 layer.

4.1 Growth kinetics limited by solid state diffusion– parabolic regime 4.1.1 Cu/Sn couple: simultaneous growth of two phase layers (e and g) in parabolic regime Let us start with reminding the well known equations for single-phase and two-phase growth of intermetallic compounds continuous layers with narrow concentration ranges between two mutually almost insoluble (at the temperature of reaction) components [16]. Let C be a mole fraction of copper (Cu): Cg ¼ 6=11; Ce ¼ 3=4. So far we assume that the so-called Wagner integrated diffusivities (products DiDCi of average interdiffusion coefficient Di and of concentration range DCi) remain constant during reaction. This corresponds to bulk diffusion through them or to grain boundary diffusion with constant lateral grain size. In such case the growth equations for two phase layers growing simultaneously in diffusion-controlled regime are determined by the flux balance equations at the moving boundaries Cu/e, e/g and g/Sn (see Fig. 4a) and by Fick’s law inside each phase:

dDXg 1 Ce Dg DCg De DCe ¼ Ce Cg Cg DXg dt DXe dDXe 1 Dg DCg 1 Cg De DCe ¼ þ Ce Cg dt DXg 1 Ce DXe

ð1Þ ð2Þ

. If we consider the diffusion-controlled growth of both phases with constant effective diffusivities, one can assume from the very beginning that: pffi pffi DXg ¼ kg t; DXe ¼ ke t; DXe =DXg ¼ ke =kg r ¼ const ð3Þ Substituting it into initial equations, one gets: kg 1 Ce Dg DCg De DCe ke ¼ ; Ce Cg Cg kg 2 ke 2 1 Dg DCg 1 Cg De DCe þ ¼ Ce Cg kg 1 Ce ke

ð4Þ

This set of two algebraic equations for the growth constants ke and kg has simple but clumsy solution. In our case, the Wagner diffusivity of e-phase is close to that of g-phase, De DCe =Dg DCg 1 [17], which leads to ke =kg 1:15, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ke 1:22 2Dg DCg and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð5Þ kg 1:06 2Dg DCg 4.1.2 Incremental e/Sn couple:single g layer growing parabolically between e-phase and Sn At first we formulate the general case when the marginal e-phase has concentration in between the homogeneity range and has higher copper content than needed for equilibrium with g-phase at the e/g phase boundary (see

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Fig. 4b and Appendix). From calculation performed in ‘‘Appendix’’, one can conclude that the smaller is the deviation of Cu concentration in the bulk e-phase from equilibrium eq with g-phase, Ce=g , the faster will be the growth of g-phase. Thus, by assuming that Ceq e=g ¼ Ce ¼ 3=4, we calculate the highest value of growth rate of g-phase growing between e-phase and liquid (see Fig. 4b). If we consider the diffusion-controlled growth of g-phase with constant effective diffusivities, one can assume from the very beginning that: pffi DXg0 ¼ kg0 t ð6Þ By writing the mass balance at e/g and g/liquid Sn inter0 faces (see Fig. 4b), DX 0 g ¼ X 0 g=Sn Xe=g , one gets: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 0 0 0 e 2Dg DCg kg kg=Sn kg=e ¼ Cg Ce Cg pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2:6 2Dg DCg ð7Þ Comparison of Eq. (5) with Eq. (7) shows that the transition from Cu/Sn couple to incremental e-Cu3Sn/Sn couple will make the growth rate of g-phase faster (k0 1/2 [ k1/2), by a factor 2.6/1.06 = 2.45 if Wagner integrated diffusivities are equal in both e and g phases. Note that this case corresponds to the Cu/Sn system for which DeDCe * DgDCg [17]. 4.2 Reaction of Cu with liquid Sn occurs by Cu penetration through the liquid intergranular channels of g-phase 4.2.1 Cu/Sn couple: simultaneous growth of two phase layers (e and g) In some studies, it is shown that the much faster growth kinetics of the g-Cu6Sn5 compared to that of e-Cu3Sn phase, is due to the fact that during the reaction between molten solder and copper, the growth of Cu6Sn5 scallops takes place at the solder/metal interface by rapid liquid state diffusion through nanometric liquid channels between Cu6Sn5 scallops, leading thus to a rapid growth rate of this phase [18, 19]. In order to evaluate the evolution of the average scallop size (see Fig. 5a) we use a simplified version of FDR model developed in Ref. [8]. We assume that all scallops have the same size a = R = H (see Fig. 5b), the liquid is homogeneous in concentration and already saturated by copper for X [ R and a constant concentration gradient exists in the liquid phase (channels) from the top of the liquid/g interface (X = Xl/g) where the copper concentration corresponds to the liquid/g equilibrium concentration (Clg) to the e/g interface at the bottom of the channels (X = Xe/g) where the copper concentration in the liquid

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channels corresponds to the liquid/e metastable equilibrium (Cl/e)–see Figs. 4c and 5c. For the fluxes of Cu atoms (number of atoms per unit area per unit time) through both intermetallic phases (evidently, through e-phase layer and liquid channels) we can use the following expressions [8]: XJðeÞ ¼

~ðeÞ DCeq D e DXe

XJðgÞ ¼ Dmelt Cu

ð8Þ

Cl=e Cl=g Sfree d ¼ 2 Dmelt Cl=e Cl=g Cu total R R S ð9Þ

~ðeÞ DCeq is the integrated where X is an atomic volume, D e diffusion coefficient in e-phase, Dmelt Cu is the diffusion coefficient of Cu in liquid Sn-Cu solution, Cl=e and Cl=g are the equilibrium concentration of Cu at the liquid/e-phase and liquid/g phase interface correspondingly (see Fig. 4c). According to the constraint that the interface between the scallops and Cu is occupied completely by scallops except the thin channels, we have: N pR2 & Stotal = constant, where N is the number of scallops. The free surface (the cross-sectional area of channels at the bottom) for the supply of Cu from the substrate is: Sfree = 2pRN(d/2) = (d/R)Stotal where d is the channel width. Note that the number of liquid channels per unit area of reaction interface (proportional to Sfree/Stotal = d/R) is higher for a scallop-form g-phase compared to a semispherical-form g-phase. The error on d/R value can be estimated to be given by the a/H ratio (about 20–30 %) where a and H are the average values of base radius and height of a g-scallop respectively (a = h = R in the case of a semispherical-form). Then the basic kinetic equations for simultaneous growth of both phases are: ~ðeÞ DCeeq D dXCu=e ¼ dt DXe D ~ðeÞ Dceq dXe=g dDmelt l=e e ¼ Cu Ce Cg C Cl=g þ 2 dt H DXe dXg=Sn dDmelt l=e l=g Cu ¼ Cg 0 C C dt H2 ð10a; b; cÞ ð1 Ce Þ

The growth rate of each layer can be found as the difference in the velocities of two interfaces, for example for e=g

Cu=e

dX dX e e-phase layer: dDX dt ¼ dt dt . Equations for phase thicknesses are then: ðeÞ eq ~ Dce D dDXe 1 1 ¼ þ Ce Cg 1 Ce dt DXe melt dDCu l=e l=g C C ð11aÞ Ce Cg H 2


(a)

(b)

(c)

(d)

Fig. 5 Model system: a Schematic morphology of g-Cu6Sn5 phase formed at the solid Cu/liquid Sn interface. b Simplified schematic morphology of g-Cu6Sn5 phase formed at the solid Cu/liquid Sn interface (a = H = R). c Schematic presentation of variation of Cu

dH ¼ dt

melt 1 1 dDCu l=e l=g þ C C Cg Ce Cg H2 ðeÞ eq ~ Dce D 1 Ce Cg DXe

ð11bÞ

From experiment we know that during soldering e-phase grows slowly. Let us use a steady-state approximation for the e-phase: ðeÞ eq ~ Dce D 1 1 þ Ce Cg 1 Ce DXe melt ~ðeÞ Dceq dDCu D l=e l=g e ð12Þ C C 0 ) DXe Ce Cg H 2 1 Ce dDmelt Cu C l=e C l=g 2 1 Cg H Then, substituting Eq. (12) into Eq. (11b), one obtains: dH 1 dDmelt l=e l=g Cu C C dt Cg 1 Cg H 2 Simple integration gives: H at Cu substrate ¼

H at Cu substrate ¼

3dDmelt Cu Cl=e Cl=g t Cg 1 Cg 3dDmelt Cu

Cg 1 Cg ¼ k1=3 calc t1=3

Cl=e Cl=g

!1=3

!1=3 t1=3 ð13Þ

concentration through the solid Cu/liquid Sn and d through the Cu3Sn/liquid system considering that liquid channels exist inside the g-Cu6Sn5 layer

where k1=3 calc is the calculated growth constant of the g-phase in the case of Cu/Sn couple by Cu penetration through the liquid intergranular channels of g-phase: ! 1=3 3dDmelt l=e l=g Cu C C k1=3 calc ¼ ð130 Þ Cg 1 Cg 4.2.2 Incremental e/Sn couple: single g layer growing between e-phase and Sn by copper penetration through the liquid intergranular channels of g-phase If e-phase becomes marginal (as was realized in our experiment) and, as above, there is no flux inside this phase, then one obtains for incremental couple e-Cu3Sn/liquid Sn saturated with copper (see Fig. 5d). Only one phase is growing (e-phase) in the form of scallops. Once more by making the approximations used in Eq. (9) the basic kinetic equations for growth of e-phase at e/liquid interface become: dX e=g dDmelt l=e Ce Cg ¼ Cu C Cl=g 2 dt H dX g=Sn ð14Þ dDmelt l=e l=g Cu Cg 0 ¼ C C 2 dt H Then, Eq. (14) for H = Xg/Sn – Xe/g lead to: melt dH 1 1 dDCu ¼ þ Cl=e Cl=g 2 dt Cg Ce Cg H

ð15Þ

Simple integration of Eq. (15) gives:

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H

incremental

! 1=3 Ce melt l=e l=g 3dDCu C C ¼ t1=3 Cg Ce Cg 0 ¼ k1=3 t1=3 calc

ð16Þ 0 where k1=3

calc

is the calculated growth constant of the -

phase in the case of incremental e/Sn couple by Cu penetration through the liquid intergranular channels of g-phase: ! 0 1=3 Ce melt l=e l=g 3dDCu C C k1=3 ¼ calc Cg Ce Cg ð160 Þ Thus the ratio between the average values of H = R (but also between the average thicknesses of the g-layer) obtained for both cases can be calculated from Eqs. (13) and (16): 0 k incremental 1=3 H ð1Cg ÞCe 1=3 calc ¼ ¼ 1:19: H atCusubstrate ðCe Cg Þ k1=3 calc ð17Þ

variation with time of the average thickness of the g-Cu6Sn5 phase (eg) is given in Sect. 3 (see Fig. 3b) by: 0 eg ¼ ðk1=3 Þexp t1=3

whereas the calculated variation with time of the average height of scallops (see Fig. 5) is evaluated in Sect. 4.2.2 by: 0 H incremental ¼ R ¼ ðk1=3 Þ3calc t1=3

From Eqs. (20) and (21) one obtains: 0 Þcalc ¼ ðk1=3

R 0 ðk Þ eg 1=3 exp

We can see that the experimental value of the ratio of cubic root dependencies for full Cu/Sn and incremental e/Cu3Sn couples (k0 1/3/k1/3)exp & 1.18 practically coincides with our theoretical prediction: (k0 1/3/k1/3)calc & 1.19, see Eq. (17). This is much better than if one tries to represent the data as square root dependences. Indeed, in this last case we have (k0 1/2/k1/2)calc & 2.45, value much higher than the experimental one (k0 1/2/k1/2)exp & 1.21 (see Fig. 3). Moreover, Eq. (160 ) gives the possibility to estimate the product of copper diffusivity in channel and of channel width: 0 ðk1=3 Þ3calc Cg Ce Cg dDmelt ¼ ð18Þ Cu 3 Ce ðCl=e Cl=g Þ where Cg = 6/11 and Ce = 3/4 are the copper concentrations in g-Cu6Sn5 and e-Cu3Sn respectively. The difference of copper concentrations Cl/e – Cl/g & 8.75 9 10-3 in liquid tin between equilibrium with e-Cu3Sn and with g-Cu6Sn5 phases is obtained from thermodynamic data [15, 19, 20]. With these data, Eq. (18) becomes: ð19Þ 0

In the following, we evaluate ðk1=3 Þcalc from experimental results obtained in Sect. 3.We recall that the experimental

123

ð22Þ

The relation between the average thickness of g layer (eg) and R = H (see Fig. 5b) can be evaluated basing on the concept of the same volume, i.e., pR2e = 2pR3/3, which gives: R 3 ¼ eg 2

ð23Þ

Combination of Eqs. (22) and (23) leads to: 3 0 0 ðk1=3 Þcalc ¼ ðk1=3 Þexp 2 0

0

ð21Þ

ð24Þ

Finally, from Eqs. (19) and (24) one obtains:

5 Comparison of theory and experiment

3 dDmelt Cu 5:7ðk1=3 Þcalc

ð20Þ

3 dDmelt Cu 8:6:ðk1=3 Þexp

ð25Þ

Taking the data on kinetics for e-liquid solder interaction from Sect. 3, Fig. 3b: (k0 1/3)exp & 7.2 lm h-1/3 & 4.7 9 10-7m s-1/3, one can estimate the product dDmelt Cu from -19 3 -1 m s . Eq. (25) as approximately 9 9 10 Knowing the estimate for diffusivity in liquid solder as approximately 10-9 m2 s-1, one can estimate the channel width as about 1 nm, which is not far from the estimate of Jong-ook Suh et al. in [11] for tin–lead solder (about 2.5 nm).

6 Conclusion For the first time the reaction with liquid tin-based solder was studied by direct interaction between specially prepared e-Cu3Sn phase and liquid tin saturated with copper at 250 °C. The reaction product formed at the e-Cu3Sn/liquid Sn-Cu alloy interface consists of the single g-Cu6Sn5 phase with a scallop morphology similar to the classical morphology of the g-Cu6Sn5 phase formed at Cu/liquid Sn interface. Comparison of growth kinetics of the g-Cu6Sn5 phase in Cu/e/g/liquid Sn-Cu and e/g/liquid Sn-Cu systems shows that in the second case (incremental couple) the average thickness of the g layer is about only 20 % greater. This


relatively small difference in the growth kinetics of g-Cu6Sn5 phase between the two couples can be explained if the g-Cu6Sn5 phase growth occurs by liquid state diffusion via the liquid channels between scallops of g-phase and thus supports the FDR model. The average width of liquid channel in the g-Cu6Sn5 phase is estimated to be about 1 nm.

Appendix At first let us formulate the general case when the marginal e-phase has concentration in between the homogeneity range and has lower tin content than needed for equilibrium with g-phase at the phase boundary (see Fig. 5d). In this case a Cu concentration profile is formed in the e-phase: eq Ce=g Ce1 x ; C ðt; xÞ ¼ A þ Berf pffiffiffiffiffiffiffi ; B ¼ k 2 De t 1 þ erf 2pe=gffiffiffiffi D e

A ¼ B þ Ce1 The corresponding flux of copper from e-phase towards g-phase appears within e-phase, partially suppressing the g-phase growth

Ce Cg

ke=g 2

Dg DCg kg=Sn ke=g rffiffiffiffiffi 2 ! eq Ce=g Ce11 ke=g De þ exp ke=g p 4De 1 þ erf pffiffiffiffi

¼

2 De

ðA1Þ Cg

kg=Sn Dg DCg 0 ¼ 2 kg=Sn ke=g

ðA2Þ

De and Dg are the Cu diffusion coefficient in e and g phases respectively and ke/g and kg/Sn are kinetic constants. From Eqs. (A1) and (A2) one obtains: 1 1 2Dg DCg kg=Sn ke=g ¼ þ Cg Ce Cg kg=Sn ke=g rffiffiffiffiffi 2 ! : ðA3Þ eq Ce=g Ce1 ke=g 2 De exp p 4De Ce Cg 1 þ erf kpe=gffiffiffiffi

From the last equation one can conclude that the smaller is the deviation of Cu concentration in the bulk e-phase from equilibrium with g-phase, Ceq e=g , the faster will be the growth of g-phase. Fastest growth corresponds to eq Ce1 ¼ 0. In this case one gets: Ce=g sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C e 2Dg DCg kg kg=Sn ke=g ¼ Cg Ce Cg pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2:6 2Dg DCg : ðA4Þ

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