Morphological Stability of Diffusion Couples Under Electric Current

Transcription

1 Journal of ELECTRONIC MATERIALS, Vol. 39, No. 2, 2 DOI:.7/s Ó 2 TMS Morphologial Staility of Diffusion Couples Under Eletri Current PERRY LEO,2 ANA RASETTI,3. Department of Aerospae Engineering Mehanis, University of Minnesota, Minneapolis, MN 55455, USA phleo@aem.umn.edu rasetti@aem.umn.edu We onsider the growth morphologial staility of an intermediate phase growing in a inary diffusion ouple under eletromigration onditions. The growth rate of the intermediate phase depends primarily on the diretion of the eletromigration urrent. Current flow that drives the diffusing speies enhanes growth of the intermediate phase, while urrent flow in the opposite diretion slows growth. The morphologial staility of the interfaes etween the intermediate phase the terminal phases depends on the urrent diretion, the relative ondutivities of the phases, the thikness of the intermediate phase. We find that, when the intermediate phase has a higher ondutivity than the terminal phases, the urrent diretion that enhanes growth of the intermediate phase an also ause an instaility. Alternatively, when the ondutivity of the intermediate phase is lower than the surrounding phases, the urrent diretion that slows growth an ause an instaility. Instaility also requires that the thikness of the intermediate phase e larger than some ritial value. Key words: Eletromigration, morphologial staility, diffusion INTRODUCTION Eletromigration arises when an eletri urrent is applied to a metal. It is aused y the interation etween the applied eletri field the positive ions, the susequent sattering of these ions the ondution eletrons (wind fore). There is also a Coulom fore ating on the ions in the opposite diretion to the wind fore. The total fore an e expressed as F em ¼jejz E, where e is the eletroni unit harge, E is the applied eletri field, z* is the apparent effetive harge. The total fore leads diretly to mass transport of ions through the metal. Eletromigration is elieved to e the main ause of failure of integrated iruits miroeletroni devies. For example, in metalli interonnets (made of Al or Cu) the order of magnitude of eletri urrent densities 6 A/m 2 the range of temperature at whih the devie operates Cor higher drives eletromigration flux large enough (Reeived Marh 26, 2; aepted August 3, 2; pulished online Septemer 4, 2) to ause void nuleation growth that ultimately leads to an opening in the interonnet. 2 Eletromigration is also important in solder joints, even though the average urrent densities arried are orders of magnitude lower than in interonnets. This is eause the urrent densities needed to drive eletromigration in solder an e muh lower than in interonnets, owing to the higher resistivity, effetive harge, diffusivity of solder ompared with interonnets. 2 Eletromigration in solder joints leads to exess growth of intermetalli ompounds that an ause miroraks to initiate. 3 In this paper, we onsider the eletromigrationdriven growth morphologial staility of an intermediate phase growing from two metals in a inary diffusion ouple. This work is motivated y the experimental work of Chen Chen, 4 6 who desrie measure the growth rate of intermediate phases in Sn/Ag, Sn/Ni, other systems at different temperatures as a funtion of applied urrent. Chen Chen notied that the diretion of applied urrent an either enhane or retard the growth of the intermediate phase. Their pitures also show that one of the interfaes etween the 2687

2 2688 Leo Rasetti metal the intermetalli is slightly orrugated, reminisent of morphologial instailities at planar two-phase interfaes. Motivated y these pitures, we onsider the planar growth of an intermediate phase etween two metals, as a funtion of alloy phase diagram, system size, the ondutivities of the phases, the diretion magnitude of the applied urrent. We then onsider small perturations of the planar interfaes etween the intermediate phase the surrounding metals, analyze how the applied urrent will affet their morphologial staility. Morphologial staility in eletromigrating systems has een studied theoretially y Deuzzi, 7 Klinger Levin, 8 Klinger et al. 9 These studies onsider interfaial mass flux driven y gradients of eletri potential, stress, urvature along the interfae. Staility onditions are found for different system geometries system parameters suh as elasti moduli, urrent density, atomi moility. Maroudas oworkers have onsidered similar prolems where eletri urrent an stailize a stress-driven (Asaro Tiller Grinfeld) instaility., In ontrast here we onsider ulk mass transport rather than surfae diffusion. Orhard Greer have onsidered the role of ulk eletromigration in the growth of a planar interfae in the presene of interfaial reation arriers. 2 Their analysis follows Chen Chen s analyses, as does our unpertured prolem. In the next setion, we set up the prolem of intermediate phase growth under an applied eletri urrent, when the interfaes separating the phases are planar. We then onsider small perturations of the interfaes perform a linear staility analysis. We present disuss the results ompare them with the experimental results of Chen Chen. 5,6 ANALYSIS We study the growth of an intermediate phase etween two metals a, when an eletri urrent is applied aross the system (Fig. ). The system is inary, with all phases onsisting of the same onstituents in different ratios. Speifially we onsider an isothermal system at some temperature T suh that the a phase is A rih, the phase is B rih, the phase is an A B ompound. At T the tie-line ompositions of speies B are a a a for a/ equilirium; for / equilirium. We take the x-axis (with unit vetor ^e x ) along the diffusion ouple, the y- z-axes perpendiular to the ouple. The applied eletri field or urrent is along the ±x-diretion. We onsider diffusion of speies B only. Also we assume that diffusion ours only in the intermediate phase that the ompositions in the a phases are uniform. Hene the omposition gradient drives the flow of speies B from to a as indiated in Fig.. The mass flux of B in the phase may e written as 4 J ¼ Dr MF: () Here, the first term is mass flux in owing to ulk diffusion, with diffusivity D onentration. The seond term aounts for the total eletromigration flux, with moility M (related to the diffusivity) eletromigration driving fore F ¼jejz E ; (2) where e is the unit harge of an eletron, z* is the apparent effetive harge of the diffusing atoms, E ¼ r/ is the eletri field as found from the eletri potential /. The sign of the eletromigration term in Eq. is ruial in determining the urrent ΔV β (B rih) γ flow diretion of speies B from omposition gradient α (B poor) L β x () x () L α Fig.. Shemati of the diffusion ouple under eletromigration onditions. In the asene of eletromigration, flow of solute B in the intermetalli phase is from the solute-rih phase towards the solute-poor a phase.

3 Morphologial Staility of Diffusion Couples Under Eletri Current 2689 diretion that enhanes the intermediate phase growth. We assume that the motion of the interfaes is slow ompared with the diffusion speed in the system, so the flux J satisfies mass alane in the the quasistati limit, rj ¼. Finally, we take oundary onditions for the omposition onsistent with the equilirium phase diagram. We also onsider a orretion for apillarity onsistent with the Gis Thomsen equation (Eq. 2), though our primary fous is on whether there are any irumstanes under whih urrent an destailize an interfae. The eletri urrent I ¼ re where r is ondutivity. The eletri potentials / in eah phase must satisfy Laplae s equation together with oundary onditions expressing ontinuity of oth the potential urrent: at the / interfae / ¼ / r r/ ^n ¼ r r/ ^n (3) / ¼ / a r r/ ^n ¼ r a r/ a ^n (4) at the /a interfae, where ^n ^n are the unit normals to the / /a interfaes. Finally, eause the eletri potential satisfies Laplae s equation, we must take a finite system size, so we speify far-field onditions / ð L Þ¼V / a ðl a Þ¼V þ (5) : Beause we onsider diffusion in only, we omit susripts distinguishing the phase in terms related solely to diffusion, though we retain them on quantities assoiated with the eletri field. Also we use a supersript () to denote quantities assoiated with the unpertured prolem, a supersript () for quantities assoiated with the pertured prolem. Unpertured Prolem We first onsider planar interfaes, with x (t) x (t) giving the positions at time t of the / /a interfaes, respetively. We hoose as the origin the initial position of the unpertured / interfae, i.e., x () =. For the unpertured prolem, the / i are linear in x so the field an e easily omputed. As we only onsider mass flow in the phase, we alulate d/ /dx = a, with d a d DV a ¼ (6) d a L þ d L a þ d a x ð d Þþd x ðd a Þ DV = V + V, d a = r a /r, d = r /r. We take the omposition in the a phases to e onstant, at a a, respetively. We find the omposition field in y setting rj ¼ with J given y Eq.. For the unpertured prolem, depends on x only, so this redues to where d 2 ðþ dx 2 d Q ¼ ; (7) dx Q ¼ jejz Ma : (8) D as E ¼ r/ ¼ a ^e x r 2 / ¼. The ompositions at the planar / interfae the /a interfae are taken as their equilirium phase diagram values, i.e., ðx Þ¼ ðx Þ¼ a. In the analysis that follows we use Q as our ontrol parameter, that is, the sign magnitude of Q determine the diretion magnitude of the eletri urrent. In partiular note that for the unpertured prolem the flux of solute in the x-diretion is J ¼ Dð d QÞ; that is, when dx Q >, solute flux from eletromigration is from to a, as is flux from the onentration gradient (Fig. ). By solving Eq. 7 for applying the oundary onditions, we find a ðþ ðxþ ¼ e Qx e Qx e Qx þ a e Qx eqx : (9) e Qx e Qx We find the veloities of the / /a interfaes through the flux alanes dx dt ¼ J () a dx a a dt ¼ J ; () where for the unpertured prolem J ¼ J ^e x as given y Eq., we have used J a ¼ J ¼. After some algera, we find dx dt ¼ D Q h i e Qx eqx a eqx ð e Qx Þ (2) dx dt ¼ D Q h i a a a ð e Qx e Qx Þ eqx a eqx : (3) These flux alanes are onsistent with the formulation of Shatynski et al. for the growth of a single intermetalli produt phase formed from two phases with limited terminal soluility. 3 In the Results: Unpertured Prolem setion, we present results for how the motion of the unpertured interfaes depends on system parameters espeially the

4 269 Leo Rasetti magnitude diretion of the applied urrent through the parameter Q. Pertured Prolem We onsider perturations of the planar interfaes of the form x ðþ ðy; tþ ¼xðÞ ðtþþ ðtþ os x y x ðþ ðy; tþ ¼xðÞ ðtþþ ðtþ os x y, where x (t) x (t) are the unpertured (planar) solutions, (t) (t) are the perturation amplitudes x x are the perturation frequenies. Our goal is to determine whether perturations grow or deay y alulating the time rate of hange of under different eletromigration onditions. We assume the magnitudes of are small ompared with some harateristi length sale of the system (e.g., the initial thikness of the phase) so that terms of order 2 higher may e negleted. Aordingly, we seek solutions of the form /ðx; y; tþ ¼/ ðþ ðx; tþþ/ ðþ ðx; y; tþ (4) ðx; y; tþ ¼ ðþ ðx; tþþ ðþ ðx; y; tþ; (5) where the pertured solutions / () () have terms proportional to os(x y) os(x y). Beause we perform a linear staility analysis, we aim to derive sets of independent equations orresponding to the unpertured solution the two perturations. As usual in this type of analysis, the unpertured solution ontriutes to oth the solutions eause we evaluate at the pertured interfaes, though the solutions do not interat with eah other. Consider first the eletri potentials. The pertured potentials / () for all three phases must satisfy Laplae s equation, so it is straightforward to show that the potential in the phase is / ¼ / ðþ ðxþþ a C ex x þ C 2 e x x osðx yþ þ a C 3 e x x þ C (6) 4 ex x osðx yþ; where we have suppressed the time variale we have expliitly shown the linear dependene on the unpertured field strength a. The potentials for the a phases have the same form ut with different onstants C C 4. The onstants are found from the oundary far-field onditions given y Eqs The oundary onditions are evaluated at the pertured interfaes with normals ^n ¼ ^e x osðx yþ^e y ^n ¼ ^e x osðx yþ^e y to first order in. After evaluating the / () at either x = x + os(x y) or x ¼ x þ osðx yþ exping to first order in, we find six equations for the terms (two at eah interfae plus two farfield) six for the terms. The solution of these equations give the 2 onstants C i Ci 4, i = a,,. Beause we assume quasistati diffusion, the omposition field assoiated with the pertured interfae is found y ensuring that the eletromigration-modified diffusion flux () is divergene free. As in the unpertured ase we only onsider the omposition in the phase; however, we must onsider the perturations from oth the interfaes at x x. That is, the pertured omposition () is split up as ðþ ¼ ðþ þ ðþ. By using Eqs. 4 9, we find that, to order, the pertured flux assoiated with the perturation at the /a interfae is J ðþ ¼ D r ðþ Qr ðþ Q ðþ r/ ðþ ; (7) a the omposition ðþ a ¼ x Q 2 e Qx e Qx ðc e Qx 3 e x x C 4 ex x Þ; (8) with similar equations for J ðþ ðþ. The solution to Eq. 8 is p ffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffi ðþ ¼ A e ðqþ Q 2 þ4x 2 Þx=2 þ B e ðq Q 2 þ4x 2 Þx=2 Q os x y: a e Qx e Qx C 3 eðq x Þx þ C 4 eðqþx Þx A similar result holds for ðþ : p ffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffi ðþ ¼ A e ðqþ Q 2 þ4x 2 Þx=2 þ B e ðq Q 2 þ4x 2 os x y: Q a e Qx e Qx Þx=2 ð9þ C eðqþx Þx þ C 2 eðq x Þx ð2þ The four new onstants A, A, B, B are found from the oundary onditions at the / /a interfaes. At the pertured interfaes, we modify the equilirium ompositions a y using the Gis Thomsen oundary ondition ¼ ð þ CjÞ; (2) where is the equilirium omposition at a planar interfae (i.e., the phase diagram omposition used in the unpertured prolem), C is a apillary onstant proportional to the surfae energy of the interfae, j is the mean urvature of the interfae. 4 For interfaes that are slightly pertured from planar, j r 2 x, so for the interfaes x ¼ x þ osðx yþ x ¼ x þ osðx yþ, the oundary onditions may e written ðþ ðx ¼ x þ osðx yþ; yþþ ðþ ðx ; yþ ¼ þ C x 2 osðx yþ

5 Morphologial Staility of Diffusion Couples Under Eletri Current 269 ðþ ðx ¼ x þ osðx yþ; yþþ ðþ ðx ; yþ ¼ a þ C x 2 osðx yþ ; where for onveniene we use the same apillary onstant C for oth interfaes. By exping () to first order in mathing orders, one an find a system of equations to solve for the onstants A,A B,B. Finally, we ompute the veloity of the pertured / /a interfaes through the flux alanes dx dt ¼ J ^n at x ¼ x ðtþþ ðtþ osðx yþ a a a dx dt ¼ J ^n at x ¼ x ðtþþ ðtþ osðx yþ; where J is evaluated y using Eqs., 9, 9, 2. After evaluating the terms exping to first order in, we find the speed of the unpertured interfaes as in Eqs. 2 3, the growth rate of the perturations d dt ¼ dln dt ¼ h k A e kþ x þ k þ B e k x þcqx C eðqþx Þx C 2 eðq x Þx þ Qx C ex x C i 2 e x x (22) d dt ¼ dln dt h ¼ a a a k A e kþ x k þ B e k x þcqx C 3 eðq x Þx C 4 eðqþx Þx þ a Qx C 3 e x x C i 4 ex x ; (23) where in addition to the onstants defined aove, we have C¼ a e Qx e Qx qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ð;þ ¼ Q Q 2 þ 4x 2 ð;þ: RESULTS We now onsider how eletri urrent affets oth the motion of the planar interfaes etween the phases as well as any perturations of those interfaes. As mentioned in the Introdution, results for the unpertured prolem have also een presented y Chen Chen 4 6 Orhard Greer. 2,5 Unpertured Prolem We integrate Eqs. 2 3 to ompute the growth of the unpertured phase. We ompute the thikness t = x x of the phase for different parameters. These parameters are normalized using a length sale L, hosen elow to e the initial thikness of the phase, a time sale T = L 2 /D. Compositions are saled y the differene a a. The normalization is straightforward so in the following we do not distinguish etween dimensionless dimensional quantities. We use the (now dimensionless) parameter Q as the ontrol parameter. From Eqs. 6 8 we see that Q is proportional to the voltage drop so an e varied independently for any hoie of system parameters. The ondutivity ratios d a d affet the unpertured solution only through their role in Q. The dependene of Q on x (t) x (t) is negligile when L a L are large ompared with L so will not e onsidered. Figure 2 shows t for different values of Q. Other parameters are d a = d =, L a = L = 5, ¼ :42; a ¼ :58. We oserve that, when Q >, the growth rate of is enhaned. On the other h, when Q<, the growth rate of is retarded. These oservations hold for all values of the system parameters, are onsistent with the oservation that, when Q >, eletromigration drives flux in the same diretion as the onentration gradient, i.e., from to a. The result oupling the diretion of the urrent to an enhaned or diminished growth rate of the intermediate phase is independent of the other system parameters. However, these parameters affet the growth of the intermediate phase in the asene of urrent, also alter the magnitude of thikness of time t Q.5 Q Q.5 Fig. 2. Thikness of the intermediate phase versus time for d a = d = different values of Q. Other parameters are L a = L = 5, ¼ :58; a ¼ :42. The initial thikness of the phase is.

6 2692 Leo Rasetti the eletromigration effet. Phase diagram ompositions influene the growth of the phase as seen in Eqs Note that in Fig. 2 we hose the omposition of the intermediate phase to e entered at.5. Moving the enter of the intermediate phase to either higher or lower ompositions inreases the eletromigration effet. Also, dereasing the misiility gap of the phase region lowers the omposition gradient aross so slows its growth, while inreasing that gap enhanes its growth (figures not shown). Pertured Prolem The morphologial staility of the / /a interfaes is determined y the signs of d ln =dt d ln =dt in Eqs , respetively. If these growth rates are positive the interfae is unstale with respet to the perturations, while if they are negative the interfae is stale. As we know that surfae energy will stailize the interfaes, we fous primarily on the ase with zero surfae energy, C =, to see if there are any onditions under whih eletri urrent will destailize the interfae. Consider first the aseline ase where the system is eletrially homogeneous, d a = d =. In this ase the perturation has no effet on the eletri field, whih remains uniform over the diffusion ouple. The urrent an still affet the morphologial staility through its role in the unpertured prolem note for example that Q appears in the left-h side of Eq. 8 for () even though the right-h side vanishes when d a = d =. However, this effet is always stailizing, that is, oth dln =dt d ln =dt are less than zero for all values of Q. We also find that the diffusion prolem in the asene of eletri urrent (Q = ) leads to morphologially stale interfaes. The situation is more interesting when the ondutivities of the phases differ. We fous first on the / (x ) interfae. Consistent with the linear analysis, we find that the ondutivity ratio d a does not affet the ehavior of the x interfae so is set to. (Similarly, the ratio d does not affet the ehavior of the /a interfae.) Figure 3 shows dln =dt plotted against Q when d is The thikness t of the intermediate phase is 2. When d =.25, so the ondutivity of the phase is less than that of the phase, d ln =dt is positive for large enough positive Q, is negative for negative Q; that is, when Q<, the x interfae is stailized y urrent, while for large enough positive Q, the interfae is destailized y urrent. In ontrast, when d = 4 (also shown in Fig. 3), urrent destailizes the interfae when Q is negative (less than aout.5), while urrent stailizes the interfae when Q is positive. The results are similar at the /a (x ) interfae. Figure 4 shows d ln =dt plotted against Q when d = d a = Again, the sign of dln dt Q Fig. 3. Growth rate d ln =dt of the perturation at the / interfae as a funtion of Q for d =.25 d = 4. Other parameters are C =, d a =, t = 2, x =, L a = L = 5, ¼ :58; a ¼ :42. dln dt Q Fig. 4. Growth rate d ln =dt of the perturation at the /a interfae as a funtion of Q for d a =.25 d a = 4. Other parameters are C =, d =, t = 2, x =, L a = L = 5, ¼ :58; a ¼ :42. dln =dt depends on oth Q d a. When d a =.25, dln =dt is positive for large enough positive Q. When d a =4, dln =dt is positive for large enough negative Q. We onlude that eletri urrent an destailize an interfae for ertain values of Q the ondutivity ratios d a d. The details depend on these parameters as well as the thikness t of the intermediate phase. Figure 5 shows a staility diagram for the / interfae as a funtion of Q d with t = 2, while Fig. 6 shows a staility diagram for the same interfae as a funtion of Q t with d =.25 (so Q > is needed for instaility). The orresponding figures for the /a interfae are very similar. We see that, as the eletrial ontrast inreases (d moves away from ), the magnitude of the urrent (in the appropriate diretion) needed to destailize the interfae dereases. Also, as t inreases, the magnitude of urrent required to drive an instaility dereases. Alternatively, for a given urrent, there is some ritial thikness elow whih the interfae is stale. This is espeially relevant when d >, as this ase requires negative Q to drive the instaility. Reall from the

7 Morphologial Staility of Diffusion Couples Under Eletri Current 2693 Q Q Stale Unstale unpertured prolem that negative Q slows the aseline growth of the intermediate phase. Hene trying to drive an instaility y inreasing the magnitude of Q < ould e ounterprodutive, as it would also limit the growth of t. The remaining parameters the values of the phase diagram ompositions, the instaility wavelengths, the apillary onstant C do not hange the asi requirements for instaility, though they do affet the staility oundaries. Phase diagram parameters an e redued to the average omposition ð þ a Þ=2 width a of the intermediate. Reall that gives the omposition of the diffusing speies B. We find that, as the phase eomes B rih ( so the average omposition inreases), it eomes somewhat easier for urrent to destailize the x interfae, though there is little effet on the x interfae. Similarly, onditions for instaility are at est weakly dependent on the width a Stale Unstale Unstale Fig. 5. Staility diagram showing the staility range of the / interfae as a funtion of the parameters Q d. Other parameters are C =, d a =, t = 2, x =, L a = L = 5, ¼ :58; a ¼ : Fig. 6. Staility diagram showing the staility range of the / interfae as a funtion of the parameters Q t. Other parameters are C =, d a =, d =.25, x =, L a = L = 5, ¼ :58; a ¼ :42. of the phase region. The roles of the perturation frequeny surfae tension are straightforward. In the asene of surfae tension, the magnitude of d ln =dt inreases linearly with x at oth the / /a interfaes. In ontrast, the surfae energy ontriution sales as x 3. Hene, as expeted, surfae energy stailizes high-frequeny perturations would give rise to a fastest-growing wavelength when urrent is destailizing. DISCUSSION It is well known that eletri urrent an either inrease or derease solute diffusion through the eletromigration term in the diffusion flux (). This in turn an enhane or retard the growth of intermediate phases in inary diffusion ouples. 4 Eletromigration enhanes intermediate phase growth when urrent drives solute flux from the high-solute phase towards the low-solute phase (Q > in our notation), slows growth in the alternative ase. Current an in some ases also drive a morphologial instaility of the interfaes etween the intermediate the terminal phases. As shown in the results, the onditions for instaility depend on the sign of Q the ondutivity ratios d a d. The reason is as follows. It is straightforward to show that the most signifiant ontriutor to the pertured flux in Eq. 7 is the term assoiated with the pertured eletri field r/ ðþ. At the / interfae this term is proportional to Qðd Þ os x y, while at the /a interfae it is proportional to Qðd a Þ os x y. Consider first the /a interfae. If Q > d a >, solute flux is redued at the peaks of the perturation (os x y ¼ ) enhaned at the valleys. Beause this interfae moves via solute inorporation (to the right as in Fig. ), this means the valleys grow relative to the peaks the interfae is stailized. Alternatively, if Q > d a <, solute flux is enhaned at the peaks redued at the valleys so the interfae is destailized. A similar situation arises at the / interfae, though in this ase the interfae moves to the left (towards ) via solute rejetion. Hene, if Q > d >, solute flux is enhaned at peaks (os x y ¼ ) redued at the valleys. This auses the peaks to move left faster than the valleys, whih redues the perturation amplitude so stailizes the interfae. We an qualitatively ompare our results with the experimental oservations of Chen Chen. 4 6 Chen Chen ran a series of experiments on the role of urrent in the growth of intermetallis in several different inary diffusion ouples, inluding Sn/Ag Sn/Ni. In oth systems, Sn is the primary diffusing speies, so Sn would e the phase Ag (or Ni) the a phase in our notation. These ouples were allowed to evolve at moderate temperatures in the asene of urrent, suh that an intermetalli phase (Ag 3 Sn or Ni 3 Sn 4 ) grew at the two-phase interfae over the ourse of several

8 2694 Leo Rasetti hundred hours. Eah ouple was then retested under urrent, oth from Sn to Ag from Ag to Sn, the measured thiknesses of the intermetalli were ompared with the thikness in the asene of urrent. To ompare their results with theory, Chen Chen defined a parameter U C=C ¼ DF RT z E; (24) f where D is the diffusion onstant of the diffusing speies, F is Faraday s onstant, R is the gas onstant, T is temperature, z* f are the effetive harge orrelation oeffiient of the diffusing speies. Also E ¼ Iq A is the eletri field in the intermediate layer, with I eing the urrent, q the resistivity of that layer, A its ross-setional area. Sine E ¼ r/ ¼ a for the unpertured prolem, we see from Eq. 8 that our Q ¼ U C=C D ; so in partiular has the same sign as U C/C. As here, Chen Chen showed that growth of the phase is enhaned when U C/C > retarded when U C/C <. This agrees with their experimental oservations on Sn/Ag Sn/Ni diffusion ouples. For example, in the Sn/Ag system, 6 Chen Chen find that, for a 5 A/mm 2 urrent density from Sn to Ag at a temperature of 4 C, the thikness of the intermetalli Ag 3 Sn phase is aout 2% higher than in the asene of urrent. When the urrent diretion is reversed, the thikness of the intermetalli is aout 2% less than in the asene of urrent. We ompare our unpertured results to Chen Chen s results. Chen Chen s mirographs onsistently show enhaned growth of intermetalli with the lael that the flow diretion of eletrons is from Sn to Ag (Ni); in the text they refer to the same situation as eletron urrent flow from Sn to Ag (Ni). Based on this oservation the aove disussion of the signs of U C/C Q, we assoiate our Q > ase with eletron urrent flow from Sn to Ag (Ni), i.e., urrent flow to the right in Fig.. For the Sn/Ag system, Chen Chen report phase diagram values ¼ :237 a ¼ :25. Also, at 2 C, the diffusivity of Sn in is D =.734 lm 2 /h, the resistivity of is q =.82 X lm. From their experiments they infer a value of U C/C =.256 lm/h a orresponding z* = 2 (the sign of z* is inluded in our disussion of the sign of Q the diretion of urrent). This gives the dimensional Q = 34.5/m. If we take a length sale L ~ ¼ lm, then the dimensionless Q ~ ¼ :345. Also the time sale T ~ ¼ L ~ 2 =D ¼ :36 h. These values are input into the differential equations (2) (3) solved to find the thikness of the interfae as a funtion of time. As expeted, we find exellent agreement with Chen Chen s results, with only small quantitative differenes owing to the fat that Chen Chen only onsider motion of the Ag 3 Sn/Ag (/a) interfae. It is more hallenging to ompare our results on the morphologial staility of the Sn/Ag 3 Sn (/) Ag 3 Sn/Ag (/a) interfaes with Chen Chen s pulished mirographs. Using the resistivity of as reported aove, values of q =. X lm for tin ().47 X lm for silver (a), 6 we alulate d a = 2.4 d =.7. We also take t =, roughly onsistent with Chen Chen s mirographs. We do not expliitly onsider surfae energy, though we remark that for Q =.35 a dimensional value of the apillary length of C = 7 m (roughly a surfae energy of J/m 2 ) effetively suppresses the instaility for all ut very long-wavelength instailities. For the parameters aove, we find that eletromigration an potentially drive instaility of oth the / /a interfaes for large enough negative Q. Also the eletromigration effet is stronger at the /a interfae eause the ontrast in ondutivities is larger aross that interfae. In ontrast, in Chen Chen s mirographs of the Sn/Ag system, one oserves small perturations of the Sn/Ag 3 Sn (/) interfaes for oth the ases Q > Q<. Also the Ag 3 Sn/Ag (/a) interfae is planar independent of Q. Similar oservations an e made in other ases, for example, in the Sn/Ni system 6,5 (though the roughness of the Sn/Ni 3 Sn 4 interfae in that ase is muh more rom than in the Sn/Ag ase). Also we do not find any onditions in our analysis under whih the same interfae (i.e., the Ag 3 Sn/Sn interfae) is unstale for oth Q > Q<. We onlude that the roughness of the interfaes in Chen Chen s mirographs is not the result of a morphologial instaility of the type onsidered here. CONCLUSIONS The role of eletromigration in the morphologial staility of an interfae in a inary diffusion ouple is determined y the diretion of urrent (sign of Q) the ratio of ondutivities on either side of the interfae. Eletromigration an drive a morphologial instaility during the growth of an intermediate phase in a inary diffusion ouple. Instaility of an interfae requires that the sign of Q(d ) e negative, where d is the ratio of the eletrial ondutivity of the terminal phase to that of the intermediate phase. Instaility also requires that the thikness of the intermediate phase e larger than some ritial value that depends on oth the relative ondutivities the magnitude of the urrent. We expet that suh onditions may e possile in pratie. However, in many ases the ondutivity of the growing intermediate phase is less than that of the surrounding pure metals. In this ase d > instaility requires Q<, whih slows the overall growth of the intermediate phase. Hene, it may e diffiult to grow the intermediate phase thik enough to oserve instaility experimentally in suh systems.

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