an analytical method to predict electromigration-induced finger-shaped void growth in snagcu solder...

Q1

20

30

Q2

40

Available online at www.sciencedirect.com

SMM 10378 No. of Pages 4

30 September 2014

ScienceDirectScripta Materialia xxx (2014) xxx–xxx

www.elsevier.com/locate/scriptamat

An analytical method to predict electromigration-induced finger-shapedvoid growth in SnAgCu solder interconnect

Yao Yao,a,⇑ Yuexing Wang,a Leon M. Keerb and Morris E. Finec

aSchool of Mechanics and Civil Engineering, Northwestern Polytechnical University, Xi’an 710072, People’s Republic of ChinabDepartment of Mechanical Engineering, Northwestern University, 2145 Sheridan Rd., Evanston, IL 60208, USA

cDepartment of Materials Science and Engineering, Northwestern University, 2220 Campus Drive, Evanston, IL 60208, USA

Received 1 August 2014; accepted 31 August 2014

An analytical solution to predict electromigration-induced finger-shaped void growth in SnAgCu solder interconnect is developed based on massdiffusion theory. A quantitative nonlinear relation between the void propagation velocity and the shape evolution parameter is obtained. It is foundthat a circular void will grow at the lowest velocity, but as it collapses to a finger-shaped void it will grow at a faster velocity that is inversely pro-portional to the width. The void growth velocity predicted is consistent with the experimental observation.� 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Electromigration; High current density; Solder; Void; Analytical solution

50

60

70

Due to the reduction in size of electronic devices andsolder interconnects, more electromigration-induced failurein their solder joints has been observed by researchers. Oneof the representative failure mechanisms is a circular voidcollapsing into a finger-shaped void [1], which differs fromthe traditional metallic lines. The mechanism of electromi-gration-driven void growth has attracted great researchinterest with regard to the fundamental understanding ofthe microstructural evolution and the failure mechanismsin the solder joints. When the electric current increases toa certain magnitude, voids will nucleate around the currentcrowding zone, then migrate, bifurcate and collapse to afinger-like void [2]. Figure 1 shows a void in the currentcrowding zone as it collapses to a finger-shaped (pancake-shaped) void, resulting in the failure of the solder bump.

To predict the electromigration-driven void growth in thesolder joints, one of the key challenges is to calculate the voidwidth and the propagation velocity. An analytical solutionfor void propagation after it grows to a narrow slit in the alu-minum lines was obtained by Suo et al. [3]. A mass diffusionmodel was developed to predict void width and velocityafter the void has grown to the finger shape. However,the model ignored the atom flux at the tip of the voidand assumed that the void would move at the same speedas the circular one. The boundary conditions in a solderinterconnect are more complicated when compared withaluminum lines. The tangent at the void tip is not verticalto the horizontal line (as shown in Fig. 1) because of grain

http://dx.doi.org/10.1016/j.scriptamat.2014.08.0281359-6462/� 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights

⇑Corresponding author. Tel.: +86 29 88495935; e-mail: [email protected]

Please cite this article in press as: Y. Yao et al., Scripta Mater. (2014), ht

boundary or interface diffusion. Cho et al. [4] derived anonlinear “shape function” which deduced a relationshipfor the migration speed as a function of void size. However,they did not discuss the case when the circular voidcollapses to the finger-like void. Recently, Yao et al. [5]proposed a kinetic mass diffusion model to predict the fin-ger-like void width and propagation speed. The model isobtained by the bulk diffusion at the fracture zone in frontof the void tip. However, as the fracture zone effect cannotbe quantized, the method needs further improvement.

In this letter, a new mass diffusion model is developedthat takes into account the change of velocity during thevoid evolution. A nonlinear ordinary differential equationwith two important parameters is set up to describe thevoid shape evolution characters: one represents the voidbifurcation and the other determines the void velocitychanges under high current density.

It is noted that a rounded void can bifurcate when asmall shape perturbation occurs resulting from the chemi-cal potential around the void surface changes [6]. In this let-ter, the mechanism of a rounded void collapsing to thefinger-like void is studied. Figure 2 illustrates the simplifiedmodel. A semi-infinite finger-shape-void extends in an infi-nite (x, y) plane, subjected to an electric field E1 parallel tothe void. The coordinate system is assumed to movethrough the interconnect with the same velocity as the void.Surface diffusion is assumed to be the only process operat-ing in the void evolution. Denoting the chemical potentialper atom on the surface by l, it can be shown that:

l ¼ l0 � Xcsj ð1Þ

reserved.

tp://dx.doi.org/10.1016/j.scriptamat.2014.08.028

http://dx.doi.org/10.1016/j.scriptamat.2014.08.028

mailto:[email protected]

mailto:[email protected]



80

90

100

110

120

130

140

150

160

170

Figure 1. SEM micrographs of the void collapsing to a finger-shapedvoid in the 95.5Sn–4Ag–0.5Cu solder [1].

2 Y. Yao et al. / Scripta Materialia xxx (2014) xxx–xxx


30 September 2014

where l0 is the reference value of the potential, X is theatom volume, cs is the surface energy and j is the local cur-vature of the surface. The number of the atoms passing perlength of the void surface during unit time, namely theatomic flux, is given by the Einstein-type relation [7] (herewe just consider electromigration and capillary forces;other forces will be discussed later):

J ¼ � Dsds

XKTeZ�s Et þ

dldl

� �

¼ � Dsds

XKTeZ�s Et � Xcs

djdl

� �ð2Þ

where Ds is the surface diffusivity, ds is the thickness of thesurface layer taking part in the diffusion process, e is theelectron charge, Z�s is the effective valence, K is the Boltz-mann constant, T is the absolute temperature, E1 is theelectrical applied field and Et is the component of the elec-tric field tangential to the void surface. We define the anglebetween the surface tangent and the Y-axis, denoted by h(Fig. 2), thus [6]:

E ¼ E1 sin h ð3ÞAfter the void collapses to a finger shape, the void will

extend steadily with an invariant shape and every point willmove at the same velocity, denoted as V. Mass conserva-tion requires:

dJds¼ Vn

Xð4Þ

where Vn is the velocity of the void surface in the normaldirection. Defining Vn = Vsinh and dY = sin hds, Eq. (4)becomes:

dJdY¼ V

Xð5Þ

Integration of Eq. (5) gives:

J ¼ V

XY þ C ð6Þ

where C is a constant. Considering the boundary condi-tions, in the flat wake, where Y = 0 and dj

dl ¼ 0, only theelectronic wind force exists, thus:

180

Figure 2. A semi-infinite finger-shape-void extends in an infinite (x, y)plane.


C ¼ � Dsds

XKTZ�s eEt ð7Þ

Denote F = sin h, dYdl ¼ cos h and j ¼ � dh

dl ¼ � dFdY . The

curvature gradient can be obtained from the chain rule:

djdl¼ dj

dl

� �dYdl

� �¼ � d2F

dY 2cos h ¼ � 1� F 2

� �1=2 d2F

dY 2ð8Þ

From Eq. (1), a dimensionless number can be defined as[5]:

g ¼ Z�s eEtu2

csXð9Þ

Here, g represents the competition between the electro-migration effect term (Z�s eEt) and the surface energy term

(csu2

X ). When g is small, the surface energy effect dominates,and the void will remain rounded. When g becomes large,the electromigration effect will prevail over the surfaceenergy and the void will bifurcate and collapse to a fingershape. Yang et al. [6] have shown that when g drops from10.65 to 5.74, the void evaluates from the circular shape tothe finger shape.

Considering the void speed as it extends steadily, and ifthe void does not bifurcate and keeps its circular shape withradius u (here u also represents the width of the void whenit collapses to a void), the translation velocity can be deter-mined by using j = 1/u, and dj

dl ¼ 0 [8]:

V0 ¼Dsds

uKTZ�s eE1 ð10Þ

When the void shape evolves and collapses to a finger-like void, the void propagation velocity changes simulta-neously. The parameter n is introduced to reflect the shapeevolution effect on the void velocity. Thus the finger-likevoid propagation velocity can be defined as:

V ¼ nV0 ð11ÞWhen n > 1, Eq. (11) shows that the void moves faster

after collapsing to a finger-shaped void, whereas when0 < n < 1, it shows that the void moves more slowly. Wesubstitute Eq. (1) and Eqs. (3)–(11) into Eq. (2), and forsymmetry only half of the void shape is taken into consid-eration. A non-linear second-order ordinary differentialequation with two parameters is derived:

1

gð1� F 2Þ2 d2F

dy2þ F þ ny ¼ 1 ð12Þ

where y = Y/u. In the flat wake of the void, the initial con-dition can be defined by:

when y ¼ 0; F ¼ 1; dF =dy ¼ 0.For the variables and parameters in Eq. (12), the range

of the variables and parameters with respect to the con-straint condition is:

0 < F ¼ sin h < 1; 0 < y < 1; g > 0; n > 0 ð13ÞThe differential equation is solved using the fourth-order

Runge–Kutta method for different values of g and n. Notall combinations of g and n are able to satisfy the constraintcondition. In the numerical calculation, it is found that,when n < 1, there are no solutions for the equation for allvalues of g. When h is small, 1� F 2 ¼ 1� sin2h in the limitgoes to 1, and the equation is approximately linear. Basedon the superposition principle, n is greater than one. The



190

200

210

Q3

220

230

240

250

260

270

Table 1. The Sn data adopted in the model [1,5,16–18].

Atomic volume X = 1.66 � 10�29 m3

Surface diffusivity Ds = 3.99 � 10�10 m2/sEffect surface thickness ds = 2..86 � 10�10 mSurface energy cs = 1.6 J/m2

Temperature T = 419 kThe electron charge e = 1.6 � 10�16 cBoltzmann constant K = 1.38 � 10�23 J/kElectrical resistivity q = 13.25 lX cmThe effective charge Z* = 17

Y. Yao et al. / Scripta Materialia xxx (2014) xxx–xxx 3


30 September 2014

physical meaning of n > 1 is that the circular shape is thelowest energy state with a minimum speed for the void.When it begins to bifurcate and collapse to a void, the voidvelocity must become greater than the initial circular shape.

The solution of the Eq. (12) is determined by g and n. Itis noted that the shape evolution parameter g and the veloc-ity parameter n are closely related. In order to investigatethe relationship, consider an additional boundary conditionas shown in Figure 1; assuming the angle at the tip isapproximately 50 gives:

When y = 1, F ¼ sin h ¼ sin 50 ¼ 0:089In the following numerical calculation, one fixes the

value of g and then searches for the other parameter n usingthe traversal method combined with Runge–Kutta methodin the interval to find the solution that satisfies all the con-straint and boundary conditions for Eq. (12). The predictedrelationship between g and n is illustrated in Figure 3.

g = 5.74 is the critical value of the void collapsing to afinger-like void [5]. The void buckles when g = 10.65 witha circular shape, and becomes infinitely long wheng = 10.65 drops to 5.74, which is the critical value for thevoid collapsing to a finger shape. When g = 5.74,n = 1.001, which indicates that the velocity changes littlecompared with the circular one. It is consistent with thework of Suo et al. [3], who discuss the case wheng = 5.74 using the circular void speed. The small changein velocity can be explained by the fact that a very smallperturbation can hardly disturb the void stability withoutenough energy, as stated by Yang et al. [6]. In a constantelectrical field, when the void begins to bifurcate and col-lapses to a finger shape, g is only related to the void widthu and the other parameters remain constant, so the voidwidth u and the parameter g vary simultaneously according

to the equation g ¼ Z�s eEtu2

csX. Thus, the relationship between g

and n corresponds to the void width u and the velocity V.

From Figure 3, it is known that when the finger-shaped

void width u decreases, the velocity increases. The narrower

the finger-shaped void, the faster it grows. Meanwhile, since

the two parameters are in a nonlinear relationship, when

1:5 < g < 5:74, the velocity increases slowly with reducing

width, while the velocity will increase rapidly when the

value of drops below 1.5, and the critical void width for

velocity mutation is approximate half of the initial width

based on g ¼ Z�s eEtu2

csX.

The relationship between the two parameters g and n isfitted using an exponential function, as shown in Figure 3:

n ¼ 6:406e�4:039g þ 1:608e�0:093g ð14Þ

0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

η

ζ

numerical solution

the fitting curve

Figure 3. The numerical solution of Eq. (12) and the fitting curve of gand f.


Substituting Eq. (14) and Eq. (9) into Eq. (11) gives:

V ¼ 6:406e�4:039Z�s eEt u2

csX þ1:608e�0:093Z�s eEt u2

csX

� �Dsds

uKTeZ�s E1 ð15Þ

The proposed model is verified by using experimentaldata from Zhang et al. [9]. In their experiments, the exper-iment temperature is 419 K. The applied current density is3.67 � 103 A cm�2, the finger-shaped void length is 26.4 lmand the average void propagating velocity is about4.4 lm h�2. The properties of materials adopted in themodel are listed in Table 1. To determine the electron windforce and the void speed, the electric field is solved for agiven void shape.

For a 95.5 Sn–4Ag–0.5Cu eutectic solder interconnectwith a void defect, the effective current density aroundthe void surface can be derived by a computational method.Considering the current crowding effect at the void surface,the actual current density is greater than the applied cur-rent. The electric field in a conducting material is governedby Maxwell’s conservation of charge equation. Assuming asteady-state direct current gives:Z

dj!� n!dS ¼

ZV

rcdV ð16Þ

where V is any control volume with surface S, n! is the out-ward normal to S, j

!is the electrical current density and rc

is the internal volumetric current source per unit volume.The equivalent weak form, which is useful for finite ele-

ment analysis, is obtained by introducing an arbitrary elec-trical potential field du and integrating over the volume.Applying Green’s formula and Ohm’s law, the governingconservation of charge, Eq. (16), becomes:

�Z

V

@du

@ x!rE @u

@ x!dV ¼

ZSdujdS þ

ZVdurcdV ð17Þ

where rE is the electrical conductivity matrix.Based on Eq. (17), a 3-D coupled thermal–electric finite

element analysis was conducted on 95.5Sn–4Ag–0.5Cu sol-der interconnect with a void propagation in the current

Figure 4. The current distribution in the solder joint with a void in thecurrent crowding zone.



280

290

300

310

320

330

340

350

360

Table 2. Effects of different parameters to the theoretical predictedvoid propagation speed v.

v (lm h�1)

Experimental value [1] 4.4Ds varies from 10�10 to 10�8 m2 s�1

(with j = 3 � 3.67 � 103 A cm�2, u = 1.22 lm)0.34–34

u varies from 1.22 to 0.5 lm(with j = 3 � 3.67 � 103 A cm�2,Ds = 3.99 � 10�10 m2 s�1

1.35–17.56

4 Y. Yao et al. / Scripta Materialia xxx (2014) xxx–xxx


30 September 2014

crowding zone. The current density distribution in the solderis shown in Figure 4. The average current density at the voidsurface (at the flat wake) is about 3 � 3.67 � 103 A cm–2, thesame as in Ref. [5] (i.e. 3 times the applied current density).The relationship between the current density and the electricfield is taken as E = qj, where q is the electrical resistivity.The electric wind force is thus 9:433� 10�17N .

However, in the solder joints the void is subjected notonly to the electron wind force, but also to other forces,such as the stress migration force [10,11], which is causedby the mismatch of thermal expansion coefficients of differ-ent materials, and the thermomigration force, which iscaused by the temperature gradient due to the Joule heatingeffect at the void tip [12]. The effect of these forces upon thevoid evolution in thin films has been studied by manyresearchers [13,14]. It is necessary for the thermomigrationand stress migration effects in the solder joints to be evalu-ated first before an approximate prediction for the finger-shaped void speed can be given.

It has been proved that the electromigration force can bestronger than the stress migration force if the applied cur-rent density is high [5]. Based on the experimental data[5], the electromigration force is about 10 times the stressmigration force. The thermomigration force can be definedby: F tm ¼ Q�

T ð� @T@xÞ where Q� is the heat of transport. @T

@x isthe temperature gradient. The molar heat of transport ofSn is 1.36 kJ mol�1 [15] and the thermal gradient is2829 K cm–1. In the developed coupled thermal–electricfinite element model the temperature gradient around thevoid is about 1700 K cm�1, whereas in the experiment fromTu’s group [19] the temperature gradient is 2392 K cm�1.Here, taking the maximum value 3000 K cm–1, the thermo-migration force can be calculated as F tm ¼ Q�

T ð� @T@xÞ

¼ 1:618� 10�29N , which is much smaller than the electro-migration force. Thus, in the proposed model, the effectsof stress migration and thermomigration can be ignored.

Substituting the material parameter and the electric currentdensity into Eq. (15), the finger-shaped void speed is1.35 lm h–1. This is consistent with the prediction by Zhang

370

Figure 5. The phenomenon that a finger-shaped void width enlargessuddenly in solder joints [1].


et al. [20] and is of the same order of magnitude as theexperiment speed. Table 2 shows the predicted void velocityby varying different parameters, including the surface diffu-sion coefficient and the width of the void. With the samevariation, the void width u mostly affects the velocity. Itcan be concluded that the void width u is the main param-eter that affects the void speed. It is worthwhile applyingthis conclusion to explain other phenomena qualitatively.For example, it is observed that a finger-like void in the sol-der joint will not always keep moving with a constant widthbut will become wider at its tip, as shown in Figure 5. It cannow be understood that this is just a way of slowing thevoid speed and reducing the system energy based on theconclusion that the narrower the finger-like void becomes,the faster it moves.

In summary, a quantitative relationship between the voidpropagation velocity and the shape evolution parameter isobtained that describes how the velocity of an original roundvoid changes when it collapses to a finger-shaped void. Thetheoretical calculations are in reasonable agreement withthe experiments. For future study, more attention shouldbe paid to the angle h at the finger-shaped void tip. It isbelieved that this angle plays an important role during thevoid collapsing to the finger shape. In this letter, the valueof this angle is taken from the experiment. The theoreticalsolution has been obtained by Chuang and Rice [21] withoutconsidering electromigration. When taking the electromigra-tion effects into consideration, there is still no representablesolution; future studies should take this into account.

The authors acknowledge the funding support of: ChinaYoung Thousand Talents Program W099104; National NaturalScience Foundation of China 51301136; Northwestern Polytechni-cal University; Northwestern University; SRC1393.

[1] S. Gee, N. Kellkar, J. Huang, K.N. Tu, Lead-Free and PbSnBump Electromigration Testing, ASME InterPACK, ASME,San Francisco, CA, 2005, pp. 1313–1321.

[2] M. Schimschak, J. Krug, Phys. Rev. Lett. 80 (1998) 1674.[3] Z. Suo, W. Wang, M. Yang, Appl. Phys. Lett. 64 (1994) 1944.[4] J. Cho, M. Rauf, D. Maroudas, Appl. Phys. Lett. 85 (2004) 2214.[5] Y. Yao, L.M. Keer, M.E. Fine, J. Appl. Phys. 105 (2009)

063710.[6] W. Yang, Wang, Z. Suo, J. Mech. Phys. Solids 42 (1994) 897.[7] A. Einstein, Ann. Phys. 322 (1905) 549.[8] E. Arzt, O. Kraft, W.D. Nix, J.E. Sanchez, J. Appl. Phys. 76

(1994) 1563.[9] L. Zhang, S. Ou, J. Huang, K.N. Tu, Appl. Phys. Lett. 88

(2006) 012106.[10] Y. Li, Z. Li, X. Wang, J. Sun, J. Mech. Phys. Solids 58 (2010) 1001.[11] Y. Jung, J. Yu, Appl. Phys. Lett. 115 (2014) 083708.[12] C. Chen, H.Y. Hsiao, Y.W. Chang, F. Quyang, K.N. Tu,

Mater. Sci. Eng. R 73 (2010) 85.[13] G. Sfyris, D. Dasgupata, D. Maroudas, J. Appl. Phys. 114

(2013) 023503.[14] Y. Qin, X. Wang, Thin Solid Films 531 (2013) 137.[15] H.Y. Hsiao, C. Chen, App. Phys. Lett. 94 (2009) 029217.[16] M. Sellers, A. Schultz, C. Basaran, D. Kofke, Appl. Surf. Sci.

256 (2010) 4402.[17] W. Qing, Z. Suo, J. Appl. Phys. 79 (1995) 8979.[18] Y. Yao, J. Fry, L. M. Keer, M.E. Fine, Acta Mater. v61, 5

(2013) 1525–1536.[19] C. Chen, H.M. Tong, K.N. Tu, Annu. Rev. Mater. Res. 40

(2010) 531.[20] L. Zhang, S. Qu, J. Huang, K.N. Tu, S. Gee, Appl. Phys.

Lett. 88 (2006) 012106.[21] T.J. Chuang, J.R. Rice, Acta Metall. 21 (1973) 1625.



an analytical method to predict electromigration-induced finger-shaped void growth in snagcu solder...

Documents