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Highlights

COMMAT 4984 No. of Pages 1, Model 5G

30 January 2013

" Vacancy, stress and damage evolution of solder joints under PDC is investigated. " An electromigration and thermomigration damagemodel is proposed. " Frequency, duty cycle and current density dependence of damage is investigated. " SEM images show that vacancyaccumulates only at skin layer of solders. " A mean time to failure equation is proposed for solder joints under PDC loadings.

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Computational Materials Science xxx (2013) xxx–xxx

le

l M

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COMMAT 4984 No. of Pages 14, Model 5G

30 January 2013

Electromigration damage mechanics ofunder pulsed DC: A computational mod

Wei Yao, Cemal Basaran ⇑Electronic Packaging Laboratory, State University of New York at Buffalo, NY 14260,

a r t i c l e i n f o

Article history:Received 16 September 2012Received in revised form 17 November 2012Accepted 5 January 2013Available online xxxx

Keywords:Pulsed current electromigrationThermomigrationSAC solder jointDuty factor

a b s t r a c t

A numerical method forin 95.5Sn–4Ag–0.5Cu (Stron wind force, chemicEntropy based damage clize irreversible entropyEM damage results weobserved that increasintion. A mean time to fawhich incorporates botour experimental resultfactor exponent m equa1.96. This equation is els range.

1. Introduction

Electromigration (EM) is a mass transportation mechanismdriven by electron wind force, thermal gradient, chemical potentiaand stress gradient. According to Moore’s law, number of transistors on integrated circuits (ICs) doubles approximately ever2 years. Moore’s law holds true since its introduction in 1970s. Thinsatiable demand for smaller ICs size, larger integration and higher Input/Output (IO) count of microelectronics has made ball griarray (BGA) the most promising connection type in electronpackaging industry. This trend, however, renders EM reliability osolders joints a major bottleneck to hinder further developmenof electronics industry. Although EM reliability of metal interconnects under direct current (DC) load has drawn increasing attention in recent years, our current understanding of degradatiophysics of solder alloys under time varying high current densitloading is still quite limited. As a result of broad pulse signaapplication in IC semiconductors, investigation of pulsed direccurrent (PDC) EM reliability of flip-chip solder joints and corresponding morphology evolution and failure mechanisms have become an essential topic of study in the electronic packaginreliability field [1–6].

0927-0256/$ - see front matter � 2013 Published by Elsevier B.V.http://dx.doi.org/10.1016/j.commatsci.2013.01.016

⇑ Corresponding author.E-mail address: cjb@buffalo.edu (C. Basaran).

1

Contents lists availab

Computationa

journal homepage: www

Please cite this article in press as: W. Yao, C. Basaran, Comput. Mater. Sci. (

ad-free solder jointsl

ted States

dying migration of voids driven by pulse electric current and thermal gradien05) solder joints is developed. The theoretical model involves coupling eleotential, joule heating and stress gradient driving mass diffusion processeria is adopted to characterize the mass transportation mechanism, which utduction rate as a measure of material degradation. The pulse current induceompared with DC EM response under otherwise the same conditions. Itty factor, current density and frequency leads to a faster damage accumula

e (MTF) equation for solder joints under pulse current loadings is proposeermomigration (TM) and EM damage. The failure mechanism is verified bis observed that MTF is inversely proportional to rm , f p, and jn, where dut

o 1.1, frequency exponent p equals to 1.43 and current density exponent

at SciVerse ScienceDirect

aterials Science

sevier .com/locate /commatsci

� 2013 Published by Elsevier B.V

Two types of failure mechanisms are related to the EM processhort-circuit failure induced by anode side mass extrusion anopen-circuit induced by void coalescence at cathode side. UndeDC loadings, mass transport from cathode side toward the anodside due to scattering of conduction band electrons with ions agrain boundaries, which is referred as electron wind force. Durinthis diffusion process, when steady state is reached, chemical potential and stress gradient driving force retard the mass diffusioby electron wind force [7–9]. For the pulse DC (PDC) loading, thsame holds true during on-portion of the pulse. During the loadoff period, however, only chemical potential and stress gradiencome into play. This back diffusion in absence of an electric fielhas an opposite effect on electron wind force, and heals part othe damage induced during load-on time, thus interconnects undePDC is expected to have a longer lifetime comparing to those subjected to DC loading under otherwise identical conditions. Thdamage healing mechanism under PDC loading strongly dependon pulse frequency, current density and duty factor defined aload-on time over one loading period. Some researchers studiepulse DC and AC electromigration in deposited Al or Cu thin filmboth experimentally and theoretically [10–19]. It was found thalifetime of conductors under time varying current is times longethan those subjected to DC current under otherwise the samconditions. There is, however, very limited research on EM osolder alloys under PDC loadings. This study is intended to fill ithe gap of EM behavior of solder alloys subjected to PDC loadin

2013), http://dx.doi.org/10.1016/j.commatsci.2013.01.016

88 by89 ag90 iou91 so92 an93 ar94 Fo95 m96 we97 in98 de

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104 2.1

105

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111 wh112 ce113 co114 va115 ce116 pe117

q ¼119119

120 wh121 be122 tu123 as124 um125 fin126 in127 law128 eff129 gr130 loa131

132 va133

G135135

136 wh

137 2.2

138

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146 wh

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148

149Cu150hi151se152m153Th154fer155

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V 175175

1762.5

177

178m179ce180or181pl182te183m184sp185er186hi187th188re189m190probability relationship between entropy production and micro-191scopic disorder as follows:192

s ¼ k ln W ð8Þ 194194

195where k is Boltzmann’s constant, W is a disorder parameter which196gives the number of micro states corresponding to given macro-197state. The relationship between entropy per unite mass and the dis-198order parameter is given by the following equation:199

s ¼ Rms

ln W ¼ N0k ln W ð9Þ 201201

202where R is universal gas constant, ms is specific mass, and N0 is Avo-203gadro’s constant. By a simple transformation, the disorder function204can be written as:205

W ¼ es

N0k ð10Þ 207207

208Degradation metric is further defined as a ratio of change in dis-209order parameter with respect to an initial disorder state as follows:

Q3

2 W. Yao, C. Basaran / Computational Materials Science xxx (2013) xxx–xxx

COMMAT 4984 No. of Pages 14, Model 5G

30 January 2013

Pl

considering both EM and thermomigration (TM) induced dam-e. In this work, we employ the pulsed current loadings with var-s duty factors and frequencies on 95.5Sn–4Ag–0.5Cu (SAC405)

lder joints. Results of vacancy concentration, volumetric stressd damage evolution with respect to duty factors and frequenciese plotted and compared with those under DC current stressing.r solder joints, Basaran et al. [20–24], proposed a comprehensiveean time to failure (MTF) equation for solder joints. In this study,

extend this model for SAC solder joints subjected to PDC load-gs by incorporating current density, duty factor and frequencypendence.

Governing equations

Basaran group proposed damage mechanics formulation for sol-r joints subjected to electrical and thermal loadings by couplingcancy conservation, force equilibrium, heat transfer and electricnduction processes as shown below [25]:

. Vacancy conservation equation

Electromigration is a diffusion controlled mass transportationocess governed by the following vacancy conservationuations:

0@c@tþrq� G ¼ 0 ð1Þ

ere Cv0 is the initial thermodynamic equilibrium vacancy con-ntration in absence of any loadings; c is the normalized vacancyncentration defined as Cv=Cv0 with Cv as vacancy concentration;cancy flux due to the combined effect of gradient of vacancy con-ntration, electric wind force, mechanical stress gradient and tem-rature gradient, respectively, is given by the following equation:

�DvCv0Z�ekTðrUÞc þrC þ cfX

kTrrsp þ c

kT2 Q �� �

ð2Þ

ere Dv is effective vacancy diffusivity; Z� is effective charge num-r; e is electron charge; k is Boltzmann’s constant; T is tempera-re, U is electrical potential, f is vacancy relaxation ratio definedratio of vacancy volume over volume of an atom, X is atomic vol-e. rsp is spherical part of stress tensor. Q � is heat of transport de-

ed as heat transmitted by an atom jumping a lattice site less thetrinsic enthalpy. The mass diffusion process is governed by Fick’s

of diffusion. In PDC loadings, electron wind forces terms isective only at load-on time, while vacancy concentration, stressadient and thermal gradient terms are active during the wholeding history.In a site of flux divergence, vacancy will accumulate, nucleate or

nish, and this vacancy dynamics is given by:

¼ �Cv0

c � exp ð1�f ÞXrsp

kT

h iss

ð3Þ

ere ss is characteristic vacancy generation or annihilation time.

. Force equilibrium equation

Mass accumulation at anode side induces a compression stress;ile vacancy formation at cathode side generates tensile stressally. Thus, a stress gradient is created. In absence of body forces,

e force equilibrium equation during current loading is given bye static equilibrium equation:

;j ¼ 0 ð4Þ

ere rij;j is derivative of stress with respect to degree of j.

ease cite this article in press as: W. Yao, C. Basaran, Comput. Mater. Sci. (2013)

. Heat transfer equation

In flip-chip BGA connections, current crowding happens at Al/interconnecting solder joint corners. Joule heating produces a

gher local temperature than the nominal value at center cross-ction of solder joints. It induces a thermal gradient, which drivesass diffuse from high temperature to low temperature area [26].e governing equation of joule heating production and heat trans-is given by:

p@T@t�rðkhrTÞ � qQ ¼ 0 ð5Þ

ere is material density; Cp is specific heat; kh is coefficient of heatnsfer and Q / I2R is joule heating generated within conductor.ce the current density distribution is independent of time vary-

g electric field, thermal gradient driving mass diffusion is in con-nt direction during PDC loading history.

. Electrical conduction equation

The electric field in a conducting material is governed by Max-ll’s equation of conservation of charge given by:�Jn dS ¼ 0 ð6Þ

ere S is surface of a control volume, n is outward normal to S, J isrrent density in A/cm2. Applying the divergence theorem to con-rt the surface integral into volume integral, we obtain followingctrical conduction equation:� @

@xJdV ¼ 0 ð7Þ

. Entropy based damage evolution model

Statistical thermodynamics provides a general framework foracroscopic description of irreversible microscopic damage pro-sses. Entropy is a macroscopic measure of number of ways atomsmolecules can be arranged in a given volume. It can be decou-

ed into reversible and irreversible production inside a closed sys-m. The irreversible entropy production in the closed system is aeasure of microscopic disorder. Moreover, energy always flowsontaneously from regions of higher temperature to those of low-temperature, thus to reduce the initial state of order and to a

gher level of disorder. It has been shown by Basaran [27,28] thate change in disorder parameter in Boltzmann’s equation whichlates entropy production to system disorder can be used as aetric for material degradation. Boltzmann’s equation gives the

, http://dx.doi.org/10.1016/j.commatsci.2013.01.016

210

D ¼ DcrW �W0

W0¼ Dcr

DWW0¼ Dcr 1� e

s0�sN0k

� �ð11Þ

212212

213 where Dcr is the critical damage used to define failure. The damage214 parameter D varying from 0 to 1 is a scalar value utilized to map215 degradation of the material. It can easily be calculated separately216 for each direction. Zero indicates no damage happened, while 1 cor-217 responds to broken of material. It is used to map degradation of218 both isotropic and anisotropic material property like elastic modu-219 lus, isotropic or kinematic hardening parameters. Detailed deriva-220 tion and experimental validation of this entropy based damage221 model has been reported by Basaran et al. [29–39]. D is applied to222 the elasticity constitutive relationship as [40]:223

dr ¼ ð1� DÞCedee ð12Þ225225

226 Entropy production during EM process is generated by several227 factors as shown below:228

Ds ¼Z t

t0

1t2 crT : rT þ CvDv

kT2 F : F þ 1Tq

r : e0vp

� �dt ð13Þ

230230

231 where 1t2 crT : rT is joule heating generated entropy; Cv Dv

kT2 F : F and232 1

T r : e0vp correspond to mass diffusion and viscoplastic deformation233 produced entropy respectively.234 By substituting Eq. (13) into Eq. (11), damage parameter235 evolution can be given by:236

D ¼ Dcr 1� e�R t

t0f 1t2crT:rTþCv Dv

kT2 F:Fþ 1Tqr:e0vpgdt

kN0

264

375 ð14Þ

238238

239 where F, is diffusion driving force given by

240 Fk ¼ �ðZ�eruþ fXrrsp þ Q�

T rTþ kTc rcÞ. SAC solders have a

241 252 �C melting temperature. As a result, it undergoes viscoplastici-242 ty, Basaran et al. [41] proposed the following creep rate

243 _evpij ¼

Ad0EbkT

hFiE

� �nbd

� pe�QIRT @F

@rij. Where A is a dimensionless material

244 parameter to describe the strain rate sensitivity, D0 is a diffusion245 frequency factor, E is Young’s modulus, b is characteristic length246 of crystal dislocation, k is Boltzmann’s constant, T is absolute tem-247 perature, d is average phase size, p is phase size exponent, Q is creep248 s249250

Þ252252

253 o254 t,255 -256257

Þ259259

260 t261262

Þ264264

265 ic266 h267 -

268 l-

269270

Þ272272

273 e274

2753. Finite element modeling

276An ABAQUS UMAT subroutine was developed to conduct EM277analysis of lead free SAC405 solder joints under pulse current load-278ing. The program is verified to produce converged stable numerical279results in our previous published works [30,33,35,42–45]. Duty280factor, frequency and maximum current density are used as control281parameters in the simulation. The finite element analysis (FEA)282mesh is shown in Fig. 1.283Solder joint has a nominal diameter of 116 lm and stand-off284height of 100 lm. The Al interconnect located above the solder285joint has thickness of 2 lm and Cu trace below the solder joint286has thickness of 10 lm. Current comes from the top-left Al thin287film, flows through the solder joint, and goes out from the bot-288tom-right Cu trace. Current distribution is not uniform throughout289the whole cross section of the solder joint. Current density at alu-290minum/copper traces interconnect solder joint corner is much lar-291ger than the average current density as shown in Fig. 1, which292makes current crowding corners most vulnerable to EM and joule293heating induced damages [46,47]. Eight-node user defined ther-294mal-electrical elements with unit thickness are used to mesh Al/295Cu traces and the solder joint.296Fig. 2 shows PDC current loading applied between left-top and297right-bottom of the FEA model. Duty factor r is defined as 2t=T. PDC298current frequency varies from 0.01 to 20 Hz, duty factors from 0.38299to 1.0 and maximum current densities from 8.1 � 105 A/cm2 and3004:8� 106 A=cm2. EM in solder joints subjected to DC (duty factor301of 1.0) loadings is studied for comparison purpose. Average operat-302ing ambient temperature for most IC devices is about 330.2K. The303vacancy relaxation time for SAC405 is 1:8� 10�3 s, effective charge304number is 10, grain boundary diffusivity is 2:72� 106lm2=s, ther-305mal conductivity is 57.3 W/(m k), initial vacancy concentration is3061.1 � 106 lm�3, and average vacancy relaxation ratio is 0.2. Since307the working temperature of solder joint is about two third of its308melting point in absolute temperature scale, 330K/(273 + 252)309K = 0.63, viscoplastic material behavior is considered in this re-310search. The linear kinematic hardening constant of SAC405 solder311alloy is 9:63� 109 kg s=m, nonlinear kinematic hardening constant312is 7:25� 108kg s=ðK mÞ and isotropic hardening constant is 383.3313

314

315e316

317i-318-

d

W. Yao, C. Basaran / Computational Materials Science xxx (2013) xxx–xxx 3

COMMAT 4984 No. of Pages 14, Model 5G

30 January 2013

activation energy for viscoplastic flow, R is the universal gaconstant, F is viscoplastic yield function define as:

Fnþ1ðn;aÞ ¼ jjnnþ1jj ¼ ð1� DÞffiffiffi23

rrðanÞ ð15

where n is effective yield stress given by equation which takes intaccount of linear and nonlinear kinematic hardening componenand rðanÞ represents isotropic hardening component. D reflects isotropic hardening degradation.

ntrialnþ1 ¼ Strial

nþ1 ¼ Xn ð16

Xn is back stress tensor or kinematic hardening componendefined as:

@X@t¼ 2

3ð1� DÞcH0ðaÞ n

jjnjj ð17

where H0(a) is kinematic hardening modulus; a is equivalent plaststrain given by

R ffiffiffiffiffiffiffiffiffiffiffiffi23

_e _pije

pij

qdt; c is the consistency parameter whic

has the following relationship with a as _r ¼ffiffi23

qc. D maps degrada

tion of kinematic hardening. The isotropic hardening model has the fo

lowing form of:

rðrÞ ¼ 23ry0 þ R1½1� e�ca� ð18

where ry0 is the initial yield stress at uniaxial tension, and R1 is thisotropic hardening saturation value.

Please cite this article in press as: W. Yao, C. Basaran, Comput. Mater. Sci. (

[41,48–51].

4. Numerical results and discussion

Current density distribution during load-on time along thdiagonal section A–B of solder ball is shown in Fig. 3.

It can be seen that the current density distribution is not unform along cross-section A–B. Current crowding behavior is ob

Fig. 1. Finite element mesh and electric potential gradient distribution of simulatemodel.

2013), http://dx.doi.org/10.1016/j.commatsci.2013.01.016

319 served at both upper-left corner A and lower-right corner B with320 jA ¼ 1:2� 106 A=cm2 and jB ¼ 1:99� 106 A=cm2, which are more321 than 3 times of the average value at center cross-section of the sol-322 der ball. Current crowding effect induces a higher than usual tem-323 perature locally. In this study, highest temperature of 381 K324 happens at corner B and lowest temperature of 370 K near bottom325 of solder bump, which creates a thermal gradient of 1571 K/cm326 across the solder joint. This thermal gradient is large enough to327 drive mass transport from high temperature region into low tem-328 perature regions [26,52,53]. Depending on direction of thermal329 gradient, TM may hinder or accelerate the EM process. Both EM330 and TM driven mass diffusions are considered in this work.

331 4.1. Vacancy concentration evolution in solder joint under PDC current332 loading

333 Fig. 4a shows normalized vacancy concentration evolution at334 anode and cathode sides of the solder joint. It can be seen that335 voids form at cathode side while mass accumulates at the anode336 side. Normalized vacancy concentration fluctuates up and down337 forming a band during current loading history. It grows to 1.068338 at cathode side and goes down to 0.957 at the anode side after339 6.7 h of PDC current loading. An exponential relationship exists be-340 tween vacancy concentration and current loading time. Fig. 4b341 shows vacancy concentration distribution along diagonal cross-342 section A–B and center cross-section C–D of the solder bump after

3436.7 h PDC loading with a frequency of 0.1 Hz and a duty factor of34450%. We can see that mass piles up at corner A, which induces local345compression stress. Vacancy concentration then increases along346diagonal A–B, and reaches its largest value at corner B which pro-347duces tensile stress locally. Stress gradient and vacancy concentra-348tion gradient are thus created in the solder joint. The vacancy349concentration gradient is often referred to as chemical potential.350According to Fick’s law, concentration gradient transport mass351from high concentration area to the low concentration area, which352retard the electron wind force driving mass diffusion. The stress353gradient has similar effect. It can also be seen from Fig. 4b that va-354cancy concentration along center line is not uniform as well, with355vacancy accumulating at center and mass piling up at outer surface356of the solder joint. This phenomenon is due to the thermal gradient357driving mass flux between center of solder joint and the outer skin,358which is consistent with experimental observations [54].359Normalized vacancy concentration evolution at anode side of360solder joints for various duty factors at a constant PDC frequency361of 0.05 Hz is shown in Fig. 5. It can be seen higher duty factor in-362duces a larger vacancy depletion/mass accumulation rate. Vacancy363concentration goes up and down continuously forming a fluctua-364tion band that gradually decreases during the current loading his-365tory for PDC cases. For DC loading, however, the vacancy366concentration is observed to decrease continuously without such367band formation, and rate of depletion is much faster than those368subjected to PDC loading with the same maximum current density

Fig. 2. Pulse current load applied in simulations.

Fig. 3. Current density distribution along diagonal cross-section A–B during PDC loading history.

4 W. Yao, C. Basaran / Computational Materials Science xxx (2013) xxx–xxx

COMMAT 4984 No. of Pages 14, Model 5G

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Please cite this article in press as: W. Yao, C. Basaran, Comput. Mater. Sci. (2013), http://dx.doi.org/10.1016/j.commatsci.2013.01.016

Fig. 4. Vacancy concentration evolution history under pulse current loading of 0.1 Hz and duty factor of 0.5: (a) vacancy concentration evolution at corner A and B and (b)vacancy concentration along diagonal cross-section A–B and center cross-section C–D after 6.7 h pulse current loading.

Fig. 5. Duty factor r dependence of vacancy concentration during current loading history at corner A (anode) of solder bump, with frequency of 0.05 Hz.

W. Yao, C. Basaran / Computational Materials Science xxx (2013) xxx–xxx 5

COMMAT 4984 No. of Pages 14, Model 5G

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Please cite this article in press as: W. Yao, C. Basaran, Comput. Mater. Sci. (2013), http://dx.doi.org/10.1016/j.commatsci.2013.01.016

Fig(b)

6 W. Yao, C. Basaran / Computational Materials Science xxx (2013) xxx–xxx

COMMAT 4984 No. of Pages 14, Model 5G

30 January 2013

Pl

Fig. 6. Frequency dependence of vacancy concentration evolution at corner B (cathode)

. 7. Volumetric and shear stress evolution at anode and cathode of solder bump during pushear stress.

ease cite this article in press as: W. Yao, C. Basaran, Comput. Mater. Sci. (2013)

6

, with maximum current density jB ¼ 2:0� 10 A=cm2 and duty factor of 0.5.

lse current loading history of 0.05 Hz, duty factor of 0.5: (a) volumetric stress and

, http://dx.doi.org/10.1016/j.commatsci.2013.01.016

369 and ambient temperature. Frequency dependence of normalized370 vacancy concentration evolution history at cathode side is shown371 in Fig. 6.372 It can be observed that vacancy concentration grows exponen-373 tially during loading history. Higher frequency introduces a larger374 vacancy accumulation rate. At 20 Hz, the vacancy concentration375 accumulates so fast that fails the solder joint in just 1000 s. High376 frequency comes with a shortened thermal dissipation time, which377 introduces more thermal damage to material under tests.378 From the above observations, it can be concluded that both TM379 and EM happen during the pulse current loading history. The mass380 diffusion mechanism in solder joint is complicate in the sense that381 four types of driving forces couple with each other: electric wind382 force, thermal gradient, chemical potential and stress gradient.383 During the PDC load-on time, the electric wind force drives mass384 from cathode side to the anode side; depending on thermal gradi-385 ent direction, thermal gradient may accelerate or impede the EM386 process; while stress gradient and chemical potential always387 drives the mass in the opposite direction of electron wind force.388 During the load-off time, however, only thermal gradient, stress389 gradient and vacancy concentration come to play, which transport

390mass backward to cathode side and heal part of damage triggered391during loading-on time [18–20]. This material healing effect exhib-392its as a fluctuation response in the damage evolution history, and393in turn makes lifetime of solder joint under PDC load much longer394than those under DC loading. Effectiveness of this damage healing395mechanism depends on pulse current frequency, duty factor and396ambient temperature. Increasing frequency and duty factor leads397to more serious damage in the solder joints.

3984.2. Stresses in flip-chip solder bumps during pulse current stressing

399Volumetric and shear stress evolution history at current crowd-400ing corners of the solder ball subjected to PDC with duty factor of4010.5 and frequency of 0.05 Hz is shown in Fig. 7.402It can be seen that compression stress develops at anode side403due to the mass pile-up, while tension stress develops at cathode404side due to void formation shown above. Exponential relationship405exists between stresses development and loading time. After 6.7 h406of current loading, both compression and tension stress reaches407about 20 MPa. From Fig. 7b, we can see that shear stress develops408at both anode and cathode side of the solder joint. After 6.7 h of

d t or

W. Yao, C. Basaran / Computational Materials Science xxx (2013) xxx–xxx 7

COMMAT 4984 No. of Pages 14, Model 5G

30 January 2013

Fig. 8. Volumetric stress evolution at cathode side of the solder joint subjectedependence and (b) frequency dependence.

Please cite this article in press as: W. Yao, C. Basaran, Comput. Mater. Sci. (

o 0.05 Hz PDC with maximum current density jB ¼ 2:0� 106 A=cm2: (a) duty fact

2013), http://dx.doi.org/10.1016/j.commatsci.2013.01.016

409 current loading, shear stress at anode is 9.8 MPa, which is one and410 half times of that at cathode side, 6.5 MPa. Room temperature yield411 stress is at 7 MPa. Duty factor and frequency dependence of volu-412 metric stress evolution is shown in Fig. 8.413 Volumetric stress growth rate strongly depends on duty factor.414 Increasing duty factor is associated with a faster volumetric stress415 accumulation rate. After 3.3 h, the volumetric stress for DC loading,416 which correspond to duty factor of 1.0, has reached 19 MPa; while417 it takes 5 h for duty factor of 0.72 PDC loading at the same maxi-418 mum current density to reach this value. For the duty factor of419 0.38, the volumetric stress is only 9 MPa after 6.7 h of current load-420 ing, which is a far less than larger duty factor loading induced421 stress. Fig. 8b is the frequency dependence of volumetric stress422 evolution. It can be seen that volumetric stress grow exponentially423 with loading time, and higher frequency causes a larger stress424 growth rate. However when the frequency is below 1 Hz, stress425 is no longer frequency dependent. For frequency low enough, ther-426 mal fatigue governs the damage process instead of electric current427 loa428 (1429 m430 ev

431Same as volumetric stress development, higher duty factor and432frequency induce a larger shear stress growth rate. The duty factor433dependence of shear stress over DC loading cases, however, is434smaller than the volumetric stress dependence, and the relation-435ship is given by Sshearpdc

¼ r0Ssheardcwith r0 ranging between r and

436r2 based on our statistical analysis. Nonlinear regression results437(with R-square of 0.9936) indicate that both volumetric stress438and shear stress grow exponentially vs. the loading time in hours,439and the exponential parameters for volumetric stress and shear440stress are 1.03 and 1.8 respectively, as shown in following441equation:442

S ¼ a� ebt ð19Þ 444444

445where S represents volumetric stress or shear stress, a is parameter446depending on PDC frequency and duty factor; b is stress exponent447which is between 1 and 2.

4484.3

449

450so

Fig. 9. Shear stress evolution at cathode side of the solder joint subjected to to 0.05 Hz PDC with maximum current density jB ¼ 2:0� 106 A=cm2: (a) duty factor dependenceand (b) frequency dependence.

8 W. Yao, C. Basaran / Computational Materials Science xxx (2013) xxx–xxx

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Pl

ding. Only when the frequency goes above a threshold valueHz here), volumetric stress growth rate tends to increase dra-

atically. Duty factor and frequency dependence of shear stressolution is shown in Fig. 9.

ease cite this article in press as: W. Yao, C. Basaran, Comput. Mater. Sci. (2013)

. EM damage in solder joints under PDC and DC current stresses

Due to the back-flow healing mechanism at loading-off time,lder joints subjected to PDC loading has a longer lifespan com-

, http://dx.doi.org/10.1016/j.commatsci.2013.01.016

451 e452 o453

454)455r456f

Fig. 10. Damage evolutions at cathode and anode sides of the solder joint subjected to 0.05 Hz PDC loading with duty factor of 0.5.

f th

W. Yao, C. Basaran / Computational Materials Science xxx (2013) xxx–xxx 9

COMMAT 4984 No. of Pages 14, Model 5G

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pared to those subjected to DC current loading under otherwise thsame conditions. Entropy based damage evolution is used tapproximate EM induced failure mechanism.

Fig. 11. Damage distribution along diagonal cross-section o

Please cite this article in press as: W. Yao, C. Basaran, Comput. Mater. Sci. (

It can be seen from Fig. 10 that damage as designed by Eq. (13grows exponentially at both anode and cathode sides of the soldejoint when subjected to PDC current loading at frequency o

e solder joint after 6.7 h 0.1 Hz PDC loading at duty factor of 0.5.

2013), http://dx.doi.org/10.1016/j.commatsci.2013.01.016

457 0.0458 pr459 ca460 stu461 ca462 loa463 lar464 di465 of466 de467 tio468 joi469 da470

471 pr472 cu473 sp474 so

475m476tio477[5478in479

480at481Da482du483loa484PD485da486te487

488lat4891 H490

491be492

Fig eque3.3

10 W. Yao, C. Basaran / Computational Materials Science xxx (2013) xxx–xxx

COMMAT 4984 No. of Pages 14, Model 5G

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Pl

5 Hz and duty factor of 0.5. However, the irreversible entropyoduction at anode side develops much faster than that at thethode side. This is due to the damage mechanism adopted in thisdy. Mass accumulation tends to produce more entropy than va-

ncy concentration under the same conditions. After 6.7 h currentding, the damage at cathode is 0.012, which is about 8 timesger than that at anode side. The damage distribution along

agonal and center cross-sections of the solder joint after 6.7 hPDC loading is shown in Fig. 11. It can be seen that damage firstcreases along diagonal A–B, then increases. For center cross-sec-n C–D, maximum damage happens at outer skin of the soldernt, where compression stress developed locally. The minimummage happens at center of the solder joint.In modern electronic packaging technology, the following

ocedures are often used to protect solder joints from short-cir-it failures: 1. Minimum clearance between conductors isecified in PC board prevents conducting between neighboring

. 12. Damage at cathode side of the solder joint for various duty factors with pulse frh of current loadings.

lder joints. Although entropy production at anode side develops stu

ease cite this article in press as: W. Yao, C. Basaran, Comput. Mater. Sci. (2013)

uch faster than that at the cathode side, only vacancy accumula-n induced open-circuit failure is of concern in electronic devices

5–58]. Thus only damage at cathode side is shown in the follow-g discussions.

Fig. 12 shows the duty factor dependence of damage evolutioncathode side of the solder joint subjected to 0.05 Hz PDC loading.mage grows exponentially during current loading history. Largerty factor induces a higher damage growth rate. After 3.3 h ofding, the damage in DC loading has accumulated to 2 times ofC loading with a duty factor at 0.38, and nearly 1.5 times ofmage with duty factor at 0.5. Damage vs. PDC duty factor is pot-d in Fig. 12b.

For frequency dependence, it is observed that damage accumu-ion is frequency independent when applied PDC signal is belowz as shown in Fig. 13.It can be that damage grows faster at higher frequency. This canexplained by the fact that pulse current period used in thisdy is much larger than the thermal relaxation time (typically

ncy of 0.05 Hz: (a) damage evolution history and (b) damage vs. duty factor after

, http://dx.doi.org/10.1016/j.commatsci.2013.01.016

493 e494 -495 s496 n497

498 r499 r500 d501 d502 f503

504 s.505 e506 a507 -508 i-509 e510511

Þ513513

514t515depend on duty factor, pulse frequency and maximum current den-516f517-518l-519520

Þ 522522

523d524y525t526d527d528n529e530r531f

532533

Þ 535535

e si

ode

W. Yao, C. Basaran / Computational Materials Science xxx (2013) xxx–xxx 11

COMMAT 4984 No. of Pages 14, Model 5G

30 January 2013

in order of 10�6 s for SAC solder alloy). In this frequency range, thsolder has more time to relax from the joule heating induced thermal damage than at lower frequencies, thus more damage gethealed. The current density dependence of EM damage evolutiois shown in Fig. 14.

Damage accumulates exponentially with loading time. Largecurrent density causes much more serious EM damage. Afte1.3 h of current loading, the damage of the solder joint subjecteto a maximum current density of 4:8� 106 A=cm2 has reache2:3� 10�3, while the one with maximum current density o81� 105 A=cm2 is only 2:1� 10�4.

Table 1 shows the nonlinear regression results of damage vduty factors, frequencies and current densities. With an R-squarvalue (confidence value, which means representative of datpoints) as large as 99%, it has been found that the duty factor, frequency and maximum current density dependence of electromgration damage follows an exponential relationship when thloading period is larger than the thermal relaxation time:

D ¼ a� ebt � c � e�dt ð20

Fig. 13. Frequency dependence of damage evolution at cathod

Fig. 14. Maximum current density dependence of damage evolution at cath

Please cite this article in press as: W. Yao, C. Basaran, Comput. Mater. Sci. (

de of the solder joint subject to PDC loading with duty factor of 0.5.

where D is the damage; a, b, c and d are all positive parameters tha

side of the solder joint subjected to 0.05 Hz PDC loading with duty factor of 0.5.

sity. For a loading period long enough (over 6 h), the second part oEq. (16) will be only 1% of the first part, thus we can ignore the second parts influence, which will simplify above equation into the folowing representation:

D ¼ a� ebr bf bjt ð21

where a is the general parameter that depends on duty factor anfrequency, br is duty factor exponent parameter, bf is frequencexponent parameter, bj is current density parameter and t is currenloading times in hours. For a frequency of 0.05 Hz, it has been founthat both parameter a and br grow with increasing duty factor r, anthe value of br is approximately r1:1=9. For a duty factor of 0.5, it cabe seen from Table 1 that with increasing frequency magnitudparameter a decreases, but parameter bf increase quickly. Nonlinearegression gives us the following relationship between parameter band pulse frequency f (with R-square of 0.999):

bf ¼ 0:161� f 1:431 ð22

2013), http://dx.doi.org/10.1016/j.commatsci.2013.01.016

536

537 tw538 0.9539

bj541541

542

543 ob544 qu545

D547547

548

549 da550 th551 M552 lab553 in554 th555

M557557

558 wh559 in560 an561 ob562 pe563 eq564 cu565

M567567

568 wh569 du570 is571 tem572 m573 wi574 pa575

M577577

578

579 du580 ne581 de582 ve583 de

5845.

585

586ag587cu588stu589th590

591an592or593tio594loa595ag596to597at598an

TabNo t de

y (A/

12 W. Yao, C. Basaran / Computational Materials Science xxx (2013) xxx–xxx

COMMAT 4984 No. of Pages 14, Model 5G

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Pl

In the same manner, we obtain the following relationship be-een bj and current density j (A=mm2) with confidence value of998:

¼ 2:280� 10�12 � j1:9 ð23Þ

After we put equation bj, bf and br back into Eq. (17), we finallytain the relationship between damage, duty factor and fre-ency as follows:

¼ a� eb�r1:1 �f 1:43 �j1:9 �t ð24Þ

We further define mean time to failure (MTTF) as entropy basedmage accumulated to a critical value, which is taken as 0.01 inis work for sake of simplicity. Basaran et al. [28] proposed aTTF equation for solder joint subjected to DC based on intensiveoratory testing of test vehicles with SAC405 solder joints. By

le 1nlinear regression results of duty factor, frequency dependences and maximum curren

Dependence Duty factor Frequency (Hz) Current densit

Duty factor 0.38 0.05 2:0� 106

0.50 0.05 2:0� 106

0.72 0.05 2:0� 106

1.00 0.05 2:0� 106

Frequency 0.50 0.05 2:0� 106

0.50 0.50 2:0� 106

0.50 5.00 2:0� 106

0.50 20.00 2:0� 106

Current density 0.50 0.05 8:1� 105

0.50 0.05 2:0� 106

0.50 0.05 4:8� 106

troducing duty factor and frequency influence on this equation,e following MTTF equation is obtained:

TTF ¼ ar1:1 � f 1:43 � j1:9 ð25Þ

ere a is a general parameter. We can see that MTF for PDC case isversely proportional with duty factor r1:1, pulse frequency f 1:43

d maximum current density j1:9. Since the above relationship istained at constant ambient temperature, we can incorporate tem-rature effect by Arrhenius temperature dependence into aboveuation, which gives us as the modified MTTF equation for pulserrent loading cases as follows:

TTF ¼ ar1:1 � f 1:43 � j1:9 eðDE=kTÞ ð26Þ

ere a is general parameter that depends on material, geometry,ty factor, frequency and the assumed critical damage value, DEdiffusion activation energy, k is Boltzmann’s constant and T is

perature. To verify the computer simulation results, experi-ents are conducted on solders subjected to pulsed current loadingth duty factor, frequency and current density as controllingrameters. The following failure mechanism is obtained:

TTF ¼ ar1:2 � f 1:8 � j1:7 erE=kT ð27Þ

We have less than 10% discrepancy for the current density andty cycle exponent, 20% discrepancy for the frequency compo-nt, which is acceptable considering manufacture induced initialfects in our test vehicle. It can be seen that our program gives ary good approximation to failure mechanism of solder joints un-r PDC current loadings.

599tia600re

bamcuelear

Ac

ofDi

Re

[1[2

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[4

[5

[6[7[8[9

[10

[11

ease cite this article in press as: W. Yao, C. Basaran, Comput. Mater. Sci. (2013)

Conclusions

A fully coupled thermal–electrical–mechanical–chemical dam-e model is proposed for SAC405 solder joints subjected to pulserrent loading. For comparison purpose, DC electromigrationdy is also conducted. By inspection of the simulation results,

e following findings are reported.During pulse current loading history, vacancy concentration

d stress fluctuate up and down in a gradually growing (cathode)depleting (anode) band. By using irreversible entropy produc-n rate as a metric to quantify material degradation under PDCding, it is observed that higher frequency induces a faster dam-

e growth rate. This is due to the fact that material has more timerelax from joule heating induced damage at low frequency thanhigher frequency. It is further observed that larger duty factord current density leads to an increased EM damage. An exponen-l relationship exists between EM/TM damage, duty factor, cur-

nt density and PDC frequency. After incorporating the entropy

nsity dependence of damage at cathode side of the solder joint.

cm2) Damage vs. time (h) Adjust R-square

D ¼ 0:0006� e0:04t � 0:0005� e�0:7t 0.999

D ¼ 0:0009� e0:06t � 0:00075� e�0:75t 0.998

D ¼ 0:00092� e0:08t � 0:0008� e�0:95t 0.998

D ¼ 0:0011� e0:11t � 0:00096� e�1:5t 0.996

D ¼ 0:0009� e0:06t � 0:00075� e�0:75t 0.998

D ¼ 0:0005� e0:34t � 0:00038� e�2:5t 0.998

D ¼ 0:0003� e1:5t � 0:0002� e�2:3t 0.999

D ¼ 0:00013� e11:7t 0.996

D ¼ 0:0003� e0:014t � 0:00024� e�0:1t 0.920

D ¼ 0:0009� e0:06t � 0:00075� e�0:75t 0.998

D ¼ 0:0014� e0:34t � 0:0012� e�3:7t 0.997

601sed damage model into Basaran’s MTTF equation, we propose a602odified MTTF equation for lead-free solder joints under pulse603rrent loading, which incorporates both thermomigration and604ctromigration damages. The finite element simulation results605e verified by our experiments.

606knowledgments

607This research project has been sponsored by the US Navy, Office608Naval Research, Advanced Electrical Power Program under the609rection of Terry Ericsen.

610ferences

611] X. Chen et al., Applied Physics Letters 87 (23) (2005) 233506.612] R. Frankovic, G.H. Bernstein, J.J. Clement, IEEE Electron Device Letters 17 (5)613(1996) 244–246.614] M.I. Amir Averbuch, Igor Ravve, Irad Yavneh, Journal of Computational Physics615167 (2) (2001).616] C. Basaran, Shidong Li, D.C. Hopkins, Wei Yao, Mean time to failure of617SnAgCuNi solder joints under DC, in: Thirteenth IEEE Intersociety Conference618on Thermal and Thermomechanical Phenomena in Electronic Systems619(ITherm), 2012.620] Wei Yao, Cemal Basaran, Skin effect and material degradation of lead-free621solder joint under AC. in: Thirteenth IEEE Intersociety Conference on Thermal622and Thermomechanical Phenomena in Electronic Systems (ITherm), 2012.623] Yao. Wei, Electronic Materials Letters 8 (5) (2012) 503–505.624] J.R. Lioyd, Journal of Physics D: Applied Physics 32(109R)(1999).625] R. Peierls, A.K. Das, Journal of Physics C: Solid State Physics, 8(20) (1975).626] H.Z. Zhilin Li, Huajian Gao, Journal of Computational Physics 152(1)(1999).627] B.K. Liew, N.W. Cheung, C. Hu, Electromigration interconnect lifetime under AC628and pulse DC stress. in: Twenty Seventh Annual Proceedings: Reliability629Physics, Piscataway, NJ, United States: Publ by IEEE, April 11 1989.630] B.K. Liew, N.W. Cheung, C. Hu, Electromigration interconnect failure under631pulse test conditions, Japan Soc. Appl. Phys., Tokyo, Japan, 1988.

Q4

, http://dx.doi.org/10.1016/j.commatsci.2013.01.016

632 [12] B.J. Root, S.J. Nagalingam, Electromigration Failure On Fine Line Conductors633 Under Pulse Test Conditions, IEEE, New York, NY, USA, 1985.634 [13] T. Jiang, N.W. Cheung, H. Chenming, IEEE Transactions on Electron Devices 41635 (4) (1994) 539–545.636 [14] L. Zhihong et al., IEEE Transactions on Electron Devices 46 (1) (1999) 70–77.637 [15] L.R. C Pennetta, Gy Trefan, A Scorzoni FFantini, I De Munan, Journal of Physics638 D: Applied Physics 34 (1421) 2001.639 [16] Z. Ya-Xiong, N. Li-Sha, Computational Materials Science 46 (2) (2009)640 443–446.641 [17] H. Nguyen Van, C. Salm, Computational Materials Science 49 (4) (2010)642 235–238.643 [18] H.L. Hartnagel, B.R. Sethi, Journal of Physics D: Applied Physics 18(2) 1985.644 [19] M.I. Amir Averbuch, Igor Ravve, Journal of Computational Physics 186(2)645 (2003).646 [20] W. Yao, C. Basaran, Journal of Applied Physics 111 (6) (2012) 063703.647 [21] C. Basaran, Shidong Li, Douglas C. Hopkins, Damien Veychard,648 Electromigration Time to Failure of SnAgCuNi Solder Joints, ASME InterPack,649 2009.650 [22] S. Li, M.F. Abdulhamid, C. Basaran, Simulation 84 (8–9) (2008) 391–401.651 [23] S. Li, Michael S. Sellers, Cemal Basaran, Andrew J. Schultz, David A. Kofke,652 Computational Materials Science, 6, 2798–2808.653 [24] Y. Wei, C. Basaran. Damage of SAC405 solder joint under PDC. in: Thirteenth654 IEEE Intersociety Conference on Thermal and Thermomechanical Phenomena655 in Electronic Systems (ITherm), 2012.656 [25] M. Abdulhamid, C. Basaran, Y. Lai, IEEE Trans. on Advanced Packaging, in press.657 [26] H. Ye, C. Basaran, D. Hopkins, Applied Physics Letters 82 (7) (2003) 1045–1047.658 [27] C. Basaran, S. Nie, International Journal of Solids and Structures 44 (3–4)659 (2007) 1099–1114.660 [28] Y. Lee, C. Basaran, A viscoplasticity model for solder alloys. in: ASME 2010661 International Mechanical Engineering Congress & Exposition IMECE, Vancover,662 British Columbia, Canada, 2010.663 [29] C. Basaran, C.Y. Yan, Journal of Electronic Packaging 120 (4) (1998) 379–384.664 [30] C. Basaran, L. Minghui, Y. Hua, A Thermodynamic Model For Electrical Current665 Induced Damage, IEEE, Piscataway, NJ, USA, 2004.666 [31] C. Basaran, S. Nie, International Journal of Damage Mechanics 13 (3) (2004)667 205–223.668 [32] C. Basaran, Y. Zhao, H. Tang, J. Gomez, Journal of Electronic Packaging, 2005.669 [33] C. Basaran, Minghui Lin, Shidong Li, Computational Simulation of670 Electromigration Induced Damage in Copper Interconnects, 2007.671 [34] C. Basaran, H. Ye, D.C. Hopkins, D. Frear, J.K. Lin, Journal of Electronic Packaging672 127 (2) (2005) 157–163.673 [35] C. Basaran, M. Lin, Mechanics of Materials 40 (1–2) (2008) 66–79.674 [36] C. Basaran, M. Lin, International Journal of Materials and Structural Integrity 1675 (2007) 16–39.

676[37] J. Gomez, C. Basaran, International Journal of Solids and Structures 42 (13)677(2005) 3744–3772.678[38] H. Ye, C. Basaran, D.C. Hopkins, IEEE Transactions on Components and679Packaging Technologies 26(3) (2003) 673–681 (see also IEEE Transactions on680Components, Packaging and Manufacturing Technology, Part A: Packaging681Technologies).682[39] Y. Zhao, C. Basaran, A. Cartwright, T. Dishongh, Mechanics of Materials 32683(2000) 161–173.684[40] H. Tang, A Thermodynamic Damage Mechanics Theory and Experimental685Verification for Thermomechanical Fatigue Life Prediction of Microelectronics686Solder Joints, Buffalo, University at Buffalo, The State University of New York,6872002.688[41] C. Basaran, S. Li, M.F. Abdulhamid, Journal of Applied Physics 103 (12) (2008)689123520–123529.690[42] C. Basaran, M. Lin, H. Ye, International Journal of Solids and Structures 40 (26)691(2003) 7315–7327.692[43] C. Basaran, et al., Journal of Applied Physics 106(1) (2009) (013707–013707-69310).694[44] B. Cemal, Multidiscipline Modeling in Materials and Structures 2 (2007) 309–695326.696[45] S. Li, C. Basaran, Mechanics of Materials 41 (3) (2009) 271–278.697[46] K.N. Tu et al., Applied Physics Letters 76 (8) (2000) 988–990.698[47] R. Fei et al., Effect of Electromigration on Mechanical Behavior of Solder Joints,699Materials Research Society, Warrendale, PA, USA, 2005.700[48] M.E. Sarychev et al., Journal of Applied Physics 86 (6) (1999) 3068–3075.701[49] M. Lin, C. Basaran, Computational Materials Science 34 (1) (2005) 82–98.702[50] H. Ye, Mechanical Behavior of Microelectronics and Power Electronics Solder703Joints under High Current Density: Analytical Modeling and Experimental704Investigation, Buffalo, University at Buffalo, State University of New York,7052004.706[51] R. Balzer, H. Sigvaldason, Physica Status Solidi B: Basic Research 92 (1) (1979)707143–147.708[52] M. Abdulhamid, C. Basaran, ASME Journal of Electronic Packaging 131 (12)709(2009). 011002.710[53] Y. Dan et al., Journal of Applied Physics 102 (4) (2007). 043502.711[54] S. Li, M.F. Abdulhamid, C. Basaran, IEEE Transactions on Advanced Packaging71232 (2008) 478–485.713[55] J.M. Schoen, Journal of Applied Physics 51 (1) (1980) 508–512.714[56] L. Brooke, Pulsed Current Electromigration Failure Model, IEEE, New York, NY,715USA, 1987.716[57] S.L. Cemal Basaran, Douglas C. Hopkins, Damien Veychard, Electromigration717Time to Failure of SnAgCuNi Solder Joints. ASME InterPack, 2009.718[58] H. Ye, C. Basaran, Douglas C. Hopkins, International Journal of Solids and719Structures 40(15) (2003) 4021–4032.

720

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