Electromigration analysis of solder joints under ac load: A mean time tofailure modelWei Yao and Cemal Basaran Citation: J. Appl. Phys. 111, 063703 (2012); doi: 10.1063/1.3693532 View online: http://dx.doi.org/10.1063/1.3693532 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v111/i6 Published by the American Institute of Physics. Related ArticlesElectromigration induced fast L10 ordering phase transition in perpendicular FePt films Appl. Phys. Lett. 102, 022411 (2013) Feedback-controlled electromigration for the fabrication of point contacts Appl. Phys. Lett. 102, 023105 (2013) Rapid reversible electromigration of intercalated K ions within individual MoO3 nanobundle J. Appl. Phys. 113, 024311 (2013) Temperature and pressure dependent Mott potentials and their influence on self-limiting oxide film growth Appl. Phys. Lett. 101, 171605 (2012) Electromigration-driven complex dynamics of void surfaces in stressed metallic thin films under a general biaxialmechanical loading J. Appl. Phys. 112, 083523 (2012) Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

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Electromigration analysis of solder joints under ac load: A mean timeto failure model

Wei Yao and Cemal BasaranElectronic Packaging Laboratory, Department of Civil, Structural and Environmental Engineering,University at Buffalo, The State University of New York, Buffalo, New York 14260, USA

(Received 28 October 2011; accepted 13 February 2012; published online 19 March 2012;

publisher error corrected 9 April 2012)

In this study, alternating current (ac) electromigration (EM) degradation simulations were carried

out for Sn95.5%Ag4.0%Cu0.5 (SAC405- by weight) solder joints. Mass transport analysis was

conducted with viscoplastic material properties for quantifying damage mechanism in solder joints.

Square, sine, and triangle current wave forms ac were used as input signals. dc and pulsed dc

(PDC) electromigration analysis were conducted for comparison purposes. The maximum current

density ranged from 2:2� 106 A=cm2 to 5:0� 106 A=cm2, frequency ranged from 0.05 Hz to 5 Hz

with ambient temperature varying from 350 K to 450 K. Because the room temperature is nearly

two-thirds of SAC solder joint’s melting point on absolute temperature scale (494.15 K),

viscoplastic material model is essential. Entropy based damage evolution model was used to

investigate mean time to failure (MTF) behavior of solder joints subjected to ac stressing. It was

observed that MTF was inversely proportional to ambient temperature T1.1 in Celsius and also

inversely proportional to current density j0.27 in A/cm2. Higher frequency will lead to a shorter

lifetime with in the frequency range we studied, and a relationship is proposed as MTF / f�0:41.

Lifetime of a solder joint subjected to ac is longer compared with dc and PDC loading conditions.

By introducing frequency, ambient temperature and current density dependency terms, a modified

MTTF equation was proposed for solder joints subjected to ac current stressing. VC 2012 AmericanInstitute of Physics. [http://dx.doi.org/10.1063/1.3693532]

I. INTRODUCTION

Electromigration (EM) is a mass transport process within

metals due to scattering of electrons with atoms. This mecha-

nism causes atom depletion at cathode side and accumulation

at anode side, leading to open or short circuit failure at the

two extremities of conductors. EM-induced damage has

drawn a major interest in microelectronic field. Black pro-

posed one of the earlier eletromigration mean-time-to-failure

(MTTF) equations. The Black empirical expression1 for mean

time to failure (MTF) is well known as a standard direct cur-

rent (dc) electromigration failure equation for thin films.

Blech discovered in his experiments2–4 the critical product of

conductor line length and current density below which no

electromigration failure can be observed. With the advance of

vary large scale integrated (VLSI) circuits and insatiate

demand for miniaturization in the industry, electromigration

reliability is studied widely.5–8 However, the most commonly

used Black’s electromigration time to failure equation

does not work for solder joints. As a result, Basaran and Li9

proposed a comprehensive MTTF equation for solder joint

subjected to dc stressing based on extensive testing.

Most of the research in the literature focuses mainly on

reliability physics of thin film interconnects under dc loadings.

However, alternating current (ac) stress, on the other hand,

whether in hot-carrier effect, electromigration, or gate oxide

wear-out, is the norm in actual integrated circuit (IC) opera-

tion.10 The obvious different degradation mechanism between

ac and dc loading can lead to very different prediction in

electronics reliability, e.g., lifetime under ac stress may be

orders of magnitude longer than that of dc condition.11–13 This

result has been attributed to the damage healing effect, i.e.,

the damage induced during the positive cycle of current stress

is partially healed by the following negative-cycle. Liew14

proposed an average current density model to project the ac

lifetime in thin film interconnects. By introducing a damage

healing factor c into the Liew’s modeling, Tao et al.15 further

proposed an effective current density based MTTF equation

under ac stressing. Tao’s MTTF model is limited due to the

fact that it only works for metal thin film interconnects. With

the development of VLSI chip, ball grid array (BGA) solder

joints are used to replace traditional aluminum/copper wiring

for the sake of connecting the ever-increasing high density

input/output (I/O) pads on the chip to its substrate. Thus the

EM failure of lead-free solder joint has become a major reli-

ability issue. More research needs to be done on the damage

mechanisms of solder joints subjected to ac loads in modern

flip-chip technology to understand and to model this process

accurately.

Low melting point, high atomic diffusivity, and current

crowding effect in solder joints are the main factors that con-

tribute to the electromigration failure of solder joint

in electronics packaging, whether it is subjected to dc, PDC,

or ac loadings. For SAC solder alloy, the room temperature

is about two-thirds of its melting point on absolute tempera-

ture scale. At this temperature scale, moving dislocations ini-

tiate inter-crystalline gliding, and viscoplasticity becomes

the dominant degradation mechanism in solder alloy; this

0021-8979/2012/111(6)/063703/12/$30.00 VC 2012 American Institute of Physics111, 063703-1

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tends to induce permanent damage in the material. In solder

joints, EM can occur under a current density as low as

5� 103 A=cm2, which is about two orders of magnitude less

than that required to cause electromigration failure in Al or

Cu interconnects.16 Third, due to the line to bump geometry

of flip-chip packaging, an abrupt change of current density

exists between aluminum trace on the chip side and the sol-

der joint; this makes current density at aluminum trace and

solder joint interconnect several times higher than nominal

current density inside solder joint. This phenomenon is

called current crowding and can lead to localized overheat-

ing and formation of thermal hot spots, in extreme cases

leading to thermomigration (TM). Moreover, because Al or

Cu thin film trace on the chip acts as the major heat source,

the side of solder joint close to the chip will have a higher

temperature than that near the substrate, and it is subjected to

the thermal gradient as large as 1500 �C/cm,17,18 which will

lead to mass transport atoms down the gradient.

EM during ac is different than dc case because the elec-

tric current continuously alters its magnitude and direction

during the loading history. As a result, both chemical poten-

tial and stress gradient variations are function of time.

Because the current density distribution is independent of

current direction, for the joule heating part, there is no differ-

ence whether ac or dc is applied to solder bump.16 After

reaching the steady state, the chemical potential and the

stress gradient will stay in the same direction. However, the

electric wind force, which is the major driving force of

atomic diffusion during EM process, changes its direction

every half period for perfect ac loads. During the negative-

cycle of current stressing, most of the mass diffusion that

took place in the previous positive current loading period

will get restored.19 This retroactive material healing effect

under ac stressing contributes to the orders of magnitude lon-

ger EM lifetime compared with dc stressing under otherwise

the same conditions. ac electromigration lifetime varies with

frequency, maximum current density, ambient temperature,

and electric current waveform.

II. GOVERNING EQUATION OF ACELECTROMIGRATION

According to original formulation developed by Basaran

group,6,17,20–50 damage mechanism of solder joint subjected

to electric current load and thermal gradient is governed by

coupled vacancy conservation equation, force equilibrium

equation, heat transfer equation and electric conduction

equation. Material properties of SAC405, copper, and alumi-

num can be found in Table I.

A. Vacancy conservation equation

Electromigration is diffusion controlled mass transport

process, which is governed by the following equation:

Cv0

@c

@tþrq� G ¼ 0; (1)

where Cv0 is the initial thermodynamic equilibrium vacancy

concentration in absence of any loads; c is the normalized

vacancy concentration defined as cv=cv0 with Cv as vacancy

concentration. The vacancy flux due to the combined effect

of gradient of vacancy concentration, electric wind force,

mechanical stress gradient, and temperature gradient, respec-

tively, is given by the following equation:

q ¼ �DvCvo rCþ Z�e

kTðr;Þcþ cf X

kTrrsp þ c

kT2Q�rT

� �;

(2)

where Dv is effective vacancy diffusivity; Z�, effective

charge number; e, electron charge; k, Boltzmann’s constant.

T is temperature. ; is electric potential that reverses its direc-

tion every half cycle of ac loading. f , is vacancy relaxation

ratio defined as ratio of vacancy volume over volume of an

atom. X, is atomic volume. rsp is spherical part of stress ten-

sor. Q� is heat of transport defined as heat transmitted by an

atom jumping a lattice site less the intrinsic enthalpy.

TABLE I. Material property for SAC405, aluminum and copper at 353 K.

Material properties SAC 405 Copper Aluminum

Vacancy relaxation time (s) 1:80� 103 (Ref. 56) 1:80� 10�3 1:80� 10�3

Effective charge number 10 (Ref. 57) 0.7 (Ref. 58) 4 (Ref. 59)

Grain boundary diffusivity (lm2/s) 2:72� 106 1:2� 103 (Ref. 60) 6:0� 103

Resistivity (Xm) 1:52� 10�7 1:678� 10�8 2:82� 10�8

Density (kg/m3) 7:39� 103 8:94� 103 2:70� 103

Thermal conductivity (W/(mK)) 57:3 4:16� 102 2:37� 102

Thermal expansion coefficient (K�1) 1:89� 10�6 1:65� 10�5 2:4� 10�5

Atomic volume (lm3) 2:71� 10�11 (Ref. 61) 1:19� 10�11 1:66� 10�11

Initial vacancy concentration (lm�3) 1:107� 106 (Ref. 62) 8:413� 103 6:02� 103 (Ref. 63)

Initial average grain size 1.0 1.0 1.0

Young’s modulus (GPa) 57.7 120.5 70

Poisson ratio 0.33 0.34 0.35

Linear kinematic hardening cons. (kg/m) 9:63� 109 (Ref. 29) 2:04� 109 7:4� 108

Nonlinear kinematic hardening cons. (kg/km) 7:25� 108 (Ref. 29) 1:8� 108 4:2� 106

Isotropic hardening cons. 383.3 (Ref. 35) 383.3 13.6

Creep activation energy (kgm2/s2mol) 7:95� 104 1:97� 105 (Ref. 64) 1:6� 105

063703-2 W. Yao and C. Basaran J. Appl. Phys. 111, 063703 (2012)

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The first term rC in Eq. (2) is vacancy concentration

gradient. According to Fick’s second law, material will dif-

fuse from a region of high concentration to a region of low

concentration with a magnitude that is proportional to the

concentration gradient. After the diffusion reaches steady

states, the vacancy concentration gradient will not change

the direction with the alternating current. Z � e=kTðr;Þc is

the electron wind force term, which is the main EM mecha-

nism and changes its direction every half period during ac

loading history. cf X=kTrrsp is mechanical driving force.

Voids will also form due to the vacancy diffusion induced by

the hydrostatic stress gradient; this is often known as stress

induced voiding (SIV). The mechanical stress changes its

direction during the initial ac stressing. After a period of time,

compressive stress will form at one side, and tensile stress at

the other side; the mechanical driving force will not change its

direction after steady state is reached. ðc=kT2ÞQ�rT is the

thermal driving force (thermomigration). Material will acquire

higher kinematic energy at high joule heating region, e.g., the

current crowding corner where Al/Cu trace interconnect meets

with solder joint. When atoms of those regions collide with

colder slow-moving particles, they push the latter atoms down

the thermal gradient.

In a site of flux divergence, vacancy will accumulate,

nucleate or vanish, and this vacancy dynamics is vacancy

generation/annihilation rate G is given by:

G ¼ �Cv0

c� expð1�f ÞXrsp

kT

h iss

; (3)

where ss is characteristic vacancy generation/annihilation time.

B. Force equilibrium equation

In a metal conductor under high current density, a stress

gradient is created due to mass transport. This mass transport

mechanism induces tensile stress at the vacancy-rich end,

and compressive stress at the other end. In absence of body

forces, the force equilibrium equation during current loading

can be given by the static equation:

rij; j ¼ 0; (4)

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

C. Heat transfer equation

Governing equation of joule heating production and heat

transfer is given in the following text:

qCp@T

@t�rðkhrTÞ � qQ ¼ 0; (5)

where q is material density; Cp is specific heat; kh is coefficient

of heat transfer, and Q / I2R is joule heating generated within

conductor. Because the current density always has higher value

at the corners where Cu/Al trace interconnect meets with sol-

der joint and decreases toward center region of the solder joint

during the whole ac stressing history, the heat will flow from

current crowding corner into center of solder joint.

D. Electrical conduction equation

The electric field in a conducting material is governed by

Maxwell’s equation of conservation of charge expressed as:

ð�S

Jn dS ¼ 0; (6)

where S is surface of a control volume, n is outward normal

to S, J is current density in A=cm2.

Applying the divergence theorem to convert the surface

integral into volume integral, we obtain following electrical

conduction equation:

ð�V

@

@xJdV ¼ 0: (7)

E. Viscoplastic material model

The stress-strain constitutive model can be given by

r ¼ Hðetotal � eviscoplastic � ediffusion � ethermalÞ

where

H ¼ K1� 1þ 2l I � 1

31� 1

� �: (8)

K is bulk modulus, l is shear modulus, � is matrix product

operator, 1 is unit vector and I is unit vector.

The viscoplastic flow rule for rate dependent material is

shown in the following text:

_evp ¼ cn; (9)

where n is direction vector normal to yield surface F, and cis defined as:

c ¼ AD0Eb

khhFiE

� �n b

d

� �p

e�Q=Rh: (10)

The viscoplastic strain rate is thus

_evpij ¼

AD0Eb

kh

�hFiE

�n�b

d

�p

e�Q=Rh @F

@rij; (11)

where A is dimensionless material parameter to describe the

strain rate sensitivity, D0 is a diffusion frequency factor, E is

Young’s modulus, b is characteristic length of crystal dislocation,

k Boltzmann’s constant, h is absolute temperature in Kelvin, d is

average phase size, Q is creep activation energy for viscoplastic

flow, R is universal gas constant, and F is the yield surface,

which combined both isotropic and kinematic hardening effect:

Fðn; aÞ ¼ ks� Xk �ffiffiffi2

3

r�r; (12)

where X is back stress tensor, and r is isotropic stress which

represent radius of yield surface. The isotropic hardening

model is defined as:

063703-3 W. Yao and C. Basaran J. Appl. Phys. 111, 063703 (2012)

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�rðaÞ ¼ffiffiffi2

3

rry0 þ R1½1� e�ca�; (13)

where ry0 is initial yield stress and R1 is isotropic hardening

saturation value.

The back stress tensor (kinematic hardening) is defined

as:

@X

@t¼ c

2

3H0ðaÞ n

knk ; (14)

where H0(a) is kinematic hardening modulus, c is expressed

in Eq. (11), n is effective stress, a is plastic strain given byÐ ffiffiffiffiffiffiffiffiffiffiffi23

_epij _e

pij

qdt.

F. Entropy damage model

Boltzmann’s equation gives the probability relationship

between entropy production and microscopic disorder as

follows:

s ¼ klnW; (15)

where k is Boltzmann’s constant, W is a disorder parameter

that gives the number of micro states corresponding to given

macro state. The relationship between entropy per unite

mass and the disorder parameter is given by Eq. (16):

s ¼ R

mslnW ¼ N0klnW; (16)

where R is universal gas constant, ms is specific mass, and

N0 is Avogadro’s constant. By some simple transformation,

the disorder function can be written as:

W ¼ e

s

N0k: (17)

Degradation metric is further defined as a ratio of change in

disorder parameter with respect to an initial disorder state as

follows:

D ¼ DcrW �W0

W0

¼ DcrDW

W0

¼ Dcr 1� e

s0 � s

N0k

0@

1A; (18)

where Dcr is the critical damage used to define failure or to

map degradation of a material property like elastic modulus

to damage parameter D. Entropy production during the EM

process is generated by several factors as shown in Eq. (19):

Ds ¼ðt

t0

�1

t2crT : rT þ CvDv

kT2F : Fþ 1

Tr : e

0vp

�dt; (19)

where ð1=t2ÞcrT : rT is joule heating generated entropy;

ðCvDv=kT2ÞF : F and ð1=TÞr : e0vp correspond to mass

diffusion and viscoplastic deformation produced entropy,

respectively.

By substituting Eq. (19) into (18), damage parameter

evolution can be given by:50

D¼Dcr 1� e

�Рt

t0

�1t2 crT :rTþ CvDv

kT2 F : Fþ 1T r : e

0vp

�dt

kN0

2666664

3777775;

(20)

where C is vacancy concentration, Cv is vacancy concentra-

tion, and Dv is effective diffusivity.

III. FINITE ELEMENT MODELING

Finite element model of flip-chip solder joint that con-

nects aluminum trace on the chip surface and copper under

bump metallization (UBM) on the substrate is shown in

Fig. 1. The solder bump has a nominal diameter of 120 lm

and stand-off height of 100 lm. The aluminum trace located

above solder bump has a thickness of 2 lm, while the copper

trace is 10 lm thick. Film coefficient of air is taken as 24.4

w=m2K, with sink temperature of 353 K. Electric current

comes from aluminum trace on the top-left side into the sol-

der joint. After passing through the solder ball, it goes out

from the bottom-right copper trace. Eight node coupled

thermal-electrical elements with unit thickness are used to

mesh the flip-chip solder joint. Frequency, maximum current

density, ambient temperature, and electronic wave form are

used as controlling parameters in the simulation. Material

properties of SAC405 solder alloy, aluminum, and copper

used in the modeling are given in Table I.

Alternate current with frequencies of 0.0110 Hz and a

maximum current density up to 106 A/cm2 were applied on

the model. Triangle, square, and sine current wave forms are

used as input signals, as shown in Fig. 2. Simulations were

also conducted under dc and PDC loading for comparison.

The ambient temperature varied from 353.15 K to 453.15 K.

IV. NUMERICAL RESULTS AND DISCUSSION

A. Vacancy concentration

Flip-chip solder joint is subjected to ac stressing of 104 to

106 A/cm2. Figure 3 shows the current density profile of flip-

chip solder joint under ac stressing. Due to the unique line-to-

bump geometry of flip-chip package, it can be seen that current

density at solder joint and Al/Cu trace intersection has the

FIG. 1. (Color online) Finite element mesh of solder joint connection.

063703-4 W. Yao and C. Basaran J. Appl. Phys. 111, 063703 (2012)

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maximum current density 2:2� 106 A=cm2, which is more

than 100 times the nominal value 2:0� 104 A=cm2 in the sol-

der joint. This is the the current crowding effect. Actually, we

can identify in Fig. 3 that there is a singularity point with current

density as high as 2:7� 107 A=cm2. In reality, there will not

be such a point because it will melt immediately by the high

localized joule heating and get reshaped by the surface ten-

sion.52 With a thickness of 2lm, the Al trace has a current den-

sity of 6:7� 106 A=cm2, which is about seven times of that in

Cu trace 9:0� 105 A=cm2. The high current density and small

cross section of the Al trace usually induces a higher tempera-

ture locally; on the other hand, due to the lower current density

and good heat conductivity, the copper trace works like a ther-

mal sink. Figure 4 is the temperature profile of solder joint sub-

jected to ac current loading, with maximum current density of

2:2� 106 A=cm2 (at points A and B in Fig. 1) and frequency

of 0.05 Hz. It can be seen that aluminum trace and current

crowding corner (point A in Fig. 1) are the main heat source,

which is 2 K higher than the nominal temperature in the solder

joint and copper trace. This will create a temperature gradient

about 1000 K/cm at the current crowding corner and large

enough for themomigration (TM) to happen.17 Both electrical

and thermal loads make the current crowding corner most vul-

nerable part in the flip-chip package.

The initial vacancy concentration value is 1:11� 107lm�3.

The successive number of atomic jumps is given by,

v ¼ v0expð�rE=kTÞ; (21)

where v0 is the Debye frequency that defined as theoretical

maximum vibration frequency of the atoms in the crystal.

Because tin makes up 95.5% of the SAC405 solder alloy, we

can reasonably assume solder joint’s Debye temperature the

same as pure tin, 200 K. For metals above their Debye tem-

perature, the Debye frequency is about 103 Hz.16 Taking va-

cancy activation energy E as 0.72 eV,9,52 we obtain

v ¼ 103 jump=s. This means thousands of atoms/vacancies

are expected to jump every second. Theoretically speaking,

the vacancy jump in current-reverse half cycle is the same as

the first half cycle; this should cancel each other. However,

FIG. 2. (Color online) Current density wave

forms used as input signal.

FIG. 3. (Color online) FEM mesh and current

density profile of flip-chip solder joint subjected

to ac loading, with nominal current density at

j ¼ 2:0� 104 sinð0:314tÞA=cm2.

063703-5 W. Yao and C. Basaran J. Appl. Phys. 111, 063703 (2012)

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in a flip-chip solder joint, current distribution is non-uniform

and asymmetric. At the current crowding corner (point A

and B of Fig. 1), current density will be 10 times higher than

the surrounding area as shown in Fig. 3. Moreover, skin-

effect during ac current stressing will cause most current

passing through the skin of solder joint with current density

decreasing into the center part. Due to this non-uniform dis-

tribution of current density, mass diffusion during positive-

cycle will not be totally reversed by the following negative

current cycle; this leads to vacancy accumulation in the sol-

der joint.

The hot spot due to joule heating is at the current crowd-

ing corner. Hence local temperature is higher than the vicin-

ity. According to the original representation developed by

Huntington,53 there will be a force that driving mass flow

down the temperature gradient according to the following

relation:

Ft ¼Q�TrT; (22)

where Ft is driving force due to temperature gradient; Q* is

the heat of transport; T and rT are temperature and thermal

gradient, respectively.

This force will drive the atoms from corner of solder

joint (hot spot) toward the center part; this accelerates the

EM effect at current crowding corner. Technically, there is

no specific cathode and anode during ac stressing because

the current continually varies its direction. However, for the

purpose of labeling, we define vacancy accumulation area as

cathode (point B of Fig. 1) and mass concentrating area as

anode (point A of Fig. 1). The initial vacancy concentration

is set to be unit everywhere. It can be seen from Fig. 5 that

the vacancy concentration depletes at point A and accumu-

lates at point B for all triangle, square and sine wave forms

of current inputs. At point A, the vacancy concentration fluc-

tuates up and down, but the average value depletes continu-

ously. However, at cathode, the vacancy concentration

fluctuates up and down in a band without changing its aver-

age value for a period of time, then jumps to a higher band.

The fluctuation period grows with increasing loading time.

This is a unique behavior that only observed at cathode side

of ac electromigration. It can also be observed that vacancy/

mass accumulation develops the fastest for square current

wave form, a little slower for sine waves, and the slowest for

the triangle waveform current stressing case. This agrees

with our expectation because the energy transferred to solder

joint per period is Esquare ¼ 2:47Esine ¼ 4Etriangle. Figure 6

shows the normalized vacancy concentration distribution

along diagonal A-B and central C-D line in solder joint after

FIG. 4. (Color online) Temperature profile of

flip-chip solder joint subjected to ac loading,

nominal current density in the solder joint at

j ¼ 2:0� 104 sinð0:314tÞA=cm2.

FIG. 5. (Color online) Vacancy concentration evolution history under ac

current loading of 0.05 Hz, with nominal current density at j ¼ 2:0�104 sinð0:314tÞA=cm2: (a) point A of Fig. 1, (b) point B.

063703-6 W. Yao and C. Basaran J. Appl. Phys. 111, 063703 (2012)

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6 h of ac stressing, at a frequency of 0.05 Hz and maximum cur-

rent density of 2:2� 106 A=cm2. It can be seen that vacancy

concentration increases along diagonal A-B in solder joint with

mass accumulating at anode (point A). Figure 6(b) shows that

atoms tend to accumulate at surface of solder joint and vacan-

cies form at the center. Both Figs. 6(a) and 6(b) show that

square current loading will cause most serious mass diffusion.

Figure 7 compares the vacancy concentration evolution behavior

at both cathode and anode side of solder joint for PDC, ac

and dc under the same maximum current density of

2:2� 106 A=cm2. The PDC loading has a square wave form

with 0.5 duty factor; ac loading has square, sine and triangle cur-

rent wave forms with frequency of 0.05 Hz. It can be seen that

the normalized vacancy concentration value fluctuates up and

down and decreases continuously in both PDC and ac loading

cases at anode, reaching 0.93 and 0.980.99, respectively, after

3 h of current stressing. For the dc loadings case, however, the

vacancy concentration decreases smoothly and reach 0.89 after

3 h current loading, which is 4% more than PDC and 9% more

than the ac case. The same behavior is observed at point B too,

and the vacancy concentrating corner is shown in Fig. 7(b).

B. Stress analysis of flip-chip solder joint under acstressing

Figure 8 shows the volumetric stress development at cor-

ner A and B of solder joint subjected to ac stressing with max-

imum current density of 2:2� 106 A=cm2, and a frequency

of 0.05 Hz. It can be seen that compressive stress develops at

point A and tensile stress at point B. At point A, the volumet-

ric stress for square, sine, and triangle wave form ac inputs

with the maximum current density of 2:2� 106 A=cm2 are

�9.5 MPa, �7 MPa, and �4.5 MPa, respectively, after 5 h of

stressing. At point B, the values of volumetric stress for

square, sine and triangle wave are 8 MPa, 5.5 MPa, and

3.8 MPa, respectively. The volumetric stress ratio among

square, sine, and triangle waveform is 2:1.5:1, which is equal

to the ratio comparing integration of square, sine, and triangle

current energy input for each period. This phenomenon could

be explained by the mass transport mechanism during EM.

The mass diffusion that causes compressive stress at anode

and tensile stress at cathode is mainly due to momentum

exchange between valance electrons and ions. Square current

wave supplies the largest amount of electrons per loading pe-

riod, a smaller amount for the sine wave form, and about half

of the square current for the triangle wave current loading

case. It can be seen in Fig. 8 that the volumetric stress at point

A reaches 22 MPa for dc loading case, 14 MPa for PDC with

0.5 duty factor, and only 25 MPa for solder joints under ac

loadings after 3 h of current stressing, at frequency of 0.05 Hz.

We see that mechanical stress development after 5 h ac current

FIG. 6. (Color online) Normalized vacancy concentration of solder joint after

6 h ac loading at frequency of 0.05 Hz, maximum current density

2:2� 106 A=cm2: (a) vacancy of diagonal A-B, (b) vacancy of central C-D.

FIG. 7. (Color online) Normalized vacancy concentration evolution history

for ac and PDC loading at frequency of 0.05 Hz, maximum current density

of 2:2� 106 A=cm2 with dc under otherwise same condition for compari-

son: (a) at point A, (b) at point B.

063703-7 W. Yao and C. Basaran J. Appl. Phys. 111, 063703 (2012)

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stressing is only 1/10 of that in dc case and about 1/5 of stress

during PDC loading.

The shear stress evolution history is shown in Fig. 9. It

can be seen that shear stress also fluctuates up and down

with an average value decreasing continuously. Shear stress

develops at both corner A and corner B and the magnitudes

at both regions are 2.5 MPa, 1.6 MPa, and 1.2 MPa for

square, sine, and triangle ac current loading, respectively.

However, the fluctuation band at point B (vacancy concen-

trating area) is larger than at point A (mass accumulation

area). After 3 h loading, shear stress reaches �11 MPa for dc

loadings, �6 MPa for PDC case, and only �1�2 MPa for

ac stressing at point A; �7.5 MPa for dc, �3.5 MPa for PDC

and �0.5�1.5 MPa for ac stressing at point B. Shear stress

developed at current crowding corner of flip-chip solder joint

during ac stressing is only 25% of under PDC and 10% of

the case subjected to dc loadings under otherwise the same

conditions.

C. Damage mechanics under ac stressing

The mass transport mechanism during electromigration

will induce two types of failure. On the anode side (mass

rich side), the mass accumulation will cause compressive

stress locally and eventually squeeze material out of the sur-

face to form protrusions called hillock. If the protrusions long

enough to contact with surrounding circuit, it will cause short

circuit failure. On the other side, however, mass depletion will

cause tensile stress and void accumulation, which will coa-

lesce into a crack eventually. With the direct banning of

Pb-containing solders in all electronic consumer products in

2006, Sn-rich eutectic SnAgCu, eutectic SnAg and eutectic

SnCu solders become the most promising replacements. At

room temperature, the diffusivity of Cu along a- and c-axes of

bSn (the metallic form or white tin) is about 0:5� 10�8 and

2� 10�6 cm2=s. It indicates that the mobility of Cu in Sn is

high enough that the growth of Cu-Sn inter-metallic com-

pound (IMC) can occur at room temperature.

In this work, the entropy based damage evolution model

originally proposed by Basaran51,54,55 was used to quantify

damage induced by thermal and electrical loadings. Due to

the retroactive healing effect in reverse-direction current

loading period, lifetime is expected to be higher in a solder

joint subjected to ac current loading than those of dc current

loadings. Figure 10 shows the damage evolution at both an-

ode and cathode sides of the solder joint. After applying

electric current load, damage will jump to an initial value,

FIG. 8. (Color online) Volumetric stress evolution in solder joint subjected

to ac (PDC and dc for comparison) stressing with maximum current density

2:2� 106 A=cm2, and frequency of 0.05 Hz: (a) at point A, (b) at point B.

FIG. 9. (Color online) Shear stress evolution in solder joint subjected to ac

(PDC with 0.5 duty factor and dc for comparison) stressing with maximum

current density 2:2� 106 A=cm2, and frequency of 0.05 Hz: (a) at point A,

(b) at point B.

063703-8 W. Yao and C. Basaran J. Appl. Phys. 111, 063703 (2012)

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then increase during current loading history. It can be seen

that the damage accumulates to 3:3� 10�4, 2:9� 10�4, and

1:4� 10�4 at anode after 5.5 h of square, sine, and triangle

current loadings, respectively. At cathode it is 2:2� 10�5,

1:4� 10�5, and 1:5� 10�5. We can see that damage at mass

accumulating area (point A) is nearly 10 times of at vacancy

concentrating area (point B). However, it is generally

accepted that vacancy accumulation is a bigger problem for

most solder joints, thus point B is what we are concerned

with in this study.

Figure 11 is the damage distribution after 6 h of ac cur-

rent stressing with frequency of 0.05 Hz and nominal current

density of 2:0� 104 A=cm2. It can be seen that most serious

damage happens at point A and decreases along diagonal A-

B. Because of the skin effect, damage along outer surface of

solder joint has a larger value than inside of the solder joint

as shown in Fig. 11(b). Square alternating current loads

induce the largest damage; sine wave causes a little smaller

damage and least damage for triangle wave loadings at

almost all parts of solder joint. Figure 12 compares the ac

electromigration induced damage with damage due to PDC

and dc at vacancy accumulation point B. It can be seen that

after 3 h of current stressing, the damage is about 1:2� 10�5

under ac loading condition, which is only 0.8% of the dam-

age 1:5� 10�3 observed in dc loading case, and 1.2% of

damage 1:0� 10�3 in PDC condition. This indicates lifetime

under ac electromigration could be hundreds times longer

than those under dc current loading with otherwise the same

conditions. It is attributed to the damage healing effect, i.e.,

the damage induced during the positive half-cycle is partially

FIG. 10. (Color online) Damage evolution of solder joint subjected to ac

stressing with maximum current density 2:2� 106 A=cm2, and frequency of

0.05 Hz: (a) at point A, (b) at point B. FIG. 11. (Color online) Damage map of solder joint subjected to 6.5 h of ac

loading of 0.05 Hz and maximum current density of 2:2� 106 A=cm2: (a)

damage distribution along diagonal A-B, (b) damage distribution along cen-

tral section C-D.

FIG. 12. (Color online) Damage evolution in solder joint at point B (cath-

ode) subjected to dc, PDC at duty factor of 0.5, and ac current stressing,

with maximum current density 2:2� 106 A=cm2 and frequency of 0.05 Hz

for PDC and ac loadings.

063703-9 W. Yao and C. Basaran J. Appl. Phys. 111, 063703 (2012)

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healed during the following negative half-cycle current

stressing.

Figure 13 is the ambient temperature dependence of dam-

age evolution at cathode side of solder joint during ac current

stressing. It can be seen that higher ambient temperature induces

a larger initial damage as expected, and grows at a faster speed.

After 5 h of current stressing, damage reaches 4:9� 10�5 for

ambient temperature of 450 K, 2:5� 10�5 for ambient tempera-

ture of 400 K, and only 1:0� 10�5 for ambient temperature of

350 K. Figure 14 shows current density dependence of damage

of solder joint subjected to sine ac current loading at frequency

of 0.05 Hz. High current density will induce a high initial dam-

age and faster damage growth rate. Figure 15 is the ac current

frequency dependence of EM damage. It can be observed that

higher frequency results a larger initial damage and higher

growth rate.

The non-linear regression analysis of damage in the

SAC405 solder joint as a function of ambient temperature, fre-

quency, and maximum current density is shown in Table II with

an R-square value (confidence value, which means representa-

tive of data points) as large as 0.99. It has been found that the

ambient temperature, frequency, and current density dependence

of damage follows the following exponential model:

D ¼ a� ebt � c� e�dt; (23)

where D is the entropy production rate based material damage

metric; a–d are all positive parameters that depend on ambient

temperature, frequency, and maximum current density. For a

loading period long enough (over 6 h), the second part of the

above equation will be only 1% of the first part, thus we can

ignore the second parts’ influence, which will simplify the

preceding equation into the following representation:

D ¼ a� ebtbf bjt; (24)

where a is a general parameter that depends on current wave

form, ambient temperature, and frequency, bt is a temperature

exponent parameter, bf is a frequency exponent parameter, bj

is a current density exponent parameter, and t is current load-

ing time in hours. From Table II, it can be seen that the tem-

perature parameter bt grows with increasing temperature.

Non-linear regression gives us following relationship between

bt and temperature T in Celsius with R-square of 0.95:

bt ¼ 0:00052� T1:1: (25)

For constant ambient temperature at 350 K and frequency

of 0.05 Hz, relationship between bj and current density jin A=cm2 could be given as in Eq. (15) with a confidence

parameter of 0.98:

bj ¼ 0:0015� j0:27: (26)

By following the same procedure, we obtain the following

relationship between ac frequency f and parameter bf with a

confidence parameter R-square as 0.985:

bf ¼ 0:344� f 0:41: (27)

By combing all three parameters together, we obtain the EM

and TM damage model of solder joint under ac current

stressing as:

D ¼ a� ebT1:1f 0:41j0:27t: (28)

We further define the failure of solder joint when D¼ 0.01 in

this study for simplicity. We realize that different application

FIG. 13. (Color online) Temperature dependence of damage evolution in

solder joint at point B (cathode) subjected to ac current stressing, with maxi-

mum current density 2:2� 106 A=cm2 and frequency of 0.05 Hz.

FIG. 14. (Color online) Current density dependence of damage evolution of

solder joint at point B (cathode) subjected to ac current stressing, with ambi-

ent temperature at 350 K and frequency of 0.05 Hz.

FIG. 15. (Color online) Frequency dependence of damage evolution of sol-

der joint at point B (cathode) subjected to ac current stressing, with maxi-

mum current density 2:2� 106 A=cm2 and frequency of 0.05 Hz.

063703-10 W. Yao and C. Basaran J. Appl. Phys. 111, 063703 (2012)

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will require different failure definition. Basaran et al.9 pro-

posed a MTF equation for solder joint subjected to dc based

on extensive laboratory testing. By introducing ambient tem-

perature, current density and frequency influence on this

equation, the following MTF equation is obtained:

MTF ¼ aT1:1 � f 0:41 � j0:27

; (29)

where a is parameter that depends on ac input waveform and

material property, T is temperature in Celsius, f is loading

frequency, j is current density in A=cm2, and MTF is in

hours. After incorporating the activation energy, we can

obtain the MTF for ac electromigration as follow:

MTF ¼ aT1:1 � f 0:41 � j0:27

eðrE=kTÞ; (30)

where E is vacancy activation energy and k is Boltzmann’s

constant. We are in the process of conducting extensive

experiments will be conducted to verify this model.

V. CONCLUSIONS

A fully coupled thermal-electrical-mechanical simula-

tion of ac electromigration and thermomigration was con-

ducted for SAC405 solder joints. Thermodynamics based

damage mechanics model as well as a viscoplastic material

model were used to simulate the material degradation mech-

anism during thermal and electrical loading. Square, sine,

and triangle current wave forms were used as ac input sig-

nals. Solder joints EM analysis subjected to dc and PDC cur-

rent stressing were also performed for comparison purpose.

The vacancy concentration, mechanical stress evolution and

damage accumulation behavior were studied.

Contrary to popular belief that ac damage is completely

reversible, it is shown that due to nonhomogeneous and

asymmetric current distribution in the solder joint, ac will

lead to electromigration failure, albeit with a longer lifetime

compared to dc loading. It is also observed that because of

skin effect most of the damage accumulates near the surface.

Vacancy concentration fluctuates at anode side during the ac

loading history with average vacancy concentration decreasing

continually. However, at cathode (vacancy concentrating area),

vacancy accumulation will oscillate in a band for a period of

time with a constant average value, then jump to a higher band

and continue its fluctuation. It is also noticed that the oscillation

period grows during loading history. Mass hillock will happens

at surface of solder joint and vacancy concentrates at the center.

Both mass diffusion and stress developments under ac are slower

than PDC and dc stressing.

Irreversible entropy production rate during electromigra-

tion was used as a metric to quantify the damage. It was found

that damage under ac stressing is only 1% of that for dc load-

ings under otherwise the same conditions. Increasing ambient

temperature induces a higher damage, which follows relation-

ship of DaT1:1. Higher current density will result in higher

damage following D / j0:27. When the ac period is far above

the thermal relaxation time of the material that is in the order of

10�6 s, it has been found that increasing frequency will cause

more serious damage and the relationship is obtained between

damage and frequency as D / f 0:41. Square current wave input

induces the most serious damage and less for sine current wave

form, and triangle current loading causes least damage. A new

mean time to failure model is proposed for solder joint sub-

jected to ac current stressing based on Basaran’s MTTF empiri-

cal model, which takes into consideration thermal effect and

viscoplastic material behavior of SAC solder alloy.

ACKNOWLEDGMENTS

This research project has been sponsored by the U.S.

Navy, Office of Naval Research, Advanced Electrical Power

Program under the direction of Terry Ericsen.

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TABLE II. Nonlinear regression analysis of ambient temperature, frequency and maximum current dencity dependence of damage at cathode of solder joint

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Dependence

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Frequency

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Current density (A/cm2)

Damage vs

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Adjust

R-square

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350 0.05 2:2� 106 D ¼ 8:5� 10�6 � e0:078t � 6:8� 10�6 � e�0:63t 0.998

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Frequency

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Current density

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350 0.05 5:0� 106 D ¼ 2:2� 10�5 � e0:098t � 1:3� 10�5 � e�1:2t 0.999

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