A computational damage mechanics model for thermomigration
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Author's personal copy Mechanics of Materials 41 (2009) 271–278
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A computational damage mechanics model for thermomigration Shidong Li, Cemal Basaran * Electronic Packaging Laboratory, State University of New York at Buffalo, 212 Ketter Hall, Buffalo, NY 14260, USA
a r t i c l e
i n f o
Article history: Received 24 January 2008 Received in revised form 26 October 2008
a b s t r a c t Miniaturization of electronics to nanoscale leads to significantly higher current density levels and larger thermal gradients in electronics packaging. Laboratory test data show that thermomigration plays a significant role in high current density induced failure in solder joints and interconnects. In this paper, a computational damage mechanics model for thermomigration process is proposed and implemented in finite element method. This model is based on thermodynamics and formulated by continuum mechanics equations, mass transport principals and heat transfer equations. A damage evolution model using entropy production rate as a metric is utilized to evaluate the degradation in solder joints subjected to high temperature gradients. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction The number of integrated circuits in a package increases with Moore’s Law. Interconnect and solder joint sizes are reduced accordingly to enable a higher degree of integration. By transmitting the same amount of data through smaller features at a faster speed, current density and temperature gradients are increased. Moreover, in power electronics devices, especially SiC devices which operate at higher temperatures, significant thermal gradients occur. This increased current density and temperature gradients can reach a critical level especially within the solder joints of an electronic package quickly, which leads to failure of whole devices Joule heating generated under a high electrical current density is highly localized in current crowding points, especially in SiC power electronics devices. Therefore, there is a significant thermal gradient in the medium, which leads to thermomigration induced failure(TM). Thermomigration is a mass transport of compound material caused by a temperature gradient. If an initially homogeneous two-component phase is placed within a temperature gradient, mass diffusion can lead to decomposition of the constituents. One component diffuses preferentially faster to the hot end, and as a result the hot * Corresponding author. Tel.: +1 716 645 2114; fax: +1 716 645 3733. E-mail address: [email protected] (C. Basaran). 0167-6636/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2008.10.013
region becomes enriched in that constituent. This effect is called the Soret effect (also known as Ludwig–Soret effect) in fluids. In solids this resulting diffusion is called thermomigration. (Platten, 2006). Although the TM phenomenon in fluids has been observed for about 150 years, almost all previous works ignored the role of TM in electromigration (EM) failure of electronics packaging. This is perhaps because thus far the magnitude of TM flux has been assumed to be much smaller than EM. Recently scientists theoretically and experimentally deduced that when the thermal gradient is large enough, the thermomigration can be the dominant mass transport (Ru, 2000; Ye et al., 2003a; Abdulhamid and Basaran, in press). Additionally, some experimental work has been performed to show that indeed TM can play a significant role in EM-induced failure in thin film cracking (Bastawros and Kim, 1998). Ye et al. (2003a) was the first to show that TM forces under high current density can be larger than EM in flip chip solder joints. 1.1. Physics of thermomigration process The atomic level physics behind thermomigration in solids is not well understood, but the most established hypothesis is that it is very much similar to electromigration. It is assumed that physical mechanism used to model electromigration can also be used to model thermomigra-
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tion process. In the absence of atomic level observations, this approach is justified. In metals the primary force acting on the diffusing atom (or ion) is the momentum exchange from the collisions of scattering valence electrons. When there is no electrical current flow, the driving force in the direction of the temperature gradient results from the fact that the energy of the valence electrons at higher temperatures is greater than that of the valence electrons in the lower temperature region. This gradient in energy drives electrons from hot to cold region. During this process due to scattering of electrons they bump into ions and transfer part of their momentum to ions, which start moving with the ions from hot to cold region. This momentum exchange produces a driving force for mass transport, which is usually noted as Q* (Lloyd and Tu, 2004; Basaran et al., 2003). In alloys, due to different diffusivity and atom size of the constituents, thermomigration happens at different rates for different components. In most metals, Q* is small, and thermomigration is generally negligible. However, this is not true for solder alloys because of their very low melting temperature (around 200 °C). This effect must be taken into consideration when investigating the reliability of solder joint as is shown by Ye et al., 2003b. In SiC based power electronics due to high operating temperature, Q* becomes considerable also for interconnects and vias. 1.2. Thermomigration induced damage Damage, in the context of continuum damage mechanics is defined as the progressive deterioration of materials before final failure (Basaran and Yan, 1998; Basaran and Nie, 2004). Damage mechanics establishes fundamental principles for prediction of fatigue life of solid structures. A thermodynamic based formulation (Basaran et al., 2003; Basaran and Nie, 2004; Gomez and Basaran, 2005; Tang and Basaran, 2001; Gomez and Basaran, 2006; Tang and Basaran, 2003) has been proposed by Basaran et al. and has been successfully implemented for low cycle fatigue failure of solder joints and particle filled composite. This is a model based on thermodynamics and continuum damage mechanics and it uses entropy production rate as damage metric. This concept has also been proposed by Crooks for irreversible thermodynamic systems (Crooks, 2007). The objective of the model presented in this paper is to unify thermo mechanical induced stress, thermal field induced damage in the context of this theory. In the mean time, this damage model will include the most important features which will affect material damage process including stress, inelastic strain, diffusion, boundary conditions, etc.
2.1. Mass conservation Vacancy diffusion is governed by the following conservation equation
C V0
@c þ rq G ¼ 0 @t
ð2:1Þ
where CV0, equilibrium vacancy concentration in the absence of any stress field; c, normalized vacancy concentration and c ¼ CCVV0 ; CV, vacancy concentration; t, time; q, vacancy flux (Basaran et al., 2003; Lin and Basaran, 2005; Sarychev et al., 1999) is given by the following equations.
cf X c q ¼ DV C V0 rc þ rrspherical þ 2 Q rT kT kT
ð2:2Þ
where DV, vacancy diffusivity; f, vacancy relaxation ratio, ratio of vacancy volume to the volume of an atom; X, atomic volume; k, boltzman’s constant; T, absolute temperaturerspherical, spherical part of stress tensor, rspherical ¼ traceðrij Þ=3. Q* is heat of transport, the isothermal heat transmitted by the moving atom in the process of jumping a lattice site less the intrinsic enthalpy. G Vacancy generation/annihilation rate
G ¼ L v lv
ð2:3Þ
where Lv, is a rate parameter, which can be found from characteristic generation time; lv, is the vacancy chemical potential. By assuming isothermal condition it can be written as (Sarychev et al., 1999).
lv ¼ kT ln
Cv C ve
ð2:4Þ
where Cve, is the temperature and stress dependent equilibrium vacancy concentration, given by.
C ve ¼ C v0 exp f 0 Xrspherical =kT
ð2:5Þ
where f = 1 f, is the volumetric strain at an atomic lattice site. By inserting (2.5) into (2.4), after some algebraic transformation the following can be written,
G ¼ Lv kT ln
Cv Lv f 0 Xrspherical C v0
ð2:6Þ
2.2. Force equilibrium The differential equations of force equilibrium can be expressed as
rij;j þ fiB ¼ 0
ð2:7Þ
where rij is the stress tensor and fiB stands for the boundary conditions.
2. Model formation 2.3. Heat transfer Thermomigration is diffusion controlled mass transport process. It is governed by the mass and energy conservation equations. The mechanical equilibrium of the medium is governed by the force equilibrium equation and heat transfer is governed by the Fourier’s law.
Heat transfer is governed by the following equation.
qC
@T rðkh rT Þ qQ ¼ 0 @t
ð2:8Þ
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where q, density of the material; C, specific heat; kh, coefficient of heat transfer; Q, heat density 3. Thermodynamics of damage evolution due to thermomigration It has been shown by Basaran et al. (2003), Basaran and Yan (1998), Gomez and Basaran (2005), Tang and Basaran (2001) that entropy production can be used as a damage metric. This model exploits Boltzmann equation to correlate damage and irreversible entropy production. A variable D is defined as the ratio of the change in disorder parameter in Boltzmann equation to the original reference state as follows:
D ¼ Dcr 1 e
s0 s N 0 k N0 k
!
h
i Ds
¼ Dcr 1 eN0 k
ð3:1Þ
In this equation, Dcr is the critical damage parameter used to define the failure for specific application, whose value depends on the definition of failure threshold. Dcr can also be used in a function form to correlate entropy production with specific mechanical property degradation. s0 and s are the entropy of a unit volume at the initial reference state and current state, respectively. k is the Boltzmann’s constant. N0 is Avogadro’s number. D value starts as zero and reaches D = Dcr at the final stage. It is shown that the entropy production in an irreversible process caused by thermomigration can be expressed by
Ds ¼
Z t t0
2 ! C V Deffective kT Q rT dt rC f Xrrspherical þ 2 C T kT ð3:2Þ
The damage evolution formula can be obtained by plugging (3.2) into (3.1)
2 6 D ¼ Dcr 6 41 e
R t CV Deffective t0
kT 2
2
ðkTC rCf Xrrspherical þQ TrT Þ N0 k
3 dt
7 7 5
ð3:3Þ
Derivation of (3.2) and (3.3) is rather long. Here it is omitted for brevity. 4. Finite element analysis Finite element model of an actual flip chip solder joint is shown in Fig. 1. The solder alloy is lead free 95.5Sn–4Ag– 0.5Cu (SAC405). 8-node plain strain elements with a unit thickness are used to mesh the solder ball model. The diameter of the solder bump is 140 lm while the standoff height is 100 lm. The Al trace interconnect on the die is 1 lm thick while the Cu trace is 15 lm thick (Lai and Kao, 2006). In a finite element mesh, all interfaces have 100% perfect adhesion. However, in actual solder joints, it is well known that even in a good joint, the effective contact area is no more than 50% of the gross interface area (Un-Byoung and Young-Ho, 2004; Nah and Suh, 2005). Usually the percentage of interface actually in contact can be as low as 10%. This topic has been extensively
Fig. 1. Finite element mesh for solder joint.
studied in electronic packaging literature. Hence it is outside the scope of this paper. To consider the imperfect interface contact in between solder ball and Al/Cu interconnect traces, two interface layers of 1lm thick with 50% electrical conductivity are introduced individually. These assumptions are supported by experimental finding reported by Ye et al. (2003a). Material properties for SAC405 used in analysis are listed below. Young’s Modulus E = (57.7–0.056T) GPa (Hong, 1998), Poisson ratio m = 0.33 Hong, 1998, Yield stress ry = (140.67–0.2133T) MPa (Siviour et al., 2005), Equilibrium Vacancy concentration at a stress free state is 1.107 106 lm3 (Ye, 2004; Balzer and Sigvaldason, 1979), vacancy relaxation time is 0.0018 s(Sarychev et al., 1999), average vacancy relaxation ratio is 0.2(Sellers et al., 2007; Bassman, 1999). The thermal properties are listed in Table 1. The molar heat of transport Q* of eutectic SAC405 alloy is 22.16 kJ/mol (Chaung and Lin, 2003). An average current density of 1 104 A/cm2 is applied. The ambient temperature is set to 25 °C. 5. Numerical simulation results A steady state Joule Heating analysis is conducted. The current comes from the Aluminum trace from the right side into the solder joint. After passing through the solder ball, it goes out into the copper trace. Due to the different conductivity of various materials and the discrepancy in thickness, the current distribution is not uniform throughout the section. The current density at the Aluminum interconnect solder joint corner point can be 100 times larger than the average current density (Tu et al., 2000). This is
Table 1 Material properties.
Density (kg/lm3) Thermal conductivity (W/lm K) Electrical conductivity (O1lm1) Specific heat (J/kg K)
SAC405
Copper
Aluminum
7.39 1015 57.3 106
8.92 1015 4.16 104
2.70 1015 2.38 104
6.67
59.17
38.17
200
385
902
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due to so-called current crowding effect. Although current crowding effect has been known for a long period, so far there has been no effective method to quantify it. The reason is mathematical singularity. Theoretically speaking the current density at this singular point should be infinite. Such singular point does not exist in reality because it will melt immediately by the highly localized Joule heat and reshaped by the surface tension. In finite element analysis, it will lead to a mesh sensitivity problem (Lai and Kao, 2006; Basaran and Zhao, 2001), which means the current crowding factor at the corner point obtained by FEM cannot converge to a constant even if an extremely fine mesh is used. It should be pointed out that due to Joule heating effect and melting, classical asymptotic analysis used for singularities in stress fields is no longer suitable here (Basaran and Zhao, 2001). However, the results obtained using certain mesh density though may not be accurate quantitatively but still are valuable in the study of current crowding effect and the induced Joule heating in the near region around the corner point at present. The analysis is divided into two steps. By using the FEM mesh shown in Fig. 1, ABAQUS Standard is first utilized to conduct a coupled thermal-electrical analysis. However, ABAQUS and other commercially available finite element codes do not have the capability to solve the general thermomigration problem directly. In the second step, ABAQUS is adapted to analyze coupled mass diffusion, heat transfer and mechanical stressing process and their induced damage by coding with ABAQUS user element and material (UEL and UMAT). To minimize the influence of thermal expansion induced stress, both aluminum trace and copper trace are free to deform. Also we compulsorily remove the current to study the effect of thermomigration in the second step. Fig. 2 shows the current density map in the solder ball. It can be seen that current crowding occurs in the right upper corner where the current enters from the aluminum trace and the left lower corner where the current exits to the copper trace. The current density distribution along the section from left lower corner (point A in Fig. 1) to the right upper corner (point B in Fig. 1) is shown in Fig. 3. From Fig. 3, it is observed that the highest current density appears at point B, jB = 1.768 105 A/cm2, which is more than 10 times higher than the average current den-
Fig. 2. Current density distribution map.
sity in the middle of the solder ball. The left lower corner is another current crowding region. At point A, jA = 5.06 104 A/cm2, which is 5 times the average nominal current density. It is widely accepted that the following two factors are the major sources of high temperature gradient in solder joints. Because it is very thin, the high current density in the aluminum trace produces much more Joule heating than in solder joint and copper pad. A considerable temperature gradient will thus be created from the hot side (aluminum trace side) to the cold side (copper pad side). Another important source of highly localized temperature gradient is the imperfect contact between interfaces. The thermal and electrical resistance in interface is far higher than Al interconnect or solder joint. It will accordingly produce more Joule heating in the interfaces than in the other place, which results in a localized temperature gradient in the nearby region. The temperature map for Joule heating analysis is shown in Fig. 4. The temperature and temperature gradient distribution in the cross section of diagonal line A–B, and the cross section in the center line from the bottom side to the top side of the solder ball, as are shown in Figs. 5 and 6. We found that although the temperature in corner B is only a couple degrees above the other locations in
Current Density (104A/cm2)
20
15
10
5
0
0
50
100
Distance (micron) Fig. 3. Current density along section A–B.
150
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1.048992
40
1.040232 1.031472 20 1.022712 1.013952 0 1.005192 0.996432 -20
0.987672 0.978912 0.9684
-40 280
300
320
340
360
380
400
Fig. 7. Vacancy concentration contour map. Fig. 4. Temperature distribution map.
35
Temperature(°C)
Diagonal Line Center Line
33 31 29 27 25
0
0.2
0.4
0.6
0.8
1
Relative Distance Fig. 5. Comparison of temperature distribution.
Temperature Gradient (×102°C/cm)
the top side, the temperature gradient at B is much higher. Therefore, we can say that Joule heating induced by current crowding is the major source of the high temperature gradient in solder joint. Fig. 6 shows that the temperature gradient at point B is about 1500 °C/cm. The vacancy concentration contour map after 60 h of temperature gradient stressing is shown in Fig. 7. It is very similar to that of temperature distribution.
15
From Fig. 7 we can find that the atoms at the right upper part are depleting and migrating to the left lower part. Thus in the hot region, the vacancies are accumulating. In the cold region, the vacancies are filled by the immigrating atoms. As is seen in Fig. 7, the number of vacancies at corner point A goes down by 5% after applying thermal loads for 100 h. Fig. 8 shows the vacancy concentration distribution in the cross section A–B after applying thermal loads for 100 h. Here we can clearly see that vacancy concentration increases in the hot side (corner B) and it decreases in the cold side (corner A). By investigation into the time history of vacancy concentration (Fig. 9), we found that over 90% thermomigration happens in the first 20 h. This phenomenon can be explained by looking into the driving force mechanism in the thermomigration process. In the absence of electrical current, the vacancy transport is governed by three driving forces, (1) the vacancy concentration gradient, (2) temperature gradient and (3) back stress gradient. In our example, the initial assumed vacancy concentration is uniform throughout the solder joint. The initial stress gradient is also zero everywhere. So at first the only driving force comes from the temperature gradient, which will move the atoms from the hot side to the cold side. After a period of time, the accumulating vacancy concentration gradient and back stress gradient induce driving force that will counteract on the process by pushing the atoms back.
Diagonal Line Center Line
10
5
0
0
0.2
0.4
0.6
0.8
Relative Distance Fig. 6. Comparison of temperature gradient distribution.
1
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Normalized vacancy concentration
1.06
1.04
1.02
1
0.98
0.96 0
20
40
60
80
100
120
140
Distance (micron) Fig. 8. Vacancy concentration distribution.
Normalized Vacancy Concentration
1.08 Region B Region A
1.04
1.00
0.96 0
20
40
60
80
100
Time (hour) Fig. 9. Vacancy concentration evolution with time.
When the driving forces reach balance, the migration will stop. The strain and Von Mises stress field after 100 h are shown in Fig. 10 and Fig. 11. From Fig. 10 and 11, we can tell that the hot region is under tension and the cold region is in compression. This is due to the movement of atoms from the hot side to the cold side. The mass accumulation
at the cold side will result in compressive stresses, while the vacancy accumulation at the hot side will result in tensile field. The stress induced by thermomigration cannot be overlooked. The damage evolution contour, as defined by (3.3), across the solder joint after 100 h is shown in Fig. 12. It shows that the thermomigration induced damage grows
0.001079 0.000828 0.000577 0.000326 7.5E-005 -0.000176 -0.000427 -0.000678 -0.000929 -0.00118 -0.001431 -0.001682 -0.001933 -0.002184 -0.002435 -0.002686 -0.002937 -0.003188 -0.003439 -0.00369
40
20
0
-20
-40 280
300
320
340
360
380
Fig. 10. Volumetric strain after 100 h.
400
Fig. 11. Von Mises stress (MPa).
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balanced by the stress gradient and concentration gradient very quickly. Therefore the damage reaches a steady state very quickly in these regions. However, the same cannot be applied for the higher temperature gradient regions where far more damage will be created. When the temperature gradient exceeds a critical threshold, the counter action cannot balance the thermomigration driving force. Damage will keep accumulating until it fails, as can be seen in Fig. 14. In order to determine the critical damage value for void nucleation, which is called incubation time (Basaran et al., 2003), a regression method should be developed to determine the total time as a function of damage. From Fig. 14, we can find that under high temperature gradient, damage evolves exponentially with time. In this model, when the damage value reaches 1, the material will lose all the load carrying capacity. Definition of failure is application specific. If we define failure by 50% loss of the stiffness (Stephens et al., 2001), as can be seen from Fig. 14, the time to failure prediction will be 2168 h.
0.0007 40
0.0006 0.0005 0.0004 0.0003 0.00025
20
0.0002 0.00015 0.0001 9E-005
0
8E-005 7E-005 6E-005 5E-005
-20
4E-005 3E-005 2E-005 1E-005 0
-40 280
300
320
340
360
380
400
Fig. 12. Damage variable contour.
in the hot region. The maximum damage occurs at the right upper corner, where the peak temperature gradient locates. By investigating the time history of the damage growth at point B, C, D, E and F, (Fig. 13) can be developed. In this study, we assume that vacancy accumulation leads to failure, however, mass accumulation does not. Therefore, damage is only plotted for hot side. This is in agreement with experimental observations (Ye et al., 2003a,b; Abdulhamid, and Basaran, In press; Tu et al., 2000). In Fig. 13, we can see that at low temperature gradient region (C, D, E, F), the thermomigration driving force can be 8
A fully coupled temperature-diffusion-displacement damage mechanics model for thermomigration process is presented in this paper and associated UMAT and UEL codes have been developed and implemented in
B C D E F
7
Damage (×10-4)
6. Conclusions
6 5 4 3 2 1 0 0
10
20
30
40
50
60
70
80
90
100
Time (hour) Fig. 13. Damage evolution.
0.6
Damage
0.5 0.4 0.3 0.2 0.1 0 0
500
1000
1500
2000
Time (hour) Fig. 14. Damage evolution at B in 2500 h.
2500
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ABAQUS. By analyzing the thermomigration process in a SAC405 solder joint under the Joule heating induced high temperature gradient, it is shown that thermomigration can play a very important role in electromigration induced failure as has been proven by Ye et al., 2003a experimentally. The important findings of this paper can be listed as follows: (1) Joule heating due to current crowding effect can be the dominant factor that causes thermomigration. For instance, in this example, the current density in the right upper region is much larger than elsewhere. Hereby temperature gradient in point B is more than 10 times of the average temperature gradient across the solder joint. (2) Most of the thermomigration happens in the first period before counteracting stress gradient and vacancy gradient forces develop. The vacancy flux will lessen with time due to counterbalance forces and finally it will approach to a steady state. In this example, over 90% thermomigration happens in the first 20 h. (3) When the temperature gradient is small, the thermomigration driving force can be balanced by the spherical stress gradient and concentration gradient. The damage will reach a steady state quickly. However, when the temperature gradient is large, the damage will keep growing until it fails. (4) Complex conditions like high current density, high temperature gradient, and damage accumulation occur simultaneously in the corner of the solder connection, which make it critical region for design. (5) Proposed damage model does not utilize a phenomenological damage potential surface. Instead entropy production is used as a metric.
Acknowledgement This project has been sponsored by US Navy Office of Naval Research Advanced Electrical Power Systems under the direction of Terry Ericsen. References Abdulhamid, M., Basaran, C., in press. Influence of Thermomigration on Lead-Free Solder Joint Mechanical Properties. ASME Journal of Electronic Packaging. Balzer, R., Sigvaldason, H., 1979. Equilibrium Vacancy Concentration Measurements on Tin Single Crystals. Physica Status Solidi B: Basic Research 92 (1), 143–147. Basaran, C., Nie, S., 2004. An irreversible thermodynamics theory for damage mechanics of solids. International Journal of Damage Mechanics 13 (3), 205–223. Basaran, C., Yan, C., 1998. A thermodynamic framework for damage mechanics of solder joints. Journal of Electronic Packaging, Transactions of the ASME 120, 379–384. Basaran, C., Zhao, Y., 2001. Mesh sensitivity and Fea for multi-layered electronic packaging. Journal of Electronic Packaging 123 (3), 218– 224. Basaran, C., Lin, M., Ye, H., 2003. A thermodynamic model for electrical current induced damage. International Journal of Solids and Structures 40 (26), 7315–7327.
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A computational damage mechanics model for thermomigration Shidong Li, Cemal Basaran * Electronic Packaging Laboratory, State University of New York at Buffalo, 212 Ketter Hall, Buffalo, NY 14260, USA
a r t i c l e
i n f o
Article history: Received 24 January 2008 Received in revised form 26 October 2008
a b s t r a c t Miniaturization of electronics to nanoscale leads to significantly higher current density levels and larger thermal gradients in electronics packaging. Laboratory test data show that thermomigration plays a significant role in high current density induced failure in solder joints and interconnects. In this paper, a computational damage mechanics model for thermomigration process is proposed and implemented in finite element method. This model is based on thermodynamics and formulated by continuum mechanics equations, mass transport principals and heat transfer equations. A damage evolution model using entropy production rate as a metric is utilized to evaluate the degradation in solder joints subjected to high temperature gradients. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction The number of integrated circuits in a package increases with Moore’s Law. Interconnect and solder joint sizes are reduced accordingly to enable a higher degree of integration. By transmitting the same amount of data through smaller features at a faster speed, current density and temperature gradients are increased. Moreover, in power electronics devices, especially SiC devices which operate at higher temperatures, significant thermal gradients occur. This increased current density and temperature gradients can reach a critical level especially within the solder joints of an electronic package quickly, which leads to failure of whole devices Joule heating generated under a high electrical current density is highly localized in current crowding points, especially in SiC power electronics devices. Therefore, there is a significant thermal gradient in the medium, which leads to thermomigration induced failure(TM). Thermomigration is a mass transport of compound material caused by a temperature gradient. If an initially homogeneous two-component phase is placed within a temperature gradient, mass diffusion can lead to decomposition of the constituents. One component diffuses preferentially faster to the hot end, and as a result the hot * Corresponding author. Tel.: +1 716 645 2114; fax: +1 716 645 3733. E-mail address: [email protected] (C. Basaran). 0167-6636/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2008.10.013
region becomes enriched in that constituent. This effect is called the Soret effect (also known as Ludwig–Soret effect) in fluids. In solids this resulting diffusion is called thermomigration. (Platten, 2006). Although the TM phenomenon in fluids has been observed for about 150 years, almost all previous works ignored the role of TM in electromigration (EM) failure of electronics packaging. This is perhaps because thus far the magnitude of TM flux has been assumed to be much smaller than EM. Recently scientists theoretically and experimentally deduced that when the thermal gradient is large enough, the thermomigration can be the dominant mass transport (Ru, 2000; Ye et al., 2003a; Abdulhamid and Basaran, in press). Additionally, some experimental work has been performed to show that indeed TM can play a significant role in EM-induced failure in thin film cracking (Bastawros and Kim, 1998). Ye et al. (2003a) was the first to show that TM forces under high current density can be larger than EM in flip chip solder joints. 1.1. Physics of thermomigration process The atomic level physics behind thermomigration in solids is not well understood, but the most established hypothesis is that it is very much similar to electromigration. It is assumed that physical mechanism used to model electromigration can also be used to model thermomigra-
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tion process. In the absence of atomic level observations, this approach is justified. In metals the primary force acting on the diffusing atom (or ion) is the momentum exchange from the collisions of scattering valence electrons. When there is no electrical current flow, the driving force in the direction of the temperature gradient results from the fact that the energy of the valence electrons at higher temperatures is greater than that of the valence electrons in the lower temperature region. This gradient in energy drives electrons from hot to cold region. During this process due to scattering of electrons they bump into ions and transfer part of their momentum to ions, which start moving with the ions from hot to cold region. This momentum exchange produces a driving force for mass transport, which is usually noted as Q* (Lloyd and Tu, 2004; Basaran et al., 2003). In alloys, due to different diffusivity and atom size of the constituents, thermomigration happens at different rates for different components. In most metals, Q* is small, and thermomigration is generally negligible. However, this is not true for solder alloys because of their very low melting temperature (around 200 °C). This effect must be taken into consideration when investigating the reliability of solder joint as is shown by Ye et al., 2003b. In SiC based power electronics due to high operating temperature, Q* becomes considerable also for interconnects and vias. 1.2. Thermomigration induced damage Damage, in the context of continuum damage mechanics is defined as the progressive deterioration of materials before final failure (Basaran and Yan, 1998; Basaran and Nie, 2004). Damage mechanics establishes fundamental principles for prediction of fatigue life of solid structures. A thermodynamic based formulation (Basaran et al., 2003; Basaran and Nie, 2004; Gomez and Basaran, 2005; Tang and Basaran, 2001; Gomez and Basaran, 2006; Tang and Basaran, 2003) has been proposed by Basaran et al. and has been successfully implemented for low cycle fatigue failure of solder joints and particle filled composite. This is a model based on thermodynamics and continuum damage mechanics and it uses entropy production rate as damage metric. This concept has also been proposed by Crooks for irreversible thermodynamic systems (Crooks, 2007). The objective of the model presented in this paper is to unify thermo mechanical induced stress, thermal field induced damage in the context of this theory. In the mean time, this damage model will include the most important features which will affect material damage process including stress, inelastic strain, diffusion, boundary conditions, etc.
2.1. Mass conservation Vacancy diffusion is governed by the following conservation equation
C V0
@c þ rq G ¼ 0 @t
ð2:1Þ
where CV0, equilibrium vacancy concentration in the absence of any stress field; c, normalized vacancy concentration and c ¼ CCVV0 ; CV, vacancy concentration; t, time; q, vacancy flux (Basaran et al., 2003; Lin and Basaran, 2005; Sarychev et al., 1999) is given by the following equations.
cf X c q ¼ DV C V0 rc þ rrspherical þ 2 Q rT kT kT
ð2:2Þ
where DV, vacancy diffusivity; f, vacancy relaxation ratio, ratio of vacancy volume to the volume of an atom; X, atomic volume; k, boltzman’s constant; T, absolute temperaturerspherical, spherical part of stress tensor, rspherical ¼ traceðrij Þ=3. Q* is heat of transport, the isothermal heat transmitted by the moving atom in the process of jumping a lattice site less the intrinsic enthalpy. G Vacancy generation/annihilation rate
G ¼ L v lv
ð2:3Þ
where Lv, is a rate parameter, which can be found from characteristic generation time; lv, is the vacancy chemical potential. By assuming isothermal condition it can be written as (Sarychev et al., 1999).
lv ¼ kT ln
Cv C ve
ð2:4Þ
where Cve, is the temperature and stress dependent equilibrium vacancy concentration, given by.
C ve ¼ C v0 exp f 0 Xrspherical =kT
ð2:5Þ
where f = 1 f, is the volumetric strain at an atomic lattice site. By inserting (2.5) into (2.4), after some algebraic transformation the following can be written,
G ¼ Lv kT ln
Cv Lv f 0 Xrspherical C v0
ð2:6Þ
2.2. Force equilibrium The differential equations of force equilibrium can be expressed as
rij;j þ fiB ¼ 0
ð2:7Þ
where rij is the stress tensor and fiB stands for the boundary conditions.
2. Model formation 2.3. Heat transfer Thermomigration is diffusion controlled mass transport process. It is governed by the mass and energy conservation equations. The mechanical equilibrium of the medium is governed by the force equilibrium equation and heat transfer is governed by the Fourier’s law.
Heat transfer is governed by the following equation.
qC
@T rðkh rT Þ qQ ¼ 0 @t
ð2:8Þ
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where q, density of the material; C, specific heat; kh, coefficient of heat transfer; Q, heat density 3. Thermodynamics of damage evolution due to thermomigration It has been shown by Basaran et al. (2003), Basaran and Yan (1998), Gomez and Basaran (2005), Tang and Basaran (2001) that entropy production can be used as a damage metric. This model exploits Boltzmann equation to correlate damage and irreversible entropy production. A variable D is defined as the ratio of the change in disorder parameter in Boltzmann equation to the original reference state as follows:
D ¼ Dcr 1 e
s0 s N 0 k N0 k
!
h
i Ds
¼ Dcr 1 eN0 k
ð3:1Þ
In this equation, Dcr is the critical damage parameter used to define the failure for specific application, whose value depends on the definition of failure threshold. Dcr can also be used in a function form to correlate entropy production with specific mechanical property degradation. s0 and s are the entropy of a unit volume at the initial reference state and current state, respectively. k is the Boltzmann’s constant. N0 is Avogadro’s number. D value starts as zero and reaches D = Dcr at the final stage. It is shown that the entropy production in an irreversible process caused by thermomigration can be expressed by
Ds ¼
Z t t0
2 ! C V Deffective kT Q rT dt rC f Xrrspherical þ 2 C T kT ð3:2Þ
The damage evolution formula can be obtained by plugging (3.2) into (3.1)
2 6 D ¼ Dcr 6 41 e
R t CV Deffective t0
kT 2
2
ðkTC rCf Xrrspherical þQ TrT Þ N0 k
3 dt
7 7 5
ð3:3Þ
Derivation of (3.2) and (3.3) is rather long. Here it is omitted for brevity. 4. Finite element analysis Finite element model of an actual flip chip solder joint is shown in Fig. 1. The solder alloy is lead free 95.5Sn–4Ag– 0.5Cu (SAC405). 8-node plain strain elements with a unit thickness are used to mesh the solder ball model. The diameter of the solder bump is 140 lm while the standoff height is 100 lm. The Al trace interconnect on the die is 1 lm thick while the Cu trace is 15 lm thick (Lai and Kao, 2006). In a finite element mesh, all interfaces have 100% perfect adhesion. However, in actual solder joints, it is well known that even in a good joint, the effective contact area is no more than 50% of the gross interface area (Un-Byoung and Young-Ho, 2004; Nah and Suh, 2005). Usually the percentage of interface actually in contact can be as low as 10%. This topic has been extensively
Fig. 1. Finite element mesh for solder joint.
studied in electronic packaging literature. Hence it is outside the scope of this paper. To consider the imperfect interface contact in between solder ball and Al/Cu interconnect traces, two interface layers of 1lm thick with 50% electrical conductivity are introduced individually. These assumptions are supported by experimental finding reported by Ye et al. (2003a). Material properties for SAC405 used in analysis are listed below. Young’s Modulus E = (57.7–0.056T) GPa (Hong, 1998), Poisson ratio m = 0.33 Hong, 1998, Yield stress ry = (140.67–0.2133T) MPa (Siviour et al., 2005), Equilibrium Vacancy concentration at a stress free state is 1.107 106 lm3 (Ye, 2004; Balzer and Sigvaldason, 1979), vacancy relaxation time is 0.0018 s(Sarychev et al., 1999), average vacancy relaxation ratio is 0.2(Sellers et al., 2007; Bassman, 1999). The thermal properties are listed in Table 1. The molar heat of transport Q* of eutectic SAC405 alloy is 22.16 kJ/mol (Chaung and Lin, 2003). An average current density of 1 104 A/cm2 is applied. The ambient temperature is set to 25 °C. 5. Numerical simulation results A steady state Joule Heating analysis is conducted. The current comes from the Aluminum trace from the right side into the solder joint. After passing through the solder ball, it goes out into the copper trace. Due to the different conductivity of various materials and the discrepancy in thickness, the current distribution is not uniform throughout the section. The current density at the Aluminum interconnect solder joint corner point can be 100 times larger than the average current density (Tu et al., 2000). This is
Table 1 Material properties.
Density (kg/lm3) Thermal conductivity (W/lm K) Electrical conductivity (O1lm1) Specific heat (J/kg K)
SAC405
Copper
Aluminum
7.39 1015 57.3 106
8.92 1015 4.16 104
2.70 1015 2.38 104
6.67
59.17
38.17
200
385
902
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due to so-called current crowding effect. Although current crowding effect has been known for a long period, so far there has been no effective method to quantify it. The reason is mathematical singularity. Theoretically speaking the current density at this singular point should be infinite. Such singular point does not exist in reality because it will melt immediately by the highly localized Joule heat and reshaped by the surface tension. In finite element analysis, it will lead to a mesh sensitivity problem (Lai and Kao, 2006; Basaran and Zhao, 2001), which means the current crowding factor at the corner point obtained by FEM cannot converge to a constant even if an extremely fine mesh is used. It should be pointed out that due to Joule heating effect and melting, classical asymptotic analysis used for singularities in stress fields is no longer suitable here (Basaran and Zhao, 2001). However, the results obtained using certain mesh density though may not be accurate quantitatively but still are valuable in the study of current crowding effect and the induced Joule heating in the near region around the corner point at present. The analysis is divided into two steps. By using the FEM mesh shown in Fig. 1, ABAQUS Standard is first utilized to conduct a coupled thermal-electrical analysis. However, ABAQUS and other commercially available finite element codes do not have the capability to solve the general thermomigration problem directly. In the second step, ABAQUS is adapted to analyze coupled mass diffusion, heat transfer and mechanical stressing process and their induced damage by coding with ABAQUS user element and material (UEL and UMAT). To minimize the influence of thermal expansion induced stress, both aluminum trace and copper trace are free to deform. Also we compulsorily remove the current to study the effect of thermomigration in the second step. Fig. 2 shows the current density map in the solder ball. It can be seen that current crowding occurs in the right upper corner where the current enters from the aluminum trace and the left lower corner where the current exits to the copper trace. The current density distribution along the section from left lower corner (point A in Fig. 1) to the right upper corner (point B in Fig. 1) is shown in Fig. 3. From Fig. 3, it is observed that the highest current density appears at point B, jB = 1.768 105 A/cm2, which is more than 10 times higher than the average current den-
Fig. 2. Current density distribution map.
sity in the middle of the solder ball. The left lower corner is another current crowding region. At point A, jA = 5.06 104 A/cm2, which is 5 times the average nominal current density. It is widely accepted that the following two factors are the major sources of high temperature gradient in solder joints. Because it is very thin, the high current density in the aluminum trace produces much more Joule heating than in solder joint and copper pad. A considerable temperature gradient will thus be created from the hot side (aluminum trace side) to the cold side (copper pad side). Another important source of highly localized temperature gradient is the imperfect contact between interfaces. The thermal and electrical resistance in interface is far higher than Al interconnect or solder joint. It will accordingly produce more Joule heating in the interfaces than in the other place, which results in a localized temperature gradient in the nearby region. The temperature map for Joule heating analysis is shown in Fig. 4. The temperature and temperature gradient distribution in the cross section of diagonal line A–B, and the cross section in the center line from the bottom side to the top side of the solder ball, as are shown in Figs. 5 and 6. We found that although the temperature in corner B is only a couple degrees above the other locations in
Current Density (104A/cm2)
20
15
10
5
0
0
50
100
Distance (micron) Fig. 3. Current density along section A–B.
150
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1.048992
40
1.040232 1.031472 20 1.022712 1.013952 0 1.005192 0.996432 -20
0.987672 0.978912 0.9684
-40 280
300
320
340
360
380
400
Fig. 7. Vacancy concentration contour map. Fig. 4. Temperature distribution map.
35
Temperature(°C)
Diagonal Line Center Line
33 31 29 27 25
0
0.2
0.4
0.6
0.8
1
Relative Distance Fig. 5. Comparison of temperature distribution.
Temperature Gradient (×102°C/cm)
the top side, the temperature gradient at B is much higher. Therefore, we can say that Joule heating induced by current crowding is the major source of the high temperature gradient in solder joint. Fig. 6 shows that the temperature gradient at point B is about 1500 °C/cm. The vacancy concentration contour map after 60 h of temperature gradient stressing is shown in Fig. 7. It is very similar to that of temperature distribution.
15
From Fig. 7 we can find that the atoms at the right upper part are depleting and migrating to the left lower part. Thus in the hot region, the vacancies are accumulating. In the cold region, the vacancies are filled by the immigrating atoms. As is seen in Fig. 7, the number of vacancies at corner point A goes down by 5% after applying thermal loads for 100 h. Fig. 8 shows the vacancy concentration distribution in the cross section A–B after applying thermal loads for 100 h. Here we can clearly see that vacancy concentration increases in the hot side (corner B) and it decreases in the cold side (corner A). By investigation into the time history of vacancy concentration (Fig. 9), we found that over 90% thermomigration happens in the first 20 h. This phenomenon can be explained by looking into the driving force mechanism in the thermomigration process. In the absence of electrical current, the vacancy transport is governed by three driving forces, (1) the vacancy concentration gradient, (2) temperature gradient and (3) back stress gradient. In our example, the initial assumed vacancy concentration is uniform throughout the solder joint. The initial stress gradient is also zero everywhere. So at first the only driving force comes from the temperature gradient, which will move the atoms from the hot side to the cold side. After a period of time, the accumulating vacancy concentration gradient and back stress gradient induce driving force that will counteract on the process by pushing the atoms back.
Diagonal Line Center Line
10
5
0
0
0.2
0.4
0.6
0.8
Relative Distance Fig. 6. Comparison of temperature gradient distribution.
1
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Normalized vacancy concentration
1.06
1.04
1.02
1
0.98
0.96 0
20
40
60
80
100
120
140
Distance (micron) Fig. 8. Vacancy concentration distribution.
Normalized Vacancy Concentration
1.08 Region B Region A
1.04
1.00
0.96 0
20
40
60
80
100
Time (hour) Fig. 9. Vacancy concentration evolution with time.
When the driving forces reach balance, the migration will stop. The strain and Von Mises stress field after 100 h are shown in Fig. 10 and Fig. 11. From Fig. 10 and 11, we can tell that the hot region is under tension and the cold region is in compression. This is due to the movement of atoms from the hot side to the cold side. The mass accumulation
at the cold side will result in compressive stresses, while the vacancy accumulation at the hot side will result in tensile field. The stress induced by thermomigration cannot be overlooked. The damage evolution contour, as defined by (3.3), across the solder joint after 100 h is shown in Fig. 12. It shows that the thermomigration induced damage grows
0.001079 0.000828 0.000577 0.000326 7.5E-005 -0.000176 -0.000427 -0.000678 -0.000929 -0.00118 -0.001431 -0.001682 -0.001933 -0.002184 -0.002435 -0.002686 -0.002937 -0.003188 -0.003439 -0.00369
40
20
0
-20
-40 280
300
320
340
360
380
Fig. 10. Volumetric strain after 100 h.
400
Fig. 11. Von Mises stress (MPa).
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balanced by the stress gradient and concentration gradient very quickly. Therefore the damage reaches a steady state very quickly in these regions. However, the same cannot be applied for the higher temperature gradient regions where far more damage will be created. When the temperature gradient exceeds a critical threshold, the counter action cannot balance the thermomigration driving force. Damage will keep accumulating until it fails, as can be seen in Fig. 14. In order to determine the critical damage value for void nucleation, which is called incubation time (Basaran et al., 2003), a regression method should be developed to determine the total time as a function of damage. From Fig. 14, we can find that under high temperature gradient, damage evolves exponentially with time. In this model, when the damage value reaches 1, the material will lose all the load carrying capacity. Definition of failure is application specific. If we define failure by 50% loss of the stiffness (Stephens et al., 2001), as can be seen from Fig. 14, the time to failure prediction will be 2168 h.
0.0007 40
0.0006 0.0005 0.0004 0.0003 0.00025
20
0.0002 0.00015 0.0001 9E-005
0
8E-005 7E-005 6E-005 5E-005
-20
4E-005 3E-005 2E-005 1E-005 0
-40 280
300
320
340
360
380
400
Fig. 12. Damage variable contour.
in the hot region. The maximum damage occurs at the right upper corner, where the peak temperature gradient locates. By investigating the time history of the damage growth at point B, C, D, E and F, (Fig. 13) can be developed. In this study, we assume that vacancy accumulation leads to failure, however, mass accumulation does not. Therefore, damage is only plotted for hot side. This is in agreement with experimental observations (Ye et al., 2003a,b; Abdulhamid, and Basaran, In press; Tu et al., 2000). In Fig. 13, we can see that at low temperature gradient region (C, D, E, F), the thermomigration driving force can be 8
A fully coupled temperature-diffusion-displacement damage mechanics model for thermomigration process is presented in this paper and associated UMAT and UEL codes have been developed and implemented in
B C D E F
7
Damage (×10-4)
6. Conclusions
6 5 4 3 2 1 0 0
10
20
30
40
50
60
70
80
90
100
Time (hour) Fig. 13. Damage evolution.
0.6
Damage
0.5 0.4 0.3 0.2 0.1 0 0
500
1000
1500
2000
Time (hour) Fig. 14. Damage evolution at B in 2500 h.
2500
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ABAQUS. By analyzing the thermomigration process in a SAC405 solder joint under the Joule heating induced high temperature gradient, it is shown that thermomigration can play a very important role in electromigration induced failure as has been proven by Ye et al., 2003a experimentally. The important findings of this paper can be listed as follows: (1) Joule heating due to current crowding effect can be the dominant factor that causes thermomigration. For instance, in this example, the current density in the right upper region is much larger than elsewhere. Hereby temperature gradient in point B is more than 10 times of the average temperature gradient across the solder joint. (2) Most of the thermomigration happens in the first period before counteracting stress gradient and vacancy gradient forces develop. The vacancy flux will lessen with time due to counterbalance forces and finally it will approach to a steady state. In this example, over 90% thermomigration happens in the first 20 h. (3) When the temperature gradient is small, the thermomigration driving force can be balanced by the spherical stress gradient and concentration gradient. The damage will reach a steady state quickly. However, when the temperature gradient is large, the damage will keep growing until it fails. (4) Complex conditions like high current density, high temperature gradient, and damage accumulation occur simultaneously in the corner of the solder connection, which make it critical region for design. (5) Proposed damage model does not utilize a phenomenological damage potential surface. Instead entropy production is used as a metric.
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