Download Local Analysis

Large-Scaled 3-D Area Array Electronic Packaging Analysis
Kuo-Ning Chiang1
Department of Power Mechanical Engineering, National Tsing Hua University
HsinChu, Taiwan 300, R.O.C.
E-Mail: [email protected]
Hsien-Chie Cheng2
National Center for High-Performance Computing, Taiwan 300, R.O.C.
Wen-Hwa Chen3
Department of Power Mechanical Engineering, National Tsing Hua University
HsinChu, Taiwan 300, R.O.C.
Abstract
As integrated circuit functionality, performance and density continue to increase, innovative next
generation packaging approaches are in great demand. The ball grid array (BGA) type packaging technology
such as flip chip (FC), plastic/ceramic BGA (PBGA/CBGA, see Figure 1), and chip scale package (CSP) has
been gaining world-wide interest and commitment as the potentially lowest-cost package for high-I/O
devices and even for lower-pincount (e.g., Tessera’s  BGA) applications. Drivers include the I/O density
advantages of an area array, as well as the potential for excellent electrical and thermal performance, etc.
However, some of the reliability issues of the BGA type package are not eradicated, research is critically
needed in the area of next generation electronic package analysis and design, e.g., improvement of the
package reliability, increase of the assembly yields rate and reduction of the solder bump pitches, etc. A
substructuring like local-global finite element method with multi-point constraints boundary condition is
developed for prediction of the fatigue life of the solder joint, this methodology could significantly reduce
the analysis CPU time and make the 3-D large-scaled electronic packaging analysis possible.
Keyword: Ball Grid Array (BGA), Flip Chip (FC), Chip Scale Packaging (CSP), Solder bump pitch.
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Associate Professor, ***Corresponding Author
Associate Research Scientist
Professor
1
Large-Scaled 3-D Area Array Electronic Packaging Analysis
Kuo-Ning Chiang1
Department of Power Mechanical Engineering, National Tsing Hua Universit
Hsien-Chie Cheng2
National Center for High-Performance Computing, Taiwan 300, R.O.C.
Wen-Hwa Chen3
Department of Power Mechanical Engineering, National Tsing Hua University
Taiwan 300, R.O.C.
Summary
As integrated circuit functionality, performance and density continue to increase, innovative next
generation packaging approaches are in great demand. The ball grid array (BGA) type packaging technology
such as flip chip (FC), plastic/ceramic BGA (PBGA/CBGA, see Figure 1), and chip scale package (CSP) has
been gaining world-wide interest and commitment as the potentially lowest-cost package for high-I/O
devices and even for lower-pincount (e.g., Tessera’s  BGA) applications. Drivers include the I/O density
advantages of an area array, as well as the potential for excellent electrical and thermal performance, etc.
However, some of the reliability issues of the BGA type package are not eradicated, research is critically
needed in the area of next generation electronic package analysis and design, e.g., improvement of the
package reliability, increase of the assembly yields rate and reduction of the solder bump pitches, etc. A
substructuring like local-global finite element method with multi-point constraints boundary condition is
developed for prediction of the fatigue life of the solder joint, this methodology could significantly reduce
the analysis CPU time and make the 3-D large-scaled electronic packaging analysis possible.
Introduction
The low cycle fatigue-induced failure of solder balls in surface mounted electronic devices has
become one of the most critical reliability issues in the ball grid array type packages. Solder ball reliability
performance was found to be highly dependent on the configuration of the package, such as the combination
of substrates and geometry/material properties of die, etc., which in turn, is governed by bond pad geometry,
solder ball configuration, thermal behaviors of each component, moisture condition, as well as the solder
reflow characteristics, etc. Using finite element methods with multiphysics capabilities to predict the package
reliability and to reduce the time-to-market is imperative in the area of electronic packaging industry.
One of the dominant failure mechanism of the electronic packages is the low cycle fatigue when
subjected to thermal/power cycling loading. In general, predicting the fatigue life of the solder joint requires
the detailed thermal-mechanics and package geometry information. Application of a simplified,
approximation model, such as a “slice” or “2-D plain strain/stress” model provides a very effective
alternative due to its ease of modeling and efficiency of computation. However, Pao et. al. [1] has pointed
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Associate Professor
Associate Research Scientist
Professor
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out that the “slice” model could originate a significant error in comparison with the three-dimensional,
full-scaled models for some particular package’s analyses. As a consequence, a 3-D, full-scaled model is
favored for obtaining a more reliable, robust solution. On the other hand, due to that a BGA typed package
often consists of 300-700 solder balls and additionally, the aspect ratio of some components is extremely
exclusive, its finite element model will likely come out with enormously huge degree of freedoms. For such a
large-scaled model undergoing a nonlinear/transient analysis is in fact impractical for most computing
resources currently available. In order to resolve this difficulty, the most common approach in the literature
is to apply the substructuring based techniques. One typical option is to adopt a fine mesh to the most
reliability-concerned solder ball and apply a coarser mesh elsewhither. However, in most cases, the most
reliability concerned solder ball is not known a priori. The other way that is preferred for the current study is
to apply a substructuring typed local-global (namely, micro-macro) analysis technique (see, e.g., Voleti [2]).
This technique has been significantly applied in the past to deal with various engineering problems,
particularly in the applications of composite materials (e.g., Griffin and Vidussoni [3]). In the area array
packaging analysis, the local-global finite element approach as well has played an important role (see, Corbin
[4]; Ju [5]). In their study, an equivalent beam model is adopted to elastically/plastically simulate the specific
3-D local model, as shown in Figure 1, such that similar stiffness characteristics will be yielded when
subjected to the same force. However, their approaches may present some disadvantages. For example, their
3-D local models contain one single solder joint, sandwiched by a half thickness of substrate (i.e., BT or
ceramic) on top and PCB on bottom as shown in Figure 1, and more importantly, in the global analysis a
plate model is respectively used to simulate the substrate and PCB. Obviously, this modeling is likely
oversimplified and less accurate, and lacks of feasibility for expansion to include other components of the
package for analysis, such as molding compound etc. The main purpose of this study is to propose an
improved equivalent beam model using a FEM based method together with an optimization approach. In
addition, the solution to a realistic 3-D PBGA structure made up of multi-materials system will be also
investigated to substantiate the proposed methodology. The computational time as well as the accuracy of the
equivalent model will be considerably compared with those of the full-scaled model.
Heat Sink
Gold Wire
Die
Cap
C-4 Solder
Ceramic
90Pb/10Sn
Ceramic
Ball
Molding Compound
Eutectic Solder
Joint
Die Attach
BT
Solder Mask
Ball Pad
Solder Ball
FR-4
FR-4
Module mid-plane
Module
mid-plane
Module eutectic
solder fillet
Ceramic module
90%Pb/10%Sn
solder ball
Card eutectic
Solder fillet
60Sn/40Pb
eutectic solder
Molybdenum pad
BT module
Solder pad
FR-4 card
Copper pad
FR-4 card
Card
mid-plane
Card mid-plane
(a) CBGA/Corbin’s Local Model [4]
Fig. 1
(b) PBGA/Ju’s Local Model [5]
Existing Local Models of Ball Grid Array Typed Packages
3
Furthermore, if anyone of the solder joints is fatigued, the package can be considered as failure.
Hence, the most stressed/strained one shall be located from the global analysis, and the corresponding net
displacements of the beam will be extracted. These displacements will be again applied to the local model to
further examine the stress/strain information. However, a significant geometry change involved in the local
model will result in a stress/strain concentration field that is very sensitive to the mesh size. In the literature,
there are several methods that can be applied to investigate the problem. A nonlinear analysis with nonlinear
material properties or detailed geometry descriptions, such as a recommended corner fillet, may ease the
stress concentration but the strain concentration. It is important to note that the maximum cyclic effective
strain range is extremely crucial to the low cycle fatigue life of the solder ball as well as overall reliability
based on an empirical relationship, such as Coffin [6], Manson, Cheng et. al. [7] or Darveaux et. al. [9]
techniques. It is, therefore, vital to effectively characterize the strain concentration. In this study, an
engineering approximation approach using a volume-weighted averaging strain in a finite local zone,
proposed by Clark and Mcgregor [10], will be applied for characterizing the strain concentration field. To the
end, a finite local zone with a specific dimension will be characterized. Based on the average plastic strain in
the specific local zone, the fatigue life of the most susceptible ball can be then rationally predicted.
Local/Global Analysis
In general, the solution procedure of local/global analysis used in this study can be briefly
characterized into the following main steps. A single solder ball (i.e., the local model) is first defined, and
moreover, its elastic-plastic characteristics is extensively simulated using an equivalent beam model. The
second step is to perform the global analysis, in which the solder joints will be substituted by the equivalent
beams. Finally, the detailed stress/strain information of the solder joint subjected to the net displacements
extracted from the most susceptible equivalent beam in the global analysis will be calculated. The success of
this approach relies on reliable calculation of the effective properties of the prescribed local model. In the
literature, several approaches can be employed, such as the rule of mixture, the self-consistent method
(Willis [11]), the homogenization method (Sanchez-Palencia [12]; Guedes and Kikuchi [13]), or the FEM
based method (Cheng et. al. [7]). However, the first three methods are not appropriate for the current study
due to that solder joints are discontinuous. As a result, the FEM based method becomes a preferred choice.
The other implied advantages of the FEM based method are in that an existing finite element code is simply
required for obtaining the effective properties without the necessity of conducting a complicated modeling.
Effective Properties and Geometry Data of the Equivalent Beam
The equivalent beam theory has been fully discussed by Cheng et. al. [7], and this research will apply
their approach and formulations to a realistic 3-D plastic BGA package reliability analyses. Furthermore, to
maintain the readability of this work, some equations from Cheng et. al. [7] are included. Figure 2 presents
the local model used in this study, simply including a single near eutectic solder joint (60%Sn/40%Pb)
without comprising of any BT substrate and FR-4 PWB. The major advantage of this model in comparison
with Corbin’s and Ju’s stems from its flexibility and generality in 3-D modeling the entire assembly
associated with the area array electronic packaging. The geometry profile of the 3-D solder ball is obtained
from the simulation result of the Surface Evolver (Racz and Szekely [14]; Brakke [15]). The Evolver is for
the simulation of surfaces whose geometry is determined by surface tension and other energies, such as
gravity, and external forces, etc. It evolves a surface towards its equilibrium configuration by minimizing
total system free energy. The surface is represented as a set of interconnected triangular facets and then
iterating this initial surface combined with geometry constraints toward a minimal energy configuration by
using a gradient descent method. The Surface Evolver has been successfully applied in predicting the final
shape of the BGA joint after reflow according to various pad sizes and shape, solder volume, specific solder
height, and surface tension, etc. The advantages of using the Evolver for BGA reflow shape predictions are
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several and clearly, it is more economical and less time consuming than laboratory experiments.
Do
nL/2
mDo
(1-n)L/2
L
nL/2
Fig. 2
A Typical Cantilever Beam with Variable Cross Sections
First of all, it should be noticed that the following derivations are based on the fact that the effect of
local CTE mismatch between solder joints and substrate/PCB is negligible in contrast to the global effect
between silicon chip and PWB. Furthermore, in order to derive the corresponding equivalent beam using a
FEM based technique, the following relations are required: shear resultant forces (P) vs. shear displacements
( ws ), axial resultant forces (F) vs. axial displacements ( wa ), and moments (M) vs. shear displacements ( ws )
of the solder joint. Once they are obtained, the elastic and plastic properties as well as the corresponding
geometry of the equivalent beam can be derived in the following. Consider an equivalent beam, as shown in
Figure 2, with Young's modulus E , shear modulus G , and Poisson's ratio  . This cantilever beam is fixed
at the left-hand side and subjected to a concentrated force, either a tension force F or a shear force P, at the
right-hand side. Note that n and m are both the geometry parameters of the equivalent beam used in the
design optimization process.
The shear correction factor (see Cowper [16]) corresponding to a solid circular beam can be deduced
as:

61   
7  6 
(1)
Based on the governing equations of the beam theory as well as the shear correction factor, see Eq. (1), the
elastic shear displacement  s (or the axial displacement  a ) at the most right-hand side of the cantilever
beam subjected to a shear (or axial) concentrated force P (or F) can be obtained:
7  6 n1  12   12  PL
m  m 
 
s 

3EA
1  1 
 
n1  m 2   m 2  FL



a  
AE
 1
1

 4  1  4
m

 3m
2
n 3  3
 n n
  PL
 
 2 4 12 
EI
(2)
(3)
5
where A and I, respectively, represent the area and moment of inertia of the base beam. On the other hand,
based on Eq. (2), with a given P, L, n, m, and  s at the tip of the right-hand side of the cantilever beam, the
effective elastic material property E can be eventually derived as:
7  6 n1  12   12  PL
  m  m 
E

3 s A
 1
1  n n 2 n 3  3

 4  1  4     PL
 3m  m  2 4 12 
s I
(4)
Note that in the case of m equal to 1, it becomes a uniform equivalent beam.
Furthermore, in order to define the configuration of the equivalent beam,  a ,  s , F, P, and M
corresponding to the solder joint should be obtained in advance. Based on the shear resultant force (P) and
the resultant moment (M) associated with the solder joint, the length of the equivalent beam can be first
defined:
L M /P
(5)
Subsequently, with a given, particular geometry parameter (i.e., m and n), the base diameter Do of the
equivalent beam can be derived as follows:
 1
1  n n 2

 4  1  4  
3m  m  2 4
Do  4 L 
 F   s
 
 P   a
n 3 
 
12 
1  1 
 
n 1  m 2   m 2 

 

 (7  6 )
 
3

(6)
The effective properties of the equivalent beam, including the elastic and plastic material properties
as well as the geometry configuration, can be fully developed using an iterative method. In addition, an
optimization technique, the design of experiment method, incorporating with a proper tolerance prescribed is
applied for attaining accuracy to a great extend using the simple beam model. The tolerances defined in
Corbin’s work will be applied in this study, in which 6% of discrepancy for the shear displacement and 16%
for the axial displacement due to that the axial displacement of the entire module is estimated to be
considerably smaller than the shear displacement. For simplicity, this study selects only five representative
sets (i.e., forces vs. displacements) to determine the properties of the equivalent beam and to simulate the
linear/nonlinear stiffness characteristics of the solder joint: one from the elastic region and the other four
from the plastic zone. The elastic pair is used to determine the elastic properties, and the other four pairs are
used to define the equivalent beam's plastic properties. For optimization, the objective is to minimize the
least square mean error of the five representative stiffness characteristic sets between the solder joint and the
equivalent beam model. In addition, the length of the equivalent beam L and the geometry data n and m are
considered as the design variables in the optimization process, each of which is subjected to an appropriate
side constraint:
1  L / Lo  1.2 ,
0  n  1,
1 m  2,
(7)
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Concerning application of the design of experiment method, an appropriate number of trial tests
based on variable discrete combinations of these design variables in the feasible domain will be defined and
performed. Once all these trials are tested, a regression model can be constructed and analyzed. Note that the
definition of the test trials is completely case-dependent. From the regression analysis, the trial comprising
the minimum objective and satisfying all the specified tolerances will be chosen as the solution. The above
approaches can, in general, lead to a considerably optimal solution. Furthermore, if other temperatures are
considered, (e.g., the temperature-dependent material properties used in this study), the whole iterative
procedure can be completely reduced to one time of estimation of the corresponding material properties
using the geometry data derived formerly.
Application
Once the effective characteristics of the equivalent beam are derived, analysis of the 3-D equivalent
package would be very efficient and present no difficulties. In order to verify the previously proposed
methodology, a realistic PBGA package is substantially studied. In this application, the characteristics of the
equivalent beam will be first derived. Based on the characterized equivalent beam model, a global solution to
a typical package subjected to a given thermal loading can be then achieved, and more importantly, the
performance and accuracy of the 3-D equivalent model using the equivalent beam will be extensively
compared with those of the original 3-D full-scaled model. Furthermore, the beam that is the most
reliability-concerned will be sought and the corresponding net displacements will be extracted and placed in
the 3-D solder joint as enforced boundary conditions to further investigate the detailed stress/strain
information. Due to that a significant geometry change is involved in the solder joint, a stress/strain
concentration field that is critical to the mesh size will be resulted, in which it is crucial to the prediction of
the solder joint’s fatigue life. Hence, in this application, an engineering approach based on the
volume-weighted averaging strain in a finite zone will be applied for characterizing the strain field
neighboring the singularity point.
Stress (MPa)
30.0
o
20 C
20.0
10.0
100 oC
0.0
0.000 0.002 0.004 0.006 0.008 0.010
Strain
Fig. 3
Tensile Stress-Strain Relationships.
Linear/Nonlinear Characteristics of the Equivalent Beam
Consider that the a typical eutectic 60Sn/40Pb solder joint holds a tensile stress-strain relationship, as
shown in Figure 3. The Sn/Pb solder joint is considered as an elastic-plastic, temperature-dependent, and
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time-dependent material. Before further constructing the equivalent beam, the elastic/plastic characteristics
of the solder joint at temperature 20 o C and 100o C are first derived in Table 1 based on Figure 3. The
effective properties of the equivalent beam can be optimally obtained using the previously proposed solution
procedure. Table 1 also presents the obtained linear/nonlinear stiffness characteristics of the equivalent beam
in comparison with those of the solder joint, including shear reaction forces vs. shear displacements,
moments vs. shear displacements, as well as axial reaction forces vs. axial displacements at temperature
20o C and 100 o C .
It is found that the maximum differences that are 6% for the shear displacements and 16% for axial
displacements all occur in the elastic region (the first representative set). In other words, it implies the
differences of the shear displacements and the axial displacements in the plastic region are all much less than
the prescribed tolerances (i.e., 6% and 16% respectively). The mean error of these two models corresponding
to shear resultant forces, shear resultant moment, and axial resultant forces are 3.0%, 6.6%, and 8.5%
respectively, which are considerably satisfactory.
Table 1: Resultant Forces/Moments versus Displacements.
20o C
Shear Displacements (mm) -vs.- Shear Reaction Forces (N)
Displacements
Solder Joint
Equivalent Beam
Solder Joint
0.00015
0.962
0.907
0.435
0.0004
2.424
2.295
1.091
0.0010
4.171
4.099
1.678
0.0025
6.212
6.229
2.006
0.006
7.591
7.337
2.152
Shear Displacements (mm)-vs.-Shear Reaction Moments (N-mm)
0.00015
0.223
0.231
0.101
0.0004
0.562
0.585
0.253
0.0010
0.967
1.045
0.389
0.0025
1.441
1.589
0.466
0.006
1.764
1.871
0.502
Axial Displacements (mm)-vs.-Axial Reaction Forces (N)
0.0001
2.306
2.675
1.044
0.0003
6.180
7.031
2.764
0.0008
11.293
10.780
4.186
0.0015
14.661
13.560
4.700
0.003
17.423
17.110
5.008
100 o C
Equivalent Beam
0.411
1.028
1.675
1.963
2.156
0.105
0.262
0.467
0.501
0.550
1.214
3.133
4.390
4.857
5.177
3-D Plastic BGA Analysis
The solution to a realistic 3-D PBGA structure, as shown in Figure 4(a), made up of multi-materials
system is then pursued. The full-scaled finite element model consists of a silicon chip, a piece of BT
substrate, FR-4 printed circuit board, molding compound, and 72 eutectic solder bumps. The top view of a
quarter of the package is also shown in Figure 4(b). The result obtained from this full-scaled model is
extensively compared from that of the equivalent model, in which the eutectic solder joints with thousands
of three-dimensional solid elements is replaced by a 10-linear-beam-element model. The material properties
corresponding to these components are shown in Table 2. It should be noted that all these components are
considered as elastic except of the eutectic solder joint or the equivalent beam, which is elastic-plastic.
8
SILICON
Y
EPOXY
BT
5.5
4.0
2.5
1.0
0.0
(a) Finite Element Model
Fig. 4
1.5
3.0
4.5
6.0
X
(b) Top View
A Typical PBGA Structure
Table 2: The Material Properties and Geometry Data Used in the PBGA Package
Silicon Chip
BT
FR4
EPOXY
Young’s Modulus(MPa)
130000
19000
18200
8960
Possion Ratio
0.28
0.2
0.19
0.35
CTE(ppm)
2.62
15
16
19
Geometry Data(mm)
9  8  0.36
14  13  0.36
14  13  1.71 13.4  12.4  0.72
This entire package is subjected to a 100o C temperature change with the initial stress-free condition
set at 25o C . Since the package is symmetric in the X- and Y- planes, only a quarter of the package is
analyzed. The finite element approximation of the 1/4 full-scaled model comprises 53540 nodes and 52420
elements, and that for the equivalent model contains 22212 nodes and 18226 elements. The solution to the
package can be eventually obtained using a commercial, finite element analysis code-- ABAQUS R . The net
shear displacements of the equivalent beams and the solder joints associated with the equivalent model and
the full-scaled model, respectively, are shown in Figure 5(a). It is observed that the equivalent approach can
as well position the most critical joint, in which it locates at (X,Y)=(4.5, 4.0), right beneath the chip and in
the farthest diagonal from the neutral point. Basically, the result matches closely with the experimental
finding (see, Nagarajand and Mahalingam [17]). In addition, the differences between these two models are
shown in Figure 5(b) in percentage. It is apparent that the accuracy for most of the joints is sufficient except
of those in the first column/row, in which the discrepancy can be up to 12%. This is because ABAQUS R as
well as the currently proposed equivalent beam model lack capabilities in handling a symmetric circular
beam. An approximation is employed for this particular situation, and as a result, a larger error is induced.
Furthermore, the total computational CPU time for the 3-D full-scaled model is 237,567 (second); on the
other hand, it is 14,253 (second) for the equivalent model. Significantly, the performance increases up to 17
times.
9
Er ro r i n Pe r c e nt a g e
S he a r Di s p l a c e m e n t
1.5e- 3
1.0e- 3
Y=1.0
Y=2.5
Y=4.0
Y=5.5
Y=1.0
Y=2.5
Y=4.0
Y=5.5
5.0e- 4
( 3D F ul l- S c a le d)
( 3D F ul l- S c a le d)
( 3D F ul l- S c a le d)
( 3D F ul l- S c a le d)
( Eq uiva le nt )
( Eq uiva le nt )
( Eq uiva le nt )
( Eq uiva le nt )
14 .0
Y= 1.0
Y= 2.5
Y= 4.0
Y= 5.5
12 .0
10 .0
8.0
6.0
4.0
2.0
0.0
0.0e+0
0.0
2.0
4.0
6.0
0
(a) The Net Shear Displacement
Fig. 5
1.5
3
4.5
6
X -d is t a nc e F r o m t h e S y mm e tr y
X -d is t a nc e F r o m t h e S y mm e tr y
(b) The Error Analysis
The Net Shear Displacements and Error Analysis Using Two Various Approaches
Cyclic Thermal Fatigue Analysis Using a Local/Global Technique
Cyclic Thermal Fatigue Analysis Using a Local/Global Technique
Since the low-cycle fatigue is one of the major failure mechanism of the electronic packages when
subjected to thermal/power cycling loading, of great importance is to investigate the thermally induced
complex viscoplastic deformation of the solder joints in a package. A typical accelerated thermal cycling
curve is shown at figure 6. In this study, the local/global finite element analysis will be applied for
characterizing the detailed stress/strain information of the solder joint. Assume that in the global analysis,
only time-independent plastic deformation implies in the ramp time of the thermal cycling loading, and only
time-dependent creep deformation is involved in the dwell time. The deformation of the most
reliability-concerned solder joint in the dwell time (i.e., during the upper/lower temperature) can be extracted
by performing a nonlinear finite element analysis of the global model with equivalent beam inside subjected
to a net temperature swing loading. The magnitude of the net temperature swing depends on the range
between the upper/lower temperature in the thermal cycling loading and the stress-free temperature. Once
the thermal displacement boundary conditions of the most reliability-concerned solder joint at these two
temperatures are obtained, detailed thermal cycling simulation can be performed in the local level.
Temperature (oC)
125
-55
0
15
30
45
60
75
Time (Min.)
Fig. 6 Accelerated Thermal Cycling Curve
10
Local Analysis
Based on the net deformations of the most reliability-concerned beam in the global analysis, a
detailed stress/strain analysis can be performed on the solder joint with as the highest mesh density as
possible in order to achieve the most converged solution. Once the stress/strain information is obtained, the
fatigue life of the solder joint as well as the overall package can be predicted by directly plugging them into
the empirical Coffin-Manson relationship. However, an abrupt geometry change does occur in the
circumference of both top and bottom surfaces, in which it significantly induces a stress/strain concentration
field. To verify the great effect of the geometry and material singularity on the stress/strain field, a typical
eutectic solder joint shown in figure 7 will be practiced. Four different finite element models, each with a
different mesh density (i.e., 768, 2592, 6144, 14520 elements), are applied for modeling the solder joint. In
addition, the solder joint is subjected to a prescribed displacement boundary condition in both top and
bottom sides. Using a finite element analysis code-- ABAQUS R , the maximum Von Mises stress, plastic
strain, and plastic dissipation can be derived, shown in figure 8 to 10, respectively, corresponding to these
four finite element models.
Max. Von Mises Stress (MPa)
Based on figure 8 to 10, it can be detected that all these structural responses are mesh-dependent.
Note that the maximum plastic strain and dissipation are much sensitive to the mesh density than the
maximum Von Mises stress. This is due to the fact that a nonlinear material modeling is implemented in the
analysis. There are two empirical formulations that are intensively used by the electronic packaging
researchers for the solder joint reliability prediction, which are, the equivalent plastic strain based
Coffin-Manson relation [6] for predicting the cycles to failure and the viscoplastic strain energy density
based formulation that proposed by Darveaux et. al [9] for estimation of the number of cycles to crack
initiation. The finite element mesh density effect for these two methods are shown at figure 11 and 12, it is
clear to see that reliability results of the Coffin-Manson relation and the Darveaux model both are very
sensitive to the mesh density. Furthermore, the prediction of the Coffin-Manson relation with different mesh
density is illustrated in figure 11, it shown that the prediction of cycles to failure of plastic BGA package
may vary from 48 cycles (14,520 elements) to 4997 cycles (768 elements). The similar tendency is also
shown at figure 12, the number of cycles to crack initiation that predicted by Darveaux model could change
from 757 cycles (768 element) to 143 cycles (14,520 elements) as the finite element mesh become denser.
These results indicate the methods that can be applied to the characterization of the singularity are in critical
demand. In the literature, several existing techniques are available to resolve the stress concentration
problem; however, there still lacks an effective way to discharge or to characterize the strain concentration.
Z
Y
X
33
32
31
30
29
28
27
26
25
24
768
2592
6144
14520
Mesh Density
Fig. 7 A 3-D Eutectic Solder FE Model
Fig. 8 The Maximum Von Mises Stress
11
Max. Plastic Dissipation
(N/mm^2)
Max. Pastic Strain
0.025
Nf = C *
(eq)n
0.02
0.015
0.01
0.005
0
768
2592
6144
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
No = C * (W)n
768
14520
2592
6144
14520
Mesh Density
Mesh Density
Fig. 9 The Maximum Plastic Strain
Fig. 10 The Maximum Plastic Dissipation
757
420
290
Mesh Density
143
(14,520)
2.1
(6,144)
(768)
(2,592)
(2,592)
1.15
0.84
(6,144)
0.34
(14,520)
Mesh
(768)
1018.7 27.14 54.73
48
227 500
4,997
Density
.4
Fig. 11 Coffin-Manson Relation
Fig. 12 Darveaux et. al. Model
An engineering approximation approach based on the volume-weighted averaging equivalent plastic
~
strain  pl in a finite zone is used in the study:
n
~ pl  Ve epl
e 1
n
V
e 1
(8)
e
in which Ve is the volume of an element and  epl is the corresponding equivalent plastic strain. The radius
of the finite zone is determined in an observed manner that it should be small enough to capture the maximal
strain field; on the other hand, large enough to obtain a converging solution as the mesh density increases. In
this research, the engineering approach defines a total of four various types of finite element approximations
for the solder joint, each of which corresponds to a different element size. In addition, three different
12
semi-circular-ring zones (i.e., A, B, and C) around the circumference of the top surface of the solder joint,
each of which comprises a particular radius (i.e., 0.02, 0.062, and 0.12 (mm), respectively), are defined.
figure 13 presents the volume-weighted averaging equivalent plastic strains of these three different zones
as well as the maximum equivalent plastic strains with respect to these four finite element models. It can be
easily seen that the maximum equivalent plastic strain is singular and extremely mesh-sensitive, and
moreover, zone A seems to be able to provide better agreement to the proposed criterion in the selection of
the finite zone than the others. However, a further study on the dimension of the finite zone, such as use of a
finer zone or verification of the predicted life using experimental data, should be conducted in the future
work.
Equivalent Plastic Strain
0.010
0.008
Average at Zone A
0.006
Average at Zone B
Average at Zone C
Maximum
0.004
0.002
0.000
0
4000
8000
12000
16000
Mesh Size
Fig. 13
The Convergence of the Equivalent Plastic Strains in Various Zones
Conclusions
An effective local/global methodology is proposed for analysis of the thermally-induced,
elastic-plastic deformations of three-dimensional, full-scaled area array typed electronic packages. In
addition, an engineering approximation based on a volume-weighted averaging technique incorporating with
an observed criterion is applied to discharge the stress/strain concentration problem. It is shown that the
proposed equivalent approach can not only derive a considerable accurate global solution but also
significantly reduce the computational CPU time. Note that the proposed methodology is based on the fact
that the local CTE mismatch between solder and PCB/BT is negligible; however, as a matter of fact, for
some applications, this by no means sustains. For these particular cases, the local CTE mismatch effect
should be studied and incorporated into the modeling. In addition, only the temperature-dependent,
elastic-plastic material property is considered in the modeling of the equivalent beam while the
time-dependent viscous effect, such as creep mechanisms, can still be integrated into the model during the
detailed stress/strain analysis of the most reliability-concerned solder joint. The typical thermal-cycling
analysis of the package can be as well fully conducted using the proposed methodology as soon as the net
displacements of the most critical solder joint in both low- and high-temperature ranges in the thermal
cycling are derived. Finally, based on the obtained averaging viscoplastic equivalent strain, the low-cycle
thermal fatigue life of the solder joint can then be predicted using a Coffin-Manson’s or Darveaux’s
empirical relationship.
Acknowledgment
13
The support of the research by the National Science Council of the Republic of China through the
grants NSC-87-2212-E007-004 and NSC87-2212-E321-001 is gratefully acknowledged.
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