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THREE-DIMENSIONAL VS. TWO-DIMENSIONAL FINITE ELEMENT MODELING
OF FLIP CHIP PACKAGES
Qizhou Yao and Jianmin Qu
G.W. Woodruff School of Mechanical EngineeringGeorgia Institute of Technology
Atlanta, GA [email protected]
RevisedDecember 1998
To Appear in J. Electronic Packaging
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THREE-DIMENSIONAL VS. TWO-DIMENSIONAL FINITE ELEMENT MODELINGOF FLIP CHIP PACKAGES
Qizhou Yao and Jianmin Qu
G.W. Woodruff School of Mechanical EngineeringGeorgia Institute of Technology
Atlanta, GA [email protected]
Abstract
In this study, both two-dimensional and three-dimensional finite element analyses were used
to study the stress distribution in and deflection of the flip chip assembly under thermal loading.
It is found that the three-dimensional results compared favorably with experimental
measurements, while the two-dimensional results consistently overestimate both stresses and
deflection. Among the two-dimensional models, the plane stress assumption seems to yield
results closer to the full three-dimensional predictions. Furthermore, three-dimensional models
were used to investigate the effect of printed wiring board size on the overall deflection of the
flip-chip assembly. This size effect of the printed wiring board has significant implications on
the design of multi-chip modules. The results indicate that a square array placement pattern is
preferable to a staggered array for multiple chip modules in order to reduce mechanical
interaction between chips. For square arrays, such mechanical interaction between chips can be
neglected when the minimum distance between adjacent chips is more than 2 times the chip size.
Introduction
Flip chip attach (FCA) technology is expected to grow at an accelerated rate in the near
future, and it is going to constitute a large portion of the microelectronics industry as the next
century comes (Wesselmann, 1996). More and more evidence shows that FCA is an appealing
way to implement multi-chip module (MCM) technology (Baker, 1996). However, the
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coefficient of thermal expansion (CTE) mismatch between the printed wiring board (PWB) and
the silicon chip introduces a new concern. Under processing or operating conditions, the CTE
mismatch not only subjects the solder joints to extremely large strains, but it also causes the
overall bending of the flip chip assembly. Such CTE mismatch-induced stresses, manifested by
the increasing die size and temperature excursions, poses a great challenge to the
thermomechanical reliability of FCA packages. Although the introduction of a rigid encapsulant
layer between the chip and the substrate has considerably enhanced the solder fatigue life,
thermomechanical reliability issues still exist.
Failure analysis of flip chip devices subjected to thermal shock testing has revealed that the
typical failure mode is delamination at the encapsulant/chip interface, followed shortly by fatigue
failure of the flip chip solder joints. It should be noted that failure often initiates at the outside
perimeter of the chip. Once adhesion between these two surfaces is lost, the flip chip joints are
subjected directly to the strain resulting from the thermal mismatch of the chip and the board.
Electrical failure is the result of solder fatigue cracking. The number of thermal shock cycles
that can be tolerated before this failure mechanism occurs has been found to vary and to be
dependent on the materials in the assembly and their respective mechanical and chemical
properties. And more importantly, it depends on the stresses in the FCA assembly.
There are at least two major systems of thermal stresses and strains acting on the solder joints
and encapsulant. The first system is due to the local CTE mismatch between the adjacent
components. This is of particular concern at the chip-encapsulant and PWB-encapsulant
interfaces where the large interfacial shear stresses act as a driving force for encapsulant
delamination. The second system is due to the global CTE mismatch between the silicon chip
and the PWB. The overall bending stems largely from this global CTE mismatch. To prevent
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premature thermomechanical failure and to ensure the reliability of a FCA package, these
thermomechanical stresses, which are the primary driving force to failure, must be understood.
Furthermore, design and processing technologies must be developed to minimize such stresses.
Due to its geometrical and material complexity, analytical solutions to the stress distribution
in a FCA assembly are often unattainable. Numerical methods, such as the finite element
method (FEM), have been used extensively to analyze the stresses, strains and deformations in
FCA (LeGall, 1996; Schubet, 1997; Iannuzzelli, 1996). However, very few of these studies used
three-dimensional (3D) FEM models (Lee, 1997). For example, Dasgupta et. al. (1997)
compared 3D model with both “sector” model and 2D model, found that stress results for the
“sector” model are not comparable with those for the 3D 1/8 model when PWB warpage is a
significant factor, and the solder joint fatigue life predicted by 2D model is several times higher
than that predicted by 3D models. In most cases, either the plane stress or the plane strain
assumption is used to simplify the calculation. Inevitably, the use of such two-dimensional
simplifications introduces errors in the predicted results. For example, within the framework of
2D modeling, it is well known that the thermal stress and deformation of silicon die are
independent of the PWB size. However, this is not true in a real 3D assembly, since the
deformation resistance increases with increasing PWB size due to the 3D constrain effects. This
PWB size dependency of the stress and deformation is of great interest, for it is necessary to
understand the mechanical interactions between adjacent chips mounted on the same board, so
that a guideline for multi-chip design may be provided.
The objectives of this paper are (1) to develop a 3D FEM model for the flip-chip assembly
and validate the model with experimental data, (2) to investigate the difference between the 2D
5
and 3D FEM predictions, and (3) to understand the PWB size effect on the overall wrapage, and
its implications on packaging design.
Finite Element Modeling
The model assembly used in this study consists of a Sandia ATC04 (SNL, 1993; Peterson,
1997) stress chip directly attached to a FR-4 substrate. The chip is a 6.35x6.35 mm2 silicon die
with 35 I/O's. For the baseline model schematically shown in Fig. 1, the substrate is a 9x9 mm2
FR-4 PWB. To compare the 2D and 3D results, three FEM models were constructed, namely,
2D plane stress, 2D plane strain and full 3D. Due to symmetry, only one-half and one-eighth of
the assembly (see Figs. 2 -3) were needed for the 2D and 3D models, respectively. The 4-node
plane strain elements and the plane stress quadrilateral elements were used, respectively, in the
2D plane strain and plane stress analyses, while the 4-node linear tetrahedron elements were used
in the 3D analysis. Mesh refinement was conducted to ensure the convergence of numerical
results. It was found that satisfactory accuracy can be achieved using 11945 elements with
12157 nodes in the 2D case and 25665 elements with 5305 nodes in the 3D case. The finite
element calculations were carried out by ABAQUS Standard version 5.5.
Six different materials were considered in the finite element models. They are the silicon
chip, FR-4 substrate (PWB), encapsulant (underfill, Hysol 4520), solder mask, eutectic solder
(63Sn/37Pb, wt.%) and copper solder pads. Although the materials were assumed linearly
elastic, their properties were considered to be temperature dependent, as given in Tables 1- 6
(LeGall, 1996). Furthermore, silicon chip, underfill, solder, copper and solder mask were
assumed to be isotropic, while the FR-4 substrate was treated as an orthotropic material. The
silica filler loaded in the Hysol 4520 underfill makes the underfill a composite material in nature.
However, due to the fine spherical silica particle size, it is justifiable to treat the underfill as
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isotropic material with calculated effective properties. More comprehensive evaluation of the
underfill properties can be found in ref. (Hu, 1997). For the purpose of the comparison of 2D
and 3D modeling, the complicated filler settling effect is not considered. Caution should be
taken when the main interest is in the underfill material, as a resin rich top layer and a filler rich
bottom layer result in different local material properties.
To study the PWB size effect, two additional 3D models were constructed with the same die
size (6.35x6.35 mm2), but with different PWB sizes of 16x16 mm2 and 24x24 mm2, respectively.
In those models, the silicon die, the PWB and the solder mask were treated the same as in the
baseline models discussed in previous paragraphs. The under-die material, however, was
considered as an homogeneous material with its effective properties determined from the
mixiture of underfill and solder joints by using the Mori-Tanaka Method (Mura, 1991) and the
rule of mixture, see the Appendix. Values of the effective properties are listed in Table 7. It is
seen that the effective properties of the underfill-solder mixture are not much different from
those of the underfill itself, for the solder constitutes less than 10% of the total under-die material
volume. By using such effective properties, the computational time is significantly reduced
since modeling of individual solder joints is avoided. Although such simplification may not
yield accurate results for stress distribution within the individual solder joints, it will not affect
the overall deformation of the assembly. Therefore, it is justifiable to homogenize the under-die
solder/underfill mixiture, since our interest for this portion of the study is to investigate the
overall warpage of the assembly.
In all finite element models, the assembly was assumed stress-free at the typical reflow
temperature of 165 0C. Thermal load is applied by cooling the assembly from 165 0C to 20 0C.
For comparison with experimental data, stress distributions at the intermediate temperatures 74
7
0C and 36 0C were also obtained. For simplicity, uniform temperature distribution was assumed
throughout the analysis.
Validation of the FEM Results
To validate the FEM model, numerical predictions of the stress distribution during cool down
were compared with experimental data (Palaniappan, 1997). Figs. 4a - 4b show the comparison
between the 3D FEM results and experimental measurements for stress components σxx, σyy and
σxy at 74 0C and 36 0C post-underfill cure, respectively. The comparison was made at particular
locations corresponding to points of experimental measurement near the active surface on the
silicon die. It is seen from these figures that the numerical results compare very well with the
experimental data, considering that not a single parameter was used to adjust any of the data.
This comparison of the stress distributions indicates that the 3D FEM model developed here
provides a reasonably accurate prediction of the actual stress distribution and overall deformation
of the FCA assembly.
Before closing this section, it needs to be pointed out that, although every effort has been
made to mimic the ACT04 test vehicle, some approximations had to be made. One is to idealize
the shape of the encapsulant fillet which seems to affect the stress distribution in the assembly to
a certain extent. The other is to assume that the silicon die is at a stress-free state before the
encapsulant starts to cure at 165 0C. This seems to be reasonable because of the large
compliance of the eutectic solder joints at the temperature close to its melting point, 183 0C. It is
also worth mentioning that after the encapsulant is applied and cured, solder joints are no longer
the dominant load bearer to couple the thermal mismatch between the die and the PWB. Instead,
the rigid underfill encapsulant takes over. This is because the solder joints constitute less than
10% of the total under-die material (solder and underfill) volume. Therefore, the effective
8
properties of the under-die material are very similar to those of the underfill itself, see Table 5
and Table 7.
2D vs. 3D FEM Modeling
To investigate the applicability and limitations of the 2D finite element analysis, comparisons
were made between the stress distributions predicted by the 2D plane stress, the 2D plane strain
and the full 3D finite element calculations. Figs. 5a - 5b show the contour plots of the
distribution of the von Mises effective stress predicted by various models at 40 oC. For the 3D
model, both normal and diagonal cross-sections are shown. The extreme values of the stresses
are shown in Table 8.
It is observed from these results that both the 2D plane strain and plane stress models yield
an overall higher stress level than the 3D model does. The von Mises stress obtained from the
2D plane strain model is much higher than that obtained from the 3D model. Furthermore, in the
2D models, the 'hot spots', meaning locations with high stresses, are limited to silicon-solder
interfaces. However, in the 3D models, the hot spots are found not only in the silicon-solder
interface, but also in the encapsulant, at the corners of the silicon die, and the silicon-underfill
interface. Therefore, results from the 3D model can better explain the observed failure modes in
flip chip assemblies, such as cracks in silicon die and encapsulant, delamination in the silicon-
underfill interface, and cracks in the silicon-solder interfaces.
As for the von Mises stress, it seems that results from the plane stress model are closer to
the 3D results than to the plane strain results. Since von Mises stress is a good characterization
parameter for plastic deformation, one may therefore conclude that plane stress should yield
better prediction for plasticity induced failure in comparison with plane strain assumptions.
9
In addition to the differences in stress distributions, the 2D and 3D models also predict
significantly different overall deformation of the FCA assembly (warpage). Such comparisons
are shown in Fig. 6 for the assembly at 20 0C, where the deflection of the lower surface of the
PWB along both a normal cross-section and a diagonal cross-section are plotted. It is seen that
the 2D models give an overestimated deformation, which is consistent with the overestimated
stresses discussed earlier. Between the 2D plane stress and 2D plane strain models, the results
are not very different, although the former seems to be a slightly better approximation.
Effects of the PWB Size
To investigate the effect of PWB size on the overall assembly warpage, the deformation of
three assemblies with PWB sizes ranging from 9x9, 16x16 to 24x24 mm2 was considered.
Shown in Fig. 7a is the deflection curve along a normal cross-section of the die for different
PWB sizes. It is seen that the die deflection is almost identical for all the PWB sizes considered.
One may then conclude that as far as the stresses in the silicon die are concerned, PWB size
effects may be negligible.
However, deformation of the PWB outside the die is significantly affected by the overall
PWB size. Shown in Figs. 7b – 7c are the deflection curves of the PWB along a normal cross-
section and a diagonal cross-section, respectively. It is seen that the portion outside the silicon
die deforms differently for different PWB sizes, while the portion under the die is almost
identical for all PWB sizes.
The 3D deformed shape of the PWB is shown in Fig. 8a for the 9x9 mm2 and in Fig. 9a for
the 24x24 mm2. These deformed surfaces are obtained by plotting the deformed z-coordinates of
all nodes located on the lower surface of the PWB.
10
Note that there is an inflection region near the corners. This can be seen from the diagonal
cross section deflection curve in Fig. 6 and the deformed shape in Fig 9a. The particular shape
of the encapsulant at corners of the silicon die is believed to be responsible for this inflection.
Two encapsulant fillets on two perpendicular sides of the silicon die join at this point forming an
extended sharp ridge, which prevents the overhang of the PWB from free bending in the z-
direction while the shrinkage in the x-y plane still exists. The cause of this inflection is the
substantial z-direction CTE mismatch between the encapsulant and PWB.
Finally, it is observed that the contour density beyond the silicon die area is much more
uniform in the normal direction than in the diagonal direction, as seen from Fig 8b and Fig. 9b.
This indicates that the curvature is greater in the diagonal direction. Therefore, for designing
multi-chip modules, placing multiple chips in staggered array should be avoided because this
type of placement produces maximum mechanical interaction between the chips. In addition, to
avoid mechanical interaction, the minimum distance between adjacent dies should be
approximately 1.8 times the die size (for the PWB used in this study). If other design specifics
require a staggered array or closer distances between chips, then the mechanical interaction must
be considered. That is, the influence of the deformation of the PWB produced by one die on the
reliability of other chips needs to be considered.
Concluding Remarks
In this study, both two-dimensional and three-dimensional finite element analyses were used
to study the stress distribution in and deflection of the flip chip assembly under thermal loading.
The following conclusions were drawn.
11
(i) By comparing the numerical results with experimental data, it is demonstrated that the 3D
finite element analysis based on thermoelasticity theory provides a reasonably accurate tool for
stress analysis in FCA.
(ii) 2D plane stress and plane strain models yield similar results. Both over estimate the
deformation and stresses in FCA, although plane stress seems to be a better model.
(iii) As far as single chip modules are concerned, the size of the PWB has very little influence on
the assembly stress and deformation.
(iv) To reduce mechanical interaction between chips, a square array is preferable to a staggered
array for multiple chip modules. For square arrays, such mechanical interaction between chips
can be neglected when the minimum distance between two chips is more than 2 times the chip
size.
Acknowledgments
The research was supported by the National Science Foundation through the Packaging
Research Center at Georgia Tech.
References
Baker, D. and Kao, V., 1996, “MCMs Take Off”, Advanced Packaging, January/February,pp.16-18.
Dasgupta, A., et. al., 1997, “Miscellaneous Modeling Issues in Thermomechanical StressAnalysis of Surface-Mount Interconnects”, INTERPACK 97, Proc., EEP-Vol. 19-2, Ed. Suhir et.al., pp.1095-1100.
Hu, K.X., et al., 1997, “Electo- thermo-mechanical responses of conductive adhesivesystems,” IEEE CPMT-A, Vol.20, pp.470-477.
Iannuzzelli, R. J., et al., 1996, “Solder Joint Reliability Prediction by the Integrated MatrixCreep Method”, J. of Electronic Packaging, Vol.118, pp.55-61.
12
Lee, S. B. and Kim, J. K., 1997, “A Mechanistic Model for Fatigue Life Prediction of SolderJoints for Electronic Packages”, Int. J. Fatigue, Vol. 19, No. 1, pp.85-91.
LeGall, C.A., 1996, “Thermalmechanical Stress Analysis of Flip Chip Packages”, Master’sthesis, School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA.
Mura, T., 1991, “Micromechanics of Defects in Solids”, Kluwer Academic Publishers, 2nded.
Palaniappan, P. and Baldwin, D.F., 1997, “Preliminary In-Process Stress Analysis of Flip-Chip Assemblies During Underfill”, 30th International Symposium on Microelectronics,Philadelphia, PA.
Peterson, D. W., et al., 1997, “Stresses From Flip-Chip Assembly and Underfill;Measurements with the ATC4.1 Assembly Test Chip and Analysis by Finite Element Method”,Proceedings of the 47th Electronic Components and Technology Conference, San Jose, CA,pp.134-143.
Qu, J., 1993, "Effects of Slightly Weakened Interfaces on the Overall Elastic Properties ofComposite Materials," Mechanics of Materials, 14, pp.269 - 281.
Schubert, A., et al., 1997, “Materials Mechanics and Mechanical Reliability of Flip ChipAssemblies on Organic Substrates”. Proceedings of International Symposium on AdvancedPackaging Materials, pp.106-109.
SNL (Sandia National Laboratories), 1993, “Assembly Test Chip Ver. 04 (ACT04)Description and User’s Guide”.
Wesselmann, C., 1996, “Flip Chip Off the Dime?”, Advanced Packaging, March/April, pp.7.
13
Table 1. Material properties of silicon
Temperature (0C) Tensile Modulus (GPa) α (10-6/0C) Poisson’s Ratio, ν0 162 2.8 0.2350 150 2.8 0.23100 140 2.8 0.23
14
Table 2. Material properties of FR-4 PWBProperty 30 0C 95 0C 125 0C 150 0CE1 (GPa) 22.4 20.7 19.3 17.9E2 (GPa) 1.6 1.2 1.0 0.6E3 (GPa) 22.4 20.7 19.3 17.9G12 (GPa) 0.20 0.19 0.16 0.14G13 (GPa) 0.63 0.60 0.50 0.45G23 (GPa) 0.20 0.19 0.16 0.14
ν12 0.14 0.14 0.14 0.14ν32 0.14 0.14 0.14 0.14ν13 0.002 0.002 0.002 0.002
α1 (10-6/0C) 20.0 20.0 20.0 20.0α2 (10-6/0C) 86.5 86.5 400.0 400.0α3 (10-6/0C) 20.0 20.0 20.0 20.0
15
Table 3. Material properties of solder (63 Sn/37Pb)Temperature (0C) Tensile Modulus (GPa) α (10-6/0C) Poisson’s Ratio, ν
0 44.8 24.0 0.3720 43.2 24.0 0.37100 36.8 24.0 0.37130 34.5 24.0 0.37
16
Table 4. Material properties of copper padTemperature (0C) Tensile Modulus (GPa) α (10-6/0C) Poisson’s Ratio, ν
0 127.0 16.0 0.3420 126.0 16.0 0.34100 123.8 16.0 0.34130 121.2 16.0 0.34
17
Table 5. Material properties of underfill (Hysol 4520)Temperature (0C) Tensile Modulus (GPa) α (10-6/0C) Poisson’s Ratio, ν
25 10.0 23.1 0.33350 9.7 23.1 0.329125 8.1 23.1 0.306150 7.0 23.1 0.296
18
Table 6. Material properties of solder maskTemperature (0C) Tensile Modulus (GPa) α (10-6/0C) Poisson’s Ratio, ν
0 2.0 70 0.35100 0.7 70 0.35150 0.1 70 0.35
19
Table 7. Effective properties of the under-die solder-underfill mixtureTemperature (0C) Tensile Modulus (GPa) α (10-6/0C) Poisson’s Ratio, ν
25 12.52 23.1 0.3350 11.96 23.1 0.33100 10.28 23.1 0.31150 8.32 23.1 0.30
20
Table 8. Maximum stresses and their corresponding locationsStresses(MPa)
2D Plane Strain(xy)
2D Plane Stress(xy)
3D
xxσ -227 177 -181 146 -212 195
Location B A B A B A
yyσ -15 353 0 0 -212 195
Location F E all all B A
zzσ -47 57 -40 56 -58 56
Location D C B C H, E C
xzτ -100 111 -62 71 -43 56
Location B A G A E H
yzτ -26 47
Location I E
xyτ 16
Location B
σ (Mises) 348 244 226
Location E B A, B
A Solder side of die-solder interfaceB Die side of die-solder interfaceC Upper corner of the encapsulant filletD Solder jointsE Copper padsF Solder maskG Underfill side of die-underfill interface near the jointsH Lower corner of the dieI Upper corner of the die
21
Appendix: Effective Properties of the Under-Die Solder-Underfill Mixture
The mixture of underfill and solder joints between the die and the substrate can be viewed as
a composite material. The solders can be considered cylindrical with the axis in the z-direction.
Therefore, effective properties of the mixture can be assumed transversely isotropic with the axis
of symmetry in the z-direction.
Making use of the Mori-Tanaka method (Mura, 1991), the effective properties in the x-y
plane can be estimated as
µµ+
=K
KE39
, KE
621 −=ν , (A.1)
where
−−+−+=
))(1(2)(1
010
010 µµδµ
µµµµc
c , (A.2)
−−+−+=
))(1(3)(1
010
010 KKcK
KKcKKγ
, (A.3)
)1(2 n
nn
Eν
µ+
= , )21(3 n
nn
EKν−
= , n = 0, 1 . (A.4)
In the above equations, c is the solder volume fraction of the entire under-die mixture, and nE ,
nν are, respectively, the Young's modulus and Poisson's ratio for the underfill material (n = 0)
and for the solder materials (n = 1). The non-dimensional constants δ and γ are related to the
underfill properties through (Qu, 1993)
)1(2443
0
0
ννδ
−+= ,
)1(843
0
0
ννγ
−−= . (A.5)
22
The values listed in Table 7 were computed from (A.1).
As for the effective properties in the z-direction, the rule of mixture can be used,
01 )1( EccEE −+= , (A.6)
01 )1( ννν cc −+= . (A.7)
23
Figure Captions
Fig. 1 Baseline configuration of the flip-chip under consideration.
Fig. 2 Two-dimensional finite element model (one half).
Fig. 3 Three-dimensional finite element model (one eighth).
Fig. 4a Comparison of stresses between numerical results and experimental measurements at740C.
Fig. 4b Comparison of stresses between numerical results and experimental measurements at360C.
Fig. 5a von Mises stress contour from 2D analysis at 400C.
Fig. 5b von Mises stress contour from 3D analysis at 400C.
Fig. 6 Deflection of the lower surface of the PWB along either a normal cross-section or adiagonal cross-section at 200C.
Fig. 7a Top surface deflection of the silicon die along a normal cross-section for various PWBsizes at 200C.
Fig. 7b Lower surface deflection of the PWB along a normal cross-section for various PWBsizes at 200C.
Fig. 7c Lower surface deflection of the PWB along a diagonal cross-section for various PWBsizes at 200C.
Fig. 8a Deformed shape of the PWB of 9x9 mm2 at 200C
Fig. 8b z-contour plot (µm) for the lower surface of the PWB of 9x9 mm2 at 200C.
Fig. 9a Deformed shape of the PWB of 24x24 mm2 at 200C.
Fig. 9b z-contour plot (µm) for the lower surface of the PWB of 24x24 mm2 at 200C.
24
Fig. 1 Baseline configuration of the flip-chip under consideration.
25
z
x
3.175 mm
4.50 mm
0.60 mm
0.61 mm
Fig. 2 Two-dimensional finite element model (one half).
26
X
Y
Z
Fig. 3 Three-dimensional finite element model (one eighth).
27
LocationA B C D E F G
Stre
ss (M
Pa)
-100
-80
-60
-40
-20
0
20σxx, Num.σxx, Exp.
σyy, Num.
σyy, Exp.
σxy, Num.
σxy, Exp.
AB
C
DEF
G
chip
Fig. 4a Comparison of stresses between numerical results and experimental measurements at740C.
28
LocationA B C D E F G
Stre
ss (M
Pa)
-120
-100
-80
-60
-40
-20
0
20σxx, Num.σxx, Exp.σyy, Num.σyy, Exp.σxy, Num.σxy, Exp.
AB
C
DEF
G
chip
Fig. 4b Comparison of stresses between numerical results and experimental measurements at360C.
29
plane strain plane stress
Fig. 5a von Mises stress contour from 2D analysis at 400C.
30
diagonal cross-section normal cross section
Fig. 5b von Mises stress contour from 3D analysis at 400C.
31
Distance from the Chip Center (mm)0 1 2 3 4 5 6 7
Def
lect
ion
(µm
)
-30
-20
-10
0
3D Normal3D Diagonal2D Plane Strain2D Plane Stress
Fig. 6 Deflection of the lower surface of the PWB along either a normal cross-section or adiagonal cross-section at 200C.
32
Distance from the Chip Center (mm)0 1 2 3
Def
lect
ion
(µm
)
-8
-6
-4
-2
0
PWB Size = 9X9 mm2
PWB Size = 16X16 mm2
PWB Size = 24X24 mm2
Fig. 7a Top surface deflection of the silicon die along a normal cross-section for various PWBsizes at 200C.
33
Distance from the Chip Center (mm)0 2 4 6 8 10 12
Def
lect
ion
(µm
)
-60
-40
-20
0
PWB Size = 9X9 mm2
PWB Size = 16X16 mm2
PWB Size = 24X24 mm2
Fig. 7b Lower surface deflection of the PWB along a normal cross-section for various PWBsizes at 200C.
34
Distance from the Chip Center (mm)0 5 10 15
Def
lect
ion
(µm
)
-60
-40
-20
0
PWB Size = 9X9 mm2
PWB Size = 16X16 mm2
PWB Size = 24X24 mm2
Fig. 7c Lower surface deflection of the PWB along a diagonal cross-section for various PWBsizes at 200C.
35
-30
-20
-10
0
-6-4
-20
2 4 6
-6-4-2024Def
lect
ion
(µm
)
y-coordinate
(mm)
x-coordinate (mm)
Fig. 8a Deformed shape of the PWB of 9x9 mm2 at 200C.
36
x-coordinate (mm)-6 -4 -2 0 2 4 6
y-co
ordi
nate
(mm
)
-6
-4
-2
0
2
4
6
-28-21
-21-28
-14-14
-14
-14
-7
-7
-7
-21
-21
-28
-28
Fig. 8b z-contour plot (µm) for the lower surface of the PWB of 9x9 mm2 at 200C.
37
-60
-40
-20
0
-12-8
-40 4 8 12
-12-8-4048Def
lect
ion
(µm
)
y-coordinate
(mm)
x-coordinate (mm)
Fig. 9a Deformed shape of the PWB of 24x24 mm2 at 200C.
38
x-coordinate (mm)-15 -10 -5 0 5 10 15
y-co
ordi
nate
(mm
)
-15
-10
-5
0
5
10
15
-50
-50-50
-50-50
-40-40
-40
-30
-30
-30
-20
-20
-10
-10
Fig. 9b z-contour plot (µm) for the lower surface of the PWB of 24x24 mm2 at 200C.
