**H X Shang, J X Gao and P I Nicholson**

Paper presented at 2008 ASME International Mechanical Engineering Conference and Exposition, November 2-6, 2008, Boston, Massachusetts, USA. Paper no. IMECE2008-66417.

## Abstract

In this study, an analytical model to obtain a closed-form solution for thermomechanical behaviours of BGA (Ball Grid Array) package was derived and experimentally verified. In the theoretical analysis, the BGA package was represented with a three-layer axisymmetrical model: two dissimilar materials jointed by a graded interlayer. Based on the classical bending theory, the thermal stresses induced by temperature changes were accurately calculated. 2-D FE (Finite Element) models of BGA packages subjected to high temperature were used to verify the theoretical solutions. Furthermore, two kinds of BGA samples, with eutectic (63wt%Sn/37wt%Pb) and Pb-free SAC387 (95.5wt%Sn/3.8wt%Ag/0.7wt%Cu) solder joints respectively, were experimentally investigated by the high resolution Moiré Interferometry (MI). The thermal cycling tests were performed on each package from 25°C to 125°C. It was proved that the thermal deformation obtained from moiré tests corroborated the analytical solutions and FE analysis. Through the perditions of fatigue life of solder joints based on the shear strain values, the reliability characteristics of BGA assemblies were also identified.

## Introduction

In the service of microelectronic packages, thermal stresses have been identified as a main problem which can induce crack initiation and affect operational functionality. Because of the large thermal expansion (CTE) mismatch between constituent materials, a package is subjected to thermal deformation under thermal cycling conditions during its application. Particular attention has been attracted to the solder joints in packages since the thermomechanical performances of solder joints is the basis to assure the long-term reliability of electronic packages, not only because the solder joints provide the electrical interconnection, but also they are the sole mechanical attachment of the electronic components to the Printed Circuit Board (PCB).^{[1-4]} The constrained solder joints are cycled between the maximum and minimum temperature limits depending on the environment of use, which result in most of the failures of microelectronic packages.

Various techniques have been developed for the assessment of thermal behaviours of microelectronic packages, including experimental techniques such as mechanical fatigue testing^{[3-5]} and MI,^{[1-6]} and statics based numerical simulation like Finite Element Analysis (FEA).^{[7-8]} However, MI is sensitive to environmental vibration, and numerical method cannot exactly simulate the continuous variation of material properties within the solder joints; instead, a stepwise change in properties is used.

In this study, an analytical model to obtain a closed-form solution for BGA packages subjected to high temperature was derived based on Timoshenko's classic bending theory. The solutions were then numerically and experimentally verified. Comparisons between the performances of eutectic 63Sn/37Pb and Pb-free SAC387 solder joint were conducted with the same thermal conditions on individual component, and the fatigue life of each type package was predicted.

## Methodology

### Theoretical Analysis

Timoshenko first derived the general solution for elastic thermal stresses in a bi-metal system in 1925, in which a bi-layer beam subjected to a temperature change was studied.^{[9]} The analysis was based on classical beam theory and started by assuming that both layers have the same curvature (ρ) during bending. The individual force and bending moment in each layer were related to the curvature of the layer. By balancing the forces *F* and moments *M* in the system and by including the temperature-induced (ε_{T}), the force-induced (ε_{F}) and the bending-induced (ε_{M}) strains to satisfy the strain continuity condition at the interface between the two layers, the solution was obtained. This approach has been adopted and developed by many others to analyze the thermal stresses in multilayered systems. ^{[10-12]}

Here the BGA package is simplified and represented with a three-layers axisymmetrical model, as shown schematically in *Fig.1*, where layers with individual thicknesses, *di*, are bonded. The subscript, *i*, denotes the layer number ranging from 1 to 3. Corresponding to a real BGA package, the top layer (layer 1) is the substrate attached to the solder joints, the centre layer (layer 2) represents the solder joint and the bottom layer (layer 3) represents the FR-4 board. This elastic layered system is subjected to a uniform temperature change Δ*T*. The CTE, the shear module, Young's module, and Poisson's ratio of three materials are αi, *µ*i, *E _{i}* ,

*v*respectively.

_{i}**Fig.1. Schematic drawing of a three-layer bending system**

According to Timoshenko's bending theory, the normal stress distribution in a vertical section of the beam is a linear function of depth. e.g., in an upwards bending, the bending stress values ranges from maximum tension + σ_{m} at the top free surface, down to maximum compression - σ_{m} at the bottom free surface. At positions away from the edges, the stresses induced by the thermal mismatch are in-plane (i.e. parallel to the interface) and the stresses normal to the interface are zero. The neutral axis (the line in the cross section of the bending system where the normal stress is zero) is defined as y = 0. The free surfaces of the substrate and board are located at *y* = *h*_{1} and *y* = *-h _{3}* , respectively. The interface of layers 1 and 2 is located at

*y*=

*h*. With these definitions, the relation between

_{2}*h*and

_{i}*d*is described by:

_{i}To determine the deformation and stresses in this model under a thermal condition, firstly, the position of the neutral axis of the system needs to be identified. According to the resultant force due to the bending strain at *y*=0 is zero; i.e.

From which the distance of *h _{2}* can be determined, as:

Secondly, the force-induced strain ε_{F} and the bending-induced strain ε_{M} can be derived by satisfying the following two boundary conditions: (i) the resultant force *F* due to the force-induced strain component is zero; (ii) the resultant bending moment *M* due to the bending-induced strain is zero:

From which, the force-induced strain ε_{F} can be expressed as:

and the curvature of the system ρ can also be obtained by,

The strain distribution in the system can be formulated as

where

, (i=1, 2, 3).

The axial stresses and displacements distribution in each layer can be expressed as:

(i=1, 2, 3)

From analysis above, it can be seen that if the material and geometric parameters of a BGA package are given (*Table 1*), with the values of *h _{2}* , ε

_{F}and the curvature ρ, the strain/stress distributions subjected to a temperature change Δ

*T*are completed , as listed in

*Table 2*. Note both strain distribution ε and stress distribution σ given by equations (7) and (8) are functions of

*y*. Although the bending axis define in his paper is different from Hsueh,

^{[12]}which has been defined as the interface between layer 2 and layer 3, the final results should be the same. In all these discussions, the positions remote from the free edges are considered, and all the layers have the comparable thicknesses. If the temperature dependence of CTE of solder joint materials is concerned, the thermal strain α Δ

*T*should be replaced by an integral of the CTE with respect to the temperature, or by using the average CTE within the temperature range.

**Table.1 General solution of BGA packages with increased temperature**

| Sn-37Pb( h =0.516mm)_{2} | SAC387 solder( h =0.491mm)_{2} | ||
---|---|---|---|---|

ΔT(°C) |
1/ρ (m^{-1}) |
ε (x10 )^{-3} |
1/ρ (m^{-1}) |
ε (x10 )^{-3} |

25 | 1.509 | 1.045 | 0.938 | 0.862 |

50 | 3.019 | 2.091 | 1.877 | 1.724 |

75 | 4.528 | 3.136 | 2.815 | 2.586 |

100 | 6.037 | 4.181 | 3.753 | 3.449 |

### BGA Samples

2 kinds (3 samples for each type) of BGA packages with the same geometric parameters subjected to thermal cycle were analyzed. The BGA packages are 28 mm square and 3.16 mm in height, with a 16 x 16 array of solder balls (0.6 mm in diameter and 1.0 mm soldering pitch), as shown in *Fig.2*. Prior to the thermal cycle tests, the BGA assemblies were powered to test the electrical continuity. After that the BGA packages were cut long the centreline and polished through the rows of solder ball to expose the cut surface. A high temperature grating with frequency of 1200 lines/mm was replicated on each polished surface of package for further MI tests.

**Fig.2. Cross-sectional view of the BGA package**

### Finite-element analysis (FEA)

Before main comparative experiments, FEA was performed to simulate the thermal behaviours of BGA packages with temperature ranging from 25-125°C. In order to examine the effect of soldering materials to the thermal deformation of BGA packages, 2-D plane strain models with eutectic 63/37Pb and Pb-free SAC387 solder joints were investigated and compared. Due to the symmetry of the package, only half of the package was modelled. The mesh was refined at the significant stress/strain concentration field (i.e. multi-material junctures near solder balls). The symmetry boundary condition was imposed on the left most edge of the package and all the displacement components at the bottom left-corner node of the symmetry plane were constrained to prevent rigid body motion. The elastic material properties of these components are listed in *Table 2* based on the previous studies.^{[13,14]} Note that all the materials in the package were assumed to be linearly elastic except the solder joint. The solder joint materials were considered as a viscoelastic and temperature-dependent material because of their creep behaviours.

**Table 2 Material properties of BGA packages**

Materials |
Young'sModulus (GPa) |
PoissonRatio |
CTE(ppm/°C) |

Copper pad | 121 | 0.35 | 17 |

SAC387 solder |
25°C: 46 50°C: 44 100°C: 35 |
0.3 | 17 |

63Sn/37Pb eutectic solder |
25°C: 29.97 65°C: 20.629 105°C: 12.455 125 °C:12.450 |
0.35 | 25.1 |

BT substrate |
18.6 | 0.36 | 15.0 (x, z)57.0 ( y) |

Silicon chip | 162.4 | 0.23 | 3.3 |

Overmold | 8.96 | 0.35 | 19.0 |

FR4-PCB | 22 | 0.28 | 16.0 (x, z)72.0 ( y) |

### Moiré Interferometry (MI)

MI allows measurement of the whole-field deformation in a package during fatigue testing with a typical sensitivity of 0.417 *µm*.^{[6]} The principle of moiré interferometry is depicted schematically in *Fig.3*, where a high frequency crossed-line diffraction grating is replicated onto the surface of the specimen and deforms together with the underlying specimen. Coherent laser beams 'A' and 'B' create a virtual reference grating in their zone of intersection and interfere with the specimen grating to produce a moiré fringe pattern. The moiré fringe represents contours of constant *U* and *V* displacements (displacement components in orthogonal *x* and *y* directions, respectively), which can be determined by^{[6]}:

**Fig.3: Optical setup of Moiré interferometry (Light diffracted in the ± diffraction orders of the specimen grating admitted to the CCD to create interference patterns**

The corresponding strain components derived from displacement are:

where *fs* is the frequency of the specimen grating, and *Nx* and *Ny* are the fringe orders in the *U* and *V* field moiré patterns, respectively.

A 3-D moiré interferometer was used to measure the real-time deformation of BGA samples subjected to thermal cycle. It comprises an optical system to form moiré fringe, a high temperature chamber to produce a thermal cycling environment, and controlling software to perform image processing. Moiré fringe patterns were captured every 25°C increasing or decreasing temperature levels during heating-up and cooling-down procedures from 25 to 125°C. A ramp rate of 5°C/min and a dwell time of 10 minutes at each stage were applied to make sure that the sample would reach the designated temperature.

## Experimental results

### Displacements

Selected Moiré patterns in *x*-axis direction (*U*) and *y*-axis direction (*V*) along the cross sections of eutectic and SAC387 package by MI are shown in *Fig.4* to *Fig.5*. The stress-free temperature is assumed to be 25°C, at which the grating is replicated to the specimen. The solder ball in the centre of the ball arrays is used as a reference point and only half of the balls are investigated. It can be seen that the density of fringes increased with increasing temperature, meaning that the displacement also increased. The variation in the number of fringes between the substrate and FR-4 board is caused by the difference of CTE between the two materials. The displacement mismatch between these two materials introduces severe shear deformation in solder joints that connect the chip and substrate mechanically. This deformation becomes more significant as the temperature increases. Since the solder joints constrain the thermal deformation of the substrate and the board, an effect is verified by the shape of the fringe patterns in *Fig.4* and *5*: the fringe is not exactly vertical but slightly bends. Due to the shape and materials variations in the structure of a BGA, the *U* and *V* displacement are larger at multi-material junctures near solder balls and free edges than other areas.

**Fig.4. Moiré fringes of a BGA package with 63Sn/37Pb solder balls**

**a) U field at 25°C;**

**b) V field at 25°C;**

**c) U field at 125°C;**

**d) V field at 125°C**

**Fig.5. Moiré fringes of a BGA package with SAC 387 solder balls:**

**a) U field at 25°C; **

**b) V field at 25°C;**

**c) U field at 125°C;**

**d) V field at 125°C**

*Fig.6* shows the contour plot of the displacement fringes of a eutectic BGA package at 125°C obtained by FEM simulation. Comparing to *Fig.5*, the orientation of the FEA fringe representing regions of constant displacement coincides with the orientation of the moiré patterns. Note for a BGA package which has *x*-invariant material property distribution and geometry under thermal conditions, the edge effect occurs in deformation field when the length dimension is finite, as seen from *Fig.6*, that the high temperature induced edge effect in the vicinity of the right end. In addition, since the FE models exhibit abrupt deformation discontinuities across cell interfaces, the nodal deformation values of cell located at the same vertical location need to be averaged in the calculations. *U* displacement along the cross-sectional specimen was calculated from the moiré fringes by Eq.9. *Fig.7* shows the *U* displacement field of each solder ball generated at 100°C. The *U* deformation increases gradually by the location of the solder joints from centre to outside, meaning that bending arises in the chip and the board. This bending is related to the deformation gradient in the chip, substrate, and board along the thickness direction, due to the constraint of solder joint. The experiential results were also compared with theoretical results calculated by Eq.8 and FE simulations. As seen in *Fig.7*, the deformation measured by MI is in good agreement with those obtained by FEA for each package. The average relative error of all solders is less than 10%, except the deviation around the outermost solder ball. This may be due to the results of FE analysis which highly depends on a precise modelling of the nonlinear and complex mechanical behaviours of materials. The approximation is not as accurate as the FE 3D solid plane strain FE approximation. While 2D FEA results still support the validity of the deformation measurement within experimental accuracy. In the theoretical calculation for *U* displacement, the solder balls are replaced by the elements at the corresponding position, since no isolated solder joint exists in the model. The theoretical solution also matches with experimental result and FEA in a reasonable range. If eutectic and SAC387 solder balls are compared, the former has up to 25% larger deformation values than the latter.

**Fig.6. U- and V-displacements of the BGA package with 63Sn/37Pb solder balls from FEA**

**Fig.7. Comparison of the U- displacement from Moiré measurement, FEA and theoretical analysis at T=100°C**

Usually the deformation calculated by the theoretical method is larger than those from FEM and moiré tests at the same temperature, because more heat generate in the soldering layer during heating-up in the analytical model than the separate solder balls in an a real sample. This limits its application for precisely analyzing tiny and complex structures. However, as an easy and economic tool, theoretical solution is ideal for understanding the sophisticated mechanical behaviours and failure mechanism of the electronic packaging.

### Shear strain

Many practical analyses of solder joint lifetimes and reliability require the shear strain distribution. When the electronic package experiences a temperature change, the silicon chip and over mold tends to expand relative to the FR4-board. The dominant constraint that restricts the relative expansion is a system of shear force acted on the interface, i.e. solder joints. Strong fringe gradients appear in solder joints corresponding to plasticity or yielding of maximum shear stress/strain. The reliability of BGA package relies on the solder ball with the maximum shear strain value.^{[1,14]} Strong fringe gradients appear in solder joints corresponding to plasticity or yielding of maximum shera stress/strain. In *Fig.7*, the outermost solder ball experiences the maximum deformation among all solder joints; therefore it will have the maximum shear strain and fail easily by thermal fatigue. The shear strains for the outmost solder balls were calculated using the moiré fringes and FEA at different temperatures, as shown in *Fig.8*. Comparing plots with temperature, we first notice, regardless of the soldering materials, that the variation trends of strain distribution with temperature are similar. However, the different soldering materials does produce difference in shear strain values, which is resulted naturally from the enforcement of harder and stiffer solder joints of SAC387. The Young's modulus of SAC387 is much higher than that of the eutectic solder joint therefore it has stronger resistance to deformation. This is consistent well with the preliminary results given in Section 3.1. Shear strain values obtained from MI show a good agreement with the finite-element approach, for both BGA packages in *Fig.8*.

**Fig.8. Comparison of the shear strain values of the outermost balls in the BGA packages with temperature**

### Reliability

The shear strain of the outermost solder ball determined by MI is applied to predict the fatigue life of BGA packages. The fatigue life of the solder joint is predicted using an empirical Coffin-Manson relationship.^{[15,16]} The total number of cycles to failure *Nf* can be expressed as:

_{p}is the plastic shear strain range after one thermal cycle, and the constants

*m*and

*C*are found by performing isothermal fatigue test on a solder sample at 1Hz with the temperature range of -40°C to 150°C. The average values of constants are found to be:

*m*=0.70,

*C*= 1.69 for eutectic solders

^{[1]}and

*m*=0.853,

*C*=9.2 for Pb-free solders.

^{[14]}With this life prediction model, the fatigue cycles are estimated for eutectic and SAC387 BGA packages were 4037 and 9868, respectively. In comparison to acceleration thermal cycling tests performed by Clech

^{[17]}(with temperature range of 0-100°C, 10 minute dwell times at 100°C), the failure life ratio 2.44 of SAC387 to Sn37Pb solders in this study was close to 2.15 in Clech's study. It proves that the Pb-free solder has the potential benefit of more reliable solder joints than the Sn-37Pb eutectic solder joints under a same testing condition, since Pb-free solders have different physical and better mechanical resistance to fatigue fracture from the Sn-37Pb eutectic solder.

^{[18]}

## Conclusions

To investigate the interaction of BGA package under thermal cycling condition, the thermal stress/strain of BGA packages were analyzed theoretically, numerically and experimentally. Closed-form approach to evaluate the thermal mechanical behaviours of BGA package was developed. Obtained results can be summarized as follows:

- Analytical solution is an easy and fast approach to show the thermal stress/strain developed in a BGA assembly during thermal cycle. However, the accuracy of the method will decrease with the increase of the complexity lever of electronic packages.
- High temperature MI validated the theoretical and numerical solutions, through recording the subtle and complex deformation information of the samples. Good agreements were obtained from both the theoretical and the experimental solutions.
- The maximum shear strain of solder joints was used to predict the reliability of the packages based on Coffin-Manson equation. A ~41% increase in fatigue life was found in the Pb-free package compared to a eutectic one.

By this study, the design engineers can better understand the response of BGA assemblies under thermal conditions, access the reliability of their basic design plan at the concept design stage, and rate each factor that can affect the fatigue life of the packages.

## Acknowledgements

The work presented was carried out with the financial support of Marie Curie Fellowship of EC FP6 (Project No. 039893). The authors also thank National Microeletronics Research Centre (Cork, Ireland) and Pac-Tech (Surrey, U.K.) for manufacturing samples.

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