PRINCIPAL COMPONENT REGRESSION MODELS FOR LIFE PREDICTION OF PLASTIC BALL GRID ARRAYS ON COPPER-CORE AND NO-CORE ASSEMBLIES Pradeep Lall, Aniket Shirgaokar, Luke Drake, Timothy Moore, Jeff Suhling Auburn University Department of Mechanical Engineering and NSF Center for Advanced Vehicle Electronics Auburn, AL 36849 Tele: (334) 844-3424 E-mail:
[email protected] Milan Shah Northrop Grumman Corporation, Rolling Meadows, IL, USA
ABSTRACT In this paper, principal component regression models (PCR) have been investigated for reliability prediction and part selection of area package architectures under thermomechanical loads in conjunction with stepwise regression methods. Package architectures studied include, BGA packages mounted on CU-CORE and NO-CORE printed circuit assemblies in harsh environments. The models have been developed based on thermo-mechanical reliability data acquired on copper-core and no-core assemblies in four different thermal cycling conditions. Solder alloys examined include SnPb and SAC Alloys. The models presented in this paper provide decision guidance for smart selection and substitution to address component obsolescence by perturbing product designs for minimal risk insertion of new packaging technologies. It is conceivable for commercial off the shelf parts to become unavailable during the production-life of a product. Typical Commercial-of-the-Shelf parts are manufactured for a period of two to four years, and IC manufacturing processes are available for five to six years. It is envisioned that the reliability assessment models will enable turn-key evaluation of geometric architecture, material properties, and operating conditions effects on thermo-mechanical reliability. The presented approach enables the evaluation of qualitative parameter interaction effects, which are often ignored in closed-form modeling, have been incorporated in this work. Previously, the feasibility of using multiple linear regression models for reliability prediction has been demonstrated for flex-substrate BGA packages [1, 2], flip-chip packages [3, 4] and ceramic BGA packages [5] Convergence of statistical models with experimental data and finite element models has been demonstrated using a single factor design of experiment study. In addition, the power-law dependencies of individual variables have been correlated with established failure mechanics models. PCR approach uses the potentially important variables from stepwise regression. The statistics models are based on
accelerated test data acquired as part of this paper, in harsh environments, while finite-element models are based on damage mechanics and material constitutive behavior. Sensitivity relations for geometry, materials, and architectures based on statistical models, and FEA models have been developed.
KEY WORDS Thermo-Mechanical Reliability, Ball-Grid Arrays, Statistical Models, Part Obsolescence.
{D} o {E} {E*} E0, E1, …, Ek Hi O1, O2,…, Ok
[0
{b} bj,pc* bj,pc 1/TmeanK
978-1-4244-1701-8/08/$25.00 ©2008 IEEE
770
NOMENCLATURE The vector of new regression coefficients for the principal components The initial guess of the constant power-law dependence of predictor variable. n x 1 Vector of the n-regression coefficients n x 1 vector of transformed regression coefficients for scaled and centered variables. The regression coefficients The model error for the ith data-set. The eigenvalues of the Correlation Matrix Predictor Variable after Power-Law D Transformation ( [0 x 0 ) Least Squares Estimator of Regression Coefficients Regression Coefficients of the Principal Components Regression for the centered and scaled principal components. Regression Coefficients of the Principal Components Regression for the Original Variables Inverse of the mean temperature in Kelvin
[A]
Matrix of Predictor Variables, of full column rank 1/TmeanK Inverse of the mean temperature in Kelvin BGA Ball Grid Array BallDiaMM Diameter of the solder ball in millimeters BallHtMM Height of the solder ball in millimeters C Correlation Matrix. CABGA Chip array BGA Coeff Coefficient Cu Copper DeltaTdegC Temperature cycle range in degree centigrade DieLengthMM Chip Length in millimeters DietoBodyRatio Ratio of the length of the chip to the length of the package ENIG Electroless Nickel Immersion Gold f-ratio (Mean square between groups)/(mean square within groups) HASL Hot Air Solder Leveling k Number of predictors MSres Mean Square of residuals n number of data points p Number of variables PBGA Plastic Ball Grid Array PCR Principal Component Regression PkgPadDiaMM Diameter of the package pad in millimeters PkgWtGMS. Weight of the package in grams R2 Multiple coefficient of determination, 1-(SS Error/ SS Total) R-sq(adj) R-sq adjusted for degrees of freedom S Standard Deviation SE Coeff tandard error of the coefficient SolderVolCuMil Volume of the solder in cubic mils SSres Sum of Squares of residuals [V] The k x k eigenvector matrix consisting of normalized eigenvectors VIF Variance Inflation Factor [X] Matrix of Predictor Variables, of full column rank [X*] Scaled and Centered Predictor Variable Matrix. x1, x2,…,xk The k-predictor variables, Y Regressor Variable [Z] The n x k matrix of principal components z1, z2,…,zk The k new variables called principal components of the correlation matrix [C].
INTRODUCTION In this work, risk-management models for reliability prediction of BGA packages on NO-CORE and CU-CORE assemblies in harsh environments have been presented. The models presented in this paper provide decision guidance for selection of component packaging technologies. In addition, qualitative parameter interaction effects, which are often ignored in closed-form modeling, have been incorporated in this work. Previous studies have focused on deterministic prediction of reliability.
Methodology presented in this paper, is intended as a tool-set for doing “what-if” studies on COTS (commercial-offthe-shelf) components. The trend of the defense applications toward usage of COTS components as a cost mitigation measure and its impact on component reliability is immense. COTS components have gained visibility over the past decade. The COTS philosophy decrees that, when you can obtain commercial components and subsystems that meet military specifications, you can use them in military systems. The hybrid approach is intended as an aid for designers wrestling with the problem of “component obsolescence” and in identification of equivalent reliability packaging architectures. Component obsolescence has been deemed a leading problem for designers of systems requiring long-term reliability (e.g. 10 to 20 years). The “mil-spec” parts have special status which involves long-term availability, beyond the point at which a commercial part would be discontinued. If one were to use a commercial part, then commercial rules apply: no special treatment and there is no guarantee of supply outside normal practices. Therefore, it is essential that future packaging architectures for advanced military applications be insensitive to obsolescence. Substitution of components on the fly as they become obsolete to maintaining the highest level of operational readiness without extensive experimental testing for each change will require a fundamental understanding of the materials and interfaces in drop and shock impact loading scenarios. Methodology presented in this paper, addresses scope beyond that addressed by published standards [6, 7, 8]. The hybrid approach presented in this paper addresses parameters, beyond those addressed by finite-element models and closed form models. In this paper effort has been made to include a comprehensive list of the design parameters to address the thermal reliability. Previously, several efforts [9, 10, 11, 12, 13, 14] have been made to investigate and understand the failure mechanics and thermal reliability of these packages under harsh environment. Previous studies have demonstrated that thermal-fatigue reliability of area-array packages depends on the design parameters (i.e. I/O pitch, I/O count, die size, package size, substrate thickness etc), the material sets (properties) used in the construction of a package, and the environmental conditions such as extreme temperatures, dwell times, and ramp-rates encountered by the package during its life. Design of robust assemblies that meet product-life requirements are often addressed by accelerated testing or by using reliability models. Several approaches are available today for reliability prediction including nonlinear finite element models [11, 15, 16] and first and second-order closed form models [17, 18, 19]. There is need for a turnkey approach for making trade-offs between geometry and materials, and quantitatively evaluating the impact on reliability. A statistics-based approach has been presented, which targets a probabilistic life prediction for component wear-out under thermo-mechanical stresses. Models developed have been correlated with experimental data and non-linear finite element models. Factor effects for geometry, materials, and architectures based on statistical models, and FEA models have been developed. Convergence of statistical, failure
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mechanics, and FEA based model sensitivities with experimental data has been demonstrated. Validation of model predictions with accelerated test data has been presented. Advantages of the proposed approach include the ability to address both single and coupled effects in a decisionsupport framework, in addition to addressing tacit factors such as board finish and diagonal length, which do not lend themselves to a mathematical description easily in first-order formulations. In addition, parameter interaction effects, which are often ignored in closed form modeling, have been incorporated in the presented approach.
TEST VEHICLES Test boards were designed to include a multitude of package architectures on both Copper-Core Assemblies and No-Core Assemblies. Individual Package level parameters were varied to estimate the effect in reliability. Variables studied include, I/O count, I/O pitch, die size, package size, substrate thickness, solder ball composition, material sets. PWB construction parameters include, number of layers, material sets. Package architectures investigated include, PBGA, FCPBGA, MCM-PBGA, Flex-BGA, Flip-Chip BGA, CSPs, CBGA, and QFPs. Package I/O ranged from 49 to 1508, Package Pitch ranged from 0.254 mm to 1.27 mm. Both fullarray and perimeter-array package configurations were tested (Table 1). Figure 1 and Figure 2 illustrate the multitude of package architectures on either side of a standard test board configuration. Board-Level interconnects with two solder alloys were tested in each package configuration including, 63Sn37Pb and Sn3Ag0.5Cu. Fifty one types of packages (design, construction, and material attributes) with the sample size of 18 for each variable were tested. In this study, two PWB constructions including (1). integral copper core and (2). No integral copper core-PWBs were evaluated for each package type. Each PWB construction consisted of 18 signal and ground layers. Figure 3 illustrates the stack-up of both integral copper core and noncopper PWBs.
Figure 2: Side B of typical test board.
(a)
(b) Figure 3: Stack-up of Integral Cu core and No Cu Core PWBs.
Figure 1: Side A of typical test board.
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Table 1: Scope of the Packaging Architectures. MicroLeadFrame
QFP/LQFP
Full Perimeter
Perimeter
Perimeter
0.5 - 0.8
0.254 - 0.45
0.40 - 0.65
0.4 - 0.5
483
132 - 228
48 -317
44 - 100
100- 176
Sn-Pb Pb-Free: SAC305
Sn-Pb
Sn-Pb Pb-Free: SAC305
Sn-Pb Pb-Free: SAC305
Sn-Pb Pb-Free: Sn Finish
Sn-Pb Pb-Free: Sn Finish
25.0
29.0
7.0 - 12.0
5.08 - 6.35
9.0 - 12.0
14.0 - 20.0
Full Perimeter
MCMPBGA Full Perimeter
Hi-TCE CBGA Full
0.8 - 1.00
1.00
1.27
49 - 900
532 - 1508
128 -324
Solder Alloy
Sn-Pb Pb-Free: SAC305
Sn-Pb Pb-Free: SAC305
Pacakge Size (mm)
7.0 - 31.0
23.0 - 40.0
Package Type Array Type I/O Pitch (mm) I/O Count Range:
PBGA
FC-PBGA
Full Perimeter Mixed 0.5 - 1.00
CBGA
CSP
Flip Chip
Full
Full Perimeter
1.27
360
Sn-Pb
22.0
Die Size Range
4.00 - 24.00
Package to Die Size Ratio
1.00 to 3.94
TEST CONDITIONS Four test conditions in the experimental accelerated test matrix include, TC1 (-40 to 95qC), TC2 (-55 to 125qC), TC3 (3 to 100qC), and TC4 (-20 to 60qC). Each cycle has a dwell time of 30 minutes and a ramp rate of 3°C/min. The chambers were profiled with full-load, and the temperatures measured at various locations on the test boards in the stack, to ensure that the packages experience uniform temperature exposure. All packages are daisy-chained and the resistance monitored to identify failures. Table 2: Test Conditions Low High Low High Dwell Ramp Rate Temp Temp Dwell Profile (min) (°C/min) (°C ) (°C ) (min) TC1 -40 95 3 30 30 TC2 -55 125 3 30 30 TC3 3 100 3 30 30 TC4 -20 60 3 30 30
STATISTICAL MODELING APPROACH A combination of statistics-based, and finite element methodology has been used to identify the critical parameters and their sensitivity on the thermal reliability of the BGA packages. Sensitivities of reliability to design, material, architecture, and environment parameters have been developed from statistics, and finite element models and validated versus experimental data. Experimental data was accumulated for all package types and failure data was input into Weibull plots for each package. Figure 4 shows the Weibull plot for a PBGA228 package on a no-copper core PCB. Figure 5 shows the Weibull plot for a PBGA728 package on cu-core PCB. Output values for the Weibull distribution were analyzed for each package including, K (number of thermal cycles to first failure), E (slope of failure trend line), R2 (a measure of how much variability in the model is properly explained by the data), and n/s (a value of total test packages / packages that have not yet failed). The N1% values have been used as the response variable for model development.
Figure 4: Weibull Plot of PBGA228 Package on No-Copper Core PCB.
Figure 5: Weibull plot of PBGA728 Package on Cu-Core PCB. The relative influence of various architecture and material parameters on the thermal reliability has been determined by using multivariate regression and analysis of variance techniques. MINITAB™ has been used for statistical analysis.
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SMD Pads on Single-Sided Substrate
SMD Pads on Multilayer Substrate
SMD Pads on Double-Sided Substrate Via-in-Pad Figure 6: Failure Analysis of Solder Interconnect Samples showing Crack Propagation for SMD, NSMD, and Via-in-Pads. Parameters investigated for thermal reliability of the BGAs include: die size, die to package ratio, solder ball composition, solder ball size (ball diameter and height), solder volume, ball pitch, ball count, pad size, local CTE mismatch between mold compound, global CTE mismatch between component and PCB board, PCB thickness, board finish (HASL and ENIG), PCB material, substrate material and temperature cycle conditions such as ramp rate, dwell time, and temperature extremities. Multivariate regression, analysis of variance techniques have been used to develop mathematical equations for parametric sensitivities. The parameters are given a physical basis through correlation with failure mechanics. In each case, the samples have been cross-sectioned and failure mode verified to ensure that the predicted reliability was relevant to solder joint thermo-mechanical fatigue (Figure 6).
VARIABLE-SELECTION Predictor variables for model building have been selected by developing a super-set of variables that are known to influence the characteristic life of an area array package and then selecting the potentially important variables using stepwise regression and method of best subsets. The adjusted R2, Variance inflation factor (VIF), residual mean squares and induced bias has been used as criteria for variable selection.
Coefficient of multiple determination, adjusted R2, residual mean squares and induced bias has also been used as criteria for variable selection. Coefficient of multiple determination (R2), given by Equation (1), measures the overall adequacy of the regression model and variables that create a significant increase in coefficient of multiple determination are retained in the model. The residual mean square is calculated for all possible subset of variables and the subset that minimizes residual mean square is selected for model building. [20, 21] SSRe sidual (1) R2 SSTotal § n 1 · (2) ¸¸ 1 R 2p R 2Adj 1 ¨¨ ©n p¹ Variance inflation factor (VIF) measures the impact of colinearity among the regressors in a regression model on the precision of estimation. It expresses the degree to which colinearity among the predictors degrades the precision of an estimate. Let x1 , x 2 ,........, x k be the k predictors. The
predictor variables
x j have been regressed on the remaining 2
k-1 predictors and the R-squared from this regression, R j has been computed. Then the variance inflation factor for x i
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is
1 . In the situation that x j is highly correlated with 2 1 R j
the remaining predictors, its variance inflation factor will be very large. When x j is orthogonal to the remaining predictors, its variance inflation factor will be 1. The variable selection was done based on the stepwise regression procedure and the partial F-test.
PRINCIPAL COMPONENTS REGRESSION Multiple linear regression methods assume the predictor variables to be independent of each other. Linearly dependent variables result in multi-colinearity, instability and variability of the regression coefficients [22]. Principal components models have been used for dealing with multi-colinearity and producing stable and meaningful estimates for regression coefficients [23]. The principal components technique determines a linear transformation for transforming the set of X predictor variables into new set Z predictor variables known as the principal components. The set of new Z variables are uncorrelated with each other and together account for much of variation in X. The principal components correspond to the principal axes of the ellipsoid formed by scatter of simple points in the n dimensional space having X as a basis. The principal component transformation is thus a rotation from the original x coordinate system to the system defined by the principal axes of this ellipsoid [24]. The principal component transformation has been used to rank the new orthogonal principal components in the order of their importance. Scree plots, Eigen values and proportion of total variance explained by each principal component are then used to eliminate the least important principal components. Multiple linear regressions have been performed with the original response variable and reduced set of principal components. The principal components estimators are then transformed back to original predictor variables using the same linear transformation. Since the ordinary least square method has been used on principal components, which are pair wise independent, the new set of predictor coefficients are more reliable. Consider a thermo-mechanical reliability dataset which consists of time-to-failure as the response variable, y which depends on k-predictor variables including geometry, architecture, material properties and operating conditions. Assume that the data-set spans n-sets from the same package architecture. Assume that the regression model is of the form, (3) y i E 0 E1 x1i E 2 x 2i ... E k x ki H i Where, x1, x2,…,xk are the k-predictor variables, E0, E1, E2,…, Ek are the regression coefficients, and Hi is the model error for the ith data-set. The model can be written in matrix notation as follows,
^y` >X@^E` ^H`
(4)
where,
^y`
^E`
predictor var iables
k ª1 x 11 x 21 . . . x k1 º «1 x x 22 . . . x k 2 »» 12 « «. . » . « » . «. . » «. . » . « » «¬1 x 1n x 2 n . . . x kn »¼
y1 ½ °y ° ° 2° °° . °° ® ¾ X °. ° °. ° ° ° °¯ y n °¿ E 0 ½ °E ° ° 1° °° . °° ^H` ® ¾ °.° °.° ° ° °¯E n °¿
½ ° ° °° ¾n ° ° ° °¿
data sets
H1 ½ °H ° ° 2° °° . °° ® ¾ °.° °.° ° ° °¯H n °¿
(5) The least squares estimator, {b}, of the regression coefficients, {E}, assuming that [X] is of full column rank,
^b` ¬b 0
b1
b2 . . . bk ¼
T
>X@ >X@ T
1
>X @T ^y`
(6) The variance and co-variance matrix of the estimated regression coefficients in vector {b} is, 1 T (7) var^b` V 2 >X@ >X@ Where, each column of [X] indicates measurement of a particular predictor variable. The multiple linear regressions can be written in alternative forms by either centering or scaling or standardizing the independent variables. Such transformation of the geometry, architecture, material properties and operating conditions predictor variables has merit in electronic packaging reliability in that it allows results from different studies to be comparable. Once the independent variables are centered and scaled, then the variable, xji, is transformed as follows,
x *ji
§ x ji x j · ¸ ¨ ¸ ¨ s j ¹ ©
(8)
Where,
¦ x n
sj
xj
2
ji
(9)
i 1
The process of centering and scaling has been used to develop an alternative formulation as follows, y i E*0 E1* x 1*i E*2 x *2i ... E*k x *ki H i (10) § x xk · § x x1 · ¸¸ H i ¸¸ ... E*k ¨¨ ki E*0 E1* ¨¨ 1i © sk ¹ © s1 ¹ The equation may be written in matrix format as follows, (11) ^y` E*0 ^1` >X* @^E* ` ^H` Where, {1} is the unit vector, of size n x 1, and E* is the vector of transformed coefficients. Centering and scaling T makes >X * @ >X * @ the k x k correlation matrix of the independent variables. T (12) C >X * @ >X * @ Where, C is the correlation matrix. Principal components regression has been used to combat multicollinearity, and
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> @> @
Where, O1, O2,…, Ok are the eigenvalues of the correlation matrix, and [V] is a k x k eigenvector matrix consisting of normalized eigenvectors associated with each eigenvalues. Since, the eigenvector are orthogonal, >V @>V @T I . The regression equation of centered and scaled variables can be written as follows, ^y` E*0 ^1` X * ^E* ` ^H` T (14) ^y` E* ^1` X * >V @>V @ ^E* ` ^H` 0
^D`k x 1 >V@T ^E* ` @>V
@T ^E* ` >V @^D`k x 1 > V ^E`k x 1
> @ > @
^y` E ^1` >Z@^D` ^H` Where, [Z] is an n x k matrix of principal components and {D} is a vector of new regression coefficients. The new model formulation is then written as follows, (15) yi E*0 D1z1 D 2 z 2 ... D k z k H i * 0
1 i n
Where, z1, z2,…,zk are the k new variables called principal components of the correlation matrix [C]. The principal components are orthogonal to each other. Each principal component is a linear combination of the transformed predictor variables. v1 j ½ °v ° ° 2j° (16) °° . °° where, z j >x 1* x *2 . . . x *k @ ® ¾ ° . ° ° . ° ° ° °¯ v kj °¿
Eigenvector associated with O j
Principal components have been eliminated by discarding the component associated with the smallest eigenvalue. The rationale is that the principal component with smallest eigenvalue is the least informative. Using this procedure, principal components are eliminated until the remaining components explain some pre-selected percentage of the total variance (for example, 85 percent or more). The second approach involved retaining principal components associated with eigenvalues greater than 1.00 based on the “KaiserGutman Rule” [25, 26]. The coefficients for the centered and scaled variables are obtained as follows,
>V @>V @T
(17)
1
>V@k x k ^D`k x 1
Where [V] is the eigenvector matrix, and {D} is a vector of new regression coefficients. Assume that r-variables have been dropped. A principal component analysis has been performed on this original predictor variable matrix X and its eigenvalues and corresponding eigen vectors have been extracted. The transformation to coefficients of the natural variables is done as follows, b*j,pc (18) b j,pc where, j 1,2,..., k sj The first four eigen vectors explained more than 85% of the original matrix and had eigen values greater than 1. A kink in the scree plot (Figure 7) supports the selection of first four eigen vectors. A transformation matrix, V has been created with first four eigen vectors.
Cumulative Percentage Contribution
achieve better predictions than ordinary least squares. The original k predictor-variables have been transformed into a new set of orthogonal or uncorrelated variables called principal components of the correlation matrix. The transformation has been used to rank the new orthogonal variables in the order of their importance. Some of the principal components have been eliminated for reduction in variance. Multiple regression analysis of response variables using ordinary least squares has been done against a reduced set of principal components. The regression coefficients of the reduced set of principal components have then been transformed into new set of coefficients that correspond to the original correlated variables. The new coefficients are called principal component estimators. Eigenvalues of the Correlation Matrix, [C], have been calculated using the following determinant equation, >C@ O>I@ >V @ (13) T >C@ O>I@ 0 or X * X * O>I@ 0
100 90 80 70 60 50 40 30 20 10 0 0
2
4
6
Principal Component Figure 7: Scree plot for selecting the number of principal components
Table 3: Multiple Linear Regression Model using Principal Components of PBGAs on Cu-Core Predictor Variables Predictor Coef SE Coef T P VIF Constant 638.45 17.94 35.59 0 Z1 -3133.4 819.8 -3.82 0 1 Z2 2033.2 517.3 3.93 0 1 Z3 1458.6 374.5 3.89 0 1 Z4 -924.7 202.1 -4.57 0 1 Z5 1352.4 178.1 7.59 0 1 Z6 -325 165.6 -1.96 0.053 1 Z7 -218.74 97.92 -2.23 0.028 1
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Table 4: Multiple Linear Regression Model using Principal Components of PBGAs on No-Core Predictor Variables Predictor Coef SE Coef T P VIF Constant 5.917 0.0538 109.93 0 Z1 0.4198 0.2214 1.9 0.06 1 Z2 1.0361 0.1954 5.3 0 1 Z3 0.2106 0.1087 1.94 0.05 1 Z4 -0.4455 0.0610 -7.3 0 1 Z5 -0.1403 0.0523 -2.68 0.01 1 Z6 -0.1093 0.0280 -3.89 0 1 A multiple linear regression analysis has been performed with characteristic life as response variable and the seven principal components as predictor variables. The regression equation of the principal components is given by Table 3 and Table 4. An Alpha matrix with coefficients of principal components is then created. The overall adequacy of the model has been tested using ANOVA table given by Table 5 and Table 6. Small P value of the ANOVA table rejects the null hypothesis proving the overall adequacy of the model. Table 5: Analysis of Variance of Multiple Linear Regression Model with Principal Components as Variables for PBGAs on Cu-Core Predictor Variables Source DF SS MS F P Regression 7 4440552 634365 18.95 0 Residual Error 96 3213140 33470 Total 103 7653691 Table 6: Analysis of Variance of Multiple Linear Regression Model with Principal Components as Variables for PBGAs on No-Core Predictor Variables Source DF SS MS F P Regression 6 18.3478 3.058 18.52 0 Residual Error 50 8.2564 0.1651 Total 56 26.6041 The coefficients of regression of the original variables are obtained by transforming the coefficients of regression of the principal components using the transformation equations. The prediction equation with original variables is given by Table 7.
Table 8: Principal Component Regression Model using Original PBGAs on No-Core Predictor Variables Coeff SE T Statistic PPredictors (bk) Coeff Value (a0, fk) Predictor Constant LnDieLengthMM LnDietoBodyRatio LnBallCount LnPkgPadDiaMM LnSdrVolCuMM LnDeltaT
Coef 5.197 -0.7830 -0.0490 0.1089 0.8148 -0.3971 -0.2739
SE Coef 0.0472 -0.413 -0.0092 0.0562 -0.1116 0.14811 0.0703
T 109.93 1.8959 5.3024 1.9381 -7.301 -2.682 -3.893
P 0 0.064 0 0.058 0 0.01 0
Individual T tests on the coefficients of regression of principal components yielded very small P values indicating the statistical significance of all the four variables. The individual T test values of principal components regression components have then been used for conducting individual T test on the coefficients of regression of original variables. The test statistic proposed by Mansfield, et. al. [27] and Gunst, et al. [28] for obtaining the significance of coefficients of regression of original variables is given by (19) b j,pc t 1 ª § l 1 2 ·º 2 «MSE u ¨ ¦ O m v jm ¸» ¹¼ ©m 1 ¬ Where, bj,pc is the coefficient of regression of the jth principal component, MSE is the mean square error of the regression model with l principal components as its predictor variables, vjm is the jth element of the eigen vector vm and m are its corresponding eigen value. M takes the values from 1 to l, where l is the number of principal components in the model. The test statistic follows a students T distribution with (n-k-1) degrees of freedom. The P values of individual T tests given by Table 7 are very small proving the statistical significance of individual regression coefficients of original predictor variables
Table 7: Principal Component Regression Model using Original PBGAs on Cu-Core Predictor Variables Predictors Coeff SE T Statistic P-Value (a0, fk) (bk) Coeff Constant 638.4481 17.9395 35.5887 0 BrdFinish 18.7443 -4.9043 -3.8219 0 DieLenMM -112.4851 -28.6213 3.9301 0 DieToBodyRatio -38.8615 -9.9785 3.8945 0 PkgPdArSqMM 322.2658 -70.4448 -4.5747 0 PkgWtG -95.4576 -12.5690 7.5946 0 SdrVol -181.7895 92.5923 -1.9633 0.052 DeltaT -152.0943 68.0877 -2.2338 0.027 Figure 8:Residual Plot of Principal Components Regression Model for Cu-Core Assemblies
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principal component has been checked using Chi-Square plots (Figure 10) and Q-Q plot (Figure 11). Straight line variation of both the plots shows cumulative multivariate normal distribution.
Figure 9:Residual Plot of Principal Components Regression Model for No-Core Assemblies
(a) (b) Figure 11: Q-Q Plot Of Principal Components Regression Model Multicollinearity of the principal component variables has been studied using Pearson’s correlation matrix and variance inflation values. Since the principal components are orthogonal to each other the multi-collinearity problem ceased to exist. Addition of fifth principal component however created serious multi-collinearity problems justifying the decision of retaining only first four variables.
FINITE ELEMENT MODELS In this section, non-linear finite element models have been developed to correlate the failure mechanics with life predictions from statistical models.
(a) (b) Figure 10: Chi Square Plot Of Principal Components Regression Model The principal components regression model has been checked for underlying model assumptions such as normality, constant variance and independence. Residual plots have been used for studying the violations of model assumptions. The residual plots studied include normal probability plot, histogram plot of residuals, plot of residuals against fitted values, plot of residual against regressor and plot of residual in time sequence. Straight line variation of normal probability plot shows cumulative normal distribution. Presence of residuals in a horizontal band shows no violation of constant variance assumption. Since each principal component is a linear combination of original predictor variables, they become multivariate variables. Multivariate normality of each
Figure 12: Isometric View of Diagonal-Symmetry PBGA532 Model Mesh. 3D Symmetry Models Solder ball elements were meshed in ANSYS with VISCO107 elements, whereas all other package materials were meshed with SOLID45 elements. Figure 12 and Figure 13 shows the 3D diagional-symmetry finite element models for the PBGA532 mounted on No-Core PCB and PBGA728 mounted on Cu-Core PCB Assemblies respectively.
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ANSYS was used to simulate the thermal cycle fatigue. The quarter symmetry model was used with mapped finite element mesh that varied from approximately 40,000 nodes and 35,000 elements to 90,000 nodes and 81,000 elements depending on the complexity of the geometry and the number of solder balls. The damage prediction is based on crack propagation relationships presented in Lall, et. al [11].
1
Y
VOLUMES MAT
Z
NUM
MAY 29 2007 09:13:18
X
Figure 13: Close-Up View of Diagonal-Symmetry PBGA728 Model Mesh Material Properties Linear, nonlinear, elastic, plastic, temperature, timedependent, and time-independent material properties have been incorporated in the finite element models. It is well known that solder is above half its melting point at room temperature, which is why time-dependent creep phenomena dominate solder joint fatigue. The combined plastic deformation and creep of solder joints is the greatest contributor to thermal solder-joint fatigue failures. The Anand-Viscoplasticity model which is a widely used to model the constitutive behavior of solder, and has been used in this study. The modeling methodology utilizes finite-element analysis to calculate the viscoplastic strain-energy density that is accumulated per cycle during thermal cycling. The strainenergy density is then utilized with crack growth data to calculate time to crack initiation and propagation-to-failure. Anand-Viscoplasticity constitutive law has been used by Darveaux [29, 30, 31] and several other researchers in the development of damage relationships. Flow and evolution equations from Anand’s model, shown in Equations 1-4, describe the strain hardening or softening of the materials [32]. Material properties used in the finite element models are displayed in Table 9 and Table 10 shows the constants used for Anand’s model for eutectic and lead-free solder. Flow Equation:
dH p dt
1 §Q· A sinh([V / s 0 m exp¨ ¸ © kT ¹
(20)
Evolution Equation:
° a B½ ° dH p ®h 0 B ¾ B °¿ dt °¯ s B 1 0* s ds 0 dt
(21)
(22)
s*
º ª dH p « § Q ·» s ^ « dt exp¨ ¸» © kT ¹» « A »¼ «¬
n
(23)
Table 9: Material Properties Used in Finite Element Models Poisson's CTE Material E (Mpa) Ratio (ppm/K) Copper Pad 129000 0.34 16.3 Solder Mask 3100 0.3 30 17890 0.39 (XY) (XZ, YZ) 12.42 (XY) BT Substrate 7846 (Z) 0.11 (XY) 57 (Z) Die Adhesive 6769 0.35 52 Silicon Die 163000 0.28 2.50 Ceramic 37000 0.22 7.40 62Sn36Pb2Ag 75842-152T 0.35 24 SAC305 51000 0.35 25 Mold 23520 0.3 15 27924-37T 0.39 (XY) (XZ, YZ) 14.5(XY) PCB 12204-16T (Z) 0.11 (XY) 67.2(Z) Table 10: Anand’s Constants for Leaded and Leadfree Solders SAC305 62Sn36Pb2Ag ANSYS Model [33] [15, 30, 31] Input Parameter 45.9 12.41 C1 So (MPa) 7460 9400 C2 Q/k (1/K) 5.87e+6 4.00E+06 C3 A (1/sec) 2.0 1.5 C4 0.0942 0.303 C5 m 9350 1.38E+03 C6 ho (MPa) 58.3 13.79 C7 s^ (MPa) 0.015 0.07 C8 n 1.5 1.3 C9 a Life Prediction Several researchers have established inelastic strain energy density (ISED to be the damage proxy for solder joint thermomechanical reliability of electronic packages. Damage relationships correlating the solder joint life with the ISED accumulated per thermal cycle by the solder joint have also been developed. These relationships can be used for the life prediction of the electronic packages with solder joint cracking as the failure mode. The ISED can be calculated from the simulation. In ANSYSTM VISCO107 element has plastic work (PLWK) as a standard output, ISED has been calculated in the present study by volume averaging PLWK over the whole solder bump volume. Some researchers volume average PLWK over the few layers in the vicinity of the interface instead of the whole solder volume. The plastic work accumulated in the solder joint over the complete thermal cycle has been found to be stabilized after the first cycle. Consequently the simulation is run for only two cycles
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and the ISED is calculated based on the plastic work accumulated during the second thermal cycle. In most of the cases the corner solder ball is the critical joint and of highest deformation (Figure 14). Hysteresis Plots for 7x7 mm PBGA of 49 I/O, SAC305 solder, no-copper core PCB, are shown in Figure 15, and Figure 16.
factors quantifying the effect of design, material, architecture, and environment parameters on thermal fatigue reliability, have been used to compute life. The predictions from the statistical model have also been compared with the experimental data. Copper Core Data Validation In this section, individual parameter variations for Cu-Core PCB assemblies have been validated. The data-sets selected for validation are part of the original experiment, but have not been used for the original model development. This exercise has been undertaken to ensure scalability of the models. A. Die to Package Ratio The reliability of a ball-grid array package generally decreases with the increase in the die-to-package ratio, for a identical package size, because of increase in distance of chip-corner from the center of the package. Thermo-mechanical failure predominantly initiate under the die-edge shadow region.
Figure 14: Plastic work plot for all solder joints for lead-free 728 BGA on Cu-Core PCB Assembly under 30 minute low and high dwell temperature condition. 1.60E-02
Plastic XZ Shear Strain
1.40E-02 1.20E-02 1.00E-02 8.00E-03
TC2 TC3 TC4
6.00E-03 4.00E-03 2.00E-03 -1.E+07
0.00E+00 -5.E+06 0.E+00 -2.00E-03
5.E+06
1.E+07
2.E+07
(a)
Plastic XZ Shear Stress (Pa)
1200
1% Failure Cycles
Figure 15: Hysteresis loops in XZ plane for various temperature conditions for PBGAs on no-copper core PCBs.
Plastic YZ Shear Strain
4.00E-03 3.50E-03 3.00E-03 2.50E-03 2.00E-03
TC2
1.50E-03
TC3
1.00E-03
1000 800 Predicted Experimental
600 400 200 0
TC4
0.42
0.46
0.52
0.74
Die to Body Ratio
5.00E-04 0.00E+00 -4.E+06 -2.E+06 0.E+00 2.E+06 4.E+06 6.E+06 8.E+06 -5.00E-04
Plastic YZ Shear Stress (Pa)
Figure 16: Hysteresis loops in YZ plane for various temperature conditions for PBGAs on no-copper core PCBs. MODEL VALIDATION The effect of the various design parameters on the thermal reliability of the package has been presented. Statistics, failure mechanics, and finite element analysis based sensitivity
(b) Figure 17: (a) Weibull Graph of Effect of Die-to-Package Ratio on Fatigue Life, -40 to 95°C. (b) Correlation of Measured and Predicted Effect of die-to-package ratio on thermal fatigue reliability of PBGA on Cu-Core PCB subjected to -55 to 125°C This effect is demonstrated in Figure 17, and captured by the model predictions. This effect has been demonstrated in both rigid-substrate and flex-substrate packages. The hybrid model
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has been used to evaluate sensitivity to die-to-body. This ratio is often referred to as the packaging ratio in literature. B. Ball Count Experimental data indicates that thermal reliability of the PBGAs increases with the increase in the ball count of the package. The experimental data used correlation includes 7 mm, 12 mm, 15 mm, and 17mm PBGA packages. The die-tobody ratio has been maintained near constant for all the packages. Figure 18 indicates a close agreement between the experimental data and the statistical model predictions. The general trend of decrease in thermal reliability with the increase in the ball count is in agreement with the failure mechanics theory. Increase in the number of solder balls for the same package size distributes the thermal deformation over a smaller solder interconnect at much finer pitch, increasing the plastic-work per unit volume in the individual ball.
The plot in Figure 19 shows correlation of 1% failure cycles for PBGA with various package pad diameters. Experimental data indicates that the increase in the package pad diameters leads to overall better thermal reliability of the package because of increase in the crack propagation path for failure. D. Surface Finish Experimental data from previous studies indicates that HASL and ENIG surface finishes have produced relatively similar reliability results. Figure 20 shows the effects of these two surface finishes coupled with T, ball count, die size, ball diameter, and ball height. Model predictions are consistent with this theory and the statistical model coincides with the experimental data.
900
1% Failure Cycles
800 700 600 500
Predicted Experiment
400 300 200 100
(a)
0
800
256
324
456
700
1% Failure Cycles
196
Ball Count
Figure 18: Effect of Ball Count on Thermal Fatigue Reliability of PBGAs subjected to -55 to 125°C C. Package Pad Diameter The solder joint package pad diameter has a pronounced effect on the thermal reliability of the BGA packages. Experimental thermal reliability data has been compared with the statistical model predictions. BGA packages evaluated include package pad diameters of 0.3 mm, and 0.35 mm.
1% Failure Cycles
1400
600 500 Predicted Experimental
400 300 200 100 0 HASL
ENIG
Board Finish
(b) Figure 20: (a) Effect of PCB pad finish on 196 I/O PBGA thermal reliability. (b) Correlation of Model Predictions of Effect of PCB pad finish on 196 I/O PBGA thermal reliability.
1200 1000 800
Predicted Experimental
600 400 200 0 0.30
0.35
Pkg Pad Diameter (mm)
Figure 19: Effect of Package Pad Diameter on Thermal Fatigue Reliability of PBGA subjected to -55 to 125°C
E. Temperature Cycle Condition Temperature cycle condition affects PBGA packages reliability immensely. The sensitivity of the package thermal reliability to the thermal cycling temperature range has been quantified using the multivariate regression analysis in the statistical model. T is the thermal cycling temperature magnitude. Figure 21 shows the predicted 1% failures and the experimental 1% failures for three temperature cycles. Three different temperature cycle conditions used for the comparison
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Characteristic Life
are: (a) -55°C to 125°C with 30 min dwells (b) -20°C to 60°C with 30 min dwells (c) -40°C to 95°C with 30 min dwells
3000 2500 2000
Experiment Predicted
1500 1000 500 0 135
180
T °C (a)
(a)
1600
1% Failure Cycles
1400 1200 1000 Predicted Experimental
800 600 400 200 0 135
180
T °C (b) Figure 21: (a) Effect of Thermal Cycling Test Condition TC1 (-40 to 95C), TC2 (-55 to 125C), TC3 (3 to 100C), TC4 (-20 to 60C) on Thermo-Mechanical Reliability (b) Correlation between Model Predictions and Experimental Data on Effect of Temperature Cycle Condition on Cu Core PCB PBGA49 packages. No Copper Core Data Validation In this section, individual parameter variations for No-Core PCB assemblies have been validated. The data-sets selected for validation are part of the original experiment, but have not been used for the original model development. This exercise has been undertaken to ensure scalability of the models. A. Temperature Cycle Condition Temperature cycle condition affects BGA packages reliability immensely. The sensitivity of the package thermal reliability to the thermal cycling temperature range has been quantified using the multivariate regression analysis in the statistical model. T is the thermal cycling temperature magnitude. The cycles for 1% failure for 7 mm, 84 I/O BGA with Die/Body ratio of 0.5814 predicted by the statistical model have been plotted (Figure 22) with the experimental data. Two different temperature cycle conditions used for the comparison are: (a) 40 C to 95 C (TC1), and (b) -55 C to 125 C (TC2). This model prediction validates the thermal sensitivity of the package seen in the experimental data.
(b) Figure 22: Effect of temperature cycling condition on 7 mm, 84 I/O BGA thermal reliability. B. Ball Diameter The solder joint ball diameter has a prominent effect on the thermal reliability of the PBGA packages. Experimental thermal reliability data has been compared with the statistical model predictions. 100 I/O, 12mm BGA packages evaluated include solder ball diameter of 0.48 mm, and 0.50 mm.
Characteristic Life
80
2500 2000 1500
Experiment Predicted
1000 500 0 0.48
0.5
Ball Diameter (mm) Figure 23: Effect of solder ball diameter on reliability of 100 I/O, 12mm BGA subjected to -40 C to 95qC thermal cycle.
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Characteristic Life Cycles
Die-to-Body Ratio
(a)
2500 2000 1500 1000 500 0 232.26
99.60
84.46
76.56
Experiment Predicted
56.25
C. Die to Package Ratio The reliability of a ball-grid array package generally decreases with the increase in the die-to-package ratio. Experimental data on reliability effect of die-to-body ratio on the solder joint thermal fatigue life of different sizes of BGA packages has been evaluated. Figure 24 shows the experimental vs. predicted correlation of the thermal fatigue reliability variance that occurs from various die to package ratios. The packages were all subjected to a thermal cycle of -55º C to 125º C for 144 I/O BGA packages. The cycles for 63.2% failure from the experimental data and the statistical model have been plotted against the die to body ratio of the various packages. The predicted follow the experimental values quite accurately and show the same trend. 1400 1200 1000 800 Experiment 600 Predicted 400 200 0 0.5000 0.7407
predicted data indicates that the increase in die size leads to decreased thermal reliability of the package, which is displayed in Figure 25. The die area has been computed from the DiagLenMM and the DietoBodyRatio parameters in the model.
Characteristic Life
Experimental data indicates that the increase in the ball diameter leads to improved thermal reliability of the package, which is shown in Figure 23. This trend is in compliance with the theory of failure mechanics as the increase in the solder ball diameter decreases solder joint compliance, resulting in more plastic work at the interfaces and reduction in thermal fatigue life.
Die Area (mm2) Figure 25: Effect of die size on thermal fatigue reliability of BGA subjected to -40 C to 95qC thermal cycle CONVERGENCE OF STATISTICS AND FAILURE MECHANICS MODELS Convergence of model predictions from statistics and failure mechanics has been investigated. Figure 26 shows the correlation between the finite element and statistical models with the experimental data for Cu-Core PBGA packages with 728 and 49 I/O counts and various thermal cycling temperature conditions. One can see that these models are highly correlated with the experimental data. The correlation is especially important, if one considers the vastly different approaches used in arriving at predicted life. The statistics models are based on considerations of multivariate regression and analysis of variance techniques and therefore account for mechanics of stress-strain only implicitly. The failure mechanics approach are based on mechanics-of-materials, damage mechanics, material constitutive behavior and do not address the statistical significance and distribution of parameters explicitly. 3500 1% Failure Cycles
3000 2500 FEM Model
2000
Experimental
1500
Statistical Model
1000 500 0 PBGA 728 (TC2)
(b)
PBGA 49 (TC1)
PBGA 49 (TC2)
PBGA 49 (TC3)
PBGA 49 (TC4)
Figure 24: Effect of die to package ratio on reliability of 144 I/O BGA subjected to -55 C to 125qC thermal cycle.
Figure 26: Life predictions for various Cu Core PBGA packages under different thermal cycling conditions.
D. Die Size The die size of a package has an important effect on the thermal reliability of the PBGA packages. Experimental thermal reliability data was juxtaposed with the statistical model predictions for package die sizes of 19.184 mm, 31.136 mm, 56.25 mm, 64 mm, and 99.6 mm. Experimental and
Figure 27 shows convergence of Statistical and FE Models for No-Core Assemblies. The effect of temperature cycle magnitude on thermal reliability of 7 mm, 84 I/O BGA packages under thermal cycling from -55 C to 125 C with 30 min ramp and 30 min dwell and 3qC to 100qC with 30 min ramp and 30 min dwell. Comparison of experimental data
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1% Failure Cycles
with that of predicted values for 1% failure cycles from both statistics and failure mechanics model shows good agreement. All the values for the package show a same trend of decreased thermal reliability with the increase in the thermal cycle magnitude.
2500
Experiment
2000 1500
FEA Model
1000 Statistical Model
500 0 97
180 T °C
Figure 27: Effect of T on thermal reliability of 7 mm PBGA package.
SUMMARY AND CONCLUSIONS A statistics-based principal components regression modeling methodology has been presented in this paper. The method provides an extremely cost effective and time effective solution for doing trade-offs and the thermo-mechanical reliability assessment of the area-array packages mounted on copper-core and no-core PCB assemblies subjected to extreme environments. The methodology as been developed by subjecting a broad array of package architectures mounted on copper-core and no-core assemblies to four thermal cycle conditions including TC1 (-40 to 95C), TC2 (-55 to 125C), TC3 (3 to 100C), TC4 (-20 to 60C). Nine Test-boards with 60-120 Packages on each board were tested in each configuration. Weibull distributions have been developed for each package. The developed methodology also allows the user to understand the relative impact of the various geometric parameters, material properties and thermal environment on the thermo-mechanical reliability of the different configurations of area array devices with leaded as well as lead-free solder joints. The model predictions from statistical models have been validated with the actual ATC test failure data, not used for the model development. The convergence between experimental results and the model predictions with higher order of accuracy than achieved by any first order closed form models has been demonstrated, which develops the confidence for the application of the models for comparing the reliability of the area array packages for various parametric variations. The current approach allows the user to analyze independent as well as coupled effects of the various parameters on the package reliability under harsh environment. It is recommended to use these models for analyzing the relative influence of the parametric variations on the thermomechanical reliability of the package instead of using them for absolute life calculations.
ACKNOWLEDGMENTS The work presented here in this paper has been supported by a research grant from the Northrop Grumman Corporation. REFERENCES 1. Lall, P., Singh, N., Suhling, J., Strickland, M., Blanche, J., Thermal Reliability Considerations for Deployment of Area Array Packages in Harsh Environments, IEEE Transactions on Components and Packaging Technologies, Volume 28, Number 3, pp. 457-466, September 2005. 2. Lall, P., Singh, N., Suhling, J., Strickland, M., Blanche, J., Thermal Reliability Considerations for Deployment of Area Array Packages in Harsh Environments, Proceedings of the ITherm 2004, 9th Intersociety Conference on Thermal and Thermo-mechanical Phenomena, Las Vegas, Nevada, pp. 259-267, June 1-4, 2004 3. Lall, P., Singh, N., Strickland, M., Blanche, J., Suhling, J., Decision-Support Models for Thermo-Mechanical Reliability of Leadfree Flip-Chip Electronics in Extreme Environments, Proceedings of the 55th IEEE Electronic Components and Technology Conference, Orlando, FL, pp. 127-136, June 1-3, 2005. 4. Lall, P., Hariharan, G., Strickland, M., Blanche, J., Suhling, J., Risk Management Models for Flip-Chip Electronics in Extreme Environment, Proceedings of the ASME International Mechanical Engineering Congress and Exposition, Chicago, Illinois, Paper IMECE2006-15443, November 5-10, 2006. 5. Lall, P., G. Hariharan, A. Shirgaokar, J. Suhling, M. Strickland, J. Blanche, Thermo-Mechanical Reliability Based Part Selection Models for Addressing Part Obsolescence in CBGA, CCGA, FLEXBGA, and FlipChip Packages, ASME InterPACK Conference, Vancouver, British Columbia, Canada, IPACK200733832, pp. 1-18, July 8-12, 2007. 6. IPC-7095 Specification, Design and assembly process implementation for BGAs, ANSI October 25, 2000. 7. IPC-SM-785 Specification, Guidelines for Accelerated Reliability Testing of Surface Mount Solder Attachments, November 1992. 8. J-STD-013 Specification, Joint Industry StandardImplementation of Ball Grid Array and other High Density Technology, July1996. 9. Adams, R. M., A. Glovatsky, T. Lindley, J. L. Evans, and A. Mawer, PBGA Reliability Study for Automotive Applications,” Proceedings of the SAE International Congress Expo, Detroit, MI, pp. 11–19, Feb. 23–26, 1998. 10. Evans, J. L., R. Newberry, L. Bosley, S. G. McNeal, A. Mawer, R. W. Johnson, and J. C. Suhling, PBGA Reliability for Under-the-Hood Automotive Applications, Proceedings of the ASME InterPACK, Kohala, HI, pp. 215–219, Jun. 15–19, 1997. 11. Lall, P., N. Islam, J. Suhling, and R. Darveaux, Model for BGA and CSP Reliability in Automotive Underhood Applications, Proceedings of the Electronic Components Technology Conference, New Orleans, LA, pp. 189–196, May 27–30, 2003.
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