Microelectronics Reliability 78 (2017) 319–330

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Microelectronics Reliability

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Assembly yield prediction of plastically encapsulated packages with alarge number of manufacturing variables by advanced approximateintegration method

Hsiu-Ping Wei a, Bongtae Han a,⁎, Byeng D. Youn b, Hyuk Shin c, Ilho Kim c, Hojeong Moon ca Mechanical Engineering Department, University of Maryland College Park, MD 20742, USAb Mechanical and Aerospace Engineering, Seoul National University, Republic of Koreac Package Development Team, Semiconductor R&D Center, Samsung Electronics, Republic of Korea

⁎ Corresponding author.E-mail address: bthan@umd.edu (B. Han).

https://doi.org/10.1016/j.microrel.2017.09.0060026-2714/© 2017 Elsevier Ltd. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 20 February 2017Received in revised form 3 August 2017Accepted 5 September 2017Available online xxxx

An advanced approximate integration scheme called eigenvector dimension reduction (EDR) method is imple-mented to predict the assembly yield of a plastically encapsulated package. A total of 12 manufacturing input var-iables are considered during the yield prediction, which is based on the JEDEC reflow flatness requirements. Themethod calculates the statisticalmoments of a system response (i.e., warpage)first throughdimensional reductionand eigenvector sampling, and a probability density function (PDF) of random responses is constructed subse-quently from the statistical moments by a probability estimation method. Only 25 modeling runs are needed toproduce an accurate PDF for 12 input variables. The results prove that the EDR provides the numerical efficiencyrequired for the tail-end probability prediction of manufacturing problemswith a large number of input variables,while maintaining high accuracy.

© 2017 Elsevier Ltd. All rights reserved.

Keywords:Approximate integration schemeDimension reductionProbability density functionAssembly yieldWarpageStacked die thin flat ball grid array

1. Introduction

Epoxy molding compound (EMC) has been used extensively as aprotection layer in various semiconductor packaging components. Themismatch of coefficient of thermal expansion (CTE) causes thewarpageof components after molding, which is one of the most critical issues toboard assembly yield. The warpage issue has become more critical asPackage-on-Package (PoP) and fan-out wafer level package (FO-WLP)are widely adopted for portable devices.

The computer-aided engineering (CAE) tools, such as the finite ele-mentmethod (FEM), have been used extensively to predict thewarpage.Typically, the CAE tools provide deterministic outputs, which establishquantitative relationships between the system response (i.e., warpage)and the input parameters such as geometries, material properties, pro-cess and/or environmental conditions, etc. The deterministic approacheshave been proven effective for comparing competitive designs. In reality,the package warpage behavior shows statistical variations (or probabili-ty distributions) due to inherent manufacturing variabilities. The proba-bilistic aspect should be incorporated in prediction if the assembly yieldis to be predicted.

The yield loss is in general a small probability event (i.e., tail-endprobability) [1–3], especially for the large production volume. In manycases, even 0.1% yield loss would cause a significant profit loss. Basedon the Six Sigma concept, the target is often to control the yield losswith-in 3 to 6 sigma, i.e., 6.67% to 3.4 ppm [4].

Fig. 1 shows a schematic illustration of the tail-end probability,wherethe statistical property of system performance (e.g., warpage) is repre-sented by a probability density function (PDF). When a component hasthe performance exceeding or falling behind a certain specification, itcannot be processed further and is regarded as a failure. The probabilityof all possible failure, i.e., yield loss, is the area under the PDF where theperformance does not satisfy the specification.

A technical approach critically required for the yield loss prediction isthe uncertainty propagation analysis, which enables the intrinsically de-terministic computational model to characterize the output distributionin the presence of input uncertainties. The most popular uncertaintypropagation methods are “random sampling method” and “responsesurface method (RSM)”. When they are applied to complexmanufactur-ing problems with a large number of input variables, however, they be-come impractical due to their own limitations.

Due to its random nature, the failure probability estimated from therandom sampling method, e.g., Monte Carlo simulation (MCS), exhibitsstatistical variations [5]. The variations can be substantial when the tail-end probability is to be predicted. In order to ensure that the tail-end

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Fig. 1. Illustration of a yield loss (tail-end probability).

320 H.-P. Wei et al. / Microelectronics Reliability 78 (2017) 319–330

probability prediction falls within the specified accuracy tolerance, anextremely large number of model computations is required. This com-putational burden makes the random sampling impractical for thecases that require complex nonlinear computationalmodels (e.g., visco-elastic analysis required for warpage prediction of plastically encapsu-lated components) [6].

The RSMhas also beenwidely used in conjunctionwith theMCS [7,8]to reduce the computational burden. The RSM relies on Design of Exper-iments (DOE) to build computationally inexpensive mathematical re-sponse surface models, which can be used for the direct MCS. Twocommonly used types of DOE are the Full Factorial Design (FFD) [9–11]and the Central Composite Design (CCD) [12–14]. Although the CCDcan reduce the sample size of the FFD substantially, both types cannotavoid the challenge known as the curse of dimensionality (i.e., the com-putational costs increase exponentially as the number of random inputvariables increases). Due to this inherent limitation, the RSM has beenapplied to the designs with only a few input variables.

Another method for the uncertainty propagation analysis is “approx-imate integration scheme”. The scheme calculates the statistical mo-ments of the output response by performing a multi-dimensionalintegration. Seo and Kwak proposed a numerical algorithm to performthe integration [15]. The algorithmalso suffered from the curse of dimen-sionality as the FFDwas used to select integration points. Rahman andXuproposed the univariate dimension-reduction (UDR) method to copewith the curse of dimensionality [16]. With the method, a multi-dimen-sional integration is transformed into a series of one-dimensional inte-grations, and thus the computational cost increases only additivelywith the increased number of input variables. This additive increasemakes the method attractive to the problems with a large number ofinput variables.

In a typical UDR implementation, however, a large number of nu-merical integration points are still required to ensure the accuracy ofeach one-dimensional integration result. For a large number of inputvariables, the method also can be computationally expensive. Youn etal. developed a method called “eigenvector dimension-reduction(EDR)” method [17] to relax the requirement of the UDR method. Inthe EDRmethod, the eigenvector sampling schemewas proposed to se-lect a few sample points along the eigenvectors of the covariancematrixof the input variables, and the stepwisemoving least square (SMLS)wasimplemented to interpolate and extrapolate the numerical integrationpoints. As a result, the accuracy of statistical moment estimation byEDR remained virtually unaffected although the number of simulationswas reduced substantially.

In this paper, the EDRmethod is implemented to predict the assem-bly yield of a plastically encapsulated package. A total of 12manufactur-ing input variables are considered during the yield prediction, which isbased on the JEDEC reflow flatness requirements. Section 2 provides abrief introduction of the EDR method. In Section 3, the details of anEDR implementation are described. The accuracy of the yield predictionis verified by the direct MCS in Section 4. Section 5 concludes the paper.

2. Eigenvector dimension reduction method

The eigenvector dimension-reduction (EDR) method estimates thecomplete probability density function (PDF) of a system response by(1) calculating the statistical moments and (2) constructing the corre-sponding PDF using the probability estimation methods.

The statistical moments are the characteristics of a distribution. The1st moment, μ, is the mean, which represents the central tendency ofthe distribution, and the 2nd moment is the standard deviation, σ,which represents the spread of the distribution. The 3rd and 4th mo-ments are skewness, β1, and kurtosis, β2, which indicate the symmetryand the peakedness of the distribution, respectively. Themth-order sta-tistical moment of a system response is defined as

E Y X1;…;XNð Þ½ �m� �

≡Zþ∞−∞

⋯Zþ∞−∞

Y X1;…;XNð Þf gm f X1 ;…;XN X1;…;XNð ÞdX1⋯dXN

ð1Þ

where E(·) is the expectation operator; Y(X1,…,XN)is the system re-sponse with N random input variables, X1 ,… ,XN (i.e., N dimensions);and fX1,… ,XN(X1,…,XN) is the joint probability density function. In thispaper, the capital letters are used to denote the input variables.

To tackle the mathematical challenge associated with the multidi-mensional integration in Eq. (1), Rahman and Xu proposed the additivedecomposition [16] to transform the multidimensional response func-tion Y(X1,…,XN) into a series of one-dimensional functions. The approx-imated system response function, then, can be expressed as [16]:

Y X1;…;XNð Þ ≈ Ya X1;…;XNð Þ ¼XNj¼1

Y μ1;…; μ j−1;X j; μ jþ1;…; μN� �

− N−1ð Þ � Y μ1;…; μNð Þ

ð2Þ

where Ya is the approximated system response function obtained by theadditive decomposition, μj is the mean value of an input variable, Xj,Y(μ1,…,μj−1,Xj,μj+1,…,μN) is the system response of the input variable,Xj, while the other input variables are kept as their respective meanvalues, and Y(μ1,…,μN) is the system response with all input variablesare fixed as their mean values.

Substituting Eq. (2) into Eq. (1) yields:

E Y X1;…;XNð Þ½ �m� �

≅E Ya X1;…;XNð Þ½ �m� �

¼ E ∑N

j¼1Y μ1;…; μ j−1;X j; μ jþ1;…; μN� �

− N−1ð Þ � Y μ1;…; μNð Þ" #m( )

ð3Þ

Using the binomial formula, the right-hand side of Eq. (3) can be re-written as [16]:

E ∑N

j¼1Y μ1;…; μ j−1;X j; μ jþ1;…; μN� �

− N−1ð Þ � Y μ1;…; μNð Þ" #m( )

¼ EXmi¼0

m!i! m−ið Þ! ∑

N

j¼1Y μ1;…; μ j−1;X j; μ jþ1;…; μN� �" #i

− N−1ð Þ � Y μ1;…; μNð Þ½ �m−i8<:

9=;

¼Xmi¼0

m!i! m−ið Þ! E ∑

N

j¼1Y μ1;…; μ j−1;X j; μ jþ1;…; μN� �" #i8<

:9=; − N−1ð Þ � Y μ1;…; μNð Þ½ �m−i

ð4Þ

Fig. 2. Schematic illustration of numerical integration.

Fig. 3. Schematic illustration of eigenvectors of two random variables.

321H.-P. Wei et al. / Microelectronics Reliability 78 (2017) 319–330

The recursive formula is further employed to simplify the expecta-tion operation in Eq. (4), which yields [16]:

Xmi¼0

m!i! m−ið Þ! E ∑

N

j¼1Y μ1;…; μ j−1;X j; μ jþ1;…; μN� �" #i8<

:9=; − N−1ð Þ � Y μ1;…; μNð Þ½ �m−i

¼Xmi¼0

m!i! m−ið Þ! S

iN − N−1ð ÞY μ1;…; μNð Þ½ �m−i

ð5Þ

where

Si1 ¼Zþ∞−∞

Y X1; μ2;…; μNð Þ� �i f X1 ;…;XN X1jX2 ¼ μ2;…;XN ¼ μNð ÞdX1

Sij ¼Xik¼0

i!k! i−kð Þ! S

kj−1

Zþ∞−∞

Y μ1;…; μ j−1;X j; μ jþ1…; μN� �n oi−k

� f X1 ;…;XN X jjX1 ¼ μ1;…;X j−1 ¼ μ j−1;X jþ1 ¼ μ jþ1;…;XN ¼ μN� �

dX j

SiN ¼Xik¼0

i!k! i−kð Þ! S

kN−1

Zþ∞−∞

Y μ1;…;XNð Þ� �i−k f X1 ;…;XN XN jX1 ¼ μ1;…;XN−1 ¼ μN−1ð ÞdXN

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

ð6Þ

From Eqs. (1) to (6), a total of m × N one-dimensional integrationsare needed to obtain the mth-order statistical moment. These integra-tions in general can be done by numerical integration algorithms,which perform the calculations at the integration points, xj , i, withweights, wj ,i, as [17]:

Zþ∞−∞

Y μ1;…; μ j−1;X j; μ jþ1…; μN� �n or

� f X1 ;…;XN X jjX1 ¼ μ1;…;X j−1 ¼ μ j−1;X jþ1 ¼ μ jþ1;…;XN ¼ μN� �

dX j≅Xni¼1

wj;i Y μ1;…; μ j−1; xj;i; μ jþ1…; μN� �n or

ð7Þ

where xj , i is the ith integration point of an input variable, Xj,Y(μ1,…,μj−1,xj , i,μj+1…,μN) is the system response at xj , i, while theother input variables are kept as their respective mean values, and wj ,iis the corresponding weight which approximates the area under thePDF of Xj from xj ,i−(xj ,i−xj ,i−1)/2 to xj ,i+(xj ,i+1−xj ,i)/2, respectively.

Fig. 2 illustrates the numerical integration with 5 integration points.The PDF of Xj and the system response along Xj are shown as the blueand red dashed curves, respectively. The products of these two curvesare integrated along the variable, Xj. More specifically, the products ofthe responses and their correspondingweights (the bars) at the integra-tion points (the black cross) are added to complete the numericalintegration.

High accuracy of the 1-D numerical integration in Eq. (7) can beachieved by selecting numerous integration points, which can be com-putationally challenging for the applications with a large number ofinput variables, especially those that require nonlinear modeling. Twomajor improvementsweremade in the EDRmethod to reduce the num-ber of simulations while maintaining the accuracy.

First, the eigenvector sampling scheme was proposed to handlethe statistical correlation and variation of the input variables. Bysolving the eigenvalue problem of the covariance matrix, the eigen-values and the corresponding eigenvectors are obtained. The ei-genvector associated with the largest eigenvalue is the directionof the largest variation, wherein the square root of the eigenvalueis the standard deviation along this direction. The eigenvector as-sociated with the second largest eigenvalue is the orthogonal di-rection with the next highest variation. The sample points areselected along the eigenvectors, and the simulations are conductedonly at the sample points. Eigenvectors of two random variablesare illustrated in Fig. 3, where the 1st and 2nd eigenvector direc-tions are shown with a joint PDF.

The three sample-point scheme is typically used in practice. The loca-tions of the three sample points aremean and themean±3 standardde-viations, which are expressed as [17]:

0V ¼ μ1;…; μN½ �1Vi ¼ μ1;…; μ i−3

ffiffiffiffiffiλi

p;…; μN

h i2Vi ¼ μ1;…; μ i þ 3

ffiffiffiffiffiλi

p;…; μN

h i ð8Þ

where μi andλi are themean and eigenvalue along the ith eigenvector. Thelocations of sample points in Eq. (8) were suggested based on a paramet-ric study reported in Ref. [17]. Fig. 4(a) illustrates the three sample-pointscheme of an input variable Xj following the normal distribution. It can beobserved that the PDF values outside of the range of mean ± 3 standarddeviations are very small (i.e., 0.27% of all the possible values of Xj),which suppresses their contribution during the numerical integration.

Fig. 4. Schematic illustration of locations of sample points: (a) three and (b) five sample-point schemes.

Fig. 5. TFBGA package: (a) cross-sectional view and (b) bottom view showing themeasuring zone.

322 H.-P. Wei et al. / Microelectronics Reliability 78 (2017) 319–330

The above condition is no longer valid if the system response is highlynonlinear within the range of mean± 3 standard deviations. More sam-ple points are needed to capture the nonlinear response. Fig. 4(b) illus-trates the five sample-point scheme. The additional sample points canbe expressed as [17]:

3Vi ¼ μ1;…; μ i−1:5ffiffiffiffiffiλi

p;…; μN

h i4Vi ¼ μ1;…; μ i þ 1:5

ffiffiffiffiffiλi

p;…; μN

h i ð9Þ

The nonlinear behavior can be captured accurately using the two ad-ditional points. By excluding the repeated runs of themean value, a totalof (2N+1) or (4N+1) runs are required for the three and five sample-point schemes, respectively.

Once the corresponding system responses are obtained at thesample points, the moving least square (MLS) or stepwise moving

least square (SMLS) method [17] is utilized to interpolate and ex-trapolate the responses at the integration points. Using the approx-

imated system responses, Ŷ, at the integration points, the numericalintegration of the one-dimensional integrations finally takes thefollowing form [17]:

Zþ∞−∞

Y μ1;…; μ j−1;X j; μ jþ1…; μN� �n or

� f X1 ;…;XN X jjX1 ¼ μ1;…;X j−1 ¼ μ j−1;X jþ1 ¼ μ jþ1;…;XN ¼ μN� �

dX j≅Xni¼1

wj;i Ŷ μ1;…; μ j−1; xj;i; μ jþ1…; μN� �n or

ð10Þ

3. Assembly yield loss prediction of thin flat ball grid array

The assembly yield of a plastically encapsulated package is deter-mined. A viscoelastic analysis to predict the warpage is described afterdefining thewarpage. The uncertainty propagation analysis and PDF es-timation are followed.

3.1. Package description

A stacked die thin flat ball grid array (TFBGA) package is often used asthe top package of a Package-on-Package (PoP). Fig. 5(a) shows the sche-matic of a typical TFBGA package. The encapsulation of the TFBGA pack-age is done by the transfer molding process. For successful PoP stackingwith the high assembly yield, the package warpage at the solder reflowtemperature must be controlled [18–20].

Typically, the TFBGA packages are produced by memory manufac-turers and shipped to the outsourced semiconductor assembly andtest services (OSAT) companies for the PoP assembly. Therefore, theTFBGA packages are required to meet the warpage specification (e.g.,±0.1 mm for the package body size of 15 mm × 15 mm and the ballpitch of 0.5 mm [21]) before shipment. The packages with warpage ex-ceeding the specification cannot be processed further, which becomes a

Fig. 7. Viscoelastic FE model of TFBGA package: (a) boundary conditions, (b) die stackconfiguration, and (c) enlarged cross-sectional view.

Fig. 6. Package warpage definition and sign convention: (a) convex and (b) concave.

323H.-P. Wei et al. / Microelectronics Reliability 78 (2017) 319–330

yield loss. It is suggested by JEDEC and JEITA [22] that the packagewarp-age at solder reflow temperature be measured over the area where sol-der joints are located (will be referred to as “measuring zone”). Fig. 5(b)shows the measuring zone of the TFBGA package used in this paper.

Fig. 6 shows the definition of package warpage and the sign conven-tion. Fig. 6(a) shows a convex package (corners down during assembly– a positivewarpage),while Fig. 6(b) shows a concave package (cornersup during assembly – a negative warpage). The red dashed line shownin Fig. 6 indicates the reference plane; the coefficients of the equationof the reference plane are calculated by the least square method withthe out-of-plane deformation across the x-y spatial dimensions of thespecimen in the measuring zone. The distance between the highestpoint in the measuring zone and the reference plane is denoted as A,whereas the distance between the lowest point in the measuring zoneand the reference plane is denoted as B. The magnitude of the packagewarpage is defined as the sum of A and B.

3.2. Numerical analysis: warpage prediction

A quarter symmetry was used to build a finite element model withthe element type SOLID185 in the commercial FEA package (ANSYS®),which supports the viscoelastic and elastic material properties. It takesabout 1.5 h for a single model run using an advanced workstation. Fig.7 shows the details of the model. The boundary condition and the diestack configuration are shown in (a) and (b), respectively. The enlargedview of the cross-section in (c) shows the details of the chip and the dieattach film (DAF) configuration.

The material properties and the nominal dimensions used in themodel are summarized in Table 1 and Table 2, respectively. The temper-ature dependent Young's modulus of DAF measured bythermomechanical analysis (TMA) is shown in Fig. 8. The EMC wasmodeled as a linear viscoelastic material. The master curves used in themodel are shown in Fig. 9 [23]. The Williams-Landel-Ferry (WLF) func-tion was used to fit the shift factors at different temperatures, which

can be expressed as:

log aTð Þ ¼C1 T−Tref

� �C2 þ T−Tref

� � ð11Þwhere aT is the shift factor, Tref is the reference temperature (115 °C), andC1, C2 are the material constants. The values of C1 and C2 were −20.16and−111.38, respectively.

The EMCmolding process is done at 175 °C, which can be assumed asthe stress free temperature. The package is then subjected to the solderreflow process during the assembly. Fig. 10 shows the complete thermalexcursion of the package used for warpage prediction. The conventionallead-free solder reflow profile is considered [24], where the peak tem-perature is 260 °C.

The deformed configuration of the TFBGA package with the nominaldesign at the peak reflow temperature is shown in Fig. 11(a). The lightpink area indicates the measuring zone and the white circles representthe solder ball locations. Fig. 11(b) shows the reference plane deter-mined based on the z-direction displacements of the nodes in the mea-suring zone. The package warpage was calculated based on the

Table 1Material properties of TFBGA.

Material Young's modulus(GPa)

Poisson's ratio CTE (ppm/°C) Tg (°C)

α1 (bTg) α2 (NTg)

Silicon die 130 0.23 2.8 –DAF Temp. dependent 0.3 65.3 162.9 138Substrate 46.794 0.3 16.2 (in-plane)

61.5 (out-of-plane)–

EMC Viscoelastic 0.21 9.12 35.13 137.5

Table 2Dimensions of TFBGA.

Length × width × thickness

1st Die (mm) 13 × 11 × 0.5751st DAF (mm) 13 × 11 × 0.0252nd Die (mm) 11 × 9 × 0.5752nd DAF (mm) 11 × 9 × 0.025Substrate (mm) 15 × 15 × 0.13EMC (mm) 15 × 15 × 0.59

Fig. 9.Master curves of bulk modulus and shear modulus of EMC [23].

324 H.-P. Wei et al. / Microelectronics Reliability 78 (2017) 319–330

definition described in Section 3.1, i.e., the sum of (1) the distance of thehighest point in the measuring zone to the reference plane and (2) thedistance of the lowest point in the measuring zone to the referenceplane. The package warpage of the nominal design was 40.16 μm.

3.3. Uncertainty propagation analysis by EDR

3.3.1. Input random variablesIt is well known that the manufacturing variables tend to follow a

normal distribution according to the central limit theorem [25]. The 12random input variables with the means and standard deviations usedin the study are listed in Table 3. The material properties (X9 to X12)were measured, and the dimensions of X1 to X8 were obtained fromthe manufacturing specifications [26,27].

Among the 12 input variables, two pairs of properties have the statis-tical correlation: (1) the EMC thickness (tEMC), X3, and the PCB thickness(tPCB), X4, and (2) the EMC CTE below and above Tg, X9 and X10. To definethe joint PDF of the correlated and normally distributed input variables,an additional parameter called the correlation coefficient is required.The joint PDF with these 5 parameters (i.e., the means and standard de-viations of two variables, and the correlation coefficient) is called the bi-variate normal distribution.

The package thickness, tPKG, is equal to tEMC plus tPCB, i.e.,tPKG= tEMC+ tPCB. The statistical distributions of tPKG, tEMC, and

Fig. 8. Temperature dependent Young's modulus of DAF.

tPCB should also satisfy this relationship. Based on the manufactur-ing specification [26], the package thickness, tPKG, is given as 0.72 ±0.08 mm. The package thickness is expected to follow a normal dis-tribution, and it can be expressed with the mean and standard de-viations of μPKGt = 0.72 mm and σPKGt = 0.027 mm. It is to be notedthat σPKGt is set to be one third of the tolerance, which makes99.73% of tPKG lie within the tolerance.

The correlation coefficient of tEMC and tPCB was determined using thedistribution of the package thickness. A sweeping analysis was conduct-ed over the theoretical range of correlation coefficient, [−1, 1] with astep size of 0.05, which produced a total of 41 correlation coefficients.For each correlation coefficient, ρt , i (i = 1 to 41), random sampling ofthe bivariate normal distribution of tEMC and tPCB was conducted to gen-erate 100,000 pairs of tEMC and tPCB data, which subsequently produced100,000 tPKG data. The tPKG datawas then fitted into a normal distributionto calculate the mean, μρt,i, and standard deviation, σρt,i. The least squareerror was calculated to represent the degree of μρt ,iand σρt ,i deviatedfrom μPKGt and σPKGt. The results are shown in Fig. 12. The correlation co-efficient of X3 and X4 was determined as −0.35.

Fig. 10. Completed thermal excursion.

Fig. 11. (a) Deformed configuration of TFBGA package at 260 °C (20× magnification) and(b) package warpage calculated from on reference plane.

325H.-P. Wei et al. / Microelectronics Reliability 78 (2017) 319–330

Unlike the above case, the EMC CTEs above and below Tg always in-crease or decrease in the same direction since any pair of these two var-iables are measured from the same sample. Accordingly, it is reasonableto assume that the EMC CTEs above and below Tg have perfect positivecorrelation (i.e., the correlation coefficient of “unity”).

Table 3Input variables.

Variables Physical meaning Mean

X1 PKG length (mm) 15X2 PKG width (mm) 15X3 EMC thickness (mm) 0.59X4 PCB thickness (mm) 0.13X5 1st Chip thickness (mm) 0.0575X6 2nd Chip thickness (mm) 0.0575X7 1st DAF thickness (mm) 0.025X8 2nd DAF thickness (mm) 0.025X9 EMC CTE above Tg (ppm/°C) 35.13X10 EMC CTE below Tg (ppm/°C) 9.12X11 PCB CTE (ppm/°C) 16.2X12 PCB modulus (MPa) 46,794

3.3.2. Eigenvector sampling and sample pointsThe covariance of input variables Xi and Xj, which quantify the linear

dependency between these two variables, is defined as:

Cov Xi; X j� � ¼ Σij ¼ E Xi−μ ið Þ X j−μ j� �h i ð12Þ

where E(·) is the expectation operator; μi and μj are the mean values ofthe input variable Xi and Xj, respectively.

After calculating the covariance between each pair of theNnumber ofinput variables, the covariance matrix can be obtained as [17]:

Σ ¼

σ21 Σ12 Σ13 ⋯ Σ1NΣ21 σ22 Σ23 ⋯ Σ2NΣ31 Σ32 σ23 ⋯ Σ3N⋮ ⋮ ⋮ ⋱ ⋮

ΣN1 ΣN2 ΣN3 ⋯ σ2N

266664

377775 ¼

0:0332 0 0 0 0 00 0:0332 0 0 0 00 0 :0292 −0:0001015 0 00 0 −0:0001015 0:012 0 00 0 0 0 0:0012 00 0 0 0 0 0:0012

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

26666666666666666664

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

0:003752 0 0 0 0 00 0:003752 0 0 0 00 0 4:242 4:664 0 00 0 4:664 1:12 0 00 0 0 0 0:812 00 0 0 0 0 1592

37777777777777777775

ð13Þ

whereΣii=σi2 is the variance of the input variableXi andΣij=Σji. By solv-ing the eigenvalue problem of the covariance matrix (i.e., ΣXE=λXE), theeigenvaluesλ and the corresponding eigenvectorsXE are obtained. The ei-genvalues, λ, and the corresponding eigenvectors, XE, of the covariance

Std. dev. Distribution Correlation coefficient

0.033 Normal –0.033 Normal0.029 Bivariate normal −0.350.010.001 Normal –0.001 Normal –0.00375 Normal –0.00375 Normal –4.24 Bivariate normal 11.10.81 Normal 0159 Normal 0

Fig. 13.Marginal joint PDF of package length and width (uncorrelated case) and locationsof sampling points.

326 H.-P. Wei et al. / Microelectronics Reliability 78 (2017) 319–330

matrix of Eq. (13) are:

λ ¼

0:0332

0:0332

0:0008550:0000860:0012

0:0012

0:003752

0:003752

4:382

00:812

1592

26666666666666666664

37777777777777777775

;

XE ¼1 0 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 0 00 0 0:9911 0:1333 0 0 0 0 0 0 0 00 0 −0:1333 0:9911 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 1 0 0 0 00 0 0 0 0 0 0 0 0:9680 ‐0:2511 0 00 0 0 0 0 0 0 0 0:2511 0:9680 0 00 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 1

26666666666666666664

37777777777777777775

ð14Þ

The multi-dimensional integration of the joint PDF with 12 dimen-sions (i.e., 12 input variables) can be decomposed into 12 one dimen-sional integration along the 12 eigenvectors directions shown in Eq.(14). The marginal joint PDF of two input variables is used to illustratethe locations of the sampling points since it is difficult to show graphical-ly the joint PDFwithmore than 3 input variables. Themarginal joint PDFof Xi and Xj provides the probability of each (xi, xj) pair, which is calculat-ed by integrating the joint probability distribution of the 12 input vari-ables over the 10 input variables other than Xi and Xj. For example, themarginal joint PDF of X3 and X4 can be expressed as:

f X3X4 X3;X4ð Þ ¼Z∞−∞

⋯Z∞−∞

f X1 ;…;X12 X1; …; X12ð ÞdX1dX2dX5⋯dX12 ð15Þ

Fig. 13 shows the marginal joint PDF of an uncorrelated case, wherethe package length (X1) and package width (X2) are the uncorrelatedinput variables. After the additive decomposition is completed, this bi-variate marginal joint PDF is further transformed into two one-

Fig. 12.MCS sweeping analysis results for correlation coefficient of EMC thickness and PCBthickness.

dimensional marginal joint PDFs along the eigenvector directions(green and black lines). It is to be noted that for this uncorrelatedcase, the eigenvector directions are just the directions of the input var-iables. Fig. 13 also shows the sample points of 2N+1 sampling scheme(Eq. (8)) as well as the additional sample points required for 4N + 1sampling scheme (Eq. (9)).

Fig. 14 shows themarginal joint PDFs of the two correlated cases. Themarginal joint PDF of EMC thickness (X3) and PCB thickness (X4) isshown in (a), and the marginal joint PDF of EMC CTE above and belowTg (X9 and X10) in (b). Due to the statistical correlation, the eigenvectordirections are different from the original variable directions. It shouldbenoted that, in Fig. 14(b), the variation along the second eigenvector di-rection was zero due to the perfect correlation between X9 and X10.

The warpage at the each sample point was calculated by the sameprocedure described in Section 3.2. The results are summarized inTable 4. Based on the 2N+1 sampling scheme, a total of 25 simulationswere conducted for the 12 input variables.

3.3.3. Statistical momentsAs described in Section 2, the statistical moments can be obtained by

calculating the multiple one-dimensional integrations in Eq. (6). In thisstudy, each one-dimensional integration was calculated by the numeri-cal integration algorithm called the moment-based quadrature rule[16]. It was demonstrated that it can calculate the one-dimensional inte-gration for an arbitrary distribution of the input variable Xjwith good ac-curacy and efficiency, compared with other conventional integrationmethods such as Gauss-Legendre and Gauss-Hermite quadratures [16].

Fig. 15 shows the weights and the approximated package warpagesof the integration points for two representative variables: (a) X11 (CTEof PCB) and (b) X3 (EMC thickness). The red dots represent the packagewarpages at the sample points. A total of 21 integration points were sug-gested in Ref. [17] for several examples with nonlinear system response.The 21 integration pointswere also used in this study and expected to besufficient since the system response of this study is less nonlinear. As de-scribed in Section 2, the weight of each integration point is representedby the area of the corresponding bar. For example, the central bar inFig. 15(a) approximates the area under the PDF of X11 from x11;11−x11;11−x11;10

2 to x11;11 þx11;12−x11;11

2 .In Fig. 15(a), the package warpages at the three sample points (μ11

andμ11 � 3ffiffiffiffiffiffiffiλ11

p) linearly decrease along the direction of X11. Therefore,

the package warpages of the integration points (black cross) can be ac-curately interpolated and extrapolated byMLS,whichwas confirmed by

Fig. 14.Marginal joint PDF and locations of sampling points for correlated input variables:(a) EMC and PCB thickness and (b) EMC CTE below Tg and above Tg.

Table 4Package warpage simulation results at sample points.

Variable Package warpage (μm)

−3ffiffiffiffiffiλi

p−1:5

ffiffiffiffiffiλi

pμi þ1:5

ffiffiffiffiffiλi

pþ3

ffiffiffiffiffiλi

pX1 40.00 40.08 40.16 40.25 40.33X2 40.06 40.11 40.16 40.22 40.27X3 52.94 47.27 40.16 31.16 19.07X4 27.87 33.67 40.16 47.44 55.62X5 41.24 40.70 40.16 39.63 39.11X6 40.83 40.49 40.16 39.83 39.51X7 44.31 42.31 40.16 37.94 35.69X8 41.49 40.88 40.16 39.39 38.57X9/X10 −12.78 14.40 40.16 65.26 90.13X11 55.99 48.11 40.16 32.28 24.33X12 40.47 40.31 40.16 40.01 39.86

Fig. 15. Integration points and weights for 1-D numerical integration: (a) X11 and (b) X3.

327H.-P. Wei et al. / Microelectronics Reliability 78 (2017) 319–330

four additional simulations conducted at μ11 � 1:5ffiffiffiffiffiffiffiλ11

pand μ11 � 6ffiffiffiffiffiffiffi

λ11p

. Most of the input variables in this study show linear responsecurves similar to X11.

The most nonlinear response curve is observed with EMC thickness,

X3 (Fig. 15(b)). Due to the fact that the warpage within μ3 � 3ffiffiffiffiffiffiλ3

pis

Table 5First four statistical moments.

Mean Std. dev. Skewness Kurtosis

EDR sampling scheme 2N + 1 39.68 19.48 −0.0488 3.00534N + 1 39.76 19.25 −0.0478 3.0332

Table 6Coefficients of Pearson system.

c0 c1 c2

EDR sampling scheme 2N + 1 378.96 −0.4745 0.00034N + 1 365.03 −0.4515 0.0048

Table 7Predicted statistical moments and yield loss by MCS and EDR.

Mean Std. dev. Skewness Kurtosis Yield loss (ppm)

MCSa 54.8721 7.9191 −0.0006 2.9995 762EDR 2N + 1 54.8720 7.9202 0.0000 3.0000 754EDR 4N + 1 54.8720 7.9201 0.0000 3.0000 758

a Average value of 30 repetitions with sample size of 1000,000.

328 H.-P. Wei et al. / Microelectronics Reliability 78 (2017) 319–330

intermediately nonlinear, two additional simulations conducted at μ3 �1:5

ffiffiffiffiffiffiλ3

pindicate that the interpolation was accurate, whereas the ex-

trapolation results deviated from the simulations at μ3 � 6ffiffiffiffiffiffiλ3

p. Howev-

er, as expected, it is clearly shown in Fig. 15 that the contribution ofweights are negligible for Xbμ i−3

ffiffiffiffiffiλi

pand Xbμ i þ 3

ffiffiffiffiffiλi

p, and thus,

even the extrapolation by MLS contains error, the effect on the integra-tion results is minimal.

Once all the one-dimensional integrations in Eq. (10) are completed,they are combined to calculate the statistical moments. The first fourstatistical moments are listed in Table 5.

Fig. 16. PDFs of and assembly yield loss predicted by two schemes of EDR: (a) entire PDFand (b) enlarged view of the tail-end region.

3.3.4. PDF estimationAfter obtaining the statistical moments, the probability estimation

method such as themethod ofmoments (MOM) and the Pearson systemcan be used to construct the PDF of random response, which is the finaloutcome of the EDR method for an uncertainty propagation analysis.

In this study, the PDF of package warpage was constructed using thePearson system [28]. The Pearson system is a family of continuous prob-ability distributions, which offers flexibility in constructing the PDFbased on the first four statistical moments (mean, standard deviation,skewness and kurtosis). The Pearson distributions of the system re-sponse, Y, are defined by the following differential equation [28]:

1p Yð Þ

dp Yð ÞdY

¼ − aþ Yc0 þ c1Y þ c2Y2

ð16Þ

where a, c0, c1 and c2 are four parameters to describe the PDF. Based onthe theoretical derivation, the four parameters can be determined bythe first four moments, which can be expressed as [28]:

c0 ¼ 4β2−3β21� �

10β2−12β21−18

� �−1σ2

a ¼ c1 ¼ β1 β2 þ 3ð Þ 10β2−12β21−18� �−1

σ

c2 ¼ 2β2−3β21−6� �

10β2−12β21−18

� �−1

8>>>><>>>>:

ð17Þ

As denoted in Section 2, σ is the standard deviation, β1 is the skew-ness, and β2 is the kurtosis. It is worth noting that the 1st moment (i.e.,mean, μ) is not shown in Eq. (17) since the Pearson system first con-structs the PDF about the mean of zero and then shifts the PDF to thetrue mean.

The coefficients determined from the statistical moments obtained inSection 3.3.3 are summarized in Table 6. Fig. 16(a) depicts the predictedPDFs. Fig. 16(b) shows the enlarged view of the tail-end region. It is clear

Fig. 17. Yield loss distribution predicted by MCS with the sample size of 1000,000.

Table 8Effect of sample size on MCS predictions.

Samples size = 10,000 Samples size = 100,000 Samples size = 1000,000

Theoretical probability MCS repetitions Theoretical probability MCS repetitions Theoretical probability MCS repetitions

±1% error bounds (762 ± 7.62 ppm) 2.3% 0% (0/30) 7.3% 3.3% (1/30) 22.3% 26.7% (8/30)±10% error bounds (762 ± 76.2 ppm) 21.7% 30% (9/30) 61.6% 70% (21/30) 99.4% 100% (30/30)

329H.-P. Wei et al. / Microelectronics Reliability 78 (2017) 319–330

that the results of 2N + 1 and 4N + 1 sampling schemes are virtuallyidentical, which is attributed to the fact that most of the response curvesare linear. By applying the specification of JEDEC [21], i.e., ±0.1 mm forthe TFBGA package in this study, the prediction yield losses of the twoschemes are 765 ppm and 751 ppm, respectively.

4. Validity of the proposed approach

It has been shown that the yield loss (i.e., tail-end probability) of apackage with 12 input variables can be predicted accurately by as fewas 25 simulations. The direct MCS is used to evaluate the accuracy ofthe EDR results quantitatively.

The direct application of theMCS to the current problemwas imprac-tical due to excessive computational time. Instead, an empirical modelobtained from the coplanarity data at room temperature was used forthe verification. The empirical coplanarity model produced by usingthe stepwise regression with 43,358 coplanarity data points is given as:

YCopl ¼ 2:6366X1 þ 0:4564X2−74:4052X3 þ 36:0588X4−0:06189 X5 þ X6ð Þþ 0:0274 X7 þ X8ð Þ þ 1:7329X9 þ 1:73875X10 þ 0:6273X11−0:00116X12−0:49186 X5 þ X6ð Þ= 0:4746X1X2X3−0:036 1:913 X5 þ X6ð Þ þ 0:8 X7 þ X8ð Þð Þð Þð Þþ 2:0361 X5 þ X6ð Þ=X1X2X4ð Þ þ 14:662

ð18Þ

where YCopl is the coplanarity and X1 to X12 are input variables listed inTable 3. The two pairs of input variables – (1) the EMC thickness, X3,and the PCB thickness, X4, and (2) the EMC CTE below and above Tg, X9and X10 – are still considered to be statistically correlated with thesame correlation coefficients as−0.35 and 1, respectively.

As mentioned earlier, the statistical moments and the yield lossestimated from the MCS exhibit statistical variations. It is well-known that the true values can be obtained by employing the unbi-ased estimators in multiple repetitions. According to the centrallimit theorem, 30 repetitions will produce a good approximation ofthe true value [29,30]. More details about the unbiased estimatorsof the 1st to 4th statistical moments and the yield loss can be foundin Ref. [31,6], respectively.

For the yield loss prediction, the coplanarity criterion was set as 80μm according to the room temperature coplanarity criterion suggestedby JEDEC [32,33]. It is to be noted that this is different from the packagewarpage specification at reflow temperature discussed in Section 3.

In this study, theMCSwith the sample size of 1,000,000was conduct-ed for 30 repetitions to estimate the true statistical moments and thetrue yield loss. The comparison between the MCS and EDR is summa-rized in Table 7. As expected, the results from 2N+ 1 and 4N+ 1 sam-pling schemes are nearly identical. Differences of statistical momentsbetween the MCS result and the EDR results are very small (only fourthdecimal place). The effect of these differences on the PDF constructionare minimal, which produces only 8 ppm difference in yield lossprediction.

A quantitative comparison of the yield loss was made using the re-sults of the MCS sample size of 1,000,000. The yield loss prediction byMCS is expressed as [6]:

1−p̂ð Þ ¼ 1− kNMCS

ð19Þ

where NMCS is the sample size used in MCS; and k is the number of pre-dicted coplanarity less than or equal to the coplanarity criterion.When k

and NMCS are sufficiently large, ð1−p̂Þcan be approximated by the nor-mal distribution with the mean of (1−p) and the standard deviationof

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipð1−pÞ=NMCS

p[6], where p is the true yield loss.

Fig. 17 shows the yield loss distribution of NMCS = 1000,000. Theyield loss predicted by EDR 2N+1 scheme is 754 ppm,which falls with-in the ±1% tolerance of the true yield loss (762 ± 7.62 ppm). The prob-ability that theMCS predictions satisfy the±1% tolerance is theoretically22.3% (the light green area in Fig. 17). This theoretical probability is con-firmed by the single MCS repetition results, which are also shown inFig. 17; only 7 out of 30 repetitions (23.3% probability) fall within the±1% error bound (±7.62 ppm). In otherwords, for any yield loss predict-ed byMCSwith a sample size of 1,000,000 has only 22.3% probability thatthe error will be smaller than or equal to than the yield loss predicted byEDR. When the tolerance is relaxed to ±10% error of the true yield loss,the MCS has 99.4% probability to fulfill the tolerance. It is also confirmedby that all the 30 repetitions fall within the ±10% error bounds (Fig. 17).

The yield loss by the MCS is affected obviously by the sample size.Table 8 summarizes the results obtained from different sample sizes.When the sample size is reduced to 10,000, the probability that theMCS prediction is to be comparable to the EDR is only 2.3%. Even the tol-erance is relaxed to ±10%, the sample size of 10,000 has only 21.7%probability.

Considering the fact that it takes approximately 1.5 h to run the vis-coelastic model used in this case study using an advanced workstationwith 24 cores, the MCS with 10,000 model runs would take 2 months,which is impractical for most of the semiconductor packaging applica-tions. Conversely, the EDRprovides yield loss predictionwith uncertain-ty less than 1% with only 25 model runs and results in 37.5 hcomputational time. The comparison result clearly confirms the effec-tiveness of EDR in the tail-end probability prediction.

In this study, the accuracy of the proposed approach was confirmedfor symmetric input distributions. It was confirmed in Ref. [17] that theEDR method is effective for both symmetric and asymmetric distribu-tions. It is expected that the proposed approachwill alsowork effectivelywith any asymmetric input distributions as long as the input variableshave linear-dependency.

5. Conclusion

The eigenvector dimension reduction (EDR) method was imple-mented to predict the assembly yield of a stacked die thin flat ball gridarray (TFBGA) package. A total of 12 manufacturing input variableswere considered during the yield prediction, among which two pairs ofproperties had the statistical correlation. The method calculated the sta-tistical moments of warpage distribution first through dimension reduc-tion and eigenvector sampling. The probability density function (PDF) ofthe warpage was constructed from the statistical moments by the Pear-son system. The assembly yieldwas predicted from the PDF based on theJEDEC reflow flatness requirements.

In this case study, only 25 modeling runs were needed to predict theassembly yield with uncertainty less than 1% despite the fact that theprediction dealt with a tail-end probability (less than 1000 ppm) with12 input variables. The number of input variables was much largerthan that has been conceived as the practical limit of the uncertaintypropagation analysis. More applications of the EDRmethod are expectedto provide solutions to tail-end probability problems which have notbeen feasible due to a computational burden.

330 H.-P. Wei et al. / Microelectronics Reliability 78 (2017) 319–330

Acknowledgment

Thisworkwas supported by Samsung Electronics throughUMD: KFS4326670. Their support is greatly appreciated and graciouslyacknowledged.

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Assembly yield prediction of plastically encapsulated packages with a large number of manufacturing variables by advanced a...1. Introduction2. Eigenvector dimension reduction method3. Assembly yield loss prediction of thin flat ball grid array3.1. Package description3.2. Numerical analysis: warpage prediction3.3. Uncertainty propagation analysis by EDR3.3.1. Input random variables3.3.2. Eigenvector sampling and sample points3.3.3. Statistical moments3.3.4. PDF estimation

4. Validity of the proposed approach5. Conclusionsection13AcknowledgmentReferences