IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECHNOLOGY, VOL. 9, NO. 4, APRIL 2019 797

Advanced Statistical Model Calibration toDetermine Manufacturing-Induced Variationsof Effective Elastic Properties of SAC SolderJoints in Leadless Chip Resistor Assemblies

Hsiu-Ping Wei , Yu-Hsiang Yang, and Bongtae Han

Abstract— The unknown statistical distributions of two effec-tive elastic properties of Sn-3.0Ag-0.5Cu solder joint of leadlesschip resistors (LCRs), induced by an assembly condition,are determined by the advanced statistical model calibration.Two key elements are involved in the model calibration: an uncer-tainty propagation (UP) analysis and an optimization processusing a calibration metric. In this paper, the UP analysis utilizesan advanced approximate integration method, which allows us totake into account the statistical variations of six additional knowninput variables, including die thickness, solder joint height,termination length, and thickness and elastic moduli of a printedcircuit board. The cyclic bending test results of LCR assembliesare used in conjunction with the maximum-likelihood metricto obtain the statistical distributions of the effective properties.The cycles-to-failure distribution of the identical LCR assembliessubjected to a different loading level is predicted accuratelyby the calibrated model, which corroborates the validity of theproposed approach.

Index Terms— Anisotropic SnAgCu (SAC) properties, bendingtest, Eigenvector dimension reduction method, probability densityfunction, SAC, statistical model calibration.

I. INTRODUCTION

IN RECENT years, Sn-rich solders, such as SnAgCu (SAC)alloys, have been used widely in electronics packaging to

comply with the regulations. It has been known that the Sn-richsolders have large variations in grain sizes and orientations,intermetallic compound sizes, and distributions of anisotropicSn crystals [1]. These microstructural variations occur evenunder the same assembly condition, which results in inher-ent mechanical property variations of solder joints; not onlypackage-to-package variations but also joint-to-joint variationsin the same package [2]–[8]. The mechanical properties ofSn-rich solder materials are attributed significantly to thestatistical variation of a solder joint life.

The solder joint life is assessed typically by: 1) testing a“statistically significant” number of components under accel-

Manuscript received June 30, 2018; revised September 27, 2018; acceptedOctober 4, 2018. Date of publication October 8, 2018; date of current versionApril 1, 2019. Recommended for publication by Associate Editor J. Punchupon evaluation of reviewers’ comments. This work was supported by theCenter for Life Cycle Engineering (CALCE) of the University of Maryland.(Corresponding author: Bongtae Han.)

The authors are with the Mechanical Engineering Department, Universityof Maryland, College Park, MD 20742 USA (e-mail: bthan@umd.edu).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCPMT.2018.2874638

Fig. 1. Durability versus reliability.

erated loading conditions and 2) determining the characteristicnumber of cycles-to-failure (CTF) from the test results, whichhave a statistical distribution due to the inherent manufacturingvariability. Fig. 1 illustrates the statistical distribution of CTFdata and the characteristic life. The characteristic life is oftenreferred to as “durability,” which represents the length of thetime over which a product will provide its intended functionunder operating conditions, i.e., the reliability of nominal parts.Durability is important when competitive designs are com-pared for design-for-reliability. On the other hand, the distri-bution of test results can be used to calculate the “reliability,”which is the probability that a product will perform its intendedfunction under operating conditions, for a specific periodof time. The reliability becomes critical when early failureprobability has to be evaluated, e.g., the probability that theproduct will fail before a warranty period ends.

Numerous physics-based lifetime models have beendeveloped for durability assessment. More recently, the modelshave been extended into a probabilistic domain to predictreliability, while taking into account inherent manufacturingvariability [9]–[12]. The inherent variations can be categorizedinto “known input variables” and “unknown input variables.”The “known input variables” are the variables whosestatistical distributions are known or can be measured.The variables are called “unknown” when their statisticaldistributions are extremely difficult or impractical to beobtained experimentally. These unknown distributions can beobtained by “statistical model calibration,” which identifiesthe statistical distribution of the unknown input variables that

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798 IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECHNOLOGY, VOL. 9, NO. 4, APRIL 2019

produces the best agreement between the experimental dataand the predicted distribution through an optimization process.

In this paper, the unknown statistical distributions oftwo effective elastic properties of Sn-3.0Ag-0.5Cu (SAC305)solder joint of leadless chip resistors (LCRs) assemblies aredetermined by the advanced statistical model calibration, whileconsidering the statistical variations of serval other knowninput variables including die thickness, solder joint height,termination length, and thickness and elastic moduli of aprinted circuit board (PCB).

The background of the statistical model calibration isdescribed first. The cyclic bending test results of the LCRassemblies are followed, and the results are subsequently usedto obtain the statistical distributions of the effective elasticproperties of SAC305 solder. Validity of the calibrated modelis corroborated by comparing the predicted probability densityfunction (PDF) of CTF of the identical LCR assembliessubjected to a different loading level with the CTF distributionof actual testing data.

II. BACKGROUND: STATISTICAL MODEL CALIBRATION

Statistical model calibration is an optimization processto inversely identify the statistical distributions of unknowninput variables that produce the best agreement between theexperimental data and predicted distribution [13]. It utilizes afull set of experimental data containing multiple samples sothat the calibration results can be obtained after consideringthe statistical variations of performance responses induced bythe intrinsic variabilities of input variables. Fig. 2 shows thecalibration procedure. The uncertainties in the known andunknown input variables are propagated through the modelto acquire the predicted PDF of performance responses. Then,the predicted PDF of performance response is compared withthe experimental data. The degree of agreement is determinedby a “calibration metric,” which acts as the objective functionof the optimization process. The unknown input variablesare iteratively updated until the calibration metric is mini-mized or maximized.

The most critical element of the calibration procedure isthe uncertainty propagation (UP) analysis. The models ofsemiconductor packages are usually computationally intensive.Thus, the classical UP methods such as the random sam-pling method, e.g., Monte Carlo simulation (MCS), and theresponse surface method (RSM) are often inadequate when alarge number of input variables are considered. An advancedapproximate integration method, called eigenvector dimensionreduction (EDR) method, is implemented for the UP analysisin this paper.

The EDR method [14] calculates the statistical momentsof the PDF of performance responses (the CTF of solderjoint in this paper). The statistical moments describe differentstatistical characteristics of the PDF of performance responsessubjected to input uncertainties by different “order.” Thefirst moment, μ, is the mean, which represents the centraltendency of the distribution and the second moment is thestandard deviation, σ , which represents the spread of the dis-tribution. The third and fourth moments are skewness, β1,

Fig. 2. Procedure of statistical model calibration.

and kurtosis, β2, which indicate the symmetry and the peaked-ness of the distribution, respectively.

In the EDR method, the additive decomposition [15] is usedfor statistical moments calculation, and thus, only 4N + 1modeling runs are required, where N is the number of inputvariables; the computational cost increases “additively” withthe number of input variables. Then, the PDF can be con-structed using these statistical moments.

Recently, the EDR method was implemented successfullyfor a thin flat ball grid array package to predict an assem-bly yield (tail-end probability less than 1000 ppm) with anuncertainty less than 1% despite the fact that the problem had12 input variables [9].

The other key element of the calibration procedure is acalibration metric, which has to account for the statisticalcomparison between the predicted PDF and the experimentaldata. The likelihood function is the most widely used calibra-tion metric. Fig. 3 illustrates the cases with the maximum-likelihood value and a low-likelihood value. In practice,the logarithm operation is applied to the likelihood functionto convert the multiplication form of the likelihood functionto a summation form. In addition, most of the commercialoptimization algorithms minimize a function, and thus, a neg-ative log likelihood (NLL) is used; minimizing the NLL isequivalent to maximizing the log likelihood.

The NLL function can be expressed as [16]

L(y1, . . . , yn |⇀θ x_known,⇀

θ x_unknown)

= −n∑

i=1

log[ fpre(yi |⇀

θ x_known,⇀

θ x_unknown)] (1)

WEI et al.: ADVANCED STATISTICAL MODEL CALIBRATION TO DETERMINE MANUFACTURING-INDUCED VARIATIONS 799

Fig. 3. Illustration of likelihood function with the maximum- and minimum-likelihood values.

Fig. 4. Detail of a test coupon and a single LCR assembly.

where y1, . . . , yn are the n number of the experimental data,⇀

θ x_known and⇀

θ x_unknown are the vectors of the parametersof PDFs of the known and the unknown input variables,respectively, and fpre is the predicted PDF of performance

response, y. The⇀

θ x_unknown is calibrated by minimizing thevalue of the NLL.

III. IMPLEMENTATION USING VIBRATION-INDUCED

SOLDER FATIGUE FAILURE

The experimental setup of a cyclic bend test is describedin Section III-A. The physics-based lifetime prediction is fol-lowed in Section III-B. In Section III-C, the statistical modelcalibration for the effective elastic properties of SAC305 solderis provided.

A. Setup of Four-Point Bend Test

Fig. 4 shows the test coupon, which was fabricatedby mounting multiple 6332-type LCRs on a PCB withSAC305 solder. The pad length was 1 mm.

A four-point bend test setup used in this paper is shownin Fig. 5. A strain gauge was attached to the PCB, and it wasaligned with the longitudinal side of the PCB to determine the

Fig. 5. Test coupon mounted on four-point bend fixture.

Fig. 6. Weibull plot of the accelerated life testing results of LCR assembliessubjected to strain levels of 2000 and 1200 με.

Fig. 7. Cross section of failed assembly.

board level strain. Two sets of LCR assemblies were testedat a frequency of 2.5 Hz for the maximum board strainsof 2000 and 1200 με, respectively.

Fig. 6 shows the two-parameter Weibull plots of the testresults. The characteristic CTFs and the standard deviations oftest results were (7,723 and 2,847) and (78,618 and 43,827)for the 2000 με stain level and the 1200 με stain level,respectively. Fig. 7 shows the observed failure mode, i.e., thebulk solder failure along the copper pad.

B. Physics-Based Solder Joint Fatigue Life Modeling

The Basquin power law relation is widely used for the life-time prediction of solder joints under mechanical vibrations,

800 IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECHNOLOGY, VOL. 9, NO. 4, APRIL 2019

Fig. 8. Finite-element model used in the damage analysis.

TABLE I

MATERIAL PROPERTIES

which can be expressed as [17]

σa = σ f (2N f )b (2)

where σa is the stress amplitude, σ f is the fatigue strengthcoefficient, b is the fatigue strength exponent, and N f isthe CTF. The constants from [17] were used in this paper:σ f = 64.8 MPa and b = −0.1443.

Fig. 8 shows a finite-element model constructed by ANSYSusing quarter symmetry. The condition of periodicity wasimposed on the boundary to simulate the effect of multiplecomponents. Table I lists the material properties used in theanalysis. Since only stress magnitudes were required for (2),linear elastic material properties were used for the packagingmaterials.

The displacement loading was iteratively adjusted to findthe accurate loading condition, which made the x-directionstrain in the strain gauge area reached 2000 με.

Fig. 9 shows the von Mises stress distribution on the solderjoint. The volume averaged von Mises stress over the criticalregion on the top layer of the solder joint was selected as thedamage index, which can be expressed as [17]

σa =∑

σ · V∑V

. (3)

C. Statistical Model Calibration

The known and unknown input variables are described first.Calibration results and discussion are followed.

1) Known and Unknown Input Variables: The statisticalcalibration begins with the selection of known and unknownvariables in the solder joint fatigue life model. Six knowninput variables (x1 to x6) were identified from [11],and [18]–[20]: ceramic die thickness, solder joint height,

Fig. 9. Von Mises stress distribution in the solder joint.

TABLE II

INPUT VARIABLES

TABLE III

CORRELATION COEFFICIENTS OF CORRELATED INPUT VARIABLES

termination length, as well as thickness and two effectivemoduli of PCB. Their variations are listed in Table II.Among these variables, the effective moduli of PCB andPCB thickness have statistical correlations, because theirvariations come from the thickness variation of each layerin the PCB stack up. Their correlation coefficients wereestimated from [20] and they are listed in Table III.

Two unknown variables (x7 and x8) are theeffective Young’s modulus and Poisson’s ratio of theSAC305 solder joint. An unknown number of tin grains

WEI et al.: ADVANCED STATISTICAL MODEL CALIBRATION TO DETERMINE MANUFACTURING-INDUCED VARIATIONS 801

Fig. 10. NLL values as a function of iteration number.

and their orientations within the solder joint were attributedto their statistical variations. Both unknown variables wereassumed to follow a lognormal distribution since a negativevalue was not permitted.

Two statistical parameters are required to describe a lognor-mal distribution. The representative values of [17] were usedfor the initial guess of the mean and standard deviation of twounknown input variables: (51 and 2.55 GPa) for the Young’smodulus and (0.36 and 0.018) for the Poisson’s ratio.

2) Calibration Results and Discussion: Since the calibrationmetric was an implicit objective function, the finite differencemethod was used to determine the convergence direction bycalculating the gradients of the objective function with respectto the mean (μ) and standard deviation (σ) of the unknownvariables.

Fig. 10 shows the value of an NLL as a function ofthe iteration number. The calibration converged after fiveiterations. The iteration was terminated when the change ofthe input variables (the Young’s modulus and the Poisson’sratio) to the next iteration was smaller than 0.1 and 0.01 GPa,respectively.

Fig. 11 compares the initial PDFs of two unknownvariables with the calibrated PDFs. The mean value ofYoung’s modulus increases from 51 to 62.8 GPa, and thestandard deviation slightly decreases from 2.55 to 2.38 GPa.Based on the calibrated PDF of Young’s modulus, 99%of the Young’s modulus values are distributed within therange of [56.9–69.2 GPa], which is close to the variationreported in [21]. In Fig. 11(b), the Poisson’s ratio also showsan increasing mean (0.36 to 0.42) and decreasing standarddeviation (0.036 to 0.015). Accordingly, there are 99% of thePoisson’s ratio values are located within [0.38 and 0.46].

Fig. 12 compares the experimentally obtained CTF his-togram with the predicted PDFs of CTF using the initialguesses of input variables as well as the calibrated inputvariables. The result clearly shows an improved agreementafter calibration.

To evaluate the validity of the calibration results,the CTF distribution of the identical LCR assembliessubjected to a strain level of 1200 με was predicted by the

Fig. 11. Initial guesses and calibrated values of (a) Young’s modulus and(b) Poisson’s ratio of SAC305 solder.

Fig. 12. PDFs of CTF before and after calibration are compared with thehistogram of the experimental data obtained at 2000 με strain level.

calibrated model. The results are compared with the actual testdata in Fig. 13. This confirms that the calibrated model canaccurately predict the CTF distribution of the LCR assemblies

802 IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECHNOLOGY, VOL. 9, NO. 4, APRIL 2019

Fig. 13. PDF of CTF predicted by the calibrated model is compared withthe histogram of the experimental data obtained at 1200 με strain level.

subjected to other loading conditions, provided that the assem-blies are fabricated by the same manufacturing condition.

IV. DISCUSSION

A. Modeling Runs

The number of modeling runs required to complete theiterations is described in the following.

1) In each iteration, the gradients with respect to μ andσ of each unknown variable had to be determined. Forthe gradient of each μ, a total of 4N + 1 modelingruns (33 = 4 × 8 + 1) had to be performed to constructthe PDF. For the gradient of each σ , however, onlyfour additional modeling runs related to each unknownvariable had to be performed because the center did notchange. It is worth recalling that, with the EDR method,all of the 1-D integrations pass through the same cutcenter, which is a vector of the mean values of all inputvariables. The eigenvector sampling selects this point asthe center, and as long as the mean does not change, onlyfour additional points of μ±1.5σ and μ±3σ are neededto calculate the PDF. Thus, a total of 74 modeling runs(= 33 × 2 + 4 × 2) were performed for each iteration.

2) After the gradients were determined, the μ and σvalues of two unknown variables were updated, andan additional 4N + 1 modeling runs were performedsubsequently to calculate the calibration metric with theupdated unknown input variables: 33 modeling runs.

3) Considering the modeling runs (33) that were performedto evaluate the calibration metric with the initial guessesof two unknown variables, a total of 568 modeling runs(107×5+33) were required to complete the calibration.

A linear elastic analysis was used in this paper, andthe required modeling runs were handled readily by ahigh-performance workstation. If a more complex assemblythat requires a nonlinear analysis is considered, the requiredcomputational burden can exceed a practical limit inspite of the effectiveness provided by the EDR method.

Fig. 14. (a) PDF of CTF under the usage condition, predicted by thecalibrated model and (b) left tail end of the PDF.

A more efficient calibration metric and/or a more accuratemetamodeling scheme that does not suffer from the curse ofdimensionality are warranted for those applications.

B. Reliability Prediction Under Usage Condition

Reliability of the LCR assemblies under a usage conditioncan be predicted by the calibrated model. For this illustration,the maximum board strain and the requirement of keypresses were assumed to be 550 με and 1 million presses,respectively, in the usage condition.

The displacement loading of the model shown in Fig. 8 wasadjusted until the board strain matched to the maximumstrain. Then, an UP analysis was conducted using the originalproperties (Tables II and III) as well as the calibrated solderproperties (Fig. 11). The PDF of the solder joint fatigue lifeof the LCR assemblies under the usage condition is shownin Fig. 14(a). Fig. 14(b) shows the left tail end of the PDF,which shows the reliability of LCR assemblies satisfyingthe requirement. The reliability was estimated to be 41 ppm(i.e., the yellow area).

WEI et al.: ADVANCED STATISTICAL MODEL CALIBRATION TO DETERMINE MANUFACTURING-INDUCED VARIATIONS 803

V. CONCLUSION

The unknown statistical distributions of two effectiveelastic properties of SAC305 solder were determined by theadvanced statistical model calibration using the cyclic bendingtest results of LCR assemblies. The calibration procedureemployed the advanced approximate integration method for anUP analysis to be able to take into account the statistical vari-ations of six additional known input variables. The CTF distri-bution of the identical LCR assemblies subjected to a differentloading level was predicted accurately by the calibrated model,which corroborated the validity of the proposed approach.

It is important to note that the conventional UP analysiswould require a lot more modeling runs, which would makethe statistical calibration impractical; for example, the MCSwould need thousands to tens of thousands of modeling runsfor the PDF construction of each gradient calculation, and theRSM would need 58 modeling runs for the response surfaceconstruction with the same level of sampling resolution of the4N + 1 scheme used in this paper.

The calibrated model was further utilized to predict thereliability of the LCR assemblies under a usage condition.It is anticipated that the proposed methodology can be usedeffectively to predict a field failure rate and/or a warrantyperiod of electronics products using Sn-rich solders.

ACKNOWLEDGMENT

The authors gratefully acknowledge the support of theCenter for Life Cycle Engineering (CALCE), University ofMaryland.

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[11] Y. Aoues, A. Makhloufi, P. Pougnet, and A. El-Hami, “Robustnessstudy of solder joints of different compositions by using stochastic finiteelement modeling,” in Proc. 8th Int. Conf. Integr. Power Syst. (CIPS),2014, pp. 1–7.

[12] Y. Hsu, C.-Y. Su, and W.-F. Wu, “Quantitative reliability analysisof flip–chip packages under thermal–cyclic loading and in considera-tion of parameter uncertainties,” Microelectron. Rel., vol. 51, no. 12,pp. 2284–2289, 2011.

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Hsiu-Ping Wei received the B.S. and M.S. degreesfrom the Department of Power Mechanical Engi-neering, National Tsing Hua University, Hsinchu,Taiwan, in 2005 and 2007, respectively, and thePh.D. degree from the University of Marylandat College Park, College Park, MD, USA, in2018.

He was a Senior Engineer with the IntegratedInterconnect and Package Development Division,Taiwan Semiconductor Manufacturing Company,Hsinchu, where he was involved in zeroth-level

interconnection reliability under chip-packaging interaction. He is currentlyan IC Packaging Engineer with Apple, Inc. His current research interestsinclude the prediction of statistical variations of reliability performance andthe statistical model calibration of microelectronics.

804 IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECHNOLOGY, VOL. 9, NO. 4, APRIL 2019

Yu-Hsiang Yang received the B.S. degree from theDepartment of Mechanical Engineering, NationalChiao Tung University, Hsinchu, Taiwan, in 2011,and the M.S. degree from the Department ofPower Mechanical Engineering, National TsingHua University, Hsinchu, in 2013. He is currentlypursuing the Ph.D. degree with the Department ofMechanical Engineering, University of Maryland atCollege Park, College Park, MD, USA.

He was an Engineer with the IntegratedInterconnect and Package Development Division,

Taiwan Semiconductor Manufacturing Company, Hsinchu, where hewas involved in zeroth-level interconnection reliability under chip-packaging interaction. His current research interests include electronicpackaging and the prediction of statistical variations of reliabilityperformance.

Bongtae Han received the B.S. and M.S. degreesin mineral and petroleum engineering from SeoulNational University, Seoul, South Korea, in 1981 and1983, respectively, and the Ph.D. degree in engineer-ing mechanics from Virginia Tech, Blacksburg, VA,USA, in 1991.

He is currently a Keystone Professor of engineer-ing with the University of Maryland at College Park,College Park, MD, USA.

Dr. Han was elected as a fellow of the Society forExperimental Mechanics and the American Society

of Mechanical Engineers in 2006 and 2007, respectively. He has served as anAssociate Technical Editor for Experimental Mechanics from 1999 to 2001and the Journal of Electronic Packaging, Transaction of the ASME, from 2003to 2012. He is currently serving as an Associate Editor for MicroelectronicsReliability.