Weibull Statistics of Silicon Die Fracture

C. Bohm*, T. Hauck*, A. Juritza**, W.H. Müller***) Freescale Halbleiter Deutschland GmbHSchatzbogen 7, 81829 München, Germany**) Technische Universität Berlin, LKMEinsteinufer 5, 10587 Berlin Germany

wolfgang.h.mueller@tu-berlin.de, tel. 49-30-314-27682, fax. 49-30-314-24499

AbstractIn order to guarantee reliability of semiconductor devices

for automotive applications an optimized package design isrequired. Clearly the design must be based on the bestchoice of geometry, materials and manufacturing processes.It is well known that the various process steps involved dur-ing package manufacturing occur at relatively high tem-peratures. Consequently, due to the mismatch of thermalexpansion coefficients of the package materials, high thermalstresses arise at operating temperatures and may lead to fail-ure of the device. Typical device failure modes include de-lamination of material interfaces or bulk material fracture.In this paper we focus on the prediction of silicon fracture inthe microchips of electronic devices. Due to their brittlenature the strength data of silicon dies will scatter and aprobabilistic approach to failure is required. For this pur-pose Weibull theory will be used and combined with analyti-cal as well as numerical tools in order to describe the state ofstress in the package and in particular within the silicon die.As a result of the analysis the probability of fracture in amicrochip can now be assessed and used for further qualityassurance purposes.

The paper will start with a brief introduction to Weibulltheory and present the 3-Point-Bending (3PB) experimentsthat were used to obtain the Weibull parameters for charac-terization of a combination of surface and edge flaw inducedsilicon die fracture. Moreover, the ball-on-edge and ball-on-ring tests will be described and used to separately character-ize edge or surface flaws, respectively. Particular emphasiswill also be given to the transferability of Weibull probabil-ity results from one specimen configuration to another re-sulting in a change of surface size and stress. In this contextit will be described how a stress distribution and, in particu-lar, a multiaxial state of stress influences the variability instrength. Moreover it will be shown how the correspondingWeibull integrals can be implemented numerically and usedfor postprocessing of finite element results. The paper con-cludes by demonstrating how these procedures can be usedfor optimizing the design of a real silicon die package.

1. Flaw Population Selective Fracture TestsDepending on the preparation and processing of the sili-

con die different flaw populations will be induced whichessentially lead to different failure modes during use, in par-ticular when the die is embedded in a molded package and,therefore, subjected to a multiaxial state of stress acting onthe edges as well as on the surfaces. We may say that sur-face defects are essentially induced during wafer processing,such as etching and grinding. On the other hand, edge de-

fects will originate from cutting processes. Figure 1 showsthe chipping of a silicon wafer caused by the singulationprocedure. During a bending test the different flaw distribu-tions on the chip result in cracks that statistically emanatefrom either the surface or the edge (cf., Figure 2).

Figure 1: Silicon chipping at sawing street (acceptableand poor quality)

Figure 2: Fracture surface of bending test specimen

Consequently, the question arises as to whether and howthe different flaw populations can be distinguished andquantitatively assessed. Due to the statistical nature of brit-tle fracture, which is also pertinent to silicon, Weibull statis-tics will preferably be used, which, however, must first beput in context with experimental setups characteristic of thevarious flaw populations in question. In particular we shalluse the so-called Ball-On-Edge (BOE) test to quantify edgeflaws populations, whereas the Ball-On-Ring (BOR) (cf.,[1]) test will serve as a tool for surface flaw analysis. Itshould be noted that the traditionally used 3 and 4 Point-Bending (3PB/4PB) tests always lead to a combination ofsurface and edge induced failure. This becomes evident ifone realizes that both tests show a maximum tensile stress ofequal magnitude along the edges as well as on the surface ofthe tested specimen. In what follows we shall briefly sum-marize some essentials on Weibull statistics, comment onresults from 3PB tests, and will then discuss in detail theexperimental setup and strength data stemming from BOE-tests.

2. Elements of Weibull TheoryDuring fracture of a batch of specimens made of brittle

materials some will fail at very low loads, others are capableof withstanding very high loads, but most of them showsome intermediate strength. Clearly, the strength of a highquality material should not show high variability in strength.

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Consequently, such products are characterized by a narrowstrength distribution, whereas a broad distribution indicates aless advanced material, or even a processing fault. By cu-mulative summation of the probability data, iP , the cumula-tive distribution of strength in Figure 3 can be constructed.If the ordinate is normalized with respect to the total numberof specimens in the batch, N, the resulting plot presents ameasure for the probability of failure, iP , where i refers tothe specimen of strength iσ :

.ln1

1lnln0.4

0.3 = x ,P

= y ,+N - i =P ii

iii σ

−(1)

Experience shows that for brittle materials ( )iiP σ can befitted reasonably well using a two-parameter Weibull distri-bution:

( )��

�

�

��

�

�

��

��

σσ−−σ

0exp1 = P

m

(2)

High variability of strength within a particular batch ofspecimens is characterized by a small Weibull modulus or m-value. 0σ is known as the scale parameter.

Figure 3: Distribution of strength for brittle matter

In practice these two parameters are determined from ex-periments using a particular specimen and loading type. Theresulting strength data are either processed using the methodof least squares or the maximum likelihood procedure (see[2] for details): Figure 3.

As mentioned above the mean strength of a specimenmade of brittle materials depends on its volume, its surface,its loading conditions and the flaw distribution. Intuitivelyspeaking the more volume or surface exists the higher thechances for a critical flaw to be present will be. For exam-ple, a 4PB specimen will break more easily than a 3PB one,because in the case of the latter high tensile stresses are re-stricted to a much smaller region.

Based on three principles, viz., (i) statistical homogeneityand isotropy of the material, (ii) statistical independence ofsubvolumina and subsurfaces, and (iii) the weakest link con-cept the probability of failure for a body subjected to a spa-tially varying, but uniaxial, state of stress can, according toWeibull [3], be written for surface dominated flaws:

( )��

�

�

��

�

���

�

�− ��

A

m

eff

yx yx A

- P = dd,1exp10σ

σ , (3)

( )�� �

�

���

�

A

m

eff yx yx = A dd,

maxσσ . (4)

effA and A denote the effective and total surface of thespecimen. ( )yx,σ and σmax are the arbitrary, uniaxial tensile(positive) stress distribution and the maximum stress at thespecimen surface, respectively. It is assumed that only ten-sile stresses influence the probability of failure. Some in-formation on how the surface integration is performed can befound in [2].

3. Selected Results on 3-PB-Tests3PB tests were performed at the Technische Universität

in Berlin using a Tytron 250 provided by MTS. The 3PB-test setup is shown in Figure 4. The specimens were loadeduntil fracture occurred and the loads at fracture wererecorded. A typical fracture pattern is shown in Figure 5.

Figure 4: 3-PB-test setup (top and side view)

Figure 5: 3PB-test specimen before and after test

Typically strength measurements are presented in aWeibull plot, where the ordinate and abscissa are dividedaccording to ln(ln(1/(1-P))) and ln(σ), respectively. Theresulting curve usually matches a straight line that is definedby the Weibull distribution with modulus m and scale pa-rameter σ0. Figure 6 shows the Weibull plot for 3PB-testswith the Weibull distribution, where the parameter were de-termined using the maximum likelihood method.

783 2004 Electronics Packaging Technology Conference

100.00 1000.00

1.00

5.00

10.00

50.00

90.00

99.00

Maximum Stress [MPa]

Fracture Probability [%]

Weibull Distribution

3PB Measurement

m=5.4s0=446

Figure 6: Weibull plot for 3PB-tests

4. The Ball-on-Edge TestExperimental Setup: The realization of the edge test is

shown in Figure 7. The test rig consists of two metal blocksinto which spheres have been inserted. The left block (cf.,Figure 7) contains three spheres whereas the right blockholds only one sphere onto which the force is acting (in hori-zontal direction). A schematic of the arrangement is shownin Figure 8.

Figure 7: BOE-test setup top and bottom view

Figure 8: Schematic of BOE rig

The position of the spheres within the blocks defines thestress distribution in the specimen. The tensile stress con-centration on the lower specimen surface directly underneaththe single ball triggers brittle failure at this position. Thecloser the single sphere is moved toward the specimen edgethe more the edge will be subjected to tension and the morethe edge flaw population will be assessed.

Experimental Results: BOE-tests were performed at theTechnische Universität in Berlin according to the describedexperimental setup. The specimens were singulated fromwafer by standard sawing procedure at Freescale GmbH.Each specimen consists of an array of 3x3 silicon chips (cf.,Figure 9). A batch size of 20 samples was subjected to thetesting procedure. The resulting probability of failure vs. thefailure load is presented in Figure 10. The graph doessignificantly deviate from the typical S-shape of a Weibulldistribution (cf., Figure 3). It indicates that two active flawdistributions exist. Consequently, fractographic examinationis required in order to characterize the fracture origin in eachspecimen.

Figure 9: Silicon specimen – array of 3x3 chips

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

0 5 10 15 20

failure load [N]

Pro

babi

lity

of f

ailu

re,

P

Figure 10: Probability of failure vs. the failure load

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Indeed, the first three data points below a force of 15 N cor-relate with a crack initiation at the edge of the specimen,whereas the other data points above 15 N correlate with acrack initiation from the bottom surface underneath the ball(cf., Figure 10). The corresponding fragments are shown inFigure 11 and Figure 12. Thus we detect two distinct flawdistributions for edge and surface failure, respectively.

Figure 11: Specimen fragments indicating edge failure mode

Figure 12: Specimen fragments indicating surface failuremode

Stress-Analysis and Weibull Parameters: The resultingstress distribution at the failure load was calculated by meansof finite element analysis. Figure 13 shows the meshes ofthe specimen and the test rig. Note that the deformations

shown in the pictures were scaled and in fact remained verysmall. The balls were represented with elastic half spheresand discretized with tetrahedral elements. The elasticspecimen was discretized with hexahedral elements. Thefailure load was applied to the upper ball, whereas the sup-porting balls were completely fixed. A contact problem wasdefined between the balls and specimen. All materials weretreated elastically and geometric nonlinear effects were con-sidered. The analysis of the stress state at different loadlevel revealed a linear relationship between load and stressvalues (cf., Figure 14). Factors of proportionality whereextracted from the finite element analysis for easy calcula-tion of maximum stresses on specimen surface and edge atarbitrary loads:

���=σ −

−

surfacespecimen mm5.17edgespecimen mm2.12, 2

2

max kFk= . (5)

Figure 13: Deformed FE mesh of BOE rig and specimen

0

50

100

150

200

250

300

0 4 8 12 16Force F [N]

Str

ess

σ [M

Pa]

Maximum tensile stressunderneath the ballMaximum tensile stressat the edge

k2 = 17.5 mm-2

k1 = 12.2 mm-2

Figure 14: Maximum tensile stresses vs. applied force

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The resulting tensile stress distribution in the specimen isillustrated in Figure 15. The applied failure load at the upperball causes a stress concentration on the bottom surface un-derneath the ball (red color). Due to the proximity of thesingle sphere to the specimen edge, high stress regions ap-pear at the edge of the specimen as well (yellow color).Maximum tensile stresses were taken from underneath theball in order to determine the silicon strength for the surfaceflaw distribution, whereas maximum stresses were takenfrom the edge in order to determine the silicon strength forthe edge flaw distribution.The statistical evaluation of the strength values were per-formed separately for each failure mode. The sampledstrength data were split into two sets according to the fracto-graphic classification (cf., Figure 11and Figure 12). Each setwere treated as an independent discrete strength distribution.The corresponding Weibull parameters were determined bymeans of the maximum likelihood method. Figure 16 showsthe Weibull plot with the overall specimen failure probabil-ity for the BOE-test and the Weibull parameters for bothflaw distributions.

Figure 15: Tensile stress distribution in specimen

Figure 16: Weibull plot of the BOE-test

4. ConclusionsA ball-on-edge (BOE) test procedure was developed with

the intention of a local strength measurement in regions ofthe silicon chip surface and the silicon chip edge.

The test rig was designed such, that the fracture originwas triggered at a point close to the die edge in order tocapture flaw distributions in this region.

The evaluation of the strength data revealed two inde-pendent active flaw distributions, one on the chip surfaceand another one on the chip edge. Weibull parameters weredetermined for both distributions.

Further work is required in order to confirm the resultsand to achieve a sufficient statistical confidence.

Comparative tests with three-point-bending and ball-on-ring setup are planned in order to validate the procedure offailure probability prediction for arbitrary specimen geome-tries.

The Weibull parameter will be used for the prediction offailure probability for a silicon chip situated in a moldedpackage.

AcknowledgmentsThe authors wish to acknowledge Jörg Feige, Freescale

Gmbh for specimen preparation, Betty Yeung and VernHause, Freescale Inc., for silicon strength tests andfractographic analysis.

References1. Jahresbericht 2003, Fraunhofer Institut für Werkstoffme-

chanik IWM, (Freiburg, 2003), pp.38-39.2. Bohm, C, Hauck, T., Müller, W.H., Juritza, A., “Prob-

ability of Silicon Fracture in Molded Packages,” Proc.EuroSimE 2004, Brussels, May 2004, pp. 75-82.

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