Development of a new cluster index for wafer defects

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Int J Adv Manuf Technol (2007) 31: 705–715 DOI 10.1007/s00170-005-0240-5

O R I G I N A L A RT I C L E

Lee-Ing Tong . Chung-Ho Wang . Da-Lun Chen

Development of a new cluster index for wafer defects

Received: 24 May 2005 / Accepted: 26 August 2005 / Published online: 13 April 2006 # Springer-Verlag London Limited 2006

Abstract Defect number and defect clustering are two key determinants of wafer yield. Preventing and detecting wafer defects thus is an important issue in integrated circuit manufacturing. Defect clustering tends to grow with increasing wafer size. Methods have been developed for assessing defect clustering on wafers. However, these methods require either statistical assumptions regarding defect distribution or complex computations. This study develops a new cluster index, utilizing the rotating axis technique from multivariate analysis to accurately quantify defect clusters on a wafer. The developed defect-clustering index does not require making assumptions regarding defect distribution. Thus, the proposed method can be efficiently used by engineers with little statistical back-ground. A simulation experiment is conducted to demon-strate the effectiveness of the proposed defect-clustering index.

Keywords Integrated circuit . Wafer . Defect clustering . Rotation of axes . Multivariate analysis

1 Introduction

Fierce global competition has led integrated circuit (IC) manufacturers to strive to increase product yield. Because wafer yield significantly influences productivity,

improv-ing wafer yield is important for IC manufacturers. The number of defects per wafer is the primary determinant of wafer yield. Wafer yield decreases with increasing number of defects. However, if wafer defects display clustering, the number of defects per wafer may not reflect actual yield. Consequently, besides the number of defects, the degree of defect clustering on a wafer also must be assessed to accurately reflect defect distribution and thus enhance wafer yield.

Defect clustering increases with increasing wafer size. Consequently, the conventional Poisson yield model based on the Poisson distribution tends to underestimate yield. Stapper [6] thus proposed a negative binomial yield model that can estimate wafer yield more accurately than the conventional yield model. The negative binomial yield model is used extensively in the IC industry because it simultaneously accounts for the number of defects and defect clustering (using clustering indexα). However, the range ofα values is scattered and sometimes is negative [1] making it difficult for engineers to assess degree of defect clustering. Jun et al. [3] designed a clustering index (denoted by CI_J) for assessing the degree of defect clustering on a wafer and confirmed that the CI_J can represent defect clustering more accurately than the α obtained from a binomial negative distribution. Although the CI_J does not require any statistical assumptions regarding the defect distribution,CI_Jmay yield consistent values for various types of defect distributions formed on wafers. These consistent values confirm that the CI_J may address erroneous judgments when assessing the degree of defect clustering.

This study designed a new clustering index,CI_M, using the rotating axis technique from multivariate analysis [7], to overcome the drawbacks of theCI_J. TheCI_Mpossesses the same advantage as theCI_Jin not requiring assumptions regarding the defect distribution. Moreover, the CI_M can also represent defect clustering on a wafer more accurately than the CI_J. Simulation experiments were conducted to compare the proposed CI_M and other existing cluster indices for application in IC fabrication. These simulations verified the effectiveness of the proposed method.

L.-I. Tong

Department of Industrial Engineering and Management, National Chiao Tung University,

HsinChu, Taiwan, Republic of China C.-H. Wang (*)

Department of Computer Science, Chung Cheng Institute of Technology, National Defense University,

Taoyuan, Taiwan, Republic of China e-mail: [email protected]

D.-L. Chen

Total Quality Management Committee, United Microelectronic Corporation, HsinChu, Taiwan, Republic of China

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2 Literature review

2.1 Defect clustering indices

The conventional yield model assumes that the number of defects on a wafer follows a Poisson distribution. This assumption implies that the defects display a random distribution. However, defects tend to be clustered rather than dispersed randomly over a wafer. Consequently, certain spatial distributions related to wafer defects, including a compound Poisson distribution and generalized Poisson distribution, were considered in establishing a yield model. The negative binomial model, as follows, is probably the best known.

Y ¼ 1

1þ D0A=α

ð Þα (1)

where Y denotes the yield, D0 represents the average number of defects per unit area,A is the chip area, and α is the clustering parameterα ¼ λ2

. σ2_λ

; where λ and σ2 _{denote the mean and variance of defects per die,} respectively. A smaller α value corresponds to a larger variation in the defect density on a wafer.

Cunningham [1] noted that the α values obtained directly from the equation α ¼ λ2

. σ2_λ

can be quite scattered and sometimes negative. Cunningham [1] also showed that the compound yield equations could be closely imitated by selectingα values. Table1lists theα values and corresponding imitated yield models. For more developed compound yield models, see Raghavachari [5]. Rogers [4] developed the quadrat method for analyzing the point spatial distribution. The quadrat method involves initially dividing the plane surface into numerous equal-sized grids. The random defect distribution in each grid

then is assessed. If the defects are randomly distributed on the plane surface, the defects on each grid then are also randomly distributed and follow a Poisson distribution. The variance-mean ratio (V/M) value associated with each grid is one when the defects follow a Poisson distribution. TheV/M value exceeds one if the defects are clustered. The V/M values can be proven to possess a t distribution with n−1 degrees of freedom. The defects are clustered on each grid once theV/M value exceeds the critical tn−1point.tn−1 can be expressed as follows.

tn1 ¼ V=M 1ð Þ ffiffiffiffiffiffiffiffiffiffiffi 2 n 1 r (2) where n denotes number, while V and M represent the variance and mean of defects per wafer, respectively. Besides the V/M index, other clustering indices based on the quadrat method also have been developed. The performance of these clustering indices varies markedly with grid size, grid determination method and grid shape given a particular defect distribution.

Jun et al. [3] designed a clustering index (CI_J) to assess the degree of defect clustering on a wafer. The simulation results showed that the CI_J represents defect clustering more accurately than theα from the well-known negative binomial distribution. TheCI_Jvalue is independent of the chip area, requires no assumptions regarding the defect distribution, and is easy to calculate. Suppose thatn defects exist on a wafer, employing thex and y coordinates, the ith defect can be denoted using (x_i, y_i), where i=1, 2, ..., n. Accordingly, all of the defects can be projected onto thex andy coordinates. Figure1illustrates examples of defect maps and the corresponding projectedx and y coordinates. The defect intervals on the x and y coordinates are defined as follows [3].

Vi¼ xi xi1; i ¼ 1; 2; :::; n (3)

Wi¼ yi yi1; i ¼ 1; 2; :::; n (4)

wherex₀=y₀=0 andx_i andy_i_{represent the} _{ith and the yth} smallest values, respectively. From Fig.1d, defect cluster-ing tends to display burstiness or clumps in the x and y coordinates. This type of defect cluster creates a large Table 1 α values from the negative binomial model and the

corresponding imitated yield models [1]

Clustering α value Yield model

None 10 or greater Poisson

Some 4.2 Murphy

Some 3 Dingwall

Much 1 Seeds

Fig. 1 a–d Defect maps and projectedx and y coordinates [3] 706

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variance regarding the defect intervals. However, Fig. 1b and c show that burstiness on either thex- or y-axis does not necessarily represent clustered defects. Based on the relationship between the defect clustering and cluster

burstiness or clumps in thex and y coordinates, Jun et al. [3] designed theCI_J, as follows.

CIJ ¼ min S 2 v V2; S2 w W2 (5) where V ¼PV_in; S2 v ¼ P Vi V 2. n 1 ð Þ; W ¼ P Win and Sw2 ¼PWi W2 . n 1 ð Þ: If the defect

locations are assumed to be uniform random variables, the CIJapproaches 1. A largeCI_Jvalue corresponds to highly clustered defects. Despite the advantage that the CI_J requires no statistical assumptions regarding defect distri-bution, it still suffers some drawbacks, as shown in Fig.2. Figure 2a,b display consistent CI_J values based on Eq. 5, but display different types of defect distributions. Specifically, one distribution is randomly distributed and the other has a patterned distribution. Accordingly, theCI_J cannot accurately assess the degree of defect clustering. This inability occurs because theCI_Jvalues are calculated based on the projected defect coordinates associated with thex- and y-axes. Consequently, defect locations on a wafer cannot be depicted accurately. In this case, the CI_J could underestimate the degree of defect clustering.

2.2 Rotating axis technique [7]

Figure 3 presents an example of the application of the rotating axis technique. From Fig. 3, some points are located in a two-dimensional space (that is,x1andx2) and a new coordinate x1* is obtained by rotating the x1 axis counterclockwise using θ0, where 0 θ 180. Accord-ingly, these points in two-dimensional space can be projected into the new axis x1*. The corresponding coordinates are determined as follows.

x

1 ¼ cos θ x1þ sin θ x2 (6)

a

b

Fig. 2 a, b Defect maps and projectedx and y coordinates

Fig. 3 Expression of rotating axis [7]

Fig. 4 Wafer map illustration. a Defect distribution I. b Defect distribution II

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3 Developed clustering index procedure

The developedCI_Mincludes the following five steps. Step 1.

Project the defect coordinates (x_i, y_i) into a new axis obtained by rotating the x-axis counterclockwise usingθ0.

Suppose that a wafer hasn defects, and (x_i,y_i) denotes thex and y coordinates of the ith defect location in a two-dimensional space,i=1, ...,n. These n defects then can be projected onto a new axis x_i;θobtained by rotating thex-axis counterclockwise using θ0. The new coordinates for theith defect with respect to θ then can be calculated as follows [7].

x

i;θ¼ cos θ xiþ sin θ yi (7)

where i denotes the ith defect and θ represents a rotating angle, where 0 θ 180.

Step 2.

Sort thex_i;θ values in ascending order and calculate the intervals between each adjacent coordinate valuex_i;θ.

The intervals between each adjacent coordinate value x

i;θthen can be calculated as follows.

vi;θ¼ xð Þi;θ xði1;θÞ (8)

where x_{ð Þ}_0;θ ¼ 0 and v_i;θ represents the ith interval betweenx_{ð Þ}_i;θ andx_ð_i1;θ_Þ.

Step 3.

Calculate the squared coefficient of variation (SCV) forv_i;θ:

The SCV forv_i;θ can be determined as follows. SCVθ¼s 2 v;θ v2 θ (9)

where SCV_θ represents the squared coefficient of variation for v_i;θ_; v_θ¼ Pn

i¼1vi;θ n; and s2 v;θ¼ P n i¼1ðvi;θ vθÞ 2 n 1 ð Þ.

Fig. 5 Diagram ofSCV versus θ. a Defect distribution I. b Defect distribution II -80 -80 -60 -60 -40 -40 -20 -20 0 0 20 20 40 40 60 60 80 80 Y X

Bulls Eye Pattern

-80 -80 -60 -60 -40 -40 -20 -20 0 0 20 20 40 40 60 60 80 80 Y Y X X

Bottom Pattern Crescent Moon Pattern

100 100 -100 -100 -80 -80 -60 -60 -40 -40 -20 -20 0 0 20 20 40 40 60 60 80 80 100 100 -100 -100

Fig. 6 Defect clustering maps for the bull’s eye, bottom, and crescent moon patterns 708

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Step 4.

Change the angle ofθ and calculate the corresponding θ=10_value.

The number of 180 SCV_θ values with respect to θ, increased byθ=10, can be obtained through Steps 1–3. Step 5.

Calculate the developedCI_M.

According to theSCV_θ values obtained from Step 4, the average SCV_θ value determines the clustering indexCI_M, as follows. CIM ¼ P 180 θ¼0SCVθ 180 (10)

whereCI_Mrepresents the developed clustering index. A largerCI_Mvalue indicates a stronger degree of defect clustering formed on a wafer.

4 Comparison of theCI_Mand CI_J

The CI_M and CI_J were compared using Fig. 4 to

demonstrate the superiority of CI_M compared to CI_J. Figure 4a,b display clustered defects and a random defect distribution on respective wafer maps. The method

devel-oped by Jun et al. [3] produces a consistentCI_J value of 0.8354 while quantifying the degree of defects regarding these two wafer maps. From theCI_Jproperties, the uniform defect distribution yields aCI_Jvalue of nearly 1. These two wafers exhibit consistent defect distributions. However, the defect distribution differs significantly between the two wafers. Consequently, theCI_J cannot accurately quantify defect clustering on a wafer.

The CI_Mwas also used to assess the degree of defect clustering in Fig.4a,b. From the procedure developed for establishing theCI_M, the 180SCV_θ values can be obtained via Steps 1–4. Figure 5a,b display the diagrams of corresponding SCV_θ values versus various θ values. Clearly, theSCV_θ values vary withθ. The maximum SCV_θ values in Fig.5a,b are 41 and 3.7, and the corresponding CIMare 6.6925 and 1.6632, respectively. Accordingly, this study concludes that the degree of defect clustering in Fig. 4a exceeds that in Fig. 4b. The developed CI_M therefore can more accurately assess defect clustering on a wafer thanCI_J. Notably,CI_Jis determined usingCI_M, with θ=00_and_θ=900_{, using the minimum squared coefficient of} variation. The proposed procedure not only explains the SCV0andSCV90, but also considers 0 θ 180 to obtain the CI_M for assessing defect clustering on a wafer. Therefore, the CI_M represents defect clustering more accurately than other indices.

Table 2 Factors/levels of the

designed experiment Pattern Number of defects Chip size Defect percentage

Random 20, 50, 100, 150, 200, 300 5, 10, 20 –

Bull’s eye pattern 20, 50, 100, 150, 200, 300 5, 10, 20 75%, 90%, 95%

Bottom pattern Crescent moon pattern

Plot of Means (unweighted) Defect Point Main Effect

Defect Count V/M 0.5 0.7 0.9 1.1 1.3 1.5 20 50 100 150 200 300

Defect Count CI 0.5 0.7 0.9 1.1 1.3 1.5 20 50 100 150 200 300

Defect Count ALPHA -15 -5 5 15 25 35 20 50 100 150 200 300

a

b

c

d

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5 Simulation experiments

An efficient defect-clustering index must be robust against various chip size and defect distributions and correctly assess defect clustering degree. Accordingly, an experi-ment design involving three common defect clustering patterns, including the bull’s eye pattern, bottom pattern, crescent moon pattern [2] and no pattern (namely a random defect distribution following a Poisson distribution) were simulated on eight inch wafers to demonstrate the effec-tiveness of the developedCI_M. Figure6displays these three common defect clustering patterns. The experiment design included three factors, six levels for the number of defects,

three levels for the chip size and three levels for the defect percentage contained in the clustering area. Table 2 lists these factors and levels, where 234 factor level combina-tions (runs) were generated using this approach; ten replicate runs were conducted, and 2,340 runs with different types of wafer maps thus were simulated. Accordingly, the α, CI_J, and CI_M values for each run were determined using Eqs.1,5and10, respectively.

Figures7and8 display the defect count and chip size effects associated with a random defect distribution. From Figs. 7a–c and 8a–c, the V/M, CI_J and CI_M indices displayed the exceptional properties of being insensitive to defect count and chip size. The values of all of these indices Plot of Means (unweighted)

Chip Size Main Effect

a

Chip Size V/M 0.5 0.7 0.9 1.1 1.3 1.5 5 10 20

Plot of Means (unweighted) Chip Size Main Effect

b

Chip Size CI 0.5 0.7 0.9 1.1 1.3 1.5 5 10 20

c

Chip Size CI 0.5 0.7 0.9 1.1 1.3 1.5 5 10 20

d

Chip Size ALPHA -15 -5 5 15 25 35 5 10 20

Fig. 8 a–d Effect of chip size for a random defect distribution on four clustering indices

a

Defect Count V/M 0 5 10 15 20 25 30 35 20 50 100 150 200 300

b

Defect Count CI 0 5 10 15 20 20 50 100 150 200 300

c

Defect Count CI 0 5 10 15 20 20 50 100 150 200 300

d

Defect Count ALPHA 0.048 0.050 0.052 0.054 0.056 0.058 0.060 20 50 100 150 200 300

Fig. 9 a–d Effect of defect count for the bull’s eye pattern on four clustering indices 710

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approached one. However, from Figs. 7d and 8d, the α value had a strongly scattered pattern that varied with the defect count and chip size and sometimes was negative. Theα value is inadequate for assessing the degree of defect clustering and is not robust to the defect count and chip size when the defects are randomly distributed.

Figures9,10and11display the defect count, chip size and defect percentage effects associated with the bull’s eye pattern, respectively. From Figs.9a and11a, theV/M index value increased with clustering degree. However, From

Fig.10a, theV/M index value altered with the chip size and did not display exceptional properties for a robust defect-clustering index. Figures 9d and 11d display that the α index values are located within a narrow range, creating difficulty in assessing the degree of defect clustering. From Fig. 10d, the α index varied with chip size and did not display exceptional properties. Figures 9c,d and 11c,d display that the CI_J and CI_R can efficiently assess the degree of defect clustering. Moreover, Fig.10c and d reveal that both indices are robust to the chip size.

Plot of Means (unweighted) CHIPSIZE Main Effect

a

Chip Size V/ M 0 5 10 15 20 25 30 35 5 10 20

b

Chip Size CI 0 5 10 15 20 5 10 20

c

Chip Size CI 0 5 10 15 20 5 10 20

d

Chip Size ALPH A 0.02 0.04 0.06 0.08 0.10 5 10 20

Fig. 10 a–d Effect of chip size for the bull’s eye pattern on four clustering indices

Plot of Means (unweighted) Percent Main Effect

Percent V/M 11 13 15 17 19 75 90 95

Percent CI 0 5 10 15 20 75 90 95

Percent ALPHA 0.040 0.045 0.050 0.055 0.060 0.065 75 90 95

a

b

c

d

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Figures12,13and14display the defect count, chip size and defect percentage effects associated with the bottom pattern, respectively. From Figs. 12c and 14c, the CI_M values increased with the defect count and percentage in the clustering area. The CI_M was insensitive to the chip size. The CI_M can accurately assess the degree of defect clustering. From Figs.12b and14b, although theCI_Jvalues are insensitive to chip size, this index is insensitive to the percentage of defects in the clustering area. This index thus

does not satisfy the required properties. From Fig.13a,d, the V/M and α indices values vary with chip size. Consequently, CI_M is superior to CI_J, V/M and α for assessing defect clustering associated with the bottom pattern.

Figures15, 16and 17 illustrate the defect count, chip size and defect percentage effects associated with the crescent moon pattern, respectively. From Figs. 15c and 17c, theCI_Mvalues are insensitive to the chip size. TheCI_M

a

b

c

d

Defect Count V/M 0 2 4 6 8 10 20 50 100 150 200 300

Defect Count CI 0 2 4 6 8 10 20 50 100 150 200 300

Defect Point ALPHA 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 20 50 100 150 200 300

Fig. 12 a–d Effect of defect count for the bottom pattern on four clustering indices

Chip Size V/M 0 2 4 6 8 10 5 10 20

Chip Size CI 0 2 4 6 8 10 5 10 20

Chip Size ALPHA 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 5 10 20

a

b

c

d

Fig. 13 a–d Effect of chip size for the bottom pattern on four clustering indices 712

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values increase with the defect count and percentage of defects contained in the clustering area. Consequently,CI_M can accurately assess the degree of defect clustering in the crescent moon pattern. Figures 15b and17b indicate that the CI_J values approach one. Accordingly, this study concludes that the defects are randomly distributed based on the discrimination ofCI_J. However, actual defects on a

wafer are not randomly distributed (instead displaying a crescent moon pattern). Therefore,CI_Jcannot discriminate the degree of defect clustering and makes inaccurate judgments regarding the crescent moon pattern. From Fig.16a,d, the values of theV/M and α indices vary with the chip size. Therefore, theCI_Mis superior to theCI_J,V/M

Percent V/M 0 2 4 6 8 10 75 90 95

Percent CI J 0 2 4 6 8 10 75 90 95

Percent CI M 0 2 4 6 8 10 75 90 95

Percent ALPHA 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 75 90 95

a

b

c

d

Fig. 14 a–d Effect of defect percentage for the bottom pattern on four clustering indices

a

b

c

d

Defect Count V/M 0 1 2 3 4 5 6 7 20 50 100 150 200 300

Defect Count CI 0 1 2 3 4 5 6 7 20 50 100 150 200 300

Defect Count ALPHA -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 20 50 100 150 200 300

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and α indices for assessing defect clustering associated with the crescent moon pattern.

Table 3 summarizes comparisons of these four defect clustering indices, namely CI_M, CI_J, α and V/M, for assessing the random pattern, bull’s eye pattern, bottom pattern and crescent moon pattern under three designed factors. To summarize, CI_M is superior to the other three

indices. Consequently, the effectiveness of the developed clustering index for assessing the degree of defect clustering on a wafer is confirmed.

Chip Size V/M 0 1 2 3 4 5 6 7 5 10 20

Chip Size CI 0 1 2 3 4 5 6 7 5 10 20

Chip Size ALPHA -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 5 10 20

a

b

c

d

Fig. 16 a–d Effect of chip size for the crescent moon pattern on four clustering indices

Percent V/M 0 1 2 3 4 5 6 7 75 90 95

Percent CIJ 0 1 2 3 4 5 6 7 75 90 95

Percent CI M 0 1 2 3 4 5 6 7 75 90 95

Percent ALPHA -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 75 90 95

a

b

c

d

Fig. 17 a–d Effect of defect percentage for the crescent moon pattern on four clustering indices 714

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6 Conclusion

The defect distribution, including the number of defects and the clustering pattern on a wafer, significantly influences wafer yield. Besides the number of defects, defect clustering also must be monitored in IC fabrication process control to enhance wafer yield. Defect clustering tends to increase with wafer size. This study developed a CIM based on the rotating axis technique to assess the degree of defect clustering. TheCI_Mand other extensively used clustering indices were compared using a simulation experiment to confirm the superiority of theCI_M.

To summarize, this study possesses the following merits: 1. The CI_M has the same advantage as the CI_J in not requiring any statistical assumptions regarding the defects. TheCI_Mis also sensitive to the percentage of defects in the clustering area.

2. TheCI_Mvalues assess the degree of defect clustering accurately, allowing engineers to accurately monitor defects in the IC fabrication process.

3. The CI_M calculation is easy and simple to use for engineers with minimal statistical knowledge.

4. The developed CI_Mcan be generalized to any wafer size.

References

1. Cunningham JA (1990) The use and evaluation of yield models in integrated circuit manufacturing. IEEE T Semiconduct M 3 (2):60–71

2. Friedman DJ, Hansen MH, Nair VN, James DA (1997) Model-free estimation of defect clustering in integrated circuit fabrication. IEEE T Semiconduct M 10(3):344–359

3. Jun CH, Hong Y, Kim SY, Park KS, Park H (1999) A simulation-based semiconductor chip yield model incorporating a new defect cluster index. Microelectron Reliab 39:451–456 4. Rogers A (1974) Statistical analysis of spatial dispersion: the

quadrat method. Pion, London

5. Raghavachari M, Srinivasan A, Sullo P (1997) Poisson mixture yield models for integrated circuits: a critical review. Micro-electron Reliab 37(4):565–580

6. Stapper CH (1973) Defect density distribution for LSI yield calculations. IEEE Trans Electron Dev ED-20:655–657 7. Sharma S (1996) Applied multivariate techniques. Wiley, New

York

Table 3 Comparisons of four defect clustering indices

Number of defects Chip size Defects percentage

CIM CIJ α V/M CIM CIJ α V/M CIM CIJ α V/M

Random pattern ○ ○ × □ ○ ○ × □ – – – –

Bull’s eye pattern ○ ○ × ○ ○ □ × × ○ ○ × ○

Bottom pattern ○

▵

×

▵

○ ○ × × ○ × × ▵

Crescent moon pattern ○ × × ○ ○ ○ × × ○ × × ▵

The○ symbol represents the index with exceptional properties for assessing defect clustering The

▵

symbol represents the index with acceptable properties for assessing defect clustering The × symbol represents the index with poor properties for assessing defect clustering