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Control Charts for Attributes

P-chart (fraction non-conforming) C-chart (number of defects) U-chart (non-conformities per unit) The rest of the “magnificent seven”

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Yield

Yield Control 100

100

80

80

60

60

40

40

20

20

0

0

10

20

30

0

0

10

20

30

Months of Production

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The fraction non-conforming The most inexpensive statistic is the yield of the production line. Yield is related to the ratio of defective vs. non-defective, conforming vs. non-conforming or functional vs. nonfunctional. We often measure: • Fraction non-conforming (P) • Number of defects on product (C) • Average number of non-conformities per unit area (U)

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The P-Chart The P chart is based on the binomial distribution: n-x n x P{D = X} = x = 0,1,...,n p (1-p) x mean np variance np(1-p) the sample fraction p= D n mean p variance p(1-p) n

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The P-chart (cont.) p must be estimated. Limits are set at +/- 3 sigma.

m

Σ pi

p=

i=1

m

mean p variance p(1-p) nm

(in (in this this and and the the following following discussion, discussion, "n" "n" isis the the number number of of samples samples in in each each group group and and "m" "m" isis the the number number of of groups groups that thatwe weuse usein inorder orderto todetermine determinethe thecontrol controllimits) limits) Lecture 11: Attribute Charts

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Designing the P-Chart In general, the control limits of a chart are: UCL= µ + k σ LCL= µ - k σ where k is typically set to 3. These formulae give us the limits for the P-Chart (using the binomial distribution of the variable): p can be estimated: p(1-p) n p(1-p) LCL = p - 3 n

UCL = p + 3

Di pi = i = 1,...,m (m = 20 ~25) n m

Σ pi p=

Lecture 11: Attribute Charts

i=1

m 6

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Example: Defectives (1.0 minus yield) Chart

% non-conforming

0.5 UCL 0.411

0.4 0.3

0.232

0.2 0.1

LCL 0.053 0.0

0

10

20

30

Count

"Out "Out of of control control points" points" must must be be explained explained and and eliminated eliminated before before we we recalculate recalculatethe thecontrol controllimits. limits. This Thismeans meansthat thatsetting settingthe thecontrol controllimits limitsisisan aniterative iterativeprocess! process! Special Specialpatterns patternsmust mustalso alsobe beexplained. explained. Lecture 11: Attribute Charts

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Example (cont.) After the original problems have been corrected, the limits must be evaluated again.

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Operating Characteristic of P-Chart In order to calculate type I and II errors of the P-chart we need a convenient statistic. Normal approximation to the binomial (DeMoivre-Laplace):

if n large and np(1-p) >> 1, then P{D = x} =

n p x (1-p) (n-x) ~ x

1 2πnp(1-p)

(x-np) 2 e 2np(1-p)

In other words, the fraction nonconforming can be treated as having a nice normal distribution! (with μ and σ as given). This can be used to set frequency, sample size and control limits. Also to calculate the OC.

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Binomial distribution and the Normal bin 10, 0.1

bin 100, 0.5

bin 5000 0.007

4.0

3.0

2.0

65

50

60

45

55

40

50 45

1.0

40 35

0.0

35 30 25 20 15

As Assample samplesize sizeincreases, increases,the theNormal Normalapproximation approximationbecomes becomesreasonable... reasonable...

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Designing the P-Chart Assuming that the discrete distribution of x can be approximated by a continuous normal distribution as shown, then we may: • choose n so that we get at least one defective with 0.95 probability. • choose n so that a given shift is detected with 0.50 probability. or • choose n so that we get a positive LCL. Then, the operating characteristic can be drawn from: β = P { D < n UCL /p } - P { D < n LCL /p } Lecture 11: Attribute Charts

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The Operating Characteristic Curve (cont.) The OCC can be calculated two distributions are equivalent and np=λ).

p = 0.20, LCL=0.0303, UCL=0.3697

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In reality, p changes over time

(data from the Berkeley Competitive Semiconductor Manufacturing Study) Lecture 11: Attribute Charts

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The C-Chart Sometimes we want to actually count the number of defects. This gives us more information about the process. The basic assumption is that defects "arrive" according to a Poisson model: -c cx e x = 0,1,2,.. p(x) = x! μ = c, σ 2 = c This assumes that defects are independent and that they arrive uniformly over time and space. Under these assumptions: UCL = c + 3 c center at c LCL = c - 3 c and c can be estimated from measurements. Lecture 11: Attribute Charts

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Poisson and the Normal poisson 2

poisson 20

poisson 100 130

10 30 8

120 110

6 20

100

4

90

2

80 10

70

0

As Asthe themean meanincreases, increases,the theNormal Normalapproximation approximationbecomes becomesreasonable... reasonable...

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Example: "Filter" wafers used in yield model Fraction Nonconforming (P-chart) 0.5

UCL 0.454

Fraction Nonconforming

0.4

¯ 0.306

0.3

0.2

LCL 0.157 0.1 0

2

4

6

8

10

12

14

Defect Count (C-chart)

Number of Defects

200

UCL 99.90

100

Ý 74.08 LCL 48.26

0 0

2

4

6

8

10

12

14

Wafer No

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Counting particles Scanning a “blanket” monitor wafer. Detects position and approximate size of particle. y x

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Scanning a product wafer

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Typical Spatial Distributions

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The Problem with Wafer Maps Wafer maps often contain information that is very difficult to enumerate

A simple particle count cannot convey what is happening.

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Special Wafer Scan Statistics for SPC applications • Particle Count • Particle Count by Size (histogram) • Particle Density • Particle Density variation by sub area (clustering) • Cluster Count • Cluster Classification • Background Count Whatever Whatever we we use use (and (and we we might might have have to to use use more more than thanone), one),must mustfollow followaaknown, known,usable usabledistribution. distribution. Lecture 11: Attribute Charts

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In Situ Particle Monitoring Technology Laser light scattering system for detecting particles in exhaust flow. Sensor placed down stream from valves to prevent corrosion. Laser chamber to pump

Detector Assumed to measure the particle concentration in vacuum Lecture 11: Attribute Charts

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Progression of scatter plots over time The endpoint detector failed during the ninth lot, and was detected during the tenth lot.

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Time series of ISPM counts vs. Wafer Scans

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The U-Chart We could condense the information and avoid outliers by using the “average” defect density u = Σc/n. It can be shown that u obeys a Poisson "type" distribution with:

μu = u, σ2u = u n so UCL =u + 3 u n LCL =u - 3 u n where

u is the estimated value of the unknown u.

The sample size n may vary. This can easily be accommodated.

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The Averaging Effect of the u-chart poisson 2

average 5 5.0

10 4.0 8 6

3.0

4

2.0

2

1.0

0

0.0

Quantiles

Quantiles

Moments

Moments

By Byexploiting exploitingthe thecentral centrallimit limittheorem, theorem,ififsmall-sample small-samplepoisson poissonvariables variables can canbe bemade madeto toapproach approachnormal normalby bygrouping groupingand andaveraging averaging Lecture 11: Attribute Charts

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Filter wafer data for yield models (CMOS-1): 0.5

Fraction Nonconforming (P-chart) UCL 0.454

Fraction Nonconforming

0.4

¯ 0.306

0.3

0.2

LCL 0.157 0.1 0

2

4

6

8

10

12

14

200

Number of Defects

Defect Count (C-chart)

UCL 99.90

100

Ý 74.08 LCL 48.26

0 0

2

4

6

8

10

12

14

6

Defect Density (U-chart)

Defects per Unit

5

4

UCL 3.76 3

× 2.79

2

LCL 1.82

1 0

2

4

6

8

10

12

14

Wafer No

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The Use of the Control Chart The control chart is in general a part of the feedback loop for process improvement and control. Input

Output Process

Measurement System

Verify and follow up Implement corrective action Lecture 11: Attribute Charts

Detect assignable cause Identify root cause of problem 28

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Choosing a control chart... ...depends very much on the analysis that we are pursuing. In general, the control chart is only a small part of a procedure that involves a number of statistical and engineering tools, such as: • experimental design • trial and error • pareto diagrams • influence diagrams • charting of critical parameters

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The Pareto Diagram in Defect Analysis Typically, a small number of defect types is responsible for the largest part of yield loss. The most cost effective way to improve the yield is to identify these defect types.

figure 3.1 pp 21 Kume

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Pareto Diagrams (cont) Diagrams by Phenomena • defect types (pinholes, scratches, shorts,...) • defect location (boat, lot and wafer maps...) • test pattern (continuity etc.) Diagrams by Causes • operator (shift, group,...) • machine (equipment, tools,...) • raw material (wafer vendor, chemicals,...) • processing method (conditions, recipes,...)

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Example: Pareto Analysis of DCMOS Process DCMOS Defect Classification 100

80

Percentage

60 occurence cummulative 40

20

others

se particles

ern bridging

sed contacts

ontamination

s scratches

ss problems

evious layer

0

Though the defect classification by type is fairly easy, the classification by cause is not... Lecture 11: Attribute Charts

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Cause and Effect Diagrams (Also known as Ishikawa,fish bone or influence diagrams.)

figure 4.1 pp 27 Kume

Creating such a diagram requires good understanding of the process.

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An Actual Example

Lecture 11: Attribute Charts

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Example: DCMOS Cause and Effect Diagram Past Steps

Parametric Control

Particulate Control

rec. handling inspection SPC

automation

calibration

cassettes

SPC

equipment

monitoring cleaning Defect skill vendor experience shift

Operator Lecture 11: Attribute Charts

chemicals loading transport vendor

Wafers

boxes Handling

utilities filters

Contamination Control 35

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Example: Pareto Analysis of DCMOS (cont) DCMOS Defect Causes 100

80

percentage

60 occurence cummulative 40

20

others

smiff boxes

inspection

loading

utilities

equipmnet

0

Once classification by cause has been completed, we can choose the first target for improvement. Lecture 11: Attribute Charts

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Defect Control In general, statistical tools like control charts must be combined with the rest of the "magnificent seven": • Histograms • Check Sheet • Pareto Chart • Cause and effect diagrams • Defect Concentration Diagram • Scatter Diagram • Control Chart

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Logic Defect Density is also on the decline

Y = [ (1-e-AD)/AD ]2

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What Drives Yield Learning Speed?

Lecture 11: Attribute Charts

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