lecture 11 attribute charts

8/2/2019 Lecture 11 Attribute Charts

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Lecture 11: Attribute Charts

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P-chart (fraction non-conforming)C-chart (number of defects)

U-chart (non-conformities per unit)The rest of the magnificent seven

Control Charts for Attributes


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Yield Control

30201000

20

40

60

80

100

Months of Production

30201000

20

40

60

80

100

Yield


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The fraction non-conforming

The most inexpensive statistic is the yield of the productionline.

Yield is related to the ratio of defective vs. non-defective,conforming vs. non-conforming or functional vs. non-functional.

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

P{D = X} = nx

px(1-p)n-x x = 0,1,...,n

mean npvariance np(1-p)

the sample fract ion p=Dn

mean p

variance p(1-p)n

The P chart is based on the binomial distribution:


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The P-chart (cont.)

p =

m

i=1

p i

m

mean p

variancep(1-p)

nm

(in this and the following discussion, "n" is the number ofsamples in each group and "m" is the number of groupsthat we use in order to determine the control limits)

(in this and the following discussion, "n" is the number ofsamples in each group and "m" is the number of groupsthat we use in order to determine the control limits)

p must be estimated. Limits are set at +/- 3 sigma.


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Designing the P-Chart

p can be estimated:

p i =D in

i = 1,...,m (m = 20 ~25)

p =

m

i=1

p i

m

In general, the control limits of a chart are:

UCL= + k

LCL= - k

where k is typically set to 3.

These formulae give us the limits for the P-Chart (using thebinomial distribution of the variable):

UCL = p + 3p(1-p)

n

LCL = p - 3p(1-p)

n


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Example: Defectives (1.0 minus yield) Chart

"Out of control points" must be explained and eliminated before werecalculate the control limits.

This means that setting the control limits is an iterative process!

Special patterns must also be explained.

"Out of control points" must be explained and eliminated before werecalculate the control limits.

This means that setting the control limits is an iterative process!

Special patterns must also be explained.

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0.1

0.2

0.3

0.4

0.5

Count

LCL 0.053

0.232

UCL 0.411

%

non-conform

ing


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Example (cont.)

After the original problems have been corrected, thelimits must be evaluated again.


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Operating Characteristic of P-Chart

if n large and np(1-p) >> 1, then

P{D = x} =n

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

2np(1-p)e

-(x-np) 2

2np(1-p)

In order to calculate type I and II errors of the P-chart weneed a convenient statistic.

Normal approximation to the binomial (DeMoivre-Laplace):

In other words, the fraction nonconforming can be treated ashaving a nice normal distribution! (with and as given).

This can be used to set frequency, sample size and controllimits. Also to calculate the OC.


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Binomial distribution and the Normal

bin 10, 0.1

0.0

1.0

2.0

3.0

4.0

bin 100, 0.5

35

40

45

50

55

60

65

bin 5000 0.007

15

20

2530

35

40

45

50

As sample size increases, the Normal approximation becomes reasonable...As sample size increases, the Normal approximation becomes reasonable...


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Designing the P-Chart

= P { D < n UCL /p } - P { D < n LCL /p }

Assuming that the discrete distribution of x can beapproximated by a continuous normal distribution asshown, then we may:

choose n so that we get at least one defective with0.95 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:


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The Operating Characteristic Curve (cont.)

p = 0.20,LCL=0.0303,UCL=0.3697

The OCC can be calculated two distributions areequivalent and np=).


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In reality, p changes over time

(data from the Berkeley Competitive Semiconductor Manufacturing Study)


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The C-Chart

UCL = c + 3 ccenter at c

LCL = c - 3 c

p(x) = e-c cx

x!x = 0,1,2,..

= c, 2 = c

Sometimes we want to actually countthe number of defects.This gives us more information about the process.

The basic assumption is that defects "arrive" according to a

Poisson model:

This assumes that defects are independent and that theyarrive uniformly over time and space. Under theseassumptions:

and c can be estimated from measurements.


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Poisson and the Normalpoisson 2

0

2

4

6

8

10

poisson 20

10

20

30

poisson 100

70

80

90

100

110

120

130

As the mean increases, the Normal approximation becomes reasonable...As the mean increases, the Normal approximation becomes reasonable...


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Example: "Filter" wafers used in yield model

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0.1

0.2

0.3

0.4

0.5Fraction Nonconforming (P-chart)

FractionNonconforming

LCL 0.157

0.306

UCL 0.454

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100

200

Defect Count (C-chart)

Wafer No

NumberofDe

fects

LCL 48.26

74.08

UCL 99.90


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Counting particles

Scanning a blanket monitor wafer.

Detects position and approximate size of particle.

x

y


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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 verydifficult 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 we use (and we might have to use morethan one), must follow a known, usable distribution.

Whatever we use (and we might have to use morethan one), must follow a known, usable distribution.


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In Situ Particle Monitoring Technology

Laser light scattering system for detecting particles inexhaust flow. Sensor placed down stream from

valves to prevent corrosion.

chamber

Laser

Detector

to pump

Assumed to measure the particle concentration invacuum


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Progression of scatter plots over timeThe endpoint detector failed during the ninth lot, and wasdetected 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 = c/n. It can beshown that u obeys a Poisson "type" distribution with:

where is the estimated value of the unknown u.

The sample size n may vary. This can easily beaccommodated.

u = u, u2 = unso

UCL =u + 3

u

nLCL =u - 3 un

u


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The Averaging Effect of the u-chart

poisson 2

0

2

4

6

8

10

Quantiles

Moments

average 5

0.0

1.0

2.0

3.0

4.0

5.0

Quantiles

Moments

By exploiting the central limit theorem, if small-sample poisson variablescan be made to approach normal by grouping and averaging

By exploiting the central limit theorem, if small-sample poisson variablescan be made to approach normal by grouping and averaging


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

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0.1

0.2

0.3

0.4

0.5

Fraction Nonconforming (P-chart)

FractionNonconforming

LCL 0.157

0.306

UCL 0.454

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100

200

Defect Count (C-chart)

NumberofDefects

LCL 48.26

74.08

UCL 99.90

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2

3

4

5

6

Defect Density (U-chart)

Wafer No

DefectsperUnit

LCL 1.82

2.79

UCL 3.76


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The Use of the Control Chart

The control chart is in general a part of the feedback loopfor process improvement and control.

ProcessInput Output

Measurement System

Verify andfollow up

Implementcorrective

action

Detectassignablecause

Identify rootcause of problem

S


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Choosing a control chart...

...depends very much on the analysis that we arepursuing. In general, the control chart is only a small

part of a procedure that involves a number of statisticaland 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

figure 3.1 pp 21 Kume

Typically, a small number of defect types is responsiblefor the largest part of yield loss.

The most cost effective way to improve the yield is to

identify these defect types.

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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

viouslayer

sproblem

s

scratches

ntaminatio

n

edcontacts

rnbridging

separticles

othe

rs

0

20

40

60

80

100

occurence

cummulative

DCMOS Defect Classification

Percentage

Though the defect classification by type is fairly easy, the

classification by cause is not...

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Cause and Effect Diagrams

figure 4.1 pp 27 Kume

(Also known as Ishikawa,fish boneor influencediagrams.)

Creating such a diagram requires good understanding ofthe process.

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An Actual Example

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Example: DCMOS Cause and Effect Diagram

Past Steps Parametric Control Particulate Control

Operator Handling Contamination Control

inspection

rec. handling

transport

loading

chemicals

utilities

cassettes

equipment

cleaning

vendor

Wafers

Defect

skill

experience

vendor

calibration

SPC SPC

boxes

shift

monitoring

automation

filters

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Lecture 11: Attribute Charts 36

Example: Pareto Analysis of DCMOS (cont)

equip

mnet

utilities

lo

ading

inspe

ction

smiffboxes

o

thers

0

20

40

60

80

100

occurence

cummulative

DCMOS Defect Causes

percentage

Once classification by cause has been completed,we can choose the first target for improvement.

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Defect Control

In general, statistical tools like control charts must becombined 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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