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

30

20

10

0

0

20

40

60

80

100

Months of Production

30

20

10

0

0

20

40

60

80

100

Yield

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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. 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 } =

n

x

p

x

(1-p)

n-x

x = 0,1,...,n

mean np

variance np(1-p)

the sample fraction p=

D

n

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

Σ

i=1

p

i

m

mean p

variance

p(1-p)

nm

(in this and the following discussion, "n" is the number of

samples in each group and "m" is the number of groups
that we use in order to determine the control limits)

(in this and the following discussion, "n" is the number of

samples in each group and "m" is the number of groups
that 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

i

n

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

p =

m

Σ

i=1

p

i

m

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):

UCL = p + 3

p(1-p)

n

LCL = p - 3

p(1-p)

n

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Example: Defectives (1.0 minus yield) Chart

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

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

Special patterns must also be explained.

30

20

10

0

0.0

0.1

0.2

0.3

0.4

0.5

Count

LCL 0.053

0.232

UCL 0.411

% non-conforming

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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

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

P{D = x} =

n

x

p

x

(1-p)

(n-x)

~

1

2

πnp(1-p)

e

-

(x-np)

2

2np(1-p)

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):

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

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

25

30

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

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

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:

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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 are
equivalent and np=

λ).

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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 c

center at c

LCL = c - 3 c

p(x) = e

-c

c

x

x!

x = 0,1,2,..

μ = c, σ

2

= c

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:

This assumes that defects are independent and that they
arrive uniformly over time and space. Under these
assumptions:

and c can be estimated from measurements.

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Poisson and the Normal

poisson 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

14

12

10

8

6

4

2

0

0.1

0.2

0.3

0.4

0.5

Fraction Nonconforming (P-chart)

Fract

ion Nonconf

orming

LCL 0.157

¯ 0.306

UCL 0.454

14

12

10

8

6

4

2

0

0

100

200

Defect Count (C-chart)

Wafer No

Number of Defects

LCL 48.26

Ý 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 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 we use (and we might have to use more
than one), must follow a known, usable distribution.

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

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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.

chamber

Laser

Detector

to pump

Assumed to measure the particle concentration in
vacuum

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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 =

Σ

c/n. It can be

shown 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 be
accommodated.

μ

u

= u,

σ

u

2

= u

n

so

UCL = u + 3 u

n

LCL = u - 3 u

n

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 variables
can be made to approach normal by grouping and averaging

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

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

14

12

10

8

6

4

2

0

0.1

0.2

0.3

0.4

0.5

Fraction Nonconforming (P-chart)

Fr

action Nonconfor

ming

LCL 0.157

¯ 0.306

UCL 0.454

14

12

10

8

6

4

2

0

0

100

200

Defect Count (C-chart)

Number of

Def

ect

s

LCL 48.26

Ý 74.08

UCL 99.90

14

12

10

8

6

4

2

0

1

2

3

4

5

6

Defect Density (U-chart)

Wafer No

Def

ect

s p

e

r Un

it

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 loop
for process improvement and control.

Process

Input

Output

Measurement System

Verify and
follow up

Implement
corrective
action

Detect
assignable
cause

Identify root
cause of problem

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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

figure 3.1 pp 21 Kume

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.

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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

e

vio

u

s

la

ye

r

s

s

problems

s

s

c

ratc

hes

o

ntamination

s

ed c

ontac

ts

e

rn bridging

se particles

others

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 bone or influence diagrams.)

Creating such a diagram requires good understanding of
the 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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Example: Pareto Analysis of DCMOS (cont)

equipmnet

utilities

loading

inspection

smiff boxes

others

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 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?