P-chart (fraction non-conforming) C-chart (number of ...ee290h/fa05/Lectures/PDF/lecture 11...P-chart (fraction non-conforming) C-chart ... As sample sizeAs sample sizeincreases,

$Page 1: P-chart (fraction non-conforming) C-chart (number of ...ee290h/fa05/Lectures/PDF/lecture 11...P-chart (fraction non-conforming) C-chart ... As sample sizeAs sample sizeincreases,$
Lecture 11: Attribute Charts

SpanosEE290H F05

1

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


SpanosEE290H F05

2

Yield Control

30201000

20

40

60

80

100

Months of Production

30201000

20

40

60

80

100

Yiel

d


SpanosEE290H F05

3

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)


SpanosEE290H F05

4

The P-Chart

P{D = X} = nx px(1-p)n-x x = 0,1,...,n

mean npvariance np(1-p)

the sample fraction p=Dn

mean pvariance p(1-p)

n

The P chart is based on the binomial distribution:


SpanosEE290H F05

5

The P-chart (cont.)

p =

mΣi=1

pi

m

mean pvariance 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.


SpanosEE290H F05

6

Designing the P-Chart

p can be estimated:

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

p =

m

Σi=1

pi

m

In general, the control limits of a chart are:UCL= µ + k σLCL= µ - k σwhere 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


SpanosEE290H F05

7

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.

30201000.0

0.1

0.2

0.3

0.4

0.5

Count

LCL 0.053

0.232

UCL 0.411%

non

-con

form

ing


SpanosEE290H F05

8

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


SpanosEE290H F05

9

Operating Characteristic of P-Chart

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

P{D = x} = nx

px(1-p) (n-x) ~ 12πnp(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.


SpanosEE290H F05

10

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...


SpanosEE290H F05

11

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:


SpanosEE290H F05

12

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).


SpanosEE290H F05

13

In reality, p changes over time

(data from the Berkeley Competitive Semiconductor Manufacturing Study)


SpanosEE290H F05

14

The C-Chart

UCL = c + 3 ccenter at c

LCL = c - 3 c

p(x) = e-c cx

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

μ = c, σ2 = 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.


SpanosEE290H F05

15

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...


SpanosEE290H F05

16

Example: "Filter" wafers used in yield model

141210864200.1

0.2

0.3

0.4

0.5Fraction Nonconforming (P-chart)

Frac

tion

Non

conf

orm

ing

LCL 0.157

¯ 0.306

UCL 0.454

141210864200

100

200

Defect Count (C-chart)

Wafer No

Num

ber o

f Def

ects

LCL 48.26

Ý 74.08

UCL 99.90


SpanosEE290H F05

17

Counting particles

Scanning a “blanket” monitor wafer.

Detects position and approximate size of particle.

x

y


SpanosEE290H F05

18

Scanning a product wafer


SpanosEE290H F05

19

Typical Spatial Distributions


SpanosEE290H F05

20

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

A simple particle count cannot convey what is happening.


SpanosEE290H F05

21

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.


SpanosEE290H F05

22

In Situ Particle Monitoring Technology

Laser light scattering system for detecting particles in exhaust flow. Sensor placed down stream from valves to prevent corrosion.

chamberLaser

Detector

to pump

Assumed to measure the particle concentration in vacuum


SpanosEE290H F05

23

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


SpanosEE290H F05

24

Time series of ISPM counts vs. Wafer Scans


SpanosEE290H F05

25

The U-ChartWe 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:

where is the estimated value of the unknown u.The sample size n may vary. This can easily be accommodated.

μu = u, σu2 = unso

UCL = u + 3 un

LCL = u - 3 un

u


SpanosEE290H F05

26

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


SpanosEE290H F05

27

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

141210864200.1

0.2

0.3

0.4

0.5Fraction Nonconforming (P-chart)

Frac

tion

Non

conf

orm

ing

LCL 0.157

¯ 0.306

UCL 0.454

141210864200

100

200

Defect Count (C-chart)

Num

ber o

f Def

ects

LCL 48.26

Ý 74.08

UCL 99.90

141210864201

2

3

4

5

6Defect Density (U-chart)

Wafer No

Def

ects

per

Uni

t

LCL 1.82

× 2.79

UCL 3.76


SpanosEE290H F05

28

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

ProcessInput Output

Measurement System

Verify andfollow up

Implementcorrectiveaction

Detectassignablecause

Identify rootcause of problem


SpanosEE290H F05

29

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


SpanosEE290H F05

30

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.


SpanosEE290H F05

31

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,...)


SpanosEE290H F05

32

Example: Pareto Analysis of DCMOS Process

evio

us la

yer

ss p

robl

ems

s sc

ratc

hes

onta

min

atio

n

sed

cont

acts

ern

brid

ging

se p

artic

les

othe

rs

0

20

40

60

80

100

occurence

cummulative

DCMOS Defect Classification

Perc

enta

ge

Though the defect classification by type is fairly easy, the classification by cause is not...


SpanosEE290H F05

33

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.


SpanosEE290H F05

34

An Actual Example


SpanosEE290H F05

35

Example: DCMOS Cause and Effect Diagram

Past Steps Parametric Control Particulate Control

Operator Handling Contamination Control

inspectionrec. handling

transport

loadingchemicals

utilities

cassettes

equipment

cleaning

vendor

Wafers

Defectskill

experiencevendor

calibration

SPC SPC

boxes

shift

monitoring

automation

filters


SpanosEE290H F05

36

Example: Pareto Analysis of DCMOS (cont)

equi

pmne

t

utilit

ies

load

ing

insp

ectio

n

smiff

box

es

othe

rs

0

20

40

60

80

100

occurence

cummulative

DCMOS Defect Causespe

rcen

tage

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


SpanosEE290H F05

37

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


SpanosEE290H F05

38

Logic Defect Density is also on the decline

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


SpanosEE290H F05

39

What Drives Yield Learning Speed?

Documents

P-chart (fraction non-conforming) C-chart (number of ...ee290h/fa05/Lectures/PDF/lecture 11...P-chart (fraction non-conforming) C-chart ... As sample sizeAs sample sizeincreases,