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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
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100
Months of Production
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Yiel
d
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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} = 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:
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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.
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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
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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.
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Count
LCL 0.053
0.232
UCL 0.411%
non
-con
form
ing
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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} = 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.
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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 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.
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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
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0.3
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0.5Fraction Nonconforming (P-chart)
Frac
tion
Non
conf
orm
ing
LCL 0.157
¯ 0.306
UCL 0.454
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Defect Count (C-chart)
Wafer No
Num
ber o
f Def
ects
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 MapsWafer 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.
chamberLaser
Detector
to pump
Assumed to measure the particle concentration in vacuum
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Progression of scatter plots over timeThe 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-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
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The Averaging Effect of the u-chart
poisson 2
0
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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):
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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
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200
Defect Count (C-chart)
Num
ber o
f Def
ects
LCL 48.26
Ý 74.08
UCL 99.90
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3
4
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6Defect Density (U-chart)
Wafer No
Def
ects
per
Uni
t
LCL 1.82
× 2.79
UCL 3.76
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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
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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
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...
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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
inspectionrec. handling
transport
loadingchemicals
utilities
cassettes
equipment
cleaning
vendor
Wafers
Defectskill
experiencevendor
calibration
SPC SPC
boxes
shift
monitoring
automation
filters
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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.
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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?