C. J. Spanos
Control Charts for Attributes
The p (fraction non-conforming),
c (number of defects) and the
u (non-conformities per unit) charts.
The rest of the magnificent seven.
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C. J. Spanos
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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C. J. Spanos
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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C. J. Spanos
The p-Chart
The p chart is based on the binomial distribution:
n-x
n 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
P{D = X} =
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C. J. Spanos
The p-chart (cont.)
p must be estimated. Limits are set at +/- 3 sigma.
m
mean p
variance p(1-p)
nm
Σ pi
p=
i=1
m
(in
(in this
this and
and the
the following
following discussion,
discussion, "n"
the number
number of
of
"n" isis the
samples
in
each
group
and
is
the
number
"m"
samples in each group and "m" is the number of
of groups
groups
that
thatwe
weuse
usein
in order
orderto
todetermine
determinethe
the control
controllimits)
limits)
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C. J. Spanos
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 formulas 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
pi =
Di
i = 1,...,m (m = 20 ~25)
n
m
Σ pi
p=
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i=1
m
6
C. J. Spanos
Example: Defectives (1.0 minus yield) Chart
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 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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C. J. Spanos
Example (cont.)
After the original problems have been corrected, the
limits must be evaluated again.
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C. J. Spanos
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
(x-np) 2
1
e 2np(1-p)
2πnp(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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C. J. Spanos
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
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30
25
20
15
Quantiles
Quantiles
Quantiles
Moments
Moments
Moments
10
C. J. Spanos
Designing the p-Chart
Assuming that the discrete distribution of x can be
approximated by a continuous normal distribution as
shown, then we must:
• 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 }
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C. J. Spanos
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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C. J. Spanos
In reality, p changes over time
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(data from the Berkeley Competitive Semiconductor Manufacturing Study)
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C. J. Spanos
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 x
x = 0,1,2,..
p(x) = e c
x!
µ = c, σ2 = 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.
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C. J. Spanos
Poisson and the Normal
poisson 2
poisson 20
poisson 100
130
10
30
120
8
110
6
100
20
4
90
80
2
10
70
0
Quantiles
Quantiles
Quantiles
Moments
Moments
Moments
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C. J. Spanos
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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C. J. Spanos
Counting particles
Scanning a “blanket” monitor wafer.
Detects position and approximate size of particle.
y
x
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C. J. Spanos
Scanning a product wafer
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C. J. Spanos
Typical Spatial Distributions
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C. J. Spanos
The Problem with Wafer Maps
Wafer maps contain information that is very difficult to
enumerate
A simple particle
happening.
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count
cannot
convey
what
is
20
C. J. Spanos
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
one),
must
follow
a
known,
usable
distribution.
than one), must follow a known, usable distribution.
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C. J. Spanos
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
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C. J. Spanos
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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C. J. Spanos
Time series of ISPM counts vs. Wafer Scans
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C. J. Spanos
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
u
LCL =u - 3 n
where u is the estimated value of the unknownn of u.
The sample size n may vary. This can easily be
accommodated.
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C. J. Spanos
The Averaging Effect of the u-chart
poisson 2
average 5
5.0
10
4.0
8
6
4
2
0
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3.0
2.0
1.0
0.0
Quantiles
Quantiles
Moments
Moments
26
C. J. Spanos
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
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10
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Wafer No
C. J. Spanos
The Use of the Control Chart
The control chart is in general a part of the feedback loop
for process improvement and control.
Input
Process
Output
Measurement System
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Verify and
follow up
Detect
assignable
cause
Implement
corrective
action
Identify root
cause of problem
28
C. J. Spanos
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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C. J. Spanos
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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C. J. Spanos
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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C. J. Spanos
Example: Pareto Analysis of DCMOS Process
DCMOS Defect Classification
100
80
Percentage
60
occurence
cummulative
40
20
others
se particles
rn bridging
sed contacts
ontamination
scratches
s problems
vious layer
0
Though the defect classification by type is fairly easy, the
classification by cause is not...
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C. J. Spanos
Cause and Effect Diagrams
(Also known as Ishikawa,fish bone or influencediagrams.)
figure 4.1 pp 27 Kume
Creating such a diagram requires good understanding of
the process.
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C. J. Spanos
An Actual Example
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C. J. Spanos
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
chemicals
loading
transport vendor
boxes
Handling
Wafers
utilities
filters
Contamination Control
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C. J. Spanos
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.
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C. J. Spanos
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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C. J. Spanos
Logic Defect Density is also on the decline
Y = [ (1-e-AD)/AD ]2
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C. J. Spanos
What Drives Yield Learning Speed?
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