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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.
30201000.0
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
14121086420
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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0.2
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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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?