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SPC: A Practitioner’s Perspective 2Tim Conway
Agenda
Introduction (High Volume Semiconductor Manufacturing)
SPC Charts
SPC Control Limits, Simplified
Control Chart Risks
SPC Performance Metrics
Sample Size
Key Take-Aways
SPC: A Practitioner’s Perspective 3Tim Conway
Production Line (Old Style)
Lucy and Ethel wrap chocolate
© CBS
SPC: A Practitioner’s Perspective 4Tim Conway
Production Line
From this.
To this.
SPC: A Practitioner’s Perspective 5Tim Conway
Production Line
To this.
� Numerous Steps
� Definition of “unit” can change
� Batching
SPC: A Practitioner’s Perspective 6Tim Conway
Production Line
Semiconductor Production Line
� Batching, Multiple Tools, Multiple Substeps in a Step, Nested
Variance Structure
SPC: A Practitioner’s Perspective 7Tim Conway
Production Line (Metal Deposition)
Multiple Processing Chambers
� Example: aluminum deposition
• Collimated Titanium (CoTi)
• Aluminum (HAL)
• Titanium-Tungsten (TiW)
• Degas, Etch, Cool
• Multiple times for each wafer
SPC: A Practitioner’s Perspective8Tim Conway
SPC Charts
SPC: A Practitioner’s Perspective 9Tim Conway
Control Chart
SPC was founded by Walter Shewhart on three main
concepts
� Use of simple, graphical, time-series presentations of the data
� The notion of chance causes versus assignable cause variation
� Process output averages are normally distributed even though the
individual measurements are not
Walter Andrew Shewhart (1891-1967) was an American physicist, engineer,
and statistician. The “father of statistical quality control,” in May of 1924 he
described the first control chart, which launched statistical process control and
quality improvement.
SPC: A Practitioner’s Perspective 10Tim Conway
Types Of Variation
SPC is a Signal-to-Noise Analysis System
Common Cause Variation
� “Noise”
� Variation that is consistent, natural, predictable and random
� We hope this is how our process normally behaves
Assignable Cause Variation, or Special Cause Variation
� “Signal”
� Variation that is unusual or not typical of the process
� Usually we can assign a root cause to the variation
� Typically easier to remove than common cause variation
SPC: A Practitioner’s Perspective 11Tim Conway
Rational Subgroups
Classical SPC uses “Rational Subgroups” to improve the
signal-to-noise performance
� Select samples such that the variation within the sample is small
(i.e., homogeneous sampling) while the variation between
samples is large when assignable cause is present
� Then use the within-sample variation to estimate the between-
sample variation, assuming only common-cause variation is
present
Potential Issue
� What if the points in the sample are highly correlated?
SPC: A Practitioner’s Perspective 12Tim Conway
Control Chart Usage
General
� Monitor stability of one or multiple components of variation
(between-sample, within-sample, etc.)
� Estimate capability and predict future performance of the system
� Quickly detect and fix problems
� Identify opportunities for improvement
Variable Data (measured on continuous scale)
� Monitor process, product output parameters (dim, thickness)
� Monitor input, state variables (temps, pressures, flow rates)
� Assess matching (tool-to-tool, lot-to-lot, wafer-to-wafer)
Attribute Data (counts of defects , defective units or events)
� Monitor tool or process state variables (particles, events, etc.)
� Monitor product performance (yields, failure counts, etc.)
SPC: A Practitioner’s Perspective 13Tim Conway
Control Charts for Variables Data
Variables Control Charts
� There are many variations of control charts. Typical variables
charts include:
• Xbar/R, Xbar/S, Xbar/%S. The Xbar chart plots the sample average
and monitors the between-sample variation. The R, S or %S chart
plots the dispersion of the sample and thus monitors the within-
sample variation.
• X/MR (Individual / Moving Range). Used when sample size is one.
• Delta-to-Target, Z, Zbar/E. Also known as “short-run” charts, these
plot standardized or normalized data that allow multiple process
streams to be placed on the same chart even if the streams have
different targets.
• EWMA, CUSUM. Specialized charts that are sensitive to small drifts
or shifts.
• Multivariate (Hotelling T2). Combine multiple variables on one
chart.
SPC: A Practitioner’s Perspective 14Tim Conway
Control Charts for Attribute Data
Attribute Control Charts
� Typical charts for attribute data include:
• Proportion Defective (p). Monitors proportion or percent
nonconforming units in a group of units. Example is a yield chart for
die per wafer or die per lot.
• Number Defective (np). Monitors the count of nonconforming units in
a group.
• Number of Defects (c). Monitors defect or particle counts in a group
of units.
• Defects per Unit (u). Monitors defect or particle counts per unit.
� “p” and “np” charts assume a Binomial distribution
� “c” and “u” charts assume a Poisson distribution
� Normal distribution can approximate Binomial if np and n(1-p) > 5
� Normal distribution can approximate Poisson if λ ≥ 15
SPC: A Practitioner’s Perspective 15Tim Conway
Control Chart Decision Tree
SPC: A Practitioner’s Perspective 16Tim Conway
SPC Charts
Classical Model for Variables Control Charts
Where:
� Errors ε are normally distributed with mean = 0 and variance = σ2
� Process data is assumed to be stationary and uncorrelated
Implications
� Two components of variation (between-sample, within-sample)
� Rational subgrouping to
• Minimize chance of differences within subgroups
• Maximize chance of differences between subgroups, if assignable
causes are present
�� = � + ��
SPC: A Practitioner’s Perspective 17Tim Conway
SPC Charts
Nested Model for Variables Control Charts (Semiconductor)
Where:
� Errors ε are normally distributed with mean = 0 and variance = σ2
� But the errors are centered on the mean of the next higher level of
nesting
� Other components exist (device, tool, chamber, etc.)
Implications
� Multiple components of variation (lot-lot, wafer-to-wafer, site-to-
site, etc.)
� Still want to minimize chance of differences within samples while
maximizing chance of differences between samples, if assignable
causes are present
�� = � +��� +��� �� � � +���� �� �� �
SPC: A Practitioner’s Perspective 18Tim Conway
Recommendations
Recommendations (Variables Data)
� Characterize / understand the major components of variation
� Monitor the major components (use > 2 charts if needed)
• Don’t be limited by traditional two-chart software systems
• Innovate on visual display of variation (e.g., box plots on trend chart)
� Or have quick method to visually break out the components
• Variability (multi-vari) charts
• Probability plots
SPC: A Practitioner’s Perspective 19Tim Conway
Components of Variance
Multi-vari Example
SPC: A Practitioner’s Perspective 20Tim Conway
Components of Variance
Probability Plot Example
SPC: A Practitioner’s Perspective21Tim Conway
SPC Control Limits, Simplified
SPC: A Practitioner’s Perspective 22Tim Conway
SPC Control Limits
General Form of the Control Limit Equations (Traditional)
Where:
� "mean" is the average of the plotted points
� "StdErr of the mean" is the estimate of the common-cause sigma
of the plotted points
� Typically, ��/2 is set to 3 to provide a false alarm rate of 1/370 for
normally distributed data. (Why 3? Because it works well.)
In layman’s terms
� When the plotted points exhibit random, common-cause variation
mean ±��/�*StdErr of the mean
Avg of Plotted Points ± 3*Sigma of Plotted Points
SPC: A Practitioner’s Perspective 23Tim Conway
SPC Control Limits
Traditional Charts for Variable Data
� Note the general form
SPC: A Practitioner’s Perspective 24Tim Conway
SPC Control Limits
Traditional Charts for Attribute Data
SPC: A Practitioner’s Perspective 25Tim Conway
SPC Control Limit Factors
SPC: A Practitioner’s Perspective 26Tim Conway
SPC Control Limits
Issue: Within-Sample Correlated Data
� For Xbar charts, using the �� ��� or �� � ⁄ methods result in overly
tight control limits if the within-sample data is correlated.
Alternately, plot summarized data (e.g., sample averages) on an
individuals "X" chart and calculate the sigma estimate "#$� using:
1. Median MR (Clifford's robust sigma estimate, recommended).
2. Average MR. Risk that flyers may inflate the control limits.
3. IQR (InterQuartile Range). Sets control limits to outlier box plot
whisker ends
4. Percentile Method. Set UCL to the P99 or P99.865 for example, if data
highly skewed.
5. The standard deviation of the plotted sample averages (Levey-
Jennings). Risk that assignable-cause variation may inflate the
control limits.
SPC: A Practitioner’s Perspective 27Tim Conway
SPC Control Limits (Non-Traditional)
Median Moving Range (recommended)
� Also known as Clifford’s Robust Sigma Estimate
� Robust to outliers and to mean shifts
� Treats the summary statistic (e.g., sample average) as individuals
data; uses X/MR chart and estimates the process sigma using
median moving range
%&'()**+,- =.
-/∗ 123
%&'()**+,- = 1.05 ∗ 189:;< =) − =)?.
Where MR3 = MedianMovingAbsoluteRange
SPC: A Practitioner’s Perspective 28Tim Conway
SPC Control Limits (Non-Traditional)
Average Moving Range
� Sensitive to non-normality
%&PQ =.
-�∗ MR
%&PQ = 0.866 ∗ TU8V;W8 =) − =)?. , for MR of span 2
Where MR = AverageMovingAbsoluteRange
SPC: A Practitioner’s Perspective 29Tim Conway
SPC Control Limits (Non-Traditional)
InterQuartile Range (IQR)
� Control limits correspond to whiskers on outlier box plot
Percentile Limits
� P99.865 & P0.135 correspond to 3.0 sigma limits for normal
distribution
YZ[\]Q = ^_` + 1.5 ∗ a^2[Z[\]Q= ^�` − 1.5 ∗ a^2
Where a^2 = ^_` − ^�`
SPC: A Practitioner’s Perspective 30Tim Conway
SPC Control Limits (Non-Traditional)
Levey-Jennings
� Sigma estimate is the standard deviation of the plotted points
� Caution: assignable-cause variation inflates the control limits
� If used, check for and remove assignable-cause variation
Eyeball method
� Count out two “sigma's” from the center to the edge of the data,
then count out one more “sigma” and put the control limit there
� Assumes normally distributed, common-cause data (should check
assumptions)
� Based on idea that roughly 95% of the data (19 of 20 points) is
contained within +/- two sigma's of the average
SPC: A Practitioner’s Perspective 31Tim Conway
SPC Control Limits (Variables Data)
SPC: A Practitioner’s Perspective 32Tim Conway
SPC Control Limits
Recommendations (Variables Data)
� Plot sample statistics that are sensitive to assignable causes
• X�, R, S
• X� takes advantage of central limit theorem � improved normality
� Base control limits on outlier-resistant methods
• Median Moving Range (recommended)
� Individuals (X) charts have poor ability to detect small shifts
• Example: Average Run Length (ARL) = 44 to detect 1-sigma shift
• For improved shift detection, use smoothed data (EWMA:
Exponentially Weighted Moving Average)
SPC: A Practitioner’s Perspective 33Tim Conway
SPC Control Limits
Recommendations (Attribute Data)
� Use variables data if possible (much smaller sample sizes)
� If data approximately normal then can use variables charts
� Try transforming skewed data (SQRT, LOG) to approximate
normal
� Traditional attribute charts (p, np, c and u) work well for non-
conformity or defect monitoring, when chart not dominated by
zero-valued data
� But for rare non-conformities or defects, consider “time between
failure.” Transform time between failure using y= d./e.f
transformation to make data approximately normal.
SPC: A Practitioner’s Perspective34Tim Conway
Control Chart Risks
SPC: A Practitioner’s Perspective 35Tim Conway
Control Chart Have Risks
Caution
� Control charts have decision risks
• Calling a process out-of-control when the process has not changed
• Calling a process in control when the process has shifted
� Control charts have sampling risk
• A shift may not be detected for a number of subsequent samples
� Mixing of data from different sources can hide signals
• If multiple process streams are placed on the same chart then stream-
specific signals may be hidden.
SPC: A Practitioner’s Perspective 36Tim Conway
Decision Risk
Example: SPC Chart with mean shift
� Null hypothesis (H0): process is stable, µ = µ0
� α = false alarm rate; out-of-control point when process is stable
� β is the risk of not detecting a given shift in the mean
� 1- β is the power of detecting the given shift in the mean
SPC: A Practitioner’s Perspective 37Tim Conway
Sampling Risk: Average Run Length (ARL)
“Power” is the probability of detecting a given shift on the
next sample
� Power = 1- β
Average Run Length (ARL) can be used two ways
� ARL = 1/α gives the false alarm rate
� ARL = 1/(1- β) is the number of samples on average that it will
take to detect a given shift
SPC: A Practitioner’s Perspective 38Tim Conway
Sampling Risk: Average Run Length (ARL)
False Alarm Average Run Length (ARL), Normal Dist
SPC: A Practitioner’s Perspective 39Tim Conway
Sampling Risk: Average Run Length (ARL)
Shift Detection Average Run Length (ARL), Normal Dist
Table is for n=1, larger samples improve the ARL
� Example: n=4, 3-sigma limit, 1-sigma shift, ARL 44 � 4
SPC: A Practitioner’s Perspective40Tim Conway
SPC Performance Metrics
SPC: A Practitioner’s Perspective 41Tim Conway
Detect potential problems
� Want to monitor effectively (measure the things that are critical)
� Want to monitor efficiently (don’t over-sample)
� Want to detect issues quickly (react to real signals)
� Do not want to intervene unless the process tells us to (don’t react
to noise)
Reduce variation
� Ensure that our processes are stable (predictable)
� Ensure that our processes are capable (meet the spec)
� Ensure that our processes are targeted (loss is minimized)
The Golden Rule of SPCIntervene in a timely and efficient manner
Why Do SPC?
Lo
ss
Spec Limit Spec Limit
Target
SPC: A Practitioner’s Perspective 42Tim Conway
Performance Metrics: RV1
� RV1 monitors stability
• # of WECO Rule 1 violations in # of SPC points or timespan
• This is the headache metric (is the process or equipment creating
excessive headaches)
SPC: A Practitioner’s Perspective 43Tim Conway
Performance Metrics: Cpk, Ppk, Cp, Pp
� Cpk monitors short-term capability, where only common-cause
variation is present. Cpk is the minimum of CPU (upper spec limit
applies) and CPL (lower spec limit applies).
� Ppk monitors long-term capability and thus includes both
common-cause and assignable-cause variation. Ppk is more
representative of the quality level of the process.
SPC: A Practitioner’s Perspective 44Tim Conway
Performance Metrics: Z, Z’
� Z and Z’ monitor targeting
• Z and Z’ are the delta from target, in sigma units
• Typical requirement is |Z| < 1.0
• Potential issue: metric is penalized by goodness (small sigma)
� Alternate metric: the “k” in Cpk
• k is the deviation of the average from the center, as a proportion of
the half-spec window
g =$h − i��j�
�$�g′ =
$h − i��j�
"#lm� ��
k=$h?(o��p���)/�
(o��?���)/�(CPU case)
k=(o��p���)/�?$h
(o��?���)/�(CPL case)
SPC: A Practitioner’s Perspective45Tim Conway
Sample Size
SPC: A Practitioner’s Perspective 46Tim Conway
Sample Size: Shrinkage Factor
Using the Central Limit Theorem
� Variance of the means equals
variance of individuals divided by
sample size
Solve the equation for “n”
� “n” is the shrinkage factor
Use of the shrinkage factor
� Quick assessment of correlated
within-sample data
� If shrinkage factor is less than the
sample size then some data values
are correlated and the sample size
maybe can be reduced
n
22
X
Xσ
σ =
2
X
2
σ
σX
n =
SPC: A Practitioner’s Perspective 47Tim Conway
Sample Size: Shrinkage Factor
Conclusion:
Use n=3 as sample
size
What other
considerations should
be taken into account?
Sample
Sample Site 1 Site 2 Site 3 Site 4 Site 5 Averages
1 106 158 82 22 122 98.0
2 70 102 72 34 106 76.8
3 20 134 66 78 68 73.2
4 68 8 20 156 64 63.2
5 54 72 74 50 98 69.6
6 24 50 36 48 52 42.0
7 10 76 62 30 32 42.0
8 76 16 26 44 60 44.4
9 68 100 90 16 88 72.4
10 54 76 66 54 62 62.4
11 20 134 80 4 40 55.6
12 56 16 6 90 60 45.6
13 80 148 100 34 112 94.8
14 48 30 86 76 56 59.2
15 12 98 42 42 26 44.0
16 70 106 42 210 120 109.6
17 70 134 74 28 102 81.6
18 44 10 60 66 44 44.8
19 12 68 22 50 16 33.6
20 88 120 86 172 146 122.4
Variance (individuals) 1668
Variance (of sample averages) 610
Shrinkage factor 2.7
Measurement
2
X
2
Factor Shrinkageσ
σX=
SPC: A Practitioner’s Perspective 48Tim Conway
Sample Size: Clustering Analysis
What is Clustering Analysis?
� Assesses the within-sample data to determine if there is significant
correlation among the data points.
• One application is to determine if the sites on the wafer can be clustered.
� Correlation implies that the points are not independent of each other.
• The correlated points can be grouped into clusters.
• Each cluster contains values that are “not much different” from each other.
• Picking one value from each cluster thus efficiently represents the cluster.
SPC: A Practitioner’s Perspective49Tim Conway
Key Take-Aways
SPC: A Practitioner’s Perspective 50Tim Conway
Key Take-Aways
� Quality decreases as variability increases. Set up the SPC
system to efficiently and quickly detect and attack variation.
� SPC is a signal-to-noise system. Assignable-cause variation is
the signal and common-cause variation is the noise.
� Don’t respond to the noise (with control actions).
� Proper control limits balance the risks of false alarms vs risks of
missing an assignable cause. Use sample averages to induce
normality into your data and reduce the risks.
� Using within-sample variation to estimate between-sample
variation (classical approach) is problematic with highly-correlated
within-sample data, such as is seen in semiconductor processing.
� Use of robust-estimators of the process sigma helps mitigate the
above problem.
� But if your control limits look reasonable when the data looks
random, then that is likely OK from an engineering standpoint.
SPC: A Practitioner’s Perspective 51Tim Conway
Key Take-Aways (cont.)
� There are many types of charts. Avoid the confusion by keeping
in mind the “my control limits look reasonable” concept.
� Don’t just SPC the outputs; monitor, control and improve the
inputs.
� Determine and monitor the major components of variation.
� Graphical displays of the components of variation are hugely
beneficial.
� If you put spec limits on your chart, make sure they use the same
basis as the data (raw spec for individual data, average spec for
average data).
� Keep aware of the Average Run Length (ARL) concept. Avoid too
many rules as that will increase the false-alarm rate.
� Out-of-Control Action Plans (OCAPs) are huge! You need
methods to troubleshoot and resolve OOC points.
SPC: A Practitioner’s Perspective 52Tim Conway
Key Take-Aways (cont.)
� Stable, capable and targeted is always a good thing. Monitor the
performance of the SPC system.
� SPC performance metrics should consider capability (Cpk, Ppk),
targeting (Z) and headaches caused by OOC points (RV).
� Make sure to check your measurement system. A noisy
measurement makes process improvement much, much more
difficult.
SPC: A Practitioner’s Perspective 53Tim Conway
References
� Montgomery, D. C., (2009), Introduction to Statistical Quality
Control, 6th ed., Wiley, New York
� Clifford, P. C., (1959), “Control Charts Without Calculations,”
Industrial Quality Control, Vol. 15(11), pp. 40-44