Required TextGoetsch, D. L., & Davis, S. B. (2016). Quality management for organizational excellence. Upper Saddle River, NJ: pearson. 8th Edition 2016; ISBN: 9780133791853Supplementary TextBesterfield, D. H., Besterfield-Michna, C., Besterfield, G. H, & Besterfield-Sacre, M. (2003). TotalQuality Management, Upper Saddle River, NJ: Prentice Hall.Pyzdek, T., & Keller, P. (2010). Six Sigma Handbook: A Complete Guide for Green Belts, BlackBelts, and Managers at All Levels. McGraw-Hill, New YorkCase Study
Control Charts for Batch Processing Environment
(BPE Charts)
Semiconductor
processing
creates
multiple
sources
of
variability
to
monitor
One of the assumptions in using classical Shewhart SPC
charts is that the only source of variation is from part to part
(or within subgroup variation). This is the case for most
continuous processing situations. However, many of
today’s processing situations have different sources of
variation. The semiconductor industry is one of the areas
where the processing creates multiple sources of variation.
In semiconductor processing, the basic experimental unit is
a silicon wafer. Operations are performed on the wafer, but
individual wafers can be grouped multiple ways. In the
diffusion area, up to 150 wafers are processed in one time
in a diffusion tube. In the etch area, single wafers are
processed individually. In the lithography area, the light
exposure is done on sub-areas of the wafer. There are
many times during the production of a computer chip where
the experimental unit varies and thus there are different
sources of variation in this batch processing environment.
The following is a case study of a lithography process. Five
sites are measured on each wafer, three wafers are
measured in a cassette (typically a grouping of 24 – 25
wafers) and thirty cassettes of wafers are used in the study.
The width of a line is the measurement under study. There
are two line width variables. The first is the original data and
the second has been cleaned up somewhat. This case
study uses the raw data. The entire data table is 450 rows
long with six columns.
The first step in analyzing the
data is to generate some
simple plots of the response
and then of the response
versus the various factors.
4-Plot of Data
Interpretation
This 4-plot shows the following.
1. The run sequence plot (upper left) indicates that..
……………………………………………………….
……………………………………………………….
2. The lag plot (upper right) indicates that …………
…………………………………………………………
………………………………………………………….
…………………………………………………………
3. The histogram (lower left) shows that……………..
…………………………………………………………
4. Due to the non-constant location and scale and
autocorrelation in the data, ………………………..
…………………………………………………………
The run sequence plot is shown at full size to
………………………………………………………………………
………………………………………………………………….
Run Sequence Plot of Data
Numerical
Summary
Sample size
= 450
Mean
= 2.53228
Median
= 2.45334
Minimum
= 0.74655
Maximum
= 5.16867
Range
= 4.42212
Stan. Dev.
= 0.69376
Autocorrelation = 0.60726
We are primarily interested in the mean and standard deviation. From the
summary, we see that the mean is ………. and the standard deviation is………
Plot
response
against
individual
factors
The next step is to plot the response against each individual factor. For
comparison, we generate both a scatter plot and abox plot of the data. The scatter
plot shows more detail. However, comparisons are usually easier to see with the
box plot, particularly as the number of data points and groups become larger.
Scatter plot
of width
versus
cassette
Box plot of
width versus
cassette
Interpretation
We can make the following conclusions based on the above scatter and box plots.
1. There is considerable variation in the location for the various cassettes.
The medians vary from about ……. to ……..
2. There is also some ………………in the scale.
3. There are a number of …………..
Scatter plot
of width
versus wafer
Box plot of
width versus
wafer
Interpretation
We can make the following conclusions based on the above scatter and box
plots.
1. The locations for the three wafers are relatively ……………….
2. The scales for the three wafers are relatively ……………..
3. There are a few ……………on the high side.
4. It is reasonable to treat the wafer factor as …………………………
Scatter plot
of width
versus site
Box plot of
width versus
site
Interpretation
We can make the following conclusions based on the above scatter and box
plots.
1. There is some variation in location based on site. The center site in
particular has a lower …………….
2. The scales are relatively ………………….. across sites.
3. There are a few ………….
DOE mean
and sd plots
We can use the DOE mean plot and the DOE standard deviation plot to show
the factor means and standard deviations together for better comparison.
DOE mean
plot
DOE sd plot
Summary
The above graphs show that there are differences between the lots and the
sites.
There are various ways we can create subgroups of this dataset: each lot could
be a subgroup, each wafer could be a subgroup, or each site measured could be
a subgroup (with only one data value in each subgroup).
Recall that for a classical Shewhart means chart, the average within subgroup
standard deviation is used to calculate the control limits for the means chart.
However, with a means chart you are monitoring the subgroup mean-to-mean
variation. There is no problem if you are in a continuous processing situation – this
becomes an issue if you are operating in a batch processing environment.
We will look at various control charts based on different subgroupings.
Choosing
the right
control
charts to
monitor the
process
The largest source of variation in this data is the lot-to-lot variation. So, using classical
Shewhart methods, if we specify our subgroup to be anything other than lot, we will be
ignoring the known lot-to-lot variation and could get out-of-control points that already have
a known, assignable cause – the data comes from different lots. However, in the lithography
processing area the measurements of most interest are the site level measurements, not
the lot means. How can we get around this seeming contradiction?
Chart
sources of
variation
separately
One solution is to chart the important sources of variation separately. We would then be
able to monitor the variation of our process and truly understand where the variation is
coming from and if it changes. For this dataset, this approach would require having two sets
of control charts, one for the individual site measurements and the other for the lot means.
This would double the number of charts necessary for this process (we would have 4 charts
for line width instead of 2).
Chart only
most
important
source of
variation
Another solution would be to have one chart on the largest source of variation. This would
mean we would have one set of charts that monitor the lot-to-lot variation. From a
manufacturing standpoint, this would be unacceptable.
Use boxplot
type chart
We could create a non-standard chart that would plot all the individual data values and
group them together in a boxplottype format by lot. The control limits could be generated
to monitor the individual data values while the lot-to-lot variation would be monitored by
the patterns of the groupings. This would take special programming and management
intervention to implement non-standard charts in most floor shop control systems.
Alternate
form for
mean
control
chart
A commonly applied solution is the first option; have multiple charts on this process. When
creating the control limits for the lot means, care must be taken to use the lot-to-lot
variation instead of the within lot variation. The resulting control charts are: the standard
individuals/moving range charts (as seen previously), and a control chart on the lot means
that is different from the previous lot means chart. This new chart uses the lot-to-lot
variation to calculate control limits instead of the average within-lot standard deviation. The
accompanying standard deviation chart is the same as seen previously.
Mean
control
chart using
lot-to-lot
variation
The control limits labeled with “UCL” and “LCL” are the standard control limits. The
control limits labeled with “UCL: LL” and “LCL: LL” are based on the lot-to-lot
variation.
Your conclusion?……………………………………………………………………………………………..
………………………………………………………………………………………..
………………………………………………………………………………………..
The width of a line on a silicon wafer is measured for
several factor variables.
Response variable
= Line width in a silicon wafer (micrometers)
Number of observations = 450
Order of variables on a line image-1. Factor 1
= Cassette number = grouping of silicon
wafers (1 to 30)
2. Factor 2
= Wafer number in a given cassette (there
are 3 wafers per cassette)
3. Factor 3
= Site on the wafer
1 = Top
2 = Left
3 = Center
4 = Right
5 = Bottom
4. Response variable = Width of a line
5. Sequence Number
6. Response variable = Cleaned up version of line width
To read this file into Dataplot-SKIP 25
Cast. Wafer Site Raw
Seq. Cleaned
Line W.
Line W.
——————————————————————————1. 1. 1. 3.199275 1. 3.197275
1. 1. 2. 2.253081 2. 2.249081
1. 1. 3. 2.074308 3. 2.068308
1. 1. 4. 2.418206 4. 2.410206
1. 1. 5. 2.393732 5. 2.383732
1. 2. 1. 2.654947 6. 2.642947
1. 2. 2. 2.003234 7. 1.989234
1. 2. 3. 1.861268 8. 1.845268
1. 2. 4. 2.136102 9. 2.118102
1. 2. 5. 1.976495 10. 1.956495
1. 3. 1. 2.887053 11. 2.865053
1. 3. 2. 2.061239 12. 2.037239
1. 3. 3. 1.625191 13. 1.599191
1. 3. 4. 2.304313 14. 2.276313
1. 3. 5. 2.233187 15. 2.203187
2. 1. 1. 3.160233 16. 3.128233
2. 1. 2. 2.518913 17. 2.484913
2. 1. 3. 2.072211 18. 2.036211
2. 1. 4. 2.287210 19. 2.249210
2. 1. 5. 2.120452 20. 2.080452
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3.218655
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2.956036
2.423235
3.302224
2.808816
2.340386
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2.952106
2.149299
2.448046
2.507733
3.530112
2.940489
2.598357
2.905165
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2.548180
1.970436
2.588036
2.053235
2.930224
2.434816
1.964386
2.417120
2.485800
2.610217
2.568106
1.763299
2.060046
2.117733
3.138112
2.546489
2.202357
2.507165
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2.773653
3.109704
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1.511781
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1.491730
1.268580
2.433994
2.045667
1.612699
2.082860
1.887341
1.923003
2.124461
1.945048
2.210698
1.985225
3.131536
2.405975
2.206320
3.012211
2.628723
2.802486
2.185010
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2.102560
1.961968
3.330183
2.464046
1.687408
2.043322
2.570657
3.352633
2.691645
1.942410
2.366055
2.500987
2.886284
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1.644860
1.447341
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1.499048
1.762698
1.535225
2.679536
1.951975
1.750320
2.554211
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2.340486
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1.695802
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1.491968
2.858183
1.990046
1.211408
1.565322
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2.535618
3.067851
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2.079477
2.669512
2.105103
4.293889
3.888826
2.960655
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3.562480
3.451872
3.285934
2.638294
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3.076231
3.879683
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3.382833
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3.617621
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2.400277
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2.862084
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1.648662
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1.663096
1.843187
2.277813
1.764940
1.358137
2.065713
1.885897
3.126184
2.843505
2.041466
2.816967
2.635127
3.049442
2.446904
1.793442
2.676519
2.187865
2.758416
2.405744
1.580387
2.508542
2.574564
3.294288
2.641762
2.105774
2.655097
2.622482
4.066631
3.389733
2.993666
3.613128
3.213809
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3.972279
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3.024903
3.099192
2.048139
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3.176292
2.852873
3.026064
3.071975
3.496634
3.087091
2.517673
2.547344
2.971948
3.371306
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2.025674
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2.755446
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2.275192
1.222139
2.099978
2.322570
2.726060
2.342292
2.016873
2.188064
2.231975
2.654634
2.243091
1.671673
1.699344
2.121948
2.519306
1.321046
1.084111
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1.568069
2.079041
1.430009
1.159674
1.343540
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