Case 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 subareas 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.  
4Plot of Data  
Interpretation  This 4plot shows the following.
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………………………………………………………… 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.

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.

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.

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 meantomean 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 lottolot variation. So, using classical Shewhart methods, if we specify our subgroup to be anything other than lot, we will be ignoring the known lottolot variation and could get outofcontrol 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 lottolot variation. From a manufacturing standpoint, this would be unacceptable. 
Use boxplot type chart  We could create a nonstandard 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 lottolot variation would be monitored by the patterns of the groupings. This would take special programming and management intervention to implement nonstandard 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 lottolot 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 lottolot variation to calculate control limits instead of the average withinlot standard deviation. The accompanying standard deviation chart is the same as seen previously. 
Mean control chart using lottolot 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 lottolot variation.
Your conclusion?…………………………………………………………………………………………….. ……………………………………………………………………………………….. ……………………………………………………………………………………….. 
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