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Multiple Steam Processes, Part 2

by George Runger, Ph.D.

My previous article mentioned how common multiple stream processes are in many industries.

Let’s look at an example. Semiconductor manufacturing often processes a set of wafer together in a furnace. Each furnace run might contain anywhere from a few to dozens of wafers. In Figure 1, we show an example with four wafers in the run. After processing, oxide thickness is measured at nine sites on each wafer. Therefore, the data consist of 36 thickness measurements for each run. This is a classical multiple stream data set.

Traditionally, people control chart the run mean (the average of all 36 thickness measurements) and the run standard deviation. Although, these are worthwhile summaries, the standard deviation is an aggregate of all 36 readings and it is not sensitive process assignable causes that might be expected in such a process. For example, if these wafers are stacked top to bottom in the furnace, gas flow problems might change the top wafers relative to the bottom ones. In a process with the wafer positioned differently, one might expect differences between the center and edge sites on the wafers—again based on some knowledge of typical problems.

A set of control charts can be developed to be much more sensitive to anticipated problems, yet maintain the ability to signal unforeseen problems. A simple, but effective approach is to calculate a set of contrasts from the measurements in each run and control chart each one. A contrast is a linear combination (a weighted average, but negative weights are allowed) of the 36 measurements such that the weights sum to zero. That is, the positive weights cancel the negative weights. For example, the simple, but very effective control chart for the top versus bottom problem is a chart of the average of the nine sites from the top wafer minus the average of the nine sites from the bottom wafer.

Figures 2 and 3 illustrate control charts from a real, but prototype dataset of nine runs. This data is from

Czitrom, V. and Reece, J. E., Virgin Versus Recycled Wafers for Furnace Qualification: Is the Expense Justified?, Statistical Case Studies for Industrial Process Improvement, V. Czitrom and P. D. Spagon editors, Ch. 8, 87-104 (1997).

This dataset is too small to define control limits but it is adequate to demonstrate the method. Figure 2 illustrates an S chart for the runs and it appears fine. If we stopped there, we would be misled. We want to look more closely at within run uniformity with contrasts. Individuals charts were constructed for three contrasts. Each one can be thought of a separate measure of within run uniformity. We compared the center sites of the wafers to the edge sites with one chart and the left-to-right sites with another chart. Both of these charts were fine and are not shown. Figure 3 compares the top-to-bottom wafers and although the last point is not quite over the control limit, there is cause for concern (even with this limited data). The moral of the example is that the chart of the top-to-bottom contrast detects a problem that the aggregate of all of the data into an S chart fails to identify. As the size of our dataset continue to increase, this type of partitioning is more and more important. For more information of these types of charts contact George Runger at runger@dataengineering.com.

Figure 1

Figure 2

Figure 3