Minimizing piece-to-piece variation is critical in virtually any manufacturing situation, but sometimes within-piece variation is just as important. In our example, we are evaluating the consistency of both piece-to-piece and within-piece variability of oxide thickness on an integrated circuit wafer (see Figure 1).
To understand the silicon wafer oxide thickness, one might take measurements at each of the wafer's 5 locations, erroneously treat the 5 measurements as a subgroup, and create an Xbar and range control chart. Doing so would not be a rational way of sampling data from the process. In fact, it would be logical to expect that the variability in oxide thickness would vary less within a wafer than from wafer to wafer.
If used in conjunction with a traditional Xbar and range control chart, this "within wafer" sampling plan would result in an incorrect estimate of piece-to-piece variation and invalidate both the interpretation of the control limits as well as Cp and Cpk values. In fact, the result typically is Xbar control limits that are far too narrow, and Cp and Cpkvalues that are far too inflated (see Figure 2).
However, this same sampling plan can be used appropriately in conjunction with the 3D chart (see Figure 3).
Figure 2: Traditional Xbar and Range chart with erroneous sampling strategy and control limits. Sampling was performed by taking 5 oxide values from within an individual wafer. Since within-piece variability is typically less than piece-to-piece variation, the control limits on the Xbar chart incorrectly reflect within-piece variability rather than piece-to-piece variation. The result is an Xbar chart with abnormally close control limits and phantom indications of multiple assignable causes.
Figure 3: 3D Chart featuring an individual X chart (average oxide thickness from individual wafers), a moving range chart (wafer-to-wafer variability) and an Rwithin chart (within-wafer oxide thickness).
Interpreting the 3D Chart
The Rwithin chart is in-control indicating the within-piece variability in wafer oxide thickness is consistent from wafer to wafer. Unlike the centerline values for other range charts, the centerline on the Rwithin chart is not used in the calculation of control limits for other charts. The Moving Range chart shows no indications of assignable causes. One must be careful to remember that these moving ranges represent differences between average oxide thickness from wafer to wafer. Therefore, this MR chart shows that the variation in average oxide thickness does not change significantly from one wafer to the next. The centerline for this MR chart is used, appropriately, in control limit calculations for the IX chart. The IX chart is in-control showing no indications of a significant change in theaverage oxide thickness for the wafers. The Cp and Cpk values at the right of the charts correctly compare the variability components to specifications.
The 3D chart is the correct tool for evaluating within-piece and piece-to-piece variability on a single chart. Placing within-piece variation data on a traditional X-bar and range chart violates rational sampling procedures and invalidates calculation and interpretation of control limits and process capability ratios.
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