Articles
RSSSplitting Wafer Lots in Semiconductor Manufacturing, Part III: Comparing Two Processes – A Better Approach
May 31, 2000
Continuous process improvement is essential if a semiconductor manufacturing facility is to prosper. Tweaks in processes to improve device performance or yield are a way of life. Testing those improvements, though, demands an experimental design that minimizes the risk to production lots while gathering enough data for valid inferences. Part I of this article discussed the relevant statistical tools, Part II points out common analytical errors, and Part III explains a better approach.
•Redesigning the Experiment
•Comparing More than Two Alternatives in the Same Process Step
•Summary
•Software
The experiment discussed in Part II used two lots of 25 production wafers to compare the production process (NOMINAL) to some proposed alternative (TEST). The production wafers move together as a lot through any steps preceding the test step and move together as a lot in any subsequent processing steps. In the step involving the test, the engineer splits the wafers in each lot into two groups; one group (20 wafers) receives the nominal treatment, while the other (five wafers) receives the alternative or test treatment. Part II showed that, as designed, this experiment has only one replication of the test conditions per lot of wafers. A better experiment would have more replications, but that inherently suggests applying the splits to still more production lots – not a particularly attractive option.
Table 1 reproduces Table 6 of Part II, and is discussed in more detail there. Notice that the "df" column accounts for 50 units (identical to the number of wafers used), and that 46 of them are in the residual. Since the residual is not the proper error term to use in evaluating the significance of the effect of the splits, using that many wafers in this fashion is a waste of resources. This inefficiency is the result of the manner in which the investigator divided the wafers between the two splits. In reality, any more than two wafers in a particular trial is not the most efficient use of the wafers available. In fact the only reason for using even two wafers in a trial is to help assure that the data from that trial will survive in case one wafer is lost during processing.
In order to increase the power of this type of experiment, make better use of the wafers available from each lot. Figure 1 shows how an investigator might better execute this experiment.
Notice in Figure 1 that most of the trial "boxes" account for three wafers and that each box contains at least two. This approach uses no more wafers than the experiment previously discussed and subjects the same number of wafers to the TEST split. Later analyses will show, however, that this approach markedly increases the power of the experiment in detecting differences between splits. Each box or trial represents an independent execution of one of the two processes. That is, Trial 1 selects three wafers at random from Lot 100 and processes them using the NOMINAL process. Trial 2 selects another three wafers at random from Lot 100 and processes them in a separate run independent of the Trial 1. Particularly if the wafers are all being run in the same equipment, one should not run the individual trials involving NOMINAL, then those involving TEST. Obviously with only two examples of TEST in each lot, rigorous randomization of the trials is not possible, but the investigator should insert the TEST runs somewhere in the sequence of processing each of the two lots. Controlling this experiment to assure that the correct batches are run will be a logistics challenge, but the enhanced precision of the analysis is a major benefit.
Figure 2 compares the power curve for this experiment to that described in Part II. The graph shows that the b-risk for failing to detect an effect due to SPLIT twice the size of the experimental error is negligible. In fact the risk is only about 0.12 (power = 0.88) for an effect approximately the same size as the experimental error. Placing more than three wafers in each trial will increase the b-risk; placing only two wafers in each trial will lower it further. Of course, adding a third lot of 25 will decrease the risk still further and give a still better estimate of how the process line is behaving.
Table 2 represents a further modification that reflects the design structure shown in Figure 1. In the table the trial numbers are in numerical sequence for illustration. In conducting an experiment of this type one should randomize the trials so that not all the process nominal wafers are processed before the test wafers. Since lots arrive at processing stations at different times, randomizing among the lots is usually not practical. Table 3 summarizes an analysis of this data that assumes the engineer actually conducted the trials in the fashion outlined in the design structure in Figure 1.
Notice in Table 3 that the sum in the degrees of freedom column (df) is the same as that in Table 1 (50), but that the distribution of those degrees of freedom has changed dramatically. Now the correct error term for estimating the difference between the splits is TRIAL(LOT SPLIT), which is based on 15 df rather than the 1 in Table 1. Without increasing the number of wafers being committed to the experiment or risking any additional production wafers to the TEST split, this approach has provided a much more powerful test of significance. To the engineer this means that the decisions made regarding the competing processes are much more reliable than before – without increasing the wafer cost of the experiment. The power of the experiment increases still further if each trial contains only two wafers instead of the three used in this example.
The analysis reported in Table 3 is from RS/Explore Mulreg and uses what is known as Type II Sum of Squares in the calculations. Different software packages may default to different approaches in analyzing this data, but the effects of the different approaches will not materially affect the decisions from the analyses in most cases.
Comparing More than Two Alternatives in the Same Process Step
If the number of alternatives in a given process step increases, the design of the experiment is analogous to this one for two alternatives. Divide the wafers in each lot processed among the alternatives, populating each independent trial with at least two wafers. For a 25-wafer lot, this means that three splits in a given process can each involve at least eight wafers. But those wafers should be spread so that the experiment repeats each trial within a lot and split three or four times. Again, process at least two lots, preferably three through the splitting scheme.
Because so much depends on the interpretation of the results of comparing alternative approaches to a given process step, making certain those comparisons are statistically valid is imperative. The example in this paper demonstrated that different interpretations of data easily result when doing an analysis incorrectly.
The convention of splitting a lot of wafers into large blocks and processing those blocks as single units produces interpretations with high b-risks (failing to detect an effect). Improperly analyzing the data by assuming the results from individual wafers within a trial are useful replicates can lead to extremely misleading decisions that can be costly on a production floor.
No software that the authors have used provides convenient algorithms to design the nested experiment illustrated here.. Generating a complicated nested design of this type in this fashion is the method of choice because it illustrates how raw materials should be allocated in an experimental study.
RS/1 (Release 6.01) with RS/Explore (Release 4.1) provided the analyses described in this report. This software is available from Brooks Automation Inc., Burlington, MA. Alternative programs include SAS (SAS Institute, Cary, NC) and JMP 4.0 (SAS Institute, Cary, NC). These three software platforms are among the most sophisticated available for applied statistics studies. Other less sophisticated or popular programs may require a "balanced" data set (same number of wafers per trial).
For more information
"Statistically Speaking," is intended to help readers use statistical methods to solve process problems. Readers can pose questions for future columns through a companion Discussion Forum
Jack Reece is a member of Semiconductor Online's advisory board. He can be reached at:
PO Box 308
Lake George, CO 80827 USA
Voice: +1 719-748-8641
FAX: +1 719-748-8642
jreece@pcisys.net
George A. Milliken is a Professor of Statistics at Kansas State University, specializing in the analysis of "messy data." He has extensive consulting contracts in agricultural and biological sciences (pharmaceuticals) as well as in conventional manufacturing. He is co-author, with fellow Kansas State University professor Dallas Johnson, of a landmark text on"Analysis of Messy Data." He can be reached at:
Department of Statistics
Kansas State University
Manhattan, KS 66506 USA
Voice: +1 (785) 532-0514
milliken@ksu.edu
