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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 the correct approach.

Contents

•A Naïve Analysis (Doing it Wrong is Convenient, Intuitive, and Commonplace
•Interpretation and Critique of the Analysis
•An Alternative Analysis Using Student's t
•A Split Plot Design Structure

Figure 1 illustrates how an engineer might organize an experiment to determine whether or not an alternative process (TEST) produces better results than the current one (NOMINAL).

In this case a lot contains 25 wafers. The experiment involves production wafers, and they 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 into two groups; one group (20 wafers) receives the nominal treatment, while the other (five wafers) receives the alternative or test treatment. Since this experiment involves production wafers, electing to risk only five in the TEST branch is a prudent choice. The experiment involves repeating the process with two lots of wafers, for a total of 50 wafers. Table 1 presents a summary of the results. In the table, the parameter reported is the average of several measurements per wafer. The objective is to increase the parameter value.

Multiple measurements on individual wafers affect only the precision of the estimate of the average value. Only the average observation for a particular wafer becomes part of the analysis. The variation associated with multiple measurements plays no significant role in determining the significance of effects due to splits.

The assignments of wafers to a split have been randomized. Conventional practice of assigning odd wafers to one split and even to another or of assigning the first few wafers to one split and the next to another invites complications if a previous or future step produces some drift in results as a function of number of wafers processed.

A Naïve Analysis (Doing it Wrong is Convenient, Intuitive, and Commonplace)

The nominal split contains a total of 40 wafers, while the test split contains a total of 10 wafers. A naïve approach compares the average response of the wafers found in the nominal splits to that found for the wafers in the test splits using the Student's t calculations. Table 2 summarizes the output from a suitable software program.

Based on this analysis the engineer would conclude that the "TEST" split gave slightly higher results than the "NOMINAL" split, and that the difference is statistically significant with an a-risk far less than five percent.

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Interpretation and Critique of the Analysis

Validity of this analysis requires that the multiple observations of the parameter in each split represent a set of independent replicated trials. While the number of points for each split, respectively, was 40 and 10, those numbers are not necessarily synonymous with replication, depending on the manner in which the engineer conducted the experiment. That is, the number of replications in the experiment depends on the design structure of that experiment – the procedure used in conducting the trials. During the test step, the engineer processed a single batch of 20 wafers under nominal conditions and a single batch of five wafers under test conditions for the first lot. The same processing occurred when the second lot arrived at the test step later. Therefore, the experiment did not contain 10's of replicates: rather, each lot provided one replicate of each processing condition.

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An Alternative Analysis Using Student's t

An approximate alternative analysis involves only the averages of the NOMINAL and TEST results in each lot. Table 3 summarizes the data in this fashion. Repeating the simple comparison of these results using the Student's t statistical test produces Table 4.

This analysis could indicate no difference between the two splits at a = 0.05. However, by accepting a slightly larger a-risk (> 0.055), one could conclude that the differences seen between the two splits are significant and amount to about eight units in favor of the TEST split, assuming the objective is to increase the value of the parameter.

Arguably, this is not a major concession, and this analysis is slightly closer to the truth than the one illustrated in Table 2. Unfortunately this analysis is also misleading because of the method used to execute the trials. Regardless of the scenario, the correct analysis and interpretation of statistical tests requires the correct estimate of error. The more appropriate estimate of error in this case (SE of Diff of Means) is actually a blending of two sources of variation and their associated degrees of freedom (observations) in a manner that is not immediately obvious.

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A Split Plot Design Structure

Figure 2 is a schematic of the design structure with the wafer usage actually employed in this example. In Figure 2 the lines in the bottom row of blocks (numbers one to four) represent the wafers dedicated to each split from each lot. This diagram shows that the wafers in block (or trial) one received nominal processing and were from Lot 100. Similarly those in block four received the test process and were from Lot 101. Careful study of Figure 2 shows that a particular trial was associated with a particular split and a particular lot. That is, the trial is nested in the split used and in the lot of wafers available.

Table 5 contains the data from Table 1 with an additional column indicating the trial associated with a particular wafer as indicated in Figure 2.

Notice in Table 5 that Trial 1 is unique to the combination of Lot 100 and NOM Split. Similarly, Trial 4 is unique to the combination Lot 101 and TEST Split. This unique relationship indicates that the variables are "nested" as opposed to "crossed." In a crossed design structure, Trial 1 would appear in all splits with all lots.

One could easily number the trials 1 and 2 in both lots. Assigning unique numbers to each trial as illustrated here helps the analyst and some software algorithms complete the analysis correctly.

A nested analysis of variance applied to this data evaluates the effect of split on the parameter using the Fisher's F-ratio discussed in Part I and takes into account any variation between treatments of a similar type as well as any variation between the wafer lots investigated. The analysis of variance table compares the variation in the parameter introduced by placing wafers in the two splits to the amount of variation inherently present that applies directly to the splits. Table 6 reproduces the anova table related to this experiment generated from a software package.

In conventional analyses, R-square is a measure of the amount of variation in data that a particular model explains. Calculation of this value assumes that a single error term (residual) applies to the model. This example actually involves three error terms, so the value reported is meaningless.

Lot-to-lot variation in processing is a source of random noise. Similarly, variation between trials of a particular split is also a source of random noise. Therefore, the analysis treats both of these variables as random effects. The only fixed effect in the experiment is the possible difference between the two methods for doing the process – split. The "Signif." column is a calculation of the a-risk one would take in assuming that the effect of the splits is different from 0 (reject the null hypothesis of no difference). Clearly, this value is considerably > 0.05, so one would conclude that the differences seen between the splits are not statistically significant, despite their apparent magnitude.

In Table 6 since there are two TRIALS within each combination of LOTxSPLIT, the term TRIAL(LOT SPLIT) is equivalent to the interaction term LOTxSPLIT. This term has 1 degree of freedom (df) based on the number of levels of lots and splits in the experiment. The notation used here more directly reflects the relationship between trial number, lot, and split.

Notice also in Table 6 that the software has identified the denominator involved in computing each F-Ratio in the table. Line 4, the random effect due to TRIAL nested in LOT and SPLIT is the correct error term for evaluating the effect of SPLIT on the parameter in this experiment.

Appropriate software will support further evaluation of this experiment and allow calculation of the b-risk involved in determining an effect, given the design structure. Figure 3 shows both the a- and b-risks for this experiment, given its configuration.

The minimum point in Figure 3 near the left of the diagram at 0.0 represents the risk one takes in believing an effect might exist when one does not (a-risk). The reported power of the test – that is, the ability to detect an effect due to SPLIT when one twice the size of the experimental error is present – is 1-b or 0.445. Therefore, the experiment as constructed provides very poor likelihood of detecting a fairly large effect due to SPLIT. This could easily be the reason why the correct statistical test--analysis of variance using Fisher's F-ratio-- failed to detect a difference between splits.

The only option for increasing the power of statistical tests generally available to the typical experimenter is to increase the number of replications in a particular investigation. The obvious way to increase replication is to increase the number of lots and trials involved in the experiment – at considerable increase in cost. Part III of this article shows how the existing resources, properly used, can produce an experiment with excellent power to detect an effect.

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"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