Example 2 - Two Level Full Factorial Design
[Download DOE++ Example File (*.rdoe)]
Background
Consider that one of the initial steps in fabricating integrated circuit (IC) devices is to grow an epitaxial layer on polished silicon wafers. The wafers are mounted on a six-faceted cylinder (two wafers per facet), called a susceptor, which is spun inside a metal bell jar. The jar is injected with chemical vapors through nozzles at the top of the jar and heated. The process continues until the epitaxial layer grows to a desired thickness. The nominal value for thickness is 14.5 µm with specification limits of 14.5 ± 0.5.
The current settings caused variations that exceeded the specification by 1.0 µm. Thus, the experimenters first need to find the factors that affect the process. Then further experiments will be conducted to optimize the process. There are four experimental factors.
Experiment Design
To find the important factors, the experimenters use DOE++ to design a two level full factorial design. The design-specific settings, the factor properties and the response properties used are shown next.
The design matrix and the response data are given in the " 2 Level Full Factorial Design" Folio. To view the data set, click here.
Analysis Part I
Step 1: After performing the experiment according to the design and recording the results, the experimenters enter the data set into the Standard Folio, as shown next.
The data set as entered in the DOE++ Folio.
Note that not all rows are shown in the figure above. There are 96 rows of data in the Folio.
Step 2: The data set is analyzed with the default risk (significance) level of 0.1, using individual terms and including all effects up to four-way interactions in the analysis.
Step 3: A Pareto chart is created, as shown next.
The Pareto chart shows that effects B, D and CD are significant.
Analysis Part II
The results for the reduced model and the optimization are given in the "Optimization" Folio.
Step 1: The design Folio is duplicated and the copy is named "Optimization."
Step 2: Only the significant effects are selected to calculate the new model, as shown next.
Note that selections in the Effects window must be hierarchical (i.e. for any second order or greater effect selected for inclusion, all related main effects must also be included). Thus, effect C must be included in order to include CD in the model.
Step 3: The reduced model is calculated. The Regression Information table from the Analysis tab is shown next.
Step 4: Optimization is performed using the settings shown next.
The optimization results are:
Analysis Part III
Although both optimum solutions give the same predicted thickness, the variability at these two settings may be different. It is important to identify which solution is better in terms of variability. To do this, the experimenters study the variability of each setting. In variability analysis, the response is the standard deviation of the observations at each setting.
The results of the variability analysis are given in the "Variability Analysis" Folio.
Step 1: The design Folio is duplicated and the copy is named "Variability Analysis."
Step 2: The Thickness response is selected for variability analysis and all factors are selected to be considered in the variability analysis, as shown next.
This results in a new response column called Thickness Std. being added to the Folio. The values in this column are standard deviations of the observations at each factor setting.
Step 3: The Thickness Std. column is selected for inclusion in the analysis by selecting the checkbox in the column header and the checkbox in the Thickness column header is cleared.
Step 4: The log transformation is applied to the Thickness Std. response, as shown next. This is generally appropriate if a response is standard deviation.
Step 5: The model is calculated. The Regression Information table from the Analysis tab is shown next.
Applying this model to the two optimal solutions in the analysis for the Thickness response shows that the first solution has less variability.
The predicted results for the Thickness and Thickness Std responses are given in the "Optimal Solution" Spreadsheet, as shown next.
Conclusions
The Pareto chart shows that effects B, D and CD are significant for thickness. Using the reduced model for the optimization, the optimum settings for factors B, C and D are:
By considering the expected variability for these two solutions, the first solution is found to be the best one.
For more discussion on how to optimize both mean response and variability of response, please refer to Wu and Hamada’s book, Experiments: Planning, Analysis, and Parameter Design Optimization, John Wiley & Sons, New York, 2000.
