Two Level Factorial Designs: Example

The data set used in this example is available in the example database installed with the software (called "Weibull20_DOE_Examples.rsgz20"). To access this database file, choose File > Help, click Open Examples Folder, then browse for the file in the Weibull sub-folder.

The name of the example project is "Factorial - Two Level Full Factorial Design."

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 µm.

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. 96 units are available for testing, and the factors and levels shown next will be used.

Factor

Name

Low Level

High Level

A

Susceptor-Rotation Method

Continuous

Oscillating

B

Nozzle Position

2

6

C

Deposition Temperature

1210

1220

D

Deposition Time

Low

High

Designing the Experiment

To find the important factors, the experimenters create a two level full factorial design folio, perform the experiment according to the design, and then enter the response values into the folio for analysis. The design matrix and the response data are given in the "Two Level Full Factorial Design" folio. The following steps describe how to create this folio on your own.

Analysis and Results

The data set for this example is given in the "Two Level Full Factorial" folio of the example project. After you enter the data from the folio, you can perform the analysis by doing the following:

Note: To minimize the effect of unknown nuisance factors, the run order is randomly generated when you create the design. Therefore, if you followed these steps to create your own folio, the order of runs on the Data tab may be different from that of the folio in the example file. This can lead to different results. To ensure that you get the very same results described next, show the Standard Order column in your folio, then click a cell in that column and choose Sheet > Sheet Actions > Sort > Sort Ascending. This will make the order of runs in your folio the same as that of the example file. Then copy the response data from the example file and paste it into the Data tab of your folio.

In the window that appears, select the All Terms check box. Then click OK.

Then choose Pareto Charts - Regression from the Plot Type drop-down list. The following plot appears. The blue line in the plot marks the critical value determined by the risk level specified on the Analysis Settings page of the control panel. If the bar goes past the blue line, then the effect is considered significant.

Optimization

To optimize the response, you should work with a model that only includes the significant terms. The results for the reduced model of the analysis are given in the "Reduced Model" folio of the example project. The following steps describe how to create this folio on your own.

To automatically select only those terms that were found to be significant, click the Select Significant Terms icon. Since the interaction of factors C and D is significant, C must also be included in the model. So select the C:Deposition Temperature check box as well. Then click OK.

The factor level combinations that will produce the optimum response value are given in the "Optimal Solution Plot" folio. The following steps describe how to create this folio on your own.

This plot shows one combination of factor settings that will produce the target response value (marked with a horizontal dotted line). Factors B and D are qualitative, so the plot uses dots (i.e., a discrete function) to show what the response level would be given either one of the two factor levels used in the experiment. Factor C is quantitative, so a continuous function is used to calculate the optimal temperature.

Two optimal solutions for factors B, C and D were found, and they are shown next.

Conclusion

Two optimum solutions were found that give the same predicted thickness. In choosing between these two combinations of factor levels, it is important to identify which is better in terms of variability. To do this, you can perform a variability analysis, where the response is the standard deviation of the observations at each setting. (See Variability Analysis Example.)

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