Reliability HotWire

Issue 101, July 2009

Hot Topics

Variability Analysis in DOE++ and its Application

DOE++ offers the option of variability analysis for two level factorial designs and Response Surface Method designs. Variability analysis can be used to study the natural variation in the response for a particular combination of the levels of the investigated factors. This article illustrates variability analysis in DOE++ using a two level full factorial experiment as an example. Note that to conduct a variability analysis, it is important to have a replicated experiment (or multiple readings taken at all factor level combinations in a random order [1]).

To download the complete *.rdoe file for this example, right-click here to save the file to your computer. You must have DOE++ installed in order to be able to open this file. Free demonstration copies of the software are available for download from http://Download.ReliaSoft.com.

Full Factorial Experiment Example  
Consider the fabrication of integrated circuit (IC) devices. One of the initial steps in the fabrication 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 the thickness is 14.5 μm with specification limits of 14.5 ± 0.5. [2]

It is found that current operation settings are causing variations that exceed the specification of 1.0 μm. It is therefore decided to conduct an experiment to find the factors that affect the variation. Four experimental factors are investigated in the experiment as follows:

Experimental settings

The design used is the 24 design. Six replicates of the experiment are conducted, giving a total of 96 observations of the epitaxial layer thickness. The observations are recorded in a random order by following the run order column generated by DOE++. A portion of the DOE++ design sheet is shown in Figure 1. To see the complete design, you can download the DOE++ file via the link in the introduction to this article. The design is available in the "2 Level Full Factorial Design" Folio.

 DOE++ Folio
Figure 1: The Full Factorial Experiment Design

Experiment Results 
The results of the analysis of the full factorial experiment are available on the Analysis tab as ANOVA and Regression Information tables. Results can also be viewed using plots, the most commonly used being the Pareto chart that is shown in Figure 2.

Pareto Chart
Figure 2: Pareto Chart for the Full Factorial Experiment

From this plot, you can see that factors B and D and the interaction CD are significant. Therefore, the factors affecting the thickness of the epitaxial layer are Nozzle Position, Deposition Time and the interaction of Deposition Temperature and Deposition Time.

The experiment can now be re-analyzed by including only the significant effects in the model. To do this, click the Select Effects icon.

Select Effects Icon

 In the Effects window, click the Select Significant Effects button to include only the significant effects, as shown next.

Effects Window 
Figure 3: Selection of Significant Effects

The Regression Information table obtained in this manner includes information on coefficients of the significant effects, as shown in Figure 4.

Regression Information Table
Figure 4: Regression Information Table

 Based on these results, the relation between the epitaxial layer thickness (y) and the significant effects is as follows:

y = 14.1613 + 0.0836B – 0.0387C + 0.245D – 0.1725CD

Note that DOE++ includes the main effect, C, in the model. This is based on the principle of hierarchy, which states that if an interaction is significant, the corresponding main effects will also be included in the model.[1] Therefore, since the interaction CD is significant, the main effects of C and D have to be included in the model.

Optimization
The specification for the epitaxial layer thickness is 14.5 ± 0.5 μm. The relation shown above can be optimized to obtain the settings of B, C and D that give the required nominal thickness of 14.5. To do this, click the Optimization icon and use the settings shown in Figure 5. The values in the figure indicate that for the present optimization a value of 14.5 is highly desired, but that solutions that give values in the range of 14 to 15 are also acceptable.

Optimization Icon

Optimization Settings Window
Figure 5: Optimization Settings

The results of the optimization are shown in Figure 6. These results can be viewed by clicking the View All Solutions icon.

View All Solutions Icon

Optimization Results
Figure 6: Optimization Results

There are two possible solutions:

  • Using a setting of 2 for the Nozzle Position (factor B) along with a Deposition Temperature (factor C) setting of 1210.7 C and a Deposition Time (factor D) setting of high.
  • Using a setting of 6 for the Nozzle Position (factor B) along with a Deposition Temperature (factor C) setting of 1214.8 C and a Deposition Time (factor D) setting of high.

To decide which of the two solutions is more appropriate for the present case, it is important to conduct a variability analysis. The variability analysis investigates which factors affect the variation seen in the epitaxial layer thickness. It can therefore provide an indication as to the choice of the solution that could minimize the variations.

Variability Analysis
The variability analysis for the present example can be conducted by clicking the Variability Analysis icon and using the settings shown in Figure 7.

Variability Analysis Icon

Variability Analysis Settings
Figure 7: Variability Analysis Settings

The analysis calculates the standard deviation for all of the factor level combinations used in the full factorial experiment. The calculated standard deviation is a measure of the variability in the original response (i.e. the epitaxial layer thickness) and is used as the response of interest in the variability analysis, as shown in Figure 8.

Doe++ Folio with Standard Deviation
Figure 8: Standard Deviation Values as the New Response

This new response is now investigated for the effect of the factors A, B, C and D. Note that a logarithmic transformation is typically applied to the calculated standard deviation values, because the logarithm of these values usually are normally distributed.[1]

The results of the variability analysis are shown in the Regression Information table in Figure 9. The results indicate that factors B and C affect variability in the epitaxial layer thickness.

Variability Analysis Results
Figure 9: Variability Analysis Results

Based on the coefficients obtained, we can conclude that using the high level of both these factors decreases variability in the epitaxial layer thickness, since the relationship between the standard deviations of the epitaxial layer thickness and the significant factors is as follows:

LN Thickness STD = –0.7225 –0.0293B – 0.028C

Keeping this in mind, the optimal solution to reach an epitaxial layer thickness of 14.5 and minimize variability is the second of the two possible solutions shown in Figure 6. Therefore, to reach the specification of 14.5 ± 0.5 μm for the epitaxial layer thickness, the Nozzle Position should be set at position 6, the Deposition Temperature should be set at 1214.8 C and the Deposition Time should be set at the high level.

Conclusion
This article illustrates variability analysis in DOE++ using the example of a two level full factorial experiment. Variability analysis can be used to study the variation in the response and obtain the settings of the investigated factors to reduce the variation while meeting the goal of the response.

References
[1] ReliaSoft Corporation, Experiment Design and Analysis Reference, Tucson, AZ: ReliaSoft Publishing, 2008.
[2] Wu, C.F. J. and Hamad, M., Experiments: Planning, Analysis, and Parameter Design Optimization, New York: John Wiley & Sons, Inc., 2000.

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