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.
[Click
to Enlarge]
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. | 
