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, rightclick
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
sixfaceted 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:
The design used is the 2^{4} 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.
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.
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 reanalyzed by
including only the significant effects in the model. To do this,
click the Select Effects icon.
In the Effects window, click the
Select Significant Effects button to include only the
significant effects, as shown next.
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.
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.
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.
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.
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.
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.
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.
