5.4. Analysis of DOE data
5.4.7. Examples of DOE's
|A CCD with two responses||This example uses experimental data published in Czitrom and Spagon, (1997), Statistical Case Studies for Industrial Process Improvement. The material is copyrighted by the American Statistical Association and the Society for Industrial and Applied Mathematics, and is used with their permission. Specifically, Chapter 15, titled "Elimination of TiN Peeling During Exposure to CVD Tungsten Deposition Process Using Designed Experiments", describes a semiconductor wafer processing experiment (labeled Experiment 2).|
|Goal, response variables, and factor variables||
The goal of this experiment was to fit response surface models to the
two responses, deposition layer Uniformity and deposition
layer Stress, as a function of two particular controllable factors
of the chemical vapor deposition (CVD) reactor process. These factors
were Pressure (measured in torr) and the ratio of the gaseous
reactants H2 and WF6 (called
H2/WF6). The experiment also included
an important third (categorical) response - the presence or absence of
titanium nitride (TiN) peeling. The third response has been
omitted in this example in order to focus on the response surface
aspects of the experiment.
To summarize, the goal is to obtain a response surface model for two responses, Uniformity and Stress. The factors are: Pressure and H2/WF6.
|The design is a 13-run CCI design with 3 centerpoints||
The minimum and maximum values chosen for Pressure were 4 torr and 80
torr (0.5333 kPa and 10.6658 kPa). Although the international system
of units indicates that the standard unit for pressure is Pascal, or
1 N/m2, we use torr to be consistent with the analysis
appearing in the paper by Czitrom and Spagon.
The minimum and maximum H2/WF6 ratios were chosen to be 2 and 10. Since response curvature, especially for Uniformity, was a distinct possibility, an experimental design that allowed estimating a second order (quadratic) model was needed. The experimenters decided to use a central composite inscribed (CCI) design. For two factors, this design is typically recommended to have 13 runs with 5 centerpoint runs. However, the experimenters, perhaps to conserve a limited supply of wafer resources, chose to include only 3 centerpoint runs. The design is still rotatable, but the uniform precision property has been sacrificed.
|Table containing the CCI design points and experimental responses||
The table below shows the CCI design and experimental responses, in
the order in which they were run (presumably randomized). The last two
columns show coded
values of the factors.
(The reader can download the data as a
|Low values of both responses are better than high||Uniformity is calculated from four-point probe sheet resistance measurements made at 49 different locations across a wafer. The value in the table is the standard deviation of the 49 measurements divided by their mean, expressed as a percentage. So a smaller value of Uniformity indicates a more uniform layer - hence, lower values are desirable. The Stress calculation is based on an optical measurement of wafer bow, and again lower values are more desirable.|
Analysis of DOE Data
The analyses shown in this page can be generated using R code.
|Steps for fitting a response surface model||
The steps for fitting a response surface (second-order or quadratic)
model are as follows:
|Fitting a Model to the Uniformity Response, Simplifying the Model and Checking Residuals|
|Fit full quadratic model to Uniformity response||
Source Estimate Std. Error t value Pr(>|t|) ------ -------- ---------- ------- -------- Intercept 5.86613 0.41773 14.043 3.29e-05 Pressure -1.90967 0.36103 -5.289 0.00322 H2/WF6 -0.22408 0.36103 -0.621 0.56201 Pressure*H2/WF6 1.68617 0.71766 2.350 0.06560 Pressure^2 0.13373 0.60733 0.220 0.83442 H2/WF6^2 0.03373 0.60733 0.056 0.95786 Residual standard error: 0.7235 based on 5 degrees of freedom Multiple R-squared: 0.8716 Adjusted R-squared: 0.7431 F-statistic: 6.787 based on 5 and 5 degrees of freedom p-value: 0.0278
|Stepwise regression for Uniformity||
Start: AIC=-3.79 Model: Uniformity ~ Pressure + H2/WF6 + Pressure*H2/WF6 + Pressure^2 + H2/WF6^2 Step 1: Remove H2/WF6^2, AIC=-5.79 Model: Uniformity ~ Pressure + H2/WF6 + Pressure*H2/WF6 + Pressure^2 Step 2: Remove Pressure^2, AIC=-7.69 Model: Uniformity ~ Pressure + H2/WF6 + Pressure*H2/WF6 Step 3: Remove H2/WF6, AIC=-8.88 Model: Uniformity ~ Pressure + Pressure*H2/WF6The stepwise routine selects a model containing the intercept, Pressure, and the interaction term. However, many statisticians do not think an interaction term should be included in a model unless both main effects are also included. Thus, we will use the model from Step 2 that included Pressure, H2/WF6, and the interaction term. Interaction plots confirm the need for an interaction term in the model.
|Analysis of model selected by stepwise regression for Uniformity||
Source DF Sum of Sq Mean Sq F value Pr(>F) ------ -- --------- ------- ------- ------ Model 3 17.739 5.9130 15.66 0.0017 Total error 7 2.643 0.3776 Lack-of-fit 5 1.4963 0.2993 0.52 0.7588 Pure error 2 1.1467 0.5734 Residual standard error: 0.6145 based on 7 degrees of freedom Multiple R-squared: 0.8703 Adjusted R-squared: 0.8148 Source Estimate Std. Error t value Pr(>|t|) ------ -------- ---------- ------- -------- Intercept 5.9273 0.1853 31.993 7.54e-09 Pressure -1.9097 0.3066 -6.228 0.000433 H2/WF6 -0.2241 0.3066 -0.731 0.488607 Pressure*H2/WF6 1.6862 0.6095 2.767 0.027829
A contour plot and perspective plot of Uniformity provide a visual display of the response surface.
We perform a residuals analysis to validate the model assumptions.
We generate a normal plot, a box plot, a histogram and a
run-order plot of the residuals.
The residual plots do not indicate problems with the underlying assumptions.
|Conclusions from the analysis||
From the above output, we make the following conclusions.
|Fitting a Model to the Stress Response, Simplifying the Model and Checking Residuals|
|Fit full quadratic model to Stress response||
Source Estimate Std. Error t value Pr(>|t|) ------ -------- --------- ------- -------- Intercept 8.056791 0.179561 44.869 1.04e-07 Pressure 0.735933 0.038524 19.103 7.25e-06 H2/WF6 0.852099 0.198192 4.299 0.00772 Pressure*H2/WF6 0.069431 0.076578 0.907 0.40616 Pressure^2 -0.528848 0.064839 -8.156 0.00045 H2/WF6^2 -0.007414 0.004057 -1.827 0.12722 Residual standard error: 0.07721 based on 5 degrees of freedom Multiple R-squared: 0.9917 Adjusted R-squared: 0.9834 F-statistic: 119.8 based on 5 and 5 degrees of freedom p-value: 3.358e-05
|Stepwise regression for Stress||
Start: AIC=-53.02 Model: Stress ~ Pressure + H2/WF6 + Pressure*H2/WF6 + Pressure^2 + H2/WF6^2 Step 1: AIC=-53.35 Model: Stress ~ Pressure + H2/WF6 + Pressure^2 + H2/WF6^2The stepwise routine identifies a model containing the intercept, the main effects, and both squared terms. However, the fit of the full quadratic model indicates that neither the H2/WF6 squared term nor the interaction term are significant. A comparison of the full model and the model containing just the main effects and squared pressure terms indicates that there is no significant difference between the two models.
Model 1: Stress ~ Pressure + H2/WF6 + Pressure^2 Model 2: Stress ~ Pressure + H2/WF6 + Pressure^2 + Pressure*H2/WF6 + H2/WF6^2 Source DF Sum of Sq Mean Sq F value Pr(>F) ------ -- --------- ------- ------- ------- Model 1 2 0.024802 0.01240 2.08 0.22 Model 2 5 0.029804 0.00596In addition, interaction plots do not indicate any significant interaction.
The fact that the stepwise procedure selected a model for Stress containing a term that was not significant indicates that all output generated by statistical software should be carefully examined. In this case, the stepwise procedure identified the model with the lowest AIC (Akaike information criterion), but did not take into account contributions by individual terms. Other software using a different criteria may identify a different model, so it is important to understand the algorithms being used.
|Analysis of reduced model for Stress||
Source DF Sum of Sq Mean Sq F value Pr(>F) ------ -- --------- ------- ------- ------- Model 3 3.5454 1.1818 151.5 9.9e-07 Total error 7 0.0546 0.0078 Lack-of-fit 5 0.032405 0.00065 0.58 0.73 Pure error 2 0.022200 0.01110 Residual standard error: 0.0883 based on 7 degrees of freedom Multiple R-squared: 0.9848 Adjusted R-squared: 0.9783 Source Estimate Std. Error t value Pr(>|t|) ------ -------- ---------- ------- -------- Intercept 7.73410 0.03715 208.185 1.56e-14 Pressure 0.73593 0.04407 16.699 6.75e-07 H2/WF6 0.49686 0.04407 11.274 9.65e-06 Pressure^2 -0.49426 0.07094 -6.967 0.000218A contour plot and perspective plot of Stress provide a visual representation of the response surface.
We perform a residuals analysis to validate the model by
generating a run-order plot, box plot, histogram, and normal
probability plot of the residuals.
From the above output, we make the following conclusions.
|Response Surface Contours for Both Responses|
|Overlay contour plots||
We overlay the contour plots for the two responses to visually
compare the surfaces over the region of interest.
|Final response surface models||
The response surface models fit to (coded) Uniformity and Stress
Uniformity = 5.93 - 1.91*Pressure - 0.22*H2/WF6 + 1.70*Pressure*H2/WF6
Stress = 7.73 + 0.74*Pressure + 0.50*H2/WF6 - 0.49*Pressure2
|Trade-offs are often needed for multiple responses||The models and the corresponding contour plots show that trade-offs have to be made when trying to achieve low values for both Uniformity and Stress since a high value of Pressure is good for Uniformity while a low value of Pressure is good for Stress. While low values of H2/WF6 are good for both responses, the situation is further complicated by the fact that the Peeling response (not considered in this analysis) was unacceptable for values of H2/WF6 below approximately 5.|
|Uniformity was chosen as more important||In this case, the experimenters chose to focus on optimizing Uniformity while keeping H2/WF6 at 5. That meant setting Pressure at 80 torr.|
|Confirmation runs validated the model projections||A set of 16 verification runs at the chosen conditions confirmed that all goals, except those for the Stress response, were met by this set of process settings.|