5.
Process Improvement
5.4. Analysis of DOE data 5.4.7. Examples of DOE's


Data Source  
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 H_{2} and WF_{6} (called
H_{2}/WF_{6}). 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 H_{2}/WF_{6}. 

Experiment Description  
The design is a 13run 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/m^{2}, we use torr to be consistent with the analysis
appearing in the paper by Czitrom and Spagon.
The minimum and maximum H_{2}/WF_{6} 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.


Low values of both responses are better than high  Uniformity is calculated from fourpoint 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  
Steps for fitting a response surface model 
The steps for fitting a response surface (secondorder 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 3.29e05 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 Rsquared: 0.8716 Adjusted Rsquared: 0.7431 Fstatistic: 6.787 based on 5 and 5 degrees of freedom pvalue: 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 Lackoffit 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 Rsquared: 0.8703 Adjusted Rsquared: 0.8148 Source Estimate Std. Error t value Pr(>t)      Intercept 5.9273 0.1853 31.993 7.54e09 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. 

Residual plots 
We perform a residuals analysis to validate the model assumptions.
We generate a normal plot, a box plot, a histogram and a
runorder 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 1.04e07 Pressure 0.735933 0.038524 19.103 7.25e06 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 Rsquared: 0.9917 Adjusted Rsquared: 0.9834 Fstatistic: 119.8 based on 5 and 5 degrees of freedom pvalue: 3.358e05 

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 9.9e07 Total error 7 0.0546 0.0078 Lackoffit 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 Rsquared: 0.9848 Adjusted Rsquared: 0.9783 Source Estimate Std. Error t value Pr(>t)      Intercept 7.73410 0.03715 208.185 1.56e14 Pressure 0.73593 0.04407 16.699 6.75e07 H2/WF6 0.49686 0.04407 11.274 9.65e06 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. 

Residual plots 
We perform a residuals analysis to validate the model by
generating a runorder plot, box plot, histogram, and normal
probability plot of the residuals.


Conclusions 
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.


Summary  
Final response surface models 
The response surface models fit to (coded) Uniformity and Stress
were:
Uniformity = 5.93  1.91*Pressure  0.22*H_{2}/WF_{6} + 1.70*Pressure*H_{2}/WF_{6} Stress = 7.73 + 0.74*Pressure + 0.50*H_{2}/WF_{6}  0.49*Pressure^{2} 

Tradeoffs are often needed for multiple responses  The models and the corresponding contour plots show that tradeoffs 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 H_{2}/WF_{6} 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 H_{2}/WF_{6} below approximately 5.  
Uniformity was chosen as more important  In this case, the experimenters chose to focus on optimizing Uniformity while keeping H_{2}/WF_{6} 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. 