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

5.4.7.3.

Response surface model example
Data Source
A CCD DOE with two responses This example uses experimental data published in Czitrom and Spagon, (1997), Statistical Case Studies for Industrial Process Improvement. This material is copyrighted by the American Statistical Association and the Society for Industrial and Applied Mathematics, and 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. That part of the experiment has been omitted in this example, in order to focus on the response surface model aspects.

To summarize, the goal is to obtain a response surface model for each response where the responses are: "Uniformity" and "Stress". The factors are: "Pressure" and "H2/WF6".

Experiment Description
The design is a 13-run CCI design with 3 centerpoint runs The maximum and minimum values chosen for pressure were 4 torr and 80 torr. The lower and upper 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 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.

Run
Pressure
H2/WF6
Uniformity
Stress
Coded
Pressure
Coded
H2/WF6
1
 80
 6 
4.6 
8.04 
1 
 0 
2
42
6
6.2
7.78
0
0
3
     68.87
      3.17 
3.4 
7.58 
      0.71 
    -0.71 
4
     15.13
      8.83 
6.9 
7.27 
    -0.71 
     0.71 
5
  4 
6 
7.3 
 6.49 
-1 
0 
6
42
6
 6.4
 7.69
 0
0 
7
    15.13
      3.17 
 8.6
 6.66
    -0.71
    -0.71 
8
 42
 2
 6.3
 7.16
 0
-1 
9
      68.87
      8.83
 5.1
 8.33
      0.71
      0.71 
10
 42
 10
 5.4
 8.19
 0
1 
11
 42
 6
 5.0
 7.90
 0
0 

Low values of both responses are better than high Note: "Uniformity" is calculated from four-point probe sheet resistance measurements made at 49 different locations across a wafer. The value used 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 Using JMP 4.02
Steps for fitting a response surface model using JMP 4.02 (other software packages generally have similar procedures) The steps for fitting a response surface (second-order or quadratic) model using the JMP 4.02 software for this example are as follows:
  1. Specify the model in the "Fit Model" screen by inputting a response variable and the model effects (factors) and using the macro labeled "Response Surface".
  2. Choose the "Stepwise" analysis option and select "Run Model".
  3. The stepwise regression procedure allows you to select probabilities (p-values) for adding or deleting model terms. You can also choose to build up from the simplest models by adding and testing higher-order terms (the "forward" direction), or starting with the full second-order model and eliminating terms until the most parsimonious, adequate model is obtained (the "backward" direction). In combining the two approaches, JMP tests for both addition and deletion, stopping when no further changes to the model can be made. A choice of p-values set at 0.10 generally works well, although sometimes the user has to experiment here. Start the stepwise selection process by selecting "go".
  4. "Stepwise" will generate a screen with recommended model terms checked and p-values shown (these are called "Prob>F" in the output). Sometimes, based on p-values, you might choose to drop, or uncheck, some of these terms. However, follow the hierarchy principle and keep all main effects that are part of significant higher-order terms or interactions, even if the main effect p-value is higher than you would like (note that not all analysts agree with this principle).
  5. Choose "make model" and "run model" to obtain the full range of JMP graphic and analytical outputs for the selected model.
  6. Examine the fitted model plot, normal plot of effects, interaction plots, residual plots, and ANOVA statistics (R2, R2 adjusted, lack of fit test, etc.). By saving the residuals onto your JMP worksheet you can generate residual distribution plots (histograms, box plots, normal plots, etc.). Use all these plots and statistics to determine whether the model fit is satisfactory.
  7. Use the JMP contour profiler to generate response surface contours and explore the effect of changing factor levels on the response.
  8. Repeat all the above steps for the second response variable.
  9. Save prediction equations for each response onto your JMP worksheet (there is an option that does this for you). After satisfactory models have been fit to both responses, you can use "Graph" and "Profiler" to obtain overlaid surface contours for both responses.
  10. "Profiler" also allows you to (graphically) input a desirability function and let JMP find optimal factor settings.
The displays below are copies of JMP output screens based on following the above 10 steps for the "Uniformity" and "Stress" responses. Brief margin comments accompany the screen shots.
Fitting a Model to the "Uniformity" Response, Simplifying the Model and Checking Residuals
Model specification screen and stepwise regression (starting from a full second-order model) output We start with the model specification screen in which we input factors and responses and choose the model we want to fit. We start with a full second-order model and select a "Stepwise Fit". We set "prob" to 0.10 and direction to "Mixed" and then "Go".

JMP menus and output for the Uniformity response

The stepwise routine finds the intercept and three other terms (the main effects and the interaction term) to be significant.

JMP output for analyzing the model selected by the stepwise regression for the Uniformity response The following is the JMP analysis using the model selected by the stepwise regression in the previous step. The model is fit using coded factors, since the factor columns were given the property "coded".

JMP output for analyzing model selected by stepwise regression for the Uniformity response
Conclusions from the JMP output From the above output, we make the following conclusions.
  • The R2 is reasonable for fitting "Uniformity" (well known to be a hard response to model).
  • The lack of fit test does not have a problem with the model (very small "Prob > F " would question the model).
  • The residual plot does not reveal any major violations of the underlying assumptions.
  • The normal plot of main effects and interaction effects provides a visual confirmation of the significant model terms.
  • The interaction plot shows why an interaction term is needed (parallel lines would suggest no interaction).
Plot of the residuals versus run order We next perform a residuals analysis to validate the model. We first generate a plot of the residuals versus run order.

Plot of the residuals versus run order
Normal plot, box plot, and histogram of the residuals Next we generate a normal plot, a box plot, and a histogram of the residuals.

Normal plot, box plot, and histogram of residuals

Viewing the above plots of the residuals does not show any reason to question the model.

Fitting a Model to the "Stress" Response, Simplifying the Model and Checking Residuals
Model specification screen and stepwise regression (starting from a full second-order model) output We start with the model specification screen in which we input factors and responses and choose the model we want to fit. This time the "Stress" response will be modeled. We start with a full second-order model and select a "Stepwise Fit". We set "prob" to 0.10 and direction to "Mixed" and then "Go".

JMP menus and output for the Stress response
The stepwise routine finds the intercept, the main effects, and Pressure squared to be signficant terms.
JMP output for analyzing the model selected by the stepwise regression for the Stress response The following is the JMP analysis using the model selected by the stepwise regression, which contains four significant terms, in the previous step. The model is fit using coded factors, since the factor columns were given the property "coded".

Conclusions from the JMP output From the above output, we make the following conclusions.
  • The R2 is very good for fitting "Stress".
  • The lack of fit test does not have a problem with the model (very small "Prob > F " would question the model).
  • The residual plot does not reveal any major violations of the underlying assumptions.
  • The interaction plot shows why an interaction term is needed (parallel lines would suggest no interaction).
Plot of the residuals versus run order We next perform a residuals analysis to validate the model. We first generate a plot of the residuals versus run order.

Plot of the residuals versus run order
Normal plot, box plot, and histogram of the residuals Next we generate a normal plot, a box plot, and a histogram of the residuals.

Normal plot, box plot, and histogram of residuals

Viewing the above plots of the residuals does not show any reason to question the model.

Response Surface Contours for Both Responses
"Contour Profiler" and "Prediction Profiler" JMP has a "Contour Profiler" and "Prediction Profiler" that visually and interactively show how the responses vary as a function of the input factors. These plots are shown here for both the Uniformity and the Stress response.

Contour Profiler and Prediction Profiler plots for both the Uniformity and the Stress response

Prediction Profiles Desirability Functions for Both Responses
Desirability function: Pressure should be as high as possible and H2/WF6 as low as possible You can graphically construct a desirability function and let JMP find the factor settings that maximize it - here it suggests that Pressure should be as high as possible and H2/WF6 as low as possible.

JMP plot of desirability function

Summary
Final response surface models The response surface models fit to (coded) "Uniformity" and "Stress" were:

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 These models and the corresponding profiler 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.
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