5.4.7.3. Response surface model example

The goal is to obtain a response surface model for  each response
 
 

The responses are: "Uniformity" and "Stress"

The factors are: "Pressure" and "H₂/WF₆"
 
 
 
 

The design is a 13 run CCI design with 3 center point runs 
 
 
 
 
 
 

Low values of both responses are better than high
 
 
 
 
 

Steps for fitting a response surface model using JMP 4.04 (other software packages generally  have similar procedures) 

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 Society 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) run in order 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.

Experiment Description

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. This design is typically recommended to have 13 runs with 5 center point runs. However, the experimenters, perhaps to conserve a limited supply of wafer resources, chose to include only 3 center point runs. The design is still rotatable, but the uniform precision property has been sacrificed.

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

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

The steps for fitting a response surface (second order or quadratic) model using 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 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). By selecting the "mixed" direction, JMP tries both directions until converging on the needed model terms. 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.
5. Choose "make model" and "run model" to get 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 (R squared, R squared 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 on to 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 overlayed surface contours for both responses.
10. "Profiler" also allows you to (graphically) input a desirability function and let JMP find optimal factor settings.

We start with a full second order model and select a "Stepwise Fit" 

Set "prob" to 0.10 and direction to "Mixed" and then "Go"

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

The model is fit using coded factors, since the factor columns were given the property "coded" 

R squared is reasonable for fitting "Uniformity" (well known to be a hard response to model)
 
 

Lack of fit test does not have a problem with the model (very small "Prob > F " would question the model)
 
 
 
 
 
 
 
 
 
 
 
 
 

Residual plot looks OK
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

A visual confirmation of the significant model terms using a Normal plot
 
 
 
 
 
 
 
 
 
 

The interaction plot shows why an interaction term is needed (parallel lines would suggest no interaction)
 
 

A contour plot for the "Uniformity" response
 
 
 
 
 
 
 
 
 

We input a full second order model and choose "Stepwise Fit" 

Set "prob" to 0.10 and direction to "mixed" and then "Go"

The stepwise routine finds the intercept,  main effects and Pressure squared to be significant terms

Note: the model is fit to coded values of the factors, since the factor columns were given the property "coded" 
 

No lack of fit and a very good R squared for this model
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Residual plot looks OK
 
 
 
 
 
 
 
 
 
 
 
 

The interaction plot shows why no interaction term was needed (parallel lines)
 
 
 
 
 
 
 
 
 

A contour plot of "Stress" as a function of the factors Pressure and H₂/WF₆
 
 
 
 
 
 

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

Prediction Profiles and a Desirability Functions Example for Both Responses 
 

In this case "Uniformity" was chosen as more important

Confirmation runs validated the model projections

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

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 around 5. The experimenters chose to focus on optimizing "Uniformity" while keeping H₂/WF₆ at 5. That meant setting "Pressure" at 80 torr. 

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.