The art of what’s possible in semiconductor: Narrowing the variable list in design of experiments (DOE)
Last Modified: Feb 21, 2025 3:00 PM
Is it possible to optimize a plasma-enhanced chemical vapor deposition (PECVD) with just 25 test wafers? Absolutely! In semiconductor manufacturing, constraints such as a 25-wafer run on a tool are common, but they don’t have to limit success. Leveraging process data to inform experiment design is a winning strategy for getting effective results with limited resources.
Design of experiments (DOE) is the gold standard for optimizing the performance of equipment or tools that are used to manufacture semiconductor devices. Running an effective experiment requires a well thought-out data collection plan that generates the information needed to improve tool performance while staying within budgetary limitations.
Semiconductor fabs process hundreds to thousands of semiconductor devices (die) on a single silicon wafer. It can take roughly three months and a thousand process steps to fabricate a wafer. Experiments involving many wafers can be costly in manufacturing time and expense, as well as the loss of any die that cannot be sold. Due to these costs, it is common to have limits on the number of test wafers available to run experiments in a semiconductor fab.
In our scenario, a process engineer is tasked with optimizing a PECVD tool that deposits an insulative film on a wafer. To do this, our engineer will use experimental data from a DOE to develop a statistical model of the process that can be used to adjust the tool to achieve a desired deposition thickness and uniformity. Our process engineer is limited to one lot of 25 test wafers to run the experiment. Each wafer represents one test or run in a PECVD experiment.
There are dozens of variables that control the thickness and uniformity of a deposition film in a PECVD tool. Experiments with many factors require many wafers when using a DOE data collection strategy. For instance, if we assume the tool in our experiment has 10 adjustable continuous variables and all 10 variables interact with each other, the experiment would require a minimum of 56 test wafers to obtain a regression model. With only 25 wafers available, our engineer needs to narrow the factor list to the top four.
Fortunately, only a handful of the controls or variables available to our engineer have a big impact on deposition thickness and uniformity. Out of the many, which are important? Which variables interact? Which have a curvilinear contribution to deposition thickness? What is the best way to narrow variables from a large set of options to those that are crucial to the success of the experiment?
Answering all these questions is possible with process data and tools available in JMP and JMP Pro. The following process can serve as a guide.
Drawing on an understanding of business objectives, practical experience, and physics, our engineer has refined the goal and come up with a useful list of responses and factors.
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Goal: Define a PECVD tool recipe that achieves a uniform 6500 Å insulative deposition layer with no more than 25 wafers. |
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Next, the goal is to use observational process data to select both factors and model terms for our DOE regression model. Model term selection is a motivation for using a regression approach in lieu of other machine learning strategies.
It is important to note that in observational data, factors are not being manipulated. They are set at the level that the process requires. However, there is enough fluctuation in the process that signal from factors is observable with the right modeling strategy. If least squares regression is used, a coefficient for every term in the proposed model is determined. Figuring out which terms are important when all are included is challenging. Issues with observational data, such as multicollinearity, can make things even more difficult. The generalized regression personality in JMP Pro offers term selection techniques that are more effective when using observational data.
When using regression, the choice of the starting model requires some reasoning. In our case, a response surface model provides a good starting point as interactions and curvature terms are expected. Lasso and elastic net penalized estimation strategies are used in this exercise. In both methods, some coefficients can be estimated as zero, making them useful in term selection.
Both estimation methods perform similarly by AICc. AICc and BIC statistics are useful when assessing model fit for methods other than standard least squares methods. Models with a difference of four, where lower is better, are desirable.
Many of the model terms have been penalized to zero. Of the remaining terms, our engineer has chosen the top 10 ranked by Wald chi-square which is the square of the estimate/standard error. Wald chi-square is a measure of signal to noise for the terms in the model. The remaining terms have relatively low signal to noise and should not be considered for experimentation.
Statistical ranking should only be part of the consideration. Process understanding and experimental limitations should factor in as well. Since factors are not deliberately manipulated, it is possible that the process data may not contain enough variation to catch all the significant terms. It is also possible that some significant terms may not be practical to experiment with. In our example, data were available for electrostatic chuck change overs. An electrostatic chuck holds the wafer while it is processed. Changing a chuck in a randomized experiment, given only 25 wafers, is often not feasible. Also, pressure is a significant factor with significant interactions in the model, but it is a function of gas flow. Pressure can be left out of the experiment knowing that gas flow is included. Finally, spacing squared does not show up in the ranking but was included based on experience. Overall, our engineer’s understanding of the process has reduced the number of terms and the number of runs for the experiment.
The final experiment, a four-factor custom design, requires only 21 runs! There are four additional wafers. A good choice is to let the custom designer choose the remaining four runs optimally by setting the user-specified option to 25 in the custom designer.
In summary, process data, process expertise, and the Generalized Regression platform in JMP Pro can lead to effective outcomes for semiconductor tool optimization in situations where resources are limited.
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