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unit8

# unit8 - Stat 4220 Unit 8 Full Factorial Experiments at Two...

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Stat 4220 Unit 8 Full Factorial Experiments at Two Levels 1 2 k Full Factorial Designs In many scientific investigations, the interest lies in the study of effects of two or more factors simultaneously. Factorial designs are most commonly used for this type of investigation. This chapter considers the important class of factorial designs for k factors each at two levels. Because this class of designs requires 2 × 2 × · · · × 2 = 2 k observations, it is referred to as 2 k factorial designs . To distinguish it from the class of fractional factorial designs (which will be considered in the next chapter), it is also called the class of 2 k full factorial designs. 2 An Epitaxial Layer Growth Experiment One of the initial steps in fabricating integrated circuit (IC) devices is to grow an epitaxial layer on polished silicon wafers. The wafers are mounted on a six-faceted cylinder (two wafers per facet), called a susceptor, which is spun inside a metal bell jar. The jar is injected with chemical vapors through nozzles at the top of the jar and heated. The process continues until the epitaxial layer grows to a desired thickness. The nominal value for thickness is 14 . 5 μ m with specification limits of 14 . 5 ± 0 . 5 μ m. In other words, it is desirable that the actual thickness be as close to 14.5 as possible and within the limits [14, 15]. The current settings caused variation to exceed the specification of 1 . 0 μ m. Thus, the experimenters needed to find process factors that could be set to minimize the epitaxial layer nonuniformity while maintaining average thickness as close to the nominal value as possible. In this section, a simplified version of this experiment is discussed. The adapted epitaxial layer growth experiment uses a 16-run full factorial design as given in Table 1. In the epitaxial layer growth process, suppose that the four experimental factors, susceptor rotation method, nozzle position, deposition temperature, and deposition time (labeled A, B, C and D ) are to be investigated at the two levels given in Table 2. The purpose of this experiment is to find process conditions, i.e., combinations of factor levels for A, B, C , and D , under which the average thickness is close to the target 14 . 5 μ m with variation as small as possible. The most basic experimental design or plan is the full factorial design, which studies all possible combinations of factors at two levels. In Table 1, y denotes the average 1

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Table 1: Design Matrix and Thickness Data, Adapted Epitaxial Layer Growth Experiment Factor Run A B C D y z 1 - - - + 14.59 - 1 . 309 2 - - - - 13.59 - 1 . 234 3 - - + + 14.24 - 1 . 317 4 - - + - 14.05 - 1 . 625 5 - + - + 14.65 - 1 . 510 6 - + - - 13.94 - 1 . 585 7 - + + + 14.40 - 1 . 505 8 - + + - 14.14 - 1 . 537 9 + - - + 14.67 - 1 . 313 10 + - - - 13.72 - 1 . 302 11 + - + + 13.84 - 1 . 514 12 + - + - 13.90 - 1 . 474 13 + + - + 14.56 - 1 . 483 14 + + - - 13.88 - 1 . 374 15 + + + + 14.30 - 1 . 386 16 + + + - 14.11 - 1 . 650 thickness and z denotes the log of sample variance. In this chapter, we shall analyze the response y .
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