Question 9.10: The thickness of the silicon dioxide layer on a semiconducto...
The thickness of the silicon dioxide layer on a semiconductor wafer is crucial to its performance. In the article “Virgin Versus Recycled Wafers for Furnace Qualification: Is the Expense Justified?” (V. Czitrom and J. Reece, Statistical Case Studies for Process Improvement, SIAM-ASA, 1997:87- 103), oxide layer thicknesses were measured for three types of wafers: virgin wafers, wafers recycled in-house, and wafers recycled by an external supplier. In addition, several furnace locations were used to grow the oxide layer. A two-way ANOVA for three runs at one wafer site for the three types of wafers at three furnace locations was performed. The data are presented in the following table, followed by the results (from MINITAB).
\begin{array}{cllll}\hline \text{Furnace Location}& \text{Wafer Type}& Oxide& Layer& Thickness\\\hline 1 & \text{Virgin}& 90.1 & 90.7 & 89.4 \\1 & \text{In-house}& 90.4 & 38.8 & 90.6 \\1 & \text{External}& 92.6 & 90.0 & 93.3 \\2 & \text{Virgin}& 91.9 & 88.6 & 89.7 \\2 & \text{In-house}& 90.3 & 91.9 & 91.5 \\2 & \text{External}& 88.3 & 88.2 & 89.4 \\3 & \text{Virgin}& 88.1 & 90.2 & 86.6 \\3 & \text{In-house}& 91.0 & 90.4 & 90.2 \\3 & \text{External}& 91.5 & 89.8 & 89.8 \\\hline\end{array}
Since recycled wafers are cheaper, the company hopes that there is no difference in the oxide layer thickness among the three types of chips. If possible, determine whether the data are consistent with the hypothesis of no difference. If not possible, explain why not.
Two-way ANOVA for Thickness versus Wafer , Location
\begin{array}{lrrrrr}\text{Source}& \text{DF}& \text{SS}& \text{MS}& F & P \\\text{Wafer}& 2 & 5.8756 & 2.9378 & 2.07 & 0.155 \\\text{Location}& 2 & 4.1089 & 2.0544 & 1.45 & 0.262 \\\text{Interaction}& 4 & 21.349 & 5.3372 & 3.76 & 0.022 \\\text{Error}& 18 & 25.573 & 1.4207 & & \\\text{Total}& 26 & 56.907 & & &\end{array}
The P-value for the interactions is 0.022, which is small. Therefore the additive model is not plausible, so we cannot interpret the main effects. A good thing to do is to make a table of the cell means. Table 9.6 presents the sample mean for each treatment.
From Table 9.6, it can be seen that the thicknesses do vary among wafer types, but no one wafer type consistently produces the thickest, or the thinnest, oxide layer. For example, at furnace location 1 the externally recycled wafers produce the thickest layer while the in-house recycled wafers produce the thinnest. At furnace location 2 the order is reversed: The in-house wafers produce the thickest layer while the external ones produce the thinnest. This is due to the interaction of furnace location and wafer type.
| TABLE 9.6 Sample means for e ach treatment | ||||
| Furnace Location |
Wafer Type | |||
| Virgin | In-House | External | Row Mean | |
| 1 | 90.067 | 89.933 | 91.967 | 90.656 |
| 2 | 90.067 | 91.233 | 88.633 | 89.978 |
| 3 | 88.3 | 90.533 | 90.367 | 89 .733 |
| Column Mean | 89.478 | 90.566 | 90.322 | |