## Controlling and Optimizing Wafer Yields

Semiconductors (chips) are produced on wafers that contain hundreds of chips. The wafer yield is defined to be the proportion of these chips that are acceptable for use, and clearly control engineers aim to maximise this yield during manufacturing. This yield is greatest when the thickness of the coating material to the wafer is uniform. Control engineers are having difficulties producing a uniform coating (Y) at their plant. The main process variables that the engineers need to contend with are speed (X1), pressure (X2) and distance (X3).

Experimentation and Data Collection

To get an understanding of how to produce a uniform thickness, experimentation was carried out within the laboratories at Swansea University. Using wafer samples supplied by the company, coating thickness was measured at a number of different locations along each
wafer and the standard deviation of these measurements was taken as a measure of the uniformity of the coating thickness for each wafer. More specifically, two separate experiments were carried out.
Experiment 1.
Twenty wafer samples were tested at a low speed and a further twenty wafers were tested at high speed. In each of these forty tests, both the pressure and distance were held. The standard deviation in the thickness measurements made along each of the 40 wafers is shown
in Sheet1 of the Data Sheet.
Experiment 2.
In this experiment, the effect of the 3 process parameters (speed X1, pressure X2 and distance X3) on the standard deviation in wafer thickness was studied. Three different values were used for these process variables and were coded -1, 0 and +1 to signify low, medium
and high amounts for these variables. These test conditions and the corresponding standard deviations (Y) are shown in Sheet2 of Data Sheet.
Objectives.
You are required to write a project report (template available on blackboard) that carries out a detailed statistical investigation on the experimental data discussed above. The project should provide an answer as to what processing conditions produce the most uniform
coating thickness for the wafer and therefore which maximises the wafer yield.
When writing the report for the project, structure it in a way that allows you to cover and address all the following questions and issues in a way that reveals a progressively greater understanding of the two experimental data sets.
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1. Describe the variability present in each data set of Experiment 1. Then using appropriate data displays, describe each data set in Experiment 1, highlighting any similarities or differences that may exist between the two speeds.
2. Using the data sets collected in Experiment 1; construct an appropriate parametric and/or non-parametric test to assess the claim that the uniformity in coating thickness is not the same for each speed. When writing up your analysis of this claim state any assumptions that need to be made in conducting these tests and, if appropriately, carry out tests to validate these assumptions. Discuss also the advantages and disadvantages of each test.
3. Using the data set collected in Experiment 2 and the technique of multiple least squares, estimate the  parameters of the following second-order response surface model:
       
    
j
i
i
j i
i i j
i
i i i
i
Y iX X X X
2
1
3
1
2
3
1
3
1
0
where Y is the standard deviation in coating thickness, X1 is the speed, X2 is the pressure and X3 is the distance.  is the prediction error or residual.
When writing up your analysis of this model, describe how well this model fits the data, which variables are statistically significant (important) and what meanings can be attached to the  parameters. State any assumptions that need to be made in assessing such
statistical significance, and if appropriate carryout tests or construct scatter plots to validate these assumptions.
4. Derive a simplified version of the above model that includes only the statistically significant variables.
When writing up your analysis, describe how well this simplified model fits the data, the meaning of the parameters, the degree of accuracy achievable when predicting coating thickness uniformity using this simplified model (as described by a 95-confidence interval on the actual v prediction plot). Make full use of any suitable 2D or 3D scatter plots when writing your final report.