Controlling Wafer Yields

Background.
Semiconductors (chips) are produced on wafers that contains 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. Usingwafer samples supplied by the company, coating thickness
was measured at a number of different locations along each wafer and the standard deviation in these
measurements was taken as a measures of the uniformity of the coating thickness for each wafer. More
specifically, two separate experiments were carried out.
Experiment 1.
Twenty wafersamples were tested at a low speed and a further twenty wafers were tested at a 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 are shown in Sheet1 of ElecProject.xlsx.
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
ElecProject.xlsx.
Objectives.
You are required to write a mini project 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 this mini 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.
1. Describe the sources of 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. [20
%]

2. Using the data sets collected in Experiment 1, construct an appropriate parametric and 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 appropriate carry out tests to validate these assumptions. Discuss also
the advantages and disadvantages of each test. [20 %]

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:
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 is assessing such statistical significance, and
if appropriate carry out tests or construct scatter plots to validate these assumptions.
[30 %]
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 an actual v prediction plot). Use
the model to find the process conditions required to minimise coating thickness, making full use of any
suitable 2D or 3D scatter plots to explain these conditions.
[30 %]
Pressure and distance are unchaning at a low value in these 40 tests
Sample number Low Speed Sample number High Speed
1 74 21 151.1
2 78.9 22 136.8
3 90.3 23 161.8
4 82.1 24 147
5 89.4 25 154.7
6 90.9 26 143.1
7 90.2 27 150.2
8 69.2 28 150.3
9 76.2 29 135.9
10 82.2 30 141.1
11 81.9 31 145.5
12 87.3 32 149.2
13 104.9 33 152.3
14 74.4 34 168.7
15 87.7 35 152.9
16 87.8 36 146.1
17 88.9 37 161.4
18 102.7 38 178.9
19 95.9 39 160.5
20 95.5 40 142.2