Controlling and Maximising Semiconductor Wafer Yields

Background:

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 control engineers aim to maximise this yield during manufacturing. This yield is
highest 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:

Experimentation was carried out within the laboratories at Swansea University

in order to get an understanding of how to produce a uniform thickness. Using wafer
samples supplied by a company, coating thickness was measured at a number of
different locations along each wafer, and the fractional 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 three 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 the Data Sheet.

Note: In Data Sheet1 and Sheet2, replace the letters A - F with the last six digits of
your student number, for example, if your student is 1234567, then A = 2,B = 3,C =
4,D = 5, E = 6, and F =7 Alternatively, if your student number is 123456, then A = 1,
B = 2, C = 3, D = 4, E = 5,and F = 6.
Objectives:
You are required to write a project report (template available on Canvas) 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.

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 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 more significant understanding of the two experimental data sets, for
example:

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 β parameters of the following second-order
response surface model:

3 3 2 3
Y = 0 +  i X i +  ii X i +   ij X i X j + 
2

i =1 i =1 i =1 j = i +1

where Y is the standard deviation in coating thickness, X1 is the speed, X2 is

the pressure, and X3 is the distance. 𝜀 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 β parameters. State any assumptions that
need to be made in assessing such statistical significance, and if appropriate
carry out 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 vs prediction plot). Make
full use of any suitable 2D or 3D scatter plots when writing your final report.

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Individual project: Summary of briefing

• Issue date: 2nd November 2020 by 9:00 am.

• Submission deadline: 11th December 2020 by 16:00pm.
• Individual project to be submitted via Canvas submission portal. Zero
tolerance for late submission ditto, for plagiarism and collusion- all parties
involved will be given zero marks at least and may lead to further grave
consequences.
• Format: pdf file. Maximum of 8 pages (excluding the appendix and cover page).
• Abstract: Not more than 150 words.
• Introduction: Not more than 250 words.
• Appendix page: not more than two pages (single column).
• No screenshot of codes in the main report.
• Concisely format all code in the appendix page.
• Format all figures using the appropriate techniques (refer to Unit 1), add suitable
titles, X/Y labels, legends to all figures. For tables, clearly title columns and give
the units of the measurement.
• The overall report should be concise, complete, informative and written in a
precise scientific tone highlighting clear objectives, methodology, a detailed
discussion of main findings and a comprehensive reflection. All presented figures
should be thoroughly and insightfully analysed using statistical concepts and
themes. Presented results should logically address the research problem.
Distribution of Marks:

• Abstract → 5%
• Introduction → 10%
• Results and Discussion → 60%
• Reflection → 15 %
• General Presentation → 5%
• Lab Diary Inspection → 5%
Expected Learning Outcomes:

• Ability to use statistical software to compute and visualise statistical functions.

• Ability to apply common statistic methodologies to their field of study.
• Statistical thinking and structured problem-solving capabilities.
• Think about, understand and deal with variability.

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