Controlling and Optimizing Wafer Yields

Page | 1
Controlling and Optimizing 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 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.
Page | 2
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 ij
i
i ii
i
Y i X 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.
Page | 3
Individual project: Summary of briefing
• Issue date: 30th October 2018 by 12:00 noon.
• Submission deadline: 13th December 2018 by 16:00pm.
• Individual project to be submitted via Blackboard. Zero tolerance for late submission
ditto, for plagiarism- all parties involved will be given zero marks at least and may
lead to further grave consequences.
• 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.
• The overall report should be concise, complete, informative and written in a clear
scientific tone highlighting clear objectives, methodology, descriptive 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%
• Attendance  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.