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Chapter 1: Problem 61

Three different \(\mathrm{C}_{2} \mathrm{~F}_{6}\) flow rates \((\mathrm{SCCM})\) were considered in an experiment to investigate the effect of flow rate on the uniformity \((\%)\) of the etch on a silicon wafer used in the manufacture of integrated circuits, resulting in the following data: \(\begin{array}{lllllll}\text { Flow rate } & & & & & \\ 125 & 2.6 & 2.7 & 3.0 & 3.2 & 3.8 & 4.6 \\ 160 & 3.6 & 4.2 & 4.2 & 4.6 & 4.9 & 5.0 \\ 200 & 2.9 & 3.4 & 3.5 & 4.1 & 4.6 & 5.1\end{array}\) Compare and contrast the uniformity observations resulting from these three different flow rates.

Short Answer

Expert verified
160 SCCM provides the most uniform etch results with the highest mean and lowest range.

Step by step solution

01

Organize the Data

We have three different flow rates: 125 SCCM, 160 SCCM, and 200 SCCM with their associated uniformity percentages. Let's list them separately: - 125 SCCM: 2.6, 2.7, 3.0, 3.2, 3.8, 4.6 - 160 SCCM: 3.6, 4.2, 4.2, 4.6, 4.9, 5.0 - 200 SCCM: 2.9, 3.4, 3.5, 4.1, 4.6, 5.1.
02

Calculate Mean Uniformity

Calculate the mean of the uniformity observations for each flow rate.- For 125 SCCM: \[ \text{Mean} = \frac{2.6 + 2.7 + 3.0 + 3.2 + 3.8 + 4.6}{6} = 3.32 \]- For 160 SCCM:\[ \text{Mean} = \frac{3.6 + 4.2 + 4.2 + 4.6 + 4.9 + 5.0}{6} = 4.25 \]- For 200 SCCM:\[ \text{Mean} = \frac{2.9 + 3.4 + 3.5 + 4.1 + 4.6 + 5.1}{6} = 3.93 \]
03

Determine the Range of Uniformity

Calculate the range of observations for each flow rate as a measure of variability.- For 125 SCCM:\[ \text{Range} = 4.6 - 2.6 = 2.0 \]- For 160 SCCM:\[ \text{Range} = 5.0 - 3.6 = 1.4 \]- For 200 SCCM:\[ \text{Range} = 5.1 - 2.9 = 2.2 \]
04

Analyze and Compare

Compare both the mean and the range of uniformity percentages for each flow rate: - 125 SCCM has a mean of 3.32 and a range of 2.0, indicating moderate uniformity. - 160 SCCM has the highest mean of 4.25 and the lowest range of 1.4, suggesting the most uniform results. - 200 SCCM has a mean of 3.93 and the highest range of 2.2, indicating a wider spread of observations.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Flow Rates
Flow rates are essential in experimental designs involving etching processes. They relate to the volume or mass of C2F6 gas flowing through the system per unit time, measured in SCCM (Standard Cubic Centimeters per Minute).
In the experiment, three different flow rates were evaluated to determine their impact on the uniformity of the etch process on silicon wafers. Each flow rate鈥�125 SCCM, 160 SCCM, and 200 SCCM鈥攃an affect the speed and coverage of the process.
Choosing the right flow rate is vital. Too low, and the etch might be uneven. Too high, and you risk over-etching or causing other inconsistencies.
Uniformity Analysis
Uniformity analysis investigates how consistently the etch process performs across the wafer surface.
Any variations in uniformity can substantially impact the quality and functionality of the silicon wafer in integrated circuits.
In this exercise, the uniformity is expressed as a percentage, with higher percentages indicating more consistent performance. - At 125 SCCM, uniformity percentages ranged from 2.6% to 4.6%. - At 160 SCCM, percentages were slightly higher and tighter, from 3.6% to 5.0%. - At 200 SCCM, the uniformity percentages varied from 2.9% to 5.1%. A lower variation in uniformity values suggests a more reliable procedure.
Data Organization
Organizing data efficiently helps in understanding and processing information in experiments.
For this etching experiment, data is organized based on flow rates. - Each flow rate has a list of uniformity percentages associated with it. - This separation allows for easier comparison and statistical analysis. Clear data organization offers a structured approach to visualize and interpret results, leading to more informed decisions.
For example, it's immediately apparent that the 160 SCCM flow offers the highest mean uniformity, an important insight for optimizing processes.
Statistical Calculations
Statistical calculations allow us to properly evaluate experimental results.
Key metrics used include calculating the mean and the range of uniformity percentages for each flow rate. - **Mean Uniformity** gives the average uniformity percentage for each flow rate. - Calculating the range gives insight into the variability and spread of data points. In this experiment, the mean uniformities were:
  • 125 SCCM: 3.32
  • 160 SCCM: 4.25
  • 200 SCCM: 3.93
These means, combined with range measures, indicate which flow rate offers more consistent and reliable results.
Variability
Variability refers to the extent of differences in data points in an experimental dataset. In this etching process, variability helps analyze how consistent or inconsistent each flow rate is concerning uniformity. - **Range as Variability Measure**: The range (difference between maximum and minimum percentages) highlights variability at each flow rate. The ranges calculated in the study were:
  • 125 SCCM: 2.0
  • 160 SCCM: 1.4
  • 200 SCCM: 2.2
A smaller range, like with the 160 SCCM, indicates less variability and more consistent etching across the wafer. Higher variability could signal issues with the process that need addressing.

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Most popular questions from this chapter

Anxiety disorders and symptoms can often be effectively treated with benzodiazepine medications. It is known that animals exposed to stress exhibit a decrease in benzodiazepine receptor binding in the frontal cortex. The paper "Decreased Benzodiazepine Receptor Binding in Prefrontal Cortex in Combat- Related Posttraumatic Stress Disorder" (Amer. J. Psychiatry, 2000: 1120-1126) described the first study of benzodiazepine receptor binding in individuals suffering from PTSD. The accompanying data on a receptor binding measure (adjusted distribution volume) was read from a graph in the paper. PTSD: \(10,20,25,28,31,35,37,38,38,39,39,42\), 46 Healthy: \(23,39,40,41,43,47,51,58,63,66,67\), 69,72 Use various methods from this chapter to describe and summarize the data.

Consider the following data on type of health complaint \((J=\) joint swelling, \(F=\) fatigue, \(B=\) back pain, \(\mathrm{M}=\) muscle weakness, \(\mathrm{C}=\) coughing, \(\mathrm{N}=\) nose running/irritation, \(\mathbf{O}\) - other) made by tree planters. Obtain frequencies and relative frequencies for the various categories, and draw a histogram. (The data is consistent with percentages given in the article "Physiological Effects of Work Stress and Pesticide Exposure in Tree Planting by British Columbia Silviculture Workers," Ergonomics, 1993: 951-961.) \(\begin{array}{llllllllllllll}\mathrm{O} & \mathrm{O} & \mathrm{N} & \mathrm{J} & \mathrm{C} & \mathrm{F} & \mathrm{B} & \mathrm{B} & \mathrm{F} & \mathrm{O} & \mathrm{J} & \mathrm{O} & \mathrm{O} & \mathrm{M} \\ \mathrm{O} & \mathrm{F} & \mathrm{F} & \mathrm{O} & \mathrm{O} & \mathrm{N} & \mathrm{O} & \mathrm{N} & \mathrm{J} & \mathrm{F} & \mathrm{J} & \mathrm{B} & \mathrm{O} & \mathrm{C} \\ \mathrm{J} & \mathrm{O} & \mathrm{J} & \mathrm{J} & \mathrm{F} & \mathrm{N} & \mathrm{O} & \mathrm{B} & \mathrm{M} & \mathrm{O} & \mathrm{J} & \mathrm{M} & \mathrm{O} & \mathrm{B} \\ \mathrm{O} & \mathrm{F} & \mathrm{J} & \mathrm{O} & \mathrm{O} & \mathrm{B} & \mathrm{N} & \mathrm{C} & \mathrm{O} & \mathrm{O} & \mathrm{O} & \mathrm{M} & \mathrm{B} & \mathrm{F} \\\ \mathrm{J} & \mathrm{O} & \mathrm{F} & \mathrm{N} & & & & & & & & & & \end{array}\)

Exercise 33 in Section \(1.3\) presented a sample of 26 escape times for oil workers in a simulated escape exercise. Calculate and interpret the sample standard deviation. [Hint: \(\sum x_{i}=9638\) and \(\left.\sum x_{i}^{2}=3,587,566\right]\).

The article "Determination of Most Representative Subdivision" (J. Energy Engrg., 1993: 43-55) gave data on various characteristics of subdivisions that could be used in deciding whether to provide electrical power using overhead lines or underground lines. Here are the values of the variable \(x=\) total length of streets within a subdivision: \(\begin{array}{rrrrrrr}1280 & 5320 & 4390 & 2100 & 1240 & 3060 & 4770 \\ 1050 & 360 & 3330 & 3380 & 340 & 1000 & 960 \\ 1320 & 530 & 3350 & 540 & 3870 & 1250 & 2400 \\ 960 & 1120 & 2120 & 450 & 2250 & 2320 & 2400 \\ 3150 & 5700 & 5220 & 500 & 1850 & 2460 & 5850 \\ 2700 & 2730 & 1670 & 100 & 5770 & 3150 & 1890 \\ 510 & 240 & 396 & 1419 & 2109 & & \end{array}\) a. Construct a stem-and-leaf display using the thousands digit as the stem and the hundreds digit as the leaf, and comment on the various features of the display. b. Construct a histogram using class boundaries \(0,1000,2000,3000,4000,5000\), and 6000 . What proportion of subdivisions have total length less than 2000 ? Between 2000 and 4000 ? How would you describe the shape of the histogram?

The accompanying data set consists of observations on shower-flow rate (L/min) for a sample of \(n=129\) houses in Perth, Australia ("An Application of Bayes Methodology to the Analysis of Diary Records in a Water Use Study," J. Amer. Statist. Assoc., 1987: 705-711): \(\begin{array}{rrrrrrrrrr}4.6 & 12.3 & 7.1 & 7.0 & 4.0 & 9.2 & 6.7 & 6.9 & 11.5 & 5.1 \\ 11.2 & 10.5 & 14.3 & 8.0 & 8.8 & 6.4 & 5.1 & 5.6 & 9.6 & 7.5 \\\ 7.5 & 6.2 & 5.8 & 2.3 & 3.4 & 10.4 & 9.8 & 6.6 & 3.7 & 6.4 \\ 8.3 & 6.5 & 7.6 & 9.3 & 9.2 & 7.3 & 5.0 & 6.3 & 13.8 & 6.2 \\ 5.4 & 4.8 & 7.5 & 6.0 & 6.9 & 10.8 & 7.5 & 6.6 & 5.0 & 3.3 \\ 7.6 & 3.9 & 11.9 & 2.2 & 15.0 & 7.2 & 6.1 & 15.3 & 18.9 & 7.2 \\ 5.4 & 5.5 & 4.3 & 9.0 & 12.7 & 11.3 & 7.4 & 5.0 & 3.5 & 8.2 \\ 8.4 & 7.3 & 10.3 & 11.9 & 6.0 & 5.6 & 9.5 & 9.3 & 10.4 & 9.7 \\ 5.1 & 6.7 & 10.2 & 6.2 & 8.4 & 7.0 & 4.8 & 5.6 & 10.5 & 14.6 \\ 10.8 & 15.5 & 7.5 & 6.4 & 3.4 & 5.5 & 6.6 & 5.9 & 15.0 & 9.6 \\ 7.8 & 7.0 & 6.9 & 4.1 & 3.6 & 11.9 & 3.7 & 5.7 & 6.8 & 11.3 \\ 9.3 & 9.6 & 10.4 & 9.3 & 6.9 & 9.8 & 9.1 & 10.6 & 4.5 & 6.2 \\ 8.3 & 3.2 & 4.9 & 5.0 & 6.0 & 8.2 & 6.3 & 3.8 & 6.0 & \end{array}\) a. Construct a stem-and-leaf display of the data. b. What is a typical, or representative, flow rate? c. Does the display appear to be highly concentrated or spread out? d. Does the distribution of values appear to be reasonably symmetric? If not, how would you describe the departure from symmetry? e. Would you describe any observation as being far from the rest of the data (an outlier)?
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