91Ó°ÊÓ

In their book Empirical Model Building and Response Surfaces (John Wiley, 1987 ), Box and Draper described an experiment with three factors. The data in the following table are a variation of the original experiment from their book. Suppose that these data were collected in a semiconductor manufacturing process. (a) The response \(y\), is the average of three readings on resistivity for a single wafer. Fit a quadratic model to this response. (b) The response \(y_{2}\) is the standard deviation of the three resistivity measurements. Fit a linear model to this response. (c) Where would you recommend that we set \(x_{1}, x_{2},\) and \(x_{3}\) if the objective is to hold mean resistivity at 500 and minimize the standard deviation? $$ \begin{array}{rrrrr} \hline x_{1} & x_{2} & x_{3} & y_{1} & y_{2} \\ \hline-1 & -1 & -1 & 24.00 & 12.49 \\ 0 & -1 & -1 & 120.33 & 8.39 \\ 1 & -1 & -1 & 213.67 & 42.83 \\ -1 & 0 & -1 & 86.00 & 3.46 \\ 0 & 0 & -1 & 136.63 & 80.41 \\ 1 & 0 & -1 & 340.67 & 16.17 \\ -1 & 1 & -1 & 112.33 & 27.57 \\ 0 & 1 & -1 & 256.33 & 4.62 \\ 1 & 1 & -1 & 271.67 & 23.63 \\ -1 & -1 & 0 & 81.00 & 0.00 \\ 0 & -1 & 0 & 101.67 & 17.67 \\ 1 & -1 & 0 & 357.00 & 32.91 \\ -1 & 0 & 0 & 171.33 & 15.01 \\ 0 & 0 & 0 & 372.00 & 0.00 \\ 1 & 0 & 0 & 501.67 & 92.50 \\ -1 & 1 & 0 & 264.00 & 63.50 \\ 0 & 1 & 0 & 427.00 & 88.61 \\ \hline 1 & 1 & 0 & 730.67 & 21.08 \\ -1 & -1 & 1 & 220.67 & 133.82 \\ 0 & -1 & 1 & 239.67 & 23.46 \\ 1 & -1 & 1 & 422.00 & 18.52 \\ -1 & 0 & 1 & 199.00 & 29.44 \\ 0 & 0 & 1 & 485.33 & 44.67 \\ 1 & 0 & 1 & 673.67 & 158.21 \\ -1 & 1 & 1 & 176.67 & 55.51 \\ 0 & 1 & 1 & 501.00 & 138.94 \\ 1 & 1 & 1 & 1010.00 & 142.45 \end{array} $$

Step by step solution

01

Understanding the Model Structure

A quadratic model includes the linear terms \(x_1, x_2, x_3\), the squared terms \(x_1^2, x_2^2, x_3^2\), and the interaction terms \(x_1x_2, x_1x_3, x_2x_3\). Thus, we will fit a model of the form: \(y_1 = \beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3 + \beta_4x_1^2 + \beta_5x_2^2 + \beta_6x_3^2 + \beta_7x_1x_2 + \beta_8x_1x_3 + \beta_9x_2x_3 + \epsilon\).

02

Fitting the Quadratic Model for \(y_1\)

Using regression analysis tools, determine the coefficients \(\beta_0\) through \(\beta_9\). After fitting, you'll obtain coefficients that best describe the variation in \(y_1\) based on \(x_1, x_2, x_3\) and their interactions.

03

Linear Model Structure for \(y_2\)

A linear model only includes the linear terms of the factors \(x_1, x_2, x_3\). Thus, it takes the form \(y_2 = \gamma_0 + \gamma_1x_1 + \gamma_2x_2 + \gamma_3x_3 + \delta\).

04

Fitting the Linear Model for \(y_2\)

Perform a linear regression with \(x_1, x_2, x_3\) as predictors for \(y_2\). This will give you the coefficients \(\gamma_0\) through \(\gamma_3\) that best describe \(y_2\).

05

Meeting Objective Conditions

To hold mean resistivity (\(y_1\)) at 500, you must adjust \(x_1, x_2, x_3\) so that the predicted value from the quadratic model equals 500. Using a method like Lagrange multipliers or trial-and-error with the model, minimize \(y_2\) at this condition. Refer to both model coefficients derived from steps 2 and 4.

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Key Concepts

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

Regression Analysis

Regression analysis is a fundamental technique used in statistics to understand relationships between variables. The goal is to model the relationship of a dependent variable with one or more independent variables. In our exercise, we utilize both a quadratic and a linear regression model.

For the quadratic model, the dependent variable is denoted as \(y_1\) which represents the average resistivity of a semiconductor wafer. This model seeks to capture not only the individual effects of each factor \(x_1, x_2, x_3\) but also their interactions and higher-order effects through terms like \(x_1^2\) and \(x_1x_2\). The model form is:

• \(y_1 = \beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3 + \beta_4x_1^2 + \beta_5x_2^2 + \beta_6x_3^2 + \beta_7x_1x_2 + \beta_8x_1x_3 + \beta_9x_2x_3 + \epsilon\)

By adjusting the coefficients \(\beta_0\) through \(\beta_9\), we aim to precisely describe the behavior of \(y_1\) with respect to the changes in \(x_1, x_2,\) and \(x_3\).

In contrast, the linear model applied to \(y_2\) (the standard deviation of resistivity) only considers direct effects of the input variables. Here, the form is straightforward:

• \(y_2 = \gamma_0 + \gamma_1x_1 + \gamma_2x_2 + \gamma_3x_3 + \delta\)

These models assist in understanding how variations in \(x_1, x_2,\) and \(x_3\) can influence resistivity outcomes.

Semiconductor Manufacturing

Semiconductor manufacturing is a sophisticated process essential for producing electronic components. Understanding and controlling process parameters is crucial to ensure quality and performance of semiconductor devices. During manufacturing, factors like temperature, pressure, and chemical concentrations are carefully adjusted to achieve desired material properties like resistivity.

In the context of our exercise, resistivity measurements help determine the electrical properties of the semiconductor wafer. The controllable factors \(x_1, x_2,\) and \(x_3\) could represent variables such as the level of doping, temperature settings, and time of exposure during a specific manufacturing cycle. Each factor individually, or in combination, can significantly affect the resistivity and consistency of the wafer.

By applying a structured approach with models like quadratic and linear regression, engineers can analyze the impact of these variables and fine-tune the manufacturing process. Such optimizations lead to better product reliability and higher yield rates, ultimately making semiconductor devices more efficient and cost-effective.

Response Surface Methodology

Response Surface Methodology (RSM) is an advanced statistical approach used to optimize processes with multiple variables. It helps in understanding the relationships between several input factors and the resulting outputs. This is particularly useful in complex systems like semiconductor manufacturing where interactions between factors are common.

RSM combines design of experiments with regression analysis to build a predictive model for response variables. In our exercise, the quadratic model used for resistivity averages is an example of an RSM application. Through this model, we seek a relationship that explains how the inputs \(x_1, x_2,\) and \(x_3\) influence the output \(y_1\).

• A key objective in applying RSM here is to identify optimal settings for \(x_1, x_2,\) and \(x_3\) that maintain the average resistivity near 500 while minimizing variation.
• This is done by analyzing the response surface - a graphical representation of the relation between inputs and outputs.

Using RSM enables deeper insights and effective control over complex industrial processes, paving the way for enhanced product performance and process efficiency.