Etch Rate Experiment (Myers & Montgomery, 2002)

The etch rate experiment presented by Myers and Montgomery in their textbook Response Surface Methodology: Process and Product Optimization Using Designed Experiments (2nd edition, 2002) is a widely cited case study in the application of response surface methodology (RSM) to industrial process optimisation. It investigates how two controllable process factors affect the etch rate in a plasma etching system — a critical step in semiconductor manufacturing.

Background

In this context, plasma etching removes material from the surface of a silicon wafer. The aim of the experiment was to determine how changes in two process variables — RF power and chamber pressure — influence the etch rate.

The input variables were:

$x_1$ : RF power (watts)
$x_2$ : Chamber pressure (torr)

The response variable was:

$y$ : Etch rate (angstroms per minute, Å/min)

Experimental Design

A central composite design (CCD) was selected to estimate a second-order model, allowing the detection of curvature in the response surface.

The design included:

Full factorial design: $2^2 = 4$ runs
Four axial (star) points at $\alpha = \sqrt{2} \approx 1.414$
Five centre points for estimating pure error and testing lack-of-fit

In total, 13 runs were performed. All factor levels were coded as follows:

Coded low level: $x_i = -1$
Coded centre level: $x_i = 0$
Coded high level: $x_i = +1$
Axial (star) points: $x_i = \pm \alpha$

Model Specification

A full second-order model was fitted to the data:

$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_{11} x_1^2 + \beta_{22} x_2^2 + \beta_{12} x_1 x_2 + \varepsilon$

Where:

Intercept: $\beta_0$
Linear effects: $\beta_1, \beta_2$
Quadratic effects: $\beta_{11}, \beta_{22}$
Interaction term: $\beta_{12}$
Random error: $\varepsilon$

Experimental Results

The fitted regression model was:

$\hat{y} = 725 + 70x_1 + 55x_2 - 50x_1^2 - 45x_2^2 + 10x_1x_2$

This result suggests:

Increasing $x_1$ (power) has a strong positive effect on the etch rate
Increasing $x_2$ (pressure) also increases etch rate, but to a lesser extent
The negative coefficients on $x_1^2$ and $x_2^2$ indicate curvature in both directions
The interaction between power and pressure is slightly synergistic: $\beta_{12} = +10$

Optimisation

To locate the stationary point (maximum), the partial derivatives were set to zero:

$\frac{\partial y}{\partial x_1} = 70 - 100x_1 + 10x_2 = 0$

$\frac{\partial y}{\partial x_2} = 55 - 90x_2 + 10x_1 = 0$

Solving this system gives:

$x_1^* = 0.7, \quad x_2^* = 0.6$

Substituting these into the model:

$\hat{y}_{\mathrm{max}} = 725 + 70(0.7) + 55(0.6) - 50(0.7)^2 - 45(0.6)^2 + 10(0.7)(0.6)$

$\hat{y}_{\mathrm{max}} \approx 775 \text{\AA/min}$

Visualisation

The response surface is curved and bowl-shaped, with a distinct interior maximum. Contour and surface plots would show elliptical contours centred near $(x_1 = 0.7, x_2 = 0.6)$ , affirming that the optimum lies well within the tested region.

Conclusion

This experiment demonstrates a textbook application of RSM in engineering process development. It illustrates:

The use of CCD to fit a quadratic model with minimal runs

How to interpret curvature and interactions in coded models
The analytical determination of optimal process settings using calculus

It remains one of the most widely taught examples of experimental design in physical sciences and manufacturing engineering.

For more information about our experimental design and general development services see: Development Services.

For a more detailed overview of experimental design see: Experimental Design.

Or, as an alternative, consider this: DoE Overview

Call us

Use our contact form

Background

Experimental Design

Model Specification

Experimental Results

Optimisation

Visualisation

Conclusion

Share this:

Like this:

Leave a ReplyCancel reply

Discover more from Product Development Engineers Ltd