central composite design

Central Composite Design (CCD) is the most widely used Response Surface Methodology (RSM) experimental design, combining factorial points, axial (star) points, and center points to efficiently fit a full second-order (quadratic) model that captures curvature and interaction effects.

Design Structure

A CCD consists of three components:

• Factorial Points ($2^k$ or $2^{k-p}$): The standard factorial design — all combinations of factors at their low (−1) and high (+1) levels. These estimate main effects and interactions.
• Axial (Star) Points ($2k$ points): One factor at a time is set to an extreme value ($\pm \alpha$) while all other factors are at center (0). These estimate the quadratic (curvature) terms.
• Center Points ($n_c$, typically 3–6): All factors at their center level (0). These estimate pure error and provide the baseline.

Total runs = $2^k + 2k + n_c$. For 3 factors: $8 + 6 + 6 = 20$ runs.

The α (Alpha) Value

• $\alpha$ determines how far the axial points extend beyond the factorial range.
• Face-Centered (α = 1): Axial points are on the faces of the cube — only 3 levels needed per factor. Simple but prediction quality varies across the design space.
• Rotatable (α = $2^{k/4}$): Provides uniform prediction variance at equal distances from the center — the most statistically desirable option.
• For 3 factors: $\alpha = 2^{3/4} \approx 1.682$.
• For 4 factors: $\alpha = 2^{4/4} = 2.0$.

CCD Variants

• Circumscribed (CCC): α > 1. Axial points extend beyond the factorial range — requires the ability to run at more extreme conditions.
• Inscribed (CCI): The entire design is scaled to fit within the original factor range — axial points are at ±1 and factorial points are pulled inward. Useful when the original range represents hard limits.
• Face-Centered (CCF): α = 1. All points within the cube. Only 3 levels per factor. Slightly less efficient but practically simpler.

Why CCD Is Popular

• Sequential: Can build from a factorial design. Run the factorial first, check for curvature with center points, then add axial points only if curvature is significant.
• Flexible: Different α values accommodate different experimental constraints.
• Complete: Fits the full second-order model including all linear, quadratic, and interaction terms.

Semiconductor Applications

• Etch Optimization: Model etch rate, CD, uniformity, and selectivity as functions of RF power, pressure, and gas flow ratios.
• Lithography: Map the full dose-focus-PEB response surface for CD and process window optimization.
• Deposition: Optimize film properties (thickness, stress, composition) across temperature, pressure, and gas flow space.

CCD is the gold standard RSM design — its sequential nature, flexibility, and statistical efficiency make it the default choice for detailed process optimization in semiconductor manufacturing.