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.