Fractal Dimension of Surfaces is a mathematical metric quantifying the self-similar complexity of surface roughness — a fractal dimension between 2 (perfectly smooth plane) and 3 (volume-filling roughness) that characterizes how roughness scales across different measurement scales.

Fractal Surface Analysis

• Self-Similarity: Fractal surfaces look statistically similar at different magnifications — "zooming in" reveals similar roughness patterns.
• PSD Slope: For fractal surfaces, $PSD(f) propto f^{-alpha}$ — the exponent $alpha$ relates to the fractal dimension: $D = (7-alpha)/2$ (for 2D surfaces).
• Box-Counting: Estimate fractal dimension by counting how many boxes of size $epsilon$ are needed to cover the surface.
• Typical Values: Polished silicon: $D approx 2.1-2.3$; etched surfaces: $D approx 2.3-2.6$; deposited films: $D approx 2.2-2.5$.

Why It Matters

• Scale-Invariant: Fractal dimension captures roughness behavior across ALL scales — complementary to Rq (which is scale-dependent).
• Process Indicator: Different processes produce surfaces with characteristic fractal dimensions — useful for process monitoring.
• Adhesion: Fractal dimension affects real contact area, adhesion, and friction — important for bonding and CMP.

Fractal Dimension is the complexity of the surface — a scale-invariant metric that characterizes how rough a surface is across all measurement scales.