Modeling IC yield as a path-dependent rough process — not a static average.
Simple yield models collapse a month of production into one number: the average defect density D₀. That is fast to compute and easy to report — but it hides the most dangerous reality in manufacturing.
A fab that averages D₀ = 0.12 by spending half the month at 0.06 and half at 0.18 is not the same as one that holds steady at 0.12. The path matters. Excursions cluster. Bad states persist. Yield is destroyed during the spikes and never fully recovered in the calm.
The Rough Volatility Hypothesis (RVH) borrows from financial modeling of rough stochastic processes to give defectivity a memory, a roughness parameter, and a realistic excursion structure. The result: a simulation that separates three distinct realities a static model cannot distinguish.
Murphy–Poisson yield equation. D₀ = defect density (defects/cm²), A = die area (cm²), α = clustering. RVH makes D₀ a dynamic rough path rather than a constant.
A perfectly flat defect path and a low-volatility real fab produce nearly identical output. Normal process variation is not the economic threat.
Die area drives yield independently of fab stability. A large chip in a perfect fab still yields poorly — product complexity and process instability are separable causes.
Random daily fluctuations are tolerable. Persistent bad states with long memory — rough volatility — are not. The Hurst exponent captures which regime you are in.
| Parameter | Symbol | Effect | Stable Fab | Unstable Fab |
|---|---|---|---|---|
| Hurst Exponent | H | Memory & roughness of D₀ path. Lower = rougher. | 0.45 | 0.10 |
| Process Volatility | ν | Magnitude of day-to-day swings in D₀. | 0.05 | 1.00 |
| Excursion Probability | pexc | Daily probability of a major tool failure or process shock. | 0.0 | 0.30 |
| Excursion Impact | kexc | Multiplier applied to D₀ during an excursion event. | 1.0× | 10.0× |
| Defect Clustering | α | Spatial clustering of defects. Lower = more spread (worse yield). | 2.5 (baseline) | |