SECD Defect Probability Demo
Where photon shot noise becomes a yield problem: if a feature needs a minimum number of absorbed photons to print, the Poisson tail below that threshold is the stochastic defect rate. Watch how dose, feature size, and absorption move that failure tail across many orders of magnitude.
The idea in one line
Combine the two previous tools. From shot noise we get the mean number of absorbed photons in a feature, μ = dose × area × absorbed_fraction (per nm² at 13.5 nm). Assume the feature prints correctly only if it absorbs at least N_min photons — below that, development leaves a missing or broken feature. Because the count is Poisson-distributed, the defect probability is exactly the lower tail:
computed here with the regularized incomplete gamma function (numerically stable for large μ). The readout also reports the margin (μ − N_min)/√μ in units of σ. This is the "stochastic cliff": because the tail is exponential in σ-margin, shaving a couple of nanometres off a feature — or a few mJ off the dose — can swing the defect rate from one-in-a-trillion to one-in-a-thousand.
Why this is a demo and SECD-CRF is the real work
A single-feature Poisson tail is the simplest possible stochastic model. It has no notion of the actual layout, of neighbouring features, of the real (non-Gaussian) blur, or of the resist's chemical response — and its N_min is assumed, not fit. Predicting real hotspots means learning the mapping from a layout to a defect field, calibrated against measured data. That physics-informed prediction problem — not this toy tail — is what the SECD-CRF project is being built to solve. This demo exists to make the intuition concrete and to show the target the algorithm is aiming at.
What the model deliberately ignores
- Layout. No neighbours, no 2-D geometry, no proximity — a single isolated feature.
- Blur & chemistry. The threshold hides all of development, quencher, and acid statistics in one number.
- Calibration. Nothing here is fit to measured defectivity, so absolute rates are illustrative only. Relative trends (dose ↑ → defects ↓, area ↓ → defects ↑) are the point.
Method & numerics
The mean absorbed count is μ = photons_per_nm²(dose, 13.5 nm) × area × absorbed_fraction. The failure probability P(N < N_min) is evaluated as the regularized upper incomplete gamma Q(N_min, μ), which equals the Poisson CDF and is validated in this project against direct summation to 1 part in 10⁶. The distribution chart shows the Poisson probability mass around μ with the failing tail (N < N_min) shaded, and defects-per-billion is simply P × 10⁹.