Shot noise tells you the photon count is uncertain. Yield engineers need the next step: given that uncertainty, what fraction of features will actually fail? The answer turns out to be a clean piece of probability — the tail of a Poisson distribution — and it behaves like a cliff. A change of a nanometre or a few millijoules can move the defect rate by orders of magnitude.
The threshold idea
Picture a single feature that needs to absorb at least Nmin photons to develop correctly. Absorb more and it prints; absorb fewer and it comes out missing or broken. The mean number it absorbs is set by the process — dose, area, and how efficiently the resist soaks up EUV:
μ = photons/nm² × feature area × absorbed fractionBut the actual count on any given feature is a Poisson draw around that mean. So the probability that a feature fails is simply the probability that its draw lands below the threshold:
P(defect) = P(N < Nmin) = ∑ₖ₌₀ᵐⁱⁿ⁻¹ e⁻ᵀ μᵇ / k!The tail is the defect rate
That sum is the lower tail of the Poisson distribution, and it is the whole game. Plot the distribution of photon counts around the mean and shade everything below the threshold: the shaded area is the defect probability. When the mean sits comfortably above the threshold the tail is vanishingly thin; as the mean slides toward the threshold, the tail fattens fast.
A useful way to read the risk is the σ-margin: how many standard deviations separate the mean from the threshold, (μ − Nmin) / √μ. Because the tail of a distribution shrinks roughly exponentially as you move away from the mean, each extra σ of margin cuts the defect rate by a large factor.
Defect probability is exponential in the σ-margin. Going from a 3σ margin to a 5σ margin can drop the failure rate from roughly one-in-a-thousand to one-in-a-million. That steepness is why stochastic defects appear so suddenly as features shrink.
Why "defects per billion" matters
These per-feature probabilities look reassuringly tiny until you remember the scale. A modern chip has on the order of tens of billions of features, and a wafer holds hundreds of chips. A defect probability of 10⁻⁹ per feature — one in a billion — is not "safe"; multiplied across a whole design it can still mean failures on every chip. That is why the industry tracks defectivity in defects per billion features, and why a model that is off by even one order of magnitude in the tail is off by a lot in yield.
SECD Defect Probability Demo
Move dose, feature size, absorption, and the threshold and watch the failure tail — and the defects-per-billion number — swing across the full range.
Why an average isn't a prediction
The single-feature Poisson tail is the simplest possible stochastic model, and it has a real limitation: it gives you an average failure rate for an isolated feature with an assumed threshold. It knows nothing about the actual layout — where features sit relative to their neighbours, how the local pattern density shifts the effective dose, or how the real (non-Gaussian) blur couples adjacent edges. Real yield work needs to predict where on a specific design the hotspots are, calibrated against measured data.
That is exactly the leap the SECD-CRF project is built to make: it feeds a physics-informed stochastic model into a conditional random field that learns spatial patterns, so it can flag defect hotspots directly from a layout instead of quoting a single tail probability. This demo exists to make the underlying intuition concrete — and to show the target that the full predictor is aiming at.