What Secondary-Electron Blur Means for Lithography

A photon does not print a chip. When an EUV photon is absorbed in the resist, it hands its energy to electrons — and those electrons travel a few nanometres before they trigger the chemistry that actually develops the pattern. That short flight is why no EUV edge is ever perfectly sharp, and it sets a hard limit on how small a feature can be before it simply stops resolving.

The electron that does the work

Absorb a 92 eV EUV photon and it ejects a fast photoelectron, which in turn knocks loose a cascade of lower-energy secondary electrons. It is these secondaries that drive the acid generation and chemical reactions that define exposed versus unexposed resist. Crucially, they do this not at the point where the photon landed but wherever they come to rest — typically a few nanometres away. The measured secondary-electron range in EUV photoresist is on the order of 2–5 nm.

So the pattern that the resist chemistry "sees" is not the crisp optical image. It is that image smeared out by the distance electrons wander. Mathematically, the exposure is the ideal image convolved with a point-spread function that describes how far the energy spreads from each absorption event.

Modelling it as a blur

The simplest useful model treats that point-spread as a Gaussian of width σ and convolves it with an ideal line-and-space grating:

I_blurred(x) = ∫ I_ideal(x′) · Gₛ(x − x′) dx′

The effect is intuitive. Where the ideal image is a sharp on/off square wave, the blurred image is a smooth ripple. As σ grows toward the half-pitch of the pattern, the peaks and troughs of that ripple move toward each other until the lines and spaces are indistinguishable — the pattern has washed out entirely.

NILS: the number lithographers actually watch

Contrast tells you whether a pattern exists at all; NILS — the Normalized Image Log-Slope — tells you whether it will print reliably. NILS measures how steeply the intensity crosses the print threshold, scaled by the feature width:

NILS = w · d(ln I)/dx  at the threshold crossing  (w = line width)

A steep crossing (high NILS) means a small error in dose or focus moves the printed edge only a little — the process has latitude. A shallow crossing (low NILS) means the same small error moves the edge a lot. The rough rule of thumb: NILS ≥ 2 is healthy, NILS below ~1 means the feature is losing its process window and every stochastic wobble in the photon count now translates directly into an edge that lands in the wrong place.

Where the simple Gaussian stops being enough

A single Gaussian is a convenient first approximation, and it is exactly the "prior art" baseline that more sophisticated models are measured against. But the real secondary-electron point-spread is not Gaussian: it has an energy-dependent range distribution with heavier tails, and treating it as a bell curve underestimates how far a rare, energetic electron can reach. Getting that tail right is what separates a teaching model from a predictor — and it is the heart of the physics-informed, non-Gaussian kernel at the core of the SECD-CRF project, whose blur term is grounded in the measured electron range rather than a fitted width.

Blur and shot noise are two halves of the same picture: shot noise makes the photon count uncertain, and blur decides how much of that uncertainty survives to move a printed edge. Put them together and you get a stochastic defect probability.