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Sandbox Physics

E25 · Electron writing

Electron Lithography: Draw, Expose, Develop

Follow the written dose through scattering and a development threshold. Correct the exposure, then compare the result with the design.

Interactive modelElectron Lithography: Draw, Expose, Develop
Design/output intersection over union—\text{—}
Unintended fraction of open area—\text{—}
Isolated width on fixed center row—\text{—}
Middle dense-line width on fixed center row—\text{—}
Expected incident electron number—\text{—}
Expected writing duration—\text{—}
Selected-cell deposited reference dose—\text{—}
Experiment target—\text{—}

Physics tutorial

Electron Lithography: Draw, Expose, Develop

BackgroundFollow the written dose through scattering and a development threshold. Correct the exposure, then compare the result with the design.

Why it mattersReach at least 85 percent overlap, at most 5 percent unintended opening, isolated width 20–40 nm and dense/isolated width difference at most 10 nm.

Start with the essentials

Focus question
Why do isolated and dense lines develop differently?
One-sentence intuition
The developed pattern follows deposited dose, not the drawing.

Core mathematical model

Scatter the dose before development

K(r)=Gsf(r)+ηGsb(r)1+ηDdep=K∗DwriteOi=1[Ddep,i≥Dc]\begin{aligned}K(r)&=\frac{G_{s_f}(r)+\eta G_{s_b}(r)}{1+\eta}\\D_{\rm dep}&=K*D_{\rm write}\\O_i&=\mathbf1[D_{{\rm dep},i}\ge D_c]\end{aligned}

Each Gaussian has unit area. Finite-field tails leave the computed field and are not renormalized back. Development uses deposited dose, never the target mask.

Correct where exposure is written

wi(k+1)=min⁡ ⁣(6,1.05 wi(k)(K∗w(k))i)w_i^{(k+1)}=\min\!\left(6,\frac{1.05\,w_i^{(k)}}{(K*w^{(k)})_i}\right)

Eighteen nonnegative updates only at authored write sites. This equalizes on-pattern dose; off-pattern leakage and coarse sampling remain.

Compare achieved openings with the design

J=∣O∩M∣∣O∪M∣T=eNincidentI\begin{aligned}J&=\frac{|O\cap M|}{|O\cup M|}\\T&=\frac{eN_{\rm incident}}{I}\end{aligned}

The design is used only for an independent overlap audit after development. Dose is measured in reference units of 200 microcoulombs per square centimetre.

Common difficulties

Model parameters need calibration

Typical misconceptionA successful preset is a recipe for a real instrument.

Better mental modelExpected-dose teaching model, not an experimental resist image. A normalized two-Gaussian surrogate with an assigned broad-to-narrow weight of 0.6; its widths are not measured material parameters. Positive resist opens at a hard threshold with 10 nm spatial quantization. On-pattern dose equalization precedes coarse shot sampling; it is not an exact inverse or a universal proximity correction. No Monte Carlo transport, electron shot noise, developer kinetics, charging, drift, line-edge roughness or pattern transfer. Display gain does not change data. Current changes time at fixed dose. Beam routes show initial exposure geometry; relief shows developed resist. Hardware, lateral and vertical scales are independently enlarged.

Run the experiment

  1. 01

    Predict density-dependent exposure

    Use the proximity preset and compare isolated and dense line widths in the same developed image.

    What to observe: Nearby writes add deposited dose even outside their target boundaries.
  2. 02

    Correct the input, then develop

    Apply the corrected preset. Change the development threshold while watching the deposited-dose card.

    What to observe: The dose map stays fixed while the developed openings change.
  3. 03

    Test coarse sampling

    Try a 40 nm write step. Compare sampled writes with the design and development.

    What to observe: Dose correction cannot restore unvisited sites after coarse sampling.
  4. 04

    Check the full target

    Reach at least 85 percent overlap, at most 5 percent unintended opening, isolated width 20–40 nm and dense/isolated width difference at most 10 nm.

    What to observe: At fixed dose, current changes write time; display gain cannot change openings.