E25 · Electron writing

Electron Lithography: Draw, Expose, Develop

Why do isolated and dense lines develop differently? Electrons scatter inside the resist and the substrate, so each line also exposes its neighbors a little. In a dense pattern those extra doses add up. The developed pattern follows the dose the resist actually received, not the drawing, which is why proximity correction adjusts the exposure before development.

Follow the written dose as scattering spreads it out and a development threshold turns it into openings. Correct the exposure, then compare the result with the design.

Line-edge RMS over 160 nm $\text{—}$

Rows with both edges, out of 16 $\text{—}$

Design/output intersection over union $\text{—}$

Isolated width on fixed center row $\text{—}$

More measurements and target

Unintended fraction of open area $\text{—}$

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 $\text{—}$

Simulated examples; each plot identifies its data source.

Nominal write dose / reference

D_0

1.8

On-pattern dose equalization

c

0\,\mathrm{\%}

Resist-response correlation length

\ell

5\,\mathrm{nm}

Shot sampling step

\Delta x

10\,\mathrm{nm}

Advanced settings

Incident energy

E

30\,\mathrm{keV}

Writing current

I

100\,\mathrm{pA}

Initial resist thickness

t_r

200\,\mathrm{nm}

Stochastic exposure: 1 on

1

Independent write number

17

Development threshold relative to reference dose

D_c

1

Selected cell index

k

1162

Dose/depth display gain only

G

1

Learning goal and model limits

Predict · write · inspect · explain

Make the opening match the drawing

Target: the opening must overlap the design by at least 85 percent, with at most 5 percent unintended opening. The isolated line must be 20 to 40 nm wide, and the dense and isolated widths must differ by at most 10 nm. The reference dose is 200 microcoulombs per square centimetre.

Model assumptions

This teaching model stores both expected dose and a seeded stochastic exposure. In stochastic mode, electron arrivals at write sites and resist reactions are sampled separately; the assigned reaction yield is 0.02, followed by a Gaussian resist-response correlation. Development thresholds this response on a 10 nm grid. A new write number produces an independent exposure. Line-edge positions use linear interpolation along 16 rows over 160 nm, and RMS is computed about each edge mean; interrupted lines do not receive a complete-line roughness value. These are calculated resist responses, without microscope metrology noise or material calibration. Scattering is modeled by two normalized Gaussians, with the broad one assigned a weight of 0.6 relative to the narrow one, and their widths are not measured material parameters. The positive resist opens wherever the resist response reaches a sharp threshold, on a 10 nm grid. The correction evens out the dose inside the pattern before coarse shot sampling is applied; it is neither an exact inversion nor a general proximity correction. The model leaves out Monte Carlo electron transport, chemical development kinetics, charging, drift and pattern transfer. Display gain does not change the data. At fixed dose, the beam current changes the writing time. The beam paths show the geometry at the start of exposure, and the relief shows the developed resist. The hardware and the lateral and vertical scales are enlarged independently.

Start with the essentials

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

Core mathematical model

Scattering spreads the dose before development

\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[R_i\ge D_c]\end{aligned}

Each Gaussian has unit area. Any part of a tail that falls outside the computed field is lost and is not added back. The resist response is formed from the deposited exposure, stochastic reactions and the selected correlation length. Development reads only that response. At zero correlation with stochastic exposure off, it recovers the expected-dose reference.

Correcting the dose at the write sites

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

The correction makes eighteen nonnegative updates, only at the sites the design writes. It evens out the dose inside the pattern, but dose leaking outside the pattern and the effects of coarse sampling remain.

Compare achieved openings with the design

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

The design is used only after development, as a separate check of the overlap. Dose is given in units of a reference dose, 200 microcoulombs per square centimetre.

Common difficulties

Model parameters need calibration

Typical misconceptionA preset that works here is a recipe I can use on a real instrument.

Better mental modelThis teaching model stores both expected dose and a seeded stochastic exposure. In stochastic mode, electron arrivals at write sites and resist reactions are sampled separately; the assigned reaction yield is 0.02, followed by a Gaussian resist-response correlation. Development thresholds this response on a 10 nm grid. A new write number produces an independent exposure. Line-edge positions use linear interpolation along 16 rows over 160 nm, and RMS is computed about each edge mean; interrupted lines do not receive a complete-line roughness value. These are calculated resist responses, without microscope metrology noise or material calibration. Scattering is modeled by two normalized Gaussians, with the broad one assigned a weight of 0.6 relative to the narrow one, and their widths are not measured material parameters. The positive resist opens wherever the resist response reaches a sharp threshold, on a 10 nm grid. The correction evens out the dose inside the pattern before coarse shot sampling is applied; it is neither an exact inversion nor a general proximity correction. The model leaves out Monte Carlo electron transport, chemical development kinetics, charging, drift and pattern transfer. Display gain does not change the data. At fixed dose, the beam current changes the writing time. The beam paths show the geometry at the start of exposure, and the relief shows the developed resist. The hardware and the lateral and vertical scales are enlarged independently.

Run the experiment

01
Repeat the same exposure
Choose One stochastic exposure, then press Write another specimen. Keep the dose and sampling step fixed while comparing the edge traces. Try a different correlation length.
What to observe: The target stays fixed, while each written edge differs. A wider correlated response smooths local fluctuations and can also blur the opening.
02
Predict how line density changes exposure
Choose Isolated versus dense and compare the widths of the isolated and dense lines in the same developed image.
What to observe: Neighboring lines add dose to each other, even outside the areas they were meant to expose.
03
Correct the input, then develop
Choose Correct before development. Change the development threshold while watching the deposited-dose card.
What to observe: The dose map stays fixed while the developed openings change.
04
Try coarse write sampling
Set the shot sampling step to 40 nm, then compare the sampled writes with the design and the developed result.
What to observe: With coarse sampling, some sites are never written, and dose correction cannot bring them back.
05
Check the full target
Aim for at least 85 percent overlap, at most 5 percent unintended opening, an isolated width of 20–40 nm and a dense-to-isolated width difference of at most 10 nm.
What to observe: At fixed dose, the current changes the writing time, and display gain cannot change the openings.

Core mathematical model

Scattering spreads the dose before development

Correcting the dose at the write sites

Compare achieved openings with the design

Common difficulties

Model parameters need calibration

Run the experiment

Repeat the same exposure

Predict how line density changes exposure

Correct the input, then develop

Try coarse write sampling

Check the full target