---
title: "Electron Lithography: Draw, Expose, Develop Interactive Lab | Sandbox Physics"
description: "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."
canonical_url: "https://sandboxphysics.com/labs/electron-lithography/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/electron-lithography/"
last_modified: "2026-10-27"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

Physics tutorial

## Electron Lithography: Draw, Expose, Develop

**Background** 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.

**Why it matters** The goal is a pattern that overlaps the design by at least 85 percent, with at most 5 percent of the open area outside the design, an isolated line 20–40 nm wide, and dense and isolated lines whose widths differ by 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 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 misconception** A preset that works here is a recipe I can use on a real instrument.

**Better mental model** 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.

### Run the experiment

1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

Physical conclusion

Dose correction and shot sampling act before the development threshold and change the result; display gain acts afterward and only changes how the dose looks.

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)