🔬 applied physics

Measurements, simulations, and models of the point-spread function of electron-beam lithography

This paper demonstrates that the traditional double-Gaussian model for electron-beam lithography point-spread functions is highly inaccurate at modern high voltages, proposing instead a superior power-law plus Gaussian model validated by measurements and Monte Carlo simulations to enable precise proximity effect correction and improved nanoscale resolution.

Original authors: Nikolaj B. Hougs, Kristian S. Knudsen, Marcus Albrechtsen, Taichi Suhara, Christian A. Rosiek, Søren Stobbe

Published 2026-09-07

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CC BY 4.0

Original authors: Nikolaj B. Hougs, Kristian S. Knudsen, Marcus Albrechtsen, Taichi Suhara, Christian A. Rosiek, Søren Stobbe

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

To build the tiny circuits that power our modern world, scientists often use a technique called electron-beam lithography. Imagine trying to draw the most intricate map imaginable, but instead of ink, you use a focused beam of electrons to "write" patterns onto a special light-sensitive material called a resist. Once the writing is done, the material is developed, revealing the pattern, which is then transferred to the underlying silicon or other substrate to create the final device. This process is the workhorse for creating the master templates used to mass-produce computer chips and advanced sensors. However, the electrons do not behave like perfect, obedient arrows that stop exactly where they are aimed. When they strike the material, they scatter, bouncing around deep inside the substrate and spreading out in all directions. This scattering deposits extra energy in places where it is not wanted, blurring the sharp edges of the intended pattern. This phenomenon, known as the proximity effect, is a major hurdle for creating the incredibly small features required by modern technology. For decades, researchers have relied on a simplified mathematical description to predict how these electrons spread, a model that assumes the scattering follows a specific, smooth curve. But as the technology has pushed toward higher speeds and finer details, this old description has begun to fail, leaving engineers guessing about the true shape of the blur.

A team of researchers at the Technical University of Denmark, working with collaborators in Japan, has now taken a fresh look at this problem. They set out to measure exactly how the energy from an electron beam spreads out when it hits a resist layer on different types of substrates, specifically silicon and indium phosphide. Using a powerful electron-beam system operating at a very high voltage of 150 kilovolts, they performed a series of precise experiments. Instead of trying to guess the pattern, they exposed tiny, isolated dots of the resist with varying amounts of energy. After developing the material, they measured the size of the holes left behind. By carefully analyzing how the size of these holes changed with the amount of energy used, they were able to reconstruct the true map of how the energy was distributed, a profile known as the point-spread function. This function tells them exactly how much energy lands at any given distance from the center of the beam.

The results of their measurements revealed a startling truth about the old model. The traditional description, which relies on a combination of two smooth curves, was woefully inaccurate. When the researchers compared this old model to their actual measurements, they found that it could be off by as much as four orders of magnitude. In practical terms, this means the old model was missing the vast majority of the energy distribution, particularly in the middle range of distances from the center of the beam. It failed to capture a specific region where the energy drops off in a way that the old curves simply could not describe. This inaccuracy is critical because the software used to correct for the proximity effect relies entirely on these models. If the model is wrong, the corrections applied to the writing process will be wrong, leading to distorted patterns and failed devices.

To solve this, the researchers proposed a new way of describing the electron scattering. They found that the energy distribution is best understood as a combination of two distinct behaviors: a steep, power-law drop-off that dominates the short and middle distances, and a smooth, bell-shaped curve that takes over at very long distances. This new model, which they call a power-law plus Gaussian model, matched their experimental data with remarkable precision across a vast range of distances, from just one nanometer out to one hundred micrometers. It also aligned perfectly with complex computer simulations that track the path of individual electrons as they bounce through the material. The researchers confirmed that this new description holds true regardless of whether the substrate is silicon or indium phosphide, and it remains consistent even when they changed the development time or added a thin metal layer to the surface.

The study also explored how changing the speed of the electrons, or the acceleration voltage, affects this scattering. They found that while the long-range spread of the electrons grows significantly as the voltage increases, the behavior of the electrons in the short and middle ranges remains surprisingly constant. This is a crucial insight because it means that the most difficult part of the scattering—the part that distorts the finest details of the pattern—does not change much even when the machine is turned up to its highest settings. The old model assumed that this middle range was negligible, but the new measurements show it is actually the dominant factor for the smallest features. The researchers concluded that continuing to rely on the old, simplified model has become a limiting factor for the resolution of modern nanofabrication. By adopting this new, more accurate description, engineers can finally correct the proximity effect with the precision needed to build the next generation of microscopic devices, ensuring that the patterns written by the electron beam match the designs intended by the scientists.

Technical Summary: Measurements, Simulations, and Models of the Point-Spread Function of Electron-Beam Lithography

Problem Statement
Electron-beam lithography (EBL) is a cornerstone of modern nanofabrication, yet its resolution is fundamentally limited by the proximity effect. This phenomenon arises when incident electrons scatter deep and far within the substrate, depositing unwanted energy doses at both nano- and micro-scales. Accurate proximity effect correction (PEC) relies on precise models of the electron point-spread function (PSF). Historically, the field has relied on a double-Gaussian (2G) model, which separates dose contributions into forward-scattered and backscattered electrons. However, the authors argue that this model is an oversimplification, particularly for modern high-acceleration voltage systems (e.g., 150 kV) and narrow beam widths. The 2G model fails to accurately describe the intermediate range of electron scattering, leading to significant discrepancies between theoretical predictions and experimental reality, thereby limiting the fidelity of nanoscale device fabrication.

Methodology
The study employs a combination of experimental measurements, Monte Carlo simulations, and empirical modeling to characterize the process point-spread function (PPSF).

Experimental Setup: Experiments were conducted using a 150 kV electron-beam lithography system on chemically semi-amplified resist (CSAR) coated on silicon and indium phosphide (InP) substrates.
Measurement Technique: The PPSF was extracted using single-shot exposures. First, the clearing dose ( $D_0$ ) was determined by exposing large squares with uniform doses until the resist was fully cleared. Subsequently, sequences of isolated point exposures were performed with varying shot charges ( $q_{in}$ ). The resulting developed hole radii ( $r_0$ ) were measured via scanning electron microscopy (SEM) and optical microscopy (OM). The PPSF was derived using the relationship $\Psi(r_0) = D_0 / q_{in}$ .
Simulation: Monte Carlo simulations were used to model electron scattering and deposited dose distributions, providing a theoretical baseline for comparison.
Modeling: The authors tested the standard 2G model against their data and proposed a new empirical model comprising the sum of a power-law term and a Gaussian term (PLG model).

Key Contributions

High-Precision PPSF Measurements: The authors present extensive measurements of the PPSF covering 10 orders of magnitude in deposited dose and 5 orders of magnitude in distance (1 nm to 100 µm) for both silicon and InP substrates at 150 kV.
Validation of the PLG Model: They propose a power-law plus Gaussian (PLG) model that replaces the forward-scattered Gaussian of the 2G model with a power-law term. This model demonstrates excellent agreement with both experimental data and Monte Carlo simulations across the entire measured range.
Quantification of Model Deviations: The study quantifies the failure of the 2G model, showing it deviates from experimental results by up to four orders of magnitude in the intermediate range.
Parameter Dependence Analysis: The paper systematically analyzes how PPSF parameters vary with acceleration voltage (5–150 kV), substrate material, development time, and the presence of an aluminum decharging layer.

Results

Model Accuracy: The PLG model accurately captures the PPSF behavior across different material stacks and processing conditions. In contrast, the 2G model significantly underestimates the proximity effect in the intermediate range (tens to hundreds of nanometers).
Voltage Dependence:
- For acceleration voltages $V > 50$ kV, the short- to intermediate-range parameters (power-law slope and normalization) remain remarkably constant.
- The long-range scattering parameter ( $\beta$ ) follows an approximately quadratic dependence on voltage, increasing as voltage increases.
- The clearing dose ( $D_0$ ) scales nearly linearly with voltage, inversely related to the total deposited energy.
Substrate and Process Effects:
- Changing the substrate from Silicon to InP (higher density) significantly increases the backscattering ratio ( $\eta$ ) and reduces the long-range characteristic radius ( $\beta$ ), indicating stronger but shorter-range scattering.
- Increasing development time lowers the clearing dose but has minimal effect on the PPSF shape.
- Adding a 20 nm aluminum decharging layer resulted in negligible changes to the PPSF at high voltages.
Impact on Lithography: Simulations of mask patterns convolved with the PLG versus 2G PPSFs reveal that the 2G model significantly underestimates the magnitude of the proximity effect for features in the 10 nm–1 µm range. This leads to distorted patterns and poor agreement with the input mask when using 2G-based PEC.

Significance
The paper concludes that the reliance on the double-Gaussian approximation for proximity effect correction has become a limiting factor for resolution in modern high-resolution EBL. As EBL technology moves toward higher acceleration voltages and narrower beams, the inaccuracies of the 2G model in the intermediate range become increasingly pronounced. The proposed PLG model offers a simple yet accurate framework for quantifying electron scattering and optimizing PEC parameters. By providing a model that aligns with physical simulations and experimental data across a wide range of conditions, this work enables more precise fabrication of nanoscale structures, moving beyond the limitations of previous empirical approximations. The authors emphasize that accurate PPSF modeling is essential for advancing device fabrication, particularly where the 2G model's assumption of a negligible intermediate range is no longer valid.

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