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DeepONet-assisted angle-resolved scatterometry for sub-nanometre in situ metrology of nanowire geometry

This paper presents a DeepONet-assisted angle-resolved scatterometry framework that integrates operator learning for spectral correction, particle swarm optimization for RCWA inversion, and SAM-enhanced SEM validation to achieve sub-nanometre precision in the in situ metrology of Si-doped GaN nanowire geometry.

Original authors: Fei Han, Yichi Pan, Qimeng Sun, Gai Wu, Jingyuan Ni, Lijie Li, Chong Shen, Wei Shen, Dekun Yang

Published 2026-10-04

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CC BY 4.0

Original authors: Fei Han, Yichi Pan, Qimeng Sun, Gai Wu, Jingyuan Ni, Lijie Li, Chong Shen, Wei Shen, Dekun Yang

Original paper licensed under CC BY 4.0 (https://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

In the world of modern electronics, the most advanced chips rely on tiny structures called nanowires. These are not the thick wires found in a household lamp, but slender columns of semiconductor material so thin that hundreds could fit across the width of a human hair. To make these wires work, scientists grow them inside a vacuum chamber using a technique called molecular beam epitaxy. This process is like a high-precision oven where atoms of gallium, nitrogen, and silicon are shot at a hot surface, sticking together to build the wires layer by layer. The challenge is that these wires are incredibly sensitive. A tiny shift in temperature or a slight change in the amount of silicon added can alter their shape, size, and how they conduct electricity. If the wires are not the right size, the devices built from them will fail. For decades, scientists have struggled to measure these wires while they are still growing. Traditional methods require taking the sample out of the chamber to look at it under a powerful electron microscope, but by then, the growth process is over, and any mistakes cannot be fixed.

A team of researchers has developed a new way to measure these nanowire arrays with extreme precision while they are still inside the growth chamber, without ever touching them. They created a system that uses light to "see" the shape of the wires in real time. Instead of taking a photograph, the system shines a beam of light onto the growing wires and captures the pattern of light that bounces back. Because the wires are so small, they scatter the light in a specific way that depends entirely on their diameter and height. The researchers call this angle-resolved scatterometry. The problem is that the raw light patterns captured by the camera do not match the perfect mathematical models scientists use to predict how light should behave. The real world is messy, with imperfections in the lenses and the materials, causing a gap between what is measured and what the computer models expect. This mismatch usually makes it impossible to calculate the wire dimensions accurately.

To bridge this gap, the researchers introduced a clever software tool based on a type of artificial intelligence called DeepONet. Think of this tool as a translator that learns to convert the messy, real-world light patterns into a clean, idealized format that the computer models can understand. The system first captures the light scattered by the nanowires and turns it into a spectrum, which is essentially a graph showing how much light is reflected at different angles. The DeepONet software then takes this graph and corrects it, smoothing out the experimental errors and aligning it with the theoretical predictions. Once the light pattern is corrected, the system uses a search algorithm to find the exact diameter and height that would produce that specific pattern. It tests millions of possible combinations of size and height until it finds the one that matches the corrected light data perfectly.

To prove that this method actually works, the team needed a way to check their results against the truth. They built a separate, automated system to analyze images taken by an electron microscope after the growth was finished. This system used a powerful image recognition model to identify thousands of individual nanowires in the photos and measure their sizes automatically. By comparing the measurements taken by the light-based system during growth with the measurements taken by the electron microscope after growth, they could see how accurate their new method was. The results were remarkably precise. For the diameter of the wires, their system was off by an average of only 0.22 nanometers. For the height, the error was even smaller, at just 0.12 nanometers. To put this in perspective, a single nanometer is one-billionth of a meter; the error in their measurement is less than the width of a single atom.

The researchers tested this framework on a set of nanowire samples that the computer had never seen before during its training. Even with these completely new samples, the system maintained its high level of accuracy. This suggests that the method is not just memorizing specific patterns but has learned a general rule for translating light into shape. The study demonstrates that it is possible to monitor the growth of these delicate structures with sub-nanometer precision in real time. This capability could allow engineers to adjust the growth conditions instantly if a wire starts to form incorrectly, ensuring that every chip produced is perfect. By combining advanced optics, artificial intelligence, and rigorous validation, the team has turned a difficult optical puzzle into a reliable tool for the future of semiconductor manufacturing.

Technical Summary: DeepONet-assisted Angle-Resolved Scatterometry for Sub-nanometre In Situ Metrology of Nanowire Geometry

Problem Statement
Accurate in situ metrology of nanowire arrays grown by Molecular Beam Epitaxy (MBE) is critical for controlling semiconductor nanostructure fabrication. However, existing monitoring techniques face significant limitations. Ex situ methods (e.g., SEM) lack real-time feedback, while in situ techniques like Reflection High-Energy Electron Diffraction (RHEED) are sensitive to surface reconstruction but struggle with the quantitative extraction of nanowire geometric parameters. Optical scatterometry offers a non-contact, high-speed alternative; yet, under high Numerical Aperture (NA) conditions, direct electromagnetic inversion based on Rigorous Coupled-Wave Analysis (RCWA) suffers from non-uniqueness, instability, and systematic discrepancies between experimental spectra and ideal simulations. These mismatches degrade the robustness of geometric inversion, preventing sub-nanometre precision.

Methodology
The authors propose an integrated framework combining four synergistic components to reconstruct the morphology of Si-doped GaN nanowire arrays:

ARS-Enhanced Measurement: An angle-resolved scatterometry (ARS) system integrated into the MBE chamber captures wide-angle far-field scattering distributions. These are mapped onto the back focal plane (BFP) and converted into morphology-sensitive, normalized angle-resolved reflectance spectra, $R(\theta)$ .
Wavelength Selection: A supervised 1D-CNN classifier was employed to evaluate the discriminability of $R(\theta)$ across different wavelengths and doping temperatures. $\lambda = 574$ nm was selected as the optimal operating wavelength due to its superior feature separability and stable convergence.
DeepONet-Based Spectral Pre-correction: To address the systematic mismatch between measured spectra ( $R_{exp}$ ) and ideal RCWA simulations ( $R_{sim}$ ), a Deep Operator Network (DeepONet) was trained. This module learns a function-to-function mapping from the experimental spectral domain to the RCWA-compatible domain, producing a corrected spectrum ( $R_{corr}$ ). Crucially, DeepONet does not predict geometric parameters directly but corrects the spectral representation to align with physical simulations.
PSO-Driven RCWA Inversion: Particle Swarm Optimization (PSO) is used to retrieve the nanowire diameter ( $D$ ) and height ( $H$ ) by minimizing the Root Mean Square Error (RMSE) between the RCWA-simulated spectrum $R_{sim}(\theta; D, H)$ and the DeepONet-corrected target $R_{corr}(\theta)$ .
SAM-Assisted SEM Validation: For independent validation, an automated workflow utilizing the Segment Anything Model (SAM) was developed. This workflow processes top-view and cross-sectional SEM images to extract ensemble-averaged geometric parameters (equivalent diameter and height) from thousands of nanowire instances, providing a statistically robust reference without manual delineation.

Key Results
The framework was validated using a held-out dataset (T1080) consisting of samples grown under conditions completely excluded from all model training and hyperparameter tuning.

Spectral Correction: The DeepONet module successfully reduced the systematic domain shift between experimental and simulated spectra. Ablation studies showed that using raw measured spectra resulted in large reconstruction errors (RMSE of 4.098 nm for diameter and 26.530 nm for height). In contrast, using the DeepONet-corrected spectra reduced the RMSE to 0.219 nm for diameter and 0.118 nm for height.
Inversion Accuracy: On the T1080 test set, the proposed framework achieved sample-level mean RMSEs of 0.22 nm for diameter and 0.12 nm for height when compared to the SAM-derived SEM references.
Convergence: The spectral pre-correction significantly improved the stability of the PSO inversion, reducing the mean number of iterations required for convergence from 213 (raw data) to 156 (corrected data).
Generalization: The accurate reconstruction of the unseen T1080 condition demonstrates that the framework learns a generalizable spectral-alignment operator rather than memorizing specific training spectra.

Significance and Claims
The paper claims that this work demonstrates the feasibility of achieving sub-nanometre-precision optical reconstruction of nanowire ensemble geometry through the synergistic integration of:

Angularly resolved optical measurement.
Operator-learning-based spectral correction (DeepONet).
Physics-driven RCWA inversion (PSO).
Statistically rigorous, automated SEM validation (SAM).

The authors emphasize that by defining reconstructed quantities as "sample-level equivalent diameter and height," the method aligns with the periodic unit-cell assumptions of RCWA and the ensemble-averaged nature of the SEM reference. This approach addresses the long-standing challenge of systematic discrepancies in optical scatterometry, offering a practical, non-contact route for real-time monitoring of nanowire growth in MBE environments. The work establishes a design principle that operating wavelengths must be rigorously matched to morphology-sensitive spectral responses and that separating spectral correction from geometric inversion is essential for stable, high-precision metrology.

Technical Summary: DeepONet-assisted Angle-Resolved Scatterometry for Sub-nanometre In Situ Metrology of Nanowire Geometry

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