Abstract
As a single layer of carbon arranged in a honeycomb, graphene has garnered significant scientific interest due to its unique physical, mechanical, and chemical properties resulting from its reduced dimensionality. The discovery of graphene as the first truly two-dimensional (2D) crystal challenged the Mermin-Wagner theorem, which posits that 3D fluctuations destroy long-range order in 2D crystals. Subsequent experimental observations and theoretical calculations revealed that monolayer graphene is characterized by intrinsic three-dimensional (3D) rippling, driven by thermodynamic instability, structural defects, grain boundaries, and interactions with supporting substrates. These structural distortions significantly influence critical properties, such as electronic transport, making it essential to determine the 3D positions of all atoms and their dynamic behavior to fully understand graphene's stability and electronic properties.Recent advancements in aberration-corrected electron optics and high-speed data acquisition have enabled transmission electron microscopy (TEM) to achieve sub-angstrom resolution and detect individual atoms. However, for low-dimensional materials like graphene, which are highly sensitive to electron irradiation, maintaining a low electron dose is crucial to prevent structural damage. This requirement limits the signal-to-noise ratio (SNR) of TEM images, creating substantial challenges for high-resolution structural and dynamic analyses. In addition, electron beam-sample interactions introduce further complexities. Experimental evidence shows that the electron beam can not only alter atomic structures but also induce phenomena such as atomic dynamics, phase transformations, and radiolytic artifacts at the nanoscale. These effects highlight the dual role of the electron beam as both a pump to stimulate atomic dynamics and a probe for imaging at the atomic scale. To minimize damage while enabling dynamic studies, the electron dose must be carefully controlled to remain sufficiently low. However, this often creates a trade-off, where resolution is limited by shot noise, and while extending exposure time can improve image quality, it sacrifices time-resolved information. Conversely, higher imaging speeds capture fast dynamics but require lower doses per image, further exacerbating the challenges.
To overcome these limitations, we developed a novel methodology for determining the 3D atomic structure of free-standing single-layer graphene from a single low-dose TEM image. Our approach leverages Simulated Annealing (SA), a probabilistic optimization method designed to identify the global minimum of a cost function with multiple local minima. By defining the energy function as the difference between the reconstructed and experimental images and treating the 3D atomic positions as the system's state, we achieved highly accurate reconstructions of the structure of graphene. Given the inherent low SNR in low-dose TEM imaging, atomic positions can deviate significantly during the reconstruction process. To address this, we integrated Molecular Dynamics (MD) simulations into each iteration, ensuring that the predicted atomic positions remain physically plausible. This innovative approach enabled precise 3D structural determination of graphene with sub-nanometer accuracy in the z-direction, even under challenging imaging conditions, achieving results significantly more accurate than prior methods.
Furthermore, by combining single-shot reconstruction with sequential TEM imaging, we successfully captured the 3D dynamics of ripples in free-standing graphene excited by the electron beam. These ripples, which arise due to electron stimulation during imaging, were reconstructed with high temporal resolution, revealing how the topographical features of graphene evolve dynamically. Using Density Functional Theory (DFT), we also explored how these structural distortions influence electronic properties, providing a deeper understanding of the interplay between 3D geometry of graphene and its electronic behavior. Additionally, we integrated Kullback-Leibler (KL) divergence, a mathematical measure of differences between probability distributions, to analyze and compare TEM images under varying conditions. This statistical framework enabled robust quantification of pixel intensity distributions, offering a powerful tool for assessing image quality and dose-dependent resolution.
In addition to intact graphene structures, our work extended to defected regions, including the analysis of defected “bridge” structures. These regions exhibited rapid and complex dynamics over time, challenging traditional imaging techniques. Our method successfully captured the evolution of these defects, providing critical insights into their formation, growth, and reorganization. By systematically evaluating the influence of electron dose on structural reconstruction, we identified a critical threshold dose, below which structural information becomes indistinguishable due to noise. This result establishes the minimum dose required for meaningful analysis and serves as a guideline for optimizing low-dose TEM imaging protocols.
Overall, this research introduces a powerful and versatile framework for studying the 3D atomic structure and dynamics of graphene under low-dose imaging conditions. By addressing the challenges posed by low SNR in TEM imaging, our method enables the precise characterization of both intact and defected graphene structures. It also lays the foundation for future investigations into the dynamic properties of low-dimensional materials, offering a pathway to study their structural evolution, defect formation, and electronic behavior. These findings contribute to advancing nanotechnology and materials science, facilitating the development of innovative applications in 2D materials research and beyond.
| Date of Award | 12 Sept 2025 |
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| Original language | English |
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| Supervisor | Alice HU (Supervisor) |