Modelling Bulk and Edge Phonons in 2D Materials Observed with High-Resolution Transmission Electron Microscopy

When simulating High-Resolution Transmission Electron Microscopy (HR-TEM) images, atomic vibrations are usually taken into account either by convolving the electrostatic potential with a Debye-Waller factor, or by the Frozen Phonon method. The latter gives the most correct description of the vibrations, as it correctly reflects that the transition time of the electron through the sample is short compared to the atomic vibration time: the calculated image is an average of a sequence of images calculated with slightly perturbed atomic positions. Usually, perturbations are drawn from a normal distribution with a variance calculated with the Debye model.<br/> <br/>In a recent publication, Chen et al. were able to measure the vibrational amplitudes of the atoms in a graphite-supported cobalt-doped molybdenum disulfide (MoS₂) nanoparticle, using HR-TEM and exit wave reconstructions [1]. They observe a larger vibrational amplitude of the atoms near the edge of the nanoparticle.<br/> <br/>We use machine-learning potentials to study graphite-supported MoS₂ nanoparticles with molecular dynamics (MD). The MD simulations confirm an increase in vibrational amplitude, not only of the undercoordinated atoms right at the edge of the nanoparticles, but also of atoms situated a few lattice constants away from the edge. Simulations of the HR-TEM exit wave show a vibrational signature in the exit wave similar to what was observed by Chen et al., supporting their interpretation that it is induced by atomic vibrations. We interpret the enhanced vibrational amplitudes near the edges as the signature of a phonon edge state, similar to the electronic edge state that has been observed in MoS₂. <br/> <br/>The MD simulations were performed using Equivariant Neural Network Potentials trained on Density Functional Theory (DFT) calculations of similar but much smaller systems, using the NequIP [2] package. We demonstrate how a systematic approach to increasing the training set while limiting redundancy can result in a stable and reliable machine learning potential with only a very modest training set of around 400 DFT calculations.<br/> <br/> <br/>REFERENCES:<br/> <br/>1. Chen, F.-R. <i>et al.</i> Probing atom dynamics of excited Co-Mo-S nanocrystals in 3D. <i>Nat. Commun.</i> <b>12</b>, 5007 (2021).<br/>2. S. Batzner <i>et al.</i> E(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials. <i>Nat</i><i>.</i><i> Commun</i><i>.</i> <b>13</b>, 2453 (2022).