Visualizing phonon edge states on molybdenum disulphide

Jakob Schi{\o}tz; Mathias Stokkebye Nissen; Patrick Giese; Stig Helveg

REVIEW 4 major objections 5 minor 25 references

Visualizing phonon edge states on molybdenum disulphide

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Low-energy phonon modes pinned to the edges of MoS2 nanoparticles explain the enhanced atomic vibrations and the contrast fade seen in high-resolution electron microscopy near the boundary.

desk verdict A plausible and well-communicated edge-phonon explanation for enhanced vibrations at MoS2 edges, with a genuinely useful active-learning ENNP workflow, but the central mechanism rests on an unvalidated ML potential for low-frequency phonons. read the letter →

arxiv 2509.08497 v1 pith:F33U3VCH submitted 2025-09-10 cond-mat.mtrl-sci

Patrick Giese , Mathias Stokkebye Nissen , Stig Helveg , Jakob Schi{o}tz This is my paper · ORCID

classification cond-mat.mtrl-sci

keywords phononedgestatesmolybdenumdisulphidemoleculardynamicsequivariantneuralnetworkpotentialsfrozenapproximationhigh-resolutiontransmissionelectronmicroscopyvibrationalamplitudesactivelearning

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that the enhanced atomic vibrations seen in high-resolution electron microscopy near MoS2 nanoparticle edges are caused by low-energy phonon modes confined to the edges. Using molecular dynamics driven by a machine-learned potential trained on about 400 density-functional calculations, the authors reproduce the falloff of image contrast toward edges seen experimentally and show that it persists for roughly a nanometer. Phonon calculations on a strip of MoS2 reveal edge-localized modes that shift the vibrational density of states to low frequencies and extend a few lattice constants inward. The work makes the case that correlated molecular dynamics, rather than an independent-atom model, is needed to interpret quantitative HR-TEM images of such particles.

What carries the argument

The central object is the edge-localized phonon mode, identified by projecting the phonon density of states of a finite strip onto the formula units at the edges and coloring band-structure modes by the squared vibration amplitude weight on the edge atoms. The argument is carried by the frozen-phonon imaging model: atomic configurations sampled from molecular dynamics are used to generate and average exit waves, and the same machine-learned potential, an equivariant graph neural network trained on roughly 400 DFT configurations, is used for both the molecular dynamics and the phonon calculations. The mechanism is that low-energy edge modes, with energies below roughly $k_B T$, increase the mean-square vibrational amplitude near the edge, which smears the projected electrostatic potential and lowers the exit-wave peaks.

What would settle it

Compute the phonon band structure and edge-projected density of states of the same 20-unit-cell MoS2 strip using direct DFT force constants and compare with the machine-learned potential; if the low-frequency edge modes are not reproduced, the explanation fails. Also, measure the temperature dependence of the HR-TEM contrast decay and check that it follows the occupation of the predicted edge-mode energies.

Watch

Extended reading notes

Core claim

On the paper's own terms: the enhanced and correlated vibrational amplitudes near the edges of carbon-supported MoS2 nanoparticles, observed in molecular dynamics and in a recent HR-TEM experiment, are due to low-energy phonon modes localized at the nanoparticle edges. Projecting the phonon density of states onto edge formula units shows a strong shift toward low frequencies relative to the bulk sheet, and the band structure of a 20-unit-cell strip contains bands whose weight is concentrated on the Mo or S edge, some extending a few lattice constants into the interior. These modes suppress the imaginary part of the simulated electron exit wave at atomic columns up to a nanometer from the edge, matching the experimental contrast decay. The paper further claims that a workflow combining active learning with equivariant neural network potentials can deliver an accurate potential for this supported-nanoparticle system from only slightly more than 400 DFT configurations, with an ensemble of five potentials providing a posteriori error estimates.

Load-bearing premise

The machine-learned potential, trained on about 400 DFT configurations, reproduces the true low-energy vibrational modes of MoS2 edges even though its phonon accuracy is not separately validated against DFT.

Editorial extensions

If this is right

HR-TEM image interpretation for MoS2 nanoparticles must include correlated atomic motion: the independent-atom Einstein model cannot reproduce the observed contrast decay.
The vibrational amplitude enhancement extends roughly a nanometer from the edge, so quantitative image simulation must include atoms that are not under-coordinated.
Edge phonon modes, not just the low coordination of the outermost atoms, are the cause of the enhanced amplitudes.
The active-learning and equivariant-neural-network workflow can produce usable potentials from about 400 DFT configurations and can flag out-of-distribution configurations through ensemble disagreement.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If edge-localized low-energy phonons are generic in finite 2D crystals, HR-TEM and STEM contrast simulations of other 2D materials will need the same correlated-molecular-dynamics treatment.
The ensemble disagreement among the five potentials could be turned into a per-atom uncertainty map during molecular dynamics, giving a practical way to detect when the phonon picture itself is unreliable.
A testable extension is temperature dependence: the edge modes' occupation grows with temperature, so the edge-to-bulk contrast decay should steepen at higher temperatures in a way a static disorder model would not predict.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read this one if you care about quantitative HR-TEM interpretation or ML potentials for 2D materials. The real news is the explanation: the enhanced vibrational amplitude near MoS2 edges seen by Chen et al. is attributed to low-energy phonon modes localized at the edge, and these modes extend a few lattice constants into the sheet. That attribution is new as far as I know, and it matters because it argues that correlated MD, not an independent-atom Einstein model, is needed for quantitative image interpretation.

What the paper does well: the active-learning ENNP workflow is a genuine methodological contribution. Roughly 400 DFT configurations, query-by-committee with ensemble disagreement, and an honest account of the unphysical Mo migration event they caught and fixed. The phonon projected DOS on a strip shows a clear low-energy enhancement at the edges, with modes penetrating a few unit cells, consistent with the MD amplitudes and the simulated exit-wave decay.

The soft spot is the one the stress-test flags, and it is the load-bearing one: the ENNP's accuracy for low-frequency phonons is never quantified. There are no force/energy RMSEs on the held-out set, and no comparison of the ML phonon DOS or force constants against DFT. The training data are mostly relaxation endpoints with forces below 0.2 eV/A, which does not specifically constrain low-frequency modes. Finite-displacement phonon calculations amplify small force errors at low energies. Since the same potential drives the MD, the exit-wave simulation, and the phonon analysis, any bias in its low-energy force constants propagates directly into the edge-state claim. This is addressable: report test-set errors, and compute at least bulk and strip phonons at the DFT level for a few configurations, or compare against the experimental phonon DOS.

Smaller issues: the MD statistics are thin—one 5 ps collection window per condition—and the phonon analysis uses a free-standing strip while the MD includes the graphite support, a mismatch the authors acknowledge. The comparison to the Chen et al. data is qualitative; they do not overlay the experimental decay curve. None of these are fatal, but they should be asked about in review.

Who is this for? Electron microscopists modeling images, and anyone building targeted ML potentials for nanosystems. It deserves a serious referee. I would send it to review and ask for potential validation data and a quantitative comparison to experiment before accepting. The core idea is plausible, but I want to see the potential's low-frequency fidelity demonstrated.

Referee Report

4 major / 5 minor

Summary. The manuscript presents a combined molecular dynamics (MD) and phonon study of carbon-supported MoS2 nanoparticles, using an equivariant neural network potential (ENNP) trained on a limited set of DFT (PBE-D3) calculations via an active-learning workflow. The authors report enhanced vibrational amplitudes near the nanoparticle edges, both for Mo and S atoms, and show that this enhancement extends roughly a nanometer from the edge. They attribute this behavior to low-energy phonon modes localized at the edges, identified by projecting the phonon density of states (DOS) onto the edge formula units of a 20-unit-cell-wide strip. They further simulate HR-TEM exit waves using the frozen-phonon method with MD snapshots and observe a decay of the imaginary-part peak values near the edges, consistent with the experimental findings of Chen et al. The paper also emphasizes the value of correlated MD over independent-atom models for quantitative image interpretation and presents the active-learning workflow as a way to obtain accurate machine-learning potentials with built-in error estimation.

Significance. If the central claim holds, the paper makes a useful contribution by explaining the experimentally observed edge-induced contrast decay through phonon edge states, and by demonstrating the necessity of correlated atomic motion in HR-TEM simulations. The active-learning framework for training an ENNP with a few hundred DFT configurations is a practical step toward simulating realistic nanostructures beyond DFT. The manuscript is clearly written and the workflow is described in sufficient detail to be reproduced. However, the central explanation rests on the reliability of the ENNP for low-frequency phonons, a point that is not quantitatively validated. The paper therefore has clear potential but requires strengthening of the evidence before its main claims can be considered fully supported.

major comments (4)

[Generation of the machine learning potentials and Phonon band structure simulations] The central claim that low-energy phonon edge states cause the enhanced vibrational amplitudes is not supported by any quantitative validation of the ENNP for phonon properties. The manuscript reports neither force/energy test-set errors nor a comparison of the ENNP phonon DOS, band structure, or force constants against DFT results. Because the same potential is used for the MD simulations, the exit-wave simulations, and the phonon analysis, any inaccuracy in the low-frequency force constants propagates directly into the edge-state conclusion. Please provide test-set error metrics and a direct DFT comparison of the low-energy phonon modes (for example, a DFT phonon calculation on a small strip or cluster) to demonstrate that the edge states are not an artifact of the machine-learning potential.
[Phonon band structure simulations (Appendix)] The phonon calculations are performed on a free-standing 2D sheet, which possesses a quadratic transverse acoustic mode (the bending mode) that the authors acknowledge may distort the lowest parts of the DOS. This is a serious concern because the edge-induced DOS increase appears precisely at low energies. Without a correction, a constraining of the out-of-plane motion, or a comparison to a supported strip, it remains possible that the low-energy enhancement is dominated by the free-standing bending mode rather than by localized edge states. The authors should quantify the effect of this artifact on the edge-projected DOS, for instance by repeating the calculation with the out-of-plane directions constrained or by applying a weak restoring force.
[Molecular Dynamics and Nanoparticle simulations] The vibrational amplitudes underlying the comparison with experiment are computed from a single 5 ps production trajectory per condition, following 5 ps of thermalization. This sampling may be insufficient to converge the variance of atomic positions, especially for the low-frequency edge modes whose periods are on the order of several picoseconds. The manuscript does not report block averaging, independent trajectories, or any convergence analysis with respect to simulation length. Given that the exit-wave comparison and the central interpretation rest on these amplitudes, the quantitative conclusions are not yet statistically supported. Additional analysis of statistical convergence is needed.
[Abstract and Nanoparticle simulations] The claim of a 'built-in error estimation' is stronger than what is actually demonstrated. The ensemble of five potentials is used during active learning and for validation of shorter sequences, but the production MD and the phonon calculations use a single potential trained on the full generation-5 dataset. No uncertainty estimates or error bars are provided for the central results (vibrational amplitude profiles, exit-wave peak values, or phonon DOS). Please clarify the role of the ensemble in the final results and provide quantitative uncertainty measures for the main figures.

minor comments (5)

[Abstract] The parenthetical citation 'by Chen et al (Nature Communications 12, 5007 (2021).' is missing a closing parenthesis.
[Exit wave simulations] The text refers to 'the 3 nm wide MoS2 nanoparticle in Fig. 1(b)', but Fig. 1 shows a nanoparticle with radius 3 nm; the terminology is ambiguous and should be consistent.
[Molecular Dynamics] The description of the trajectory collection could state explicitly that only a single trajectory per condition was used and that no block or ensemble averaging was performed, which would make the sampling limitations transparent.
[Figure 3] The caption of Figure 3 mentions that bands are 'colored by their weight' but does not define the weight; the definition is given in the text, but adding it to the caption would improve readability.
[Conclusion] The conclusion states that the potential 'shows that the vibrational amplitudes are significantly enhanced in a region near the edge' without acknowledging the potential limitations regarding trajectory length and phono validation; a brief caveat would be appropriate.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the edge-state explanation emerges from a DFT-trained ML potential and is not fitted to the target observation.

full rationale

The paper's derivation chain is: PBE-D3 DFT energies and forces feed an active-learning equivariant neural network potential (ENNP); the ENNP drives MD simulations from which vibrational amplitudes and frozen-phonon exit waves are computed; the same ENNP is then used for phonon band-structure and DOS calculations on a finite strip, revealing low-energy modes localized near the edges. None of these links is fitted to the experimental observation of Chen et al.; that experiment is used only as a qualitative comparison after the simulations are already produced. The ENNP is trained exclusively on DFT data, not on phonon spectra, edge-state weights, or the experimental exit-wave decay, so the enhanced edge vibrational amplitude and the edge-localized phonon modes are emergent properties of the fitted interatomic forces rather than quantities encoded by construction. Using the same potential for MD and phonon analysis is internally consistent, not circular, because the potential was not constructed to reproduce the target phenomenon. The only self-citations (refs. 15 and 16) support general methodological claims about active learning and the data efficiency of equivariant neural networks; they are not load-bearing for the edge-state mechanism and no uniqueness theorem is imported from the authors' prior work. The paper's main weakness is the absence of a quantitative phonon validation of the ENNP, such as force RMSEs or a DFT comparison of low-frequency modes; that is a correctness and robustness risk, not a circularity. No equation or fitted parameter is renamed as a prediction, and no reduction of the central claim to its inputs can be exhibited. Accordingly, no circular step is identified and the circularity score is 0.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The paper's central explanation is emergent from an ML-potential-based MD and phonon calculation; no ad hoc physical entities are introduced. The main cost is the assumption that the ML potential accurately captures low-frequency edge modes and that a free-standing strip represents the supported nanoparticle.

free parameters (4)

ENNP model parameters (weights and biases of NequIP networks) = trained on ~250-300 DFT configurations (generation 5); 5 ensemble members
The ML potential is fit to PBE-D3 DFT energies and forces. All physical predictions (vibrational amplitudes, phonon DOS) inherit the accuracy of this fit. The paper does not report force/energy test errors for the final potential.
Active learning hyperparameters: force disagreement threshold and the 100 eV/A filter cutoff = 100 eV/A; configurations with larger forces discarded
Chosen by hand to reject unphysical structures; affects the training set composition and hence the potential.
MD collection window = 5 ps after 5 ps thermalization
Vibrational amplitudes are computed from a single 5 ps trajectory; no convergence check or independent runs are reported.
Edge-cutoff difference for S-rich vs Mo-rich nanoparticles = 1 A smaller cutoff for Mo than S
Used to create S-rich vs Mo-rich edges; this alters edge stoichiometry and thus edge vibrational amplitudes.

assumptions (4)

domain assumption PBE-D3 DFT is an accurate reference for the energetics and forces of MoS2/graphite interfaces and edges.
All training labels come from PBE+D3 (GPAW, 600 eV cutoff, single k-point). The central claim inherits any DFT error, especially in low-frequency modes.
domain assumption The NequIP equivariant neural network can extrapolate to the nanoparticle sizes and temperatures used in the MD after active learning on smaller clusters.
The workflow assumes that the active learning loops (generations 0-5) sample the relevant configuration space. The observed Mo-migration event shows this assumption can fail; the authors patched it with targeted retraining.
domain assumption Phonon modes of a free-standing 20-unit-cell MoS2 strip are a valid proxy for the edge modes of the circular supported nanoparticles and of the experimental particles.
The authors state the supported system cannot be computed due to unit-cell mismatch. The strip lacks the substrate and the circular geometry, and the buckling ZA mode may distort the lowest DOS.
domain assumption The frozen phonon approximation with 100 snapshots converges the simulated exit wave.
Used in abTEM multislice simulations; no convergence test with respect to the number of snapshots is reported.

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Cite this review

Pith. "Pith review of Visualizing phonon edge states on molybdenum disulphide." pith.science (2026). https://pith.science/paper/F33U3VCH

@misc{pith2026250908497,
  author       = {Pith},
  title        = {Pith review of: Visualizing phonon edge states on molybdenum disulphide},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F33U3VCH}},
  note         = {Machine review of arXiv:2509.08497}
}

read the original abstract

We employ Molecular Dynamics (MD) simulations to study atom vibrational amplitudes in carbon-supported Molybdenum Disulphide (MoS2) nanoparticles. Enhanced and correlated atom vibrational amplitudes are observed as the nanoparticle edges are approached from the bulk, consistent with recent experimental High-Resolution Transmission Electron Microscopy (HR-TEM) observations by Chen et al (Nature Communications 12, 5007 (2021). Analysis of phonon modes in finite systems explains the experimental observation by low-energy phonon modes confined at the nanoparticle edge, underscoring the need of full MD modeling for accurate HR-TEM image interpretation. Noticeably, we introduce a workflow for training Equivariant Neural Network-based machine learning potentials using limited Density Functional Theory (DFT) calculations. This approach effectively captures both covalent and van der Waals interactions, enabling accurate extrapolations of DFT calculations to larger systems with built-in error estimation.

Figures

Figures reproduced from arXiv: 2509.08497 by the authors.

**Figure 3.** FIG. 3. The phonon state (DOS) in MoS [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** FIG. 4. Phonon modes localized at either the sulphur edge [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗

Figures from the paper (5 more)

**Figure 5.** Figure 5: FIG. 5. (a) An example of an error produce by the model [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 6.** Figure 6: FIG. 6. Initial MoS [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

**Figure 7.** Figure 7: FIG. 7. The phonon state (DOS) in MoS [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

**Figure 3.** Figure 3: The DOS projected on the formula units at the edge of the strip is significantly altered, as already shown in [PITH_FULL_IMAGE:figures/full_fig_p007_3.png]

**Figure 8.** Figure 8: FIG. 8. The vibrational amplitude of Sulphur atoms for [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Kisielowskiet al., Detection of single atoms and buried defects in three dimensions by aberration-corrected elec- tron microscope with 0.5- ˚A information limit, Microsc

C. Kisielowskiet al., Detection of single atoms and buried defects in three dimensions by aberration-corrected elec- tron microscope with 0.5- ˚A information limit, Microsc. Microanal.14, 469 (2008)

work page 2008

[2] [2]

R. Erni, M. D. Rossell, C. Kisielowski, and U. Dah- men, Atomic-resolution imaging with a sub-50-pm elec- tron probe, Phys. Rev. Lett.102, 096101 (2009)

work page 2009

[3] [3]

F.-R. Chen, D. V. Dyck, C. Kisielowski, L. P. Hansen, B. Barton, and S. Helveg, Probing atom dynamics of ex- cited Co-Mo-S nanocrystals in 3D, Nature Comm.12, 5007 (2021)

work page 2021

[4] [4]

Z. Chen, Y. Jiang, Y.-T. Shao, M. E. Holtz, M. Odstrˇ cil, M. Guizar-Sicairos, I. Hanke, S. Ganschow, D. G. Schlom, and D. A. Muller, Electron ptychography achieves atomic-resolution limits set by lattice vibrations, Science372, 826 (2021)

work page 2021

[5] [5]

R. F. Loane, P. Xu, and J. Silcox, Thermal vibrations in convergent-beam electron diffraction, Acta Crystallogr. A47, 267 (1991)

work page 1991

[6] [6]

Hillyard and J

S. Hillyard and J. Silcox, Detector geometry, thermal dif- fuse scattering and strain effects in ADF STEM imaging, Ultramicroscopy58, 6 (1995)

work page 1995

[7] [7]

Van Dyck, Persistent misconceptions about incoher- ence in electron microscopy, Ultramicroscopy111, 894 (2011)

D. Van Dyck, Persistent misconceptions about incoher- ence in electron microscopy, Ultramicroscopy111, 894 (2011)

work page 2011

[8] [8]

J. S. Smith, B. Nebgen, N. Lubbers, O. Isayev, and A. E. Roitberg, Less is more: Sampling chemical space with active learning, J. Chem. Phys.148, 241733 (2018)

work page 2018