Anisotropic Tensile Strength and Fracture Mechanism of $\theta$-TaN: A Machine-Learning Potential Molecular Dynamics Study

Chenyang Cao; Hongfei Li; Shuo Cao

REVIEW 2 major objections 4 minor 71 references

Anisotropic Tensile Strength and Fracture Mechanism of $\theta$-TaN: A Machine-Learning Potential Molecular Dynamics Study

T0 review · 2 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read The paper establishes that theta-TaN, a metallic nitride candidate for chip thermal management, fractures in a strongly anisotropic, purely brittle way: 80.10 GPa tensile strength along the c-axis and 56.87 GPa along the a-axis, with cleava

desk verdict A careful NEP-MD tensile study that is probably right but whose headline numbers are partly in-sample; worth reviewing as a solid materials-data paper. read the letter →

arxiv 2607.27608 v1 pith:3NAYSDVQ submitted 2026-07-30 cond-mat.mtrl-sci

Chenyang Cao , Hongfei Li , Shuo Cao This is my paper · ORCID

classification cond-mat.mtrl-sci

keywords theta-tantalumnitridemachinelearningpotentialmoleculardynamicstensileanisotropybrittlefracturecleavageplanethermalmanagement

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

This paper aims to provide the missing mechanical-reliability data for theta-phase tantalum nitride, a material proposed for device thermal management because of its metallic conductivity and ultrahigh thermal conductivity. Using a machine-learned interatomic potential trained on first-principles data and large-scale molecular dynamics simulations, it claims that the material fails purely by cleavage, with strength 80.10 GPa and modulus 748.63 GPa along the c-axis, versus 56.87 GPa and 570.74 GPa along the a-axis, and fracture strains of 15.02% and 17.71% respectively. The paper also finds nearly linear thermal softening between 300 and 900 K, with more than 73% of the room-temperature strength retained at 900 K. These numbers matter because they let engineers judge whether theta-TaN can survive the thermal stress and repeated loading of real electronic devices.

What carries the argument

The central tool is the neuroevolution potential (NEP), a machine-learned interatomic potential built from first-principles calculations and trained with an active-learning loop that includes uniaxially strained configurations along both tensile axes from 1% to 25% strain. In molecular dynamics, this potential allows simulations with millions of atoms, reaching a converged supercell size of about 20 nm. The paper uses surface energies and generalized stacking fault energies for the basal, m-prismatic, and a-prismatic planes to explain why cleavage, not dislocation emission, controls fracture.

What would settle it

Perform static first-principles tensile calculations along [0001] and [2-1-10] at fine strain increments around the peaks (for instance 14-16% for c-axis and 17-19% for a-axis) and compare peak stresses to 80.10 and 56.87 GPa; a deviation beyond a few GPa would refute the strength claim. An experimental tensile test on a theta-TaN film would provide the ultimate check.

Watch

Extended reading notes

Core claim

The paper establishes that theta-TaN fails by brittle cleavage without dislocation activity, and that the cleavage plane depends on loading direction: {10-10} prismatic planes under a-axis tension and the (0001) basal plane under c-axis tension. It reports ideal tensile strengths of 80.10 GPa (c-axis) and 56.87 GPa (a-axis), with corresponding moduli of 748.63 GPa and 570.74 GPa, and fracture strains of 15.02% and 17.71%. It also shows that from 300 to 900 K the modulus, strength, and fracture strain decrease almost linearly, retaining more than 73% of the 300 K strength at 900 K.

Load-bearing premise

The results rest on the trained machine-learned potential remaining accurate in the rare, disordered bond-breaking configurations near a nucleating crack, which the training set samples only indirectly through uniform strains and thermal vibrations.

Editorial extensions

If this is right

If theta-TaN is used in thermal management or interconnects, it will tolerate high tensile stresses but will fail suddenly and without plastic warning.
The strong anisotropy means device designers can orient films so that the main tensile load falls along the c-axis to exploit the higher 80 GPa strength, at the cost of a lower fracture strain.
The near-linear thermal softening and retention of more than 73% of 300 K strength at 900 K suggest mechanical usability well above typical operating temperatures.
Because the isostructural tungsten carbide shows a similar anisotropy pattern, the WC-type crystal geometry appears to control the directional ranking of strength, suggesting design routes through chemical substitution.

Reading between the lines

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

The cleavage-limited behavior implies low fracture toughness and sensitivity to pre-existing flaws; combining the reported surface energies with a Griffith criterion could predict critical crack sizes in theta-TaN films.
The training set covers uniaxial tension along only two axes, but service conditions such as biaxial or cyclic loading might activate different failure modes not yet probed.
The comparison with tungsten carbide suggests that substituting other metals in the WC-type nitride structure could tune absolute strength while preserving anisotropy, offering an untested design handle.

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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

Quick read: this is a solid, carefully executed NEP-MD tensile study of θ-TaN, and it fills a real gap in the mechanical data for this material. I'd trust the numbers as a well-converged simulation, but the validation is partly in-sample, so the abstract's 'predicted' should be read as 'MLP simulation consistent with DFT on the training manifold.'

What the paper does well: the size-convergence study up to 40 nm is thorough, the strain-rate and temperature sweeps are standard and properly replicated, and the NEP fitting errors are small. The surface-energy and generalized-stacking-fault checks are a good idea, because they test the energetics most relevant to crack nucleation and cleavage. The comparison with isostructural WC is useful context, and the fracture-plane analysis (prismatic cleavage under a-axis tension, basal cleavage under c-axis tension) is clearly presented.

The main soft spot is the one the stress-test note flags. The training set explicitly includes uniaxially strained configurations along the same a and c axes over 1–25% strain, which is exactly the window from which the headline strengths and fracture strains are extracted. The Fig. 3 DFT comparison on trajectories extracted from the NEP-MD runs is nice, but the paper never states that those configurations were held out from active learning. If they were added during training, the agreement is an in-sample check, not independent validation. The only clearly independent test set is hydrostatic compression, which does not exercise the near-fracture tensile states. This does not invalidate the central claim, but it does mean the reported strengths are interpolations of the fitted manifold, and the paper should say so plainly.

A couple of minor things: the data-availability statement mentions Zenodo but gives no URL or hash, which should be fixed; and the 'defect atom' criterion is a qualitative visualization threshold, which is fine but should not be mistaken for a rigorous defect analysis. The absence of experimental data is unsurprising for a new material and is not by itself a flaw.

Net: this is a competent, honest study with a clearly bounded claim. The authors need to clarify the training/validation split and soften the language from 'prediction' to 'MLP simulation validated against DFT,' ideally adding a held-out tensile test if they can. It deserves peer review, not desk rejection. Send it out.

Referee Report

2 major / 4 minor

Summary. The paper develops a neuroevolution potential (NEP) for θ-TaN from DFT-PBE data, including active-learning sampling of uniaxial tensile configurations along the a and c axes at 300–900 K, and uses it in large-scale MD tensile simulations on supercells up to 40 nm. The central quantitative results are strongly anisotropic: c-axis [0001] tensile strength 80.10 GPa, modulus 748.63 GPa, fracture strain 15.02%; a-axis [2-1-10] strength 56.87 GPa, modulus 570.74 GPa, fracture strain 17.71%. The paper reports nearly linear thermal softening from 300 to 900 K, weak strain-rate dependence over 10^7–10^9 s^-1, and brittle fracture via microvoid formation and cleavage: {10-10} prismatic planes under a-axis tension and (0001) under c-axis tension, with no observable dislocation activity. Validation includes NEP-DFT stress–strain comparison on selected configurations, surface energies, and stacking-fault energies.

Significance. The manuscript addresses a genuine gap: tensile and fracture properties of θ-TaN have not been characterized. Its strengths include systematic size convergence (5–40 nm, five independent runs per condition), strain-rate and temperature studies, and a public data-availability statement for training/testing files. If the tensile validation is genuinely independent, the results provide a credible first atomistic estimate of θ-TaN ideal tensile strengths and fracture planes, with clear relevance for thermal-management reliability. However, the tensile validation in Fig. 3 risks being an in-sample check because the training set was actively enriched from the same tensile paths; this issue must be resolved before the headline strength values can be treated as robust predictions.

major comments (2)

[§II.B.2 and §III.A (Fig. 3)] The only direct tensile validation of the NEP potential is Fig. 3, but the manuscript does not establish that those configurations were excluded from training. Section II.B.2 states that the training set includes uniaxial tensile configurations along [0001] and [2-1-10] over 1–25% strain at 300–900 K, and that active learning added high-uncertainty snapshots from MD tensile simulations under exactly these conditions. Fig. 3 uses 128-atom configurations extracted from 300 K NEP-MD tensile trajectories over the same 1–25% strain range. If these configurations participated in training, the agreement is an interpolation check, not an independent validation. The only explicitly independent test set is hydrostatic compression, which does not exercise the tensile fracture path. Please state clearly that the Fig. 3 configurations were held out, or replace/supplement Fig. 3 with a held-out tensil
[§III.A (Table II) and §II.B.2] The transferability claims for the fracture regime rest on surface energies and unstable stacking-fault energies, but these are necessary rather than sufficient for crack nucleation: fracture involves strained bond networks, microvoids, and undercoordinated atoms near the crack tip. The active-learning dataset was built from homogeneous tensile deformation, so local crack/void configurations may be underrepresented. Since the a-axis crack path runs along {10-10} prismatic planes, the 1.3 J/m^2 deviation (11%) in m-plane γ_us and the 0.14 J/m^2 deviation in the m-plane surface energy are directly relevant. A quantitative statement of how these errors propagate into the predicted cleavage-plane selection and peak stress would strengthen the claim; ideally, the DFT/NEP stress-strain comparison should also include held-out configurations containing the actual crack and microvoid geometry.

minor comments (4)

[Introduction, Ref. [20]] The moment tensor potential (MTP) is cited to Ref. [20], which appears to be a paper on diamond/Al composites rather than on MTP. Please replace this with the appropriate MTP reference.
[§III.E] The term 'normalized displacement larger than 0.5' is used to define defect atoms, but the normalization is not defined. State the reference length (e.g., lattice constant or nearest-neighbor distance) used to normalize displacements; otherwise the displacement maps are not interpretable quantitatively.
[§III.C] The strain-rate sensitivity exponents m_a = 0.0015 and m_c = 0.0024 are fitted from five points over a total strength change of less than 1.5%. Please report fit uncertainties or explicitly state that these exponents are only order-of-magnitude indicators; as presented, the difference between m_a and m_c may not be statistically meaningful.
[Throughout] Several typographical and formatting issues remain: '10 7' and '109' should be 10^7 and 10^9, 'θ-T aN' appears with spacing artifacts, and some equation/caption formatting is inconsistent. A careful copyedit is needed.

Circularity Check

2 steps flagged · score 6.0 of 10

Tensile 'prediction' is an interpolation of the training manifold: the NEP was trained on a/c uniaxial tension at 1–25% strain and 300–900 K, then reported as a prediction with a validation drawn from the same tensile MD trajectories.

fitted input called prediction [Section II.B.2 (Datasets for training the NEP model); Section III.A (Benchmark), Fig. 3]
"Since the main objective of this work is to study the anisotropic tensile behavior of θ-TaN along the a and c axes, the training dataset further included uniaxially strained configurations along the a axis ([2-1-10]) and the c axis ([0001]) at different temperatures from 300 to 900 K. The strain range was set to 1%–25%. ... To further verify the reliability of the NEP model under tensile loading, representative configurations were extracted from the 300 K NEP-MD tensile trajectories along increasing strain."

The central quantities—80.10/56.87 GPa strength and 15.02/17.71% fracture strain—are obtained from MD along exactly the a/c loading paths and temperatures that were used to construct the training set, including active-learning snapshots from MD tensile simulations in the 1–25% strain window. Those strains bracket the reported fracture strains, so the stress-strain response, including the peak, is interpolation of fitted DFT virials. Fig. 3 then validates NEP against DFT on configurations extracted from the same 300 K tensile MD trajectories without stating they were excluded from training; the agreement is therefore an in-sample check, not a held-out test. The only explicitly independent test set is hydrostatic compression, a different loading mode that does not test the tensile fracture p
fitted input called prediction [Section II.B.2; Section III.A, Table II]
"The remaining 407 configurations, accounting for 38.7%, consisted of lattice-perturbation configurations, shear-strain configurations, uniaxially strained configurations along the a and c axes over the 1%–25% strain range, including those added through active learning, and generalized stacking-fault configurations."

The paper validates the potential's stacking-fault energetics (Table II) and uses the high γ_us values to argue that dislocation nucleation is unfavorable. But generalized stacking-fault configurations are explicitly part of the training set, so the NEP/DFT agreement on γ_us is a fit to those labels rather than an independent confirmation. This weakens the dislocation-absence argument, though it is secondary to the main tensile-strength claim.

full rationale

The derivation chain is not circular in the mathematical sense: NEP is a standard ML fit to DFT energies/forces/virials, and the crack paths are emergent from large-scale MD. The circularity is in the validation architecture. The training set was deliberately focused on the same uniaxial tensile modes (a and c axes, 1–25% strain, 300–900 K) that the paper then reports as predictions; active learning even added MD tensile snapshots back into the training data. The reported fracture strains and strengths lie inside this fitted manifold, and the Fig. 3 DFT comparison uses configurations drawn from the same 300 K tensile MD trajectories, with no statement that they were held out. Thus the central claim is not independently tested by tension data. Some independent evidence exists—the hydrostatic-compression test set, the external DP force RMSE, and possibly surface energies—but none of these exercises the tensile fracture path. The fracture-plane selection and brittle-vs-ductile character are more emergent and less directly fitted, so a score of 6 (partial circularity) is appropriate rather than 8 or 10. No self-citation chain or definitional identity is at play.

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

The reported numbers are not derived analytically; they are outputs of an MLP fitted to DFT data. The physical content rests on PBE accuracy, training-set coverage near fracture, the MD loading protocol, and hand-set visualization thresholds. No new physical entities are introduced, and no experimental mechanical data for θ-TaN is provided as an external benchmark.

free parameters (4)

NEP network weights and biases = SNES-optimized, not enumerated
All energy, force, and virial predictions, including those used in fracture simulations, come from this fitted neural network.
NEP descriptor cutoffs, expansion orders, hidden width = r_c=5 Å; n_R,max=n_A,max=8; l_3b,max=4; l_4b,max=2; N_neu=50
Chosen via convergence tests; these hyperparameters define the learned potential-energy surface used for every reported mechanical value.
Defect-atom displacement threshold = 0.5 (normalized displacement)
Hand-set visualization criterion used to identify 'defect atoms' and support the fracture-mechanism maps; not used in strength/modulus numbers.
Strain-rate sensitivity exponents m_a, m_c = 0.0015, 0.0024
Power-law fits to the simulated strength vs strain-rate data; descriptive, not load-bearing for the central claim.

assumptions (4)

domain assumption DFT-GGA-PBE is an accurate reference for θ-TaN ideal tensile strength and fracture
All NEP labels come from VASP/PBE; if PBE misjudges Ta-N bond rupture at 15-18% strain, every reported strength and cleavage plane inherits the error. Used throughout Sections II.B.2-III.A.
ad hoc to paper Training coverage over 1-25% uniaxial strain along [0001] and [2-1-10] implies predictive accuracy in the fracture window
The 1,053-configuration dataset with active learning is assumed to span fracture-nucleation configurations; the external test set is hydrostatic compression, not tension. See Section II.B.2.
domain assumption Quasi-uniaxial MD with periodic boundaries, thermostat, and transverse barostat reproduces ideal tensile fracture of a defect-free crystal
Tensile protocol in Section II.C; real films contain defects and surfaces that would lower strengths below the reported ideal values.
domain assumption NEP descriptors (radial and angular, Eqs. 1-3) are sufficiently complete to represent the θ-TaN PES near fracture
The representational capacity of the descriptor + neural network is assumed; only fitting RMSE and limited DFT spot checks are provided.

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Cite this review

Pith. "Pith review of Anisotropic Tensile Strength and Fracture Mechanism of $\theta$-TaN: A Machine-Learning Potential Molecular Dynamics Study." pith.science (2026). https://pith.science/paper/3NAYSDVQ

@misc{pith2026260727608,
  author       = {Pith},
  title        = {Pith review of: Anisotropic Tensile Strength and Fracture Mechanism of $\theta$-TaN: A Machine-Learning Potential Molecular Dynamics Study},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3NAYSDVQ}},
  note         = {Machine review of arXiv:2607.27608}
}

read the original abstract

theta-phase tantalum nitride (theta-TaN) combines metallic conductivity with exceptionally high thermal conductivity, making it a potential material for device thermal management and interconnect applications. However, its tensile strength and fracture behavior remain unclear. Here, we investigate the anisotropic tensile response and fracture mechanism of theta-TaN using neuroevolution-potential molecular dynamics simulations. Size-convergence tests show that a 20 nm long model is sufficient for reliable prediction, and the mechanical parameters vary by less than 3.5% over the strain-rate range of 10^7 to 10^9 s^-1. The results reveal strong tensile anisotropy. The c-axis direction ([0001]) shows a higher strength of 80.10 GPa and modulus of 748.63 GPa, but a lower fracture strain of 15.02%. In contrast, the a-axis direction ([2-1-10]) shows a lower strength of 56.87 GPa and modulus of 570.74 GPa, but a higher fracture strain of 17.71%. From 300 to 900 K, the mechanical properties decrease nearly linearly, while more than 73% of the 300 K strength is retained at 900 K. Fracture occurs without observable dislocation activity and is governed by cleavage-plane selection: {10-10} prismatic planes under a-axis tension and the (0001) basal plane under c-axis tension. Atomic displacement analysis shows that local separation and microvoid formation precede macroscopic crack growth, indicating a brittle fracture process driven by local bond-network instability. These results provide atomic-scale mechanical data for assessing the reliability of theta-TaN in thermal management applications.

Figures

Figures reproduced from arXiv: 2607.27608 by the authors.

**Figure 2.** (a), the loss function decreases continuously with increasing training generations and then approaches convergence. This indicates that the SNES optimization is stable and that the model parameters are well converged. Figs. 2(b)–(d) compare the NEP predictions with the DFT reference values for energy, force, and virial in both datasets. For the training dataset, the root-meansquare errors (RMSEs) of energy, force… view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: shows the results of large-scale MD tensile simulations using the NEP potential. As shown in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: (a) presents the stress–strain curves at different strain rates. In the elastic regime, the curves almost overlap, indicating that the mechanical response in this stage is insensitive to strain rate. With increasing strain rate, the a-axis curves show a slight increa…

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 8.** Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

**Figure 9.** Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]

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Works this paper leans on

71 extracted references · 1 canonical work pages

[1]

This method combines a single- hidden-layer feedforward neural network with the sep- arable natural evolution strategy (SNES) optimization algorithm[37]

The NEP Formalism In this work, the potential energy surface ofθ-TaN was constructed using the neuroevolution potential (NEP) framework [25–28]. This method combines a single- hidden-layer feedforward neural network with the sep- arable natural evolution strategy (SNES) optimization algorithm[37]. The model parameters are iteratively op- timized during tr...
[2]

For the perfect crystal, configurations were sampled from NPT, NVT, and selected NVE ensembles over the temperature range of 300–900 K

Datasets for training the NEP model To accurately evaluate the mechanical response ofθ- TaN under tensile loading, a training dataset specifically designed for tensile simulations was constructed. For the perfect crystal, configurations were sampled from NPT, NVT, and selected NVE ensembles over the temperature range of 300–900 K. To improve the ability o...
[3]

defect atoms

Trained NEP model The NEP model was trained using theGPUMDpack- age [45]. The hyperparameters used in the training are summarized in Table I. The model parameters were opti- mized using the SNES algorithm. The loss function was defined as a weighted sum of the root-mean-square errors (RMSEs) of energy, force, and virial, with the weightsλe, λf , andλ v se...
[4]

M. Li, S. Li, Z. Zhang, C. Su, B. Wong, and Y. Hu, Ad- vancing thermal management technology for power semi- conductors through materials and interface engineering, Accounts of Materials Research6, 563–576 (2025)

2025
[5]

Mengesha and K

W. Mengesha and K. Nagessar, Advanced thermal and magnetic materials for high-power and high-temperature applications: a comprehensive review, Discover Materials 5(2025)

2025
[6]

Y. Cui, M. Li, and Y. Hu, Emerging interface materials for electronics thermal management: experiments, mod- eling, and new opportunities, J. Mater. Chem. C8, 10568 (2020)

2020
[7]

B. Wang, Y. He, N. Rodionov, and J. Zhu, First- principles predictions of thermal conductivity of bulk di- amond under isotropic and uniaxial (100) strains, Mate- rials Today Physics36, 101182 (2023)

2023
[8]

S. Li, Q. Zheng, Y. Lv, X. Liu, X. Wang, P. Y. Huang, D. G. Cahill, and B. Lv, High thermal conductivity in cubic boron arsenide crystals, Science361, 579 (2018)

2018

Show all 71 references

[9]

S. Yue, F. Tian, B. Song, Y. Zhong, J. Bao, and X. Liu, Cubic boron arsenide: An emerging semiconductor with exceptional thermal conductivity and high carrier mobil- ity, Matter8, 102131 (2025)

2025
[10]

Waqas, H

M. Waqas, H. M. Chong, P.-J. Sazio, and I. Zeimpekis, Cubic boron arsenide: A transformative material reshap- ing modern electronics and thermal management, Micro and Nano Engineering32, 100377 (2026)

2026
[11]

Q. Cai, D. Scullion, W. Gan, A. Falin, S. Zhang, K. Watanabe, T. Taniguchi, Y. Chen, E. J. G. Santos, and L. H. Li, High thermal conductivity of high-quality monolayer boron nitride and its thermal expansion, Sci- ence Advances5, eaav0129 (2019)

2019
[12]

K. Chen, B. Song, N. K. Ravichandran, Q. Zheng, X. Chen, H. Lee, H. Sun, S. Li, G. A. G. U. Gamage, F. Tian, Z. Ding, Q. Song, A. Rai, H. Wu, P. Koirala, A. J. Schmidt, K. Watanabe, B. Lv, Z. Ren, L. Shi, D. G. Cahill, T. Taniguchi, D. Broido, and G. Chen, Ultrahigh thermal co...

2020
[13]

Van Nghia, H

N. Van Nghia, H. K. Hieu, P. T. M. Hanh, and L. T. Lam, Study of thermodynamic properties and elastic constants of cubic boron nitride by statistical moment method, Vacuum230, 113663 (2024)

2024
[14]

M. Kim, J. Kim, W. Park, and J. S. Kang, A review in thermal management for advanced chip packaging from chip to heat sink, Microelectronics Reliability170, 115782 (2025)

2025
[15]

S. Li, C. Su, Z. Qin, A. Alatas, M. Kunz, T. Ya- mada, S. D. Kelly, M. H. Upton, A. Gironda, J. Zhao, B. Kalkan, W. Yang, T. Aoki, and Y. Hu, Metallicθ- phase tantalum nitride has a thermal conductivity triple that of copper, Science391, 707 (2026)

2026
[16]

Kundu, X

A. Kundu, X. Yang, J. Ma, T. Feng, J. Carrete, X. Ruan, G. K. H. Madsen, and W. Li, Ultrahigh thermal conduc- tivity ofθ-phase tantalum nitride, Phys. Rev. Lett.126, 115901 (2021)

2021
[17]

Kundu, Y

A. Kundu, Y. Chen, X. Yang, F. Meng, J. Carrete, M. Kabir, G. K. H. Madsen, and W. Li, Electron-induced nonmonotonic pressure dependence of the lattice ther- mal conductivity ofθ-tan, Phys. Rev. Lett.132, 116301 (2024)

2024
[18]

S. A. Humphry-Baker, T. A. Mellan, M. Finnis, P. Pol- cik, W. E. Lee, M. Reece, and S. Grasso, Thermal conductivity of wc: Microstructural design driven by first-principles simulations, Acta Materialia283, 120517 (2025)

2025
[19]

M. Zhao, Q. Zhu, J. Zou, W. Wang, R. Zeng, L. Zhao, W. Ji, and Z. Fu, Binderless nanocrystalline tungsten carbide with enhanced hardness induced by high-pressure sintering, Journal of the European Ceramic Society44, 4875 (2024)

2024
[20]

M. Hu, X. Gao, Z. Wang, X. Dang, C. Yang, X. Li, M. Li, Y. Li, X. Hu, and Z. Huang, One-step synthesis of wc-xtac powders improves the mechanical properties of binderless wc cemented carbides, Ceramics International 51, 9960 (2025)

2025
[21]

H. Wang, L. Zhang, J. Han, and E. Weinan, DeePMD- kit: A deep learning package for many-body potential en- ergy representation and molecular dynamics, Computer Physics Communications.228, 178 (2018)

2018
[22]

A. P. Bart´ ok, M. C. Payne, R. Kondor, and G. Cs´ anyi, Gaussian approximation potentials: The accuracy of 15 quantum mechanics, without the electrons, Physical Re- view Letters.104, 136403 (2010)

2010
[23]

N. Li, Y. Zhang, Y. Zhang, X. Wang, H. Wu, L.-D. Zhao, and H. Zhang, Realizing ultrahigh thermal conductivity in bimodal-diamond/al composites via interface engineer- ing, Materials Today Physics28, 100901 (2022)

2022
[24]

Zhang, J

Y. Zhang, J. Xia, and B. Jiang, Physically motivated re- cursively embedded atom neural networks: Incorporating local completeness and nonlocality, Phys. Rev. Lett.127, 156002 (2021)

2021
[25]

C. W. Tan, M. L. Descoteaux, M. Kotak, G. de Mi- randa Nascimento, S. R. Kavanagh, L. Zichi, M. Wang, A. Saluja, Y. R. Hu, T. Smidt, A. Johansson, W. C. Witt, B. Kozinsky, and A. Musaelian, High-performance training and inference for deep equivariant interatomic potentials, Di...

2026
[26]

Batzner, A

S. Batzner, A. Musaelian, L. Sun, M. Geiger, J. P. Mailoa, M. Kornbluth, N. Molinari, T. E. Smidt, and B. Kozinsky, E(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials, Nature Communications13, 2453 (2022)

2022
[27]

Batatia, D

I. Batatia, D. P. Kovacs, G. Simm, C. Ortner, and G. Cs´ anyi, MACE: Higher order equivariant message passing neural networks for fast and accurate force fields, Advances in Neural Information Processing Systems.35, 11423 (2022)

2022
[28]

K. Song, R. Zhao, J. Liu, Y. Wang, E. Lindgren, Y. Wang, S. Chen, K. Xu, T. Liang, P. Ying,et al., General-purpose machine-learned potential for 16 ele- mental metals and their alloys, Nature Communications 15, 10208 (2024)

2024
[29]

Z. Fan, H. Dong, A. Harju, and T. Ala-Nissila, Homo- geneous nonequilibrium molecular dynamics method for heat transport and spectral decomposition with many- body potentials, Phys. Rev. B99, 064308 (2019)

2019
[30]

Z. Fan, Z. Zeng, C. Zhang, Y. Wang, K. Song, H. Dong, Y. Chen, and T. Ala-Nissila, Neuroevolution machine learning potentials: Combining high accuracy and low cost in atomistic simulations and application to heat transport, Phys. Rev. B104, 104309 (2021)

2021
[31]

Z. Fan, W. Chen, V. Vierimaa, and A. Harju, Efficient molecular dynamics simulations with many-body poten- tials on graphics processing units, Computer Physics Communications218, 10 (2017)

2017
[32]

W. Liu, H. Deng, J. Qi, X. Zhang, Y. Liu, Z. Zhao, J. Zhu, H. Luo, H. Yin, W. Xiang, N. Zhang, Y. Qin, L. Zhou, L. Chen, and Y. Ma, Heterogeneous crss evo- lution and multistage twinning in nearβtitanium alloys during tensile deformation at cryogenic temperature, Ma- terials S...

2026
[33]

G. Ren, J. Che, H. Zhang, H. Cai, W. Li, W. Hao, Q. Hu, X. Zhao, and F. Yang, Tailoring the fracture toughness of fluorite-type rare-earth tantalates: From atomic scale fracture mechanism to bond-mediated toughening design, Acta Materialia304, 121820 (2026)

2026
[34]

Y. Wang, X. Jia, R. Chen, and E. Gao, Data-driven extreme-value statistics for fracture size effects, Journal of Applied Physics138, 114303 (2025)

2025
[35]

Shi and Z

P. Shi and Z. Xu, Exploring fracture of h-bn and graphene by neural network force fields, Journal of Physics: Condensed Matter36, 415401 (2024)

2024
[36]

M. Yu, Z. Zhao, W. Guo, and Z. Zhang, Fracture tough- ness of two-dimensional materials dominated by edge en- ergy anisotropy, Journal of the Mechanics and Physics of Solids186, 105579 (2024)

2024
[37]

P. Ying, H. Dong, T. Liang, Z. Fan, Z. Zhong, and J. Zhang, Atomistic insights into the mechanical anisotropy and fragility of monolayer fullerene networks using quantum mechanical calculations and machine- learning molecular dynamics simulations, Extreme Me- chanics Letters58...

2023
[38]

Z. Qin, L. Zhu, J. Zhu, P. Ying, E. P. Busso, D. Li, and L. Sun, High strain rate performance of ni–co alloys with hierarchical nanotwins, International Journal of Mechan- ical Sciences324, 111741 (2026)

2026
[39]

S. Lin, L. Casillas-Trujillo, F. Tasn´ adi, L. Hultman, P. H. Mayrhofer, D. G. Sangiovanni, and N. Koutn´ a, Machine- learning potentials for nanoscale simulations of tensile deformation and fracture in ceramics, npj Computational Materials10, 67 (2024)

2024
[40]

Schaul, T

T. Schaul, T. Glasmachers, and J. Schmidhuber, High Di- mensions and Heavy Tails for Natural Evolution Strate- gies, inProceedings of the 13th Annual Conference on Genetic and Evolutionary Computation., GECCO ’11 (Association for Computing Machinery, New York, NY, USA, 2011) p...

2011
[41]

Behler and M

J. Behler and M. Parrinello, Generalized Neural-Network Representation of High-Dimensional Potential-Energy Surfaces, Physical Review Letters.98, 146401 (2007)

2007
[42]

A. P. Bart´ ok, R. Kondor, and G. Cs´ anyi, On representing chemical environments, Physical Review B.87, 184115 (2013)

2013
[43]

M. A. Caro, Optimizing many-body atomic descrip- tors for enhanced computational performance of machine learning based interatomic potentials, Phys. Rev. B100, 024112 (2019)

2019
[44]

Kresse and J

G. Kresse and J. Furthm¨ uller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Physical Review B.54, 11169 (1996)

1996
[45]

P. E. Bl¨ ochl, Projector augmented-wave method, Physi- cal Review B.50, 17953 (1994)

1994
[46]

Kresse and D

G. Kresse and D. Joubert, From ultrasoft pseudopoten- tials to the projector augmented-wave method, Physical Review B.59, 1758 (1999)

1999
[47]

J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized Gradient Approximation Made Simple, Physical Review Letters.77, 3865 (1996)

1996
[48]

Z. Fan, Y. Wang, P. Ying, and K. e. a. Song, Gpumd: A package for constructing accurate machine-learned po- tentials and performing highly efficient atomistic simu- lations, The Journal of Chemical Physics157, 114801 (2022)

2022
[49]

Z. Zong, Y. Qin, J. Zhan, H. Fang, and N. Yang, Machine learning potential for predicting thermal conductivity of θ-phase and amorphous tantalum nitride, International Journal of Heat and Mass Transfer257, 128155 (2026)

2026
[50]

V. Wang, N. Xu, J.-C. Liu, G. Tang, and W.-T. Geng, Vaspkit: A user-friendly interface facilitating high- throughput computing and analysis using vasp code, Computer Physics Communications267, 108033 (2021)

2021
[51]

Hirel, Atomsk: A tool for manipulating and convert- ing atomic data files, Computer Physics Communications 197, 212 (2015)

P. Hirel, Atomsk: A tool for manipulating and convert- ing atomic data files, Computer Physics Communications 197, 212 (2015)

2015
[52]

A. Stukowski, Visualization and analysis of atomistic simulation data with ovito–the open visualization tool, Modelling and simulation in materials science and engi- neering18, 015012 (2009)

2009
[53]

C. Chen, Y. Li, R. Zhao, Z. Liu, Z. Fan, G. Tang, and 16 Z. Wang, Neptrain and neptrainkit: Automated active learning and visualization toolkit for neuroevolution po- tentials, Computer Physics Communications317, 109859 (2025)

2025
[54]

G. E. Dieter, D. Bacon,et al.,Mechanical metallurgy, Vol. 3 (McGraw-hill New York, 1976)

1976
[55]

A. S. M. J. Islam, M. S. Akbar, M. S. Islam, and J. Park, Temperature- and defect-induced uniaxial ten- sile mechanical behaviors and the fracture mechanism of two-dimensional silicon germanide, ACS Omega6, 21861 (2021)

2021
[56]

D. G. Sangiovanni, F. Tasn´ adi, L. J. S. Johnson, M. Od´ en, and I. A. Abrikosov, Strength, transformation toughening, and fracture dynamics of rocksalt-structure Ti1−xAlxN (0≤x≤0.75) alloys, Phys. Rev. Mater.4, 033605 (2020)

2020
[57]

D. Yin, Y. Yang, X. Peng, Y. Qin, and Z. Wang, Tensile and fracture process of the tin/vn interface from first principles, Ceramics International40, 14453 (2014)

2014
[58]

Zhang, G

Z. Zhang, G. He, and J. Chen, First-principles study of the stability, electronic and mechanical properties of cr2n and crn with the variation of ni doping concentration, Ceramics International47, 24430 (2021)

2021
[59]

C. Lu, C. Cui, and F. Peng, Stress–strain relations and deformation mechanisms of zrn and hfn superconductors, Crystal Growth & Design22, 1104 (2022)

2022
[60]

Zhang, X

S. Zhang, X. Zhao, J. Zhang, J. Sheng, J. Ren, X. Lu, and X. Tang, Investigation on the influence of vacancy and alloying element content on the performance of fe/nbn interface, Metals15, 10.3390/met15070759 (2025)

2025 doi
[61]

T. Liao, J. Wang, and Y. Zhou, Deformation modes and ideal strengths of ternary layered ti2AlC and ti2AlN from first-principles calculations, Phys. Rev. B73, 214109 (2006)

2006
[62]

Huang, S

S. Huang, S. Xu, Y. Xu, Z. Zhou, and J. Li, Doping effects on mechanical and thermodynamic properties of zirconium carbide systems: a first-principles study, Ma- terials Research Express8, 065012 (2021)

2021
[63]

Hua and D

G. Hua and D. Li, A first-principles study on the me- chanical and thermodynamic properties of (Nb1-XTix)C complex carbides based on virtual crystal approximation, RSC Advances5, 103686 (2015)

2015
[64]

Sun, C.-Y

M. Sun, C.-Y. Wang, and J.-P. Liu, Anisotropic elastic properties and ideal uniaxial compressive strength of tib2 from first principles calculations, Chinese Physics B27, 077103 (2018)

2018
[65]

Zhang, X

X. Zhang, X. Luo, J. Li, P. Hu, and J. Han, The ideal strength of transition metal diborides tmb2 (tm=ti, zr, hf): Plastic anisotropy and the role of prismatic slip, Scripta Materialia62, 625 (2010)

2010
[66]

Wang and Y.-J

J. Wang and Y.-J. Wang, Mechanical and electronic properties of 5d transition metal diborides mb2 (m=re, w, os, ru), Journal of Applied Physics105, 083539 (2009)

2009
[67]

Z. Wang, X. Wang, X. Song, X. Zhang, H. Liu, and M. Zhang, Computational prediction of ideal strength for a material, Computational Materials Science231, 112600 (2024)

2024
[68]

Zhang, H

Y. Zhang, H. Sun, and C. Chen, Structural deformation, strength, and instability of cubic bn compared to dia- mond: A first-principles study, Phys. Rev. B73, 144115 (2006)

2006
[69]

Zhang, H

Y. Zhang, H. Sun, and C. Chen, Ideal tensile and shear strength ofβ−c 3n4 from first-principles calculations, Phys. Rev. B76, 144101 (2007)

2007
[70]

Zhang, H

Y. Zhang, H. Sun, and C. Chen, Influence of carbon con- tent on the strength of cubic bc xN: A first-principles study, Phys. Rev. B77, 094120 (2008)

2008
[71]

Jiang, Z

C. Jiang, Z. Lin, J. Zhang, and Y. Zhao, First-principles prediction of mechanical properties of gamma-boron, Ap- plied Physics Letters94, 191906 (2009)

2009

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

This method combines a single- hidden-layer feedforward neural network with the sep- arable natural evolution strategy (SNES) optimization algorithm[37]

The NEP Formalism In this work, the potential energy surface ofθ-TaN was constructed using the neuroevolution potential (NEP) framework [25–28]. This method combines a single- hidden-layer feedforward neural network with the sep- arable natural evolution strategy (SNES) optimization algorithm[37]. The model parameters are iteratively op- timized during tr...

[2] [2]

For the perfect crystal, configurations were sampled from NPT, NVT, and selected NVE ensembles over the temperature range of 300–900 K

Datasets for training the NEP model To accurately evaluate the mechanical response ofθ- TaN under tensile loading, a training dataset specifically designed for tensile simulations was constructed. For the perfect crystal, configurations were sampled from NPT, NVT, and selected NVE ensembles over the temperature range of 300–900 K. To improve the ability o...

[3] [3]

defect atoms

Trained NEP model The NEP model was trained using theGPUMDpack- age [45]. The hyperparameters used in the training are summarized in Table I. The model parameters were opti- mized using the SNES algorithm. The loss function was defined as a weighted sum of the root-mean-square errors (RMSEs) of energy, force, and virial, with the weightsλe, λf , andλ v se...

[4] [4]

M. Li, S. Li, Z. Zhang, C. Su, B. Wong, and Y. Hu, Ad- vancing thermal management technology for power semi- conductors through materials and interface engineering, Accounts of Materials Research6, 563–576 (2025)

2025

[5] [5]

Mengesha and K

W. Mengesha and K. Nagessar, Advanced thermal and magnetic materials for high-power and high-temperature applications: a comprehensive review, Discover Materials 5(2025)

2025

[6] [6]

Y. Cui, M. Li, and Y. Hu, Emerging interface materials for electronics thermal management: experiments, mod- eling, and new opportunities, J. Mater. Chem. C8, 10568 (2020)

2020

[7] [7]

B. Wang, Y. He, N. Rodionov, and J. Zhu, First- principles predictions of thermal conductivity of bulk di- amond under isotropic and uniaxial (100) strains, Mate- rials Today Physics36, 101182 (2023)

2023

[8] [8]

S. Li, Q. Zheng, Y. Lv, X. Liu, X. Wang, P. Y. Huang, D. G. Cahill, and B. Lv, High thermal conductivity in cubic boron arsenide crystals, Science361, 579 (2018)

2018