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REVIEW 3 major objections 5 minor 73 references

Thermal Conductivity Of Monolayer Hexagonal Boron Nitride: Four-Phonon Scattering And Quantum Sampling Effects

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The lattice thermal conductivity of monolayer h-BN drops to about 150 W/mK when four-phonon scattering is included, showing that 3-phonon-only calculations overestimate heat transport in planar 2D materials.

desk verdict A careful TDEP+MLIP study confirming the large 4-phonon suppression of monolayer h-BN thermal conductivity and adding a new PIMD comparison showing negligible nuclear quantum effects; the headline 150 W/mK value remains conditional on the standard energy-conservation rule, which the authors themselves flag as open. read the letter →

arxiv 2506.14547 v1 pith:JOPSNM53 submitted 2025-06-17 cond-mat.mtrl-sci

José Pedro Alvarinhas Batista , Matthieu J. Verstraete , Aloïs Castellano This is my paper · ORCID
classification cond-mat.mtrl-sci
keywords monolayerhexagonalboronnitridelatticethermalconductivityfour-phononscatteringnuclearquantumeffectspath-integralmoleculardynamicstemperaturedependenteffectivepotentialflexuralphononsmirror-planesymmetry
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

Monolayer hexagonal boron nitride is a light, planar crystal whose reported thermal conductivities span 218 to 1060 W/mK. The paper argues that the spread comes from missing physics: including four-phonon scattering cuts the room-temperature value from 1085 W/mK (with isotope and three-phonon scattering) to about 150 W/mK. The reason is structural: the mirror-plane symmetry of the sheet suppresses three-phonon scattering, so even the smaller fourth-order force constants dominate the resistance. The paper also tests nuclear quantum effects and finds them negligible for this observable, so classical sampling suffices. If right, prior calculations that stop at three-phonon scattering systematically overestimate heat conduction in planar 2D materials.

What carries the argument

The central objects are the four-phonon scattering phase space opened by mirror-plane symmetry in strictly planar 2D materials, and the temperature dependent effective potential (TDEP) method, which extracts temperature-dependent interatomic force constants from molecular dynamics snapshots. The fourth-order force constants are smaller in amplitude than the third-order ones, but the three-phonon phase space is so restricted that four-phonon events carry most of the scattering, particularly for the low-frequency acoustic flexural mode. The numerical work uses a moment tensor potential machine-learning interatomic potential to sample configurations, and solves the iterative Boltzmann transport equation with the TDEP solver.

What would settle it

Recompute the room-temperature thermal conductivity of monolayer h-BN with the fluctuation-dissipation-compliant energy-conservation rule for four-phonon processes (the same treatment used for boron arsenide). If the four-phonon contribution drops dramatically and the value moves back toward the 1000 W/mK range, the paper's central claim that four-phonon scattering is essential would be falsified.

Watch

Extended reading notes

Core claim

Using temperature-dependent effective potentials fitted to classical and path-integral molecular dynamics, the authors compute the lattice thermal conductivity of monolayer h-BN with isotope, three-phonon, and four-phonon scattering. Within the standard (on-shell) energy-conservation rule, they obtain 150 W/mK at room temperature, a factor-of-seven reduction from the 1085 W/mK obtained with only isotope and three-phonon scattering. Four-phonon processes cut the flexural acoustic phonon lifetimes by up to two orders of magnitude, especially at low temperature, because the mirror-plane symmetry forbids odd numbers of flexural modes in three-phonon events. The classical and path-integral sampling results agree at all temperatures, showing nuclear quantum effects are not important for this system's thermal transport.

Load-bearing premise

The headline numbers rest on the standard on-shell energy-conservation rule for four-phonon scattering; if the fluctuation-dissipation-compliant energy-conservation rule (which the authors applied to boron arsenide) is the correct one for h-BN, the importance of four-phonon scattering would shrink and the computed thermal conductivity would rise.

Editorial extensions

If this is right

  • Room-temperature lattice thermal conductivity of monolayer h-BN is about 150 W/mK, roughly seven times lower than the value obtained from three-phonon plus isotope scattering alone.
  • Prior first-principles calculations that omit four-phonon scattering overestimate heat conduction in planar 2D materials with mirror-plane symmetry.
  • Classical molecular dynamics sampling is sufficient for h-BN thermal transport; path-integral sampling adds no accuracy for this observable at 150 K and above.
  • The large spread of experimental and theoretical values (218-1060 W/mK) is attributed to missing four-phonon channels, plus a possible experimental overestimate from air exposure.
  • Isotope scattering lowers the four-phonon-inclusive value by about 20% at room temperature, so it remains a significant channel.

Reading between the lines

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

  • The mirror-plane argument generalizes: any strictly planar 2D material with a flexural mode should show the same dominance of four-phonon scattering, so graphene's thermal conductivity may also be sensitive to the same energy-conservation correction tested here.
  • The fluctuation-dissipation-compliant correction applied to boron arsenide could change the h-BN numbers upward; the paper's own outlook flags this as the next step, and it is the natural falsifying test.
  • A testable prediction: measurements in high vacuum of high-quality suspended monolayer h-BN should land near 150 W/mK, whereas measurements in air or on supported samples should read higher.
  • The TDEP least-squares argument implies that machine-learning force errors do not bias phonon lifetimes, a practical advantage that makes this workflow easier to port to other 2D materials.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper computes the lattice thermal conductivity of monolayer hexagonal boron nitride using temperature-dependent effective potentials (TDEP) fitted to classical MD and path-integral MD (PIMD) sampling driven by a machine-learned moment-tensor potential, with the iterative Boltzmann transport equation including isotope, three-phonon, and four-phonon scattering. The central results are that four-phonon scattering reduces the room-temperature conductivity from 1085 W/mK (3-phonon plus isotope) to 150 W/mK (3+4-phonon plus isotope), that nuclear quantum effects are negligible for this property, and that the previously reported spread of theoretical and experimental values can be explained by the omission of one or both of these channels. The paper explicitly acknowledges in the Conclusions/Outlook that the four-phonon numbers are obtained within the standard on-shell energy-conservation rule and that a fluctuation-dissipation-compliant treatment has been shown, for boron arsenide, to drastically reduce the importance of four-phonon scattering.

Significance. If the reported value kappa(300 K) ~ 150 W/mK is correct, the paper provides a resolution to a long-standing discrepancy in monolayer h-BN thermal transport and establishes four-phonon scattering as the dominant resistive process in a planar 2D material. The strength of the work is that the thermal conductivity is not fitted to any target value: the IFCs are obtained from first-principles-driven sampling, and the MLIP accuracy is documented. The comprehensive cross-checks (lattice parameter, pair distribution functions, R2 of TDEP fits, comparison with previous BTE and MD results) give internal consistency to the qualitative conclusions. The conclusion that classical sampling suffices is also useful for future studies of light-element 2D materials. However, the headline quantitative claim is conditional on the on-shell energy-conservation approximation, which the authors themselves identify as unresolved for this class of materials, and the four-phonon q-grid convergence is not demonstrated. Both points must be addressed before the specific value of 150 W/mK can be regarded as established.

major comments (3)
  1. [Section III C and Section IV (Conclusions/Outlook)] The paper's central quantitative claim -- kappa(300 K) = 150 W/mK with isotope plus 3-phonon and 4-phonon scattering, versus 1085 W/mK without 4-phonon scattering -- is computed "within the standard energy conservation approach" (Section III C). In the Conclusions/Outlook the authors state that for boron arsenide a fluctuation-dissipation-compliant treatment (Ref. [75]) drastically reduces the importance of four-phonon scattering relative to isotope and three-phonon scattering. Since monolayer h-BN also has large phonon band gaps that restrict the three-phonon phase space, the same correction could substantially raise the four-phonon-inclusive conductivity and weaken the claim that four-phonon scattering is essential. The manuscript needs either an FD-compliant four-phonon calculation for h-BN or a quantitative sensitivity estimate; as written, the central claim is conditional on a rule the authors themselves flag as unresolved.
  2. [Section II (Methods) and Section III C] For four-phonon scattering the manuscript uses the kappa obtained at the largest q-grid (64x64x1) without extrapolation, stating that the extrapolation can be noisy at the lowest temperatures (Section II). No convergence data for the four-phonon kappa as a function of q-grid density are shown. The headline order-of-magnitude reduction at 150 K and the factor-of-seven reduction at 300 K are the main quantitative results, so a grid-convergence study for the four-phonon channel is load-bearing; without it, the reported 150 W/mK is not established as the converged value.
  3. [Section III B and Fig. 5] The conclusion that nuclear quantum effects are negligible for kappa rests on the near-equality of the classical-MD and PIMD results, but the PIMD sampling at 150 K shows noticeably lower R2 values in the TDEP fits, which the authors attribute to bead-averaging noise (Fig. 5 and Section III B). The manuscript should demonstrate that increasing the number of beads beyond the N_beads x T = 12000 scaling does not change the low-temperature kappa or the phonon lifetimes; otherwise the "classical suffices" conclusion may be affected by sampling noise rather than being a fully physical statement.
minor comments (5)
  1. [Section III C, last paragraph] The Conclusions state that previous theoretical results "range from 550-650 W/mK at room temperature," but Section III C cites published values spread from 218 to 1060 W/mK (after renormalization); the summary range should be made consistent with the values actually listed.
  2. [Section II (Methods)] The statement that MLIP force errors do not affect the TDEP fit assumes the errors are zero-mean and independent of the configuration; this assumption should be stated explicitly, since MLIP errors can be correlated with the local environment.
  3. [Section II (Methods)] The effective thickness normalization uses c/a = 1.317 from bulk h-BN; a one-sentence sensitivity analysis to this convention would help comparisons, since all literature values are renormalized to this choice.
  4. [Fig. 8 caption] The caption contains a typo: "it's PIMD counterpart" should be "its PIMD counterpart."
  5. [Section III A] The text describes the MTP as "order 22"; it would be clearer to specify whether this is the polynomial degree or the number of basis functions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: κ is computed from DFPT-validated IFCs via the IBTE with no parameter fitted to the target value; the energy-conservation caveat is an acknowledged modeling uncertainty, not a circular reduction.

full rationale

The derivation chain is self-contained and predictive. The MLIP is validated against DFT forces (R2 = 0.9998, Section III A) and used only to generate canonical-ensemble snapshots; the TDEP method then least-squares fits second-, third-, and fourth-order IFCs from those snapshots (Section II), and the lattice thermal conductivity is obtained by solving the iterative Boltzmann transport equation with isotope scattering included via the Tamura model. No parameter is fitted to the target κ: the values 1303/1085 W/mK (3-phonon ± isotope) and 180/150 W/mK (3+4-phonon ± isotope) at room temperature are emergent outputs of the scattering phase space and IFC amplitudes, so the claimed factor-of-~7 reduction from four-phonon scattering is a computed comparison, not a fitted input renamed as a prediction. Self-citations to the method papers (Refs. [36], [51], [75]) are not load-bearing in a circular sense: the PIMD+TDEP formalism was validated on fcc 4He against inelastic neutron scattering and on silicon, and the IBTE solver is a standard approach whose inputs do not include the h-BN result. The one substantive caveat, that all headline numbers are computed 'within the standard energy conservation approach' while the authors' own Ref. [75] shows that a fluctuation-dissipation-compliant treatment greatly reduces four-phonon importance in BAs, is a limitation and a modeling choice about the scattering rule, explicitly deferred to future work in the Outlook; it does not reduce the h-BN claim to its own inputs. The effective-thickness convention is applied transparently to the authors' own results and to prior literature values. Overall, no circular step could be exhibited; the central claim rests on an open physical approximation, which is a correctness risk, not circularity.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard DFT, a fitted machine-learned potential, and several numerical convergence choices; no new physical entities are introduced. The most consequential input is the energy-conservation rule for four-phonon scattering, which is a domain assumption rather than a fitted parameter, along with the numerical cutoffs that control the anharmonic force constants.

free parameters (5)
  • Effective thickness ratio c/a = 1.317
    Used to normalize thermal conductivity to an effective thickness; the value is taken from bulk h-BN (Ref [56]). Different thickness conventions shift absolute kappa values, and the paper renormalizes literature values to this convention.
  • Fourth-order IFC cutoff = 2.51 Å
    TDEP fit truncates fourth-order force constants at 2.51 Å; long-range four-phonon interactions beyond this cutoff are omitted, which could affect scattering rates.
  • Third-order IFC cutoff = 5.1 Å
    Third-order force constants are fitted to 5.1 Å; this choice influences 3-phonon scattering and therefore the baseline against which the four-phonon reduction is measured.
  • Q-point grid for 4-phonon kappa = 64x64x1
    The four-phonon thermal conductivity is taken at the largest grid without extrapolation because extrapolation is noisy; values may not be fully converged.
  • PIMD bead scaling = Nbeads x T = 12000 (80 beads at 150 K)
    Bead number is set by converging the 150 K case and scaling linearly with temperature; insufficient beads can introduce noise in the centroid surface, as the R2 drop at 150 K is attributed to this.
assumptions (5)
  • domain assumption On-shell energy conservation is the correct selection rule for four-phonon scattering.
    All quantitative kappa values are computed within the standard energy conservation approach (Section III C); the authors' related work [75] shows that a fluctuation-dissipation-compliant rule greatly reduces four-phonon effects in boron arsenide. If the same applies to h-BN, the headline numbers change.
  • domain assumption The PBE-D3 exchange-correlation functional describes the anharmonic potential of monolayer h-BN accurately enough for thermal transport.
    DFT with PBE and D3 corrections generates the reference forces used to train the MLIP and to validate it (Fig. 1); exchange-correlation errors propagate into the IFCs.
  • domain assumption The moment tensor potential faithfully reproduces DFT forces in out-of-sample configurations.
    All MD and PIMD sampling and TDEP fits use the MLIP; the force RMSE is 0.0252 eV/A, and fourth-order IFCs are highly sensitive to force accuracy.
  • domain assumption The Tamura model with natural isotope abundance captures isotope scattering in h-BN.
    Isotope effects are included via the Tamura mass-disorder model (Refs [52,53]); alternative isotope-scattering models could shift the reported isotope correction.
  • domain assumption TDEP extraction of IFCs from PIMD centroids correctly approximates the quantum Kubo correlation function.
    The method relies on the mode-coupling theory justification in the authors' previous paper [36]; any formal error in that treatment would affect the nuclear-quantum-effects comparison.

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Cite this review

Pith. "Pith review of Thermal Conductivity Of Monolayer Hexagonal Boron Nitride: Four-Phonon Scattering And Quantum Sampling Effects." pith.science (2026). https://pith.science/paper/JOPSNM53

@misc{pith2026250614547,
  author       = {Pith},
  title        = {Pith review of: Thermal Conductivity Of Monolayer Hexagonal Boron Nitride: Four-Phonon Scattering And Quantum Sampling Effects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JOPSNM53}},
  note         = {Machine review of arXiv:2506.14547}
}
read the original abstract

Monolayer hexagonal boron nitride is a prototypical planar 2-dimensional system material and has been the subject of many investigations of its exceptional vibrational, spectroscopic and transport properties. The lattice thermal conductivity remains quite uncertain, with theoretical and experimental reports varying between 218 and 1060 Wm-1K-1. It has a strong temperature evolution and is sensitive to strain effects and isotope concentrations. While the impact of isotope scattering has been widely studied and is well understood, nuclear quantum effects and 4-phonon scattering have so far been neglected. Monolayer hexagonal boron nitride is composed of light elements, and further has its 3-phonon scattering phase space restricted by mirror plane symmetry, so these effects may be of similar order as isotope scattering, and would lead to a completely different understanding of the fundamental processes limiting the lattice thermal conductivity for this system. In this work, we use both classical and path-integral molecular dynamics, in conjunction with the Temperature Dependent Effective Potential method, to compute temperature-dependent renormalized phonons including isotope scattering, 3-phonon scattering, 4-phonon scattering and nuclear quantum effects. We show the impact of the latter two on the lattice thermal conductivity for a large temperature range, as well as their impact on the phonon lifetimes. Overall, our work provides a robust framework for calculations of the lattice thermal conductivity in solids, providing quantitative improvements and physical understanding that help explain the variety of results found in the literature.

Figures

Figures reproduced from arXiv: 2506.14547 by the authors.

Figure 1
Figure 1. FIG. 1. Correlation plots for training and test sets. The x-axis pertains to the energies, forces and stresses calculated with [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Convergence of the temperature evolution of the lat [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Comparison between pair-distribution functions us [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figures from the paper (4 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Temperature evolution of the R [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Temperature evolution of the phonon band structure. [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Phonon lifetimes for the classical MD sampling case. [PITH_FULL_IMAGE:figures/full_fig_p007_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Phonon lifetimes including 3+4-phonon scattering [PITH_FULL_IMAGE:figures/full_fig_p007_10.png]

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Works this paper leans on

73 extracted references · 58 canonical work pages

  1. [66]

    R. Kubo, M. Yokota, and S. Nakajima, Statistical- mechanical theory of irreversible processes. ii. response to thermal disturbance, Journal of the Physical Society of Japan12, 1203–1211 (1957)

    work page 1957

  2. [75]

    N. K. Ravichandran and D. Broido, Exposing the hidden influence of selection rules on phonon–phonon scattering by pressure and temperature tuning, Nature Communi- cations12, 10.1038/s41467-021-23618-7 (2021)

    work page doi:10.1038/s41467-021-23618-7 2021

  3. [1]

    Lindsay, D

    L. Lindsay, D. A. Broido, and N. Mingo, Flexural phonons and thermal transport in graphene, Physical Re- view B82, 10.1103/physrevb.82.115427 (2010)

    work page doi:10.1103/physrevb.82.115427 2010

  4. [2]

    Lindsay, W

    L. Lindsay, W. Li, J. Carrete, N. Mingo, D. A. Broido, and T. L. Reinecke, Phonon thermal transport in strained and unstrained graphene from first principles, Phys. Rev. B89, 155426 (2014)

    work page 2014

  5. [3]

    Alofi and G

    A. Alofi and G. P. Srivastava, Thermal conductivity of graphene and graphite, Phys. Rev. B87, 115421 (2013)

    work page 2013

  6. [4]

    Fugallo, A

    G. Fugallo, A. Cepellotti, L. Paulatto, M. Lazzeri, N. Marzari, and F. Mauri, Thermal conductivity of graphene and graphite: Collective excitations and mean free paths, Nano Letters14, 6109 (2014), pMID: 25343716, https://doi.org/10.1021/nl502059f

    work page doi:10.1021/nl502059f 2014

  7. [5]

    C. Yuan, J. Li, L. Lindsay, D. Cherns, J. W. Pomeroy, S. Liu, J. H. Edgar, and M. Kuball, Modulating the thermal conductivity in hexagonal boron nitride via controlled boron isotope concentration, Communications Physics2, 10.1038/s42005-019-0145-5 (2019)

    work page doi:10.1038/s42005-019-0145-5 2019

  8. [6]

    Lindsay and D

    L. Lindsay and D. A. Broido, Enhanced thermal conduc- tivity and isotope effect in single-layer hexagonal boron nitride, Phys. Rev. B84, 155421 (2011)

    work page 2011

Show all 73 references
  1. [7]

    Sevik, A

    C. Sevik, A. Kinaci, J. B. Haskins, and T. C ¸ a˘ g ın, Charac- terization of thermal transport in low-dimensional boron nitride nanostructures, Phys. Rev. B84, 085409 (2011)

    2011

  2. [8]

    Wu and Q

    X. Wu and Q. Han, Thermal conductivity of monolayer hexagonal boron nitride: From defective to amorphous, Computational Materials Science184, 109938 (2020)

    2020

  3. [9]

    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, 10.1126/sciadv.aav0129 (2019)

    2019 doi

  4. [10]

    H. Ying, A. Moore, J. Cui, Y. Liu, D. Li, S. Han, Y. Yao, Z. Wang, L. Wang, and S. Chen, Tailoring the thermal transport properties of monolayer hexagonal boron ni- tride by grain size engineering, 2D Materials7, 015031 (2019)

    2019

  5. [11]

    A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Tewelde- brhan, F. Miao, and C. N. Lau, Superior thermal conduc- tivity of single-layer graphene, Nano Letters8, 902–907 (2008)

    2008

  6. [12]

    J.-U. Lee, D. Yoon, H. Kim, S. W. Lee, and H. Cheong, Thermal conductivity of suspended pristine graphene measured by raman spectroscopy, Physical Review B83, 10.1103/physrevb.83.081419 (2011). 10

    2011 doi

  7. [13]

    X. Xu, L. F. C. Pereira, Y. Wang, J. Wu, K. Zhang, X. Zhao, S. Bae, C. Tinh Bui, R. Xie, J. T. L. Thong, B. H. Hong, K. P. Loh, D. Donadio, B. Li, and B. ¨Ozyilmaz, Length-dependent thermal conductiv- ity in suspended single-layer graphene, Nature Commu- nications5, 10.1038/nc...

    2014 doi

  8. [14]

    D. L. Nika and A. A. Balandin, Two-dimensional phonon transport in graphene, Journal of Physics: Condensed Matter24, 233203 (2012)

    2012

  9. [15]

    Han and X

    Z. Han and X. Ruan, Thermal conductivity of monolayer graphene: Convergent and lower than diamond, Physical Review B108, 10.1103/physrevb.108.l121412 (2023)

    2023 doi

  10. [16]

    Feng and X

    T. Feng and X. Ruan, Four-phonon scattering reduces intrinsic thermal conductivity of graphene and the con- tributions from flexural phonons, Physical Review B97, 10.1103/physrevb.97.045202 (2018)

    2018 doi

  11. [17]

    Y. Han, C. Yang, X. Cheng, D. Han, W. Ding, and X. Wang, Investigation of the effect of four-phonon scat- tering on thermal transport in two-dimensional group- iv materials, ACS Applied Energy Materials7, 649–656 (2023)

    2023

  12. [18]

    Mariani and F

    E. Mariani and F. von Oppen, Flexural phonons in free-standing graphene, Physical Review Letters100, 10.1103/physrevlett.100.076801 (2008)

    2008 doi

  13. [19]

    Illera, M

    S. Illera, M. Pruneda, L. Colombo, and P. Ordej´ on, Thermal and transport properties of pristine single-layer hexagonal boron nitride: A first principles investigation, Phys. Rev. Mater.1, 044006 (2017)

    2017

  14. [20]

    Y. Xie, Z. Xu, S. Xu, Z. Cheng, N. Hashemi, C. Deng, and X. Wang, The defect level and ideal thermal conductivity of graphene uncovered by residual thermal reffusivity at the 0 k limit, Nanoscale7, 10101–10110 (2015)

    2015

  15. [21]

    Errea, M

    I. Errea, M. Calandra, and F. Mauri, First-principles theory of anharmonicity and the inverse isotope effect in superconducting palladium-hydride compounds, Phys- ical Review Letters111, 10.1103/physrevlett.111.177002 (2013)

    2013 doi

  16. [22]

    Errea, M

    I. Errea, M. Calandra, and F. Mauri, Anharmonic free energies and phonon dispersions from the stochastic self- consistent harmonic approximation: Application to plat- inum and palladium hydrides, Physical Review B89, 10.1103/physrevb.89.064302 (2014)

    2014 doi

  17. [23]

    Bianco, I

    R. Bianco, I. Errea, L. Paulatto, M. Calandra, and F. Mauri, Second-order structural phase transitions, free energy curvature, and temperature-dependent anhar- monic phonons in the self-consistent harmonic approxi- mation: Theory and stochastic implementation, Physical Review ...

    2017 doi

  18. [25]

    Monacelli, R

    L. Monacelli, R. Bianco, M. Cherubini, M. Calandra, I. Errea, and F. Mauri, The stochastic self-consistent har- monic approximation: calculating vibrational properties of materials with full quantum and anharmonic effects, Journal of Physics: Condensed Matter33, 363001 (2021)

    2021

  19. [26]

    I. R. Craig and D. E. Manolopoulos, Quantum statistics and classical mechanics: Real time correlation functions from ring polymer molecular dynamics, The Journal of Chemical Physics121, 3368–3373 (2004)

    2004

  20. [27]

    B. J. Braams and D. E. Manolopoulos, On the short-time limit of ring polymer molecular dynamics, The Journal of Chemical Physics125, 10.1063/1.2357599 (2006)

    2006 doi

  21. [28]

    Morresi, L

    T. Morresi, L. Paulatto, R. Vuilleumier, and M. Casula, Probing anharmonic phonons by quantum correlators: A path integral approach, The Journal of Chemical Physics 154, 224108 (2021)

    2021

  22. [29]

    Hellman, I

    O. Hellman, I. A. Abrikosov, and S. I. Simak, Lattice dynamics of anharmonic solids from first principles, Phys. Rev. B84, 180301(R) (2011)

    2011

  23. [30]

    Hellman, P

    O. Hellman, P. Steneteg, I. A. Abrikosov, and S. I. Simak, Temperature dependent effective potential method for accurate free energy calculations of solids, Phys. Rev. B 87, 104111 (2013)

    2013

  24. [31]

    Hellman and I

    O. Hellman and I. A. Abrikosov, Temperature-dependent effective third-order interatomic force constants from first principles, Phys. Rev. B88, 144301 (2013)

    2013

  25. [32]

    Knoop, N

    F. Knoop, N. Shulumba, A. Castellano, J. P. A. Batista, R. Farris, M. J. Verstraete, M. Heine, D. Broido, D. S. Kim, J. Klarbring, I. A. Abrikosov, S. I. Simak, and O. Hellman, Tdep: Temperature dependent effective po- tentials, Journal of Open Source Software9, 6150 (2024)

    2024

  26. [33]

    Shulumba, O

    N. Shulumba, O. Hellman, and A. J. Minnich, Intrin- sic localized mode and low thermal conductivity of pbse, Physical Review B95, 10.1103/physrevb.95.014302 (2017)

    2017 doi

  27. [34]

    H. Y. Geng, Full temperature-dependent potential and anharmonicity in metallic hydrogen: Colossal nqe and the consequences, The Journal of Physical Chemistry C 126, 19355–19366 (2022)

    2022

  28. [35]

    D. A. Folkner, Z. Chen, G. Barbalinardo, F. Knoop, and D. Donadio, Elastic moduli and thermal conductivity of quantum materials at finite temperature, Journal of Ap- plied Physics136, 10.1063/5.0238723 (2024)

    2024 doi

  29. [36]

    Castellano, J

    A. Castellano, J. P. A. Batista, and M. J. Verstraete, Mode-coupling theory of lattice dynamics for classical and quantum crystals, The Journal of Chemical Physics 159, 10.1063/5.0174255 (2023)

    2023 doi

  30. [37]

    A. V. Shapeev, Moment tensor potentials: A class of systematically improvable interatomic potentials, Multiscale Modeling & Simulation14, 1153 (2016), https://doi.org/10.1137/15M1054183

    2016 doi

  31. [38]

    I. S. Novikov, K. Gubaev, E. V. Podryabinkin, and A. V. Shapeev, The mlip package: moment tensor potentials with mpi and active learning, Machine Learning: Science and Technology2, 025002 (2021)

    2021

  32. [39]

    Castellano, F

    A. Castellano, F. m. c. Bottin, J. Bouchet, A. Levitt, and G. Stoltz,abinitio canonical sampling based on varia- tional inference, Phys. Rev. B106, L161110 (2022)

    2022

  33. [40]

    Castellano, R

    A. Castellano, R. B´ ejaud, P. Richard, O. Nadeau, C. Du- val, G. Geneste, G. Antonius, J. Bouchet, A. Levitt, G. Stoltz, and F. Bottin, Machine learning assisted canonical sampling (mlacs) (2024)

    2024

  34. [41]

    Gonze, B

    X. Gonze, B. Amadon, G. Antonius, F. Arnardi, L. Baguet, J.-M. Beuken, J. Bieder, F. Bottin, J. Bouchet, E. Bousquet, N. Brouwer, F. Bruneval, G. Brunin, T. Cavignac, J.-B. Charraud, W. Chen, M. Cˆ ot´ e, S. Cottenier, J. Denier, G. Geneste, P. Ghosez, M. Giantomassi, Y. Gille...

    2020

  35. [42]

    A. H. Romero, D. C. Allan, B. Amadon, G. An- tonius, T. Applencourt, L. Baguet, J. Bieder, F. Bottin, J. Bouchet, E. Bousquet, F. Bruneval, G. Brunin, D. Caliste, M. Cˆ ot´ e, J. Denier, C. Dreyer, P. Ghosez, M. Giantomassi, Y. Gillet, O. Gingras, D. R. Hamann, G. Hautier, F. ...

    2020

  36. [43]

    J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996)

    1996

  37. [44]

    Grimme, J

    S. Grimme, J. Antony, S. Ehrlich, and H. Krieg, A con- sistent and accurate ab initio parametrization of den- sity functional dispersion correction (DFT-d) for the 94 elements h-pu, The Journal of Chemical Physics132, 154104 (2010)

    2010

  38. [45]

    A. P. Thompson, H. M. Aktulga, R. Berger, D. S. Bolin- tineanu, W. M. Brown, P. S. Crozier, P. J. in ’t Veld, A. Kohlmeyer, S. G. Moore, T. D. Nguyen, R. Shan, M. J. Stevens, J. Tranchida, C. Trott, and S. J. Plimpton, Lammps - a flexible simulation tool for particle-based mat...

    2022

  39. [46]

    Grønbech-Jensen and O

    N. Grønbech-Jensen and O. Farago, A simple and effective verlet-type algorithm for simulating langevin dynamics, Molecular Physics111, 983 (2013), https://doi.org/10.1080/00268976.2012.760055

    2013

  40. [47]

    Omini and A

    M. Omini and A. Sparavigna, Beyond the isotropic-model approximation in the theory of thermal conductivity, Phys. Rev. B53, 9064 (1996)

    1996

  41. [48]

    Fugallo, M

    G. Fugallo, M. Lazzeri, L. Paulatto, and F. Mauri, Ab initio variational approach for evaluating lattice thermal conductivity, Phys. Rev. B88, 045430 (2013)

    2013

  42. [49]

    D. A. Broido, M. Malorny, G. Birner, N. Mingo, and D. A. Stewart, Intrinsic lattice thermal conductivity of semiconductors from first prin- ciples, Applied Physics Letters91, 231922 (2007), https://pubs.aip.org/aip/apl/article- pdf/doi/10.1063/1.2822891/14384383/231922 1 online.pdf

    2007 doi

  43. [50]

    Hellman and D

    O. Hellman and D. A. Broido, Phonon thermal transport in bi 2te3 from first principles, Phys. Rev. B90, 134309 (2014)

    2014

  44. [51]

    Castellano, J

    A. Castellano, J. P. A. Batista, O. Hellman, and M. J. Verstraete, Mode-coupling formulation of heat transport in anharmonic materials (2024)

    2024

  45. [52]

    Tamura, Isotope scattering of dispersive phonons in ge, Phys

    S.-i. Tamura, Isotope scattering of dispersive phonons in ge, Phys. Rev. B27, 858 (1983)

    1983

  46. [53]

    N. H. Protik and C. Draxl, Beyond the tamura model of phonon-isotope scattering, Phys. Rev. B109, 165201 (2024)

    2024

  47. [54]

    Materials for the Quan- tum Age

    and electronic [55] properties. q-point grids of up to 360x360x1 for 3-phonon scattering cases and 64x64x1 for 4-phonon scattering cases are used. The lattice thermal conductivity is then obtained for the infinitely dense q- point grid as the y-intersect of the linear extrapol...

    1900

  48. [55]

    Janzen, H

    E. Janzen, H. Schutte, J. Plo, A. Rousseau, T. Michel, W. Desrat, P. Valvin, V. Jacques, G. Cassabois, B. Gil, and J. H. Edgar, Boron and nitrogen isotope effects on hexagonal boron nitride properties, Advanced Materials 36, 10.1002/adma.202306033 (2023)

    2023 doi

  49. [56]

    T. Q. P. Vuong, S. Liu, A. Van der Lee, R. Cusc´ o, L. Art´ us, T. Michel, P. Valvin, J. H. Edgar, G. Cass- abois, and B. Gil, Isotope engineering of van der waals interactions in hexagonal boron nitride, Nature Materials 17, 152–158 (2017)

    2017

  50. [57]

    Cepellotti, G

    A. Cepellotti, G. Fugallo, L. Paulatto, M. Lazzeri, F. Mauri, and N. Marzari, Phonon hydrodynamics in two-dimensional materials, Nature Communications6, 10.1038/ncomms7400 (2015)

    2015 doi

  51. [58]

    X. Wu, W. Zhou, H. Dong, P. Ying, Y. Wang, B. Song, Z. Fan, and S. Xiong, Correcting force error-induced un- derestimation of lattice thermal conductivity in machine learning molecular dynamics, The Journal of Chemical Physics161, 10.1063/5.0213811 (2024)

    2024 doi

  52. [59]

    Carter Hill, W

    R. Carter Hill, W. E. Griffiths, and G. C. Lim,Principles of econometrics, 5th ed. (John Wiley & Sons, Nashville, TN, 2024)

    2024

  53. [60]

    M. A. Kriegel, K. M. Omambac, S. Franzka, F.-J. Meyer zu Heringdorf, and M. Horn-von Hoegen, Incom- mensurability and negative thermal expansion of single layer hexagonal boron nitride, Applied Surface Science 624, 157156 (2023)

    2023

  54. [61]

    Knoop, T

    F. Knoop, T. A. R. Purcell, M. Scheffler, and C. Car- bogno, Anharmonicity measure for materials, Physical Review Materials4, 10.1103/physrevmaterials.4.083809 (2020)

    2020 doi

  55. [64]

    J. S. Kang, M. Li, H. Wu, H. Nguyen, and Y. Hu, Ex- perimental observation of high thermal conductivity in boron arsenide, Science361, 575–578 (2018)

    2018

  56. [65]

    S. Bi, Z. Chang, K. Yuan, Z. Sun, X. Zhang, Y. Gao, and D. Tang, First-principles prediction of the lat- tice thermal conductivity of two-dimensional (2d) h-bx (x=p, as, sb) considering the effects of fourth-order and all-order scattering, Journal of Applied Physics132, 10.106...

    2022 doi

  57. [67]

    G. Sun, J. Ma, C. Liu, Z. Xiang, D. Xu, T.-H. Liu, and X. Luo, Four-phonon and normal scattering in 2d hexag- onal structures, International Journal of Heat and Mass Transfer215, 124475 (2023)

    2023

  58. [68]

    Mortazavi, L

    B. Mortazavi, L. F. C. Pereira, J.-W. Jiang, and T. Rabczuk, Modelling heat conduction in polycrys- talline hexagonal boron-nitride films, Scientific Reports 5, 10.1038/srep13228 (2015)

    2015 doi

  59. [69]

    A. I. Khan, I. A. Navid, M. Noshin, and S. Subrina, Ther- mal transport characterization of hexagonal boron nitride nanoribbons using molecular dynamics simulation, AIP Advances7, 10.1063/1.4997036 (2017)

    2017 doi

  60. [70]

    Mortazavi and Y

    B. Mortazavi and Y. R´ emond, Investigation of ten- sile response and thermal conductivity of boron-nitride nanosheets using molecular dynamics simulations, Phys- 12 ica E: Low-dimensional Systems and Nanostructures44, 1846–1852 (2012)

    2012

  61. [71]

    H. Fan, H. Wu, L. Lindsay, and Y. Hu, Ab initio inves- tigation of single-layer high thermal conductivity boron compounds, Physical Review B100, 10.1103/phys- revb.100.085420 (2019)

    2019 doi

  62. [72]

    Saleta Reig, S

    D. Saleta Reig, S. Varghese, R. Farris, A. Block, J. D. Mehew, O. Hellman, P. Wo´ zniak, M. Sledzinska, A. El Sachat, E. Ch´ avez-´Angel, S. O. Valenzuela, N. F. van Hulst, P. Ordej´ on, Z. Zanolli, C. M. Sotomayor Tor- res, M. J. Verstraete, and K.-J. Tielrooij, Unraveling he...

    2022 doi

  63. [73]

    S. Chen, A. L. Moore, W. Cai, J. W. Suk, J. An, C. Mishra, C. Amos, C. W. Magnuson, J. Kang, L. Shi, and R. S. Ruoff, Raman measurements of thermal trans- port in suspended monolayer graphene of variable sizes in vacuum and gaseous environments, ACS Nano5, 321–328 (2010)

    2010

  64. [74]

    N. K. Ravichandran and D. Broido, Phonon-phonon in- teractions in strongly bonded solids: Selection rules and higher-order processes, Phys. Rev. X10, 021063 (2020)

    2020

  65. [76]

    Castellano, J

    A. Castellano, J. P. A. Batista, and M. J. Verstraete, Fluctuation-dissipation and virtual processes in interact- ing phonon systems (2025)

    2025

Pith tools

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