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REVIEW 3 major objections 7 minor 40 references

Electronic conductivity in anharmonic crystals: Phonon dephasing in the electron-phonon interaction

T0 review · 3 major / 7 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Anharmonic phonon dephasing substantially enhances electron-phonon scattering in MgB2 and brings calculated conductivity closer to experiment.

desk verdict Important new correction to electron-phonon transport, but the printed formulas have sign problems and the derivation lives in a companion paper—worth reviewing, not yet self-contained. read the letter →

arxiv 2608.09050 v1 pith:N2R7JVVY submitted 2026-08-10 cond-mat.mtrl-sci physics.app-phphysics.comp-ph

Mingran Kong , Bartomeu Monserrat This is my paper · ORCID
classification cond-mat.mtrl-sciphysics.app-phphysics.comp-ph
keywords electron-phononinteractionanharmonicdephasingfinitephononlifetimeelectricalconductivityMgB2first-principlescalculationphonon-phononscatteringBoltzmanntransport
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

All first-principles calculations of electron-phonon coupling so far assume phonons have infinite lifetimes. This paper claims that when phonons can decay into other phonons, a process called anharmonic dephasing, electrons scatter much more strongly, and ignoring this misses real physics in metals. The authors derive a lowest-order correction that combines the standard electron-phonon vertex with a phonon-phonon bubble, implement it from first principles, and show that for MgB2 the resulting scattering rates rise sharply near the Fermi level. The corrected electrical conductivity drops by roughly a factor of two and lands substantially closer to measured values. If the claim is right, finite phonon lifetimes are essential for quantitative transport predictions in anharmonic metals.

What carries the argument

The load-bearing object is a one-loop self-energy diagram in which an electron emits or absorbs a phonon that then splits into two phonons via the three-phonon vertex, namely an electron coupled to a phonon dressed by a phonon-phonon bubble. The resulting scattering rate is the sum of three terms, one-emission-one-absorption, two-emission, and two-absorption, given in Eqs. (2) through (5). This machinery replaces the infinite-lifetime phonon picture with a dephasing picture: phonons have finite lifetimes, and energy-conserving phonon-phonon combinations enlarge the phase space available for electron-phonon scattering. The sign and occupation structure of the three channels determine whether the total anharmonic correction is positive, which the paper argues is always the case for the dominant one-emission-one-absorption term.

What would settle it

Recompute the MgB2 electron-phonon scattering rates and conductivity using an independent derivation of the electron-anharmonic-phonon self-energy from the same Hamiltonian, and check whether the anharmonic correction near the Fermi level is positive and of the reported size; a sign flip or a factor-of-two discrepancy in the two-phonon-absorption channel would overturn the agreement with experiment.

Watch

Extended reading notes

Core claim

The paper's central claim is that anharmonic phonon dephasing is a first-order correction to the electron-phonon self-energy, not a small footnote. In MgB2 the anharmonic electron-phonon scattering rate, written as the sum of one-emission-one-absorption, two-emission, and two-absorption channels, no longer dips near the Fermi level the way the harmonic rate does; phonon-phonon scattering supplies new energy- and momentum-conserving channels. The anharmonic rates therefore exceed the harmonic ones across a broad energy window, and the Boltzmann-transport conductivity falls from $1.62\times10^8$ to $8.70\times10^7$ S/m at 100 K and from $1.33\times10^7$ to $9.47\times10^6$ S/m at 300 K, approaching the experimental $3.57\times10^7$ and $7.14\times10^6$ S/m. The authors take this as evidence that finite phonon lifetimes are essential for electron transport in MgB2 and likely in other anharmonic metals.

Load-bearing premise

The central conclusion rests on the printed equations (2) through (5) faithfully reproducing the companion derivation, with the correct signs and occupation factors in each scattering channel; if that transcription is wrong, the MgB2 rates and the conductivity suppression do not follow from this paper as written.

Editorial extensions

If this is right

  • Finite phonon lifetimes must be included in first-principles transport calculations of MgB2; harmonic-only calculations overestimate conductivity by roughly a factor of two at 100 to 300 K.
  • Anharmonic dephasing removes the harmonic dip in scattering rate near the Fermi level, meaning metals with strong phonon-phonon coupling have extra scattering channels that standard theory misses.
  • The same formalism can be applied to other strongly anharmonic metals and to carrier mobilities in semiconductors, where phonon dephasing is usually ignored.
  • The enhanced scattering near the Fermi level may renormalize the electron-phonon pairing interaction in MgB2, with possible consequences for its superconductivity, a direction the authors leave for future work.
  • Comparisons between calculated and measured resistivities that ignore anharmonicity will systematically overestimate conductivity in anharmonic metals.

Reading between the lines

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

  • If the mechanism is generic, the anharmonic correction should grow with temperature faster than the harmonic rate, so the gap between harmonic theory and experiment should widen with temperature in other light-element metals.
  • A testable extension: applying pressure or alloying to shift the energy of the phonon valleys near 400 cm$^{-1}$ should move the anharmonic contribution in a way that a harmonic calculation cannot reproduce.
  • The same anharmonic electron-phonon rates feed hot-carrier relaxation and ultrafast dynamics, so time-resolved experiments on MgB2 could provide an independent check of the predicted enhancement.
  • Because the two-phonon-absorption channel carries an unusual negative prefactor, an independent implementation of Eqs. (2)-(5) would reveal whether the reported conductivity suppression is sensitive to numerical details of the phonon-phonon summation.
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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 / 7 minor

Summary. The manuscript proposes a theory for electron-phonon scattering in crystals where phonons have finite lifetimes due to anharmonic three-phonon interactions. The central result is a set of expressions, Eqs. (2)-(5), for the electron-anharmonic-phonon scattering rate within the self-energy relaxation time approximation, combining the Fan-Migdal electron-phonon self-energy with a phonon-phonon bubble. The authors report a first-principles implementation in an in-house code called DaoQuantum and apply it to metallic MgB2. They find that anharmonic dephasing increases electron-phonon scattering rates, especially near the Fermi level, and consequently suppresses the calculated electrical conductivity, bringing it closer to experimental values than the standard harmonic calculation. The authors argue that finite phonon lifetimes are essential for describing electron transport in anharmonic metals such as MgB2.

Significance. If correct, this work would fill a real gap in ab initio transport calculations, where phonon lifetimes are normally assumed infinite. The proposed mechanism, that anharmonic phonon-phonon interactions open new electron-phonon scattering channels, is physically plausible and could be relevant to many anharmonic metals and superconductors. The comparison to experiment is used as a benchmark rather than for fitting, which is a strength. However, the central formulas are not derived in the manuscript and are deferred to a companion paper, and the printed Eq. (5) has a clear sign issue that makes the two-phonon absorption rate negative. Because the numerical MgB2 results depend directly on these unverified expressions, the main claim is not currently supported by the manuscript as written. The paper is therefore not yet suitable for publication, although the underlying idea may be salvageable with a rigorous revision.

major comments (3)
  1. [Eqs. (4)-(5)] The printed two-phonon absorption rate in Eq. (5) is negative for all finite-temperature processes. The bracket (Nλ2+Nλ3+1)(fμ1−1)+Nλ2Nλ3+Nλ2+Nλ3+1 simplifies to (Nλ2+Nλ3+1)fμ1+Nλ2Nλ3, which is non-negative for physical occupations. Since the prefactor is −π/N^2 and the remaining factors are non-negative, Γ(2a) is negative wherever the energy-conserving delta function can be satisfied. A scattering rate cannot be negative. The same concern applies to Eq. (4): for Nλ2=Nλ3=0 and fμ1=0.5, the bracket is −0.5, so the 'two-phonon emitted' term is also negative. If these objects are intermediate self-energy contributions rather than physical rates, the text must state this and justify why the total rate remains positive. As printed, the equations do not describe a valid anharmonic electron-phonon scattering rate.
  2. [Derivation and implementation (pp. 2-3)] The central equations (2)-(5) are not derived in this manuscript. The text repeatedly defers to the companion reference [30] for the derivation, and the numerical implementation in DaoQuantum is described only as 'full details of which will be reported elsewhere.' This makes the manuscript non-self-contained: a reader cannot verify the sign and occupation factors discussed above, nor the convergence and accuracy of the MgB2 calculation, from the submitted material. The central claim of dramatically enhanced electron-phonon scattering depends entirely on these unverifiable inputs. The derivation should be included in the manuscript or a supplement, or at minimum the diagrammatic rules and algebraic reduction to Eqs. (2)-(5) should be stated explicitly.
  3. [Fig. 3 and Ref. [36]] The experimental comparison in Fig. 3 is cited to Ref. [36], which is Sologubenko et al., 'Thermal conductivity of single-crystalline MgB2' (Phys. Rev. B 66, 014504). Thermal conductivity is not electrical conductivity, so this reference does not support the 'Exp.' data plotted in the electrical-conductivity figure. Please cite the correct electrical-resistivity or electrical-conductivity data and indicate sample details and the treatment of crystal directions. Without a correct experimental source, the conclusion that the anharmonic calculation is 'substantially closer to experiment' cannot be evaluated.
minor comments (7)
  1. [Fig. 1 caption] The caption reads 'Feynman digram'; this should be 'Feynman diagram'.
  2. [p.2] The text reads 'Fan-Midgal'; the standard name is 'Fan-Migdal'.
  3. [Fig. 3 caption] The caption contains the typo 'expreimental'; it should be 'experimental'.
  4. [p.4] The sentence beginning 'Amoug these results' contains a typo; 'Amoug' should be 'Among'.
  5. [Fig. 3, p.4] The text states the calculations cover 'from 80 K to 340 K', but the figure axes span 100 K to 350 K; these should be reconciled.
  6. [Ref. [30]] Reference [30] is listed as 'accompanying manuscript' with no journal or arXiv identifier; if the companion paper is available, a full citation should be provided, and if it is not yet public, the authors should clearly state its status.
  7. [Fig. 2(f)] The label '2e phonon-phonon scattering rate' is ambiguous; clarify whether this quantity is a three-phonon scattering rate that underlies the 2e electron-scattering channel, and define how it is computed from the anharmonic coupling.

Circularity Check

1 steps flagged · score 4.0 of 10

No fitted-input circularity, but the central scattering-rate formulas are deferred to a same-author companion manuscript, making the MgB2 conclusion depend on load-bearing self-citation.

  1. self citation load bearing [Theory section, Eqs. (2)-(5), and the paragraph introducing them]
    "Within many-body perturbation theory, the lowest-order term exhibiting anharmonic phonon dephasing is captured by the Feynman diagram depicted in Fig. 1 (see companion work). This diagram leads to an electron-anharmonic-phonon self-energy ... The associated scattering rate under the self-energy relaxation time approximation is given by (see companion work): ..."

    Eqs. (2)-(5) are the entire theory used to compute the MgB2 scattering rates and the conductivity suppression that is the central claim. These equations are not derived in this manuscript; each is introduced with '(see companion work)', and the step-by-step derivation is explicitly assigned to [30], an accompanying manuscript by the same two authors. The numerical result therefore follows only if the unpublished companion work contains exactly these formulas, sign conventions, and occupation factors. This is load-bearing self-citation: the present paper contributes the MgB2 application, but the predicted effect itself rests on an unverified same-author citation rather than on a derivation shown here.

full rationale

The paper does not fit any parameter to the experimental conductivity; the experimental comparison is used only as a benchmark, so there is no fitted-input-called-prediction circularity. The conductivity result is a genuinely numerical first-principles application using external codes (Quantum ESPRESSO, EPW, ShengBTE) and independent structural and experimental inputs. The only circularity-type concern is the deferral of the central scattering-rate derivation to companion work [30] by the same authors: Eqs. (2)-(5) are load-bearing and unshown in this manuscript. This merits a score of 4 rather than 0 or 2 because the self-citation is not minor: the entire anharmonic-dephasing mechanism is imported from the authors' own companion work. However, it is not a reduction-by-construction circularity, so the score is not 6 or higher. Separately, the printed Eq. (5) appears to have a sign problem: its occupation bracket simplifies to a positive quantity while the overall term carries a minus sign, yielding a negative two-phonon absorption rate at T=0. This is a correctness/verifiability defect that prevents reproduction from the manuscript as written, but it is not itself a circularity and is therefore not scored as one here.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The paper uses standard many-body perturbation theory and DFT machinery and introduces no new physical entities. Its main external debts are the companion derivation and the private code. The numerical broadening is the only explicit but unspecified numerical parameter. All other assumptions are domain assumptions about perturbation theory, BTE transport, and DFT accuracy.

free parameters (1)
  • Broadening eta in the denominators and delta functions of Eqs. (3)-(5) = Not reported
    The rates require numerical regularization of the 1/(... + i eta) poles and the delta functions, but the manuscript does not state the value or the grid convergence. This is a numerical parameter that affects the printed rates.
assumptions (4)
  • domain assumption The Hamiltonian in Eq. (1), with linear electron-phonon coupling and cubic three-phonon coupling, is a sufficient model for the electron-lattice interaction in MgB2.
    Higher-order couplings such as quartic anharmonicity and higher-order electron-phonon terms are neglected. The paper argues that three-phonon interactions dominate, but does not quantify the neglected terms.
  • domain assumption The Feynman diagram in Fig. 1 is the lowest-order correction and captures the dominant effect of finite phonon lifetimes.
    The derivation, deferred to the companion manuscript [30], assumes that this diagram is the leading contribution and that the perturbation series converges. This is a standard but load-bearing assumption.
  • domain assumption The self-energy relaxation time approximation and the Boltzmann transport equation are valid for normal-state MgB2 transport at 80 to 340 K.
    The authors apply the formalism within the Boltzmann transport framework without vertex corrections or electron-electron scattering. This is common practice but not verified here.
  • domain assumption The DFT and DFPT inputs from Quantum ESPRESSO, EPW, and ShengBTE are consistent and converged.
    The paper does not report the exchange-correlation functional, pseudopotentials, plane-wave cutoffs, or k/q-point grids. The numerical conductivity depends on these choices.

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Pith. "Pith review of Electronic conductivity in anharmonic crystals: Phonon dephasing in the electron-phonon interaction." pith.science (2026). https://pith.science/paper/N2R7JVVY

@misc{pith2026260809050,
  author       = {Pith},
  title        = {Pith review of: Electronic conductivity in anharmonic crystals: Phonon dephasing in the electron-phonon interaction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N2R7JVVY}},
  note         = {Machine review of arXiv:2608.09050}
}
read the original abstract

The electron-phonon interaction underpins many material properties, for example, the conductivity of metals and the optoelectronic response of semiconductors. First-principles calculations of the electron-phonon interaction are a powerful tool to quantitatively describe many of these properties in increasingly complex materials. However, one key assumption of all calculations is that phonons have infinite lifetimes, an approximation that may break down when anharmonic phonon-phonon interactions are strong. In this work, we present a theory for the interaction of electrons with finite-lifetime phonons experiencing dephasing. Using a first-principles implementation of the theory, we find that anharmonic dephasing dramatically enhances electron-phonon scattering rates in metallic MgB2. Microscopically, phonon-phonon interactions create new scattering channels that increase the phase space available for electron-phonon scattering. As a result, anharmonic dephasing strongly suppresses conductivity in MgB2, bringing the calculated values substantially closer to experiment within the Boltzmann transport equation framework. This example establishes the importance of finite phonon lifetimes in the evaluation of electron-phonon scattering, and the microscopic mechanism suggests that anharmonic dephasing could play an important role in the conductivity of many metals. More broadly, our theory and first-principles implementation of anharmonic dephasing in the electron-phonon interaction provides a solid foundation to explore this regime in other materials.

Figures

Figures reproduced from arXiv: 2608.09050 by the authors.

Figure 1
Figure 1. FIG. 1. Feynman digram for electron-anharmonic-phonon [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. MgB [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Electrical conductivity of MgB [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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Reference graph

Works this paper leans on

40 extracted references · 30 canonical work pages

  1. [30]

    Kong and B

    M. Kong and B. Monserrat, accompanying manuscript

    work page

  2. [36]

    A. V. Sologubenko, J. Jun, S. M. Kazakov, J. Karpinski, and H. R. Ott, Thermal conductivity of single-crystalline mgb2, Phys. Rev. B66, 014504 (2002)

    work page 2002

  3. [1]

    M. C. Payne, R. A. Davies, J. C. Inkson, and M. Pepper, Energy loss rate in silicon inversion layers, Journal of Physics C: Solid State Physics16, L291 (1983)

    work page 1983

  4. [2]

    M. C. Payne and J. C. Inkson, Inelastic electron tun- nelling spectroscopy, Journal of Physics C: Solid State Physics16, 4259 (1983)

    work page 1983

  5. [3]

    Cui, G.-H

    X. Cui, G.-H. Lee, Y. D. Kim, G. Arefe, P. Y. Huang, C.- H. Lee, D. A. Chenet, X. Zhang, L. Wang, F. Ye, F. Piz- zocchero, B. S. Jessen, K. Watanabe, T. Taniguchi, D. A. Muller, T. Low, P. Kim, and J. Hone, Multi-terminal transport measurements of mos2 using a van der waals heterostructure device platform, Nature Nanotechnology 10, 534 (2015)

    work page 2015

  6. [4]

    Waldecker, R

    L. Waldecker, R. Bertoni, R. Ernstorfer, and J. Vor- berger, Electron-phonon coupling and energy flow in a simple metal beyond the two-temperature approxima- tion, Phys. Rev. X6, 021003 (2016)

    work page 2016

  7. [5]

    Maldonado, T

    P. Maldonado, T. Chase, A. H. Reid, X. Shen, R. K. Li, K. Carva, T. Payer, M. Horn von Hoegen, K. Sokolowski- Tinten, X. J. Wang, P. M. Oppeneer, and H. A. D¨ urr, Tracking the ultrafast nonequilibrium energy flow be- tween electronic and lattice degrees of freedom in crys- talline nickel, Phys. Rev. B101, 100302 (2020)

    work page 2020

  8. [6]

    Maxwell, Isotope effect in the superconductivity of mercury, Phys

    E. Maxwell, Isotope effect in the superconductivity of mercury, Phys. Rev.78, 477 (1950)

    work page 1950

Show all 40 references
  1. [7]

    C. A. Reynolds, B. Serin, W. H. Wright, and L. B. Nesbitt, Superconductivity of isotopes of mercury, Phys. Rev.78, 487 (1950)

    1950

  2. [8]

    Bardeen, L

    J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of superconductivity, Phys. Rev.108, 1175 (1957)

    1957

  3. [9]

    G. M. Eliashberg, Interactions between electrons and lat- tice vibrations in a superconductor, Sov. Phys. - JETP (Engl. Transl.); (United States)11:3(1960)

    1960

  4. [10]

    W. L. McMillan, Transition temperature of strong- coupled superconductors, Phys. Rev.167, 331 (1968)

    1968

  5. [11]

    Profeta, M

    G. Profeta, M. Calandra, and F. Mauri, Phonon- mediated superconductivity in graphene by lithium de- position, Nature Physics8, 131 (2012)

    2012

  6. [12]

    Monserrat, N

    B. Monserrat, N. D. Drummond, C. J. Pickard, and R. J. Needs, Electron-phonon coupling and the metallization of solid helium at terapascal pressures, Phys. Rev. Lett. 112, 055504 (2014)

    2014

  7. [13]

    Antonius, S

    G. Antonius, S. Ponc´ e, P. Boulanger, M. Cˆ ot´ e, and X. Gonze, Many-body effects on the zero-point renor- malization of the band structure, Phys. Rev. Lett.112, 215501 (2014)

    2014

  8. [14]

    Errea, M

    I. Errea, M. Calandra, C. J. Pickard, J. Nelson, R. J. Needs, Y. Li, H. Liu, Y. Zhang, Y. Ma, and F. Mauri, High-pressure hydrogen sulfide from first principles: A strongly anharmonic phonon-mediated superconductor, Phys. Rev. Lett.114, 157004 (2015)

    2015

  9. [15]

    Faber, P

    C. Faber, P. Boulanger, C. Attaccalite, E. Cannuccia, I. Duchemin, T. Deutsch, and X. Blase, Exploring ap- proximations to thegwself-energy ionic gradients, Phys. Rev. B91, 155109 (2015)

    2015

  10. [16]

    Antonius and S

    G. Antonius and S. G. Louie, Temperature-induced topo- logical phase transitions: Promoted versus suppressed nontrivial topology, Phys. Rev. Lett.117, 246401 (2016)

    2016

  11. [17]

    Monserrat and D

    B. Monserrat and D. Vanderbilt, Temperature effects in the band structure of topological insulators, Phys. Rev. Lett.117, 226801 (2016)

    2016

  12. [18]

    Coulter, R

    J. Coulter, R. Sundararaman, and P. Narang, Micro- scopic origins of hydrodynamic transport in the type-ii weyl semimetal wp 2, Phys. Rev. B98, 115130 (2018)

    2018

  13. [19]

    H.-Y. Chen, D. Sangalli, and M. Bernardi, Exciton- phonon interaction and relaxation times from first prin- ciples, Phys. Rev. Lett.125, 107401 (2020)

    2020

  14. [20]

    Miglio, V

    A. Miglio, V. Brousseau-Couture, E. Godbout, G. Anto- nius, Y.-H. Chan, S. G. Louie, M. Cˆ ot´ e, M. Giantomassi, and X. Gonze, Predominance of non-adiabatic effects in zero-point renormalization of the electronic band gap, npj Computational Materials6, 167 (2020)

    2020

  15. [21]

    H. Yang, M. Govoni, A. Kundu, and G. Galli, Com- bined first-principles calculations of electron–electron 6 and electron–phonon self-energies in condensed systems, Journal of Chemical Theory and Computation17, 7468 (2021)

    2021

  16. [22]

    H. Lee, S. Ponc´ e, K. Bushick, S. Hajinazar, J. Lafuente- Bartolome, J. Leveillee, C. Lian, J.-M. Lihm, F. Macheda, H. Mori, H. Paudyal, W. H. Sio, S. Tiwari, M. Zacharias, X. Zhang, N. Bonini, E. Kioupakis, E. R. Margine, and F. Giustino, Electron–phonon physics from first p...

    2023

  17. [23]

    D. J. Abramovitch, J. Mravlje, J.-J. Zhou, A. Georges, and M. Bernardi, Respective roles of electron-phonon and electron-electron interactions in the transport and quasi- particle properties of srvo3, Phys. Rev. Lett.133, 186501 (2024)

    2024

  18. [24]

    Li and S

    Z. Li and S. G. Louie, Two-gap superconductivity and the decisive role of rare-earthdelectrons in infinite-layer nickelates, Phys. Rev. Lett.133, 126401 (2024)

    2024

  19. [25]

    Dolui, L

    K. Dolui, L. J. Conway, C. Heil, T. A. Strobel, R. P. Prasankumar, and C. J. Pickard, Feasible route to high- temperature ambient-pressure hydride superconductiv- ity, Phys. Rev. Lett.132, 166001 (2024)

    2024

  20. [26]

    Garmroudi, J

    F. Garmroudi, J. Coulter, I. Serhiienko, S. Di Cataldo, M. Parzer, A. Riss, M. Grasser, S. Stockinger, S. Khmelevskyi, K. Pryga, B. Wiendlocha, K. Held, T. Mori, E. Bauer, A. Georges, and A. Pustogow, Topo- logical flat-band-driven metallic thermoelectricity, Phys. Rev. X15, 0...

    2025

  21. [27]

    D. Duan, Y. Liu, F. Tian, D. Li, X. Huang, Z. Zhao, H. Yu, B. Liu, W. Tian, and T. Cui, Pressure-induced metallization of dense (h2s)2h2 with high-tc supercon- ductivity, Sci. Rep.4, 6968 (2014)

    2014

  22. [28]

    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

  23. [29]

    J. M. Skelton, L. A. Burton, S. C. Parker, A. Walsh, C.- E. Kim, A. Soon, J. Buckeridge, A. A. Sokol, C. R. A. Catlow, A. Togo, and I. Tanaka, Anharmonicity in the high-temperaturecmcmphase of snse: Soft modes and three-phonon interactions, Phys. Rev. Lett.117, 075502 (2016)

    2016

  24. [31]

    A. Y. Liu, I. I. Mazin, and J. Kortus, Beyond eliash- berg superconductivity in mgb 2: Anharmonicity, two- phonon scattering, and multiple gaps, Phys. Rev. Lett. 87, 087005 (2001)

    2001

  25. [32]

    Lee, J.-J

    N.-E. Lee, J.-J. Zhou, H.-Y. Chen, and M. Bernardi, Ab initio electron-two-phonon scattering in gaas from next- to-leading order perturbation theory, Nature Communi- cations11, 1607 (2020)

    2020

  26. [33]

    Y. Kong, O. V. Dolgov, O. Jepsen, and O. K. Andersen, Electron-phonon interaction in the normal and supercon- ducting states of mgb 2, Phys. Rev. B64, 020501 (2001)

    2001

  27. [34]

    Delaire, J

    O. Delaire, J. Ma, K. Marty, A. F. May, M. A. McGuire, M.-H. Du, D. J. Singh, A. Podlesnyak, G. Ehlers, M. D. Lumsden, and B. C. Sales, Giant anharmonic phonon scattering in pbte, Nature Materials10, 614 (2011)

    2011

  28. [35]

    Nagamatsu, N

    J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, and J. Akimitsu, Superconductivity at 39 k in magnesium diboride, Nature410, 63 (2001)

    2001

  29. [37]

    See Supplemental Material at [URL will be inserted by publisher] for the iterative solution of the Boltzmann transport equation and convergence tests

  30. [38]

    A. Wang, J. Yin, F. A. Goudreault, M. Cˆ ot´ e, O. Hellman, and S. Ponc´ e, Opposite impact of thermal expansion and phonon anharmonicity on the phonon-limited resistivity of elemental metals from first principles, Physical Review B113, L060302 (2026)

    2026

  31. [39]

    Y. Luo, D. Desai, B. K. Chang, J. Park, and M. Bernardi, Data-driven compression of electron-phonon interactions, Phys. Rev. X14, 021023 (2024)

    2024

  32. [40]

    Y. Luo, D. Mangtani, S. Peng, J. Yao, S. Kliavinek, and M. Bernardi, Tensor learning and compression of n-phonon interactions, Phys. Rev. Lett.135, 126101 (2025)

    2025

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