Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Unified theory of thermal transport in crystals and glasses

Nature Physics volume 15, pages 809–813 (2019)Cite this article

Subjects

Abstract

Crystals and glasses exhibit fundamentally different heat conduction mechanisms: the periodicity of crystals allows for the excitation of propagating vibrational waves that carry heat, as first discussed by Peierls, while in glasses the lack of periodicity breaks Peierls’s picture and heat is mainly carried by the coupling of vibrational modes, often described by a harmonic theory introduced by Allen and Feldman. Anharmonicity or disorder are thus the limiting factors for thermal conductivity in crystals or glasses. Hitherto, no transport equation has been able to account for both. Here, we derive such an equation, resulting in a thermal conductivity that reduces to the Peierls and Allen–Feldman limits, respectively, in anharmonic crystals or harmonic glasses, while also covering the intermediate regimes where both effects are relevant. This approach also solves the long-standing problem of accurately predicting the thermal properties of crystals with ultralow or glass-like thermal conductivity, as we show with an application to a thermoelectric material representative of this class.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Prices vary by article type

from$1.95

to$39.95

Learn more

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Bulk thermal conductivity of CsPbBr3 as a function of temperature.
Fig. 2: Vibrational properties and heat conduction mechanisms in CsPbBr3.

Data availability

Raw data were generated using the SCITAS High Performance Computing facility at the École Polytechnique Fédérale de Lausanne. Derived data supporting the findings of this study are available at https://doi.org/10.24435/materialscloud:2019.0001/v2.

Code availability

Quantum ESPRESSO is available at www.quantum-espresso.org; the scripts related to the computation of the third-order force constants using the finite-difference method are available at bitbucket.org/sousaw/thirdorder; the D3Q package for Quantum ESPRESSO is available at sourceforge.net/projects/d3q/. The custom code developed in this work will be made available in a next release of the D3Q package.

References

  1. Peierls, R. Zur Kinetischen Theorie der Wärmeleitung in Kristallen. Ann. Phys. (NY) 395, 1055–1101 (1929).

    Article  ADS  Google Scholar 

  2. Garg, J., Bonini, N., Kozinsky, B. & Marzari, N. Role of disorder and anharmonicity in the thermal conductivity of silicon–germanium alloys: a first-principles study. Phys. Rev. Lett. 106, 045901 (2011).

    Article  ADS  Google Scholar 

  3. Omini, M. & Sparavigna, A. An iterative approach to the phonon Boltzmann equation in the theory of thermal conductivity. Phys. B Condens. Matter 212, 101–112 (1995).

    Article  ADS  Google Scholar 

  4. Broido, D., Malorny, M., Birner, G., Mingo, N. & Stewart, D. Intrinsic lattice thermal conductivity of semiconductors from first principles. Appl. Phys. Lett. 91, 231922 (2007).

    Article  ADS  Google Scholar 

  5. Carrete, J. et al. almaBTE: a solver of the space–time dependent Boltzmann transport equation for phonons in structured materials. Comput. Phys. Commun. 220, 351–362 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  6. Fugallo, G., Lazzeri, M., Paulatto, L. & Mauri, F. Ab initio variational approach for evaluating lattice thermal conductivity. Phys. Rev. B 88, 045430 (2013).

    Article  ADS  Google Scholar 

  7. Chaput, L. Direct solution to the linearized phonon Boltzmann equation. Phys. Rev. Lett. 110, 265506 (2013).

    Article  ADS  Google Scholar 

  8. Cepellotti, A. & Marzari, N. Thermal transport in crystals as a kinetic theory of relaxons. Phys. Rev. X 6, 041013 (2016).

    Google Scholar 

  9. Baroni, S., de Gironcoli, S., Dal Corso, A. & Giannozzi, P. Phonons and related crystal properties from density-functional perturbation theory. Rev. Mod. Phys. 73, 515–562 (2001).

    Article  ADS  Google Scholar 

  10. Paulatto, L., Mauri, F. & Lazzeri, M. Anharmonic properties from a generalized third-order ab initio approach: theory and applications to graphite and graphene. Phys. Rev. B 87, 214303 (2013).

    Article  ADS  Google Scholar 

  11. Esfarjani, K., Chen, G. & Stokes, H. T. Heat transport in silicon from first-principles calculations. Phys. Rev. B 84, 085204 (2011).

    Article  ADS  Google Scholar 

  12. Luckyanova, M. N. et al. Coherent phonon heat conduction in superlattices. Science 338, 936–939 (2012).

    Article  ADS  Google Scholar 

  13. Cahill, D. G. et al. Nanoscale thermal transport. II. 2003–2012. Appl. Phys. Rev. 1, 011305 (2014).

    Article  ADS  Google Scholar 

  14. Hardy, R. J. Energy-flux operator for a lattice. Phys. Rev. 132, 168–177 (1963).

    Article  ADS  MathSciNet  Google Scholar 

  15. Allen, P. B. & Feldman, J. L. Thermal conductivity of glasses: theory and application to amorphous Si. Phys. Rev. Lett. 62, 645–648 (1989).

    Article  ADS  Google Scholar 

  16. Allen, P. B., Feldman, J. L., Fabian, J. & Wooten, F. Diffusons, locons and propagons: character of atomic vibrations in amorphous Si. Philos. Mag. B 79, 1715–1731 (1999).

    Article  ADS  Google Scholar 

  17. Lv, W. & Henry, A. Non-negligible contributions to thermal conductivity from localized modes in amorphous silicon dioxide. Sci. Rep. 6, 35720 (2016).

    Article  ADS  Google Scholar 

  18. Li, W. & Mingo, N. Ultralow lattice thermal conductivity of the fully filled skutterudite YbFe4Sb12 due to the flat avoided-crossing filler modes. Phys. Rev. B 91, 144304 (2015).

    Article  ADS  Google Scholar 

  19. Lee, W. et al. Ultralow thermal conductivity in all-inorganic halide perovskites. Proc. Natl Acad. Sci. USA 114, 8693–8697 (2017).

    Article  ADS  Google Scholar 

  20. Chen, X. et al. Twisting phonons in complex crystals with quasi-one-dimensional substructures. Nat. Commun. 6, 6723 (2015).

    Article  ADS  Google Scholar 

  21. Weathers, A. et al. Glass-like thermal conductivity in nanostructures of a complex anisotropic crystal. Phys. Rev. B 96, 214202 (2017).

    Article  ADS  Google Scholar 

  22. Lory, P.-F. et al. Direct measurement of individual phonon lifetimes in the clathrate compound Ba7.81Ge40.67Au5.33. Nat. Commun. 8, 491 (2017).

    Article  ADS  Google Scholar 

  23. Mukhopadhyay, S. et al. Two-channel model for ultralow thermal conductivity of crystalline Tl3VSe4. Science 360, 1455–1458 (2018).

    Article  ADS  Google Scholar 

  24. Shenogin, S., Bodapati, A., Keblinski, P. & McGaughey, A. J. H. Predicting the thermal conductivity of inorganic and polymeric glasses: the role of anharmonicity. J. Appl. Phys. 105, 034906 (2009).

    Article  ADS  Google Scholar 

  25. Ziman, J. M. Electrons and Phonons: The Theory of Transport Phenomena in Solids (Oxford University Press, 1960).

  26. Tamura, S.-i Isotope scattering of dispersive phonons in Ge. Phys. Rev. B 27, 858–866 (1983).

    Article  ADS  Google Scholar 

  27. Gebauer, R. & Car, R. Kinetic theory of quantum transport at the nanoscale. Phys. Rev. B 70, 125324 (2004).

    Article  ADS  Google Scholar 

  28. Frensley, W. R. Boundary conditions for open quantum systems driven far from equilibrium. Rev. Mod. Phys. 62, 745–791 (1990).

    Article  ADS  Google Scholar 

  29. Cohen-Tannoudji, C., Dupont-Roc, J., Grynberg, G. & Thickstun, P. Atom–Photon Interactions: Basic Processes and Applications (Wiley-VCH, 2004).

  30. Kané, G., Lazzeri, M. & Mauri, F. Zener tunneling in the electrical transport of quasimetallic carbon nanotubes. Phys. Rev. B 86, 155433 (2012).

    Article  ADS  Google Scholar 

  31. Cepellotti, A. et al. Phonon hydrodynamics in two-dimensional materials. Nat. Commun. 6, 6400 (2015).

    Article  ADS  Google Scholar 

  32. Wang, Y. et al. Cation dynamics governed thermal properties of lead halide perovskite nanowires. Nano Lett. 18, 2772–2779 (2018).

    Article  ADS  Google Scholar 

  33. Sun, T. & Allen, P. B. Lattice thermal conductivity: computations and theory of the high-temperature breakdown of the phonon-gas model. Phys. Rev. B 82, 224305 (2010).

    Article  ADS  Google Scholar 

  34. Voneshen, D. et al. Suppression of thermal conductivity by rattling modes in thermoelectric sodium cobaltate. Nat. Mater. 12, 1028–1032 (2013).

    Article  ADS  Google Scholar 

  35. Cahill, D. G., Watson, S. K. & Pohl, R. O. Lower limit to the thermal conductivity of disordered crystals. Phys. Rev. B 46, 6131–6140 (1992).

    Article  ADS  Google Scholar 

  36. Iotti, R. C., Ciancio, E. & Rossi, F. Quantum transport theory for semiconductor nanostructures: a density-matrix formulation. Phys. Rev. B 72, 125347 (2005).

    Article  ADS  Google Scholar 

  37. Agne, M. T., Hanus, R. & Snyder, G. J. Minimum thermal conductivity in the context of diffuson-mediated thermal transport. Energy Environ. Sci. 11, 609–616 (2018).

    Article  Google Scholar 

  38. Clarke, D. R. Materials selection guidelines for low thermal conductivity thermal barrier coatings. Surf. Coat. Technol. 163–164, 67–74 (2003).

    Article  Google Scholar 

  39. Marcolongo, A., Umari, P. & Baroni, S. Microscopic theory and quantum simulation of atomic heat transport. Nat. Phys. 12, 80–84 (2016).

    Article  Google Scholar 

  40. Seyf, H. R. et al. Rethinking phonons: the issue of disorder. npj Comp. Mat. 3, 49 (2017).

    Article  Google Scholar 

  41. Carbogno, C., Ramprasad, R. & Scheffler, M. Ab initio Green–Kubo approach for the thermal conductivity of solids. Phys. Rev. Lett. 118, 175901 (2017).

    Article  ADS  Google Scholar 

  42. Puligheddu, M., Gygi, F. & Galli, G. First-principles simulations of heat transport. Phys. Rev. Mater. 1, 060802 (2017).

    Article  Google Scholar 

  43. Marzari, N., Mostofi, A. A., Yates, J. R., Souza, I. & Vanderbilt, D. Maximally localized Wannier functions: theory and applications. Rev. Mod. Phys. 84, 1419–1475 (2012).

    Article  ADS  Google Scholar 

  44. Hardy, R. J. Phonon Boltzmann equation and second sound in solids. Phys. Rev. B 2, 1193 (1970).

    Article  ADS  Google Scholar 

  45. Allen, P. B. & Feldman, J. L. Thermal conductivity of disordered harmonic solids. Phys. Rev. B 48, 12581–12588 (1993).

    Article  ADS  Google Scholar 

  46. Auerbach, A. & Allen, P. B. Universal high-temperature saturation in phonon and electron transport. Phys. Rev. B 29, 2884–2890 (1984).

    Article  ADS  Google Scholar 

  47. Feldman, J. L., Kluge, M. D., Allen, P. B. & Wooten, F. Thermal conductivity and localization in glasses: numerical study of a model of amorphous silicon. Phys. Rev. B 48, 12589–12602 (1993).

    Article  ADS  Google Scholar 

  48. Stoumpos, C. C. et al. Crystal growth of the perovskite semiconductor CsPbBr3: a new material for high-energy radiation detection. Cryst. Growth Des. 13, 2722–2727 (2013).

    Article  Google Scholar 

  49. Gražulis, S. et al. Crystallography open database—an open-access collection of crystal structures. J. Appl. Crystallogr. 42, 726–729 (2009).

    Article  Google Scholar 

  50. Momma, K. & Izumi, F. VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data. J. Appl. Crystallogr. 44, 1272–1276 (2011).

    Article  Google Scholar 

  51. Miyata, K. et al. Large polarons in lead halide perovskites. Sci. Adv. 3, e1701217 (2017).

    Article  Google Scholar 

  52. Giannozzi, P. et al. Advanced capabilities for materials modelling with quantum ESPRESSO. J. Phys. Condens. Matter 29, 465901 (2017).

    Article  Google Scholar 

  53. Garrity, K. F., Bennett, J. W., Rabe, K. M. & Vanderbilt, D. Pseudopotentials for high-throughput DFT calculations. Comput. Mater. Sci. 81, 446–452 (2014).

    Article  Google Scholar 

  54. Li, W., Carrete, J., Katcho, N. A. & Mingo, N. ShengBTE: a solver of the Boltzmann transport equation for phonons. Comput. Phys. Commun. 185, 1747–1758 (2014).

    Article  ADS  Google Scholar 

  55. Paulatto, L., Errea, I., Calandra, M. & Mauri, F. First-principles calculations of phonon frequencies, lifetimes and spectral functions from weak to strong anharmonicity: the example of palladium hydrides. Phys. Rev. B 91, 054304 (2015).

    Article  ADS  Google Scholar 

  56. Togo, A., Chaput, L. & Tanaka, I. Distributions of phonon lifetimes in Brillouin zones. Phys. Rev. B 91, 094306 (2015).

    Article  ADS  Google Scholar 

  57. Chernatynskiy, A. & Phillpot, S. R. Phonon transport simulator (PhonTS). Comput. Phys. Commun. 192, 196–204 (2015).

    Article  ADS  Google Scholar 

  58. Tadano, T., Gohda, Y. & Tsuneyuki, S. Anharmonic force constants extracted from first-principles molecular dynamics: applications to heat transfer simulations. J. Phys. Condens. Matter 26, 225402 (2014).

    Article  Google Scholar 

  59. Bianco, R., Errea, I., Paulatto, L., Calandra, M. & Mauri, F. Second-order structural phase transitions, free energy curvature and temperature-dependent anharmonic phonons in the self-consistent harmonic approximation: theory and stochastic implementation. Phys. Rev. B 96, 014111 (2017).

    Article  ADS  Google Scholar 

  60. Tadano, T. & Tsuneyuki, S. Quartic anharmonicity of rattlers and its effect on lattice thermal conductivity of clathrates from first principles. Phys. Rev. Lett. 120, 105901 (2018).

    Article  ADS  Google Scholar 

  61. van Roekeghem, A., Carrete, J. & Mingo, N. Anomalous thermal conductivity and suppression of negative thermal expansion in ScF3. Phys. Rev. B 94, 020303 (2016).

    Article  Google Scholar 

  62. Yang, F. C. et al. Temperature dependence of phonons in Pd3Fe through the Curie temperature. Phys. Rev. B 98, 024301 (2018).

    Article  ADS  Google Scholar 

  63. Zhang, D.-B., Sun, T. & Wentzcovitch, R. M. Phonon quasiparticles and anharmonic free energy in complex systems. Phys. Rev. Lett. 112, 058501 (2014).

    Article  ADS  Google Scholar 

  64. Ravichandran, N. K. & Broido, D. Unified first-principles theory of thermal properties of insulators. Phys. Rev. B 98, 085205 (2018).

    Article  ADS  Google Scholar 

  65. Romero, A. H., Gross, E. K. U., Verstraete, M. J. & Hellman, O. Thermal conductivity in PbTe from first principles. Phys. Rev. B 91, 214310 (2015).

    Article  ADS  Google Scholar 

  66. Aseginolaza, U. et al. Phonon collapse and second-order phase transition in thermoelectric SnSe. Phys. Rev. Lett. 122, 075901 (2019).

    Article  ADS  Google Scholar 

  67. Hellman, O., Abrikosov, I. A. & Simak, S. I. Lattice dynamics of anharmonic solids from first principles. Phys. Rev. B 84, 180301 (2011).

    Article  ADS  Google Scholar 

  68. Hellman, O. & Abrikosov, I. A. Temperature-dependent effective third-order interatomic force constants from first principles. Phys. Rev. B 88, 144301 (2013).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  70. Souvatzis, P., Eriksson, O., Katsnelson, M. I. & Rudin, S. P. Entropy driven stabilization of energetically unstable crystal structures explained from first principles theory. Phys. Rev. Lett. 100, 095901 (2008).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This research was partially supported by the NCCR MARVEL, funded by the Swiss National Science Foundation.

Author information

Authors and Affiliations

  1. Theory and Simulation of Materials (THEOS) and National Centre for Computational Design and Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Michele Simoncelli & Nicola Marzari

  2. Dipartimento di Fisica, Università di Roma La Sapienza, Rome, Italy

    Francesco Mauri

Authors
  1. Michele Simoncelli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicola Marzari
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Francesco Mauri
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The project was conceived by all authors. M.S. performed the numerical calculations and prepared the figures with inputs from N.M. and F.M. All authors contributed to the editing of the manuscript.

Corresponding author

Correspondence to Francesco Mauri.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Journal peer review information: Nature Physics thanks David Cahill and Marco Buongiorno Nardelli for their contribution to the peer review of this work.

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Simoncelli, M., Marzari, N. & Mauri, F. Unified theory of thermal transport in crystals and glasses. Nat. Phys. 15, 809–813 (2019). https://doi.org/10.1038/s41567-019-0520-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-019-0520-x

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing