Heat transfer by phonons in the generalized transport model

Authors: Kruglyak Yu. A., Shtefan N. Z.

Year: 2017

Issue: 22

Pages: 107-119


On the basis of Landauer – Datta – Lundstrom transport model the generalized model of heat transfer by phonons is formulated. Similarly to the Fermi window for electron conductivity the concept of the Fermi window for phonon conductivity is introduced and used to obtain the general expression for the lattice thermal conductivity with the quantum of thermoconductance appearing at the very beginning. There are emphasized the similarity and differences in the construction of the theory of electron conductivity and the theory of heat conduction. There are discussed the thermal conductivity of the conductors, the physical sense of proportionality between the thermal conductivity and specific heat capacity at constant volume, the relationship between the transmission coefficient and the mean-free-path, the calculation of the number of phonon modes and density of phonon states, the Debye model of heat conductivity and scattering of phonons, the temperature dependence of the lattice thermal conductivity, the difference between the lattice thermal conductivity and electron conduction, and quantization of thermal conductivity.

In the present review it is emphasized that the concepts used to describe the electron transport can be successfully transferred to the phonon transport. And in both cases the Landauer approach, generalized subsequently by Datta and Lundstrom, allows a quantitative description of transport processes in conductors of any dimension and in all modes of transport from diffusive to ballistic. Lattice thermal conductivity and electronic conductivity are described very similar as far as the shape of corresponding equations. There are, however, two significant differences related to the physics of electron and phonon processes which are also emphasized in the review.

Tags: Debye model; nanoelectronics; nanophysics; phonon modes; phonon scattering; phonon transport; quantum thermal conductivity; transmission coefficient; дифузійно-дрейфова модель; дифузійно-дрейфова модель; молекулярна електроніка; молекулярна електроніка; нанофизика; нанофизика; наноэлектроника; наноэлектроника; роль контактів; роль контактів; струм насичення; струм насичення


  1. Datta Supriyo. Lessons from Nanoelectronics: A New Perspective on Transport. Hackensack, New Jersey: World Sci-entific Publishing Company, 2012, 473 p. www.nanohub.org/courses/FoN1.
  2. Lundstrom Mark, Jeong Changwook. Near-Equilibrium Transport: Fundamentals and Applications. Hackensack, New Jersey: World Scientific Publishing Company, 2013, 227 p. www.nanohub.org/resources/11763.
  3. Kruglyak Yu. A. Fizicheskoe obrazovanie v vuzakh – Physical education in Universities, 2013, vol. 19, no. 4, pp. 70 – 85. (In Russian).
  4. Kruglyak Yu. A. Nanoelektronika «znyzu – vverkh» [Nanoelectronics “bottom-up”]. Odessa: TES, 2015.
  5. Zayman Dzh. Elektrony i fonony. Teoriya yavleniy perenosa v tverdykh telakh [Electrons and phonons. The theory of transport phenomena in solids]. Moscow: IIL, 1962. 488 p.
  6. Zayman Dzh. Printsipy teorii tverdogo tela [Principles of solid state theory]. Moscow: Vysshaya shkola, 1974.
  7. Kittel’ Ch. Vvedenie v fiziku tverdogo tela [Introduction to Solid State Physics]. Moscow: Nauka, 1978.
  8. Ashkroft N., Mermin N. Fizika tverdogo tela, toma 1 i 2 [Solid State Physics vol. 1and 3]. Moscow: Mir, 1979.
  9. Mohr M., Maultzsch J., Dobardžić E., Reich S., Milošević I., Damnjanović M., Bosak A., Krisch M., Thomsen C. Phonon dispersion of graphite by inelastic x-ray scattering. Phys. Rev. B., 2007, vol. 76, no. 3, p. 035439/7.
  10. Eletskiy A. V., Iskandarova I. M., Knizhnik A. A., Krasikov D. N. UFN – Successes of physical sciences, 2011, vol. 181, pp. 227—258. (In Russian)
  11. Katsnelson M. I. Graphene: Carbon in Two Dimensions. New York: Cambridge University Press, 2012. 366 p.
  12. Kruglyak Yu. A., Nanosystemy, nanomaterіaly, nanotekhnolohіi — Nanosystems, nanomaterials, nanotechnologies, 2013, vol. 11, №3, pp. 519–549. (In Russian).
  13. Schwab K., Henriksen E. A., Worlock J. M., Roukes M. L. Measurement of the quantum of thermal conductance. Na-ture, 2000, vol. 404, pp. 974–977.
  14. Jeong C., Kim R., Luisier M., Datta S., Lundstrom M. On Landauer versus Boltzmann and full band versus effective mass evaluation of thermoelectric transport coefficients. J.Appl.Phys., 2010, vol. 107, p. 023707.
  15. Mark Lundstrom. Fundamentals of Carrier Transport. Cambridge UK: Cambridge University Press, 2012. 440 р.
  16. Jeong C., Datta S., Lundstrom M. Full Dispersion versus Debye Model Evaluation of Lattice Thermal Conductivity with a Landauer Approach. J.Appl.Phys, 2011, vol. 109, p. 073718/8.
  17. Callaway J. Model for lattice thermal conductivity at low tevpretures. Phys. Rev, 1959, vol. 11, no. 4, pp. 1046–1051.
  18. Holland M. G. Analysis of lattice thermal conductivity. Phys. Rev, 1963, vol. 132, no. 6, pp. 2461–2471.
  19. Jeong C., Datta S., Lundstrom M. . Thermal Conductivity of Bulk and Thin-Film Silicon: A Landauer Approach. J. Appl. Phys, 2012,vol. 111, p. 093708.
  20. Gang Chen. Nanoscale Energy Transport and Conversion: A Parallel Treatment of Electrons, Molecules, Phonons, and Photons. New York: Oxford University Press: 2005, p. 560.
  21. Glassbrenner C. J., Slack G. A. Thermal Conductivity of Silicon and Germanium from 3º K to the Melting Point. Phys. Rev.,1964, vol. 134, no. 4A, p. A1058.
  22. Pendry J. B. Quantum limits to the flow of information and entropy. J.Phys.A.,1983, vol. 16, p. 2161–2171.
  23. Angelescu D. E., Cross M. C., Roukes M. L. Heat transport in mesoscopic systems. Superlatt. Microstruct, 1998, vol. 23, pp. 673–689.
  24. Rego L.G.C., Kirczenow G.. Quantized thermal conductance of dielectric quantum wires. Phys.Rev.Lett, 1998, vol. 81, pp. 232–235.
  25. Blencowe M. P. Quantum energy flow in mesoscopic dielectric Structures. Phys. Rev. B., 1999, vol. 59, pp. 4992–4998.
  26. Rego L.G.C., Kirczenow G. Fractional exclusion statistics and the universal quantum of thermal conductance: A uni-fying approach. Phys.Rev. B., 1999, vol. 59, pp. 13080–13086.
  27. Krive I. V., Mucciolo E. R. . Transport properties of quasiparticles with fractional exclusion statistics. Phys. Rev. B.,1999, vol. 60, pp. 1429 – 1432.
  28. Caves C. M., Drummond P. D. Quantum limits on bosonic communication rates. Rev.Mod.Phys., 1994, vol. 66, pp. 481–537.
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