The European Physical Journal B

, 85:372

Landauer formula for phonon heat conduction: relation between energy transmittance and transmission coefficient

Regular Article
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Abstract

The heat current across a quantum harmonic system connected to reservoirs at different temperatures is given by the Landauer formula, in terms of an integral over phonon frequencies ω, of the energy transmittance \hbox{$\mathcal{T}(\omega)$}𝒯(ω). There are several different ways to derive this formula, for example using the Keldysh approach or the Langevin equation approach. The energy transmittance \hbox{$\mathcal{T}(\omega)$} 𝒯(ω) is usually expressed in terms of nonequilibrium phonon Green’s function and it is expected that it is related to the transmission coefficient τ(ω) of plane waves across the system. In this paper, for a one-dimensional set-up of a finite harmonic chain connected to reservoirs which are also semi-infinite harmonic chains, we present a simple and direct demonstration of the relation between \hbox{$\mathcal{T}(\omega)$}𝒯(ω) and τ(ω). Our approach is easily extendable to the case where both system and reservoirs are in higher dimensions and have arbitrary geometries, in which case the meaning of τ and its relation to \hbox{$\mathcal{T}$}𝒯are more non-trivial.

Keywords

Mesoscopic and Nanoscale Systems 

References

  1. 1.
    D.E. Angelescu, M.C. Cross, M.L. Roukes, Superlatt. Microsctruct. 23, 673 (1998)ADSCrossRefGoogle Scholar
  2. 2.
    L.G.C. Rego, G. Kirczenow, Phys. Rev. Lett. 81, 232 (1998)ADSCrossRefGoogle Scholar
  3. 3.
    M.P. Blencowe, Phys. Rev. B 59, 4992 (1999)ADSCrossRefGoogle Scholar
  4. 4.
    A. Dhar, D. Roy, J. Stat Phys. 125, 801 (2006)MathSciNetADSMATHCrossRefGoogle Scholar
  5. 5.
    A. Dhar, Adv. Phys. 57, 457 (2008)ADSCrossRefGoogle Scholar
  6. 6.
    J.-S. Wang, J. Wang, N. Zeng, Phys. Rev. B 74, 033408 (2006)ADSCrossRefGoogle Scholar
  7. 7.
    T. Yamamoto, K. Watanabe, Phys. Rev. Lett. 96, 255503 (2006)ADSCrossRefGoogle Scholar
  8. 8.
    Y. Imry, R. Landauer, Rev. Mod. Phys. 71, S306 (1999)CrossRefGoogle Scholar
  9. 9.
    C. Caroli, R. Combescot, D. Lederer, P. Nozières, D. Saint-James, J. Phys. C 4, 916 (1971)ADSCrossRefGoogle Scholar
  10. 10.
    Y. Meir, N.S. Wingreen, Phys. Rev. Lett. 68, 2512 (1992)ADSCrossRefGoogle Scholar
  11. 11.
    A. Dhar, D. Sen, Phys. Rev. B 73, 085119 (2006)ADSCrossRefGoogle Scholar
  12. 12.
    T.N. Todorov, G.A.D. Briggs, A.P. Sutton, J. Phys.: Condens. Matter 5, 2389 (1993)ADSCrossRefGoogle Scholar
  13. 13.
    P.A. Khomyakov, G. Brocks, V. Karpan, M. Zwierzycki, P.J. Kelly, Phys. Rev. B 72, 035450 (2005)ADSCrossRefGoogle Scholar
  14. 14.
    J.-S. Wang, J. Wang, J.T. Lü, Eur. Phys. J. B 62, 381 (2008)ADSCrossRefGoogle Scholar
  15. 15.
    Y.A. Kosevich, Phys. Rev. B 52, 1017 (1995)ADSCrossRefGoogle Scholar
  16. 16.
    N. Mingo, Liu Yang, Phys. Rev. B 68, 245406 (2003)ADSCrossRefGoogle Scholar
  17. 17.
    M.A. Panzer, K.E. Goodson, J. Appl. Phys. 103, 094301 (2008)ADSCrossRefGoogle Scholar
  18. 18.
    L. Zhang, P. Keblinski, J.S. Wang, B. Li, Phys. Rev. B 83, 064303 (2011)ADSCrossRefGoogle Scholar
  19. 19.
    R.J. Rubin, W.L. Greer, J. Math. Phys. 12, 1686 (1971)ADSCrossRefGoogle Scholar
  20. 20.
    H. Spohn, J.L. Lebowitz, Commun. math. Phys. 54, 97 (1977)MathSciNetADSCrossRefGoogle Scholar
  21. 21.
    M.E. Lumpkin, W.M. Saslow, W.M. Visscher, Phys. Rev. B 17, 4295 (1978)ADSCrossRefGoogle Scholar
  22. 22.
    D. Segal, A. Nitzan, P. Hanggi, J. Chem. Phys. 119, 6840 (2003)ADSCrossRefGoogle Scholar
  23. 23.
    A. Dhar, K. Saito, P. Hanggi, Phys. Rev. E 85, 011126 (2012)ADSCrossRefGoogle Scholar
  24. 24.
    A. Dhar, D. Sen, Phys. Rev. B 73, 085119 (2006)ADSCrossRefGoogle Scholar
  25. 25.
    A. Chaudhuri, A. Kundu, D. Roy, A. Dhar, J.L. Lebowitz, H. Spohn, Phys. Rev. B 81, 064301 (2010)ADSCrossRefGoogle Scholar
  26. 26.
    A. Casher, J.L. Lebowitz, J. Math. Phys. 12, 1701 (1971)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Raman Research InstituteBangaloreIndia

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