Alternative representation of the Kubo formula for the optical conductivity: A shortcut to transport properties

Physical Review B - Condensed Matter and Materials Physics

Volumn 90, Issue 1, 2014, Pages

  De Filippis, G ^a   Cataudella, V ^a   De Candia, A ^a   Mishchenko, A S ^b   Nagaosa, N ^b,c  

^a UNIVERSITY OF NAPLES FEDERICO II   (Italy)

^b INFN SPIN CNR Università di Napoli Federico II ^*  (Japan)

^c UNIVERSITY OF TOKYO   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84905395962     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.90.014310     Document Type: Article

References (44)

1. The Boltzmann Equation, Proc. Intern. Symp. 100 years Boltzmann Equation, edited by E. G. D. Cohen and W. Thirringh (Springer, Verlag Wein, 1973).
2. N. R. Mott and H. Jones, The Theory of the Properties of Metals and Alloys (Dover, New York, 1958).
3. R. Kubo, J. Phys. Soc. Japan 12, 570 (1957). JUPSAU 0031-9015 10.1143/JPSJ.12.570
4. H. Mori, Prog. Theor. Phys. 33, 423 (1965) PTPKAV 0033-068X 10.1143/PTP.33.423;
5. H. Mori, Prog. Theor. Phys. 34, 399 (1965). PTPKAV 10.1143/PTP.34.399 0033-068X
6. An in-depth study of the effects of phonons on the properties of electrons in normal metals is contained in R. E. Prange and L. P. Kadanoff, Phys. Rev. 134, A566 (1964); PHRVAO 0031-899X 10.1103/PhysRev.134.A566
7. T. Holstein, Ann. Phys. (NY) 29, 410 (1964) 10.1016/0003-4916(64)90008-9.
8. In particular, Boltzmann equations have been reproduced through a perturbative treatment of the Kubo formula to leading order in the Migdal expansion (Equation presented), where (Equation presented) (Equation presented) is the electron (ion) mass. Then, in the metals, the phenomenological theory works well even if the parameter giving the mass enhancement is not small, provided the states near the Fermi level are propagating quasiparticle states. In any case, the Fermi energy has to be much larger than other energies such as (Equation presented) or (Equation presented), where (Equation presented) is the Debye frequency.
9. The coupling strength to phonon or impurity (Equation presented) differs from the dimensionless parameter introduced by A. B. Migdal, Zh. Eksp. Teor. Fiz. 34, 1438 (1958)
10. A. B. Migdal, [Sov. Phys. JETP 7, 996 (1958)], which, in particular, is quadratic in the coupling strength.
11. G. D. Mahan, Many Particle Physics (Plenum, New York, 1981).
12. L. Van Hove, Physica (Utrecht) 21, 517 (1955). PHYSAG 0031-8914 10.1016/S0031-8914(54)92646-4
13. E. Verboven, Physica 26, 1091 (1960). PHYSAG 0031-8914 10.1016/0031-8914(60)90143-9
14. G. V. Chester and A. Thellung, Proc. Phys. Soc. (London) 73, 745 (1959). 10.1088/0370-1328/73/5/308
15. R. Zwanzig, Phys. Rev. 124, 983 (1961). PHRVAO 0031-899X 10.1103/PhysRev.124.983
16. W. Götze and P. Wölfle, Phys. Rev. B 6, 1226 (1972). 0556-2805 10.1103/PhysRevB.6.1226
17. V. M. Kenkre and M. Dresden, Phys. Rev. A 6, 769 (1972). 0556-2791 10.1103/PhysRevA.6.769
18. P. N. Argyres and J. L. Sigel, Phys. Rev. Lett. 31, 1397 (1973). PRLTAO 0031-9007 10.1103/PhysRevLett.31.1397
19. S. Mukerjee and B. S. Shastry, Phys. Rev. B 77, 245131 (2008). PRBMDO 1098-0121 10.1103/PhysRevB.77.245131
20. In particular, it has been conjectured [17] that (i) integrable systems exhibit ballistic behavior, i.e., dissipationless finite-temperature conductivity, if the Drude coefficient is different from zero at (Equation presented), and remain ideal insulators, i.e., (Equation presented), at all temperatures if (Equation presented); (ii) a generic nonintegrable system displays finite conductivity at finite temperature, i.e., it is characterized by diffusive motion, being (Equation presented). We will restrict our attention to normal conductors, i.e., (Equation presented). On the other hand, we emphasize that in these systems, (Equation presented) only if the exact eigenstates are used in Eq. (5), so that we will retain this contribution up to the end of the calculation. We also emphasize that at (Equation presented), the Drude coefficient provides the criterion to distinguish between conductors and insulators. In particular, it has recently been shown [see B. Hetenyi, Phys. Rev. B 87, 235123 (2013) PRBMDO 1098-0121 10.1103/PhysRevB.87.235123] that wave functions, i.e., eigenfunctions of the total current operator, give rise to a finite (Equation presented) and are therefore metallic.
21. H. Castella, X. Zotos, and P. Prelovsek, Phys. Rev. Lett. 74, 972 (1995) PRLTAO 0031-9007 10.1103/PhysRevLett.74.972;
22. X. Zotos and P. Prelovsek, Phys. Rev. B 53, 983 (1996). PRBMDO 10.1103/PhysRevB.53.983 0163-1829
23. P. F. Maldague, Phys. Rev. B 16, 2437 (1977). 0556-2805 10.1103/PhysRevB.16.2437
24. L. P. Kadanoff and P. C. Martin, Ann. Phys. (NY) 24, 419 (1963). APNYA6 0003-4916 10.1016/0003-4916(63)90078-2
25. This function allows one to relate the dc conductivity to physically meaningful quantities as, for example, the Einstein diffusion coefficient [3].
26. F. M. Peeters and J. T. Devreese, in Solid State Physics, edited by F. Seitz and D. Turnbull, Vol. 38, (Academic, New York, 1984), p. 81.
27. K. M. Van Vliet, J. Math. Phys. 20, 2573 (1979). JMAPAQ 0022-2488 10.1063/1.524020
28. H. Fröhlich, Phys. Rev. 79, 845 (1950). PHRVAO 0031-899X 10.1103/PhysRev.79.845
29. G. De Filippis, V. Cataudella, A. S. Mishchenko, C. A. Perroni, and J. T. Devreese, Phys. Rev. Lett. 96, 136405 (2006) PRLTAO 0031-9007 10.1103/PhysRevLett.96.136405;
30. V. Cataudella, G. De Filippis, and G. Iadonisi, Eur. Phys. J. B 12, 17 (1999) EPJBFY 10.1007/s100510050970 1434-6028;
31. G. Iadonisi, V. Cataudella, G. De Filippis, and D. Ninno, Europhys. Lett. 41, 309 (1998). EULEEJ 10.1209/epl/i1998-00148-y 0295-5075
32. D. C. Langreth and L. P. Kadanoff, Phys. Rev. 133, A1070 (1964) PHRVAO 0031-899X 10.1103/PhysRev.133.A1070;
33. G. D. Mahan, Phys. Rev. 142, 366 (1966) PHRVAO 10.1103/PhysRev.142.366 0031-899X;
34. Y. Osaka, J. Phys. Soc. Japan 35, 381 (1973) JUPSAU 10.1143/JPSJ.35.381 0031-9015;
35. K. K. Thornber and R. P. Feynman, Phys. Rev. B 1, 4099 (1970). 0556-2805 10.1103/PhysRevB.1.4099
36. L. P. Kadanoff, Phys. Rev. 130, 1364 (1963). PHRVAO 0031-899X 10.1103/PhysRev.130.1364
37. D. Sels and F. Brosens, Phys. Rev. E 89, 012124 (2014). PLEEE8 1539-3755 10.1103/PhysRevE.89.012124
38. R. P. Feynman, R. W. Hellwarth, C. K. Iddings, and P. M. Platzman, Phys. Rev. 127, 1004 (1962). PHRVAO 0031-899X 10.1103/PhysRev.127.1004
39. F. M. Peeters and J. T. Devreese, Phys. Rev. B 28, 6051 (1983). PRBMDO 0163-1829 10.1103/PhysRevB.28.6051
40. H. Fröhlich, Proc. R. Soc. Lond. A 160, 230 (1937). 1364-5021 10.1098/rspa.1937.0106
41. H. Fröhlich and N. F. Mott, Proc. R. Soc. London A 171, 496 (1939). We note that within this reference, the equations determining the transport scattering time coincide with Eqs. (28) and (29), except for the factor (Equation presented) that is replaced with (Equation presented). Fröhlich and Mott demonstrated, by using the energy conservation, that the factor (Equation presented) is equivalent to the quantity (Equation presented) [(Equation presented) (Equation presented) for one phonon emission (absorption)], where (Equation presented). On the other hand, the inspection of the Boltzmann equation reveals that the contribution (Equation presented) is correct only in the elastic-scattering problems (where (Equation presented) and, in this case, all the approaches provide the same result). We also note that the term (Equation presented) has been neglected by Frohlich in Ref. [30]. Actually it is not clear which is the appropriate factor in the inelastic-scattering problems, and, in general, it is discarded (in this way, transport scattering time and electron lifetime coincide). Then the problem is open and, so far, not solved. This does not allow one to establish which is the correct perturbative limit for the mobility. Within BWFOC, the appropriate factor both in elastic- and inelastic-scattering problems turns out to be (Equation presented). We note, in particular, that the validity of our approach is not related to the nature of the scattering events. 1364-5021 10.1098/rspa.1939.0080
42. Note the strange fact that in the FHIP [28] treatment, mobility is dominated by phonon emission processes [21].
43. We emphasize that within the SKF, the second vertex correction provides a further factor in Eq. (27) of the order of (Equation presented) [7], in agreement, apart from a numerical factor, with the prediction of WBFOC (we note that WBFOC obtains this result at the lowest order).
44. W. Götze and P. Wölfle, J. Low Temp. Phys. 5, 575 (1971). JLTPAC 0022-2291 10.1007/BF00628186