Strong-Coupling Superconductivity. I

D. J. Scalapino; J. R. Schrieffer; J. W. Wilkins

doi:10.1103/PhysRev.148.263

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Strong-Coupling Superconductivity. I

D. J. Scalapino^*, J. R. Schrieffer, and J. W. Wilkins^†

Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania

^*Alfred P. Sloan Foundation Fellow.
^†Present address: Department of Physics, Cornell University, Ithaca, New York.

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Phys. Rev. 148, 263 – Published 5 August, 1966

DOI: https://doi.org/10.1103/PhysRev.148.263

Abstract

The pairing theory of superconductivity is extended to treat systems having strong electron-phonon coupling. In this regime the Landau quasiparticle approximation is invalid. In the theory we treat phonon and Coulomb interactions on the same basis and carry out the analysis using the nonzero-temperature Green's functions of the Nambu formalism. The generalized energy-gap equation thus obtained is solved (at $T = 0$ °K) for a model which closely represents lead and the complex energy-gap parameter $Δ (ω)$ ) is plotted as a function of energy for several choices of phonon and Coulomb interaction strengths. An expression for the single-particle tunneling density of states is derived, which, when combined with $Δ (ω)$ , gives excellent agreement with experiment, if the phonon interaction strength is chosen to give the observed energy gap $Δ_{0}$ at zero temperature. The tunneling experiments therefore give a detailed justification of the phonon mechanism of superconductivity and of the validity of the strong-coupling theory. In addition, by combining theory and the tunneling experiments, much can be learned about the electron-phon interaction and the phonon density of states. The theory is accurate to terms of order the square root of the electron-ion mass ratio, ${\sim 10}^{- 2}$ - $10^{- 3}$ .

References (30)

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[c1] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957)

[c2] L. D. Landau, Zh. Eksperim. i Teor. Fiz. 30, 1058 (1956)

[c3] J. Bardeen and J. R. Schrieffer, Progress in Low Temperature Physics (North-Holland Publishing Company, Amsterdam, 1961), Vol. III

[c4] G. M. Eliashberg, Zh. Eksperim. i Teor. Fiz. 38, 966 (1960) [English transl.: Soviet Phys.—JETP 11, 696 (1960)]

[c5] H. Fröhlich, Phys. Rev. 79, 845 (1950)

[c6] J. Bardeen and D. Pines, Phys. Rev. 99, 1140 (1955)

[c7] N. N. Bogoliubov, Zh. Eksperim. i Teor. Fiz. 34, 58 (1958) [English transl.: Soviet Phys.—JETP 7, 41 (1958)]

[c8] A. B. Migdal, Zh. Eksperim. i Teor. Fiz. 34, 1438 (1958) [English transl.: Soviet Phys.—JETP 7, 996 (1958)]

[c9] Y. Nambu, Phys. Rev. 117, 648 (1960)

[c10] P. Morel and P. W. Anderson [Phys. Rev. 125, 1263 (1962)] G. J. Culler, B. C. Fried, R. W. Huff, and J. R. Schrieffer, Phys. Rev. Letters 8, 399 (1962)

[c11] J. R. Schrieffer, D. J. Scalapino, and J. W. Wilkins, Phys. Rev. Letters 10, 336 (1963)

[c12] L. P. Gorkov, Zh. Eksperim. i Teor. Fiz. 34, 735 (1958) [English transl.: Soviet Phys.—JETP 7, 505 (1958)]

[c13] R. E. Prange, Phys. Rev. 129, 2495 (1963)

[c14] D. J. Scalapino, Y. Wada, and J. C. Swihart, Phys. Rev. Letters 14, 502 (1964)

[c15] T. M. Rice (private communication)

[c16] A. J. Bennett (to be published)

[c17] D. Markowitz and L. P. Kadanoff, Phys. Rev. 131, 563 (1963)

[c18] I. Giaever, H. R. Hart, Jr., and K. Megerle, Phys. Rev. 126, 941 (1962)

[c19] J. M. Rowell, A. G. Chynoweth, and J. C. Phillips, Phys. Rev. Letters 9, 59 (1962)

[c20] J. Bardeen, Phys. Rev. Letters 6, 57 (1961)

[c21] M. H. Cohen, L. M. Falicov, and J. C. Phillips, Phys. Rev. Letters 8, 316 (1962)

[c22] W. A. Harrison, Phys. Rev. 123, 85 (1961)

[c23] L. Esaki and P. J. Stiles, Phys. Rev. Letters 14, 902 (1965) W. L. McMillan J. M. Rowell (private communication)

[c24] B. N. Brockhouse, T. Arase, G. Caglioti, K. R. Rao, and A. D. B. Woods, Phys. Rev. 128, 1099 (1962)

[c25] D. J. Scalapino, Y. Wada, and J. C. Swihart, Phys. Rev. Letters 14, 502 (1964) J. C. Swihart, D. J. Scalapino, and Y. Wada, ibid. 14, 106 (1965)

[c26] W. L. McMillan and J. M. Rowell, Phys. Rev. Letters 14, 108 (1965)

[c27] J. M. Rowell, P. W. Anderson, and D. E. Thomas, Phys. Rev. Letters 10, 334 (1963)

[c28] D. J. Scalapino and P. W. Anderson, Phys. Rev. 133, A921 (1964) D. J. Scalapino, Rev. Mod. Phys. 36, 205 (1964)

[c29] M. Fibich, Phys. Rev. Letters 14, 561 (E) (1965) ibid.14, 621 (E) (1965)

[c30] V. Anbegoakar and L. Tewordt, Phys. Rev. 134, 805 (1964) V. Ambegoakar and J. Woo, ibid. 139, A1818 (1965)

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