Advances in Condensed Matter Physics

Advances in Condensed Matter Physics / 2010 / Article
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Phonons and Electron Correlations in High-Temperature and Other Novel Superconductors

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Volume 2010 |Article ID 167985 |

G. A. Ummarino, "The Phenomenology of Iron Pnictides Superconductors Explained in the Framework of -Wave Three-Band Eliashberg Theory", Advances in Condensed Matter Physics, vol. 2010, Article ID 167985, 6 pages, 2010.

The Phenomenology of Iron Pnictides Superconductors Explained in the Framework of 𝑠 -Wave Three-Band Eliashberg Theory

Academic Editor: Igor Mazin
Received25 May 2009
Revised10 Aug 2009
Accepted03 Sep 2009
Published22 Nov 2009


The s-wave three-band Eliashberg theory can simultaneously reproduce the experimental critical temperatures and the gap values of the superconducting materials , and as exponent of the more important families of iron pnictides. In this model the dominant role is played by interband interactions and the order parameter undergoes a sign reversal between hole and electron bands (-wave symmetry). The values of all the gaps (with the exact experimental critical temperature) can be obtained by using high values of the electron-boson coupling constants and small typical boson energies (in agreement with experiments).

The discovery of Fe-based pnictide superconductors [13] has aroused great interest in the scientific community. For the first time noncuprate superconductor shows high critical temperature. In these systems, as in cuprates, the superconductivity occurs upon charge doping of a magnetic parent compound above a certain critical value. The more relevant difference is that in cuprates the parent compound is a Mott insulator with localized charge carriers and a strong Coulomb repulsion between electrons, while in the pnictides it is a bad metal and shows a tetragonal to orthorhombic structural transition below 140 K, followed by the onset of an antiferromagnetic spin-density-wave ordering [4]. Charge doping gives rise to superconductivity and, at the same time, inhibits the occurrence of both the static magnetic order and the structural transition. The Fermi surface consists of two or three hole-like sheets around the point in the first Brillouin zone and two electron-like sheets around point. Up to now, the most intensively studied systems are the 1111 compounds, ReFeAs (Re = La, Sm, Nd, Pr, etc.) and the 122 ones, hole- or electron-doped AFA (A = Ba, Sr, Ca).

At present it is not completely clear what is the microscopic pairing mechanism responsible for superconductivity. The conventional phonon-mediated coupling mechanism is too week and cannot explain the observed high within the standard Migdal-Eliashberg theory [5, 6]. The calculated increases only marginally with the inclusion of multiband effects and remains far from experimental values. On the other hand, the magnetic nature of the parent compound seems to favor a coupling mechanism based on nesting-related antiferromagnetic spin fluctuations [7]. In this case an interband sign reversal of the order parameter between different sheets of the Fermi surface ( symmetry) is predicted. The number, amplitude, and symmetry of the superconducting energy gaps are indeed fundamental physical quantities that any microscopic model of superconductivity has to account for. Experiments with powerful techniques such as ARPES, point-contact spectroscopy, and STM, have been carried out to study the superconducting gaps in pnictides (for a review see [8]). Although the results are sometimes in disagreement with each other, a multigap scenario is emerging with evidence for rather high gap ratios, [8]. A two-band BCS model cannot account either for the amplitude of the experimental gaps or for their ratio. Three-band BCS models have been investigated [911] which can reproduce the experimental gap ratio but not the exact experimental gap values when the experimental is exactly reproduced. In this regard, a reliable study has to be carried out within the framework of the Eliashberg theory for strong coupling superconductors [1218], due to the possible high values of the coupling constants necessary to explain the experimental data.

By using this strong-coupling approach, I show here that the superconducting iron pnictides represent a case of dominant negative interband-channel superconductivity (-wave symmetry) with high values of the electron-boson coupling constants and small typical boson energies. Furthermore I prove that a small contribution of intraband coupling does not significantly affect the obtained results and that the contribution of the Coulomb pseudopotential is negligible.

The electronic structure of pnictides can be approximately reproduced by using a three-band model (Figure 1) with two hole bands (1 and 2) and one equivalent electron band (3) [9]. The -wave order parameters of the hole bands and have opposite sign with respect to that of the electron [7]. In such systems intraband coupling could be provided by phonons while interband coupling by antiferromagnetic spin fluctuations which in a one-band system are always pair breaking but here, in a multiband system, the interband terms can contribute to increase the critical temperature. In the multiband Eliashberg equations the spin fluctuations term in the intraband channel has positive sign for the renormalization functions and negative sign for the superconducting order parameters thus leading to a strong reduction of . However, if we consider negative interband contributions in the equations, the final result can be an increase of the critical temperature [19].

The generalization of the Eliashberg theory [1218] for multiband systems has already been used with success to study the MgB superconductor [2023]. To obtain the gaps and the critical temperature within the -wave, three-band Eliashberg equations, one has to solve six coupled equations for the gaps and the renormalization functions , where is a band index that ranges between and (see Figure 1) and are the Matsubara frequencies. For completeness we included in the equations the nonmagnetic and magnetic impurity scattering rates in the Born approximation, and . In the imaginary-axis formulation [24] the equations are

where , . is the Heaviside function and is a cut-off energy. In particular, , where means “phonon” and “spin fluctuations.” Finally, and .

In the real axis formulation the multiband -wave Eliashberg equations [25, 26] are

where now , . In particular, , and and .

In principle the solution of the three-band Eliashberg equations shown in (1) (or (2)) requires a huge number of input parameters: (i) nine electron-phonon spectral functions, ; (ii) nine electron-SF spectral functions, ; (iii) nine elements of the Coulomb pseudopotential matrix, ; (iv) nine nonmagnetic and nine paramagnetic impurity scattering rates.

It is obvious that a practical solution of these equations requires a drastic reduction in the number of free parameters of the model. According to the work of Mazin et al. [7] I know that (i) that is, phonons mainly provide intraband coupling but the total electron-phonon coupling constant should be very small [5, 6], (ii) , that is, SF mainly provide interband coupling. I include these features in the simplest three-band model by posing , , and . Here, of course, it is . Moreover, I set in (1)-(2) and (3)-(4).

Within these approximations, the electron-boson coupling-constant matrix becomes [9]

where , , and is the normal density of states at the Fermi level for the -band ( according to Figure 1).

I initially solved the Eliashberg equations on the imaginary axis to calculate the critical temperature and, by means of the technique of the Padè approximants [27, 28], to obtain the low-temperature value of the gaps because in presence of a strong coupling interaction or of impurities, the value of obtained by solving the imaginary-axis Eliashberg equations can be very different from the value of obtained from the real-axis Eliashberg equations [29]. I also solved the three-band Eliashberg equations in the real-axis formulation.

I reproduced the critical temperature and the gap values in three representative cases: (i) the compound LaFeAsOF with  K where point-contact spectroscopy measurements gave meV and  meV [30]; (ii) the compound BaKFeAs with K where ARPES measurements gave  meV,  meV, and  meV [31]; (iii) the compound SmFeAsOF with  K where, according to point-contact spectroscopy measurements,  meV and  meV [32, 33].

Inelastic neutron-scattering experiments suggest that the typical boson energy possibly responsible for superconductivity ranges roughly between 10 and 30 meV [34, 35]. In the numerical simulations I used spectral functions with Lorentzian shape, that is, , where , are the normalization constants necessary to obtain the proper values of while and are the peak energies and half-widths, respectively. In all the calculations I always set , with ranging between 5 and 35 meV and  meV. The cut-off energy is and the maximum quasiparticle energy is .

In the  K case I know that and [36] while in the  K case and [9] and in the  K case I have and [37]. Once the energy of the boson peak is set, only two free parameters are left in the model: and .

By properly selecting the values of these parameters it is relatively easy to obtain the experimental values of the critical temperature and of the small gap, which is well known. It is more difficult to reproduce the values of the large gaps of bands 1 and 3 since, due to the high ratio (of the order of 8-9), high values of the coupling constants and small boson energies are required. Figures 2, 3, and 4 show the values of the calculated gaps (full symbols, left axis) as a function of the boson peak energy, . The corresponding values of and , chosen in order to reproduce the values of and of the small gap, , are also shown in the figure (open symbols, right axis). In all materials examined, only when  meV the values of the large gap correspond to the experimental data. Indeed, when increases, the values of and strongly decrease. As a consequence, a rather small energy of the boson peak together with a very strong coupling (particularly in the 3-1 channel) is needed in order to obtain the experimental and the correct gap values. In this regard, it is worth noticing that the absolute values of the large gaps cannot be reproduced in a interband-only, two-band Eliashberg model [38], as well as within a three-band BCS model. In the latter case it is only possible to obtain a ratio of the gaps close to the experimental one [10, 11].

I also tested the effect into the model of a small intraband coupling (possibly of phonon origin). In the case of BaKFeAs ( K) I fixed  meV and since I know indeed that this coupling cannot be very high [5, 6]. Then I determined the free parameters and in order to obtain  K. It might be thought that this term can sensibly contribute to increase the gap values but, as can be seen in Table 1, this is not the case as the gap values only show a slight increase.

Pure interband  meV
Free parameters
Results  meV  meV  meV
Intraband  meV
Free parameters
Results  meV  meV  meV
Intraband and Coulomb  meV
Free parameters
Results  meV  meV  meV

The effect of Coulomb interaction was also investigated for the case shown in Table 1 where a weak intraband coupling is included. I chose and, as expected, I found that the intraband Coulomb pseudopotential has a negligible effect while the interband one [19] strongly contributes to raise and reduces in a considerable way [24] the value of . In this case, as shown in Table 1, it is only possible to obtain the correct value of the small gap because the electron-boson coupling is now too small and it is impossible to reproduce the value of the big gap. As a consequence, this result seems to exclude a strong interband Coulomb interaction in these compounds.

I also calculated the superconductive density of states in all three cases. The parameters used for the K case are  meV, , and , for the  K case  mev, , and and for  K case  mev, , and .

The value of coupling constant is in the range 1–6 and this fact, at a first glance, may seem very unusual but these systems have some peculiarities in common with the heavy fermions superconductors. For example, in the compound LaFeAsOF, the normal state at is asymmetric and pseudogapped, with two broad maxima that are progressively smoothed out on increasing the temperature [30]. This shape is very similar to that observed by point contact spectroscopy in materials with long-range spin-density-wave order, like URuSi [39, 40]. The calculated superconductive normalized conductances are shown in Figure 5; the presence of a hump at is a typical strong coupling effect [41]. This feature, of course, is more evident when is bigger. By cause of thermal broadening it is impossible to separate the peaks of the gaps and .

The penetration depth as a function of temperature has been calculated in the three cases and is reported in Figure 6. It is in qualitative agree with the experimental data [42]. The inset shows the behaviour of at low temperature. Although at sufficiently low temperature an exponential fit may be certainly possible the inset shows that, on a larger range (up to ) these curves can be best fitted by a third-order polynomial as experimentally observed [42].

In conclusion, I have shown that the newly discovered iron pnictides very likely represent a case of dominant negative interband-channel pairing superconductivity where an electron-boson coupling, such as the electron-spin fluctuactions one, can become a fundamental ingredient to increase in a multiband strong-coupling picture. In particular, the present results prove that a simple three-band model in strong-coupling regime can reproduce in a quantitative way the experimental and the energy gaps of the pnictide superconductors with only two free parameters, and , provided that the typical energies of the spectral functions are of the order of 10 meV and the coupling constants are very high ().


The author thanks I. I. Mazin for useful discussions, Mauro Tortello and Dario Daghero for manuscript revision.


  1. Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, “Iron-based layered superconductor La[O1xFx]FeAs(x=0.050.12) with Tc=26K,” Journal of the American Chemical Society, vol. 130, no. 11, pp. 3296–3297, 2008. View at: Publisher Site | Google Scholar
  2. Z.-A. Ren, W. Lu, J. Yang et al., “Superconductivity at 55 K in iron-based F-doped layered quaternary compound Sm[O1xFx]FeAs,” Chinese Physics Letters, vol. 25, no. 6, pp. 2215–2216, 2008. View at: Publisher Site | Google Scholar
  3. M. Rotter, M. Tegel, and D. Johrendt, “Superconductivity at 38 K in the iron arsenide (Ba1xKx)Fe2As2,” Physical Review Letters, vol. 101, no. 10, Article ID 107006, 2008. View at: Publisher Site | Google Scholar
  4. C. de La Cruz, Q. Huang, J. W. Lynn et al., “Magnetic order close to superconductivity in the iron-based layered LaO1xFxFeAs systems,” Nature, vol. 453, no. 7197, pp. 899–902, 2008. View at: Publisher Site | Google Scholar
  5. L. Boeri, O. V. Dolgov, and A. A. Golubov, “Is LaFeAsO1xFx an electron-phonon superconductor?” Physical Review Letters, vol. 101, no. 2, Article ID 026403, 4 pages, 2008. View at: Publisher Site | Google Scholar
  6. L. Boeri, O. V. Dolgov, and A. A. Golubov, “Electron-phonon properties of pnictide superconductors,” Physica C, vol. 469, no. 9–12, pp. 628–634, 2009. View at: Publisher Site | Google Scholar
  7. I. I. Mazin, D. J. Singh, M. D. Johannes, and M. H. Du, “Unconventional superconductivity with a sign reversal in the order parameter of LaFeAsO1xFx,” Physical Review Letters, vol. 101, no. 5, Article ID 057003, 2008. View at: Publisher Site | Google Scholar
  8. Physica C 469, 2009, Special Issue on Pnictides.
  9. I. I. Mazin and J. Schmalian, “Pairing symmetry and pairing state in ferropnictides: theoretical overview,” Physica C, vol. 469, no. 9–12, pp. 614–627, 2009. View at: Publisher Site | Google Scholar
  10. L. Benfatto, M. Capone, S. Caprara, C. Castellani, and C. di Castro, “Multiple gaps and superfluid density from interband pairing in a four-band model of the iron oxypnictides,” Physical Review B, vol. 78, no. 14, Article ID 140502, 4 pages, 2008. View at: Publisher Site | Google Scholar
  11. E. Z. Kuchinskii and M. V. Sadovskii, “Multiple bands: a key to high-temperature superconductivity in iron arsenides?” Journal of Experimental and Theoretical Physics Letters, vol. 11, no. 3, pp. 156–160, 2009. View at: Publisher Site | Google Scholar
  12. G. M. Eliashberg, “Interactions between electrons and lattice vibrations in a superconductor,” Soviet Physycs. JETP Letters, vol. 11, no. 3, pp. 696–702, 1960. View at: Google Scholar
  13. D. J. Scalapino, “The Electron-phonon interaction and strong-coupling superconductors,” in Superconductivity, R. D. Parks, Ed., p. 449, Marcel Dekker, New York, NY, USA, 1969. View at: Google Scholar
  14. J. P. Carbotte, “Properties of boson-exchange superconductors,” Reviews of Modern Physics, vol. 62, no. 4, pp. 1027–1157, 1990. View at: Publisher Site | Google Scholar
  15. P. B. Allen and B. Mitrovich, “Theory of superconducting Tc,” in Solid State Physics, H. Ehrenreich, F. Seitz, and D. Turnbull, Eds., vol. 37, Academic Press, New York, NY, USA, 1982. View at: Google Scholar
  16. F. Marsiglio and J. P. Carbotte, “Electron-phonon superconductivity,” in The Physics of Conventional and Unconventional Superconductors, K. H. Bennemann and J. B. Ketterson, Eds., p. 233, Springer, New York, NY, USA, 2003. View at: Google Scholar
  17. F. Marsiglio, “Eliashberg theory of the critical temperature and isotope effect. Dependence on bandwidth, band-filling, and direct Coulomb repulsion,” Journal of Low Temperature Physics, vol. 87, no. 5-6, pp. 659–682, 1992. View at: Publisher Site | Google Scholar
  18. C. R. Leavens and E. Talbot, “Effect of thermal phonons on the superconducting transition temperature,” Physical Review B, vol. 28, no. 3, pp. 1304–1313, 1983. View at: Publisher Site | Google Scholar
  19. G. A. Ummarino, “Effects of magnetic impurities on two-band superconductor,” Journal of Superconductivity and Novel Magnetism, vol. 20, no. 7-8, pp. 639–642, 2007. View at: Publisher Site | Google Scholar
  20. S. V. Shulga, S.-L. Drechsler, G. Fuchs et al., “Upper critical field peculiarities of superconducting YnI2B2C and LuNi2B2C,” Physical Review Letters, vol. 80, no. 8, pp. 1730–1733, 1998. View at: Google Scholar
  21. S. D. Adrian, S. A. Wolf, O. V. Dolgov, S. Shulga, and V. Z. Kresin, “Density of states and the energy gap in superconducting cuprates,” Physical Review B, vol. 56, no. 13, pp. 7878–7881, 1997. View at: Publisher Site | Google Scholar
  22. G. A. Ummarino, R. S. Gonnelli, S. Massidda, and A. Bianconi, “Two-band Eliashberg equations and the experimental Tc of the diboride Mg1-xAIxB2,” Physica C, vol. 407, no. 3-4, pp. 121–127, 2004. View at: Publisher Site | Google Scholar
  23. E. J. Nicol and J. P. Carbotte, “Properties of the superconducting state in a two-band model,” Physical Review B, vol. 71, no. 5, Article ID 054501, 18 pages, 2005. View at: Publisher Site | Google Scholar
  24. G. A. Ummarino, “Iron-based layered compounds: the effect of negative interband coupling,” Journal of Superconductivity and Novel Magnetism, vol. 22, no. 6, pp. 603–607, 2009. View at: Publisher Site | Google Scholar
  25. O. V. Dolgov, R. K. Kremer, J. Kortus, A. A. Golubov, and S. V. Shulga, “Thermodynamics of two-band superconductors: the case of MgB2,” Physical Review B, vol. 72, no. 2, Article ID 024504, 2005. View at: Publisher Site | Google Scholar
  26. D. Parker, O. V. Dolgov, M. M. Korshunov, A. A. Golubov, and I. I. Mazin, “Extended s± scenario for the nuclear spin-lattice relaxation rate in superconducting pnictides,” Physical Review B, vol. 78, no. 13, Article ID 134524, 5 pages, 2008. View at: Publisher Site | Google Scholar
  27. H. J. Vidberg and J. W. Serene, “Solving the Eliashberg equations by means of N-point Padé approximants,” Journal of Low Temperature Physics, vol. 29, no. 3-4, pp. 179–192, 1977. View at: Publisher Site | Google Scholar
  28. C. R. Leavens and D. S. Ritchie, “Extension of the N-point Padé approximants solution of the Eliashberg equations to TTC,” Solid State Communications, vol. 53, no. 2, pp. 137–142, 1985. View at: Google Scholar
  29. G. A. Ummarino and R. S. Gonnelli, “Real-axis direct solution of the d-wave Eliashberg equations and the tunneling density of states in optimally doped Bi2Sr2CaCu2O8+x,” Physica C, vol. 328, no. 3, pp. 189–194, 1999. View at: Publisher Site | Google Scholar
  30. R. S. Gonnelli, D. Daghero, M. Tortello et al., “Coexistence of two order parameters and a pseudogaplike feature in the iron-based superconductor LaFeAsO1xFx,” Physical Review B, vol. 79, no. 18, Article ID 184526, 11 pages, 2009. View at: Publisher Site | Google Scholar
  31. H. Ding, P. Richard, K. Nakayama et al., “Observation of Fermi-surface-dependent nodeless superconducting gaps in Ba0.6K0.4Fe2As2,” Europhysics Letters, vol. 83, no. 4, Article ID 47001, 2008. View at: Publisher Site | Google Scholar
  32. D. Daghero, M. Tortello, R. S. Gonnelli, V. A. Stepanov, N. D. Zhigadlo, and J. Karpinski, “Evidence for two-gap nodeless superconductivity in SmFeAsO1xFx from point-contact Andreev-reflection spectroscopy,” Physical Review B, vol. 80, no. 6, 2009. View at: Publisher Site | Google Scholar
  33. D. Daghero, M. Tortello, R. S. Gonnelli et al., “Point-contact Andreev-reflection spectroscopy in ReFeAsO1xFx(Re=La,Sm): possible evidence for two nodeless gaps,” Physica C, vol. 469, no. 9–12, pp. 512–520, 2009. View at: Publisher Site | Google Scholar
  34. A. D. Christianson, E. A. Goremychkin, R. Osborn et al., “Unconventional superconductivity in Ba0.6K0.4Fe2As2 from inelastic neutron scattering,” Nature, vol. 456, no. 7224, pp. 930–932, 2008. View at: Publisher Site | Google Scholar
  35. R. Osborn, S. Rosenkranz, E. A. Goremychkin, and A. D. Christianson, “Inelastic neutron scattering studies of the spin and lattice dynamics in iron arsenide compounds,” Physica C, vol. 469, no. 9–12, pp. 498–506, 2009. View at: Publisher Site | Google Scholar
  36. M. V. Sadovskii, “High-temperature superconductivity in iron-based layered iron compounds,” Physics-Uspekhi, vol. 51, no. 12, pp. 1201–1227, 2008. View at: Publisher Site | Google Scholar
  37. I. I. Mazin, private communication.
  38. O. V. Dolgov, I. I. Mazin, D. Parker, and A. A. Golubov, “Interband superconductivity: contrasts between Bardeen-Cooper-Schrieffer and Eliashberg theories,” Physical Review B, vol. 79, no. 6, Article ID 060502, 2009. View at: Publisher Site | Google Scholar
  39. K. Hasselbach, J. R. Kirtley, and P. Lejay, “Point-contact spectroscopy of superconducting URu2Si2,” Physical Review B, vol. 46, no. 9, pp. 5826–5829, 1992. View at: Publisher Site | Google Scholar
  40. R. Escudero, F. Morales, and P. Lejay, “Temperature dependence of the antiferromagnetic state in URu2Si2 by point-contact spectroscopy,” Physical Review B, vol. 49, no. 21, pp. 15271–15275, 1994. View at: Publisher Site | Google Scholar
  41. G. A. Ummarino, R. S. Gonnelli, and D. Daghero, “Tunneling conductance of SIN junctions with different gap symmetries and non-magnetic impurities by direct solution of real-axis Eliashberg equations,” Physica C, vol. 377, no. 3, pp. 292–303, 2002. View at: Publisher Site | Google Scholar
  42. R. Prozorov, M. A. Tanatar, R. T. Gordon et al., “Anisotropic London penetration depth and superfluid density in single crystals of iron-based pnictide superconductors,” Physica C, vol. 469, no. 9–12, pp. 582–589, 2009. View at: Publisher Site | Google Scholar

Copyright © 2010 G. A. Ummarino. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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