EUROPHYSICS LETTERS

10 November 1994

Europhys. Lett., 28 (5), pp. 351-356 (1994)

Superconductivity beyond Migdal’s Theorem and High-Tc Phenomenology. P. BENEDETTI (*), C. GRIMALDI (**), L. PIETRONERO (*) and G. VARELOGIANNIS (*) (*) Dipartimento di Fisica, Universit& degli Studi di Roma .La Sapienzau Piazzale Aldo Mom 2, 00185 Roma, Italy (**) Max-Planck Institute - Heissenbergstrasse 1, 70569 Stuttgart, Germany (received 12 August 1994; accepted in final form 4 October 1994)

PACS. 74.20 - Theory. PACS. 74.20F - BCS theory and ita applications.

Abstract. - Many properties of the superconducting state of high-T, materials have been interpreted within the Eliashberg framework with a very large coupling ( A 2 3). Here we point out that the inclusion of non-adiabatic corrections acta as an effective coupling for various properties that can, therefore, be reproduced with a much smaller coupling ( A = 1) within the generalized theory. This can provide a new perspective for the analysis of superconductivitydata with moderate couplings but within a theory that goes beyond Migdal’s theorem.

While initially the high-T, superconducting state was interpreted as completely exotic, the accumulation of accurate experimental data points now to a situation corresponding to a relatively conventional superconducting state. The proposed models for nonconventional superconductivity are in general motivated by anomalies in the normal-state properties [l]. The extension of those concepts to the superconducting state is not obvious and the conventional Eliashberg theory [2] remains the only complete framework for the analysis of the superconducting properties. Several groups have investigated the relevance of Eliashberg theory for the description of the high-temperature superconducting state [3-’71. A large part of the high-T, superconducting phenomenology is now reasonably well understood within the conventional Eliashberg framework of strong-coupling boson exchange superconductivity. Some characteristic anomalies with respect to the conventional BCS-like behaviour can be considered as simple effects of a stronger electron-boson coupling. The absence of coherence effects [8,9] and other parts of the behaviour of the low-energy dynamics of YBa&u30, can be considered as evidence for conventional superconductivity [10,111. Characteristic anomalies in the tunnelling and photoemission data of Bi2Sr2CaCu208like a diplike structure and a second-peak structure can also be obtained within Eliashberg theory[121. The understanding of the previously cited anomalies within the Eliashberg framework requires a very strong electron-boson coupling A 3 3. The gap ratio values [131 and the disagreement of the gap measurements by infrared spectroscopy with the other gap measurements in YBa$2u307[14] can also be understood with the same couplings. Therefore,

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the Eliashberg theory description of those parts of the phenomenology appears to be coherent. Nevertheless important questions remain open, in particular concerning the symmetry of the pairing, or the behaviour of the scattering rate near Tc (for this last point see, for example, the introduction in [lo]). Such questions can only be clarified when contradictions in the experimental data will disappear. Considering the characteristic dip-like and second-peak structures in the quasi-particle spectrum reported from tunnelling and photoemission, they provide a strong indication that the system is at the beginning of a crossover from BCS superconductivity to Bose condensation [U]. On the other hand, for couplings of the order of A = 3 there are clear signs withing Eliashberg theory that the crossover from BCS superconductivity might be appreciable and the need to go beyond Migdal’s theorem is manifest [15]. In addition, for those couplings there are also signs of a critical fluctuation regime [ll,151. This last point is surprising since Eliashberg theory is a mean-field theory. Another reason to consider deviations from Migdal’s theorem even if the electron-boson coupling is weak is the fact that the boson energies are not negligible compared to the electronic energies. This last situation is certainly realized in fullerides and non-adiabatic corrections have been proposed to be potentially relevant for the understanding of high-Tc superconductivity in those materials [16]. In addition, the data of Uemura and collaborators [17] and various other experiments point to a generalization of this situation also to the oxides. Some insight about the question of the robustness of Eliashberg theory can be found considering first-order corrections to Migdal’s theorem. Since the crossover from BCS superconductivity is expected to be smooth, those first-order corrections are expected to be the most relevant at the beginning of this crossover where there is good evidence that high-T, materials are situated [12,15,18]. On the other hand, although Eliashberg theory appears to be robust, first-order non-adiabatic corrections lead to important effects. Under some conditions they can lead, for example, to a significant enhancement of T, E191 compared to the conventional Eliashberg framework. This situation can provide a new perspective to phonon-mediated superconductivity. Recently there have been detailed studies of the first-order non-adiabatic corrections to Migdal’s theorem [19,20]. In the critical region the Eliashberg equations can be generalized in the following way:

where w o is the energy of the considered phonon Einstein spectrum, and the adiabatic corrections were included in the effective coupling functions A A and A Z defined as follows: A ~ ( i w n i,w m ; Qc) =

+ 2 f l ~ ( i w n ,i w m ; Qc) + f l c ( i w n , i w m ; Qd1,

A ~ ( i w , ,i w m ;

Qc)

=A[1+

flv(iw,, i w m ;

&,)I.

(3) (4)

The functions Pv and P, refer to the first-order adiabatic corrections in the diagonal and off-diagonal sectors, respectively, and A is the electron-phonon coupling in the conventional Eliashberg-theory framework.

P. BENEDETTI

et al.: SUF'ERCONDUCTMTY

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The function Pv corresponds to the vertex correction term and it depends in a complex way on the exchanged energy ( w ) and momentum (q). In particular, it is positive for small-q scattering and negative for large-q scattering. A similar behaviour can be obtained for the function P, that corresponds to the cross-phonon diagram [19,20]. This situation implies that the effect of these non-adiabatic terms will be strongly dependent on the structure of the electron-phonon coupling as a function of q. In particular, the existence of an upper cut-off q, for the scattering will lead to mainly positive contributions and a corresponding enhancement of T, For this reason we discuss our results as a function of this momentum cut-off that can arise from different physical reasons. Therefore we consider a momentum-independent scattering up to a cut-off Q, = qc/ 2 b , and in the case of half-filling E = 2Ef (for the detailed model considered, the approximations and the specific structure of the functions Pv and P, see ref. [20]). From those equations it is possible to obtain the critical temperature T , . An interesting remark can be done about the behaviour of this quantity. The frequency structure of the considered adiabatic corrections appears to be marginal, compared, in particular, to the momentum cut-off dependence of T,. It is, therefore, possible for certain properties to neglect this frequency structure and the coupling functions Ad and AZ can be considered as effective couplings. Then, if the considered corrections Pv and P, are positive, one obtains an enhancement of T , . This is the case, in particular, when the momentum cut-off is rather low[19], a characteristic that arises naturally if one includes the effect of electronic correlations in the electron-phonon coupling [19-23]. For example, in fig. 1 we show the behaviour of T , / w o as a function of A for the usual Eliashberg theory compared with the non-adiabatic generalized theory. This is presented with the full frequency dependence and with the approximation of effective couplings. The comparison between the two sets of data gives an idea of the validity of the effective-coupling approximation. Then the function Pv and P, are only dependent on the momentum cut-off Q, and on the ratio m = wo/Ef. In the approximation of effective couplings the non-adiabatic corrections are considered as frequency independent and eqs. (1) and (2) have a form very similar to the conventional Eliashberg equations. This is clearer for weak couplings where the renormalization function Z(iw,) can be replaced by the unity in the module of arctg in eqs. (1)and (2). On the other

.

Fig. 1. - Behaviour of T,/o0as a function of the coupling A for various theoretical schemes (m= 0.2, Q, = 0.1). The stars correspond to the Eliashberg limit (m4 0 ) . The other two cases correspond to two different approximation schemes for the generalized Eliashberg equations that include the first non-adiabatic effects. The black dots correspond to a numerical solution taking into account the full frequency dependence while the squares correspond to the approximation of the effective couplings.

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hand the effective couplings AA(m,Q,) and AZ(m,Q,) are different in the diagonal and off-diagonal sector (eqs. (1) and (2), respectively). When both non-adiabatic contributions are positive, AA > A Z and the effects on the superconducting parameters of the additional couplings are expected to be amplified. As has been discussed in detail in[20], the parameters Pv and P, can take significant positive values especially when Q, << 1. This last situation can have important implications. In fact we have seen in the introduction that a large part of the superconducting phenomenology can be understood within Eliashberg theory provided that the couplings are of the order of A = 3 or higher. However, if we used instead the generalized theory with non-adiabatic effects we would actually reproduce some of the essential properties with a much smaller coupling. For example, for the critical temperature, a coupling of the order of A = 3 is only apparent if non-adiabatic corrections are important, and the corresponding real coupling is much lower if the corrections are positive. We illustrate in the following how one can reinterpret the superconducting phenomenology when non-adiabatic effects beyond hfigdal’s theorem are included. Let us consider, for example, the case where T,= 0 . 1 2 0 ~ .Within the conventional Eliashberg framework this corresponds to A = 1. When non-adiabatic corrections are taken into account, the analysis is more complex. To a given critical temperature we cannot associate a given value of the coupling but a function A(m, Q,). We show in fig. 2 the couplings A which within the generalized equations (1)-(4) give the same critical temperature T,= 0 . 1 2 0 ~ .We give these last couplings as a function of Migdal‘s ratio m = w o /Ef for different values of the momentum cut-off Q, .We can see that for realistic values of Migdal’s ratio m = 0.1 or 0.2 and for small Q, we can obtain the same T,with a coupling two times lower than that needed in the adiabatic case. In fact we see in fig. 2 that if m = 0.1 and Q, = 0.1, A 2: 0.45 gives within the generalized framework the same T, as A = 1 within the adiabatic framework. We then consider in fig. 3 the case T,= 0 . 2 8 ~ Within ~. the conventional adiabatic framework this corresponds to A = 3. Solving eqs. (1)-(4) in the optimal situation m = 0.1 and Q, = 0.1, it is possible to reproduce the same critical temperature with a coupling strength three times smaller (A = 1). We remark here that eqs. (1)-(4) are relevant for rather weak real couplings A S 1 [19,20]; that is why the A > 1 part of fig. 3 must be considered with 1.0

a

0.9 0.8

0.7 0.6

0.5 0.0

0.2

0.4

0.6

0.5 0.0

0.8 m 1.0

Fig. 2.

0.2

0.4

0.6

0.8 m 1 0

Fig. 3.

Fig. 2. - We consider a situation in which T , / w , = 0.12 (A = 1 in Eliashberg theory) and study which coupling A corresponds to this value in the generalized theory as a function of the Migdal parameter m, for different values of the momentum cut-off Q, We can observe that the generalized theory requires much smaller couplings than the usual Eliashberg theory in order to lead to the same T , . Fig. 3. - Same as in fig. 2 but with T , / w o = 0.28 (A = 3 in Eliashberg theory). Again the generalized theory requires a much smaller value of A, up to a factor of three, to produce the same value of T , .

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caution. In particular, the fact that for m > 0.7 it appears that within the generalized framework we need a coupling higher than that needed in the adiabatic case (A = 3) is just due to the non-validity of eqs. (1)-(4)for those couplings. The whole deformation of the curves in fig. 3 with respect to those in fig. 2 reflects precisely the limits of validity of the approach considered here. However, even with A S 1 the generalized framework can lead to a phenomenology which, within the conventional framework, corresponds to A = 3. In summary the small Fermi energy observed in all high-T, superconductors implies a breakdown of Migdal's theorem and a corresponding generalization of Eliashberg equations [19,20]. In general one has a much more complex situation with additional frequencydependent terms. For certain properties, however, like T, it is plausible to average over frequencies and consider the non-adiabatic effects as an effective change of the coupling. This provides a new perspective of the analysis that has been made of the superconducting phenomenology using Eliashberg equations with a very strong coupling. In particular: 1) Within the generalized theory what appears to be A = 3 from standard Eliashberg theory can actually be reproduced with a much smaller coupling (A = 1). 2) Non-adiabatic effects should lead to different corrections for different properties. At the simplest level this situation may manifest as apparent couplings that are different for different properties. However new qualitative behaviours are also expected for some properties like the isotope effect [19,20]and the dynamic properties of the normal state that one should investigate. 3) The fact that non-adiabatic corrections may act for some properties as an effective coupling may also give a clue to understand the robustness of Eliashberg theory even at couplings for which the scheme should become inconsistent [151.

*** GV acknowledges support from the Human Capital and Mobility program of EEC under contract ERBCHBICT930906.

REFERENCES

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