Elsevier

Physics Reports

Volume 338, Issues 1–2, November 2000, Pages 1-264
Physics Reports

Interplay of electron–phonon interaction and strong correlations: the possible way to high-temperature superconductivity

https://doi.org/10.1016/S0370-1573(00)00008-9Get rights and content

Abstract

The pairing mechanism in high-Tc-superconductors (HTS) is still, 13 years after the discovery of HTS, under dispute. However, there are experimental evidences that the electron–phonon (E–P) interaction together with strong electronic correlations plays a decisive role in the formation of the normal state and superconductivity. Tunneling spectroscopy shows clear phonon features in the conductance and together with infrared and Raman optic measurements give strong support for the electron–phonon interaction as the pairing mechanism in HTS oxides. The tunneling experiments show also that almost all phonons contribute to the pairing interaction and the E–P interaction is sufficiently large to produce Tc∼100 K. The strong E–P interaction is due to (a) the layered and almost ionic-metallic structure of HTS oxides; (b) the almost two-dimensional motion of conduction carriers, which give rise to large contribution of the Madelung energy in the E–P interaction, especially for axial phonons.

On the other hand, a variety of phase-sensitive measurements give support for d-wave pairing in HTS oxides, which has been usually interpreted to be due to the spin-fluctuation mechanism.

We argue in this review that contrary to low-Tc-superconductors (LTS), where the phonon mechanism leads to s-wave pairing, strong electronic correlations in HTS oxides renormalize the electron–phonon (E–P) interaction, as well as other electron–boson scattering processes related to charge fluctuations, in such a way that the forward scattering peak (FSP) appears, while the backward scattering is suppressed. The FSP mechanism is also supported by the long-range Madelung E–P interaction and the former is pronounced for smaller hole doping δ⪡1.

The renormalization of the E–P interaction and other charge scattering processes (like impurity scattering) by strong correlations gives rise to (i) a significant (relative) increase of the coupling constant for d-wave pairing λd making λdλs for δ≤0.2, where λs is the coupling for s-wave pairing. The residual Coulomb repulsion between quasiparticles (or the interaction via spin fluctuations, which is peaked in the “backward” scattering at Q≈(π,π)) triggers the system to d-wave pairing, while Tc is dominantly due to the E–P interaction; (ii) a reduction (with respect to the pairing coupling constant λ) of the transport E–P coupling constant λtr(≲λ/3), i.e. to the quenching of the resistivity ρ(T) where ρλtrT for T>ΘD/5; (iii) a suppression of the residual quasiparticle scattering on nonmagnetic impurities; (iv) robustness of d-wave pairing in the presence of nonmagnetic impurities and (v) nonadiabatic corrections to the E–P interaction and accordingly to a possible increase of Tc in systems with ωDEF.

Furthermore, the development of the forward scattering peak in the E–P interaction of the optimally hole-doped HTS oxides gives rise, besides the d-wave superconductivity, also to (a) the small isotope effect; and (b) the strong temperature dependence of the gap anisotropy.

In the overdoped oxides the FSP mechanism and spin fluctuations are suppressed which leads to (a) anisotropic s-wave pairing with moderate gap anisotropy, and (b) an increase of the isotope effect.

Introduction

The remarkable discovery by Bednorz and Müller of high-temperature superconductivity (HTS) in the Ba–La–Cu–O system with Tc≈30K [1] has opened a new era in the solid-state physics, not only because of the high critical temperature Tc in HTS oxides and its possible application, but also because of diversity of peculiar transport, magnetic, thermal and superconducting properties. During this period the critical temperature is drastically increased from Tc≈30K up to Tc≈134K in mercurocuprates as it is seen in Table 1.1.

In this period many experimental results were published studying transport, magnetic and lattice properties of HTS materials in superconducting and non-superconducting state. Several review articles were also published – see for instance [3], [4], [7], [223], [8], [10], [11], [12], [13] – which deal with various properties of HTS systems related to magnetic excitations [7], [8], [9], [11], [12] to lattice effects [3], [4], [5], [13] and to optical properties [74]. Therefore, it is not our aim to cover here all these subjects in detail but it is more to elucidate an interplay of the rather strong electron–phonon (E–P) interaction and strong electronic correlations (SEC) in HTS oxides. One should also stress that in the past most of the theoretical articles and reviews have dealt with strong correlations and one of their manifestations, magnetic fluctuations as a possible origin for peculiar normal-state properties and for the pairing mechanism in HTS systems, while much less studies were related to the E–P mechanism of pairing. This disproportion in the research intensity is partly due to some earlier imprecise statements on the possible limit of the superconducting critical temperature in the E–P mechanism of pairing [15]. Namely, in an electron–ion system besides the E–P interaction there is also the repulsive Coulomb interaction and these are not independent. In the case of an isotropic and homogeneous system with weak (quasi)particle interaction the effective interaction Veff(k,ω) in the leading approximation looks like as for two external charges (e) embedded in the medium with the total longitudinal dielectric function εtot(k,ω) (k is the momentum and ω is the frequency) [16], i.e.Veff(k,ω)=Vext(k)εtot(k,ω)=4πe2k2εtot(k,ω).Note that in case of strong interaction between quasiparticles, the state of embedded quasiparticles, changes significantly due to interaction with other quasiparticles giving rise to Veff(k,ω)≠4πe2/k2εtot(k,ω). In fact, in that case Veff depends on other (than εtot(k,ω)) response functions. However, in the case when Eq. (1.1) holds the superconducting critical temperature Tc in the weak-coupling limit is given by [14]Tc=ω̄e−1/(λ−μ),where λ is the E–P coupling constant, ω̄ is the average phonon frequency and μ is the Coulomb pseudo-potentialμ=μ1+μlnEF/ω̄.EF is the Fermi energy. The parameters λ and μ are expressed via the total static dielectric function of the crystal [16] (a more precise definition of λ is given in Eq. (2.133))〈N(0)Veff(k,ω=0)〉≡μ−λ=N(0)02kFkdk2kF24πe2k2εtot(k,ω=0),where Veff(k,ω=0) is the effective interaction which includes screened Coulomb and (screened) E–P interaction, and N(0) is the density of states at the Fermi surface. In [15] it was claimed that because of the alleged violation of the lattice stability with respect to the charge density wave formation the total dielectric constant εtot(k,ω=0) must fulfill the condition εtot(k,ω=0)>1 for all k. If this condition were indeed correct then from Eq. (1.4a) it follows inevitably that the inequality μ>λ holds [15], which limits the maximal value of Tc given by [15]Tcmax=EFexp(−4−3/λ).In typical metals EF<(1−10)eV and if one accepts λμ≲0.5 it gives Tc∼(1−10)K. Note that for μ>0.5 there is a magnetic instability of the system [16]. If this way of thinking were correct then the E–P mechanism would be ineffective in reaching not only high Tc but also Tc≈24K. The latter value is the highest value obtained in low-temperature superconductors (LTS). However, it is well known that in PbBi alloys λ≈2.5 is realized, which is definitely much higher than μ(≈0.5), thus contradicting the statement in [15].

In that respect there are two powerful arguments against this statement. The first one is a rigorous and theoretical one, which tells us that the total dielectric function εtot(k,ω) is not the response function, while 1/εtot(k,ω=0) is. Namely, if a small external potential δVext(k,ω) is applied to the system it induces screening by charges of the medium and the total potential is given by δVtot(k,ω)=δVext(k,ω)/εtot(k,ω) which means that the inverse of the total longitudinal function 1/εtot(k,ω) is the response function. It obeys the Kramers–Kronig dispersion relation which implies that 1/εtot(k,ω=0) must fulfill the following stability condition: 1/εtot(k,ω=0)<1 for k≠0. The latter means that either εtot(k≠0,ω=0)>1 or εtot(k≠0,ω=0)<0. This important theorem has been first proved in the seminal article by David Abramovich Kirzhnits [6]. Combined with Eq. (1.4b) this theorem means that in fact there is no restriction on the maximal value of Tc.

The question is – in which systems is εtot(k≠0,ω=0)<0 realized? This question was thoroughly studied in Ref. [18] and here we discuss it briefly for the case of a crystal. The latter is an inhomogeneous system where the total longitudinal dielectric function is matrix in the space of reciprocal lattice vectors (Q), i.e. ε̂tot(k+Q,k+Q′,ω). The total macroscopic longitudinal dielectric constant εtot(k,ω) is defined byεtot−1(k,ω)=ε̂tot−1(k+0,k+0,ω),where ε̂tot−1(k+Q,k+Q′,ω) is the inverse matrix of ε̂tot(k+Q,k+Q′,ω). In the crystal εtot−1(k,ω) has the form [18], [13]εtot−1(k,0)=ε̂e−1(k+0,k+0,0)−4πe2Ωλ,α,β,γ,κ,κ′Zκαγeλγ(κ,k)Zκ′αβeλβ(κ′,k)MκMκ′ωλ2(k,κ).Here, ε̂e−1(k+0,k+0,0) is the diagonal matrix element of the inverse static electronic dielectric function ε̂e−1(k+Q,k+Q′,0),ωλ(k,κ) and eλγ(κ,k) are frequency and polarization vector in the λth mode for the ion κ, respectively, Mκ is the mass of the ion κ and Ω is the volume of the unit cell. The summation goes over all phonon modes λ in the crystal. The second term in Eq. (1.6) represents the phonon contribution (in εtot−1(k,0)) where the effective charge tensor Zκαγ(k) is given by (α,β,γ=1,2,3)Zκαβ(k)=Qkα(k+Q)βε̂e−1(k+0,k+0,0)Ve−i(k+Q)eiQκ.Vei is the bare electron–phonon (ion) interaction potential. In the following we assume that ε̂e−1(k+0, k+0,0)>0 (the possibility for negative value ε̂e−1(k+0,k+0,0)<0 is analyzed in [17], [18], [13]). If one applies Eq. (1.6), for instance, to dense metallic hydrogen (with one ion per cell) then after some simplifications one obtains [17], [18]εtot(k,0)=εel(k,0)1−1εel(k,0)GEP(k),ωl2(k)=Ωpl2εel(k,0)[1−εel(k,0)GEP(k)],where the local field correction GEP is given byGEP(k)=Q(k(k+Q))2(k)2(k+Q)2εel(k+Q,0)(kQ)2(k)2(Q)2εel(Q,0).

In fact, the correct condition for the lattice stability requires that the phonon frequency must be positive, ωl2(k)>0, which implies that εel(k,0)>0 must be fulfilled as well as εel(k,0)GEP(k)<1. The latter condition automatically leads to negative value of the total longitudinal dielectric function εtot(k,0)<0 – see Eq. (1.8). So, in the case of metallic hydrogen crystal εtot(k,0)<0 for all k≠0.

The sign of εtot(k,0) for complexer crystals is thoroughly analyzed in [18], where it is shown that negative value of the total static dielectric function, εtot(k,0)<0, is more the rule than an exception.

The physical reason for negative values of the longitudinal total (or electronic) dielectric function lies in local field effects described by the function GEP(k). (We point out that in [15], in spite of misinterpretation of the dielectric function and its sign, the importance of the local field effects in increasing Tc above the value given by Eq. (1.4b) was realized.) Whenever local electric fields Eloc acting on electrons (and ions) are different from the average electric field E, i.e. ElocE there are additional corrections to εtot(k,0) (or in the case of the electronic subsystem to εe(k,0)) which may lead to εtot(k,0)<0 (or εe(k,0)<0). To illustrate the local field effects let us consider an ionic cubic crystal with the ionic polarizability α and with the ion polarization P=Eloc=E(ε−1)/4π. n is the ion density and ε is the (electronic) dielectric function. From the electrostatics we have [19] thatEloc=E+4π3P=ε+23E,where the longitudinal dielectric function ε(k=0)(≡ε) is given byε(k=0)=1+4π1−4πnα/3=1+4π1−4πnαG.In this case the local field correction is G=13. However, in this specific case one has ε(k=0)>0 due to the thermodynamic stability of the system.

In an electronic subsystem (of a crystal) electronic correlations, which are responsible for the formation of the correlation hole by depleting charge around each electron, give rise to local field effects at finite k≠0 (where αα(k) and GG(k) in Eq. (1.12)), i.e. ElocE, G(k)≠0 and εe(k≠0)<0. In complicated crystals with many atoms in the unit cell the negative electronic dielectric function does not lead necessarily to lattice instability (like in the one-component system where the simple Eq. (1.9) holds), while it can lead to a non-phonon (excitonic) mechanism of superconductivity – see the discussion in Section 3.1. The problem of the excitonic mechanism of superconductivity in many component systems, like for instance HTS oxides, is still unclear and it is out of the scope of this article. One should add that in the case of the excitonic pairing mechanism the effective electron–electron pairing interaction is much more complicated, i.e. Veff(q)≠V(q)/ε(q) which means that Veff(q) should be calculated beyond the RPA approximation [18].

The above analysis tells us that in real crystals the total longitudinal dielectric function can be negative, due to local field effects, in the large portion of the Brillouin zone giving rise to λμ>0, i.e. from that point of view there is no limitation on Tc in the E–P mechanism of pairing, contrary to the statement in [15]. We mention in advance that in HTS oxides, due to their layered structure with ionic-metallic binding, the local field effects play an important and decisive role thus giving rise to large E–P interaction – for more see 3 E–P interaction in HTS oxides, 4 Theory of strong electronic correlations, 5 E–P interaction and strong correlations.

The second argument against the statement in [15] comes from tunneling experiments on LTS systems [218], which gave the definite proof that the E–P interaction is responsible for superconductivity in these systems (including fullerenes except probably heavy fermions). It is clear from the above analysis that there are no theoretical arguments for ignoring the problem of the E–P coupling in HTS oxides.

Moreover, it is necessary to answer several important questions which are also related to experimental finding in HTS oxides: (1) If the E–P interaction is responsible for pairing of electrons in HTS oxides, and if the superconductivity is of d-wave type, how are these two facts compatible? (2) If the answer on this question is affirmative, then why is the transport coupling constant (λtr), which is extracted from resistivity measurements, (much) smaller than the pairing coupling constant λ(>1), which is obtained from tunneling measurements, i.e. why is it realized that λtr(≈0.4–1.0)⪡λ(∼2)? (3) In which way does the E–P interaction interfere with existing spin fluctuations and strong electronic correlations? (4) Is the high Tc value possible for a moderate E–P coupling constant, let us say λ≲1? (5) Finally, if the E–P interaction is ineffective for pairing in HTS oxides, why is it so?

It is now a well-established fact that metallic compounds of HTS oxides are obtained from insulating parent compounds by doping with small number of carriers, which are usually called holes. It should be stressed that the parent insulating state is far from being conventional band insulator where usually an even number of electrons (holes) per lattice site fill Bloch bands completely. From that point of view the parent compounds of copper oxides (for instance La2CuO4 and YBa2Cu3O6) should be metallic, because in the unit cell there is an odd (nine) number of d -electrons per Cu2+ ion – for more see below. The way out of this controversy is in the presence of strong electronic correlations (SEC). Namely, because the d-orbital of the Cu2+ ion is localized there is strong Coulomb repulsion U of two electrons (or holes) with opposite spins in the same orbital (at a given lattice site). This repulsion keeps electrons apart making them to be localized on the lattice, but with localized spins (S=12). This type of insulating state, which is due to SEC, is called the Mott–Hubbard insulator. Speaking in language of electronic bands, for large on-site repulsion UW and for one electron per lattice site the original conduction band with the width W is split into lower Hubbard band with localized spins and the empty upper band separated by U from the lower one – see Section 4.

Additionally, the relevance of SEC is well documented by other experiments:

(i) The electron-energy-loss spectroscopy (EELS) [27] shows a transfer of intensity (which is a measure of the number of states) from higher to lower energies by doping the system. This is clearly seen in Fig. 1 where by doping (x grows) the peak in the EELS spectrum at 530 eV – interpreted as the upper Hubbard band – decreases in intensity. It is known that such a property is characteristic for the class of Hubbard models which describe strong repulsion of electrons with opposite spins at the same lattice site. In that case the number of states in the upper Hubbard band decreases by increasing the hole doping – a similar behavior to that shown in Fig. 1. For comparison, in typical semiconductors the number of states in conduction (or valence) band is determined by the number of atoms, i.e. it is fixed and doping independent.

(ii) The self-consistent band-structure calculations and the photoemission experiments (PES) have given that the effective Hubbard interaction (U) for the Cu ions is of the order U≈6–10 eV [28], which is much larger than the observed width of the conduction band W (∼2 eV) [29].

(iii) A rather direct evidence for strong correlations comes also from the doping dependence of the dynamic conductivity σ(ω) in La2−xSrxCuO4 and Nd2−xCexCuO4−y, particularly from the observed shift of the spectral weight from large to low energies with doping [30] – see Fig. 2, Fig. 3. Besides the development of the Drude peak around ω=0 in the underdoped systems, the so-called mid-infrared (MIR) peak is also developed around 0.4 eV. It is interesting to note that despite the fact that both compounds exhibit a similar doping dependence of σ(ω) it seems that anisotropic s-wave superconductivity is realized in Nd2−xCexCuO4−y (and the pairing is probably due to the E–P interaction), while there are evidences that La2−xSrxCuO4 might be a d-wave superconductor.

Regarding the E–P interaction one can put forward an “old fashioned” question: Does the E–P interaction make (contribute to) the superconducting pairing in HTS oxides? Surprisingly, most of the researchers in the field of superconductivity believe that the E–P interaction is irrelevant and that the pairing mechanism is due to spin fluctuations (or more generally due to strong correlations alone [31]); see for instance [32]. This belief is mainly based on the above-mentioned incorrect stability criterion (which would limit strongly Tc in the E–P mechanism) and also on various experimental results which give evidences for strong anisotropic (d-wave like) pairing with gapless regions (lines) on the Fermi surface [9], [276]. Moreover, the phase-sensitive SQUID measurements of the Josephson effect [33], [34] in the orthorhombic material YBa2Cu3O6+x are strongly in favor of an “orthorhombic” d-wave superconducting order parameter, for instance Δ(k)=Δsd(coskxcosky). As experiments of Tsuei et al. [33], [34] show, one has Δs<0.1Δd in optimally doped YBa2Cu3O6+x, which means that zeros of Δ(k) are near intersections of the Fermi surface and the lines kx≈±ky. Note, here we use the terminology “orthorhombic” d-wave in order to stress that the superconducting order parameter changes sign by the π/2 rotation of the lattice, although this rotation is not the symmetry operation in the orthorhombic crystal with the D2h point symmetry group. Recent experiments on the single-layer crystals Tl2Ba2CuO6+x and on Bi2Sr2CaCu2O8+x (Bi2212) done by Tsuei and co-workers [35], [36], as well as [37], prove the existence of pure d-wave pairing in these systems.

Let us make here a digression and for the sake of further clarity introduce useful terminology. Currently, various types of pairings and order parameters are under consideration in connection with high-temperature superconductivity. They are considered as quasi-two-dimensional and usually classified with respect to the tetragonal point group symmetry:

(a) s-wave pairing:Δs(k)=Δ0,orΔ(k)=Δ0|cos(kxa)−cos(kya)|p.Δ0 and Δ(k) are isotropic and anisotropic (p=1,2,4) gaps, respectively;Δext(k)=Δ0[cos(kxa)+cos(kya)]

  • extended s-wave pairing. All these order parameters transform according to the A1g irreducible representation;

(b) Pure d-wave pairing:Δx2−y2(k)=Δ0[cos(kxa)−cos(kya)],

  • the B1g representation;

Δxy0sin(kxa)sin(kya),
  • the B2g representation;

(c) d+s (or ds) pairing:Δd+s(k)=s+d[cos(kxa)−cos(kya)];

(d) s+idx2y2 (or s−idx2y2) pairing:Δ(k)=s+id[cos(kxa)−cos(kya)];

(e) dx2y2+idxy (or dx2y2−idxy) pairing:Δ(k)=dx2−y2[cos(kxa)−cos(kya)]+idxysin(kxa)sin(kya).In the following we put the lattice constant a=1.

Note that the (d) and (e) order parameters are complex and break time reversal symmetry. However, at present there is no experimental indication of these states in the bulk samples of HTS oxides but there are various cases where the tunneling and interference experiments can be interpreted in terms of (d) and (e) order parameters, which appear near sample surfaces (interfaces) – for details see Section 2.5.2.

After this digression we point out that there is also a widespread (and erroneous) belief that d-wave is incompatible with the E–P pairing mechanism. This point will be a subject of Section 5.

An additional (alleged) argument against the E–P interaction as an origin of superconductivity in HTS oxides is based on the small value of the total isotope effect (α=αO+αCu+αY+αBa) in optimally doped YBCO with highest critical temperature Tc≈92K where αO≈0.05 [284], instead of the canonical value α=12 which would be in case of the E–P pairing mechanism alone.

These facts, as well as the experience in the physics of low-temperature superconductors (with Tcmax<24K) – where s-wave pairing is realized with the finite gap everywhere on the Fermi surface and where the pairing interaction is mediated by phonons [38] – seemingly disqualify the E–P pairing mechanism in HTS oxides. Even if this conclusion were correct it would be, from the scientific viewpoint, important to answer the above-addressed question on the role of the E–P interaction in HTS oxides?

However, there are good experimental evidences that the E–P coupling is sufficiently large in order to produce superconductivity in HTS oxides, i.e. λ>1. Let us quote some of them: (1) The superconductivity-induced phonon renormalization [4], [39], [5], [6] is much larger in HTS oxides than in LTS superconductors; (2) the line-shape in the phonon Raman scattering is very asymmetric (Fano line), which points to a substantial interaction of the lattice with some quasiparticle (electronic liquid) continuum. For instance, the recent phonon Raman measurements [5] on HgBa2Ca3Cu4O10+x at temperatures below Tc give very large softening (self-energy effects) of the A1g phonons with frequencies 240 and 390cm−1 by 6% and 18%, respectively. At the same time there is a dramatic increase of the line-width immediately below Tc, while above Tc the line-shape is strongly asymmetric. Similar results (substantial phonon renormalization) are obtained on (Cu,C)Ba2Ca3Cu4O10+x [6]; (3) the large isotope coefficients (αO>0.4) in YBCO systems away from the optimal doping [284] and αO≈0.15–0.2 in the optimally doped La1.85Sr0.15CuO4; (4) the most important evidence that the E–P interaction plays an important and probably decisive role in pairing comes from tunneling spectra in HTS oxides, where phonon-related features have been clearly seen in the IV characteristics [41], [42], [43], [44], [45].

On the theoretical side there are self-consistent LDA band-structure calculations which in spite of their shortcomings give a rather large bare E–P coupling constant λ∼1.5 in La1.85Sr0.15CuO4 [51], [54]. The nonadiabatic effects due to poor metallic screening along the c-axis may increase λ additionally [53], [54]. All these facts are in favor of a substantial E–P coupling in HTS oxides.

On the other hand, if properties of the normal and superconducting state in HTS oxides are interpreted only in terms of the standard E–P coupling, like in LTS systems, some puzzles arise. One of them is related to the normal-state conductivity – in optimally doped systems the width of the Drude peak and the temperature dependence of the resistivity ρ(T) are not incompatible with the strong-coupling theory with λ∼3 and λtr∼1, where λtr is the transport E–P coupling constant [49]. However, large values of λtr and λ give rather short mean-free path l which at T∼900K takes the value l≈3Å. The latter leads to a saturation in resistivity ρ(T) at that temperature, which is not seen experimentally [55]. On the other hand, the combined resistivity and low-frequency conductivity (Drude part) measurements give for λtr≈0.4–1.0 and for the plasma frequency ωp∼2 eV. If one assumes that λtrλ, which is the case in most low-temperature superconductors (LTS), then the conclusion is that such a small λ cannot give large Tc(≈100K). Based on such a qualitative (but in fact inadequate) estimation, the E–P interaction was eliminated as the pairing interaction in HTS oxides.

There are also doubts on the ability of the E–P interaction to explain the linear temperature dependence of the resistivity in the underdoped system [56] Bi2+xSr2−yCuOδ, which starts at T>10–20 K. Because the asymptotic T5 behavior of ρ(T) (for TΘD) is absent in this sample, then it seems that this experiment is questioning seriously the role of the E–P scattering in resistivity. However, there are measurements [57] on Bi2+xSr2−yCuOδ which show that the linear behavior starts at higher temperature, i.e. at T>50K and therefore it is premature to exclude the E–P interaction. To this point it is interesting that resistivity measurements [20] on Bi2SrCuOx samples with low Tc≃3K show saturation to finite value at T=0K. After subtraction of this constant part one gets (clear) Bloch–Grüneisen behavior between Tc≃3K up to 300 K, which might be an evidence for relevance of the E–P interaction in HTS oxides.

The above facts imply that in HTS oxides one of the following possibilities is realized: (a) λtrλ but the pairing is due to the E–P interaction, i.e. λ>1 or (b) λtrλ≈0.4–1.0 and the E–P interaction is ineffective (although present) in pairing; (c) λtrλ but the E–P interaction is responsible for pairing at the expense of some peculiarities of equations describing superconductivity. In Section 5 we shall present a theory of the E–P interaction renormalized by strong electronic correlations, which is in favor of the case (a), while the possibility for the case (c) will be discussed in Section 7. It is interesting that a similar puzzling situation (λtrλ) is realized in BaxK1−xBiO3 compounds, where optical measurements give λtr≈0.1–0.3 [21], while tunneling measurements give λ≥1 [22]. Note that in BaxK1−xBiO3 there are no magnetic fluctuations (or magnetic order) and no signs of strong electronic correlations in this compound and therefore the E–P interaction is favored as the pairing mechanism. It seems that in this compound the long-range forces, in conjunction with some nesting effects, can explain this discrepancy.

One can summarize that the E–P theory, which pretends to explain normal metallic state and superconductivity in HTS oxides, is confronted with the problem of explaining why the E–P coupling is present in self-energy effects (which are governed by the coupling constant λ>1) but it is suppressed in transport properties (which depend on λtr<1), i.e. why λtr is (much) smaller than λ. One of the possibilities is that strong electronic correlations as well as long-range Madelung forces affect the E–P coupling significantly. This will be discussed thoroughly in forthcoming sections. In the light of the above discussion it is also important to know the role of the E–P coupling in the formation of d-wave superconducting state in HTS oxides, i.e. does it contribute constructively or suppress it?

It is the aim of this review to present and discuss theoretical and also (some) experimental achievements in the physics of HTS oxides, especially those which are related to strong quasiparticle scattering in the normal state, to the pairing mechanism, to the symmetry of the order parameter and to the interplay of strong correlations and the E–P interaction [23], [24], [25], [26].

In order to provide the readers with some basic and necessary knowledge of the HTS physics we present in Section 2 a review of some physical properties of HTS oxides in the normal and superconducting states whose understanding will give a clue for the microscopic theory of superconductivity. We stress, that only those experiments and theoretical interpretations are discussed here which are in our opinion most important in getting the following information: (1) on the type of pairing and (2) on the mechanism of pairing in HTS oxides. It turns out that tunneling measurements in superconducting state are powerful tools in discerning these two questions, especially the first one. That is the reason that in this review we will devote enough space to this method.

The general theory of the E–P interaction and some relevant approximations are presented in Section 3. In this section some approximative schemes for calculating the E–P coupling constant λ in HTS oxides are also critically analyzed, which point to large E–P interaction in these materials. The important role of nonadiabatic effects in increasing λ, which are due to the small quasiparticle hopping along the c-axis, low-energy c-axis plasmon and nonadiabatic screening, is discussed in Section 3.3.

The theory of strong electronic correlations (SEC) is studied extensively in Section 4, where much space is devoted to a systematic, recently elaborated, method for strongly correlated electrons [23], [24], [25] – the X-method. This method considers strongly interacting quasiparticles as composite objects, contrary to the slave-boson method which at some stage assumes spin and charge separation [31]. A systematic theory of the renormalization of the E–P coupling by strong electronic correlations (SEC) [23], [24], [25] is exposed in Section 5, where it is shown that the forward scattering peak (FSP) appears in the E–P interaction, while the latter is suppressed at large transfer momenta (the backward scattering).

In the past there were a lot of proposals on the spin-fluctuation mechanism of superconductivity in HTS oxides, which is the subject of several review articles [11], [9] and therefore we shall not discuss it here. However, in Section 6 an interplay of the spin fluctuation (S–F) and the E–P interaction is studied by assuming that the pairing is due to the repulsive S–F interaction which is strongly peaked at large Q≈(π,π) (pronounced backward scattering peak (BSP)). In this Section it will be demonstrated that when the E–P interaction is isotropic or weakly anisotropic these two interactions interfere destructively leading to a decrease of Tc. However, if there is an unrealistically pronounced FSP in the E–P interaction then these two mechanisms can interfere constructively by increasing Tc and giving rise to d-wave pairing. This means that the role of the spin-fluctuation mechanism in pairing is overestimated.

The additional nonadiabatic E–P effects due to the Migdal vertex corrections and which also take into account specific renormalization of the E–P coupling by strong correlations, and the possibility to increase the effective E–P coupling are the subjects of Section 7. In Section 8 the renormalization by strong correlations of other charge scattering processes, like the scattering on nonmagnetic impurities, is discussed. The latter renormalization gives rise to a robustness of d-wave superconductivity in HTS oxides in the presence of nonmagnetic impurities and defects. Summary and discussion of the subject are presented in Section 9.

Necessary derivations of some formulas, as well as some additional theoretical considerations, are given in several appendices.

Section snippets

Phase diagram and crystal structure

As a result of intensive experimental and theoretical investigations a very reach phase diagram [58] of HTS oxides has been obtained, which is shown in Fig 4 for the representative HTS compound YBa2Cu3O6+x – 0≲x≲1. From this phase diagram, which is generic for all high-temperature superconductors, it is seen that in the undoped (parent) compound with x=0 the antiferromagnetic order sets in below the Neel temperature TN(<500K), while superconductivity (SC) appears below Tc and above some

E–P interaction in HTS oxides

In this section we shall present some elements of the E–P theory which are necessary for an understanding of the following sections. In the second part we shall discuss some theoretical methods in evaluating the E–P coupling constant λ and superconducting critical temperature Tc as well as the main achieved results.

In the years after the discovery of HTS oxides a number of articles have been published which were devoted to the evaluation of the E–P coupling constant. Unfortunately, this problem

Theory of strong electronic correlations

According to the previous sections the superconductivity in HTS materials is due to the E–P interaction, being large because of the quasi-two-dimensional electronic structure in conjunction with the ionic structure along the c-axis. However, in all the above calculations one very important ingredient of the physics in HTS oxides is missing – strong electronic correlations. Let us remind the reader that the LDA method is based on the local exchange-correlation potential which is certainly

Renormalization of the E–P coupling by strong correlations

In preceding sections arguments (which are based on experimental results) are given in favor of substantial renormalization of the E–P interaction in HTS oxides in the presence of strong electronic correlations. It is explained there that if one wants to explain both properties: (i) the linear temperature dependence of resistivity ρ(T) – which gives a rather small transport coupling constant λtr=0.4–0.6, and (ii) the high critical temperature Tc∼100K – which needs (not very) large coupling

Interplay of spin fluctuations and E–P interaction

At present one of the possible candidates in explaining experimental results in HTS oxides appears to be the theory based on the spin-fluctuation pairing mechanism (the S–F pairing). The latter is usually described by the single-band Hubbard model, or on the phenomenological level by the postulated form of the self-energy (written below) [32], [385], [384], [128], [129], [386]. On the other hand, a number of experiments give evidence for a substantial and decisive contribution of the E–P

E–P Interaction beyond the Migdal approximation

HTS oxides are characterized not only by strong correlations but also by a rather small Fermi energy EF which is not much larger than the characteristic (maximal) phonon ωD(ωphmax) frequency, i.e. EF≃0.1–0.3eV,ωphmax≃80meV. The situation is even more pronounced in fullerene compounds A3C60, with Tc=20–35 K, where EF≃0.2 eV and ωphmax≃0.16 eV. This fact implies a possible breakdown of Migdal's theorem [394], [14], which asserts that the relevant vertex corrections due to the E–P interaction are

Nonmagnetic impurities in HTS oxides

In 4 Theory of strong electronic correlations, 5 E–P interaction and strong correlations we have analyzed effects of strong correlations on the E–P interaction where the charge vertex (which describes the screening of the E–P interaction) is calculated. The self-energy due to a nonmagnetic impurity scattering in the Born limit has a similar structure as the E–P interaction, except the retardation effects which are absent in case of elastic impurity scattering. Therefore, the impurity

Summary and conclusions

In this review the intention was to elucidate the present status of our understanding of normal state properties and pairing mechanism in HTS oxides. The relevant experimental results related to transport and tunneling properties of HTS oxides in the normal and superconducting states are analyzed in order to support our attitude, that strong correlations in conjunction with strong E–P interaction are the main ingredients in the pairing mechanism of cuprates and for their normal-state properties.

Acknowledgements

I devote this work to two outstanding men and physicists, my friends and colleagues, Marko Jarić and David Abramovich Kirzhnits, who passed away recently.

I am deeply grateful to Alexander Buzdin – University of Bordeaux, Lars Hedin – Max-Planck-Institute, Yves Leroyer – University of Bordeaux, Michael Mehring– University of Stuttgart and Dierk Rainer – University of Bayreuth for their generous and decisive support during my work on various subjects presented in this review, as well as on the

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