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Advances in Condensed Matter Physics
VolumeΒ 2010Β (2010), Article IDΒ 423725, 64 pages
http://dx.doi.org/10.1155/2010/423725
Review Article

Bosonic Spectral Function and the Electron-Phonon Interaction in HTSC Cuprates

1I. E. Tamm Theoretical Department, Lebedev Physical Institute, 119991 Moscow, Russia
2Institute for Theoretical Physics, Goethe University, 60438 Frankfurt am Main, Germany
3Max-Born-Institut fΓΌr Nichtlineare Optik und Kurzzeitspektroskopie, 12489 Berlin, Germany
4Theoretische Abteilung, Max-Planck-Institut fΓΌr FestkΓΆrperphysik, 70569 Stuttgart, Germany

Received 20 July 2009; Revised 1 November 2009; Accepted 24 February 2010

Academic Editor: CarloΒ Di Castro

Copyright Β© 2010 E. G. Maksimov et al. 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.

Abstract

In this paper we discuss experimental evidence related to the structure and origin of the bosonic spectral function 𝛼2𝐹(πœ”) in high-temperature superconducting (HTSC) cuprates at and near optimal doping. Global properties of 𝛼2𝐹(πœ”), such as number and positions of peaks, are extracted by combining optics, neutron scattering, ARPES and tunnelling measurements. These methods give evidence for strong electron-phonon interaction (EPI) with 1<πœ†π‘’π‘β‰²3.5 in cuprates near optimal doping. We clarify how these results are in favor of the modified Migdal-Eliashberg (ME) theory for HTSC cuprates near optimal doping. In Section 2 we discuss theoretical ingredientsβ€”such as strong EPI, strong correlationsβ€”which are necessary to explain the mechanism of d-wave pairing in optimally doped cuprates. These comprise the ME theory for EPI in strongly correlated systems which give rise to the forward scattering peak. The latter is supported by the long-range part of EPI due to the weakly screened Madelung interaction in the ionic-metallic structure of layered HTSC cuprates. In this approach EPI is responsible for the strength of pairing while the residual Coulomb interaction and spin fluctuations trigger the d-wave pairing.

1. Experimental Evidence for Strong EPI

1.1. Introduction

In spite of an unprecedented intensive experimental and theoretical study after the discovery of high-temperature superconductivity (HTSC) in cuprates, there is, even twenty-three years after, no consensus on the pairing mechanism in these materials. At present there are two important experimental facts which are not under dispute: (1) the critical temperature 𝑇𝑐 in cuprates is high, with the maximum 𝑇𝑐max∼160K in the Hg-1223 compounds; (2) the pairing in cuprates is 𝑑-wave like, that is, Ξ”(𝐀,πœ”)β‰ˆΞ”π‘‘(πœ”)(cosπ‘˜π‘₯βˆ’cosπ‘˜π‘¦). On the contrary there is a dispute concerning the scattering mechanism which governs normal state properties and pairing in cuprates. To this end, we stress that in the HTSC cuprates, a number of properties can be satisfactorily explained by assuming that the quasiparticle dynamics is governed by some electron-boson scattering and in the superconducting state bosonic quasiparticles are responsible for Cooper pairing. Which bosonic quasiparticles are dominating in the cuprates is the subject which will be discussed in this work. It is known that the electron-boson (phonon) scattering is well described by the Migdal-Eliashberg theory if the adiabatic parameter π΄β‰‘π›Όβ‹…πœ†(πœ”π΅/π‘Šπ‘) fulfills the condition 𝐴β‰ͺ1, where πœ† is the electron-boson coupling constant, πœ”π΅ is the characteristic bosonic energy, π‘Šπ‘ is the electronic band width, and 𝛼 depends on numerical approximations [1, 2]. The important characteristic of the electron-boson scattering is the Eliashberg spectral function 𝛼2𝐹(𝐀,π€ξ…ž,πœ”) (or its average 𝛼2𝐹(πœ”)) which characterizes scattering of quasiparticle from 𝐀 to π€ξ…ž by exchanging bosonic energy πœ”. Therefore, in systems with electron-boson scattering the knowledge of the spectral function is of crucial importance.

There are at least two approaches differing in assumed pairing bosons in the HTSC cuprates. The first one is based on the electron-phonon interaction (EPI), with the main proponents in [3–11], where mediating bosons are phonons and where the average spectral function 𝛼2𝐹(πœ”) is similar to the phonon density of states 𝐹ph(πœ”). Note that 𝛼2𝐹(πœ”) is not the product of two functions although sometimes one defines the function 𝛼2(πœ”)=𝛼2𝐹(πœ”)/𝐹(πœ”) which should approximate the energy dependence of the strength of the EPI coupling. There are numerous experimental evidences in cuprates for the importance of the EPI scattering mechanism with a rather large coupling constant in the normal scattering channel 1<πœ†π‘’π‘β‰²3, which will be discussed in detail below. In the EPI approach 𝛼2𝐹ph(πœ”) is extracted from tunnelling measurements in conjunction with IR optical measurements. The HTSC cuprates are on the borderline and it is a natural questionβ€”under which condition can high 𝑇𝑐 be realized in the nonadiabatic limit π΄β‰ˆ1?

The second approach [12–17] assumes that EPI is too weak to be responsible for high 𝑇𝑐 in cuprates and it is based on a phenomenological model for spin-fluctuation interaction (𝑆𝐹𝐼) as the dominating scattering mechanism, that is, it is a nonphononic mechanism. In this (phenomenological) approach the spectral function is proportional to the imaginary part of the spin susceptibility Imπœ’(π€βˆ’π€ξ…ž,πœ”), that is, 𝛼2𝐹(𝐀,π€ξ…ž,πœ”)βˆΌπ‘”2sfImπœ’(π€βˆ’π€ξ…ž,πœ”) where 𝑔sf is the SFI coupling constant. NMR spectroscopy and magnetic neutron scattering give evidence that in HTSC cuprates πœ’(πͺ,πœ”) is peaked at the antiferromagnetic wave vector 𝑄=(πœ‹/π‘Ž,πœ‹/π‘Ž) and this property is favorable for 𝑑-wave pairing. The SFI theory roots basically on the strong electronic repulsion on Cu atoms, which is usually studied by the Hubbard model or its (more popular) derivative the 𝑑-𝐽 model. Regarding the possibility to explain high 𝑇𝑐  solely by strong correlations, as it is reviewed in [18], we stress two facts. First, at present there is no viable theory as well as experimental facts which can justify these (nonphononic) mechanisms of pairing with some exotic pairing mechanism such as RVB pairing [18], fractional statistics, anyon superconductivity, and so forth. Therefore we will not discuss these, in theoretical sense interesting approaches. Second, the central question in these nonphononic approaches is the followingβ€”do models based solely on the Hubbard Hamiltonian show up superconductivity at sufficiently high critical temperatures (π‘‡π‘βˆΌ100K)? Although the answer on this important question is not definitely settled, there are a number of numerical studies of these models which offer negative answers. For instance, the sign-free variational Monte Carlo algorithm in the 2D repulsive (π‘ˆ>0) Hubbard model gives no evidence for superconductivity with high 𝑇𝑐, neither the BCS-like nor the Berezinskii-Kosterlitz-Thouless- (BKT-) like [19]. At the same time, similar calculations show that there is a strong tendency to superconductivity in the attractive (π‘ˆ<0) Hubbard model for the same strength of π‘ˆ, that is, at finite temperature in the 2D model with π‘ˆ<0 the BKT superconducting transition is favored. Concerning the possibility of HTSC in the 𝑑-𝐽 model, various numerical calculations such as Monte Carlo calculations of the Drude spectral weight [20] and high-temperature expansion for the pairing susceptibility [21] give evidence that there is no superconductivity at temperatures characteristic for cuprates and if it exists 𝑇𝑐 must be rather lowβ€”few Kelvins. These numerical results tell us that the lack of high 𝑇𝑐 (even in 2D BKT phase) in the repulsive (π‘ˆ>0) single-band Hubbard model and in the 𝑑-𝐽 model is not only due to thermodynamical 2D-fluctuations (which at finite 𝑇 suppress and destroy superconducting phase coherence in large systems) but it is also mostly due to an inherent ineffectiveness of strong correlations to produce solely high 𝑇𝑐  in cuprates. These numerical results signal that the simple single-band Hubbard and its derivative the 𝑑-𝐽 model are insufficient to explain solely the pairing mechanism in cuprates and some additional ingredients must be included.

Since EPI is rather strong in cuprates, then it must be accounted for. As it will be argued in the following, the experimental support for the importance of EPI in cuprates comes from optics, tunnelling, and recent ARPES measurements [22, 23]. It is worth mentioning that recent ARPES activity was a strong impetus for renewed experimental and theoretical studies of EPI in cuprates. However, in spite of accumulating experimental evidence for importance of EPI with πœ†π‘’π‘>1, there are occasionally reports which doubt its importance in cuprates. This is the case with recent interpretation of some optical measurements in terms of SFI only [24–27] and with the LDA-DFT (local density approximation-density functional theory) band-structure calculations [28, 29], where both claim that EPI is negligibly small, that is, πœ†π‘’π‘<0.3. The inappropriateness of these statements will be discussed in the following sections.

The paper is organized as follows. In Section 1 we will mainly discuss experimental results in cuprates at and near optimal doping by giving also minimal theoretical explanations which are related to the bosonic spectral function 𝛼2𝐹(πœ”) as well as to the transport spectral function 𝛼2tr𝐹(πœ”) and their relations to EPI. The reason that we study only cuprates at and near optimal doping is that in these systems there are rather well-defined quasiparticlesβ€”although strongly interactingβ€”while in highly underdoped systems the superconductivity is perplexed and possibly masked by other phenomena, such as pseudogap effects, formation of small polarons, interaction with spin and (possibly charge) order parameters, pronounced inhomogeneities of the scattering centers, and so forth. As the ARPES experiments confirm, there are no polaronic effects in systems at and near the optimal doping, while there are pronounced polaronic effects due to EPI in undoped and very underdoped HTSC [8–11]. In this work we consider mainly those direct one-particle and two-particle probes of low-energy quasiparticle excitations and scattering rates which give information on the structure of the spectral functions 𝛼2𝐹(𝐀,π€ξ…ž,πœ”) and 𝛼2tr𝐹(πœ”) in systems near optimal doping. These are angle-resolved photoemission (ARPES), various arts of tunnelling spectroscopy such as superconductor/insulator/normal metal (SIN) junctions, break junctions, scanning-tunnelling microscope spectroscopy (STM), infrared (IR) and Raman optics, inelastic neutron and X-ray scattering, and so forth. We will argue that these direct probes give evidence for a rather strong EPI in cuprates. Some other experiments on EPI are also discussed in order to complete the arguments for the importance of EPI in cuprates. The detailed contents of Section 1 are the following. In Section 1.2 we discuss some prejudices related to the strength of EPI as well as on the Fermi-liquid behavior of HTSC cuprates. We argue that any nonphononic mechanism of pairing should have very large bare critical temperature 𝑇𝑐0≫𝑇𝑐 in the presence of the large EPI coupling constant, πœ†π‘’π‘β‰₯1, if the EPI spectral function is weakly momentum dependent, that is, if 𝛼2𝐹(𝐀,π€ξ…ž,πœ”)β‰ˆπ›Ό2𝐹(πœ”) like in low-temperature superconductors. The fact that EPI is large in the normal state of cuprates and the condition that it must be conform with 𝑑-wave pairing imply that EPI in HTSC cuprates should be strongly momentum dependent. In Section 1.3 we discuss direct and indirect experimental evidences for the importance of EPI in cuprates and for the weakness of SFI in cuprates. These are the following.

(a) Magnetic Neutron Scattering Measurements. These measurements provide dynamic spin susceptibility πœ’(πͺ,πœ”) which is in the SFI phenomenological approach [12–17] related to the Eliashberg spectral function, that is, 𝛼2𝐹sf(𝐀,π€ξ…ž,πœ”)βˆΌπ‘”2sfImπœ’(πͺ=π€βˆ’π€ξ…ž,πœ”). We stress that such an approach can be theoretically justified only in the weak coupling limit, 𝑔sfβ‰ͺπ‘Šπ‘, where π‘Šπ‘ is the band width and 𝑔sf is the phenomenological SFI coupling constant. Here we discuss experimental results on YBCO which give evidence for strong rearrangement (with respect to πœ”) of Imπœ’(πͺ,πœ”) (with πͺ at and near 𝐐=(πœ‹,πœ‹)) by doping toward the optimal doped HTSC [30, 31]. It turns out that in the optimally doped cuprates with 𝑇𝑐=92.5KImπœ’(𝐐,πœ”) is drastically suppressed compared to that in slightly underdoped ones with 𝑇𝑐=91K. This fact implies that the SFI coupling constant 𝑔sf must be small.

(b) Optical Conductivity Measurements. From these measurements one can extract the transport relaxation rate 𝛾tr(πœ”) and indirectly an approximative shape of the transport spectral function 𝛼2tr𝐹(πœ”). In the case of systems near optimal doping we discuss the following questions. (i) First is the physical and quantitative difference between the optical relaxation rate 𝛾tr(πœ”) and the quasiparticle relaxation rate 𝛾(πœ”). It was shown in the past that equating these two (unequal) quantities is dangerous and brings incorrect results concerning the quasiparticle dynamics in most metals by including HTSC cuprates too [3–6, 32–38]. (ii) Second are methods of extraction of the transport spectral function 𝛼2tr𝐹(πœ”). Although these methods give at finite temperature 𝑇 a blurred 𝛼2tr𝐹(πœ”) which is (due to the ill-defined methods) temperature dependent, it turns out that the width and the shape of the extracted 𝛼2tr𝐹(πœ”) are in favor of EPI. (iii) Third is the restricted sum rule for the optical weight as a function of 𝑇 which can be explained by strong EPI [39, 40]. (iv) Fourth is the good agreement with experiments of the 𝑇-dependence of the resistivity 𝜌(𝑇) in optimally doped YBCO, where 𝜌(𝑇) is calculated by using the spectral function from tunnelling experiments. Recent femtosecond time-resolved optical spectroscopy in La2βˆ’π‘₯Srπ‘₯CuO4 which gives additional evidence for importance of EPI [41] will be shortly discussed.

(c) ARPES Measurements and EPI. From these measurements the self-energy Ξ£(𝐀,πœ”) is extracted as well as some properties of 𝛼2𝐹(𝐀,π€ξ…ž,πœ”). Here we discuss the following items: (i) the existence of the nodal and antinodal kinks in optimally and slightly underdoped cuprates, as well as the structure of the ARPES self-energy (Ξ£(𝐀,πœ”)) and its isotope dependence, which are all due to EPI; (ii) the appearance of different slopes of Ξ£(𝐀,πœ”) at low (πœ”β‰ͺπœ”ph) and high energies (πœ”β‰«πœ”ph ) which can be explained by the strong EPI; (iii) the formation of small polarons in the undoped HTSC which was interpreted to be due to strong EPIβ€”this gives rise to phonon side bands which are clearly seen in ARPES of undoped HTSC [10, 11].

(d) Tunnelling Spectroscopy. It is well known that this method is of an immense importance in obtaining the spectral function 𝛼2𝐹(πœ”) from tunnelling conductance. In this part we discuss the following items: (i) the extracted Eliashberg spectral function 𝛼2𝐹(πœ”) with the coupling constant πœ†(tun)=2-3.5 from the tunnelling conductance of break-junctions in optimally doped YBCO and Bi-2212 [42–55] which gives that the maxima of 𝛼2𝐹(πœ”) coincide with the maxima in the phonon density of states 𝐹ph(πœ”); (ii) the existence of eleven peaks in βˆ’π‘‘2𝐼/𝑑𝑉2 in superconducting La1.84Sr0.16CuO4 films [56], where these peaks match precisely with the peaks in the intensity of the existing phonon Raman scattering data [57]; (iii) the presence of the dip in 𝑑𝐼/𝑑𝑉 in STM which shows the pronounced oxygen isotope effect and important role of these phonons.

(e) Inelastic Neutron and X-Ray Scattering Measurements. From these experiments one can extract the phonon density of state 𝐹ph(πœ”) and in some cases the strengths of the quasiparticle coupling with various phonon modes. These experiments give sufficient evidence for quantitative inadequacy of LDA-DFT calculations in HTSC cuprates. Here we argue that the large softening and broadening of the half-breathing Cu–O bond-stretching phonon, of the apical oxygen phonons and of the oxygen 𝐡1𝑔 buckling phonons (in LSCO, BSCO, YBCO), cannot be explained by LDA-DFT. It is curious that the magnitude of the softening can be partially obtained by LDA-DFT but the calculated widths of some important modes are an order of magnitude smaller than the neutron scattering data show. This remarkable fact confirms that additionally the inadequacy of LDA-DFT in strongly correlated systems and a more sophisticated many-body theory for EPI is needed. The problem of EPI will be discussed in more details in Section 2.

In Section 1.4 brief summary of Section 1 is given. Since we are dealing with the electron-boson scattering in cuprates near the optimal doping, then in Appendix A (and in Section 2) we introduce the reader briefly to the Migdal-Eliashberg theory for superconductors (and normal metals) where the quasiparticle spectral function 𝛼2𝐹(𝐀,π€ξ…ž,πœ”) and the transport spectral function 𝛼2tr𝐹(πœ”) are defined.

Finally, one can pose a questionβ€”do the experimental results of the above enumerated spectroscopic methods allow a building of a satisfactory and physically reasonable microscopic theory for basic scattering and pairing mechanism in cuprates? The posed question is very modest compared to the much stringent request for the theory of everythingβ€”which would be able to explain all properties of HTSC materials. Such an ambitious project is not realized even in those low-temperature conventional superconductors where it is definitely proved that in most materials the pairing is due to EPI and many properties are well accounted for by the Migdal-Eliashberg theory. For an illustration, let us mention only two examples. First, the experimental value for the coherence peak in the microwave response πœŽπ‘ (𝑇<𝑇𝑐,πœ”=const) at πœ”=17GHz in the superconducting Nb is much higher than the theoretical value obtained by the strong coupling Eliashberg theory [58]. So to say, the theory explains the coherence peak at 17GHz in Nb qualitatively but not quantitatively. However, the measurements at higher frequency πœ”βˆΌ60GHz are in agreement with the Eliashberg theory [59]. Then one can say that instead of the theory of everything we deal with a satisfactory theory, which allows us qualitative and in many aspects quantitative explanation of phenomena in superconducting state. Second example is the experimental boron (B) isotope effect in MgB2 (π‘‡π‘β‰ˆ40K) which is smaller than the theoretical value, that is, 𝛼expBβ‰ˆ0.3<𝛼Bth=0.5, although the pairing is due to EPI for boron vibrations [60]. Since the theory of everything is impossible in the complex materials such as HTSC cuprates in Section 1, we will not discuss those phenomena which need much more microscopic details and/or more sophisticated many-body theory. These are selected by chance: (i) large ratio 2Ξ”/𝑇𝑐 which is on optimally doped YBCO and BSCO β‰ˆ5 and 7, respectively, while in underdoped BSCO one has even (2Ξ”/𝑇𝑐)β‰ˆ20; (ii) peculiarities of the coherence peak in the microwave response 𝜎(𝑇) in HTSC cuprates, which is peaked at 𝑇 much smaller than 𝑇𝑐, contrary to the case of LTSC where it occurs near 𝑇𝑐; (iii) the dependence of 𝑇𝑐 on the number of CuO2 in the unit cell; (iv) the temperature dependence of the Hall coefficient; (v) distribution of states in the vortex core, and so forth.

The microscopic theory of the mechanism for superconducting pairing in HTSC cuprates will be discussed in Section 2. In Section 2.1 we introduce an ab initio many-body theory of superconductivity which is based on the fundamental (microscopic) Hamiltonian and the many-body technique. This theory can in principle calculate measurable properties of materials such as the critical temperature 𝑇𝑐, the critical fields, the dynamic and transport properties, and so forth. However, although this method is in principle exact, which needs only some fundamental constants 𝑒,ℏ,π‘šπ‘’,𝑀ion,π‘˜π΅ and the chemical composition of superconducting materials, it was practically never realized in practice due to the complexity of many-body interactionsβ€”electron-electron and electron-latticeβ€”as well as of structural properties. Fortunately, the problem can be simplified by using the fact that superconductivity is a low-energy phenomenon characterized by the very small energy parameters (𝑇𝑐/𝐸𝐹,Ξ”/𝐸𝐹,πœ”ph/𝐸𝐹)β‰ͺ1. It turns out that one can integrate high-energy electronic processes (which are not changed by the appearance of superconductivity) and then solve the low-energy problem by the (so-called) strong-coupling Migdal-Eliashberg theory. It turns out that in such an approach the physics is separated into the following: (1) the solution of the ideal band-structure Hamiltonian with the nonlocal exact crystal potential (sometimes called the excitation potential) 𝑉IBS(𝐫,π«ξ…ž) (IBSβ€”the ideal band structure) which includes the static self-energy (Ξ£(β„Ž)𝑐0(𝐫,π«ξ…ž,πœ”=0)) due to high-energy electronic processes, that is, 𝑉IBS(𝐫,π«ξ…ž)=[𝑉𝑒-𝑖(𝐫)+𝑉𝐻(𝐫)]𝛿(π«βˆ’π«ξ…ž)+Ξ£(β„Ž)𝑐0(𝐫,π«ξ…ž,πœ”=0), with 𝑉𝑒-𝑖 and 𝑉𝐻 being the electron-ion and Hartree potential, respectively; (2) solving the low-energy Eliashberg equations. However, the calculation of the (excited) potential 𝑉IBS(𝐫,π«ξ…ž) and the real EPI coupling 𝑔𝑒𝑝(𝐫,π«ξ…ž)=𝛿𝑉IBS(𝐫,π«ξ…ž)/𝛿𝐑𝑛, which include high-energy many-body electronic processesβ€”for instance, the large Hubbard π‘ˆ effectsβ€”is extremely difficult at present, especially in strongly correlated systems such as HTSC cuprates. Due to this difficulty the calculations of the EPI coupling in the past were usually based on the LDA-DFT method which will be discussed in Section 2.2 in the contest of HTSC cuprates, where the nonlocal potential is replaced by the local potential 𝑉LDA(𝐫)β€”the ground-state potentialβ€”and the real EPI coupling by the β€œlocal” LDA one 𝑔𝑒𝑝(𝐫)=𝛿𝑉LDA(𝐫)/𝛿𝐑𝑛. Since the exchange-correlation effects enter 𝑉LDA(𝐫)=𝑉𝑒-𝑖(𝐫)+𝑉𝐻(𝐫)+𝑉XC(𝐫) via the local exchange-correlation potential 𝑉XC(𝐫), it is clear that the LDA-DFT method describes strong correlations scarcely and it is inadequate in HTSC cuprates (and other strongly correlated systems such as heavy fermions) where one needs an approach beyond the LDA-DFT method. In Section 2.3 we discuss a minimal theoretical model for HTSC cuprates which takes into account minimal number of electronic orbitals and strong correlations in a controllable manner [6]. This theory treats the interplay of EPI and strong correlations in systems with finite doping in a systematic and controllable way. The minimal model can be further reduced (in some range of parameters) to the single-band 𝑑-𝐽 model, which allows the approximative calculation of the excited potential 𝑉IBS(𝐫,π«ξ…ž) and the nonlocal EPI coupling 𝑔𝑒𝑝(𝐫,π«ξ…ž). As a result one obtains the momentum-dependent EPI coupling 𝑔𝑒𝑝(𝐀𝐹,πͺ) which is for small hole-doping (𝛿<0.3) strongly peaked at small transfer momentaβ€”the forward scattering peak. In the framework of this minimal model it is possible to explain some important properties and resolve some puzzling experimental results, like the following, for instance. (a) Why is 𝑑-wave pairing realized in the presence of strong EPI? (b) Why is the transport coupling constant (πœ†tr) rather smaller than the pairing one πœ†, that is, πœ†trβ‰²πœ†/3? (c) Why is the mean-field (one-body) LDA-DFT approach unable to give reliable values for the EPI coupling constant in cuprates and how many-body effects can help? (d) Why is 𝑑-wave pairing robust in the presence of nonmagnetic impurities and defects? (e) Why are the ARPES nodal and antinodal kinks differently renormalized in the superconducting states, and so forth? In spite of the encouraging successes of this minimal model, at least in a qualitative explanation of numerous important properties of HTSC cuprates, we are at present stage rather far from a fully microscopic theory of HTSC cuprates which is able to explain high 𝑇𝑐. In that respect at the end of Section 2.3 we discuss possible improvements of the present minimal model in order to obtain at least a semiquantitative theory for HTSC cuprates.

Finally, we would like to point out that in real HTSC materials there are numerous experimental evidences for nanoscale inhomogeneities. For instance, recent STM experiments show rather large gap dispersion, at least on the surface of BSCO crystals [61–63], giving rise to a pronounced inhomogeneity of the superconducting order parameter Ξ”(𝐀,𝐑), where 𝐀 is the relative momentum of the Cooper pair and 𝐑 is the center of mass of Cooper pairs. One possible reason for the inhomogeneity of Ξ”(𝐀,𝐑) and disorder on the atomic scale can be due to extremely high doping level of ∼(10–20)% in HTSC cuprates which is many orders of magnitude larger than in standard semiconductors (1021 versus 1015 carrier concentration). There are some claims that high 𝑇𝑐 is exclusively due to these inhomogeneities (of an extrinsic or intrinsic origin) which may effectively increase pairing potential [64], while some others try to explain high 𝑇𝑐 solely within the inhomogeneous Hubbard or 𝑑-𝐽 model. Here we will not discuss this interesting problem but mention only that the concept of 𝑇𝑐 increase by inhomogeneity is not well-defined, since the increase of 𝑇𝑐 is defined with respect to the average value 𝑇𝑐. However, 𝑇𝑐 is experimentally not well defined quantity and the hypothesis of an increase of 𝑇𝑐 by material inhomogeneities cannot be tested at all. In studying and analyzing HTSC cuprates near optimal doping we assume that basic effects are realized in nearly homogeneous systems and inhomogeneities are of secondary role, which deserve to be studied and discussed separately.

1.2. EPI versus Nonphononic Mechanisms

Concerning the high 𝑇𝑐 values in cuprates, two dilemmas have been dominating after its discovery: (i) which interaction is responsible for strong quasiparticle scattering in the normal state? This question is related also to the dilemma of Fermi versus non-Fermi liquid; (ii) What is the mediating (gluing) boson responsible for the superconducting pairing, that is, phonons or nonphonons? In the last twenty-three years, the scientific community was overwhelmed by numerous proposed pairing mechanisms, most of which are hardly verifiable in HTSC cuprates.

(1) Fermi versus Non-Fermi Liquid in Cuprates
After discovery of HTSC in cuprates there was a large amount of evidence on strong scattering of quasiparticles which contradicts the canonical (popular but narrow) definition of the Fermi liquid, thus giving rise to numerous proposals of the so called non-Fermi liquids, such as Luttinger liquid, RVB theory, marginal Fermi liquid, and so forth. In our opinion there is no need for these radical approaches in explaining basic physics in cuprates at least in optimally, slightly underdoped and overdoped metallic and superconducting HTSC cuprates. Here we give some clarifications related to the dilemma of Fermi versus non-Fermi liquid. The definition of the canonical Fermi liquid (based on the Landau work) in interacting Fermi systems comprises the following properties: (1) there are quasiparticles with charge π‘ž=±𝑒, spin 𝑠=1/2, and low-energy excitations πœ‰π€(=πœ–π€βˆ’πœ‡) which are much larger than their inverse life-times, that is, πœ‰π€β‰«1/πœπ€βˆΌπœ‰2𝐀/π‘Šπ‘. Since the level width Ξ“=2/𝜏𝐀 of the quasiparticle is negligibly small, this means that the excited states of the Fermi liquid are placed in one-to-one correspondence with the excited states of the free Fermi gas; (2) at 𝑇=0K there is an energy level πœ‰π€πΉ=0 which defines the Fermi surface on which the Fermi quasiparticle distribution function 𝑛𝐹(πœ‰π€) has finite jump at π‘˜πΉ; (3) the number of quasiparticles under the Fermi surface is equal to the total number of conduction particles (we omit here other valence and core electrons)β€”the Luttinger theorem; (4) the interactions between quasiparticles are characterized by the set of Landau parameters which describe the low-temperature thermodynamics and transport properties. Having this definition in mind one can say that if fermionic quasiparticles interact with some bosonic excitation, for instance, with phonons, and if the coupling is sufficiently strong, then the former are not described by the canonical Fermi liquid since at energies and temperatures of the order of the characteristic (Debye) temperature π‘˜π΅Ξ˜π·(≑ β„πœ”π·) (for the Debye spectrum ~Θ𝐷/5), that is, for πœ‰π€βˆΌΞ˜π·, one has πœπ€βˆ’1β‰³πœ‰π€ and the quasiparticle picture (in the sense of the Landau definition) is broken down. In that respect an electron-boson system can be classified as a noncanonical Fermi liquid for sufficiently strong electron-boson coupling. It is nowadays well known that, for instance, Al, Zn are weak coupling systems since for πœ‰π€βˆΌΞ˜π· one has πœπ€βˆ’1β‰ͺπœ‰π€ and they are well described by the Landau theory. However, in (the noncanonical) cases where for higher energies πœ‰π€βˆΌΞ˜π· one has πœπ€βˆ’1β‰³πœ‰π€, the electron-phonon system is satisfactory described by the Migdal-Eliashberg theory and the Boltzmann theory, where thermodynamic and transport properties depend on the spectral function 𝛼2𝐹sf(𝐀,π€ξ…ž,πœ”) and its higher momenta. Since in HTSC cuprates the electron-boson (phonon) coupling is strong and 𝑇𝑐 is large, then it is natural that in the normal state (at 𝑇>𝑇𝑐) we deal with a strong interacting noncanonical Fermi liquid which is for modest nonadiabaticity parameter 𝐴<1 described by the Migdal-Eliashberg theory, at least qualitatively and semiquantitatively. In order to justify this statement we will in the following elucidate some properties in more details by studying optical, ARPES, tunnelling and other experiments in HTSC oxides.

(2) Is There Limit of the EPI Strength?
In spite of the reached experimental evidence in favor of strong EPI in HTSC oxides, there was a disproportion in the research activity (especially theoretical) in the past, since the investigation of the SFI mechanism of pairing prevailed in the literature. This trend was partly due to an incorrect statement in [65, 66] on the possible upper limit of 𝑇𝑐 in the phonon mechanism of pairing. Since in the past we have discussed this problem thoroughly in numerous papersβ€”for the recent one see [67]β€”we will outline here the main issue and results only.

It is well known that in an electron-ion crystal, besides the attractive EPI, there is also repulsive Coulomb interaction. In case of an isotropic and homogeneous system with weak quasiparticle interaction, the effective potential 𝑉eff(𝐀,πœ”) in the leading approximation looks like as for two external charges (𝑒) embedded in the medium with the total longitudinal dielectric function πœ€tot(𝐀,πœ”) (𝐀 is the momentum and πœ” is the frequency) [68, 69], that is, 𝑉eff𝑉(𝐀,πœ”)=ext(𝐀)πœ€tot=(𝐀,πœ”)4πœ‹π‘’2π‘˜2πœ€tot.(𝐀,πœ”)(1) In case of strong interaction between quasiparticles, the state of embedded quasiparticles changes significantly due to interaction with other quasiparticles, giving rise to 𝑉eff(𝐀,πœ”)β‰ 4πœ‹π‘’2/π‘˜2πœ€tot(𝐀,πœ”). In that case 𝑉eff depends on other (than πœ€tot(𝐀,πœ”)) response functions. However, in the case when (1) holds, that is, when the weak-coupling limit is realized, 𝑇𝑐 is given by π‘‡π‘β‰ˆπœ”exp(βˆ’1/(πœ†π‘’π‘βˆ’πœ‡βˆ—)) [68–70]. Here, πœ†π‘’π‘ is the EPI coupling constant, πœ” is an average phonon frequency, and πœ‡βˆ— is the Coulomb pseudopotential, πœ‡βˆ—=πœ‡/(1+πœ‡ln𝐸𝐹/πœ”) (𝐸𝐹 is the Fermi energy). The couplings πœ†π‘’π‘ and πœ‡ are expressed by πœ€tot(𝐀,πœ”=0): πœ‡βˆ’πœ†π‘’π‘=βŸ¨π‘(0)𝑉eο¬€ξ€œ(𝐀,πœ”=0)⟩=𝑁(0)2π‘˜πΉ0π‘˜π‘‘π‘˜2π‘˜2𝐹4πœ‹π‘’2π‘˜2πœ€tot,(𝐀,πœ”=0)(2) where 𝑁(0) is the density of states at the Fermi surface and π‘˜πΉ is the Fermi momentumβ€”see more in [3–5]. In [65, 66] it was claimed that the lattice stability of the system with respect to the charge density wave formation implies the condition πœ€tot(𝐀,πœ”=0)>1 for all 𝐀. If this were correct, then from (2) it would follow that πœ‡>πœ†π‘’π‘, which limits the maximal value of 𝑇𝑐 to the value 𝑇𝑐maxβ‰ˆπΈπΉexp(βˆ’4βˆ’3/πœ†π‘’π‘). In typical metals 𝐸𝐹<(1-10)eV, and if one accepts the statement in [65, 66] that πœ†π‘’π‘β‰€πœ‡(≀0.5), one obtains π‘‡π‘βˆΌ(1-10)K. The latter result, if it would be correct, means that EPI is ineffective in producing not only high-𝑇𝑐 superconductivity but also low-temperature superconductivity (LTS with 𝑇𝑐≲20K). However, this result is in conflict first of all with experimental results in LTSC, where in numerous systems one has πœ‡β‰€πœ†π‘’π‘ and πœ†π‘’π‘>1. For instance, πœ†π‘’π‘β‰ˆ2.6 is realized in PbBi alloy which is definitely much higher than πœ‡(<1), and so forth.

Moreover, the basic theory tells us that πœ€tot(𝐀≠0,πœ”) is not the response function [68, 69] (contrary to the assumption in [65, 66]). Namely, if a small external potential 𝛿𝑉ext(𝐀,πœ”) is applied to the system (of electrons and ions in solids), it induces screening by charges of the medium and the total potential is given by 𝛿𝑉tot(𝐀,πœ”)=𝛿𝑉ext(𝐀,πœ”)/πœ€tot(𝐀,πœ”), which means that 1/πœ€tot(𝐀,πœ”) is the response function. The latter obeys the Kramers-Kronig dispersion relation which implies the following stability condition [68, 69]: 1πœ€tot(𝐀,πœ”=0)<1,𝐀≠0,(3) that is, either πœ€tot(𝐀≠0,πœ”=0)>1,(4) or πœ€tot(𝐀≠0,πœ”=0)<0.(5) This important theorem invalidates the restriction on the maximal value of 𝑇𝑐 in the EPI mechanism given in [65, 66]. We stress that the condition πœ€tot(𝐀≠0,πœ”=0)<0 is not in conflict with the lattice stability at all. For instance, in inhomogeneous systems such as crystal, the total longitudinal dielectric function is matrix in the space of reciprocal lattice vectors (𝐐), that is, Μ‚πœ€tot(𝐀+𝐐,𝐀+πξ…ž,πœ”), and πœ€tot(𝐀,πœ”) is defined by πœ€βˆ’1tot(𝐀,πœ”)=Μ‚πœ€βˆ’1tot(𝐀+𝟎,𝐀+𝟎,πœ”). In dense metallic systems with one ion per cell (such as metallic hydrogen) and with the electronic dielectric function πœ€el(𝐀,0), the macroscopic total dielectric function πœ€tot(𝐀,0) is given by [71–73] πœ€totπœ€(𝐀,0)=el(𝐀,0)1βˆ’1/πœ€el(𝐀,0)𝐺𝑒𝑝.(𝐀)(6) At the same time the energy of the longitudinal phonon πœ”π‘™(𝐀) is given by πœ”2𝑙Ω(𝐀)=2π‘πœ€el(𝐀,0)1βˆ’πœ€el(𝐀,0)𝐺𝑒𝑝,(𝐀)(7) where Ξ©2𝑝 is the ionic plasma frequency, and 𝐺𝑒𝑝 is the local (electric) field correctionβ€”see [71–73]. The right condition for lattice stability requires that πœ”2𝑙(𝐀)>0, which implies that for πœ€el(𝐀,0)>0 one has πœ€el(𝐀,0)𝐺𝑒𝑝(𝐀)<1. The latter condition gives automatically πœ€tot(𝐀,0)<0. Furthermore, the calculations [71–73] show that in the metallic hydrogen (H) crystal πœ€tot(𝐀,0)<0 for all π€β‰ πŸŽ. Note that in metallic H the EPI coupling constant is very large, that is, πœ†π‘’π‘β‰ˆ7 and 𝑇𝑐 may reach very large value π‘‡π‘β‰ˆ600K [74]. Moreover, the analyses of crystals with more ions per unit cell [71–73] give that πœ€tot(π€β‰ πŸŽ,0)<0 is more a rule than an exceptionβ€”see Figure 1. The physical reason for πœ€tot(π€β‰ πŸŽ,0)<0 is local field effects described by 𝐺𝑒𝑝(𝐀). Whenever the local electric field 𝐄loc acting on electrons (and ions) is different from the average electric field 𝐄, that is, 𝐄loc≠𝐄, there are corrections to πœ€tot(𝐀,0) which may lead to πœ€tot(𝐀,0)<0.

423725.fig.001
Figure 1: Inverse total static dielectric function πœ€βˆ’1(𝐩) for normal metals (K, Al, Pb) and metallic H in 𝐩=(1,0,0) direction. 𝐆 is the reciprocal lattice vector.

The above analysis tells us that in real crystals πœ€tot(𝐀,0)  can be negative in the large portion of the Brillouin zone thus giving rise to πœ†π‘’π‘βˆ’πœ‡>0 in (2). This means that analytic properties of the dielectric function πœ€tot(𝐀,πœ”)  do not limit 𝑇𝑐 in the phonon mechanism of pairing. This result does not mean that there is no limit on 𝑇𝑐 at all. We mention in advance that the local field effects play important role in HTSC cuprates, due to their layered structure with very unusual ionic-metallic binding, thus opening a possibility for large EPI.

In conclusion, we point out that there are no serious theoretical and experimental arguments for ignoring EPI in HTSC cuprates. To this end it is necessary to answer several important questions which are related to experimental findings in HTSC cuprates. (1) If EPI is important for pairing in HTSC cuprates and if superconductivity is of 𝑑-wave type, how are these two facts compatible? (2) Why is the transport EPI coupling constant πœ†tr (entering resistivity) rather smaller than the pairing EPI coupling constant πœ†π‘’π‘(>1) (entering 𝑇𝑐), that is, why one has πœ†tr(β‰ˆ0.6–1.4)β‰ͺπœ†π‘’π‘(∼2–3.5)? (3) If EPI is ineffective for pairing in HTSC oxides, in spite of πœ†π‘’π‘>1, why is it so?

(3) Is a Nonphononic Pairing Realized in HTSC?
Regarding EPI one can pose a question about whether it contributes significantly to 𝑑-wave pairing in cuprates. Surprisingly, despite numerous experiments in favor of EPI, there is a belief that EPI is irrelevant for pairing [12–17]. This belief is mainly based, first, on the above discussed incorrect lattice stability criterion related to the sign of πœ€tot(𝐀,0), which implies small EPI and, second, on the well-established experimental fact that 𝑑-wave pairing is realized in cuprates [75], which is believed to be incompatible with EPI. Having in mind that EPI in HTSC at and near optimal doping is strong with 2<πœ†π‘’π‘<3.5 (see below), we assume for the moment that the leading pairing mechanism in cuprates, which gives 𝑑-wave pairing, is due to some nonphononic mechanism. For instance, let us assume an exitonic mechanism due to the high-energy pairing boson (Ξ©nphβ‰«πœ”ph) and with the bare critical temperature 𝑇𝑐0 and look for the effect of EPI on 𝑇𝑐. If EPI is approximately isotropic, like in most LTSC materials, then it would be very detrimental for 𝑑-wave pairing. In the case of dominating isotropic EPI in the normal state and the exitonic-like pairing, then near 𝑇𝑐 the linearized Eliashberg equations have an approximative form for a weak nonphonon interaction (with the large characteristic frequency Ξ©nph) π‘ξ€·πœ”π‘›ξ€ΈΞ”π‘›(𝐀)β‰ˆπœ‹π‘‡π‘Ξ©nphξ“π‘šξ“πͺ𝑉nphΞ”(𝐀,πͺ,𝑛,π‘š)π‘š(πͺ)||πœ”π‘š||,π‘ξ€·πœ”π‘›ξ€ΈΞ“β‰ˆ1+π‘’π‘πœ”π‘›.(8) For pure 𝑑-wave pairing with the pairing potential 𝑉nph=𝑉nph(πœƒπ€,πœƒπͺ)β‹…Ξ˜(Ξ©nphβˆ’|πœ”π‘›|)Θ(Ξ©nphβˆ’|πœ”π‘›β€²|) with 𝑉nph(𝐀,πͺ)=𝑉0β‹…π‘Œπ‘‘(πœƒπ€)π‘Œπ‘‘(πœƒπͺ) and π‘Œπ‘‘(πœƒπ€)=πœ‹βˆ’1/2cos2πœƒπ€, one obtains Δ𝑛(𝐀)=Ξ”π‘‘β‹…Ξ˜(Ξ©nphβˆ’|πœ”π‘›|)π‘Œπ‘‘(πœƒπ€) and the equation for 𝑇𝑐—see [3–5] 𝑇ln𝑐𝑇𝑐0ξ‚€1β‰ˆΞ¨21βˆ’Ξ¨2+Γ𝑒𝑝2πœ‹π‘‡π‘ξ‚Ά.(9) Here Ξ¨ is the di-gamma function. At temperatures near 𝑇𝑐 one has Ξ“π‘’π‘β‰ˆ2πœ‹πœ†π‘’π‘π‘‡π‘ and the solution of (9) is approximately π‘‡π‘β‰ˆπ‘‡π‘0exp{βˆ’πœ†π‘’π‘} with 𝑇𝑐0β‰ˆΞ©nphexp{βˆ’πœ†nph}, πœ†nph=𝑁(0)𝑉0. This means that for 𝑇𝑐max∼160K and πœ†π‘’π‘>1 the bare 𝑇𝑐0 due to the nonphononic interaction must be very large, that is, 𝑇𝑐0>500K.

Concerning other nonphononic mechanisms, such as the SFI one, the effect of EPI in the framework of Eliashberg equations was studied numerically in [76]. The latter is based on (A.1) in Appendix A with the kernels in the normal and superconducting channels πœ†π‘π€π©(π‘–πœˆπ‘›) and πœ†Ξ”π€π©, respectively. Usually, the spin-fluctuation kernel πœ†sf,𝐀𝐩(π‘–πœˆπ‘›) is taken in the FLEX approximation [77]. The calculations [76] confirm the very detrimental effect of the isotropic (𝐀-independent) EPI on 𝑑-wave pairing due to SFI. For the bare SFI critical temperature 𝑇𝑐0∼100K and for πœ†π‘’π‘>1 the calculations give very small (renormalized) critical temperature 𝑇𝑐β‰ͺ100K. These results tell us that a more realistic pairing interaction must be operative in cuprates and that EPI must be strongly momentum dependent and peaked at small transfer momenta [78–80]. Only in that case does strong EPI conform with 𝑑-wave pairing, either as its main cause or as a supporter of a nonphononic mechanism. In Section 2 we will argue that the strongly momentum-dependent EPI is important scattering mechanism in cuprates providing the strength of the pairing mechanism, while the residual Coulomb interaction (by including weaker SFI) triggers it to 𝑑-wave pairing.

1.3. Experimental Evidence for Strong EPI

In the following we discuss some important experiments which give evidence for strong electron-phonon interaction (EPI) in cuprates. However, before doing it, we will discuss some indicative inelastic magnetic neutron scattering  (IMNS)  measurements in cuprates whose results in fact seriously doubt in the effectiveness of the phenomenological SFI mechanism of pairing which is advocated in [12–17, 81]. First, the experimental results related to the pronounced imaginary part of the susceptibility Imπœ’(𝐀,π‘˜π‘§,πœ”) in the normal state at and near the AF wave vector 𝐀=𝐐=(πœ‹,πœ‹) were interpreted in a number of papers as a support for the SFI mechanism for pairing [12–17, 81]. Second, the existence of the so called magnetic resonance peak of Imπœ’(𝐀,π‘˜π‘§,πœ”) (at some energies πœ”<2Ξ”) in the superconducting state was also interpreted in a number of papers either as the origin of superconductivity or as a mechanism strongly affecting superconducting gap at the antinodal point.

1.3.1. Magnetic Neutron Scattering and the Spin-Fluctuation Spectral Function

(a) Huge Rearrangement of the SFI Spectral Function and Small Change of 𝑇𝑐
Before discussing experimental results in cuprates on the imaginary part of the spin susceptibility Imπœ’(𝐀,πœ”) we point out that in the (phenomenological) theories based on the spin-fluctuation interaction (SFI) the quasiparticle self-energy Σsf(𝐀,πœ”π‘›) (πœ”π‘› is the Matsubara frequency and Μ‚πœ0 is the Nambu matrix) in the normal and superconducting state and the effective (repulsive) pairing potential 𝑉sf(𝐀,πœ”) (where π‘–πœ”π‘›β†’πœ”+π‘–πœ‚) are assumed in the form [12–17] Σsf𝐀,πœ”π‘›ξ€Έ=𝑇𝑁𝐀′,π‘šπ‘‰sfξ€·π€βˆ’π€ξ…ž,πœ”βˆ’π‘›π‘šξ€ΈΜ‚πœ0ξπΊξ€·π€ξ…ž,πœ”π‘šξ€ΈΜ‚πœ0,𝑉sf𝐀,πœ”βˆ’π‘›π‘šξ€Έ=𝑔2sfξ€œβˆžβˆ’βˆžπ‘‘πœˆπœ‹ξ€·Imπœ’πͺ,𝜈+𝑖0+ξ€Έπœˆβˆ’π‘–πœ”βˆ’π‘›π‘š,(10) where πœ”βˆ’π‘›π‘šβ‰‘πœ”π‘›βˆ’πœ”π‘š. Although the form of 𝑉sf cannot be justified theoretically, except in the weak coupling limit (𝑔sfβ‰ͺπ‘Šπ‘) only, it is often used in the analysis of the quasiparticle properties in the normal and superconducting state of cuprates where the spin susceptibility (spectral function) Imπœ’(πͺ,πœ”) is strongly peaked at and near the AF wave vector 𝐐=(πœ‹/π‘Ž,πœ‹/π‘Ž).

Can the pairing mechanism in HTSC cuprates be explained by such a phenomenology and what is the prise for it is? The best answer is to look at the experimental results related to the inelastic magnetic neutron scattering (IMNS) which gives Imπœ’(πͺ,πœ”). In that respect very indicative and impressive IMNS measurements on YBa2Cu3O6+π‘₯, which are done by Bourges group [30], demonstrate that the normal-state susceptibility Imπœ’(odd)(πͺ,πœ”) (the odd part of the spin susceptibility in the bilayer system) at πͺ=𝐐=(πœ‹,πœ‹) is strongly dependent on the hole-doping as it is shown in Figure 2.

423725.fig.002
Figure 2: Magnetic spectral function Imπœ’(βˆ’)(𝐀,πœ”) in the normal state of YBa2Cu3O6+π‘₯ at 𝑇=100K and at 𝑄=(πœ‹,πœ‹). 100 counts in the vertical scale correspond to πœ’(βˆ’)maxβ‰ˆ350πœ‡2𝐡/eV. The superconducting critical temperature 𝑇𝑐(π‘₯) by increasing doping (π‘₯) from the underdoped system with π‘₯=0.5 (top) to the optimally doped one with π‘₯=0.97 (bottom): 𝑇𝑐(π‘₯)=45K (π‘₯=0.5), 85K (π‘₯=0.83), 91K (π‘₯=0.92), and 92.5K (π‘₯=0.97). From [30].

The most pronounced result for our discussion is that by varying doping there is a huge rearrangement of Imπœ’(odd)(𝐐,πœ”) in the normal state, especially in the energy (frequency) region which might be important for superconducting pairing, let us say 0meV<πœ”<60meV. This is clearly seen in the last two curves in Figure 2 where this rearrangement is very pronounced, while at the same time there is only small variation of the critical temperature 𝑇𝑐. It is seen in Figure 2 that in the underdoped YBa2Cu3O6.92 crystal Imπœ’(odd)(𝐐,πœ”) and 𝑆(𝐐)=𝑁(0)𝑔2sf∫060π‘‘πœ”Imπœ’(odd)(𝐐,πœ”) are much larger than that in the near optimally doped YBa2Cu3O6.97, that is, one has 𝑆6.92(𝐐)≫𝑆6.97(𝐐), although the difference in the corresponding critical temperatures 𝑇𝑐 is very small, that is, 𝑇𝑐(6.92)=91K (in YBa2Cu3O6.92) and 𝑇𝑐(6.97)=92.5K (in YBa2Cu3O6.97). This pronounced rearrangement and suppression of Imπœ’(odd)(𝐐,πœ”) in the normal state of YBCO by doping (toward the optimal doping) but with the negligible change in 𝑇𝑐 is strong evidence that the SFI pairing mechanism is not the dominating one in HTSC cuprates. This insensitivity of 𝑇𝑐, if interpreted in terms of the SFI coupling constant πœ†sf(~𝑔2sf), means that the latter is small, that is, πœ†(exp)sfβ‰ͺ1. We stress that the explanation of high 𝑇𝑐 in cuprates by the SFI phenomenological theory [12–17] assumes very large SFI coupling energy with 𝑔(th)sfβ‰ˆ0.7eV while the frequency (energy) dependence of Imπœ’(𝐐,πœ”) is extracted from the fit of the NMR relaxation rate 𝑇1βˆ’1 which gives 𝑇(NMR)π‘β‰ˆ100K [12–17]. To this point, the NMR measurements (of 𝑇1βˆ’1) give that there is an anticorrelation between the decrease of the NMR spectral function 𝐼𝐐=limπœ”β†’0Imπœ’(NMR)(𝐐,πœ”)/πœ” and the increase of 𝑇𝑐 by increasing doping toward the optimal oneβ€”see [6] and references therein. The latter result additionally disfavors the SFI model of pairing [12–17] since the strength of pairing interaction is little affected by SFI. Note that if instead of taking Imπœ’(𝐐,πœ”) from NMR measurements one takes it from IMNS measurements, as it was done in [82], than for the same value 𝑔(th)sf one obtains much smaller 𝑇𝑐. For instance, by taking the experimental values for Imπœ’(IMNS)(𝐐,πœ”) in underdoped YBa2Cu3O6.6 with π‘‡π‘β‰ˆ60K one obtains 𝑇(IMNS)𝑐<𝑇(NMR)𝑐/3 [82], while 𝑇(IMNS)𝑐→50K for 𝑔(th)sf≫1. The situation is even worse if one tries to fit the resistivity with Imπœ’(IMNS)(𝐐,πœ”) in YBa2Cu3O6.6 since this fit gives 𝑇(IMNS)𝑐<7K. These results point to a deficiency of the SFI phenomenology (at least that based on (10)) to describe pairing in HTSC cuprates.

Having in mind the results in [82], the recent theoretical interpretation in [81] of IMNS experiments [83, 84] and ARPES measurements [85, 86] on the underdoped YBa2Cu3O6.6 in terms of the SFI phenomenology deserve to be commented. The IMNS experiments [83, 84] give evidence for the β€œhourglass” spin excitation spectrum (in the superconducting state) for the momenta πͺ at, near and far from 𝐐, which is richer than the common spectrum with magnetic resonance peaks measured at 𝐐. In [81] the self-energy of electrons due to their interaction with spin excitations is calculated by using (10) with 𝑔2sfξ‚π‘ˆ=(3/2)2 and Imπœ’(πͺ,πœ”) taken from [83, 84]. However, in order to fit the ARPES self-energy and low-energy kinks (see discussion in Section 1.3.3) the authors of [81] use very large value ξ‚π‘ˆ=1.59eV, that is, much larger than the one used in [82]. Such a large value of ξ‚π‘ˆ has been obtained earlier within the Monte Carlo simulation of the Hubbard model [87]. In our opinion this value for ξ‚π‘ˆ is unrealistically large in the case of strongly correlated systems where spin fluctuations are governed by the effective electron-exchange interaction 𝐽Cu-Cu≲0.15eV [88]. This implies that ξ‚π‘ˆβ‰ͺ1eV and 𝑇𝑐β‰ͺ60K. Note that this value for 𝐽Cu-Cu(~0.15 eV) comes out also from the theory of strongly correlated electrons in the three-band Emery model which gives 𝐽Cu-Cuβ‰ˆ[4𝑑4𝑝𝑑/(Δ𝑑𝑝+π‘ˆπ‘π‘‘)2,(1/π‘ˆπ‘‘)+2/(π‘ˆπ‘+2Ξ”)]β€”for parameters see Section 2.3. We would like to emphasize here that an additional richness of the spin-fluctuations spectrum (the hourglass instead of the spin resonance) does not change the situation with the smallness of the exchange coupling constant ξ‚π‘ˆ (and 𝑔sf).

Concerning the problem related to the rearrangement of the SFI spectral function Imπœ’(𝐐,πœ”) in YB2Cu3O6+π‘₯ [30] we would like to stress that despite the fact that the latter results were obtained ten years ago they are not disputed by the new IMNS measurements [31] on high quality samples of the same compound (where much longer counting times were used in order to reduce statistical errors). In fact the results in [30] are confirmed in [31] where the magnetic intensity 𝐼(πͺ,πœ”)(~Imπœ’(πͺ,πœ”)) (for πͺ at and in the broad range of 𝐐) for the optimally doped YBa2Cu3O6.95 (with 𝑇𝑐=93K) is at least three times smaller than in the underdoped YBa2Cu3O6.6 with 𝑇𝑐=60K. This result is again very indicative sign of the weakness of SFI since such a huge reconstruction would decrease 𝑇𝑐 in the optimally doped YBa2Cu3O6.95 if analyzed in the framework of the phenomenological SFI theory based on (10). It also implies that due to the suppression of Imπœ’(πͺ,πœ”) by increasing doping toward the optimal one a straightforward extrapolation of the theoretical approach in [81] to the explanation of 𝑇𝑐 in the optimally doped YBa2Cu3O6.95 would require an increase of ξ‚π‘ˆ to the value even larger than 4eV, which is highly improbable.

(b) Ineffectiveness of the Magnetic Resonance Peak
A less direct argument for smallness of the SFI coupling constant, that is, 𝑔expsf≀0.2eV and 𝑔expsfβ‰ͺ𝑔sf, comes from other experiments related to the magnetic resonance peak in the superconducting state, and this will be discussed next. In the superconducting state of optimally doped YBCO and BSCO, Imπœ’(𝐐,πœ”) is significantly suppressed at low frequencies except near the resonance energy πœ”resβ‰ˆ41meV where a pronounced narrow peak appearsβ€”the magnetic resonance peak. We stress that there is no magnetic resonance peak in some families of HTSC cuprates, for instance, in LSCO, and consequently one can question the importance of the resonance peak in the scattering processes. The experiments tell us that the relative intensity of this peak (compared to the total one) is small, that is, 𝐼0∼(1-5)%β€”see Figure 3. In underdoped cuprates this peak is present also in the normal state as it is seen in Figure 2.

423725.fig.003
Figure 3: Magnetic spectral function Imπœ’(βˆ’)(𝐀,πœ”) in the superconducting state of YBa2Cu3O6+π‘₯ at 𝑇=5K and at 𝑄=(πœ‹,πœ‹). 100 counts in the vertical scale correspond to πœ’(βˆ’)maxβ‰ˆ350πœ‡2𝐡/eV. From [30].

After the discovery of the resonance peak there were attempts to relate it, first, to the origin of the superconducting condensation energy and, second, to the kink in the energy dispersion or the peak-dimp structure in the ARPES spectral function. In order that the condensation energy is due to the magnetic resonance, it is necessary that the peak intensity 𝐼0 is small [89]. 𝐼0 is obtained approximately by equating the condensation energy 𝐸conβ‰ˆπ‘(0)Ξ”2/2 with the change of the magnetic energy 𝐸mag in the superconducting state, that is, 𝛿𝐸magβ‰ˆ4𝐼0⋅𝐸mag: 𝐸mag=π½π‘‘πœ”π‘‘2π‘˜(2πœ‹)3ξ€·1βˆ’cosπ‘˜π‘₯βˆ’cosπ‘˜π‘¦ξ€Έπ‘†(𝐀,πœ”),(11) where 𝑆(𝐀,πœ”)=(1/πœ‹)[1+𝑛(πœ”)]Imπœ’(𝐀,πœ”) is the spin structure factor and 𝑛(πœ”) is the Bose distribution function. By taking Ξ”β‰ˆ2𝑇𝑐 and the realistic value 𝑁(0)∼1/(10𝐽)∼1states/eVβ‹…spin, one obtains 𝐼0∼10βˆ’1(𝑇𝑐/𝐽)2∼10βˆ’3. However, such a small intensity cannot be responsible for the anomalies in ARPES and optical spectra since it gives rise to small coupling constant πœ†sf,res for the interaction of holes with the resonance peak, that is, πœ†sf,resβ‰ˆ(2𝐼0𝑁(0)𝑔2sf/πœ”res)β‰ͺ1. Such a small coupling does not affect superconductivity at all. Moreover, by studying the width of the resonance peak one can extract an order of magnitude of the SFI coupling constant 𝑔sf. Since the magnetic resonance disappears in the normal state of the optimally doped YBCO, it can be qualitatively understood by assuming that its broadening scales with the resonance energy πœ”res, that is, 𝛾res<πœ”res, where the line width is given by 𝛾res=4πœ‹(𝑁(0)𝑔sf)2πœ”res [89]. This condition limits the SFI coupling to 𝑔sf<0.2eV. We stress that in such a way obtained 𝑔sf is much smaller (at least by factor three) than that assumed in the phenomenological spin-fluctuation theory [12–17, 81] where 𝑔sf∼0.6-0.7eV and ξ‚π‘ˆβ‰ˆ1.6eV, but much larger than estimated in [89] (where 𝑔sf<0.02eV). The smallness of 𝑔sf comes out also from the analysis of the antiferromagnetic state in underdoped metals of LSCO and YBCO [90], where the small (ordered) magnetic moment πœ‡(<0.1πœ‡π΅) points to an itinerant antiferromagnetism with small coupling constant 𝑔sf<0.2eV. The conclusion from this analysis is that in the optimally doped YBCO the sharp magnetic resonance is a consequence of the onset of superconductivity and not its cause. There is also one principal reason against the pairing due to the magnetic resonance peak at least in optimally doped cuprates. Since the intensity of the magnetic resonance near 𝑇𝑐 is vanishingly small, though not affecting pairing at the second-order phase transition at 𝑇𝑐, then, if it would be solely the origin for superconductivity, the phase transition at 𝑇𝑐 would be first order, contrary to experiments. Recent ARPES experiments give evidence that the magnetic resonance cannot be related to the kinks in ARPES spectra [91, 92]β€”see the discussion below.

Finally, we would like to point out that the recent magnetic neutron scattering measurements on optimally doped large-volume crystals Bi2Sr2CaCu2O8+𝛿 [93], where the absolute value of Imπœ’(πͺ,πœ”) is measured, are questioning also the interpretation of the electronic magnetism in cuprates in terms of the itinerant magnetism. This experiment shows a lack of temperature dependence of the local spin susceptibility βˆ‘Imπœ’(πœ”)=π‘žImπœ’(πͺ,πœ”) across the superconducting transition 𝑇𝑐=91K, that is, there is only a minimal change in Imπœ’(πœ”) between 10K and 100K. Note that if the magnetic excitations were due to itinerant quasiparticles we should have seen dramatic changes of Imπœ’(πœ”) as a function of 𝑇 over the whole energy range. This 𝑇-independence of Imπœ’(πœ”) strongly opposes the theoretical results in [24–27] which assume that the bosonic spectral function is proportional to Imπœ’(πœ”) and that the former can be extracted from optic measurements. Namely, the fitting procedure in [24–27] gives that Imπœ’(πœ”) is strongly 𝑇-dependent contrary to the experimental results in [93]β€”see more in Section 1.3.2 on optical conductivity.

1.3.2. Optical Conductivity and EPI

Optical spectroscopy gives information on optical conductivity 𝜎(πœ”) and on two-particle excitations, from which one can indirectly extract the transport spectral function 𝛼2tr𝐹(πœ”). Since this method probes bulk sample (on the skin depth), contrary to ARPES and tunnelling methods which probe tiny regions (10-15 Å) near the sample surface, this method is indispensable. However, one should be careful not to overinterpret the experimental results since 𝜎(πœ”)  is not a directly measured quantity but it is derived from the reflectivity βˆšπ‘…(πœ”)=|(πœ€π‘–π‘–βˆš(πœ”)βˆ’1)/(πœ€π‘–π‘–(πœ”)+1)|2 with the transversal dielectric tensor πœ€π‘–π‘–(πœ”)=πœ€π‘–π‘–,∞+πœ€π‘–π‘–,latt+4πœ‹π‘–πœŽπ‘–π‘–(πœ”)/πœ”. Here, πœ€π‘–π‘–,∞ is the high-frequency dielectric function, πœ€π‘–π‘–,latt describes the contribution of the lattice vibrations, and πœŽπ‘–π‘–(πœ”) describes the optical (dynamical) conductivity of conduction carriers. Since 𝑅(πœ”) is usually measured in the limited-frequency interval πœ”min<πœ”<πœ”max, some physical modelling for 𝑅(πœ”) is needed in order to guess it outside this rangeβ€”see more in reviews in [3–6]. This was the reason for numerous misinterpretations of optic measurements in cuprates, which will be uncovered below. An illustrative example for this claim is large dispersion in the reported value of πœ”plβ€”from 0.06 to 25eVβ€”that is, almost three orders of magnitude. However, it turns out that IR measurements of 𝑅(πœ”) in conjunction with elipsometric measurements of πœ€π‘–π‘–(πœ”) at high frequencies allow more reliable determination of 𝜎(πœ”) [94].

(1) Transport and Quasiparticle Relaxation Rates
The widespread misconception in studying the quasiparticle scattering in cuprates was an ad hoc assumption that the transport relaxation rate 𝛾tr(πœ”) is equal to the quasiparticle relaxation rate 𝛾(πœ”), in spite of the well-known fact that the inequality 𝛾tr(πœ”)≠𝛾(πœ”) holds in a broad-frequency (energy) region Allen. This (incorrect) assumption was one of the main arguments against the relevance of the EPI scattering mechanism in cuprates. Although we have discussed this problem several times before, we do it again due to the importance of this subject.

The dynamical conductivity 𝜎(πœ”) consists of two parts, that is, 𝜎(πœ”)=𝜎inter(πœ”)+𝜎intra(πœ”) where 𝜎inter(πœ”) describes interband transitions which contribute at higher than intraband energies, while 𝜎intra(πœ”) is due to intraband transitions which are relevant at low energies πœ”<(1-2)eV. (Note that in the IR measurements the frequency is usually given in cmβˆ’1, where the following conversion holds: 1cmβˆ’1=29.98GHz=0.123985meV=1.44K.) The experimental data for 𝜎(πœ”)=𝜎1+π‘–πœŽ2 in cuprates are usually processed by the generalized (extended) Drude formula [32–36, 95]: πœ”πœŽ(πœ”)=2𝑝14πœ‹π›Ύtr(πœ”)βˆ’π‘–πœ”π‘štr(πœ”)/π‘šβˆžβ‰‘π‘–πœ”2𝑝4πœ‹ξ‚πœ”tr,(πœ”)(12) where π‘š is the mass of the band electrons while the quantity ξ‚πœ”tr(πœ”) is defined in (19). The expression (12) is a useful representation for systems with single-band electron-boson scattering which is justified in HTSC cuprates. However, this procedure is inadequate for interpreting optical data in multiband systems such as new high-temperature superconductors Fe-based pnictides since even in absence of the inelastic intra- and interband scattering the effective optic relaxation rate may be strongly frequency dependent [96]. (The usefulness of introducing the optic relaxation ξ‚πœ”tr(πœ”) will be discussed below.) Here, 𝑖=π‘Ž,𝑏 enumerates the plane axis; πœ”π‘,𝛾tr(πœ”,𝑇), and π‘šop(πœ”) are the electronic plasma frequency, the transport (optical) scattering rate, and the optical mass, respectively. Very frequently it is analyzed the quantity π›Ύβˆ—tr(πœ”,𝑇) given by [95] π›Ύβˆ—trπ‘š(πœ”,𝑇)=βˆžπ‘štr(π›Ύπœ”)tr(πœ”,𝑇)=πœ”Im𝜎(πœ”).Re𝜎(πœ”)(13) In the weak coupling limit πœ†π‘’π‘<1, the formula for conductivity given in Appendix A, equations (A.20) and (A.21) can be written in the form of (12) where 𝛾tr reads [33–36] 𝛾tr(πœ”,𝑇)=πœ‹π‘™ξ€œβˆž0π‘‘πœˆπ›Ό2tr,𝑙𝐹𝑙×2ξ€·(𝜈)1+2π‘›π΅ξ€Έπœˆ(𝜈)βˆ’2πœ”βˆ’πœ”+πœˆπœ”π‘›π΅+(πœ”+𝜈)πœ”βˆ’πœˆπœ”π‘›π΅ξ‚„.(πœ”βˆ’πœˆ)(14) Here 𝑛𝐡(πœ”) is the Bose distribution function. For completeness we give also the explicit form of the transport mass π‘štr(πœ”), see [3–6, 32–36]: π‘štr(πœ”)π‘šβˆž2=1+πœ”ξ“π‘™ξ€œ0π‘₯0200π‘‘βˆž0π‘‘πœˆπ›Ό2tr,π‘™πΉπ‘™ξ‚€πœ”(𝜈)Re𝐾,𝜈2πœ‹π‘‡ξ‚,2πœ‹π‘‡(15) with the Kernel 𝐾(π‘₯,𝑦)=(𝑖/𝑦)+{((π‘¦βˆ’π‘₯)/π‘₯)[πœ“(1βˆ’π‘–π‘₯+𝑖𝑦)βˆ’πœ“(1+𝑖𝑦)]}βˆ’{π‘¦β†’βˆ’π‘¦} where πœ“ is the di-gamma function. In the presence of impurity scattering one should add 𝛾imp,tr to 𝛾tr. It turns out that (14) holds within a few percents also for large πœ†π‘’π‘(>1). Note that 𝛼2tr,𝑙𝐹𝑙(𝜈)≠𝛼2𝑙𝐹𝑙(𝜈) and the index 𝑙  enumerates all scattering bosonsβ€”phononsβ€”spin fluctuations, and so forth. For comparison, the quasiparticle scattering rate 𝛾(πœ”,𝑇) is given by ξ€œπ›Ύ(πœ”,𝑇)=2πœ‹βˆž0π‘‘πœˆπ›Ό2×𝐹(𝜈)2𝑛𝐡(𝜈)+𝑛𝐹(𝜈+πœ”)+𝑛𝐹(πœˆβˆ’πœ”)+𝛾imp,(16) where 𝑛𝐹 is the Fermi distribution function. For completeness we give also the expression for the quasiparticle effective mass π‘šβˆ—(πœ”): π‘šβˆ—(πœ”)π‘š1=1+πœ”ξ“π‘™ξ€œβˆž0π‘‘πœˆπ›Ό2π‘™πΉπ‘™ξ‚†πœ“ξ‚€1(𝜈)Γ—Re2+π‘–πœ”+πœˆξ‚ξ‚€12πœ‹π‘‡βˆ’πœ“2βˆ’π‘–πœ”βˆ’πœˆ.2πœ‹π‘‡ξ‚ξ‚‡(17) The term 𝛾imp is due to the impurity scattering. By comparing (14) and (16), it is seen that 𝛾tr and 𝛾 are different quantities, that is, 𝛾tr≠𝛾, where the former describes the relaxation of Bose particles (excited electron-hole pairs) while the latter one describes the relaxation of Fermi particles. This difference persists also at 𝑇=0K where one has (due to simplicity we omit in the following summation over 𝑙) [32] 𝛾tr(πœ”)=2πœ‹πœ”ξ€œπœ”0π‘‘πœˆ(πœ”βˆ’πœˆ)𝛼2trξ€œ(𝜈)𝐹(𝜈),𝛾(πœ”)=2πœ‹πœ”0π‘‘πœˆπ›Ό2(𝜈)𝐹(𝜈).(18) In the case of EPI with the constant electronic density of states, the above equations give that 𝛾𝑒𝑝(πœ”)=const for πœ”>πœ”maxph while 𝛾𝑒𝑝,tr(πœ”) (as well as π›Ύβˆ—π‘’π‘,tr) is monotonic growing for πœ”>πœ”maxph, where πœ”maxph is the maximal phonon frequency. So, the growing of 𝛾𝑒𝑝,tr(πœ”) (and π›Ύβˆ—π‘’π‘,tr) for πœ”>πœ”maxph is ubiquitous and natural for the EPI scattering and has nothing to do with some exotic scattering mechanism. This behavior is clearly seen by comparing 𝛾(πœ”,𝑇),𝛾tr(πœ”,𝑇), and π›Ύβˆ—tr which are calculated for the EPI spectral function 𝛼2𝑒𝑝(πœ”)𝐹ph(πœ”) extracted from tunnelling experiments in YBCO (with πœ”maxph∼80meV ) [42–45]β€”see Figure 4.

fig4
Figure 4: (a) Scattering rates 𝛾(πœ”,𝑇),𝛾tr(πœ”,𝑇), and π›Ύβˆ—trβ€”from top to bottomβ€”for the Eliashberg function in (b). From [33–35]. (b) Eliashberg spectral function 𝛼2𝑒𝑝(πœ”)𝐹ph(πœ”) obtained from tunnelling experiments on break junctions [42–45]. Inset shows π›Ύβˆ—tr with (full line) and without (dashed line) interband transitions [3–5].

The results shown in Figure 4 clearly demonstrate the physical difference between two scattering rates 𝛾𝑒𝑝 and 𝛾𝑒𝑝,tr (or π›Ύβˆ—tr). It is also seen that π›Ύβˆ—tr(πœ”,𝑇) is even more linear function of πœ” than 𝛾tr(πœ”,𝑇). From these calculations one concludes that the quasilinearity of 𝛾tr(πœ”,𝑇) (and π›Ύβˆ—tr) is not in contradiction with the EPI scattering mechanism but it is in fact a natural consequence of EPI. We stress that such behavior of 𝛾𝑒𝑝 and 𝛾𝑒𝑝,tr (and π›Ύβˆ—π‘’π‘,tr), shown in Figure 4, is in fact not exceptional for HTSC cuprates but it is generic for many metallic systems, for instance, 3D metallic oxides, low-temperature superconductors such as Al, Pb, and so forthβ€”see more in [3–6] and references therein.

Let us discuss briefly the experimental results for 𝑅(πœ”) and π›Ύβˆ—tr(πœ”,𝑇) and compare these with theoretical predictions obtained by using a single-band model with 𝛼2𝑒𝑝(πœ”)𝐹ph(πœ”) extracted from the tunnelling data with the EPI coupling constant πœ†π‘’π‘β‰³2 [42–45]. In the case of YBCO the agreement between measured and calculated 𝑅(πœ”) is very good up to energies πœ”<6000cmβˆ’1, which confirms the importance of EPI in scattering processes. For higher energies, where a mead-infrared peak appears, it is necessary to account for interband transitions [3–5]. In optimally doped Bi2Sr2CaCu2O6 (𝐡𝑖2212) [97, 98] the experimental results for π›Ύβˆ—tr(πœ”,𝑇) are explained theoretically by assuming that the EPI spectral function 𝛼2𝑒𝑝(πœ”)𝐹(πœ”)∼𝐹ph(πœ”), where 𝐹ph(πœ”) is the phononic density of states in BSCO, with πœ†π‘’π‘=1.9 and 𝛾impβ‰ˆ320cmβˆ’1β€”see Figure 5(a). At the same time the fit of π›Ύβˆ—tr(πœ”,𝑇) by the marginal Fermi liquid phenomenology fails as it is evident in Figure 5(b).

fig5
Figure 5: (a) Experimental transport scattering rate π›Ύβˆ—tr (solid lines) for BSCO and the theoretical curve by using (A.20) and transport mass π‘šβˆ—tr with 𝛼2𝐹(πœ”) due to EPI which is described in text (dashed lines). (b) Comparison with the marginal Fermi liquid theoryβ€”dashed lines. From [3–5, 99].

Now we will comment the so called pronounced linear behavior of 𝛾tr(πœ”,𝑇) (and π›Ύβˆ—tr(πœ”,𝑇)) which was one of the main arguments for numerous inadequate conclusions regarding the scattering and pairing bosons and EPI. We stress again that the measured quantity is reflectivity 𝑅(πœ”) and derived ones are 𝜎(πœ”),𝛾tr(πœ”,𝑇), and π‘štr(πœ”), which are very sensitive to the value of the dielectric constant πœ€βˆž. This sensitivity is clearly demonstrated in Figure 6 for Bi-2212 where it is seen that 𝛾tr(πœ”,𝑇) (and π›Ύβˆ—tr(πœ”,𝑇)) for πœ€βˆž=1 is linear up to much higher πœ” than in the case πœ€βˆž>1.

423725.fig.006
Figure 6: Dependence of π›Ύβˆ—tr(πœ”,𝑇) on πœ€βˆž in Bi2Sr2CaCu2O8 for different temperatures: πœ€βˆž=4 (solid lines) and πœ€βˆž=1 (dashed lines). On the horizontal axis is πœ” in units cmβˆ’1. From [99].

However, in some experiments [100–103] the extracted 𝛾tr(πœ”,𝑇) (and π›Ύβˆ—tr(πœ”,𝑇)) is linear up to very high πœ”β‰ˆ1500cmβˆ’1. This means that the ion background and interband transitions (contained in πœ€βˆž) are not properly taken into account since too small πœ€βˆž (≳1) is assumed. The recent elipsometric measurements on YBCO [104] give the value πœ€βˆžβ‰ˆ4-6, which gives much less spectacular linearity in the relaxation rates 𝛾tr(πœ”,𝑇) (and π›Ύβˆ—tr(πœ”,𝑇)) than it was the case immediately after the discovery of HTSC cuprates, where much smaller πœ€βˆž was assumed.

Furthermore, we would like to comment two points related to 𝜎,𝛾tr, and 𝛾. First, the parametrization of 𝜎(πœ”) with the generalized Drude formula in (12) and its relation to the transport scattering rate 𝛾tr(πœ”,𝑇) and the transport mass π‘štr(πœ”,𝑇) is useful if we deal with electron-boson scattering in a single-band problem. In [36, 96] it is shown that 𝜎(πœ”) of a two-band model with only elastic impurity scattering can be represented by the generalized (extended) Drude formula with πœ” and 𝑇 dependence of effective parameters 𝛾efftr(πœ”,𝑇), π‘šefftr(πœ”,𝑇) despite the fact that the inelastic electron-boson scattering is absent. To this end we stress that the single-band approach is justified for a number of HTSC cuprates such as LSCO, BSCO, and so forth. Second, at the beginning we said that 𝛾tr(πœ”,𝑇) and 𝛾(πœ”,𝑇) are physically different quantities and it holds that 𝛾tr(πœ”,𝑇)≠𝛾(πœ”,𝑇). In order to give the physical picture and qualitative explanation for this difference we assume that 𝛼2tr𝐹(𝜈)β‰ˆπ›Ό2𝐹(𝜈). In that case the renormalized quasiparticle frequency ξ‚πœ”(πœ”)=𝑍(πœ”)πœ”=πœ”βˆ’Ξ£(πœ”) and the transport one ξ‚πœ”tr(πœ”)β€”defined in (12)β€”are related and at 𝑇=0 they are given by [32, 36] ξ‚πœ”tr1(πœ”)=πœ”ξ€œπœ”0π‘‘πœ”ξ…žξ€·πœ”2ξ‚πœ”ξ…žξ€Έ.(19) (For the definition of 𝑍(πœ”) see Appendix A.) It gives the relation between 𝛾tr(πœ”) and 𝛾(πœ”) as well asπ‘štr(πœ”) and π‘šβˆ—(πœ”), respectively: 𝛾tr1(πœ”)=πœ”ξ€œπœ”0π‘‘πœ”ξ…žπ›Ύξ€·πœ”ξ…žξ€Έ,πœ”π‘štr1(πœ”)=πœ”ξ€œπœ”0π‘‘πœ”ξ…ž2πœ”ξ…žπ‘šβˆ—ξ€·πœ”ξ…žξ€Έ.(20) The physical meaning of (19) is the following: in optical measurements one photon with the energy πœ” is absorbed and two excited particles (electron and hole) are created above and below the Fermi surface. If the electron has energy πœ”ξ…ž and the hole πœ”βˆ’πœ”ξ…ž, then they relax as quasiparticles with the renormalized frequency ξ‚πœ”. Since πœ”ξ…ž takes values 0<πœ”ξ…ž<πœ”, then the optical relaxation ξ‚πœ”tr(πœ”) is the energy-averaged ξ‚πœ”(πœ”) according to (19). The factor 2 is due to the two quasiparticlesβ€”electron and hole. At finite 𝑇, the generalization reads [32, 36] ξ‚πœ”tr1(πœ”)=πœ”ξ€œβˆž0π‘‘πœ”ξ…žξ€Ί1βˆ’π‘›πΉξ€·πœ”ξ…žξ€Έβˆ’π‘›πΉξ€·πœ”βˆ’πœ”ξ…žξ€·πœ”ξ€Έξ€»2ξ‚πœ”ξ…žξ€Έ.(21)

(2) Inversion of the Optical Data and 𝛼2tr(πœ”)𝐹(πœ”)
In principle, the transport spectral function 𝛼2tr(πœ”)𝐹(πœ”) can be extracted from 𝜎(πœ”) (or 𝛾tr(πœ”)) only at 𝑇=0K, which follows from (18) as 𝛼2trπœ”(πœ”)𝐹(πœ”)=2𝑝8πœ‹2πœ•2πœ•πœ”2ξ‚Έ1πœ”Reξ‚Ή,𝜎(πœ”)(22) or equivalently as 𝛼2tr(πœ”)𝐹(πœ”)=(1/2πœ‹)πœ•2(πœ”π›Ύtr(πœ”))/πœ•πœ”2. However, real measurements are performed at finite 𝑇 (at 𝑇>𝑇𝑐 which is rather high in HTSC cuprates) and the inversion procedure is an ill-posed problem since 𝛼2tr(πœ”)𝐹(πœ”) is the deconvolution of the inhomogeneous Fredholm integral equation of the first kind with the temperature-dependent Kernel 𝐾2(πœ”,𝜈,𝑇)β€”see (14). It is known that an ill-posed mathematical problem is very sensitive to input since experimental data contain less information than one needs. This procedure can cause, first, that the fine structure of 𝛼2tr(πœ”)𝐹(πœ”)  get blurred (most peaks are washed out) in the extraction procedures and, second, the extracted 𝛼2tr(πœ”)𝐹(πœ”)  be temperature dependent even when the true 𝛼2tr(πœ”)𝐹(πœ”) is 𝑇 independent. This artificial 𝑇 dependence is especially pronounced in HTSC cuprates because 𝑇𝑐(~100 K) is very high. In the context of HTSC cuprates, this problem was first studied in [33–36] where this picture is confirmed by the following results: (1) the extracted shape of 𝛼2tr(πœ”)𝐹(πœ”) in YBa2Cu3O7βˆ’π‘₯ as well as in other cuprates is not unique and it is temperature dependent, that is, at higher 𝑇>𝑇𝑐 the peak structure is smeared and only a single peak (slightly shifted to higher πœ”) is present. For instance, the experimental data of 𝑅(πœ”) in YBCO were reproduced by two different spectral functions 𝛼2tr(πœ”)𝐹(πœ”), one with single peak and the other one with three-peak structure as it is shown in Figure 7, where all spectral functions give almost identical 𝑅(πœ”). The similar situation is realized in optimally doped BSCO as it is seen in Figure 8 where again different functions 𝛼2(πœ”)𝐹(πœ”) reproduce very well curves for 𝑅(πœ”) and 𝜎(πœ”). However, it is important to stress that the obtained width of the extracted 𝛼2tr(πœ”)𝐹(πœ”) in both compounds coincide with the width of the phonon density of states 𝐹ph(πœ”) [33–36, 99]. (2) The upper energy bound for 𝛼2tr(πœ”)𝐹(πœ”) is extracted in [33–36] and it coincides approximately with the maximal phonon frequency in cuprates πœ”maxph≲80meV as it is seen in Figures 7 and 8.

423725.fig.007
Figure 7: Experimental (solid lines) and calculated (dashed lines) data of 𝑅(πœ”) in optimally doped YBCO [105] at 𝑇=100,200,300K (from top to bottom). Inset: the two (solid and dashed lines) reconstructed 𝛼2tr(πœ”)𝐹(πœ”)’s at 𝑇=100K. The phonon density of states 𝐹(πœ”)β€”dotted line in the inset. From [33–35].
423725.fig.008
Figure 8: Experimental (solid line) and calculated (dashed line) data of 𝑅(πœ”) in optimally doped BSCO [106] at 𝑇=100K. Inset: the reconstructed 𝛼2tr(πœ”)𝐹(πœ”)β€”solid line. The phonon density of states 𝐹(πœ”)β€”dotted line. From [33–35].

These results demonstrate the importance of EPI in cuprates [33–36]. We point out that the width of 𝛼2tr(πœ”)𝐹(πœ”) which is extracted from the optical measurements [33–36] coincides with the width of the quasiparticle spectral function 𝛼2(πœ”)𝐹(πœ”) obtained in tunnelling and ARPES spectra (which we will discuss below), that is, both functions are spread over the energy interval 0<πœ”<πœ”maxph(≲80meV). Since in cuprates this interval coincides with the width in the phononic density of states 𝐹(πœ”) and since the maxima of 𝛼2(πœ”)𝐹(πœ”) and 𝐹(πœ”) almost coincide, this is further evidence for the importance of EPI.

To this end, we would like to comment two aspects which appear from time to time in the literature. First, in some reports [24–27] it is assumed that 𝛼2tr(πœ”)𝐹(πœ”) of cuprates can be extracted also in the superconducting state by using (22). However, (22) holds exclusively in the normal state (at 𝑇=0) since 𝜎(πœ”) can be described by the generalized (extended) Drude formula in (12) only in the normal state. Such an approach does not hold in the superconducting state since the dynamical conductivity depends not only on the electron-boson scattering but also on coherence factors and on the momentum and energy dependent order parameter Ξ”(𝐀,πœ”). Second, if 𝑅(πœ”)’s (and 𝜎(πœ”)’s) in cuprates are due to some other bosonic scattering which is pronounced up to much higher energies πœ”π‘β‰«πœ”maxph, this should be seen in the width of the extracted spectral function 𝛼2tr(πœ”)𝐹(πœ”). In that respect in [25–27] it is assumed that SFI dominates in the quasiparticle scattering and that 𝛼2tr(πœ”)𝐹(πœ”)βˆΌπ‘”2sfImπœ’(πœ”) where βˆ«π‘‘Imπœ’(πœ”)=2π‘˜πœ’(𝐀,πœ”). This claim is based on reanalyzing of some IR measurements [25–27] and the transport spectral function 𝛼2tr(πœ”)𝐹(πœ”) is extracted in [25] by using the maximum entropy method in solving the Fredholm equation. However, in order to exclude negative values in the extracted 𝛼2tr(πœ”)𝐹(πœ”), which is an artefact and due to the chosen method, in [25] it is assumed that 𝛼2tr(πœ”)𝐹(πœ”) has a rather large tail at large energiesβ€”up to 400 meV. It turns out that even such an assumption in extracting 𝛼2tr(πœ”)𝐹(πœ”) does not reproduce the experimental curve Imπœ’(πœ”) [107] in some important aspects. First, the relative heights of the two peaks in the extracted spectral function 𝛼2tr(πœ”)𝐹(πœ”) at lower temperatures are opposite to the experimental curve Imπœ’(πœ”) [107]β€”see [25, Figure  1]. Second, the strong temperature dependence of the extracted 𝛼2tr(πœ”)𝐹(πœ”) found in [25–27] is not an intrinsic property of the spectral function but it is an artefact due to the high sensitivity of the extraction procedure on temperature. As it is already explained before, this is due to the ill-posed problem of solving the Fredholm integral equation of the first kind with strong 𝑇-dependent kernel. Third, the extracted spectral weight 𝐼𝐡(πœ”)=𝛼2tr(πœ”)𝐹(πœ”) in [25] has much smaller values at larger frequencies (πœ”>100meV) than it is the case for the measured Imπœ’(πœ”), that is, (𝐼𝐡(πœ”>100meV)/𝐼𝐡(πœ”max))β‰ͺImπœ’(πœ”>100meV)/Imπœ’(πœ”max)β€”see [25, Figure  1]. Fourth, the recent magnetic neutron scattering measurements on optimally doped large-volume crystals Bi2Sr2CaCu2O8+𝛿 [93] (where the absolute value of Imπœ’(πͺ,πœ”) is measured) are not only questioning the theoretical interpretation of magnetism in HTSC cuprates in terms of itinerant magnetism but also opposing the finding in [25–27]. Namely, this experiment shows that the local spin susceptibility βˆ‘Imπœ’(πœ”)=π‘žImπœ’(πͺ,πœ”) is temperature independent across the superconducting transition 𝑇𝑐=91K, that is, there is only a minimal change in Imπœ’(πœ”) between 10K and 100K. This 𝑇-independence of Imπœ’(πœ”) strongly opposes the (above discussed) results in [24–27], where the fit of optic measurements gives strong 𝑇 dependence of Imπœ’(πœ”).

Fifth, the transport coupling constant πœ†tr extracted in [25] is too large, that is, πœ†tr>3 contrary to the previous findings that πœ†tr≲1.5 [33–36, 99]. Since in HTSC one has πœ†>πœ†tr, this would probably give πœ†β‰ˆ6, which is not confirmed by other experiments. Sixth, the interpretation of 𝛼2tr(πœ”)𝐹(πœ”) in LSCO and BSCO solely in terms of Imπœ’(πœ”) is in contradiction with the magnetic neutron scattering in the optimally doped and slightly underdoped YBCO [30]. The latter was discussed in Section 1.3.1, where it is shown that Imπœ’(𝐐,πœ”) is small in the normal state and its magnitude is even below the experimental noise. This means that if the assumption that 𝛼2tr(πœ”)𝐹(πœ”)β‰ˆπ‘”2sfImπœ’(πœ”) were correct then the contribution to Imπœ’(πœ”) from the momenta 0<π‘˜β‰ͺ𝑄 would be dominant, which is detrimental for 𝑑-wave superconductivity.

Finally, we point out that very similar (to cuprates) properties, of 𝜎(πœ”),𝑅(πœ”) (and 𝜌(𝑇) and electronic Raman spectra), were observed in 3D isotropic metallic oxides La0.5Sr0.5CoO3 and Ca0.5Sr0.5RuO3 which are nonsuperconducting [108] and in Ba1βˆ’π‘₯Kπ‘₯BiO3 which is superconducting below 𝑇𝑐≃30K at π‘₯=0.4. This means that in all of these materials the scattering mechanism might be of similar origin. Since in these compounds there are no signs of antiferromagnetic fluctuations (which are present in cuprates), then the EPI scattering plays important role also in other oxides.

(3) Restricted Optical Sum Rule
The restricted optical sum rule was studied intensively in HTSC cuprates. It shows some peculiarities not present in low-temperature superconductors. It turns out that the restricted spectral weight π‘Š(Ω𝑐,𝑇) is strongly temperature dependent in the normal and superconducting state, which was interpreted either to be due to EPI [39, 40] or to some nonphononic mechanisms [109]. In the following we demonstrate that the temperature dependence of π‘Š(Ω𝑐,𝑇)=π‘Š(0)βˆ’π›½π‘‡2 in the normal state can be explained in a natural way by the 𝑇 dependence of the EPI transport relaxation rate 𝛾𝑒𝑝tr(πœ”,𝑇) [39, 40]. Since the problem of the restricted sum rule has attracted much interest, it will be considered here in some details. In fact, there are two kinds of sum rules related to 𝜎(πœ”). The first one is the total sum rule which in the normal state reads ξ€œβˆž0π‘‘πœ”πœŽπ‘1πœ”(πœ”)=2pl8=πœ‹π‘›π‘’2,2π‘š(23) while in the superconducting state it is given by the Tinkham-Ferrell-Glover (TFG) sum rule ξ€œβˆž0π‘‘πœ”πœŽπ‘†1𝑐(πœ”)=28πœ†2𝐿+ξ€œβˆž+0π‘‘πœ”πœŽπ‘†1πœ”(πœ”)=2pl8.(24) Here,𝑛 is the total electron density, 𝑒 is the electron charge, π‘š is the bare electron mass, and πœ†πΏ is the London penetration depth. The first (singular) term 𝑐2/8πœ†2𝐿 in (24) is due to the superconducting condensate which contributes πœŽπ‘†1,cond(πœ”)=(𝑐2/4πœ†2𝐿)𝛿(πœ”). The total sum rule represents the fundamental property of matterβ€”the conservation of the electron number. In order to calculate it one should use the total Hamiltonian 𝐻tot=𝑇𝑒+𝐻int where all electrons, electronic bands, and their interactions 𝐻int (Coulomb, EPI, with impurities, etc.) are accounted for. Here, 𝑇𝑒 is the kinetic energy of bare electrons: 𝑇𝑒=ξ“πœŽξ€œπ‘‘0π‘₯0200𝑑3π‘₯ξπœ“β€ πœŽΜ‚π©(π‘₯)22π‘šξπœ“πœŽξ“(π‘₯)=𝐩,𝜎𝐩22π‘šπ‘’Μ‚π‘β€ π©πœŽΜ‚π‘π©πœŽ.(25)

The partial sum rule is related to the energetics solely in the conduction (valence) band which is described by the Hamiltonian of the conduction (valence) band electrons: 𝐻𝑣=𝐩,πœŽπœ‰π©Μ‚π‘β€ π‘£,π©πœŽΜ‚π‘π‘£,𝐩𝜎+𝑉𝑣,𝑐.(26)𝐻𝑣 contains the band energy with the dispersion πœ–π© (πœ‰π©=πœ–π©βˆ’πœ‡) and the effective Coulomb interaction of the valence electrons 𝑉𝑣,𝑐. In this case the partial sum rule in the normal state reads [110] (for a general form of πœ–π©) ξ€œβˆž0π‘‘πœ”πœŽπ‘1,𝑣(πœ”)=πœ‹π‘’22𝑉𝐩0π‘₯0200𝑑̂𝑛𝑣,π©ξ¬π»π‘£π‘šπ©,(27) where the number operator ̂𝑛𝑣,𝐩=βˆ‘πœŽΜ‚π‘β€ π©πœŽΜ‚π‘π©πœŽ;1/π‘šπ©=πœ•2πœ–π©/πœ•π‘2π‘₯ is the momentum-dependent reciprocal mass and 𝑉 is volume. In practice, the optic measurements are performed up to finite frequency and the integration over πœ” goes up to some cutoff frequency Ω𝑐 (of the order of the band plasma frequency). In this case the restricted sum rule has the form π‘Šξ€·Ξ©π‘ξ€Έ=ξ€œ,𝑇Ω𝑐0π‘‘πœ”πœŽπ‘1,𝑣=πœ‹(πœ”)2ξ€ΊπΎπ‘‘ξ€»βˆ’ξ€œ+Ξ (0)Ω𝑐0π‘‘πœ”ImΞ (πœ”)πœ”,(28) where 𝐾𝑑 is the diamagnetic Kernel given by (30) below and Ξ (πœ”) is the paramagnetic (current-current) response function. In the perturbation theory without vertex correction Ξ (π‘–πœ”π‘›) (at the Matsubara frequency πœ”π‘›=πœ‹π‘‡(2𝑛+1)) is given by [39, 40] Π(π‘–πœ”)=2π©ξ‚΅πœ•πœ–π©ξ‚Άπœ•π©2ξ“πœ”π‘šπΊξ€·π©,π‘–πœ”+π‘›π‘šξ€ΈπΊξ€·π©,π‘–πœ”π‘šξ€Έ,(29) where πœ”+π‘›π‘š=πœ”π‘›+πœ”π‘š and 𝐺(𝐩,π‘–πœ”π‘›)=(π‘–πœ”π‘›βˆ’πœ‰π©βˆ’Ξ£(𝐩,π‘–πœ”π‘›))βˆ’1 is the electron Green's function. In the case when the interband gap 𝐸𝑔 is the largest scale in the problem, that is, when π‘Šπ‘<Ω𝑐<𝐸𝑔, in this region one has approximately ImΞ (πœ”)β‰ˆ0 and the limit Ξ©π‘β†’βˆž in (28) is justified. In that case one has ∫Π(0)β‰ˆΞ©π‘0(ImΞ (πœ”)/πœ”)π‘‘πœ” which gives the approximate formula for π‘Š(Ω𝑐,𝑇): π‘Šξ€·Ξ©π‘ξ€Έ=ξ€œ,𝑇Ω𝑐0π‘‘πœ”πœŽπ‘1,π‘£πœ‹(πœ”)β‰ˆ2𝐾𝑑=𝑒2πœ‹ξ“π©πœ•2πœ–π©πœ•π©2𝑛𝐩,(30) where 𝑛𝐩=βŸ¨Μ‚π‘›π‘£,𝐩⟩ is the quasiparticle distribution function in the interacting system. Note that π‘Š(Ω𝑐,𝑇) is cutoff dependent while 𝐾𝑑 in (30) does not depend on Ω𝑐. So, one should be careful not to overinterpret the experimental results in cuprates by this formula. In that respect the best way is to calculate π‘Š(Ω𝑐,𝑇) by using the exact result in (28) which apparently depends on Ω𝑐. However, (30) is useful for appropriately chosen Ω𝑐, since it allows us to obtain semiquantitative results. In most papers related to the restricted sum rule in HTSC cuprates it was assumed, due to simplicity, the tight-binding model with nearest neighbors (n.n.) with the energy πœ–π©=βˆ’2𝑑(cos𝑝π‘₯π‘Ž+cosπ‘π‘¦π‘Ž) which gives 1/π‘šπ©=βˆ’2π‘‘π‘Ž2cos𝑝π‘₯π‘Ž. It is straightforward to show that in this case (30) is reduced to a simpler one: π‘Šξ€·Ξ©π‘ξ€Έ=ξ€œ,𝑇Ω𝑐0π‘‘πœ”πœŽπ‘1,π‘£β‰ˆ(πœ”)πœ‹π‘’2π‘Ž22π‘‰βŸ¨βˆ’π‘‡π‘£βŸ©,(31) where βŸ¨π‘‡π‘£βŸ©π»π‘£=βˆ‘π©πœ–π©βŸ¨π‘›π‘£βŸ©π»π‘£ is the average kinetic energy of the band electrons, π‘Ž is the Cu–Cu lattice distance, and 𝑉 is the volume of the system. In this approximation π‘Š(Ω𝑐,𝑇) is a direct measure of the average band (kinetic) energy. In the superconducting state the partial band sum rule reads π‘Šπ‘ ξ€·Ξ©π‘ξ€Έ=𝑐,𝑇28πœ†2𝐿+ξ€œΞ©π‘+0π‘‘πœ”πœŽπ‘†1,𝑣=(πœ”)πœ‹π‘’2π‘Ž22π‘‰βŸ¨βˆ’π‘‡π‘£βŸ©π‘ .(32) In order to introduce the reader to (the complexity of) the problem of the 𝑇 dependence of π‘Š(Ω𝑐,𝑇), let us consider the electronic system in the normal state and in absence of the quasiparticle interaction. In that case one has 𝑛𝐩=𝑓𝐩 (𝑓𝐩 is the Fermi distribution function) and π‘Šπ‘›(Ω𝑐,𝑇) increases with the decrease of the temperature, that is, π‘Šπ‘›(Ω𝑐,𝑇)=π‘Šπ‘›(0)βˆ’π›½π‘π‘‡2 where π›½π‘βˆΌ1/π‘Šπ‘. To this end, let us mention in advance that the experimental value 𝛽exp is much larger than 𝛽𝑏, that is, 𝛽exp≫𝛽𝑏, thus telling us that the simple Sommerfeld-like smearing of 𝑓𝐩 by the temperature effects cannot explain quantitatively the 𝑇 dependence of π‘Š(Ω𝑐,𝑇). We stress that the smearing of 𝑓𝐩 by temperature lowers the spectral weight compared to that at 𝑇=0K, that is, π‘Šπ‘›(Ω𝑐,𝑇)<π‘Šπ‘›(Ω𝑐,0). In that respect it is not surprising that there is a lowering of π‘Šπ‘ (Ω𝑐,𝑇) in the BCS superconducting state, π‘ŠBCS𝑠(Ω𝑐,𝑇β‰ͺ𝑇𝑐)<π‘Šπ‘›(Ω𝑐,𝑇β‰ͺ𝑇𝑐) since at low temperatures 𝑓𝐩 is smeared mainly due to the superconducting gap, that is, 𝑓𝐩=[1βˆ’(πœ‰π©/𝐸𝐩)th(𝐸𝐩/2𝑇)]/2, 𝐸𝐩=ξ”πœ‰2𝐩+Ξ”2, πœ‰π©=πœ–π©βˆ’πœ‡. The maximal decrease of π‘Šπ‘ (Ω𝑐,𝑇) is at 𝑇=0.

Let us enumerate and discuss the main experimental results for π‘Š(Ω𝑐,𝑇) in HTSC cuprates. (1) In the normal state (𝑇>𝑇𝑐) of most cuprates, one has π‘Šπ‘›(Ω𝑐,𝑇)=π‘Šπ‘›(0)βˆ’π›½ex𝑇2 with 𝛽exp≫𝛽𝑏, that is, π‘Šπ‘›(Ω𝑐,𝑇) is increasing by decreasing 𝑇, even at 𝑇 below π‘‡βˆ—β€”the temperature for the opening of the pseudogap. The increase of π‘Šπ‘›(Ω𝑐,𝑇) from the room temperature down to 𝑇𝑐 is no more than 5%. (2) In the superconducting state (𝑇<𝑇𝑐) of some underdoped and optimally doped Bi-2212 compounds [111, 113, 114] (and underdoped Bi-2212 films [115]) there is an effective increase of π‘Šπ‘ (Ω𝑐,𝑇) with respect to that in the normal state, that is, π‘Šπ‘ (Ω𝑐,𝑇)>π‘Šπ‘›(Ω𝑐,𝑇) for 𝑇<𝑇𝑐. This is a non-BCS behavior which is shown in Figure 9. Note that in the tight binding model the effective band (kinetic) energy βŸ¨π‘‡π‘£βŸ© is negative (βŸ¨π‘‡π‘£βŸ©<0) and in the standard BCS case (32) gives that π‘Šπ‘ (𝑇<𝑇𝑐)  decreases due to the increase of βŸ¨π‘‡π‘£βŸ©. Therefore the experimental increase of π‘Šπ‘ (𝑇<𝑇𝑐) by decreasing 𝑇 is called the non-BCS behavior. The latter means a lowering of the kinetic energy βŸ¨π‘‡π‘£βŸ© which is frequently interpreted to be due either to strong correlations or to a Bose-Einstein condensation (BEC) of the preformed tightly bound Cooper pair-bosons, for instance, bipolarons [116]. It is known that in the latter case the kinetic energy of bosons is decreased below the BEC critical temperature 𝑇𝑐. In [117] it is speculated that the latter case might be realized in underdoped cuprates.

423725.fig.009
Figure 9: Measured spectral weight π‘Šπ‘ (Ω𝑐,𝑇)(~𝐴𝑙+𝐷 in figures) for Ξ©π‘β‰ˆ1.25eV in two underdoped Bi2212 (with 𝑇𝑐=88K and 𝑇𝑐=66K). From [111].

However, in some optimally doped and in most overdoped cuprates, there is a decrease of π‘Šπ‘ (Ω𝑐,𝑇) at 𝑇<𝑇𝑐 (π‘Šπ‘ (Ω𝑐,𝑇)<π‘Šπ‘›(Ω𝑐,𝑇)) which is the BCS-like behavior [112] as it is seen in Figure 10.

fig10
Figure 10: (a) Spectral weight π‘Šπ‘›(Ω𝑐,𝑇) of the overdoped Bi2212 for Ω𝑐=1eV. Closed symbolsβ€”normal state. Open symbolsβ€”superconducting state. (b) Change of the kinetic energy Δ𝐸kin=𝐸kin,π‘†βˆ’πΈkin,𝑁 in meV per Cu site versus the charge 𝑝 per Cu with respect to the optimal value 𝑝opt. From [112].

We stress that the non-BCS behavior of π‘Šπ‘ (Ω𝑐,𝑇) for underdoped (and in some optimally doped) systems was obtained by assuming that Ξ©π‘β‰ˆ(1-1.2)eV. However, in [104] these results were questioned by claiming that the conventional BCS-like behavior was observed (π‘Šπ‘ (Ω𝑐,𝑇)<π‘Šπ‘›(Ω𝑐,𝑇)) in the optimally doped YBCO and slightly underdoped Bi-2212 by using larger cutoff energy Ω𝑐=1.5eV. This discussion demonstrates how risky is to make definite conclusions on some fundamental physics based on the parameter- (such as the cutoff energy Ω𝑐) dependent analysis. Although the results obtained in [104] look very trustfully, it is fair to say that the issue of the reduced spectral weight in the superconducting state of the underdoped cuprates is still unsettled and under dispute. In overdoped Bi-2212 films, the BCS-like behavior π‘Šπ‘ (Ω𝑐,𝑇)<π‘Šπ‘›(Ω𝑐,𝑇) was observed, while in LSCO it was found that π‘Šπ‘ (Ω𝑐,𝑇)β‰ˆconst, that is, π‘Šπ‘ (Ω𝑐,𝑇<𝑇𝑐)β‰ˆπ‘Šπ‘›(Ω𝑐,𝑇𝑐).

The first question is the following. How to explain the strong temperature dependence of π‘Š(Ω𝑐,𝑇) in the normal state? In [39, 40] π‘Š(𝑇) is explained solely in the framework of the EPI physics where the EPI relaxation 𝛾𝑒𝑝(𝑇) plays the main role in the 𝑇 dependence of π‘Š(Ω𝑐,𝑇). The main theoretical results of [39, 40] are the following: the calculations of π‘Š(𝑇) based on the exact (30) give that for Ω𝑐≫Ω𝐷 (the Debye energy) the difference in spectral weights of the normal and superconducting states is small, that is, π‘Šπ‘›(Ω𝑐,𝑇)β‰ˆπ‘Šπ‘ (Ω𝑐,𝑇) since π‘Šπ‘›(Ω𝑐,𝑇)βˆ’π‘Šπ‘ (Ω𝑐,𝑇)βˆΌΞ”2/Ξ©2𝑐. (2) In the case of large Ω𝑐 the calculations based on (30) give π‘Šξ€·Ξ©π‘ξ€Έβ‰ˆπœ”,𝑇2pl8𝛾1βˆ’(𝑇)π‘Šπ‘βˆ’πœ‹22𝑇2π‘Š2𝑏.(33) In the case when EPI dominates one has 𝛾=𝛾𝑒𝑝(𝑇)+𝛾imp where π›Ύπ‘’π‘βˆ«(𝑇)=∞0𝑑𝑧𝛼2(𝑧)𝐹(𝑧)coth(𝑧/2𝑇). It turns out that for 𝛼2(πœ”)𝐹(πœ”), shown in Figure 4, one obtains (i) 𝛾𝑒𝑝(𝑇)βˆΌπ‘‡2 in the temperature interval 100K<𝑇<200K as it is seen in Figure 11 for the 𝑇 dependence of π‘Š(Ω𝑐,𝑇) [39, 40]; (ii) the second term in (33) is much larger than the last one (the Sommerfeld-like term). For the EPI coupling constant πœ†π‘’π‘,tr=1.5 one obtains rather good agreement between the theory in [39, 40] and experiments in [104, 111, 113, 114]. At lower temperatures, 𝛾𝑒𝑝(𝑇) deviates from the 𝑇2 behavior and this deviation depends on the structure of the spectrum in 𝛼2(πœ”)𝐹(πœ”). It is seen in Figure 11 that, for a softer Einstein spectrum (with Ω𝐸=200K), π‘Š(Ω𝑐,𝑇) lies above the curve with the 𝑇2 asymptotic behavior, while the curve with a harder phononic spectrum (with Ω𝐸=400K) lies below it. This result means that different behavior of π‘Š(Ω𝑐,𝑇) in the superconducting state of cuprates for different doping might be simply related to different contributions of low- and high-frequency phonons. We stress that such a behavior of π‘Š(<