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

Authors: E. G. Maksimov; M. L. Kulić; O. V. Dolgov
Journal: Advances in Condensed Matter Physics
 (2010)

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## Abstract

In this paper we discuss experimental evidence related to the
structure and origin of the bosonic spectral functionα2F(ω) in high-temperature superconducting (HTSC) cuprates at
and near optimal doping. Global properties of α2F(ω), 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<λep≲3.5 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 ingredients—such as strong EPI, strong
correlations—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.


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## Body

## 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, Δ(k,ω)≈Δd(ω)(coskx-cosky). 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 A≡α·λ(ωB/Wb) fulfills the condition A≪1, where λ is the electron-boson coupling constant, ωB is the characteristic bosonic energy, Wb 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(k,k′,ω) (or its average α2F(ω)) which characterizes scattering of quasiparticle from k to k′ by exchanging bosonic energy ω. 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. Thefirst one is based on the electron-phonon interaction (EPI), with the main proponents in [3–11], where mediating bosons are phononsand where the average spectral function α2F(ω) is similar to the phonon density of states Fph(ω). Note that α2F(ω) is not the product of two functions although sometimes one defines the function α2(ω)=α2F(ω)/F(ω) 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<λep≲3, which will be discussed in detail below. In the EPI approach α2Fph(ω) is extracted from tunnelling measurements in conjunction with IR optical measurements. The HTSC cuprates are on the borderline and it is a natural question—under which condition can high Tc be realized in the nonadiabatic limit A≈1?Thesecond 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 Imχ(k-k′,ω), that is,  α2F(k,k′,ω)~gsf2Imχ(k-k′,ω) where gsf is the SFI coupling constant. NMR spectroscopy and magnetic neutron scattering give evidence that in HTSC cuprates χ(q,ω) is peaked at the antiferromagnetic wave vector Q=(π/a,π/a) 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 t-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 following—do models based solely on the Hubbard Hamiltonian show up superconductivity at sufficiently high critical temperatures (Tc~100K)? 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 (U>0) Hubbard model gives no evidence for superconductivity with highTc, 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 (U<0) Hubbard model for the same strength of U, that is, at finite temperature in the 2D model with U<0 the BKT superconducting transition is favored. Concerning the possibility of HTSC in the t-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 low—few Kelvins. These numerical results tell us that the lack of high Tc (even in 2D BKT phase) in the repulsive (U>0) single-band Hubbard model and in the t-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 highTc  in cuprates. These numerical results signal that the simple single-band Hubbard and its derivative the t-J model are insufficient to explain solely the pairing mechanism in cuprates and some additional ingredients must be included.SinceEPI 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 λep>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, λep<0.3. The inappropriateness of these statements will be discussed in the following sections.The paper is organized as follows. In Section1 we will mainly discuss experimental results in cuprates at and near optimal doping by giving also minimal theoretical explanations which are related to thebosonic spectral functionα2F(ω) as well as to the transport spectral function αtr2F(ω) 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 quasiparticles—although strongly interacting—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(k,k′,ω) and αtr2F(ω) 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, λep≥1, if the EPI spectral function is weakly momentum dependent, that is, if α2F(k,k′,ω)≈α2F(ω) 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 χ(q,ω) which is in the SFI  phenomenological approach [12–17] related to the Eliashberg spectral function, that is, α2Fsf(k,k′,ω)~gsf2Imχ(q=k-k′,ω). We stress that such an approach can be theoretically justified only in the weak coupling limit, gsf≪Wb, where Wb 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 ω) of Imχ(q,ω) (with q at and near Q=(π,π)) by doping toward the optimal doped HTSC [30, 31]. It turns out that in the optimally doped cuprates with Tc=92.5KImχ(Q,ω) 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(ω) and indirectly an approximative shape of the transport spectral function αtr2F(ω). 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(ω) and the quasiparticle relaxation rate γ(ω). 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 αtr2F(ω). Although these methods give at finite temperature T a blurred αtr2F(ω) which is (due to the ill-defined methods) temperature dependent, it turns out that the width and the shape of the extracted αtr2F(ω) 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 La2-xSrxCuO4 which gives additional evidence for importance of EPI [41] will be shortly discussed.(c)ARPES  Measurements and EPI. From these measurements the self-energy Σ(k,ω) is extracted as well as some properties of α2F(k,k′,ω). 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 (Σ(k,ω)) and its isotope dependence, which are all due to EPI; (ii) the appearance of different slopes of Σ(k,ω) at low (ω≪ωph) and high energies (ω≫ωph ) 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 EPI—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(ω) from tunnelling conductance. In this part we discuss the following items: (i) the extracted Eliashberg spectral function α2F(ω) with the coupling constant λ(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(ω) coincide with the maxima in the phonon density of states Fph(ω); (ii) the existence of eleven peaks in -d2I/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(ω) 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 B1g 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 Section1.4 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(k,k′,ω) and the transport spectral function αtr2F(ω) are defined.Finally, one can pose a question—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 thetheory of everything—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 σs(T<Tc,ω=const) at ω=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 ω~60GHz 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 (Tc≈40K) which is smaller than the theoretical value, that is, αBexp≈0.3<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 2Δ/Tc which is on optimally doped YBCO and BSCO ≈5 and 7, respectively, while in underdoped BSCO one has even (2Δ/Tc)≈20; (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 Section2. 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,me,Mion,kB and the chemical composition of superconducting materials, it was practically never realized in practice due to the complexity of many-body interactions—electron-electron and electron-lattice—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,Δ/EF,ωph/EF)≪1. 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,r′) (IBS—the ideal band structure) which includes the static self-energy (Σc0(h)(r,r′,ω=0)) due to high-energy electronic processes, that is, VIBS(r,r′)=[Ve-i(r)+VH(r)]δ(r-r′)+Σc0(h)(r,r′,ω=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,r′) and the real EPI coupling gep(r,r′)=δVIBS(r,r′)/δRn, which include high-energy many-body electronic processes—for instance, the large Hubbard U effects—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 potentialVLDA(r)—the ground-state potential—and the real EPI coupling by the “local” 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 correlationsin 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 t-J model, which allows the approximative calculation of the excited potential VIBS(r,r′) and the nonlocal EPI coupling gep(r,r′). As a result one obtains the momentum-dependent EPI coupling gep(kF,q) which is for small hole-doping (δ<0.3) strongly peaked at small transfer momenta—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 (λtr) rather smaller than the pairing one λ, that is, λtr≲λ/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 Δ(k,R), where k 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 Δ(k,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 t-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 T¯c. However, T¯c 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 highTc 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 leastin 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 q=±e, spin s=1/2, and low-energy excitations ξk(=ϵk-μ) which are much larger than their inverse life-times, that is, ξk≫1/τk~ξk2/Wb. Since the level width Γ=2/τk 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 ξkF=0 which defines the Fermi surface on which the Fermi quasiparticle distribution function nF(ξk) has finite jump at kF; (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)—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 kBΘD(≡ℏωD) (for the Debye spectrum ~ΘD/5), that is, for ξk~ΘD, one has τk-1≳ξk 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 liquidfor sufficiently strong electron-boson coupling. It is nowadays well known that, for instance, Al, Zn are weak coupling systems since for ξk~ΘD one has τk-1≪ξk and they are well described by the Landau theory. However, in (the noncanonical) cases where for higher energies ξk~ΘD one has τk-1≳ξk, 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(k,k′,ω) 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 papers—for the recent one see [67]—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 potentialVeff(k,ω) in the leading approximation looks like as for two external charges (e) embedded in the medium with the total longitudinal dielectric functionɛtot(k,ω) (k is the momentum and ω is the frequency) [68, 69], that is,
(1)Veff(k,ω)=Vext(k)ɛtot(k,ω)=4πe2k2ɛtot(k,ω).
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/k2ɛtot(k,ω). In that case Veff depends on other (than ɛtot(k,ω)) response functions. However, in the case when (1) holds, that is, when the weak-coupling limit is realized, Tc is given by Tc≈ω¯exp(-1/(λep-μ*)) [68–70]. Here, λep is the EPI coupling constant, ω¯ is an average phonon frequency, and μ* is the Coulomb pseudopotential, μ*=μ/(1+μlnEF/ω¯) (EF is the Fermi energy). The couplings λep and μ are expressed by ɛtot(k,ω=0):
(2)μ-λep=〈N(0)Veff(k,ω=0)〉=N(0)∫02kFkdk2kF24πe2k2ɛtot(k,ω=0),
where N(0) is the density of states at the Fermi surface and kF is the Fermi momentum—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 ɛtot(k,ω=0)>1 for all k. If this were correct, then from (2) it would follow that μ>λep, which limits the maximal value of Tc to the value Tcmax≈EFexp(-4-3/λep). In typical metals EF<(1–10)eV, and if one accepts the statement in [65, 66] that λep≤μ(≤0.5), one obtains Tc~(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 μ≤λep and λep>1. For instance, λep≈2.6 is realized in PbBi alloy which is definitely much higher than μ(<1), and so forth.Moreover, the basic theory tells us thatɛtot(k≠0,ω) is not the response function [68, 69] (contrary to the assumption in [65, 66]). Namely, if a small external potential δVext(k,ω) 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(k,ω)=δVext(k,ω)/ɛtot(k,ω), which means that 1/ɛtot(k,ω) is the response function. The latter obeys the Kramers-Kronig dispersion relation which implies the following stability condition [68, 69]:
(3)1ɛtot(k,ω=0)<1,k≠0,
that is, either
(4)ɛtot(k≠0,ω=0)>1,
or
(5)ɛtot(k≠0,ω=0)<0.
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 ɛtot(k≠0,ω=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, ɛ̂tot(k+Q,k+Q′,ω), and ɛtot(k,ω) is defined by ɛtot-1(k,ω)=ɛ̂tot-1(k+0,k+0,ω). In dense metallic systems with one ion per cell (such as metallic hydrogen) and with the electronic dielectric function ɛel(k,0), the macroscopic total dielectric function ɛtot(k,0) is given by [71–73]
(6)ɛtot(k,0)=ɛel(k,0)1-1/ɛel(k,0)Gep(k).
At the same time the energy of the longitudinal phonon ωl(k) is given by
(7)ωl2(k)=Ωp2ɛel(k,0)[1-ɛel(k,0)Gep(k)],
where Ωp2 is the ionic plasma frequency, and Gep is the local (electric) field correction—see [71–73]. The right condition for lattice stability requires that ωl2(k)>0, which implies that for ɛel(k,0)>0 one has ɛel(k,0)Gep(k)<1. The latter condition gives automatically ɛtot(k,0)<0. Furthermore, the calculations [71–73] show that in the metallic hydrogen (H) crystal ɛtot(k,0)<0 for all k≠0. Note that in metallic H the EPI coupling constant is very large, that is, λep≈7 and Tc may reach very large value Tc≈600K [74]. Moreover, the analyses of crystals with more ions per unit cell [71–73] give that ɛtot(k≠0,0)<0 is more a rule than an exception—see Figure 1. The physical reason for ɛtot(k≠0,0)<0 is local field effects described by Gep(k). 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 ɛtot(k,0) which may lead to ɛtot(k,0)<0.Figure 1
Inverse total static dielectric functionɛ-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 crystalsɛtot(k,0)  can be negative in the large portion of the Brillouin zone thus giving rise to λep-μ>0 in (2). This means that analytic properties of the dielectric function ɛtot(k,ω)  do not limitTcin 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 unusualionic-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 λtr (entering resistivity) rather smaller than the pairing EPI coupling constant λep(>1) (entering Tc), that is, why one has λtr(≈0.6–1.4)≪λep(~2–3.5)? (3) If EPI is ineffective for pairing in HTSC oxides, in spite of λep>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 tod-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 ɛtot(k,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<λep<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 (Ωnph≫ωph) 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 Ωnph)
(8)Z(ωn)Δn(k)≈πTc∑mΩnph∑qVnph(k,q,n,m)Δm(q)|ωm|,Z(ωn)≈1+Γepωn.
For pure d-wave pairing with the pairing potential Vnph=Vnph(θk,θq)·Θ(Ωnph-|ωn|)Θ(Ωnph-|ωn′|) with Vnph(k,q)=V0·Yd(θk)Yd(θq) and Yd(θk)=π-1/2cos2θk, one obtains Δn(k)=Δd·Θ(Ωnph-|ωn|)Yd(θk) and the equation for Tc—see [3–5]
(9)lnTcTc0≈Ψ(12)-Ψ(12+Γep2πTc).
Here Ψ is the di-gamma function. At temperatures near Tc one has Γep≈2πλepTc and the solution of (9) is approximately Tc≈Tc0exp{-λep} with Tc0≈Ωnphexp{-λnph}, λnph=N(0)V0. This means that for Tcmax~160K and λep>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 λkpZ(iνn) and λkpΔ, respectively. Usually, the spin-fluctuation kernel λsf,kp(iνn) is taken in the FLEX approximation [77]. The calculations [76] confirm the very detrimental effect of the isotropic  (k-independent) EPI on d-wave pairing due to SFI. For the bare SFI critical temperature Tc0~100K and for λep>1 the calculations give very small (renormalized) critical temperature Tc≪100K. These results tell us that a more realistic pairing interaction must be operative in cuprates and that EPI must be strongly momentum dependentand 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 indicativeinelastic 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 Imχ(k,kz,ω) in the normal state at and near the AF wave vector k=Q=(π,π) 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 Imχ(k,kz,ω) (at some energies ω<2Δ) 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 ofTc
Before discussing experimental results in cuprates on the imaginary part of the spin susceptibilityImχ(k,ω) we point out that in the (phenomenological) theories based on the spin-fluctuation interaction (SFI) the quasiparticle self-energy Σ̂sf(k,ωn) (ωn is the Matsubara frequency and τ̂0 is the Nambu matrix) in the normal and superconducting state and the effective (repulsive) pairing potential Vsf(k,ω) (where iωn→ω+iη) are assumed in the form [12–17]
(10)Σ̂sf(k,ωn)=TN∑k′,mVsf(k-k′,ωnm-)τ̂0Ĝ(k′,ωm)τ̂0,Vsf(k,ωnm-)=gsf2∫-∞∞dνπImχ(q,ν+i0+)ν-iωnm-,
where ωnm-≡ωn-ωm. Although the form of Vsf cannot be justified theoretically, except in the weak coupling limit (gsf≪Wb) 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) Imχ(q,ω) is strongly peaked at and near the AF wave vector Q=(π/a,π/a).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 givesImχ(q,ω). In that respect very indicative and impressive IMNS measurements on YBa2Cu3O6+x, which are done by Bourges group [30], demonstrate that the normal-state susceptibility Imχ(odd)(q,ω) (the odd part of the spin susceptibility in the bilayer system) at q=Q=(π,π) is strongly dependent on the hole-doping as it is shown in Figure 2.Figure 2
Magnetic spectral functionImχ(-)(k,ω) in the normal state of YBa2Cu3O6+x at T=100K and at Q=(π,π). 100 counts in the vertical scale correspond to χmax(-)≈350μB2/eV. The superconducting critical temperature Tc(x) by increasing doping (x) from the underdoped system with x=0.5 (top) to the optimally doped one with x=0.97 (bottom): Tc(x)=45K (x=0.5), 85K (x=0.83), 91K (x=0.92), and 92.5K (x=0.97). From [30].The most pronounced result for our discussion is that by varying dopingthere is a huge rearrangement of Imχ(odd)(Q,ω) in the normal state, especially in the energy (frequency) region which might be important for superconducting pairing, let us say 0meV<ω<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 temperatureTc. It is seen in Figure 2 that in the underdoped YBa2Cu3O6.92 crystal Imχ(odd)(Q,ω) and S(Q)=N(0)gsf2∫060dωImχ(odd)(Q,ω) 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 suppressionof Imχ(odd)(Q,ω) 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 λsf(~gsf2), means that the latter is small, that is, λsf(exp)≪1. We stress that the explanation of high Tc in cuprates by the SFI phenomenological theory [12–17] assumes very large SFI coupling energy with gsf(th)≈0.7eV while the frequency (energy) dependence of Imχ(Q,ω) is extracted from the fit of the NMR relaxation rate T1-1 which gives Tc(NMR)≈100K [12–17]. To this point, the NMR measurements (of T1-1) give that there is an anticorrelation between the decrease of the NMR spectral function IQ=limω→0Imχ(NMR)(Q,ω)/ω and the increase of Tc by increasing doping toward the optimal one—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 Imχ(Q,ω) from NMR measurements one takes it from IMNS measurements, as it was done in [82], than for the same value gsf(th) one obtains much smaller Tc. For instance, by taking the experimental values for Imχ(IMNS)(Q,ω) in underdoped YBa2Cu3O6.6 with Tc≈60K one obtains Tc(IMNS)<Tc(NMR)/3 [82], while Tc(IMNS)→50K for gsf(th)≫1. The situation is even worse if one tries to fit the resistivity with Imχ(IMNS)(Q,ω) in YBa2Cu3O6.6 since this fit gives Tc(IMNS)<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 “hourglass” spin excitation spectrum (in the superconducting state) for the momenta q 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 gsf2=(3/2)Ũ2 and Imχ(q,ω) 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 valueŨ=1.59eV, that is, much larger than the one used in [82]. Such a large value of Ũ has been obtained earlier within the Monte Carlo simulation of the Hubbard model [87]. In our opinion this value for Ũ 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 Ũ≪1eV and Tc≪60K. 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–Cu≈[4tpd4/(Δdp+Upd)2,(1/Ud)+2/(Up+2Δ)]—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 Ũ (and gsf).Concerning the problem related to the rearrangement of the SFI spectral functionImχ(Q,ω) in YB2Cu3O6+x [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(q,ω)(~Imχ(q,ω)) (for q 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 Imχ(q,ω) 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 Ũ to the value even larger than 4eV, which is highly improbable.(b) Ineffectiveness of the Magnetic Resonance Peak
A less direct argument forsmallness of the SFI coupling constant, that is,gsfexp≤0.2eV and gsfexp≪gsf, 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, Imχ(Q,ω) is significantly suppressed at low frequencies except near the resonance energy ωres≈41meV where a pronounced narrow peak appears—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)%—see Figure 3. In underdoped cuprates this peak is present also in the normal state as it is seen in Figure 2.Figure 3
Magnetic spectral functionImχ(-)(k,ω) in the superconducting state of YBa2Cu3O6+x at T=5K and at Q=(π,π). 100 counts in the vertical scale correspond to χmax(-)≈350μB2/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 intensityI0 is small [89]. I0 is obtained approximately by equating the condensation energy Econ≈N(0)Δ2/2 with the change of the magnetic energy Emag in the superconducting state, that is, δEmag≈4I0·Emag:
(11)Emag=J∬dωd2k(2π)3(1-coskx-cosky)S(k,ω),
where S(k,ω)=(1/π)[1+n(ω)]Imχ(k,ω) is the spin structure factor and n(ω) is the Bose distribution function. By taking Δ≈2Tc and the realistic value N(0)~1/(10J)~1states/eV·spin, one obtains I0~10-1(Tc/J)2~10-3. However, such a small intensity cannot be responsible for the anomalies in ARPES and optical spectra since it gives rise to small coupling constant λsf,res for the interaction of holes with the resonance peak, that is, λsf,res≈(2I0N(0)gsf2/ωres)≪1. 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 ωres, that is, γres<ωres, where the line width is given by γres=4π(N(0)gsf)2ωres [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 Ũ≈1.6eV, 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 μ(<0.1μB) 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]—see the discussion below.Finally, we would like to point out that the recent magnetic neutron scattering measurements on optimally doped large-volume crystalsBi2Sr2CaCu2O8+δ [93], where the absolute value of Imχ(q,ω) 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 Imχ(ω)=∑qImχ(q,ω) across the superconducting transition Tc=91K, that is, there is only a minimal change in Imχ(ω) between 10K and 100K. Note that if the magnetic excitations were due to itinerant quasiparticles we should have seen dramatic changes of Imχ(ω) as a function of T over the whole energy range. This T-independence of Imχ(ω) strongly opposes the theoretical results in [24–27] which assume that the bosonic spectral function is proportional to Imχ(ω) and that the former can be extracted from optic measurements. Namely, the fitting procedure in [24–27] gives that Imχ(ω) is strongly T-dependent contrary to the experimental results in [93]—see more in Section 1.3.2 on optical conductivity.


#### 1.3.2. Optical Conductivity and EPI

Optical spectroscopy gives information onoptical conductivityσ(ω) and on two-particle excitations, from which one can indirectly extract the transport spectral function αtr2F(ω). 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(ω)  is not a directly  measured quantity but it is derived from the reflectivity R(ω)=|(ɛii(ω)-1)/(ɛii(ω)+1)|2 with the transversal dielectric tensor ɛii(ω)=ɛii,∞+ɛii,latt+4πiσii(ω)/ω. Here, ɛii,∞ is the high-frequency dielectric function, ɛii,latt describes the contribution of the lattice vibrations, and σii(ω) describes the optical (dynamical) conductivity of conduction carriers. Since R(ω) is usually measured in the limited-frequency interval ωmin<ω<ωmax, some physical modelling for R(ω) is needed in order to guess it outside this range—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 ωpl—from 0.06 to 25eV—that is, almost three orders of magnitude. However, it turns out that IR measurements of R(ω) in conjunction with elipsometric measurements of ɛii(ω) at high frequencies allow more reliable determination of σ(ω) [94].(1) Transport and Quasiparticle Relaxation Rates
The widespread misconception in studying the quasiparticle scattering in cuprates was an ad hoc assumption that thetransport relaxation rateγtr(ω) is equal to the quasiparticle relaxation rateγ(ω), in spite of the well-known fact that the inequality γtr(ω)≠γ(ω) 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 conductivityσ(ω) consists of two parts, that is, σ(ω)=σinter(ω)+σintra(ω) where σinter(ω) describes interband  transitions which contribute at higher than intraband energies, while σintra(ω) is due to intraband transitions which are relevant at low energies ω<(1–2)eV. (Note that in the IR measurements the frequency is usually given in cm-1, where the following conversion holds: 1cm-1=29.98GHz=0.123985meV=1.44K.) The experimental data for σ(ω)=σ1+iσ2 in cuprates are usually processed by the generalized (extended) Drude formula [32–36, 95]:
(12)σ(ω)=ωp24π1γtr(ω)-iωmtr(ω)/m∞≡iωp24πω̃tr(ω),
where m is the mass of the band electrons while the quantity ω̃tr(ω) 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 ω̃tr(ω) will be discussed below.) Here, i=a,b enumerates the plane axis; ωp,γtr(ω,T), and mop(ω) are the electronic plasma frequency, the transport (optical) scattering rate, and the optical mass, respectively. Very frequently it is analyzed the quantity γtr*(ω,T) given by [95]
(13)γtr*(ω,T)=m∞mtr(ω)γtr(ω,T)=ωImσ(ω)Reσ(ω).
In the weak coupling limit λep<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]
(14)γtr(ω,T)=π∑l∫0∞dναtr,l2Fl(ν)×[2(1+2nB(ν))-2νω-ω+νωnB(ω+ν)+ω-νωnB(ω-ν)].
Here nB(ω) is the Bose distribution function. For completeness we give also the explicit form of the transport mass mtr(ω), see [3–6, 32–36]:
(15)mtr(ω)m∞=1+2ω∑l‍∫0∞dναtr,l2Fl(ν)ReK(ω2πT,ν2πT),
with the Kernel K(x,y)=(i/y)+{((y-x)/x)[ψ(1-ix+iy)-ψ(1+iy)]}-{y→-y} where ψ 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 λep(>1). Note that αtr,l2Fl(ν)≠αl2Fl(ν) and the index l  enumerates all scattering bosons—phonons—spin fluctuations, and so forth. For comparison, the quasiparticle scattering rate γ(ω,T) is given by
(16)γ(ω,T)=2π∫0∞dνα2F(ν)×{2nB(ν)+nF(ν+ω)+nF(ν-ω)}+γimp,
where nF is the Fermi distribution function. For completeness we give also the expression for the quasiparticle effective mass m*(ω):
(17)m*(ω)m=1+1ω∑l∫0∞dναl2Fl(ν)×Re{ψ(12+iω+ν2πT)-ψ(12-iω-ν2πT)}.
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]
(18)γtr(ω)=2πω∫0ωdν(ω-ν)αtr2(ν)F(ν),γ(ω)=2π∫0ωdνα2(ν)F(ν).
In the case of EPI with the constant electronic density of states, the above equations give that γep(ω)=const for ω>ωphmax while γep,tr(ω) (as well as γep,tr*) is monotonic growing for ω>ωphmax, where ωphmax is the maximal phonon frequency. So, the growing of γep,tr(ω) (and γep,tr*) for ω>ωphmax 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(ω,T),γtr(ω,T), and γtr* which are calculated for the EPI spectral function αep2(ω)Fph(ω) extracted from tunnelling experiments in YBCO (with ωphmax~80meV ) [42–45]—see Figure 4.(a) Scattering ratesγ(ω,T),γtr(ω,T), and γtr*—from top to bottom—for the Eliashberg function in (b). From [33–35]. (b) Eliashberg spectral function αep2(ω)Fph(ω) obtained from tunnelling experiments on break junctions [42–45]. Inset shows γtr* with (full line) and without (dashed line) interband transitions [3–5].
(a)(b)The results shown in Figure4 clearly demonstrate the physical difference between two scattering rates γep and γep,tr (or γtr*). It is also seen that γtr*(ω,T) is even more linear function of ω than γtr(ω,T). From these calculations one concludes that the quasilinearity of γtr(ω,T) (and γtr*) 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 γep,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 forth—see more in [3–6] and references therein.Let us discuss briefly the experimental results forR(ω) and γtr*(ω,T) and compare these with theoretical predictions obtained by using a single-band model with αep2(ω)Fph(ω) extracted from the tunnelling data with the EPI coupling constant λep≳2 [42–45]. In the case of YBCO the agreement between measured and calculated R(ω) is very good up to energies ω<6000cm-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 (Bi2212) [97, 98] the experimental results for γtr*(ω,T) are explained theoretically by assuming that the EPI spectral function αep2(ω)F(ω)~Fph(ω), where Fph(ω) is the phononic density of states in BSCO, with λep=1.9 and γimp≈320cm-1—see Figure 5(a). At the same time the fit of γtr*(ω,T) by the marginal Fermi liquid phenomenology fails as it is evident in Figure 5(b).(a) Experimental transport scattering rateγtr* (solid lines) for BSCO and the theoretical curve by using (A.20) and transport mass mtr* with α2F(ω) due to EPI which is described in text (dashed lines). (b) Comparison with the marginal Fermi liquid theory—dashed lines. From [3–5, 99].
(a)(b)Now we will comment the so called pronounced linear behavior ofγtr(ω,T) (and γtr*(ω,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(ω) and derived ones are σ(ω),γtr(ω,T), and mtr(ω), which are very sensitive to the value of the dielectric constant ɛ∞. This sensitivity is clearly demonstrated in Figure 6 for Bi-2212 where it is seen that γtr(ω,T) (and γtr*(ω,T)) for ɛ∞=1 is linear up to much higher ω than in the case ɛ∞>1.Figure 6
Dependence ofγtr*(ω,T) on ɛ∞ in Bi2Sr2CaCu2O8 for different temperatures: ɛ∞=4 (solid lines) and ɛ∞=1 (dashed lines). On the horizontal axis is ω in units cm-1. From [99].However, in some experiments [100–103] the extracted γtr(ω,T) (and γtr*(ω,T)) is linear up to very high ω≈1500cm-1. This means that the ion background and interband transitions (contained in ɛ∞) are not properly taken into account since too small ɛ∞ (≳1) is assumed. The recent elipsometric measurements on YBCO [104] give the value ɛ∞≈4–6, which gives much less spectacular linearity in the relaxation rates γtr(ω,T) (and γtr*(ω,T)) than it was the case immediately after the discovery of HTSC cuprates, where much smaller ɛ∞ was assumed.Furthermore, we would like to comment two points related toσ,γtr, and γ. First, the parametrization of σ(ω) with the generalized Drude formula in (12) and its relation to the transport scattering rate γtr(ω,T) and the transport mass mtr(ω,T) is useful if we deal with electron-boson scattering in a single-band problem. In [36, 96] it is shown that σ(ω) of a two-band model with only elastic impurity scattering can be represented by the generalized (extended) Drude formula with ω and T dependence of effective parameters γtreff(ω,T), mtreff(ω,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(ω,T) and γ(ω,T) are physically different quantities and it holds that γtr(ω,T)≠γ(ω,T). In order to give the physical picture and qualitative explanation for this difference we assume that αtr2F(ν)≈α2F(ν). In that case the renormalized quasiparticle frequency ω̃(ω)=Z(ω)ω=ω-Σ(ω) and the transport one ω̃tr(ω)—defined in (12)—are related and at T=0 they are given by [32, 36]
(19)ω̃tr(ω)=1ω∫0ωdω′2ω̃(ω′).
(For the definition of Z(ω) see Appendix A.) It gives the relation between γtr(ω) and γ(ω) as well asmtr(ω) and m*(ω), respectively:
(20)γtr(ω)=1ω∫0ωdω′γ(ω′),ωmtr(ω)=1ω∫0ωdω′2ω′m*(ω′).
The physical meaning of (19) is the following: in optical measurements one photon with the energy ω is absorbed and two excited particles (electron and hole) are created above and below the Fermi surface. If the electron has energy ω′ and the hole ω-ω′, then they relax as quasiparticles with the renormalized frequency ω̃. Since ω′ takes values 0<ω′<ω, then the optical relaxation ω̃tr(ω) is the energy-averaged ω̃(ω) according to (19). The factor 2 is due to the two quasiparticles—electron and hole. At finite T, the generalization reads [32, 36]
(21)ω̃tr(ω)=1ω∫0∞dω′[1-nF(ω′)-nF(ω-ω′)]2ω̃(ω′).(2) Inversion of the Optical Data andαtr2(ω)F(ω)
In principle, the transport spectral functionαtr2(ω)F(ω) can be extracted from σ(ω) (or γtr(ω)) only at T=0K, which follows from (18) as
(22)αtr2(ω)F(ω)=ωp28π2∂2∂ω2[ωRe1σ(ω)],
or equivalently as αtr2(ω)F(ω)=(1/2π)∂2(ωγtr(ω))/∂ω2. 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 αtr2(ω)F(ω) is the deconvolution of the inhomogeneous Fredholm integral equation of the first kind with the temperature-dependent Kernel K2(ω,ν,T)—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 αtr2(ω)F(ω)  get blurred (most peaks are washed out) in the extraction procedures and, second, the extractedαtr2(ω)F(ω)  be temperature dependent even when the true αtr2(ω)F(ω) 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 αtr2(ω)F(ω) in YBa2Cu3O7-x 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 ω) is present. For instance, the experimental data of R(ω) in YBCO were reproduced by two different spectral functions αtr2(ω)F(ω), 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(ω). The similar situation is realized in optimally doped BSCO as it is seen in Figure 8 where again different functions α2(ω)F(ω) reproduce very well curves for R(ω) and σ(ω). However, it is important to stress that the obtained width of the extracted αtr2(ω)F(ω) in both compounds coincide with the width of the phonon density of states Fph(ω) [33–36, 99]. (2) The upper energy bound for αtr2(ω)F(ω) is extracted in [33–36] and it coincides approximately with the maximal phonon frequency in cuprates ωphmax≲80meV as it is seen in Figures 7 and 8.Figure 7
Experimental (solid lines) and calculated (dashed lines) data ofR(ω) in optimally doped YBCO [105] at T=100,200,300K (from top to bottom). Inset: the two (solid and dashed lines) reconstructed αtr2(ω)F(ω)’s at T=100K. The phonon density of states F(ω)—dotted line in the inset. From [33–35].Figure 8
Experimental (solid line) and calculated (dashed line) data ofR(ω) in optimally doped BSCO [106] at T=100K. Inset: the reconstructed αtr2(ω)F(ω)—solid line. The phonon density of states F(ω)—dotted line. From [33–35].These results demonstrate the importance of EPI in cuprates [33–36]. We point out that the width of αtr2(ω)F(ω) which is extracted from the optical measurements [33–36] coincides with the width of the quasiparticle spectral function α2(ω)F(ω) obtained in tunnelling and ARPES spectra (which we will discuss below), that is, both functions are spread over the energy interval 0<ω<ωphmax(≲80meV). Since in cuprates this interval coincides with the width in the phononic density of states F(ω) and since the maxima of α2(ω)F(ω) and F(ω) 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 αtr2(ω)F(ω) 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(ω) 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 Δ(k,ω). Second, if R(ω)’s (and σ(ω)’s) in cuprates are due to some other bosonic scattering which is pronounced up to much higher energies ωc≫ωphmax, this should be seen in the width of the extracted spectral function αtr2(ω)F(ω). In that respect in [25–27] it is assumed that SFI dominates in the quasiparticle scattering and that αtr2(ω)F(ω)~gsf2Imχ(ω) where Imχ(ω)=∫d2kχ(k,ω). This claim is based on reanalyzing of some IR measurements [25–27] and the transport spectral function αtr2(ω)F(ω) 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 αtr2(ω)F(ω), which is an artefact and due to the chosen method, in [25] it is assumed that αtr2(ω)F(ω) has a rather large tail at large energies—up to 400 meV. It turns out that even such an assumption in extracting αtr2(ω)F(ω) does not reproduce the experimental curve Imχ(ω) [107] in some important aspects. First, the relative heights of the two peaks in the extracted spectral function αtr2(ω)F(ω) at lower temperatures are opposite to the experimental curve Imχ(ω) [107]—see [25, Figure  1]. Second, the strong temperature dependence of the extracted αtr2(ω)F(ω) 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 IB(ω)=αtr2(ω)F(ω) in [25] has much smaller values at larger frequencies (ω>100meV) than it is the case for the measured Imχ(ω), that is, (IB(ω>100meV)/IB(ωmax))≪Imχ(ω>100meV)/Imχ(ωmax)—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 Imχ(q,ω) 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 Imχ(ω)=∑qImχ(q,ω) is temperature independent across the superconducting transition Tc=91K, that is, there is only a minimal change in Imχ(ω) between 10K and 100K. This T-independence of Imχ(ω) strongly opposes the (above discussed) results in [24–27], where the fit of optic measurements gives strong T dependence of Imχ(ω).Fifth, the transport coupling constant λtr extracted in [25] is too large, that is, λtr>3 contrary to the previous findings that λtr≲1.5 [33–36, 99]. Since in HTSC one has λ>λtr, this would probably give λ≈6, which is not confirmed by other experiments. Sixth, the interpretation of αtr2(ω)F(ω) in LSCO and BSCO solely in terms of Imχ(ω) 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 Imχ(Q,ω) is small in the normal state and its magnitude is even below the experimental noise. This means that if the assumption that αtr2(ω)F(ω)≈gsf2Imχ(ω) were correct then the contribution to Imχ(ω) from the momenta 0<k≪Q would be dominant, which is detrimental for d-wave superconductivity.Finally, we point out that very similar (to cuprates) properties, ofσ(ω),R(ω) (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 Ba1-xKxBiO3 which is superconducting below Tc≃30K at x=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
Therestricted  optical sum rulewas studied intensively in HTSC cuprates. It shows some peculiarities not present in low-temperature superconductors. It turns out that the restricted spectral weight W(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 W(Ωc,T)=W(0)-βT2 in the normal state can be explained in a natural way by the T dependence of the EPI transport relaxation rate γtrep(ω,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(ω). The first one is the total sum rule which in the normal state reads
(23)∫0∞dωσ1N(ω)=ωpl28=πne22m,
while in the superconducting state it is given by the Tinkham-Ferrell-Glover (TFG) sum rule
(24)∫0∞dωσ1S(ω)=c28λL2+∫+0∞dωσ1S(ω)=ωpl28.
Here,n is the total electron density, e is the electron charge, m is the bare electron mass, and λL is the London penetration depth. The first (singular) term c2/8λL2 in (24) is due to the superconducting condensate which contributes σ1,condS(ω)=(c2/4λL2)δ(ω). The total sum rule represents the fundamental property of matter—the conservation of the electron number. In order to calculate it one should use the total Hamiltonian Ĥtot=T̂e+Ĥint where all electrons, electronic bands, and their interactions Ĥint (Coulomb, EPI, with impurities, etc.) are accounted for. Here, Te is the kinetic energy of bare electrons:
(25)T̂e=∑σ‍∫d3xψ̂σ†(x)p̂22mψ̂σ(x)=∑p,σp22meĉpσ†ĉpσ.Thepartial sum rule is related to the energetics solely in the conduction (valence) bandwhich is described by the Hamiltonian of the conduction (valence) band electrons:
(26)Ĥv=∑p,σξpĉv,pσ†ĉv,pσ+V̂v,c.Ĥv contains the band energy with the dispersion ϵp (ξp=ϵp-μ) and the effective Coulomb interaction of the valence electrons V̂v,c. In this case the partial sum rule in the normal state reads [110] (for a general form of ϵp)
(27)∫0∞dωσ1,vN(ω)=πe22V∑p‍〈n̂v,p〉Hvmp,
where the number operator n̂v,p=∑σĉpσ†ĉpσ;1/mp=∂2ϵp/∂px2 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 ω 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
(28)W(Ωc,T)=∫0Ωcdωσ1,vN(ω)=π2[Kd+Π(0)]-∫0ΩcdωImΠ(ω)ω,
where Kd is the diamagnetic Kernel given by (30) below and Π(ω) is the paramagnetic (current-current) response function. In the perturbation theory without vertex correction Π(iωn) (at the Matsubara frequency ωn=πT(2n+1)) is given by [39, 40]
(29)Π(iω)=2∑p(∂ϵp∂p)2∑ωmG(p,iωnm+)G(p,iωm),
where ωnm+=ωn+ωm and G(p,iωn)=(iωn-ξp-Σ(p,iωn))-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 Wb<Ωc<Eg, in this region one has approximately ImΠ(ω)≈0 and the limit Ωc→∞ in (28) is justified. In that case one has Π(0)≈∫0Ωc(ImΠ(ω)/ω)dω which gives the approximate formula for W(Ωc,T):
(30)W(Ωc,T)=∫0Ωcdωσ1,vN(ω)≈π2Kd=e2π∑p∂2ϵp∂p2np,
where np=〈n̂v,p〉 is the quasiparticle distribution function in the interacting system. Note that W(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 W(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 ϵp=-2t(cospxa+cospya) which gives 1/mp=-2ta2cospxa. It is straightforward to show that in this case (30) is reduced to a simpler one:
(31)W(Ωc,T)=∫0Ωcdωσ1,vN(ω)≈πe2a22V〈-Tv〉,
where 〈Tv〉Hv=∑pϵp〈nv〉Hv is the average kinetic energy of the band electrons, a is the Cu–Cu lattice distance, and V is the volume of the system. In this approximation W(Ωc,T) is a direct measure of the average band (kinetic) energy. In thesuperconducting state the partial band sum rule reads (32)Ws(Ωc,T)=c28λL2+∫+0Ωcdωσ1,vS(ω)=πe2a22V〈-Tv〉s.
In order to introduce the reader to (the complexity of) the problem of the T dependence of W(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 Wn(Ωc,T) increases with the decrease of the temperature, that is, Wn(Ωc,T)=Wn(0)-βbT2 where βb~1/Wb. 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 W(Ωc,T). We stress that the smearing of fp by temperature lowers the spectral weight compared to that at T=0K, that is, Wn(Ωc,T)<Wn(Ωc,0). In that respect it is not surprising that there is a lowering of Ws(Ωc,T) in the BCS superconducting state, WsBCS(Ωc,T≪Tc)<Wn(Ωc,T≪Tc) since at low temperatures fp is smeared mainly due to the superconducting gap, that is, fp=[1-(ξp/Ep)th(Ep/2T)]/2, Ep=ξp2+Δ2, ξp=ϵp-μ. The maximal decrease of Ws(Ωc,T) is at T=0.Let us enumerate and discuss themain experimental results forW(Ωc,T) in HTSC cuprates. (1) In the normal state (T>Tc) of most cuprates, one has Wn(Ωc,T)=Wn(0)-βexT2 with βexp≫βb, that is, Wn(Ωc,T) is increasing by decreasing T, even at T below T*—the temperature for the opening of the pseudogap. The increase of Wn(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 Ws(Ωc,T) with respect to that in the normal state, that is, Ws(Ωc,T)>Wn(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 〈Tv〉 is negative (〈Tv〉<0) and in the standard BCS case (32) gives that Ws(T<Tc)  decreases due to the increase of 〈Tv〉. Therefore the experimental increase of Ws(T<Tc) by decreasing T is called the non-BCS behavior. The latter means a lowering of the kinetic energy 〈Tv〉 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.Figure 9
Measured spectral weightWs(Ωc,T)(~Al+D in figures) for Ωc≈1.25eV 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 ofWs(Ωc,T) at T<Tc (Ws(Ωc,T)<Wn(Ωc,T)) which is the BCS-like behavior [112] as it is seen in Figure 10.(a) Spectral weightWn(Ωc,T) of the overdoped Bi2212 for Ωc=1eV. Closed symbols—normal state. Open symbols—superconducting state. (b) Change of the kinetic energy ΔEkin=Ekin,S-Ekin,N in meV per Cu site versus the charge p per Cu with respect to the optimal value popt. From [112].
(a)(b)We stress that the non-BCS behavior ofWs(Ωc,T) for underdoped (and in some optimally doped) systems was obtained by assuming that Ωc≈(1–1.2)eV. However, in [104] these results were questioned by claiming that the conventional BCS-like behavior was observed (Ws(Ωc,T)<Wn(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 Ws(Ωc,T)<Wn(Ωc,T) was observed, while in LSCO it was found that Ws(Ωc,T)≈const, that is, Ws(Ωc,T<Tc)≈Wn(Ωc,Tc).The first question is the following. How to explain the strong temperature dependence ofW(Ωc,T) in the normal state? In [39, 40] W(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 W(Ωc,T). The main theoretical results of [39, 40] are the following: the calculations of W(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, Wn(Ωc,T)≈Ws(Ωc,T) since Wn(Ωc,T)-Ws(Ωc,T)~Δ2/Ωc2. (2) In the case of large Ωc the calculations based on (30) give
(33)W(Ωc,T)≈ωpl28[1-γ(T)Wb-π22T2Wb2].
In the case when EPI dominates one has γ=γep(T)+γimp where γep(T)=∫0∞dzα2(z)F(z)coth(z/2T). It turns out that for α2(ω)F(ω), shown in Figure 4, one obtains (i) γep(T)~T2 in the temperature interval 100K<T<200K as it is seen in Figure 11 for the T dependence of W(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 λep,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(ω)F(ω). It is seen in Figure 11 that, for a softer Einstein spectrum (with ΩE=200K), W(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 W(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 W(Ωc,T) was observed in experiments in [104, 111, 113, 114].Figure 11
Spectral weightW(Ωc,T) in the normal state for Einstein phonons with ΩE=200K (full triangles) and ΩE=400K (open circles, left axis). Dashed lines show T2 asymptotic. From [40].To summarize, the above analysis demonstrates that the theory based on EPI is able to explain in a satisfactory way the temperature behavior ofW(Ωc,T) above and below Tc in systems at and near the optimal doping.(4) α2(ω)F(ω) and the In-Plane Resistivity ρab(T)
The temperature dependence of the in-plane resistivityρab(T) in cuprates is a direct consequence of the quasi-2D motion of quasiparticles and of the inelastic scattering which they experience. At present, there is no consensus on the origin of the linear temperature dependence of the in-plane resistivity ρab(T) in the normal state. Our intention is not to discuss this problem, but only to demonstrate that the EPI spectral function α2(ω)F(ω), which is obtained from tunnelling experiments in cuprates (see Section 1.3.4), is able to explain the temperature dependence of ρab(T) in the optimally doped YBCO. In the Boltzmann theory ρab(T) is given by
(34)ρab(T)=ρimp+4πωp2γtr(T),
where
(35)γtr(T)=πT∫0∞dωωsinh2(ω/2T)αtr2(ω)F(ω).
The residual resistivity ρimp is due to the impurity scattering. Since ρ(T)=1/σ(ω=0,T) and having in mind that the dynamical conductivity σ(ω,T) in YBCO (at and near the optimal doping) is satisfactory explained by the EPI scattering, then it is to expect that ρab(T) is also dominated by EPI in some temperature region T>Tc. This is indeed confirmed in the optimally dopedYBCO, where ρimp is chosen appropriately and the spectral function αtr2(ω)F(ω) is taken from the tunnelling experiments in [42–45]. The very good agreement with the experimental results [118] is shown in Figure 12. We stress that in the case of EPI there is always a temperature region where γtr(T)~T for T>αΘD, α<1 depending on the shape of αtr2(ω)F(ω) (for the simple Debye spectrum α≈0.2). In the linear regime one has ρ(T)≃ρimp+8π2λep,tr(kBT/ℏωp2)=ρimp+ρ′T.(a) Calculated resistivityρ(T) for the EPI spectral function αtr2(ω)F(ω) in [118]. (b) Measured resistivity in a(x)- and b(y)-crystal direction of YBCO [119] and calculated Bloch-Grüneisen curve (points) for λep=1 [120].
(a)(b)There is experimental constraint onλtr since λtr≈0.25ωpl2(eV)ρ′(μΩcm/K). For instance, for ωpl≈(2-3)eV [108] and ρ′≈0.6 in the oriented YBCO films and ρ′≈0.3-0.4 in single crystals of BSCO, one obtains λtr≈0.6–1.4. In case of YBCO single crystals, there is a pronounced anisotropy in ρab(T) [119] which gives ρx′(T)=0.6μΩcm/K and ρy′(T)=0.25μΩcm/K. The function λtr(ωpl) is shown in Figure 13 where the plasma frequency ωpl can be calculated by LDA-DFT and also extracted from the width (~ωpl*) of the Drude peak at small frequencies, where ωpl=ɛ∞ωpl*. We stress that the rather good agreement of theoretical and experimental results for ρab(T), in some optimally doped HTSC cuprates such as YBCO, should not be overinterpreted in the sense that the above rather simple electron-phonon approach can explain the resistivity in other HTSC cuprates and for various doping. For instance, in highly underdoped systems ρab(T) is very different from the behavior in Figure 12 and the simple Migdal-Eliashberg theory based on the EPI spectral function is inadequate. In this case one should certainly take into account polaronic effects [8–11], strong correlations, and so forth. The above analysis on the resistivity in the optimally doped YBCO demonstrates only that in this case if in (35) one uses the EPI spectral function α2(ω)F(ω) obtained from the tunnelling experiments (and optics) one obtains the correct T dependence of ρa,b(T). This result is an additional evidence for the importance of EPI.Figure 13
Transport EPI spectral function coupling constant in YBCO as a function of plasma frequencyωp as derived from the experimental slope of resistivity ρ′(T).λx for ρx′(T)=0.6μΩcm/K and λy for ρy′(T)=0.25μΩcm/K [119]. Squares are LDA values [121].Concerning the temperature dependence of the resistivity in other (than YBCO) families of the optimally doped HTSC cuprates we would like to point out that there is some evidence that the linear (inT) resistivity is observed in some of them even at temperatures T<0.2ΘD [122, 123]. This possibility is argued also theoretically in [124] where it is shown that in two-dimensional systems with a broad interval of phonon spectra the quasilinear behavior of ρab(T) is realized even at T<0.2ΘD. The quasilinear behavior of the resistivity at T≪0.2ΘD has been observed in Bi2(Sr0.97Pr0.003)2CuO6 [125], in LSCO, and in 1-layer Bi-2201 [122, 123, 126, 127], where in all these systems the critical temperature is rather small, Tc≈10K. In that respect all existing theories based on the electron-boson scattering are plagued and having difficulties to explain this low-temperature behavior of ρab(T). To this point, we would like to emphasize here that some of these (experimental) observations are contradictory. For example, the results obtained by the Vedeneev group [127] show that some samples demonstrate the quasilinear behavior of the resistivity up to T=10K but some others with approximately the same Tc have the usual Bloch-Grüneisen-type behavior characteristic for the EPI scattering. In that respect it is very unlikely that the linear resistivity up to T=10K can be simply explained in the standard way by interactions of electrons with some known bosons either by phonons or spin fluctuations (magnons). The question why in some cuprates the linear resistivity is observed up to T=10K is still a mystery and its explanation is a challenge for all kinds of the electron-boson scattering, not only for EPI. In that respect it is interesting to mention that the existence of the forward scattering peak in EPI (with the width qc≪kF), which is due to strong correlations, may give rise to the linear behavior of ρ(T) down to very low temperatures T~ΘD/30 [6, 128, 129]—see more in Section 2.3.4, item (6).  We will argue in Section 1.3.4 that if one interprets the tunnelling experiments in systems near optimal doping [42–54] in the framework of the Eliashberg theory one obtains the large EPI coupling constant λep≈2–3.5 which implies that λtr~(λ/3). This means that EPI is reduced much more in transport properties than in the self-energy. We stress that such a large reduction of λtr cannot be obtained within the LDA-DFT band-structure calculations, which means that λep and λtr contain renormalization which do not enter in the LDA-DFT theory. In Section 2 we will argue that the strong suppression of λtr may have its origin in strong electronic correlations [78–80, 130] and in the long-range Madelung energy [3–6].(5) Femtosecond Time-Resolved Optical Spectroscopy
The femtosecond time-resolved optical spectroscopy (FTROS) has been developed in the last couple of years and applied to HTSC cuprates. In this method a femtosecond (1fs=10-15sec) laser pump excites in materials electron-hole pairs via interband transitions. These hot carriers release their energy via electron-electron (with the relaxation time τee) and electron-phonon scattering reaching states near the Fermi energy within 10–100fs—see [131]. The typical energy density of the laser pump pulses with the wavelength λ≈810 nm (ℏω=1.5eV) was around F~1μJ/cm2 (the excitation fluenceF) which produces approximately 3×1010 carriers per pulse (by assuming that each photon produces ℏω/Δ carriers, Δ is the superconducting gap). By measuring photoinduced changes of the reflectivity in time, that is, ΔR(t)/R0, one can extract information on the relaxation dynamics of the low-laying electronic excitations. Since ΔR(t) relax to equilibrium, the fit with exponential functions is used as
(36)ΔR(t)R0=f(t)[Ae-t/τA+Be-t/τB+⋯],
where f(t)=H(t)[1-exp{-t/τee}] (H(t) is the Heavyside function) describes the finite rise-time. The parameters A, B depend on the fluence F. This method was used in studying the superconducting phase of La2-xSrxCuO4, with x=0.1 and 0.15 and Tc=30K and 38K, respectively [41]. In that case one has A≠0 for T<Tc and A=0 for T>Tc, while the signal B was present also at T>Tc. It turns out that the signal A is related to the quasiparticle recombination across the superconducting gap Δ(T) and has a relaxation time of the order τA>10ps at T=4.5K. At the so called threshold fluence (FT=4.2±1.7μJ/cm2 for x=0.1 and FT=5.8±2.3μJ/cm2 for x=0.15) the vaporization (destroying) of the superconducting phase occurs, where the parameter A saturates. This vaporization process takes place at the time scala τr≈0.8ps. The external fluence is distributed in the sample over the excitation volume which is proportional to the optical penetration depth λop(≈150 nm at λ≈810nm) of the pump. The energy densities stored in the excitation volume at the vaporization threshold for x=0.1 and x=0.15 are Up=FT/λop=2.0±0.8K/Cu and 2.6±1.0K/Cu, respectively. The important fact is that Up is much larger than the superconducting condensation energy which is Ucond≈0.12K/Cu for x=0.1 and Ucond≈0.3K/Cu for x=0.15, that is, Up≫Ucond. This means that the energy difference Up-Ucond must be stored elsewhere on the time scale τr. The only present reservoir which can absorb the difference in energy is the bosonic baths of phonons and spin fluctuations. The energy required to heat the spin reservoir from T=4.5K to Tc is Usf=∫TTcCsf(T)dT. The measured specific heat Csf(T) in La2CuO4 [41] gives very small value Usf≈0.01K. In the case of the phonon reservoir one obtains Uph=∫TTcCph(T)dT=9K/Cu for x=0.1 and 28K/Cu for x=0.15, where Cph is the phonon specific heat. Since Usf≪Up-Ucond, the spin reservoir cannot absorb the rest energy Up-Ucond. The situation is opposite with phonons since Uph≫Up-Ucond  and phonon can absorb the rest energy in the excitation volume. The complete vaporization dynamics can be described in the framework of the Rothwarf-Taylor model which describes approaching of electrons and phonons to quasiequilibrium on the time scale of 1 ps [132]. We will not go into details but only summarize by quoting the conclusion in [132] that only phonon-mediated vaporization is consistent with the experiments, thus ruling out spin-mediated quasiparticle recombination and pairing in HTSC cuprates. The FTROS method tells us that at least for nonequilibrium processes EPI is more important than SFI. It gives also some opportunities for obtaining the strength of EPI but at present there is no reliable analysis.In conclusion, optics and resistivity measurements in the normal state of cuprates give evidence that EPI is important while the spin-fluctuation scattering is weaker than it is believed. However, some important questions related to the transport properties remain to be answered. (i) What are the values of λtr and ωpl? (ii) What is the reason that λtr≪λ is realized in cuprates? (iii) What is the role of the Coulomb scattering in σ(ω) and ρ(T)? Later on we will argue that ARPES measurements in cuprates give evidence for an appreciable contribution of the Coulomb scattering at higher frequencies, where γ(ω)≈γ0+λcω for ω>ωmaxph with λc~1. One should stress that despite the fact that EPI is suppressed in transport properties it is sufficiently strong in the quasiparticle self-energy, as it comes out from tunnelling measurements discussed below.


#### 1.3.3. ARPES and the EPI Self-Energy

The angle-resolved photoemission spectroscopy (ARPES) is nowadays one of leading spectroscopy methods in the solid-state physics [22, 23]. In some favorable conditions it provides direct information on the one-electron removal spectrum in a complex many-body system. The method involves shining light (photons) with energies between Ei=5–1000eV on samples and by detecting momentum (k)- and energy(ω)-distribution of the outgoing electrons. The resolution of ARPES has been significantly increased in the last decade with the energy resolution of ΔE≈1-2meV (for photon energies ~20 eV) and angular resolution of Δθ≲0.2∘. On the other side the ARPES method is surface-sensitive technique, since the average escape depth (lesc) of the outgoing electrons is of the order of lesc~10 Å, depending on the energy of incoming photons. Therefore, very good surfaces are needed in order that the results be representative for bulk samples. The most reliable studies were done on the bilayer Bi2Sr2CaCu2O8 (Bi2212) and its single-layer counterpart Bi2Sr2CuO6 (Bi2201), since these materials contain weakly coupled BiO planes with the longest interplane separation in the cuprates. This results in a natural cleavage plane making these materials superior to others in ARPES experiments. After a drastic improvement of sample quality in other families of HTSC materials, the ARPES technique has became an important method in theoretical considerations. The ARPES can indirectly give information on the momentum and energy dependence of the pairing potential. Furthermore, the electronic spectrum of the (abovementioned) cuprates is highly quasi-2D which allows rather unambiguous determination of the initial state momentum from the measured final state momentum, since the component parallel to the surface is conserved in photoemission. In this case, the ARPES probes (under some favorable conditions) directly the single-particle spectral function A(k,ω). In the following we discuss mainly those ARPES experiments which give evidence for the importance of the EPI in cuprates—see more in [22, 23].ARPES measures a nonlinear response function of the electron system and it is usually analyzed in the so-calledthree-step model, where the total photoemission intensity Itot(k,ω)≈I1·I2·I3 is the product of three independent terms: (1) I1 that describes optical excitation of the electron in the bulk, (2) I2 that describes the scattering probability of the travelling electrons, and (3) I3 that describes the transmission probability through the surface potential barrier. The central quantity in the three-step model is I1(k,ω) and it turns out that for k=k∥ it can be written in the form I1(k,ω)≃I0(k,υ)f(ω)A(k,ω) [22, 23] with I0(k,υ)~|〈ψf|pA|ψi〉|2 and the quasiparticle spectral function A(k,ω)=-ImG(k,ω)/π:
(37)A(k,ω)=-1πImΣ(k,ω)[ω-ξ(k)-ReΣ(k,ω)]2+ImΣ2(k,ω).
Here, 〈ψf|p·A|ψi〉 is the dipole matrix element which depends on k, polarization, and energy Ei of the incoming photons. The knowledge of the matrix element is of a great importance and its calculation from first principles was done in [133]. f(ω) is the Fermi function; G and Σ=ReΣ+iImΣ are the quasiparticle Green's function and the self-energy, respectively. We summarize and comment here some important ARPES results which were obtained in the last several years and which confirm the existence of the Fermi surface and importance of EPI in the quasiparticle scattering [22, 23].ARPES in the Normal State
(N1) There is well-defined Fermi surface in the metallic state of optimally and near optimally doped cuprates with the topology predicted by the LDA-DFT. However, the bands are narrower than LDA-DFT predicts which points to a strong quasiparticle renormalization. (N2) The spectral lines are broad with |ImΣ(k,ω)|~ω (or ~T for T>ω) which tells us that the quasiparticle liquid is a noncanonical Fermi liquid for larger values of T,ω. (N3) There is a bilayer band splitting in Bi2212 (at least in the overdoped state), which is also predicted by LDA-DFT. In the case when the coherent hopping t⊥ between two layers in the bilayer dominates, then the antibonding and bonding bands ξka,b=ξk±tk⊥ with tk⊥=[t⊥(cos2kx-cos2ky)+⋯] have been observed. It is worth to mention that the previous experiments did not show this splitting which was one of the reasons for various speculations on some exotic electronic scattering and non-Fermi liquid scenarios. (N4) In the underdoped cuprates and at temperatures Tc<T<T* there is a d-wave-like pseudogap Δpg(k)~Δpg,0(coskx-cosky) in the quasiparticle spectrum where Δpg,0 increases by lowering doping. We stress that the pseudogap phenomenon is not well understood at present and since we are interested in systems near optimal doping where the pseudogap phenomena are absent or much less pronounced we will not discuss this problem here. Its origin can be due to a precursor superconductivity or due to a competing order, such as spin- or charge-density wave, strong correlations, and so forth. (N5) The ARPES self-energy gives evidence that EPI interaction is rather strong. The arguments for the latter statement are the following: (i) at T>Tc there are kinks in the quasiparticle dispersion ω(ξk) in the nodal direction (along the (0,0)-(π,π) line) at the characteristic phonon energy ωph(70)~(60–70)meV [91], see Figure 14 (top), and near the antinodal point(π,0) at 40meV [134]—see Figure 14 (bottom).Figure 14
(top) Quasiparticle dispersion ofBi2212, Bi2201, and LSCO along the nodal direction, plotted versus the momentum k for (a)–(c) different doping, and (d)-(e) different T; black arrows indicate the kink energy; the red arrow indicates the energy of the q=(π,0) oxygen stretching phonon mode; inset of (e) shows T-dependent Σ′ for optimally doped Bi2212; (f) shows doping dependence of the effective coupling constant λ′ along (0,0)-(π,π) for the different HTSC oxides. From [91]. (bottom) Quasiparticle dispersion E(k) in the normal state (a1, b1, c), at 107 K and 115 K, along various directions ϕ around the antinodal point. The kink at E=40meV is shown by the horizontal arrow. (a2 and b2) are E(k) in the superconducting state at 10 K with the shifted kink to 70meV. (d) kink positions as a function of ϕ in the antinodal region. From [134].(ii) The kink structure is observed in a variety of the hole-doped cuprates such asLSCO, Bi2212, Bi2201, Tl2201 (Tl2Ba2CuO6), Na–CCOC (Ca2-xNaxCuO2Cl2). These kinks exist also above Tc, which excludes the scenario with the magnetic resonance peak in Imχs(Q,ω). Moreover, since the tunnelling and magnetic neutron scattering measurements give small SFI coupling constant gsf<0.2eV, then the kinks are not due to SFI. (iii) The position of the nodal kink is practically doping independent which points towards phonons as the scattering and pairing boson. (N6) The quasiparticles (holes) at and near the nodal-point kN couple practically to a rather broad spectrum of phonons since at least three groups of phonons were extracted in the bosonic spectral function α2F(kN,ω) from the ARPES effective self-energy in La2-xSrxCuO4 [135]—Figure 15.(a) Effective real self-energy for the nonsuperconductingLa2-xSrxCuO4, x=0.03. Extracted αeff2(ω)F(ω) is in the inset. (b) Top: the total MDC width—open circles. Bottom: the EPI contribution shows saturation, impurity contribution—dotted black line. The residual part is growing ~ω1.3. From [135].
(a)(b)The latter result is in a qualitative agreement with numerous tunnelling measurements [42–54] which apparently demonstrate that the very broad spectrum of phonons couples with holes without preferring any particular phonons—see discussion below. (N7) Recent ARPES measurements in Bi2212 [92] show very different slope dω/dξk of the quasiparticle energy ω(ξk) at small |ξk|≪ωph and at large energies |ξk|≫ωph—see Figure 16. The theoretical analysis [137] of these results gives the total coupling constant λZ=λepZ+λcZ≈3, and for the EPI coupling λepZ≈2, while the Coulomb coupling (SFI is a part of it) is λcZ≈1 [137]—see Figure 16. (Note that the upper index Z in the coupling constants means the quasiparticle renormalization in the normal part of the self-energy.) To this end let us mention some confusion which is related to the value of the EPI coupling constant extracted from ARPES. Namely, [22, 23, 138, 139] the EPI self-energy was obtained by subtracting the high-energy slope of the quasiparticle spectrum ω(ξk) at ω~0.3eV. The latter is apparently due to the Coulomb interaction. Although the position of the low-energy kink is not affected by this procedure (if ωphmax≪ωc), this subtraction procedure gives in fact an effective EPI self-energyΣeffep(k,ω) andthe effective coupling constantλep,effZ(k) only. We demonstrate below that the λep,effZ(k) is smaller than the real EPI coupling constant λepZ(k). The total self-energy is Σ(k,ω)=Σep(k,ω)+Σc(k,ω) where Σc is the contribution due to the Coulomb interaction. At very low energies ω≪ωc one has usually Σc(k,ω)=-λcZ(k)ω, where ωc(~1 eV) is the characteristic Coulomb energies and λcZ is the Coulomb coupling constant. The quasiparticle spectrum ω(k) is determined from the condition ω-ξ(k)-Re[Σep(k,ω)+Σc(k,ω)]=0 where ξ(k) is the bare band-structure energy. At low energies ω<ωphmax≪ωc it can be rewritten in the form
(38)ω-ξren(k)-ReΣeffep(k,ω)=0,
with ξren(k)=[1+λcZ(k)]-1ξ(k),
(39)ReΣeffep(k,ω)=ReΣeffep(k,ω)1+λcZ(k).
Since at very low energies ω≪ωphmax one has ReΣep(k,ω)=-λepZ(k)ω and ReΣeffep(k,ω)=-λep,effZ(k)ω, then the real coupling constant is related to the effective one by
(40)λepZ(k)=[1+λcZ(k)]λep,effZ(k).
As a result one has λepZ(k)>λep,effZ(k). At higher energies ωphmax<ω<ωc, where the EPI effects are suppressed and Σep(k,ω) stops growing, one has ReΣ(k,ω)≈ReΣep(k,ω)-λcZ(k)ω. The measured ReΣexp(k,ω) at T=10K near and slightly away from the nodal point in the optimally doped Bi-2212 with Tc=91K [136] is shown in Figure 16.Figure 16
Figure  4b from [136]: ReΣ(ω) measured in Bi2212 (thin line) and model ReΣ(ω) (bold line) obtained in [136]. The three thin lines (λ1,λ2,λ3) are the slopes of ReΣ(ω) in different energy regions—see the text.It is seen thatReΣexp(k,ω) has two kinks—the first one at low energyω1≈ωphhigh≈50–70meV which is (as we already argued) most probably of the phononic origin [22, 23, 138, 139], while the second kink at higher energyω2≈ωc≈350meV which is due to the Coulomb interaction. However, the important results in [136] are that the slopes of ReΣexp(k,ω) at low (ω<ωphhigh) and high energies (ωphhigh<ω<ωc) are very different. The low-energy and high-energy slope near the nodal point are shown in Figure 16 schematically (thin lines). From Figure 16 it is obvious that EPI prevails at low energies ω<ωphhigh. More precisely digitalization of ReΣexp(k,ω) in the interval ωphhigh<ω<0.4eV gives the Coulomb coupling λcZ≈1.1 while the same procedure at 20meV≈ωphlow<ω<ωphhigh≈50–70meV gives the total coupling constant (λ2≡)λZ=λepZ+λcZ≈3.2 and the EPI coupling constant λepZ(≡λep,highZ)≈2.1>2λep,effZ(k), that is, the EPI coupling is at least twice larger than the effective EPI coupling constant obtained in the previous analysis of ARPES results [22, 23, 138, 139]. This estimation tells us that at (and near) the nodal point, the EPI interaction dominatesin the quasiparticle scattering at low energies since λepZ(≈2.1)≈2λzc>2λsfZ, while at large energies (compared to ωph) the Coulomb interaction with λcZ≈1.1 dominates. We point out that EPI near the antinodal point can be even larger than in the nodal point, mostly due to the higher density of states near the antinodal point. (N8) Recent ARPES spectra in the optimally doped Bi2212 near the nodal and antinodal point [139] show a low-energy isotope effect in ReΣexp(k,ω), which can be well described in the framework of the Migdal-Eliashberg theory for EPI [140]. At higher energies ω>ωph obtained in [139] very pronounced isotope effect cannot be explained by the simple Migdal-Eliashberg theory [140]. However, there are controversies with the strength of the high-energy isotope effect since it was not confirmed in other measurements [141, 142]—see the discussion in Section 1.3.6(2) related to the isotope effects in HTSC cuprates.  (N9) The ARPES experiments in Ca2CuO2Cl2 give strong evidence for the formation of small polarons in undoped cuprates which are due to phonons and strong EPI, while in the doped systems quasiparticles are formed  and there are no small polarons [143]. Namely, in [143] a broad peak around -0.8eV is observed at the top of the band (k=(π/2,π/2)) with the dispersion similar to that predicted by the t-J model—see Figure 17.(a) The ARPES spectrum of undopedCa2CuO2Cl2 at k=(π/2,π/2). Gaussian shape—solid line, Lorentzian shape—dashed line. (b) Dispersion of the polaronic band—A—and of the quasiparticle band—B—along the nodal direction. Horizontal lines are the chemical potentials for a large number of samples. From [143].
(a)(b)However, the peak in Figure17(a) is of Gaussian shape and can be described only by coupling to bosons, that is, this peak is a boson side band—see more in [10, 11] and references therein. The theory based on the t-J model (in the antiferromagnetic state of the undoped compound) by including coupling to several (half-breathing, apical oxygen, low-lying) phonons, which is given in [144–146], explains successfully this broad peak of the boson side band by the formation of small polarons due to the EPI coupling (λep≈1.2). Note that this value of λep is for the polaron at the bottom of the band while in the case where the Fermi surface exists (in doped systems) this coupling is even larger due to the larger density of states at the Fermi surface [144–146]. In [144–146] it was stressed that even when the electron-magnon interaction is stronger than EPI the polarons in the undoped systems are formed due to EPI. The latter mechanism involves excitation of many phonons at the lattice site (where the hole is seating), while it is possible to excite only one magnon at the given site. (N10) Recent soft X-ray ARPES measurements on the electron-doped HTSC Nd1.85Ce0.15CuO4 [147], and Sm(2-x)CexCuO4 (x=0.1,0.15,0.18), Nd1.85Ce0.15CuO4, and Eu1.85Ce0.15CuO4 [148] show kink at energies 50–70meV in the quasiparticle dispersion relation along both the nodal and antinodal, directions as it is shown in Figure 18.NCCO electron-doped: (a)ImΣ(ω) measured in the nodal point. Curves are offsets by 50 meV for clarity. The change of the slope in the last bottom curve is at the phonon energy. (b) ImΣ(ω) for the antinodal direction with 30meV offset. (c) Experimental phonon dispersion of the bond stretching modes. (d) Estimated λeffep from ImΣ(ω). From [148].
(a)(b)(c)(d)It is seen from this figure that the effective EPI coupling constantλep,eff(<λep) is isotropic and λep,eff≈0.8–1. It seems that the kink in the electron-doped cuprates is due solely to EPI and in that respect the situation is similar to the one in the hole-doped cuprates.ARPES Results in the Superconducting State
(S1) There is an anisotropic superconducting gap in most HTSC compounds [22, 23], which is predominately d-wave like, that is, Δ(k)≈Δ0(coskx-cosky) with 2Δ0/Tc≈5-6 in the optimally doped systems. (S2) The particle-hole coherence in the superconducting state which is expected for the BCS-like theory of superconductivity has been observed first in [149] and confirmed with better resolution in [150], where the particle-hole mixing is clearly seen in the electron and hole quasiparticle dispersion. To remind the reader, the excited Bogoliubov-Valatin quasiparticles (α̂k,±) with energies Ekα±=ξk2+|Δk|2 are a mixture of electron (ĉk,σ) and hole (ĉ-k,-σ†), that is, α̂k,+=ukĉk↑+vkĉ-k↓†, α̂k,-=ukĉ-k↓+vkĉk↑† where the coherence factors uk, vk are given by |uk|2=1-|vk|2=(1+ξk/Ek)/2. Note that |uk|2+|vk|2=1, which is exactly observed, together with d-wave pairing Δ(k)=Δ0(coskx-cosky), in experiments in [150]. This is very important result since it proves that the pairing in HTSC cuprates is of the BCS type and not exotic one as it was speculated long time after the discovery of HTSC cuprates. (S3) The kink at (60–70)meV in the quasiparticle energy around the nodal point is not shifted (in energy) while the antinodal kink at ωph(40)~40meV is shifted (in energy) in the superconducting state by Δ0(=(25–30)meV), that is, ωph(40)→ωph(40)+Δ0=(65–70)meV [22, 23]. To remind the reader, in the standard Eliashberg theory the kink in the normal state at ω=ωph should be shifted in the superconducting state to ωph+Δ0 at all k-points at the Fermi surface. This puzzling result (that the quasiparticle energy around the nodal point is not shifted in the superconducting state) might be a smoking gun result since it makes an additional constraint on the quasiparticle interaction in cuprates. Until now there is only one plausible explanation [151] of this nonshift puzzle which is based on an assumption of the forward scattering peak (FSP) in EPI—see more in Section 2. The FSP in EPI means that electrons scatter into a narrow region (q<qc≪kF) around the initial point in the k-space, so that at the most part of the Fermi surface there is practically no mixing of states with different signs of the order parameter Δ(k). In that case the EPI bosonic spectral function (which is defined in Appendix A) α2F(k,k′,Ω)≈α2F(φ,φ′,Ω) (φ is the angle on the Fermi surface) has a pronounced forward scattering peak (at δφ=φ-φ′=0) due to strong correlations—see Section 2. Its width δφc is narrow, that is, δφc≪2π and the angle integration goes over the region δφc around the point φ. In that case the kink is shifted (approximately) by the local gap Δ(φ)=Δmaxcos2φ—for more details see [151]. As a consequence, the antinodal kink is shifted by the maximal gap, that is, |Δ(φAN≈π/2)|=Δmax while the nodal gap is practically unshifted since |Δ(φAN≈π/4)|≈0. (S4) The recent ARPES spectra [152] in the undoped single crystalline 4-layered cuprate with the apical fluorine (F), Ba2Ca3Cu4O8F2 (F0234) give rather convincing evidence against the SFI mechanism of pairing—see Figure 19.Figure 19
Crystal structure ofBa2Ca3Cu4O8(OδF1-δ)2. There are four CuO2 layers in a unit cell with the outer having apical F atoms. CRL—charge reservoir layer; SCL/OP—superconducting layer/outer plane; SCL/IP—superconducting layer/inner plane. From [152].First, F0234 is not a Mott insulator—as expected from valence charge counting which puts Cu valence as 2+, but it is a superconductor with Tc=60K. Moreover, the ARPES data [152] reveal at least two metallic Fermi-surface sheets with corresponding volumes equally below and above half-filling—see Figure 20.(a) Fermi surface (FS) contours from two samples of F0234.N—electron-like; P—hole-like. Bold arrow is (π,π) scattering vector. Angle θ defines the horizontal axis in (b). (b) Leading gap edge along k-space angle from the two FS contours. From [152].
(a)(b)Second, one of the Fermi surfaces is due to the electron-like (N) band (with 20±6% electron-doping) and the other one due to the hole-like (P) band (with 20±8% hole-doping) and their splitting along the nodal direction is significant and cannot be explained by the LDA-DFT calculations [153]. This electron and hole self-doping of inner and outer layers is in an appreciable contrast to other multilayered cuprates where there is only hole self-doping. For instance, in HgBa2CanCun+1O2n+2 (n=2,3) and (Cu,C)Ba2CanCun+1O3n+2 (n=2,3,4), the inner CuO2 layers are less hole-doped than outer layers. It turns out, unexpectedly, that the superconducting gap on theN-band Fermi surface is significantly larger than on the P-one, where in Ba2Ca3Cu4O8F2 the ratio is anomalous (ΔN/ΔP)≈2 and ΔN is an order of magnitude larger than in the electron-doped cuprate Nd2-xCexCuO4. Third, the N-band Fermi surface  is rather far from the antinodal point at (π,0). This is very important result which means that the antiferromagnetic spin fluctuations with the AF wave-vector Q=(π,π), as well as the van Hove singularity, are not dominant in the pairing in the N-band. To remind the reader, the SFI scenario assumes that the pairing is due to spin fluctuations with the wave-vector Q (and near it) which connects two antinodal points which are near the van Hove singularity at the hole-surface (at (π,0) and (0,π)) giving rise to large density of states. This is apparently not the case for the N-band Fermi surface—see Figure 20. The ARPES data give further that there is a kink at ~85 meV in the quasiparticle dispersion of both bands, while the kink in the N-band is stronger than that in the P-band. This result, together with the anomalous ratio (ΔN/ΔP)≈2, disfavors SFI as a pairing mechanism. (S5) Despite the presence of significant elastic quasiparticle scattering in a number of samples of optimally doped Bi-2212, there are dramatic sharpenings of the spectral function near the antinodal point (π,0) at T<Tc (in the superconducting state) [154]. This effect can be explained by assuming that the small q-scattering (the forward scattering peak) dominates in the elastic impurity scattering as it is pointed in [78–80, 130, 155, 156]. As a result, one finds that the impurity scattering rate in the superconducting state is almost zero, that is, γimp(k,ω)=γn(k,ω)+γa(k,ω)≈0 for |ω|<Δ0 for any kind of pairing (s-, p-, d-wave, etc.) since the normal (γn) and the anomalous (γa) scattering rates compensate each other. This collapse of the elastic scattering rate is elaborated in details in [154] and it is a consequence of the Anderson-like theorem for unconventional superconductors which is due to the dominance of the small q-scattering [78–80, 130, 155, 156]. In such a case d-wave pairing is weakly unaffected by nonmagnetic impurities and as a consequence there is small reduction in Tc [156, 157]. The physics behind this result is rather simple. The small q-scattering (usually called forward scattering) means that electrons scatter into a small region in the k-space, so that at the most part of the Fermi surface there is no mixing of states with different signs of the order parameter Δ(k). In such a way the detrimental effect of nonmagnetic impurities on d-wave pairing is significantly reduced. This result points to the importance of strong correlations in the renormalization of the nonmagnetic impurity scattering too—see discussion in Section 2.In conclusion, in order to explain the ARPES results in cuprates it is necessary to take into account (1) the electron-phonon interaction (EPI) since it dominates in the quasiparticle scattering in the energy region important for pairing, (2) the elastic nonmagnetic impurities with the forward scattering peak (FSP) due to strong correlations, and (3) the Coulomb interaction which dominates at higher energies ω>ωph. In this respect, the presence of ARPES kinks and the knee-like shape of the T dependence of the spectral width are important constraints on the scattering and pairing mechanism in HTSC cuprates.


#### 1.3.4. Tunnelling Spectroscopy and Spectral Functionα2F(ω)

By measuring current-voltageI-V characteristics in NIS (normal metal-insulator-superconductor) tunnelling junctions with large tunnelling barrier one obtains from tunnelling conductance GNS(V)=dI/dV the so called tunnelling density of states in superconductors NT(ω). Moreover, by measuring of GNS(V) at voltages eV>Δ it is possible to determine the Eliashberg spectral function α2F(ω) and finally to confirm the phonon mechanism of pairing in LTSC materials. Four tunnelling techniques were used in the study of HTSC cuprates: (1) vacuum tunnelling by using the STM technique—scanning tunnelling microscope; (2) point-contact tunnelling; (3) break-junction tunnelling; (4) planar-junction tunnelling. Each of these techniques has some advantages although in principle the most potential one is the STM technique since it measures superconducting properties locally [158]. Since tunnelling measurements probe a surface region on the scale of the superconducting coherence length ξ0, then this kind of measurements in HTSC materials with small coherence length ξ0 (ξab~20 Å in the a-b plane and ξc~1–3 Å along the c-axis) depends strongly on the surface quality and sample preparation. Nowadays, many of the material problems in HTSC cuprates are understood and as a result consistent picture of tunnelling features is starting to emerge.From tunnelling experiments one obtains the (energy-dependent) gap functionΔ(ω) in the superconducting state. Since we have already discussed this problem in [6], we will only briefly mention some important result. For instance, in most systems GNS(V) has V-shape in all families of HTSC hole- and electron-doped cuprates. The V-shape is characteristic for d-wave pairing with gapless spectrum, which is also confirmed in the interference experiments on hole- and electron-doped cuprates [75]. Some experiments give a U-shape of GNS(V) which resembles s-wave pairing. This controversy is explained to be the property of the tunnelling matrix element which filters out states with the maximal gap.Here we are interested in thebosonic spectral functionα2F(ω) of HTSC cuprates near optimal doping which can be extracted by using tunnelling spectroscopy. We inform the reader in advance that the shape and the energy width of α2F(ω), which are extracted from the second derivative d2I/dV2 at voltages above the superconducting gap, in most HTSC cuprates resemble the phonon density of states Fph(ω). This result is strong evidence for the importance of EPI in the pairing potential of HTSC cuprates. For instance, plenty of break junctions made from Bi2212 single crystals [42–45] show that the peaks (and shoulders) in -d2I/dV2 (or dips-negative peaks in d2I/dV2) coincide with the peaks (and shoulders) in the phonon density of states Fph(ω) measured by neutron scattering—see Figure 21.(a) Second derivative ofI(V) for a Bi2212 break junction in various magnetic fields (from 0–20T). The structure of dips (minima) in d2I/dV2 can be compared with the phonon density of states F(ω); (b) the spectral functions α2F(ω) in various magnetic fields. From [42–45].
(a)(b)The tunnelling spectra in Bi-2212 break junctions [42–45], which are shown in Figure 21 indicates that the spectral function α2F(ω) is independent of magnetic field, which is in contradiction with the theoretical prediction based on the SFI pairing mechanism where this function should be sensitive to magnetic field. The reported broadening of the peaks in α2F(ω) is partly due to the gapless spectrum of d-wave pairing in HTSC cuprates. Additionally, the tunnelling density of states NT(ω) at very low T and for ω>Δ shows a pronounced gap structure. It was found that 2Δ¯/Tc=6.2–6.5, where Tc=74–85Kand Δ¯ is some average value of the gap. In order to obtain α2F(ω) the inverse procedure was used by assuming s-wave superconductivity and the effective Coulomb parameter μ*≈0.1 [42–45]. The obtained α2F(ω) gives large EPI coupling constant λep≈2.3. Although this analysis [42–45] was done in terms of s-wave pairing, it mimics qualitatively the case of d-wave pairing, since one expects that d-wave pairing does not change significantly the global structure of d2I/dV2 at eV>Δ albeit introducing a broadening in it—see the physical meaning in Appendix A. We point out that the results obtained in [42–45] were reproducible on more than 30 junctions. In that respect very important results on slightly overdoped Bi2212–GaAs and on Bi2212–Au planar tunnelling junctions are obtained in [46, 47]—see Figure 22.Figure 22
The spectral functionsα2F(ω) and the calculated density of states at 0K (upper solid line) obtained from the conductance measurements, the Bi(2212)–Au planar junctions. GPDS—generalized phonon density of states. From [46, 47].These results show very similar features to those obtained in [42–45] on break junctions. It is worth mentioning that several groups [48–52] have obtained similar results for the shape of the spectral function α2F(ω) from the I-V measurements on various HTSC cuprates—see the comparison in Figure 23. These facts leave no much doubts about the importance of the EPI in pairing mechanism of HTSC cuprates.Figure 23
The spectral functionsα2F(ω) from measurements of various groups: Vedeneev et al. [42–45], Gonnelli et al. [52], Miyakawa et al. [48, 49], and Shimada et al. [46, 47]. The generalized density of states GPDS for Bi2212 is at the bottom. From [46, 47].In that respect, the tunnelling measurements onslightly overdopedBi2Sr2CaCu2O8 [46, 47, 53, 54] give impressive results. The Eliashberg spectral function α2F(ω) of this compound was extracted from the measurements of d2I/dV2 and by solving the inverse problem—see Appendix A. The extracted α2F(ω) has several peaks in broad energy region up to 80meV as it is seen in Figures 22 and 23, which coincide rather well with the peaks in the phonon density of states Fph(ω)—more precisely the generalized phonon density of states GPDS(ω) defined in Appendix A. In [53, 54] numerous peaks, from P1–P13, in α2F(ω) are discerned as shown in Figure 24, which correspond to various groups of phonon modes—laying in (and around) these peaks. Moreover, in [46, 47, 53, 54] the coupling constants for these modes are extracted as well as their contribution (ΔTc) to Tc as it is seen in Table 1. Note that due to the nonlinearity of the problem the sum of (ΔTc)i,i=1,2,…,13, due to various modes is not equal to Tc.Table 1
Phonon frequencyω, EPI coupling constant λi of the peaks P1–P13, and contribution ΔTc to Tc of each peak in α2F(ω)—as shown in Figure 24—obtained from the tunnelling conductance of Bi2Sr2CaCu2O8. ΔTc is the decrease in Tc when the peak in α2F(ω) is eliminated. From [53, 54].
No. of peakω [meV]λiΔTc [K]P114.31.267.4P220.80.9511.0P331.70.4810.5P435.10.286.7P539.40.247.0P645.30.3010.0P758.30.156.5P863.90.010.6P969.90.073.6P1073.70.063.3P1177.30.010.8P1282.10.010.7P1387.10.031.8Figure 24
The spectral functionsα2F(ω) from the tunnelling conductance of Bi2Sr2CaCu2O8 for the positive and the negative bias voltages, and the averaged one [46, 47]. The averaged one is divided into 13 components. The origin of the ordinate is 2,1,0, and -0.5 from the top down. From [46, 47, 53, 54].The next remarkable result is that the extractedEPI coupling constant is very large, that is, λep(=2∫dωα2F(ω)/ω)=∑iλi≈3.5—see Table 1. It is obvious from Figure 24 and Table 1 that almost all phonon modes contribute to λep and Tc, which means that on the average each particular phonon mode is not too strongly coupled to electrons since λi<1.3,i=1,2,…,13, thus keeping the lattice stable.Let us discuss the content of Table1 in more details where it is shown the strength of the EPI coupling and the relative contribution of different phononic modes to Tc. In Table 1 it is seen that lower-frequency modes from P1–P3, corresponding to Cu,Sr, and Ca vibrations, are rather strongly coupled to electrons (with λκ~1) which give appreciable contributions to Tc. It is also seen in Table 1 that the coupling constants λi of the high-energy phonons (P9–P13 with ω≥70meV) have λi≪1 and give moderate contribution to Tc—around 10%. These results give solid evidence for the importance of the low-energy modes related to the change of the Madelung energy in the ionic-metallic structure of HTSC cuprates—the idea advocated in [3–6] and discussed in Section 2. If confirmed in other HTSC families, these results are in favor of the moderate oxygen isotope effect in cuprates near the optimal doping since the oxygen modes are higher-energy modes and give smaller contribution to Tc. We stress that each peak P1–P13 in α2F(ω) corresponds to many modes. For a better understanding of the EPI coupling in these systems we show in Figure 25 the total and partial density of phononic states. It is seen that the low-energy phonons are due to the vibrations of the Ca, Sr, and Cu ions which correspond to the peaks P1-P2 in Figures 23 and 24. In order to obtain information on the structure of vibrations which are strongly involved in pairing, we show in Figures 26 and 27 the structure of these vibrations at special points in the Brillouin zone. It is seen in Figure 26 that the low-frequency phonons P1-P2 are dominated by Cu, Sr, Ca vibrations.Figure 25
The phonon density of statesF(ω) (PDOS) of Bi2Sr2CaCu2O8 compared with the generalized density of states (GPDOS) [159]. Atomic vibrations: O1—O in the CuO2 plane; O2—apical O; O3—O in the BiO plane. From [46, 47].Atomic polarization vectors and their frequencies (inmeV) at special points in the Brillouin zone. The larger closed circles in the lattice are O-ions. Γ-X is along the Cu-O-Cu direction. Arrows indicate displacements. The modes in square and round brackets are the transverse and longitudinal optical modes, respectively. (a) Modes of the P1 peak. (b) Modes of the P2 peak. From [46, 47, 53, 54].
(a)(b)Atomic polarization vectors and their frequencies (inmeV) at special points in the Brillouin zone. The larger closed circles in the lattice are O-ions. Γ-X is along the Cu-O-Cu direction. Arrows indicate displacements. The modes in square and round brackets are the transverse and longitudinal optical modes, respectively. (a) Modes of the P3 peak. (b) Modes of the P4 peak. From [46, 47, 53, 54].
(a)(b)Further, based on Table1 one concludes that the P3 modes are strongerly coupled to electrons than the P4 ones, although the density of state for the P4 modes is larger. The reason for such an anomalous behavior might be due to symmetries of the corresponding phonons as it is seen in Figure 27. Namely, to the P3 peak contribute axial vibrations of O(1) in the CuO2 plane which are odd under inversion, while in the P4 peak these modes are even. The in-plane modes of Ca and O(1) are present in P3 which are in-phase and out-of-phase modes, while in P4 they are all out-of-phase modes. For more information on other modes, P5–P13, see [53, 54]. We stress that the Eliashberg equations based on the extracted α2F(ω) of the slightly overdoped Bi2Sr2CaCu2O8 with the ratio (2Δ/Tc)≈6.5 describe rather well numerous optical, transport, and thermodynamic properties [53, 54]. However, in underdoped systems with (2Δ/Tc)≈10, where the pseudogap phenomena are pronounced, there are serious disagreements between experiments and the Migdal-Eliashberg theory [53, 54]. We would like to stress that the contribution of the high-frequency modes (mostly the oxygen modes) to α2F(ω) may be underestimated in tunnelling measurements due to their sensitivity to the surface contamination and defects. Namely, the tunnelling current probes a superconductor to a depth of order of the quasiparticle mean-free path l(ω)=vFγ-1(ω). Since the relaxation time γ-1(ω) decreases with increasing ω, the mean-free path can be rather small and the effects of the high-energy phonons are sensitive to the surface contamination.Similar conclusion regarding the structure of theEPI spectral function α2F(ω) in HTSC cuprates comes out from tunnelling measurements on Andreev junctions (the BTK parameter Z≪1—low barrier) and Giaver junctions (Z≫1—high barrier) in La2-xSrxCuO4 and YBCO compounds [160], where the extracted α2F(ω) is in good accordance with the phonon density of states Fph(ω)—see Figure 28.(a)d2I/dV2 of a Giaver-like contact in La2-xSrxCuO4—note the large structure below 50meV; (b) d2I/dV2 of an Andreev- and Giaver-like contact compared to the peaks in the phonon density of states. From [160].
(a)(b)Note that the BTK parameterZ is related to the transmission and reflection coefficients for the normal metal (1+Z2)-1 and Z2(1+Z2)-1, respectively.Although most of the peaks inα2F(ω) of HTSC cuprates coincide with the peaks in the phonon density of states, it is legitimate to put the following question. Can the magnetic resonance in the superconducting state give significant contribution to α2F(ω)? In that respect the inelastic magnetic neutron scattering measurements of the magnetic resonance as a function of doping [161] give that the resonance energy Er scales with Tc, that is, Er=(5-6)Tc as shown in Figure 29.Figure 29
Doping dependence of the energyEr of the magnetic resonance peak at Q=(π,π) in YBCO and Bi2212 measured at low temperatures by inelastic neutron scattering. From [161].This means that if one of the peaks inα2F(ω) is due to the magnetic resonance at ω=Er, then it must shift strongly with doping as it is observed in [161]. This is contrary to phonon peaks (energies) whose positions are practically doping independent. To this end, recent tunnelling experiments on Bi-2212 [55] show clear doping independenceof α2F(ω) as it is seen in Figure 30. This remarkable result is an additional evidence in favor of EPI and against the SFI mechanism of pairing in HTSC cuprates which is based on the magnetic resonance peak in the superconducting state. In that respect the analysis in [162] of the tunneling spectra of the electron-doped cuprate Pr0.88Ce0.12CuO4 with Tc=24K shows the existence of the bosonic mode at ωB=16meV which is significantly larger than the magnetic-resonance mode with ωr=(10-11)meV. This result excludes the magnetic-resonance mode as an important factor which modifies superconductivity.Figure 30
Second derivative ofI(V) for Bi2212 tunnelling junctions for various doping: UD—underdoped; OD—optimally doped; OVD—overdoped system. The structure of minima in d2I/dV2 can be compared with the phonon density of states F(ω). The full and vertical lines mark the positions of the magnetic resonance energy Er≈5.4Tc for various doping taken from Figure 29. Red tiny arrows mark positions of the magnetic resonance Er in various doped systems. Dotted vertical lines mark various phonon modes. From [55].The presence of pronounced phononic structures (and the importance of EPI) in theI(V) characteristics was quite recently demonstrated by the tunnelling measurements on the very good La1.85Sr0.15CuO4 films prepared by the molecular beam epitaxy on the [001]-symmetric SrTiO3 bicrystal substrates [56]. They give unique evidence for eleven peaks in the (negative) second derivative, that is, -d2I/dV2. Furthermore, these peaks coincide with the peaks in the intensities of the phonon Raman scattering data measured at 30K in single crystals of LSCO with 20% of Sr [57]. These results are shown in Figure 31. In spite of the lack of a quantitative analysis of the data in the framework of the Eliashberg equations, the results in [56] are important evidence that phonons are relevant pairing bosons in HTSC cuprates.Figure 31
Second derivative datad2I(V)/dV2 of the tunnelling spectra on thin films of La1.85Sr0.15CuO4 are shown along with phonon Raman scattering data on single crystals of LSCO with 20% Sr. The polarization of the incident and scattered light in the Raman spectra is parallel to the CuO2 planes. From [56].It is interesting that in thec-axis vacuum tunnelling STM measurements [163] the fine structure in d2I/dV2 at eV>Δ was not seen below Tc, while the pseudogap structure is observed at temperatures near and above Tc. This result could mean that the STM tunnelling is likely dominated by the nontrivial structure of the tunnelling matrix element (along the c-axis), which is derived from the band-structure calculations [164]. However, recent STM experiments on Bi2212 [61–63] give information on the possible nature of the bosonic mode which couples with electrons. In [61–63] the local conductance dI/dV(r,E) is measured where it is found that d2I/dV2(r,E) has peak at E(r)=Δ(r)+Ω(r) where dI/dV(r,E) has the maximal slope—see Figure 32(a).(a) Typical conductancedI/dV(r,E). The ubiquitous features at eV>Δ(gap) with maximal slopes, which give peaks in d2I/dV2(r,E), are indicated by arrows. (b) The histograms of all values of Ω(r) for samples with O16—right curve and with O18—left curve. From [61–63].
(a)(b)It turns out that the corresponding average phonon energyΩ¯ depends on the oxygen mass, that is, Ω¯~MO-1/2, with Ω¯16=52meV and Ω¯18≈48meV—as it is seen in Figure 32(b). This result is interpreted in [61–63] as an evidence that the oxygen phonons are strongly involved in the quasiparticle scattering. A possible explanation is put forward in [61–63] by assuming that this isotope effect is due to the B1g phonon which interacts with the antinodal quasiparticles. However, this result requires a reanalysis since the energy of the bosonic mode in fact coincides with the dip and not with the peak of d2I/dV2(r,E)—as reported in [61–63].The important message of numerous tunnelling experiments inHTSC cuprates near and at the optimal doping is that there is strong evidence for the importance of EPI in the quasiparticle scattering and that no particular phonon mode can be singled out in the spectral function α2F(ω) as being the only one which dominates in pairing mechanism. This important result means that the high Tc is not attributable to a particular phonon mode in the EPI mechanism but all phonon modes contribute to λep. Having in mind that the phonon spectrum in HTSC cuprates is very broad (up to 80meV), then the large EPI constant (λep≳2) obtained in the tunnelling experiments is not surprising at all. Note that similar conclusion holds for some other oxide superconductors such as Ba9.6K0.4BiO3 with Tc=30K where the peaks in the bosonic spectral function/extracted from tunnelling measurements coincide with the peaks in the phononic density of states [165–167].


#### 1.3.5. Phonon Spectra and EPI

Although experiments related to phonon spectra and their renormalization byEPI, such as inelastic neutron, inelastic X-ray, and Raman scattering, do not give the spectral function α2F(ω), they nevertheless can give useful, but indirect, information on the strength of EPI for some particular phonons. We stress in advance that the interpretation of the experimental results in HTSC cuprates by the theory of EPI for weakly correlated electrons is inadequate since in strongly correlated systems, such as HTSC cuprates, the phonon renormalization due to EPI is different than in weakly correlated metals [168]. Since these questions are reviewed in [168], we will briefly enumerate the main points: (1) in strongly correlated systems the EPI coupling for a number of phononic modes can be significantly larger than the LDA-DFT and Hartree-Fock methods predict. This is due to many-body effects not contained in LDA-DFT [168, 169]. The lack of the LDA-DFT calculations in obtaining phonon line-widths is clearly demonstrated, for instance, in experiments on L2-xSrxCuO4—see review in [170] and references therein, where the bond-stretching phonons at q=(0.3,0,0) are softer and much broader than the LDA-DFT calculations predict. (Note the wave vector q is in units (2π/a,2π/b,2π/c)—for instance, in these units (π,π) corresponds to (0.5,0.5).) (2) The calculation of phonon spectra is in principle very difficult problem since besides the complexity of structural properties in a given material one should take into account appropriately the long-range Coulomb interaction of electrons as well as strong short-range repulsion. Our intention is not to discuss this complexity here—for that see, for instance, [69]—but we only stress some important points which will help to understand problems with which is confronted the theory of phonons in cuprates.The phonon Green's functionD(q,ω) depends on the phonon self-energy Π(q,ω) which takes into account all the enumerated properties (note that D-1(q,ω)=D0-1(q,ω)-Π(q,ω)). In cases when the EPI coupling constant gep(k,k′) is a function on the transfer momentum q=k-k′ only, then Π(q) (q=(q,iωn)) depends on the quasiparticle charge susceptibility χc(q)=P(q)/ɛe(q):
(41)Π(q)=|gep(q)|2χc(q),
and P(q) is the irreducible electronic polarization given by
(42)P(q)=-∑pG(p+q)Γc(p,q)G(p).
The screening due to the long-range Coulomb interaction is contained in the electronic dielectric function ɛe(q) while the “screening” due to (strong) correlations is described by the charge vertex function Γc(p,q). Due to complexity of the physics of strong correlations the phonon dynamics was studied in the t-J model but without the long-range Coulomb interaction [168, 169, 171], in which case one has ɛe=1 and χc(q)=P(q). However, in studying the phonon spectra in HTSC cuprates it is believed that this deficiency might be partly compensated by choosing the bare phonon frequency ω0(q) (contained in D0-1(q,ω)) to correspond to the undoped compounds [168, 171]. It is a matter of future investigations to incorporate all relevant interactions in order to obtain a fully microscopic and reliable theory of phonons in cuprates. Additionally, the electron-phonon interaction (with the bare coupling constant gep(q)) is dominated by the change of the energy of the Zhang-Rice singlet—see more in Section 2.3—and (41) for Π(q) is adequate one [6, 168, 169]. Since the charge fluctuations in HTSC cuprates are strongly suppressed (no doubly occupancy of the Cu 3d9 state) due to strong correlations, and since the suppressed value of χc(q) cannot be obtained by the band-structure calculations, this means that LDA-DFT underestimates the EPI coupling constant significantly. In the following we discuss this important result briefly.(1) Inelastic Neutron and X-Ray Scattering—The Phonon Softening and the Line-Width due to EPI
The appreciable softening and broadening of numerous phonon modes has been observed in the normal state of HTSC cuprates, thus giving evidence for pronounced EPI effects and for inadequacy of the LDA-DFT calculations in treating strong correlations and suppression of the charge susceptibility [6, 10, 11, 168, 171]. There are several relevant reviews on this subject [10, 11, 168, 170, 172] and here we discuss briefly two important examples which demonstrate the inefficiency of the LDA-DFT-band structure calculations to treat quantitatively EPI in HTSC cuprates. For instance, the Cu–O bond-stretching phonon mode shows a substantial softening at qhb=(0.3,0,0) by doping  of La1.85Sr0.15CuO4 and YBa2Cu3O7 [170, 172]—called the half-breathing phonon, and a large broadening by 5meV at 15% doping [173–175] as it is seen in Figure 33. While the softening can be partly described by the LDA-DFT method [176], the latter theory predicts an order of magnitude smaller broadening than the experimental one. This failure of LDA-DFT is due to the incorrect treatment of the effects of strong correlations on the charge susceptibility χc(q) and due to the absence of many-body effects which can increase the coupling constant gep(q)—see more in Section 2. The neutron scattering measurements in La1.85Sr0.15CuO4 give evidence for large (30%) softening of the OZZ with Λ1 symmetry with the energy ω≈60meV, which is theoretically predicted in [177], and for the large line-width about 17meV which also suggests strong EPI. These apex modes are favorable for d-wave pairing since their coupling constants are peaked at small momentum q [10, 11]. Having in mind the above results, then it is not surprising that the recent calculations of the EPI coupling constant λep in the framework of LDA-DFT give very small EPI coupling constant λep≈0.3 [28, 29]. The critical analysis of the LDA-DFT results in HTSC cuprates is done in [6] and additionally argued in [10, 11, 178] by pointing their disagreement with the inelastic neutron and X-ray scattering measurements—as it is shown in Figure 33.Comparison of DFT calculations with experimental results of inelastic X-ray scattering: (a) phonon energies inLa1.85Sr0.15CuO4 and (b) in YBa2Cu3O7; (c) phonon line-widths in La1.85Sr0.15CuO4. DFT calculations [28] give much smaller width than experiments [173–175]. From [178].
(a)(b)(c)In Section2 we will discuss some theoretical approaches related to EPI in strongly correlated systems but without discussing the phonon renormalization. The latter problem was studied in more details in the review articles in [10, 11, 170]. Here, we point out only three (for our purposes) relevant results. First, there is an appreciable difference in the phonon renormalization in strongly and weakly correlated systems. Namely, the change of the phonon frequencies in the presence of the conduction electrons is proportional to the squared coupling constant |gq| and charge susceptibility χc, that is, δω(q)~|gep(q)|2Reχc, while the line-width is given by Γω(q)~|gep(q)|2|Imχc|. All these quantities can be calculated in LDA-DFT and as we discussed above, where for some modes one obtains that Γω(q)(LDA)≪Γω(q)(exp). However, it turns out that in strongly correlated systems doped by holes (with the concentration δ≪1) the charge fluctuations are suppressed in which case the following sum rule holds [10, 11, 171]:
(43)1πN∑q≠0‍∫-∞∞dω|Imχc(q,ω)|=2δ(1-δ)N,
while in the LDA-DFT method one has
(44)1πN∑q≠0∫-∞∞dω|Imχc(q,ω)|(LDA)=(1-δ)N.
The inequality Γω(q)(LDA)≪Γω(q)(exp) (for some phonon modes) together with (43)-(44) means that for low doping δ≪1 the LDA calculations strongly underestimate the EPI coupling constantin the large portion of the Brillouin zone, that is, one has |gep(LDA)(q)|≪|gep(exp)(q)|. The large softening and the large line-width of the half-breathing mode at q=(0.5,0), but very moderate effects for the breathing mode at q=(0.5,0.5), are explained in the framework of the one slave-boson (SB) theory (for U=∞) in [171], where χc(q,ω) (i.e., Γc(p,q)=Γc(p,q)) is calculated in leading O(1/N) order. We stress that there is another method for studying strong correlations—the X-method—where the controllable 1/N expansion is performed in terms of the Hubbard operators and where the charge vertex Γc(p,q) is calculated [6, 78–80, 130, 179, 180]. It turns out that in the adiabatic limit (ω=0) the vertex functions Γc(pF,q) in these two methods have important differences. For instance, Γc(X)(pF,q) (in the X-method) is peaked at q=0—the so called forward scattering peak (FSP)—while Γc(SB)(pF,q) has maximum at finite |q|≠0 [181]—see Section 2.3.5. The enumerated properties of Γc(X)(pF,q) are confirmed by the numerical Monte Carlo calculations in the finite-U Hubbard model [182], where it is found that FSP exists for all U, but it is especially pronounced in the limit U≫t. These results are also confirmed in [183] where the calculations are performed in the four-slave-boson technique—see more in Section 2.3.5. Having in mind this difference it would be useful to have calculations of χc(q,ω) in the framework of the X-method which are unfortunately not done yet. Second, the many-body theory gives that for coupling to some modes the coupling constant |gep(q)| in HTSC cuprates can be significantly larger than the LDA-DFT calculations predict [10, 11], which is due to some many-body effects not present in the latter [169]. In Section 2 it will be argued that for some phonon modes one has |gep(q)|2≫|gep(LDA)(q)|2. For instance, for the half-breathing mode, one has |gep(q)|2≈3|gep(LDA)(q)|2 [10, 11, 169]—see Section 2. These two results point to an inadequacy of LDA-DFT in calculations of EPI effects in HTSC cuprates. Third, the phonon self-energy (Π(q)) and quasiparticle self-energy Σ(k) are differently renormalized by strong correlations [6, 10, 11, 78–80, 130, 179, 180], which is the reason that Π(q) is much more suppressed than Σ(k)—see Section 2. The effects of the charge vertex on Π(q) and Σ(k) are differently manifested. Namely, the vertex function enters quadratically in Σ(k) and the presence of the forward scattering peak in the charge vertex strongly affects the EPI coupling constant gep(q) in Σ(k):
(45)Σ(k)=-∑q|gep(q)γc(k,q)|2D(q)g(k+q),
where g(k)(≡G(k)/Q) is the quasiparticle Green's function, γc(k,q)=Γc(k,q)/Q is the quasiparticle vertex, and Q(~δ) is the Hubbard quasiparticle spectral weight—see Section 2.3. In the adiabatic limit |q|>qω=ωph/vF one has γc(k,q)≈γc(k,q) and for q≫qc(≈δ·π/a) the charge vertex is strongly suppressed (γc(k,q)≪1) making the effective EPI coupling (which also enters the pairing potential) small at large (transfer) momenta q. This has strong repercussion on the pairing due to EPI since for small doping it makes the d-wave pairing coupling constant to be of the order of the s-one (λd≈λs). Then in the presence of the residual Coulomb interaction EPI gives rise to d-wave pairing. On the other side the charge vertex Γc(k,q) enters Π(q)  linearlyand it is additionally integrated over the quasiparticle momentum k—see (42). Therefore, one expects that the effects of the forward scattering peak on Π(q) are less pronounced than on Σ(k). Nevertheless, the peak of Γc(k,q) at q=0 may be (partly) responsible that the maximal experimental softening and broadening of the stretching (half-breathing) mode in La1.85Sr0.15CuO4 and YBa2Cu3O7 is at qhb(exp)=(0.3,0,0) [170] and not at qhb=(0.5,0) for which gep(qhb) reaches maximum. This means that the charge vertex function pushes the maximum of the renormalized EPI coupling constant to smaller momenta q. It would be very interesting to have calculations for other phonons by including the vertex function obtained by the X-method—see Section 2.3.(2) The Phonon Raman Scattering
Thephonon Raman scattering gives an indirect evidence for importance of EPI in cuprates [184–188]. We enumerate some of them—see more in [6] and references therein. (i) There is a pronounced asymmetric line-shape (of the Fano resonance) in the metallic state. For instance, in YBa2Cu3O7 two Raman modes at 115cm-1 (Ba dominated mode) and at 340cm-1 (O dominated mode in the CuO2 planes) show pronounced asymmetry which is absent in YBa2Cu3O6. This asymmetry means that there is an appreciable interaction of Raman active phonons with continuum states (quasiparticles). (ii) The phonon frequencies for some A1g and B1g are strongly renormalized in the superconducting state, between (6–10)%, pointing again to the importance of EPI [188]—see also [6, 37, 38]. To this point we mention that there is a remarkable correlation between the electronic Raman cross-section S̃exp(ω) and the optical conductivity in the a-b plane σab(ω), that is, S̃exp(ω)~σab(ω) [6]. In previous subsections it is argued that EPI with the very broad spectral function α2F(ω) (0<ω≲80meV) explains in a natural way the ω and T dependence of σab(ω). This means that the electronic Raman spectra in cuprates can be explained by EPI in conjunction with strong correlations. This conclusion is supported by the calculations of the Raman cross-section [189] which take into account EPI with α2F(ω) extracted from the tunnelling measurements in YBa2Cu3O6+x and Bi2Sr2CaCu2O8+x [6, 42–54]. Quite similar properties (to cuprates) of the electronic Raman scattering, as well as of σ(ω), R(ω), and ρ(T), were observed in experiments [108] on isotropic 3D metallic oxides La0.5Sr0.5CoO3 and Ca0.5Sr0.5RuO3 where there are no signs of antiferromagnetic fluctuations. This means that low dimensionality and antiferromagnetic spin fluctuations cannot be a prerequisite for anomalous scattering of quasiparticles and EPI must be inevitably taken into account since it is present in all these compounds.


#### 1.3.6. Isotope Effect inTc and Σ(k,ω)

The isotope effectαTC in the critical temperature Tc was one of the very important proofs for the EPI pairing mechanism in low-temperature superconductors (LTSCs). As a curiosity the isotope effect in LTSC systems was measured almost exclusively in mono-atomic systems and in few polyatomic systems: the hydrogen isotope effect in PdH, the Mo and Se isotope shift of Tc in Mo6Se8, and the isotope effect in Nb3Sn and MgB2. We point out that very small (αTC≈0 in Zr and Ru) and even negative (in PdH) isotope effects in some polyatomic systems of LTSC materials are compatible with the EPI pairing mechanism but in the presence of substantial Coulomb interaction or lattice anharmonicity. The isotope effect αTC cannot be considered as the smoking gun effect since it is sensitive to numerous influences. For instance, in MgB2 it is with certainty proved that the pairing is due to EPI and strongly dominated by the boron vibrations, but the boron isotope effect is significantly reduced, that is, αTC≈0.3 and the origin for this smaller value is still unexplained. The situation in HTSC cuprates is much more complicated because they are strongly correlated systems and contain many atoms in unit cell. Additionally, the situation is complicated with the presence of intrinsic and extrinsic inhomogeneities, low dimensionality which can mask the isotope effects. On the other hand new techniques such as ARPES, STM, and μSR allow studies of the isotope effects in quasiparticle self-energies, that is, αΣ, which will be discussed below.(1) Isotope Effect αTC in Tc
This problem will be discussed only briefly since more extensive discussion can be found in [6]. It is well known that in the pure EPI pairing mechanism the total isotope coefficient α is given by αTC=∑i,pαi(p)=-∑i,pdlnTc/dlnMi(p), where Mi(p) is the mass of the ith element in the pth crystallographic position. We stress that the total isotope effect is not measured in HTSC cuprates but only some partial ones. Note that, in the case when the screened Coulomb interaction is negligible, that is, μc*=0, the theory predicts αTC=1/2. From this formula one can deduce that the relative change of Tc,δTc/Tc, for heavier elements should be rather small—for instance, it is 0.02 for Ba135→Ba138, 0.03 for Cu63→Cu65, and 0.07 for La138→La139. This means that the measurements of αi for heavier elements are confronted with the ability of the present experimental techniques. Therefore most isotope effect measurements were done by substituting light atoms O16 by O18 only. It turns out that in most optimally doped HTSC cuprates αO is rather small. For instance, αO≈0.02–0.05 in YBa2Cu3O7 with Tc,max≈91K, but it is appreciable in La1.85Sr0.15CuO4 with Tc,max≈35K where αO≈0.1-0.2. In Bi2Sr2CaCu2O8 with Tc,max≈76K one has αO≈0.03–0.05 while αO≈0.03 and even negative (-0.013) in Bi2Sr2Ca2Cu2O10 with Tc,max≈110K. The experiments on Tl2Can-1BaCunO2n+4 (n=2,3) with Tc,max≈121K are still unreliable and αO is unknown. In the electron-doped (Nd1-xCex)2CuO4 with Tc,max≈24K one has αO<0.05 while in the underdoped materials αO increases. The largest αO is obtained even in the optimally doped compounds like in systems with substitution, such as La1.85Sr0.15Cu1-xMxO4, M=Fe,Co, where αO≈1.3 for x≈0.4%. In La2-xMxCuO4 there is a Cu-isotope effect which is of the order of the oxygen one, that is, αCu≈αO giving αCu+αO≈0.25–0.35 for optimally doped systems (x=0.15). In case when x=0.125 with Tc≪Tc,max one hasαCu≈0.8-1 with αCu+αO≈1.8 [190, 191]. The appreciable copper isotope effect in La2-xMxCuO4 tells us that vibrations other than oxygen ions are important in giving high Tc. In that sense one should have in mind the tunnelling experiments discussed above, which tell us that all phonons contribute to the Eliashberg pairing function α2F(k,ω) and according to these results the oxygen modes give moderate contribution to Tc [53, 54]. Hence the small oxygen isotope effect αTc(O) in optimally doped cuprates, if it is an intrinsic property at all (due to pronounced local inhomogeneities of samples and quasi-two-dimensionality of the system), does not exclude the EPI mechanism of pairing.(2) Isotope Effect αΣ in the Self-Energy
The fine structure of the quasiparticle self-energyΣ(k,ω), such as kinks and slopes, can be resolved in ARPES measurements and in some respect in STM measurements. It turns out that there is isotope effect in the self-energy in the optimally doped Bi2212 samples [139, 141, 142]. In the first paper on this subject [139] it is reported a red shiftδωk,70~-(10–15)meV of the nodal kink at ωk,70≃70meV for the O16→O18 substitution. In [139] it is reported that the isotope shift of the self-energy δΣ=Σ16-Σ18~10meV is very pronounced at large energies ω=100–300meV. Concerning the latter result, there is a dispute since it is not confirmed in other experiments [141, 142]. However, the isotope effect in ReΣ(k,ω) at low energies [141, 142] is well described in the framework of the Migdal-Eliashberg theory for EPI [140] which is in accordance with the recent ARPES measurements with low-energy photons ~7 eV [192]. The latter allowed very good precision in measuring the isotope effect in the nodal point of Bi-2212 with Tc16=92.1K and Tc18=91.1K [192]. They observed a shift in the maximum of ReΣ(kN,ω)—at ωk,70≈70meV (it corresponds to the half-breathing or to the breathing phonon)—by δωk,70≈3.4±0.5meV as shown in Figure 34.(a) EffectiveReΣ for five samples for O16 (blue) and O18 (red) along the nodal direction. (b) Effective ImΣ determined from MDC full widths. An impurity term is subtracted at ω=0. From [192].
(a)(b)By analyzing the shift in ImΣ(kN,ω)—shown in Figure 34—one finds similar result for δωk,70≈3.2±0.6meV. The similar shift was obtained in STM measurements [61–63] which is shown in Figure 32(b) and can have its origin in different phonons. We would like to stress two points: (i) in compounds with Tc~100K the oxygen isotope effect in Tc is moderate, that is, αTc(O)<0.1 [192]. If we consider this value to be intrinsic, then even in this case it is not in conflict with the tunnelling experiments [53, 54] since the latter give evidence that vibrations of heavier ions contribute significantly to Tc—see the discussion in Subsection 1.3.4 on the tunnelling spectroscopy. (ii) In ARPES measurements of [192] the effective EPI coupling constant λep,eff≳0.6 is extracted, while the theory in Subsection 1.3.3 gives that the real coupling constant is larger, that is, λep>1.2. This value is significantly larger than the LDA-DFT theory predicts λep,LDA<0.3 [28, 29]. This again points that the LDA-DFT method does not pick up the many-body effects due to strong correlations—see Section 2.


### 1.4. Summary of Section1

The analysis of experimental data in HTSC cuprates which are related to optics, tunnelling, and ARPES measurementsnear and at the optimal doping gives evidence for the large electron-phonon interaction (EPI) with the coupling constant 1<λep<3.5. We stress that this analysis is done in the framework of the Migdal-Eliashberg theory for EPI which is a reliable approach for systems near the optimal doping. The spectral function α2F(ω), averaged over the Fermi surface, is extracted from various tunnelling measurements on bulk materials and tin films. It contains peaks at the same energies as the phonon density of states Fph(ω). So obtained spectral function when inserted in the Eliashberg equations provides sufficient strength for obtaining high critical temperature Tc~100K. These facts are a solid proof for the important role of EPI in the normal-state scattering and pairing mechanism of cuprates. Such a large (experimental) value of the EPI coupling constant and the robustness of the d-wave superconductivity in the presence of impurities imply that the EPI potential and the impurity scattering amplitude must be strongly momentum dependent. The IR optical reflectivity data provide additional but indirect support for the importance of EPI since by using the spectral function (extracted from tunnelling measurements) one can quantitatively explain frequency dependence of the dynamical conductivity, optical relaxation rate, and optical mass. These findings related to EPI are additionally supported by ARPES measurements on BSCO compounds. The ARPES kinks, the phononic features and the isotope effect in the quasiparticle self-energy in the nodal and antinodal points at low energies (ω≪ωc) persist in the normal and superconducting state. They are much more in favor of EPI than for the spin fluctuation (SFI) scattering mechanism. The transport EPI coupling constant in HTSC cuprates is much smaller than λep, that is, λtr~λep/3, which points to some peculiar scattering mechanism not met in low-temperature superconductors. The different renormalization of the quasiparticle and transport self-energies by the Coulomb interaction (strong correlations) hints to the importance of the small-momentum scattering in EPI. This will be discussed in Section 2.The ineffectiveness of SFI to solely provide pairing mechanism in cuprates comes out also from the magnetic neutron scattering on YBCO and BSCO. As a result, the imaginary part of the susceptibility is drastically reduced in the low-energy region by going from slightly underdoped toward optimally doped systems, whileTc is practically unchanged. This implies that the real SFI coupling constant λsf(~gsf2) is small since the experimental value gsf(exp)<0.2eV is much smaller than the assumed theoretical value gsf(th)≈(0.7–1.5)eV.Inelastic neutron and X-ray scattering measurements in HTSC cuprates show that the broadening of some phonon lines is by an order of magnitude larger than the LDA-DFA methods predict. Since the phonon line-widths depend on the EPI coupling and the charge susceptibility, it is evident that calculations of both quantities are beyond the range of applicability of LDA-DFT. As a consequence, the LDA-DFT calculations overestimate the electronic screening and thus underestimate the EPI coupling, since many-body effects due to strong correlations are not contained in this mean-field theory. However, in spite of the promising and encouraging experimental results about the dominance of EPI in cuprates, the theory is still confronted with difficulties in explaining sufficiently large coupling constant in thed-channel. At present there is not such a satisfactory microscopic theory although some concepts, such as the the dominant EPI scattering at small transfer momenta, are understood at least qualitatively. This set of problems and questions will be discussed in Section 2.


## 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, Δ(k,ω)≈Δd(ω)(coskx-cosky). 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 A≡α·λ(ωB/Wb) fulfills the condition A≪1, where λ is the electron-boson coupling constant, ωB is the characteristic bosonic energy, Wb 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(k,k′,ω) (or its average α2F(ω)) which characterizes scattering of quasiparticle from k to k′ by exchanging bosonic energy ω. 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. Thefirst one is based on the electron-phonon interaction (EPI), with the main proponents in [3–11], where mediating bosons are phononsand where the average spectral function α2F(ω) is similar to the phonon density of states Fph(ω). Note that α2F(ω) is not the product of two functions although sometimes one defines the function α2(ω)=α2F(ω)/F(ω) 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<λep≲3, which will be discussed in detail below. In the EPI approach α2Fph(ω) is extracted from tunnelling measurements in conjunction with IR optical measurements. The HTSC cuprates are on the borderline and it is a natural question—under which condition can high Tc be realized in the nonadiabatic limit A≈1?Thesecond 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 Imχ(k-k′,ω), that is,  α2F(k,k′,ω)~gsf2Imχ(k-k′,ω) where gsf is the SFI coupling constant. NMR spectroscopy and magnetic neutron scattering give evidence that in HTSC cuprates χ(q,ω) is peaked at the antiferromagnetic wave vector Q=(π/a,π/a) 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 t-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 following—do models based solely on the Hubbard Hamiltonian show up superconductivity at sufficiently high critical temperatures (Tc~100K)? 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 (U>0) Hubbard model gives no evidence for superconductivity with highTc, 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 (U<0) Hubbard model for the same strength of U, that is, at finite temperature in the 2D model with U<0 the BKT superconducting transition is favored. Concerning the possibility of HTSC in the t-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 low—few Kelvins. These numerical results tell us that the lack of high Tc (even in 2D BKT phase) in the repulsive (U>0) single-band Hubbard model and in the t-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 highTc  in cuprates. These numerical results signal that the simple single-band Hubbard and its derivative the t-J model are insufficient to explain solely the pairing mechanism in cuprates and some additional ingredients must be included.SinceEPI 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 λep>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, λep<0.3. The inappropriateness of these statements will be discussed in the following sections.The paper is organized as follows. In Section1 we will mainly discuss experimental results in cuprates at and near optimal doping by giving also minimal theoretical explanations which are related to thebosonic spectral functionα2F(ω) as well as to the transport spectral function αtr2F(ω) 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 quasiparticles—although strongly interacting—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(k,k′,ω) and αtr2F(ω) 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, λep≥1, if the EPI spectral function is weakly momentum dependent, that is, if α2F(k,k′,ω)≈α2F(ω) 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 χ(q,ω) which is in the SFI  phenomenological approach [12–17] related to the Eliashberg spectral function, that is, α2Fsf(k,k′,ω)~gsf2Imχ(q=k-k′,ω). We stress that such an approach can be theoretically justified only in the weak coupling limit, gsf≪Wb, where Wb 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 ω) of Imχ(q,ω) (with q at and near Q=(π,π)) by doping toward the optimal doped HTSC [30, 31]. It turns out that in the optimally doped cuprates with Tc=92.5KImχ(Q,ω) 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(ω) and indirectly an approximative shape of the transport spectral function αtr2F(ω). 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(ω) and the quasiparticle relaxation rate γ(ω). 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 αtr2F(ω). Although these methods give at finite temperature T a blurred αtr2F(ω) which is (due to the ill-defined methods) temperature dependent, it turns out that the width and the shape of the extracted αtr2F(ω) 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 La2-xSrxCuO4 which gives additional evidence for importance of EPI [41] will be shortly discussed.(c)ARPES  Measurements and EPI. From these measurements the self-energy Σ(k,ω) is extracted as well as some properties of α2F(k,k′,ω). 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 (Σ(k,ω)) and its isotope dependence, which are all due to EPI; (ii) the appearance of different slopes of Σ(k,ω) at low (ω≪ωph) and high energies (ω≫ωph ) 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 EPI—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(ω) from tunnelling conductance. In this part we discuss the following items: (i) the extracted Eliashberg spectral function α2F(ω) with the coupling constant λ(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(ω) coincide with the maxima in the phonon density of states Fph(ω); (ii) the existence of eleven peaks in -d2I/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(ω) 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 B1g 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 Section1.4 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(k,k′,ω) and the transport spectral function αtr2F(ω) are defined.Finally, one can pose a question—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 thetheory of everything—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 σs(T<Tc,ω=const) at ω=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 ω~60GHz 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 (Tc≈40K) which is smaller than the theoretical value, that is, αBexp≈0.3<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 2Δ/Tc which is on optimally doped YBCO and BSCO ≈5 and 7, respectively, while in underdoped BSCO one has even (2Δ/Tc)≈20; (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 Section2. 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,me,Mion,kB and the chemical composition of superconducting materials, it was practically never realized in practice due to the complexity of many-body interactions—electron-electron and electron-lattice—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,Δ/EF,ωph/EF)≪1. 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,r′) (IBS—the ideal band structure) which includes the static self-energy (Σc0(h)(r,r′,ω=0)) due to high-energy electronic processes, that is, VIBS(r,r′)=[Ve-i(r)+VH(r)]δ(r-r′)+Σc0(h)(r,r′,ω=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,r′) and the real EPI coupling gep(r,r′)=δVIBS(r,r′)/δRn, which include high-energy many-body electronic processes—for instance, the large Hubbard U effects—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 potentialVLDA(r)—the ground-state potential—and the real EPI coupling by the “local” 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 correlationsin 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 t-J model, which allows the approximative calculation of the excited potential VIBS(r,r′) and the nonlocal EPI coupling gep(r,r′). As a result one obtains the momentum-dependent EPI coupling gep(kF,q) which is for small hole-doping (δ<0.3) strongly peaked at small transfer momenta—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 (λtr) rather smaller than the pairing one λ, that is, λtr≲λ/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 Δ(k,R), where k 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 Δ(k,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 t-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 T¯c. However, T¯c 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 highTc 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 leastin 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 q=±e, spin s=1/2, and low-energy excitations ξk(=ϵk-μ) which are much larger than their inverse life-times, that is, ξk≫1/τk~ξk2/Wb. Since the level width Γ=2/τk 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 ξkF=0 which defines the Fermi surface on which the Fermi quasiparticle distribution function nF(ξk) has finite jump at kF; (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)—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 kBΘD(≡ℏωD) (for the Debye spectrum ~ΘD/5), that is, for ξk~ΘD, one has τk-1≳ξk 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 liquidfor sufficiently strong electron-boson coupling. It is nowadays well known that, for instance, Al, Zn are weak coupling systems since for ξk~ΘD one has τk-1≪ξk and they are well described by the Landau theory. However, in (the noncanonical) cases where for higher energies ξk~ΘD one has τk-1≳ξk, 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(k,k′,ω) 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 papers—for the recent one see [67]—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 potentialVeff(k,ω) in the leading approximation looks like as for two external charges (e) embedded in the medium with the total longitudinal dielectric functionɛtot(k,ω) (k is the momentum and ω is the frequency) [68, 69], that is,
(1)Veff(k,ω)=Vext(k)ɛtot(k,ω)=4πe2k2ɛtot(k,ω).
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/k2ɛtot(k,ω). In that case Veff depends on other (than ɛtot(k,ω)) response functions. However, in the case when (1) holds, that is, when the weak-coupling limit is realized, Tc is given by Tc≈ω¯exp(-1/(λep-μ*)) [68–70]. Here, λep is the EPI coupling constant, ω¯ is an average phonon frequency, and μ* is the Coulomb pseudopotential, μ*=μ/(1+μlnEF/ω¯) (EF is the Fermi energy). The couplings λep and μ are expressed by ɛtot(k,ω=0):
(2)μ-λep=〈N(0)Veff(k,ω=0)〉=N(0)∫02kFkdk2kF24πe2k2ɛtot(k,ω=0),
where N(0) is the density of states at the Fermi surface and kF is the Fermi momentum—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 ɛtot(k,ω=0)>1 for all k. If this were correct, then from (2) it would follow that μ>λep, which limits the maximal value of Tc to the value Tcmax≈EFexp(-4-3/λep). In typical metals EF<(1–10)eV, and if one accepts the statement in [65, 66] that λep≤μ(≤0.5), one obtains Tc~(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 μ≤λep and λep>1. For instance, λep≈2.6 is realized in PbBi alloy which is definitely much higher than μ(<1), and so forth.Moreover, the basic theory tells us thatɛtot(k≠0,ω) is not the response function [68, 69] (contrary to the assumption in [65, 66]). Namely, if a small external potential δVext(k,ω) 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(k,ω)=δVext(k,ω)/ɛtot(k,ω), which means that 1/ɛtot(k,ω) is the response function. The latter obeys the Kramers-Kronig dispersion relation which implies the following stability condition [68, 69]:
(3)1ɛtot(k,ω=0)<1,k≠0,
that is, either
(4)ɛtot(k≠0,ω=0)>1,
or
(5)ɛtot(k≠0,ω=0)<0.
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 ɛtot(k≠0,ω=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, ɛ̂tot(k+Q,k+Q′,ω), and ɛtot(k,ω) is defined by ɛtot-1(k,ω)=ɛ̂tot-1(k+0,k+0,ω). In dense metallic systems with one ion per cell (such as metallic hydrogen) and with the electronic dielectric function ɛel(k,0), the macroscopic total dielectric function ɛtot(k,0) is given by [71–73]
(6)ɛtot(k,0)=ɛel(k,0)1-1/ɛel(k,0)Gep(k).
At the same time the energy of the longitudinal phonon ωl(k) is given by
(7)ωl2(k)=Ωp2ɛel(k,0)[1-ɛel(k,0)Gep(k)],
where Ωp2 is the ionic plasma frequency, and Gep is the local (electric) field correction—see [71–73]. The right condition for lattice stability requires that ωl2(k)>0, which implies that for ɛel(k,0)>0 one has ɛel(k,0)Gep(k)<1. The latter condition gives automatically ɛtot(k,0)<0. Furthermore, the calculations [71–73] show that in the metallic hydrogen (H) crystal ɛtot(k,0)<0 for all k≠0. Note that in metallic H the EPI coupling constant is very large, that is, λep≈7 and Tc may reach very large value Tc≈600K [74]. Moreover, the analyses of crystals with more ions per unit cell [71–73] give that ɛtot(k≠0,0)<0 is more a rule than an exception—see Figure 1. The physical reason for ɛtot(k≠0,0)<0 is local field effects described by Gep(k). 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 ɛtot(k,0) which may lead to ɛtot(k,0)<0.Figure 1
Inverse total static dielectric functionɛ-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 crystalsɛtot(k,0)  can be negative in the large portion of the Brillouin zone thus giving rise to λep-μ>0 in (2). This means that analytic properties of the dielectric function ɛtot(k,ω)  do not limitTcin 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 unusualionic-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 λtr (entering resistivity) rather smaller than the pairing EPI coupling constant λep(>1) (entering Tc), that is, why one has λtr(≈0.6–1.4)≪λep(~2–3.5)? (3) If EPI is ineffective for pairing in HTSC oxides, in spite of λep>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 tod-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 ɛtot(k,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<λep<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 (Ωnph≫ωph) 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 Ωnph)
(8)Z(ωn)Δn(k)≈πTc∑mΩnph∑qVnph(k,q,n,m)Δm(q)|ωm|,Z(ωn)≈1+Γepωn.
For pure d-wave pairing with the pairing potential Vnph=Vnph(θk,θq)·Θ(Ωnph-|ωn|)Θ(Ωnph-|ωn′|) with Vnph(k,q)=V0·Yd(θk)Yd(θq) and Yd(θk)=π-1/2cos2θk, one obtains Δn(k)=Δd·Θ(Ωnph-|ωn|)Yd(θk) and the equation for Tc—see [3–5]
(9)lnTcTc0≈Ψ(12)-Ψ(12+Γep2πTc).
Here Ψ is the di-gamma function. At temperatures near Tc one has Γep≈2πλepTc and the solution of (9) is approximately Tc≈Tc0exp{-λep} with Tc0≈Ωnphexp{-λnph}, λnph=N(0)V0. This means that for Tcmax~160K and λep>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 λkpZ(iνn) and λkpΔ, respectively. Usually, the spin-fluctuation kernel λsf,kp(iνn) is taken in the FLEX approximation [77]. The calculations [76] confirm the very detrimental effect of the isotropic  (k-independent) EPI on d-wave pairing due to SFI. For the bare SFI critical temperature Tc0~100K and for λep>1 the calculations give very small (renormalized) critical temperature Tc≪100K. These results tell us that a more realistic pairing interaction must be operative in cuprates and that EPI must be strongly momentum dependentand 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 indicativeinelastic 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 Imχ(k,kz,ω) in the normal state at and near the AF wave vector k=Q=(π,π) 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 Imχ(k,kz,ω) (at some energies ω<2Δ) 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 ofTc
Before discussing experimental results in cuprates on the imaginary part of the spin susceptibilityImχ(k,ω) we point out that in the (phenomenological) theories based on the spin-fluctuation interaction (SFI) the quasiparticle self-energy Σ̂sf(k,ωn) (ωn is the Matsubara frequency and τ̂0 is the Nambu matrix) in the normal and superconducting state and the effective (repulsive) pairing potential Vsf(k,ω) (where iωn→ω+iη) are assumed in the form [12–17]
(10)Σ̂sf(k,ωn)=TN∑k′,mVsf(k-k′,ωnm-)τ̂0Ĝ(k′,ωm)τ̂0,Vsf(k,ωnm-)=gsf2∫-∞∞dνπImχ(q,ν+i0+)ν-iωnm-,
where ωnm-≡ωn-ωm. Although the form of Vsf cannot be justified theoretically, except in the weak coupling limit (gsf≪Wb) 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) Imχ(q,ω) is strongly peaked at and near the AF wave vector Q=(π/a,π/a).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 givesImχ(q,ω). In that respect very indicative and impressive IMNS measurements on YBa2Cu3O6+x, which are done by Bourges group [30], demonstrate that the normal-state susceptibility Imχ(odd)(q,ω) (the odd part of the spin susceptibility in the bilayer system) at q=Q=(π,π) is strongly dependent on the hole-doping as it is shown in Figure 2.Figure 2
Magnetic spectral functionImχ(-)(k,ω) in the normal state of YBa2Cu3O6+x at T=100K and at Q=(π,π). 100 counts in the vertical scale correspond to χmax(-)≈350μB2/eV. The superconducting critical temperature Tc(x) by increasing doping (x) from the underdoped system with x=0.5 (top) to the optimally doped one with x=0.97 (bottom): Tc(x)=45K (x=0.5), 85K (x=0.83), 91K (x=0.92), and 92.5K (x=0.97). From [30].The most pronounced result for our discussion is that by varying dopingthere is a huge rearrangement of Imχ(odd)(Q,ω) in the normal state, especially in the energy (frequency) region which might be important for superconducting pairing, let us say 0meV<ω<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 temperatureTc. It is seen in Figure 2 that in the underdoped YBa2Cu3O6.92 crystal Imχ(odd)(Q,ω) and S(Q)=N(0)gsf2∫060dωImχ(odd)(Q,ω) 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 suppressionof Imχ(odd)(Q,ω) 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 λsf(~gsf2), means that the latter is small, that is, λsf(exp)≪1. We stress that the explanation of high Tc in cuprates by the SFI phenomenological theory [12–17] assumes very large SFI coupling energy with gsf(th)≈0.7eV while the frequency (energy) dependence of Imχ(Q,ω) is extracted from the fit of the NMR relaxation rate T1-1 which gives Tc(NMR)≈100K [12–17]. To this point, the NMR measurements (of T1-1) give that there is an anticorrelation between the decrease of the NMR spectral function IQ=limω→0Imχ(NMR)(Q,ω)/ω and the increase of Tc by increasing doping toward the optimal one—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 Imχ(Q,ω) from NMR measurements one takes it from IMNS measurements, as it was done in [82], than for the same value gsf(th) one obtains much smaller Tc. For instance, by taking the experimental values for Imχ(IMNS)(Q,ω) in underdoped YBa2Cu3O6.6 with Tc≈60K one obtains Tc(IMNS)<Tc(NMR)/3 [82], while Tc(IMNS)→50K for gsf(th)≫1. The situation is even worse if one tries to fit the resistivity with Imχ(IMNS)(Q,ω) in YBa2Cu3O6.6 since this fit gives Tc(IMNS)<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 “hourglass” spin excitation spectrum (in the superconducting state) for the momenta q 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 gsf2=(3/2)Ũ2 and Imχ(q,ω) 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 valueŨ=1.59eV, that is, much larger than the one used in [82]. Such a large value of Ũ has been obtained earlier within the Monte Carlo simulation of the Hubbard model [87]. In our opinion this value for Ũ 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 Ũ≪1eV and Tc≪60K. 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–Cu≈[4tpd4/(Δdp+Upd)2,(1/Ud)+2/(Up+2Δ)]—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 Ũ (and gsf).Concerning the problem related to the rearrangement of the SFI spectral functionImχ(Q,ω) in YB2Cu3O6+x [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(q,ω)(~Imχ(q,ω)) (for q 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 Imχ(q,ω) 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 Ũ to the value even larger than 4eV, which is highly improbable.(b) Ineffectiveness of the Magnetic Resonance Peak
A less direct argument forsmallness of the SFI coupling constant, that is,gsfexp≤0.2eV and gsfexp≪gsf, 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, Imχ(Q,ω) is significantly suppressed at low frequencies except near the resonance energy ωres≈41meV where a pronounced narrow peak appears—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)%—see Figure 3. In underdoped cuprates this peak is present also in the normal state as it is seen in Figure 2.Figure 3
Magnetic spectral functionImχ(-)(k,ω) in the superconducting state of YBa2Cu3O6+x at T=5K and at Q=(π,π). 100 counts in the vertical scale correspond to χmax(-)≈350μB2/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 intensityI0 is small [89]. I0 is obtained approximately by equating the condensation energy Econ≈N(0)Δ2/2 with the change of the magnetic energy Emag in the superconducting state, that is, δEmag≈4I0·Emag:
(11)Emag=J∬dωd2k(2π)3(1-coskx-cosky)S(k,ω),
where S(k,ω)=(1/π)[1+n(ω)]Imχ(k,ω) is the spin structure factor and n(ω) is the Bose distribution function. By taking Δ≈2Tc and the realistic value N(0)~1/(10J)~1states/eV·spin, one obtains I0~10-1(Tc/J)2~10-3. However, such a small intensity cannot be responsible for the anomalies in ARPES and optical spectra since it gives rise to small coupling constant λsf,res for the interaction of holes with the resonance peak, that is, λsf,res≈(2I0N(0)gsf2/ωres)≪1. 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 ωres, that is, γres<ωres, where the line width is given by γres=4π(N(0)gsf)2ωres [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 Ũ≈1.6eV, 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 μ(<0.1μB) 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]—see the discussion below.Finally, we would like to point out that the recent magnetic neutron scattering measurements on optimally doped large-volume crystalsBi2Sr2CaCu2O8+δ [93], where the absolute value of Imχ(q,ω) 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 Imχ(ω)=∑qImχ(q,ω) across the superconducting transition Tc=91K, that is, there is only a minimal change in Imχ(ω) between 10K and 100K. Note that if the magnetic excitations were due to itinerant quasiparticles we should have seen dramatic changes of Imχ(ω) as a function of T over the whole energy range. This T-independence of Imχ(ω) strongly opposes the theoretical results in [24–27] which assume that the bosonic spectral function is proportional to Imχ(ω) and that the former can be extracted from optic measurements. Namely, the fitting procedure in [24–27] gives that Imχ(ω) is strongly T-dependent contrary to the experimental results in [93]—see more in Section 1.3.2 on optical conductivity.


### 1.3.2. Optical Conductivity and EPI

Optical spectroscopy gives information onoptical conductivityσ(ω) and on two-particle excitations, from which one can indirectly extract the transport spectral function αtr2F(ω). 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(ω)  is not a directly  measured quantity but it is derived from the reflectivity R(ω)=|(ɛii(ω)-1)/(ɛii(ω)+1)|2 with the transversal dielectric tensor ɛii(ω)=ɛii,∞+ɛii,latt+4πiσii(ω)/ω. Here, ɛii,∞ is the high-frequency dielectric function, ɛii,latt describes the contribution of the lattice vibrations, and σii(ω) describes the optical (dynamical) conductivity of conduction carriers. Since R(ω) is usually measured in the limited-frequency interval ωmin<ω<ωmax, some physical modelling for R(ω) is needed in order to guess it outside this range—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 ωpl—from 0.06 to 25eV—that is, almost three orders of magnitude. However, it turns out that IR measurements of R(ω) in conjunction with elipsometric measurements of ɛii(ω) at high frequencies allow more reliable determination of σ(ω) [94].(1) Transport and Quasiparticle Relaxation Rates
The widespread misconception in studying the quasiparticle scattering in cuprates was an ad hoc assumption that thetransport relaxation rateγtr(ω) is equal to the quasiparticle relaxation rateγ(ω), in spite of the well-known fact that the inequality γtr(ω)≠γ(ω) 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 conductivityσ(ω) consists of two parts, that is, σ(ω)=σinter(ω)+σintra(ω) where σinter(ω) describes interband  transitions which contribute at higher than intraband energies, while σintra(ω) is due to intraband transitions which are relevant at low energies ω<(1–2)eV. (Note that in the IR measurements the frequency is usually given in cm-1, where the following conversion holds: 1cm-1=29.98GHz=0.123985meV=1.44K.) The experimental data for σ(ω)=σ1+iσ2 in cuprates are usually processed by the generalized (extended) Drude formula [32–36, 95]:
(12)σ(ω)=ωp24π1γtr(ω)-iωmtr(ω)/m∞≡iωp24πω̃tr(ω),
where m is the mass of the band electrons while the quantity ω̃tr(ω) 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 ω̃tr(ω) will be discussed below.) Here, i=a,b enumerates the plane axis; ωp,γtr(ω,T), and mop(ω) are the electronic plasma frequency, the transport (optical) scattering rate, and the optical mass, respectively. Very frequently it is analyzed the quantity γtr*(ω,T) given by [95]
(13)γtr*(ω,T)=m∞mtr(ω)γtr(ω,T)=ωImσ(ω)Reσ(ω).
In the weak coupling limit λep<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]
(14)γtr(ω,T)=π∑l∫0∞dναtr,l2Fl(ν)×[2(1+2nB(ν))-2νω-ω+νωnB(ω+ν)+ω-νωnB(ω-ν)].
Here nB(ω) is the Bose distribution function. For completeness we give also the explicit form of the transport mass mtr(ω), see [3–6, 32–36]:
(15)mtr(ω)m∞=1+2ω∑l‍∫0∞dναtr,l2Fl(ν)ReK(ω2πT,ν2πT),
with the Kernel K(x,y)=(i/y)+{((y-x)/x)[ψ(1-ix+iy)-ψ(1+iy)]}-{y→-y} where ψ 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 λep(>1). Note that αtr,l2Fl(ν)≠αl2Fl(ν) and the index l  enumerates all scattering bosons—phonons—spin fluctuations, and so forth. For comparison, the quasiparticle scattering rate γ(ω,T) is given by
(16)γ(ω,T)=2π∫0∞dνα2F(ν)×{2nB(ν)+nF(ν+ω)+nF(ν-ω)}+γimp,
where nF is the Fermi distribution function. For completeness we give also the expression for the quasiparticle effective mass m*(ω):
(17)m*(ω)m=1+1ω∑l∫0∞dναl2Fl(ν)×Re{ψ(12+iω+ν2πT)-ψ(12-iω-ν2πT)}.
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]
(18)γtr(ω)=2πω∫0ωdν(ω-ν)αtr2(ν)F(ν),γ(ω)=2π∫0ωdνα2(ν)F(ν).
In the case of EPI with the constant electronic density of states, the above equations give that γep(ω)=const for ω>ωphmax while γep,tr(ω) (as well as γep,tr*) is monotonic growing for ω>ωphmax, where ωphmax is the maximal phonon frequency. So, the growing of γep,tr(ω) (and γep,tr*) for ω>ωphmax 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(ω,T),γtr(ω,T), and γtr* which are calculated for the EPI spectral function αep2(ω)Fph(ω) extracted from tunnelling experiments in YBCO (with ωphmax~80meV ) [42–45]—see Figure 4.(a) Scattering ratesγ(ω,T),γtr(ω,T), and γtr*—from top to bottom—for the Eliashberg function in (b). From [33–35]. (b) Eliashberg spectral function αep2(ω)Fph(ω) obtained from tunnelling experiments on break junctions [42–45]. Inset shows γtr* with (full line) and without (dashed line) interband transitions [3–5].
(a)(b)The results shown in Figure4 clearly demonstrate the physical difference between two scattering rates γep and γep,tr (or γtr*). It is also seen that γtr*(ω,T) is even more linear function of ω than γtr(ω,T). From these calculations one concludes that the quasilinearity of γtr(ω,T) (and γtr*) 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 γep,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 forth—see more in [3–6] and references therein.Let us discuss briefly the experimental results forR(ω) and γtr*(ω,T) and compare these with theoretical predictions obtained by using a single-band model with αep2(ω)Fph(ω) extracted from the tunnelling data with the EPI coupling constant λep≳2 [42–45]. In the case of YBCO the agreement between measured and calculated R(ω) is very good up to energies ω<6000cm-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 (Bi2212) [97, 98] the experimental results for γtr*(ω,T) are explained theoretically by assuming that the EPI spectral function αep2(ω)F(ω)~Fph(ω), where Fph(ω) is the phononic density of states in BSCO, with λep=1.9 and γimp≈320cm-1—see Figure 5(a). At the same time the fit of γtr*(ω,T) by the marginal Fermi liquid phenomenology fails as it is evident in Figure 5(b).(a) Experimental transport scattering rateγtr* (solid lines) for BSCO and the theoretical curve by using (A.20) and transport mass mtr* with α2F(ω) due to EPI which is described in text (dashed lines). (b) Comparison with the marginal Fermi liquid theory—dashed lines. From [3–5, 99].
(a)(b)Now we will comment the so called pronounced linear behavior ofγtr(ω,T) (and γtr*(ω,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(ω) and derived ones are σ(ω),γtr(ω,T), and mtr(ω), which are very sensitive to the value of the dielectric constant ɛ∞. This sensitivity is clearly demonstrated in Figure 6 for Bi-2212 where it is seen that γtr(ω,T) (and γtr*(ω,T)) for ɛ∞=1 is linear up to much higher ω than in the case ɛ∞>1.Figure 6
Dependence ofγtr*(ω,T) on ɛ∞ in Bi2Sr2CaCu2O8 for different temperatures: ɛ∞=4 (solid lines) and ɛ∞=1 (dashed lines). On the horizontal axis is ω in units cm-1. From [99].However, in some experiments [100–103] the extracted γtr(ω,T) (and γtr*(ω,T)) is linear up to very high ω≈1500cm-1. This means that the ion background and interband transitions (contained in ɛ∞) are not properly taken into account since too small ɛ∞ (≳1) is assumed. The recent elipsometric measurements on YBCO [104] give the value ɛ∞≈4–6, which gives much less spectacular linearity in the relaxation rates γtr(ω,T) (and γtr*(ω,T)) than it was the case immediately after the discovery of HTSC cuprates, where much smaller ɛ∞ was assumed.Furthermore, we would like to comment two points related toσ,γtr, and γ. First, the parametrization of σ(ω) with the generalized Drude formula in (12) and its relation to the transport scattering rate γtr(ω,T) and the transport mass mtr(ω,T) is useful if we deal with electron-boson scattering in a single-band problem. In [36, 96] it is shown that σ(ω) of a two-band model with only elastic impurity scattering can be represented by the generalized (extended) Drude formula with ω and T dependence of effective parameters γtreff(ω,T), mtreff(ω,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(ω,T) and γ(ω,T) are physically different quantities and it holds that γtr(ω,T)≠γ(ω,T). In order to give the physical picture and qualitative explanation for this difference we assume that αtr2F(ν)≈α2F(ν). In that case the renormalized quasiparticle frequency ω̃(ω)=Z(ω)ω=ω-Σ(ω) and the transport one ω̃tr(ω)—defined in (12)—are related and at T=0 they are given by [32, 36]
(19)ω̃tr(ω)=1ω∫0ωdω′2ω̃(ω′).
(For the definition of Z(ω) see Appendix A.) It gives the relation between γtr(ω) and γ(ω) as well asmtr(ω) and m*(ω), respectively:
(20)γtr(ω)=1ω∫0ωdω′γ(ω′),ωmtr(ω)=1ω∫0ωdω′2ω′m*(ω′).
The physical meaning of (19) is the following: in optical measurements one photon with the energy ω is absorbed and two excited particles (electron and hole) are created above and below the Fermi surface. If the electron has energy ω′ and the hole ω-ω′, then they relax as quasiparticles with the renormalized frequency ω̃. Since ω′ takes values 0<ω′<ω, then the optical relaxation ω̃tr(ω) is the energy-averaged ω̃(ω) according to (19). The factor 2 is due to the two quasiparticles—electron and hole. At finite T, the generalization reads [32, 36]
(21)ω̃tr(ω)=1ω∫0∞dω′[1-nF(ω′)-nF(ω-ω′)]2ω̃(ω′).(2) Inversion of the Optical Data andαtr2(ω)F(ω)
In principle, the transport spectral functionαtr2(ω)F(ω) can be extracted from σ(ω) (or γtr(ω)) only at T=0K, which follows from (18) as
(22)αtr2(ω)F(ω)=ωp28π2∂2∂ω2[ωRe1σ(ω)],
or equivalently as αtr2(ω)F(ω)=(1/2π)∂2(ωγtr(ω))/∂ω2. 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 αtr2(ω)F(ω) is the deconvolution of the inhomogeneous Fredholm integral equation of the first kind with the temperature-dependent Kernel K2(ω,ν,T)—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 αtr2(ω)F(ω)  get blurred (most peaks are washed out) in the extraction procedures and, second, the extractedαtr2(ω)F(ω)  be temperature dependent even when the true αtr2(ω)F(ω) 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 αtr2(ω)F(ω) in YBa2Cu3O7-x 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 ω) is present. For instance, the experimental data of R(ω) in YBCO were reproduced by two different spectral functions αtr2(ω)F(ω), 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(ω). The similar situation is realized in optimally doped BSCO as it is seen in Figure 8 where again different functions α2(ω)F(ω) reproduce very well curves for R(ω) and σ(ω). However, it is important to stress that the obtained width of the extracted αtr2(ω)F(ω) in both compounds coincide with the width of the phonon density of states Fph(ω) [33–36, 99]. (2) The upper energy bound for αtr2(ω)F(ω) is extracted in [33–36] and it coincides approximately with the maximal phonon frequency in cuprates ωphmax≲80meV as it is seen in Figures 7 and 8.Figure 7
Experimental (solid lines) and calculated (dashed lines) data ofR(ω) in optimally doped YBCO [105] at T=100,200,300K (from top to bottom). Inset: the two (solid and dashed lines) reconstructed αtr2(ω)F(ω)’s at T=100K. The phonon density of states F(ω)—dotted line in the inset. From [33–35].Figure 8
Experimental (solid line) and calculated (dashed line) data ofR(ω) in optimally doped BSCO [106] at T=100K. Inset: the reconstructed αtr2(ω)F(ω)—solid line. The phonon density of states F(ω)—dotted line. From [33–35].These results demonstrate the importance of EPI in cuprates [33–36]. We point out that the width of αtr2(ω)F(ω) which is extracted from the optical measurements [33–36] coincides with the width of the quasiparticle spectral function α2(ω)F(ω) obtained in tunnelling and ARPES spectra (which we will discuss below), that is, both functions are spread over the energy interval 0<ω<ωphmax(≲80meV). Since in cuprates this interval coincides with the width in the phononic density of states F(ω) and since the maxima of α2(ω)F(ω) and F(ω) 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 αtr2(ω)F(ω) 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(ω) 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 Δ(k,ω). Second, if R(ω)’s (and σ(ω)’s) in cuprates are due to some other bosonic scattering which is pronounced up to much higher energies ωc≫ωphmax, this should be seen in the width of the extracted spectral function αtr2(ω)F(ω). In that respect in [25–27] it is assumed that SFI dominates in the quasiparticle scattering and that αtr2(ω)F(ω)~gsf2Imχ(ω) where Imχ(ω)=∫d2kχ(k,ω). This claim is based on reanalyzing of some IR measurements [25–27] and the transport spectral function αtr2(ω)F(ω) 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 αtr2(ω)F(ω), which is an artefact and due to the chosen method, in [25] it is assumed that αtr2(ω)F(ω) has a rather large tail at large energies—up to 400 meV. It turns out that even such an assumption in extracting αtr2(ω)F(ω) does not reproduce the experimental curve Imχ(ω) [107] in some important aspects. First, the relative heights of the two peaks in the extracted spectral function αtr2(ω)F(ω) at lower temperatures are opposite to the experimental curve Imχ(ω) [107]—see [25, Figure  1]. Second, the strong temperature dependence of the extracted αtr2(ω)F(ω) 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 IB(ω)=αtr2(ω)F(ω) in [25] has much smaller values at larger frequencies (ω>100meV) than it is the case for the measured Imχ(ω), that is, (IB(ω>100meV)/IB(ωmax))≪Imχ(ω>100meV)/Imχ(ωmax)—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 Imχ(q,ω) 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 Imχ(ω)=∑qImχ(q,ω) is temperature independent across the superconducting transition Tc=91K, that is, there is only a minimal change in Imχ(ω) between 10K and 100K. This T-independence of Imχ(ω) strongly opposes the (above discussed) results in [24–27], where the fit of optic measurements gives strong T dependence of Imχ(ω).Fifth, the transport coupling constant λtr extracted in [25] is too large, that is, λtr>3 contrary to the previous findings that λtr≲1.5 [33–36, 99]. Since in HTSC one has λ>λtr, this would probably give λ≈6, which is not confirmed by other experiments. Sixth, the interpretation of αtr2(ω)F(ω) in LSCO and BSCO solely in terms of Imχ(ω) 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 Imχ(Q,ω) is small in the normal state and its magnitude is even below the experimental noise. This means that if the assumption that αtr2(ω)F(ω)≈gsf2Imχ(ω) were correct then the contribution to Imχ(ω) from the momenta 0<k≪Q would be dominant, which is detrimental for d-wave superconductivity.Finally, we point out that very similar (to cuprates) properties, ofσ(ω),R(ω) (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 Ba1-xKxBiO3 which is superconducting below Tc≃30K at x=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
Therestricted  optical sum rulewas studied intensively in HTSC cuprates. It shows some peculiarities not present in low-temperature superconductors. It turns out that the restricted spectral weight W(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 W(Ωc,T)=W(0)-βT2 in the normal state can be explained in a natural way by the T dependence of the EPI transport relaxation rate γtrep(ω,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(ω). The first one is the total sum rule which in the normal state reads
(23)∫0∞dωσ1N(ω)=ωpl28=πne22m,
while in the superconducting state it is given by the Tinkham-Ferrell-Glover (TFG) sum rule
(24)∫0∞dωσ1S(ω)=c28λL2+∫+0∞dωσ1S(ω)=ωpl28.
Here,n is the total electron density, e is the electron charge, m is the bare electron mass, and λL is the London penetration depth. The first (singular) term c2/8λL2 in (24) is due to the superconducting condensate which contributes σ1,condS(ω)=(c2/4λL2)δ(ω). The total sum rule represents the fundamental property of matter—the conservation of the electron number. In order to calculate it one should use the total Hamiltonian Ĥtot=T̂e+Ĥint where all electrons, electronic bands, and their interactions Ĥint (Coulomb, EPI, with impurities, etc.) are accounted for. Here, Te is the kinetic energy of bare electrons:
(25)T̂e=∑σ‍∫d3xψ̂σ†(x)p̂22mψ̂σ(x)=∑p,σp22meĉpσ†ĉpσ.Thepartial sum rule is related to the energetics solely in the conduction (valence) bandwhich is described by the Hamiltonian of the conduction (valence) band electrons:
(26)Ĥv=∑p,σξpĉv,pσ†ĉv,pσ+V̂v,c.Ĥv contains the band energy with the dispersion ϵp (ξp=ϵp-μ) and the effective Coulomb interaction of the valence electrons V̂v,c. In this case the partial sum rule in the normal state reads [110] (for a general form of ϵp)
(27)∫0∞dωσ1,vN(ω)=πe22V∑p‍〈n̂v,p〉Hvmp,
where the number operator n̂v,p=∑σĉpσ†ĉpσ;1/mp=∂2ϵp/∂px2 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 ω 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
(28)W(Ωc,T)=∫0Ωcdωσ1,vN(ω)=π2[Kd+Π(0)]-∫0ΩcdωImΠ(ω)ω,
where Kd is the diamagnetic Kernel given by (30) below and Π(ω) is the paramagnetic (current-current) response function. In the perturbation theory without vertex correction Π(iωn) (at the Matsubara frequency ωn=πT(2n+1)) is given by [39, 40]
(29)Π(iω)=2∑p(∂ϵp∂p)2∑ωmG(p,iωnm+)G(p,iωm),
where ωnm+=ωn+ωm and G(p,iωn)=(iωn-ξp-Σ(p,iωn))-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 Wb<Ωc<Eg, in this region one has approximately ImΠ(ω)≈0 and the limit Ωc→∞ in (28) is justified. In that case one has Π(0)≈∫0Ωc(ImΠ(ω)/ω)dω which gives the approximate formula for W(Ωc,T):
(30)W(Ωc,T)=∫0Ωcdωσ1,vN(ω)≈π2Kd=e2π∑p∂2ϵp∂p2np,
where np=〈n̂v,p〉 is the quasiparticle distribution function in the interacting system. Note that W(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 W(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 ϵp=-2t(cospxa+cospya) which gives 1/mp=-2ta2cospxa. It is straightforward to show that in this case (30) is reduced to a simpler one:
(31)W(Ωc,T)=∫0Ωcdωσ1,vN(ω)≈πe2a22V〈-Tv〉,
where 〈Tv〉Hv=∑pϵp〈nv〉Hv is the average kinetic energy of the band electrons, a is the Cu–Cu lattice distance, and V is the volume of the system. In this approximation W(Ωc,T) is a direct measure of the average band (kinetic) energy. In thesuperconducting state the partial band sum rule reads (32)Ws(Ωc,T)=c28λL2+∫+0Ωcdωσ1,vS(ω)=πe2a22V〈-Tv〉s.
In order to introduce the reader to (the complexity of) the problem of the T dependence of W(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 Wn(Ωc,T) increases with the decrease of the temperature, that is, Wn(Ωc,T)=Wn(0)-βbT2 where βb~1/Wb. 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 W(Ωc,T). We stress that the smearing of fp by temperature lowers the spectral weight compared to that at T=0K, that is, Wn(Ωc,T)<Wn(Ωc,0). In that respect it is not surprising that there is a lowering of Ws(Ωc,T) in the BCS superconducting state, WsBCS(Ωc,T≪Tc)<Wn(Ωc,T≪Tc) since at low temperatures fp is smeared mainly due to the superconducting gap, that is, fp=[1-(ξp/Ep)th(Ep/2T)]/2, Ep=ξp2+Δ2, ξp=ϵp-μ. The maximal decrease of Ws(Ωc,T) is at T=0.Let us enumerate and discuss themain experimental results forW(Ωc,T) in HTSC cuprates. (1) In the normal state (T>Tc) of most cuprates, one has Wn(Ωc,T)=Wn(0)-βexT2 with βexp≫βb, that is, Wn(Ωc,T) is increasing by decreasing T, even at T below T*—the temperature for the opening of the pseudogap. The increase of Wn(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 Ws(Ωc,T) with respect to that in the normal state, that is, Ws(Ωc,T)>Wn(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 〈Tv〉 is negative (〈Tv〉<0) and in the standard BCS case (32) gives that Ws(T<Tc)  decreases due to the increase of 〈Tv〉. Therefore the experimental increase of Ws(T<Tc) by decreasing T is called the non-BCS behavior. The latter means a lowering of the kinetic energy 〈Tv〉 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.Figure 9
Measured spectral weightWs(Ωc,T)(~Al+D in figures) for Ωc≈1.25eV 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 ofWs(Ωc,T) at T<Tc (Ws(Ωc,T)<Wn(Ωc,T)) which is the BCS-like behavior [112] as it is seen in Figure 10.(a) Spectral weightWn(Ωc,T) of the overdoped Bi2212 for Ωc=1eV. Closed symbols—normal state. Open symbols—superconducting state. (b) Change of the kinetic energy ΔEkin=Ekin,S-Ekin,N in meV per Cu site versus the charge p per Cu with respect to the optimal value popt. From [112].
(a)(b)We stress that the non-BCS behavior ofWs(Ωc,T) for underdoped (and in some optimally doped) systems was obtained by assuming that Ωc≈(1–1.2)eV. However, in [104] these results were questioned by claiming that the conventional BCS-like behavior was observed (Ws(Ωc,T)<Wn(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 Ws(Ωc,T)<Wn(Ωc,T) was observed, while in LSCO it was found that Ws(Ωc,T)≈const, that is, Ws(Ωc,T<Tc)≈Wn(Ωc,Tc).The first question is the following. How to explain the strong temperature dependence ofW(Ωc,T) in the normal state? In [39, 40] W(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 W(Ωc,T). The main theoretical results of [39, 40] are the following: the calculations of W(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, Wn(Ωc,T)≈Ws(Ωc,T) since Wn(Ωc,T)-Ws(Ωc,T)~Δ2/Ωc2. (2) In the case of large Ωc the calculations based on (30) give
(33)W(Ωc,T)≈ωpl28[1-γ(T)Wb-π22T2Wb2].
In the case when EPI dominates one has γ=γep(T)+γimp where γep(T)=∫0∞dzα2(z)F(z)coth(z/2T). It turns out that for α2(ω)F(ω), shown in Figure 4, one obtains (i) γep(T)~T2 in the temperature interval 100K<T<200K as it is seen in Figure 11 for the T dependence of W(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 λep,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(ω)F(ω). It is seen in Figure 11 that, for a softer Einstein spectrum (with ΩE=200K), W(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 W(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 W(Ωc,T) was observed in experiments in [104, 111, 113, 114].Figure 11
Spectral weightW(Ωc,T) in the normal state for Einstein phonons with ΩE=200K (full triangles) and ΩE=400K (open circles, left axis). Dashed lines show T2 asymptotic. From [40].To summarize, the above analysis demonstrates that the theory based on EPI is able to explain in a satisfactory way the temperature behavior ofW(Ωc,T) above and below Tc in systems at and near the optimal doping.(4) α2(ω)F(ω) and the In-Plane Resistivity ρab(T)
The temperature dependence of the in-plane resistivityρab(T) in cuprates is a direct consequence of the quasi-2D motion of quasiparticles and of the inelastic scattering which they experience. At present, there is no consensus on the origin of the linear temperature dependence of the in-plane resistivity ρab(T) in the normal state. Our intention is not to discuss this problem, but only to demonstrate that the EPI spectral function α2(ω)F(ω), which is obtained from tunnelling experiments in cuprates (see Section 1.3.4), is able to explain the temperature dependence of ρab(T) in the optimally doped YBCO. In the Boltzmann theory ρab(T) is given by
(34)ρab(T)=ρimp+4πωp2γtr(T),
where
(35)γtr(T)=πT∫0∞dωωsinh2(ω/2T)αtr2(ω)F(ω).
The residual resistivity ρimp is due to the impurity scattering. Since ρ(T)=1/σ(ω=0,T) and having in mind that the dynamical conductivity σ(ω,T) in YBCO (at and near the optimal doping) is satisfactory explained by the EPI scattering, then it is to expect that ρab(T) is also dominated by EPI in some temperature region T>Tc. This is indeed confirmed in the optimally dopedYBCO, where ρimp is chosen appropriately and the spectral function αtr2(ω)F(ω) is taken from the tunnelling experiments in [42–45]. The very good agreement with the experimental results [118] is shown in Figure 12. We stress that in the case of EPI there is always a temperature region where γtr(T)~T for T>αΘD, α<1 depending on the shape of αtr2(ω)F(ω) (for the simple Debye spectrum α≈0.2). In the linear regime one has ρ(T)≃ρimp+8π2λep,tr(kBT/ℏωp2)=ρimp+ρ′T.(a) Calculated resistivityρ(T) for the EPI spectral function αtr2(ω)F(ω) in [118]. (b) Measured resistivity in a(x)- and b(y)-crystal direction of YBCO [119] and calculated Bloch-Grüneisen curve (points) for λep=1 [120].
(a)(b)There is experimental constraint onλtr since λtr≈0.25ωpl2(eV)ρ′(μΩcm/K). For instance, for ωpl≈(2-3)eV [108] and ρ′≈0.6 in the oriented YBCO films and ρ′≈0.3-0.4 in single crystals of BSCO, one obtains λtr≈0.6–1.4. In case of YBCO single crystals, there is a pronounced anisotropy in ρab(T) [119] which gives ρx′(T)=0.6μΩcm/K and ρy′(T)=0.25μΩcm/K. The function λtr(ωpl) is shown in Figure 13 where the plasma frequency ωpl can be calculated by LDA-DFT and also extracted from the width (~ωpl*) of the Drude peak at small frequencies, where ωpl=ɛ∞ωpl*. We stress that the rather good agreement of theoretical and experimental results for ρab(T), in some optimally doped HTSC cuprates such as YBCO, should not be overinterpreted in the sense that the above rather simple electron-phonon approach can explain the resistivity in other HTSC cuprates and for various doping. For instance, in highly underdoped systems ρab(T) is very different from the behavior in Figure 12 and the simple Migdal-Eliashberg theory based on the EPI spectral function is inadequate. In this case one should certainly take into account polaronic effects [8–11], strong correlations, and so forth. The above analysis on the resistivity in the optimally doped YBCO demonstrates only that in this case if in (35) one uses the EPI spectral function α2(ω)F(ω) obtained from the tunnelling experiments (and optics) one obtains the correct T dependence of ρa,b(T). This result is an additional evidence for the importance of EPI.Figure 13
Transport EPI spectral function coupling constant in YBCO as a function of plasma frequencyωp as derived from the experimental slope of resistivity ρ′(T).λx for ρx′(T)=0.6μΩcm/K and λy for ρy′(T)=0.25μΩcm/K [119]. Squares are LDA values [121].Concerning the temperature dependence of the resistivity in other (than YBCO) families of the optimally doped HTSC cuprates we would like to point out that there is some evidence that the linear (inT) resistivity is observed in some of them even at temperatures T<0.2ΘD [122, 123]. This possibility is argued also theoretically in [124] where it is shown that in two-dimensional systems with a broad interval of phonon spectra the quasilinear behavior of ρab(T) is realized even at T<0.2ΘD. The quasilinear behavior of the resistivity at T≪0.2ΘD has been observed in Bi2(Sr0.97Pr0.003)2CuO6 [125], in LSCO, and in 1-layer Bi-2201 [122, 123, 126, 127], where in all these systems the critical temperature is rather small, Tc≈10K. In that respect all existing theories based on the electron-boson scattering are plagued and having difficulties to explain this low-temperature behavior of ρab(T). To this point, we would like to emphasize here that some of these (experimental) observations are contradictory. For example, the results obtained by the Vedeneev group [127] show that some samples demonstrate the quasilinear behavior of the resistivity up to T=10K but some others with approximately the same Tc have the usual Bloch-Grüneisen-type behavior characteristic for the EPI scattering. In that respect it is very unlikely that the linear resistivity up to T=10K can be simply explained in the standard way by interactions of electrons with some known bosons either by phonons or spin fluctuations (magnons). The question why in some cuprates the linear resistivity is observed up to T=10K is still a mystery and its explanation is a challenge for all kinds of the electron-boson scattering, not only for EPI. In that respect it is interesting to mention that the existence of the forward scattering peak in EPI (with the width qc≪kF), which is due to strong correlations, may give rise to the linear behavior of ρ(T) down to very low temperatures T~ΘD/30 [6, 128, 129]—see more in Section 2.3.4, item (6).  We will argue in Section 1.3.4 that if one interprets the tunnelling experiments in systems near optimal doping [42–54] in the framework of the Eliashberg theory one obtains the large EPI coupling constant λep≈2–3.5 which implies that λtr~(λ/3). This means that EPI is reduced much more in transport properties than in the self-energy. We stress that such a large reduction of λtr cannot be obtained within the LDA-DFT band-structure calculations, which means that λep and λtr contain renormalization which do not enter in the LDA-DFT theory. In Section 2 we will argue that the strong suppression of λtr may have its origin in strong electronic correlations [78–80, 130] and in the long-range Madelung energy [3–6].(5) Femtosecond Time-Resolved Optical Spectroscopy
The femtosecond time-resolved optical spectroscopy (FTROS) has been developed in the last couple of years and applied to HTSC cuprates. In this method a femtosecond (1fs=10-15sec) laser pump excites in materials electron-hole pairs via interband transitions. These hot carriers release their energy via electron-electron (with the relaxation time τee) and electron-phonon scattering reaching states near the Fermi energy within 10–100fs—see [131]. The typical energy density of the laser pump pulses with the wavelength λ≈810 nm (ℏω=1.5eV) was around F~1μJ/cm2 (the excitation fluenceF) which produces approximately 3×1010 carriers per pulse (by assuming that each photon produces ℏω/Δ carriers, Δ is the superconducting gap). By measuring photoinduced changes of the reflectivity in time, that is, ΔR(t)/R0, one can extract information on the relaxation dynamics of the low-laying electronic excitations. Since ΔR(t) relax to equilibrium, the fit with exponential functions is used as
(36)ΔR(t)R0=f(t)[Ae-t/τA+Be-t/τB+⋯],
where f(t)=H(t)[1-exp{-t/τee}] (H(t) is the Heavyside function) describes the finite rise-time. The parameters A, B depend on the fluence F. This method was used in studying the superconducting phase of La2-xSrxCuO4, with x=0.1 and 0.15 and Tc=30K and 38K, respectively [41]. In that case one has A≠0 for T<Tc and A=0 for T>Tc, while the signal B was present also at T>Tc. It turns out that the signal A is related to the quasiparticle recombination across the superconducting gap Δ(T) and has a relaxation time of the order τA>10ps at T=4.5K. At the so called threshold fluence (FT=4.2±1.7μJ/cm2 for x=0.1 and FT=5.8±2.3μJ/cm2 for x=0.15) the vaporization (destroying) of the superconducting phase occurs, where the parameter A saturates. This vaporization process takes place at the time scala τr≈0.8ps. The external fluence is distributed in the sample over the excitation volume which is proportional to the optical penetration depth λop(≈150 nm at λ≈810nm) of the pump. The energy densities stored in the excitation volume at the vaporization threshold for x=0.1 and x=0.15 are Up=FT/λop=2.0±0.8K/Cu and 2.6±1.0K/Cu, respectively. The important fact is that Up is much larger than the superconducting condensation energy which is Ucond≈0.12K/Cu for x=0.1 and Ucond≈0.3K/Cu for x=0.15, that is, Up≫Ucond. This means that the energy difference Up-Ucond must be stored elsewhere on the time scale τr. The only present reservoir which can absorb the difference in energy is the bosonic baths of phonons and spin fluctuations. The energy required to heat the spin reservoir from T=4.5K to Tc is Usf=∫TTcCsf(T)dT. The measured specific heat Csf(T) in La2CuO4 [41] gives very small value Usf≈0.01K. In the case of the phonon reservoir one obtains Uph=∫TTcCph(T)dT=9K/Cu for x=0.1 and 28K/Cu for x=0.15, where Cph is the phonon specific heat. Since Usf≪Up-Ucond, the spin reservoir cannot absorb the rest energy Up-Ucond. The situation is opposite with phonons since Uph≫Up-Ucond  and phonon can absorb the rest energy in the excitation volume. The complete vaporization dynamics can be described in the framework of the Rothwarf-Taylor model which describes approaching of electrons and phonons to quasiequilibrium on the time scale of 1 ps [132]. We will not go into details but only summarize by quoting the conclusion in [132] that only phonon-mediated vaporization is consistent with the experiments, thus ruling out spin-mediated quasiparticle recombination and pairing in HTSC cuprates. The FTROS method tells us that at least for nonequilibrium processes EPI is more important than SFI. It gives also some opportunities for obtaining the strength of EPI but at present there is no reliable analysis.In conclusion, optics and resistivity measurements in the normal state of cuprates give evidence that EPI is important while the spin-fluctuation scattering is weaker than it is believed. However, some important questions related to the transport properties remain to be answered. (i) What are the values of λtr and ωpl? (ii) What is the reason that λtr≪λ is realized in cuprates? (iii) What is the role of the Coulomb scattering in σ(ω) and ρ(T)? Later on we will argue that ARPES measurements in cuprates give evidence for an appreciable contribution of the Coulomb scattering at higher frequencies, where γ(ω)≈γ0+λcω for ω>ωmaxph with λc~1. One should stress that despite the fact that EPI is suppressed in transport properties it is sufficiently strong in the quasiparticle self-energy, as it comes out from tunnelling measurements discussed below.


### 1.3.3. ARPES and the EPI Self-Energy

The angle-resolved photoemission spectroscopy (ARPES) is nowadays one of leading spectroscopy methods in the solid-state physics [22, 23]. In some favorable conditions it provides direct information on the one-electron removal spectrum in a complex many-body system. The method involves shining light (photons) with energies between Ei=5–1000eV on samples and by detecting momentum (k)- and energy(ω)-distribution of the outgoing electrons. The resolution of ARPES has been significantly increased in the last decade with the energy resolution of ΔE≈1-2meV (for photon energies ~20 eV) and angular resolution of Δθ≲0.2∘. On the other side the ARPES method is surface-sensitive technique, since the average escape depth (lesc) of the outgoing electrons is of the order of lesc~10 Å, depending on the energy of incoming photons. Therefore, very good surfaces are needed in order that the results be representative for bulk samples. The most reliable studies were done on the bilayer Bi2Sr2CaCu2O8 (Bi2212) and its single-layer counterpart Bi2Sr2CuO6 (Bi2201), since these materials contain weakly coupled BiO planes with the longest interplane separation in the cuprates. This results in a natural cleavage plane making these materials superior to others in ARPES experiments. After a drastic improvement of sample quality in other families of HTSC materials, the ARPES technique has became an important method in theoretical considerations. The ARPES can indirectly give information on the momentum and energy dependence of the pairing potential. Furthermore, the electronic spectrum of the (abovementioned) cuprates is highly quasi-2D which allows rather unambiguous determination of the initial state momentum from the measured final state momentum, since the component parallel to the surface is conserved in photoemission. In this case, the ARPES probes (under some favorable conditions) directly the single-particle spectral function A(k,ω). In the following we discuss mainly those ARPES experiments which give evidence for the importance of the EPI in cuprates—see more in [22, 23].ARPES measures a nonlinear response function of the electron system and it is usually analyzed in the so-calledthree-step model, where the total photoemission intensity Itot(k,ω)≈I1·I2·I3 is the product of three independent terms: (1) I1 that describes optical excitation of the electron in the bulk, (2) I2 that describes the scattering probability of the travelling electrons, and (3) I3 that describes the transmission probability through the surface potential barrier. The central quantity in the three-step model is I1(k,ω) and it turns out that for k=k∥ it can be written in the form I1(k,ω)≃I0(k,υ)f(ω)A(k,ω) [22, 23] with I0(k,υ)~|〈ψf|pA|ψi〉|2 and the quasiparticle spectral function A(k,ω)=-ImG(k,ω)/π:
(37)A(k,ω)=-1πImΣ(k,ω)[ω-ξ(k)-ReΣ(k,ω)]2+ImΣ2(k,ω).
Here, 〈ψf|p·A|ψi〉 is the dipole matrix element which depends on k, polarization, and energy Ei of the incoming photons. The knowledge of the matrix element is of a great importance and its calculation from first principles was done in [133]. f(ω) is the Fermi function; G and Σ=ReΣ+iImΣ are the quasiparticle Green's function and the self-energy, respectively. We summarize and comment here some important ARPES results which were obtained in the last several years and which confirm the existence of the Fermi surface and importance of EPI in the quasiparticle scattering [22, 23].ARPES in the Normal State
(N1) There is well-defined Fermi surface in the metallic state of optimally and near optimally doped cuprates with the topology predicted by the LDA-DFT. However, the bands are narrower than LDA-DFT predicts which points to a strong quasiparticle renormalization. (N2) The spectral lines are broad with |ImΣ(k,ω)|~ω (or ~T for T>ω) which tells us that the quasiparticle liquid is a noncanonical Fermi liquid for larger values of T,ω. (N3) There is a bilayer band splitting in Bi2212 (at least in the overdoped state), which is also predicted by LDA-DFT. In the case when the coherent hopping t⊥ between two layers in the bilayer dominates, then the antibonding and bonding bands ξka,b=ξk±tk⊥ with tk⊥=[t⊥(cos2kx-cos2ky)+⋯] have been observed. It is worth to mention that the previous experiments did not show this splitting which was one of the reasons for various speculations on some exotic electronic scattering and non-Fermi liquid scenarios. (N4) In the underdoped cuprates and at temperatures Tc<T<T* there is a d-wave-like pseudogap Δpg(k)~Δpg,0(coskx-cosky) in the quasiparticle spectrum where Δpg,0 increases by lowering doping. We stress that the pseudogap phenomenon is not well understood at present and since we are interested in systems near optimal doping where the pseudogap phenomena are absent or much less pronounced we will not discuss this problem here. Its origin can be due to a precursor superconductivity or due to a competing order, such as spin- or charge-density wave, strong correlations, and so forth. (N5) The ARPES self-energy gives evidence that EPI interaction is rather strong. The arguments for the latter statement are the following: (i) at T>Tc there are kinks in the quasiparticle dispersion ω(ξk) in the nodal direction (along the (0,0)-(π,π) line) at the characteristic phonon energy ωph(70)~(60–70)meV [91], see Figure 14 (top), and near the antinodal point(π,0) at 40meV [134]—see Figure 14 (bottom).Figure 14
(top) Quasiparticle dispersion ofBi2212, Bi2201, and LSCO along the nodal direction, plotted versus the momentum k for (a)–(c) different doping, and (d)-(e) different T; black arrows indicate the kink energy; the red arrow indicates the energy of the q=(π,0) oxygen stretching phonon mode; inset of (e) shows T-dependent Σ′ for optimally doped Bi2212; (f) shows doping dependence of the effective coupling constant λ′ along (0,0)-(π,π) for the different HTSC oxides. From [91]. (bottom) Quasiparticle dispersion E(k) in the normal state (a1, b1, c), at 107 K and 115 K, along various directions ϕ around the antinodal point. The kink at E=40meV is shown by the horizontal arrow. (a2 and b2) are E(k) in the superconducting state at 10 K with the shifted kink to 70meV. (d) kink positions as a function of ϕ in the antinodal region. From [134].(ii) The kink structure is observed in a variety of the hole-doped cuprates such asLSCO, Bi2212, Bi2201, Tl2201 (Tl2Ba2CuO6), Na–CCOC (Ca2-xNaxCuO2Cl2). These kinks exist also above Tc, which excludes the scenario with the magnetic resonance peak in Imχs(Q,ω). Moreover, since the tunnelling and magnetic neutron scattering measurements give small SFI coupling constant gsf<0.2eV, then the kinks are not due to SFI. (iii) The position of the nodal kink is practically doping independent which points towards phonons as the scattering and pairing boson. (N6) The quasiparticles (holes) at and near the nodal-point kN couple practically to a rather broad spectrum of phonons since at least three groups of phonons were extracted in the bosonic spectral function α2F(kN,ω) from the ARPES effective self-energy in La2-xSrxCuO4 [135]—Figure 15.(a) Effective real self-energy for the nonsuperconductingLa2-xSrxCuO4, x=0.03. Extracted αeff2(ω)F(ω) is in the inset. (b) Top: the total MDC width—open circles. Bottom: the EPI contribution shows saturation, impurity contribution—dotted black line. The residual part is growing ~ω1.3. From [135].
(a)(b)The latter result is in a qualitative agreement with numerous tunnelling measurements [42–54] which apparently demonstrate that the very broad spectrum of phonons couples with holes without preferring any particular phonons—see discussion below. (N7) Recent ARPES measurements in Bi2212 [92] show very different slope dω/dξk of the quasiparticle energy ω(ξk) at small |ξk|≪ωph and at large energies |ξk|≫ωph—see Figure 16. The theoretical analysis [137] of these results gives the total coupling constant λZ=λepZ+λcZ≈3, and for the EPI coupling λepZ≈2, while the Coulomb coupling (SFI is a part of it) is λcZ≈1 [137]—see Figure 16. (Note that the upper index Z in the coupling constants means the quasiparticle renormalization in the normal part of the self-energy.) To this end let us mention some confusion which is related to the value of the EPI coupling constant extracted from ARPES. Namely, [22, 23, 138, 139] the EPI self-energy was obtained by subtracting the high-energy slope of the quasiparticle spectrum ω(ξk) at ω~0.3eV. The latter is apparently due to the Coulomb interaction. Although the position of the low-energy kink is not affected by this procedure (if ωphmax≪ωc), this subtraction procedure gives in fact an effective EPI self-energyΣeffep(k,ω) andthe effective coupling constantλep,effZ(k) only. We demonstrate below that the λep,effZ(k) is smaller than the real EPI coupling constant λepZ(k). The total self-energy is Σ(k,ω)=Σep(k,ω)+Σc(k,ω) where Σc is the contribution due to the Coulomb interaction. At very low energies ω≪ωc one has usually Σc(k,ω)=-λcZ(k)ω, where ωc(~1 eV) is the characteristic Coulomb energies and λcZ is the Coulomb coupling constant. The quasiparticle spectrum ω(k) is determined from the condition ω-ξ(k)-Re[Σep(k,ω)+Σc(k,ω)]=0 where ξ(k) is the bare band-structure energy. At low energies ω<ωphmax≪ωc it can be rewritten in the form
(38)ω-ξren(k)-ReΣeffep(k,ω)=0,
with ξren(k)=[1+λcZ(k)]-1ξ(k),
(39)ReΣeffep(k,ω)=ReΣeffep(k,ω)1+λcZ(k).
Since at very low energies ω≪ωphmax one has ReΣep(k,ω)=-λepZ(k)ω and ReΣeffep(k,ω)=-λep,effZ(k)ω, then the real coupling constant is related to the effective one by
(40)λepZ(k)=[1+λcZ(k)]λep,effZ(k).
As a result one has λepZ(k)>λep,effZ(k). At higher energies ωphmax<ω<ωc, where the EPI effects are suppressed and Σep(k,ω) stops growing, one has ReΣ(k,ω)≈ReΣep(k,ω)-λcZ(k)ω. The measured ReΣexp(k,ω) at T=10K near and slightly away from the nodal point in the optimally doped Bi-2212 with Tc=91K [136] is shown in Figure 16.Figure 16
Figure  4b from [136]: ReΣ(ω) measured in Bi2212 (thin line) and model ReΣ(ω) (bold line) obtained in [136]. The three thin lines (λ1,λ2,λ3) are the slopes of ReΣ(ω) in different energy regions—see the text.It is seen thatReΣexp(k,ω) has two kinks—the first one at low energyω1≈ωphhigh≈50–70meV which is (as we already argued) most probably of the phononic origin [22, 23, 138, 139], while the second kink at higher energyω2≈ωc≈350meV which is due to the Coulomb interaction. However, the important results in [136] are that the slopes of ReΣexp(k,ω) at low (ω<ωphhigh) and high energies (ωphhigh<ω<ωc) are very different. The low-energy and high-energy slope near the nodal point are shown in Figure 16 schematically (thin lines). From Figure 16 it is obvious that EPI prevails at low energies ω<ωphhigh. More precisely digitalization of ReΣexp(k,ω) in the interval ωphhigh<ω<0.4eV gives the Coulomb coupling λcZ≈1.1 while the same procedure at 20meV≈ωphlow<ω<ωphhigh≈50–70meV gives the total coupling constant (λ2≡)λZ=λepZ+λcZ≈3.2 and the EPI coupling constant λepZ(≡λep,highZ)≈2.1>2λep,effZ(k), that is, the EPI coupling is at least twice larger than the effective EPI coupling constant obtained in the previous analysis of ARPES results [22, 23, 138, 139]. This estimation tells us that at (and near) the nodal point, the EPI interaction dominatesin the quasiparticle scattering at low energies since λepZ(≈2.1)≈2λzc>2λsfZ, while at large energies (compared to ωph) the Coulomb interaction with λcZ≈1.1 dominates. We point out that EPI near the antinodal point can be even larger than in the nodal point, mostly due to the higher density of states near the antinodal point. (N8) Recent ARPES spectra in the optimally doped Bi2212 near the nodal and antinodal point [139] show a low-energy isotope effect in ReΣexp(k,ω), which can be well described in the framework of the Migdal-Eliashberg theory for EPI [140]. At higher energies ω>ωph obtained in [139] very pronounced isotope effect cannot be explained by the simple Migdal-Eliashberg theory [140]. However, there are controversies with the strength of the high-energy isotope effect since it was not confirmed in other measurements [141, 142]—see the discussion in Section 1.3.6(2) related to the isotope effects in HTSC cuprates.  (N9) The ARPES experiments in Ca2CuO2Cl2 give strong evidence for the formation of small polarons in undoped cuprates which are due to phonons and strong EPI, while in the doped systems quasiparticles are formed  and there are no small polarons [143]. Namely, in [143] a broad peak around -0.8eV is observed at the top of the band (k=(π/2,π/2)) with the dispersion similar to that predicted by the t-J model—see Figure 17.(a) The ARPES spectrum of undopedCa2CuO2Cl2 at k=(π/2,π/2). Gaussian shape—solid line, Lorentzian shape—dashed line. (b) Dispersion of the polaronic band—A—and of the quasiparticle band—B—along the nodal direction. Horizontal lines are the chemical potentials for a large number of samples. From [143].
(a)(b)However, the peak in Figure17(a) is of Gaussian shape and can be described only by coupling to bosons, that is, this peak is a boson side band—see more in [10, 11] and references therein. The theory based on the t-J model (in the antiferromagnetic state of the undoped compound) by including coupling to several (half-breathing, apical oxygen, low-lying) phonons, which is given in [144–146], explains successfully this broad peak of the boson side band by the formation of small polarons due to the EPI coupling (λep≈1.2). Note that this value of λep is for the polaron at the bottom of the band while in the case where the Fermi surface exists (in doped systems) this coupling is even larger due to the larger density of states at the Fermi surface [144–146]. In [144–146] it was stressed that even when the electron-magnon interaction is stronger than EPI the polarons in the undoped systems are formed due to EPI. The latter mechanism involves excitation of many phonons at the lattice site (where the hole is seating), while it is possible to excite only one magnon at the given site. (N10) Recent soft X-ray ARPES measurements on the electron-doped HTSC Nd1.85Ce0.15CuO4 [147], and Sm(2-x)CexCuO4 (x=0.1,0.15,0.18), Nd1.85Ce0.15CuO4, and Eu1.85Ce0.15CuO4 [148] show kink at energies 50–70meV in the quasiparticle dispersion relation along both the nodal and antinodal, directions as it is shown in Figure 18.NCCO electron-doped: (a)ImΣ(ω) measured in the nodal point. Curves are offsets by 50 meV for clarity. The change of the slope in the last bottom curve is at the phonon energy. (b) ImΣ(ω) for the antinodal direction with 30meV offset. (c) Experimental phonon dispersion of the bond stretching modes. (d) Estimated λeffep from ImΣ(ω). From [148].
(a)(b)(c)(d)It is seen from this figure that the effective EPI coupling constantλep,eff(<λep) is isotropic and λep,eff≈0.8–1. It seems that the kink in the electron-doped cuprates is due solely to EPI and in that respect the situation is similar to the one in the hole-doped cuprates.ARPES Results in the Superconducting State
(S1) There is an anisotropic superconducting gap in most HTSC compounds [22, 23], which is predominately d-wave like, that is, Δ(k)≈Δ0(coskx-cosky) with 2Δ0/Tc≈5-6 in the optimally doped systems. (S2) The particle-hole coherence in the superconducting state which is expected for the BCS-like theory of superconductivity has been observed first in [149] and confirmed with better resolution in [150], where the particle-hole mixing is clearly seen in the electron and hole quasiparticle dispersion. To remind the reader, the excited Bogoliubov-Valatin quasiparticles (α̂k,±) with energies Ekα±=ξk2+|Δk|2 are a mixture of electron (ĉk,σ) and hole (ĉ-k,-σ†), that is, α̂k,+=ukĉk↑+vkĉ-k↓†, α̂k,-=ukĉ-k↓+vkĉk↑† where the coherence factors uk, vk are given by |uk|2=1-|vk|2=(1+ξk/Ek)/2. Note that |uk|2+|vk|2=1, which is exactly observed, together with d-wave pairing Δ(k)=Δ0(coskx-cosky), in experiments in [150]. This is very important result since it proves that the pairing in HTSC cuprates is of the BCS type and not exotic one as it was speculated long time after the discovery of HTSC cuprates. (S3) The kink at (60–70)meV in the quasiparticle energy around the nodal point is not shifted (in energy) while the antinodal kink at ωph(40)~40meV is shifted (in energy) in the superconducting state by Δ0(=(25–30)meV), that is, ωph(40)→ωph(40)+Δ0=(65–70)meV [22, 23]. To remind the reader, in the standard Eliashberg theory the kink in the normal state at ω=ωph should be shifted in the superconducting state to ωph+Δ0 at all k-points at the Fermi surface. This puzzling result (that the quasiparticle energy around the nodal point is not shifted in the superconducting state) might be a smoking gun result since it makes an additional constraint on the quasiparticle interaction in cuprates. Until now there is only one plausible explanation [151] of this nonshift puzzle which is based on an assumption of the forward scattering peak (FSP) in EPI—see more in Section 2. The FSP in EPI means that electrons scatter into a narrow region (q<qc≪kF) around the initial point in the k-space, so that at the most part of the Fermi surface there is practically no mixing of states with different signs of the order parameter Δ(k). In that case the EPI bosonic spectral function (which is defined in Appendix A) α2F(k,k′,Ω)≈α2F(φ,φ′,Ω) (φ is the angle on the Fermi surface) has a pronounced forward scattering peak (at δφ=φ-φ′=0) due to strong correlations—see Section 2. Its width δφc is narrow, that is, δφc≪2π and the angle integration goes over the region δφc around the point φ. In that case the kink is shifted (approximately) by the local gap Δ(φ)=Δmaxcos2φ—for more details see [151]. As a consequence, the antinodal kink is shifted by the maximal gap, that is, |Δ(φAN≈π/2)|=Δmax while the nodal gap is practically unshifted since |Δ(φAN≈π/4)|≈0. (S4) The recent ARPES spectra [152] in the undoped single crystalline 4-layered cuprate with the apical fluorine (F), Ba2Ca3Cu4O8F2 (F0234) give rather convincing evidence against the SFI mechanism of pairing—see Figure 19.Figure 19
Crystal structure ofBa2Ca3Cu4O8(OδF1-δ)2. There are four CuO2 layers in a unit cell with the outer having apical F atoms. CRL—charge reservoir layer; SCL/OP—superconducting layer/outer plane; SCL/IP—superconducting layer/inner plane. From [152].First, F0234 is not a Mott insulator—as expected from valence charge counting which puts Cu valence as 2+, but it is a superconductor with Tc=60K. Moreover, the ARPES data [152] reveal at least two metallic Fermi-surface sheets with corresponding volumes equally below and above half-filling—see Figure 20.(a) Fermi surface (FS) contours from two samples of F0234.N—electron-like; P—hole-like. Bold arrow is (π,π) scattering vector. Angle θ defines the horizontal axis in (b). (b) Leading gap edge along k-space angle from the two FS contours. From [152].
(a)(b)Second, one of the Fermi surfaces is due to the electron-like (N) band (with 20±6% electron-doping) and the other one due to the hole-like (P) band (with 20±8% hole-doping) and their splitting along the nodal direction is significant and cannot be explained by the LDA-DFT calculations [153]. This electron and hole self-doping of inner and outer layers is in an appreciable contrast to other multilayered cuprates where there is only hole self-doping. For instance, in HgBa2CanCun+1O2n+2 (n=2,3) and (Cu,C)Ba2CanCun+1O3n+2 (n=2,3,4), the inner CuO2 layers are less hole-doped than outer layers. It turns out, unexpectedly, that the superconducting gap on theN-band Fermi surface is significantly larger than on the P-one, where in Ba2Ca3Cu4O8F2 the ratio is anomalous (ΔN/ΔP)≈2 and ΔN is an order of magnitude larger than in the electron-doped cuprate Nd2-xCexCuO4. Third, the N-band Fermi surface  is rather far from the antinodal point at (π,0). This is very important result which means that the antiferromagnetic spin fluctuations with the AF wave-vector Q=(π,π), as well as the van Hove singularity, are not dominant in the pairing in the N-band. To remind the reader, the SFI scenario assumes that the pairing is due to spin fluctuations with the wave-vector Q (and near it) which connects two antinodal points which are near the van Hove singularity at the hole-surface (at (π,0) and (0,π)) giving rise to large density of states. This is apparently not the case for the N-band Fermi surface—see Figure 20. The ARPES data give further that there is a kink at ~85 meV in the quasiparticle dispersion of both bands, while the kink in the N-band is stronger than that in the P-band. This result, together with the anomalous ratio (ΔN/ΔP)≈2, disfavors SFI as a pairing mechanism. (S5) Despite the presence of significant elastic quasiparticle scattering in a number of samples of optimally doped Bi-2212, there are dramatic sharpenings of the spectral function near the antinodal point (π,0) at T<Tc (in the superconducting state) [154]. This effect can be explained by assuming that the small q-scattering (the forward scattering peak) dominates in the elastic impurity scattering as it is pointed in [78–80, 130, 155, 156]. As a result, one finds that the impurity scattering rate in the superconducting state is almost zero, that is, γimp(k,ω)=γn(k,ω)+γa(k,ω)≈0 for |ω|<Δ0 for any kind of pairing (s-, p-, d-wave, etc.) since the normal (γn) and the anomalous (γa) scattering rates compensate each other. This collapse of the elastic scattering rate is elaborated in details in [154] and it is a consequence of the Anderson-like theorem for unconventional superconductors which is due to the dominance of the small q-scattering [78–80, 130, 155, 156]. In such a case d-wave pairing is weakly unaffected by nonmagnetic impurities and as a consequence there is small reduction in Tc [156, 157]. The physics behind this result is rather simple. The small q-scattering (usually called forward scattering) means that electrons scatter into a small region in the k-space, so that at the most part of the Fermi surface there is no mixing of states with different signs of the order parameter Δ(k). In such a way the detrimental effect of nonmagnetic impurities on d-wave pairing is significantly reduced. This result points to the importance of strong correlations in the renormalization of the nonmagnetic impurity scattering too—see discussion in Section 2.In conclusion, in order to explain the ARPES results in cuprates it is necessary to take into account (1) the electron-phonon interaction (EPI) since it dominates in the quasiparticle scattering in the energy region important for pairing, (2) the elastic nonmagnetic impurities with the forward scattering peak (FSP) due to strong correlations, and (3) the Coulomb interaction which dominates at higher energies ω>ωph. In this respect, the presence of ARPES kinks and the knee-like shape of the T dependence of the spectral width are important constraints on the scattering and pairing mechanism in HTSC cuprates.


### 1.3.4. Tunnelling Spectroscopy and Spectral Functionα2F(ω)

By measuring current-voltageI-V characteristics in NIS (normal metal-insulator-superconductor) tunnelling junctions with large tunnelling barrier one obtains from tunnelling conductance GNS(V)=dI/dV the so called tunnelling density of states in superconductors NT(ω). Moreover, by measuring of GNS(V) at voltages eV>Δ it is possible to determine the Eliashberg spectral function α2F(ω) and finally to confirm the phonon mechanism of pairing in LTSC materials. Four tunnelling techniques were used in the study of HTSC cuprates: (1) vacuum tunnelling by using the STM technique—scanning tunnelling microscope; (2) point-contact tunnelling; (3) break-junction tunnelling; (4) planar-junction tunnelling. Each of these techniques has some advantages although in principle the most potential one is the STM technique since it measures superconducting properties locally [158]. Since tunnelling measurements probe a surface region on the scale of the superconducting coherence length ξ0, then this kind of measurements in HTSC materials with small coherence length ξ0 (ξab~20 Å in the a-b plane and ξc~1–3 Å along the c-axis) depends strongly on the surface quality and sample preparation. Nowadays, many of the material problems in HTSC cuprates are understood and as a result consistent picture of tunnelling features is starting to emerge.From tunnelling experiments one obtains the (energy-dependent) gap functionΔ(ω) in the superconducting state. Since we have already discussed this problem in [6], we will only briefly mention some important result. For instance, in most systems GNS(V) has V-shape in all families of HTSC hole- and electron-doped cuprates. The V-shape is characteristic for d-wave pairing with gapless spectrum, which is also confirmed in the interference experiments on hole- and electron-doped cuprates [75]. Some experiments give a U-shape of GNS(V) which resembles s-wave pairing. This controversy is explained to be the property of the tunnelling matrix element which filters out states with the maximal gap.Here we are interested in thebosonic spectral functionα2F(ω) of HTSC cuprates near optimal doping which can be extracted by using tunnelling spectroscopy. We inform the reader in advance that the shape and the energy width of α2F(ω), which are extracted from the second derivative d2I/dV2 at voltages above the superconducting gap, in most HTSC cuprates resemble the phonon density of states Fph(ω). This result is strong evidence for the importance of EPI in the pairing potential of HTSC cuprates. For instance, plenty of break junctions made from Bi2212 single crystals [42–45] show that the peaks (and shoulders) in -d2I/dV2 (or dips-negative peaks in d2I/dV2) coincide with the peaks (and shoulders) in the phonon density of states Fph(ω) measured by neutron scattering—see Figure 21.(a) Second derivative ofI(V) for a Bi2212 break junction in various magnetic fields (from 0–20T). The structure of dips (minima) in d2I/dV2 can be compared with the phonon density of states F(ω); (b) the spectral functions α2F(ω) in various magnetic fields. From [42–45].
(a)(b)The tunnelling spectra in Bi-2212 break junctions [42–45], which are shown in Figure 21 indicates that the spectral function α2F(ω) is independent of magnetic field, which is in contradiction with the theoretical prediction based on the SFI pairing mechanism where this function should be sensitive to magnetic field. The reported broadening of the peaks in α2F(ω) is partly due to the gapless spectrum of d-wave pairing in HTSC cuprates. Additionally, the tunnelling density of states NT(ω) at very low T and for ω>Δ shows a pronounced gap structure. It was found that 2Δ¯/Tc=6.2–6.5, where Tc=74–85Kand Δ¯ is some average value of the gap. In order to obtain α2F(ω) the inverse procedure was used by assuming s-wave superconductivity and the effective Coulomb parameter μ*≈0.1 [42–45]. The obtained α2F(ω) gives large EPI coupling constant λep≈2.3. Although this analysis [42–45] was done in terms of s-wave pairing, it mimics qualitatively the case of d-wave pairing, since one expects that d-wave pairing does not change significantly the global structure of d2I/dV2 at eV>Δ albeit introducing a broadening in it—see the physical meaning in Appendix A. We point out that the results obtained in [42–45] were reproducible on more than 30 junctions. In that respect very important results on slightly overdoped Bi2212–GaAs and on Bi2212–Au planar tunnelling junctions are obtained in [46, 47]—see Figure 22.Figure 22
The spectral functionsα2F(ω) and the calculated density of states at 0K (upper solid line) obtained from the conductance measurements, the Bi(2212)–Au planar junctions. GPDS—generalized phonon density of states. From [46, 47].These results show very similar features to those obtained in [42–45] on break junctions. It is worth mentioning that several groups [48–52] have obtained similar results for the shape of the spectral function α2F(ω) from the I-V measurements on various HTSC cuprates—see the comparison in Figure 23. These facts leave no much doubts about the importance of the EPI in pairing mechanism of HTSC cuprates.Figure 23
The spectral functionsα2F(ω) from measurements of various groups: Vedeneev et al. [42–45], Gonnelli et al. [52], Miyakawa et al. [48, 49], and Shimada et al. [46, 47]. The generalized density of states GPDS for Bi2212 is at the bottom. From [46, 47].In that respect, the tunnelling measurements onslightly overdopedBi2Sr2CaCu2O8 [46, 47, 53, 54] give impressive results. The Eliashberg spectral function α2F(ω) of this compound was extracted from the measurements of d2I/dV2 and by solving the inverse problem—see Appendix A. The extracted α2F(ω) has several peaks in broad energy region up to 80meV as it is seen in Figures 22 and 23, which coincide rather well with the peaks in the phonon density of states Fph(ω)—more precisely the generalized phonon density of states GPDS(ω) defined in Appendix A. In [53, 54] numerous peaks, from P1–P13, in α2F(ω) are discerned as shown in Figure 24, which correspond to various groups of phonon modes—laying in (and around) these peaks. Moreover, in [46, 47, 53, 54] the coupling constants for these modes are extracted as well as their contribution (ΔTc) to Tc as it is seen in Table 1. Note that due to the nonlinearity of the problem the sum of (ΔTc)i,i=1,2,…,13, due to various modes is not equal to Tc.Table 1
Phonon frequencyω, EPI coupling constant λi of the peaks P1–P13, and contribution ΔTc to Tc of each peak in α2F(ω)—as shown in Figure 24—obtained from the tunnelling conductance of Bi2Sr2CaCu2O8. ΔTc is the decrease in Tc when the peak in α2F(ω) is eliminated. From [53, 54].
No. of peakω [meV]λiΔTc [K]P114.31.267.4P220.80.9511.0P331.70.4810.5P435.10.286.7P539.40.247.0P645.30.3010.0P758.30.156.5P863.90.010.6P969.90.073.6P1073.70.063.3P1177.30.010.8P1282.10.010.7P1387.10.031.8Figure 24
The spectral functionsα2F(ω) from the tunnelling conductance of Bi2Sr2CaCu2O8 for the positive and the negative bias voltages, and the averaged one [46, 47]. The averaged one is divided into 13 components. The origin of the ordinate is 2,1,0, and -0.5 from the top down. From [46, 47, 53, 54].The next remarkable result is that the extractedEPI coupling constant is very large, that is, λep(=2∫dωα2F(ω)/ω)=∑iλi≈3.5—see Table 1. It is obvious from Figure 24 and Table 1 that almost all phonon modes contribute to λep and Tc, which means that on the average each particular phonon mode is not too strongly coupled to electrons since λi<1.3,i=1,2,…,13, thus keeping the lattice stable.Let us discuss the content of Table1 in more details where it is shown the strength of the EPI coupling and the relative contribution of different phononic modes to Tc. In Table 1 it is seen that lower-frequency modes from P1–P3, corresponding to Cu,Sr, and Ca vibrations, are rather strongly coupled to electrons (with λκ~1) which give appreciable contributions to Tc. It is also seen in Table 1 that the coupling constants λi of the high-energy phonons (P9–P13 with ω≥70meV) have λi≪1 and give moderate contribution to Tc—around 10%. These results give solid evidence for the importance of the low-energy modes related to the change of the Madelung energy in the ionic-metallic structure of HTSC cuprates—the idea advocated in [3–6] and discussed in Section 2. If confirmed in other HTSC families, these results are in favor of the moderate oxygen isotope effect in cuprates near the optimal doping since the oxygen modes are higher-energy modes and give smaller contribution to Tc. We stress that each peak P1–P13 in α2F(ω) corresponds to many modes. For a better understanding of the EPI coupling in these systems we show in Figure 25 the total and partial density of phononic states. It is seen that the low-energy phonons are due to the vibrations of the Ca, Sr, and Cu ions which correspond to the peaks P1-P2 in Figures 23 and 24. In order to obtain information on the structure of vibrations which are strongly involved in pairing, we show in Figures 26 and 27 the structure of these vibrations at special points in the Brillouin zone. It is seen in Figure 26 that the low-frequency phonons P1-P2 are dominated by Cu, Sr, Ca vibrations.Figure 25
The phonon density of statesF(ω) (PDOS) of Bi2Sr2CaCu2O8 compared with the generalized density of states (GPDOS) [159]. Atomic vibrations: O1—O in the CuO2 plane; O2—apical O; O3—O in the BiO plane. From [46, 47].Atomic polarization vectors and their frequencies (inmeV) at special points in the Brillouin zone. The larger closed circles in the lattice are O-ions. Γ-X is along the Cu-O-Cu direction. Arrows indicate displacements. The modes in square and round brackets are the transverse and longitudinal optical modes, respectively. (a) Modes of the P1 peak. (b) Modes of the P2 peak. From [46, 47, 53, 54].
(a)(b)Atomic polarization vectors and their frequencies (inmeV) at special points in the Brillouin zone. The larger closed circles in the lattice are O-ions. Γ-X is along the Cu-O-Cu direction. Arrows indicate displacements. The modes in square and round brackets are the transverse and longitudinal optical modes, respectively. (a) Modes of the P3 peak. (b) Modes of the P4 peak. From [46, 47, 53, 54].
(a)(b)Further, based on Table1 one concludes that the P3 modes are strongerly coupled to electrons than the P4 ones, although the density of state for the P4 modes is larger. The reason for such an anomalous behavior might be due to symmetries of the corresponding phonons as it is seen in Figure 27. Namely, to the P3 peak contribute axial vibrations of O(1) in the CuO2 plane which are odd under inversion, while in the P4 peak these modes are even. The in-plane modes of Ca and O(1) are present in P3 which are in-phase and out-of-phase modes, while in P4 they are all out-of-phase modes. For more information on other modes, P5–P13, see [53, 54]. We stress that the Eliashberg equations based on the extracted α2F(ω) of the slightly overdoped Bi2Sr2CaCu2O8 with the ratio (2Δ/Tc)≈6.5 describe rather well numerous optical, transport, and thermodynamic properties [53, 54]. However, in underdoped systems with (2Δ/Tc)≈10, where the pseudogap phenomena are pronounced, there are serious disagreements between experiments and the Migdal-Eliashberg theory [53, 54]. We would like to stress that the contribution of the high-frequency modes (mostly the oxygen modes) to α2F(ω) may be underestimated in tunnelling measurements due to their sensitivity to the surface contamination and defects. Namely, the tunnelling current probes a superconductor to a depth of order of the quasiparticle mean-free path l(ω)=vFγ-1(ω). Since the relaxation time γ-1(ω) decreases with increasing ω, the mean-free path can be rather small and the effects of the high-energy phonons are sensitive to the surface contamination.Similar conclusion regarding the structure of theEPI spectral function α2F(ω) in HTSC cuprates comes out from tunnelling measurements on Andreev junctions (the BTK parameter Z≪1—low barrier) and Giaver junctions (Z≫1—high barrier) in La2-xSrxCuO4 and YBCO compounds [160], where the extracted α2F(ω) is in good accordance with the phonon density of states Fph(ω)—see Figure 28.(a)d2I/dV2 of a Giaver-like contact in La2-xSrxCuO4—note the large structure below 50meV; (b) d2I/dV2 of an Andreev- and Giaver-like contact compared to the peaks in the phonon density of states. From [160].
(a)(b)Note that the BTK parameterZ is related to the transmission and reflection coefficients for the normal metal (1+Z2)-1 and Z2(1+Z2)-1, respectively.Although most of the peaks inα2F(ω) of HTSC cuprates coincide with the peaks in the phonon density of states, it is legitimate to put the following question. Can the magnetic resonance in the superconducting state give significant contribution to α2F(ω)? In that respect the inelastic magnetic neutron scattering measurements of the magnetic resonance as a function of doping [161] give that the resonance energy Er scales with Tc, that is, Er=(5-6)Tc as shown in Figure 29.Figure 29
Doping dependence of the energyEr of the magnetic resonance peak at Q=(π,π) in YBCO and Bi2212 measured at low temperatures by inelastic neutron scattering. From [161].This means that if one of the peaks inα2F(ω) is due to the magnetic resonance at ω=Er, then it must shift strongly with doping as it is observed in [161]. This is contrary to phonon peaks (energies) whose positions are practically doping independent. To this end, recent tunnelling experiments on Bi-2212 [55] show clear doping independenceof α2F(ω) as it is seen in Figure 30. This remarkable result is an additional evidence in favor of EPI and against the SFI mechanism of pairing in HTSC cuprates which is based on the magnetic resonance peak in the superconducting state. In that respect the analysis in [162] of the tunneling spectra of the electron-doped cuprate Pr0.88Ce0.12CuO4 with Tc=24K shows the existence of the bosonic mode at ωB=16meV which is significantly larger than the magnetic-resonance mode with ωr=(10-11)meV. This result excludes the magnetic-resonance mode as an important factor which modifies superconductivity.Figure 30
Second derivative ofI(V) for Bi2212 tunnelling junctions for various doping: UD—underdoped; OD—optimally doped; OVD—overdoped system. The structure of minima in d2I/dV2 can be compared with the phonon density of states F(ω). The full and vertical lines mark the positions of the magnetic resonance energy Er≈5.4Tc for various doping taken from Figure 29. Red tiny arrows mark positions of the magnetic resonance Er in various doped systems. Dotted vertical lines mark various phonon modes. From [55].The presence of pronounced phononic structures (and the importance of EPI) in theI(V) characteristics was quite recently demonstrated by the tunnelling measurements on the very good La1.85Sr0.15CuO4 films prepared by the molecular beam epitaxy on the [001]-symmetric SrTiO3 bicrystal substrates [56]. They give unique evidence for eleven peaks in the (negative) second derivative, that is, -d2I/dV2. Furthermore, these peaks coincide with the peaks in the intensities of the phonon Raman scattering data measured at 30K in single crystals of LSCO with 20% of Sr [57]. These results are shown in Figure 31. In spite of the lack of a quantitative analysis of the data in the framework of the Eliashberg equations, the results in [56] are important evidence that phonons are relevant pairing bosons in HTSC cuprates.Figure 31
Second derivative datad2I(V)/dV2 of the tunnelling spectra on thin films of La1.85Sr0.15CuO4 are shown along with phonon Raman scattering data on single crystals of LSCO with 20% Sr. The polarization of the incident and scattered light in the Raman spectra is parallel to the CuO2 planes. From [56].It is interesting that in thec-axis vacuum tunnelling STM measurements [163] the fine structure in d2I/dV2 at eV>Δ was not seen below Tc, while the pseudogap structure is observed at temperatures near and above Tc. This result could mean that the STM tunnelling is likely dominated by the nontrivial structure of the tunnelling matrix element (along the c-axis), which is derived from the band-structure calculations [164]. However, recent STM experiments on Bi2212 [61–63] give information on the possible nature of the bosonic mode which couples with electrons. In [61–63] the local conductance dI/dV(r,E) is measured where it is found that d2I/dV2(r,E) has peak at E(r)=Δ(r)+Ω(r) where dI/dV(r,E) has the maximal slope—see Figure 32(a).(a) Typical conductancedI/dV(r,E). The ubiquitous features at eV>Δ(gap) with maximal slopes, which give peaks in d2I/dV2(r,E), are indicated by arrows. (b) The histograms of all values of Ω(r) for samples with O16—right curve and with O18—left curve. From [61–63].
(a)(b)It turns out that the corresponding average phonon energyΩ¯ depends on the oxygen mass, that is, Ω¯~MO-1/2, with Ω¯16=52meV and Ω¯18≈48meV—as it is seen in Figure 32(b). This result is interpreted in [61–63] as an evidence that the oxygen phonons are strongly involved in the quasiparticle scattering. A possible explanation is put forward in [61–63] by assuming that this isotope effect is due to the B1g phonon which interacts with the antinodal quasiparticles. However, this result requires a reanalysis since the energy of the bosonic mode in fact coincides with the dip and not with the peak of d2I/dV2(r,E)—as reported in [61–63].The important message of numerous tunnelling experiments inHTSC cuprates near and at the optimal doping is that there is strong evidence for the importance of EPI in the quasiparticle scattering and that no particular phonon mode can be singled out in the spectral function α2F(ω) as being the only one which dominates in pairing mechanism. This important result means that the high Tc is not attributable to a particular phonon mode in the EPI mechanism but all phonon modes contribute to λep. Having in mind that the phonon spectrum in HTSC cuprates is very broad (up to 80meV), then the large EPI constant (λep≳2) obtained in the tunnelling experiments is not surprising at all. Note that similar conclusion holds for some other oxide superconductors such as Ba9.6K0.4BiO3 with Tc=30K where the peaks in the bosonic spectral function/extracted from tunnelling measurements coincide with the peaks in the phononic density of states [165–167].


### 1.3.5. Phonon Spectra and EPI

Although experiments related to phonon spectra and their renormalization byEPI, such as inelastic neutron, inelastic X-ray, and Raman scattering, do not give the spectral function α2F(ω), they nevertheless can give useful, but indirect, information on the strength of EPI for some particular phonons. We stress in advance that the interpretation of the experimental results in HTSC cuprates by the theory of EPI for weakly correlated electrons is inadequate since in strongly correlated systems, such as HTSC cuprates, the phonon renormalization due to EPI is different than in weakly correlated metals [168]. Since these questions are reviewed in [168], we will briefly enumerate the main points: (1) in strongly correlated systems the EPI coupling for a number of phononic modes can be significantly larger than the LDA-DFT and Hartree-Fock methods predict. This is due to many-body effects not contained in LDA-DFT [168, 169]. The lack of the LDA-DFT calculations in obtaining phonon line-widths is clearly demonstrated, for instance, in experiments on L2-xSrxCuO4—see review in [170] and references therein, where the bond-stretching phonons at q=(0.3,0,0) are softer and much broader than the LDA-DFT calculations predict. (Note the wave vector q is in units (2π/a,2π/b,2π/c)—for instance, in these units (π,π) corresponds to (0.5,0.5).) (2) The calculation of phonon spectra is in principle very difficult problem since besides the complexity of structural properties in a given material one should take into account appropriately the long-range Coulomb interaction of electrons as well as strong short-range repulsion. Our intention is not to discuss this complexity here—for that see, for instance, [69]—but we only stress some important points which will help to understand problems with which is confronted the theory of phonons in cuprates.The phonon Green's functionD(q,ω) depends on the phonon self-energy Π(q,ω) which takes into account all the enumerated properties (note that D-1(q,ω)=D0-1(q,ω)-Π(q,ω)). In cases when the EPI coupling constant gep(k,k′) is a function on the transfer momentum q=k-k′ only, then Π(q) (q=(q,iωn)) depends on the quasiparticle charge susceptibility χc(q)=P(q)/ɛe(q):
(41)Π(q)=|gep(q)|2χc(q),
and P(q) is the irreducible electronic polarization given by
(42)P(q)=-∑pG(p+q)Γc(p,q)G(p).
The screening due to the long-range Coulomb interaction is contained in the electronic dielectric function ɛe(q) while the “screening” due to (strong) correlations is described by the charge vertex function Γc(p,q). Due to complexity of the physics of strong correlations the phonon dynamics was studied in the t-J model but without the long-range Coulomb interaction [168, 169, 171], in which case one has ɛe=1 and χc(q)=P(q). However, in studying the phonon spectra in HTSC cuprates it is believed that this deficiency might be partly compensated by choosing the bare phonon frequency ω0(q) (contained in D0-1(q,ω)) to correspond to the undoped compounds [168, 171]. It is a matter of future investigations to incorporate all relevant interactions in order to obtain a fully microscopic and reliable theory of phonons in cuprates. Additionally, the electron-phonon interaction (with the bare coupling constant gep(q)) is dominated by the change of the energy of the Zhang-Rice singlet—see more in Section 2.3—and (41) for Π(q) is adequate one [6, 168, 169]. Since the charge fluctuations in HTSC cuprates are strongly suppressed (no doubly occupancy of the Cu 3d9 state) due to strong correlations, and since the suppressed value of χc(q) cannot be obtained by the band-structure calculations, this means that LDA-DFT underestimates the EPI coupling constant significantly. In the following we discuss this important result briefly.(1) Inelastic Neutron and X-Ray Scattering—The Phonon Softening and the Line-Width due to EPI
The appreciable softening and broadening of numerous phonon modes has been observed in the normal state of HTSC cuprates, thus giving evidence for pronounced EPI effects and for inadequacy of the LDA-DFT calculations in treating strong correlations and suppression of the charge susceptibility [6, 10, 11, 168, 171]. There are several relevant reviews on this subject [10, 11, 168, 170, 172] and here we discuss briefly two important examples which demonstrate the inefficiency of the LDA-DFT-band structure calculations to treat quantitatively EPI in HTSC cuprates. For instance, the Cu–O bond-stretching phonon mode shows a substantial softening at qhb=(0.3,0,0) by doping  of La1.85Sr0.15CuO4 and YBa2Cu3O7 [170, 172]—called the half-breathing phonon, and a large broadening by 5meV at 15% doping [173–175] as it is seen in Figure 33. While the softening can be partly described by the LDA-DFT method [176], the latter theory predicts an order of magnitude smaller broadening than the experimental one. This failure of LDA-DFT is due to the incorrect treatment of the effects of strong correlations on the charge susceptibility χc(q) and due to the absence of many-body effects which can increase the coupling constant gep(q)—see more in Section 2. The neutron scattering measurements in La1.85Sr0.15CuO4 give evidence for large (30%) softening of the OZZ with Λ1 symmetry with the energy ω≈60meV, which is theoretically predicted in [177], and for the large line-width about 17meV which also suggests strong EPI. These apex modes are favorable for d-wave pairing since their coupling constants are peaked at small momentum q [10, 11]. Having in mind the above results, then it is not surprising that the recent calculations of the EPI coupling constant λep in the framework of LDA-DFT give very small EPI coupling constant λep≈0.3 [28, 29]. The critical analysis of the LDA-DFT results in HTSC cuprates is done in [6] and additionally argued in [10, 11, 178] by pointing their disagreement with the inelastic neutron and X-ray scattering measurements—as it is shown in Figure 33.Comparison of DFT calculations with experimental results of inelastic X-ray scattering: (a) phonon energies inLa1.85Sr0.15CuO4 and (b) in YBa2Cu3O7; (c) phonon line-widths in La1.85Sr0.15CuO4. DFT calculations [28] give much smaller width than experiments [173–175]. From [178].
(a)(b)(c)In Section2 we will discuss some theoretical approaches related to EPI in strongly correlated systems but without discussing the phonon renormalization. The latter problem was studied in more details in the review articles in [10, 11, 170]. Here, we point out only three (for our purposes) relevant results. First, there is an appreciable difference in the phonon renormalization in strongly and weakly correlated systems. Namely, the change of the phonon frequencies in the presence of the conduction electrons is proportional to the squared coupling constant |gq| and charge susceptibility χc, that is, δω(q)~|gep(q)|2Reχc, while the line-width is given by Γω(q)~|gep(q)|2|Imχc|. All these quantities can be calculated in LDA-DFT and as we discussed above, where for some modes one obtains that Γω(q)(LDA)≪Γω(q)(exp). However, it turns out that in strongly correlated systems doped by holes (with the concentration δ≪1) the charge fluctuations are suppressed in which case the following sum rule holds [10, 11, 171]:
(43)1πN∑q≠0‍∫-∞∞dω|Imχc(q,ω)|=2δ(1-δ)N,
while in the LDA-DFT method one has
(44)1πN∑q≠0∫-∞∞dω|Imχc(q,ω)|(LDA)=(1-δ)N.
The inequality Γω(q)(LDA)≪Γω(q)(exp) (for some phonon modes) together with (43)-(44) means that for low doping δ≪1 the LDA calculations strongly underestimate the EPI coupling constantin the large portion of the Brillouin zone, that is, one has |gep(LDA)(q)|≪|gep(exp)(q)|. The large softening and the large line-width of the half-breathing mode at q=(0.5,0), but very moderate effects for the breathing mode at q=(0.5,0.5), are explained in the framework of the one slave-boson (SB) theory (for U=∞) in [171], where χc(q,ω) (i.e., Γc(p,q)=Γc(p,q)) is calculated in leading O(1/N) order. We stress that there is another method for studying strong correlations—the X-method—where the controllable 1/N expansion is performed in terms of the Hubbard operators and where the charge vertex Γc(p,q) is calculated [6, 78–80, 130, 179, 180]. It turns out that in the adiabatic limit (ω=0) the vertex functions Γc(pF,q) in these two methods have important differences. For instance, Γc(X)(pF,q) (in the X-method) is peaked at q=0—the so called forward scattering peak (FSP)—while Γc(SB)(pF,q) has maximum at finite |q|≠0 [181]—see Section 2.3.5. The enumerated properties of Γc(X)(pF,q) are confirmed by the numerical Monte Carlo calculations in the finite-U Hubbard model [182], where it is found that FSP exists for all U, but it is especially pronounced in the limit U≫t. These results are also confirmed in [183] where the calculations are performed in the four-slave-boson technique—see more in Section 2.3.5. Having in mind this difference it would be useful to have calculations of χc(q,ω) in the framework of the X-method which are unfortunately not done yet. Second, the many-body theory gives that for coupling to some modes the coupling constant |gep(q)| in HTSC cuprates can be significantly larger than the LDA-DFT calculations predict [10, 11], which is due to some many-body effects not present in the latter [169]. In Section 2 it will be argued that for some phonon modes one has |gep(q)|2≫|gep(LDA)(q)|2. For instance, for the half-breathing mode, one has |gep(q)|2≈3|gep(LDA)(q)|2 [10, 11, 169]—see Section 2. These two results point to an inadequacy of LDA-DFT in calculations of EPI effects in HTSC cuprates. Third, the phonon self-energy (Π(q)) and quasiparticle self-energy Σ(k) are differently renormalized by strong correlations [6, 10, 11, 78–80, 130, 179, 180], which is the reason that Π(q) is much more suppressed than Σ(k)—see Section 2. The effects of the charge vertex on Π(q) and Σ(k) are differently manifested. Namely, the vertex function enters quadratically in Σ(k) and the presence of the forward scattering peak in the charge vertex strongly affects the EPI coupling constant gep(q) in Σ(k):
(45)Σ(k)=-∑q|gep(q)γc(k,q)|2D(q)g(k+q),
where g(k)(≡G(k)/Q) is the quasiparticle Green's function, γc(k,q)=Γc(k,q)/Q is the quasiparticle vertex, and Q(~δ) is the Hubbard quasiparticle spectral weight—see Section 2.3. In the adiabatic limit |q|>qω=ωph/vF one has γc(k,q)≈γc(k,q) and for q≫qc(≈δ·π/a) the charge vertex is strongly suppressed (γc(k,q)≪1) making the effective EPI coupling (which also enters the pairing potential) small at large (transfer) momenta q. This has strong repercussion on the pairing due to EPI since for small doping it makes the d-wave pairing coupling constant to be of the order of the s-one (λd≈λs). Then in the presence of the residual Coulomb interaction EPI gives rise to d-wave pairing. On the other side the charge vertex Γc(k,q) enters Π(q)  linearlyand it is additionally integrated over the quasiparticle momentum k—see (42). Therefore, one expects that the effects of the forward scattering peak on Π(q) are less pronounced than on Σ(k). Nevertheless, the peak of Γc(k,q) at q=0 may be (partly) responsible that the maximal experimental softening and broadening of the stretching (half-breathing) mode in La1.85Sr0.15CuO4 and YBa2Cu3O7 is at qhb(exp)=(0.3,0,0) [170] and not at qhb=(0.5,0) for which gep(qhb) reaches maximum. This means that the charge vertex function pushes the maximum of the renormalized EPI coupling constant to smaller momenta q. It would be very interesting to have calculations for other phonons by including the vertex function obtained by the X-method—see Section 2.3.(2) The Phonon Raman Scattering
Thephonon Raman scattering gives an indirect evidence for importance of EPI in cuprates [184–188]. We enumerate some of them—see more in [6] and references therein. (i) There is a pronounced asymmetric line-shape (of the Fano resonance) in the metallic state. For instance, in YBa2Cu3O7 two Raman modes at 115cm-1 (Ba dominated mode) and at 340cm-1 (O dominated mode in the CuO2 planes) show pronounced asymmetry which is absent in YBa2Cu3O6. This asymmetry means that there is an appreciable interaction of Raman active phonons with continuum states (quasiparticles). (ii) The phonon frequencies for some A1g and B1g are strongly renormalized in the superconducting state, between (6–10)%, pointing again to the importance of EPI [188]—see also [6, 37, 38]. To this point we mention that there is a remarkable correlation between the electronic Raman cross-section S̃exp(ω) and the optical conductivity in the a-b plane σab(ω), that is, S̃exp(ω)~σab(ω) [6]. In previous subsections it is argued that EPI with the very broad spectral function α2F(ω) (0<ω≲80meV) explains in a natural way the ω and T dependence of σab(ω). This means that the electronic Raman spectra in cuprates can be explained by EPI in conjunction with strong correlations. This conclusion is supported by the calculations of the Raman cross-section [189] which take into account EPI with α2F(ω) extracted from the tunnelling measurements in YBa2Cu3O6+x and Bi2Sr2CaCu2O8+x [6, 42–54]. Quite similar properties (to cuprates) of the electronic Raman scattering, as well as of σ(ω), R(ω), and ρ(T), were observed in experiments [108] on isotropic 3D metallic oxides La0.5Sr0.5CoO3 and Ca0.5Sr0.5RuO3 where there are no signs of antiferromagnetic fluctuations. This means that low dimensionality and antiferromagnetic spin fluctuations cannot be a prerequisite for anomalous scattering of quasiparticles and EPI must be inevitably taken into account since it is present in all these compounds.


### 1.3.6. Isotope Effect inTc and Σ(k,ω)

The isotope effectαTC in the critical temperature Tc was one of the very important proofs for the EPI pairing mechanism in low-temperature superconductors (LTSCs). As a curiosity the isotope effect in LTSC systems was measured almost exclusively in mono-atomic systems and in few polyatomic systems: the hydrogen isotope effect in PdH, the Mo and Se isotope shift of Tc in Mo6Se8, and the isotope effect in Nb3Sn and MgB2. We point out that very small (αTC≈0 in Zr and Ru) and even negative (in PdH) isotope effects in some polyatomic systems of LTSC materials are compatible with the EPI pairing mechanism but in the presence of substantial Coulomb interaction or lattice anharmonicity. The isotope effect αTC cannot be considered as the smoking gun effect since it is sensitive to numerous influences. For instance, in MgB2 it is with certainty proved that the pairing is due to EPI and strongly dominated by the boron vibrations, but the boron isotope effect is significantly reduced, that is, αTC≈0.3 and the origin for this smaller value is still unexplained. The situation in HTSC cuprates is much more complicated because they are strongly correlated systems and contain many atoms in unit cell. Additionally, the situation is complicated with the presence of intrinsic and extrinsic inhomogeneities, low dimensionality which can mask the isotope effects. On the other hand new techniques such as ARPES, STM, and μSR allow studies of the isotope effects in quasiparticle self-energies, that is, αΣ, which will be discussed below.(1) Isotope Effect αTC in Tc
This problem will be discussed only briefly since more extensive discussion can be found in [6]. It is well known that in the pure EPI pairing mechanism the total isotope coefficient α is given by αTC=∑i,pαi(p)=-∑i,pdlnTc/dlnMi(p), where Mi(p) is the mass of the ith element in the pth crystallographic position. We stress that the total isotope effect is not measured in HTSC cuprates but only some partial ones. Note that, in the case when the screened Coulomb interaction is negligible, that is, μc*=0, the theory predicts αTC=1/2. From this formula one can deduce that the relative change of Tc,δTc/Tc, for heavier elements should be rather small—for instance, it is 0.02 for Ba135→Ba138, 0.03 for Cu63→Cu65, and 0.07 for La138→La139. This means that the measurements of αi for heavier elements are confronted with the ability of the present experimental techniques. Therefore most isotope effect measurements were done by substituting light atoms O16 by O18 only. It turns out that in most optimally doped HTSC cuprates αO is rather small. For instance, αO≈0.02–0.05 in YBa2Cu3O7 with Tc,max≈91K, but it is appreciable in La1.85Sr0.15CuO4 with Tc,max≈35K where αO≈0.1-0.2. In Bi2Sr2CaCu2O8 with Tc,max≈76K one has αO≈0.03–0.05 while αO≈0.03 and even negative (-0.013) in Bi2Sr2Ca2Cu2O10 with Tc,max≈110K. The experiments on Tl2Can-1BaCunO2n+4 (n=2,3) with Tc,max≈121K are still unreliable and αO is unknown. In the electron-doped (Nd1-xCex)2CuO4 with Tc,max≈24K one has αO<0.05 while in the underdoped materials αO increases. The largest αO is obtained even in the optimally doped compounds like in systems with substitution, such as La1.85Sr0.15Cu1-xMxO4, M=Fe,Co, where αO≈1.3 for x≈0.4%. In La2-xMxCuO4 there is a Cu-isotope effect which is of the order of the oxygen one, that is, αCu≈αO giving αCu+αO≈0.25–0.35 for optimally doped systems (x=0.15). In case when x=0.125 with Tc≪Tc,max one hasαCu≈0.8-1 with αCu+αO≈1.8 [190, 191]. The appreciable copper isotope effect in La2-xMxCuO4 tells us that vibrations other than oxygen ions are important in giving high Tc. In that sense one should have in mind the tunnelling experiments discussed above, which tell us that all phonons contribute to the Eliashberg pairing function α2F(k,ω) and according to these results the oxygen modes give moderate contribution to Tc [53, 54]. Hence the small oxygen isotope effect αTc(O) in optimally doped cuprates, if it is an intrinsic property at all (due to pronounced local inhomogeneities of samples and quasi-two-dimensionality of the system), does not exclude the EPI mechanism of pairing.(2) Isotope Effect αΣ in the Self-Energy
The fine structure of the quasiparticle self-energyΣ(k,ω), such as kinks and slopes, can be resolved in ARPES measurements and in some respect in STM measurements. It turns out that there is isotope effect in the self-energy in the optimally doped Bi2212 samples [139, 141, 142]. In the first paper on this subject [139] it is reported a red shiftδωk,70~-(10–15)meV of the nodal kink at ωk,70≃70meV for the O16→O18 substitution. In [139] it is reported that the isotope shift of the self-energy δΣ=Σ16-Σ18~10meV is very pronounced at large energies ω=100–300meV. Concerning the latter result, there is a dispute since it is not confirmed in other experiments [141, 142]. However, the isotope effect in ReΣ(k,ω) at low energies [141, 142] is well described in the framework of the Migdal-Eliashberg theory for EPI [140] which is in accordance with the recent ARPES measurements with low-energy photons ~7 eV [192]. The latter allowed very good precision in measuring the isotope effect in the nodal point of Bi-2212 with Tc16=92.1K and Tc18=91.1K [192]. They observed a shift in the maximum of ReΣ(kN,ω)—at ωk,70≈70meV (it corresponds to the half-breathing or to the breathing phonon)—by δωk,70≈3.4±0.5meV as shown in Figure 34.(a) EffectiveReΣ for five samples for O16 (blue) and O18 (red) along the nodal direction. (b) Effective ImΣ determined from MDC full widths. An impurity term is subtracted at ω=0. From [192].
(a)(b)By analyzing the shift in ImΣ(kN,ω)—shown in Figure 34—one finds similar result for δωk,70≈3.2±0.6meV. The similar shift was obtained in STM measurements [61–63] which is shown in Figure 32(b) and can have its origin in different phonons. We would like to stress two points: (i) in compounds with Tc~100K the oxygen isotope effect in Tc is moderate, that is, αTc(O)<0.1 [192]. If we consider this value to be intrinsic, then even in this case it is not in conflict with the tunnelling experiments [53, 54] since the latter give evidence that vibrations of heavier ions contribute significantly to Tc—see the discussion in Subsection 1.3.4 on the tunnelling spectroscopy. (ii) In ARPES measurements of [192] the effective EPI coupling constant λep,eff≳0.6 is extracted, while the theory in Subsection 1.3.3 gives that the real coupling constant is larger, that is, λep>1.2. This value is significantly larger than the LDA-DFT theory predicts λep,LDA<0.3 [28, 29]. This again points that the LDA-DFT method does not pick up the many-body effects due to strong correlations—see Section 2.


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

(a) Huge Rearrangement of the SFI Spectral Function and Small Change ofTc
Before discussing experimental results in cuprates on the imaginary part of the spin susceptibilityImχ(k,ω) we point out that in the (phenomenological) theories based on the spin-fluctuation interaction (SFI) the quasiparticle self-energy Σ̂sf(k,ωn) (ωn is the Matsubara frequency and τ̂0 is the Nambu matrix) in the normal and superconducting state and the effective (repulsive) pairing potential Vsf(k,ω) (where iωn→ω+iη) are assumed in the form [12–17]
(10)Σ̂sf(k,ωn)=TN∑k′,mVsf(k-k′,ωnm-)τ̂0Ĝ(k′,ωm)τ̂0,Vsf(k,ωnm-)=gsf2∫-∞∞dνπImχ(q,ν+i0+)ν-iωnm-,
where ωnm-≡ωn-ωm. Although the form of Vsf cannot be justified theoretically, except in the weak coupling limit (gsf≪Wb) 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) Imχ(q,ω) is strongly peaked at and near the AF wave vector Q=(π/a,π/a).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 givesImχ(q,ω). In that respect very indicative and impressive IMNS measurements on YBa2Cu3O6+x, which are done by Bourges group [30], demonstrate that the normal-state susceptibility Imχ(odd)(q,ω) (the odd part of the spin susceptibility in the bilayer system) at q=Q=(π,π) is strongly dependent on the hole-doping as it is shown in Figure 2.Figure 2
Magnetic spectral functionImχ(-)(k,ω) in the normal state of YBa2Cu3O6+x at T=100K and at Q=(π,π). 100 counts in the vertical scale correspond to χmax(-)≈350μB2/eV. The superconducting critical temperature Tc(x) by increasing doping (x) from the underdoped system with x=0.5 (top) to the optimally doped one with x=0.97 (bottom): Tc(x)=45K (x=0.5), 85K (x=0.83), 91K (x=0.92), and 92.5K (x=0.97). From [30].The most pronounced result for our discussion is that by varying dopingthere is a huge rearrangement of Imχ(odd)(Q,ω) in the normal state, especially in the energy (frequency) region which might be important for superconducting pairing, let us say 0meV<ω<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 temperatureTc. It is seen in Figure 2 that in the underdoped YBa2Cu3O6.92 crystal Imχ(odd)(Q,ω) and S(Q)=N(0)gsf2∫060dωImχ(odd)(Q,ω) 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 suppressionof Imχ(odd)(Q,ω) 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 λsf(~gsf2), means that the latter is small, that is, λsf(exp)≪1. We stress that the explanation of high Tc in cuprates by the SFI phenomenological theory [12–17] assumes very large SFI coupling energy with gsf(th)≈0.7eV while the frequency (energy) dependence of Imχ(Q,ω) is extracted from the fit of the NMR relaxation rate T1-1 which gives Tc(NMR)≈100K [12–17]. To this point, the NMR measurements (of T1-1) give that there is an anticorrelation between the decrease of the NMR spectral function IQ=limω→0Imχ(NMR)(Q,ω)/ω and the increase of Tc by increasing doping toward the optimal one—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 Imχ(Q,ω) from NMR measurements one takes it from IMNS measurements, as it was done in [82], than for the same value gsf(th) one obtains much smaller Tc. For instance, by taking the experimental values for Imχ(IMNS)(Q,ω) in underdoped YBa2Cu3O6.6 with Tc≈60K one obtains Tc(IMNS)<Tc(NMR)/3 [82], while Tc(IMNS)→50K for gsf(th)≫1. The situation is even worse if one tries to fit the resistivity with Imχ(IMNS)(Q,ω) in YBa2Cu3O6.6 since this fit gives Tc(IMNS)<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 “hourglass” spin excitation spectrum (in the superconducting state) for the momenta q 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 gsf2=(3/2)Ũ2 and Imχ(q,ω) 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 valueŨ=1.59eV, that is, much larger than the one used in [82]. Such a large value of Ũ has been obtained earlier within the Monte Carlo simulation of the Hubbard model [87]. In our opinion this value for Ũ 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 Ũ≪1eV and Tc≪60K. 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–Cu≈[4tpd4/(Δdp+Upd)2,(1/Ud)+2/(Up+2Δ)]—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 Ũ (and gsf).Concerning the problem related to the rearrangement of the SFI spectral functionImχ(Q,ω) in YB2Cu3O6+x [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(q,ω)(~Imχ(q,ω)) (for q 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 Imχ(q,ω) 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 Ũ to the value even larger than 4eV, which is highly improbable.(b) Ineffectiveness of the Magnetic Resonance Peak
A less direct argument forsmallness of the SFI coupling constant, that is,gsfexp≤0.2eV and gsfexp≪gsf, 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, Imχ(Q,ω) is significantly suppressed at low frequencies except near the resonance energy ωres≈41meV where a pronounced narrow peak appears—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)%—see Figure 3. In underdoped cuprates this peak is present also in the normal state as it is seen in Figure 2.Figure 3
Magnetic spectral functionImχ(-)(k,ω) in the superconducting state of YBa2Cu3O6+x at T=5K and at Q=(π,π). 100 counts in the vertical scale correspond to χmax(-)≈350μB2/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 intensityI0 is small [89]. I0 is obtained approximately by equating the condensation energy Econ≈N(0)Δ2/2 with the change of the magnetic energy Emag in the superconducting state, that is, δEmag≈4I0·Emag:
(11)Emag=J∬dωd2k(2π)3(1-coskx-cosky)S(k,ω),
where S(k,ω)=(1/π)[1+n(ω)]Imχ(k,ω) is the spin structure factor and n(ω) is the Bose distribution function. By taking Δ≈2Tc and the realistic value N(0)~1/(10J)~1states/eV·spin, one obtains I0~10-1(Tc/J)2~10-3. However, such a small intensity cannot be responsible for the anomalies in ARPES and optical spectra since it gives rise to small coupling constant λsf,res for the interaction of holes with the resonance peak, that is, λsf,res≈(2I0N(0)gsf2/ωres)≪1. 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 ωres, that is, γres<ωres, where the line width is given by γres=4π(N(0)gsf)2ωres [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 Ũ≈1.6eV, 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 μ(<0.1μB) 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]—see the discussion below.Finally, we would like to point out that the recent magnetic neutron scattering measurements on optimally doped large-volume crystalsBi2Sr2CaCu2O8+δ [93], where the absolute value of Imχ(q,ω) 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 Imχ(ω)=∑qImχ(q,ω) across the superconducting transition Tc=91K, that is, there is only a minimal change in Imχ(ω) between 10K and 100K. Note that if the magnetic excitations were due to itinerant quasiparticles we should have seen dramatic changes of Imχ(ω) as a function of T over the whole energy range. This T-independence of Imχ(ω) strongly opposes the theoretical results in [24–27] which assume that the bosonic spectral function is proportional to Imχ(ω) and that the former can be extracted from optic measurements. Namely, the fitting procedure in [24–27] gives that Imχ(ω) is strongly T-dependent contrary to the experimental results in [93]—see more in Section 1.3.2 on optical conductivity.


## 1.3.2. Optical Conductivity and EPI

Optical spectroscopy gives information onoptical conductivityσ(ω) and on two-particle excitations, from which one can indirectly extract the transport spectral function αtr2F(ω). 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(ω)  is not a directly  measured quantity but it is derived from the reflectivity R(ω)=|(ɛii(ω)-1)/(ɛii(ω)+1)|2 with the transversal dielectric tensor ɛii(ω)=ɛii,∞+ɛii,latt+4πiσii(ω)/ω. Here, ɛii,∞ is the high-frequency dielectric function, ɛii,latt describes the contribution of the lattice vibrations, and σii(ω) describes the optical (dynamical) conductivity of conduction carriers. Since R(ω) is usually measured in the limited-frequency interval ωmin<ω<ωmax, some physical modelling for R(ω) is needed in order to guess it outside this range—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 ωpl—from 0.06 to 25eV—that is, almost three orders of magnitude. However, it turns out that IR measurements of R(ω) in conjunction with elipsometric measurements of ɛii(ω) at high frequencies allow more reliable determination of σ(ω) [94].(1) Transport and Quasiparticle Relaxation Rates
The widespread misconception in studying the quasiparticle scattering in cuprates was an ad hoc assumption that thetransport relaxation rateγtr(ω) is equal to the quasiparticle relaxation rateγ(ω), in spite of the well-known fact that the inequality γtr(ω)≠γ(ω) 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 conductivityσ(ω) consists of two parts, that is, σ(ω)=σinter(ω)+σintra(ω) where σinter(ω) describes interband  transitions which contribute at higher than intraband energies, while σintra(ω) is due to intraband transitions which are relevant at low energies ω<(1–2)eV. (Note that in the IR measurements the frequency is usually given in cm-1, where the following conversion holds: 1cm-1=29.98GHz=0.123985meV=1.44K.) The experimental data for σ(ω)=σ1+iσ2 in cuprates are usually processed by the generalized (extended) Drude formula [32–36, 95]:
(12)σ(ω)=ωp24π1γtr(ω)-iωmtr(ω)/m∞≡iωp24πω̃tr(ω),
where m is the mass of the band electrons while the quantity ω̃tr(ω) 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 ω̃tr(ω) will be discussed below.) Here, i=a,b enumerates the plane axis; ωp,γtr(ω,T), and mop(ω) are the electronic plasma frequency, the transport (optical) scattering rate, and the optical mass, respectively. Very frequently it is analyzed the quantity γtr*(ω,T) given by [95]
(13)γtr*(ω,T)=m∞mtr(ω)γtr(ω,T)=ωImσ(ω)Reσ(ω).
In the weak coupling limit λep<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]
(14)γtr(ω,T)=π∑l∫0∞dναtr,l2Fl(ν)×[2(1+2nB(ν))-2νω-ω+νωnB(ω+ν)+ω-νωnB(ω-ν)].
Here nB(ω) is the Bose distribution function. For completeness we give also the explicit form of the transport mass mtr(ω), see [3–6, 32–36]:
(15)mtr(ω)m∞=1+2ω∑l‍∫0∞dναtr,l2Fl(ν)ReK(ω2πT,ν2πT),
with the Kernel K(x,y)=(i/y)+{((y-x)/x)[ψ(1-ix+iy)-ψ(1+iy)]}-{y→-y} where ψ 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 λep(>1). Note that αtr,l2Fl(ν)≠αl2Fl(ν) and the index l  enumerates all scattering bosons—phonons—spin fluctuations, and so forth. For comparison, the quasiparticle scattering rate γ(ω,T) is given by
(16)γ(ω,T)=2π∫0∞dνα2F(ν)×{2nB(ν)+nF(ν+ω)+nF(ν-ω)}+γimp,
where nF is the Fermi distribution function. For completeness we give also the expression for the quasiparticle effective mass m*(ω):
(17)m*(ω)m=1+1ω∑l∫0∞dναl2Fl(ν)×Re{ψ(12+iω+ν2πT)-ψ(12-iω-ν2πT)}.
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]
(18)γtr(ω)=2πω∫0ωdν(ω-ν)αtr2(ν)F(ν),γ(ω)=2π∫0ωdνα2(ν)F(ν).
In the case of EPI with the constant electronic density of states, the above equations give that γep(ω)=const for ω>ωphmax while γep,tr(ω) (as well as γep,tr*) is monotonic growing for ω>ωphmax, where ωphmax is the maximal phonon frequency. So, the growing of γep,tr(ω) (and γep,tr*) for ω>ωphmax 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(ω,T),γtr(ω,T), and γtr* which are calculated for the EPI spectral function αep2(ω)Fph(ω) extracted from tunnelling experiments in YBCO (with ωphmax~80meV ) [42–45]—see Figure 4.(a) Scattering ratesγ(ω,T),γtr(ω,T), and γtr*—from top to bottom—for the Eliashberg function in (b). From [33–35]. (b) Eliashberg spectral function αep2(ω)Fph(ω) obtained from tunnelling experiments on break junctions [42–45]. Inset shows γtr* with (full line) and without (dashed line) interband transitions [3–5].
(a)(b)The results shown in Figure4 clearly demonstrate the physical difference between two scattering rates γep and γep,tr (or γtr*). It is also seen that γtr*(ω,T) is even more linear function of ω than γtr(ω,T). From these calculations one concludes that the quasilinearity of γtr(ω,T) (and γtr*) 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 γep,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 forth—see more in [3–6] and references therein.Let us discuss briefly the experimental results forR(ω) and γtr*(ω,T) and compare these with theoretical predictions obtained by using a single-band model with αep2(ω)Fph(ω) extracted from the tunnelling data with the EPI coupling constant λep≳2 [42–45]. In the case of YBCO the agreement between measured and calculated R(ω) is very good up to energies ω<6000cm-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 (Bi2212) [97, 98] the experimental results for γtr*(ω,T) are explained theoretically by assuming that the EPI spectral function αep2(ω)F(ω)~Fph(ω), where Fph(ω) is the phononic density of states in BSCO, with λep=1.9 and γimp≈320cm-1—see Figure 5(a). At the same time the fit of γtr*(ω,T) by the marginal Fermi liquid phenomenology fails as it is evident in Figure 5(b).(a) Experimental transport scattering rateγtr* (solid lines) for BSCO and the theoretical curve by using (A.20) and transport mass mtr* with α2F(ω) due to EPI which is described in text (dashed lines). (b) Comparison with the marginal Fermi liquid theory—dashed lines. From [3–5, 99].
(a)(b)Now we will comment the so called pronounced linear behavior ofγtr(ω,T) (and γtr*(ω,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(ω) and derived ones are σ(ω),γtr(ω,T), and mtr(ω), which are very sensitive to the value of the dielectric constant ɛ∞. This sensitivity is clearly demonstrated in Figure 6 for Bi-2212 where it is seen that γtr(ω,T) (and γtr*(ω,T)) for ɛ∞=1 is linear up to much higher ω than in the case ɛ∞>1.Figure 6
Dependence ofγtr*(ω,T) on ɛ∞ in Bi2Sr2CaCu2O8 for different temperatures: ɛ∞=4 (solid lines) and ɛ∞=1 (dashed lines). On the horizontal axis is ω in units cm-1. From [99].However, in some experiments [100–103] the extracted γtr(ω,T) (and γtr*(ω,T)) is linear up to very high ω≈1500cm-1. This means that the ion background and interband transitions (contained in ɛ∞) are not properly taken into account since too small ɛ∞ (≳1) is assumed. The recent elipsometric measurements on YBCO [104] give the value ɛ∞≈4–6, which gives much less spectacular linearity in the relaxation rates γtr(ω,T) (and γtr*(ω,T)) than it was the case immediately after the discovery of HTSC cuprates, where much smaller ɛ∞ was assumed.Furthermore, we would like to comment two points related toσ,γtr, and γ. First, the parametrization of σ(ω) with the generalized Drude formula in (12) and its relation to the transport scattering rate γtr(ω,T) and the transport mass mtr(ω,T) is useful if we deal with electron-boson scattering in a single-band problem. In [36, 96] it is shown that σ(ω) of a two-band model with only elastic impurity scattering can be represented by the generalized (extended) Drude formula with ω and T dependence of effective parameters γtreff(ω,T), mtreff(ω,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(ω,T) and γ(ω,T) are physically different quantities and it holds that γtr(ω,T)≠γ(ω,T). In order to give the physical picture and qualitative explanation for this difference we assume that αtr2F(ν)≈α2F(ν). In that case the renormalized quasiparticle frequency ω̃(ω)=Z(ω)ω=ω-Σ(ω) and the transport one ω̃tr(ω)—defined in (12)—are related and at T=0 they are given by [32, 36]
(19)ω̃tr(ω)=1ω∫0ωdω′2ω̃(ω′).
(For the definition of Z(ω) see Appendix A.) It gives the relation between γtr(ω) and γ(ω) as well asmtr(ω) and m*(ω), respectively:
(20)γtr(ω)=1ω∫0ωdω′γ(ω′),ωmtr(ω)=1ω∫0ωdω′2ω′m*(ω′).
The physical meaning of (19) is the following: in optical measurements one photon with the energy ω is absorbed and two excited particles (electron and hole) are created above and below the Fermi surface. If the electron has energy ω′ and the hole ω-ω′, then they relax as quasiparticles with the renormalized frequency ω̃. Since ω′ takes values 0<ω′<ω, then the optical relaxation ω̃tr(ω) is the energy-averaged ω̃(ω) according to (19). The factor 2 is due to the two quasiparticles—electron and hole. At finite T, the generalization reads [32, 36]
(21)ω̃tr(ω)=1ω∫0∞dω′[1-nF(ω′)-nF(ω-ω′)]2ω̃(ω′).(2) Inversion of the Optical Data andαtr2(ω)F(ω)
In principle, the transport spectral functionαtr2(ω)F(ω) can be extracted from σ(ω) (or γtr(ω)) only at T=0K, which follows from (18) as
(22)αtr2(ω)F(ω)=ωp28π2∂2∂ω2[ωRe1σ(ω)],
or equivalently as αtr2(ω)F(ω)=(1/2π)∂2(ωγtr(ω))/∂ω2. 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 αtr2(ω)F(ω) is the deconvolution of the inhomogeneous Fredholm integral equation of the first kind with the temperature-dependent Kernel K2(ω,ν,T)—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 αtr2(ω)F(ω)  get blurred (most peaks are washed out) in the extraction procedures and, second, the extractedαtr2(ω)F(ω)  be temperature dependent even when the true αtr2(ω)F(ω) 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 αtr2(ω)F(ω) in YBa2Cu3O7-x 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 ω) is present. For instance, the experimental data of R(ω) in YBCO were reproduced by two different spectral functions αtr2(ω)F(ω), 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(ω). The similar situation is realized in optimally doped BSCO as it is seen in Figure 8 where again different functions α2(ω)F(ω) reproduce very well curves for R(ω) and σ(ω). However, it is important to stress that the obtained width of the extracted αtr2(ω)F(ω) in both compounds coincide with the width of the phonon density of states Fph(ω) [33–36, 99]. (2) The upper energy bound for αtr2(ω)F(ω) is extracted in [33–36] and it coincides approximately with the maximal phonon frequency in cuprates ωphmax≲80meV as it is seen in Figures 7 and 8.Figure 7
Experimental (solid lines) and calculated (dashed lines) data ofR(ω) in optimally doped YBCO [105] at T=100,200,300K (from top to bottom). Inset: the two (solid and dashed lines) reconstructed αtr2(ω)F(ω)’s at T=100K. The phonon density of states F(ω)—dotted line in the inset. From [33–35].Figure 8
Experimental (solid line) and calculated (dashed line) data ofR(ω) in optimally doped BSCO [106] at T=100K. Inset: the reconstructed αtr2(ω)F(ω)—solid line. The phonon density of states F(ω)—dotted line. From [33–35].These results demonstrate the importance of EPI in cuprates [33–36]. We point out that the width of αtr2(ω)F(ω) which is extracted from the optical measurements [33–36] coincides with the width of the quasiparticle spectral function α2(ω)F(ω) obtained in tunnelling and ARPES spectra (which we will discuss below), that is, both functions are spread over the energy interval 0<ω<ωphmax(≲80meV). Since in cuprates this interval coincides with the width in the phononic density of states F(ω) and since the maxima of α2(ω)F(ω) and F(ω) 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 αtr2(ω)F(ω) 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(ω) 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 Δ(k,ω). Second, if R(ω)’s (and σ(ω)’s) in cuprates are due to some other bosonic scattering which is pronounced up to much higher energies ωc≫ωphmax, this should be seen in the width of the extracted spectral function αtr2(ω)F(ω). In that respect in [25–27] it is assumed that SFI dominates in the quasiparticle scattering and that αtr2(ω)F(ω)~gsf2Imχ(ω) where Imχ(ω)=∫d2kχ(k,ω). This claim is based on reanalyzing of some IR measurements [25–27] and the transport spectral function αtr2(ω)F(ω) 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 αtr2(ω)F(ω), which is an artefact and due to the chosen method, in [25] it is assumed that αtr2(ω)F(ω) has a rather large tail at large energies—up to 400 meV. It turns out that even such an assumption in extracting αtr2(ω)F(ω) does not reproduce the experimental curve Imχ(ω) [107] in some important aspects. First, the relative heights of the two peaks in the extracted spectral function αtr2(ω)F(ω) at lower temperatures are opposite to the experimental curve Imχ(ω) [107]—see [25, Figure  1]. Second, the strong temperature dependence of the extracted αtr2(ω)F(ω) 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 IB(ω)=αtr2(ω)F(ω) in [25] has much smaller values at larger frequencies (ω>100meV) than it is the case for the measured Imχ(ω), that is, (IB(ω>100meV)/IB(ωmax))≪Imχ(ω>100meV)/Imχ(ωmax)—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 Imχ(q,ω) 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 Imχ(ω)=∑qImχ(q,ω) is temperature independent across the superconducting transition Tc=91K, that is, there is only a minimal change in Imχ(ω) between 10K and 100K. This T-independence of Imχ(ω) strongly opposes the (above discussed) results in [24–27], where the fit of optic measurements gives strong T dependence of Imχ(ω).Fifth, the transport coupling constant λtr extracted in [25] is too large, that is, λtr>3 contrary to the previous findings that λtr≲1.5 [33–36, 99]. Since in HTSC one has λ>λtr, this would probably give λ≈6, which is not confirmed by other experiments. Sixth, the interpretation of αtr2(ω)F(ω) in LSCO and BSCO solely in terms of Imχ(ω) 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 Imχ(Q,ω) is small in the normal state and its magnitude is even below the experimental noise. This means that if the assumption that αtr2(ω)F(ω)≈gsf2Imχ(ω) were correct then the contribution to Imχ(ω) from the momenta 0<k≪Q would be dominant, which is detrimental for d-wave superconductivity.Finally, we point out that very similar (to cuprates) properties, ofσ(ω),R(ω) (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 Ba1-xKxBiO3 which is superconducting below Tc≃30K at x=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
Therestricted  optical sum rulewas studied intensively in HTSC cuprates. It shows some peculiarities not present in low-temperature superconductors. It turns out that the restricted spectral weight W(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 W(Ωc,T)=W(0)-βT2 in the normal state can be explained in a natural way by the T dependence of the EPI transport relaxation rate γtrep(ω,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(ω). The first one is the total sum rule which in the normal state reads
(23)∫0∞dωσ1N(ω)=ωpl28=πne22m,
while in the superconducting state it is given by the Tinkham-Ferrell-Glover (TFG) sum rule
(24)∫0∞dωσ1S(ω)=c28λL2+∫+0∞dωσ1S(ω)=ωpl28.
Here,n is the total electron density, e is the electron charge, m is the bare electron mass, and λL is the London penetration depth. The first (singular) term c2/8λL2 in (24) is due to the superconducting condensate which contributes σ1,condS(ω)=(c2/4λL2)δ(ω). The total sum rule represents the fundamental property of matter—the conservation of the electron number. In order to calculate it one should use the total Hamiltonian Ĥtot=T̂e+Ĥint where all electrons, electronic bands, and their interactions Ĥint (Coulomb, EPI, with impurities, etc.) are accounted for. Here, Te is the kinetic energy of bare electrons:
(25)T̂e=∑σ‍∫d3xψ̂σ†(x)p̂22mψ̂σ(x)=∑p,σp22meĉpσ†ĉpσ.Thepartial sum rule is related to the energetics solely in the conduction (valence) bandwhich is described by the Hamiltonian of the conduction (valence) band electrons:
(26)Ĥv=∑p,σξpĉv,pσ†ĉv,pσ+V̂v,c.Ĥv contains the band energy with the dispersion ϵp (ξp=ϵp-μ) and the effective Coulomb interaction of the valence electrons V̂v,c. In this case the partial sum rule in the normal state reads [110] (for a general form of ϵp)
(27)∫0∞dωσ1,vN(ω)=πe22V∑p‍〈n̂v,p〉Hvmp,
where the number operator n̂v,p=∑σĉpσ†ĉpσ;1/mp=∂2ϵp/∂px2 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 ω 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
(28)W(Ωc,T)=∫0Ωcdωσ1,vN(ω)=π2[Kd+Π(0)]-∫0ΩcdωImΠ(ω)ω,
where Kd is the diamagnetic Kernel given by (30) below and Π(ω) is the paramagnetic (current-current) response function. In the perturbation theory without vertex correction Π(iωn) (at the Matsubara frequency ωn=πT(2n+1)) is given by [39, 40]
(29)Π(iω)=2∑p(∂ϵp∂p)2∑ωmG(p,iωnm+)G(p,iωm),
where ωnm+=ωn+ωm and G(p,iωn)=(iωn-ξp-Σ(p,iωn))-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 Wb<Ωc<Eg, in this region one has approximately ImΠ(ω)≈0 and the limit Ωc→∞ in (28) is justified. In that case one has Π(0)≈∫0Ωc(ImΠ(ω)/ω)dω which gives the approximate formula for W(Ωc,T):
(30)W(Ωc,T)=∫0Ωcdωσ1,vN(ω)≈π2Kd=e2π∑p∂2ϵp∂p2np,
where np=〈n̂v,p〉 is the quasiparticle distribution function in the interacting system. Note that W(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 W(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 ϵp=-2t(cospxa+cospya) which gives 1/mp=-2ta2cospxa. It is straightforward to show that in this case (30) is reduced to a simpler one:
(31)W(Ωc,T)=∫0Ωcdωσ1,vN(ω)≈πe2a22V〈-Tv〉,
where 〈Tv〉Hv=∑pϵp〈nv〉Hv is the average kinetic energy of the band electrons, a is the Cu–Cu lattice distance, and V is the volume of the system. In this approximation W(Ωc,T) is a direct measure of the average band (kinetic) energy. In thesuperconducting state the partial band sum rule reads (32)Ws(Ωc,T)=c28λL2+∫+0Ωcdωσ1,vS(ω)=πe2a22V〈-Tv〉s.
In order to introduce the reader to (the complexity of) the problem of the T dependence of W(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 Wn(Ωc,T) increases with the decrease of the temperature, that is, Wn(Ωc,T)=Wn(0)-βbT2 where βb~1/Wb. 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 W(Ωc,T). We stress that the smearing of fp by temperature lowers the spectral weight compared to that at T=0K, that is, Wn(Ωc,T)<Wn(Ωc,0). In that respect it is not surprising that there is a lowering of Ws(Ωc,T) in the BCS superconducting state, WsBCS(Ωc,T≪Tc)<Wn(Ωc,T≪Tc) since at low temperatures fp is smeared mainly due to the superconducting gap, that is, fp=[1-(ξp/Ep)th(Ep/2T)]/2, Ep=ξp2+Δ2, ξp=ϵp-μ. The maximal decrease of Ws(Ωc,T) is at T=0.Let us enumerate and discuss themain experimental results forW(Ωc,T) in HTSC cuprates. (1) In the normal state (T>Tc) of most cuprates, one has Wn(Ωc,T)=Wn(0)-βexT2 with βexp≫βb, that is, Wn(Ωc,T) is increasing by decreasing T, even at T below T*—the temperature for the opening of the pseudogap. The increase of Wn(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 Ws(Ωc,T) with respect to that in the normal state, that is, Ws(Ωc,T)>Wn(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 〈Tv〉 is negative (〈Tv〉<0) and in the standard BCS case (32) gives that Ws(T<Tc)  decreases due to the increase of 〈Tv〉. Therefore the experimental increase of Ws(T<Tc) by decreasing T is called the non-BCS behavior. The latter means a lowering of the kinetic energy 〈Tv〉 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.Figure 9
Measured spectral weightWs(Ωc,T)(~Al+D in figures) for Ωc≈1.25eV 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 ofWs(Ωc,T) at T<Tc (Ws(Ωc,T)<Wn(Ωc,T)) which is the BCS-like behavior [112] as it is seen in Figure 10.(a) Spectral weightWn(Ωc,T) of the overdoped Bi2212 for Ωc=1eV. Closed symbols—normal state. Open symbols—superconducting state. (b) Change of the kinetic energy ΔEkin=Ekin,S-Ekin,N in meV per Cu site versus the charge p per Cu with respect to the optimal value popt. From [112].
(a)(b)We stress that the non-BCS behavior ofWs(Ωc,T) for underdoped (and in some optimally doped) systems was obtained by assuming that Ωc≈(1–1.2)eV. However, in [104] these results were questioned by claiming that the conventional BCS-like behavior was observed (Ws(Ωc,T)<Wn(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 Ws(Ωc,T)<Wn(Ωc,T) was observed, while in LSCO it was found that Ws(Ωc,T)≈const, that is, Ws(Ωc,T<Tc)≈Wn(Ωc,Tc).The first question is the following. How to explain the strong temperature dependence ofW(Ωc,T) in the normal state? In [39, 40] W(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 W(Ωc,T). The main theoretical results of [39, 40] are the following: the calculations of W(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, Wn(Ωc,T)≈Ws(Ωc,T) since Wn(Ωc,T)-Ws(Ωc,T)~Δ2/Ωc2. (2) In the case of large Ωc the calculations based on (30) give
(33)W(Ωc,T)≈ωpl28[1-γ(T)Wb-π22T2Wb2].
In the case when EPI dominates one has γ=γep(T)+γimp where γep(T)=∫0∞dzα2(z)F(z)coth(z/2T). It turns out that for α2(ω)F(ω), shown in Figure 4, one obtains (i) γep(T)~T2 in the temperature interval 100K<T<200K as it is seen in Figure 11 for the T dependence of W(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 λep,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(ω)F(ω). It is seen in Figure 11 that, for a softer Einstein spectrum (with ΩE=200K), W(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 W(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 W(Ωc,T) was observed in experiments in [104, 111, 113, 114].Figure 11
Spectral weightW(Ωc,T) in the normal state for Einstein phonons with ΩE=200K (full triangles) and ΩE=400K (open circles, left axis). Dashed lines show T2 asymptotic. From [40].To summarize, the above analysis demonstrates that the theory based on EPI is able to explain in a satisfactory way the temperature behavior ofW(Ωc,T) above and below Tc in systems at and near the optimal doping.(4) α2(ω)F(ω) and the In-Plane Resistivity ρab(T)
The temperature dependence of the in-plane resistivityρab(T) in cuprates is a direct consequence of the quasi-2D motion of quasiparticles and of the inelastic scattering which they experience. At present, there is no consensus on the origin of the linear temperature dependence of the in-plane resistivity ρab(T) in the normal state. Our intention is not to discuss this problem, but only to demonstrate that the EPI spectral function α2(ω)F(ω), which is obtained from tunnelling experiments in cuprates (see Section 1.3.4), is able to explain the temperature dependence of ρab(T) in the optimally doped YBCO. In the Boltzmann theory ρab(T) is given by
(34)ρab(T)=ρimp+4πωp2γtr(T),
where
(35)γtr(T)=πT∫0∞dωωsinh2(ω/2T)αtr2(ω)F(ω).
The residual resistivity ρimp is due to the impurity scattering. Since ρ(T)=1/σ(ω=0,T) and having in mind that the dynamical conductivity σ(ω,T) in YBCO (at and near the optimal doping) is satisfactory explained by the EPI scattering, then it is to expect that ρab(T) is also dominated by EPI in some temperature region T>Tc. This is indeed confirmed in the optimally dopedYBCO, where ρimp is chosen appropriately and the spectral function αtr2(ω)F(ω) is taken from the tunnelling experiments in [42–45]. The very good agreement with the experimental results [118] is shown in Figure 12. We stress that in the case of EPI there is always a temperature region where γtr(T)~T for T>αΘD, α<1 depending on the shape of αtr2(ω)F(ω) (for the simple Debye spectrum α≈0.2). In the linear regime one has ρ(T)≃ρimp+8π2λep,tr(kBT/ℏωp2)=ρimp+ρ′T.(a) Calculated resistivityρ(T) for the EPI spectral function αtr2(ω)F(ω) in [118]. (b) Measured resistivity in a(x)- and b(y)-crystal direction of YBCO [119] and calculated Bloch-Grüneisen curve (points) for λep=1 [120].
(a)(b)There is experimental constraint onλtr since λtr≈0.25ωpl2(eV)ρ′(μΩcm/K). For instance, for ωpl≈(2-3)eV [108] and ρ′≈0.6 in the oriented YBCO films and ρ′≈0.3-0.4 in single crystals of BSCO, one obtains λtr≈0.6–1.4. In case of YBCO single crystals, there is a pronounced anisotropy in ρab(T) [119] which gives ρx′(T)=0.6μΩcm/K and ρy′(T)=0.25μΩcm/K. The function λtr(ωpl) is shown in Figure 13 where the plasma frequency ωpl can be calculated by LDA-DFT and also extracted from the width (~ωpl*) of the Drude peak at small frequencies, where ωpl=ɛ∞ωpl*. We stress that the rather good agreement of theoretical and experimental results for ρab(T), in some optimally doped HTSC cuprates such as YBCO, should not be overinterpreted in the sense that the above rather simple electron-phonon approach can explain the resistivity in other HTSC cuprates and for various doping. For instance, in highly underdoped systems ρab(T) is very different from the behavior in Figure 12 and the simple Migdal-Eliashberg theory based on the EPI spectral function is inadequate. In this case one should certainly take into account polaronic effects [8–11], strong correlations, and so forth. The above analysis on the resistivity in the optimally doped YBCO demonstrates only that in this case if in (35) one uses the EPI spectral function α2(ω)F(ω) obtained from the tunnelling experiments (and optics) one obtains the correct T dependence of ρa,b(T). This result is an additional evidence for the importance of EPI.Figure 13
Transport EPI spectral function coupling constant in YBCO as a function of plasma frequencyωp as derived from the experimental slope of resistivity ρ′(T).λx for ρx′(T)=0.6μΩcm/K and λy for ρy′(T)=0.25μΩcm/K [119]. Squares are LDA values [121].Concerning the temperature dependence of the resistivity in other (than YBCO) families of the optimally doped HTSC cuprates we would like to point out that there is some evidence that the linear (inT) resistivity is observed in some of them even at temperatures T<0.2ΘD [122, 123]. This possibility is argued also theoretically in [124] where it is shown that in two-dimensional systems with a broad interval of phonon spectra the quasilinear behavior of ρab(T) is realized even at T<0.2ΘD. The quasilinear behavior of the resistivity at T≪0.2ΘD has been observed in Bi2(Sr0.97Pr0.003)2CuO6 [125], in LSCO, and in 1-layer Bi-2201 [122, 123, 126, 127], where in all these systems the critical temperature is rather small, Tc≈10K. In that respect all existing theories based on the electron-boson scattering are plagued and having difficulties to explain this low-temperature behavior of ρab(T). To this point, we would like to emphasize here that some of these (experimental) observations are contradictory. For example, the results obtained by the Vedeneev group [127] show that some samples demonstrate the quasilinear behavior of the resistivity up to T=10K but some others with approximately the same Tc have the usual Bloch-Grüneisen-type behavior characteristic for the EPI scattering. In that respect it is very unlikely that the linear resistivity up to T=10K can be simply explained in the standard way by interactions of electrons with some known bosons either by phonons or spin fluctuations (magnons). The question why in some cuprates the linear resistivity is observed up to T=10K is still a mystery and its explanation is a challenge for all kinds of the electron-boson scattering, not only for EPI. In that respect it is interesting to mention that the existence of the forward scattering peak in EPI (with the width qc≪kF), which is due to strong correlations, may give rise to the linear behavior of ρ(T) down to very low temperatures T~ΘD/30 [6, 128, 129]—see more in Section 2.3.4, item (6).  We will argue in Section 1.3.4 that if one interprets the tunnelling experiments in systems near optimal doping [42–54] in the framework of the Eliashberg theory one obtains the large EPI coupling constant λep≈2–3.5 which implies that λtr~(λ/3). This means that EPI is reduced much more in transport properties than in the self-energy. We stress that such a large reduction of λtr cannot be obtained within the LDA-DFT band-structure calculations, which means that λep and λtr contain renormalization which do not enter in the LDA-DFT theory. In Section 2 we will argue that the strong suppression of λtr may have its origin in strong electronic correlations [78–80, 130] and in the long-range Madelung energy [3–6].(5) Femtosecond Time-Resolved Optical Spectroscopy
The femtosecond time-resolved optical spectroscopy (FTROS) has been developed in the last couple of years and applied to HTSC cuprates. In this method a femtosecond (1fs=10-15sec) laser pump excites in materials electron-hole pairs via interband transitions. These hot carriers release their energy via electron-electron (with the relaxation time τee) and electron-phonon scattering reaching states near the Fermi energy within 10–100fs—see [131]. The typical energy density of the laser pump pulses with the wavelength λ≈810 nm (ℏω=1.5eV) was around F~1μJ/cm2 (the excitation fluenceF) which produces approximately 3×1010 carriers per pulse (by assuming that each photon produces ℏω/Δ carriers, Δ is the superconducting gap). By measuring photoinduced changes of the reflectivity in time, that is, ΔR(t)/R0, one can extract information on the relaxation dynamics of the low-laying electronic excitations. Since ΔR(t) relax to equilibrium, the fit with exponential functions is used as
(36)ΔR(t)R0=f(t)[Ae-t/τA+Be-t/τB+⋯],
where f(t)=H(t)[1-exp{-t/τee}] (H(t) is the Heavyside function) describes the finite rise-time. The parameters A, B depend on the fluence F. This method was used in studying the superconducting phase of La2-xSrxCuO4, with x=0.1 and 0.15 and Tc=30K and 38K, respectively [41]. In that case one has A≠0 for T<Tc and A=0 for T>Tc, while the signal B was present also at T>Tc. It turns out that the signal A is related to the quasiparticle recombination across the superconducting gap Δ(T) and has a relaxation time of the order τA>10ps at T=4.5K. At the so called threshold fluence (FT=4.2±1.7μJ/cm2 for x=0.1 and FT=5.8±2.3μJ/cm2 for x=0.15) the vaporization (destroying) of the superconducting phase occurs, where the parameter A saturates. This vaporization process takes place at the time scala τr≈0.8ps. The external fluence is distributed in the sample over the excitation volume which is proportional to the optical penetration depth λop(≈150 nm at λ≈810nm) of the pump. The energy densities stored in the excitation volume at the vaporization threshold for x=0.1 and x=0.15 are Up=FT/λop=2.0±0.8K/Cu and 2.6±1.0K/Cu, respectively. The important fact is that Up is much larger than the superconducting condensation energy which is Ucond≈0.12K/Cu for x=0.1 and Ucond≈0.3K/Cu for x=0.15, that is, Up≫Ucond. This means that the energy difference Up-Ucond must be stored elsewhere on the time scale τr. The only present reservoir which can absorb the difference in energy is the bosonic baths of phonons and spin fluctuations. The energy required to heat the spin reservoir from T=4.5K to Tc is Usf=∫TTcCsf(T)dT. The measured specific heat Csf(T) in La2CuO4 [41] gives very small value Usf≈0.01K. In the case of the phonon reservoir one obtains Uph=∫TTcCph(T)dT=9K/Cu for x=0.1 and 28K/Cu for x=0.15, where Cph is the phonon specific heat. Since Usf≪Up-Ucond, the spin reservoir cannot absorb the rest energy Up-Ucond. The situation is opposite with phonons since Uph≫Up-Ucond  and phonon can absorb the rest energy in the excitation volume. The complete vaporization dynamics can be described in the framework of the Rothwarf-Taylor model which describes approaching of electrons and phonons to quasiequilibrium on the time scale of 1 ps [132]. We will not go into details but only summarize by quoting the conclusion in [132] that only phonon-mediated vaporization is consistent with the experiments, thus ruling out spin-mediated quasiparticle recombination and pairing in HTSC cuprates. The FTROS method tells us that at least for nonequilibrium processes EPI is more important than SFI. It gives also some opportunities for obtaining the strength of EPI but at present there is no reliable analysis.In conclusion, optics and resistivity measurements in the normal state of cuprates give evidence that EPI is important while the spin-fluctuation scattering is weaker than it is believed. However, some important questions related to the transport properties remain to be answered. (i) What are the values of λtr and ωpl? (ii) What is the reason that λtr≪λ is realized in cuprates? (iii) What is the role of the Coulomb scattering in σ(ω) and ρ(T)? Later on we will argue that ARPES measurements in cuprates give evidence for an appreciable contribution of the Coulomb scattering at higher frequencies, where γ(ω)≈γ0+λcω for ω>ωmaxph with λc~1. One should stress that despite the fact that EPI is suppressed in transport properties it is sufficiently strong in the quasiparticle self-energy, as it comes out from tunnelling measurements discussed below.


## 1.3.3. ARPES and the EPI Self-Energy

The angle-resolved photoemission spectroscopy (ARPES) is nowadays one of leading spectroscopy methods in the solid-state physics [22, 23]. In some favorable conditions it provides direct information on the one-electron removal spectrum in a complex many-body system. The method involves shining light (photons) with energies between Ei=5–1000eV on samples and by detecting momentum (k)- and energy(ω)-distribution of the outgoing electrons. The resolution of ARPES has been significantly increased in the last decade with the energy resolution of ΔE≈1-2meV (for photon energies ~20 eV) and angular resolution of Δθ≲0.2∘. On the other side the ARPES method is surface-sensitive technique, since the average escape depth (lesc) of the outgoing electrons is of the order of lesc~10 Å, depending on the energy of incoming photons. Therefore, very good surfaces are needed in order that the results be representative for bulk samples. The most reliable studies were done on the bilayer Bi2Sr2CaCu2O8 (Bi2212) and its single-layer counterpart Bi2Sr2CuO6 (Bi2201), since these materials contain weakly coupled BiO planes with the longest interplane separation in the cuprates. This results in a natural cleavage plane making these materials superior to others in ARPES experiments. After a drastic improvement of sample quality in other families of HTSC materials, the ARPES technique has became an important method in theoretical considerations. The ARPES can indirectly give information on the momentum and energy dependence of the pairing potential. Furthermore, the electronic spectrum of the (abovementioned) cuprates is highly quasi-2D which allows rather unambiguous determination of the initial state momentum from the measured final state momentum, since the component parallel to the surface is conserved in photoemission. In this case, the ARPES probes (under some favorable conditions) directly the single-particle spectral function A(k,ω). In the following we discuss mainly those ARPES experiments which give evidence for the importance of the EPI in cuprates—see more in [22, 23].ARPES measures a nonlinear response function of the electron system and it is usually analyzed in the so-calledthree-step model, where the total photoemission intensity Itot(k,ω)≈I1·I2·I3 is the product of three independent terms: (1) I1 that describes optical excitation of the electron in the bulk, (2) I2 that describes the scattering probability of the travelling electrons, and (3) I3 that describes the transmission probability through the surface potential barrier. The central quantity in the three-step model is I1(k,ω) and it turns out that for k=k∥ it can be written in the form I1(k,ω)≃I0(k,υ)f(ω)A(k,ω) [22, 23] with I0(k,υ)~|〈ψf|pA|ψi〉|2 and the quasiparticle spectral function A(k,ω)=-ImG(k,ω)/π:
(37)A(k,ω)=-1πImΣ(k,ω)[ω-ξ(k)-ReΣ(k,ω)]2+ImΣ2(k,ω).
Here, 〈ψf|p·A|ψi〉 is the dipole matrix element which depends on k, polarization, and energy Ei of the incoming photons. The knowledge of the matrix element is of a great importance and its calculation from first principles was done in [133]. f(ω) is the Fermi function; G and Σ=ReΣ+iImΣ are the quasiparticle Green's function and the self-energy, respectively. We summarize and comment here some important ARPES results which were obtained in the last several years and which confirm the existence of the Fermi surface and importance of EPI in the quasiparticle scattering [22, 23].ARPES in the Normal State
(N1) There is well-defined Fermi surface in the metallic state of optimally and near optimally doped cuprates with the topology predicted by the LDA-DFT. However, the bands are narrower than LDA-DFT predicts which points to a strong quasiparticle renormalization. (N2) The spectral lines are broad with |ImΣ(k,ω)|~ω (or ~T for T>ω) which tells us that the quasiparticle liquid is a noncanonical Fermi liquid for larger values of T,ω. (N3) There is a bilayer band splitting in Bi2212 (at least in the overdoped state), which is also predicted by LDA-DFT. In the case when the coherent hopping t⊥ between two layers in the bilayer dominates, then the antibonding and bonding bands ξka,b=ξk±tk⊥ with tk⊥=[t⊥(cos2kx-cos2ky)+⋯] have been observed. It is worth to mention that the previous experiments did not show this splitting which was one of the reasons for various speculations on some exotic electronic scattering and non-Fermi liquid scenarios. (N4) In the underdoped cuprates and at temperatures Tc<T<T* there is a d-wave-like pseudogap Δpg(k)~Δpg,0(coskx-cosky) in the quasiparticle spectrum where Δpg,0 increases by lowering doping. We stress that the pseudogap phenomenon is not well understood at present and since we are interested in systems near optimal doping where the pseudogap phenomena are absent or much less pronounced we will not discuss this problem here. Its origin can be due to a precursor superconductivity or due to a competing order, such as spin- or charge-density wave, strong correlations, and so forth. (N5) The ARPES self-energy gives evidence that EPI interaction is rather strong. The arguments for the latter statement are the following: (i) at T>Tc there are kinks in the quasiparticle dispersion ω(ξk) in the nodal direction (along the (0,0)-(π,π) line) at the characteristic phonon energy ωph(70)~(60–70)meV [91], see Figure 14 (top), and near the antinodal point(π,0) at 40meV [134]—see Figure 14 (bottom).Figure 14
(top) Quasiparticle dispersion ofBi2212, Bi2201, and LSCO along the nodal direction, plotted versus the momentum k for (a)–(c) different doping, and (d)-(e) different T; black arrows indicate the kink energy; the red arrow indicates the energy of the q=(π,0) oxygen stretching phonon mode; inset of (e) shows T-dependent Σ′ for optimally doped Bi2212; (f) shows doping dependence of the effective coupling constant λ′ along (0,0)-(π,π) for the different HTSC oxides. From [91]. (bottom) Quasiparticle dispersion E(k) in the normal state (a1, b1, c), at 107 K and 115 K, along various directions ϕ around the antinodal point. The kink at E=40meV is shown by the horizontal arrow. (a2 and b2) are E(k) in the superconducting state at 10 K with the shifted kink to 70meV. (d) kink positions as a function of ϕ in the antinodal region. From [134].(ii) The kink structure is observed in a variety of the hole-doped cuprates such asLSCO, Bi2212, Bi2201, Tl2201 (Tl2Ba2CuO6), Na–CCOC (Ca2-xNaxCuO2Cl2). These kinks exist also above Tc, which excludes the scenario with the magnetic resonance peak in Imχs(Q,ω). Moreover, since the tunnelling and magnetic neutron scattering measurements give small SFI coupling constant gsf<0.2eV, then the kinks are not due to SFI. (iii) The position of the nodal kink is practically doping independent which points towards phonons as the scattering and pairing boson. (N6) The quasiparticles (holes) at and near the nodal-point kN couple practically to a rather broad spectrum of phonons since at least three groups of phonons were extracted in the bosonic spectral function α2F(kN,ω) from the ARPES effective self-energy in La2-xSrxCuO4 [135]—Figure 15.(a) Effective real self-energy for the nonsuperconductingLa2-xSrxCuO4, x=0.03. Extracted αeff2(ω)F(ω) is in the inset. (b) Top: the total MDC width—open circles. Bottom: the EPI contribution shows saturation, impurity contribution—dotted black line. The residual part is growing ~ω1.3. From [135].
(a)(b)The latter result is in a qualitative agreement with numerous tunnelling measurements [42–54] which apparently demonstrate that the very broad spectrum of phonons couples with holes without preferring any particular phonons—see discussion below. (N7) Recent ARPES measurements in Bi2212 [92] show very different slope dω/dξk of the quasiparticle energy ω(ξk) at small |ξk|≪ωph and at large energies |ξk|≫ωph—see Figure 16. The theoretical analysis [137] of these results gives the total coupling constant λZ=λepZ+λcZ≈3, and for the EPI coupling λepZ≈2, while the Coulomb coupling (SFI is a part of it) is λcZ≈1 [137]—see Figure 16. (Note that the upper index Z in the coupling constants means the quasiparticle renormalization in the normal part of the self-energy.) To this end let us mention some confusion which is related to the value of the EPI coupling constant extracted from ARPES. Namely, [22, 23, 138, 139] the EPI self-energy was obtained by subtracting the high-energy slope of the quasiparticle spectrum ω(ξk) at ω~0.3eV. The latter is apparently due to the Coulomb interaction. Although the position of the low-energy kink is not affected by this procedure (if ωphmax≪ωc), this subtraction procedure gives in fact an effective EPI self-energyΣeffep(k,ω) andthe effective coupling constantλep,effZ(k) only. We demonstrate below that the λep,effZ(k) is smaller than the real EPI coupling constant λepZ(k). The total self-energy is Σ(k,ω)=Σep(k,ω)+Σc(k,ω) where Σc is the contribution due to the Coulomb interaction. At very low energies ω≪ωc one has usually Σc(k,ω)=-λcZ(k)ω, where ωc(~1 eV) is the characteristic Coulomb energies and λcZ is the Coulomb coupling constant. The quasiparticle spectrum ω(k) is determined from the condition ω-ξ(k)-Re[Σep(k,ω)+Σc(k,ω)]=0 where ξ(k) is the bare band-structure energy. At low energies ω<ωphmax≪ωc it can be rewritten in the form
(38)ω-ξren(k)-ReΣeffep(k,ω)=0,
with ξren(k)=[1+λcZ(k)]-1ξ(k),
(39)ReΣeffep(k,ω)=ReΣeffep(k,ω)1+λcZ(k).
Since at very low energies ω≪ωphmax one has ReΣep(k,ω)=-λepZ(k)ω and ReΣeffep(k,ω)=-λep,effZ(k)ω, then the real coupling constant is related to the effective one by
(40)λepZ(k)=[1+λcZ(k)]λep,effZ(k).
As a result one has λepZ(k)>λep,effZ(k). At higher energies ωphmax<ω<ωc, where the EPI effects are suppressed and Σep(k,ω) stops growing, one has ReΣ(k,ω)≈ReΣep(k,ω)-λcZ(k)ω. The measured ReΣexp(k,ω) at T=10K near and slightly away from the nodal point in the optimally doped Bi-2212 with Tc=91K [136] is shown in Figure 16.Figure 16
Figure  4b from [136]: ReΣ(ω) measured in Bi2212 (thin line) and model ReΣ(ω) (bold line) obtained in [136]. The three thin lines (λ1,λ2,λ3) are the slopes of ReΣ(ω) in different energy regions—see the text.It is seen thatReΣexp(k,ω) has two kinks—the first one at low energyω1≈ωphhigh≈50–70meV which is (as we already argued) most probably of the phononic origin [22, 23, 138, 139], while the second kink at higher energyω2≈ωc≈350meV which is due to the Coulomb interaction. However, the important results in [136] are that the slopes of ReΣexp(k,ω) at low (ω<ωphhigh) and high energies (ωphhigh<ω<ωc) are very different. The low-energy and high-energy slope near the nodal point are shown in Figure 16 schematically (thin lines). From Figure 16 it is obvious that EPI prevails at low energies ω<ωphhigh. More precisely digitalization of ReΣexp(k,ω) in the interval ωphhigh<ω<0.4eV gives the Coulomb coupling λcZ≈1.1 while the same procedure at 20meV≈ωphlow<ω<ωphhigh≈50–70meV gives the total coupling constant (λ2≡)λZ=λepZ+λcZ≈3.2 and the EPI coupling constant λepZ(≡λep,highZ)≈2.1>2λep,effZ(k), that is, the EPI coupling is at least twice larger than the effective EPI coupling constant obtained in the previous analysis of ARPES results [22, 23, 138, 139]. This estimation tells us that at (and near) the nodal point, the EPI interaction dominatesin the quasiparticle scattering at low energies since λepZ(≈2.1)≈2λzc>2λsfZ, while at large energies (compared to ωph) the Coulomb interaction with λcZ≈1.1 dominates. We point out that EPI near the antinodal point can be even larger than in the nodal point, mostly due to the higher density of states near the antinodal point. (N8) Recent ARPES spectra in the optimally doped Bi2212 near the nodal and antinodal point [139] show a low-energy isotope effect in ReΣexp(k,ω), which can be well described in the framework of the Migdal-Eliashberg theory for EPI [140]. At higher energies ω>ωph obtained in [139] very pronounced isotope effect cannot be explained by the simple Migdal-Eliashberg theory [140]. However, there are controversies with the strength of the high-energy isotope effect since it was not confirmed in other measurements [141, 142]—see the discussion in Section 1.3.6(2) related to the isotope effects in HTSC cuprates.  (N9) The ARPES experiments in Ca2CuO2Cl2 give strong evidence for the formation of small polarons in undoped cuprates which are due to phonons and strong EPI, while in the doped systems quasiparticles are formed  and there are no small polarons [143]. Namely, in [143] a broad peak around -0.8eV is observed at the top of the band (k=(π/2,π/2)) with the dispersion similar to that predicted by the t-J model—see Figure 17.(a) The ARPES spectrum of undopedCa2CuO2Cl2 at k=(π/2,π/2). Gaussian shape—solid line, Lorentzian shape—dashed line. (b) Dispersion of the polaronic band—A—and of the quasiparticle band—B—along the nodal direction. Horizontal lines are the chemical potentials for a large number of samples. From [143].
(a)(b)However, the peak in Figure17(a) is of Gaussian shape and can be described only by coupling to bosons, that is, this peak is a boson side band—see more in [10, 11] and references therein. The theory based on the t-J model (in the antiferromagnetic state of the undoped compound) by including coupling to several (half-breathing, apical oxygen, low-lying) phonons, which is given in [144–146], explains successfully this broad peak of the boson side band by the formation of small polarons due to the EPI coupling (λep≈1.2). Note that this value of λep is for the polaron at the bottom of the band while in the case where the Fermi surface exists (in doped systems) this coupling is even larger due to the larger density of states at the Fermi surface [144–146]. In [144–146] it was stressed that even when the electron-magnon interaction is stronger than EPI the polarons in the undoped systems are formed due to EPI. The latter mechanism involves excitation of many phonons at the lattice site (where the hole is seating), while it is possible to excite only one magnon at the given site. (N10) Recent soft X-ray ARPES measurements on the electron-doped HTSC Nd1.85Ce0.15CuO4 [147], and Sm(2-x)CexCuO4 (x=0.1,0.15,0.18), Nd1.85Ce0.15CuO4, and Eu1.85Ce0.15CuO4 [148] show kink at energies 50–70meV in the quasiparticle dispersion relation along both the nodal and antinodal, directions as it is shown in Figure 18.NCCO electron-doped: (a)ImΣ(ω) measured in the nodal point. Curves are offsets by 50 meV for clarity. The change of the slope in the last bottom curve is at the phonon energy. (b) ImΣ(ω) for the antinodal direction with 30meV offset. (c) Experimental phonon dispersion of the bond stretching modes. (d) Estimated λeffep from ImΣ(ω). From [148].
(a)(b)(c)(d)It is seen from this figure that the effective EPI coupling constantλep,eff(<λep) is isotropic and λep,eff≈0.8–1. It seems that the kink in the electron-doped cuprates is due solely to EPI and in that respect the situation is similar to the one in the hole-doped cuprates.ARPES Results in the Superconducting State
(S1) There is an anisotropic superconducting gap in most HTSC compounds [22, 23], which is predominately d-wave like, that is, Δ(k)≈Δ0(coskx-cosky) with 2Δ0/Tc≈5-6 in the optimally doped systems. (S2) The particle-hole coherence in the superconducting state which is expected for the BCS-like theory of superconductivity has been observed first in [149] and confirmed with better resolution in [150], where the particle-hole mixing is clearly seen in the electron and hole quasiparticle dispersion. To remind the reader, the excited Bogoliubov-Valatin quasiparticles (α̂k,±) with energies Ekα±=ξk2+|Δk|2 are a mixture of electron (ĉk,σ) and hole (ĉ-k,-σ†), that is, α̂k,+=ukĉk↑+vkĉ-k↓†, α̂k,-=ukĉ-k↓+vkĉk↑† where the coherence factors uk, vk are given by |uk|2=1-|vk|2=(1+ξk/Ek)/2. Note that |uk|2+|vk|2=1, which is exactly observed, together with d-wave pairing Δ(k)=Δ0(coskx-cosky), in experiments in [150]. This is very important result since it proves that the pairing in HTSC cuprates is of the BCS type and not exotic one as it was speculated long time after the discovery of HTSC cuprates. (S3) The kink at (60–70)meV in the quasiparticle energy around the nodal point is not shifted (in energy) while the antinodal kink at ωph(40)~40meV is shifted (in energy) in the superconducting state by Δ0(=(25–30)meV), that is, ωph(40)→ωph(40)+Δ0=(65–70)meV [22, 23]. To remind the reader, in the standard Eliashberg theory the kink in the normal state at ω=ωph should be shifted in the superconducting state to ωph+Δ0 at all k-points at the Fermi surface. This puzzling result (that the quasiparticle energy around the nodal point is not shifted in the superconducting state) might be a smoking gun result since it makes an additional constraint on the quasiparticle interaction in cuprates. Until now there is only one plausible explanation [151] of this nonshift puzzle which is based on an assumption of the forward scattering peak (FSP) in EPI—see more in Section 2. The FSP in EPI means that electrons scatter into a narrow region (q<qc≪kF) around the initial point in the k-space, so that at the most part of the Fermi surface there is practically no mixing of states with different signs of the order parameter Δ(k). In that case the EPI bosonic spectral function (which is defined in Appendix A) α2F(k,k′,Ω)≈α2F(φ,φ′,Ω) (φ is the angle on the Fermi surface) has a pronounced forward scattering peak (at δφ=φ-φ′=0) due to strong correlations—see Section 2. Its width δφc is narrow, that is, δφc≪2π and the angle integration goes over the region δφc around the point φ. In that case the kink is shifted (approximately) by the local gap Δ(φ)=Δmaxcos2φ—for more details see [151]. As a consequence, the antinodal kink is shifted by the maximal gap, that is, |Δ(φAN≈π/2)|=Δmax while the nodal gap is practically unshifted since |Δ(φAN≈π/4)|≈0. (S4) The recent ARPES spectra [152] in the undoped single crystalline 4-layered cuprate with the apical fluorine (F), Ba2Ca3Cu4O8F2 (F0234) give rather convincing evidence against the SFI mechanism of pairing—see Figure 19.Figure 19
Crystal structure ofBa2Ca3Cu4O8(OδF1-δ)2. There are four CuO2 layers in a unit cell with the outer having apical F atoms. CRL—charge reservoir layer; SCL/OP—superconducting layer/outer plane; SCL/IP—superconducting layer/inner plane. From [152].First, F0234 is not a Mott insulator—as expected from valence charge counting which puts Cu valence as 2+, but it is a superconductor with Tc=60K. Moreover, the ARPES data [152] reveal at least two metallic Fermi-surface sheets with corresponding volumes equally below and above half-filling—see Figure 20.(a) Fermi surface (FS) contours from two samples of F0234.N—electron-like; P—hole-like. Bold arrow is (π,π) scattering vector. Angle θ defines the horizontal axis in (b). (b) Leading gap edge along k-space angle from the two FS contours. From [152].
(a)(b)Second, one of the Fermi surfaces is due to the electron-like (N) band (with 20±6% electron-doping) and the other one due to the hole-like (P) band (with 20±8% hole-doping) and their splitting along the nodal direction is significant and cannot be explained by the LDA-DFT calculations [153]. This electron and hole self-doping of inner and outer layers is in an appreciable contrast to other multilayered cuprates where there is only hole self-doping. For instance, in HgBa2CanCun+1O2n+2 (n=2,3) and (Cu,C)Ba2CanCun+1O3n+2 (n=2,3,4), the inner CuO2 layers are less hole-doped than outer layers. It turns out, unexpectedly, that the superconducting gap on theN-band Fermi surface is significantly larger than on the P-one, where in Ba2Ca3Cu4O8F2 the ratio is anomalous (ΔN/ΔP)≈2 and ΔN is an order of magnitude larger than in the electron-doped cuprate Nd2-xCexCuO4. Third, the N-band Fermi surface  is rather far from the antinodal point at (π,0). This is very important result which means that the antiferromagnetic spin fluctuations with the AF wave-vector Q=(π,π), as well as the van Hove singularity, are not dominant in the pairing in the N-band. To remind the reader, the SFI scenario assumes that the pairing is due to spin fluctuations with the wave-vector Q (and near it) which connects two antinodal points which are near the van Hove singularity at the hole-surface (at (π,0) and (0,π)) giving rise to large density of states. This is apparently not the case for the N-band Fermi surface—see Figure 20. The ARPES data give further that there is a kink at ~85 meV in the quasiparticle dispersion of both bands, while the kink in the N-band is stronger than that in the P-band. This result, together with the anomalous ratio (ΔN/ΔP)≈2, disfavors SFI as a pairing mechanism. (S5) Despite the presence of significant elastic quasiparticle scattering in a number of samples of optimally doped Bi-2212, there are dramatic sharpenings of the spectral function near the antinodal point (π,0) at T<Tc (in the superconducting state) [154]. This effect can be explained by assuming that the small q-scattering (the forward scattering peak) dominates in the elastic impurity scattering as it is pointed in [78–80, 130, 155, 156]. As a result, one finds that the impurity scattering rate in the superconducting state is almost zero, that is, γimp(k,ω)=γn(k,ω)+γa(k,ω)≈0 for |ω|<Δ0 for any kind of pairing (s-, p-, d-wave, etc.) since the normal (γn) and the anomalous (γa) scattering rates compensate each other. This collapse of the elastic scattering rate is elaborated in details in [154] and it is a consequence of the Anderson-like theorem for unconventional superconductors which is due to the dominance of the small q-scattering [78–80, 130, 155, 156]. In such a case d-wave pairing is weakly unaffected by nonmagnetic impurities and as a consequence there is small reduction in Tc [156, 157]. The physics behind this result is rather simple. The small q-scattering (usually called forward scattering) means that electrons scatter into a small region in the k-space, so that at the most part of the Fermi surface there is no mixing of states with different signs of the order parameter Δ(k). In such a way the detrimental effect of nonmagnetic impurities on d-wave pairing is significantly reduced. This result points to the importance of strong correlations in the renormalization of the nonmagnetic impurity scattering too—see discussion in Section 2.In conclusion, in order to explain the ARPES results in cuprates it is necessary to take into account (1) the electron-phonon interaction (EPI) since it dominates in the quasiparticle scattering in the energy region important for pairing, (2) the elastic nonmagnetic impurities with the forward scattering peak (FSP) due to strong correlations, and (3) the Coulomb interaction which dominates at higher energies ω>ωph. In this respect, the presence of ARPES kinks and the knee-like shape of the T dependence of the spectral width are important constraints on the scattering and pairing mechanism in HTSC cuprates.


## 1.3.4. Tunnelling Spectroscopy and Spectral Functionα2F(ω)

By measuring current-voltageI-V characteristics in NIS (normal metal-insulator-superconductor) tunnelling junctions with large tunnelling barrier one obtains from tunnelling conductance GNS(V)=dI/dV the so called tunnelling density of states in superconductors NT(ω). Moreover, by measuring of GNS(V) at voltages eV>Δ it is possible to determine the Eliashberg spectral function α2F(ω) and finally to confirm the phonon mechanism of pairing in LTSC materials. Four tunnelling techniques were used in the study of HTSC cuprates: (1) vacuum tunnelling by using the STM technique—scanning tunnelling microscope; (2) point-contact tunnelling; (3) break-junction tunnelling; (4) planar-junction tunnelling. Each of these techniques has some advantages although in principle the most potential one is the STM technique since it measures superconducting properties locally [158]. Since tunnelling measurements probe a surface region on the scale of the superconducting coherence length ξ0, then this kind of measurements in HTSC materials with small coherence length ξ0 (ξab~20 Å in the a-b plane and ξc~1–3 Å along the c-axis) depends strongly on the surface quality and sample preparation. Nowadays, many of the material problems in HTSC cuprates are understood and as a result consistent picture of tunnelling features is starting to emerge.From tunnelling experiments one obtains the (energy-dependent) gap functionΔ(ω) in the superconducting state. Since we have already discussed this problem in [6], we will only briefly mention some important result. For instance, in most systems GNS(V) has V-shape in all families of HTSC hole- and electron-doped cuprates. The V-shape is characteristic for d-wave pairing with gapless spectrum, which is also confirmed in the interference experiments on hole- and electron-doped cuprates [75]. Some experiments give a U-shape of GNS(V) which resembles s-wave pairing. This controversy is explained to be the property of the tunnelling matrix element which filters out states with the maximal gap.Here we are interested in thebosonic spectral functionα2F(ω) of HTSC cuprates near optimal doping which can be extracted by using tunnelling spectroscopy. We inform the reader in advance that the shape and the energy width of α2F(ω), which are extracted from the second derivative d2I/dV2 at voltages above the superconducting gap, in most HTSC cuprates resemble the phonon density of states Fph(ω). This result is strong evidence for the importance of EPI in the pairing potential of HTSC cuprates. For instance, plenty of break junctions made from Bi2212 single crystals [42–45] show that the peaks (and shoulders) in -d2I/dV2 (or dips-negative peaks in d2I/dV2) coincide with the peaks (and shoulders) in the phonon density of states Fph(ω) measured by neutron scattering—see Figure 21.(a) Second derivative ofI(V) for a Bi2212 break junction in various magnetic fields (from 0–20T). The structure of dips (minima) in d2I/dV2 can be compared with the phonon density of states F(ω); (b) the spectral functions α2F(ω) in various magnetic fields. From [42–45].
(a)(b)The tunnelling spectra in Bi-2212 break junctions [42–45], which are shown in Figure 21 indicates that the spectral function α2F(ω) is independent of magnetic field, which is in contradiction with the theoretical prediction based on the SFI pairing mechanism where this function should be sensitive to magnetic field. The reported broadening of the peaks in α2F(ω) is partly due to the gapless spectrum of d-wave pairing in HTSC cuprates. Additionally, the tunnelling density of states NT(ω) at very low T and for ω>Δ shows a pronounced gap structure. It was found that 2Δ¯/Tc=6.2–6.5, where Tc=74–85Kand Δ¯ is some average value of the gap. In order to obtain α2F(ω) the inverse procedure was used by assuming s-wave superconductivity and the effective Coulomb parameter μ*≈0.1 [42–45]. The obtained α2F(ω) gives large EPI coupling constant λep≈2.3. Although this analysis [42–45] was done in terms of s-wave pairing, it mimics qualitatively the case of d-wave pairing, since one expects that d-wave pairing does not change significantly the global structure of d2I/dV2 at eV>Δ albeit introducing a broadening in it—see the physical meaning in Appendix A. We point out that the results obtained in [42–45] were reproducible on more than 30 junctions. In that respect very important results on slightly overdoped Bi2212–GaAs and on Bi2212–Au planar tunnelling junctions are obtained in [46, 47]—see Figure 22.Figure 22
The spectral functionsα2F(ω) and the calculated density of states at 0K (upper solid line) obtained from the conductance measurements, the Bi(2212)–Au planar junctions. GPDS—generalized phonon density of states. From [46, 47].These results show very similar features to those obtained in [42–45] on break junctions. It is worth mentioning that several groups [48–52] have obtained similar results for the shape of the spectral function α2F(ω) from the I-V measurements on various HTSC cuprates—see the comparison in Figure 23. These facts leave no much doubts about the importance of the EPI in pairing mechanism of HTSC cuprates.Figure 23
The spectral functionsα2F(ω) from measurements of various groups: Vedeneev et al. [42–45], Gonnelli et al. [52], Miyakawa et al. [48, 49], and Shimada et al. [46, 47]. The generalized density of states GPDS for Bi2212 is at the bottom. From [46, 47].In that respect, the tunnelling measurements onslightly overdopedBi2Sr2CaCu2O8 [46, 47, 53, 54] give impressive results. The Eliashberg spectral function α2F(ω) of this compound was extracted from the measurements of d2I/dV2 and by solving the inverse problem—see Appendix A. The extracted α2F(ω) has several peaks in broad energy region up to 80meV as it is seen in Figures 22 and 23, which coincide rather well with the peaks in the phonon density of states Fph(ω)—more precisely the generalized phonon density of states GPDS(ω) defined in Appendix A. In [53, 54] numerous peaks, from P1–P13, in α2F(ω) are discerned as shown in Figure 24, which correspond to various groups of phonon modes—laying in (and around) these peaks. Moreover, in [46, 47, 53, 54] the coupling constants for these modes are extracted as well as their contribution (ΔTc) to Tc as it is seen in Table 1. Note that due to the nonlinearity of the problem the sum of (ΔTc)i,i=1,2,…,13, due to various modes is not equal to Tc.Table 1
Phonon frequencyω, EPI coupling constant λi of the peaks P1–P13, and contribution ΔTc to Tc of each peak in α2F(ω)—as shown in Figure 24—obtained from the tunnelling conductance of Bi2Sr2CaCu2O8. ΔTc is the decrease in Tc when the peak in α2F(ω) is eliminated. From [53, 54].
No. of peakω [meV]λiΔTc [K]P114.31.267.4P220.80.9511.0P331.70.4810.5P435.10.286.7P539.40.247.0P645.30.3010.0P758.30.156.5P863.90.010.6P969.90.073.6P1073.70.063.3P1177.30.010.8P1282.10.010.7P1387.10.031.8Figure 24
The spectral functionsα2F(ω) from the tunnelling conductance of Bi2Sr2CaCu2O8 for the positive and the negative bias voltages, and the averaged one [46, 47]. The averaged one is divided into 13 components. The origin of the ordinate is 2,1,0, and -0.5 from the top down. From [46, 47, 53, 54].The next remarkable result is that the extractedEPI coupling constant is very large, that is, λep(=2∫dωα2F(ω)/ω)=∑iλi≈3.5—see Table 1. It is obvious from Figure 24 and Table 1 that almost all phonon modes contribute to λep and Tc, which means that on the average each particular phonon mode is not too strongly coupled to electrons since λi<1.3,i=1,2,…,13, thus keeping the lattice stable.Let us discuss the content of Table1 in more details where it is shown the strength of the EPI coupling and the relative contribution of different phononic modes to Tc. In Table 1 it is seen that lower-frequency modes from P1–P3, corresponding to Cu,Sr, and Ca vibrations, are rather strongly coupled to electrons (with λκ~1) which give appreciable contributions to Tc. It is also seen in Table 1 that the coupling constants λi of the high-energy phonons (P9–P13 with ω≥70meV) have λi≪1 and give moderate contribution to Tc—around 10%. These results give solid evidence for the importance of the low-energy modes related to the change of the Madelung energy in the ionic-metallic structure of HTSC cuprates—the idea advocated in [3–6] and discussed in Section 2. If confirmed in other HTSC families, these results are in favor of the moderate oxygen isotope effect in cuprates near the optimal doping since the oxygen modes are higher-energy modes and give smaller contribution to Tc. We stress that each peak P1–P13 in α2F(ω) corresponds to many modes. For a better understanding of the EPI coupling in these systems we show in Figure 25 the total and partial density of phononic states. It is seen that the low-energy phonons are due to the vibrations of the Ca, Sr, and Cu ions which correspond to the peaks P1-P2 in Figures 23 and 24. In order to obtain information on the structure of vibrations which are strongly involved in pairing, we show in Figures 26 and 27 the structure of these vibrations at special points in the Brillouin zone. It is seen in Figure 26 that the low-frequency phonons P1-P2 are dominated by Cu, Sr, Ca vibrations.Figure 25
The phonon density of statesF(ω) (PDOS) of Bi2Sr2CaCu2O8 compared with the generalized density of states (GPDOS) [159]. Atomic vibrations: O1—O in the CuO2 plane; O2—apical O; O3—O in the BiO plane. From [46, 47].Atomic polarization vectors and their frequencies (inmeV) at special points in the Brillouin zone. The larger closed circles in the lattice are O-ions. Γ-X is along the Cu-O-Cu direction. Arrows indicate displacements. The modes in square and round brackets are the transverse and longitudinal optical modes, respectively. (a) Modes of the P1 peak. (b) Modes of the P2 peak. From [46, 47, 53, 54].
(a)(b)Atomic polarization vectors and their frequencies (inmeV) at special points in the Brillouin zone. The larger closed circles in the lattice are O-ions. Γ-X is along the Cu-O-Cu direction. Arrows indicate displacements. The modes in square and round brackets are the transverse and longitudinal optical modes, respectively. (a) Modes of the P3 peak. (b) Modes of the P4 peak. From [46, 47, 53, 54].
(a)(b)Further, based on Table1 one concludes that the P3 modes are strongerly coupled to electrons than the P4 ones, although the density of state for the P4 modes is larger. The reason for such an anomalous behavior might be due to symmetries of the corresponding phonons as it is seen in Figure 27. Namely, to the P3 peak contribute axial vibrations of O(1) in the CuO2 plane which are odd under inversion, while in the P4 peak these modes are even. The in-plane modes of Ca and O(1) are present in P3 which are in-phase and out-of-phase modes, while in P4 they are all out-of-phase modes. For more information on other modes, P5–P13, see [53, 54]. We stress that the Eliashberg equations based on the extracted α2F(ω) of the slightly overdoped Bi2Sr2CaCu2O8 with the ratio (2Δ/Tc)≈6.5 describe rather well numerous optical, transport, and thermodynamic properties [53, 54]. However, in underdoped systems with (2Δ/Tc)≈10, where the pseudogap phenomena are pronounced, there are serious disagreements between experiments and the Migdal-Eliashberg theory [53, 54]. We would like to stress that the contribution of the high-frequency modes (mostly the oxygen modes) to α2F(ω) may be underestimated in tunnelling measurements due to their sensitivity to the surface contamination and defects. Namely, the tunnelling current probes a superconductor to a depth of order of the quasiparticle mean-free path l(ω)=vFγ-1(ω). Since the relaxation time γ-1(ω) decreases with increasing ω, the mean-free path can be rather small and the effects of the high-energy phonons are sensitive to the surface contamination.Similar conclusion regarding the structure of theEPI spectral function α2F(ω) in HTSC cuprates comes out from tunnelling measurements on Andreev junctions (the BTK parameter Z≪1—low barrier) and Giaver junctions (Z≫1—high barrier) in La2-xSrxCuO4 and YBCO compounds [160], where the extracted α2F(ω) is in good accordance with the phonon density of states Fph(ω)—see Figure 28.(a)d2I/dV2 of a Giaver-like contact in La2-xSrxCuO4—note the large structure below 50meV; (b) d2I/dV2 of an Andreev- and Giaver-like contact compared to the peaks in the phonon density of states. From [160].
(a)(b)Note that the BTK parameterZ is related to the transmission and reflection coefficients for the normal metal (1+Z2)-1 and Z2(1+Z2)-1, respectively.Although most of the peaks inα2F(ω) of HTSC cuprates coincide with the peaks in the phonon density of states, it is legitimate to put the following question. Can the magnetic resonance in the superconducting state give significant contribution to α2F(ω)? In that respect the inelastic magnetic neutron scattering measurements of the magnetic resonance as a function of doping [161] give that the resonance energy Er scales with Tc, that is, Er=(5-6)Tc as shown in Figure 29.Figure 29
Doping dependence of the energyEr of the magnetic resonance peak at Q=(π,π) in YBCO and Bi2212 measured at low temperatures by inelastic neutron scattering. From [161].This means that if one of the peaks inα2F(ω) is due to the magnetic resonance at ω=Er, then it must shift strongly with doping as it is observed in [161]. This is contrary to phonon peaks (energies) whose positions are practically doping independent. To this end, recent tunnelling experiments on Bi-2212 [55] show clear doping independenceof α2F(ω) as it is seen in Figure 30. This remarkable result is an additional evidence in favor of EPI and against the SFI mechanism of pairing in HTSC cuprates which is based on the magnetic resonance peak in the superconducting state. In that respect the analysis in [162] of the tunneling spectra of the electron-doped cuprate Pr0.88Ce0.12CuO4 with Tc=24K shows the existence of the bosonic mode at ωB=16meV which is significantly larger than the magnetic-resonance mode with ωr=(10-11)meV. This result excludes the magnetic-resonance mode as an important factor which modifies superconductivity.Figure 30
Second derivative ofI(V) for Bi2212 tunnelling junctions for various doping: UD—underdoped; OD—optimally doped; OVD—overdoped system. The structure of minima in d2I/dV2 can be compared with the phonon density of states F(ω). The full and vertical lines mark the positions of the magnetic resonance energy Er≈5.4Tc for various doping taken from Figure 29. Red tiny arrows mark positions of the magnetic resonance Er in various doped systems. Dotted vertical lines mark various phonon modes. From [55].The presence of pronounced phononic structures (and the importance of EPI) in theI(V) characteristics was quite recently demonstrated by the tunnelling measurements on the very good La1.85Sr0.15CuO4 films prepared by the molecular beam epitaxy on the [001]-symmetric SrTiO3 bicrystal substrates [56]. They give unique evidence for eleven peaks in the (negative) second derivative, that is, -d2I/dV2. Furthermore, these peaks coincide with the peaks in the intensities of the phonon Raman scattering data measured at 30K in single crystals of LSCO with 20% of Sr [57]. These results are shown in Figure 31. In spite of the lack of a quantitative analysis of the data in the framework of the Eliashberg equations, the results in [56] are important evidence that phonons are relevant pairing bosons in HTSC cuprates.Figure 31
Second derivative datad2I(V)/dV2 of the tunnelling spectra on thin films of La1.85Sr0.15CuO4 are shown along with phonon Raman scattering data on single crystals of LSCO with 20% Sr. The polarization of the incident and scattered light in the Raman spectra is parallel to the CuO2 planes. From [56].It is interesting that in thec-axis vacuum tunnelling STM measurements [163] the fine structure in d2I/dV2 at eV>Δ was not seen below Tc, while the pseudogap structure is observed at temperatures near and above Tc. This result could mean that the STM tunnelling is likely dominated by the nontrivial structure of the tunnelling matrix element (along the c-axis), which is derived from the band-structure calculations [164]. However, recent STM experiments on Bi2212 [61–63] give information on the possible nature of the bosonic mode which couples with electrons. In [61–63] the local conductance dI/dV(r,E) is measured where it is found that d2I/dV2(r,E) has peak at E(r)=Δ(r)+Ω(r) where dI/dV(r,E) has the maximal slope—see Figure 32(a).(a) Typical conductancedI/dV(r,E). The ubiquitous features at eV>Δ(gap) with maximal slopes, which give peaks in d2I/dV2(r,E), are indicated by arrows. (b) The histograms of all values of Ω(r) for samples with O16—right curve and with O18—left curve. From [61–63].
(a)(b)It turns out that the corresponding average phonon energyΩ¯ depends on the oxygen mass, that is, Ω¯~MO-1/2, with Ω¯16=52meV and Ω¯18≈48meV—as it is seen in Figure 32(b). This result is interpreted in [61–63] as an evidence that the oxygen phonons are strongly involved in the quasiparticle scattering. A possible explanation is put forward in [61–63] by assuming that this isotope effect is due to the B1g phonon which interacts with the antinodal quasiparticles. However, this result requires a reanalysis since the energy of the bosonic mode in fact coincides with the dip and not with the peak of d2I/dV2(r,E)—as reported in [61–63].The important message of numerous tunnelling experiments inHTSC cuprates near and at the optimal doping is that there is strong evidence for the importance of EPI in the quasiparticle scattering and that no particular phonon mode can be singled out in the spectral function α2F(ω) as being the only one which dominates in pairing mechanism. This important result means that the high Tc is not attributable to a particular phonon mode in the EPI mechanism but all phonon modes contribute to λep. Having in mind that the phonon spectrum in HTSC cuprates is very broad (up to 80meV), then the large EPI constant (λep≳2) obtained in the tunnelling experiments is not surprising at all. Note that similar conclusion holds for some other oxide superconductors such as Ba9.6K0.4BiO3 with Tc=30K where the peaks in the bosonic spectral function/extracted from tunnelling measurements coincide with the peaks in the phononic density of states [165–167].


## 1.3.5. Phonon Spectra and EPI

Although experiments related to phonon spectra and their renormalization byEPI, such as inelastic neutron, inelastic X-ray, and Raman scattering, do not give the spectral function α2F(ω), they nevertheless can give useful, but indirect, information on the strength of EPI for some particular phonons. We stress in advance that the interpretation of the experimental results in HTSC cuprates by the theory of EPI for weakly correlated electrons is inadequate since in strongly correlated systems, such as HTSC cuprates, the phonon renormalization due to EPI is different than in weakly correlated metals [168]. Since these questions are reviewed in [168], we will briefly enumerate the main points: (1) in strongly correlated systems the EPI coupling for a number of phononic modes can be significantly larger than the LDA-DFT and Hartree-Fock methods predict. This is due to many-body effects not contained in LDA-DFT [168, 169]. The lack of the LDA-DFT calculations in obtaining phonon line-widths is clearly demonstrated, for instance, in experiments on L2-xSrxCuO4—see review in [170] and references therein, where the bond-stretching phonons at q=(0.3,0,0) are softer and much broader than the LDA-DFT calculations predict. (Note the wave vector q is in units (2π/a,2π/b,2π/c)—for instance, in these units (π,π) corresponds to (0.5,0.5).) (2) The calculation of phonon spectra is in principle very difficult problem since besides the complexity of structural properties in a given material one should take into account appropriately the long-range Coulomb interaction of electrons as well as strong short-range repulsion. Our intention is not to discuss this complexity here—for that see, for instance, [69]—but we only stress some important points which will help to understand problems with which is confronted the theory of phonons in cuprates.The phonon Green's functionD(q,ω) depends on the phonon self-energy Π(q,ω) which takes into account all the enumerated properties (note that D-1(q,ω)=D0-1(q,ω)-Π(q,ω)). In cases when the EPI coupling constant gep(k,k′) is a function on the transfer momentum q=k-k′ only, then Π(q) (q=(q,iωn)) depends on the quasiparticle charge susceptibility χc(q)=P(q)/ɛe(q):
(41)Π(q)=|gep(q)|2χc(q),
and P(q) is the irreducible electronic polarization given by
(42)P(q)=-∑pG(p+q)Γc(p,q)G(p).
The screening due to the long-range Coulomb interaction is contained in the electronic dielectric function ɛe(q) while the “screening” due to (strong) correlations is described by the charge vertex function Γc(p,q). Due to complexity of the physics of strong correlations the phonon dynamics was studied in the t-J model but without the long-range Coulomb interaction [168, 169, 171], in which case one has ɛe=1 and χc(q)=P(q). However, in studying the phonon spectra in HTSC cuprates it is believed that this deficiency might be partly compensated by choosing the bare phonon frequency ω0(q) (contained in D0-1(q,ω)) to correspond to the undoped compounds [168, 171]. It is a matter of future investigations to incorporate all relevant interactions in order to obtain a fully microscopic and reliable theory of phonons in cuprates. Additionally, the electron-phonon interaction (with the bare coupling constant gep(q)) is dominated by the change of the energy of the Zhang-Rice singlet—see more in Section 2.3—and (41) for Π(q) is adequate one [6, 168, 169]. Since the charge fluctuations in HTSC cuprates are strongly suppressed (no doubly occupancy of the Cu 3d9 state) due to strong correlations, and since the suppressed value of χc(q) cannot be obtained by the band-structure calculations, this means that LDA-DFT underestimates the EPI coupling constant significantly. In the following we discuss this important result briefly.(1) Inelastic Neutron and X-Ray Scattering—The Phonon Softening and the Line-Width due to EPI
The appreciable softening and broadening of numerous phonon modes has been observed in the normal state of HTSC cuprates, thus giving evidence for pronounced EPI effects and for inadequacy of the LDA-DFT calculations in treating strong correlations and suppression of the charge susceptibility [6, 10, 11, 168, 171]. There are several relevant reviews on this subject [10, 11, 168, 170, 172] and here we discuss briefly two important examples which demonstrate the inefficiency of the LDA-DFT-band structure calculations to treat quantitatively EPI in HTSC cuprates. For instance, the Cu–O bond-stretching phonon mode shows a substantial softening at qhb=(0.3,0,0) by doping  of La1.85Sr0.15CuO4 and YBa2Cu3O7 [170, 172]—called the half-breathing phonon, and a large broadening by 5meV at 15% doping [173–175] as it is seen in Figure 33. While the softening can be partly described by the LDA-DFT method [176], the latter theory predicts an order of magnitude smaller broadening than the experimental one. This failure of LDA-DFT is due to the incorrect treatment of the effects of strong correlations on the charge susceptibility χc(q) and due to the absence of many-body effects which can increase the coupling constant gep(q)—see more in Section 2. The neutron scattering measurements in La1.85Sr0.15CuO4 give evidence for large (30%) softening of the OZZ with Λ1 symmetry with the energy ω≈60meV, which is theoretically predicted in [177], and for the large line-width about 17meV which also suggests strong EPI. These apex modes are favorable for d-wave pairing since their coupling constants are peaked at small momentum q [10, 11]. Having in mind the above results, then it is not surprising that the recent calculations of the EPI coupling constant λep in the framework of LDA-DFT give very small EPI coupling constant λep≈0.3 [28, 29]. The critical analysis of the LDA-DFT results in HTSC cuprates is done in [6] and additionally argued in [10, 11, 178] by pointing their disagreement with the inelastic neutron and X-ray scattering measurements—as it is shown in Figure 33.Comparison of DFT calculations with experimental results of inelastic X-ray scattering: (a) phonon energies inLa1.85Sr0.15CuO4 and (b) in YBa2Cu3O7; (c) phonon line-widths in La1.85Sr0.15CuO4. DFT calculations [28] give much smaller width than experiments [173–175]. From [178].
(a)(b)(c)In Section2 we will discuss some theoretical approaches related to EPI in strongly correlated systems but without discussing the phonon renormalization. The latter problem was studied in more details in the review articles in [10, 11, 170]. Here, we point out only three (for our purposes) relevant results. First, there is an appreciable difference in the phonon renormalization in strongly and weakly correlated systems. Namely, the change of the phonon frequencies in the presence of the conduction electrons is proportional to the squared coupling constant |gq| and charge susceptibility χc, that is, δω(q)~|gep(q)|2Reχc, while the line-width is given by Γω(q)~|gep(q)|2|Imχc|. All these quantities can be calculated in LDA-DFT and as we discussed above, where for some modes one obtains that Γω(q)(LDA)≪Γω(q)(exp). However, it turns out that in strongly correlated systems doped by holes (with the concentration δ≪1) the charge fluctuations are suppressed in which case the following sum rule holds [10, 11, 171]:
(43)1πN∑q≠0‍∫-∞∞dω|Imχc(q,ω)|=2δ(1-δ)N,
while in the LDA-DFT method one has
(44)1πN∑q≠0∫-∞∞dω|Imχc(q,ω)|(LDA)=(1-δ)N.
The inequality Γω(q)(LDA)≪Γω(q)(exp) (for some phonon modes) together with (43)-(44) means that for low doping δ≪1 the LDA calculations strongly underestimate the EPI coupling constantin the large portion of the Brillouin zone, that is, one has |gep(LDA)(q)|≪|gep(exp)(q)|. The large softening and the large line-width of the half-breathing mode at q=(0.5,0), but very moderate effects for the breathing mode at q=(0.5,0.5), are explained in the framework of the one slave-boson (SB) theory (for U=∞) in [171], where χc(q,ω) (i.e., Γc(p,q)=Γc(p,q)) is calculated in leading O(1/N) order. We stress that there is another method for studying strong correlations—the X-method—where the controllable 1/N expansion is performed in terms of the Hubbard operators and where the charge vertex Γc(p,q) is calculated [6, 78–80, 130, 179, 180]. It turns out that in the adiabatic limit (ω=0) the vertex functions Γc(pF,q) in these two methods have important differences. For instance, Γc(X)(pF,q) (in the X-method) is peaked at q=0—the so called forward scattering peak (FSP)—while Γc(SB)(pF,q) has maximum at finite |q|≠0 [181]—see Section 2.3.5. The enumerated properties of Γc(X)(pF,q) are confirmed by the numerical Monte Carlo calculations in the finite-U Hubbard model [182], where it is found that FSP exists for all U, but it is especially pronounced in the limit U≫t. These results are also confirmed in [183] where the calculations are performed in the four-slave-boson technique—see more in Section 2.3.5. Having in mind this difference it would be useful to have calculations of χc(q,ω) in the framework of the X-method which are unfortunately not done yet. Second, the many-body theory gives that for coupling to some modes the coupling constant |gep(q)| in HTSC cuprates can be significantly larger than the LDA-DFT calculations predict [10, 11], which is due to some many-body effects not present in the latter [169]. In Section 2 it will be argued that for some phonon modes one has |gep(q)|2≫|gep(LDA)(q)|2. For instance, for the half-breathing mode, one has |gep(q)|2≈3|gep(LDA)(q)|2 [10, 11, 169]—see Section 2. These two results point to an inadequacy of LDA-DFT in calculations of EPI effects in HTSC cuprates. Third, the phonon self-energy (Π(q)) and quasiparticle self-energy Σ(k) are differently renormalized by strong correlations [6, 10, 11, 78–80, 130, 179, 180], which is the reason that Π(q) is much more suppressed than Σ(k)—see Section 2. The effects of the charge vertex on Π(q) and Σ(k) are differently manifested. Namely, the vertex function enters quadratically in Σ(k) and the presence of the forward scattering peak in the charge vertex strongly affects the EPI coupling constant gep(q) in Σ(k):
(45)Σ(k)=-∑q|gep(q)γc(k,q)|2D(q)g(k+q),
where g(k)(≡G(k)/Q) is the quasiparticle Green's function, γc(k,q)=Γc(k,q)/Q is the quasiparticle vertex, and Q(~δ) is the Hubbard quasiparticle spectral weight—see Section 2.3. In the adiabatic limit |q|>qω=ωph/vF one has γc(k,q)≈γc(k,q) and for q≫qc(≈δ·π/a) the charge vertex is strongly suppressed (γc(k,q)≪1) making the effective EPI coupling (which also enters the pairing potential) small at large (transfer) momenta q. This has strong repercussion on the pairing due to EPI since for small doping it makes the d-wave pairing coupling constant to be of the order of the s-one (λd≈λs). Then in the presence of the residual Coulomb interaction EPI gives rise to d-wave pairing. On the other side the charge vertex Γc(k,q) enters Π(q)  linearlyand it is additionally integrated over the quasiparticle momentum k—see (42). Therefore, one expects that the effects of the forward scattering peak on Π(q) are less pronounced than on Σ(k). Nevertheless, the peak of Γc(k,q) at q=0 may be (partly) responsible that the maximal experimental softening and broadening of the stretching (half-breathing) mode in La1.85Sr0.15CuO4 and YBa2Cu3O7 is at qhb(exp)=(0.3,0,0) [170] and not at qhb=(0.5,0) for which gep(qhb) reaches maximum. This means that the charge vertex function pushes the maximum of the renormalized EPI coupling constant to smaller momenta q. It would be very interesting to have calculations for other phonons by including the vertex function obtained by the X-method—see Section 2.3.(2) The Phonon Raman Scattering
Thephonon Raman scattering gives an indirect evidence for importance of EPI in cuprates [184–188]. We enumerate some of them—see more in [6] and references therein. (i) There is a pronounced asymmetric line-shape (of the Fano resonance) in the metallic state. For instance, in YBa2Cu3O7 two Raman modes at 115cm-1 (Ba dominated mode) and at 340cm-1 (O dominated mode in the CuO2 planes) show pronounced asymmetry which is absent in YBa2Cu3O6. This asymmetry means that there is an appreciable interaction of Raman active phonons with continuum states (quasiparticles). (ii) The phonon frequencies for some A1g and B1g are strongly renormalized in the superconducting state, between (6–10)%, pointing again to the importance of EPI [188]—see also [6, 37, 38]. To this point we mention that there is a remarkable correlation between the electronic Raman cross-section S̃exp(ω) and the optical conductivity in the a-b plane σab(ω), that is, S̃exp(ω)~σab(ω) [6]. In previous subsections it is argued that EPI with the very broad spectral function α2F(ω) (0<ω≲80meV) explains in a natural way the ω and T dependence of σab(ω). This means that the electronic Raman spectra in cuprates can be explained by EPI in conjunction with strong correlations. This conclusion is supported by the calculations of the Raman cross-section [189] which take into account EPI with α2F(ω) extracted from the tunnelling measurements in YBa2Cu3O6+x and Bi2Sr2CaCu2O8+x [6, 42–54]. Quite similar properties (to cuprates) of the electronic Raman scattering, as well as of σ(ω), R(ω), and ρ(T), were observed in experiments [108] on isotropic 3D metallic oxides La0.5Sr0.5CoO3 and Ca0.5Sr0.5RuO3 where there are no signs of antiferromagnetic fluctuations. This means that low dimensionality and antiferromagnetic spin fluctuations cannot be a prerequisite for anomalous scattering of quasiparticles and EPI must be inevitably taken into account since it is present in all these compounds.


## 1.3.6. Isotope Effect inTc and Σ(k,ω)

The isotope effectαTC in the critical temperature Tc was one of the very important proofs for the EPI pairing mechanism in low-temperature superconductors (LTSCs). As a curiosity the isotope effect in LTSC systems was measured almost exclusively in mono-atomic systems and in few polyatomic systems: the hydrogen isotope effect in PdH, the Mo and Se isotope shift of Tc in Mo6Se8, and the isotope effect in Nb3Sn and MgB2. We point out that very small (αTC≈0 in Zr and Ru) and even negative (in PdH) isotope effects in some polyatomic systems of LTSC materials are compatible with the EPI pairing mechanism but in the presence of substantial Coulomb interaction or lattice anharmonicity. The isotope effect αTC cannot be considered as the smoking gun effect since it is sensitive to numerous influences. For instance, in MgB2 it is with certainty proved that the pairing is due to EPI and strongly dominated by the boron vibrations, but the boron isotope effect is significantly reduced, that is, αTC≈0.3 and the origin for this smaller value is still unexplained. The situation in HTSC cuprates is much more complicated because they are strongly correlated systems and contain many atoms in unit cell. Additionally, the situation is complicated with the presence of intrinsic and extrinsic inhomogeneities, low dimensionality which can mask the isotope effects. On the other hand new techniques such as ARPES, STM, and μSR allow studies of the isotope effects in quasiparticle self-energies, that is, αΣ, which will be discussed below.(1) Isotope Effect αTC in Tc
This problem will be discussed only briefly since more extensive discussion can be found in [6]. It is well known that in the pure EPI pairing mechanism the total isotope coefficient α is given by αTC=∑i,pαi(p)=-∑i,pdlnTc/dlnMi(p), where Mi(p) is the mass of the ith element in the pth crystallographic position. We stress that the total isotope effect is not measured in HTSC cuprates but only some partial ones. Note that, in the case when the screened Coulomb interaction is negligible, that is, μc*=0, the theory predicts αTC=1/2. From this formula one can deduce that the relative change of Tc,δTc/Tc, for heavier elements should be rather small—for instance, it is 0.02 for Ba135→Ba138, 0.03 for Cu63→Cu65, and 0.07 for La138→La139. This means that the measurements of αi for heavier elements are confronted with the ability of the present experimental techniques. Therefore most isotope effect measurements were done by substituting light atoms O16 by O18 only. It turns out that in most optimally doped HTSC cuprates αO is rather small. For instance, αO≈0.02–0.05 in YBa2Cu3O7 with Tc,max≈91K, but it is appreciable in La1.85Sr0.15CuO4 with Tc,max≈35K where αO≈0.1-0.2. In Bi2Sr2CaCu2O8 with Tc,max≈76K one has αO≈0.03–0.05 while αO≈0.03 and even negative (-0.013) in Bi2Sr2Ca2Cu2O10 with Tc,max≈110K. The experiments on Tl2Can-1BaCunO2n+4 (n=2,3) with Tc,max≈121K are still unreliable and αO is unknown. In the electron-doped (Nd1-xCex)2CuO4 with Tc,max≈24K one has αO<0.05 while in the underdoped materials αO increases. The largest αO is obtained even in the optimally doped compounds like in systems with substitution, such as La1.85Sr0.15Cu1-xMxO4, M=Fe,Co, where αO≈1.3 for x≈0.4%. In La2-xMxCuO4 there is a Cu-isotope effect which is of the order of the oxygen one, that is, αCu≈αO giving αCu+αO≈0.25–0.35 for optimally doped systems (x=0.15). In case when x=0.125 with Tc≪Tc,max one hasαCu≈0.8-1 with αCu+αO≈1.8 [190, 191]. The appreciable copper isotope effect in La2-xMxCuO4 tells us that vibrations other than oxygen ions are important in giving high Tc. In that sense one should have in mind the tunnelling experiments discussed above, which tell us that all phonons contribute to the Eliashberg pairing function α2F(k,ω) and according to these results the oxygen modes give moderate contribution to Tc [53, 54]. Hence the small oxygen isotope effect αTc(O) in optimally doped cuprates, if it is an intrinsic property at all (due to pronounced local inhomogeneities of samples and quasi-two-dimensionality of the system), does not exclude the EPI mechanism of pairing.(2) Isotope Effect αΣ in the Self-Energy
The fine structure of the quasiparticle self-energyΣ(k,ω), such as kinks and slopes, can be resolved in ARPES measurements and in some respect in STM measurements. It turns out that there is isotope effect in the self-energy in the optimally doped Bi2212 samples [139, 141, 142]. In the first paper on this subject [139] it is reported a red shiftδωk,70~-(10–15)meV of the nodal kink at ωk,70≃70meV for the O16→O18 substitution. In [139] it is reported that the isotope shift of the self-energy δΣ=Σ16-Σ18~10meV is very pronounced at large energies ω=100–300meV. Concerning the latter result, there is a dispute since it is not confirmed in other experiments [141, 142]. However, the isotope effect in ReΣ(k,ω) at low energies [141, 142] is well described in the framework of the Migdal-Eliashberg theory for EPI [140] which is in accordance with the recent ARPES measurements with low-energy photons ~7 eV [192]. The latter allowed very good precision in measuring the isotope effect in the nodal point of Bi-2212 with Tc16=92.1K and Tc18=91.1K [192]. They observed a shift in the maximum of ReΣ(kN,ω)—at ωk,70≈70meV (it corresponds to the half-breathing or to the breathing phonon)—by δωk,70≈3.4±0.5meV as shown in Figure 34.(a) EffectiveReΣ for five samples for O16 (blue) and O18 (red) along the nodal direction. (b) Effective ImΣ determined from MDC full widths. An impurity term is subtracted at ω=0. From [192].
(a)(b)By analyzing the shift in ImΣ(kN,ω)—shown in Figure 34—one finds similar result for δωk,70≈3.2±0.6meV. The similar shift was obtained in STM measurements [61–63] which is shown in Figure 32(b) and can have its origin in different phonons. We would like to stress two points: (i) in compounds with Tc~100K the oxygen isotope effect in Tc is moderate, that is, αTc(O)<0.1 [192]. If we consider this value to be intrinsic, then even in this case it is not in conflict with the tunnelling experiments [53, 54] since the latter give evidence that vibrations of heavier ions contribute significantly to Tc—see the discussion in Subsection 1.3.4 on the tunnelling spectroscopy. (ii) In ARPES measurements of [192] the effective EPI coupling constant λep,eff≳0.6 is extracted, while the theory in Subsection 1.3.3 gives that the real coupling constant is larger, that is, λep>1.2. This value is significantly larger than the LDA-DFT theory predicts λep,LDA<0.3 [28, 29]. This again points that the LDA-DFT method does not pick up the many-body effects due to strong correlations—see Section 2.


## 1.4. Summary of Section1

The analysis of experimental data in HTSC cuprates which are related to optics, tunnelling, and ARPES measurementsnear and at the optimal doping gives evidence for the large electron-phonon interaction (EPI) with the coupling constant 1<λep<3.5. We stress that this analysis is done in the framework of the Migdal-Eliashberg theory for EPI which is a reliable approach for systems near the optimal doping. The spectral function α2F(ω), averaged over the Fermi surface, is extracted from various tunnelling measurements on bulk materials and tin films. It contains peaks at the same energies as the phonon density of states Fph(ω). So obtained spectral function when inserted in the Eliashberg equations provides sufficient strength for obtaining high critical temperature Tc~100K. These facts are a solid proof for the important role of EPI in the normal-state scattering and pairing mechanism of cuprates. Such a large (experimental) value of the EPI coupling constant and the robustness of the d-wave superconductivity in the presence of impurities imply that the EPI potential and the impurity scattering amplitude must be strongly momentum dependent. The IR optical reflectivity data provide additional but indirect support for the importance of EPI since by using the spectral function (extracted from tunnelling measurements) one can quantitatively explain frequency dependence of the dynamical conductivity, optical relaxation rate, and optical mass. These findings related to EPI are additionally supported by ARPES measurements on BSCO compounds. The ARPES kinks, the phononic features and the isotope effect in the quasiparticle self-energy in the nodal and antinodal points at low energies (ω≪ωc) persist in the normal and superconducting state. They are much more in favor of EPI than for the spin fluctuation (SFI) scattering mechanism. The transport EPI coupling constant in HTSC cuprates is much smaller than λep, that is, λtr~λep/3, which points to some peculiar scattering mechanism not met in low-temperature superconductors. The different renormalization of the quasiparticle and transport self-energies by the Coulomb interaction (strong correlations) hints to the importance of the small-momentum scattering in EPI. This will be discussed in Section 2.The ineffectiveness of SFI to solely provide pairing mechanism in cuprates comes out also from the magnetic neutron scattering on YBCO and BSCO. As a result, the imaginary part of the susceptibility is drastically reduced in the low-energy region by going from slightly underdoped toward optimally doped systems, whileTc is practically unchanged. This implies that the real SFI coupling constant λsf(~gsf2) is small since the experimental value gsf(exp)<0.2eV is much smaller than the assumed theoretical value gsf(th)≈(0.7–1.5)eV.Inelastic neutron and X-ray scattering measurements in HTSC cuprates show that the broadening of some phonon lines is by an order of magnitude larger than the LDA-DFA methods predict. Since the phonon line-widths depend on the EPI coupling and the charge susceptibility, it is evident that calculations of both quantities are beyond the range of applicability of LDA-DFT. As a consequence, the LDA-DFT calculations overestimate the electronic screening and thus underestimate the EPI coupling, since many-body effects due to strong correlations are not contained in this mean-field theory. However, in spite of the promising and encouraging experimental results about the dominance of EPI in cuprates, the theory is still confronted with difficulties in explaining sufficiently large coupling constant in thed-channel. At present there is not such a satisfactory microscopic theory although some concepts, such as the the dominant EPI scattering at small transfer momenta, are understood at least qualitatively. This set of problems and questions will be discussed in Section 2.


## 2. Theory of EPI in HTSC

The experimental results in Section1 give evidence that the electron-phonon interaction (EPI) in HTSC cuprates is strong and in order to be conform with d-wave pairing EPI must be peaked at small transfer momenta. A number of other experiments in HTSC cuprates give evidence that these are strongly correlated systemswith large on-site Coulomb repulsion of electrons on the Cu-ions. However, at present there is no satisfactory microscopic theory of pairing in HTSC cuprates which is able to calculate Tc and the order parameter. This is due to mathematical difficulties in obtaining a solution of the formally exact ab initio many-body equations which take into account two important ingredients—EPI and strong correlations [6]. In Section 2.1 we discuss first the ab initio many-body theory of superconductivity in order to point places which are most difficult to be solved. Since the superconductivity is low energy phenomenon (also in HTSC cuprates), one can simplify the structure of the ab initio equations in the low-energy sector (the Migdal-Eliashberg theory), where the high-energy processes are incorporated in the (so called) ideal band-structure (nonlocal) potential VIBS(x,y) and the vertex functionΓ. This program of calculations of VIBS(x,y),Γ, and the EPI coupling (matrix elements) gep(x,y) is not realized in HTSC superconductors due to its complexity. However, one pragmatical way out is to calculate gep in the framework of the LDA-DFT method which is at present stage unable to treat strong correlations in a satisfactory manner. Some achievements and results of the LDA-DFT methods which are related to HTSC cuprates are discussed in Section 2.2.In the case of very complicated systems, such as the HTSC cuprates, the standard (pragmatical) procedure in physics is to formulate a minimal theoretical model—sometimes called toy model—which includes minimal set of important ingredients necessary for qualitative and semiquantitative study of a phenomenon. As a consequence of the experimental results,the minimal theoretical model must comprise two important ingredients: (1) EPI and (2) strong correlations. In Section 2.3 we will formulate such a minimal theoretical model—calledthet-Jmodel which includes EPItoo. In the framework of this model we will discuss the renormalization of EPI by strong correlations. In recent years the interest in these problems is increased and numerous numerical calculations were done mostly on small clusters with n×n atoms (n<8). We will not discuss this subject which is fortunately covered in the recent comprehensive review in [10, 11]. The analytical approaches in studying the renormalization of EPI by strong correlations, which are based on a controllable and systematic theory, are rather scarce. We will discuss such a systematic and controllable theory in the framework of  the t-J model with EPI, which is formulated and solved in terms of Hubbard operators. The theory of this (toy) model predicts some interesting effects which might be important for understanding the physics of HTSC cuprates. It predicts that the high-energy processes (due to the suppression of doubly occupancy for U≫Wb) give rise to a nonlocal contribution to the band-structure potential (self-energy Σ(x,y,ω=0)) as well as to EPI. This nonlocality in EPI is responsible for the peak in the effective pairing potential (Vep,eff(q,ω)) at small transfer momenta q(⩽qc≪kF) [6, 78–80, 130]. The latter property allows that the (strong) EPI is conform with d-wave pairing in HTSC cuprates. Furthermore, the peculiar structural properties of HTSC cuprates and corresponding electronic quasi-two-dimensionality give an additional nonlocality in EPI. The latter is due to the change of the weakly screened Madelung energy which is involved in most of the lattice vibrations along the c-axis. Since at present there is no quantitative theory for the latter effect, we tackle this problem here only briefly. The next task for the future studies of the physics of HTSC cuprates is to incorporate these structural properties in the minimal theoretical t-J model.Finally, by writing this chapter our intention is not to overview the theoretical studies of EPI in HTSC cuprates—which is an impossible task—but first to elucidate the descending way from the (old) well-defined ab initio microscopic theory of superconductivity to the one of the minimal model which treats the interplay of EPI and strong correlations. Next, we would like to encourage the reader to further develop the theory of HTSC cuprates.


### 2.1. Microscopic Theory of Superconductivity

#### 2.1.1. Ab Iniitio Many-Body Theory

The many-body theory of superconductivity is based on the fully microscopic electron-ion Hamiltonian for electrons and ions in the crystal—see, for instance, [193, 194]. It comprises mutually interacting electrons which interact also with the periodic lattice and with the lattice vibrations. In order to pass continually to the problem of the interplay of EPI and strong correlations and also to explain why the LDA-DFT method is inadequate for HTSC cuprates, we discuss this problem here with restricted details—more extended discussion can be found in [6, 194]. In order to describe superconductivity the Nambu-spinor ψ̂†(r)=(ψ̂↑†(r)ψ̂↓(r)) is introduced which operates in the electron-hole space (ψ̂(r)=(ψ̂†(r))†) where ψ̂↑(r), ψ̂↑†(r) are annihilation and creation operators for spin up, respectively, and so forth. The microscopic Hamiltonian of the system under consideration contains three parts: Ĥ=Ĥe+Ĥi+Ĥe-i. The electronic HamiltonianĤe, which describes the kinetic energy and the Coulomb interactions of electrons, is given by
(46)Ĥe=∫d3rψ̂†(r)τ̂3ϵ0(p̂)ψ̂(r)+12∫d3rd3r′ψ̂†(r)τ̂3ψ̂(r)Vc(r-r′)ψ̂†(r′)τ̂3ψ̂(r′),
where ϵ0(p̂)=p̂2/2m is the kinetic energy of electron and Vc(r-r′)=e2/|r-r′| is the electron-electron Coulomb interaction. Note that in the electron-hole space the pseudospin (Nambu) matrices τ̂i, i=0,1,2,3 are Pauli matrices. Since we will discuss only the electronic properties, the explicit form of the lattice HamiltonianĤi [6, 194] is omitted here. The electron-ion Hamiltonian describes the interaction of electrons with the equilibrium lattice and with its vibrations, respectively:
(47)Ĥe-i=∑n‍∫d3rVe-i(r-Rn0)ψ̂†(r)τ̂3ψ̂(r)+∫d3rΦ̂(r)ψ̂†(r)τ̂3ψ̂(r).
Here, Ve-i(r-Rn0) is the electron-ion potential and its form depends on the level of description of the electronic subsystem. For instance, in the all-electron calculations one has Ve-i(r-Rn0)=-Ze2/|r-Rn0| where Ze is the ionic charge. The second term which is proportional to the lattice distortion operator Φ̂(r)=-∑n,αûαn∇αVe-i(r-Rn0)+Φ̂anh(r) (because of convenience it includes also the EPI coupling ∇αVe-i) describes the interaction of electrons with harmonic (~ûαn) (or anharmonic ~Φ̂anh(r)) lattice vibrations.Dyson's equations for the electron and phonon Green's functionsĜ(1,2)=-〈Tψ̂(1)ψ̂†(2)〉, D̃(1-2)=-〈TΦ̂(1)Φ̂(2)〉 are Ĝ-1(1,2)=Ĝ0-1(1,2)-Σ̂(1,2) and D̃-1(1,2)=D̃0-1(1,2)-Π̃(1,2), where the Ĝ0-1(1,2)=[(-∂/∂τ1-ϵ0(p1)+μ)τ̂0-ueff(1)τ̂3]δ(1-2) is the bare inverse electronic Green's function. Here, 1=(r1,τ1), where τ1 is the imaginary time in the Matsubara technique, and the effective one-body potential ueff(1)=Ve-i(1)+VH++〈Φ̂(1)〉, where VH is the Hartree potential. The electron and phonon self-energies Σ̂(1,2) and Π̃(1,2) take into account many-body dynamics of the interacting system. The electronic self-energyΣ̂(1,2)=Σ̂c(1,2)+Σ̂ep(1,2) is obtained in the form
(48)Σ̂(1,2)=-Veff(1,1¯)τ̂3Ĝ(1,2¯)Γ̂eff(2¯,2;1¯),
where integration (summation) over the bar indices is understood. The effective retarded potential Veff(1,1¯) in (48) contains the screened (by the electron dielectric function ɛe(1,2)) Coulomb and EPI interactions:
(49)Veff(1,2)=Vc(1-1¯)ɛe-1(1¯,2)+ɛe-1(1,1¯)D̃(1¯,2¯)ɛe-1(2¯,2).
The inverse electronic dielectric permeability ɛe-1(1,2) = δ(1-2)+Vc(1-1¯)P(1¯,2¯)ɛe-1(2¯,2) is defined via the irreducible electronic polarization operator P(1,2) = -Sp{τ̂3Ĝ(1,2¯)Γ̂eff(2¯,3¯;2)Ĝ(3¯,1+)}. The vertex functionΓ̂eff(1,2;3) = -δĜ(1,2)/δueff(3) in (48) is the solution of the complicated (and practically unsolvable) integro-differential functional equation
(50)Γ̂eff(1,2;3)=τ̂3δ(1-2)δ(1-3)+δΣ̂(1,2)δĜ(1¯,2¯)Ĝ(1¯,3¯)Ĝ(4¯,2¯)Γ̂eff(3¯,4¯;3).
Note that the effective vertex function Γ̂eff(1,2;3), which takes into account all renormalizations going beyond the simple Coulomb (RPA) screening, is the functional of both the electronic and phononic Green's functions Ĝ and D̃, thus making at present the ab initio microscopic equations practically unsolvable.


#### 2.1.2. Low-Energy Migdal-Eliashberg Theory

If the vertex functionΓ̂eff would be known, we would have a closed set of equations for Green's functions which describe dynamics of the interacting electrons and lattice vibrations (phonons) in the normal and superconducting state. However, this is a formidable task and at present far from any practical realization. Fortunately, we are mostly interested in low-energy phenomena (with energies |ωn|,ξ≪ωc and for momenta k=kF+δk in the shell δk≪δkc near the Fermi momentum kF;ωc and δkc are some cutoffs), which allows us further simplification of equations [1, 2]. Therefore, the strategy is to integrate high-energy processes—see more in [194]. Here, we sketch this procedure briefly. Namely, Green's function Ĝ(k,ωn)=[iωn-(k2/2m-μ)τ̂3-Σ̂(k,ωn)]-1 can be formally written in the form
(51)Ĝ(k,ωn)=Ĝlow(k,ωn)+Ĝhigh(k,ωn),
where Ĝlow(k,ωn)=Ĝ(k,ωn)Θ(ωc-|ωn|)Θ(δkc-δk) is the low-energy Green's function and Ĝhigh(k,ωn)=Ĝ(k,ωn)Θ(|ωn|-ωc)Θ(δk-δkc) is the high-energy one and analogously D=Dlow(k,ωn)+Dhigh(k,ωn). By introducing the small parameter of the theory s~(ω/ωc)~(δk/δkc)≪1 one has in leading order Ĝlow(k,ωn)~s-1,  Ĝhigh(k,ωn)≲1 and Dlow(k,ωn)~s0,  Dhigh(k,ωn)~s2. Note that the coupling constants (Vei,∇Vei,Vii, etc.) are of the order s0=1.The procedure of separating low-energy and high-energy processes lies also behind theadiabatic approximation since in most materials the characteristic phonon (Debye) energy ωD of lattice vibrations is much smaller than the characteristic electronic Fermi energy EF (ωD≪EF). In the small s(≪1) limit the Migdal theory [1, 2] keeps in the total self-energy Σ linear terms in the phonon propagator D̃ (D) only. In that case the effective vertex function can be written in the form Γ̂eff(1,2;3)≅Γ̂c(1,2;3)+δΓ̂ep(1,2;3) [1, 2], where the Coulomb charge vertex Γ̂c(1,2;3)=τ̂3δ(1-2)δ(1-3)+δΣ̂c(1,2)/δueff(3) contains correlations due to the Coulomb interaction only but does not contain EPI  and phonon propagator D̃ explicitly. The part δΓ̂ep(1,2;3)=δΣ̂ep(1,2)/δueff(3) contains all linear terms with respect to EPI. Note that in these diagrams enters the dressed Green's function which contains implicitly EPI up to infinite order. By careful inspection of all (explicit) contributions to δΓ̂ep(1,2;3) which is linear in D̃ one can express the self-energy in terms of the charge (Coulomb) vertex Γ̂c(1,2;3) only. As a result of this approximation, the part of the self-energy due to Coulomb interaction is given by
(52)Σ̂c(1,2)=-Vcsc(1,1¯)τ̂3Ĝ(1,2¯)Γ̂c(2¯,2;1¯),
where Vcsc(1,2)=Vc(1,2¯)ɛe-1(2¯,2) is the screened Coulomb interaction. The part which is due to EPI has the following form:
(53)Σ̂ep(1,2)=-Vep(1¯,2¯)Γ̂c(1,3¯;1¯)Ĝ(3¯,4¯)Γ̂c(4¯,2;2¯),
where Vep(1,2)=ɛe-1(1,1¯)D̃(1¯,2¯)ɛe-1(2¯,2) is the screened EPI potential. Note that Σ̂ep(1,2) depends now quadratically on the charge vertex Γ̂c, which is due to the adiabatic theorem.It is well known that the Coulomb self-energyΣ̂c(1,2) is the most complicating part of the electronic dynamics, but since we are interested in low-energy physics when s≪1, then the term Σ̂c(1,2) can be further simplified by separating it in two parts:
(54)Σ̂c(1,2)=Σ̂c(h)(1,2)+Σ̂c(l)(1,2).
The term Σ̂c(h)(1,2) is due to high-energy processes contained in the product Ĝhigh(1,2¯)Γ̂chigh(2¯,2;1¯) (e.g., due to the large Hubbard U in strongly correlated systems) and Σ̂c(l)(1,2) is due to low-energy processes. The leading part of Σ̂c(h)(1,2) is 1, that is, Σ̂c(h)(1,2)~s0, while Σ̂c(l)(1,2) is small of order 1, that is, Σ̂c(l)(1,2)~s1. For further purposes we define the quantity V̂0 as
(55)V̂0(1,2)={Ve-i(1)+VH(1)}τ3δ(1-2)+Σ̂c(h)(1,2),
where Ve-i, VH are also of order s0. After the Fourier transform with respect to time (and for small |ωn|≪ωc) Σ̂c(h) is given by
(56)Σ̂c(h)(x1,x2,ωn)≃Σc0(h)(x1,x2,0)τ̂3+(Σ̂c0(h))′(x1,x2,0)·iωn.
As we said, Σc0(h)~s0 while (Σ̂c0(h))′·ωn~s1 because ωn~s1. From (52) it is seen that the part Σ̂c(l)(1,2) contains the low-energy Green's function Ĝlow(1,2¯) and this skeleton diagram is of order s1. The similar analysis based on (53) for Σ̂ep(1,2) gives that the leading order is s1 which describes the low-energy part of EPI. After the separations of terms (of s0 and s1 orders) the Dyson equation in the low-energy region has the form
(57)[iωnZc(x,x¯)-Ĥ0(x,x¯)-Σ̂c(l)(x,x¯,ωn)-Σ̂ep(x,x¯,ωn)]×Ĝlow(x¯,y,ωn)=δ(x-y)τ̂0,
where x¯ means integration ∫d3x¯ over the crystal volume. The Coulomb renormalization function Zc(x,y)=δ(x-y)-(Σ0c(h))′(x,y,0) and the single-particle Hamiltonian Ĥ0(x,y) collect formally all high-energy processes which are unaffected by superconductivity (which is low-energy process) where
(58)Ĥ0(x,y)={(-12m∇x2-μ)δ(x-y)+V0(h)(x,y,0)}τ̂3
with
(59)V0(h)(x,y,0)={Ve-i(x)+VH(x)}δ(x-y)+Σc0(h)(x,y,0).
One can further absorb Zc(x,y) into the renormalized Green's function
(60)Ĝr(x,y,ωn)=Zc1/2(x,x¯)Ĝlow(x¯,y¯,ωn)Zc1/2(y¯,y),
the renormalized vertex function Γ̂ren(1,2;3)=Zc-1/2Γ̂cZc-1/2, and the renormalized self-energies
(61)Σ̂r;c,ep(l)(x,y,ωn)=Zc-1/2(x,x¯)Σ̂c,ep(l)(x¯,y¯,ωn)Zc-1/2(y¯,y)
and introduce the ideal band-structure Hamiltonianĥ0(x,y)=Zc-1/2(x,x¯)Ĥ0(x¯,y¯)Zc-1/2(y¯,y) given by
(62)ĥ0(x,y)={(-12m∇x2-μ)δ(x-y)+VIBS(x,y)}τ̂3.
Here,
(63)VIBS(x,y)=Zc-1/2(x,x¯)V0(h)(x¯,y¯)Zc-1/2(y¯,y)
is the ideal band-structure potential (sometimes called the excitation potential) and apparently nonlocal quantity, which is contrary to the standard local potential Vg(x) in the LDA-DFT theories—see Section 2.2. The static potential VIBS(x,y) is of order s0 and includes high-energy processes.Finally, we obtain the matrix Dyson equation for the renormalized Green's functionĜr(x,y,ωn) which is the basis for the (strong-coupling) Migdal-Eliashberg theory in the low-energy region
(64)[iωnδ(x-x¯)-ĥ0(x,x¯)-Σ̂c,r(l)(x,x¯,ωn)-Σ̂ep,r(x,x¯,ωn)]×Ĝr(x¯,y,ωn)=δ(x-y)τ̂0,
where Σ̂c,r(l) and Σ̂ep,r have the same form as (52)-(53) but with the renormalized Green's and vertex functions Ĝr,Γ̂r instead of Ĝ,Γ̂. We stress that (64) holds in the low-energy region only. In the superconducting state the set of Eliashberg equations in (64) are written explicitly in Appendix A, where it is seen that the superconducting properties depend on the Eliashberg spectral function αkp2F(ω). The latter function is defined also in Appendix A, (A.4), and it depends on material properties of the system.The important ingredients of the low-energy Migdal-Eliashberg theory are the ideal band-structure Hamiltonianĥ0(x,y)—given by (62) which contains many-body (excitation) ideal band-structure nonlocal periodic crystal potential VIBS(x,y). The Hamiltonian ĥ0(x,y) determines the ideal energy spectrumϵ(k) of the conduction electrons and the wave function ψi,p(x) through
(65)ĥ0(x,y¯)ψi,k(y¯)=[ϵi(k)-μ]ψi,k(x),
where μ is the chemical potential. We stress that the Hamiltonian ĥ0(x,y) also governs transport properties of metals in low-energy region.After solving (65) the next step is to expand all renormalized Green's function, self-energies, vertices, and the renormalized EPI matrix element (written symbolically as  gep,r=gep,0Γrenɛe-1) in the basis of ψi,p(x) and to write down the Eliashberg equations in this basis. We will not elaborate further this program and refer the reader to the relevant literature in [193, 194]. We point out that even such simplified program of the low-energy Migdal-Eliashberg theory was never fully realized in low-temperature superconductors, because the nonlocal potential VIBS(x,y) (enters the ideal band-structure Hamiltonian h0(x,y)) and the renormalized vertex function (entering the EPI coupling constant gep,r) which include electronic correlations are difficult to calculate especially in strongly correlated metals. Therefore, it is not surprising at all that the situation is even more difficult in HTS materials which are strongly correlated systems with complex structural and material properties. Due to these difficulties the calculations of the electronic band structure and the EPI coupling are usually done in the framework of LDA-DFT where the many-body excitation potential VIBS(x,y) is replaced by some (usually local) potential VLDA(x) which in fact determines the ground-state properties of the crystal. In the next section we briefly describe (i) the LDA-DFT procedure in calculating the EPI coupling constant and (ii) some results of the LDA-DFT calculations related to HTSC cuprates. We will also discuss why this approximation is inappropriate when applied to HTS materials.


### 2.2. LDA-DFT Calculations of the EPI Matrix Elements

We point out again two results which are important for the future microscopic theory of pairing in HTSC cuprates.First, numerous experiments (discussed in Part I) give evidence that the EPI coupling constant which enters the normal part of the quasiparticle self-energy λepZ=λs+λd+⋯ is rather large, that is, 1<λepZ<3.5. In order to be conform with d-wave pairing the effective EPI potential must be nonlocal (and peaked at small transfer momenta q), which implies that the s-wave and d-wave coupling constants are of the same order, that is, λd≈λs. Second, the theory based on the minimal t-J model, which will be discussed in Section 2.3, gives that strong electronic correlations produce a peak at small transfer momenta in the effective EPI pairing potential thus giving rise to λd≈λs. This is a striking property which allows that EPI is conform with d-wave pairing. However, the theory is seriously confronted with the problem of calculation of the coupling constants λepZ. It turns out that at present it is an illusory task to calculate λepZ and λd since it is extremely difficult (if possible at all) to incorporate the peculiar structural properties of HTSC cuprates (layered structure, ionic-metallic system, etc.) and strong correlations effects in a consistent and reliable microscopic theory which is described in Section 2.1. As it is stressed several times, the LDA-DFT methods miss some important many-body effects (especially in the band-structure potential) and therefore fail to describe correctly screening properties of HTSC cuprates and the strength of EPI. However, the LDA-DFT methods are able to incorporate diverse structural properties of HTSC cuprates much better than the simplified minimal t-J (toy) model. Here, we discuss briefly some achievements of the advanced LDA-DFT calculations which are able to take partially into account some nonlocal effects in the EPI. The latter are mainly due to the almost ionic structure along thec-axis which is reflected in the very small c-axis plasma frequency (ωc≪ωab).The main task of the LDA-DFT theory in obtaining the EPI matrix elements is to calculate the change of the ground-state (self-consistent) potentialδVg(r)/δRα and the EPI coupling constant (matrix element) gαLDA(k,k′) (see its definition below), which is the most difficult part of calculations. Since in the LDA-DFT method the EPI scattering cannot be formulated, then the recipe is that the calculated gαLDA(k,k′) is inserted into the many-body Eliashberg equations. By knowing gαLDA(k,k′) one can define the total (λ) and partial (λq,ν) EPI coupling constants for the νth mode, respectively [195], as
(66)λ=1Np∑q,νλq,ν,λq,ν=pγq,νπN(0)ωq,ν,
where p=3κ is the number of phonon branches (κ is the number of atoms in the unit cell) and N(0) is the density of states at the Fermi energy (per spin and unit cell). The phonon line-widthγq,ν is defined in the Migdal-Eliashberg theory by
(67)γq,ν=2πωq,ν1N∑ll′k12Mωk-q,ν|eνα(q)·gα,ll′(k,k-q)|2×[nF(ξl,k)-nF(ξl,k+ℏωq,ν)ℏωq,ν]×δ(ξl′,k-q-ξl,k-ℏωq,ν).
Here, eνα(q) is the phonon polarization vectors; nF is the Fermi function. Since the ideal energy spectrum from (65) ξl,k=El,k-μ and the corresponding eigenfunctions ψkl are unknown, then instead of these one sets in (67) the LDA-DFT eigenvalues for the lth band ξl,k(LDA) and ψkl(LDA). In the LDA-DFT method the EPI matrix elementgα,ll′(LDA) is defined by the change of the ground-state potential δVg(r)/δRα:
(68)gα,ll′(LDA)(k,k′)=〈ψkl(LDA)|∑nδVg(r)δRnα|ψk′l′(LDA)〉.
The index n means summation over the lattice sites; α=x,y,z and the wave function ψkl(LDA) are the solutions of the Kohn-Sham equation—see [6]. In the past various approximations within the LDA-DFT method have been used in calculating δVg(r)/δRα and λ while here we comment some of them only. (i) In most calculations in LTS systems and in HTSC cuprates the rigid-ion (RI) approximation was used as well as its further simplifications which inevitably (due to its shortcomings and obtained small λ) deserves to be commented. The RI approximation is based on the very specific assumption that the ground-state (crystal) potential Vg(r) can be considered as a sum of ionic potentials Vg(r)=∑nVg(r-Rn) where the ion potential Vg(r-Rn) and the electron density ρe(r) are carried rigidly with the ion at Rn during the ion displacement (Rn=Rn0+ûαn). In the RI approximation the change of Vg(r) is given by
(69)δVg(r)=∑n‍∇αVg(r-Rn0)uαn,δVg(r)δRnα=∇αVg(r-Rn0),
which means that RI does not take into account changes of the electron density during the ion displacements. In numerous calculations applied to HTSC cuprates the rigid-ion model is even further simplified by using the rigid muffin-tin approximation (RMTA) (or similar version with the rigid-atomic sphere)—see discussions in [195–198]. The RMTA assumes that the ground-state potential and the electron density follow ion displacements rigidly inside the Wigner-Seitz cell while outside it Vg(r) is not changed because of the assumed very good metallic screening (e.g., in simple metals):
(70)∇αVg(r-Rn)={∇αVg(r-Rn),rincelln,0,outside.
This means that the dominant EPI scattering is due to the nearby atoms only and that the scattering potential is isotropic. All nonlocal effects related to the interaction of electrons with ions far away are neglected in the RMTA. In this case gα,nLDA(k,k′) is calculated by the wave function centered at the given ion Rn0 which can be expanded inside the muffin-tin sphere (outside it the potential is assumed to be constant) in the angular momentum basis {l,m}, that is,
(71)〈r∣ψk(RMTA)〉=∑lmnClm(k,Rn0)ρl(|r-Rn0|)Ylm(ϕ,θ)
(the angles ϕ,θ are related to the vector r̂=(r-Rn0)/|r-Rn0|). The radial function ρl(|r-Rn0|) is zero outside the muffin-tin sphere. In that case the EPI matrix element is given by gα,nRMTA(k,k′)~〈Ylm|r̂|Yl′m′〉 and because r̂ is vector the selection rule implies that only terms with Δl≡l′-l=±1 contribute to the EPI coupling constant in the RMTA. This result is an immediate consequence of the assumed locality of the EPI potential in RMTA. However, since nonlocal effects, such as the long-range Madelung-like interaction, are important in HTSC cuprates, then additional terms contribute also to the coupling constant gα,n, that is, gα,n(k,k′)=gα,nRMTA(k,k′)+gα,nnonloc(k,k′), where a part (δgα,nnonloc) of the nonlocal contribution to gα,nnonloc is represented schematically:(72)δgα,nnonloc(k,k′)~〈Ylm|(Rn0-Rm0)α|Yl′m′〉.
From (72) comes out the selection rule Δl=l′-l=0 for the nonlocal part of the E-P interaction. We stress that the Δl=0 (nonlocal) terms are omitted in the RMTA approach and therefore it is not surprising that this approximation works satisfactorily in elemental (isotropic) metals only. The latter are characterized by the large density of states at the Fermi surface which makes electronic screening very efficient. This gives rise to a local EPI where an electron feels potential changes of the nearby atom only. One can claim with certainty that the RMTA method is not suitable for HTSC cuprates which are highly anisotropic systems with pronounced ionic character of binding and pronounced strong electronic correlations. The RMTA method applied to HTSC cuprates misses just this important part—the long-range part EPI due to the change of the long-range Madelung energy in the almost ionic structure of HTSC cuprates. For instance, the first calculations done in [199] which are based on the RMTA give very small EPI coupling constant λRMTA~0.1 in YBCO, which is in apparent contradiction with the experimental finding that λep is large—see Section 1.However, these nonlocal effects are taken into account in [195] by using the frozen-in phonon (FIP) method in evaluating of λep in La2-xMxCuO4. In this method some symmetric phonons are considered and the band structure is calculated for the system with the super-cell which is determined by the periodicity of the phonon displacement. By comparing the unperturbed and perturbed energies the corresponding EPI coupling λν (for the considered phonon νth mode) is found. More precisely speaking, in this approach the matrix elements of δVg(r)/δR0,ακ are determined from the finite difference of the ground-state potential ΔVg,q,ν(r)=Vg(R0,Lκ+Δτq,νκ(L))-Vg(R0,Lκ)=∑L,κΔτq,νκ(L)∂Vg(R0,Lκ)/∂R0,Lκ, where L,κ enumerate elementary lattice cells and atoms in the unit cell, respectively. The frozen-in atomic displacements of the phonon Δτq,νκ(L) of the νth mode are given by Δτq,νκ(L)=Δuq,ν(ℏ/2Mκωq,ν)1/2Re[eκ,ν(q)eiq·R] where Δuq,ν is the dimensionless phonon amplitude and the phonon polarization (eigen)vector eκ,ν(q) fulfills the condition ∑κeκ,ν*(q)·eκ,ν′(q)=δν,ν′. Based on this approach various symmetric Ag (and some B3g) modes of La2-xMxCuO4 were studied [195], where it was found that the large matrix elements are due to unusually long-range Madelung-like, especial for the c-axis phonon modes. The obtained large λep≈1.37 is the consequence of the following three main facts. (i) The electronic spectrum in HTSC cuprates is highly anisotropic, that is, it is quasi-two-dimensional. This is an important fact for pairing because if the conduction electrons would be uniformly spread over the whole unit cell then due to the rather low electron density (n~1021cm-3) the density of states on the Cu and O in-plane atoms would be an order of magnitude smaller than the real value. This would further give an order of magnitude smaller EPI coupling constant λep. Note that the calculated density of states on the (heavy) Cu and (light) O in-plane atoms, NCu(0)≅0.54states/eV  and NOxy(0)≅0.35states/eV, is of same order of magnitude as in some LTS materials. For instance, in NbC where Tc≈11K one has on (the heavy) Nb atom NNb(0)≅0.58states/eV and on (the light) C atom NC(0)≅0.25states/eV. So, the quasi-two-dimensional character of the spectrum is crucial in obtaining appreciable density of states on the light O atoms in the CuO2 planes. (ii) In HTSC cuprates there is strong Cu–O hybridization leading to good in-plane metallic properties. This large covalency in the plane is due to the (fortunately) small energy separation of the electron levels on Cu and Oxy atoms which comes out from the band-structure calculations [200], that is, Δ=|ϵCu-ϵOxy|≈3eV. The latter value gives rise to strong covalent mixing (the hybridization parameter tpd) of the Cudx2-y2 and Opx,y states, that is, tpd=-1.85eV. It is interesting that the small value of Δ is not due to the ionic structure (crystal field effect) of the system but it is mainly due to the natural falling of the Cudx2-y2 states across the transition-metal series. So, the natural closeness of the atomic energy levels of the Cudx2-y2 and Opx,py states is this distinctive feature of HTSC cuprates which basically allows achievement of high Tc. (iii) The ionic structure of HTSC  cuprates which is very pronounced along the c-axis is responsible for the weak electronic screening along this axis and according to that for the significant contribution of the nonlocal (long-range) Madelung-like interaction to EPI. It turns out that because of the ionicity of the structure the La and Oz  axial modes are strongly coupled with charge carriers in the CuO2 planes despite the fact that the local density of states on these atoms is very small [195], that is, NLa(0)=0.13states/eV and NOz(0)=0.016states/eV. (For comparison, on planar atoms Cu and Oxy one has NCu(0)=0.54states/eV  and NOxy(0)=0.35states/eV.) These calculations show that the lanthanum mode (with ωq,ν=202cm-1) at the q=(0,0.2π/c) zone boundary (fully symmetric Z-point) has ten times larger coupling constant λq,νLa(FIP)=4.8 than it is predicted in the RMT approximation λq,νLa(RMT)=0.48. The similar increase holds for the average coupling constant, where λν,averageLa(FIP)=1.0 but λν,averageLa(RMT)=0.1. Note that for the q≈0 La-mode one obtains λνLa(FIP)=4.54 compared  to λνLa(RMT)=0.12. Similar results hold for the axial apex-oxygen q=(0,0.2π/c) mode (Oz) with ωq,ν=396cm-1 where the large (compared to the RMT method) coupling constant is obtained: λq,νOz=14 and λν,averageOz=4.7, while for q≈0 axial apex-oxygen modes with ωq,ν=415cm-1 one has λν,averageOz=11.7. After averaging over all calculated modes it was estimated that λ=1.37 and ωlog≈400K. By assuming that μ*=0.1 one obtains Tc=49K by using Allen-Dynes formula for Tc≈0.83ωlogexp{-1.04(1+λ)/[λ-μ*(1+0.62λ)]} with ωlog=2∫dωdωα2(ω)F(ω)lnω/λω.  For μ*=0.15 and 0.2 one obtains Tc=41 and 32K, respectively. We stress that the rather large λep (and Tc) is due to the nonlocal (long range) effects of the metallic-ionic structure of HTSC cuprates and non-muffin-tin corrections in EPI, as was first proposed in [201, 202]. However, we would like to stress that the optimistic results for λep obtained in [195] are in fact based on the calculation of the EPI coupling for some wave vectors q with symmetric vibration patterns and in fact the obtained λep is an extrapolated value. The all-q calculations of λep,q which take into account long-range effects are a real challenge for the LDA-DFT calculations and are still awaiting.Finally, it is worth to mention important calculations of the EPI coupling constant in the framework of thelinear-response full-potential linear-muffin-tin-orbital  method (LRFP-LMTO) invented in [203, 204] and applied to the doped HTSC cuprate (Ca1-xSrx)1-yCuO2 for x~0.7 and y~0.1 with Tc=110K [205]. Namely, these calculations give strong evidence that the structural properties of HTSC cuprates already alone make the dominance of small-q scattering in EPI, whose effect is additionally increased by strong correlations. In order to analyze this compound in [205] the calculations are performed for CaCuO2 doped by holes in a uniform, neutralizing back-ground charge. The momentum (q=(q∥,q⊥)) dependent EPI coupling constant (summed over all phonon branches ν) in different L channels (s,p,d.) is calculated by using a standard expression
(73)λL(q)=M∑k,νYL(k+q)|gk,q,ν|2YL(k)δ(ξk+q)δ(ξk).
Here, ξk is the quasiparticle energy, gk,q,ν is the EPI coupling constant (matrix element) with the νth branch, YL(k) is the L-channel wave function, and the normalization factor M∝NL-1(0) with the partial density of states is NL(0)∝∑kYL2(k)δ(ξk). The total coupling constant in the L-channel is an average of λL(q) over the whole 2D Brillouin zone (over q∥), that is, λL(q⊥)=〈λL(q∥)〉BZ. We stress three important results of [205]. First, the s- and d-coupling constants, λs(q), λd(q), are peaked at small transfer momentaq~(π/3a,0,0) as it is shown in [205, Figure  3]. This result is mainly caused by the nesting properties of the Fermi surface shown in [205, Figure  1]. Second, the q-dependence of the integrated EPI matrix elements |g¯L,q|2=λL(q)/χL′′(q) (with χL′′(q)∝∑kYL(k+q)YL(k)δ(ξk+q)δ(ξk)) for L=s,d  is similar to that of λL(q), that is, these are peaked at small transfer momenta q≪2kF. Both of these results mean that the structural properties of HTSC cuprates imply the dominance of small-q EPI scattering. Third, the calculations give similar values for λs(q⊥=0) and λd(q⊥=0), that is, λs=0.47 for s-wave and λd=0.36 for d-wave pairing [205]. The result that λd≈λs is due to the dominance of the small q-scattering in EPI, which means that the nonlocal effects (long-range forces) in EPI of HTSC cuprates are very important. This result together with the finding of the dominance of the small-q scattering in EPI due to strong correlations [78–80, 130, 179, 180] mean that strong correlations and the peculiar structural properties of HTSC cuprates make EPI conform with d-wave pairing, either as its main cause or as its supporter. We stress that the obtained coupling constant λd=0.36 is rather small to give d-wave pairing with large Tc and on the first glance this result is against the EPI mechanism of pairing in cuprates. However, it is argued throughout this paper that the LDA methods applied to strongly correlated systems overestimate the screening effects and underestimate the coupling constant and therefore their quantitative predictions are not reliable.


### 2.3. EPI and Strong Correlations in HTSC Uprates

#### 2.3.1. Minimal Model Hamiltonian

Theminimal microscopic model for HTSC cuprates must include at least three orbitals: one dx2-y2-orbital of the Cu-ion and two p-orbitals (px,y) of the O-ion since they participate in transport properties of these materials—see more in [6] and references therein. The electronic part of the Hamiltonian (of the minimal model) is Ĥ=Ĥ0+Ĥint—usually called the Emery model (or the p-dmodel) [206], where the one-particle tight-binding Hamiltonian Ĥ0 describes the lowering of the kinetic energy in the p-d model (with three bands or orbitals):
(74)Ĥ0=∑i,σ(ϵd0-μ)diσ†diσ+∑j,α,σ(ϵpα0-μ)pjασ†pjασ+∑i,j,α,σtijαpddiσ†pjασ+∑j,j′,α,β,σtjj′,αβpppjασ†pj′βσ.
Here tijαpd (i,j enumerate the Cu- and O-sites, resp.) is the hopping integral between the pα(α=x,y)—and d-states and tjj′αβpp between the pα- and pβ-states—while ϵd0 and ϵpα0 are the bare d- and p-local energy levels and μ is the chemical potential. This tight-binding Hamiltonian is written in the electronic notation where the charge-transfer energy Δdp,0≡ϵd0-ϵp0>0 by assuming that there is one 3dx2-y2 electron on the copper (Cu2+) while electrons in the p-levels of the O2- ions occupy filled bands. Ĥ0 contains the main ingredients coming from the comparison with the LDA-DFT band-structure calculations. The LDA-DFT results are reproduced by assuming that tpp≪tpd (and ϵpα0=ϵp0 ) where the good fit to the LDA-DFT band structure is found for Δdp,0≡ϵd0-ϵp0≈3.2eV and tpd(≡tpd)=(3/2)(pdσ),(pdσ)=-1.8eV. The total LDA bandwidth Wb=(42)|tpd|≅9eV [207].The electron interaction is described byĤint:
(75)Ĥint=Ud∑ini↑dni↓d+Up∑j,αnjα↑pnjα↓p+V̂c+V̂ep,
where Ud and Up are the on-site Coulomb repulsion energies at Cu and O sites, respectively, while V̂c and V̂ep describe the long-range part of the Coulomb interaction of electrons (holes) and EPI, respectively. Note that the Hubbard repulsion Ud on the Cu-ion is different from its bare atomic value Ud0(≈16 eV for Cu) due to various kinds of screening effects in solids [208–210]. It turns out that in most transition metal oxides one has Ud≪Ud0. This problem is thoroughly studied in [208–210] and applied to HTSC cuprates. The estimation from the numerical cluster calculations [211] gives Ud=9–11eV and Up=4–6eV but because nidUd≫njpUp the on-site repulsion on the oxygen ion is usually neglected at the first stage of the analysis.Note that in the case of largeUd(≫tpd,Δdp,0) the hole notation is usually used where in the parent compound (and for |tpd|≪Δdp,0) one has 〈nid〉=1, that is, one hole in the 3D-shell (in the 3dx2-y2 state) in the ground state. In the limit of large Ud→∞ the doubly occupancy on the Cu atoms is forbidden and only two copper states are possible: Cu2+—described by the quantum state diσ†|0〉 with one hole in the 3D shell and Cu1+—described by |0〉 with zero holes in the filled 3D shell. In this (hole) notation the oxygen p-level is fully occupied by electrons, that is, there are no holes (〈njp〉=0) in the occupied oxygen 2p-shell of O2-. In this notation the vacuum state |0v〉 (not the ground state) of the Hamiltonian Ĥ for large Ud corresponds to the closed-shell configuration Cu1+O2-. In the hole notation the hole p-level ϵph0 lies higher than the hole d-level ϵdh0, that is, Δpd,0≡ϵph0-ϵdh0>0 (note that in the electron notation it is opposite) and Ud means repulsion of two holes (in the 3dx2-y2 orbital) with opposite spins—3d8 configuration of the Cu3+ ion. Note that ϵph0=-ϵp0,ϵdh0=-ϵd0, and tpd,h=-tpd. In the following the index h in tpd,h is omitted. The reason for ϵph0>ϵdh0 is partly in different energies for the hole sitting on the oxygen and copper, respectively [207]. From this model one can derive in the limit U→∞ the t-J model for the 2D lattice in the CuO2 plane [212, 213], where now each lattice site corresponds to a Cu-atom. In the presence of one hole in the 3D-shell then in the undoped (no oxygen holes) HTSC cuprate each lattice site is occupied by one hole. By doping the system with holes the additional holes go onto O-sites. Furthermore, due to the strong Cu–O covalent binding the energetics of the system implies that an O-hole forms a Zhang-Rice singlet with a Cu-hole [212]. In the t-J model the Zhang-Rice singlet is described by an empty site.Since in the t-J model the doubly occupancy is forbidden, one introduces annihilation (Hubbard) operator of the composite fermion X̂iσ0=ciσ†(1-ni,-σ) which describes creation of a hole (in the 3D-shell of the Cu-atoms) on the ith site if this site is previously empty (thus excluding doubly occupancy), that is, the constraint ni,σ+ni,-σ≤1 must be fulfilled on each lattice site.  In this picture the doped-hole concentration δ means at the same time the concentration of the oxygen holes, that is, of the Zhang-Rice singlets.In order not to confuse the reader we stress the difference in the meaning of the hole in the (p-d) three-band Emery model and in the single-band (effective) t-J model. In the Emery model the hole means the absence of the electron in the filled shell—the 3D shell for Cu atoms(ions) and 2p shell for O atoms(ions). On the other side the hole on the ith lattice site in the t-J model means the presence of the Zhang-Rice singlet on this site.Thebosonic-like operatorsX̂iσ1σ2=X̂iσ10X̂i0σ2 for  σ1≠σ2 create a spin fluctuation at the ith site and the spin operator is given by S=X̂iσ¯10(σ⃗)σ¯1σ¯2X̂i0σ¯2 where summation over the bar indices is understood. The operator ∑σX̂iσσ has the meaning of thehole number on theith site. It is useful to introduce the operator X̂i00=X̂i0σX̂iσ0 at the ith lattice site which is the number of Zhang-Rice singlets on theith site. For X̂i00|0〉=1|0〉 the ith site is occupied by the Zhang-Rice singlet, while for X̂i00|1〉=0|1〉 there is no Zhang-Rice singlet on the ith site (i.e., this site is occupied only by one 3d9 hole on the Cu site). This property of X̂i00 is due to the local constraint(76)X̂i00+∑σ=↑↓X̂iσσ=1,
which forbids doubly occupancy of the ith site by holes. By projecting out doubly occupied (high-energy) states the t-J model reads
(77)Ĥt-j=∑i,σϵi0X̂iσσ-∑i,j,σtijX̂iσ0X̂j0σ+∑i,jJij(Si·Sj-14n̂in̂j)+Ĥ3.
The first term (~ϵi0) describes an effective local energy of the hole (or the Zhang-Rice singlet), the second one (~tij) describes hopping of the holes, and the third one (~Jij) is the Heisenberg-like exchange energy between two holes. The theory [212] predicts that |ϵi0|≫|tij|. This property is very important in the study of EPI. Ĥ3 contains three-site term which is usually omitted believing that it is not important. For charge fluctuation processes it is plausible to omit it, while for spin-fluctuation processes it is questionable approximation. If one introduces the enumeration α,β,γ,λ=0,↑,↓, then the Hubbard operators satisfy the following algebra:
(78)[X̂iαβ,X̂jγλ]±=δij[δγβX̂iαλ±δαλX̂iγβ],
where δij is the Kronecker symbol. Note that the Hubbard operators possess the projection properties with X̂iαβX̂iγλ=δβγX̂iαλ. The (anti)commutation relations in (78) are more complicated than the canonical Fermi and Bose (anti)commutation relations, which complicates the mathematical structure of the theory. To escape these complications some novel techniques have been used, such as the one slave boson-technique. In this technique X̂i0σ=fiσbi†,X̂iσ1σ2=fiσ1†fiσ2 are represented in terms of the fermion (spinon) operator fiσ which annihilates the spin on the ith and the boson (holon) operator bi† which creates the Zhang-Rice singlet.In the minimal theoretical model the electron-phonon interaction (EPI) contains in principle two leading terms:(79)Ĥep=Ĥepion+Ĥepcov,
which are the “ionic” one (Ĥepion) and the “covalent” one (Ĥepcov). The “ionic” term describes the change of the energy of the hole (or the Zhang-Rice singlet) at the ith site due to lattice vibrations and it reads [6, 78–80, 130]
(80)Ĥepion=∑i,σΦ̂iX̂iσσ,
where the “displacement” operator
(81)Φ̂i=∑Lκ[ϵ(Ri0-RLκ0+ûi-ûLκ)-ϵ(Ri0-RLκ0)]
(which as in Section 2.1 includes the bare coupling constant) describes the change of the hole (or Zhang-Rice singlet) energy ϵa,i0 by displacing atoms in the lattice by the vector ûLκ. In the harmonic approximation the EPI potential is given by Φ̂i=∑gi(q,λ)exp{iqRi}[bq,λ+b-q,λ†] where bq,λ and bq,λ† are the annihilation and creation operator of phonons with the polarization λ, respectively. This term describes in principle the following processes: (1) the change of the O-hole and Cu-hole bare energies ϵph0,ϵdh0 in the three-band model due to lattice vibrations, (2) the change of the long-range Madelung energy (which is due to the ionicity of the structure) by lattice vibrations along the c-axis, and (3) the change of the Cu–O hopping parameter tpd in the presence of vibrations, and so forth. Here, L and κ enumerate unit lattice vectors and atoms in the unit cell, respectively. Usually, the EPI scattering is studied in the harmonic approximation where the phonon operator Φ̂i is calculated in the harmonic approximation (Φ̂~û) for the EPI interaction of holes with some specific phononic modes, such as the breathing and half-breathing ones [10, 11, 169]. The theory which includes also all other (than oxygen) vibrations in Φ̂i is still awaiting.It is interesting to make comparison of the EPI coupling constants in thet-J model and in the Hartree-Fock (HF) approximation (which is the analogous of the LDA-DFT method) of the three-band Emery (p-d) model in (74)-(75) when the problem is projected on the single band. For instance, the coupling constant with thehalf-breathing mode at the zone boundary in the HF approximation (which mimics the LDA-DFT approach) is given by
(82)ghbHF=±4tpd∂tpd∂RCu–O1ϵd-ϵpu0,
while the coupling constant in thet-Jmodelghbt-J(=∂ϵ0/∂RCu–O) is given by
(83)ghbt-J=±4tpd∂tpd∂RCu–O[2p2-1ϵd-ϵp+2p2Ud-|ϵd-ϵp|]u0,
where p=0.96—see [10, 11, 169] and references therein. It is obvious that in the t-J model the electron-phonon coupling is different from the HF one, since the former contains an additional term coming from the many-body effects, which are not comprised by the HF (LDA-DFT) calculations. The first term in (83) describes the hopping of a 3D hole into the O 2p-states and this term exists also in the LDA-DFT coupling constant—see (82). However, the second term in (83), which is due to many-body effects, describes the hopping of an O 2p-hole into the (already) single occupied Cu 3D state and it does not exist in the LDA-DFT approach. Since the corresponding dimensionless coupling constant λhb is proportional to |ghb|2, one obtains that the bare t-J coupling constant is almost three times larger than the LDA-DFT one:
(84)λhbt-J≈3λhbHF.
This example demonstrates clearly that the LDA-DFT method is inadequate for calculating the EPI coupling constant in HTSC cuprates.Note that there is also a covalent contribution toEPI which comes from the change of the effective hopping (t) in of the t-J model (77) and the exchange energy (J) in the presence of atomic displacements:
(85)Ĥepcov=-∑i,j,σ∂tij∂(Ri0-Rj0)(ûi-ûj)X̂iσ0X̂j0σ+∑i,j,∂Jij∂(Ri0-Rj0)(ûi-ûj)Si·Sj.
Here, we will not go into details but only stress that since |ϵi0|≫|tij| then the covalent term in the effectivet-Jmodel is much smaller than the ionic term—see more in [6, 10, 11, 169] and references therein—and in the following only the term Ĥepion will be considered [6, 78–80, 130].


#### 2.3.2. Controllable X-Method for the Quasiparticle Dynamics

The minimal model Hamiltonian for strongly correlated holes with EPI (discussed above) is expressed via the Hubbard operators which obey “ugly” noncanonical commutation relations. The latter property is rather unpleasant for making a controllable theory in terms of Feynmann diagrams (for these “ugly” operators) and some other approaches are required. A possible way out is to express the Hubbard operators in terms of fermions and bosons (which must be confined) as, for instance, in the slave boson (SB) method. However, in real calculations which are based on some approximations the SB method is confronted with some subtle constraints whose fulfillments require very sophisticated mathematical treatment. Fortunately, there is amathematically controllable approach for treating the problem directly with Hubbard operators and without using slave-boson (or fermion) techniques. This method—we call it the X-method—is based on the general Baym-Kadanoff technique which allows to treat the problem by the well-defined and controllable 1/N expansion for the Green's functions in terms of Hubbard operators. This approach is formulated in [214] while the important refinement of the method is done in [78–80, 130]. In the paramagnetic and homogeneous state (with finite doping) the Green's function Gσ1σ2(1-2) is diagonal, that is, Gσ1σ2(1-2)=δσ1σ2G(1-2) where
(86)G(1-2)=-〈TX̂0σ(1)X̂σ0(2)〉=g(1-2)Q,
with the Hubbard spectral weight Q=〈X̂00〉+〈X̂σσ〉. The function g(1-2) plays the role of the quasiparticle Green's function—see more in [6, 78–80, 130, 179, 180]. It turns out that in order to have a controllable theory (1/N expansion) one way is to increase the number of spin components from two to N by changing the constraint (76) into the new one
(87)X̂i00+∑σ=1NX̂iσσ=N2.
In order to reach the convergence of physical quantities in the limit N→∞ the hopping and exchange energy are also rescaled, that is, tij=t0,ij/N and Jij=J0,ij/N. In order to eliminate possible misunderstandings we stress that in the case N>2 the constraint in (87) spoils some projection properties of the Hubbard operators. Fortunately, these (lost) projection properties are not used at all in the refined theory. As a result one obtains the functional integral equation for G(1,2), thus allowing unambiguous mathematical and physical treatment of the problem. In [78–80, 130, 179, 180] it is developed a systematic 1/N expansion for the quasiparticle Green's function g(1-2)(=g0+g1/N+⋯), Q(=Nq0+q1+⋯) (also for G(1-2)) and the self-energy. For large N(→∞) the leading term is G0(1-2)=g0(1-2)Q0=O(N) with g0=O(1) and Q0=〈X̂i00〉=Nδ/2. Here, δ is the concentration of the oxygen holes (that is, of the Zhang-Rice singlets) which is related to the chemical potential by the equation 1-δ=2∑pnF(p) with nF(p)=(eϵ0(k)-μ+1)-1. The quasiparticle Green's function g0(k,ω) and the quasiparticle spectrum ϵ0(k) in the leading order are given by
(88)g0(k,ω)≡G0(k,ω)Q0=1ω-[ϵ0(k)-μ],(89)ϵ0(k)=ϵc-δ·t(k)-∑pJ0(k+p)nF(p).
The level shift is ϵc=ϵ0+2∑pt(p)nF(p) and t(p) is the Fourier transform of the hopping integral tij—see more in [6].Let us summarize themain results of the X-method in leading O(1)-order for the quasiparticle properties in the t-J model [6, 78–80, 130, 179, 180]. (i) The Green's function g0(k,ω) describes the coherent motion  of quasiparticles whose contribution to the total spectral weight of the Green's function G0(k,ω) is Q0=Nδ/2. The coherent motion of quasiparticles is described in leading order by G0(k,ω)=Q0g0(k,ω) and the quasiparticle residuumQ0 disappears in the undoped Mott insulating state (δ=0). This result is physically plausible since in the Mott insulating state the coherent motion of quasiparticles, which is responsible for finite conductivity, vanishes. (ii) The quasiparticle spectrum ϵ0(k) plays the same role as the eigenvalues of the ideal band-structure Hamiltonian ĥ0(x,y) (it contains the excitation potential VIBS(x,y) which is due to high-energy processes of the Coulomb interaction). So, if we would consider ϵtb(k)=-t(k) as the tight-binding parametrization of the LDA-DFT band-structure spectrum which takes int account only weak correlations (with the local potential Vxc(x)δ(x-y)), then one can define a nonlocal excitation potentialVIBStJ(x,y)=ṼIBStJ(x,y)+Vxc(x)δ(x-y) which mimics strong correlations in the t-J model
(90)ṼIBStJ(x,y)≈V0δ(x-y)+(1-δ)t(x-y)-J̃(x-y).
Here, V0=2∑pt(p)nF(p) and t(x-y) is the Fourier transform of t(k) while J̃(x-y) is the Fourier transform of the third term in (89). The relative excitation potential ṼIBStJ(x,y) is due to strong correlations (suppression of doubly occupancy on each lattice site) and as we will see below it is responsible for the short-range screening of EPI in such a way that the forward scattering peak appears in the effective EPI interaction—see discussion below. (iii) For the very low doping ϵ0(k) is dominated by the exchange parameter if J0>δ·t0. However, in the case when J0≪δ·t0 there is a band narrowing by lowering the hole-doping δ, where the band width is proportional to the hole-concentration δ, that is, Wb=z·δ·t0. (iv) The O(1)-order quasiparticle Green's function g0(k,ω) and the quasiparticle spectrum ϵ0(k) in the X-method have similar form as the spinon Green's function g0,f(k,ω)=-〈Tfσfσ†〉k,ω and the spinon energy ϵs(k) in the SB method. However, in the SB method there is a broken gauge symmetry in the metallic state (with δ≠0) which is characterized by 〈b̂i〉≠0. This broken local gauge symmetry in the slave-boson method in O(1) order, which is due to the local decoupling of spinon and holon, is in fact forbidden by Elitzur's theorem. On the other side the local gauge invariance is not broken in the X-method where Green's function G0(k,ω) describes motion of the composite object, that is, simultaneous creation of the hole and annihilation of the spin at a given lattice site, while in the SB theory there is a spin-charge separation because of the broken symmetry (〈b̂i〉≠0). The assumption of the broken symmetry 〈b̂i〉≠0 gives qualitative satisfactory results for the quasiparticle energy for the case N=∞ in D>2 dimensions. However, the analysis of response functions and of higher-order 1/N corrections to the self-energies very delicate in the SB theory and special techniques must be implemented in order to restore the gauge invariance of the theory. On the other side the X-method is intrinsically gauge invariant and free of spurious effects in all orders of the 1/N expansion. Therefore, one expects that these two methods may deliver different results in O(1) and higher order in response functions. This difference is already manifested in the calculation of EPI where the charge vertex in these two methods is peaked at different wave vectors q, that is, at q=0 in the X-method and |q|≠0 in the SB method—see Section 2.3.5. (v) In [215, 216] it is shown that in the superconducting state the anomalous self-energies (which are of O(1/N)-order in the 1/N expansion) of the X- and SB-methods differ substantially. As a consequence, the SB method [217] predicts false superconductivity in the t-J model (for J=0) with large Tc (due to the kinematical interaction), while the X-method gives extremely small Tc(≈0) [215, 216]. So, although the two approaches yield some similar results in leading O(1)-order they, are different at least in next to leading O(1/N)-order.


#### 2.3.3. EPI Effective Potential in thet-J Model

The theory of EPI in the minimalt-J model based on the X-method predicts that the leading term in the EPI self-energy Σep is given by the expression [6, 78–80, 130]
(91)Σep(1,2)=-Vep(1¯-2¯)γc(1,3¯;1¯)g0(3¯-4¯)γc(4¯,2;2¯),
where the screened (by the dielectric constant) EPI potential
(92)Vep(1-2)=ɛe-1(1-1¯)Vep0(1¯-2¯)ɛe-1(2¯-2)
and Vep0(1-2)=-〈TΦ̂(1)Φ̂(2)〉 is the “phonon” propagator which may also describe an anharmonic EPI. It is obvious that (91) is equivalent to (53) in spite the fact that the theory is formulated in terms of the Hubbard operators. The charge vertexγc(1,2;3)=-δg0-1(1,2)/δueff(3) corresponds to the the renormalized vertex Γc,r in (53) and it describes the screening by strong correlations. It depends on the relative excitation potential ṼIBStJ(x,y). The electronic dielectric function ɛe(1-2) describes the screening of EPI by the long-range part of the Coulomb interaction. Note that in the harmonic approximation Φ̂(1) contains the bare EPI coupling constant gep0 and lattice displacement û, that is, Φ̂~gep0û—see more in [6]. (Note that in the above equations summation and integration over bar indices are understood.) The self-energy Σep(k,ω) due to EPI reads
(93)Σep(k,ω)=∫0∞dν〈α2F(k,k′,ν)〉k′R(ω,ν),
with R(ω,ν)=-2πi(nB(ν)+1/2)+ψ(1/2+i)-ψ(1/2-i(ν+ω)/2πT) where nB(ν) is the Bose distribution function and ψ is di-gamma function. The Eliashberg spectral function is given by
(94)α2F(k,k′,ω)=N(0)∑ν|gν(k,k-k′)|2×δ(ω-ων(k-k′))γc2(k,k-k′),
where gν(k,p) is the EPI coupling constant for the νth mode, where the renormalization by long-range Coulomb interaction is included, that is, gν(k,p)=gep,ν0(k,p)/ɛe(p). 〈⋯〉k denotes Fermi-surface average with respect to the momentum k and N(0) is the density of states renormalized by strong correlations. The effect of strong correlations in the adiabatic limit is stipulated in the charge vertex functionγc(k,k-k′) which, as we will see in Section 2.3.4, changes the properties of Vep(q,ν) drastically compared to weakly correlated systems. In fact the charge vertex depends on frequency ω but in the adiabatic limit (ωph≪W) and for qvF>ωph it is practically frequency independent, that is, γc(ad)(k,q,ω)≈γc(k,q,ω=0) where the latter is real quantity. For J=0 in the t-t′ model the 1/N expansion gives N(0)=N0(0)/q0 where q0=δ/2. For J≠0 the density of states N(0) does not diverge for δ→0 where N(0)(~1/J0)>N0(0). The bare density of statesN0(0) is calculated in absence of strong correlations, for instance, by the LDA-DFT method.Depending on the symmetry of the superconducting order parameterΔ(k,ω) (s- and d-wave pairing) various projected averages (over the Fermi surface) of α2F(k,k′,ω) enter the Eliashberg equations. Assuming that the superconducting order parameter transforms according to the representation Γi of the point group C4v of the square lattice (in the CuO2 planes) the appropriate symmetry-projected spectral function is given by
(95)α2Fi(k̃,k̃′,ω)=N(0)8∑ν,j|gν(k̃,k̃-Tjk̃′)|2×δ(ω-ων(k̃-Tjk̃′))×γc2(k̃,k̃-Tjk̃′)Di(j)
where k̃ and k̃′ are momenta on the Fermi line in the irreducible Brillouin zone (1/8 of the total Brillouin zone). Tj,j=1,…,8 denotes the eight point-group transformations forming the symmetry group of the square lattice. This group has five irreducible representations which we distinguish by the label i=1,2,…,5. In the following we discuss the representations i=1  and i=3, which correspond to the s- and d-wave symmetry of the full rotation group, respectively. Di(j) is the representation matrix of the jth transformation for the representation i. Assuming that the superconducting order parameter Δ(k,ω) does not vary much in the irreducible Brillouin zone, one can average over k̃ and k̃′ in the Brillouin zone. For each symmetry one obtains the corresponding pairing spectral functionα2Fi(ω):
(96)α2Fi(ω)=〈〈α2Fi(k̃,k̃′,ω)〉k̃〉k̃′,
which governs the transition temperature for the order parameter with the symmetry Γi. For instance, α2F3(ω) is the pairing spectral function in the d-channel and it gives the coupling for d-wave superconductivity (the irreducible representation Γ3—sometimes labelled as B1g). Performing similar calculations for the phonon-limited resistivity, one finds that the resistivity is related to the transport spectral functionα2Ftr(ω):
(97)αtr2F(ω)=〈〈α2F(k,k′,ω)[v(k)-v(k′)]2〉〉kk′2〈〈v2(k)〉〉kk′.
The effect of strong correlations on EPI was discussed in [130] within the model where gν(k,p) and the phonon frequencies ων(k̃-k̃′) are weakly momentum dependent. In order to elucidate the main effect of strong correlations on EPI and α2Fi(ω) we consider the latter functions for a simple model with Einstein phonon, where these functions are proportional to the (so called) relative coupling constantΛi:
(98)Λi=18N(0)N0(0)∑j=18〈〈|γc(k̃,k̃-Tjk̃′)|2〉〉k̃k̃′Di(j).
Similarly, the resistivity ρ(T)(~λtr~Λtr) is renormalized by the correlation effects where the transport coupling constant Λtr is given by
(99)Λtr=N(0)N0(0)〈〈|γc(k̃,k̃-Tjk̃′)|2[v(k)-v(k′)]2〉〉kk′2〈〈v2(k)〉〉kk′.As we see, all projected spectral functionsαi2F(ω) depend on the charge vertex functionγc(k,q) which describes the screening (renormalization) of EPI due to strong correlations (suppression of doubly occupancy) [78–80, 130]. This important ingredient (which respects also the Ward identities) is a decisive step beyond the MFA renormalization of EPI in strongly correlated systems which was previously studied in connection with heavy fermions—see review in [218].


#### 2.3.4. Charge Vertex and the EPI Coupling

The charge vertex functionγc(k,q) (in the adiabatic approximation) has been calculated in [78–80, 130, 179, 180] in the framework of the 1/N expansion in the X-method—see also [6]—and here we discuss only the main results. Note that γc(k,q) renormalizes all charge fluctuation processes, such as the EPI interaction, the long-range Coulomb interaction, the nonmagnetic impurity scattering, and so forth. In fact γc(k,q) describes specific screening due to the vanishing of doubly occupancy in strongly correlated systems. Note that the latter constraint is at present impossible to incorporate into the LDA-DFT band-structure calculations, thus making the latter method unreliable in highly correlated systems. In [78–80, 130, 179, 180] γc(k,q,ω) was calculated as a function of the model parameters t,t′,δ,J in leading O(1) order of the t-J model:
(100)γc(k,q)=1-∑α=16‍∑β=16‍Fα(k)[1̂+χ̂(q)]αβ-1χβ2(q),
where χαβ(q)=∑pGα(p,q)Fβ(p),Fα(k)=[t(k),1, 2J0coskx,2J0sinkx,2J0cosky,2J0sinky], and Gα(p,q) = [1,t(p+q),cospx,sinpx,cospy,sinpy]Π(p,q). Here, Π(k,q) = -g(k)g(k+q) and q=(q,iqn),qn=2πnT,p=(p,ipm),pm=πT(2m+1). The physical meaning of the vertex function γc(k,q) is following: in the presence of an external (or internal) charge perturbation there is screening due to the change of the excitation potential VIBStJ(x,y), that is, of the change of the bandwidth, as well as of the local chemical potential. The central result is that for momenta k lying at (and near) the Fermi surface the vertex function γc(k,q,ω=0) has very pronounced forward scattering peak(at q=0) especially at very low doping concentration δ(≪1), while the backward scattering is substantially suppressed,as it is seen in Figure 35 where γc(kF,q,ω=0) is shown. The peak at q=0 is very narrow at very small doping since its width qc is proportional to the doping δ, that is, qc~δ(π/a) where a is the lattice constant. It is interesting that γc(k,q), as well as the dynamics of charge fluctuations, depend only weakly on the exchange energy J and are mainly dominated by the constraint of having no doubly occupancy of sites, as it is shown in [78–80, 130, 179, 180].Figure 35
Adiabatic (ω=0) vertex function γ(kF,q) of the t-J model as a function of the momentum aq with q=(q,q) for three different doping levels δ. From [130].The existence of the forward scattering peak inγc(k,q) at q=0 is confirmed by numerical calculations in the Hubbard model,which show that this peak is very pronounced at large U [182]. This is important result since it proves that the 1/N expansion in the X-method is reliable method in studying charge fluctuation processes in strongly correlated systems. The strong suppression of γc(k,q) at large q(~kF) means that at small distances the charge fluctuations are strongly suppressed (correlated). Such a behavior of the vertex function means that a quasiparticle moving in the strongly correlated medium digs up a giant correlation hole with the radius ξch(~π/qc)≈a/δ, where a is the lattice constant. As a consequence of this effect the renormalized EPI becomes long ranged which is contrary to the weakly correlated systems where it is short ranged.By knowingγc(k,q) one can calculate the relative coupling constants Λ1≡Λs, Λ3≡Λd, Λtr, and so forth. In the absence of correlations and for an isotropic band one has Λ1=Λtr=1, Λi=0 for i>1. The averages in Λs, Λd, and Λtr were performed numerically in [130] by using the realistic anisotropic band dispersion in the t-t′-J model and the results are shown in Figure 36. For convenience, the three curves are multiplied with a common factor so that Λs approaches 1 in the empty-band limit δ→1, when strong correlations are absent. Note that the superconducting critical temperature Tc in the weak coupling limit and in the ith channel scales like Tc(i)~exp(-1/(λ0Λi-μi*) where λ0 is some effective coupling constant which depends on microscopic details. The parameter μi* is the effective residual Coulomb repulsion in the ith superconducting channel. We stress here several interesting results which come out from the above theory and which are partially presented in Figures 35 and 36.Figure 36
Normalizeds-wave Λs, d-wave Λd, and transport Λtr·δ coupling constants as a function of doping δ for t′=0 and J=0. From [179, 180].(1) In principle the bare EPI coupling constant gλ0(k,q) depends on the quasiparticle momentum k and the transfer momentum q. In the t-J model the EPI coupling is dominated by the ionic coupling Ĥepion (see (80)) and corresponding EPI depends only on the momentum transfer q, that is, gλ0(k,q)≈gλ0(q) while for the much smaller covalent coupling Ĥepcov depends on both k and q [6, 10, 11]. However, the EPI couplings for most phonon modes are renormalized by the charge vertex and since the latter is peaked at small momentum transfer q=|k-k′| then the maxima of the corresponding effective potentials are pushed toward smaller values of q. The further consequence of the vertex renormalization is that in the absence of strong correlations the bare EPI coupling |g0(k,q)|2 for some phonon modes (which enters in the effective t-J model) is detrimental for d-wave pairing; it can be less detrimental or even supports it in the presence of strong correlations (since the maximum is pushed toward smaller q). To illustrate this let us consider the in-plane oxygen breathing mode with the frequency ωbr which is supposed to be important in HTSC cuprates. The bare coupling constant (squared) for this mode is approximately given by |gbr0(k,q)|2=|gbr0|2[sin2(qxa/2)+sin2(qya/2)] which reaches maximum for large q=(π/a,π/a). By extracting the component in the d-channel one has
(101)|gbr0(k-k′)|2=|gbr0|2[1-(14)ψd(k)ψd(k′)+⋯]
with
(102)ψd(k)=coskxa-coskya.
This gives the repulsive coupling constant λd0 in the d-channel, that is,
(103)λd0=2ωbr〈ψd(k)|gbr0(k-k′)|2ψd(k′)〉<0.
However, in the presence of strong correlations one expects that the effective coupling constant is given approximately by |gbreff(k,k-k′)|≈|gbr0(k-k′)|2γc2(kF,k-k′) which is at small doping δ suppressed substantially at large q since γc2 starts to fall off drastically at q~qc~δ(π/a). The latter property makes the effective coupling constant (in the d-channel) λdeff for these modes less negative or even positive (depending on the ratio ξch/a~1/δ), that is, one has λdeff>λd0. We stress again that this analysis is only qualitative (and semiquantitative) since it is based on the t-J model while the better quantitative results are expected in the strongly correlated three-band Emery model with Ud≫t,Δpd—see [6, Appendix  D]. Unfortunately, these calculations are not finalized until now.(2) In weakly correlated systems (or, e.g., in the empty-band limit δ→1) the relative d-wave coupling constant Λd is much smaller than the s-wave coupling constant Λs, that is, Λd≪Λs as it is seen in Figure 36. Furthermore, Λs decreases with decreasing doping.(3) It is indicative that independently on the value of t′≠0 or t′=0 the coupling constant Λs and Λd  meet each other (note that Λs>Λd for all δ) at some small doping δ≈0.1–0.2 where Λs≈Λd. We would like to stress that such a unique situation (with Λs≈Λd) was practically never realized in low-temperature and weakly correlated superconductors and in that respect the strong momentum-dependent EPI in HTSC cuprates is an exclusive but very important phenomenon.(4) By taking into account the residual Coulomb repulsion of quasiparticles then the s-wave superconductivity (which is governed by Λs) is suppressed, while the d-wave superconductivity (which is governed by Λd) stays almost unaffected, since μs*≫μd*. In that case the d-wave superconductivity which is mainly governed by EPI becomes more stable than the s-wave one at sufficiently low doping δ. This transition between s- and d-wave superconductivity is triggered by electronic correlations because in the model calculations [78–80, 130] the bare EPI coupling is assumed to be momentum independent, that is, the bare coupling constant contains the s-wave symmetry only.(5) The calculations of the charge vertex γc are performed in the adiabatic limit, that is, for ω<q·vF(q) the frequency ω in γc can be neglected. In the nonadiabatic regime, that is, for ω>q·vF(q), the function γc2(kF,q,ω) may be substantially larger compared to the adiabatic case because γc(kF,q,ω) tends to the bare value 1 for q=0. This means that EPI for different phonons (with different energies ω) is differently affected by strong correlations. For a given ω the EPI coupling to those phonons with momenta q<qω=ω/vF will be (relatively) enhanced since γc(kF,q,ω)≈1, while the coupling to those with q>qω=ω/vF will be substantially reduced due to the suppression of the backward scattering by strong correlations [37, 38]. These results are a consequence of the Ward identities and generally hold in the Landau-Fermi liquid theory [219].(6) The transport EPI coupling constant Λtr is significantly reduced in the presence of strong correlations especially for low doping (δ≪1) where Λtr<Λ/3. This result is physically plausible since the resistivity is dominated by the backward scattering processes (large q~kF) which are suppressed by strong correlations—the suppression of γc(kF,q,ω) at large q.The theory based on the forward scattering peak in EPI is a good candidate to explain the linear temperature behavior of the resistivity down to very small temperaturesT(~ΘD/30)≈10K in some cuprates with low Tc(≈10 K) [6, 128, 129]. One physically rather plausible model, which is based on the forward scattering peak in EPI, is elaborated in [128]. It takes into account (i) the quasiparticle scattering on acoustic (a) and on optic (o) phonons, (ii) the extended van Hove singularity in the quasiparticle density of states N(ξ) which in some cuprates is very near the Fermi surface, and (iii) the umklapp and “undulation” (due to the flat regions at the Fermi surface) processes with vk′≅-vk—this condition can partly increase the EPI coupling. The transport Eliashberg function αtr2F(ω) is calculated similarly to (97) by using the definition of α2F(k,k′,ω) in (95) with the renormalized coupling constantgν(r)(k-k′)=gν(k-k′)γc(k-k′) of the ν=a,o mode, respectively. In [128] it is assumed a phenomenological form for the forward scattering peak in γc(k-k′) with the cutoff  qc≪kF (and which mimics the exact results from [78–80, 130, 179, 180]). Since the scattering of the quasiparticles on phonons (with the sound velocity vs) is limited to small-q transfer processes (with q<qc), then the maximal energy of the acoustic branch is not the Debye energy ΘD(≈kFvs) but the effective Debye energyΘA(≈qcvs)≪ΘD. In the case of Bi2201 in [128] it is taken (from the numerical results in [78–80, 130, 179, 180]) that qc≈kF/10 which gives ΘA≈(30–50)K. As a result the calculated αtr2F(ω) gives that ρab(T)~T down to very low T(~0.2ΘA)≈10K. The slope (dρab/dT) is governed by the effective EPI coupling constant for acoustic phonons. In systems with the extended van Hove singularity (in N(ξ)) near the Fermi surface, which is the case in Bi-2201, the effective coupling constant for acoustic phonons can be sufficiently large to give experimental values for the slope (dρab/dT)~(0.5–1)μΩcm/K—for details see [128]. This physical picture is applicable also to cuprates near and at the optimal doping but since in these systems Tc is large the linearity of ρab(T) down to very low T is “screened” by the appearance of superconductivity.(7) The width of the forward scattering peak in γc(kF,q) is very narrow in underdoped cuprates—with the width qc~δ(π/a)—which may have further interesting consequences. For instance, HTSC cuprates are characterized not only by strong correlations but also by the relatively small Fermi energy EF, which is in underdoped systems not much larger than the characteristic (maximal) phonon frequency ωphmax, that is, EF≃0.1–0.3eV, ωphmax≃80meV. Due to the appreciable magnitude of ωD/EF it is necessary to correct the Migdal-Eliashberg theory by the non-Migdal vertex corrections due to the EPI. It is well known that these vertex corrections lower Tc in systems with the isotropic EPI. However, the non-Migdal vertex corrections in systems with the forward scattering peak in the EPI coupling with the cutoff qc≪kF may increase Tc which can be appreciable. The corresponding calculations [220, 221] give two interesting results: (i) there is an appreciable increase of Tc by lowering Qc=qc/2kF, for instance, Tc(Qc=0.1)≈4Tc(Qc=1); (ii) even small values of λep<1 can give large Tc. The latter results open a new possibility in reaching high Tc in systems with appreciable ratio ωD/EF and with the forward scattering peak in EPI. The difference between the Migdal-Eliashberg and the non-Migdal theory can be explained qualitatively in the framework of an approximative McMillan formula for Tc (for not too large λ) which reads Tc≈〈ω〉e-1/(λ̃-μ*). The Migdal-Eliashberg theory predicts
(104)λ̃(ME)≈λ1+λ,
while the non-Migdal theory [220, 221] gives
(105)λ̃(n-ME)≈λ(1+λ).
For instance, Tc~100K in HTSC oxides can be explained by the Migdal-Eliashberg theory for λ(ME)~2, while in the non-Migdal theory much smaller coupling constant is needed, that is, λ(n-ME)~0.5.(8) The existence of the forward scattering peak in EPI can in a plausible way explain the ARPES puzzle that the antinodal kink is shifted by the maximal superconducting gap Δmax while the nodal kink is unshifted. The reason is (as explained in Section 1.3.3) that due to strong correlations the EPI spectral function α2F(k,k′,Ω)≈α2F(φ-φ′,Ω) is strongly peaked at φ-φ′=0 [151].(9) The scattering potential on nonmagnetic impurities is renormalized by strong correlations giving also the forward scattering peak in the impurity scattering potential (amplitude) [155]. The latter effect gives large d-wave channel in the renormalized impurity potential, which is the reason that d-wave pairing in HTSC cuprates is robust in the presence of nonmagnetic impurities (and defects) [6, 155].


#### 2.3.5. EPI and Strong Correlations—Other Methods

The calculations of the static (adiabatic) charge-vertexγc(kF,q) in the X-method are done for the case U=∞ [78–80, 130, 179, 180] where it is found that it is peaked at q=0—the forward scattering peak (FSP). This result is confirmed by the numerical Monte Carlo calculations for the finite-U Hubbard model [182], where it is found that FSP exists for all U, but it is especially pronounced in the limit t≪U. These results are additionally confirmed in the calculations [183] within the four slave-boson method of Kotliar-Rückenstein where γc(kF,q) is again peaked at q=0 and the peak is also pronounced at t≪U.There are several calculations of the charge vertex in theone slave-boson method [219, 222–224] which is invented to study the limit U→∞. It is interesting to compare the results for the charge vertex in the X-method [78–80, 130, 179, 180] and in the one slave-boson theory [222] which are calculated in O((1/N)0) order. For instance, for J=0 one has
(106)γc(X)(k,q)=1+b(q)-a(q)t(k)[1+b(q)]2-a(q)c(q),γc(SB)(k,q)=1+b(q)-a(q)[t(k)+t(k+q)]/2[1+b(q)]2-a(q)c(q).
The explicit expressions for the “bare” susceptibilities a(q),b(q), and c(q) can be found in [78–80, 130]. It is obvious from (106) that γc(X)(k,q=0)=γc(SB)(k,q=0) but the calculations give that max{γc(X)(k,q)} is for q=0, while max{γc(SB)(k,q)} is for |q|≠0 [181]. So, the SB vertex is peaked at finite q which is in contradiction with the numerical Monte Carlo results for the Hubbard model [182] and with the four slave-boson theory [183]. The reason for the discrepancy of the one slave-boson (SB) in studying EPI with the numerical results and the X-method is not quite clear and might be due to the symmetry breaking of the local gauge invariance in leading order of the SB theory.


### 2.4. Summary of Section2

The experimental results in HTSC cuprates which are exposed in Section1 imply that the EPI coupling constant is large and in order to be conform with d-wave pairing this interaction must be very nonlocal (long range), that is, weakly screened and peaked at small transfer momenta. In absence of quantitative calculations in the framework of the ab initio microscopic many-body theory the effects of strong correlations on EPI are studied within the minimal t-J model where this pronounced nonlocality is due to two main reasons: (1) strong electronic correlations and (2) the combinedmetallic-ionic layered structure in these materials. In case (1) the pronounced nonlocality of EPI, which is found in the t-J model system, is due to the suppression of doubly occupancy at the Cu lattice sites in the CuO2 planes, which drastically weakens the screening effect in these systems. The pronounced nonlocality and suppression of the screening are mathematically expressed by the charge vertex function γc(kF,q,ω) which multiplies the bare EPI matrix element. The vertex function is peaked at q=0 and strongly suppressed at large q, especially for low (oxygen) hole-doping δ≪1 near the Mott-Hubbard transition. Such a structure of γc gives that the d-wave and s-wave coupling constants are of the same order of magnitude around and below some optimal doping δop≈0.1, that is, λd≈λs. This is very peculiar situation never met before. Since the residual effective (low-energy) Coulomb interaction is much smaller in the d-channel than in the s-channel, that is, μs*≫μd* (with the possibility that μd*<0), then the critical temperature for d-wave pairing is much larger than for the s-wave one, that is, Tc(d)≫Tc(s). Since all charge fluctuation processes are modified by strong correlations, then the quasiparticle scattering on nonmagnetic impurities is also drastically changed; the pair-breaking effect on d-wave pairing is drastically reduced. This nonlocal effect, which is not discussed here—see more in [6] and references therein—is one of the main reasons for the robustness of d-wave pairing in HTSC oxides in the presence of nonmagnetic impurities and numerous local defects. The development of the forward scattering peak in γc(kF,q) and suppression at large q(≫qc=δ(π/a)) give rise to the suppression of the transport coupling constant λtr making it much smaller than the self-energy coupling constant λ, that is, one has λtr≈λ/3 near the optimal doping δ=0.1–0.2. Thus the behavior of the vertex function and the dominance of EPI in the quasiparticle scattering resolve the experimental puzzle that the transport and the self-energy coupling constant take very different values, λtr,ep≪λep. Note that this is not the case with the SFI mechanism which is dominant at large q≈Q=(π,π) thus giving λtr,sf≈λsf. This result means that if in the SFI mechanism one fits the temperature-dependent resistivity (governed by λtr,sf) then one obtains very low Tc.We stress that the strength of the EPI coupling constantsλep,λep,d is at present impossible to calculate since it is difficult to incorporate strong correlations and numerous structural effects in a tractable microscopic theory.


### 2.5. Discussions and Conclusions

Numerous experimental results related to tunnelling, optics, ARPES, inelastic neutron, and X-ray scattering measurements in HTSC cupratesat and near the optimal doping give evidence for strong electron-phonon interaction (EPI) with the coupling constant 1<λep<3.5. The tunnelling measurements furnish evidence for strong EPI which give that the peaks in the bosonic spectral functionα2F(ω)  coincide well with the peaks in the phonon density of statesFph(ω). The tunnelling spectra show that almost all phonons contribute to Tc and that no particular phonon mode can be singled out in the spectral function α2F(ω) as being the only one which dominates in pairing mechanism. In light of these results the small oxygen isotope effect in optimally doped systems can be partly due to this effect, thus not disqualifying the important role of EPI in pairing mechanism. The compatibility of the strong EPI with d-wave pairing implies an important constraint on the EPI pairing potential—it must be nonlocal, that is, peaked at small transfer momenta. The latter is due to (a) strong electronic correlations and (b) the combined metallic-ionic structure of these materials. If the EPI scattering is the main player in pairing in HTSC cuprates, then this nonlocality implies that at and below some optimal doping (δop~0.1) the magnitude of the EPI coupling constants in d-wave and s-wave channel must be of the same order, that is, λep,d≈λep,s. This result in conjunction with the fact that the residual effective Coulomb coupling in d-wave channel is much smaller than in the s-wave one, that is, μs*≫μd* (with the possibility that μd*<0) gives that the critical temperature for d-wave pairing is much larger than for s-wave pairing.The numerous tunnelling, ARPES, optics, and magnetic neutron scattering measurements give sufficient evidence that the spin-fluctuation interaction (SFI) plays a secondary role in pairing in HTSC cuprates. Especially important evidence for the smallness of SFI (in pairing) comes from the magnetic neutron scattering measurements which show that by varying doping slightly around the optimal one there is a huge reconstruction of the SFI spectral functionImχ(q,ω) (imaginary part of the spin susceptibility) for q≈Q, while there is very small change in the critical temperature Tc. These experimental results imply important constraints on the pairing scenario  for systems at and near optimal doping: (1) the strength of the d-wave pairing potential is provided by EPI (i.e., one has λep,d≈λep,s) while the role of the residual Coulomb interaction and SFI, together, is to trigger d-wave pairing; (2) the Migdal-Eliashberg theory, but with the pronounced momentum dependent of EPI, is a rather good starting theory.The ab initio microscopic theory of pairing in HTSC cuprates fails at present to calculateTc and to predict the magnitude of the d-wave order parameter. From that point of view it is hard to expect a significant improvement of this situation at least in the near future. However, the studies of some minimal (toy) models, such as the single-band t-J model, allow us to understand part of the physics in cuprates on a qualitative and in some cases even on a semiquantitative level. In that respect the encouraging results come from the theoretical studies of the EPI scattering in the t-J model by using controllable mathematical methods in the X-method formulated in terms of the Hubbard operators [78–80, 130, 179, 180]. This theory predicts dressing of quasiparticles by strong correlations which dig up a large-scale correlation hole of the size ξch~a/δ for δ≪1. These quasiparticles respond to lattice vibrations in such a way to produce an effective long-range electron interaction (due to EPI), that is, the effective pairing potential Veff(q,ω) is peaked at small transfer momenta q—the forward scattering peak. This theory (of the toy model) is conform with the experimental scenario by predicting the following results: (i) the EPI coupling constants in d-wave and s-wave channels are of the same order, that is, λep,d≈λep,s, at some optimal doping δop~0.1; (ii) the transport coupling is much smaller than the pairing one, that is, λtr≈λ/3; (iii) due to strong correlations there is forward scattering peak in the potential for scattering on nonmagnetic impurities, thus making d-wave pairing robust in materials with a lot of defects and impurities. Applied to HTSC superconductors at and near the optimal doping, this theory is a realization of the Migdal-Eliashberg theory but with strongly momentum dependent EPI coupling, which is conform with the proposed experimental pairing scenario. This scenario which is also realized in the t-J toy model may be useful in making a (phenomenological) theory of pairing in cuprates. However, all present theories are confronted with the unsolved and challenging task—the calculation ofTc. From that point of view we do not have at present a proper microscopic theory of pairing in HTSC cuprates.


## 2.1. Microscopic Theory of Superconductivity

### 2.1.1. Ab Iniitio Many-Body Theory

The many-body theory of superconductivity is based on the fully microscopic electron-ion Hamiltonian for electrons and ions in the crystal—see, for instance, [193, 194]. It comprises mutually interacting electrons which interact also with the periodic lattice and with the lattice vibrations. In order to pass continually to the problem of the interplay of EPI and strong correlations and also to explain why the LDA-DFT method is inadequate for HTSC cuprates, we discuss this problem here with restricted details—more extended discussion can be found in [6, 194]. In order to describe superconductivity the Nambu-spinor ψ̂†(r)=(ψ̂↑†(r)ψ̂↓(r)) is introduced which operates in the electron-hole space (ψ̂(r)=(ψ̂†(r))†) where ψ̂↑(r), ψ̂↑†(r) are annihilation and creation operators for spin up, respectively, and so forth. The microscopic Hamiltonian of the system under consideration contains three parts: Ĥ=Ĥe+Ĥi+Ĥe-i. The electronic HamiltonianĤe, which describes the kinetic energy and the Coulomb interactions of electrons, is given by
(46)Ĥe=∫d3rψ̂†(r)τ̂3ϵ0(p̂)ψ̂(r)+12∫d3rd3r′ψ̂†(r)τ̂3ψ̂(r)Vc(r-r′)ψ̂†(r′)τ̂3ψ̂(r′),
where ϵ0(p̂)=p̂2/2m is the kinetic energy of electron and Vc(r-r′)=e2/|r-r′| is the electron-electron Coulomb interaction. Note that in the electron-hole space the pseudospin (Nambu) matrices τ̂i, i=0,1,2,3 are Pauli matrices. Since we will discuss only the electronic properties, the explicit form of the lattice HamiltonianĤi [6, 194] is omitted here. The electron-ion Hamiltonian describes the interaction of electrons with the equilibrium lattice and with its vibrations, respectively:
(47)Ĥe-i=∑n‍∫d3rVe-i(r-Rn0)ψ̂†(r)τ̂3ψ̂(r)+∫d3rΦ̂(r)ψ̂†(r)τ̂3ψ̂(r).
Here, Ve-i(r-Rn0) is the electron-ion potential and its form depends on the level of description of the electronic subsystem. For instance, in the all-electron calculations one has Ve-i(r-Rn0)=-Ze2/|r-Rn0| where Ze is the ionic charge. The second term which is proportional to the lattice distortion operator Φ̂(r)=-∑n,αûαn∇αVe-i(r-Rn0)+Φ̂anh(r) (because of convenience it includes also the EPI coupling ∇αVe-i) describes the interaction of electrons with harmonic (~ûαn) (or anharmonic ~Φ̂anh(r)) lattice vibrations.Dyson's equations for the electron and phonon Green's functionsĜ(1,2)=-〈Tψ̂(1)ψ̂†(2)〉, D̃(1-2)=-〈TΦ̂(1)Φ̂(2)〉 are Ĝ-1(1,2)=Ĝ0-1(1,2)-Σ̂(1,2) and D̃-1(1,2)=D̃0-1(1,2)-Π̃(1,2), where the Ĝ0-1(1,2)=[(-∂/∂τ1-ϵ0(p1)+μ)τ̂0-ueff(1)τ̂3]δ(1-2) is the bare inverse electronic Green's function. Here, 1=(r1,τ1), where τ1 is the imaginary time in the Matsubara technique, and the effective one-body potential ueff(1)=Ve-i(1)+VH++〈Φ̂(1)〉, where VH is the Hartree potential. The electron and phonon self-energies Σ̂(1,2) and Π̃(1,2) take into account many-body dynamics of the interacting system. The electronic self-energyΣ̂(1,2)=Σ̂c(1,2)+Σ̂ep(1,2) is obtained in the form
(48)Σ̂(1,2)=-Veff(1,1¯)τ̂3Ĝ(1,2¯)Γ̂eff(2¯,2;1¯),
where integration (summation) over the bar indices is understood. The effective retarded potential Veff(1,1¯) in (48) contains the screened (by the electron dielectric function ɛe(1,2)) Coulomb and EPI interactions:
(49)Veff(1,2)=Vc(1-1¯)ɛe-1(1¯,2)+ɛe-1(1,1¯)D̃(1¯,2¯)ɛe-1(2¯,2).
The inverse electronic dielectric permeability ɛe-1(1,2) = δ(1-2)+Vc(1-1¯)P(1¯,2¯)ɛe-1(2¯,2) is defined via the irreducible electronic polarization operator P(1,2) = -Sp{τ̂3Ĝ(1,2¯)Γ̂eff(2¯,3¯;2)Ĝ(3¯,1+)}. The vertex functionΓ̂eff(1,2;3) = -δĜ(1,2)/δueff(3) in (48) is the solution of the complicated (and practically unsolvable) integro-differential functional equation
(50)Γ̂eff(1,2;3)=τ̂3δ(1-2)δ(1-3)+δΣ̂(1,2)δĜ(1¯,2¯)Ĝ(1¯,3¯)Ĝ(4¯,2¯)Γ̂eff(3¯,4¯;3).
Note that the effective vertex function Γ̂eff(1,2;3), which takes into account all renormalizations going beyond the simple Coulomb (RPA) screening, is the functional of both the electronic and phononic Green's functions Ĝ and D̃, thus making at present the ab initio microscopic equations practically unsolvable.


### 2.1.2. Low-Energy Migdal-Eliashberg Theory

If the vertex functionΓ̂eff would be known, we would have a closed set of equations for Green's functions which describe dynamics of the interacting electrons and lattice vibrations (phonons) in the normal and superconducting state. However, this is a formidable task and at present far from any practical realization. Fortunately, we are mostly interested in low-energy phenomena (with energies |ωn|,ξ≪ωc and for momenta k=kF+δk in the shell δk≪δkc near the Fermi momentum kF;ωc and δkc are some cutoffs), which allows us further simplification of equations [1, 2]. Therefore, the strategy is to integrate high-energy processes—see more in [194]. Here, we sketch this procedure briefly. Namely, Green's function Ĝ(k,ωn)=[iωn-(k2/2m-μ)τ̂3-Σ̂(k,ωn)]-1 can be formally written in the form
(51)Ĝ(k,ωn)=Ĝlow(k,ωn)+Ĝhigh(k,ωn),
where Ĝlow(k,ωn)=Ĝ(k,ωn)Θ(ωc-|ωn|)Θ(δkc-δk) is the low-energy Green's function and Ĝhigh(k,ωn)=Ĝ(k,ωn)Θ(|ωn|-ωc)Θ(δk-δkc) is the high-energy one and analogously D=Dlow(k,ωn)+Dhigh(k,ωn). By introducing the small parameter of the theory s~(ω/ωc)~(δk/δkc)≪1 one has in leading order Ĝlow(k,ωn)~s-1,  Ĝhigh(k,ωn)≲1 and Dlow(k,ωn)~s0,  Dhigh(k,ωn)~s2. Note that the coupling constants (Vei,∇Vei,Vii, etc.) are of the order s0=1.The procedure of separating low-energy and high-energy processes lies also behind theadiabatic approximation since in most materials the characteristic phonon (Debye) energy ωD of lattice vibrations is much smaller than the characteristic electronic Fermi energy EF (ωD≪EF). In the small s(≪1) limit the Migdal theory [1, 2] keeps in the total self-energy Σ linear terms in the phonon propagator D̃ (D) only. In that case the effective vertex function can be written in the form Γ̂eff(1,2;3)≅Γ̂c(1,2;3)+δΓ̂ep(1,2;3) [1, 2], where the Coulomb charge vertex Γ̂c(1,2;3)=τ̂3δ(1-2)δ(1-3)+δΣ̂c(1,2)/δueff(3) contains correlations due to the Coulomb interaction only but does not contain EPI  and phonon propagator D̃ explicitly. The part δΓ̂ep(1,2;3)=δΣ̂ep(1,2)/δueff(3) contains all linear terms with respect to EPI. Note that in these diagrams enters the dressed Green's function which contains implicitly EPI up to infinite order. By careful inspection of all (explicit) contributions to δΓ̂ep(1,2;3) which is linear in D̃ one can express the self-energy in terms of the charge (Coulomb) vertex Γ̂c(1,2;3) only. As a result of this approximation, the part of the self-energy due to Coulomb interaction is given by
(52)Σ̂c(1,2)=-Vcsc(1,1¯)τ̂3Ĝ(1,2¯)Γ̂c(2¯,2;1¯),
where Vcsc(1,2)=Vc(1,2¯)ɛe-1(2¯,2) is the screened Coulomb interaction. The part which is due to EPI has the following form:
(53)Σ̂ep(1,2)=-Vep(1¯,2¯)Γ̂c(1,3¯;1¯)Ĝ(3¯,4¯)Γ̂c(4¯,2;2¯),
where Vep(1,2)=ɛe-1(1,1¯)D̃(1¯,2¯)ɛe-1(2¯,2) is the screened EPI potential. Note that Σ̂ep(1,2) depends now quadratically on the charge vertex Γ̂c, which is due to the adiabatic theorem.It is well known that the Coulomb self-energyΣ̂c(1,2) is the most complicating part of the electronic dynamics, but since we are interested in low-energy physics when s≪1, then the term Σ̂c(1,2) can be further simplified by separating it in two parts:
(54)Σ̂c(1,2)=Σ̂c(h)(1,2)+Σ̂c(l)(1,2).
The term Σ̂c(h)(1,2) is due to high-energy processes contained in the product Ĝhigh(1,2¯)Γ̂chigh(2¯,2;1¯) (e.g., due to the large Hubbard U in strongly correlated systems) and Σ̂c(l)(1,2) is due to low-energy processes. The leading part of Σ̂c(h)(1,2) is 1, that is, Σ̂c(h)(1,2)~s0, while Σ̂c(l)(1,2) is small of order 1, that is, Σ̂c(l)(1,2)~s1. For further purposes we define the quantity V̂0 as
(55)V̂0(1,2)={Ve-i(1)+VH(1)}τ3δ(1-2)+Σ̂c(h)(1,2),
where Ve-i, VH are also of order s0. After the Fourier transform with respect to time (and for small |ωn|≪ωc) Σ̂c(h) is given by
(56)Σ̂c(h)(x1,x2,ωn)≃Σc0(h)(x1,x2,0)τ̂3+(Σ̂c0(h))′(x1,x2,0)·iωn.
As we said, Σc0(h)~s0 while (Σ̂c0(h))′·ωn~s1 because ωn~s1. From (52) it is seen that the part Σ̂c(l)(1,2) contains the low-energy Green's function Ĝlow(1,2¯) and this skeleton diagram is of order s1. The similar analysis based on (53) for Σ̂ep(1,2) gives that the leading order is s1 which describes the low-energy part of EPI. After the separations of terms (of s0 and s1 orders) the Dyson equation in the low-energy region has the form
(57)[iωnZc(x,x¯)-Ĥ0(x,x¯)-Σ̂c(l)(x,x¯,ωn)-Σ̂ep(x,x¯,ωn)]×Ĝlow(x¯,y,ωn)=δ(x-y)τ̂0,
where x¯ means integration ∫d3x¯ over the crystal volume. The Coulomb renormalization function Zc(x,y)=δ(x-y)-(Σ0c(h))′(x,y,0) and the single-particle Hamiltonian Ĥ0(x,y) collect formally all high-energy processes which are unaffected by superconductivity (which is low-energy process) where
(58)Ĥ0(x,y)={(-12m∇x2-μ)δ(x-y)+V0(h)(x,y,0)}τ̂3
with
(59)V0(h)(x,y,0)={Ve-i(x)+VH(x)}δ(x-y)+Σc0(h)(x,y,0).
One can further absorb Zc(x,y) into the renormalized Green's function
(60)Ĝr(x,y,ωn)=Zc1/2(x,x¯)Ĝlow(x¯,y¯,ωn)Zc1/2(y¯,y),
the renormalized vertex function Γ̂ren(1,2;3)=Zc-1/2Γ̂cZc-1/2, and the renormalized self-energies
(61)Σ̂r;c,ep(l)(x,y,ωn)=Zc-1/2(x,x¯)Σ̂c,ep(l)(x¯,y¯,ωn)Zc-1/2(y¯,y)
and introduce the ideal band-structure Hamiltonianĥ0(x,y)=Zc-1/2(x,x¯)Ĥ0(x¯,y¯)Zc-1/2(y¯,y) given by
(62)ĥ0(x,y)={(-12m∇x2-μ)δ(x-y)+VIBS(x,y)}τ̂3.
Here,
(63)VIBS(x,y)=Zc-1/2(x,x¯)V0(h)(x¯,y¯)Zc-1/2(y¯,y)
is the ideal band-structure potential (sometimes called the excitation potential) and apparently nonlocal quantity, which is contrary to the standard local potential Vg(x) in the LDA-DFT theories—see Section 2.2. The static potential VIBS(x,y) is of order s0 and includes high-energy processes.Finally, we obtain the matrix Dyson equation for the renormalized Green's functionĜr(x,y,ωn) which is the basis for the (strong-coupling) Migdal-Eliashberg theory in the low-energy region
(64)[iωnδ(x-x¯)-ĥ0(x,x¯)-Σ̂c,r(l)(x,x¯,ωn)-Σ̂ep,r(x,x¯,ωn)]×Ĝr(x¯,y,ωn)=δ(x-y)τ̂0,
where Σ̂c,r(l) and Σ̂ep,r have the same form as (52)-(53) but with the renormalized Green's and vertex functions Ĝr,Γ̂r instead of Ĝ,Γ̂. We stress that (64) holds in the low-energy region only. In the superconducting state the set of Eliashberg equations in (64) are written explicitly in Appendix A, where it is seen that the superconducting properties depend on the Eliashberg spectral function αkp2F(ω). The latter function is defined also in Appendix A, (A.4), and it depends on material properties of the system.The important ingredients of the low-energy Migdal-Eliashberg theory are the ideal band-structure Hamiltonianĥ0(x,y)—given by (62) which contains many-body (excitation) ideal band-structure nonlocal periodic crystal potential VIBS(x,y). The Hamiltonian ĥ0(x,y) determines the ideal energy spectrumϵ(k) of the conduction electrons and the wave function ψi,p(x) through
(65)ĥ0(x,y¯)ψi,k(y¯)=[ϵi(k)-μ]ψi,k(x),
where μ is the chemical potential. We stress that the Hamiltonian ĥ0(x,y) also governs transport properties of metals in low-energy region.After solving (65) the next step is to expand all renormalized Green's function, self-energies, vertices, and the renormalized EPI matrix element (written symbolically as  gep,r=gep,0Γrenɛe-1) in the basis of ψi,p(x) and to write down the Eliashberg equations in this basis. We will not elaborate further this program and refer the reader to the relevant literature in [193, 194]. We point out that even such simplified program of the low-energy Migdal-Eliashberg theory was never fully realized in low-temperature superconductors, because the nonlocal potential VIBS(x,y) (enters the ideal band-structure Hamiltonian h0(x,y)) and the renormalized vertex function (entering the EPI coupling constant gep,r) which include electronic correlations are difficult to calculate especially in strongly correlated metals. Therefore, it is not surprising at all that the situation is even more difficult in HTS materials which are strongly correlated systems with complex structural and material properties. Due to these difficulties the calculations of the electronic band structure and the EPI coupling are usually done in the framework of LDA-DFT where the many-body excitation potential VIBS(x,y) is replaced by some (usually local) potential VLDA(x) which in fact determines the ground-state properties of the crystal. In the next section we briefly describe (i) the LDA-DFT procedure in calculating the EPI coupling constant and (ii) some results of the LDA-DFT calculations related to HTSC cuprates. We will also discuss why this approximation is inappropriate when applied to HTS materials.


## 2.1.1. Ab Iniitio Many-Body Theory

The many-body theory of superconductivity is based on the fully microscopic electron-ion Hamiltonian for electrons and ions in the crystal—see, for instance, [193, 194]. It comprises mutually interacting electrons which interact also with the periodic lattice and with the lattice vibrations. In order to pass continually to the problem of the interplay of EPI and strong correlations and also to explain why the LDA-DFT method is inadequate for HTSC cuprates, we discuss this problem here with restricted details—more extended discussion can be found in [6, 194]. In order to describe superconductivity the Nambu-spinor ψ̂†(r)=(ψ̂↑†(r)ψ̂↓(r)) is introduced which operates in the electron-hole space (ψ̂(r)=(ψ̂†(r))†) where ψ̂↑(r), ψ̂↑†(r) are annihilation and creation operators for spin up, respectively, and so forth. The microscopic Hamiltonian of the system under consideration contains three parts: Ĥ=Ĥe+Ĥi+Ĥe-i. The electronic HamiltonianĤe, which describes the kinetic energy and the Coulomb interactions of electrons, is given by
(46)Ĥe=∫d3rψ̂†(r)τ̂3ϵ0(p̂)ψ̂(r)+12∫d3rd3r′ψ̂†(r)τ̂3ψ̂(r)Vc(r-r′)ψ̂†(r′)τ̂3ψ̂(r′),
where ϵ0(p̂)=p̂2/2m is the kinetic energy of electron and Vc(r-r′)=e2/|r-r′| is the electron-electron Coulomb interaction. Note that in the electron-hole space the pseudospin (Nambu) matrices τ̂i, i=0,1,2,3 are Pauli matrices. Since we will discuss only the electronic properties, the explicit form of the lattice HamiltonianĤi [6, 194] is omitted here. The electron-ion Hamiltonian describes the interaction of electrons with the equilibrium lattice and with its vibrations, respectively:
(47)Ĥe-i=∑n‍∫d3rVe-i(r-Rn0)ψ̂†(r)τ̂3ψ̂(r)+∫d3rΦ̂(r)ψ̂†(r)τ̂3ψ̂(r).
Here, Ve-i(r-Rn0) is the electron-ion potential and its form depends on the level of description of the electronic subsystem. For instance, in the all-electron calculations one has Ve-i(r-Rn0)=-Ze2/|r-Rn0| where Ze is the ionic charge. The second term which is proportional to the lattice distortion operator Φ̂(r)=-∑n,αûαn∇αVe-i(r-Rn0)+Φ̂anh(r) (because of convenience it includes also the EPI coupling ∇αVe-i) describes the interaction of electrons with harmonic (~ûαn) (or anharmonic ~Φ̂anh(r)) lattice vibrations.Dyson's equations for the electron and phonon Green's functionsĜ(1,2)=-〈Tψ̂(1)ψ̂†(2)〉, D̃(1-2)=-〈TΦ̂(1)Φ̂(2)〉 are Ĝ-1(1,2)=Ĝ0-1(1,2)-Σ̂(1,2) and D̃-1(1,2)=D̃0-1(1,2)-Π̃(1,2), where the Ĝ0-1(1,2)=[(-∂/∂τ1-ϵ0(p1)+μ)τ̂0-ueff(1)τ̂3]δ(1-2) is the bare inverse electronic Green's function. Here, 1=(r1,τ1), where τ1 is the imaginary time in the Matsubara technique, and the effective one-body potential ueff(1)=Ve-i(1)+VH++〈Φ̂(1)〉, where VH is the Hartree potential. The electron and phonon self-energies Σ̂(1,2) and Π̃(1,2) take into account many-body dynamics of the interacting system. The electronic self-energyΣ̂(1,2)=Σ̂c(1,2)+Σ̂ep(1,2) is obtained in the form
(48)Σ̂(1,2)=-Veff(1,1¯)τ̂3Ĝ(1,2¯)Γ̂eff(2¯,2;1¯),
where integration (summation) over the bar indices is understood. The effective retarded potential Veff(1,1¯) in (48) contains the screened (by the electron dielectric function ɛe(1,2)) Coulomb and EPI interactions:
(49)Veff(1,2)=Vc(1-1¯)ɛe-1(1¯,2)+ɛe-1(1,1¯)D̃(1¯,2¯)ɛe-1(2¯,2).
The inverse electronic dielectric permeability ɛe-1(1,2) = δ(1-2)+Vc(1-1¯)P(1¯,2¯)ɛe-1(2¯,2) is defined via the irreducible electronic polarization operator P(1,2) = -Sp{τ̂3Ĝ(1,2¯)Γ̂eff(2¯,3¯;2)Ĝ(3¯,1+)}. The vertex functionΓ̂eff(1,2;3) = -δĜ(1,2)/δueff(3) in (48) is the solution of the complicated (and practically unsolvable) integro-differential functional equation
(50)Γ̂eff(1,2;3)=τ̂3δ(1-2)δ(1-3)+δΣ̂(1,2)δĜ(1¯,2¯)Ĝ(1¯,3¯)Ĝ(4¯,2¯)Γ̂eff(3¯,4¯;3).
Note that the effective vertex function Γ̂eff(1,2;3), which takes into account all renormalizations going beyond the simple Coulomb (RPA) screening, is the functional of both the electronic and phononic Green's functions Ĝ and D̃, thus making at present the ab initio microscopic equations practically unsolvable.


## 2.1.2. Low-Energy Migdal-Eliashberg Theory

If the vertex functionΓ̂eff would be known, we would have a closed set of equations for Green's functions which describe dynamics of the interacting electrons and lattice vibrations (phonons) in the normal and superconducting state. However, this is a formidable task and at present far from any practical realization. Fortunately, we are mostly interested in low-energy phenomena (with energies |ωn|,ξ≪ωc and for momenta k=kF+δk in the shell δk≪δkc near the Fermi momentum kF;ωc and δkc are some cutoffs), which allows us further simplification of equations [1, 2]. Therefore, the strategy is to integrate high-energy processes—see more in [194]. Here, we sketch this procedure briefly. Namely, Green's function Ĝ(k,ωn)=[iωn-(k2/2m-μ)τ̂3-Σ̂(k,ωn)]-1 can be formally written in the form
(51)Ĝ(k,ωn)=Ĝlow(k,ωn)+Ĝhigh(k,ωn),
where Ĝlow(k,ωn)=Ĝ(k,ωn)Θ(ωc-|ωn|)Θ(δkc-δk) is the low-energy Green's function and Ĝhigh(k,ωn)=Ĝ(k,ωn)Θ(|ωn|-ωc)Θ(δk-δkc) is the high-energy one and analogously D=Dlow(k,ωn)+Dhigh(k,ωn). By introducing the small parameter of the theory s~(ω/ωc)~(δk/δkc)≪1 one has in leading order Ĝlow(k,ωn)~s-1,  Ĝhigh(k,ωn)≲1 and Dlow(k,ωn)~s0,  Dhigh(k,ωn)~s2. Note that the coupling constants (Vei,∇Vei,Vii, etc.) are of the order s0=1.The procedure of separating low-energy and high-energy processes lies also behind theadiabatic approximation since in most materials the characteristic phonon (Debye) energy ωD of lattice vibrations is much smaller than the characteristic electronic Fermi energy EF (ωD≪EF). In the small s(≪1) limit the Migdal theory [1, 2] keeps in the total self-energy Σ linear terms in the phonon propagator D̃ (D) only. In that case the effective vertex function can be written in the form Γ̂eff(1,2;3)≅Γ̂c(1,2;3)+δΓ̂ep(1,2;3) [1, 2], where the Coulomb charge vertex Γ̂c(1,2;3)=τ̂3δ(1-2)δ(1-3)+δΣ̂c(1,2)/δueff(3) contains correlations due to the Coulomb interaction only but does not contain EPI  and phonon propagator D̃ explicitly. The part δΓ̂ep(1,2;3)=δΣ̂ep(1,2)/δueff(3) contains all linear terms with respect to EPI. Note that in these diagrams enters the dressed Green's function which contains implicitly EPI up to infinite order. By careful inspection of all (explicit) contributions to δΓ̂ep(1,2;3) which is linear in D̃ one can express the self-energy in terms of the charge (Coulomb) vertex Γ̂c(1,2;3) only. As a result of this approximation, the part of the self-energy due to Coulomb interaction is given by
(52)Σ̂c(1,2)=-Vcsc(1,1¯)τ̂3Ĝ(1,2¯)Γ̂c(2¯,2;1¯),
where Vcsc(1,2)=Vc(1,2¯)ɛe-1(2¯,2) is the screened Coulomb interaction. The part which is due to EPI has the following form:
(53)Σ̂ep(1,2)=-Vep(1¯,2¯)Γ̂c(1,3¯;1¯)Ĝ(3¯,4¯)Γ̂c(4¯,2;2¯),
where Vep(1,2)=ɛe-1(1,1¯)D̃(1¯,2¯)ɛe-1(2¯,2) is the screened EPI potential. Note that Σ̂ep(1,2) depends now quadratically on the charge vertex Γ̂c, which is due to the adiabatic theorem.It is well known that the Coulomb self-energyΣ̂c(1,2) is the most complicating part of the electronic dynamics, but since we are interested in low-energy physics when s≪1, then the term Σ̂c(1,2) can be further simplified by separating it in two parts:
(54)Σ̂c(1,2)=Σ̂c(h)(1,2)+Σ̂c(l)(1,2).
The term Σ̂c(h)(1,2) is due to high-energy processes contained in the product Ĝhigh(1,2¯)Γ̂chigh(2¯,2;1¯) (e.g., due to the large Hubbard U in strongly correlated systems) and Σ̂c(l)(1,2) is due to low-energy processes. The leading part of Σ̂c(h)(1,2) is 1, that is, Σ̂c(h)(1,2)~s0, while Σ̂c(l)(1,2) is small of order 1, that is, Σ̂c(l)(1,2)~s1. For further purposes we define the quantity V̂0 as
(55)V̂0(1,2)={Ve-i(1)+VH(1)}τ3δ(1-2)+Σ̂c(h)(1,2),
where Ve-i, VH are also of order s0. After the Fourier transform with respect to time (and for small |ωn|≪ωc) Σ̂c(h) is given by
(56)Σ̂c(h)(x1,x2,ωn)≃Σc0(h)(x1,x2,0)τ̂3+(Σ̂c0(h))′(x1,x2,0)·iωn.
As we said, Σc0(h)~s0 while (Σ̂c0(h))′·ωn~s1 because ωn~s1. From (52) it is seen that the part Σ̂c(l)(1,2) contains the low-energy Green's function Ĝlow(1,2¯) and this skeleton diagram is of order s1. The similar analysis based on (53) for Σ̂ep(1,2) gives that the leading order is s1 which describes the low-energy part of EPI. After the separations of terms (of s0 and s1 orders) the Dyson equation in the low-energy region has the form
(57)[iωnZc(x,x¯)-Ĥ0(x,x¯)-Σ̂c(l)(x,x¯,ωn)-Σ̂ep(x,x¯,ωn)]×Ĝlow(x¯,y,ωn)=δ(x-y)τ̂0,
where x¯ means integration ∫d3x¯ over the crystal volume. The Coulomb renormalization function Zc(x,y)=δ(x-y)-(Σ0c(h))′(x,y,0) and the single-particle Hamiltonian Ĥ0(x,y) collect formally all high-energy processes which are unaffected by superconductivity (which is low-energy process) where
(58)Ĥ0(x,y)={(-12m∇x2-μ)δ(x-y)+V0(h)(x,y,0)}τ̂3
with
(59)V0(h)(x,y,0)={Ve-i(x)+VH(x)}δ(x-y)+Σc0(h)(x,y,0).
One can further absorb Zc(x,y) into the renormalized Green's function
(60)Ĝr(x,y,ωn)=Zc1/2(x,x¯)Ĝlow(x¯,y¯,ωn)Zc1/2(y¯,y),
the renormalized vertex function Γ̂ren(1,2;3)=Zc-1/2Γ̂cZc-1/2, and the renormalized self-energies
(61)Σ̂r;c,ep(l)(x,y,ωn)=Zc-1/2(x,x¯)Σ̂c,ep(l)(x¯,y¯,ωn)Zc-1/2(y¯,y)
and introduce the ideal band-structure Hamiltonianĥ0(x,y)=Zc-1/2(x,x¯)Ĥ0(x¯,y¯)Zc-1/2(y¯,y) given by
(62)ĥ0(x,y)={(-12m∇x2-μ)δ(x-y)+VIBS(x,y)}τ̂3.
Here,
(63)VIBS(x,y)=Zc-1/2(x,x¯)V0(h)(x¯,y¯)Zc-1/2(y¯,y)
is the ideal band-structure potential (sometimes called the excitation potential) and apparently nonlocal quantity, which is contrary to the standard local potential Vg(x) in the LDA-DFT theories—see Section 2.2. The static potential VIBS(x,y) is of order s0 and includes high-energy processes.Finally, we obtain the matrix Dyson equation for the renormalized Green's functionĜr(x,y,ωn) which is the basis for the (strong-coupling) Migdal-Eliashberg theory in the low-energy region
(64)[iωnδ(x-x¯)-ĥ0(x,x¯)-Σ̂c,r(l)(x,x¯,ωn)-Σ̂ep,r(x,x¯,ωn)]×Ĝr(x¯,y,ωn)=δ(x-y)τ̂0,
where Σ̂c,r(l) and Σ̂ep,r have the same form as (52)-(53) but with the renormalized Green's and vertex functions Ĝr,Γ̂r instead of Ĝ,Γ̂. We stress that (64) holds in the low-energy region only. In the superconducting state the set of Eliashberg equations in (64) are written explicitly in Appendix A, where it is seen that the superconducting properties depend on the Eliashberg spectral function αkp2F(ω). The latter function is defined also in Appendix A, (A.4), and it depends on material properties of the system.The important ingredients of the low-energy Migdal-Eliashberg theory are the ideal band-structure Hamiltonianĥ0(x,y)—given by (62) which contains many-body (excitation) ideal band-structure nonlocal periodic crystal potential VIBS(x,y). The Hamiltonian ĥ0(x,y) determines the ideal energy spectrumϵ(k) of the conduction electrons and the wave function ψi,p(x) through
(65)ĥ0(x,y¯)ψi,k(y¯)=[ϵi(k)-μ]ψi,k(x),
where μ is the chemical potential. We stress that the Hamiltonian ĥ0(x,y) also governs transport properties of metals in low-energy region.After solving (65) the next step is to expand all renormalized Green's function, self-energies, vertices, and the renormalized EPI matrix element (written symbolically as  gep,r=gep,0Γrenɛe-1) in the basis of ψi,p(x) and to write down the Eliashberg equations in this basis. We will not elaborate further this program and refer the reader to the relevant literature in [193, 194]. We point out that even such simplified program of the low-energy Migdal-Eliashberg theory was never fully realized in low-temperature superconductors, because the nonlocal potential VIBS(x,y) (enters the ideal band-structure Hamiltonian h0(x,y)) and the renormalized vertex function (entering the EPI coupling constant gep,r) which include electronic correlations are difficult to calculate especially in strongly correlated metals. Therefore, it is not surprising at all that the situation is even more difficult in HTS materials which are strongly correlated systems with complex structural and material properties. Due to these difficulties the calculations of the electronic band structure and the EPI coupling are usually done in the framework of LDA-DFT where the many-body excitation potential VIBS(x,y) is replaced by some (usually local) potential VLDA(x) which in fact determines the ground-state properties of the crystal. In the next section we briefly describe (i) the LDA-DFT procedure in calculating the EPI coupling constant and (ii) some results of the LDA-DFT calculations related to HTSC cuprates. We will also discuss why this approximation is inappropriate when applied to HTS materials.


## 2.2. LDA-DFT Calculations of the EPI Matrix Elements

We point out again two results which are important for the future microscopic theory of pairing in HTSC cuprates.First, numerous experiments (discussed in Part I) give evidence that the EPI coupling constant which enters the normal part of the quasiparticle self-energy λepZ=λs+λd+⋯ is rather large, that is, 1<λepZ<3.5. In order to be conform with d-wave pairing the effective EPI potential must be nonlocal (and peaked at small transfer momenta q), which implies that the s-wave and d-wave coupling constants are of the same order, that is, λd≈λs. Second, the theory based on the minimal t-J model, which will be discussed in Section 2.3, gives that strong electronic correlations produce a peak at small transfer momenta in the effective EPI pairing potential thus giving rise to λd≈λs. This is a striking property which allows that EPI is conform with d-wave pairing. However, the theory is seriously confronted with the problem of calculation of the coupling constants λepZ. It turns out that at present it is an illusory task to calculate λepZ and λd since it is extremely difficult (if possible at all) to incorporate the peculiar structural properties of HTSC cuprates (layered structure, ionic-metallic system, etc.) and strong correlations effects in a consistent and reliable microscopic theory which is described in Section 2.1. As it is stressed several times, the LDA-DFT methods miss some important many-body effects (especially in the band-structure potential) and therefore fail to describe correctly screening properties of HTSC cuprates and the strength of EPI. However, the LDA-DFT methods are able to incorporate diverse structural properties of HTSC cuprates much better than the simplified minimal t-J (toy) model. Here, we discuss briefly some achievements of the advanced LDA-DFT calculations which are able to take partially into account some nonlocal effects in the EPI. The latter are mainly due to the almost ionic structure along thec-axis which is reflected in the very small c-axis plasma frequency (ωc≪ωab).The main task of the LDA-DFT theory in obtaining the EPI matrix elements is to calculate the change of the ground-state (self-consistent) potentialδVg(r)/δRα and the EPI coupling constant (matrix element) gαLDA(k,k′) (see its definition below), which is the most difficult part of calculations. Since in the LDA-DFT method the EPI scattering cannot be formulated, then the recipe is that the calculated gαLDA(k,k′) is inserted into the many-body Eliashberg equations. By knowing gαLDA(k,k′) one can define the total (λ) and partial (λq,ν) EPI coupling constants for the νth mode, respectively [195], as
(66)λ=1Np∑q,νλq,ν,λq,ν=pγq,νπN(0)ωq,ν,
where p=3κ is the number of phonon branches (κ is the number of atoms in the unit cell) and N(0) is the density of states at the Fermi energy (per spin and unit cell). The phonon line-widthγq,ν is defined in the Migdal-Eliashberg theory by
(67)γq,ν=2πωq,ν1N∑ll′k12Mωk-q,ν|eνα(q)·gα,ll′(k,k-q)|2×[nF(ξl,k)-nF(ξl,k+ℏωq,ν)ℏωq,ν]×δ(ξl′,k-q-ξl,k-ℏωq,ν).
Here, eνα(q) is the phonon polarization vectors; nF is the Fermi function. Since the ideal energy spectrum from (65) ξl,k=El,k-μ and the corresponding eigenfunctions ψkl are unknown, then instead of these one sets in (67) the LDA-DFT eigenvalues for the lth band ξl,k(LDA) and ψkl(LDA). In the LDA-DFT method the EPI matrix elementgα,ll′(LDA) is defined by the change of the ground-state potential δVg(r)/δRα:
(68)gα,ll′(LDA)(k,k′)=〈ψkl(LDA)|∑nδVg(r)δRnα|ψk′l′(LDA)〉.
The index n means summation over the lattice sites; α=x,y,z and the wave function ψkl(LDA) are the solutions of the Kohn-Sham equation—see [6]. In the past various approximations within the LDA-DFT method have been used in calculating δVg(r)/δRα and λ while here we comment some of them only. (i) In most calculations in LTS systems and in HTSC cuprates the rigid-ion (RI) approximation was used as well as its further simplifications which inevitably (due to its shortcomings and obtained small λ) deserves to be commented. The RI approximation is based on the very specific assumption that the ground-state (crystal) potential Vg(r) can be considered as a sum of ionic potentials Vg(r)=∑nVg(r-Rn) where the ion potential Vg(r-Rn) and the electron density ρe(r) are carried rigidly with the ion at Rn during the ion displacement (Rn=Rn0+ûαn). In the RI approximation the change of Vg(r) is given by
(69)δVg(r)=∑n‍∇αVg(r-Rn0)uαn,δVg(r)δRnα=∇αVg(r-Rn0),
which means that RI does not take into account changes of the electron density during the ion displacements. In numerous calculations applied to HTSC cuprates the rigid-ion model is even further simplified by using the rigid muffin-tin approximation (RMTA) (or similar version with the rigid-atomic sphere)—see discussions in [195–198]. The RMTA assumes that the ground-state potential and the electron density follow ion displacements rigidly inside the Wigner-Seitz cell while outside it Vg(r) is not changed because of the assumed very good metallic screening (e.g., in simple metals):
(70)∇αVg(r-Rn)={∇αVg(r-Rn),rincelln,0,outside.
This means that the dominant EPI scattering is due to the nearby atoms only and that the scattering potential is isotropic. All nonlocal effects related to the interaction of electrons with ions far away are neglected in the RMTA. In this case gα,nLDA(k,k′) is calculated by the wave function centered at the given ion Rn0 which can be expanded inside the muffin-tin sphere (outside it the potential is assumed to be constant) in the angular momentum basis {l,m}, that is,
(71)〈r∣ψk(RMTA)〉=∑lmnClm(k,Rn0)ρl(|r-Rn0|)Ylm(ϕ,θ)
(the angles ϕ,θ are related to the vector r̂=(r-Rn0)/|r-Rn0|). The radial function ρl(|r-Rn0|) is zero outside the muffin-tin sphere. In that case the EPI matrix element is given by gα,nRMTA(k,k′)~〈Ylm|r̂|Yl′m′〉 and because r̂ is vector the selection rule implies that only terms with Δl≡l′-l=±1 contribute to the EPI coupling constant in the RMTA. This result is an immediate consequence of the assumed locality of the EPI potential in RMTA. However, since nonlocal effects, such as the long-range Madelung-like interaction, are important in HTSC cuprates, then additional terms contribute also to the coupling constant gα,n, that is, gα,n(k,k′)=gα,nRMTA(k,k′)+gα,nnonloc(k,k′), where a part (δgα,nnonloc) of the nonlocal contribution to gα,nnonloc is represented schematically:(72)δgα,nnonloc(k,k′)~〈Ylm|(Rn0-Rm0)α|Yl′m′〉.
From (72) comes out the selection rule Δl=l′-l=0 for the nonlocal part of the E-P interaction. We stress that the Δl=0 (nonlocal) terms are omitted in the RMTA approach and therefore it is not surprising that this approximation works satisfactorily in elemental (isotropic) metals only. The latter are characterized by the large density of states at the Fermi surface which makes electronic screening very efficient. This gives rise to a local EPI where an electron feels potential changes of the nearby atom only. One can claim with certainty that the RMTA method is not suitable for HTSC cuprates which are highly anisotropic systems with pronounced ionic character of binding and pronounced strong electronic correlations. The RMTA method applied to HTSC cuprates misses just this important part—the long-range part EPI due to the change of the long-range Madelung energy in the almost ionic structure of HTSC cuprates. For instance, the first calculations done in [199] which are based on the RMTA give very small EPI coupling constant λRMTA~0.1 in YBCO, which is in apparent contradiction with the experimental finding that λep is large—see Section 1.However, these nonlocal effects are taken into account in [195] by using the frozen-in phonon (FIP) method in evaluating of λep in La2-xMxCuO4. In this method some symmetric phonons are considered and the band structure is calculated for the system with the super-cell which is determined by the periodicity of the phonon displacement. By comparing the unperturbed and perturbed energies the corresponding EPI coupling λν (for the considered phonon νth mode) is found. More precisely speaking, in this approach the matrix elements of δVg(r)/δR0,ακ are determined from the finite difference of the ground-state potential ΔVg,q,ν(r)=Vg(R0,Lκ+Δτq,νκ(L))-Vg(R0,Lκ)=∑L,κΔτq,νκ(L)∂Vg(R0,Lκ)/∂R0,Lκ, where L,κ enumerate elementary lattice cells and atoms in the unit cell, respectively. The frozen-in atomic displacements of the phonon Δτq,νκ(L) of the νth mode are given by Δτq,νκ(L)=Δuq,ν(ℏ/2Mκωq,ν)1/2Re[eκ,ν(q)eiq·R] where Δuq,ν is the dimensionless phonon amplitude and the phonon polarization (eigen)vector eκ,ν(q) fulfills the condition ∑κeκ,ν*(q)·eκ,ν′(q)=δν,ν′. Based on this approach various symmetric Ag (and some B3g) modes of La2-xMxCuO4 were studied [195], where it was found that the large matrix elements are due to unusually long-range Madelung-like, especial for the c-axis phonon modes. The obtained large λep≈1.37 is the consequence of the following three main facts. (i) The electronic spectrum in HTSC cuprates is highly anisotropic, that is, it is quasi-two-dimensional. This is an important fact for pairing because if the conduction electrons would be uniformly spread over the whole unit cell then due to the rather low electron density (n~1021cm-3) the density of states on the Cu and O in-plane atoms would be an order of magnitude smaller than the real value. This would further give an order of magnitude smaller EPI coupling constant λep. Note that the calculated density of states on the (heavy) Cu and (light) O in-plane atoms, NCu(0)≅0.54states/eV  and NOxy(0)≅0.35states/eV, is of same order of magnitude as in some LTS materials. For instance, in NbC where Tc≈11K one has on (the heavy) Nb atom NNb(0)≅0.58states/eV and on (the light) C atom NC(0)≅0.25states/eV. So, the quasi-two-dimensional character of the spectrum is crucial in obtaining appreciable density of states on the light O atoms in the CuO2 planes. (ii) In HTSC cuprates there is strong Cu–O hybridization leading to good in-plane metallic properties. This large covalency in the plane is due to the (fortunately) small energy separation of the electron levels on Cu and Oxy atoms which comes out from the band-structure calculations [200], that is, Δ=|ϵCu-ϵOxy|≈3eV. The latter value gives rise to strong covalent mixing (the hybridization parameter tpd) of the Cudx2-y2 and Opx,y states, that is, tpd=-1.85eV. It is interesting that the small value of Δ is not due to the ionic structure (crystal field effect) of the system but it is mainly due to the natural falling of the Cudx2-y2 states across the transition-metal series. So, the natural closeness of the atomic energy levels of the Cudx2-y2 and Opx,py states is this distinctive feature of HTSC cuprates which basically allows achievement of high Tc. (iii) The ionic structure of HTSC  cuprates which is very pronounced along the c-axis is responsible for the weak electronic screening along this axis and according to that for the significant contribution of the nonlocal (long-range) Madelung-like interaction to EPI. It turns out that because of the ionicity of the structure the La and Oz  axial modes are strongly coupled with charge carriers in the CuO2 planes despite the fact that the local density of states on these atoms is very small [195], that is, NLa(0)=0.13states/eV and NOz(0)=0.016states/eV. (For comparison, on planar atoms Cu and Oxy one has NCu(0)=0.54states/eV  and NOxy(0)=0.35states/eV.) These calculations show that the lanthanum mode (with ωq,ν=202cm-1) at the q=(0,0.2π/c) zone boundary (fully symmetric Z-point) has ten times larger coupling constant λq,νLa(FIP)=4.8 than it is predicted in the RMT approximation λq,νLa(RMT)=0.48. The similar increase holds for the average coupling constant, where λν,averageLa(FIP)=1.0 but λν,averageLa(RMT)=0.1. Note that for the q≈0 La-mode one obtains λνLa(FIP)=4.54 compared  to λνLa(RMT)=0.12. Similar results hold for the axial apex-oxygen q=(0,0.2π/c) mode (Oz) with ωq,ν=396cm-1 where the large (compared to the RMT method) coupling constant is obtained: λq,νOz=14 and λν,averageOz=4.7, while for q≈0 axial apex-oxygen modes with ωq,ν=415cm-1 one has λν,averageOz=11.7. After averaging over all calculated modes it was estimated that λ=1.37 and ωlog≈400K. By assuming that μ*=0.1 one obtains Tc=49K by using Allen-Dynes formula for Tc≈0.83ωlogexp{-1.04(1+λ)/[λ-μ*(1+0.62λ)]} with ωlog=2∫dωdωα2(ω)F(ω)lnω/λω.  For μ*=0.15 and 0.2 one obtains Tc=41 and 32K, respectively. We stress that the rather large λep (and Tc) is due to the nonlocal (long range) effects of the metallic-ionic structure of HTSC cuprates and non-muffin-tin corrections in EPI, as was first proposed in [201, 202]. However, we would like to stress that the optimistic results for λep obtained in [195] are in fact based on the calculation of the EPI coupling for some wave vectors q with symmetric vibration patterns and in fact the obtained λep is an extrapolated value. The all-q calculations of λep,q which take into account long-range effects are a real challenge for the LDA-DFT calculations and are still awaiting.Finally, it is worth to mention important calculations of the EPI coupling constant in the framework of thelinear-response full-potential linear-muffin-tin-orbital  method (LRFP-LMTO) invented in [203, 204] and applied to the doped HTSC cuprate (Ca1-xSrx)1-yCuO2 for x~0.7 and y~0.1 with Tc=110K [205]. Namely, these calculations give strong evidence that the structural properties of HTSC cuprates already alone make the dominance of small-q scattering in EPI, whose effect is additionally increased by strong correlations. In order to analyze this compound in [205] the calculations are performed for CaCuO2 doped by holes in a uniform, neutralizing back-ground charge. The momentum (q=(q∥,q⊥)) dependent EPI coupling constant (summed over all phonon branches ν) in different L channels (s,p,d.) is calculated by using a standard expression
(73)λL(q)=M∑k,νYL(k+q)|gk,q,ν|2YL(k)δ(ξk+q)δ(ξk).
Here, ξk is the quasiparticle energy, gk,q,ν is the EPI coupling constant (matrix element) with the νth branch, YL(k) is the L-channel wave function, and the normalization factor M∝NL-1(0) with the partial density of states is NL(0)∝∑kYL2(k)δ(ξk). The total coupling constant in the L-channel is an average of λL(q) over the whole 2D Brillouin zone (over q∥), that is, λL(q⊥)=〈λL(q∥)〉BZ. We stress three important results of [205]. First, the s- and d-coupling constants, λs(q), λd(q), are peaked at small transfer momentaq~(π/3a,0,0) as it is shown in [205, Figure  3]. This result is mainly caused by the nesting properties of the Fermi surface shown in [205, Figure  1]. Second, the q-dependence of the integrated EPI matrix elements |g¯L,q|2=λL(q)/χL′′(q) (with χL′′(q)∝∑kYL(k+q)YL(k)δ(ξk+q)δ(ξk)) for L=s,d  is similar to that of λL(q), that is, these are peaked at small transfer momenta q≪2kF. Both of these results mean that the structural properties of HTSC cuprates imply the dominance of small-q EPI scattering. Third, the calculations give similar values for λs(q⊥=0) and λd(q⊥=0), that is, λs=0.47 for s-wave and λd=0.36 for d-wave pairing [205]. The result that λd≈λs is due to the dominance of the small q-scattering in EPI, which means that the nonlocal effects (long-range forces) in EPI of HTSC cuprates are very important. This result together with the finding of the dominance of the small-q scattering in EPI due to strong correlations [78–80, 130, 179, 180] mean that strong correlations and the peculiar structural properties of HTSC cuprates make EPI conform with d-wave pairing, either as its main cause or as its supporter. We stress that the obtained coupling constant λd=0.36 is rather small to give d-wave pairing with large Tc and on the first glance this result is against the EPI mechanism of pairing in cuprates. However, it is argued throughout this paper that the LDA methods applied to strongly correlated systems overestimate the screening effects and underestimate the coupling constant and therefore their quantitative predictions are not reliable.


## 2.3. EPI and Strong Correlations in HTSC Uprates

### 2.3.1. Minimal Model Hamiltonian

Theminimal microscopic model for HTSC cuprates must include at least three orbitals: one dx2-y2-orbital of the Cu-ion and two p-orbitals (px,y) of the O-ion since they participate in transport properties of these materials—see more in [6] and references therein. The electronic part of the Hamiltonian (of the minimal model) is Ĥ=Ĥ0+Ĥint—usually called the Emery model (or the p-dmodel) [206], where the one-particle tight-binding Hamiltonian Ĥ0 describes the lowering of the kinetic energy in the p-d model (with three bands or orbitals):
(74)Ĥ0=∑i,σ(ϵd0-μ)diσ†diσ+∑j,α,σ(ϵpα0-μ)pjασ†pjασ+∑i,j,α,σtijαpddiσ†pjασ+∑j,j′,α,β,σtjj′,αβpppjασ†pj′βσ.
Here tijαpd (i,j enumerate the Cu- and O-sites, resp.) is the hopping integral between the pα(α=x,y)—and d-states and tjj′αβpp between the pα- and pβ-states—while ϵd0 and ϵpα0 are the bare d- and p-local energy levels and μ is the chemical potential. This tight-binding Hamiltonian is written in the electronic notation where the charge-transfer energy Δdp,0≡ϵd0-ϵp0>0 by assuming that there is one 3dx2-y2 electron on the copper (Cu2+) while electrons in the p-levels of the O2- ions occupy filled bands. Ĥ0 contains the main ingredients coming from the comparison with the LDA-DFT band-structure calculations. The LDA-DFT results are reproduced by assuming that tpp≪tpd (and ϵpα0=ϵp0 ) where the good fit to the LDA-DFT band structure is found for Δdp,0≡ϵd0-ϵp0≈3.2eV and tpd(≡tpd)=(3/2)(pdσ),(pdσ)=-1.8eV. The total LDA bandwidth Wb=(42)|tpd|≅9eV [207].The electron interaction is described byĤint:
(75)Ĥint=Ud∑ini↑dni↓d+Up∑j,αnjα↑pnjα↓p+V̂c+V̂ep,
where Ud and Up are the on-site Coulomb repulsion energies at Cu and O sites, respectively, while V̂c and V̂ep describe the long-range part of the Coulomb interaction of electrons (holes) and EPI, respectively. Note that the Hubbard repulsion Ud on the Cu-ion is different from its bare atomic value Ud0(≈16 eV for Cu) due to various kinds of screening effects in solids [208–210]. It turns out that in most transition metal oxides one has Ud≪Ud0. This problem is thoroughly studied in [208–210] and applied to HTSC cuprates. The estimation from the numerical cluster calculations [211] gives Ud=9–11eV and Up=4–6eV but because nidUd≫njpUp the on-site repulsion on the oxygen ion is usually neglected at the first stage of the analysis.Note that in the case of largeUd(≫tpd,Δdp,0) the hole notation is usually used where in the parent compound (and for |tpd|≪Δdp,0) one has 〈nid〉=1, that is, one hole in the 3D-shell (in the 3dx2-y2 state) in the ground state. In the limit of large Ud→∞ the doubly occupancy on the Cu atoms is forbidden and only two copper states are possible: Cu2+—described by the quantum state diσ†|0〉 with one hole in the 3D shell and Cu1+—described by |0〉 with zero holes in the filled 3D shell. In this (hole) notation the oxygen p-level is fully occupied by electrons, that is, there are no holes (〈njp〉=0) in the occupied oxygen 2p-shell of O2-. In this notation the vacuum state |0v〉 (not the ground state) of the Hamiltonian Ĥ for large Ud corresponds to the closed-shell configuration Cu1+O2-. In the hole notation the hole p-level ϵph0 lies higher than the hole d-level ϵdh0, that is, Δpd,0≡ϵph0-ϵdh0>0 (note that in the electron notation it is opposite) and Ud means repulsion of two holes (in the 3dx2-y2 orbital) with opposite spins—3d8 configuration of the Cu3+ ion. Note that ϵph0=-ϵp0,ϵdh0=-ϵd0, and tpd,h=-tpd. In the following the index h in tpd,h is omitted. The reason for ϵph0>ϵdh0 is partly in different energies for the hole sitting on the oxygen and copper, respectively [207]. From this model one can derive in the limit U→∞ the t-J model for the 2D lattice in the CuO2 plane [212, 213], where now each lattice site corresponds to a Cu-atom. In the presence of one hole in the 3D-shell then in the undoped (no oxygen holes) HTSC cuprate each lattice site is occupied by one hole. By doping the system with holes the additional holes go onto O-sites. Furthermore, due to the strong Cu–O covalent binding the energetics of the system implies that an O-hole forms a Zhang-Rice singlet with a Cu-hole [212]. In the t-J model the Zhang-Rice singlet is described by an empty site.Since in the t-J model the doubly occupancy is forbidden, one introduces annihilation (Hubbard) operator of the composite fermion X̂iσ0=ciσ†(1-ni,-σ) which describes creation of a hole (in the 3D-shell of the Cu-atoms) on the ith site if this site is previously empty (thus excluding doubly occupancy), that is, the constraint ni,σ+ni,-σ≤1 must be fulfilled on each lattice site.  In this picture the doped-hole concentration δ means at the same time the concentration of the oxygen holes, that is, of the Zhang-Rice singlets.In order not to confuse the reader we stress the difference in the meaning of the hole in the (p-d) three-band Emery model and in the single-band (effective) t-J model. In the Emery model the hole means the absence of the electron in the filled shell—the 3D shell for Cu atoms(ions) and 2p shell for O atoms(ions). On the other side the hole on the ith lattice site in the t-J model means the presence of the Zhang-Rice singlet on this site.Thebosonic-like operatorsX̂iσ1σ2=X̂iσ10X̂i0σ2 for  σ1≠σ2 create a spin fluctuation at the ith site and the spin operator is given by S=X̂iσ¯10(σ⃗)σ¯1σ¯2X̂i0σ¯2 where summation over the bar indices is understood. The operator ∑σX̂iσσ has the meaning of thehole number on theith site. It is useful to introduce the operator X̂i00=X̂i0σX̂iσ0 at the ith lattice site which is the number of Zhang-Rice singlets on theith site. For X̂i00|0〉=1|0〉 the ith site is occupied by the Zhang-Rice singlet, while for X̂i00|1〉=0|1〉 there is no Zhang-Rice singlet on the ith site (i.e., this site is occupied only by one 3d9 hole on the Cu site). This property of X̂i00 is due to the local constraint(76)X̂i00+∑σ=↑↓X̂iσσ=1,
which forbids doubly occupancy of the ith site by holes. By projecting out doubly occupied (high-energy) states the t-J model reads
(77)Ĥt-j=∑i,σϵi0X̂iσσ-∑i,j,σtijX̂iσ0X̂j0σ+∑i,jJij(Si·Sj-14n̂in̂j)+Ĥ3.
The first term (~ϵi0) describes an effective local energy of the hole (or the Zhang-Rice singlet), the second one (~tij) describes hopping of the holes, and the third one (~Jij) is the Heisenberg-like exchange energy between two holes. The theory [212] predicts that |ϵi0|≫|tij|. This property is very important in the study of EPI. Ĥ3 contains three-site term which is usually omitted believing that it is not important. For charge fluctuation processes it is plausible to omit it, while for spin-fluctuation processes it is questionable approximation. If one introduces the enumeration α,β,γ,λ=0,↑,↓, then the Hubbard operators satisfy the following algebra:
(78)[X̂iαβ,X̂jγλ]±=δij[δγβX̂iαλ±δαλX̂iγβ],
where δij is the Kronecker symbol. Note that the Hubbard operators possess the projection properties with X̂iαβX̂iγλ=δβγX̂iαλ. The (anti)commutation relations in (78) are more complicated than the canonical Fermi and Bose (anti)commutation relations, which complicates the mathematical structure of the theory. To escape these complications some novel techniques have been used, such as the one slave boson-technique. In this technique X̂i0σ=fiσbi†,X̂iσ1σ2=fiσ1†fiσ2 are represented in terms of the fermion (spinon) operator fiσ which annihilates the spin on the ith and the boson (holon) operator bi† which creates the Zhang-Rice singlet.In the minimal theoretical model the electron-phonon interaction (EPI) contains in principle two leading terms:(79)Ĥep=Ĥepion+Ĥepcov,
which are the “ionic” one (Ĥepion) and the “covalent” one (Ĥepcov). The “ionic” term describes the change of the energy of the hole (or the Zhang-Rice singlet) at the ith site due to lattice vibrations and it reads [6, 78–80, 130]
(80)Ĥepion=∑i,σΦ̂iX̂iσσ,
where the “displacement” operator
(81)Φ̂i=∑Lκ[ϵ(Ri0-RLκ0+ûi-ûLκ)-ϵ(Ri0-RLκ0)]
(which as in Section 2.1 includes the bare coupling constant) describes the change of the hole (or Zhang-Rice singlet) energy ϵa,i0 by displacing atoms in the lattice by the vector ûLκ. In the harmonic approximation the EPI potential is given by Φ̂i=∑gi(q,λ)exp{iqRi}[bq,λ+b-q,λ†] where bq,λ and bq,λ† are the annihilation and creation operator of phonons with the polarization λ, respectively. This term describes in principle the following processes: (1) the change of the O-hole and Cu-hole bare energies ϵph0,ϵdh0 in the three-band model due to lattice vibrations, (2) the change of the long-range Madelung energy (which is due to the ionicity of the structure) by lattice vibrations along the c-axis, and (3) the change of the Cu–O hopping parameter tpd in the presence of vibrations, and so forth. Here, L and κ enumerate unit lattice vectors and atoms in the unit cell, respectively. Usually, the EPI scattering is studied in the harmonic approximation where the phonon operator Φ̂i is calculated in the harmonic approximation (Φ̂~û) for the EPI interaction of holes with some specific phononic modes, such as the breathing and half-breathing ones [10, 11, 169]. The theory which includes also all other (than oxygen) vibrations in Φ̂i is still awaiting.It is interesting to make comparison of the EPI coupling constants in thet-J model and in the Hartree-Fock (HF) approximation (which is the analogous of the LDA-DFT method) of the three-band Emery (p-d) model in (74)-(75) when the problem is projected on the single band. For instance, the coupling constant with thehalf-breathing mode at the zone boundary in the HF approximation (which mimics the LDA-DFT approach) is given by
(82)ghbHF=±4tpd∂tpd∂RCu–O1ϵd-ϵpu0,
while the coupling constant in thet-Jmodelghbt-J(=∂ϵ0/∂RCu–O) is given by
(83)ghbt-J=±4tpd∂tpd∂RCu–O[2p2-1ϵd-ϵp+2p2Ud-|ϵd-ϵp|]u0,
where p=0.96—see [10, 11, 169] and references therein. It is obvious that in the t-J model the electron-phonon coupling is different from the HF one, since the former contains an additional term coming from the many-body effects, which are not comprised by the HF (LDA-DFT) calculations. The first term in (83) describes the hopping of a 3D hole into the O 2p-states and this term exists also in the LDA-DFT coupling constant—see (82). However, the second term in (83), which is due to many-body effects, describes the hopping of an O 2p-hole into the (already) single occupied Cu 3D state and it does not exist in the LDA-DFT approach. Since the corresponding dimensionless coupling constant λhb is proportional to |ghb|2, one obtains that the bare t-J coupling constant is almost three times larger than the LDA-DFT one:
(84)λhbt-J≈3λhbHF.
This example demonstrates clearly that the LDA-DFT method is inadequate for calculating the EPI coupling constant in HTSC cuprates.Note that there is also a covalent contribution toEPI which comes from the change of the effective hopping (t) in of the t-J model (77) and the exchange energy (J) in the presence of atomic displacements:
(85)Ĥepcov=-∑i,j,σ∂tij∂(Ri0-Rj0)(ûi-ûj)X̂iσ0X̂j0σ+∑i,j,∂Jij∂(Ri0-Rj0)(ûi-ûj)Si·Sj.
Here, we will not go into details but only stress that since |ϵi0|≫|tij| then the covalent term in the effectivet-Jmodel is much smaller than the ionic term—see more in [6, 10, 11, 169] and references therein—and in the following only the term Ĥepion will be considered [6, 78–80, 130].


### 2.3.2. Controllable X-Method for the Quasiparticle Dynamics

The minimal model Hamiltonian for strongly correlated holes with EPI (discussed above) is expressed via the Hubbard operators which obey “ugly” noncanonical commutation relations. The latter property is rather unpleasant for making a controllable theory in terms of Feynmann diagrams (for these “ugly” operators) and some other approaches are required. A possible way out is to express the Hubbard operators in terms of fermions and bosons (which must be confined) as, for instance, in the slave boson (SB) method. However, in real calculations which are based on some approximations the SB method is confronted with some subtle constraints whose fulfillments require very sophisticated mathematical treatment. Fortunately, there is amathematically controllable approach for treating the problem directly with Hubbard operators and without using slave-boson (or fermion) techniques. This method—we call it the X-method—is based on the general Baym-Kadanoff technique which allows to treat the problem by the well-defined and controllable 1/N expansion for the Green's functions in terms of Hubbard operators. This approach is formulated in [214] while the important refinement of the method is done in [78–80, 130]. In the paramagnetic and homogeneous state (with finite doping) the Green's function Gσ1σ2(1-2) is diagonal, that is, Gσ1σ2(1-2)=δσ1σ2G(1-2) where
(86)G(1-2)=-〈TX̂0σ(1)X̂σ0(2)〉=g(1-2)Q,
with the Hubbard spectral weight Q=〈X̂00〉+〈X̂σσ〉. The function g(1-2) plays the role of the quasiparticle Green's function—see more in [6, 78–80, 130, 179, 180]. It turns out that in order to have a controllable theory (1/N expansion) one way is to increase the number of spin components from two to N by changing the constraint (76) into the new one
(87)X̂i00+∑σ=1NX̂iσσ=N2.
In order to reach the convergence of physical quantities in the limit N→∞ the hopping and exchange energy are also rescaled, that is, tij=t0,ij/N and Jij=J0,ij/N. In order to eliminate possible misunderstandings we stress that in the case N>2 the constraint in (87) spoils some projection properties of the Hubbard operators. Fortunately, these (lost) projection properties are not used at all in the refined theory. As a result one obtains the functional integral equation for G(1,2), thus allowing unambiguous mathematical and physical treatment of the problem. In [78–80, 130, 179, 180] it is developed a systematic 1/N expansion for the quasiparticle Green's function g(1-2)(=g0+g1/N+⋯), Q(=Nq0+q1+⋯) (also for G(1-2)) and the self-energy. For large N(→∞) the leading term is G0(1-2)=g0(1-2)Q0=O(N) with g0=O(1) and Q0=〈X̂i00〉=Nδ/2. Here, δ is the concentration of the oxygen holes (that is, of the Zhang-Rice singlets) which is related to the chemical potential by the equation 1-δ=2∑pnF(p) with nF(p)=(eϵ0(k)-μ+1)-1. The quasiparticle Green's function g0(k,ω) and the quasiparticle spectrum ϵ0(k) in the leading order are given by
(88)g0(k,ω)≡G0(k,ω)Q0=1ω-[ϵ0(k)-μ],(89)ϵ0(k)=ϵc-δ·t(k)-∑pJ0(k+p)nF(p).
The level shift is ϵc=ϵ0+2∑pt(p)nF(p) and t(p) is the Fourier transform of the hopping integral tij—see more in [6].Let us summarize themain results of the X-method in leading O(1)-order for the quasiparticle properties in the t-J model [6, 78–80, 130, 179, 180]. (i) The Green's function g0(k,ω) describes the coherent motion  of quasiparticles whose contribution to the total spectral weight of the Green's function G0(k,ω) is Q0=Nδ/2. The coherent motion of quasiparticles is described in leading order by G0(k,ω)=Q0g0(k,ω) and the quasiparticle residuumQ0 disappears in the undoped Mott insulating state (δ=0). This result is physically plausible since in the Mott insulating state the coherent motion of quasiparticles, which is responsible for finite conductivity, vanishes. (ii) The quasiparticle spectrum ϵ0(k) plays the same role as the eigenvalues of the ideal band-structure Hamiltonian ĥ0(x,y) (it contains the excitation potential VIBS(x,y) which is due to high-energy processes of the Coulomb interaction). So, if we would consider ϵtb(k)=-t(k) as the tight-binding parametrization of the LDA-DFT band-structure spectrum which takes int account only weak correlations (with the local potential Vxc(x)δ(x-y)), then one can define a nonlocal excitation potentialVIBStJ(x,y)=ṼIBStJ(x,y)+Vxc(x)δ(x-y) which mimics strong correlations in the t-J model
(90)ṼIBStJ(x,y)≈V0δ(x-y)+(1-δ)t(x-y)-J̃(x-y).
Here, V0=2∑pt(p)nF(p) and t(x-y) is the Fourier transform of t(k) while J̃(x-y) is the Fourier transform of the third term in (89). The relative excitation potential ṼIBStJ(x,y) is due to strong correlations (suppression of doubly occupancy on each lattice site) and as we will see below it is responsible for the short-range screening of EPI in such a way that the forward scattering peak appears in the effective EPI interaction—see discussion below. (iii) For the very low doping ϵ0(k) is dominated by the exchange parameter if J0>δ·t0. However, in the case when J0≪δ·t0 there is a band narrowing by lowering the hole-doping δ, where the band width is proportional to the hole-concentration δ, that is, Wb=z·δ·t0. (iv) The O(1)-order quasiparticle Green's function g0(k,ω) and the quasiparticle spectrum ϵ0(k) in the X-method have similar form as the spinon Green's function g0,f(k,ω)=-〈Tfσfσ†〉k,ω and the spinon energy ϵs(k) in the SB method. However, in the SB method there is a broken gauge symmetry in the metallic state (with δ≠0) which is characterized by 〈b̂i〉≠0. This broken local gauge symmetry in the slave-boson method in O(1) order, which is due to the local decoupling of spinon and holon, is in fact forbidden by Elitzur's theorem. On the other side the local gauge invariance is not broken in the X-method where Green's function G0(k,ω) describes motion of the composite object, that is, simultaneous creation of the hole and annihilation of the spin at a given lattice site, while in the SB theory there is a spin-charge separation because of the broken symmetry (〈b̂i〉≠0). The assumption of the broken symmetry 〈b̂i〉≠0 gives qualitative satisfactory results for the quasiparticle energy for the case N=∞ in D>2 dimensions. However, the analysis of response functions and of higher-order 1/N corrections to the self-energies very delicate in the SB theory and special techniques must be implemented in order to restore the gauge invariance of the theory. On the other side the X-method is intrinsically gauge invariant and free of spurious effects in all orders of the 1/N expansion. Therefore, one expects that these two methods may deliver different results in O(1) and higher order in response functions. This difference is already manifested in the calculation of EPI where the charge vertex in these two methods is peaked at different wave vectors q, that is, at q=0 in the X-method and |q|≠0 in the SB method—see Section 2.3.5. (v) In [215, 216] it is shown that in the superconducting state the anomalous self-energies (which are of O(1/N)-order in the 1/N expansion) of the X- and SB-methods differ substantially. As a consequence, the SB method [217] predicts false superconductivity in the t-J model (for J=0) with large Tc (due to the kinematical interaction), while the X-method gives extremely small Tc(≈0) [215, 216]. So, although the two approaches yield some similar results in leading O(1)-order they, are different at least in next to leading O(1/N)-order.


### 2.3.3. EPI Effective Potential in thet-J Model

The theory of EPI in the minimalt-J model based on the X-method predicts that the leading term in the EPI self-energy Σep is given by the expression [6, 78–80, 130]
(91)Σep(1,2)=-Vep(1¯-2¯)γc(1,3¯;1¯)g0(3¯-4¯)γc(4¯,2;2¯),
where the screened (by the dielectric constant) EPI potential
(92)Vep(1-2)=ɛe-1(1-1¯)Vep0(1¯-2¯)ɛe-1(2¯-2)
and Vep0(1-2)=-〈TΦ̂(1)Φ̂(2)〉 is the “phonon” propagator which may also describe an anharmonic EPI. It is obvious that (91) is equivalent to (53) in spite the fact that the theory is formulated in terms of the Hubbard operators. The charge vertexγc(1,2;3)=-δg0-1(1,2)/δueff(3) corresponds to the the renormalized vertex Γc,r in (53) and it describes the screening by strong correlations. It depends on the relative excitation potential ṼIBStJ(x,y). The electronic dielectric function ɛe(1-2) describes the screening of EPI by the long-range part of the Coulomb interaction. Note that in the harmonic approximation Φ̂(1) contains the bare EPI coupling constant gep0 and lattice displacement û, that is, Φ̂~gep0û—see more in [6]. (Note that in the above equations summation and integration over bar indices are understood.) The self-energy Σep(k,ω) due to EPI reads
(93)Σep(k,ω)=∫0∞dν〈α2F(k,k′,ν)〉k′R(ω,ν),
with R(ω,ν)=-2πi(nB(ν)+1/2)+ψ(1/2+i)-ψ(1/2-i(ν+ω)/2πT) where nB(ν) is the Bose distribution function and ψ is di-gamma function. The Eliashberg spectral function is given by
(94)α2F(k,k′,ω)=N(0)∑ν|gν(k,k-k′)|2×δ(ω-ων(k-k′))γc2(k,k-k′),
where gν(k,p) is the EPI coupling constant for the νth mode, where the renormalization by long-range Coulomb interaction is included, that is, gν(k,p)=gep,ν0(k,p)/ɛe(p). 〈⋯〉k denotes Fermi-surface average with respect to the momentum k and N(0) is the density of states renormalized by strong correlations. The effect of strong correlations in the adiabatic limit is stipulated in the charge vertex functionγc(k,k-k′) which, as we will see in Section 2.3.4, changes the properties of Vep(q,ν) drastically compared to weakly correlated systems. In fact the charge vertex depends on frequency ω but in the adiabatic limit (ωph≪W) and for qvF>ωph it is practically frequency independent, that is, γc(ad)(k,q,ω)≈γc(k,q,ω=0) where the latter is real quantity. For J=0 in the t-t′ model the 1/N expansion gives N(0)=N0(0)/q0 where q0=δ/2. For J≠0 the density of states N(0) does not diverge for δ→0 where N(0)(~1/J0)>N0(0). The bare density of statesN0(0) is calculated in absence of strong correlations, for instance, by the LDA-DFT method.Depending on the symmetry of the superconducting order parameterΔ(k,ω) (s- and d-wave pairing) various projected averages (over the Fermi surface) of α2F(k,k′,ω) enter the Eliashberg equations. Assuming that the superconducting order parameter transforms according to the representation Γi of the point group C4v of the square lattice (in the CuO2 planes) the appropriate symmetry-projected spectral function is given by
(95)α2Fi(k̃,k̃′,ω)=N(0)8∑ν,j|gν(k̃,k̃-Tjk̃′)|2×δ(ω-ων(k̃-Tjk̃′))×γc2(k̃,k̃-Tjk̃′)Di(j)
where k̃ and k̃′ are momenta on the Fermi line in the irreducible Brillouin zone (1/8 of the total Brillouin zone). Tj,j=1,…,8 denotes the eight point-group transformations forming the symmetry group of the square lattice. This group has five irreducible representations which we distinguish by the label i=1,2,…,5. In the following we discuss the representations i=1  and i=3, which correspond to the s- and d-wave symmetry of the full rotation group, respectively. Di(j) is the representation matrix of the jth transformation for the representation i. Assuming that the superconducting order parameter Δ(k,ω) does not vary much in the irreducible Brillouin zone, one can average over k̃ and k̃′ in the Brillouin zone. For each symmetry one obtains the corresponding pairing spectral functionα2Fi(ω):
(96)α2Fi(ω)=〈〈α2Fi(k̃,k̃′,ω)〉k̃〉k̃′,
which governs the transition temperature for the order parameter with the symmetry Γi. For instance, α2F3(ω) is the pairing spectral function in the d-channel and it gives the coupling for d-wave superconductivity (the irreducible representation Γ3—sometimes labelled as B1g). Performing similar calculations for the phonon-limited resistivity, one finds that the resistivity is related to the transport spectral functionα2Ftr(ω):
(97)αtr2F(ω)=〈〈α2F(k,k′,ω)[v(k)-v(k′)]2〉〉kk′2〈〈v2(k)〉〉kk′.
The effect of strong correlations on EPI was discussed in [130] within the model where gν(k,p) and the phonon frequencies ων(k̃-k̃′) are weakly momentum dependent. In order to elucidate the main effect of strong correlations on EPI and α2Fi(ω) we consider the latter functions for a simple model with Einstein phonon, where these functions are proportional to the (so called) relative coupling constantΛi:
(98)Λi=18N(0)N0(0)∑j=18〈〈|γc(k̃,k̃-Tjk̃′)|2〉〉k̃k̃′Di(j).
Similarly, the resistivity ρ(T)(~λtr~Λtr) is renormalized by the correlation effects where the transport coupling constant Λtr is given by
(99)Λtr=N(0)N0(0)〈〈|γc(k̃,k̃-Tjk̃′)|2[v(k)-v(k′)]2〉〉kk′2〈〈v2(k)〉〉kk′.As we see, all projected spectral functionsαi2F(ω) depend on the charge vertex functionγc(k,q) which describes the screening (renormalization) of EPI due to strong correlations (suppression of doubly occupancy) [78–80, 130]. This important ingredient (which respects also the Ward identities) is a decisive step beyond the MFA renormalization of EPI in strongly correlated systems which was previously studied in connection with heavy fermions—see review in [218].


### 2.3.4. Charge Vertex and the EPI Coupling

The charge vertex functionγc(k,q) (in the adiabatic approximation) has been calculated in [78–80, 130, 179, 180] in the framework of the 1/N expansion in the X-method—see also [6]—and here we discuss only the main results. Note that γc(k,q) renormalizes all charge fluctuation processes, such as the EPI interaction, the long-range Coulomb interaction, the nonmagnetic impurity scattering, and so forth. In fact γc(k,q) describes specific screening due to the vanishing of doubly occupancy in strongly correlated systems. Note that the latter constraint is at present impossible to incorporate into the LDA-DFT band-structure calculations, thus making the latter method unreliable in highly correlated systems. In [78–80, 130, 179, 180] γc(k,q,ω) was calculated as a function of the model parameters t,t′,δ,J in leading O(1) order of the t-J model:
(100)γc(k,q)=1-∑α=16‍∑β=16‍Fα(k)[1̂+χ̂(q)]αβ-1χβ2(q),
where χαβ(q)=∑pGα(p,q)Fβ(p),Fα(k)=[t(k),1, 2J0coskx,2J0sinkx,2J0cosky,2J0sinky], and Gα(p,q) = [1,t(p+q),cospx,sinpx,cospy,sinpy]Π(p,q). Here, Π(k,q) = -g(k)g(k+q) and q=(q,iqn),qn=2πnT,p=(p,ipm),pm=πT(2m+1). The physical meaning of the vertex function γc(k,q) is following: in the presence of an external (or internal) charge perturbation there is screening due to the change of the excitation potential VIBStJ(x,y), that is, of the change of the bandwidth, as well as of the local chemical potential. The central result is that for momenta k lying at (and near) the Fermi surface the vertex function γc(k,q,ω=0) has very pronounced forward scattering peak(at q=0) especially at very low doping concentration δ(≪1), while the backward scattering is substantially suppressed,as it is seen in Figure 35 where γc(kF,q,ω=0) is shown. The peak at q=0 is very narrow at very small doping since its width qc is proportional to the doping δ, that is, qc~δ(π/a) where a is the lattice constant. It is interesting that γc(k,q), as well as the dynamics of charge fluctuations, depend only weakly on the exchange energy J and are mainly dominated by the constraint of having no doubly occupancy of sites, as it is shown in [78–80, 130, 179, 180].Figure 35
Adiabatic (ω=0) vertex function γ(kF,q) of the t-J model as a function of the momentum aq with q=(q,q) for three different doping levels δ. From [130].The existence of the forward scattering peak inγc(k,q) at q=0 is confirmed by numerical calculations in the Hubbard model,which show that this peak is very pronounced at large U [182]. This is important result since it proves that the 1/N expansion in the X-method is reliable method in studying charge fluctuation processes in strongly correlated systems. The strong suppression of γc(k,q) at large q(~kF) means that at small distances the charge fluctuations are strongly suppressed (correlated). Such a behavior of the vertex function means that a quasiparticle moving in the strongly correlated medium digs up a giant correlation hole with the radius ξch(~π/qc)≈a/δ, where a is the lattice constant. As a consequence of this effect the renormalized EPI becomes long ranged which is contrary to the weakly correlated systems where it is short ranged.By knowingγc(k,q) one can calculate the relative coupling constants Λ1≡Λs, Λ3≡Λd, Λtr, and so forth. In the absence of correlations and for an isotropic band one has Λ1=Λtr=1, Λi=0 for i>1. The averages in Λs, Λd, and Λtr were performed numerically in [130] by using the realistic anisotropic band dispersion in the t-t′-J model and the results are shown in Figure 36. For convenience, the three curves are multiplied with a common factor so that Λs approaches 1 in the empty-band limit δ→1, when strong correlations are absent. Note that the superconducting critical temperature Tc in the weak coupling limit and in the ith channel scales like Tc(i)~exp(-1/(λ0Λi-μi*) where λ0 is some effective coupling constant which depends on microscopic details. The parameter μi* is the effective residual Coulomb repulsion in the ith superconducting channel. We stress here several interesting results which come out from the above theory and which are partially presented in Figures 35 and 36.Figure 36
Normalizeds-wave Λs, d-wave Λd, and transport Λtr·δ coupling constants as a function of doping δ for t′=0 and J=0. From [179, 180].(1) In principle the bare EPI coupling constant gλ0(k,q) depends on the quasiparticle momentum k and the transfer momentum q. In the t-J model the EPI coupling is dominated by the ionic coupling Ĥepion (see (80)) and corresponding EPI depends only on the momentum transfer q, that is, gλ0(k,q)≈gλ0(q) while for the much smaller covalent coupling Ĥepcov depends on both k and q [6, 10, 11]. However, the EPI couplings for most phonon modes are renormalized by the charge vertex and since the latter is peaked at small momentum transfer q=|k-k′| then the maxima of the corresponding effective potentials are pushed toward smaller values of q. The further consequence of the vertex renormalization is that in the absence of strong correlations the bare EPI coupling |g0(k,q)|2 for some phonon modes (which enters in the effective t-J model) is detrimental for d-wave pairing; it can be less detrimental or even supports it in the presence of strong correlations (since the maximum is pushed toward smaller q). To illustrate this let us consider the in-plane oxygen breathing mode with the frequency ωbr which is supposed to be important in HTSC cuprates. The bare coupling constant (squared) for this mode is approximately given by |gbr0(k,q)|2=|gbr0|2[sin2(qxa/2)+sin2(qya/2)] which reaches maximum for large q=(π/a,π/a). By extracting the component in the d-channel one has
(101)|gbr0(k-k′)|2=|gbr0|2[1-(14)ψd(k)ψd(k′)+⋯]
with
(102)ψd(k)=coskxa-coskya.
This gives the repulsive coupling constant λd0 in the d-channel, that is,
(103)λd0=2ωbr〈ψd(k)|gbr0(k-k′)|2ψd(k′)〉<0.
However, in the presence of strong correlations one expects that the effective coupling constant is given approximately by |gbreff(k,k-k′)|≈|gbr0(k-k′)|2γc2(kF,k-k′) which is at small doping δ suppressed substantially at large q since γc2 starts to fall off drastically at q~qc~δ(π/a). The latter property makes the effective coupling constant (in the d-channel) λdeff for these modes less negative or even positive (depending on the ratio ξch/a~1/δ), that is, one has λdeff>λd0. We stress again that this analysis is only qualitative (and semiquantitative) since it is based on the t-J model while the better quantitative results are expected in the strongly correlated three-band Emery model with Ud≫t,Δpd—see [6, Appendix  D]. Unfortunately, these calculations are not finalized until now.(2) In weakly correlated systems (or, e.g., in the empty-band limit δ→1) the relative d-wave coupling constant Λd is much smaller than the s-wave coupling constant Λs, that is, Λd≪Λs as it is seen in Figure 36. Furthermore, Λs decreases with decreasing doping.(3) It is indicative that independently on the value of t′≠0 or t′=0 the coupling constant Λs and Λd  meet each other (note that Λs>Λd for all δ) at some small doping δ≈0.1–0.2 where Λs≈Λd. We would like to stress that such a unique situation (with Λs≈Λd) was practically never realized in low-temperature and weakly correlated superconductors and in that respect the strong momentum-dependent EPI in HTSC cuprates is an exclusive but very important phenomenon.(4) By taking into account the residual Coulomb repulsion of quasiparticles then the s-wave superconductivity (which is governed by Λs) is suppressed, while the d-wave superconductivity (which is governed by Λd) stays almost unaffected, since μs*≫μd*. In that case the d-wave superconductivity which is mainly governed by EPI becomes more stable than the s-wave one at sufficiently low doping δ. This transition between s- and d-wave superconductivity is triggered by electronic correlations because in the model calculations [78–80, 130] the bare EPI coupling is assumed to be momentum independent, that is, the bare coupling constant contains the s-wave symmetry only.(5) The calculations of the charge vertex γc are performed in the adiabatic limit, that is, for ω<q·vF(q) the frequency ω in γc can be neglected. In the nonadiabatic regime, that is, for ω>q·vF(q), the function γc2(kF,q,ω) may be substantially larger compared to the adiabatic case because γc(kF,q,ω) tends to the bare value 1 for q=0. This means that EPI for different phonons (with different energies ω) is differently affected by strong correlations. For a given ω the EPI coupling to those phonons with momenta q<qω=ω/vF will be (relatively) enhanced since γc(kF,q,ω)≈1, while the coupling to those with q>qω=ω/vF will be substantially reduced due to the suppression of the backward scattering by strong correlations [37, 38]. These results are a consequence of the Ward identities and generally hold in the Landau-Fermi liquid theory [219].(6) The transport EPI coupling constant Λtr is significantly reduced in the presence of strong correlations especially for low doping (δ≪1) where Λtr<Λ/3. This result is physically plausible since the resistivity is dominated by the backward scattering processes (large q~kF) which are suppressed by strong correlations—the suppression of γc(kF,q,ω) at large q.The theory based on the forward scattering peak in EPI is a good candidate to explain the linear temperature behavior of the resistivity down to very small temperaturesT(~ΘD/30)≈10K in some cuprates with low Tc(≈10 K) [6, 128, 129]. One physically rather plausible model, which is based on the forward scattering peak in EPI, is elaborated in [128]. It takes into account (i) the quasiparticle scattering on acoustic (a) and on optic (o) phonons, (ii) the extended van Hove singularity in the quasiparticle density of states N(ξ) which in some cuprates is very near the Fermi surface, and (iii) the umklapp and “undulation” (due to the flat regions at the Fermi surface) processes with vk′≅-vk—this condition can partly increase the EPI coupling. The transport Eliashberg function αtr2F(ω) is calculated similarly to (97) by using the definition of α2F(k,k′,ω) in (95) with the renormalized coupling constantgν(r)(k-k′)=gν(k-k′)γc(k-k′) of the ν=a,o mode, respectively. In [128] it is assumed a phenomenological form for the forward scattering peak in γc(k-k′) with the cutoff  qc≪kF (and which mimics the exact results from [78–80, 130, 179, 180]). Since the scattering of the quasiparticles on phonons (with the sound velocity vs) is limited to small-q transfer processes (with q<qc), then the maximal energy of the acoustic branch is not the Debye energy ΘD(≈kFvs) but the effective Debye energyΘA(≈qcvs)≪ΘD. In the case of Bi2201 in [128] it is taken (from the numerical results in [78–80, 130, 179, 180]) that qc≈kF/10 which gives ΘA≈(30–50)K. As a result the calculated αtr2F(ω) gives that ρab(T)~T down to very low T(~0.2ΘA)≈10K. The slope (dρab/dT) is governed by the effective EPI coupling constant for acoustic phonons. In systems with the extended van Hove singularity (in N(ξ)) near the Fermi surface, which is the case in Bi-2201, the effective coupling constant for acoustic phonons can be sufficiently large to give experimental values for the slope (dρab/dT)~(0.5–1)μΩcm/K—for details see [128]. This physical picture is applicable also to cuprates near and at the optimal doping but since in these systems Tc is large the linearity of ρab(T) down to very low T is “screened” by the appearance of superconductivity.(7) The width of the forward scattering peak in γc(kF,q) is very narrow in underdoped cuprates—with the width qc~δ(π/a)—which may have further interesting consequences. For instance, HTSC cuprates are characterized not only by strong correlations but also by the relatively small Fermi energy EF, which is in underdoped systems not much larger than the characteristic (maximal) phonon frequency ωphmax, that is, EF≃0.1–0.3eV, ωphmax≃80meV. Due to the appreciable magnitude of ωD/EF it is necessary to correct the Migdal-Eliashberg theory by the non-Migdal vertex corrections due to the EPI. It is well known that these vertex corrections lower Tc in systems with the isotropic EPI. However, the non-Migdal vertex corrections in systems with the forward scattering peak in the EPI coupling with the cutoff qc≪kF may increase Tc which can be appreciable. The corresponding calculations [220, 221] give two interesting results: (i) there is an appreciable increase of Tc by lowering Qc=qc/2kF, for instance, Tc(Qc=0.1)≈4Tc(Qc=1); (ii) even small values of λep<1 can give large Tc. The latter results open a new possibility in reaching high Tc in systems with appreciable ratio ωD/EF and with the forward scattering peak in EPI. The difference between the Migdal-Eliashberg and the non-Migdal theory can be explained qualitatively in the framework of an approximative McMillan formula for Tc (for not too large λ) which reads Tc≈〈ω〉e-1/(λ̃-μ*). The Migdal-Eliashberg theory predicts
(104)λ̃(ME)≈λ1+λ,
while the non-Migdal theory [220, 221] gives
(105)λ̃(n-ME)≈λ(1+λ).
For instance, Tc~100K in HTSC oxides can be explained by the Migdal-Eliashberg theory for λ(ME)~2, while in the non-Migdal theory much smaller coupling constant is needed, that is, λ(n-ME)~0.5.(8) The existence of the forward scattering peak in EPI can in a plausible way explain the ARPES puzzle that the antinodal kink is shifted by the maximal superconducting gap Δmax while the nodal kink is unshifted. The reason is (as explained in Section 1.3.3) that due to strong correlations the EPI spectral function α2F(k,k′,Ω)≈α2F(φ-φ′,Ω) is strongly peaked at φ-φ′=0 [151].(9) The scattering potential on nonmagnetic impurities is renormalized by strong correlations giving also the forward scattering peak in the impurity scattering potential (amplitude) [155]. The latter effect gives large d-wave channel in the renormalized impurity potential, which is the reason that d-wave pairing in HTSC cuprates is robust in the presence of nonmagnetic impurities (and defects) [6, 155].


### 2.3.5. EPI and Strong Correlations—Other Methods

The calculations of the static (adiabatic) charge-vertexγc(kF,q) in the X-method are done for the case U=∞ [78–80, 130, 179, 180] where it is found that it is peaked at q=0—the forward scattering peak (FSP). This result is confirmed by the numerical Monte Carlo calculations for the finite-U Hubbard model [182], where it is found that FSP exists for all U, but it is especially pronounced in the limit t≪U. These results are additionally confirmed in the calculations [183] within the four slave-boson method of Kotliar-Rückenstein where γc(kF,q) is again peaked at q=0 and the peak is also pronounced at t≪U.There are several calculations of the charge vertex in theone slave-boson method [219, 222–224] which is invented to study the limit U→∞. It is interesting to compare the results for the charge vertex in the X-method [78–80, 130, 179, 180] and in the one slave-boson theory [222] which are calculated in O((1/N)0) order. For instance, for J=0 one has
(106)γc(X)(k,q)=1+b(q)-a(q)t(k)[1+b(q)]2-a(q)c(q),γc(SB)(k,q)=1+b(q)-a(q)[t(k)+t(k+q)]/2[1+b(q)]2-a(q)c(q).
The explicit expressions for the “bare” susceptibilities a(q),b(q), and c(q) can be found in [78–80, 130]. It is obvious from (106) that γc(X)(k,q=0)=γc(SB)(k,q=0) but the calculations give that max{γc(X)(k,q)} is for q=0, while max{γc(SB)(k,q)} is for |q|≠0 [181]. So, the SB vertex is peaked at finite q which is in contradiction with the numerical Monte Carlo results for the Hubbard model [182] and with the four slave-boson theory [183]. The reason for the discrepancy of the one slave-boson (SB) in studying EPI with the numerical results and the X-method is not quite clear and might be due to the symmetry breaking of the local gauge invariance in leading order of the SB theory.


## 2.3.1. Minimal Model Hamiltonian

Theminimal microscopic model for HTSC cuprates must include at least three orbitals: one dx2-y2-orbital of the Cu-ion and two p-orbitals (px,y) of the O-ion since they participate in transport properties of these materials—see more in [6] and references therein. The electronic part of the Hamiltonian (of the minimal model) is Ĥ=Ĥ0+Ĥint—usually called the Emery model (or the p-dmodel) [206], where the one-particle tight-binding Hamiltonian Ĥ0 describes the lowering of the kinetic energy in the p-d model (with three bands or orbitals):
(74)Ĥ0=∑i,σ(ϵd0-μ)diσ†diσ+∑j,α,σ(ϵpα0-μ)pjασ†pjασ+∑i,j,α,σtijαpddiσ†pjασ+∑j,j′,α,β,σtjj′,αβpppjασ†pj′βσ.
Here tijαpd (i,j enumerate the Cu- and O-sites, resp.) is the hopping integral between the pα(α=x,y)—and d-states and tjj′αβpp between the pα- and pβ-states—while ϵd0 and ϵpα0 are the bare d- and p-local energy levels and μ is the chemical potential. This tight-binding Hamiltonian is written in the electronic notation where the charge-transfer energy Δdp,0≡ϵd0-ϵp0>0 by assuming that there is one 3dx2-y2 electron on the copper (Cu2+) while electrons in the p-levels of the O2- ions occupy filled bands. Ĥ0 contains the main ingredients coming from the comparison with the LDA-DFT band-structure calculations. The LDA-DFT results are reproduced by assuming that tpp≪tpd (and ϵpα0=ϵp0 ) where the good fit to the LDA-DFT band structure is found for Δdp,0≡ϵd0-ϵp0≈3.2eV and tpd(≡tpd)=(3/2)(pdσ),(pdσ)=-1.8eV. The total LDA bandwidth Wb=(42)|tpd|≅9eV [207].The electron interaction is described byĤint:
(75)Ĥint=Ud∑ini↑dni↓d+Up∑j,αnjα↑pnjα↓p+V̂c+V̂ep,
where Ud and Up are the on-site Coulomb repulsion energies at Cu and O sites, respectively, while V̂c and V̂ep describe the long-range part of the Coulomb interaction of electrons (holes) and EPI, respectively. Note that the Hubbard repulsion Ud on the Cu-ion is different from its bare atomic value Ud0(≈16 eV for Cu) due to various kinds of screening effects in solids [208–210]. It turns out that in most transition metal oxides one has Ud≪Ud0. This problem is thoroughly studied in [208–210] and applied to HTSC cuprates. The estimation from the numerical cluster calculations [211] gives Ud=9–11eV and Up=4–6eV but because nidUd≫njpUp the on-site repulsion on the oxygen ion is usually neglected at the first stage of the analysis.Note that in the case of largeUd(≫tpd,Δdp,0) the hole notation is usually used where in the parent compound (and for |tpd|≪Δdp,0) one has 〈nid〉=1, that is, one hole in the 3D-shell (in the 3dx2-y2 state) in the ground state. In the limit of large Ud→∞ the doubly occupancy on the Cu atoms is forbidden and only two copper states are possible: Cu2+—described by the quantum state diσ†|0〉 with one hole in the 3D shell and Cu1+—described by |0〉 with zero holes in the filled 3D shell. In this (hole) notation the oxygen p-level is fully occupied by electrons, that is, there are no holes (〈njp〉=0) in the occupied oxygen 2p-shell of O2-. In this notation the vacuum state |0v〉 (not the ground state) of the Hamiltonian Ĥ for large Ud corresponds to the closed-shell configuration Cu1+O2-. In the hole notation the hole p-level ϵph0 lies higher than the hole d-level ϵdh0, that is, Δpd,0≡ϵph0-ϵdh0>0 (note that in the electron notation it is opposite) and Ud means repulsion of two holes (in the 3dx2-y2 orbital) with opposite spins—3d8 configuration of the Cu3+ ion. Note that ϵph0=-ϵp0,ϵdh0=-ϵd0, and tpd,h=-tpd. In the following the index h in tpd,h is omitted. The reason for ϵph0>ϵdh0 is partly in different energies for the hole sitting on the oxygen and copper, respectively [207]. From this model one can derive in the limit U→∞ the t-J model for the 2D lattice in the CuO2 plane [212, 213], where now each lattice site corresponds to a Cu-atom. In the presence of one hole in the 3D-shell then in the undoped (no oxygen holes) HTSC cuprate each lattice site is occupied by one hole. By doping the system with holes the additional holes go onto O-sites. Furthermore, due to the strong Cu–O covalent binding the energetics of the system implies that an O-hole forms a Zhang-Rice singlet with a Cu-hole [212]. In the t-J model the Zhang-Rice singlet is described by an empty site.Since in the t-J model the doubly occupancy is forbidden, one introduces annihilation (Hubbard) operator of the composite fermion X̂iσ0=ciσ†(1-ni,-σ) which describes creation of a hole (in the 3D-shell of the Cu-atoms) on the ith site if this site is previously empty (thus excluding doubly occupancy), that is, the constraint ni,σ+ni,-σ≤1 must be fulfilled on each lattice site.  In this picture the doped-hole concentration δ means at the same time the concentration of the oxygen holes, that is, of the Zhang-Rice singlets.In order not to confuse the reader we stress the difference in the meaning of the hole in the (p-d) three-band Emery model and in the single-band (effective) t-J model. In the Emery model the hole means the absence of the electron in the filled shell—the 3D shell for Cu atoms(ions) and 2p shell for O atoms(ions). On the other side the hole on the ith lattice site in the t-J model means the presence of the Zhang-Rice singlet on this site.Thebosonic-like operatorsX̂iσ1σ2=X̂iσ10X̂i0σ2 for  σ1≠σ2 create a spin fluctuation at the ith site and the spin operator is given by S=X̂iσ¯10(σ⃗)σ¯1σ¯2X̂i0σ¯2 where summation over the bar indices is understood. The operator ∑σX̂iσσ has the meaning of thehole number on theith site. It is useful to introduce the operator X̂i00=X̂i0σX̂iσ0 at the ith lattice site which is the number of Zhang-Rice singlets on theith site. For X̂i00|0〉=1|0〉 the ith site is occupied by the Zhang-Rice singlet, while for X̂i00|1〉=0|1〉 there is no Zhang-Rice singlet on the ith site (i.e., this site is occupied only by one 3d9 hole on the Cu site). This property of X̂i00 is due to the local constraint(76)X̂i00+∑σ=↑↓X̂iσσ=1,
which forbids doubly occupancy of the ith site by holes. By projecting out doubly occupied (high-energy) states the t-J model reads
(77)Ĥt-j=∑i,σϵi0X̂iσσ-∑i,j,σtijX̂iσ0X̂j0σ+∑i,jJij(Si·Sj-14n̂in̂j)+Ĥ3.
The first term (~ϵi0) describes an effective local energy of the hole (or the Zhang-Rice singlet), the second one (~tij) describes hopping of the holes, and the third one (~Jij) is the Heisenberg-like exchange energy between two holes. The theory [212] predicts that |ϵi0|≫|tij|. This property is very important in the study of EPI. Ĥ3 contains three-site term which is usually omitted believing that it is not important. For charge fluctuation processes it is plausible to omit it, while for spin-fluctuation processes it is questionable approximation. If one introduces the enumeration α,β,γ,λ=0,↑,↓, then the Hubbard operators satisfy the following algebra:
(78)[X̂iαβ,X̂jγλ]±=δij[δγβX̂iαλ±δαλX̂iγβ],
where δij is the Kronecker symbol. Note that the Hubbard operators possess the projection properties with X̂iαβX̂iγλ=δβγX̂iαλ. The (anti)commutation relations in (78) are more complicated than the canonical Fermi and Bose (anti)commutation relations, which complicates the mathematical structure of the theory. To escape these complications some novel techniques have been used, such as the one slave boson-technique. In this technique X̂i0σ=fiσbi†,X̂iσ1σ2=fiσ1†fiσ2 are represented in terms of the fermion (spinon) operator fiσ which annihilates the spin on the ith and the boson (holon) operator bi† which creates the Zhang-Rice singlet.In the minimal theoretical model the electron-phonon interaction (EPI) contains in principle two leading terms:(79)Ĥep=Ĥepion+Ĥepcov,
which are the “ionic” one (Ĥepion) and the “covalent” one (Ĥepcov). The “ionic” term describes the change of the energy of the hole (or the Zhang-Rice singlet) at the ith site due to lattice vibrations and it reads [6, 78–80, 130]
(80)Ĥepion=∑i,σΦ̂iX̂iσσ,
where the “displacement” operator
(81)Φ̂i=∑Lκ[ϵ(Ri0-RLκ0+ûi-ûLκ)-ϵ(Ri0-RLκ0)]
(which as in Section 2.1 includes the bare coupling constant) describes the change of the hole (or Zhang-Rice singlet) energy ϵa,i0 by displacing atoms in the lattice by the vector ûLκ. In the harmonic approximation the EPI potential is given by Φ̂i=∑gi(q,λ)exp{iqRi}[bq,λ+b-q,λ†] where bq,λ and bq,λ† are the annihilation and creation operator of phonons with the polarization λ, respectively. This term describes in principle the following processes: (1) the change of the O-hole and Cu-hole bare energies ϵph0,ϵdh0 in the three-band model due to lattice vibrations, (2) the change of the long-range Madelung energy (which is due to the ionicity of the structure) by lattice vibrations along the c-axis, and (3) the change of the Cu–O hopping parameter tpd in the presence of vibrations, and so forth. Here, L and κ enumerate unit lattice vectors and atoms in the unit cell, respectively. Usually, the EPI scattering is studied in the harmonic approximation where the phonon operator Φ̂i is calculated in the harmonic approximation (Φ̂~û) for the EPI interaction of holes with some specific phononic modes, such as the breathing and half-breathing ones [10, 11, 169]. The theory which includes also all other (than oxygen) vibrations in Φ̂i is still awaiting.It is interesting to make comparison of the EPI coupling constants in thet-J model and in the Hartree-Fock (HF) approximation (which is the analogous of the LDA-DFT method) of the three-band Emery (p-d) model in (74)-(75) when the problem is projected on the single band. For instance, the coupling constant with thehalf-breathing mode at the zone boundary in the HF approximation (which mimics the LDA-DFT approach) is given by
(82)ghbHF=±4tpd∂tpd∂RCu–O1ϵd-ϵpu0,
while the coupling constant in thet-Jmodelghbt-J(=∂ϵ0/∂RCu–O) is given by
(83)ghbt-J=±4tpd∂tpd∂RCu–O[2p2-1ϵd-ϵp+2p2Ud-|ϵd-ϵp|]u0,
where p=0.96—see [10, 11, 169] and references therein. It is obvious that in the t-J model the electron-phonon coupling is different from the HF one, since the former contains an additional term coming from the many-body effects, which are not comprised by the HF (LDA-DFT) calculations. The first term in (83) describes the hopping of a 3D hole into the O 2p-states and this term exists also in the LDA-DFT coupling constant—see (82). However, the second term in (83), which is due to many-body effects, describes the hopping of an O 2p-hole into the (already) single occupied Cu 3D state and it does not exist in the LDA-DFT approach. Since the corresponding dimensionless coupling constant λhb is proportional to |ghb|2, one obtains that the bare t-J coupling constant is almost three times larger than the LDA-DFT one:
(84)λhbt-J≈3λhbHF.
This example demonstrates clearly that the LDA-DFT method is inadequate for calculating the EPI coupling constant in HTSC cuprates.Note that there is also a covalent contribution toEPI which comes from the change of the effective hopping (t) in of the t-J model (77) and the exchange energy (J) in the presence of atomic displacements:
(85)Ĥepcov=-∑i,j,σ∂tij∂(Ri0-Rj0)(ûi-ûj)X̂iσ0X̂j0σ+∑i,j,∂Jij∂(Ri0-Rj0)(ûi-ûj)Si·Sj.
Here, we will not go into details but only stress that since |ϵi0|≫|tij| then the covalent term in the effectivet-Jmodel is much smaller than the ionic term—see more in [6, 10, 11, 169] and references therein—and in the following only the term Ĥepion will be considered [6, 78–80, 130].


## 2.3.2. Controllable X-Method for the Quasiparticle Dynamics

The minimal model Hamiltonian for strongly correlated holes with EPI (discussed above) is expressed via the Hubbard operators which obey “ugly” noncanonical commutation relations. The latter property is rather unpleasant for making a controllable theory in terms of Feynmann diagrams (for these “ugly” operators) and some other approaches are required. A possible way out is to express the Hubbard operators in terms of fermions and bosons (which must be confined) as, for instance, in the slave boson (SB) method. However, in real calculations which are based on some approximations the SB method is confronted with some subtle constraints whose fulfillments require very sophisticated mathematical treatment. Fortunately, there is amathematically controllable approach for treating the problem directly with Hubbard operators and without using slave-boson (or fermion) techniques. This method—we call it the X-method—is based on the general Baym-Kadanoff technique which allows to treat the problem by the well-defined and controllable 1/N expansion for the Green's functions in terms of Hubbard operators. This approach is formulated in [214] while the important refinement of the method is done in [78–80, 130]. In the paramagnetic and homogeneous state (with finite doping) the Green's function Gσ1σ2(1-2) is diagonal, that is, Gσ1σ2(1-2)=δσ1σ2G(1-2) where
(86)G(1-2)=-〈TX̂0σ(1)X̂σ0(2)〉=g(1-2)Q,
with the Hubbard spectral weight Q=〈X̂00〉+〈X̂σσ〉. The function g(1-2) plays the role of the quasiparticle Green's function—see more in [6, 78–80, 130, 179, 180]. It turns out that in order to have a controllable theory (1/N expansion) one way is to increase the number of spin components from two to N by changing the constraint (76) into the new one
(87)X̂i00+∑σ=1NX̂iσσ=N2.
In order to reach the convergence of physical quantities in the limit N→∞ the hopping and exchange energy are also rescaled, that is, tij=t0,ij/N and Jij=J0,ij/N. In order to eliminate possible misunderstandings we stress that in the case N>2 the constraint in (87) spoils some projection properties of the Hubbard operators. Fortunately, these (lost) projection properties are not used at all in the refined theory. As a result one obtains the functional integral equation for G(1,2), thus allowing unambiguous mathematical and physical treatment of the problem. In [78–80, 130, 179, 180] it is developed a systematic 1/N expansion for the quasiparticle Green's function g(1-2)(=g0+g1/N+⋯), Q(=Nq0+q1+⋯) (also for G(1-2)) and the self-energy. For large N(→∞) the leading term is G0(1-2)=g0(1-2)Q0=O(N) with g0=O(1) and Q0=〈X̂i00〉=Nδ/2. Here, δ is the concentration of the oxygen holes (that is, of the Zhang-Rice singlets) which is related to the chemical potential by the equation 1-δ=2∑pnF(p) with nF(p)=(eϵ0(k)-μ+1)-1. The quasiparticle Green's function g0(k,ω) and the quasiparticle spectrum ϵ0(k) in the leading order are given by
(88)g0(k,ω)≡G0(k,ω)Q0=1ω-[ϵ0(k)-μ],(89)ϵ0(k)=ϵc-δ·t(k)-∑pJ0(k+p)nF(p).
The level shift is ϵc=ϵ0+2∑pt(p)nF(p) and t(p) is the Fourier transform of the hopping integral tij—see more in [6].Let us summarize themain results of the X-method in leading O(1)-order for the quasiparticle properties in the t-J model [6, 78–80, 130, 179, 180]. (i) The Green's function g0(k,ω) describes the coherent motion  of quasiparticles whose contribution to the total spectral weight of the Green's function G0(k,ω) is Q0=Nδ/2. The coherent motion of quasiparticles is described in leading order by G0(k,ω)=Q0g0(k,ω) and the quasiparticle residuumQ0 disappears in the undoped Mott insulating state (δ=0). This result is physically plausible since in the Mott insulating state the coherent motion of quasiparticles, which is responsible for finite conductivity, vanishes. (ii) The quasiparticle spectrum ϵ0(k) plays the same role as the eigenvalues of the ideal band-structure Hamiltonian ĥ0(x,y) (it contains the excitation potential VIBS(x,y) which is due to high-energy processes of the Coulomb interaction). So, if we would consider ϵtb(k)=-t(k) as the tight-binding parametrization of the LDA-DFT band-structure spectrum which takes int account only weak correlations (with the local potential Vxc(x)δ(x-y)), then one can define a nonlocal excitation potentialVIBStJ(x,y)=ṼIBStJ(x,y)+Vxc(x)δ(x-y) which mimics strong correlations in the t-J model
(90)ṼIBStJ(x,y)≈V0δ(x-y)+(1-δ)t(x-y)-J̃(x-y).
Here, V0=2∑pt(p)nF(p) and t(x-y) is the Fourier transform of t(k) while J̃(x-y) is the Fourier transform of the third term in (89). The relative excitation potential ṼIBStJ(x,y) is due to strong correlations (suppression of doubly occupancy on each lattice site) and as we will see below it is responsible for the short-range screening of EPI in such a way that the forward scattering peak appears in the effective EPI interaction—see discussion below. (iii) For the very low doping ϵ0(k) is dominated by the exchange parameter if J0>δ·t0. However, in the case when J0≪δ·t0 there is a band narrowing by lowering the hole-doping δ, where the band width is proportional to the hole-concentration δ, that is, Wb=z·δ·t0. (iv) The O(1)-order quasiparticle Green's function g0(k,ω) and the quasiparticle spectrum ϵ0(k) in the X-method have similar form as the spinon Green's function g0,f(k,ω)=-〈Tfσfσ†〉k,ω and the spinon energy ϵs(k) in the SB method. However, in the SB method there is a broken gauge symmetry in the metallic state (with δ≠0) which is characterized by 〈b̂i〉≠0. This broken local gauge symmetry in the slave-boson method in O(1) order, which is due to the local decoupling of spinon and holon, is in fact forbidden by Elitzur's theorem. On the other side the local gauge invariance is not broken in the X-method where Green's function G0(k,ω) describes motion of the composite object, that is, simultaneous creation of the hole and annihilation of the spin at a given lattice site, while in the SB theory there is a spin-charge separation because of the broken symmetry (〈b̂i〉≠0). The assumption of the broken symmetry 〈b̂i〉≠0 gives qualitative satisfactory results for the quasiparticle energy for the case N=∞ in D>2 dimensions. However, the analysis of response functions and of higher-order 1/N corrections to the self-energies very delicate in the SB theory and special techniques must be implemented in order to restore the gauge invariance of the theory. On the other side the X-method is intrinsically gauge invariant and free of spurious effects in all orders of the 1/N expansion. Therefore, one expects that these two methods may deliver different results in O(1) and higher order in response functions. This difference is already manifested in the calculation of EPI where the charge vertex in these two methods is peaked at different wave vectors q, that is, at q=0 in the X-method and |q|≠0 in the SB method—see Section 2.3.5. (v) In [215, 216] it is shown that in the superconducting state the anomalous self-energies (which are of O(1/N)-order in the 1/N expansion) of the X- and SB-methods differ substantially. As a consequence, the SB method [217] predicts false superconductivity in the t-J model (for J=0) with large Tc (due to the kinematical interaction), while the X-method gives extremely small Tc(≈0) [215, 216]. So, although the two approaches yield some similar results in leading O(1)-order they, are different at least in next to leading O(1/N)-order.


## 2.3.3. EPI Effective Potential in thet-J Model

The theory of EPI in the minimalt-J model based on the X-method predicts that the leading term in the EPI self-energy Σep is given by the expression [6, 78–80, 130]
(91)Σep(1,2)=-Vep(1¯-2¯)γc(1,3¯;1¯)g0(3¯-4¯)γc(4¯,2;2¯),
where the screened (by the dielectric constant) EPI potential
(92)Vep(1-2)=ɛe-1(1-1¯)Vep0(1¯-2¯)ɛe-1(2¯-2)
and Vep0(1-2)=-〈TΦ̂(1)Φ̂(2)〉 is the “phonon” propagator which may also describe an anharmonic EPI. It is obvious that (91) is equivalent to (53) in spite the fact that the theory is formulated in terms of the Hubbard operators. The charge vertexγc(1,2;3)=-δg0-1(1,2)/δueff(3) corresponds to the the renormalized vertex Γc,r in (53) and it describes the screening by strong correlations. It depends on the relative excitation potential ṼIBStJ(x,y). The electronic dielectric function ɛe(1-2) describes the screening of EPI by the long-range part of the Coulomb interaction. Note that in the harmonic approximation Φ̂(1) contains the bare EPI coupling constant gep0 and lattice displacement û, that is, Φ̂~gep0û—see more in [6]. (Note that in the above equations summation and integration over bar indices are understood.) The self-energy Σep(k,ω) due to EPI reads
(93)Σep(k,ω)=∫0∞dν〈α2F(k,k′,ν)〉k′R(ω,ν),
with R(ω,ν)=-2πi(nB(ν)+1/2)+ψ(1/2+i)-ψ(1/2-i(ν+ω)/2πT) where nB(ν) is the Bose distribution function and ψ is di-gamma function. The Eliashberg spectral function is given by
(94)α2F(k,k′,ω)=N(0)∑ν|gν(k,k-k′)|2×δ(ω-ων(k-k′))γc2(k,k-k′),
where gν(k,p) is the EPI coupling constant for the νth mode, where the renormalization by long-range Coulomb interaction is included, that is, gν(k,p)=gep,ν0(k,p)/ɛe(p). 〈⋯〉k denotes Fermi-surface average with respect to the momentum k and N(0) is the density of states renormalized by strong correlations. The effect of strong correlations in the adiabatic limit is stipulated in the charge vertex functionγc(k,k-k′) which, as we will see in Section 2.3.4, changes the properties of Vep(q,ν) drastically compared to weakly correlated systems. In fact the charge vertex depends on frequency ω but in the adiabatic limit (ωph≪W) and for qvF>ωph it is practically frequency independent, that is, γc(ad)(k,q,ω)≈γc(k,q,ω=0) where the latter is real quantity. For J=0 in the t-t′ model the 1/N expansion gives N(0)=N0(0)/q0 where q0=δ/2. For J≠0 the density of states N(0) does not diverge for δ→0 where N(0)(~1/J0)>N0(0). The bare density of statesN0(0) is calculated in absence of strong correlations, for instance, by the LDA-DFT method.Depending on the symmetry of the superconducting order parameterΔ(k,ω) (s- and d-wave pairing) various projected averages (over the Fermi surface) of α2F(k,k′,ω) enter the Eliashberg equations. Assuming that the superconducting order parameter transforms according to the representation Γi of the point group C4v of the square lattice (in the CuO2 planes) the appropriate symmetry-projected spectral function is given by
(95)α2Fi(k̃,k̃′,ω)=N(0)8∑ν,j|gν(k̃,k̃-Tjk̃′)|2×δ(ω-ων(k̃-Tjk̃′))×γc2(k̃,k̃-Tjk̃′)Di(j)
where k̃ and k̃′ are momenta on the Fermi line in the irreducible Brillouin zone (1/8 of the total Brillouin zone). Tj,j=1,…,8 denotes the eight point-group transformations forming the symmetry group of the square lattice. This group has five irreducible representations which we distinguish by the label i=1,2,…,5. In the following we discuss the representations i=1  and i=3, which correspond to the s- and d-wave symmetry of the full rotation group, respectively. Di(j) is the representation matrix of the jth transformation for the representation i. Assuming that the superconducting order parameter Δ(k,ω) does not vary much in the irreducible Brillouin zone, one can average over k̃ and k̃′ in the Brillouin zone. For each symmetry one obtains the corresponding pairing spectral functionα2Fi(ω):
(96)α2Fi(ω)=〈〈α2Fi(k̃,k̃′,ω)〉k̃〉k̃′,
which governs the transition temperature for the order parameter with the symmetry Γi. For instance, α2F3(ω) is the pairing spectral function in the d-channel and it gives the coupling for d-wave superconductivity (the irreducible representation Γ3—sometimes labelled as B1g). Performing similar calculations for the phonon-limited resistivity, one finds that the resistivity is related to the transport spectral functionα2Ftr(ω):
(97)αtr2F(ω)=〈〈α2F(k,k′,ω)[v(k)-v(k′)]2〉〉kk′2〈〈v2(k)〉〉kk′.
The effect of strong correlations on EPI was discussed in [130] within the model where gν(k,p) and the phonon frequencies ων(k̃-k̃′) are weakly momentum dependent. In order to elucidate the main effect of strong correlations on EPI and α2Fi(ω) we consider the latter functions for a simple model with Einstein phonon, where these functions are proportional to the (so called) relative coupling constantΛi:
(98)Λi=18N(0)N0(0)∑j=18〈〈|γc(k̃,k̃-Tjk̃′)|2〉〉k̃k̃′Di(j).
Similarly, the resistivity ρ(T)(~λtr~Λtr) is renormalized by the correlation effects where the transport coupling constant Λtr is given by
(99)Λtr=N(0)N0(0)〈〈|γc(k̃,k̃-Tjk̃′)|2[v(k)-v(k′)]2〉〉kk′2〈〈v2(k)〉〉kk′.As we see, all projected spectral functionsαi2F(ω) depend on the charge vertex functionγc(k,q) which describes the screening (renormalization) of EPI due to strong correlations (suppression of doubly occupancy) [78–80, 130]. This important ingredient (which respects also the Ward identities) is a decisive step beyond the MFA renormalization of EPI in strongly correlated systems which was previously studied in connection with heavy fermions—see review in [218].


## 2.3.4. Charge Vertex and the EPI Coupling

The charge vertex functionγc(k,q) (in the adiabatic approximation) has been calculated in [78–80, 130, 179, 180] in the framework of the 1/N expansion in the X-method—see also [6]—and here we discuss only the main results. Note that γc(k,q) renormalizes all charge fluctuation processes, such as the EPI interaction, the long-range Coulomb interaction, the nonmagnetic impurity scattering, and so forth. In fact γc(k,q) describes specific screening due to the vanishing of doubly occupancy in strongly correlated systems. Note that the latter constraint is at present impossible to incorporate into the LDA-DFT band-structure calculations, thus making the latter method unreliable in highly correlated systems. In [78–80, 130, 179, 180] γc(k,q,ω) was calculated as a function of the model parameters t,t′,δ,J in leading O(1) order of the t-J model:
(100)γc(k,q)=1-∑α=16‍∑β=16‍Fα(k)[1̂+χ̂(q)]αβ-1χβ2(q),
where χαβ(q)=∑pGα(p,q)Fβ(p),Fα(k)=[t(k),1, 2J0coskx,2J0sinkx,2J0cosky,2J0sinky], and Gα(p,q) = [1,t(p+q),cospx,sinpx,cospy,sinpy]Π(p,q). Here, Π(k,q) = -g(k)g(k+q) and q=(q,iqn),qn=2πnT,p=(p,ipm),pm=πT(2m+1). The physical meaning of the vertex function γc(k,q) is following: in the presence of an external (or internal) charge perturbation there is screening due to the change of the excitation potential VIBStJ(x,y), that is, of the change of the bandwidth, as well as of the local chemical potential. The central result is that for momenta k lying at (and near) the Fermi surface the vertex function γc(k,q,ω=0) has very pronounced forward scattering peak(at q=0) especially at very low doping concentration δ(≪1), while the backward scattering is substantially suppressed,as it is seen in Figure 35 where γc(kF,q,ω=0) is shown. The peak at q=0 is very narrow at very small doping since its width qc is proportional to the doping δ, that is, qc~δ(π/a) where a is the lattice constant. It is interesting that γc(k,q), as well as the dynamics of charge fluctuations, depend only weakly on the exchange energy J and are mainly dominated by the constraint of having no doubly occupancy of sites, as it is shown in [78–80, 130, 179, 180].Figure 35
Adiabatic (ω=0) vertex function γ(kF,q) of the t-J model as a function of the momentum aq with q=(q,q) for three different doping levels δ. From [130].The existence of the forward scattering peak inγc(k,q) at q=0 is confirmed by numerical calculations in the Hubbard model,which show that this peak is very pronounced at large U [182]. This is important result since it proves that the 1/N expansion in the X-method is reliable method in studying charge fluctuation processes in strongly correlated systems. The strong suppression of γc(k,q) at large q(~kF) means that at small distances the charge fluctuations are strongly suppressed (correlated). Such a behavior of the vertex function means that a quasiparticle moving in the strongly correlated medium digs up a giant correlation hole with the radius ξch(~π/qc)≈a/δ, where a is the lattice constant. As a consequence of this effect the renormalized EPI becomes long ranged which is contrary to the weakly correlated systems where it is short ranged.By knowingγc(k,q) one can calculate the relative coupling constants Λ1≡Λs, Λ3≡Λd, Λtr, and so forth. In the absence of correlations and for an isotropic band one has Λ1=Λtr=1, Λi=0 for i>1. The averages in Λs, Λd, and Λtr were performed numerically in [130] by using the realistic anisotropic band dispersion in the t-t′-J model and the results are shown in Figure 36. For convenience, the three curves are multiplied with a common factor so that Λs approaches 1 in the empty-band limit δ→1, when strong correlations are absent. Note that the superconducting critical temperature Tc in the weak coupling limit and in the ith channel scales like Tc(i)~exp(-1/(λ0Λi-μi*) where λ0 is some effective coupling constant which depends on microscopic details. The parameter μi* is the effective residual Coulomb repulsion in the ith superconducting channel. We stress here several interesting results which come out from the above theory and which are partially presented in Figures 35 and 36.Figure 36
Normalizeds-wave Λs, d-wave Λd, and transport Λtr·δ coupling constants as a function of doping δ for t′=0 and J=0. From [179, 180].(1) In principle the bare EPI coupling constant gλ0(k,q) depends on the quasiparticle momentum k and the transfer momentum q. In the t-J model the EPI coupling is dominated by the ionic coupling Ĥepion (see (80)) and corresponding EPI depends only on the momentum transfer q, that is, gλ0(k,q)≈gλ0(q) while for the much smaller covalent coupling Ĥepcov depends on both k and q [6, 10, 11]. However, the EPI couplings for most phonon modes are renormalized by the charge vertex and since the latter is peaked at small momentum transfer q=|k-k′| then the maxima of the corresponding effective potentials are pushed toward smaller values of q. The further consequence of the vertex renormalization is that in the absence of strong correlations the bare EPI coupling |g0(k,q)|2 for some phonon modes (which enters in the effective t-J model) is detrimental for d-wave pairing; it can be less detrimental or even supports it in the presence of strong correlations (since the maximum is pushed toward smaller q). To illustrate this let us consider the in-plane oxygen breathing mode with the frequency ωbr which is supposed to be important in HTSC cuprates. The bare coupling constant (squared) for this mode is approximately given by |gbr0(k,q)|2=|gbr0|2[sin2(qxa/2)+sin2(qya/2)] which reaches maximum for large q=(π/a,π/a). By extracting the component in the d-channel one has
(101)|gbr0(k-k′)|2=|gbr0|2[1-(14)ψd(k)ψd(k′)+⋯]
with
(102)ψd(k)=coskxa-coskya.
This gives the repulsive coupling constant λd0 in the d-channel, that is,
(103)λd0=2ωbr〈ψd(k)|gbr0(k-k′)|2ψd(k′)〉<0.
However, in the presence of strong correlations one expects that the effective coupling constant is given approximately by |gbreff(k,k-k′)|≈|gbr0(k-k′)|2γc2(kF,k-k′) which is at small doping δ suppressed substantially at large q since γc2 starts to fall off drastically at q~qc~δ(π/a). The latter property makes the effective coupling constant (in the d-channel) λdeff for these modes less negative or even positive (depending on the ratio ξch/a~1/δ), that is, one has λdeff>λd0. We stress again that this analysis is only qualitative (and semiquantitative) since it is based on the t-J model while the better quantitative results are expected in the strongly correlated three-band Emery model with Ud≫t,Δpd—see [6, Appendix  D]. Unfortunately, these calculations are not finalized until now.(2) In weakly correlated systems (or, e.g., in the empty-band limit δ→1) the relative d-wave coupling constant Λd is much smaller than the s-wave coupling constant Λs, that is, Λd≪Λs as it is seen in Figure 36. Furthermore, Λs decreases with decreasing doping.(3) It is indicative that independently on the value of t′≠0 or t′=0 the coupling constant Λs and Λd  meet each other (note that Λs>Λd for all δ) at some small doping δ≈0.1–0.2 where Λs≈Λd. We would like to stress that such a unique situation (with Λs≈Λd) was practically never realized in low-temperature and weakly correlated superconductors and in that respect the strong momentum-dependent EPI in HTSC cuprates is an exclusive but very important phenomenon.(4) By taking into account the residual Coulomb repulsion of quasiparticles then the s-wave superconductivity (which is governed by Λs) is suppressed, while the d-wave superconductivity (which is governed by Λd) stays almost unaffected, since μs*≫μd*. In that case the d-wave superconductivity which is mainly governed by EPI becomes more stable than the s-wave one at sufficiently low doping δ. This transition between s- and d-wave superconductivity is triggered by electronic correlations because in the model calculations [78–80, 130] the bare EPI coupling is assumed to be momentum independent, that is, the bare coupling constant contains the s-wave symmetry only.(5) The calculations of the charge vertex γc are performed in the adiabatic limit, that is, for ω<q·vF(q) the frequency ω in γc can be neglected. In the nonadiabatic regime, that is, for ω>q·vF(q), the function γc2(kF,q,ω) may be substantially larger compared to the adiabatic case because γc(kF,q,ω) tends to the bare value 1 for q=0. This means that EPI for different phonons (with different energies ω) is differently affected by strong correlations. For a given ω the EPI coupling to those phonons with momenta q<qω=ω/vF will be (relatively) enhanced since γc(kF,q,ω)≈1, while the coupling to those with q>qω=ω/vF will be substantially reduced due to the suppression of the backward scattering by strong correlations [37, 38]. These results are a consequence of the Ward identities and generally hold in the Landau-Fermi liquid theory [219].(6) The transport EPI coupling constant Λtr is significantly reduced in the presence of strong correlations especially for low doping (δ≪1) where Λtr<Λ/3. This result is physically plausible since the resistivity is dominated by the backward scattering processes (large q~kF) which are suppressed by strong correlations—the suppression of γc(kF,q,ω) at large q.The theory based on the forward scattering peak in EPI is a good candidate to explain the linear temperature behavior of the resistivity down to very small temperaturesT(~ΘD/30)≈10K in some cuprates with low Tc(≈10 K) [6, 128, 129]. One physically rather plausible model, which is based on the forward scattering peak in EPI, is elaborated in [128]. It takes into account (i) the quasiparticle scattering on acoustic (a) and on optic (o) phonons, (ii) the extended van Hove singularity in the quasiparticle density of states N(ξ) which in some cuprates is very near the Fermi surface, and (iii) the umklapp and “undulation” (due to the flat regions at the Fermi surface) processes with vk′≅-vk—this condition can partly increase the EPI coupling. The transport Eliashberg function αtr2F(ω) is calculated similarly to (97) by using the definition of α2F(k,k′,ω) in (95) with the renormalized coupling constantgν(r)(k-k′)=gν(k-k′)γc(k-k′) of the ν=a,o mode, respectively. In [128] it is assumed a phenomenological form for the forward scattering peak in γc(k-k′) with the cutoff  qc≪kF (and which mimics the exact results from [78–80, 130, 179, 180]). Since the scattering of the quasiparticles on phonons (with the sound velocity vs) is limited to small-q transfer processes (with q<qc), then the maximal energy of the acoustic branch is not the Debye energy ΘD(≈kFvs) but the effective Debye energyΘA(≈qcvs)≪ΘD. In the case of Bi2201 in [128] it is taken (from the numerical results in [78–80, 130, 179, 180]) that qc≈kF/10 which gives ΘA≈(30–50)K. As a result the calculated αtr2F(ω) gives that ρab(T)~T down to very low T(~0.2ΘA)≈10K. The slope (dρab/dT) is governed by the effective EPI coupling constant for acoustic phonons. In systems with the extended van Hove singularity (in N(ξ)) near the Fermi surface, which is the case in Bi-2201, the effective coupling constant for acoustic phonons can be sufficiently large to give experimental values for the slope (dρab/dT)~(0.5–1)μΩcm/K—for details see [128]. This physical picture is applicable also to cuprates near and at the optimal doping but since in these systems Tc is large the linearity of ρab(T) down to very low T is “screened” by the appearance of superconductivity.(7) The width of the forward scattering peak in γc(kF,q) is very narrow in underdoped cuprates—with the width qc~δ(π/a)—which may have further interesting consequences. For instance, HTSC cuprates are characterized not only by strong correlations but also by the relatively small Fermi energy EF, which is in underdoped systems not much larger than the characteristic (maximal) phonon frequency ωphmax, that is, EF≃0.1–0.3eV, ωphmax≃80meV. Due to the appreciable magnitude of ωD/EF it is necessary to correct the Migdal-Eliashberg theory by the non-Migdal vertex corrections due to the EPI. It is well known that these vertex corrections lower Tc in systems with the isotropic EPI. However, the non-Migdal vertex corrections in systems with the forward scattering peak in the EPI coupling with the cutoff qc≪kF may increase Tc which can be appreciable. The corresponding calculations [220, 221] give two interesting results: (i) there is an appreciable increase of Tc by lowering Qc=qc/2kF, for instance, Tc(Qc=0.1)≈4Tc(Qc=1); (ii) even small values of λep<1 can give large Tc. The latter results open a new possibility in reaching high Tc in systems with appreciable ratio ωD/EF and with the forward scattering peak in EPI. The difference between the Migdal-Eliashberg and the non-Migdal theory can be explained qualitatively in the framework of an approximative McMillan formula for Tc (for not too large λ) which reads Tc≈〈ω〉e-1/(λ̃-μ*). The Migdal-Eliashberg theory predicts
(104)λ̃(ME)≈λ1+λ,
while the non-Migdal theory [220, 221] gives
(105)λ̃(n-ME)≈λ(1+λ).
For instance, Tc~100K in HTSC oxides can be explained by the Migdal-Eliashberg theory for λ(ME)~2, while in the non-Migdal theory much smaller coupling constant is needed, that is, λ(n-ME)~0.5.(8) The existence of the forward scattering peak in EPI can in a plausible way explain the ARPES puzzle that the antinodal kink is shifted by the maximal superconducting gap Δmax while the nodal kink is unshifted. The reason is (as explained in Section 1.3.3) that due to strong correlations the EPI spectral function α2F(k,k′,Ω)≈α2F(φ-φ′,Ω) is strongly peaked at φ-φ′=0 [151].(9) The scattering potential on nonmagnetic impurities is renormalized by strong correlations giving also the forward scattering peak in the impurity scattering potential (amplitude) [155]. The latter effect gives large d-wave channel in the renormalized impurity potential, which is the reason that d-wave pairing in HTSC cuprates is robust in the presence of nonmagnetic impurities (and defects) [6, 155].


## 2.3.5. EPI and Strong Correlations—Other Methods

The calculations of the static (adiabatic) charge-vertexγc(kF,q) in the X-method are done for the case U=∞ [78–80, 130, 179, 180] where it is found that it is peaked at q=0—the forward scattering peak (FSP). This result is confirmed by the numerical Monte Carlo calculations for the finite-U Hubbard model [182], where it is found that FSP exists for all U, but it is especially pronounced in the limit t≪U. These results are additionally confirmed in the calculations [183] within the four slave-boson method of Kotliar-Rückenstein where γc(kF,q) is again peaked at q=0 and the peak is also pronounced at t≪U.There are several calculations of the charge vertex in theone slave-boson method [219, 222–224] which is invented to study the limit U→∞. It is interesting to compare the results for the charge vertex in the X-method [78–80, 130, 179, 180] and in the one slave-boson theory [222] which are calculated in O((1/N)0) order. For instance, for J=0 one has
(106)γc(X)(k,q)=1+b(q)-a(q)t(k)[1+b(q)]2-a(q)c(q),γc(SB)(k,q)=1+b(q)-a(q)[t(k)+t(k+q)]/2[1+b(q)]2-a(q)c(q).
The explicit expressions for the “bare” susceptibilities a(q),b(q), and c(q) can be found in [78–80, 130]. It is obvious from (106) that γc(X)(k,q=0)=γc(SB)(k,q=0) but the calculations give that max{γc(X)(k,q)} is for q=0, while max{γc(SB)(k,q)} is for |q|≠0 [181]. So, the SB vertex is peaked at finite q which is in contradiction with the numerical Monte Carlo results for the Hubbard model [182] and with the four slave-boson theory [183]. The reason for the discrepancy of the one slave-boson (SB) in studying EPI with the numerical results and the X-method is not quite clear and might be due to the symmetry breaking of the local gauge invariance in leading order of the SB theory.


## 2.4. Summary of Section2

The experimental results in HTSC cuprates which are exposed in Section1 imply that the EPI coupling constant is large and in order to be conform with d-wave pairing this interaction must be very nonlocal (long range), that is, weakly screened and peaked at small transfer momenta. In absence of quantitative calculations in the framework of the ab initio microscopic many-body theory the effects of strong correlations on EPI are studied within the minimal t-J model where this pronounced nonlocality is due to two main reasons: (1) strong electronic correlations and (2) the combinedmetallic-ionic layered structure in these materials. In case (1) the pronounced nonlocality of EPI, which is found in the t-J model system, is due to the suppression of doubly occupancy at the Cu lattice sites in the CuO2 planes, which drastically weakens the screening effect in these systems. The pronounced nonlocality and suppression of the screening are mathematically expressed by the charge vertex function γc(kF,q,ω) which multiplies the bare EPI matrix element. The vertex function is peaked at q=0 and strongly suppressed at large q, especially for low (oxygen) hole-doping δ≪1 near the Mott-Hubbard transition. Such a structure of γc gives that the d-wave and s-wave coupling constants are of the same order of magnitude around and below some optimal doping δop≈0.1, that is, λd≈λs. This is very peculiar situation never met before. Since the residual effective (low-energy) Coulomb interaction is much smaller in the d-channel than in the s-channel, that is, μs*≫μd* (with the possibility that μd*<0), then the critical temperature for d-wave pairing is much larger than for the s-wave one, that is, Tc(d)≫Tc(s). Since all charge fluctuation processes are modified by strong correlations, then the quasiparticle scattering on nonmagnetic impurities is also drastically changed; the pair-breaking effect on d-wave pairing is drastically reduced. This nonlocal effect, which is not discussed here—see more in [6] and references therein—is one of the main reasons for the robustness of d-wave pairing in HTSC oxides in the presence of nonmagnetic impurities and numerous local defects. The development of the forward scattering peak in γc(kF,q) and suppression at large q(≫qc=δ(π/a)) give rise to the suppression of the transport coupling constant λtr making it much smaller than the self-energy coupling constant λ, that is, one has λtr≈λ/3 near the optimal doping δ=0.1–0.2. Thus the behavior of the vertex function and the dominance of EPI in the quasiparticle scattering resolve the experimental puzzle that the transport and the self-energy coupling constant take very different values, λtr,ep≪λep. Note that this is not the case with the SFI mechanism which is dominant at large q≈Q=(π,π) thus giving λtr,sf≈λsf. This result means that if in the SFI mechanism one fits the temperature-dependent resistivity (governed by λtr,sf) then one obtains very low Tc.We stress that the strength of the EPI coupling constantsλep,λep,d is at present impossible to calculate since it is difficult to incorporate strong correlations and numerous structural effects in a tractable microscopic theory.


## 2.5. Discussions and Conclusions

Numerous experimental results related to tunnelling, optics, ARPES, inelastic neutron, and X-ray scattering measurements in HTSC cupratesat and near the optimal doping give evidence for strong electron-phonon interaction (EPI) with the coupling constant 1<λep<3.5. The tunnelling measurements furnish evidence for strong EPI which give that the peaks in the bosonic spectral functionα2F(ω)  coincide well with the peaks in the phonon density of statesFph(ω). The tunnelling spectra show that almost all phonons contribute to Tc and that no particular phonon mode can be singled out in the spectral function α2F(ω) as being the only one which dominates in pairing mechanism. In light of these results the small oxygen isotope effect in optimally doped systems can be partly due to this effect, thus not disqualifying the important role of EPI in pairing mechanism. The compatibility of the strong EPI with d-wave pairing implies an important constraint on the EPI pairing potential—it must be nonlocal, that is, peaked at small transfer momenta. The latter is due to (a) strong electronic correlations and (b) the combined metallic-ionic structure of these materials. If the EPI scattering is the main player in pairing in HTSC cuprates, then this nonlocality implies that at and below some optimal doping (δop~0.1) the magnitude of the EPI coupling constants in d-wave and s-wave channel must be of the same order, that is, λep,d≈λep,s. This result in conjunction with the fact that the residual effective Coulomb coupling in d-wave channel is much smaller than in the s-wave one, that is, μs*≫μd* (with the possibility that μd*<0) gives that the critical temperature for d-wave pairing is much larger than for s-wave pairing.The numerous tunnelling, ARPES, optics, and magnetic neutron scattering measurements give sufficient evidence that the spin-fluctuation interaction (SFI) plays a secondary role in pairing in HTSC cuprates. Especially important evidence for the smallness of SFI (in pairing) comes from the magnetic neutron scattering measurements which show that by varying doping slightly around the optimal one there is a huge reconstruction of the SFI spectral functionImχ(q,ω) (imaginary part of the spin susceptibility) for q≈Q, while there is very small change in the critical temperature Tc. These experimental results imply important constraints on the pairing scenario  for systems at and near optimal doping: (1) the strength of the d-wave pairing potential is provided by EPI (i.e., one has λep,d≈λep,s) while the role of the residual Coulomb interaction and SFI, together, is to trigger d-wave pairing; (2) the Migdal-Eliashberg theory, but with the pronounced momentum dependent of EPI, is a rather good starting theory.The ab initio microscopic theory of pairing in HTSC cuprates fails at present to calculateTc and to predict the magnitude of the d-wave order parameter. From that point of view it is hard to expect a significant improvement of this situation at least in the near future. However, the studies of some minimal (toy) models, such as the single-band t-J model, allow us to understand part of the physics in cuprates on a qualitative and in some cases even on a semiquantitative level. In that respect the encouraging results come from the theoretical studies of the EPI scattering in the t-J model by using controllable mathematical methods in the X-method formulated in terms of the Hubbard operators [78–80, 130, 179, 180]. This theory predicts dressing of quasiparticles by strong correlations which dig up a large-scale correlation hole of the size ξch~a/δ for δ≪1. These quasiparticles respond to lattice vibrations in such a way to produce an effective long-range electron interaction (due to EPI), that is, the effective pairing potential Veff(q,ω) is peaked at small transfer momenta q—the forward scattering peak. This theory (of the toy model) is conform with the experimental scenario by predicting the following results: (i) the EPI coupling constants in d-wave and s-wave channels are of the same order, that is, λep,d≈λep,s, at some optimal doping δop~0.1; (ii) the transport coupling is much smaller than the pairing one, that is, λtr≈λ/3; (iii) due to strong correlations there is forward scattering peak in the potential for scattering on nonmagnetic impurities, thus making d-wave pairing robust in materials with a lot of defects and impurities. Applied to HTSC superconductors at and near the optimal doping, this theory is a realization of the Migdal-Eliashberg theory but with strongly momentum dependent EPI coupling, which is conform with the proposed experimental pairing scenario. This scenario which is also realized in the t-J toy model may be useful in making a (phenomenological) theory of pairing in cuprates. However, all present theories are confronted with the unsolved and challenging task—the calculation ofTc. From that point of view we do not have at present a proper microscopic theory of pairing in HTSC cuprates.



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*Source: 423725-2010-06-07.xml*