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E. G. Maksimov, M. L. Kulić, O. V. Dolgov, "Bosonic Spectral Function and the Electron-Phonon Interaction in HTSC Cuprates", Advances in Condensed Matter Physics, vol. 2010, Article ID 423725, 64 pages, 2010. https://doi.org/10.1155/2010/423725
Bosonic Spectral Function and the Electron-Phonon Interaction in HTSC Cuprates
In this paper we discuss experimental evidence related to the structure and origin of the bosonic spectral function in high-temperature superconducting (HTSC) cuprates at and near optimal doping. Global properties of , 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 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.
1. Experimental Evidence for Strong EPI
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: () the critical temperature in cuprates is high, with the maximum in the Hg-1223 compounds; () the pairing in cuprates is -wave like, that is, . 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 fulfills the condition , where is the electron-boson coupling constant, is the characteristic bosonic energy, is the electronic band width, and depends on numerical approximations [1, 2]. The important characteristic of the electron-boson scattering is the Eliashberg spectral function (or its average ) which characterizes scattering of quasiparticle from to by exchanging bosonic energy . Therefore, in systems with electron-boson scattering the knowledge of the spectral function is of crucial importance.
There are at least two approaches differing in assumed pairing bosons in the HTSC cuprates. The first one is based on the electron-phonon interaction (EPI), with the main proponents in [3–11], where mediating bosons are phonons and where the average spectral function is similar to the phonon density of states . Note that is not the product of two functions although sometimes one defines the function 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 , which will be discussed in detail below. In the EPI approach 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 be realized in the nonadiabatic limit ?
The second approach [12–17] assumes that EPI is too weak to be responsible for high in cuprates and it is based on a phenomenological model for spin-fluctuation interaction () 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 , that is, where is the SFI coupling constant. NMR spectroscopy and magnetic neutron scattering give evidence that in HTSC cuprates is peaked at the antiferromagnetic wave vector and this property is favorable for -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 model. Regarding the possibility to explain high solely by strong correlations, as it is reviewed in , 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 , 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 ()? 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 () Hubbard model gives no evidence for superconductivity with high , neither the BCS-like nor the Berezinskii-Kosterlitz-Thouless- (BKT-) like . At the same time, similar calculations show that there is a strong tendency to superconductivity in the attractive () Hubbard model for the same strength of , that is, at finite temperature in the 2D model with the BKT superconducting transition is favored. Concerning the possibility of HTSC in the model, various numerical calculations such as Monte Carlo calculations of the Drude spectral weight  and high-temperature expansion for the pairing susceptibility  give evidence that there is no superconductivity at temperatures characteristic for cuprates and if it exists must be rather low—few Kelvins. These numerical results tell us that the lack of high (even in 2D BKT phase) in the repulsive () single-band Hubbard model and in the model is not only due to thermodynamical 2D-fluctuations (which at finite suppress and destroy superconducting phase coherence in large systems) but it is also mostly due to an inherent ineffectiveness of strong correlations to produce solely high in cuprates. These numerical results signal that the simple single-band Hubbard and its derivative the model are insufficient to explain solely the pairing mechanism in cuprates and some additional ingredients must be included.
Since EPI is rather strong in cuprates, then it must be accounted for. As it will be argued in the following, the experimental support for the importance of EPI in cuprates comes from optics, tunnelling, and recent ARPES measurements [22, 23]. It is worth mentioning that recent ARPES activity was a strong impetus for renewed experimental and theoretical studies of EPI in cuprates. However, in spite of accumulating experimental evidence for importance of EPI with , 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, . The inappropriateness of these statements will be discussed in the following sections.
The paper is organized as follows. In Section 1 we will mainly discuss experimental results in cuprates at and near optimal doping by giving also minimal theoretical explanations which are related to the bosonic spectral function as well as to the transport spectral function 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 and 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 in the presence of the large EPI coupling constant, , if the EPI spectral function is weakly momentum dependent, that is, if 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 -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 which is in the SFI phenomenological approach [12–17] related to the Eliashberg spectral function, that is, . We stress that such an approach can be theoretically justified only in the weak coupling limit, , where is the band width and is the phenomenological SFI coupling constant. Here we discuss experimental results on YBCO which give evidence for strong rearrangement (with respect to ) of (with at and near ) by doping toward the optimal doped HTSC [30, 31]. It turns out that in the optimally doped cuprates with is drastically suppressed compared to that in slightly underdoped ones with . This fact implies that the SFI coupling constant must be small.
(b) Optical Conductivity Measurements. From these measurements one can extract the transport relaxation rate and indirectly an approximative shape of the transport spectral function . 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 and the quasiparticle relaxation rate . 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 . Although these methods give at finite temperature a blurred which is (due to the ill-defined methods) temperature dependent, it turns out that the width and the shape of the extracted are in favor of EPI. (iii) Third is the restricted sum rule for the optical weight as a function of which can be explained by strong EPI [39, 40]. (iv) Fourth is the good agreement with experiments of the -dependence of the resistivity in optimally doped YBCO, where is calculated by using the spectral function from tunnelling experiments. Recent femtosecond time-resolved optical spectroscopy in which gives additional evidence for importance of EPI  will be shortly discussed.
(c) ARPES Measurements and EPI. From these measurements the self-energy is extracted as well as some properties of . 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 () and its isotope dependence, which are all due to EPI; (ii) the appearance of different slopes of at low () and high energies ( ) 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 from tunnelling conductance. In this part we discuss the following items: (i) the extracted Eliashberg spectral function with the coupling constant from the tunnelling conductance of break-junctions in optimally doped YBCO and Bi-2212 [42–55] which gives that the maxima of coincide with the maxima in the phonon density of states ; (ii) the existence of eleven peaks in in superconducting films , where these peaks match precisely with the peaks in the intensity of the existing phonon Raman scattering data ; (iii) the presence of the dip in 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 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 buckling phonons (in LSCO, BSCO, YBCO), cannot be explained by LDA-DFT. It is curious that the magnitude of the softening can be partially obtained by LDA-DFT but the calculated widths of some important modes are an order of magnitude smaller than the neutron scattering data show. This remarkable fact confirms that additionally the inadequacy of LDA-DFT in strongly correlated systems and a more sophisticated many-body theory for EPI is needed. The problem of EPI will be discussed in more details in Section 2.
In Section 1.4 brief summary of Section 1 is given. Since we are dealing with the electron-boson scattering in cuprates near the optimal doping, then in Appendix A (and in Section 2) we introduce the reader briefly to the Migdal-Eliashberg theory for superconductors (and normal metals) where the quasiparticle spectral function and the transport spectral function 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 the theory 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 at in the superconducting is much higher than the theoretical value obtained by the strong coupling Eliashberg theory . So to say, the theory explains the coherence peak at in qualitatively but not quantitatively. However, the measurements at higher frequency are in agreement with the Eliashberg theory . 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 () which is smaller than the theoretical value, that is, , although the pairing is due to EPI for boron vibrations . 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 which is on optimally doped YBCO and BSCO and , respectively, while in underdoped BSCO one has even ; (ii) peculiarities of the coherence peak in the microwave response in HTSC cuprates, which is peaked at much smaller than , contrary to the case of LTSC where it occurs near ; (iii) the dependence of on the number of in the unit cell; (iv) the temperature dependence of the Hall coefficient; (v) distribution of states in the vortex core, and so forth.
The microscopic theory of the mechanism for superconducting pairing in HTSC cuprates will be discussed in Section 2. In Section 2.1 we introduce an ab initio many-body theory of superconductivity which is based on the fundamental (microscopic) Hamiltonian and the many-body technique. This theory can in principle calculate measurable properties of materials such as the critical temperature , 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 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 . 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: () the solution of the ideal band-structure Hamiltonian with the nonlocal exact crystal potential (sometimes called the excitation potential) (IBS—the ideal band structure) which includes the static self-energy () due to high-energy electronic processes, that is, , with and being the electron-ion and Hartree potential, respectively; () solving the low-energy Eliashberg equations. However, the calculation of the (excited) potential and the real EPI coupling , which include high-energy many-body electronic processes—for instance, the large Hubbard 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 potential —the ground-state potential—and the real EPI coupling by the “local” LDA one . Since the exchange-correlation effects enter via the local exchange-correlation potential , 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 . This theory treats the interplay of EPI and strong correlations in systems with finite doping in a systematic and controllable way. The minimal model can be further reduced (in some range of parameters) to the single-band model, which allows the approximative calculation of the excited potential and the nonlocal EPI coupling . As a result one obtains the momentum-dependent EPI coupling which is for small hole-doping () 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 -wave pairing realized in the presence of strong EPI? (b) Why is the transport coupling constant () rather smaller than the pairing one , that is, ? (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 -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 . 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 , where is the relative momentum of the Cooper pair and is the center of mass of Cooper pairs. One possible reason for the inhomogeneity of 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 ( versus carrier concentration). There are some claims that high is exclusively due to these inhomogeneities (of an extrinsic or intrinsic origin) which may effectively increase pairing potential , while some others try to explain high solely within the inhomogeneous Hubbard or model. Here we will not discuss this interesting problem but mention only that the concept of increase by inhomogeneity is not well-defined, since the increase of is defined with respect to the average value . However, is experimentally not well defined quantity and the hypothesis of an increase of by material inhomogeneities cannot be tested at all. In studying and analyzing HTSC cuprates near optimal doping we assume that basic effects are realized in nearly homogeneous systems and inhomogeneities are of secondary role, which deserve to be studied and discussed separately.
1.2. EPI versus Nonphononic Mechanisms
Concerning the high values in cuprates, two dilemmas have been dominating after its discovery: (i) which interaction is responsible for strong quasiparticle scattering in the normal state? This question is related also to the dilemma of Fermi versus non-Fermi liquid; (ii) What is the mediating (gluing) boson responsible for the superconducting pairing, that is, phonons or nonphonons? In the last twenty-three years, the scientific community was overwhelmed by numerous proposed pairing mechanisms, most of which are hardly verifiable in HTSC cuprates.
(1) Fermi versus Non-Fermi Liquid in Cuprates
After discovery of HTSC in cuprates there was a large amount of evidence on strong scattering of quasiparticles which contradicts the canonical (popular but narrow) definition of the Fermi liquid, thus giving rise to numerous proposals of the so called non-Fermi liquids, such as Luttinger liquid, RVB theory, marginal Fermi liquid, and so forth. In our opinion there is no need for these radical approaches in explaining basic physics in cuprates at least in optimally, slightly underdoped and overdoped metallic and superconducting HTSC cuprates. Here we give some clarifications related to the dilemma of Fermi versus non-Fermi liquid. The definition of the canonical Fermi liquid (based on the Landau work) in interacting Fermi systems comprises the following properties: () there are quasiparticles with charge , spin , and low-energy excitations which are much larger than their inverse life-times, that is, . Since the level width 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; () at there is an energy level which defines the Fermi surface on which the Fermi quasiparticle distribution function has finite jump at ; () 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; () 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 ≡ (for the Debye spectrum ~), that is, for , one has and the quasiparticle picture (in the sense of the Landau definition) is broken down. In that respect an electron-boson system can be classified as a noncanonical Fermi liquid for sufficiently strong electron-boson coupling. It is nowadays well known that, for instance, Al, Zn are weak coupling systems since for one has and they are well described by the Landau theory. However, in (the noncanonical) cases where for higher energies one has , 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 and its higher momenta. Since in HTSC cuprates the electron-boson (phonon) coupling is strong and is large, then it is natural that in the normal state (at ) we deal with a strong interacting noncanonical Fermi liquid which is for modest nonadiabaticity parameter 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 in the phonon mechanism of pairing. Since in the past we have discussed this problem thoroughly in numerous papers—for the recent one see —we will outline here the main issue and results only.
It is well known that in an electron-ion crystal, besides the attractive EPI, there is also repulsive Coulomb interaction. In case of an isotropic and homogeneous system with weak quasiparticle interaction, the effective potential in the leading approximation looks like as for two external charges () embedded in the medium with the total longitudinal dielectric function ( is the momentum and is the frequency) [68, 69], that is, In case of strong interaction between quasiparticles, the state of embedded quasiparticles changes significantly due to interaction with other quasiparticles, giving rise to . In that case depends on other (than ) response functions. However, in the case when (1) holds, that is, when the weak-coupling limit is realized, is given by [68–70]. Here, is the EPI coupling constant, is an average phonon frequency, and is the Coulomb pseudopotential, ( is the Fermi energy). The couplings and are expressed by : where is the density of states at the Fermi surface and 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 for all . If this were correct, then from (2) it would follow that , which limits the maximal value of to the value . In typical metals , and if one accepts the statement in [65, 66] that 0.5), one obtains . The latter result, if it would be correct, means that EPI is ineffective in producing not only high- superconductivity but also low-temperature superconductivity (LTS with ). However, this result is in conflict first of all with experimental results in LTSC, where in numerous systems one has and . For instance, is realized in PbBi alloy which is definitely much higher than 1), and so forth.
Moreover, the basic theory tells us that is not the response function [68, 69] (contrary to the assumption in [65, 66]). Namely, if a small external potential 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 , which means that is the response function. The latter obeys the Kramers-Kronig dispersion relation which implies the following stability condition [68, 69]: that is, either or This important theorem invalidates the restriction on the maximal value of in the EPI mechanism given in [65, 66]. We stress that the condition 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 (), that is, , and is defined by . In dense metallic systems with one ion per cell (such as metallic hydrogen) and with the electronic dielectric function , the macroscopic total dielectric function is given by [71–73] At the same time the energy of the longitudinal phonon is given by where is the ionic plasma frequency, and is the local (electric) field correction—see [71–73]. The right condition for lattice stability requires that , which implies that for one has . The latter condition gives automatically . Furthermore, the calculations [71–73] show that in the metallic hydrogen (H) crystal for all . Note that in metallic H the EPI coupling constant is very large, that is, and may reach very large value . Moreover, the analyses of crystals with more ions per unit cell [71–73] give that is more a rule than an exception—see Figure 1. The physical reason for is local field effects described by . Whenever the local electric field acting on electrons (and ions) is different from the average electric field , that is, , there are corrections to which may lead to .
The above analysis tells us that in real crystals can be negative in the large portion of the Brillouin zone thus giving rise to in (2). This means that analytic properties of the dielectric function do not limit in the phonon mechanism of pairing. This result does not mean that there is no limit on at all. We mention in advance that the local field effects play important role in HTSC cuprates, due to their layered structure with very unusual ionic-metallic binding, thus opening a possibility for large EPI.
In conclusion, we point out that there are no serious theoretical and experimental arguments for ignoring EPI in HTSC cuprates. To this end it is necessary to answer several important questions which are related to experimental findings in HTSC cuprates. () If EPI is important for pairing in HTSC cuprates and if superconductivity is of -wave type, how are these two facts compatible? () Why is the transport EPI coupling constant (entering resistivity) rather smaller than the pairing EPI coupling constant 1) (entering ), that is, why one has ≈0.6–1.42–3.5)? () If EPI is ineffective for pairing in HTSC oxides, in spite of , why is it so?
(3) Is a Nonphononic Pairing Realized in HTSC?
Regarding EPI one can pose a question about whether it contributes significantly to -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 , which implies small EPI and, second, on the well-established experimental fact that -wave pairing is realized in cuprates , which is believed to be incompatible with EPI. Having in mind that EPI in HTSC at and near optimal doping is strong with (see below), we assume for the moment that the leading pairing mechanism in cuprates, which gives -wave pairing, is due to some nonphononic mechanism. For instance, let us assume an exitonic mechanism due to the high-energy pairing boson () and with the bare critical temperature and look for the effect of EPI on . If EPI is approximately isotropic, like in most LTSC materials, then it would be very detrimental for -wave pairing. In the case of dominating isotropic EPI in the normal state and the exitonic-like pairing, then near the linearized Eliashberg equations have an approximative form for a weak nonphonon interaction (with the large characteristic frequency ) For pure -wave pairing with the pairing potential with and , one obtains and the equation for —see [3–5] Here is the di-gamma function. At temperatures near one has and the solution of (9) is approximately with , . This means that for and the bare due to the nonphononic interaction must be very large, that is, .
Concerning other nonphononic mechanisms, such as the SFI one, the effect of EPI in the framework of Eliashberg equations was studied numerically in . The latter is based on (A.1) in Appendix A with the kernels in the normal and superconducting channels and , respectively. Usually, the spin-fluctuation kernel is taken in the FLEX approximation . The calculations  confirm the very detrimental effect of the isotropic (-independent) EPI on -wave pairing due to SFI. For the bare SFI critical temperature and for the calculations give very small (renormalized) critical temperature . These results tell us that a more realistic pairing interaction must be operative in cuprates and that EPI must be strongly momentum dependent and peaked at small transfer momenta [78–80]. Only in that case does strong EPI conform with -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 -wave pairing.
1.3. Experimental Evidence for Strong EPI
In the following we discuss some important experiments which give evidence for strong electron-phonon interaction (EPI) in cuprates. However, before doing it, we will discuss some indicative inelastic magnetic neutron scattering (IMNS) measurements in cuprates whose results in fact seriously doubt in the effectiveness of the phenomenological SFI mechanism of pairing which is advocated in [12–17, 81]. First, the experimental results related to the pronounced imaginary part of the susceptibility in the normal state at and near the AF wave vector 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 (at some energies ) in the superconducting state was also interpreted in a number of papers either as the origin of superconductivity or as a mechanism strongly affecting superconducting gap at the antinodal point.
1.3.1. Magnetic Neutron Scattering and the Spin-Fluctuation Spectral Function
(a) Huge Rearrangement of the SFI Spectral Function and Small Change of
Before discussing experimental results in cuprates on the imaginary part of the spin susceptibility we point out that in the (phenomenological) theories based on the spin-fluctuation interaction (SFI) the quasiparticle self-energy ( is the Matsubara frequency and is the Nambu matrix) in the normal and superconducting state and the effective (repulsive) pairing potential (where ) are assumed in the form [12–17] where . Although the form of cannot be justified theoretically, except in the weak coupling limit () 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) is strongly peaked at and near the AF wave vector .
Can the pairing mechanism in HTSC cuprates be explained by such a phenomenology and what is the prise for it is? The best answer is to look at the experimental results related to the inelastic magnetic neutron scattering (IMNS) which gives . In that respect very indicative and impressive IMNS measurements on , which are done by Bourges group , demonstrate that the normal-state susceptibility (the odd part of the spin susceptibility in the bilayer system) at is strongly dependent on the hole-doping as it is shown in Figure 2.
The most pronounced result for our discussion is that by varying doping there is a huge rearrangement of in the normal state, especially in the energy (frequency) region which might be important for superconducting pairing, let us say . This is clearly seen in the last two curves in Figure 2 where this rearrangement is very pronounced, while at the same time there is only small variation of the critical temperature . It is seen in Figure 2 that in the underdoped crystal and are much larger than that in the near optimally doped , that is, one has , although the difference in the corresponding critical temperatures is very small, that is, (in ) and (in ). This pronounced rearrangement and suppression of in the normal state of YBCO by doping (toward the optimal doping) but with the negligible change in is strong evidence that the SFI pairing mechanism is not the dominating one in HTSC cuprates. This insensitivity of , if interpreted in terms of the SFI coupling constant (~, means that the latter is small, that is, . We stress that the explanation of high in cuprates by the SFI phenomenological theory [12–17] assumes very large SFI coupling energy with while the frequency (energy) dependence of is extracted from the fit of the NMR relaxation rate which gives [12–17]. To this point, the NMR measurements (of ) give that there is an anticorrelation between the decrease of the NMR spectral function and the increase of by increasing doping toward the optimal one—see  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 from NMR measurements one takes it from IMNS measurements, as it was done in , than for the same value one obtains much smaller . For instance, by taking the experimental values for in underdoped with one obtains , while for . The situation is even worse if one tries to fit the resistivity with in since this fit gives . 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 , the recent theoretical interpretation in  of IMNS experiments [83, 84] and ARPES measurements [85, 86] on the underdoped 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 at, near and far from , which is richer than the common spectrum with magnetic resonance peaks measured at . In  the self-energy of electrons due to their interaction with spin excitations is calculated by using (10) with and 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  use very large value , that is, much larger than the one used in . Such a large value of has been obtained earlier within the Monte Carlo simulation of the Hubbard model . In our opinion this value for is unrealistically large in the case of strongly correlated systems where spin fluctuations are governed by the effective electron-exchange interaction . This implies that and . Note that this value for (~0.15 eV) comes out also from the theory of strongly correlated electrons in the three-band Emery model which gives —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 (and ).
Concerning the problem related to the rearrangement of the SFI spectral function in  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  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  are confirmed in  where the magnetic intensity (~ (for at and in the broad range of ) for the optimally doped (with ) is at least three times smaller than in the underdoped with . This result is again very indicative sign of the weakness of SFI since such a huge reconstruction would decrease in the optimally doped if analyzed in the framework of the phenomenological SFI theory based on (10). It also implies that due to the suppression of by increasing doping toward the optimal one a straightforward extrapolation of the theoretical approach in  to the explanation of in the optimally doped would require an increase of to the value even larger than , which is highly improbable.
(b) Ineffectiveness of the Magnetic Resonance Peak
A less direct argument for smallness of the SFI coupling constant, that is, and , 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, is significantly suppressed at low frequencies except near the resonance energy 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, —see Figure 3. In underdoped cuprates this peak is present also in the normal state as it is seen in Figure 2.
After the discovery of the resonance peak there were attempts to relate it, first, to the origin of the superconducting condensation energy and, second, to the kink in the energy dispersion or the peak-dimp structure in the ARPES spectral function. In order that the condensation energy is due to the magnetic resonance, it is necessary that the peak intensity is small . is obtained approximately by equating the condensation energy with the change of the magnetic energy in the superconducting state, that is, : where is the spin structure factor and is the Bose distribution function. By taking and the realistic value , one obtains . However, such a small intensity cannot be responsible for the anomalies in ARPES and optical spectra since it gives rise to small coupling constant for the interaction of holes with the resonance peak, that is, . 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 . 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 , that is, , where the line width is given by . This condition limits the SFI coupling to . We stress that in such a way obtained is much smaller (at least by factor three) than that assumed in the phenomenological spin-fluctuation theory [12–17, 81] where and , but much larger than estimated in  (where ). The smallness of comes out also from the analysis of the antiferromagnetic state in underdoped metals of LSCO and YBCO , where the small (ordered) magnetic moment 0.1 points to an itinerant antiferromagnetism with small coupling constant . 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 is vanishingly small, though not affecting pairing at the second-order phase transition at , then, if it would be solely the origin for superconductivity, the phase transition at 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 crystals , where the absolute value of 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 across the superconducting transition , that is, there is only a minimal change in between and . Note that if the magnetic excitations were due to itinerant quasiparticles we should have seen dramatic changes of as a function of over the whole energy range. This -independence of strongly opposes the theoretical results in [24–27] which assume that the bosonic spectral function is proportional to and that the former can be extracted from optic measurements. Namely, the fitting procedure in [24–27] gives that is strongly -dependent contrary to the experimental results in —see more in Section 1.3.2 on optical conductivity.
1.3.2. Optical Conductivity and EPI
Optical spectroscopy gives information on optical conductivity and on two-particle excitations, from which one can indirectly extract the transport spectral function . Since this method probes bulk sample (on the skin depth), contrary to ARPES and tunnelling methods which probe tiny regions ( Å) near the sample surface, this method is indispensable. However, one should be careful not to overinterpret the experimental results since is not a directly measured quantity but it is derived from the reflectivity with the transversal dielectric tensor . Here, is the high-frequency dielectric function, describes the contribution of the lattice vibrations, and describes the optical (dynamical) conductivity of conduction carriers. Since is usually measured in the limited-frequency interval , some physical modelling for 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 —from to —that is, almost three orders of magnitude. However, it turns out that IR measurements of in conjunction with elipsometric measurements of at high frequencies allow more reliable determination of .
(1) Transport and Quasiparticle Relaxation Rates
The widespread misconception in studying the quasiparticle scattering in cuprates was an ad hoc assumption that the transport relaxation rate is equal to the quasiparticle relaxation rate , in spite of the well-known fact that the inequality holds in a broad-frequency (energy) region Allen. This (incorrect) assumption was one of the main arguments against the relevance of the EPI scattering mechanism in cuprates. Although we have discussed this problem several times before, we do it again due to the importance of this subject.
The dynamical conductivity consists of two parts, that is, where describes interband transitions which contribute at higher than intraband energies, while is due to intraband transitions which are relevant at low energies . (Note that in the IR measurements the frequency is usually given in , where the following conversion holds: .) The experimental data for in cuprates are usually processed by the generalized (extended) Drude formula [32–36, 95]: where is the mass of the band electrons while the quantity 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 . (The usefulness of introducing the optic relaxation will be discussed below.) Here, enumerates the plane axis; , and are the electronic plasma frequency, the transport (optical) scattering rate, and the optical mass, respectively. Very frequently it is analyzed the quantity given by  In the weak coupling limit , the formula for conductivity given in Appendix A, equations (A.20) and (A.21) can be written in the form of (12) where reads [33–36] Here is the Bose distribution function. For completeness we give also the explicit form of the transport mass , see [3–6, 32–36]: with the Kernel where is the di-gamma function. In the presence of impurity scattering one should add to . It turns out that (14) holds within a few percents also for large 1. Note that and the index enumerates all scattering bosons—phonons—spin fluctuations, and so forth. For comparison, the quasiparticle scattering rate is given by where is the Fermi distribution function. For completeness we give also the expression for the quasiparticle effective mass : The term is due to the impurity scattering. By comparing (14) and (16), it is seen that and are different quantities, that is, , 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 where one has (due to simplicity we omit in the following summation over )  In the case of EPI with the constant electronic density of states, the above equations give that for while (as well as ) is monotonic growing for , where is the maximal phonon frequency. So, the growing of (and ) for 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 , and which are calculated for the EPI spectral function extracted from tunnelling experiments in YBCO (with ) [42–45]—see Figure 4.
The results shown in Figure 4 clearly demonstrate the physical difference between two scattering rates and (or ). It is also seen that is even more linear function of than . From these calculations one concludes that the quasilinearity of (and ) 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 and (and ), 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 for and and compare these with theoretical predictions obtained by using a single-band model with extracted from the tunnelling data with the EPI coupling constant [42–45]. In the case of YBCO the agreement between measured and calculated is very good up to energies , 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 () [97, 98] the experimental results for are explained theoretically by assuming that the EPI spectral function , where is the phononic density of states in BSCO, with and —see Figure 5(a). At the same time the fit of by the marginal Fermi liquid phenomenology fails as it is evident in Figure 5(b).
Now we will comment the so called pronounced linear behavior of (and ) 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 and derived ones are , and , which are very sensitive to the value of the dielectric constant . This sensitivity is clearly demonstrated in Figure 6 for Bi-2212 where it is seen that (and ) for is linear up to much higher than in the case .
However, in some experiments [100–103] the extracted (and ) is linear up to very high . This means that the ion background and interband transitions (contained in ) are not properly taken into account since too small (1) is assumed. The recent elipsometric measurements on YBCO  give the value , which gives much less spectacular linearity in the relaxation rates (and ) than it was the case immediately after the discovery of HTSC cuprates, where much smaller was assumed.
Furthermore, we would like to comment two points related to , and . First, the parametrization of with the generalized Drude formula in (12) and its relation to the transport scattering rate and the transport mass is useful if we deal with electron-boson scattering in a single-band problem. In [36, 96] it is shown that of a two-band model with only elastic impurity scattering can be represented by the generalized (extended) Drude formula with and dependence of effective parameters , 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 and are physically different quantities and it holds that . In order to give the physical picture and qualitative explanation for this difference we assume that . In that case the renormalized quasiparticle frequency and the transport one —defined in (12)—are related and at they are given by [32, 36] (For the definition of see Appendix A.) It gives the relation between and as well as and , respectively: The physical meaning of (19) is the following: in optical measurements one photon with the energy is absorbed and two excited particles (electron and hole) are created above and below the Fermi surface. If the electron has energy and the hole , then they relax as quasiparticles with the renormalized frequency . Since takes values , then the optical relaxation is the energy-averaged according to (19). The factor is due to the two quasiparticles—electron and hole. At finite , the generalization reads [32, 36]
(2) Inversion of the Optical Data and
In principle, the transport spectral function can be extracted from (or ) only at , which follows from (18) as or equivalently as . However, real measurements are performed at finite (at which is rather high in HTSC cuprates) and the inversion procedure is an ill-posed problem since is the deconvolution of the inhomogeneous Fredholm integral equation of the first kind with the temperature-dependent Kernel —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 get blurred (most peaks are washed out) in the extraction procedures and, second, the extracted be temperature dependent even when the true is independent. This artificial dependence is especially pronounced in HTSC cuprates because (~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: () the extracted shape of in as well as in other cuprates is not unique and it is temperature dependent, that is, at higher the peak structure is smeared and only a single peak (slightly shifted to higher ) is present. For instance, the experimental data of in YBCO were reproduced by two different spectral functions , 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 . The similar situation is realized in optimally doped BSCO as it is seen in Figure 8 where again different functions reproduce very well curves for and . However, it is important to stress that the obtained width of the extracted in both compounds coincide with the width of the phonon density of states [33–36, 99]. () The upper energy bound for is extracted in [33–36] and it coincides approximately with the maximal phonon frequency in cuprates as it is seen in Figures 7 and 8.
These results demonstrate the importance of EPI in cuprates [33–36]. We point out that the width of which is extracted from the optical measurements [33–36] coincides with the width of the quasiparticle spectral function obtained in tunnelling and ARPES spectra (which we will discuss below), that is, both functions are spread over the energy interval . Since in cuprates this interval coincides with the width in the phononic density of states and since the maxima of and 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 of cuprates can be extracted also in the superconducting state by using (22). However, (22) holds exclusively in the normal state (at ) since 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 . Second, if ’s (and ’s) in cuprates are due to some other bosonic scattering which is pronounced up to much higher energies , this should be seen in the width of the extracted spectral function . In that respect in [25–27] it is assumed that SFI dominates in the quasiparticle scattering and that where . This claim is based on reanalyzing of some IR measurements [25–27] and the transport spectral function is extracted in  by using the maximum entropy method in solving the Fredholm equation. However, in order to exclude negative values in the extracted , which is an artefact and due to the chosen method, in  it is assumed that has a rather large tail at large energies—up to 400 meV. It turns out that even such an assumption in extracting does not reproduce the experimental curve  in some important aspects. First, the relative heights of the two peaks in the extracted spectral function at lower temperatures are opposite to the experimental curve —see [25, Figure 1]. Second, the strong temperature dependence of the extracted 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 -dependent kernel. Third, the extracted spectral weight in  has much smaller values at larger frequencies () than it is the case for the measured , that is, —see [25, Figure 1]. Fourth, the recent magnetic neutron scattering measurements on optimally doped large-volume crystals  (where the absolute value of 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 is temperature independent across the superconducting transition , that is, there is only a minimal change in between and . This -independence of strongly opposes the (above discussed) results in [24–27], where the fit of optic measurements gives strong dependence of .
Fifth, the transport coupling constant extracted in  is too large, that is, contrary to the previous findings that [33–36, 99]. Since in HTSC one has , this would probably give , which is not confirmed by other experiments. Sixth, the interpretation of in LSCO and BSCO solely in terms of is in contradiction with the magnetic neutron scattering in the optimally doped and slightly underdoped YBCO . The latter was discussed in Section 1.3.1, where it is shown that is small in the normal state and its magnitude is even below the experimental noise. This means that if the assumption that were correct then the contribution to from the momenta would be dominant, which is detrimental for -wave superconductivity.
Finally, we point out that very similar (to cuprates) properties, of (and and electronic Raman spectra), were observed in 3D isotropic metallic oxides and which are nonsuperconducting  and in which is superconducting below at . This means that in all of these materials the scattering mechanism might be of similar origin. Since in these compounds there are no signs of antiferromagnetic fluctuations (which are present in cuprates), then the EPI scattering plays important role also in other oxides.
(3) Restricted Optical Sum Rule
The restricted optical sum rule was studied intensively in HTSC cuprates. It shows some peculiarities not present in low-temperature superconductors. It turns out that the restricted spectral weight 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 . In the following we demonstrate that the temperature dependence of in the normal state can be explained in a natural way by the dependence of the EPI transport relaxation rate [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 . The first one is the total sum rule which in the normal state reads while in the superconducting state it is given by the Tinkham-Ferrell-Glover (TFG) sum rule Here, is the total electron density, is the electron charge, is the bare electron mass, and is the London penetration depth. The first (singular) term in (24) is due to the superconducting condensate which contributes . 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 where all electrons, electronic bands, and their interactions (Coulomb, EPI, with impurities, etc.) are accounted for. Here, is the kinetic energy of bare electrons:
The partial sum rule is related to the energetics solely in the conduction (valence) band which is described by the Hamiltonian of the conduction (valence) band electrons: contains the band energy with the dispersion () and the effective Coulomb interaction of the valence electrons . In this case the partial sum rule in the normal state reads  (for a general form of ) where the number operator is the momentum-dependent reciprocal mass and is volume. In practice, the optic measurements are performed up to finite frequency and the integration over goes up to some cutoff frequency (of the order of the band plasma frequency). In this case the restricted sum rule has the form where is the diamagnetic Kernel given by (30) below and is the paramagnetic (current-current) response function. In the perturbation theory without vertex correction (at the Matsubara frequency ) is given by [39, 40] where and is the electron Green's function. In the case when the interband gap is the largest scale in the problem, that is, when , in this region one has approximately and the limit in (28) is justified. In that case one has which gives the approximate formula for : where is the quasiparticle distribution function in the interacting system. Note that is cutoff dependent while in (30) does not depend on . 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 by using the exact result in (28) which apparently depends on . However, (30) is useful for appropriately chosen , 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