Bosonic Spectral Function in HTSC Cuprates:
Part I - Experimental Evidence for Strong Electron-Phonon Interaction
In Part I we discuss accumulating experimental evidence related to the structure and origin of the bosonic spectral function in high-temperature superconducting (HTSC) cuprates near optimal doping. Some global properties of , such as number and positions of peaks, are extracted by combining optics, neutron scattering, ARPES and tunnelling measurements. These methods give convincing evidence for strong electron-phonon interaction (EPI) with in cuprates near optimal doping. Here we clarify how these results are in favor of the Eliashberg-like theory for HTSC cuprates near optimal doping.We argue that the neglect of EPI in some previous studies of HTSC was based on a number of deceptive prejudices related to the strength of EPI, on some physical misconceptions and misleading interpretation of experimental results.
In Part II we discuss some theoretical ingredients which are necessary to explain the experimental results related to pairing mechanism in optimally doped cuprates. These comprise the Migdal-Eliashberg theory for EPI in strongly correlated systems which give rise to the forward scattering peak. The latter is due to the combined effects of the weakly screened Madelung interaction in the ionic-metallic structure of layered cuprates and many body effects of strong correlations. While EPI is responsible for the strength of pairing the residual Coulomb interaction (by including spin fluctuations) triggers the d-wave pairing.
In spite of an unprecedentedly intensive experimental and theoretical study after the discovery of high-temperature superconductivity (HTSC) in cuprates there is, even twenty-two 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 in cuprates is high, with the maximum ; (2) the pairing in cuprates is d-wave like, i.e. with . 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 quasi-particle dynamics is governed by some electron-boson scattering and in the superconducting state bosonic quasi-particles are gluing electrons in Cooper pairs. Which bosonic quasi-particles 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 and is the electronic band width. The important characteristic of the electron-boson scattering is the Eliashberg spectral function (or its average ) which characterizes scattering of quasi-particle from to by exchanging bosonic energy . Therefore, in systems with electron-boson scattering the knowledge of this function is of crucial importance. There are at least two approaches differing in assumed gluing bosons. The first one is based on the electron-phonon interaction (EPI) Maksimov-Review (), Kulic-Review (), Alexandrov (), Gunnarsson-review-2008 (), Falter () where mediating bosons are phonons and where the the average spectral function is similar to the phonon density of states . Note, 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 evidence in cuprates which support the dominance of the EPI scattering mechanism with a rather large coupling constant and which will be discussed in detail below. In the EPI approach is extracted from tunnelling measurements in conjunction with IR optical measurements. We stress again that the Migdal-Eliashberg theory is well justified framework for EPI since in most superconductors the condition is fulfilled. The HTSC are on the borderline and it is a natural question - under which condition can high T be realized in the non-adiabatic limit ? The second approach Pines () 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, i.e. it is a non-phononic mechanism. In this approach the spectral function is proportional to the imaginary part of the spin susceptibility , i.e. . NMR spectroscopy and magnetic neutron scattering give that in HTSC cuprates is peaked at the antiferromagnetic wave vector and this property is favorable for d-wave pairing. The 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 T solely by strong correlations, as it is reviewed in Patrick-Lee (), we stress two facts. First, at present there is no viable theory which can justify these (non-phononic) mechanisms of pairing. Second, the central question in this approach is - do models based on the Hubbard Hamiltonian show up superconductivity at sufficiently high critical temperatures ( ) ? A number of numerical studies of these models offer a negative answer. For instance, the sign-free variational Monte Carlo algorithm in the 2D repulsive () Hubbard model gives no evidence for HTSC, neither the BCS- nor Berezinskii-Kosterlitz-Thouless (BKT)-like Imada-MC (). 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 , i.e. at finite temperature in the 2D model with the BKT superconducting transition is favored. Concerning HTSC in the model, various numerical calculations such as Monte Carlo calculations of the Drude spectral weight Scalapino-Drude-weight () and high temperature expansion Pryadko () have shown 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 BKT phase) in the repulsive () single-band Hubbard model and in the model is not only due to thermodynamical -fluctuations (which at finite T suppress and destroy superconducting phase coherence in large systems) but it is mostly due to an inherent ineffectiveness of strong correlations to produce solely high in cuprates. These numerical results certainly mean that the simple single-band Hubbard, as well as its derivative the t-J model, are insufficient to explain the pairing mechanism in cuprates and some other ingredients must be included. Having in mind these facts there is no room at present for any kind of celebration of the victory of non-phononic mechanisms of pairing as some prefer to do.
Since is as a rule strong in oxides, then it is plausible that it should be accounted for in cuprates at least in the normal metallic state. As it will be argued later on, the experimental support for the importance of EPI comes from optics, tunnelling, and recent ARPES measurements Shen-review (). 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 Hwang-Timusk-1 () and with LDA band structure calculations Bohnen-Cohen (), Giuistino (), where both claim that EPI is negligibly small, i.e. .
The paper is organized as follows. There are two parts - in Part I we will mainly discuss experimental results in cuprates near optimal doping and minimal theoretical explanations which are related to the spectral function as well to the transport spectral function and how these are related to EPI in cuprates. In this work we consider mainly those direct one-particle and two-particles probes of low energy quasi-particle excitations (by including gap and pseudogap) and scattering rates which give informations on the structure of the spectral functions and in systems near optimal doping. These are angle-resolved photoemission (), various arts of tunnelling spectroscopies such as superconductor/insulator/ normal metal () junctions and break junctions, scanning-tunnelling microscope spectroscopy (), infrared () and Raman optics, inelastic neutron scattering () etc. We shall argue that these direct probes give evidence for a rather strong EPI in cuprates as dominating scattering mechanism of quasi-particles. 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 Part I is the following. In Section II we discuss some prejudices related to and the Fermi-liquid behavior of HTSC cuprates. We argue that any non-phononic mechanism of pairing should have very large bare critical temperature in the presence of the large EPI coupling constant, , if its spectral function is weakly momentum dependent, i.e. if . The fact that EPI is large in the normal state of cuprates and the condition that it must conform with d-wave pairing implies inevitably that EPI in cuprates must be strongly momentum dependent. In Section III we discuss direct and indirect experimental evidence for the importance of EPI and for the weakness of SFI in cuprates. These are:
(A) Magnetic neutron scattering measurements - These measurements provide dynamic spin susceptibility which is in the phenomenological approach Pines () related to the Eliashberg spectral function, i.e. . We stress that such an approach can be qualitatively justified only in the weak coupling limit, , where is the band width. Here we discuss experimental results which give evidence for strong rearrangement (with respect to ) of by doping toward the optimal doped HTSC Bourges (). It turns out that in the optimally doped cuprates with is drastically suppressed compared to that in slightly underdoped ones with , and this is strong evidence for the smallness of the SFI coupling constant.
(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 that respect we discuss: (i) the misleading concept concerning the relation between the optical relaxation rate and the quasi-particle relaxation rate . This (misleading) concept has been appearing repeatedly in the last twenty years despite the fact that this controversy is resolved many years ago Allen (), Dolgov-Shulga (), Shulga (), Maksimov-Review (), Kulic-Review (), Kulic-AIP (); (ii) some methods of extraction of the optical spectral function from optical reflectivity measurements. It turns out that the width and the shape of the extracted favor EPI; (iii) the restricted sum-rule for the optical weight as a function of temperature which can also be explained by strong Maks-Karakoz-1 (), Maks-Karakoz-2 (); (iv) good agreement with experiments of the temperature dependence of the resistivity calculated with the extracted . Recent femtosecond time-resolved optical spectroscopy in gives additional evidence for importance of EPI and ineffectiveness of SFI Kusar-2008 ().
(C) ARPES measurements and EPI - From these measurements one can extract the self-energy from which one can extract some properties of . Here we discuss the following items: (i) appearance of the nodal and anti-nodal 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) appearance of different slopes of at low () and high energies () which can be explained with strong EPI; (iii) formation of small polarons in the undoped HTSC which is due to strong EPI - this gives rise to phonon side bands which are clearly seen in ARPES of undoped HTSC.
(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) extraction of the Eliashberg spectral function with from the tunnelling conductance of break-junctions Tunneling-Vedeneev ()-Ponomarev-Tunnel () which gives that the maxima of coincide with the maxima in the phonon density of states; (ii) the presence of the dip in dI/dV in STM which shows the pronounced oxygen isotope effect and important role of these phonons: (iii) the presence of fine and doping independent structure in I(V) characteristics due to phonon emission by the Josephson current in layered HTSC cuprates with intrinsic Josephson junctions.
(E) Phonon neutron scattering measurements - From these experiments one can extract the phonon density of state and strengths of the quasi-particle coupling with various phonon modes. Here we argue, that the large softening and broadening of the half-breathing Cu-O bond-stretching phonon, of apical oxygen phonons and of oxygen buckling phonons (in LSCO, BISCO,YBCO) cannot be explained by LDA. It is curious that the magnitude of softening can be partially obtained by LDA but the calculated widths of some important modes are an order of magnitude smaller than the neutron scattering data show. This remarkable fact implies the inadequacy of LDA in strongly correlated systems and a more sophisticated many body theory for EPI is needed. This problem will be discussed in more details in Part II MaKuDoAk (). In Section IV brief summary of the Part I is given. Since we are dealing with electron-boson scattering in cuprate near optimal doping, then in Section V - Appendix we introduce the reader briefly into the Migdal-Eliashberg theory for superconductors (and normal metals) where the quasi-particle spectral function and the transport spectral function are defined.
Finally, at the end of the day one poses 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 with a 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 the pairing is due to EPI and many properties are well accounted for by the Migdal-Eliashberg theory. Let us mention only two examples. First, the experimental value for the coherence peak in the microwave response at in is much higher than the theoretical value obtained by the Migdal-Eliashberg theory Marsiglio-1994 (). 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 Migdal-Eliashberg theory Klein-1994 (). Second, the experimental boron (B) isotope effect in (with ) is much smaller than the theoretical value, i.e. , although the pairing is due solely by EPI for boron vibrations MgB2-isotop (). Since the theory of everything is impossible in the complex materials such as HTSC cuprates in Part I and II we shall not discuss those phenomena which need much more microscopic details and/or more sophisticated many-body theory. These are selected by chance: (i) 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 ; (ii) dependence on the number of in the unit cell; (iii) temperature dependence of the Hall coefficient; (iv) distribution of states in the vortex core, etc. However, in a separate paper - Part II MaKuDoAk () we shall discuss some minimal theoretical concepts which can explain at least qualitatively and semi-quantitatively results related to the above enumerated spectroscopic methods. Due to the presence of strong correlations and quasi-2D electronic structure some of these concepts go beyond the LDA approach. In our opinion at this stage of the HTSC physics some important ingredients of the future theory are already recognized. These are: (1) very peculiar quasi-2D ionic-metallic structure with a rather weak screening along the c-axis, which is a prerequisite for strong EPI; (2) strong Coulomb interaction and correlations which are responsible for strong magnetism in undoped cuprates and for important renormalizations of EPI. Since both ingredients belong to the class of strong coupling problems, at present there is no quantitative theory and therefore we must rely on approximative and model theories. Even these approaches allow us qualitative (and semi-quantitative) explanations of some important properties which are due to the interplay of EPI and strong correlations. The latter two cause the appearance of momentum dependent EPI - peaked at small transfer momenta Kulic-Review (). Based on such an approach we are able to explain (understand) at least qualitatively some very puzzling experimental results, for instance: (a) why is d-wave pairing realized in the presence of strong EPI? (b) why is the transport coupling constant () smaller than the pairing one , i.e. ? (c) Why is the mean-field (one-body) LDA approach unable to give reliable values for the EPI coupling constant in cuprates and how many-body effects help; (d) why is d-wave pairing robust in presence of non-magnetic impurities and defects? (e) why the ARPES nodal and antinodal kinks are differently renormalized in the superconducting states?
In real materials there are numerous experimental evidence for nanoscale inhomogeneities in HTSC oxides . For instance recent STM experiments show rather large gap dispersion at least on the surface of BISCO crystals Davis () giving rise for a pronounced inhomogeneity of the superconducting order parameter, i.e. 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 in HTSC cuprates which is many orders of magnitude larger than in standard semiconductors ( vs carrier concentration). There are some claims that high is exclusively due to these inhomogeneities (of an extrinsic or intrinsic origin) which may increase EPI Phillips (), while other try to explain high within the inhomogeneous Hubbard or t-J model. In Part II MaKuDoAk () we argue that the concept of an increase of T by inhomogeneity is ill-defined, since the increase of is defined with respect to the average value . However, the latter quantity is experimentally not well defined and an alleged increase of by the material inhomogeneity cannot be tested at all.
Ii EPI vs non-phononic mechanisms - facts vs prejudices
Concerning the high in cuprates, one of the central questions is - which interaction(s) is(are) responsible for strong quasi-particle scattering in the normal state and for the superconducting pairing? In the last twenty-two years, the scientific community was overwhelmed by all kinds of (im)possible proposed pairing mechanisms, most of which are hardly verifiable in any material, if at all. This trend is still continuing nowadays (although with smaller slope), in spite of the fact that the accumulated experimental results eliminate all but few.
A. Fermi vs non-Fermi liquid in cuprates
After discovery of HTSC in cuprates there was a large amount of evidence on strong scattering of quasi-particles 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, etc. 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. This subject will be discussed more in Part II and here we give some clarifications related to the dilemma of Fermi vs 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 quasi-particles with charge , spin and low-laying energy excitations which are much larger than their inverse life-times, i.e. . Since the level width of the quasi-particle 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 there is an energy level with the Fermi surface at which and the Fermi quasi-particle distribution function has finite jump; (3) the number of quasi-particles 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 interaction between quasi-particles are characterized with a few (Landau) parameters which describe low-temperature thermodynamics and transport properties. Having this definition in mind one can say that if fermionic quasi-particles 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 Debay temperature (more precisely ), i.e. for one has and the quasi-particle picture (in the sense of the Landau definition) is broken down. In that respect an electron-boson system can be classified as a non-canonical 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, 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 rather strong and is large, i.e. of the order of characteristic boson energies (), , then it is natural that in the normal state (at ) we deal inevitably with a strong interacting non-canonical Fermi liquid which is at least qualitatively and semi-quantitatively described by the Migdal-Eliashberg theory. In order to justify this statement we shall in the following elucidate some properties in more details by studying optical, ARPES, tunnelling and other experiments.
B. Prejudice on the limitation of the strength of EPI
In spite of reach 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 retrograde trend was partly due to an incorrect statement in Cohen () on the possible upper limit of T in the phonon mechanism of pairing. Since in the past we have discussed this problem thoroughly in numerous papers - for the recent one see Maksimov-Dolgov-2007 (), we shall 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 quasi-particle 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) Kirzhnitz (), i.e.
In case of strong interaction between quasi-particles, the state of embedded quasi-particles changes significantly due to interaction with other quasi-particles, giving rise to . In that case depends on other (than ) response functions. However, in the case when Eq.(1) holds, i. e. when the weak-coupling limit is realized, is given by Allen-Mitrovic (), Kirzhnitz (). Here, is the EPI coupling constant, is an average phonon frequency and is the Coulomb pseudo-potential, ( 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 Maksimov-Review (). In Cohen () it was claimed that lattice stability of the system with respect to the charge density wave formation implies the condition for all . If this was correct then from Eq.(2) it follows that , which limits the maximal value of T to the value . In typical metals and if one accepts the statement in Cohen (), i.e. that , one obtains . The latter result, if it would be true, means that EPI is ineffective in producing not only high-T superconductivity but also low-temperature superconductivity (LTS). However, this result is apparently in conflict first of all with experimental results in LTSC, where in numerous systems and . For instance, is realized in alloy which is definitely much higher than .
Moreover, the basic theory tells us that is not the response function Kirzhnitz (). Namely, if a small external potential is applied to the system 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: for , i.e. either or . This important theorem invalidates the above restriction on the maximal value of T in the EPI mechanism. We stress that the condition is not in conflict with the lattice instability. For instance, in inhomogeneous systems such as crystal, the total longitudinal dielectric function is matrix in the space of reciprocal lattice vectors (), i.e. , and is defined by . It is well known that in dense metallic systems with one ion per cell (such as metallic hydrogen ) and with the electronic dielectric function , one has DKM ()
At the same time the frequency of the longitudinal phonon is given by
where is the ionic plasma frequency, is the local (electric) field correction - see Ref. DKM (). The real condition for lattice stability requires that , which implies that for one has . The latter condition gives automatically . Furthermore, the calculations DKM () show that in the metallic hydrogen crystal, for all . Moreover, the analyzes of crystals with more ions per unit cell DKM () gives that is more a rule than an exception - see Fig. 1. The physical reason for are local field effects described above by . Whenever the local electric field acting on electrons (and ions) is different from the average electric field , i.e. , there are corrections to (and 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 giving rise to in Eq.(2). This means that the dielectric function does not limit in the phonon mechanism of pairing. This result does not mean that there is no limit on T at all - see more in Maksimov-Dolgov-2007 () and references therein. We mention in advance that the local field effects play important role in HTSC oxides, due to their layered structure with very unusual ionic-metallic binding, thus giving rise to large - see more in the subsequent sections. It is pertinent to note that one of the author of Cohen () recognizes the possibility and in Cohen2 () even makes interesting proposals for compounds with large EPI and , while the other author Anderson2 () still ignores rigors of scientific arguments and negates importance of EPI in HTSC cuprates.
In conclusion we point out that there are no 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 (oxides): (1) if EPI is responsible for pairing in HTSC cuprates and if superconductivity is of type, how are these two facts compatible? (2) why is the transport EPI coupling constant (entering resistivity) much smaller than the pairing EPI coupling constant (entering T), i.e. why one has ? (3) is high T possible for a moderate EPI coupling constant, let say for , and under which conditions? (4) if EPI is ineffective for pairing in HTSC oxides, inspite of , why it is so?
C. Is a non-phononic pairing realized in HTSC?
Regarding EPI one can pose a question - whether it contributes significantly to d-wave pairing in cuprates? Surprisingly, despite numerous experiments in favor of EPI, a number of researchers still believe that EPI is irrelevant for pairing Pines (). This belief is mainly based: (i) on the, previously discussed, incorrect lattice stability criterion, which implies small EPI; (ii) on the well established experimental fact that d-wave pairing is realized in cuprates Tsui-Kirtley (), which is believed to be incompatible with EPI. Having in mind that EPI in HTSC is strong with (see below), we assume for the moment that the leading pairing mechanism in cuprates, which gives d-wave pairing, is due to some non-phononic mechanism, like the exitonic one, with the high energy gluing boson () and with the bare critical temperature and look for the effect of EPI. If EPI is isotropic, like in most LTSC materials, then it would be very detrimental for d-wave pairing - the pair breaking effect. 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 weak non–phonon interaction (with the characteristic frequency )
For pure d-wave pairing one has and which gives the equation for T - see Maksimov-Review ()
Here is the di-gamma function. At temperatures near one has and the solution of Eq. (6) is approximately , which means that for and the bare due to the non-phononic interaction must be very large, i.e. .
Concerning other non-phononic mechanisms, such as the SFI one, the effect of the isotropic EPI in the framework of Eliashberg equations was studied numerically in Licht (). The latter is based on Eqs.(42-44) in Appendix A. with kernels
where is taken in the FLEX approximation Scalapino-Review (). The calculations Licht () confirm the very detrimental effect of the isotropic EPI on the d-wave pairing due to SFI. For the bare SFI and the calculations give very small . These results tell us that a more realistic pairing interaction must be operative in cuprates and that EPI is strongly momentum dependent Kulic1 (). Only in that case is strong EPI conform with d-wave pairing, either as its main cause or as a supporter of a non-phononic mechanism. In Part II we shall argue that the strongly momentum dependent EPI is the main player in cuprates providing the strength of the pairing mechanism, while the residual Coulomb interaction and SF, although weaker, trigger it to d-wave pairing.
Iii Experimental evidence for strong EPI and weak SFI
In the following we discuss some important experiments which give evidence for strong EPI in cuprates. Before doing it; we shall discuss some magnetic neutron scattering measurements in cuprates whose results are against the SFI mechanism of pairing. The experimental results related to the pronounced imaginary part of the susceptibility at the AF wave vector were interpreted in a number of papers as a support for the SFI mechanism for pairing Pines (). We briefly explain inadequacy of such an interpretation.
iii.1 Magnetic neutron scattering and the spin fluctuation spectral function
A. SFI affects very little
Before discussing experimental results in cuprates on the imaginary part of the spin susceptibility we point out that in theories based on spin fluctuations the effective pairing potential , which is repulsive, is assumed in the form Pines ()
This form of can be theoretically justified in the weak coupling limit () only. This mechanism of pairing could be effective in cuprates only if the spin susceptibility (spectral function) Im is strongly peaked at the AF wave vector . What is the experimental situation? The breakthrough came from magnetic neutron scattering experiments on by Bourges group Bourges (). They showed that (the odd part of the spin susceptibility in the bilayer system) at is strongly dependent on the hole doping as it is shown in Fig. 2.
By varying doping there is a huge rearrangement of in the frequency interval which is important for superconducting pairing, let say as it is seen in the last two curve in Fig. 2(top). At the same time there is only a small variation of the corresponding critical temperature ! For instance, in the underdoped crystal Im, and , is much larger than that in the near optimally doped , i.e. , although the difference in the corresponding critical temperatures is very small, i.e. (in ) and (in ). This pronounced rearrangement and decrease of Im by doping, but a negligible change in in YBCO is clearly seen in Fig. 2(top), which is strong evidence against the mechanism of pairing. These results in fact mean that the coupling constant is small, i.e. , and the pairing mechanism is ineffective in cuprates. We stress that in the phenomenological theory of the SFI pairing Pines (), an unrealistically large coupling was assumed which gives . The latter value cannot be justified neither experimentally nor theoretically. Let us add that the anti-correlation between the decrease of and increase of by increasing doping toward the optimal value is also present in the NMR spectral function which determines the longitudinal relaxation rate - see Kulic-Review (). This result additionally disfavors the SFI model of pairing Pines (), i.e. the strength of pairing interaction is little affected by SFI. As we shall discuss below the role of SFI together with the stronger direct Coulomb interaction is to trigger d-wave pairing.
A less direct argument for smallness of the SFI coupling constant, i.e. and comes from other experiments related to the magnetic resonance peak in the superconducting state, and this will be discussed next.
B. Ineffectiveness of the magnetic resonance peak
In the superconducting state of optimally doped YBCO and BISCO, 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 LSCO sand consequently one can question the importance of the resonance peak in the scattering processes. The relative intensity of this peak (compared to the total one) is small, i.e. - see Fig 2 (bottom). In underdoped cuprates this peak is present also in the normal state as it is seen in Fig 2 (top). After the discovery of the resonance peak there were attempts to relate it: (i) to the origin of the superconducting condensation energy and (ii) to the kink in the energy dispersion or the peak-dimp structure in the ARPES spectral function. In order that the property (i) holds it is necessary that the peak intensity is small Kivelson (). is obtained by equating the condensation energy with the change of the magnetic energy in the superconducting state, i.e. , where
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, i.e. . Such a small coupling does not affect superconductivity. Moreover, by studying the width of the resonance peak one can extract the SFI coupling constant . Thus, the magnetic resonance disappears in the normal state of the optimally doped YBCO, which can be qualitatively understood by assuming that its broadening scales with the resonance energy , i.e. , where the line-width is given by Kivelson (). This limits to . We stress that the obtained is much smaller than the one assumed in the phenomenological spin-fluctuation theory Pines () where , but much larger than in Kivelson () (where ). The smallness of comes out also from the analysis of the antiferromagnetic state in underdoped metals of LSCO and YBCO Kulic-Kulic (), where the small magnetic moment points to an itinerant antiferromagnetism with small coupling constant . The conclusion is that the magnetic resonance in the optimally doped YBCO is a consequence of the onset of superconductivity and not its cause.
There is also a principal reason against the pairing due to the resonance peak at least in optimally doped cuprates. Since its intensity near is vanishingly small, though not affecting pairing at the second order phase transition at , then if it would be the origin for superconductivity the phase transition at T 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 Lanzara (), Valla () - see the discussion below.
We shall argue below that despite its smallness, spin fluctuations can, together with other contributions of the residual Coulomb interaction, trigger d-wave pairing, while the strength of pairing is due to EPI which is peaked at small transfer momenta - see more below and in Kulic-Review (), Kulic-AIP ().
iii.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 very indispensable. However, 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. was usually measured in the limited frequency interval . Therefore, some physical modelling for is needed in order to guess it outside this range - see more in reviews Maksimov-Review (), Kulic-Review (). This was the reason for numerous inadequate interpretations of optic measurements in cuprates, as well as the misconceptions and misinterpretations that will be uncover below. An illustrative example for this claim is large dispersion in the reported value of - from to , i.e. almost three orders of magnitude - see discussion in Bozovic-Plasma (). This tells us also that in some periods science suffers from a lack of rigorousness and objectiveness. However, it turns out that measurements of in conjunction with elipsometric measurements of at high frequencies allows reliable determination of .
1. Transport and quasiparticle relaxation rates
The widespread misconception in studying the quasi-particle scattering in cuprates was an ad hoc assumption that the transport relaxation rate is equal to the quasi-particle relaxation rate , in spite of the well known fact that Allen (). This incorrect assumption led to the abandoning of EPI as relevant scattering mechanism in cuprates. Although we have discussed this problem several times before, we want to do it again, since the correct understanding of the scattering mechanism in cuprates will take us forward in understanding of the pairing mechanism.
The dynamical conductivity consists of two parts, i.e. where describes interband transitions which contribute at higher frequencies, while is due to intraband transitions which are relevant at low frequencies . (In 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 Allen (), Schlesinger (), Dolgov-Shulga (), Shulga ()
which is a useful representation for systems with single band electron-boson scattering which is justified in HTSC cuprates - see the discussion below. (The usefulness of introducing the optic relaxation will be discussed in Appendix B.) Here, enumerates the plane axis, , and are the electronic plasma frequency, the transport (optical) scattering rate and the optical mass, respectively. Very frequently, the quantity Schlesinger (), which is determined from the half-width of the Drude-like expression for , was analyzed since it is independent of . In the weak coupling limit , the formula for conductivity given in Eqs. (64-67) can be written in the form of Eq.(11) where reads Dolgov-Shulga ()-Shulga ()
Here is the Bose distribution function. (For the explicit form of the transport mass see Allen (), Dolgov-Shulga (), Shulga (), Maksimov-Review (), Kulic-Review ().) In the presence of impurity scattering one should add to . It turns out that Eq.(12) holds within a few percents also for large . Note, that and the index enumerates all scattering bosons - phonons, spin fluctuations, etc. For comparison, we give the quasi-particle scattering rate
where is the Fermi distribution function. By comparing Eq.(13) and Eq.(12), it is seen that and are different quantities, , i.e. the former describes the relaxation of Bose particles (electron-hole pairs) while the latter one the relaxation of Fermi particles. This difference persists also at where one has (due to simplicity we omit in the following summation over ) Allen ()
In the case of EPI, the above equations give that for while (as well as ) is monotonic growing for , where is the maximal phonon frequency. This is clearly seen by comparing , and which are calculated for the EPI spectral function extracted from tunnelling experiments in YBCO (with ) Tunneling-Vedeneev () - see Fig. 3.
The results shown in Fig. 3 clearly demonstrate the physical difference between two scattering rates and . It is also seen that is more a linear function of than . From these calculations one concludes that the quasi-linearity 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 , shown in Fig. 3, 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 , , etc. - see more in Maksimov-Review (), Kulic-Review ().
Let us discuss briefly the experimental results for and and compare these with theoretical predictions obtained by using a single band model and from tunnelling data with the EPI coupling Tunneling-Vedeneev (). In the case of YBCO the agreement between measured and calculated is very good up to frequencies which confirms the importance of EPI in scattering processes. For higher frequencies, where a mead infrared peak appears, it is necessary to account for interband transitions Maksimov-Review (). In optimally doped Romero92 () the experimental results for are explained theoretically by assuming that the EPI spectral function , where is the phononic DOS in BISCO while , and - see Fig. 4(a). The agreement is rather good. At the same time the fit of by the marginal Fermi liquid fails as it is evident in Fig. 4(b).
Now we will comment on the so called pronounced linear behavior of (and ) which served in the past for numerous inadequate conclusions. We stress that the measured quantity is reflectivity and derived ones are , and , which are very sensitive to the value of the dielectric constant . This is clearly demonstrated in Fig. 5 for Bi2212 where it is seen that (and ) for is linear up to much higher than in the case .
In some experiments Puchkov (), Timusk-old () (and ) is linear up to very high which means that the ion background and interband transitions (contained in ) are not properly taken into account since it is assumed too small . The recent elipsometric measurements on YBCO BorisMPI () give the reliable value for . The latter gives rise to a much less spectacular linearity in the relaxation rates than it was the case immediately after the discovery of HTSC cuprates.
Furthermore, we would like to comment on two points concerning , , and their interrelations. First, the parametrization of with the generalized Drude formula in Eq.(11) 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 Shulga () 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 fully justified for a number of HTSC cuprates such as LSCO, BISCO etc. Second, at the beginning we said that and are physically different quantities and it holds . In order to give the physical picture and qualitative explanation we assume that . In that case the renormalized frequencies, the quasi-particle one (for the definition of see Appendix A.) and the transport one - defined above, are related and at , they are given by Allen (), Shulga ()
It gives the relation between and , and respectively
The physical meaning of Eq.(16) 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 quasi-particles with the renormalized . Since takes values then the optical relaxation is the energy-averaged according to Eq.(16). The factor 2 is due to the two quasi-particles, electron+hole. At finite , the generalization reads Allen (), Shulga ()
2. Inversion of the optical data and
In principle, the transport spectral function can be precisely extracted from , i.e. , only at , which follows from Eq.( 14)
However, real measurements are performed at finite (and also at ) and the inversion procedure is in principle an ill-posed problem since is the deconvolution of the inhomogeneous Fredholm integral equation of the first kind with the temperature dependent Kernel in Eq.(12). An ill-posed mathematical problem, like this one, is very sensitive to input since experimental data contain less information than one needs. This can cause the fine structure of gets blurred in the extraction procedures and it can be temperature dependent even when the true is independent. In the context of HTSC cuprates, this problem was first studied in Dolgov-Shulga (), Shulga () with the following results: (1) the extracted shape of in is not unique and it is temperature dependent, i.e. 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 with three peaks structure as it is shown in Fig. 6 The similar situation is realized in optimally doped BISCO as it is seen in Fig. 7. It is important to stress that the width of the extracted