Abstract

The far- and mid-infrared reflectivity R(ω) of e-doped single crystals belonging to the family M2−xCexCuO4−y(M = Pr, Nd, Gd; 0<x<0.15; 0<y<0.04) has been studied between 300 and 20 K. In addition to the phonons predicted for the T' structure, R(ω) shows local modes in the far infrared as well as a broad infrared absorption centered at about 0.1 eV (d or J band). These features depend strongly on both T and y. We have resolved the d band at low T, in samples doped by oxygen vacancies. We demonstrate its polaronic origin by showing that it is made up of intense overtones of the local modes observed in the far infrared. We also find that Ce-doped superconductors (x> 0.12,y = 0.03) have the same polaronic structure as the semiconducting ones, partially superimposed on a weak Drude term.

Introduction

Since the discovery of high-Tc superconductors (HTSC), infrared reflectivity measurements have been largely employed to investigate both electronic and transport properties of these cuprates [1]. Early spectra already showed several intriguing features common to all HTSC families, which were then attributed to the peculiar properties of the Cu–O plane. Those features are well reproducible and could be studied in greater detail as soon as large single crystals became available. However, their interpretation is still being debated. In the insulating parent compounds of HTSC, one observes phonon modes in the far infrared, and a broad band at high frequencies (from about 1.5 to 2.5 eV). This latter has been unanimously assigned to charge-transfer (CT) transitions between O 2p and Cu 3d orbitals.

Abstract

A Feynman path-integral type of treatment is developed to determine under which conditions the energy of a bipolaron is lower than the energy of two polarons. A detailed analytical and numerical study of the Fröhlich bipolaron is presented, resulting in a phase diagram for the stability of the bipolaron in terms of the electron–phonon coupling strength α and the strength U of the Coulomb repulsion. The stability region for two- and three-dimensional bipolarons is examined for several materials.

It is shown that the bipolaron binds more easily in 2D than in 3D and that its radius is only a few ångström units. Alexandrov, Bratkovsky and Mott have recently stressed the importance of this confinement, as derived by the present authors, for high-Tc superconductivity. We analyze as an example the occurence of bipolarons in La2CuO4. First results on optical absorption of bipolarons are also presented.

Bednorz and Müller's discovery of the high-temperature superconductors stimulated both experimental and theoretical efforts to determine the mechanism responsible for superconductivity in these new materials. Bipolarons (dielectric and spin) have been invoked as possible ‘Cooper pairs’ at the basis of high-Tc superconductivity (Alexandrov, Bratkovsky and Mott [1]).

Bipolarons (large and small) had been studied before [2, 5–10] also in the context of superconductivity [3]. Emin proposed Bose–Einstein condensation of large two-dimensional bipolarons as a possible mechanism responsible for superconductivity in these materials.

In the present paper a path-integral study of large bipolarons is presented in two and three dimensions. Conditions will be discussed under which bipolarons can exist in the copper oxides. Some experimental difficulties in determining material parameters, e.g. the band mass, are also discussed.

Abstract

The physical properties of polarons and bipolarons in WO3− x are reviewed and compared with characteristics of carriers in YBa2Cu3O7 and several other high-temperature superconductors, namely (Ca1−x Yx)Sr2(Tl0.5Pb0.5)Cu2O7, Bi2Sr2(Ca0.9Y0.1)Cu2O8+δ and La2CuO4+δ. The fingerprint for (bi)polarons is optical excitations in the spectral near-infrared region. The absorption cross section is drastically reduced in the superconducting phase. The temperature evolution is analysed quantitatively in terms of Bose–Einstein condensation of bipolarons.

Introduction

The physics of polarons and bipolarons has recently been reconsidered because it is believed that condensation of bipolarons is closely related to, or may even be the origin of, superconductivity in oxide materials [1–12]. The justification for such belief is based on several experimental observations such as the absence of the Korringa law in the nuclear spin relaxation rate [3], the heat capacity anomaly [4, 5] and the softening of phonons above the (pseudo-) gap in the superconducting phase [7, 8, 13–15]. Few experimental results point directly to the existence of polarons and/or bipolarons in these materials, however. Probably the most direct indication for the existence of such particles stems from observation of their internal excitations in the infrared and visible spectral range. Such excitations were firmly established in WO3−x (in its ɛ-phase) and related transition metal oxides [16–30]. Similar excitations were recently observed in YBa2Cu3O7 and other high-temperature superconductors [31–37]. Although their original discovery by Dewing and Salje [32] was contested on experimental grounds, it is now confirmed that the apparently contradictory result that such excitations were not seen in reflection spectra of YBa2Cu3O7 crystals lies in the insensitivity of early reflection measurements and statistical errors introduced by subsequent Kramers–Kronig analysis [38, 39].

Abstract

This paper presents a scenario in which large (multi-site) bipolarons form and give rise to superconductivity. First the physical circumstances in which large bipolarons can form are elucidated. Then several identifying properties of large bipolarons are discussed. Finally, a model of how interactions between large bipolarons lead to their superconductivity is presented. I emphasize the existence of a phonon-mediated intermediate-range attraction between large bipolarons. With attractive interactions between large bipolarons, they can condense into a liquid phase. This liquid is a quantum liquid if the ground state of the interacting large bipolarons is a fluid rather than a solid. This quantum liquid of charged bosons is analogous to the quantum liquid of neutral bosons envisioned for superfluid He. As such, the superconductivity of large bipolarons can be understood (or rationalized) in a similar manner to that employed in addressing the superfluidity of liquid He.

Introduction

This article begins by describing electron–lattice interactions of ionic solids. A long-range electron–lattice interaction results from the dependence of the Coulombic potential energy of a carrier on the positions of the solid's ions. [1] Short-range electron–lattice interactions reflect the sensitivity of the energy of a carrier's local state (e.g., bonding or antibonding state) to the positions of nearby atoms [2,3].

The notions of self-trapping and bipolaron formation are then reviewed. Since a self-trapped carrier can only move when atoms move, the adiabatic approach is employed to discuss polaron and bipolaron formation.

Abstract

The possibility of bipolaron formation is studied taking into account both electron–phonon interaction and direct Coulomb repulsion. Starting from the Bethe–Salpeter equation with a rather general interaction of the form V(k,ω) = 4πe2/ε(k,ω), and using spectral representations for both the bipolaron wave function and the interaction, the effective Schrödinger-type equation is obtained with the new effective potential, which is non-local and which parametrically depends on the binding energy. It is shown that, if the static response function l/ε(K:,0) is non-negative, then there is no bound state, i.e. in this case bipolarons do not form (electron–lattice interaction is not sufficient to overcome direct Coulomb repulsion). Possible ways out are discussed, among them the possibility of a negative static dielectric function or more general form of the effective electron–electron interaction.

Introduction

The problem of the state of electrons in crystals with strong electron–phonon interaction is now attracting considerable attention. It is well known that one of the possibilities in this case is the formation of polarons [1,2]. The conditions for their existence and their properties have been studied in numerous publications.

Much less studied (and more controversial) is the next possible step – formation of bipolarons in certain cases. The possibility of bipolaron formation was probably first pointed out by Vinetskii and Giterman [3]; later, bipolrons were suggested as possible candidates to explain some properties of Ti4O7 [4]. The problem of the existence of bipolarons has recently acquired special significance in view of the suggestions that they may exist in cuprates and may possibly explain the phenomenon of high-temperature superconductivity in them (see especially [5–7]).

Abstract

It is explained why, despite the strong exothermicity of the reaction generating two Bi4+ ions from one Bi3+ and one Bi5+ ion in the gas phase, solid barium bismuthate (Ba2BivBiIIIO6) contains bismuth in two different oxidation states and has a structure distorted from that of a perfect cubic perovskite by displacements of the oxide ions towards the Biv species. The predicted difference between the energy of this and that of an undistorted undisproportionated cubic perovskite (BaBiO3) containing only Biiv is in good agreement with the activation energy measured for conductivity by thermal hopping. It is also shown why, for x≳O.3, all materials of stoichiometry KxBa1-xBiO3 contain only one type of bismuth (BiIV) in a perovskite structure without the distorting oxide displacements of the undoped material. The distorted disproportionated structure is predicted to be degenerate with the undistorted undisproportionated structure when x = 0.34. A possible connection between this degeneracy and high-temperature superconductivity is discussed.

Observations for explanation

The primary object of this paper is to explain two related experimental observations concerning the structure of the semi-conductor barium bismuthate and the materials that result from its doping with K+ ions. We then comment on possible connections with the high-temperature superconductivity (HTS) exhibited by the latter materials at sufficiently high levels of doping.

The first observation requiring an explanation is that there are two different types of bismuth site in the undoped material of stoichiometric formula BaBiO3[l–7]. This compound thus has the formula Ba2BvBiIIIO6[l–7], where the superscript denotes formal oxidation state.

Abstract

There is ample experimental evidence for localized polaronic charge carriers in high-Tc materials in the insulating phase as well as in the metallic phase at high temperatures. This would rule out a priori any condensation of bipolarons, since for that purpose they should be in free-particle-like states in the longwavelength limit. Yet, provided that the localized bipolarons hybridize with a band of itinerant electrons, such a mixture of Bosons (bipolarons) and Fermion pairs (pairs of conduction electrons) can undergo an instability towards a superconducting ground state in which at high temperatures the initially localized bipolarons become superfluid upon lowering of the temperature. The experimental situation leading up to such a picture and its physical consequences are discussed.

Introduction

The large values of the critical temperature Tc, the small number of charge carriers together with the short coherence length, the strong dependence of Tc on n/m (n being the carrier concentration and m their mass) and the large temperature regime near Tc (with a Ginzburg temperature of TG 0.1−0.01) controlled by X−Y universality strongly suggest that high-Tc superconductivity is more closely related to Bose–Einstein condensation of real-space pairs than to a BCS state of Cooper pairs. The polaronic nature of at least part of the charge carriers in these materials has been experimentally established in both the insulating and the metallic phase of these compounds. On theoretical grounds one expects small polarons to interact with each other over short distances and in a practically unretarded fashion.

Abstract

We provide strong evidence that cuprate superconductors and uncharged superfluids like 4He share the universal 3D x–y properties in the fluctuation dominated regimes. The universal relations between critical amplitudes and Tc, supplemented by the empirical phase diagram (doping-dependence of Tc) also imply – in agreement with recent kinetic-induction measurements on La2−xSrxCuO4 films and muon spin resonance (μSR) data – unconventional behavior of the penetration depth in the overdoped regime. The evidence of uncharged superfluid of 3D x–y behavior is completed in terms of the asymptotic low-temperature behavior of the penetration depth, because the sound-wave contribution of the uncharged superfluid accounts remarkably well for the experimental data. The dominant role of 3D x–y fluctuations, implying tightly bound and interacting pairs even above Tc, together with the doping-dependent specific heat singularity point uniquely to Bose condensation of hard pairs on a lattice as the mechanism that drives the transition from the normal to the superconducting state.

Considerable debate has arisen over the nature of superconductivity and the symmetry of the order parameter in high-Tc superconductors [1–9]. In view of the fact that thermal fluctuations reflect the structure of the order parameter and that these extreme type II materials exhibit pronounced fluctuation effects [10–17], we discuss in this paper the use of thermal fluctuations to elucidate the nature of the superconducting state.

The organization of this paper is as follows. First we provide strong evidence that cuprate superconductors and uncharged superfluids like 4He share the universal three-dimensional (3D) x–y properties in the fluctuation-dominated regimes.

Abstract

The electrical features in the normal phase of high-Tc superconducting materials can be explained by the coexistence model of small polarons and Anderson-localized carriers. According to this model, with increasing carrier concentration, the degree of Anderson localization decreases, and then the concentration of coexisting small polarons increases, attains a maximum and decreases with this variation. If Tc is determined by the concentration of bosons (bipolarons) as in Shafroth's formula for Bose condensation, then the shape of the superconducting phase can be explained by this behavior of small polarons. The degree of localization in the oxides without superconductivity is too large for coexistence to be attained.

Introduction

In the high-Tc superconducting oxides, the superconducting phase appears just in the composition region in which a metal–insulator transition takes place. This fact leads to the idea that the electronic states proper to this transition are responsible for the origin of superconductivity. This transition accompanying a gradual change in electronic nature in accordance with stoichiometric variation or with carrier doping has the following characteristic features [1].

1. The electrical conductivity can be described by the variable-range hopping (VRH) mechanism at low temperatures in the insulator (semiconductor) region, which means that the carriers in this region are Anderson-localized. The degree of localization decreases with increasing carrier concentration resulting in the occurrence of the metallic phase. So this transition should be classified as an Anderson transition.

2. Plural types of carriers, itinerant and localized ones, coexist in both the metallic and the semiconducting phase.

Abstract

An energy scale of the superconducting condensate, which we call the effective Fermi temperature TF, can be derived from the magnetic field penetration depth λ determined by muon spin relaxation (μSR) measurements. We classify various superconductors in the crossover from Bose–Einstein (BE) to BCS condensation, based on the ratio Tc/TF. The phase diagram of high- Tc cuprate superconductors, as a function of carrier doping, can be understood in the context of this BE–BCS crossover, if we identify the ‘pseudo-gap’ as the signature of pair formation in the normal state. In particular, the universal linear relation between Tc and ns/m* (superconducting carrier density/effective mass), found in the underdoped region, comes from a general feature expected in the BE condensation of pre-formed pairs. The optimal Tc occurs around the doping concentration at which the condensate energy scale TF becomes comparable to the energy scale hωB of the pair-mediating interaction. A surprising decrease of ns/m* with increasing carrier doping was found in the overdoped Tl 2201 system. This behavior suggests that evolution to the BCS region does not occur in a simple way, but rather is associated with a possible microscopic separation between superconducting and residual normal metallic phases.

During the past several years, we have performed measurements of the magnetic field penetration depth λ of high- Tc cuprate and other superconducting systems using the muon spin relaxation (μSR) technique [1–5].

Abstract

An analytically solvable Ansatz set of Migdal-type self-consistent equations is proposed for the coupling between spin and charge degrees of freedom in the strongly correlated electron system described by the t–t′–J(t–J) Hamiltonian. The small parameter validating Migdal's approximation for this problem is found to be 1/ln(U/t) (when U»t), where U and t are on-site Coulomb repulsion energy and the bare electron hopping integral of the basic Hubbard Hamiltonian (t′ = J =4t2/U). The analytical results, obtained for electron concentrations close to half-filling, demonstrate strong enhancement of the quasi-particle mass, accompanied by a depletion of the particle density of states at the Fermi-level (EF). The spectral density is pushed away from EF into a broad range of energies (of order t) and possesses a sawtooth form. Theoretical predictions for the optical conductivity are derived, which could provide a qualitative explanation of the mid-infrared anomaly observed experimentally in high-Tc cuprates. The variation in quasi-particle energy over the Brillouin zone is found to be of order J only, in good accord with previous theoretical work and the dispersion observed in recent ARPES experiments in 2212 high- Tc compounds.

Introduction and summary of our previous work

It is by now generally accepted that strong Coulomb correlations should play an essential role in the physics of the high-temperature superconductors (HTS). A great number of experimental results obtained since the discovery [1] of the HTSs raise strong doubts about the applicability of ‘classical’ BCS theory to the description of the superconductivity phenomena in these compounds.

Abstract

The spectral function and momentum distribution for holes in an antiferromagnet are calculated on the basis of the t–J model in a slave-fermion representation. The self-consistent Born approximation for a two-time Green function is used to study the dependences on temperature and doping (δ) of the self-energy operator. The numerical calculations show weak dependences on concentration and temperature of the spectral function (quasi-particle hole spectrum) while the momentum distribution function changes dramatically with increasing temperature for T>Td with Td⋍Jδ.

Introduction

The problem of hole motion in an antiferromagnetic (AF) background has attracted much attention in recent years. That is mainly due to the hope of elucidating the nature of the carriers involved in high-Tc superconductivity in copper oxides. It is believed that the essential features of the problem are described by the t–J model with a Hamiltonian written as

Here 〈ij〉 indicates nearest-neighbor pairs, c+iσ = c+iσ (1 − ni − σ) are the electron operators with the constraint of no double occupancy. The properties of a single hole doped in the Néel spin background have been analyzed intensively with various numerical and analytical methods. Among them are exact diagonalization of small clusters [1] and variational calculations [2]. A rather transparent description within a ‘string’ picture has been developed by several authors [3]. A perturbative approach to the problem was proposed by Schmitt–Rink, Varma and Ruckenstein [4] and developed further by Kane, Lee and Read [5] and Martinez and Horsch [6].

Abstract

Two-polaron states on a square lattice are studied in the presence of on-site repulsion U and inter-site attraction V. In the limit of infinite U the exact critical value Vcr for bipolaron formation is obtained as a function of the attraction radius R. The results are compared with the continuum limit of the same model. It is shown that if R≃(2–3) lattice constants then Vcr is of the order of the characteristic phonon frequency in the high-temperature superconductors.

The temperature-dependence of the upper critical field [1,2], the resistivity and Seebeck coefficient [3], and the universal correlation between the critical temperature and the hole content [4] in the p-type oxide superconductors unambiguously support the validity of the local pair conception for these compounds at low doping 0.06≤n≤0.12. Phonons are the most natural candidates for the bosonic field whose interaction with the carriers (polarons) results in the effective interpolaron attraction. However, there are several arguments against the phonon pairing mechanism. One of them is that the phonon-mediated attraction between polarons is much weaker than the shortrange Coulomb repulsion, hence creation of local pairs is inhibited. The typical estimates for the on-site copper, copper–oxygen, and inter-site copper Coulomb potentials are Udd≃10 eV, Upd≃1 eV and U′dd≃Q.l eV correspondingly [5, 6]. Since the typical phonon frequency ω is of the order of 0.1 eV or less, ω » U/dd, Upd and ω≃U′dd and therefore the existence of the local pair in which the polarons are localized on the same lattice site or on the nearest neighbours is impossible.

Abstract

The spectrum of collective (pair) excitations in the ground state of a dilute twodimensional (2D) attractive Fermi-gas is studied within the functional integral formalism. The linearized equations for the fluctuations about the non-trivial saddle point are analyzed for all coupling regimes, which are characterized by the ratio ε0/εF, where ε0 is the two-fermion binding energy and εF is the Fermi energy. The approximation takes into account propagation of the fluctuations and their interaction with the condensate. In the strong-coupling, or ‘;Bosegas’, regime (ε0/εF> 1) the spectrum is continuous and has the Bogolubov form, but in the weak-coupling limit (ε0/εF«1) there are two types of excitations (different from the two-fermion scattering states): (i) long-wavelength sound-like excitations with the cut-off at momentum qc≃ l/ξ0 (where ξ0 is the Cooper pair size), and (ii) pair excitations with q≃ (8mμ)½, where μ is the chemical potential and m is the fermion mass. The crossover between weakand strong-coupling behavior of the excitation spectrum is found to occur at the value of the coupling parameter ε0/εF≃¼.

Introduction

Since the discovery of high-Tc superconductivity there has been growing interest in studying 2D Fermi gases with attractive interaction, especially in the crossover regime, when the pair size ξ0 is of the order of the interparticle distance. The importance of such a model, which can be regarded as a semiphenomenological model of a 2D superconductor, is highlighted by experimental evidence that the high- Tc superconductors, most of which have a layered structure, have a short coherence length, so that kFξ0≃1, where kF is the Fermi momentum.

Abstract

We report on experimental evidence for different electronic phases in the superconductor YBa2Cu4O8: simple metallic Cu–O chains and highly correlated CuO2 planes. YBa2Cu4O8 is a genuine untwinned compound; hence, we were able to determine the anisotropy of the resistivity along the a and b directions. Along the b direction (chain and plane conduction channels), a normal metallic temperature behavior of the chain dominates. For the a direction (only the plane conduction channel), the resistivity reveals unconventional behavior with a kink at 160 K, becoming linear at higher temperatures. Further, we present results of the dynamical (optical) conductivity of the CuO2 plane as a function of frequency and temperature. The frequency-dependence of the optical conductivity is consistent with a model of ferromagnetic polarons in an antiferromagnetic matrix. This is further confirmed by a depolarization experiment, which is sensitive to crystal regions with different spin polarizations. The temperature behavior of the thermal occupation of the ground state is in agreement with a real-space condensation.

Introduction

From the beginning [1] the high-Tc superconductors (HTSC) have been very challenging systems. Regarding their solid state chemistry, the defect structure with its wide variability in stoichiometry gives rise to homogeneity problems: the samples may show a chemical phase separation of different oxygenated regions. So, proper preparation of homogeneous samples is crucial. Concerning the physics, the normal as well as the superconducting state show unusual features. In addition, as we will show at least for YBa2Cu3O7 (123) or YBa2Cu4O8 (124), one has to take into account the coexistence of different physical phases.

Abstract

Using picosecond pulses we excite a large number of carriers in YBa2Cu3O7−δ and drive the system through an insulator-to-metal transition. As we do this, we study the carrier interaction with the apical O and the 340 cm−1 planar O buckling vibrations by counting the number of non-equilibrium phonons that the carriers generate as they relax towards equilibrium. Counting is done directly by photoexcited anti-Stokes Raman scattering (PEARS). Carrier transport is found to be thermally activated; presumably the carriers relax by hopping between localized states. The in-plane and chain activation energies are found to be very different, suggesting that different carriers are involved: the chain carriers form polarons, which are strongly coupled to apical O vibrations, while the planar holes, which have a smaller activation energy, are suggested to interact less strongly with the lattice, especially in the metallic state.

Introduction

One may ask a very simple question about charge carrier transport in a cuprate superconductor: are the carriers moving through the crystal as in an extended band, or are they hopping from site to site? Given that the structure of these materials is composed of two distinct parts – the CuO2 planes plus charge reservoirs – one may perhaps also wonder whether the behaviour of carriers in the two parts is different. In order to answer such questions, we need a microscopic probe, which, it is to be hoped, might tell us something about both the type of interaction between the carriers and the symmetry and location of the interaction.

Abstract

The BSC theory is extended to strong electron–phonon coupling for λ > 1. In this limit carriers are charged 2e bosons (singlet and triplet inter-site bipolarons). The Anderson localisation of the bosons resulting from disorder is also considered. Several non Fermi-liquid features of copper-based high-Tc oxides, in particular the spin gap in NMR and neutron scattering, the temperature dependent Hall effect, linear resistivity and divergent Hc2(T) are explained.

The strong-coupling extension of the BCS theory

The electron–phonon coupling constant λ in the BCS theory is the ratio of the characteristic interaction energy V = 2Ep of carriers with a bosonic field, for instance of phonons, which is responsible for the coupling to their kinetic energy EF, λ ⋍ V/(2EF). At the point λ ⋍ 1 the characteristic potential energy due to the local lattice deformation exceeds the kinetic energy. This is a condition of small-polaron formation which has been known for a long time as a solution for a single electron on a lattice coupled with lattice vibrations. So long as λ > 1 the kinetic energy remains smaller than the interaction energy and a self-consistent treatment of a many-electron system strongly coupled with phonons is possible with the ‘1/λ’ expansion technique [1]. This possibility results from the fact, which has been known for a long time, that there is an exact solution for a single electron in the strong-coupling limit λ rarr; infin. Following Lang and Firsov (1962) one can apply the canonical transformation exp (S1) to diagonalise the single-electron Fröhlich Hamiltonian (under the ‘Fröhlich Hamiltonian’ we assume that any electron–phonon interaction occurs with its matrix element depending on the phonon momentum).