Our purpose in this chapter is to describe an approach (often referred to as the Landauer approach) that has proved to be very useful in describing mesoscopic transport. In this approach, the current through a conductor is expressed in terms of the probability that an electron can transmit through it. The earliest application of current formulas of this type was in the calculation of the current-voltage characteristics of tunneling junctions where the transmission probability is usually much less than unity (see J. Frenkel (1930), Phys. Rev., 36, 1604 or W. Ehrenberg and H. Honl (1931), Z. Phys., 68, 289). Landauer [2.1] related the linear response conductance to the transmission probability and drew attention to the subtle questions that arise when we apply this relation to conductors having transmission probabilities close to unity. For example, if we impress a voltage across two contacts to a ballistic conductor (that is, one having a transmission probability of unity) the current is finite indicating that the resistance is not zero. But can a ballistic conductor have any resistance? If not, where does this resistance come from? These questions were clarified by Imry [2.2], enlarging upon earlier notions due to Engquist and Anderson [2.3]. Büttiker extended the approach to describe multi-terminal measurements in magnetic fields and this formulation (generally referred to as the Landauer–Büttiker formalism) has been widely used in the interpretation of mesoscopic experiments.

In Section 2.6 we briefly studied the effects of electron scattering on resonant tunnelling which are inevitable in a real system operating at room temperature. The phenomenological Breit–Wigner formula was introduced to describe the incoherent aspect of the electron tunnelling which in general results in a broadening of the transmission peak and thus degraded current P/V ratios in RTDs. In this chapter we look in more detail at various scattering processes, both elastic and inelastic, which have been of great interest not only from a quantum transport physics point of view but also because of the possibility of controlling and even engineering these interactions in semiconductor microstructures. The inelastic longitudinal–optical (LO) phonon scattering, introduced in the preceding chapter, is the most influential process, with Г–X-intervalley scattering and impurity scattering also affecting the resonant tunnelling electrons. Section 3.1 describes the dominant electron–LO-phonon interactions. Both theoretical and experimental studies of a postresonant current peak are presented, which provide much information about the electron–phonon interactions in the quantum well. Section 3.2 then discusses the effects of the upper X-valley which become more significant in AlxGa1−xAs/GaAs systems with an Al mole fraction, x, higher than 0.45 since the energy of the X-valley then becomes lower than that of the Г-valley. Finally, in Section 3.3, we study elastic impurity scattering, which may be caused by residual background impurities or those diffused from the heavily doped contact regions.

One of the most significant discoveries of the 1980s is the quantum Hall effect (see K. von Klitzing, G. Dorda and M. Pepper (1980), Phys. Rev. Lett., 45, 494). Normally in solid state experiments, scattering processes introduce enough uncertainty that most results have an ‘error bar’ of plus or minus several per cent. For example, the conductance of a ballistic conductor has been shown (see Fig. 2.1.2) to be quantized in units of (h/2e2). But this is true as long as we are not bothered by deviations of a few per cent, since real conductors are usually not precisely ballistic. On the other hand, at high magnetic fields the Hall resistance has been observed to be quantized in units of (h/2e2) with an accuracy that is specified in parts per million. Indeed the accuracy of the quantum Hall effect is so impressive that the National Institute of Standards and Technology is interested in utilizing it as a resistance standard.

This impressive accuracy arises from the near complete suppression of momentum relaxation processes in the quantum Hall regime resulting in a truly ballistic conductor of incredibly high quality. Mean free paths of several millimeters have been observed. These unusually long mean free paths do not arise from any unusual purity of the samples. They arise because, at high magnetic fields, the electronic states carrying current in one direction are localized on one side of the sample while those carrying current in the other direction are localized on the other side of the sample. Due to the formation of this ‘divided highway’ there is hardly any overlap between the two groups of states and backscattering cannot take place even though impurities are present.

Abstract

If lattice polarons exist in high-temperature superconducting oxides then there must be evidence of local lattice distortion associated with polarons. While the distortions are dynamic and subtle, making direct observation difficult, there are numerous indications that some anomalous local deviations from the crystallographic lattice structure exist in superconducting oxides. Based largely upon the results of pulsed neutron scattering measurements, we present an argument in favor of the presence of local lattice distortions consistent with lattice polarons. A few implications of the observation in relation to other physical properties are discussed.

Introduction

Even though polarons have been known for a long time, direct experimental observation of lattice distortions associated with them is surprisingly scarce, largely because the density of polarons is usually low and consequently the lattice distortion is small on average, making observation very difficult. While some observations of lattice distortion associated with polarons have been made for low-dimensional organic conductors in which the periodic lattice distortion (Peierls distortion) can be regarded as an array of localized polarons [1], there are very few such reports for oxides [2]. Moreover, most known cases of polarons are heavy, small polarons, while in high-temperature superconducting (HTSC) oxides the presence of mobile large polarons is suspected. For those reasons, local lattice distortion has been observed so far mostly by nontraditional methods of structural study, while the crystallographic community has largely been skeptical. In this paper we discuss why observation is difficult, whether there is sufficient experimental evidence to support the presence of polarons in high-temperature superconducting oxides or not, and the implications of these observations.

Abstract

Motivated by aspects of layered high-temperature superconductors and related quasi-one-dimensional materials, we consider polaron structure, dynamics, and coherence in certain extended Peierls–Hubbard models. We emphasize the qualitative importance of electron–lattice interactions even in the presence of dominant electron–electron correlations, the signatures of polaron structure and dynamics in energy-resolved pair-distribution structure functions, and the effect of disorder on polaron propagation and stability.

Introduction

The formation and dynamics of polarons (and bipolarons), despite a halfcentury of theoretical and experimental study, remain fascinating topics in many-body physics, combining as they do (often competing) aspects of coupled fields with distinct natural time scales (e.g., electron–phonon, spin–phonon, exciton–phonon), electron–electron interactions, lattice discreteness, nonadiabaticity, collective quantum tunneling, thermal fluctuations, competitions between disorder and polaronic localization, etc. Direct observation of polarons through real-space imaging of any of the coupled fields is rare even with the advent of STEM, AFM techniques, etc. Likewise, global measurements such as that of electronic band-structure are insensitive to polaron features. Thus, experiments and theoretical techniques have had to focus on indirect effects on microscopic probes such as transport coefficients, electronic absorption, vibrational spectroscopy, and so forth.

Our purpose here is to briefly review three qualitative effects in polaron physics that have arisen in modeling two components of the lattice structure of the layered cuprate superconductors [1] – namely, extended multi-band Peierls–Hubbard models of (a) the active CuO2 planes and (b) polarizable Cu–O clusters in the axial direction (polarizable interplanar medium).

Abstract

The spin-polaron concept is introduced in analogy to ionic and electronic polarons and the assumptions underlying the author's approach to spinpolaron-mediated high-Tc superconductivity are discussed. Elementary considerations about the spin-polaron formation energy are reviewed and the possible origin of the pairing mechanism illustrated schematically. The electronic structure of the CuO2 planes is treated from the standpoint of antiferromagnetic band calculations that lead directly to the picture of holes predominantly on the oxygen sublattice in a Mott–Hubbard/charge transfer insulator. Assuming the holes to be described in a Bloch representation but with the effective mass renormalized by spin-polaron formation, equations for the superconducting gap, Δ, and transition temperature, Tc, are developed and the symmetry of Δ discussed. After further simplifications, Tc is calculated as a function of the carrier concentration, x. It is shown that the calculated behavior of Tc(x) follows the experimental results closely and leads to a natural explanation of the effects of under- and over-doping. The paper concludes with a few remarks about the evidence for the carriers being fermions (polarons) or bosons (bipolarons).

Introduction

A carrier (electron or hole) moving through an ionic lattice will induce displacements of the ions and, under certain conditions, the carrier plus ionic displacements may form a good quasi-particle, i.e., the ionic polaron [1]. Similarly, electronic polarons may form when the carriers induce polarization of localized or quasi-localized electronic distributions. In an analagous manner, a spin polaron is a spin ½ carrier moving in a magnetic medium accompanied by deviations of localized ionic spins.

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.