Abstract

In small-polaron models the hopping amplitude for a carrier from a site to a neighboring site is reduced due to ‘dressing’ by a background degree of freedom. Electron–hole symmetry is broken if this reduction is different for a carrier in a singly occupied site and one in a doubly occupied site. Assuming that the reduction is smaller in the latter case, the implication is that a gradual ‘undressing’ of the carriers takes place as the system is doped and the carrier concentration increases. A similar ‘undressing’ will occur at fixed (low) carrier concentration as the temperature is lowered, if the carriers pair below a critical temperature and as a result the ‘local’ carrier concentration increases (and the system becomes a superconductor). In both cases the ‘undressing’ can be seen in a transfer of spectral weight in the frequency-dependent conductivity from high frequencies (corresponding to non-diagonal transitions) to low frequencies (corresponding to diagonal transitions), as the carrier concentration increases or the temperature is lowered repectively. This experimental signature of electron–hole asymmetric polaronic superconductors as well as several others have been seen in high-temperature superconducting oxides. Other experimental signatures predicted by electron–hole-asymmetric polaron models remain to be tested.

The physics of high-Tc oxides

From the beginning of the high-Tc era there have been indications that small polarons may play an important role in the physics of these materials [1–10]. Among the workers that have not completely abandoned the Fermi liquid framework for this problem most would agree that the physics of the normal state may be described in terms of heavily dressed quasi-particles [11].

Abstract

Recent experimental findings concerning the temperature-dependence of the penetration depth and strong empirical evidence for a universal 3D X–Y critical behaviour in several classes of high- Tc superconductors suggest that Bose condensation of ‘weakly charged’ bosons can be a driving mechanism for the phase transition in extreme type II superconductors. We explore the occurence of analogous behaviour in a simple model of local electron pairs, which is that of hard-core charged bosons on a lattice. We examine the electromagnetic properties of the model in the superfiuid phase as a function of boson concentration and density–density interaction. In the low-density limit, with the use of the exact scattering length, we present new results for the ground state characteristics. The relevance of the obtained results to interpretation of experimental data on high-Tc oxides is discussed.

Introduction

Some recent experimental results concerning temperature-dependence of the London penetration depth and empirical evidence for a universal 3D X–Y critical behaviour in several families of high- Tc superconductors indicate that Bose condensation can be a driving mechanism for the phase transition in these extreme type II superconductors [1–3]. Motivated by these experimentally established universal trends, we examine the electromagnetic properties of the effective pseudo-spin model of the local pair superconductor, equivalent to a hard-core charged Bose gas on a lattice, for the case of arbitrary electron concentration n. From linear response theory, the current response is obtained and expressions for the paramagnetic and diamagnetic kernels are evaluated in the RPA (random phase approximation) and SWA (spin-wave approximation).

With the advent of high-temperature superconductors, research on polarons and bipolarons has gained renewed attention. It appears that carriers in some high-temperature superconductors are strongly correlated both in the normal and in the superconducting state. In the strong-coupling limit the Fermi-liquid ground state may be destroyed by the formation of small polarons and bipolarons. Experimental and theoretical analysis of such particles is a central issue for current research in superconductivity.

This book contains a series of authoritative articles on the most advanced research on polarons and bipolarons. They were invited for presentation during a workshop on ‘Polarons and Bipolarons in High-Tc Superconductors and Related Materials’, which was held at the Interdisciplinary Research Centre (IRC) in Superconductivity, Cambridge, UK, between 7th and 9th April 1994. Over 50 participants from ten countries took part in this workshop, representing the major research centres currently working in this field. The workshop was held in honour of Sir Nevill Mott in recognition of his important contributions to the physics of polarons.

The editors are grateful to all contributors for their overwhelmingly positive response to the idea of publishing this book. We are indebted to Ken Diffey and Margaret Hilton who ensured the smooth running of the workshop, to all of the referees and to William Beere for their help editing the manuscripts. We also thank Simon Capelin of Cambridge University Press for his encouragement and help throughout the publication of this book.

Abstract

We review a series of exact, analytical and numerical results obtained on the adiabatic Holstein–Hubbard model, many of which are new and non-trivial. We study next the role of the quantum lattice fluctuations that were initially neglected. The possibility of having high-Tc bipolaronic superconductors is analysed on the basis of these results.

We suggest that models that involve only electron–phonon coupling are very unlikely to produce bipolaronic superconductivity, which is prevented by the spatial ordering of the bipolarons associated with a lattice instability. This is due to a very large effective mass of the bipolarons that can be related to the large Peierls–Nabarro energy barrier required to move these bipolarons through the lattice in the adiabatic limit.

We conjecture that high-Tc superconductivity originates specifically from the exceptionally well-balanced competition between electron–phonon coupling and electron–electron repulsion. In the restricted region in which, within the mean field approach, the energy of a bipolaron is close to those of two polarons, a new type of electron pairing occurs by formation of pairs of polarons in the spin singlet state. Such a polaron pair, called a spin resonant bipolaron (SRB), is not the standard bipolaron (both could exist in the adiabatic case). Its Peierls–Nabarro energy barrier can be sharply depressed, almost to zero. As a result of quantum lattice fluctuations, its tunnelling energy is sharply enhanced. Then, superconductivity could persist for a relatively large electron–phonon coupling, with an unusually large critical temperature as a Bose condensate of SRBs before becoming, at larger coupling and in any case, an insulating spin–Peierls polaronic phase.

Abstract

The near infrared and optical absorption of several high-Tc superconductors is investigated using the KBr powder method. The spectra show broad NIR absorption peaks. Examples of absorption spectra obtained using transmission data of thin film work for Ba1−xKxBiO3 (BKB) and single-crystal measurements for Bi2Sr2Ca1−xYxCu2O8+δ (BSCC) are given for comparison. They confirm the results of the KBr spectra. An example of similar investigations on NiO also shows good agreement over the whole spectral range under investigation. These facts provide some evidence that the line profiles deduced by the KBr technique may indicate a correct absorption line shape. A possible explanation of the NIR line profile is discussed.

Introduction

The absorption of various high- Tc systems measured using Kubelka-Munk [1, 2] or standard KBr powder techniques [3–7] is known to possess broad peaks in the near infrared spectral range. Similar features have earlier been discussed in connection with a small-polaron absorption mechanism in the non-stoichiometric compounds of systems like TiO2−x [8], WO3−x [9], NbO2.5−x [10,11] and the ternary compounds NbxW1−xOy [12]. For the superconductors of the system YBa2Cu3O7−δ, La2CuO4+δ and Bi2Sr2CaCu2O8+δ [3–5] it has been shown that the NIR absorption cross-sections increase approximately linearly with decreasing temperature down to the superconducting transition temperatures (Tc). Below Tc the slope decreases to smaller values. Dewing et al. [3, 4] have explained this effect as evidence for Bose–Einstein condensation of small bipolarons for YBa2Cu3O7.

Abstract

The small polaron has proved useful in understanding the transport properties of such low-mobility solids as oxides, glasses, and amorphous semiconductors. Polarons and bipolarons are of interest in high-Tc superconductors. I will first briefly review the basic mechanism for the Hall effect found in the localized regime where transport is due to multi-phonon-assisted transitions between localized small polaron states. The temperature-dependence of the Hall mobility will be reviewed for the non-adiabatic, adiabatic and three- and four site cases. I will then indicate how the magnetic phase factors in the localized regime give the conventional magnetic Lorentz force in a description of polaron band motion or of purely electronic bands of narrow width. This narrow-band regime is more relevant to the normal state of high- Tc materials in which carrier motion is itinerant. I will then survey experimental evidence for the Hall effect due to small polarons and in the narrow-band regime for several materials and conclude with an example of the Hall effect in the normal state of the cuprate superconductors taken from David Emin.

The basic mechanism of the Hall effect in the localized regime

The model used is a straightforward two-dimensional generalization of the molecular crystal model of Holstein [1]. (This case admits only a small-polaron and free-particle solution and no large-polaron solution.) Briefly, the model consists of a site occupied by diatomic molecules with fixed centres of gravity and orientation but variable internuclear separation so that each acts like an Einstein oscillator with fixed frequency, ω0. The oscillators are subject to weak coupling giving rise to dispersion of the vibrational frequencies.

Abstract

We describe the influence of local magnetization on electron localization in concentrated and diluted magnetic semiconductors. This includes a review of transport and optical evidence such as the magnetic-field-induced insulator metal transition in concentrated systems, i.e. Eu chalcogenides and Gd3-xvxS4 (where v = vacancy). It also includes a brief review of salient experimental evidence for polaron formation in the diluted magnetic semiconductor Cd1-xMnxTeiln. In addition, static and dynamic photo-induced magnetic measurements in ZnTe/Cd1-xMnxSe heterostructures are presented and their relevance to high-Tc materials discussed.

Introduction

This paper deals with the effects of local exchange due to interaction of carrier spins with the ionic spins in magnetic semiconductors. We are specifically concerned with the 4f shell of Gd3+ and Eu2+ (S) and the 3d shell of Mn2+ (S), which contribute to the magnetic character of the concentrated and diluted systems respectively. Towards this end, the body of the paper begins with a brief review of the relevant exchange mechanisms leading to polaron formation. In the concentrated systems, this leads to giant magneto-resistive effects in the Eu chalcogenides [1] and very distinctive luminescence [2]. It alsoleads to the spectacular magnetic-field-induced metal-insulator phase transition in the Gd3-xvxS4 compounds [3–5]. In the dilute system, polarons account for novel effects in spin-flip Raman scattering [6, 7] and low-temperature transport properties [8, 9], and for the best examples to date of photo-induced magnetism. The latter properties have been studied both by magneto-optical [10] and by very sensitive magnetic techniques [11, 12]. Furthermore, new femtosecond dynamic studies have given information about the formation and decay of the polarons themselves [10,12].

Abstract

There exists a well-established empirical trend, namely that the best superconductors are among the bad conductors, in which electrons are essentially localized on atomic orbitals. Just this electronic structure is requisite for the metal-insulator transition predicted by N. Mott. I show in this paper that the coexistence of these two remarkable phenomena within the same set of materials is not accidental.

Introduction

It was understood long ago that there exists some relationship between superconductivity and Bose–Einstein condensation. According to Schafroth et al. to get the Bose particles the electrons should be bound somehow into quasimolecules (pairs) [1], and ‘the only obstacle’ to achieving an explanation of superconductivity was the nature of these quasi-molecules. The original belief was that they should have an atomic size to maintain Bose statistics when their concentration is high (of the order of one per unit cell), and the problem of how to overcome Coulomb repulsion seemed insurmountable. However, this had nothing to do with local pairs in the case of the superconductors known in the middle of the 1950s. Being good metals, these superconductors above Tc = 1−10 K display almost-free-electron behaviour with the Fermi energy not less than 5 eV, and it was clear that the superconducting transition here concerns only a very thin shell around the Fermi surface. BCS theory [2] was addressed precisely to these superconductors. It was shown that, at T = 0, the normal state is unstable with respect to formation of Cooper pairs [3], when in the vicinity of the Fermi surface there exists an arbitrary weak attraction between electrons.

Abstract

The observed characteristics of mid infrared (MIR) spectra in doped semiconductors are discussed. These characteristics were explained by Reik and coworkers on the basis of hopping motion of small polarons from a localized site to a neighbouring localized site. The success and limitations of this model are pointed out. Emin, on the other hand, showed the importance of large polarons for the conductivity. The recently observed features of MIR spectra in high-Tc cuprates are then summarized. The low-frequency peak in many cuprates with frequency 0.1−0.2 eV has been ascribed by many investigators to polaronic origin. We have undertaken in the present work numerical studies of polaronic conductivity in the two-site and four-site cluster model by diagonalization of the dynamical matrix. Broadening of the phonon spectra due to damping has been taken into account by considering a small but finite phonon lifetime. For intermediate and strong coupling, a number of peaks in the optical conductivity appear due to bound states with different numbers of phonons. We have also studied the importance of Hubbard U by calculating the optical conductivity as a function of U with two electrons in a two-site model. The experimental results of MIR spectra for the cuprates can be better understood on the basis of the present calculations.

Introduction

It was Landau who first introduced the idea of polarons for explaining the F centres in NaCl as due to self-trapping of electrons [1]. Polarons are the quasiparticles formed by the accompanying self-consistent polarization field and are generated due to the dynamical electron–phonon interaction. As a consequence there is extra scattering of the charge carriers, phonon energies are renormalized and the charge carriers are heavy [2].

Abstract

A review is presented of a recent theory of strong two-band electron selftrapping in a semiconductor, for which hybridization of the related electron state with the band states is essential and gives rise to new features of both electron and atomic dynamics. Pressure-induced phenomena in such materials predicted in the theory are discussed.

Introduction

As demonstrated in many works (see, e.g. [1–9]), the type of self-trapping of quasi-particles, e.g., electrons or holes in semiconductors, depends on properties of the materials. Whatever the origin of self-trapping, in most papers [1–4, 9] generally important contributions come from states of a single energy band, e.g. of the conduction band for electrons. Hence, single-band self-trapping has largely been considered in most works, which holds true, insofar as the characteristic self-trapping energy WST (< 0) is substantially less in magnitude than the interband, or mobility, gap width E(0)g, WST|<<E(0)g. However, there are realistic semiconducting materials in which two-band self-trapping occurs, in the sense that contributions from states of both conduction and valence bands are important and hybridization of the states in the gap gives rise to new effects [5–8]. For instance, a single-band self-trapping energy for a single electron in a harmonic atomic lattice WST= W1≃ – W∂Q2d/(2ka20) may be comparable in magnitude to E(0)g/2, as self-trapping occurs at a soft ‘defect’ exhibiting a small effective atomic spring constant k<<k0, for typical values Qd≈3–5 eV, E(0)g ≈ 1–3 eV, and K0 ≈ 30–50 eV Å2 (hole self-trapping can be treated in a similar way, with trivial substitutions of conduction band states for valence band ones and vice versa).

Condensed matter physics by its very nature deals with systems with a large number of degrees of freedom, i.e. systems for which a statistical description is essential. In this chapter, we will present a rather comprehensive review of the fundamentals of thermodynamics and statistical mechanics. Much of this chapter, especially the parts dealing with homogeneous fluids (ideal and interacting gases and liquids), should be familiar to everyone. They are included here mostly to establish a basis for discussing more complicated ordered systems. As we saw in the preceding chapter, a great deal of useful and experimentally accessible information is contained in correlation functions such as the density-density correlation function Cnn(x,x′). This chapter will define these functions in a statistical mechanical context and develop a method of calculating them using functional differentiation with respect to spatially varying external fields. Functional differentiation will allow us to calculate position-dependent correlation and response functions by a simple generalization of the familiar technique used to calculate the magnetic susceptibility by differentiating the free energy with respect to the magnetic field. This is a very powerful tool that will be used throughout this book. It is presented here first in a familiar context that should make it easy to grasp.

After reviewing the properties of homogeneous fluids, we will introduce order parameters in Sec. 3.5 and show how they modify both thermodynamics and statistical mechanics.

Much of what we observe in nature is either time- or frequency-dependent. In this chapter, we will introduce language to describe time- and frequency-dependent phenomena in condensed matter systems near thermal equilibrium. We will focus on dynamic correlations and on linear response to time-dependent external fields that are described by time-dependent generalizations of correlation functions and susceptibilities introduced in Chapters 2 and 3. These functions, whose definitions are detailed in Sec. 7.1, contain information about the nature of dynamical modes. To understand how and why, we will consider linear response in damped harmonic oscillators in Sees. 7.2 and 7.3, and in systems whose dynamics are controlled by diffusion in Sec. 7.4. These examples show that complex poles in a complex, frequency-dependent response function determine the frequency and damping of system modes. Furthermore, the imaginary part of this response function is a measure of the rate of dissipation of energy of external forces.

A knowledge of phenomenological equations of motion in the presence of external forces is sufficient to determine dynamical response functions. The calculation of dynamical correlation functions in dissipative systems requires either a detailed treatment of many degrees of freedom or some phenomenological model for how thermal equilibrium is approached. In Sec. 7.5, we follow the latter approach and introduce Langevin theory, in which thermal equilibrium is maintained by interactions with random forces with well prescribed statistical properties. Frequency-dependent correlation functions for a diffusing particle and a damped harmonic oscillator are proportional to the imaginary part of a response function.

In the preceding several chapters, we have seen that the order established below a phase transition breaks the symmetry of the disordered phase. In many cases, the broken symmetry is continuous. For example, the vector order parameter m of the ferromagnetic phase breaks the continuous rotational symmetry of the paramagnetic phase, the tensor order parameter Qij of the nematic phase breaks the rotational symmetry of the isotropic fluid phase, and the set of complex order parameters ρG of the solid phase breaks the translational symmetry of the isotropic liquid. In these cases, there are an infinite number of equivalent ordered phases that can be transformed one into the other by changing a continuous variable θ. If rotational symmetry is broken, θ specifies the angle (or angles) giving the direction of the order parameter; if translational symmetry is broken, θ specifies the origin of a coordinate system. Uniform changes in θ do not change the free energy. Spatially non-uniform changes in θ, however, do. In the absence of evidence to the contrary, one expects the free energy density f to have an analytic expansion in gradients of θ. Thus we expect a term in f that is proportional to (∇θ)2 for θ varying slowly in space. We refer to this as the elastic free energy, fel, since it produces a restoring force against distortion, and we will refer to θ as an elastic or hydrodynamic variable.

In Chapter 9, we studied topological defects in ordered systems with a broken continuous symmetry. In this chapter, we will study fundamental defects in systems with discrete symmetry such as the Ising model. These defects are surfaces, of one dimension less than the dimension of space, that separate regions of equal free energy but with different values of the order parameter. They are variously called domain walls, kinks, solitons, discommensurations, or simply walls, depending on the particular system and context. They can also be regarded as interfaces, such as, for example, the interface separating coexisting liquid and gas phases. They play an important, if not dominant, role in determining the physical properties of systems with discrete symmetry.

We begin this chapter (Sec. 10.1) with a number of examples of walls. In Sec. 10.2, we study the continuum mean-field theory for kinks and solitons. Then, in Sec. 10.3, we discuss in some detail the Frenkel-Kontorowa model for atoms adsorbed on a periodic substrate. This will introduce in a natural way a lattice of interacting kinks (called discommensurations in this case) to describe the incommensurate phase of adsorbed monolayers. After investigating the properties of interacting kinks at zero temperature, we will in Sec. 10.4 study thermal fluctuations of walls in dimensions greater than one and show that structureless walls in three dimensions or less have divergent height fluctuations that render them macroscopically rough.

Mean-field theory, presented in the preceding chapter, correctly describes the qualitative features of most phase transitions and, in some cases, the quantitative features. Since mean-field theory replaces the actual configurations of the local variables (e.g. spins) by their average value, it neglects the effects of fluctuations about this mean. These fluctuations may or may not be important. The more spins that interact with a particular test spin, the more the test spin sees an effective average or mean field. If the test spin interacts with two neighbors, the averaging is minimal and the fluctuations are large and important. The number of spins producing the effective field increases with the range of the interaction and with the dimension. Thus we will find that mean-field theory is a good approximation in high dimensions but fails to provide a quantitatively correct description of second-order critical points in low dimensions. This chapter will be devoted to the study of second-order phase transitions when mean-field theory is not a good approximation.

We will begin by considering fluctuations of the order parameter about its spatially uniform mean-field value. We will find that for spatial dimensions below an upper critical dimension dc (typically dc = 4), fluctuations always become important and invalidate mean-field theory for temperatures sufficiently close to Tc. This will motivate a generalization of Landau theory that incorporates spatial fluctuations about a mean-field free energy minimum.

Thermodynamics provides a description of the equilibrium states of systems with many degrees of freedom. It focuses on a small number of macroscopic degrees of freedom, such as internal energy, temperature, number density, or magnetization, needed to characterize a homogeneous equilibrium state. In systems with a broken continuous symmetry, thermodynamics can be extended to include slowly varying elastic degrees of freedom and to provide descriptions of spatially nonuniform states produced by boundary conditions or external fields. Since the wavelengths of the elastic distortions are long compared to any microscopic length, the departure from ideal homogeneous equilibrium is small. In this chapter, we will develop equations governing dynamical disturbances in which the departure from ideal homogeneous equilibrium of each point in space is small at all times.

Conserved and broken-symmetry variables

Thermodynamic equilibrium is produced and maintained by collisions between particles or elementary excitations that occur at a characteristic time interval τ. In classical fluids, τ is of order 10−10 to 10−14 seconds. In low-temperature solids or in quantum liquids, τ can be quite large, diverging as some inverse power of the temperature T. The mean distance λ between collisions (mean free path) of particles or excitations is a characteristic velocity v times τ. In fluids, v is determined by the kinetic energy, v ~ (T/m)½, where m is a mass. In solids, v is typically a sound velocity. Imagine now a disturbance from the ideal equilibrium state that varies periodically in time and space with frequency ω and wave number q.

In Chapter 6, we studied the states of systems with broken continuous symmetry in which the slowly varying elastic variables described distortions from a spatially constant ground state configuration. These distortions arose from the imposition of boundary conditions, from external fields, or from thermal fluctuations. In this chapter, we will consider a class of defects, called topological defects, in systems with broken continuous symmetry. A topological defect is in general characterized by some core region (e.g., a point or a line) where order is destroyed and a far field region where an elastic variable changes slowly in space. Like an electric point charge, it has the property that its presence can be determined by measurements of an appropriate field on any surface enclosing its core. Topological defects have different names depending on the symmetry that is broken and the particular system in question. In superfluid helium and xy-models, they are called vortices; in periodic crystals, dislocations; and in nematic liquid crystals, disclinations.

Topological defects play an important role in determining properties of real materials. For example, they are responsible to a large degree for the mechanical properties of metals like steel. They are particularly important in two dimensions, where they play a pivotal role in the transition from low-temperature phases characterized by a non-vanishing rigidity to a high-temperature disordered phase.

This chapter begins (Sec. 9.1) with a discussion of how topological defects are characterized and a brief introduction to the concepts of homotopy theory.