Polarons and Bipolarons

The following sections are included:

• Dedication
• Preface
• Contents

The concept of polarons was first introduced by Landau (1933). If an electron is placed into the conduction band of an ionic crystal the force on another electron at a distance r from it would be e²/ϵ₀r². But if the ions did not move the force would be e²/ϵr², where ϵ₀ and ϵ are the static and high-frequency dielectric constants, respectively. Thus an electron is acted upon by a potential energy

The following sections are included:

• Strong-coupling large polaron
• Effective mass of large strong-coupling polaron
• Fröhlich large polaron
• Intermediate-coupling large polaron
• Small polaron: two-site Holstein model
• Nonadiabatic small polaron
• Adiabatic small polaron
• From large to small polaron: numerical calculations

The following sections are included:

• Strong-coupling large bipolaron
• Intermediate-coupling large bipolaron
• Path integral approach to large bipolaron
• Small bipolaron
• Small bipolaron in perovskite structures
• Effect of the kinetic energy on small-bipolaron formation

The following sections are included:

• Electrons, phonons and the Fröhlich interaction
• Bare phonons and sound in a metal
• Effect of the Fröhlich interaction on electrons in a metal
• Broken gauge symmetry and the BCS ground state
• Bloch states in semiconductors: Effective mass approximation
• Tight-binding approximation

The following sections are included:

• Small-polaron instability within the Migdal approach
• Exact solution of the multi-polaron problem for λ → ∞
• Polaron band and self-energy
• Temperature collapse of the polaron band
• Phonons in a strong-coupling system
• Screening and polaronic plasmon
• Many-body polaronic effect in the phonon spectrum
• Polaron thermodynamics
• Polaron kinetics: hopping transport
• MIR conductivity of polarons
• Polaron kinetics: band tunneling transport
• Electron Green's function and ARPES of polaronic systems

At low temperatures and (or) at high densities the residual polaron-phonon H_p–ph and polaron-polaron H_p–p interactions lead to phase transformations of the polaronic Fermi-liquid. There are several major instabilities. The polaronic Fermi-liquid is unstable versus the bipolaron formation followed by the Bose-Einstein condensation if H_p–p is attractive at short distances. If the density of carriers is of the order of the atomic density the commensurate or incommensurate charge-density wave (CDW) develops competing with the superfluid ground state. If the Coulomb repulsion is strong the polaronic Fermi liquid undergoes the metal-insulator Mott transition. The ground state for the half-filled band is an antiferromagnetic insulator, which, if doped, has spin-lattice small (bi)polarons as the charge carriers.

As we have mentioned in Chapter 5 the onsite or intersite attractive energy of two small polarons is generally larger than the polaron bandwidth. At this condition real space pairs of polarons i.e. small bipolarons form if temperature is not very high, T < T**. Then the polaronic Fermi liquid transforms into a bipolaronic liquid. The latter is stable at low density of bipolarons because the effective bipolaron-bipolaron interaction is repulsive for all distances (see below). At large density the bipolaronic CDW state competes with the superfluid.

A charged Coulomb Bose-gas (CBG) is a fundamental reference system in many-particle physics with a superfluid phase transition. It has been studied by several authors and recently became of particular interest motivated by the bipolaron theory of high temperature superconductivity. As we have discussed in Chapter 6 the long-wave excitations of the bipolaronic liquid are those of charged bosons with the Coulomb repulsion between them. Schafroth (1955) demonstrated that an ideal gas of charged bosons exhibits the Meissner-Ochsenfeld effect (expulsion of a magnetic field) below the ideal Bose-gas condensation temperature. Later on the one-particle excitation spectrum at T = 0 was calculated by Foldy (1961), who worked at zero temperature using the Bogoliubov (1947) approach. The Bogoliubov method leads to the result that the ground state of the system has a negative correlation energy, whose magnitude increases with the density of bosons. Perhaps more interesting is the fact that the elementary excitations of the system have, for small momenta, energies characteristic of plasma oscillations which pass over smoothly for large momenta to the energies characteristic of single particle excitations. Further investigations have been carried out at or near T_c, the transition temperature for the gas. These works has been concerned with the critical exponents (Bishop (1974)) and the change in the transition temperature from that of the ideal gas (Bishop (1974), Fetter (1971)). The RPA dielectric response function and screening in CBG have been studied in the high-density limit (Hore and Frankel (1975, 1976), including a low-dimensional (2D) CBG (Hines and Frankel (1979), Gold (1991)). In this Chapter we verify superfluid properties of charged bosons with the Bogoliubov-de Genes type equations derived for CBG by Alexandrov and Beere (1995) and discuss superconducting as well as normal state CBG kinetics.

The following sections are included:

• Small versus large polarons
• Small-polaron transport in TiO₂ and NiO
• Mobile polarons and bipolarons in WO_3−x
• Small polarons in high-T_c oxides
  • High-T_c oxides are doped semiconductors
  • Low mobility
  • MIR conductivity of high-T_c oxides
• Small bipolarons in high-T_c oxides
  • λ-point in high-T_c oxides
  • Doping dependence of R_H, T_c and λ_H
  • NIR absorption in YBCO
  • Upper critical field of high-T_c oxides
• Polaronic ARPES in high-T_c oxides and fullerene

The following sections are included:

• Conclusion
• References
• Subject index