## ABSTRACT

The electron–phonon coupling constant λ is about the ratio of the electron–phonon interaction energy E_{p}
to the half-bandwidth D ∝ N(E
_{F})^{–1} (appendix A). We expect [11] that when the coupling is strong (λ > 1), all electrons in the Bloch band are ‘dressed’ by phonons because their kinetic energy (<D) is small compared with the potential energy due to a local lattice deformation, E_{p}
, caused by an electron. If phonon frequencies are very low, the local lattice deformation traps the electron. This self-trapping phenomenon was predicted by Landau [60]. It has been studied in greater detail by Pekar [61], Fröhlich [62], Feynman [63], Devreese [64] and other authors in the effective mass approximation, which leads to the so-called large polaron. The large polaron propagates through the lattice like a free electron but with the enhanced effective mass. In the strong-coupling regime (λ > 1), the finite bandwidth becomes important, so that the effective mass approximation cannot be applied. The electron is called a small polaron in this regime. The self-trapping is never ‘complete’, that is any polaron can tunnel through the lattice. Only in the extreme adiabatic limit, when the phonon frequencies tend to zero, is the self-trapping complete and the polaron motion no longer translationally continuous (section 4.2). The main features of the small polaron were understood by Tjablikov [65], Yamashita and Kurosava [66], Sewell [67], Holstein [68] and his school [69, 70], Lang and Firsov [71], Eagles [72] and others and described in several review papers and textbooks [13, 64, 73–76]. The exponential reduction of the bandwidth at large values of λ is one of those features (section 4.3). The small polaron bandwidth decreases with increasing temperature up to a crossover region from the coherent small polaron tunnelling to a thermally activated hopping. The crossover from the polaron Bloch states to the incoherent hopping takes place at temperatures T ≈ ω_{0}/2 or higher, where ω_{0} is the characteristic phonon frequency. In this chapter, we extend BCS theory to the strong-coupling regime (λ > 1) with the itinerant (Bloch) states of polarons and bipolarons.