Briefly, polaron is the quasi-particle (i.e. a particle-like elementary excitation) of a system of electrons coupled with the lattice vibrations, or phonons. Considering for simplicity a system of independent electrons, in an ideal rigid periodic lattice electrons move unperturbed through the lattice in Bloch states. When the lattice is allowed to vibrate, with each atom oscillating around its average position, at any given instance the lattice is no longer in its ideal periodic form, resulting in the scattering of electrons from the Bloch states corresponding to the ideal rigid lattice. In turn, the movement of electrons through the lattice polarises the lattice, which like an assembly of electrons is polarisable. Taking account of this mutual interaction (electrons on phonons and phonons on electrons), the elementary excitations of the system are no longer describable in terms of a mere redistribution of electrons from the occupied Bloch states in the ground state into the unoccupied Bloch states, both corresponding to the rigid lattice.

The original paper by LD Landau with the English title Electron Motion in Crystal Lattices [1] is an example of simplicity. You may wish to consult this paper. You may also wish to consult the pedagogical paper by Landau and Pekar [2], on the effective mass of a polaron. Electron-phonon theory in general and the polaron theory in particular have both been dealt with in Chapter 7 of the book by Mahan [3].

[1] D Ter Haar, translator and editor, Collected Papers of L.D. Landau (Gordon & Breach, New York, 1965), pp. 67 and 68. (Originally published in 1933.)

[2] LD Landau, and SI Pekar, Effective Mass of a Polaron, reprinted in Ukr. J. Phys. 53 [Special Issue], 71-74 (2008). (Originally published in 1948.)

[3] GD Mahan, Many-Particle Physics, 3rd edition (Kluwer, New York, 2000).

   Historically, the polaron wa introduced to describe the polar interaction between longitudinal (optical) phonons ( producing a self-induced polarization on the charges) and electrons (or holes) moving in a periodic lattice (Landau. Pekar and Fröhlich). But nowadays, the concept of polaron has been extended to systems where fermions interact with a bath of bosons: Jahn-Teller polaron, piezoelectric polaron,spin polaron,small polaron, bipolaron or many-polaron systems.

   Broadly spaking, the polaron is a quasiparticle formed phonon and electron (or hole) characterized by having an effective mass very different than the effective mass associated to the band, a sef-energy or binding energy and by its response to the magnetic or the electric fields. The first Hamiltonian to describe the was due to Fröhlich (1954) and it corresponds to a "large" polaron because it extends several times the lattice constant.The polarons confined at a length around the constant lattice are known as "small" polarons (Holstain,..), which have very different behaviour than the large polarons. Small polarons obliges the charges to do hopping transport and to have a thermally activated mobility (Arrhenius type). Both polarons can coexist depending of the material structure.

1. A.S. Alexandrov and N.F. Mott, Reports on Progress in Physics 57, 1197 (1994)

2.J.T. Devreese, Polarons in Ionic Crystals and Polar semiconductors, Amsterdarm, North Holland (1972)

see this

https://www.fkf.mpg.de/4434975/Polarons-in-complex-materials

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