# Introduction

• Alexandre S. Alexandrov
• Jozef T. Devreese
Chapter
Part of the Springer Series in Solid-State Sciences book series (SSSOL, volume 159)

## Abstract

Charge carriers in inorganic and organic matter interact with ion vibrations. The corresponding electron–phonon interaction (EPI) dominates transport and other properties of many poor metals and semiconductors. EPI causes also phase transformations, including superconductivity. When EPI is sufficiently strong, electron Bloch states are affected even in the normal phase. Phonons are also affected by conduction electrons. In doped insulators, including the advanced materials discussed in this review, bare phonons are well defined in insulating parent compounds, but microscopic separation of electrons and phonons is not so straightforward in metals and heavily doped insulators [1], where the Born and Oppenheimer [2] and density functional [3,4] methods are used. Here, we have to start with the first-principle Hamiltonian describing conduction electrons and ions coupled by the Coulomb forces:
$$\begin{array}{rll}H = &&- \sum\limits_{i} \frac{\Delta^2_{i}}{2m_{{\rm e}}} + \frac{e^{2}}{2} \sum\limits_{i\neq i^{{\prime}}} \frac{1}{|{\rm r}_{i} - {\rm r}_{i}^{\prime}|} - Ze^{2} \sum\limits_{ij} \frac{1}{|{\rm r}_{i} - {\rm R}_{j}|}\\ &&+ \frac{Z^{2}e^{2}}{2} \sum\limits_{j\neq j^{\prime}} \frac{1}{|{\rm R}_{j} - {\rm R}_{j^{\prime}}|} - \sum\limits_{j} \frac{\Delta^{2}_{j}}{2M},\end{array}$$
(1.1)
where $${\rm r}_{i}, {\rm R}_{j}$$ are the electron and ion coordinates, respectively, $$i = 1,\ldots,N_{{\rm e}}; j = 1,\ldots,N; \Delta_i = \partial/\partial{\rm r}_{i}, \Delta_{j} = \partial/\partial{\rm R}_{j}, Ze$$ is the ion charge, and M is the ion mass. The system is neutral, $$N_{{\rm e}} = ZN$$. The inner electrons are strongly coupled to the nuclei and follow their motion, so the ions can be considered as rigid charges. To account for their “high-energy” electron degrees of freedom, we can replace the elementary charge in (1.1) by $$e/\sqrt{\epsilon}$$, where $$\epsilon$$ is the high-frequency dielectric constant, or introduce an electron–ion “pseudopotential” [5] instead of the bare Coulomb electron–nuclear interaction.

## Keywords

Local Density Approximation Dope Semiconductor Phonon Branch Canonical Linear Transformation Site Representation
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