Electrons and Phonons

Abstract

The aim of this Chapter is not to produce a complete course on electrons and phonons, but rather to give a concise outline of a certain number of their basic properties in order to be able to describe transport phenomena in crystalline solids. Indeed, transport phenomena will provide the guiding motivation.

Transport phenomena in dilute media were discussed with the help of the Boltzmann equation in the Chapter on Transport in Dilute Media by Carminati in this volume, in the context of the kinetic approach. The idea is to establish an expression for the particle flux through a given area. We then deduce the flux of any quantity transported by each particle. The Boltzmann equation is used to determine the velocity distribution of the particles. It can be solved fairly straightforwardly if the system is close to equilibrium. We thus introduce the idea of local thermodynamic equilibrium (LTE). As we have seen, this notion can only be defined for length and time scales greater than the mean free path and the average time between consecutive collisions, respectively.

In this Chapter, we explain how to transpose this kinetic approach¹ to the case of electrons and phonons. The first step is to define the fluxes. The second is to obtain the counterpart of the velocity distribution function. A difficulty arises because we can no longer apply classical mechanics. The system is described using the wave functions of quantum mechanics. We must first revise the notion of flux using the language of waves. It is no longer useful to introduce the particle aspect when expressing the fluxes in this context. The next step is to find the counterpart of the velocity distribution function. The velocity is not an observable for an electron in a crystal. What plays the role of the velocity distribution function is the average occupation number of a state. At equilibrium, this is given by the Fermi–Dirac distribution. We are still in the framework of the wave description, since the stationary states are described by wave functions. However, it is the Boltzmann equation which provides a way of determining the correction required to take into account an imbalance due to the application of a temperature or potential gradient. This is done by returning to a particle view of electrons or phonons. The problem will thus be to see how to revert to a so-called semi-classical approach in terms of particles in order to describe these objects.