Electronic structure of solids

To understand this article, the reader needs to be familiar with the principles of quantum mechanics, which can be found in the corresponding article. of this treatise. A basic knowledge of atomic physics and crystallography (article Geometrical crystallography in the present treatise), while not strictly indispensable, are nonetheless extremely useful.

A solid is a collection of positively charged nuclei and electrons. Its theoretical description, both in a state of equilibrium and under the influence of external perturbations, involves the use of quantum mechanics applied to the motions of both nuclei and electrons. But nuclei are much heavier than electrons, so their wave functions are more localized, as Born and Oppenheimer have shown. As a first approximation, it's enough to consider nuclei as fixed, i.e. as classical particles, and to describe electrons in the electrostatic potential they create. With a few important exceptions, such as superconductivity, or less important ones, such as the Jahn-Teller effect, this description is sufficient and will not be exceeded in this article.

It is logical, in the spirit of this approximation, to leave the positions of the nuclei free, then choose these positions to make the total energy minimal. In principle, this program is too ambitious, so for a long time we were content to calculate the total energy (when we did so) only for the observed positions of the nuclei or for neighboring positions, in order to deduce the restoring forces. Recently, we've sought to go even further, using the power of modern computing tools. But this extension still raises a number of difficulties, which will only be alluded to briefly here.

In most cases, we know, but can't really demonstrate, that the minimum energy state is crystalline. In the first part of this article, we look at electrons in a crystal lattice from the point of view of formal properties. The second part describes the main types of crystalline solids. The third part deals with the problem of the electronic structure of disordered systems, starting with the effect of an isolated point defect and moving on to what we call strong and weak localization. Finally, in a fourth section on recent developments, we mention, without too much emphasis, incommensurable structures, quasicrystals and charge density waves, the quantum Hall effect and its metrological importance, as well as a few phenomena where correlations between electrons play a fundamental role and have spectacular effects; these are also the properties of conducting polymers, the fractional quantum Hall effect and a theme that is still very controversial from a theoretical point of view, high critical temperature superconductors.