Electrons in the Periodic Potential of a Crystal

Abstract

The discussion of the properties of metals in the previous chapter was based on a free-electron model (or rather: a gas of neutral fermionic particles) in an empty box. The classical Drude model and the quantum mechanical Sommer-feld model (based on the Fermi-Dirac statistics) were introduced, and it was shown that a suitable choice of certain parameters leads to a good description of several properties of simple metals. We have to specify which electrons of the atoms remain bound in the ion core and which can be considered free (and hence can participate in conduction in the solid state). The number of conduction electrons is an important parameter even in the Drude model. In much the same way, in the Sommerfeld model various properties of metals are determined through the Fermi energy, by the number of conduction electrons per atom. Since core electrons are ignored, these models obviously cannot account for the electrical properties of ionically or covalently bonded materials, in which electrons are fairly well localized to the ions and covalent bonds. Therefore not even the quantum mechanical model can explain the existence of insulators and semiconductors. However, the conduction electrons are not perfectly free even in metals, since they move through the regular crystalline array of ions, thus their motion is determined by the periodic crystal potential. To resolve these difficulties and contradictions, the behavior of the electrons has to be studied in the presence of the atoms (ions) that make up the crystal lattice. As a first approximation, we shall consider ions to be fixed at the lattice points, and ignore their vibrations, the phonons. The justification of this approximation and the influence of the motion of ions on the electrons will be discussed in Chapter 23. In the present chapter we shall lump the effects of ions into a local static potential U ion (r) that can be taken as the sum of the individual atomic potentials v a (r − R m) of periodically spaced ions. Taking into account the influence of other electrons is much more difficult. Only by employing the methods of the many-body problem can electron-electron interactions be treated more or less precisely. We shall delve into this complex subject in Volume 3. Below we shall assume that electrons feel the