Determination of the dynamic behavior of free electrons in the crystal lattice
Quantitatively, the dynamical behavior of free electrons in the crystal lattice is determined by a wave function satisfying the Schrödinger equation which, given the characteristics of the potential created by the lattice atoms, is very complex (W. Pauli, 1927; A. Sommerfeld, 1928; F. Bloch, 1928). Therefore, simplified models of the potential are generally introduced.
If a crystal lattice is considered one-dimensional, the solution of Schrödinger’s equation shows that electrons can have only energies of value within well-defined intervals, called allowed bands, separated by forbidden energy bands, called precisely forbidden bands. In a representation in which the electron energy is expressed as a function of the wave number, that is the inverse of the wavelength associated with the electron, these bands are called Brillouin zones.
Similar results are obtained for a three-dimensional crystal lattice if you consider the wave number as a vector quantity. Within each band, electrons can only take on a certain number of discrete energy values, each of which constitutes an energy level. Given a band with N levels, this, for the Pauli exclusion principle, can contain 2N electrons.
In complete bands, i.e. containing 2N electrons, the average velocity of these electrons is zero and therefore they do not contribute to the current circulating in the body; in bands occupied only partially the absence of an electron is equivalent to the presence of an identical charge, but of opposite sign, i.e. positive, called hole.
The distribution of electrons in a band is regulated by Fermi-Dirac statistic and in it is fundamental the Fermi level, which corresponds to the maximum value of energy that can be reached by electrons at a temperature of 0 K.