9.A.1 Energy Levels in a Semiconductor

A semiconductor can be considered as a collection of negative charges (electrons) and of positive charges (holes) free to move inside a crystal. Because of Quantum Mechanics, the energy of the electrons and holes cannot take any value and should be equal to some allowed values. Figure 9.A.1 shows the diagram of the energy levels that are allowed for electrons and holes in a semiconductor: energy levels gather in energy bands.

Figure 9.A.1: Diagram of the energy levels in a semiconductor. The energy is plotted vertically. ? _C(E) and ? _V(E) are the densities of the allowed states of energy. The Fermi function indicates which levels are occupied; E _F is the energy of the Fermi level.

The upper band is called the conduction band and corresponds to electrons; the lower band is called the valence band and corresponds to holes. The repartition of the electrons and holes among the different levels is fixed by the temperature and follows the Fermi-Dirac statistic; it is described by a Fermi function, see Figure 9.A.1. This is to be compared with the case of atoms and molecules that followed a Maxwell-Boltzmann distribution, see formula (9.20) in Section 9.2.2.1.

We now intend to show that, if a piece of sample of semiconductor is placed inside a blackbody, the three basic Einstein processes are necessary to ensure that:

• The photons are distributed according to a Bose-Einstein distribution.

• The electrons and holes are distributed according to...