The Fermi distribution

This is the ground state of the N electron system at absolute zero. What happens if the temperature is increased? The kinetic energy of the electron gas increases with temperature. Therefore, some energy levels become occupied which were vacant at zero temperature, and some levels become vacant which were occupied at absolute zero. The distribution of electrons among the levels is usually described by the distribution function, f(E), which is defined as the probability that the level E is occupied by an electron. Thus if the level is certainly empty, then, f(E) = 0, while if it is certainly full, then f(E) = 1. In general, f(E) has a value between zero and unity.

Fig. 2 (a) Occupation of energy levels according to the Pauli exclusionprinciple, (b) The distribution function f(E), at T = 0°K and T> 0°K.

It follows from the preceding discussion that the distribution function for electrons at T = 0°K has the form

Eq. 5

That is, all levels below EF are completely filled, and all those above EF are completely empty. This function is plotted in Fig. 2(b), which shows the discontinuity at the Fermi energy.

When the system is heated (T>0°K), thermal energy excites the electrons. However, all the electrons do not share this energy equally, as would be the case in the classical treatment, because the electrons lying well below the Fermi level EF cannot absorb energy. If they did so, they would move to a higher level, which would be already occupied, and hence the exclusion principle would be violated.

Recall in this context that the energy which an electron may absorb thermally is of the order kBT ( =0.025 eV at room temperature), which is much smaller than EF, this being of the order of 5 eV. Therefore only those electrons close to the Fermi level can be excited, because the levels above EF are empty, and hence when those electrons move to a higher level there is no violation of the exclusion principle. Thus only these electrons - which are a small fraction of the total number - are capable of being thermally excited.

The distribution function at non-zero temperature is given by the Fermi distribution function. The Fermi distribution function determines the probability that an orbital of energy E is occupied at thermal equilibrium

Eq. 6

This function is also plotted in Fig. 2(b), which shows that it is substantially the same as the distribution at T = 0°K, except very close to the Fermi level, where some of the electrons are excited from below EF to above it.

The quantity μ is called the chemical potential. The chemical potential can be determined in a way that the total number of electrons in the system is equal to N. At absolute zero μ= EF