This highest filled electron energy is known as Fermi energy.

Substituting typical values of parameters corresponding to electrons in metals in the above equation yields E_F values in the range of ~1.5 to 15 eV or E_F/ k_B = ~20 × 10³ K to 100 × 10³ K, which is very much higher than room temperature. The velocity of electrons at the Fermi surface is Although the Fermi surface at T = 0 is the ground state of the electron system, the electrons present are enormously energetic.

A function which describes the number of electron states in a particular energy range is very useful. To arrive at this expression let us consider eqn. (6.4) with a general value for k rather than k_F. Such an expression gives n(k), the number of states per unit volume of r -space with wavevector less than | k |. For free electrons, with energy , one can then define a density of states g (E ), where g (E ) dE is the number of electrons per unit volume of r -space with energies between E and E + dE :

(6.7)

Since only the electrons within ~ k_B T are capable of taking part in thermal processes, only the density of electron states (DOS) at the Fermi energy E_F will be of importance. Taking natural logarithm of the expression for E _F [eqn.(6.6)] gives

(6.8)

Differentiating the above expression yields,

(6.9)

Rearranging and using eqn.(6.7) gives

(6.10)

which is the electronic DOS at E_F.

References:

[1]. D. Jiles, Introduction to the electronic properties of materials, 2e, Nelson Thomas Ltd., Cheltenham, UK, 2001.

[2] J. Singleton, Band theory an electronic properties of solids, Oxford Univ. Press, New York, 2001.

[3] C. Kittel, Introduction to solid state physics, Wiley, New York, 1996.