Q. Define Fermi energy. Write down the expression for Fermi Dirac distribution law. Deduce an expression for Fermi energy of a system of free particles.
Sol. The Fermi Energy : it is important to determine how the electrons may distribute themselves in a band among the energies from zero to E. The number of energy levels in the energy rage dE available to a free particle enclosed in a box of volume V is given by each level can accommodate two electrons (one with spin up and one with spin down). Therefore if we refer to a unit volume, the total number of electrons per unit volume with energy between E and E + dE, is. The quality is shown in fig. for a given band the curve should stop at E. The number of electrons per unit volume that can be accommodated up to an energy E is given by
If the metal is in its ground state (which takes place at absolute zero), all electrons occupy the lowest probable energy levels in accordance with the exclusion principle. If the total number of electrons per unit volume is less than the total number of energy levels available in the band, the electrons will then occupy all energy states up to a maximum energy designated by E, and called the Fermi energy. If we put E = EF in equation (3), we must have n = n0. Therefore for the Fermi energy we obtain the value.
Thus the energy distribution of electrons in the metal, ground state corresponds to the shaded area in fig. when the Fermi energy is equal to the bandwidths the band is completely occupied. When a band is not completely filled, a small amount of energy is sufficient to excite the uppermost electrons to nearby energy levels.