Size Dependence of Fermi Energy

SIZE DEPENDENCE OF FERMI ENERGY

In terms of the distribution of energy, solids have thick energy bands, whereas atoms have thin, discrete energy states.

It seems clear that to make a solid behave electronically more like an atom, we need to make it about the same size as an atom.

The electron density in a conductor at T = 0 K is

Rearranging the Eqn. (1), we get

Fermi energy of a conductor at T = 0 K

In the above equation 'n' is the only variable.

Eqn. (2) suggests that the Fermi energy of a conductor depends on the number of free electrons 'N' per unit volume 'V'.

Hint n = N/V

approximately

 E_F (0) ∝ N/V    

   ...(3)

Since electron density is the property of the material, the Fermi energy does not vary with material's size.

E_F is the same for a particle of copper as it is for a brick of copper.

Hence, we can say that the energy states will have the same range for small volume and large volume of atoms. But for small volume of atoms we get larger spacing between states.

This is not only for conductors, but also for semiconductors and insulautors too.

Let us consider that all states upto E_F(0) are occupied by a total of free electrons (N).

The average spacing between energy states is given by

 ΔΕ ≈ E_F(0) / N            ...(4)

From Eqn. (3) and Eqn.(4) we get.

 ∆E ∝ 1/V          ...(5)

Thus, the spacing between energy states is inversely proportional to the volume of the solid.

The energy sublevel and the spacing between energy states within it will depend on the number of atoms as shown in Fig. 3.1..

Fig. 3.1 The spacing between energy states gets larger as the volume gets smaller. At the bottom, a single atom has just one energy state per sublevel. As atoms group together to form particles, there are as many splits per sublevel as there are many atoms in the particle. If the particle is small enough, we can notice these splits within the sublevel. As the volume increases to the size of a solid, the spacing between splits gets so tight that the sublevel is best characterized as a band.

At one point, we know that an energy sublevel must be divided as many times as there are atoms in a solid, which eventually results too many splits to differentiate. Hence, we just refer to each sublevel as a solid energy band.

On the other hand, a single atom in the sublevel contain only one discrete energy state.

If we reduce the volume of a solid, the tiny piece of material behaves electronically like an artificial atom.