Brillouin Zones

BRILLOUIN ZONES

Definition: Brillouin Zones are the boundaries that are marked by the values of propagation wave vector k in which the electrons can have allowed energy values without diffraction. Since k is a vector, it has different values along different directions. 

Explanation: The relation between the wave vector and the energy of the electron in a constant potential field can be got as follows.

We know, in a length of potential box ‘Ɩ' the energy of the electron in a constant potential field

............(1)

Also,we know wave vector............(2)

(or) ............(3)

Substituting equation (3) in (1), we get

............(4)

A plot is made between the total energy 'E' and the wave vector k, for various values of k with n = ±1, ±2,... (since k = n π /Ɩ)

(i.e) for n = ± 1k=± π/Ɩ

for n= ± 2k = ± 2π/Ɩ

for n= ±3k = ± 3π/Ɩ

For the above values of k, the curve is obtained as shown in Fig. 1.21, which is in the form of a parabola with discontinuities.

From the figure it can be seen that the energy of the electron increase continuously from zero to ± π/Ɩ then the electron meets the wall and is reflected.

This range of allowed energy values in the region between to is called First Brillouin Zone.

The second allowed energy values consists of two parts; one from π/l to 2π/Ɩ and another for-π/Ɩ to -2π/Ɩ is called Second Brillouin Zone.

Similarly there will be 3^rd, 4^th etc, Brillouin Zones for various ranges of k values.

Also from Fig. 1.21 we can see that each Brillouin Zone is separated by breakup of energy values known as Forbidden Zone (or) Energy Gap.

Conclusion

Therefore, we can conclude that the electron can go from one brillouin zone to the other only if it is supplied with an energy equal to forbidden gap energy. Thus the forbidden gap is the one which decides whether the solid is a conductor (or) insulator (or) semiconductor.