Energy in Brilloun Zone representation.

We learned about energy bands E or values for which has a solution and between these bands where it is not possible to find a value of E or are called forbidden gaps. Discrete number of such bands are separated by band gaps.

Now for

If we plot the allowed values of energy as a function of k, we obtain the E-k diagram for the one dimensional lattice.

Brillouin zone

When we plot the expanded E-k diagram of Periodic potential perturbation we notice the dissimilarity with the free space E-K diagram (given by the dotted line),

Free Particle Solution

How in particular can the periodic potential solution with an adjustable k approach the free particle solution with a fixed k in the limit where E > >U 0 ?

In this regard it must be remembered that the wave function for an electron in a crystal is the product of two and Q(x) where Q(x) is the wave function in the unit cell. Q(x) is also a function of k. Increasing or decreasing k by modifies both and Q(x) in such a way that the product of approach the free particle.

The k-value associated with given energy band is called a Brillouin Zone.

1 st Brillouin Zone

2 nd Brillouin Zone

One way of drawing is to k between in basically range. In the eigenvalue equation you notice that increasing or decreasing by has no effect on the allowed electron energy of E(k) is periodic with a period of .

Fig. 3.8

Therefore all the electron energies can be represented within , by changing the k values by , where n is an integer. This representation of the electron is called the reduced Brillouin zone representation as shown in Fig. 3.9 where the bands of energies are identified. As the number of electrons in the system increases the bands starts to be filled up from the lowest available energies. Normally most of the bands are completely full of electrons as allowed by Pauli's Exclusion Principle. At low temperature, there could be a band completely empty. The one below it is usually completely full, called the valence band. None of these electrons can now conduct electricity. If now a condition arises that some of the electrons from the completely filled band can be excited into the completely empty band, then current can be conducted by the electrons in the empty band called the conduction band.

Also according to the Bloch theorem there are two and only two k values associated with each allowed energy, one for the electron moving in the +ve direction and the other for the electron moving in the -ve direction.

Also note that at & k = 0. This is a property of all E-k plots.