Tight Binding Approximation

TIGHT BINDING APPROXIMATION

Before discussing about the tight binding approximation, Let us know about free electron approximation.

Free electron approximation

We know in solids, there exist the Ionic core which are tightly bounded to the lattice location, while the electrons are free to move here and there throughout the solid. This is called the free-electron approximation.

In free electron approximation the following points are observed, viz.

1. The potential energy of the electron is assumed to be lesser than its total energy.

2. The width of the forbidden bands (E_g) are smaller than the allowed bands as shown in fig 1.20(a).

3. Therefore, the interaction between the neighbouring atoms will be very strong.

4. As the atoms are closer to each other, the inter atomic distance decreases and hence the wave functions overlap with each other as shown in fig 1.20.

Tight binding approximation

Tight binding approximation is exactly an opposite approach of discussing the atomic arrangements, when compared to free electron approximation.

Here instead of beginning with the solid core, we begin with the electrons, i.e., All the electrons are bounded to the atoms. In other way we can say that the atoms are free, while the electrons are tightly bounded. Hence, this is called tight binding approximation.

The following points are observed in tight binding approximation, viz.

1. The potential energy of the electrons is nearly equal to the total energy.

2. The width of the forbidden bands (E_g)) are larger than the allowed bands, as shown in fig 1.20(b).

3. Therefore, the interactions between the neighbouring atoms will be weak.

4. As the atoms are not closer, the interatomic distance increases and hence the wave functions will not overlap, as shown in fig 1.20.

Explanation

Let us consider the atoms with larger inter atomic distance (a₂) as shown in fig 1.20. Here the atoms are far apart, and all the bounded electrons have fixed energy levels. Therefore when a solid is formed by using the same element, then the energy levels occupied by the electrons in each atom will be identical, which lead to tight binding approximation.

Now, when we bring the atoms closer to each other to form the solid, then inter atomic distance (a₁), decreases. Therefore, the outer shell electrons begin to overlap and the energy levels also splits as shown in fig 1.20.

If the inter atomic distance is further reduced, then the splitting of energy level happens for the inner shall electrons also, which lead to free electron approximation.