Overlapping of Energy Bands in Metals

Question

As per Energy Band Diagram of metals, the conduction band and valence bands overlap, But if we assume there is overlapping then the total number of energy bands will reduce as hence electron will have less number of energy states to occupy, which will violate Pauli exclusion principle.

For Example, suppose, if there are N states in Valance band and M states in Conduction band and total M+N electrons. Assume 10% of overlapping of states, we will then have 0.9M+N states remaining, but since the total electron is still M+N even after overlapping, Electrons will have to occupy the same energy state i.e. more than 1 electron will be occupied in the same energy state, But this is not allowed by Pauli exclusion principle...

My question is do overlapping of energy band is allowed? If allowed then where I am Wrong?

Answer 1

The fact that two bands overlap does not mean that there are less states.

One electron can be in the valence band while the other is "overlapped". The fact that they are overlapped only means that both states have the same energy, but they are still different states.

The fact that they are overlapped only means that the electron can pass from one band to another without changing its energy. In other words, without an external energy supply.

Answer 2

I think the problem arises from the way you visualise bands.

Bands appears as lines and overlap means when the lines intercept each other. But only the states below the Fermi energy can be occupied (red lines here), and if you sum over all the occupied states it gives you total number of electrons.

Personally, I think the overlapping picture is a bit confusing. The correct definition of metal is when Fermi energy cross a band. For example, here is a metal band structure without overlap.