Consider an isolated atom. Its energy levels are discrete.

Now consider a metal. In this case, every energy level of the isolated atom is replaced by a band which is composed of very close discrete energy levels. So, because the energy levels of the isolated atom are discrete, in a metal we have bands with allowed discrete energy levels (and these energy levels are very close one to another, as I've said) separated by bands with not allowed energy levels.

My question. Are these bands with not allowed energy levels small as the difference between the allowed discrete energy levels within the band with allowed discrete energy levels? So can we consider that the electronic band of a metal is a single broad electronic band? Or the difference is rather large, and in the textbooks they only show us the band where the Fermi level is (without showing us the bands that are below the Fermi level band)?

I attached a picture. In the first figure, we see that the difference is rather large. But in the second figure, I see that the electrons occupy everything from energy 0 to the Fermi energy level (so there are no not allowed energy levels like the first figure shows) So, what is happening?

 

A metal is no different than any other kind of atom for the electron orbitals close to the nucleus. The outermost electron is the one which forms an energy band, because it can easily be pulled away from one nucleus and come under the influence of a neighboring nucleus. Valence electrons in Silicon and Germanium behave in a similar fashion except it is a bit harder to get them to move. Using III-V impurities also creates "holes" in the crystal lattice which are places that an electron can migrate to, leaving a "hole" behind.

 

I understand this.

But I am asking about the contradiction between the two figures.

In the first figure, I see that the electrons do not occupy all the energy levels from energy 0 to Fermi energy level. There are visible forbidden gaps.
In the second figure, I see that the electrons occupy all the energy levels from energy 0 to Fermi energy level. So no forbidden band gaps.

Is it a poor drawing?

 

I understand this.

But I am asking about the contradiction between the two figures.

In the first figure, I see that the electrons do not occupy all the energy levels from energy 0 to Fermi energy level. There are visible forbidden gaps.
In the second figure, I see that the electrons occupy all the energy levels from energy 0 to Fermi energy level. So no forbidden band gaps.

Is it a poor drawing?

I think it might be a poor drawing, or we are not interpreting it correctly. In a semiconductor the band gap is small, on the order of .72 to 1.1 eV, and some fraction of the electrons can tunnel through it. In metals the valence band and the conduction band overlap. In an insulator the gap is much wider.

Even in a metal there can be energy gaps below the valence band, but I don't see how there can be gaps above the conduction band.

http://hyperphysics.phy-astr.gsu.edu/hbase/solids/band.html

 

I think you are right. Thanks for the answer.
I think the 0 on the axis is the 0 for the probability, and not the energy.

Maybe in the figure, we see only the valence and conduction band. We don't see everything. And at 0K, the valence band is fully occupied like the figure shows.

I mistaken the 0 to belong to the energy axis.

 

A value of 0 for the energy would be tantamount to the end of the universe. There is however a concept known as the "ground state", which is an atom in which all the electrons occupy the lowest possible energy states.