Energy bands in semiconductors

Question

I'm trying to understand the basic generalisations of how an atom works. I understand there are discrete energy levels or shells (in a simplified atomic model), in which only a specific number of electrons can occupy. The innermost shell (closest to the nucleus) is the lowest energy, and the outermost, the valence shell, has the highest energy.

I've read that, at absolute zero, only the valence shell is filled. I don't understand why this is the case and not the other way round. If you reduce the temperature of an atom down to 0K, then wouldn't that mean the electrons lose energy, instead of gaining it to hop away from the nucleus?

Also, in terms of energy bands, I'm confused as to why on band gap diagrams the valence band is shown as having the lower energy compared with the conduction band.

Finally, could someone please clarify in simple terms the difference between the quantum state of an electron and the energy level of a shell.

I know this is probably quite basic but any help would be appreciated!

Answer 1

I will try to solve your doubts in order:

• It is fundamental that you understand that electrons, as every other particle with spin 1/2, obey the Pauli exclusion principle, which states that two electrons cannot be in the same quantum state simultaneously. In order to understand things properly you need to study how the wave functions of the electrons are characterized by a variety of quantum number (spin, orbital, principal), but for the following you can just accept that you cannot have all the electrons of an atom in the lowest energetic state. They will occupy all the possible energy states in order, until the last electron has found a "place". The last shell that gets filled is the valence shell.

• Therefore, as you correctly point out, the electrons in the valence shell are those with the highest energy in the unexcited atom. However, there is no way they can reach a state with lower energy, since all of them are occupied by other electrons. Since these valence electrons are the outermost ones, they are less attracted by the nucleus and therefore are easier to "take away".

• When you talk about bands you refer to a material, and not about a single atom. However the intuitive picture is somewhat similar: there are electrons which belong to the valence band, which has the highest energy among the bands that are filled. These electrons are the ones that participate the most in the interactions that occur inside the material, since they are not so strictly bounded to the atom. At non-zero temperature some of them might get excited and jump to the conduction band, where they are even more free and contribute significantly to the electrical conductivity. But even if you decrease the temperature to $0$ Kelvin, they will not be able to decrease their energy more than the one they have in the valence band, since the lower energetic bands are already completely filled.

• In some pictures that you see only the valence and conduction bands are shown, because they are relevant to the physics of a solid. But below the valence bands there are a lot of other bands, that are completely filled by electrons which will hardly be removed from their place. This might be the source of your confusion.

• A single shell can contain electrons in multiple states. As an example, two electrons with opposite spin but otherwise same quantum numbers belong to the same shell.