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I was trying to understand the band theory of solids and came across this graph for a semi-conductor.

In between the areas between b and c, why does the graph display two curves? From the description that I've read it says that when the atoms of the semi-conductor lattice approach, the 3s and 3p energy levels show no difference. How is that possible and if it is what is the reason for the other curve in that region?

Also, it is said that between B and C, the number of energy gaps are no longer what it would normally dictate as I've learn in Chemistry. It says that if there were N atoms in the lattice, there would be 2N energy levels just corresponding to the 3s electrons though I don't know if they would even be called 3s electrons anymore. How is it possible for exactly 2 electrons to fill 2 of each of these levels and why do so many levels come into play simply because of atomic interaction?

Does the same graph apply to the impurity atoms of a doped semi-conductor? Because from what I've read they still have their own discrete energy levels.

Also, veering off topic a bit, why does the valence and conduction band of a conductor overlap? Doesn't that mean that at 0K there would be electrons in the conduction band? But even in a conductor like silver or copper isn't excitation required implying energy is required and hence no electrons would be present in the conduction band at 0K?

Also, I'm in grade 12 so I would appreciate it if you could keep the answers at a level I could understand. This is the first I'm learning of semi-conductors

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    $\begingroup$ "Veering off topic" looks like a completely different question. It should be addressed separately — maybe after you do get answer on this one. Also, "Band Theory of Solids" is not very descriptive of a title. It'd be better if you summarize your question in the title instead of just saying "This Particular Topic". $\endgroup$ – Ruslan Nov 11 '16 at 12:05
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In a lattice there already many atoms. But when they are far away from each other, or when they interact weakly, their energy levels are almost the same as those of free atoms. This is why you see "single" levels in the c-d region. They are not actually single there — just very close to each other.

When the interatomic spacing gets smaller, the levels of individual atoms split due to their interaction, and as there's a huge number of atoms, these levels create a very dense band of allowed energies.

As the lattice constant gets smaller, the splitting becomes more pronounced, so the allowed bands expand. Eventually they may overlap, which is what you see between conduction and valence bands in conductors. This corresponds to the right-hand half of a-b region in your picture.

Overlapping of two bands just means that an electron may go from one band to another without changing its energy. It may happen e.g. due to scattering on a phonon or some other interaction which can change quasimomentum without much affecting energy.

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  • $\begingroup$ What do you mean by lattice constant? $\endgroup$ – LeroyJD Nov 11 '16 at 14:06
  • $\begingroup$ @LeroyJD see wikipedia article about it. Basically it's distance between atoms in the lattice. $\endgroup$ – Ruslan Nov 11 '16 at 14:07
  • $\begingroup$ Also is there is a mathematical derivation to find all possible energy states belonging to this "dense band"? $\endgroup$ – LeroyJD Nov 11 '16 at 14:10
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    $\begingroup$ @LeroyJD for some mathematics see Nearly free electron model and links therein. $\endgroup$ – Ruslan Nov 11 '16 at 14:10

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