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In most introductions to solid state physics that I've read, they explain the origin, basically, of the band gap as follows: By doing a superposition of the standing waves inside a BZ, one can find two wave equations with different energy bands, with a possible gap between them (the band gap).

However, I have a difficulty relating this to the conduction/valence bands. Is the "lower energy" band always the valence band and the higher one always the conduction band? Or asked in a different way: How do we know that the electrons fill up to the top of the valence band at zero degrees K (which overlaps with the conduction band in metals)? There does not seem to be some inherent mathematical property in the theory which guarantees this?

I hope I am making sense, otherwise I will be happy to clarify my question. Thanks.

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  • $\begingroup$ There is also no guarantee that a particular element or compound or mixture (alloy) at particular thermodynamic conditions is a metal. Some are, some are not. Those who are can be explained with one configuration of electrons and insulators can be explained with another. The question whether one can actually predict which it will be from first principles is a lot harder to answer, but I think we are getting quite good at that, too (I might be wrong about that). $\endgroup$ – CuriousOne Feb 11 '16 at 16:49
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In insulators, the valence band and lower bands comprise just enough quantum states for the number of electrons in uncharged material. If there's no thermal energy to promote electrons into higher bands, the valence band will, at equilibrium, be full. And since it is an insulator, there's no electrons in the next higher (conduction) band.

If the material is a metal, some (but not all) the states in the conduction band are also occupied.

Semiconductors, at absolute zero temperature, turn to insulators.

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