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.