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In the textbook examples, the band character (VB and CB) of semiconductors is explained by their location above (CB) or below (VB) the Fermi level. In metals however, the valence and conduction bands might overlap at the Fermi level. How can I distinguish these bands, or how can I attribute them the predicate 'valence' or 'conduction' if the aforementioned definition does not work anymore?
I would like to keep the whole discussion under the restriction of T = 0K.
A single atom has a discrete spectrum of bound states and the (true) continuum of free states. The discrete states are only filled up to a certain level, which marks the valence electrons of the atom. Taking a valence electron up one level requires a discrete energy.
If you now conceptually join together a lot of atoms into a crystal lattice, it turns out that the formerly discrete levels of the individual atoms group together to form clusters of levels that are close together, i.e. the single level X of the single atom becomes a group of levels that all together represent the former X in the crystal. If the crystal becomes bigger (the number of joined atoms gets larger), each of these groups is comprised of so many levels that are so close together energy-wise, that it doesn't really make sense to consider them discrete anymore. So one speaks of energy bands in that limit.
Due to the nature of the numerical approximation procedure while solving the many particle Schrödinger equation for the lattice, one is more or less able to relate energy bands in the crystal to discrete energy levels of the single atoms. If an energy band corresponds to valence electrons (i.e. just occupied states) in a single atom, it is considered the valence band. If the band relates to unoccupied states in the single atom, it is considered the conduction band. If there is still a discrete gap between valence and conduction band, the valence band is filled at T=0K (a matter of simple counting of states), so it takes finite energy to liberate electrons to the (quasi, because it is different from the true continuum of free electrons) continuum of the conduction band (electron gas).
But since each of the originally discrete states of the single atom widen to bands with a certain energy width when the lattice is composed, the possibility also exists that those bands overlap formally. Then it is called a metal. However, as I said, the correspondence between single levels and bands is a result of a specific (although very reasonable or useful) solution procedure, and therefore, artificial to some extent.
Hence, it is absolutely correct that you doubt the universality of the term "overlapping bands". In the end, there is only a higher state density where bands supposedly overlap, and you cannot practicably distinguish between a state in the "valence band", that is just unoccupied, and a state in the "conduction band" that is just unoccupied.
