In this pdf it says
Because the number of electron states in a Brillouin zone is twice the number of primitive cells in a Bravais lattice (the factor two comes from spin), the zone can be completely occupied only if there are an odd number electrons per primitive cell in the lattice. Thus, all materials, with an odd number of electrons in a primitive cell, are metals. [...]
I have often read the result that (in a simplified model) materials with an odd number of electrons per primitive cells do conduct while those with an even number do not. This is, however, the first time I've read an explanation. And I don't understand it. Why does it take an odd number of electrons to fill an even number of states, and why does this determine the conductivity?
Alternatively, if anyone can provide another explanation, I'll be glad to hear it.