Is there only one electron for each $k$ value allowed in the electronic band structure?

Question

When we look at the electron band structure of a material we see the energy as a function of the wavevector k. But what I don't understand is if there is more than one electron in the same band with the same $k$ value, since according to pauli principle electrons are not allowed to have the same quantum number and $k$ is a well-defined number here.

If this is true i am not able to imagine two electrons in a material,one in one edge and the other one in the opposite edge, being force to have different wavevectors. Or maybe it is a consequence of the fact that electrons in the solid are delocalized??

Answer 1

You're allowed to have two electrons in the same band (one spin up, one spin down)$^*$. You're not allowed to have any more than that! The Pauli principle says that each state cannot be double occupied.

The reason your intuition breaks down is because each of these states is completely delocalized. If you want to have a definite k-vector, your wavefunction has to be spread out over the whole system. So there's no way you can say an electron is "on one edge of the material with wavevector k". It either has wavevector k or is on one edge of the material, but not both!

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$^*$I am assuming the bands are not spin-polarized, in which case each band only holds on electron, but there are twice as many bands, so the counting still works out.