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Suppose I have ZnS that's described via an fcc lattice with basis [$\frac{1}{4}, \frac{1}{4},\frac{1}{4}$], therefore we have 4 Zn and 4 S per unit cell. Zn has 2 valence electrons and S has 6 valence electrons, therefore we have (4$\cdot$2 + 4$\cdot$6) = 32 electrons per unit cell. We know that every 2 electron per unit cell will fill a band therefore we'll have 16 filled bands. Is this line of reasoning right or there's something that I've misunderstood?

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  • $\begingroup$ You would have 16 bands worth of electrons, but as one answer below indicates, this does not necessarily mean all 16 bands are filled completely (for an arbitrary material). But, in your case, ZnS is a semiconductor at STP, so yes you would have 16 filled bands for this material at STP. $\endgroup$
    – hft
    Commented Sep 22, 2023 at 15:52

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Unfortunately, there is a flaw in your argument. If correct, no metal would exist made by elements with an even number of valence electrons.

Instead, two or more bands may overlap, resulting in partly filled bands (and metallic behavior). Only in 1D, where it is impossible to have more than two states with the same energy, band overlap is impossible, and the argument works.

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