This isn't a duplicate of What does Fermi level in the band gap mean?. I read it and I'm still confused, and moreover I'm not allowed to add another comment to it anyway because I don't have enough reputation.
- the Fermi level $E_F$ is the chemical potential at 0 Kelvin.
- chemical potential is the change in energy per particle added
- the zero of the potential is at the top of the valence band
If I have a full valence band and I add another electron, I need at least the energy of the band-gap to promote it into the conduction band. So surely, that must mean $E_F$ can't be smaller than the band-gap.
How then, is the Fermi level ever inside the band-gap? I feel as though I'm missing something incredibly obvious here.