Does the (intrinsic) Fermi level of an insulator HAVE to lie very near the middle of the band gap? I know it might deviate slightly if electrons and holes have different effective masses, e.g. in Si. But can it be radically different? For example, are there insulators with 5 eV band gaps that have the Fermi level 4 eV above the valence band (so 1 eV from the conduction band)? Or is it a very general result that the Fermi level ends up very close to the middle?