Fermi energy in conduction band while valence band and conduction band have no overlap

I often see band diagrams of solids and in cases of semiconductors, I often see a band gap separating the valence band and the conduction band, while the Fermi energy lies in the conduction band as shown below.

I understand that the doping can shift the position of the Fermi energy.

If you have the Fermi energy lying in the conduction band while the valence band and conduction band have no overlap, is this a really bad conductor, but a conductor nonetheless?

*Diagram from (R. Dombrowski et al Phys. Rev. B 59, 8043)

• Note that the spectrum is of a quantum dot, while the indicated bands and band gap are those of the bulk, as noted in the figure caption. Jan 7, 2020 at 18:05
• When the Fermi level is located in the conduction band, it will be just a metal! The partially filled conduction band allows the electrons to be free carriers. Jan 7, 2020 at 19:36
• @Simon Since the Fermi energy is the highest energy level occupied at 0 kelvin, if the Fermi energy is in the conduction band, you have the electrons in the energy level equal to the Fermi level, free to move around as charge carriers, regardless of the fact that there is no overlap between the valence band and conduction band, it is a metal. Is this correct? Jan 10, 2020 at 15:15

So to answer the question whether we need overlap of valence and conduction band. Well, when $$E_F$$ is located in the above energy band (in your figure), we will say that the valence band becomes to some extent the part below your $$E_F$$ (so no longer the one from the figure) and the part above is the conduction band to some extent, and in this way there is an overlap. But don't pin yourself too much on these notions. Keep in mind the physical intuition from above.