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

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)

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    $\begingroup$ 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. $\endgroup$ – Jon Custer Jan 7 at 18:05
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    $\begingroup$ 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. $\endgroup$ – Simon Jan 7 at 19:36
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    $\begingroup$ @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? $\endgroup$ – stratofortress Jan 10 at 15:15
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I think it is better to really think physically about how energy bandstructures determine the electronic transport.

First, let's answer the question which electrons participate in the electronic transport of the material? Well, transport requires some empty available energy levels throughout the material in which electrons can move "freely". The electrons that can be excited into these empty states are the highest energy states, i.e. the electrons closest to the Fermi level. So indeed, the electrons close to the Fermi level determine the electronic transport properties.

Secondly, we must ask the question where the Fermi level is located. If it is located in an energy gap, than the electrons will be located in the valence band, the closest energy band below the Fermi level. Electrons can be excited and overcome the energy gap. Depending on the size of the energy gap we speak of insulators or semiconductors. However, if you Fermi level lies inside an energy band such that there are sufficient available (non-occupied) energy states above the Fermi level which lie still inside the energy band. Well, in that case, electrons can be easily excited into these available states where they are considered as being "free carriers". So in this case, we speak about a metal.

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 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.

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