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At this point, I understand that at least one fundamental difference between conductors and semiconductors is that in conductors, there is typically no band gap because the valence band and conductance band overlap one another; in semiconductors, there is no such overlap meaning there is an energy hurdle to be overcome before an electron can break free from the valence band and move about in the conduction band.

What I still don't get is WHY is that the case? Is this just a simple fact of electron configuration such that I should be asking the phys-chemists at Chemistry Stackexchange about this? Is there something going on at the fundamental particle level that accounts for the differences in behaviors? Do we even know the "why" of this yet?

Also, this may be related: as you pare down the size of a sample of conductive material and get into the range of perhaps a couple dozen atoms, you achieve quantum confinement which means that you've created a band gap between valence and conductance bands. So what property of matter is it that accounts for this?

This is all a single question, I'm just trying to approach it from a few different angles to underscore my confusion. Thanks for your input!

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  • $\begingroup$ It depends on the energy of the Molecular orbitals formed due to overlapping of the atomic orbitals of different materials. This is generally observed as the band gap. $\endgroup$ – Lelouch Jul 7 '16 at 14:43
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    $\begingroup$ The valence and conduction bands do not (necessarily) overlap in conductors. Conductors conduct because they have a band that is only partially full. That means there can be a non-zero net electron momentum, as discussed in the answer to your pervious question. There is still a gap between the valence and conduction bands. $\endgroup$ – John Rennie Jul 7 '16 at 14:46
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    $\begingroup$ There can be a big gap, a small gap, or no gap. This pretty much covers the whole range of possible gaps. We then name any materials that fall in each group isolators, semiconductors, and conductors. A larger question would be if the whole range of gaps wasn't covered - that would be weird. $\endgroup$ – Steeven Jul 7 '16 at 14:49
  • $\begingroup$ And don't forget semi-metals, where the conduction band minimum is below the valence band maximum but offset in k-space. Then you get conduction in both bands, but not very well because of the limited range in k space for each. $\endgroup$ – Jon Custer Jul 7 '16 at 14:54
  • $\begingroup$ @John Rennie: when you say they have a partially full band, you are talking about the partially filled valence band, correct? In other words the non-full valence band can give up an electron into the conduction band and in so doing shift down into a full valence band which is energetically favorable. Is that shift to an energetically "favorable" configuration what undergirds the reason that conductors conduct while insulators don't? $\endgroup$ – 1John5vs7 Jul 7 '16 at 14:55
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If you just take the empty bandstructure, you will see that any periodic arrangement of atoms (conductors, semiconductors, insulators) features a set of allowed bands and forbidden regions, so called bandgaps. Fully occupied bands can not contribute to electrical current. There are no free places, where carriers could move. Only partially occupied levels allow current.

The difference between these groups of materials is, where the Fermi level lies.

In the case of a conductor, the Fermi level is in one of the bands, therefore at least one of the bands is partially filled and can carry current.

In the case of a semiconductor, the Fermi energy lies within the bandgap. In an ideal case of a defect and contamination-free (undoped) semiconductor, it would be in the center of the bandgap. Carriers can thermally be excited and lead to some intrinsic conductivity as there are some electrons in the conduction band, which naturally leave holes in the valence band. Both can carry current. The amount of free carriers can also be engineered through doping with impurities (e.g. P or B in Si, Ge or Si in GaAs).

Insulators are basically semiconductors, where the bandgap is so large that only a negligible amount of carriers is excited thermally.

As indicated by the comments to your question, there are many special cases with zero bandgap, negative bandgaps, overlapping bands, ...

What is the "origin" of these bands?

You start from Schrödinger's equation with a periodic potential and apply Bloch's theorem to solve with a periodic wavefunction. I found a good explanation at physicspages.com, but there are also several articles in wikipedia, which explain this. Not to forget, this is treated in about any solid-state physics textbook.

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