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If the band theory is complete right, or in other words, if we have indeed free electrons, then there cannot be exciton. We just fill the energy levels.

In other words, in a tight-binding model, we cannot get binding between an electron in the conduction band and a hole in the valence band.

Only in a realistic material, where the model is not a whole story, can we get the exciton, right? But what exactly is missed by the tight-binding model? This is not clear in textbooks.

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  • $\begingroup$ The exciton is a local phenomenon, not part of the band structure. I’m not quite sure what you mean by ‘failure’ here. $\endgroup$ – Jon Custer Dec 14 '17 at 5:17
  • $\begingroup$ I mean you cannot get it within the band theory framework. At least in many textbooks, concepts from the band theory are heavily used, like the effective mass. $\endgroup$ – poisson Dec 14 '17 at 5:28
  • $\begingroup$ Sure, but an exciton is not a Bloch function, so it does not exist in a band structure. $\endgroup$ – Jon Custer Dec 14 '17 at 6:29
  • $\begingroup$ Perhaps a bad sports analogy, but think of band theory constructing the field. Now, that field could be an American football field, a field hockey field, or even an ice hockey rink. Once the field is made (band structure, density of states, effective mass, ...), though, your find that there are now new possibilities. You can do curling on an ice hockey rink, but not on a football field. Those possibilities are not a requirement to get the field proper, but are instead enabled by it. To get excitons, you need a particular band structure and occupancy - they don't occur in metals, for example. $\endgroup$ – Jon Custer Dec 14 '17 at 19:19
  • $\begingroup$ Band theory is not valid for excited states and since an exciton is a manifestation of an excited state, it does not exist in band theory. $\endgroup$ – fgoudra Dec 15 '17 at 13:51
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Conventional band theory ignores many correlations between electrons. They would just move independently in a common potential. In aluminum for example, there would be on average three valence electrons in a Wigner-Seitz cell (the volume of one atom). But because electrons are supposed to move independently, there are large probabilities that there would be two or four electrons. And the chance that there would be zero valence electrons would not be negligible, in theory. In reality, this is nonsense because of the on-site Coulomb interaction $U$.

The Coulomb interaction $U$ is also the reason for Mott insulators, for example the insulators with an odd number of electrons per unit cell.

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