If the particle size decreases to the dimensions of the Bohr radius, a good model of the exciton behavior is the particle model in the potential box. The solution to this problem is the mathematical relationship between energy states and the dimension of space. This formula shows that reducing the dimensions of the available space increases the energy of states. Thus, the energy gap increases. It is described in such a way that the valence level decreases and the conductivity level increases. But is it always like that? Maybe one of them is e.g. the constant? What exactly happens with valence and conductivity band when there is a quantum size effect?
Łukasz Jarosiński
Your first point is physically impossible: 'If the particle size decreases to the dimensions of the Bohr radius'
Conference Paper First-principle study of quantum confinement effect on small...
https://pubs.rsc.org/en/content/articlelanding/2018/cp/c8cp01403e#!divAbstract
If you get down to such a size that your are close to or below the Bohr radius, the excitons will squish together in quantum confinement effect. And indeed, the valence band will decrease and conduction band will increase in the squeeze and result in an increased band gap. This explains why smaller quantum dots emit shorter wavelengths of light than larger quantum dots, as smaller dots are in greater confinement and need more energy to overcome the greater band gap. Since the 'squeezing' occurs due to narrowing of the band width, both bands will change rather than only one band. Take a look at the highly recommended answer here: https://www.researchgate.net/post/Why_does_the_band_gap_increase_when_decreasing_the_size_of_nanostructures
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