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Atoms are in the range of $1$ Angstrom while Quantum dots are in the range of $2$-$10$ nm. In any atom, $99.9$% is unoccupied. So if I have a Quantum dots of size $3$ nm and suppose in my Quantum dot, I have $20$ atoms, however when I do the math, I find that electrons can never be confined $3$ dimensionally in that quantum dot.

So how can electrons be confined in Quantum dots?

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And where is the question? – Misha Feb 3 '13 at 6:07

In a metal or n-doped semiconductor the conduction electrons are not localised to any particular atom. Instead they are delocalised across the whole crystal. The same is true of a quantum dot. There are lots of ways of making quantum dots, but typically you can think of them as a very small piece of a semiconductor. Within the dot the electrons in the outermost shells are delocalised across the dot, just as they are in a macroscopic semiconductor, so the electron is only confined by the edges of the dot.

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My answer is not very different from John's and maybe a little tangential, but just offering my take. Electrons are confined in any system,e.g., a packet of liquid, solid etc. Just the fact that there is phase boundary or a system boundary that confines the electron (think electron in some potential well) the energy levels become discretized. Depending on the number of atoms in the phase/system the number of energy levels becomes huge and the energy-level gap very small. In crystalline systems the periodic potential allows bands (groups of energy level) with large gaps in between that has attractive implications. Anyway going back, the quantum dots is just an idea to shrink the size of the phase, therefore increasing the separation between energy levels, as though you were approaching a single atom (probably the best quantum dot). If your quantum dots were 20 atoms the levels will be more separated than if your dot had 100 atoms. Remember also that the elementary band theory is based on single non-interacting electrons. There are other effects (electron-electron, electron-phonon) that one might need to account for. But in any case this confinement is not special to systems that have an underlying crystalline periodicity. That just allows more interesting features.

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