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I often read in introductions about quantum dots that depending on the fabrication method, a single dot contains about 100 - 100000 atoms.

Assuming a self-assembled dot of lens or pyramid (cone) shape, I did some rough estimation by considering the volume (or the area) of a dot divided by the volume (or area of a cell) of a single atom, taken to be $10^{-3}$$m^{-3}$. The estimation falls within the range of $100-100000$, which is not surprising given how wide this range is.

Is there a typical and vigorous method that people use to estimate the number of atoms in a single dot?

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The most accurate way to assess this number is by counting the relevant lattice sites with TEM;

TEM of SAQDs

(from Chamard et al. Phys. Rev. B 69 (2004) 125327.)

But it is ambiguous which sites at the edge of the dot to include, so the number of atoms is not well defined, at least in these self assembled dots.

One could perhaps ask the question - is the number of atoms the important thing? The key thing that makes a quantum dot is the existence of a strong confining potential. In a gate defined quantum dot in, say, a GaAs/AlGaAs heterostructure (as located at the red dots in the SEM below), the confining potential is (very roughly) parabolic, so where is the edge of the dot?

AlGaAs gate defined double dot

(From LMU-Munich)

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Is there a typical and vigorous method that people use to estimate the number of atoms in a single dot?

I doubt it. It is a matter of definition. And quantum dots dont appear to be defined any more strictly. Our free online encyclopedia provides the same numbers:

These quantum dots can contain as few as 100 to 100,000 atoms within the quantum dot volume, with a diameter of 10 to 50 atoms. This corresponds to about 2 to 10 nanometers, and at 10 nm in diameter, nearly 3 million quantum dots could be lined up end to end and fit within the width of a human thumb.

https://en.wikipedia.org/wiki/Quantum_dot

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