Do nano-metals become semiconductors?

[Sonderval]

Consider the Sommerfeld-model of a metal. We have a discrete but very large number of possible states, bounded by the Fermi energy.
Since the distance between the levels in a potential well scales as 1/L^2, for a very small specimen the number of states becomes small as well. Taking the electron density of copper, the Fermi radius should be about 2000 (i.e., the state with quantum numbers (2000,0,0) - or (2000,1,1), depending on your boundary conditions is at the edge) in a cube with a side length of 1 micron.
This means that the distance between the highest occupied and the lowest unoccupied level is about 7meV (The Fermi energy is 7eV, corresponding to level 2000, the next-highest corresponds to 2001, so the difference is 7eV (2001^2-2000^2)/2000^2=7meV.
7meV is of the order of the thermal energy at room temperature. So I would expect that at low temperature the gap between the highest occupied and the lowest unoccupied state cannot be easily crossed by thermal excitations, rendering the metal a semiconductor.

So here's my two simple questions:
1. Is this reasoning correct?
2. Has this ever been observed experimentally or are other effects obscuring this?

 

[mfb]

That is the idea of quantum dots.

 

[Sonderval]

@mfb
Thanks a lot, I wasn't aware of that simple connection.
Glad that my reasoning wasn't faulty.