  In the recent work of Ref.\cite{Vlaic2017-bs}, it has been shown that Pb
nanocrystals grown on the electron accumulation layer at the (110) surface of
InAs are in the regime of Coulomb blockade. This enabled the first scanning
tunneling spectroscopy study of the superconducting parity effect across the
Anderson limit. The nature of the tunnel barrier between the nanocrystals and
the substrate has been attributed to a quantum constriction of the electronic
wave-function at the interface due to the large Fermi wavelength of the
electron accumulation layer in InAs. In this manuscript, we detail and review
the arguments leading to this conclusion. Furthermore, we show that, thanks to
this highly clean tunnel barrier, this system is remarkably suited for the
study of discrete electronic levels induced by quantum confinement effects in
the Pb nanocrystals. We identified three distinct regimes of quantum
confinement. For the largest nanocrystals, quantum confinement effects appear
through the formation of quantum well states regularly organized in energy and
in space. For the smallest nanocrystals, only atomic-like electronic levels
separated by a large energy scale are observed. Finally, in the intermediate
size regime, discrete electronic levels associated to electronic wave-functions
with a random spatial structure are observed, as expected from Random Matrix
Theory.
