Scanning tunneling microscopy and spectroscopy have been employed to investigate the superconductivity in
single unit-cell FeSe nanoflakes on SrTiO3 substrates. We find that the differential conductance dI/dV spectra are
spatially nonuniform and fluctuate within the flakes as their area is reduced to below ∼150 nm2. An enhancement
in the superconductivity-related gap size as large as 25% is observed. The superconductivity behavior disappears
when the FeSe nanoflakes reduce to ∼40 nm2. Compared to a previous report [Wang et al., Chin. Phys. Lett.
29, 037402 (2012)], the gap is asymmetric relative to the Fermi energy EF. All the features, particularly the
fluctuating gap and quenched superconductivity, could be accounted for by quantum size effects. Our study helps
to understand nanoscale superconductivity in low-dimensional systems.