  There is an ever increasing interest in the development of plasmonic 2D
nanomaterials, with widespread applications in optoelectronics, high resolution
microscopy, imaging and sensing, among others. With the current ability of
ultrathin noble metal film deposition down to a few monolayers in thickness,
there is a need for an analytical expression of the thickness dependent complex
dielectric function for predicting optical properties for arbitrary
thicknesses. The free and bound electron contributions to the dielectric
function are dealt with independently, since their influences affect separate
wavelengths ranges. The former is dealt within the Drude model framework for
large wavelengths with appropriately addressed damping constant and plasma
frequency parameters to account for thickness dependence. Applying our
previously developed method, we determine these parameters for specific film
thicknesses, based on refractive index experimental values for Ag and Au thin
films. Fitting separately each one of these parameters allowed us to find an
analytical expression for their dependence on arbitrary film thickness and
consequently for the free electron contribution. Concerning bound electrons, it
is seen that its contribution for small wavelengths is the same for all
analyzed thicknesses and may be set equal to the bulk bound contribution.
Taking all these facts into account, the complex dielectric function can be
rewritten analytically, in terms of the bulk dielectric function plus
corrective film thickness dependent terms. In particular, the fitting process
for the damping constant allows us to determine that the electron scattering at
the film boundary is mainly diffusive (inelastic) for both silver and gold thin
films. It is also shown that, in accordance with theoretical studies, plasma
frequency shows a red shift as the film thickness decreases.
