Do the Raman or SERS spectrum of full-width half-maximum depend on, the concentration of adsorbed molecules (considering as self-assembled monolayer of adsorbate ie., about the 10-12M concentration)? or on the size of noble metal nanoparticles?
Everything in SERS depends on the kind of substrate you use, even the appearance or non-appearance of whole bands. Therefore the size of the nanostructures of the SERS material has a massive effect on everything.
When it comes to regular Raman, at least for my fullerene films back when I was doing that I did not see a dependence. However, for a SAM it would be plausible that isolated molecules scatter differently than when they are within the assembly. That may not be the actual linewidth, but you may have a component distribution which may have an envelope with a seemingly increased FWHM.
The full-width half-maximum (FWHM) of the Raman or SERS spectrum does not depend on the concentration of adsorbed molecules or the size of noble metal nanoparticles. The FWHM is determined by the spectral resolution of the instrument used to measure the spectrum. The concentration of adsorbed molecules and the size of noble metal nanoparticles can affect the intensity of the Raman or SERS signal, but not the FWHM.
In bulk Raman the peak width can be approximately viewed as measure of amorphousness (thicker peaks less crystallinity). In true SERS, such as using a self-organized monolayer, the peaks will always be very thin as they come from (relatively) small number of molecules in exactly same condition. However, in real life the surface coverage can be far from optimal and as you integrate over several (many) different orientations/bindings of the molecules w.r.t the surface, the peaks will also get broader. This is because each orientation feels slightly different force field, which produces slightly shifted peak. The sum of such shifted peaks (albeit thin) appears as a broader band (think about how amide bands come into being for conventional Raman spectra of proteins).