  We investigate the frontier between classical and quantum plasmonics in
highly doped semiconductor layers. The choice of a semiconductor platform
instead of metals for our study permits an accurate description of the quantum
nature of the electrons constituting the plasmonic response, which is a crucial
requirement for quantum plasmonics. Our quantum model allows us to calculate
the collective plasmonic resonances from the electronic states determined by an
arbitrary one-dimensional potential. Our approach is corroborated with
experimental spectra, realized on a single quantum well, in which higher order
longitudinal plasmonic modes are present. We demonstrate that their energy
depends on the plasma energy, as it is also the case for metals, but also on
the size confinement of the constituent electrons. This work opens the way
towards the applicability of quantum engineering techniques for semiconductor
plasmonics.
