Semiclassical Quantum Theory in Nanoscale Plasmon Excitation

We introduce several quantum theories of plasmon excitations in nanostructures and explain our semiclassical quantum theory that uses the classical electromagnetism and quantum theory in the random-phase approximation (RPA) and retarded RPA (r-RPA). Local electron densities at ground states in nanostructures play essential roles in our theory. Coupling terms between bulk and surface plasmon excitations appear for the scalar potential, which derive non-local dielectric functions, while local dielectric functions are derived for electric fields. Integral equations for the scalar and vector potentials in plasmon excitations are derived, which are transformed into Maxwell equations with induced charge and current terms. They are applied to investigate plasmon excitations in Na nanospheres where quantum oscillations of the electron density are formed. Multiple resonance peaks are produced by quantum oscillations, indicating the importance of quantum effects on nanoscale plasmon excitations.