Differential Equations for Localized Plasmons in the Random Phase Approximation

Abstract

Differential equations are derived for localized bulk and surface plasmons in metal nanostructures in the random phase approximation (RPA) at the high frequency condition. A differential equation for the scalar potential in the RPA gives a position-dependent Drude-like dielectric function. Using the dielectric function, a differential equation for the vector potential is derived in the Lorentz gauge. A corrected RPA differential equation for the scalar potential is then derived by using the differential equation for the vector potential and the Lorentz condition. The vector potential contribution to the electric field in the Lorentz gauge is found negligible compared with the scalar potential one in metal nanostructures. This indicates that the scalar potential plays important roles in analyzing the localized plasmons in metal nanostructures as reported previously. The corrected RPA differential equations are found equivalent to the Maxwell equations using the position-dependent Drude-like dielectric function. [DOI: 10.1380/ejssnt.2017.103]