Semiclassical theory for plasmons in spatially inhomogeneous media

Recent progress in experimental techniques has made the quantum regime plasmonics accessible. Since plasmons correspond to collective electron excitations, electron–electron interaction plays an essential role their theoretical description. Within Random Phase Approximation, this is incorporated through a system of equations motion, which be solved self-consistently. For homogeneous media, analytical solution can found using Fourier transform, giving rise Lindhard theory. When medium spatially inhomogeneous, no longer possible and one often uses numerical approaches, are however limited smaller systems. In paper, we present novel semi-analytical approach for bulk inhomogeneous media based on semiclassical (or WKB) approximation, applicable when charge density varies smoothly. By solving motion self-consistently, obtain expressions theory with varying Fermi wavevector. The derivation involves passing from operators symbols, thought as classical observables phase space. way effective (Hamiltonian) plasmons. We then find quantized energy levels plasmon spectrum Einstein–Brilllouin–Keller quantization. Our results provide basis describe different setups plasmonics, such nanoparticles, dots waveguides.