Dielectric function of spherical dome shells with quantum size effects

Metallic spherical dome shells have received much attention in recent years because they have proven to possess highly impressive optical properties. The expected distinctive changes occurring owing to quantum confinement of conduction electrons in these nanoparticles as their thickness is reduced, have not been properly investigated. Here we carry out a detailed analytical derivation of the quantum contributions by introducing linearly shifted Associated Legendre Polynomials, which form an approximate orthonormal eigenbasis for the single-electron Hamiltonian of a spherical dome shell. Our analytical results clearly show the contribution of different elements of a spherical dome shell to the effective dielectric function. More specifically, our results provide an accurate, quantitative correction for the dielectric function of metallic spherical dome shells with thickness below 10 nm.