General Approach to Study Geometric Effects on Classical & Quantum Fields & Eigenmodes of Particles with Finite & Infinite Extend for Applications in Plasmonics & Scanning Probe Microscopy

Abstract

This manuscript provides a general approach to the investigation of field quantization in high-curvature geometries. The models and calculations can help with understanding the elastic and inelastic scattering of photons and electrons in nanostructures and probe-like metallic domains. The results find important applications in high-resolution photonic and electronic modalities of scanning probe microscopy, nano-optics, plasmonics, and quantum sensing. Quasistatic formulation, leading to nonretarded quantities, is employed and justified on the basis of the nanoscale, here subwavelength, dimensions of the considered domains of interest. Within the quasistatic framework, the nanostructure material domains with frequency-dependent dielectric functions are presented. Quantities associated with the normal modes of the electronic systems, the nonretarded plasmon dispersion relations, eigenmodes, and fields are then calculated for several geometric entities of use in nanoscience and nanotechnology. From the classical energy of the charge density oscillations in the modeled nanoparticle, the Hamiltonian of the system, which is used for quantization, is derived. The quantized plasmon field is obtained and, employing an interaction Hamiltonian derived from the first-order perturbation theory within the hydrodynamic model of the electron gas, an analytical expression for the radiative decay rate of the plasmons could be obtained. The established treatment could be applied to multiple geometries to investigate the quantized charge density oscillations on their bounding surfaces.