Theory of Localized Plasmons for Multiple Metal Nanostructures in the Random Phase Approximation

Abstract

A theory is derived for localized plasmons in multiple metal nanostructures by developing the theory of localized bulk and surface plasmons for metal nanostructures in the random phase approximation (RPA) at the high frequency condition. The local electron density in multiple metal nanostructures is expressed as the sum of the electron density of each metal nanostructure. Self-consistent integral equations derived in the RPA give determinants to calculate the localized surface plasmon frequencies for multiple metal nanospheres with step-function-like electron density at their surfaces. The frequencies are analytically calculated in the dipole approximation for a dimer and chain of metal nanospheres. The frequencies are red-or blue-shifted depending on the spacing between the metal nanospheres. A light emission formula is also derived for multiple metal nanostructures in the dipole approximation. The light emission intensities from the dimer and chain are analytically calculated using the step-function-like electron density model. The retardation effect on the localized plasmons for multiple metal nanostructures is then investigated by applying the structural Green's function method, which is used to calculate the electronic structures of condensed matter.

A Quantum Chemistry Approach Based on the Analogy with π-System in Polymers for a Rapid Estimation of the Resonance Wavelength of Nanoparticle Systems

“…This results in a notable change of the system’s resonance energy. Several experiments and models have been performed in order to quantify this phenomenon and, thus, determine the resonance wavelength of the NPs system [11,12]. Although various compositions of multi-NPs have been investigated [13,14], due to the mathematical approach complexity and the difficult experimental realization of the controlled structure at the nanoscale level, most studies have focused on the dimeric system of NPs.…”

“…Although various compositions of multi-NPs have been investigated [13,14], due to the mathematical approach complexity and the difficult experimental realization of the controlled structure at the nanoscale level, most studies have focused on the dimeric system of NPs. In this frame, the nature and the dependence on the particle size, as well as on interparticle distances, in the dimeric system, has been largely investigated and numerous articles have been published [11,15] outlining a full description of NPs dimer both from theoretical and experimental point of views. The classical approach for studying plasmon effects is based on the determination of the permittivity function.…”

“…They enable us to obtain information on more complex NPs systems without having to resort to time consuming calculations. Apart from self-consistent methodologies [11] for multiple metal NPs where the random phase approximation in the secular determinants is also used to estimate the surface plasmon resonant energies that, in turn, depend on the nanoparticle spacing, this approach requires the energy shift of simple systems as initial input data, such as dimers and single NPs on the surface that has already been reported in literature [15,20]. These input data of the energy, which contains the dependence on the nanoparticle distance and surrounding medium, enable us to obtain information on more complex NPs systems without having to resort to time-consuming calculations.…”

Theory of localized plasmons for multiple metal nanostructures in dielectrics

“…[22][23][24][25] The author has also developed several theories of localized plasmons using the random phase approximation (RPA) at high frequency conditions. [26][27][28][29] In these theories, the local electron density in the metal nanostructures plays an essential role in the plasmon excitation, and the coupling between the localized bulk and surface plasmons is properly considered. These theories have been applied to calculate the inelastic scattering cross sections of incident electrons by the plasmon excitations, 26) the light emission intensity from a single metal nanostructure by considering the retardation of the scalar potentials for localized plasmons 27) and to investigate localized plasmon excitations for multiple metal nanostructures.…”

“…These theories have been applied to calculate the inelastic scattering cross sections of incident electrons by the plasmon excitations, 26) the light emission intensity from a single metal nanostructure by considering the retardation of the scalar potentials for localized plasmons 27) and to investigate localized plasmon excitations for multiple metal nanostructures. 28) A differential equation formula for the localized plasmons has been also derived, where the validity of the RPA formula has been investigated, leading to the corrected RPA formula where the vector potential contribution has been considered in the electric field. 29) These theories, however, are limited to metal nanostructures in vacuum or air where the dielectric polarizations at the metal surfaces can be ignored.…”