Beyond the Maxwell Garnett approximation for interacting plasmonic nanoparticles: An analytical and numerical study

Abstract

We revisit the issue of building a precise mixing formula for the effective permittivity of interacting assemblies of plasmonic nanoparticles. More precisely, we reconsider the analytical expressions rendered by the Maxwell Garnett and Torquato et al. approximation formulas and compare them to each other and to a numerical approach based on the boundary element method applied to interacting assemblies of metallic (gold or silver) nanoparticles. For efficient numerical simulations of interacting assemblies of relatively large sizes, we set up an algorithm with adaptive surface meshing that depends on the particle’s position within the assembly. Next, we derive expressions for the resonance frequency of the assembly from the analytical formulas, which are valid for gold and silver particle assemblies embedded in matrices with large optical indices. We then compare the analytical results with our numerical findings. We find that the Maxwell Garnett approximation formula underestimates the resonance wavelength and that its validity range in terms of inclusion fraction strongly depends on the nature of the metal and the embedding matrix. In the case of silver particles embedded in high-permittivity matrices, the Maxwell Garnett formula should only be used for low particle concentrations. Torquato’s formula, on the other hand, which accounts for multipolar interactions and the assembly spatial arrangement, renders a better agreement with the numerical simulations.