Surface-plasmon dispersion relations in chains of metallic nanoparticles: An exact quasistatic calculation

Abstract

We calculate the surface-plasmon dispersion relations for a periodic chain of spherical metallic nanoparticles in an isotropic host, including all multipole modes, in a generalized tight-binding approach. For sufficiently small particles $(kd\ensuremath{\ll}1,$ where k is the wave vector and d is the interparticle separation), the calculation is exact. The lowest bands differ only slightly from previous point-dipole calculations provided the particle radius $a\ensuremath{\lesssim}d/3,$ but differ substantially at smaller separation. We also calculate the dispersion relations for many higher bands, and estimate the group velocity ${v}_{g}$ and the exponential decay length ${\ensuremath{\xi}}_{D}$ for energy propagation for the lowest two bands due to single-grain damping. For $a/d=0.33,$ the result for ${\ensuremath{\xi}}_{D}$ is in qualitative agreement with experiments on gold nanoparticle chains, while for smaller separation, such as $a/d=0.45,$ ${v}_{g}$ and ${\ensuremath{\xi}}_{D}$ are expected to be strongly k dependent because of the multipole corrections. When the particles touch, we predict percolation effects in the spectrum, and find surprising symmetry in the plasmon band structure. Finally, we reformulate the band-structure equations for a Drude metal in the time domain, and suggest how to include localized driving electric fields in the equations of motion.