Surface Plasmon Dispersion Relations in Chains of Metallic Nanoparticles: 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 ($kd \ll 1$, 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 $a \lesssim d/3$, 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 $ξ_D$ for energy propagation for the lowest two bands due to single-grain damping. For $a/d=0.33$, the result for $ξ_D$ is in qualitative agreement with experiments on gold nanoparticle chains, while for larger $a/d$, such as $a/d=0.45$, $v_g$ and $ξ_D$ are expected to be strongly $k$-dependent because of the multipole corrections. When $a/d \sim 1/2$, we predict novel 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.