Plasmon-Plasmon Interaction in Nanoparticle Assemblies: Role of the Dipole-Quadrupole Coupling

TL;DR

The paper develops a minimal, tractable description of plasmon-plasmon interactions in a 1D chain of metallic nanoparticles by explicitly including both dipole ($\ell=1$) and quadrupole ($\ell=2$) modes. Using a truncated multipole Hamiltonian ($\ell_{\max}=2$) in the quasistatic limit, and a Bogoliubov transformation, the authors derive a semi-analytical dispersion for the low-energy bands and identify when quadrupoles critically modify the spectrum. They demonstrate that dipole-only models fail for dense lattices while the dipole+quadrupole model closely reproduces the full spectrum ($\ell_{\max}=20$) with errors below a few percent across the Brillouin zone, and that quadrupole contributions become dominant near the Brillouin zone center for small interparticle spacing. The work concludes that quadrupole effects are essential for understanding low-energy plasmonic bands in dense nanoparticle assemblies and sets up a framework for future coupling to photons and to other quantum systems.

Abstract

The synthesis of metallic nanoparticle assemblies is nowadays well-controlled, such that these systems offer the possibility of controlling light at a sub-wavelength scale, thanks, for instance, to surface plasmons. Determining the energy dispersion of plasmons likely to couple to light within these nanostructures is, therefore, a necessary preliminary task on the way to understanding both their photonic properties and their physical nature, namely the role of the quadrupole contribution. Starting with a general model that takes account of all energy modes, we show that its low-lying energy dispersion gained numerically, can be compared to that of a minimal model that treats dipoles and quadrupoles on the same footing. The main advantage of the latter relies on the fact that its formulation is tractable, such that a semi-analytical Bogoliubov transformation allows one to access the experimentally relevant energy bands. Based on this semi-analytical derivation, we determine quantitatively the limit of validity of the dipole-only model, the presently proposed dipole and quadrupole model, compared to a full-plasmon-mode Hamiltonian. The results show that the dispersion relation, which includes dipoles and quadrupoles, is sufficient to capture the low-energy physics at play in most experimental situations. Besides, we show that at small lattice spacing, the contribution of quadrupoles is dominant around the Brillouin zone center.

Figures (6)

• Figure 1: Schematic representation of a 1D chain of spherical particles of radius $R$ with center-to-center distance $d$. The chain runs along the $z$ axis.
• Figure 2: Terms appearing in the dispersion relation of the model with schematic descriptions of the surface charge densities associated to the $k=0$ and $k=\pi/d$ modes. The dotted blue line corresponds to interacting dipoles, while the orange solid line corresponds to a chain of interacting quadrupoles. As pointed out in the text, the bands cross for dense enough chains, here the values correspond to $d/R=2.4$ and $m=1$.
• Figure 3: Plasmonic band diagram of a chain of spherical metallic nanoparticles of radius $R=10$ nm with three different lattice spacing (a) $d=30\ {\rm nm}$, (b) $d=26\ {\rm nm}$ and (c) $d=24\ {\rm nm}$. Black lines are calculated with $\ell_{max}=20$, red dashed lines are the dipole model eigenvalues with $\ell_{max}=1$, and the blue crosses are the $\ell_{max}=2$ model taking into account quadrupoles. The orange solid lines in (c) correspond to $\ell_{max}=3$ for comparison. In (a), (b) and (c) the high energy bands calculated with $\ell_{max}=20$ are grayed out for readability. Insets show the corresponding chain densities.
• Figure 4: Relative energy differences $\Delta$ with regard to the converged model $\ell_{max}=20$ across the Brillouin zone for (a) $d/R=3$, (b) $d/R=2.6$ and (c) $d/R=2.4$ in the same order as in Fig.\ref{['fig:bands']} and for both $m=0$ and $m=\pm1$. The expression for $\Delta$ is $\Delta=\left|\left(\omega(k)-\omega_{\ell=20}(k)\right)/\omega_{\ell=20}(k)\right|$.
• Figure 5: (a) Components of the eigenfunctions associated with the band dispersion (\ref{['eqn:wk']}) for the $m=0$ band of a chain of spherical nanoparticles of radius $R=10$nm with three different lattice spacing (a) $d=30nm$, (b) $d=26$nm and (c) $d=24$nm. Red dashed lines are the dipole component $a_{\ell=1,m=0}(k)$, and the blue lines are the quadrupolar components $a_{\ell=2,m=0}(k)$