Plasmon–plasmon interaction in nanoparticle assemblies: role of the dipole–quadrupole coupling

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, in parts to surface plasmons. Determining the energy dispersion of plasmons likely to couple to light in these nanostructures is, therefore, a necessary preliminary task on the way to understanding both their photonic properties and their physical nature, in particular 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 both the dipole-only model and the presently proposed dipole and quadrupole model, compared to a full-plasmon-mode Hamiltonian. The results show that the dispersion relation, which accounts for dipoles and quadrupoles, is sufficient to capture the low-energy physics in most experimental situations. Besides, we show that at small lattice spacing, the contribution of quadrupoles is dominant around the Brillouin zone center.