Interaction of molecules with localized surface plasmons in metallic nanoparticles

A theory is developed to model the interaction of molecules with the localized surface plasmon resonances in metallic nanoparticles that are used for single-molecule sensing. Each molecule is represented by a simple point-like dipole based on a dielectric sphere, taken in the limit of a small radius. The surface-charge and surface-dipole eigenfunctions of a small spherical particle are represented analytically and it is shown that these are natural extensions of the electrostatic coupling theory of Davis et al. [Phys. Rev. B 79, 155423 (2009)]. The effect of a molecule on the surface plasmon resonances is described in terms of an effective background permittivity and formulas for the frequency and phase shifts of the resonances are obtained that depend on the polarizability of the molecule, the eigenvalues associated with the nanoparticle resonances and the strength of the geometric coupling. The interaction of the point-like dipoles with surface charges of different distributions is studied and it is shown that for molecules that cannot approach closer to the nanoparticle than a fixed distance, there is an optimum dimension of the nanoparticle to obtain the maximum coupling. This is important for the optimum design of nanoparticle-based sensors. Analytical expressions for the coupling of molecules to nanoparticles are obtained for some simple geometries and the results are compared with numerical simulations.