Substrates

Interaction of plasmonic nanostructures with substrates

One of the problems with modelling the localized surface plasmon (LSP) resonances is taking account of the interactions with the substrate. In most experiments, the nanoparticles are fabricated on a substrate, often glass or quartz. This is necessary so that the resonances can be measured optically. However, it can be quite difficult including the effects of the substrate in the numerical models (an example is the Discrete Dipole Approximation numerical method.)

In my group we had been playing around with a simple mathematical description of LSPs in terms of an "electrostatic" model. In the situation where the nanoparticle is much smaller than the wavelength of light, the electric and magnetic fields decouple and, mathematically, the problem has the same form as in electrostatics. In effect, the time dependent terms cancel from both sides of the equation leaving only the space-dependent terms to solve for.

A standard problem in electrostatics is working out the electric fields due to a charge near a dielectric surface. This is done using the method of images where the interaction is modelled by an image of the charge in the dielectric.

We applied this same method to LSPs in nanoparticles on a substrate. The surface charge associated with the LSP has a mirror image in the substrate. But this mirror image looks just like another nanoparticle so we can use our method of coupling between nanoparticles to solve the problem [1]. We found that the interaction of a nanoparticle with a substrate can be modelled by a nanoparticle embedded in a medium with a particular electric permittivity. For a nanorod on glass, the effective permittivity is about 1.3. With our collaborators we compared the model with experimental data collected from chemically synthesized nanorods of various sizes. As can be seen from the graph, the model works very well. The blue dots are the experimental points, the red curve our model based on an effective medium and the dashed line our model used to create a permittivity correction for the resonance.

[1] K. C. Vernon, A. M. Funston, C. Novo, D. E. Gomez, P. Mulvaney, T. J. Davis: "Influence of Particle-Substrate Interaction on Localized Plasmon Resonances" Nano Letters 10, 2080-2086 (2010)

A comparison between calculated and experimental resonances for gold nanorods of different aspect ratio [1].