# DGTD solver for non-local Drude dispersion model

We study an elementary nanophotonic setup that consists of two metal nanospheres. Positioning these two spheres as depicted in Figure 1 leads to a strongly coupled nanosphere dimer if the gap size undershoots the sphere’s radius. Such configurations are well known to show extreme field enhancements in the vicinity of the gap (the term gap is used for the area where the spheres are closest).

Figure 1. Dimer sphere setup and sketched of incident plane wave.

Combining length dimensions in a range less than 100 nm and electromagnetic wavelengths in the optical regime requires an appropriate material model for the metal, i.e. perfectly conducting (PEC) assumption as in the microwave regime is not valid anymore du to the relatively large skin-depth. Well-known dispersion models are e.g. local Drude and Drude-Lorentz dispersion models that have been comprehensively studied in the last decades. If geometric details of the structure under investigation approach length dimensions below approximately 25 nm (this value strongly depends on the material model and the actual geometry), spatial dispersion, i.e. the non-local response of the electron gas significantly increases its influence.

Our simulation results show a non-negligible blue shift in the scattering cross-section spectrum if non-locality is taken into account. For this simulation, we have used gold spheres with a radius of 20 nm and a gap size of 2 nm. The incident plane wave is parallel to the z-axis and its electric field is polarized parallel to the x-axis.

Figure 2 shows the resulting scattering cross-sections for the local and non-local dispersion model. A comparison of two numerical fluxes, i.e. centered fluxes and upwind fluxes is provided on this figure.

Figure 2. Logarithmic scattering cross-section of the sphere dimer (kz,Ex). Both material models show the same resonances. However, we observe a significant blue shift in the nonlocal case (DGTD-ce: centered flux DGTD solver - DGTD-up: upwind flux DGTD solver).

The simulations exploit third order spatial polynomials and an explicit fourth order low storage Runge-Kutta time discretization scheme. Perfectly matched layers mimic the infinite open space and truncate the tetrahedral mesh. A total field/scattered field formulation permits the evaluation of the scattering cross-section.

Figure 3. Tetrahedral mesh of the sphere dimer.

All simulations have been performed with a dedicated DGDT solver developed in the framework of the DIOGENeS software suite.