Plasmon tunability in metallodielectric metamaterials

The dielectric properties of metamaterials consisting of periodically arranged metallic nanoparticles of spherical shape are calculated by rigorously solving Maxwell’s equations. Effective dielectric functions are obtained by comparing the reflectivity of planar surfaces limiting these materials with Fresnel’s formulas for equivalent homogeneous media, showing mixing and splitting of individual-particle modes due to interparticle interaction. Detailed results for simple-cubic and fcc crystals of aluminum spheres in vacuum, silver spheres in vacuum, and silver spheres in a silicon matrix are presented. The filling fraction of the metal $f$ is shown to determine the position of the plasmon modes of these metamaterials. Significant deviations are observed with respect to Maxwell-Garnett effective-medium theory for large $f$, and multiple plasmons are predicted to exist in contrast to Maxwell-Garnett theory.

Figure 1

Plasmon chemistry in interacting nanoparticles. The nonretarded plasmon energies of two identical metallic particles are shown along with schematic representations of the corresponding oscillation modes. The particles have radius $R$, their center-to-center distance is $d$, and the metal is described by a Drude dielectric function of bulk-plasma frequency ${\omega }_p$ [Eq. (1)].

Figure 2

Total scattering cross section $\sigma$ of a system formed by two metallic spheres as a function of incident photon energy $\omega$ and center-to-center distance $d$ between two aluminum spheres described by the Drude dielectric function of Eq. (1) with ${\omega }_p=15\mathrm{eV}$ and $\eta =0.6\mathrm{eV}$. Two different polarization directions of the external electric field have been considered in (a) and (b), as shown in the insets. $\sigma$ is normalized to the projected area of the two spheres, $2\pi R^2$. The solid curves represent the modes given by the expressions of Fig. 1.

Figure 3

Contour plot of $\mathrm{Im}\{-1∕{ϵ}_{\mathrm{eff}}\}$ as a function of incident photon energy $\omega$ and filling fraction of the metal for aluminium spheres in simple-cubic (a) and fcc (b) configurations, surrounded by vacuum. Brighter regions correspond to higher values of $\mathrm{Im}\{-1∕{ϵ}_{\mathrm{eff}}\}$. The solid curves correspond to Maxwell-Garnett theory as given by Eq. (6).

Figure 4

Contour plot of $\mathrm{Im}\{-1∕{ϵ}_{\mathrm{eff}}\}$ as a function of incident photon energy $\omega$ and filling fraction of the metal for silver spheres in an fcc configuration, surrounded by (a) vacuum or (b) silicon. Brighter regions correspond to higher values of $\mathrm{Im}\{-1∕{ϵ}_{\mathrm{eff}}\}$. The solid curves correspond to Maxwell-Garnett theory as given by Eq. (6).

Figure 5

Real and imaginary parts of the effective dielectric function ${ϵ}_{\mathrm{eff}}$ for an fcc arrangement of Ag spheres surrounded by Si (solid and broken curves, respectively). Two different filling fractions of the metal have been considered: (a) $f=0.2$ and (b) $f=0.5$. The dielectric function has been divided by 50 in the low-$\omega$ region to improve readability.