Scaling behavior and surface-plasmon modes in metal-insulator composites

The ac dielectric response of metal-insulator composites is studied numerically, using the transfer-matrix algorithm of Derrida and Vannimenus. For two-dimensional random composites with site percolation, we verify numerically that the effective dielectric function can be written numerically in the form ${\mathrm{\epsilon }}_{\mathrm{e}}$/${\mathrm{\epsilon }}_1$=${\xi }^{\mathrm{-}\mathrm{t}/\nu }$ ${\mathrm{G}}_{\pm }$((${\mathrm{\epsilon }}_2$/${\mathrm{\epsilon }}_1$)${\xi }^{\mathrm{}(\mathrm{t}+\mathrm{s})/\nu }$ξ/L, where ${\epsilon }_1$ and ${\epsilon }_2$ are the dielectric functions, ξ is the correlation length, L is the system size (or wavelength of the electric field), $G_{+}$ and $G_{\mathrm{-}}$ are universal functions above and below percolation, and t, s, and ν are standard percolation exponents. A similar form has been previously verified for bond percolation by Bug et al. We also study surface-plasmon resonances in a two-dimensional lattice model of a composite of Drude metal and insulator. The effective conductivity of the composite in this case is found to consist of a Drude peak which disappears below the metal percolation threshold, plus a band of surface-plasmon states separated from zero frequency by a gap which appears to vanish near the percolation threshold. The results in this case agree qualitatively with effective-medium predictions. The potential relation of these results to experiment, and the possibility of a Lifshitz tail in the surface-plasmon density of states, are briefly discussed.