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

Abstract

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 ${\mathrm{\ensuremath{\epsilon}}}_{\mathrm{e}}$/${\mathrm{\ensuremath{\epsilon}}}_{1}$=${\ensuremath{\xi}}^{\mathrm{\ensuremath{-}}\mathrm{t}/\ensuremath{\nu}}$${\mathrm{G}}_{\ifmmode\pm\else\textpm\fi{}}$((${\mathrm{\ensuremath{\epsilon}}}_{2}$/${\mathrm{\ensuremath{\epsilon}}}_{1}$)${\ensuremath{\xi}}^{(\mathrm{t}+\mathrm{s})/\ensuremath{\nu}}$\ensuremath{\xi}/L, where ${\ensuremath{\epsilon}}_{1}$ and ${\ensuremath{\epsilon}}_{2}$ are the dielectric functions, \ensuremath{\xi} is the correlation length, L is the system size (or wavelength of the electric field), ${G}_{+}$ and ${G}_{\mathrm{\ensuremath{-}}}$ are universal functions above and below percolation, and t, s, and \ensuremath{\nu} 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.