Nonlinear susceptibilities of granular matter

TLDR

Abstract

We discuss the nonlinear behavior of a random composite material in which current density and electric field are related by J=\ensuremath{\sigma}E+a\ensuremath{\Vert}E${\ensuremath{\Vert}}^{2}$E, with \ensuremath{\sigma} and a position dependent. To first order in the nonlinear coefficient a, the effective nonlinear conductivity of the composite material is shown to be expressible as ${a}_{e}$=〈a\ensuremath{\Vert}E${\ensuremath{\Vert}}^{4}$〉/${E}_{0}^{4}$, where ${E}_{0}$ is the magnitude of the applied field, the angular brackets denote a volume average, and E is the electric field in the linear limit (a=0). To the same order, the coefficient ${a}_{e}$ is also shown to be related to the mean-square conductivity fluctuation in an analogous problem in which the composite is linear but the conductivity fluctuates: The connection is \ensuremath{\lambda}${a}_{e}$=V(\ensuremath{\delta}${\ensuremath{\sigma}}_{\mathrm{rms}{)}^{2}}$, where V is the volume, \ensuremath{\delta}${\ensuremath{\sigma}}_{\mathrm{rms}}$ is the rms conductivity fluctuation, and \ensuremath{\lambda} is a constant with dimensions of energy. In the low-concentration regime (p\ensuremath{\ll}1, where p is the concentration of nonlinear material), an expression for ${a}_{e}$ is derived which is exact to first order in p. The ratio ${a}_{e}$/${\ensuremath{\sigma}}_{e}^{2}$ (where ${\ensuremath{\sigma}}_{e}$ is the conductivity of the composite) is shown to diverge near the percolation threshold for both a metal-insulator composite and a normal-metal--perfect-conductor composite; the power law characterizing the divergence is estimated. The results are generalized to nonlinear dielectric response at finite frequencies.At low concentrations, the cubic nonlinear dielectric susceptibility is found to be ${\mathrm{\ensuremath{\chi}}}_{\mathit{e}}$=p${\mathrm{\ensuremath{\chi}}}_{\mathit{i}}$\ensuremath{\Vert}3${\mathrm{\ensuremath{\epsilon}}}_{\mathit{h}}$/ (${\mathrm{\ensuremath{\epsilon}}}_{\mathit{i}}$+2${\mathrm{\ensuremath{\epsilon}}}_{\mathit{h}}$)${\mathrm{\ensuremath{\Vert}}}^{2}$[3${\mathrm{\ensuremath{\epsilon}}}_{\mathit{h}}$/(\ensuremath{\epsilon}${/}_{\mathit{i}}$+2${\mathrm{\ensuremath{\epsilon}}}_{\mathit{h}}$)${]}^{2}$ (plus terms of higher order in ${\ensuremath{\chi}}_{i}$), where p is the volume fraction of inclusion, ${\ensuremath{\epsilon}}_{i}$ and ${\ensuremath{\epsilon}}_{h}$ are the dielectric constants of the nonlinear inclusion and of the host, and ${\ensuremath{\chi}}_{i}$ is the nonlinear electric susceptibility of the inclusion. This expression becomes very large near a Maxwell-Garnett resonance, in analogy with similar local-field effects in surface-enhanced Raman scattering.