Nonlinear susceptibilities of granular matter

We discuss the nonlinear behavior of a random composite material in which current density and electric field are related by J={σ}E+a{}E²E, with {σ} 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 aₑ=〈a{}E⁴〉/E₀⁴, where E₀ 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 aₑ 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 {λ}aₑ=V({δ}σ_rms)², where V is the volume, {δ}σᵣₘₛ is the rms conductivity fluctuation, and {λ} is a constant with dimensions of energy. In the low-concentration regime (p{}1, where p is the concentration of nonlinear material), an expression for aₑ is derived which is exact to first order in p. The ratio aₑ/σₑ² (where σₑ 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 χₑ=pχᵢ{}3εₕ/ (εᵢ+2εₕ)²[3εₕ/({ε}/ᵢ+2εₕ)]² (plus terms of higher order in χᵢ), where p is the volume fraction of inclusion, εᵢ and εₕ are the dielectric constants of the nonlinear inclusion and of the host, and χᵢ 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.