Crossover electric field in percolating perfect-conductor–nonlinear-normal-metal composites

The nonlinear response is studied in a two-component composite with concentration p of perfect conductor (S) and concentration (1-p) of normal metal (N) with nonlinear response of the form J=${\mathrm{\sigma }}_{\mathit{A}}$ E+${\mathrm{\chi }}_{\mathit{A}}$‖E ${\mathrm{\Vert }}^2$ E. Below the percolation threshold ${\mathit{p}}_{\mathit{c}}$ of the perfect conductor, the response of the composite can be represented by 〈J〉=${\mathrm{\sigma }}_{\mathit{e}}$〈E〉+${\mathrm{\chi }}_{\mathit{e}}$‖〈E〉${\mathrm{\Vert }}^2$〈E〉, where 〈 . . .〉 represents spatial averages. The magnitude of the crossover field ‖${\mathbf{E}}_{\mathit{c}}$‖, defined as the electric field at which the linear and nonlinear response of the composite become comparable, is found to have a power-law dependence ‖${\mathbf{E}}_{\mathit{c}}$‖∼(${\mathit{p}}_{\mathit{c}}$-p${)}^{\mathit{M}}$ as the percolation threshold is approached from below. Within the effective-medium approximation, the exponent M is found to be 1/2 for all spatial dimensions. By relating the cubic nonlinear composite problem to the noise problem in linear composites, the exponent M is found to be M=(κ’+s)/2, where κ’ and s are the noise and conductivity exponents in a S/N composite, respectively. Previously derived bounds on the noise exponent κ’ thus lead to bounds on the exponent M.