Crossover exponents in percolating nonlinear normal conductor–insulator mixtures

The critical behavior of nonlinear properties is studied in d-dimensional percolating weakly nonlinear normal conductor (N)–insulator (I) composites near the percolation threshold, in which the concentration p of normal conductor (N) obeys weakly nonlinear current density (J)-electric field (E) characteristics J=σ1E+χ1|E|βE and concentration 1−p of insulator (I) has σ2=0. Above the percolation threshold pc of the normal conductor, the effective response of the composites can be represented by 〈J〉=σe〈E〉+χe|〈E〉|β〈E〉, where 〈⋯〉 represents spatial averages, σe and χe are the effective linear and nonlinear response of the system. The magnitudes of the crossover field |Ec| and current density |Jc|, defined as the electric field and the current density at which the effective linear and nonlinear response become comparable, are found to have the power-law dependence |Ec|∼(p−pc)M(β) and |Jc|∼(p−pc)N(β) as the percolation threshold is approached from above. Here M(β) and N(β) are critical exponents and depend on dimensionality d and nonlinearity β in general. By means of effective medium approximation, “Nodes-Links-Blobs” picture and the relation to resistance fluctuation, respectively, we investigate the influence of nonlinearity β and dimensionality d on these crossover exponents in detail. Interesting crossover behavior is found in the two-dimensional system.