Crossover exponents in percolating superconductor endash nonlinear-conductor mixtures

The nonlinear response is studied in a two-component composite with concentration p of superconductor (S) and concentration 1-p of normal conductor (N) with nonlinear response of the form J=σ1E+χ1Eβ (β approx-gt 1). Below the percolation threshold pc of the superconductor, the response of the composite can be represented by left-angle J right-angle=σeE+χeEβ, where left-angle...right-angle represents spatial averages. The magnitude of the crossover field Ec, 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 Ec∼(pc-p)M(β), and the corresponding crossover current Ic is found to have similar dependence Ec∼(pc-p)W(β) as the percolation threshold is approached from below. By using a connection between the nonlinear response of the random nonlinear composite problem and the conductance fluctuation of the corresponding random linear composite problem the explicit expressions for M(β) and W(β) are calculated. We prove that both M(β) and W(β) are monotonically decreasing functions of β, which have the special values M(1+)=W(1+)=+∞, M(3)=[κ'(2)+s/2], W(3)=[κ'(2)-s/2], M(+∞)=(2-d)ν/2≤0, W(+∞)=-(s+ζG)/2<0. The magnitude of the crossover field Ec and the corresponding crossover current Ic are discussed as functions of β and some interesting crossover effects are reported. copyright 1996 The American Physical Society