Crossover exponents in a superconductor-nonlinear-normal-conductor network below the percolation threshold

Abstract

The crossover behaviour in a d-dimensional percolating superconductor (S)-nonlinear-normal-conductor (N) random network is studied. The system is composed of a volume fraction f of superconductors, and fraction 1-f of nonlinear normal conductors with the current (i)-voltage (v) relation of the form i = g1v + 1v+1, where g1 and 1 are the linear and nonlinear response of normal conductors. As the percolating threshold fc of the superconductor is approached from below, the crossover electric field |L-NL| and corresponding current density |L - NL|, defined as the field and current density at which the linear and nonlinear response of the random network become comparable, are found to have power-law dependence |L - NL| ~ ( fc - f ) M(), |L - NL| ~ ( fc - f ) N() respectively. Within the effective medium approximation (EMA), critical exponents M() and N() are estimated to be ½ and -½ for all spatial dimensions d and arbitrary nonlinearity . By means of the multifractal approach, explicit expressions for M() and N() as a function of are obtained. For d = 2, we investigate the influence of nonlinearity on the crossover properties analytically and numerically; while for d = 3, we present such special values as M(2)0.74 and N(2) -0.01; M(4)0.79 and N(4)0.04; N ()0.88 and N ()0.13. Careful examination of exponents M() and N() gives interesting crossover behaviour. Numerical results are also compared with previous bounds and good agreement is found.