Percolation effects in two-component nonlinear composites: Crossover from linear to nonlinear behavior.

Abstract

The nonlinear response of a two-component composite is studied. The first component is assumed to be nonlinear and obeys a current-voltage (I-V) characteristic of the form I=${\mathrm{g}}_{1}$V+${\mathrm{\ensuremath{\chi}}}_{1}$${\mathrm{V}}^{3}$, while the second component is linear with I=${\mathrm{g}}_{2}$V, where ${\mathrm{g}}_{1}$ and ${\mathrm{g}}_{2}$ are linear conductances of the constituents and ${\mathrm{\ensuremath{\chi}}}_{1}$ is the nonlinear susceptibility. The volume fractions of the two components are p and q, respectively, and p+q=1. Near the percolation threshold, we identify two important limits: (i) the conductor-insulator (N/I) limit in which ${\mathrm{g}}_{2}$=0 and (ii) the superconductor-conductor (S/N) limit in which ${\mathrm{g}}_{2}$=\ensuremath{\infty}. For the S/N case and below the percolation threshold (q${\mathrm{q}}_{\mathrm{c}}$), the crossover voltage ${\mathrm{V}}_{\mathrm{c}}$, defined as the voltage at which the linear and nonlinear response become comparable, is found to have a power-law dependence ${\mathrm{V}}_{\mathrm{c}}$\ensuremath{\approx}(${\mathrm{q}}_{\mathrm{c}}$-q${)}^{\mathrm{v}\ensuremath{'}}$ as the percolation threshold is approached from below. For the N/I limit and above the percolation threshold (p>${\mathrm{p}}_{\mathrm{c}}$), the crossover current ${\mathrm{I}}_{\mathrm{c}}$ is found to have a similar dependence ${\mathrm{I}}_{\mathrm{c}}$\ensuremath{\approx}(p-${\mathrm{p}}_{\mathrm{c}}$${)}^{\mathrm{w}}$ as the percolation threshold is approached from above. By a connection between the nonlinear response of the random nonlinear composite problem and the relative conductance fluctuations of the corresponding random linear composite problem, the exponents w and v\ensuremath{'} are found to be w=(\ensuremath{\kappa}+t)/2 and v\ensuremath{'}=(\ensuremath{\kappa}\ensuremath{'}+s)/2, respectively where \ensuremath{\kappa} and \ensuremath{\kappa}\ensuremath{'} are noise exponents and t and s are conductivity exponents. Previously derived bounds and estimates on \ensuremath{\kappa} and \ensuremath{\kappa}\ensuremath{'} were used to give reasonable estimates of the exponents w and v\ensuremath{'}. For a small but finite ratio h of poor to good conductances and right at the percolation threshold, the linear and nonlinear response functions are found to cross over from the fractal (h=0) to homogeneous (h=1) behavior. The scaling functions of the crossover voltage and current are obtained within the effective-medium approximation and numerical simulations. An excellent agreement with general scaling arguments is found.