2D-to-3D percolation crossover of metal-insulator composites

Percolation properties and d.c. conductivity were determined for an L2×h-random resistor network model of metal-insulator composite films. The effects of the thickness h on the percolation threshold and conductivity were studied numerically in the limit of an infinite size of the L2-plane parallel to the film. For thicknesses ranging from h/L=0.01 to h/L=0.24, a crossover between a finite-size regime and a saturation regime was observed at h/L≈0.1. In the finite-size regime (h/L⩾0.01), the percolation threshold scales as pc(h)−pc3∝h−1/x, the exponent x being compatible with that of the critical exponent of the 3D correlation length, ν3. The conductivity exponent t appeared to depend linearly on the ratio h/L with a slope νD compatible with 2+ν2, where ν2 is the 2D correlation length exponent. In the saturation regime, a scaling correction for the percolation threshold was found with an exponent 1+1/ν3. In this regime we also observed a logarithmic dependence of the conductivity exponent on h/L.