Conductivity of a mixture of conducting and insulating grains: Dimensionality effects

The percolation behavior of films of thickness $d$ composed of a compact mixture of conducting (concentration $p$) and insulating (concentration $1-p$) beads is studied by measuring their conductivity $\sigma (d, p)$. A theoretical model based on a renormalization operation to the thickness of the film leads to predictions on the threshold $p_c\mathrm{}\left(d\right)\mathrm{}$, the crossover values between two- and three-dimensional behavior $p_{\pm }\mathrm{}\left(d\right)\mathrm{}$, and the amplitude of the conductivity above threshold. The experiments are analyzed in terms of the model and lead to the determination of several critical exponents.