Optical Properties of Granular Metallic Media Near the Percolation Transition
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PROPERTIES OF GRANULAR TRANSITION THE PERCOLATION MEDIA NEAR METALLICOPTICAL J. LAFAIT, S. BERTHIER, M. GADENNE, P. GADENNE Laboratoire d'Optique des Solides, CNRS, Universit6 Pierre et Marie Curie, 4 place Jussieu, 75252 Paris C~dex 05 - France
ABSTRACT
This review covers most of new physical aspects opened by the recent interest in the optical properties of heterogeneous media close to the percolation transition: relevant parameters and lengths, caracteristic optical behaviour (transmittance, reflectance and absorptance), theoretical models, dimensionali ty effects, discrimination between optical, electrical and morphological percolation transitions. Morphological aspects are emphasized as the key of optical properties at percolation.
I. INTRODUCTION. The concept of percolation has been built up by Broadbent and Ham mersley between 1954 and 1957 /1/ to account for gas diffusion in porous carbon granules. The formal analogy between a percolation transition and a second-order phase transition was pointed out by Fortuin and Kasteleyn in 1972 /2/: the percolation volume fraction Pc is analogous to the critical temperature Tc. A few years later, Kirkpatrick /3/, Straley /4/ and Stauffer /5/ incorporated the scaling concept to the core of this theory :physical properties of a system at percolation exhibit a power-law dependence on IP-PcI. The exponents of these power laws depend only on the dimensionality of the system, provided the constituents be randomly distributed in the sample, itself assumed as infinite. A full an alytic theory was now available to account for the percolation transition. As a matter of fact, numerical calculations followed at a good rate, performed on computer-simulated resistor networks /3, 4/, the accuracy in the determination of critical exponents increasing with computer capacity. Due to the experimental difficulty mention..,ed above few experimental studies were performed, most measuring the d.c. electrical conductivity on 3D samples /6, 7, 8/. Results were in good agreement with numerical calculations. Grannan et al /9/ published the first results concerning the critical behaviour of the low frequency dielectric constant (below Pc) in 1981, on Ag-KCI samples. They obtained a critical exponent s = 0.73 + 0.07 (see fig. 1). Experimentalists, at that time, be~n to be conscious of the essential role of the cluster morphology of the medium on the physical properties at the percolation transition. The first attempt to correlate morphology and conductiviy at percolation may be due to Laibowitz et al. /10/ on A1-AI 2 0 3 films. Among other results they point out that the geometric percolation (as deduced from TEM micrographs) and the metal-insulating transition do not exactly coincide due to tunneling.
A full analytic approach is thus available.
Unfortunately,
critical
exponents and the analytic nature of scaling fail to provide a comprehensive description of the cluster morphology. Fractal theory as synthetized by Mandelbrot /11/ provides this intuitive geometrical basis for the scaling be
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