David J. Bergman

Numerical study of transport properties in continuum percolation

The authors present numerical simulations of AC conductance for a random resistor-capacitor network. The conductance obeys a probability density function p(g) varies as g- alpha (0( alpha (1). They use a highly efficient propagation algorithm to calculate the effective conductance of a long strip of a lattice. At low frequencies, they find that for the concentration p of conducting bonds less than the percolation threshold pc, the imaginary part of conductance is proportional to frequency Im(geff) approximately= omega and the real part of conductance shows an anomalous frequency dependence Re(geff) approximately= omega 2- alpha . The results of simulations in such a continuum system are in agreement with the predictions of the effective medium and the Maxwell-Garnett approximation. They also calculate the non-universal DC conductivity exponents in continuum percolation; the results are consistent with earlier theoretical predictions and numerical calculations.

High-field magnetotransport in composite conductors: Effective-medium approximation

The self-consistent effective-medium approximation ~SEMA! is used to study three-dimensional random conducting composites under the influence of a strong magnetic field B, in the case where all constituents exhibit isotropic response. Asymptotic analysis is used to obtain almost closed-form results for the strong-field magnetoresistance and Hall resistance in various types of two- and three-constituent isotropic mixtures for the entire range of compositions. Numerical solutions of the SEMA equations are also obtained, in some cases, and compared with those results. In two-constituent free-electron-metal/perfect-insulator mixtures, the magnetoresistance is asymptotically proportional to uBu at all concentrations above the percolation threshold . In threeconstituent metal/insulator/superconductor mixtures a line of critical points is found, where the strong-field magnetoresistance switches abruptly from saturating to nonsaturating dependence on uBu, at a certain value of the insulator-to-superconductor concentration ratio. This transition appears to be related to the phenomenon of anisotropic percolation.

Discrete network models for the low-field Hall effect near a percolation threshold: Theory and simulations

The critical behavior of the weak-field Hall effect near a percolation threshold is studied with the help of two discrete random network models. Many finite realizations of such networks at the percolation threshold are produced and solved to yield the potentials at all sites. A new algorithm for doing that was developed that is based on the transfer matrix method. The site potentials are used to calculate the bulk effective Hall conductivity and Hall coefficient, as well as some other properties, such as the Ohmic conductivity, the size of the backbone, and the number of binodes. Scaling behavior for these quantities as power laws of the network size is determined and values of the critical exponents are found.

Mean field theory for weakly nonlinear composites

We discuss the nonlinear behavior of a random composite material characterized by a weakly nonlinear relation between the electric displacement of the form D = ϵE + χ|E|2E, where ϵ and χ are position dependent. A general expression for the effective nonlinear susceptibility to first order in the nonlinear susceptibility of the constituents in the composite is given. A general method of approximation is introduced which gives the effective nonlinear susceptibility in terms of the solution of the linear dielectric function of the random composite. Various applications of the proposed approximation are demonstrated.

Averaged local field intensities in composite materials

Abstract Averaged local field intensities are calculated for isotropic composites in the Maxwell-Garnett and in the effective medium theories. Exact upper and lower bounds on these intensities are also found. Implications for photophysical properties of molecules embedded in the composites are discussed.

Electrical transport properties near a classical conductivity or percolation threshold

This review emphasizes progress in understanding electrical conduction and related physical properties of a composite medium near a classical conductivity threshold - mostly but not entirely in the context of a percolation threshold. Besides dc conductivity, we discuss the dielectric susceptibility in a metal-dielectric composite, Hall effect and magnetoresistance in a good conductor - bad conductor mixture, conductivity fluctuations (flicker noise) in such a composite, and weakly nonlinear behavior in a composite dielectric. Scaling theories are described for the dielectric properties and for the magnetotransport near a percolation threshold. Because the conductivity fluctuations and weak nonlinearities are sensitive to microgeometric features that do no affect the linear conductivity, measuring them can provide new information about the microstructure.

Magnetoresistance of three-constituent composites: Percolation near a critical line

Scaling theory, duality symmetry, and numerical simulations of a random network model are used to study the magnetoresistance of a metal/insulator/perfect conductor composite with a disordered columnar microstructure. The phase diagram is found to have a critical line which separates regions of saturating and nonsaturating magnetoresistance. The percolation problem which describes this line is a generalization of anisotropic percolation. We locate the percolation threshold and determine the values of the critical exponents

Critical point in the strong-field magnetoresistance of a normal conductor/perfect insulator/perfect conductor composite with a random columnar microstructure

{t}_{\ensuremath{\Vert}}{=t}_{\ensuremath{\perp}}{=s}_{\ensuremath{\Vert}}{=s}_{\ensuremath{\perp}}=1.30\ifmmode\pm\else\textpm\fi{}0.02,

Frequency dependence of the polarization catastrophe at a metal-insulator transition and related problems

\ensuremath{\nu}=4/3\ifmmode\pm\else\textpm\fi{}0.02,