THE DIAGRAM TECHNIQUE FOR CALCULATION OF TRANSPORT CONSTANTS OF RANDOM INHOMOGENEOUS MATERIALS

Diagram methods are applied for evaluating the off-diagonal (Hall) components of the ac effective magnetoconductivity tensor sigma(yx)*(omega) in an inhomogeneous material for the case of low magnetic field. Closed expressions for sigma(ik)*(omega) are obtained in two approximations, namely in the self-consistent cumulant approximation and in the effective-medium approximation (EMA). Our expression for sigma(ik)*(omega) in the EMA coincides with the one obtained earlier by Fishchuk. The obtained results are applied to the model of a random binary mixture consisting of two conducting materials 1 and 2 with conductivity tensors sigma(ik)(1) and sigma(ik)(2) and volume fractions x and 1 - x, respectively. In the special case of sigma(ik)(2) = 0 and omega = 0 the (dc) Hall conductivity sigma(yk)* in both above-mentioned approximations has a percolation threshold at some critical value x(c). In each approximation the value of x(c) coincides with the one for the dc diagonal (ohmic) conductivity sigma* taken in the same approximation: the self-consistent cumulant approximation gives x(c) = 1 - exp (- 1/3), while in the EMA x(c) is equal to 1/3.