The Diagram Technique for Calculation of Transport Constants of Random Inhomogeneous Materials. I. The AC Effective Conductivity

Diagram methods are applied for evaluating the ac effective conductivity σ^*(ω) of a disordered semiconductor (e.g., a highly doped compensated semiconductor, a composite, etc.). Expressions for σ^*(ω) are obtained in a number of approximations — the cumulant approximation, the self‐consistent moment approximation, and the effective medium approximation. For a random binary mixture which consists of two materials with conductivities σ₁ and σ₂, the dependences of real and imaginary parts of σ^*(ω) on the frequency ω of the external electric field, volume fraction x of the most conducting material and the ratio of the conductivities σ₂/σ₁ are presented for all approximations mentioned. For a metal‐insulator mixture (σ₂ = 0) on the onset of percolation the cumulant and effective medium approximations predict the following behaviour of σ MI*(ω): σ MI*(0) ∼ (x — x_c)^t, σ MI*(ω) ∼ ω^s, here t = 1, s = 1/2. The results, obtained for σ MI*(ω) are compared with those of computer simulation in the model of fully penetrating spheres (Balberg and Binenbaum). The cumulant approximation is in good agreement with the computer results.