Universal low-temperature ac conductivity of macroscopically disordered nonmetals

This paper discusses a macroscopic model for ac conduction in electronically or ionically conducting disordered solids. The model considers ac conduction in an inhomogeneous solid that is characterized by a spatially randomly varying thermally activated (frequency-independent) conductivity. Discretizing Maxwell's equations leads to an equivalent electrical circuit that is a simple-cubic lattice where each pair of nodes are linked by a resistor and a capacitor in parallel. The values of the resistors are determined by the local resistivity while the capacitors are all equal, given by the infinite-frequency dielectric constant. It is shown that the capacitor currents are Maxwell's displacement currents. Assuming uncorrelated resistances, the model is solved analytically at low temperatures in the effective-medium approximation (EMA) and in a naive percolation-path approximation. Both approximations predict similar universal ac responses as T{→}0, where the macroscopic frequency-dependent conductivity becomes independent of the activation-energy probability distribution. The universality represents an unusual type of regularity appearing in the extreme disorder limit. The universality prediction is tested by computer simulations of 200×{}200 lattices in two dimensions and of 50×{}50×{}50 lattices in three dimensions. The computer simulations show that the EMA works very well in two dimensions in the whole temperature range studied; in particular, the low-temperature universality prediction is confirmed. In three dimensions the universality prediction is confirmed as well.