Correlated hopping in a disordered medium

This paper discusses random walks with memory on a percolating network as a model of correlated hopping transport through a disordered system. Correlations can arise from such sources as hard-core and Coulomb repulsions, correlated hops of groups of particles, or lattice-relaxation effects. In general these correlations will result in a difference between the hopping probability for return to the previously visited site and the probability to jump to another nearest neighbor of the currently occupied site. Thus the hopping process possesses a memory of its previous hop. Such a random walk is investigated in this paper for the case of bond percolation on a regular lattice. The frequency-dependent conductivity σ(ω) is calculated using a generalized effective-medium approximation. Results are presented for the linear chain and the hexagonal lattice. New features appear in both the real and the imaginary part of σ. These depend on the strength of the correlations and on the concentration of bonds. As an example, the possibility of a pronounced maximum in the real part of σ(ω) at finite frequencies is found, which is sometimes accompanied by a change of sign in the imaginary part. The results are found to agree qualitatively with experimental data on ionic transport in ${\mathrm{Na}}^{+}$–β-alumina, where both disorder and correlations are known to be important.