Effective one-dimensionality of AC hopping conduction in the extreme disorder limit

It is argued that in the limit of extreme disorder AC hopping is dominated by ”percolation paths”. Modelling a percolation path as a one-dimensional path with a sharp jump rate cut-off leads to an expression for the universal AC conductivity, that fits computer simulations in two and three dimensions better than the effective medium approximation. 72.20.-i; 72.80.Ng; 66.30.Dn Typeset using REVTEX 1 While ordered solids show no frequency-dependence of their conductivity at frequencies below phonon frequencies, disordered solids are characterized by AC conductivity that varies as an approximate power-law of frequency [1–6]. The exponent is usually less than one, but often quite close to one. As the frequency goes to zero the conductivity stabilizes and becomes frequency-independent. The characteristic frequency marking the onset of DC conduction has roughly the same temperature-dependence as the DC conductivity. These features are observed universally for electronically conducting disordered solids like amorphous semiconductors [1,2,4,7], polymers [8,9], doped crystalline semiconductors at helium temperatures [10] (where the random positions of the dopant atoms becomes important), or high temperature superconductors above Tc [11], as well as for ionically conducting disordered solids like glasses or polymers [2,3,5,6]. The standard model for AC conduction in disordered solids is the hopping model [12–18]. The simplest version is hopping of non-interacting charge carriers on a regular lattice with random symmetric nearest neighbor jump rates; this is the model to be studied below. Alternatively, a macroscopic approach may be adopted by considering Maxwell’s equations for a solid with spatially randomly varying conductivity [19–21]. For both models the limit of extreme disorder may be studied by letting the temperature go to zero when the jump rates/the macroscopic conductivities are thermally activated with randomly varying activation energies. It has recently been shown by computer simulations [20–22] that in the low temperature limit, the AC conductivity becomes universal in both models, i.e., independent of the activation energy probability distribution, p(E). The effective medium approximation (EMA) for both models predicts the same universal AC conductivity in the extreme disorder limit (in more than one dimension). If σ̃ denotes the conductivity relative to the DC conductivity and s̃ is a suitably scaled dimensionless imaginary frequency (“Laplace frequency”), the EMA universality equation [20–22], first derived by Bryksin for the model of electrons tunnelling between randomly localized positions [13,23], is