The a.c. conductivity of variable range hopping systems such as amorphous semiconductors

Abstract

A new theory for the frequency-dependent conductivity of variable range hopping (VRH) systems is presented. The physical basis of the theory is that local relaxation processes dominate at high frequencies, non-local processes at low frequencies. The loss peak in the imaginary part of the dielectric constant represents the onset of percolation, defining approximately the boundary between these two regimes. At high frequencies, the pair approximation is modified to allow the density of contributing pairs to be frequency dependent. At lower frequencies, a cluster expansion consistent with percolation theory is used to calculate σ(ω). The relaxation times of clusters of impedances are strongly enhanced through the necessity to transport a large amount of charge over large distances. The theory gives the correct analytical expressions for the frequency and the temperature dependence of σ(ω) in every frequency regime, and the pair approximation joins smoothly on to the low-frequency results at the loss-peak frequency. Experimental confirmation has been found in a-Si (Long et al. 1988), and a-Si:H:Au (Long and Hansmann 1990) variable-range hopping systems. Numerical values in a-Si agree with experiment within a factor of 2 or 3. The results represent a strong confirmation of the role of percolation theory and the importance of slow relaxation phenomena in transport in highly disordered media.