Understanding Mott's law from scaling of variable-range-hopping currents and intrinsic current fluctuations

摘要

We have used the master equation to simulate variable-range hopping (VRH) of charges in a strongly disordered d-dimensional energy landscape (d=1,2,3). The current distribution over hopping distances and hopping energies gives a clear insight into the difference between hops that occur most frequently, dominate quantitatively in the integral over the mobility distribution, or are critical ones that still need to be considered in that integral to recover the full low-temperature mobility. The recently reported scaling with temperature of the VRH-current distribution over hopping distances and hopping energies is quantitatively analyzed in 1D and 2D, and accurately confirmed. Based on this, we present an analytical scaling theory of VRH, which distinguishes between a scaling part of the distribution and an exponential tail, separated by critical currents that set the scale and that follow self-consistently at each temperature. This naturally renders Mott's law for the low-temperature mobility, in a way and with a physical picture different from that of the established critical-percolation-network approach to VRH. We argue that current fluctuations as observed in simulations are intrinsic to VRH and play an essential role in this distinction.