Hopping transport in systems of finite thickness or length

Variable-range hopping transport along short one-dimensional wires and across the shortest dimension of thin three-dimensional films and narrow two-dimensional ribbons is studied theoretically. Geometric and transport characteristics of the hopping resistor network are shown to depend on temperature T and the dimensionality of the system. In two and three dimensions, the usual Mott law applies at high T where the correlation length of the network is smaller than the sample thickness. As T decreases, the network breaks into sparse filamentary paths while the Mott law changes to a different T dependence, which is derived using the percolation theory methods. In one dimension, deviations from the Mott law are known to exist at all temperatures because of rare fluctuations. The evolution of such fluctuations from highly resistive breaks at high T to highly conducting shorts at low T is elucidated.