Size effect in the longitudinal hopping conduction of a narrow two-dimensional channel

The variable-range-hopping conduction of a narrow two-dimensional (2D) strip is studied analytically. The gradual transition from 2D to 1D behavior of conductivity with the narrowing of the strip is traced. It is shown that, in the wide range of the width of the strip, the intermediate regime of conduction takes place. In this regime the passage of current through a typical cross section is determined by the subnetwork of the 2D infinite cluster, but the total resistance is dominated by ``cuts''---sparse high-Ohmic regions in which the subnetwork is broken. These cuts are responsible for the size effect---an exponentially strong dependence of the resistance on the width. They are also responsible for the fluctuations of the resistance of a sufficiently short strip with a change of the Fermi-level position. The amplitude of the fluctuations as a function of temperature and the parameters of the strip is evaluated.