Theory of hopping transport through a constriction dominated by a single hop

Hopping transport through a point contact between two bulk semiconductors is considered for the case when the contact size is much smaller than a typical hopping length in the bulk. In this case the conductance is controlled by a single hop between the two sites located the opposite banks of the contact. For the variable range hopping (VRH) the choice of the pair depends on temperature, and the temperature dependence of the conductance exhibits exponentially large mesoscopic fluctuations. For large enough voltage the conductance is strongly nonlinear and also exhibits giant fluctuations as a result of “switching” between different “critical pairs.” It also exhibits regions of negative differential resistance due to resonant tunneling effective at some “resonant” values of the bias. At higher biases the nonlinear $I\left(V\right)\mathrm{}$ curve tends to a smooth one $\left(I\propto V^2\right).\mathrm{}$ A comparison to existing theoretical and experimental works relevant to the problem is made.