Resonant-tunnelling conductance of a finite-size amorphous sample

The conductivity of an amorphous sample at low temperatures is calculated. While Mott's variable range hopping theory considers infinite samples, the proposed formalism treats finite ones. It turns out that this is a crucial difference. The model predicts a transition temperature (T_c) between two conductivity behaviours: ln(σ) ∼ -(T₂^L/T)^1/3 for T < T_c, and ln(σ) ∼ -(T₂^H/T)^1/2 for T > T_c (the transition temperature, T_c, depends on the Fermi energy and on the sample's characteristics). The former resembles the simple two-dimensional Mott conductivity behaviour, while the latter resembles the Efrös and Shklovskiǐ conductivity theory. We also show a simple connection between these temperatures: T_c = (T₂^H)³/(T₂^L)².