Variable range hopping in thin film with large dielectric constant

In a film with large dielectric constant κ the electric field of an electron spreads inside the film before exiting the film at large distances of order of κd (d is the film width). This leads to the logarithmic Coulomb repulsion between electrons and modifies the shape of the Coulomb gap in the density of localized states in a doped film. As a result the variable range hopping conductivity in such a film has a peculiar temperature dependence, where the domain of the in σ(T) ∝ (T₀/T)^p dependence, with the index p ∼ 0.7, is sandwiched between the two domains with p = 1/2.