Variable range hopping in thin film with large dielectric constant

In a film with large dielectric constant $\kappa$ the electric field of an electron spreads inside the film before exiting the film at large distances of order of $\kappa d$ ($d$ 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 $\ln \sigma (T) \propto (T_0/T)^{p}$ dependence, with the index $p \simeq 0.7$, is sandwiched between the two domains with $p = 1/2$.