Conductivity of two-dimensional narrow gap semiconductors subjected to strong Coulomb disorder

In the ideal disorder-free situation, a two-dimensional band-gap insulator has an activation energy for conductivity equal to half the band gap Δ. But transport experiments usually exhibit a much smaller activation energy at low temperature, and the relation between this activation energy and Δ is unclear. Here we consider the temperature-dependent conductivity of a two-dimensional insulator on a substrate containing Coulomb impurities, with random potential amplitude Γ≫Δ. We show that the conductivity generically exhibits three regimes of conductivity, and only the highest-temperature regime exhibits an activation energy that reflects the band gap. At lower temperatures, the conduction proceeds through activated hopping or Efros-Shklovskii variable-range hopping between electron and hole puddles created by the disorder. We show that the activation energy and characteristic temperature associated with these processes steeply collapse near a critical impurity concentration. Larger concentrations lead to an exponentially small activation energy and exponentially long localization length, which in mesoscopic samples can appear as a disorder-induced insulator-to-metal transition. We also arrive at a similar steep disorder driven insulator-metal transition in thin films of three-dimensional topological insulators with large dielectric constant, for which Coulomb impurities inside the film create a large disorder potential due to confinement of their electric field inside the film.