The relationship between activation energy and band gap in a disordered 2D insulator

We study how the activation energy for conductivity is related to the band gap in a 2D band insulator that experiences a disorder potential created by charged impurities. This problem can be mapped to a problem of classical percolation, since the conductivity is limited by the rate at which electrons are activated from the chemical potential to a percolating energy contour in real space (the classical mobility edge). When impurities are sparse, we find that the activation energy is the same as in the disorder-free case, even though there is significant band bending. On the other hand, when the Coulomb disorder is strong enough to induce large, closely-spaced electron puddles, the activation energy reflects the Coulomb charging energy of large puddles, and has a critical dependence on the band gap that is characteristic of 2D percolation.