What is the relation between activation energy and band gap in a 2D insulator?

What can one actually tell about the band gap from the activation energy for conductivity in a 2D material? At first glance, it seems like the activation energy should be equal to half the band gap if the Fermi level is in the middle of the gap. But this simple relation is often strongly violated in experiments, where it is common to observe a much smaller activation energy. In this poster, I will review some examples of relevant experiments in topological insulators thin films, bilayer graphene, and Mott insulators in twisted moiré bilayers. We will show theoretically how disorder, even when present at a very low level, almost inevitably lowers the activation energy to a nonuniversal value that is parametrically smaller than the band gap. We will further show how a sufficiently large disorder can produce an apparent insulator-to-metal transition.