Theory of the random potential and conductivity at the surface of a topological insulator

We study the disorder potential induced by random Coulomb impurities at the surface of a topological insulator (TI). We use a simple model in which positive and negative impurities are distributed uniformly throughout the bulk of the TI, and we derive the magnitude of the disorder potential at the TI surface using a self-consistent theory based on the Thomas-Fermi approximation for screening by the Dirac mode. Simple formulas are presented for the mean squared potential both at the Dirac point and far from it, as well as for the characteristic size of electron/hole puddles at the Dirac point and the total concentration of electrons/holes that they contain. We also derive an expression for the autocorrelation function for the potential at the surface and show that it has an unusually slow decay, which can be used to verify the bulk origin of disorder. The implications of our model for the electron conductivity of the surface are also presented.