Dielectric screening in two-dimensional insulators: Implications for excitonic and impurity states in graphane

For atomic thin layer insulating materials we provide an exact analytic form of the two-dimensional (2D) screened potential. In contrast to three-dimensional systems where the macroscopic screening can be described by a static dielectric constant, in 2D systems the macroscopic screening is nonlocal (q dependent) showing a logarithmic divergence for small distances and reaching the unscreened Coulomb potential for large distances. The crossover of these two regimes is dictated by 2D layer polarizability that can be easily computed by standard first-principles techniques. The present results have strong implications for describing gap-impurity levels and also exciton binding energies. The simple model derived here captures the main physical effects and reproduces well, for the case of graphane, the full many-body GW plus Bethe-Salpeter calculations. As an additional outcome we show that the impurity hole-doping in graphane leads to strongly localized states, which hampers applications in electronic devices. In spite of the inefficient and nonlocal two-dimensional macroscopic screening we demonstrate that a simple k . p approach is capable to describe the electronic and transport properties of confined 2D systems.