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For atomic thin layer insulating materials we provide an exact analytic formof the two-dimensional screened potential. In contrast to three-dimensionalsystems where the macroscopic screening can be described by a static dielectricconstant in 2D systems the macroscopic screening is non local (q-dependent)showing a logarithmic divergence for small distances and reaching theunscreened Coulomb potential for large distances. The cross-over of these tworegimes is dictated by 2D layer polarizability that can be easily computed bystandard first-principles techniques. The present results have strongimplications for describing gap-impurity levels and also exciton bindingenergies. The simple model derived here captures the main physical effects andreproduces well, for the case of graphane, the full many-body GW plusBethe-Salpeter calculations. As an additional outcome we show that the impurityhole-doping in graphane leads to strongly localized states, what hampersapplications in electronic devices. In spite of the inefficient and nonlocaltwo-dimensional macroscopic screening we demonstrate that a simple$\mathbf{k}\cdot\mathbf{p}$ approach is capable to describe the electronic andtransport properties of confined 2D systems.

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