Analytical solutions for the idealized “strict 2D” limit do exist, but a fundamental theory that accounts for the “quasi-2D” nature of real layers is still missing. Here, we develop an exact theory of the long-range electrostatics in quasi-2D systems, enabling an accurate modeling of any physical property (such as interatomic forces at large distances) that depends on them.

Our main mathematical result consists of achieving a sound and compact decomposition of the Coulomb interactions between long-range and short-range contributions in the quasi-2D case, thereby extending and generalizing existing approaches and providing them with a solid foundation. To do this, we rely on only two simple and intuitive formal devices: the well-known image-charge method of classical electrostatics and the bisection formula of the hyperbolic trigonometric functions.

Our formalism provides a general platform for describing both intralayer and extralayer electrostatic interactions in 2D systems, with an applicability that goes well beyond the specifics of lattice dynamics. We expect our method to greatly facilitate the calculation of electron-phonon couplings and any other physical property (such as flexoelectricity) that requires a careful treatment of screening in two dimensions.