Electrons in insulating crystals polarize in response to an externally applied electric field, resulting in a partial suppression of the field amplitude; such a phenomenon is known as “dielectric screening.” While much effort has gone into developing a quantitative understanding of this behavior and its impact on materials properties, 2D crystals remain challenging to describe by means of established modeling strategies. 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. As a first application, we demonstrate its usefulness in describing higher-order electromechanical couplings, such as flexoelectricity. (The latter refers to the macroscopic polarization induced by strain gradients, e.g., flexural deformations of a 2D layer.) We find that both the direct and converse effect are described by a universal coupling constant, encompassing purely electronic and lattice-mediated contributions. Within the former, we identify a key metric term, consisting in the quadrupolar moment of the unperturbed charge density. We propose a simple continuum model to connect our findings with the available experimental measurements.