Orbital-Free Density Functional Theory of Out-of-Plane Charge Screening in Graphene

We propose a density functional theory of Thomas–Fermi–Dirac–von Weizsäcker type to describe the response of a single layer of graphene resting on a dielectric substrate to a point charge or a collection of charges some distance away from the layer. We formulate a variational setting in which the proposed energy functional admits minimizers, both in the case of free graphene layers and under back-gating. We further provide conditions under which those minimizers are unique and correspond to configurations consisting of inhomogeneous density profiles of charge carrier of only one type. The associated Euler–Lagrange equation for the charge density is also obtained, and uniqueness, regularity and decay of the minimizers are proved under general conditions. In addition, a bifurcation from zero to nonzero response at a finite threshold value of the external charge is proved. Communicated by Felix Otto. B Cyrill B. Muratov [email protected] Jianfeng Lu [email protected] Vitaly Moroz [email protected] 1 Departments of Mathematics, Physics, and Chemistry, Duke University, Box 90320, Durham, NC 27708, USA 2 Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, Wales, UK 3 Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ 07102, USA 123 Author's personal copy