Orbital-Free Density Functional Theory of Out-of-Plane Charge Screening in Graphene
- 258 Downloads
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.
KeywordsMassless relativistic fermions Non-linear response Ground state Fractional Laplacian
The authors wish to thank an anonymous referee for helpful suggestions. JL would like to acknowledge support from the Alfred P. Sloan Foundation and the National Science Foundation under award DMS-1312659. CBM was supported, in part, by the National Science Foundation via grants DMS-0908279 and DMS-1313687.
- du Plessis, N.: An introduction to potential theory. University Mathematical Monographs, No. 7. Hafner Publishing Co., Darien (1970)Google Scholar
- Landkof, N.S.: Foundations of modern potential theory. Springer, New York (1972). Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band. 180Google Scholar
- Lieb, E.H., Loss, M.: Analysis, second, graduate studies in mathematics, vol. 14. American Mathematical Society, Providence (2001)Google Scholar
- Lu, J., Moroz, V., Muratov, C.B.: In: preparation (2015)Google Scholar
- Yu, G.L., Jalil, R., Bell, B., Mayorov, A.S., Blake, P., Schedin, F., Morozov, S.V., Ponomarenko, L.A., Chiappini, F., Wiedmann, S., Zeitler, U., Katsnelson, M.I., Geim, A.K., Novoselov, K.S., Elias, D.C.: Interaction phenomena in graphene seen through quantum capacitance. Proc. Natl. Acad. Sci. USA 110, 3282–3286 (2013)CrossRefGoogle Scholar