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

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Abstract

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.

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Notes

  1. Note that in Brey and Fertig (2009), Das Sarma et al. (2011) and some other papers in the physics literature, a factor of \({{\mathrm{sgn}}}(\rho )\) was mistakenly added to the integrand of the Thomas–Fermi term. The resulting energy functional is then not bounded from below and is inconsistent with the Thomas–Fermi equation.

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Acknowledgments

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.

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Authors and Affiliations

  1. Departments of Mathematics, Physics, and Chemistry, Duke University, Box 90320, Durham, NC, 27708, USA

    Jianfeng Lu

  2. Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, Wales, UK

    Vitaly Moroz

  3. Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ, 07102, USA

    Cyrill B. Muratov

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  1. Jianfeng Lu
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  2. Vitaly Moroz
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Correspondence to Cyrill B. Muratov.

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Communicated by Felix Otto.

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Lu, J., Moroz, V. & Muratov, C.B. Orbital-Free Density Functional Theory of Out-of-Plane Charge Screening in Graphene. J Nonlinear Sci 25, 1391–1430 (2015). https://doi.org/10.1007/s00332-015-9259-4

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