Journal of Nonlinear Science

, Volume 25, Issue 6, pp 1391–1430 | Cite as

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

  • Jianfeng Lu
  • Vitaly Moroz
  • Cyrill B. MuratovEmail author


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.


Massless 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.


  1. Abergel, D.S.L., Pietiläinen, P., Chakraborty, T.: Electronic compressibility of graphene: the case of vanishing electron correlations and the role of chirality. Phys. Rev. B 80, 081408 (2009)CrossRefGoogle Scholar
  2. Abergel, D.S.L., Apalkov, V., Berashevich, J., Ziegler, K., Chakraborty, T.: Properties of graphene: a theoretical perspective. Adv. Phys. 59, 261–482 (2010)CrossRefGoogle Scholar
  3. Ando, T.: Screening effect and impurity scattering in monolayer graphene. J. Phys. Soc. Jpn. 75, 074716 (2006)CrossRefGoogle Scholar
  4. Armitage, D.H.: A counter-example in potential theory. J. Lond. Math. Soc. 10(2), 16–18 (1975)zbMATHMathSciNetCrossRefGoogle Scholar
  5. Barlas, Y., Pereg-Barnea, T., Polini, M., Asgari, R., MacDonald, A.H.: Chirality and correlations in graphene. Phys. Rev. Lett. 98, 236601 (2007)CrossRefGoogle Scholar
  6. Benguria, R.D., Brezis, H., Lieb, E.H.: The Thomas–Fermi–von Weizsäcker theory of atoms and molecules. Commun. Math. Phys. 79, 167–180 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  7. Benguria, R.D., Loss, M., Siedentop, H.: Stability of atoms and molecules in an ultrarelativistic Thomas–Fermi–Weizsäcker model. J. Math. Phys. 49, 012302 (2008)MathSciNetCrossRefGoogle Scholar
  8. Brey, L., Fertig, H.A.: Linear response and the Thomas–Fermi approximation in undoped graphene. Phys. Rev. B 80, 035406 (2009)CrossRefGoogle Scholar
  9. Brézis, H., Browder, F.: A property of Sobolev spaces. Commun. Partial Differ. Equ. 4, 1077–1083 (1979)zbMATHCrossRefGoogle Scholar
  10. Cancès, E., Ehrlacher, V.: Local defects are always neutral in the Thomas–Fermi–von Weiszäcker theory of crystals. Arch. Ration. Mech. Anal. 202, 933–973 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  11. Carmona, R., Masters, W.C., Simon, B.: Relativistic Schrödinger operators: asymptotic behavior of the eigenfunctions. J. Funct. Anal. 91, 117–142 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  12. Cartan, H.: Théorie du potentiel newtonien: énergie, capacité, suites de potentiels. Bull. Soc. Math. Fr. 73, 74–106 (1945)zbMATHMathSciNetGoogle Scholar
  13. Castro Neto, A.H., Guinea, F., Peres, N.M.R., Novoselov, K.S., Geim, A.K.: The electronic properties of graphene. Rev. Mod. Phys. 81, 109–162 (2009)CrossRefGoogle Scholar
  14. Das Sarma, S., Adam, S., Hwang, E.H., Rossi, E.: Electronic transport in twodimensional graphene. Rev. Mod. Phys. 83, 407–470 (2011)CrossRefGoogle Scholar
  15. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  16. DiVincenzo, D.P., Mele, E.J.: Self-consistent effective-mass theory for intralayer screening in graphite intercalation compounds. Phys. Rev. B 29, 1685–1694 (1984)CrossRefGoogle Scholar
  17. du Plessis, N.: An introduction to potential theory. University Mathematical Monographs, No. 7. Hafner Publishing Co., Darien (1970)Google Scholar
  18. Engel, E., Dreizler, R.M.: Field-theoretical approach to a relativistic Thomas–Fermi–Weizsäcker model. Phys. Rev. A. 35, 3607–3618 (1987)CrossRefGoogle Scholar
  19. Engel, E., Dreizler, R.M.: Solution of the relativistic Thomas–Fermi–Dirac–Weizsäcker model for the case of neutral atoms and positive ions. Phys. Rev. A. 38, 3909–3917 (1988)CrossRefGoogle Scholar
  20. Fefferman, C.L., Weinstein, M.I.: Honeycomb lattice potentials and Dirac points. J. Am. Math. Soc. 25, 1169–1220 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  21. Fefferman, C.L., Weinstein, M.I.: Wave packets in honeycomb structures and two-dimensional Dirac equations. Commun. Math. Phys. 326, 251–286 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  22. Felmer, P., Quaas, A., Tan, J.: Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian. Proc. R. Soc. Edinb. Sect. A Math. 142, 1237–1262 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  23. Fogler, M.M., Novikov, D.S., Shklovskii, B.I.: Screening of a hypercritical charge in graphene. Phys. Rev. B 76, 233402 (2007)CrossRefGoogle Scholar
  24. Frank, R.L., Lieb, E.H., Seiringer, R.: Hardy–Lieb–Thirring inequalities for fractional Schrödinger operators. J. Am. Math. Soc. 21, 925–950 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  25. García-Cuerva, J., Gatto, A.E.: Boundedness properties of fractional integral operators associated to non-doubling measures. Studia Math. 162, 245–261 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  26. Geim, A.K., Novoselov, K.S.: The rise of graphene. Nat. Mater. 6, 183–191 (2007)CrossRefGoogle Scholar
  27. González, J., Guinea, F., Vozmediano, M.A.H.: Non-fermi liquid behavior of electrons in the half-filled honeycomb lattice (a renormalization group approach). Nucl. Phys. B 424, 595–618 (1994)CrossRefGoogle Scholar
  28. Hainzl, C., Lewin, M., Sparber, C.: Ground state properties of graphene in Hartree–Fock theory. J. Math. Phys. 63, 095220 (2012)MathSciNetCrossRefGoogle Scholar
  29. Hwang, E.H., Das Sarma, S.: Dielectric function, screening, and plasmons in twodimensional graphene. Phys. Rev. B 75, 205418 (2007)CrossRefGoogle Scholar
  30. Kaleta, K., Lörinczi, J.: Fractional \(P(\phi )_{1}\)-processes and Gibbs measures. Stoch. Process. Appl. 122, 3580–3617 (2012)zbMATHCrossRefGoogle Scholar
  31. Katsnelson, M.I.: Nonlinear screening of charge impurities in graphene. Phys. Rev. B 74, 201401(R) (2006)CrossRefGoogle Scholar
  32. Kotov, V.N., Uchoa, B., Pereira, V.M., Guinea, F., Castro Neto, A.H.: Electron–electron interactions in graphene: current status and perspectives. Rev. Mod. Phys. 84, 1067–1125 (2012)CrossRefGoogle Scholar
  33. 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
  34. Le Bris, C., Lions, P.-L.: From atoms to crystals: a mathematical journey. Bull. Am. Math. Soc. (N.S.) 42, 291–363 (2005)zbMATHCrossRefGoogle Scholar
  35. Lieb, E.H.: Thomas–Fermi and related theories of atoms and molecules. Rev. Mod. Phys. 53, 603–641 (1981)MathSciNetCrossRefGoogle Scholar
  36. Lieb, E.H., Loss, M., Siedentop, H.: Stability of relativistic matter via Thomas–Fermi theory. Helv. Phys. Acta 69, 974–984 (1996)zbMATHMathSciNetGoogle Scholar
  37. Lieb, E.H., Loss, M.: Analysis, second, graduate studies in mathematics, vol. 14. American Mathematical Society, Providence (2001)Google Scholar
  38. Lieb, E.H., Simon, B.: The Thomas–Fermi theory of atoms, molecules and solids. Adv. Math. 23, 22–116 (1977)zbMATHMathSciNetCrossRefGoogle Scholar
  39. Lieb, E.H., Yau, H.-T.: The stability and instability of relativistic matter. Commun. Math. Phys. 118, 177–213 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  40. Lu, J., Moroz, V., Muratov, C.B.: In: preparation (2015)Google Scholar
  41. Martin, J., Akerman, N., Ulbricht, G., Lohmann, T., Smet, J.H., von Klitzing, K., Yacoby, A.: Observation of electron–hole puddles in graphene using a scanning single-electron transistor. Nat. Phys. 4, 144–148 (2008)CrossRefGoogle Scholar
  42. Maz’ja, V.G., Havin, V.P.: A nonlinear potential theory. Uspehi Mat. Nauk 27, 67–138 (1972)MathSciNetGoogle Scholar
  43. Novoselov, K.S., Geim, A.K., Morozov, S.V., Jiang, D., Zhang, Y., Dubonos, S.V., Grigorieva, I.V., Firsov, A.A.: Electric field effect in atomically thin carbon films. Science 306, 666–669 (2004)CrossRefGoogle Scholar
  44. Polini, M., Tomadin, A., Asgari, R., MacDonald, A.H.: Density functional theory of graphene sheets. Phys. Rev. B 78, 115426 (2008)CrossRefGoogle Scholar
  45. Reed, J.P., Uchoa, B., Joe, Y.I., Gan, Y., Casa, D., Fradkin, E., Abbamonte, P.: The effective fine-structure constant of freestanding graphene measured in graphite. Science 330, 805–808 (2010)CrossRefGoogle Scholar
  46. Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. Academic Press, New York (1978)zbMATHGoogle Scholar
  47. Rempel, S.: Über die Nichtvollständigkeit eines Raumes von Ladungen mit endlicher Energie. Math. Nachr. 72, 87–91 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  48. Ruiz, D.: On the Schrödinger–Poisson–Slater system: behavior of minimizers, radial and nonradial cases. Arch. Ration. Mech. Anal. 198, 349–368 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  49. Shung, K.W.K.: Dielectric function and plasmon structure of stage-1 intercalated graphite. Phys. Rev. B 34, 979–993 (1986)CrossRefGoogle Scholar
  50. Shytov, A.V., Katsnelson, M.I., Levitov, L.S.: Vacuum polarization and screening of supercritical impurities in graphene. Phys. Rev. Lett. 99, 236801 (2007)CrossRefGoogle Scholar
  51. Silvestre, L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60, 67–112 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  52. Sodemann, I., Fogler, M.M.: Interaction corrections to the polarization function of graphene. Phys. Rev. B 86, 115408 (2012)CrossRefGoogle Scholar
  53. Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton mathematical series, vol. 30. Princeton University Press, Princeton (1970)zbMATHGoogle Scholar
  54. Struwe, M.: Variational methods. Springer, Berlin (1990)zbMATHCrossRefGoogle Scholar
  55. Wallace, P.R.: The band theory of graphite. Phys. Rev. 71, 622–634 (1947)zbMATHCrossRefGoogle Scholar
  56. Wang, J., Fertig, H.A., Murthy, G., Brey, L.: Excitonic effects in two-dimensional massless Dirac fermions. Phys. Rev. B 83, 035404 (2011)CrossRefGoogle Scholar
  57. Wang, Y., Brar, V.W., Shytov, A.V., Wu, Q., Regan, W., Tsai, H.-Z., Zettl, A., Levitov, L.S., Crommie, M.F.: Mapping Dirac quasiparticles near a single Coulomb impurity on graphene. Nat. Phys. 8, 653–657 (2012)CrossRefGoogle Scholar
  58. 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
  59. Zhang, L.M., Fogler, M.M.: Nonlinear screening and ballistic transport in a graphene p–n junction. Phys. Rev. Lett. 100, 116804 (2008)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Departments of Mathematics, Physics, and ChemistryDuke UniversityDurhamUSA
  2. 2.Department of MathematicsSwansea UniversitySwanseaWales, UK
  3. 3.Department of Mathematical Sciences, New Jersey Institute of TechnologyUniversity HeightsNewarkUSA

Personalised recommendations