Ground-state Properties of Inhomogeneous Graphene Sheets

When inter-valley scattering is weak and gauge fields due to {\it e.g.} ripples are neglected, doped and gated graphene sheets can be described using an envelope-function Hamiltonian with a new sublattice pseudospin degree-of freedom, an ultrarelativistic massless-Dirac free-fermion term, a pseudospin {\it scalar} disorder potential, and a non-relativistic instantaneous Coulombic interaction term. There is considerable evidence from experiment that this simplified description of a honeycomb lattice of Carbon atoms is usually a valid starting point for theories of those observables that depend solely on the electronic properties of $\pi$-electrons near the graphene Dirac point [1]. Although the use of this model simplifies the physics considerably it still leaves us with a many-body problem without translational invariance, which we do not know how to solve. In this talk we present a Kohn-Sham-Dirac density-functional-theory (DFT) scheme for graphene sheets that treats slowly-varying inhomogeneous scalar external potentials and electron-electron interactions on an equal footing [2]. The theory is able to account for the unusual property that the exchange-correlation contribution to chemical potential increases with carrier density in graphene [3,4]. Consequences of this property, and advantages and disadvantages of using the DFT approach to describe it, are discussed. The approach is illustrated by solving the Kohn-Sham-Dirac equations self-consistently for a model random potential describing charged point-like impurities located close to the graphene plane. The influence of electron-electron interactions on these non-linear screening calculations is discussed at length, in the light of recent experiments [5,6] reporting evidence for the presence of electron-hole puddles in nearly-neutral graphene sheets. \\[4pt] [1] A.K. Geim and K.S. Novoselov, Nature Mater. {\bf 6}, 183 (2007); A.K. Geim and A.H. MacDonald, Phys. Today {\bf 60}, 35 (2007); A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, and A.K. Geim, arXiv:0709.1163v2 (2007).\\[0pt] [2] M. Polini, A. Tomadin, R. Asgari, and A.H. MacDonald, Phys. Rev. B {\bf 78}, 115426 (2008).\\[0pt] [3] Y. Barlas, T. Pereg-Barnea, M. Polini, R. Asgari, and A.H. MacDonald, Phys. Rev. Lett. {\bf 98}, 236601 (2007); M. Polini, R. Asgari, Y. Barlas, T. Pereg-Barnea, and A.H. MacDonald, Solid State Commun. {\bf 143}, 58 (2007). \\[0pt] [4] E.H. Hwang, B.Y.-K. Hu, and S. Das Sarma, Phys. Rev. Lett. {\bf 99}, 226801 (2007).\\[0pt] [5] J. Martin, N. Akerman, G. Ulbricht, T. Lohmann, J.H. Smet, K. von Klitzing, and A. Yacoby, Nature Phys. {\bf 4}, 144 (2008).\\[0pt] [6] V.W. Brar, Y. Zhang, C. Girit, F. Wang, A. Zettl, and M. Crommie, Bull. Am. Phys. Soc. {\bf 53} (2), 443 (2008).