We study the inverse scattering for Schrodinger operators on locally perturbed periodic lattices. We show that the associated scattering matrix is equivalent to the Dirichlet-to-Neumann map for a boundary value problem on a finite part of the graph, and reconstruct scalar potentials as well as the graph structure from the knowledge of the S-matrix. In particular, we give a procedure for probing defects in hexagonal lattices (graphene).
In physical literatures, the 2-dimensional Dirac operator is usually adopted as a mathematical model for the graphene (see e.g. @cite_18 , @cite_48 , @cite_10 ). Therefore, our discrete Laplacian on the hexagonal lattice is regarded as a discretization of this Dirac operator. For the mathematical model of carbon nano-tube, see @cite_44 , @cite_32 . An experimental result for the defects in graphite is seen in @cite_55 .
@cite_18 @cite_48 @cite_55 @cite_32 @cite_44 @cite_10
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The electronic spectrum of sheets of graphite (plane honeycomb lattice) folded into regular polyhedra is studied. A continuum limit valid for sufficiently large molecules and based on the tight-binding approximation is derived. It is found that a Dirac equation describes the flat graphite lattice. Curving the lattice by insertion of odd-numbered rings can be mimicked by coupling effective gauge fields. In particular the C60 and related molecules are well described by the Dirac equation on the surface of a sphere coupled to a color monopole sitting at its center. This article reviews the basic theoretical aspects of graphene, a one-atom-thick allotrope of carbon, with unusual two-dimensional Dirac-like electronic excitations. The Dirac electrons can be controlled by application of external electric and magnetic fields, or by altering sample geometry and or topology. The Dirac electrons behave in unusual ways in tunneling, confinement, and the integer quantum Hall effect. The electronic properties of graphene stacks are discussed and vary with stacking order and number of layers. Edge (surface) states in graphene depend on the edge termination (zigzag or armchair) and affect the physical properties of nanoribbons. Different types of disorder modify the Dirac equation leading to unusual spectroscopic and transport properties. The effects of electron-electron and electron-phonon interactions in single layer and multilayer graphene are also presented. We report on the local atomic and electronic structures of a nitrogen-doped graphite surface by scanning tunneling microscopy, scanning tunneling spectroscopy, x-ray photoelectron spectroscopy, and first-principles calculations. The nitrogen-doped graphite was prepared by nitrogen ion bombardment followed by thermal annealing. Two types of nitrogen species were identified at the atomic level: pyridinic-N (N bonded to two C nearest neighbors) and graphitic-N (N bonded to three C nearest neighbors). Distinct electronic states of localized π states were found to appear in the occupied and unoccupied regions near the Fermi level at the carbon atoms around pyridinic-N and graphitic-N species, respectively. The origin of these states is discussed based on experimental results and theoretical simulations. An explicit derivation of dispersion relations and spectra for periodic Schrodinger operators on carbon nano-structures (including graphene and all types of single-wall nano-tubes) is provided. We consider the Schrodinger operator on nanoribbons (tight-binding models) in an external electric potential V on the plane. The corresponding electric field is perpendicular to the axis of the nanoribbon. If V = 0, then the spectrum of the Schrodinger operator consists of two non-flat bands and one flat band (an eigenvalue with infinite multiplicity) between them. If we switch on a weak electric potential V → 0, then there are two cases: (1) this eigenvalue splits into the small spectral band. We determine the asymptotics of the spectral bands for small fields. (2) the unperturbed eigenvalue remains the flat band. We describe all potentials when the unperturbed eigenvalue remains the flat band and when one splits into the small band of the continuous spectrum. We show that in graphene, modelled by the two-dimensional Dirac operator, charge distributions with non-vanishing dipole moment have infinitely many bound states. The corresponding eigenvalues accumulate at the edges of the gap faster than any power.