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.

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The electronic properties of graphene

To realize graphene-based electronics, various types of graphene are required; thus, modulation of its electrical properties is of great importance. Theoretic studies show that intentional doping is a promising route for this goal, and the doped graphene might promise fascinating properties and widespread applications. However, there is no experimental example and electrical testing of the substitutionally doped graphene up to date. Here, we synthesize the N-doped graphene by a chemical vapor deposition (CVD) method. We find that most of them are few-layer graphene, although single-layer graphene can be occasionally detected. As doping accompanies with the recombination of carbon atoms into graphene in the CVD process, N atoms can be substitutionally doped into the graphene lattice, which is hard to realize by other synthetic methods. Electrical measurements show that the N-doped graphene exhibits an n-type behavior, indicating substitutional doping can effectively modulate the electrical properties of graphene. Our finding provides a new experimental instance of graphene and would promote the research and applications of graphene.

Synthesis of N-Doped Graphene by Chemical Vapor Deposition and Its Electrical Properties

Graphene quantum dots: emergent nanolights for bioimaging, sensors, catalysis and photovoltaic devices

and Applications of Fullerenes, Carbon Dots, Nanotubes, Graphene, Nanodiamonds, and Combined Superstructures Vasilios Georgakilas,† Jason A. Perman,‡ Jiri Tucek,‡ and Radek Zboril*,‡ †Material Science Department, University of Patras, 26504 Rio Patras, Greece ‡Regional Centre of Advanced Technologies and Materials, Department of Physical Chemistry, Faculty of Science, Palacky University in Olomouc, 17 listopadu 1192/12, 771 46 Olomouc, Czech Republic

Broad family of carbon nanoallotropes: classification, chemistry, and applications of fullerenes, carbon dots, nanotubes, graphene, nanodiamonds, and combined superstructures.

We study the electronic states of narrow graphene ribbons (``nanoribbons'') with zigzag and armchair edges. The finite width of these systems breaks the spectrum into an infinite set of bands, which we demonstrate can be quantitatively understood using the Dirac equation with appropriate boundary conditions. For the zigzag nanoribbon we demonstrate that the boundary condition allows a particlelike and a holelike band with evanescent wave functions confined to the surfaces, which continuously turn into the well-known zero energy surface states as the width gets large. For armchair edges, we show that the boundary condition leads to admixing of valley states, and the band structure is metallic when the width of the sample in lattice constant units has the form $3M+1$, with $M$ an integer, and insulating otherwise. A comparison of the wave functions and energies from tight-binding calculations and solutions of the Dirac equations yields quantitative agreement for all but the narrowest ribbons.

https://arxiv.org/pdf/cond-mat/0603107.pdf

Electronic states of graphene nanoribbons studied with the Dirac equation

Author(s): Roldan, R; Tsai, SW; Lopez-Sancho, MP | Abstract: The superconducting instability of the Fermi-liquid state is investigated by considering anisotropic electron-boson couplings. Both electron-electron interactions and anisotropic electron-boson couplings are treated with a renormalization group method that takes into account retardation effects. Considering a noninteracting circular Fermi surface, we find analytical solutions for the flow equations and derive a set of generalized Eliashberg equations. Electron-boson couplings with different momentum dependences are studied and we find superconducting instabilities of the metallic state with competition between order parameters of different symmetries. Numerical solutions for some couplings are given to illustrate the frequency dependence of the vertices at different coupling regimes. © 2009 The American Physical Society.

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Renormalization group approach to anisotropic superconductivity

Ferromagnetism and Disorder in Graphene

The dynamics of a single hole (or electron) in the two dimensional Hubbard model is investigated. The antiferromagnetic background is described by a Neel state, and the hopping of the carrier is analyzed within a configuration interaction approach. Results are in agreement with other methods and with experimental data when available. All data are compatible with the opening of a mean field gap in a Fermi liquid of spin polarons, the so called Slater type of transition. In particular, this hypothesis explains the unusual dispersion relation of the quasiparticle bands near the transition. Recent photoemission data for Ca$_2$CuO$_2$Cl$_2$ are analyzed within this context.

Universality Class of the Antiferromagnetic Transition in the Two Dimensional Hubbard Model

We analyze the deformations of the Fermi surface induced by electron-electron interactions in anisotropic two-dimensional systems. We use perturbation theory to treat, on the same footing, the regular and singular regions of the Fermi surface. It is shown that, even for weak local coupling, the self-energy presents a nontrivial behavior showing momentum dependence and interplay with the Fermi surface shape. Our scheme gives simple analytical expressions based on local features of the Fermi surface.

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Deformation of anisotropic Fermi surfaces due to electron-electron interactions

We study the properties of heterostructures formed by two layers of diluted magnetic semiconductor separated by a nonmagnetic semiconductor layer. We find that there is a RKKY-type exchange coupling between the magnetic layers that oscilles between ferromagnetic and antiferromagnetic as a function of the different parameters in the problem. The different transport properties of these phases make that this heterostructure presents strong magnetoresistive effects. The coupling can be also modified by an electric field. We propose that it is possible to alter dramatically the electrical resistance of the heterostructure by applying an electric field. Our results indicate that in a single gated sample the magnetoresistance could be modulated by with an electrical bias voltage.

M. P. López-Sancho

Papers

Renormalization group approach to anisotropic superconductivity

Ferromagnetism and Disorder in Graphene

Universality Class of the Antiferromagnetic Transition in the Two Dimensional Hubbard Model

Deformation of anisotropic Fermi surfaces due to electron-electron interactions

Electromodulation of the Magnetoresistance in Diluted Magnetic Semiconductors Based Heterostructures