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

Raman spectroscopy of graphene and graphite: Disorder, electron phonon coupling, doping and nonadiabatic effects

We investigate Raman spectra of graphite oxide and functionalized graphene sheets with epoxy and hydroxyl groups and Stone−Wales and 5−8−5 defects by first-principles calculations to interpret our experimental results. Only the alternating pattern of single−double carbon bonds within the sp2 carbon ribbons provides a satisfactory explanation for the experimentally observed blue shift of the G band of the Raman spectra relative to graphite. To obtain these single−double bonds, it is necessary to have sp3 carbons on the edges of a zigzag carbon ribbon.

Raman Spectra of Graphite Oxide and Functionalized Graphene Sheets

Molybdenum disulfide (MoS2) of single- and few-layer thickness was exfoliated on SiO2/Si substrate and characterized by Raman spectroscopy. The number of S−Mo−S layers of the samples was independently determined by contact-mode atomic force microscopy. Two Raman modes, E12g and A1g, exhibited sensitive thickness dependence, with the frequency of the former decreasing and that of the latter increasing with thickness. The results provide a convenient and reliable means for determining layer thickness with atomic-level precision. The opposite direction of the frequency shifts, which cannot be explained solely by van der Waals interlayer coupling, is attributed to Coulombic interactions and possible stacking-induced changes of the intralayer bonding. This work exemplifies the evolution of structural parameters in layered materials in changing from the three-dimensional to the two-dimensional regime.

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Anomalous lattice vibrations of single- and few-layer MoS2.

Every few years, a new material with unique properties emerges and fascinates the scientific community, typical recent examples being high-temperature superconductors and carbon nanotubes. Graphene is the latest sensation with unusual properties, such as half-integer quantum Hall effect and ballistic electron transport. This two-dimensional material which is the parent of all graphitic carbon forms is strictly expected to comprise a single layer, but there is considerable interest in investigating two-layer and few-layer graphenes as well. Synthesis and characterization of graphenes pose challenges, but there has been considerable progress in the last year or so. Herein, we present the status of graphene research which includes aspects related to synthesis, characterization, structure, and properties.

Graphene: The New Two-Dimensional Nanomaterial

We present Raman spectroscopy measurements on single- and few-layer graphene flakes. By using a scanning confocal approach, we collect spectral data with spatial resolution, which allows us to directly compare Raman images with scanning force micrographs. Single-layer graphene can be distinguished from double- and few-layer by the width of the D' line: the single peak for single-layer graphene splits into different peaks for the double-layer. These findings are explained using the double-resonant Raman model based on ab initio calculations of the electronic structure and of the phonon dispersion. We investigate the D line intensity and find no defects within the flake. A finite D line response originating from the edges can be attributed either to defects or to the breakdown of translational symmetry.

Spatially Resolved Raman Spectroscopy of Single- and Few-Layer Graphene

We report on Coulomb blockade and Coulomb diamond measurements on an etched, tunable single-layer graphene quantum dot. The device consisting of a graphene island connected via two narrow graphene constrictions is fully tunable by three lateral graphene gates. Coulomb blockade resonances are observed and from Coulomb diamond measurements, a charging energy of ≈3.5meV is extracted. For increasing temperatures, we detect a peak broadening and a transmission increase of the nanostructured graphene barriers.

https://people.phys.ethz.ch/~ihn/papers/Stampfer08.pdf

Tunable Coulomb blockade in nanostructured graphene

We present spatially resolved Raman images of the G and 2D lines of single-layer graphene flakes. The spatial fluctuations of G and 2D lines are correlated and are thus shown to be affiliated with local doping domains. We investigate the position of the 2D line—the most significant Raman peak to identify single-layer graphene—as a function of charging up to ∣n∣≈4×1012cm−2. Contrary to the G line which exhibits a strong and symmetric stiffening with respect to electron and hole doping, the 2D line shows a weak and slightly asymmetric stiffening for low doping. Additionally, the linewidth of the 2D line is, in contrast to the G line, doping independent making this quantity a reliable measure for identifying single-layer graphene.

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Raman imaging of doping domains in graphene on SiO2

We report on Coulomb blockade and Coulomb diamond measurements on an etched, tunable single-layer graphene quantum dot. The device consisting of a graphene island connected via two narrow graphene constrictions is fully tunable by three lateral graphene gates. Coulomb blockade resonances are observed and from Coulomb diamond measurements a charging energy of ~3.5 meV is extracted. For increasing temperatures we detect a peak broadening and a transmission increase of the nanostructured graphene barriers.

We present spatially resolved Raman images of the G and 2D lines of single-layer graphene flakes. The spatial fluctuations of G and 2D lines are correlated and are thus shown to be affiliated with local doping domains. We investigate the position of the 2D line -- the most significant Raman peak to identify single-layer graphene -- as a function of charging up to |n|~4 10^12 cm^-2. Contrary to the G line which exhibits a strong and symmetric stiffening with respect to electron and hole-doping, the 2D line shows a weak and slightly asymmetric stiffening for low doping. Additionally, the line width of the 2D line is, in contrast to the G line, doping-independent making this quantity a reliable measure for identifying single-layer graphene.

/pdf/raman-imaging-of-doping-domains-in-graphene-on-sio2-2u9sbp9hc0.pdf

D. Graf

Papers

Spatially Resolved Raman Spectroscopy of Single- and Few-Layer Graphene

Tunable Coulomb blockade in nanostructured graphene

Raman imaging of doping domains in graphene on SiO2

Tunable Coulomb blockade in nanostructured graphene

Raman imaging of doping domains in graphene on SiO2