Graphene is a rapidly rising star on the horizon of materials science and condensed-matter physics. This strictly two-dimensional material exhibits exceptionally high crystal and electronic quality, and, despite its short history, has already revealed a cornucopia of new physics and potential applications, which are briefly discussed here. Whereas one can be certain of the realness of applications only when commercial products appear, graphene no longer requires any further proof of its importance in terms of fundamental physics. Owing to its unusual electronic spectrum, graphene has led to the emergence of a new paradigm of 'relativistic' condensed-matter physics, where quantum relativistic phenomena, some of which are unobservable in high-energy physics, can now be mimicked and tested in table-top experiments. More generally, graphene represents a conceptually new class of materials that are only one atom thick, and, on this basis, offers new inroads into low-dimensional physics that has never ceased to surprise and continues to provide a fertile ground for applications.

The rise of graphene

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

CARBON nanotubes1 are expected to have a wide variety of interesting properties. Capillarity in open tubes has already been demonstrated2–5, while predictions regarding their electronic structure6–8 and mechanical strength9 remain to be tested. To examine the properties of these structures, one needs tubes with well defined morphologies, length, thickness and a number of concentric shells; but the normal carbon-arc synthesis10,11 yields a range of tube types. In particular, most calculations have been concerned with single-shell tubes, whereas the carbon-arc synthesis produces almost entirely multi-shell tubes. Here we report the synthesis of abundant single-shell tubes with diameters of about one nanometre. Whereas the multi-shell nanotubes are formed on the carbon cathode, these single-shell tubes grow in the gas phase. Electron diffraction from a single tube allows us to confirm the helical arrangement of carbon hexagons deduced previously for multi-shell tubes1.

Single-shell carbon nanotubes of 1-nm diameter

The use of individual molecules as functional electronic devices was first proposed in the 1970s (ref 1) Since then, molecular electronics2,3 has attracted much interest, particularly because it could lead to conceptually new miniaturization strategies in the electronics and computer industry The realization of single-molecule devices has remained challenging, largely owing to difficulties in achieving electrical contact to individual molecules Recent advances in nanotechnology, however, have resulted in electrical measurements on single molecules4,5,6,7 Here we report the fabrication of a field-effect transistor—a three-terminal switching device—that consists of one semiconducting8,9,10 single-wall carbon nanotube11,12 connected to two metal electrodes By applying a voltage to a gate electrode, the nanotube can be switched from a conducting to an insulating state We have previously reported5 similar behaviour for a metallic single-wall carbon nanotube operated at extremely low temperatures The present device, in contrast, operates at room temperature, thereby meeting an important requirement for potential practical applications Electrical measurements on the nanotube transistor indicate that its operation characteristics can be qualitatively described by the semiclassical band-bending models currently used for traditional semiconductor devices The fabrication of the three-terminal switching device at the level of a single molecule represents an important step towards molecular electronics

Room-temperature transistor based on a single carbon nanotube

We investigate electronic transport in lithographically patterned graphene ribbon structures where the lateral confinement of charge carriers creates an energy gap near the charge neutrality point. Individual graphene layers are contacted with metal electrodes and patterned into ribbons of varying widths and different crystallographic orientations. The temperature dependent conductance measurements show larger energy gaps opening for narrower ribbons. The sizes of these energy gaps are investigated by measuring the conductance in the nonlinear response regime at low temperatures. We find that the energy gap scales inversely with the ribbon width, thus demonstrating the ability to engineer the band gap of graphene nanostructures by lithographic processes.

Energy band-gap engineering of graphene nanoribbons.

Finite graphite systems having a zigzag edge exhibit a special edge state. The corresponding energy bands are almost flat at the Fermi level and thereby give a sharp peak in the density of states. The charge density in the edge state is strongly localized on the zigzag edge sites. No such localized state appears in graphite systems having an armchair edge. By utilizing the graphene ribbon model, we discuss the effect of the system size and edge shape on the special edge state. By varying the width of the graphene ribbons, we find that the nanometer size effect is crucial for determining the relative importance of the edge state. We also have extended the graphene ribbon to have edges of a general shape, which is defined as a mixture of zigzag and armchair sites. Examining the relative importance of the edge state for graphene ribbons with general edges, we find that a non-negligible edge state survives even in graphene ribbons with less developed zigzag edges. We demonstrate that such an edge shape with three or four zigzag sites per sequence is sufficient to show an edge state, when the system size is on a nanometer scale. The special characteristics of the edge state play a large role in determining the density of states near the Fermi level for graphite networks on a nanometer scale.

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Edge state in graphene ribbons: Nanometer size effect and edge shape dependence.

We study the electronic states of graphite ribbons with edges of two typical shapes, armchair and zigzag, by performing tight binding band calculations, and find that the graphite ribbons show striking contrast in the electronic states depending on the edge shape. In particular, a zigzag ribbon shows a remarkably sharp peak of density of states at the Fermi level, which does not originate from infinite graphite. We find that the singular electronic states arise from the partly flat bands at the Fermi level, whose wave functions are mainly localized on the zigzag edge. We reveal the puzzle for the emergence of the peculiar edge state by deriving the analytic form in the case of semi-infinite graphite with a zigzag edge. Applying the Hubbard model within the mean-field approximation, we discuss the possible magnetic structure in nanometer-scale micrographite.

Peculiar Localized State at Zigzag Graphite Edge

The electronic structure for graphenemonolayer tubules is predicted as a function of the diameter and helicity of the constituent graphene tubules. The calculated results show that approximately 1/3 of these tubules are a one‐dimensional metal which is stable against a Peierls distortion, and the other 2/3 are one‐dimensional semiconductors. The implications of these results are discussed.

Electronic structure of chiral graphene tubules

The electronic structures of some possible carbon fibers nucleated from the hemisphere of a ${\mathrm{C}}_{60}$ molecule are presented. A one-dimensional electronic band-structure model of such carbon fibers, having not only rotational symmetry but also screw axes, is derived by folding the two-dimensional energy bands of graphite. A simple tight-binding model shows that some fibers are metallic and are stable against perturbations of the one-dimensional energy bands and the mixing of \ensuremath{\sigma} and \ensuremath{\pi} bands due to the curvature of the circumference of the fiber.

Electronic structure of graphene tubules based on C 60

Electronic and magnetic properties of ribbon-shaped nanographite systems with zigzag and armchair edges in a magnetic field are investigated by using a tight-binding model. One of the most remarkable features of these systems is the appearance of edge states, strongly localized near zigzag edges. The edge state in a magnetic field, generating a rational fraction of the magnetic flux ( f5 p/q) in each hexagonal plaquette of the graphite plane, behaves like a zero-field edge state with q internal degrees of freedom. The orbital diamagnetic susceptibility strongly depends on the edge shapes. The reason is found in the analysis of the ring currents, which are very sensitive to the lattice topology near the edge. Moreover, the orbital diamagnetic susceptibility is scaled as a function of the temperature, Fermi energy, and ribbon width. Because the edge states lead to a sharp peak in the density of states at the Fermi level, the graphite ribbons with zigzag edges show Curie-like temperature dependence of the Pauli paramagnetic susceptibility. Hence, there is a crossover from hightemperature diamagnetic to low-temperature paramagnetic behavior in the magnetic susceptibility of nanographite ribbons with zigzag edges. @S0163-1829~99!02111-6#

https://arxiv.org/pdf/cond-mat/9809260.pdf

Mitsutaka Fujita

Papers

Edge state in graphene ribbons: Nanometer size effect and edge shape dependence.

Peculiar Localized State at Zigzag Graphite Edge

Electronic structure of chiral graphene tubules

Electronic structure of graphene tubules based on C 60

Electronic and magnetic properties of nanographite ribbons