A. H. Castro Neto

The electronic properties 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.

Making graphene visible

Microfabrication of graphene devices used in many experimental studies currently relies on the fact that graphene crystallites can be visualized using optical microscopy if prepared on top of Si wafers with a certain thickness of SiO2. The authors study graphene’s visibility and show that it depends strongly on both thickness of SiO2 and light wavelength. They have found that by using monochromatic illumination, graphene can be isolated for any SiO2 thickness, albeit 300nm (the current standard) and, especially, ≈100nm are most suitable for its visual detection. By using a Fresnel-law-based model, they quantitatively describe the experimental data.

Strong Light-Matter Interactions in Heterostructures of Atomically Thin Films

Atomic Layer Heterostructures—More Is More The isolation of stable layers of various materials, only an atom or several atoms thick, has provided the opportunity to fabricate devices with novel functionality and to probe fundamental physics. Britnell et al. (p. 1311, published online 2 May; see the Perspective by Hamm and Hess) sandwiched a single layer of the transition metal dichalcogenide WS2 between two sheets of graphene. The photocurrent response of the heterostructure device was enhanced, compared to that of the bare layer of WS2. The prospect of combining single or several-atom-thick layers into heterostructures should help to develop materials with a wide range of properties. Transition metal dichalcogenides sandwiched between two layers of graphene produce an enhanced photoresponse. [Also see Perspective by Hamm and Hess] The isolation of various two-dimensional (2D) materials, and the possibility to combine them in vertical stacks, has created a new paradigm in materials science: heterostructures based on 2D crystals. Such a concept has already proven fruitful for a number of electronic applications in the area of ultrathin and flexible devices. Here, we expand the range of such structures to photoactive ones by using semiconducting transition metal dichalcogenides (TMDCs)/graphene stacks. Van Hove singularities in the electronic density of states of TMDC guarantees enhanced light-matter interactions, leading to enhanced photon absorption and electron-hole creation (which are collected in transparent graphene electrodes). This allows development of extremely efficient flexible photovoltaic devices with photoresponsivity above 0.1 ampere per watt (corresponding to an external quantum efficiency of above 30%).

Biased bilayer graphene: Semiconductor with a gap tunable by the electric field effect

We demonstrate that the electronic gap of a graphene bilayer can be controlled externally by applying a gate bias. From the magnetotransport data (Shubnikov-de Haas measurements of the cyclotron mass), and using a tight-binding model, we extract the value of the gap as a function of the electronic density. We show that the gap can be changed from zero to midinfrared energies by using fields of less, approximately < 1 V/nm, below the electric breakdown of SiO2. The opening of a gap is clearly seen in the quantum Hall regime.

Gate-tuning of graphene plasmons revealed by infrared nano-imaging

Surface plasmons are collective oscillations of electrons in metals or semiconductors that enable confinement and control of electromagnetic energy at subwavelength scales. Rapid progress in plasmonics has largely relied on advances in device nano-fabrication, whereas less attention has been paid to the tunable properties of plasmonic media. One such medium—graphene—is amenable to convenient tuning of its electronic and optical properties by varying the applied voltage. Here, using infrared nano-imaging, we show that common graphene/SiO2/Si back-gated structures support propagating surface plasmons. The wavelength of graphene plasmons is of the order of 200 nanometres at technologically relevant infrared frequencies, and they can propagate several times this distance. We have succeeded in altering both the amplitude and the wavelength of these plasmons by varying the gate voltage. Using plasmon interferometry, we investigated losses in graphene by exploring real-space profiles of plasmon standing waves formed between the tip of our nano-probe and the edges of the samples. Plasmon dissipation quantified through this analysis is linked to the exotic electrodynamics of graphene. Standard plasmonic figures of merit of our tunable graphene devices surpass those of common metal-based structures.

Electronic properties of disordered two-dimensional carbon

Two-dimensional carbon, or graphene, is a semimetal that presents unusual low-energy electronic excitations described in terms of Dirac fermions. We analyze in a self-consistent way the effects of localized (impurities or vacancies) and extended (edges or grain boundaries) defects on the electronic and transport properties of graphene. On the one hand, point defects induce a finite elastic lifetime at low energies with the enhancement of the electronic density of states close to the Fermi level. Localized disorder leads to a universal, disorder independent, electrical conductivity at low temperatures, of the order of the quantum of conductance. The static conductivity increases with temperature and shows oscillations in the presence of a magnetic field. The graphene magnetic susceptibility is temperature dependent (unlike an ordinary metal) and also increases with the amount of defects. Optical transport properties are also calculated in detail. On the other hand, extended defects induce localized states near the Fermi level. In the absence of electron-hole symmetry, these states lead to a transfer of charge between the defects and the bulk, the phenomenon we call self-doping. The role of electron-electron interactions in controlling self-doping is also analyzed. We also discuss the integer and fractional quantum Hall effect in graphene, the role played by the edge states induced by a magnetic field, and their relation to the almost field independent surface states induced at boundaries. The possibility of magnetism in graphene, in the presence of short-range electron-electron interactions and disorder is also analyzed.

Strain-induced Pseudo-Magnetic Fields Greater Than 300 Tesla in Graphene Nanobubbles

Straining Graphenes Electronic States The conduction electrons in graphene, single sheets of graphite, can have very high mobilities. Under the influence of an applied magnetic field, a series of energy steps, or Landau levels, can be observed that correspond to the conduction electrons traveling in cyclotron orbits. Recent theoretical work has indicated that if graphene layers are strained, the strain field creates a pseudomagnetic field that should also lead to observable Landau levels. Levy et al. (p. 544) used scanning tunneling microscopy to probe the energy levels of graphene grown on a platinum surface, which forms highly strained “nanobubbles.” The strain is equivalent to applying very high magnetic fields (in excess of 300 tesla). Thus, the electronic properties of graphene can indeed be modified using applied strain. Strain creates energy levels in graphene that are similar to those seen in very high applied magnetic fields. Recent theoretical proposals suggest that strain can be used to engineer graphene electronic states through the creation of a pseudo–magnetic field. This effect is unique to graphene because of its massless Dirac fermion-like band structure and particular lattice symmetry (C3v). Here, we present experimental spectroscopic measurements by scanning tunneling microscopy of highly strained nanobubbles that form when graphene is grown on a platinum (111) surface. The nanobubbles exhibit Landau levels that form in the presence of strain-induced pseudo–magnetic fields greater than 300 tesla. This demonstration of enormous pseudo–magnetic fields opens the door to both the study of charge carriers in previously inaccessible high magnetic field regimes and deliberate mechanical control over electronic structure in graphene or so-called “strain engineering.”

Electric field effect in ultrathin black phosphorus

Black phosphorus exhibits a layered structure similar to graphene, allowing mechanical exfoliation of ultrathin single crystals. Here, we demonstrate few-layer black phosphorus field effect devices on Si/SiO2 and measure charge carrier mobility in a four-probe configuration as well as drain current modulation in a two-point configuration. We find room-temperature mobilities of up to 300 cm2/Vs and drain current modulation of over 103. At low temperatures, the on-off ratio exceeds 105, and the device exhibits both electron and hole conduction. Using atomic force microscopy, we observe significant surface roughening of thin black phosphorus crystals over the course of 1 h after exfoliation.

Tight-binding approach to uniaxial strain in graphene

We analyze the effect of tensional strain in the electronic structure of graphene. In the absence of electron-electron interactions, within linear elasticity theory, and a tight-binding approach, we observe that strain can generate a bulk spectral gap. However, this gap is critical, requiring threshold deformations in excess of 20% and only along preferred directions with respect to the underlying lattice. The gapless Dirac spectrum is robust for small and moderate deformations and the gap appears as a consequence of the merging of the two inequivalent Dirac points only under considerable deformations of the lattice. We discuss how strain-induced anisotropy and local deformations can be used as a means to affect transport characteristics and pinch off current flow in graphene devices.

Strain engineering of graphene's electronic structure.

We propose a route to all-graphene integrated electronic devices by exploring the influence of strain on the electronic structure of graphene. We show that strain can be easily tailored to generate electron beam collimation, 1D channels, surface states and confinement, the basic elements for allgraphene electronics. In addition this proposal has the advantage that patterning can be made on substrates rather than on the graphene sheet, thereby protecting the integrity of the latter. PACS numbers: 81.05.Uw,85.30.Mn,73.90.+f 1 ar X iv :0 81 0. 45 39 v3 [ co nd -m at .m es -h al l] 1 9 Fe b 20 09 Notwithstanding its atomic thickness, graphene sheets have been shown to accommodate a wealth of remarkable fundamental properties, and to hold sound prospects in the context of a new generation of electronic devices and circuitry [1]. The exciting prospect about graphene is that, not only can we have extremely good conductors, but also most active devices made out of graphene. One of the current difficulties with respect to this lies in that conventional electronic operations require the ability to completely pinch-off the charge transport on demand. Although the electric field effect is impressive in graphene [2], the existence of a minimum of conductivity poses a serious obstacle towards desirable on/off ratios. A gapped spectrum would certainly be instrumental. The presence of a gap is implicitly related to the problem of electron confinement, which for Dirac fermions is not easily achievable by conventional means (like electrostatic potential wells) [3]. Geometrical confinement has been achieved in graphene ribbons and dots [4, 5], but the sensitivity of transport to the edge profile [6], and the inherent difficulty in the fabrication of such microstructures with sharply defined edges remains a problem. The ultimate goal would be an all-graphene circuit. This could be achieved by taking a graphene sheet and patterning the different devices and leads by means of appropriate cuts that would generate leads ribbons, dots, etc.. This papercutting electronics can have serious limitations with respect to reliability, scalability, and is prone to damaging and inducing disorder in the graphene sheet [7]. Therefore, in keeping with the paper art analogy, we propose an alternative origami electronics [8]. We show here that all the characteristics of graphene ribbons and dots (viz. geometrical quantization, 1D channels, surface modes) might be locally obtained by patterning, not graphene, but the substrate on which it rests. The essential aspect of our approach is the generation of strain in the graphene lattice capable of changing the in-plane hopping amplitude in an anisotropic way. This can be achieved by means of appropriate geometrical patterns in an homogeneous substrate (grooves, creases, steps or wells), or by means of an heterogeneous substrate in which different regions interact differently with the graphene sheet, generating different strain profiles [Fig. 1(b)]. Another design alternative consists in depositing graphene onto substrates with regions that can be controlably strained on demand [9]. Through a combination of folding and/or clamping a graphene sheet onto such substrate patterns, one might generate local strain profiles suitable for the applications discussed in detail below, while preserving a whole graphene sheet.