Electric Field Effect in Atomically Thin Carbon Films

Graphene nanoFlakes with large spin

We investigate, using benzenoid graph theory and first-principles calculations, the magnetic properties of arbitrarily shaped finite graphene fragments to which we refer as graphene nanoflakes (GNFs). We demonstrate that the spin of a GNF depends on its shape due to topological frustration of the pi-bonds. For example, a zigzag-edged triangular GNF has a nonzero net spin, resembling an artificial ferrimagnetic atom, with the spin value scaling with its linear size. In general, the principle of topological frustration can be used to introduce large net spin and interesting spin distributions in graphene. These results suggest an avenue to nanoscale spintronics through the sculpting of graphene fragments.

Molecular doping of graphene

Graphene is considered as one of the most promising materials for post silicon electronics, as it combines high electron mobility with atomic thickness [Novoselov et al. Science 2004, 306, 666-669. Novoselov et al. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 10451-10453]. The possibility of chemical doping and related excellent chemical sensor properties of graphene have been demonstrated experimentally [Schedin et al. Nat. Mater. 2007, 6, 652-655], but a microscopic understanding of these effects has been lacking, so far. In this letter, we present the first joint experimental and theoretical investigation of adsorbate-induced doping of graphene. A general relation between the doping strength and whether adsorbates are open- or closed-shell systems is demonstrated with the NO2 system: The single, open shell NO2 molecule is found to be a strong acceptor, whereas its closed shell dimer N2O4 causes only weak doping. This effect is pronounced by graphene's peculiar density of states (DOS), which provides an ideal situation for model studies of doping effects in semiconductors. We show that this DOS is ideal for "chemical sensor" applications and explain the recently observed [Schedin et al. Nat. Mater. 2007, 6, 652-655] NO2 single molecule detection.

Transparent, conductive graphene electrodes for dye-sensitized solar cells

Transparent, conductive, and ultrathin graphene films, as an alternative to the ubiquitously employed metal oxides window electrodes for solid-state dye-sensitized solar cells, are demonstrated. These graphene films are fabricated from exfoliated graphite oxide, followed by thermal reduction. The obtained films exhibit a high conductivity of 550 S/cm and a transparency of more than 70% over 1000-3000 nm. Furthermore, they show high chemical and thermal stabilities as well as an ultrasmooth surface with tunable wettability.

Graphene bilayer with a twist: electronic structure

We consider a graphene bilayer with a relative small angle rotation between the layers--a stacking defect often seen in the surface of graphite--and calculate the electronic structure near zero energy in a continuum approximation. Contrary to what happens in an AB stacked bilayer and in accord with observations in epitaxial graphene, we find: (a) the low energy dispersion is linear, as in a single layer, but the Fermi velocity can be significantly smaller than the single-layer value; (b) an external electric field, perpendicular to the layers, does not open an electronic gap.

Measurement of scattering rate and minimum conductivity in graphene

The conductivity of graphene samples with various levels of disorder is investigated for a set of specimens with mobility in the range of 1-20x10(3) cm2/V sec. Comparing the experimental data with the theoretical transport calculations based on charged impurity scattering, we estimate that the impurity concentration in the samples varies from 2-15x10(11) cm(-2). In the low carrier density limit, the conductivity exhibits values in the range of 2-12e2/h, which can be related to the residual density induced by the inhomogeneous charge distribution in the samples. The shape of the conductivity curves indicates that high mobility samples contain some short-range disorder whereas low mobility samples are dominated by long-range scatterers.

Magnetic and quantum confinement effects on electronic and optical properties of graphene ribbons

Through the tight-binding calculation, we demonstrate that magnetic and quantum confinements have a great influence on the low-energy band structures of one-dimensional (1D) armchair graphene ribbons. The magnetic field first changes 1D parabolic bands into the Hall-edge states which originate in the Landau wavefunctions deformed by one or two ribbon edges. The quantum confinement dominates the characteristics of the Hall-edge states only when the Landau wavefunctions touch two ribbon edges. Then, some of the Hall-edge states evolve as the Landau states when the field strength grows. The partial flat bands (Landau levels), related to the Landau states, appear. The magnetic field dramatically modifies the energy dispersions and it changes the size of the bandgap, shifts the band-edge states, destroys the degeneracy of the energy bands, induces the semiconductor-metal transition and generates the partial flat bands. The above-mentioned magneto-electronic properties are completely reflected in the low-frequency absorption spectra--the shift of peak position, the change of peak symmetry, the alteration of peak height, the generation of new peaks and the change of absorption edges. As a result, there are magnetic-field-dependent absorption frequencies. The findings show that the magnetic field could be used to modulate the electronic properties and the absorption spectra.

Valley-contrasting physics in graphene: magnetic moment and topological transport

We investigate physical properties that can be used to distinguish the valley degree of freedom in systems where inversion symmetry is broken, using graphene systems as examples. We show that the pseudospin associated with the valley index of carriers has an intrinsic magnetic moment, in close analogy with the Bohr magneton for the electron spin. There is also a valley dependent Berry phase effect that can result in a valley contrasting Hall transport, with carriers in different valleys turning into opposite directions transverse to an in-plane electric field. These effects can be used to generate and detect valley polarization by magnetic and electric means, forming the basis for the valley-based electronics applications.

Graphene nanostrip digital memory device

In equilibrium, graphene nanostrips, with hydrogens sp2-bonded to carbons along their zigzag edges, are expected to exhibit a spin-polarized ground state. However, in the presence of a ballistic current, we find that there exists a voltage range over which both spin-polarized and spin-unpolarized nanostrip states are stable. These states can represent a bit in a binary memory device that could be switched through the applied bias and read by measuring the current through the nanostrip.

Quantum critical scaling in graphene

We show that the emergent relativistic symmetry of electrons in graphene near its quantum critical point (QCP) implies a crucial importance of the Coulomb interaction. We derive scaling laws, valid near the QCP, that dictate the nontrivial magnetic and charge response of interacting graphene. Our analysis yields numerous predictions for how the Coulomb interaction will be manifested in experimental observables such as the diamagnetic response and electronic compressibility.

Sample-size effects in the magnetoresistance of graphite

Conduction electrons in graphite are expected to have micrometer large de Broglie wavelength as well as mean free path. A direct influence of these lengths in the electric transport properties of finite-size samples was neglected in the past. We provide a direct evidence of this effect through the size dependence of the magnetoresistance, which decreases with the sample size even for samples hundreds of micrometers large. Our findings may explain the absence of magnetoresistance in small few graphene layers samples and ask for a general revision of the experimental and theoretical work on the transport properties of this material.

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.

Minimum electrical and thermal conductivity of graphene: a quasiclassical approach

We investigate the minimum conductivity of graphene within a quasiclassical approach taking into account electron-hole coherence effects which stem from the chiral nature of low energy excitations. Relying on an analytical solution of the kinetic equation in the electron-hole coherent and incoherent cases, we study both the electrical and the thermal conductivity whose relation satisfies the Wiedemann-Franz law. We find that most of the previous findings based on the Boltzmann equation are restricted to only high mobility samples where electron-hole coherence effects are not sufficient.

Quantum-Hall activation gaps in graphene

We have measured the quantum-Hall activation gaps in graphene at filling factors nu=2 and nu=6 for magnetic fields up to 32 T and temperatures from 4 to 300 K. The nu=6 gap can be described by thermal excitation to broadened Landau levels with a width of 400 K. In contrast, the gap measured at nu=2 is strongly temperature and field dependent and approaches the expected value for sharp Landau levels for fields B>20 T and temperatures T>100 K. We explain this surprising behavior by a narrowing of the lowest Landau level.

A self-consistent theory for graphene transport

We demonstrate theoretically that most of the observed transport properties of graphene sheets at zero magnetic field can be explained by scattering from charged impurities. We find that, contrary to common perception, these properties are not universal but depend on the concentration of charged impurities n(imp). For dirty samples (250 x 10(10) cm(-2) < n(imp) < 400 x 10(10) cm(-2)), the value of the minimum conductivity at low carrier density is indeed 4e(2)/h in agreement with early experiments, with weak dependence on impurity concentration. For cleaner samples, we predict that the minimum conductivity depends strongly on n(imp), increasing to 8e(2)/h for n(imp) approximately 20 x 10(10) cm(-2). A clear strategy to improve graphene mobility is to eliminate charged impurities or use a substrate with a larger dielectric constant.

Odd-integer quantum Hall effect in graphene: interaction and disorder effects

We study the competition between the long-range Coulomb interaction, disorder scattering, and lattice effects in the integer quantum Hall effect (IQHE) in graphene. By direct transport calculations, both nu=1 and nu=3 IQHE states are revealed in the lowest two Dirac Landau levels. However, the critical disorder strength above which the nu=3 IQHE is destroyed is much smaller than that for the nu=1 IQHE, which may explain the absence of a nu=3 plateau in recent experiments. While the excitation spectrum in the IQHE phase is gapless within numerical finite-size analysis, we do find and determine a mobility gap, which characterizes the energy scale of the stability of the IQHE. Furthermore, we demonstrate that the nu=1 IQHE state is a Dirac valley and sublattice polarized Ising pseudospin ferromagnet, while the nu=3 state is an xy plane polarized pseudospin ferromagnet.

Quasiparticle energies and band gaps in graphene nanoribbons

We present calculations of the quasiparticle energies and band gaps of graphene nanoribbons (GNRs) carried out using a first-principles many-electron Green's function approach within the GW approximation. Because of the quasi-one-dimensional nature of a GNR, electron-electron interaction effects due to the enhanced screened Coulomb interaction and confinement geometry greatly influence the quasiparticle band gap. Compared with previous tight-binding and density functional theory studies, our calculated quasiparticle band gaps show significant self-energy corrections for both armchair and zigzag GNRs, in the range of 0.5-3.0 eV for ribbons of width 2.4-0.4 nm. The quasiparticle band gaps found here suggest that use of GNRs for electronic device components in ambient conditions may be viable.

Intrinsic ripples in graphene

The stability of two-dimensional (2D) layers and membranes is the subject of a long-standing theoretical debate. According to the so-called Mermin-Wagner theorem, long-wavelength fluctuations destroy the long-range order of 2D crystals. Similarly, 2D membranes embedded in a 3D space have a tendency to be crumpled. These fluctuations can, however, be suppressed by anharmonic coupling between bending and stretching modes meaning that a 2D membrane can exist but will exhibit strong height fluctuations. The discovery of graphene, the first truly 2D crystal, and the recent experimental observation of ripples in suspended graphene make these issues especially important. Besides the academic interest, understanding the mechanisms of the stability of graphene is crucial for understanding electronic transport in this material that is attracting so much interest owing to its unusual Dirac spectrum and electronic properties. We address the nature of these height fluctuations by means of atomistic Monte Carlo simulations based on a very accurate many-body interatomic potential for carbon. We find that ripples spontaneously appear owing to thermal fluctuations with a size distribution peaked around 80 A which is compatible with experimental findings (50-100 A). This unexpected result might be due to the multiplicity of chemical bonding in carbon.

A chemical route to graphene for device applications

Oxidation of graphite produces graphite oxide, which is dispersible in water as individual platelets. After deposition onto Si/SiO2 substrates, chemical reduction produces graphene sheets. Electrical conductivity measurements indicate a 10000-fold increase in conductivity after chemical reduction to graphene. Tapping mode atomic force microscopy measurements show one to two layer graphene steps. Electrodes patterned onto a reduced graphite oxide film demonstrate a field effect response when the gate voltage is varied from +15 to -15 V. Temperature-dependent conductivity indicates that the graphene-like sheets exhibit semiconducting behavior.

Formation of single-walled carbon nanotube via the interaction of graphene nanoribbons: ab initio density functional calculations

The interaction of bare graphene nanoribbons (GNRs) was investigated by ab initio density functional theory calculations with both the local density approximation (LDA) and the generalized gradient approximation (GGA). Remarkably, two bare 8-GNRs with zigzag-shaped edges are predicted to form an (8, 8) armchair single-wall carbon nanotube (SWCNT) without any obvious activation barrier. The formation of a (10, 0) zigzag SWCNT from two bare 10-GNRs with armchair-shaped edges has activation barriers of 0.23 and 0.61 eV for using the LDA and the revised PBE exchange correlation functional, respectively. Our results suggest a possible route to control the growth of specific types SWCNT via the interaction of GNRs.

Electronic transport properties of individual chemically reduced graphene oxide sheets

Individual graphene oxide sheets subjected to chemical reduction were electrically characterized as a function of temperature and external electric fields. The fully reduced monolayers exhibited conductivities ranging between 0.05 and 2 S/cm and field effect mobilities of 2-200 cm2/Vs at room temperature. Temperature-dependent electrical measurements and Raman spectroscopic investigations suggest that charge transport occurs via variable range hopping between intact graphene islands with sizes on the order of several nanometers. Furthermore, the comparative study of multilayered sheets revealed that the conductivity of the undermost layer is reduced by a factor of more than 2 as a consequence of the interaction with the Si/SiO2 substrate.

Random resistor network model of minimal conductivity in graphene

Transport in undoped graphene is related to percolating current patterns in the networks of n- and p-type regions reflecting the strong bipolar charge density fluctuations. Finite transparency of the p-n junctions is vital in establishing the macroscopic conductivity. We propose a random resistor network model to analyze scaling dependencies of the conductance on the doping and disorder, the quantum magnetoresistance and the corresponding dephasing rate.

Magnetism in graphene nanoislands

We study the magnetic properties of nanometer-sized graphene structures with triangular and hexagonal shapes terminated by zigzag edges. We discuss how the shape of the island, the imbalance in the number of atoms belonging to the two graphene sublattices, the existence of zero-energy states, and the total and local magnetic moment are intimately related. We consider electronic interactions both in a mean-field approximation of the one-orbital Hubbard model and with density functional calculations. Both descriptions yield values for the ground state total spin S consistent with Lieb's theorem for bipartite lattices. Triangles have a finite S for all sizes whereas hexagons have S=0 and develop local moments above a critical size of approximately 1.5 nm.

Electronic transport and quantum hall effect in bipolar graphene p-n-p junctions

We have developed a device fabrication process to pattern graphene into nanostructures of arbitrary shape and control their electronic properties using local electrostatic gates. Electronic transport measurements have been used to characterize locally gated bipolar graphene p-n-p junctions. We observe a series of fractional quantum Hall conductance plateaus at high magnetic fields as the local charge density is varied in the p and n regions. These fractional plateaus, originating from chiral edge states equilibration at the p-n interfaces, exhibit sensitivity to interedge backscattering which is found to be strong for some of the plateaus and much weaker for other plateaus. We use this effect to explore the role of backscattering and estimate disorder strength in our graphene devices.

Coulomb blockade in graphene nanoribbons

We propose that recent transport experiments revealing the existence of an energy gap in graphene nanoribbons may be understood in terms of Coulomb blockade. Electron interactions play a decisive role at the quantum dots which form due to the presence of necks arising from the roughness of the graphene edge. With the average transmission as the only fitting parameter, our theory shows good agreement with the experimental data.

Algebraic solution of a graphene layer in transverse electric and perpendicular magnetic fields

We present an exact algebraic solution of a single graphene plane in transverse electric and perpendicular magnetic fields. The method presented gives both the eigenvalues and the eigenfunctions of the graphene plane. It is shown that the eigenstates of the problem can be cast in terms of coherent states, which appears in a natural way from the formalism.

Topological delocalization of two-dimensional massless Dirac fermions

The beta function of a two-dimensional massless Dirac Hamiltonian subject to a random scalar potential, which, e.g., underlies theoretical descriptions of graphene, is computed numerically. Although it belongs to, from a symmetry standpoint, the two-dimensional symplectic class, the beta function monotonically increases with decreasing conductance. We also provide an argument based on the spectral flows under twisting boundary conditions, which shows that none of the states of the massless Dirac Hamiltonian can be localized.

Dirac fermions and conductance oscillations in s- and d-wave superconductor-graphene junctions

We investigate quantum transport in a normal-superconductor graphene heterostructure, including the possibility of an anisotropic pairing potential in the superconducting region. We find that under certain circumstances, the conductance displays an undamped, oscillatory behavior as a function of applied bias voltage. Also, we investigate how the conductance spectra are affected by a d-wave pairing symmetry. These results combine unusual features of the electronic structure of graphene with the unconventional pairing symmetry found for instance in high-Tc superconductors.

Excitonic effects in the optical spectra of graphene nanoribbons

We present a first-principles calculation of the optical properties of armchair-edged graphene nanoribbons (AGNRs) with many-electron effects included. The reduced dimensionality of the AGNRs gives rise to an enhanced electron-hole binding energy for both bright and dark exciton states (0.8-1.4 eV for GNRs with width approximately 1.2 nm) and dramatically changes the optical spectra owing to a near complete transfer of oscillator strength to the exciton states from the continuum transitions. The characteristics of the excitons of the three distinct families of AGNRs are compared and discussed. The enhanced excitonic effects found here are expected to be of importance in optoelectronic applications of graphene-based nanostructures.

Substrate-induced bandgap opening in epitaxial graphene

Graphene has shown great application potential as the host material for next-generation electronic devices. However, despite its intriguing properties, one of the biggest hurdles for graphene to be useful as an electronic material is the lack of an energy gap in its electronic spectra. This, for example, prevents the use of graphene in making transistors. Although several proposals have been made to open a gap in graphene's electronic spectra, they all require complex engineering of the graphene layer. Here, we show that when graphene is epitaxially grown on SiC substrate, a gap of approximately 0.26 eV is produced. This gap decreases as the sample thickness increases and eventually approaches zero when the number of layers exceeds four. We propose that the origin of this gap is the breaking of sublattice symmetry owing to the graphene-substrate interaction. We believe that our results highlight a promising direction for bandgap engineering of graphene.

Carbon-based electronics

The semiconductor industry has been able to improve the performance of electronic systems for more than four decades by making ever-smaller devices. However, this approach will soon encounter both scientific and technical limits, which is why the industry is exploring a number of alternative device technologies. Here we review the progress that has been made with carbon nanotubes and, more recently, graphene layers and nanoribbons. Field-effect transistors based on semiconductor nanotubes and graphene nanoribbons have already been demonstrated, and metallic nanotubes could be used as high-performance interconnects. Moreover, owing to the excellent optical properties of nanotubes it could be possible to make both electronic and optoelectronic devices from the same material.

First principles study of magnetism in nanographenes

Magnetism in nanographenes [also known as polycyclic aromatic hydrocarbons (PAHs)] is studied with first principles density functional calculations. We find that an antiferromagnetic (AFM) phase appears as the PAH reaches a certain size. This AFM phase in PAHs has the same origin as the one in infinitely long zigzag-edged graphene nanoribbons, namely, from the localized electronic state at the zigzag edge. The smallest PAH still having an AFM ground state is identified. With increased length of the zigzag edge, PAHs approach an infinitely long ribbon in terms of (1) the energetic ordering and difference among the AFM, ferromagnetic, and nonmagnetic phases and (2) the average local magnetic moment at the zigzag edges. These PAHs serve as ideal targets for chemical synthesis of nanographenes that possess magnetic properties. Moreover, our calculations support the interpretation that experimentally observed magnetism in activated carbon fibers originates from the zigzag edges of the nanographenes.