Recent experiments show that a substantial energy gap in graphene can be induced via patterned hydrogenation on an iridium substrate. Here, we show that the energy gap is roughly proportional to $\sqrt{N_{H}}/N_{C}$ when disorder is accounted for, where $N_H$ and $N_C$ denote concentration of hydrogen and carbon atoms, respectively. The dispersion relation, obtained through calculation of the momentum-energy resolved density of states, is shown to agree with previous angle-resolved photoemission spectroscopy results. Simulations of electronic transport in finite size samples also reveal a similar transport gap, up to 1eV within experimentally achievable $\sqrt{N_{H}}/N_{C}$ value.
In this paper, we propose a practical way to stabilize half-hydrogenated graphene (graphone). We show that the dipole moments induced by an hexagonal-boron nitride (h-BN) substrate on graphene stabilize the hydrogen atoms on one sublattice of the graphene layer and suppress the migration of the absorbed hydrogen atoms. Based upon first principle spin polarized density of states (DOS) calculations, we show that the half hydrogenated graphene (graphone) obtained in different graphene-h-BN heterostructures exhibits a half metallic state. We propose to use this new exotic material for spin valve a…
We calculate the spin transport of hydrogenated graphene using the Landauer-Büttiker formalism with a spin-dependent tight-binding Hamiltonian. The advantages of using this method is that it simultaneously gives information on sheet resistance and localization length as well as spin relaxation length. Furthermore, the Landauer-Büttiker formula can be computed very efficiently using the recursive Green's function technique. Previous theoretical results on spin relaxation time in hydrogenated graphene have not been in agreement with experiments. Here, we study magnetic defects in graphene with r…
Spin-orbit coupling (SOC) can provide essential tools to manipulate electron spins in two-dimensional materials like graphene, which is of great interest for both fundamental physics and spintronics application. In this paper, we report the low-field magnetotransport of in situ hydrogenated graphene where hydrogen atoms are attached to the graphene surface in continuous low temperature and vacuum environment. Transition from weak localization to weak antilocalization with increasing hydrogen adatom density is observed, indicating enhancing Bychkov-Rashba-type SOC in a mirror symmetry broken sy…
In this work, we present a performance analysis of Field Effect Transistors based on recently fabricated 100% hydrogenated graphene (the so-called graphane) and theoretically predicted semi-hydrogenated graphene (i.e. graphone). The approach is based on accurate calculations of the energy bands by means of GW approximation, subsequently fitted with a three-nearest neighbor (3NN) sp3 tight-binding Hamiltonian, and finally used to compute ballistic transport in transistors based on functionalized graphene. Due to the large energy gap, the proposed devices have many of the advantages provided by …
In this work we present a fully atomistic reactive (ReaxFF force field) molecular dynamics study of the structural and dynamical aspects of the one-side hydrogenation of graphene membranes, leading to the formation of the so-called graphone structure. We have considered different substrates: graphene, few-layers graphene, graphite and platinum at different temperatures. Our results showed that the hydrogenation rates are very dependent on the substrate and thermal effects. Our results also showed that, similarly to graphane, large hydrogenated domains are unlikely to be formed. These hydrogena…
Conventional in-gap Yu-Shiba-Rusinov states require two ingredients: magnetic atoms and a superconducting host that, in the normal phase, has a finite density of states at the Fermi energy. Here we show that hydrogenated graphene can host Yu-Shiba-Rusinov states without any of those two ingredients. Atomic hydrogen chemisorbed in graphene is known to act as paramagnetic center with a weakly localized magnetic moment. Our calculations for hydrogenated graphene in proximity to a superconductor show that individual adatoms induce in-gap Yu-Shiba-Rusinov states with an exotic spectrum whereas chai…
The weak spin-orbit interaction in graphene was predicted to be increased, e.g., by hydrogenation. This should result in a sizable spin Hall effect (SHE). We employ two different methods to examine the spin Hall effect in weakly hydrogenated graphene. For hydrogenation we expose graphene to a hydrogen plasma and use Raman spectroscopy to characterize this method. We then investigate the SHE of hydrogenated graphene in the H-bar method and by direct measurements of the inverse SHE. Although a large nonlocal resistance can be observed in the H-bar structure, comparison with the results of the ot…
We propose a new class of semiconducting graphene-based nanostructures: hydrogenated graphene nanoripples (HGNRs), based on continuum-mechanics analysis and first principles calculations. They are formed via a two-step combinatorial approach: first by strain engineered pattern formation of graphene nanoripples, followed by a curvature-directed self-assembly of H adsorption. It offers a high level of control of the structure and morphology of the HGNRs, and hence their band gaps which share common features with graphene nanoribbons. A cycle of H adsorption/desorption at/from the same surface lo…
We have investigated by means of first principles calculations the structural and electronic properties of hydrogenated graphene structures with distinct grain boundary defects. Our total energy results reveal that the adsorption of a single H is more stable at grain boundary defect. The electronic structure of the grains boundaries upon hydrogen adsorption have been examined. Further total energy calculations indicate that the adsorption of two H on two neighbor carbons, forming a basic unit of graphane, is more stable at the defect region. Therefore, we expect that these extended defects wou…
Spin-dependent features in the conductivity of graphene, chemically modified by a random distribution of hydrogen adatoms, are explored theoretically. The spin effects are taken into account using a mean-field self-consistent Hubbard model derived from first-principles calculations. A Kubo-Greenwood transport methodology is used to compute the spin-dependent transport fingerprints of weakly hydrogenated graphene-based systems with realistic sizes. Conductivity responses are obtained for paramagnetic, antiferromagnetic, or ferromagnetic macroscopic states, constructed from the mean-field soluti…
We report the observation of a giant positive magnetoresistance in millimetre scale hydrogenated graphene with magnetic field oriented in the plane of the graphene sheet. A positive magnetoresistance in excess of 200\% at a temperature of 300 mK was observed in this configuration, reverting to negative magnetoresistance with the magnetic field oriented normal to the graphene plane. We attribute the observed positive, in-plane, magnetoresistance to Pauli-blockade of hopping conduction induced by spin polarization. Our work shows that spin polarization in concert with electron-electron interacti…
We report a metal-insulator transition in disordered graphene with low coverages of hydrogen atoms. Hydrogen interacting with graphene creates short-range disorder and localizes states near the neutrality point. The energy range of localization grows with increasing of H concentration. Calculations show that the conductances through low-energy propagating channels decay exponentially with sample size and are well fitted by one-parameter scaling function, similar to a disorder-driven metal-insulator transition in 2-dimensional disordered systems.
We calculate the spin-dependent zero-bias conductance $G_{σσ'}$ in armchair graphene nanoribbons with hydrogen adsorbates employing a DFT-based ab initio transport formalism including spin-orbit interaction. We find that the spin-flip conductance $G_{σ\barσ}$ can reach the same order of magnitude as the spin-conserving one, $G_{σσ}$, due to exchange-mediated spin scattering. In contrast, the genuine spin-orbit interaction appears to play a secondary role, only.
A lot of efforts have been devoted to understanding the origin and effects of magnetic moments induced in graphene with carbon atom vacancy, or light adatoms like hydrogen or fluorine. At the meantime, the large negative magnetoresistance (MR) widely observed in these systems is not well understood, nor had it been associated with the presence of magnetic moments. In this paper, we study the systematic evolution of the large negative MR of in-situ hydrogenated graphene in ultra-high vacuum (UHV) environment. We find for most combination of electron density ($n_e$) and hydrogen density ($n_H$),…
We show how hydrogenation of graphene nanoribbons at small concentrations can open new venues towards carbon-based spintronics applications regardless of any especific edge termination or passivation of the nanoribbons. Density functional theory calculations show that an adsorbed H atom induces a spin density on the surrounding $π$ orbitals whose symmetry and degree of localization depends on the distance to the edges of the nanoribbon. As expected for graphene-based systems, these induced magnetic moments interact ferromagnetically or antiferromagnetically depending on the relative adsorption…
We show that for graphene with any finite difference in the on-site energy between the two sub-lattices ($Δ$), The optical absorption edge is determined by the $Δ$. The universal conductance will be broken and the conductance near the band edge varies with frequency as $1/ω^2$. Moreover, we have identified another universal conductance for such systems without inversion symmetry, i.e., the onset conductance at the band edge is $% σ_c=2σ_0=πe^2/2h$, independent of the size of the band gap. The total integrated optical response is nearly conserved despite of the opening of the band gap.
We show that the $d^{0}$ ferromagnetism with high Curie temperature ($T_c$) can be achieved in the electron doped hydrogenated epitaxial graphene on some certain SiC substrates through first-principles calculations. The pristine systems are found to be a Mott insulator independent of SiC polytypes (2$H$, 4$H$ or 6$H$) which, however, play a significant role in the modulation of magnetic interaction. Carrier doping enhances the ferromagnetic coupling due to the double exchange mechanism and thus realizes the phase transition from antiferromagnetism to ferromagnetism. A $T_c$ of around 400 K is …
There exist three conformers of hydrogenated graphene, referred to as chair-, boat-, or washboard-graphane. These systems have a perfect two-dimensional periodicity mapped onto the graphene scaffold, but they are characterized by a $sp^3$ orbital hybridization, have different crystal symmetry, and otherwise behave upon loading. By first principles calculations we determine their structural and phonon properties, as well as we establish their relative stability. Through continuum elasticity we define a simulation protocol addressed to measure by a computer experiment their linear and nonlinear …
First-principles calculations of the spin-orbit coupling in graphene with hydrogen adatoms in dense and dilute limits are presented. The chemisorbed hydrogen induces a giant local enhancement of spin-orbit coupling due to $sp^3$ hybridization which depends strongly on the local lattice distortion. Guided by the reduced symmetry and the local structure of the induced dipole moments we use group theory to propose realistic minimal Hamiltonians that reproduce the relevant spin-orbit effects for both single-side semihydrogenated graphene (graphone) and for a single hydrogen adatom in a large super…
The spin-orbit couplings (SOCs) of hydrogenated and fluorinated graphenes are calculated from the first principles method. It is found that the SOC-induced band splittings near their Fermi energies can be significantly enhanced to the order of 10$^{-2}$ eV from the original about 10$^{-6}$ eV of the pure raphene, which is comparable to those found in the diamond and even the archetypal semiconductors. And two different mechanisms are proposed to explain the SOC enhancements in these two systems. The huge SOC enhancements are found to come not only from the sp$^3$ hybridization of carbon atoms,…
Geometry, stability, and thermal transport of graphene nanoquilts folded by hydrogenation are studied using molecular dynamics simulations. The hydrogenated graphene nanoquilts show increased thermodynamic stability and better transport properties than folded graphene structures without hydrogenation. For the two-fold graphene nanoquilt, both geometry and thermal conductivity are very sensitive to the adsorbed hydrogen chains, which is interpreted by the red-shift of PDOS. For the multi-fold nanoquilts, their thermal conductivities can be tuned from 100% to 15% of pristine graphene, by varying…
Hydrogenated graphene edges are assumed to be either armchair, zigzag or a combination of the two. We show that the zigzag is not the most stable fully hydrogenated structure along the <2-1-10> direction. Instead hydrogenated Klein and reconstructed Klein based edges are found to be energetically more favourable, with stabilities approaching that of armchair edges. These new structures "unify" graphene edge topology, the most stable flat hydrogenated graphene edges always consisting of pairwise bonded C2H4 edge groups, irrespective the edge orientation. When edge rippling is included, CH3 edge…
Hydrogenation has proven to be an effective tool to open the bandgap of graphene. In the present density functional study we demonstrate that single-side-hydrogenated graphene is a semiconductor with an indirect bandgap of 1.89 eV, in between the gapless graphene and wide bandgap graphane. We show that its electronic structure and lattice characteristics are substantially different from those of graphene, graphone, or graphane. The lattice parameter and C-C bond length are found to be lengthened by 15% of those of graphene. Our binding energy analysis confirms that such a single sided hydrogen…
Hydrogen adsorbates in graphene are interesting as they are not only strong Coulomb scatterers but they also induce a change in orbital hybridization of the carbon network from sp^2 into sp^3. This change increases the spin-orbit coupling and is expected to largely modify spin relaxation. In this work we report the change in spin transport properties of graphene due to plasma hydrogenation. We observe an up to three-fold increase of spin relaxation time tau_S after moderate hydrogen exposure. This increase of tau_S is accompanied by the decrease of charge and spin diffusion coefficients, resul…
We present a theoretical study of spin-velocity injection (SVI) of a pure spin current (PSC) induced by a linearly polarized light that impinges normally on the surface of two 50\% hydrogenated noncentrosymmetric two-dimensional (2D) graphene structures. The first structure, hydrogenated at only one side, labeled Up, also known as graphone, and the second, labelled Alt, is 25\% hydrogenated at both sides. The hydrogenation opens an energy gap in both structures. We analyze two possibilities: in the first, the spin is fixed along a chosen direction, and the resulting SVI is calculated; in the s…
The instability of a free-standing one sided hydrogenated/fluorinated graphene nano-ribbon, i.e. graphone/fluorographene, is studied using ab-initio, semiempirical and large scale molecular dynamics simulations. Free standing semi-infinite arm-chair like hydrogenated/fluorinated graphene (AC-GO/AC-GF) and boat like hydrogenated/fluorinated graphene (B-GO/B-GF) (nano-ribbons which are periodic along the zig-zag direction) are unstable and spontaneously transform into spiral structures. We find that rolled, spiral B-GO and B-GF are energetically more favorable than spiral AC-GO and AC-GF which i…
Hydrogenation of graphene on the α-quartz (0001) SiO_2 substrate is studied, considering different surface terminations in order to take into account the amorphic nature of the material. Our ab initio calculations show that the formation of graphane by hydrogen adsorption on graphene is energetically favored on hydroxyl and oxygen terminated surfaces, whereas silicon termination and reconstruction of the oxygen termination hinder adsorption. Our results indicate that in order to fabricate graphane on SiO_2, it is beneficial to oxygenize the surface and saturate it with hydrogen. For the pristi…
Magnetism in single-side hydrogenated (C$_2$H) and fluorinated (C$_2$F) graphene is analyzed in terms of the Heisenberg model with parameters determined from first principles. We predict a frustrated ground state for both systems, which means the instability of collinear spin structures and sheds light on the absence of a conventional magnetic ordering in defective graphene demonstrated in recent experiments. Moreover, our findings suggest a highly correlated magnetic behavior at low temperatures offering the possibility of a spin-liquid state.
Hydrogen adatoms are one of the most the promising proposals for the functionalization of graphene. Hydrogen induces narrow resonances near the Dirac energy, which lead to the formation of magnetic moments. Furthermore, they also create local lattice distortions which enhance the spin-orbit coupling. The combination of magnetism and spin-orbit coupling allows for a rich variety of phases, some of which have non trivial topological features. We analyze the interplay between magnetism and spin-orbit coupling in ordered arrays of hydrogen on graphene monolayers, and classify the different phases …
We show that charge doping can induce transitions between three distinct adsorbate phases in hydrogenated and fluorinated graphene. By combining ab initio, approximate density functional theory and tight binding calculations we identify a transition from islands of C$_8$H$_2$ and C$_8$F$_2$ to random adsorbate distributions around a doping level of $\pm 0.05$ e/C-atom. Furthermore, in situations with random adsorbate coverage, charge doping is shown to trigger an ordering transition where the sublattice symmetry is spontaneously broken when the doping level exceeds the adsorbate concentration.…
Motivated by recent experimental and calculational investigations of bilayer, hydrogenated and fluorinated graphene, we apply the formalisms of U(1) QED (quantum electrodynamics) and SU(2) QCD (quantum chromodynamics) theories of strongly correlated state. Unlike non-bipartite triangular lattice, on bipartite honeycomb lattice there always exists a monopole that transforms trivially under all the microscopic symmetries, destabilizing the Dirac spin liquid (DSL), so that one can continuously tune the DSL state to the state with parent SU(2) instead of U(1) gauge group. The SU(2) theory describe…
We describe a mechanism that explains the formation of hydrocarbons and hydrocarbyls from hydrogenated graphene/graphite; hard C-C bonds are weakened and broken by the synergistic effect of chemisorbed hydrogen and high temperature vibrations. Total energies, optimized structures, and transition states are obtained from Density Functional Theory simulations. These values have been used to determine the Boltzman probability for a thermal fluctuation to overcome the kinetic barriers, yielding the time scale for an event to occur. This mechanism can be used to rationalize the possible routes for …
The quantum Hall effect is observed in a two-dimensional electron gas formed in millimeter-scale hydrogenated graphene, with a mobility less than 10 $\mathrm{cm^{2}/V\cdot s}$ and corresponding Ioffe-Regel disorder parameter $(k_{F}λ)^{-1}\gg1$. In zero magnetic field and low temperatures, the hydrogenated graphene is insulating with a two-point resistance of order of $250 h/e^2$. Application of a strong magnetic field generates a negative colossal magnetoresistance, with the two-point resistance saturating within 0.5% of $h/2e^{2}$ at 45T. Our observations are consistent with the opening of a…
Using helium-3 atom scattering, we have studied the adsorption kinetics, the structure and the diffusional dynamics of atomic hydrogen on the surface of a graphene monolayer on Ni(111). Diffraction measurements reveal a 4$^\circ$ rotated rectangular hydrogen overstructure. Hydrogen adsorption and desorption exhibit activation barriers of $E_a=(89\pm7)\,$meV and $E_d=(1.8\pm0.2)\,$eV, respectively. Helium-3 spin-echo measurements showed no decay of the spatial correlation function (or intermediate scattering function) within the time range of the spectrometer. Hence, we are able to set lower li…
Electron correlation effects caused by the topological zero mode of a hydrogenated graphene vacancy, $V_{111}$, with three adsorbed hydrogen atoms is discussed theoretically. A Kondo model is derived from the multi-reference representation of the density functional theory, where exchange scattering processes between the zero mode and low-energy modes in the Dirac cones are estimated. Even when the Dirac cone is slightly off from the charge neutral point, a finite on-site correlation energy, $U_0$, for the zero mode of an isolated $V_{111}$ allows the half-filling of the localized level giving …
The effects of tensile strain on the current-voltage (I-V) characteristics of hydrogenated-edge armchair graphene nanoribbons (HAGNRs) are investigated by using DFT theory. The strain is introduced in two different ways related to the two types of systems studied in this work: in-plane strained systems (A), and out-of-plane strained systems due to bending (B). These two kinds of strain lead to make a distinction among three cases: in-plane strained systems with strained electrodes (A1) and with unstrained electrodes (A2), and out-of-plane homogeneously strained systems with unstrained, fixed e…
Graphene nanoribbons' electronic transport properties strongly depend on the type of edge, armchair, zigzag or other, and on edge functionalization that can be used for band-gap engineering. For only partly hydrogenated edges interesting magnetic properties are predicted. Electric charge accumulates at edges and corners. Scanning force microscopy has so far shown the centre of graphene nanoribbons with atomic resolution using a quartz crystal tuning fork sensor of high stiffness. Weak long-range electrostatic forces related to the charge accumulation on the edges of graphene nanoribbons could …
While graphene is a semi-metal, recently synthesized hydrogenated graphene called graphane, turns out to be an insulator. We have probed the metal insulator Transition in graphene-graphane system within the framework of density functional theory. By analysing the evolutionary trends in the electronic structure for fifteen different hydrogen concentrations on graphene, we unravel some novel features of this transition. As hydrogen coverage increases the semi-metal turns first into a metal, then transforms into an insulator. The metallic phase is spatially inhomogeneous in the sense, it contains…
In the paper a theoretical study of critical (Curie) temperature of diluted ferromagnet based on multilayer graphene (or graphite) with hydrogen adatoms deposited over carbon atoms belonging to single sublattice is presented. The calculations are performed within Pair Approximation (PA) for diluted ferromagnetic system with long-range interactions. The method is able to take into account the spin-space anisotropy of coupling. The results obtained within Mean Field Approximation (MFA) are also presented for comparison. The assumed interaction between hydrogen adatom spins is inversely proportio…
We revisit the problem of local moment formation in graphene due to chemisorption of individual atomic hydrogen or other analogous sp$^3$ covalent functionalizations. We describe graphene with the single orbital Hubbard model, so that the H chemisorption is equivalent to a vacancy in the honeycomb lattice. In order to circumvent artefacts related to periodic unit cells, we use either huge simulation cells of up to $8\times10^5$ sites, or an embedding scheme that allows the modelling of a single vacancy in an otherwise pristine infinite honeycomb lattice. We find three results that stress the a…
We study the band gap of finite $N_A=7$ armchair graphene nanoribbons (7-AGNRs) on Au(111) through scanning tunneling microscopy/spectroscopy combined with density functional theory calculations. The band gap of 7-AGNRs with lengths of 6 nm and more is converged to within 0.1 eV of its bulk value of 2.3 eV, while the band gap opens by several hundred meV in very short 7-AGNRs. The termination has a significant effect on the band gap, doubly hydrogenated termini yielding a lower band gap than singly hydrogenated ones.
We have developed a novel method for crystalline hydrogenation of graphene on the nanoscale. Molecular hydrogen was physisorbed at 5 K onto pristine graphene islands grown on Cu(111) in ultrahigh vacuum. Field emission local to the tip of a scanning tunneling microscope dissociates H$_2$ and results in hydrogenated graphene. At lower coverage, isolated point defects are found on the graphene and are attributed to chemisorbed H on top and bottom surfaces. Repeated H$_2$ exposure and field emission yielded patches and then complete coverage of a crystalline $\sqrt{3}$ $\times$ $\sqrt{3}$ R30° ph…
We propose using graphene electrodes with hydrogenated edges for solid-state nanopore-based DNA sequencing, and perform molecular dynamics simulations in conjunction with electronic transport calculations to explore the potential merits of this idea. The results of our investigation show that, compared to the unhydrogenated system, edge-hydrogenated graphene electrodes facilitate the temporary formation of H-bonds with suitable atomic sites in the translocating DNA molecule. As a consequence, the average conductivity is drastically raised by about 3 orders of magnitude while exhibiting signifi…
While graphene is a semi-metal, a recently synthesized hydrogenated graphene called graphane, is an insulator. We have probed the transformation of graphene upon hydrogenation to graphane within the framework of density functional theory. By analyzing the electronic structure for eighteen different hydrogen concentrations, we bring out some novel features of this transition. Our results show that the hydrogenation favors clustered configurations leading to the formation of compact islands. The analysis of the charge density and electron localization function (ELF) indicates that as hydrogen co…
The effect of hydrogenation on the topography and the electronic properties of graphene and graphite surfaces are studied by scanning tunneling microscopy and spectroscopy. The surfaces are chemically modified using Ar/H2 plasma. Analyzing thousands of scanning tunneling spectroscopy measurements we determine that the hydrogen chemisorption on the surface of graphite/graphene opens on average an energy band gap of 0.4 eV around the Fermi level. We find that although the plasma treatment modifies the surface topography in a non-reversible way, the change in the electronic properties can be reve…
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.
We study the Coulomb drag transresistivity between graphene layers employing the finite temperature density response function. We analyze the Coulomb coupling between the two layers and show that a universal scaling behavior, independent of the interlayer distance can be deduced. We argue that this universal behavior may be experimentally observable in a system of two carbon nanotubes of large radius.
Electrodynamics of single-layer graphene is studied in the scaling regime. At any finite temperature, there is a weakly damped collective thermo-plasma polariton mode whose dispersion and wavelength dependent damping is determined analytically. The electric and magnetic fields associated with this mode decay exponentially in the direction perpendicular to the graphene layer, but unlike the surface plasma polariton modes of metals, the decay length and the mode frequency are strongly temperature dependent. This may lead to new ways of generation and manipulation of these modes.
We report an experimental investigation of the edge effect on the room-temperature transport in graphene nanoribbon and graphene sheet (both single-layer and bilayer). By measuring the resistance scaling behaviors at both low and high carrier densities, we show that the transport of single-layer nanoribbons lies in a strong localization regime, which can be attributed to an edge effect. We find that this edge effect can be weakened by enlarging the width, decreasing the carrier densities or adding an extra layer. From graphene nanoribbon to graphene sheet, the data show a dimensional crossover…
We present a calculation of the free energy, the surface free energy and the elastic constants ("Lam'e parameters" i.e, Poisson ratio, Young's modulus) of graphene flakes on the level of the density functional theory employing different standard functionals. We observe that the Lam'e parameters in small flakes can differ from the bulk values by 30% for hydrogenated zig-zag edges. The change results from the edge of the flake that compresses the interior. When including the vibrational zero point motion, we detect a decrease in the bending rigidity by ~26%. This correction is depending on the f…
We numerically calculate the conductivity $σ$ of an undoped graphene sheet (size $L$) in the limit of vanishingly small lattice constant. We demonstrate one-parameter scaling for random impurity scattering and determine the scaling function $β(σ)=d\lnσ/d\ln L$. Contrary to a recent prediction, the scaling flow has no fixed point ($β>0$) for conductivities up to and beyond the symplectic metal-insulator transition. Instead, the data supports an alternative scaling flow for which the conductivity at the Dirac point increases logarithmically with sample size in the absence of intervalley scatteri…
