Dirac Point Of Graphene Research Articles

Electronic transport properties of graphene nanoribbons

We will present a brief overview of the electronic and transport properties of graphene nanoribbons focusing on the effect of edge shapes and impurity scattering. The low-energy electronic states of graphene have two non-equivalent massless Dirac spectra. The relative distance between these two Dirac points in the momentum space and edge states due to the existence of zigzag-type graphene edges is a deciding factor in the electronic and transport properties of graphene nanoribbons. In graphene nanoribbons with zigzag edges (zigzag nanoribbons), two valleys related to each Dirac spectrum are well separated in momentum space. The propagating modes in each valley contain a single chiral mode originating from a partially flat band at the band center. This feature gives rise to a perfectly conducting channel in the disordered system, if impurity scattering does not connect the two valleys, i.e. for long-range impurity (LRI) potentials. Ribbons with short-range impurity potentials, however, display ordinary localization behavior through inter-valley scattering. On the other hand, the low-energy spectrum of graphene nanoribbons with armchair edges (armchair nanoribbons) is described as the superposition of two non-equivalent Dirac points of graphene. In spite of the lack of two well separated valley structures, the single-channel transport subjected to LRIs is nearly perfectly conducting, where the backward scattering matrix elements in the lowest order vanish as a manifestation of internal phase structures of the wave function. For the multi-channel energy regime, however, conventional exponential decay of the averaged conductance occurs. Symmetry considerations lead to the classification of disordered zigzag ribbons into the unitary class for LRIs, and the orthogonal class for short-range impurities. Since inter-valley scattering is not completely absent, armchair nanoribbons can be classified into the orthogonal universality class irrespective of the range of impurities.

Landau Levels and Quantum Hall Effect in Graphene Superlattices

We show that, when graphene is subjected to an appropriate one-dimensional external periodic potential, additional branches of massless fermions are generated with nearly the same electron-hole crossing energy as that at the original Dirac point of graphene. Because of these new zero-energy branches, the Landau levels at charge neutral filling become 4(2N + 1)-fold degenerate (with N = 0, 1, 2, ..., tunable by the potential strength and periodicity) with the corresponding Hall conductivity sigma_{xy} showing a step of size 4(2N + 1)e;{2}/h. These theoretical findings are robust against variations in the details of the external potential and provide measurable signatures of the unusual electronic structure of graphene superlattices.

The enigma of the [formula omitted] quantum Hall effect in graphene

We apply Laughlin’s gauge argument to analyze the ν = 0 quantum Hall effect observed in graphene when the Fermi energy lies near the Dirac point, and conclude that this necessarily leads to divergent bulk longitudinal resistivity in the zero temperature thermodynamic limit. We further predict that in a Corbino geometry measurement, where edge transport and other mesoscopic effects are unimportant, one should find the longitudinal conductivity vanishing in all graphene samples which have an underlying ν = 0 quantized Hall effect. We argue that this ν = 0 graphene quantum Hall state is qualitatively similar to the high field insulating phase (also known as the Hall insulator) in the lowest Landau level of ordinary semiconductor two-dimensional electron systems. We establish the necessity of having a high magnetic field and high mobility samples for the observation of the divergent resistivity as arising from the existence of disorder-induced density inhomogeneity at the graphene Dirac point.

Divergent resistance at the Dirac point in graphene: Evidence for a transition in a high magnetic field

We have investigated the behavior of the resistance of graphene at the $n=0$ Landau level in an intense magnetic field $H$. Employing a low-dissipation technique (with power $P<3\text{ }\text{fW}$), we find that at low temperature $T$, the resistance at the Dirac point ${R}_{0}(H)$ undergoes a 1000-fold increase from $\ensuremath{\sim}10\text{ }\text{k}\ensuremath{\Omega}$ to $40\text{ }\text{M}\ensuremath{\Omega}$ within a narrow interval of field. The abruptness of the increase suggests that a transition to an insulating ordered state occurs at the critical field ${H}_{c}$. Results from five samples show that ${H}_{c}$ depends systematically on the disorder, as measured by the offset gate voltage ${V}_{0}$. Samples with small ${V}_{0}$ display a smaller critical field ${H}_{c}$. Empirically, the steep increase in ${R}_{0}$ fits accurately a Kosterlitz-Thouless-type correlation length over three decades. The curves of ${R}_{0}$ vs $T$ at fixed $H$ approach the thermal-activation form with a gap $\ensuremath{\Delta}\ensuremath{\sim}15\text{ }\text{K}$ as $H\ensuremath{\rightarrow}{H}_{c}^{\ensuremath{-}}$, consistent with a field-induced insulating state.

Nearly perfect single-channel conduction in disordered armchair nanoribbons

The low-energy spectrum of graphene nanoribbons with armchair edges (armchair nanoribbons) is described as the superposition of two nonequivalent Dirac points of graphene. In spite of the lack of well-separated two valley structures, the single-channel transport subjected to long-ranged impurities is nearly perfectly conducting, where the backward-scattering matrix elements in the lowest order vanish as a manifestation of internal phase structures of the wave function. For multichannel energy regime, however, the conventional exponential decay of the averaged conductance occurs. Since the intervalley scattering is not completely absent, armchair nanoribbons can be classified into orthogonal universality class irrespective of the range of impurities. The nearly perfect single-channel conduction dominates the low-energy electronic transport in rather narrow nanorribbons.

Disorder-driven splitting of the conductance peak at the Dirac point in graphene

The electronic properties of a bricklayer model, which shares the same topology as the hexagonal lattice of graphene, are investigated numerically. We study the influence of random magnetic-field disorder in addition to a strong perpendicular magnetic field. We found a disorder-driven splitting of the longitudinal conductance peak within the narrow lowest Landau band near the Dirac point. The energy splitting follows a relation which is proportional to the square root of the magnetic field and linear in the disorder strength. We calculate the scale invariant peaks of the two-terminal conductance and obtain the critical exponents as well as the multifractal properties of the chiral and quantum Hall states. We found approximate values $\nu\approx 2.5$ for the quantum Hall states, but $\nu=0.33\pm 0.1$ for the divergence of the correlation length of the chiral state at E=0 in the presence of a strong magnetic field. Within the central $n=0$ Landau band, the multifractal properties of both the chiral and the split quantum Hall states are the same, showing a parabolic $f[\alpha(s)]$ distribution with $\alpha(0)=2.27\pm 0.02$. In the absence of the constant magnetic field, the chiral critical state is determined by $\alpha(0)=2.14\pm 0.02$.

Quantum critical scaling in magnetic field near the Dirac point in graphene

Motivated by the recent measurement of the activation energy at the quantum Hall state at the filling factor f=1 in graphene we discuss the scaling of the interaction-induced gaps in vicinity of the Dirac point with the magnetic field. The gap at f=1 is shown to be bounded from above by E(1)/C, where E(n) are the Landau level energies and C = 5.985 + O(1/N) is a universal number. The universal scaling functions are computed exactly for a large number of Dirac fermions N. We find a sublinear dependence of the gap at the laboratory magnetic fields for realistic values of short-range repulsion between electrons, and in quantitative agreement with observation.

Density functional study of graphene overlayers on SiC

AbstractDespite the ongoing “graphene boom” of the last three years our understanding of epitaxial graphene grown on SiC substrate is only beginning to emerge. Along with experimental methods such as low energy electron diffraction (LEED), scanning tunneling microscopy (STM) and angle resolved photoemission spectroscopy (ARPES), ab initio calculations help to uncover the geometric and electronic structure of the graphene/SiC interface. In this chapter we describe the density‐functional calculations we performed for single and double graphene layers on Si‐ and C‐terminated 6H‐SiC surfaces. Experimental data reveal a pronounced difference between the two surface terminations. On a Si‐terminated surface the interface adopts a 6√3 × 6√3 unit cell whereas the C‐face supports misoriented (turbostratic) graphene layers. It has been recently realized that, on the Si‐face, the large commensurate cell is subdivided into patches of coherently matching to the substrate carbon atoms. In our calculations we assumed the “coherent match” geometry for the whole interface plane. This reduces the periodic unit to the √3 × √3R 30° cell but requires a substantial stretching of the graphene sheet. Although simplified, the model provides a qualitative picture of the bonding and of the interface electron energy spectrum. We find that the covalent bonding between the carbon layer and the substrate destroys the massless “relativistic” electron energy spectrum, the hallmark of a freestanding graphene. Hence the first carbon layer cannot be responsible for the graphene‐type electron spectrum observed by ARPES and rather plays a role of a buffer between the substrate and the subsequent carbon sheets. The “true” graphene spectrum appears with the second carbon layer which exhibits a weak van der Waals bonding to the underlying structure. For Si‐terminated substrate, we find that the Fermi level is pinned by the interface state at 0.45 eV above the graphene Dirac point, in agreement with experimental data. This renders the interface metallic. On the contrary, for a C‐face the “coherent match” model predicts the Fermi level exactly at the Dirac point. However, this does not necessarily apply to the turbostratic graphene layers that normally grow on the C‐terminated substrate. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Zero-Energy State in Graphene in a High Magnetic Field

The fate of the charge-neutral Dirac point in graphene in a high magnetic field H has been investigated at low temperatures (T approximately 0.3 K). In samples with small gate-voltage offset V0, the resistance R0 at the Dirac point diverges steeply with H, signaling a crossover to a state with a very large R0. The approach to this state is highly unusual. Despite the steep divergence in R0, the profile of R0 vs T in fixed H saturates to a T-independent value below 2 K, consistent with gapless charge-carrying excitations.

Shot Noise in Ballistic Graphene

We have investigated shot noise in graphene field effect devices in the temperature range of 4.2-30 K at low frequency (f=600-850 MHz). We find that for our graphene samples with a large width over length ratio W/L, the Fano factor F reaches a maximum F ~ 1/3 at the Dirac point and that it decreases strongly with increasing charge density. For smaller W/L, the Fano factor at Dirac point is significantly lower. Our results are in good agreement with the theory describing that transport at the Dirac point in clean graphene arises from evanescent electronic states.

Origin of Anomalous Electronic Structures of Epitaxial Graphene on Silicon Carbide

On the basis of first-principles calculations, we report that a novel interfacial atomic structure occurs between graphene and the surface of silicon carbide, destroying the Dirac point of graphene and opening a substantial energy gap there. In the calculated atomic structures, a quasiperiodic 6x6 domain pattern emerges out of a larger commensurate 6 sqrt [3] x 6 sqrt [3]R30 degrees periodic interfacial reconstruction, resolving a long standing experimental controversy on the periodicity of the interfacial superstructures. Our theoretical energy spectrum shows a gap and midgap states at the Dirac point of graphene, which are in excellent agreement with the recently observed anomalous angle-resolved photoemission spectra. Beyond solving unexplained issues in epitaxial graphene, our atomistic study may provide a way to engineer the energy gaps of graphene on substrates.

Effect of atomic-scale defects on the low-energy electronic structure of graphene: Perturbation theory and local-density-functional calculations

Based on perturbation theory and local-density-functional calculations, we study the effect of atomic-scale defects, whose potentials vary on the scale of interatomic distance, on the electronic structure of graphene in the region of low energies. If defects are identical, for example, vacancies at the same sublattice sites or the Stone-Wales defects with the same orientations, the degeneracy at the Dirac point of graphene is removed with an energy splitting proportional to $\ensuremath{\lambda}$ for low disorder densities $(\ensuremath{\lambda})$, which is attributed to the breaking of the intrinsic symmetry of the honeycomb lattice by the presence of atomic-scale defects. However, the degeneracy at the Dirac point is nearly restored if physically equivalent disorders, which are generated by the symmetry operations of graphene, such as reflection and rotation, coexist with similar concentrations.

Demonstration of a new transport regime of photon in two-dimensional photonic crystal

A new transport regime of photon in two-dimensional photonic crystal near the Dirac point has been demonstrated by exact numerical simulation. In this regime, the conductance of photon is inversely proportional to the thickness of sample, which can be described by Dirac equation very well. Both of bulk and surface disorders always reduce the transmission, which is in contrast to the previous theoretical prediction that they increase the conductance of electron at the Dirac point of graphene. However, regular tuning of interface structures can cause the improvement of photon conductance. Furthermore, large conductance fluctuations of photon have also been observed, which is similar to the case of electron in graphene.

Quantum Hall States near the Charge-Neutral Dirac Point in Graphene

We investigate the quantum Hall (QH) states near the charge-neutral Dirac point of a high mobility graphene sample in high magnetic fields. We find that the QH states at filling factors nu=+/-1 depend only on the perpendicular component of the field with respect to the graphene plane, indicating that they are not spin related. A nonlinear magnetic field dependence of the activation energy gap at filling factor nu=1 suggests a many-body origin. We therefore propose that the nu=0 and +/-1 states arise from the lifting of the spin and sublattice degeneracy of the n=0 Landau level, respectively.

One-Parameter Scaling at the Dirac Point in Graphene

We numerically calculate the conductivity sigma of an undoped graphene sheet (size L) in the limit of a vanishingly small lattice constant. We demonstrate one-parameter scaling for random impurity scattering and determine the scaling function beta(sigma)=dlnsigma/dlnL. Contrary to a recent prediction, the scaling flow has no fixed point (beta>0) for conductivities up to and beyond the symplectic metal-insulator transition. Instead, the data support an alternative scaling flow for which the conductivity at the Dirac point increases logarithmically with sample size in the absence of intervalley scattering--without reaching a scale-invariant limit.

Substrate-induced band gap in graphene on hexagonal boron nitride:Ab initiodensity functional calculations

We determine the electronic structure of a graphene sheet on top of a lattice-matched hexagonal boron nitride $(h\text{\ensuremath{-}}\mathrm{B}\mathrm{N})$ substrate using ab initio density functional calculations. The most stable configuration has one carbon atom on top of a boron atom, and the other centered above a BN ring. The resulting inequivalence of the two carbon sites leads to the opening of a gap of $53\phantom{\rule{0.3em}{0ex}}\mathrm{meV}$ at the Dirac points of graphene and to finite masses for the Dirac fermions. Alternative orientations of the graphene sheet on the BN substrate generate similar band gaps and masses. The band gap induced by the BN surface can greatly improve room temperature pinch-off characteristics of graphene-based field effect transistors.

XeF 2 and F-atom reactions with Si: their significance for plasma etching : Daniel L. Flamm, Dale E. Ibbotson, John A. Mucha and Vincent M. Donnelly. Solid St. Technol., 117 (April 1983)

Necessity of opening of energy gap in the band structure of, otherwise a zero gap semiconductor, graphene, is a must for its use in fabrication of high speed electronic devices. One such technique, for opening of energy gap in graphene, is by way of doping the pristine graphene with boron or nitrogen. Besides many important applications, to which B- and N-doped graphene has been put, the one very important for solving the global energy crisis is by way of its capacity for hydrogen storage. In this paper electronic structure of B- and N-doped graphene has been studied by using Density Functional Theory, as implemented in WIEN2K code. PBE-GGA (Perdew–Burke–Ernzerhof 96) pseudo-potential is used for solving the Schrodinger equation in self-consistent manner and to account for the exchange and correlation effects through the use of Generalized Gradient Approximation. The band structure calculations reveal that whereas a band gap opens at the symmetry point K both for B- and N-doped graphene, the center of the gap (i.e., the Dirac point in pristine graphene) is shifted above the Fermi level by about 2.20 eV for B-doped graphene, and shifted down the Fermi level by about the same amount for N-doped graphene, when the doping level was kept 25% in each case. The energy gap opened was found to be about 0.30 eV for B-doped graphene and 0.45 eV for N-doped graphene. The linear dispersion characteristics obtained at Dirac point for pristine graphene, almost vanish for B-/N-doped graphene, due to symmetry breaking and opening of energy gap. The band structure and Density of States of B-/N-doped graphene are found also to depend on the choice of cell parameters.