Electronic Spectrum Of Graphene Research Articles

Localized and quasi-localized energy levels in the electron spectrum of graphene with isolated boron and nitrogen substitutions

Based on the calculation and analysis of local Green’s functions of impurity atoms of low concentration in a two-dimensional graphene lattice, the conditions for the formation and characteristics of local discrete levels with energies lying outside the band of the quasi-continuous spectrum and quasi-localized states with energies near the Fermi one are determined. Specific calculations were performed for boron and nitrogen impurity atoms, which can actually replace carbon in graphite and graphene nanostructures. For a boron impurity that forms local discrete levels outside the band of the quasi-continuous spectrum, sufficiently simple analytical expressions for the conditions for their formation, energy, intensity at the impurity atom, and damping parameter are obtained. An analysis of the formation of states quasi-localized on nitrogen impurities with energy near the Fermi level in graphene nanostructures was carried out.

Interaction effects in graphene in a weak magnetic field

A weak perpendicular magnetic field, $B$, breaks the chiral symmetry of each valley in the electron spectrum of graphene, preserving the overall chiral symmetry in the Brillouin zone. We explore the consequences of this symmetry breaking for the interaction effects in graphene. In particular, we demonstrate that the electron-electron interaction lifetime acquires an anomalous $B$ dependence. Also, the ballistic zero-bias anomaly, $\ensuremath{\delta}\ensuremath{\nu}(\ensuremath{\omega})$, where $\ensuremath{\omega}$ is the energy measured from the Fermi level, emerges at a weak $B$ and has the form $\ensuremath{\delta}\ensuremath{\nu}(B)\ensuremath{\sim}{B}^{2}/{\ensuremath{\omega}}^{2}$. Temperature dependence of the magnetic-field corrections to the thermodynamic characteristics of graphene is also anomalous. We discuss experimental manifestations of the effects predicted. The microscopic origin of the $B$-field sensitivity is an extra phase acquired by the electron wave function resulting from the chirality-induced pseudospin precession.

Local spectra at impurity and neighboring sites in graphene: Resonance manifestation

The electronic spectrum of graphene with a single point impurity is considered. Local densities of states at the impurity site and its nearest neighbors are calculated analytically. Their evolution with increasing the impurity potential is examined. It is shown that in the domain of the well-defined impurity resonance the local density of states at the first-nearest neighbor of the impurity site is approximately a scaled-up copy of the local density of states at the impurity site. The corresponding factor is found to be proportional to the squared impurity potential.

Electronic properties of graphene with point defects

This article attempts a point-by-point review of the electron spectrum of graphene containing defects (adsorbed atoms, substitutional atoms, vacancies) that can be adequately described within the Lifshitz model. In this regard, the known Hamiltonian for this model is calculated for the case of two-dimensional relativistic electrons, and the criteria for occurrence of an impurity resonance near a Dirac point are given. The theory of concentration band structure transformation of graphene is presented, from which it follows that a transport gap is opened upon reaching a strictly defined value of impurity concentration in the neighborhood of the energy of impurity resonance. At the same time, the question of the possibility (or impossibility) of localization in such a {spatially disordered system} of Dirac quasiparticles is analyzed. Based on this, it is possible to provide an explanation and present a description of a phenomenon recently observed in {impure graphene}, the metal-dielectric transition that appears as a direct consequence of the decrease of the Fermi energy of the system in the region of a transport gap. The concept of local spectrum rearrangement of graphene, which also occurs during the process of increased concentration of defects in it, is introduced and substantiated. Physical reasons are formulated, by which the position of the minimum of the low-temperature conductivity of graphene as a function of the Fermi energy of electrons corresponds to the impurity resonance energy rather than to the Dirac point, as has been validated in a number of theoretical and test studies. Here, it appears that the minimum value is not a universal magnitude, but depends on the concentration of defects. Analytical examination of impurity effects is accompanied by numerical modeling of the system under study, as a result of which complete correspondence has been established between these two approaches. In particular, the overall picture of spectrum rearrangement, localization of electron states, and also the effects having local nature are confirmed.

Photoinduced Dynamical Band Gap in Graphene: The Effects of Electron–Phonon and Spin–Orbit Interaction

The combined effects of electron–A1g phonon and Rashba spin–orbit couplings on the electronic spectrum of graphene in the presence of Haldane interaction induced by photo‐irradiation have been studied. A Fröhlich‐type Hamiltonian has been proposed to model these interactions, and then a diagonalization procedure based on the Lee‐Low‐Pines (LLP) theory which includes two successive unitary transformations has been applied. The results show that combined effects arising from the Rashba spin–orbit interaction and electron–A1g phonon interaction in the presence of Haldane interaction remove the valley degeneracy of electronic bands, and thus allow one to tune the resulting band gap just by adjusting the associated strengths of the interactions.

Artificial graphenes: Dirac matter beyond condensed matter

After the discovery of graphene and its many fascinating properties, there has been a growing interest for the study of "artificial graphenes". These are totally different and novel systems which bear exciting similarities with graphene. Among them are lattices of ultracold atoms, microwave or photonic lattices, "molecular graphene" or new compounds like phosphorene. The advantage of these structures is that they serve as new playgrounds for measuring and testing physical phenomena which may not be reachable in graphene, in particular: the possibility of controlling the existence of Dirac points (or Dirac cones) existing in the electronic spectrum of graphene, of performing interference experiments in reciprocal space, of probing geometrical properties of the wave functions, of manipulating edge states, etc. These cones, which describe the band structure in the vicinity of the two connected energy bands, are characterized by a topological "charge". They can be moved in reciprocal space by appropriate modification of external parameters (pressure, twist, sliding, stress, etc.). They can be manipulated, created or suppressed under the condition that the total topological charge be conserved. In this short review, I discuss several aspects of the scenarios of merging or emergence of Dirac points as well as the experimental investigations of these scenarios in condensed matter and beyond.

Effect of Rashba type spin–orbit interaction on the electronic spectrum of graphene in the presence of a hydrogenic impurity

The effect of Rashba spin–orbit interaction on the electronic spectrum of gapped graphene with a hydrogenic impurity in the presence of topological defects is analyzed analytically. Degenerate perturbation theory is used to investigate the dependence of electronic spectrum of gapped graphene on the strengths of impurity and Rashba spin–orbit coupling. The results show that, as the strength of Rashba spin–orbit coupling increases, pseudo-Zeeman splitting of energy levels induced by topological defects is enhanced. Therefore, it is possible to tune this pseudo-Zeeman splitting through the strength of Rashba spin–orbit coupling and of the strength of hydrogenic impurity.

Impurity scattering and size quantization effects in a single graphene nanoflake

By using Fourier-transform scanning tunneling spectroscopy we measure the interference patterns produced by the impurity scattering of confined Dirac quasiparticles in epitaxial graphene nanoflakes. Upon comparison of the experimental results with tight-binding calculations of realistic model flakes, we show that the characteristic features observed in the Fourier-transformed local density of states are related to scattering between different transverse modes (sub-bands) of a graphene nanoflake and allow direct insight into the electronic spectrum of graphene. We also observe a strong reduction of quasiparticle lifetime which is attributed to the interaction with the underlying substrate. In addition, we show that the distribution of the onsite energies at flower defects leads to an effectively broken pseudospin selection rule, where intravalley back-scattering is allowed.

An analytical solution for the graphene electronic spectrum in the presence of external fields and confinement potential

An algebraic solution of [Formula: see text]-dimensional Dirac equation describing the massless Dirac fermions in the presence of both magnetic field perpendicular to the graphene plane and in-plane homogeneous electric field together with a quadratic confinement potential is investigated. The associated energy eigenvalues are obtained within the framework of unitary transformation scheme. It is shown that the decreasing quadratic confinement potential strength leads to the enhancement of the Landau levels.

On the Energy Gaps Induced by a Semiconducting Substrate in the Graphene Density of States

<p class="zhengwen"><span lang="EN-GB">The analytical expressions for the densities of states for graphene formed on semiconducting substrates are obtained. The problem on the induced gap is studied thoroughly. It is shown, that graphene electronic spectrum according to the relation between the system’s parameters can contain two gaps or one gap, overlapping with the energy gap of substrate. The gaps width dependences on the coupling regimes (tight and weak) are obtained. Numerical estimations are fulfilled for the epitaxial graphene on 6H-SiC{0001}.</span></p>

Spontaneous strains and gap in graphene on boron nitride

The interaction between a graphene layer and a hexagonal Boron Nitride (hBN) substrate induces lateral displacements and strains in the graphene layer. The displacements lead to the appearance of commensurate regions and the existence of an average gap in the electronic spectrum of graphene. We present a simple, but realistic model, by which the displacements, strains and spectral gap can be derived analytically from the adhesion forces between hBN and graphene. When the lattice axes of graphene and the substrate are aligned, strains reach a value of order 2\%, leading to effective magnetic fields above 100T. The combination of strains and induced scalar potential gives a sizeable contribution to the electronic gap. Commensuration effects are negligible, due to the large stiffness of graphene.

Raman Fingerprint of Aligned Graphene/h-BN Superlattices

Graphene placed on hexagonal-boron nitride (h-BN) experiences a superlattice (Moiré) potential, which leads to a strong reconstruction of graphene's electronic spectrum with new Dirac points emerging at sub-eV energies. Here we study the effect of such superlattices on graphene's Raman spectrum. In particular, the 2D Raman peak is found to be exquisitely sensitive to the misalignment between graphene and h-BN lattices, probably due to the presence of a strain distribution with the same periodicity of the Moiré potential. This feature can be used to identify graphene superlattices with a misalignment angle smaller than 2°.

On rearrangement of spectrum of epitaxial graphene formed on thin metallic film

A simple model has been applied to the investigation of electron states of epitaxial graphene formed on the surface of dimensionally quantized film. It has been shown that quantum dimensional effects in the substrate influence the electron spectrum of graphene, i.e., Fermi velocity is renormalized and energy forbidden gap is generated in the density of states. The gap and Fermi velocity have been estimated for various values of interaction constant and substrate parameters.

Continuous wafer-scale graphene on cubic-SiC(001)

The atomic and electronic structure of graphene synthesized on commercially available cubic-SiC(001)/Si(001) wafers have been studied by low energy electron microscopy (LEEM), scanning tunneling microscopy (STM), low energy electron diffraction (LEED), and angle resolved photoelectron spectroscopy (ARPES). LEEM and STM data prove the wafer-scale continuity and uniform thickness of the graphene overlayer on SiC(001). LEEM, STM and ARPES studies reveal that the graphene overlayer on SiC(001) consists of only a few monolayers with physical properties of quasi-freestanding graphene. Atomically resolved STM and micro-LEED data show that the top graphene layer consists of nanometersized domains with four different lattice orientations connected through the 〈110〉-directed boundaries. ARPES studies reveal the typical electron spectrum of graphene with the Dirac points close to the Fermi level. Thus, the use of technologically relevant SiC(001)/Si(001) wafers for graphene fabrication represents a realistic way of bridging the gap between the outstanding properties of graphene and their applications.

Cloning of Dirac fermions in graphene superlattices

Superlattices have attracted great interest because their use may make it possible to modify the spectra of two-dimensional electron systems and, ultimately, create materials with tailored electronic properties. In previous studies (see, for example, refs 1-8), it proved difficult to realize superlattices with short periodicities and weak disorder, and most of their observed features could be explained in terms of cyclotron orbits commensurate with the superlattice. Evidence for the formation of superlattice minibands (forming a fractal spectrum known as Hofstadter's butterfly) has been limited to the observation of new low-field oscillations and an internal structure within Landau levels. Here we report transport properties of graphene placed on a boron nitride substrate and accurately aligned along its crystallographic directions. The substrate's moiré potential acts as a superlattice and leads to profound changes in the graphene's electronic spectrum. Second-generation Dirac points appear as pronounced peaks in resistivity, accompanied by reversal of the Hall effect. The latter indicates that the effective sign of the charge carriers changes within graphene's conduction and valence bands. Strong magnetic fields lead to Zak-type cloning of the third generation of Dirac points, which are observed as numerous neutrality points in fields where a unit fraction of the flux quantum pierces the superlattice unit cell. Graphene superlattices such as this one provide a way of studying the rich physics expected in incommensurable quantum systems and illustrate the possibility of controllably modifying the electronic spectra of two-dimensional atomic crystals by varying their crystallographic alignment within van der Waals heterostuctures.

Electronic and Transport Properties of Unbalanced Sublattice N-Doping in Graphene

Using both first-principles techniques and a real-space Kubo-Greenwood approach, electronic and transport properties of nitrogen-doped graphene with a single sublattice preference are investigated. Such a breaking of the sublattice symmetry leads to the appearance of a true band gap in graphene electronic spectrum even for a random distribution of the N dopants. More surprisingly, a natural spatial separation of both types of charge carriers at the band edge is predicted, leading to a highly asymmetric electronic transport. Both the presence of a band gap, allowing large on/off ratio, and an asymmetric transport pave a new route toward efficient graphene-based field-effect transistors.

Possible formation of chiral polarons in graphene

A theoretical investigation of the possible existence of chiral polaron formation in graphene is reported. We present an analytical method to calculate the ground-state of the electron–phonon system within the framework of the Lee–Low–Pines theory. On the basis of our model, the influence of electron–optical phonon interaction on the graphene electronic spectrum is investigated. We considered only the doubly degenerate optical phonon modes of E2g symmetry near the zone center Γ. It is analytically shown that the energy dispersions of both valence and conduction bands of the pristine graphene differ significantly from those obtained through the standard electron self-energy calculations arising from the electron–optical phonon interactions. In this paper, we also show for the first time that the degenerate band structure of the graphene promotes the chiral polaron formation. Furthermore, due to the k-dependent nature of the polaronic self-energy, in analogy with quantum chromodynamics, we also propose a running electron–phonon coupling constant as a function of energy.

Graphene on cubic and hexagonal SiC: A comparative theoretical study

Epitaxial graphene grows on different SiC polytypes possessing distinct bulk band gaps. In this work we systematically investigate the influence of polytypes on the graphene electronic spectrum employing local-density approximation (LDA)/Heyd-Scuseria-Ernzerhof (HSE) ab initio calculations including different buffer layer--graphene layer stackings. We find a variation of the Dirac point position with respect to the valence-band edge as a function of the polytype hexagonality. HSE values are in good agreement with recent experimental results, while LDA corroborates the trends. Since the Dirac point, interface-related states, and the Fermi level follow similar polytype-induced shifts, the doping of the epilayer stays practically the same. For the $AB$ stacked buffer and epilayer on a Si-terminated SiC substrate the graphene spectrum exhibits a polytype-dependent energy gap ${\ensuremath{\epsilon}}_{g}$ which ranges $25--40$ meV for different polytypes. On the contrary, for the $AA$ stacking the Dirac cone remains intact. We suggest a symmetry-based analytical model which explains the origin of the gap and its absence for the $AA$ geometry and provides a direct connection between ${\ensuremath{\epsilon}}_{g}$ and the buffer-epilayer interaction potential.

Tuning Electronic Structure of Graphene: A First-Principles Study

Based on first-principles study, tuning of electronic structure of graphene is reported. The emergence of band gap in this semimetal can be accomplished through different mechanisms. In this study, we have reported on the band gap modulations in graphene through chemical functionalization with oxygen, under the application of external stress, and through the creation of vacancies. Our study suggests that all these mechanisms alter either electronic properties or both structural and electronic properties of graphene. As a result, these mechanisms completely destroy the nature of massless Dirac fermions of graphene. Also, we report on the effect of static electric field on the band gap in hydrogenated graphene (graphane). The combined action of structural modifications that involves stretching and compression of C-C bonds in the hexagonal network and charge transfer mechanism are responsible for the gap opening in electronic spectrum of graphene, which is essential for the future application of graphene in electronics. The introduction of strain is a nondestructive method when compared to other methods for band gap engineering in graphene.

Graphene coupling to circularly polarized photons

AbstractWe have demonstrated theoretically that the Dirac electron spectrum of graphene leads to the strong electron in‐teraction with circularly polarized photons for low light intensities. This results in bound electron‐photon states (charged quasiparticles) which should be considered as a substantially new kind of light‐matter coupling.The energy spectrum of the quasiparticles is of dielectric type, that leads to the photon‐induced metal‐insulator transition in graphene. (© 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)