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The electronic properties of graphene
Abstract
This article reviews the basic theoretical aspects of graphene, a one atom thick allotrope of carbon, with unusual two-dimensional Dirac-like electronic excitations. The Dirac electrons can be controlled by application of external electric and magnetic fields, or by altering sample geometry and/or topology. We show that the Dirac electrons behave in unusual ways in tunneling, confinement, and integer quantum Hall effect. We discuss the electronic properties of graphene stacks and show that they vary with stacking order and number of layers. Edge (surface) states in graphene are strongly dependent on the edge termination (zigzag or armchair) and affect the physical properties of nanoribbons. We also discuss how different types of disorder modify the Dirac equation leading to unusual spectroscopic and transport properties. The effects of electron-electron and electron-phonon interactions in single layer and multilayer graphene are also presented.
... At the same time, however, the velocity of the electrons and holes is less than the speed of light. This makes graphene a test bed for high-energy physics: some quantum relativistic effects that are hardly reachable in experiments with subatomic particles using particle accelerators have clear analogy in the physics of electrons and holes in graphene, which can be measured and studied more easily because of their lower velocity [13,14]. An example is the Klein paradox, in which ultra-relativistic quantum particles, contrary to intuition, penetrate easily through very high and broad energy barriers [15,16]. ...
... At the same time, however, the velocity of the electrons and holes is less than the speed of light. This makes graphene a test bed for high-energy physics: some quantum relativistic effects that are hardly reachable in experiments with subatomic particles using particle accelerators have clear analogy in the physics of electrons and holes in graphene, which can be measured and studied more easily because of their lower velocity [13,14]. An example is the Klein paradox, in which ultrarelativistic quantum particles, contrary to intuition, penetrate easily through very high and broad energy barriers [15,16]. ...
Little research has been conducted to determine the thermal properties and phenomena of graphane and fluorographene. A clear understanding of the thermal problems involved is needed, which may provide a basis for further research on other material properties. In the present study, molecular dynamics simulations were performed to investigate the thermal properties of graphane and fluorographene and especially the phenomena involved, including thermal fluctuations and bending rigidities. Furthermore, comparisons of thermal properties and the phenomena involved were made computationally between pristine and functionalised graphene. The thermal fluctuations and bending rigidities were determined at different temperatures. The present study aims to provide a clear understanding of the thermal problems involved in hydrogenated and fluorinated graphene. The results indicated that while thermally excited ripples spontaneously appear in graphene, fully hydrogenated or fluorinated graphene is substantially unrippled due to their very high bending rigidities. There is no significant effect of thermal rippling throughout graphane and fluorographene due to their very high bending rigidities. However, partially hydrogenated or fluorinated graphene exhibits strong thermal fluctuations. Graphene behaves differently from graphane and fluorographene with regard to the dependence of bending rigidity on temperature. Furthermore, significant out-of-plane fluctuations may occur in partially fluorinated graphene. Thermal fluctuations of graphene are more sensitive to temperature than those of graphane and fluorographene.
Keywords: Thermal properties; Molecular dynamics; Thermal rippling; Thermal fluctuations; Bending rigidities; Thermal phenomena
... At energies below a few eV, the electronic properties of graphene are well described by a set of massless or very light quasiparticles with spin 1=2 obeying the Dirac equation, where the speed of light c is replaced with the Fermi velocity v F ≈ c=300 [12][13][14][15]. (In the following text, we use the system of units where ℏ ¼ c ¼ 1.) ...
... Note that for graphene the Dirac cones are located at the two points at the corners of the Brillouin zone [13]. Then, after taking the trace over the gamma matrices, the resulting polarization tensor in the momentum representation is given by [30] Π μν ðik Dl ; k; ...
In this paper, we consider the convergence properties of the polarization tensor of graphene obtained in the framework of thermal quantum field theory in three-dimensional spacetime. During the last years, this problem attracted much attention in connection with the calculation of the Casimir force in graphene systems and the investigation of the electrical conductivity and reflectance of graphene sheets. There are contradictory statements in the literature, especially on whether this tensor has an ultraviolet divergence in three dimensions. Here, we analyze this problem using the well-known method of dimensional regularization. It is shown that the thermal correction to the polarization tensor is finite at any D , whereas its zero-temperature part behaves differently for D = 3 and 4. For D = 3 , it is obtained by analytic continuation with no subtracting of infinitely large terms. As for the spacetime of D = 4 , the finite result for the polarization tensor at zero temperature is found after subtracting the pole term. Our results are in agreement with previous calculations of the polarization tensor at both zero and nonzero temperature. This opens the possibility for a wider application of the quantum-field-theoretical approach in investigations of graphene and other two-dimensional novel materials.
Published by the American Physical Society 2024
... Boosted by the exceptional friction and wear properties of graphene and its derivatives, the application of nanomaterials in tribology has gained momentum [8]. Graphene serves as the fundamental building component for several well-known carbon compounds, including zerodimensional fullerene, one-dimensional carbon nanotubes, and threedimensional (3D) graphite. ...
... Although there are several review articles on different 2D materials in general [6,8,9,11,[17][18][19][20][21][22][23] as well as when employed for friction and wear improvement [1,3,[27][28][29][33][34][35][36][37][38][39][40][41][42][43][44][45] or protection against corrosion or oxidation [46][47][48], to the best of our knowledge, the combination of these two aspects is yet to be covered by a comprehensive review article. Therefore, this article aims at critically assessing the existing state-ofthe-art regarding the application of 2D nanomaterials to improve the tribo-corrosion and -oxidation behavior. ...
... The discovery of graphene has provided a groundbreaking platform for the application of nanomaterials, directing significant research attention toward the field of two-dimensional materials [1][2]. Among these materials, borophene is a notable member, anticipated to serve as a functional group or precursor for constructing boron nanotubes [3]. ...
The element boron has long been central to two-dimensional superconducting materials, and numerous studies have demonstrated the presence of superconductivity in various boron-based structures. Recent work introduced a new variant: Bilayer Kagome borophene, characterized by its bilayer Kagome lattice with van Hove singularity. Using first-principles calculations, our research investigates the unique electronic structure and superconducting properties of Bilayer Kagome borophene (BK-borophene) through first-principles calculations. BK-borophene is identified as a single-gap superconductor with an initial superconducting transition temperature (Tc) of 11.0 K. By strategically doping the material to align its Fermi level with the Van Hove singularity, Tc is significantly enhanced to 30.0 K. The results contribute to the existing understanding of BK-borophene, highlighting its potential as a member of the expanding family of two-dimensional superconducting materials.
... While there is still debate on whether or not this behavior can be realized in simulations [15,16,20,21], there is a resurgent interest in the properties of low dimensional adsorbed phases of helium. Adding to this is the discovery of graphene [22], an atomically thin substrate with reduced (and possibly tunable [23]) van der Waals interactions (compared to graphite), which should enhance delocalization and thus quantum effects near the substrate [9,21,24,25]. Yet, there is still a question of whether the current effective and empirical models used in simulations to describe both helium-graphite and helium-graphene [26][27][28][29], constructed from the superposition of 4 He-C interactions, are sufficient to adequately represent the physics of adsorption under experimental conditions, and in particular, what role corrugation plays in the first and higher adsorbed layers. ...
Adsorption of 4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^4$$\end{document}He on graphene substrates has been a topic of great interest due to the intriguing effects of graphene corrugation on the manifestation of commensurate solid and exotic phases in low-dimensional systems. In this study, we employ worm algorithm quantum Monte Carlo to study helium adsorbed on a graphene substrate to explore corrugation effects in the grand canonical ensemble. We utilized a Szalewicz potential for helium–helium interactions and a summation of isotropic interactions between helium and carbon atoms to construct a helium–graphene potential. We implement different levels of approximation to achieve a smooth potential, three partially corrugated potentials, and a fully ab initio potential to test the effects of corrugation on the first and second layers. We demonstrate that the omission of corrugation within the helium–graphene potential could lead to finite-size effects in both the first and second layers. Thus, a fully corrugated potential should be used when simulating helium in this low-dimensional regime.
... Recent advancements in TMDCs, fabricated at the few-atomiclayer scale, have expanded our understanding of lowdimensional physics [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. Unlike monolayer graphene, TMDCs possess a band gap, enabling optoelectronic devices [3,[14][15][16]. ...
Single-layer quantum dot gate potential causes type-II band alignment, i.e. electrostatically confines holes and repels electrons, or vice versa. Hence, the confinement of excitons in gated type II quantum dots involves a delicate balance of the repulsion of electrons due to the gate potential with the attraction caused by the Coulomb interaction with a hole localized in the quantum dot. This work presents a theory for neutral excitonic complexes within gated $\text{WSe}_\text{2}$ quantum dots, considering spin, valley, electronic orbitals, and many-body interactions. We analyze how the electron-hole attraction depends on a range of system parameters, such as screened Coulomb interaction, strength of confinement of holes, and repulsion of electrons. Using an atomistic tight binding model we compute valence and conduction band states within a computational box comprising over one million atoms with applied gate potential. The atomistic wavefunctions are then used to calculate direct and exchange Coulomb matrix elements for a fictitious type I quantum dot, and to obtain a spectrum of interacting electron-hole pairs. Next, we study the effect of repulsive potential, pulling away electrons from the valence hole. We determine whether electron-hole pairs are sufficiently attracted to overcome electron repulsion by the confinement potential. Finally, we compute the dipole transition between hole and electron states to obtain the absorption spectrum.
We study the bulk-boundary correspondences for zigzag ribbons (ZRs) of massless Dirac fermion in two-dimensional $\alpha$-${T}_3$ lattice. By tuning the hopping parameter $\alpha\in[0,1]$, the $\alpha$-${T}_3$ lattice interpolates between pseudospin $S=1/2$ (graphene) and $S=1$ (${T}_3$ or dice lattice), for $\alpha=0$ and $1$, respectively, which is followed by continuous change of the Berry phase from $\pi$ to $0$. The range of existence for edge states in the momentum space is determined by solving tight-binding equations at the boundaries of the ZRs. We find that the transitions of in-gap bands from bulk to edge states in the momentum space do not only occur at the positions of the Dirac cones but also at additional points depending on $\alpha\notin \{0,1\}$. The $\alpha$-${T}_3$ ZRs are mapped into stub Su-Schrieffer-Heeger chains by performing unitary transforms of the bulk Hamiltonian. The non-trivial topology of the bulk bands is revealed by the Majorana representation of the eigenstates, where the $\mathbb{Z}_2$ topological invariant is manifested by the winding numbers on the complex plane and the Bloch sphere.
We explore the electronic transport characteristics of gapped graphene subjected to a perpendicular magnetic field and scalar potential barriers. Employing the Dirac–Weyl Hamiltonian and the transfer-matrix method, we calculate the transmission and conductance of the system. Our investigation delves into the impact of the energy, the gap energy parameter ( $$\Delta $$ Δ ) and the magnetic flux parameters, including the number of magnetic barriers ( N ), the magnetic field strength ( B ) and the width of the magnetic barriers. We demonstrate that manipulating energy and total magnetic flux parameters allow precise control over the range of incident angle variation. Moreover, adjusting the tunable parameter $$\Delta $$ Δ effectively confines quasiparticles within the magnetic system under study. Notably, an increase in N results in a strong wave vector filtering effect. The resonance effects and the peaks in the transmission and conductance versus $$\Delta $$ Δ are observed for $$N>1$$ N > 1 . The tunability of the system’s transport properties, capable of being toggled on or off, is demonstrated by adjusting $$\Delta $$ Δ and B . As $$\Delta $$ Δ or B increases, we observe suppression of the transmission and conductance beyond critical parameter values.
Starting from a microscopic tight-binding model and using second-order perturbation theory, we derive explicit expressions for the intrinsic and Rashba spin-orbit interaction induced gaps in the Dirac-like low-energy band structure of an isolated graphene sheet. The Rashba interaction parameter is first order in the atomic carbon spin-orbit coupling strength and first order in the external electric field E perpendicular to the graphene plane, whereas the intrinsic spin-orbit interaction which survives at E = 0 is second order in . The spin-orbit terms in the low-energy effective Hamiltonian have the form proposed recently by Kane and Mele. Ab initio electronic structure calculations were performed as a partial check on the validity of the tight-binding model.
Modern electronic devices and novel materials often derive their extraordinary properties from the intriguing, complex behavior of large numbers of electrons forming what is known as an electron liquid. This book provides an in-depth introduction to the physics of the interacting electron liquid in a broad variety of systems, including metals, semiconductors, artificial nano-structures, atoms and molecules. One, two and three dimensional systems are treated separately and in parallel. Different phases of the electron liquid, from the Landau Fermi liquid to the Wigner crystal, from the Luttinger liquid to the quantum Hall liquid are extensively discussed. Both static and time-dependent density functional theory are presented in detail. Although the emphasis is on the development of the basic physical ideas and on a critical discussion of the most useful approximations, the formal derivation of the results is highly detailed and based on the simplest, most direct methods.
Krätschmer and Huffman's revolutionary discovery of a new solid phase of carbon, solid C60, in 1990 opened the way to an entire new class of materials with physical properties so diverse that their richness has not yet been fully exploited. Moreover, as a by-product of fullerene research, carbon nanotubes were later identified, from which novel nanostructures originated that are currently fascinating materials scientists worldwide. Rivers of words have been written on both fullerenes and nanotubes, in the form of journal articles, conference proceedings and books. The present book offers, in a concise and self-contained manner, the basics of the science of these materials as well as detailed information on those aspects that have so far been better explored. Structural, electronic and dynamical properties are described as obtained from various measurements and state-of-the-art calculations. Their interrelation emerges as well as their possible dependence on, for example, preparation conditions or methods of investigation. By presenting and comparing data from different sources, experiment and theory, this book helps the reader to rapidly master the basic knowledge, to grasp important issues and critically discuss them. Ultimately, it aims to inspire him or her to find novel ways to approach still open questions. As such, this book is addressed to new researchers in the field as well as experts.
The construction of a quantum theory of gravity is the most fundamental challenge confronting contemporary theoretical physics. The different physical ideas which evolved while developing a theory of quantum gravity require highly advanced mathematical methods.
This book presents different mathematical approaches to formulate a theory of quantum gravity. It represents a carefully selected cross-section of lively discussions about the issue of quantum gravity which took place at the second workshop "Mathematical and Physical Aspects of Quantum Gravity" in Blaubeuren, Germany. This collection covers in a unique way aspects of various competing approaches. A unique feature of the book is the presentation of different approaches to quantum gravity making comparison feasible. This feature is supported by an extensive index.
The book is mainly addressed to mathematicians and physicists who are interested in questions related to mathematical physics. It allows the reader to obtain a broad and up-to-date overview on a fascinating active research area.
It is shown that a strong impurity potential induces short-range antiferromagnetic (ferrimagnetic) order around itself in a Hubbard model on a half-filled honeycomb lattice. This implies that short-range magnetic order is induced in monolayer graphene by a nonmagnetic defect such as a vacancy with full hydrogen termination or a chemisorption defect.
The uses of graphene in various industrial applications due to its unique electronic properties are discussed. Graphene is a one-atom-thick sheet of carbon and produced by gently pushing small graphite crystals along a hard surface. It can be also developed by an epitaxial growth process that may be suitable for mass-producing graphene for industrial applications. It has unique properties which arise from its honeycomb-lattice structure that could allow to observe strange relativistics effects at speeds much slower than the speed of light and from the collective behavior of electrons. The electrons in graphene can travel large distance without being scattered enable it to be in very fast electronic components and also modify the quantum Hall effect in metals and semiconductors. The interaction between electrons and the honeycomb lattice causes the electrons to behave as if they have absolutely no mass because of this the electrons in graphene are governed by the Dirac equation.
A review on the variety of intercalants of graphite intercalation compounds (GIC) as well as on the peculier behavior of intercalants is given. About a handred kinds of intercalants are listed in the case of single-species intercalation. Intercalation takes place as the gaining of free enthalpy by the attraction between intercalants and graphens mainly by the electrostatic interaction due to the charge transfer. The donor type and acceptor type intercalants make GICs electron rich and hole rich conductors respectively. The charge transfer rate varies from unity to, e.g., 0.02. Staging is the most notable behavior of GIC which can make a c-axis superlattice as large periodic distance as more than 50Å. Staging kinetics due to Daumas-Hérold model, a ‘sliding model’ as the movement of inter-calant sheet and intra-stage fine phase transitions are discussed. Various methods of syntheses of GICs are described. Finally the multi-species intercalation is reviewed giving examples of different methods and different combination of species.
1. Introduction 2. Quantum field theory in Minkowski space 3. Quantum field theory in curved spacetime 4. Flat spacetime examples 5. Curved spacetime examples 6. Stress-tensor renormalization 7. Applications of renormalization techniques 8. Quantum black holes 9. Interacting fields References Index.
Professor Ziman's classic textbook on the theory of solids was first pulished in 1964. This paperback edition is a reprint of the second edition, which was substantially revised and enlarged in 1972. The value and popularity of this textbook is well attested by reviewers' opinions and by the existence of several foreign language editions, including German, Italian, Spanish, Japanese, Polish and Russian. The book gives a clear exposition of the elements of the physics of perfect crystalline solids. In discussing the principles, the author aims to give students an appreciation of the conditions which are necessary for the appearance of the various phenomena. A self-contained mathematical account is given of the simplest model that will demonstrate each principle. A grounding in quantum mechanics and knowledge of elementary facts about solids is assumed. This is therefore a textbook for advanced undergraduates and is also appropriate for graduate courses.