Graphene is a rapidly rising star on the horizon of materials science and condensed-matter physics. This strictly two-dimensional material exhibits exceptionally high crystal and electronic quality, and, despite its short history, has already revealed a cornucopia of new physics and potential applications, which are briefly discussed here. Whereas one can be certain of the realness of applications only when commercial products appear, graphene no longer requires any further proof of its importance in terms of fundamental physics. Owing to its unusual electronic spectrum, graphene has led to the emergence of a new paradigm of 'relativistic' condensed-matter physics, where quantum relativistic phenomena, some of which are unobservable in high-energy physics, can now be mimicked and tested in table-top experiments. More generally, graphene represents a conceptually new class of materials that are only one atom thick, and, on this basis, offers new inroads into low-dimensional physics that has never ceased to surprise and continues to provide a fertile ground for applications.

The rise of graphene

다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

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. The Dirac electrons behave in unusual ways in tunneling, confinement, and the integer quantum Hall effect. The electronic properties of graphene stacks are discussed and vary with stacking order and number of layers. Edge (surface) states in graphene depend on the edge termination (zigzag or armchair) and affect the physical properties of nanoribbons. 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.

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The electronic properties of graphene

Quantum electrodynamics (resulting from the merger of quantum mechanics and relativity theory) has provided a clear understanding of phenomena ranging from particle physics to cosmology and from astrophysics to quantum chemistry. The ideas underlying quantum electrodynamics also influence the theory of condensed matter, but quantum relativistic effects are usually minute in the known experimental systems that can be described accurately by the non-relativistic Schrodinger equation. Here we report an experimental study of a condensed-matter system (graphene, a single atomic layer of carbon) in which electron transport is essentially governed by Dirac's (relativistic) equation. The charge carriers in graphene mimic relativistic particles with zero rest mass and have an effective 'speed of light' c* approximately 10(6) m s(-1). Our study reveals a variety of unusual phenomena that are characteristic of two-dimensional Dirac fermions. In particular we have observed the following: first, graphene's conductivity never falls below a minimum value corresponding to the quantum unit of conductance, even when concentrations of charge carriers tend to zero; second, the integer quantum Hall effect in graphene is anomalous in that it occurs at half-integer filling factors; and third, the cyclotron mass m(c) of massless carriers in graphene is described by E = m(c)c*2. This two-dimensional system is not only interesting in itself but also allows access to the subtle and rich physics of quantum electrodynamics in a bench-top experiment.

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Two-dimensional gas of massless Dirac fermions in graphene

Single-layer metal dichalcogenides are two-dimensional semiconductors that present strong potential for electronic and sensing applications complementary to that of graphene.

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Electronics and optoelectronics of two-dimensional transition metal dichalcogenides.

GVA - Generalitat Valenciana(Prometeo2017/139); European Commission through the project 'Graphene- Driven Revolutions in ICT and Beyond' (reference No. 881603 – Core 3), and the Portuguese Foundation for Science and Technology (FCT) in the framework of the Strategic Financing UID/FIS/04650/2013, COMPETE2020, PORTUGAL2020, FEDER and the Portuguese Foundation for Science and Technology (FCT) through projects PTDC/FIS-NAN/3668/2013 and POCI-01-0145-FEDER-028114. J F-R acknowledges financial support from FCT UTAPEXPL/NTec/0046/2017 project, as well as Generalitat Valenciana funding Prometeo2017/139 and MINECO Spain (Grant No. MAT2016-78625-C2). D L R S thankfully acknowledges the use of HPC resources provided by the National Laboratory for Scientific Computing (LNCC/MCTI, Brazil). A T C thankfully acknowledges the use of computer resources at MareNostrum and the technical support provided by Barcelona Supercomputing Center (RES-FI-2019-2-0034)

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Topological magnons in CrI3 monolayers: an itinerant fermion description

CNPq (Brazil), FCT (Portugal), EC Graphene Flagship Project (Contract No. CNECTICT-
604391)

Exciton polaritons in two-dimensional dichalcogenide layers placed in a planar microcavity: Tunable interaction between two Bose-Einstein condensates

We study the effect of electron-electron interactions in the electronic properties of a biased graphene bilayer. This system is a semiconductor with conduction and valence bands characterized by an unusual “mexican-hat” dispersion. We focus on the metallic regime where the chemical potential lies in the “mexican-hat” in the conduction band, leading to a topologically non-trivial Fermi surface in the shape of a ring. We show that due to the unusual topology of the Fermi surface electron-electron interactions are greatly enhanced. We show that the ferromagnetic instability can occur provided a low density of carriers. We compute the electronic polarization function in the random phase approximation and show that, while at low energies the system behaves as a Fermi liquid (albeit with peculiar Friedel oscillations), at high frequencies it shows a highly anomalous response when compare to ordinary metals.

http://absimage.aps.org/image/MAR07/MWS_MAR07-2006-004986.pdf

Fermi Liquid Theory of a Fermi Ring

We study the infrared conductivity of graphene at finite chemical potential and temperature taking into account the effect of phonons and disorder due to charged impurities and unitary scatterers. The screening of the long-range Coulomb potential is treated using the random phase approximation coupled to the coherent potential approximation. The effect of the electron-phonon coupling is studied in second-order perturbation theory. The theory has essentially one free parameter, namely, the number of charge impurities per carbon, n^{{\rm C}}_i. We find an anomalous enhancement of the conductivity in a frequency region that is blocked by Pauli exclusion and an impurity broadening of the conductivity threshold. We also find that phonons induce Stokes and anti-Stokes lines that produce an excess conductivity, when compared to the far infrared value of \sigma_0 = (\pi/2) e^2/h.

The infrared conductivity of graphene

In this paper we analyse the electronic properties of Dirac electrons in finite-size ribbons and in circular and hexagonal quantum dots. We show that due to the formation of sub-bands in the ribbons it is possible to spatially localize some of the electronic modes using a p-n-p junction. We also show that scattering of confined Dirac electrons in a narrow channel by an infinitely massive wall induces mode mixing, giving a qualitative reason for the fact that an analytical solution to the spectrum of Dirac electrons confined in a square box has not yet been found. A first attempt to solve this problem is presented. We find that only the trivial case k = 0 has a solution that does not require the existence of evanescent modes. We also study the spectrum of quantum dots of graphene in a perpendicular magnetic field. This problem is studied in the Dirac approximation, and its solution requires a numerical method whose details are given. The formation of Landau levels in the dot is discussed. The inclusion of the Coulomb interaction among the electrons is considered at the self-consistent Hartree level, taking into account the interaction with an image charge density necessary to keep the back-gate electrode at zero potential. The effect of a radial confining potential is discussed. The density of states of circular and hexagonal quantum dots, described by the full tight-binding model, is studied using the Lanczos algorithm. This is necessary to access the detailed shape of the density of states close to the Dirac point when one studies large systems. Our study reveals that zero-energy edge states are also present in graphene quantum dots. Our results are relevant for experimental research in graphene nanostructures. The style of writing is pedagogical, in the hope that newcomers to the subject will find this paper a good starting point for their research.

http://iopscience.iop.org/article/10.1088/0953-8984/21/34/344202/pdf

Nuno M. R. Peres

Papers

Topological magnons in CrI3 monolayers: an itinerant fermion description

Exciton polaritons in two-dimensional dichalcogenide layers placed in a planar microcavity: Tunable interaction between two Bose-Einstein condensates

Fermi Liquid Theory of a Fermi Ring

The infrared conductivity of graphene

Dirac electrons in graphene-based quantum wires and quantum dots.