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.

A. J. Chaves acknowledges the scholarship from the Brazilian agency CNPq (Conselho Nacional de Desenvolvimento Cientiﬁco e Tecnologico). N.M.R. Peres acknowledges support from the European Commission through the project “Graphene-Driven Revolutions in ICT and Beyond" (Ref. No. 696656) and the Portuguese Foundation for Science and Technology (FCT) in the framework of the Strategic Financing UID/FIS/04650/2013. The authors acknowledge Jose Carlos Viana Gomes for discussions that led to the estimations made in Sec. VIII.

https://repositorium.sdum.uminho.pt/bitstream/1822/43424/1/1610.00162.pdf

Pumping electrons in graphene to the M point in the Brillouin zone: Emergence of anisotropic plasmons

We consider electronic transport accross one-dimensional heterostructures described by the Dirac equation. We discuss the cases where both the velocity and the mass are position dependent. We show how to generalize the Dirac Hamiltonian in order to obtain a Hermitian problem for spatial dependent velocity. We solve exactly the case where the position dependence of both velocity and mass is linear. In the case of velocity profiles, it is shown that there is no backscattering of Dirac electrons. In the case of the mass profile backscattering exists. In this case, it is shown that the linear mass profile induces less backscattering than the abrupt step-like profile. Our results are a first step to the study of similar problems in graphene.

Scattering in one-dimensional heterostructures described by the Dirac equation

The band structure and the optical conductivity of an ABA (Bernal-type) stacked graphene trilayer are calculated. It is shown that, under appropriate doping, a strong resonant peak develops in the optical conductivity, located at the frequency corresponding to approximately 1.4 times the interlayer hopping energy and caused by the 'nesting' of two nearly parabolic bands in the electronic spectrum. The intensity of this resonant absorption can be controlled by adjusting the gate voltage. The effect is robust with respect to increasing temperature.

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Optical conductivity of ABA stacked graphene trilayer: mid-IR resonance due to band nesting

We obtain an analytical expression for the linewidth of the $1s$ exciton as a function of temperature in transition metal dichalcogenides The total linewidth, as a function of temperature, is dominated by three contributions: (i) the radiative decay (essentially temperature independent), (ii) the phonon-induced intravalley scattering, and (iii) the phonon-induced intervalley scattering Our approach uses a variational Ansatz to solve the Wannier equation, allowing for an analytical treatment of the excitonic problem, including rates of the decay dynamics Our results are in good agreement with experimental data already present in the literature and can be used to readily predict the value of the total linewidth at any temperature in the broad class of excitonic two-dimensional materials

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Analytical description of the 1 s exciton linewidth temperature dependence in transition metal dichalcogenides

In this paper we study analytically a one-dimensional model for a semiconductor-metal junction. We study the formation of Tamm states and how they evolve when the semi-infinite semiconductor and metal are coupled together. The non-linear current, as a function of the bias voltage, is studied using the non-equilibrium Green's function method and the density matrix of the interface is given. The electronic occupation of the sites defining the interface has strong non-linearities as a function of the bias voltage due to strong resonances present in the Green's functions of the junction sites. The surface Green's function is computed analytically by solving a quadratic matrix equation, which does not require adding a small imaginary constant to the energy. The wave function for the surface states is given.

Nuno M. R. Peres

Papers

Pumping electrons in graphene to the M point in the Brillouin zone: Emergence of anisotropic plasmons

Scattering in one-dimensional heterostructures described by the Dirac equation

Optical conductivity of ABA stacked graphene trilayer: mid-IR resonance due to band nesting

Analytical description of the 1 s exciton linewidth temperature dependence in transition metal dichalcogenides

Analytical study of non-linear transport across a semiconductor-metal junction. Resonances, surface states, and non-linear transport