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

There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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

Topological insulators are new states of quantum matter which cannot be adiabatically connected to conventional insulators and semiconductors. They are characterized by a full insulating gap in the bulk and gapless edge or surface states which are protected by time-reversal symmetry. These topological materials have been theoretically predicted and experimentally observed in a variety of systems, including HgTe quantum wells, BiSb alloys, and Bi2Te3 and Bi2Se3 crystals. Theoretical models, materials properties, and experimental results on two-dimensional and three-dimensional topological insulators are reviewed, and both the topological band theory and the topological field theory are discussed. Topological superconductors have a full pairing gap in the bulk and gapless surface states consisting of Majorana fermions. The theory of topological superconductors is reviewed, in close analogy to the theory of topological insulators.

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Topological insulators and superconductors

The physics of Anderson transitions between localized and metallic phases in disordered systems is reviewed The term ``Anderson transition'' is understood in a broad sense, including both metal-insulator transitions and quantum-Hall-type transitions between phases with localized states The emphasis is put on recent developments, which include: multifractality of critical wave functions, criticality in the power-law random banded matrix model, symmetry classification of disordered electronic systems, mechanisms of criticality in quasi-one-dimensional and two-dimensional systems and survey of corresponding critical theories, network models, and random Dirac Hamiltonians Analytical approaches are complemented by advanced numerical simulations

Anderson Transitions

We study the conductivity $\ensuremath{\sigma}(T)$ of interacting electrons in a low-dimensional disordered system at low temperature $T$ For weak interactions, the weak-localization regime crosses over with lowering $T$ into a dephasing-induced ``power-law hopping'' As $T$ is further decreased, the Anderson localization in Fock space crucially affects $\ensuremath{\sigma}(T)$, inducing a transition at $T={T}_{c}$, so that $\ensuremath{\sigma}(Tl{T}_{c})=0$ The critical behavior of $\ensuremath{\sigma}(T)$ above ${T}_{c}$ is $\mathrm{ln}﻿\ensuremath{\sigma}(T)\ensuremath{\propto}\ensuremath{-}(T\ensuremath{-}{T}_{c}{)}^{\ensuremath{-}1/2}$ The mechanism of transport in the critical regime is many-particle transitions between distant states in Fock space

Interacting electrons in disordered wires: Anderson localization and low-T transport

http://inis.jinr.ru/sl/P_Physics/PT_Thermodynamics,%20statistical%20physics/Mirlin%20Statistics.pdf

Statistics of energy levels and eigenfunctions in disordered systems

We study the electron transport properties of a monoatomic graphite layer (graphene) with different types of disorder. We show that the transport properties of the system depend strongly on the character of disorder. Away from half filling, the concentration dependence of conductivity is linear in the case of strong scatterers, in line with recent experimental observations, and logarithmic for weak scatterers. At half filling the conductivity is of the order of ${e}^{2}∕h$ if the randomness preserves one of the chiral symmetries of the clean Hamiltonian, whereas for generic disorder the conductivity is strongly affected by localization effects.

Electron transport in disordered graphene

We study statistical properties of the ensemble of large N\ifmmode\times\else\texttimes\fi{}N random matrices whose entries ${\mathit{H}}_{\mathit{ij}}$ decrease in a power-law fashion ${\mathit{H}}_{\mathit{ij}}$\ensuremath{\sim}|i-j${\mathrm{|}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$. Mapping the problem onto a nonlinear \ensuremath{\sigma} model with nonlocal interaction, we find a transition from localized to extended states at \ensuremath{\alpha}=1. At this critical value of \ensuremath{\alpha} the system exhibits multifractality and spectral statistics intermediate between the Wigner-Dyson and Poisson statistics. These features are reminiscent of those typical of the mobility edge of disordered conductors. We find a continuous set of critical theories at \ensuremath{\alpha}=1, parametrized by the value of the coupling constant of the \ensuremath{\sigma} model. At \ensuremath{\alpha}g1 all states are expected to be localized with integrable power-law tails. At the same time, for 13/2 the wave packet spreading at a short time scale is superdiffusive: 〈|r|〉\ensuremath{\sim}${\mathit{t}}^{1/(2\mathrm{\ensuremath{\alpha}}\mathrm{\ensuremath{-}}1)}$, which leads to a modification of the Altshuler-Shklovskii behavior of the spectral correlation function. At 1/21 the statistical properties of eigenstates are similar to those in a metallic sample in d=(\ensuremath{\alpha}-1/2${)}^{\mathrm{\ensuremath{-}}1}$ dimensions. Finally, the region \ensuremath{\alpha}1/2 is equivalent to the corresponding Gaussian ensemble of random matrices (\ensuremath{\alpha}=0). The theoretical predictions are compared with results of numerical simulations. \textcopyright{} 1996 The American Physical Society.

Alexander D. Mirlin

Papers

Anderson Transitions

Interacting electrons in disordered wires: Anderson localization and low-T transport

Statistics of energy levels and eigenfunctions in disordered systems

Electron transport in disordered graphene

Transition from localized to extended eigenstates in the ensemble of power-law random banded matrices