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

We calculate the average magnetopolarizability of an isolated metallic sample at a frequency $\ensuremath{\omega}$ comparable to the mean level spacing $\ensuremath{\Delta}.$ The frequency dependence of the magnetopolarizability is described by a universal function of $\ensuremath{\omega}/\ensuremath{\Delta}.$

Frequency dependence of the magnetopolarizability of mesoscopic grains

We present a theoretical description of the semiclassical kinetics of two-dimensional fermions in a smoothly varying random magnetic field (RMF), with emphasis on the composite-fermion (CF) approach to the half-filled Landau level. We demonstrate that the Drude picture of the CF kinetics is only marginally valid at ν= 1 2 and becomes totally inadequate already at a small deviation from half-filling. We show that the non-Markovian character of the transport leads to a strong positive magnetoresistance ρxx at small ν− 1 2 . At larger deviations from ν= 1 2 , the positive magnetoresistance is followed by a sharp falloff of ρxx (“adiabatic localization”). We show that the AC conductivity σxx(ω) in the long-range RMF exhibits distinct non-Drude features. In particular, it has a sharp kink [σxx(ω)−σxx(0)∝|ω|] at zero ω and falls off exponentially at higher ω.

Semiclassical transport in a random magnetic field

We study the quasiclassical magnetoresistance ρxx(B) of a two-dimensional electron gas scattered by a random ensemble of antidots and, additionally, by a smooth random potential. We demonstrate that the combination of the two types of disorder yields qualitatively new behavior of ρxx(B). In particular, (i) it induces a novel quasiclassical memory effect which leads to a strong negative magnetoresistance, with ρxx(B)∝B−4, followed with increasing B by saturation at a value determined solely by the background smooth disorder; (ii) for larger B, the interplay of drift in smooth inhomogeneities and scattering by antidots gives rise to a “diffusion-controlled percolation”, which yields a positive magnetoresistance and ρxx(B) diverging as a power law in the limit of large B. Experimental relevance to the transport in semiconductor heterostructures is discussed.

Magnetotransport in a random array of antidots

Erratum: ``Analytical derivation of the scaling law for the inverse participation ratio in quasi-one-dimensional disordered systems'' [Phys. Rev. Lett. 69, 1093 (1992)]

We consider the problem of finite resistance $R$ in superconducting films with geometry of a strip of width $W$ near zero temperature. The resistance is generated by vortex configurations of the phase field. In the first type of process, quantum phase slip, the vortex worldline in 2+1 dimensional space-time is space-like (i.e., the superconducting phase winds in time and space). In the second type, vortex tunneling, the worldline is time-like (i.e., the phase winds in the two spatial directions) and connects opposite edges of the film. For moderately disordered samples, processes of second type favor a train of vortices, each of which tunnels only across a fraction of the sample. Optimization with respect to the number of vortices yields a tunneling distance of the order of the coherence length $\xi$, and the train of vortices becomes equivalent to a quantum phase slip. Based on this theory, we find the resistance $\ln R \sim -g W/\xi$, where $g$ is the dimensionless normal-state conductance. Incorporation of quantum fluctuations indicates a quantum phase transition to an insulating state for $g \lesssim 1$.

Alexander D. Mirlin

Papers

Frequency dependence of the magnetopolarizability of mesoscopic grains

Semiclassical transport in a random magnetic field

Magnetotransport in a random array of antidots

Erratum: ``Analytical derivation of the scaling law for the inverse participation ratio in quasi-one-dimensional disordered systems'' [Phys. Rev. Lett. 69, 1093 (1992)]

Resistance of 2D superconducting films