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

A quantized Hall plateau of ${\ensuremath{\rho}}_{\mathrm{xy}}=\frac{3h}{{e}^{2}}$, accompanied by a minimum in ${\ensuremath{\rho}}_{\mathrm{xx}}$, was observed at $Tl5$ K in magnetotransport of high-mobility, two-dimensional electrons, when the lowest-energy, spin-polarized Landau level is $\frac{1}{3}$ filled. The formation of a Wigner solid or charge-density-wave state with triangular symmetry is suggested as a possible explanation.

https://link.aps.org/pdf/10.1103/PhysRevLett.48.1559

Two-Dimensional Magnetotransport in the Extreme Quantum Limit

Topological insulators are a new phase of matter, with the striking property that conduction of electrons occurs only on their surfaces. In two dimensions, electrons on the surface of a topological insulator are not scattered despite defects and disorder, providing robustness akin to that of superconductors. Topological insulators are predicted to have wide-ranging applications in fault-tolerant quantum computing and spintronics. Substantial effort has been directed towards realizing topological insulators for electromagnetic waves. One-dimensional systems with topological edge states have been demonstrated, but these states are zero-dimensional and therefore exhibit no transport properties. Topological protection of microwaves has been observed using a mechanism similar to the quantum Hall effect, by placing a gyromagnetic photonic crystal in an external magnetic field. But because magnetic effects are very weak at optical frequencies, realizing photonic topological insulators with scatter-free edge states requires a fundamentally different mechanism-one that is free of magnetic fields. A number of proposals for photonic topological transport have been put forward recently. One suggested temporal modulation of a photonic crystal, thus breaking time-reversal symmetry and inducing one-way edge states. This is in the spirit of the proposed Floquet topological insulators, in which temporal variations in solid-state systems induce topological edge states. Here we propose and experimentally demonstrate a photonic topological insulator free of external fields and with scatter-free edge transport-a photonic lattice exhibiting topologically protected transport of visible light on the lattice edges. Our system is composed of an array of evanescently coupled helical waveguides arranged in a graphene-like honeycomb lattice. Paraxial diffraction of light is described by a Schrodinger equation where the propagation coordinate (z) acts as 'time'. Thus the helicity of the waveguides breaks z-reversal symmetry as proposed for Floquet topological insulators. This structure results in one-way edge states that are topologically protected from scattering.

Photonic Floquet topological insulators

The recent emergence of two-dimensional layered materials — in particular the transition metal dichalcogenides — provides a new laboratory for exploring the internal quantum degrees of freedom of electrons and their potential for new electronics. These degrees of freedom are the real electron spin, the layer pseudospin, and the valley pseudospin. New methods for the quantum control of the spin and these pseudospins arise from the existence of Berry phase-related physical properties and strong spin–orbit coupling. The former leads to the versatile control of the valley pseudospin, whereas the latter gives rise to an interplay between the spin and the pseudospins. Here, we provide a brief review of both theoretical and experimental advances in this field. Understanding the physics of two-dimensional materials beyond graphene is of both fundamental and practical interest. Recent theoretical and experimental advances uncover the interplay between real spin and pseudospins in layered transition metal dichalcogenides.

Spin and pseudospins in layered transition metal dichalcogenides

A broad review of recent research work on the preparation and the remarkable properties of intercalation compounds of graphite, covering a wide range of topics from the basic chemistry, physics and materials science to engineering applications.

Intercalation compounds of graphite

Characteristics of level broadening and transverse conductivity of a two-dimensional electron system under extremely strong fields have been theoretically investigated in the simplest approximation without the difficulty of divergence, i e. in the so-called damping theoretical one. Those of various cases of short- and long-ranged scatterers have been obtained. To see the dependence on the range explicitly, numerical calculation has been performed for the system with scatterers with the Gaussian potential. Especially in case of short-ranged ones the peak value of the transverse conductivity has been shown to be $(N+1/2)e^{2}/\pi^{2}\hbar$ which depends only on the natural constants and the Landau level index. It has been argued from general point of view that this fact is approximately true without reference to kinds of approximations. Such characteristic of the conductivity was confirmed experimentally, but there still remain some problems as to the absolute value of the level width.

Theory of Quantum Transport in a Two-Dimensional Electron System under Magnetic Fields. I. Characteristics of Level Broadening and Transport under Strong Fields

The oscillatory enhancement of the g factor caused by the exchange interaction among electrons is calculated. The degree of the spin splitting of each Landau level observed in Schubnikov-de Haas oscillation is shown to be explained by the theory of the level broadening of Landau levels if such enhancement is taken into account. The g factor is also calculated under tilted magnetic fields, and the theoretical one is in excellent agreement with the experiment by Fang and Stiles, and with the similar one performed recently by Kobayashi and Komatsubara. It is proposed that the valley splitting is enhanced by just the same mechanism and that such enhancement is a possible reason why it is observed experimentally in low-lying Landau levels.

Theory of Oscillatory g Factor in an MOS Inversion Layer under Strong Magnetic Fields

Hall conductivity σ XY is studied in various approximations. Characteristics of σ XY are obtained for the case of both short- and long-ranged scatterers in the self-consistent Born approximation, which is the simplest one free form the difficulty of divergence. In case of short-ranged scatterers, a relation is shown to hold between σ XY and σ XX within this approximation, if one uses a relaxation time under magnetic fields. Under strong magnetic fields, effects of higher Born scattering become important in low-lying Landau levels. They depend on the sign of scatterers and strongly on their concentrations. Effects of simultaneous scattering from many scatterers are calculated to the lowest order.

Theory of Hall Effect in a Two-Dimensional Electron System

Quantized surface states in an inversion later of narrow-gap semiconductor with the zinc-blend structure are investigated theoretically by the effective mass approximation for nearly degenerate bands. When the band gap is small, and the spin-orbit interaction is large, the inversion asymmetry of surface potentioal causes large k -linear term in the two dimensional dispersion relation, which removes the spin degeneracy. Also, the g -factor becomes so large that the reversal of the ordering of the Landau levels is expected. The theory is applied to the analysis of the Schubnikov-de Haas measurements of Hg 0.79 Cd 0.21 ( e G =68 meV, m 0 * =0.006 m). The inversion asymmetry splitting of the spin degeneracy is as large as several meV, and the inversion in the order of Landau levels is expected to be observed. The surface potentioal of the sample is expressed in the parametrized form, and values of the parameters are estimated by using the experimental data.

Quantized Surface States of a Narrow-Gap Semiconductor

The theories based upon the multi-valley effective mass equation proposed by Twose are criticized and it is concluded that they can not explain the observed valley splittings. In order to treat valley splittings correctly, a formulation based upon the extended zone scheme is presented with the main interest in the valley splitting in an n -channel (100) inversion layer of Si.

Yasutada Uemura

Papers

Theory of Quantum Transport in a Two-Dimensional Electron System under Magnetic Fields. I. Characteristics of Level Broadening and Transport under Strong Fields

Theory of Oscillatory g Factor in an MOS Inversion Layer under Strong Magnetic Fields

Theory of Hall Effect in a Two-Dimensional Electron System

Quantized Surface States of a Narrow-Gap Semiconductor

Theory of Valley Splitting in an N -Channel (100) Inversion Layer of Si I. Formulation by Extended Zone Effective Mass Theory