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

Topological insulators are materials with a bulk excitation gap generated by the spin-orbit interaction that are different from conventional insulators. This distinction is characterized by ${Z}_{2}$ topological invariants, which characterize the ground state. In two dimensions, there is a single ${Z}_{2}$ invariant that distinguishes the ordinary insulator from the quantum spin-Hall phase. In three dimensions, there are four ${Z}_{2}$ invariants that distinguish the ordinary insulator from ``weak'' and ``strong'' topological insulators. These phases are characterized by the presence of gapless surface (or edge) states. In the two-dimensional quantum spin-Hall phase and the three-dimensional strong topological insulator, these states are robust and are insensitive to weak disorder and interactions. In this paper, we show that the presence of inversion symmetry greatly simplifies the problem of evaluating the ${Z}_{2}$ invariants. We show that the invariants can be determined from the knowledge of the parity of the occupied Bloch wave functions at the time-reversal invariant points in the Brillouin zone. Using this approach, we predict a number of specific materials that are strong topological insulators, including the semiconducting alloy ${\mathrm{Bi}}_{1\ensuremath{-}x}{\mathrm{Sb}}_{x}$ as well as $\ensuremath{\alpha}\text{\ensuremath{-}}\mathrm{Sn}$ and HgTe under uniaxial strain. This paper also includes an expanded discussion of our formulation of the topological insulators in both two and three dimensions, as well as implications for experiments.

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Topological insulators with inversion symmetry

We provide a broad review of fundamental electronic properties of two-dimensional graphene with the emphasis on density and temperature dependent carrier transport in doped or gated graphene structures. A salient feature of our review is a critical comparison between carrier transport in graphene and in two-dimensional semiconductor systems (e.g. heterostructures, quantum wells, inversion layers) so that the unique features of graphene electronic properties arising from its gap- less, massless, chiral Dirac spectrum are highlighted. Experiment and theory as well as quantum and semi-classical transport are discussed in a synergistic manner in order to provide a unified and comprehensive perspective. Although the emphasis of the review is on those aspects of graphene transport where reasonable consensus exists in the literature, open questions are discussed as well. Various physical mechanisms controlling transport are described in depth including long- range charged impurity scattering, screening, short-range defect scattering, phonon scattering, many-body effects, Klein tunneling, minimum conductivity at the Dirac point, electron-hole puddle formation, p-n junctions, localization, percolation, quantum-classical crossover, midgap states, quantum Hall effects, and other phenomena.

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Electronic transport in two-dimensional graphene

The transport properties of disordered solids have been the subject of much work since at least the 1950s, but with a new burst of activity during the 1980s which has survived up to the present day. There have been numerous reviews of a more or less specialized nature. The present review aims to fill the niche for a non-specialized review of this very active area of research. The basic concepts behind the theory are introduced with more detailed sections covering experimental results, one-dimensional localization, scaling theory, weak localization, magnetic field effects and fluctuations.

https://iopscience.iop.org/article/10.1088/0034-4885/56/12/001/pdf

Localization: theory and experiment

Weak localization in thin films: a time-of-flight experiment with conduction electrons

Effect of the spin-orbit interaction is studied for the random potential scattering in two dimensions by the renormalization group method. It is shown that the localization behaviors are classified in the three different types depending on the symmetry. The recent observation of the negative magnetoresistance of MOSFET is discussed. In recent experiments on MOSFET by Kawaguchi et al.,u it was found that electrons confined in the MOS inversion layer exhibit the negative magnetoresist­ ance. This effect is closely related to the localization problem in a random potential. In two dimensions, the quantum inter­ ference is important and, if the impurity scattering is spin-independent, the con­ ductivity vanishes at zero temperature even when the scattering is very weak. 2>

Spin-Orbit Interaction and Magnetoresistance in the Two Dimensional Random System

Localized Magnetic States in Dilute Alloys

A theory of the superlattice transitions of TTF-TCNQ is presented with strict regard for its quasi-one-dimensional characters. The model investigated is a two-dimensional array of two kinds of linear chains weakly coupled with one another. The fluctuation of phase of charge density waves on each chain is treated carefully, while the interchain couplings are approximated by mean fields. The properties of the transitions at 54 K and 49 K are well explained, and the expressions of the transition temperatures and the temperature dependence of order parameters are derived.

Theory of the Structural Phase Transitions in TTF-TCNQ. II

The spectrum of spin waves in ferromagnetic metals where ferromagnetism is caused by the so·called s-d exchange interaction between localized spins and conduction electrons is investigated theoretically without taking an adiabatic approximation which is usually used. The dynamical susceptibility of the system is calculated by making use of the two·time Green's function, and the spectrum is determined as divergent points of the susceptibility. As non-adiabatic effects, there appear some features in the spectrum which have not been considered in usual adiabatic theories. First, if the wave number· of spin waves is not so large, there are two or more mod~s of collective oscillations of the system, among which one may well be called acoustical mode where two spin systems, localized spins and conduc­ tion electron spins, oscillate in phase, and another an optical mode where they oscillate in antiphase. Further, as to the acoustical mode, the spectrum has a dip in the vicinity of a certain wave number where an adiabatic approximation becomes incorrect.

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Dynamical Theory of Spin Waves in Ferromagnetic Metals with s-d Exchange Interaction

A single electron tunneling device consisting of a double junction with inductance, resistance and capacitance (Z-SET) is considered. Current–voltage characteristics are calculated using a self-consistent treatment of the environmental impedance. It is shown that the single electron tunneling can be controlled by modulating the resistive environmental impedance. It is also shown that the controllability of the device is strongly degraded when the capacitance becomes large. This suggests the importance of the careful design of Z-SET against the stray capacitance inevitable in the actual device. The critical value of stray capacitance for controllability of Z-SET is estimated.

Yosuke Nagaoka

Papers

Spin-Orbit Interaction and Magnetoresistance in the Two Dimensional Random System

Localized Magnetic States in Dilute Alloys

Theory of the Structural Phase Transitions in TTF-TCNQ. II

Dynamical Theory of Spin Waves in Ferromagnetic Metals with s-d Exchange Interaction

Single electron tunneling device controlled by environmental impedance modulation