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 show that the quantum spin Hall (QSH) effect, a state of matter with topological properties distinct from those of conventional insulators, can be realized in mercury telluride–cadmium telluride semiconductor quantum wells. When the thickness of the quantum well is varied, the electronic state changes from a normal to an “inverted” type at a critical thickness d c . We show that this transition is a topological quantum phase transition between a conventional insulating phase and a phase exhibiting the QSH effect with a single pair of helical edge states. We also discuss methods for experimental detection of the QSH effect.

Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells

Graphene-Like Two-Dimensional Materials

Electrical current can be completely spin polarized in a class of materials known as half-metals, as a result of the coexistence of metallic nature for electrons with one spin orientation and insulating nature for electrons with the other. Such asymmetric electronic states for the different spins have been predicted for some ferromagnetic metals--for example, the Heusler compounds--and were first observed in a manganese perovskite. In view of the potential for use of this property in realizing spin-based electronics, substantial efforts have been made to search for half-metallic materials. However, organic materials have hardly been investigated in this context even though carbon-based nanostructures hold significant promise for future electronic devices. Here we predict half-metallicity in nanometre-scale graphene ribbons by using first-principles calculations. We show that this phenomenon is realizable if in-plane homogeneous electric fields are applied across the zigzag-shaped edges of the graphene nanoribbons, and that their magnetic properties can be controlled by the external electric fields. The results are not only of scientific interest in the interplay between electric fields and electronic spin degree of freedom in solids but may also open a new path to explore spintronics at the nanometre scale, based on graphene.

https://arxiv.org/pdf/cond-mat/0611600.pdf

Half-metallic graphene nanoribbons

Starting from a microscopic tight-binding model and using second-order perturbation theory, we derive explicit expressions for the intrinsic and Rashba spin-orbit interaction induced gaps in the Dirac-like low-energy band structure of an isolated graphene sheet. The Rashba interaction parameter is first order in the atomic carbon spin-orbit coupling strength $\ensuremath{\xi}$ and first order in the external electric field $E$ perpendicular to the graphene plane, whereas the intrinsic spin-orbit interaction which survives at $E=0$ is second order in $\ensuremath{\xi}$. The spin-orbit terms in the low-energy effective Hamiltonian have the form proposed recently by Kane and Mele. Ab initio electronic structure calculations were performed as a partial check on the validity of the tight-binding model.

Intrinsic and Rashba spin-orbit interactions in graphene sheets

We study the gate-voltage induced gap that occurs in graphene bilayers using ab initio density functional theory. Our calculations confirm the qualitative picture suggested by phenomenological tight-binding and continuum models. We discuss enhanced screening of the external interlayer potential at small gate voltages, which is more pronounced in the ab initio calculations, and quantify the role of crystalline inhomogeneity using a tight-binding model self-consistent Hartree calculation.

Ab initio theory of gate induced gaps in graphene bilayers

The $ABC$-stacked $N$-layer-graphene family of two-dimensional electron systems is described at low energies by two remarkably flat bands with Bloch states that have strongly momentum-dependent phase differences between carbon $\ensuremath{\pi}$-orbital amplitudes on different layers and large associated momentum-space Berry phases. These properties are most easily understood using a simplified model with only nearest-neighbor interlayer hopping which leads to gapless semiconductor electronic structure and ${p}^{N}$ dispersion in both conduction and valence bands. We report on a study of the electronic band structures of trilayers which uses ab initio density-functional theory and $\mathbit{k}\ensuremath{\cdot}\mathbit{p}$ theory to fit the parameters of a $\ensuremath{\pi}$-band tight-binding model. We find that when remote interlayer hopping is retained, the triple Dirac point of the simplified model is split into three single Dirac points located along the three KM directions. External potential differences between top and bottom layers are strongly screened by charge transfer within the trilayer but still open an energy gap at overall neutrality.

Band structure of ABC-stacked graphene trilayers

The structures of pure gold and silver clusters (Auk, Agk, k = 1−13) and neutral and anionic gold−silver binary clusters (AumAgn, 2 ≤ k = m + n ≤ 7) have been investigated by using density functional theory (DFT) with generalized gradient approximation (GGA) and high level ab initio calculations including coupled cluster theory with relativistic ab initio pseudopotentials. Pure Auk clusters favor 2-D planar configurations, while pure Agk clusters favor 3-D structures. In the case of Au, the valence orbital energies of 5d are close to that of 6s. This allows the hybridization of 6s and 5d orbitals in favor of planar structures of Auk clusters. Even 1-D linear structures show reasonable stability as local minima (or as global minima in a few small anionic clusters). This explains the ductility of gold. On the other hand, the Ag-4d orbital has a much lower energy than the 5s. This prevents hybridization, and so the coordination number (Nc) of Ag in Agk tends to be large in s-like spherical 3-D coordination i...

Geometrical and electronic structures of gold, silver, and gold-silver binary clusters: Origins of ductility of gold and gold-silver alloy formation

Using a first-principles density-functional electronic structure method, we study the energy gaps and magnetism in bilayer graphene nanoribbons as a function of the ribbon width and the strength of an external electric field between the layers. We assume AB (Bernal) stacking and consider both armchair and zigzag edges and two edge alignments distinguished by different ways of shifting the top layer with respect to the other. Armchair ribbons exhibit three classes of bilayer gaps which decrease with increasing ribbon width. An external electric field between the layers increases the gap in narrow ribbons and decreases the gap for wide ribbons, a property which can be understood semianalytically using a $\ensuremath{\pi}$-band tight-binding model and perturbation theory. The magnetic properties of zigzag edge ribbons are different for the two different edge alignments, and not robust for all exchange-correlation approximations considered. Bilayer ribbon gaps are sensitive to the presence or absence of magnetism.

Bhagawan Sahu

Papers

Intrinsic and Rashba spin-orbit interactions in graphene sheets

Ab initio theory of gate induced gaps in graphene bilayers

Band structure of ABC-stacked graphene trilayers

Geometrical and electronic structures of gold, silver, and gold-silver binary clusters: Origins of ductility of gold and gold-silver alloy formation

Energy gaps, magnetism, and electric-field effects in bilayer graphene nanoribbons