We study the effects of spin orbit interactions on the low energy electronic structure of a single plane of graphene. We find that in an experimentally accessible low temperature regime the symmetry allowed spin orbit potential converts graphene from an ideal two dimensional semimetallic state to a quantum spin Hall insulator. This novel electronic state of matter is gapped in the bulk and supports the transport of spin and charge in gapless edge states that propagate at the sample boundaries. The edge states are non chiral, but they are insensitive to disorder because their directionality is correlated with spin. The spin and charge conductances in these edge states are calculated and the effects of temperature, chemical potential, Rashba coupling, disorder and symmetry breaking fields are discussed.PACS numbers: 72.25.Hg, 73.61.Wp, The generation of spin currents solid state systems has been a focus of intense recent interest. It has been argued that in doped semiconductors the spin orbit (SO) interaction leads to a spin-Hall effect [1,2], in which a spin current flows perpendicular to an applied electric field. The spin Hall effect has been observed in GaAs [3,4]. Murakami et al. [5] have identified a class of cubic materials that are insulators, but nonetheless exhibit a finite spin Hall conductivity. Such a "spin Hall insulator" would be of intrinsic interest, since it would allow for spin currents to be generated without dissipation.In this paper we show that at sufficiently low energy a single plane of graphene exhibits a quantum spin-Hall (QSH) effect with an energy gap that is generated by the SO interaction. Our motivation is twofold. First, Novoselov et al. [6] have recently reported progress in the preparation of single layer graphene films. These films exhibit the expected ambipolar behavior when gated and have mobilities up to 10 4 cm 2 /Vs. Thus, the detailed experimental study of graphene now appears feasible. We believe the QSH effect in graphene is observable below a low but experimentally accessible temperature. Secondly, we will show the QSH effect in graphene is different from the spin hall effects studied for three dimensional cubic systems in Ref. 5 because it leads to a phase which is topologically distinct from a band insulator. The QSH effect in graphene resembles the charge quantum Hall effect, and we will show that spin and charge currents can be transported in gapless edge states. As a model system, graphene thus identifies a new class of spin Hall insulator. It may provide a starting point for the search for other spin-Hall insulators in two dimensional or in layered materials with stronger SO interaction.SO effects in graphite have been known for over 40 years [7], and play a role in the formation of minority hole pockets in the graphite Fermi surface [8]. However, these effects have largely been ignored because they are predicted to be quite small and they are overwhelmed by the larger effect of coupling between the graphene planes. Unlike graphite which has a finite Fermi surfa...

We study three dimensional generalizations of the quantum spin Hall (QSH) effect. Unlike two dimensions, where the QSH effect is distinguished by a single Z2 topological invariant, in three dimensions there are 4 invariants distinguishing 16 "topological insulator" phases. There are two general classes: weak (WTI) and strong (STI) topological insulators. The WTI states are equivalent to layered 2D QSH states, but are fragile because disorder continuously connects them to band insulators. The STI states are robust and have surface states that realize the 2+1 dimensional parity anomaly without fermion doubling, giving rise to a novel "topological metal" surface phase. We introduce a tight binding model which realizes both the WTI and STI phases, and we discuss the relevance of this model to real three dimensional materials, including bismuth.

The quantum spin Hall (QSH) phase is a time reversal invariant electronic state with a bulk electronic band gap that supports the transport of charge and spin in gapless edge states. We show that this phase is associated with a novel Z2 topological invariant, which distinguishes it from an ordinary insulator. The Z2 classification, which is defined for time reversal invariant Hamiltonians, is analogous to the Chern number classification of the quantum Hall effect. We establish the Z2 order of the QSH phase in the two band model of graphene and propose a generalization of the formalism applicable to multi band and interacting systems.PACS numbers: 73. 72.25.Hg, 73.61.Wp, The classification of electronic states according to topological invariants is a powerful tool for understanding many body phases which have bulk energy gaps. This approach was pioneered by Thouless et al.[1] (TKNN), who identified the topological invariant for the non interacting integer quantum Hall effect. The TKNN integer, n, which gives the quantized Hall conductivity for each band σ xy = ne 2 /h, is given by an integral of the Bloch wavefunctions over the magnetic Brillouin zone, and corresponds to the first Chern class of a U (1) principal fiber bundle on a torus [2,3]. An equivalent formulation, generalizable to interacting systems, is to consider the sensitivity of the many body ground state to phase twisted periodic boundary conditions [4,5]. This topological classification distinguishes a simple insulator from a quantum Hall state, and explains the insensitivity of the Hall conductivity to weak disorder and interactions. Nonzero TKNN integers are also intimately related to the presence of gapless edge states on the sample boundaries [6].Since the Hall conductivity violates time reversal (TR) symmetry, the TKNN integer must vanish in a TR invariant system. Nonetheless, we have recently shown that the spin orbit interaction in a single plane of graphene leads to a TR invariant quantum spin Hall (QSH) state which has a bulk energy gap, and a pair of gapless spin filtered edge states on the boundary [7]. In the simplest version of our model (a π electron tight binding model with mirror symmetry about the plane) the perpendicular component of the spin, S z , is conserved. Our model then reduces to independent copies for each spin of a model introduced by Haldane[8], which exhibits an integer quantum Hall effect even though the average magnetic field is zero. When S z is conserved the distinction between graphene and a simple insulator is thus easily understood. Each spin has an independent TKNN integer n ↑ , n ↓ . TR symmetry requires n ↑ + n ↓ = 0, but the difference n ↑ − n ↓ is nonzero and defines a quantized spin Hall conductivity.This characterization breaks down when S z non conserving terms are present. Such terms will inevitably arise due to coupling to other bands, mirror symmetry breaking terms, interactions or disorder. Though these perturbations destroy the quantization of the spin Hall conductance, we argued that they do not ...

We show that the pseudorelativistic physics of graphene near the Fermi level can be extended to three dimensional (3D) materials. Unlike in phase transitions from inversion symmetric topological to normal insulators, we show that particular space groups also allow 3D Dirac points as symmetry protected degeneracies. We provide criteria necessary to identify these groups and, as an example, present ab initio calculations of β-cristobalite BiO(2) which exhibits three Dirac points at the Fermi level. We find that β-cristobalite BiO(2) is metastable, so it can be physically realized as a 3D analog to graphene.

A theory of the long wavelength low energy electronic structure of graphite-derived nanotubules is presented. The propagating π electrons are described by wrapping a massless two dimensional Dirac Hamiltonian onto a curved surface. The effects of the tubule size, shape and symmetry are included through an effective vector potential which we derive for this model. The rich gap structure for all straight single wall cylindrical tubes is obtained analytically in this theory, and the effects of inhomogeneous shape deformations on nominally metallic armchair tubes are analyzed.Since the discovery of a new family of carbon based structures formed by folding graphite sheets into compact tube-shaped objects, there has been interest in the electronic properties which can be realized with these structures [1]. It is now understood that these tubes exhibit insulating, semimetallic or metallic behavior depending on the helicity of the mapping of the graphite sheet onto the surface of the tube [2][3][4]. Discrete microscopic defects, in the form of disclination pairs, provide an interface between neighboring straight tubule segments of different helicities with different electronic gaps, providing a novel class of elemental heterojunctions [5].In this Letter we investigate the effects of shape fluctuations on the electronic properties of the carbon nanotubes. We present a new formulation of this problem which allows us to study the effects of geometry on the quantum dynamics for a π electron propagating within the surface of the wrapped graphite sheet. We show that the very rich phenomenology already well established for straight single wall cylindrical tubules can be derived directly from this geometrical theory [2][3][4]. We then extend the model to consider the effects of inhomogeneous deformations in the form of local twists and bends of the tubule on the low energy electronic structure. These are important low energy structural degrees of freedom of the tubules, and indeed one finds that these deformations are easily quenched into any three dimensional network composed of tubules. We show that these shape fluctuations also have a very strong effect on the low energy electronic and transport properties.An isolated two dimensional sheet of graphite is a semimetal, with the Fermi energy residing at a critical point in the two dimensional π electron spectrum. The Fermi surface is collapsed to a point for this system; there are two distinct Fermi points at K (K ′ ) points of the zone (±4π/3a, 0), where a is the length of the primitive translation vector (a = √ 3d where d is the nearest neighbor bond length of the graphite lattice). Expanding the π electron Hamiltonian around either of these points, and linearizing in spatial derivatives, one finds that the low energy electronic states are described by a massless two dimensional Dirac Hamiltonian, H eff = v σ · p, where p denotes a two dimensional momentum in the graphite plane, and the σ's are the 2 × 2 Pauli matrices [6]. Here the two spin polarizations of the particle refe...

The low energy electronic spectra of rotationally faulted graphene bilayers are studied using a long wavelength theory applicable to general commensurate fault angles. Lattice commensuration requires low energy electronic coherence across a fault and preempts massless Dirac behavior near the neutrality point. Sublattice exchange symmetry distinguishes two families of commensurate faults that have distinct low energy spectra which can be interpreted as energy-renormalized forms of the spectra for the limiting Bernal and AA stacked structures. Sublattice-symmetric faults are generically fully gapped systems due to a pseudospin-orbit coupling appearing in their effective low energy Hamiltonians.

We report broadband visible photoluminescence from solid graphene oxide, and modifications of the emission spectrum by progressive chemical reduction. The data suggest a gapping of the two-dimensional electronic system by removal of π-electrons. We discuss possible gapping mechanisms, and propose that a Kekule pattern of bond distortions may account for the observed behavior.

We demonstrate the existence of topological superconductors (SCs) protected by mirror and time-reversal symmetries. D-dimensional (D=1, 2, 3) crystalline SCs are characterized by 2(D-1) independent integer topological invariants, which take the form of mirror Berry phases. These invariants determine the distribution of Majorana modes on a mirror symmetric boundary. The parity of total mirror Berry phase is the Z(2) index of a class DIII SC, implying that a DIII topological SC with a mirror line must also be a topological mirror SC but not vice versa and that a DIII SC with a mirror plane is always time-reversal trivial but can be mirror topological. We introduce representative models and suggest experimental signatures in feasible systems. Advances in quantum computing, the case for nodal SCs, the case for class D, and topological SCs protected by rotational symmetries are pointed out.