Dirac Dispersion Research Articles

Spatial fluctuations of helical Dirac fermions on the surface of topological insulators

Surfaces of topological insulators host a new class of states with Dirac dispersion and helical spin texture. Potential quantum computing and spintronic applications using these states require manipulation of their electronic properties at the Dirac energy of their band structure by inducing magnetism or superconductivity through doping and proximity effect. Yet, the response of these states near the Dirac energy in their band structure to various perturbations has remained unexplored. Here we use spectroscopic mapping with the scanning tunneling microscope to study their response to magnetic and non-magnetic bulk dopants in Bi2Te3 and Bi2Se3. Far from the Dirac energy helicity provides remarkable resilience to backscattering even in the presence of ferromagnetism. However, approaching the Dirac point, where the surface states' wavelength diverges bulk doping results in pronounced nanoscale spatial fluctuations of energy, momentum, and helicity. Our results and their connection with similar studies of Dirac electrons in graphene demonstrate that while backscattering and localization are absent for Dirac topological surface states, reducing charge defects is required for both tuning the chemical potential to Dirac energy and achieving high electrical mobility for these novel states.

Anisotropic Dirac Fermions in a Bi Square Net ofSrMnBi2

We report the observation of highly anisotropic Dirac fermions in a Bi square net of SrMnBi(2), based on a first-principles calculation, angle-resolved photoemission spectroscopy, and quantum oscillations for high-quality single crystals. We found that the Dirac dispersion is generally induced in the (SrBi)(+) layer containing a double-sized Bi square net. In contrast to the commonly observed isotropic Dirac cone, the Dirac cone in SrMnBi(2) is highly anisotropic with a large momentum-dependent disparity of Fermi velocities of ~8. These findings demonstrate that a Bi square net, a common building block of various layered pnictides, provides a new platform that hosts highly anisotropic Dirac fermions.

Anomalous magnetic transport in ferromagnetic graphene junctions

We investigate magnetotransport in a ferromagnetic-normal-ferromagnetic graphene junction where a gate electrode is attached to the normal segment. It is shown that the charge conductance can be maximal at an antiparallel configuration of the magnetizations. Moreover, we demonstrate that both the magnitude and the sign of the spin-transfer torque can be controlled by means of the gate voltage in the normal segment. In this way, the present system constitutes a spin-transfer torque transistor. These anomalous phenomena are attributed to the combined effect of the exchange field and the Dirac dispersion of graphene. Our prediction opens up the possibility of moving domain walls parallel or antiparallel to the current in a controllable fashion by means of a local gate voltage.

Three-dimensional Dirac-like fermions in an optical lattice

We present a scheme to realize three-dimensional Dirac-like fermions with an edge-centered cubic optical lattice. We propose a method to construct the optical edge-centered cubic crystals and then determine the parameters required for the description of the systems with a tight-binding model. Interestingly, we show that the low-energy quasiparticles are three-dimensional massive or massless Dirac-like fermions determined by the parameters of the system. In addition, two three-dimensional degenerate flat bands lie between the upper and the lower branches of the Dirac dispersion. Furthermore, we demonstrate that the Dirac dispersion relation and the flat bands can be verified by the density profile measurement.

The Effective Fine-Structure Constant of Freestanding Graphene Measured in Graphite

Electrons in graphene behave like Dirac fermions, permitting phenomena from high-energy physics to be studied in a solid-state setting. A key question is whether or not these fermions are critically influenced by Coulomb correlations. We performed inelastic x-ray scattering experiments on crystals of graphite and applied reconstruction algorithms to image the dynamical screening of charge in a freestanding graphene sheet. We found that the polarizability of the Dirac fermions is amplified by excitonic effects, improving screening of interactions between quasiparticles. The strength of interactions is characterized by a scale-dependent, effective fine-structure constant, α(g)* (k,ω), the value of which approaches 0.14 ± 0.092 ~ 1/7 at low energy and large distances. This value is substantially smaller than the nominal α(g) = 2.2, suggesting that, on the whole, graphene is more weakly interacting than previously believed.

Weakly Interacting Graphene

Many unusual properties of graphene are a consequence of the Dirac dispersion of its electrons—a linear relationship between an electron's momentum and energy. Naively, this dispersion leads to the conclusion that electrons in graphene are strongly affected by mutual electrostatic interactions; however, there is little experimental evidence for strong interaction. Reed et al. (p. [805][1]) resolved this discrepancy by using inelastic x-ray scattering spectra of graphite (which consists of loosely bound layers of graphene) to estimate how much the electric field was damped by the presence of mobile charge carriers. In fact, damping was strong at distances in excess of 1 nanometer, suggesting that graphene is more weakly interacting than was assumed. [1]: /lookup/doi/10.1126/science.1190920

Guided modes near the Dirac point in negative-zero-positive index metamaterial waveguide

Motivated by the realization of the Dirac point (DP) with a double-cone structure for optical field in the negative-zero-positive index metamaterial (NZPIM), we make theoretical investigations of the guided modes in the NZPIM waveguide near the DP by using the graphical method. Due to the linear Dirac dispersion, the fundamental mode is absent when the angular frequency is smaller than the DP, while the behaviors of NZPIM waveguide are similar to the conventional dielectric waveguide when the angular frequency is larger than the DP. The unique properties of the guided modes are analogous to the propagation of electron waves in graphene waveguide [Appl. Phys. Lett., 94, 212105 (2009)], corresponding to the classical motion and the Klein tunneling. These results suggest that many exotic phenomena in graphene can be simulated by the relatively simple optical NZPIM.

Thermal emission from layered structures containing a negative-zero-positive index metamaterial

Motivated by the realization of the optical Dirac dispersion in the homogenous negative-zero-positive index (NZPI) material, we make a theoretical investigation on the properties of thermal emission in layered structures containing the NZPI medium. We find that when the thermal emission frequency is close to the Dirac point of the NZPI medium, the spectral hemispherical power of thermal emission in such a structure is strongly suppressed and the emission can become a high directional source with large spatial coherence.

Electronic structure near an impurity and terrace on the surface of a three-dimensional topological insulator

Motivated by recent scanning tunneling microscopy experiments on surfaces of ${\text{Bi}}_{1\ensuremath{-}x}{\text{Sb}}_{{x}^{\ensuremath{'}}}$ and ${\text{Bi}}_{2}{\text{Te}}_{3}$, we theoretically study the electronic structure of a three-dimensional (3D) topological insulator in the presence of a local impurity or a domain wall on its surface using a 3D lattice model. While the local density of states (LDOS) oscillates significantly in space at energies above the bulk gap, the oscillation due to the in-gap surface Dirac fermions is very weak. The extracted modulation wave number as a function of energy satisfies the Dirac dispersion for in-gap energies and follows the border of the bulk continuum above the bulk gap. We have also examined analytically the effects of the defects by using a pure Dirac fermion model for the surface states and found that the LDOS decays asymptotically faster at least by a factor of $1/r$ than that in normal metals, consistent with the results obtained from our lattice model.

Photovoltaic Berry curvature in the honeycomb lattice

Photovoltaic Hall effect — the Hall effect induced by intense, circularly-polarized light in the absence of static magnetic fields — has been proposed in Phys. Rev. B 79, 081406R (2009) for graphene where a massless Dirac dispersion is realized. The photovoltaic Berry curvature (a nonequilibrium extension of the standard Berry curvature) is the key quantity to understand this effect, which appears in the Kubo formula extended to Hall transport in the presence of strong AC field backgrounds. Here we elaborate the properties of the photovoltaic curvature such as the frequency and field strength dependence in the honeycomb lattice.

Dirac nodal pockets in the antiferromagnetic parent phase of FeAs superconductors

We show that previously measured small Fermi surface pockets within the antiferromagnetic phase of SrFe2As2 and BaFe2As2 are consistent with a Dirac dispersion modulated by interlayer hopping, giving rise to a Dirac point in k-space and a cusp in the magnetic field angle-dependent magnetic quantum oscillation frequencies. These findings support the existence of a nodal spin-density wave in these materials, which could play an important role in protecting the metallic state against localization effects. The speed of the Dirac fermions in SrFe2As2 and BaFe2As2 is found to be 14-20 times slower than in graphene, suggesting that the pnictides provide a laboratory for exploring the effects of strongly interacting Dirac fermions.

Transmission gap, Bragg-like reflection, and Goos-Hänchen shifts near the Dirac point inside a negative-zero-positive index metamaterial slab

Motivated by the realization of the Dirac point (DP) with a double-cone structure for optical field in the negative-zero-positive index metamaterial (NZPIM), the reflection, transmission, and Goos-H\"anchen (GH) shifts inside the NZPIM slab are investigated. Due to the linear Dirac dispersion, the transmission as the function of the frequency has a gap, thus, the corresponding reflection has a frequency or wavelength window for the perfect reflection, which is similar to the Bragg reflection in the one-dimensional photonic crystals. Near the DP, the associated GH shifts in the transmission and reflection can be changed from positive to negative with increasing the wavelength. These negative and positive shifts can also be enhanced by transmission resonances when the frequency is far from that at the DP. All these phenomena will lead to some potential applications in the integrated optics and optical devices.

Optical Hall conductivity in QHE systems

As a natural extension of the static Hall conductivity to ac regimes, the optical Hall conductivity σxy(ω) is studied theoretically in the THz region. We found that the Hall plateaus remain in the optical (THz) region when the disorder is not too large, although the quantization of the plateau height does not hold. We have also calculated σxy(ω) in the graphene QHE system to find that the Hall plateaus also remain with σxy(ω) reflecting the massless Dirac dispersion in graphene. We predict that the optical Hall conductivity should be measurable through an accurate detection of the Hall angle in the THz regime.

Photo-induced Hall Effect in graphene — effect of boundary types

We theoretically predict, with the Keldysh Green's function combined with the Floquet method, that the AC electric field of a circularly polarized light should induce a Hall effect, in the absence of uniform magnetic fields, in graphene with a pair of Dirac dispersions. Although the Hall coefficient is not quantized, the dynamical gap generated by the AC field and the associated Hall effect bear a topological origin which can be traced back to the Dirac cones. We also study the dependence on boundary conditions. The required AC field strength is estimated to be realistic.

Restricted Wiedemann-Franz Law and Vanishing Thermoelectric Power in One-Dimensional Conductors

In one-dimensional conductors with linear E-k dispersion (Dirac systems), intra-branch thermalization is favored by elastic electron-electron interaction in contrast with electron systems with a nonlinear (parabolic) dispersion. We show that under external electric fields or thermal gradients, the carrier populations of different branches, treated as Fermi gases, have different temperatures as a consequence of self-consistent carrier-heat transport. Specifically, in the presence of elastic phonon scattering, the Wiedemann-Franz law is restricted to each branch with its specific temperature. In addition, thermoelectric power vanishes due to electron-hole symmetry, which is validated by experiment.

Berry phase in graphene: Semiclassical perspective

We derive a semiclassical expression for the Green’s function in graphene, in which the presence of a semiclassical phase is made apparent. The relationship between this semiclassical phase and the adiabatic Berry phase, usually referred to in this context, is discussed. These phases coincide for the perfectly linear Dirac dispersion relation. They differ however when a gap is opened at the Dirac point. We furthermore present several applications of our semiclassical formalism. In particular we provide, for various configurations, a semiclassical derivation of the electron’s Landau levels, illustrating the role of the semiclassical “Berry-like” phase.

Local spectrum rearrangement in impure graphene

The evolution of the averaged local density of states at the impurity site with increasing the impurity concentration is studied by the cluster expansion method for the self-energy. It is demonstrated that the general shape of the local density of states and the decrease rate of the resonance peak height undergo a qualitative change at a certain critical impurity concentration. As a result, the distinctive features in the single-impurity local spectrum are completely smeared out when this critical concentration is exceeded. Within the Lifshitz impurity model, this local spectrum rearrangement is described in detail for the two-dimensional system with the Dirac dispersion of electrons, which is characteristic of graphene. The correspondent critical impurity concentration is related to the spatial overlap of individual impurity states.

Topological Aspects of Quantum Hall Effect in Graphene

We study the recently observed quantum Hall effect (QHE) in graphene from a theoretical viewpoint of topological nature of the QHE to pose questions: (i) The zero-mass Dirac dispersion, which is the origin of the anomalous QHE, exists only around the zero gap, so a natural question is what happens to the QHE topological numbers over the entire energy spectrum. (ii) How the property that the bulk QHE topological number is equal to the edge QHE topological number, shown for the ordinary QHE, applies to the honeycomb lattice. We have shown that (a) the anomalous QHE ∝ (2N + 1) persists, surprisingly, all the way up to the van-Hove singularities, at which the normal behaviour abruptly takes over. (b) The edge-bulk correspondence persists as shown from the result for finite systems. All these properties hold for the entire sequence of lattice Hamiltonians that interpolate between square↔honeycomb↔ π-flux lattices, so the anomalous QHE is on a quantum critical line.

Topological aspects of graphene

We discuss topological aspects of electronic properties of graphene, including edge effects, with the tight-binding model on a honeycomb lattice and its extensions to show the following: (i) Appearance of the pairn of massless Dirac dispersions, which is the origin of anomalous properties including a peculiar quantum Hall effect (QHE), is not accidental to honeycomb, but is rather generic for a class of two-dimensional lattices that interpolate between square and $\pi$-flux lattices. Persistence of the peculiar QHE is interpreted as a topological stability. (ii) While we have the massless Dirac dispersion only around E=0, the anomalous QHE associated with the Dirac cone unexpectedly persists for a wide range of the chemical potential. The range is bounded by van Hove singularities, at which we predict a transition to the ordinary fermion behavior acompanied by huge jumps in the QHE with a sign change. (iii) For edges we establish a coincidence between the quantum Hall effect in the bulk and the quantum Hall effect for the edge states, which is a manifestation of the topological bulk-edge correspondence. We have also explicitly shown that the E=0 edge states in honeycomb in zero magnetic field persist in magnetic field.

On modified Weyl–Heisenberg algebras, noncommutativity, matrix-valued Planck constant and QM in Clifford spaces

A novel Weyl–Heisenberg algebra in Clifford spaces is constructed that is based on a matrix-valued extension of Planck's constant. As a result of this modified Weyl–Heisenberg algebra one will no longer be able to measure, simultaneously, the pairs of variables (x, px), (x, py), (x, pz), (y, px), ... with absolute precision. New Klein–Gordon and Dirac wave equations and dispersion relations in Clifford spaces are presented. The latter Dirac equation is a generalization of the Dirac–Lanczos–Barut–Hestenes equation. We display the explicit isomorphism between Yang's noncommutative spacetime algebra and the area-coordinates algebra associated with Clifford spaces. The former Yang's algebra involves noncommuting coordinates and momenta with a minimum Planck scale λ (ultraviolet cutoff) and a minimum momentum p = /R (maximal length R, infrared cutoff). The double-scaling limit of Yang's algebra λ → 0, R → ∞, in conjunction with the large n → ∞ limit, leads naturally to the area quantization condition λR = L2 = nλ2 (in Planck area units) given in terms of the discrete angular-momentum eigenvalues n. It is shown how modified Newtonian dynamics is also a consequence of Yang's algebra resulting from the modified Poisson brackets. Finally, another noncommutative algebra which differs from Yang's algebra and related to the minimal length uncertainty relations is presented. We conclude with a discussion of the implications of noncommutative QM and QFT's in Clifford spaces.