Anisotropic Fermi Velocity Research Articles

Magneto-optical conductivity of anisotropic two-dimensional Dirac–Weyl materials

In the presence of an external magnetic field, the optical response of two-dimensional materials, whose charge carriers behave as massless Dirac fermions with arbitrary anisotropic Fermi velocity, is investigated. Using Kubo formalism, we obtain the magneto-optical conductivity tensor for these materials, which allows to address the magneto-optical response of anisotropic Dirac fermions from the well known magneto-optical conductivity of isotropic Dirac fermions. As an application, we analyse the combined effects of strain-induced anisotropy and magnetic field on the transmittance, as well as on the Faraday rotation, of linearly polarized light after passing strained graphene. The reported analytical expressions can be a useful tool to predict the absorption and the Faraday angle of strained graphene under magnetic field. Finally, our study is extended to anisotropic two-dimensional materials with Dirac fermions of arbitrary pseudospin.

Low-energy theory for strained graphene: an approach up to second-order in the strain tensor

An analytical study of low-energy electronic excited states in uniformly strained graphene is carried out up to second-order in the strain tensor. We report a new effective Dirac Hamiltonian with an anisotropic Fermi velocity tensor, which reveals the graphene trigonal symmetry being absent in first-order low-energy theories. In particular, we demonstrate the dependence of the Dirac-cone elliptical deformation on the stretching direction with respect to graphene lattice orientation. We further analytically calculate the optical conductivity tensor of strained graphene and its transmittance for a linearly polarized light with normal incidence. Finally, the obtained analytical expression of the Dirac point shift allows a better determination and understanding of pseudomagnetic fields induced by nonuniform strains.

Three-dimensional Dirac cone carrier dynamics inNa3BiandCd3As2

Optical measurements and band structure calculations are reported on three-dimensional Dirac materials. The electronic properties associated with the Dirac cone are identified in the reflectivity spectra of ${\mathrm{Cd}}_{3}{\mathrm{As}}_{2}$ and ${\mathrm{Na}}_{3}\mathrm{Bi}$ single crystals. In ${\mathrm{Na}}_{3}\mathrm{Bi}$, the plasma edge is found to be strongly temperature dependent due to thermally excited free carriers in the Dirac cone. The thermal behavior provides an estimate of the Fermi level ${E}_{F}=25$ meV and the $z$-axis Fermi velocity ${v}_{z}=0.3$ eV \AA{} associated with the heavy bismuth Dirac band. At high energies above the $\mathrm{\ensuremath{\Gamma}}$-point Lifshitz gap energy, a frequency- and temperature-independent ${\ensuremath{\epsilon}}_{2}$ indicative of Dirac cone interband transitions translates into an ab-plane Fermi velocity of 3 eV \AA{}. The observed number of IR phonons rules out the $P{6}_{3}/mmc$ space-group symmetry but is consistent with the $P\overline{3}c1$ candidate symmetry. A plasmaron excitation is discovered near the plasmon energy that persists over a broad range of temperature. The optical signature of the large joint density of states arising from saddle points at $\mathrm{\ensuremath{\Gamma}}$ is strongly suppressed in ${\mathrm{Na}}_{3}\mathrm{Bi}$, consistent with band structure calculations that show the dipole transition-matrix elements to be weak due to the very small $s$-orbital character of the Dirac bands. In ${\mathrm{Cd}}_{3}{\mathrm{As}}_{2}$, a distinctive peak in reflectivity due to the logarithmic divergence in ${\ensuremath{\epsilon}}_{1}$ expected at the onset of Dirac cone interband transitions is identified. The center frequency of the peak shifts with temperature quantitatively consistent with a linear dispersion and a carrier density of $n=1.3\ifmmode\times\else\texttimes\fi{}{10}^{17}\phantom{\rule{4.pt}{0ex}}{\text{cm}}^{\ensuremath{-}3}$. The peak width gives a measure of the Fermi-velocity anisotropy of $10%$, indicating a nearly spherical Fermi surface. The line shape gives an upper bound estimate of 7 meV for the potential fluctuation energy scale.

Hybridization and anisotropy in the exchange interaction in three-dimensional Dirac semimetals

We study the Ruderman-Kittel-Kasuya-Yosida interaction in 3D Dirac semimetals. Using retarded Green's functions in real space, we obtain and analyze asymptotic expressions for the interaction, with magnetic impurities at different distances and relative angle with respect to high symmetry directions on the lattice. We show that the Fermi velocity anisotropy in these materials produces a strong renormalization of the magnitude of the interaction, as well as a correction to the frequency of oscillation in real space. Hybridization of the impurities to different conduction electron orbitals are shown to result in interesting anisotropic spin-spin interactions which can generate spiral spin structures in doped samples.

Spin textures of topological surface states at side surfaces ofBi2Se3from first principles

We investigate the spin and spin-orbital textures and electronic structures of topologically protected surface states at side surfaces of ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ by using slab models within density-functional theory. This is motivated by recent experiments on nanowires, nanoribbons, and nanoplates of ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ with side surfaces. In particular, two representative surfaces normal to the (111) surface, such as $(1\overline{1}0)$ and $(11\overline{2})$ surfaces, are examined, in the presence of time-reversal symmetry and inversion symmetry. The $(1\overline{1}0)$ surface lying in the mirror plane has twofold $({\mathrm{C}}_{2})$ rotational symmetry, whereas the $(11\overline{2})$ surface has only mirror symmetry. For the $(1\overline{1}0)$ surface, we find that a Dirac cone with strongly anisotropic Fermi velocity is formed at $\mathrm{\ensuremath{\Gamma}}$ with the Dirac point at the Fermi level, and that the spin texture reveals features of Rashba-type combined with Dresselhaus-type spin-orbit coupling. For the $(11\overline{2})$ surface, a Dirac cone is found at either $\mathrm{\ensuremath{\Gamma}}$ or the $Y$ point (along the mirror symmetry axis) below the Fermi level. In this case, the spin texture of the surface states strikingly differs from that of the (111) and $(1\overline{1}0)$ surfaces: (i) the in-plane spin polarization dominantly aligned perpendicular to the [111] direction or the mirror symmetry axis, (ii) the Dresselhaus-type spin texture, and (iii) significant out-of-plane spin polarization away from the mirror symmetry axis. Our findings distinctively differ from the previous works based on the effective bulk model Hamiltonian. Our calculated spin and spin-orbital textures and band structures can be observed by spin-resolved angle-resolved photoemission spectroscopy.

Emergent gravity in the cubic tight-binding model of Weyl semimetal in the presence of elastic deformations

We consider the tight-binding model with cubic symmetry that may be relevant for the description of a certain class of Weyl semimetals. We take into account elastic deformations of the semimetal through the modification of hopping parameters. This modification results in the appearance of emergent gauge field and the coordinate dependent anisotropic Fermi velocity. The latter may be interpreted as emergent gravitational field.

Generalizing the Fermi velocity of strained graphene from uniform to nonuniform strain

The relevance of the strain-induced Dirac point shift to obtain the appropriate anisotropic Fermi velocity of strained graphene is demonstrated. Then a critical revision of the available effective Dirac Hamiltonians is made by studying in detail the limiting case of a uniform strain. An effective Dirac Hamiltonian for nonuniform strain is thus reported, which takes into account all strain-induced effects: changes in the nearest-neighbor hopping parameters, the reciprocal lattice deformation and the true shift of the Dirac point. Pseudomagnetic fields are thus explained by means of position-dependent Dirac cones, whereas complex gauge fields appear as a consequence of a position-dependent Fermi velocity. Also, position-dependent Fermi velocity effects on the spinor wavefunction are considered for interesting cases of deformations such as flexural modes.

Strained graphene Josephson junction with anisotropic d-wave superconductivity

Effect of proximity-induced superconductivity in the new two-dimensional structures, as graphene and topological insulator on the Andreev bound states (ABSs) and Josephson supercurrent has attracted much efforts. Motivated by this subject, we study, in particular, the influence of anisotropic Fermi velocity and unconventional d-wave pairing in a strained graphene-based superconductor/normal/ superconductor junction. Strain is applied in the zigzag direction of graphene sheet. In this process, effect of zero energy states and Fermi wavevector mismatch are investigated. It is shown, that strain up to 22% in graphene lattice differently affects Josephson currents in parallel and perpendicular directions of strain. Strain causes to exponentially decrease the supercurrent in the strain direction, whereas increase for other direction. We find that, in one hand, the ABSs strongly depend on strain and, on the other hand, a gap opens in the states with respect to non-zero incidence angle of quasiparticles, where a period of 2π is obtained for Andreev states. Moreover, we observe no gap for θs≠0, when the zero energy states (ZESs) occur in α=π/4 due to anisotropic superconducting gap. In this case, ABSs have a period of 4π.S

(111)surface states of SnTe

The characterization and applications of topological insulators depend critically on their protected surface states, which, however, can be obscured by the presence of trivial dangling bond states. Our first principle calculations show that this is the case for the pristine $(111)$ surface of SnTe. Yet, the predicted surface states unfold when the dangling bond states are passivated in proper chemisorption. We further extract the anisotropic Fermi velocities, penetration lengths and anisotropic spin textures of the unfolded $\bar\Gamma$- and $\bar M$-surface states, which are consistent with the theory in http://dx.doi.org/10.1103/PhysRevB.86.081303 Phys. Rev. B 86, 081303 (R). More importantly, this chemisorption scheme provides an external control of the relative energies of different Dirac nodes, which is particularly desirable in multi-valley transport.

Interacting Weyl semimetals on a lattice

We perform an exact renormalization group analysis of a fermionic system on a three-dimensional lattice, showing the persistence of the Weyl semimetallic phase up the quantum critical point discriminating between the semimetallic and insulating phases. The interaction produces finite modifications of the anisotropic Fermi velocity and of the overlap of the true (dressed) quasiparticle with the free electron wave function. The optical conductivity is the one of Weyl fermions with anisotropic Fermi velocities replacing the light velocity, provided that the frequencies are below a certain energy scale, which is vanishing at the quantum critical point; above such scale, a different behavior is found, dominated by the quadratic corrections to the dispersion relation.

Stability of multinode Dirac semimetals against strong long-range correlations

We study the stability of Dirac semimetals with $N$ nodes in three spatial dimensions against strong $1/r$ Coulomb interactions. We particularly study the cases of $N=4$ and $N=16$, where the $N=4$ Dirac semimetal is described by the staggered fermions and the $N=16$ Dirac semimetal is described by the doubled lattice fermions. We take into account the $1/r$ long-range Coulomb interactions between the bulk electrons. Based on the U(1) lattice gauge theory, we analyze the system from the strong coupling limit. It is shown that the Dirac semimetals survive in the strong coupling limit when the out-of-plane Fermi velocity anisotropy of the Dirac cones is weak, whereas they change to Mott insulators when the anisotropy is strong. A Possible global phase diagram of correlated multinode Dirac semimetals is presented. Implications of our result to the stability of Weyl semimetals and three-dimensional topological insulators are discussed.

Geometric and Electronic Structures of Two-Dimensional SiC3 Compound

Although graphene is an unconventional semimetallic nanosheet, the graphene-like binary sheets, such as BN and SiC with 1:1 stoichiometry, usually discard the semimetallicity. Here, we report the geometric structures and electronic properties of the two-dimensional SiC3 sheet by the particle-swarm optimization method and density functional calculations. The unbiased global search reveals three lowest-energy structures for SiC3 ones, all of which are Si-graphene hybrid honeycomb lattices with robust dynamical stabilities. Depending on the ordered arrangement of Si atoms, the SiC3 sheets could be direct-band-gap semiconductors or zero-band-gap semimetals. More interestingly, the semiconducting SiC3 sheet has a strong adsorption ability in the visible-light region, and the semimetallic one even possesses a distorted Dirac cone, which induces an anisotropic Fermi velocity. These rich electronic properties of SiC3 nanosheets endow the system with promising applications in nanoelectronics and photovoltaics.

Rotation of Triangular Vortex Lattice in the Two-Band Superconductor MgB2

To identify the contributions of the multiband nature and the anisotropy of a microscopic electronic structure to a macroscopic vortex lattice morphology, we develop a method based on the Eilenberger theory near \(H_{\text{c2}}\) combined with the first-principles band calculation to estimate the stable vortex lattice configuration. For a typical two-band superconductor MgB2, successive transitions of vortex lattice orientation that have been observed recently by small angle neutron scattering [Das et al.: Phys. Rev. Lett. 108 (2012) 167001] are explained by the characteristic field-dependence of two-band superconductivity and the competition of sixfold anisotropy between the σ- and π-bands. The reentrant transition at low temperature reflects the Fermi velocity anisotropy of the σ-band.

Low-Energy Electronic Properties of Graphene and Armchair Ribbon Superlattices

The geometric structures and electronic properties of graphene and armchair ribbon superlattices are investigated by the first-principles calculations. Parallel armchair ribbons are periodically placed upon a graphene sheet. The interlayer atomic interactions strongly influence the interlayer distance, binding energy, energy dispersion, state degeneracy, band gap, Fermi velocity, and Fermi momentum except for the intralayer bond length. These properties depend on the stacking configurations, ribbon width, and existence of hydrogen atoms at the ribbon edges. Such interactions also result in the distortion of graphene and ribbons, which is more obvious in the AA-stacked systems than the AB-stacked ones. All low-energy bands come from the 2pz orbitals of graphene or ribbon or from those of both. Most AA-stacked systems possess two intersecting low-lying bands, which exhibit highly anisotropic Fermi velocities. AB-stacked systems mainly have a direct band gap that is monotonously widened as ribbon width expands.

Electronic ground-state properties of strained graphene

We consider the effect of the Coulomb interaction in strained graphene using tight-binding approximation together with the Hartree-Fock interactions. The many-body energy dispersion relation, anisotropic Fermi velocity renormalization and charge compressibility in the presence of uniaxial strain are calculated. We show that the quasiparticle quantities are sensitive to homogenous strain and indeed, to its sign. The charge compressibility is enhanced by stretching and suppressed by compressing a graphene sheet. We find a reduction of Fermi velocity renormalization along the direction of graphene deformation, in good agreement with the recent experimental observation.

Transport Phenomena in Multilayered Massless Dirac Fermion System α-(BEDT-TTF)2I3

A zero-gap state with a Dirac cone type energy dispersion was discovered in an organic conductor α-(BEDT-TTF)2I3 under high hydrostatic pressures. This is the first two-dimensional (2D) zero-gap state discovered in bulk crystals with a layered structure. In contrast to the case of graphene, the Dirac cone in this system is highly anisotropic. The present system, therefore, provides a new type of massless Dirac fermion system with anisotropic Fermi velocity. This system exhibits remarkable transport phenomena characteristic to electrons on the Dirac cone type energy structure.

Many-electron effects on optical absorption spectra of strained graphene

Abstract

Screening and band structure effects on quasi-one-dimensional transport in periodically modulated graphene

Using ab-initio density functional theory the electronic response is studied for graphene subjected to an external superlattice potential. Different potential strengths and crystallographic directions forming one-dimensional (1D) superlattices with zigzag and armchair edges are investigated. The graphene band structure and the screening effect are self-consistently taken into account. The Dirac points appear slightly displaced in reciprocal space with a strongly anisotropic Fermi velocity. The screening tends to weaken the potential action and renormalization of the Fermi velocity. We demonstrate that the vanishing group velocity perpendicular to the superlattice as predicted for non-interacting massless Dirac fermions still occurs although somewhat modified by the self-consistent reaction of the system. Features of a quasi-1D metal remain observable.

Generic First-Order Orientation Transition of Vortex Lattices in Type II Superconductors

First order transition of vortex lattices (VL) observed in various superconductors with four-fold symmetry is explained microscopically by quasi-classical Eilenberger theory combined with nonlocal London theory. This transition is intrinsic in the generic successive VL phase transition due to either gap or Fermi velocity anisotropies. This is also suggested by the electronic states around vortices. Ultimate origin of this phenomenon is attributed to some what hidden frustrations of a spontaneous symmetry broken hexagonal VL on the underlying four-fold crystalline symmetry.

Upper critical field of RbOs2O6 analyzed within Eliashberg theory

The upper critical field Hc2(T) of the β-pyrochlore superconductor RbOs2O6 is analyzed within the framework of s-wave Eliashberg theory including anisotropy and scattering effects, yielding a small electron–phonon coupling anisotropy parameter of about 〈a2〉=0.005, and a Fermi velocity anisotropy parameter 〈b2〉=0.3–0.35. Agreement between theory and experiment is achieved for these parameters and a model spectral function α2F(ω) consisting of two Einstein peaks and a Debye contribution. A renormalization of the Fermi velocity by an enhancement factor due to electron–electron correlations 〈vF〉ee=1 is considered in the analysis of the experimental Hc2(T)-data. Comparison between experiment and theory of thermodynamic properties such as the specific-heat difference between superconducting and normal states Cs–Cn, and the thermodynamic critical field Hc(T) support the results from the upper critical field analysis.