Two-dimensional Brillouin Zone Research Articles

Line nodes and surface Majorana flat bands in static and kicked p -wave superconducting Harper model

We investigate the effect of introducing nearest-neighbor $p$-wave superconducting pairing to both the static and kicked extended Harper model with two periodic phase parameters acting as artificial dimensions to simulate three-dimensional systems. It is found that in both the static model and the kicked model, by varying the $p$-wave pairing order parameter, the system can switch between a fully gapped phase and a gapless phase with point nodes or line nodes. The topological property of both the static and kicked model is revealed by calculating corresponding topological invariants defined in the one-dimensional lattice dimension. Under open boundary conditions along the physical dimension, Majorana flat bands at energy zero (quasienergy zero and $\pi$) emerge in the static (kicked) model at the two-dimensional surface Brillouin zone. For certain values of pairing order parameter, (Floquet) Su-Schrieffer-Heeger-like edge modes appear in the form of arcs connecting different (Floquet) Majorana flat bands. Finally, we find that in the kicked model, it is possible to generate two controllable Floquet Majorana modes, one at quasienergy zero and the other at quasienergy $\pi$, at the same parameter values.

Shubnikov-de Haas quantum oscillations reveal a reconstructed Fermi surface near optimal doping in a thin film of the cuprate superconductorPr1.86Ce0.14CuO4±δ

We study magnetotransport properties of the electron-doped superconductor Pr$_{2-x}$Ce$_x$CuO$_{4\pm\delta}$ with $x$ = 0.14 in magnetic fields up to 92~T, and observe Shubnikov de-Haas magnetic quantum oscillations. The oscillations display a single frequency $F$=255$\pm$10~T, indicating a small Fermi pocket that is $\sim$~1\% of the two-dimensional Brillouin zone and consistent with a Fermi surface reconstructed from the large hole-like cylinder predicted for these layered materials. Despite the low nominal doping, all electronic properties including the effective mass and Hall effect are consistent with overdoped compounds. Our study demonstrates that the exceptional chemical control afforded by high quality thin films will enable Fermi surface studies deep into the overdoped cuprate phase diagram.

Vibrational properties at the ordered metallic surface alloy system Au(110)-1×2-Pd

We present a calculation for the vibrational properties of the ordered surface alloy Au(110)-1×2-Pd on a crystalline substrate of Au. The surface phonon dispersion curves and the local vibrations densities of states (LDOS) are calculated in the harmonic approximation for the system, using the phase field matching theory (PFMT) method and associated real space Green’s functions. In particular, it is shown that the surface alloy presents optic vibrational modes above the Au bulk bands, along the directions of high-symmetry [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] of the corresponding two-dimensional Brillouin zone. Measurements of the surface phonon dispersion branches can hence be made by different techniques such as helium atom scattering (HAS) to compare with. The calculated LDOS for Au and Pd atomic sites in the four top surface atomic layers span a wider range of frequencies than those for the individual Au(110) or Pd(110) metallic surfaces. These LDOS provide a spectral signature for the progressive transition from the surface dynamics to that of the Au crystal bulk. Knowledge of these LDOS for the surface alloy can also serve as an input for modeling the diffusion and reaction rates of chemical species at its surface.

Correlated oscillations of the magnetic anisotropy energy and orbital moment anisotropy in thin films: The role of quantum well states

We report the first-principles study of the correlated behavior of the magnetic anisotropy energy (MAE) and orbital moment anisotropy (OMA) as the functions of the thickness $N$ of the Fe film. The work is motivated by recent experimental studies combining photoemission, x-ray magnetic circular dichroism, and magnetic anisotropy measurements. In agreement with experiment, the correlated oscillations of $\mathrm{MAE}(N)$ and $\mathrm{OMA}(N)$ are obtained that have their origin in the formation of the 3d quantum well states (QWS) confined in the films. The main contribution to the oscillation amplitude comes from the surface layer. This is an interesting feature of the phenomenon consisting in the peculiar dependence of the physical quantities on the thickness of the film. We demonstrate that the band structure of the bulk Fe does not reflect adequately the properties of the 3d QWS in thin films and, therefore, does not provide the basis for understanding the oscillations of $\mathrm{MAE}(N)$ and $\mathrm{OMA}(N)$. A detailed point-by-point analysis in the two-dimensional (2D) Brillouin zone (BZ) of the film shows that the contribution of the $\mathrm{\ensuremath{\Gamma}}$ point, contrary to a rather common expectation, does not play an important role in the formation of the oscillations. Instead, the most important contributions come from a broad region of the 2D BZ distant from the center of the BZ. Combining symmetry arguments and direct calculations we show that orbital moments of the electronic states possess nonzero transverse components orthogonal to the direction of the spin magnetization. The account for this feature is crucial in the point-by-point analysis of the OMA. On the basis of the calculations for noncollinear spin configurations we suggest interpretations of two interesting experimental findings: fast temperature decay of the oscillation amplitude in $\mathrm{MAE}(N)$ and unexpectedly strong spin mixing of the initial states of the photoemission process.

Quantum spin Hall effect in IV-VI topological crystalline insulators

We envision that the quantum spin Hall effect should be observed in (111)-oriented thin films of SnSe and SnTe topological crystalline insulators. Using a tight-binding approach supported by first-principles calculations of the band structures, we demonstrate that in these films the energy gaps in the two-dimensional band spectrum depend in an oscillatory fashion on the layer thickness. These results as well as the calculated topological invariant indexes and edge state spin polarizations show that for films ∼20–40 monolayers thick a two-dimensional topological insulator phase appears. In this range of thicknesses in both SnSe and SnTe, (111)-oriented films edge states with Dirac cones with opposite spin polarization in their two branches are obtained. While in the SnTe layers a single Dirac cone appears at the projection of the point of the two-dimensional Brillouin zone, in the SnSe (111)-oriented layers three Dirac cones at points projections are predicted.

Electronic structure of stacking faults in rhombohedral graphite

The electronic structure of stacking faults and surfaces without and with an additional displaced layer is calculated for the case of rhombohedral (ABC) graphite. The full-potential local-orbital code and the generalized gradient approximation to density functional theory are used. All considered surfaces and interfaces induce surface/interface bands. All discovered surface and interface bands are restricted to the vicinity of the symmetry line $K\ensuremath{-}M$ in the two-dimensional Brillouin zone. There are groups of localized band pairs around $\ifmmode\pm\else\textpm\fi{}0$, $\ifmmode\pm\else\textpm\fi{}0.2$, and $\ifmmode\pm\else\textpm\fi{}0.6$ eV for one of the two considered types of stacking faults; $\ifmmode\pm\else\textpm\fi{}0$ and $\ifmmode\pm\else\textpm\fi{}0.5$ eV for the other type and for a displaced surface layer. At the $K$ point in the Brillouin zone, there is a one-to-one correspondence between these localized bands and the eigenvalues of those linear atomic clusters, which are produced by the perturbation of periodicity due to the displaced surface layer or due to the stacking faults. Some of the localized bands produce strong van Hove singularities in the local density of states near the surface or interface at energies up to several 0.1 eV. It is suggested to check these findings experimentally by appropriate spectroscopic methods. Undisturbed bulk (ABC) graphite is virtually a zero-gap semiconductor with a minute density of states at the Fermi energy. Both the surface and any of the considered stacking faults produce sharp peaks in the local density of states near the perturbation at energies of about 10 meV around the Fermi energy. This should provide a considerable contribution to the conductivity and its temperature dependence for samples with stacking faults or large surface-to-volume fraction.

Role of surface passivation in the formation of Dirac states at polar surfaces of topological crystalline insulators: The case of SnTe(111)

We present ab initio density functional theory (DFT) calculation results for electronic and spin structures of both the Te- and Sn-terminated SnTe(111) polar surfaces. Rocksalt narrow-gap semiconductor SnTe belongs to the recently discovered class of topological crystalline insulators in which the topological nature of surface electronic states arises from the crystal symmetry combined with band inversion at the $L$ point. We demonstrate that in contrast to earlier model calculations only trivial spin-split states propagating over the entire two-dimensional Brillouin zone emerge at the SnTe(111) surfaces owing to the surface potential effect which destroys weakly protected topological states. We show that the surface passivation eradicates the trivial surface states and recovers the even number of the helical spin-polarized topological Dirac cones centered at the $\overline{\ensuremath{\Gamma}}$ and $\overline{\mathrm{M}}$ points prescribed for the topological crystalline insulator by the crystal symmetry.

Three-dimensional symmetry-breaking nontrivial topological states

We discuss topological electronic states described by the Dirac Hamiltonian plus an additional one in three-dimension. When the additional Hamiltonian is an element of an Abelian group, electronic states become topologically nontrivial even in the absence of fundamental symmetries such as the time-reversal and the particle-hole symmety. The symmetry-breaking topological states are charercterized by the Chern number defined in the two-dimensional partial Brillouin zone. The topological insulators under Zeeman field are an example of the symmetry-breaking topological matters. We show the transision from the topological insulating phase to the topological semimetal one under the strong Zeeman field.

Inversion Symmetry and Wave-Function Nodal Lines of Dirac Electrons in Organic Conductor α-(BEDT-TTF)2I3

By examining the organic conductor α-(BEDT-TTF)2I3, which is described by a nearest neighbor tight-binding model, it is shown that, owing to inversion symmetry, each component of a wave function (WF) exhibits nodal lines (NLs) (i.e., lines in the two-dimensional Brillouin zone where the WF component vanishes). In the absence of any band crossing, each NL connects two time reversal invariant momenta (TRIMs) as partners. In the presence of a pair of Dirac points (band crossing), there are NLs that connect the pair of Dirac points via a TRIM without a partner. This second kind of NL leads to a discontinuous sign change for nonvanishing components of the WF. Such a property is the origin of the \(\pm\)π Berry phase accumulated on a contour integral encircling one Dirac point. The results are exemplified by the numerical calculation of WF components for the above conductor with a 3/4 filled band.

On a new type of massless Dirac fermions in crystalline topological insulators

A new type of massless Dirac fermions in crystalline three-dimensional topological insulators (three-dimensional → two-dimensional situation) has been predicted. The spectrum has fourfold degeneracy at the top of the two-dimensional Brillouin zone (M point) and twofold degeneracy near the M point. Crystal symmetry along with the time reversal invariance in three-dimensional topological insulators allows fourfold degenerate Dirac cones, which are absent in the classification of topological features in R.-J. Slager et al., Nat. Phys. 9, 98 (2013). The Hamiltonian in the cited work does not contain Dirac singularities with more than twofold degeneracy. For this reason, the corresponding topological classification is incomplete. The longitudinal magnetic field in the spinless case holds the massless dispersion law of fermions and does not lift fourfold degeneracy. In the spinor case, the magnetic field lifts fourfold degeneracy, holding only twofold degeneracy, and results in the appearance of a band gap in the spectrum of fermions.

Bulk-Edge Correspondence for Two-Dimensional Topological Insulators

Topological insulators can be characterized alternatively in terms of bulk or edge properties. We prove the equivalence between the two descriptions for two-dimensional solids in the single-particle picture. We give a new formulation of the $${\mathbb{Z}_{2}}$$ -invariant, which allows for a bulk index not relying on a (two-dimensional) Brillouin zone. When available though, that index is shown to agree with known formulations. The method also applies to integer quantum Hall systems. We discuss a further variant of the correspondence, based on scattering theory.

Zone-boundary phonon induced mini band gap formation in graphene

We investigate the effect of electron–A1g phonon coupling on the gapless electronic band dispersion of the pristine graphene. The electron–phonon interaction is introduced through a Kekulé-type distortion giving rise to inter-valley scattering between K and K′ points in graphene. We develop a Fröhlich type Hamiltonian within the continuum model in the long-wave length limit. By presenting a fully theoretical analysis, we show that the interaction of charge carriers with the highest frequency zone-boundary phonon mode of A1g-symmetry induces a mini band gap at the corners of the two-dimensional Brillouin zone of the graphene in the THz region. Since electron–electron interactions favor this type of lattice distortion, it is expected to be enhanced, and thus its quantitative implications might be measurable in graphene.

Chemical Vapor Deposition and Characterization of Aligned and Incommensurate Graphene/Hexagonal Boron Nitride Heterostack on Cu(111)

Two limiting factors for a new technology of graphene-based electronic devices are the difficulty of growing large areas of defect-free material and the integration of graphene with an atomically flat and insulating substrate material. Chemical vapor deposition (CVD) on metal surfaces, in particular on copper, may offer a solution to the first problem, while hexagonal boron nitride (h-BN) has been identified as an ideal insulating substrate material. The bottom-up growth of graphene/h-BN stacks on copper surfaces appears therefore as a promising route for future device fabrication. As an important step, we demonstrate the consecutive growth of well-aligned graphene on h-BN, both as single layers, by low-pressure CVD on Cu(111) in an ultrahigh vacuum environment. The resulting films show a largely predominant orientation, defined by the substrate, where the graphene lattice aligns parallel to the h-BN lattice, while each layer maintains its own lattice constant. The lattice mismatch of 1.6% between h-BN and graphene leads to a moiré pattern with a periodicity of about 9 nm, as observed with scanning tunneling microscopy. Accordingly, angle-resolved photoemission data reveal two slightly different Brillouin zones for electronic states localized in graphene and in h-BN, reflecting the vertical decoupling of the two layers. The graphene appears n-doped and shows no gap opening at the K[overline] point of the two-dimensional Brillouin zone.

Signature of the two-dimensional phonon dispersion in graphene probed by double-resonant Raman scattering

In this article we unravel the origin of the two-phonon D + D �� peak and its asymmetric line shape by combining experimental data of single-layer graphene with a full twodimensional calculation of the double-resonant Raman process based on fourth-order perturbation theory. We show that the main peak originates from phonons along the K highsymmetry line and that the asymmetry is due to phonons from the two-dimensional Brillouin zone. The analysis of the asymmetric line shape in experiment provides a direct probe of the two-dimensional phonon dispersion. We further show how the D + D �� peak evolves with the number of graphene layers.

Transport Properties of Fe/GaAs/Ag(001) System

We present results of ab initio transport calculations for epitaxial magnetic tunnel junctions Fe/GaAs/Ag(001). The electronic structure is calculated by means of the tight-binding linear muffin-tin orbital method and the ballistic conductances are evaluated within the Kubo-Landauer formalism which includes the effect of the spin-orbit interaction. Particular attention is paid to the dependence of the conductances on the orientation of magnetization direction of the Fe electrode with respect to the crystal lattice and on the thickness of the tunneling barrier. We have found that the in-plane tunneling anisotropic magnetoresistance (TAMR) exhibits a non-monotonic thickness dependence with a maximum around 7.5 nm of GaAs. This behavior is ascribed to a hybridization of interface resonances formed on both sides of the junction and manifested as hot spots in k[]-resolved conductances. For thicker GaAs barriers, the relative intensity of the hot spots is reduced on account of the contribution from a narrow central region of the two-dimensional Brillouin zone which leads to the final decrease of the TAMR effect.

Tunnelling anisotropic magnetoresistance of Fe/GaAs/Ag(001) junctions from first principles: effect of hybridized interface resonances

Results of first-principles calculations of the Fe/GaAs/Ag(001) epitaxial tunnel junctions reveal that hybridization of interface resonances formed at both interfaces can enhance the tunnelling anisotropic magnetoresistance (TAMR) of the systems. This mechanism is manifested by a non-monotonic dependence of the TAMR effect on the thickness of the tunnel barrier, with a maximum for intermediate thicknesses. A detailed scan of k∥-resolved transmissions over the two-dimensional Brillouin zone proves an interplay between a few hybridization-induced hot spots and a contribution to the tunnelling from the vicinity of the point. This interpretation is supported by calculated properties of a simple tight-binding model of the junction, which reproduce qualitatively most of the features of the first-principles theory.

First-principles study of tunneling magnetoresistance in Fe/MgAl2O4/Fe(001) magnetic tunnel junctions

We investigated the spin-dependent transport properties of Fe/MgAl2O4/Fe(001) magnetic tunneling junctions (MTJs) on the basis of first-principles calculations of the electronic structures and the ballistic conductance. The calculated tunneling magnetoresistance (TMR) ratio of a Fe/MgAl2O4/Fe(001) MTJ was about 160%, which was much smaller than that of a Fe/MgO/Fe(001) MTJ (1600%) for the same barrier thickness. However, there was an evanescent state with delta 1 symmetry in the energy gap around the Fermi level of normal spinel MgAl2O4, indicating the possibility of a large TMR in Fe/MgAl2O4/Fe(001) MTJs. The small TMR ratio of the Fe/MgAl2O4/Fe(001) MTJ was due to new conductive channels in the minority spin states resulting from a band-folding effect in the two-dimensional (2-D) Brillouin zone of the in-plane wave vector (k//) of the Fe electrode. Since the in-plane cell size of MgAl2O4 is twice that of the primitive in-plane cell size of bcc Fe, the bands in the boundary edges are folded, and minority-spin states coupled with the delta 1 evanescent state in the MgAl2O4 barrier appear at k//=0, which reduces the TMR ratio of the MTJs significantly.

Asymmetric velocities of Dirac particles and Vernier spectrum in metallic single-wall carbon nanotubes

Left- and right-going Dirac particles near the $K$ (or ${K}^{\ensuremath{'}}$) point in the two-dimensional Brillouin zone have different velocities in metallic single-wall carbon nanotubes. The metallic linear energy band is tilted by a curvature effect of the nanotube. The difference of the velocities is calculated by a numerical calculation and by an effective mass theory. Tilting as well as warping of the Dirac cones explains the diameter and chirality dependences of the velocities quantitatively. The asymmetric velocities give a ``vernier'' spectrum of an electron in a nanotube quantum dot with a sharp confinement potential containing two different sequences of levels, which is relevant to the lifting of the fourfold degeneracy of the energy levels.

Emergent many-body translational symmetries of Abelian and non-Abelian fractionally filled topological insulators

The energy and entanglement spectrum of fractionally filled interacting topological insulators exhibit a peculiar manifold of low energy states separated by a gap from a high energy set of spurious states. In the current manuscript, we show that in the case of fractionally filled Chern insulators, the topological information of the many-body state developing in the system resides in this low-energy manifold. We identify an emergent many-body translational symmetry which allows us to separate the states in quasi-degenerate center of mass momentum sectors. Within one center of mass sector, the states can be further classified as eigenstates of an emergent (in the thermodynamic limit) set of many-body relative translation operators. We analytically establish a mapping between the two-dimensional Brillouin zone for the Fractional Quantum Hall effect on the torus and the one for the fractional Chern insulator. We show that the counting of quasi-degenerate levels below the gap for the Fractional Chern Insulator should arise from a folding of the states in the Fractional Quantum Hall system at identical filling factor. We show how to count and separate the excitations of the Laughlin, Moore-Read and Read-Rezayi series in the Fractional Quantum Hall effect into two-dimensional Brillouin zone momentum sectors, and then how to map these into the momentum sectors of the Fractional Chern Insulator. We numerically check our results by showing the emergent symmetry at work for Laughlin, Moore-Read and Read-Rezayi states on the checkerboard model of a Chern insulator, thereby also showing, as a proof of principle, that non-Abelian Fractional Chern Insulators exist.

Design of [formula omitted] electron network in graphene using atomic Pt adsorbates

Based on a first-principles total-energy calculation, we demonstrate the possibility of π electron network design in graphene using atomic Pt adsorbate. The d state of the adsorbed Pt atoms hybridizes with the π state of graphene, which effectively creates a boundary in the two-dimensional π electron network in graphene. By adsorbing Pt atoms along the line parallel to the zigzag direction in graphene, we found that a characteristic flat dispersion band emerges at the zone boundary of the two-dimensional Brillouin zone. This flat dispersion band is partially filled by electrons, which induces a magnetic moment in this system of non-magnetic elements.