type-II Dirac Points Research Articles

Type-II Dirac phonons in a two-dimensional phononic crystal

We explore the distinctive properties associated with a type-II Dirac point in a simply structured phononic crystal with a lattice deformation. This type-II Dirac point emerges at the Brillouin zone boundary, resulting from the lifting of two degenerate bands and featuring a conical-like Fermi surface in the equi-frequency curve. A practical implementation of such a phononic crystal is achieved with LEGO bricks. Upon introducing a periodic parity-time (PT) symmetric non-Hermitian perturbation, the phononic crystal undergoes a transition from PT-symmetric phase to PT-broken phase, causing the deformation of type-II Dirac point into an oval of exceptional points in the band structure. Based on the eigenmodes of the type-II Dirac point, a k⃗⋅p⃗ perturbation theory can be used to characterize these systems before and after the phase transition. Using a scattering matrix, we analyze the symmetric and broken phases and demonstrate that broadband unidirectional transparency and a coherent perfect absorber and laser can be realized with such a phononic crystal slab.

Bound-in-continuum-like corner states in the type-II Dirac photonic lattice

Dirac points are special points in the energy band structure of various materials, around which the dispersion is linear. If the corresponding Fermi surface is projected as a pair of crossing lines—or touching cones in two dimensions, the Dirac point is known as the type-II; such points violate the Lorentz invariance. Until now, thanks to its unique characteristics, the Klein tunneling is successfully mimicked and the topological edge solitons are obtained in the type-II Dirac photonic lattices that naturally possess type-II Dirac points. However, the interplay between these points and corner states is still not investigated. Here, we report both linear and nonlinear corner states in the type-II Dirac photonic lattice with elaborate boundaries. The states, classified as in-phase and out-of-phase, hide in the extended bands that are similar to the bound states in the continuum (BIC). We find that the nonlinear BIC-like corner states are remarkably stable. In addition, by removing certain sites, we establish the fractal Sierpiński gasket structure in the type-II Dirac photonic lattice, in which the BIC-like corner states are also demonstrated. The differences between results in the fractal and nonfractal lattices are rather small. Last but not least, the corner breather states are proposed. Our results provide a novel view on the corner states and may inspire fresh ideas on how to manipulate/control the localized states in different photonic lattices.

Photonic Dirac cone and topological transition in a moving dielectric slab

The moving media theory is applied to a photonic confined structure which is a continuous dielectric slab waveguide with the uniaxial anisotropy and without the discrete translational symmetry. The moving effect not only brings about non-reciprocity to the whole photonic band structure in the co-moving and counter-moving directions, but also leads to the topological transition of local degenerate points within the band diagram. We demonstrate through calculation that the type-II Dirac point can be turned into type-I Dirac point when the uniaxial slab is moving over certain speed. Our results provide a new approach to regulate the topology of degeneracy for two-dimensional photonic bands in the continuous translational symmetry condition.

Topological semimetal phases in a family of monolayer X3YZ6 (X=Nb,Ta, Y=Si,Ge,Sn, Z=S,Se,Te) with abundant nodal lines and nodes

The electronic and topological properties of single-layer X3YZ6 (X=Nb,Ta, Y=Si,Ge,Sn, Z=S,Se,Te) materials have been studied with the aid of first principles calculations. This kind of materials belong to topological semimetals (TMs) with abundant nodal lines and nodes. Considering their similar properties, we focus on the analysis of Ta3SnTe6 and Ta3SiSe6. The presence of spin–orbit coupling (SOC) leads to the transition from type-I nodal lines to Dirac points as well as the disappear of type-II Dirac points. The three-dimensional (3D) plot of the band structure reproduces vividly the characteristics of nodes and nodal lines. The appearance of the flat bands in (110) edge states further confirm their nontrivial topological properties. We also explore the relationship among different nodal lines (nodes), crystal symmetry and SOC. The type-I nodal lines are protected by Mz and Mỹ symmetry in the absence of SOC. Symmetry breaking leads to band splitting even in the presence of SOC. The single-layer X3YZ6 can be used as candidates for two-dimensional (2D) Dirac materials and provide a platform for further study of interesting physical phenomena.

Type-II Dirac points and topological Fermi arcs in nonlocal metamaterials

Dirac semimetals are topological materials with degenerated Dirac points in their band structures. In particular, the Dirac points play a central role in the topologically protected phases. Recently, it was reported that the peculiar type-II Dirac points could be realized in photonics by means of photonic crystals (PCs). Nevertheless, the practical implementation is rather challenging. In this paper, we demonstrate that the nonlocal effect in the interior of a homogeneous photonic metamaterial (HPM) enables the existence of type-II Dirac points in the HPM's band structure. With theoretical formulations and numerical evaluations, we discuss the evolution of the HPM's band type and present the interesting transition between type-I and type-II Dirac degeneracy of the HPMs. Remarkably, a pair of the type-II Dirac points can be generated at two different frequencies, which is supported by the mode overlapping in the HPMs with weak nonlocality. The Fermi arc surface states connect the projections of the type-II Dirac points at the boundary between the vacuum and HPMs. The topological protection of the Fermi arc surface states is verified by the reflectionless transmission with sharp corners. Our paper provides a fundamental understanding of topological photonics by type-II Dirac points in HPMs.

Emergent topological states via digital (001) oxide superlattices

Oxide heterostructures exhibit many intriguing properties. Here we provide design principles for inducing multiple topological states in (001) (AMO3)1/(AM′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$AM^{\\prime}$$\\end{document}O3)1 oxide superlattices. Aided by first-principles calculations and model analysis, we show that a (SrMO3)1/(SrM′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M^{\\prime}$$\\end{document}O3)1 superlattice (M = Nb, Ta and M′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M^{\\prime}$$\\end{document} = Rh, Ir) is a strong topological insulator with Z2 index (1;001). More remarkably, a (SrMoO3)1/(SrIrO3)1 superlattice exhibits multiple coexisting topological insulator (TI) and topological Dirac semi-metal (TDS) states. The TDS state has a pair of type-II Dirac points near the Fermi level and symmetry-protected Dirac node lines. The surface TDS Dirac cone is sandwiched by two surface TI Dirac cones in the energy-momentum space. The non-trivial topological properties arise from the band inversion between d orbitals of two dissimilar transition metal atoms and a particular parity property of (001) superlattice geometry. Our work demonstrates how to induce non-trivial topological states in (001) perovskite oxide heterostructures by rational design.

Linear-in-frequency optical conductivity over a broad range in the three-dimensional Dirac semimetal candidate Ir2In8Se

The optical conductivity of the new Dirac semimetal candidate ${\mathrm{Ir}}_{2}{\mathrm{In}}_{8}\mathrm{Se}$ is measured in a frequency range from 40 to 30 000 ${\mathrm{cm}}^{\ensuremath{-}1}$ at temperatures from 300 K down to 10 K. The measurement reveals that the compound is a low carrier density metal. We find that the real part of the conductivity ${\ensuremath{\sigma}}_{1}(\ensuremath{\omega})$ is linear in frequency over a broad range from 500 to 4000 ${\mathrm{cm}}^{\ensuremath{-}1}$ at 300 K and varies slightly with cooling. This linearity strongly suggests the presence of three-dimensional linear electronic bands with band crossings near the Fermi level. Band-structure calculations indicate the presence of type-II Dirac points. By comparing our data with the optical conductivity computed from the band structure, we conclude that the observed linear dependence mainly originates from the Dirac cones and the transition between the Dirac cones and the next lower bands. In addition, a weak energy gap feature is resolved below the charge density wave phase transition temperature in the reflectivity spectra. An enhanced structure arising from imperfect Fermi surface nesting is identified in the electronic susceptibility function, suggesting a Fermi surface nesting driven instability.

Possible realization of optical Dirac points in woodpile photonic crystals.

The simulation of fermionic relativistic physics, e.g., Dirac and Weyl physics, has led to the discovery of many unprecedented phenomena in photonics, of which the optical-frequency realization is, however, still challenging. Here, surprisingly, we discover that the woodpile photonic crystals commonly used for optical frequency applications host exotic fermion-like relativistic degeneracies: a Dirac nodal line and a fourfold quadratic point, as protected by the nonsymmorphic crystalline symmetry. Deforming the woodpile photonic crystal leads to the emergence of type-II Dirac points from the fourfold quadratic point. Such type-II Dirac points can be detected by its anomalous refraction property which is manifested as a giant birefringence in a slab setup. Our findings provide a promising route towards 3D optical Dirac physics in all-dielectric photonic crystals.

Spin-Orbit Interaction of Light Enabled by Negative Coupling in High-Quality-Factor Optical Metasurfaces

We study the negative couplings amid local resonances of photonic metasurfaces and their radiation polarizations. In our analysis, we discover fully circularly polarized, wave-vector variant, radiational eigenstates that are attributed to interorbit coupling-induced negative couplings. Our theoretical model is exemplified via a guided resonance dielectric metasurface that possesses type-II non-Hermitian Dirac points, from where the circular polarization lines emanate. The designed metasurface carries circularly polarized radiation eigenstates in both the zeroth and the first diffraction orders, allowing spin-selective light deflections. The high quality factor and field enhancement of the designed metasurface could lead to applications for spin-selective sensing, beam control, and nonlinear optics. Our findings open a gateway for the design of near-field coupling-assisted metasurfaces.

Remarkable Topological Features of Electronic Band Dispersion of IrGa and RhGa Compounds from First Principles

Exotic behavior of linearly dispersed electronic bands near the Fermi level implies advanced physical properties in a material. In this paper, we present an ab initio study of the electronic properties of IrGa and RhGa, with and without spin-orbit interaction, using first-principles calculations. Linearly dispersed band crossings, reminiscent of topological semimetallic band structures, were identified near the Fermi energy. These include type-I and type-II Dirac points and nodal lines. By applying compressive and tensile stress to the lattice along x, y, and z, the response to the band structure near the Fermi level has been studied.

Nonlinear topological valley Hall edge states arising from type-II Dirac cones

Type-II Dirac/Weyl points, although impermissible in particle physics due to Lorentz covariance, were uncovered in condensed matter physics, driven by fundamental interest and intriguing applications of topological materials. Recently, there has been a surge of exploration of such generic points using various engineered platforms including photonic crystals, waveguide arrays, metasurfaces, magnetized plasma and polariton micropillars, aiming towards relativistic quantum emulation and understanding of exotic topological phenomena. Such endeavors, however, have focused mainly on linear topological states in real or synthetic Dirac/Weyl materials. Here, we demonstrate nonlinear valley Hall edge states (VHESs) in laser-writing anisotropic photonic lattices hosting type-II Dirac points. These self-trapped VHESs, manifested as topological gap quasi-solitons, are fundamentally distinct from their linear counterparts in type-I lattices and from all previously found topological solitons. Our finding may provide a route for understanding nonlinear phenomena in Lorentz-violating topological systems and for developing advanced light sources from topological insulator lasers.

Predicting MnB6 monolayer with room temperature ferromagnetism and high magnetic anisotropy

Searching for two-dimensional (2D) magnetic materials with high critical temperature and intrinsic magnetism have attracted significant research interests recently. By using swarm-intelligence structure search and first-principles calculations, we predict two low-energy MnB 6 monolayers, namely MnB 6 -I and MnB 6 -II, with high stabilities and intriguing electronic and magnetic properties . Specifically, the phonon spectrum and ab initio molecular dynamics simulations indicate the two sheets are dynamically and thermally stable. The MnB 6 -I and MnB 6 -II layers are both metallic with ferromagnetic (FM) and antiferromagnetic (AFM) ground state, respectively. When spin-orbital coupling is considered, the type-II Dirac points and a sizable magnetocrystalline anisotropy energy (938 μ eV per Mn atom) can be found in the MnB 6 -I sheet. The Monte Carlo simulation based on the Heisenberg model suggests the Curie temperature for MnB 6 -I sheet is up to 470 K, which is above room temperature. The superior electronic and magnetic properties of the MnB 6 monolayers render them promising candidates for spintronic applications. • Two stable phases of MnB 6 monolayers were predicted with ferromagnetic and antiferromagnetic ground states. • The calculated Curie temperature for MnB 6 -I sheet is up to 470 K, which is above room temperature. • The calculated Néel temperature for MnB 6 -II sheet is 180 K, higher than many reported antiferromagnetic materials. • When spin-orbital coupling is considered, MnB 6 -I sheet shows type-II Dirac points and a sizable magnetic anisotropy energy.

Type-II Dirac point in RbAg5Se3

Using first-principles calculations and low-energy effective model analysis, we propose that silver selenide-based ${\mathrm{RbAg}}_{5}{\mathrm{Se}}_{3}$ compound is an ideal type-II Dirac semimetal. There is a pair of Dirac points appearing at about 25.68 meV below the Fermi energy. The Dirac points existing on the ${k}_{z}$ axis are protected by ${C}_{4z}$ rotation symmetry in addition to $\mathcal{PT}$ symmetry. Meanwhile, the system is also characterized by a nontrivial ${\mathbb{Z}}_{2}$ topological invariant. The long clean Fermi arcs exist on the (010) surface, connecting the projections of the Dirac points. In addition, the Landau levels of the type-II Dirac points under magnetic field are also calculated. Our discovery not only adds one promising candidate into the type-II Dirac semimetals family but also provides an ideal platform for the study of topological phase transitions.

Noncentrosymmetric topological Dirac semimetals in three dimensions

Topological Dirac semimetals are a class of semimetals that host symmetry-protected Dirac points near the Fermi level, which arise due to a band inversion of the conduction and valence bands. In this work, we study the less explored class of \emph{noncentrosymmetric} topological Dirac semimetals in three dimensions. We identify the noncentrosymmetric crystallographic point groups required to stabilize fourfold degenerate band crossings and derive model Hamiltonians for all distinct types of band inversions allowed by symmetry. Using these model Hamiltonians, which emphasize the physical nature of the allowed couplings, we establish the generic electronic phase diagram noncentrosymmetric Dirac semimetals and show that it generically includes phases with coexistent Weyl point nodes or Weyl line nodes. In particular, for one specific type of band inversion in sixfold symmetric systems we show that Weyl line nodes are always present. Based on first-principles calculations, we predict that BiPd$_2$O$_4$ is a noncentrosymmetric Dirac semimetal under 20 Gpa pressure and hosts topological type-II Dirac points on the fourfold rotation axis. Furthermore, we propose that the hexagonal polar alloy LiZnSb$_{x}$Bi$_{1-x}$ realizes a Dirac semimetal with coexistent Weyl points. Interestingly, the emergence and location of the Weyl points is highly tunable and can be controlled by the alloy concentration $x$. More generally, our results not only establish band-inverted noncentrosymmetric systems as a broad and versatile class of topological semimetals, but also provide a framework for studying the quantum nonlinear Hall effect and nonlinear optical properties in the Dirac semimetals.

Superconductivity in centrosymmetric topological superconductor candidate TaC

We report the synthesis and physical properties of the single crystals of TaC, which are proposed to hold topological band structure as a topological superconductor (TSC) candidate. Magnetization, resistivity and specific heat measurements are performed and indicate that TaC is bulk superconductor with critical temperature of 10.3 K. TaC is a strongly coupled type-II superconductor and the superconducting state can be well described by s-wave Bardeen–Cooper–Schrieffer theory with a single gap. The upper critical field (H c2) of TaC shows linear temperature dependence, which is quite different from most conventional superconductors and isostructural NbC, which is proposed to manifest topological nodal-loops or type-II Dirac points as well as superconductivity. Our results suggest that TaC would be a new candidate for further research of TSCs.

Type-II Dirac points and edge transports in phononic crystal plates

The accidentally degenerate type-II Dirac points in sonic crystal has been realized recently. However, elastic phononic crystals with type-II Dirac points have not yet been explored. In this work, we design a two-dimensional phononic crystal plate in square lattice with type-II Dirac points for elastic waves. The type-II Dirac points, different from the type-I counterparts, have the tiled dispersions and thus the iso-frequency contours become crossed lines. By tuning structures to break the mirror symmetry, the degeneracies of the type-II Dirac points are lifted, leading to a band inversion. In order to have a further explanation, we also calculate the Berry curvatures of phononic crystals with opposite structure parameters, and it turns out that these two crystals hold opposite signs around the valley. The phononic crystal plates before and after the band inversion belong to different topological valley phases, whose direct consequence is that the topologically protected gapless interface states exist between two distinct topological phases. Topologically protected interface states are found by calculating the projected band structures of a supercell that contains two kinds of interfaces between two topological phases. Robustness of the interface transport is verified by comparing the transmission rate for perfect interface with that for defective interface. Moreover, owing to the special stress field distributions of the elastic plate waves, the boundaries of a single phononic crystal phase can similarly host the gapless boundary states, which is found by calculating the projected band structures of a supercell with a single phase, thus having two free boundaries on the edges. This paper extends the two-dimensional Dirac points and valley states in graphene-like systems to the type-II cases, and obtains in the same structure the gapless interface and boundary propagations. Owing to the simple design scheme of the structure, the phononic crystal plates can be fabricated and scaled to a small size. Our system provides a feasible way of constructing high-frequency elastic wave devices.

Strongly correlated superconductor with polytypic 3D Dirac points

Topological superconductors should be able to provide essential ingredients for quantum computing, but are very challenging to realize. Spin–orbit interaction in iron-based superconductors opens the energy gap between the p-states of pnictogen and d-states of iron very close to the Fermi level, and such p-states have been recently experimentally detected. Density-functional theory predicts existence of topological surface states within this gap in FeTe1−xSex making it an attractive candidate material. Here we use synchrotron-based angle-resolved photoemission spectroscopy and band structure calculations to demonstrate that FeTe1−xSex (x = 0.45) is a superconducting 3D Dirac semimetal hosting type-I and type-II Dirac points and that its electronic structure remains topologically trivial. We show that the inverted band gap in FeTe1−xSex can possibly be realized by further increase of Te content, but strong correlations reduce it to a sub-meV size, making the experimental detection of this gap and corresponding topological surface states very challenging, not to mention exact matching with the Fermi level. On the other hand, the p–d and d–d interactions are responsible for the formation of extremely flat band at the Fermi level pointing to its intimate relation with the mechanism of high-Tc superconductivity in IBS.

Fermi-crossing Type-II Dirac fermions and topological surface states in NiTe2

Transition-metal dichalcogenides (TMDs) offer an ideal platform to experimentally realize Dirac fermions. However, typically these exotic quasiparticles are located far away from the Fermi level, limiting the contribution of Dirac-like carriers to the transport properties. Here we show that NiTe2 hosts both bulk Type-II Dirac points and topological surface states. The underlying mechanism is shared with other TMDs and based on the generic topological character of the Te p-orbital manifold. However, unique to NiTe2, a significant contribution of Ni d orbital states shifts the energy of the Type-II Dirac point close to the Fermi level. In addition, one of the topological surface states intersects the Fermi energy and exhibits a remarkably large spin splitting of 120 meV. Our results establish NiTe2 as an exciting candidate for next-generation spintronics devices.

De Haas-van Alphen Quantum Oscillations in BaSn3 Superconductor with Multiple Dirac Fermions**Supported by the National Natural Science Foundation of China (Grant No. 11874264), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDA18000000), the Starting Grant of ShanghaiTech University, the Shenzhen Peacock Team Plan (Grant No. KQTD20170809110344233), the Bureau of Industry and Information Technology of Shenzhen through

Characterization of Fermi surface of the BaSn3 superconductor (Tc ∼ 4.4 K) by de Haas–van Alphen (dHvA) effect measurement reveals its non-trivial topological properties. Analysis of non-zero Berry phase is supported by the ab initio calculations, which reveals a type-II Dirac point setting and tilting along the high symmetric K–H line of the Brillouin zone, about 0.13 eV above the Fermi level, and other two type-I Dirac points on the high symmetric Γ–A direction, but slightly far below the Fermi level. The results demonstrate BaSn3 as an excellent example hosting multiple Dirac fermions and an outstanding platform for studying the interplay between nontrivial topological states and superconductivity.

Type-II Dirac Semimetal State in a Superconductor Tantalum Carbide**Supported by the National Natural Science Foundation of China (Grant Nos. 11974076, 11674369 and 11925408), and the Natural Science Foundation of Fujian Province of China (Grant No. 2018J06001), the Beijing Natural Science Foundation (Grant No. Z180008), Beijing Municipal Science and Technology Commission (Grant No. Z191100007219013), the National Key Research and Development Program

The exploration of topological Dirac semimetals with intrinsic superconductivity can be a most plausible way to discover topological superconductors. We propose that type-II Dirac semimetal states exist in the band structure of TaC, a well-known s-wave superconductor, by using the first-principles calculations and the k ⋅ p effective model. The tilted gapless Dirac cones, which are composed of Ta d and C p orbitals and are protected by C4v symmetry, are found to be below the Fermi level. The bands from Ta d orbitals are greatly coupled with the acoustic modes around the zone boundary, indicating their significant contribution to the superconductivity. The relatively high transition temperature ∼10.5 K is estimated to be consistent with the experimental data. To bring the type-II Dirac points close to chemical potential, hole doping is needed. This seems to decrease the transition temperature a lot, making the realization of topological superconductivity impossible.