Multiple Dirac Cones Research Articles

Topologically protected bound modes in non-neighboring hexagonal lattices

In the last decade, the study of wave localization and guiding has gained a renewed interest thanks to the introduction of the concept of topological protection. Originally developed in the field of quantum mechanics, topological protection has successively inspired the search for its analogue in other physical domains, including elasticity. In this context, periodic hexagonal lattices have been proposed as ideal candidates to realize these concepts. In this work, we explore the dynamic behavior of discrete hexagonal lattices with third nearest neighbor connections. First, the formation and transition of multiple Dirac cones above a critical value of the relative stiffness between the first and third nearest neighbor connections is derived. Then, the formation of extremely localized modes (i.e., bound modes) at the interface between regions formed by lattices which are inverted copies of each other is demonstrated. Our results show that this localization derives from a type of interface satisfying a point reflection symmetry for which the considered chain decouples into two identical copies at the high symmetry point of the Brillouin zone. Our findings open novel avenues for topological waveguiding and confinement with potential applications in surface acoustic wave devices.

Transport evidence of superlattice Dirac cones in graphene monolayer on twisted boron nitride substrate

Strong band engineering in two-dimensional (2D) materials can be achieved by introducing moiré superlattices, leading to the emergence of various novel quantum phases with promising potential for future applications. Presented works to create moiré patterns have been focused on a twist embedded inside channel materials or between channel and substrate. However, the effects of a twist inside the substrate materials on the unaligned channel materials are much less explored. In this work, we report the realization of superlattice multi-Dirac cones with the coexistence of the main Dirac cone in a monolayer graphene (MLG) on a ∼0.14° twisted double-layer boron nitride (tBN) substrate. Transport measurements reveal the emergence of three pairs of superlattice Dirac points around the pristine Dirac cone, featuring multiple metallic or insulating states surrounding the charge neutrality point. Displacement field tunable and electron–hole asymmetric Fermi velocities are indicated from temperature dependent measurements, along with the gapless dispersion of superlattice Dirac cones. The experimental observation of multiple Dirac cones in MLG/tBN heterostructure is supported by band structure calculations employing a periodic moiré potential. Our results unveil the potential of using twisted substrate as a universal band engineering technique for 2D materials regardless of lattice matching and crystal orientations, which might pave the way for a new branch of twistronics.

Massless Dirac Fermions on a Space‐Time Lattice with a Topologically Protected Dirac Cone

AbstractThe symmetries that protect massless Dirac fermions from a gap opening may become ineffective if the Dirac equation is discretized in space and time, either because of scattering between multiple Dirac cones in the Brillouin zone (fermion doubling) or because of singularities at zone boundaries. Here an implementation of Dirac fermions on a space‐time lattice that removes both obstructions is introduced. The quasi‐energy band structure has a tangent dispersion with a single Dirac cone that cannot be gapped without breaking both time‐reversal and chiral symmetries. It is shown that this topological protection is absent in the familiar single‐cone discretization with a linear sawtooth dispersion, as a consequence of the fact that there the time‐evolution operator is discontinuous at Brillouin zone boundaries.

Reflectionless Klein tunneling of Dirac fermions: comparison of split-operator and staggered-lattice discretization of the Dirac equation

Massless Dirac fermions in an electric field propagate along the field lines without backscattering, due to the combination of spin-momentum locking and spin conservation. This phenomenon, known as ‘Klein tunneling’, may be lost if the Dirac equation is discretized in space and time, because of scattering between multiple Dirac cones in the Brillouin zone. To avoid this, a staggered space-time lattice discretization has been developed in the literature, with one single Dirac cone in the Brillouin zone of the original square lattice. Here we show that the staggering doubles the size of the Brillouin zone, which actually contains two Dirac cones. We find that this fermion doubling causes a spurious breakdown of Klein tunneling, which can be avoided by an alternative single-cone discretization scheme based on a split-operator approach.

Engineering of the Topological Surface States and Topological Dangling Bond States in the (0001) Surface of Bi2Se3 via Structural Distortion

The (0001) surface band structure evolution of Bi2Se3 is studied for different broken symmetries in the top quintuple layer (QL) of 8QLs slab. Each surface electronic structure shows dangling bond states (DBS) irrespective of broken symmetries introduced by the surface termination or layer vacancy. The DBS become topological (TDBS) in the presence of spin–orbit coupling (SOC) effect, which form multiple Dirac cones (DC) at the Γ and M points in the conduction band due to Bi1 termination in top QL. For Bi1 and Se2 terminations, the contribution of p‐orbital of Bi2 atom to DBS in the gap is negligible. The DBS eliminates the topological state of Bi2Se3 for all broken symmetries except Se2 layer vacancy. The band structure with only Se2 and Se3 layer vacancies shows a narrow‐bandgap semiconducting nature with a small bandgap value of 0.18 eV and 0.08 eV, respectively. The DC feature is observed in vacant atomic layer configurations with SOC. The layer vacancies created by Bi atoms form a metallic state, whereas those produced by Se atoms form a narrow‐bandgap semiconducting state. The introduction of broken symmetries in Bi2Se3 can be utilized in the tuning of surface states and TDBS for possible technological applications.

Micrometre-scale single-crystalline borophene on a square-lattice Cu(100) surface.

Borophene, a crystalline monolayer boron sheet, has been predicted to adopt a variety of structures-owing to its high polymorphism-that may possess physical properties that could serve in flexible electronics, energy storage and catalysis. Several borophene polymorphs have been synthesized on noble metal surfaces but for device fabrication larger single-crystal domains are typically needed with, ideally, weak borophene-substrate interactions. Here we report the synthesis of borophene on a square-lattice Cu(100) surface and show that incommensurate coordination reduces the borophene-substrate interactions and also leads to a borophene polymorph different from those previous reported. Micrometre-scale single-crystal domains formed as isolated faceted islands or merged together to achieve full monolayer coverage. The crystal structure of this phase has ten boron atoms and two hexagonal vacancies in its unit cell. First-principles calculations indicate that charge transfer, rather than covalent bonding, binds this two-dimensional boron to the Cu(100) surface. The electronic band structure of this material features multiple anisotropic tilted Dirac cones.

Dirac Magic and Lifshitz Transitions in AA-Stacked Twisted Multilayer Graphene

We uncover a new type of magic-angle phenomena when an AA-stacked graphene bilayer is twisted relative to another graphene system with band touching. In the simplest case this constitutes a trilayer system formed by an AA-stacked bilayer twisted relative to a single layer of graphene. We find multiple anisotropic Dirac cones coexisting in such twisted multilayer structures at certain angles, which we call "Dirac magic." We trace the origin of Dirac magic angles to the geometric structure of the twisted AA-bilayer Dirac cones relative to the other band-touching spectrum in the moiré reciprocal lattice. Theanisotropy of the Dirac cones and a concomitant cascade of saddle points induce a series of topological Lifshitz transitions that can be tuned by the twist angle and perpendicular electric field. We discuss the possibility of direct observation of Dirac magic as well as its consequences for the correlated states of electrons in this moiré system.

Ferromagnetic coupling in a two-dimensional Cairo pentagonal Ni2(TCNQ)2 lattice

Magnetism has revolutionized important technologies, and continues to bring forth new phenomena in emergent materials and reduced dimensions. Here, using first-principles calculations, we demonstrate that the already-synthesized two-dimensional (2D) Ni-tetracyanoquinodimethane (Ni2(TCNQ)2) lattice is a stable ferromagnetism material with multiple spin-polarized Dirac cones. The conical bands in proximity of the Fermi level can be tuned by external tensile strain and show the fourfold degenerate electronic states at the critical tensile strain of ∼2.35 %, whose energy dispersion is consistent with 2D Cairo pentagonal lattice. In addition, spin-orbital coupling can open a band gap at the Dirac point of A, leading to topologically nontrivial electronic states characterized by the non-zero Chern number and the edge states of nanoribbon. Our results offer versatile platforms for the realization of massless spintronics with full-spin polarization in 2D Cairo pentagonal Ni2(TCNQ)2 Lattice.

Fragile topological insulators protected by rotation symmetry without spin-orbit coupling

We present a series of models of three-dimensional rotation-symmetric fragile topological insulators in class AI (time-reversal symmetric and spin-orbit-free systems), which have gapless surface states protected by time-reversal ($T$) and $n$-fold rotation ($C_n$) symmetries ($n=2,4,6$). Our models are generalizations of Fu's model of a spinless topological crystalline insulator, in which orbital degrees of freedom play the role of pseudo-spins. We consider minimal surface Hamiltonian with $C_n$ symmetry in class AI and discuss possible symmetry-protected gapless surface states, i.e., a quadratic band touching and multiple Dirac cones with linear dispersion. We characterize topological structure of bulk wave functions in terms of two kinds of topological invariants obtained from Wilson loops: $\mathbb{Z}_2$ invariants protected by $C_n$ ($n=4,6$) and time-reversal symmetries, and $C_2T$-symmetry-protected $\mathbb{Z}$ invariants (the Euler class) when the number of occupied bands is two. Accordingly, our models realize two kinds of fragile topological insulators. One is a fragile $\mathbb{Z}$ topological insulator whose only nontrivial topological index is the Euler class that specifies the number of surface Dirac cones. The other is a fragile $\mathbb{Z}_2$ topological insulator having gapless surface states with either a quadratic band touching or four (six) Dirac cones, which are protected by time-reversal and $C_4$ ($C_6$) symmetries. Finally, we discuss the instability of gapless surface states against the addition of $s$-orbital bands and demonstrate that surface states are gapped out through hybridization with surface-localized $s$-orbital bands.

Photoemission Spectroscopic Evidence of Multiple Dirac Cones in Superconducting BaSn3 Supported by the National Key R&D Program of China (Grant No. 2016YFA0300204), the National Natural Science Foundation of China (Grant Nos. U2032208 and 11874264), and the Natural Science Foundation of Shanghai (Grant No. 14ZR1447600). Y. F. Guo acknowledges the starting grant of ShanghaiTech University and the Program for Professor of

Signatures of topological superconductivity (TSC) in superconducting materials with topological nontrivial states prompt intensive researches recently. Utilizing high-resolution angle-resolved photoemission spectroscopy and first-principles calculations, we demonstrate multiple Dirac fermions and surface states in superconductor BaSn3 with a critical transition temperature of about 4.4 K. We predict and then unveil the existence of two pairs of type-I topological Dirac fermions residing on the rotational axis. Type-II Dirac fermions protected by screw axis are confirmed in the same compound. Further calculation for the spin helical texture of the observed surface states originating from the Dirac fermions gives an opportunity for realization of TSC in one single material. Hosting multiple Dirac fermions and topological surface states, the intrinsic superconductor BaSn3 is expected to be a new platform for further investigation of topological quantum materials as well as TSC.

Multiple Dirac cones and Lifshitz transition in a two-dimensional Cairo lattice as a Hawking evaporation analogue

The linear energy–momentum dispersion of Dirac cones offers a unique platform for mimicking the fantastical phenomena in high energy physics, such as Dirac fermions and black hole (BH) horizons. Three types of Dirac cones (I, III, and II) with different tilts have been proposed individually in specific materials but lack of integral lattice model. Here, we demonstrated the three types of Dirac cones inherited in a π-conjugated Cairo lattice of double-degenerate d π and p z orbitals by means of tight-binding (TB) approach, which paves a way for the design of two-dimensional (2D) Dirac materials. From first-principles calculations, we predicted a candidate material, penta-NiSb2 monolayer, to achieve these multiple Dirac cones and the Lifshitz transition between different Dirac cones driven by external biaxial strain. The coexistence of the three types of Dirac cones renders penta-NiSb2 monolayer a promising 2D fermionic analogue to simulate the event-horizon evaporation with a high Hawking temperature.

Quantum Size Effects, Multiple Dirac Cones, and Edge States in Ultrathin Bi(110) Films

The presence of inherently strong spin-orbit coupling in bismuth, its unique layer-dependent band topology and high carrier mobility make it an interesting system for both fundamental studies and applications. Theoretically, it has been suggested that strong quantum size effects should be present in the Bi(110) films, with the possibility of Dirac Fermion states in the odd-bilayer (BL) films, originating from dangling pz orbitals and quantum-spin hall (QSH) states in the even-bilayer films. However, the experimental verification of these claims has been lacking. Here, we study the electronic structure of Bi(110) films grown on a high-Tc superconductor, Bi2Sr2CaCu2O8+δ (Bi2212) using angle-resolved photoemission spectroscopy (ARPES). We observe an oscillatory behavior of electronic structure with the film thickness and identify the Dirac-states in the odd-bilayer films, consistent with the theoretical predictions. In the even-bilayer films, we find another Dirac state that was predicted to play a crucial role in the QSH effect. In the low thickness limit, we observe several extremely one-dimensional states, probably originating from the edge-states of Bi(110) islands. Our results provide a much needed experimental insight into the electronic and structural properties of Bi(110) films.

Anisotropic ferroelectric distortion effects on the RKKY interaction in topological crystalline insulators

Manipulation of electronic and magnetic properties of topological materials is a topic of much interest in spintronic and valleytronic applications. Perturbation tuning of multiple Dirac cones on the (001) surface of topological crystalline insulators (TCIs) is also a related topic of growing interest. Here we show the numerical evidence for the ferroelectric structural distortion effects on the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between two magnetic impurity moments on the SnTe (001) and related alloys. The mirror symmetry breaking between Dirac cones induced by the ferroelectric distortion could be divided into various possible configurations including the isotropically gapped, coexistence of gapless and gapped, and anisotropically gapped phases. Based on the retarded perturbed Green’s functions of the generalized gapped Dirac model, we numerically find the RKKY response for each phase. The distortion-induced symmetry breaking constitutes complex and interesting magnetic responses between magnetic moments compared to the pristine TCIs. In the specific case of coexisted gapless and gapped phases, a nontrivial behavior of the RKKY interaction is observed, which has not been seen in other Dirac materials up until now. For two impurities resided on the same sublattices, depending on the distortion strength, magnetic orders above of a critical impurity separation exhibit irregular ferromagnetic ⇔ antiferromagnetic phase transitions. However, independent of the impurity separation and distortion strength, no phase transition emerges for two impurities resided on different sublattices. This essential study sheds light on magnetic properties of Dirac materials with anisotropic mass terms and also makes TCIs applications relatively easy to understand.

Regularly multiple double Dirac cones in photonic bands and topological transitions of all-dielectric photonic crystals

Research on the remarkable characteristics of topological phases and topological phase transitions in various topological systems has been performed. In this paper, we propose a two-dimensional topological system composed of triangle photonic crystals (PCs) with ${C}_{3}$ point group symmetry, which hosts both regularly multiple fourfold-degenerate double Dirac cones and topological transitions in all-dielectric PC systems. By applying the tight-binding model to our system, we reveal the mechanism for topological transitions in the present configuration manufactured of conventional silicon material. Topological edge states at the common boundary of two different topological regions are demonstrated, which indicates that electromagnetic wave can propagate unidirectionally and robustly along the boundary with eximious transmission performance. Further, we manifest that topological edge states characterized by pseudospin-dependent unidirectional edge transport and have robustness against commonly minor defects. This flexible topological system demonstrates more abundant physics principles compared with previous research, which can be used to explore different topological phenomena and be implemented in integrated optical devices.

Anisotropic band flattening in graphene with one-dimensional superlattices.

Patterning graphene with a spatially periodic potential provides a powerful means to modify its electronic properties1-3. In particular, in twisted bilayers, coupling to the resulting moiré superlattice yields an isolated flat band that hosts correlated many-body phases4,5. However, both the symmetry and strength of the effective moiré potential are constrained by the constituent crystals, limiting its tunability. Here, we have exploited the technique of dielectric patterning6 to subject graphene to a one-dimensional electrostatic superlattice (SL)1. We observed the emergence of multiple Dirac cones and found evidence that with increasing SL potential the main and satellite Dirac cones are sequentially flattened in the direction parallel to the SL basis vector, behaviour resulting from the interaction between the one-dimensional SL electric potential and the massless Dirac fermions hosted by graphene. Our results demonstrate the ability to induce tunable anisotropy in high-mobility two-dimensional materials, a long-desired property for novel electronic and optical applications7,8. Moreover, these findings offer a new approach to engineering flat energy bands where electron interactions can lead to emergent properties9.

Multi-branch valley-chiral edge states of antisymmetric plate wave in phononic crystal plates with double-sided symmetric pillars

We build up phononic crystals (PCs) by attaching triangular pillars symmetrically to the double sides of a thin plate in the triangular lattice. The degeneracy of multiple Dirac cones is simultaneously lifted for the same PC plate, due to the mirror-inversion symmetry breaking caused by the rotation of triangular pillars; opposite Berry curves and valley Chern number are got at the lower and upper branches of each valley band. The multi-branch topological protected edge states are then created on the interface between PC plates with opposite topological phase. Interestingly enough, those edge states present various transverse symmetries and distinct robustness against sample imperfection such as cavity defect, rotation angle disorder, or sharp bending. Our results present the rich behaviors of multiple branches of topological edges.

Band engineering of non-hexagonal 2D tetragonal-silicene sheet and nanoribbons: A theoretical approach

We report combined first-principle and tight-binding (TB) calculation to investigate the mechanical and electronic properties of non-hexagonal two-dimensional (2D) tetragonal-silicene (TS) sheet and nanoribbons (NRs). The results obtained by the TB method are consistent with the density functional theory (DFT) results. The elastic constants of the TS sheet are comparable to silicene. The existence of two Dirac cones is observed at different k→ points in the irreducible Brillouin Zone. A spin-orbit bandgap larger than silicene is observed in TS. The criteria for the stability and robustness of the Dirac Cones on the TB parameters are extensively investigated. Both the Dirac points are robust and stable under a wide range of hopping parameters. The degeneracy of the Dirac cones can be removed by introducing asymmetry in the on-site energy. Comparing the DFT and TB band structures, the values of the TB hopping parameters are estimated. The electronic properties of the TSNRs show a strong dependence on the width and the edge states. The symmetric armchair TSNRs show a width dependent multiple Dirac cones in the Brillouin zone. The number of Dirac cones of the NR is found to be dependent on the TB hopping parameters as well. On the other hand, asymmetric armchair TSNRs are semiconducting. The tuning of the bandgap for asymmetric armchair TSNRs by modulating the TB hopping parameters is also explored. Zigzag TSNR behaves as a degenerate semiconductor with the presence of Dirac cones just below and above EF. We subsequently expect that these theoretical findings will give a better comprehension of Dirac materials and their NRs with its potential application in nano-electronics.

Multiple robust Dirac states in hexagonal lattice induced by p−d electron-counting rule and bilayer stacking

In this paper, we predict two-dimensional transition metal boride $\mathrm{Ti}{\mathrm{B}}_{2}{\mathrm{C}}_{2}$ and $\mathrm{Hf}{\mathrm{B}}_{2}{\mathrm{C}}_{2}$ nanosheets, which exhibit favorable mechanical and thermal properties. Young's moduli of $\mathrm{Ti}{\mathrm{B}}_{2}{\mathrm{C}}_{2}$ and $\mathrm{Hf}{\mathrm{B}}_{2}{\mathrm{C}}_{2}$ are 137.8 and 128.9 N/m, respectively. Both nanosheets could sustain up to 500 K based on our ab initio molecular dynamics results. In addition, both nanosheets are semimetallic, exhibiting 12 Dirac cones in the first Brillouin zone, six pairs of which form compensated electron-hole pockets. Under different types of external strains, both $\mathrm{Ti}{\mathrm{B}}_{2}{\mathrm{C}}_{2}$ and $\mathrm{Hf}{\mathrm{B}}_{2}{\mathrm{C}}_{2}$ are robust, and the Dirac cones could be tuned to be anisotropic. We propose that the multiple Dirac cones are induced by a combination of the $p\ensuremath{-}d$ electron-counting rule and bilayer stacking; to verify our opinion, we also predicted and proved that $T{\mathrm{B}}_{2}{\mathrm{Si}}_{2}$ ($T$ = Ti, Zr, Hf) are mechanical stable two-dimensional nanosheets with multiple Dirac cones based on phonon and electronic band structure calculations.

Tunable Dirac cones in single-layer selenium

Dirac cone, one of the main characters of topological materials, provides us an approach to explore topological phase transitions and topological states. Single-element 2D-Xenes are prominent candidates for hosting Dirac cones. Till now, the multiple Dirac cones, Dirac-like cones, and semi-metal Dirac point have been discovered in them. However, it is still difficult to realize the tunable Dirac cones due to the lack of appropriate materials. Using first-principles calculations, this paper proposes that monolayer selenium with square lattice could achieve tunable Dirac cones and a topological phase transition. Double structural phases of the monolayer selenium can be distinguished according to strain applied, i.e., buckled square and buckled rectangular phases, which have rich Dirac physics. There exist four anisotropic Dirac cones in the buckled square phase, owing to fourfold symmetry. The buckled rectangular phase hosts a topological phase transition from a 2D topological insulator with double Dirac cones to a simple insulator, with a Dirac semi-metal having single Dirac point as the phase transition point. Moreover, the topological insulator has a global band gap of 0.16 eV, suggesting its potential utilizations in room-temperature devices. These studies will greatly promote the development of the Dirac physics and widen the application ranges of 2D-Xenes.

Serendipity for Topological Insulator as Multifunctional Electrocatalyst

The design for multifunctional two-dimensional (2D) materials is considered as one of the main strategies to expand its application. Here, we propose that the already-synthesized organometallic lattice (Ni3(CNH)12) concurrently hosts topological and catalytic properties. Using first-principles calculations combined with tight-binding (TB) model, we demonstrate that the 2D Ni3(CNH)12 intrinsically possesses multiple Dirac cones along with flat bands passing through the Dirac bands. Taking the spin–orbital coupling (SOC) into account, all the degenerate points open gaps, spawning abundant topologically nontrivial insulating phases characterized by a nonzero Z2 topological invariant and edge states. Furthermore, because of its special geometry and electronic states, the 2D lattice can serve as a bifunctional electrocatalytic activity for overall water splitting with low overpotentials. Our current results offer a viable approach for the discovery of 2D organometallic materials with multifunctional application in spintronics devices and catalysts from the emerging field of topological materials.