Highly Anisotropic Dirac Fermions in Square Graphynes

Dirac cones in bipartite square-octagon lattice: A theoretical approach

Dirac cones are difficult to achieve in a square lattice with full symmetry. Here, we have theoretically investigated a bipartite tetragonal lattice composed of tetragons and octagons using both Tight-Binding (TB) model and density functional theory (DFT) calculations. The TB model predicts that the system exhibits nodal line semi-metallic properties when the on-site energies of all atoms are identical. When the on-site energies differ, the formation of an elliptical Dirac cone is predicted. Its physical properties (anisotropy, tilting, merging, and emerging) can be regulated by the hopping energies. An exact analytical formula is derived to determine the position of the Dirac point by the TB parameters, and a criterion for the existence of Dirac cones is obtained. The "divide-and-coupling" method is applied to understand the origin of the Dirac cone, which involves dividing the bands into several groups and examining the couplings among inter-groups and intra-groups. Various practical systems computed by DFT methods, e.g., t-BN, t-Si, 4,12,2-graphyne, and t-SiC, are also examined, and they all possess nodal lines or Dirac cones as predicted by the TB model. The results provide theoretical foundation for designing novel Dirac materials with tetragonal symmetry.

A primer on twistronics: a massless Dirac fermion's journey to moiré patterns and flat bands in twisted bilayer graphene

The recent discovery of superconductivity in magic-angle twisted bilayer graphene (TBLG) has sparked a renewed interest in the strongly-correlated physics ofsp²carbons, in stark contrast to preliminary investigations which were dominated by the one-body physics of the massless Dirac fermions. We thus provide a self-contained, theoretical perspective of the journey of graphene from its single-particle physics-dominated regime to the strongly-correlated physics of the flat bands. Beginning from the origin of the Dirac points in condensed matter systems, we discuss the effect of the superlattice on the Fermi velocity and Van Hove singularities in graphene and how it leads naturally to investigations of the moiré pattern in van der Waals heterostructures exemplified by graphene-hexagonal boron-nitride and TBLG. Subsequently, we illuminate the origin of flat bands in TBLG at the magic angles by elaborating on a broad range of prominent theoretical works in a pedagogical way while linking them to available experimental support, where appropriate. We conclude by providing a list of topics in the study of the electronic properties of TBLG not covered by this review but may readily be approached with the help of this primer.

Quantum transmission through the n-p-n heterojunction of massive 8-Pmmnborophene

We investigate the quantum transmission through the n-p-n heterojunction of massive 8-Pmmnborophene. It is found that the Dirac mass of the electron interacts nontrivially with the anisotropy of the 8-Pmmnborophene, leading to the occurrence of new transmission behaviors in this n-p-n heterojunction. Firstly, the effective energy range of nonzero transmission can be reduced but deviates from the mass amplitude, which induces the further controllability of the transmission property. Secondly, even if the equal-energy surfaces in the p and n parts do not encounter in thek-space, finite transmission is allowed to occur as well. In addition, the existence of Dirac mass can change the reflection manner from the retroreflection to the specular reflection under appropriate conditions. The findings in this work can be helpful in describing the quantum transport properties of the heterojunction based on 8-Pmmnborophene.

Emerging properties of carbon based 2D material beyond graphene

Graphene turns out to be the pioneering material for setting up boulevard to a new zoo of recently proposed carbon based novel two dimensional (2D) analogues. It is evident that their electronic, optical and other related properties are utterly different from that of graphene because of the distinct intriguing morphology. For instance, the revolutionary emergence of Dirac cones in graphene is particularly hard to find in most of the other 2D materials. As a consequence the crystal symmetries indeed act as a major role for predicting electronic band structure. Since tight binding calculations have become an indispensable tool in electronic band structure calculation, we indicate the implication of such method in graphene's allotropes beyond hexagonal symmetry. It is to be noted that some of these graphene allotropes successfully overcome the inherent drawback of the zero band gap nature of graphene. As a result, these 2D nanomaterials exhibit great potential in a broad spectrum of applications, viz nanoelectronics, nanooptics, gas sensors, gas storages, catalysis, and other specific applications. The miniaturization of high performance graphene allotrope based gas sensors to microscopic or even nanosized range has also been critically discussed. In addition, various optical properties like the dielectric functions, optical conductivity, electron energy loss spectra reveal that these systems can be used in opto-electronic devices. Nonetheless, the honeycomb lattice of graphene is not superconducting. However, it is proposed that the tetragonal form of graphene can be intruded to form new hybrid 2D materials to achieve novel superconducting device at attainable conditions. These dynamic experimental prospects demand further functionalization of these systems to enhance the efficiency and the field of multifunctionality. This topical review aims to highlight the latest advances in carbon based 2D materials beyond graphene from the basic theoretical as well as future application perspectives.

Tunable topologically nontrivial states in newly discovered graphyne allotropes: from Dirac nodal grid to Dirac nodal loop

By means of quotient-graph associated crystal prediction method, a new graphyne allotrope with unique Dirac nodal grid state is reported in this work. It is named as 191-E24Y24-1 according to its hexagonal lattice (with P6/mmm symmetry, No. 191) containing 24 sp²-hybridized carbon atoms and 24 sp-hybridized ones. The first-principles results show that the total energy of 191-E24Y24-1 is more favorable than that of recent synthesizedβ-graphdiyne and carbon ene-yne. It is also demonstrated to be dynamically, thermally, and mechanically stable. Interestingly, the 191-E24Y24-1 harbors intrinsic semimetal features showing intriguing hexagonal Dirac nodal grid state in the reciprocal space. Such unique electronic state is stable against small external tensile strains, and it is tunable under compression strains which will transform to new triangle Dirac nodal grid state. Moreover, a new metastable graphyne allotrope named 191-E12Y36-4 with Dirac nodal loop state is also observed in the process of stretching 191-E24Y24-1 with large tensile strains. The results presented in this work reveal two novel graphyne allotropes with exotic electronic properties. These discoveries are not only physical interesting, but also provide potential material candidates for carbon-based high performance electronic nanodevices.

Newly discovered graphyne allotrope with rare and robust Dirac node loop

Two-dimensional (2D) carbon allotropes with topologically nontrivial states are drawing considerable attention owing to their unique physical properties and great potential applications in the next generation of micro-nano devices. In contrast to the numerous Dirac points predicted in 2D carbon allotropes, systems featuring Dirac nodal lines (loops) are still quite rare. Here, by means of first-principles calculation, we report our newly discovered carbon monolayer 123-E8Y24-1 with robust Dirac nodal line states, which possesses a tetragonal lattice with P4/mmm symmetry and contains 8 sp² carbon atoms (graphene: E8) and 24 sp carbon atoms (grapheyne: Y24) in the crystalline cell. This 2D material is as energetically stable as the recently experimentally synthesized β-graphdiyne, and it is further predicted to be dynamically, mechanically, and also thermodynamically stable. Owing to its intrinsic geometric characteristics, 123-E8Y24-1 also exhibits obvious Young's modulus anisotropy, with a sizable ratio between the maximum and minimum value of up to 5.8. Remarkably, 123-E8Y24-1 presents a semimetal nature and possesses Dirac nodal line states in the electronic band structure, and such behavior could be kept well under external strain between -10.0% and 8.0%. The electronic properties of 123-E8Y24-1 can be carefully confirmed by constructing a tight-binding (TB) model. The findings presented in this paper reveal a novel 2D Dirac nodal loop carbon sheet, providing a new candidate for carbon-based high-speed electronic devices.

Exotic Topological Bands and Quantum States in Metal-Organic and Covalent-Organic Frameworks

ConspectusMetal-organic and covalent-organic frameworks (MOFs/COFs) have been extensively studied for fundamental interests and their promising applications, taking advantage of their unique structural properties, i.e., high porosity and large surface-to-volume ratio. However, their electronic and magnetic properties have been somewhat overlooked because of their relatively poor performance as conductive and/or magnetic materials. Recent experimental breakthroughs in synthesizing two-dimensional (2D) π-conjugated MOFs/COFs with high conductivity and robust magnetism through doping have generated renewed and increasing interest in their electronic properties. Meanwhile, comprehensive theoretical studies of the underlying physical principles have led to discovery of many exotic quantum states, such as topological insulating states, which were only observed in inorganic systems. Especially, the diversity and high tunability of MOFs/COFs have provided a playground to explore novel quantum physics and quantum chemistry as well as promising applications.The band theory has empowered us to understand the most fundamental electronic properties of inorganic crystalline materials, which can also be used to better understand MOFs/COFs. The first obvious difference between the two is that instead of atomic orbitals residing at lattice sites of inorganic crystals, molecular orbitals of organic ligands are predominant in MOFs/COFs. The second key difference is that usually all atomic orbitals in an inorganic crystal are subject to one common group of lattice symmetry, while atomic orbitals of metal ion and molecular orbitals of different organic ligands in MOFs/COFs belong to different subgroups of lattice symmetries. Both these differences will impact the band structure of MOFs/COFs, in particular making it more complex. Consequently, which subset of bands are of most importance depends strongly on the location of Fermi level, i.e., electron counting and charge doping. Furthermore, there are usually two types of characteristic electrons coupled in MOFs, i.e., strongly correlated localized d and f electrons and diffusive s and p electrons, which interplay with lattice, orbital, and spin degrees of freedom, leading to more exotic topological and magnetic band structures.In this Account, we present an up-to-date review of recent theoretical developments to better understand the exotic band structures of MOFs/COFs. Starting from three fundamental 2D lattice models, i.e., honeycomb, Kagome, and Lieb lattices, exotic Dirac and flat bands as well as the intriguing topological quantum states they host, e.g., quantum spin Hall and quantum anomalous Hall states, are outlined. In addition to the single-lattice models, we further elaborate on combined lattice model Hamiltonians, which give rise to overlapping bands hosting novel quantum states, such as nodal-line Dirac semimetal and unconventional superconducting states. Also, first-principles predictions of candidate MOFs/COFs that host these exotic bands and hence quantum phases are reviewed, which greatly extends the pool of materials beyond inorganic crystals for hosting exotic band structures.

Me-graphene: a graphene allotrope with near zero Poisson's ratio, sizeable band gap, and high carrier mobility

The exploration of new two-dimensional (2D) allotropes of carbon has attracted great research attention after graphene, but experiment-feasible graphene allotropes with novel properties are still rare. Here, we predict a new allotrope of graphene, named Me-graphene, composed of both sp2- and sp3-hybridized carbon by topological assembly of C-(C3H2)4 molecules. With a transitional ratio of sp2- and sp3-hybridized carbon atoms (12 : 1) between those of graphene (1 : 0) and penta-graphene (2 : 1), Me-graphene has transition properties between those of graphene and penta-graphene, such as energy, band gap, and Poisson's ratio. Unusually, Me-graphene exhibits a near zero Poisson's ratio of from -0.002 to 0.009 in the xy-plane (or called "anepirretic"), different from that of graphene (0.169) and penta-graphene (-0.068). More importantly, the near zero Poisson's ratio behavior remains in a large strain range, being less than ±0.02 for strain from -15% to +3%. Me-graphene possesses an indirect band gap of 2.04 eV, as a transition of graphene (semimetal) and penta-graphene (wide band gap), and turns into a direct-bandgap semiconductor with an enlarged band gap of 2.62 eV under compressive strain. It possesses high hole mobility of 1.60 × 105 cm2 V-1 s-1 at 300 K. Me-Graphene has potential applications in electronic, photoelectric and high-speed mechatronic devices. The transitional properties related to the ratio of sp2- and sp3-hybridized carbon atoms are inspiring for searching for new graphene allotropes with combinational properties.

A novel SiO monolayer with a negative Poisson's ratio and Dirac semimetal properties

Although a number of interesting physical properties such as a negative Poisson's ratio (NPR) and Dirac semimetal (DS) properties have been recently predicted in two-dimensional (2D) materials, the realization of a 2D material that exhibits both of these DS and NPR features has rarely been reported. Here by adopting particle swarm optimization (PSO) algorithms combined with first-principles methods, we successfully construct a novel SiO monolayer (Pmna), the dynamic and thermal stability of which was characterized using phonon spectrum calculations and molecular dynamics simulations. In particular, Young's modulus and Poisson's ratio calculations showed that the Pmna monolayer exhibits high mechanical anisotropy with an in-plane NPR originating from its puckered atomic arrangement. More notably, the band structure of the Pmna monolayer possesses zero bandgap with four Dirac cones in its first Brillouin zone, exhibiting a DS feature. From the calculations of orbital-resolved band structures, the Dirac cone was revealed to originate from the orbital hybridization of Si and O atoms. The Pmna monolayer is the first 2D structure in the Si-O system that has both an NPR and Dirac semi-metal properties, providing a new model for exploring 2D multifunctional materials.

Peculiar symmetry-protected electronic dispersions in two-dimensional materials

Symmetry indicates that low energy spectra of materials could be richer than well-known Dirac, semi-Dirac, or quadratic, hosting some unusual quasiparticles. Performing the systematic study of exact forms of low energy effective Hamiltonians and dispersions in high-symmetry points with fourfold degeneracy of bands, we found new, previously unreported dispersion, which we named poppy flower (PF) after its shape. This massless fermion exists in non-magnetic two-dimensional (2D) crystals with spin-orbit coupling (SOC), which are invariant under one of the proposed ten noncentrosymmetric layer groups. We suggest real three-dimensional (3D) layered materials suitable for exfoliation, having layers that belong to these symmetry groups as candidates for realization of PF fermions. In 2D systems without spin-orbit interaction, fortune teller (FT)-like fermions were theoretically predicted, and afterward experimentally verified in the electronic structure of surface layer of silicon. Herein, we show that such fermions can also be hosted in 2D crystals with SOC, invariant under additional two noncentrosymmetric layer groups. This prediction is confirmed by density functional based calculation: layered BiIO4, which has been synthesized already as a 3D crystal, exfoliates to stable monolayer with symmetrypb21a, and FT fermion is observed in the band structure. Analytically calculated density of states (DOS) of the PF shows semimetallic characteristic, in contrast to metallic nature of FT having non-zero DOS at the bands contact energy. We indicate possibilities for symmetry breaking patterns which correspond to the robustness of the proposed dispersions as well as to the transition from Dirac centrosymmetric semimetal to PF.

Nano-makisu: highly anisotropic two-dimensional carbon allotropes made by weaving together nanotubes

Graphene and carbon nanotubes (CNT) are the representatives of two-dimensional (2D) and one-dimensional (1D) forms of carbon, both exhibiting unique geometric structures and peculiar physical and chemical properties. Herein, we propose a family or series of 2D carbon-based highly anisotropic Dirac materials by weaving together an array of CNTs by direct C-C bonds or by graphene ribbons. By employing first-principles calculations, we demonstrate that these nano-makisus are thermally and dynamically stable and possess unique electronic properties. These 2D carbon allotropes are all metals and some nano-makisus show largely anisotropic Dirac cones, causing very different transport properties for the Dirac fermions along different directions. The Fermi velocities in the k_x direction could be ∼170 times higher than those in the k_y direction, which is the strongest anisotropy among 2D carbon allotropes to the best of our knowledge. This intriguing feature of the electronic structure has only been observed in heavy element materials with strong spin-orbit coupling. These results indicate that carbon based materials may have much broader applications in future nanoelectronics.

Semimetallic carbon honeycombs: new three-dimensional graphene allotropes with Dirac cones

Classic two-dimensional (2D) graphene possesses outstanding properties due to Dirac cone structures. When scaling up to three-dimensional (3D) structures, their high porosity and large surface-area-to-volume ratio made them have more promising engineering perspectives. However, the currently synthesized and density-functional-theory-predicted 3D graphene structures, termed as carbon honeycombs (CHCs), are metallic. Herein, we propose new families of stable semimetallic CHC structures, which have lower energies than the previous experimentally reported structure and they would be realized experimentally. Results from density functional theory (DFT) and tight binding (TB) model showed that multiple Dirac cones with massless Dirac Fermions are present in both pristine and strained CHCs. Dirac cones in pristine CHCs originated from interactions between sp²-hybridized carbon atoms along the zigzag direction (denoted as C, i = α, β,…), while strain-induced direction-dependent Dirac cones primarily stemmed from interactions (i) between the two C atoms bonded to a selected sp³-hybridized carbon atom or (ii) between C and C (α carbon atoms at the armchair direction) atoms. The largest Fermi velocity achieved is 1.204 × 10⁶ m s^-1, which is approximately 44.7% larger than that of graphene. These results open up a new direction in carbon-based 3D porous materials and these findings provide significant insights on numerous applications, ranging from nanoelectronics and nanomechanics to gas and liquid separations.

Discovery of 2D Anisotropic Dirac Cones

2D anisotropic Dirac cones are observed in χ₃ borophene, a monolayer boron sheet, using high-resolution angle-resolved photoemission spectroscopy. The Dirac cones are centered at the X and X' points. The data also reveal that the hybridization between borophene and Ag(111) is very weak, which explains the preservation of the Dirac cones. As χ₃ borophene has been predicated to be a superconductor, the results may stimulate further research interest in the novel physics of borophene, such as the interplay between Cooper pairs and the massless Dirac fermions.

Square transition-metal carbides MC6 (M = Mo, W) as stable two-dimensional Dirac cone materials

We identify the existence of Dirac cones in 2D square transition-metal carbides MC₆ (M = Mo, W) with an ultrahigh Fermi velocity comparable to that of graphene.

Cp-Graphyne: A Low-Energy Graphyne Polymorph with Double Distorted Dirac Points

In the present investigation, we have proposed a novel form of two-dimensional (2D) carbon allotropes with the aid of first-principle density functional theory-based calculations. The carbon polymorph is mainly composed of carbon pentagons (cp) and acetylenic linkers and hence named cp-graphyne. This 2D material is energetically more preferable than the rest of the semimetals of graphyne family, including graphdiyne monolayer. Close inspection of lattice dynamics and thermal and mechanical properties demonstrates the excellent dynamic, thermal, and mechanical stabilities of cp-graphyne. Interestingly, cp-graphyne exhibits a semimetallic nature and possesses double distorted Dirac points in the electronic band spectrum. The Fermi velocities (v _f) of cp-graphyne are highly anisotropic and are predicted to be in the range of 1.50-8.20 × 10⁵ m/s. Furthermore, the analysis of structural and electronic properties of the cp-graphyne bilayer discloses the presence of self-doped Dirac-like points nearer to the Fermi level in the electronic spectrum.

Origins of Dirac cone formation in AB3 and A3B (A, B = C, Si, and Ge) binary monolayers

Compared to the pure two-dimensional (2D) graphene and silicene, the binary 2D system silagraphenes, consisting of both C and Si atoms, possess more diverse electronic structures depending on their various chemical stoichiometry and arrangement pattern of binary components. By performing calculations with both density functional theory and a Tight-binding model, we elucidated the formation of Dirac cone (DC) band structures in SiC₃ and Si₃C as well as their analogous binary monolayers including SiGe₃, Si₃Ge, GeC₃, and Ge₃C. A “ring coupling” mechanism, referring to the couplings among the six ring atoms, was proposed to explain the origin of DCs in AB₃ and A₃B binary systems, based on which we discussed the methods tuning the SiC₃ systems into self-doped systems. The first-principles quantum transport calculations by non-equilibrium Green’s function method combined with density functional theory showed that the electron conductance of SiC₃ and Si₃C lie between those of graphene and silicene, proportional to the carbon concentrations. Understanding the DC formation mechanism and electronic properties sheds light onto the design principles for novel Fermi Dirac systems used in nanoelectronic devices.

Dirac Fermions in Borophene

Honeycomb structures of group IV elements can host massless Dirac fermions with nontrivial Berry phases. Their potential for electronic applications has attracted great interest and spurred a broad search for new Dirac materials especially in monolayer structures. We present a detailed investigation of the β_{12} sheet, which is a borophene structure that can form spontaneously on a Ag(111) surface. Our tight-binding analysis revealed that the lattice of the β_{12} sheet could be decomposed into two triangular sublattices in a way similar to that for a honeycomb lattice, thereby hosting Dirac cones. Furthermore, each Dirac cone could be split by introducing periodic perturbations representing overlayer-substrate interactions. These unusual electronic structures were confirmed by angle-resolved photoemission spectroscopy and validated by first-principles calculations. Our results suggest monolayer boron as a new platform for realizing novel high-speed low-dissipation devices.

Body-Centered Tetragonal C16 : A Novel Topological Node-Line Semimetallic Carbon Composed of Tetrarings

The present work not only predicts the existence of 3D topological semimetallic carbon allotropes composed of tetrarings, but also provides a likely crystalline structure for the unknown phase produced in the detonation soot.

Origins of Dirac cones and parity dependent electronic structures of α-graphyne derivatives and silagraphynes

Compared with graphene, graphyne and its derivatives possess more diversified atomic configurations and richer electronic structures including Dirac cones (DCs) and metallic features depending on the parity of the number of sp carbon atoms of graphynes. This report described conceptually the process of DC formation of α-graphyne within a tight-binding framework parameterized from density functional calculations. We propose a "triple coupling" mechanism elucidating the DC formation and some flat bands of α-graphynes where the couplings among the three sp carbon chain atoms are critical. The extension of this mechanism further explains the origins of DCs of silagraphynes and the parity dependent electronic structures of α-graphyne derivatives with extended sp carbon chains. Understanding these origins helps in tuning electronic properties in the design of C or C-Si based nanoelectronic devices.

Graphene-like Two-Dimensional Ionic Boron with Double Dirac Cones at Ambient Condition

Recently, partially ionic boron (γ-B28) has been predicted and observed in pure boron, in bulk phase and controlled by pressure [ Nature 2009 , 457 , 863 ]. By using ab initio evolutionary structure search, we report the prediction of ionic boron at a reduced dimension and ambient pressure, namely, the two-dimensional (2D) ionic boron. This 2D boron structure consists of graphene-like plane and B2 atom pairs with the P6/mmm space group and six atoms in the unit cell and has lower energy than the previously reported α-sheet structure and its analogues. Its dynamical and thermal stability are confirmed by the phonon-spectrum and ab initio molecular dynamics simulation. In addition, this phase exhibits double Dirac cones with massless Dirac Fermions due to the significant charge transfer between the graphene-like plane and B2 pair that enhances the energetic stability of the P6/mmm boron. A Fermi velocity (vf) as high as 2.3 × 10(6) m/s, which is even higher than that of graphene (0.82 × 10(6) m/s), is predicted for the P6/mmm boron. The present work is the first report of the 2D ionic boron at atmospheric pressure. The unique electronic structure renders the 2D ionic boron a promising 2D material for applications in nanoelectronics.