Dirac Cones and Nodal Line in Borophene

Two-dimensional bilayer blue phosphorus Dirac-like material: a multi-orbital tight-binding investigation

This study presents a theoretical examination of the electronic band structure of AA (AB) stacked bilayer blue phosphorus system within the fifth intralayer (5NN) and second interlayer nearest-neighbor (2NN) multi-orbital tight-binding (MOTB) approach. The variation of energy levels has been investigated through the symmetrical tensile strain of the low-buckled honeycomb lattice. Here, the primary objective is to examine the existence of Dirac electronic features in hexagonal stacked bilayer geometry. Our theoretical calculations predict that the AA bilayer is a new hexagonal two-dimensional material with px,y-orbital Dirac-like states at the high-symmetry K point. Consequently, these systems can host massless (massive) Dirac fermions. In particular, the AA bilayer exhibits zero-gap Dirac-like properties and manifests distinguishable Dirac-like cones in the presence of weak spin-orbit coupling when a modest stretch of 2.30% is achieved with a remarkably high Fermi velocity of approximately vf ≈ 0.12 × 105 m s-1. The behavior of the dispersion bands aligns reasonably well with recent experimental observations. Moreover, a stretch of 7.17% breaks some of the sublattice equivalence and enhances the spin-orbit interaction, resulting in the emergence of an electronic band gap of approximately ≈ 0.27 eV in the proximity of the high-symmetry K point. Furthermore, the tiny gap induced by the spin-orbit interaction implies topological nontriviality in the electronic state (quantum anomalous Hall state) of the honeycomb lattice. These findings categorize the AA bilayer as a rare two-dimensional Dirac-like material. This work provides, to the extent of our knowledge, a pioneering investigation into the existence of Dirac electronic properties in bilayer blue phosphorus. In addition, we present the first derivation of the MOTB model. However, the identified electronic characteristics designate this two-dimensional system as an ideal candidate for high-performance nanoelectronic devices.

Borophene: Synthesis, Chemistry, and Electronic Properties

As a neighbor of carbon in the periodic table, boron exhibits versatile structural and electronic configurations, with its allotropes predicted to possess intriguing structures and properties. Since the experimental realization of two-dimensional (2D) boron sheets (borophene) on Ag(111) substrates in 2015, the experimental study of the realization and characteristics of borophene has drawn increasing interest. In this review, we summarize the synthesis and properties of borophene, which are mainly based on experimental results. First, the synthesis of borophene on different substrates, as well as borophane and bilayer borophene, featuring unique phases and properties, are discussed. Next, the chemistry of borophene, such as oxidation, hydrogenation, and its integration into heterostructures with other materials, is summarized. We also mention a few works focused on the physical properties of borophene, specifically its electronic properties. Lastly, the brief outlook addresses challenges toward practical applications of borophene and possible solutions.

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 two-dimensional borophene monolayer with ideal Dirac nodal-line fermions

As a relatively new member of two-dimensional materials, borophene has gained huge interest over the past years, especially in the field of discovering new topological materials, such as Dirac nodal line semimetals. Here, based on first-principles calculations, for the first time, we find a completely flat borophene monolayer (named χ_2/9) with ideal Dirac nodal line states around the Fermi level. A tight-binding model using the Slater-Koster approach is proposed to demonstrate that the unique electronic feature of χ_2/9 that mainly originated from the first-nearest neighbor interactions of the p_z orbitals of boron. According to our symmetry analysis, the Dirac nodal line in χ_2/9 is guaranteed by the out-of-plane mirror or C₂ rotational symmetry and the negligible p_z orbital coupling. The chemical bonding analysis reveals the rare electronic properties of this material, which can be attributed to the multicentered π bonds.

High temperature phases of borophene: borophene glass and liquid

Borophene is a family of two-dimensional (2D) boron materials containing many isomers with different hole concentrations and distributions in a triangular lattice. Although it has been widely studied theoretically and some have been synthesized experimentally, their thermodynamic properties are still unexplored. Based on density functional theory (DFT), we developed an accurate potential for the kinetic Monte Carlo (kMC) simulations of borophene. Through extensive kMC simulations, new phases were discovered, such as the glass state of borophene, liquid borophene and borophene with large holes. A phase diagram of borophene is constructed to guide future experiments on borophene materials at high temperature.

Substrate-Mediated Borophane Polymorphs through Hydrogenation of Two-Dimensional Boron Sheets

The two-dimensional boron monolayer (borophene) stands out from the two-dimensional atomic layered materials due to its structural flexibility and tunable electronic and mechanical properties from a large number of allotropic materials. The stability of pristine borophene polymorphs could possibly be improved via hydrogenation with atomic hydrogen (referred to as borophane). However, the precise adsorption structures and the underlying mechanism are still elusive. Employing first-principles calculations, we demonstrate the optimal configurations of freestanding borophanes and the ones grown on metallic substrates. For freestanding borophenes, the energetically favored hydrogen adsorption sites are sensitive to the polymorphs and corresponding coordination numbers of boron atoms. With various metal substrates, the hydrogenation configurations of borophenes are modulated significantly, attributed to the overlap between B p_z and H s orbitals. These findings provide a deep insight into the hydrogenating borophenes and facilitate the stabilization of two-dimensional boron polymorphs by engineering hydrogen adsorption sites and concentrations.

Prediction of novel two-dimensional Dirac nodal line semimetals in Al2B2 and AlB4 monolayers

Topological semimetal phases in two-dimensional (2D) materials have gained widespread interest due to their potential applications in novel nanoscale devices. Despite the growing number of studies on 2D topological nodal lines (NLs), candidates with significant topological features that combine nontrivial topological semimetal phase with superconductivity are still rare. Herein, we predict Al₂B₂ and AlB₄ monolayers as new 2D nonmagnetic Dirac nodal line semimetals with several novel features. Our extensive electronic structure calculations combined with analytical studies reveal that, in addition to multiple Dirac points, these 2D configurations host various highly dispersed NLs around the Fermi level, all of which are semimetal states protected by time-reversal and in-plane mirror symmetries. The most intriguing NL in Al₂B₂ encloses the K point and crosses the Fermi level, showing a considerable dispersion and thus providing a fresh playground to explore exotic properties in dispersive Dirac nodal lines. More strikingly, for the AlB₄ monolayer, we provide the first evidence for a set of 2D nonmagnetic open type-II NLs coexisting with superconductivity at a rather high transition temperature. The coexistence of superconductivity and nontrivial band topology in AlB₄ not only makes it a promising material to exhibit novel topological superconducting phases, but also a rather large energy dispersion of type-II nodal lines in this configuration may offer a platform for the realization of novel topological features in the 2D limit.

Dirac Fermions in the Boron Nitride Monolayer with a Tetragon

Two-dimensional (2D) boron nitride (BN) is a promising candidate for aerospace materials due to its excellent mechanical and thermal stability properties. However, its unusually prominent band gap limits its application prospects. In this work, we report a gapless monolayer BN, t-BN, which has four anisotropic Dirac cones in the first Brillouin zone exactly at the Fermi level. To further confirm the semimetallic character, the nontrivial topological properties are proven through the topologically protected edge states and the invariant non-zero Z₂. Additionally, the Young's modulus and Poisson ratio characterize the strong mechanical strength of t-BN. Our theoretical predictions provide more possibilities for exploring the Dirac cone in BN, which will enhance the 2D boron derivative materials.

Step-Edge Epitaxy for Borophene Growth on Insulators

Borophene─a monatomic layer of boron atoms─stands out among two-dimensional (2D) materials, with its versatile properties tantalizing for physics exploration and next-generation devices. Yet its phases are all synthesized on and stay bound to metal substrates, hampering both characterization and use. Borophene growth on an inert insulator would allow postsynthesis exfoliation, but the weak adhesion to such a substrate results in a high 2D nucleation barrier, preventing clean borophene growth. This challenge can be circumvented in a strategy devised and demonstrated here with ab initio calculations. Naturally present 1D-defects, the step-edges on an h-BN substrate surface, enable boron epitaxial assembly, reduce the nucleation dimensionality, and lower the barrier by an order of magnitude (to 1.1 eV or less), yielding a v_1/9 phase. Weak borophene adhesion to the insulator makes it readily accessible for comprehensive property tests or transfer into the device setting.

The Emergence and Evolution of Borophene

Neighboring carbon and sandwiched between non-metals and metals in the periodic table of the elements, boron is one of the most chemically and physically versatile elements, and can be manipulated to form dimensionally low planar structures (borophene) with intriguing properties. Herein, the theoretical research and experimental developments in the synthesis of borophene, as well as its excellent properties and application in many fields, are reviewed. The decade-long effort toward understanding the size-dependent structures of boron clusters and the theory-directed synthesis of borophene, including bottom-up approaches based on different foundations, as well as up-down approaches with different exfoliation modes, and the key factors influencing the synthetic effects, are comprehensively summarized. Owing to its excellent chemical, electronic, mechanical, and thermal properties, borophene has shown great promise in supercapacitor, battery, hydrogen-storage, and biomedical applications. Furthermore, borophene nanoplatforms used in various biomedical applications, such as bioimaging, drug delivery, and photonic therapy, are highlighted. Finally, research progress, challenges, and perspectives for the future development of borophene in large-scale production and other prospective applications are discussed.

Polymorphism in Post-Dichalcogenide Two-Dimensional Materials

Two-dimensional (2D) materials exhibit a wide range of atomic structures, compositions, and associated versatility of properties. Furthermore, for a given composition, a variety of different crystal structures (i.e., polymorphs) can be observed. Polymorphism in 2D materials presents a fertile landscape for designing novel architectures and imparting new functionalities. The objective of this Review is to identify the polymorphs of emerging 2D materials, describe their polymorph-dependent properties, and outline methods used for polymorph control. Since traditional 2D materials (e.g., graphene, hexagonal boron nitride, and transition metal dichalcogenides) have already been studied extensively, the focus here is on polymorphism in post-dichalcogenide 2D materials including group III, IV, and V elemental 2D materials, layered group III, IV, and V metal chalcogenides, and 2D transition metal halides. In addition to providing a comprehensive survey of recent experimental and theoretical literature, this Review identifies the most promising opportunities for future research including how 2D polymorph engineering can provide a pathway to materials by design.

Borophene: Current Status, Challenges and Opportunities

Borophenes (2D boron sheets) have triggered a surge of interest both theoretically and experimentally because of its distinct structural, optical and electronic properties for extensive potential applications. Although theoretical efforts have guided the research directions of borophene, only few synthetic borophene sheets have been demonstrated experimentally. Borophene sheets have been successfully synthesized experimentally on metal substrates until 2015. Afterwards, more efforts were put on the controlled synthesis of crystalline and semiconducting borophene sheets as well as on the investigation of their novel and fascinating physical properties. This report provides a brief review on theoretical and experimental progress in borophene research. Some typical structures and properties of borophenes have been reviewed. The focus is laid on summarizing the experimental synthesis of borophene in recent years, and on showing some ultrastable and semiconducting borophenes which have been applied in electronic information devices. Finally, the future challenges and opportunities regarding experimental realization and practical applications of borophenes are presented.

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.

Unique topological nodal line states and associated exceptional thermoelectric power factor platform in Nb3GeTe6 monolayer and bulk

To date, ideal topological nodal line semimetal (TNLS) candidates in high dynamically stable and high thermally stable two-dimensional (2D) materials are still extremely scarce. Herein, by performing first-principles calculations, on the one hand, we found that three-dimensional Nb₃GeTe₆ bulk possesses a single closed TNL in the k_x = 0 plane and a fourfold TNL in the S-R direction without considering spin-orbit coupling (SOC). Under the SOC effect, a new topological signature, i.e., hourglass-like Dirac nodal line, occurs in Nb₃GeTe₆ bulk. On the other hand, we found that the 2D Nb₃GeTe₆ monolayer features a doubly degenerate TNL along surface X-S paths. Importantly, this monolayer enjoys the following advantages: (i) it has high thermal stability at room temperature and above; (ii) its TNL is nearly flat in energy and is very close to the Fermi level (E_F), which provides a fantastic maximum value platform of the thermoelectric power factor around the E_F; and (iii) no extraneous bands are close to the TNL, near the Fermi level. Moreover, we explore the entanglement between the topological states and thermolectric properties for the 2D Nb₃GeTe₆ monolayer. Our work not only reports the discovery of a novel TNL material, but also builds the link between the TNL and thermoelectric properties.

Borophene with Large Holes

In two-dimensional (2D) borophene, the structural transition from triangular lattice to hexagonal lattice with an increase in vacancy concentration is a basic principle of constructing various borophene isomers. Here, by performing an extensive structural search of 4239 borophene isomers with both hexagonal holes (HHs) and large holes (LHs), we show that the structural transformation from triangular lattice to borophene with large holes is energetically more favorable. Borophene isomers with LHs are more stable than those with only HHs at high vacancy concentrations (>20%) and are just slightly less stable than those with only HHs at low vacancy concentrations. This discovery greatly expands the family of 2D borophene and opens a route for synthesizing new borophene isomers.

Pnma metal hydride system LiBH: a superior topological semimetal with the coexistence of twofold and quadruple degenerate topological nodal lines

To date, a handful of topological semimetals (TMs) with multiple types of topological nodal line (TNL) states have been theoretically predicted in novel materials. However, their TNLs are often affected by many factors, such as spin-orbit coupling (SOC) effect, strain, and the extraneous bands near the band crossing points, and therefore, the TNL states cannot be easily verified by experiments. Here, by using first-principles calculations, we report that the Pnma LiBH is a potential TM with twofold and quadruple degenerate topological nodal lines. These TNLs situate very close to the Fermi level, and do not coexist with other extraneous bands. More importantly, the TNLs of this material are very robust to the effect of SOC, uniform strain, and biaxial strain. The nontrivial band structure in LiBH produces drum-head-like surface states in the (001) surface projection. Our result reveals that LiBH material is an excellent candidate to study the multiple kinds of TNLs.

Rich novel zero-dimensional (0D), 1D, and 2D topological elements predicted in the P63/m type ternary boride HfIr3B4

Topological semimetals, including topological nodal point semimetals (TNPSs), topological nodal line state semimetals (TNLSs), and topological nodal surface semimetals (TNSSs), featuring zero-dimensional (0D), one-dimensional (1D), and two-dimensional (2D) topological elements (TEs), respectively, have attracted widespread attention in recent years. In this work, based on first-principles calculations, we propose for the first time that three different (0D, 1D, and 2D) TEs are simultaneously present in a synthetic compound, HfIr₃B₄, with a P6₃/m type structure. In detail, HfIr₃B₄ hosts a Dirac point (DP) state at the K point, a TNL state in the k_z = 0 plane, and a 2D TNS state in the k_z = π plane, respectively. All sorts of topological elements, 0D, 1D, and 2D TEs, coexisting in the P6₃/m type HfIr₃B₄, provide an ideal platform to study the rich fermionic states and their related physical properties in this type of compound. In addition, because the 0D, 1D, and 2D TEs of HfIr₃B₄ are equally distributed in different energy ranges relative to the Fermi level, an approach is proposed to utilize individual TEs to build on-demand devices.

Borophosphene: A New Anisotropic Dirac Cone Monolayer with a High Fermi Velocity and a Unique Self-Doping Feature

Two-dimensional (2D) Dirac cone materials exhibit linear energy dispersion at the Fermi level, where the effective masses of carriers are very close to zero and the Fermi velocity is ultrahigh, only 2-3 orders of magnitude lower than the light velocity. Such Dirac cone materials have great promise in high-performance electronic devices. Herein, we have employed the genetic algorithm methods combined with first-principles calculations to propose a new 2D anisotropic Dirac cone material, an orthorhombic boron phosphide (BP) monolayer named borophosphene. Molecular dynamics simulation and phonon dispersion have been used to evaluate the dynamic and thermal stability of borophosphene. Because of the unique arrangements of B-B and P-P dimers, the mechanical and electronic properties are highly anisotropic. Of great interest is the fact that the Dirac cone of the borophosphene is robust, independent of in-plane biaxial and uniaxial strains, and can also be observed in its one-dimensional zigzag nanoribbons and armchair nanotubes. The Fermi velocities are ∼10⁵ m/s, on the same order of magnitude as that of graphene. By using a tight-binding model, the origin of the Dirac cone of borophosphene is analyzed. Moreover, a unique feature of self-doping can be induced by the in-plane biaxial and uniaxial strains of borophosphene and the curvature effect of nanotubes, which is greatly beneficial for realizing high-speed carriers (holes). Our results suggest that the borophosphene holds great promise for high-performance electronic devices, which could promote experimental and theoretical studies for further exploring the potential applications of other 2D Dirac cone sheets.

Flat Boron: A New Cousin of Graphene

The mechanical exfoliation of graphene from graphite provides the cornerstone for the synthesis of other 2D materials with layered bulk structures, such as hexagonal boron nitride, transition metal dichalcogenides, black phosphorus, and so on. However, the experimental production of 2D flat boron is challenging because bulk boron has very complex spatial structures and a rich variety of chemical properties. Therefore, the realization of 2D flat boron marks a milestone for the synthesis of 2D materials without layered bulk structures. The historical efforts in this field, particularly the most recent experimental progress, such as the growth of 2D flat boron on a metal substrate by chemical vapor deposition and molecular beam epitaxy, or liquid exfoliation from bulk boron, are described.

Near-equilibrium growth from borophene edges on silver

Two-dimensional boron, borophene, was realized in recent experiments but still lacks an adequate growth theory for guiding its controlled synthesis. Combining ab initio calculations and experimental characterization, we study edges and growth kinetics of borophene on Ag(111). In equilibrium, the borophene edges are distinctly reconstructed with exceptionally low energies, in contrast to those of other two-dimensional materials. Away from equilibrium, sequential docking of boron feeding species to the reconstructed edges tends to extend the given lattice out of numerous polymorphic structures. Furthermore, each edge can grow via multiple energy pathways of atomic row assembly due to variable boron-boron coordination. These pathways reveal different degrees of anisotropic growth kinetics, shaping borophene into diverse elongated hexagonal islands in agreement with experimental observations in terms of morphology as well as edge orientation and periodicity. These results further suggest that ultrathin borophene ribbons can be grown at low temperature and low boron chemical potential.

From Two- to Three-Dimensional van der Waals Layered Structures of Boron Crystals: An Ab Initio Study

A remarkable recent advancement has been the successful synthesis of two-dimensional boron monolayers on metal substrates. However, although up to 16 possible bulk allotropes of boron have been reported, none of them possess van der Waals (vdW) layered structures. In this work, starting from the experimentally synthesized monolayer boron sheet (β12 borophene), we explored the possibility for forming vdW layered bulk boron. We found that two β12 borophene sheets cannot form a stable vdW bilayer structure, as covalent-like B-B bonds are formed between them because of the peculiar bonding. Interestingly, when the covalently bonded bilayer borophene sheets are stacked on top of each other, three-dimensional (3D) layered structures are constructed via vdW interlayer interactions, rather than covalent. The 3D vdW layered structures were found to be dynamically stable. The interlayer binding energy is about 20 meV/Å2, which is close to the weakly bound graphene layers in graphite (∼16 meV/Å2). Furthermore, the density functional theory predicted electronic band structure testifies that these vdW bulk boron crystals can behave as good conductors. The insights obtained from this work suggest an opportunity to discover new vdW layered structures of bulk boron, which is expected to be crucial to numerous applications ranging from microelectronic devices to energy storage devices.

Two-dimensional honeycomb borophene oxide: strong anisotropy and nodal loop transformation

The search for topological semimetals is mainly focused on heavy-element compounds by following the footsteps of previous research on topological insulators, with less attention on light-element materials. However, the negligible spin orbit coupling with light elements may turn out to be beneficial for realizing topological band features. Here, using first-principles calculations, we propose a new two-dimensional light-element material-the honeycomb borophene oxide (h-B2O), which has nontrivial topological properties. The proposed structure is based on the recently synthesized honeycomb borophene on an Al (111) substrate [W. Li, L. Kong, C. Chen, J. Gou, S. Sheng, W. Zhang, H. Li, L. Chen, P. Cheng and K. Wu, Sci. Bull., 2018, 63, 282-286]. The h-B2O monolayer is completely flat, unlike the oxides of graphene or silicene. We systematically investigate the structural properties of h-B2O, and find that it has very good stability and exhibits significant mechanical anisotropy. Interestingly, the electronic band structure of h-B2O hosts a nodal loop centered around the Y point in the Brillouin zone, protected by the mirror symmetry. Furthermore, under moderate lattice strain, the single nodal loop can be transformed into two loops, each penetrating through the Brillouin zone. The loops before and after the transition are characterized by different [Doublestruck Z] × [Doublestruck Z] topological indices. Our work not only predicts a new two-dimensional material with interesting physical properties, but also offers an alternative approach to search for new topological phases in 2D light-element systems.

Recent advancements in Pt-nanostructure-based electrocatalysts for the oxygen reduction reaction

A comprehensive evaluation of Pt-nanostructure-based electrocatalysts for the oxygen reduction reaction.