Organic topological insulators in organometallic lattices

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.

On-Surface Self-Assembly Kinetic Study of Cu-Hexaazatriphenylene 2D Conjugated Metal-Organic Frameworks on Coinage Metals and MoS2 Substrates

Supramolecular coordination self-assembly on solid surfaces provides an effective route to form two-dimensional (2D) metal-organic frameworks (MOFs). In such processes, surface-adsorbate interaction plays a key role in determining the MOFs' structural and chemical properties. Here, we conduct a systematic study of Cu-HAT (HAT = 1,4,5,8,9,12-hexaazatriphenylene) 2D conjugated MOFs (c-MOFs) self-assembled on Cu(111), Au(111), Ag(111), and MoS2 substrates. Using scanning tunneling microscopy and density functional theory calculations, we found that the as-formed Cu3HAT2 c-MOFs on the four substrates exhibit distinctive structural features including lattice constant and molecular conformation. The structural variations can be attributed to the differentiated substrate effects on the 2D c-MOFs, including adsorption energy, lattice commensurability, and surface reactivity. Specifically, the framework grown on MoS2 is nearly identical to its free-standing counterpart. This suggests that the 2D van der Waals (vdW) materials are good candidate substrates for building intrinsic 2D MOFs, which hold promise for next-generation electronic devices.

Large-scale 2D heterostructures from hydrogen-bonded organic frameworks and graphene with distinct Dirac and flat bands

The current strategies for building 2D organic-inorganic heterojunctions involve mostly wet-chemistry processes or exfoliation and transfer, leading to interface contaminations, poor crystallizing, or limited size. Here we show a bottom-up procedure to fabricate 2D large-scale heterostructure with clean interface and highly-crystalline sheets. As a prototypical example, a well-ordered hydrogen-bonded organic framework is self-assembled on the highly-oriented-pyrolytic-graphite substrate. The organic framework adopts a honeycomb lattice with faulted/unfaulted halves in a unit cell, resemble to molecular "graphene". Interestingly, the topmost layer of substrate is self-lifted by organic framework via strong interlayer coupling, to form effectively a floating organic framework/graphene heterostructure. The individual layer of heterostructure inherits its intrinsic property, exhibiting distinct Dirac bands of graphene and narrow bands of organic framework. Our results demonstrate a promising approach to fabricate 2D organic-inorganic heterostructure with large-scale uniformity and highly-crystalline via the self-lifting effect, which is generally applicable to most of van der Waals materials.

Chiral Honeycomb Lattices of Nonplanar π-Conjugated Supramolecules with Protected Dirac and Flat Bands

The honeycomb lattice is a fundamental two-dimensional (2D) network that gives rise to surprisingly rich electronic properties. While its expansion to 2D supramolecular assembly is conceptually appealing, its realization is not straightforward because of weak intermolecular coupling and the strong influence of a supporting substrate. Here, we show that the application of a triptycene derivative with phenazine moieties, Trip-Phz, solves this problem due to its strong intermolecular π-π pancake bonding and nonplanar geometry. Our scanning tunneling microscopy (STM) measurements demonstrate that Trip-Phz molecules self-assemble on a Ag(111) surface to form chiral and commensurate honeycomb lattices. Electronically, the network can be viewed as a hybrid of honeycomb and kagome lattices. The Dirac and flat bands predicted by a simple tight-binding model are reproduced by total density functional theory (DFT) calculations, highlighting the protection of the molecular bands from the Ag(111) substrate. The present work offers a rational route for creating chiral 2D supramolecules that can simultaneously accommodate pristine Dirac and flat bands.

Identifying topological corner states in two-dimensional metal-organic frameworks

Due to the diversity of molecular building blocks, the two-dimensional (2D) metal-organic frameworks (MOFs) are ideal platforms to realize exotic lattice models in condensed matter theory. In this work, we demonstrate the universal existence of topological corner states in 2D MOFs with a star lattice configuration, and confirm the intriguing higher-order nontrivial topology in the energy window between two Kagome-bands, or between Dirac-band and four-band. Furthermore, combining first-principles calculations and scanning tunneling microscopy measurements, the unique topological corner state is directly identified in monolayer Ni3(HITP)2 (HITP = 2,3,6,7,10,11-hexaiminotriphenylene) grown on the Au(111) substrate. Our results not only illustrate the first organic topological state in the experiments, but also offer an exciting opportunity to study higher-order topology in 2D MOFs with the large insulating band gap.

Two-Dimensional Metal-Organic Frameworks Towards Spintronics

The intrinsic properties of predesignable topologies and tunable electronic structures, coupled with the increase of electrical conductivity, make two-dimensional metal-organic frameworks (2D MOFs) highly prospective candidates for next-generation electronic/spintronic devices. In this Minireview, we present an outline of the design principles of 2D MOF-based spintronics materials. Then, we highlight the spin-transport properties of 2D MOF-based organic spin valves (OSVs) as a notable achievement in the progress of 2D MOFs for spintronics devices. After that, we discuss the potential for spin manipulation in 2D MOFs with bipolar magnetic semiconductor (BMS) properties as a promising field for future research. Finally, we provide a brief summary and outlook to encourage the development of novel 2D MOFs for spintronics applications.

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.

Tunable electronic and magnetic properties of planar and corrugated phases of two-dimensional metal-organic frameworks

The stability of two-dimensional (2D) metal-organic frameworks (MOFs) and their physical and chemical properties for potential applications are contentious. We herein investigated geometric, electronic and magnetic properties of the planar (p-) and corrugated (c-) phases of nickel ions with hexathiolbenzene (HTB)-based coordination nanosheets (Ni₃HTB). The c-Ni₃HTB is an antiferromagnetic semiconductor with a direct band gap of 0.33 eV, while the p-Ni₃HTB is a ferromagnetic metal. This indicates that the electronic and magnetic properties of c-Ni₃HTB and p-Ni₃HTB depend on their geometric pattern. Furthermore, we applied biaxial strain and molecular adsorption to control their electronic and magnetic properties. In addition, we have proved that the corrugated phase in some kinds of 2D MOFs is common. Our work not only demonstrates that the potential applications of 2D MOFs should be scrupulously explored but also offers a new platform to investigate the physical and chemical properties of 2D MOFs.

Correlation-driven organic 3D topological insulator with relativistic fermions

Exploring new topological phenomena and functionalities induced by strong electron correlation has been a central issue in modern condensed-matter physics. One example is a topological insulator (TI) state and its functionality driven by the Coulomb repulsion rather than a spin-orbit coupling. Here, we report a 'correlation-driven' TI state realized in an organic zero-gap system α-(BETS)₂I₃. The topological surface state and chiral anomaly are observed in temperature and field dependences of resistance, indicating a three-dimensional TI state at low temperatures. Moreover, we observe a topological phase switching between the TI state and non-equilibrium Dirac semimetal state by a dc current, which is a unique functionality of a correlation-driven TI state. Our findings demonstrate that correlation-driven TIs are promising candidates not only for practical electronic devices but also as a field for discovering new topological phenomena and phases.

A p-orbital honeycomb-Kagome lattice realized in a two-dimensional metal-organic framework

The experimental realization of p-orbital systems is desirable because p-orbital lattices have been proposed theoretically to host strongly correlated electrons that exhibit exotic quantum phases. Here, we synthesize a two-dimensional Fe-coordinated bimolecular metal-organic framework which constitutes a honeycomb lattice of 1,4,5,8,9,12-hexaazatriphenylene molecules and a Kagome lattice of 5,15-di(4-pyridyl)-10,20-diphenylporphyrin molecules on a Au(111) substrate. Density-functional theory calculations show that the framework features multiple well-separated spin-polarized Kagome bands, namely Dirac cone bands and Chern flat bands, near the Fermi level. Using tight-binding modelling, we reveal that these bands are originated from two effects: the low-lying molecular orbitals that exhibit p-orbital characteristics and the honeycomb-Kagome lattice. This study demonstrates that p-orbital Kagome bands can be realized in metal-organic frameworks by using molecules with molecular orbitals of p-orbital like symmetry.

Topological Insulator Metamaterials

Confinement of electromagnetic fields at the subwavelength scale via metamaterial paradigms is an established method to engineer light-matter interaction in most common material systems, from insulators to semiconductors and from metals to superconductors. In recent years, this approach has been extended to the realm of topological materials, providing a new avenue to access nontrivial features of their electronic band structure. In this review, we survey various topological material classes from a photonics standpoint, including crystal growth and lithographic structuring methods. We discuss how exotic electronic features such as spin-selective Dirac plasmon polaritons in topological insulators or hyperbolic plasmon polaritons in Weyl semimetals may give rise to unconventional magneto-optic, nonlinear, and circular photogalvanic effects in metamaterials across the visible to infrared spectrum. Finally, we dwell on how these effects may be dynamically controlled by applying external perturbations in the form of electric and magnetic fields or ultrafast optical pulses. Through these examples and future perspectives, we argue that topological insulator, semimetal and superconductor metamaterials are unique systems to bridge the missing links between nanophotonic, electronic, and spintronic technologies.

Recent advances in n-type and ambipolar organic semiconductors and their multi-functional applications

Organic semiconductors have received broad attention and research interest due to their unique integration of semiconducting properties with structural tunability, intrinsic flexibiltiy and low cost. In order to meet the requirements of organic electronic devices and their integrated circuits, p-type, n-type and ambipolar organic semiconductors are all necessary. However, due to the limitation in both material synthesis and device fabrication, the development of n-type and ambipolar materials is quite behind that of p-type materials. Recent development in synthetic methods of organic semiconductors greatly enriches the range of n-type and ambipolar materials. Moreover, the newly developed materials with multiple functions also put forward multi-functional device applications, including some emerging research areas. In this review, we give a timely summary on these impressive advances in n-type and ambipolar organic semiconductors with a special focus on their synthesis methods and advanced materials with enhanced properties of charge carrier mobility, integration of high mobility and strong emission and thermoelectric properties. Finally, multi-functional device applications are further demonstrated as an example of these developed n-type and ambipolar materials.

The Promise of Soft-Matter-Enabled Quantum Materials

The field of quantum materials has experienced rapid growth over the past decade, driven by exciting new discoveries with immense transformative potential. Traditional synthetic methods to quantum materials have, however, limited the exploration of architectural control beyond the atomic scale. By contrast, soft matter self-assembly can be used to tailor material structure over a large range of length scales, with a vast array of possible form factors, promising emerging quantum material properties at the mesoscale. This review explores opportunities for soft matter science to impact the synthesis of quantum materials with advanced properties. Existing work at the interface of these two fields is highlighted, and perspectives are provided on possible future directions by discussing the potential benefits and challenges which can arise from their bridging.

Intrinsic Second-Order Topological Insulator in Two-Dimensional Covalent Organic Frameworks

As an intriguing topological phase, higher-order topological insulators have attracted tremendous attention, but the candidate materials are limited in artificial and inorganic systems. In this work, we propose a universal approach to search for two-dimensional (2D) second-order topological insulators (SOTIs) in covalent organic frameworks (COFs) with C₃ symmetric cores. The underlying mechanism is illustrated through tight-binding calculations in a star lattice, showing the 2D SOTI in an overlooked energy window between two Kagome-bands with four types of nontrivial band structures. The emergence of the unique topological edge and corner states can be understood from the Su-Schrieffer-Heeger model. Furthermore, using the frontier orbital of the monomer building block as an indicator, the 2D SOTI is directly confirmed in three realistic COFs by first-principles calculations. Our results not only extend the concept of organic topological insulators from first-order to second-order but also demonstrate the universal existence of intrinsic higher-order topology in 2D COFs.

Quantum anomalous Hall effect in M2X3honeycomb Kagome lattice

The quantum anomalous Hall (QAH) effect has recently drawn great attention in spintronics with extraordinary property of chiral edge states without dissipation in absence of magnetic field. In M₂X₃honeycomb Kagome lattice, numerous two-dimensional materials are predicted to be QAH insulators including metal oxides/sulfides and metal organic lattice. In this work, we proposed a general model to explain the mechanism of Dirac half metal with absence of spin orbital coupling and the nontrivial topological property with spin orbital coupling, which could be induced by combination of electron counting rule, crystal field effect anddxz,dyzorbitals hybridization. Based on the mechanism, we further predict that triphenyl-metal lattice M₂(C₆H₄)₃(M= V, Nb, Ta) are all QAH insulators with high Curie temperature and large nontrivial band gap for triphenyl-Nb and triphenyl-Ta lattice.

On-Surface Synthesis toward Two-Dimensional Polymers

Two-dimensional (2D) polymers have garnered widespread interest because of their intriguing physicochemical properties. Envisaged applications in fields including nanodevices, solid-state chemistry, physical organic chemistry, and condensed matter physics, however, demand high-quality and large-scale production. In this perspective, we first introduce exotic band structures of organic frameworks holding honeycomb, kagome, and Lieb lattices. We further discuss how mesoscale ordered 2D polymers can be synthesized by means of choosing suitable monomers and optimizing growth conditions. We describe successful polymerization strategies to introducing a non-benzenoid subunit into a π-conjugated carbon lattice via delicately designed monomer precursors. Also, to obviate transfer and restore the intrinsic properties of π-conjugated polymers, new paradigms of aryl-aryl coupling on inert surfaces are discussed. Recent achievements in the photopolymerization demonstrate the need for monomer design. We conclude the potential applications of these organic networks and project the future possibilities in providing new insights into on-surface polymerization.

3D-Printed Möbius Microring Lasers: Topology Engineering in Photonic Microstructures

Manipulating photons in artificially structured materials is highly desired in modern photonic technology. Nontrivial topological structures are rapidly emerging as a state-of-art platform for achieving unprecedented fascinating phenomena of photon manipulation. However, the current studies mainly focus on planar structures, and the fabrication of photonic microstructures with specific topological geometric features still remains a great challenge. Extending the topological photonics to 3D microarchitectures is expected to enrich the photon manipulation capabilities and further advance the topological photonic devices. Here, a femtosecond laser direct writing technique is employed to fabricate 3D topological Möbius microring resonators from dye-doped polymer. The high-quality-factor Möbius microring resonator supports a unique spin-orbit coupled lasing at very low threshold. Due to the spin-orbit coupling induced geometric/Berry phase, the Möbius microrings, in striking contrast with ordinary microrings, output laser signals with all polarization states. The manipulation of miniaturized coherent light sources in the fabricated Möbius microrings represents a significant step forward toward 3D topological photonics that offers a novel design philosophy for functional photonic and optoelectronic devices.

Rashba Splitting and Electronic Valley Characteristics of Janus Sb and Bi Topological Monolayers

Janus Sb and Bi monolayers as a new class of 2D topological insulator materials, which could be fulfilled by asymmetrical functionalizations with methyl or hydroxyl, are demonstrated by first-principles spin–orbit coupling (SOC) electronic structure calculations to conflate nontrivial topology, Rashba splitting and valley-contrast circular dichroism. Cohesive energies and phonon frequency dispersion spectra indicate that all Janus Sb and Bi monolayers possess a structural stability in energetic statics but represent virtual acoustic phonon vibrations of the hydrogen atoms passivating on monolayer surfaces. Band structures of Janus Sb and Bi monolayers and their nanoribbons demonstrate they are nontrivial topological insulators. Rashba spin splitting at G point in Brillouin zone of Janus Bi monolayers arises from the strong SOC p_x and p_y orbitals of Bi bonding atoms together with the internal out-of-plane electric field caused by asymmetrical functionalization. Janus Sb and Bi monolayers render direct and indirect giant bandgaps, respectively, which are derived from the strong SOC p_x and p_y orbitals at band-valley Brillouin points K and K′ where valley-selective circular dichroism of spin valley Hall insulators is also exhibited.

Synthesis of Single-Layer Two-Dimensional Metal-Organic Frameworks M3 (HAT)2 (M=Ni, Fe, Co, HAT=1,4,5,8,9,12-hexaazatriphenylene) Using an On-Surface Reaction

1,4,5,8,9,12-Hexaazatriphenylene (HAT) is one of the smallest polyheterocyclic aromatic building blocks for forming conjugated metal-organic frameworks (cMOFs). However, the strong inter-molecular steric hindrance impedes the growth of HAT-based cMOFs. Here we employ on-surface synthesis to grow single-layer two-dimensional cMOFs of M₃ (HAT)₂ (M=Ni, Fe, Co). Using scanning tunnelling microscopy and density-functional theory (DFT) analysis, we resolve that the frameworks comprise a hexagonal lattice of HAT molecules and a Kagome lattice of metal atoms. The DFT analysis indicates that Ni, Co and Fe carry a magnetic moment of 1.1, 2.5, and 3.7 μB, respectively. We anticipate that the small π-conjugated core of HAT and strong bidentate chelating coordination give rise to appealing electronic and magnetic properties.

Topological band transition between hexagonal and triangular lattices with (px,py) orbitals

By combining tight-binding modelling with density functional theory based first-principles calculations, we investigate the band evolution of two-dimensional (2D) hexagonal lattices with (p_x,p_y) orbitals, focusing on the electronic structures and topological phase transitions. The (p_x,p_y)-orbital hexagonal lattice model possesses two flat bands encompassing two linearly dispersive Dirac bands. Breaking the A/B sublattice symmetry could transform the model into two triangular lattices, each featuring a flat band and a dispersive band. Inclusion of the spin-orbit coupling and magnetization may give rise to quantum spin Hall and quantum anomalous Hall (QAH) states. As a proof of concept, we demonstrate that half-hydrogenated stanene is encoded by a triangular lattice with (p_x,p_y) orbitals, which exhibits ferromagnetism and QAH effect with a topological gap of ∼0.15 eV, feasible for experimental observation. These results provide insights into the structure-property relationships involving the orbital degree of freedom, which may shed light on future design and preparation of 2D topological materials for novel electronic/spintronic and quantum computing devices.

Tunable Dirac states in doped B2S3 monolayers

Two-dimensional (2D) Dirac materials have been a research hotspot due to their intriguing properties, such as high carrier mobility and ballistic charge transport. Here, we demonstrate that the B₂S₃ monolayer with a hexagonal structure, which has been reported as a photocatalyst, can be tuned to new 2D Dirac materials by doping atoms. The Young's modulus can reach 65.23 N m^-1, indicating that the monolayer can be used as a buffer materials. The electronic structures of the pristine B₂S₃ monolayer show that some Dirac points appear but do not occur exactly on the Fermi level (E_F). Fortunately, we find that the Dirac cone can be tuned to the E_F by doping C, N, or Sn atoms. The C-doped B₂S₃ monolayer can be a half-metallic Dirac material, which has significant potential application in spintronics. For N- and Sn-doped B₂S₃ monolayers, the typical kagome bands are formed near the E_F, which arise from three molecular orbitals hybridized by B, S, and N (Sn) atoms. These outstanding properties render the doped B₂S₃ monolayers promising 2D Dirac materials for future nanoelectronic devices.

On-Surface Synthesis: A New Route Realizing Single-Layer Conjugated Metal-Organic Structures

Recently, both experimental and theoretical advances have demonstrated that two-dimensional conjugated metal-organic frameworks (2D-cMOFs) exhibit interesting electronic and magnetic properties, such as high conductivity and ferromagnetism. Theoretical studies have predicted that exotic quantum states, including topological insulating states and superconductivity, emerge in some 2D-MOFs. The high design tunability of MOFs' structure and composition provides great opportunities to realize these structures. However, most conventional synthesis methods yield multilayer structures of the 2D-cMOFs, in which the predicted exotic quantum phases are often quenched because of interlayer interactions. It is highly desirable to synthesize single-layer cMOFs. On-surface synthesis represents a novel strategy toward this goal. In this Perspective, we discuss the recent developments in on-surface synthesis of 1D- and 2D-cMOFs.

Trigonal multivalent polonium monolayers with intrinsic quantum spin Hall effects

Two-dimensional (2D) topological insulators, a type of the extraordinary quantum electronic states, have attracted considerable interest due to their unique electronic properties and promising potential applications. Recently, the successful fabrication of 2D Te monolayers (i.e. tellurene) in experiments (Zhu et al., Phys Rev Lett 119:106101, 2017) has promoted researches on the group-VI monolayer materials. With first-principles calculations and tight-binding (TB) method, we investigate the structures and electronic states of 2D polonium (poloniumene), in which Po is a congener of Te. The poloniumene is found to have the tendency of forming a three-atomic-layer 1 T-MoS₂-like structure (called trigonal poloniumene), namely, the central-layer Po atoms behave metal-like, while the two-outer-layer Po atoms are semiconductor-like. This unique multivalent behavior of the Po atoms is conducive to the structural stability of the monolayer, which is found to be an intrinsic quantum spin Hall insulator with a large band gap. The nontrivial topology originates from the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{x,y} - p_{z}$$\end{document}px,y-pz band inversion, which can be understood based on a built TB model. The poloniumene with different congener elements doped is also explored. Our results provide a thorough understanding of structures and electronic states of 2D polonium-related materials.

Robust Dirac spin gapless semiconductors in a two-dimensional oxalate based organic honeycomb-kagome lattice

Two-dimensional (2D) ferromagnetic materials with intrinsic and robust spin-polarized Dirac cones are of great interest in exploring exciting physics and in realizing spintronic devices. Using comprehensive ab initio calculations, herein we reveal a family of 2D oxalate-based metal-organic frameworks (MOFs) that possess the desired characteristics. We propose that these 2D oxalate-based MOFs may be assembled by oxalate ions (C₂O₄^2-) and two homo-transition metal atoms. We demonstrate that 2D MOFs of Ni₂(C₂O₄)₃ and Re₂(C₂O₄)₃ are intrinsic Dirac spin gapless semiconductors with linear band dispersion, low energy dissipation and high electron carrier velocity. As robust ferromagnets, they also possess large magnetic moments, large perpendicular magnetic anisotropy, and high Curie temperatures, e.g. 208 K for Ni₂(C₂O₄)₃. In particular, spin-orbit coupling triggers a topologically nontrivial band gap of 143 meV in Re₂(C₂O₄)₃, which is promising to realize the quantum anomalous Hall effect at high temperatures.

Epitaxial Growth of Ultraflat Bismuthene with Large Topological Band Inversion Enabled by Substrate-Orbital-Filtering Effect

Quantum spin Hall (QSH) systems hold promises of low-power-consuming spintronic devices, yet their practical applications are extremely impeded by the small energy gaps. Fabricating QSH materials with large gaps, especially under the guidance of design principles, is essential for both scientific research and practical applications. Here, we demonstrate that large on-site atomic spin-orbit coupling can be directly exploited via the intriguing substrate-orbital-filtering effect to generate large-gap QSH systems and experimentally realized on the epitaxially synthesized ultraflat bismuthene on Ag(111). Theoretical calculations reveal that the underlying substrate selectively filters Bi p_z orbitals away from the Fermi level, leading p_xy orbitals with nonzero magnetic quantum numbers, resulting in large topological gap of ∼1 eV at the K point. The corresponding topological edge states are identified through scanning tunneling spectroscopy combined with density functional theory calculations. Our findings provide general strategies to design large-gap QSH systems and further explore their topology-related physics.

Resistive Memory Devices Based on Reticular Materials for Electrical Information Storage

Recently, reticular materials, such as metal-organic frameworks and covalent organic frameworks, have been proposed as an active insulating layer in resistive switching memory systems through their chemically tunable porous structure. A resistive random access memory (RRAM) cell, a digital memristor, is one of the most outstanding emergent memory devices that achieves high-density electrical information storage with variable electrical resistance states between two terminals. The overall design of the RRAM devices comprises an insulating layer sandwiched between two metal electrodes (metal/insulator/metal). RRAM devices with fast switching speeds and enhanced storage density have the potential to be manufactured with excellent scalability owing to their relatively simple device architecture. In this review, recent progress on the development of reticular material-based RRAM devices and the study of their operational mechanisms are reviewed, and new challenges and future perspectives related to reticular material-based RRAM are discussed.

Steering Metal-Organic Network Structures through Conformations and Configurations on Surfaces

Molecular adsorption conformations and arrangement configurations on surfaces are important structural aspects of surface stereochemistry, but their roles in steering the structures of metal-organic networks (MONs) remain vague and unexplored. In this study, we constructed MONs by the coordination self-assembly of isocyanides on Cu(111) and Ag(111) surfaces and demonstrated that the MON structures can be steered by surface stereochemistry, including the adsorption conformations of the isocyanide molecules and the arrangement configurations of the coordination nodes and subunits. The coordination self-assembly of 1,4-phenylene diisocyanobenzene afforded a honeycomb MON consisting of 3-fold (isocyano)₃-Cu motifs on a Cu(111) surface. In contrast, geometrically different chevron-shaped 1,3-phenylene diisocyanobenzene (m-DICB) failed to generate a MON, which is ascribable to its standing conformation on the Cu(111) surface. However, m-DICB was adsorbed in a flat conformation on a Ag(111) surface, which has a larger lattice constant than a Cu(111) surface, and smoothly underwent coordination self-assembly to form a MON consisting of (isocyano)₃-Ag motifs. Interestingly, only C₃-Ag nodes with heterotactic configurations could grow into larger subunits; those subunits with heterotactic configurations further grew into Sierpiński triangle fractals (up to fourth order), while subunits with homotactic configurations afforded a triangular MON.

Simple lattice model of self-assembling metal-organic layers of pyridyl-substituted porphyrins and copper on Au(111) surface

A simple lattice model of metal-organic adsorption layers self-assembling on a Au(111) surface and based on pyridyl-substituted porphyrins differing in the number of functional groups and their position has been proposed. The model has been parameterized using DFT methods. The ground state analysis of the considered model demonstrates the variety of surface-confined metal-organic networks (SMONs) containing square, linear, and discrete elements appearing in the adsorption layer depending on the partial pressure of the components. The SMONs comprising more symmetrical molecules with a greater number of pyridyl substituents in the porphyrin core exhibit more diverse phase behavior. Structures of the phase diagrams were verified at nonzero temperatures using Grand Canonical Monte Carlo simulations. It was found that the continuous SMONs have higher thermal stability at relatively low partial pressures of the organic component, while the linear and discrete SMONs are more thermally stable at high pressure. Depending on the partial pressure of the organic component, thermal destruction of continuous SMONs occur either through the formation of defects/islands having structures of the linear SMONs, or through the sublimation of individual structural elements. Melting of linear SMONs reveals the appearance of 2D pores or islands of a purely organic phase. The latter fact is confirmed by the experimentally observed coexistence of these phases.

Designing Intrinsic Topological Insulators in Two-Dimensional Metal-Organic Frameworks

The connection between electronic structures of metal-organic frameworks (MOFs) and their building subunits is a key cornerstone for rational MOF material design. Some two-dimensional conjugated MOFs were reported to be topological insulators. However, many of them are not intrinsic as the Fermi levels are far from the topological gaps. The subunit-to-MOF electronic orbital correspondence should be established to bridge their chemical structure and physical properties, thus understanding the design rules toward intrinsic topological insulators. Herein we reveal the fundamental role of the subunit-to-MOF symmetry relation in determining their orbital interaction and hybridization and, consequently, topological characteristics. In particular, such honeycomb-kagome MOFs possess delocalized symmetry-enforced nonbonding electronic states with the topological spin-orbit gap. The nonbonding nature of these states allows tailored band structure modulation through molecular structure and strain engineering, with the potential realization of an intrinsic metal-organic topological insulator.

Understanding Topological Insulators in Real Space

A real space understanding of the Su–Schrieffer–Heeger model of polyacetylene is introduced thanks to delocalization indices defined within the quantum theory of atoms in molecules. This approach enables to go beyond the analysis of electron localization usually enabled by topological insulator indices—such as IPR—enabling to differentiate between trivial and topological insulator phases. The approach is based on analyzing the electron delocalization between second neighbors, thus highlighting the relevance of the sublattices induced by chiral symmetry. Moreover, the second neighbor delocalization index, δi,i+2, also enables to identify the presence of chirality and when it is broken by doping or by eliminating atom pairs (as in the case of odd number of atoms chains). Hints to identify bulk behavior thanks to δ1,3 are also provided. Overall, we present a very simple, orbital invariant visualization tool that should help the analysis of chirality (independently of the crystallinity of the system) as well as spreading the concepts of topological behavior thanks to its relationship with well-known chemical concepts.

Searching for kagome multi-bands and edge states in a predicted organic topological insulator

Recently, mixed honeycomb-kagome lattices featuring metal-organic networks have been theoretically proposed as topological insulator materials capable of hosting nontrivial edge states. This new family of so-called "organic topological insulators" are purely two-dimensional and combine polyaromatic-flat molecules with metal adatoms. However, their experimental validation is still pending given the generalized absence of edge states. Here, we generate one such proposed network on a Cu(111) substrate and study its morphology and electronic structure with the purpose of confirming its topological properties. The structural techniques reveal a practically flawless network that results in a kagome network multi-band observed by angle-resolved photoemission spectroscopy and scanning tunneling spectroscopy. However, at the network island borders we notice the absence of edge states. Bond-resolved imaging of the network exhibits an unexpected structural symmetry alteration that explains such disappearance. This collective lifting of the network symmetry could be more general than initially expected and provide a simple explanation for the recurrent experimental absence of edge states in predicted organic topological insulators.

Moiré-Potential-Induced Band Structure Engineering in Graphene and Silicene

A moiré pattern results from the projection of one periodic pattern to another with relative lattice constant or misalignment and provides great periodic potential to modify the electronic properties of pristine materials. In this Review, recent research on the effect of the moiré superlattice on the electronic structures of graphene and silicene, both of which possess a honeycomb lattice, is focused on. The moiré periodic potential is introduced by the interlayer interaction to realize abundant phenomena, including new generation of Dirac cones, emergence of Van Hove singularities (vHs) at the cross point of two sets of Dirac cones, Mott-like insulating behavior at half-filling state, unconventional superconductivity, and electronic Kagome lattice and flat band with nontrivial edge state. The role of interlayer coupling strength, which is determined by twist angle and buckling degree, in these exotic properties is discussed in terms of both the theoretical prediction and experimental measurement, and finally, the challenges and outlook for this field are discussed.

Half-Dirac semimetals and the quantum anomalous Hall effect in Kagome Cd2N3 lattices

Half-Dirac semimetals (HDSs), which possess 100% spin-polarizations for Dirac materials, are highly desirable for exploring various topological phases of matter as low-dimensionality opens unprecedented opportunities for manipulating the quantum state of low-cost electronic nanodevices. The search for high-temperature HDSs is still a current hotspot and yet challenging experimentally. Herein based on first-principles calculations, we propose the realization of Half Dirac semimetals (HDS) in two-dimensional (2D) Kagome transition-metal nitride Cd₂N₃, which is robust against strain engineering. Monte Carlo simulations reveal that Cd₂N₃ possesses a Curie temperature reaching up to T _C = 225 K, which is much higher than that of the reported monolayers CrI₃ (T _C = 45 K) and Cr₂Ge₂Te₆ (T _C = 20 K). The band crossings in Cd₂N₃ are gapped out by the spin-orbit coupling, which brings about the quantum anomalous Hall (QAH) effect with a sizeable band gap of E _g = 4.9 meV, characterized by the nonzero Chern number (C = 1) and chiral edge states. A tight-binding model is further used to clarify the origin of HDSs and nontrivial electronic properties. The results suggest monolayer transition-metal nitrides as a promising platform to explore fascinating physical phenomena associated with novel 2D emergent HDSs and QAH insulators toward realistic spintronics devices, thus stimulating experimental interest.

Functionalization induced quantum spin Hall to quantum anomalous Hall phase transition in monolayer jacutingaite

As novel states of quantum matter, quantum spin Hall (QSH) and quantum anomalous Hall (QAH) states have attracted considerable interest in condensed matter and materials science communities. Recently, a monolayer of the naturally occurring mineral jacutingaite (Pt2HgSe3), was theoretically proposed to be a large-gap QSH insulator and experimentally confirmed. Here, based on first-principles calculations and tight-binding modeling, we demonstrate QSH to QAH phase transition in jacutingaite by chemical functionalization with chalogen. We show that two-dimensional (2D) chalogenated jacutingaite, Pt2HgSe3-X (X = S, Se, Te), is ferromagnetic with Curie temperature up to 316 K, and it exhibits QAH effect with chiral edge states inside a sizeable topological gap. The physical mechanism lies in the adsorption induced transformation of the original Kane-Mele model into an effective four-band model involving (px, py) orbitals on a hexagonal lattice, so that the topological gap size can be controlled by spin-orbit coupling strength of the chalogen (0.28 eV for Pt2HgSe3-Te). These results not only show the promise of functionalization in orbital-engineering of 2D functional structures, but also provide an ideal and practical platform for achieving exotic topological phases for dissipationless transport and quantum computing.

Two dimensional honeycomb-kagome Be3Pb2: a mechanically flexible topological insulator with high intrinsic carrier mobilities

We theoretically predict a stable 2D nanosheet consisting of Pb and Be atoms sited at the honeycomb and kagome sites, respectively, forming a mixed honeycomb-kagome phase of Be3Pb2. Without the spin-orbit interaction, its band structure resembles that of honeycomb-structured graphene, namely, the valence and conduction bands touch at isolated points, whose energies linearly depend on the momentum. The presence of spin-orbit coupling (SOC), however, would result in a small bandgap opening, ∼116 meV. So, the SOC induces an electronic phase transition from a semimetal to a semiconductor. A coarse estimation based on the deformation potential method gives rise to very high carrier mobilities which are at least comparable to those of black phosphorene. Most interestingly, the 2D Be3Pb2 shows a non-trivial topology in the electronic structure accompanying the SOC induced band gap opening. Hence, 2D Be3Pb2 would be a versatile candidate for many applications, e.g., nanoelectronic 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.

Metal-organic frameworks: possible new two-dimensional magnetic and topological materials

Finding new two-dimensional (2D) materials with novel quantum properties is highly desirable for technological innovations. In this work, we studied a series of metal-organic frameworks (MOFs) with different metal cores and discovered various attractive properties, such as room-temperature magnetic ordering, strong perpendicular magnetic anisotropy, huge topological band gap (>200 meV), and excellent spin-filtering performance. As many MOFs have been successfully synthesized in experiments, our results suggest realistic new 2D functional materials for the design of spintronic nanodevices.

Isolated flat bands and physics of mixed dimensions in a 2D covalent organic framework

We demonstrate that it is possible to rationally incorporate both an isolated flat band and the physics of zero dimensions (0D), one dimension (1D), and two dimensions (2D) in a single 2D material. Such unique electronic properties are present in a recently synthesized 2D covalent organic framework (COF), where "I"-shaped building blocks and "T"-shaped connectors result in quasi-1D chains that are linked by quasi-0D bridge units arranged in a stable 2D lattice. The lowest unoccupied conduction band is an isolated flat band, and electron-doping gives rise to novel quantum phenomena, such as magnetism and Mott insulating phases. The highest occupied valence band arises from wave functions in the quasi-1D chains. Examples of mixed dimensional physics are illustrated in this system. The strong electron-hole asymmetry in this material results in a large Seebeck coefficient, while the quasi-1D nature of the chains leads to linear dichroism, in conjunction with strongly bound 2D excitons. We elucidate strategies to design and optimize 2D COFs to host both isolated flat bands and quantum-confined 1D subsystems. The properties of the 2D COF discussed here provide a taste of the intriguing possibilities in this open research field.

Non-Collinear Orbital-induced Planar Quantum Anomalous Hall Effect

Based on the 2D star lattice with noncollinear orbital, we present an intriguing six-band tight-binding model for planar quantum anomalous Hall effect (PQAHE) with in-plane magnetization. Depending on the orientation of the noncollinear orbital, the characterized Dirac-, Kagome-, and Four-band are all realizable. Moreover, including the intrinsic spin-orbital coupling, a topological phase diagram versus angle of both in-plane magnetization and noncollinear orbital is mapped out, showing a tunable PQAHE with Chern number of either +1 or -1. Most remarkably, using first-principles calculations, this exact model of PQAHE is identified in an experimentally synthesized 2D metal-organic framework (MOF) of Pr₂(C₆O₄Cl₂)₃. These notable results not only demonstrate the significance of noncollinear orbital for designing PQAHE but also provide a MOF platform for its realization, greatly enriching the fundamental topological physics and extending the candidate topological materials.

Magnetism, half-metallicity, and topological signatures in Fe2-xVxPO5 (x = 0, 0.5, 1, 1.5, 2) materials: a potential class of advanced spintronic materials

Novel spintronic materials combining both magnetism and nontrivial topological electronic structures have attracted increasing attention recently. Here, we systematically studied the doping effects, magnetism, half-metallicity, and topological properties in the family of Fe_2-xV_xPO₅ (x = 0, 0.5, 1, 1.5, 2) compounds. Our results show that Fe₂PO₅ takes an antiferromagnetic (AFM) ordering with a zero total magnetic moment. Meanwhile, the material hosts a Dirac nodal line and a Weyl nodal line near the Fermi level. V₂PO₅ is a ferromagnetic (FM) nodal line half-metal with a 100% spin-polarized Weyl nodal line. After doping, we find that Fe_1.5V_0.5PO₅, Fe₁V₁PO₅ and Fe_0.5V_1.5PO₅ all take ferrimagnetic (FiM) ordering, with the Fe and V atoms taking opposite spin directions. Both Fe_1.5V_0.5PO₅ and Fe_0.5V_1.5PO₅ are FiM half-metals. Meanwhile, they show several pairs of fully spin-polarized Weyl points near the Fermi level. Fe₁V₁PO₅ is a FiM semiconductor with different sizes of band gaps in different spin channels. These Fe_2-xV_xPO₅ materials not only provide a good research platform to study the novel properties combining magnetism and nontrivial band topology, but also have promising applications in spintronic applications.

Two-Dimensional Dirac Semimetals without Inversion Symmetry

Realizing stable two-dimensional (2D) Dirac points against spin-orbit coupling (SOC) has attracted much attention because it provides a platform to study the unique transport properties. In previous work, Young and Kane [Phys. Rev. Lett. 115, 126803 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.126803 proposed stable 2D Dirac points with SOC, in which the Berry curvature and edge states vanish due to the coexistence of inversion and time-reversal symmetries. Herein, using the tight-binding model and k·p effective Hamiltonian, we present that 2D Dirac points can survive in the presence of SOC without inversion symmetry. Such 2D Dirac semimetals possess nonzero Berry curvature near the crossing nodes, and two edge states are terminated at one pair of Dirac points. In addition, according to symmetry arguments and high-throughput first-principles calculations, we identify a family of ideal 2D Dirac semimetals, which has nonzero Berry curvature in the vicinity of Dirac points and visible edge states, thus facilitating the experimental observations. Our work shows that 2D Dirac points can emerge without inversion symmetry, which not only enriches the classification of 2D topological semimetals but also provides a promising avenue to observe exotic transport phenomena beyond graphene, e.g., nonlinear Hall effect.

Electronic Structure Modeling of Metal-Organic Frameworks

Owing to their molecular building blocks, yet highly crystalline nature, metal-organic frameworks (MOFs) sit at the interface between molecule and material. Their diverse structures and compositions enable them to be useful materials as catalysts in heterogeneous reactions, electrical conductors in energy storage and transfer applications, chromophores in photoenabled chemical transformations, and beyond. In all cases, density functional theory (DFT) and higher-level methods for electronic structure determination provide valuable quantitative information about the electronic properties that underpin the functions of these frameworks. However, there are only two general modeling approaches in conventional electronic structure software packages: those that treat materials as extended, periodic solids, and those that treat materials as discrete molecules. Each approach has features and benefits; both have been widely employed to understand the emergent chemistry that arises from the formation of the metal-organic interface. This Review canvases these approaches to date, with emphasis placed on the application of electronic structure theory to explore reactivity and electron transfer using periodic, molecular, and embedded models. This includes (i) computational chemistry considerations such as how functional, k-grid, and other model variables are selected to enable insights into MOF properties, (ii) extended solid models that treat MOFs as materials rather than molecules, (iii) the mechanics of cluster extraction and subsequent chemistry enabled by these molecular models, (iv) catalytic studies using both solids and clusters thereof, and (v) embedded, mixed-method approaches, which simulate a fraction of the material using one level of theory and the remainder of the material using another dissimilar theoretical implementation.

Organic topological insulators (OTI): a dream coming true?

Experimental discovery of organic topological insulators (OTI) is a dream for both topological matters and organic materials. Despite great challenges, we anticipate that the dream will become a reality by engineered studies on materials chemistry, characterization techniques and device physics of conjugated molecules.

Orbital design of topological insulators from two-dimensional semiconductors

Two-dimensional (2D) materials have attracted much attention because they exhibit various intrinsic properties, which are, however, usually not interchangeable. Here we propose a generic approach to convert 2D semiconductors to 2D topological insulators (TIs) via atomic adsorption. The approach is underlined by an orbital design principle that involves introducing an extrinsic s-orbital state of the adsorbate into the intrinsic sp-bands of a 2D semiconductor, so as to induce s-p band inversion for a TI phase, as demonstrated by tight-binding model analyses. Based on first-principles calculations, we successfully apply this approach to convert CuS, CuSe and CuTe into TIs by adsorbing one adatom per unit cell of Na, Na_0.5K_0.5 and K as well as Rb and Cs. Moreover, if the chalcogens in the 2D semiconductor have a decreasing ability of accepting electrons, the adsorbates should have an increasing ability of donating electrons. Our findings open a new door to discovering TIs by predictive material design beyond finding preexisting TIs.