A bird's eye view on the flat and conic band world of the honeycomb and Kagome lattices: towards an understanding of 2D metal-organic frameworks electronic structure

Recent progress in on-surface synthesis of nanoporous graphene materials

Nanoporous graphene (NPG) materials are generated by removing internal degree-3 vertices from graphene and introducing nanopores with specific topological structures, which have been widely explored and exploited for applications in electronic devices, membranes, and energy storage. The inherent properties of NPGs, such as the band structures, field effect mobilities and topological properties, are crucially determined by the geometric structure of nanopores. On-surface synthesis is an emerging strategy to fabricate low-dimensional carbon nanostructures with atomic precision. In this review, we introduce the progress of on-surface synthesis of atomically precise NPGs, and classify NPGs from the aspects of element types, topological structures, pore shapes, and synthesis strategies. We aim to provide a comprehensive overview of the recent advancements, promoting interdisciplinary collaboration to further advance the synthesis and applications of NPGs.

2D Co-Directed Metal-Organic Networks Featuring Strong Antiferromagnetism and Perpendicular Anisotropy

Antiferromagnetic spintronics is a rapidly emerging field with the potential to revolutionize the way information is stored and processed. One of the key challenges in this field is the development of novel 2D antiferromagnetic materials. In this paper, the first on-surface synthesis of a Co-directed metal-organic network is reported in which the Co atoms are strongly antiferromagnetically coupled, while featuring a perpendicular magnetic anisotropy. This material is a promising candidate for future antiferromagnetic spintronic devices, as it combines the advantages of 2D and metal-organic chemistry with strong antiferromagnetic order and perpendicular magnetic anisotropy.

Diboraperylene Diborinic Acid Self-Assembly on Ag(111)-Kagome Flat Band Localized States Imaged by Scanning Tunneling Microscopy and Spectroscopy

Replacement of sp2-hybridized carbon in polycyclic aromatic hydrocarbons (PAHs) by boron affords electron-deficient π-scaffolds due to the vacant pz-orbital of three-coordinate boron with the potential for pronounced electronic interactions with electron-rich metal surfaces. Using a diboraperylene diborinic acid derivative as precursor and a controlled on-surface non-covalent synthesis approach, we report on a self-assembled chiral supramolecular kagome network on an Ag(111) surface stabilized by intermolecular hydrogen-bonding interactions at low temperature. Scanning tunneling microscopy (STM) and spectroscopy (STS) reveal a flat band at ca. 0.33 eV above the Fermi level which is localized at the molecule center, in good agreement with tight-binding model calculations of flat bands characteristic for kagome lattices.

Understanding the Electronic Structure of Magnetic Trinuclear Complexes Based on the Tris-Dioxolene Triphenylene Non-Innocent Bridging Ligand, a Theoretical Study

A complete theoretical analysis using first the simple Hückel model followed by more sophisticated multi-reference calculations on a trinuclear Ni(II) complex (Tp#Ni3 HHTP), bearing the non-innocent bridging ligand HHTP3- , is carried out. The three semiquinone moieties of HHTP3- couple antiferromagnetically and lead to a single unpaired electron localized on one of the moieties. The calculated exchange coupling integrals together with the zero-field parameters allow, when varied within a certain range, reproducing the experimental data. These results are generalized for two similar other trinuclear complexes containing Ni(II) and Cu(II). The electronic structure of HHTP3- turns out to be independent of both the chemical nature and the geometry of the metal ions. We also establish a direct correlation between the geometrical and the electronic structures of the non-innocent ligand that is consistent with the results of calculations. It allows experimentalists to get insight into the magnetic behavior of this type of complexes by an analysis of their X-ray structure.

Designer artificial chiral kagome lattice with tunable flat bands and topological boundary states

The kagome lattice is a well-known model system for the investigation of strong correlation and topological electronic phenomena due to the intrinsic flat band, magnetic frustration, etc. Introducing chirality into the kagome lattice would bring about new physics due to the unique symmetry, which is still yet to be fully explored. Here we report the investigation on a two-dimensional chiral kagome lattice utilizing tight binding band calculation and topological index analysis. It is found that the periodic chiral kagome lattice would bring about a robust zero-energy flat band. Furthermore, in the Su-Schrieffer-Heeger type dimer-/trimerized breathing chiral kagome lattice with particular edge terminations, topological corner states or metallic edge states would appear, implying new candidates for the second-order topological insulator. We also proposed the construction strategy for such lattices employing the scanning tunneling microscope atom manipulation technique.

Coupled topological flat and wide bands: Quasiparticle formation and destruction

Flat bands amplify correlation effects and are of extensive current interest. They provide a platform to explore both topology in correlated settings and correlation physics enriched by topology. Recent experiments in correlated kagome metals have found evidence for strange-metal behavior. A major theoretical challenge is to study the effect of local Coulomb repulsion when the band topology obstructs a real-space description. In a variant to the kagome lattice, we identify an orbital-selective Mott transition in any system of coupled topological flat and wide bands. This was made possible by the construction of exponentially localized and Kramers-doublet Wannier functions, which, in turn, leads to an effective Kondo-lattice description. Our findings show how quasiparticles are formed in such coupled topological flat-wide band systems and, equally important, how they are destroyed. Our work provides a conceptual framework for the understanding of the existing and emerging strange-metal properties in kagome metals and beyond.

Understanding the electronic pi-system of 2D covalent organic frameworks with Wannier functions

We investigate a family of hexagonal 2D covalent organic frameworks (COFs) with phenyl and biphenyl spacer units and different chemical linker species. Chemical trends are elucidated and attributed to microscopic properties of the [Formula: see text]-electron-system spanned by atomic [Formula: see text]-orbitals. We systematically investigate the electronic structure, delocalization of electronic states, effects of disorder, bond torsion, and doping, and correlate these with variable [Formula: see text]-conjugation and nucleus-independent chemical shift (NICS) aromaticity. Molecular orbitals are obtained from maximally localized Wannier functions that have [Formula: see text]- and [Formula: see text]-character, forming distinct [Formula: see text]- and [Formula: see text]-bands for all valence states. The Wannier-orbital description goes beyond simple tight-binding models and enables a detailed understanding of the electronic topology, effective electronic coupling and delocalization. It is shown that a meaningful comparison between COFs with different chemical elements can only be made by examining the entire [Formula: see text]-electron system, while a comparison of individual bands (e.g., bands near the Fermi energy) can be a insufficient to derive general design rules for linker and spacer monomer selection. We further identify delocalized states that are spread across tens or hundreds of pores of the 2D COFs and analyze their robustness against structural and energetic disorders like out-of-plane rotations of molecular fragments, different strength of energetic disorder and energetic shifts due to chemical doping.

Orbital Design of Two-Dimensional Transition-Metal Peroxide Kagome Crystals with Anionogenic Dirac Half-Metallicity

Assembling p orbital ferromagnetic half-metallicity and a topological element, such as a Dirac point at the Fermi level, in a single nanomaterial is of particular interest for long-distance, high-speed, and spin-coherent transportation in nanoscale spintronic devices. On the basis of the tight-binding model, we present an orbital design of a two-dimensional (2D) anionogenic Dirac half-metal (ADHM) by patterning cations with empty d orbitals and anions with partially filled p-type orbitals into a kagome lattice. Our first-principles calculations show that 2D transition-metal peroxides h-TM₂(O₂)₃ (TMO₃, TM = Ti, Zr, Hf), containing group IVB transition-metal cations [TM]⁴⁺ bridged with dioxygen anions [O₂]^8/3- in a kagome structure, are stable ADHMs with a Curie temperature over 103 K. The 2/3 filled π* orbitals of dioxygen anions are ferromagnetically coupled, leading to p orbital ferromagnetism and a half-metallic Dirac point right at the Fermi level with a Fermi velocity reaching 2.84 × 10⁵ m/s. We proposed that 2D h-TM₂(O₂)₃ crystals may be extracted from ABO₃ bulk materials containing 2D TMO₃ layers.

On-Surface Synthesis of Nitrogen-Doped Kagome Graphene

Nitrogen-doped Kagome graphene (N-KG) has been theoretically predicted as a candidate for the emergence of a topological band gap as well as unconventional superconductivity. However, its physical realization still remains very elusive. Here, we report on a substrate-assisted reaction on Ag(111) for the synthesis of two-dimensional graphene sheets possessing a long-range honeycomb Kagome lattice. Low-temperature scanning tunneling microscopy (STM) and atomic force microscopy (AFM) with a CO-terminated tip supported by density functional theory (DFT) are employed to scrutinize the structural and electronic properties of the N-KG down to the atomic scale. We demonstrate its semiconducting character due to the nitrogen doping as well as the emergence of Kagome flat bands near the Fermi level which would open new routes towards the design of graphene-based topological materials.

Hybrid topological guiding mechanisms for photonic crystal fibers

We create hybrid topological-photonic localisation of light by introducing concepts from the field of topological matter to that of photonic crystal fiber arrays. S-polarized obliquely propagating electromagnetic waves are guided by hexagonal, and square, lattice topological systems along an array of infinitely conducting fibers. The theory utilises perfectly periodic arrays that, in frequency space, have gapped Dirac cones producing band gaps demarcated by pronounced valleys locally imbued with a nonzero local topological quantity. These broken symmetry-induced stop-bands allow for localised guidance of electromagnetic edge-waves along the crystal fiber axis. Finite element simulations, complemented by asymptotic techniques, demonstrate the effectiveness of the proposed designs for localising energy in finite arrays in a robust manner.

Synthesis of mesoscale ordered two-dimensional π-conjugated polymers with semiconducting properties

Two-dimensional materials with high charge carrier mobility and tunable band gaps have attracted intense research effort for their potential use in nanoelectronics. Two-dimensional π-conjugated polymers constitute a promising subclass because the band structure can be manipulated by varying the molecular building blocks while preserving key features such as Dirac cones and high charge mobility. The major barriers to the application of two-dimensional π-conjugated polymers have been the small domain size and high defect density attained in the syntheses explored so far. Here, we demonstrate the fabrication of mesoscale ordered two-dimensional π-conjugated polymer kagome lattices with semiconducting properties, Dirac cone structures and flat bands on Au(111). This material has been obtained by combining a rigid azatriangulene precursor and a hot dosing approach, which favours molecular diffusion and eliminates voids in the network. These results open opportunities for the synthesis of two-dimensional π-conjugated polymer Dirac cone materials and their integration into devices.

Designing Kagome Lattice from Potassium Atoms on Phosphorus-Gold Surface Alloy

Materials with flat bands are considered as ideal platforms to explore strongly correlated physics such as the fractional quantum hall effect, high-temperature superconductivity, and more. In theory, a Kagome lattice with only nearest-neighbor hopping can give rise to a flat band. However, the successful fabrication of Kagome lattices is still very limited. Here, we provide a new design principle to construct the Kagome lattice by trapping atoms into Kagome arrays of potential valleys, which can be realized on a potassium-decorated phosphorus-gold surface alloy. Theoretical calculations show that the flat band is less correlated with the neighboring trivial electronic bands, which can be further isolated and dominate around the Fermi energy with increased Kagome lattice parameters of potassium atoms. Our results provide a new strategy for constructing Kagome lattices, which serve as an ideal platform to study topological and more general flat band phenomena.

Transformation between 2D and 3D Covalent Organic Frameworks via Reversible [2 + 2] Cycloaddition

We report the first transformation between crystalline vinylene-linked two-dimensional (2D) polymers and crystalline cyclobutane-linked three-dimensional (3D) polymers. Specifically, absorption-edge irradiation of the 2D poly(arylenevinylene) covalent organic frameworks (COFs) results in topological [2 + 2] cycloaddition cross-linking of the π-stacked layers in 3D COFs. The reaction is reversible, and heating to 200 °C leads to a cycloreversion while retaining the COF crystallinity. The resulting difference in connectivity is manifested in the change of mechanical and electronic properties, including exfoliation, blue-shifted UV-vis absorption, altered luminescence, modified band structure, and different acid-doping behavior. The Li-impregnated 2D and 3D COFs show a significant room-temperature ion conductivity of 1.8 × 10^-4 S/cm and 3.5 × 10^-5 S/cm, respectively. Even higher room-temperature proton conductivity of 1.7 × 10^-2 S/cm and 2.2 × 10^-3 S/cm was found for H₂SO₄-treated 2D and 3D COFs, respectively.

Stable Flatbands, Topology, and Superconductivity of Magic Honeycomb Networks

We propose a new principle to realize flatbands which are robust in real materials, based on a network superstructure of one-dimensional segments. This mechanism is naturally realized in the nearly commensurate charge-density wave of 1T-TaS_{2} with the honeycomb network of conducting domain walls, and the resulting flatband can naturally explain the enhanced superconductivity. We also show that corner states, which are a hallmark of the higher-order topological insulators, appear in the network superstructure.

Staggered potential and magnetic field tunable electronic switch in a kagome nanoribbon junction

We propose a possible electronic switch on a two dimensional (2D) kagome lattice by applying a perpendicular inhomogeneous magnetic field and a staggered sublattice potential. By means of the tight-binding lattice model and the non-equilibrium Green's function method, we calculate the quantum Hall conductance of the device at zero temperature limit. The numerical results demonstrate that when a staggered lattice potential is considered, the conventional integer Hall effect is changed into discrete fractional conductance peaks, and a finite energy gap can be opened in the system, which may induce a metal-insulator transition and can be designed as a 2D electronic valve. The conductance valve phenomena mainly come from the interplay between the asymmetry energy band induced by the magnetic field and a band gap opened by the staggered potential. The ON(OFF) state of the electron transport is efficiently controlled by the device parameters such as the magnetic field, the staggered lattice potential and the Fermi level. Our findings might be useful for designing efficient current valves in 2D nano-electronic devices.

Emergence of an Antiferromagnetic Mott Insulating Phase in Hexagonal π-Conjugated Covalent Organic Frameworks

While the search for 2D organic semimetallic Dirac materials displaying, like graphene, a Dirac cone at the Fermi level remains active, attention is also being paid to the quantum phase transition from semimetal to antiferromagnet. Such a transition in graphene-like materials is predicted based on theoretical investigations of the 2D honeycomb lattice; it occurs (within a Hubbard model) when the on-site electron-electron Coulomb repulsion (U) is much larger than the nearest-neighbor inter-site electronic coupling (t). Here, monomers carrying long-lived radicals are considered and used as building blocks to design 2D hexagonal π-conjugated covalent organic frameworks (COFs). Both the nonmagnetic semimetallic phase and magnetically ordered phases are evaluated. It is found that the electronic coupling between adjacent radical centers in these COFs is more than an order of magnitude smaller than in graphene while the on-site Coulomb repulsion is reduced to a lesser extent. The resulting large U/t ratio drives these COFs into the antiferromagnetic side of the phase diagram. This work provides a first theoretical evidence of the realization of an antiferromagnetic Mott insulating phase in 2D π-conjugated COFs and allows a strategy to achieve quantum phase transitions from antiferromagnet to spin liquid and to semimetal to be outlined.