Realization of Lieb lattice in covalent-organic frameworks with tunable topology and magnetism

Room-temperature ferromagnetism, half-metallicity and spin transport in monolayer CrSc2Te4-based magnetic tunnel junction devices

The discovery of novel two-dimensional (2D) half-metallic materials with a robust ferromagnetic (FM) order and a high Curie temperature (Tc) is attractive for the advancement of next-generation spintronic devices. Here, we propose a monolayer with stable 2D intrinsic FM half-metallicity, i.e., the CrSc2Te4 monolayer, which was constructed by intercalating a monolayer of 1T-CrTe2-type sandwiched between two layers of 2H-ScTe2 monolayers. Our calculations reveal that it exhibits exceptional dynamical, thermal, and mechanical stabilities accompanied by a robust half-metallicity characterized by a wide bandgap of 1.02 eV and FM ordering with a high Tc of 326 K. Notably, these properties remain intact in almost the entire range of the biaxial strain from -5% to 5%. Furthermore, our investigations demonstrate excellent spin transport capabilities, including an outstanding spin-filtering effect, and a remarkably high tunneling magnetoresistance ratio peaking at 6087.07%. The remarkable magnetic features of the 2D CrSc2Te4 monolayer with room temperature FM, intrinsic half-metallicity, and 100% spin-polarization make it a promising candidate for the next-generation high-performance spintronic nanodevices as well as high-density magnetic recording and sensors.

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.

Topology and giant circular dichroism of enantiomorphic Kagome bands in a designed covalent organic framework

Covalent organic frameworks (COFs) are an emerging class of crystalline organic materials that have shown potential to be a new physical platform. In this work, a designed COF named AB-COF, which has novel enantiomorphic Kagome bands, is proposed and a feasible route to synthesize it is given. Via a combination of first-principles calculations and tight-binding analysis, we investigate the electronic structures and the phase interference of the COF. It becomes topologically nontrivial when doping one iodine atom in a unit cell. The Berry curvatures of the valence band (VB) and conduction band (CB) of the iodine-doped AB-COF show opposite values and different distributions. This provides an opportunity to study the new mechanism of circular dichroism from the different Berry curvatures of the VB and CB. Surprisingly, the circular-dichroism dissymmetry factor of AB-COF reaches a theoretical maximum value, and the oscillator strength data are in agreement with this result. When two iodine atoms are doped in a unit cell, the Berry curvatures of the VB and CB also have different values, but with more symmetry and similar distributions. This behavior enhances the circular dichroism with a wider range of dissymmetric absorption, and the circular dichroism dissymmetry factor also reaches its theoretical maximum value.

Electronic Lieb lattice signatures embedded in two-dimensional polymers with a square lattice

Exotic band features, such as Dirac cones and flat bands, arise directly from the lattice symmetry of materials. The Lieb lattice is one of the most intriguing topologies, because it possesses both Dirac cones and flat bands which intersect at the Fermi level. However, the synthesis of Lieb lattice materials remains a challenging task. Here, we explore two-dimensional polymers (2DPs) derived from zinc-phthalocyanine (ZnPc) building blocks with a square lattice (sql) as potential electronic Lieb lattice materials. By systematically varying the linker length (ZnPc-xP), we found that some ZnPc-xP exhibit a characteristic Lieb lattice band structure. Interestingly though, fes bands are also observed in ZnPc-xP. The coexistence of fes and Lieb in sql 2DPs challenges the conventional perception of the structure-electronic structure relationship. In addition, we show that manipulation of the Fermi level, achieved by electron removal or atom substitution, effectively preserves the unique characteristics of Lieb bands. The Lieb Dirac bands of ZnPc-4P shows a non-zero Chern number. Our discoveries provide a fresh perspective on 2DPs and redefine the search for Lieb lattice materials into a well-defined chemical synthesis task.

Effect of a flat band on a multiband two-dimensional Lieb lattice with intra- and interband interactions

In this study, we explore the effect of a single flat band in the electronic properties of a ferromagnetic two-dimensional Lieb lattice using the multiband Hubbard model with polarized carriers, spin-up and spin-down. We employ the self-consistent dynamical mean field theory and a Green functions cumulant expansion around the atomic limit to obtain the correlated densities of states while varying the intra- and interband interactions. Our findings demonstrate a renormalization of the correlated density of states in both the spin-up and spin-down carriers as we varied the intra- and interband interactions. We conclude that the presence of a flat band enables the system to maintain a metal state with itinerant ferromagnetism in the spin-up carrier.

Enantiomorphic kagome bands in a two-dimensional covalent organic framework with non-trivial magnetic and topological properties

The kagome lattice is one of the most intriguing topics to study. It has a frustrated flat band touching a set of Dirac bands and can possess various promising properties, such as ferromagnetism, superconductivity, and a non-trivial topology. Covalent organic frameworks (COFs) are a rare type of inorganic material, however, they can provide a platform for generating certain required lattices. Based on first-principles density functional theory calculations, we show that a newly synthesized two-dimensional COF named COF-SH has novel enantiomorphic kagome bands, which include two sets of flat bands touching the Dirac bands around the Fermi level. The Bloch wave of the flat-valence band at the K-point shows the kagome nature of the phase interference. Under charge doping, the COF-SH exhibits a ferromagnetic ground state. Moreover, when COF-SH is doped with iodine atoms, a sizable gap in the system is opened between the flat bands and the Dirac bands due to the spin-orbit coupling (SOC) effect. Meanwhile, the spin degeneracy is lifted since the organic layer loses electrons due to the oxidizing property of iodine. In addition, our tight-binding analysis with the SOC effect shows that the flat valence band separates from the Dirac bands and holds a nonzero Chern number. Consequently, this I-doped COF can give rise to a quantum anomalous Hall effect.

TMB12: a newly designed 2D transition-metal boride for spintronics and electrochemical catalyst applications

Exploring two-dimensional (2D) ferromagnetic materials with a high transition temperature and large magnetic anisotropy is extremely essential for highly efficient spintronic applications. With the density functional theory method, we predicted planar hypercoordinate transition-metal borides, TMB₁₂ (TM = Ti, V, Cr, Mn, Fe; B = boron), by the condensation of TM@B₈ and B₄ units. The results showed that these transition-metal borides possess superior thermal, dynamic and mechanical stabilities. Interestingly, the TMB₁₂ monolayer with TM = (V, Cr) is confirmed as a robust ferromagnetic metal with a high Curie temperature of ∼335 K and ∼221 K, respectively. In addition, the system with TM = (Mn, Fe) is found to be an antiferromagnetic metal with a Néel temperature of ∼173 K and ∼91 K, respectively. In particular, large perpendicular magnetic anisotropy is identified for CrB₁₂, MnB₁₂, and FeB₁₂ monolayers, around 198-623 μeV. Furthermore, four TMB₁₂ (TM = Ti, V, Cr, Mn) systems are determined to be candidate catalysts for the hydrogen evolution reaction, with nearly zero free energy of hydrogen adsorption (ΔG_H = -0.0003 to -0.03 eV). Our study highlighted potential 2D metal borides for spintronic devices and high efficiency electrochemical catalysts.

Growth of Mesoscale Ordered Two-Dimensional Hydrogen-Bond Organic Framework with the Observation of Flat Band

Flat bands (FBs), presenting a strongly interacting quantum system, have drawn increasing interest recently. However, experimental growth and synthesis of FB materials have been challenging and have remained elusive for the ideal form of monolayer materials where the FB arises from destructive quantum interference as predicted in 2D lattice models. Here, we report surface growth of a self-assembled monolayer of 2D hydrogen-bond (H-bond) organic frameworks (HOFs) of 1,3,5-tris(4-hydroxyphenyl)benzene (THPB) on Au(111) substrate and the observation of FB. High-resolution scanning tunneling microscopy or spectroscopy shows mesoscale, highly ordered, and uniform THPB HOF domains, while angle-resolved photoemission spectroscopy highlights a FB over the whole Brillouin zone. Density-functional-theory calculations and analyses reveal that the observed topological FB arises from a hidden electronic breathing-kagome lattice without atomically breathing bonds. Our findings demonstrate that self-assembly of HOFs provides a viable approach for synthesis of 2D organic topological materials, paving the way to explore many-body quantum states of topological FBs.

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.

Superatom Regiochemistry Dictates the Assembly and Surface Reactivity of a Two-Dimensional Material

The area of two-dimensional (2D) materials research would benefit greatly from the development of synthetically tunable van der Waals (vdW) materials. While the bottom-up synthesis of 2D frameworks from nanoscale building blocks holds great promise in this quest, there are many remaining hurdles, including the design of building blocks that reliably produce 2D lattices and the growth of macroscopic crystals that can be exfoliated to produce 2D materials. Here we report the regioselective synthesis of the cluster [trans-Co₆Se₈(CN)₄(CO)₂]^3-/4-, a "superatomic" building block designed to polymerize and assemble into a 2D cyanometalate lattice whose surfaces are chemically addressable. The resulting vdW material, [Co(py)₄]₂[trans-Co₆Se₈(CN)₄(CO)₂], grows as bulk single crystals that can be mechanically exfoliated to produce flakes as thin as bilayers, with photolabile CO ligands on the exfoliated surface. As a proof of concept, we show that these surface CO ligands can be replaced by 4-isocyanoazobenzene under blue light irradiation. This work demonstrates that the bottom-up assembly of layered vdW materials from superatoms is a promising and versatile approach to create 2D materials with tunable physical and chemical properties.

Engineering of flat bands and Dirac bands in two-dimensional covalent organic frameworks (COFs): relationships among molecular orbital symmetry, lattice symmetry, and electronic-structure characteristics

Two-dimensional covalent organic frameworks (2D-COFs), also referred to as 2D polymer networks, display unusual electronic-structure characteristics, which can significantly enrich and broaden the fields of electronics and spintronics. In this Focus article, our objective is to lay the groundwork for the conceptual description of the fundamental relationships among the COF electronic structures, the symmetries of their 2D lattices, and the frontier molecular orbitals (MOs) of their core and linker components. We focus on monolayers of hexagonal COFs and use tight-binding model analyses to highlight the critical role of the frontier-MO symmetry, in addition to lattice symmetry, in determining the nature of the electronic bands near the Fermi level. We rationalize the intriguing feature that, when the core unit has degenerate highest occupied MOs [or lowest unoccupied MOs], the COF highest valence band [or lowest conduction band] is flat but degenerate with a dispersive band at a high-symmetry point of the Brillouin zone; the consequences of having such band characteristics are briefly described. Multi-layer and bulk 2D COFs are found to maintain the salient features of the monolayer electronic structures albeit with a reduced bandgap due to the interlayer coupling. This Focus article is thus meant to provide an effective framework for the engineering of flat and Dirac bands in 2D polymer networks.

Geometric characterization of anomalous Landau levels of isolated flat bands

According to the Onsager’s semiclassical quantization rule, the Landau levels of a band are bounded by its upper and lower band edges at zero magnetic field. However, there are two notable systems where the Landau level spectra violate this expectation, including topological bands and flat bands with singular band crossings, whose wave functions possess some singularities. Here, we introduce a distinct class of flat band systems where anomalous Landau level spreading (LLS) appears outside the zero-field energy bounds, although the relevant wave function is nonsingular. The anomalous LLS of isolated flat bands are governed by the cross-gap Berry connection that measures the wave-function geometry of multi bands. We also find that symmetry puts strong constraints on the LLS of flat bands. Our work demonstrates that an isolated flat band is an ideal system for studying the fundamental role of wave-function geometry in describing magnetic responses of solids.

Landau levels are normally bounded by upper and lower band edges at zero magnetic field. Here, the authors predict a system with isolated flat bands, where anomalous Landau level spreading appears outside the zero-field energy bounds.

Ag2S monolayer: an ultrasoft inorganic Lieb lattice

Lieb lattice, a two-dimensional edge-centered square lattice, has attracted considerable interest due to its exotic electronic and topological properties. Although various optical and photonic Lieb lattices have been experimentally demonstrated, it remains challenging for an electronic Lieb lattice to be realized in real material systems. Here, based on first-principles calculations and tight-binding modeling, a silver sulfide (Ag₂S) monolayer is reported as a long-sought-after inorganic electronic Lieb lattice. This Lieb-lattice Ag₂S is further found to be ultrasoft, which enables its electronic properties and topological states near the Fermi level to be finely tuned, as evidenced by the strain-induced topologically non-trivial edge states near the valence band edge. These results not only provide an ideal platform to further explore and harvest interesting quantum properties but also pave a way to pursue other inorganic electronic Lieb lattices in a broader material domain.

Synthesis of Boroxine and Dioxaborole Covalent Organic Frameworks via Transesterification and Metathesis of Pinacol Boronates

Boroxine and dioxaborole are the first and some of the most studied synthons of covalent organic frameworks (COFs). Despite their wide application in the design of functional COFs over the last 15 years, their synthesis still relies on the original Yaghi's condensation of boronic acids (with itself or with polyfunctional catechols), some of which are difficult to prepare, poorly soluble, or unstable in the presence of water. Here, we propose a new synthetic approach to boroxine COFs (on the basis of the transesterification of pinacol aryl boronates (aryl-Bpins) with methyl boronic acid (MBA) and dioxaborole COFs (through the metathesis of pinacol boronates with MBA-protected catechols). The aryl-Bpin and MBA-protected catechols are easy to purify, highly soluble, and bench-stable. Furthermore, the kinetic analysis of the two model reactions reveals high reversibility (K_eq ∼ 1) and facile control over the equilibrium. Unlike the conventional condensation, which forms water as a byproduct, the byproduct of the metathesis (MBA pinacolate) allows for easy kinetic measurements of the COF formation by conventional ¹H NMR. We show the generality of this approach by the synthesis of seven known boroxine/dioxaborole COFs whose crystallinity is better or equal to those reported by conventional condensation. We also apply metathesis polymerization to obtain two new COFs, Py4THB and B2HHTP, whose synthesis was previously precluded by the insolubility and hydrolytic instability, respectively, of the boronic acid precursors.

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-Free Magnetism in Chemically Doped Covalent Organic Frameworks

Organic and molecule-based magnets are not easily attainable, as introduction of stable paramagnetic centers to pure organic systems is particularly challenging. Crystalline covalent organic frameworks (COFs) with high designability and chemical diversity constitute ideal platforms to access intriguing magnetic phenomena of organic materials. In this work, we proposed a general approach to attain unpaired electron spin and metal-free magnetism in narrow-band COFs by chemical doping. By using density functional theory calculations, we found that dopants with energy-matched frontier orbitals to COFs not only inject charges but also further localize them through orbital hybridization and the formation of a supramolecular charge-transfer complex. The localized electronic states ensure that stable paramagnetic centers can be introduced to nonmagnetic COFs. On the basis of these discoveries, we designed two new COFs with narrow valence bands, which show prospective magnetism after doping with iodine. Further, we unraveled the magnetic anisotropy in two-dimensional COFs and demonstrated that both spin-conduction and magnetic interactions can be effectively modulated by manipulating the building blocks of COFs. Our work highlights a practical route to attain magnetism in COFs and other organic materials, which show great potential for applications in organic spintronic devices.

Topological Band Engineering of Lieb Lattice in Phthalocyanine-Based Metal-Organic Frameworks

Topological properties of the Lieb lattice, i.e., the edge-centered square lattice, have been extensively studied and are, however, mostly based on theoretical models without identifying real material systems. Here, based on tight-binding and first-principles calculations, we demonstrate the Lieb-lattice features of the experimentally synthesized phthalocyanine-based metal-organic framework (MPc-MOF), which holds various intriguing topological phase transitions through band engineering. First, we show that the MPc-MOFs indeed have a peculiar Lieb band structure with 1/3 filling, which has been overlooked because of its unconventional band structure deviating from the ideal Lieb band. The intrinsic MPc-MOF presents a trivial insulating state, with its gap size determined by the on-site energy difference (ΔE) between the corner and edge-center sites. Through either chemical substitution or physical strain engineering, one can tune ΔE to close the gap and achieve a topological phase transition. Specifically, upon closing the gap, topological semimetallic/insulating states emerge from nonmagnetic MPc-MOFs, while magnetic semimetal/Chern insulator states arise from magnetic MPc-MOFs, respectively. Our discovery greatly enriches our understanding of the Lieb lattice and provides a guideline for experimental observation of the Lieb-lattice-based topological states.