2D Kagome Lattice Research Articles

Synthesis and characterization of low-dimensional N-heterocyclic carbene lattices.

The covalent interaction of N-heterocyclic carbenes (NHCs) with transition metal atoms gives rise to distinctive frontier molecular orbitals (FMOs). These emergent electronic states have spurred the widespread adoption of NHC ligands in chemical catalysis and functional materials. Although formation of carbene-metal complexes in self-assembled monolayers on surfaces has been explored, design and electronic structure characterization of extended low-dimensional NHC-metal lattices remains elusive. Here we demonstrate a modular approach to engineering one-dimensional (1D) metal-organic chains and two-dimensional (2D) Kagome lattices using the FMOs of NHC-Au-NHC junctions to create low-dimensional molecular networks exhibiting intrinsic metallicity. Scanning tunneling spectroscopy and first-principles density functional theory reveal the contribution of C-Au-C π-bonding states to dispersive bands that imbue 1D- and 2D-NHC lattices with exceptionally small work functions.

The type-I, III nodal ring, type-I, III quadratic nodal point, and Dirac valley phonons in 2D kagome lattices M2C3 (M = As, Bi, Cd, Hg, P, Sb, Zn)

Topological phases in kagome systems have garnered considerable interest since the introduction of the colloidal kagome lattice. Our study employs first-principle calculations and symmetry analysis to predict the existence of ideal type-I, III nodal rings (NRs), type-I, III quadratic nodal points (QNPs), and Dirac valley phonons (DVPs) in a collection of two-dimensional (2D) kagome lattices M2C3 (M = As, Bi, Cd, Hg, P, Sb, Zn). Specifically, the Dirac valley points (DVPs) can be observed at two inequivalent valleys with Berry phases of +π and −π , connected by edge arcs along the zigzag and armchair directions. Additionally, the QNP is pinned at the Γ point, and two edge states emerge from its projections. Notably, these kagome lattices also exhibit ideal type-I and III nodal rings protected by time inversion and spatial inversion symmetries. Our work examines the various categories of nodal points and nodal ring phonons within the 2D kagome systems and presents a selection of ideal candidates for investigating topological phonons in bosonic systems.

Asymptotic, second-order homogenization of linear elastic beam networks

We propose a general approach to the higher-order homogenization of discrete elastic networks made up of linear elastic beams or springs in dimension 2 or 3. The network may be nearly (rather than exactly) periodic: its elastic and geometric properties are allowed to vary slowly in space, in addition to being periodic at the scale of the unit cell. The reference configuration may be prestressed. A homogenized strain energy depending on both the macroscopic strain ɛ and its gradient ∇ɛ is obtained by means of a two-scale expansion. The homogenized energy is asymptotically exact two orders beyond that obtained by classical homogenization. The homogenization method is implemented in a symbolic calculation language and applied to various types of networks, such as a 2D honeycomb, a 2D Kagome lattice, a 3D truss and a 1D pantograph. It is validated by comparing the predictions of the microscopic displacement to that obtained by full, discrete simulations. This second-order method remains highly accurate even when the strain gradient effects are significant, such as near the lips of a crack tip or in regions where a gradient of pre-strain is imposed.

Computation insights of kagome structure for probing electronic and thermodynamic properties by applying external factors such as strain, spin-orbital coupling, and external magnetic field

In this study, we explored the electronic properties of a two-dimensional (2D) kagome lattice in the presence of an external magnetic field, spin-orbit coupling (SOC), and strain. Our focus was on investigating the energy-dependent behavior of the electronic heat capacity and paramagnetic spin susceptibility in response to changing factors. Employing the Green's function approach, we successfully determined the energy dependence of the electronic heat capacity and paramagnetic susceptibility using the tight-binding model. Notably, at low energies, the electronic heat capacity almost reaches the Schottky anomaly peak, while the paramagnetic susceptibility sharply decreases. Furthermore, we conducted a detailed investigation of the energy-dependent paramagnetic susceptibility and electronic heat capacity of a kagome lattice monolayer, considering the influences of SOC and magnetic field factors. Our investigation of the density of states (DOS) in the presence of SOC suggests that semimetal-to-insulator phase transitions occur. Additionally, throughout our study, we observed variations in the DOS spectrum, indicating the influence of both compressive and tensile strains. We found that these alterations, along with the splitting of levels under different conditions, led to changes in the material's electrical properties.

Chiral breathing-valley locking in two-dimensional kagome lattice Ta3I8

The exploration of valley-related physics is essential to the development and refinement of valleytronics. Here, a paradigm called chiral breathing-valley locking (CBVL) is proposed, in which two chiral “breathing” phases are completely locked to their valley indexes. Utilizing first-principles calculations, a two-dimensional (2D) kagome lattice with large spontaneous valley polarization (VP), namely, monolayer Ta3I8, is proposed to realize CBVL. There are two breathing phases with chiral symmetry in Ta3I8, and they can change to each other under some conditions. The valley indexes change between “−K” and “K” along the breathing of the two chiral symmetric phases, so CBVL can be obtained. Remarkably, the VP is up to 199.7 meV, so CBVL can cause the obvious switching of the anomalous valley Hall effect (AVHE). Additionally, a 2D Janus Kagome structure TaI4X4 is constructed to confirm experimentally the feasibility of electric field modulation on the CBVL. The built-in electric field and the breathing mode can mutually influence each other in TaI4X4, which provides a synergistic regulation of the AVHE. Our findings broaden the horizon for exploring AVHE materials and provide a platform for future valleytronic applications.

On-Surface Synthesis of Novel Kagome Lattices Coordinated via Four-Fold N-Ag Bonding.

The Kagome lattice structures based on metal-organic coordination have garnered widespread interest because of their topologically Dirac/flat bands and other exotic electronic structures. However, the experimental fabrication of large-area two-dimensional (2D) Kagome lattice structures of metal-organic frameworks (MOFs) via on-surface synthesis remains limited. Herein, we successfully construct two kinds of large-scale 2D Kagome-type lattices stabilized by 4-fold N-Ag coordination on the Ag(111) surface. With the aid of scanning tunneling microscopy (STM) and synchrotron radiation photoemission spectroscopy (SRPES), we clearly elucidate the reaction pathway and mechanism of fabrication of the two Kagome lattices. This work provides a novel platform for investigating related intriguing physical properties.

Crystal Structures and Physicochemical Properties of Be2N and Mg2N as Electride Materials

Electrides are a special family of materials with exotic physical properties and various crystal structures in different dimensions. Among them, alkaline-earth metal nitrides (${\mathrm{AE}}_{2}\mathrm{N}$) have attracted enormous attention, since the synthesis of ${\mathrm{Ca}}_{2}\mathrm{N}$ as a two-dimensional (2D) electride. However, apart from ${\mathrm{Ca}}_{2}\mathrm{N}$, the physicochemical properties of other ${\mathrm{AE}}_{2}\mathrm{N}$ are not well understood, and in some cases, their crystal structures remain unknown. To overcome these issues, here, we conduct systematic investigations into the crystal structures, electronic structures, and optical properties of two typical ${\mathrm{AE}}_{2}\mathrm{N}$, i.e., ${\mathrm{Mg}}_{2}\mathrm{N}$ and ${\mathrm{Be}}_{2}\mathrm{N}$, based on first-principles calculations. By symmetry analysis, we derive a crystal structure with $R\overline{3}m$ symmetry for ${\mathrm{Be}}_{2}\mathrm{N}$, which is energetically more stable than all the other structures proposed previously. Our calculations show that $R\overline{3}m$ ${\mathrm{Be}}_{2}\mathrm{N}$ is a 2D electride with the anionic electrons located in the interstitial space, forming a 2D Kagome lattice. Moreover, distinct from traditional 2D electrides, it exhibits anomalous physicochemical properties, such as having a small interstitial space, a high work function, and a large cleavage energy, mainly due to the relatively strong bonding effect between anionic electrons and cationic framework. For ${\mathrm{Mg}}_{2}\mathrm{N}$, on the other hand, we find that the most stable structure has R3m symmetry, which shows a total energy of 0.005 eV per formula unit lower than the previously predicted Cmcm structure. Moreover, different from $R\overline{3}m$ ${\mathrm{Be}}_{2}\mathrm{N}$, R3m ${\mathrm{Mg}}_{2}\mathrm{N}$ is identified as a zero-dimensional (0D) electride with a semiconducting band structure, since its anionic electrons are confined in separated 0D cavities and cannot interact with each other. These studies thus provide a deeper understanding of the crystal structures and physicochemical properties of ${\mathrm{Be}}_{2}\mathrm{N}$ and ${\mathrm{Mg}}_{2}\mathrm{N}$ as potential electride materials.

Inverse design of triblock Janus spheres for self-assembly of complex structures in the crystallization slot via digital alchemy.

The digital alchemy framework is an extended ensemble simulation technique that incorporates particle attributes as thermodynamic variables, enabling the inverse design of colloidal particles for desired behavior. Here, we extend the digital alchemy framework for the inverse design of patchy spheres that self-assemble into target crystal structures. To constrain the potentials to non-trivial solutions, we conduct digital alchemy simulations with constant second virial coefficient. We optimize the size, range, and strength of patchy interactions in model triblock Janus spheres to self-assemble the 2D kagome and snub square lattices and the 3D pyrochlore lattice, and demonstrate self-assembly of all three target structures with the designed models. The particles designed for the kagome and snub square lattices assemble into high quality clusters of their target structures, while competition from similar polymorphs lower the yield of the pyrochlore assemblies. We find that the alchemically designed potentials do not always match physical intuition, illustrating the ability of the method to find nontrivial solutions to the optimization problem. We identify a window of second virial coefficients that result in self-assembly of the target structures, analogous to the crystallization slot in protein crystallization.

Delocalization and re-entrant localization of flat-band states in non-Hermitian disordered lattice models with flat bands

Abstract We present a numerical study of Anderson localization in disordered non-Hermitian lattice models with flat bands. Specifically, we consider 1D stub and 2D kagome lattices that have a random scalar potential and a uniform imaginary vector potential and calculate the spectra of the complex energy, the participation ratio, and the winding number as a function of the strength of the imaginary vector potential, h. The flat-band states are found to show a double transition from localized to delocalized and back to localized states with h, in contrast to the dispersive-band states going through a single delocalization transition. When h is sufficiently small, all flat-band states are localized. As h increases above a certain critical value h1, some pairs of flat-band states become delocalized. The participation ratio associated with them increases substantially and their winding numbers become nonzero. As h increases further, more and more flat-band states get delocalized until the fraction of the delocalized states reaches a maximum. For larger h values, a re-entrant localization takes place and, at another critical value h2, all flat-band states return to compact localized states with very small participation ratios and zero winding numbers. This re-entrant localization transition, which is due to the interplay among disorder, non-hermiticity, and the flat band, is a phenomenon occurring in many models having an imaginary vector potential and a flat band simultaneously. We explore the spatial characteristics of the flat-band states by calculating the local density distribution.

Evolution of ultraflat band in the van der Waals kagome semiconductor Pd3P2(S1−xSex)8

We investigate the evolutions of structural parameters, optical properties, and electronic structures of a van der Waals kagome semiconductor Pd$_{3}$P$_{2}$S$_{8}$ with Se doping. When the doping level of Se increases, the bandgaps of Pd$_{3}$P$_{2}$(S$_{1-x}$Se$_{x}$)$_{8}$ single crystals decrease gradually, accompanying with the expanded unit cells. The first-principles calculations show that there is a flat band (FB) near the Fermi level in bulk Pd$_{3}$P$_{2}$S$_{8}$. This FB mainly originates from the $d_{z^2}$-like orbitals of Pd atoms in the Pd kagome lattice, which have a finite interlayer electron hopping perpendicular to the PdS$_4$ square plane. The interlayer hopping can be reinforced with Se doping, inducing a stronger interlayer coupling via the chalcogen atoms at apical sites, which reduces the bandgap and enhances the cleavage energy. In contrast, the varnishing interlayer hopping in the two-dimensional limit results in the formation of ultra-FB in the monolayers of these compounds. The easy exfoliation and the existence of unique ultra-FB near $E_{\rm F}$ make Pd$_{3}$P$_{2}$(S$_{1-x}$Se$_{x}$)$_{8}$ a model system to explore the exotic physics of FB in 2D kagome lattice.

Emergent Kagome Electrides.

In a two-dimensional (2D) Kagome lattice, the ideal Kagome bands including Dirac cones, van Hove singularities, and a flat band are highly expected, because they can provide a promising platform to investigate novel physical phenomena. However, in the reported Kagome materials, the complex 3D and multiorder electron hoppings result in the disappearance of the ideal Kagome bands in these systems. Here, we propose an alternative way to achieve the ideal Kagome bands in non-Kagome materials by confining excess electrons in the system to the crystal interstitial sites to form a 2D Kagome lattice, coined as a Kagome electride. Then, we predict two novel stable 2D Kagome electrides in hexagonal materials Li5Si and Li5Sn, whose band structures are similar to the ideal Kagome bands, including topological Dirac cones with beautiful Fermi arcs in their surface states, van Hove singularities, and a flat band. In addition, Li5Si is revealed to be a low-temperature superconductor at ambient pressure, and its superconducting transition temperature Tc can be increased from 1.1 K at 0 GPa to 7.2 K at 100 GPa. The high Tc is unveiled to be the consequence of strong electron-phonon coupling originated from the sp-hybridized phonon-coupled bands and phonon softening caused by strong Fermi nesting. Due to the strong Fermi nesting, the charge density wave phase transition occurs at 110 GPa with the lattice reconstructed from hexagonal to orthorhombic, accompanied with the increase of Tc to 10.5 K. Our findings pave an alternative way to fabricate more real materials with Kagome bands in electrides.

Recent Progress on 2D Kagome Magnets: Binary TmSnn (T = Fe, Co, Mn)

AbstractDue to the particularity of the lattice geometry, intriguing electronic characteristics of Dirac cone and flat band can coexist in the two‐dimensional (2D) kagome lattice. The 2D kagome materials, therefore, have attracted tremendous attention, as a platform for studying the topological non‐trivial band structures and strong correlated phenomena. In this paper, the recent experimental progress on binary kagome metals Fe3Sn2, FeSn, CoSn, and Mn3Sn, mainly by the methods of scanning tunneling microscopy (STM), scanning tunneling spectroscopy (STS), and angle‐resolved photoemission spectroscopy (ARPES), are reviewed. In those compounds, the kagome layers give rise to the electronic structure of flat bands and Dirac cones, which induce a wide range of novel electron behaviors. Their exposed surface structure, electronic band structure, and topological properties are therefore highlighted, aiming at illustrating the determining role of their 2D kagome layers. In addition, the interplay between frustrated kagome geometry, frustrated magnetism, and topological magnetism, which brings new opportunities and broad prospects to the research of strongly correlated topological kagome magnetic materials, is introduced through reviewing recent experimental works by electronic and magnetic transport measurements.

On Computations of Topological Descriptors of Kagome Lattice

The analysis of graphs and networks is a multidisciplinary research field from its origin. As such, it has been extensively used also in biomedicine, cheminformatics, and in bioinformatics, where approaches based on graph descriptors have been offered for communicating with the numerous challenging tasks. In quantitative structure-activity (QSAR) and quantitative structure-property (QSPR) relationships studies, graph invariants are used to forecast the biological activities and properties of nanomaterials. In these studies, degree-based topological descriptors have founded comprehensive acceptance among the different types of descriptors, because of the ease of generation and the speed with which these calculations can be performed. In this article, we first compute the M-polynomials of 2D kagome lattice and then derive the closed form of various significant topological degree-based indices in terms of the M-polynomials and partition techniques. Furthermore, we sketch surfaces that show the dependence of some topological indices on the parameters of the structure.

Magnetic properties of the David star in the 2D-Kagome lattice

ABSTRACT We investigate the magnetisation and internal energy of the David star in the Kagome lattice by the effective field theory developed by Kaneyoshi. First-order, Second-order and successive phase transitions are observed for different spin-down orientations. Spin-down orientation of identical atoms causes a second-order phase transition, spin-down orientation of different atoms causes a first-order and successive phase transitions. Minimum internal energy is obtained as U David star = −5 for all spin-up orientations, whereas it increases (U David star > −5) for at least one spin-down orientation. For spin-down orientation of A↑–A↓, internal energy becomes zero at a certain temperature and we call this temperature ‘zero internal energy temperature (Tzie)’. At T < Tzie = 1.99, internal energy is positive, whereas it is negative for other spin states. Therefore, spin-induced minimum internal energy broken is observed except for all spin-up (A↑, B↑ and C↑) cases.

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.

Bilayer twisting as a mean to isolate connected flat bands in a kagome lattice throughWigner crystallization* *Project supported by the National Natural Science Foundation of China (Grant No. 11874314). Sheng-Bai Zhang was supported by U.S. DOE under Grant No. DE-SC0002623.

The physics of flat band is novel and rich but difficult to access. In this regard, recently twisting of bilayer van der Waals (vdW)-bounded two-dimensional (2D) materials has attracted much attention, because the reduction of Brillouin zone will eventually lead to a diminishing kinetic energy. Alternatively, one may start with a 2D kagome lattice, which already possesses flat bands at the Fermi level, but unfortunately these bands connect quadratically to other (dispersive) bands, leading to undesirable effects. Here, we propose, by first-principles calculation and tight-binding modeling, that the same bilayer twisting approach can be used to isolate the kagome flat bands. As the starting kinetic energy is already vanishingly small, the interlayer vdW potential is always sufficiently large irrespective of the twisting angle. As such the electronic states in the (connected) flat bands become unstable against a spontaneous Wigner crystallization, which is expected to have interesting interplays with other flat-band phenomena such as novel superconductivity and anomalous quantum Hall effect.

Performance Potential of 2D Kagome Lattice Interconnects

Kagome lattice materials are layered two-dimensional (2D) materials in which atoms are arranged in a trihexagonal tiling lattice pattern. It has been suggested that the Kagome lattice can possess topologically non-trivial band structures. By performing atomistic quantum transport simulations, we show that the topological edge modes of a Fe3Sn Kagome nanoribbon have excellent carrier transport properties, with a mean free path several orders of magnitude larger than that of the bulk modes. Vertical stacking of intercalated Kagome layers can further boost the conductance per unit width. As a result, the 2D Kagome lattice materials offer low resistivity and promising potential for interconnect applications in the sub-10nm regime.

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.

Two-Dimensional Kagome Lattices Made of Hetero Triangulenes Are Dirac Semimetals or Single-Band Semiconductors.

Here we discuss, based on first-principles calculations, two-dimensional (2D) kagome lattices composed of polymerized heterotriangulene units, planar molecules with D3h point group containing a B, C, or N center atom and CH2, O, or CO bridges. We explore the design principles for a functional lattice made of 2D polymers, which involves control of π-conjugation and electronic structure of the knots. The former is achieved by the chemical potential of the bridge groups, while the latter is controlled by the heteroatom. The resulting 2D kagome polymers have a characteristic electronic structure with a Dirac band sandwiched by two flat bands and are either Dirac semimetals (C center), or single-band semiconductors-materials with either exclusively electrons (B center) or holes (N center) as charge carriers of very high mobility, reaching values of up to ∼8 × 103 cm2 V-1 s-1, which is comparable to crystalline silicon.

Chiral Kagome Lattices from On‐Surface Synthesized Molecules

Kagome lattices have attracted much attention owing to their potential applications in spin-frustrated magnetism and host-guest chemistry. Examples toward the fabrication of 2D Kagome lattices reported previously have in common that the precursor molecules were typically deposited on the surface structurally intact with no chemical reactions accompanied. Herein, by using a combination of synchrotron radiation photoelectron spectroscopy (SRPES) and scanning tunneling microscopy (STM), we demonstrated the fabrication of two types of chiral Kagome lattices from on-surface synthesized organometallic compounds, which are known as intermediates of Glaser coupling on silver single crystal surfaces. These Kagome lattices are stabilized by the interplay of various intermolecular interactions, including Br⋅⋅⋅Br bonds, C-Br⋅⋅⋅π bonds and π-π stacking. The chiral transference and host-guest supramolecular structure in the novel Kagome lattices were also studied. Our studies may pave a new way to engineer complex supramolecular networks through on-surface reactions.