Two-dimensional Hexagonal Lattice Research Articles

Observation of Kekulé vortices around hydrogen adatoms in graphene

Fractional charges are one of the wonders of the fractional quantum Hall effect. Such objects are also anticipated in two-dimensional hexagonal lattices under time reversal symmetry—emerging as bound states of a rotating bond texture called a Kekulé vortex. However, the physical mechanisms inducing such topological defects remain elusive, preventing experimental realization. Here, we report the observation of Kekulé vortices in the local density of states of graphene under time reversal symmetry. The vortices result from intervalley scattering on chemisorbed hydrogen adatoms. We uncover that their 2π winding is reminiscent of the Berry phase π of the massless Dirac electrons. We can also induce a Kekulé pattern without vortices by creating point scatterers such as divacancies, which break different point symmetries. Our local-probe study thus confirms point defects as versatile building blocks for Kekulé engineering of graphene’s electronic structure.

Topological design of hexagonal lattice phononic crystals for vibration attenuation combined fast plane wave expansion method with elite seed strategy genetic algorithm

The characteristics of manipulating elastic wave propagation in phononic crystals (PnCs) have been applied in various fields. A triangular element discrete method for two-dimensional (2D) hexagonal lattice PnCs is proposed, which combines with the fast plane wave expansion method (FPWEM) to obtain the band structure. As compared to the finite element method (FEM), time consumption is one order of magnitude faster while ensuring accuracy. To design the wider band gap (BG) of PnCs, the elite seed strategy genetic algorithm (ESS-GA) is used to optimize the topology of PnCs for in-plane mode and out-of-plane mode, and based on this, the proposed method is first applied to optimize hexagonal lattice PnCs, which is extended to the BG design of mixed mode. The relationship between the optimized individual under different propagate modes, and the volume fraction of PnCs at each BG is explained, physical mechanism of optimized unit cells is also estimated through iso-frequency contours and dynamic effective mass. The numerical and experimental results of a hexagonal lattice composed of optimized unit cells indicate that elastic waves can be suppressed within the BG, fully demonstrating the effectiveness of the method. In addition, this method is expected to explore its potential applications in the reverse design of PnCs.

A novel 2D-PhC based ring resonator design with flexible structural defects for CWDM applications

In this article, a chain-shaped photonic crystal-based ring resonator (PhCRR), which can function as a channel drop filter (CDF), is designed using two-dimensional photonic crystals. Silicon rods with a refractive index of 3.44 are chosen, and they are perforated in the air with a refractive index of 1 for the PhCRR layout design. Silicon material is selected for realizing the PhCRR-based channel drop filter because it exhibits nearly zero absorption in the C-band (1530–1565 nm) spectral region. The PhCRR structure is established within two-dimensional hexagonal lattices. Transmission efficiency, quality factor, and sensitivity are critical parameters in the design of optical components. Our proposed ring resonator achieves a transmission efficiency of 99.7% with a quality factor of 4550 at a wavelength of 1551 nm. To analyze the drop filter’s functionality, we calculate the electric field distribution of two-dimensional photonic crystals at 1551 nm and 1553 nm. We employ the FDTD numerical analyser to extract simulation results, and the plane wave expansion solver method is used to estimate the photonic band gap of the designed resonator. The chain-shaped photonic crystal-based ring resonator is designed to operate in the third optical window wavelength, which offers very low loss (less than 0.2 dB). The proposed PhCRR is designed to operate within the conventional band range of 1530 to 1565 nm and can be used for various applications such as tunable add/drop multiplexing, optical filters, signal routing and switching in CWDM applications, free space communication systems, and photonic integrated circuits.

Investigating valley-dependent current generation due to asymmetric energy dispersion for charge-transfer from a quantum dot to single-walled carbon nanotube

Single-wall carbon nanotubes (SWCNT), which consist of a two-dimensional hexagonal lattice of carbon atoms, possess unique mechanical, electrical, optical and thermal properties. SWCNT can be synthesized in diverse chiral indexes to determine certain attributes. This work theoretically investigates electron transport in different directions along SWCNT. The electron studied in this research transfers from the quantum dot that can possibly move to the right or left direction in SWCNT with different valley-dependent probability. These results show that valley polarized current is present. The valley current in the right and left directions has a composition of valley degrees of freedom where its components (K and K′) are not identical. Such a result can be traced theoretically by certain effects. That firstly is the curvature effect on SWCNT in which the hopping integral between pi electrons from the flat graphene is altered, and another is curvature-inducing sigma -pi mixture. Due to these effects, the band structure of SWCNT is asymmetric in certain chiral indexes leading to the asymmetry of valley electron transport. Our results exhibit that the zigzag chiral indexes is the only type making electron transport symmetrical that is different to the result from the other chiral index types which are the armchair and chiral. This work also illustrates the characteristic of the electron wave function propagating from the initial point to the tip of the tube over time, and the current density of the probability in specific times. Additionally, our research simulates the result from the dipole interaction between the electron in QD and the tube that impacts the lifetime of the electron being in QD. The simulation portrays that more dipole interaction encourages the electron transfer to the tube, thereby shortening the lifetime. We as well suggest the reversed electron transfer from the tube to QD that the time duration of such transfer is much less than the opposite transfer owing to the different orbital of the electron’s states. Valley polarized current in SWCNTs may also be used in the development of energy storage devices such as batteries and supercapacitors. The performance and effectiveness of nanoscale devices, including transistors, solar cells, artificial antennas, quantum computers, and nano electronic circuits, must be improved in order to achieve a variety of benefits.

Monte Carlo studies of noncollinear magnetic phases in multiferroic Cu2OSeO3

Monte Carlo simulations based on a first-principles-derived Hamiltonian are conducted to study the finite-temperature properties of chiral-lattice multiferroic insulator ${\mathrm{Cu}}_{2}{\mathrm{OSeO}}_{3}$. The use of this numerical technique (i) reveals basic features of the phase diagram as a function of temperature and external magnetic field, including the long-range helical phase at low temperature and zero magnetic field and the skyrmion lattice phase, in which skyrmions are arranged in a two-dimensional hexagonal lattice, and (ii) leads to the discovery of an overlooked vortex lattice phase in a narrow pocket of the phase diagram near the fluctuations' disordered-helical phase transition. The scheme also provides strong numerical evidence that the transition to a helical state in ${\mathrm{Cu}}_{2}{\mathrm{OSeO}}_{3}$ is of first order driven by critical fluctuations.

Spin-gapless van der Waals heterostructure for spin gating through magnetic injection devices.

Spin-gapless semiconductors (SGSs) are new magnetic zero-bandgap materials whose band structure is extremely sensitive to external influences (pressure or magnetic fields) and have great potential for high-speed and low-energy spintronics applications. The first-principles method was used to systematically study the heterostructures constructed of an asymmetric surface-functionalized Janus MXene material, Cr2NOF, and a two-dimensional hexagonal lattice (2DH) semiconductor material and to study the effects of the electronic structure, Curie temperature, magnetism, and the design of unusual band structures and magnetic injection in the bilayer to obtain an SGS structure. Through the design and construction of Cr2NOF/2DH van der Waals heterojunction spintronic devices, the spin-filtering effect of the devices can reach 100%, especially, realizing spin gating through magnetic injection. We report the transport mechanism of the heterojunction spintronic devices to achieve the goal of a controllable optimization of the device functions, which provides a theoretical basis for the design of MXene van der Waals heterojunctions for high-efficiency and low-power-consumption spintronic devices.

Quantum anomalous valley Hall effect in ferromagnetic MXenes with asymmetric functionalization.

The potential to detect and manipulate the valley degree of freedom within two-dimensional hexagonal lattices possessing both inversion asymmetry and time-reversal symmetry is theoretically feasible. Intrinsic ferrovalley polarization in MXenes could be induced by asymmetric surface functionalization to break their inversion symmetry and the presence of spin-orbital coupling ensures their time-reversal symmetry. Our results indicate that the ferromagnetic Cr2COF MXene with Janus functionalization becomes an intrinsic Chern insulator with large spin-valley polarization and belongs to the family of quantum anomalous valley Hall effect (QAVHE) materials, based on Berry curvature and edge state calculations. Applying chemical engineering of functionalization to magnetic MXenes allows us to tune the structure-property relationship in 2D layers to obtain desirable spin-valley coupling. Our theoretical insight into the QAVHE on magnetic MXenes with asymmetry functionalization provides a new opportunity for valleytronics and spintronics.

Giant and tunable Rashba spin splitting and quantum spin Hall effect in H-Pb-Cl

Rashba spin splitting and quantum spin Hall effect have attracted enormous interest due to their great significance in the application of spintronics. According to the first-principles calculation, we propose a two-dimensional hexagonal lattice material H-Pb-Cl, which realizes the coexistence of giant Rashba spin splitting and quantum spin Hall effect. Owing to the break of space inversion symmetry and the existence of intrinsic electric field, H-Pb-Cl has a huge Rashba spin splitting phenomenon (<i>α</i><sub>R</sub> = 3.78 eV·Å), and the Rashba spin splitting of H-Pb-Cl(–16%—16%) can be adjusted by changing the biaxial stress. By analyzing the electronic properties of H-Pb-Cl, we find that H-Pb-Cl has a huge band gap near the Fermi surface (1.31 eV), and the topological invariant <i>Z</i><sub>2</sub> = 1 of the system is caused by the inversion of s-p orbit, which indicates that H-Pb-Cl is a two-dimensional topological insulator with a huge topological band gap, and the gap is large enough to observe the topological edge states at room temperature. In addition, we further consider the effect of BN and graphane substrates on the topological band gap of H-Pb-Cl by using the H-Pb-Cl (111)-(1×1) /BN (111)-(2×2) and H-Pb-Cl(1×1)/ graphane (2×2) system, and find that the lattice mismatch between H-Pb-Cl (5.395 Å) and BN (2.615 Å) and between H-Pb-Cl (5.395 Å) and graphane (2.575 Å) are about 3% and 4.5%, respectively. According to our calculation results, H-Pb-Cl still retains the properties of topological insulator under the effect of spin orbit coupling, and is not affected by BN nor graphane. Our results show that the nontrivial topological band gap of H-Pb-Cl can be well preserved under both biaxial stress effect and substrate effect. In addition, H-Pb-Cl can well retain the nontrivial topological band gap under the stress of –16%–16%, and thus there are many kinds of substrate materials used to synthesize this material, which is very helpful in successfully realizing preparation experimentally. Our research provides a promising candidate material for exploring and realizing the coexistence of Rashba spin splitting and quantum spin Hall effect. And the coexistence of giant Rashba spin splitting and quantum spin Hall effect greatly broadens the scope of potential applications of H-Pb-Cl in the field of spintronic devices.

Spontaneous Valley Polarization Caused by Crystalline Symmetry Breaking in Nonmagnetic LaOMX2 Monolayers.

In order to achieve valley polarization, breaking the time-reversal symmetry in two-dimensional hexagonal lattices with inversion asymmetry is the heart of current valleytronic research, which, however, has caused studies to stagnate due to the inevitable drawbacks. In this work, we go beyond the conventional paradigm and demonstrate the novel valley physics caused by lowering the crystalline symmetry instead of breaking the time-reversal symmetry. In particular, we translate our concept into concrete nonmagnetic LaOMX2 monolayers with a tetragonal lattice, confirming that a spontaneous structure distortion can cause the long-sought, considerably large valley polarization. In detail, the physics of valley-orbital coupling, valley-orbital-layer coupling, valley-contrasting linear dichroism, and interlayer exciton valleytronics are discussed.

Minimum-Perimeter Lattice Animals and the Constant-Isomer Conjecture

We consider minimum-perimeter lattice animals, providing a set of conditions which are sufficient for a lattice to have the property that inflating all minimum-perimeter animals of a certain size yields (without repetitions) all minimum-perimeter animals of a new, larger size. We demonstrate this result on the two-dimensional square and hexagonal lattices. In addition, we characterize the sizes of minimum-perimeter animals on these lattices that are not created by inflating members of another set of minimum-perimeter animals.

Metastability of Ising and Potts Models Without External Fields in Large Volumes at Low Temperatures

In this article, we investigate the energy landscape and metastable behavior of Ising and Potts models on two-dimensional square or hexagonal lattices in the low-temperature regime, especially in the absence of an external magnetic field. The energy landscape of these models without an external field is known to have a very large and complex saddle structure between the ground states. In the small-volume regime where the lattice is finite and fixed, the aforementioned complicated saddle structure has been successfully analyzed in Kim and Seo (Metastability of stochastic Ising and Potts models on lattices without external fields. arXiv: 2102.05565 , 2021) for two- or three-dimensional square lattices when the inverse temperature tends to infinity. In this article, we consider the large-volume regime where the size of the lattice grows to infinity. First, we establish an asymptotically sharp threshold such that the ground states are metastable if and only if the inverse temperature exceeds a threshold in a suitable sense. Then, we carry out a detailed analysis of the energy landscape and rigorously establish the Eyring–Kramers formula when the inverse temperature is sufficiently greater than the aforementioned sharp threshold. The proof relies on a detailed characterization of dead-ends appearing in the vicinity of optimal transitions between the ground states and on a combinatorial estimation of the number of configurations lying on a certain energy level.

Self-organization and tunable characteristic lengths of two-dimensional hexagonal superlattices of nanowires directly grown on substrates

The organization of nano-objects on macroscopic surfaces is a key challenge for the technological improvement and implementation of nanotechnologies. For achieving operational functions, it is required to assemble nano-objects as controllable building blocks in highly ordered superstructures. Herein, we demonstrate the growth and self-organization of metallic nanowires on surfaces into hexagonal superlattices with tunable characteristic lengths depending of the stabilizing surfactants employed. Starting from a reacting mixture containing a Pt(111) substrate, a Co organometallic precursor, an amine, and an acid dissolved in a solvent, we quantify the structural evolution of superlattices of vertical single-crystalline Co nanowires on Pt, using a combined analysis of small angle neutron scattering, transmission, and scanning electron microscopies. We show the concerted steps of a spontaneous growth and self-organization of the nanowires into two-dimensional (2D) hexagonal lattice on Pt, at intervals starting from a few hours of reaction to a highly ordered superlattice at longer times. Furthermore, it is shown that apart from long-chain acid and long-chain aliphatic amine pairs used as stabilizers, the combination of a long-chain aliphatic and a short-chain aromatic ligand in the synthesis can also be employed for the nanowire superlattices development. Interestingly, the possibility to employ different pairs allows quantitative modulation of the nanowire arrays, such as the interwire distance and the packing fraction.

Optimization of configurations of atomic species on two-dimensional hexagonal lattices for copper-based systems

We explore the stable configurations of atomic species on a hexagonal plane in two-dimensional (2D) binary systems. We investigate configurations for 9 and 16 atomic sites in the unit cell (3 × 3 and 4 × 4 supercells) by using the density-functional theory approach, where the configuration search is accelerated by the Bayesian optimization. Our results, which assume the 2D copper-based systems of Cu–X, show that the structures having a hexagonal or elongated ring of Cu (or X) atoms containing a X (or Cu) cluster in the unit cell are relatively stable, in particular, for X = Be, Pd, and Au. Assuming a 4 × 4 supercell, we also find such structures in binary Lennard-Jones crystals with moderately weak bonding between different atomic species. The present finding indicates that binary systems may show complex crystal structures if they are neither phase-separated nor strongly bonded systems.

(Digital Presentation) Graphene Meets Ionic Liquids: Fermi Level Engineering via Electrostatic Forces

Graphene, a single layer of carbon atoms in a two-dimensional (2D) hexagonal lattice, showcases key properties suitable for electrode materials such as high specific surface area, good electrical and thermal conductivities.1 Ionic liquids (ILs) are a class of molten salts that are entirely composed of ions. These liquids possess exceptional properties appropriate for electrolytes, such as high electrochemical window, electrochemical stabilities and extremely low vapor pressure. Graphene-based 2D materials are promising candidates for a number of different energy applications. A particularly interesting one is in next generation supercapacitors, where graphene is being explored as an electrode material in combination with ionic liquids as electrolytes. Because the amount of energy that can be stored in such supercapacitors critically depends on the electrode − electrolyte interface, there is considerable interest in understanding the structure and properties of the graphene/IL interface. Here, we present the changes in the properties of graphene upon adsorption of ILs and ILs – organic solvent mixtures using a combination of experimental and theoretical tools. For example, Raman spectroscopy reveals that imidazolium tetrafluoroborate ILs (CnMIM BF4) cause n-type doping of graphene, and the magnitude of doping increases with increasing cation chain length despite the expected decrease in the density of surface-adsorbed ions (Fig 1). Molecular modeling simulations show that doping originates from the changes in the electrostatic potential at the graphene/IL interface. The findings described here represent an important step in developing a comprehensive understanding of the graphene/IL interface.2

Evolution and collision of wave fronts in two-dimensional hexagonal packing granular lattices

Evolution and collision of wave fronts in two-dimensional hexagonal packing granular lattices

Small-world disordered lattices: spectral gaps and diffusive transport

We investigate the dynamic behavior of lattices with disorder introduced through non-local network connections. Inspired by the Watts–Strogatz small-world model, we employ a single parameter to determine the probability of local connections being re-wired, and to induce transitions between regular and disordered lattices. These connections are added as non-local springs to underlying periodic one-dimensional (1D) and two-dimensional (2D) square, triangular and hexagonal lattices. Eigenmode computations illustrate the emergence of spectral gaps in various representative lattices for increasing degrees of disorder. These gaps manifest themselves as frequency ranges where the modal density goes to zero, or that are populated only by localized modes. In both cases, we observe low transmission levels of vibrations across the lattice. Overall, we find that these gaps are more pronounced for lattice topologies with lower connectivity, such as the 1D lattice or the 2D hexagonal lattice. We then illustrate that the disordered lattices undergo transitions from ballistic to super-diffusive or diffusive transport for increasing levels of disorder. These properties, illustrated through numerical simulations, unveil the potential for disorder in the form of non-local connections to enable additional functionalities for metamaterials. These include the occurrence of disorder-induced spectral gaps, which is relevant to frequency filtering devices, as well as the possibility to induce diffusive-type transport which does not occur in regular periodic materials, and that may find applications in dynamic stress mitigation.

Two-dimensional tetragonal and hexagonal lattices of transition metal carbides MC (M = Ti, Zr, Hf): Observation of two nodal loops and strong light-harvesting ability

Two-dimensional (2D) transition metal carbides (TMCs) have captured immense interest for their fascinating physical and chemical properties. Among them, bare carbide MXenes exhibit a metallic feature, greatly limiting the potential applications in optoelectronic devices. Herein, based on the particle swarm optimization and first-principles calculations, we predicted the existence of two single layer (SL) TMCs, namely, tetragonal β-MC and hexagonal γ-MC (M = Ti, Zr, Hf). The two structures are more stable than previously reported α-MC and possess semimetal and semiconductor properties, respectively. We further reveal that the β-MC monolayers have two nodal loops around the Fermi level. SL γ-TiC, ZrC, and HfC have a bandgap of 1.59, 1.24, and 1.57 eV, respectively, and display strong light-harvesting ability in the visible light range. Our research expands the 2D TMCs family beyond the extensively existing MXenes and greatly enriches the carbon chemistry.

Growth of Two-Dimensional Hexagonal Lattices in the Phase-Field Crystal Model

Growth of Two-Dimensional Hexagonal Lattices in the Phase-Field Crystal Model

Topological band transition between hexagonal and triangular lattices with (p x , p y ) 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.

Multi-orbital topolectrical circuit for topological quantum states

Remarkable progress has been made in using electric circuits as a powerful platform to realize a plethora of exotic topological quantum states, even of higher orders and/or dimensions. So far the proposed circuits are restricted to a single-orbital tight-binding model with different lattices. Here, we introduce the concept of a multi-orbital topolectrical circuit and construct practical LC circuits to demonstrate its superiorities. As a proof of concept, we assemble two sets of inductors in one plaquette to simulate a (px, py )-orbital model within a two-dimensional hexagonal lattice. In the presence of spin–orbit coupling, as generated by mixing voltage degrees of freedom, a quantum spin Hall (QSH) state emerges with spin-resolved edge modes propagating along the boundary in the time domain. Implementation of negative impedance converters (NICs) with nonreciprocal links transforms the circuit into a quantum anomalous Hall (QAH) state. Remarkably, we demonstrate that QSH/QAH states can be reversibly switched by tuning the resistance of NIC, and an experimental observable-edge distance ratio is proposed to facilitate the phase transition detection. This work provides an exciting playground for exploring multi-orbital physics in topolectrical circuits, paving the way for future applications in nanoelectronics, telecommunications, signal processing and quantum computing.