Zigzag Boundary Research Articles

Convergence of generalized MIT bag models to Dirac operators with zigzag boundary conditions

Convergence of generalized MIT bag models to Dirac operators with zigzag boundary conditions

Isotropy of fatigue crack growth in hybrid additive manufactured Ti6Al4V ELI titanium alloy

In the present paper, the effects of macrostructure and microstructure on fatigue crack growth (FCG) behavior of wire and arc additive manufacturing (WAAM) and hybrid additive manufacturing with in-situ rolling (HAMR) were investigated in three sampling directions for Ti6Al4V ELI. The results demonstrate that WAAMed samples exhibit coarse columnar grains with zigzag boundaries, interlayer heat affected band (HAB), and basketweave microstructures. The fatigue crack growth rate (FCGR) is lowest in the Y-X direction while being similar between the X–Y and X-Z directions, indicating significant anisotropy. This difference can be attributed to changes in plastic region size and internal slip characteristics near the crack tip caused by variations in macrostructure and microstructure. After applying HAMR with quasi-β heat treatment, interlayer HABs are eliminated, resulting in uniform equiaxed grains and colony microstructures. The FCGR curves in three directions significantly overlap without any observed anisotropy, which are lower than those obtained for traditionally forged materials. Therefore, the isotropic FCGR of plastic materials can be reduced by achieving a nearly flawless metallurgical quality and ensuring uniform equiaxed grain matching with the corresponding critical colony size and fine thickness lamellae. The findings suggest that the application of quasi-β heat treatment in conjunction with HAMR offers a promising high-efficiency manufacturing technique for producing isotropic titanium alloy forgings characterized by exceptional damage tolerance.

Evaluation of topological protection in kagome lattice-based thermal diffusion systems

We evaluated topological protection for edge and higher-order corner states in topological diffusion systems based on the breathing kagome lattice. In the kagome lattice, the corner states appear at the corner boundary where all three Wannier centers in nontrivial unit cells are located. The three Wannier centers in a unit cell can be placed on the obtuse- and acute-corner boundaries utilizing the armchair boundary, generating topological acute- and obtuse-corner states. For another representative zigzag boundary, only the acute-corner unit cell has three Wannier centers located at the boundary; hence, only the acute-corner state appears. Our band analysis and numerical studies show that the topologically protected decay behavior for armchair boundaries is as robust as that for zigzag boundaries, unlike wave phenomena with space and time periodicities. Our findings can guide the flexible design of topological diffusion applications such as heat localization and recovery systems.

Nernst and seebeck effects in α−T3 lattice

By using the tight-binding Hamiltonian and non-equilibrium Green’s function methods, the Seebeck and Nernst effects of α−T3 lattice are investigated, in which the lattice interpolates between graphene and the dice lattice via the parameter α. For α = 0 (graphene), flat bands are always present in the band structure. The Seebeck and Nernst coefficients are consistent with those in graphene. When α is non-zero at zero magnetic field, the Seebeck coefficient is an odd function of the Fermi energy. It produces a very large and wide first peak within the band gap for the zigzag boundary. Under the influence of magnetic fields, the first peak of the Seebeck coefficient in the gap region increases with α increasing. The Nernst effect occurs under the influence of a magnetic field. The height of the zeroth peak of the Nernst coefficient increases with α increasing. When α reaches a certain value, the zeroth peak splits. The post-split peak decreases with α increasing for the zigzag boundary, but continues to become wider and higher for the armchair boundary.

Detecting the spin-polarization of edge states in graphene nanoribbons

Low dimensional carbon-based materials can show intrinsic magnetism associated to p-electrons in open-shell π-conjugated systems. Chemical design provides atomically precise control of the π-electron cloud, which makes them promising for nanoscale magnetic devices. However, direct verification of their spatially resolved spin-moment remains elusive. Here, we report the spin-polarization of chiral graphene nanoribbons (one-dimensional strips of graphene with alternating zig-zag and arm-chair boundaries), obtained by means of spin-polarized scanning tunnelling microscopy. We extract the energy-dependent spin-moment distribution of spatially extended edge states with π-orbital character, thus beyond localized magnetic moments at radical or defective carbon sites. Guided by mean-field Hubbard calculations, we demonstrate that electron correlations are responsible for the spin-splitting of the electronic structure. Our versatile platform utilizes a ferromagnetic substrate that stabilizes the organic magnetic moments against thermal and quantum fluctuations, while being fully compatible with on-surface synthesis of the rapidly growing class of nanographenes.

Explicit derivation of the chiral and generic helical edge states for the Kane-Mele model: Closed expressions for the wave function, dispersion relation, and spin rotation

Although one of the most important and intriguing features of the topological insulators is the presence of edge states, the closed-form expressions for the edge states of some famous topological models are still lacking. Here, we focus on the Kane-Mele model with and without Rashba spin-orbit coupling as a well-known model to describe a two-dimensional version of the ${\mathbb{Z}}_{2}$ topological insulator to study the properties of its edge states analytically. By considering the tight-binding model on a honeycomb lattice with zigzag boundaries and introducing a perturbative approach, we derive explicit expressions for the wave functions, energy dispersion relations, and the spin rotations of the (generic) helical edge states. To this end, we first map the edge states of the ribbon geometry into an effective two-leg ladder model with momentum-dependent energy parameters. Then, we split the Hamiltonian of the system into an unperturbed part and a perturbation. The unperturbed part has a flat-band energy spectrum and can be solved exactly, which allows us to consider the remaining part of the Hamiltonian perturbatively. The resulting energy dispersion relation within the first-order perturbation, surprisingly, is in excellent agreement with the numerical spectra over a very wide range of wave numbers. Our perturbative framework also allows deriving an explicit form for the rotation of the spins of the momentum edge states in the absence of axial spin symmetry due to the Rashba spin-orbit interaction.

A 2.5D Finite Element Method Combined with Zigzag-Paraxial Boundary for Long Tunnel under Obliquely Incident Seismic Wave

Seismic waves propagation with an oblique angle to the tunnel axis will cause asynchronous tunnel motions and have a significant effect on the axial response. A high-precision 2.5D finite element method is established in the frequency domain to simulate the 3D seismic response of the tunnel. This method avoids the disturbance caused by the truncation of the tunnel in the longitudinal direction. Meanwhile, a 2.5D zigzag-paraxial boundary is derived to further improve the calculation efficiency of the 2.5D finite element model. Moreover, by combining the 2.5D finite element method, 2.5D zigzag boundary condition and seismic motion input methods, an obliquely incident substructure method for plane seismic waves is built by converting the plane seismic wave into equivalent nodal forces. The proposed 2.5D finite element method is verified by comparing with a reference solution. Finally, the 2.5D finite element method is applied to study the seismic response of the long lined tunnel. Parameter analyses illustrate that the wave propagation effect to the tunnel axis has a non-negligible influence on the axil deformation of long tunnels.

Microstructure Evolution in a GOES Thin Strip

This paper focuses on the evolution of the microstructure in a grain-oriented electrical steel (GOES) thin strip after casting. After solidification, the microstructure consisted of delta-ferrite. A small fraction of austenite was formed during the cooling of the thin strip in the two-phase region (gamma+delta). Fine Cr2CuS4 particles precipitated in the ferrite and along the delta/gamma interfaces. Laths of primary Widmanstätten austenite (WA) nucleated directly on the high-angle delta-ferrite grain boundaries. The formation of WA laths in both adjacent ferritic grains resulted in a zig-zag shape of delta-ferrite grain boundaries due to their local rotation during austenite nucleation. Based on the EBSD results, a mechanism of the formation of the zig-zag grain boundaries has been proposed. Besides the Widmanstätten morphology, austenite also formed as films along the delta-ferrite grain boundaries. Sulfide precipitation along the delta/gamma interfaces made it possible to prove that austenite decomposition upon a drop in temperature was initiated by the formation of epitaxial ferrite. Further cooling brought the decay of austenite to either pearlite or a mixture of plate martensite and some retained austenite.

Smoothing topology optimization results using pre-built lookup tables

Topology optimization techniques are typically performed on a design domain discretized with finite element meshes to generate efficient and innovative structural designs. The optimized structural topologies usually exhibit zig-zag boundaries formed from straight element edges. Existing techniques to obtain smooth structural topologies are limited. Most methods are computationally expensive, as they are performed iteratively with topology optimization. Other methods, such as post-processing methods, are applied after topology optimization, but they cannot guarantee to obtain equivalent structural designs, as the volume and geometric features may be changed. This study presents a new method that uses pre-built lookup tables to transform the shape of boundary elements obtained from topology optimization to create smoothed structural topologies. The new method is developed based on the combination of the bi-directional evolutionary structural optimization (BESO) technique and marching geometries to determine structural topologies and lookup tables, respectively. An additional step is used to ensure that the generated result meets a target volume. A variety of 2D and 3D examples are presented to demonstrate the effectiveness of the new method. This research shows that the new method is highly efficient, as it can be directly added to the last step of topology optimization with a low computational cost, and the volume and geometric features can be preserved in smoothed topologies. Finite element models are also created for original and smoothed structural topologies to show that the structural stiffness can be significantly enhanced after smoothing.

Polarization Raman spectra of graphene nanoribbons

The on-surface synthesis method allows the fabrication of atomically precise narrow graphene nanoribbons (GNRs), which bears great potential in electronic applications. Here, we synthesize armchair graphene nanoribbons (AGNRs) and chevron-type graphene nanoribbons (CGNRs) array on a vicinal Au(11 11 12) surface using 10,10′-dibromo-9,9′-bianthracene (DBBA) and 6,12-dibromochrysene (DBCh) as precursors, respectively. This process creates spatially well-aligned GNRs, as characterized by scanning tunneling microscopy. AGNRs show strong Raman linear polarizability for application in optical modulation devices. Different from the distinct polarization of AGNRs, only weak polarization exists in CGNRs polarized Raman spectrum, which suggests that the presence of the zigzag boundary in the nanoribbon attenuates the polarization rate as an important factor affecting the polarization. We analyze the Raman activation mode of CGNRs using the peak polarization to expand the application of the polarization Raman spectroscopy in nanoarray analysis.

Electronic characteristics of SbBi binary nanoflakes

The structural and electronic characteristics of the Antimony–Bismuth binary monolayer is presented here. The electronic gap is reported as a function of the nanoflake dimensions. Tunable photocatalytic properties are observed. The stability of the SbBi binary nanoflakes falls by widening the nanosheet from both sides of the Zigzag and Armchair edges. The dependency of the energy bandgap between frontier orbital levels concerning the nanoflakes width across either of the Zigzag and Armchair boundaries are explored for structures having maximum of 388 atoms. Regions of chemical activity in SbBi monolayers are mainly focused on their middling points, leading to negligible chemical activity along the edges. Furthermore, some of the considered nanoflakes ions exhibit strong spin-exchange splitting.

In-Plane Magnetization-Induced Corner States in Bismuthene

We theoretically demonstrate that the electronic second-order topological insulator with robust corner states, having a buckled honeycomb lattice, can be realized in bismuthene by inducing in-plane magnetization. Based on the sp 3 Slater–Koster tight-binding model with parameters extracted from first-principles results, we show that spin-helical edge states along zigzag boundaries are gapped out by the in-plane magnetization whereas four robust in-gap electronic corner states at the intersection between two zigzag boundaries arise. By regulating the orientation of in-plane magnetization, we show different position distribution of four corner states with different energies. Nevertheless, it respects some spatial symmetries and thus can protect the higher-order topological phase. Combined with the Kane–Mele model, we discuss the influence of the magnetization orientation on the position distribution of corner states.

A lattice Boltzmann model for shallow water equations on the smoothed quadtree grid

AbstractIn hydraulic simulation, the fine land‐water boundary often means more accurate result but more lattices and time cost. In this study, we developed a smoothed quadtree grid and introduced it into the lattice Boltzmann model for the shallow water equations (LABSWE) to balance this problem. The recommended grid is meticulously designed to own a minimum width of advantage over a uniform grid or multi‐block grid, and can describe the zigzag boundary via smaller size lattices with acceptable total number of lattice. As an improved multi‐block grid, the lattices are numbered in serialization for facilitating data storage and parallel computation, and a standardized process is developed to employ this grid in the LABSWE. The spatial interpolation is modified based on the water surface and fluxes instead of the water depth and velocities, in order to satisfy the necessary property (N‐property: the numerical scheme can replicate the exact solution to a stationary case in which there is a nonzero force or source term). The proposed model is verified by two benchmark tests and then applied to the long‐term simulation of unsteady flows in large aquatorium. The agreement between predictions and analytical results or other solutions are satisfactory. Three popular collision schemes, the Bhatnagar–Gross–Krook (BGK) scheme, the multiple relaxation time (MRT) scheme and the central moments (CMs) scheme, have also been tested, showing that the stability of the MRT and CMs schemes are better than the BGK scheme for the flows with higher Reynolds number, but cost only 4%–6% more computational load.

Shearing Characteristics of Cu-Cu Joints Fabricated by Two-Step Process Using Highly <111>-Oriented Nanotwinned Cu

Cu-Cu bonding has the potential to break through the extreme boundary of scaling down chips’ I/Os into the sub-micrometer scale. In this study, we investigated the effect of 2-step bonding on the shear strength and electrical resistance of Cu-Cu microbumps using highly <111>-oriented nanotwinned Cu (nt-Cu). Alignment and bonding were achieved at 10 s in the first step, and a post-annealing process was further conducted to enhance its bonding strength. Results show that bonding strength was enhanced by 2–3 times after a post-annealing step. We found 50% of ductile fractures among 4548 post-annealed microbumps in one chip, while the rate was less than 20% for the as-bonded counterparts. During the post-annealing, interfacial grain growth and recrystallization occurred, and the bonding interface was eliminated. Ductile fracture in the form of zig-zag grain boundary was found at the original bonding interface, thus resulting in an increase in bonding strength of the microbumps.

Effect of asymmetry on terahertz transmissions in topological photonic crystals comprising of dielectric rod structures

We numerically demonstrate robust terahertz (THz) transmissions in a topological photonic crystal comprised of dielectric rod structures. Topological edge states (i.e., the vortex states) in the structure are excited via a zig-zag domain boundary at the interface of two nontrivial valley photonic crystals (VPCs). A comprehensive study of the effect of asymmetry (defined as δd=d1−d2) on transmission loss and bandwidth is performed for three different kinds of domain boundaries: straight, Z-shape, and Ω-shape. An increase in the bandwidth of the topological THz transmission is achieved with the increasing asymmetry. The study further demonstrates blue shifting of the topological transmission bandwidth with the increase in asymmetry. An optimized value of the asymmetry parameter is obtained based on the bandwidth as well as the transmission loss. The most robust propagation in terms of the widest bandwidth and low loss transmission is achieved with asymmetry ≥0.25a. The robust THz transmission is attributed to the strong edge modes confinement at the domain boundary of the VPC due to intervalley scattering suppression at this optimal value. Our observations open the path for a strong electromagnetic wave flow in the THz regime with a high transmission, which could be useful in photonic broadband communication and other on-chip applications at THz frequencies, particularly for 6G communications.

Failure Mechanisms of Cu-Cu Bumps under Thermal Cycling.

The failure mechanisms of Cu–Cu bumps under thermal cycling test (TCT) were investigated. The resistance change of Cu–Cu bumps in chip corners was less than 20% after 1000 thermal cycles. Many cracks were found at the center of the bonding interface, assumed to be a result of weak grain boundaries. Finite element analysis (FEA) was performed to simulate the stress distribution under thermal cycling. The results show that the maximum stress was located close to the Cu redistribution lines (RDLs). With the TiW adhesion layer between the Cu–Cu bumps and RDLs, the bonding strength was strong enough to sustain the thermal stress. Additionally, the middle of the Cu–Cu bumps was subjected to tension. Some triple junctions with zig-zag grain boundaries after TCT were observed. From the pre-existing tiny voids at the bonding interface, cracks might initiate and propagate along the weak bonding interface. In order to avoid such failures, a postannealing bonding process was adopted to completely eliminate the bonding interface of Cu–Cu bumps. This study delivers a deep understanding of the thermal cycling reliability of Cu–Cu hybrid joints.

Topology optimization and de-homogenization of graded lattice structures based on asymptotic homogenization

This paper presents a novel two-step homogenization-based topology optimization and de-homogenization method for the design of graded lattice structures. The lattice orientation and material layout are first optimized for square base cells in the macro scale. Then by introducing the lattice stretching design variables of micro base cells, which bridge the base cell distortion with lattice stretch, the error residual of mapping functions is integrated with compliance formulation to form a novel mixed optimization formulation, concurrently optimizing structural performance and mapping functions. The advantage of this formulation is two-fold. First, the micro design space is relaxed from square base cells to rectangular ones so that performance improvement is further expected. Second, an excellent agreement, in both shape and performance, between the projected single-scale lattice structures with the homogenization results is secured, as compared to the frequently adopted post-process procedure of constructing single-scale lattices, where performance deviation could arise for specific microstructural patterns. With the optimized mapping functions, de-homogenization procedure is carried out to construct single-scale spatially graded lattice structures, where a simple filter-projection operation is proposed to obtain fine-scale smoothed boundaries from coarse-scale homogenization results with zig-zag boundaries. Several numerical examples are presented and compared with conventional post-process treatment results to show the validity of the proposed method, and different kinds of lattice patterns adopted in this work show its versatility for a broad range of lattice patterns.

Discrete cosine and sine transforms generalized to honeycomb lattice II. Zigzag boundaries

The discrete cosine and sine transforms are generalized to a triangular fragment of the honeycomb lattice with zigzag boundaries. The zigzag honeycomb point sets are constructed by subtracting the weight lattice from the refined root lattice points of the crystallographic root system A2. The two-variable (anti)symmetric orbit functions of the Weyl group of A2, discretized simultaneously on the triangular fragments of the root and weight lattices, induce a novel parametric family of zigzag extended Weyl and Hartley orbit functions. As specific linear combinations of the original orbit functions, the zigzag extended orbit functions retain the Neumann and Dirichlet boundary conditions. Three types of discrete complex Fourier–Weyl transforms and real-valued Hartley–Weyl transforms are detailed. The corresponding unitary transform matrices and interpolating behavior of the discrete transforms are exemplified.

Effect of various combinations of Ti and Zr interlayers on the tensile properties of laser welded joints of molybdenum

Molybdenum (Mo) is of great potential in the nuclear energy field, however, the embrittlement and the lower tensile strength limit the application of the fusion welded joints. Titanium (Ti) and zirconium (Zr) were synchronously added into the fusion zone (FZ) to improve the tensile strength and certain ductility of laser beam welding joints. By utilizing high-resolution scanning electron microscopy (SEM), electron backscatter diffraction (EBSD) and nanoindentation, the microstructures and mechanical properties of FZs in the joints under different alloying strategies with Ti and Zr were analyzed. Zr is mainly present in Mo as Mo2Zr, playing the second-phase strengthening effect; Ti is mainly present in Mo in an atomic state, playing a role of the solid - solution strengthening. The addition of Ti and Zr with different mass proportions can control the relative contents of Mo2Zr phase and solid - solution atoms in the FZ, control the distribution of zigzag and flat grain boundaries and influence the relative distributions of low - angle grain boundaries (LAGBs) and high - angle grain boundaries (HAGBs). In the end, combined addition of Ti and Zr is possible to improve the tensile strength while decrease the loss of ductility of the joint.

Current distribution and group velocities for electronic states on lattice ribbons in a magnetic field

We study the group velocities of electronic states and distributions of currents in lattice ribbons under a uniform perpendicular magnetic field. Using the effective low-energy model we analyze all possible simple configurations of lattice termination with zigzag and armchair boundaries. We show that the edge current depends on the type of zigzag termination, and can be zero or finite near the edge. Also similar dependence is observed in the case of armchair termination and is related to the size of the ribbon. The nonzero current flowing along the edge can be used a signature of formation of propagating edge states. In addition, we show the qualitative difference in the distribution of the edge current between the case of α = 1 (dice model) and other values of model parameter α ≠ 1 for armchair-terminated ribbons.