Lattice Type Research Articles

Shiba states in systems with density of states singularities

Magnetic impurities placed in the superconductor can lead to emergence of the Yu-Shiba-Rusinov bound states. Coupling between the impurity and the substrate depends on density of states (DOS) at the Fermi level and can be tuned by DOS singularities. In this paper, we study the role of DOS singularities using the real space Bogoliubov-de Gennes equations for chosen lattice models. To uncover the role of these singularities (Dirac point, van Hove singularity, or the flat band), we study honeycomb, kagome, and Lieb lattices. We show that the properties of the Shiba state strongly depends on the type of lattice. Nevertheless some behaviors are generic, e.g. dependence of the critical magnetic coupling on the DOS at the Fermi level. However, the Shiba states realized in the Lieb lattice exhibit extraordinary properties, which can be explained by the presence of a few nonequivalent sublattices. Depending on the location of the magnetic impurity in the chosen sublattice, the value of critical magnetic coupling $J_\text{c}$ can be reduced or enhanced when the flat band is located at the Fermi level. In this context, we also present differences in the local DOS and coherence lengths for different sublattices in the Lieb lattice.

Uniaxial tensile behaviour of additively manufactured elastically isotropic truss lattices made of 316L

Metal additive manufacturing facilitates the production of complex light-weight structures, such as lattice structures, using a wide range of materials. In this work, elastically isotropic truss lattice structures are investigated, produced by laser powder bed fusion (LPBF) from austenitic 316L stainless steel. The lattice specimens are used to explore the effect of different unit cell sizes, orientations and volume fractions on the mechanical behaviour.Quasi-static, uni-axial tensile experiments are carried out with 3D digital image correlation (DIC) measurements. The experiments are validated by finite element (FE) simulations in order to unravel the governing mechanism of the tensile strength and deformation behaviour for the tested specimens. Comparisons are performed by means of force–displacement curves and strain distribution maps.For all lattice types, the geometrical deviations due to manufacturing are quantified and their impact on the mechanical properties is investigated using numerical models. The high resolution DIC measurements provide a detailed insight into the failure mechanisms of the specimens, which are also captured numerically for specific cases. The capabilities of experimental and computational methods are combined to reveal the uncertainties in the mechanical properties of LPBF produced truss lattice structures.

Numerical hybrid thermal MRT-LBM for condensation and boiling phenomena on horizontal walls of different wettability

In this paper, boiling and condensation phenomena are described on walls with mixed wettability using a hybrid thermal model with multiple relaxation times of the Boltzmann lattice type. We start this study by validating our code. To do so, different simulations are performed namely the spreading of a liquid droplet on an adjustable wettability surface by varying its density, the surface tension for different temperatures around the critical point using Laplace’s law which is used to compute the characteristic parameters (Taylor wavelength , characteristic length and characteristic time ), the evaporation of a liquid droplet is evaluated in compraison with the literature work for constant thermal conductivity values. Subsequently, the power law is verified by processing the growth of a liquid droplet for three different cases of wettability. On the other hand, we study the behavior of condensation and boiling processes and their interactions between the boundaries of the solid surface in which they occur. For this purpose, the cavity walls are considered wetting in some areas and non-wetting in others. The results show different behaviors depending on the zones of the walls.

Congruence permutability is prime

We give a combinatorial proof that congruence permutability is prime in the lattice of interpretability types of varieties. Thereby, we settle a 1984 conjecture of Garcia and Taylor.

Improving coarse-grained models of protein folding through weighting of polar-polar/hydrophobic-hydrophobic interactions into crowded spaces.

Herein were tested 7 hydrophobic-polar sequences in two types of 2D-square space lattices, homogeneous and correlated, the latter simulating molecular crowding included as a geometric boundary restriction. Optimization of 2D structures was carried out using a variant of Dill's model, inspired by convex function, taking into account both hydrophobic (Dill's model) and polar interactions, including more structural information to reach better folding solutions. While using correlated networks, degrees of freedom in the folding of sequences were limited; as a result in all cases, more successful structural trials were found in comparison to a homogeneous lattice. The majority of employed sequences were designed by our workgroup, two of them were folded with other approaches, and another is a modified version of a previous sequence, initial forms of the other two have been employed but without taking into account polar-polar contributions. Three of them are newly proposed, intended to test the conjoint hydrophobic-hydrophobic and polar-polar contributions in crowded spaces. One sequence turned out to be the most difficult of the seven folded, this perhaps due to intrinsic (i) degrees of freedom and (ii) motifs of the expected 2D HP structure. Meanwhile two-sequence, although optimal folding was not achieved for neither of the two approaches, folding with correlated network approach not only produced better results than homogeneous space, but for them the best values found with crowding were very close to the expected optimal fitness. In general, five sequences were better folded with medium lattice units for correlated media; instead, another two sequences were better folded with a bit larger degree of lattice unit, revealing that depending on the degrees of freedom and particular folding, motifs in each sequence would require tuned crowding to achieve better folding. Therefore, the main goal herein was to obtain a modified 2D HP lattice model to mimic folding of proteins or secondary structures, like β-sheets, taking into account both hydrophobic-hydrophobic and polar-polar interactions, and fold them in a crowded environment. This simple but enough construction would be conducted to determine the needed information to fold sequences in a sort of a minimal but complete heuristic model. Finally, we claim that all folded sequences into crowded spaces achieve better results than homogeneous ones.

Manipulation of photon states in photonic crystals based on silicon

Coupling between the active centers generated by localized states and the two-dimensional photonic crystals (2D-PCs) has been used to select model in the nano-laser and the optical interconnects on silicon chip. The photonic crystals have been investigated in the simulating calculation, and the physical model has been built in the manipulation of photon states. New quantum phenomena appear in the photonic crystals as well as in the atomic crystals. We have numerically studied on the band structure of 2D-PCs with different types of lattice and various shapes in the optical communication windows. The simulating calculation results show that the band gap of photon states can be effectively manipulated in 2D-PCs by using the competition between the quantum confinement effect and the lattice symmetry effect, in which the band gap width of the photon states will obviously increase by changing the lattice symmetry and by increasing the filling rate of the air hole to restrain photons in 2D-PCs on silicon. It is interesting to make a comparison between the transverse electric (TE) polarization and the transverse magnetic (TM) polarization, in which a wider band gap occurs in the TM mode. By adjusting the defect mode in 2D-PCs, the photon states can be localized into the second and third windows of optical communication for selecting modes and resonating on silicon chip.

Monte Carlo studies of the Blume–Capel model on nonregular two- and three-dimensional lattices: phase diagrams, tricriticality, and critical exponents

We perform Monte Carlo simulations, combining both the Wang–Landau and the Metropolis algorithms, to investigate the phase diagrams of the Blume–Capel model on different types of nonregular lattices (Lieb lattice (LL), decorated triangular lattice (DTL), and decorated simple cubic lattice (DSC)). The nonregular character of the lattices induces a double transition (reentrant behavior) in the region of the phase diagram at which the nature of the phase transition changes from first-order to second-order. A physical mechanism underlying this reentrance is proposed. The large-scale Monte Carlo simulations are performed with the finite-size scaling analysis to compute the critical exponents and the critical Binder cumulant for three different values of the anisotropy , showing thus no deviation from the standard Ising universality class in two and three dimensions. We report also the location of the tricritical point to considerable precision: (Δ t /J = 1.3457(1); k B T t /J = 0.309(2)), (Δ t /J = 1.5766(1); k B T t /J = 0.481(2)), and (Δ t /J = 1.5933(1); k B T t /J = 0.569(4)) for LL, DTL, and DSC, respectively.

Phase shift, ellipticity, angle, and topological number in skyrmion lattices

We theoretically study skyrmion lattices realized in a Kondo lattice model on a triangular lattice, focusing on the phase, ellipticity, and angle of the constituent multiple-Q waves. Analyzing the numerical data obtained in the previous study [Ozawa R, Hayami S and Motome Y 2017 Phys. Rev. Lett. 118 147205], we extract these parameters for the two types of skyrmion lattices with the skyrmion number of 1 and 2. We show that the topological transition between the two skyrmion lattices driven by an external magnetic field is accompanied by significant modulations of all three parameters.

In silico evaluation of lattice designs for additively manufactured total hip implants

Additive manufacturing restructures the fabrication of custom medical implants and transforms the design, topology optimization, and material selection perspectives in biomechanical applications. Additionally, it facilitated the design and fabrication of patient-oriented hip implants. Selection of proper lattice type is critical in additive manufacturing of hip implants. The lattice types reduce the implant mass and, due to higher stress distribution and deformations as compared to the rigid implants, it brings down the stress shielding issues. This study introduces a rigid shell structure and infill lattice hip implant.Additionally, the effect of various lattice unit cell thickness (0.2–1 mm) and elemental size (2.5–5 mm) while applying 2300 N axial force is explored numerically. A cubic structure with two rigid surfaces on the top and bottom is outlined to separate the effect of the hip implant cross-sectional area variations. The stress distribution and deformation characteristics are validated with the hip implant design. The Finite Element Analysis (FEA) demonstrated that the Weaire-Phelan lattice structure exhibits the least stress and deformation among the other types at various design parameters. Additionally, the same methodology is applied to three biocompatible hip implant materials as Ti–6Al–4V, TA15 (Ti–6Al–2Zr–1Mo–1V), and CoCr28Mo6. Finally, the effect of the unit cell thickness and size on the implant's mass reduction considering the lattice's safety factor is investigated for the mentioned materials. The selection of a Weaire–Phelan lattice with the optimized safety factor and mass reduction is represented considering all the results. The optimized parameters for Titanium-based alloys are approximately 3.5 mm unit cell size with 0.6 mm beam thickness. However, the CoCr Mo-based alloy requires a thicker beam size (about 0.8 mm) due to lower safety factors.

Effects of Lattice Types on the Frequency Band Widths in Photonic Crystals

The plane wave expansion method was implemented in modeling and simulating the band structures of two dimensional photonic crystals (PhCs) with square, triangular and honeycomb lattices with circular air holes in dielectric or circular GaAs rods in air. The eigen value equation of TE and TM modes will presented. The Fourier transform of dielectric constant will analyzed and the effects of lattice type will discussed. The eigenvalue equations of TE and TM modes will solved using Matlab environment. Our results show that the first type of PhCs is more flexible than the second type in controlling the band widths and the achieved frequency positions, and the triangular lattice showed the greatest flexibility. On the other hand, this work showed the significant effect of the dielectric constant and the lattice constant on the achieved bandwidth.

Preparation and thermophysical properties of Lu3TaO7

Lu3TaO7 was prepared through a modified sol-gel technique and high-temperature sintering method to investigate its lattice type, micro-morphology, phase stability, thermal expansion coefficient, and conductivity. The results indicate that the obtained powder and bulk samples are composed of a single fluorite structure. The obtained bulk sample shows a dense microstructure, and uniform element distribution. The thermal conductivity of the Lu3TaO7 oxide slightly increases with temperature due to typical amorphous-like behaviour, which is much lower than those of 7 wt% yttrium oxide–stabilized zirconium oxide (7YSZ) and Sm2Zr2O7 ceramics. Furthermore, the enhanced phonon scattering leads to a lower thermal conductivity of Lu3TaO7 than that of Gd3TaO7. The thermal expansion coefficient of the Lu3TaO7 oxide reaches 10.72 × 10−6 K−1 at 1200 °C, which is slightly greater than that of 7YSZ. The outstanding phase stability up to 1200 °C satisfies the requirements for thermal barrier coatings.

Revisiting orthogonal lattice attacks on approximate common divisor problems

In this paper, we revisit three existing types of orthogonal lattice (OL) attacks and propose optimized cases to solve approximate common divisor (ACD) problems. In order to reduce both space and time costs, we also make an improved lattice using the rounding technique. Further, we present asymptotic formulas of the time complexities on our optimizations as well as three known OL attacks. Besides, we give specific conditions that the optimized OL attacks can work and show how the attack ability depends on the blocksize β in the BKZ-β algorithm.

Multivariate Orthogonal Laurent Polynomials and Integrable Systems

An ordering for Laurent polynomials in the algebraic torus (\mathbb{C}^*)^D , inspired by the Cantero–Moral–Vel´azquez approach to orthogonal Laurent polynomials in the unit circle, leads to the construction of a moment matrix for a given Borel measure in the unit torus \mathbb{T}^D . The Gauss–Borel factorization of this moment matrix allows for the construction of multivariate biorthogonal Laurent polynomials in the unit torus, which can be expressed as last quasi-determinants of bordered truncations of the moment matrix. The associated second-kind functions are expressed in terms of the Fourier series of the given measure. Persymmetries and partial persymmetries of the moment matrix are studied and Cauchy integral representations of the second-kind functions are found, as well as Plemelj-type formulae. Spectral matrices give string equations for the moment matrix, which model the three-term relations as well as the Christoffel–Darboux formulae. Christoffel-type perturbations of the measure given by the multiplication by Laurent polynomials are studied. Sample matrices on poised sets of nodes, which belong to the algebraic hypersurface of the perturbing Laurent polynomial, are used to find a Christoffel formula that expresses the perturbed orthogonal Laurent polynomials in terms of a last quasi-determinant of a bordered sample matrix constructed in terms of the original orthogonal Laurent polynomials. Poised sets exist only for prepared Laurent polynomials, which are analyzed from the perspective of Newton polytopes and tropical geometry. Then, an algebraic geometrical characterization of prepared Laurent polynomial perturbation and poised sets is given; full-column-rankness of the corresponding multivariate Laurent–Vandermonde matrices and a product of different prime prepared Laurent polynomials leads to such sets. Some examples are constructed in terms of perturbations of the Lebesgue–Haar measure. Discrete and continuous deformations of the measure lead to a Toda-type integrable hierarchy, being the corresponding flows described through Lax and Zakharov–Shabat equations; bilinear equations and vertex operators are found. Varying size matrix nonlinear partial difference and differential equations of two-dimensional Toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials. The discrete flows are connected with a Gauss–Borel factorization of the Jacobi-type matrices and its quasi-determinants allow for expressions for the multivariate orthogonal polynomials in terms of shifted quasi-tau matrices, which generalize those that relate the Baker functions with ratios of Miwa shifted \tau -functions in the one-dimensional scenario. It is shown that the discrete and continuous flows are deeply connected and determine nonlinear partial difference-differential equations that involve only one site in the integrable lattice behaving as a Kadomtsev–Petviashvili-type system.

Electronic Quantum Materials Simulated with Artificial Model Lattices.

The band structure and electronic properties of a material are defined by the sort of elements, the atomic registry in the crystal, the dimensions, the presence of spin–orbit coupling, and the electronic interactions. In natural crystals, the interplay of these factors is difficult to unravel, since it is usually not possible to vary one of these factors in an independent way, keeping the others constant. In other words, a complete understanding of complex electronic materials remains challenging to date. The geometry of two- and one-dimensional crystals can be mimicked in artificial lattices. Moreover, geometries that do not exist in nature can be created for the sake of further insight. Such engineered artificial lattices can be better controlled and fine-tuned than natural crystals. This makes it easier to vary the lattice geometry, dimensions, spin–orbit coupling, and interactions independently from each other. Thus, engineering and characterization of artificial lattices can provide unique insights. In this Review, we focus on artificial lattices that are built atom-by-atom on atomically flat metals, using atomic manipulation in a scanning tunneling microscope. Cryogenic scanning tunneling microscopy allows for consecutive creation, microscopic characterization, and band-structure analysis by tunneling spectroscopy, amounting in the analogue quantum simulation of a given lattice type. We first review the physical elements of this method. We then discuss the creation and characterization of artificial atoms and molecules. For the lattices, we review works on honeycomb and Lieb lattices and lattices that result in crystalline topological insulators, such as the Kekulé and “breathing” kagome lattice. Geometric but nonperiodic structures such as electronic quasi-crystals and fractals are discussed as well. Finally, we consider the option to transfer the knowledge gained back to real materials, engineered by geometric patterning of semiconductor quantum wells.

Control and Optimization of Lattice Converters

Multilevel converters continue their upward trend in renewable generation, electric vehicles, and power quality conditioning applications. Despite having satisfactory voltage capabilities, mainstream multilevel converters suffer from poor current sharing performances, thereby leading to the development of lattice converters, i.e., a strong and versatile type of future multilevel power converters. This article addresses two problems faced by lattice converters. First, we propose and detail how to optimize the efficiency of a given lattice converter by controlling the on/off states of H-bridge submodules. Second, we introduce the method that determines the voltage at each node of the converter in order to satisfy output voltage and current requirements. Design and analysis of lattice converters need a different mathematical toolbox than routinely exercised in power electronics. By use of graph theory, this article provides control methods of 3 × 3 and 4 × 4 lattice converters, satisfying various control objectives such as input/output terminals and output voltages. We further validate the methods with simulation results. The methodologies, algorithms, and special cases described in the article will aid further design and refinement of more efficient and easy-to-control lattice converters.

Correlated Electrons and Mass Enhancement on the β-Mn Type Lattice

Electronic structure and electron correlations of the Hubbard model on the Beta-Mn lattice have been investigated on the basis of the momentum dependent local ansatz wavefunction method. It is found that the electrons on the Beta-Mn lattice are significantly correlated as compared with those on the face centered cubic (fcc) and the body centered cubic (bcc) lattices; the correlation energy gain of the Beta-Mn lattice structure is the largest among the Beta-Mn, the fcc, and the bcc structures, and the charge fluctuations show a significant suppression. In particular, it is found that the mass enhancement factor on the Beta-Mn lattice shows a large peak around the electron number per atom n=1.1 when n is varied at the typical Coulomb interaction energy strength of transition metals. It is pointed that the large mass enhancement may explain anomalous electronic specific heat coefficient in Beta-Mn.

Iron(II) Complexes of 4-(Alkyldisulfanyl)-2,6-di(pyrazolyl)pyridine Derivatives. Correlation of Spin-Crossover Cooperativity with Molecular Structure Following Single-Crystal-to-Single-Crystal Desolvation.

The complex salts [Fe(L1)2]X2 (1X2; L1 = 4-(isopropyldisulfanyl)-2,6-bis(pyrazolyl)pyridine; X– = BF4–, ClO4–) form solvated crystals from common organic solvents. Crystals of 1X2·Me2CO show abrupt spin transitions near 160 K, with up to 22 K thermal hysteresis. 1X2·Me2CO cocrystallizes with other, less cooperative acetone solvates, which all transform into the same solvent-free materials 1X2·sf upon exposure to air, or mild heating. Conversion of 1X2·Me2CO to 1X2·sf proceeds in a single-crystal to single-crystal fashion. 1X2·sf are not isomorphous with the acetone solvates, and exhibit abrupt spin transitions at low temperature with hysteresis loops of 30–38 K (X– = BF4–) and 10–20 K (X– = ClO4–), depending on the measurement method. Interestingly, the desolvation has an opposite effect on the SCO temperature and hysteresis in the two salts. The hysteretic spin transitions in 1X2·Me2CO and 1X2·sf do not involve a crystallographic phase change but are accompanied by a significant rearrangement of the metal coordination sphere. Other solvates 1X2·MeNO2, 1X2·MeCN, and 1X2·H2O are mostly isomorphous with each other and show more gradual spin-crossover equilibria near room temperature. All three of these lattice types have similar unit cell dimensions and contain cations associated into chains through pairwise, intermolecular S···π interactions. Polycrystalline [Fe(L2)2][BF4]2·MeNO2 (2[BF4]2·MeNO2; L2 = 4-(methyldisulfanyl)-2,6-bis(pyrazolyl)pyridine) shows an abrupt spin transition just above room temperature, with an unsymmetrical and structured hysteresis loop, whose main features are reversible upon repeated thermal scanning.

Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions.

Extended-range percolation on various regular lattices, including all 11 Archimedean lattices in two dimensions and the simple cubic (sc), body-centered cubic (bcc), and face-centered cubic (fcc) lattices in three dimensions, is investigated. In two dimensions, correlations between coordination number z and site thresholds p_{c} for Archimedean lattices up to 10th nearest neighbors (NN) are seen by plotting z versus 1/p_{c} and z versus -1/ln(1-p_{c}) using the data of d'Iribarne etal. [J. Phys. A 32, 2611 (1999)JPHAC50305-447010.1088/0305-4470/32/14/002] and others. The results show that all the plots overlap on a line with a slope consistent with the theoretically predicted asymptotic value of zp_{c}∼4η_{c}=4.51235, where η_{c} is the continuum threshold for disks. In three dimensions, precise site and bond thresholds for bcc and fcc lattices with 2nd and 3rd NN, and bond thresholds for the sc lattice with up to the 13th NN, are obtained by Monte Carlo simulations, using an efficient single-cluster growth method. For site percolation, the values of thresholds for different types of lattices with compact neighborhoods also collapse together, and linear fitting is consistent with the predicted value of zp_{c}∼8η_{c}=2.7351, where η_{c} is the continuum threshold for spheres. For bond percolation, Bethe-lattice behavior p_{c}=1/(z-1) is expected to hold for large z, and the finite-z correction is confirmed to satisfy zp_{c}-1∼a_{1}z^{-x}, with x=2/3 for three dimensions as predicted by Frei and Perkins [Electron. J. Probab. 21, 56 (2016)1083-648910.1214/16-EJP6] and by Xu etal. [Phys. Rev. E 103, 022127 (2021)2470-004510.1103/PhysRevE.103.022127]. Our analysis indicates that for compact neighborhoods, the asymptotic behavior of zp_{c} has universal properties, depending only on the dimension of the system and whether site or bond percolation but not on the type of lattice.

Diffusive topological transport in spatiotemporal thermal lattices

Topological phases have been studied in photonic, acoustic and phononic metamaterials, promising a range of applications. Such topological modes usually stem from collective resonant effects in periodic lattices. One may, therefore, expect similar features to be forbidden for thermal diffusion that is purely dissipative and mostly incoherent, prohibiting collective resonances. Here we report the discovery of diffusion-based topological states supported by spatiotemporally modulated advections stacked over a fluidic surface. This arrangement imitates a periodic propagating potential in an effective thermal lattice. We observe edge states in topologically non-trivial and bulk states in topologically trivial lattices. Interface states form at boundaries between these two types of lattice, manifesting inhomogeneous thermal properties on the fluidic surface. Our findings establish a framework for topological diffusion and thermal edge or bulk states, and it may allow a distinct mechanism for the flexible manipulation of diffusive phenomena for robust heat and mass transfer.

Thermophysical performances of high-entropy (La0.2Nd0.2Yb0.2Y0.2Sm0.2)2Ce2O7 and (La0.2Nd0.2Yb0.2Y0.2Lu0.2)2Ce2O7 oxides

Two High-entropy ceramics, (La 0.2 Nd 0.2 Yb 0.2 Y 0.2 Sm 0.2 ) 2 Ce 2 O 7 (LNYSC) and (La 0.2 Nd 0.2 Yb 0.2 Y 0.2 Lu 0.2 ) 2 Ce 2 O 7 (LNYLC) were synthesized using a sol-gel technique and sintering at high temperatures. The lattice type, micro-morphology, and thermophysical and mechanical properties of these oxides were investigated. The results showed that the obtained high-entropy powders and bulk samples have single fluorite lattice, dense microstructure, and clear grain boundary. These oxides have less thermal conductivities than (7YSZ), and the thermal conductivity of LNYLC is lower than that of LNYSC. The thermal expansion coefficients of these high-entropy oxides are larger than that of 7YSZ. Owing to the relative less negative electrical difference in LNYLC that in LNYSC, the thermal expansion coefficient of LNYLC is larger than that of LNYSC. Further, these oxides exhibit outstanding lattice stability up to 1200 °C, which can help to improve the durability of thermal barrier coatings. However, these oxides have lower mechanical properties compared to 7YSZ.