Two-dimensional Rectangular Lattice Research Articles

Recurrence solution of monomer-polymer models on two-dimensional rectangular lattices

Recurrence solution of monomer-polymer models on two-dimensional rectangular lattices

Magnetic ground state and strain-mediated chiral-like atomic distortion behavior in a two-dimensional rectangular spin lattice

Magnetic ground state and strain-mediated chiral-like atomic distortion behavior in a two-dimensional rectangular spin lattice

Eigenstate thermalization hypothesis in two-dimensional XXZ model with or without SU(2) symmetry.

We investigate the eigenstate thermalization properties of the spin-1/2 XXZ model in two-dimensional rectangular lattices of size L_{1}×L_{2} under periodic boundary conditions. Exploiting the symmetry property, we can perform an exact diagonalization study of the energy eigenvalues up to system size 4×7 and of the energy eigenstates up to 4×6. Numerical analysis of the Hamiltonian eigenvalue spectrum and matrix elements of an observable in the Hamiltonian eigenstate basis supports that the two-dimensional XXZ model follows the eigenstate thermalization hypothesis. When the spin interaction is isotropic, the XXZ model Hamiltonian conserves the total spin and has SU(2) symmetry. We show that the eigenstate thermalization hypothesis is still valid within each subspace where the total spin is a good quantum number.

Enhancement of third harmonic generation induced by surface lattice resonances in plasmonic metasurfaces.

We investigate experimentally third harmonic generation (THG) from plasmonic metasurfaces consisting of two-dimensional rectangular lattices of centrosymmetric gold nanobars. By varying the incidence angle and the lattice period, we show how surface lattice resonances (SLRs) at the involved wavelengths are the major contributors in determining the magnitude of the nonlinear effects. A further boost on THG is observed when we excite together more than one SLR, either at the same or at different frequency. When such multiple resonances take place, interesting phenomena are observed, such as maximum THG enhancement for counter-propagating surface waves along the metasurface, and cascading effect emulating a third-order nonlinearity.

Ordered structure constructed from C2-symmetric hexa-peri-hexabenzocoronene linked with oligo(dimethylsiloxane).

The preparation of sub-5-nm ordered structures is very important to the development of today's nanotechnology. Block molecules have the potential to form structures with significantly small characteristic dimensions. Herein two novel organic-inorganic block molecules composed of a hexa-peri-hexabenzocoronene (HBC) core and two oligo(dimethylsiloxane) (ODMS) tails with C2 symmetry are reported. A hierarchical lamello-columnar structure with a two-dimensional rectangular lattice where HBC cores adopt a tilted arrangement was obtained from their bulk self-assembly. The feature sizes are all below 5 nm and can be regulated via the number of ODMS chains. Sub-5-nm line structures were obtained through spin-coating of the block molecules onto silicon substrates modified with poly(dimethylsiloxane). As organic-inorganic hybrid materials, these block molecules may be further applied in sub-5-nm nanopatterning.

Metastability phenomena in two-dimensional rectangular lattices with nearest-neighbour interaction

We study analytically the dynamics of two-dimensional rectangular lattices with periodic boundary conditions. We consider anisotropic initial data supported on one low-frequency Fourier mode. We show that, in the continuous approximation, the resonant normal form of the system is given by integrable PDEs. We exploit the normal form in order to prove the existence of metastability phenomena for the lattices. More precisely, we show that the energy spectrum of the normal modes attains a distribution in which the energy is shared among a packet of low-frequencies modes; such distribution remains unchanged up to the time-scale of validity of the continuous approximation.

A First Order Phase Transition Studied by an Ising-Like Model Solved by Entropic Sampling Monte Carlo Method

Two-dimensional (2D) square, rectangular and hexagonal lattices and 3D parallelepipedic lattices of spin crossover (SCO) compounds which represent typical examples of first order phase transitions compounds are studied in terms of their size, shape and model through an Ising-like Hamiltonian in which the fictitious spin states are coupled via the respective short and long-range interaction parameters J, and G. Furthermore, an environmental L parameter accounting for surface effects is also introduced. The wealth of SCO transition properties between its bi-stable low spin (LS) and high spin (HS) states are simulated using Monte Carlo Entropic Sampling (MCES) method which favors the scanning of macro states of weak probability occurrences. For given J and G, the focus is on surface effects through parameter L. It is shown that the combined first-order phase transition effects of the parameters of the Hamiltonian can be highlighted through two typical temperatures, TO.D., the critical order-disorder temperature and Teq the equilibrium temperature that is fixed at zero effective ligand field. The relative positions of TO.D. and Teq control the nature of the transition and mediate the width and position of the thermal hysteresis curves with size and shape. When surface effects are negligible (L = 0), the equilibrium transition temperature, Teq. becomes constant, while the thermal hysteresis’ width increases with size. When surface effects are considered, L ≠ 0, Teq. increases with size and the first order transition vanishes in favor of a gradual transition until reaching a threshold size, below which a reentrance phenomenon occurs and the thermal hysteresis reappears again, as shown for hexagonal configuration.

Dynamics of curved travelling fronts for the discrete Allen-Cahn equation on a two-dimensional lattice

<p style='text-indent:20px;'>In this paper we consider the discrete Allen-Cahn equation posed on a two-dimensional rectangular lattice. We analyze the large-time behaviour of solutions that start as bounded perturbations to the well-known planar front solution that travels in the horizontal direction. In particular, we construct an asymptotic phase function <inline-formula><tex-math id="M1">\begin{document}$ \gamma_j(t) $\end{document}</tex-math></inline-formula> and show that for each vertical coordinate <inline-formula><tex-math id="M2">\begin{document}$ j $\end{document}</tex-math></inline-formula> the corresponding horizontal slice of the solution converges to the planar front shifted by <inline-formula><tex-math id="M3">\begin{document}$ \gamma_j(t) $\end{document}</tex-math></inline-formula>. We exploit the comparison principle to show that the evolution of these phase variables can be approximated by an appropriate discretization of the mean curvature flow with a direction-dependent drift term. This generalizes the results obtained in [<xref ref-type="bibr" rid="b47">47</xref>] for the spatially continuous setting. Finally, we prove that the horizontal planar wave is nonlinearly stable with respect to perturbations that are asymptotically periodic in the vertical direction.

Transforming a C 3-Symmetrical Liquid Crystal to a π-Gelator by Alkoxy Chain Variation.

Rational understanding of the structural features involving different noncovalent interactions is necessary to design a liquid crystal (LC) or an organogelator. Herein, we report the effect of the number and positions of alkoxy chains on the self-assembly induced physical properties of a few π-conjugated molecules. For this purpose, we designed and synthesized three C3-symmetrical molecules based on oligo(p-phenylenevinylene), C3OPV1–3. The self-assembly properties of these molecules are studied in the solid and solution states. All of the three molecules follow the isodesmic self-assembly pathway. Upon cooling from isotropic melt, C3OPV1 having nine alkoxy chains (−OC12H25) formed a columnar phase with two-dimensional rectangular lattice and retained the LC phase even at room temperature. Interestingly, when one of the −OC12H25 groups from each of the end benzene rings is knocked out, the resultant molecule, C3OPV2 lost the LC property, however, transformed as a gelator in toluene and n-decane. Surprisingly, when the −OC12H25 group from the middle position is removed, the resultant molecule C3OPV3 failed to form either the LC or the gel phases.

Asymptotic behavior for a long-range Domany–Kinzel model

We consider a long-range Domany–Kinzel model proposed by Li and Zhang (1983), such that for every site (i,j) in a two-dimensional rectangular lattice there is a directed bond present from site (i,j) to (i+1,j) with probability one. There are also m+1 directed bounds present from (i,j) to (i−k+1,j+1), k=0,1,…,m with probability pk∈[0,1), where m is a non-negative integer. Let τm(M,N) be the probability that there is at least one connected-directed path of occupied edges from (0,0) to (M,N). Defining the aspect ratio α=M∕N, we derive the correct critical value αm,c∈R such that as N→∞, τm(M,N) converges to 1, 0 and 1∕2 for α>αm,c, α<αm,c and α=αm,c, respectively, and we study the rate of convergence. Furthermore, we investigate the cases in the infinite m limit. Specifically, we discuss in details the case such that pn∈[0,1) with n∈Z+ and pn≈n→∞pn−s for p∈(0,1) and s>0. We find that the behavior of limm→∞τm(M,N) for this case highly depends on the value of s and how fast one approaches to the critical aspect ratio. The present study corrects and extends the results given in Li and Zhang (1983).

Multiple nonlinear Bragg diffraction of femtosecond laser pulses in a photonic lattice with hexagonal domains

The frequency doubling of femtosecond laser pulses in a two-dimensional (2D) rectangular nonlinear photonic lattice with hexagonal domains is studied experimentally and theoretically. The broad fundamental spectrum enables frequency conversion under nonlinear Bragg diffraction for a series of transverse orders at a fixed longitudinal quasi-phase-matching order. The consistent nonstationary theory of the frequency doubling of femtosecond laser pulses is developed using the representation based on the reciprocal lattice of the structure. The calculated spatial distribution of the second-harmonic spectral intensity agrees well with the experimental data. The condition for multiple nonlinear Bragg diffraction in a 2D nonlinear photonic lattice is offered. The hexagonal shape of the domains contributes to multibeam second harmonic excitation. The maximum conversion efficiency for a series of transverse orders in the range 0.01%–0.03% is obtained.

Tricriticality for dimeric Coulomb molecular crystals in ground state

We study the ground-state properties of a system of dimers. Each dimer consists in a pair of equivalent charges at a fixed distance, immersed in a neutralizing homogeneous background. All charges interact pairwisely by Coulomb potential. The dimer centers form a two-dimensional rectangular lattice with the aspect ratio and each dimer is allowed to rotate around its center. The previous numerical simulations, made for the more general Yukawa interaction, indicate that only two basic dimer configurations can appear: either all dimers are parallel or they have two different angle orientations within alternating (checkerboard) sublattices. As the dimer size increases, two second-order phase transitions, related to two kinds of the symmetry breaking in dimer’s orientations, were reported. In this paper, we use a recent analytic method based on an expansion of the interaction energy in Misra functions which converges quickly and provides an analytic derivation of the critical behaviour. Our main result is that there exists a specific aspect ratio of the rectangular lattice which divides the space of model’s phases onto two distinct regions. If the lattice aspect ratio , we recover both types of the second-order phase transitions and find that they are of mean-field type with the critical exponent . If , the phase transition associated with the discontinuity of dimer’s angles on alternating sublattices becomes of first order. For , the first- and second-order phase transitions meet at the tricritical point, characterized by the different critical index . Such phenomenon is known from literature about the Landau theory of one-component fields, but in our two-component version the scenario is more complicated: the component which is already in the symmetry-broken state at the tricritical point also interferes and exhibits unexpectedly the mean-field singular behaviour.

Relativistic energy-dispersion relations of 2D rectangular lattices

An exactly solvable relativistic approach based on inseparable periodic well potentials is developed to obtain energy-dispersion relations of spin states of a single-electron in two-dimensional (2D) rectangular lattices. Commutation of axes transfer matrices is exploited to find energy dependencies of the wave vector components. From the trace of the lattice transfer matrix, energy-dispersion relations of conductance and valence states are obtained in transcendental form. Graphical solutions of relativistic and nonrelativistic transcendental energy-dispersion relations are plotted to compare how lattice parameters [Formula: see text], core and interstitial size of the rectangular lattice affects to the energy-band structures in a situation core and interstitial diagonals are of equal slope.

Two-Dimensional Rectangular and Honeycomb Lattices of NbN: Emergence of Piezoelectric and Photocatalytic Properties at Nanoscale.

Using first-principles calculations, we predict that monolayered honeycomb and rectangular two-dimensional (2D) lattice forms of NbN are metastable and naturally derivable from different orientations of its rocksalt structure. While the rectangular form is shown to retain the metallic and superconducting (SC) properties of the bulk, spectacularly contrasting properties emerge in the honeycomb form of NbN: it exhibits (a) semiconducting electronic structure suitable for valleytronics and photocatalysis of water splitting, (b) piezoelectricity with a spontaneous polarization originating from a rare sd(2)-sp(2) type hybridization, and (c) a wide gap in its phonon spectrum making it suitable for use in hot carrier solar cells. Our work demonstrates how low coordination numbers and associated strong bonding stabilize 2D nanoforms of covalently bonded solids and introduce novel functionalities of technological importance.

Topological Control of Columnar Stacking Made of Liquid-Crystalline Thiophene-Fused Metallonaphthalocyanines.

The spontaneous organization of two‐dimensional polyaromatic molecules into well‐defined nanostructures through noncovalent interactions is important in the development of organic‐based electronic and optoelectronic devices. Two regioisomers of thiophene‐fused zinc naphthalocyanines ZnTNcendo and ZnTNcexo have been designed and synthesized to obtain photo‐ and electroactive liquid crystalline materials. Both compounds exhibited liquid crystalline behavior over a wide temperature range through intermolecular π–π interactions and local phase segregation between the aromatic cores and peripheral side chains. The structural differences between ZnTNcendo and ZnTNcexo affected the stacking mode in self‐assembled columns, as well as symmetry of the two‐dimensional rectangular columnar lattice. The columnar structure in liquid crystalline phase exhibited an ambipolar charge‐transport behavior.

A Natural Transition Between Equilibrium Patterns of Dislocation Dipoles

The equilibrium configurations of a row of uniformly distributed dislocation dipoles are first studied. The analysis is then generalised to study dipoles in two-dimensional rectangular periodic lattices. By examining the stability of the equilibrium configurations we find that the system may undergo a natural transition from the Taylor lattice to a row of dipole walls. This bifurcation may be involved in the transition from channel-vein to persistent slip band (PSB) structures in the early stage of metal fatigue.

Interference effects during the reradiation of ultrashort electromagnetic pulses by polyatomic systems

A theory of the reradiation of ultrashort electromagnetic pulses by arbitrary polyatomic systems of isolated complex atoms has been developed. The technique used allows the spatial inhomogeneity of the field of an ultrashort pulse and photon momenta in reradiation processes to be accurately taken into account. The angular distributions of the reradiation spectra have been obtained for an arbitrary number of atoms in the system. The processes of interference between the photon emission amplitudes are shown to give rise to characteristic “diffraction” maxima. We consider one-dimensional, two-dimensional, and three-dimensional rectangular lattices as examples as well as planar and cylindrical structures as models of planar nanosystems and nanotubes.

Spin configurations of the four-sublattice model on rectangular lattice

The spin configurations of the four-sublattice model with seven exchange interaction parameters (four of them are taken as direct-exchange interactions, the other three as super-exchange interactions) on a two-dimensional rectangular lattice have been investigated using a matrix method. For the four sublattices, using the two sets of the exchange parameters, we obtain collinear and non-collinear spin configurations for the given propagation vectors in the ground and the first excited states. When k=0, spin configuration is collinear ferromagnetic mode in the ground state and collinear antiferromagnetic mode in the first excited state. When k=[0.5, 0.5], spin configurations are non-collinear, that is, canted structures.

Sr2FeO3 with Stacked Infinite Chains of FeO4 Square Planes

The synthesis of Sr2FeO3 through a hydride reduction of the Ruddlesden-Popper layered perovskite Sr2FeO4 is reported. Rietveld refinements using synchrotron and neutron powder diffraction data revealed that the structure contains corner-shared FeO4 square-planar chains running along the [010] axis, being isostructural with Sr2CuO3 (Immm space group). Fairly strong Fe-O-Fe and Fe-Fe interactions along [010] and [100], respectively, make it an S = 2 quasi two-dimensional (2D) rectangular lattice antiferromagnet. This compound represents the end-member (n = 1) of the serial system Sr(n+1)FenO(2n+1), together with previously reported Sr3Fe2O5 (n = 2) and SrFeO2 (n = ∞), thus giving an opportunity to study the 2D-to-3D dimensional crossover. Neutron diffraction and Mössbauer spectroscopy show the occurrence of G-type antiferromagnetic order below 179 K, which is, because of dimensional reduction, significantly lower than those of the other members, 296 K in Sr3Fe2O5 and 468 K in SrFeO2. However, the temperature dependence of magnetic moment shows a universal behavior.

First-principles study of oxygen-induced copper segregation in Cu3Au(1 1 1)

Using ab initio calculations based on density-functional theory, we study oxygen-induced copper segregation at the Cu3Au(111) surface. We find that high-oxygen concentrations lead to complete segregation of copper at the topmost surface plane. This result is explained by the larger oxygen affinity of copper with respect to gold. Depending on concentration, oxygen forms two-dimensional hexagonal or rectangular lattices at the surface.