Two-dimensional Periodic Lattice Research Articles

On acoustic wave beaming in two-dimensional structural lattices

We discuss directional energy flow, often referred to as wave beaming, in two-dimensional periodic truss lattices under infinitesimal harmonic excitation. While the phenomenon of directional wave guiding is well-known and commonly treated in the context of dispersion relations, the theoretical and computational tools to predict beaming are limited, which is why a fundamental understanding for complex lattices is incomplete. Here, we present a new strategy to identify partial band gaps and wave beaming in a simple fashion, covering wide frequency ranges and distinguishing in-plane and out-of-plane vibrational modes in lattices composed of linear elastic Euler-Bernoulli beams. By calculating group velocities that provide insight into the frequency-dependent directional energy flow, we show that dispersion surfaces overlap in frequency and beaming direction, elucidating the need to consider multiple surfaces when predicting global system response – in contrast to many prior approaches that focused on the lowest surface(s) individually. These concepts are demonstrated for three examples of two-dimensional structural lattices (of rectangular, sheared, and hexagonal architecture), for each of which we study the influence of geometry on wave dispersion. Direct numerical simulations validate directional energy flow predictions, demonstrate directional frequency dispersion, and highlight conventional dispersion analysis limitations.

Towards a standardized setup for surface energy calculations

High-throughput design of new materials with desired electronic properties, based on screening of large collections of crystal structures organized in the from of libraries or databases require fast, widely applicable, consistent and unsupervised methods to calculate the property of interest. In this work we present an approach for the calculation of surface energies of two-dimensional periodic crystal lattices which meets all these requirements. For materials slabs which are terminated with two identical surfaces, the task of calculating the surface energy is trivial. More problematic are the cases where both terminating surfaces are different, as there is no single established method allowing for equal treatment of a wide range of surface morphologies and orientations. Our proposed approach addresses this problem. It relies on appropriately chosen capping atoms, whose bonding energy contributions are used to approximate the total energy of the surface. The choice of the capping atoms is governed by a set of simple guidelines that are applicable for surfaces with different terminations. We present the results for different semiconductor materials and show that our approach leads to surface energies with errors that are below 10%, and that are as low as 2% in many cases. We show that hydrogen is not always the best choice for a capping atom if accurate surface energies are the target of the calculations.

Self-Organized Societies: On the Sakoda Model of Social Interactions

We characterize the behavior and the social structures appearing from a model of general social interaction proposed by Sakoda. The model consists of two interacting populations in a two-dimensional periodic lattice with empty sites. It contemplates a set of simple rules that combine attitudes, ranges of interactions, and movement decisions. We analyze the evolution of the 45 different interaction rules via a Potts-like energy function which drives the system irreversibly to an equilibrium or a steady state. We discuss the robustness of the social structures, dynamical behaviors, and the existence of spatial long range order in terms of the social interactions and the equilibrium energy.

Transition from two-dimensional photonic crystals to dielectric metasurfaces in the optical diffraction with a fine structure.

We study experimentally a fine structure of the optical Laue diffraction from two-dimensional periodic photonic lattices. The periodic photonic lattices with the C4v square symmetry, orthogonal C2v symmetry, and hexagonal C6v symmetry are composed of submicron dielectric elements fabricated by the direct laser writing technique. We observe surprisingly strong optical diffraction from a finite number of elements that provides an excellent tool to determine not only the symmetry but also exact number of particles in the finite-length structure and the sample shape. Using different samples with orthogonal C2v symmetry and varying the lattice spacing, we observe experimentally a transition between the regime of multi-order diffraction, being typical for photonic crystals to the regime where only the zero-order diffraction can be observed, being is a clear fingerprint of dielectric metasurfaces characterized by effective parameters.

Universality of the Hall Conductivity in Interacting Electron Systems

We prove the quantization of the Hall conductivity for general weakly interacting gapped fermionic systems on two-dimensional periodic lattices. The proof is based on fermionic cluster expansion techniques combined with lattice Ward identities, and on a reconstruction theorem that allows us to compute the Kubo conductivity as the analytic continuation of its imaginary time counterpart.

Optical fabrication of two-dimensional photorefractive periodic photonic lattices and quasicrystal microstructures by multi-lens board

We experimentally demonstrate a convenient approach to fabricate two-dimensional periodic photonic lattices and quasicrystal microstructures in iron-doped lithium niobate photorefractive crystal by multi-lens board. The method does not require complicated optical adjustment system. We analyze the induced square photonic lattices, elliptic photonic lattices and quasicrystal microstructures by plane wave guiding, far-field diffraction pattern imaging and Brillouin-zone spectroscopy. Induced photonic microstructures can be fixed or erased and re-recorded in the crystal. We also numerically demonstrate that the method can be easily extended to form other complex photonic microstructures by designing the multi-lens board appropriately.

Nonequilibrium random-field Ising model on a diluted triangular lattice.

We study critical hysteresis in the random-field Ising model on a two-dimensional periodic lattice with a variable coordination number z(eff) in the range 3≤z(eff)≤6. We find that the model supports critical behavior in the range 4<z(eff)≤6, but the critical exponents are independent of z(eff). The result is discussed in the context of the universality of nonequilibrium critical phenomena and extant results in the field.

Modeling selective local interactions with memory: Motion on a 2D lattice

We consider a system of particles that simultaneously move on a two-dimensional periodic lattice at discrete times steps. Particles remember their last direction of movement and may either choose to continue moving in this direction, remain stationary, or move toward one of their neighbors. The form of motion is chosen based on predetermined stationary probabilities. Simulations of this model reveal a connection between these probabilities and the emerging patterns and size of aggregates. In addition, we develop a reaction–diffusion master equation from which we derive a system of ODEs describing the dynamics of the particles on the lattice. Simulations demonstrate that solutions of the ODEs may replicate the aggregation patterns produced by the stochastic particle model. We investigate conditions on the parameters that influence the locations at which particles prefer to aggregate. This work is a two-dimensional generalization of Galante and Levy (2012), in which the corresponding one-dimensional problem was studied.

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.

Optical properties of two-dimensional magnetoelectric point scattering lattices

We explore the electrodynamic coupling between a plane wave and an infinite two-dimensional periodic lattice of magneto-electric point scatterers, deriving a semi-analytical theory with consistent treatment of radiation damping, retardation, and energy conservation. We apply the theory to arrays of split ring resonators and provide a quantitive comparison of measured and calculated transmission spectra at normal incidence as a function of lattice density, showing excellent agreement. We further show angle-dependent transmission calculations for circularly polarized light and compare with the angle-dependent response of a single split ring resonator, revealing the importance of cross coupling between electric dipoles and magnetic dipoles for quantifying the pseudochiral response under oblique incidence of split ring lattices.

Theoretical and numerical investigation of HF elastic wave propagation in two-dimensional periodic beam lattices

The elastic wave propagation phenomena in two-dimensional periodic beam lattices are studied by using the Bloch wave transform. The numerical modeling is applied to the hexagonal and the rectangular beam lattices, in which, both the in-plane (with respect to the lattice plane) and out-of-plane waves are considered. The dispersion relations are obtained by calculating the Bloch eigenfrequencies and eigenmodes. The frequency bandgaps are observed and the influence of the elastic and geometric properties of the primitive cell on the bandgaps is studied. By analyzing the phase and the group velocities of the Bloch wave modes, the anisotropic behaviors and the dispersive characteristics of the hexagonal beam lattice with respect to the wave propagation are highlighted in high frequency domains. One important result presented herein is the comparison between the first Bloch wave modes to the membrane and bending/transverse shear wave modes of the classical equivalent homogenized orthotropic plate model of the hexagonal beam lattice. It is shown that, in low frequency ranges, the homogenized plate model can correctly represent both the in-plane and out-of-plane dynamic behaviors of the beam lattice, its frequency validity domain can be precisely evaluated thanks to the Bloch modal analysis. As another important and original result, we have highlighted the existence of the retropropagating Bloch wave modes with a negative group velocity, and of the corresponding “retro-propagating” frequency bands.

Critical dispersion of advecting-diffusing tracers in periodic landscapes of hard-wall symmetric potentials

Large-scale, time-asymptotic dispersion properties of diffusing tracers dragged by a uniform drive through a two-dimensional periodic lattice of hard-wall symmetric potentials are investigated. Dispersion is quantified by a typically anisotropic effective diffusivity tensor D, whose eigenvalues and eigenvectors depend on the dimensionless bare diffusivity 1/Pe for each given lattice geometry. Attention is focused on critical lattice geometries yielding sustained macroscale dispersion D([perpendicular]) along the direction orthogonal to the uniform drive in the limit where Pe→∞. A simple one-dimensional model is proposed, which predicts the anomalous scaling D([perpendicular])~1/[A(1)+A(2)log(Pe)].

Fabrication of Large Surface Area Semitransparent Monocrystalline Si Solar Cells

An experimental technique is presented for fabricating large surface area semitransparent monocrystalline Si solar cells. The semitransparency is achieved by laser-cutting two-dimensional periodic lattices of ellipses through industrially manufactured Si cells. A novel two-level metal base was invented for minimizing cell breakages during laser cutting. The periodic lattices consist of a large number of ellipses, so as to achieve uniform distribution of the transmitted solar radiation. High efficiencies over 13% are deduced from the I–V curves, recorded under physical sunlight, for all fabricated cells with a 5% transparency.

Surface field excitation by an obliquely incident wave

Observation of surface field excitation by an obliquely incident wave inside a cylindrical two-dimensional periodic surface lattice is a step forward toward THz Cherenkov amplifiers. Here we observe and discuss this phenomenon, investigating it using different approaches. The results are compared, and it is shown that an increase in the lattice contrast results in excitation of surface fields by an obliquely incident wave resulting in the eigenmode formation. Measurements were conducted by studying forward transmission of the signal, and it is shown that the measured gaps in the spectra are associated with the eigenmode formation rather than band gap establishment.

Anisotropy-induced broadband stress wave steering in periodic lattices

This paper investigates two-dimensional periodic lattices as a means to induce strongly directional stress wave propagation in solids in the low-frequency broadband regime. The real 2D structure of the lattice is considered as opposed to traditional truss-like approaches, allowing the characterization of the system seamlessly between the extremes of grid-like systems, porous systems, and a continuum material. Wave directionality in structural lattices is associated to their effective anisotropic mechanical properties. When the wavelength of deformation is larger than the representative unit cell, the system behaves as an equivalent medium with orthotropic properties that are correlated to the lattice skewness and void volume fraction. Polar plots of wave velocities, computed through the application of Bloch’s theorem, provide insights into the effective anisotropy and wave directionality (or “steering”) of the structure. Such directional response is also correlated to the stiffness properties of an equivalent medium by means of proper homogenization techniques. The effects of the resulting anisotropy on wave propagation are also illustrated through wave propagation simulations for problems involving both harmonic excitations and impulsive loads.

Broad-Band Giant Circular Dichroism in Metamaterials of Twisted Chains of Metallic Nanoparticles

By using a discrete-dipole approximation for two-dimensional periodic lattices of scatterers, we show that three-dimensional layer-by-layer metamaterials consisting of twisted arrays of metal-nanoparticle chains exhibit giant circular dichroism spanning over the entire visible regime. The evident broad-band circular dichroism of the metamaterial is not a result of the intrinsic handeness of the constituent meta-atoms as in mainstream chiral designs but arises from the combined action of the anisotropy of each array of chains of nanoparticles, the gradual rotational twist of the metamaterial, and the strong electromagnetic coupling between adjacent layers of nanoparticle chains. As such, it is shown numerically that the degree of circular dichroism of the twisted metamaterial under study depends strongly on the twist angle between two successive lattices of nanoparticle chains and on the degree of anisotropy of each lattice.

High-Rotational Symmetry Lattices Fabricated by Moiré Nanolithography

This paper describes a new nanofabrication method, moiré nanolithography, that can fabricate subwavelength lattices with high-rotational symmetries. By exposing elastomeric photomasks sequentially at multiple offset angles, we created arrays with rotational symmetries as high as 36-fold, which is three times higher than quasiperiodic lattices (≤12-fold) and six times higher than two-dimensional periodic lattices (≤6-fold). Because these moiré nanopatterns can be generated over wafer-scale areas, they are promising for a range of photonic applications, especially those that require broadband, omnidirectional absorption of visible light.

Fractal Landau-Level Spectra in Twisted Bilayer Graphene

The Hofstadter butterfly spectrum for Landau levels in a two-dimensional periodic lattice is a rare example exhibiting fractal properties in a truly quantum system. However, the observation of this physical phenomenon in a conventional material will require a magnetic field strength several orders of magnitude larger than what can be produced in a modern laboratory. It turns out that for a specific range of rotational angles twisted bilayer graphene serves as a special system with a fractal energy spectrum under laboratory accessible magnetic field strengths. This unique feature arises from an intriguing electronic structure induced by the interlayer coupling. Using a recursive tight-binding method, we systematically map out the spectra of these Landau levels as a function of the rotational angle. Our results give a complete description of LLs in twisted bilayer graphene for both commensurate and incommensurate rotational angles and provide quantitative predictions of magnetic field strengths for observing the fractal spectra in these graphene systems.

Pattern-integrated interference lithography: single-exposure fabrication of photonic-crystal structures

Multibeam interference represents an approach for producing one-, two-, and three-dimensional periodic optical-intensity distributions with submicrometer features and periodicities. Accordingly, interference lithography (IL) has been used in a wide variety of applications, typically requiring additional lithographic steps to modify the periodic interference pattern and create integrated functional elements. In the present work, pattern-integrated interference lithography (PIIL) is introduced. PIIL is the integration of superposed pattern imaging with IL. Then a pattern-integrated interference exposure system (PIIES) is presented that implements PIIL by incorporating a projection imaging capability in a novel three-beam interference configuration. The purpose of this system is to fabricate, in a single-exposure step, a two-dimensional periodic photonic-crystal lattice with nonperiodic functional elements integrated into the periodic pattern. The design of the basic system is presented along with a model that simulates the resulting optical-intensity distribution at the system sample plane where the three beams simultaneously interfere and integrate a superposed image of the projected mask pattern. Appropriate performance metrics are defined in order to quantify the characteristics of the resulting photonic-crystal structure. These intensity and lattice-vector metrics differ markedly from the metrics used to evaluate traditional photolithographic imaging systems. Simulation and experimental results are presented that demonstrate the fabrication of example photonic-crystal structures in a single-exposure step. Example well-defined photonic-crystal structures exhibiting favorable intensity and lattice-vector metrics demonstrate the potential of PIIL for fabricating dense integrated optical circuits.

Vortex-type elastic structured media and dynamic shielding

The paper presents an approach to modelling a novel elastic metamaterial structure that possesses non-trivial dispersion features. A system of spinners has been embedded into a two-dimensional periodic lattice system. The analysis of the motion of the spinners is used to derive an expression for a ‘chiral term’ in the equations describing the dynamics of the lattice. Dispersion of elastic waves is shown to possess innovative filtering and polarization properties induced by the vortex-type nature of the structured media. The related effective behaviour in a continuous medium is implemented to build a shielding cloak around an obstacle. Analytical work is accompanied by numerical illustrations.