Diatomic Lattice Research Articles

A blueprint for truncation resonance placement in elastic diatomic lattices with unit cell asymmetrya).

Elastic periodic lattices act as mechanical filters of incident vibrations. By and large, they forbid wave propagation within bandgaps and resonate outside them. However, they often encounter "truncation resonances" (TRs) inside bandgaps when certain conditions are met. In this study, we show that the extent of unit cell asymmetry, its mass and stiffness contrasts, and the boundary conditions all play a role in the TR location and wave profile. The work is experimentally supported via two examples that validate the methodology, and a set of design charts is provided as a blueprint for selective TR placement in diatomic lattices.

An Exploration of the Phonon Frequency Spectrum and Born–von Karman Periodic Boundary Conditions in 1D and 2D Lattice Systems Using a Computational Approach

Periodic boundary conditions (PBCs) are a pivotal concept in the treatment of ideal lattices of infinite extent as a finite lattice. Most undergraduate texts that delve into the analytical treatment of lattice dynamics solve the problem in the reciprocal space with an implicit assumption of PBCs. While the traditional approach centered in the reciprocal or [Formula: see text]-space is elegant for simple one-dimensional lattice structures, the method becomes mathematically cumbersome for more complex lattices with real-world applications, such as a honeycomb lattice involving a [Formula: see text] secular determinant. In this work, we have explored the solution of lattice dynamics in direct space numerically, by constructing unit cells through an explicit implementation of Born–von Karman PBCs and solving the lattice dynamical equations in the displacement–time domain using the fourth-order Runge–Kutta algorithm. The Fast Fourier Transform technique is then used to obtain the phonon frequency spectrum corresponding to the computed instantaneous displacements. The fidelity of the model is validated by the agreement of the computational results obtained for the phonon spectrum at high symmetry points in the Brillouin zone with the analytical values for the given problem. Initially, the dynamics and the phonon spectrum are explored for monatomic and diatomic linear lattices which serve to act as introductory cases for the approach before investigating the 2D monatomic square and honeycomb lattices using the nearest and next-nearest neighbor approximations. A short-range force constant model employing the central forces and angular forces is used for the monatomic square lattice to account for the interatomic interactions. Our computational approach is expected to be physically more intuitive for a student for visualizing the periodicity of a given lattice and understanding its related dynamics. The approach demonstrates the usage of Born–von Karman PBCs in constructing a unit cell that effectively captures the dynamics of a lattice system, complementing the traditional secular determinant formalism. The work also illustrates an instance of how the computer programming can be used as a tool to empower the students to explore abstract concepts in physics. A honing of programming and computational skills of students enables them to interact with dynamic models in physics and gain deeper insight into the real-world applications of physics principles. Target group of this work is the students and educators at undergraduate level who seek a thorough and didactic comprehension of lattice dynamics.

On the Determination of Elastic Properties of Single-Walled Nitride Nanotubes Using Numerical Simulation.

In recent years, tubular nanostructures have been related to immense advances in various fields of science and technology. Considerable research efforts have been centred on the theoretical prediction and manufacturing of non-carbon nanotubes (NTs), which meet modern requirements for the development of novel devices and systems. In this context, diatomic inorganic nanotubes formed by atoms of elements from the 13th group of the periodic table (B, Al, Ga, In, Tl) and nitrogen (N) have received much research attention. In this study, the elastic properties of single-walled boron nitride, aluminium nitride, gallium nitride, indium nitride, and thallium nitride nanotubes were assessed numerically using the nanoscale continuum modelling approach (also called molecular structural mechanics). The elastic properties (rigidities, surface Young's and shear moduli, and Poisson's ratio) of nitride nanotubes are discussed with respect to the bond length of the corresponding diatomic hexagonal lattice. The results obtained contribute to a better understanding of the mechanical response of nitride compound-based nanotubes, covering a broad range, from the well-studied boron nitride NTs to the hypothetical thallium nitride NTs.

On Aspects of Continuous Approximation of Diatomic Lattice

This paper is devoted to the continualization of a diatomic lattice, taking into account natural intervals of wavenumber changes. Continualization refers to the replacement of the original pseudo-differential equations by a system of PDEs that provides a good approximation of the dispersion relations. In this regard, the Padé approximants based on the conditions for matching the values of the dispersion relations of the discrete and continuous models at several characteristic points are utilized. As a result, a sixth-order unconditionally stable system with modified inertia is obtained. Appropriate boundary conditions are formulated. The obtained continuous approximation accurately describes the amplitude ratios of neighboring masses. It is also shown that the resulting continuous system provides a good approximation for the natural frequencies.

Supratransmission phenomenon in a Fermi-Pasta-Ulam diatomic lattice.

The nonlinear supratransmission phenomenon in a Fermi-Pasta-Ulam (FPU) diatomic lattice with two forbidden bands is investigated. Using a decoupling ansatz for the motion of the two different sublattices combined with the continuum (quasidiscrete) approximation, we derived analytically the threshold amplitudes of supratransmission occurrence when a sinusoidal driving with frequency in the upper forbidden band (lower forbidden band gap between acoustic and optical modes) is applied at one end. The resulting estimate of the threshold of a lattice with a first heavy particle is different to the one obtained from a lattice with a first light particle, showing the influence of the driven particle and giving also the possibility to have two thresholds on each forbidden gap of a diatomic lattice by switching the order of light (m) and heavy (M) masses. In the lower forbidden band, the dependence of the supratransmission threshold on the mass ratio (μ=m/M) has been evidenced and it appears that for large (small) values of μ, that is μ>60%, the coupling between the two modes must (must not) be considered. Numerical explorations were subsequently performed with an emphasis on the dependence of the threshold on the driving frequency and also on the mass of the driven particle (light or heavy). A good agreement is found between the numerical and analytical thresholds. For the limit case where all the masses are identical, the results of the monoatomic FPU previously found in the literature are recovered.

Designing Topological Acoustic Lattices via Electroacoustic Analogies

Abstract Topological acoustics has recently witnessed a spurt in research activity, owing to their unprecedented properties transcending typical wave phenomena. In recent years, the use of coupled arrays of acoustic chambers has gained popularity in designing topological acoustic systems. In their common form, an array of acoustic chambers with relatively large volume is coupled via narrow channels. Such configuration is generally modeled as a full three-dimensional system, requiring extended computational time for simulating its harmonic response. To this end, this article establishes a comprehensive mathematical treatment of the use of electroacoustic analogies for designing topological acoustic lattices. The potential of such analytical approach is demonstrated via two types of topological systems: (i) edge states with quantized winding numbers in an acoustic diatomic lattice and (ii) valley Hall transition in an acoustic honeycomb lattice that leads to robust waveguiding. In both cases, the established analytical approach exhibits an excellent agreement with the full three-dimensional model, whether in dispersion analyses or the response of an acoustic system with a finite number of cells. The established analytical framework is invaluable for designing a variety of acoustic topological insulators with minimal computational cost.

Excited quantum anomalous and spin Hall effect: dissociation of flat-bands-enabled excitonic insulator state

Quantum anomalous Hall effect (QAHE) and quantum spin Hall effect (QSHE) are two interesting physical manifestations of 2D materials that have an intrinsic nontrivial band topology. In principle, they are ground-state equilibrium properties characterized by Fermi level lying in a topological gap, below which all the occupied bands are summed to a non-zero topological invariant. Here, we propose theoretical concepts and models of ‘excited’ QAHE (EQAHE) and EQSHE generated by dissociation of an excitonic insulator (EI) state with complete population inversion (CPI), a unique many-body ground state enabled by two yin-yang flat bands (FBs) of opposite chirality hosted in a diatomic Kagome lattice. The two FBs have a trivial gap in between, i.e. the system is a trivial insulator in the single-particle ground-state, but nontrivial gaps above and below, so that upon photoexcitation the quasi-Fermi levels of both electrons and holes will lie in a nontrivial gap achieved by the CPI-EI state, as demonstrated by exact diagonalization calculations. Then dissociation of singlet and triplet EI state will lead to EQAHE and EQSHE, respectively. Realizations of yin-yang FBs in real materials are also discussed.

Lamb waves in discrete homogeneous and heterogeneous systems: Dispersion properties, asymptotics and non-symmetric wave propagation

In this paper, we study Lamb waves propagating in a discrete strip, whose microstructure is represented by either a monatomic or a diatomic triangular lattice. In considering the in-plane vector problem, we derive an analytical solution for the dispersion relation of Lamb waves. Additionally, we investigate the main features of the eigenmodes of the system, which describe how the lattice strip vibrates at different frequencies. Further, we discuss how the dispersion properties depend on the number of the lattice’s rows and on the chosen boundary conditions. For heterogeneous systems, we focus the attention on the internal stop-band and on the flat bands appearing in the dispersion diagram. Different asymptotic models are employed to approximate the low-frequency behaviour of the lattice strip, starting from the classical Euler–Bernoulli beam. The effective behaviour of a lattice strip with dense microstructure is also investigated, and we present a comparative numerical analysis with the analogous continuum for which the classical Lamb wave problem is posed. The theory developed is exploited here to design a structured medium capable of manipulating wavemodes, and, through conversion and selection, generating uni-directional wave phenomena. We envisage that the present work can fill a gap in the research field related to the analytical study of dispersive waves in microstructured media, whose dynamic performance is influenced by the presence of multiple external boundaries.

Existence and Multiplicity of Wave Trains in a 2D Diatomic Face-Centered Lattice

Existence and Multiplicity of Wave Trains in a 2D Diatomic Face-Centered Lattice

The anti-Fermi–Pasta–Ulam–Tsingou problem in one-dimensional diatomic lattices

We study the thermalization dynamics of one-dimensional diatomic lattices (which represents the simplest system possessing multi-branch phonons), exemplified by the famous Fermi–Pasta–Ulam–Tsingou (FPUT)-β and the Toda models. Here we focus on how the system relaxes to the equilibrium state when part of highest-frequency optical modes are initially excited, which is called the anti-FPUT problem comparing with the original FPUT problem (low frequency excitations of the monatomic lattice). It is shown numerically that the final thermalization time T eq of the diatomic FPUT-β chain depends on whether its acoustic modes are thermalized, whereas the T eq of the diatomic Toda chain depends on the optical ones; in addition, the metastable state of both models have different energy distributions and lifetimes. Despite these differences, in the near-integrable region, the T eq of both models still follows the same scaling law, i.e. T eq is inversely proportional to the square of the perturbation strength. Finally, comparisons of the thermalization behavior between different models under various initial conditions are briefly summarized.

Exact strain gradient modelling of prestressed nonlocal diatomic lattice metamaterials

In this article, an exact strain gradient (SG) continuum model to capture the broadband bandgap characteristics of the prestressed diatomic lattice (PDL) with local/nonlocal interactions has been proposed. By considering the periodicity of wave motion, the wavelength-dependent Taylor expansion is established, leading to a desirable prediction when the wavelength is close to the lattice scale. For PDL, the dispersion curves of the proposed continuum model are always consistent with those of discrete model in the first Brillouin zone. The proposed SG continuum model is used to investigate the effects of structural parameters, orders of SG continua and various nonlocal interactions on bandgap properties of PDL. Highlights Continuum modelling of prestressed acoustic diatomic lattices. Perfect agreements achieved between dispersion diagrams by discrete and proposed continuum models. Effects of prestress, nonlocal interactions, mass and stiffness ratios on band gap structures analyzed. Proper strain gradient (SG) orders determined to guarantee a satisfactory accuracy.

Interface effects on the energy spectrum and quantum transport in two-dimensional topological heterostructures

We theoretically investigate the spectral and quantum transport properties of two-dimensional heterostructures build up of different topological materials. Surprising behavior arises due to particular tuning of the edge states, provided by internal symmetries or an external magnetic field, in the different constituents of the heterostructure. Specifically, we use the Chern-type topological insulator and the Weyl semimetals with one or two interfaces interposed between these systems. The energy spectrum may contain gaps which exhibit edge states of alternating chirality or, on the contrary, of a given chirality everywhere in the spectrum. Then, the calculation predicts that the quantum Hall effect may lack the negative plateaus or show asymmetric plateaus in the lower and upper part of the spectrum. Also, the channel distribution on the finite size system with interfaces and attached leads may give rise to fractional values of the quantum Hall resistance. The calculations are based on a diatomic lattice model, with hopping to the nearest- and next-nearest-neighbors, exposed to an internal periodic flux (of Haldane-type), and also to an external magnetic field.

On band gaps of nonlocal acoustic lattice metamaterials: a robust strain gradient model

We have proposed an “exact” strain gradient (SG) continuum model to properly predict the dispersive characteristics of diatomic lattice metamaterials with local and nonlocal interactions. The key enhancement is proposing a wavelength-dependent Taylor expansion to obtain a satisfactory accuracy when the wavelength gets close to the lattice spacing. Such a wavelength-dependent Taylor expansion is applied to the displacement field of the diatomic lattice, resulting in a novel SG model. For various kinds of diatomic lattices, the dispersion diagrams given by the proposed SG model always agree well with those given by the discrete model throughout the first Brillouin zone, manifesting the robustness of the present model. Based on this SG model, we have conducted the following discussions. (I) Both mass and stiffness ratios affect the band gap structures of diatomic lattice metamaterials, which is very helpful for the design of metamaterials. (II) The increase in the SG order can enhance the model performance if the modified Taylor expansion is adopted. Without doing so, the higher-order continuum model can suffer from a stronger instability issue and does not necessarily have a better accuracy. The proposed SG continuum model with the eighth-order truncation is found to be enough to capture the dispersion behaviors all over the first Brillouin zone. (III) The effects of the nonlocal interactions are analyzed. The nonlocal interactions reduce the workable range of the well-known long-wave approximation, causing more local extrema in the dispersive diagrams. The present model can serve as a satisfactory continuum theory when the wavelength gets close to the lattice spacing, i.e., when the long-wave approximation is no longer valid. For the convenience of band gap designs, we have also provided the design space from which one can easily obtain the proper mass and stiffness ratios corresponding to a requested band gap width.

Manipulation of wave motion in smart nonlinear phononic crystals made of shape memory alloys

Thanks to the functional role of shape memory alloys (SMAs) in controlling the mechanical behavior of structures, researchers have started investigating the possibility of manipulating wave motion in phononic crystals using SMAs. While SMAs were used before to tune the wave propagation in linear phononic crystals, in this work, we aim to extend their utilization to nonlinear lattices. For this purpose, SMA helical springs are used to manipulate the dispersion curves and the location of stop-bands in weakly nonlinear monoatomic and diatomic lattice chains. Using Brinson’s formulation to describe the thermo-mechanical behavior of SMA wires and Lindstedt-Poincaré method to solve the derived governing equations, closed-form nonlinear dispersion relations in monoatomic and diatomic lattice chains are obtained and the effects of temperature-induced phase transformation and stiffness nonlinearity on the wave propagation are investigated. The results reveal that the dispersion curves of a weakly nonlinear monoatomic chain are formed at lower frequencies through the austenite-to-martensite phase transformation. Similarly, both the acoustic and optical branches of a diatomic lattice are moved to lower frequencies during the phase transformation in the cooling process. Therefore, the generated stop-bands in nonlinear diatomic lattices are also moved to lower frequencies. In addition, using auxiliary SMA ground springs, new classes of nonlinear monoatomic and diatomic chains exhibiting additional low-frequency attenuation zones are introduced. These low-frequency stop-bands are tunable and their frequency range can be modulated by exploiting the temperature-induced phase transformation in the SMA springs. The results obtained from analytic formulations are verified by numerical calculations and an excellent agreement is observed. Such tunability and the potential for adding stop-bands in low frequencies reveal that SMAs can be very helpful in designing nonlinear phononic and acoustic devices, such as vibration mitigators and wave filters with pre-defined attenuation zones.

Arbitrary-Order Sensitivity Analysis in Phononic Metamaterials Using the Multicomplex Taylor Series Expansion Method Coupled with Bloch's Theorem

Abstract Phononic metamaterials (PMs) exhibit frequency ranges at which elastic waves are attenuated. However, this phenomenon is highly sensitive to geometrical variations and the metamaterial's mechanical properties. It is useful to have accurate sensitivity information to identify the variables that produce the highest impact on band gaps and guide the design of PMs with a desired wave propagation behavior. Current methodologies for sensitivity analysis in PMs, such as the finite difference method (FDM), are computationally inefficient, subjected to subtraction cancelation errors, and their accuracy is highly dependent on the magnitude of the perturbation step-size. In this work, we introduce a new computational methodology to perform parameter sensitivity in the dynamic behavior of PMs using the multicomplex Taylor series expansion (ZTSE) coupled with Bloch's theorem. The methodology allows one to obtain arbitrary order sensitivities with high accuracy. In contrast to FDM, this methodology is computationally more efficient, eliminates the step-size selection issue, and is not subjected to subtractive cancelation errors. Also, we show how the method can be applied using real algebra solvers. The analysis is limited to linear undamped PMs.The methodology using ZTSE with Bloch's theorem is presented in numerical examples for the di-atomic lattice and a 2D square lattice, where we compute up to third-order sensitivities. The results normalized errors in the order of 10(−9) for sensitivities of the di-atomic lattice, and in the order of 10(−8) for the 2D square lattice when compared to the analytical solutions.

Flat-Band-Enabled Triplet Excitonic Insulator in a Diatomic Kagome Lattice.

The excitonic insulator (EI) state is a strongly correlated many-body ground state, arising from an instability in the band structure toward exciton formation. We show that the flat valence and conduction bands of a semiconducting diatomic Kagome lattice, as exemplified in a superatomic graphene lattice, can possibly conspire to enable an interesting triplet EI state, based on density-functional theory calculations combined with many-body GW and Bethe-Salpeter equation. Our results indicate that massive carriers in flat bands with highly localized electron and hole wave functions significantly reduce the screening and enhance the exchange interaction, leading to an unusually large triplet exciton binding energy (∼1.1 eV) exceeding the GW band gap by ∼0.2 eV and a large singlet-triplet splitting of ∼0.4 eV. Our findings enrich once again the intriguing physics of flat bands and extend the scope of EI materials.

Micropteron traveling waves in diatomic Fermi–Pasta–Ulam–Tsingou lattices under the equal mass limit

The diatomic Fermi–Pasta–Ulam–Tsingou (FPUT) lattice is an infinite chain of alternating particles connected by identical nonlinear springs. We prove the existence of micropteron traveling waves in the diatomic FPUT lattice in the limit as the ratio of the two alternating masses approaches 1, at which point the diatomic lattice reduces to the well-understood monatomic FPUT lattice. These are traveling waves whose profiles asymptote to a small periodic oscillation at infinity, instead of vanishing like the classical solitary wave. We produce these micropteron waves using a functional-analytic method, originally due to Beale, that was successfully deployed in the related long wave and small mass diatomic problems. Unlike the long wave and small mass problems, this equal mass problem is not singularly perturbed, and so the amplitude of the micropteron’s oscillation is not necessarily small beyond all orders (i.e., the traveling wave that we find is not necessarily a nanopteron). The central challenge of this equal mass problem hinges on a hidden solvability condition in the traveling wave equations, which manifests itself in the existence and fine properties of asymptotically sinusoidal solutions (Jost solutions) to an auxiliary advance–delay differential equation. The novelty compared to previous approaches is that this operator is neither a Fourier multiplier nor a small nonlocal perturbation of a classical differential operator. This causes fundamental technical obstructions, which we overcome by developing new functional-analytic techniques to uncover the asymptotic phase shifts of the Jost solutions.

Nonreciprocal transmission of nonlinear elastic wave metamaterials by incremental harmonic balance method

Based on the granular diatomic lattice chain model, the band gap properties and nonreciprocal transmission of elastic wave metamaterials with the nonlinearity are studied. Extending the incremental harmonic balance (IHB) method to the nonlinear elastic wave, the traditional approach for the nonlinear periodic structures is improved. The influences on the band gap properties are discussed and the diode phenomenon can be observed. Then the nonreciprocal transmission of elastic wave metamaterials is realized. Both the theoretical and numerical results are presented to show the band gap structure and nonreciprocal phenomenon. And the experiments are performed to support these predictions. This study is expected to extend the applications of phononic crystals and elastic wave metamaterials with nonlinear properties.

π-Orbital Yin-Yang Kagome bands in anilato-based metal-organic frameworks.

π-Orbital bonding plays an important role not only in traditional molecular science and solid-state chemistry but also in modern quantum physics and materials, such as the relativistic Dirac states formed by bonding and antibonding π-bands in graphene. Here, we disclose an interesting manifestation of π-orbitals in forming the Yin-Yang Kagome bands, which host potentially a range of exotic quantum phenomena. Based on first-principles calculations and tight-binding orbital analyses, we show that the frontier π2- and π3-orbitals in anilato-based metal-organic frameworks form concurrently a conduction and valence set of Kagome bands, respectively, but with opposite signs of lattice hopping to constitute a pair of enantiomorphic Yin and Yang Kagome bands, as recently proposed in a diatomic Kagome lattice. The twisted configuration of neighboring benzene-derived organic ligands bridged by an octahedrally O-coordinated metal ion is found to play a critical role in creating the opposite sign of lattice hopping for the π2- versus π3-orbitals. Our finding affords a new material platform to study π-orbital originated quantum chemistry and physics.

1D topological phases in transition-metal monochalcogenide nanowires.

The Su-Schrieffer-Heeger (SSH) model is a prototypical one-dimensional (1D) diatomic lattice model for non-trivial topological phases and topological excitations. Theoretically, many variations and extensions of the SSH model have been proposed and explored to better understand the novel aspects of topological physics in low dimensions on the nanoscale. However, the outstanding challenge remains to find real nanomaterials with robust structural stability for realizing the 1D topological states. Here, we develop an extended version of the SSH model with multi-atomic bases of four, six and eight atoms and an imposed screw rotation symmetry. Furthermore, based on first-principles calculations, we demonstrate the realization of this model in transition metal monochalcogenide M6X6 (M = Mo and W; X = S, Se and Te) nanowires. The topological features of the doped M6X6 nanowires are confirmed with non-trivial edge modes and e/2 fractional charges, representative of the 1D non-trivial Zak phase. Our finding not only sheds new light on our fundamental understanding of 1D topological physics, but also significantly extends the scope of 1D topological materials that will attract immediate experimental interest, since isolated M6X6 nanowires have already been synthesized in experiments.