One-dimensional Diatomic Lattice Research Articles

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.

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.

Formulation of an efficient continuum mechanics-based model to study wave propagation in one-dimensional diatomic lattices

In this paper, we formulate an efficient continuum mechanics-based model on the basis of a discrete lattice model. First, the dispersion relation of a lattice wave in a one-dimensional diatomic crystal lattice is derived. Then, the second- and fourth-order continuum models are obtained from the differential-difference equations of motion by using the Padé approximations. The results show that the proposed fourth-order continuum model can predict the dispersion behaviour of the one-dimensional diatomic crystal lattice very well in the first Brillouin zone. Furthermore, the applicability of the present model to the prediction of the dispersion behaviour of the one-dimensional diatomic lattice with internal resonator and inerter is examined. Finally, the vibration frequencies of finite diatomic lattices are calculated by both the discrete and the proposed continuum models.

Nonintegrability and thermalization of one-dimensional diatomic lattices.

Nonintegrability is a necessary condition for the thermalization of a generic Hamiltonian system. In practice, the integrability can be broken in various ways. As illustrating examples, we numerically studied the thermalization behaviors of two types of one-dimensional (1D) diatomic chains in the thermodynamic limit. One chain was the diatomic Toda chain whose nonintegrability was introduced by unequal masses. The other chain was the diatomic Fermi-Pasta-Ulam-Tsingou-β chain whose nonintegrability was introduced by quartic nonlinear interaction. We found that these two different methods of destroying the integrability led to qualitatively different routes to thermalization in the near-integrable region, but the thermalization time, T_{eq}, followed the same scaling law; T_{eq} was inversely proportional to the square of the perturbation strength. This law also agreed with the existing results of 1D monatomic lattices. All these results imply that there is a universal scaling law of thermalization that is independent of the method of breaking integrability.

Wave propagation properties of one-dimensional acoustic metamaterials with nonlinear diatomic microstructure

Acoustic metamaterials are artificial microstructured media, typically characterized by a periodic locally resonant cell. The cellular microstructure can be functionally customized to govern the propagation of elastic waves. A one-dimensional diatomic lattice with cubic inter-atomic coupling—described by a Lagrangian model—is assumed as minimal mechanical system simulating the essential undamped dynamics of nonlinear acoustic metamaterials. The linear dispersion properties are analytically determined by solving the linearized eigenproblem governing the free wave propagation in the small-amplitude oscillation range. The dispersion spectrum is composed by a low-frequency acoustic branch and a high-frequency optical branch. The two frequency branches are systematically separated by a stop band, whose amplitude is analytically derived. Superharmonic 3:1 internal resonances can occur within a wavenumber-dependent locus defined in the mechanical parameter space. A general asymptotic approach, based on the multiple scale method, is employed to determine the nonlinear dispersion properties. Accordingly, the nonlinear frequencies and waveforms are obtained for the two fundamental cases of non-resonant and superharmonically 3:1 resonant or nearly resonant lattices. Moreover, the invariant manifolds associated with the nonlinear waveforms are parametrically determined in the space of the two principal coordinates. Finally, some examples of non-resonant and resonant lattices are selected to discuss their nonlinear dispersion properties from a qualitative and quantitative viewpoint.

Continuum model of acoustic metamaterials with diatomic crystal lattice

ABSTRACTThe continuum model of one-dimensional acoustic metamaterial with diatomic crystal lattice is studied in this article. First, the dispersive relation of lattice wave in one-dimensional diatomic crystal lattice of metamaterial is established and compared with that of the classic material. Then, the continuum model of the acoustic metamaterial leads to the classical continuum model, the stain gradient continuum model, and the nonlocal gradient continuum model based on different assumptions. The dispersive curves, which correspond to the three kinds of models, are shown graphically and compared with that of discrete crystal lattice of metamaterial. The disadvantage of the classic continuum model and the strain gradient continuum model are discussed. The nonlocal gradient continuum model is derived based on the nonlocal assumption of continuous displacement field. The stability of dispersive curves is guaranteed. However, the actual prediction effects are still dependent upon the appropriate selection of the nonlocal parameter in the nonlocal gradient continuum model.

Size Dependent Heat Conduction in One-Dimensional Diatomic Lattices

We study the size dependency of heat conduction in one-dimensional diatomic FPU-β lattices and establish that for low dimensional material, contribution from optical phonons is found more effective to the thermal conductivity and enhance heat transport in the thermodynamic limit N → ∞. For the finite size, thermal conductivity of 1D diatomic lattice is found to be lower than 1D monoatomic chain of the same size made up of the constituent particle of the diatomic chain. For the present 1D diatomic chain, obtained value of power divergent exponent of thermal conductivity 0.428±0.001 and diffusion exponent 1.2723 lead to the conclusions that increase in the system size, increases the thermal conductivity and existence of anomalous energy diffusion. Existing numerical data supports our findings.

Solving dispersion relations of one-dimensional diatomic chain with on-site potential by invariant eigen-operator method

The dispersion law of a one-dimensional diatomic lattice with on-site potential cross on its dispersion relation is solved under the harmonic approximation with quantized invariant eigen-operator method (IEO) and the influences of on-site potential and force constant are discussed. It is shown that due to the existence of on-site potential the lattice vibration spectra induced are shifted. Under certain conditions, the point of intersection exist between optical branch and acoustic branch in the boundary of Brillioun zone, which means the frequency gap between optical branch and acoustic branch is zero.

Soliton excitations in a one-dimensional nonlinear diatomic chain of split-ring resonators

We present a systematic analytical study of the dynamics of nonlinear magnetoinductive waves in a one-dimensional diatomic lattice of split ring resonators (SRRs) with Kerr nonlinear interaction between nearest neighbors. The linear spectrum of this model have two branches and exhibits a gap, which is proportional to the difference between two types of SRRs. We analyze the nonlinear excitations genuine of the discreteness and nonlinearity in such a diatomic chain based on an extended quasidiscreteness approach. Gap solitons (with vibrating frequency lying in the gap), resonant kinks (with the vibrating frequency lying in the frequency bands), and intrinsic localized modes (with the vibrating frequency being above all the frequency bands) are obtained explicitly. It is also shown that the existence of different localized structures depend strongly on the type of nonlinearity of the embedded medium (a self-focusing or defocusing one).

Anti-Fermi-Pasta-Ulam energy recursion in diatomic lattices at low energy densities

We study the dynamics of one- and two-dimensional diatomic lattices with the interatomic Morse potentials for the initial conditions selected at the edge of the Brillouin zone of the dispersion spectrum, when only light atoms are excited with the staggered mode while all heavy atoms remain at rest (the so-called anti-Fermi-Pasta-Ulam problem). We demonstrate that modulational instability of such a nonlinear state may result in almost periodic temporal dynamics of the lattice with spatial localization and delocalization of energies. Such a recursion occurs many times with a very slow decay, especially for the initial states with low energy. The energy recursion results in the formation of highly localized, large-amplitude gap discrete breathers. For one-dimensional diatomic lattices, we describe the periodic energy recursion analytically for a simple model with the nearest-neighbor interaction and cubic anharmonicity.

Coupling of intrinsic localized modes in doped polar semiconductors with plasmons

We present a detailed analysis of the interaction between intrinsic localized modes andplasmons in a doped polar semiconductor. The investigation has been performed for ananharmonic one-dimensional diatomic lattice with alternating interactions couplingsuccessive neighbours. The system simulates a row of atoms in the direction of a III–V semiconductor. Specific calculations have been performed for GaN,because it has a large gap between the acoustic and optical phonon branches. Thecalculations of the intrinsic localized modes have been performed by using two-bodypotentials to describe the interactions. We have used the rotating wave approximation andwe have found the intrinsic localized modes in the phonon gap. The interaction with theplasmon has been studied by adding to the equations of motion the alternating electric fieldwhich is related to the electron density of the plasmons. We obtain an expression for theelectric dynamical polarization associated with the intrinsic localized modes and with theplasmons. We derive an expression for the dielectric function of the coupled system. Thezeros of the dielectric function give the frequency of the combined modes. We havefound two regimes in which combined modes are possible. One is related to smallanharmonicity of the potential. The combined mode has a frequency above the top ofthe optical branch and can be explained in terms of the theory of the harmonicdielectric response of polar lattice vibrations. The second regime is related to highanharmonicity. The combined modes exist only for a finite slab. We show that onincreasing the anharmonicity, i.e. the amplitude of the intrinsic localized mode,the width of the slab increases. The frequency of the combined mode is insidethe phonon gap. We have also studied the dynamical stability of these modes.

Variational method for the generation of localized Wannier functions on the basis of Bloch functions

A simple and universal variational method for constructing localized Wannier functions from Bloch functions is proposed. The variational procedure is preceded by a symmetry analysis based on the induced representation theory and succeeded by a suitable orthogonalization procedure. The reliability of the method is demonstrated by computations of localized displacements in a one-dimensional diatomic lattice and a germanium lattice, of localized electronic states in a one-dimensional Kronig-Penney model, for the upper valence bands of Si and MgO crystals.

Moving nonlinear localized modes for one-dimensional Klein-Gordon diatomic lattice

We study analytically the moving nonlinear localized vibrational modes (discrete breathers) for a one-dimensional Klein-Gordon diatomic lattice in the whole omega (q) plane of the system by means of a semi-discrete approximation, in which the carrier wave of the modes is treated explicitly while the envelope is described in the continuum approximation. We find that both pulse and kink envelope moving modes for this lattice system can occur with certain carrier wave vectors and vibrational frequencies in separate regions of the omega (q) plane. However, the kink envelope moving modes have not been reported previously for this lattice system.

Parametric simultons in one-dimensional nonlinear lattices

Parametric simultaneous solitary wave (simulton) excitations are shown to be possible in nonlinear lattices. Taking a one-dimensional diatomic lattice with a cubic potential as an example, we consider the nonlinear coupling between the upper cut-off mode of acoustic branch (as a fundamental wave) and the upper cut-off mode of optical branch (as a second harmonic wave). Based on a quasi-discreteness approach the Karamzin-Sukhorukov equations for two slowly varying amplitudes of the fundamental and the second harmonic waves in the lattice are derived when the condition of second harmonic generation is satisfied. The lattice simulton solutions are given explicitly and the results show that these lattice simultons can be nonpropagating when the wave vectors of the fundamental wave and the second harmonic waves are exactly at π/a (where a is the lattice constant) and zero, respectively.

Surface and gap intrinsic localized modes in one-dimensional III-V semiconductors

A theoretical investigation has been made of intrinsic localized vibrational modes in an anharmonic one-dimensional diatomic lattice with alternating force constants coupling successive neighbours. This system simulates a row of atoms in the 111 direction of a III-V semiconductor. Specific calculations have been carried out for GaN, because it has a large gap between acoustic and optical branches. We study small-amplitude atom vibrations (up to 0.4 Å), accessible to experimental detection, in order to legitimize the expansion of the full potential to include cubic and quartic terms. We consider then nearest-neighbour interactions through harmonic as well as cubic and quartic anharmonic interactions to study the interplay between cubic and quartic terms in the frequencies of the localized modes. The force constants were determined empirically by fitting the longitudinal branches in the -L direction of GaN. We have studied both gap and surface intrinsic localized modes. Zinc-blende-structure chains are of particular interest, because the lack of inversion symmetry prevents the classification of the modes as even or odd parity. Nevertheless, modes were found that closely resemble the even- or odd-parity modes of an NaCl-structure chain. Their frequencies lie inside the gap for GaN. The absence of inversion symmetry permits a variety of surface modes to exist, depending on whether the bond at the surface is strong or weak and the atom at the surface is light or heavy. All surface mode frequencies for GaN lie inside the gap as found with the use of the full potential.

Asymmetric gap soliton modes in diatomic lattices with cubic and quartic nonlinearity

Nonlinear localized excitations in one-dimensional diatomic lattices with cubic and quartic nonlinearity are considered analytically by a quasi-discreteness approach. The criteria for the occurence of asymmetric gap solitons (with vibrating frequency lying in the gap of phonon bands) and small-amplitude, asymmetric intrinsic localized modes (with the vibrating frequency being above all the phonon bands) are obtained explicitly based on the modulational instabilities of corresponding linear lattice plane waves. The expressions of particle displacement for all these nonlinear localized excitations are also given. The result is applied to standard two-body potentials of the Toda, Born-Mayer-Coulomb, Lennard-Jones, and Morse type. The comparison with previous numerical study of the anharmonic gap modes in diatomic lattices for the standard two-body potentials is made and good agreement is found.

Odd and even intrinsic modes in diatomic nonlinear lattices

Systematic simulations of a one-dimensional diatomic dynamical lattice with cubic and quartic anharmonicity demonstrate that the odd (Sievers-Takeno) intrinsic mode is stable at relatively low frequencies, being changed by the even (Page) one at some critical frequency. The transition between the two modes is hysteretic, i.e., it depends upon the direction of change of the frequency. The critical frequency is a growing function of the mass difference between the particles with the odd and even numbers, but it proves to be practically independent of the ratio between the quadratic and cubic terms in the equations of motion.

Intrinsic resonant modes for a one-dimensional lattice with a soft optic mode

Localization of lattice vibrations is investigated for a soft zone-center optic mode overlapping the acoustic spectrum of an anharmonic one-dimensional diatomic lattice. An intrinsic resonant mode is found below the plane-wave optic spectrum. Using numerical methods the frequency and eigenvector of the intrinsic resonant mode has been measured as a function of its amplitude and the optic-mode frequency. Molecular-dynamics simulations show that these localized modes are stable.

Erratum: Soliton excitations in one-dimensional diatomic lattices

Erratum: Soliton excitations in one-dimensional diatomic lattices

Soliton excitations in one-dimensional diatomic lattices.

We present a systematic analytical study of the soliton excitations in a one-dimensional diatomic lattice with nonlinear on-site potential and a quartic interaction between nearest neighbors. We show that (1) the decoupling ansatz widely used in literature for the motion of two different masses is unncessary and can be naturally derived in our approach; (2) the system may support some new types of gap solitons and resonant kinks, two of which have been observed recently in a nonlinear diatomic pendulum lattice experiment; (3) for nonlinear on-site potential when the wave number of carrier waves is near the edge of the Brillouin zone and the difference of mass between two kinds of atoms becomes small, the results coincide with that of Kivshar and Flytzanis about the gap solitons in diatomic lattices; and (4) the theoretical results, being without any divergence, are valid in the whole Brillouin zone and can be applied to other nonlinear lattices.