Diatomic Lattice Research Articles

Localized solutions of a nonlinear diatomic lattice

We consider a coupled system of differential-difference nonlinear equations. We study the dynamics of such a diatomic lattice showing global existence and uniqueness in an appropriate function space. Our approach based on energy estimates allows us to prove the result only in the case where nonlinear force constants are positive and equal. All other situations remain at this point as open problems.

Intrinsic localized modes and their interaction with the plasmon electric field in one-dimensional zinc-blende lattice

Abstract In this paper we study the behavior of a diatomic nonlinear one-dimensional lattice which represents the 〈1 1 1〉 direction of a zinc-blende three-dimensional crystal. In these nonlinear lattices are present intrinsic localized modes due to the anharmonicity. These modes lie in the vibrational gap between acoustic and optical linear spectrum. Considering a highly doped GaN semiconductor we examine the coexistence of the localized modes with the electronic plasma oscillations. We derive an expression for the plasmon electric field, we study the dynamics of the anharmonic lattice in presence of this electric field and we calculate the dielectric function of the coupled system. The zeroes of the dielectric function give the frequencies of the coupled modes.

The interaction of intrinsic localized modes in the gap of doped semiconductors withplasmons

We present a theoretical investigation of the interaction of intrinsic localized modeswith plasmons. The anharmonic localized modes for a 1D diatomic lattice aredetermined using a two-body potential which describes the interactions amongparticles in a zinc-blende structure material. The localized mode frequency isinside the gap between acoustic and optical phonons. Calculations have beenperformed for GaN because it has a large phonon gap, can be highly n-doped, and theplasma frequency of free carriers is in the range of phonon and intrinsic localizedmode frequencies. To study the coupling we add to the equations of motion anelectric field to simulate the plasmon. Solving the system we obtain the dynamicaldisplacements pattern from which we evaluate the total polarization. From thepolarization we determine the frequency of the combined mode for which the dielectricfunction is zero. In this investigation we have analysed both the case of smalllocalized mode amplitudes and the case of larger amplitudes, obtaining differentbehaviour. In the first case the mixed mode has a frequency above the top of theoptical branch, which can be explained in terms of the theory of the harmonicdielectric response of polar lattice vibrations. In the second case the coupledmode exists only for a finite slab, and its frequency is inside the phonon gap.

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.

The ground-state phase diagrams of the spin-3/2 Ising model

The ground-state spin configurations are obtained for the spin-3/2 Ising model Hamiltonian with bilinear and biquadratic exchange interactions and a single-ion crystal field. The interactions are assumed to be only between nearest-neighbors. The calculated ground-state phase diagrams are presented on diatomic lattices, such as the square, honeycomb and sc lattices, and triangular lattice in the ( Δ/ z| J|, K/| J|) and ( H/ z| J|, K/| J|) planes.

Three-Wave Parametric Simultons in Nonlinear Lattices

A new type of nonlinear excitations, i.e. three simultaneous lattice solitons (simultons), in a nonlinear diatomic lattice is predicted. We show that three-wave resonance condition can be fulfilled in the diatomic lattice. Using a quasi-discrete multi-scale method we derive nonlinear amplitude equations for the three-wave resonance with the dispersion of the system taken into account. We provide several types of exact lattice simultons solutions and show that the lattice simultons can be non propagating and their oscillating frequencies may be within the gap of phonon spectrum bands.

Dynamics of nonlinear diatomic lattices

Dynamics of nonlinear diatomic lattices

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.

Quasi-continuum approximations for travelling kinks in diatomic lattices

We extend the quasi-continuum method of approximation to the case of a diatomic lattice. This is illustrated by a lattice in which both small and large atoms interact with first- and second-nearest neighbours. We show that highly accurate quasi-continuum techniques may be generalized to determine the shape of nontopological kinks. Many previous analyses of such systems have found travelling wave solutions only for one particular speed using a second-order continuum theory; we show (i) how wave shapes can be found for arbitrary speeds, and (ii) how solution profiles can be calculated from fourth-order partial differential equations which approximate the lattice. Alongside the theoretical analysis, we also present numerical simulations of the lattice which demonstrate propagation of the predicted waves through the lattice. We show that the particular speed for which solutions have been found in previous studies is a special speed for waves in the lattice, but waves can travel for long periods of time at faster or slower speeds, whilst slowly relaxing to this critical speed.

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.

Nonlinear Gap Modes in a 1D Alternating Bond Monatomic Lattice with Anharmonicity

We analytically study the nonlinear localized gap modes in a one-dimensional atomic chain with uniform atomic mass but two periodically alternating force constants between the nearest neighbors by means of a quasi-continuum approximation. This model simulates a row of atoms in the direction of a diamond-structure type of crystals or molecular crystals with alternating double and single bonds. For this lattice system, we find that the harmonic plus quartic anharmonic terms of inter-site potential produce a new type of nonlinear localized gap modes with a slightly asymmetry distribution of atomic displacements. These localized gap modes are somewhat different from widely studied localized gap modes with a symmetry atomic displacement distribution in diatomic ion lattices.

Solitary waves in a diatomic lattice: analytic approximations for a wide range of speeds by quasi-continuum methods

We extend the quasi-continuum method of approximation to find waves of general speed in a diatomic lattice. Our results include as a special case the one speed which previous methods have found; we find the shape of nonlinear waves of arbitrary velocity up to some maximum speed.

Second harmonic generation in a two-dimensional diatomic lattice

Resonant interactions of modes in two-dimensional (2D) diatomic lattices are studied within the framework of the multiscale expansion. Equations governing three-wave processes and second harmonic generation are derived for a lattice having a finite number of cells. A particular example of the second harmonic generation in a two-dimensional lattice with nearest-neighbor interactions is considered in detail. It is shown that the 2D case reveals new features like, for example, possibility of simultaneous resonant interaction of one mode with two other ones, existence of solitary wave pulses localized in space and moving along the lattice, and creation of stationary spatial structures subject to proper boundary conditions.

Properties of one-dimensional anharmonic lattice solitons

The existence of bell- and kink-shaped solitons moving at constant velocity while keeping a permanent profile is studied in infinite periodic monoatomic chains of arbitrary anharmonicity by taking advantage of the equation of motion being integrable with respect to solitons. A second-order, non-linear differential equation involving advanced and retarded terms must be solved, which is done by implementing a scheme based on the finite element and Newton's methods. If the potential has a harmonic limit, the asymptotic time-decay behaves exponentially and there is a dispersion relation between propagation velocity and decay time. Inversely if the potential has no harmonic limit, the asymptotic regime shows up either as a power-law or faster than exponential. Excellent agreement is achieved with Toda's model. Illustrative examples are also given for the Fermi–Pasta–Ulam and sine-Gordon potentials. Owing to integrability an effective one-body potential is worked out in each case. Lattice and continuum solitons differ markedly from one another as regards the amplitude versus propagation velocity relationship and the asymptotic time behavior. The relevance of the linear stability analysis when applied to solitons propagating in an infinite crystal is questioned. The reasons preventing solitons from arising in a diatomic lattice are discussed.

Dynamics of coupled gap solitons in diatomic lattices with cubic and quartic nonlinearities

The dynamics of coupled gap solitons in diatomic lattices with cubic and quartic nonlinearities is considered analytically based on an extended quasidiscreteness approach. For various mass differences (and thus different gap widths of the phonon spectrum), the coupled gap solitons are shown to display very rich dynamical behavior and their properties are strongly dependent on the force-constant ratio K(2)(3)/(K2K4), where K(j)(j=1,2,3) are the force constants for the quadratic, cubic, and quartic parts of the intersite interaction potential, respectively. Several previous theoretical approaches for studying gap soliton dynamics in diatomic lattices are recovered in our scheme, and the relations between these methods are elucidated in a systematic way.

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.

Discrete breathers in aperiodic diatomic FPU lattices with long range order

This study presents various aspects of discrete breathers in diatomic FPU lattices with masses varying according to the aperiodic Fibonacci and Thue–Morse sequences. We investigate the existence of time-periodic breathers, starting from the anti-continuous limit and consider excitations of isolated light atoms, hence obtaining the domain of unique existence for these breathers. We also perform a linear stability analysis by studying the Floquet operator. The found exact solutions are used, slightly perturbed, as initial conditions for long-time simulations of the breathers. These breathers turn out to be robust. Finally we consider how initial excitations of two consecutive light atoms evolve. Depending on the properties of the phase space for the two-atom system at the anti-continuous limit, we obtain different behaviour of these excitations. Especially we find that the aperiodic lattices can support localized excitations with a continuous frequency distribution for the timescales we consider, while a periodic lattice is unable to. These excitations are referred to as chaotic breathers.

Envelope solitons and companion modes in diatomic lattices

Dependence of the dynamics of envelope solitons in diatomic lattices on the gap width is studied analytically and numerically. It is shown that depending on lattice parameters, there are four different cases of the soliton behavior. (i) At a sufficiently large gap solitons with zero group velocity can be excited. In this case the soliton is accompanied by a mode associated with the other branch of the spectrum. (ii) If the gap is negligible then the soliton behavior closely resembles the behavior of the envelope soliton in a monoatomic lattice. (iii) Small enough but non-negligible gap results in the creation of a soliton tail associated with the companion mode. (iv) There are intermediate values of the gap which do not allow creation of envelope solitons of the given amplitude at the boundary of the Brillouin zone. Direct evidence of the existence of the companion modes is presented.

Non–existence of truly solitary waves in water with small surface tension

This paper considers permanent capillary–gravity waves on the free surface of a two–dimensional incompressible inviscid fluid of finite depth. It is shown that there are no solitary-wave solutions of the exact governing equations of the flow that decay to zero exponentially at infinity if the surface tension coefficient is less than its critical value and lies in some intervals. The proof is based upon an estimate of a constant that is related to the approximation of the solution, if it exists, near its singularity. The approximation satisfies a fourth–order nonlinear ordinary differential equation when the solution is extended to the complex plane. Then the non–existence of truly solitary waves is obtained by using a contradiction on this constant.