Elastic Superlattice Research Articles

Pseudo-spin polarized one-way elastic wave eigenstates in one-dimensional phononic superlattices

We use a one-dimensional discrete binary elastic superlattice bridging continuous models of superlattices that showcase one-way propagation character and the discrete elastic Su-Schrieffer-Heeger model that does not. By considering Bloch wave solutions of the superlattice wave equation, we demonstrate conditions supporting elastic eigenmodes that do not satisfy translational invariance of Bloch waves over the entire Brillouin zone, unless their amplitude vanishes for some wave number. These modes are characterized by a pseudo-spin, and occur only on one side of the Brillouin zone for given spin, leading to spin-selective one-way wave propagation. We demonstrate how these features result from the interplay of translational invariance of Bloch waves, pseudo-spin, and a Fabry-Pérot resonance condition in the superlattice unit cell.

Pseudo-Spin Polarized One-Way Elastic Wave Eigenstates in One-Dimensional Phononic Superlattices

We investigate a one-dimensional discrete binary elastic superlattice bridging continuous models of superlattices that showcase a one-way propagation character, as well as the discrete elastic Su–Schrieffer–Heeger model, which does not exhibit this character. By considering Bloch wave solutions of the superlattice wave equation, we demonstrate conditions supporting elastic eigenmodes that do not satisfy the translational invariance of Bloch waves over the entire Brillouin zone, unless their amplitude vanishes for a certain wave number. These modes are characterized by a pseudo-spin and occur only on one side of the Brillouin zone for a given spin, leading to spin-selective one-way wave propagation. We demonstrate how these features result from the interplay of the translational invariance of Bloch waves, pseudo-spins, and a Fabry–Pérot resonance condition in the superlattice unit cell.

Phononic crystal with rigid scatterers: Infinite density versus infinite elasticity

Models involving materials with extreme parameters, while usually unreachable in practice, play important role in different areas of physics, for example, black body and ideal gas in thermodynamics, perfect conductor, and ideal diamagnetic in electrodynamics. In acoustics, the concept of rigid (or hard) scatterer is widely used to study propagation of sound in heterogeneous media with high-acoustic contrast between the constituents. Here, we report the results obtained for band structure calculations of phononic crystals with rigid scatterers. A scatterer with infinite acoustic impedance is modeled by approaching either the mass density (ρ) or the elastic modulus (λ) to infinity. It is shown, using the plane-wave expansion method, that in both cases the dispersion equation contains singular matrices with elements related to the form factor of the crystal lattice. However, this singularity does not affect calculations of the band structure in the case λ → □. Unlike this, in the limiting case of infinite density, the dispersion equation becomes meaningless. We explain the mathematical reason of this drastic difference and propose a regularization numerical procedure. Our general results are illustrated by a particular case of elastic superlattice when the dispersion relation is available analytically.

Geometric phase invariance in spatiotemporal modulated elastic system

We study the topological characteristics of elastic waves in a one-dimensional mass and spring elastic superlattice that exhibits non-reciprocal elastic wave propagation due to extrinsic application of a single sinusoidal spatiotemporal modulation of its spring stiffness. We employ a computational procedure to generate the band structure, traveling modes' amplitudes and phases, and subsequently the geometric phases to characterize the global vibrational behavior of the system and its topological character. The elastic superlattice demonstrates the notion of non-conventional band structure, where hybridization gaps arise as bands appear and cross for certain wavenumber values. We estimate these hybridization points using multiple time scale perturbation theory for low modulation velocity. At the hybridization points, both the theoretical and numerical analyses display a discontinuity in the traveling modes’ amplitudes. Consequently, we find a multiple of π, sometimes zero, geometric phase value in the spatiotemporal modulated elastic system, if the unit cell has inversion symmetry as imposed by the values of the spring constant at the initial time. The temporal modulation, which creates hybridization gaps, changes the nature of the geometric phase from a closed loop geometric phase when the modulation is only spatial, to an open loop geometric phase. This open loop geometric phase is invariant to the temporal modulation.

Spectral analysis of amplitudes and phases of elastic waves: Application to topological elasticity.

The topological characteristics of waves in elastic structures are determined by the geometric phase of waves and, more specifically, by the Berry phase, as a characterization of the global vibrational behavior of the system. A computational procedure for the numerical determination of the geometrical phase characteristics of a general elastic structure is introduced: the spectral analysis of amplitudes and phases method. Molecular dynamics simulation is employed to computationally generate the band structure, traveling modes' amplitudes and phases, and subsequently the Berry phase associated with each band of periodic superlattices. In an innovative procedure, the phase information is used to selectively excite a particular mode in the band structure. It is shown analytically and numerically, in the case of one-dimensional elastic superlattices composed of various numbers of masses and spring stiffness, how the Berry phase varies as a function of the spatial arrangement of the springs. A symmetry condition on the arrangement of springs is established, which leads to bands with Berry phase taking the values of 0 or π. Finally, it is shown how the Berry phase may vary upon application of unitary operations that mathematically describe transformations of the structural arrangement of masses and springs within the unit cells.

Dynamical effective parameters of elastic superlattice with strong acoustic contrast between the constituents

Analytical formulas are obtained for frequency-dependent effective elastic modulus and effective mass density for a periodic layered structure. The proposed homogenization procedure is valid at sufficiently high frequencies well above the lowest band gap in the acoustic spectrum of the structure. It is shown that frequency-dependent effective parameters may take negative values either in different regions of frequencies or in the same quite narrow region. This property demonstrates that the 1D elastic structure may behave in the limit of small Bloch wave vectors as a double-negative acoustic metamaterial.

Bulk elastic waves with unidirectional backscattering-immune topological states in a time-dependent superlattice

Recent progress in electronic and electromagnetic topological insulators has led to the demonstration of one way propagation of electron and photon edge states and the possibility of immunity to backscattering by edge defects. Unfortunately, such topologically protected propagation of waves in the bulk of a material has not been observed. We show, in the case of sound/elastic waves, that bulk waves with unidirectional backscattering-immune topological states can be observed in a time-dependent elastic superlattice. The superlattice is realized via spatial and temporal modulation of the stiffness of an elastic material. Bulk elastic waves in this superlattice are supported by a manifold in momentum space with the topology of a single twist Möbius strip. Our results demonstrate the possibility of attaining one way transport and immunity to scattering of bulk elastic waves.

Elastic superlattices with simultaneously negative effective mass density and shear modulus

We investigate the vibrational properties of superlattices with layers of rubber and polyurethane foam, which can be either conventional or auxetic. Phononic dispersion calculations show a second pass band for transverse modes inside the lowest band gap of the longitudinal modes. In such a band, the superlattices behave as a double-negative elastic metamaterial since the effective dynamic mass density and shear modulus are both negative. The pass band is associated to a Fabry-Perot resonance band which turns out to be very narrow as a consequence of the high contrast between the acoustic impedances of the superlattice components.

An Elastic “Sieve” to Probe Momentum Space: Gd Chains on W(110)

Electron scattering conditions of one-dimensional nanostructures are explored in angle-resolved photoelectron spectroscopy. Tiny increments of the Gd submonolayer coverage on W(110) lead to strong modifications in the spectra. It is shown that the Gd overlayer represents an elastic superlattice of chains which performs a systematic mapping of the electronic band dispersion of the W substrate through a quasicontinuous series of umklapp wave vectors. Conversely, a single valence-band spectrum contains essential and precise structural information readily accessible by comparison to the band dispersion.

Linear and nonlinear vibrations and waves in optical or acoustic superlattices (photonic or phonon crystals)

A model of a dielectric or an elastic superlattice is proposed which describes quite simply the frequency spectrum of electromagnetic or acoustic waves. The frequency band spectrum of a one-dimensional lattice consists of minibands, which narrow down with increasing frequency (so that the forbidden bands in the spectrum broaden with increasing frequency). An elementary analysis of the spectrum of a one-dimensional lattice reveals the presence of many forbidden frequency bands in this case as well. It is shown that dynamic equations for superlattices can be generalized to the nonlinear case, leading to equations of the type of the nonlinear Schrodinger equation for the lattice. Soliton excitations are described and the particle-like dynamics of solitons is demonstrated. Local vibrations near point defects of different complexity in superlattices are studied and graphically illustrated. The existence of Bloch oscillations of a wave packet in a superlattice in a homogeneous external field is discussed.

On a simple model of the photonic or phononic crystal

A model is proposed for a one-dimensional dielectric or elastic superlattice (SL) that relatively simply describes the frequency spectrum of electromagnetic or acoustic waves. The band frequency spectrum is reduced to minibands contracting with increasing frequency. A procedure is suggested for obtaining local states near a defect in a SL, and the simplest of these states is described. Conditions for the initiation of Bloch oscillations of a wave packet in a SR are discussed.

Sound velocity in a model elastic superlattice composite with two dimensional periodic modulations

In this article, I study, by means of numerical solution to the wave equation, the long-wavelength sound velocities for a model of two component elastic superlattice composite. The composite is defined by a periodic array of parallel infinite rods of one component embedded in another elastic medium. This is a superlattice with periodic modulations in two directions. The wave equation is solved numerically with a finite plane wave basis and is used to calculate the long-wavelength sound velocities for the lowest and second lowest energy modes. I study the convergence in the number of Fourier components used and extrapolate with a power law. The deviations of the sound velocity squares from the uniform limits are observed to be proportional to the squares of the elastic constant modulation amplitude for the small amplitude regime. I observe similar results for a range of parameters and composite geometries.

Transverse elastic waves in superlattices: The Brewster acoustic angle

We use a plane wave basis to study the transverse acoustic waves in elastic superlattices---artificial binary structures of periodic mass density $\ensuremath{\rho}(z)$ and periodic transverse elastic velocity ${c}_{t}(z).$ The bulk frequency bands for oblique propagation support minigaps that can close due to the edge crossing of subsequent bands. Some of these crossings are explained in terms of the acoustic Brewster angle ${\ensuremath{\theta}}_{a\mathrm{B}}$ that is a function of the material parameters only. A second condition for the complete shrinking of the gaps that involves the structural parameters is also discussed. The surface waves in the corresponding truncated superlattices are obtained by use of the supercell method. The dispersion curves of these waves are strongly dependent on the material of the layer at the surface---the surface waves that appear for a selected layer at the surface can completely disappear when the last layer is the other one. Specific applications are given for Zn/Fe, Pb/Ge, and Si/Au superlattices.

Wave propagation in polar elastic superlattices : W. A. Green & E. R. Green, Geophysical Journal International, 118(2), 1994, pp 459–465

Wave propagation in polar elastic superlattices : W. A. Green & E. R. Green, Geophysical Journal International, 118(2), 1994, pp 459–465

Wave propagation in polar elastic superlattices

This paper examines the passband and stop band regions for time-periodic waves travelling normal to the layering through an infinite medium composed of alternating layers of two different elastic materials. The materials are such that the elastic energy density is a function of the strains and the strain gradients and, in consequence, a deformation gives rise to both the usual Cauchy stress and to a hyperstress or couple-stress. Such materials can exhibit a non-uniform wrinkling deformation at a free surface and similar non-uniform deformations can arise at interfaces between two different media. The presence of the strain derivatives in the elastic energy function introduces a natural length scale l into the material and the depth of the non-uniform deformation is of the order of this length scale. This model can give rise to enhanced elastic response when the layer depths are comparable with l and it is of interest as a possible mathematical model of nanolayered structures. The model also includes a non-standard set of continuity conditions at material interfaces. These arise from the elastic interaction energy of the two materials at the boundary and their effect is localized in a boundary layer whose depth is of order l. The periodic layering gives rise to displacements which are periodic with a frequency-dependent wave number, the Floquet wave number. Dispersion curves, relating circular frequency to the Floquet wave number, are obtained for different ratios of the layer depth to the natural length l and for different values of the elastic interface coupling parameters.