Theory of surface and interface transverse elastic waves in N-layer superlattices

Characteristics of truncation resonances in periodic bilayer rods and beams with symmetric and asymmetric unit cellsa)

Truncation resonances are resonant frequencies that occur within bandgaps and are a prominent feature of finite phononic crystals. While recent studies have shed light on the existence conditions and modal characteristics of truncation resonances in discrete systems, much remains to be understood about their behavior in continuous structures. To address this knowledge gap, this paper investigates the existence and modal characteristics of truncation resonances in periodic bilayer beams, both numerically and experimentally. Specifically, the effect of symmetry of the unit cells, boundary conditions, material/geometric properties, and the number of unit cells are studied. To this end, we introduce impedance and phase velocity ratios based on the material and geometric properties and show how they affect the existence of truncation resonances, relative location of the truncation resonances within the bandgap, and spatial attenuation or degree of localization of the truncation resonance mode shapes. Finally, the existence and mode shapes of truncation resonances are experimentally validated for both longitudinal and flexural cases using three-dimensional (3D) printed periodic beams. This paper highlights the potential impact of these results on the design of finite phononic crystals for various applications, including energy harvesting and passive flow control.

Closed-form existence conditions for bandgap resonances in a finite periodic chain under general boundary conditions

Bragg scattering in periodic media generates bandgaps, frequency bands where waves attenuate rather than propagate. Yet, a finite periodic structure may exhibit resonance frequencies within these bandgaps. This is caused by boundary effects introduced by the truncation of the nominal infinite medium. Previous studies of discrete systems determined existence conditions for bandgap resonances, although the focus has been limited to mainly periodic chains with free-free boundaries. In this paper, we present closed-form existence conditions for bandgap resonances in discrete diatomic chains with general boundary conditions (free-free, free-fixed, fixed-free, or fixed-fixed), odd or even chain parity (contrasting or identical masses at the ends), and the possibility of attaching a unique component (mass and/or spring) at one or both ends. The derived conditions are consistent with those theoretically presented or experimentally observed in prior studies of structures that can be modeled as linear discrete diatomic chains with free-free boundary conditions. An intriguing case is a free-free chain with even parity and an arbitrary additional mass at one end of the chain. Introducing such an arbitrary mass underscores a transition among a set of distinct existence conditions, depending on the type of chain boundaries and parity. The proposed analysis is applicable to linear periodic chains in the form of lumped-parameter models, examined across the frequency spectrum, as well as continuous granular media models, or similar configurations, examined in the low-frequency regime.

Stoneley-type waves in anisotropic periodic superlattices

The paper investigates the existence of interfacial (Stoneley-type) acoustic waves localised at the internal boundary between two semi-infinite superlattices which are adjoined with each other to form one-dimensional phononic bicrystal. Each superlattice is a periodic sequence of perfectly bonded homogeneous and/or functionally graded layers of general anisotropy. Given any asymmetric arrangement of unit cells (periods) of superlattices, it is found that the maximum number of interfacial waves, which can emerge at a fixed tangential wavenumber for the frequency varying within a stopband, is three for the lowest stopband and six for any upper stopband. Moreover, we show that this number of three or six waves in the lowest or upper stopband, is actually the maximum for the number of waves occurring per stopband in a given bicrystal plus their number in the "complementary" bicrystal, which is obtained by swapping upper and lower superlattices of the initial one (so that both bicrystals have the same band structure). An example is provided demonstrating attainability of this upper bound, i.e. the existence of six interfacial waves in a stopband. The results obtained under no assumptions regarding the material anisotropy are also specified to the case of monoclinic symmetry leading to acoustic mode decoupling.

Graphene-Based One-Dimensional Terahertz Phononic Crystal: Band Structures and Surface Modes

In this paper, we provide a theoretical and numerical study of the acoustic properties of infinite and semi-infinite superlattices made out of graphene-semiconductor bilayers. In addition to the band structure, we emphasize the existence and behavior of localized and resonant acoustic modes associated with the free surface of such structures. These modes are polarized in the sagittal plane, defined by the incident wavevector and the normal to the layers. The surface modes are obtained from the peaks of the density of states, either inside the bulk bands or inside the minigaps of the superlattice. In these structures, the two directions of vibrations (longitudinal and transverse) are coupled giving rise to two bulk bands associated with the two polarizations of the waves. The creation of the free surface of the superlattice induces true surface localized modes inside the terahertz acoustic forbidden gaps, but also pseudo-surface modes which appear as well-defined resonances inside the allowed bands of the superlattice. Despite the low thickness of the graphene layer, and though graphene is a gapless material, when it is inserted periodically in a semiconductor, it allows the opening of wide gaps for all values of the wave vector k// (parallel to the interfaces). Numerical illustrations of the band structures and surface modes are given for graphene-Si superlattices, and the surface layer can be either Si or graphene. These surface acoustic modes can be used to realize liquid or bio-sensors graphene-based phononic crystal operating in the THz frequency domain.

Interfacial acoustic waves in one-dimensional anisotropic phononic bicrystals with a symmetric unit cell

The paper is concerned with the interfacial acoustic waves localized at the internal boundary of two different perfectly bonded semi-infinite one-dimensional phononic crystals represented by periodically layered or functionally graded elastic structures. The unit cell is assumed symmetric relative to its midplane, whereas the constituent materials may be of arbitrary anisotropy. The issue of the maximum possible number of interfacial waves per full stop band of a phononic bicrystal is investigated. It is proved that, given a fixed tangential wavenumber, the lowest stop band admits at most one interfacial wave, while an upper stop band admits up to three interfacial waves. The results obtained for the case of generally anisotropic bicrystals are specialized for the case of a symmetric sagittal plane.

Non-leaky surface acoustic waves in the passbands of one-dimensional phononic crystals

The paper theoretically investigates the occurrence of non-leaky surface acoustic waves (SAWs) in the passbands of the Floquet-Bloch spectra of half-infinite one-dimensional phononic crystals. The phononic crystal is represented by a periodic structure of perfectly bonded anisotropic elastic layers. The traction-free boundary plane truncates the phononic crystal at the edge of a period. For the general case of unrestricted anisotropy of the constitutive layers, it is shown that if the passband allows only two partial bulk modes, then the non-leaky SAW must simultaneously satisfy three real equations imposed on problem parameters such as the SAW frequency and tangential wavenumber, the angle of its propagation direction along the given boundary plane, and the characteristics of the medium. If there are four bulk modes in the passband, then the non-leaky SAW must satisfy at least five real equations. In the case where the layers possess a common plane of crystallographic symmetry which is either parallel to the layer interfaces or perpendicular to the direction of propagation, the number of real equations conditioning the occurrence of non-leaky SAWs in the passbands with two and with four bulk modes reduces to two and three, respectively. If the sagittal plane is a plane of symmetry, then the sagittally polarized non-leaky passband SAW must satisfy three real equations; however, if this sagittal symmetry plane coexists with another plane of symmetry which is either parallel to the layer interfaces or perpendicular to the direction of propagation, then the existence of non-leaky passband SAW requires the fulfillment of two conditions only. In particular, this is always the case when the constitutive layers are elastically isotropic. The general conclusions are illustrated by numerical examples.

Surface acoustic waves in one-dimensional piezoelectric-metallic phononic crystal: Effect of a cap layer

We study the propagation of transverse acoustic waves associated with the surface of a semi-infinite superlattice (SL) composed of piezoelectric-metallic layers and capped with a piezoelectric layer. We present closed-form expressions for localized surface waves, the so-called Bleustein-Gulyaev (BG) waves depending on whether the cap layer is open-circuited or short-circuited. These expressions are obtained by means of the Green's function method which enables to deduce also the densities of states. These theoretical results are illustrated by a few numerical applications to SLs made of piezoelectric layers of hexagonal symmetry belonging to the 6 mm class such as PZT4 and ZnO in contact with metallic layers such as Fe, Al, Au, Cu and boron-doped-diamond. We demonstrate a rule about the existence of surface modes when considering two complementary semi-infinite SLs obtained by the cleavage of an infinite SL along a plane parallel to the piezoelectric layers. Indeed, when the surface layers are open-circuited, one obtains one surface mode per gap, this mode is associated with one of the two complementary SLs. However, when the surface layers are short-circuited, this rule is not fulfilled and one can obtain zero, one or two modes inside each gap of the two complementary SLs depending on the position of the plane where the cleavage is produced. We show that in addition to the BG surface waves localized at the surface of the cap layer, there may exist true guided waves and pseudo-guided waves (i.e. leaky waves) induced by the cap layer either inside the gaps or inside the bands of the SL respectively. Also, we highlight the possibility of existence of interface modes between the SL and a cap layer as well as an interaction between these modes and the BG surface mode when both modes fall in the same band gaps of the SL. The strength of the interaction depends on the width of the cap layer. Finally, we show that the electromechanical coupling coefficient (ECC) is very sensitive to the cap layer thickness, in particular we calculate and discuss the behavior of the ECC as a function of the adlayer thickness for the low velocity surface modes of the SL which exhibit the highest ECC values. The effect of the nature of the metallic layers inside the SL on the ECC is also investigated. The different surface modes discussed in this work should have applications in sensing applications.

Localized transversal-rotational modes in linear chains of equal masses

The propagation and localization of transversal-rotational waves in a two-dimensional granular chain of equal masses are analyzed in this study. The masses are infinitely long cylinders possessing one translational and one rotational degree of freedom. Two dispersive propagating modes are predicted in this granular crystal. By considering the semi-infinite chain with a boundary condition applied at its beginning, the analytical study demonstrates the existence of localized modes, each mode composed of two evanescent modes. Their existence, position (either in the gap between the propagating modes or in the gap above the upper propagating mode), and structure of spatial localization are analyzed as a function of the relative strength of the shear and bending interparticle interactions and for different boundary conditions. This demonstrates the existence of a localized mode in a semi-infinite monatomic chain when transversal-rotational waves are considered, while it is well known that these types of modes do not exist when longitudinal waves are considered.

The viscous effects on shear horizontal surface acoustic waves in semi-infinite superlattices

In this paper, by a semi-analytical method, the propagation characteristics of shear horizontal surface acoustic waves in semi-infinite superlattices containing viscous materials are investigated. The factors that influence the attenuation and phase velocity of the surface waves are analyzed in detail. The results may be useful for the design of acoustic wave devices.

Shear horizontal surface acoustic waves in semi-infinite piezoelectrics/metal superlattices

In this paper, an analytical method is presented for the study of shear horizontal surface acoustic waves in semi-infinite piezoelectrics/metal superlattices. The results show that the high electromechanical coupling coefficient and wide usable frequency range can be obtained in these superlattices.

Existence and spectral properties of shear horizontal surface acoustic waves in vertically periodic half-spaces

The paper is concerned with the propagation of shear horizontal surface waves (SHSW) in semi-infinite elastic media with vertically periodic continuous and/or discrete variation of material properties. The existence and spectral properties of the SHSW are shown to be intimately related to the shape of the properties variation profile. Generally, the SHSW dispersion branches represent randomly broken spectral intervals on the ( ω ,  k ) plane. They may, however, display a particular regularity in being confined to certain distinct ranges of slowness s = ω / k , which can be predicted and estimated directly from the profile shape. The SHSW spectral regularity is especially prominent when the material properties at the opposite edge points of a period are different. In particular, a unit cell can be arranged so that the SHSW exists within a single slowness window, narrow in the measure of material contrast between the edges, and does not exist elsewhere or vice versa. Explicit analysis in the ( ω ,  k ) domain is complemented and verified through the numerical simulation of the SH wave field in the time–space domain. The results also apply to a longitudinally periodic semi-infinite strip with a homogeneous boundary condition at the faces.

Two types of modes in finite size one-dimensional coaxial photonic crystals: general rules and experimental evidence

We demonstrate analytically and experimentally the existence and behavior of two types of modes in finite size one-dimensional coaxial photonic crystals made of N cells with vanishing magnetic field on both sides. We highlight the existence of N-1 confined modes in each band and one mode by gap associated to either one or the other of the two surfaces surrounding the structure. The latter modes are independent of N . These results generalize our previous findings on the existence of surface modes in two semi-infinite superlattices obtained from the cleavage of an infinite superlattice between two cells. The analytical results are obtained by means of the Green's function method, whereas the experiments are carried out using coaxial cables in the radio-frequency regime.

Electromagnetic wave propagation in quasi-periodic photonic circuits

We study theoretically and experimentally the properties of quasiperiodic one-dimensional serial loop structures made of segments and loops arranged according to a Fibonacci sequence (FS). Two systems are considered. (i) By inserting the FS horizontally between two waveguides, we give experimental evidence of the scaling behaviour of the amplitude and the phase of the transmission coefficient. (ii) By grafting the FS vertically along a guide, we obtain from the maxima of the transmission coefficient the eigenmodes of the finite structure (assuming the vanishing of the magnetic field at the boundaries of the FS). We show that these two systems (i) and (ii) exhibit the property of self-similarity of order three at certain frequencies where the quasiperiodicity is most effective. In addition, because of the different boundary conditions imposed on the ends of the FS, we show that horizontal and vertical structures give different information on the localization of the different modes inside the FS. Finally, we show that the eigenmodes of the finite FS coincide exactly with the surface modes of two semi-infinite superlattices obtained by the cleavage of an infinite superlattice formed by a periodic repetition of a given FS.