One-Dimensional Phononic Crystals

Abstract

In this chapter, we discuss the vibrational properties of one-dimensional (1D) phononic crystals of both discrete and continuous media. These properties include the dispersion curves of infinite crystals as well as the confined modes and localized (surface, cavity) modes of finite and semi-infinite crystals. A general rule about the existence of localized surface modes in finite and semi-infinite superlattices with free surfaces is presented. We also present the calculations of reflection and transmission coefficients, particularly in view of selective filtering through localized modes. Most of the results presented in this chapter deal with waves propagating along the axis of the superlattice. However, in the last part of the chapter, we also discuss wave propagation out of the normal incidence and, more particularly, we demonstrate the possibility of omnidirectional transmission gap and selective filtering for any incidence angle. A comparison of the theoretical results with experimental data available in the literature is also presented and the reliability of the theoretical predictions is indicated. IntroductionThe one-dimensional (1D) phononic crystals called superlattices (SLs) are of great importance in material science. These structures are, in general, composed of two or several layers repeated periodically along the direction of growth. The layers constituting each cell of the SL can be made of a combination of solid-solid or solid-fluid-layered media. These materials enter now in the category of so-called phononic crystals (see the first chapter of this book and references therein) [1][2][3] constituted by inclusions (spheres, cylinders, etc.) arranged in a host matrix along two-dimensional (2D) and three-dimensional (3D) of the space. After the proposal of SLs by Esaki [4], the study of elementary excitations in multilayered systems has been very active. Among these excitations, acoustic phonons have received increased attention after the first observation by Colvard et al.[5] of a doublet associated to folded longitudinal acoustic phonons by means of Raman scattering. The essential property of these structures is the existence of forbidden frequency bands induced by the difference in acoustic properties of the constituents and the periodicity of these systems leading to unusual physical phenomena in these heterostructures in comparison with bulk materials [6][7][8].With regard to acoustic waves in solid-solid SLs, a number of theoretical and experimental works have been devoted to the study of the band gap structures of periodic SLs [6-10] composed of crystalline, amorphous semi-conductors, or metallic multilayers at the nanometric scale. The theoretical models used are essentially the transfer matrix [7,[11][12][13] and the Green's function methods [6,[14][15][16], whereas the experimental techniques include Raman scattering [5,17,18], ultrasonics [19][20][21][22][23][24][25][26][27][28][29], and time-resolved X-ray diffraction [30]. Besides the existence of the band-gap structures in perfe...

Existence of surface acoustic waves in one-dimensional piezoelectric phononic crystals of general anisotropy

“…In recent years much effort has been made for studying bulk and surface acoustic waves in both the purely elastic and the piezoelectric man-made periodic structures called the phononic crystals [31][32][33][34]. In particular, many studies were devoted to the wave propagation in one-dimensional (1D) phononic crystals, also termed superlattices, which consist of periodically arranged layers [35]. The most numerous results were obtained for the shear horizontally polarized SAWs considered under different settings in phononic crystals of various compositions [36][37][38][39][40][41][42][43][44][45].…”

“…35) The above link allows us to take advantage of the signdefiniteness property of the frequency derivative of the admittance and impedance matrices, which is established in the Appendix [see Eqs. (A3) and (A4)].…”

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

“…periodically layered elastic media; see [12][13][14] and the bibliographies therein. Having intensified later on (see for example [15][16][17][18][19][20][21][22][23]), this research trend is nowadays associated with one-dimensional (1D) elastic phononic crystals, which remain highly topical side by side with piezoelectric and solid-fluid analogues and with two-and three-dimensional models. The combination of anisotropy and (periodic) spatial inhomogeneity essentially prevents explicit solutions to the boundary problem (which is already complicated enough in the case of homogeneous crystals) being reached, so SAW properties used to be investigated mainly by numerical means.…”

“…and Z j,23 of the impedance matricesẐ j of the media j = 1, 2 vanish, so there IAWs and the SH-IAWs waves, respectively. Note that the full stop bands for decoupled S and SH modes, in which the solutions of, respectively, (3.11) 1 and (3.11) 2 are sought, are different and independent of each other.Consider the SH-IAWs.…”

Surface acoustic waves in one-dimensional piezoelectric phononic crystals with symmetric unit cell

“…Throughout recent years, there has been a steady interest to the propagation of bulk and surface acoustic waves in phononic crystals [1][2][3]. In particular, much attention has been paid to the wave propagation in one-dimensional (1D) phononic crystals, otherwise termed superlattices, which represent periodic sequences of multilayers [4]. Reflection of bulk waves was investigated in piezoelectric [5][6][7] and solid-fluid superlattices [8][9][10], as well as in solid-solid and solid-fluid Fibonacci structures [11][12][13][14].…”