Acoustic excitations in a non-ideal one-dimensional superlattice with anisotropic impurity layers

The virtual crystal approximation is used to study the specifics of propagation of acoustic excitations through a non-ideal one-dimensional superlattice. Numerical modeling is performed to evaluate the dependence of the lowermost acoustic band gap in a two-sublattice phonon crystal (disordered in composition and widths of constituent layers) on impurity layer concentration.


Introduction
Investigations into propagation of sound in matter, which constitute the subject matter of physical acoustics [1,2] come to the forefront in numerous applied problems, such as the elimination of extraneous sounds (noise abatement), the search for extraction methods of useful acoustic signals, as well as the problem of acoustic object detection through the use of a sonic depth finder. In this connection, a significant attention is devoted to development and enhancement of acoustic devices intended for sound-measuring of physical properties of various media, as well as to fabrication of novel meta-materials for the purpose of controllable propagation of acoustic waves. Perfection of experimental techniques along with the broadening of theoretical perspectives result in expansion of the studied frequency ranges and make acoustic methods an indispensable tool for structural investigations of solids.
At present, there is a significant number of works [3][4][5][6][7][8] devoted to calculation of electromagnetic and acoustic excitations in superlattices based on the T-matrix method, which ultimately involve a solution of a system of equations for the Fourier-expansion coefficients of corresponding fields. The problem of finding of certain specific physical characteristics (such as e.g. the transmission coefficients of electromagnetic excitation, the band spectrum etc.) requires in the general case an expensive and often unfeasible calculation, which compels one to resort to some approximation methods. For instance, it is shown in reference [9] that in the vicinity of the Brillouin zone the dependence of the corresponding frequencies on Bloch wave vector can be approximated by a certain analytical expression. It is worthwhile reminding that the real-life acoustic superlattices are non-ideal systems [10][11][12]. A widely used method of calculation of normal modes in disordered superlattices with randomly distributed (over an entire volume) structural defects is the virtual crystal approximation, whose essence consists [13][14][15] in replacing the configurationally dependent Hamiltonian parameters by their configurationally averaged values. In references [7,8] we have utilized an approach (initially developed for ideal superlattices [9]) to study electromagnetic excitations in non-ideal one-dimensional systems with impurity (defect) layers randomly distributed  [7,8] to obtain the desired optical characteristics of non-ideal superlattices.
In the present work (which continues the lines of reference [8]) this approach is extended to investigation of the specifics of propagation of acoustic excitations through a non-ideal onedimensional phonon crystal constituted by a system of plane-parallel layers containing anisotropic impurity layers with different elastic characteristics (in contrast to reference [16] devoted to arrays of isotropic layers).

Elastic waves in non-ideal one-dimensional superlattices
In the general case of a non-uniform medium the matter density    r and the elastic moduli    r are functions of coordinates, whereas the field of elastic displacements   ,t ur is described by a system of equations [1,17]: where   L r is a differential operator of the form Let us consider acoustic excitations in a one-dimensional inhomogeneous (along the z-axis) layered medium. It is assumed to be comprised by an array of plane-parallel uniform layers obtained by a random replacement of layers of a certain given one-dimensional superlattice with extrinsic uniform uniaxial impurity layers. The structural "cells" of the so transformed "superlattice" differ from the corresponding cells of an ideal system both in widths and composition. Strictly speaking, such a system does not allow invoking the concept of a cell due to the broken translation invariance. Nevertheless a one-to-one correspondence is preserved between the layers of the non-ideal and ideal systems. In the frames of the constructed model the matter density   z  and the elastic moduli   z  can be represented through the use of  -function: In what follows we can conveniently use the quantity      B z z z   , which by analogy with equation (4) can be written as and n a  denote, correspondingly, the layerwise characteristics (matter density and elastic moduli) and layer widths. n numerates the "cells", and  numerates elements in a "cell".
A simplest approximation, which allows examination of transformation of elementary excitation spectrum (caused by the presence of foreign layers) is the virtual crystal approximation. It permits (through configurational averaging of relevant parameters entering the problem's Hamiltonian) to obtain the acoustic excitation spectrum as a function of impurity concentration. The averaging procedure "transforms" the considered non-ideal system into the so-called "virtual crystal", where the translation symmetry is restored.
Unlike in the case of an ideal superlattice, in an imperfect one-dimensional phonon crystal of varied composition and layer widths, tensor ˆn B  and quantity n a  are configurationally dependent. In terms of random variables n    they can be written as where , 1 n     , if the n -th node of the one-dimensional crystal is occupied by a layer of the () th type (and/or by a layer of the ()  -th width), and , 0 n     otherwise. It is assumed below that these factors of disorderliness are independent of each other. The configurational averaging procedure (denoted by angular brackets) applied to equation (7) in accordance with the VCA (and by analogy with the quasi-particle approach [8,14]) yields Here C C   T C   are concentrations of impurity layers of the  -th and  -th types of composition (lower index "C") and width (lower index "T") contained in the  -th sublattice. An obvious condition must hold Similarly to the case of a non-ideal superlattice, the problem of finding phonon-polariton characteristics is reduced to a corresponding problem formulated for the "virtual crystlal", whose layerwise characteristics are described by quantities in a one-dimensional phonon "virtual crystal", which is an ideal system with lattice period d obtained as a result of configurational averaging. We shall have then For anisotropic layers of a one-dimensional phonon crystal tensor components of  (and, correspondingly of B ) are given in [18], whereas for uniform layers of a one-dimensional phonon crystal tensor components of  (and B ) have the form specified in reference [19]: where  ,  are Lame coefficients. In an arbitrary case expression (11) In such a case occurs a longitudinal-transverse splitting of the phonon excitation and the system of equations (9) breaks up into two independent subsystems, first of which contains only quantities zzzz B , and describes propagation of longitudinal acoustic excitations, whereas the second one describes transverse excitations and contains xzzx yzzzy BB  . Such splitting is obviously possible due to a clever choice of the problem's geometry. The dispersion laws of the corresponding acoustic excitations are defined by the infinite system of equations (6), which in the general case (for arbitrary K 's) is solved with the use of approximation methods (similarly to calculation of exciton-polariton excitations in dielectric superlattices [3] (similarly to (11) in reference [7]). Here  [18]. Retaining in system (9) only the terms corresponding to resonance of the specified plane waves 0, 1 p  , we arrive at the following dispersion law of acoustic excitations: ,||, ||, ||, , , ,

Results and discussion
In real-world applications of all the various quantities, which can be obtained from theoretical examinations of propagation of acoustic excitations a special role is played by the band-gap width


 are the roots of equation (12) which define the boundaries of the spectral band.  (12) in the considered case the lowermost band gap width equals to , . CT , which depend on concentration on extrinsic (as compared to an ideal superlattice) layers, are defined by the number of sublattices as well as by material characteristics such as the Lame coefficients  ,  (for homogeneous systems), the matter density  and the coefficients of elasticity (for a uniaxial subsystem). For this reason under various problem's parameters the band gap width can demonstrate diverse types of concentration dependences.
We have performed numerical modeling for the specific case of disordered (in composition and layer widths) two-sublattice one-dimensional phonon crystal. The first sublattice is assumed to be comprised by duraluminium layers (with Young modulus ). The second sublattice contains impurity layers of  -quartz whose concentration is C C . Since the second sublattice is also varied in layer widths the averaged period of the considered one-dimensional phonon crystal equals has the form   (1) (2) (1) where 1 a is the layer width of the first sublattice, (1) 2 a is the layer width of the second lattice of the ideal superlattice, (2) 2 a is the width of foreign layer in the second sublattice, whose concentration is T C .
Detailed calculation based on equation (10) yields the following expressions:   can be quite large (and so the considered multi-layer system would be weakly permeable for acoustic waves), whereas for other values of impurity concentration the band gap   may become rather small. As can be seen from figure 1, the band gap can turn to zero at a certain value of C C , and as this takes place T C may assume any arbitrary value. Once the zero value of   is attained, acoustic excitations would pass unhindered through the layered material with corresponding characteristics.

Conclusion
From the above numerical modeling follows a natural conclusion that the band gap width depends on parameters of the considered superlattice as well as on polarization of propagating acoustic waves. It is shown that concentration dependence   , CT CC   is substantially affected by the ratios Also it should be noted that for a given superlattice at a certain specific acoustic wave frequency for there won't be necessary values of , CT CC such that the transverse and/or longitudinal modes would satisfy the condition 0   (see figure 1). As can be evidently seen in the figure this condition holds only at a certain domain of C C -values. Investigation of the dependence of the lowermost acoustic band gap width on impurity layer concentration carried out in the present work may prove useful for fabrication of acoustic composite materials intended for various operative conditions.