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Elastic properties of finite three-dimensional solid phononic-crystal slabs
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Photonics and Nanostructures – Fundamentals and Applications 6 (2008) 122–126 www.elsevier.com/locate/photonics
Elastic properties of finite three-dimensional solid phononic-crystal slabs R. Sainidou *, B. Djafari-Rouhani, J.O. Vasseur Institut d’Electronique, de Microe´lectronique et de Nanotechnologie, UMR CNRS 8520, Avenue Poincare´ - BP 60069, 59652 Villeneuve d’Ascq Cedex, France Received 18 June 2007; received in revised form 20 September 2007; accepted 8 October 2007 Available online 13 October 2007
Abstract We study theoretically, by means of layer-multiple scattering techniques, the propagation of elastic waves through finite slabs of phononic crystals consisting of metallic spheres in polyester matrix, embedded in air. We focus on the study of modes localized on the surfaces of the structure, investigating the physical parameters which influence and determine their appearance. Our results reveal the existence of absolute phononic frequency gaps in these finite structures, and point out the possibility, under an appropriate choice of the parameters, of tunable regions of frequency free of propagating and/or surface-localized modes. This could be very useful in the design of devices related to frequency filtering, waveguiding, etc. # 2007 Elsevier B.V. All rights reserved. PACS : 43.20.+g; 63.22.+m; 68.35.Ja; 68.35.Iv Keywords: Phononic crystals; Finite slabs; Surface-localized waves; Multiple scattering
1. Introduction The propagation of elastic waves in inhomogeneous media has attracted a lot of attention over the years as it relates to many and varied disciplines ranging from geophysics to mechanical engineering, with applications in quantitative nondestructive evaluation, the design of ultrasound absorptive materials, etc. More recently there has been a growing interest [1] in a special type of inhomogeneous materials, the so-called phononic crystals, whose elastic coefficients vary periodically in space. These are composite materials consisting of locally homogeneous inclusions, distributed periodically in a homogeneous host medium. The interest in these materials arises mainly from the
* Corresponding author. E-mail address: [email protected] (R. Sainidou). 1569-4410/$ – see front matter # 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.photonics.2007.10.001
possibility of having frequency regions, known as absolute phononic gaps, over which there can be no propagation of elastic waves in the crystal, whatever the direction of propagation. However, in an actual experiment, one usually uses finite solid phononiccrystal slabs embedded in a fluid (air or water). And in such a case the transmission-gap is not necessarily the same with the gap obtained under the concept of the corresponding infinite solid crystal. New modes are expected to appear related to the existence of the ending surfaces of the finite structure, strongly localized in these regions. In general, surface elastic waves are of great importance, since they may lead, one hopes, to interesting properties and applications in a variety of fields, including telecommunications, waveguiding, phenomena associated with defects, etc. They are extensively studied in cases of two-dimensional (2D) phononic-crystal slabs [2–4], but as far as we know very
R. Sainidou et al. / Photonics and Nanostructures – Fundamentals and Applications 6 (2008) 122–126
little [5] has been done on the corresponding threedimensional (3D) systems. In this work, we are concerned with the study of surface-localized modes in relatively thick 3D phononic-crystal slabs and we show that such systems offer much more possibilities for tailoring these modes and their properties. 2. Method of calculation For the study of propagation of elastic waves in finite slabs of 3D phononic crystals, and especially in slabs of layered form along the z-direction, normal to their surfaces, the layer-multiple scattering method as developed for phononic crystals [6], is an ideally suited tool as it does not require periodicity in z-direction. The method can treat efficiently, besides an infinite phononic crystal, also a slab of the crystal of finite thickness which may be, in the general case, a succession of different layers which can be single interfaces, homogeneous slabs, or planes of spheres arranged with a given 2D periodicity on each plane. The only requirement of this method is that the 2D periodicity must be always the same among the different layers. This implies that kk , the component of the wavevector of the elastic wave, reduced in the corresponding surface Brillouin zone (SBZ) of the 2D periodic lattice, must be a conserved quantity. For each layer, the method calculates the transmission and reflection matrices, QI and QIII , respectively, for a plane wave incident on the layer with given frequency and kk from the left (i.e. with kz > 0), as well as the corresponding matrices QIV and QII for incidence from the right (i.e. with kz < 0). Explicit expressions for these Q matrices can be found elsewhere [6]. If one is interested to study the propagation of elastic waves of frequency v incident from the one side (e.g. from the left) of the finite slab with a given kk , then the layer-multiple scattering formalism provides us with a set of transmission and reflection matrices, for the whole finite structure, calculated from those of the constituent layers, through which the transmittance and reflectance of this slab for a wave of given v, kk and polarization, as well as the corresponding density of states of the elastic field, are obtained [6]. Moreover, the method can be also applied to calculate the dispersion diagrams (v; kk ) for a finite slab, i.e. determine the eigenfrequencies of the bound and virtual bound states of such a structure [5] from the condition det ½I
QIIL QIII R
¼ 0;
(1)
which results from the requirement of the existence of a wave field localized within the slab. Real- or imaginary-
123
frequency solutions of the above secular equation correspond to bound or virtual bound states (leaky modes) of the system under consideration. 3. Results and discussion To begin with, we consider pffiffiffi a finite slab of fcc crystal of lattice constant a0 2 (a0 is the first-neighbor distance), consisting of steel spheres (mass density: rs ¼ 7800 kg/m3, longitudinal and transverse velocities: cls ¼ 5940 m/s and cts ¼ 3200 m/s) in polyester (mass density: r ¼ 1220 kg/m3, longitudinal and transverse velocities: cl ¼ 2490 m/s and ct ¼ 1180 m/s). The spheres have pffiffiffia radius S ¼ 0:325a0 , leading to a value f ¼ ðð4p 2Þ=3ÞðS=a0 Þ3 ¼ 0:20 for the volume filling fraction of the corresponding infinite crystal. The slab is constructed as a sequence of N L successive (1 1 1) p fcc ffiffiffi planes of spheres (layers), with a distance d ¼ a0 6=3 between them, and for the present study we assume it sufficiently thick: we consider N L ¼ 8. The whole structure is embedded in air (mass density: rair ¼ 1:23 kg/m3, sound velocity: cair ¼ 340 m/s), as it is shown in Fig. 1, with the two ending surfaces at a distance ds from the center of the surface sphere planes. Of course ds can vary, following the inequality d s S (the spheres cannot be cut from the pffiffiffi air–polyester interface). We assume d s ¼ d=2 ¼ a0 6=6 ¼ 0:408a0 , except otherwise stated. This system in its infinite form (i.e. the corresponding crystal of steel spheres in polyester) presents a large absolute gap extending from va0 =cair ¼ 16:73 to va0 =cair ¼ 23:63. In Fig. 2 we show the projection of the frequency band structure of the acoustic field in this crystal on the SBZ of its (1 1 1) surface, which we
Fig. 1. The structure under study. The finite, along z-direction, slab consists of eight (1 1 1) fcc planes of steel spheres of radius S ¼ 0:325a0 (a0 is the first-neighbor distance) in polyester host and is embedded in air. d and d s denote, respectively, the distance between two successive planes of spheres and between the air–polyester interface and the center of the surface plane of spheres.
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Fig. 2. Projection of the phononic band structure of a fcc crystal of steel spheres (radius S ¼ 0:325a0 ) in polyester on the high symmetry lines of the SBZ of its (1 1 1) surface (shown in the inset). Propagating modes: shaded regions, gaps: blank regions. Solid lines: surfacelocalized modes of the corresponding finite slab consisting of eight (1 1 1) layers of spheres, embedded in air. Dotted lines: lines of propagation of longitudinal and transverse waves in infinite polyester (higher and lower branch, respectively).
calculated using the computer program of Ref. [6]. The shaded regions extend over the frequency bands of the acoustic field: at any one frequency within the shaded region, for a given kk [the component of the reduced wavevector within the SBZ of the (1 1 1) surface], there exists at least one propagating mode [the corresponding kz ðvÞ is real] of the acoustic field in the infinite crystal. The blank areas correspond to frequency gaps [all kz ðvÞ are complex]. Therefore, over this frequency region, there are no propagating modes of the acoustic field in the infinite crystal. If now we consider the corresponding finite system embedded in air (see Fig. 1), we find, apart from the propagating modes of the slab which reproduce the regions of elastic states of the infinite crystal (shaded regions in Fig. 2), additional modes originating from the presence of the two ending surfaces of the finite system (air–polyester interfaces), shown by the solid lines in Fig. 2. In the region of the absolute gap they extend from va0 =cair ¼ 22:68 to va0 =cair ¼ 23:63 and of course they continue upwards penetrating within the region of propagating modes, but we are not interested in those regions. We focus our study only on such modes lying within the frequency-gap regions of the corresponding infinite system. For slabs sufficiently thick, which is the case here, these modes are doubly degenerated each of them associated to each one of the two surfaces (apart from the low-frequency region, i.e. for va0 =cair 9 5, where a split is observed because of the strong repulsion of the two corresponding modes). Strictly speaking, the above-mentioned modes are not surface modes, in the sense that they do not decay exponentially on both sides of the air–polyester interfaces: they propagate along zdirection in air regions. However, they are strongly
localized in the surface regions, i.e. between the air– polyester interface and the first sphere planes: they attenuate in the interior of the thick phononic-crystal slab, in the gap-frequency regions of the infinite system. These surface modes manifest themselves as resonances in the transmission spectrum of a longitudinal wave incident on the slab. However, contrary to resonances originating from propagating states of the finite slab, their height decreases by increasing the thickness of the slab. A typical case is given in Fig. 3(a) for a longitudinal wave incident normally (kk ¼ 0) on the slab from the left, in which case we removed the second air–polyester interface (at the right) in order to clearly see the effect of attenuation of the waves at the corresponding frequencies inside the sufficiently thick slab. The presence of the second air–polyester interface will produce for relatively thin slabs a double resonance peak in the transmission spectrum (see Fig. 3(b)) resulting from the interaction of the two surfacelocalized modes; the thinner the slab, the stronger the interaction and the higher the resonance peak value. But this second surface has nothing to add in the physical interpretation, so we will omit it, at this point. We can say that the phononic-crystal slab under consideration behaves, for frequencies within the gap, as a homogeneous and isotropic effective medium characterized by an imaginary wavenumber Im½kz ðvÞ which determines the attenuation of the wave field over this region; for given v and kk ¼ 0 we obtain ln T ¼ 2 Im½kz dN L þ const. This is demonstrated in Fig. 3(c) for the resonance frequency within the gap shown in Fig. 3(a). One expects that appropriate modifications in the slab surfaces can produce smart tuning in frequency of the above-described surface-localized modes, while the otherwise absolute gap remains unaffected, since it is related to the thick-slab’s internal crystal structure. In Fig. 4 we demonstrate such an effect by changing the distance ds between each of the two ending surfaces of the finite slab and the center of the surface planes of spheres. By placing the two surfaces just at the position where the surface layers of spheres terminate (ds ¼ 0:325a0 ¼ S) the surface-localized modes are completely suppressed to the top of the absolute gap, and the inverse happens as we increase progressively ds : they go downwards destroying completely the full region of this gap. This can be understood as follows: the larger the d s , these modes approach more to the Lamb modes of a homogeneous slab of the same thickness (having air at its left and a phononic-crystal slab at its right), folded of course because of the 2D periodicity. The whole picture obtained is an interplay between the homogeneous slab and the influence of the presence of the spheres which
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125
Fig. 3. (a) Transmittance of a wave incident normally on the left of a slab containing only the left air–polyester interface (infinitely extended polyester at the right), for different thicknesses of the slab (N L is the number of (1 1 1) fcc layers). (b) Same as (a) for N L ¼ 5 in the case of one (solid line) and two (dashed line) air–polyester interfaces. (c) The transmittance as a function of the thickness of the slab at the resonance frequency va0 =cair ¼ 23:1896. The straight line has a slope equal to 2 Im½kz d ¼ 3:37, corresponding to an effective attenuation of the waves at the given frequency.
dominates for small values of ds , thus creating gap regions [7]. Similar behavior can be obtained through surface modifications of the surface planes of spheres (e.g. by changing the material and/or the size of the spheres), although the detailed picture will be different from those presented in Fig. 4. Possibilities of tuning such surface-localized modes can be also present at lower frequency regions, where the first gap of a phononic structure appears (below the first pass-band). In this case one needs to use phononiccrystal slabs made of smaller spheres, in order to push up the propagating modes at higher frequencies and reveal in such a way the surface-localized modes in full
detail. An example is given in Fig. 5(a), where we have limited the sketch along the KM direction of the SBZ. The slab is now made of spheres of radius S ¼ 0:10a0 resulting to a dispersion plot for the propagating modes (shaded region) almost identical to the propagation lines in the homogeneous polyester (dotted lines). Moreover, between the eight-layers thick slab and the two air–polyester interfaces we add now two surface layers of steel spheres of larger radius Ss ¼ 0:40a0 (by keeping of course the same 2D periodicity), thus obtaining surface-localized modes quite low in frequency with respect to the propagating modes, as can be seen in the corresponding dispersion diagram (see Fig. 5(a)) with the solid lines, as compared to the case shown in Fig. 2. We note the presence of a narrow gap extending from va0 =cair ¼ 8:708 to
Fig. 4. The evolution of the states in the region of the absolute gap of the system of Fig. 2 as a function of the distance d s of the two ending surfaces of the finite slab from the center of the surface planes of spheres. Propagating/surface-localized modes: light-gray/gray shaded regions. Frequency-gaps: white regions. The arrows indicate the region occupied by the surface-localized modes of the system of Fig. 2.
Fig. 5. Dispersion curves along KM direction for a slab consisting of eight (1 1 1) fcc planes of steel spheres of radius S ¼ 0:10a0 on both sides of which one additional surface layer of different sphere radius Ss ¼ 0:40a0 made of (a) steel or (b) gold is added between the air– polyester interface and the eight-layers thick phononic-crystal slab. Dotted lines: lines of propagation of transverse waves in infinite polyester. The arrows indicate the position of small gaps.
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va0 =cair ¼ 8:863. By replacing the steel spheres of the surface planes by gold-made ones (mass density: rs ¼ 19; 700 kg/m3, longitudinal and transverse velocities: cls ¼ 3240 m/s and cts ¼ 1200 m/s) the surface-localized modes are pushed down even more [Fig. 5(b)], an effect which is due mostly to the mass density difference between steel and gold. Now apart from two small gaps (for va0 =cair from 5.602 to 5.675 and from 7.323 to 7.538), a large region free of modes appears, above the surface-localized modes, extending from va0 =cair ¼ 8:37 to va0 =cair ¼ 12:43.
exist. Our findings could be useful in the construction and design of devices related to frequency filtering, waveguiding, etc.
4. Conclusions
[1] For a comprehensive list of references on phononic crystals, see the Phononic Crystal database at http://www.phys.uoa.gr/phono nics/PhononicDatabase.html. [2] Y. Tanaka, S. Tamura, Phys. Rev. B 58 (1998) 7958. [3] A. Khelif, B. Aoubiza, S. Mohammadi, A. Adibi, V. Laude, Phys. Rev. E 74 (2006) 046610. [4] J.H. Sun, T.T. Wu, Phys. Rev. B 74 (2006) 174305. [5] R. Sainidou, N. Stefanou, Phys. Rev. B 73 (2006) 184301. [6] R. Sainidou, N. Stefanou, I.E. Psarobas, A. Modinos, Comput. Phys. Commun. 166 (2005) 197. [7] R. Sainidou, B. Djafari-Rouhani, J.O. Vasseur, unpublished data.
In this work we presented results concerning the appearance of surface-localized modes inside the frequency gap regions of thick solid phononic-crystal slabs, embedded in air. We demonstrated that these modes can be tunable under an appropriate choice of the geometrical or elastic parameters of the slab, giving rise to regions where no propagating and surface modes
Acknowledgement R. Sainidou gratefully acknowledges the hospitality of the UFR de Physique, Universite´ de Lille 1, France. References
Photonics and Nanostructures – Fundamentals and Applications 6 (2008) 122–126 www.elsevier.com/locate/photonics
Elastic properties of finite three-dimensional solid phononic-crystal slabs R. Sainidou *, B. Djafari-Rouhani, J.O. Vasseur Institut d’Electronique, de Microe´lectronique et de Nanotechnologie, UMR CNRS 8520, Avenue Poincare´ - BP 60069, 59652 Villeneuve d’Ascq Cedex, France Received 18 June 2007; received in revised form 20 September 2007; accepted 8 October 2007 Available online 13 October 2007
Abstract We study theoretically, by means of layer-multiple scattering techniques, the propagation of elastic waves through finite slabs of phononic crystals consisting of metallic spheres in polyester matrix, embedded in air. We focus on the study of modes localized on the surfaces of the structure, investigating the physical parameters which influence and determine their appearance. Our results reveal the existence of absolute phononic frequency gaps in these finite structures, and point out the possibility, under an appropriate choice of the parameters, of tunable regions of frequency free of propagating and/or surface-localized modes. This could be very useful in the design of devices related to frequency filtering, waveguiding, etc. # 2007 Elsevier B.V. All rights reserved. PACS : 43.20.+g; 63.22.+m; 68.35.Ja; 68.35.Iv Keywords: Phononic crystals; Finite slabs; Surface-localized waves; Multiple scattering
1. Introduction The propagation of elastic waves in inhomogeneous media has attracted a lot of attention over the years as it relates to many and varied disciplines ranging from geophysics to mechanical engineering, with applications in quantitative nondestructive evaluation, the design of ultrasound absorptive materials, etc. More recently there has been a growing interest [1] in a special type of inhomogeneous materials, the so-called phononic crystals, whose elastic coefficients vary periodically in space. These are composite materials consisting of locally homogeneous inclusions, distributed periodically in a homogeneous host medium. The interest in these materials arises mainly from the
* Corresponding author. E-mail address: [email protected] (R. Sainidou). 1569-4410/$ – see front matter # 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.photonics.2007.10.001
possibility of having frequency regions, known as absolute phononic gaps, over which there can be no propagation of elastic waves in the crystal, whatever the direction of propagation. However, in an actual experiment, one usually uses finite solid phononiccrystal slabs embedded in a fluid (air or water). And in such a case the transmission-gap is not necessarily the same with the gap obtained under the concept of the corresponding infinite solid crystal. New modes are expected to appear related to the existence of the ending surfaces of the finite structure, strongly localized in these regions. In general, surface elastic waves are of great importance, since they may lead, one hopes, to interesting properties and applications in a variety of fields, including telecommunications, waveguiding, phenomena associated with defects, etc. They are extensively studied in cases of two-dimensional (2D) phononic-crystal slabs [2–4], but as far as we know very
R. Sainidou et al. / Photonics and Nanostructures – Fundamentals and Applications 6 (2008) 122–126
little [5] has been done on the corresponding threedimensional (3D) systems. In this work, we are concerned with the study of surface-localized modes in relatively thick 3D phononic-crystal slabs and we show that such systems offer much more possibilities for tailoring these modes and their properties. 2. Method of calculation For the study of propagation of elastic waves in finite slabs of 3D phononic crystals, and especially in slabs of layered form along the z-direction, normal to their surfaces, the layer-multiple scattering method as developed for phononic crystals [6], is an ideally suited tool as it does not require periodicity in z-direction. The method can treat efficiently, besides an infinite phononic crystal, also a slab of the crystal of finite thickness which may be, in the general case, a succession of different layers which can be single interfaces, homogeneous slabs, or planes of spheres arranged with a given 2D periodicity on each plane. The only requirement of this method is that the 2D periodicity must be always the same among the different layers. This implies that kk , the component of the wavevector of the elastic wave, reduced in the corresponding surface Brillouin zone (SBZ) of the 2D periodic lattice, must be a conserved quantity. For each layer, the method calculates the transmission and reflection matrices, QI and QIII , respectively, for a plane wave incident on the layer with given frequency and kk from the left (i.e. with kz > 0), as well as the corresponding matrices QIV and QII for incidence from the right (i.e. with kz < 0). Explicit expressions for these Q matrices can be found elsewhere [6]. If one is interested to study the propagation of elastic waves of frequency v incident from the one side (e.g. from the left) of the finite slab with a given kk , then the layer-multiple scattering formalism provides us with a set of transmission and reflection matrices, for the whole finite structure, calculated from those of the constituent layers, through which the transmittance and reflectance of this slab for a wave of given v, kk and polarization, as well as the corresponding density of states of the elastic field, are obtained [6]. Moreover, the method can be also applied to calculate the dispersion diagrams (v; kk ) for a finite slab, i.e. determine the eigenfrequencies of the bound and virtual bound states of such a structure [5] from the condition det ½I
QIIL QIII R
¼ 0;
(1)
which results from the requirement of the existence of a wave field localized within the slab. Real- or imaginary-
123
frequency solutions of the above secular equation correspond to bound or virtual bound states (leaky modes) of the system under consideration. 3. Results and discussion To begin with, we consider pffiffiffi a finite slab of fcc crystal of lattice constant a0 2 (a0 is the first-neighbor distance), consisting of steel spheres (mass density: rs ¼ 7800 kg/m3, longitudinal and transverse velocities: cls ¼ 5940 m/s and cts ¼ 3200 m/s) in polyester (mass density: r ¼ 1220 kg/m3, longitudinal and transverse velocities: cl ¼ 2490 m/s and ct ¼ 1180 m/s). The spheres have pffiffiffia radius S ¼ 0:325a0 , leading to a value f ¼ ðð4p 2Þ=3ÞðS=a0 Þ3 ¼ 0:20 for the volume filling fraction of the corresponding infinite crystal. The slab is constructed as a sequence of N L successive (1 1 1) p fcc ffiffiffi planes of spheres (layers), with a distance d ¼ a0 6=3 between them, and for the present study we assume it sufficiently thick: we consider N L ¼ 8. The whole structure is embedded in air (mass density: rair ¼ 1:23 kg/m3, sound velocity: cair ¼ 340 m/s), as it is shown in Fig. 1, with the two ending surfaces at a distance ds from the center of the surface sphere planes. Of course ds can vary, following the inequality d s S (the spheres cannot be cut from the pffiffiffi air–polyester interface). We assume d s ¼ d=2 ¼ a0 6=6 ¼ 0:408a0 , except otherwise stated. This system in its infinite form (i.e. the corresponding crystal of steel spheres in polyester) presents a large absolute gap extending from va0 =cair ¼ 16:73 to va0 =cair ¼ 23:63. In Fig. 2 we show the projection of the frequency band structure of the acoustic field in this crystal on the SBZ of its (1 1 1) surface, which we
Fig. 1. The structure under study. The finite, along z-direction, slab consists of eight (1 1 1) fcc planes of steel spheres of radius S ¼ 0:325a0 (a0 is the first-neighbor distance) in polyester host and is embedded in air. d and d s denote, respectively, the distance between two successive planes of spheres and between the air–polyester interface and the center of the surface plane of spheres.
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Fig. 2. Projection of the phononic band structure of a fcc crystal of steel spheres (radius S ¼ 0:325a0 ) in polyester on the high symmetry lines of the SBZ of its (1 1 1) surface (shown in the inset). Propagating modes: shaded regions, gaps: blank regions. Solid lines: surfacelocalized modes of the corresponding finite slab consisting of eight (1 1 1) layers of spheres, embedded in air. Dotted lines: lines of propagation of longitudinal and transverse waves in infinite polyester (higher and lower branch, respectively).
calculated using the computer program of Ref. [6]. The shaded regions extend over the frequency bands of the acoustic field: at any one frequency within the shaded region, for a given kk [the component of the reduced wavevector within the SBZ of the (1 1 1) surface], there exists at least one propagating mode [the corresponding kz ðvÞ is real] of the acoustic field in the infinite crystal. The blank areas correspond to frequency gaps [all kz ðvÞ are complex]. Therefore, over this frequency region, there are no propagating modes of the acoustic field in the infinite crystal. If now we consider the corresponding finite system embedded in air (see Fig. 1), we find, apart from the propagating modes of the slab which reproduce the regions of elastic states of the infinite crystal (shaded regions in Fig. 2), additional modes originating from the presence of the two ending surfaces of the finite system (air–polyester interfaces), shown by the solid lines in Fig. 2. In the region of the absolute gap they extend from va0 =cair ¼ 22:68 to va0 =cair ¼ 23:63 and of course they continue upwards penetrating within the region of propagating modes, but we are not interested in those regions. We focus our study only on such modes lying within the frequency-gap regions of the corresponding infinite system. For slabs sufficiently thick, which is the case here, these modes are doubly degenerated each of them associated to each one of the two surfaces (apart from the low-frequency region, i.e. for va0 =cair 9 5, where a split is observed because of the strong repulsion of the two corresponding modes). Strictly speaking, the above-mentioned modes are not surface modes, in the sense that they do not decay exponentially on both sides of the air–polyester interfaces: they propagate along zdirection in air regions. However, they are strongly
localized in the surface regions, i.e. between the air– polyester interface and the first sphere planes: they attenuate in the interior of the thick phononic-crystal slab, in the gap-frequency regions of the infinite system. These surface modes manifest themselves as resonances in the transmission spectrum of a longitudinal wave incident on the slab. However, contrary to resonances originating from propagating states of the finite slab, their height decreases by increasing the thickness of the slab. A typical case is given in Fig. 3(a) for a longitudinal wave incident normally (kk ¼ 0) on the slab from the left, in which case we removed the second air–polyester interface (at the right) in order to clearly see the effect of attenuation of the waves at the corresponding frequencies inside the sufficiently thick slab. The presence of the second air–polyester interface will produce for relatively thin slabs a double resonance peak in the transmission spectrum (see Fig. 3(b)) resulting from the interaction of the two surfacelocalized modes; the thinner the slab, the stronger the interaction and the higher the resonance peak value. But this second surface has nothing to add in the physical interpretation, so we will omit it, at this point. We can say that the phononic-crystal slab under consideration behaves, for frequencies within the gap, as a homogeneous and isotropic effective medium characterized by an imaginary wavenumber Im½kz ðvÞ which determines the attenuation of the wave field over this region; for given v and kk ¼ 0 we obtain ln T ¼ 2 Im½kz dN L þ const. This is demonstrated in Fig. 3(c) for the resonance frequency within the gap shown in Fig. 3(a). One expects that appropriate modifications in the slab surfaces can produce smart tuning in frequency of the above-described surface-localized modes, while the otherwise absolute gap remains unaffected, since it is related to the thick-slab’s internal crystal structure. In Fig. 4 we demonstrate such an effect by changing the distance ds between each of the two ending surfaces of the finite slab and the center of the surface planes of spheres. By placing the two surfaces just at the position where the surface layers of spheres terminate (ds ¼ 0:325a0 ¼ S) the surface-localized modes are completely suppressed to the top of the absolute gap, and the inverse happens as we increase progressively ds : they go downwards destroying completely the full region of this gap. This can be understood as follows: the larger the d s , these modes approach more to the Lamb modes of a homogeneous slab of the same thickness (having air at its left and a phononic-crystal slab at its right), folded of course because of the 2D periodicity. The whole picture obtained is an interplay between the homogeneous slab and the influence of the presence of the spheres which
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125
Fig. 3. (a) Transmittance of a wave incident normally on the left of a slab containing only the left air–polyester interface (infinitely extended polyester at the right), for different thicknesses of the slab (N L is the number of (1 1 1) fcc layers). (b) Same as (a) for N L ¼ 5 in the case of one (solid line) and two (dashed line) air–polyester interfaces. (c) The transmittance as a function of the thickness of the slab at the resonance frequency va0 =cair ¼ 23:1896. The straight line has a slope equal to 2 Im½kz d ¼ 3:37, corresponding to an effective attenuation of the waves at the given frequency.
dominates for small values of ds , thus creating gap regions [7]. Similar behavior can be obtained through surface modifications of the surface planes of spheres (e.g. by changing the material and/or the size of the spheres), although the detailed picture will be different from those presented in Fig. 4. Possibilities of tuning such surface-localized modes can be also present at lower frequency regions, where the first gap of a phononic structure appears (below the first pass-band). In this case one needs to use phononiccrystal slabs made of smaller spheres, in order to push up the propagating modes at higher frequencies and reveal in such a way the surface-localized modes in full
detail. An example is given in Fig. 5(a), where we have limited the sketch along the KM direction of the SBZ. The slab is now made of spheres of radius S ¼ 0:10a0 resulting to a dispersion plot for the propagating modes (shaded region) almost identical to the propagation lines in the homogeneous polyester (dotted lines). Moreover, between the eight-layers thick slab and the two air–polyester interfaces we add now two surface layers of steel spheres of larger radius Ss ¼ 0:40a0 (by keeping of course the same 2D periodicity), thus obtaining surface-localized modes quite low in frequency with respect to the propagating modes, as can be seen in the corresponding dispersion diagram (see Fig. 5(a)) with the solid lines, as compared to the case shown in Fig. 2. We note the presence of a narrow gap extending from va0 =cair ¼ 8:708 to
Fig. 4. The evolution of the states in the region of the absolute gap of the system of Fig. 2 as a function of the distance d s of the two ending surfaces of the finite slab from the center of the surface planes of spheres. Propagating/surface-localized modes: light-gray/gray shaded regions. Frequency-gaps: white regions. The arrows indicate the region occupied by the surface-localized modes of the system of Fig. 2.
Fig. 5. Dispersion curves along KM direction for a slab consisting of eight (1 1 1) fcc planes of steel spheres of radius S ¼ 0:10a0 on both sides of which one additional surface layer of different sphere radius Ss ¼ 0:40a0 made of (a) steel or (b) gold is added between the air– polyester interface and the eight-layers thick phononic-crystal slab. Dotted lines: lines of propagation of transverse waves in infinite polyester. The arrows indicate the position of small gaps.
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va0 =cair ¼ 8:863. By replacing the steel spheres of the surface planes by gold-made ones (mass density: rs ¼ 19; 700 kg/m3, longitudinal and transverse velocities: cls ¼ 3240 m/s and cts ¼ 1200 m/s) the surface-localized modes are pushed down even more [Fig. 5(b)], an effect which is due mostly to the mass density difference between steel and gold. Now apart from two small gaps (for va0 =cair from 5.602 to 5.675 and from 7.323 to 7.538), a large region free of modes appears, above the surface-localized modes, extending from va0 =cair ¼ 8:37 to va0 =cair ¼ 12:43.
exist. Our findings could be useful in the construction and design of devices related to frequency filtering, waveguiding, etc.
4. Conclusions
[1] For a comprehensive list of references on phononic crystals, see the Phononic Crystal database at http://www.phys.uoa.gr/phono nics/PhononicDatabase.html. [2] Y. Tanaka, S. Tamura, Phys. Rev. B 58 (1998) 7958. [3] A. Khelif, B. Aoubiza, S. Mohammadi, A. Adibi, V. Laude, Phys. Rev. E 74 (2006) 046610. [4] J.H. Sun, T.T. Wu, Phys. Rev. B 74 (2006) 174305. [5] R. Sainidou, N. Stefanou, Phys. Rev. B 73 (2006) 184301. [6] R. Sainidou, N. Stefanou, I.E. Psarobas, A. Modinos, Comput. Phys. Commun. 166 (2005) 197. [7] R. Sainidou, B. Djafari-Rouhani, J.O. Vasseur, unpublished data.
In this work we presented results concerning the appearance of surface-localized modes inside the frequency gap regions of thick solid phononic-crystal slabs, embedded in air. We demonstrated that these modes can be tunable under an appropriate choice of the geometrical or elastic parameters of the slab, giving rise to regions where no propagating and surface modes
Acknowledgement R. Sainidou gratefully acknowledges the hospitality of the UFR de Physique, Universite´ de Lille 1, France. References