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Transverse acoustic waves in piezoelectric-metallic Fibonacci multilayers
Available online at www.sciencedirect.com
ScienceDirect Materials Today: Proceedings 13 (2019) 541–548
www.materialstoday.com/proceedings
ICMES 2018
Transverse acoustic waves in piezoelectric-metallic Fibonacci multilayers M. Alamia, I. Quotanea, E. H. El Boudoutia*, and B. Djafari-Rouhanib a b
LPMR, Département de Physique, Faculté des Sciences, Université Mohamed Premier, 60000 Oujda, Morocco. IEMN, UMR CNRS 8520, Dpartement de Physique, Université de Lille, 59655 Villeneuve d’Ascq, France
Abstract We study the propagation of transverse acoustic waves in quasi-periodic structures following the Fibonacci sequence made of two blocs A and B where the block A is a metal layer and the bloc B is a piezoelectric layer. The phonon dynamics is described by coupled elastic equations within the static field approximation model. We use the Green's function formalism which enables to get simple analytical expression for the phonon dispersion relation and transmission coefficient. We considered two types of structures: either a given Fibonacci sequence or a periodic repetition of a given sequence called Fibonacci superlattice. The analysis of the transmission spectra of different Fibonacci generations allowed us to conclude that these spectra exhibit selfsimilarity of order three with a scaling factor F for normal and oblique incidence. Also, we present the results of bulk acoustic waves in Fibonacci superlattices composed of periodic cells where each cell consists of a given sequence. We show the property related to bulk bands such as the fragmentation of the bands as function of the generation number following a power law as well as the existence of stable and transient gaps which are due to the quasi-periodicity. © 2019 Elsevier Ltd. All rights reserved. Peer-review under responsibility of the scientific committee of the International Conference on Materials and Environmental Science, ICMES 2018. Keywords: Acoustic waves; Superlattice; Fibonacci sequence; Piezoelectric layer; Quasi-periodic systems; Self-similarity;
1. Introduction During the last three decades, much attention has been devoted to the study of wave propagation in onedimensional (1D) periodically stratified media in many contexts including acoustic, elastic and electromagnetic
* Corresponding author. E-mail address: [email protected] 2214-7853 © 2019 Elsevier Ltd. All rights reserved. Peer-review under responsibility of the scientific committee of the International Conference on Materials and Environmental Science, ICMES 2018.
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M. Alami et al / Materials Today: Proceedings 13 (2019) 541–548
systems [1, 2]. The essential property of these structures is the existence of forbidden frequency bands induced by the difference in acoustic and dielectric properties of the constituents and the periodicity of these systems. With regard to acoustic waves, number of theoretical and experimental works has been devoted to the study of the band gap structures of periodic solid-solid superlattices (SLs) [3-5] composed of crystalline, amorphous semi-conductors or metallic multilayers at the manometric scale. The theoretical models used are essentially the transfer matrix [3,6] and the Green's function methods [4,5,7,8], whereas the experimental techniques include phonon spectroscopy [911], Raman scattering [12,13], ultrasonics [14,15] and time-resolved x-ray diffraction[16]. Besides the periodic multilayer systems, quasiperiodic systems have been also intensively studied. Many theoretical studies based on simple 1D models have been performed, and interesting properties have been deduced [17,18]. The high level of control and perfection reached in the growth techniques of microstructures and nanostructures has allowed the production of some quasiperiodic systems [19–24] by means of molecular beam epitaxy techniques. It should be stressed that the theory of all the mathematical and formal properties of quasiperiodic systems holds for infinitely large sequences, and this never happens in experiments or calculations. In practice we always deal with a ‘’high’’ but finite realization. A finite realization is the n-generation which results from applying the substitution or formation rule for the given sequence n times and this is what one grows experimentally and what one calculates. If n is sufficiently large, one can think that the description of the properties of the quasiperiodic systems will be reasonably accurate. The study of these systems requires more realistic models than those simple 1D models frequently used. Elastic waves provide a very good ground for the study of the properties of the spectra of "real" quasiperiodic systems. This is because in the range of validity of the elasticity theory we have a rigorous description of the systems considered. The study of elastic waves in quasiperiodic systems presents an additional feature as compared to the simple 1D model. It is known that because of the symmetry there are cases in which the transverse elastic waves are decoupled from the sagittal elastic waves (i.e., those having components of the elastic displacement in the sagittal plane, defined by the propagation direction and the normal to the interfaces) [25]. Thus transverse elastic waves play a similar role to that of the excitations studied by means of the simple 1D model [17, 18]. Some studies of the elastic waves in quasiperiodic systems with different sequences have been performed [26–35]. The frequency spectrum of the transverse elastic waves exhibits in a clear way the succession of principal and secondary gaps (spectrum fragmentation) characteristic of the results of the simple 1D model. Amongst all the possible self-similar structures, our work is devoted to the study of the Fibonacci sequence. This iteration process combined with layered structures has the advantage of being simple and easy to implement numerically to create one-dimensional pre-fractal multilayers [36,37]. Wave propagation in these structures has been widely studied for electronic [38] and classical (elastic [37,39] or optical [40,41]) waves. The self-similarity of Fibonacci multilayered structure makes the wave transmission present self-similar features. Furthermore, they can be used to enhance the acoustic losses of structure [42,43] as well to create optical filters [44]. To our knowledge, few works have been devoted to the quasiperiodic piezoelectric superlattices [33,34,45-47]. Recently, Martinez-Gutiérrez et al. [48] studied the transverse acoustic waves in ZnO/MgO and GaN/AlN piezoelectric Fibonacci superlattices; they have considered also the case of hybrid superlattices including Fibonacci and periodic parts. In this work, we are interested in the Fibonacci-type structures [49] characterized by a mathematical substitution rule formed from two blocks A and B where the block A is a metal layer and the bloc B is a piezoelectric layer, following the substitution rule Sj+1 = SjSj-1where j is the number of the generation. For example: S1 = A, S2 = AB, S3 = ABA, S4 = ABAAB, S5 = ABAABABA... The number of blocks in each generation is Fk = Fk-1+ Fk-2 with F1 = 1 and F2 = 2. In particular, we studied the different properties of propagation and localization of acoustic waves of transverse polarization coupled to the electric potential by considering either a Fibonacci sequence alone or a given sequence repeated periodically called Fibonacci superlattice. Specific applications are given for PZT4-Fe layered materials. Lead zirconium titanate (PZT) ceramics are comprised from the inorganic oxide compounds of lead (Pb), zirconium (Zr) and titanium (Ti), they are high-performance piezoelectric materials, which are widely used in sensors, actuators, ultrasonic transducers and other electronic devices. Moreover, for select Microelectromechanical system (MEMS) resonators and filters applications [50,51], thin film PZT offers many unique advantages due to the high electromechanical coupling factors, permittivity, piezoelectric stress constants, and the DC-bias electric field dependence of these properties. Considerable progress has also been made in developing materials deposition and processing technologies that address the fabrication challenges with PZT thin films.
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Among the series of formulation-labeled PZT, the piezoelectric media PZT4 are assumed to be of hexagonal symmetry belonging to the 6mm crystallographic class with their c axis along the x3 axis and the wave vector k∥ (parallel to the layers) is along x2. In this particular geometry, the shear horizontal vibrations parallel to x3 are accompanied by an electric potential, while the sagittal vibrations (polarized in the (x1, x2) plane) are decoupled from the latter. The details of the calculation as well as the closed form expressions of the dispersion relations of bulk waves and the transmission coefficient obtained by means of the Green’s function method will be published elsewhere together with the surface modes results not treated here. 2. Results and Discussion 2.1. Quasi-periodic Fibonacci structure Figure 1 (a) shows the transmission spectrum as a function of the dimensionless frequency =dm/vm of a periodic structure composed of N=5 periods around the central frequencyc=/2. The layer thicknesses are chosen such that dm/vm=dp/vp where dm(vm) and dp(vp) represent the thickness (transverse velocity of sound) of metal and piezoelectric layers respectively. At c the layers behave like quarter-wavelength stacks. Figures 1(b) and 1(c) give the transmission spectra for the 6th and 9th Fibonacci generations. When we compared with the spectra of Fibonacci structures (Figs. 1(b)-(c)) we note that there are several areas where the transmission is growing when the generation number increases [49], this property is a characteristic of quasi-periodic structures in which the allowed bands undergo fragmentation for higher generations. An interesting result can be observed around the central frequency of the gap of the periodic structure. Indeed, the transmission spectra around c reproduce all three generations. This property of self-similarity called scaling law [49] has been interpreted as a sign of localization of waves in Fibonacci systems. Kohmoto et al. [41] have shown that the behavior of the transmission spectrum is characterized by a scale factor [41, 52] : 2 F I 4 1 I
1/ 2
2 1 I ,
(1)
where I is an invariant which remains constant at each step of the recursive procedure, its expression has the following form [49] :
I ¼ ( Z m /Z p Z p /Z m ) 2 sin 2 (d m / v m ) ,
(2)
where Zm=mvm and Zp=pvp are the impedances of the metal and piezoelectric layers respectively. Also, it has been shown that this phenomenon occurs around nwhere quasi-periodicity is most effective (n being an integer). This means that the transmission coefficient should have a self-similarity nature around the central frequency cmwith Tj+3=Tj (the transmission coefficient exhibits a third order periodicity) with a scale factor F. For the central frequency c, equations (1) and (2) give I= 0.56 and F = 6.39. This result is shown in Fig. 1(b) to be compared with Fig. 1(c). We note great resemblance between the generations S6 and S9 around the central frequency c with a periodicity of three and a scaling factor F. The same results are obtained for generations S4-S7 and S5-S8 (not presented here). This result is similar to that found by Kohmoto et al. [41] in the optical multilayers and El Boudouti et al. [53] in coaxial cables. However, in our case, because of the piezoelectricity effect, the phenomenon of self-similarity occurs around the central frequency c which is slightly different fromnas it is the case for pure transverse waves (see Figs. 1(b) and 1(c)).
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1 (a)
Transmission
0 1 S6 (b)
0 1 1
S9 (c)
c
0
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
Fig.1. Variation of the transmission coefficient as a function of the dimensionless frequency for different structures at normal incidence. (a) The periodic structure with N =5 periods. (b), (c) The same as (a) but for S6 and S9 Fibonacci generations.
In the case of oblique incidence, Figs. 2 (a) and (d) show the transmission spectrum as a function of the dimensionless frequency of a periodic structure composed of N=5 periods for two oblique incidence angles: θ = 20° (Fig. 2(a)) and θ = 50° (Fig. 2(d)). The self-similarity spectra should be expected around the center of the first gag. This is illustrated in Figs. 2(b) and 2(c) for θ=20° and Figs. 2(e) and 2(f) for θ=50° where the transmission spectra show a fragmentation around c and a self-similarity effect between S6 and S9 generations as in normal incidence (Fig. 1). However, we note that the self-similarity phenomenon shifts towards higher frequencies above the central frequency c when the incidence angle increases, indicating that the transmission spectra are sensitive to the incidence angle. Figures 2(b), 2(c), 2(e) and 2(f) clearly show a self-similarity of the S6 and S9 spectra aroundcslightlydifferent from . To give a better insight about the effect of the incidence angle on the self-similarity phenomenon, we have shown in Fig. 3 the variation of the positions of the central frequency c versus the incidence angle θ. This result clearly shows that the position of the central frequency c where the self-similarity occurs, increases as a function of the incidence angle θ. It is worth mentioning that in order to obtain the behavior of self-similarity around the same frequency as in the case of normal incidence, we have to choose the layer thicknesses such that (dm/vm) cosθ1=(dp/vp)cosθ2 where θ1 and θ2 represent the incidence and refraction angles between metal and piezoelectric materials respectively. Then we have to draw the transmission spectra versus the dimensionless frequency = (dm/vm)cos θ1. 2.2. Band structure of a Fibonacci superlattice In this section, we consider the dispersion curves, distribution of bands widths of a Fibonacci superlattice according to the generation number (power law) and for different values of the incidence angle θ. Figure 4 shows the distribution of the widths of allowed and forbidden bands of a Fibonacci superlattice for different generation numbers k until the 9th generation at normal and oblique incidence.
M. Alami et al / Materials Today: Proceedings 13 (2019) 541–548
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1 (a)
0
1
S6 (b)
0
1
Transmission
S9 (c)
c
0
1
(d)
0
1 S6 (e)
0
1
S9 (f)
c 0.63
0 0
Fig.2. Variation of the transmission coefficient as a function of the dimensionless frequency for periodic (a), (d) and two Fibonacci generations S6(b), (e) and S9 (c), (f) for two incidence angles: θ=20° ((a), (b),(c)) and θ=50°((d), (e),(f)). The periodic structure is composed of N =5 periods.
In the case of normal incidence (Fig. 4 (a)), we can notice a fragmentation of the allowed bands as far as the generation number k increases. In addition, there are stable gaps (denoted Gs) that appear for all generations (see for example gaps around the frequency c=0.35, 0.60, 0.76and , for θ=0) and transition gaps (denoted Gt) that appear every three generations (for example the gaps around the central frequency for S2, S5, S8 ...). Unlike the case of pure transverse waves, we can notice the opening of forbidden bands around due to the effect of piezoelectricity. The same results are obtained in the case of oblique incidence (see Figs. 4 (b) and 4(c)).
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0.66 0.64 0.62 0.60
C
0.58 0.56 0.54 0.52 0.50 0.48 0.46 0
10
20
30
40
50
Fig.3. Variation of the positions of the central frequency c as a function of the angle of incidence θ
The total width Δk of the allowable energy bands (Lebesgue measure of energy spectrum) decreases when the number of generation increases with a power law k∼(Fk)• as shown in Fig. 5, where Fk is the Fibonacci number (number of blocks in each sequence), and δ is the diffusion constant of the spectrum. We note that this power law is valid even for the values of θ different from zero and the parameter δ changes slightly with θ. When k becomes larger, Δk tends to zero whatever the value of θ. This result which is a characteristic of quasiperiodic structures has been demonstrated for other types of elementary excitations [53]. Normal incidence (a)
1.2
Oblique incidence (c)
Oblique incidence (b)
1.2
1.2
1
1
Gs
0.8
Gs 0.6
Gs
0.6
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Gt
Gt
Gt
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Gs
0.6
Gt
Gt
Gt
Gt
Gt
Gs 0.4
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Gs
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Gs
0.8
Gs
Gs Gs
0.2
Gs
0.2
0.2
Gs
0
0
0 2
3
4
5
6
7
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8
9
10
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2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Generation number K
Fig.4. Distribution of band widths (solid lines) as function of the generation number at θ= 0° (a), θ= 20° (b) and θ= 50° (c). Gs and Gt denote transient and stable gaps respectively.
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0.6
log (
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0.1 0.2
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1.2
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1.6
1.8
2.0
log (F Fig. 5. The log-log curves representing the width of the allowed bands Δk as a function of the generation number Fk for different values of the incidence angle θ.
3. Conclusion In this work, we studied acoustic waves in Fibonacci quasiperiodic structures composed of two blocks A (metal layer) and B (piezoelectric layer). The analysis of the transmission spectra for different Fibonacci sequences allowed us to conclude that these spectra exhibit self-similarity of order three with a scaling factor F for normal and oblique incidence. Also, we have shown a fragmentation of the allowed bands of a Fibonacci superlattice made of a periodic repetition of a given sequence following a power law. This fragmentation gives rise to stable and transient gaps separating the allowed bands. References [1] [2] [3] [4] [5] [6] [7]
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ScienceDirect Materials Today: Proceedings 13 (2019) 541–548
www.materialstoday.com/proceedings
ICMES 2018
Transverse acoustic waves in piezoelectric-metallic Fibonacci multilayers M. Alamia, I. Quotanea, E. H. El Boudoutia*, and B. Djafari-Rouhanib a b
LPMR, Département de Physique, Faculté des Sciences, Université Mohamed Premier, 60000 Oujda, Morocco. IEMN, UMR CNRS 8520, Dpartement de Physique, Université de Lille, 59655 Villeneuve d’Ascq, France
Abstract We study the propagation of transverse acoustic waves in quasi-periodic structures following the Fibonacci sequence made of two blocs A and B where the block A is a metal layer and the bloc B is a piezoelectric layer. The phonon dynamics is described by coupled elastic equations within the static field approximation model. We use the Green's function formalism which enables to get simple analytical expression for the phonon dispersion relation and transmission coefficient. We considered two types of structures: either a given Fibonacci sequence or a periodic repetition of a given sequence called Fibonacci superlattice. The analysis of the transmission spectra of different Fibonacci generations allowed us to conclude that these spectra exhibit selfsimilarity of order three with a scaling factor F for normal and oblique incidence. Also, we present the results of bulk acoustic waves in Fibonacci superlattices composed of periodic cells where each cell consists of a given sequence. We show the property related to bulk bands such as the fragmentation of the bands as function of the generation number following a power law as well as the existence of stable and transient gaps which are due to the quasi-periodicity. © 2019 Elsevier Ltd. All rights reserved. Peer-review under responsibility of the scientific committee of the International Conference on Materials and Environmental Science, ICMES 2018. Keywords: Acoustic waves; Superlattice; Fibonacci sequence; Piezoelectric layer; Quasi-periodic systems; Self-similarity;
1. Introduction During the last three decades, much attention has been devoted to the study of wave propagation in onedimensional (1D) periodically stratified media in many contexts including acoustic, elastic and electromagnetic
* Corresponding author. E-mail address: [email protected] 2214-7853 © 2019 Elsevier Ltd. All rights reserved. Peer-review under responsibility of the scientific committee of the International Conference on Materials and Environmental Science, ICMES 2018.
542
M. Alami et al / Materials Today: Proceedings 13 (2019) 541–548
systems [1, 2]. The essential property of these structures is the existence of forbidden frequency bands induced by the difference in acoustic and dielectric properties of the constituents and the periodicity of these systems. With regard to acoustic waves, number of theoretical and experimental works has been devoted to the study of the band gap structures of periodic solid-solid superlattices (SLs) [3-5] composed of crystalline, amorphous semi-conductors or metallic multilayers at the manometric scale. The theoretical models used are essentially the transfer matrix [3,6] and the Green's function methods [4,5,7,8], whereas the experimental techniques include phonon spectroscopy [911], Raman scattering [12,13], ultrasonics [14,15] and time-resolved x-ray diffraction[16]. Besides the periodic multilayer systems, quasiperiodic systems have been also intensively studied. Many theoretical studies based on simple 1D models have been performed, and interesting properties have been deduced [17,18]. The high level of control and perfection reached in the growth techniques of microstructures and nanostructures has allowed the production of some quasiperiodic systems [19–24] by means of molecular beam epitaxy techniques. It should be stressed that the theory of all the mathematical and formal properties of quasiperiodic systems holds for infinitely large sequences, and this never happens in experiments or calculations. In practice we always deal with a ‘’high’’ but finite realization. A finite realization is the n-generation which results from applying the substitution or formation rule for the given sequence n times and this is what one grows experimentally and what one calculates. If n is sufficiently large, one can think that the description of the properties of the quasiperiodic systems will be reasonably accurate. The study of these systems requires more realistic models than those simple 1D models frequently used. Elastic waves provide a very good ground for the study of the properties of the spectra of "real" quasiperiodic systems. This is because in the range of validity of the elasticity theory we have a rigorous description of the systems considered. The study of elastic waves in quasiperiodic systems presents an additional feature as compared to the simple 1D model. It is known that because of the symmetry there are cases in which the transverse elastic waves are decoupled from the sagittal elastic waves (i.e., those having components of the elastic displacement in the sagittal plane, defined by the propagation direction and the normal to the interfaces) [25]. Thus transverse elastic waves play a similar role to that of the excitations studied by means of the simple 1D model [17, 18]. Some studies of the elastic waves in quasiperiodic systems with different sequences have been performed [26–35]. The frequency spectrum of the transverse elastic waves exhibits in a clear way the succession of principal and secondary gaps (spectrum fragmentation) characteristic of the results of the simple 1D model. Amongst all the possible self-similar structures, our work is devoted to the study of the Fibonacci sequence. This iteration process combined with layered structures has the advantage of being simple and easy to implement numerically to create one-dimensional pre-fractal multilayers [36,37]. Wave propagation in these structures has been widely studied for electronic [38] and classical (elastic [37,39] or optical [40,41]) waves. The self-similarity of Fibonacci multilayered structure makes the wave transmission present self-similar features. Furthermore, they can be used to enhance the acoustic losses of structure [42,43] as well to create optical filters [44]. To our knowledge, few works have been devoted to the quasiperiodic piezoelectric superlattices [33,34,45-47]. Recently, Martinez-Gutiérrez et al. [48] studied the transverse acoustic waves in ZnO/MgO and GaN/AlN piezoelectric Fibonacci superlattices; they have considered also the case of hybrid superlattices including Fibonacci and periodic parts. In this work, we are interested in the Fibonacci-type structures [49] characterized by a mathematical substitution rule formed from two blocks A and B where the block A is a metal layer and the bloc B is a piezoelectric layer, following the substitution rule Sj+1 = SjSj-1where j is the number of the generation. For example: S1 = A, S2 = AB, S3 = ABA, S4 = ABAAB, S5 = ABAABABA... The number of blocks in each generation is Fk = Fk-1+ Fk-2 with F1 = 1 and F2 = 2. In particular, we studied the different properties of propagation and localization of acoustic waves of transverse polarization coupled to the electric potential by considering either a Fibonacci sequence alone or a given sequence repeated periodically called Fibonacci superlattice. Specific applications are given for PZT4-Fe layered materials. Lead zirconium titanate (PZT) ceramics are comprised from the inorganic oxide compounds of lead (Pb), zirconium (Zr) and titanium (Ti), they are high-performance piezoelectric materials, which are widely used in sensors, actuators, ultrasonic transducers and other electronic devices. Moreover, for select Microelectromechanical system (MEMS) resonators and filters applications [50,51], thin film PZT offers many unique advantages due to the high electromechanical coupling factors, permittivity, piezoelectric stress constants, and the DC-bias electric field dependence of these properties. Considerable progress has also been made in developing materials deposition and processing technologies that address the fabrication challenges with PZT thin films.
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Among the series of formulation-labeled PZT, the piezoelectric media PZT4 are assumed to be of hexagonal symmetry belonging to the 6mm crystallographic class with their c axis along the x3 axis and the wave vector k∥ (parallel to the layers) is along x2. In this particular geometry, the shear horizontal vibrations parallel to x3 are accompanied by an electric potential, while the sagittal vibrations (polarized in the (x1, x2) plane) are decoupled from the latter. The details of the calculation as well as the closed form expressions of the dispersion relations of bulk waves and the transmission coefficient obtained by means of the Green’s function method will be published elsewhere together with the surface modes results not treated here. 2. Results and Discussion 2.1. Quasi-periodic Fibonacci structure Figure 1 (a) shows the transmission spectrum as a function of the dimensionless frequency =dm/vm of a periodic structure composed of N=5 periods around the central frequencyc=/2. The layer thicknesses are chosen such that dm/vm=dp/vp where dm(vm) and dp(vp) represent the thickness (transverse velocity of sound) of metal and piezoelectric layers respectively. At c the layers behave like quarter-wavelength stacks. Figures 1(b) and 1(c) give the transmission spectra for the 6th and 9th Fibonacci generations. When we compared with the spectra of Fibonacci structures (Figs. 1(b)-(c)) we note that there are several areas where the transmission is growing when the generation number increases [49], this property is a characteristic of quasi-periodic structures in which the allowed bands undergo fragmentation for higher generations. An interesting result can be observed around the central frequency of the gap of the periodic structure. Indeed, the transmission spectra around c reproduce all three generations. This property of self-similarity called scaling law [49] has been interpreted as a sign of localization of waves in Fibonacci systems. Kohmoto et al. [41] have shown that the behavior of the transmission spectrum is characterized by a scale factor [41, 52] : 2 F I 4 1 I
1/ 2
2 1 I ,
(1)
where I is an invariant which remains constant at each step of the recursive procedure, its expression has the following form [49] :
I ¼ ( Z m /Z p Z p /Z m ) 2 sin 2 (d m / v m ) ,
(2)
where Zm=mvm and Zp=pvp are the impedances of the metal and piezoelectric layers respectively. Also, it has been shown that this phenomenon occurs around nwhere quasi-periodicity is most effective (n being an integer). This means that the transmission coefficient should have a self-similarity nature around the central frequency cmwith Tj+3=Tj (the transmission coefficient exhibits a third order periodicity) with a scale factor F. For the central frequency c, equations (1) and (2) give I= 0.56 and F = 6.39. This result is shown in Fig. 1(b) to be compared with Fig. 1(c). We note great resemblance between the generations S6 and S9 around the central frequency c with a periodicity of three and a scaling factor F. The same results are obtained for generations S4-S7 and S5-S8 (not presented here). This result is similar to that found by Kohmoto et al. [41] in the optical multilayers and El Boudouti et al. [53] in coaxial cables. However, in our case, because of the piezoelectricity effect, the phenomenon of self-similarity occurs around the central frequency c which is slightly different fromnas it is the case for pure transverse waves (see Figs. 1(b) and 1(c)).
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1 (a)
Transmission
0 1 S6 (b)
0 1 1
S9 (c)
c
0
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
Fig.1. Variation of the transmission coefficient as a function of the dimensionless frequency for different structures at normal incidence. (a) The periodic structure with N =5 periods. (b), (c) The same as (a) but for S6 and S9 Fibonacci generations.
In the case of oblique incidence, Figs. 2 (a) and (d) show the transmission spectrum as a function of the dimensionless frequency of a periodic structure composed of N=5 periods for two oblique incidence angles: θ = 20° (Fig. 2(a)) and θ = 50° (Fig. 2(d)). The self-similarity spectra should be expected around the center of the first gag. This is illustrated in Figs. 2(b) and 2(c) for θ=20° and Figs. 2(e) and 2(f) for θ=50° where the transmission spectra show a fragmentation around c and a self-similarity effect between S6 and S9 generations as in normal incidence (Fig. 1). However, we note that the self-similarity phenomenon shifts towards higher frequencies above the central frequency c when the incidence angle increases, indicating that the transmission spectra are sensitive to the incidence angle. Figures 2(b), 2(c), 2(e) and 2(f) clearly show a self-similarity of the S6 and S9 spectra aroundcslightlydifferent from . To give a better insight about the effect of the incidence angle on the self-similarity phenomenon, we have shown in Fig. 3 the variation of the positions of the central frequency c versus the incidence angle θ. This result clearly shows that the position of the central frequency c where the self-similarity occurs, increases as a function of the incidence angle θ. It is worth mentioning that in order to obtain the behavior of self-similarity around the same frequency as in the case of normal incidence, we have to choose the layer thicknesses such that (dm/vm) cosθ1=(dp/vp)cosθ2 where θ1 and θ2 represent the incidence and refraction angles between metal and piezoelectric materials respectively. Then we have to draw the transmission spectra versus the dimensionless frequency = (dm/vm)cos θ1. 2.2. Band structure of a Fibonacci superlattice In this section, we consider the dispersion curves, distribution of bands widths of a Fibonacci superlattice according to the generation number (power law) and for different values of the incidence angle θ. Figure 4 shows the distribution of the widths of allowed and forbidden bands of a Fibonacci superlattice for different generation numbers k until the 9th generation at normal and oblique incidence.
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1 (a)
0
1
S6 (b)
0
1
Transmission
S9 (c)
c
0
1
(d)
0
1 S6 (e)
0
1
S9 (f)
c 0.63
0 0
Fig.2. Variation of the transmission coefficient as a function of the dimensionless frequency for periodic (a), (d) and two Fibonacci generations S6(b), (e) and S9 (c), (f) for two incidence angles: θ=20° ((a), (b),(c)) and θ=50°((d), (e),(f)). The periodic structure is composed of N =5 periods.
In the case of normal incidence (Fig. 4 (a)), we can notice a fragmentation of the allowed bands as far as the generation number k increases. In addition, there are stable gaps (denoted Gs) that appear for all generations (see for example gaps around the frequency c=0.35, 0.60, 0.76and , for θ=0) and transition gaps (denoted Gt) that appear every three generations (for example the gaps around the central frequency for S2, S5, S8 ...). Unlike the case of pure transverse waves, we can notice the opening of forbidden bands around due to the effect of piezoelectricity. The same results are obtained in the case of oblique incidence (see Figs. 4 (b) and 4(c)).
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0.66 0.64 0.62 0.60
C
0.58 0.56 0.54 0.52 0.50 0.48 0.46 0
10
20
30
40
50
Fig.3. Variation of the positions of the central frequency c as a function of the angle of incidence θ
The total width Δk of the allowable energy bands (Lebesgue measure of energy spectrum) decreases when the number of generation increases with a power law k∼(Fk)• as shown in Fig. 5, where Fk is the Fibonacci number (number of blocks in each sequence), and δ is the diffusion constant of the spectrum. We note that this power law is valid even for the values of θ different from zero and the parameter δ changes slightly with θ. When k becomes larger, Δk tends to zero whatever the value of θ. This result which is a characteristic of quasiperiodic structures has been demonstrated for other types of elementary excitations [53]. Normal incidence (a)
1.2
Oblique incidence (c)
Oblique incidence (b)
1.2
1.2
1
1
Gs
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2
3
4
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10
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2
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5
6
7
8
9
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Generation number K
Fig.4. Distribution of band widths (solid lines) as function of the generation number at θ= 0° (a), θ= 20° (b) and θ= 50° (c). Gs and Gt denote transient and stable gaps respectively.
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0.6
log (
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0.1 0.2
0.4
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1.2
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log (F Fig. 5. The log-log curves representing the width of the allowed bands Δk as a function of the generation number Fk for different values of the incidence angle θ.
3. Conclusion In this work, we studied acoustic waves in Fibonacci quasiperiodic structures composed of two blocks A (metal layer) and B (piezoelectric layer). The analysis of the transmission spectra for different Fibonacci sequences allowed us to conclude that these spectra exhibit self-similarity of order three with a scaling factor F for normal and oblique incidence. Also, we have shown a fragmentation of the allowed bands of a Fibonacci superlattice made of a periodic repetition of a given sequence following a power law. This fragmentation gives rise to stable and transient gaps separating the allowed bands. References [1] [2] [3] [4] [5] [6] [7]
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