One-Dimensional Phononic Crystals

One-Dimensional Phononic Crystals

Chapter 4 One-Dimensional Phononic Crystals El Houssaine El Boudouti*, Abdellatif Akjouj† , Leonard Dobrzynski† , Bahram Djafari-Rouhani† , Yan Penne...

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Chapter 4

One-Dimensional Phononic Crystals El Houssaine El Boudouti*, Abdellatif Akjouj† , Leonard Dobrzynski† , Bahram Djafari-Rouhani† , Yan Pennec† , Housni Al-Wahsh‡ and Gaëtan Lévêque† *Laboratory of Physics of Matter and Radiation, Faculty of Science, Mohammed I University, 60000 Oujda, Morocco, † The National Center for Scientific Research-Lille University, Lille; UMR CNRS 8520, Faculty of Science, Villeneuve d’Ascq, France, ‡ Faculty of Engineering, Benha University, 11241 Cairo, Egypt

Contents 1 Introduction 140 2 Shear-Horizontal Acoustic Waves in Semiinfinite PnCs 142 2.1 Transverse Elastic Waves in 1D Phononic Material 142 2.2 Shear-Horizontal Acoustic Waves in Piezoelectric PnCs159 3 Shear-Horizontal Acoustic Waves in Finite PnCs 173 3.1 Density of States and Reflection and Transmission Coefficients 174 3.2 Effect of a Cap Layer 180 3.3 Effect of a Cavity Layer 186 3.4 Relation to Experiments 187 4 Sagittal Acoustic Waves in Finite Solid-Fluid PnCs 209 4.1 Green’s Functions, Dispersion Relations, Transmission and Reflection Coefficients 211

4.2 Application to a Finite Symmetric PnC Embedded in a Fluid 4.3 General Rule About Confined and Surface Modes in a Finite Asymmetric PnC 5 Omnidirectional Reflection and Selective Transmission in Layered Media 5.1 Model and Method of Calculation 5.2 Case of Solid-Solid PnCs 5.3 Case of Solid-Fluid PnCs 5.4 Relation to Experiments 6 Conclusions References

Phononics. http://dx.doi.org/10.1016/B978-0-12-809948-3.00004-1 Copyright © 2018 Elsevier Inc. All rights reserved.

219

226

230 232 233 246 259 262 264

139

140 Phononics

1 INTRODUCTION One-dimensional (1D) phononic crystals (PnCs) are of great importance in material science. These structures are synthetic periodic materials for controlling and manipulating the propagation of elastic and acoustic waves. They are composed of two or several layers repeated periodically along the direction of growth. The layers constituting each cell of the PnC can be made of a combination of solid-solid or solid-fluid layered media. After the proposal of 1D PnC by Esaki [1], 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. [2] of a doublet associated with folded longitudinal acoustic phonons by means of Raman scattering and the selective transmission of elastic waves by Narayanamurti et al. [3]. 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 1D PnCs in comparison with bulk materials [4, 5]. This was realized already prior to the theoretical prediction of phononic band gaps in 2D and 3D structures [6, 7]. With regard to acoustic waves in solid-solid PnCs, a number of theoretical and experimental works have been devoted to the study of the band gap structures of periodic PnCs [4, 8–18] composed of crystalline, amorphous semiconductor, metallic, and hybrid (organic/inorganic) multilayers at the nanometric scale. The theoretical models used are essentially the transfer matrix [4, 19–23] and the Green’s function methods [5, 14–16, 24–26], whereas the experimental techniques include Raman scattering [2, 12, 14–16, 18, 27, 28], ultrasonics [3, 29–38], acoustic transmission spectrum [39–46], and timeresolved X-ray diffraction [47]. Besides the existence of the band-gap structures in perfect periodic PnCs, it was shown theoretically and experimentally, that the ideal PnC should be modified to take into account the media surrounding the structure as: a free surface [15, 24, 25, 48–66], a PnC/substrate interface [24, 25, 51, 58, 59], a cavity layer [15, 39, 62, 67–76], etc., which are often used in experiments together with PnCs. In addition to the defect modes that can be introduced by such inhomogeneities inside the band gaps, some other works have shown the existence of small peaks in folded longitudinal acoustic phonons and interpreted as confined phonons of the whole finite PnC [15, 77–79]. All the earlier phenomena have been exploited to propose 1D solid-solid layered media for several interesting applications as in their 2D and 3D counterparts PnCs [6, 7]. Among these applications, one can mention: (i) omnidirectional band gaps [80–83], (ii) the possibility to engineer small-size sonic crystals with locally resonant band gaps in the audible frequency range [84], (iii) hypersonic crystals with high-frequency band gaps to enhance acousto-optical interaction [72–74] and to realize stimulated emission of acoustic phonons [85], (iv) the possibility to enhance selective transmission through guided modes of a cavity layer inserted in the periodic structure [5, 39] or by interface resonance modes induced by the PnC/substrate interface [86–88], and (v) unidirectional asymmetric acoustic transmission (acoustic diode) [89, 90] and decrease of heat

Chapter | 4 One-Dimensional Phononic Crystals 141

conductivity [91]. The advantage of 1D systems lies in the fact that their design is more feasible and they require only relatively simple analytical and numerical calculations. The analytical calculations enable us to understand deeply different physical properties related to the band gaps in such systems. In comparison with solid-solid layered media, the propagation of acoustic waves in the solid-fluid counterparts’ structures has received less attention [92]. The first works on these systems have been carried out by Rytov [93] and summarized by Brekhovskikh [92]. Rytov’s approach has been used by Schöenberg [94] together with propagator matrix formalism to account for propagation through such a periodic medium in any direction of propagation and at arbitrary frequency. Similar results are also obtained by Rousseau [95]. In the low-frequency limit, it was shown [94] that besides the existence of small gaps, there is one wave speed for propagation perpendicular to the layering and two wave speeds for propagation parallel to the layering, which are without analog in solid-solid PnCs. The two latter speeds both correspond to compressional waves and their existence is suggestive of Biot’s theory [96] of wave propagation in porous media. Alternating solid and viscous fluid layers have been proposed [97–99] as an idealized porous medium to evaluate dispersion and attenuation of acoustic waves in porous solids saturated with fluids. The experimental evidence [100] of these waves is carried out using ultrasonic techniques in Al-water and Plexiglas-water PnCs. Also, it was shown theoretically and experimentally that finite size layered structures composed of a few cells of solid-fluid layers with one [101, 102] or multiple [103] periodicity may exhibit large gaps and the presence of defect layers in these structures may give rise to well-defined defect modes in these gaps [102]. Also, solid layers separated by graded fluid layers have shown the possibility of acoustic Bloch oscillations [104] analogous to the Wannier-Stark ladders of electronic states in a biased superlattice [105]. The purpose of this chapter is threefold: (i) We shall give a detailed study on surface and interface acoustic waves of shear-horizontal polarization in semiinfinite and finite solid-solid PnCs. In the case of semiinfinite PnCs, we have demonstrated analytically a general rule about the existence of surface and interface modes associated with a PnC free-stress surface and a PnC/substrate interface, respectively. This rule predicts the existence of one mode per gap when we consider together two semiinfinite PnCs obtained from the cleavage of an infinite PnC along a plane lying inside one layer. This rule has been shown to be valid either for an elastic PnC made of N > 2 layers or for a piezoelectric PnC where shear-horizontal waves are coupled to electric potential [5]. The effect of the stiffness of a homogeneous substrate in contact with a PnC on interface modes as well as guided and pseudo-guided modes induced by a cap layer on top of the PnC is analyzed. In the case of finite PnCs, we give the detailed expressions of dispersion relations, densities of states (DOSs) as well as reflection and transmission coefficients for a PnC deposited on a substrate or embedded between two substrates. In particular, we show an

142 Phononics

exact relation between the DOS and phase times. Several applications are discussed for the effects of a cap layer deposited on top of the PnC and a cavity layer inserted at different places within the PnC [5]. (ii) Surface and interface waves of sagittal polarization in finite size solid-fluid PnCs are presented in the second fold. The wave propagation may exhibit new phenomena as compared to solid-solid PnCs such as the existence of transmission zeros which influences the origin of the band gaps, the conditions for band gap closings and the possibility of existence of an internal resonance induced by a fluid layer and lying in the vicinity of a transmission zero, the so-called Fano resonance [5]. (iii) 1D PnC made of alternating solid-solid and solid-fluid layers may exhibit total reflection of acoustic incident waves in a given frequency range for all incident angles. In general, this property cannot be fulfilled with a simple finite PnC if the incident wave is launched from an arbitrary transmitting medium. Therefore, we propose two solutions to obtain such an omnidirectional band gap, namely: by cladding of the PnC with a layer of high acoustic velocities that acts like a barrier for the propagation of phonons, or by associating in tandem two different PnCs in such a way that the superposition of their band structures exhibits an absolute acoustic band gap. We discuss the appropriate choices of the material and geometrical parameters to realize such structures. The behavior of the transmission coefficients is discussed in relation with the dispersion curves of the finite size structure. Also, these structures may be used as acoustic filters that may transmit selectively certain frequencies within the omnidirectional gaps. The transmission filtering can be achieved either through the guided modes of a defect layer inserted in the periodic structure or through the interface modes between the PnC and a homogeneous fluid medium when these two media are chosen appropriately [5].

2 SHEAR-HORIZONTAL ACOUSTIC WAVES IN SEMIINFINITE PnCs Since the pioneering work of Camley et al. [63] and Djafari-Rouhani et al. [48], surface acoustic waves in semiinfinite elastic PnCs have been the subject of intense theoretical and experimental studies [15, 24, 25, 31, 48–66].

2.1 Transverse Elastic Waves in 1D Phononic Material In the present paragraph, we study resonant and localized modes together with the variation of the DOS associated with surfaces and interfaces in 1D PnC made of two different layers. Closed form expressions are obtained for transverse elastic waves polarized perpendicular to the sagittal plane, that is, the plane containing the propagation vector k (parallel to the interfaces) and the normal to the interfaces. However, these results also remain valid in the case of longitudinal waves propagating along the axis of the PnC, which means in the limit of k = 0.

Chapter | 4 One-Dimensional Phononic Crystals 143

2.1.1 Model The PnC is formed out of an infinite repetition of two different slabs, labeled by the unit-cell index n. Each of these slabs of width di is labeled by the index i = 1 or 2, within the unit cell n. All the interfaces are taken to be parallel to the (x1 , x2 ) plane. A space position along the x3 axis in medium i belonging to the unit cell n is indicated by (n, i, x3 ), where −di /2 < x3 < di /2. The period of the PnC is called D = d1 + d2 . We limit ourselves to the simplest case of shear-horizontal vibrations where the field displacements u2 (x3 ) are along the axis x2 and the wave vector k (parallel to the interfaces) is directed along the x1 axis. We can then consider with the same general equations the two following cases. (i) A PnC built out of cubic crystals with (001) interfaces and k along the [100] crystallographic direction. The corresponding bulk equation of motion for medium i is [4]:   2 (i) d (i) 2 2 (i) (1) ρ ω − k C44 + C44 2 u2 (x3 ) = 0, dx3 (i)

where ρ (i) and C44 are, respectively, the mass density and the elastic constant and ω is the frequency of the vibrations. (ii) A PnC built out of hexagonal crystals with (0001) interfaces. The isotropy of these interfaces enables us to choose k along any direction within the (x1 , x2 ) plane. For simplicity we shall leave k along the x1 axis. In this case, the bulk equation of motion for medium i becomes [4]:   (i)   (i) 2 (i) d (i) 2 2 C11 − C12 (2) + C44 2 u2 (x3 ) = 0, ρ ω − k 2 dx3 (i) (i) (i) where C11 , C12 , and C44 are the elastic constants of medium i.

We also took advantage of the infinitesimal translational invariance in directions parallel to the interfaces and Fourier analyzed the equations of motion and all operators according to, for example [5]: g(ω; x3 , x3 ) =



 d2 k g(ω, k ; x3 , x3 )eik (x −x ) , 2 (2π )

(3)

where x ≡ (x1 , x2 ) is the component parallel to the interfaces of the realspace position x. In the following, we shall drop for simplicity the ω and k dependence of the functions g. Let us define: αi2 = k2 − ρ (i)

ω2 (i) C44

,

(4)

144 Phononics

for the cubic crystals and αi2

 =

k2

(i)

(i)

C11 − C12 (i)

2C44

 − ρ (i)

ω2 (i)

,

(5)

C44

for the hexagonal ones. Then, as mentioned earlier, Eqs. (1), (2) have the same expression:   d2 (i) 2 − αi u2 (x3 ) = 0. (6) C44 dx32 The corresponding bulk response function for medium i is defined by [106]:   d2 (i) 2 − αi Gi (x3 − x3 ) = δ(x3 − x3 ). (7) C44 dx32 The response functions associated with the different heterostructures considered here are defined in the same manner taking into account the appropriate boundary conditions. Let us recall [63] that the implicit expression giving the bulk dispersion relations of such an infinite PnC is:   F1 F2 + (8) S1 S2 , cos(k3 D) = C1 C2 + F2 F1 where Ci = cosh(αi di ), Si = sinh(αi di ), (i)

Fi = αi C44 ,

(9) (10) (11)

and k3 is the component perpendicular to the slabs of the propagation vector k ≡ (k1 , k3 ).

2.1.2 Density of States Knowing the response functions an infinite PnC, one obtains for a given value of k the local and total DOS for a semiinfinite PnC with a surface cap layer. We shall indicate at the end of this section how one can obtain from these quantities similar results for two limiting cases, namely the case of the interface between a PnC and a homogeneous substrate, and that of a semiinfinite PnC without a cap layer. 2.1.2.1 Local Densities of States The local densities of states (LDOSs) on the plane (n, i, x3 ) are given by: n(ω2 , k ; n, i, x3 ) = −

ρ (i) + 2 d (ω , k ; n, i, x3 ; n, i, x3 ), π

(12)

Chapter | 4 One-Dimensional Phononic Crystals 145

where d+ (ω2 ) = lim d(ω2 + iε),

(13)

ε→0

and d(ω2 ) is the response function whose elements are given in El Boudouti et al. [5]. The DOS can also be given as a function of ω, instead of ω2 using the well-known relation n(ω) = 2ωn(ω2 ). From the elements of the response function given in El Boudouti et al. [5], we obtained the following explicit expressions for the LDOSs on the surface of the semiinfinite PnC with a cap layer (n = 0, i = 0) of width d0 :   d0 C2 S1 1 C1 S2 2 + =−  ns ω , k ; 0, 0, 2 π F2 F1 

 S0 F1 + S1 S2 − t Δ−1 , (14) C1 C2 + F0 C0 F2 where C0 , S0 , F0 have the same definitions as Ci , Si , Fi given by Eqs. (9)–(11) for i = 0: ⎧ η < −1, ⎨ η + (η2 − 1)1/2 (15) t = η + i(1 − η2 )1/2 −1 < η < +1, ⎩ 2 1/2 η − (η − 1) η > 1, with η = C1 C2 +

1 2



F2 F1 + F2 F1

and Δ = C1 C2 +

F2 1 F0 S0 S1 S2 − − F1 t C0





S1 S2 ,

C2 S1 C1 S2 + F2 F1

(16)  .

(17)

In the same manner the LDOS at the interface between the cap layer and the semiinfinite PnC was found to be:     d0 C2 S1 1 C1 S2 2 + (18) =−  Δ−1 . ni ω , k ; 0, 0, − 2 π F2 F1 2.1.2.2 Total Density of States The total DOS for a given value of k is obtained by integrating over x3 and summing on n and i the local density n(ω2 , k ; n, i, x3 ). A particularly interesting quantity is the variation of the total DOS between the semiinfinite PnC with the cap layer n = 0 and the infinite PnC having the same number of slabs as the semiinfinite PnC without the cap layer. The variation Δn(ω2 ) can be written as the sums of the variations Δn1 (ω2 ) and Δn2 (ω2 ) of the DOS in slabs 1 and 2 and the DOS n0 (ω2 ) inside the cap layer: Δn(ω2 ) = Δn1 (ω2 ) + Δn2 (ω2 ) + n0 (ω2 ),

(19)

146 Phononics

where  + 1 0 2 ρ (1)   d [d(n, 1, x3 ; n, 1, x3 ) − g(n, 1, x3 ; n, 1, x3 )] dx3 , Δn1 (ω ) = − 1 π n=−∞ − 2 d

2

(20) Δn2 (ω2 ) = −

ρ (2) π

−1 

 

n=−∞

d + 22



d2 2

[d(n, 2, x3 ; n, 2, x3 ) − g(n, 2, x3 ; n, 2, x3 )] dx3 , (21)

n0 (ω2 ) = −

ρ (0)  π



d + 20



d0 2

d(0, 0, x3 ; 0, 0, x3 )dx3 ,

(22)

and d and g are the response function of, respectively, the semiinfinite PnC with the cap layer and of the infinite PnC. With the help of the explicit expressions of these response functions, we obtained:  

 ρ (1) 1 F2 t F1 S1 +  2 C2 S1 + C1 S2 Δn1 (ω2 ) = − π (t − 1) α1 F1 2 F2 F1   2 F d1 S2 Y + , (23) 1 − 22 2F2 Δ F1  

 ρ (2) 1 F2 t F1 S2 2 +  C1 S2 + C2 S1 Δn2 (ω ) = − π (t2 − 1) α2 F2 2 F2 F1   2 F d2 S1 Y + , (24) 1 − 12 2F1 Δ F2      ρ (0) C2 S1 F1 S0 C1 S2 2 + S1 S2 − t  + d0 C1 C2 + n0 (ω ) = − 2π α0 C0 F2 F1 F2

 S0 C1 S2 C2 S1 1 + + , (25) F0 C0 F2 F1 Δ where F0 S0 Y = C2 − C1 t − C0



S2 S1 t+ F1 F2

 .

(26)

At the limits of the bulk bands of the PnC given by t(ω0 ) = ±1, an expansion to first order in (ω − ω0 ) provides:   −1     1 t dη 1 = (27) P − iπ δ(ω − ω0 ) , 8 dω ω0 ω − ω0 t2 − 1 and then 1 Δn1 (ω2 ) + Δn2 (ω2 ) = − δ(ω − ω0 ). 4

(28)

Chapter | 4 One-Dimensional Phononic Crystals 147

So, the creation of a semiinfinite PnC from an infinite one gives rise to δ peaks of weight (−1/4) in the DOS at the edges of the PnC bulk bands.

2.1.3 Localized States When the denominator of Δn(ω2 ) vanishes for a frequency lying inside the gaps of the infinite PnC, one obtains localized states within the cap layer which decay exponentially inside the bulk of the PnC. The explicit expression giving these localized states is       F2 C0 F2 F1 C0 F0 S0 F1 F0 S0 − − − + S1 S2 + C2 S1 = 0, C1 S2 F2 C0 F0 S0 F2 F1 F1 C0 F0 S0 (29) together with the condition      C1 C2 + F2 S1 S2 − F0 S0 C1 S2 + C2 S1  > 1.  F1 C0 F2 F1 

(30)

2.1.4 Limit of a Semiinfinite PnC Without a Cap Layer In the limit when the thickness d0 of the cap layer goes to zero, S0 → 0 and the above results (Eqs. 14, 23, 24, 28–30) remain valid for a semiinfinite PnC ending with a complete i = 1 surface layer. We remark on Eq. (25) that in this limit, n0 (ω2 ) vanishes. In the limit where the cap layer i = 0 is of the same nature as the i = 2 PnC layer and d0 = ds < d2 , the same results provide the localized modes for a semiinfinite PnC ending with an incomplete i = 2 surface layer. In this case, we can calculate the variation of the DOS between such a semiinfinite PnC and the same amount of the bulk PnC, using in Eq. (19) Δn2 (ω2 ) integrated to ds /2 rather than to d2 /2 in the last layer, and taking n0 (ω2 ) = 0. A particularly interesting result can be obtained when cleaving an infinite PnC for the variation Δnc (ω2 ) of the total DOS between the two complementary semiinfinite PnCs and the infinite one. It is possible to show by using standard transformation of the trace of the response functions [106] that Δnc (ω2 ) can be obtained from the knowledge of the elements d1 (0, 2, ds ; 0, 2, ds ) and d2 (0, 2, ds ; 0, 2, ds ) of the surface response function of the two complementary semiinfinite PnCs, namely: 1 d  ln det [d1 (0, 2, ds ; 0, 2, ds ) + d2 (0, 2, ds ; 0, 2, ds )] . (31) π dω2 Using the expressions given in El Boudouti et al. [5] for these elements of the response functions, one finds that Δnc (ω2 ) is zero inside the bulk bands of the PnC and that at all edges of these bulk bands Δnc (ω2 ) display δ functions of weight (−1/2). These two facts, together with the necessary conservation of the number of states, enable us to conclude that when one considers together the two semiinfinite PnCs obtained by the cleavage of an infinite one, one has as Δnc (ω2 ) =

148 Phononics

many localized surface modes as mini-gaps for each value of k . There is only one very special exception to this general rule for a cleavage done along a plane situated exactly in the middle of a given slab.

2.1.5 Limit of an Interface Between a Semiinfinite PnC and a Homogeneous Substrate When the thickness d0 of the cap layer goes to infinity, S0 /C0 → 1 in the above expressions which remains valid and enables us to study the interface between a semiinfinite PnC and a homogeneous substrate. In particular, the results of Eqs. (29), (30) remain valid in this limit giving the localized interface states, Eq. (28) giving δ peaks of weight (−1/4) at the edges of the PnC bulk bands with Eqs. (23), (24) giving the variation of the DOS within the space of the PnC. Within the space of the substrate, rather than n0 (ω2 ) we shall calculate the variation Δ0 n(ω2 ) of the DOS between the substrate in contact with the PnC and the same volume of the infinite substrate, namely:  ρ (0) +∞ [d0 (x3 , x3 ) − G0 (x3 , x3 )] dx3 , Δ0 n(ω2 ) = − (32) π 0 where  

−1 −α0 |x3 −x | 1 1 C1 S2 C2 S1  3 + e + + e−α0 (x3 +x3 ) , d0 (x3 , x3 ) = 2F0 2F0 Δ0 F2 F1 (33)   F2 1 C2 S1 C1 S2 S1 S2 − − F0 + , (34) Δ0 = C1 C2 + F1 t F2 F1 and G0 (x3 , x3 ) = −

1 (0) 2α0 C44



e−α0 |x3 −x3 | .

(35)

We obtained like that:

    ρ (0) 1 C1 S2 C2 S1 1 1 Δ0 n(ω ) = − + +  . π 2α0 2F0 Δ0 F2 F1 2

(36)

Here also it is interesting to calculate the variation of the DOS Δ(1) nI (ω2 ) between the semiinfinite PnC and substrate on one hand and these same elements but coupled. Δ(1) nI (ω2 ) can be obtained in the same manner as above (Eq. 31) but with d2 (0, 2, ds ; 0, 2, ds ) now being the surface element of the response function of a semiinfinite homogeneous substrate. Consider now the semiinfinite PnC complementary to the one above, in the same cleavage of an infinite PnC and calculate as above the variation of the DOS Δ(2) nI (ω2 ) between this complementary semiinfinite PnC and the above substrate, on one hand, and these same elements but coupled. Such calculations provide one exact result, namely that the sum of the variation of the DOS of the two

Chapter | 4 One-Dimensional Phononic Crystals 149

complementary systems ΔnIC (ω2 ) = Δ(1) nI (ω2 ) + Δ(2) nI (ω2 ) is zero for ω belonging at the same time to the substrate and PnC bulk bands. Bearing in mind the result of Section 2.1.4 regarding the existence of surface states on these two complementary semiinfinite PnCs, we can now expect resonances, associated with the PnC/substrate interface, which fall within the PnC gaps and inside the bulk band of the substrate.

2.1.6 Applications and Discussions of the Results In what follows, specific results will be given for Y-Dy [107] or GaAs-AlAs PnCs and also for this last PnC with an Si surface cap layer. Tables 1 and 2 give the numerical values of the elastic constants and of the mass densities of these crystals. We shall first consider semiinfinite PnCs, then semiinfinite PnCs with a surface cap layer and finally semiinfinite PnCs on a semiinfinite homogeneous substrate. All the specific results presented here are given for transverse elastic waves with polarization perpendicular to the sagittal plane containing the normal to the interfaces and the propagation vector k parallel to the interfaces. 2.1.6.1 1D Semiinfinite Phononic Crystal The applications presented here refer to a GaAs-AlAs PnC with d1 = d2 and the period D = d1 + d2 . Fig. 1 shows the dispersion of bulk bands and surface modes as a function of k D. We have represented the surface modes of the two complementary semiinfinite PnCs obtained by cleaving the infinite GaAs-AlAs PnC within one GaAs slab, such that the thickness of the remaining surface GaAs layer is, respectively, ds = 0.3d2 and ds = 0.7d2 in each semiinfinite part.

TABLE 1 Elastic Constants and Mass Densities of Y and Dy C11 (1010 )(N m−2 )

C12 (1010 )(N m−2 )

C44 (1010 )(N m−2 )

ρ (kg m−3 )

Y

7.79

2.85

2.431

4450

Dy

7.31

2.53

2.4

8560

TABLE 2 Elastic Constants and Mass Densities of GaAs, AlAs, and Si C44 (1010 )(N m−2 )

ρ (kg m−3 )

GaAs

5.94

5316.9

AlAs

5.42

3721.8

Si

7.96

2330

150 Phononics

10

wD

Ct (GaAs)

8

6

4

2

0

2

4

6

k||D Fig. 1 Bulk and surface transverse elastic waves in a GaAs-AlAs phononic crystal (PnC). The curves give ωD/Ct (GaAs) as a function of k D, where ω is the frequency, k the propagation vector parallel to the interfaces, Ct (GaAs) the transverse speed of sound in GaAs, and D = dl + d2 the period of the PnC. The shaded areas represent the bulk bands. The dotted lines represent the surface phonons for the semiinfinite PnC terminated by a GaAs layer of thickness ds = 0.7d2 . The dashed lines represent the surface phonons for the complementary PnC terminated by a GaAs layer of thickness ds = 0.3d2 .

As demonstrated in Section 2.1.4, one obtains as many surface states as gaps and moreover there is one surface state in each gap associated with either one or the other of the complementary semiinfinite PnCs. One can observe that the surface modes are very dependent on the thickness of the last surface layer of GaAs. We shall come back to this point in the discussion of Fig. 3. For the moment, let us show in Fig. 2, for k D = 6, the variation of the vibrational DOS between the semiinfinite PnC terminated by a GaAs layer of width ds = 0.7d2 and the same amount of the bulk PnC, as defined in Section 2.1.4. The δ functions appearing in this figure are enlarged by the

Chapter | 4 One-Dimensional Phononic Crystals 151

L2

30 L1

Δn(w , K||D = 6)

20

10

0 T1

B2

B1

–10

6

T2 8

B3 10

wD/Ct(GaAs) Fig. 2 Variation of the DOS in units of D/Ct (GaAs) between a semiinfinite GaAs-AlAs PnC terminated by a GaAs layer of width ds = 0.7d2 and the same amount of a bulk PnC, for k D = 6 and as a function of ωD/Ct (GaAs). Bi and Ti , respectively, refer to δ peaks of weight (−1/4) situated at the bottom and the top of the bulk bands and Li indicates the localized surface modes.

addition of a small imaginary part to the frequency ω. The δ functions associated with the surface localized states are noted as Li and the δ functions of weight (−1/4) situated, respectively, at the bottom and top of the bulk bands are called Bi and Ti . The form of these enlarged δ functions Bi and Ti of weight (−1/4) is not exactly the same because of the contributions coming from the divergences in (ω − ωTi )−1/2 or (ω − ωBi )−1/2 existing in the DOS in one dimension. Apart from the above δ peaks and the particular behavior near the band edges, the variation Δn(ω, k ) of the DOS does not show any other significant effect inside the bulk bands of the PnC. Having seen that the frequencies of the surface states are very sensitive to the width ds of the last surface layer, we shown in Fig. 3 the variation of these frequencies, for k D = 3, as a function of ds /d2 , as well for a surface GaAs layer (dashed lines) as for a surface AlAs layer (full lines). One can see in this figure, for ds /d2 ≤ 1, that for all combinations of two complementary PnCs such that ds1 + ds2 = d2 , one always has a surface state in each gap. Let us also note that the same frequency of a surface state reappears with a given periodicity when ds /d2 takes values greater than 1. When ds increases, the frequencies of the existing surface modes decrease until the corresponding branches merge into the bulk bands and become resonant states; at the same time new localized branches

152 Phononics

10.77

wD

Ct (GaAs)

10.32 7.489 7.366 4.890

4.364 3.177 0

0.5

1 ds d2

1.5

2

Fig. 3 Variation of the dimensionless frequencies ωD/Ct (GaAs) of the surface modes of semiinfinite GaAs-AlAs PnCs, for k D = 3, as a function of ds /d2 , where ds is the width of the surface layer which may be GaAs (dashed lines) or AlAs (full lines). The shaded areas show the first three bulk bands of the PnC.

are extracted from the bulk bands. However, the resonant modes remain welldefined features of the DOS only as far as their frequencies remain in the vicinity of the band edges. Let us mention that the rule about the existence of one mode by gap has been shown experimentally within the first gaps [53, 56, 57, 61] by many authors in metallic PnCs using ultrasonic experiments (see Section 3.4). 2.1.6.2 1D PnCs With a Surface Cap Layer Now we assume that a cap layer of Si, of thickness d0 , is deposited on top of the GaAs-AlAs PnC terminated by a full GaAs layer. The dispersion of localized and resonant modes induced by a cap layer of relative width d0 /D = 4 is shown in Fig. 4. Depending on their frequencies, these modes may propagate along the direction perpendicular to the interfaces in both the PnC and the cap layers, or propagate in one and decay in the other, or decay on both sides of the PnCadlayer interface. The interface localized modes corresponding to this last case are labeled by the index i in Fig. 4. Note that when the Si cap layer is deposited on an AlAs layer of the PnC, different localized and resonant modes appear [64]. The variation of the DOS Δn(ω) between this PnC with the Si cap layer and the same amount of the bulk PnC without the cap layer was calculated as explained in Section 2.1.2.2. This Δn(ω) is shown in Fig. 5, for k D = 3, as a function of ωD/Ct (GaAs). Bi and Ti here also refer to δ peaks of weight (−1/4) at the edges of the PnC bulk bands; Li and Ri , respectively, indicate the localized

Chapter | 4 One-Dimensional Phononic Crystals 153

i

10

i

wD

Ct (GaAs)

8

6

4

2

0

2

4

6

k||D Fig. 4 Dispersion of localized and resonant modes (dashed lines) induced by an Si cap layer of thickness d0 = 4D, deposited on top of the GaAs-AlAs PnC terminated by a full GaAs layer. The shaded areas are the PnC bulk bands. The heavy line indicates the bottom of the bulk band of Si. The branches labeled “i” correspond to modes localized at the PnC-adlayer interface.

and resonant modes induced by the Si cap layer. The most intense resonance R2 is the lowest one situated just above the Si sound line. The next resonances are less intense, especially at higher frequencies where the separations between the successive branches increase. With the help of Eqs. (14), (18), we also studied LDOS. We found that they change with the position of the plane on which they are calculated. In particular, we found that the LDOS on the surface of the Si adlayer shows the same behavior as the total DOS shown in Fig. 5. On the contrary, the LDOS at the PnC-adlayer interface is pretty different. These behaviors can be understood by the very different boundary conditions exiting on these two planes. The frequencies of the localized and resonant modes vary with the thickness d0 of the cap layer. Fig. 6 shows these variations, for k D = 1. The first branches become closer one to each other when d0 increases, and as a consequence the

154 Phononics 40 L2

Δn (w , K||D = 3, d0/D = 4)

30

L1

R2

20

R4 R5

10

R3

R6

R1

R7

0 T1 –10

T2

B2

B1 4

B3

6

8

T3 B4 10

wD/Ct (GaAs) Fig. 5 DOS (in units of D/Ct (GaAs)) corresponding to the case shown in Fig. 4, for k D = 3. The contribution of the same amount of the bulk GaAs-AlAs PnC was subtracted. Bi , Ti , and Li have the same meaning as in Fig. 2; Ri refers to resonant modes.

intensities of the corresponding resonances increase. Let us also notice that the curves in this figure are almost horizontal when a localized branch is going to become resonant by merging into a bulk band. The variation with d0 is faster when the resonant branch penetrates deep into the band, but then the intensity of the resonant state decreases, or may even vanish in particular when d0 is small or the frequency is high. Finally, let us mention here too that for any given frequency ω in Fig. 6, there is a periodic repetition of the modes as a function of d0 . Let us mention that guided modes induced by a cap layer at the surface of the PnC have been observed in several works using Raman scattering in hybrid, amorphous, and semiconductor PnCs [15, 52, 65] and ultrasonic techniques in metallic PnCs [31]. When the thickness d0 of the cap layer goes to infinite, we find the situation of a semiinfinite PnC in contact with a homogeneous substrate. We address this case in the next section. 2.1.6.3 Semiinfinite PnCs on a Semiinfinite Substrate The possibility of shear-horizontal waves localized at the interface between a PnC and a substrate was first demonstrated [58] using a transfer-matrix method. Here we show the possibility of resonant modes, associated with this interface,

Chapter | 4 One-Dimensional Phononic Crystals 155

10

wD

Ct (GaAs)

8

6

4

2

0

1

2

3

4

5

do D Fig. 6 Dimensionless frequencies ωD/Ct (GaAs) of the localized and resonant modes induced by an Si cap layer of width d0 on the semiinfinite GaAs-AlAs PnC of Fig. 4, for k D = 1.

which appear as well-defined features of the DOS. The results will be illustrated, as in Khourdifi and Djafari-Rouhani [58], for a Y-Dy PnC such that d1 = d2 and D = 2d2 , in contact with a substrate having its transverse speed of sound equal to two times the Dy transverse speed of sound. In Khourdifi and Djafari-Rouhani [58] the existence of localized modes was (Dy) (s) discussed as a function of the parameter γ = C44 /C44 (where the index s refers to the substrate), considering either that a Dy or a Y layer of the PnC is in contact with the substrate. The interface localized modes originated in general from one of the two following extreme cases: γ = 0 or γ → ∞; in the former case the localized modes are those associated with the free surface of the PnC, whereas in the latter the amplitudes of the vibrations go to zero at the interface and remain vanishingly small in the substrate. To show the interface resonant modes in this section, we shown, respectively, in Figs. 7 and 9 two examples in (s) which the elastic constant C44 of the substrate takes two very different values, (Dy) (s) such that γ = C44 /C44 = 0.5 or 4. (i) Case γ = 0.5. Fig. 7 shows the localized and resonant interface modes for both the complementary PnCs in which the substrate is either in contact with a full Y or a full Dy layer. In the former case, the two full lines in the minigaps of the PnC are localized interface modes which continue (dashed lines) as

156 Phononics

10

w D/Ct (Dy)

8

6

4

2

0

2

4

6

k||D Fig. 7 Interface localized and resonant modes associated with the two complementary Y-Dy PnCs in which the substrate is either in contact with a Y or a Dy layer. The shaded areas are the bulk bands of the PnC. The heavy straight line indicates the bottom of the substrate bulk band. The (Dy) (s) (s) (Dy) and γ = C44 /C44 = 0.5. When the parameters of the substrate are defined as Ct = 2Ct PnC terminates with a Y layer, the localized (respectively, resonant) modes are presented by the full (respectively, dashed) lines. The dashed-dotted line is an interface branch associated with a Dy termination of the PnC.

well-defined resonances inside the bulk band of the substrate and within the PnC (s) mini-gaps. As the elastic constant C44 of the substrate has here a weak value γ = 0.5, these resonances are close to the surface states of the semiinfinite PnC (γ = 0). Their intensities, of course, decrease when γ increases. Now, if the substrate is in contact with a Dy layer, one obtains the dasheddotted branch near the bottom of the bulk bands, which is partly localized (k D ≥ 5) and partly resonant with the PnC states (k D ≤ 5). However, the dashed lines mentioned in the preceding paragraph are also associated with small resonances in this case.

Chapter | 4 One-Dimensional Phononic Crystals 157 R2

R1

R3

ΔnIC (w, K||D = 1, g = 0.5)

0

–10

–20

T2

B1 –30

Bs

B2

T1

B3

(A)

ΔNIC (w)

0.5

0

–0.5

(B)

1

2

3

4

5

6

7

8

Fig. 8 Variation of the DOS (A) and of the number of states (B), at k D = 1, for the two complementary PnCs of Fig. 7 created from the infinite PnC and the infinite substrate. Bi and Ti have the same meaning as in Fig. 2, whereas Bs refers to the δ peak of weight (−1/2) situated at the bottom of the substrate bulk band.

When one creates the two complementary PnCs used in Fig. 7 from the infinite PnC and the infinite substrate, the variation of the DOS ΔnIC (ω) can again show the new distribution of the states. We have presented such an example in Fig. 8A, for k D = 1: the loss of states due to the δ peaks of weight (−1/2) at every edge of the bulk bands is mostly compensated by the peaks associated with the resonant states (R1 , R2 , R3 ). This compensation can even be observed more easily in  ωFig. 8B showing the variation of the number of states, defined as ΔNIC (ω) = 0 ΔnIC (ω )dω . One can also check the validity of the statement presented in Section 2.1.5, namely, that ΔnIC (ω2 ) is zero for ω belonging at the same time to the substrate and PnC bulk bands. (ii) Case γ = 4. In Fig. 9 we have considered the case of a PnC terminated by a Dy layer. The two localized interface states (full lines) continue as resonances (dashed lines) lying just below the substrate bulk band. They correspond to modes localized on the side of the substrate and progressive on the side of the PnC. Note also the existence of two other resonances in the mini-gaps of the PnC; they are localized on the side of the PnC and progressive on the side of the substrate. A study of the DOS shows that the resonance appearing in the

158 Phononics

10

wD/Ct (Dy)

8

6

4

2

0

2

4

6

k||D Fig. 9

(s)

Interface localized and resonant modes, as in Fig. 7, but for a substrate such that C44 =

(Dy) (Dy) (s) 4C44 and Ct = 2Ct in contact with a Dy slab of the PnC.

lowest PnC mini-gap is as wide in frequency as the gap and is less sharp than the resonance lying in the second mini-gap. The frequencies of these last two resonances are rather close to those of the localized modes appearing on the surface of this PnC in the limit γ → ∞ [58] (this imposes on the displacements to vanish on this surface). When γ decreases, the intensities of these resonances decrease and their widths increase over the whole mini-gaps. Now if the substrate is in contact with a Y layer, the dashed curves in Fig. 9 still correspond to interface resonant states, which are, however, less intense than in the case of a PnC with Dy termination. (The localized modes are, however, different from those shown in Fig. 9.) In the above discussions, the resonances were defined as peaks in the DOS of the whole system. It is worth mentioning that these features do not necessarily appear in the LDOS at the PnC-substrate interface. This especially happens when the stiffness of the substrate (parameter γ ) is high; indeed, in this case,

Chapter | 4 One-Dimensional Phononic Crystals 159

the frequencies of the resonant modes are practically the same as in the case γ → ∞, but the Green’s function (and therefore the LDOS) at the PnC-substrate interface vanishes exactly at the latter frequencies. Let us mention that recent theoretical and experimental works [86–88] have shown the possibility of enhanced transmission between two substrates separated by a PnC. The transmission occurs through surface resonant modes induced by the interface between the PnC and one of the two substrates. In summary, we have presented in this section an analytical study of the density of transverse elastic waves for two-layer semiinfinite PnCs with or without a cap layer or in contact with a substrate. Particular attention was devoted to resonances (also called leaky waves) appearing in such heterostructures and to their relations with the localized modes. It was demonstrated in particular that when one considers together the two semiinfinite PnCs obtained by cleavage of an infinite one along a plane parallel to the interfaces, as many localized surface states as mini-gaps exist for all values of k . As a final remark, let us emphasize that the calculations presented here for the transverse elastic waves can be transposed straightforwardly to the electronic structure of PnCs in the effective-mass approximation [108, 109] or to the propagation of polaritons [110] in these heterostructures when each constituent is characterized by a local dielectric constant ε(ω). This is because both the equations of motion and the boundary conditions in the above problems involve similar mathematical equations.

2.2 Shear-Horizontal Acoustic Waves in Piezoelectric PnCs Piezoelectric materials are widely used to make electromechanical transducers for converting mechanical energy to electrical energy and vice versa. The energy converting capability of a piezoelectric material is the most important consideration for transducer design [111]. The study of acoustic vibrations in piezoelectric PnCs has received increased attention in the last three decades [4, 112–125] due to the unusual physical properties observed in these heterostructures in comparison with bulk materials [126]. The propagation of bulk and surface acoustic waves in homogeneous piezoelectric materials has found wide application, for instance, in the realization of transducers or in filtering. Electromechanical resonators are directly inserted into the circuits, the vibration being maintained by the electric field. Several years ago, the usefulness of piezoelectric PnCs as transducers in the high-frequency range was demonstrated [127–131], showing the possible application of this type of material to fabricate high-frequency acoustic resonance devices. On the other hand, it has been shown previously that the ideal PnC, which consists of an infinite repetition of two alternating layers, needs to be modified to take into account the media surrounding the PnC, such as vacuum, cap layer, substrate, etc. The presence of such inhomogeneities within the perfect piezoelectric PnC gives rise to localized modes inside the mini-gaps

160 Phononics

separating the bulk bands [4, 112–117]. One can quote also the works of Alshits et al. [118–121] on the propagation of acoustic waves in finite PnCs made up of identical piezoelectric or piezomagnetic layers separated by infinitely thin cladding layers with metallic or superconducting properties, respectively. This allows an analysis of the reflection-transmission spectrum due to multiple reflection at the boundaries between layers. Similar studies have been performed [123–125] on piezoelectric composites. The effect of a polymer bonding layer placed between each couple of piezoelectric layers is discussed [123, 125]. Stark-Ladder resonances are also evidenced in composites consisting of N piezoelectric layers whose piezoelectric properties obey a special linear solution [124, 125]. These investigations were performed using the transfer matrix [4, 112, 118, 121, 123–125], the Hamiltonian system formalism [117], and the Green’s function method [113–116]. The latter method is quite suitable for studying the spectral properties of the composite materials; in particular, it enables us to calculate the total DOS and LDOS in which the localized modes associated with the different perturbations cited previously appear as well-defined peaks. In Section 2.1, we have applied such a formalism to the case of purely shear-horizontal elastic waves associated with a semiinfinite PnC or to its interface with a substrate [24, 25]. For these waves involving only one direction of vibration, the Green’s function has been calculated analytically and the DOS has been obtained as a function of the frequency ω and the wave vector k (parallel to the interfaces). However, the direct calculation becomes very cumbersome once the shear-horizontal waves couple to the electric potential, although it remains analytical [5]. In this section, we will discuss shear-horizontal waves associated with the free surface of a semiinfinite piezoelectric PnC or with its interface with a piezoelectric substrate by means of the variation of the DOS associated with these structures [66]. In particular, we generalize the result obtained previously (see Section 2.1) in the case of nonpiezoelectric PnCs, namely, by considering together the two complementary PnCs obtained by cleavage of an infinite PnC along a plane parallel to the interfaces, one always has as many localized surface modes as mini-gaps for any value of the wave vector k (parallel to the interfaces). On the other hand, we show that in contrast to the case of an interface between two piezoelectric homogeneous media where interface modes are rather rare to find [126], localized modes with high degree of localization may exist at the interface between a piezoelectric PnC and a piezoelectric substrate. After a brief presentation of the model and the method of calculation in Section 2.2.1, we give in Section 2.2.2 some numerical results for a CdS/ZnO PnC with a free surface or in contact with a BeO substrate.

2.2.1 Model and Method of Calculation The interface PnC/substrate under consideration here is composed of a semiinfinite PnC formed out of a semiinfinite repetition of two different stabs

Chapter | 4 One-Dimensional Phononic Crystals 161

(i = 1, 2) within the unit cell n, the PnC being in contact with homogeneous substrate i = s. The interface PnC/vacuum can be obtained as a particular case of the PnC/substrate system by replacing the substrate with a vacuum. All the interfaces are taken to be parallel to the (x2 , x3 ) plane of a reference orthonormal basis set. A space position along the x1 axis (perpendicular to the interfaces) in medium i belonging to the unit cell n is indicated by (n, i, x1 ), where −di /2 ≤ x1 ≤ di /2, while x1 ≥ 0 in the substrate. The layers and the substrate are characterized by their elastic, dielectric, and piezoelectric constants and by their mass densities. The thicknesses d1 and d2 of the layers are assumed to be equal without loss of generality, the period of the PnC being D = d1 + d2 . The media forming the layers of the PnC and the substrate are assumed to be of hexagonal symmetry belonging to the 6 mm class with their c axis along the x3 axis and the wave vector k , parallel to the interfaces, 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 [4, 112] (see also El Boudouti et al. [5]). In our case, the composite material is composed of a PnC built out of alternating slabs of materials i (i = 1, 2) with thickness di , in contact with a substrate of material i = s. The details of the analysis are the same as for shearhorizontal waves (Section 2.1), but with more complicated calculations [5]. Let us emphasize that, in the geometry of the PnC/substrate structure, the elements of the Green’s function take the form gαβ (ω2 , k |n, i, x1 ; n , i , x1 ), where ω is the frequency of the acoustic wave, k the wave vector parallel to the interfaces, and α, β denote the components of a 4 × 4 matrix (α, β = 1, 2, 3, 4) representing the coupling between the acoustic waves (α, β = 1, 2, 3) and the electric field (α, β = 4). For the sake of simplicity, we shall omit in the following the parameters ω2 and k , and we note as g(n, i, x1 ; n , i , x1 ) the 4 × 4 matrix whose elements are gαβ (n, i, x1 ; n , i , x1 ) (α, β = 1, 2, 3, 4). By assuming that k is along the x2 direction, the component gαβ (α, β = 1, 2) of the Green’s function decouple [5] from the components gαβ (α, β = 3, 4) (i.e., g13 = g14 = g23 = g24 = g31 = g32 = g41 = g42 = 0); the former corresponds to vibrations polarized in the sagittal plane, whereas the latter are associated with shear-horizontal vibrations accompanied by the electric potential. For the shear-horizontal vibrations coupled to the electric potential studied here, the knowledge of the Green’s function g in the PnC/substrate system enables us to calculate the LDOS, for a given value of the wave vector k : 1 nα (ω2 , k ; n, i, x1 ) = − gαα (ω2 , k |n, i, x1 ; n, i, x1 ), π

(α = 3, 4),

(37)

2ω gαα (ω2 , k |n, i, x1 ; n, i, x1 ), π

(α = 3, 4).

(38)

or nα (ω, k ; n, i, x1 ) = −

162 Phononics

The total DOS can be obtained either by integrating directly over the space variable x1 or by using a method that involves only the interface matrix elements of the Green’s functions. In the latter method, one can calculate the difference between the total DOS of two complementary semiinfinite PnCs (terminating with layers i = 1 and i = 2) in contact with two substrates and the reference system (i.e., an infinite PnC and an infinite substrate), if we only know [5] the elements of the Green’s function at the surface Ms of the two complementary semiinfinite PnCs and of the two substrates. By calling d1 (Ms , Ms ), d2 (Ms , Ms ), and gs (Ms , Ms ), these Green’s functions matrix elements, one can write:   1 d [d1 (Ms , Ms ) + gs (Ms , Ms )][d2 (Ms , Ms ) + gs (Ms , Ms )] ln det . Δn(ω) = −  π dω 2[d1 (Ms , Ms ) + d2 (Ms , Ms )]gs (Ms , Ms ) (39)

As mentioned previously, the interface PnC/vacuum is obtained from Eq. (39) by replacing the Green’s function gs of the substrate by the Green’s function gv of vacuum. We shall distinguish also, as in usual piezoelectric materials [126], two types of surface boundary conditions depending on whether the surface is metallized (short-circuit) or not (open-circuit).

2.2.2 Discussion and Results In the following, we shall give some specific illustrations of these results. In these examples, the PnC is made of CdS and ZnO with a free surface or in contact with a BeO substrate, having a high transverse velocity. The characteristic parameters of the materials used here are listed in Table 3. For given ω and k , the wave vectors along the axis x1 of the PnC, which can be deduced from the bulk dispersion, are called k1 . In the case of piezoelectric waves involving two components of the displacement vector and the electric potential, there are two pairs of values of k1 with given k and ω, which can be written as [4, 112] ±(K1 + iL1 ) and ±(K2 + iL2 ). Now an elastic wave with a frequency ω will propagate in the PnC if L1 = 0 or L2 = 0, while it is attenuated if both L1 and L2 are different from zero. Each pair of values of k1 (the first for instance) can take four different forms [66, 112]:

TABLE 3 Elastic, Piezoelectric, and Dielectric Parameters of the Materials CdS, ZnO, and BeO Materials

C44 (1010 )(N m−2 )

ρ (kg m−3 )

e15 (C m−1 )

11 (10−11 ) (F m−1 )

CdS

1.49

4824

−0.21

7.99

ZnO

4.25

5876

−0.59

7.38

BeO

14.77

3009



5.99

Chapter | 4 One-Dimensional Phononic Crystals 163

⎧ (i) pure real (L1 = 0), ⎪ ⎪ ⎨ (ii) pure imaginary (K1 = 0), (iii) complex but with (K1 = ±π/D), ⎪ ⎪ ⎩ (iv) complex with (K1 = π/D).

(40)

However, in case (iv) the two pairs of k1 necessarily become: K  + iL,

−(K  + iL ),

(K  − iL ),

−(K  − iL ).

(41)

Fig. 10A and B shows two examples of the complex band structure (thin solid curves) in a CdS/ZnO PnC showing combinations of the above-mentioned

10 wL3

w D/Ct (Cds)

8 6

wL2

4 2 wL1, wL′1

0

p Im(k1D)

–0 Im(k1D)

(A)

Re(k1D)

10

wL6

w D/Ct (Cds)

8 wL5

6 4

wL4

2 0

(B)

4

0 Im(k1D)

Re(k1D)

p Im(k1D)

Fig. 10 Complex band structure (ω vs. complex k1 ) in a CdS/ZnO PnC with d1 = d2 = D/2 and k D = 0.2 (A), 4 (B). Thin solid curves are k1 real (middle panel of the figure). Dashed curves are k1 imaginary (left panel). Dotted curves are the imaginary part of k1 when its real part is equal to π/D (right panel). Heavy solid curves in (A) represent the complex band structure but neglecting the piezoelectricity. The horizontally full (dashed) arrows give the frequencies of the surface modes inside the mini-gaps of the PnC when the layer at the surface of the PnC is CdS (ZnO).

164 Phononics

cases for small and large values of k D, namely, k D = 0.2 (Fig. 10A) and k D = 4 (Fig. 10B). One can see the dispersion of the displacement component in Fig. 10A and B giving rise to direct gaps at the center and at the edge of the reduced Brillouin zone (BZ), while the potential component is dispersion-less and appears as a vertical line at the imaginary part of k1 such that (k1 D) = k D. However, for small values of k D, the potential curve may interact with the vibrational one leading to the lifting of the degeneracy at the crossing points as is the case in the second gap of Fig. 10A. As a matter of comparison, we have also shown in Fig. 10A, the complex band structure associated with pure transverse elastic waves (heavy solid curves), that is, when the piezoelectric constant e15 is taken equal to zero. The dispersion curves are slightly shifted toward lower frequencies at small values of k D (Fig. 10A); however, the shifting disappears for large values of k D (Fig. 10B), where we have not plotted the curves associated with the pure elastic waves. Now, the creation of the free surface of the PnC gives rise to localized modes inside the mini-gaps of the PnC; their frequencies, indicated by arrows in Fig. 10A and B, are very dependent upon the composition of the layer that terminates the PnC. The full (dashed) arrows in Fig. 10 represent the surface modes when a CdS (ZnO) film is at the surface of the PnC; moreover, if the surface is metallized, the surface modes change slightly but cannot be distinguished from the preceding ones at the scale of the figure. Let us mention that in the small range of k D (Fig. 10A), there are two surface modes (labeled ωL1 and ωL 1 ) lying below the bulk bands. These modes may exist for both CdS and ZnO terminations of the PnC depending on whether the surface is metallized or not (see details in the following paragraphs). The surface modes are poles of the Green’s function and therefore appear as δ peaks in the DOS as shown in Fig. 11. In this figure, we have presented, for k D = 0.2, the variation of the total DOS when two complementary semiinfinite PnCs are created by the cleavage of an infinite PnC in such a way that one obtains one PnC with a full CdS layer at the surface and its complementary one with a full ZnO layer at the surface. For the sake of clarity and despite the analytic nature of our calculation, the δ peaks in the DOS are broadened by adding a small imaginary part to the frequency ω (ω → ω + i ). Fig. 11A–C is, respectively, associated with the cases where the surface of the PnC is metallized (Fig. 11A), nonmetallized (Fig. 11B), and pure elastic (Fig. 11C). Li is δ peaks associated with surface localized modes, whereas Bi and Ti refer to δ peaks of weight (−1/2) situated at the bottom and the top of the mini-bands [24, 25, 59, 66] given by k1 D = 0 and k1 D = π. One can see that the positions of the surface modes are almost the same in the first two cases. However, there exists two (respectively, one) surface modes below the bulk bands in the case of metallized (respectively, nonmetallized) surfaces (see the insets in Fig. 11A and B); these modes, labeled L1 and L1 , are associated with CdS and ZnO terminations of the PnC, respectively. Fig. 11C, corresponding to the pure elastic case, shows a shift of the bulk bands and surface modes toward lower frequencies.

Chapter | 4 One-Dimensional Phononic Crystals 165

20

L1

L′1 L2

30

L3

L1, L′1 0

15

B1

–20

Variation of the total density of states (k||D = 0.2)

0.27

0.28

0 –15

B1

T1 L1

20

30

B2

B3

T2

L2

(A)

L3

0

15

B1

L1 –20 0.27

0.28

0 –15

B1

T1

B2

T2

B3

L2

30

(B)

L3

L1

15 0

B1

–15 0

T1

2

B2

4 6 wD/Ct(CdS)

T2

B3

8

(C) 10

Fig. 11 Variation of the DOS (in units of D/Ct (CdS)) when creating two complementary semiinfinite PnCs from the infinite PnC, for k D = 0.2. Panels (A), (B), and (C) correspond, respectively, to the case when the films at the surfaces of the PnCs are metallized, nonmetallized, and pure elastic. L1 , L2 , and L3 represent surface modes when CdS is at the surface of the PnC. L1 corresponds to the surface mode when ZnO is at the surface; it appears when the surface is metallized. Bi and Ti correspond to δ peaks of weight −(1/2) which appear, respectively, at the bottom and the top of each bulk band. In the low-frequency range, the surface modes are located very close to the limits of the bulk bands and therefore mask the band-edge antiresonances. Therefore, the low-frequency region is enlarged in the insets of panels (A) and (B).

In Fig. 12 we represent the so-called projected band structure of the bulk bands and surface modes, namely, ω versus k . The shaded areas represent bulk bands separated by mini-gaps. Inside these gaps, we have represented surface modes corresponding to the two complementary semiinfinite PnCs with CdS termination (dashed curves) and ZnO termination (dotted curves). As mentioned previously, the frequencies of the surface modes are almost the same in the case of metallized and nonmetallized surfaces. A particular situation

166 Phononics 10

6

(a)

4 w / k||Ct(CdS)

w D/Ct (CdS)

8

2

(c)

1,375

(b) 1,370 (d) 1,365

0,0

0,2 k||D

0,4

0 0

1

2

3

4

5

6

7

k||D Fig. 12 Bulk band and surface wave dispersion for a CdS/ZnO PnC with d1 = d2 . The curves give ωD/Ct (CdS) as a function of k D, where ω is the frequency, k the propagation vector parallel to the interfaces. The shaded areas are the bulk bands. The dashed (dotted) curves represent the surface modes when a nonmetallized CdS (ZnO) layer is at the surface. For a metallized surface, the surface modes are slightly different but cannot be distinguished from the preceding ones at the scale of the figure. The inset shows, in units of Ct (CdS), the phase velocities of the bottom of the bulk bands (a) and of the surface waves below it (b–d) over a small range of k D. The curves (b), (d), and (c), respectively, correspond to the case when the film at the surface is CdS nonmetallized, CdS metallized, and ZnO metallized.

occurs in the long wavelength limit (see the inset of Fig. 12): when the PnC terminates with a CdS layer, the surface mode below the bulk band exists whether the surface is metallized or nonmetallized; on the other hand, for a ZnO termination of the PnC, the surface mode exists over a short range of the dimensionless wave vector k D only if the surface is metallized. These results are in accordance with those presented in the insets of Fig. 11A and b. From the results given in Fig. 12, one should notice that apart from the long wavelength region, there are as many surface states as mini-gaps for each value of the wave vector k . Each surface mode is associated with either one or the other of the complementary semiinfinite PnCs. This result is obtained numerically based on the following two steps: (i) the variation of the DOS when creating two complementary semiinfinite PnCs from an infinite one is equal to

Chapter | 4 One-Dimensional Phononic Crystals 167

zero for any frequency ω falling inside a PnC bulk band; (ii) there is a loss of 1/2 state at every edge of a bulk band (see Fig. 11), that is, a loss of one state per mini-band. Therefore, to ensure the conservation of the total number of states, it is necessary to have as many localized modes as mini-gaps in the band structure. The above rule generalizes the result presented previously in Section 2.1 for nonpiezoelectric PnCs. To illustrate the spatial localization of the surface modes, we have sketched in Fig. 13 the LDOS associated with the modes labeled ωL2 , ωL3 , and ωL6 in Fig. 10. The LDOS is presented as a function of the space position x1 , at k D = 0.2 and k D = 4. Both the vibrational and the electric components of the DOS are given for two types of electric boundary conditions, namely a metallized (dotted curves) and a nonmetallized (solid lines) surfaces. The displacement field is not greatly affected by the electric boundary condition, while the electric field shows different behaviors for a metallized or a nonmetallized surface. However, these differences become less important when going to higher values of k D. For both vibrational and electric components of the surface wave, one should notice an exponential decrease of their envelopes when penetrating deep into the interior of the PnC; this attenuation is accompanied by an increasing number of oscillations in each period of the PnC as far as one is dealing with higher frequency modes. In the following we show the possibility of the existence of piezoelectric waves localized at the interface between a piezoelectric PnC and a piezoelectric substrate. The results will be illustrated for the same CdS/ZnO PnC in contact with a BeO substrate. The velocity of the transverse speed of sound in the BeO substrate is rather high, which enables us to give a qualitative description of the different localized modes induced by the PnC/substrate interface, inside the mini-gaps and below the substrate bulk band. Fig. 14 shows the interface modes for both the complementary PnCs in which the substrate is in contact either with a ZnO layer (triangles) or with a CdS layer (dotted curves). One can clearly notice that the frequencies of the interface modes are very sensitive to the type of layer, which is at the surface of the PnC. Let us mention that, to our knowledge, the piezoelectric constant e15 of the BeO substrate is not available in the literature [132]. However, we have determined that physically acceptable values of the substrate piezoelectric constant e15 [132] do not affect the frequencies of the interface modes. As in the case of a free surface, one can distinguish two types of boundary conditions (metallized or nonmetallized interface). The interface modes in Fig. 14 are very close for both boundary conditions and cannot be distinguished at the scale of Fig. 14. However, the frequencies of the interface modes slightly shift toward lower frequencies when the piezoelectric coupling is not taken into account as it can be seen in Fig. 15. In this figure we have plotted, at k D = 5, the variation of the total DOS for the metallized (Fig. 15A), nonmetallized (Fig. 15B), and pure elastic (Fig. 15C) cases. The latter case is obtained by taking the piezoelectric constants equal to zero in the media constituting the PnC and the substrate. The

168 Phononics

Vacuum

Superlattice 30

wL2, k||D = 0.2

LDOS (u.a.)

Vibrational component

(A) 0 120 Potentional component 60

(B) 0 –10

–8

LDOS (u.a.)

–4

–2

0

2

6

wL3, k||D = 0.2

Vibrational component

10

4 Vacuum

Superlattice

20

0

–6

(C)

400

Potential component

(D) 0 –10

–8

–6

–4

–2

0

2

4

6

Superlattice

20 LDOS (u.a.)

Vacuum Vibrational component

10

wL6, k||D = 4

(E) 0 40 Potential component 20

(F) 0 –10

–8

–6

–4

–2 x1/D

0

2

4

6

Fig. 13 Spatial representation of the LDOS for the modes labeled ωL2 (A, B), ωL3 (C, D), and ωL6 (E, F) in Fig. 10. The curves in (A), (C), and (E) (respectively, (B), (D), and (F)) correspond to the vibrational (respectively, potential) components of the LDOS. The full (dotted) curves correspond to the case when the film at the surface is CdS metallized (CdS nonmetallized).

interface modes labeled L1 and L2 in Fig. 15 are associated with CdS and ZnO terminations of the PnC, respectively. Let us mention finally that the interface modes between two homogeneous piezoelectric media (the so-called MaerfeldTournois modes) are rather rare to find [126]. In our case, for instance, the interface BeO/CdS and BeO/ZnO do not support interface modes. Therefore, the

Chapter | 4 One-Dimensional Phononic Crystals 169

10

w D/Ct(CdS)

8

6

4

2

0 0

1

2

3

4

5

6

7

k||D Fig. 14 Interface localized modes associated with a CdS/ZnO PnC in contact with a BeO substrate. The shaded areas are the bulk bands of the PnC. The heavy straight line indicates the bottom of the substrate bulk band. When the PnC terminates with a ZnO (respectively, CdS) layer, the localized modes are represented by the triangles (dotted curves).

PnC/substrate interface presents a useful structure for the existence of interface modes in comparison with the interface between two homogeneous media. Finally, let us mention that from the above results one can study also surface acoustic waves in piezoelectric/metal PnCs by taking the piezoelectric constant e15 in one layer equal to zero. This case has been studied by Chen et al. [133] and analytical expressions have been deduced for surface waves in semiinfinite PnCs ended with a piezoelectric layer at the surface. The free surface of the PnC can be either metallized or nonmetallized. From these two boundary conditions, one can define the electromechanical coupling coefficient as κ = (vo −vs )/vs , here vo and vs denote the phase velocity of the surface waves with open (nonmetallized) and short (metallized) circuit surfaces, respectively. Fig. 16A shows the first-band structure (the phase velocity vs. ωD) of the infinite PZT4/Fe PnC (gray areas), whereas the surface branches for open and short circuit surfaces are sketched by open and filled circles, respectively. The thickness of the layers in the PnC is supposed equal. As it was shown by Chen et al. [133], only the surface branches lying below the bulk bands give rise to significant surface electromechanical coupling and can be excited effectively by interdigital transducers (IDT). Therefore, we limited ourselves

170 Phononics 40 L2

L1

30 20 10 0 –10

T1

–20 B1

Variation of the total density of states (k||D = 5)

–30

T2

B3

B2

(A)

–40 L1

30

L2

20 10 0 –10 –20

B1

(B)

L1

30

B3

T2

B2

T1

L2

20 10 0 –10 B1

–20

T1

B2

7

8

B3

T2

(C) –30 4

5

6

9

10

11

w D/Ct(CdS) Fig. 15 Variation of the DOS at k D = 5, for the two complementary PnCs of Fig. 14 created from the infinite PnC and the infinite substrate. L1 and L2 represent interface modes associated with CdS and ZnO terminations of the PnC, respectively, whereas Bi and Ti have the same meaning as in Fig. 11.

to these branches; the corresponding electromechanical coupling is shown in Fig. 16B. These results are similar to those found by Chen et al. [133] using another method of calculation based on the boundary conditions at the different interfaces. When ωD is very large, the phase velocities of the first surface mode of the PnC with short and open circuit surfaces tend to Cs and Co , respectively

Chapter | 4 One-Dimensional Phononic Crystals 171

Phase velocity (km/s)

4.0

3.5

3.0

2.4624

2.5

2.3154

2.0 0

10

20

(A)

30

40

50

60

40

50

60

w.D 0.14 0.12

k (%)

0.10 0.08 0. 0634 0.06 0.04 0.02 0.00 0

(B)

10

20

30 w.D (GHz μm)

Fig. 16 (A) The band structure and dispersion curves of the lowest two shear-horizontal surface waves of the semiinfinite PZT4/Fe PnC. The open and filled circles denote dispersion curves of the system with open and short circuit surfaces, respectively. The gray areas represent the bulk bands. (B) Variation of the electromechanical coupling κ as function of ωD for the lowest two surface waves of the semiinfinite PZT4/Fe superlattices. (Reproduced after S. Chen, S. Lin, Z. Wang, Ultrasonics 49 (2009) 446.)

172 Phononics

(see the horizontal dashed lines in Fig. 16A). Cs and Co are, respectively, the phase velocities of the surface wave in the semiinfinite uniform PZT4 crystal with short and open circuit surfaces, their expressions are given by Eqs. (74), (75) in El Boudouti et al. [5]. These results are predicted, indeed when ωD is large, the surface waves become strongly localized near to the surface and therefore are not affected by the layers inside the PnC. At very low-frequency case, the κ of the surface modes leads to zero, which implies that the surface modes with very low frequency cannot be excited effectively by IDTs. When the frequency is not very low, the κ is larger than of the surface modes of a semiinfinite uniform crystal κPZT4 = (Co − Cs )/Cs = 6.34% for PZT4. This is a prominent advantage of the piezoelectric/metal PnC. It is worth mentioning that electromechanical coupling for surface acoustic waves in KNbO3 [134], AlN [135], GaN [136], and ceramic thin plate [137] has been measured experimentally. These studies enable to deduce the elastic, dielectric, and piezoelectric parameters of these materials. In summary, the main points, presented in this section, apart from the analytical derivation of the Green’s function and the corresponding DOS [5] are: (i) We have generalized to piezoelectric PnCs of the previous theorem (Section 2.1) about the existence and number of surface states. More precisely, we have shown that, apart from the long-wavelength limit, one obtains as many localized surface states as mini-gaps for any value of the wave vector k . This result is based on the general rule about the conservation of the total number of states and expresses a compensation between the losses of (1/2) state at every bulk band edge (due to the creation of the free surfaces) and the gain due to the occurrence of surface states. The imaginary part of the k1 wave vectors obtained in the complex band structure calculation gives the attenuation of the surface waves in a semiinfinite PnC. Although the frequencies of the localized modes associated with metallized and nonmetallized surfaces are almost the same, their spatial localization may exhibit noticeable differences, in particular, for small values of k D. (ii) We have derived the dispersion curves and the localization properties of the interface modes at the PnC/substrate boundary, with specific application to the CdS/ZnO piezoelectric PnC in contact with a BeO piezoelectric substrate. The dispersion curves of the interface modes are very dependent upon the nature of the layer in the PnC which is in contact with the substrate. (iii) We have shown that the interface between a homogeneous and a periodic layered media appears as a fruitful structure for the existence of interface modes in comparison with the interface between two homogeneous media where the existence of localized modes (called Maerfeld-Tournois modes) becomes rather hard to find [126]. This opens, at least in principle, some new opportunities for guided waves and filtering which is of potential value in the interface characterization of the multilayer/homogeneous structure.

Chapter | 4 One-Dimensional Phononic Crystals 173

(iv) We have reproduced the results of Chen et al. [133] on surface acoustic waves in piezoelectric/metal PnCs. It was shown that at some frequency ranges, the electromechanical coupling at the surface of such PnCs may exhibit large values in comparison with the one associated with a semiinfinite homogeneous piezoelectric medium. Therefore, this study may be of interest in the field of the dynamics of piezoelectric layered structures and in particular for those applications which deal with the propagation of surface and interface acoustic waves in the high-frequency range.

3 SHEAR-HORIZONTAL ACOUSTIC WAVES IN FINITE PnCs In Section 2, we have considered PnCs with infinite and semiinfinite extension to study the bulk and surface acoustic waves. However, from the practical point of view, the PnCs are constituted of a finite number of cells deposited on a substrate. Often, a buffer layer is first grown before the deposition of the PnC in order to relieve strains and defects of the underlying substrate surface caused by damage due to polishing or other imperfections. In other experiments, a defect layer is introduced at the surface of the PnC (as a cap layer) or inside the PnC (as a cavity layer) for some applications as selective filters and efficient waveguides. Therefore, it is interesting to take into account the finite-size effect of the PnC as well as the media surrounding the perfect PnC. Experimentally, very small peaks [77–79] have been observed in the folded longitudinal-acoustic phonons of Si/Ge1−x Six and GaAs-AlAs PnCs by highresolution Raman measurements and interpreted as confined phonons of the whole finite PnC. In addition, surface acoustic waves induced by a cap layer in the mini-gaps of a PnC have been shown in hybrid [15], metallic [31, 53], and semiconductor [52, 54–57, 60, 61, 65, 138] PnCs by using picosecond ultrasonic study and Raman scattering experiments, respectively. The effect of a cavity layer inserted in the middle [72–76] or at different places [71] within the PnC has been also well studied by Raman scattering. From the theoretical point of view, Raman scattering [72–74, 139, 140], DOSs [51], and reflectiontransmission coefficients [3, 19, 20, 30, 50, 80–83] have been performed to understand the propagation and localization properties of different confined modes within the finite-size PnC. The aim of the present section is to investigate localized and resonant modes of shear-horizontal polarization in finite-size PnCs, in particular we shall emphasize the effect of the different inhomogeneities cited above on acoustic waves in such PnCs. Section 3.1 presents the analytical results of dispersion relations, DOSs, as well as transmission and reflection coefficients for a finite PnC inserted between two substrates and in presence of buffer and cap layers (see Fig. 17). Sections 3.2 and 3.3 will be devoted to numerical applications concerning the effects of, respectively, the cap layer, buffer layer, and cavity layer inserted within the PnC. Section 3.4 gives some experimental results related to this work.

174 Phononics Cell 0

Substrate (s)

Buffer layer

Cell N

2

1

2

1

Cap layer

(b)

(c)

x3 = 0

x3 = db + dc + ND

db

d2

d1

Substrate (v)

x3

dc

Fig. 17 Schematic representation of a finite PnC (i = 1, 2) with a buffer layer (n = 0, i = b) and a cap layer (n = N, i = c) and sandwiched between two substrates (s) and (v). db , dc , d1 , and d2 , respectively, are the thicknesses of the buffer layer, the cap layer, and of the two different slabs out of which the unit cell of the PnC is built. D is the period of the PnC.

3.1 Density of States and Reflection and Transmission Coefficients In the present study we are dealing with a general structure, often used experimentally [141], this structure consists of a substrate-buffer layer (b), finite PnC-cap layer (c), semiinfinite medium (v) (see Fig. 17). The finite PnC has a total N unit cells, where a unit cell is composed of two different slabs denoted by the unit-cell index n. Each of these slabs is characterized by its elastic constant (i) C44 , mass density ρ (i) , and thickness di (i = 1, 2) within the unit cell n. The repetition period is called D = d1 + d2 . Similarly, the buffer layer, the cap layer, (b) the semiinfinite medium, and the substrate involve the parameters (C44 , ρ (b) , (c) (v) (s) db ), (C44 , ρ (c) , dc ), (C44 , ρ (v) ), and (C44 , ρ (s) ), respectively. A space position along the x3 axis in medium i belonging to the cell n is indicated by (n, i, x3 ), where −di /2 ≤ x3 ≤ di /2 (i = 1, 2). We limit ourselves to the simplest case of shear-horizontal vibrations where the field displacements u2 (x3 ) are along the axis x2 and the wave vector k (parallel to the interfaces) is directed along the x1 axis. We consider then a PnC built out of cubic crystals with (001) interfaces and k along the [100] direction. In this case, as it was shown in the previous sections, the transverse waves are not coupled to the other waves polarized in the sagittal plane which contains the normal to the interfaces and the wave vector k . From the expression of the Green’s function for the system [5], one obtains for a given value of k , the local and total DOS for a substrate-buffer layer (b), finite PnC-cap layer (c), semiinfinite medium composite system (see Fig. 17).

3.1.1 Local Density of States The LDOSs on the plane (n, i, x3 ) are given by: n(ω2 , k ; n, i, x3 ) = −

ρ (i) + 2 d (ω , k ; n, i, x3 ; n, i, x3 ), π

(42)

Chapter | 4 One-Dimensional Phononic Crystals 175

where d+ (ω2 ) = lim d(ω2 + iε), ε→0

(43)

and d(ω2 ) is the response function whose elements are given in the Appendix C from El Boudouti et al. [5]. As mentioned earlier, the DOS can also be given as a function of ω, instead of ω2 using the well-known relation n(ω) = 2ωn(ω2 ).

3.1.2 Total Density of States The total DOS for a given value of k is obtained by integrating over x3 and summing on n and i, the local density n(ω2 , k ; n, i, x3 ). More particularly, we are interested in this total DOS from which the substrate and semiinfinite medium contribution have been subtracted. The expression of the variation of the DOS when the buffer layer (b) and the cap layer (c) are in contact with materials 2 and 1, respectively, of the PnC, can be written as the sum of six contributions: n(ω2 ) = n1 (ω2 ) + n2 (ω2 ) + nb (ω2 ) + nc (ω2 ) + Δns (ω2 ) + Δnv (ω2 ), (44) where n1 (ω2 ) and n2 (ω2 ) are the contributions of layers 1 and 2, respectively, of the PnC; nb (ω2 ) and nc (ω2 ) are the contributions of buffer layer (b) and cap layer (c); and Δns (ω2 ) and Δnv (ω2 ) come from the substrate and semiinfinite medium, respectively. Actually, in the latter terms we subtract the bulk contributions of the substrate and the semiinfinite medium. Then the six quantities in Eq. (44) are given by [142]:  + d1 N 2 ρ (1)  2  d d(n, 1, x3 ; n, 1, x3 )dx3 , (45) n1 (ω ) = − 1 π −2 n=1

n2 (ω ) = − 2

N−1 ρ (2) 

π

 



n=0

ρ (b) nb (ω ) = −  π



2

nc (ω2 ) = −

ρ (c)  π

Δs n(ω2 ) = −

ρ (s)  π

Δv n(ω2 ) = −

ρ (v)  π

 

+

db 2

d2 2

d2 2

d(n, 2, x3 ; n, 2, x3 )dx3 ,

(46)

d(0, b, x3 ; 0, b, x3 )dx3 ,

(47)

d(N, c, x3 ; N, c, x3 )dx3 ,

(48)

[d(x3 , x3 ) − Gs (x3 , x3 )]dx3 ,

(49)

db 2 + d2c



− d2c 0

+

−∞  +∞

[d(x3 , x3 ) − Gv (x3 , x3 )]dx3 ,

(50)

xc

where xc = db + ND + dc , and N is the number of the finite PnC cells.

(51)

176 Phononics

The quantities d, Gs , and Gv are, respectively, the Green’s functions of (1) the coupled substrate-buffer layer (b), finite PnC-cap layer (c), semiinfinite medium system; (2) the infinite homogeneous substrate (s); and (3) the infinite homogeneous medium (v). Let us define the following quantities in each material i = 1, 2, b, c, s, and v: αi2 = k2 − ρ (i)

ω2 (i)

Ci = cosh(αi di ), (i) Fi = αi C44 ,

,

(52)

Si = sinh(αi di ),

(53)

C44

i = 1, 2, b, c, s, v,

(54)

Rs =

1 + Fb Sb /Fs Cb , 1 + Fs Sb /Fb Cb

(55)

Rv =

1 + Fc Sc /Fv Cc , 1 + Fv Sc /Fc Cc

(56)

t = exp(ik3 D),

(57)

where k3 is a wave vector along the axis of the PnC and that satisfies the following dispersion relation in the infinite PnC:   1 F1 F2 + (58) S1 S2 . cos(k3 D) = C1 C2 + 2 F2 F1 Therefore, with the help of the explicit expressions of the Green’s functions given in El Boudouti et al. [5] and Eqs. (45)–(50), we obtain: t ρ (1) 1 − t2N  2 (AB0 t + A0 B) t 2 π (t − 1)Δ− t − 1     

F22 S2 d1 1 F2 F1 S1 1− 2 C2 S1 + C1 S2 + + × α1 F 1 2 F2 F1 2F2 F1     

F22 S2 S1 1 F2 d1 F1 1− 2 , C2 S1 + C1 S2 + + NΔ+ + F1 2 F2 F1 2α1 F2 F

n1 (ω2 ) = −

1

(59) ρ (2)

1 − t2N

t  2 (AB0 + A0 Bt) t (t − 1)Δ− t2 − 1    

F12 S1 d2 1 F2 F1 S2 1− 2 C1 S2 + C2 S1 + + × α2 F 2 2 F2 F1 2F1 F2     

F12 S2 S1 1 F2 d2 F1 1− 2 , C1 S2 + C2 S1 + + NΔ+ + F2 2 F2 F1 2α2 F1 F

n2 (ω2 ) = −

π 

2

(60)

Chapter | 4 One-Dimensional Phononic Crystals 177

1 ρ (b)   F Sb 2π 1 + F sC Δ− b b      db Sb ZFs Sb Sb B0 t × Y + db + (Z + YFs ) − 2 − db αb C b Fb C b F b αb C b      db Sb Z 0 Fs Sb Sb 2N Y Bt, + db + (Z0 + YFs ) − 2 − db −t αb C b Fb C b αb C b F

nb (ω2 ) = −

b

(61) 1 ρ (c)   F 2π 1 + F vCSc Δ− c c   

 dc Sc KFv Sc Sc × X + dc − (K − XFv ) + 2 − dc A0 αc C c Fc C c αc C c Fc    

d K Fv S S Sc c c c − t2N X0 + dc − (K0 − X0 Fv ) + 0 2 − dc A, αc C c Fc C c αc C c Fc (62)    ⎤  ⎡ b b B0 t − t2N Y + Z0 FSC Bt Y + Z FSC ρ (s) 1 ⎣ 1 b b b b ⎦ , (63)    Δs n(ω2 ) = − + F S π 2αs 2Fs 1 + F sCb Δ− b b    ⎤  ⎡ Sc c (s) A A −X + K − t2N −X0 + K0 FSC 0 1 ρ 1 F C c c c c ⎣ ⎦,    + Δv n(ω2 ) = − π 2αv 2Fv 1 + Fv Sc Δ− nc (ω2 ) = −

Fc Cc

(64)

where



 AB A0 B0 ± t2N , X0 X = Fs Rs X0 + K0 , A = Fs Rs X + K, = Fv Rv y − Z0 , B = Fv Rv Y − Z, S1 S2 S1 S2 = t+ , X= + t, F1 F2 F1 F2 F1 1 F1 = C1 C2 + S1 S2 − , Z = C1 C2 + S1 S2 − t, F2 t F2 = C2 − C1 t, K = C2 t − C1 ,

Δ± = Yt A0 B0 X0 Z0 K0

(65) (66) (67) (68) (69) (70)

and Y=

C1 S2 C2 S1 + . F2 F1

(71)

The localized waves are given by the poles of the Green’s function, namely: Δ− = 0.

(72)

178 Phononics

The general expressions admit the following particular cases: (i) A finite PnC sandwiched between two semiinfinite substrates (i.e., db = d1 and dc = d2 ). (ii) A finite PnC deposited on the substrate (s) with only a cap layer (i.e., db = 0 and Fv = 0). (iii) A finite PnC deposited on the substrate (s) with only a buffer layer (i.e., dc = 0 and Fv = 0). We can also deduce the limiting case N → ∞ that corresponds to a semiinfinite PnC with or without a cap layer or in contact with a substrate.

3.1.3 Reflection and Transmission Waves Consider an incident wave in the substrate represented by the plane wave exp(−αs x3 ) of unit amplitude, propagating from x3 = −∞ toward the PnC. The incident waves are scattered from the interfaces between dissimilar layers constituting the system. The square modulus of the wave function of the reflected waves (denoted R) in the substrate and the transmitted waves in semiinfinite medium v (denoted T) are expressed, with the help of the interface response theory (see Chapter 1), as:   2  B0 A0−  2N BA−     1 − Fs Sb /Fb Cb Yt X0 − t X  , (73) R =   Δ−   1 + Fs Sb /Fb Cb  2  2YFs tN (t2 − 1) Fv   , T= Fs  Cb Cc (1 + Fs Sb /Fb Cb )(1 + Fv Sc /Fc Cc )Δ− 

(74)

respectively, where A0− = −Fs X0 Rs− + K0 ,

A− = −Fs XRs− + K,

(75)

and Rs− =

1 − Fb Sb /Fs Cb . 1 − Fs Sb /Fb Cb

(76)

3.1.4 Relations Between Densities of States and Phase Times Besides the information obtained from the imaginary part of the system Green’s function (namely the DOS) giving the frequencies of the different modes of the system [5], it is possible to obtain additional information by using the transmission and reflection coefficients and the corresponding phase times [142]. A detailed account for nonmagnetic dielectric media and electromagnetic waves has been presented in [142], and we shall give here only the transposition and the essential expressions for the understanding of the transverse elastic waves case. We shall consider in a general way a finite multilayer system denoted by an index i = 2 sandwiched between two different homogeneous semiinfinite

Chapter | 4 One-Dimensional Phononic Crystals 179

media having indexes i = 1 and 3, respectively. In the case of the structure studied here, 1 = s, 3 = v, and 2 represents the finite PnC with free surfaces. We have two interfaces bounding the multilayer system, we shall call them l (left) and r (right), respectively. The system inverse Green’s function projected at the interfaces can be expressed as: −1

gS (l, l) g−1 S (l, r) = . g−1 −1 S g−1 S (r, l) gS (r, r) Let us then consider an incident wave in the semiinfinite medium 1: u(x3 ) = exp(−iα1 x3 ), where

 αi =

ρ (i) ω2 (i) C44

− k2 ,

i = 1, 2, 3,

(77)

(78)

and k is the wave vector parallel to the interfaces. Following the expressions detailed in [142] it can be found that the transmitted wave in medium 3 has the form: uT (x3 ) = −2iα1 gS (l, r) exp(−iα3 x3 ),

(79)

whereas the reflected wave has the form: uR (x3 ) = −[1 + 2iα1 gS (l, l)] exp(iα1 x3 ).

(80)

It is then clear that Eqs. (79), (80) can be written as: uT (x3 ) = CT exp(−iα3 x3 ), uR (x3 ) = CR exp(iα1 x3 ),

(81)

where CT and CR are the transmission and reflection amplitudes given by: CT = −2iα1 g−1 S (l, r) det |gS |,

(82)

CR = −[1 + 2iα1 g−1 S (l, l) det |gS |]. From Eq. (82) it is possible to obtain the derivatives of the phases θT and θR of the transmission and reflection coefficients with respect to the frequency. These derivatives are an indication of the times needed by a wave packet to complete the transmission or reflection processes. These derivatives are usually called phase times [5] and are given by: dθT , dω dθR . τR = dω From Eqs. (82), (83) we obtain: τT =

τT =

d arg det|gS |. dω

(83)

(84)

180 Phononics

The general expression for τR can also be obtained, but there are two particular cases related to important physical situations, in which τR is related to −1 τT . If the system is symmetric, α1 = α3 and g−1 S (l, l) = gS (r, r), then we have d (85) arg det|gS |. dω If α3 becomes pure imaginary (total reflection), the transmission vanishes and then we have: d (86) τR = 2 arg det|gS | = 2τT . dω Let us now recall that the difference of the DOS between the present composite system and a reference system formed out of the same volumes of the bulk media 1 and 3 and the finite medium 2, can be obtained from:

1 d g2 (M, M)[G1 (0, 0)G3 (0, 0)]1/2 arg det . (87) Δ n(ω) = − π dω g(M, M) τR = τT =

If one subtracts the discrete states of the finite medium 2, then the variation of the DOS between the composite system and the same volumes of the bulk media 1 and 3 is given by: 1 d arg det [g(M, M)] . (88) π dω This relation is equivalent to the obtained one by using a direct calculation of the imaginary part of the trace of the Green’s function. Then for all frequencies, one obtains: Δn(ω) =

τT = π Δn(ω),

(89)

which is equivalent to the result of Avishai and Band [143]. Moreover, τR = τT = π Δn(ω),

(90)

when the composite system is symmetric and τR = 2π Δn(ω),

(91)

when one has total reflection. In what follows, we shall give numerical applications of the analytical expressions given above for a PnC with either a cap layer, a buffer layer, or a cavity layer.

3.2 Effect of a Cap Layer In the following we give a few illustrations related to a finite GaAs-AlAs PnC deposited on an Si substrate with or without a surface GaAs cap layer of different thickness, and for such a finite PnC sandwiched between two Si

Chapter | 4 One-Dimensional Phononic Crystals 181 11 i

10 9 8

s

wD

Ct (GaAs)

7 6 i

5 r

4 3 2 1 0 0

1

2

3

4

5

6

7

k||D Fig. 18 Dispersion of Love waves associated with the deposition of a finite GaAs-AlAs PnC on an Si substrate. The PnC contains N = 5 layer of GaAs and 6 layers of AlAs. The heavy line corresponds to the sound line of the substrate, separating the Love modes confined within the PnC from their extension as resonant waves into the substrate bulk band. When the dispersion curves belong to the PnC mini-bands, they are drawn as full lines; when falling inside the PnC mini-gaps, they are represented by dashed-dotted lines (labeled s) or dashed lines (labeled i) corresponding, respectively, to attenuated waves in PnC either from the free surface or from the PnC-substrate interface. The label r refers to a resonance obtained from the mixing of a surface and an interface mode.

substrates [144]. The thickness of the buffer layer is reduced to zero, whereas the thicknesses d1 and d2 of the layers in the PnC are assumed to be equal, the period of the PnC being D = d1 + d2 = 2d1 . Fig. 18 shows the Love modes, as deduced from the peaks of the DOS (see also Fig. 19), for a finite GaAs-AlAs PnC deposited on an Si substrate, assuming that the outermost layers in the PnC are both of AlAs type. For the sake of clarity in Fig. 18, the PnC only contains N = 5 layers of GaAs and N + 1 = 6 layers of AlAs. The branches situated below the substrate bulk band correspond

182 Phononics 9 s

i

6

(A)

Density of states

3

0 Bs 6

r

(B) 3

0 Bs –3 3

4

5

6

7

8

9

wD/Ct(GaAs) Fig. 19 DOS n(ω, k ), in units of D/Ct (GaAs), depicted for (A) k D = 2.6 and (B) k D = 1.9 in Fig. 18. The Love modes localized within the PnC give rise to δ peaks represented by arrows. The symbols s, r, and i have the same meanings as in Fig. 18. (The bulk contribution of the substrate to the DOS is subtracted. Bs is a δ function of weight −1/4 appearing at the bottom of the substrate bulk band.)

to Love waves confined in the finite PnC and decaying exponentially into the substrate; they appear as true δ functions in the DOS (Fig. 19). The extension of these curves into the substrate band represents resonant modes (also called leaky waves) associated with the deposition of the finite PnC on top of the substrate. The fine structure features observed [77–79] in the Raman experiments on SiSi1−x Gex and GaAs-AlAs PnCs seems to be analogous to these waves, in the case of longitudinal vibrations and for k D = 0. The curves in Fig. 18 can also be classified according to their oscillatory (full lines) or localized character in the PnC; the latter are decaying either from the free surface (dashed-dotted lines are also labeled s as surface) or from the PnC-substrate interface (dashed lines, also labeled i as interface). The number of branches corresponding to oscillatory waves in the PnC increases

Chapter | 4 One-Dimensional Phononic Crystals 183

with the number N of periods (see also Fig. 21A for N = 20, to be discussed following), leading to the bulk bands of an infinite PnC in the limit N → ∞. On the other hand, the surface and interface localized modes, which are already distinguishable in Fig. 18, even for such a small number of periods as N = 5, shift only slightly with N when going to the limit of a semiinfinite PnC; however, due to the finiteness of the PnC, a surface and an interface branch may interact together when falling in the same mini-gap of the PnC, as for k D ∼ = 2.3 in Fig. 18 (dotted lines). The general behavior of the resonant states in the DOS is shown in Fig. 19 where the widths of the peaks are due to the interaction between the PnC states and the substrate. In particular, we have shown the mixing of the surface and interface states, around k D = 2.3 in Fig. 18, into a resonant peak r whose weight is almost equal to two states; this peak remains very near to the free surface mode of a semiinfinite PnC. The interaction between the surface and interface states disappears by increasing N; this decoupling occurs in the present example for N of the order of 10–15. As a matter of completeness, we have also studied the spatial distribution of this resonance r in the PnC by presenting (Fig. 20) the LDOS integrated over each period of the PnC in the cases N = 5 and 20. The two sets of results present both similarities and differences. In both cases, the intensity of the peak in the DOS decreases by penetrating from the surface into the PnC, as a result of the decay of the surface acoustic wave; also, in both cases, the DOS near the interface with the substrate remains very broad, covering the whole mini-gap of the PnC. On the contrary, by increasing N, the DOS near the surface becomes very narrow and almost similar to a δ function; near the interface, the DOS loses small features, which exist for a very thin PnC as in the case N = 5. The modification of the DOS inside the substrate, due to the deposition of the PnC on its top, is a small quantity oscillating around zero. We have shown in Section 2.1 that surface localized modes in semiinfinite PnC’s are very sensitive to the nature and width of the outermost cap layer. Considering here the case of the finite PnC with N = 20 and with a GaAs cap layer of varying thickness dc , Fig. 21A shows the frequencies of the discrete modes versus dc , for a given value of k , namely, k D = 3. The classification of the curves is similar to that in Fig. 18. We shall be interested in the branches L and R (dashed-dotted lines), which are associated with the free surface of the PnC and are rather close to the surface modes of a semiinfinite PnC. The former branches L correspond to attenuated waves in the substrate; they take place when the lowest discrete mode emerges from the bulk (an effect which is periodically reproduced as a function of dc ). This branch interacts with the PnC-substrate interface branch i (dashed line) giving rise to the lifting of degeneracy at the crossing points around dc /d2 ∼ = 1.45, 3.1, etc. This interaction is more important for smaller values of N, as emphasized in Fig. 21B around dc /d2 ∼ = 1.45. The branches R are resonant with the substrate bulk band; the corresponding peaks in the DOS become wider and decrease in intensity when N decreases, as shown

184 Phononics 2.5 a

Density of states

2.0 1.5 b

1.0

c 0.5 0.0 d –0.5

3.6

3.8

4.0

4.4

wD/Ct(GaAs)

(A)

a

0.6

Density of states

4.2

× 10–4 c

0.4 d b

0.2

× 10–4

0.0

–0.2

(B)

3.7

3.8

3.9

4.0

4.1

4.2

w D/Ct(GaAs)

Fig. 20 (A) Spatial distribution of the resonance r shown in Fig. 19B: curves a and b, respectively, represent the LDOS integrated over the first and the third periods of the PnC from the free surface; curve c refers to the same quantity for the period in contact with the substrate, whereas curve d gives the modification of the substrate DOS after the deposition of the PnC on its top. (B) Same as in (A) but for a PnC containing N = 20 layers of GaAs and 21 layers of AlAs. Note that the LDOS at the surface becomes like a δ function.

in Fig. 21C in the case of dc /d2 = 1.8. This behavior can be attributed to the finiteness of the PnC and the interaction between the capped PnC modes with the substrate; indeed, in the limit of N → ∞, the resonant mode at dc /d2 = 1.8 becomes a surface localized mode of a semiinfinite PnC and appears as a δ peak in the DOS. We can push further this discussion by considering again the LDOS

Chapter | 4 One-Dimensional Phononic Crystals 185

5 w D/Ct (GaAs)

R

Ct (GaAs)

wD

7.489 7.366

4.89

L

i

4.364

4.9 i

4.8 L

4.7

L 4.6

0 1

1.5

2

2.5

3

3.5

dc d2

(A)

1

(B)

10

(c)

1.25

1.5 dc d2

1.75

2

R

Density of states

0 20 (b)

R

10 0 40 R

30 20

(a)

10 0 5.0 5.5 6.0 6.5 7.0 7.5 8.0

(C)

wD/Ct (GaAs)

Fig. 21 (A) Variations of the frequencies of the Love and pseudo-Love modes versus the thickness dc of the cap layer. The PnC contains N = 20 periods of GaAs-AlAs and it is capped with a GaAs layer of varying thickness dc . The wave vector k is chosen such that k D = 3. The symbol i refers to PnC-substrate interface modes, whereas L and R refer to the free surface modes of the PnC, respectively, evanescent in or resonant with the substrate. The heavy line indicates the sound line of the substrate; the arrows give the limits of the PnC mini-gaps. (B) The interaction between the surface and interface states near dc /d2 = 1.45, in (A), is emphasized for several values of the number N of periods in the PnC: N = 20 (—–), 10 (− − −), 7 (−.−.−.), and 5 (. . . ). (C) Density of states n(ω, k ) for k D = 3 and dc /d2 = 1.8, for several values of N: N = 20 (a), 12 (b), and 7 (c). R corresponds to the resonance depicted in (A). In this figure we have avoided the representation of the δ peaks.

186 Phononics

integrated over each period of the PnC; the surface mode R at dc /d2 = 1.8 extends over 10–15 periods of the PnC and therefore its interaction with the substrate is significant as far as N does not exceed this order of magnitude.

3.3 Effect of a Cavity Layer We have also studied the case of a defect layer inserted inside a finite-size PnC as a cavity layer [5]. The schematic illustration is shown in Fig. 22A for a finite PnC composed of N = 10 GaAs-AlAs cells deposited on a GaAs substrate, while the other surface is supposed free of stress. An Si cavity layer of thickness d0 = 3D is inserted within the PnC. Because of the existence of two perturbations (cavity and surface) within the PnC, one can expect two types of defect modes inside the band gaps and an interaction between these modes depending on the distance between these two defects. An analysis of the

x3 40 N=2

n=N

DOS

30

AlAs

d1

GaAs

d2

n = N–1

20 10

(B) 0 40

N=5

DOS

30

Si

20

0

d0

GaAs n = j–1

10

–10 40

(C)

GaAs n=0

30 DOS

GaAs

n=j

AlAs

N = 10

0

20 GaAs

10 0

(D)

–10 3.7

3.8

3.9

4.0

4.1

wD/Ct (GaAs)

4.2

4.3

(A)

Fig. 22 (A) Schematic representation of a finite GaAs-AlAs PnC terminated on both sides with AlAs layers and in presence of an Si cavity layer embedded within the cell j. The whole structure is deposited on a GaAs substrate. (B–D) DOS (in units of D/Ct (GaAs)) as a function of the reduced frequency for a finite-size PnC composed of N = 10 GaAs-AlAs cells and in presence of an Si defect layer inserted in the sites: j = 2 (B), j = 5 (C), and j = 9 (D) at k D = 2.

Chapter | 4 One-Dimensional Phononic Crystals 187

variation of the DOS as function of the reduced frequency ωD/Ct (GaAs) for different positions of the defect layer inside this structure (Fig. 22B–D) clearly shows this interaction. We have sketched these DOS for a defect layer near the surface of the PnC (j = 9, Fig. 22B), in the middle of the structure (j = 5, Fig. 22C), and near the interface PnC/substrate (j = 2, Fig. 22D). One can notice that, as expected, the interaction between the surface and defect modes falling within the first band gap of the PnC becomes important (weak) when the defect layer is closer to (far from) the surface, as it is shown in Fig. 22B–D. Let us notice that several works have been devoted to the modes induced by a cavity layer inserted in the middle [15, 72–76] or at different places [71] within the finite PnC by means of Raman scattering techniques (see Section 3.4.1). Cavity phonons in PnCs have been also investigated using the so-called asynchronous optical sampling [145] and ultrafast coherent generation of acoustic phonons in the presence of photon confinement in an optical resonator [146].

3.4 Relation to Experiments In this section, we shall give some experimental results related to the determination of bulk and defect modes in finite PnCs. In particular, we shall concentrate essentially on experimental measurements based on two technics, namely the light scattering by longitudinal acoustic phonons and laser picosecond ultrasonics.

3.4.1 Light Scattering by Longitudinal Acoustic Phonons The principle of this scattering can be summarized as follows: the propagation of an acoustic wave in the PnC excites periodic variations of strain which in turn induce a modulation of the dielectric tensor εij from the photo-elastic coupling to elastic fluctuations:    1 ∂uk ∂ul Pijkl + . (92) δεij = εii εjj 2 ∂xl ∂xk kl

Pijkl are the elements of the photo-elastic tensor and can be considered as functions of x3 . The coupling of incident light to phonons gives rise to a polarization in the PnC, which creates a scattered field. In this work we are interested in pure longitudinal phonons along the axis x3 of a multilayer structure composed of cubic materials with (001) interfaces. In this case, we can assume that all the electromagnetic fields (incident, scattered, and polarization waves) are polarized parallel to the x1 axis and propagates along x3 . Then each medium α in the structure can be characterized by an elastic constant Cα (which means C11 ), the mass density ρα , the dielectric constant εα = n2α (where n is the index of refraction in the medium α) and one photo-elastic constant pα = −εα2 Pα1133 .

188 Phononics

Eq. (92) becomes for each medium α: δεα = pα

∂uα (x3 ) . ∂x3

(93)

The calculation of the emitted electric field Es (x3 , t) when the PnC is submitted to an incident electromagnetic field can be done following the Green’s function method [147]:  ∂uα (x3 ) 0  ωi2  pα G(x3 , x3 ) Ei (x3 , t) dx3 . (94) Es (x3 , t) = − 2 ∂x3 ε0 c α Here ωi is the angular frequency of the incident wave, ε0 and c are the permittivity and the speed of light in vacuum, respectively, Ei0 (x3 , t) is the electric field in the PnC, and G(x3 , x3 ) is the Green’s function associated with the propagation of an electromagnetic filed along x3 in the vacuum/PnC system in the absence of acoustic deformation. In the particular case where the dielectric modulation of the multilayer structure can be neglected (which happens when the layers are thin as compared to the optical wavelengths), the system can be considered as a homogeneous  medium from the optical point of view, then Ei0 (x3 ) = Ei0 eiki x3 is a plane wave  (instead of being a Bloch wave) and G(x3 , x3 ) ∝ eiks (x3 −x3 ) . ki and ks are the wave vectors of the incident and scattered waves. Therefore, Eq. (94) becomes   ω2   ∂uα (x3 ) pα eiqx3 dx3 , (95) Es (x3 , t) ∝ − i 2  ∂x3 ε0 c α where q = ki − ks = 2ki = 4πneff /λ,

(96)

is the wave vector of the phonon in the backscattering geometry and neff is the effective index of refraction of the whole system. A great deal of work has been devoted to light scattering from acoustic phonons in multilayered structures, since the first observation of folded longitudinal-acoustic modes by Colvard et al. [2]. Several experimental studies have been reported on GaAs-Gax Al1−x As and Si-Gex Si1−x systems. As mentioned earlier, in an ideal PnC consisting of an infinite sequence of building blocks AB made of different semiconductors A and B, the branches of the acoustic-phonon dispersion are back-folded inside the BZ due to the periodicity of the system. In the Raman process involving longitudinal acoustic phonons in backscattering geometry along the growth direction (x3 ), crystal momentum is conserved, that is, the wave vector transfer to the phonon corresponds to the sum of the magnitudes of the wave vectors of incident and scattered photons ki and ks , respectively. Characteristic doublets are observed in the spectrum, which reflects the folding of the PnC dispersion curves in the first BZ. Crystal momentum conservation at the doublet frequencies implies that all partial waves

Chapter | 4 One-Dimensional Phononic Crystals 189

are coherently scattered, that is, all layers of the PnC contribute constructively to the total intensity. Therefore, the doublets are very sharp and pronounced. In a real PnC, the coherence of the scattering contributions from the individual layers is partly removed due to interface roughness and layer thickness fluctuations, finite-size effect of the PnC as well as the effect of different defects that may be introduced inside these systems such as surfaces, interfaces, and defect layers. In view of the relatively small thicknesses of the layers in the PnC, the acoustic phonon Raman scattering can be obtained from Eq. (95) as: 2       iqx3 ∂uα (x3 )   pα e dx3  . (97) I(ω) ∝     α ∂x3 Here we assume that the light propagates like in a homogeneous medium and u(x3 ) is the normalized lattice displacement.

3.4.2 Semiconductor PnCs Fig. 23A shows the simulated results of Raman intensity obtained by Zhang et al. [77, 78] for a PnC composed of 15 periods of 20.5 nm of Si and 4.9 nm of Si0.52 Ge0.48 epitaxially grown on a [100] oriented Si substrate. The different curves in Fig. 23A correspond to different laser wavelengths (i.e., different phonon wave vectors). These results are in very good agreement with the experimental data of Zhang et al. [77, 78]. By reporting the frequency positions of the doublets within the band gap structure (Fig. 23B), a good agreement is obtained between the dispersion curves of the infinite PnC (full curves) and the frequencies obtained from the doublets (dots). These results enable one to deduce a precise measurement of the width of the first three gaps. Besides the description of the band gap structure, the Raman spectra show also small features (not observable at the scale of the theoretical spectra in Fig. 23A), which are interpreted as confined modes (discrete modes) due to the finite-size structure of the PnC. In Fig. 23C we have calculated the intensity variation of different phonon branches labeled 1–6 in Fig. 23B within the reduced BZ. The intensities show drastic variations, especially for q close to the Brillouin edges. The Brillouin line (branch labeled 1) is the most intense mode for a large range of q values except near the zone boundary. These behaviors are similar to the theoretical predictions obtained by He et al. [139] on GaAs-Gax Al1−x As PnCs. Besides the doublets associated with folded longitudinal acoustic phonons, Lemos et al. [65] have shown the existence of additional modes between the doublets which are induced by a cap layer deposited at the surface of the PnC. These modes fall inside the gap located at ∼15 cm−1 . The PnC is composed of 20 periods of 21.5 nm of Si and 5.0 nm of Ge0.44 Si0.56 terminated by a cap layer made of Ge0.44 Si0.56 with a thickness dc = 1.5 nm. The top and bottom curves in Fig. 24 are drawn for two different wavelengths 514.5 and 496.5 nm, respectively. These results reproduce correctly those of Lemos et al. [65], except

190 Phononics

Raman intensity (a.u.)

5 541.5 nm

4

496.5 nm

3

488 nm

2

476 nm 1 457.9 nm 0 0

(A)

10

15

20

30

35

40

Frequency shift (cm ) 600 6

500

20

Intensity (a.u.)

5

25

4 3

15

1 2 3 4 5 6

400 300 200

10 2

5

100

1

0 0.0

(B)

25

–1

30

Frequency shift (cm–1)

5

0.5 qD/p

0 0.0

1.0

(C)

0.2

0.4 0.6 qD/p

0.8

1.0

Fig. 23 (A) Simulated acoustic phonon Raman spectra in the Si/Si0.52 Ge0.48 PnC, excited with the five studied laser lines. (B) Calculated dispersion curves of the infinite PnC (solid lines). The filled circles are obtained from the doublets of the theoretical spectra sketched in (A) using our theoretical model. (C) Variation of the intensities of the six first folded branches as functions of the diffusion wave vector qD/π . (The parameters in (A) are taken from P.X. Zhang, D.J. Lockwood, H.J. Labbe, J.M. Baribeau, Phys. Rev. B 46 (1992) 9881.)

for the gap-mode which greatly surpassed the observed value. To confirm that the gap mode is induced by the cap layer, we have calculated the LDOS as a function of the space position for the mode lying at ∼15 cm−1 . The spatial localization of this mode (not shown here) clearly shows that it is localized in the cap layer and decreases inside the PnC. Another example we have considered concerns a finite size mirror plane PnC with building blocks SL = GaAs-AlAs and LS = AlAs-GaAs arranged to form

Chapter | 4 One-Dimensional Phononic Crystals 191

Raman intensity (a.u.)

8

Si/Ge0.44Si0.56 514.5 nm

6

4 496.5 nm 2

0 5

10

15

20

25

Frequency shift (cm–1) Fig. 24 Simulated Raman spectra of a PnC composed of 20 periods of 21.5 nm of Si and 5.0 nm of Ge0.44 Si0.56 terminated by a cap layer made of Ge0.44 Si0.56 with a thickness dc = 1.5 nm. The top and bottom curves are drawn for two different wavelengths 514.5 and 496.5 nm, respectively. (The parameters used in the calculation are taken from V. Lemos, O. Pilla, M. Montagna, C.F. de Souza, Superlattices Microstruct. 17 (1995) 51.)

layer sequences (SL)m/2 /(LS)m/2 with different numbers of building blocks m. This leads to mirror-plane PnCs with different numbers of building blocks m. (SL)m/2 /(LS)m/2 with m = 10, 20, and 40 as well as a reference specimen (SL)20 have been investigated by Giehler et al. [75]. Fig. 25A shows the simulated Raman spectra of three mirror plane PnCs and the (SL)20 reference sample (bottom curve) in the frequency range of the first longitudinal acoustic phonon doublet. For the reference sample, the crystal momentum transfer q is about 0.3π/D, where D = 86.8 nm is the period of the PnC. The thickness d1 and d2 of the GaAs and AlAs layers are chosen such that d1 = 36.4 nm and d2 = 50.4 nm, respectively. One therefore observes a folded-phonon doublet close to the center of the PnC BZ whose components are denoted by ω±1 . In contrast to the periodic structure, the spectra of the mirror-plane PnC display pronounced splittings of the doublet lines. These splittings increase with decreasing total length of the SL and LS building blocks. These effects are interpreted as being due to the interference of the scattering contributions from the different quantum wells in the sample, and reflect a phase shift introduced by the mirror-plane symmetry. Furthermore additional small peaks appear in Fig. 25A with characteristic separations. These modes have been interpreted as confined modes induced by the finite size effects. In order to describe more precisely the experimental curves, we have introduced the absorption effect in the system by adding a small imaginary part to the effective index of refraction (see also Giehler et al. [148]). These results are reported in Fig. 25B and show very good agreement with the experimental results of Giehler et al. [75]. One can notice that light

192 Phononics

Raman intensity (a.u.)

(SL)20(LS)20

(SL)10(LS)10 (SL)5(LS)5

(SL)20 10

(A)

15

20

25 –1

Frequency shift (cm )

10

30

(B)

15

20

25

30

–1

Frequency shift (cm )

Fig. 25 (A) Simulated Raman spectra of folded longitudinal acoustic phonons of mirror-plane PnCs (SL)m/2 /(LS)m/2 with m = 10, 20, and 40 compared with that of the ideal finite size (SL)20 . The building blocks are SL = GaAs-AlAs and LS = AlAs-GaAs, respectively. The curves are sketched without taking into account the optical absorption in the layers. Notice the small peaks between the principal peaks (doublets). (B) The same as in (A) but in presence of absorption. (The parameters used in the calculation are taken from M. Giehler, T. Ruf, M. Cardona, K. Ploog, Phys. Rev. B 55 (1997) 7124.)

absorption smears out the split lines considerably because the field scattered from the (LS)m/2 unit incompletely cancels the field from the (SL)m/2 unit. A similar study has been developed by Pascual Winter et al. [76] on an acoustical phonon cavity that consists of phonon mirrors made of 10 periods (λ/4, 3λ/4) stacks of two materials, namely (Ga0.47 In0.53 As, Al0.48 In0.52 As) enclosing a 7λ/2 cavity made of InP. The whole structure is grown on InP substrate. λ refers to the sound wavelength in the specific material making the mirror or the cavity layer. The period of the PnC is estimated by X-ray diffraction to be 4.52 nm. The Raman spectra are shown to be highly sensitive to the details of the structure, allowing a proper characterization of the device. The interest of such a structure consists in reflecting back and forth the longitudinal acoustic phonon in the cavity before escaping by tunneling, thus enhancing the an-harmonic coupling with longitudinal optic phonons. A measure of this enhancement is the longitudinal acoustic phonon lifetime in the device, that for typical phonon mirror reflectivities can be augmented by a factor larger than 100. An example of the simulated Raman spectra in the acoustical domain is given in Fig. 26 for five laser wavelengths and by considering the elastic, photoelastic, and refractive index parameters of the materials as described in Pascual Winter et al. [76]. The agreement between these results and those of Pascual

Raman intensity (a.u.)

Chapter | 4 One-Dimensional Phononic Crystals 193

458

488 514 568 647 24

28

32

36

40

Frequency shift (cm–1) Fig. 26 Simulated acoustic phonon Raman spectra for five studied laser lines. The spectra are realized on 10 periods Ga0.47 In0.53 As-Al0.48 In0.52 As PnC enclosing a cavity made of InP. The intensities have been normalized. Solid (dashed) lines are guides to the eye indicating the dispersion of the main (secondary) doublets. (The parameters used in the calculation are taken from M.F. Pascual Winter, A. Fainstein, M. Trigo, T. Eckhause, R. Merlin, A. Cho, J. Chen. Phys. Rev. B 71 (2005).)

Winter et al. [76] is quite good. Several features can be highlighted from these spectra: (i) the spectra are dominated by an intense acoustical phonon doublet which disperses with laser wavelength, (ii) in addition to this main doublet, a small intensity secondary doublet shifted to smaller energies and also displaying dispersion is observed, and (iii) some weaker oscillations can be discerned toward smaller energies (not shown in the figure). These latter oscillations arise from the finite size effects as shown in Figs. 23 and 25. Here also, it was shown that the observed splitting of the longitudinal acoustic doublets can be traced down to interference effects on the Raman spectra coming from the two mirrors PnCs and the phase-shift introduced between them by the InP spacer. The case of a cavity layer embedded at different positions within a finite PnC has been examined by Schwartz et al. [71]. The structure consists on nine periods of 5/2.5 nm AlSb/GaSb layers grown on (001) GaSb in which an additional 50 Å layer of AlSb was embedded either at the substrate/PnC interface, in the center of the structure, or near the PnC/air interface (see the insets in Fig. 27). The exact placement of the defect layer can also be viewed as a 10-period PnC with a missing GaSb layer. A 10 periodic PnC without a defect was also examined for comparison. Raman scattering using 514.5 nm excitation was utilized to study the zone-folded acoustic phonon spectra. Data values appropriate for GaSb and AlSb at 514.5 nm are given in Schwartz et al. [149]. Fig. 27 shows the simulated Raman spectra of the zone-folded acoustic phonons for the periodic PnC and the three samples with a single defect layer. Weaker structures which show up as shoulders or bumps between the periodic

194 Phononics

Periodic Raman intensity (a.u.)

S DL

S

4

S

0

20

9

DL 5

8

40

DL1

60 –1

Frequency shift (cm ) Fig. 27 Simulated Raman spectra of folded longitudinal acoustic phonons in samples with and without (periodic) a single-embedded defect layer. The location of the defect layer is shown in the schematic inset in which S and DL designate the substrate and defect layer, and the integers represent the number of regular 50/25 Å AlSb/GaSb periods. (The parameters used in the calculation are taken from W.A. Sunder, Appl. Phys. Lett. 58 (1991) 971.)

zone-folded doublets are seen in the samples with the defect layer moved away from the substrate interface. The theoretical results reproduce well the four spectra observed experimentally by Schwartz et al. [71]. One can notice that when the defect layer is placed near to the substrate, the doublets resemble those of the periodic structure, whereas when the defect is inserted in the middle of the structure or near to the free surface, then each doublet splits into two peaks with different shapes depending on the position of the defect within the finite PnC. This behavior is mainly due to interference phenomenon between different layers of the system. As mentioned in Sunder [71], the lower resolution (2 cm−1 ) and inherent noise in the experimental data obscure most of the less intense features. Also, let us quote the experimental work of Rozas et al. [150] where ultrahigh-resolution Raman study of the lifetime of 1 THz acoustic phonons confined in nanocavities has been performed. The structure consists on an asymmetric acoustic mirror made of GaAs (3.5 nm)/AlAs (1.5 nm) in presence of a GaAs cavity layer (7 nm). The GaAs spacer is broadened from one side by a finite

Chapter | 4 One-Dimensional Phononic Crystals 195

PnC (20 periods), whereas the other side is constituted of a finite PnC made of N = 4, 8, 12, or 16 periods. The whole structure is deposited on an optical cavity made of AlAs/Al0.33 Ga0.67 As Bragg reflector. The presence of the optical microcavity has a twofold purpose: increasing the Raman signal and enabling the detection in backscattering geometry of the cavity peak, otherwise accessible only in a forward-scattering configuration [72, 73]. This study has shown that λ where λ is the acoustic the quality factor Q of the cavity (defined as Q = Δλ wavelength and Δλ is the full width at half-maximum of the cavity mode) can be tailored by designing and a Q factor as high as 1000 can be achieved by MBE technology in the 0.1–1 MHz range.

3.4.3 Hybrid PnCs The above-cited works on semiconductors have stimulated further studies in search of phononic bandgaps establishing the concepts of unidirectional bandgap in the case of 1D hypersonic PnCs [12–18]. The use of polymers expands the performance possibilities of nanocomposites, in opposition to rigid crystals, due to their elastic versatility, an inexpensive fabrication, and their general biocompatibility [151, 152]. The combination of polymers with inorganic materials opens new routes for the generation of quality and functional hybrid phononic structures. Hybrid (organic-inorganic) PnCs can be prepared over large areas and are excellent coatings for multiple applications in photonics such as selective mirrors [153, 154]. Through the phononic behavior of PnCs, knowledge on the elastic properties, necessary for an optimal performance, can be obtained [11]. Silica and poly(methylmethacrylate) (PMMA-SiO2 ) PnCs, studied by nondestructive Brillouin spectroscopy (BLS), have advanced our understanding of sound propagation that opens new pathways to tunable hybrid periodic films. For example, the architecture of finite PnCs with controlled defects allows the engineering of the bandgap region through the interaction of surface and cavity vibration modes [14, 15]. In the case of spin-coated SiO2 PMMA PnCs, the bandgap width of the hybrid PnC has been limited by the porosity of the inorganic layer (to Δf /fg 30%, here fg is the mid-frequency of the gap). This limitation on the elastic impedance contrast has been overcome by the fabrication of polycarbonate (PC) and tungsten (W) PnCs by pulsed laser deposition; the PC/W interface roughness is significantly low and the structure exhibits a broad (Δf /fg 70%) bandgap. The disadvantages of pulsed-laser deposited PnCs over spin-coated PnCs relate to the complex sample preparation, which involves controlled vacuum and temperature conditions. The replacement of the porous SiO2 phase by a heavier phase, such as barium titanate (BaTiO3 ), in spin-coated PnCs is expected to have an impact on the bandgap width. In what follows, we shall review some recent results of BLS on PMMA-SiO2 [14, 15] and PMMA-TiO2 [16] multilayers. In Schwartz et al. [14], stacks of PMMA and silica (p-SiO2 ) were assembled by alternating spin-coating from the respective stock solutions on a glass substrate starting with the PMMA layer. Two multilayer stacks each

196 Phononics

consisting of 20 alternating PMMA and SiO2 layers with different thicknesses dPMMA and dSiO2 and hence periodicity a(= dPMMA + dSiO2 ) were fabricated. A combination of scanning electron microscopy (SEM) and confocal microscopy was used to determine (i) the relative distribution of materials and (ii) the absolute thickness of the multilayer structure. Fig. 28A and B shows, respectively, the SEM pictures of stack A with a = 117 nm (dSiO2 = 79 ± 5 nm and dPMMA = 38 ± 5 nm) and stack B with a = 100 nm (dSiO2 = 55 ± 5 nm and dPMMA = 45 ± 5 nm). Fig. 28C shows BLS spectra for stacks A and B, both for four q⊥ -values near the edge of the first BZ appearing at G/2 = π/a, respectively, for stacks A and B. The double peak structure of the BLS spectra corresponds to the bands n = 1 (at low) and n = 2 (at high frequency) between which the Bragg gap occurs. The different spacing of the two films is manifested in the BLS spectral shape (Fig. 28C); stack A (periodicity a = 117 nm) shows only the n = 2 branch while stack B (a = 100 nm) displays both branches at the high q⊥ value (Fig. 28C). The BLS spectra correspond to different q⊥ a values and their shape sensitively depends on the proximity to the edge of the BZ at q⊥ a = π . Both stacks were scanned through the accessible q-range to obtain the dispersion relation f (q⊥ ) for on-axis phonon propagation (yellow circles/red diamonds on blue ground in Fig. 28D) (Color online). The BLS spectra were well represented by a double Lorentzian convoluted with the instrumental function (Fig. 28C). The line intensities and the peak frequencies are shown in Fig. 28D and E (blue-shaded area), respectively. Fig. 28E also includes the phonon frequencies for in-plane propagation (reddish area) in the two hybrid stacks. The opening of a hypersonic phononic stop band along the periodicity direction (at q⊥ a = π) is demonstrated with the two stacks covering different regimes in the BZ; stack A falls in the second BZ. In spite of the small differences in the elastic parameters (Table 4), the intensity ratio I(2)/I(1) of the two bands superimpose on a common curve when plotted versus q⊥ a in Fig. 28D, thereby justifying the larger values observed for stack A (Fig. 28C). Fig. 28C denotes the fit of I(ω, q⊥ ) (Eq. 97) to the BLS spectra using the values of the parameters listed in Table 4. The computed I(2)/I(1) ratio (solid symbols in Fig. 28D) and the frequencies of modes (1) and (2) (blue and green symbols on the right shaded areas in Fig. 28E) in the dispersion relations of the two stacks are in good accordance with the corresponding experimental values. The on-axis dispersion diagram in the present 1D PnCs was found to be well represented (solid lines in Fig. 28E) using Eq. (8), namely     ω d⊥PMMA ω d⊥SiO2 cos cos(ka) = cos c⊥SiO2 c⊥PMMA       1 ZSiO2 Z⊥PMMA ω d⊥SiO2 ω d⊥PMMA − + sin sin , 2 ZPMMA ZSiO2 c⊥SiO2 c⊥PMMA (98) with Z(= ρ cL ) being the acoustic impedance of two materials.

Chapter | 4 One-Dimensional Phononic Crystals 197 Stack B

Stack A

(A)

(B)

stack A

stack B

q = 0.0349 nm−1

5 4

Experiment A

B

3

Theory A

B

q = 0.0345 nm−1

q = 0.0313 nm−1

q = 0.0334 nm−1

q = 0.0305 nm−1

q = 0.0320 nm−1

q = 0.0289 nm−1

q = 0.0305 nm−1

l(1)/I(2)

2 1 0 –1 0.9 –18 –16 –14 –12 –10

8

10 12 14 16 18 –16 –14 –12 –10

8

10 12 14 16 18

(D)

Frequency (f / GHz)

(C)

1.0

1.1

1.2

1.3

q^a / π

Frequency (f / GHz)

20

20

18

stack A

stack B

18

16

a = 117 nm

a = 100 nm Experiment

16

14

Experiment Theory

12

10

10

8

8

6

6

4

qII

2

(E)

14

Theory

12

0 0.00

SiO2 PMMA SiO2

q^

PMMA SiO2 PMMA

0.01

0.02

SiO2 PMMA SiO2

4

PMMA SiO2 PMMA

2 0

0.03

0.04 q / nm–1

0.01

0.02

0.03

0.04

Fig. 28 (A, B) Cross-sectional micrograph of two Bragg stacks A and B. (C) Experimental BLS spectra of the stacks A and B superimposed with the theoretical spectra (solid lines) at different phonon wave vectors q⊥ normal to the layers. (D) The ratio I(2)/I(1) of the intensities of the highand low-frequency bands in (A). The small-size symbols indicate the values of I(2)/I(1) obtained from the theoretical spectra in (C). (E) Dispersion relation of stacks A and B (experimental data given in yellow circles/red diamonds) (Color online) for in-plane (reddish back) and out-of-plane (light blue back) propagation. The solid lines denote the theoretical dispersion curves for sound propagation along the periodicity axis (scheme for q⊥ ) and hollow symbols indicate the modes (1) and (2) frequencies obtained from the theoretical spectra in (C). The linear dispersion for phonon propagation parallel to the stacks (scheme for q|| ) is denoted as a dotted line along with the experimental data in the same direction.

198 Phononics

TABLE 4 Values of the Physical Quantities Used in the Calculation

Sample A

PMMA

p-SiO2

Substrate

cL (m s−1 )

2800a

3100

5600

cT (m s−1 )

1400a

1800

ρ (kg m−3 )

1190a

1420

d (nm)

38

79

pPMMA /pSiO2 Sample B

2

cL (m s−1 )

2800a

3030

cT (m s−1 )

1400a

1800

ρ (kg m−3 )

1190a

1500

d (nm)

45

55

pPMMA /pSiO2

2200

5600

2200

2

a These parameters were fixed to the values of bulk PMMA film [12, 14, 15].

The effective medium longitudinal sound velocity cL,⊥ amounts to 2970 ms−1 in stack A and 2890 ms−1 in stack B (slope of Eq. 98 at the low-q limit). In fact, these values can be also computed by Wood’s law [155], which is obtained by Taylor expansion of Eq. (98) around ω = 0: 1−φ φ 1 + 2 , = 2 M(A/B) ρc⊥,PMMA ρc⊥,SiO2

(99)

whereas M(= C11 = ρ c2L,⊥ ) is the bulk modulus of the whole system with an effective density of ρeff = φρPMMA + (1 − φ)ρSiO2 , and volume fraction φ = dPMMA /a. The frequencies of band (1) below the gap (reddish area in Fig. 28E) deviates from the corresponding frequencies (experimental symbols) for inplane propagation for both stacks. In the direction parallel to the layers, the effective medium sound velocity cL,|| is a different average of the elastic properties in the individual layers as it is affected by sagittal modes. In fact, the computed in-plane acoustic phonon frequency [48] (dots in the reddish area of Fig. 28E) agrees well with the experimental frequencies along the same direction. The slopes of these dotted lines, that is, the sound velocities for inplane propagation cL,|| (A) = 3020 m s−1 and cL,|| (B) = 2990 m s−1 are slightly higher than the corresponding cL,⊥ , respectively. In order to provide a measure for the width of the band gap we first look at dSiO PMMA two exemplary situations. If we suppose that dcPMMA = 2 cSiO2 (almost the case 2 in stack A). Then the frequencies of the upper/lower limit of the gap are given exactly by:

Chapter | 4 One-Dimensional Phononic Crystals 199

f1,2

c⊥PMMA = cos−1 2πd⊥PMMA



Z⊥PMMA,SiO2 Z⊥PMMA + Z⊥SiO2

 ,

(100)

and we obtain a gap width of Δf = f2 −f1 ∼ 2 GHz, which is in good agreement PMMA = with the experimental findings in stack A. Now, if we suppose that dcPMMA dSiO2 cSiO2 (almost the case in stack B), the center of the first gap lies f0 dSiO2 cSiO2 = 1/4, that is, the system acts as an “quarter wave stack”

at

f0 dPMMA cPMMA

=

exhibiting the

widest possible band gap in 1D crystals [156]   2f0 −1 Z⊥PMMA − Z⊥SiO2 f1,2 = f0 ± sin , π Z⊥PMMA + Z⊥SiO2

(101)

with f0 being the frequency of the center of the gap at kBZ = π/a. The larger gap width in stack B is Δf ∼ 3 GHz according to Eq. (101) and in good agreement with the experimentally observed gap. Aside of these two special cases a small contrast in elastic impedance ΔZ/Z (which is the case here) the width of the gap can be approximated by a general expression [27]: ΔZ π dPMMA cSiO2 (102) dPMMA cSiO2 + dSiO2 cPMMA Z √ − ZSiO2 |, Z = Z1 Z2 and f0 , allocated by the sound

Δf ∼ = 4f0 sin where ΔZ = |ZPMMA

dSiO

PMMA + cSiO2 . f0 defines velocity c0 (via f0 = c0 /2a), is obtained through 21f0 = dcPMMA 2 the middle of the gap only in the case of a “quarter wave stack,” where the gap width is maximized. The fundamental quantity of 1D PnCs, the width of their primary band gap, depends on many parameters, such as thickness and sound velocity of constituent materials and not just from the impedance mismatch ΔZ/Z, although it plays a central role. ΔZ/Z = 0.37 for the present PMMA and p-SiO2 layers. For small elastic contrast, Eqs. (100)–(102) yield two simple equations for stop band width of samples A (dPMMA /dSiO2 ≈ 0.5) and B (dPMMA /dSiO2 ≈ 1), respectively: √ 3 cPMMA ΔZ ∼ ≈ 1.8 GHz (103) Δf = 6 πdPMMA Z and 1 cPMMA ΔZ ≈ 3 GHz. (104) Δf ∼ = 2 πdPMMA Z The observed good agreement between theory and experiment leads to important conclusions: (i) the phononic dispersion is not simply scalable with the contrast of elastic impedance ΔZ/Z. In contrast to photonics, density and sound velocity of both layers enter explicitly in Eq. (98); (ii) the physical quantities, density and longitudinal elastic modulus of the porous SiO2 layer adopt lower

200 Phononics 25

q =0.0313 nm

Frequency (f/ GHz)

20

15

10

2

2

1

1

5

0 0.00 0.01 0.02 0.03 0.04

(A)

q/nm–1

DOS/a.u.

(B)

Intensity/a.u. Intensity/a.u.

(C)

(D)

Fig. 29 Dispersion curve at normal incidence (A), DOS (B), and two modeled spectra of different resolutions (C, D) at q = 0.0313 nm−1 (vertical-dashed line in a). The linewidth (FWHM) of the peaks in the DOS (b) is ∼0.25 GHz and the theoretical spectrum in (C) is a triplet spectral structure for each of the two bands (1) and (2) with maximum frequencies arising at f fixed by q⊥ in the dispersion curve (A) (red guides, black horizontal lines in the print version). The triplet structure in (C) is smeared out to the experimental doublet (D) due to the reduced resolution (Γ = 0.53 GHz).

values than in silica glass. They are slightly different in the two stacks; (iii) the average thicknesses of the individual layers are uniquely obtained, and (iv) effective medium elastic parameters (ρ, cL,⊥ ) are also obtained from the frequency of mode (1) which becomes acoustic only at low q⊥ ’s. Fig. 29 contrasts the theoretical prediction of the modes near the edge of the BZ of an ideal 1D periodic structure with the experimental spectral doublet of the BLS spectrum for stack A at a constant q⊥ . The left panel shows the theoretical dispersion relation and Fig. 29B shows the calculated DOS with 10 sharp peaks for each branch in the BZ equivalent to the number of periods in the stack. The peak separation is about 1.2 GHz in the DOS diagram, while their inherent broadening (Γ ∗ ∼ 0.25 GHz, even smaller near the gap) is due to interaction of the superlattice discrete modes with the substrate continuum as these modes are propagating within the substrate. The modeled spectrum shown in Fig. 29C for q⊥ = 0.0313 nm−1 (vertical dashed line in Fig. 29A), exhibits a triplet spectral structure with the maximum at the fixed q⊥ . It had to be convoluted with the instrumental function (Gaussian with Γ ∼ 2Γ ∗ ∼ 0.53 GHz) to match the doublet shape of the experimental spectrum in Fig. 29D associated with dispersion diagram of the infinite structure (Eq. 98 and

Chapter | 4 One-Dimensional Phononic Crystals 201

Fig. 28E). The effect of structural disorder exemplified by incoherent spacing was also examined theoretically by varying randomly the thickness of PMMA (34–44 nm) and SiO2 (73–83 nm) layers. It is remarkable that the spectral doublet remains robust and coincides with the experimental doublet of the BLS spectrum [14]. In Schneider et al. [15], we have considered the same PnC (i.e., PMMASiO2 ), but in presence of defects such a surface and a cavity layer. We have demonstrated a first unequivocal observation of surface modes and their interaction with cavity modes varying the material and the thickness of the top layer, as well as the thickness and position of the cavity layer. As mentioned in Section 2.1.6, in order to excite surface modes, the PnC should be terminated by the layer with lower impedance. Fig. 30 shows the band diagram as well as the surface modes for a PnC composed of eight bilayers (8 BL) and PMMA surface layer. A new mode with frequency around 14 GHz falls inside the band gap region of the infinite PnC indicated by solid lines in Fig. 30A. The small thickness of PnC enables mode resolution in the BLS spectrum (Fig. 30D) as the separation between the three modes (DOS in Fig. 30B) exceeds the instrumental width (∼0.5 GHz). As mentioned in Fig. 28D, modes along the longitudinal acoustic (LA) branch are more intense than along the folded FLA−1 [139] as suggested by the contour plot in Fig. 30A. Hence, suitable q values have been selected for a strong detection (white circles in Fig. 30A). The optimal design of the present PnC was based on theoretical simulations of the BLS spectra and the band diagram: thinner PnCs, for example, with 5 BL would reduce the BLS signal and the sampling quality of the dispersion relation (Fig. 30G); thicker PnCs (20 BL) would decrease the separation of the individual modes rendering their experimental resolution hard (Fig. 30H). These finite size effect calculations for PnCs have been mentioned in Section 3.2. Some of these modes are observed as sets of small satellite lines in high-resolution Raman spectra in semiconductors (see Fig. 25). However, this is the first experimental documentation utilizing the advantages of hybrid PnCs and coherent BLS. The total DOS (Fig. 30B) reveals three main contributions, the upper (e2 ) and lower (e1 ) edge modes, and the surface mode (s), whose nature can be identified by their displacement fields (Fig. 30E). This mode assignment is further supported by the pattern of the LDOS at the surface (Fig. 30C). It should be noted that the peak in the DOS associated with this s mode becomes narrower as the number of BL increases; the coupling strength of the s mode with the substrate modes weakens with surface-substrate separation. Hence, after convolution of this s peak with the instrumental function, its documentation in the experimental BLS spectrum is severely affected. In fact, the contour plots for the notional PnCs in Fig. 30G (5 BL) and Fig. 30H (20 BL) indicated by the s arrows demonstrates the suppression of the s mode with the PnC thickness. The LDOS at the surface (Fig. 30C) underlines the importance of the s mode, while the e modes have their maximum displacement centered in the middle of the PnC (Fig. 30E). Moreover, the envelope function of the displacement field

LA

(B)

s

FLA–1

0.02

15 f/GHz

20

0.03 0.04 q/nm–1

0.05

0.06

(D)

q = 0.0327 nm–1

q = 0.0304 nm–1

10

e1

10

(C)

(G)

15 f/GHz

20

mode e2

15 f/GHz

20

(E) 0

400 600 z/nm

800

20 BL+1P

s

q

200

f

f 500 nm

e2

mode e1

5 BL+1P

(F)

s

mode s

experiment theory

LA

(A)

10

LDOS at surface

q = 0.0342 nm–1

10

5 0.01

e2

e1

Displacement field

15

Intensity /a.u. 0 1 2 3 4 5

Intensity/a.u.

f/GHz

20

s

Local DOS

P

integrated over z

Same scale as in A

s

q

(H)

Fig. 30 (A) Dispersion relation for a PnC with eight bilayers (BL) and a PMMA surface layer. The theoretical Brillouin intensity is given as a color scale and the dispersion of an infinite PnC as solid lines. Peak positions of experimental (theoretical) spectra are given as white (black) circles (Color online). (B) Total DOS and LDOS at the surface (C) with indicated edge e1,2 and surface modes s. (D) Experimental (theoretical) Brillouin spectra in black (red solid line) (Color online) at three q values. (E) Displacement fields of the modes indicated in (C). (F) Cross-sectional electron micrograph of the PnC. (G, H) Mode separation and strength of the surface mode s in PnCs with 5 BL (G) and 20 BL (H); mode s (indicated by arrows) is hardly discernible in (H).

202 Phononics

8 BL+1

Total DOS

25

Chapter | 4 One-Dimensional Phononic Crystals 203

22 20 18

ƒ/GHz

C

SL5 SL3

16 S

SL4 SL1

S

14 12

C

10 8 20

Displacement /a.u.

(A)

(B)

40

60

80 100 dc /nm

120

z

mode c

q

mode s

140

Surface layer Cavity SiO2 PMMA

0

200 400 600 800 z/nm

(C)

Fig. 31 (A) Mode dispersion around the Bragg gap as a function of cavity thickness. As the central PMMA layer is enlarged the cavity mode shifts inside the band gap and interacts with a surface mode for large cavity stacks. Experimental data are given by circles. The shaded region denotes the gap region of the PnC. (B) Displacement field for modes c and s. (C) Schematic indicating the position of the surface and cavity layers.

for mode e2 almost vanishes at the surface. The latter can therefore be qualified as a surface avoiding mode (SAM) recently reported for semiconductor PnCs [157, 158]. Additional evidence of the surface mode is its dependence on the top layer thickness (see Fig. 3). As mentioned in Section 2.1.6, the frequency of mode s should decrease with ds and its frequency could then be easily tuned inside the gap. Insertion of cavity layers in the interior of the PnCs as schematically shown in Fig. 31C represents a second class of defects distinctly manifested in the band diagram. In addition, a cavity mode (c) appears which can be affected by the interactions with mode s. In the gap region, Fig. 31A shows the variation of the frequency of modes s and c with the thickness of a cavity located in the middle of

204 Phononics

an eight BL-PnC. While the frequency of mode s remains almost independent of the cavity thickness dc , the frequency of mode c decreases monotonically with dc . When the two frequencies of modes c and s approach each other, the interaction between their evanescent fields leads to an anticrossing of the dispersion curves (Fig. 31A). For this observation a relatively small spatial separation between surface and cavity layers is required. To address this predicted behavior experimentally, the thickness of the cavity layer located in the middle of the hybrid PnCs was stepwise increased (from 42 to 140 nm) while keeping the thickness of the surface layer roughly constant (ds ∼ 50 nm). At large dc , mode c anticrosses mode s, but maintains the characteristic features of a cavity mode as proven by the displacement plot (Fig. 31B). These results are in accordance with the theoretical work on the interaction of surface and cavity modes (see Fig. 21). Finally, we have studied experimentally and theoretically the directiondependent elastic and electromagnetic wave propagation in a supported film of hybrid PMMA (poly[methylmethacrylate])-TiO2 PnC. In the direction normal to the layers, this 1D periodic structure opens propagation band gaps for both hypersonic (GHz) phonons and near-UV photons. The high mismatch of elastic and optical impedance results in a large dual phoxonic band gap [16], which renders hybrid (soft-hard) periodic materials, promising simple platform for opto-acoustic interaction. To achieve large phononic and photonic band gaps, we fabricated a PMMATiO2 PnC by spin-coating subsequent layers on a glass substrate. We have chosen a spatial periodicity of d 100 nm and probing phonon wave vector q, with qd 1. The periodicity was selected to achieve dual band gap, a near-UV photonic and a hypersonic (GHz) phononic. Also, we probed phonon propagation normal to the periodicity direction using BLS, where no phononic gap but different effective medium behavior is expected. At the nanoscale, confinement and interface effects can render different effective material elastic properties than in the bulk. Experimental results along with theoretical band structure analysis allow a complete optical and nanomechanical characterization of the multilayer structure, thereby settling the necessary fundamental knowledge for applications. These results are summarized in Fig. 32 where the phononic band structure shows a large Bragg-type gap centered at 15 GHz, with a bandwidth of

5 GHz (Fig. 32B) [16]. The elastic, optic, and photoelastic parameters used in the calculation are listed in Table 5. Further characterization of the 1D photonic crystal (PC) was done via optical spectroscopy. When a monochromatic light is launched onto a periodic layered medium, with frequency in the range of the forbidden band gap, such a wave is evanescent and does not propagate through the medium. The energy is reflected, and the medium acts as a Bragg reflector. The peak reflectance for normal incidence occurs at the center of the forbidden band, given by λ = 2neff d [156]. In the present PC with an effective refractive index neff = 1.71 and lattice constant d = 97 nm, the peak should appear around 332 nm.

Chapter | 4 One-Dimensional Phononic Crystals 205 25

15

Phononic band gap

10

80 60 40 20

(A)

Wavenumber q (nm )

TiO2

0

5 0.01 0.02 0.03 0.04 0.05 0.06 –1

Photonic band gap

20

Transmittance (%)

Frequency (GHz)

100

300

400

500

PMMA 600

700

800

Wavelength (nm)

(B)

Fig. 32 (A) UV-visible spectrum of the 10 bilayer PMMA/TiO2 superlattice (SL) (solid). The PMMA (dotted) and TiO2 nanoparticle (dash-dotted line) films are also shown with a distinct transmittance than the SL. The wavelength of the probing light (532 nm) is pointed by an arrow. (B) Theoretical dispersion relation of an infinite SL (solid line) and Brillouin intensity, given as a color scale, for the PMMA-TiO2 SL.

TABLE 5 Values of the Physical Quantities Used in the Calculation PMMA

p-TiO2 a

Substrate

cL (m s−1 )

2700

2900

5660

ρ (kg m−3 )

1190

1900

2480

d

40 nm

57 nm

1 mm

n

1.5

1.85

1.5

pPMMA /pTiO2

2

a Infiltrated.

Indeed, the photonic band gap was found experimentally in the range 315– 316 nm (Fig. 32B). As the films are supported by a glass substrate, a strong absorber in the UV region, experimental curves are affected by an error for λ < 300 nm. At the laser wavelength (532 nm) used in the BLS experiment, the PC is transparent (see Fig. 32B) allowing for optimal transmittance required for strong BLS signal. The modulation of the PC optical behavior is noticeable in comparison to its constituents (dotted and dash-dotted lines), both being transparent in the visible. To account for the band gap observed in the experimental transmittance, we have computed the photonic band structure of Fig. 32B using the refractive indices nPMMA = 1.5 and nTiO2 = 1.85 for the constituent layers (Table 5). We should note that nTiO2 must be lower than the bulk anatase TiO2 (nTiO2,bulk = 2.5)

206 Phononics

due to the infilled PMMA. We have estimated nTiO2 from neff , nPMMA , and the TiO2 fraction in the PC, assuming a linear dependence. The computed photonic band structure exhibits a gap in the region of wavelengths 310–350 nm [16]. The band gap opens because of the large mismatch in the dielectric constant ε (ε = n2 ), or in other words, because of the difference in field energy location. At the edges of the band gap, the majority of the energy is localized either in the PMMA (lower part of band 2) or TiO2 (upper part of band 1). The gap located in the UV region blocks those wavelengths and lets the rest, for example, visible light passes through the structure [16]. Thus, a unidirectional phoxonic behavior (Fig. 32) at different frequencies but with almost the same wavelengths for the elastic Λ = 2π/0.032 nm and electromagnetic λ ∼ 332/neff nm (i.e., ∼200 nm) waves is realized and justified for a hybrid PC. Similar results have been found in the case of PMMA-Ba TiO3 PC [18].

3.4.4 Picosecond Ultrasonics In addition to the Raman scattering, picosecond ultrasonics (the pump-probe technique) has proven to be a useful technique for the study of the dynamics of phonons and carriers in thin films, PnCs, and other nanostructures [159– 161]. In this type of experiments, a subpicosecond light pulse is absorbed in some region of a nanostructure. The absorption of the light pulse sets up a local stress. The relaxation of this stress launches strain pulses that propagate through the structure. As these strain pulses propagate they change the optical properties of the different parts of the structure, and consequently lead to a change ΔR(t) in the overall optical reflectivity of the structure [162]. This change is then measured by a probe pulse that is time delayed relative to the light pulse used to generate the phonons (the “pump” pulse). Theoretical models for the generation and detection of normal modes are derived from the elastic continuum [35, 53, 57]. Besides the study of bulk phonon modes [37, 38, 141, 163–168], surface acoustic waves in PnCs have been studied by several groups [35, 53–57]. The first studies have been performed by Maris et al. [35, 53] on metallic Al(111)/Ag(111) PnCs grown on Si(111) substrate. Series of samples were prepared with bilayer thickness ranging from 3 to 24 nm. The ratio of the thickness of the Ag layers to Al layers was approximately 0.8 in all the samples. The total number of bilayers is about 70. The pump and prob light pulses are focused onto an area of the structure that is 20 μm in diameter. The light pulses used in the experiment were produced by hybrid mode-locked dye laser operating at 632 nm. After the absorption of the light, the stress excites the vibrational modes of the PnC structure. However, after a short time, the vibrations associated with the propagating modes will have moved deep into the PnC and can no longer be detected by the probe pulse; only the motion associated with the localized surface modes remains near to the free surface. This component makes a contribution to the reflectivity change ΔR(t), which has the form of a persistent oscillation at a definite frequency. The experiment

Chapter | 4 One-Dimensional Phononic Crystals 207

TABLE 6 Comparison of the Experimentally Measured Surface-Mode Frequencies (νexp ) With Those Calculated From the Transfer-Matrix Theory: νth (Without Taking Into Account the Al Oxidation) and νOx (Taking into Account the Oxidation) D (Å)

νth (GHz)

νexp (GHz)

νOx (GHz)

227

117

110

109

218

122

122

113

163

166

151

153

120

222

196

199

89.3

298

251

265

70.4

376

303

334

Source: After W. Chen, Y. Lu, H.J. Maris, G. Xiao, Phys. Rev. B 50 (1994) 14506.

has been performed for a sample with 75 bilayers, each consisting of 12.1 nm of Al and 9.7 nm of Ag, and the layer that is adjacent to the free surface was Al layer. The frequency of the oscillations is 122 GHz. However, if the PnC is ended with Ag layer, the optical reflectivity do not show any oscillation which means that no surface modes have been detected. A series of regular PnCs with periods varying from 5.0 to 22.7 nm have been studied. For each sample only one persistent oscillation was detected when the superlattice is ended with Al layer. The results of the surface-mode frequencies are listed in Table 6 for different bilayer thicknesses. It is worth noticing that samples with smaller periods do not exhibit such oscillation. The theoretical results of surface modes (νth ) are obtained using Eqs. (29), (30). These results agree within a few percent for the PnCs with large respect periods, but differ by an increasing amount as the period becomes smaller. The authors attributed this discrepancy to the possible oxidation of the Al surface layer (e.g., Al2 O3 ). The surface-mode frequency νOx that was calculated by taking into account this oxidized layer (with assumed thickness of 2.5 nm) is listed in Table 6. They are in much better agreement with the experimental data than those calculated assuming no oxidation. It is worth noticing that surface modes induced by cap layers with different thicknesses have been also investigated by the same group [53] and more than one mode per gap as well as modes lying inside the second gap have been detected. In addition, a Pippard’s theory of the electron-phonon interaction for bulk materials and multilayered structures has been proposed to explain the attenuation of surface phonons observed in metallic PnCs. However, the calculated attenuation rate has been found much smaller than the measured damping rate [169, 170].

208 Phononics

TABLE 7 Frequencies of the Surface Modes in the First, Second, and Sixth Gaps Determined From the Theory (νs1 th , νs2 th , νs3 th ) and Experiment (νs1 exp , νs2 exp , νs3 exp ) D (Å)

νs1 th (GHz)

νs2 th (GHz)

νs6 th (GHz)

νs1 exp (GHz)

νs2 exp (GHz)

νs6 exp (GHz)

326

103

194

584

94

179

532

196

172

323

972

158

298

904

68.4

491

923

2780

413

873

...

Source: After N.-W. Pu, J. Bokor, Phys. Rev. Lett. 91 (2003) 076101.

Similar studies have been performed on Si/Mo [56, 57], Be/Mo [57], and Cu/W [55] PnCs. Besides the surface modes lying within the two first gaps, a higher mode falling inside the sixth gap has been detected by Pu et al. in Si/Mo PnCs [56, 57]. A selection rule has been derived for symmetry considerations to provide new understanding of why certain modes are seen and not the others, and analytical expressions for the delectability as well as the spatial and temporal excitability are derived by the method of normal mode expansion [57]. The authors have measured ΔR(t) traces for samples with various periods D: 32.6, 19.6, and 6.84 nm. The number of periods is 40. In all of the samples, the surface modes in the first (lowest zone edge) gap and the second (lowest zone center) gap have been detected. In addition to these two modes, a higher-order surface mode is also seen in samples with D = 32.6 and 19.6 nm. The amplitude of this mode decreases rapidly as D is reduced from D = 32.6 to 19.6 nm, and vanishes in the thinnest sample (D = 6.84 nm). The theoretical and measured values of the first, second, and sixth surface modes are listed in Table 7. The measured frequencies have been found less than 16% than the theoretical values. This discrepancy has been attributed by the authors to the possibility of the formation of silicide at the Si/Mo interfaces or to the modification of elastic properties in thin films. A complete theoretical and experimental study on the collective excitation and transmission of the low-mini-branch phonons in Si/Mo PnCs has been presented [61]. The Fourier transform spectra of the temporal evolution of strain at different depths within the PnC has evidenced clearly the band gaps as well as the surface modes within the two first folded acoustic branches. This study enables also to deduce the effective sound velocity of the folded phonons as well as the individual velocities of the materials [171, 172]. Also, let us notice the work of Trigo et al. [157] on GaAs-AlAs PnC using the standard pump-probe set-up in the reflection geometry in which the only phonons that can be generated or scattered are longitudinal acoustic modes.

Chapter | 4 One-Dimensional Phononic Crystals 209

In this study, the authors have demonstrated theoretically and experimentally the existence of a new class of extended states in periodic media in addition to the existence of folded and surface branches. These modes fall at the vicinity of the center and edge of the BZ and have a tendency to avoid the boundaries [173], irrespective of the boundary conditions. These modes present a slowly varying envelope wave function with an amplitude minimum in the vicinity of the surface, they referred to them as surface avoiding modes. Combe et al. [158] have shown theoretically that besides surface avoiding modes, there may exist also what they called surface loving modes for which the envelope wave function exhibits a maximum at the surface. The existence of these two types of modes depends on the thickness of the layer at the surface of the PnC, but also on the position of these modes inside the band edges. Such modes have been also studied in multiquantum wells [174] and Bragg mirrors [175]. In summary, we have reviewed in this section localized and resonant acoustic modes in finite size PnCs deposited on a substrate which serves as a support. Particular attention was devoted to the effect of different defect layers on acoustic phonons in these PnCs. These layers are often introduced in a PnC as a cap layer which can serve to protect the PnC and a cavity layer that can be used for filtering certain frequencies. Different guided modes induced by these layers have been investigated and their interaction with modes localized at the free surface of the PnC has been detailed. The localized and resonant modes associated with these structures appear as well-defined peaks in the DOS, with their relative importance being very dependent on the size of the PnC, the wave vector k , and the defect layer thickness as well as on the parameters of the different constituent materials. On the other hand, we have also calculated the phonon phase time expressions and shown that they present similar behaviors to the DOS versus the frequency. We have shown that the presence of the defect layer increases strongly the magnitude of the time delay of the reflected phonon. The experimental observation of the localized and resonant modes predicted here in such finite PnCs may be possible with Raman experiments [15, 60, 65, 75–79, 138] and picosecond laser techniques [31, 53–57, 61]. Some of the Raman experiments have been reproduced by our theoretical model.

4 SAGITTAL ACOUSTIC WAVES IN FINITE SOLID-FLUID PnCs In El Hassouani et al. [176], we have shown the possibility of the existence of surface acoustic waves in semiinfinite solid-fluid PnCs with different surface terminations. Different surface modes are obtained depending on whether the PnC is terminated by a fluid layer or a solid layer. In the case of a fluid layer termination, we have generalized the rule about the existence of shear-horizontal surface modes in solid-solid PnCs (see Section 2), namely in creating two complementary semiinfinite PnCs from cutting an infinite PnC within a fluid

210 Phononics

layer, one obtains as many localized surface states as gaps for any value of k . This result is based on the general rule about the conservation of number of states and expresses a compensation between the loss of (1/2) state at every bulk band edge (due to the creation of two free surfaces) and the gain due to the occurrence of surface states. However, the results are at variance if the cleavage is carried out inside a solid layer, in particular the compensation of the loss of (1/2) state at every edge of the bulk bands can be made by the existence of zero, one, or even two surface states in each gap [176]. In addition, we have discussed the modes induced by a fluid cap layer at the surface of the PnC and discussed the resulting guided and pseudo-guided modes. When the cap layer is of semiinfinite extent, we obtain the interface modes between a PnC and a homogeneous fluid. Here also, we have shown the existence of different interface and pseudo-interface modes, which are without analog in the case of homogeneous media [176]. In this section, we are interested in sagittal acoustic waves in finite size solidfluid PnCs in contact with one or two semiinfinite fluids on both sides. Our goal is to give closed form expressions of dispersion relations, DOSs as well as the transmission and reflection coefficients associated with such systems [177]. These analytical expressions enable us to show peculiar properties related to solid-fluid PnCs as compared to solid-solid PnCs, namely: (i) the stop bands originate both from the periodicity of the system (Bragg like gaps) and the transmission zeros induced by the presence of the solid layers immersed in the fluid. The width of the band gaps strongly depends on the thickness and the contrast between the elastic parameters of the two constituting layers. (ii) In addition to the usual crossing of subsequent bands, we show that solid-fluid PnCs may present a closing of the bands giving rise to large gaps separated by flat bands for which the group velocity vanishes. Also, we give an analytical expression that relates the DOS and the transmission and reflection phase times in finite size systems embedded between two fluids. In particular, we show that the transmission zeros may give rise to a phase drop of π in the transmission phase and therefore a negative delta peak in the phase time (or equivalently a negative group velocity) when the absorption is taken into account in the system. (iii) The possibility of the existence of internal resonance induced by a fluid layer and lying at the vicinity of a transmission zero, the so-called Fano resonance. The organization of this section is as follows. Section 4.1 presents the analytical results obtained for the Green’s function, dispersion relations, transmission and reflection coefficients, and DOSs associated with different solidfluid layered media. All these quantities represent the ingredients necessary to study analytically and numerically new features on wave propagation in solidfluid layered systems like the origin of the band gaps and the conditions for band gaps closing (Section 4.2) as well as a general rule on confined and surface modes in finite solid-fluid PnCs (Section 4.3).

Chapter | 4 One-Dimensional Phononic Crystals 211

4.1 Green’s Functions, Dispersion Relations, Transmission and Reflection Coefficients 4.1.1 Surface Green’s Function of an Infinite Solid-Fluid PnC Let us emphasize that in the geometry of the structures studied, all the interfaces are taken to be parallel to (x1 , x2 ) plane. A space position along the x3 axis in medium i belonging to the unit cell n is indicated by (n, i, x3 ) where −di /2 < x3 < di /2 (i = f for the fluid and i = s for the solid, see Fig. 33A). As we are interested by the propagation of sagittal acoustic waves in such structures, the elements of the Green’s functions take the form g(ω2 , k |n, i, x3 ; n , i , x3 ), where ω is the frequency of the acoustic wave, k the wave vector parallel to the interfaces. For the sake of simplicity, we shall omit in the following the parameters ω2 and k , and we note as g(n, i, x3 ; n , i , x3 ) the x3 x3 component of the Green’s function.

Cell. 1 Fluid

–∞

Cell. 2

Solid

Fluid

Cell. N

Solid

Cell. 1

Cell. 2

Solid

Fluid

Fluid

Solid

df

ds

Fluid

Solid

x3 +∞

D

(A)

Cell. N+1

Cell. N

Cell. N–1 Fluid

Solid

Solid

Fluid

Solid

(B) Cell. 1

Cell. 2

Solid

(C)

Fluid

Cell. N

Cell. 1 Fluid

(D)

(E)

Solid

Fluid

Solid

Cell. 2 Fluid

Cell.P

Cell.1 Incident fluid

Solid

Fluid

Solid

Solid

Cell. N+1 Fluid

Cell. N Fluid

Solid

Cell.P+1 Cavit

Solid

Solid

Cell. N Fluid

Solid

Detector fluid

d0

Fig. 33 (A) Schematic representation of an infinite solid-fluid PnC. df and ds are the thicknesses of the fluid and solid layers, respectively. D = df + ds is the period of the PnC. (B) Schematic representation of a finite PnC composed of N cells with (solid, solid) terminations on both sides. (C, D) The same as (B) but for a PnC with (solid, fluid) and (fluid, solid) terminations, respectively. The two extremities are free of stress. (E) Schematic representation of a finite PnC with a cavity fluid layer in the cell p + 1. The whole system is embedded between two semiinfinite fluids.

212 Phononics

Before addressing the problem of the fluid-solid PnC, it is helpful to know the surface elements of its elementary constituents, namely, the Green’s function of an ideal fluid of thickness df , sound speed vf , and mass density ρf and an elastic isotropic solid characterized by its thickness ds , longitudinal speed v , transverse speed vt , and mass density ρs . The Green’s function of an ideal fluid layer (for which the viscous shearstress vanishes) can be reduced to only a 2 × 2 matrix composed by the x3 x3 elements. We shall call this matrix:   a b −1 , (105) [gf (MM)] = b a where Cf ω2 F , b = , F = −ρf , Sf Sf αf Cf = cosh(αf df ), Sf = sinh(αf df ), a = −F

(106) (107)

and αf2 = k2 −

ω2 . v2f

(108)

It is worthwhile to notice that the assumption of ideal fluid behavior is valid over a very broad frequency range for which the viscous skin depth σ = (2η/ρω) is much smaller than the fluid layer thickness df (η is the viscosity of the fluid). The Green’s function of a solid layer (when it is surrounded by fluids) can be reduced to only a 2 × 2 matrix composed by the x3 x3 elements. We shall call this matrix:   A B , (109) [gs (MM)]−1 = B A where Ct γ β C −β , B= + , S St S St v4t v4 γ = ρ 2 (k2 + αt2 )2 , β = −4ρ t2 αt k2 ω α ω Ct = cosh(αt ds ), C = cosh(α ds ), St = sinh(αt ds ), S = sinh(α ds ), −γ

(110) (111) (112) (113)

and αt2 = k2 −

ω2 , v2t

α2 = k2 −

ω2 . v2

(114)

The Green’s function of the infinite PnC (Fig. 33A) is obtained by a linear juxtaposition of the 2 × 2 matrices (Eqs. 105, 109) at the different interfaces,

Chapter | 4 One-Dimensional Phononic Crystals 213

leading to a tri-diagonal matrix. Taking advantage of the periodicity D in the direction x3 of the solid-fluid PnC, the Fourier transformed [g(k3 ; M, M)]−1 of the above infinite tri-diagonal matrix within one unit cell (1 ≤ i ≤ N) has the following form:   A+a B + be−jk3 D . (115) [g(k3 ; MM)]−1 = B + bejk3 D A+a The bulk bands (eigenmodes) of the infinite solid-fluid PnC are easily obtained: A2 − B2 + a2 − b2 + 2Aa = η, (116) 2Bb where k3 is the component perpendicular to the slabs of the propagation vector − → k ≡ (k , k3 ). It is also straightforward to Fourier analysis back into real space all the elements of g(k3 ; MM) and obtain all the interface elements of g in the following form      df  df df  df (A + a) t|n−n |+1 , g n, f , − ; n , f , − = g n, f , ; n , f , =− 2 2 2 2 Bb t2 − 1 (117a)    |+1  −1|+1 |n−n |n−n t df df t + , (117b) g n, f , − ; n , f , =− 2 2 2 B(t − 1) b(t2 − 1)     df t|n−n +1|+1 df t|n−n |+1 g n, f , ; n , f , − + . (117c) =− 2 2 2 B(t − 1) b(t2 − 1) cos(k3 D) =

In these expressions, t represents eik3 D and is defined by  t = η + η2 − 1 if η < −1  t = η + i 1 − η2 if |η| ≤ 1  t = η − η2 − 1 if η > −1.

(118a) (118b) (118c)

4.1.2 Inverse Surface Green’s Functions of Finite Solid-Fluid PnCs With Free Surfaces We consider in this section different finite size solid-fluid PnCs with free surfaces. The surface layers on both ends of these systems could be (solid, solid) (Fig. 33B), (solid, fluid) (Fig. 33C), or (fluid, solid) (Fig. 33D). The knowledge of the inverse of the Green’s functions on both ends of these systems constitutes the necessary ingredients to deduce easily the dispersion relations as well as the transmission and reflection coefficients through different finite size solid-fluid

214 Phononics

PnCs with or without defect layers. In what follows, we shall detail the results concerning the Green’s function calculation of the structure shown in Fig. 33B with (solid, solid) terminations and give briefly the results concerning the other structures in Fig. 33C and D with (solid, fluid) and (fluid, solid) terminations, respectively. The structure in Fig. 33B is constructed from the infinite PnC of Fig. 33A. In a first step, one suppresses the fluid layers in the cells n = 1 and n = N + 1. For this new system composed of a finite PnC and two semiinfinite PnCs on both sides (not shown here), the inverse surface Green’s function, [gs (M, M)]−1 , is an infinite tridiagonal matrix defined in the interface domain of all the sites n, (−∞ ≤ n ≤ +∞). The matrix is similar to the one associated with the infinite PnC. Only a few matrix elements differ, namely, those associated with the interface space Ms = {(n = 1, i = f , − d2f ), (n = 1, i = f , d2f ), (n = N+1, i = f , − d2f ), (n = N + 1, i = f , d2f )}. The cleavage operator: Vc (MM) = [gs (M, M)]−1 − [g(M, M)]−1 ,

(119)

is the following 4 × 4 square matrix defined in the interface domain Ms : ⎛ ⎞ −a −b 0 0 ⎜ −b −a 0 0 ⎟ ⎟. (120) Vc (Ms Ms ) = ⎜ ⎝ 0 0 −a −b ⎠ 0 0 −b −a On the other hand, using Eq. (117) one can write the elements of the surface Green’s function of the infinite PnC in the interface space Ms in the form of a 4 × 4 square matrix: ⎛ b+Bt b+Bt N ⎞ N − A+a − A+a Bb Bb Bb t Bb t ⎜ b+Bt ⎟ B+bt N−1 N ⎟ − A+a − A+a t ⎜ Bb Bb Bb t Bb t ⎟ ⎜ g(Ms Ms ) = 2 ⎜ ⎟ . (121) b+Bt A+a ⎟ t − 1 ⎜ − A+a tN B+bt tN−1 − ⎝ ⎠ Bb Bb Bb Bb b+Bt N Bb t

N − A+a Bb t

b+Bt Bb

− A+a Bb

Using Eqs. (120), (121), one obtains the matrix operator Δ(Ms Ms ) = I(Ms Ms )+Vc (Ms Ms )g(Ms Ms ) in the space Ms . For the calculation of the inverse Green’s function on both ends of the structure in Fig. 33B, we only need the matrix Δ(M0 M0 ) where M0 = {(n = 1, i = f , d2f ), (n = N + 1, i = f , − d2f )} represents the interface space corresponding to both extremities of the system in Fig. 33B: ⎛ ⎞ t Y1 t N Y2 − 1 − t2 −1 2 Bb t −1 Bb ⎠ , (122) Δ(M0 M0 ) = ⎝ t N Y2 t Y1 − t2 −1 Bb 1 − t2 −1 Bb where Y1 = b2 − a2 − aA + Bbt and Y2 = aB − Abt.

Chapter | 4 One-Dimensional Phononic Crystals 215 −1 (M M ) in the interface The inverse of the surface Green’s function dss 0 0 space M0 of the finite PnC in Fig. 33B is given by: −1 dss (M0 M0 ) = Δ(M0 M0 )g−1 (M0 M0 ),

with g(M0 M0 ) =

t 2 t −1



− A+a Bb B+bt N−1 Bb t

B+bt N−1 Bb t − A+a Bb

(123)  .

From Eqs. (122)–(124), one obtains finally:   A(N) B(N) −1 , dss (M0 M0 ) = B(N) A(N) with

(124)

(125)



  

Y1 1 Y1 1 − Bb t − , A+a t Δ   Y1 Y2 N−1 1 , t B(N) = Bb t − t (A + a)Δ A(N) =

(126) (127)

and Δ = Y12 − Y22 t2(N−1) .

(128)

By following the same procedure and after some algebraic calculations, one can obtain, respectively, the operator Δ(M0 M0 ) and the inverse Green’s function −1 dsf (M0 M0 ) in the interface space M0 = {(n = 1, i = f , d2f ), (n = N + 1, i = f , − d2f )} of the structure shown in Fig. 33C ended at the left side by a solid layer and at the right side by a fluid layer, namely:   N+1 Y t Y1 1 1 − t2 −1 − tt2 −1 Bb ← →   Bb Δ (M0 M0 ) = , (129) Y1 Y1 t tN+1 [t + 1t + Bb ] 1 − t2 −1 [t + 1t + Bb ] t2 −1 and −1 (M0 M0 ) dsf

where

 =

X(N) Y(N)

Y(N) Z(N)

 ,



Bb Y1 1 2N ) , X(N) = − − (1 − t t − t Bb (A + a)(1 − t2N ) 

 1 N Bb t , t − Y(N) = − t (A + a)(1 − t2N ) and

Y1 Bb 2N ) . (1 − t 2t + Z(N) = Bb (A + a)(1 − t2N )

(130)

(131)

216 Phononics

Now, if the structure is ended by a fluid layer on the left side and a solid −1 has the same layer on the right side (Fig. 33D), the inverse Green’s function dfs form as in Eq. (130) where we should just permit the terms X(N) and Z(N).

4.1.3 Transmission and Reflection Coefficients of a Finite PnC Embedded Between Two Fluids Consider a structure made of solid-fluid PnC and embedded between two fluids characterized by their mass densities ρ1 and ρ2 and sound velocities v1 and v2 (see Fig. 34). Consider now an incident longitudinal wave launched in the fluid 1 and polarized in the sagittal plane (x1 , x3 ) (Fig. 34). The incident, reflected, and transmitted waves can be written, respectively, as follows:   1 v1 k (132) e−α1 x3 , Ui (x3 ) = iα1 ω k   1 v1 k Ur (x3 ) = (133) eα1 x3 , −iα1 ω k and

$ where α1 = j

ω2 v21

  1 v2 k (134) Ut (x3 ) = e−α2 (x3 −L) , iα2 ω k $ 2 − k2 , α2 = j ω2 − k2 , and L is the total length of the v2

multilayered structure. The transmission coefficient can be obtained this equation: −1 ut (x3 ) = G2 (x3 , L)G−1 2 (L, L)g(L, 0)G1 (0, 0)Ui (0).

(135)

From Eqs. (132), (135), one obtains ut (x3 ) =

v1 2ρ1 ω2 g(L, 0)Ut (x3 ), v2 α2

(136)

x1

Fluid (1)

x2

q

Layered media



Fluid (2)

k ||

k

0

L

x3

Fig. 34 Schematic representation of a finite layered structure inserted between two different semiinfinite fluids labeled 1 and 2. k is the component of the wave vector k parallel to the layers. θ is the incident angle in the fluid 1.

Chapter | 4 One-Dimensional Phononic Crystals 217

where g(L, 0) is the x3 x3 component of the Green’s function that relates the interfaces L and 0 at both extremities of the finite structure. Therefore, the transmission coefficient is given by: t=−

v1 α1 F1 g(L, 0), v2 α2

(137)

2

where F1 is defined as in Eq. (106): F1 = − ρ1αω1 . By the same way, the reflection coefficient is given by: Ur (x3 ) = − G1 (x3 , 0)G−1 1 (0, 0)Ui (0) −1 + G1 (x3 , 0)G−1 1 (0, 0)g(0, 0)G1 (0, 0)Ui (0).

From Eqs. (132), (138), one obtains

2ρ1 ω2 g(0, 0) Ur (x3 ), ur (x3 ) = 1 − α1

(138)

(139)

where g(0, 0) is the x3 x3 component of the Green’s function at the interface between the fluid 1 and the multilayers. Therefore, the reflection coefficient is given by: r = 1 + 2F1 g(0, 0).

(140)

Eqs. (137), (140) show that the calculation of transmission and reflection coefficients requires the knowledge of only the x3 x3 component of the Green’s function in the space of interfaces at the extremities of the whole system. The reflection and transmission rates are given as follows: R = |r|2 ,

(141)

and T = |t|2

ρ2 α1 . ρ1 α2

(142)

The term ρρ21 αα12 is a correction term that ensures the conservation of sound power through such supposed lossless systems.

4.1.4 Relation Between the DOS and the Phase Times Consider a finite layered structure inserted between two different semiinfinite fluids labeled 1 and 2 (Fig. 34). The inverse of the Green’s function in the interface space M is formed here by the two planes separating these three media (M = {0, L}). For each sagittal mode, the x3 x3 component of the above defined [g(MM)]−1 can be obtained from the surface [gi (MM)]−1 of these three media, namely: [gi (0, 0)]−1 = −Fi ,

(143)

218 Phononics

for the two semiinfinite fluids i = 1 and 2, and   A1 B , [gL (MM)]−1 = B A2

(144)

for the layered media with free surfaces. The detailed expressions for A1 , A2 , and B are given in Section 4.1.2 for finite solid-fluid PnCs with different terminations. A1 and A2 are identical (different) for symmetrical (asymmetrical) structures (see Eqs. 125, 130). The important point to notice is that these three quantities are purely real functions in a finite system; however, F1 and F2 are pure imaginary functions for the semiinfinite fluid media (Eq. 106). Therefore, [g(MM)]−1 of the whole composite system can be obtained as follows [106]:   A1 − F1 B −1 . (145) [g(MM)] = B A2 − F2 From Eqs. (137), (145), one obtains the transmission coefficient as follows: v1 α1 2F1 B [det[g(MM)]], (146) t= v2 α2 % &−1 where det[g(MM)] = A1 A2 − B2 + F1 F2 − F1 A2 − F2 A1 . The reflection coefficient is given by Eqs. (140), (145): r = [A1 A2 − B2 − F1 F2 + F1 A2 − F2 A1 ] det[g(MM)].

(147)

From Eqs. (146), (147), one can obtain the phases θT and θR of the transmission and reflection coefficients, respectively. Of more interest are the derivatives of these phases with respect to the frequency that are indicative of the times needed by a wave packet to complete the transmission or reflection processes. These quantities, usually called phase times [5], are defined by: τT =

dθT , dω

(148a)

τR =

dθR . dω

(148b)

and

From Eqs. (146), (148), one can deduce that the transmission phase time can be written as:     d dB  sgn (149) arg det[g(MM)] + π δ(ω − ωn ). τT = dω dω ω=ωn n The reflection phase time τR can also be derived from Eqs. (147), (148) as: τR =

d d arg(det[g(MM)]) + arg(A1 A2 − B2 − F1 F2 + F1 A2 − F2 A1 ). dω dω (150)

Chapter | 4 One-Dimensional Phononic Crystals 219

Let us now recall [5] that the difference of the DOS between the present composite system and a reference system formed out of the same volumes of the semiinfinite fluids 1 and 2 and the finite structure can be obtained from: 1 d arg(det[g(MM)]). (151) Δn(ω) = π dω From Eqs. (149), (151), one can deduce two cases: (i) If the structure do not present transmission zeros (i.e., B = 0). Then arg(B ) = 0 and τT = π Δn(ω).

(152)

(ii) If the transmission zeros occur at some frequencies. Then the transmission coefficient changes sign, its phase exhibits a jump of π and τT = π Δn(ω).

(153)

Eqs. (150), (151) show that τR is in general different from Δn(ω). However, if medium 2 is evanescent (i.e., F2 is real and the incident wave is totally reflected) then τR = 2π Δn(ω).

(154)

It is worth noticing that the calculation of the phase time enables to deduce the group velocity in such structures using the relation [178]: vg = L/τ ,

(155)

where L represents the size of the structure.

4.2 Application to a Finite Symmetric PnC Embedded in a Fluid 4.2.1 Band-Gap Structure and Conditions for Band and Gap Closing In order to illustrate the general results given earlier, we present here a simple application for sagittal acoustic waves in the special case of periodic solid plates immersed in the same fluid (e.g., water). In this case, the transmission and reflection coefficients are given by Eqs. (146), (147), namely: tN = 2F

B(N) A2 (N) − B2 (N) + F 2

− 2FA(N)

,

(156)

and rN =

A2 (N) − B2 (N) − F 2 , − 2FA(N)

A2 (N) − B2 (N) + F 2

(157)

where A(N), B(N), and F are given by Eqs. (126), (127), (106), respectively.

220 Phononics

In the particular case of one solid layer inserted in the fluid (i.e., N = 1), one can show easily that A(N) = A and B(N) = B (Eqs. 126, 127) and tN and rN become, respectively, t1 = 2F

A2

− B2

B , + F 2 − 2FA

(158)

and A2 − B2 − F 2 . (159) A2 − B2 + F 2 − 2FA Also, it is worth noticing that the numerator of rN (Eq. 157) can be written after some algebraic calculation as:   2N  2 Y1 − Y22 t −1 2 2 2 2 2 2 . (160) A (N) − B (N) − F = [A − B − F ] Δ t2 r1 =

Eqs. (157), (160) clearly show that the reflection zeros associated with a finite PnC made of N plates inserted periodically in a fluid are given either by: A2 − B2 − F 2 = 0,

(161)

which coincides also with the reflection zeros associated with just one solid layer inserted in the fluid (Eq. 159) or mπ , m = 1, 2, . . . . . . , N − 1. (162) sin(NkD) = 0, i.e., kD = N The third term in the right-hand side of Eq. (160) cannot vanishes as Y1 = ± Y2 . The transmission zeros are given by B(N) = 0 (Eq. 156) or equivalently B = 0 (see Eq. 127). This result shows that the transmission zeros of the whole structure coincide exactly with the transmission zeros of just one solid layer inserted in the fluid (Eq. 158). In addition, the parameter B characterizes only the solid layer (Eq. 110); therefore, the transmission zeros are independent on the choice of the fluid surrounding the solid layer, but depend only on the thickness and the elastic parameters of the solid. Fig. 35 shows the dispersion curves (gray areas) for an infinite PnC made of Plexiglas and water layers. The dashed straight lines represent the transverse and longitudinal velocities of sound in Plexiglas, whereas the dashed-dotted line gives the longitudinal velocity of sound in water. The thin solid and dotted curves represent the dispersion curves obtained from the reflection zeros (total transmission) for a finite PnC composed of N = 5 Plexiglas layers inserted in water. The thin solid curves correspond to the N − 1 branches given by Eq. (162) whereas the dotted curves are given by Eq. (161). The open circles’ curves show the positions of the transmission zeros (total reflection). One can notice a shrinking of the N − 1 branches when they intercept the transmission zero branch around (Ω = 4.07, k D = 2.3) and (Ω = 7.64, k D = 3.8). This phenomenon reproduces for other values of the couple (Ω, k D) not shown

Chapter | 4 One-Dimensional Phononic Crystals 221 14

12

wD/vt (Plexiglas)

10

8

6

4

2

0 0

2

4

6

8

10

k||D Fig. 35 Dispersion curves for a PnC made of Plexiglas and water layers. The curves give Ω = ωD/vt(Plexiglas) as a function of k D. The widths of fluid and solid layers are supposed equal: df = ds = D/2. The gray areas represent the bulk bands for an infinite PnC. The thin solid lines and dotted curves show the positions of the reflection zeros (total transmission). Whereas, the open circles give the positions of the transmission zeros (total reflection). The dashed straight lines represent the transverse and longitudinal velocities of sound in Plexiglas. The dashed-dotted line represents the longitudinal velocity of sound in water.

here. This property of the shrinking of the modes is a characteristic of solid-fluid PnCs and is without analog in their counterpart solid-solid PnCs [5]. Now, if we compare together the different branches associated with reflection and transmission zeros and the band gap structure of the infinite Plexiglaswater PnC, one can notice (Fig. 35) the following: (i) As predicted, the thin solid and dotted curves corresponding to total transmission fall inside the allowed bands (gray areas), in particular the positions of the closing of the gaps are given by the intersection of the

222 Phononics

limits of the band gaps and the dotted curves, that is, we should have simultaneously: cos(k3 D) =

A2 − B2 + F 2 + 2Aa = ±1, Bb

(163)

and A2 − B2 − F 2 = 0.

(164)

(ii) The open circles’ curves with total reflection (zero transmission) fall inside the forbidden bands and the position of the closing of the bands should satisfy the two following conditions: cos(k3 D) =

A2 − B2 + F 2 + 2Aa = ±1, Bb

(165)

and B = 0.

(166)

These particular crossings of the gaps give rise to a no dispersive curves (flat bands) for which the group velocity vanishes. It is worth noticing that the transmission zeros (open circles) fall above a straight line, that is, below a critical angle θcr . Indeed, a simple Taylor expansion of the function B(ω) in Eqs. (110), (111) at the low-frequency limit (i.e., around ω 0 and k 0), gives:   2 vt ω = 2 vt 1 − = vcr . (167) k v However, k is related to the incident angle θ by the relation k = vωf sin(θ ) (see Fig. 34). Thus, one obtains transmission zeros for wave velocities v > vcr , or equivalently: ⎛ ⎞ vf ⎜1 ⎟ vt ⎜ ⎟ θ < θcr = arcsin ⎝ $  2 ⎠ = 39 degrees. 2 vt 1 − v

(168)

However, let us mention that at normal incidence (i.e., k = 0 or θ = 0 degree), B(ω) cannot vanishes and one obtains the well-known dispersion relation [101–103]:   1 Z Zf + (169) S Sf , cos(k3 D) = C Cf + 2 Zf Z where Z = ρs v and Zf = ρf vf are the acoustic impedances of longitudinal waves in solid and fluid layers, respectively. The previous results show that the transmission zeros occur only for incidence angles θ such that 0 degree < θ < θcr .

Chapter | 4 One-Dimensional Phononic Crystals 223

Fig. 36 shows the variation of the transmission rates T (Fig. 36A–C, E–G, and I–K) as a function of the reduced frequency Ω for a finite PnC composed of N = 1, 2, and 5 Plexiglas layers immersed in water. The left, middle, and right panels correspond to incident angles: θ = 0 degree, 25 degrees, and 40 degrees, respectively. At the bottom of these panels we plotted the corresponding dispersion curves (i.e., Ω vs. the Bloch wave vector k3 ) (Fig. 36D, H, and L). As predicted previously, for θ = 0 degree (left panel) and θ > θcr (right panel), the transmission exhibits dips at some frequency regions which transform into gaps as far as N increases. These gaps are due to the periodicity of the system (Bragg gaps) and coincide with the band gap structure of the infinite PnC shown in Fig. 36D and L. For an incident angle 0 degree < θ < θcr (middle panel), one can notice the existence of a transmission zero around Ω = 7.64 (Fig. 36E), which is due to the insertion of one Plexiglas layer (N = 1) in water. This transmission zero transforms to a large gap when N increases. Besides this gap there exists a dip around Ω = 5 for N = 2 (Fig. 36F) which also transforms to a gap when N increases; this gap is due to the periodicity of the structure. The transmission gaps map the band gap structure of the infinite PnC (Fig. 36H), where one can notice that the imaginary part of the Bloch wave vector (responsible of the attenuation of the waves associated with defect modes) is finite in the Bragg gaps and tends to infinity inside the gaps due to the transmission zeros. These latter gaps can be used to localize strongly defect modes within the structure. From all the previous results, one can conclude that for an incident angle 0 degree < θ < θcr (middle panel) there exists two types of gaps: Bragg gaps which are due to the periodicity of the structure and gaps which are induced by the transmission zeros. However, at normal incidence (θ = 0 degree) (left panel) and for θ > θcr (right panel) all the gaps are due to the periodicity of the system. The existence of these two types of gaps has been discussed also by Shuvalov and Gorkunova [179] in periodic systems of planar sliding-contact interfaces.

4.2.2 Brewster Acoustic Angle Another interesting result that may be exhibited by solid-fluid layered media is the possibility of existence of Brewster acoustic angles as for electromagnetic waves in dielectric media [156]. The Brewster angle corresponds to an incident angle between two homogenous media for which there is no reflection. By analogy with transverse magnetic waves in 1D PCs, the existence of such angles for transverse acoustic waves between two solids has been shown [5]. This angle leads to the shrinking of the PnC gaps to zero along a straight line whose slope is defined by the Brewster condition. The reflection zeros between solid and fluid media can be obtained by matching the Green’s function of a semiinfinite −1 solid (g−1 f (0, 0) = −F) with that of a semiinfinite fluid (gs (0, 0) = −γ − β), namely: ρ

ω2 v4t v4t 2 2 2 2 (k + α ) − 4ρ α k = ρ . t f t   αf ω2 α ω2

(170)

q = 0°

1

q = 25°

1

(A)

q = 40°

1

1

0

N=1

N=1

N=1 0

2

4

6

8

10

0 1

0

2

4

6

8

10

0 1

0

2

4

6

8

(B)

(J)

Transmission

(F) N=2 0 1

0

N=2

N=2 2

4

6

8

10

0 1

0

2

4

6

8

10

(C)

N=5

1

0

2

4

6

8

2

4

6

8

10

N=5 0

0

2

4

6

8

10

(D)

0

2

4

6

8

(H) p/D

10

(L) p/D

k3D

p/D

10

(K)

N=5 0

0

0

(G)

0

10

0

0

0

2

4

6

wD/vt (Plexiglas)

8

10

0

0

2

4

6

wD/vt (Plexiglas)

8

10

0

2

4

6

8

10

wD/vt (Plexiglas)

Fig. 36 Variation of the transmission coefficients as a function of the reduced frequency Ω for a finite PnC composed of N = 1 [(A), (E), and (I)], N = 2 [(B), (F), and (J)], and N = 5 [(C), (G), and (K)] Plexiglas layers immersed in water. The left, middle, and right panels correspond to incident angles: θ = 0 degree, 25 degrees, and 40 degrees, respectively. (D), (H), and (L) give the dispersion curves (i.e., Ω vs. the Bloch wave vector k3 ) inside the reduced BZ 0 < k3 < π/D. Outside this zone is represented the imaginary parts of k3 .

224 Phononics

(I)

(E)

Chapter | 4 One-Dimensional Phononic Crystals 225

In the velocity region vt < v < v , α is real (evanescent wave), whereas αt and αf are pure imaginary (propagative waves). Therefore, Eq. (170) is satisfied if: ' ( 2 vf ( =) , (171) ρ2 vt 1 + ρf2 and ω=



vf 2k vt or equivalently θB = arcsin √ , 2vt

(172)

where θB is the Brewster angle. Through such angle, the incident longitudinal wave in the fluid enters completely the solid but converts to transverse wave. By using Snell’s Law, Eq. (172) shows that the wave enters the solid at 45 degrees to the interface. Let us mention that a study of the Brewster acoustic angles at the fluid-solid interface in all the velocity regions has been performed some years ago by Sotiropoulos et al. [180, 181]. Now, if a layered structure is made from such solid-fluid interfaces, then an incident wave will be totally transmitted giving rise to the closing of the gaps along a straight line corresponding to Brewster angle. Such an angle is independent of the thickness of the layers in the PnC as well as on the longitudinal velocity of sound in the solid (see Eq. 172). In general, Eq. (171) is not easy to be satisfied by solid and fluid materials. However, in the case of Plexiglas-water structure considered here, Eq. (171) is almost satisfied and Eq. (172) gives θB = 49.77 degrees. Now, if we take an incident angle near to θB one obtains almost total transmission as it is shown in Fig. 37A and B corresponding, respectively, to N = 1 and 5 Plexiglas layers immersed in water. The dispersion curves (Fig. 37C) show clearly the closing of the gaps at the center and edges of the reduced BZ for this incidence angle.

4.2.3 Comparative Study of the DOS and Phase Times A comparative analysis of the transmission phase time and the DOS is given in Fig. 38. The phase times τ (ω) (Fig. 38A–C) and the variation of the DOS Δn(ω) (Fig. 38D–F) are plotted as function of the reduced frequency Ω for a finite PnC composed of N = 5 Plexiglas layers immersed in water and for three incident angles: θ = 0 degree (a) and (d), θ = 30 degrees (b) and (e), and θ = 60 degrees (c) and (f). The phase time gives information on the time spent by the phonon inside the structure before its transmission, while the DOS gives the weight of the modes. In Fig. 38, the DOS and the phase time are strongly reduced in the band gap regions. As predicted by the analytical results in Section 4.1.4, the phase time may give rise to delta functions around the transmission zeros as it is shown in Fig. 38B around Ω 8 for 0 degree < θ < θcr . This delta function, which does not exist in the DOS (Fig. 38E), has been enlarged by adding a small imaginary part to the pulsation ω, which plays the role of absorption in

226 Phononics q =q B 1

N=1

(A)

0 0

2

4

6

8

10

2

4

6

8

10

4

6

8

10

1

(B)

0

N=5 0

k3D

p/D

0

(C) Fig. 37

0

2

wD/vt (Plexiglas)

Same as in Fig. 36 but for the Brewster angle: θ = θB = 49 degrees.

the system. Such negative delta peaks have been shown experimentally in simple photonic [182] and phononic [183] loop waveguides, giving rise to the so-called superluminal or negative group velocity (Eq. 155). Because of the nonexistence of transmission zeros, solid-solid layered media do not exhibit such negative phase times or negative group velocities. Fig. 38A–C and D–F clearly shows, in accordance with Eqs. (149), (151), that except the frequencies lying around the transmission zeros, the DOS and the phase time exhibit exactly the same behavior.

4.3 General Rule About Confined and Surface Modes in a Finite Asymmetric PnC In the previous section, we have demonstrated that the creation of two semiinfinite PnCs from the cleavage of an infinite solid-fluid PnC, gives rise to one surface mode per gap for any value of the wave vector k . This mode belongs to one or the other of the two complementary PnCs. In this section, we give a

Chapter | 4 One-Dimensional Phononic Crystals 227 10 8

8

6

6

4

4

2

2

0

(A)

0

80

2

4

6

8

10

0

(D)

40 20

2

4

6

8

10

6

8

10

6

8

10

0
60 40 20

0

0 0

3.0 2.8 2.6 2.4 2.2 2.0 1.8 1.6

(C)

0

80

60

(B)

q = 0°

100

0
DOS

Phase time

10

q = 0°

2

4

6

8

10

q cr
2

4

6

w D/vt (Plexiglas)

8

10

(E)

0

2

4

3.0 2.8 2.6 2.4 2.2 2.0 1.8 q cr
(F)

w D/vt (Plexiglas)

Fig. 38 Transmission phase time (left panel) and DOS (right panel) (in units of D/vt (Plexiglas)) as a function of the reduced frequency Ω for a finite PnC composed of N = 5 Plexiglas layers immersed in water and for three incident angles: θ = 0 degree (A) and (D), θ = 30 degrees (B) and (E), and θ = 60 degrees (C) and (F).

generalization of these results to a finite size PnC made of N solid-fluid cells (Fig. 33C and D) with both extremities in contact with vacuum. The expression giving the eigenmodes of such a structure is given by Eqs. (130), (131) and can be written in the following form:



 a2 − b2 + Aa 1 a2 − b2 + Aa  − 1 − t2N = 0. (173) t− Bb t Bb This expression shows that there are two types of eigenmodes in this kind of finite structure: (1) If the wave vector k3 is real which corresponds to an allowed band, then the eigenmodes of the finite PnC are given by the third term in Eq. (173), namely: sin(Nk3 D) = 0,

(174)

228 Phononics

which gives mπ , m = 1, 2, . . . , N − 1, (175) N whereas the first and second terms in Eq. (173) cannot vanish in the bulk bands as t = ejk3 D is complex and a, b, A, and B are real. (2) If the wave vector k3 is imaginary (modulo π ) which corresponds to a forbidden band, then the eigenmodes are given by the two first terms of Eq. (173), namely: k3 D =

t=

a2 − b2 + Aa Bb

(176)

and 1 a2 − b2 + Aa = . t Bb

(177)

These two expressions give the localized modes associated with the two surfaces surrounding the structure. The third term in Eq. (173) cannot vanish inside the gap since t should satisfy the condition: |t| < 1,

(178)

to ensure the decaying of surface modes from the surface. In addition, we remark that if N → ∞ the term t2N vanishes and therefore the two expressions (Eqs. 176, 177) give the surface modes for two semiinfinite PnCs obtained from the cleavage of the infinite PnC between the solid and fluid layers [5]. Eq. (173) clearly shows that the surface modes are independent of the number N of cells in the finite system. Eqs. (176), (177) can be written in a unique explicit form by replacing them 1 , one obtains: in Eq. (116) and factorizing by the factor Bb a(B2 − A2 ) − A(a2 − b2 ) = 0.

(179)

Therefore, the surface modes associated with one surface are given by Eq. (179) together with the condition | aB Ab | > 1 (Eqs. 176, 178), whereas the surface modes of the other surface are given by Eq. (179) but with the condition | aB Ab | < 1 (Eqs. 177, 178). This result shows that if a surface mode appears on one surface of the finite PnC, it does not appear on the other surface. Eq. (179) with the supplementary condition is similar to those given for semiinfinite PnCs [5]. The previous results clearly show that a finite PnC made of N solid-fluid layers exhibits N − 1 modes in each allowed band and one additional mode per gap induced by one of the two surfaces surrounding the structure. These results generalize our previous findings on semiinfinite solid-fluid PnCs [184]. An example of the dispersion curves is shown in Fig. 39 for a PnC composed of N = 4 Plexiglas-water cells. The other parameters are the same as in Fig. 35.

Chapter | 4 One-Dimensional Phononic Crystals 229 10

wD/vt (Plexiglas)

8

6

4

2

0 0

2

4

6

8

10

k||D Fig. 39 Dispersion curves for a finite PnC made of N = 4 Plexiglas-water bilayers with free surfaces. The dotted curves represent N − 1 = 3 confined modes in the finite PnC. These modes fall inside the allowed bands of the PnC. The open circles (open triangles) correspond to localized modes induced by the surface of the PnC terminated by the fluid layer (solid layer). These modes fall inside the band gaps. The gray areas represent the bulk bands for the infinite PnC.

One can notice the existence of N − 1 = 3 modes in each band, these modes correspond to confined modes (stationary waves) and one surface branch in each gap induced by one or the other of the two surfaces ending the structure. The open circles and triangles correspond to surface modes induced by fluid and solid layers terminations, respectively. As mentioned earlier, these modes coincide exactly with the surface modes of two complementary PnCs obtained from the cleavage of an infinite PnC between the solid and fluid layers. When N increases, the number of branches in each band increases, whereas the surface branches fall at the same frequencies. It is worth mentioning that the above-mentioned rule on confined and surface modes has been obtained recently by Ren et al. [185] for pure transverse elastic waves in solid-solid PnCs. The same rule has been confirmed theoretically and experimentally [186] by some of the authors for electromagnetic waves in quasi1D structures made of coaxial cables.

230 Phononics

In summary we have presented in this section a theoretical analysis of the propagation of sagittal acoustic waves in finite PnCs made of alternating elastic solid and ideal fluid layers. We have developed theoretically the expressions giving the Green’s functions of different solid-fluid layered media, which enables us to deduce analytically in a closed form the expressions of the dispersion relations, the transmission and reflection coefficients and the DOS. We have shown analytically and numerically particular features of wave propagation in solid-fluid layered media in comparison with their counterparts composed only of solid media. The main features of solid-fluid PnCs are the existence of transmission zeros that are without analog in solid-solid PnCs. These transmission zeros exist only for a range of incident angles θ such that 0 < θ < θcr . The consequences of the transmission zeros are: (i) The existence of new gaps besides the gaps induced by the periodicity of the system (Bragg gaps). The imaginary part of the Bloch wave vector inside the former gaps is much higher than those in the latter ones which enables a strong localization of the waves when a defect is inserted in the system. (ii) Besides the closing of the gaps, the solid-fluid PnCs present a closing of the bands leading to flat bands for which the group velocity vanishes. (iii) The phase of the transmission exhibits a phase drop of π and therefore a negative phase time or equivalently a negative group velocity, the so-called superluminal phenomena. (iv) The possibility of the existence of Brewster acoustic angle in the velocity region between transverse and longitudinal velocities of sound in the solid. Total transmission occurs through such angles with mode conversion from longitudinal waves in the fluid to transverse waves in the solid and vice versa. These angles can have practical applications in the area of ultrasonic nondestructive evaluation. Besides these new properties specific to solid-fluid systems, we have derived exact relations between the DOS and phase times in finite systems embedded between two fluids. Also, we have presented a theoretical evidence of the existence of two types modes in a finite solid-fluid PnC made of N cells with free surfaces. In particular, we have shown the existence of N − 1 modes that fall inside the bulk bands and one additional mode by gap that is associated with one of the two surfaces surrounding the structure. These surface modes are independent of N and coincide with the surface modes of two complementary semiinfinite PnCs obtained from the cleavage of an infinite PnC between the solid and fluid layers.

5 OMNIDIRECTIONAL REFLECTION AND SELECTIVE TRANSMISSION IN LAYERED MEDIA During the last two decades, much attention has been devoted to the study of 2D and 3D periodic PnCs [6, 7]. In analogy to the more familiar PCs [187, 188], the essential property of these structures is the existence of forbidden frequency bands, where the propagation of sound and ultrasonic vibrations is inhibited in any direction of space. As mentioned earlier, such phononic band gap materials

Chapter | 4 One-Dimensional Phononic Crystals 231

can have practical applications such as: acoustic filters [189], ultrasonic silent blocks [190], acoustic mirrors, and improvements in the design of piezoelectric ultrasonic transducers [191]. The contrast in elastic properties and densities between the constituents of the composite system is a critical parameter in determining the existence and the width of absolute band gaps. In the field of photonic band gap materials, it has been argued [192–194] that 1D structures such as PtCs can also exhibit the property of omnidirectional reflection, that is, the existence of a band gap for any incident wave independent of the incidence angle and polarization. However, because the photonic band structure of a PtC does not display any absolute band gap (i.e., a gap for any value of the wave vector), the property of omnidirectional reflection holds in general when the incident light is launched from vacuum, or from a medium with relatively low index of refraction (or high velocity of light). To overcome this difficulty, when the incident light is generated in a high refraction index medium, a solution [195] that consists to associate with the a cladding layer with a low index of refraction has been proposed. This layer acts like a barrier for the propagation of light. The object of this section is to examine the possibility of realizing 1D structures that exhibit the property of omnidirectional reflection for acoustic waves. In the frequency range of the omnidirectional reflection, the structure will behave analogously to the case of 2D and 3D PnCs; that is, it reflects any acoustic wave independent of its polarization and incidence angle. We will show that a simple PnC can fulfill this property, provided the substrate from which the incident waves are launched is made of a material with relatively high acoustic velocities of sound. However, the substrate may have relatively low acoustic velocities, according to the large varieties in the elastic properties of materials. Then, we propose two alternative solutions to overcome the difficulty related to the choice of the substrate, in order to obtain a frequency domain in which the transmission of sound waves is inhibited even for a substrate with low velocities of sound. As mentioned in the case of photonic band gap materials, one solution would be to associate the PnC with a cladding layer having high velocities of sound in order to create a barrier for the propagation of acoustic waves. Another solution will consist of associating two PnCs chosen appropriately in such a way that the superposition of their band structures displays a complete acoustic band gap [81, 82]. In what follows, we shall present a comprehensive investigation of the conditions necessary for obtaining this acoustic gap and its evolution according to the physical parameters defining the 1D structure. More precisely, the transmission spectra for different polarizations of the incident waves are calculated and analyzed in relation with the dispersion curves of the modes associated with the finite structure embedded between two substrates. When a maximum threshold for transmittance is imposed, we investigate the contributions of the different modes induced by the finite structure (bulk phonons of the PnCs, modes of the cladding layer, and interface modes) to the transmission spectra,

232 Phononics

thus revealing the limitations on the existence of an absolute band gap. We discuss the dependence of the transmission coefficients upon the thickness of the clad layer and the number of cells in the PnCs, as well as upon the choice of the layers in the PnC, which are in contact with the substrates and with the clad. Specific illustrations are given for solid-solid and solid-fluid PnCs. In addition, we show that these structures can be used as an acoustic filter that may transmit selectively certain frequencies within the omnidirectional gaps. In particular, we show the possibility of filtering assisted either by cavity modes or by interface resonances. After a brief presentation of the model and method of calculation in Section 5.1, we present in Sections 5.2 and 5.3 the numerical illustrations as well as the discussion of the transmission coefficients for the occurrence of an omnidirectional band gap and the possibility of selective transmission through these gaps.

5.1 Model and Method of Calculation The geometries studied in this paper are schematically shown in Fig. 40. We consider a finite lamellar structure L sandwiched between two substrates S1 and S2 (Fig. 40A). The details about the composition of the finite structure are shown in Fig. 40B and C. In one case (Fig. 40B), the lamellar structure is composed of a finite PnC containing alternating layers of materials A and B, and a clad layer of material C. Let us notice that in our calculation, the material C can be embedded inside the PnC instead of being at its boundary. In the second geometry (Fig. 40C), two finite PnCs made, respectively, of materials (A1 , B1 ) and (A2 , B2 ) are associated together in tandem. All the interfaces are taken to be parallel to (x1 , x2 ) plane of a Cartesian coordinates system. All the media are assumed to be isotropic elastic media characterized by their mass densities, their transverse velocity Ct , and longitudinal velocity Cl of sound. The study of acoustic wave propagation in such a composite lamellar system is performed by means of total DOSs as well as transmission and reflection coefficients. These quantities are obtained in the same way as in previous sections. The dispersion curves are obtained from the peaks in the DOS which are associated with the modes of the medium L interacting with the continuum of substrate modes. We shall focus our attention on transmitted waves through the lamellar composite system, in relation with the dispersion curves. The existence of an omnidirectional acoustic gap requires that the transmission coefficients fall below a threshold value for all polarizations and any incidence angle of the incoming waves. Let us notice that the incident wave, generated in a homogeneous solid medium S1 , can have three different polarizations; namely, transverse horizontal (or shear horizontal), transverse vertical, and longitudinal. However, only longitudinal wave can be launched through a homogeneous fluid medium S1 .

Chapter | 4 One-Dimensional Phononic Crystals 233

Homogeneous medium (S1)

Finitelamellar composite (L)

Homogeneous medium (S2)

(A)

C

B1

B1 A1 B1

B1 A1 x3

(B) Cell 0

A1 B1

B1 A1 B1

Cell N1

B1 A1 A2 B2

B2 A2 B2

B2 A2 x3

(C)

Cell 0

Cell N1 Cell 0

Cell N2

Fig. 40 Geometries of the omnidirectional band gap structure. (A) A finite lamellar composite system L embedded between two homogenous media S1 and S2 . (B) The system L is constituted by a PnC cladded with a material C. (C) The system L is constituted by a combination in tandem of two different PnCs.

5.2 Case of Solid-Solid PnCs In this section, we show that omnidirectional reflection of acoustic waves can be achieved with only 1D systems instead of 2D or 3D PnCs. First, we emphasize that a single PnC can display an omnidirectional reflection band, provided the substrate is made of a material with relatively high velocities of sound. Then, in order to remove the limitation about the choice of the substrate, we consider the geometries shown in Fig. 40 where either a clad layer is added to the PnC or two different PnCs with appropriately chosen parameters are combined in tandem. The expressions of the transmission and reflection coefficients and DOSs are cumbersome. We shall avoid the details of these calculations which are given in El Boudouti et al. [5]. Let us first examine the so-called projected band structure of a PnC, that is, the frequency ω versus the wave vector k . Fig. 41 shows the phononic band

Reduced frequency

234 Phononics 8

8

6

6

4

4

2

2

0

0 5

4

3

2

1

0

1

2

3

4

5

Reduced wave vector Fig. 41 Projected band structure of sagittal (right panel) and transverse (left panel) elastic waves in a W/AL PnC. The reduced frequency Ω = ωD/Ct (Al) is presented as a function of the reduced wave vector k D. The shaded and white areas, respectively, correspond to the mini-bands and minigaps of the PnC. The heavy and thin straight lines correspond, respectively, to sound velocities equal to transverse and longitudinal velocities of sound in epoxy.

structure of an infinite PnC composed of Al and W materials with thicknesses d1 and d2 , such as d1 = d2 = 0.5D, D being the period of the PnC. We have used a dimensionless frequency Ω = ωD/Ct (Al), where Ct (Al) is the transverse velocity of sound in Al (the elastic parameters of the materials are listed in Table 8). The left and right panels, respectively, present the band structure for transverse and sagittal acoustic waves. For every value of k , the shaded and white areas in the projected band structure, respectively, correspond to the mini-bands and to the mini-gaps of the PnC, where the propagation of acoustic waves is allowed or forbidden. Due to the large contrast between the elastic parameters of Al and W, the mini-gaps of the PnC are rather large in contrast to the case of other systems such as GaAs-AlAs PnCs. Nevertheless, it can be easily noticed that the band structure shown in Fig. 41 does not display any absolute gap, this means a gap existing for every value of the wave vector k . However, the PnC can display an omnidirectional reflection band in the frequency range of the mini-gap (2.952 < Ω < 4.585) if the velocities of sound in the substrate are high enough. More precisely, let us assume that the transverse velocity of sound in the substrate Ct (s) is greater than 5543 m s−1 (the heavy line in Fig. 41 indicates the sound line with the velocity 5543 m s−1 ). For any wave launched from this substrate, the frequency will be situated above the sound line ω = Ct (s)k , that is, above the heavy line in Fig. 41. When the frequency falls in the range 2.952 < Ω < 4.585 (corresponding to the mini-gap of the PnC at k = 0), the wave cannot propagate inside the PnC and will be

Chapter | 4 One-Dimensional Phononic Crystals 235

TABLE 8 Elastic Parameters of the Materials Involved in the Calculations Materials

Mass Density (kg m−3 )

Ct (m s−1 )

Cl (m s−1 )

W

19,300

2860

5231

Al

2700

3110

6422

Si

2330

5845

8440

Fe

8133

2669

4757

Epoxy

1200

1160

2830

Pb Nylon

10,760

850

1960

1110

1100

2600

reflected back. Thus, the frequency range 2.952 < Ω < 4.585 corresponds to an omnidirectional reflection band for the chosen substrate. Generally speaking, the above condition expresses that the cone defined by the transverse velocity of sound in the substrate contains a mini-gap of the PnC. With the Al/W PnC, this condition is, for instance, fulfilled if the substrate is made of Si [80–82]. Of course, in practice, due to the finiteness of the omnidirectional mirror, one can only impose that the transmittance remains below a given threshold (e.g., 10−3 or 10−2 ). Based on these results, an experiment has been performed by Manzanares-Martinez et al. [83] on Pb/Epoxy PnCs to show the occurrence of such omnidirectional band gaps. Now, if the incident wave is initiated in a substrate made of a material with low velocities of sound such as epoxy (with Ct (s) = 1160 m s−1 , see the thin straight line in Fig. 41), the wave is not prohibited from propagation inside the PnC, whatever the frequency. Thus, the wave will be partially transmitted through the PnC, and only partially reflected back, depending upon the incidence angle (or equivalently, upon the wave vector k ). Therefore, the occurrence of an omnidirectional band gap introduces a limitation regarding the choice of the substrate material, namely, this material should have relatively high acoustic velocities as compared to the typical velocities of the materials constituting the PnC. In order to remove this limitation or at least facilitate the existence of an omnidirectional reflection band, we, respectively, present in the next two sections the solutions mentioned earlier. The first one consists of cladding the PnC with a layer of high acoustic velocities, which can act like a barrier for the propagation of phonons. The second solution consists of considering a combination of two different PnCs, provided their band structures do not overlap over the frequency range of the omnidirectional band gap.

236 Phononics

Reduced frequency

5.2.1 Cladded PnC Structure This section contains results of the transmission spectra, DOS, and dispersion curves for acoustic modes in a finite Al/W PnC cladded on one side by an Si layer of thickness dSi , and embedded between two substrates made of epoxy (Fig. 40). Fig. 42 shows an example of the dispersion curves for the above structure, together with the frequency domains in which the transmission power exceeds a threshold of 10−3 (shaded areas). In this example, the thickness of the Si layer is dSi = 8D, and the PnC contains four bilayers of Al and W; the clad layer is in contact with either an Al layer (Fig. 42A) or a W layer (Fig. 42B) in the PnC (see Fig. 40B). The branches that fall outside the mini-bands of the PnC are

5

5

4

4

3

3

2

2

1

1

(A)

0

0 2

3

1

0

1

2

3

5

5

4

4

3

3

2

2 1

1 0

(B) 3

0 2

1

0

1

2

3

Reduced wave vector Fig. 42 Dispersion curves of the cladded finite PnC embedded between two substrates. The shaded area corresponds to the frequency domain in which the transmission power can exceed a threshold of 10−3 . The thickness of the clad layer is dSi = 8D, and the PnC contains four bilayers of Al and W. The clad is in contact either with an Al layer (A) or with a W layer (B). The left and right panels, respectively, refer to shear-horizontal and sagittal acoustics modes. The horizontal dashed lines delimit the edges of the omnidirectional acoustic band gap. The heavy and thin straight lines, respectively, show the transverse sound lines of the substrate (epoxy) and the clad (Si).

Chapter | 4 One-Dimensional Phononic Crystals 237

essentially associated either with the guided modes of the Si layer or with the interface modes localized at the Si-PnC boundary (the latter are located below the sound lines of Si). As compared with the PnC mini-gap, the omnidirectional reflection band (delimited by the two horizontal dashed lines in Fig. 42) can be significantly reduced. In the case of Fig. 42B where the omnidirectional gap almost disappears, the main limitation is due to transmission through the modes belonging to a narrow mini-band of the PnC; this corresponds to a narrow range of the incidence angle in the substrate (around 14 degrees). Therefore, with the geometrical parameters chosen in this example, the clad layer is not efficient to bring the transmission below the threshold of 10−3 in a broad frequency range, for all incidence angles and all polarizations. Actually, increasing the threshold to 10−2 does not significantly improve the omnidirectional gap in this case. In the example of Fig. 42A where an Al layer in the PnC is in contact with the Si clad, the omnidirectional gap extends from Ω = 3.176 to Ω = 4. Here the upper edge of the gap is decreased as compared to the PnC mini-gap, due to transmission around the frequencies Ω = 4–4.5 (upper right corner of the figure). From right to left, the transmission occurs through bulk modes of the PnC belonging to a narrow mini-band, through an interface mode at the boundary between the PnC and the Si-clad layer, and through a guided mode of the Si layer. Although the transmittance through the latter modes exceeds the chosen threshold of 10−3 , still it remains very small (see Fig. 44). One can notice that the presence of the clad layer has two opposite effects. It decreases the transmittance in some frequency domains (essentially below the sound line defined by the transverse velocity of sound in the clad), but also introduces new modes that can contribute themselves to transmission. The transmission by the latter modes is prevented by the PnC when the corresponding branches fall inside the mini-gaps. To give a better insight into the behaviors of the transmission coefficients, we shown in Figs. 43 and 44, for two values of k D, the transmitted intensities through the cladded PnC. For the sake of comparison, we have also given the DOS. The results are presented for different polarizations, namely, the incident wave can be shear horizontal, transverse in the sagittal plane, or longitudinal, and the DOS is given for either shear-horizontal or sagittal modes. The thickness of the Si layer is again dSi = 8D, the PnC is composed of N = 4 bilayers of Al and W, and the clad layer is in contact with an Al layer. At k D = 0 (Fig. 43), corresponding to a normal incidence, there is a decoupling between waves of transverse and longitudinal polarizations. One can observe that the presence of the clad layer does not affect the band gap that is almost identical to the mini-gap of the PnC (2.95 < Ω < 4.58). The clad layer induces additional modes (see the peaks in the DOS), which are the guided modes of the Si layer, but these modes do not contribute to transmission when they fall inside the mini-gap of the PnC (see, for instance, the peaks in DOS around the frequencies 3 < Ω < 4). At k D = 2 (Fig. 44), the presence of

238 Phononics

(A)

TtransH

1

0

(B)

80 DOS

60 40 20 0 0 1

1

2

1

2

3

4

5

6

3

4

5

6

Tlong

(C)

0

(D)

80 DOS

60 40 20 0 0

Reduced frequency Fig. 43 Transmission coefficients and densities of states for the cladded PnC structure of Fig. 42A, at k D = 0. The panels (A) and (B) are plotted for acoustic waves of transverse polarization. Panels (C) and (D) refer to waves of longitudinal polarization.

the clad layer of Si prevents the propagation of sound in the frequency range that lies below the transverse sound line of Si (Ω < 3.5). Hence, in this range of frequency, the clad layer plays the role of a barrier between phonons in the substrate and the PnC, leading to a decrease in the transmitted intensity (please notice the very small scales in the vertical axes of Fig. 44C and D). On the other hand, the modes induced by the Si layer, which fall in the PnC mini-gap (around Ω = 4.5) contribute very weakly to the transmission process, as mentioned in connection with the discussion of Fig. 42A.

Chapter | 4 One-Dimensional Phononic Crystals 239

(A)

(B)

(C)

(D)

(E)

Fig. 44 Same as in Fig. 42 but for k D = 2. Panels (A), (C), and (D), respectively, give the transmission power for an incident wave of the following polarization: shear-horizontal, transverse in the sagittal plane, and longitudinal in the sagittal plane (one can notice the small scales on the vertical axes of panels (C) and (D)). Panels (B) and (E) present the densities of states, respectively, associated with shear-horizontal and sagittal waves.

To investigate the effect of PnC termination on the existence of the omnidirectional gap, we have also considered, besides the examples of Fig. 42, two other cases; namely, the case of a PnC containing N + 1 = 5 layers of Al and N = 4 layers of W (with Al termination on both sides of the PnC) and the case of a PnC containing 5 layers of W and 4 layers of Al (with W termination on both sides). These cases are less favorable than those presented in Fig. 42 and, more especially, the absolute acoustic gap disappears in the latter case.

240 Phononics

We can now briefly discuss the existence and behavior of the omnidirectional reflection band as a function of the geometrical parameters involved in our structure, namely, the thickness dSi of the Si layer and the number N of unit cells in the PnC. The maximum tolerance for transmission is chosen to be either 10−3 or 10−2 . A detailed investigation of the transmission coefficients shows that the gap stabilizes for dSi exceeding a thickness of 5.5D and N greater than 4. In Fig. 45, we present the variation of the gap as a function of dSi for three values of N; namely, N = 5, 8, and 10. In each case, the omnidirectional gap is sketched for both choices of the transmittance threshold. First, let us notice that the limitation about the width of the absolute gap comes from the waves of sagittal polarization, since the gap in the shear-horizontal polarization is relatively broad and already exists for values

5

5

4

4 N= 5

3

3

2

2

1

1

Reduced frequency

–15 –12 –9

–6

–3

0

3

6

9

12

15

5

5

4

4

3

3

N=8

2

2

1

1 –15 –12 –9

–6

–3

0

3

6

9

12

15

5

5

4

4 3

N = 10

3

2

2

1

1 –15 –12 –9

–6

–3

0

3

6

9

12

15

Clad layer thickness dSi Fig. 45 Dependence of the omnidirectional gap with the thickness dSi of the clad layer for different numbers of Al/W bilayers in the PnC: (A) N = 5, (B) N = 8, and (C) N = 10. The gray and dark areas, respectively, correspond to the frequency domains where the transmission exceeds 10−3 or 10−2 . The left and right panels, respectively, refer to shear-horizontal and sagittal acoustic modes.

Chapter | 4 One-Dimensional Phononic Crystals 241

of dSi and N of the order of 1.5D and 2D, respectively. The sagittal gap shown in Fig. 45 widens with increasing the thickness of the Si layer, although some irregular behaviors can be noticed at the edges of the gap. It is even worth mentioning that for N below or equal to 4, the gap may close when going to increasing value of dSi ; this means that the transmission through the guided modes of the clad layer are not efficiently prevented by the PnC. From Fig. 45, one can conclude that the acoustic gap almost reaches its maximum value for N = 6 and dSi = 11D. For the sake of completeness, we also present in Fig. 46 the variation of the gaps as a function of the number of unit cells in the PnC, for dSi = 8D. Finally, we have compared the behavior of the transmission coefficients when an additional Si layer is inserted at different places inside the PnC. It turns out that the best solution is obtained when the Si layer is added as a clad, that is, at the boundary of the PnC. More precisely, with N = 4 and dSi = 8D, there is no absolute gap when the Si layer is inserted inside the PnC.

5.2.2 Coupled Solid-Solid Multilayer Structures In this section, we study the transmission of acoustic waves through a layered structure composed of two coupled PnCs (Fig. 40C) chosen in such a way that the superposition of their band structure displays an absolute band gap. This means, in some frequency range, the mini-bands of one PnC overlap with the mini-gaps of the other, and vice versa.

5

5

4

4 T > 10–3

Reduced frequency

3

3

2

2

(A) 1

1 14

12

10

8

6

4

2

0

2

4

6

8

10 12

14 5

5

4

4 T > 10–2

3

3 2

2

(B)

1

1 14

12

10

8

6

4

2

0

2

4

6

8

10 12

14

Number N of (Al/W) cells Fig. 46 Dependence of the omnidirectional gap with the number N of unit cells in the Al/W PnC. The Si clad is in contact with an Al layer and has a thickness of dSi = 8D. The transmission threshold is fixed to 10−3 (A) and 10−2 (B).

242 Phononics

8

8

6

6

4

4

2

2

Reduced frequency

Reduced frequency

We have investigated several possibilities of elastic and geometric parameters for the coupled PnC structure. Among a few possibilities that give rise to the occurrence of an omnidirectional band gap, one interesting solution consists of combining the Al/W PnC with a Fe/epoxy PnC of the same period D but with d1 = 0.8D and d2 = 0.2D. The superposition of the band structures for these PnCs is shown in Fig. 47 and clearly displays a broad absolute acoustic gap in the frequency range 2.54 < Ω < 5.29 (delimited by the horizontal-dashed lines). One can expect that in this frequency domain, any wave generated in any substrate will be totally reflected. In practice, the coupled PnC structure is of finite width, and one can only impose a maximum tolerance on the transmission coefficients. In the following, we assume that the substrates are made of a low-velocity material such as epoxy. Fig. 48 shows the dispersion curves of the coupled PnC structure together with the frequency domains in which the power transmission does not exceed a threshold of 10−3 . The finite system is composed of an Al/W PnC containing nine layers of Al and eight layers of W, and an epoxy/Fe PnC with six layers of epoxy and five layers of Fe. The omnidirectional reflection band extends from 2.48 to 5.35, and practically coincides with the complete acoustic gap of Fig. 47. Let us notice that there is an interface mode at the boundary between epoxy and the Al/W PnC (see the branch in the upper right corner of Fig. 48, around k D = 4–5 and Ω around 5.5–6); however, this mode does not contribute noticeably to transmission. It is worth mentioning that the choice of the materials that are the terminal layers in each PnC is important for the omnidirectional gap to exist and to have a relatively large bandwidth. Another illustration for the occurrence of the omnidirectional gap is shown in

0

0 –5

–4

–3

–2

–1 0 1 2 Reduced wave vector

3

4

5

Fig. 47 Projected band structures of two different PnCs, namely, Al/W (bright gray) and Fe/epoxy (dark gray). The overlap between both band structures is presented as black areas. The right and left panels represent the sagittal and the transverse band structures, respectively.

Reduced frequency

Chapter | 4 One-Dimensional Phononic Crystals 243 6

6

5

5

4

4

3

3

2

2

1

1

0

0 6

5

4

3

2

1

0

1

2

3

4

5

6

Reduced wave vector

Fig. 48 Dispersion curves of two finite PnCs combined in tandem and embedded between two substrates. The shaded area corresponds to the frequency domain in which the transmission power can exceed a threshold of 10−3 . The Al/W PnC contains N + 1 = 9 layers of Al and N = 8 layers of W. The epoxy/Fe PnC contains N  + 1 = 6 layers of epoxy and N  = 5 layers of Fe.

Fig. 49 where we give, at k D = 2.5, the transmission coefficients for different polarizations of the incident wave, together with the DOS of transverse and sagittal modes. It can be seen that the modes of each PnC that fall inside a gap of the other PnC contribute only a negligible amount to the transmission power. Finally, in Fig. 50, we sketch the effect of the numbers N and N  of unit cells in each PnC on the transmission power. The absolute gap is presented by assuming that the transmission remains below the threshold of 10−3 . The absolute gap starts to open for N and N  , respectively, higher than 4 and 7 (Fig. 50A). However, it is necessary to take values such as N = 8 and N  = 6 to obtain a relatively broad reflection band. Going to higher values of N and N  stabilizes the gap width without a noticeable modification.

5.2.3 Solid-Solid Layered Acoustic Filters and Mode Conversion There exist different ways to realize selective transmission through layered solid-solid structures. One way consists to insert a defect layer (cavity) within the structure. The filtering is carried out through the resonant modes of the cavity. An example is shown in Fig. 51A for a PnC composed of five layers of Al and four layers of W. The cavity is made of epoxy and inserted in the middle of the Al-W PnC. The whole system is embedded between two Si substrates. Fig. 51A shows the dispersion curves associated with defect modes in the first gap of the PnC. Because of the low velocities of sound in epoxy as compared to Si, the defect branch is almost flat and falls around Ω 3.48, which means that the transmission filtering arises around almost the same

244 Phononics

TtransH

1.0

(A)

0.5

0.0 100

(B)

DOS

80 60 40 20 0

0

TtransV

0.0015

1

2

3

4

5

6

1

2 3 4 Reduced frequency

5

6

(C)

0.0010 0.0005 0.0000

(D)

Tlong

0.0010 0.0005

DOS

0.0000 60

(E)

40 20 0

0

Fig. 49 Transmission coefficients and DOS for the coupled (Al/W) and (epoxy/Fe) PnCs, at k D = 2.5. The other descriptions are the same as in Fig. 44.

frequency for all incident angles and polarizations of the waves. Fig. 51B shows the evolution of the maximum of the transmission coefficient as function of the incident angle along the defect branch. Depending on the polarization of the incident wave, one can have two possibilities: (i) the incident wave with shear-horizontal polarization is completely transmitted (straight-horizontal line), (ii) an incident wave with shear-vertical polarization gives rise to two transmitted waves, one longitudinal (dashed-dotted curve) and the other shearvertical (dashed curve). The two latter curves present a noticeable variation for the incident angles 0 degree < θ < 45 degrees with an important conversion of modes from transverse-vertical to longitudinal around θ 19 degrees. For 45 degrees < θ < 90 degrees (i.e., Ct (Si) < C < C (Si)), the longitudinal

Chapter | 4 One-Dimensional Phononic Crystals 245

5 4

5

(A) N = 3

3

3

2

2

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 Reduced frequency

4

0

1

2

3

4

5

6

7

8

9 10

5 4

5

(B) N = 4

4

3

3

2

2

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

0

1

2

3

4

5

6

7

8

9 10

5 4

5

(C) N = 8

4

3

3

2

2

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0

1

2

3

4

5

6

7

8

9 10

Nⴕ Fig. 50 Dependence of the omnidirectional gap with the number N  of unit cells in the epoxy/Fe PnC, for different numbers of unit cells in the Al/W PnC: (A) N = 3, (B) N = 4, and (C) N = 8. The transmission threshold is fixed to 10−3 .

component of the transmitted wave vanishes, whereas the transverse component continues to exist. The number of defect branches inside the omnidirectional gap depends on the size of the defect layer, this number increases as function of the thickness of the defect layer. Let us mention that the existence and the behavior of localized sagittal modes induced by defect layers within PnCs have been the subject of recent studies [196, 197]. Resonances and mode conversions of phonons scattered by PnCs with and without inhomogeneities have been discussed [197–199]. In addition, group velocities in the infinite and finite PnCs have been calculated [200, 201]. In a frequency gap, their magnitude in the finite PnC becomes much larger than that in the band region, and increases as the periodicity N increases [201]. This N dependence is qualitatively different depending on whether the gap in the corresponding infinite PnC is due to the intramode or intermode Bragg reflection. The frequency gaps associated with intramode and intermode reflections lay, respectively, at the edges and within the BZ. The latter modes are strongly related to the conversion mode effect.

wD/Ct(AI)

246 Phononics

(A)

6

6

5

5

4

4

3

3

2

2

1

1

0

0 3

2

1

0

1

2

3

k||D 1.0

Transmission

0.8

0.6

0.4

0.2

0.0

(B)

0

20

40

60

80

θ Fig. 51 (A) Band gap structure of transverse and sagittal modes as described in Fig. 41. The bold line inside the omnidirectional gap represents the defect branch induced by a cavity layer made of epoxy inserted in the middle of the finite Al/W PnC embedded between two Si substrates. The thick (thin) straight and dashed lines give, respectively, the transverse and longitudinal velocities of sound in Si (epoxy). (B) Amplitudes of the transmitted waves along the defect branch in (A) as a function of the incident angle θ . The horizontal line with total transmission corresponds to shear-horizontal wave, whereas dashed and dashed-dotted curves correspond to shear-vertical and longitudinal transmitted waves, respectively.

5.3 Case of Solid-Fluid PnCs As in Section 5.2, the goal of this section is to examine the condition for the existence and behavior of omnidirectional band gaps in finite solid-fluid layered media. Let us first come back to the band gap structure shown in Fig. 35 for a PnC made of Plexiglas and water with the same thickness ds = df = D/2. One can notice that the band gap structure of the infinite Plexiglas-water PnC does not display any absolute gap, this means a gap existing for every value of the wave vector k . Fig. 52A reproduces the results given in Fig. 35 for a finite Plexiglas-water PnC made of N = 8 cells. The discrete modes are

Chapter | 4 One-Dimensional Phononic Crystals 247

8

8 w D/vt (Plexiglas)

10

w D/vt (Plexiglas)

10

6 Partial gap

4

Absolute gap

4 2

2

0

0 0

(A)

6

2

4

6 k||D

8

0

10

(B)

2

4

6

8

10

k||D

Fig. 52 (A) Dispersion curves of a finite PnC composed of N = 8 Plexiglas layers immersed in water. The thicknesses of Plexiglas and water layers are equal. The discrete modes correspond to the frequencies obtained from the maxima of the transmission rate that exceeds a threshold of 10−3 . (B) The same as (A) but here the PnC is cladded with an Al layer of thickness d0 = 7D on one side.

obtained from the maxima of the transmission rate that exceeds a threshold fixed to 10−3 . One can notice that any wave launched from water will display a partial gap for an incident angle 0 degree < θ < 35 degrees in the frequency region 4.015 < Ω < 5.105 indicated by horizontal lines. However, waves with incident angles 35 degrees < θ < 90 degrees will be totally transmitted through the discrete modes of the PnC as it was discussed in Section 4.2. These results remain valid for any incident liquid medium as, in general, the velocities of sound in most liquids are of the same order or less than water. In order to overcome this limitation or at least facilitate the existence of an omnidirectional gap, we proposed, like in the previous section on solid-solid PnCs, two solutions. The first one consists to clad the PnC on one side by a buffer layer of high acoustic velocities, which can act as a barrier for the propagation of phonons. The second solution consists to associate in tandem two PnCs in such a way that their band structures do not overlap. The calculation procedure used to deduce the expressions of transmission coefficients for a PnC with or without a defect layer as well as for the association of two PnCs will be developed later in Section 5.3.3.

5.3.1 Cladded Solid-Fluid PnC Structure Fig. 52B shows the discrete modes associated with the cladded-PnC structure, that is, the frequency domains in which the transmission rate exceeds a threshold of 10−3 . In this example the clad layer is made of Al with transverse and longitudinal velocities of sound (dashed and straight lines) higher than the PnC bulk modes lying in the frequency region 4.015 < Ω < 5.105 (Fig. 52A and B).

248 Phononics

The thickness of the Al layer is d0 = 7D and the PnC contains N = 8 cells of Plexiglas-water. By combining these two systems, the allowed modes of the PnC and the guided modes induced by the Al clad layer above its velocities of sound do not overlap over the frequency range of the omnidirectional gap. This means that each system acts as a barrier for phonons of the other system. In such a way, one obtains an omnidirectional band gap indicated by the two horizontal lines in Fig. 52B in the frequency region 4.015 < Ω < 5.105. By comparing Fig. 52A and B, one can notice clearly that the presence of the clad layer has two opposite effects. It decreases the transmittance in some frequency domains (essentially below the sound line defined by the transverse velocity of sound in the clad), but also introduces new modes that can contribute themselves to transmission. The transmission by the latter modes is prevented by the PnC when the corresponding branches fall inside the mini-gaps. In the allowed frequency regions belonging to both the PnC and the clad layers, one can notice an interaction and an anticrossing of the modes associated with these two systems. To give a better insight into the behavior of the transmission coefficient, we shown in Fig. 53 the transmission rates through the cladded-PnC system for three reduced wave vectors: k D = 0 (Fig. 53A), 1 (Fig. 53B), and 3 (Fig. 53C). One can see clearly the common forbidden region in the transmission spectra indicated by the two vertical lines, showing the mirror effect played by the layered structure in the frequency region 4.015 < Ω < 5.105. For the sake of comparison, we have also shown in Fig. 53D–F the DOS (or equivalently the phase time). One can notice that the Al clad layer induces guided modes which appear as peaks in the DOS. These modes do not contribute to transmission when they fall inside the mini-gap of the PnC as it is clearly shown for the modes lying in the frequency region 4.015 < Ω < 5.105 in Fig. 53D–F in comparison to Fig. 53A–C, respectively. The existence and behavior of the omnidirectional reflection depends on the geometrical parameters involved in the structure, namely, the thickness d0 of the Al layer and the number N of unit cells in the PnC. In Fig. 54A, we present the variation of the gap width as a function of d0 for a finite PnC made of N = 8 cells. The omnidirectional gap widens with increasing the thickness of Al and reaches a constant width for d0 > 3D. Similarly, if the thickness of the Al layer is fixed to d0 = 4D, then a finite PnC made of at least N = 7 cells is required to reach a large omnidirectional gap (see Fig. 54B).

5.3.2 Coupled Solid-Fluid Multilayer Structure As mentioned in the case of solid-solid multilayered structure, the second solution which enables us to perform omnidirectional transmission gaps consists of considering a combination of two PnCs chosen in such a way that the superposition of their band structures displays an absolute band gap. This means that the mini-bands of one PnC overlap with the mini-gaps of the other PnC, and vice versa in some frequency range. An example showing this property

Chapter | 4 One-Dimensional Phononic Crystals 249

(A)

90

(D)

0.8 k||D = 0 60 0.4 30 0 90

(B)

0.8

(E)

k||D = 1

DOS

Transmission

0.0

0.4

60 30

0.0

0 90

(C)

0.8

(F)

k||D = 3

60 0.4 30 0.0

0 0

2

4 6 8 wD/vt(Plexiglas)

10

0

2

4 6 8 wD/vt(Plexiglas)

10

Fig. 53 Transmission rate (left panel) and DOS (right panel) as a function of the reduced frequency Ω for the finite PnC shown in Fig. 52B and for three values of k D: k D = 0 (A) and (D), k D = 1 (B) and (E), and k D = 2 (C) and (F).

is shown in Fig. 55A for a combination of the band structures of two PnCs made of Plexiglas-Hg and PVC-Hg. The periods of the two PnCs as well as the thickness of the corresponding layers are supposed to be equal: d(Plexiglas) = d(PVC) = d(Hg) = D/2. The band structures of Plexiglas-Hg and PVCHg PnCs are indicted by black and gray areas, respectively. The superposition of these two types of bands clearly displays two broad acoustic gaps in the frequency ranges 2.72 < Ω < 4.94 and 5.14 < Ω < 5.96. One can expect that in these frequency domains, an incident wave launched from any semiinfinite fluid will be totally reflected. In practice the two-coupled PnC structure is of finite width, and one can only impose a maximum threshold on the transmission coefficient (T > 10−3 ). An example is shown in Fig. 55B where we have considered two finite PnCs made of N1 = 4 layers of Plexiglas and N2 = 4 layers of PVC immersed in Hg. The two omnidirectional band gaps fall in the frequency ranges 2.72 < Ω < 4.94 and 5.14 < Ω < 5.96 and practically coincide with acoustic band gap of Fig. 55A. Similarly to the cladded-PnC structure discussed previously, the bulk

250 Phononics 10

wD/vt (Plexiglas)

8 6

N=8

4 2 0 0

2

4

(A)

6

8 d0(Al)

10

12

14

10

wD/vt(Plexiglas)

8 6 d0(Al) = 4D

4 2 0 0

(B)

5

10 N (number of cells)

15

20

Fig. 54 (A) Variation of the omnidirectional gap width as a function of the thickness d0 of the Al clad layer for a finite PnC made of N = 8 cells. (B) Variation of the omnidirectional gap width as a function of the number N of unit cells in the finite PnC for a fixed thickness of the Al layer such that d0 = 4D.

modes of each of the two PnCs may give rise to well-defined peaks in the DOS within the omnidirectional gaps (not shown here). However, these modes do not contribute to the transmission spectra. Obviously, the width and the position of the omnidirectional gaps depend upon the relative widths of the layers in each PnC but also upon the numbers N1 and N2 of cells in each PnC. Let us mention that a partial gap obtained from the association of two solid-fluid PnCs has been shown theoretically and experimentally [103] for normal incidence.

5.3.3 Solid-Fluid Layered Acoustic Filters In this section we shall discuss the possibility of acoustic waves filtering through the band gap of solid-fluid layered media. This selective transmission can be realized either by inserting a defect layer within the finite PnC or through the

Chapter | 4 One-Dimensional Phononic Crystals 251

8

8 wD/vt (Plexiglas)

10

wD/vt (Plexiglas)

10

6 4 2

Absolute gap p

te

4

lu so

ga

Ab

2

0

0 0

(A)

6

2

4

6 k||D

8

10

0

(B)

2

4

6

8

10

k||D

Fig. 55 (A) Band structures for two different PnCs, namely Plexiglas-Hg (dark areas) and PVCHg (gray areas). The thickness of the layers are considered to be the same: d(Plexiglas) = d(PVC) = d(Hg) = D/2. (B) Dispersion curves for two coupled finite PnCs structures made of N1 = 4 layers of Plexiglas and N2 = 4 layers of PVC immersed in Hg. The discrete modes are obtained from the maxima of the transmission rate that exceeds a threshold of 10−3 . The straight line indicates the Hg sound line.

modes induced by the interface between the PnC and a homogeneous fluid medium. 5.3.3.1 Transmission Through Resonant Cavity Modes The cavity modes can be created in the solid-fluid PnC by replacing for example a fluid layer of width df in the cell (n = P) by a different fluid of width d0 and characterized by the density ρ0 and the sound velocity v0 . Consider a solid-fluid PnC with a finite number N of cells and containing a defect fluid in the cell P (1 < P ≤ N), the whole structure is inserted between two fluids characterized by their density ρs and sound velocity vs (Fig. 33E). The transmission coefficient through the system described earlier can be obtained in the same way as for the structure without defect (Section 4.1). It consists in coupling two finite PnCs made of P and N − P cells by a fluid layer and inserting the whole system between two fluid media (Fig. 33E). The inverse of the Green’s function in the space of interfaces M = {(1, f , d2f ), (P + 1, f , − d2f ), (P + 1, f , d2f ), (N + 1, f , − d2f )} of the whole system is given by a superposition of the Green’s functions matrices associated with the different media constituting the system, namely: ⎞ ⎛ B(P) 0 0 A(P) − Fs ⎟ ⎜ b0 0 B(P) A(P) + a0 ⎟, g−1 (MM) = ⎜ ⎠ ⎝ A(N − P) + a0 B(N − P) 0 b0 0 0 B(N − P) A(N − P) − Fs (180)

252 Phononics

where A(P), B(P), A(N − P), B(N − P) are defined by Eqs. (126)–(128) for N = P and N = N − P, a0 and b0 are given by Eq. (106) for a layer fluid labeled “0” and Fs is the same as in Eq. (106) for a semiinfinite fluid labeled “s.” From Eqs. (137), (180), one can deduce the transmission coefficient as follows: 2Fs b0 B(P)B(N − P) , (181) t= Ψ (P)Ψ (N − P) − b20 [A(P) − Fs ][A(N − P) − Fs ] where Ψ (l) = B2 (l) − [A(l) − Fs ][A(l) + a0 ],

(182)

for l = P, N − P. It is well known that the introduction of a defect layer (cavity) in a periodic structure can give rise to defect modes inside the band gaps [62, 67– 74, 102]. These modes appear as well-defined peaks in the DOS; however, their contribution to the transmission rate depends strongly on the position of these defects inside the structure. Indeed, as it was shown before, a defect layer placed at the contact between the PnC and the substrate (clad layer) induces guided modes in the band gap of the PnC but without contributing to the transmission. However, the transmission through these modes can be significantly enhanced if the cavity layer is placed at the middle of the structure [5, 69, 72–74]. In general, a periodic structure made of N cells (N > 2) is needed to create a transmission gap in which a defect mode is then introduced for filtering. In this section, we are interested to show that contrary to solid-solid PnCs, it is possible to achieve large gaps as well as sharp resonances inside these gaps with a solid-fluid structure as small as a solid-fluid-solid sandwich triple layers (i.e., N = 2, see Fig. 33B). This property is associated with the existence of zeros of transmission. Fig. 56A shows the transmission rate as a function of the reduced frequency Ω for a finite Plexiglas-water PnC composed of N = 2 (solid curves) and N = 4 (dotted curves) cells and for incidence angle θ = 35 degrees. The fluid and solid layers have the same width df = ds = D/2. One can notice that the transmission rate exhibits a large dip in the frequency region 4 < Ω < 8 around the transmission zero indicated by an open circle on the abscissa. This transmission gap maps the band gap of the infinite system indicated by solid circles on the abscissa. As it was discussed earlier, the transmission gap becomes well defined as far as N increases. Now, if a fluid cavity layer of thickness d0 = D is inserted in the middle of the structure, then a resonance with total transmission can be introduced in the gap (Fig. 56B). This resonance falls at almost the same frequency and its width decreases when N increases. Let us mention that the structure shown in Fig. 56A and B with N = 2 consists on a sandwich system made of two Plexiglas layers separated by a water layer. Therefore, such a small-size structure clearly shows the possibility of obtaining a large gap and a sharp resonance inside the gap by just tailoring the width of these three layered media. This property is specific to solid-fluid structures

Chapter | 4 One-Dimensional Phononic Crystals 253 q = 35°

1.0

d0 = 0.5D

0.8 0.6 0.4

N=4 N= 2

0.2 0.0

(A)

0

2

4

6

8

10

8

10

Transmission

1.0 d0 = 1D

0.8 0.6 0.4 N=4 N= 2

0.2 0.0

(B)

0

2

4

6

1.0 d0 = 1D d0 = 0.8D d0 = 0.65D d0 = 0.6D

0.8 0.6

N= 2

0.4 0.2 0.0 4

(C)

5

6 wD/vt (Plexiglas)

7

Fig. 56 (A) Transmission rate for a finite PnC composed of N = 2 (solid curves) and N = 4 (dotted curves) Plexiglas layers immersed in water at an incidence angle θ = 35 degrees. The solid and open circles on the abscissa indicate the positions of the band gap edges and transmission zeros, respectively. (B) Same as in (A) but in presence of a defect fluid layer of thickness d0 = D at the middle of the structure (see the structure in Fig. 33E). (C) Same as in (B) for N = 2 and different values of the thickness d0 of the cavity fluid layer as indicated in the inset.

and is without analog for their counterparts solid-solid systems where at least a number N > 2 of layers is needed to achieve well-defined gaps and cavity modes. In what follows, we shall focus on the simple case of sandwich system (i.e., N = 2). An important point to notice in Fig. 56B is the shape of the resonance lying in the vicinity of the transmission zero. Such a resonance is called Fano resonance [202]. The origin and the asymmetry Fano profile of this resonance

254 Phononics

was explained as a result of the interference between the discrete resonance and the smooth continuum background in which the former is embedded. The existence of such resonances in 2D and 3D PnCs, the so-called locally resonant band gap materials [203, 204], has been shown recently [205–207]. Some analytical models have been proposed to explain the origin and the behavior of these resonances [205–207]. In the case of 1D model proposed here, the Fano resonance in Fig. 56B is just an internal resonance induced by the discrete modes of the fluid layer when these modes fall at the vicinity of the transmission zeros induced by the surrounding solid layers. By decreasing the width of the fluid layer from d0 = 1D to d0 = 0.6D (Fig. 56C), one can notice that the position of the Fano resonance moves to higher frequencies, its width decreases and vanishes for a particular value of d0 = 0.71D before increasing again. At exactly d0 = 0.71D, the transmission vanishes and the resonance collapses giving rise to the so-called ghost Fano resonance [208]. Around d0 = 0.71D, the asymmetric Fano profile of the resonance becomes symmetric and changes the shape. In Fig. 56C, the two solids surrounding the fluid layer have the same widths ds , therefore the transmission zeros induced by the solid layers fall at the same frequency. Now, if the two solids have different widths (labeled for example ds1 and ds2 ), then one can obtain two transmission zeros and a resonance that can be squeezed between these two dips if ds1 and ds2 are chosen appropriately. In this case a symmetric Fano resonance can be obtained whose width can be tuned by adjusting the frequencies of the zeros of transmission. Such resonances have been found also for acoustic [209], magnetic [210], and photonic [211] circuits formed by a guide inserted between two dangling resonators. 5.3.3.2 Transmission Enhancement Assisted by Surface Resonance The possibility of the enhanced transmission from a semiinfinite solid to a semiinfinite fluid, in spite of a large mismatch of their acoustic impedances, has been shown theoretically and experimentally [86–88]. The transmission occurs through the surface resonances induced by a 1D solid-solid layered structure inserted between these two media. These resonances are attributed to the PnC/fluid interface [87] and coincide with the surface modes of the semiinfinite PnC terminated with the layer having the lower acoustic impedance [48]. The possibility of the so-called extraordinary acoustic transmission assisted by surface resonances between two fluids has been shown [212]. The structure consists in separating the two fluids by a rigid film flanked on both sides by finite arrays of grooves. The transmission followed by a strong collimation of sound arises through a single hole perforated in the film. By analogy with the previous works on this subject [87], we show the possibility of enhanced transmission between two fluids by inserting a solidfluid layered material between these two fluids. Besides the possibility of selective transmission, this structure enables from a practical point of view to separate the two fluids which are in general miscible. We give a simple analytical expression of the effective acoustic impedance of the finite PnC that

Chapter | 4 One-Dimensional Phononic Crystals 255

enables to deduce easily the optimal value N of layers in the PnC to reach total transmission. In addition to the amplitude analysis, we study also the behavior of the phase time around the surface resonances as function of N. This study has not been performed before [87]. As in the previous work [87], we consider a structure formed by a finite solid-fluid PnC composed of N solid layers of impedance Zs separated by N − 1 fluid layers of impedance Zf and inserted between two fluids of impedances Zf1 and Zf2 . In the particular case of normal incidence (k = 0) and assuming quarter wavelength layers, that is, vω ds = vωf df = π2 , the inverse of the Green’s function of the finite PnC with free surfaces (Eq. 121) becomes: ⎛  N ⎞ Zs 0 Z f Zf ⎠,  N (183) g(MM)−1 = ⎝ Zs Zf Zf 0 which is equivalent to the inverse Green’s function of a quarter wavelength layer with an effective acoustic impedance Ze = Zf ( ZZsf )N . Then we can use the well-known relation [156] that enables to use an intermediate layer to form an antireflection coating between two different semiinfinite media, namely Zf1 Zf2 = Ze2 . Then we get easily:   Zf1 Zf2 ln Zf2 1   . (184) N= 2 ln Zs Zf This relation requires a suitable choice of the materials in order to get a positive value of N greater than unity. In particular, the solid and fluid media constituting the PnC should have close impedances. An example is shown in Fig. 57 for a PnC composed of Al and Hg and sandwiched between water (incident medium) and Hg (detector medium). The thicknesses of the layers in the PnC are chosen such that vds = dvff . One can see clearly that selective transmission occurs around the reduced frequency ωdf π s Ω0 = ωd v = vf = (2n + 1) 2 for a number of cells such that N = 11 according to Eq. (184). Far from N = 11, the transmission vanishes as it is shown in the inset of Fig. 57. As a matter of comparison, we have also sketched by horizontal line the transmission rate between water and Hg in the absence of the finite PnC. The resonances in Fig. 57 are of Breit-Wigner type [87] with a Lorentzian shape because of the absence of transmission zeros at normal incidence. Zhao et al. [88] have attributed the resonances lying in the middle of the gaps of the PnC to the interference effect of acoustic waves reflected from all periodically aligned interfaces. This explanation is of course correct but a physical interpretation is still needed. We show that the resonances are actually surface resonances induced by the interface between the PnC and water. Indeed, after some algebraic calculations, the dispersion relation giving the surface modes (Eq. 179) of a PnC ended with a solid layer in contact with vacuum

256 Phononics 1.0 1.0

N = 11 Transmission

0.8

Transmission

0.8

0.6

W = p /2

0.4 0.2

0.6 0.0 0 2 4 6 8 10 12 14 16 18 20 22

N 0.4

0.2

0.0 0

p/2

p

3p /2

w d(Al)/v1(Al) Fig. 57 Transmission rate for a finite PnC composed of N = 11 layers of Al separated by N − 1 = 10 layers of Hg. The structure is inserted between water (incident medium) and Hg (detector medium). The inset shows the variation of the maxima of the transmission as a function of the ωdAl = π2 . The straight horizontal line corresponds number of unit cells N for the mode situated at v (Al)  to the transmission rate between water and Hg (i.e., without the finite PnC).

becomes simply Cf = C = 0 where Cf and C are given by Eqs. (107), (112), respectively. Therefore, one obtains surface modes for: Ω0 =

ωds ωdf π = = (2n + 1) . v vf 2

(185)

In addition to Eq. (185), the supplementary condition (178) that ensures the decaying of surface modes from the surface, becomes Zs < Zf .

(186)

This condition is fulfilled in the case of a PnC made of Al-Hg. Now, when the Al layer of the PnC is in contact with water (instead of vacuum), this latter medium does not affect considerably the position of the surface resonances as the impedance of water is much smaller than Al. In order to confirm the above analysis, we have also sketched the LDOS as a function of the space position x3 (Fig. 58) for the mode lying at Ω0 = π/2. This figure clearly shows that this resonance is localized at the surface of the PnC and decreases inside its bulk. Let us notice that the LDOS reflects the square modulus of the displacement field. Therefore, these results show without ambiguity that the transmission is enhanced by surface resonances.

Chapter | 4 One-Dimensional Phononic Crystals 257 1.6 Water

Al-Hg Superlattice

1.2

LDOS

Hg

N = 11

0.8 0.4 0.0 –2 –1

0

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

x3 /D Fig. 58 Variation of the LDOS (in arbitrary units) as a function of the space position x3 /D for the ωdAl surface resonance situated at v (Al) = π2 in Fig. 57 and for N = 11. 

Besides the amplitude of the transmission, we have also analyzed the behavior of the phase time (Fig. 59). One can notice a strong delay time at the frequencies corresponding to surface resonances, reflecting the time spent by the phonon at the PnC/water interface before its transmission. Contrary to the amplitude (see the inset of Fig. 57), the phase time at the surface resonance goes asymptotically to a limiting value (∼110) (in units of ds /v (Al)) when

60 120

N =11 Phase time

100

Phase time

50

80 60 40 20

40

0 0 5 10 15 20 25 30 35 40 45 50

N 30

20

10 0

p/2

p

3p / 2

w d(Al)/v1(Al) Fig. 59

Same as in Fig. 57, but for the transmission phase time (in units of dAl /v (Al)).

258 Phononics

N increases. This result known as the Hartman effect [213] arises for classical waves tunneling through a barrier where the phase time saturates to a constant value for a sufficiently barrier’s thickness. This phenomenon has been observed experimentally [214] and explained theoretically [215–217] in 1D PCs. For a frequency lying in the allowed bands, the phase time (not shown here) increases linearly as a function of N. The above results can be explained in terms of the DOS. Indeed, due to the similarity between the DOS and the phase time (see Section 4.1.4), Fig. 59 reflects also the DOS where the different resonant modes are enlarged because of their interaction with the bulk waves of the surrounding media. When N increases, the number of oscillations in the bulk bands (which is related to the number of cells in the system) and the corresponding DOS increase. However, the behavior is different for the peak associated with the surface resonance. Indeed, for low values of N, the localization of this mode increases as a function of N because the mode interacts less with the second substrate. So, its width decreases and its maximum increases to ensure an area equal to unity under the resonance peak. However, the peak width cannot decrease indefinitely and reaches a threshold because of its interaction with the first substrate. Therefore, the DOS (or the phase time) saturates to a constant value. By using Eq. (155), we have also examined the group velocity vg in such structures and found that vg oscillates around the mean velocity vm = D(df /vf + ds /v )−1 inside the bands, whereas this quantity is strongly reduced around the surface resonance. Therefore, such structures can be used as a tool to reduce the speed of wave propagation. As a matter of completeness we have also checked two other cases: (i) the case where there is no surface resonance in the gap of the PnC. This can be obtained by using Hg on both sides of the structure. In this case, even if Eqs. (185), (186) are satisfied, Eq. (184) gives unacceptable value of N (N < 0). In spite of the absence of surface resonances, the phase time saturates to a constant value (∼17) (in units of ds /v (Al)) at the mid-gap frequencies, because of the Hartman effect [213, 215]. This value is much smaller than in the presence of a surface resonance. (ii) The case where there is two surface resonances in the gap of the PnC. This can be obtained by using water on both sides of the structure. In this case, Eqs. (185), (186) are satisfied and Eq. (184) gives N 22. Because of the existence of two symmetrical surfaces that can support surface modes, one obtains a large surface resonance at Ω0 = π/2, 3π/2, . . . for N = 22. For smaller values of N, this resonance splits into two distinguished resonances around Ω0 because of the interaction between the two surfaces. A total transmission is still obtained at each resonance. On the contrary, for higher values of N (N > 22), there is a single peak in the transmission because the two surface resonances become decoupled, although being enlarged due to their interaction with the substrates. In this case, the transmission peak decreases as far as N increases.

Chapter | 4 One-Dimensional Phononic Crystals 259

5.4 Relation to Experiments 5.4.1 Omnidirectional Band Gap Some years ago, Manzanares-Martinez et al. [83] have demonstrated experimentally and theoretically the occurrence of omnidirectional reflection in a finite PnC made of a few periods of Pb/epoxy and sandwiched between substrates made of Nylon. The parameters of the materials are given in Table 8. The thicknesses of the layers were chosen so that the structure has its omnidirectional gap in the working frequencies of the transducers. They took layers of the same thickness 1 mm so as to generate a gap centered at around 300 kHz. They have displayed the band gap structure for transverse and sagittal acoustic waves. An omnidirectional gap was predicted in the frequency region 273 kHz ≤ f ≤ 371 kHz, which corresponds to the normal incidence band edges of the sagittal modes. However, it is worth noticing that the proposed structure does not have the property of omnidirectional reflection for transverse waves for which the velocity is about half of that of the longitudinal waves. The transmission measurements have been performed for longitudinal incident waves at different angles of incidence for the samples analyzed. They have demonstrated that the transmission is almost negligible in the regions where a gap is predicted by the band structure. In particular, the common gap region has been found in very good agreement with the theoretical predictions. 5.4.2 Selective Transmission An experiment has been realized by Zhao et al. [88] on a layered structure that consists of an alternative stacking of Aluminum (A) and Glass (B) planar sheets, which have the same dimensions: 12×12 cm section and 3 cm thickness. They have used an experimental set-up where the emitter contact transducer is coupled to the substrate using a coupling gel and the last layer is immersed in water (Fig. 60) [88]. The receiver transducer is placed at a distance away from the interface of the last layer B and water. A pulse generator produces a shortduration pulse. The pulses transmitted through the sample were detected by an immersion transducer, which has a central frequency of 0.5 MHz and a diameter of 12.5 mm. Because of the limit of central frequency of the transducers, only the first peculiar transmission peak was studied [88]. The elastic parameters of these materials are given in Table 9. The layers in the PnC are chosen such that ωdA /vA = ωdB /vB = π/2 (i.e., quarter wavelength layers). In the case where the PnC starts with layer A and terminates with layer B, the transmission spectrum shows a peculiar peak in the first gap around f = 0.5 MHz for a finite PnC composed of N 4 periods. An analytical expression giving the number of bilayers necessary to attain the transmission unity has been derived ln(ZB /ZC ) (i.e., N 4 in the present case). The and shown to be N = 12 ln(Z A /ZB ) experimental measurements are found in good agreement with the theoretical

260 Phononics Emitter transducer Substrate B A PCs

B 5900P/R

A

Computer

B A B

Air Water

Receiver transducer

(A)

Transmission

1.0

1.0

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0.0 0.2

(B)

0.4

0.6

Frequency(MHz)

0.0 0.8 0.2

(C)

0.4

0.6

0.0 0.2

0.8

Frequency(MHz)

(D)

0.4

0.6

Water

C

Al

Glass

Glass

1.0

A B A B A B A A Glass

B

Glass

C

Water

Glass

Glass

A B A B A B A Glass

B Glass

C

Water

Glass

Glass

A B A B A B Glass

Glass

B

0.8

Frequency(MHz)

Fig. 60 (A) Configuration of a finite PnC and the liquid detector. The free surface of the sample is immersed in liquid. Two kinds of solid layers (A and B) are alternately stacked in the sample. (B–D) Transmission rate of acoustic waves in water (medium C) for three different samples as described in the insets. Dashed and solid curves represent theoretical and experimental results, respectively. (From D. Zhao, W. Wang, Z. Liu, J. Shi, W. Wen, Phys. B 390 (2007) 159.)

TABLE 9 Elastic Parameters of Aluminum, Glass, and Water Materials

Mass Density, ρ (kg m−3 )

Longitudinal Velocity (m s−1 )

Wave Impedance, Z (kg m−2 s)

Aluminum: A

2.716 × 103

6.17 × 103

16.758 × 106

Glass: B

2.427 × 103

5.40 × 103

13.106 × 106

Water: C

1.00 × 103

1.479 × 103

1.479 × 106

results. Inside the bands, the transmission coefficient oscillates around the transmission value T 0.36 when the wave is transmitted directly from the substrate B (i.e., Glass) to Water without the presence of the finite PnC. In addition to the expression giving the optimized value of N to get the total transmission, the authors have attributed this enhancement to an interference

Chapter | 4 One-Dimensional Phononic Crystals 261

of the acoustic waves reflected from all the interfaces. Whereas, Mizuno [87] has associated these peculiar transmission phenomenon to surface (or interface) waves that can exist between the last layer and the receiver medium (i.e., water). In addition to this structure (considered as the reference system), Zhao et al. [88] have studied also experimentally two other structures: the first one consists on a PnC ending by layer A (i.e. Al) on both sides and the second structure consists on a PnC starting with A layer and ending with two layers A (i.e., layer A with thickness 2 dA ). The first structure did not show any selective transmission around 0.5 MHz, whereas the second structure exhibits the reappearance of the transmission mode. The authors have explained in terms of interference phenomenon between the different materials how the selective transmission can appear and disappear depending on the nature of the layer in contact with the last substrate (i.e., water). In order to give another insight and explanation to these results, we have taken the same structure as in Fig. 60 (i.e., composed of N periods AB, but ended with a cap layer (e.g., D) in contact with the substrate C as follows: B|A|B|A|B|A|B|. . . |A|B|A|B|D|C. We suppose that A and B are quarter wavelength layers (as in Zhao’s work), whereas the cap layer D can take any thickness dD , velocity vD , density ρD , and impedance ZD . By doing the same calculation as in Section 5.3.3.2, one can show that the transmission coefficient reaches unity in three situations, namely: 1 ln(ZB /ZC ) and ZC = ZD , 2 ln(ZA /ZB ) 1 ln(ZD2 /ZB ZC ) (ii) N = and cos(ωdD /vD ) = 0, 2 ln(ZA /ZB ) 1 ln(ZB /ZC ) (iii) N = and sin(ωdD /vD ) = 0. 2 ln(ZA /ZB ) (i) N =

(187)

These three conditions can explain easily the three spectra in Fig. 60. One can see that Fig. 60B corresponds to the first situation (i) and gives N 4 as in Zhao’s work [88]. Fig. 60C corresponds to the second situation (ii) where the cap layer D = A; in this case one can check easily that N becomes negative which means that this condition cannot be fulfilled and therefore the transmission cannot reaches unity around f 0.5 MHz. Fig. 60D corresponds to the third situation (iii) where the cap layer D = 2A (i.e., the double layer dD = 2dA ). In this case D becomes a half wavelength layer (i.e., sin(ωdD /vD ) = 0 or ωdD /vD = mπ) and N 4. It is worth mentioning that Zhao et al. [88] have also studied theoretically the situation where the receiver substrate presents a high impedance like tungsten for example. In this case, it was found that contrary to the situation where the system is in contact with water, the selective transmission arises when the PnC terminates with A layer (i.e., aluminum) or B layer (i.e., glass) but with double thickness. The optimized number of periods to reach the maximum transmission when ωdA /vA = ωdB /vB has been also examined numerically.

262 Phononics

In summary, we have developed in this section the idea that 1D PnCs can exhibit an omnidirectional reflection band, analogous to the case of 2D and 3D PnCs. This property can be fulfilled with a PnC when the velocities of sound in the substrate are higher than the characteristic velocities of the PnC constituents. In the more general case when the substrate is made of a soft material, we have proposed two solutions to realize the omnidirectional mirror, namely, the cladding of a PnC with a hard material that acts like a barrier for the propagation of phonons, or a combination in tandem of two different PnCs in such a way that their band structures do not overlap over a given frequency range. The latter solution gives rise to a relatively broad band gap, provided an appropriate choice of the material and geometrical properties is made. With the former solution, the contribution of the guided modes induced by the clad layer to power transmission should be more carefully taken into account. The thickness of the clad layer, the number of unit cells in the PnCs, as well as the nature of the terminal layers in the PnCs involved in the structure are also important parameters for determining the maximum tolerance for power transmission. Also, we have shown that layered media can play the role of acoustic filters. In the case of solid-solid multilayers, we have shown the possibility of filtering through guided modes of a cavity solid layer inserted in the middle of the structure. In particular, we have shown the possibility of transmitting around almost the same frequency for all incident angles and polarizations of the waves, if the velocities of sound in the cavity layer are chosen very lower than those in the substrates. In the case of solid-fluid multilayers, we have shown two possibilities of enhanced transmission between two fluids. (i) The first solution consists to insert a cavity fluid layer inside the perfect PnC. We have evidenced that a simple structure as small as a solid-fluid-solid sandwich can exhibit a large gap with sharp resonances of Fano type. This is due to an internal resonance induced by the fluid layer when it falls in the vicinity of the transmission zeros induced by the solid layers. (ii) The second solution consists to insert a finite solid-fluid PnC between two fluids. An effective acoustic impedance of the PnC has been derived which enables to deduce the optimal value of the number N of cells needed to reach the antireflection coating. This occurs around some specific frequencies close to the free surface modes lying in the mid-gap of the PnC, the so-called surface resonances. Contrary to the amplitude, the phase time around these resonances increases monotonically as function of N before saturating at a constant value for a large value of N. These phenomena known as the Hartman effect arise in evanescent regions where waves are traveling by tunneling effect.

6 CONCLUSIONS In this chapter, we have presented a comprehensive theoretical analysis of the propagation and localization of acoustic waves in solid, nonviscous fluid, and piezoelectric layered materials. In general, we have limited ourselves to the

Chapter | 4 One-Dimensional Phononic Crystals 263

case of isotropic materials for which shear-horizontal waves are decoupled from sagittal waves polarized in the plane defined by the normal to the surface and the wave vector parallel to the surface. The phonon modes are particularly emphasized in the case of periodic multilayered systems such as PnCs. This study has been performed within the framework of the Green’s function method, which enables us to deduce the dispersion curves, DOSs as well as the transmission and reflection coefficients. The Green’s function approach used in this work is also of interest for studying the scattering of light by bulk and surface phonons. Despite the problem of acoustic waves in solid and fluid materials has been intensively studied since the beginning of the last century, this subject still attracts attention of researchers because of the high-quality level of control and perfection reached in the growth techniques of microstructures and nanostructures, but also due to the sophisticated experimental techniques used to probe different modes of these systems in different frequency domains. In addition, these systems may present several applications in guiding, stopping, and filtering waves. In practice, the PnCs are composed of finite number of cells deposited on a substrate; however, if the number of cells is big enough, one can treat the free surface independently of the PnC/substrate interface, each system being considered as semiinfinite. Even though surface acoustic waves in semiinfinite PnCs have been intensively studied since the pioneering work of DjafariRouhani et al. [48, 63]; however, a clear evidence about the conditions of their existence lacks. We have demonstrated in this work a general rule about the existence of these modes. More generally, we have shown that the eigenmodes of a finite PnC constituted of N cells with free-stress surfaces are composed of N − 1 modes in each band and one mode by gap which is associated with one of the two surfaces surrounding the system. These latter modes are independent of N and coincide with the surface modes of two complementary PnCs obtained from the cleavage of an infinite PnC along a plane parallel to the interfaces. This rule has been shown to be valid for shear-horizontal waves in elastic and piezoelectric PnCs and sagittal waves in solid-fluid PnCs when the cleavage is produced inside the fluid layer. However, this rule is not fulfilled for sagittal waves in solid-solid and solid-fluid PnCs when the cleavage occurs in the solid layers and one can have zero, one, or even two modes in each gap. Nevertheless, in these latter cases, pseudo-surface (or leaky-surface) modes may exist inside the bulk bands of the PnC. These modes are without analog for shear-horizontal waves. In addition to these rules, we have investigated interface and pseudointerface modes induced by the interface between a homogeneous substrate and a PnC depending on the stiffness of the substrate but also on the layer in the PnC which is in contact with the substrate. We have also given a detailed study about the interaction of the different modes existing in a finite-size PnC, in particular we have emphasized the interaction between the surface and interface

264 Phononics

(or cavity) modes depending on the distance between these two defects. The transmission and reflection coefficients of wave propagation through these systems have shown several new properties as the relation between the DOS and the transmission and reflection phase times, the existence of transmission zeros in solid-fluid PnCs and therefore new gaps in addition to Bragg gaps, the conditions of band gap closings, the existence of Fano resonances, and the possibility of enhanced transmission between two media through cavity modes or interface modes. The application of 1D PnC as acoustic mirrors has been shown. Some experimental results and the interest of the Green’s function calculation in explaining the Raman spectra are also reviewed.

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