Propagation study of Rayleigh surface acoustic wave in a one-dimensional piezoelectric phononic crystal covered with two homogeneous layers

Accepted Manuscript Propagation study of Rayleigh surface acoustic wave in a one-dimensional piezoelectric phononic crystal covered with two homogeneous layers Mohamed Mkaoir, Hassiba Ketata, Mohamed Hedi Ben Ghozlen PII:

S0749-6036(17)31934-1

DOI:

10.1016/j.spmi.2017.11.014

Reference:

YSPMI 5349

To appear in:

Superlattices and Microstructures

Received Date: 14 August 2017 Revised Date:

8 November 2017

Accepted Date: 10 November 2017

Please cite this article as: M. Mkaoir, H. Ketata, M.H.B. Ghozlen, Propagation study of Rayleigh surface acoustic wave in a one-dimensional piezoelectric phononic crystal covered with two homogeneous layers, Superlattices and Microstructures (2017), doi: 10.1016/j.spmi.2017.11.014. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Propagation study of Rayleigh surface acoustic wave in a one-dimensional piezoelectric phononic crystal covered with two homogeneous layers Mohamed Mkaoira, *Hassiba Ketataa,b and Mohamed Hedi Ben Ghozlena a

Materials Physics Laboratory, Faculty of Sciences , Sfax University, 3000 Sfax BP 1171. Tunisia b Preparatory Engineering Institute, Sfax University. Tunisia

*

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Corresponding author: [email protected]

ABSTRACT

In this paper, plane wave expansion and stiffness matrix methods are adopted to analyze the

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dispersion relation of Rayleigh surface acoustic waves in a piezoelectric phononic composite composed of two homogeneous layers (ZnO and AlN) deposited on a one-dimensional

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piezoelectric (111) Si/AlN phononic substrate. The effect of crystallographic orientation of silicon on the dispersion relation is discussed. We found that the width of the gap became larger when the middle layer was introduced. The influence of filling fraction, thicknesses of the film and the middle layer on the band gap width is discussed. In addition, the phase velocity and the electromechanical coupling coefficient for Rayleigh surface modes are calculated versus the filling fraction. A comparison of phononic composite with ZnO/ AlN/

substrate.

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(111)Si layered structure is presented to deduce the interest of introduction of the phononic

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Keywords: Piezoelectric phononic composite; Layered structure; Dispersion relation; Surface

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acoustic wave; Band gap; PWE method; Stiffness matrix method

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ACCEPTED MANUSCRIPT I. INTRODUCTION The propagation study of surface acoustic wave (SAW) in phononic crystals has attracted attention over the last few decades due to their applications use namely as transducers, resonators, acoustic filters etc.[1-3]. In the literature, several studies of the band gap of bulk acoustic wave (BAW) in infinite phononic structures have attracted much interest [6-10].

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Various authors have investigated the existence of surface acoustic waves localized at the free surface of a semi-infinite phononic crystal [10-14]. The dispersion relations of surface and Pseudo surface waves in a two-dimensional periodic structure consisting of AlAs circular cylinders forming a square lattice in a GaAs matrix are proposed by Tanaka and Tamura [10].

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R. Sainidou et al [15] studied the propagation of surface-localized modes through finite slabs of phononic crystals consisting of metallic spheres in a polyester matrix, embedded in air. In addition, experimental studies of surface waves in a phononic crystal have demonstrated the

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existence of band gap [4], [16]. Measurement of frequency band gap of two-dimensional air/silicon phononic crystal using layered slanted finger interdigital transducers (SFIT) was performed by Wu et al [4]. Attention has been given to the piezoelectric phononic crystal [13], [17], interdigital transducers (IDT) and piezoelectric materials are used to generate and receive surface acoustic waves [4], [5]. Hsu et Wu [18] used the plane wave method (PWE) to

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analyze the Bleustein-Gulyaev-Shimizu (BGS) and the Rayleigh surface waves in a ZnO/ CdS piezoelectric periodic structure. They observed that the BGS wave has larger band gap width than those of the Rayleigh waves. Yang et al [19] studied the propagation of Rayleigh surface wave in a one-dimensional PZT-2/ZnO piezoelectric phononic crystal covered with a Si

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elastic film using the PWE method and the finite element method (FEM), and they have obtained that the band width gap decreases as the thickness of the film increases. Pang et al

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[20] used the stiffness matrix method to investigate the characteristics of elastic waves propagation in a one-dimensional (1D) piezoelectric/piezomagnetic phononic crystal with line defect. Moreover, Farjallah et al [21] used the plane wave expansion and stiffness matrix methods to study the transmission spectrum and the guided modes of phononic plates. We know that ZnO/AlN/Si layered structures are commonly used to model SAW devices [22-25]. Generally, the use of a ZnO film makes it possible to design high frequencies SAW devices with a large electromechanical coupling. [24]. However, the introduction of a phononic substrate allows the creation of frequency band gaps with which the wave propagation can be modulated. Due to the periodicity of the structure, the dispersion curves will be folded at the reduced Brillouin zone boundary for BAW and SAW modes. It is then interesting to control

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ACCEPTED MANUSCRIPT the evolution of band gap of surface modes by varying the geometrical and physical parameters such as filling factor, material orientation and layer thicknesses which is the objective of this study. This paper investigated the propagation of Rayleigh surface acoustic waves for a half infinite one-dimensional (111)Si/AlN piezoelectric phononic crystal (P.C) and for the P.C covered with two homogeneous piezoelectric layers (ZnO and AlN) using the

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plane wave expansion (PWE) and stiffness matrix methods (SMM). The folding effect of bulk and surface modes is observed for the simple and composite phononic structures, but no band gap exists for SAW in PC. The choice of the (111) Si orientation and the introduction of an AlN as middle layer play a positive role in increasing the band gap width for the phononic

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composite surface modes. The influence of filling fraction on phase velocity and electromechanical coupling coefficient (ECC) is also determined. With comparison to the homogeneous ZnO/AlN/ (111)Si layered structure, the use of phononic substrate allows to

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obtain higher frequencies of surface modes. Moreover, the coupling coefficient, along the first Brillouin zone, appears with a different behavior when changing the filling fraction. We hope these findings could be relevant to understand the properties of the phononic composite surface modes and to provide flexible choices to meet real engineering applications.

A. System description

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II. BASIC FORMULATION

Consider a composite structure consisting of three compartments, a piezoelectric film (ZnO), a piezoelectric middle layer (AlN) and a half-infinite phononic material, as shown in Fig.1.

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The phononic crystal consists of two materials A and B with lattice constant a , which occupies the half space z>0, A is the silicon ((1 1 1) Si), and B is the piezoelectric material

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(AlN) polarized in z direction. h1 is the film thickness and h2 is the thickness of the middle layer. We consider the propagation of Rayleigh wave along x direction with a polarization in the (x, z) sagittal plane.

h1

ZnO

h2

AlN

x

A B A B A B A B A

z Fig.1. Schematic configuration of phononic composite 3

ACCEPTED MANUSCRIPT B. Plane wave expansion method We consider the two uniform layers occupying the lower space z <0 as a phononic crystal. Therefore, the plane wave expansion method is applied for each layer to calculate the dispersion relation of Rayleigh surface acoustic waves (RSAW) propagating in the xdirection. The equation governing the propagation of elastic waves in an inhomogeneous

ρ

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medium with no body force are given by: ∂ 2ui ∂σ ij = ∂t 2 ∂x j

(1)

∂Di =0 ∂xi

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(2)

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Where i, j =1, 3. ρ(x) is the position-dependent mass density u i ( x, z , t ) is the mechanical displacement, σ ij ( x, z , t ) and Di ( x, z , t ) are the stress and electric displacement vector fields, respectively. The piezoelectric constitutive equations can be expressed as the following forms

σ ij = Cijkl S kl − ekij Ek

(3)

Di = eikl S kl + ε ik Ek

(4)

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Where C ijkl , e kij , and ε ik are the elastic stiffness constants, piezoelectric constant and dielectric permittivity. E k is the electric field

S kl is the elastic deformation tensor can be given by

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S kl =

1 ∂u k ∂ul ( + ) 2 ∂xl ∂xk

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By the quasi-static approximation, the electric field E k can be given as ∂φ Ek = − ∂xk

(5)

(6)

where ( k, l =1,3), φ is the electric potential Due to the periodicity of the structure along the x-direction, the material constants can be expanded in Fourier series as follows:

α ( x) =

∑α e

iGx

(7)

G

G

where α ( x ) = ( ρ ( x ), C ijkl ( x ), e kij ( x ), ε ik ( x )) , G = {− nπ / a, nπ / a} is the one dimensional reciprocal lattice vector with integer n = 0,±1,±2,....... ± M . The summation over G is 4

ACCEPTED MANUSCRIPT truncated to N values (N=2M+1). α G is either ρ G , C Gijkl , eGkij , ε Gik , which are the corresponding Fourier coefficient of the material constants, that can be calculated by the integral as:

αG =

a

1 a

∫ α ( x) e

−iGx

(8)

dx

0

According to the Bloch theorem, the mechanical displacement, electric potential, stress and

∑e

(e iGx e ik z . z )

G

∑e

i ( k x . x −ωt )

(e

iGx ik z . z

G

e

U iG   G Φ 

(9)

TijG  )  d iG 

(10)

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σ ij   ( x, z , t ) =  Di 

i ( k x . x −ωt )

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u i   ( x, z , t ) = φ 

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electric displacement can be expanded in Fourier series, one obtains:

where k x is a Bloch wave vector, k z is the wave number along the z-direction, ω is the circular frequency, U iG and Φ G are the amplitude of the displacement vector and electric potential, TijG and d iG are the amplitude of stress and electric displacement respectively.

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C. Surface waves

It is useful to define a generalized displacement vector U = [U 1G ' , U 3G ' , Φ G ' ]T and a generalized stress vector T = [iT13G ' , iT33G ' , id 3G ' ]T , and to establish a basic eigenvalue equation, we have used

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the well-known properties of first-order ODE’s ‘‘Ordinary differential equation’’. In this approach, we have compiled the generalized displacement vector and the generalized stress

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normal to the surface vector in the state vector ξ = [U , T ] = [U1G ' ,U 3G ' , Φ G ' , iT13G ' , iT33G ' , id 3G ' ]T . T

By substituting equations (5-10) into equations (1) and (2) then into equations (3) and (4) and by separating the amplitude expansion of k z with that of other amplitudes, we obtain two sub systems (11) and (12) as:

M .U − k z .( L.U + I .T ) = 0

(11)

N .U + I .T − k z .( R.U ) = 0

(12)

Collecting the equation (11) into equation (12), we obtain an expression of generalized eigenvalues problem

M N 

0 L I [ξ ] = k z    [ξ ] I  R 0

(13) 5

ACCEPTED MANUSCRIPT The explicit expressions of the 3N×3N matrices M, N, L, and R, are given in Appendix, I is

 M 0 L I  the identity matrix. We define B =  , and C =   R 0 are 6N×6N matrices. N I    Equation (13) can be rewritten in the following form:

Aξ = k zξ With A = C −1.B

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(14)

For the case of Rayleigh surface acoustic wave, in each layer and in the substrate, the 6N eigenvalues k z of equation (14) are the wave numbers of the plane waves along the z-

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direction; hence, for each layer, we separate the set of 6N eigenvalues S (k z ) into two subsets: the S1 ( p) associated with 3N positive partial waves (propagating down) and having a positive

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imaginary part, the S 2 (m) associated with 3N negative partial waves (propagating upward) and which contains a negative imaginary part. The sum of positive and negative partial waves provides the solution of equation (14), we obtain:

 u~ ( x, z, t )   ~  = e −iωt i σ ( x , z , t )  

∑e

i ( k x +G ). x

G

p r  3N   p V ik z G . . A p e z ,r . r  +  it   r =1 r  3,G  

∑

3N

∑A e

m m ik z , r . z r

r =1

VGr  . r   it   3,G 

m

   

(15)

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 σ 13 ( x, z , t )   u1 ( x, z , t )      where u~ ( x, z , t ) =  u3 ( x, z , t )  and σ~ ( x, z , t ) =  σ 33 ( x, z , t )   D ( x, z , t )   φ ( x, z , t )     3 

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Arp , Arm are, respectively, amplitudes of 3N positive and 3N negative partial waves  V r   V r   The set   G  ,  G   represents (3N+3N) eigenvectors associated to the  r   r  p

 it 3 ,G  

{

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  it 3 ,G 

m

}

corresponding (3N+3N) eigenvalues k zp,r , k zm,r of the partial waves, whose dimensions are 6N. In the case of a semi-infinite substrate occupying the space (z >0), we must have a positive imaginary part associated with 3N positive partial waves (propagating down). In that case, we write the solution of equation (14) as follows:

 u~( x, z, t )   ~  = e −iωt  iσ ( x , z , t ) 

∑e G

i ( k x +G ). x

p  3N  r   p ik z ,r . z  VG  p  . A e . r  it    r =1 r  3,G   

∑

(16)

D. Boundary Conditions 6

ACCEPTED MANUSCRIPT The mechanical boundary conditions at free surface ( z = − h) are usually requiring the nullity of stress components σ i 3

z=− h

= 0 with ( i =1, 3), h = h1 + h2

(17)

For the electric boundary conditions at free surface ( z = − h) , they can be divided in two cases The open-circuit boundary condition: z=−h

=0

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And the shorted-circuit boundary condition: z=−h

=0

(19)

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φ

(18)

At the interface between the two homogeneous layers at ( z = −h2 ) , both generalized stress

E. Stiffness matrix method

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and displacement vectors are assumed to be continuous.

The stiffness matrix method provides a stable and effective calculation tool for reducing the number of unknown amplitudes relative to partial waves, describing the dynamic state in one layer and at the interface between two successive layers. The boundary conditions at free

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surface and the continuity conditions at the interface between two successive layers and the continuity relationship at the interface ( z = 0) lead to the total stiffness matrix of the system which reduces to a single equivalent layer. Equation (10) can be expressed in the form of a

∑

r ,l p 0  e i ( k x +G ). x   (VG )  .  0 e i ( k x +G ).x   (it r ,l ) p    3,G

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 u~ l  −iωt  =e  iσ~ l   

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matrix system related to a layer l [21], we obtain:

G

(VGr ,l ) m   e ik z ,r . z . (it 3r,,Gl ) m   0 p ,l

  Arp ,l  .  ik zm, r,l . z   m,l  A e   r 

0

Each e i ( k x +G ).x is 3Ν × 3Ν diagonal matrix (three components and N vectors

(20)

kx + G

)

(VGr ,l ) p , (VGr ,l ) m are 3Ν × 3Ν generalized displacement eigenvector matrices, (it 3r,,Gl ) p , (it 3r,,Gl ) m are 3Ν × 3Ν generalized stress eigenvector matrices, e

p,l

ik z,r .z

,e

m,l .z ik z,r

are a 3Ν × 3Ν diagonal

matrices and k zp,r and k zm,r are the eigenvalues (as specified above). The continuity conditions at the interface between (l ) th and (l + 1) th layers are written below: 7

ACCEPTED MANUSCRIPT + −  u~ l   u~ l +1    =    iσ~ l   iσ~ l +1     

(21)

Using the superscripts "+" and "-" to describe the propagation of the wave at the boundary upper (top) and lower (down) of the layer, respectively. By using Eq. (20) and continuity conditions (21), we can express generalized displacement

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= ElT . Al

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D + D − H −   Al+  Tl −1     T  =  + + D −  l  Al−   l  l  D H

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 Al+  and generalized stress vectors in terms of amplitude vector  −  by the following equations: A   l

P + P − H −   Al+  U l −1  =  + +    U  P −  l  Al−   l  l  P H = E lu . Al

H+ =e

ik zp,,rl .hl

,H− =e

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where Al+ = Arp ,l , Al− = Arm ,l , D + = (it3r,,Gl ) p , D − = (it3r,,Gl ) m , P + = (VGr ,l ) p , ik zm,r,l .hl

(22)

(23)

P − = (VGr ,l ) m

, (with hl is the thickness of the layer l )

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By eliminating the amplitude vector Al , we obtain a relationship between stress vector and

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displacement vector for the same layer (l ) :

[ ]

Tl−1  T u T  = El E l  l l

U l−1  U   l l

−1

U l −1  = Kl   U l 

(24)

The equation above defines the stiffness matrix K l of the l th layer as

[ ]

K l = ElT Elu

−1

(25)

Dividing the stiffness matrix K l into four blocks, we obtain:  K11l Kl =  l K  21

K12l   l  K 22 

(26)

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ACCEPTED MANUSCRIPT To assume the continuity of the state vector on boundary separating two successive layers (l) and (l + 1) , we apply the recursive process from the bottom surface ( z = 0) of the film to the free surface ( z = − h) . The total stiffness matrix K N which relates stress to displacement vectors at the interfaces ( z = − h) and ( z = 0) is obtained:

U −h  U   0 

(27)

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T−h  N T  = K  0 

The total stiffness matrix K N is dimensioned 6Ν × 6Ν . To simplify the calculation, K N can be N , K 22N , and the general stress and displacement vectors at divided into four blocks K11N , K12N , K 21

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different interfaces, can be deduced in terms of T0 and U 0 . We denote by Ts and U s the generalized stress and displacement vectors in the substrate. Accordingly, the results are

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written based on Asr ( A sr = A rp with r = 1.....3N ), the set of amplitudes associated with the 3N positive partial waves in the substrate. From boundary conditions, 6N dimensioned vectors T0 and U 0 are replaced by Ts and U s , respectively, which are 3N dimensioned vectors [26]. N −1 N N −1 T−h = ( K12N − K11N ∗ ( K 21 ) ∗ K 22 ) ∗ U s + K11N ∗ ( K 21 ) ∗ Ts N −1 N U − h = ( K 21 ) * (Ts − K 22 ∗U s )

(28) (29)

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As from the equations (28) and (29), one obtains the state vector at all interfaces in function of the amplitudes Asr associated with partial waves in the substrate. Hence, various Fourier amplitude amounts can be deduced according to { Asr } basis, mainly stress vectors amplitude

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( T13G , T33G ), electric displacement vector amplitude d 3G , and electrical potential amplitude Φ G . In the case of a free surface, having the open-circuit condition (OC), the SAW solutions can

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be identified by locating the zeros of the determinant of 3Ν × 3Ν matrices ∆Goc,r as:

iT13G = ∆Goc,r [ Asr ]

iT33G id

G 3

(30)

−h

In the case of a shorted-circuit boundary condition (SC), we must have a non-trivial solution only if the determinant of 3Ν × 3Ν matrix ∆Gsc,r vanishes as:

iT13G = ∆Gsc,r [ Asr ]

iT33G iΦ G

(31)

−h

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ACCEPTED MANUSCRIPT A small frequency shift exists between two zeros determinants ∆Goc,r and ∆Gsc,r , we call by Voc and V sc , the phase velocities for the case of the open-circuit and short-circuit conditions, respectively. Therefore, the electromechanical coupling coefficient (ECC) noted C of a piezoelectric medium can be approximated as [26]: Voc − Vsc Voc

F. Bulk acoustic wave

(32)

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C 2 = 2.

The dispersion relations of piezoelectric bulk acoustic waves can be obtained using the same

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calculation procedure of generalized eigenvalues problem preceding (13) by setting k z = 0 , we must have a non-trivial solution only if the determinant of matrix Bv in equation (33) vanishes.

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 P 0 Q I  [ϕ ] = Bv [ϕ ] = 0  

(33)

Where ϕ = [U1G ' , U 2G ' , U 3G ' , Φ G ' , iT13G ' , iT23G ' , iT33G ' , id 3G ' ]T is the state vector used in the calculation of bulk acoustic waves. The elements of 4N × 4N matrices P and Q are listed in the Appendix. I is the identity matrix.

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III. NUMERICAL RESULTS

A. Comparison with results from literature

To check the validity of the approach and the programs, we will consider an example taken from literature [19]. Fig.2 shows the dispersion curves of Rayleigh wave propagating in a

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half-infinite one-dimensional PZT-2/ZnO piezoelectric phononic crystal covered with an elastic film (Si). Comparing the numerical result obtained by our method with the result

well.

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obtained by Guang-ying YANG and al. [19], we find that both dispersion diagrams agree very

B. Dispersion relation of the phononic crystal Since we consider a one-dimensional piezoelectric phononic crystal with lattice constant a = 8 µ m consisting of alternating Silicon ((111) Si) and Aluminum nitride (AlN) with

filling fraction f=0.4, the elastic wave propagates along the x direction. The independent material parameters used in the calculations are given in Table 1. Fig.3.a shows the dispersion relation of BAW and SAW modes for the (111) Si/AlN phononic crystal, the vertical axis is the frequency, and the horizontal axis is the reduced wave vector k ∗ = k x a / π . k x is the wave vector along the first irreducible Brillouin zone. 10

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ACCEPTED MANUSCRIPT

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Fig.2. Dispersion curves of Rayleigh surface wave (a): result from [19], (b): obtained result

Fig.3.a: Dispersion Relations of BAW, and SAW modes for the (111) Si/AlN phononic crystal (f=0.4, ,) 11

ACCEPTED MANUSCRIPT The bold solid line represents the longitudinal (L) mode, the thin solid line corresponds to the shear vertical (SH) mode, and the thin starred line stands for the shear horizontal (SV) mode. The solid circles represent the Rayleigh surface (RSAW) mode and the open circles correspond to a pseudo-surface wave (PSAW) mode for the phononic crystal. In this figure, we find folding effect of BAW, RSAW, and PSAW modes at the Brillouin zone boundary

bulk mode. There is no band gap for RSAW mode in this model.

C. Dispersion relation of the phononic composite

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because of the periodicity of the structure. We observe that RSAW mode exists just below SV

To analyze the influence of middle layer (AlN) on dispersion relation of the phononic

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composite (Fig.1), we have considered two models. The first model (F.M) is the phononic composite with AlN layer and the second model (S.M) is the same composite without AlN layer. The Rayleigh surface and pseudo surface modes of the phononic composites with and

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without middle layer are shown in Fig.3b. The thickness of ZnO film is h1 = 1.6µm , and the thickness of AlN is h2 = 0.8µm . The blue solid circles represent the Rayleigh surface (RSAW (F.M)) modes and the blue open circles correspond to a pseudo-surface wave (PSAW (F.M) mode for the first model. Moreover, the black solid circles represent the Rayleigh surface (RSAW (S.M)) modes, and the black open circles are those for the pseudo-surface wave

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(PSAW (S.M)) mode for the second model.

Fig.3.b: Comparison of RSAW and PSAW modes of phononic composite with and without AlN (f=0.4, , , ) 12

ACCEPTED MANUSCRIPT We find folding effect of RSAW, and PSAW modes of these two models. The shear horizontal (SV) mode (thin line) of the phononic crystal is represented in Fig.3.b since there is an intersection between the modes types of these two systems. We notice the existence of band gaps width ∆f1 = 25.35 MHz and ∆f 2 = 24.86 MHz for RSAW (F.M) and RSAW (S.M) modes at the boundary X = π / a point ( k ∗ = 1) of the phononic composite with and without

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AlN, respectively. By comparing the band width ∆f 1 and ∆f 2 of these two models, we can note that the introduction of AlN provides a bigger band width. In addition, one can easily see that the second RSAW modes of these two models of phononic composite become PSAW, when there is an intersection with the SV bulk wave [10]. Thus, the PSAW occurs when a

SC

coupling between surface and transverse bulk modes appears.

D. Effect of orientation of silicon

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For further information about the choice of crystallographic orientation of silicon and their effect on the dispersion relation of the phononic composite, we re-oriented the silicon on (001) direction. Fig.3.c shows the RSAW (F.M) modes of the phononic composite with (001)Si. Compared with Fig.3.b, the band gap width ( ∆f = 5.51 MHz) of the phononic composite with (001)Si is much smaller than the band gap width for phononic composite with

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(111)Si.

Fig.3.c: Dispersion Relations of RSAW modes for the phononic composite with (001) Si (f=0.4, , , ) 13

ACCEPTED MANUSCRIPT No folding effect is seen for the RSAW modes, and no PSAW appears by the intersection of RSAW with the bulk modes (SV) in (001) silicon orientation. For this purpose, the structure of phononic composite in Fig.2 with direction (111) Si is chosen for the rest of this study since we obtain a large band gap in this orientation.

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E. Phase velocity and electromechanical coupling coefficients Filling fraction is a key parameter in controlling the band gap width, phase velocity and electromechanical coupling coefficient. Fig.4. shows the variation of band gap width for the Rayleigh surface modes of the phononic composite (Fig.2) as a function of filling fraction f.

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The maximum value of band gap width is 28 MHz at f=0.55.

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Fig.4: Band gap width of the RSAW (F.M) modes of the phononic composite as a function of the filling fraction f

Fig.5. displays the phase velocity of the RSAW (F.M) modes (fundamental and second modes) of the phononic composite calculated along the frequency edges at X = π / a point versus filling fraction f. It can be seen that the phase velocity of the fundamental and second modes increases when the filling fraction increases. Fig.6. illustrates the variation of electromechanical coupling coefficient of the RSAW (F.M) modes at X = π / a point as a function of the filling fraction f. The maximum value of ECC for the second mode is 5.8 % at f=0.35. The fundamental mode decreases progressively with the increase of filling fraction. We find that the second mode has higher coupling coefficients than that of the fundamental 14

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Fig.5: Phase velocity of the RSAW (F.M) modes calculated along the frequency edges at the point, versus filling fraction f

Fig.6: Electromechanical coupling coefficients ( frequency edges at the

) of the RSAW (F.M) modes calculated along the point versus filling fraction f 15

ACCEPTED MANUSCRIPT mode. It should be noted that the difference between the values of ECC, for both modes at filling factor 0.35 - 0.5, are larger than the value at filling factor 0.55 which corresponds to the maximum frequency band width.

F. Effect on band gap of thicknesses h1 and h2

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The values of film thickness and middle layer of phononic composite structure are very important for the control of the band gap width of surface acoustic wave. Firstly, we fixed the thickness of the middle layer ( h2 ) at a value equal to 0.8 µm and we varied the thickness width of the film ( h1 ). Then, we fixed h1 in value 1.6 µm and we varied the thickness width of the

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middle layer. Fig.7 shows the variation of band gap width of the RSAW (F.M) versus the reduced thickness ratio ( h1 / a ) of the film (ZnO), the band gap exists from the ratio h1 / a =

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0.16. We find that the band gap width decreases when the value of h1 / a increases. Fig.8 displays the variation of band gap width of the RSAW (F.M) as a function of the reduced thickness ( h2 / a ) of the middle layer (AlN). It is shown that the gap width increases by the introduction of the AlN intermediate layer, its value changes from 23.2 to 24.8 MHz by

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varying the ratio h2 / a from 0 to 0.07.

Fig.7: Variation of band gap width of RSAW (F.M) as a function of the thickness of ZnO film, for

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Fig.8: Variation of band gap width of RSAW (F.M) as a function of thickness of middle layer (AlN) for .

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G. Comparison of the phononic composite with the homogeneous layered structure Consider a homogeneous structure consisting of three layers, a piezoelectric layer (ZnO) having a thickness h f = 1.6µm , a piezoelectric middle layer (AlN) with a thickness

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hm = 0.8µm , and a semi-infinite substrate ((1 1 1) Si) as shown in Fig.9.

ZnO AlN

x

(1 1 1)Si

z Fig.9: Schematic configuration of ZnO/AlN/ (1 1 1) Si layered structure

Let us consider the propagation of Rayleigh surface acoustic wave in the ZnO/ AlN/ (1 1 1) Si layered structure in the x direction and which is polarized in plan (x, z). As an indication, in 17

ACCEPTED MANUSCRIPT the homogeneous case, the surface mode can be obtained using the same method of calculation as in the case of the phononic composite by taking a filling factor equal to zero (f=0). The reciprocal lattice vector is zero ( G = 0 ). In Fig.10, we conducted a comparison of fundamental mode of the RSAW (F.M) of ZnO/AlN/(111)Si layered structure (Fig.9) with the phononic composite (Fig.1). The black solid line represents the Rayleigh surface mode of

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layered structure, the blue, red and green points correspond to the RSAW (F.M) modes of phononic composite for filling fractions f=0.4, f=0.55 and f=0.7, respectively. We find that the RSAW (F.M) modes in the phononic composite exhibit folding effect with frequency gaps at the zone-edge X = π / a point, but the homogeneous mode will not be folded. This folding

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effect can be explained by the periodicity of the structure. In fact, in phononic structure, the surface modes will be folded and confined to the Brillouin zone edge [27]. We note that the surface mode of layered structure is located at lower frequencies than the surface modes of

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phononic composite, and when the filling factor increases, we obtain a higher surface mode.

Fig.10: Variation of RSAW (F.M) modes for different filling fraction f and with RSAW (F.M) mode of the layered structure

Fig.11 displays the electromechanical coupling coefficients (ECC) of the fundamental mode of RSAW (F.M) for different filling fractions, the horizontal axis is the reduced wave vector k ∗ = k x a / π , and the vertical axis is the coupling coefficients. The plotting in red corresponds

to an ECC for ZnO/AlN/ (1 1 1) Si, layered structure (f=0); in this case the ECC reaches a maximum of 5.4% for k ∗ =0.5. By varying the filling factor, the coupling coefficient 18

ACCEPTED MANUSCRIPT underwent changes; there are two areas of reverse behavior separated by the value of the reduced wave number k p = 0.15 : in the first area ( k ∗ < k p ), the ECC increases when f

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increases, in the second area ( k ∗ > k p ), ECC decreases when f increases.

Fig.11: Electromechanical coupling coefficients ( ) of the fundamental mode of RSAW (F.M) for different filling fractions

IV. CONCLUSION

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In this paper, we used the plane wave method and the Stiffness matrix method to calculate the dispersion relation of RSAW modes of the phononic composite structure consisting of two

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homogeneous layers deposited on a half-infinite one-dimensional piezoelectric (1 1 1) Si/AlN phononic substrate. The bulk and surface modes for the phononic crystal are calculated and discussed. Folding effect of BAW and SAW modes is observed for the simple phononic crystal and the phononic composite structure. The effect of orientation of silicon on the dispersion relation is discussed. The band gap, phase velocity and electromechanical coupling coefficient of the Rayleigh surface modes were successfully calculated. We found that the band gap width decreases when the value of ratio ( h1 / a ) increases. In addition, the effect of the thickness ratio ( h2 / a ) of the middle layer is determined. The comparison of RSAW modes and coupling coefficients for the phononic composite with a ZnO/ AlN/(1 1 1) Si layered structure is analyzed and discussed. Moreover, a higher surface mode is obtained 19

ACCEPTED MANUSCRIPT when the filling factor of phononic composite increases. We also found that, for high value of wave vector, the coupling coefficient in the homogeneous structure is larger than that in the phononic composite, and inversely in the vicinity of the center of the first Brillouin zone. Finally, the results of this paper can offer some fundamental theory for the application of piezoelectric phononic composites structures.

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Acknowledgments The authors are grateful for the funding provided to LPM laboratory by the Tunisian Ministry of Higher Education, Scientific Research.

APPENDIX

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Table 1

Materials

Mass density (Kg m-3)

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Independent parameters of the materials used in the calculations [24].

Elastic constants (1010 N m-2)

C11

C13

C33

ZnO

5720

15.7

8.3

20.8

AlN

3255

41.1

9.9

38.9

Si

2329

16.56

(C m-2)

6.39

16.56

Dielectric constants (10-11 Fm-2)

C 44

e15

e31

e33

ε 11

ε 33

3.8

-0.45

-0.51

1.22

7.57

9.03

12.5

-0.48

-0.58

1.55

9

1.1

0.0

0.0

0.0

10.36

10.36

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ρ

Piezoelectric constants

7.95

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The elements of matrices M, N, L and R in Eq. (13) are expressed as:

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M 11 = −ω 2 ρ G −G ' + C11G −G ' ( k x + G ' )( k x + G ) G −G ' M 22 = −ω 2 ρ G −G ' + C 55 ( k x + G ' )(k x + G )

M 23 = e15G −G ' (k x + G ' )( k x + G ) M 32 = e15G −G ' (k x + G ' )( k x + G ) M 33 = −ε 11G −G ' ( k x + G ' )( k x + G )

M 12 = M 13 = M 21 = M 31 = 0 G −G ' N 12 = C 55 (k x + G ' )

N 13 = e15G −G ' (k x + G ' ) 20

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N11 = N 22 = N 23 = N 32 = N 33 = 0

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L12 = −C13G −G ' ( k x + G ) G −G ' L12 = −e31 (k x + G )

L21 = −C 55G −G ' ( k x + G ) L31 = −e15G −G ' ( k x + G )

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R11 = −C 55G −G '

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L11 = L22 = L23 = L32 = L33 = 0

R22 = −C 33G −G ' G −G ' R23 = −e33 G −G ' R32 = −e33

R12 = R13 = R21 = R31 = 0

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G −G ' R33 = ε 33

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The elements of matrices P and Q in Eq. (33) are expressed as:

P11 = −ω 2 ρ G −G ' + C11G −G ' ( k x + G ' )( k x + G )

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P13 = C15G −G ' ( k x + G ' )(k x + G )

G −G ' P22 = −ω 2 ρ G −G ' + C 66 ( k x + G ' )(k x + G ) G −G ' P31 = C 51 ( k x + G ' )(k x + G )

P33 = −ω 2 ρ G −G ' + C 55G −G ' ( k x + G ' )( k x + G ) P34 = e15G −G ' ( k x + G ' )( k x + G ) P43 = e15G −G ' ( k x + G ' )( k x + G ) P44 = −ε 11G −G ' ( k x + G ' )( k x + G )

P12 = P14 = P21 = P23 = P24 = P32 = P41 = P42 = 0 21

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Q13 = C 55G −G ' ( k x + G ' ) Q14 = e15G −G ' ( k x + G ' ) G −G ' Q22 = C 46 (k x + G ' )

G −G ' Q33 = C 35 (k x + G ' ) G −G ' Q41 = e31 (k x + G ' )

Q12 = Q21 = Q24 = Q32 = Q34 = Q42 = Q43 = Q44 = 0

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REFERENCES

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Q31 = C13G −G ' ( k x + G ' )

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G −G ' Q23 = C 45 (k x + G ' )

[1] J. Pierre, B. Bonello, O. Boyko, and L. Belliard, Refraction of surface acoustic waves through 2D phononic crystals, Journal of Physics: Conference Series, 214 (2010) 012048. [2] A. Khelif, A. Choujaa, B. Djafari-Rouhani, M. Wilm, S. Ballandras, and V. Laude, Trapping and guiding of acoustic waves by defect modes in a full-band-gap ultrasonic crystal,

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Physical Review B 68 (21) (2003) 214301.

[3] T.-T. Wu, S.-M. Wang, Y.-Y. Chen, T.-Y. Wu, P.-Z. Chang, L.-S. Huang, C.-L. Wang, C.-W. Wu and C.-K. Lee, Inverse determination of coupling of modes parameters of surface acoustic wave resonators, Japanese Journal of Applied Physics 41 (11) (2002) 6610-6615.

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[4] T.-T. Wu, L.-C. Wu, and Z-G. Huang, Frequency band-gap measurement of twodimensional air/silicon phononic crystals using layered slanted finger interdigital transducers,

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Journal of Applied Physics 97 (9) (2005) 094916. [5 ] S. G. Joshi and Y. Jin, Excitation of ultrasonic waves in piezoelectric plates, Journal of Applied Physics 69 (1991) 8018-8024. [6] M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B. Djafari-Rouhani, Acoustic band structure of periodic elastic composites, Physical Review Letters 71 (13) (1993) 2022-2025. [7] M. S. Kushwaha, P. Halevi, G. Martinez, L. Dobrzynski, and B.Djafari-Rouhani, Theory of acoustic band structure of periodic elastic composites, Physical Review B 49 (4) (1994) 2313- 2322. [8] C. Goffaux and J. P. Vigneron, Theoretical study of a tunable phononic band gap system, Physical Review B 64 (7) (2001) 075118.

22

ACCEPTED MANUSCRIPT [9] X. Li, F. Wu, H. Hu, S. Zhong, and Y. Liu, Large acoustic band gaps created by rotating square rods in two-dimensional periodic composites, Journal of Physics D: Applied Physics 36 (2003) L15–L17. [10] Y. Tanaka and S. Tamura, Surface acoustic waves in two-dimensional periodic elastic structures, Physical Review B 58 (12) (1998) 7958-7965.

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[11] Tanaka, Y. Tamura, Acoustic stop bands of surface and bulk modes in two-dimensional phononic lattices consisting of aluminum and a polymer, Physical Review B 60 (1999) 13294. [12] Wu, Tsung-Tsong; Huang, Zi-Gui; Lin, S., Surface and bulk acoustic waves in twodimensional phononic crystals consisting of materials with general anisotropy, Physical

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Review B 69 (2004) 094301.

[13] T.-T. Wu, Z.-C. Hsu, and Z.-G Huang, Band gaps and the electromechanical coupling

Physical Review B 71 (2005) 064303.

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coefficient of a surface acoustic wave in a two-dimensional piezoelectric phononic crystal,

[14] Yong Li, Zhilin Hou, Mourad Oudich, and M. Badreddine Assouar, Analysis of surface acoustic wave propagation in a two-dimensional phononic crystal, Journal of Applied Physics 112 (2012) 023524.

[15] R. Sainidou, B. Djafari-Rouhani, and J. O. Vasseur, Surface acoustic waves in finite

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slabs of three-dimensional phononic crystals, Physical Review B 77 (2008) 094304. [16] S. Benchabane, A. Khelif, J.-Y. Rauch, L. Robert, and V. Laude, Evidence for complete surface wave band gap in a piezoelectric phononic crystal, Physical Review E 73 (2006) 065601.

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[17] D. Yudistira, A. Boes, B. Djafari-Rouhani, Y. Pennec, L. Y. Yeo, A. Mitchell, and J. R. Friend, Monolithic Phononic Crystals with a Surface Acoustic Band Gap from Surface Phonon-Polariton Coupling, Physical Review Letters 113 (2014) 215503.

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[18] J.-C. Hsu and T.-T. Wu, Calculations of Lamb wave band gaps and dispersions for piezoelectric phononic plates Using Mindlin’s theory-based plane wave expansion method, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 55 (2008) 431- 441. [19] G-Y. YANG, J-K. Du, and J. Wang, Propagation of surface acoustic wave in onedimensional piezoelectric phononic crystals, IEEE: Conference: Symposium on Piezoelectricity, Acoustic Waves, and Device Applications, (2014) 366-369. [20] Y. Pang, F.-Jiao, and J.-X. Liw, Propagation behavior of elastic waves in onedimensional piezoelectric/piezomagnetic phononic crystal with line defects, Acta Mechanica Sinica 30 (5) (2014) 703–713.

23

ACCEPTED MANUSCRIPT [21] Hajer Farjallah, Hassiba Ketata, Mohamed Hedi Ben Ghozlen, Application of plane wave expansion and stiffness matrix methods to study transmission properties and guided mode of phononic plates, Wave Motion 64 (2016) 68–78. [22] L. Wang, Y. Pu, Y.F. Chen, C.L. Mo, W.Q. Fang, C.B. Xiong, J.N. Dai, F.Y. Jiang, MOCVD growth of ZnO films on Si(1 1 1) substrate using a thin AlN buffer layer, Journal of

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Crystal Growth 284 (2005) 459–463. [23] Dwaraka Maneekanta A ,Amrutha Ch , Lakshmi Sowmya K , Naveen Kumar M, Simulation of AlN based SAW resonator, International Journal of Pure and Applied Mathematics 116 (5) (2017) 177-182.

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[24] S. Maouhoub, Y. Aoura, A. Mir, FEM simulation of Rayleigh waves for SAW devices based on ZnO/AlN/Si, Microelectronic Engineering 136 (2015) 22–25.

[25] Jun-Phil Jung, Jin-Bock Lee, Jung-Sun Kim, Jin-Seok Park, Fabrication and

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characterization of high frequency SAW device with IDT/ZnO/AlN/Si configuration: role of AlN buffer, Thin Solid Films 447 - 448 (2004) 605-609.

[26] Issam Ben Salah, Morched Ben Amor, Mohamed Hédi Ben Ghozlen, Effect of a functionally graded soft middle layer on Love waves propagating in layered piezoelectric systems, Ultrasonics 61 (2015) 145–150.

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[27] Edward Dechaumphai and Renkun Chen, Thermal transport in phononic crystals: The

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role of zone folding effect, Journal of Applied Physics 111 (2012) 073508.

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Analyze of dispersion relation of Rayleigh surface acoustic waves in a piezoelectric phononic composite. Effect of crystallographic orientation of silicon on the dispersion relation is discussed.

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Influences of filling fraction, thickness of the film and the middle layer in the band gap width are determinated. Phase velocity and electromechanical coupling coefficient for Rayleigh surface modes are calculated versus filling fraction.

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Comparison of phononic composite with layered structure.