Surface acoustic waves in two dimensional phononic crystal with anisotropic inclusions

An analysis is given to the band structure of the two dimensional solid phononic crystal considered as a semi infinite medium. The lattice includes an array of elastic anisotropic materials with different shapes embedded in a uniform matrix. For illustration two kinds of phononic materials are assumed. A particular attention is devoted to the computational procedure which is mainly based on the plane wave expansion (PWE) method. It has been adapted to Matlab environment. Numerical calculations of the dispersion curves have been achieved by introducing particular functions which transform motion equations into an Eigen value problem. Significant improvements are obtained by increasing reasonably the number of Fourier components even when a large elastic mismatch is assumed. Such approach can be generalized to different types of symmetry and permit new physical properties as piezoelectricity to be added. The actual semi infinite phononic structure with a free surface has been shown to support surface acoustic waves (SAW). The obtained dispersion curves reveal band gaps in the SAW branches. It has been found that the influence, of the filling factor and anisotropy on their band gaps, is different from that of bulk waves.


Introduction
The propagation of elastic or acoustic waves in periodic heterogeneous materials which is called phononic crystal (PCs) has received much attention in the last fifteen years [1,2,3].One of the properties of composite materials is the possibility of having phononic band gaps, within which sound and vibrations at certain frequencies do not propagate.To probe the acoustic band structure of these 2D composites ultrasound transmission experiments in both the bulk and surface of the structures have been performed [4] These materials are of great interest for many applications, such as transducers, elastic/acoustic filters, noise control, and vibration shields.Most of previous works concentrates on PCs made of elastic isotropic materials; however, band gaps can be enlarged by using non-isotropic materials, such as piezoelectric materials [5].At the micro-scale, phononic crystals are useful for acoustic isolation of vibrating structures [6].Rigid substrate attachment improves yield, quality factor and immunity to environmental noise sources such as vibration.
To study the elastic wave behavior in this kind of system, several numerical analytical methods such as the plane wave expansion method (PWE) [7], the multi-scattering theory (MST) [8], and the finite-different time-domain method (FDTD) [9] have been developed and extensively used.Among them, the PWE method, by which the wave equations are solved in the Fourier space. is mostly used to calculate the band structure [4].
The purpose of the present study is to elucidate theoretically and numerically the characteristics of bulk and surface acoustic waves in 2D phononic crystals.The presence of elastic anisotropy of the two chosen components materials should properly be taken into account.At low frequencies a complete band gap exists for same filling fraction and same mismatch in the elastic constants and mass densities of component materials.The behavior of surface waves with respect to the filling fraction is different from a phononic material to another.

1 Equations of wave motion
We consider a two-dimensional phononic crystal shown in Fig. 1, which is infinite in x 1 and x 2 directions, with lattice constants a and coefficient of implantation f.
The elastic wave propagating in elastic media can be described by these two equations: where i, j, k, l = 1, 2 or 3, and represents the position and time dependant displacement vector and stress tensor respectively.Considering the elastic isotropy of the studied materials, if we state propagation along x 3 direction Eqs (1) and ( 2) can be simplified as According to the periodicity of the structures, material constants (density and rigidity tensor can be expanded in the Fourier series and (5) Based on the Bloch theorem, and can be expanded in Fourier series: where is the wave circular frequency, vector position, and being the unit vectors along the x 1 , x 2 and x 3 axes, : the wave vector in the first irreducible Brillouin zone and reciprocal lattice vectors where n x and n y =0, ±1, ±2,…….±Mif the summation over G is truncated to n values (n= (2M+1) 2 )

Calculation of the band structure of bulk wave
In the case where we study the dispersion curves of the bulk waves in the first Brillouin zone the displacement vector u tress tensor are independent of z then: and Substituting Eqs.( 4) and ( 6) into Eq.(3), we can obtain the following generalized eigenvalue equation in the matrix form: (8) Where is the generalized Fourier amplitudes of displacement vector, (R) and (B) the (3n)x(3n) matrices which are functions of k 0 , G, G' and the Fourier coefficients of material constants.The expressions of matrixes blocks (3x3) are: and By solving equation ( 8) for as a function n of the wave vector k 0 in the first Brillouin zone, the band structures can be built.

Surface waves
In order to obtain the surface wave solution we further put , where solution adopted should have a positive imaginary part A plane wave expansion of Eqs (3) along x 1 and x 2 direction gives 3n equations for amplitude vectors (9) Where L, M, N, J, K are matrix (3x3) depend on, , , (k 0 +G) i , (k 0 +G') i ) Putting, further, , eq (9) is reduced to Eq (10) (10) As well we obtain an generalized eigenvalue equation presented by equation (11) with respect to , which determines the spatial variation of the wave with the distance z from the surface, P, Q are (3n)x(3n) matrix, if we truncate the expressions of Eqs ( 6) and ( 7) by choosing n reciprocal lattice vectors.Equation (11) gives 3n eigenvalues For the surface wave we are seeking solutions for which the lattice displacement may decay exponentially into the medium (z>0) away from the surface z=0.If such a set of s are found for a given frequency the displacement vector of the surface wave takes the form: Where is a unit polarization vectors and the wave amplitudes.The surface wave should satisfy the stress-free boundary condition at the surface z = 0, whether:  14) determines the relative weights of 3n wave amplitudes if the frequency is correctly chosen.Actually we do not know a priori the eigenfrequency of the surface localized acoustic mode, so Eq .(11) and the equation( 16): (16) should be solved simultaneously to obtain .

Numerical Results
We consider the structure where elastic square cylinders (W or Al) of section s=f.a 2 are embedded periodically in a background material (SiO 2 or Si), forming a square lattice with lattice spacing a=10 -2 m (Fig( 1)).The phononic materials studied in the present work are obtained from recent literature [6].Materials are cubic crystals and elastic anisotropy is fully taken into account.Mechanical characteristics are presented in table 1 (annex).The irreducible part of the Brillouin zone is displayed in the figure (2).

Dispersion diagram of bulk waves
We have presented the dispersion curves of bulk waves along the boundary of the irreducible part of the Brillouin zone.Figures (3)

Surface waves
In figure 5 and 6 bulk and surface modes are presented in the direction of propagation GX.There are a few differences among transverse and surface modes.As detailed in the previous paragraph the motion equation has been transformed in eigenvalue problem.The obtained solutions are divided into two sets, the lower and the upper frequency modes.When frequency is low the wavelength is comparatively large and the material behaves as an homogenous one.In addition the Rayleigh surface mode commonly called Ao is observed.For high frequency range several modes appear due to the complex nature of the phononic material.Surface waves frequencies w s at the X point are plotted as the function of the filling fraction f.In W/SiO 2 (figure 7) the both surface wave frequencies takes minimum value at f=0.35.In Al/Si (figure8) both surface waves decrease with f.The investigated two cases Al/Si and W/SiO 2 correspond to a low and high mismatch respectively.The mismatch is crucial to fix the number n of harmonics involved in the Fourier developments.Additionally the largest gaps are associated with W/SiO2 where the mismatch is extremely bigger [see Tab 1].
In figure 9 two branches of surface modes (ws1 and ws2) and gap, that exist only for high value of filing fraction (fig 7) in W/SiO2, are presented.By comparing our results with the results of Tanaka [2] who used the phononic materials AlAs/GaAs, we note that the behavior of phononic materials is very diverse.The dispersion curves, the surface waves, the gaps change each time we change the materials used, the geometry of the implantation, and essentially the ratio between the mechanical properties of both materials.

Conclusion
In the present work we have calculated the dispersion relations of bulk and surface acoustic waves in two-dimensional phononic crystals (PCs) consisting of periodic arrays of cylinders with square shapes embedded in a background substance.The acoustic band structures of 2D phononic crystals have been calculated with the plane-wave-expansion (PWE) method.
As examples, the band structures of Al/Si and W/SiO 2 infinite and semi-infinite PCs are presented.The numerical calculation shows the acoustic band gaps between the dispersive curves result from large contrast in the elastic constants and mass densities of component materials.The calculation also demonstrate that the change in the frequency of the surface waves as a function of filling factor is not the same for the two compounds considered.For Al/Si frequency decreases when f increases, while for W/SiO 2 curve presents a minimum when f=3.5.Finally surface waves gap is present only for high value of filing fraction in W/SiO 2 .
shoes the case W/SiO 2 we note that for f = 0.7, (Fig (a)) there's no gap, while for f=0.25, (Fig.(b)) an absolute gaps are present in three main directions of the Brillouin

Fig. 7 .Fig. 8 .
Fig.7.Frequency of the surface modes(x and •) at the X point vs filling fraction (f) of W/SiO 2 .

Fig. 9 .
Fig.9.Reduced pulsation and gap of the surface modes at the X point, f=0.9 of W/SiO 2

Table 1 .
Figure4shows dispersion curves for Al/Si with f=0.25 but there's no gap, because those materials present a low density mismatch compared for W/SiO 2 (table1) Mechanical characteristics of components phononic materials i:implantation, m:matrice