Interfacial effects in electromagnetic coupling within piezoelectric phononic crystals

Transcription

1 Acta Mech Sin (29) 25:95 99 DOI 1.17/s y RESEARCH PAPER Interfacial effects in electromagnetic coupling within pieoelectric phononic crystals F. J. Sabina A. B. Movchan Received: 14 July 28 / Accepted: 15 July 28 / Published online: 21 October 28 The Chinese Society of Theoretical Applied Mechanics Springer-Verlag GmbH 28 Abstract In this paper we discuss waves in pieoelectric periodic composite with the emphasis on the connection between the electromechanical coupling the effects of dispersion of Bloch Floquet waves. A particular attention is given to structures containing interfaces between dissimilar media localiation of the electrical fields near such interfaces. Keywords Pieoelectric periodic composites Interfaces Dispersion of Bloch Floquet waves 1 Introduction The paper deals with the influence of interfaces on the dispersion of elastic waves in periodic pieoelectric structures. The topics involving pieoelectric photonic/phononic crystals were discussed in Refs. [1 5]. In particular the article [1] presents a model for the transmission problem in a stratified medium with the emphasis on applications in acoustics. Effective stiffness of pieoelectric plate structures was evaluated in Ref. [2]. B gap structures the effects of electromechanical coupling in models of surface acoustic waves were discussed in Refs. [34] for pieoelectric phononic crystals. Surface bulk acoustic waves in two-dimensional F. J. Sabina Instituto de Investigaciones en Matematicas Aplicadas y en Sistemas Universidad Nacional Autónoma de Méico Deleg. A. Obregón Apartado Postal Méico Meico fjs@mym.iimas.unam.m A. B. Movchan (B) Department of Mathematical Sciences University of Liverpool Liverpool L69 3BX UK abm@liverpool.ac.uk phononic crystals were studied in Ref. [3]. Numerical simulations for spectral problems in phononic crystals with pieoelectric inclusions based on the plane wave epansions of the eigensolutions were published in Ref. [5]. In our paper we draw attention of the readers to composite structures where pieoelectric inclusions are embedded into the pieoelectric matri made of the same elastic material. However we construct special types of interfaces either by introducing a thin coating for each inclusion or by changing the polariation of the material in the inclusion compared to the surrounding matri. We begin by stating formal settings for a stratified medium show that an effect of discontinuity in the derivatives of elastic fields is very pronounced for the electromechanical coupling. On the other h the coupling terms present in the equations of motion are small provide no significant influence on the dispersion. We then proceed further with a more challenging setting within a two-dimensional phononic crystal. Here we show that a specially designed interface can lead to a substantial change in the dispersion diagrams appearance of partial phononic b gaps localiation of the electric field within the structure close to the interface. Numerical simulations illustrate the theoretical concept which is further supported by the analysis of the dispersion diagrams maps of sample eigensolutions ehibiting interfacial electromechanical coupling. 2 Formal settings eamples First we address the elementary models of elastic waves in inhomogeneous periodic media with a particular attention given to dispersion properties linked to the electromechanical coupling. For the sake of simplicity we consider two types of crystal symmetry of the cubic 6 mm classes.

2 96 F. J. Sabina A. B. Movchan The periodic medium is assumed to be stratified with the interface boundaries being perpendicular to the Oy ais; we use the notation Ɣ j ={( y ) : hj< y < h( j + 1) ( ) R 2 } where h is the thickness of the layers. All layers Ɣ 2k 1 k Z are occupied by the pieoelectric material of type I whereas Ɣ 2k layers are filled in with the material of type II. The aes of symmetry for both types of materials are aligned. The governing equations involve components u E of the elastic displacement the electric field for 6 mm materials the formulation incorporates u E for the materials of the cubic symmetry. We start with the materials of the cubic symmetry consider a plane wave propagating along the Oy ais. The components E u of the electric field the elastic displacement satisfy the following partial differential equations: 2 E ( j) 2 µ ( j) µ ε ( j) 2 E ( j) 33 t 2 µ ( j) µ e ( j) 3 u ( j) t 2 c ( j) 2 u ( j) 2 ρ ( j) 2 u ( j) t 2 e ( j) E ( j) = (1) = (2) where j = I for the odd numbered layers Ɣ 2k 1 k Z j = II for the even numbered layers Ɣ 2k k Z. In the above equations below j may be replaced by I or II. Then µ ( j) j = I II are magnetic permeabilities of the materials; µ is the magnetic permeability of the vacuum ρ ( j) are the mass densities of the materials; c ( j) are the transverse shear moduli of the materials measured at constant electric field; ε ( j) 33 are the electrical permittivity constants measured at constant strain; e ( j) are the shear pieoelectric stress constants. At the interface between the materials of different types we set the following transmission conditions: u (I ) = u (II) E (I ) = E (II) =: E (3) 1 E (I ) µ (I ) = 1 E (II) µ (II) c (I ) u (I ) c (II) u (II) ( ) = e (I ) e(ii) E which provide continuity of the elastic displacement electric field tractions across the interface. We assume that the fields u E are time harmonic u = U (y)e iωt E = E (y)e iωt (5) (4) of radian frequency ω the amplitudes U E satisfy the Bloch Floquet quasi-periodicity conditions U (y + 2hm) = U (y)e ik 2hm E (y + 2hm) = E (y)e ik 2hm with K being the Bloch parameter. We first look at an elementary eample where the media I II are the same hence the transmission conditions (3) (4) become trivial. In this case u E can be represented as travelling waves so that Eq. (5) is replaced by u = Ue i(ωt Ky) E = Me i(ωt Ky) (7) then Eqs. (1) (2)imply K 2 M + µµ ε 33 ω 2 M iµµ e ω 2 U = (8) K 2 c U + ρω 2 U + ie K M = (9) no confusion should arise here as the superscript indices I II have been dropped. The algebraic system (8) (9) has a non-trivial solution if only if ( µµ ε 33 ω 2 K 2 µµ e ω 2 ) K det e K ρω 2 K 2 = (1) c which is equivalent to ( ( ω ) )( 2 1 ( ω ) ) 2 c ( ω ) 2. µµ ε 33 ρ = µµ e 2 K K K (11) For non-pieoelectric materials e = hence Eq. (11) gives two independent wave speeds ω K = ω K = c ρ for elastic waves 1 µµ ε 33 for electromagnetic waves. In pieoelectric media where the right-h side in Eq. (11) is not ero the electromagnetic coupling leads to an enhancement which results in a slight increase of the speed of the enhanced electromagnetic waves correspondingly decrease in the speed of the enhanced elastic wave. However the right-h side in Eq. (11) is very small the above mentioned enhancement does not provide a significant practical effect. Net we change polariation to the opposite direction for all layers Ɣ 2K K Z without altering their elastic properties. In this case e (I ) = e(ii) =: e c (I ) = c(ii) =: c (6)

3 Interfacial effects in electromagnetic coupling within pieoelectric phononic crystals 97 µ (I ) = µ (II) so that the transmission conditions (4) become E (I ) = E(II) u (I ) u(ii) = 2e c E. (12) Such an alteration of polariation leads to a periodic structure where electro-mechanical coupling would lead to dispersion of the Bloch Floquet waves. Similar formulations are valid for 6 mm two-phase media correspondingly u E will be replaced by u E. 3 Effects of pieoelectric interfaces dispersion of Bloch Floquet waves Consider a doubly periodic array of circular cylinders in a pieoelectric medium as shown in Fig. 1. We consider the plane-strain problem for Bloch Floquet waves assume that the elementary cell of the periodic structure has a square shape [.5.5] [.5.5] contains a circular inclusion of radius a with the centre coinciding with the centre of the square. The material used in the computations is PZT-5H a pieoelectric ceramic which belongs to the cystal symmetry class 6 mm with the elasticity matri C measured at constant electric field E ingpa C = the coupling pieoelectric matri e [C/m 2 ] 17.3 e = the dielectric constant matri ɛ = The mass density is ρ = 75 kg/m 3. In the computations below we show the influence of the interfaces on the dispersion properties of the Bloch Floquet waves. It is assumed that the plane perpendicular to the cylinders is Oy. (a) First we take that both the inclusion the elastic matri are made of the PZT-5H ceramics. If the orientation of the materials is the same the ideal contact is maintained at the interface then the waves within such a medium are not dispersive. On the other h if we change the polariation of the inclusion as compared to the matri then it may bring the interfacial effect leading to the dispersion of Bloch Floquet waves. In the numerical simulation we set the material orientation in the y-plane in the interior of the inclusion in the y-plane in the surrounding matri (see Fig. 1b). The dispersion diagram is shown in Fig. 2. We emphasie that the inclusions the matri are made of the same material should the polariation be the same in both media the diagram would include just straight lines (or in other words there would be no dispersion). In our case we have changed the polariation of the ceramic material within the inclusion (y-plane) compared to the material of the elastic matri (y-plane) which has provided inhomogeneous transmission conditions in tractions at the interface between the inclusion the matri. This results in the dispersion of Bloch Floquet waves clearly visible in Fig. 2. In particular a partial b gap is present along the direction AB in the dispersion diagram. Localiation of the electric field is illustrated in Fig. 3 where we show an eigensolution corresponding to the third eigenvalue for K = (periodic solution). The colormap displays the contrast in the electric potential while the arrows pointing towards the centre of the inclusion show the localied electric field in Fig. 3. Fig. 1 Elementary cells of the periodic structure (a b)the contour ABC within the Brillouin one

4 98 F. J. Sabina A. B. Movchan 18 Dispersion diagram 18 Dispersion diagram Radian frequency Radian frequency C A B C Bloch parameter Fig. 2 Dispersion diagram for the case when materials of the inclusion the matri have opposite polariation C A B C Bloch parameter Fig. 4 Dispersion diagram for the case of inclusions with a thin coating C = the coupling matri e [C/m 2 ] in the stress-charge form.638 e = the dielectric constant matri 8.55 ɛ = Fig. 3 An eigenfield corresponding to the case of the opposite polariation of materials within the inclusion the elastic matri (b) The net eample correspond to the y- or y-plane orientation of the PZT-5H material inside outside the inclusion. In this case the change of polariation in the plane strain problem does not bring the required interfacial effect. To see the influence of the boundary we introduce an intermediate layer (see Fig. 1a) made of a different pieoelectric material (Zinc Sulfide) with the elasticity matri C measured at constant electric field E ingpa The mass density is ρ = 398 kg/m 3. The dispersion diagram is shown in Fig. 4. This figure shows the radian frequency as a function of the Bloch vector corresponding to points of the contour ABC within the Brillouin one in Fig. 1. For the homogeneous structure where inclusions have no thin coating there would be no dispersion the corresponding diagram would include a set of straight lines. The dispersion clearly visible in Fig. 4 is linked to the presence of the thin interfaces. In particular a partial b gap is present on the part CAof this dispersion diagram. We also show an eigenmode corresponding to the fourth eigenvalue K = (periodic solution). The electric potential is almost constant inside the inclusion in the

5 Interfacial effects in electromagnetic coupling within pieoelectric phononic crystals 99 between dissimilar media or a thin interfacial layer (thin coating). The ideas illustrated here are etendable to threedimensional photonic/phononic pieoelectric structures as well as pieoelectric plate structures. Localiation of the elastic field the clearly visible dispersion of Bloch Floquet waves are the main features stressed here. Acknowledgments The paper has been completed during the academic visit of Prof. F. J. Sabina to Liverpool University. The visit was supported by The Research Centre in Mathematics Modelling of Liverpool University CIC-Coordinación de la Investigación Científica Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas Universidad Nacional Autónoma de Méico Conacyt project number F. Thanks are due to Ana Pére Arteaga Ramiro Cháve for computational support. References Fig. 5 An eigenmode illustrating localiation of the electric field within the thin interface surrounding matri. The thin coating enclosing the inclusion is acting as a capacitor the localied electric field (shown by arrows) is clearly visible in Fig Conclusions The emphasis along this paper has been made in the electromechanical coupling due to the presence of interfaces in periodic composite structures either for a simple interface 1. Zhang V. Djafari-Rouhani B.: A general model for analysis of acoustic phonons in pieoelectric super-lattices. Application to (111)-AlAs/GaAs super-lattice. J. Phys. Condens Matter 19 Article No (27) 2. Huang C.H.: Transverse vibration analysis measurement for the pieoceramic annular plate with different boundary conditions. J. Sound Vib (25) 3. Wu T.T. Huang Z.G. Lin S.: Surface bulk acoustic waves in two-dimensional phononic crystal consisting of materials with general anisotropy. Phys. Rev. B 69 Article No 9431 (24) 4. Laude V. Wilm M. Benchabane S. Khelif A.: Full b gap for surface waves in a pieoelectric phononic crystal. Phys. Rev/ E 71 Article No 67 (25) 5. Hou Z. Wu F. Liu Y.: Phononic crystals containing pieoelectric material. Solid State Commun (24)