Study of full band gaps and propagation of acoustic waves in two-dimensional piezoelectric phononic plates

Transcription

1 Study o ull band gaps and propagation o acoustic waves in two-dimensional pieoelectric phononic plates J.-C. Hsu and T.-T. Wu Institute o Applied Mechanics, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, 16 Taipei, Taiwan hsu@ndt.iam.ntu.edu.tw 551

2 In this study, we investigate the band-gap and propagation properties o acoustic waves in two-dimensional (D) phononic crystal plates that consist o pieoelectric material by employing a revised ull three-dimensional (3D) plane wave expansion (PWE) method and inite element method (FEM). The considered structures are periodic air-hole array in a inc oxide plate. To apply the PWE method eiciently and to the pieoelectric solids, the equations governing the pieoelectric wave propagation are considered, and Fourier series expansions are perormed not only in the D periodic plane, but also in the thickness direction or an imaginary 3D periodic structure by stacking the phononic-crystal plates and uniorm vacuum layers alternatively, In the investigated pieoelectric phononic plates, large ull band gaps are ound. On the other hand, analyses by FEM using Bloch periodic boundary conditions is also employed to compare the results rom the PWE method and to serve as a numerical tool or a more detailed study. Eects o lattices and thickness o the phononic plate on the ull band gaps are discussed as well. 1 Introduction In recent years, there has been an increasing interest in studying the properties o acoustic wave propagation in the composite materials, called phononic crystals (PCs), whose mass densities and elastic coeicients are periodically modulated in space. The interest in these materials mainly arises rom that they can give rise to complete acoustic stop bands, which are analogous to the photonic band gaps or optical or electromagnetic waves in photonic crystals and may ind many promising applications to engineering such as acoustic wave guiding, iltering, and vibration shielding [1, ]. In addition to those phononic structures dealing with the bulk acoustic modes that travel in the interior o the medium and the acoustic modes localied near a truncated ree surace o the two-dimensional periodic plane o the structure [3-5], very recent studies show that another worthwhile category o phononic-crystal structures would be the periodic plates o inite thickness whereby the Lamb waves can propagate in [6-9]. On the other hand, it is worth noting that Lamb modes have been important in a variety o applications, such as characteriation o elastic properties o thin ilms, resonators, and sensing applications. For periodic plate structures, some released experiments [6] and theoretical work [7, 8] have demonstrated the existence o directional and complete band gaps o Lamb waves, respectively; these demonstrations may also acilitate the usages o band-gap materials rom those conducted by bulk and surace acoustic modes to the Lamb-wave modes in phononic plate structures. Moreover, Lamb waves are guided waves and well-conined in the thickness direction. As a result, the two-dimensional (D) phononic plates can be expected to serve the same properties o a orbidden band or Lamb modes as a three-dimensional phononic crystal or bulk acoustic modes, in a much more unsophisticated way. Among the existing studies, a lot o theoretical methods, or instance, plane wave expansion (PWE) method, multiplescattering theory (MST), inite element method (FEM), and inite-dierence time-domain (FDTD) method [1], have been successully applied to analye the bulk acoustic waves in ininite phononic crystals; however, it is not always a straight orward task to adapt these methods or phononic plate problems. Based on the classical plate theory and three-dimensional equations o motion with suitable boundary conditions, respectively, the PWE method is used to address the phononic plate problems [11, 1]. Applying Bloch theorem o a periodic medium to the FEM ormulation, the requency band structures o Lamb waves in phononic plate consisting o quart inclusions arranged periodically in an epoxy host were calculated and analyed [8]. Recently, the Mindlin s plate theory based PWE method has been developed to analye the requency band structures o lower-order Lamb modes in lower bands or D non-pieoelectric phononic plates [9]. As well as, a ull 3D PWE method is modiied by considering an imaginary 3D periodic layered structure to analye the waves in phononic crystal plate [13, 14]. In this paper, we derive a ull 3D PWE method or D phononic plate containing constituents with pieoelectricity. Similar to the idea by considering an imaginary 3D periodic layered structure in the Fourier expansion, the ormulation o pieoelectric version is proposed. Moreover, FEM model is also applied to serve a more detailed study. The plate considered and studied is made o inc oxide ilm with a periodic air-hole array. Methods o Calculation.1 Plane wave expansion method Consider a phononic crystal plate with thickness h as shown in Fig. 1, which is composed o an array o cylindrical inclusions A periodically illed in an ininite matrix B. Moreover, in order to apply the ull 3D PWE method to phononic plates, the phononic plate is placed on a uniorm layer C o thickness H that will be taken as a vacuum layer in the PWE calculation. Such a system is then assumed to be periodically stacked along the thickness direction (i.e., the direction) to orm an imaginary three-dimensional periodic structure. The period l along the -direction is, thereore, l=h+h. As a result, the traction-ree boundary conditions and continuity o the electric displacement at the up and down suraces o the phononic plate are implicitly satisied in the ull 3D PWE ormulation. x y FI. 1 Unit cell o an imaginary 3D periodic structure by stacking a phononic plate and a uniorm layer in the PWE method. r h H 55

3 The governing ield equations o acoustic wave propagation in a pieoelectric solid are given by uk φ ρ u j = cijkl + elij, (1) xi xl xl uk φ = eikl εil, () xi xl xl where u j is the displacement and φ is the electric potential. ρ, c ijkl, e ikl, and ε ij are position-dependent mass density, elastic stiness, pieoelectric constants, and permittivity, respectively. Since the imaginary structure is 3D periodicity in space now, the displacements and electric potential satisy the Bloch theorem that can be expressed as ollows: j j i( ) i t u = A e + k r ω, (3) φ= 4 i( ) i t Ae + k r ω, (4) where k=(k x, k y ) is the Bloch wave vectors in the D irst Brillouin one, and r=(x, y, ) and =( x, y, ) are 3D position vectors and reciprocal lattice vectors, respectively. Moreover, spatial distributions o the material properties can be expanded in the Fourier series, as ( ) i r = e r, (5) where the unction is any one o the position-dependent material properties. The Fourier coeicients o the material properties can be evaluated by the ormula ( ) = V r e dv, (6) 1 ir c Vc where V c is the volume o a unit cell o the imaginary 3D periodic structure. Substitution o Eqs. (3), (4), and (5) into Eqs. (1) and () leads to a linear system o equations in the matrix orm: L, A, L, A, L A= (7) L, A, L, A, Note that the summations o the reciprocal lattice vectors in Eqs. (3), (4), and (5) have to be truncated to obtain a initedimension square matrix L. In practicing the numerical calculations, the -component o is separated as =(, ), where is D a reciprocal lattice vector, dependent on the lattice symmetry o the phononic plate, and the component o is = nπ / l, ( n =, ± 1, ±,...). To give an explicit expression o the Fourier coeicient or the imaginary 3D structure, Eq. (6) can be rewritten as H 1 i l C e d H = + l H i i (, ) ( + H l ) 1 l pc x y e d e d (8) In this orm, we set the interval o the integration to integrate over one unit cells that is symmetric with respect to the x-y plane, which can avoid the algebraic operation o the complex numbers, and the unction pc is the distribution o material properties in the D phononic-crystal plate. The inal result is given by = C S( H ) + (8) S r S l S H when ( A B) xy( ) ( ( ) ( )) and, = S H + C ( ) ( ) ( ( ) ( ) ) S r S l S H pc xy when = and, when ( ) ( ) = l H + l h S r (9) 1 1 (1) C A B xy and =, and 1 1 l = HC + l h (11) pc when = and =. In the above expressions, r is the radius o the inclusions in the phononic plate, and πr πr pc = A + 1 B Ac Ac S xy or a square lattice, ( r ) πr = A c J ( r ) 1 r ( r ) πr J 1 Sxy ( r ) = cos a A r or a honeycomb lattice, and S ( ) c sin ( ), (1) ( y ) (13) (13) = l (14) where A c is the area on the x-y plane o a unit cell o the phononic plate, a is the distance between the centers o two nearest cylinders which is reerred to as the lattice constant o a honeycomb lattice, and J 1 is the Bessel unction o the irst kind. Finally, the eigenrequency o the phononic plate mode can be solved by examining the condition ( ) det L = (14) In the ollowing calculations, we will consider two kinds o lattices that can open up ull acoustic band gaps or an airsolid phononic crystal: square lattice and honeycomb lattice (as shown in Fig. ). 553

4 y y 3 Zinc-Oxide Phononic Crystal Plate a x a x 3.1 Acoustic band structures We consider phononic-crystal plate consisting o a periodic air-hole array in a ZnO ilm. The material constants o ZnO used in all the calculations are listed as ollows: ρ =568 kg/m 3, c 11 =9.7Pa, c 33 =1.9Pa, c 44 =4.47Pa, c 1 = 11.1pa, c 13 =15.1pa, e 15 =-.48 C/m, e 31 =-.573 C/m, e 33 =1.3 C/m, ε 11 =75.7pF/m, ε 33 =9.3pF/m. FI. Square lattice and honeycomb lattice and their corresponding deinition o lattice constant. 3 5 Air/ZnO Plate with Square Lattice, r =.47 um and h=.45 um. Finite element method In this study, analyses by using the inite element method is employed in order to compare the results obtained rom the ull 3D PWE method. We use the COMSOL MULTIPHYSICS commercial sotware to carry out the calculations o FEM. In the FEM model, a unit cell o the phononic-crystal plate is constructed, and, according to the Bloch theorem, the mechanical displacement ield and electric potential o a monochromatic wave obey the ollowing conditions on the boundary o the unit cell ( + ) = ( ) exp( ) u r R u r ik R (15) j mp j mp ( mp ) ( ) exp( i mp ) φ r+ R =φ r k R (16) where R mp is a vector displaced by a D period o the lattice multiplied by an arbitrary pair o integers (m, p) in the spatial position, and k is the Bloch wave vector conined in the irst Brillouin one. With the conditions (15) and (16) applied to the unit cell, the eigenrequency is then obtained by solving the pieoelectric wave problem o the model. Examples o the meshed unit cells o the phononic-crystal plates with square and honeycomb lattices are shown in Fig. 3, respectively. FI. 3 Examples o the meshed unit cells o the phononiccrystal plates with the square lattice and honeycomb lattice in the FEM models. Frequency (MH) Frequency (MH) X Air/ZnO Plate with Square Lattice, r =.47 um and h=.45 um X FI. 4 Acoustic band structures o waves in phononiccrystal plates composed o a square-lattice air-hole array in a inc oxide ilm. PWE method. FEM model. Figure 4 and show the acoustic band structures o ZnO phononic-crystal plate with a square lattice calculated by using the ull 3D PWE method and FEM model, respectively. The lattice constant a is 1.um. The radius o the air hole and plate thickness are r =.47um and h=.45um, respectively. The illing ratio o air is, thereore, F=.694. Moreover, the thickness o the uniorm vacuum layer assumed in the PWE calculation is.um, which is enough to segregate the inluences o the waves rom the neighboring phononic plates. In the igures, the results by the two methods exhibit good agreement. In the ull 3D PWE method, 81 Fourier terms in the expansions (i.e., 81 reciprocal lattice vectors or plane waves) are used in the D plane (i.e., x-y plane), and 5 plane waves are used along the -direction. In the FEM model, 3643 tetrahedral elements are meshed or the unit cell. The ull acoustic band gap is in the requency range rom 1.4H to 1.75H. The relative band-gap width, thereore, is about %. Figure 5 shows the vibrations o band-edge modes at the lower-edge (i.e., M M 554

5 1397 MH at X point) and upper-edge (i.e., 175 MH at point) requencies o the ull band gap. At the loweredge requency, the most o the energy is localied on the up and down suraces o the plate; however, at the upperedge requency, the energy is concentrated in the our ribs o the structure. FI. 5 Band-edge plate modes. The mode at 1397 MH o the lower edge and the mode at 175 MH o the upper edge o the ull band gap o the square lattice. Figure 6 shows the acoustic band structure or the case with a honeycomb lattice calculated by the FEM model with about 65 tetrahedral elements meshed. For this plate, the lattice constant is a=1. um, the radius o air hole is r =.45 um (F=.49), and the plate thickness h=1. um. A wide ull band gap ranges rom.8h to 1.16H. The relative band-gap width is about 36.7%. Figure 7 shows the vibrations o band-edge modes at the lower-edge (i.e., 81 MH at point) and upper-edge (i.e., 1187 MH at K point) requencies o the ull band gap. At the lower-edge requency, the most energy is localied on the up and down suraces o the plate; however, at the upper-edge requency, the energy is concentrated in two o the ribs o the structure. Frequency (MH) 15 5 Air/ZnO Plate with Honeycomb Lattice, r =.45 um and h=1. um K FI. 6 Acoustic band structure o waves in the inc-oxide phononic-crystal plate with a honeycomb-lattice. M FI. 7 Band-edge plate modes. The mode at 81 MH o the lower edge and the mode at 1187 MH o the upper edge o the ull band gap o the honeycomb lattice. 3. Eect o plate thickness For a phononic-crystal plate structure, the plate thickness can also be dominated the open and width o the ull band gaps. Figures 8 and show the distributions o the ull band gaps as a unction o plate thickness h or the square lattice and honeycomb lattice, respectively. For the square lattice, the lattice constant and radius o holes are 1.um and.47um, respectively, and or the honeycomb lattice, the lattice constant and radius o air holes are 1.um and.45um, respectively. The calculations are based on the FEM model. Note that more meshed elements in the model are needed to obtain a convergent result when the plate thickness is increased. In Fig. 8, two regions o band gaps are in the requency-thickness diagram or the square lattice. The requencies o band gaps are distributing in the range orm 118MH to 184MH. This requency range is in between the undamental plate modes and their irst olded bands, and the ull band gaps is aect by the olded bands o the lexural mode and the modes with a cuto requency that are shited by changing the plate thickness. In Fig. 8, the distributions o the ull band gaps are much more complicated or the honeycomb lattice. The region o the ull band gap is divided into ive areas by shit o the lexural bands and cuto-requency modes as the plate thickness changes in the diagram. Moreover, the band-gap requencies are in the lower requency range (about rom 79MH to 14MH) than that o the square lattice in spite o their same value o the lattice constant a in the corresponding plate thickness. 555

6 Band-ap Frequency (MH) Band-ap Frequency (MH) Square Lattice, r =.47 um Plate Thickness (um) Honeycomb Lattice, r =.45 um Plate Thickness (um) FI. 8 Band-gap distributions as a unction o the plate thickness h. Square lattice with r =.47um; honeycomb lattice with r =.45um. Conclusion By utiliing the ull 3D PWE method and FEM, we studied the acoustic band structures and ull band gaps o acoustic waves propagating in pieoelectric phononic crystal plate. The plate is composed o a periodic air-hole array in a ZnO ilm. The revised ormulation o PWE method applied to the plate structure is derived by considering an imaginary 3D periodic structure that stacks the phononic plates and uniorm vacuum layers alternatively. In the plates, two lattices are considered in this study, i.e., the square lattice and honeycomb lattice in which both exhibit wide ull band gaps according to the theoretical calculations. Comparison o the calculated results respectively by the PWE method and FEM model also shows good agreement. Finally the eect o the plate thickness on the ull band gaps is discussed or the two lattices, respectively. For the square lattice, the band-gap requencies appear at higher requency range, while the ull band gaps are in the lower requency ranges or the honeycomb lattice. Acknowledgments The authors grateully thank the National Science Council o Taiwan or the inancial support o this work (rant No. NSC 96-1-E--6-MY3). Reerences [1] J.-H Sun and T.-T. Wu, Propagation o surace acoustic waves through sharply bent two-dimensional phononic crystal waveguides using a inite-dierence time-domain method, Phys. Rev. B 74, (6). [] Y. Pennec, B. Djaari-Rouhani, J. O. Vasseur, A. Kheli, and P. A. Deymier, Tunable iltering and demultiplexing in phononic crystal with hollow cylinders, Phys. Rev. B 69, 4668 (4). [3] Y. Tanaka and S. Tamura, Surace acoustic waves in two-dimensional periodic elastic structures, Phys. Rev. B 58, (1998). [4] T.-T. Wu, J.-C. Hsu, and Z.- Huang, Band gaps and the electromechanical coupling coeicient o a surace acoustic wave in a two-dimensional pieoelectric phononic crystal, Phys. Rev. B 71, 6433 (5). [5] J.-C. Hsu and T.-T. Wu, Bleustein-ulyaev-Shimiu surace acoustic waves in two-dimensional pieoelectric phononic crystals, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 53, (6). [6] X. Zhang, T. Jackson, E. Laond, P. Deymier, and J. Vasseur, Evidence o surace acoustic wave band gaps in the phononic crystals created on thin plates, Appl. Phys. Lett. 88, (4). [7] J.-C. Hsu and T.-T. Wu, Eicient ormulation or band-structure calculations o two-dimensional phononic-crystal plates, Phys. Rev. B 74, (6). [8] A. Kheli, B. Aoubia, S. Mohammadi, A. Adibi, and V. Laude, Complete band gaps in two-dimensional phononic crystal slabs, Phys. Rev. E 74, 4661 (6). [9] J.-C. Hsu and T.-T. Wu, Lamb waves in binary locally resonant phononic plates with two-dimensional lattices, Appl. Phys. Lett. 9, 194 (7). [1] Y. Tanaka, Y. Tomoyasu, and S. Tamura, Band structure o acoustic waves in phononic lattices: Twodimensional composites with large acoustic mismatch, Phys. Rev. B 6, (). [11] M. M. Sigalas and E. N. Economou, Elastic waves in plates with periodically placed inclusions, J. Appl. Phys. 75, (1994). [1] M. Wilm, S. Ballandras, V. Laude, and T. Pastureaud, A ull 3D plane-wave-expansion model or 1-3 pieoelectric composite structures, J. Acoust. Soc. Amer. 11, (). [13] J. O. Vasseur, P. A. Deymier, B. Djaari-Rouhani, Y. Pennec, and A-C. Hladky-Hennion, Absolute orbidden bands and waveguiding in two-dimensional phononic crystal plate, Phys. Rev. B 77, (8). [14] Z. Hou and B. M. Assouar, Modeling o Lamb wave propagation in plate with two-dimensional phononic crystal layer coated on uniorm substrate using planewave-expansion method", Phys. Lett. A 37, (8) 556