Phononic Plate Waves - IEEE Xplore

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IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control ,

vol. 58, no. 10,

October

2011

Phononic Plate Waves Tsung-Tsong Wu, Jin-Chen Hsu, and Jia-Hong Sun Abstract—In the past two decades, phononic crystals (PCs) which consist of periodically arranged media have attracted considerable interest because of the existence of complete frequency band gaps and maneuverable band structures. Recently, Lamb waves in thin plates with PC structures have started to receive increasing attention for their potential applications in filters, resonators, and waveguides. This paper presents a review of recent works related to phononic plate waves which have recently been published by the authors and coworkers. Theoretical and experimental studies of Lamb waves in 2-D PC plate structures are covered. On the theoretical side, analyses of Lamb waves in 2-D PC plates using the plane wave expansion (PWE) method, finite-difference time-domain (FDTD) method, and finite-element (FE) method are addressed. These methods were applied to study the complete band gaps of Lamb waves, characteristics of the propagating and localized wave modes, and behavior of anomalous refraction, called negative refraction, in the PC plates. The theoretical analyses demonstrated the effects of PC-based negative refraction, lens, waveguides, and resonant cavities. We also discuss the influences of geometrical parameters on the guiding and resonance efficiency and on the frequencies of waveguide and cavity modes. On the experimental side, the design and fabrication of a silicon-based Lamb wave resonator which utilizes PC plates as reflective gratings to form the resonant cavity are discussed. The measured results showed significant improvement of the insertion losses and quality factors of the resonators when the PCs were applied.

I. Introduction

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ver the past two decades, acoustic wave propagation in periodic elastic structures comprised of multicomponents has received a great deal of attention because of their particular physical properties and potential applications in a variety of fields, such as noise and vibration isolation, frequency filters in wireless communication, superlens design, etc. These periodic elastic structures, which are called phononic crystals (PCs) [1], [2], give rise to forbidden bands of acoustic waves which are analogous to the band gaps of electromagnetic waves in photonic crystals. In the forbidden bands, the acoustic waves are completely reflected by the structures. Major mechanisms leading to the acoustic band gaps are Bragg origin and localized resonances [3], [4]. The former opens up the Bragg gap in the Brillouin zone (BZ), and the band-gap frequen-

Manuscript received December 12, 2010; accepted July 18, 2011. The authors thank the National Science Council of Taiwan for its continuous financial support for this research. T.-T. Wu is with the Institute of Applied Mechanics, National Taiwan University, Taipei, Taiwan (e-mail: [email protected]). J.-C. Hsu is with the Department of Mechanical Engineering, National Yunlin University of Science and Technology, Douliou, Yunlin, Taiwan. J.-H. Sun is with the Department of Mechanical Engineering, Chang Gung University, Kwei-Shan, Tao-Yuan, Taiwan Digital Object Identifier 10.1109/TUFFC.2011.2064 0885–3010/$25.00

cy corresponds to wavelength in the order of the structural period, i.e., the lattice constant, and relates to the lattice symmetry. On the other hand, the localized resonances (meaning the resonances of individual scatterer), which in general can be regarded as morphology-dependent resonances, create the resonant gaps dictated by the resonance frequencies associated with the scattering units (i.e., the periodic inclusions) and depend less on the lattice symmetry, orderliness, and periodicity of the PC structures. Moreover, in the passbands, the dispersion relations of acoustic waves in PCs revealed the existence of negative refraction which may lead to perfect focusing and subwavelength imaging. In the literature, theoretical calculations for BAWs and SAWs in 2-D PCs were reported by using the plane wave expansion (PWE) method [1], [2], [5]–[7], multiple-scattering theory (MST) [8]–[10], and finite-difference timedomain (FDTD) method [11], [12]. These methods have been used to calculate the characteristics of the dispersion relations, transmission and reflection coefficients, and wave fields of BAWs and SAWs in PCs. In recent years, finite-element (FE) method has also been generally used in the simulations of elastic waves in PCs. On the experimental side, evidences of the complete acoustic band gaps for BAW and SAW modes in PCs have been reported [13]–[17]. Negative refraction with PCs and super lensing of acoustic waves based on the negative refraction were demonstrated [18]–[21]. Transmission properties and local resonances of 2-D PCs in the low-frequency range were also demonstrated [22]. Recently, the propagation of Lamb waves in 2-D PC plate structures has received increasing interest. It is known that excitation of high-frequency Lamb wave modes can be achieved by using interdigital transducers (IDTs) deposited on a thin piezoelectric plate, similar to that of SAWs on a piezoelectric half space. High-frequency Lambwave devices have been important in acoustic resonators and sensing applications, and PC plate structures have the potential to improve the performance of the Lamb-wave devices, such as yield, quality (Q) factor, and resistance to ambient noise sources. Propagation of Lamb waves in 2-D PC plate structures with either non-piezoelectric or piezoelectric constituents has been analyzed by utilizing the PWE method [23]–[26]. It was found that, similar to the cases of the bulk modes, directional (partial) as well as complete band gaps of Lamb wave modes exist in PC plate structures. In addition to the theoretical prediction, experimental evidence of band gaps in PC plates was also demonstrated using the laser ultrasonics technique [27]. In recent years, band structures of PC plates with 3-D inclusions and/or defects were studied and discussed in the literature using the MST, FDTD, or PWE method [28]–[30].

© 2011 IEEE

wu et al.: phononic plate waves

Results of these studies showed that with the defects in PC plates, wave disturbances can be localized or propagated along the defects and used as a point resonance cavity or waveguide. In addition to flat PC plates which have two flat free surfaces, complete band gaps and wave guiding in PC plates with a periodic studded surface have been demonstrated both numerically and experimentally [31]. In particular, the results showed that by introducing the periodic cylindrical studs on the surface of a homogeneous thin plate, a low-frequency gap which is similar to the case of locally resonant structures can be formed [32], [33]. Recently, the negative refraction of bending waves in 2-D perforated PC thin plates was studied [34]. Compared with other types of PC structures, i.e., semifinite PCs for SAWs and bulk PCs for BAWs, PC plates have shown advantages in several applications of Lambwave–based PC devices such as waveguides and cavities. SAWs in semi-infinite media have many applications, for example, filters and resonators. However, investigation of the SAW modes localized near the surface of semi-infinite PCs with a 2-D periodicity found that the leaky energy behavior is general unless the SAW frequency bands are well below the BAW bands [5]–[7]. When the SAW couples to the bulk waves, it is attenuated by radiating acoustic energy into the bulk of the semi-infinite PCs and becomes a so-called pseudo-SAW mode, where the coupling of SAW to bulk bands arises from both the anisotropy and bandfolding effect in the periodic structures. This leaky characteristic results in a short lifetime of SAWs for the energy to be confined on the semi-infinite PC surface. It can be an unfavorable factor in SAW-based applications such as PC waveguides. To overcome the difficulty, full 3-D phononic structures may be an alternative, but they are hard to make. An equivalent of this alternative method is use of a PC plate of finite thickness with only a 2-D periodicity. The free surfaces of the plate can confine the acoustic waves in the thickness direction well. On the experimental side, there have been studies, though not many, on the demonstration of the band-gap properties of PC plates with lattice constants on the millimeter and micrometer scales. For the millimeter case (with operating frequencies of a few megahertz or lower), the transmission and resonance spectra of Lamb wave propagation in flat PC plates or plates with a periodic studded surface were examined experimentally using laser ultrasonics [31], [35], [36]; very good agreement with the numerical predictions on the band gaps and resonant frequencies was found. A demonstration was also conducted of the efficient waveguiding and frequency selection capability of PC plates [37]. The physical realization of Lamb wave PC micro electromechanical system (MEMS) devices has only recently begun. The first demonstration of the complete band gap of Lamb waves in a micro PC slab was given by Olsson et al. [38] The PC slab was realized by implementing periodic tungsten (W) scatterers in a SiO2 matrix. Then, wide-band AlN piezoelectric couplers were used to interrogate the devices with a center frequency of approximately 67 MHz. Results showed a 30-dB acoustic

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rejection with bandwidth exceeding 25% of the mid-gap frequency of the band gap. Another demonstration of complete band gap of a micro 2-D PC slab formed by embedding cylindrical air holes in a Si plate was given by Mohammadi et al. [39]. The PC structure was made by etching a hexagonal array of air holes through a freestanding Si plate. A pair of IDTs was utilized to measure the transmission of elastic waves through eight layers of the hexagonal-lattice PC plate in the ΓK -direction. Their results showed that more than 30-dB attenuation was observed at 134 MHz with a band-gap–to–mid-gap ratio of 23%. In their subsequent paper [40], high-Q-factor PC resonators were fabricated using a CMOS compatible process by etching a hexagonal array of holes in a 15-μmthick slab of Si. Results showed that the complete band gap of the PC plate structure supports resonant modes with Q factors of more than 6000 at 126 MHz. Recently, great progress has been made in the study of the microfabricated PC plates and PC-plate–based devices [41]– [47]. Soliman et al. [42] reported micro-fabricated PC plates operating above 1 GHz. The PC plates were made of W inclusions embedded in a 1.2-μm-thick Si membrane, with a lattice constant of 2.5 μm. Experimental data for elastic-wave transmission through the PC plates showed that a wide band gap appears between 1 and 1.8 GHz. Mohammadi et al. [45] demonstrated that by using microfabricated PC plate structures with a complete band gap to replace the supporting anchors of a conventional piezoelectric micromechanical resonator operating at 122 MHz, the support loss in the resonator was suppressed and the Q factor can be improved from 1200 to 6000. Wu et al. [47] utilized micro-fabricated PC plates with a complete band gap to form a resonant cavity for a two-port Lamb-wave resonator and showed the improvement of the Q factor of the resonator. In this article, our recent investigations of Lamb waves in 2-D PC plate structures are reviewed, including the theoretical and experimental results, with a demonstrated application of PC plates to the Lamb-wave resonators. The paper is organized as follows. In Section II, the applied analytical and numerical methods are described. In Section III, the calculated results of Lamb wave propagation and related phenomena, including the frequency band structures and band gaps, are given for several types of PC plate structures. Section IV discusses the analysis and utilization of the properties of band gaps to construct PCbased waveguides and resonant cavities for efficient guiding and confinement of acoustic energy and the study of the negative refraction with PC plates. Section V discusses experimental demonstrations of phononic band gaps for application to Lamb-wave resonators, based on the simulated results of the phononic resonant cavity shown in Section IV. Finally, a summary is given in Section VI. II. Analytical and Numerical Methods This section briefly describes the applied analytical and numerical methods, including the PWE method, FDTD

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method, and FE method, for PC plates. The significant requirements of the methods and some important technical details are provided.

ρ u j = ∂ iTij , (1)

where ρ(x) is position-dependent mass density, uj(r, t) is the displacement, and Tij(r, t) is the stress. r = (x1, x2, x3) = (x, x3) is the position vector. The elastic constitutive laws with the displacement uj(r, t) as variables are given by Tij = c ijkl∂ lu k , (2)

where cijkl(x) are the position-dependent elastic constants. Because of the spatial periodicity of the plate, the material properties α (i.e., the mass density and elastic constants) can be expanded in Fourier series: α(x) =

∑ e iG⋅xαG, (3) G

where G = (G1, G2) is the 2-D reciprocal lattice vector associated with the 2-D lattice, and αG is the corresponding Fourier coefficients. By utilizing the Bloch-Floquet theorem, the displacement vector takes the form

uj =

∑ AjGe i(G+ k)⋅x−iωte ik x , (4)

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 X 1   X  H  2  = 0, (6)    X 6n 

where H is a 6n × 6n matrix. For the existence of a nontrivial solution of Xl, the following condition must be satisfied

det(H) = 0. (7)

Eqs. (5) and (7) should be solved simultaneously to determine the dispersion relations for the Lamb waves. Note that another efficient PWE approach to obtain the dispersion relations of lower-order Lamb modes in a PC plate is based on the Mindlin plate theory [23]. This approach avoids calculating the traction-free surface boundary condition determinant so the computation time can be significantly reduced. B. Finite-Difference Time-Domain Method Eqs. (1) and (2) can be applied to PC plates by periodically arranging the mass density and elastic constants in space. With staggered grids, the differential equations (1) and (2) are discretized into the difference equations based on the Taylor series expansion to develop the finitedifference formulation. The Bloch-Floquet theorem can be applied to treat the periodic boundary conditions (PBCs) for a unit cell of the PC plates. According to the theorem, the PBCs of the displacement and stress fields of the PCs can be expressed as

3 3

G

where k = (k1, k2) is the Bloch wave vector, ω is the circular frequency, and k3 is the wave number along the x3direction. In practice, the summations in (3) and (4) are truncated by choosing n 2-D reciprocal lattice vectors. Substituting (3) and (4) into (1) and (2), a generalized eigenvalue problem is obtained

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the traction: Ti3 = (ci3kl∂luk) = 0 (i = 1, 2, 3). The boundary conditions lead to a homogeneous linear system of equations for Xl,

A. Plane Wave Expansion Method Consider an infinite plate of thickness h, exhibiting spatial periodicity on the x1-x2 plane by embedded scatterers obeying certain lattice symmetry. This arrangement means that the material properties of the plate depend on the coordinate variables x1 and x2. The governing field equations can be expressed as

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(Rk 32 + Qk 3 + P)U = 0, (5)

where U is a vector gathering the Fourier coefficients of the displacement. The explicit expressions of the matrices R, Q, and P, which are functions of k, G, ω and, αG, can be found in [48]. Solving (5) yields 6n eigenvalues and eigenvectors. For elastic wave solutions in the PC plate, we put AjG(l) = Xl ajG(l), (j = 1 to 3 and l = 1 to 6n), in (4) and sum over l, where ajG(l) is the associated eigenvector of the eigenvalue k3(l), and Xl is the undetermined weighting coefficient which can be determined by the boundary conditions on the plate surfaces. The boundary conditions at the two free surfaces of the plate require the nullity of

u j(r + a, t) = e ik ⋅au j(r, t), (8)

Tij(r + a, t) = e ik ⋅aTij(r, t), (9)

where a is a translation vector of the 2-D lattice. With the PBCs, dispersion relations of the PC plates can be analyzed by calculating the wave fields of the unit cell. In calculating the dispersion relations, the wave vector k is assigned, and a wide-band disturbance at an arbitrary position in the unit cell is set to be the initial condition. Then, all possible wave modes can be developed to propagate inside the unit cell, and the displacement is recorded and Fourier transformed. The eigenfrequencies with the given wave vector k are indicated by selecting the corresponding frequencies of the resonance peaks of the Fourier transformed spectrum. This allows the information about the possible wave modes to be found. In analyzing PC structures, the perfectly matched layer (PML) can be adopted to serve as a non-reflecting boundary condition [25]. Berenger [49] introduced the concept of PML to reduce the electromagnetic wave reflection from a boundary, and the elastic-wave counterpart of the PML


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was later developed [50]. A stretched coordinate is defined with a complex variable,

ei = ai + i

Ωi , (10) ω

to derive the FDTD code for the PML region. In (10), the real part ai is a scale factor, and Ωi/ω is the imaginary part with and attenuation factor Ωi and circular frequency ω. The differential operation in the stretched coordinate is defined and applied to the equations of motion and elastic constitutive laws. After introducing the plane wave solutions into the equations, the numerical attenuation is achieved via the factor Ωi. In addition, the non-reflecting condition at the interface between the PML region and the inner space is obtained by setting corresponding material constants and the scale factor. In the elastodynamic equations for the stretched coordinate, displacement and stress components are taken with their spatial partial differential operations in all directions; thus the variables are split into three components to realize the difference equations. Then, the actual values are obtained from summation of the split components. Finally, the PML equations can be shown to have the form,

ρu i/j + ρΩ ju i/j = Tij, j , (11)

u δ + u l,kδmk . (12) Tij /m + Ω mTij /m = c ijkl k,l ml 2

In (11) and (12), ui/j and Tij/m are the split displacement and split stress components, which satisfy ui = ui/1 + ui/2 + ui/3 and Tij = Tij/1 + Tij/2 + Tij/3. The symbol δij represents the delta function. After transforming (11) and (12) to the difference equations, the PML can be arranged outside the space boundaries as a buffer zone with a matched acoustic impedance to suppress the reflection. A numerical attenuation occurs as waves decay rapidly inside the region. With the PMLs, the reflection can be reduced to less than 1%, and PMLs can be used in the calculation of the dispersion relations and transmission. C. Finite-Element Method In the FE analysis, the commercial FE package and solver COMSOL Multiphysics (Comsol Inc., Burlington, MA) can be used to carry out the numerical calculations. To calculate the eigenfrequencies and eigenmodes of Lamb waves in PC plates, an FE structural model is constructed for the PC unit cell and then meshed. By assuming a time-harmonic solution, the wave equations lead to an eigenvalue problem in the FE formulism and yield the eigenfrequencies and eigenvectors (i.e., eigenmodes). The PBCs as (8) can be adequately applied to the PC unit cell boundaries to obtain the dispersion relations. In the calculations, the COMSOL program connected to the MATLAB script (The MathWorks Inc., Natick, MA) can be used to iteratively vary the structural dimensions, geometry, and the k-dependent PBCs. Moreover, the frequency

response analysis of the FE method can also be used to calculate the transmission and reflection in a PC plate structure. This approach also requires PMLs, which have been implemented in the COMSOL FE package, to model the non-reflecting boundaries. III. Band Structures of Phononic Crystal Plates Acoustic waves in homogeneous plates have already been well studied in the literature, including the flexural (antisymmetric, Am) and longitudinal (symmetric, Sm) Lamb wave modes and transverse (shear-horizontal, Tm or SHm) waves, where m = 0, 1, 2, 3, ... for different orders of the modes. In PC plates, the acoustic modes are similar to those in a homogeneous plate in the low-frequency range; however, as the frequency increases, the eigenmodes of PC plates are complex combinations of the Am, Sm, and Tm modes and are difficult to fully classify. One of the most noticeable properties of the PC plates is that the band structures may have a frequency gap in the surface Brillouin zone (SBZ) called the phononic band gap. Note that SBZ is used here for 2-D periodic systems, whereas the term BZ usually refers to bulk 3-D periodic systems. The frequency gap between the bands means that there are no eigenmodes in the frequency range, and thus wave propagation in the structures is not allowed. The band gap is one of the key characteristics of PC plates. In PC plates, similarly to homogeneous thin plates, the dispersion relations of the Lamb wave modes and transverse modes (including the higher-order plate modes, m = 1, 2, 3, ...) are significantly influenced by the plate thickness. Thickening the plate thickness can lower the eigenfrequencies of the higher-order modes in a PC plate. Indeed, change in the plate thickness changes the dispersion relations and affects the opening and range of the band gaps [24], [48]. As a result, the plate thickness must be properly chosen to obtain a desired band gap. On the other hand, the PC plate thickness can serve as another key parameter for tailoring the band-gap ranges as well as the band structures for manipulative purposes, e.g., negative refraction. The dispersion relations of acoustic waves are analyzed along the boundary of the first SBZ, that is, the smallest irreducible area enclosed in the wavevector space (k-space), to investigate the band gaps. If there is a frequency range in which acoustic waves cannot propagate in any direction in the structure, it is called a complete band gap. This is an important phenomenon for application of the PCs. In the following section, the properties of PCs are discussed via the dispersion curves (i.e., the band structures). Here we introduce the band structures of waves in several PC plates with different geometries. First, three flat surface plates that are composed of binary constituents and periodic air holes are introduced. Then, PC plates with a single constituent material, including a PC plate with a periodic studded surface and a drilled-holes PC plate with periodic circular thin plates, are presented.

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TABLE I. Elastic Stiffness and Mass Density. Material

c11 (1010 N/m2)

W Si Epoxy Rubber Al 6061 Al Fe

50.23 16.56 0.785 0.0001416 11.1 10.82 23.7

c12 (1010 N/m2) 6.39

c44 (1010 N/m2)

ρ (kg/m3)

14.98 7.95 0.158 0.00000325 2.5 2.85 11.6

18 700 2329 1180 1300 2965 2730 7780

A. Flat Phononic-Crystal Plates The basic type of PC plate is a plate composed of two materials. The plates discussed in this subsection have two flat free surfaces and inclusions arranged periodically. The dispersion curves and propagation modes are presented, and the key features of the band gaps in the phononic plates are discussed. 1) Binary Phononic Plates: The first illustrated case is a square-lattice PC plate consisting of W cylinders embedded in a Si matrix [29]. Table I lists the elastic stiffness and mass density of the materials discussed in this paper. A schematic of the W/Si PC plate is shown in Fig. 1(a). The cylindrical inclusions are arranged to form a square lattice on the x1-x2 plane. The normal direction of the free surfaces of the plate is along the x3-axis. The [100] direction of a (001) Si plate coincides with the x1-axis of the PC plates. The lattice constant a, which is defined as the distance between the centers of two closest cylinders, is 10 μm, and the radius r of the W cylinders is 2.5 μm. Thus, the filling fraction F of W in the square-lattice PC plate is equal to πr2/a2 = 0.196. The plate thickness h is chosen to be 10 μm to obtain an obvious complete band gap. The band structures of the PC plates are dependent on the filling fraction, material properties, geometry conditions (i.e., the lattice symmetry), and plate thickness [23], [24]. The FDTD method [25] was adopted to analyze the dispersion of acoustic waves inside the W/Si PC plate. The calculated band structure is shown in Fig. 1(b) and the inset shows the first SBZ in the k space. There are three developing bands starting from the zero frequency to a higher frequency range. On the curves, every point represents one eigenmode of the PC plate. It is observed from Fig. 1(b) that there is a range without any eigenmode, and thus the complete band gap is identified as the frequency range from 223 to 250 MHz. In addition, a partial band gap in the ΓX -direction is observed between 250 and 255 MHz. The eigenmodes of the frequency bands were investigated by calculating their displacement distribution with the FDTD method. A region consisting of four unit cells, as shown in Fig. 2(a), is defined to demonstrate the polarization of the eigenmodes. In Figs. 2(b)–2(e), 2-D vector plots were plotted to show the characteristic displacement

Fig. 1. (a) Schematic of a 2-D square-lattice PC plate. a is the lattice constant, r is the radius of the cylindrical inclusions, and h is the plate thickness. (b) The band structure of a square-lattice W/Si PC plate, where a = 10 μm, r = 2.5 μm, and h = 10 μm. The complete band gap is from 223 to 250 MHz. The inset shows the irreducible part of the first SBZ. 

distribution. Four eigenmodes are shown in the figures to present the first four bands in the ΓX -direction, respectively. These eigenmodes are marked by the points A, B, C, and D in Fig. 1(b) where the wave vector is k = (π/a, 0) and the frequencies are 121, 157, 160, and 197 MHz, respectively. The directions of the cones in the vector plots indicate the direction of the displacement vector, and the corresponding sizes of the cones represent the magnitudes of the vectors, which are normalized by the maximum value of the vector field. With the displacement distributions, these wave modes can be classified. Fig. 2(b) shows the displacement distribution of the eigenmode denoted by point A in Fig.


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Fig. 3. The band structure of an air/Si PC plate, where a = 20 μm, r = 8.9 μm, and h = 12 μm. The complete band gap is from 148.8 to 174.1 MHz [47]. 

Fig. 2. (a) A 2a × 2a segment of a square-lattice W/Si PC plate. The dashed-dot lines indicate the planes on which the following vector plots lie. (b) The displacement field of A0 mode. Dashed lines indicate the boundaries of the unit cells. (c) SH0 mode; (d) A0 mode of the folded band; (e) S0 mode [29]. 

1(b). The displacement components u1 and u3 dominate the behavior of the mode. Thus, the first band basically corresponds to the lowest flexural (A0) mode in a plate. Fig. 2(c) presents the mode of point B. The eigenmode has a primary polarization in the x2-direction, and this mode is associated with the 0th-order shear-horizontal mode (SH0). The displacement of point C is shown in Fig. 2(d). The main components, consisting of u1 and u3, indicate that this mode also belongs to the A0 mode, in the folded band of the first band. The mode of point D has a higher phase velocity than the two previous bands. Fig. 3(e) shows that the displacement is mainly along the x1-direction, and the wave field is symmetric with respect to the middle plane of the plate. This mode is associated with the lowest symmetric mode (S0). In summary, the acoustic wave fields of lower bands in the PC plate are similar to the Lamb wave modes in a flat homogeneous plate. 2) Drilled-Hole Phononic Plates: The second type of flat phononic plate is composed of only one material with periodically drilled holes. The example discussed here is the air/Si PC plate, and the holes are arranged in a square lattice [47]. The lattice constant a is 20 μm, the radius r of the cylindrical holes is 8.9 μm, and the plate thickness h is equal to 12 μm. The filling fraction is as large as 0.622. Fig. 3 shows the band structure of the PC plate, which was obtained by using the FE method. The band structure exhibits a complete band gap from 148.8 to

160 MHz and a directional band gap from 148.8 to 174.1 MHz along the ΓX -direction. Similarly, the modes of bands below the band gap can be identified according to their polarization; however, the modes of bands above the band gap are coupled and become too complex to be classified clearly in this PC plate. 3) Binary Locally Resonant Phononic Plates: The flat PC plates discussed in this sub-subsection are also binary PC plates; in this case, strong resonances are localized in the inclusions. The PC structure is an epoxy matrix plate with soft rubber cylinders periodically embedded that can result in low-frequency resonances. The rubber cylinders described are arranged in a square lattice and in a hexagonal lattice. The radius of the rubber cylinders is r = 0.4a, and plate thickness h = 0.115a. By using the PWE method based on Mindlin plate theory [23], [26], the dispersion curves were calculated and are shown in Fig. 4. It is observed that many flat bands exist in the band structure, which indicates the resonance frequencies of localized plate modes. Such PCs are called locally resonant PCs. As shown in Fig. 4(a), the square-lattice PC plate has a complete band gap in the normalized frequency range from ωa/ct = 0.0481 to 0.05095, which is two orders of magnitude lower than that resulting from Bragg scattering compared with the previous two cases, where ct = 1157 m/s is the transverse wave velocity of epoxy. In the band structure, several flexure-dominated branches of plate modes, which have a larger out-of-plane component of displacement, can be distinguished and are labeled F1 to F6. These branches are sensitive to the variation of the plate thickness, whereas the other bands with a dominated in-plane vibrations are not. In addition, three frequencies of flexure-dominated modes with a zero group velocity at the Γ point, which are denoted R1–R3, also present a localized pattern. Similarly, the band structure of the hexagonal-lattice PC plate shown in Fig. 4(b) also exhibits flat bands and a low-frequency gap for all modes, extend-

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Fig. 4. The band structures of locally resonant PC plates: (a) The PC plate is composed of a square lattice of rubber in epoxy. r = 0.4a and h = 0.115a. The complete band gap is from ωa/ct = 0.0481 to 0.05095. (b) The PC plate is composed of a hexagonal lattice of rubber in epoxy. r = 0.4a and h = 0.115a. The complete band gap is from ωa/ct = 0.0481 to 0.0515 [26]. 

ing the frequency range from 0.0481 to 0.0515. The complete band gaps in the two kinds of lattices are almost consistent, which corresponds to the fact that the complete band gaps in a locally resonant PC plate depend less on the lattice or orderliness of the inclusions. B. Studded Phononic-Crystal Plates Different from the flat PC plates, PC plates with a periodic studded surface have also been proposed [31]–[33], and the structures were reported to exhibit low-frequency band gaps. Here we introduce a studded phononic plate with single constituting material [31]. The studded PC plate is made of aluminum 6061 (Al 6061) with cylindrical studs on one of the plate surfaces. The studs are arranged in square lattice with a lattice constant a = 10 mm. The diameter of the cylindrical studs is d = 7 mm, and the thickness of the base plate h1 = 1 mm.

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Acoustic dispersion relations along the ΓX -direction of the first SBZ were analyzed to show their properties. In the calculation, the height of the studs increases from one plate thickness (h1) gradually up to 10 h1, and the results are shown in Fig. 5. It was found that as the stud height approaches about triple plate thickness, i.e., 3h1, a narrow complete band gap ranging from 167.5 to 171 kHz forms. Moreover, when the stud height is equal to 9h1, the complete band gap is the largest, with a range from 119 to 157.5 kHz and a relative band gap width equal to 0.278. When the stud height is ten times of the plate thickness, i.e., 10h1, a large complete band gap appears, ranging from 119 to 143 kHz (shaded region), and three partial band gaps also appears, as shown in Fig. 5(d). As the height of the studs gradually increases, some resonances form, which result in slower wave velocity and flatter dispersion curves near the SBZ boundary. One can also observe that as the stud height gradually increases, many bands originally crisscrossing in the dispersion curves start to separate at their intersections. The repelling of the frequency bands finally results in different individual curves in which each curve is composed of several evolved segments of old bands from the case of a uniform thin plate without the studs. Figs. 6(b)–6(g) show the corresponding eigenmodes of the points b–g, respectively, labeled in Fig. 6(a) [band structure the same as Fig. 5(d)]. Fig. 6(b) shows the 19kHz eigenmode which is derived from the coupling of the S0 and A0 modes. The cylindrical stud moves as a rigid body motion in this mode. The mode at 60 kHz and shown in Fig. 6(c) is derived from the A1 and T0 modes. In this mode, the stud vibrates as the circumferential mode of a cylinder. The mode in Fig. 6(d) shows that the displacement of the plate is governed mainly by the S0 and A1 modes, and flexural (or bending) vibration of the stud can be observed. The mode shown in Fig. 6(e) is also governed mainly by the S0 mode and A1 mode, and the high studs tremble in the transverse direction with a powerful vibration of the plate. Fig. 6(f) is the mode governed by A1 mode at the cutoff frequency 114 kHz. The stud is motionless in this mode so that its eigenfrequency does not considerably affect by the change of the stud height (see Fig. 5). Finally, the mode in Fig. 6(g) is evolved from the A1 mode, and the flexural vibration of the plate gives rise to a longitudinal wave propagating along the axial direction of the stud. C. Phononic-Crystal Plates With Periodic Circular Thin Plates The final example to be discussed is a PC plate structure consisting of periodic circular thin plates [51]. The schematic shown in Fig. 7 is a unit cell of the PC structure with periodic circular thin plates arranged in a square lattice. The unit cell contains a cylindrical hole and a much thinner circular plate at one end of the hole. The lattice constant is a = 9.52 mm, and the radii of the hole and the circular thin plate are r = 4.475 mm. The total thickness


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Fig. 5. The band structures of a PC thin plate with studs of different heights along the ΓX -direction: (a) flat thin plate, (b) stud height h2 = 0.25h1, (c) h2 = 0.5h1, and (d) h2 = 10h1. Lattice constant a = 10 mm, plate thickness h1 = 1 mm, and diameter of the cylindrical stud is 7 mm [31]. 

Fig. 6. (a) The band structure of Lamb waves in the studded PC plate, where the stud height h2 = 10h1. (b) Eigenmode shape with ka/π = 1 and f = 19 kHz, (c) with ka/π = 0.6 and f = 60 kHz, (d) with ka/π = 1 and f = 100 kHz, (e) with ka/π = 0.6 and f = 109 kHz, (f) with ka/π = 0 and f = 114 kHz, and (g) with ka/π = 0 and f = 205 kHz [31]. 

of the PC plate is H = 5.05 mm, and the thickness of the circular thin plates is h = 0.29 mm. Note that the thickness ratio h/H = 1/17.4, causing the circular thin plates to have relatively weak rigidity. The band structures of the PC plates with and without the thin plate layer covered are shown in Fig. 8(a). The

results were obtained by using the FE method with Bloch PBCs implemented. Because of the weak rigidity of the circular thin plates, there are slow propagating resonant modes appended to the band structure. It is found that there exist some nearly flat bands imposed on the band structure (labeled A, B1, B2, C1, C2, D, E1, E2, and F).

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Fig. 7. (a) Schematic of the unit cell of the square-lattice PC plate with periodic circular thin plates. (b) Photograph of the specimen; the structure is made of steel. (c) Full section view of the unit cell [51]. 

These bands, therefore, are associated with the vibration modes (flexural modes) of the circular thin plates. The interaction of the circular thin plate modes with the Bloch modes of the PC plate results in additional bands in the band structure. Moreover, the weak rigidity of the circular thin plates causes the additive bands to be nearly flat, and the original band structure is altered. Comparing the two band structures, the circular thin plates also create eigenmodes with frequencies inside the complete band gap (i.e., the F modes, where the complete band gap of the structure without circular thin plates is from 173 to 212 kHz.) These eigenmodes are not standing waves of energy localized inside the circular thin plates but are propagating modes with very slow group velocities. Furthermore, the complete band gap causes the circular thin plate modes to be isolated from the other bands, whereas others outside the complete band gap might couple to other bands such that the propagating energy is not always well confined in the circular thin plates within the entire SBZ. Figs. 8(b) and 8(c) show the calculated mode shapes of the circular thin plate modes labeled F on Fig. 8(a). It can be observed that these patterns coincide with the eigenmodes of a clamped circular thin plate so their eigenfrequencies are relevant. We classify the resonant plate modes into monopole, dipole, quadruple mode, and so on, according to their symmetries of the mode shapes. And each mode may have different orders. As a result, the F modes (the two bands denoted by dot-dashed lines in the shaded region) belong to the lower and higher second-order dipole modes. The modes, as dipole modes, having the same poles and order but belonging to different bands at different (lower and higher) frequencies have very similar mode shapes but different orientations, and the orientation of the mode shape is dependent on the Bloch wave vector. Three types of PC plates—flat PC plates, studded PC plates, and PC plates covered with a very thin plate— have been discussed. The classification of these PC plates is based on their geometrical configurations. The different configurations result in distinct differences in the band structures for Lamb wave modes. The variations of the

Fig. 8. (a) The band structures of the PC plate with (dots) and without (solid curves) the thin plate layer covered. Mode shapes of the resonant slow modes in the complete band gap (F modes). (b) The lower mode. (c) The higher mode [51]. 

band structures and the opening of the complete band gaps can be attributed to the Bragg scattering and resonances, which are dependent on the configurations of the PC plates and their constituents. In the following, we compare the three types of PC plates by discussing the mechanisms of Bragg scattering and local resonances. For the flat PC plates with two parallel free surfaces, the PC structures can be formed either by embedding periodic solid inclusions or by drilling periodic air holes in a host plate. The inclusions and the holes play the role of multiple scatterers to diffract Lamb waves. For the perforated plate, the periodic scatterers can only reflect the waves, whereas the solid scatterers can also absorb wave energy to induce resonances in the scatterers. Such resonances intensify the wave scattering with frequencies relevant to the resonance frequencies of the scatterers. It is certain that the scattering is dependent on the size and pitch of the scatterers. For the perforated PC plate, the scatterer size (i.e., the radius of the holes) must be large enough and close to the pitch between the scatterers so that the scattering can meet the Bragg frequency to open the complete band gaps. Such band gaps are called Bragg band gaps; they forbid the waves with wavelengths on the order of the lattice pitch. On the other hand, for the solid scatterers, the material properties of the scatterers also come into play. Stiffened and heavy scatterers in a softer plate benefit the opening of the complete band gaps be-


cause the resonance frequency of the scatterers can meet the frequency of Bragg scattering. From the viewpoint of fabrication, perforated PC plates are much easier to make. Thus far, PC plates with an array of air holes are relatively more likely to be adopted in MEMS devices. Furthermore, when the periodic scatterers are made of very soft material such as soft rubber, the resonance frequencies are reduced by two orders of magnitude compared with the common solids, and the resonances are relatively strong and localized in the scatterers. The structures are referred to as a locally resonant PC, where the local resonances override the lattice symmetries of the structures in forming the frequency gaps. The complete band gaps are, therefore, on the very low frequencies. This is especially useful to prevent low-frequency vibrations or noise without a structure size comparable to the wavelength. In PCs, formation of the complete band gaps is due to Bragg origin and/or local resonances. Resonances of individual scatterer coalescing with Bragg gap can create hybridization gaps [3]. By growing periodic studs as the scatterers on one of the surfaces of a plate, hybridization gaps can also appear in the studded plates because of the existence of the Bragg origin and resonances of the studs. Note that in such structures, the studs can be made of the same material as the base plate. The resonance frequencies can be determined not only by their radius but also by their height. Because the studs are not constrained on their circumferences, the resonances will be stronger and on the lower frequency range. As a result, the surface-studded PC plate structures create obvious resonance characteristics and low-frequency band gaps for the band structures. This kind of structure is also available to the MEMS method by using deep dry etching or deposition technology. The PC plate with periodic holes covered by circular thin plates results in a more complex band structure of waves. The band structure can be viewed as a result of the interaction of the circular thin plate modes with the Bloch wave modes in the perforated PC plate. Here, the periodic circular thin plates are not the key to the band gap formation but create resonant modes or slow modes in the band structures. The slow modes that exist in the complete band gap received particular interest because they are isolated from other bands which may lead to different design of phononic waveguide and cavity from the two previous types of PC plates.

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In this section, numerical simulations of Lamb wave propagations in PC-based waveguide, cavity, and negative refraction at the interface between homogenous plates and the PC region are discussed.

confined acoustic waves from leaving the top and bottom surfaces. Using the band gap property of 2-D PCs, phononic waveguides to guide waves in a specific path inside a plate can be further constructed. A PC waveguide is formed based on the complete band gaps of acoustic waves. The square-lattice PC plate consisting of Fe cylinders embedded in an epoxy matrix and proposed in [25] can be employed to construct acoustic waveguides. The lattice constant a of the PC plate is 8 mm, and the radius r of the Fe cylinders is 3 mm. The dispersion relations calculated by using the FDTD method show two complete band gaps in the frequency ranges from 89 to 101 kHz and 125 to 162 kHz. To construct a waveguide, continuous point defects are defined in the PC plate with complete band gaps. These defects form a route without scatterers (i.e., Fe cylinders), and thus, the acoustic waves can propagate freely inside the waveguiding route. Here a waveguide of total width w = 6 mm is discussed. The width w denotes the distance between two neighboring cylinders on both sides of the waveguide. The waveguide with w = 6 mm means that the route with one-half length of a unit cell is inserted into the perfect periodic structure. The supercell technique with the FDTD method was used to analyze the dispersion of the waveguide, where the supercell of the PC waveguide consists of 10.5 unit cells, including the inserted defect region of width 0.5a = 4 mm, in the analysis. The band structures of the PC waveguide are shown in Fig. 9(a) to observe the guided modes in the complete band gaps. The extended modes appearing in the pass bands are not shown in the diagram, and the regions are presented in gray. In Fig. 9(a), there are ten bands appearing in the range of the complete band gaps. The wave field of the defect mode marked as point A in Fig. 9(a) is shown in Fig. 9(b). In the calculation, a 95-kHz wave source was set at the inlet of the waveguide, and the signals on the surface of the supercell were recorded. Because the mode of point A belongs to a folded band, the wave vector k is (1.69π/a, 0), and the wavelength is about 1.18a, which agrees with the pattern shown in Fig. 9(b). The characteristic displacement field is also plotted in Fig. 9(c). The 3-D vector plot shows a full view of the defect mode in the waveguide, and the 2-D figure of the cross section at the center of waveguide helps identify the wave as a flexural mode. Further study showed that waveguides with a narrowed width have fewer modes. Moreover, waveguides modified by inserting smaller scatterers at the center of waveguides can also reduce the number of the defect modes and can keep the waveguides consisting of multiple unit cells to construct an acoustic circuit in a periodic PC, especially with a bend waveguide [25].

A. Phononic Waveguides

B. Resonant Cavities

An acoustic waveguide can confine wave propagation and avoid energy dissipation. In fact, a plate has already

The band gaps play an important role in engineering PC-based devices. For wave frequencies within the com-

IV. Waveguides, Cavities, and Negative Refraction

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plete band gaps and with defects in PCs, many applications related to the control of wave flows can be designed. In the following, we discuss another defect-contained PC plate structure, i.e., an acoustic resonant cavity, which may be used in acousto-electronic devices. The following resonant cavities were constructed based on the W/Si square-lattice PC plate that was discussed in Section III [29]. A sketch of the phononic resonant cavity is shown in Fig. 10(a). The structure is formed by inserting an extra space of ΔL into the periodic PC plate. Then a cavity length L is defined as the distance between two neighboring cylinders of the defect. The analysis of the dispersion of the cavity was obtained by using the FDTD method with the supercell technique. A cavity of L = 12.5 μm (i.e., 1.25a) is considered; this means that an extra dislocation of ΔL = 7.5 μm is defined in the periodic PC plate. The dispersion relations of this PC plate structure-based cavity are shown in Fig. 10(b). The figure is focused on the frequency range of 190 to 280 MHz to observe cavity modes in the complete band gap; the upper and lower bounds of the complete band gaps are marked with horizontal solid lines at 223 and 250 MHz. The line at 255 MHz indicates the upper limit of the partial band gap between 250 and 255 MHz along the ΓX -direction. There are two flat bands within the range of the complete band gap, which appear at 232.4 and 247.7 MHz. Because these two frequency bands are horizontally flat, they have zero group velocity and represent the resonant modes. The transmission of Lamb waves passing through the resonant cavity was computed. The waves were generated by a line source, acting as body force along the x2-direction and polarized in the x3-direction, on the top plate surface. On the right-hand side of Fig. 10(b), the transmission of the cavity is plotted as the solid line, and a comparative transmission of Lamb waves propagating through a tenlayer PC plate is shown by the dashed line. The result of the case with the cavity shows two peaks at 232.4 and 247.7 MHz, which correspond to the two flat bands in the dispersion diagram. Thus, both the dispersion and the transmission show the resonant modes inside a cavity of the PC plate. The Fabry-Perot type standing wave resonant condition, nλn/2 = Ln, is useful to understand the resonant waves inside the PC cavities, where n is the order of the resonant modes, λn is the wavelength, and Ln is the equivalent cavity length for the nth-order resonant mode. However, because the boundaries of the resonant cavity are PCs, the resonant waves can penetrate the geometric boundary. The equivalent cavity length is expected to be different for different sources. Thus, the resonance frequency cannot be simply calculated by the Fabry-Perot condition. We directly calculated the transmission of various cavities with different length L. The length is changed from a to 2.5a. The resonance frequencies of these cavities modes are identified from the peaks inside the spectra and marked in Fig. 10(c). Because the partial band gap is available above the complete band gap and the non-excitation band is below the complete band gap, the resonant

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Fig. 9. (a) Band structure of Lamb waves in the Fe/epoxy phononic waveguides of width w = 6 mm. (b) Amplitude distribution of the 95kHz Lamb wave [point A in (a)]. The darker color means larger amplitude. (c) Displacement field of eigenmode A in (a) along the central line of the waveguide [25]. 

modes in the range of 202 to 255 MHz can be identified. In Fig. 10(c), the resonance frequencies result in four curves. In each curve, it is observed that the longer the cavity length is, the lower is the resonance frequency. The variation is consistent with the Fabry-Perot condition. Furthermore, to distinguish these four curves, the polarizations of the resonant waves were calculated. An example is shown in Fig. 10(d) for the mode of point (b) at 232.4 MHz with the cavity length L = 1.25a. The displacement field shows obvious u3 components which are antisymmetric with respect to the middle plane of the plate. This is the lowest-order flexural mode (A0, n = 1) with only onehalf of a complete wavelength inside the cavity. Similarly, the mode of point (c) belongs to the second-order resonant flexural mode (A0, n = 2), point (c) is the first-order longitudinal resonant mode (S0, n = 1), and point (d) is the


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second-order longitudinal resonant mode (S0, n = 2). The result shows that PC cavities can potentially be designed as acoustic-wave resonators. A design for a magnified Lamb wave source based on a PC cavity was proposed in [29]. C. Negative Refraction

Fig. 10. (a) A resonant cavity based on a PC plate. (b) The dispersion shows resonant modes of a cavity of L = 1.25a. The transmission is on the right. The solid and dotted lines present the results of Lamb waves propagating through the cavity and a ten-layer W/Si PC plate, respectively. (c) The resonant modes excited by the source polarized in the x3-direction on the top surfaces of plates. (d) The displacement field of the 232.4 MHz mode inside a cavity of L = 1.25a [29]. 

Negative refraction in PC structures is an abnormal phenomenon based on the strong anisotropy and periodicity of PCs. In the literature, most related studies dealt with the negative refraction of BAWs in PCs. Negative refraction of Lamb waves in a PC plate has not been studied until recently [34]. Here, we adopt a PC plate which consists of a hexgonal array of drilled-holes in an Al plate to demonstrate the abnormal refraction. The lattice constant a is set to be 10 mm, the radius r of the cylindrical holes is 4.39 mm, and the plate thickness h is 1 mm. The band structure calculated by the FE method is shown in Fig. 11(a). The polarization modes in this PC plate are marked, and no complete gap is observed for this PC. Slowness curves are usually calculated to understand the anisotropic properties of acoustic waves. For PC plates, equal frequency contours (EFCs) are adopted as an equivalent property of slowness curves. Fig. 11(b) shows the EFC of acoustic waves at 40 kHz in the PC plate. The EFCs are presented in the k-space with a range larger than the first SBZ [the hexagon in Fig. 11(b)]; then the periodic contour pattern can be constructed. Obviously, the SH0 mode shows a strong anisotropy, and the contour of A0 mode shows a slightly distorted circle. The contours of S0 and SH0 modes are within the first SBZ but the A0 mode is outside the first SBZ. Thus the slowness curve of A0 mode actually consists of six pieces of curves in the second SBZ, as shown in Fig. 11(c). Based on the EFCs, Snell’s law can be used to analyze the refraction at the boundary between the homogeneous plate and the PC area. An example of the 40-kHz A0 wave with incident angle of 15° is demonstrated. The direction of group velocity is normal to the tangent of the EFC, and thus the concave curve indicates that energy velocity performs the negative refraction with a refraction angle of 21°. The calculation was achieved by using the FE method. A line source is set to excite the directional A0 mode, and the steady-state frequency responses are demonstrated. Fig. 12(a) shows the displacement field of a directional A0 wave passing a three-layer PC area, and a shift of the wave beam which verifies the abnormal refraction is observed. Further, Fig. 12(b) shows the displacement field resulting from a point source at a distance of 20 mm away from the center point of the air hole of the first layer. On the right side of the PC region, the Lamb wave is focused as an image of the point source based on the negative refraction phenomenon. V. Micro-Fabricated Phononic Plate Devices In commercial SAW filters and resonators, there are usually hundreds of periodically-spaced metal strips placed on

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both sides of IDTs which occupy a great amount of space. To reduce device size, PCs were used as high-efficiency reflectors for SAW devices in the literature [52]. Furthermore, a two-port Lamb-wave device using PC plate as the reflective gratings has also been demonstrated [47]. The numerical studies of PC cavities have been discussed in the previous section. In this section, the design, fabrication, and measured frequency responses of the Lamb-wave resonator with PC reflective gratings are discussed. A. Design For the design, the square-lattice air/Si PC plate mentioned in Section III was adopted. The acoustic band structure obtained by the FE method shows a directional band gap of 148.8 to 174.1 MHz along the ΓX -direction, and then the PC was used as acoustic-wave reflectors of the resonator. First, we numerically analyzed the reflection of Lamb waves encountering the PC plate structure. The operation frequency of 160 MHz was chosen and the corresponding wavelength of the lowest A0 mode is equal to 25 μm in the 12-μm thick Si plate. Two pairs of normal line forces with positive and negative polarization, separated by one-half of the wavelength, were used to simulate an IDT. Lamb waves generated by the line forces radiated outward in both directions, and the steady-state displacement fields were calculated by the frequency response analysis of the FE method. Fig. 13(a) shows the displacement fields of the steadystate responses due to the harmonic excitation. On the left of the source, the displacement is either enhanced or cancelled, depending on the distance between the sources and the PC grating, which is defined as D. The phenomena can be interpreted by the superposition of two harmonic waves as follows. The line source generates waves propagating toward both the right and left, and the waves toward the right will eventually be reflected by the PC grating. Thus, on the left of the source, the direct waves and the reflected waves can result in constructive or destructive interference according to their phase difference, which can be controlled by the distance D. When D is equal to 1.18λ, the reflected waves are in-phase with the direct waves, and therefore, the wave magnitude on the left of the IDT is enhanced. However, when D is equal to 1.43λ, the direct waves and reflected waves are outof-phase. Cancellation results, so the displacement field almost vanishes. In this regard, to use PC as an acoustic reflector in a Lamb-wave device, the distance D needs to be determined, and the cavity length should be equal to 2 × D plus a multiple of 1/2λ. As shown in Fig. 13(b), the Si-based Lamb-wave device includes a Si plate, two PC reflectors, a pair of IDTs, and a ZnO piezoelectric film on top of the Si plate. By considering the dispersion of the plate modes, the wavelength of the A0 mode at 160 MHz in the Si plate area is λSi = 25 μm and that in the ZnO/ Si area is λZnO/Si = 24.8 μm. Design parameters of the Lamb-wave resonator are listed in Table II.

Fig. 11. (a) Band structure of Lamb waves in the air/Al hexagonallattice PC plate. (b) EFCs of 40-kHz acoustic waves in the PC plate. (c) EFCs in k-space of the plate and PC at 40 kHz show the refraction.


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Fig. 12. Negative refraction demonstrations of A0 Lamb wave induced by (a) a directional wave source and (b) a point source.  TABLE II. Design Parameters of the Lamb-Wave Resonator. Substrate

0.9-μm ZnO/12-μm Si

Central frequency IDT line width IDT pair IDT aperture IDT thickness PC layers D DIDT

160 MHz 6.2 μm 40 80λZnO/Si 150 nm 15 2.178λSi 30.5λZnO/Si

B. Fabrication To realize the designed device, standard COMS fabrication process was adopted which includes: the deposition of thin metal films for the IDTs and bottom electrode, deposition of the piezoelectric ZnO film, lithography for patterning the IDTs and the PCs, dry etching to make the periodic air holes, and finally, wet etching to back release the thin plate structure. A silicon-on-insulator (SOI) wafer with a 12-μm Si device layer was utilized. In addition to the device layer, the thicknesses of the handle layer and the oxide layer of the SOI wafer are 300 μm and 0.5 μm, respectively. The oxide layer was used as the blocking material during the dry etching process. At the beginning, a gold layer of thickness 100 nm was evaporated on the device side of the SOI wafer. A 0.9-μm thick piezoelectric ZnO layer was then deposited and patterned using RF sputtering and wet etching, respectively. An Al thin film with 150 nm thickness was evaporated on the photoresist structure and then IDTs were formed by using the liftoff process. Fifteen layers of PC structure were fabricated

Fig. 13. (a) The displacement field component u3 for the constructive (D = 1.18λ) and destructive (D = 1.43λ) interference cases, respectively. (b) Sketch of the two-port Lamb-wave resonator with PC gratings [47]. 

by inductively coupled plasma reactive ion etching (ICP RIE). Finally, the lower handle layer and oxide layer were removed by KOH and BOE wet etching, respectively. C. Measurement Fig. 14 shows the measured frequency response of the Lamb-wave resonator with the PC reflective gratings. The result shows two resonant peaks with approximately the same insertion loss. One is at 158.15 MHz (24.2 dB), and the other one is at 157.34 MHz (24.8 dB). The deviation of the peak frequency from the designed value, i.e., 160 MHz, may contribute to the differences between the fabricated sizes and material properties and the simulated ones. The peak appearing on the lower frequency side at 156.28 MHz has a larger insertion loss of 30.1 dB. The largest Q factor of the resonant peaks (2269) appears at 158.15 MHz, which is the one closest to the designed frequency, 160 MHz. The multiple resonances inside the PC cavity can be interpreted by the Fabry-Perot resonant condition that multiple wavelengths may satisfy the condition in one cavity. The simulations show that a cavity with a larger length allows a smaller frequency difference and more resonant peaks. From the numerical investigation, it is qualitatively realized that the appearance of three resonant peaks

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Fig. 14. Measured frequency response of the two-port Lamb-wave resonator with PC gratings [47]. 

in Fig. 14 is due to the relatively large cavity length of the Lamb-wave resonator, i.e., 144.33a (2886.5 μm). To have fewer resonant peaks appearing around the frequency range of interest, the cavity length needs to be shortened accordingly. VI. Conclusion In this article, we present a review of our recent investigations of Lamb waves in 2-D PC plate structures. First, the band gaps in three types of PC plate structures are given. The variations of the band structures and the opening of the complete band gaps are discussed with the configurations of the PC plates and their constituents. A comparison of the different types of the PC plates has also been discussed. Then, several numerical calculations demonstrated the utilization of the properties of band gaps to construct PC-based waveguides and resonant cavities for efficient guiding and confinement of acoustic energy and the negative refraction in PC plates. Experimental demonstrations of phononic band gaps for the application to Lamb wave resonators are also given. Finally, we note that the focusing of Lamb waves in a gradient-index phononic plate has recently been demonstrated [53] and can potentially be utilized as an efficient beam-width compressor. Acknowledgments The authors thank the National Science Council of Taiwan for its continuous financial support for this research. References [1] M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B. Djafari-Rouhani, “Acoustic band structure of periodic elastic composites,” Phys. Rev. Lett., vol. 71, no. 13, pp. 2022–2025, 1993. [2] M. M. Sigalas and E. N. Economou, “Elastic and acoustic wave band structure,” J. Sound Vibrat., vol. 158, no. 2, pp. 377–382, 1992. [3] I. E. Psarobas, A. Modinos, R. Sainidou, and N. Stefanou, “Acoustic properties of colloidal crystals,” Phys. Rev. B, vol. 65, no. 6, art. no. 064307, 2002.

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Tsung-Tsong Wu received his B.S. degree in civil engineering from the National Taiwan University in 1977 and M.S. and Ph.D. degrees in theoretical and applied mechanics from Cornell University in 1983 and 1987, respectively. He joined the faculty of the National Taiwan University in 1987 and currently is distinguished professor of the Institute of Applied Mechanics. His research interests include surface acoustic wave devices and related sensors, phononic crystals, and nondestructive evaluation of materials. He has been awarded the outstanding research prizes and the distinguished research fellow prize by the National Science Council (NSC). He has served as the Deputy Minister of the NSC as well as the president of the Society of Theoretical and Applied Mechanics of the Republic of China from 2006 to 2008. He is a fellow of the American Society of Mechanical Engineers.

Jin-Chen Hsu received his B.S. degree in mechanical engineering from the National Taiwan University of Science and Technology in 1997 and his Ph.D. degree in applied mechanics from the National Taiwan University in 2007. From 2007 to 2009, he was a postdoctoral research fellow of the Institute of Applied Mechanics of the National Taiwan University. Currently, he is an assistant professor of Department of Mechanical Engineering, National Yunlin University of Science and Technology, Taiwan. His research interests include micromechanical acoustic-wave resonators, acoustic waves in piezoelectric media, and phononic crystal structures.

Jia-Hong Sun received his B.S. degree from the Department of Aeronautics and Astronautics of National Cheng Kung University in 1995 and his M.S. degree from the Institute of Applied Mechanics, National Taiwan University, in 1997. After serving his 2-year compulsory military service as an officer in the army, he returned to the campus and received his Ph.D. degree in applied mechanics from the National Taiwan University in 2006. From 2006 to July 2011, he was a postdoctoral research fellow of the Institute of Applied Mechanics of the National Taiwan University. Currently, he is an assistant professor in the Department of Mechanical Engineering, Chang Gung University, Taiwan. His research interests are acoustic waves in phononic crystal structures, application of SAW devices, and numerical experiments of wave propagation using the FDTD and FE methods.