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Physics Letters A www.elsevier.com/locate/pla

Surface acoustic waves in 2D-phononic crystal of laminated pillars on a semi-inﬁnite ZnO substrate Tengjiao Jiang a,1 , Chunlei Li a,b,1 , Qiang Han a,∗ a b

School of Civil Engineering and Transportation, South China University of Technology, Guangzhou, Guangdong Province, 510640, PR China State Key Laboratory of Subtropical Building Science, South China University of Technology, Guangzhou 510640, PR China

a r t i c l e

i n f o

Article history: Received 31 May 2019 Received in revised form 17 July 2019 Accepted 4 September 2019 Available online 11 September 2019 Communicated by R. Wu Keywords: Phononic crystal Surface acoustic waves Laminated pillar Damping Linear defect

a b s t r a c t In this paper a phononic crystal of laminated pillars which are composed of two materials on a semi-inﬁnite ZnO substrate is proposed. The unit cell can be regarded as a double-resonator model, which explicates the variation of band gaps. It is revealed that the band structure of the PC is highly inﬂuenced by the proportion of PZT-4 in laminated pillars. Besides, the high-frequency band gaps are the result of combined actions of the local resonance and the Bragg scattering, and the former dominates. Furthermore, the inﬂuences of the structural damping are investigated. It is observed that the bandgapratio remain unchanged and the band gaps translate to the right with isotropic structural loss factor increasing. Additionally, by introducing linear defects inside the laminated pillars, elastic energy at surface is strongly conﬁned within defect pillars, and the coupling between defective pillars and SAW is governed by the physical parameter of resonators. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Phononic crystals(PC) are a new physical concept similar to photonic crystals in the ﬁeld of condensed matter physics. As highly dispersive elastic materials, PCs are generally composed of periodic distribution of two or more kinds of materials [1,2]. The special geometric and elastic parameters of the scatters lead to the bandgap (BG) for elastic waves in certain frequency ranges. PCs have been widely used in ﬁltering, waveguides, sensing-sensors, acoustic focusing and topological phonon [3–9]. Subsequently, for the sake of the unusual physical properties, more and more interests have been turned into acoustic metamaterials [10,11], which include negative effective mass density and negative bulk modulus. These physical properties can realize low frequency BG and low frequency superabsorption of acoustic metamaterials. The limitations of conventional physics are broken and new application such as sound insulation, sub-wavelength focusing are achieved [12,13]. Discovered by Rayleigh in 1885, surface acoustic wave (SAW) is a type of acoustic wave which is conﬁned within several wavelengths of the semi-inﬁnite elastic medium [14] and attenuates rapidly away from the surface. The band gap generated by the Bragg scattering of the periodic structure are called Bragg gap. The

* 1

Corresponding author. E-mail address: [email protected] (Q. Han). These authors contributed equally to this work.

https://doi.org/10.1016/j.physleta.2019.125956 0375-9601/© 2019 Elsevier B.V. All rights reserved.

frequency range of Bragg gap depends on the lattice size of the PC. A new type of band gap which is created by introducing local resonance in unit cell has drawn considerable attention. The surface-coupled modes are resulted by the interaction between the locally resonant modes and the surface modes. Since the particular propagation characteristics of SAW in periodic structures, extensive research have been conducted on PC [15–19]. Benchabane et al. [20] introduced the concept of the sound line to explain that radiation to the bulk of the substrate dominates near the upper boundary of the BG. Meseguer et al. [21] ﬁrstly reported an experimental study SAW scattering by a periodic array of cylindrical holes. The existence of BG for surface waves in the semi-inﬁnite two-dimensional crystals were shown by the attenuation spectra. However, PC comprised of pillar arrays [22,23] have greater ﬂexibility compared with hole arrays [24,25] in designing the geometry and materials. The dispersion of SAW was investigated theoretically in Khelif et al. [26], and it revealed the possibility to open BG in a two-dimensional array of cylindrical pillars. Achaoui et al. [27] proposed the low frequency BG appeared on account of local resonance in the pillars. It also proved that the attenuation of the transmission spectrum is similar to the holes in a substrate at high frequency. However, little attention has been paid on the laminated pillar structure. Only in recent years, Oudich et al. [28,29] made some works about the case of multilayer pillars on substrates and mainly studied the interaction mechanism between surface wave and localized modes. Furthermore, a certain defect frequency [30] can be introduced into the BG when the internal structure of PC is

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the radial thickness of cladding layer are expressed as rPZT−4 and rPMMA , respectively. 2.2. Calculation of dispersion and transmission spectrum For the electro-elastic medium, there is a linear relationship between the electric ﬁeld and the stress ﬁeld, and the generalized constitutive relations are given as

S i j = siEjkl T kl + dtikl E k

(1)

T D i = dikl T kl + εik Ek

Fig. 1. A unit cell of the PC with the laminated pillar: (a) longitudinal laminated pillar, and (b) radial laminated pillar. (For interpretation of the colors in the ﬁgure(s), the reader is referred to the web version of this article.)

modiﬁed and tailored. The defect modes generated in the BG are so dispersed that the local resonance gap can be closed. This characteristic restricts the potential of designing high-quality-factor (QF) SAW modes in the sensing application. This work proposes a pillar-based PC on a semi-inﬁnite ZnO substrate, and the pillars are the laminated structure as shown in Fig. 1. The effects of the physical parameters on dispersive characteristics of SAW in PCs are investigated. A double-resonators model is proposed to explain the variation of band structure of PC with different proportion of PZT-4 in laminated pillars. Besides, the damping cannot be neglected in practical structural applications and the inﬂuence of isotropic structural loss factor on SAW propagation is analyzed. Finally, the linear defects with different volume fraction are introduced into laminated pillars and the variation of defect modes frequencies is demonstrated.

where S i j is the strain tensor, D i is the electric displacement. T i j , E k are the strain tensor, the electric ﬁeld, respectively. siEjkl , di jk ,

εikT are the elastic constants, the piezoelectric constants, and the

dielectric permittivity constants, respectively. According to Einstein summation convention, where i , j , k and l = 1, 2, 3 corresponding to x, y , z directions, respectively. According to the lattice symmetry of the substrate ZnO which the polarization direction is deﬁned along the positive direction of Z axis, the dielectric permittivity constants, the elastic constants, and the piezoelectric constants are given as

⎡

T ε11

εikT = ⎣ 0

0

⎡

E s11

0

T ε33

E s12

E s13

(2) 0

0

E s66

0

0

0

0

d15

di jk = ⎣ 0

0

0

d15

0

0⎦

d31

d33

0

0

0

φ φh φr

ηs φh1

0

(3)

⎤ ⎥

(4)

The periodicity of PC can be characterized by periodic shift op∧

erator T R ∧

T R f (r) = f (r + R) = f (r)

(5)

herein, f and r are basic physical parameters and the position vector, respectively. According to Bloch theorem, uk (r) satisﬁes the periodic shift operator and the eigenfunction can be expressed as following

uk (r + R) = uk (r), u(r) = uk (r)e ik·r

(6)

where uk (r) and k indicates the periodic vector function and wave vector, respectively.

Table 1 Calculation parameters. R H hPMMA hPZT−4 rPMMA rPZT−4

⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎦ 0 ⎥

0 d31

0

⎤

0

⎢

0

0

0

⎡

All the parameters involved in the calculations are shown in Table 1. As illustrated in Fig. 1, a unit cell of PC is parallel to the z axis and considered to be inﬁnite in both x and y directions. The lattice constant a = 10um. In the negative direction along the z axis, SAWs nearly disappear within 3-5 wavelengths below the coupling surface. Therefore, the thickness of substrate is deﬁned as 3 times the wavelength of SAW. Particularly, the radius and height of the laminated pillar are constant R = 3um and H = 6um, respectively. Fig. 1(a) shows a longitudinal laminated pillar above the coupling surface and the substrate material is considered to be ZnO. The top and bottom ends of the longitudinal laminated pillar are polymethyl methacrylate (PMMA), which the height is denoted by hPMMA . The middle material is PZT-4 with the height of hPZT−4 . Fig. 1(b) displays a radial laminated pillar which the radius of the middle layer and

T ε22

⎤

0 0 ⎦

⎢ E E E ⎢ s12 s11 s13 0 0 ⎢ ⎢ sE sE sE 0 0 ⎢ siEjkl = ⎢ 13 13 33 E ⎢ 0 0 0 s44 0 ⎢ ⎢ 0 E 0 0 0 s44 ⎣

2. Model and theory 2.1. Unit cell model

0

the radius of the laminated pillar the height of the laminated pillar the height of the top and bottom ends of the longitudinal laminated pillar the height of the middle of the longitudinal laminated pillar the radius of the middle layer of the radial laminated pillar the radial thickness of cladding layer of the radial laminated pillar volume fraction of the laminated pillar volume fraction of the longitudinal laminated pillar volume fraction of the radial laminated pillar isotropic structural loss factors volume fraction of the defect longitudinal laminated pillar

T. Jiang et al. / Physics Letters A 383 (2019) 125956

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Fig. 2. (a) The ﬁrst Brillouin zone of the crystal, and (b) a supercell with 11 laminated pillars, interdigital transducer (IDT), ZnO substrate and perfectly matched layers (PML) of the simulation.

Bloch-Floquet periodic boundary conditions are applied to the lateral sides of the unit cell, and the wave vector in the x and y directions is swept between the high symmetry points of the ﬁrst Brillouin zone ( − X − M − ) as shown in Fig. 2(a). A supercell composed of the 11 same unit cell is schematically shown in Fig. 2(b). SAWs are triggered by IDT, and the wave is transmitted in the x direction (ﬁnite periodic direction). BlochFloquet condition is added to the sides of the supercell in the y direction and perfectly matched layers (PML) are applied around the supercell to reduce reﬂections. Meanwhile, low-reﬂection and ﬁxed boundary conditions are used to the lateral sides of the x direction of PML and the bottom of perfectly matched layers, respectively. To conﬁrm the transmission of PC, the average RMS (root mean square) displacements in the x, y and z directions were then calculated along the output line which is illustrated by the red line in Fig. 2(b). 3. Results and discussion 3.1. Undamped longitudinal laminated pillar The pillars array on the substrate shown in this paper is similar to the photonic crystal slabs [31]. In the band structure of photonic crystals, guided modes exist outside the light cone, which exactly deﬁnes the boundary between the guided mode and the radiation mode. Similarly, in the band structure of phononic crystals, guided surface modes also exist outside the sound cone. Here, volume fraction is deﬁned as following

φ=

V PZT−4 V

(7)

where the volume of laminated pillars and PZT-4 of pillars are represented as V and V PZT−4 , respectively. It can be simpliﬁed h

−4 to φh = PZT for a longitudinal laminated pillar, and φh occurs H between 0 and 1. When φh = 0, the material of laminated pillar is PMMA. Similarly, it is represented as the PZT-4 pillar since φh = 1. Fig. 3(a)-(d) display the band structure calculated along high symmetry directions of the ﬁrst irreducible Brillouin zone, which has a high dependency on φh . The brown region highlights the sound cone which distinguishes the bulk modes from the surface coupling modes. The blue and red lines represent the sound line and propagating SAW bands, respectively. Meanwhile, the BGs within the ﬁrst Brillouin zone are indicated by the light blue region. The cone boundary is obtained by calculating the smallest elastic bulk wave phase velocity in ZnO along different propagation directions. The sound line along the → X direction is obtained √ from the smallest elastic wave velocity of c 44 /ρ . √ For → M, the quasishear horizontal wave velocity is deﬁned by (c 11 − c 12 )/ρ . The total displacement and deformation of the modes at the X

point are shown in Fig. 3(e). For instance, the displacement ﬁeld of the mode A, B, C1 and C2, located outside the sound cone is mainly concentrated near the pillar array and the substrate surface. The bulk mode 1 and mode 2 are located inside the sound cone, which the displacement ﬁeld is distributed in the whole pillar and substrate. It is shown that displacement is conﬁned to the pillars and the top of the substrate for surface modes, however, it completely radiates into the bulk for bulk modes. So far, two kinds of band gap mechanisms have been mainly revealed in the ﬁeld of phononic crystals: Bragg scattering and local resonance mechanism. As illustrated in Fig. 4, band gaps change obviously with the variation of volume fraction of PZT-4, however, the lattice constant and the total height of the laminated pillar are constant. It is clearly shows that the origin of the band gap is the result of local resonance. For the longitudinal laminated pillar in this paper, the stiffness of the pillar varies with the volume fraction, which results in the variation of the acoustic modes frequency of the pillar. These acoustic modes interact at certain frequency points and form collective propagation surface modes, so that acoustic energy can be propagated along the surface of the substrate. Furthermore, the coupling of SAWs with the local mode of pillar array on the substrate surface will lead to band gap, and this interaction can open the band gaps that guided SAWs are prohibits to propagate. The resonance frequencies of the pillars depend on the mechanical parameters of the resonators, which can be simpliﬁed as a spring-mass model. For the laminated pillar model proposed in this paper, a double-resonator model shown in Fig. 5 will be used to clarify the variation trend of BGs with φh increasing. For the unit cell of the PC, the equivalent spring stiffness and mass of the major-resonators are supposed as K and M, respectively. As for the laminated pillar, the equivalent spring stiffness and mass of the minor-resonators are supposed as K1 and M1 . ZnO plays a dominant effect of the equivalent mass of the resonators since the stiffness of substrate is larger than that of laminated pillar. Meanwhile, for the equivalent spring stiffness, laminated pillar dominates. Fig. 4 illustrates the trend of BGs variation with volume fraction of PZT. For φh < 1/3, PMMA plays a dominant effect of the natural frequency of the minor-resonator, and the equivalent mass of the major-resonator increases and the equivalent stiffness decreases, such that resonance frequencies decrease with φh increasing. However, for φh values higher than 1/3, the effect of PZT-4 dominates in the natural frequency of the minor-resonator. With the increase of φh , the equivalent mass of the resonator decreases and the equivalent stiffness increases, which leads to the increase of the major-resonator frequencies. Additionally, the material of laminated pillar is PZT when φh =1, and the high frequency BG is highly consistent with the Bragg gap, which occurs from f = 130.65 to 139.98 MHz. Fig. 4 illuminates that the initial frequency of high frequency BG increases from 70 MHz to 120 MHz with φh increasing, while the lattice size remains unchanged. It re-

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T. Jiang et al. / Physics Letters A 383 (2019) 125956

Fig. 3. (a)-(d) Band structure of PC composed of longitudinal laminated pillars on a ZnO substrate. (e) The total displacement and deformation of a single cell: (e) A, B, C1, C2 are surface coupling modes, and bulk modes 1 and 2 are selected in the shadow area outside the sound line.

Fig. 5. The local resonance model with longitudinal laminated pillars: the broken lines and the solids represent the spring oscillator of the resonator and the mass, respectively. Fig. 4. The low and high frequency BGs of longitudinal laminated pillar.

3.2. Undamped radial laminated pillar vealed that the high-frequency BG is the result of the combined action of the local resonance and the Bragg scattering, and the former dominates.

As for the radial laminated pillar displayed in Fig. 1(b), the r volume fraction can be simpliﬁed to φr = PZTR−4 and φr occurs between 0 to 1. Considering the similarity of the structure, we might

T. Jiang et al. / Physics Letters A 383 (2019) 125956

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Fig. 6. (a)-(d) Band structure of PC composed of radial laminated pillars on a ZnO substrate. (e) Band structure in → X direction beside the normalized transmission spectra of Rayleigh wave.

boldly predict the results coincide with section 3.1. Fig. 6(a)-(d) display the band structure with φr =0.35, 0.5, 0.65, 0.75, which has a high dependency on φr . The size of the BG can be expressed in terms of a dimensionless parameter bandgap-ratio to avoid frequency dependence [32]

BGr =

fU − fL ( f U + f L )/2

(8)

where f L and f U are upper and lower bounds, respectively. For φr =0.35, low frequency BG occurs from f = 34.98 to 37.25 MHz, which correspond to a bandgap-ratio of 6.3%. For φr =0.5, low frequency BG occurs from f = 35.02 to 37.78 MHz, which correspond to a bandgap-ratio of 7.6%. The high frequency BG occurs from f = 117.40 to 130.36 MHz, which correspond to a bandgap-ratio of 10.5%. For φr =0.65, low frequency BG occurs from f = 37.20 to 40.13 MHz, which correspond to a bandgap-ratio of 7.6%. The high frequency BG occurs from f = 123.56 to 136.69 MHz with a bandgap-ratio of 10.1%. For φr =0.75, low frequency BG occurs from f = 39.31 to 42.48 MHz, which correspond to a bandgapratio of 8.2%. The high frequency BG occurs from f = 126.78 to 139.98 MHz with a bandgap-ratio of 9.9%. Fig. 6 illustrates the transmission spectrum of radial laminated pillar (part (b)) for the

Fig. 7. The low and high frequency BG of radial laminated pillar.

→ X for Rayleigh waves. The utilized supercell with 11 pillars for calculating the transmission spectra is schematically presented in Fig. 2 (b) It is revealed that the transmission gaps are in good agreement with the band gaps in the band structure. Fig. 7 illustrates the trend of BGs variation with volume fraction of PZT. A similar conclusion can be drawn on account of

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T. Jiang et al. / Physics Letters A 383 (2019) 125956

Fig. 8. For the longitudinal laminated pillar of φh =

1 : 3

(a) Variation trend of BGs with

the similarity with the longitudinal laminated pillar. For φr < 0.35 PMMA plays a dominant effect of the natural frequency of the minor-resonator, and the equivalent mass of the major-resonator increases and the equivalent stiffness decreases, such that resonance frequencies decrease with φr increasing. However, for φr values higher than 0.35, the effect of PZT-4 dominates in the natural frequency of the minor-resonator. With the increase of φr , the equivalent mass of the major-resonator decreases and the equivalent stiffness increases, which leads to the increase of the resonance frequencies. In addition, the biggest difference between the longitudinal and radial laminated pillar is that the materials in direct contact with the substrate are different. The contact surface materials of longitudinal laminated structures are ZnO and PMMA, while the radial laminated structures are ZnO, PMMA and PZT-4. Comparing Fig. 4 and Fig. 7, it can be observed that the highfrequency BGs of the radial laminated pillar are obviously higher than that of the longitudinal laminated pillar in the case of similar volume fraction. This is mainly because of the direct contact between PZT-4 in the radial laminated pillar and the substrate, which enhances the surface coupling effect. In other words, the direct contact between PZT-4 and the substrate will lead to the equivalent stiffness of the resonator increases whereas the equivalent mass remains unchanged, which leads to the increase of the resonance frequencies. 3.3. Damped laminated pillar The structural damping caused by the internal friction and the relative slip between the connecting interfaces of the components should be taken into account. For the structural damping system, the equation of motion can be written as ..

M X +KX+ jRX = F

(9)

where M, K and R are the stiffness matrix, mass matrix and structural damping matrix, respectively. In this paper, the hysteretic damping which is a type of the structural damping is considered. For the multi-degree-of-freedom vibration system, the effect of hysteretic damping on band structure is mainly studied by introducing structural loss factor. And the eigenvalue equation can be obtained by Fourier transform

(I+ jG) K−ω2 M = 0

(10)

herein, G and (I+ jG) K are structural loss factor matrix and complex stiffness matrix, respectively, and satisfy GK = R. Furthermore, for the isotropic structural loss factor, the structural loss factor matrix is a diagonal matrix with the same values on the diagonal. Therefore, Eq. (10) can be simpliﬁed as following

ηs increasing. (b) Normalized transmission spectra of SAW with different ηs .

Fig. 9. The supercell of the PC with a middle defect pillar.

(1+ j ηs ) K−ω2 M= 0. Besides, on the basis of Eq. (10), the scale factor is deﬁned as ξ =1+ j ηs . It is clear that the isotropic loss factor has a linear relationship with the quadratic of the characteristic frequency. Fig. 8(a) display the trend of BGs variation with different isotropic structural loss factors for the longitudinal laminated pillar of φh = 1/3. The red and blue points represent the frequency bounds of the low and high bandgaps, respectively. The brown region represents the continuous variation trend of the bandgap for different isotropic structural loss factors which the uniform values is obtained in the range of 0 to 0.1. Additionally, the lowfrequency BG for the longitudinal laminated pillar with φh = 1/3 and ηs = 0.5 occurs from f 1 = 25.29 to 26.70 MHz, which correspond to a bandgap-ratio of 5.4%. Comparing with low-frequency BG shown in Fig. 3(b), the radio of f 1 2 to f 2 is equal to real part of the scale factor ξ . Consequently, it is revealed that the initial frequency points of low and high increase proportionally with ηs increasing, while the bandwidths remain unchanged. The only difference comparing with the section 3.1 and 3.2 is that the small dissipation is taken into account in each material of phononic crystal in the calculation of transmission spectrum in this section. After setting φh = 1/3, ηs =0.01, 0.02, 0.03, 0.04, 0.05, the variations of transmission spectra are shown in Fig. 8(b). Coarse red line represents undamped transmission spectra. With ηs increasing, the bandwidths remain unchanged and the BGs translate to the right. It is proven the credibility of the conclusion described in Fig. 8(a). Moreover, the amplitude decreases rapidly with the increase of ηs in high frequency regions, which is caused by the greater dissipation of damping on the vibration energy in the high frequency region. 3.4. Defect laminated pillar Here, a linear defect is introduced in pillars and the related supercell consisting of 11 pillars is displayed in Fig. 9. The Bloch-

T. Jiang et al. / Physics Letters A 383 (2019) 125956

Fig. 10. (a)-(d) Show band structure in the direction of φh1 =

1 1 1 4 , , , . 12 4 2 5

Floquet boundary condition is added to the sides of the supercell in the y direction and the direction of wave propagation is along Y direction. The parameter φh is changed to φh1 to describe the defect differently from the normal unit cells. Fig. 10(a)-(d) exhibits the band structures with a line defect of pillars for φh1 =1/12, 1/4, 1/2, 4/5, while φh =2/3, which the wave vector is swept between the and X points of the ﬁrst Brillouin zone. For φh1 =1/12, defect bands (red lines) are shown above the local-resonance band gap (blue area) which is consistent with the low frequency BG in Fig. 2(c). Then, it is observed that the defect band is below the band gap for φh1 =1/2, and the defect band is highly dispersed in the BG for φh1 =1/4 and φh1 =4/5. Moreover, the variation of defect modes frequencies at the point with different φh1 is exhibited in Fig. 11 (a). Obviously, the variation trend of defect mode is similar to that of Fig. 4. Besides, Fig. 11(b) illustrates the total displacement and deformation of the defect modes of part (a) near the X point of the ﬁrst Brillouin zone. It is observed that the displacement ﬁeld mainly concentrated near the defect pillar and the substrate surface. In other words, elastic energy at surface is conﬁned within the defect pillars. Considering the complexity of structure and material components, the modes shown in Fig. 11(b) are hybrid modes with multiple modes coupling. The Ydirection and X-direction displacement ﬁeld are dominant for the mode A1 and A2, respectively, and have the similar displacement modes shape. Meanwhile, the displacements ﬁeld of X and Y are dominant for the mode B which has the form of expansion displacement perpendicular to the Z axis. The double-resonator model shown in Fig. 5 is also applicable to explain the behavior of the defect bands. For φh1 < 1/2, PMMA plays a dominant effect of the natural frequency of the minor-resonator, and the equivalent mass of the major-resonator increases and the equivalent stiffness decreases with φh1 increasing, such that the resonance frequencies of defect modes decrease and pass through the band gap to the below of BG. However, for φh1 values higher than 1/2, the effect of PZT-4 dominates in the natural frequency of the minor-resonator. With φh1 increasing, resonance frequencies of defect modes increase and once again pass

7

Fig. 11. (a) Variation of defect mode frequency with different φh . (b) The total displacement and deformation of the defect modes: A1, A2 are symmetrical bending mode and B is expansion mode.

through the band gap to the above of BG. The dominant effect of the natural frequency of defect pillars gradually changed from PMMA to PZT-4 with φh1 increasing, such that defect modes frequencies at the X point decrease ﬁrst and then increase. 4. Conclusion In this paper, a phononic crystal of laminated pillars on a semiinﬁnite ZnO substrate is investigated. A double-resonators model is proposed to explain that the band structures of PC of laminated pillar change greatly with the variation of physical parameters. With φh increasing, the dominant effect of the natural frequency of defect pillars gradually changed from PMMA to PZT-4, such that resonance frequencies decrease ﬁrst and then increase. It is also indicated that the high-frequency BG is the result of the combined action of the local resonance and the Bragg scattering, and the former dominates. For the damping system, the initial frequency of both BGs increase proportionally with ηs increasing but the bandwidths remain unchanged. Besides, the amplitude of transmission spectrum decreases rapidly with the increase of isotropic structural loss factors in high frequency regions, which is caused by the greater dissipation of damping on the vibration energy in the high frequency region. Additionally, the linear defects inside the laminated pillar are discussed. The displacement ﬁelds of defect modes indicated that the elastic energy at surface is strongly conﬁned within the defect pillar and the substrate surface. It is also demonstrated that the coupling between pillars and SAW is governed by the physical parameter of the resonators. The calculations presented in this paper are based on the ﬁnite element methods. Acknowledgements The authors wish to acknowledge the support from the Natural Science Foundation of China (Grant No. 11772130, 11972160 and 11902117), and the Fundamental Research Funds for the Central Universities, SCUT (Grant No. D2192630).

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