Surface Acoustic Waves in Phononic Crystals

Abstract

Investigations of the surface acoustic wave (SAW) properties of phononic crystals in which the periodic modulation occurs on the traction-free surface did not take place until the late 1990s. Theoretical investigations initially focused on isotropic materials before being extended to the case of general anisotropy or to piezoelectric phononic crystals (PC) a few years later. Initially, experimental studies of SAW in 2D PCs were conducted on periodic structures at the millimeter scale exhibiting band gaps frequency in the range of a couple of MHz or lower. Surface modes as well as localization phenomena in linear and point defects were also demonstrated. In contrast, studies in the micrometer scale which may find applications in radio frequency (RF) communications or in MEMS devices have not started before the middle of the last decade. We propose in this chapter to report on some results obtained on the investigation of 2D SAW phononic band gap structures. The theoretical formulations of the plane-wave expansion (PWE) and of the finite difference time domain (FDTD) methods for SAWs are briefly summarized, and calculated results of SAW propagation and related phenomena such as band gaps and waveguiding are presented. Experiments conducted in PC at the micron scale allowing to demonstrate phononic band gaps and potential applications to SAW resonator devices are eventually reported.

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6.1 Introduction

Over the past two decades, propagation of acoustic waves in periodic structures comprised of multi-components has received much attention because of renewed physical properties and potential applications in a variety of fields, such as noise and vibration isolation, frequency filters in wireless communication, super lens design, etc. These composite materials, called phononic crystals (PCs) [1, 2], give rise to forbidden gaps for acoustic waves which are analogous to the band gaps for electromagnetic waves in photonic crystals. Major mechanisms leading to the forbidden gaps are Bragg scattering and localized resonances (LR) [3]. The former opens up the Bragg gap at the Brillouin-zone boundaries, and the band-gap frequency corresponds to the wavelength in the order of the structural period, i.e. of the lattice constant, and relates to the lattice symmetry. On the other hand, localized resonance creates resonant gaps dictated by the frequency of resonance associated with scattering units and depends less on the lattice symmetry, orderliness, and periodicity of the structure.

In the literature, theoretical calculations of bulk acoustic waves (BAWs) in phononic band structures of two-dimensional (2D) periodic composites were reported using the plane wave expansion (PWE) method [1–14], multiple scattering theory [15–18], and finite difference time domain (FDTD) method [19–21]. On the other hand, the experimental evidence for the existence of absolute acoustic band gaps (BAW modes) has also been reported [19, 22–27]. The existence of band gap, transmission properties, and local resonance of 2D phononic structures in the low frequency range were demonstrated. A comprehensive review of bulk waves in PCs can be found in a review article by Sigalas et al. [28].

Investigations into the surface acoustic wave (SAW) properties of PCs in which the periodic modulation occurs on the traction-free surface did not take place until the late 1990s [29–36]. The theoretical studies of SAWs on a square and hexagonal superlattice consisting of cubic (AlAs/GaAs) and isotropic (Al/polymer) materials were reported [29, 30] and extended to the case of general anisotropy a couple of years later by Wu et al. [31]. The case of SAW modes in 2D piezoelectric PC was then reported subsequently [32]. The surface mode of a square-lattice Y-cut lithium niobate PC with a large full band gap (a fractional bandwidth of 34 %) was found [33]. In [34], existence of the Bleustein–Gulyaev (BG) piezoelectric SAW in a 2D piezoelectric PC (ZnO(100)/CdS(100)) was reported. The authors showed that the BG surface waves have higher acoustic wave velocities, high electromechanical coupling coefficients, and larger band gap width than those of the Rayleigh surface waves and pseudosurface waves. In a recent article [35], the concept of the localization factor was introduced to study the Rayleigh surface wave propagation and localization in disordered piezoelectric PCs and the authors found that the larger the randomness degree, the stronger the degree of wave localization.

In the late 1990s, experimental studies of SAW in 2D PCs were concentrated on cases with frequency in the range of a few MHz or lower. The existence of surface states for sonic propagation in elastic band gap finite periodic systems was first demonstrated by Torres et al. [36] both theoretically and experimentally. In their experimental work, surface modes as well as localization phenomena in linear and point defects were demonstrated by drilling millimeter cylinders filled up with Hg in an Al matrix. At about the same time, experiments on the attenuation of Rayleigh surface waves by a semi-infinite 2D elastic band gap crystal made up of marble with centimeter size were reported by Meseguer et al. [37]. Vines et al. [38] conducted an experimental study on the surface waves generated by a line-focus acoustic lens at the water-loaded surfaces of a number of 2D superlattices that intersect the surface normally. Highly anisotropic transmission of SAW with frequency in the low MHz range was observed by continuously scanning the wave vector angle. Recently, further experimental demonstrations of the band gaps of surface modes in 2D PCs in the MHz range using laser ultrasonics have also been reported [39, 40].

Although demonstration of the band gap properties of SAWs in PCs at the millimeter scale has been conducted since the late 1990s, studies at the micrometer scale, which may find applications in the radio frequency (RF) communications or MEMS devices, had not started until the mid of last decade. In an effort toward the integration of PC and SAW frequency filters or oscillators, Wu et al. [41] utilized silicon micromachining to fabricate air/silicon square lattice PC with layered slanted finger interdigital transducers (SFIT) attached. The transmission of high-frequency SAW through 6 layers of PC with more than 30 dB attenuation was observed in the band gap between 183 and 215 MHz. In a subsequent paper, Benchabane et al. [42] demonstrated experimentally the existence of a complete SAW band gap in a 2D square lattice piezoelectric PC etched in lithium niobate. The SAW was generated by normal interdigital transducer (IDT) and a complete band gap extending from 203 to 226 MHz was demonstrated experimentally. Scattering and propagation of SAWs in PCs revealed by optical methods were given by Kokkonen et al. [43] and Profunser et al. [44]. Recently, a design that combines two-port SAW devices and PCs acting as reflective gratings was demonstrated [45]. The design consists of a layered ZnO/Si SAW device and a square lattice PC composed of cylindrical holes in silicon. With 15-layer PC cylinders, experimental insertion loss shows a 7 dB improvement at the central frequency of 212 MHz. As compared with the conventional metallic reflective gratings, the major advantage of PC is that the size of the CMOS compatible layered SAW filter can be reduced significantly.

In this chapter, investigation of SAW in 2D phononic band gap structures is presented. The content is organized as follows. In Sect. 6.2, the theoretical formulations of the PWE and the FDTD methods for SAW in periodic structures are briefly summarized. In Sect. 6.3, calculated results of SAW propagation and related phenomena are presented. Then, in Sect. 6.4, utilizing the properties of phononic band gaps, waveguides constructed with the band gap structures for SAW are investigated. The numerical results show that the acoustic wave energy can be efficiently guided or confined with the proposed structures. In Sects. 6.5 and 6.6, experiments are conducted to demonstrate the phononic band gaps and potential applications to SAW resonator devices. Conclusions are finally given in Sect. 6.7.

6.2 Theoretical Formulations

6.2.1 PWE Method

6.2.1.1 General Equations

Consider a PC composed of a 2D periodic array (on the x ₁ − x ₂ plane) of material A embedded in a background material B. Both materials A and B are piezoelectric or dielectric. In the quasi-static approximation, the governing field equations of piezoelectricity can be expressed as

$$ \frac{\partial {T}_{ij}}{\partial {x}_i}=\rho \frac{\partial^2{u}_j}{\partial {t}^2}, $$

(6.1)

$$ \frac{\partial {D}_i}{\partial {x}_i}=0,\;i,j=1,2,3, $$

(6.2)

where ρ(x) is the position-dependent mass density, u _j (r, t) is the displacement vector, r = (x ₁, x ₂, x ₃) = (x, x ₃) is the position vector, and T _ij (r, t) and D _i (r, t) are the stress and electric displacement, respectively. The piezoelectric constitutive equations with the displacement u _j (r, t) and the electric potential φ(r, t) as variables are given by

$$ {T}_{ij}={c}_{ijkl}\frac{\partial {u}_k}{\partial {x}_l}+{e}_{lij}\frac{\partial \varphi }{\partial {x}_l}, $$

(6.3)

$$ {D}_i={\varepsilon}_{il}\frac{\partial \varphi }{\partial {x}_l}-{e}_{ikl}\frac{\partial {u}_k}{\partial {x}_l},\;i,j,k,l=1,2,3, $$

(6.4)

where c _ijkl (x), e _lij (x), and ε _il (x) are the position-dependent elastic stiffness constants, piezoelectric constants, and permittivity, respectively. Due to the spatial periodicity, the material constants can be expanded in Fourier series with respect to the 2D reciprocal lattice vectors. On utilizing the Bloch–Floquet theorem and expanding the displacement vector and electric potential in Fourier series, a generalized eigenvalue problem with respect to the x ₃ component of the wave vector k ₃ can be derived:

$$ \left(\mathbf{P}+{k}_3\mathbf{Q}+{k}_3^2\mathbf{R}\right)\mathbf{U}=\mathbf{0}. $$

(6.5)

The dimensions of the matrices P, Q, R, and U are dependent on the number of the reciprocal lattice vectors used in the Fourier series expansions. As N reciprocal lattice vectors are used, P, Q, and R are 4 N × 4 N square matrices, and U is a 4 N × 1 column matrices. Equation (6.5) with an arbitrarily guessed frequency ω yields a set of eigenvalues k ₃ ^(m), (m = 1–8 N). The displacement field and electric potential of a SAW exponentially decay into the structure $\left({x}_3<0\right)$ away from the surface. Thus, 4 N eigenvalues with the following characteristics are picked out from the 8 N eigenvalues:

$$ \left\{\begin{array}{c}\hfill \mathrm{R}\mathrm{e}\left({k}_3^{(m)}\right)>0\kern1em \mathrm{if}\kern0.5em \mathrm{I}\mathrm{m}\left({k}_3^{(m)}\right)=0,\kern0.5em \hfill \\[6pt] {}\hfill \mathrm{I}\mathrm{m}\left({k}_3^{(m)}\right)<0\kern1em \mathrm{otherwise}.\kern3.5em \hfill \end{array}\right. $$

(6.6)

For SAWs propagating along the traction-free surface, the following mechanical boundary conditions are satisfied:

$$ {T}_{i3}={c}_{i3 kl}\frac{\partial {u}_k}{\partial {x}_l}+{e}_{li3}\frac{\partial \varphi }{\partial {x}_l}=0,\kern1em \mathrm{at}\kern0.5em {x}_3=0. $$

(6.7)

For a piezoelectric medium, the electrical boundary condition is imposed. Here, two important cases are distinguished:

• Open-circuit condition: If the adjacent medium is air, the normal component of the electric displacement across the surface is continuous, that is,

  $$ {D}_3=\left({e}_{3 kl}\frac{\partial {u}_k}{\partial {x}_l}-{\varepsilon}_{3l}\frac{\partial \varphi }{\partial {x}_l}\right)=-{\varepsilon}_0\frac{\partial {\varphi}_{\mathrm{air}}}{\partial {x}_3}, $$

  (6.8)

  giving the angular frequency ω = ω ₀. The electric potential φ _air in air can be solved by the Laplace equation together with the boundary conditions φ _air = φ at x ₃ = 0, and φ _air = 0 as x ₃ → ∞.

• Short-circuit condition: If the surface is covered by a very thin metallic film, the electric potential at the surface is equal to zero, that is, φ = 0, giving the angular frequency ω = ω _∞.

6.2.1.2 Case of Void Inclusions in a Solid Matrix

The PWE method as briefly stated above and detailed in Chap. 4 generally considers a PC with a solid–solid composition. In practice, however, a very convenient way to fabricate PCs consists in drilling holes in a solid matrix. This particularly holds in the case of periodical structures exhibiting features at the micron-scale. The resulting air/solid compositions in addition offer the highest possible impedance contrast between host and inclusion, hence contributing to the opening of wide band gaps. The case of inclusions made of air or vacuum, as propagation in air can be neglected when considering PCs in the ultrasonic regime, is therefore of definite interest.

In the literature, this issue is usually tackled by representing vacuum in a PWE model as an artificial medium with appropriate elastic constants and density. Laude et al. [33], for instance, proposed to fake vacuum by a material exhibiting zero elastic constants and mass density. In their paper, the case of a piezoelectric solid/void PC is specifically dealt with. Their starting point is to consider air or vacuum as a non-piezoelectric solid where wave propagation is governed by the equation of motion in its purely elastic form, i.e.:

$$ {T}_{ij}={c}_{ijkl}\frac{\partial {u}_k}{\partial {x}_l}, $$

(6.9)

$$ \rho \frac{\partial {u}_j}{\partial {t}^2}=\frac{\partial {T}_{ij}}{\partial {x}_j}. $$

(6.10)

Setting c _ijkl to zero imposes ${T}_{ij}={c}_{ijkl}\frac{\partial {u}_k}{\partial {x}_l}=0$ independently of the displacements, while $\rho =0$ ensures free displacements at the vacuum/solid interface. These solutions hence remain compatible with the definition of a free interface (unspecified displacement and zero-stress) while verifying the relation of continuity of the normal component of the stress tensor and of the elastic displacement field components at the interface, that is, ${u}_i^{\mathrm{air}}={u}_i^{\mathrm{solid}}$ and ${T}_{ij}^{\mathrm{air}}{n}_j={T}_{ij}^{\mathrm{solid}}{n}_j$. A simple pseudo-solid is hence defined, that will fake a vacuum in solid–solid PWE computations.

An alternative initially proposed by Tanaka et al. [20] and later refined by Manzanares-Martínez et al. [46] and Vasseur et al. [47], consists in representing vacuum as an artificial isotropic solid material characterized by physical parameters allowing for a fast decay of the elastic displacement field in the holes. The choice of these parameters is obviously governed by the boundary conditions at the vacuum–solid interface that impose $\rho {c}_{\mathrm{t},\mathrm{l}}^2\to 0$, where ρ is the mass density and c _t and c _l are the transverse and longitudinal wave velocities, respectively. A natural choice consists in imposing a very low mass density ρ ^air (of the order of 10⁻⁴ kg/m³) and longitudinal and transverse velocities much larger than the usual sound of speed in solids (typically ${c}_{\mathrm{l}}={c}_{\mathrm{t}}={10}^5$ m/s) to limit propagation of waves to the surrounding, actual solid while remaining with finite values. Under such conditions, the obtained pseudo-solid presents non-zero elastic constants C ₁₁ and C ₄₄ around 10⁶ N/m², four orders of magnitude lower than those encountered in real solids.

These two possible implementations offer stable numerical solutions by ensuring at all time that the Fourier coefficients describing the material constants of both matrix and inclusions are well-defined.

6.2.2 Finite-Difference Time-Domain Method

The FDTD method was introduced in 2000 for the study of PCs. Sigalas and Garcia [48] calculated the transmission of acoustic waves passing a PC structure. Kafesaki et al. [49] demonstrated the wave propagation and transmission inside a PC waveguide. Further, Tanaka et al. [20] applied the Bloch theorem to FDTD method and analyzed the acoustic dispersion. A directed periodic boundary condition was suggested by Hsieh et al. [50] to study the band structures of PCs. So far, the FDTD method has been well-developed to calculate the dispersion, transmission, and propagation of BAWs inside PC structures. Further application of the FDTD method to SAWs was suggested by Sun and Wu [51]. The principle of FDTD method is mentioned in the following.

In a linear elastic material, the constitutive law and equation of motion of elastic materials are expressed as Eqs. (6.9) and (6.10). These equations describe the property of an infinitesimal element of an anisotropic material in general. They are applicable to the inhomogeneous structure of PCs by arranging the density and elastic constant periodically. Further, with staggered grids, the differential equations (6.9) and (6.10) are transferred into difference equations based on the Taylor’s expansion to develop the 3D heterogeneous finite difference formulation. Then these equations are calculated recursively to study the wave propagation in PCs.

The Bloch’s theorem is introduced to treat the periodic boundary condition of a unit cell of PCs. The displacement and stress fields of PCs can be expressed by a plane wave and a periodic function as:

$$ {u}_i\left(\mathbf{x},t\right)={e}^{\mathrm{i}\mathbf{k}\cdot \mathbf{x}}{U}_i\left(\mathbf{x},t\right), $$

(6.11)

$$ {T}_{ij}\left(\mathbf{x},t\right)={e}^{\mathrm{i}\mathbf{k}\cdot \mathbf{x}}{S}_{ij}\left(\mathbf{x},t\right), $$

(6.12)

where k is a wave vector. U _i (x, t) and S _ij (x, t) are periodic functions satisfying

$$ {U}_i\left(\mathbf{x}+\mathbf{a},t\right)={U}_i\left(\mathbf{x},t\right), $$

(6.13)

$$ {S}_{ij}\left(\mathbf{x}+\mathbf{a},t\right)={S}_{ij}\left(\mathbf{x},t\right), $$

(6.14)

with a lattice translation vector a. In [20], the equation of motion and constitutive law were transformed using the periodic functions to satisfy the conditions (6.13)–(6.14). Alternatively, we combined Eqs. (6.11)–(6.12) and (6.13)–(6.14) to write a periodic boundary condition (PBC) of displacement u _i and stress T _ij directly as [50]:

$$ {u}_i\left(\mathbf{x}+\mathbf{a},t\right)={e}^{\mathrm{i}\mathbf{k}\cdot \mathbf{a}}{u}_i\left(\mathbf{x},t\right), $$

(6.15)

$$ {T}_{ij}\left(\mathbf{x}+\mathbf{a},t\right)={e}^{\mathrm{i}\mathbf{k}\cdot \mathbf{a}}{T}_{ij}\left(\mathbf{x},t\right). $$

(6.16)

With the PBC, both the 2D and 3D PC cases can be analyzed by calculating a unit cell. In the analyses of dispersions, a small disturbance in a random position of the unit cell is set as the initial condition. Thus all possible wave modes are transported inside the considered PC, and the displacement is recorded and expanded into Fourier series. Then the eigenfrequencies as a function of the wave vector k are indicated by selecting the resonance peaks of the spectrum. This procedure allows finding acoustic modes which satisfy the periodic boundary condition in PCs.

To solve the SAW modes of PCs, extra boundary conditions are needed to develop SAWs inside a PC unit cell. In this study, we set a free surface boundary condition and use the PML absorbing boundary condition to deal with the reflection from the numerical boundary.

Berenger [52] introduced the concept of PML to reduce the electromagnetic wave reflection from the boundary, and PML have also been developed in the case of elastic wave propagation [53, 54]. In the present study, a 3D PML program applicable to orthogonal materials is adopted to serve the non-reflection boundary condition. A stretched coordinate is defined with a complex variable

$$ {e}_i={a}_i+\mathrm{i}\frac{\varOmega_i}{\omega } $$

(6.17)

to derive the code of the PML region [53]. In Eq. (6.17), the real part a _i is the scale factor, and Ω _i /ω is the imaginary part with the attenuation factor Ω _i and the circular frequency ω. Then the differential operation in the stretched coordinate is defined and applied to the equation of motion and constitutive law. After introducing the plane wave solutions into the equations, the numerical attenuation is achieved via the factor Ω _i . Besides, a non-reflection condition at the interface between PML region and inner space is obtained by setting corresponding material constants and the unity scale factor. In the elastodynamic equations for the stretched coordinate, displacement and stress components are taken via spatial partial differential operation in all directions; thus, the variables are split into three components to realize the difference equations. Then actual values are obtained from the summation of splitting components. Finally, the PML equations can be written as follows:

$$ \rho \frac{\partial {u}_{i/j}}{\partial {t}^2}+\rho {\varOmega}_j\frac{\partial {u}_{i/j}}{\partial t}=\frac{\partial {T}_{ij}}{\partial {x}_j}, $$

(6.18)

$$ \frac{\partial {T}_{ij/m}}{\partial t}+{\varOmega}_m{T}_{ij/m}=\frac{c_{ijkl}}{2}\frac{\partial }{\partial t}\left(\frac{\partial {u}_k}{\partial {x}_l}{\delta}_{ml}+\frac{\partial {u}_l}{\partial {x}_k}{\delta}_{mk}\right). $$

(6.19)

In the above equations, u _i/j and T _ij/m are the splitting displacement and splitting stress which satisfy ${u}_i={u}_{i/1}+{u}_{i/2}+{u}_{i/3}$ and ${T}_{ij}={T}_{ij/1}+{T}_{ij/2}+{T}_{ij/3}$. δ _ij is the delta function and ${\delta}_{ij}=1$ when $i=j$, else ${\delta}_{ij}=0$.

After transforming equations (6.18) and (6.19) into the difference formula, the PML is arranged outside the space boundaries as a buffer zone with a matched acoustic impedance to suppress reflection. It performs a numerical attenuation so that waves decay rapidly inside the region. With this absorbing boundary condition, reflection is reduced to less than 1 % and the PML boundary is used in both the calculations of dispersion relation and transmission.

6.3 SAWs in Phononic Crystals

6.3.1 Frequency Band Structures

An example of a frequency band structure of a 2D PC is presented in Fig. 6.1. The structure is a 2D square lattice of circular cylinders of radius r ₀ embedded periodically in a background material. The lattice spacing is a, and the reciprocal lattice vectors are G = 2π(n ₁/a, n ₂/a) with integers n ₁ and n ₂. Figure 6.1 is calculated by using the PWE method with 25 Fourier components in the series expansion (i.e., −2 ≦ n ₁, n ₂ ≦ 2). The filling fraction F = πr ₀ ²/a ² is 0.6. The cylinders are assumed to be made of bismuth germanium oxide (Bi₁₂GeO₂₀), and the background material is silica (SiO₂). The three fourfold axes of Bi₁₂GeO₂₀ are aligned with the x ₁-, x ₂-, and x ₃-axis of the PC, respectively. The material constants used in the calculations are listed in Table 6.1. In Fig. 6.1, bulk and surface wave modes are presented. The thick and thin solid lines represent the quasi-longitudinal (denoted by L) and quasi-shear-horizontal modes (denoted by SH) polarized on the x ₁ − x ₂ plane, respectively. The dashed lines represent the piezoelectric shear-vertical modes (denoted by SV). The surface modes are represented by the solid dots and the open circles, respectively, for SAWs and pseudosurface acoustic waves (PSAWs), where the PSAWs leak energy into the structure.

Fig. 6.1

Frequency band structures of the bulk, SAW and PSAW modes in the Bi₁₂GeO₂₀/SiO₂ piezoelectric PC with square lattice. F = 0.6

Table 6.1 Material constants used in the calculation

Figure 6.1 shows that the SAW branch exists along the boundary Γ–X–M of the Brillouin zone and slightly below the lowest branch of the SV modes. After reaching the Brillouin zone boundary, the folded branch of SAWs turns into the PSAWs. The PSAWs have a characteristic that they are attenuated by radiating the acoustic energy into the bulk of the structure. More detail about PSAW characteristics will be given in Sect. 6.3.2. It is noted that in this case, only the SH wave is piezoelectric. The L and SH modes are not affected by the piezoelectricity since the only nonzero piezoelectric constants of Bi₁₂GeO₂₀ are e ₁₄ = e ₂₅ = e ₃₆. In this regard, Eq. (6.5) can be decoupled into the mixed in-plane modes (i.e., L and SH vibrations are coupled), which are piezoelectric inactive, and piezoelectric SV modes with polarization of the displacement along the x ₃ direction.

6.3.2 Band Gaps and SAW Characteristics

Figure 6.2 shows the angular dependence of dispersion relations at the magnitude of the reduced wave vector ka/π = 1, where $k=\sqrt{k_1^2+{k}_2^2}$. The PSAW branch exists in between the fundamental branches and the folded branches of bulk shear modes, and its frequency increases as the propagation angle θ increases. Figure 6.3 shows the SAW frequency ω _SAW and PSAW frequency ω _PSAW, as well as the magnitude of the frequency gap $\Delta \omega ={\omega}_{\mathrm{PSAW}}-{\omega}_{\mathrm{SAW}}$ at the symmetry point X as a function of filling fraction F. Both the SAW and PSAW frequencies at the lower and upper band-gap edges decrease monotonically with the increase in F; the decrease in the frequency is reasonable because the filling material Bi₁₂GeO₂₀ has a lower SAW velocity propagating along its crystallographic X-axis than that of SiO₂. Therefore, increasing the filling fraction of Bi₁₂GeO₂₀ accordingly decreases the SAW and PSAW frequencies. The maximum gap width appears at F = 0.183. The value of maximum gap width is 0.5 in normalized frequency. When the filling fraction decreases to zero, the phase velocity of SAW reduces to the SAW velocity in homogeneous SiO₂.

Fig. 6.2

Angular dependence of the frequencies of the bulk waves, surface wave, and pseudosurface wave at ka/π = 1. The dashed lines denote the branches of SV mode with the piezoelectric effect ignored

Fig. 6.3

Width of the frequency gap Δω = ω _PSAW − ω _SAW in between the SAW and PSAW (solid line) at the symmetry point X as a function of filling fraction. The frequencies ω _PSAW at the upper edge and ω _SAW at the lower edge of the gap are also plotted (dashed lines)

Figure 6.2 shows that the folded branch of surface mode exists inside the bulk bands and turns into a PSAW branch. The reason can be understood thanks to Fig. 6.4 which shows the k ₁ − k ₃ and k − k ₃ sections of the equifrequency surfaces for the bulk waves, respectively. Here, k represents the magnitude of the Bloch wave vector along Γ–M direction. Figure 6.4a shows that, at the given value of normalized frequency equal to 1.4, the wave number of the SAW, denoted by k _SAW, exists along k ₁-axis and outside the three equifrequency surfaces of the bulk wave modes. Instead, Fig. 6.4b shows a different case. At the normalized frequency equal to 1.8, which is in the frequency gap of the bulk shear modes at X point, a real wave vector K = (k _SAW = k _PSAW, k ₃) of a PSAW can be found by drawing a dashed line, from the point k ₁ = k _PSAW, parallel to the k ₃-axis and intersecting the curves of the bulk shear modes. The corresponding group-velocity directions are plotted by the outward normal of the surfaces that point inside the bulk of the medium. This means that the surface mode couples to bulk shear modes, and thus that it propagates with attenuation by radiating acoustic energy into the medium. This coupling is attributed to the simultaneous anisotropic effect of the constituents and geometry of the structure. Because of the coupling, the folded branch of the surface modes in the frequency band structure of Fig. 6.1b turns into a PSAW branch. Along the Γ–M direction, Fig. 6.5 also shows a similar phenomenon.

Fig. 6.4

The section of equifrequency surfaces of the bulk waves by the k ₁ − k ₃ plane. (a) Normalized frequency equal to 1.4. (b) Normalized frequency equal to 1.8

Fig. 6.5

The section of constant-frequency surfaces of the bulk waves by the k − k ₃ plane, where k is the magnitude of the wave vector along Γ–M direction. (a) Normalized frequency equal to 2.0. (b) Normalized frequency equal to 2.4

The coupling of a surface mode to the bulk-wave bands to become a PSAW can also be revealed by the eigenvalues solved by Eq. (6.5); that is, the energy radiating into the bulk of medium leads to a part of the 4 N eigenvalues being real, and a real eigenvalue means a non-decayed out-of-plane propagation. As the general solution of surface modes contains a part of the partial waves constructed by these real eigenvalues and their corresponding eigenvectors, the surface mode has a leaky property of propagation.

6.3.3 Bleustein–Gulyaev Waves

In a homogeneous piezoelectric crystal, a transverse SAW with no counterpart in a non-piezoelectric material was discovered by Bleustein and Gulyaev in 1968. They showed that the non-vanishing piezoelectric coupling factor in a class of transverse isotropic piezoelectric crystal (e.g., crystal symmetry class 6 mm) can result in a surface wave with a very simple mechanical displacement and electric potential fields that decay exponentially into the medium. In early 1969, Shimizu et al. theoretically and experimentally demonstrated the existence of such a unique wave in a poled lead zirconate titanate (PZT) ceramic independent of the knowledge of Bleustein and Gulyaev’s papers. Since then, the follow-up researches and related applications have made the Bleustein–Gulyaev wave theory as one of the cornerstones for the modern signal processing and electro-acoustic technology. In this section, Bleustein–Gulyaev (BG) waves in a homogeneous piezoelectric crystal and in a 2D piezoelectric PC are discussed, respectively.

First, let us consider the homogeneous case. The equations governing the BG wave with purely transverse polarization u ₂ and propagating along the x ₂-direction in a semi-infinite homogeneous piezoelectric crystal whose surface outward normal is in the x ₃-direction are

$$ {c}_{i22l}\frac{\partial^2{u}_2}{\partial {x}_i\partial {x}_l}+{e}_{l2j}\frac{\partial^2\varphi }{\partial {x}_i\partial {x}_l}=\rho \frac{\partial^2{u}_2}{\partial {t}^2}, $$

(6.20)

$$ {e}_{i2l}\frac{\partial^2{u}_2}{\partial {x}_i\partial {x}_l}-{\varepsilon}_{il}\frac{\partial^2\varphi }{\partial {x}_i\partial {x}_l}=0, $$

(6.21)

$$ \frac{\partial^2{\varphi}_{\mathrm{air}}}{\partial {x}_i\partial {x}_i}=0, $$

(6.22)

with the mechanical boundary conditions at the surface

$$ {T}_{i3}={c}_{i32l}\frac{\partial {u}_2}{\partial {x}_l}+{e}_{li3}\frac{\partial \varphi }{\partial {x}_l}=0,\kern1.25em {x}_3=0, $$

(6.23)

and the electrical boundary conditions, either

$$ \begin{array}{l}{e}_{32l}\frac{\partial {u}_2}{\partial {x}_l}-{\varepsilon}_{3l}\frac{\partial \varphi }{\partial {x}_l}=-{\varepsilon}_0\frac{\partial {\varphi}_{\mathrm{air}}}{\partial {x}_3},\kern1.25em {x}_3=0,\\ {}\kern4.62em \varphi ={\varphi}_{\mathrm{air}},\kern3.88em {x}_3=0,\\ {}\kern5.24em {\varphi}_{\mathrm{air}}=0,\kern3.4em {x}_3\to \infty, \end{array} $$

(6.24)

for open-circuit surface, or

$$ \begin{array}{l}\varphi =0,\kern1.84em {x}_3=0,\\ {}{\varphi}_{\mathrm{air}}=0,\kern1em {x}_3\to \infty, \end{array} $$

(6.25)

for short-circuit surface. The solutions for u ₂ and ϕ can be written as

$$ \begin{array}{l}{u}_2={A}_2\cdot {e}^{\beta {x}_3}\cdot {e}^{ik{x}_1-i\omega t},\\ {}\varphi ={A}_4\cdot {e}^{\beta {x}_3}\cdot {e}^{ik{x}_1-i\omega t},\kern2em \mathrm{R}\mathrm{e}\left(\beta \right)>0\end{array}$$

(6.26)

where β is a decay constant. Substituting Eq. (6.26) into the governing equations and boundary conditions accordingly solves the BG wave velocity, decay constant, and the electromechanical coupling coefficient.

As an example, consider a case for the existence of the BG wave: a hexagonal crystal of 6 mm class including the [010] propagation direction in the (100) plane. The physical properties transformed from the crystallographic coordinates are given by

$$ \left(\begin{array}{cccccc} {c}_{11} & {c}_{13} & {c}_{12} & 0 & 0 & 0 \\ {} {c}_{13} & {c}_{33} & {c}_{13} & 0 & 0 & 0 \\ {} {c}_{12} & {c}_{13} & {c}_{11} & 0 & 0 & 0 \\ {} 0 & 0 & 0 & {c}_{44} & 0 & 0 \\ {} 0 & 0 & 0 & 0 & \frac{c_{11}-{c}_{12}}{2} & 0 \\ {} 0 & 0 & 0 & 0 & 0 & {c}_{44} \end{array}\right), $$

(6.27)

$$ \left(\begin{array}{cccccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {e}_{15}\hfill \\ {}\hfill {e}_{31}\hfill & \hfill {e}_{33}\hfill & \hfill {e}_{31}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {e}_{15}\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right), $$

(6.28)

$$ \left(\begin{array}{ccc}\hfill {\varepsilon}_{11}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {\varepsilon}_{33}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill {\varepsilon}_{11}\hfill \end{array}\right). $$

(6.29)

Analytically, BG wave velocity and decay constant for this configuration are given by

$$ {V}_0={V}_T\sqrt{1-\frac{K_T^4}{{\left(1+{\varepsilon}_{11}/{\varepsilon}_0\right)}^2}}, $$

(6.30)

$$ \beta =\frac{K_T^2}{\left(1+{\varepsilon}_{11}/{\varepsilon}_0\right)}, $$

(6.31)

for open-circuit surface, and

$$ {V}_{\infty }={V}_T\sqrt{1-{K}_T^4}, $$

(6.32)

$$ \beta ={K}_T^2, $$

(6.33)

for short-circuit surface, where

$$ {V}_T=\sqrt{\frac{c_{44}+\left({e}_{15}^2/{\varepsilon}_{11}\right)}{\rho }}, $$

(6.34)

$$ {K}_T=\frac{e_{15}}{\sqrt{c_{44}{\varepsilon}_{11}+{e}_{15}^2}}, $$

(6.35)

and V _T and K _T are the piezoelectric bulk transverse wave velocity and the bulk transverse-wave electromechanical coupling factor, respectively. Equations (6.30)–(6.35) tell that the decay is more rapid, and the difference in velocity from the bulk transverse wave velocity for BG wave increases when the piezoelectric constant is large. For the case of zinc oxide (ZnO) that belongs to class 6 mm, Fig. 6.6 shows the values of boundary-condition determinant versus the phase velocity $V=\omega /k$. The solid line denotes the short-circuit condition, and the dashed line denotes the open-circuit condition. There are two sharp local minima that represent the roots of the boundary-condition determinant for each electrical condition, respectively. The one with lower phase velocity indicates the phase velocity of Rayleigh wave, and the other one indicates the phase velocity of BG wave. For Rayleigh surface wave, there is no difference in phase velocities between the open-circuit and short-circuit conditions. For BG surface wave, the difference in phase velocities between the two electrical conditions can be clearly differentiated because the BG waves are always piezoelectric; otherwise, the BG wave degenerates into a non-piezoelectric bulk SH wave. The phase velocity of BG wave under short-circuit condition is slower than that under open-circuit condition, and they are 2,869.79 m/s and 2,884.04 m/s, respectively. This always happens because the short-circuit surface eliminates the tangential electric field, and thus, partially suppresses the piezoelectricity of the material. The electromechanical coupling for BG wave in ZnO is 9.88 %.

Fig. 6.6

The boundary-condition determinant vs. phase velocity. The sharp minima indicate the eigen-solutions of surface waves, where the chosen orientation of ZnO substrate is [010] direction on (100) plane

Figures 6.7 and 6.8 show the depth dependence of the displacement and electric potential fields for Rayleigh wave and BG wave under short-circuit surface condition, respectively. In Fig. 6.7, the Rayleigh wave is polarized in the sagittal plane with zero u ₂ and φ, so the wave is a non-piezoelectric wave. This means that, in this propagation direction and plane orientation, the mechanical terms are decoupled from the electrical terms for sagittal-plane vibrations. In Fig. 6.8, the piezoelectric BG wave polarized transversely accompanies with an electric potential field. Moreover, the BG wave penetrates much deeper than the Rayleigh wave into the medium.

Fig. 6.7

The depth dependence of displacement amplitudes of Rayleigh surface wave propagating along [010] direction on (100) plane of ZnO

Fig. 6.8

The depth dependence of displacement and electric potential amplitudes of BG surface wave propagating along [010] direction on (100) plane of ZnO (short-circuit condition)

As an example of piezoelectric PC, let us consider a square array of circular ZnO cylinders, arranged with lattice constant a, embedded periodically in a cadmium sulfide (CdS) background material. In the arrangement of the constituents of the piezoelectric PC, the crystallographic plane (100) of ZnO and CdS is parallel to the x ₁ − x ₂ plane, and their sixfold symmetry axis (i.e., the crystallographic Z-axis) coincides with the x ₂-axis of the PC so that, according to BG wave theory, the piezoelectric effect can compensate for the mechanical displacement field to satisfy the stress-free boundary condition, as well as the SH displacement could decay into the medium.

Figure 6.9 shows the dispersion relations of surfaces modes under short-circuit boundary condition in the frequency range around the band gap near the X point. The filling fraction F is equal to 0.3. In the figure, another fundamental branch of surface wave solution (red dots) exists in addition to the common surface wave for this configuration. Further calculation will show that this surface wave branch is mainly composed of SH displacement, i.e., it is a BG wave in the 2D piezoelectric PC. This wave mode degenerates into the piezoelectric bulk wave as the piezoelectricity is ignored. Moreover, the folded branch of the BG wave has also been found, the folded branch of BG waves is higher than the PSAW branch. This folded BG branch is a pseudosurface-type wave (denoted by BG PSAW) with very small attenuation. The band gap width of the BG waves in the ΓX direction is 0.773, in the unit of normalized frequency.

Fig. 6.9

Dispersion relations of SAW (blue dots), BG SAW (red dots), PSAW (black circles), and BG PSAW wave (red circles) under short-circuit condition in the frequency range around the band gap and near the point X

To reveal the influence of the electrical boundary conditions on the BG surface wave in the piezoelectric PC, a detailed analysis at the symmetry point X is conducted. Figure 6.10 shows the boundary-condition determinants of the fundamental and the folded branches of surface modes as a function of frequency, respectively. In both figures, solid lines denote the short-circuit boundary condition, and dashed lines denote the open-circuit condition. In the case of the fundamental branch (Fig. 6.10a), there are two sharp minima for each boundary-condition determinant. The first one with lower frequency indicates eigenfrequency of the common SAW, while the other one with higher frequency denotes the BG surface wave. For the common SAW, the frequency difference between the open-circuit and short-circuit conditions is very small. In other words, the electromechanical coupling coefficient almost vanishes for the common SAW in this configuration. For the BG waves, there is a clear difference in frequency with the same wave vector between the two boundary conditions, and the resulted electromechanical coupling coefficient is about 0.12%. Figure 6.10b shows results for the folded branches, and the sharp minima indicate the frequencies of PSAWs. The electromechanical coupling coefficient of the folded BG PSAW is 0.63%.

Fig. 6.10

The boundary-condition determinant vs. normalized frequency. (a) The frequency range is around the fundamental branches of SAW. (b) The frequency range is around the folded branches of SAW at the point X

The depth dependence of the displacement and electric potential fields of the BG waves at the center of the zinc oxide circular cylinder at symmetry point X in the piezoelectric PC are shown in Figs. 6.11 and 6.12. Figure 6.11a, b shows the profiles of the displacement and electric potential amplitudes in relative magnitudes of the fundamental modes of the BG waves under open-circuit and short-circuit conditions, respectively. The results show that, compared with the SH component, the other two components are negligible. In addition, the decay of the displacements and the electric potential is much more rapid for the short-circuit condition than that for the open-circuit condition. From the dispersion relations, the phase velocities of fundamental BG surface modes with reduced wave vector k ₁ a/π at the symmetry point X are 1,832.09 and 1,830.97 m/s for open-circuit condition and electrical shorted condition, respectively. Figure 6.12a, b shows the profiles of relative amplitudes of the displacement and electric potential of the folded BG waves at point X under short- and open-circuit conditions, respectively. It is observed that the folded BG waves decay more rapidly into the medium than the fundamental modes. However, in this branch a non-vanishing longitudinal component exists. In other words, the BG wave in this 2D piezoelectric PC contains displacement components other than the SH component, which differs from the BG waves in homogeneous piezoelectric materials. This phenomenon is caused by the scattering effect due to the periodic cylindrical fillers in the structure and can be explained by two reasons:

Fig. 6.11

Displacement and electrical potential fields of BG surface wave at the center of the ZnO cylinder vs. the distance from the surface. (a) Under short-circuit condition. (b) Under open-circuit condition

Fig. 6.12

Displacement and electrical potential fields of BG PSAW at the center of the ZnO cylinder vs. the distance from the surface. (a) Under short-circuit condition. (b) Under open-circuit condition

• 1. The SH vibration, u ₂, is coupled with the sagittal-plane vibrations, u ₁ and u ₃, through the elastic terms in Eq. (6.5).

• 2. The electric potential is coupled with the mechanical vibration, not only with the SH vibration but also with the sagittal-plane vibrations, through the piezoelectric and dielectric terms in Eq. (6.5).

These two effects (i.e., the coupling from scattering effect of the periodic structure as well as the anisotropic property of the constituents) do not appear in the homogeneous media belonging to 6 mm crystal where the SH vibration with piezoelectricity is decoupled from the sagittal-plane vibrations (and the sagittal-plane vibrations are piezoelectric inactive) for acoustic waves propagating in the [010] direction on (100) plane of the materials belonging to 6 mm symmetry class. The coupling of u ₁ with u ₂ is stronger than u ₃; therefore, the longitudinal displacement is larger than the SV displacement which is very small. The phase velocities of folded mode of the BG waves for open- and short-circuit conditions with reduced wave vector at symmetry point X, calculated by using ω/k, are 2,270.60 and 1,163.43 m/s, respectively.

Figure 6.13 shows the band edges of the frequency gaps of the surface modes and their normalized gap widths at the point X as a function of filling fraction F. The figures show that the BG wave (open circles) band gaps in the Γ–X direction are larger than the gaps in between the common SAW and the PSAW. The maximum band gap width appears at F = 0.576 in this material configuration.

Fig. 6.13

The band edges of the frequency gaps in between the SAW and PSAW, as well as BG SAW and BG PSAW at the symmetry point X as a function of filling fraction

6.4 Phononic Crystal Waveguides for SAW

This section deals with the PC waveguides for SAW. First, a steel/epoxy PC is introduced and the band gap is investigated. According to the band gap property, line defects in the PC are defined as acoustic waveguides and the guided SAW propagating inside the structure are analyzed [51].

6.4.1 Complete Band Gaps of Steel/Epoxy Phononic Crystals

In this study, a square lattice PC consisting of steel cylinders embedded in an epoxy half space was employed. The lattice constant a of the PC is 8 mm and the radius r of steel cylinder is 3 mm, and thus the filling ratio F equals to 0.442. The density and elastic constant C₁₁ and C₄₄ of steel are assumed as 7,900 kg/m³, 280.2 GPa and 82.9 GPa, respectively, and those for epoxy are 1,180 kg/m³, 7.61 GPa and 1.59 GPa. Thus the impedance of the cylinders is more than 15 times larger than the one of the matrix.

To investigate the band gap, the dispersion of acoustic waves inside PCs was calculated first. The PC structure has a traction free surface (x ₃ = 0) to allow SAW propagation. Although the SAW propagates within the x ₁ − x ₂ plane perpendicular to the cylinders, the SAW propagation is indeed a 3D problem. Thus we defined a 3D unit cell as shown in Fig. 6.14a to calculate the characteristic SAW modes. The dimension of this unit cell is chosen as 1a × 1a of the cross section on x ₁ − x ₂ plane and the height h is 9a to develop SAWs. Then a free surface boundary is on the surface at x ₃ = 0, and a PML region is under the bottom surface to apply non-reflection boundary. Finally, the PBC is defined on the other four surfaces of the unit cell. In the FDTD method, each unit cell is divided into 24 × 24 grids and the time step interval is 20 ns to satisfy the numerical stability conditions.

Fig. 6.14

(a) The 3D unit cell of PC employed to calculate SAWs. (b) Dispersion of acoustic waves in the square lattice steel/epoxy PC with a free surface [51]

In this PC, the structure is infinite along the x ₃-direction, and the SAWs propagate in the x ₁ − x ₂ plane on the free surface. Meanwhile, BAWs accompany SAW and propagate in the x ₁ − x ₂ plane. The polarization of BAW modes are decoupled as the in-plane mode (x ₁ − x ₂ plane) and the anti-plane mode. In Fig. 6.14b, the eigenmodes of BAWs are presented in solid and hollow circles for in-plane and anti-plane modes, respectively. For the in-plane modes, there is a noticeable wide complete band gap from 90 to 204 kHz and several narrow complete band gaps at higher range 231–237 kHz, 245–255 kHz. From the curves of anti-plane modes, band gaps are located at 55–143, 153–212, 225–250, and 270–276 kHz. Considering both modes together, the complete band gaps are at 90–143, 153–204, 231–237, and 245–250 kHz.

Then we used the 3D unit cell with PBC, free surface and PML to calculate the dispersion of SAWs. The eigenmodes are obtained and marked as rhombus symbols in Fig. 6.14b. As shown in the figure, extra modes are detected by the sensors on the free surface. For example, new bands appear at 7–79.5 kHz, 199–201 kHz in ΓX direction. Also, some selected peaks overlap the circle symbol of BAW modes, such as the first and second bands of in-plane and anti-plane transverse waves in ΓX direction. In the higher frequency range, there are numerous peaks but most peak values are very small.

To investigate the modes of acoustic wave detected in the 3D PC unit cell, the displacement distributions of the eigenmodes are calculated. The calculating setup is the same as Fig. 6.14a and a wave source generating a wave package of chosen frequency was defined. Because both the wave vector and frequency are assigned, the specific eigenmode is excited and the displacement field is recorded to help identify the properties of modes. The displacement distributions of the eigenmodes are plotted in a 3D vector plot.

First, two eigenmodes which differ from those of BAW are calculated. The eigenmodes of wave vector $\mathbf{k}=\left(\pi /a,0\right)$ with frequency f = 77 kHz (point A in Fig. 6.14b) and f = 199 kHz (point B in Fig. 6.14b) are shown in Fig. 6.15a, b. In the vector plots, the direction of the cones indicates the polarization of displacement vector and the size reflects the magnitude. The unit cell has grid lines in the location of multiples of the lattice constant. These modes of wave vector $\mathbf{k}=\left(\pi /a,0\right)$ propagate along the x ₁-direction in Fig. 6.15. The wave length of first band with $\left|\mathbf{k}\right|=\pi /a$ is 2a and thus only a half of the periodic displacement field is shown is one unit cell. The wave in Fig. 6.15a has a main polarization in the sagittal plane, the x ₁ − x ₃ plane, and the wave in Fig. 6.15b has a main polarization in the x ₁ − x ₂ plane. The result shows confined displacement fields—the amplitude decays rapidly as the depth increases and presents a typical property of SAWs. An interesting phenomenon is that the band which the mode in Fig. 6.15a belongs to has a higher velocity than the transverse wave of BAW modes. That means the surface wave is a leakage type, i.e. a PSAW.

Fig. 6.15

The 3D displacement fields of eigenmodes. (a) k = (π/a, 0), f = 77 kHz (point A of Fig. 6.14b); (b) k = (π/a, 0), f = 199 kHz (point B of Fig. 6.14b) [51]

In addition, the peaks overlapped on the BAW modes were also analyzed. For example, the eigenmode of wave vector $\mathbf{k}=\left(\pi /a,0\right)$ with frequency f = 43 kHz was calculated and the distribution of displacement field showed a x ₂-direction polarization and the amplitude does not decay rapidly within nine lattice constants. This reflects the property of transverse waves, the same with the mode of in-plane transverse bulk waves. Analysis of another mode of wave vector $\mathbf{k}=\left(\pi /2a,0\right)$ with frequency f = 84 kHz also shows a rotation field which remains within the depth of the unit cell. Thus, the peaks overlapped on the modes of BAW present the property of BAW.

The analysis shows that the 2D steel/epoxy square PC with a free surface allows PSAW instead of SAW. The dispersion curves show an important fact that there exists a complete band gap for both BAW and PSAW (SAW) in this PC. The complete band gaps obtained by both results of 2D and 3D unit cell locate from 90 to 143 kHz and 154 to 199 kHz. This is the basis to design a PC waveguide of surface waves.

6.4.2 SAWs Inside a Phononic Crystal Waveguide

A PC waveguide is constructed by arranging adjacent point defects in a PC structure. These defects form a continuous region which is free of inclusions and acoustic waves are expected to propagate within. A PC waveguide of SAWs is formed based on the complete band gap of SAW. The acoustic waves in the waveguide cannot penetrate the periodic structure and thus the guided modes result from the existence of defects appearing in the range of complete band gaps.

For the steel/epoxy PC presented in the last section, the complete frequency ranges of band gaps are 90–143 and 154–199 kHz.

To understand the property of SAW inside PC waveguides, the supercell technique is used to analyze the dispersion of a waveguide. We consider a waveguide with a 10 mm width w (the distance between two neighboring cylinders on both sides) as shown in Fig. 6.16a. A 3D supercell consisting of 11 × 1 × 9 unit cells is defined, and then the waveguide is constructed of ten periodic arrayed cylinders and a dislocation in the center with the PBC surrounding it. Similarly, an initial disturbance is defined inside the waveguide, and thus acoustic waves are generated and the displacement field is recorded. Then the eigenmodes are selected from the local maximum peaks of spectra as the procedure of analyzing a unit cell. By setting corresponding boundaries, both the dispersion calculated by 2D and 3D supercells are obtained and plotted in Fig. 6.16b. We focused on the frequency of 70–220 kHz to observe defect modes in the range of complete band gaps. The extended bands outside the complete band gaps are omitted and the region is marked in gray. As shown in the result, new defect modes of acoustic waves appear in the range of complete band gaps. The eigenmodes of BAW are marked as solid and hollow circles and those obtained from the 3D supercell are rhombus symbols.

Fig. 6.16

(a) The supercell used to analyze the PC waveguide; (b) Dispersion diagram of BAW and SAW eigenmodes in the steel/epoxy PC waveguide [51]

Comparing the dispersion for the PC waveguides to the case of perfect PC, new defect modes can be obtained. In the 3D supercell, sensors on the free surface of PC waveguide receive the signal of defect modes of both BAW and SAW. The signal of BAW modes has a small peak value and make the spectrum shows a complex pattern. Thus, in the 3D waveguide calculation, the BAW modes are removed and only the SAW mode is shown in Fig. 6.16b. Inside the first complete band gap of 90–143 kHz, a band extends from wave vector $\mathbf{k}=\left(\pi /a,0\right)$ with frequency f = 87.5 kHz to $\mathbf{k}=\left(0.05\pi /a,0\right)$ with f = 141 kHz, and another band appears from $\mathbf{k}=\left(0.75\pi /a,0\right)$ with f = 154 kHz to $\mathbf{k}=\left(0.9\pi /a,0\right)$ with f = 206 kHz in the second complete band gap of 154–199 kHz.

The displacement fields of the two defect modes are plotted in 3D vector diagrams. The eigenmodes of $\mathbf{k}=\left(0.5\pi /a,0\right)$ with f = 114.5 kHz (point A in Fig. 6.16b) and $\mathbf{k}=\left(0.5\pi /a,0\right)$ with f = 180 kHz (point B in Fig. 6.16b) are calculated and shown in Fig. 6.17a, b, respectively. Since the dispersion diagram is presented in a reduced zone scheme, the actual wave vectors of these two modes in folded bands are $\mathbf{k}=\left(1.5\pi /a,0\right)$ and $\mathbf{k}=\left(2.5\pi /a,0\right)$ with wavelengths of 1.33a and 0.8a, and the consistent result is obtained from the displacement fields of x ₁ − x ₃ plane. The guided waves propagate along the x ₁-direction, and the displacement fields have a confined distribution within the region of waveguides. The maximum amplitude occurs on the free surface and the amplitude decays rapidly in a deeper location. Thus these two bands have a typical property of SAWs.

Fig. 6.17

The 3D displacement fields of defect modes inside the PC waveguide. (a) k = (0.5π/a, 0), f = 114.5 kHz (point A of Fig. 6.16b); (b) k = (0.5π/a, 0), f = 180 kHz (point B of Fig. 6.16b)[51]

A further discussion of the new defect modes in PC waveguide is presented as follows. In the calculation of eigenmode displacement fields, a Gaussian weighted wave packet with specified frequency f is set as the source and input finite energy into the supercell structure to excite the selected eigenmode. Then, the sensor on the free surface detects a monotonous displacement variation and the vertical component U ₃ recorded after the wave packet is excited. The result shows that the eigenmode has decaying amplitude. This means an energy leakage, partial acoustic waves propagates downward into the half space, and thus the defect modes in this PC waveguide are actual PSAWs. Finally, converting the period of signal into propagation distance, the attenuation coefficient is obtained as −0.049 dB/a and −0.06 dB/a for these eigenmodes, respectively.

SAW propagation inside a straight PC waveguide was further calculated. Figure 6.18 shows a top view of the free surface and the actual structure extends vertically to the surface. The periodic circles present the boundaries of steel cylinders and the epoxy matrix. The waveguide has a width of 10 mm by removing one row of adjoining cylinder inclusions along ΓX-direction, and the length of waveguide is 25a. An acoustic wave is generated in the inlet of waveguide with five line sources to simulate an IDT. The PSAW defect mode of wavelength $\lambda =1.33a$ with frequency f = 114.5 kHz is generated and the wave propagation pattern is demonstrated. The displacement field of x ₃-direction component U ₃ is presented in a gray scale to show the surface wave propagation. The scale is fixed in this study for the convenience to compare the amplitude in different cases. The result shows that the PSAW is confined well within the waveguide and only slight energy leaks into the periodic PC area. In addition, although the PSAW has energy leakage, the PSAW propagates along the waveguide and carries most energy.

Fig. 6.18

The x ₃-direction displacement field of the 114.5 kHz PSAW inside a linear PC waveguide with a width of 10 mm [51]

In acoustic waveguides, a bend section is an elementary structure to change the wave propagation direction. Here the PC waveguide with a sharp bend is also calculated to estimate the transmission efficiency. The first case is a simple junction of two ΓX-direction waveguide. The structure is a 17 a × 17 a area and two 11 a waveguides are connected as shown in Fig. 6.19a. The same PSAW of 114.5 kHz defect mode is launched into the waveguide from the inlet in the lower-left corner and the U ₃ displacement field on the surface is shown in the same gray scale. The displacement pattern shows that PSAW is still confined within the bending waveguide structure, but most energy of the incident wave is reflected at the corner of the bend. In the case of a linear straight waveguide in Fig. 6.18, the amplitude of the 114.5 kHz PSAW in the outlet is chosen as a reference value which results from the same monotonous wave source. With the identical wave source, the maximum amplitude on the surface of the vertical waveguide branch is about 59% of the reference amplitude in the bending waveguide. Another phenomenon is that BAW modes are also excited as the turned-wave, and thus considerable energy is taken away and the amplitude in the vertical branch decays rapidly. Thus a simple joint bend is not an efficient design.

Fig. 6.19

The x ₃-direction displacement field of the 114.5 kHz PSAW inside (a) a sharp bending waveguide, (b) a three-segment bending waveguide, and (c) an improved PC bending waveguide with scatterers [51]

Another bend structure is shown in Fig. 6.19b. It contains two linear waveguide in ΓX-direction and one in ΓM-direction, thus the wave turns 45° twice to the left direction. The U ₃ displacement pattern of 114.5 kHz PSAW defect mode also shows a strong reflection of the incident wave. The amplitude of turned-wave is about 52 % of the reference amplitude in ΓM-direction segment and 26 % in the vertical ΓX-direction segment. Finally, the mode of turned-wave is rebuilt to a symmetric PSAW mode. In these cases, the wavelengths of PSAW are almost the same as a lattice constant, thus there is a strong scattering phenomenon when acoustic waves encounter inclusions at their propagating direction. In addition, the anti-symmetry conjunction section in Fig. 6.19b also enhances the scattering and the mode conversion. Therefore energy disperses while the excited BAW propagates downward, and thus the surface amplitude decays when the PSAW encounters the boundary of the bending waveguides.

To raise the transmission of bending PC waveguide, an improved bend structure is introduced. In the study of photonic crystals, the scatters are used to enhance transmission through waveguide bends in two-dimensional photonic crystal slabs. The idea is adopted to raise the transmission of PSAW in PC waveguides. The design is based on the prototype shown in Fig. 6.19c and the cylinders of smaller radius 1.3 mm are inserted into ΓM-direction segment. With the inserted scatterers, the conjunction section has a better symmetric geometry property. This also changes the dispersion curve, as the result shown in the photonic crystal cases, and result in eigenmodes that propagate in the conjunction section with less loss. In Fig. 6.19c, the distance d between the center of scatters is 16.97 mm ($3\sqrt{2}a/2$). Then the U ₃ displacement pattern of PSAW of 114.5 kHz is calculated and shown. Obviously, the PSAW passing the corner remains the same eigenmode and therefore the amplitude is raised to 72 % of the reference amplitude.

Further calculation shows that the transmission of the guided PSAW is affected by the scatters obviously. If the scatters are moved toward the center of the waveguide, the dispersion relation is altered. In the case of the distance d of 11.31 mm ($\sqrt{2}a$), the amplitude of turned-PSAW at 114.5 kHz becomes 48 % of the reference amplitude. Another important fact is that waves of different frequency in the structure do not have identical performance. The PSAW of 135 kHz in the bend structure of Fig. 6.19c shifts to BAW mode and the mode decays very rapidly in the turned waveguide, but has a 62 % amplitude ratio in the structure with d of 11.31 mm. Therefore, the transmission efficient is affected by the scatters and also a function of frequency. Indeed, setting scatters properly in the PC waveguides can raise the transmission and change the propagation direction in a small region.

In brief, we introduced steel/epoxy PC with complete band gaps for both bulk and SAWs. Further, PC waveguides are designed based on the complete band gap property. The defect mode inside the PC waveguide has an energy leakage because the 3D structure allows the wave propagating downward to the half space. The eigenmodes of the 2D PC waveguide with a free surface is thus identified as PSAW. Although the PSAW has a slight energy leakage inside the 2D waveguide, it is possible to have a waveguide with a sharp bend. Adding scatters inside the PC waveguides can raise the transmission and the design depends on the frequency of PSAW.

6.5 Experiment on the SAW Band Gap

In this section, the experimental studies of SAW in the 2D PCs are presented. Using the MEMS process, micro-scale PC were realized and the band gaps were observed in the hundreds MHz frequency range. In the following, the PC fabricated on silicon wafers [41] and lithium niobate wafers [42] are introduced.

6.5.1 Band Gap for SAWs in Silicon-Based Phononic Crystals

To demonstrate the frequency band gap of SAWs in a 2D silicon PC, a square array of circular holes in a silicon substrate was realized [41]. High-frequency wideband slanted finger interdigital transducers (SFITs) on the silicon substrate were used to generate and detect the SAW signals. Since silicon is not a piezoelectric material, the layered structure SFIT/ZnO/silicon as shown in Fig. 6.20a was considered in the device, where the ZnO layer is piezoelectric, and the metal SFITs are on the top of the ZnO layer so that SAWs can be excited electrically. The design parameters of the SFITs and the geometry of the layered structure depend upon the band-gap frequency of the 2D PC. The lattice constant of the PC was 10 μm, and the radius of the circular holes r was 3.5 μm which corresponds to a filling ratio F of 0.385. Analysis with the PWE method for this structure showed that the band-gap frequencies along the ΓX direction are from 183 to 215 MHz.

Fig. 6.20

(a) The schematic of the experimental setup for measuring the band gap of SAW. Top view (b) and cross section (c) of the 2D micro silicon PC [41]

According to the calculated band-gap frequencies, the design parameters of the SFITs are listed in Table 6.2. In design 1, the excited SAW frequency range by the layered SFITs covers the band gap of the PC. Therefore, the frequency band-gap width of the PC can be measured by the device. In Fig. 6.20a, an array of cylindrical holes between the two SFITs were micromachined to act as the PC.

Table 6.2 Design parameters of the SFIT/ZnO/silicon-layered structure

The fabricated 2D silicon-based PC consisting of six rows of air cylinders is shown in Fig. 6.20b. Figure 6.20c is the cross section view of the structure. The depth of the 2D cylindrical holes is about 80 μm which is more than two times of the wave length. Figure 6.21a shows the measured frequency band gaps of the air/silicon PCs by using the SFITs of design 1. The shaded region denotes the theoretical band gap of the fabricated PCs. The dashed curves and solid curve denote the measured frequency responses of the layered structures with and without the PC slab, respectively. The two dashed curves are the results of different samples with the same design. The insertion losses of SAW propagated in the layered structure with PCs are clearly higher than that without the PC for frequency located in the band gap. This corresponds to the fact that most of the SAW energy in the band-gap range is blocked by the phononic structures. The ratio of the insertion loss with phononic structures to that without phononic structures gives the transmission coefficients. The transmission spectrum is shown in Fig. 6.21b. The transmission spectrum also clearly shows that the SAWs are attenuated greatly in the forbidden band.

Fig. 6.21

(a) The band gaps for SAW in the silicon PCs are revealed by the layered SFIT of design 1 in the range of pass-band frequency. The shadow region represents the band gap of fabricated PCs with six rows of air cylinders. (b) The acoustic transmission coefficient [41]

6.5.2 Piezoelectric Phononic Crystals

The combination of SAWs and piezoelectric monocrystalline solids such as quartz, lithium tantalate (LiTaO₃) or lithium niobate (LiNbO₃), amongst others, occupy a prominent position in the field of wireless telecommunications and signal processing. Monocrystalline substrates indeed offer piezoelectric and electro-mechanical coupling properties that remain unmatched by the currently available piezoelectric thin films. The demonstration of phononic band gaps for SAWs in such materials hence offers particularly rich applicative prospects while constituting a very good field for more fundamental investigations. Very efficient electrical SAW sources and detectors can be found in interdigital transducers and single-mode operation is made possible by the very nature of the generated surface waves. These very appealing properties are however counter-balanced by the difficulty of processing these materials that are quite often complex oxides using standard micromachining technologies. Fabrication then stands as a challenge, while design is not made easier: the strong anisotropy of acoustic wave propagation inherent to piezoelectric materials, combined with the quasi-systematic mixing of shear and longitudinal polarizations, put tighter constraints on the geometrical parameters of the periodical structure itself.

The most obvious configuration from a conceptual as well as from a fabrication point of view that can be implemented to proceed with the demonstration of a frequency band gap in a piezoelectric crystal consists in considering a square lattice of air inclusions obtained by drilling holes in a piezoelectric solid. Such a demonstration has in particular been achieved in lithium niobate [42]. LiNbO₃ benefits from rich optical properties that add up to its acoustic characteristics to extend the application field of PCs made out of this material to the field of integrated acousto-optics and dual photonic and phononic structures that will be dealt with later in Chap. 9.

The PWE theory can, for instance, be used for the design of such a PC. Figure 6.22a displays the computation of the density of surface states in the case of a perfect, infinite 2D PC for LiNbO₃ in the Y-crystallographic orientation plotted along the closed path Γ–X–M–Y–Γ in the first irreducible Brillouin zone. The considered holes have a circular cross section with a diameter d = 0.9a, where a is the lattice constant. The filling fraction then equals 63 %. Seven harmonics are used in each direction in the PWE computations, resulting in a total of N = 49 harmonics. It can be seen that a full band gap (i.e. a band gap for any direction of propagation and polarization) exists for waves propagating in-plane from f.a = 1,935 to 2,745 m/s. The fractional bandwidth is then larger than 34 %. Qualitatively, it can be observed that although in principle anisotropy makes it more difficult to open a full band gap than with isotropic materials, the free boundaries of void inclusions are very efficient scatters for elastic waves of any polarization. Another interesting observation is that the frequency band gap width and position were found to be exactly coincident with those found via PWE for in-plane propagating bulk waves [33]. Figure 6.22b reports some finite element method (FEM) simulations of bulk waves propagating in the same periodical structure that confirm this initial observation. The displayed propagation directions have here been kept to the only ones that can actually be investigated experimentally.

Fig. 6.22

(a) Band structure for surface waves along the Γ–X–M–Y–Γ path of the first Brillouin zone for a PC made of a square-lattice of air holes etched in a Y-cut lithium niobate matrix. The filling fraction has been set to 64 %. (b) Theoretical band diagrams for bulk waves in the same crystal. The dispersion relations on a free surface are indicated for the Rayleigh (short dashed line) and the leaky (long dashed line) surface waves. The grayed regions indicate the sound cone

The corresponding PC was then fabricated in a 500 μm thick Y-cut LiNbO₃ substrate. To relieve the technological constraints in this initial experiment and to be able to use optical lithography and collective fabrication methods for the patterning of the phononic structure, the operating frequency was set to 200 MHz, hence setting the lattice parameter to 10 μm and the hole diameter to about 9 μm. The FEM theory consequently predicts a complete band gap spanning from 190 to 250 MHz. It can be noted in Fig. 6.23a that the obtained holes exhibit a rather low sidewall slope angle, around 72°, which limits the depth to about 11 μm and therefore the aspect ratio to 1.1.

Fig. 6.23

(a) Scanning electron microscope photograph of the array of 10 μm deep, 9 μm diameter holes etched in a lithium niobate substrate constituting the PC, before removal of the etching mask. (b) Optical microscope image of the series of devices used to demonstrate the band gap phenomenon over the whole Brillouin zone

Interdigital transducers in a delay-line configuration were used to launch and detect surface waves propagating through the crystal. In order to keep full control of the wave vector direction in a highly anisotropic material such as LiNbO₃ and to make sure that Rayleigh waves exclusively were generated, a series of eight ten-finger-pair IDTs of varying mechanical period, corresponding to emission wavelengths ranging from 12.2 to 26 μm were used and preferred over the slanted IDTs of Sect. 6.5.1. The frequency range of interest was then covered by juxtaposing neighboring responses. The IDTs were oriented for elastic wave propagation along the ΓX, ΓM, and ΓY directions of the first Brillouin zone, as illustrated in Fig. 6.23b.

Figure 6.24 displays the measured electrical normalized transmission, computed as the ratio of the transmission scattering parameters with and without the PC for each set of delay lines. If a full band gap is definitely observed, with the crystal causing a 20 dB loss in the transmission value, no transmitted signal was measured at frequencies lying above the band gap. This phenomenon can be accounted for by considering the actual geometry of the fabricated SAW PC: since the surface wave penetration depth in the substrate is at least of the order of the hole depth, the experimental structure can be seen as a stratified medium, with a void/lithium niobate PC slab lying on a semi-infinite lithium niobate substrate. This, combined with the conicity of the holes tends to favor out-of-plane scattering, leading to coupling with bulk modes of the substrate. A sound line, by analogy with the light line of photonic crystal slabs, can then be defined, that corresponds to the dispersion relation of the slowest bulk acoustic mode in the substrate, i.e. of the slowest shear wave. Surface modes lying inside the sound cone are, possibly, highly lossy radiation modes. Only modes below the sound line are guaranteed to be guided by the surface, which agrees well with the experimental results. Another origin of the signal extinction at higher frequencies can also be found in elastic wave diffraction by the PC that can indeed be considered as a 2D diffraction grating [43]. Clearly, electrical characterizations through RF probe testing cannot provide us with a means of observing this kind of phenomena. Additional characterization tools giving access to more direct information on wave propagation such as optical mapping of the displacement fields at the surface are then required to reach a higher level of understanding of both scattering and diffraction phenomena in PCs. These techniques will be more thoroughly accounted for in Chaps. 7 and 8.

Fig. 6.24

Relative transmission, defined as the ratio of the transmittances with and without a PC. The hatched area delimits the complete band gap that extends from 203 to 226 MHz

The sound cones are superimposed on the band diagrams in Fig. 6.22b and correspond to the grayed areas. This kind of representation for both the sound and light cones is widely adopted by both the photonic and phononic community, though it can somehow be considered as misleading. It is indeed quite tempting to fully discard the dispersion properties of modes lying within the sound cone. Yet, stating that the modes lying beyond the sound line are likely to radiate to the bulk does not necessarily exclude the existence of surface or pseudosurface guided waves. This was, for example, demonstrated experimentally through the fabrication and characterization of the structure reported in Fig. 6.25. The PC considered here is a scaled-down version, by a factor of 5, of the previous one. It consists again in a square-lattice crystal with a filling fraction close to 64 %. This time, however, the period of the structure is about 2.2 μm and the LiNbO₃ substrate was chosen in the X-crystallographic orientation. A critical difference lies in the patterning method used for the 2 μm diameter air holes that were milled by focused ion beam (FIB). This fabrication technique indeed allows obtaining a nearly vertical hole profile, with with a sidewall slope angle around 85, which is a significant improvement compared to 72° previously reported.

Fig. 6.25

Scanning electron microscope image of a hypersonic PC device. The pitch of the structure is about 2.2 μm and the holes are around 2 μm in diameter, resulting in a filling fraction of about 64 %. A pair of broadband chirped transducers is used for electrical emission and detection of the elastic waves

The main drawback of the FIB fabrication method lies in the time required to etch a single one of the phononic structures presented here that was of the order of 9 hours with the equipment at hand (Orsay Physics LEO FIB 440). For this reason, the fabrication of multiple occurrences of identical PCs was this time not conceivable. The standard interdigital transducers were then replaced by chirped IDTs exhibiting a linear variation in the electrical period along their length and allowing for a broadband emission over more than an octave. Only a couple of identical PCs, surrounded by two different sets of transducers, were then required for a full characterization of the expected band gap, as opposed to eight pairs of transducers in the previous case. If using chirped IDTs instead of SFITs still allows setting the k-vector direction, the detection bandwidth along a given propagation direction is limited by possible interferences between pure Rayleigh waves and slow, shear leaky-surface waves in some cases. In X-cut LiNbO₃ for example, wide band electrical detection is only possible in the (XZ) propagation direction, while the achievable bandwidth along (XY) is limited by a transverse leaky wave of velocity v = 4,100 m/s (v = 3,680 m/s for the Rayleigh wave).

The raw transmission data for the reference delay lines and for the PCs, along with the normalized transmittance are reported in Fig. 6.26. A very good overlap between the two signals can be observed at frequencies below 620 MHz. Beyond this point, and much similarly to what has been reported in the lower frequency SAW LiNbO₃ PC, the electrical response is strongly attenuated, with an extinction ratio of the order of 13 dB. Partial re-transmission occurs from a frequency of about 1 GHz and there is once again no undisputable evidence of surface modes above the band gap, although a close study of the reflection scattering parameters seemed to indicate that the frequency band gap should be exited at a frequency of about 1 GHz [55]. This point was further confirmed by optical measurements performed via heterodyne interferometry. Maps of the amplitude wave profile across the PC were taken at frequencies supposedly lying before, within and after the expected band gap. The resulting averaged cross sections of the surface motion through the PC area are reported in Fig. 6.27. As expected, below the band gap, e.g. at 540 MHz, the elastic wave passes through the crystal and is almost unaltered. In contrast, a standing wave pattern is clearly observed at 660 MHz (band gap edge) and at 800 MHz (center), confirming that both frequencies lie within the band gap for surface guided modes. The wave penetration length inside the PC is higher at the band edge than at the central frequency and up to 25% of the elastic energy is still transmitted at 660 MHz. At 800 MHz, the plot of the averaged cross-section of the surface motion reveals an exponential decrease of the wave amplitude inside the crystal, resulting in an output wave amplitude within the noise level. The most striking results are observed at higher frequencies: at 1.05 GHz, at which frequency the electrical transmittance is weak, the optical measurements clearly demonstrate that transmission occurs above the band gap. The average amplitude at the output was evaluated to be around 0.17 nm, versus 0.22 nm for the incident wave, resulting in an amplitude transmission close to 75 %. This result shows that configurations do exist where the sound line limit can be partly overcome: the sound cone designates a frequency region where leakage from surface to bulk modes can happen, but this does not imply that surface wave propagation is fully prohibited in this part of the spectrum.

Fig. 6.26

Transmission responses of each PC, along with the response of the electro-acoustic delay lines used as references. The normalized transmittance versus frequency for surface guided modes propagating through the PCs is also plotted. The transmittance plot gathers the results obtained for the two phononic structures

Fig. 6.27

Measured vertical displacements through the PC structure obtained by optical interferometry at different frequencies: 540 MHz (below the band gap), 660 MHz (bandgap edge), 800 MHz (within the bandgap), and 1.05 GHz (above the bandgap). The amplitude data are averaged in the y-direction. In all cases, the emitting transducer is on the right-hand side

The implementation of PCs for SAW in piezoelectric materials therefore remains at an early stage and both the physical description of such structures and the technological means used for their realizations need to be improved. The current theoretical, perfect, two-dimensional representation of these periodical structures cannot even find a remote real-life equivalent. Surface structuration using micro- nano-fabrication processes remain to date the most obvious choice of realization of these PCs and these techniques will inevitably lead to limited aspect ratio structures, particularly in piezoelectric materials that usually happen to be quite resilient to usual etching processes. Pure surface-guided modes will under such configuration be clearly difficult to maintain at frequencies lying above the sound line, and this is true that in practical applications, radiation to the bulk can prevent from taking full advantage of the PC capabilities over the entire frequency range of operation. Yet, the band gaps observed for surface-guided modes lying below the sound line remain perfectly valid. The frequency range outside the sound cone should hence be favored for the design of efficient waveguides, high-Q cavities or for the implementation of RF signal processing functions where electrical signal transmission is the essential vector of information. The demonstration of a clear transmission of the elastic waves outside of the band gap for modes lying within the sound cone however shows that losses experienced by high frequency surface waves can be partly overcome, provided that enhanced sidewall verticality and aspect ratio of the holes constituting the crystal can be improved. This may open interesting prospects for both fundamental and applicative studies involving elastic wave propagation on its own and its interaction with the host medium, as in the case, for instance, of elasto-optical interactions.

6.6 Application of SAW Band Gap to SAW Devices

6.6.1 Reflective Grating for SAWs Using Phononic Crystals

SAW devices consisting of IDTs and metal gratings have been widely used as resonators for decades. In order to have good reflection, hundreds of metal strips are usually employed, and it is volume consuming. PCs consist of periodic arranged media and perform acoustic band gaps. The experimental studies in the last section showed that PCs can block SAW propagation efficiently with only several lattice periods. In other words, PC can be designed as space-saving wave reflectors. In this section, the analysis and design of PC reflective grating for SAW devices are demonstrated.

The substrate of the SAW device is a silicon wafer, and cylindrical holes in the wafer were made from a square lattice PC. For an air/Si PC with square lattice, propagation of SAWs was analyzed in previous studies [56, 41]. With a filling fraction F = 0.283, there is a partial band gap for SAW along the ΓX direction [56]. Further, the band gap enlarges with the increase of filling fraction, and reaches the widest gap for the case of F = 0.48 [41]. To study the PC grating, the air/Si PC of F = 0.283 was adopted for the convenience of specimen fabrication. The dispersion curves were analyzed by the PWE method and the band structures of acoustic waves are shown in Fig. 6.28. As marked in the figure, a partial band gap was obtained for SAW between the normalized frequencies 2.1 and 2.41. In this case, only partial band gap for bulk waves exists as well. We note that the SAW modes become a leaky SAW that radiates energy into the half space in the folded SAW band for the reduced frequency higher than 2.41 because it pierces into the bulk-wave bands and couples to the SV modes. However, the SAW modes of the lowest SAW band polarized in the sagittal plane and decoupled from the SH modes is still a true SAW. In the study, the lattice constant is 10 μm, thus the corresponding range of the partial band gap is 195–224 MHz. In the previous section, the existence of the band gap has been demonstrated experimentally.

Fig. 6.28

Dispersion relations of SAW and BAW in the air/Si PC of F = 0.283. The denotations Γ, X, and M of the inset are the symmetry points of the first Brillouin zone in the k space. The normalized frequency is defined as ω _a/C_T, where ω is angular frequency, C _T transverse wave velocity [45]

In traditional SAW devices, hundreds of metal strips are usually adopted as reflective gratings to improve insertion losses of IDTs. The distance between the IDT and the grating has to be optimized to result in coherent reflective waves in the device. To apply PCs as a reflective grating, the location of effective reflective plane of the PC grating was investigated by using the FDTD method. A 210 MHz SAW is excited in a silicon half space and encounters an air/Si square lattice PC of lattice constant a = 10 μm and filling fraction F = 0.283. Because the frequency of the SAW is inside the band gap, the evanescent wave penetrating into the PC decays rapidly and most of the incident wave is reflected. With a continuous wave source, the reflected waves interact with the incident waves and form a standing wave pattern.

A surface displacement field of in-plane component U ₁ is shown in Fig. 6.29. There are fifteen columns of cylinders in the calculation. The SAW is reflected by the PC and the incident one is attenuated inside the PC. Due to the circular boundaries of the cylinders, the reflective wave needs an extra delay distance to reconstruct the linear wavefront. As shown in Fig. 6.29a, l is defined as the distance from the centerline of the first layer of cylinders to the location of a peak of U ₁. The location where there is the peak value of U ₁ is considered as the effective reflective plane of the PC grating. We note that the distance from the effective reflective plane to the location of wave sources (on the left) satisfies the condition of multiples of a half wavelength. For the SAW of 210 MHz in silicon, we chose l as about 1.32 λ, where λ is the wavelength of the 210 MHz SAW in silicon half space, i.e. about 23.39 μm. This delay distance is adopted as a reference to design the two-port resonant SAW devices in the following section.

Fig. 6.29

The displacement field of in-plane component U ₁ shows SAW encountering the PC. (a) The picture shows an extra delay distance is needed to reconstruct the linear wavefront. (b) A profile along the dashed line in (a) shows the reflection and attenuation inside the PC. The amplitude is normalized to the U ₁ of the case without the PC [45]

6.6.2 SAW Resonator

After investigating the reflection of SAW on the PC grating, two-port layered SAW devices combining PC gratings are designed to investigate the reflection efficiency and the optimum delay distance between IDTs and PCs. A schematic diagram of the two-port layered SAW device is shown in Fig. 6.30a. The center frequency 210 MHz of the layered ZnO/Si SAW device was calculated based on the effective permittivity method [57]. The designed distance between the central line of the first two metal strips of IDT and the edge of ZnO film is 1.5 λ. There are two fifteen-layer PCs outside the IDTs as reflective gratings. To study the effect of the delay distance D between the IDT and the PC grating, four different D were used; i.e., 0.875 λ, 1.0 λ, 1.125 λ, and 1.25 λ.

Fig. 6.30

(a) Schematic of a two-port layered SAW device using PC grating. (b) SEM photo of a layered SAW device with PC gratings [45]

Herein a process realizing a two-port layered SAW device with PC gratings is briefly described. A 0.5 μm ZnO film was deposited on silicon by sputtering. Aluminum thin film of 150 nm thickness was evaporated on the ZnO film, and interdigital transducers were formed by conventional photolithography and lift-off process. The line width of IDTs is 5.5 μm, the aperture is 100 wavelengths, and IDT pair number is 70. PC of 15-layer cylinders was fabricated by using the ICP process. Figure 6.30b shows SEM images of the realized device. 15 rows of cylindrical holes acting as reflective gratings were placed on both sides of transducers. As the figure shows, the depth of the cylindrical holes is about 20 μm. Moreover, the width of the PC gratings is only about seven pairs of IDTs. Comparing to a traditional two-port SAW resonator with hundreds of metal strips, the size of phononic-crystal reflective gratings is reduced significantly. After fabricating the SAW devices, the transmission coefficient S ₂₁ was measured by an RF network analyzer. The frequency responses of the two-port SAW devices without PC gratings were measured as reference signals. Then the device was measured again after fabricating the square lattice PC.

To verify the design of the optimum delay distance between the IDTs and PC gratings, two-port SAW devices of four different delay distance D were realized. Figure 6.31 presents the results of the cases with D = 1.0 λ and D = 1.25 λ. In these cases, the transmission shows the central frequency is about 212 MHz. The shift of central frequency may be caused by the difference of material constants of ZnO between single crystals used for calculations and the deposited polycrystalline thin film. Although the frequency is not exactly the same, it does not interfere the effect of PC gratings. With PC gratings, Fig. 6.31a shows a significant improvement of insertion loss. However, Fig. 6.31b has a sharp drop-off at the central frequency of the SAW device. Thus, the delay distance in Fig. 6.31b results in destructive wave interaction inside the SAW device. Consequently, with improper delay distance, the cases of D = 0.875 λ and 1.125 λ present distorted frequency responses, whose peaks appear at higher and low frequencies. In addition, due to the wide band gap of the PC gratings, the waves of frequency ranging from 195 to 224 MHz were reflected. Several peaks and dips at different frequencies were observed in the transmission of Fig. 6.31, which result from constructive and destructive interactions affected by the delay distance. Indeed, the estimation of effective reflective plane plays an important role in designing the SAW devices with PC gratings.

Fig. 6.31

Transmission of a two-port SAW device with PC gratings. (a) D = 1.0 λ; (b) D = 1.25 λ [45]

To show the improvement of the PC grating compared to the metal reflector, a two-port SAW device with 300 metal reflectors was fabricated and measured. Since the PC grating was silicon-based, the metal reflectors were also fabricated on silicon. The measured insertion loss did not show obvious improvement with the existence of metal reflectors. This may be due to the fact that the metal reflectors deposited on silicon directly can only reflect SAW by the mechanical effect but electrical reflection.

In summary, a structure which combines a layered SAW device and PCs acting as gratings has been demonstrated. The results showed that the delay distance between the IDTs and PC grating is an important parameter and has to be designed carefully. By using PCs as wave reflectors, the experimental results show a 7 dB improvement in the central frequency. Moreover, the size of gratings is reduced significantly as compared to the traditional gratings with hundreds of metal strips.

6.7 Conclusions

In this chapter, we summarized recent theoretical and experimental studies of SAWs propagating in 2D phononic structures. The theoretical formulations of PWE method for SAWs in both non-piezoelectric and piezoelectric PC structures were briefly introduced, while the technique allowing to account for the case of void inclusions in a solid matrix was also addressed. The principle of FDTD method and useful boundary conditions were presented and employed to calculate the band structures of SAW in PCs. By using these numerical tools, acoustic waves propagating along the free surface of 2D PCs can be analyzed successfully. By analyzing the equifrequency surfaces of SAWs in general PCs, both the Rayleigh surface wave and pseudosurface waves can be identified clearly. For the case of a piezoelectric PC, the existence of BG SAWs was demonstrated and results show that the electrical boundary condition of piezoelectric media affects the phase velocities of BG SAW and PSAW. Nonetheless, the most important result is the demonstration of the existence of complete band gaps for SAW in 2D PCs with a free surface.

By utilizing the FDTD method with the supercell technique, salient features of SAW propagation in both straight and bent waveguides have been revealed. The results showed that it is quite straightforward to construct a straight SAW waveguide by removing a layer of PC from the 2D PC. However, to maintain proper transmission efficiency, the structure at the corner of a bent waveguide should be designed carefully. Results also indicated that surface waves inside straight waveguides are PSAWs which have slight energy leakage.

On the experimental side, both the silicon based and piezoelectric based substrates were studied in the past few years and the results demonstrated for the first time the existence of SAW band gap in micro-scale PCs [41, 42]. It is demonstrated that the transmission spectra could be obtained by employing IDTs to detect the penetrating SAW. In addition, based on the laser interferometer, the surface profile of the SAW field of micro-scale PC demonstrated further the band gap property of SAW in such a PC structures.

Finally, we note that the application of air/silicon PC in a two-port SAW resonator has been introduced recently. Through a detailed design of the band gap and effective reflecting distance, both numerical and experimental results showed that ten layers of PC are enough to serve as the reflectors of a SAW resonator. By using PCs as the reflective grating, the size of gratings can be reduced significantly as compared to the traditional gratings with hundreds of metal strips.